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This file is part of the Flocq formalization of floating-point arithmetic in Coq: http://flocq.gforge.inria.fr/

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Relative error of the roundings

Require Import Core.
Require Import Psatz.  (* for lra *)

Section Fprop_relative.

Variable beta : radix.
Notation bpow e := (bpow beta e).

Section Fprop_relative_generic.

Variable fexp : Z -> Z.
Context { prop_exp : Valid_exp fexp }.

Section relative_error_conversion.

Variable rnd : R -> Z.
Context { valid_rnd : Valid_rnd rnd }.

Lemma relative_error_lt_conversion :
  forall x b, (0 < b)%R ->
  (x <> 0 -> Rabs (round beta fexp rnd x - x) < b * Rabs x)%R ->
  exists eps,
  (Rabs eps < b)%R /\ round beta fexp rnd x = (x * (1 + eps))%R.beta:radix
fexp:Z -> Z
prop_exp:Valid_exp fexp
rnd:R -> Z
valid_rnd:Valid_rnd rnd
forall x b : R,
(0 < b)%R ->
(x <> 0%R ->
 (Rabs (round beta fexp rnd x - x) < b * Rabs x)%R) ->
exists eps : R,
  (Rabs eps < b)%R /\
  round beta fexp rnd x = (x * (1 + eps))%R
Proof with auto with typeclass_instances.beta:radix
fexp:Z -> Z
prop_exp:Valid_exp fexp
rnd:R -> Z
valid_rnd:Valid_rnd rnd
forall x b : R,
(0 < b)%R ->
(x <> 0%R ->
 (Rabs (round beta fexp rnd x - x) < b * Rabs x)%R) ->
exists eps : R,
  (Rabs eps < b)%R /\
  round beta fexp rnd x = (x * (1 + eps))%R
intros x b Hb0 Hxb.beta:radix
fexp:Z -> Z
prop_exp:Valid_exp fexp
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, b:R
Hb0:(0 < b)%R
Hxb:x <> 0%R ->
(Rabs (round beta fexp rnd x - x) < b * Rabs x)%R
exists eps : R,
  (Rabs eps < b)%R /\
  round beta fexp rnd x = (x * (1 + eps))%R
destruct (Req_dec x 0) as [Hx0|Hx0].beta:radix
fexp:Z -> Z
prop_exp:Valid_exp fexp
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, b:R
Hb0:(0 < b)%R
Hxb:x <> 0%R ->
(Rabs (round beta fexp rnd x - x) < b * Rabs x)%R
Hx0:x = 0%R
exists eps : R,
  (Rabs eps < b)%R /\
  round beta fexp rnd x = (x * (1 + eps))%R
beta:radix
fexp:Z -> Z
prop_exp:Valid_exp fexp
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, b:R
Hb0:(0 < b)%R
Hxb:x <> 0%R ->
(Rabs (round beta fexp rnd x - x) < b * Rabs x)%R
Hx0:x <> 0%R
exists eps : R,
  (Rabs eps < b)%R /\
  round beta fexp rnd x = (x * (1 + eps))%R
(* *)
exists 0%R.beta:radix
fexp:Z -> Z
prop_exp:Valid_exp fexp
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, b:R
Hb0:(0 < b)%R
Hxb:x <> 0%R ->
(Rabs (round beta fexp rnd x - x) < b * Rabs x)%R
Hx0:x = 0%R
(Rabs 0 < b)%R /\
round beta fexp rnd x = (x * (1 + 0))%R
beta:radix
fexp:Z -> Z
prop_exp:Valid_exp fexp
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, b:R
Hb0:(0 < b)%R
Hxb:x <> 0%R ->
(Rabs (round beta fexp rnd x - x) < b * Rabs x)%R
Hx0:x <> 0%R
exists eps : R,
  (Rabs eps < b)%R /\
  round beta fexp rnd x = (x * (1 + eps))%R
split.beta:radix
fexp:Z -> Z
prop_exp:Valid_exp fexp
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, b:R
Hb0:(0 < b)%R
Hxb:x <> 0%R ->
(Rabs (round beta fexp rnd x - x) < b * Rabs x)%R
Hx0:x = 0%R
(Rabs 0 < b)%R
beta:radix
fexp:Z -> Z
prop_exp:Valid_exp fexp
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, b:R
Hb0:(0 < b)%R
Hxb:x <> 0%R ->
(Rabs (round beta fexp rnd x - x) < b * Rabs x)%R
Hx0:x = 0%R
round beta fexp rnd x = (x * (1 + 0))%R
beta:radix
fexp:Z -> Z
prop_exp:Valid_exp fexp
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, b:R
Hb0:(0 < b)%R
Hxb:x <> 0%R ->
(Rabs (round beta fexp rnd x - x) < b * Rabs x)%R
Hx0:x <> 0%R
exists eps : R,
  (Rabs eps < b)%R /\
  round beta fexp rnd x = (x * (1 + eps))%R
now rewrite Rabs_R0.beta:radix
fexp:Z -> Z
prop_exp:Valid_exp fexp
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, b:R
Hb0:(0 < b)%R
Hxb:x <> 0%R ->
(Rabs (round beta fexp rnd x - x) < b * Rabs x)%R
Hx0:x = 0%R
round beta fexp rnd x = (x * (1 + 0))%R
beta:radix
fexp:Z -> Z
prop_exp:Valid_exp fexp
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, b:R
Hb0:(0 < b)%R
Hxb:x <> 0%R ->
(Rabs (round beta fexp rnd x - x) < b * Rabs x)%R
Hx0:x <> 0%R
exists eps : R,
  (Rabs eps < b)%R /\
  round beta fexp rnd x = (x * (1 + eps))%R
rewrite Hx0, Rmult_0_l.beta:radix
fexp:Z -> Z
prop_exp:Valid_exp fexp
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, b:R
Hb0:(0 < b)%R
Hxb:x <> 0%R ->
(Rabs (round beta fexp rnd x - x) < b * Rabs x)%R
Hx0:x = 0%R
round beta fexp rnd 0 = 0%R
beta:radix
fexp:Z -> Z
prop_exp:Valid_exp fexp
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, b:R
Hb0:(0 < b)%R
Hxb:x <> 0%R ->
(Rabs (round beta fexp rnd x - x) < b * Rabs x)%R
Hx0:x <> 0%R
exists eps : R,
  (Rabs eps < b)%R /\
  round beta fexp rnd x = (x * (1 + eps))%R
apply round_0...beta:radix
fexp:Z -> Z
prop_exp:Valid_exp fexp
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, b:R
Hb0:(0 < b)%R
Hxb:x <> 0%R ->
(Rabs (round beta fexp rnd x - x) < b * Rabs x)%R
Hx0:x <> 0%R
exists eps : R,
  (Rabs eps < b)%R /\
  round beta fexp rnd x = (x * (1 + eps))%R
(* *)
specialize (Hxb Hx0).beta:radix
fexp:Z -> Z
prop_exp:Valid_exp fexp
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, b:R
Hb0:(0 < b)%R
Hxb:(Rabs (round beta fexp rnd x - x) < b * Rabs x)%R
Hx0:x <> 0%R
exists eps : R,
  (Rabs eps < b)%R /\
  round beta fexp rnd x = (x * (1 + eps))%R
exists ((round beta fexp rnd x - x) / x)%R.beta:radix
fexp:Z -> Z
prop_exp:Valid_exp fexp
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, b:R
Hb0:(0 < b)%R
Hxb:(Rabs (round beta fexp rnd x - x) < b * Rabs x)%R
Hx0:x <> 0%R
(Rabs ((round beta fexp rnd x - x) / x) < b)%R /\
round beta fexp rnd x =
(x * (1 + (round beta fexp rnd x - x) / x))%R
split.beta:radix
fexp:Z -> Z
prop_exp:Valid_exp fexp
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, b:R
Hb0:(0 < b)%R
Hxb:(Rabs (round beta fexp rnd x - x) < b * Rabs x)%R
Hx0:x <> 0%R
(Rabs ((round beta fexp rnd x - x) / x) < b)%R
beta:radix
fexp:Z -> Z
prop_exp:Valid_exp fexp
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, b:R
Hb0:(0 < b)%R
Hxb:(Rabs (round beta fexp rnd x - x) < b * Rabs x)%R
Hx0:x <> 0%R
round beta fexp rnd x =
(x * (1 + (round beta fexp rnd x - x) / x))%R 2: now field.beta:radix
fexp:Z -> Z
prop_exp:Valid_exp fexp
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, b:R
Hb0:(0 < b)%R
Hxb:(Rabs (round beta fexp rnd x - x) < b * Rabs x)%R
Hx0:x <> 0%R
(Rabs ((round beta fexp rnd x - x) / x) < b)%R
unfold Rdiv.beta:radix
fexp:Z -> Z
prop_exp:Valid_exp fexp
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, b:R
Hb0:(0 < b)%R
Hxb:(Rabs (round beta fexp rnd x - x) < b * Rabs x)%R
Hx0:x <> 0%R
(Rabs ((round beta fexp rnd x - x) * / x) < b)%R
rewrite Rabs_mult.beta:radix
fexp:Z -> Z
prop_exp:Valid_exp fexp
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, b:R
Hb0:(0 < b)%R
Hxb:(Rabs (round beta fexp rnd x - x) < b * Rabs x)%R
Hx0:x <> 0%R
(Rabs (round beta fexp rnd x - x) * Rabs (/ x) < b)%R
apply Rmult_lt_reg_r with (Rabs x).beta:radix
fexp:Z -> Z
prop_exp:Valid_exp fexp
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, b:R
Hb0:(0 < b)%R
Hxb:(Rabs (round beta fexp rnd x - x) < b * Rabs x)%R
Hx0:x <> 0%R
(0 < Rabs x)%R
beta:radix
fexp:Z -> Z
prop_exp:Valid_exp fexp
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, b:R
Hb0:(0 < b)%R
Hxb:(Rabs (round beta fexp rnd x - x) < b * Rabs x)%R
Hx0:x <> 0%R
(Rabs (round beta fexp rnd x - x) * Rabs (/ x) *
 Rabs x < b * Rabs x)%R
now apply Rabs_pos_lt.beta:radix
fexp:Z -> Z
prop_exp:Valid_exp fexp
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, b:R
Hb0:(0 < b)%R
Hxb:(Rabs (round beta fexp rnd x - x) < b * Rabs x)%R
Hx0:x <> 0%R
(Rabs (round beta fexp rnd x - x) * Rabs (/ x) *
 Rabs x < b * Rabs x)%R
rewrite Rmult_assoc, <- Rabs_mult.beta:radix
fexp:Z -> Z
prop_exp:Valid_exp fexp
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, b:R
Hb0:(0 < b)%R
Hxb:(Rabs (round beta fexp rnd x - x) < b * Rabs x)%R
Hx0:x <> 0%R
(Rabs (round beta fexp rnd x - x) * Rabs (/ x * x) <
 b * Rabs x)%R
rewrite Rinv_l with (1 := Hx0).beta:radix
fexp:Z -> Z
prop_exp:Valid_exp fexp
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, b:R
Hb0:(0 < b)%R
Hxb:(Rabs (round beta fexp rnd x - x) < b * Rabs x)%R
Hx0:x <> 0%R
(Rabs (round beta fexp rnd x - x) * Rabs 1 <
 b * Rabs x)%R
now rewrite Rabs_R1, Rmult_1_r.
Qed.

Lemma relative_error_le_conversion :
  forall x b, (0 <= b)%R ->
  (Rabs (round beta fexp rnd x - x) <= b * Rabs x)%R ->
  exists eps,
  (Rabs eps <= b)%R /\ round beta fexp rnd x = (x * (1 + eps))%R.beta:radix
fexp:Z -> Z
prop_exp:Valid_exp fexp
rnd:R -> Z
valid_rnd:Valid_rnd rnd
forall x b : R,
(0 <= b)%R ->
(Rabs (round beta fexp rnd x - x) <= b * Rabs x)%R ->
exists eps : R,
  (Rabs eps <= b)%R /\
  round beta fexp rnd x = (x * (1 + eps))%R
Proof with auto with typeclass_instances.beta:radix
fexp:Z -> Z
prop_exp:Valid_exp fexp
rnd:R -> Z
valid_rnd:Valid_rnd rnd
forall x b : R,
(0 <= b)%R ->
(Rabs (round beta fexp rnd x - x) <= b * Rabs x)%R ->
exists eps : R,
  (Rabs eps <= b)%R /\
  round beta fexp rnd x = (x * (1 + eps))%R
intros x b Hb0 Hxb.beta:radix
fexp:Z -> Z
prop_exp:Valid_exp fexp
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, b:R
Hb0:(0 <= b)%R
Hxb:(Rabs (round beta fexp rnd x - x) <= b * Rabs x)%R
exists eps : R,
  (Rabs eps <= b)%R /\
  round beta fexp rnd x = (x * (1 + eps))%R
destruct (Req_dec x 0) as [Hx0|Hx0].beta:radix
fexp:Z -> Z
prop_exp:Valid_exp fexp
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, b:R
Hb0:(0 <= b)%R
Hxb:(Rabs (round beta fexp rnd x - x) <= b * Rabs x)%R
Hx0:x = 0%R
exists eps : R,
  (Rabs eps <= b)%R /\
  round beta fexp rnd x = (x * (1 + eps))%R
beta:radix
fexp:Z -> Z
prop_exp:Valid_exp fexp
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, b:R
Hb0:(0 <= b)%R
Hxb:(Rabs (round beta fexp rnd x - x) <= b * Rabs x)%R
Hx0:x <> 0%R
exists eps : R,
  (Rabs eps <= b)%R /\
  round beta fexp rnd x = (x * (1 + eps))%R
(* *)
exists 0%R.beta:radix
fexp:Z -> Z
prop_exp:Valid_exp fexp
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, b:R
Hb0:(0 <= b)%R
Hxb:(Rabs (round beta fexp rnd x - x) <= b * Rabs x)%R
Hx0:x = 0%R
(Rabs 0 <= b)%R /\
round beta fexp rnd x = (x * (1 + 0))%R
beta:radix
fexp:Z -> Z
prop_exp:Valid_exp fexp
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, b:R
Hb0:(0 <= b)%R
Hxb:(Rabs (round beta fexp rnd x - x) <= b * Rabs x)%R
Hx0:x <> 0%R
exists eps : R,
  (Rabs eps <= b)%R /\
  round beta fexp rnd x = (x * (1 + eps))%R
split.beta:radix
fexp:Z -> Z
prop_exp:Valid_exp fexp
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, b:R
Hb0:(0 <= b)%R
Hxb:(Rabs (round beta fexp rnd x - x) <= b * Rabs x)%R
Hx0:x = 0%R
(Rabs 0 <= b)%R
beta:radix
fexp:Z -> Z
prop_exp:Valid_exp fexp
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, b:R
Hb0:(0 <= b)%R
Hxb:(Rabs (round beta fexp rnd x - x) <= b * Rabs x)%R
Hx0:x = 0%R
round beta fexp rnd x = (x * (1 + 0))%R
beta:radix
fexp:Z -> Z
prop_exp:Valid_exp fexp
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, b:R
Hb0:(0 <= b)%R
Hxb:(Rabs (round beta fexp rnd x - x) <= b * Rabs x)%R
Hx0:x <> 0%R
exists eps : R,
  (Rabs eps <= b)%R /\
  round beta fexp rnd x = (x * (1 + eps))%R
now rewrite Rabs_R0.beta:radix
fexp:Z -> Z
prop_exp:Valid_exp fexp
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, b:R
Hb0:(0 <= b)%R
Hxb:(Rabs (round beta fexp rnd x - x) <= b * Rabs x)%R
Hx0:x = 0%R
round beta fexp rnd x = (x * (1 + 0))%R
beta:radix
fexp:Z -> Z
prop_exp:Valid_exp fexp
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, b:R
Hb0:(0 <= b)%R
Hxb:(Rabs (round beta fexp rnd x - x) <= b * Rabs x)%R
Hx0:x <> 0%R
exists eps : R,
  (Rabs eps <= b)%R /\
  round beta fexp rnd x = (x * (1 + eps))%R
rewrite Hx0, Rmult_0_l.beta:radix
fexp:Z -> Z
prop_exp:Valid_exp fexp
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, b:R
Hb0:(0 <= b)%R
Hxb:(Rabs (round beta fexp rnd x - x) <= b * Rabs x)%R
Hx0:x = 0%R
round beta fexp rnd 0 = 0%R
beta:radix
fexp:Z -> Z
prop_exp:Valid_exp fexp
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, b:R
Hb0:(0 <= b)%R
Hxb:(Rabs (round beta fexp rnd x - x) <= b * Rabs x)%R
Hx0:x <> 0%R
exists eps : R,
  (Rabs eps <= b)%R /\
  round beta fexp rnd x = (x * (1 + eps))%R
apply round_0...beta:radix
fexp:Z -> Z
prop_exp:Valid_exp fexp
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, b:R
Hb0:(0 <= b)%R
Hxb:(Rabs (round beta fexp rnd x - x) <= b * Rabs x)%R
Hx0:x <> 0%R
exists eps : R,
  (Rabs eps <= b)%R /\
  round beta fexp rnd x = (x * (1 + eps))%R
(* *)
exists ((round beta fexp rnd x - x) / x)%R.beta:radix
fexp:Z -> Z
prop_exp:Valid_exp fexp
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, b:R
Hb0:(0 <= b)%R
Hxb:(Rabs (round beta fexp rnd x - x) <= b * Rabs x)%R
Hx0:x <> 0%R
(Rabs ((round beta fexp rnd x - x) / x) <= b)%R /\
round beta fexp rnd x =
(x * (1 + (round beta fexp rnd x - x) / x))%R
split.beta:radix
fexp:Z -> Z
prop_exp:Valid_exp fexp
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, b:R
Hb0:(0 <= b)%R
Hxb:(Rabs (round beta fexp rnd x - x) <= b * Rabs x)%R
Hx0:x <> 0%R
(Rabs ((round beta fexp rnd x - x) / x) <= b)%R
beta:radix
fexp:Z -> Z
prop_exp:Valid_exp fexp
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, b:R
Hb0:(0 <= b)%R
Hxb:(Rabs (round beta fexp rnd x - x) <= b * Rabs x)%R
Hx0:x <> 0%R
round beta fexp rnd x =
(x * (1 + (round beta fexp rnd x - x) / x))%R 2: now field.beta:radix
fexp:Z -> Z
prop_exp:Valid_exp fexp
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, b:R
Hb0:(0 <= b)%R
Hxb:(Rabs (round beta fexp rnd x - x) <= b * Rabs x)%R
Hx0:x <> 0%R
(Rabs ((round beta fexp rnd x - x) / x) <= b)%R
unfold Rdiv.beta:radix
fexp:Z -> Z
prop_exp:Valid_exp fexp
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, b:R
Hb0:(0 <= b)%R
Hxb:(Rabs (round beta fexp rnd x - x) <= b * Rabs x)%R
Hx0:x <> 0%R
(Rabs ((round beta fexp rnd x - x) * / x) <= b)%R
rewrite Rabs_mult.beta:radix
fexp:Z -> Z
prop_exp:Valid_exp fexp
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, b:R
Hb0:(0 <= b)%R
Hxb:(Rabs (round beta fexp rnd x - x) <= b * Rabs x)%R
Hx0:x <> 0%R
(Rabs (round beta fexp rnd x - x) * Rabs (/ x) <= b)%R
apply Rmult_le_reg_r with (Rabs x).beta:radix
fexp:Z -> Z
prop_exp:Valid_exp fexp
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, b:R
Hb0:(0 <= b)%R
Hxb:(Rabs (round beta fexp rnd x - x) <= b * Rabs x)%R
Hx0:x <> 0%R
(0 < Rabs x)%R
beta:radix
fexp:Z -> Z
prop_exp:Valid_exp fexp
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, b:R
Hb0:(0 <= b)%R
Hxb:(Rabs (round beta fexp rnd x - x) <= b * Rabs x)%R
Hx0:x <> 0%R
(Rabs (round beta fexp rnd x - x) * Rabs (/ x) *
 Rabs x <= b * Rabs x)%R
now apply Rabs_pos_lt.beta:radix
fexp:Z -> Z
prop_exp:Valid_exp fexp
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, b:R
Hb0:(0 <= b)%R
Hxb:(Rabs (round beta fexp rnd x - x) <= b * Rabs x)%R
Hx0:x <> 0%R
(Rabs (round beta fexp rnd x - x) * Rabs (/ x) *
 Rabs x <= b * Rabs x)%R
rewrite Rmult_assoc, <- Rabs_mult.beta:radix
fexp:Z -> Z
prop_exp:Valid_exp fexp
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, b:R
Hb0:(0 <= b)%R
Hxb:(Rabs (round beta fexp rnd x - x) <= b * Rabs x)%R
Hx0:x <> 0%R
(Rabs (round beta fexp rnd x - x) * Rabs (/ x * x) <=
 b * Rabs x)%R
rewrite Rinv_l with (1 := Hx0).beta:radix
fexp:Z -> Z
prop_exp:Valid_exp fexp
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, b:R
Hb0:(0 <= b)%R
Hxb:(Rabs (round beta fexp rnd x - x) <= b * Rabs x)%R
Hx0:x <> 0%R
(Rabs (round beta fexp rnd x - x) * Rabs 1 <=
 b * Rabs x)%R
now rewrite Rabs_R1, Rmult_1_r.
Qed.

Lemma relative_error_le_conversion_inv :
  forall x b,
  (exists eps,
   (Rabs eps <= b)%R /\ round beta fexp rnd x = (x * (1 + eps))%R) ->
  (Rabs (round beta fexp rnd x - x) <= b * Rabs x)%R.beta:radix
fexp:Z -> Z
prop_exp:Valid_exp fexp
rnd:R -> Z
valid_rnd:Valid_rnd rnd
forall x b : R,
(exists eps : R,
   (Rabs eps <= b)%R /\
   round beta fexp rnd x = (x * (1 + eps))%R) ->
(Rabs (round beta fexp rnd x - x) <= b * Rabs x)%R
Proof with auto with typeclass_instances.beta:radix
fexp:Z -> Z
prop_exp:Valid_exp fexp
rnd:R -> Z
valid_rnd:Valid_rnd rnd
forall x b : R,
(exists eps : R,
   (Rabs eps <= b)%R /\
   round beta fexp rnd x = (x * (1 + eps))%R) ->
(Rabs (round beta fexp rnd x - x) <= b * Rabs x)%R
intros x b (eps, (Beps, Heps)).beta:radix
fexp:Z -> Z
prop_exp:Valid_exp fexp
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, b, eps:R
Beps:(Rabs eps <= b)%R
Heps:round beta fexp rnd x = (x * (1 + eps))%R
(Rabs (round beta fexp rnd x - x) <= b * Rabs x)%R
assert (Pb : (0 <= b)%R); [now revert Beps; apply Rle_trans, Rabs_pos|].beta:radix
fexp:Z -> Z
prop_exp:Valid_exp fexp
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, b, eps:R
Beps:(Rabs eps <= b)%R
Heps:round beta fexp rnd x = (x * (1 + eps))%R
Pb:(0 <= b)%R
(Rabs (round beta fexp rnd x - x) <= b * Rabs x)%R
rewrite Heps; replace (_ - _)%R with (eps * x)%R; [|ring].beta:radix
fexp:Z -> Z
prop_exp:Valid_exp fexp
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, b, eps:R
Beps:(Rabs eps <= b)%R
Heps:round beta fexp rnd x = (x * (1 + eps))%R
Pb:(0 <= b)%R
(Rabs (eps * x) <= b * Rabs x)%R
now rewrite Rabs_mult; apply Rmult_le_compat_r; [apply Rabs_pos|].
Qed.

Lemma relative_error_le_conversion_round_inv :
  forall x b,
  (exists eps,
   (Rabs eps <= b)%R /\ x = (round beta fexp rnd x * (1 + eps))%R) ->
  (Rabs (round beta fexp rnd x - x) <= b * Rabs (round beta fexp rnd x))%R.beta:radix
fexp:Z -> Z
prop_exp:Valid_exp fexp
rnd:R -> Z
valid_rnd:Valid_rnd rnd
forall x b : R,
(exists eps : R,
   (Rabs eps <= b)%R /\
   x = (round beta fexp rnd x * (1 + eps))%R) ->
(Rabs (round beta fexp rnd x - x) <=
 b * Rabs (round beta fexp rnd x))%R
Proof with auto with typeclass_instances.beta:radix
fexp:Z -> Z
prop_exp:Valid_exp fexp
rnd:R -> Z
valid_rnd:Valid_rnd rnd
forall x b : R,
(exists eps : R,
   (Rabs eps <= b)%R /\
   x = (round beta fexp rnd x * (1 + eps))%R) ->
(Rabs (round beta fexp rnd x - x) <=
 b * Rabs (round beta fexp rnd x))%R
intros x b.beta:radix
fexp:Z -> Z
prop_exp:Valid_exp fexp
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, b:R
(exists eps : R,
   (Rabs eps <= b)%R /\
   x = (round beta fexp rnd x * (1 + eps))%R) ->
(Rabs (round beta fexp rnd x - x) <=
 b * Rabs (round beta fexp rnd x))%R
set (rx := round _ _ _ _).beta:radix
fexp:Z -> Z
prop_exp:Valid_exp fexp
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, b:R
rx:=round beta fexp rnd x:R
(exists eps : R,
   (Rabs eps <= b)%R /\ x = (rx * (1 + eps))%R) ->
(Rabs (rx - x) <= b * Rabs rx)%R
intros (eps, (Beps, Heps)).beta:radix
fexp:Z -> Z
prop_exp:Valid_exp fexp
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, b:R
rx:=round beta fexp rnd x:R
eps:R
Beps:(Rabs eps <= b)%R
Heps:x = (rx * (1 + eps))%R
(Rabs (rx - x) <= b * Rabs rx)%R
assert (Pb : (0 <= b)%R); [now revert Beps; apply Rle_trans, Rabs_pos|].beta:radix
fexp:Z -> Z
prop_exp:Valid_exp fexp
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, b:R
rx:=round beta fexp rnd x:R
eps:R
Beps:(Rabs eps <= b)%R
Heps:x = (rx * (1 + eps))%R
Pb:(0 <= b)%R
(Rabs (rx - x) <= b * Rabs rx)%R
rewrite Heps; replace (_ - _)%R with (- (eps * rx))%R; [|ring].beta:radix
fexp:Z -> Z
prop_exp:Valid_exp fexp
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, b:R
rx:=round beta fexp rnd x:R
eps:R
Beps:(Rabs eps <= b)%R
Heps:x = (rx * (1 + eps))%R
Pb:(0 <= b)%R
(Rabs (- (eps * rx)) <= b * Rabs rx)%R
now rewrite Rabs_Ropp, Rabs_mult; apply Rmult_le_compat_r; [apply Rabs_pos|].
Qed.

End relative_error_conversion.

Variable emin p : Z.
Hypothesis Hmin : forall k, (emin < k)%Z -> (p <= k - fexp k)%Z.

Variable rnd : R -> Z.
Context { valid_rnd : Valid_rnd rnd }.

Theorem relative_error :
  forall x,
  (bpow emin <= Rabs x)%R ->
  (Rabs (round beta fexp rnd x - x) < bpow (-p + 1) * Rabs x)%R.beta:radix
fexp:Z -> Z
prop_exp:Valid_exp fexp
emin, p:Z
Hmin:forall k : Z,
(emin < k)%Z -> (p <= k - fexp k)%Z
rnd:R -> Z
valid_rnd:Valid_rnd rnd
forall x : R,
(bpow emin <= Rabs x)%R ->
(Rabs (round beta fexp rnd x - x) <
 bpow (- p + 1) * Rabs x)%R
Proof with auto with typeclass_instances.beta:radix
fexp:Z -> Z
prop_exp:Valid_exp fexp
emin, p:Z
Hmin:forall k : Z,
(emin < k)%Z -> (p <= k - fexp k)%Z
rnd:R -> Z
valid_rnd:Valid_rnd rnd
forall x : R,
(bpow emin <= Rabs x)%R ->
(Rabs (round beta fexp rnd x - x) <
 bpow (- p + 1) * Rabs x)%R
intros x Hx.beta:radix
fexp:Z -> Z
prop_exp:Valid_exp fexp
emin, p:Z
Hmin:forall k : Z,
(emin < k)%Z -> (p <= k - fexp k)%Z
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x:R
Hx:(bpow emin <= Rabs x)%R
(Rabs (round beta fexp rnd x - x) <
 bpow (- p + 1) * Rabs x)%R
assert (Hx': (x <> 0)%R).beta:radix
fexp:Z -> Z
prop_exp:Valid_exp fexp
emin, p:Z
Hmin:forall k : Z,
(emin < k)%Z -> (p <= k - fexp k)%Z
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x:R
Hx:(bpow emin <= Rabs x)%R
x <> 0%R
beta:radix
fexp:Z -> Z
prop_exp:Valid_exp fexp
emin, p:Z
Hmin:forall k : Z,
(emin < k)%Z -> (p <= k - fexp k)%Z
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x:R
Hx:(bpow emin <= Rabs x)%R
Hx':x <> 0%R
(Rabs (round beta fexp rnd x - x) <
 bpow (- p + 1) * Rabs x)%R
intros T; contradict Hx; rewrite T, Rabs_R0.beta:radix
fexp:Z -> Z
prop_exp:Valid_exp fexp
emin, p:Z
Hmin:forall k : Z,
(emin < k)%Z -> (p <= k - fexp k)%Z
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x:R
T:x = 0%R
~ (bpow emin <= 0)%R
beta:radix
fexp:Z -> Z
prop_exp:Valid_exp fexp
emin, p:Z
Hmin:forall k : Z,
(emin < k)%Z -> (p <= k - fexp k)%Z
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x:R
Hx:(bpow emin <= Rabs x)%R
Hx':x <> 0%R
(Rabs (round beta fexp rnd x - x) <
 bpow (- p + 1) * Rabs x)%R
apply Rlt_not_le, bpow_gt_0.beta:radix
fexp:Z -> Z
prop_exp:Valid_exp fexp
emin, p:Z
Hmin:forall k : Z,
(emin < k)%Z -> (p <= k - fexp k)%Z
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x:R
Hx:(bpow emin <= Rabs x)%R
Hx':x <> 0%R
(Rabs (round beta fexp rnd x - x) <
 bpow (- p + 1) * Rabs x)%R
apply Rlt_le_trans with (ulp beta fexp x)%R.beta:radix
fexp:Z -> Z
prop_exp:Valid_exp fexp
emin, p:Z
Hmin:forall k : Z,
(emin < k)%Z -> (p <= k - fexp k)%Z
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x:R
Hx:(bpow emin <= Rabs x)%R
Hx':x <> 0%R
(Rabs (round beta fexp rnd x - x) < ulp beta fexp x)%R
beta:radix
fexp:Z -> Z
prop_exp:Valid_exp fexp
emin, p:Z
Hmin:forall k : Z,
(emin < k)%Z -> (p <= k - fexp k)%Z
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x:R
Hx:(bpow emin <= Rabs x)%R
Hx':x <> 0%R
(ulp beta fexp x <= bpow (- p + 1) * Rabs x)%R
now apply error_lt_ulp...beta:radix
fexp:Z -> Z
prop_exp:Valid_exp fexp
emin, p:Z
Hmin:forall k : Z,
(emin < k)%Z -> (p <= k - fexp k)%Z
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x:R
Hx:(bpow emin <= Rabs x)%R
Hx':x <> 0%R
(ulp beta fexp x <= bpow (- p + 1) * Rabs x)%R
rewrite ulp_neq_0; trivial.beta:radix
fexp:Z -> Z
prop_exp:Valid_exp fexp
emin, p:Z
Hmin:forall k : Z,
(emin < k)%Z -> (p <= k - fexp k)%Z
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x:R
Hx:(bpow emin <= Rabs x)%R
Hx':x <> 0%R
(bpow (cexp beta fexp x) <= bpow (- p + 1) * Rabs x)%R
unfold cexp.beta:radix
fexp:Z -> Z
prop_exp:Valid_exp fexp
emin, p:Z
Hmin:forall k : Z,
(emin < k)%Z -> (p <= k - fexp k)%Z
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x:R
Hx:(bpow emin <= Rabs x)%R
Hx':x <> 0%R
(bpow (fexp (mag beta x)) <= bpow (- p + 1) * Rabs x)%R
destruct (mag beta x) as (ex, He).beta:radix
fexp:Z -> Z
prop_exp:Valid_exp fexp
emin, p:Z
Hmin:forall k : Z,
(emin < k)%Z -> (p <= k - fexp k)%Z
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x:R
Hx:(bpow emin <= Rabs x)%R
Hx':x <> 0%R
ex:Z
He:x <> 0%R -> (bpow (ex - 1) <= Rabs x < bpow ex)%R
(bpow (fexp (Build_mag_prop beta x ex He)) <=
 bpow (- p + 1) * Rabs x)%R
simpl.beta:radix
fexp:Z -> Z
prop_exp:Valid_exp fexp
emin, p:Z
Hmin:forall k : Z,
(emin < k)%Z -> (p <= k - fexp k)%Z
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x:R
Hx:(bpow emin <= Rabs x)%R
Hx':x <> 0%R
ex:Z
He:x <> 0%R -> (bpow (ex - 1) <= Rabs x < bpow ex)%R
(bpow (fexp ex) <= bpow (- p + 1) * Rabs x)%R
specialize (He Hx').beta:radix
fexp:Z -> Z
prop_exp:Valid_exp fexp
emin, p:Z
Hmin:forall k : Z,
(emin < k)%Z -> (p <= k - fexp k)%Z
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x:R
Hx:(bpow emin <= Rabs x)%R
Hx':x <> 0%R
ex:Z
He:(bpow (ex - 1) <= Rabs x < bpow ex)%R
(bpow (fexp ex) <= bpow (- p + 1) * Rabs x)%R
apply Rle_trans with (bpow (-p + 1) * bpow (ex - 1))%R.beta:radix
fexp:Z -> Z
prop_exp:Valid_exp fexp
emin, p:Z
Hmin:forall k : Z,
(emin < k)%Z -> (p <= k - fexp k)%Z
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x:R
Hx:(bpow emin <= Rabs x)%R
Hx':x <> 0%R
ex:Z
He:(bpow (ex - 1) <= Rabs x < bpow ex)%R
(bpow (fexp ex) <= bpow (- p + 1) * bpow (ex - 1))%R
beta:radix
fexp:Z -> Z
prop_exp:Valid_exp fexp
emin, p:Z
Hmin:forall k : Z,
(emin < k)%Z -> (p <= k - fexp k)%Z
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x:R
Hx:(bpow emin <= Rabs x)%R
Hx':x <> 0%R
ex:Z
He:(bpow (ex - 1) <= Rabs x < bpow ex)%R
(bpow (- p + 1) * bpow (ex - 1) <=
 bpow (- p + 1) * Rabs x)%R
rewrite <- bpow_plus.beta:radix
fexp:Z -> Z
prop_exp:Valid_exp fexp
emin, p:Z
Hmin:forall k : Z,
(emin < k)%Z -> (p <= k - fexp k)%Z
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x:R
Hx:(bpow emin <= Rabs x)%R
Hx':x <> 0%R
ex:Z
He:(bpow (ex - 1) <= Rabs x < bpow ex)%R
(bpow (fexp ex) <= bpow (- p + 1 + (ex - 1)))%R
beta:radix
fexp:Z -> Z
prop_exp:Valid_exp fexp
emin, p:Z
Hmin:forall k : Z,
(emin < k)%Z -> (p <= k - fexp k)%Z
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x:R
Hx:(bpow emin <= Rabs x)%R
Hx':x <> 0%R
ex:Z
He:(bpow (ex - 1) <= Rabs x < bpow ex)%R
(bpow (- p + 1) * bpow (ex - 1) <=
 bpow (- p + 1) * Rabs x)%R
apply bpow_le.beta:radix
fexp:Z -> Z
prop_exp:Valid_exp fexp
emin, p:Z
Hmin:forall k : Z,
(emin < k)%Z -> (p <= k - fexp k)%Z
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x:R
Hx:(bpow emin <= Rabs x)%R
Hx':x <> 0%R
ex:Z
He:(bpow (ex - 1) <= Rabs x < bpow ex)%R
(fexp ex <= - p + 1 + (ex - 1))%Z
beta:radix
fexp:Z -> Z
prop_exp:Valid_exp fexp
emin, p:Z
Hmin:forall k : Z,
(emin < k)%Z -> (p <= k - fexp k)%Z
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x:R
Hx:(bpow emin <= Rabs x)%R
Hx':x <> 0%R
ex:Z
He:(bpow (ex - 1) <= Rabs x < bpow ex)%R
(bpow (- p + 1) * bpow (ex - 1) <=
 bpow (- p + 1) * Rabs x)%R
assert (emin < ex)%Z.beta:radix
fexp:Z -> Z
prop_exp:Valid_exp fexp
emin, p:Z
Hmin:forall k : Z,
(emin < k)%Z -> (p <= k - fexp k)%Z
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x:R
Hx:(bpow emin <= Rabs x)%R
Hx':x <> 0%R
ex:Z
He:(bpow (ex - 1) <= Rabs x < bpow ex)%R
(emin < ex)%Z
beta:radix
fexp:Z -> Z
prop_exp:Valid_exp fexp
emin, p:Z
Hmin:forall k : Z,
(emin < k)%Z -> (p <= k - fexp k)%Z
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x:R
Hx:(bpow emin <= Rabs x)%R
Hx':x <> 0%R
ex:Z
He:(bpow (ex - 1) <= Rabs x < bpow ex)%R
H:(emin < ex)%Z
(fexp ex <= - p + 1 + (ex - 1))%Z
beta:radix
fexp:Z -> Z
prop_exp:Valid_exp fexp
emin, p:Z
Hmin:forall k : Z,
(emin < k)%Z -> (p <= k - fexp k)%Z
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x:R
Hx:(bpow emin <= Rabs x)%R
Hx':x <> 0%R
ex:Z
He:(bpow (ex - 1) <= Rabs x < bpow ex)%R
(bpow (- p + 1) * bpow (ex - 1) <=
 bpow (- p + 1) * Rabs x)%R
apply (lt_bpow beta).beta:radix
fexp:Z -> Z
prop_exp:Valid_exp fexp
emin, p:Z
Hmin:forall k : Z,
(emin < k)%Z -> (p <= k - fexp k)%Z
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x:R
Hx:(bpow emin <= Rabs x)%R
Hx':x <> 0%R
ex:Z
He:(bpow (ex - 1) <= Rabs x < bpow ex)%R
(bpow emin < bpow ex)%R
beta:radix
fexp:Z -> Z
prop_exp:Valid_exp fexp
emin, p:Z
Hmin:forall k : Z,
(emin < k)%Z -> (p <= k - fexp k)%Z
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x:R
Hx:(bpow emin <= Rabs x)%R
Hx':x <> 0%R
ex:Z
He:(bpow (ex - 1) <= Rabs x < bpow ex)%R
H:(emin < ex)%Z
(fexp ex <= - p + 1 + (ex - 1))%Z
beta:radix
fexp:Z -> Z
prop_exp:Valid_exp fexp
emin, p:Z
Hmin:forall k : Z,
(emin < k)%Z -> (p <= k - fexp k)%Z
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x:R
Hx:(bpow emin <= Rabs x)%R
Hx':x <> 0%R
ex:Z
He:(bpow (ex - 1) <= Rabs x < bpow ex)%R
(bpow (- p + 1) * bpow (ex - 1) <=
 bpow (- p + 1) * Rabs x)%R
apply Rle_lt_trans with (2 := proj2 He).beta:radix
fexp:Z -> Z
prop_exp:Valid_exp fexp
emin, p:Z
Hmin:forall k : Z,
(emin < k)%Z -> (p <= k - fexp k)%Z
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x:R
Hx:(bpow emin <= Rabs x)%R
Hx':x <> 0%R
ex:Z
He:(bpow (ex - 1) <= Rabs x < bpow ex)%R
(bpow emin <= Rabs x)%R
beta:radix
fexp:Z -> Z
prop_exp:Valid_exp fexp
emin, p:Z
Hmin:forall k : Z,
(emin < k)%Z -> (p <= k - fexp k)%Z
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x:R
Hx:(bpow emin <= Rabs x)%R
Hx':x <> 0%R
ex:Z
He:(bpow (ex - 1) <= Rabs x < bpow ex)%R
H:(emin < ex)%Z
(fexp ex <= - p + 1 + (ex - 1))%Z
beta:radix
fexp:Z -> Z
prop_exp:Valid_exp fexp
emin, p:Z
Hmin:forall k : Z,
(emin < k)%Z -> (p <= k - fexp k)%Z
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x:R
Hx:(bpow emin <= Rabs x)%R
Hx':x <> 0%R
ex:Z
He:(bpow (ex - 1) <= Rabs x < bpow ex)%R
(bpow (- p + 1) * bpow (ex - 1) <=
 bpow (- p + 1) * Rabs x)%R
exact Hx.beta:radix
fexp:Z -> Z
prop_exp:Valid_exp fexp
emin, p:Z
Hmin:forall k : Z,
(emin < k)%Z -> (p <= k - fexp k)%Z
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x:R
Hx:(bpow emin <= Rabs x)%R
Hx':x <> 0%R
ex:Z
He:(bpow (ex - 1) <= Rabs x < bpow ex)%R
H:(emin < ex)%Z
(fexp ex <= - p + 1 + (ex - 1))%Z
beta:radix
fexp:Z -> Z
prop_exp:Valid_exp fexp
emin, p:Z
Hmin:forall k : Z,
(emin < k)%Z -> (p <= k - fexp k)%Z
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x:R
Hx:(bpow emin <= Rabs x)%R
Hx':x <> 0%R
ex:Z
He:(bpow (ex - 1) <= Rabs x < bpow ex)%R
(bpow (- p + 1) * bpow (ex - 1) <=
 bpow (- p + 1) * Rabs x)%R
generalize (Hmin ex).beta:radix
fexp:Z -> Z
prop_exp:Valid_exp fexp
emin, p:Z
Hmin:forall k : Z,
(emin < k)%Z -> (p <= k - fexp k)%Z
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x:R
Hx:(bpow emin <= Rabs x)%R
Hx':x <> 0%R
ex:Z
He:(bpow (ex - 1) <= Rabs x < bpow ex)%R
H:(emin < ex)%Z
((emin < ex)%Z -> (p <= ex - fexp ex)%Z) ->
(fexp ex <= - p + 1 + (ex - 1))%Z
beta:radix
fexp:Z -> Z
prop_exp:Valid_exp fexp
emin, p:Z
Hmin:forall k : Z,
(emin < k)%Z -> (p <= k - fexp k)%Z
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x:R
Hx:(bpow emin <= Rabs x)%R
Hx':x <> 0%R
ex:Z
He:(bpow (ex - 1) <= Rabs x < bpow ex)%R
(bpow (- p + 1) * bpow (ex - 1) <=
 bpow (- p + 1) * Rabs x)%R
omega.beta:radix
fexp:Z -> Z
prop_exp:Valid_exp fexp
emin, p:Z
Hmin:forall k : Z,
(emin < k)%Z -> (p <= k - fexp k)%Z
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x:R
Hx:(bpow emin <= Rabs x)%R
Hx':x <> 0%R
ex:Z
He:(bpow (ex - 1) <= Rabs x < bpow ex)%R
(bpow (- p + 1) * bpow (ex - 1) <=
 bpow (- p + 1) * Rabs x)%R
apply Rmult_le_compat_l.beta:radix
fexp:Z -> Z
prop_exp:Valid_exp fexp
emin, p:Z
Hmin:forall k : Z,
(emin < k)%Z -> (p <= k - fexp k)%Z
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x:R
Hx:(bpow emin <= Rabs x)%R
Hx':x <> 0%R
ex:Z
He:(bpow (ex - 1) <= Rabs x < bpow ex)%R
(0 <= bpow (- p + 1))%R
beta:radix
fexp:Z -> Z
prop_exp:Valid_exp fexp
emin, p:Z
Hmin:forall k : Z,
(emin < k)%Z -> (p <= k - fexp k)%Z
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x:R
Hx:(bpow emin <= Rabs x)%R
Hx':x <> 0%R
ex:Z
He:(bpow (ex - 1) <= Rabs x < bpow ex)%R
(bpow (ex - 1) <= Rabs x)%R
apply bpow_ge_0.beta:radix
fexp:Z -> Z
prop_exp:Valid_exp fexp
emin, p:Z
Hmin:forall k : Z,
(emin < k)%Z -> (p <= k - fexp k)%Z
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x:R
Hx:(bpow emin <= Rabs x)%R
Hx':x <> 0%R
ex:Z
He:(bpow (ex - 1) <= Rabs x < bpow ex)%R
(bpow (ex - 1) <= Rabs x)%R
apply He.
Qed.

1+ε property in any rounding

Theorem relative_error_ex :
  forall x,
  (bpow emin <= Rabs x)%R ->
  exists eps,
  (Rabs eps < bpow (-p + 1))%R /\ round beta fexp rnd x = (x * (1 + eps))%R.beta:radix
fexp:Z -> Z
prop_exp:Valid_exp fexp
emin, p:Z
Hmin:forall k : Z,
(emin < k)%Z -> (p <= k - fexp k)%Z
rnd:R -> Z
valid_rnd:Valid_rnd rnd
forall x : R,
(bpow emin <= Rabs x)%R ->
exists eps : R,
  (Rabs eps < bpow (- p + 1))%R /\
  round beta fexp rnd x = (x * (1 + eps))%R
Proof with auto with typeclass_instances.beta:radix
fexp:Z -> Z
prop_exp:Valid_exp fexp
emin, p:Z
Hmin:forall k : Z,
(emin < k)%Z -> (p <= k - fexp k)%Z
rnd:R -> Z
valid_rnd:Valid_rnd rnd
forall x : R,
(bpow emin <= Rabs x)%R ->
exists eps : R,
  (Rabs eps < bpow (- p + 1))%R /\
  round beta fexp rnd x = (x * (1 + eps))%R
intros x Hx.beta:radix
fexp:Z -> Z
prop_exp:Valid_exp fexp
emin, p:Z
Hmin:forall k : Z,
(emin < k)%Z -> (p <= k - fexp k)%Z
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x:R
Hx:(bpow emin <= Rabs x)%R
exists eps : R,
  (Rabs eps < bpow (- p + 1))%R /\
  round beta fexp rnd x = (x * (1 + eps))%R
apply relative_error_lt_conversion...beta:radix
fexp:Z -> Z
prop_exp:Valid_exp fexp
emin, p:Z
Hmin:forall k : Z,
(emin < k)%Z -> (p <= k - fexp k)%Z
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x:R
Hx:(bpow emin <= Rabs x)%R
(0 < bpow (- p + 1))%R
beta:radix
fexp:Z -> Z
prop_exp:Valid_exp fexp
emin, p:Z
Hmin:forall k : Z,
(emin < k)%Z -> (p <= k - fexp k)%Z
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x:R
Hx:(bpow emin <= Rabs x)%R
x <> 0%R ->
(Rabs (round beta fexp rnd x - x) <
 bpow (- p + 1) * Rabs x)%R
apply bpow_gt_0.beta:radix
fexp:Z -> Z
prop_exp:Valid_exp fexp
emin, p:Z
Hmin:forall k : Z,
(emin < k)%Z -> (p <= k - fexp k)%Z
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x:R
Hx:(bpow emin <= Rabs x)%R
x <> 0%R ->
(Rabs (round beta fexp rnd x - x) <
 bpow (- p + 1) * Rabs x)%R
intros _.beta:radix
fexp:Z -> Z
prop_exp:Valid_exp fexp
emin, p:Z
Hmin:forall k : Z,
(emin < k)%Z -> (p <= k - fexp k)%Z
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x:R
Hx:(bpow emin <= Rabs x)%R
(Rabs (round beta fexp rnd x - x) <
 bpow (- p + 1) * Rabs x)%R
now apply relative_error.
Qed.

Theorem relative_error_F2R_emin :
  forall m, let x := F2R (Float beta m emin) in
  (x <> 0)%R ->
  (Rabs (round beta fexp rnd x - x) < bpow (-p + 1) * Rabs x)%R.beta:radix
fexp:Z -> Z
prop_exp:Valid_exp fexp
emin, p:Z
Hmin:forall k : Z,
(emin < k)%Z -> (p <= k - fexp k)%Z
rnd:R -> Z
valid_rnd:Valid_rnd rnd
forall m : Z,
let x := F2R {| Fnum := m; Fexp := emin |} in
x <> 0%R ->
(Rabs (round beta fexp rnd x - x) <
 bpow (- p + 1) * Rabs x)%R
Proof.beta:radix
fexp:Z -> Z
prop_exp:Valid_exp fexp
emin, p:Z
Hmin:forall k : Z,
(emin < k)%Z -> (p <= k - fexp k)%Z
rnd:R -> Z
valid_rnd:Valid_rnd rnd
forall m : Z,
let x := F2R {| Fnum := m; Fexp := emin |} in
x <> 0%R ->
(Rabs (round beta fexp rnd x - x) <
 bpow (- p + 1) * Rabs x)%R
intros m x Hx.beta:radix
fexp:Z -> Z
prop_exp:Valid_exp fexp
emin, p:Z
Hmin:forall k : Z,
(emin < k)%Z -> (p <= k - fexp k)%Z
rnd:R -> Z
valid_rnd:Valid_rnd rnd
m:Z
x:=F2R {| Fnum := m; Fexp := emin |}:R
Hx:x <> 0%R
(Rabs (round beta fexp rnd x - x) <
 bpow (- p + 1) * Rabs x)%R
apply relative_error.beta:radix
fexp:Z -> Z
prop_exp:Valid_exp fexp
emin, p:Z
Hmin:forall k : Z,
(emin < k)%Z -> (p <= k - fexp k)%Z
rnd:R -> Z
valid_rnd:Valid_rnd rnd
m:Z
x:=F2R {| Fnum := m; Fexp := emin |}:R
Hx:x <> 0%R
(bpow emin <= Rabs x)%R
unfold x.beta:radix
fexp:Z -> Z
prop_exp:Valid_exp fexp
emin, p:Z
Hmin:forall k : Z,
(emin < k)%Z -> (p <= k - fexp k)%Z
rnd:R -> Z
valid_rnd:Valid_rnd rnd
m:Z
x:=F2R {| Fnum := m; Fexp := emin |}:R
Hx:x <> 0%R
(bpow emin <= Rabs (F2R {| Fnum := m; Fexp := emin |}))%R
rewrite <- F2R_Zabs.beta:radix
fexp:Z -> Z
prop_exp:Valid_exp fexp
emin, p:Z
Hmin:forall k : Z,
(emin < k)%Z -> (p <= k - fexp k)%Z
rnd:R -> Z
valid_rnd:Valid_rnd rnd
m:Z
x:=F2R {| Fnum := m; Fexp := emin |}:R
Hx:x <> 0%R
(bpow emin <= F2R {| Fnum := Z.abs m; Fexp := emin |})%R
apply bpow_le_F2R.beta:radix
fexp:Z -> Z
prop_exp:Valid_exp fexp
emin, p:Z
Hmin:forall k : Z,
(emin < k)%Z -> (p <= k - fexp k)%Z
rnd:R -> Z
valid_rnd:Valid_rnd rnd
m:Z
x:=F2R {| Fnum := m; Fexp := emin |}:R
Hx:x <> 0%R
(0 < Z.abs m)%Z
apply lt_F2R with beta emin.beta:radix
fexp:Z -> Z
prop_exp:Valid_exp fexp
emin, p:Z
Hmin:forall k : Z,
(emin < k)%Z -> (p <= k - fexp k)%Z
rnd:R -> Z
valid_rnd:Valid_rnd rnd
m:Z
x:=F2R {| Fnum := m; Fexp := emin |}:R
Hx:x <> 0%R
(F2R {| Fnum := 0; Fexp := emin |} <
 F2R {| Fnum := Z.abs m; Fexp := emin |})%R
rewrite F2R_0, F2R_Zabs.beta:radix
fexp:Z -> Z
prop_exp:Valid_exp fexp
emin, p:Z
Hmin:forall k : Z,
(emin < k)%Z -> (p <= k - fexp k)%Z
rnd:R -> Z
valid_rnd:Valid_rnd rnd
m:Z
x:=F2R {| Fnum := m; Fexp := emin |}:R
Hx:x <> 0%R
(0 < Rabs (F2R {| Fnum := m; Fexp := emin |}))%R
now apply Rabs_pos_lt.
Qed.

Theorem relative_error_F2R_emin_ex :
  forall m, let x := F2R (Float beta m emin) in
  exists eps,
  (Rabs eps < bpow (-p + 1))%R /\ round beta fexp rnd x = (x * (1 + eps))%R.beta:radix
fexp:Z -> Z
prop_exp:Valid_exp fexp
emin, p:Z
Hmin:forall k : Z,
(emin < k)%Z -> (p <= k - fexp k)%Z
rnd:R -> Z
valid_rnd:Valid_rnd rnd
forall m : Z,
let x := F2R {| Fnum := m; Fexp := emin |} in
exists eps : R,
  (Rabs eps < bpow (- p + 1))%R /\
  round beta fexp rnd x = (x * (1 + eps))%R
Proof with auto with typeclass_instances.beta:radix
fexp:Z -> Z
prop_exp:Valid_exp fexp
emin, p:Z
Hmin:forall k : Z,
(emin < k)%Z -> (p <= k - fexp k)%Z
rnd:R -> Z
valid_rnd:Valid_rnd rnd
forall m : Z,
let x := F2R {| Fnum := m; Fexp := emin |} in
exists eps : R,
  (Rabs eps < bpow (- p + 1))%R /\
  round beta fexp rnd x = (x * (1 + eps))%R
intros m x.beta:radix
fexp:Z -> Z
prop_exp:Valid_exp fexp
emin, p:Z
Hmin:forall k : Z,
(emin < k)%Z -> (p <= k - fexp k)%Z
rnd:R -> Z
valid_rnd:Valid_rnd rnd
m:Z
x:=F2R {| Fnum := m; Fexp := emin |}:R
exists eps : R,
  (Rabs eps < bpow (- p + 1))%R /\
  round beta fexp rnd x = (x * (1 + eps))%R
apply relative_error_lt_conversion...beta:radix
fexp:Z -> Z
prop_exp:Valid_exp fexp
emin, p:Z
Hmin:forall k : Z,
(emin < k)%Z -> (p <= k - fexp k)%Z
rnd:R -> Z
valid_rnd:Valid_rnd rnd
m:Z
x:=F2R {| Fnum := m; Fexp := emin |}:R
(0 < bpow (- p + 1))%R
beta:radix
fexp:Z -> Z
prop_exp:Valid_exp fexp
emin, p:Z
Hmin:forall k : Z,
(emin < k)%Z -> (p <= k - fexp k)%Z
rnd:R -> Z
valid_rnd:Valid_rnd rnd
m:Z
x:=F2R {| Fnum := m; Fexp := emin |}:R
x <> 0%R ->
(Rabs (round beta fexp rnd x - x) <
 bpow (- p + 1) * Rabs x)%R
apply bpow_gt_0.beta:radix
fexp:Z -> Z
prop_exp:Valid_exp fexp
emin, p:Z
Hmin:forall k : Z,
(emin < k)%Z -> (p <= k - fexp k)%Z
rnd:R -> Z
valid_rnd:Valid_rnd rnd
m:Z
x:=F2R {| Fnum := m; Fexp := emin |}:R
x <> 0%R ->
(Rabs (round beta fexp rnd x - x) <
 bpow (- p + 1) * Rabs x)%R
now apply relative_error_F2R_emin.
Qed.

Theorem relative_error_round :
  (0 < p)%Z ->
  forall x,
  (bpow emin <= Rabs x)%R ->
  (Rabs (round beta fexp rnd x - x) < bpow (-p + 1) * Rabs (round beta fexp rnd x))%R.beta:radix
fexp:Z -> Z
prop_exp:Valid_exp fexp
emin, p:Z
Hmin:forall k : Z,
(emin < k)%Z -> (p <= k - fexp k)%Z
rnd:R -> Z
valid_rnd:Valid_rnd rnd
(0 < p)%Z ->
forall x : R,
(bpow emin <= Rabs x)%R ->
(Rabs (round beta fexp rnd x - x) <
 bpow (- p + 1) * Rabs (round beta fexp rnd x))%R
Proof with auto with typeclass_instances.beta:radix
fexp:Z -> Z
prop_exp:Valid_exp fexp
emin, p:Z
Hmin:forall k : Z,
(emin < k)%Z -> (p <= k - fexp k)%Z
rnd:R -> Z
valid_rnd:Valid_rnd rnd
(0 < p)%Z ->
forall x : R,
(bpow emin <= Rabs x)%R ->
(Rabs (round beta fexp rnd x - x) <
 bpow (- p + 1) * Rabs (round beta fexp rnd x))%R
intros Hp x Hx.beta:radix
fexp:Z -> Z
prop_exp:Valid_exp fexp
emin, p:Z
Hmin:forall k : Z,
(emin < k)%Z -> (p <= k - fexp k)%Z
rnd:R -> Z
valid_rnd:Valid_rnd rnd
Hp:(0 < p)%Z
x:R
Hx:(bpow emin <= Rabs x)%R
(Rabs (round beta fexp rnd x - x) <
 bpow (- p + 1) * Rabs (round beta fexp rnd x))%R
assert (Hx': (x <> 0)%R).beta:radix
fexp:Z -> Z
prop_exp:Valid_exp fexp
emin, p:Z
Hmin:forall k : Z,
(emin < k)%Z -> (p <= k - fexp k)%Z
rnd:R -> Z
valid_rnd:Valid_rnd rnd
Hp:(0 < p)%Z
x:R
Hx:(bpow emin <= Rabs x)%R
x <> 0%R
beta:radix
fexp:Z -> Z
prop_exp:Valid_exp fexp
emin, p:Z
Hmin:forall k : Z,
(emin < k)%Z -> (p <= k - fexp k)%Z
rnd:R -> Z
valid_rnd:Valid_rnd rnd
Hp:(0 < p)%Z
x:R
Hx:(bpow emin <= Rabs x)%R
Hx':x <> 0%R
(Rabs (round beta fexp rnd x - x) <
 bpow (- p + 1) * Rabs (round beta fexp rnd x))%R
intros T; contradict Hx; rewrite T, Rabs_R0.beta:radix
fexp:Z -> Z
prop_exp:Valid_exp fexp
emin, p:Z
Hmin:forall k : Z,
(emin < k)%Z -> (p <= k - fexp k)%Z
rnd:R -> Z
valid_rnd:Valid_rnd rnd
Hp:(0 < p)%Z
x:R
T:x = 0%R
~ (bpow emin <= 0)%R
beta:radix
fexp:Z -> Z
prop_exp:Valid_exp fexp
emin, p:Z
Hmin:forall k : Z,
(emin < k)%Z -> (p <= k - fexp k)%Z
rnd:R -> Z
valid_rnd:Valid_rnd rnd
Hp:(0 < p)%Z
x:R
Hx:(bpow emin <= Rabs x)%R
Hx':x <> 0%R
(Rabs (round beta fexp rnd x - x) <
 bpow (- p + 1) * Rabs (round beta fexp rnd x))%R
apply Rlt_not_le, bpow_gt_0.beta:radix
fexp:Z -> Z
prop_exp:Valid_exp fexp
emin, p:Z
Hmin:forall k : Z,
(emin < k)%Z -> (p <= k - fexp k)%Z
rnd:R -> Z
valid_rnd:Valid_rnd rnd
Hp:(0 < p)%Z
x:R
Hx:(bpow emin <= Rabs x)%R
Hx':x <> 0%R
(Rabs (round beta fexp rnd x - x) <
 bpow (- p + 1) * Rabs (round beta fexp rnd x))%R
apply Rlt_le_trans with (ulp beta fexp x)%R.beta:radix
fexp:Z -> Z
prop_exp:Valid_exp fexp
emin, p:Z
Hmin:forall k : Z,
(emin < k)%Z -> (p <= k - fexp k)%Z
rnd:R -> Z
valid_rnd:Valid_rnd rnd
Hp:(0 < p)%Z
x:R
Hx:(bpow emin <= Rabs x)%R
Hx':x <> 0%R
(Rabs (round beta fexp rnd x - x) < ulp beta fexp x)%R
beta:radix
fexp:Z -> Z
prop_exp:Valid_exp fexp
emin, p:Z
Hmin:forall k : Z,
(emin < k)%Z -> (p <= k - fexp k)%Z
rnd:R -> Z
valid_rnd:Valid_rnd rnd
Hp:(0 < p)%Z
x:R
Hx:(bpow emin <= Rabs x)%R
Hx':x <> 0%R
(ulp beta fexp x <=
 bpow (- p + 1) * Rabs (round beta fexp rnd x))%R
now apply error_lt_ulp.beta:radix
fexp:Z -> Z
prop_exp:Valid_exp fexp
emin, p:Z
Hmin:forall k : Z,
(emin < k)%Z -> (p <= k - fexp k)%Z
rnd:R -> Z
valid_rnd:Valid_rnd rnd
Hp:(0 < p)%Z
x:R
Hx:(bpow emin <= Rabs x)%R
Hx':x <> 0%R
(ulp beta fexp x <=
 bpow (- p + 1) * Rabs (round beta fexp rnd x))%R
rewrite ulp_neq_0; trivial.beta:radix
fexp:Z -> Z
prop_exp:Valid_exp fexp
emin, p:Z
Hmin:forall k : Z,
(emin < k)%Z -> (p <= k - fexp k)%Z
rnd:R -> Z
valid_rnd:Valid_rnd rnd
Hp:(0 < p)%Z
x:R
Hx:(bpow emin <= Rabs x)%R
Hx':x <> 0%R
(bpow (cexp beta fexp x) <=
 bpow (- p + 1) * Rabs (round beta fexp rnd x))%R
unfold cexp.beta:radix
fexp:Z -> Z
prop_exp:Valid_exp fexp
emin, p:Z
Hmin:forall k : Z,
(emin < k)%Z -> (p <= k - fexp k)%Z
rnd:R -> Z
valid_rnd:Valid_rnd rnd
Hp:(0 < p)%Z
x:R
Hx:(bpow emin <= Rabs x)%R
Hx':x <> 0%R
(bpow (fexp (mag beta x)) <=
 bpow (- p + 1) * Rabs (round beta fexp rnd x))%R
destruct (mag beta x) as (ex, He).beta:radix
fexp:Z -> Z
prop_exp:Valid_exp fexp
emin, p:Z
Hmin:forall k : Z,
(emin < k)%Z -> (p <= k - fexp k)%Z
rnd:R -> Z
valid_rnd:Valid_rnd rnd
Hp:(0 < p)%Z
x:R
Hx:(bpow emin <= Rabs x)%R
Hx':x <> 0%R
ex:Z
He:x <> 0%R -> (bpow (ex - 1) <= Rabs x < bpow ex)%R
(bpow (fexp (Build_mag_prop beta x ex He)) <=
 bpow (- p + 1) * Rabs (round beta fexp rnd x))%R
simpl.beta:radix
fexp:Z -> Z
prop_exp:Valid_exp fexp
emin, p:Z
Hmin:forall k : Z,
(emin < k)%Z -> (p <= k - fexp k)%Z
rnd:R -> Z
valid_rnd:Valid_rnd rnd
Hp:(0 < p)%Z
x:R
Hx:(bpow emin <= Rabs x)%R
Hx':x <> 0%R
ex:Z
He:x <> 0%R -> (bpow (ex - 1) <= Rabs x < bpow ex)%R
(bpow (fexp ex) <=
 bpow (- p + 1) * Rabs (round beta fexp rnd x))%R
specialize (He Hx').beta:radix
fexp:Z -> Z
prop_exp:Valid_exp fexp
emin, p:Z
Hmin:forall k : Z,
(emin < k)%Z -> (p <= k - fexp k)%Z
rnd:R -> Z
valid_rnd:Valid_rnd rnd
Hp:(0 < p)%Z
x:R
Hx:(bpow emin <= Rabs x)%R
Hx':x <> 0%R
ex:Z
He:(bpow (ex - 1) <= Rabs x < bpow ex)%R
(bpow (fexp ex) <=
 bpow (- p + 1) * Rabs (round beta fexp rnd x))%R
assert (He': (emin < ex)%Z).beta:radix
fexp:Z -> Z
prop_exp:Valid_exp fexp
emin, p:Z
Hmin:forall k : Z,
(emin < k)%Z -> (p <= k - fexp k)%Z
rnd:R -> Z
valid_rnd:Valid_rnd rnd
Hp:(0 < p)%Z
x:R
Hx:(bpow emin <= Rabs x)%R
Hx':x <> 0%R
ex:Z
He:(bpow (ex - 1) <= Rabs x < bpow ex)%R
(emin < ex)%Z
beta:radix
fexp:Z -> Z
prop_exp:Valid_exp fexp
emin, p:Z
Hmin:forall k : Z,
(emin < k)%Z -> (p <= k - fexp k)%Z
rnd:R -> Z
valid_rnd:Valid_rnd rnd
Hp:(0 < p)%Z
x:R
Hx:(bpow emin <= Rabs x)%R
Hx':x <> 0%R
ex:Z
He:(bpow (ex - 1) <= Rabs x < bpow ex)%R
He':(emin < ex)%Z
(bpow (fexp ex) <=
 bpow (- p + 1) * Rabs (round beta fexp rnd x))%R
apply (lt_bpow beta).beta:radix
fexp:Z -> Z
prop_exp:Valid_exp fexp
emin, p:Z
Hmin:forall k : Z,
(emin < k)%Z -> (p <= k - fexp k)%Z
rnd:R -> Z
valid_rnd:Valid_rnd rnd
Hp:(0 < p)%Z
x:R
Hx:(bpow emin <= Rabs x)%R
Hx':x <> 0%R
ex:Z
He:(bpow (ex - 1) <= Rabs x < bpow ex)%R
(bpow emin < bpow ex)%R
beta:radix
fexp:Z -> Z
prop_exp:Valid_exp fexp
emin, p:Z
Hmin:forall k : Z,
(emin < k)%Z -> (p <= k - fexp k)%Z
rnd:R -> Z
valid_rnd:Valid_rnd rnd
Hp:(0 < p)%Z
x:R
Hx:(bpow emin <= Rabs x)%R
Hx':x <> 0%R
ex:Z
He:(bpow (ex - 1) <= Rabs x < bpow ex)%R
He':(emin < ex)%Z
(bpow (fexp ex) <=
 bpow (- p + 1) * Rabs (round beta fexp rnd x))%R
apply Rle_lt_trans with (2 := proj2 He).beta:radix
fexp:Z -> Z
prop_exp:Valid_exp fexp
emin, p:Z
Hmin:forall k : Z,
(emin < k)%Z -> (p <= k - fexp k)%Z
rnd:R -> Z
valid_rnd:Valid_rnd rnd
Hp:(0 < p)%Z
x:R
Hx:(bpow emin <= Rabs x)%R
Hx':x <> 0%R
ex:Z
He:(bpow (ex - 1) <= Rabs x < bpow ex)%R
(bpow emin <= Rabs x)%R
beta:radix
fexp:Z -> Z
prop_exp:Valid_exp fexp
emin, p:Z
Hmin:forall k : Z,
(emin < k)%Z -> (p <= k - fexp k)%Z
rnd:R -> Z
valid_rnd:Valid_rnd rnd
Hp:(0 < p)%Z
x:R
Hx:(bpow emin <= Rabs x)%R
Hx':x <> 0%R
ex:Z
He:(bpow (ex - 1) <= Rabs x < bpow ex)%R
He':(emin < ex)%Z
(bpow (fexp ex) <=
 bpow (- p + 1) * Rabs (round beta fexp rnd x))%R
exact Hx.beta:radix
fexp:Z -> Z
prop_exp:Valid_exp fexp
emin, p:Z
Hmin:forall k : Z,
(emin < k)%Z -> (p <= k - fexp k)%Z
rnd:R -> Z
valid_rnd:Valid_rnd rnd
Hp:(0 < p)%Z
x:R
Hx:(bpow emin <= Rabs x)%R
Hx':x <> 0%R
ex:Z
He:(bpow (ex - 1) <= Rabs x < bpow ex)%R
He':(emin < ex)%Z
(bpow (fexp ex) <=
 bpow (- p + 1) * Rabs (round beta fexp rnd x))%R
apply Rle_trans with (bpow (-p + 1) * bpow (ex - 1))%R.beta:radix
fexp:Z -> Z
prop_exp:Valid_exp fexp
emin, p:Z
Hmin:forall k : Z,
(emin < k)%Z -> (p <= k - fexp k)%Z
rnd:R -> Z
valid_rnd:Valid_rnd rnd
Hp:(0 < p)%Z
x:R
Hx:(bpow emin <= Rabs x)%R
Hx':x <> 0%R
ex:Z
He:(bpow (ex - 1) <= Rabs x < bpow ex)%R
He':(emin < ex)%Z
(bpow (fexp ex) <= bpow (- p + 1) * bpow (ex - 1))%R
beta:radix
fexp:Z -> Z
prop_exp:Valid_exp fexp
emin, p:Z
Hmin:forall k : Z,
(emin < k)%Z -> (p <= k - fexp k)%Z
rnd:R -> Z
valid_rnd:Valid_rnd rnd
Hp:(0 < p)%Z
x:R
Hx:(bpow emin <= Rabs x)%R
Hx':x <> 0%R
ex:Z
He:(bpow (ex - 1) <= Rabs x < bpow ex)%R
He':(emin < ex)%Z
(bpow (- p + 1) * bpow (ex - 1) <=
 bpow (- p + 1) * Rabs (round beta fexp rnd x))%R
rewrite <- bpow_plus.beta:radix
fexp:Z -> Z
prop_exp:Valid_exp fexp
emin, p:Z
Hmin:forall k : Z,
(emin < k)%Z -> (p <= k - fexp k)%Z
rnd:R -> Z
valid_rnd:Valid_rnd rnd
Hp:(0 < p)%Z
x:R
Hx:(bpow emin <= Rabs x)%R
Hx':x <> 0%R
ex:Z
He:(bpow (ex - 1) <= Rabs x < bpow ex)%R
He':(emin < ex)%Z
(bpow (fexp ex) <= bpow (- p + 1 + (ex - 1)))%R
beta:radix
fexp:Z -> Z
prop_exp:Valid_exp fexp
emin, p:Z
Hmin:forall k : Z,
(emin < k)%Z -> (p <= k - fexp k)%Z
rnd:R -> Z
valid_rnd:Valid_rnd rnd
Hp:(0 < p)%Z
x:R
Hx:(bpow emin <= Rabs x)%R
Hx':x <> 0%R
ex:Z
He:(bpow (ex - 1) <= Rabs x < bpow ex)%R
He':(emin < ex)%Z
(bpow (- p + 1) * bpow (ex - 1) <=
 bpow (- p + 1) * Rabs (round beta fexp rnd x))%R
apply bpow_le.beta:radix
fexp:Z -> Z
prop_exp:Valid_exp fexp
emin, p:Z
Hmin:forall k : Z,
(emin < k)%Z -> (p <= k - fexp k)%Z
rnd:R -> Z
valid_rnd:Valid_rnd rnd
Hp:(0 < p)%Z
x:R
Hx:(bpow emin <= Rabs x)%R
Hx':x <> 0%R
ex:Z
He:(bpow (ex - 1) <= Rabs x < bpow ex)%R
He':(emin < ex)%Z
(fexp ex <= - p + 1 + (ex - 1))%Z
beta:radix
fexp:Z -> Z
prop_exp:Valid_exp fexp
emin, p:Z
Hmin:forall k : Z,
(emin < k)%Z -> (p <= k - fexp k)%Z
rnd:R -> Z
valid_rnd:Valid_rnd rnd
Hp:(0 < p)%Z
x:R
Hx:(bpow emin <= Rabs x)%R
Hx':x <> 0%R
ex:Z
He:(bpow (ex - 1) <= Rabs x < bpow ex)%R
He':(emin < ex)%Z
(bpow (- p + 1) * bpow (ex - 1) <=
 bpow (- p + 1) * Rabs (round beta fexp rnd x))%R
generalize (Hmin ex).beta:radix
fexp:Z -> Z
prop_exp:Valid_exp fexp
emin, p:Z
Hmin:forall k : Z,
(emin < k)%Z -> (p <= k - fexp k)%Z
rnd:R -> Z
valid_rnd:Valid_rnd rnd
Hp:(0 < p)%Z
x:R
Hx:(bpow emin <= Rabs x)%R
Hx':x <> 0%R
ex:Z
He:(bpow (ex - 1) <= Rabs x < bpow ex)%R
He':(emin < ex)%Z
((emin < ex)%Z -> (p <= ex - fexp ex)%Z) ->
(fexp ex <= - p + 1 + (ex - 1))%Z
beta:radix
fexp:Z -> Z
prop_exp:Valid_exp fexp
emin, p:Z
Hmin:forall k : Z,
(emin < k)%Z -> (p <= k - fexp k)%Z
rnd:R -> Z
valid_rnd:Valid_rnd rnd
Hp:(0 < p)%Z
x:R
Hx:(bpow emin <= Rabs x)%R
Hx':x <> 0%R
ex:Z
He:(bpow (ex - 1) <= Rabs x < bpow ex)%R
He':(emin < ex)%Z
(bpow (- p + 1) * bpow (ex - 1) <=
 bpow (- p + 1) * Rabs (round beta fexp rnd x))%R
omega.beta:radix
fexp:Z -> Z
prop_exp:Valid_exp fexp
emin, p:Z
Hmin:forall k : Z,
(emin < k)%Z -> (p <= k - fexp k)%Z
rnd:R -> Z
valid_rnd:Valid_rnd rnd
Hp:(0 < p)%Z
x:R
Hx:(bpow emin <= Rabs x)%R
Hx':x <> 0%R
ex:Z
He:(bpow (ex - 1) <= Rabs x < bpow ex)%R
He':(emin < ex)%Z
(bpow (- p + 1) * bpow (ex - 1) <=
 bpow (- p + 1) * Rabs (round beta fexp rnd x))%R
apply Rmult_le_compat_l.beta:radix
fexp:Z -> Z
prop_exp:Valid_exp fexp
emin, p:Z
Hmin:forall k : Z,
(emin < k)%Z -> (p <= k - fexp k)%Z
rnd:R -> Z
valid_rnd:Valid_rnd rnd
Hp:(0 < p)%Z
x:R
Hx:(bpow emin <= Rabs x)%R
Hx':x <> 0%R
ex:Z
He:(bpow (ex - 1) <= Rabs x < bpow ex)%R
He':(emin < ex)%Z
(0 <= bpow (- p + 1))%R
beta:radix
fexp:Z -> Z
prop_exp:Valid_exp fexp
emin, p:Z
Hmin:forall k : Z,
(emin < k)%Z -> (p <= k - fexp k)%Z
rnd:R -> Z
valid_rnd:Valid_rnd rnd
Hp:(0 < p)%Z
x:R
Hx:(bpow emin <= Rabs x)%R
Hx':x <> 0%R
ex:Z
He:(bpow (ex - 1) <= Rabs x < bpow ex)%R
He':(emin < ex)%Z
(bpow (ex - 1) <= Rabs (round beta fexp rnd x))%R
apply bpow_ge_0.beta:radix
fexp:Z -> Z
prop_exp:Valid_exp fexp
emin, p:Z
Hmin:forall k : Z,
(emin < k)%Z -> (p <= k - fexp k)%Z
rnd:R -> Z
valid_rnd:Valid_rnd rnd
Hp:(0 < p)%Z
x:R
Hx:(bpow emin <= Rabs x)%R
Hx':x <> 0%R
ex:Z
He:(bpow (ex - 1) <= Rabs x < bpow ex)%R
He':(emin < ex)%Z
(bpow (ex - 1) <= Rabs (round beta fexp rnd x))%R
generalize He.beta:radix
fexp:Z -> Z
prop_exp:Valid_exp fexp
emin, p:Z
Hmin:forall k : Z,
(emin < k)%Z -> (p <= k - fexp k)%Z
rnd:R -> Z
valid_rnd:Valid_rnd rnd
Hp:(0 < p)%Z
x:R
Hx:(bpow emin <= Rabs x)%R
Hx':x <> 0%R
ex:Z
He:(bpow (ex - 1) <= Rabs x < bpow ex)%R
He':(emin < ex)%Z
(bpow (ex - 1) <= Rabs x < bpow ex)%R ->
(bpow (ex - 1) <= Rabs (round beta fexp rnd x))%R
apply round_abs_abs...beta:radix
fexp:Z -> Z
prop_exp:Valid_exp fexp
emin, p:Z
Hmin:forall k : Z,
(emin < k)%Z -> (p <= k - fexp k)%Z
rnd:R -> Z
valid_rnd:Valid_rnd rnd
Hp:(0 < p)%Z
x:R
Hx:(bpow emin <= Rabs x)%R
Hx':x <> 0%R
ex:Z
He:(bpow (ex - 1) <= Rabs x < bpow ex)%R
He':(emin < ex)%Z
forall rnd0 : R -> Z,
Valid_rnd rnd0 ->
forall x0 : R,
(0 <= x0)%R ->
(bpow (ex - 1) <= x0 < bpow ex)%R ->
(bpow (ex - 1) <= round beta fexp rnd0 x0)%R
clear rnd valid_rnd x Hx Hx' He.beta:radix
fexp:Z -> Z
prop_exp:Valid_exp fexp
emin, p:Z
Hmin:forall k : Z,
(emin < k)%Z -> (p <= k - fexp k)%Z
Hp:(0 < p)%Z
ex:Z
He':(emin < ex)%Z
forall rnd : R -> Z,
Valid_rnd rnd ->
forall x : R,
(0 <= x)%R ->
(bpow (ex - 1) <= x < bpow ex)%R ->
(bpow (ex - 1) <= round beta fexp rnd x)%R
intros rnd valid_rnd x _ Hx.beta:radix
fexp:Z -> Z
prop_exp:Valid_exp fexp
emin, p:Z
Hmin:forall k : Z,
(emin < k)%Z -> (p <= k - fexp k)%Z
Hp:(0 < p)%Z
ex:Z
He':(emin < ex)%Z
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x:R
Hx:(bpow (ex - 1) <= x < bpow ex)%R
(bpow (ex - 1) <= round beta fexp rnd x)%R
rewrite <- (round_generic beta fexp rnd (bpow (ex - 1))).beta:radix
fexp:Z -> Z
prop_exp:Valid_exp fexp
emin, p:Z
Hmin:forall k : Z,
(emin < k)%Z -> (p <= k - fexp k)%Z
Hp:(0 < p)%Z
ex:Z
He':(emin < ex)%Z
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x:R
Hx:(bpow (ex - 1) <= x < bpow ex)%R
(round beta fexp rnd (bpow (ex - 1)) <=
 round beta fexp rnd x)%R
beta:radix
fexp:Z -> Z
prop_exp:Valid_exp fexp
emin, p:Z
Hmin:forall k : Z,
(emin < k)%Z -> (p <= k - fexp k)%Z
Hp:(0 < p)%Z
ex:Z
He':(emin < ex)%Z
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x:R
Hx:(bpow (ex - 1) <= x < bpow ex)%R
generic_format beta fexp (bpow (ex - 1))
now apply round_le.beta:radix
fexp:Z -> Z
prop_exp:Valid_exp fexp
emin, p:Z
Hmin:forall k : Z,
(emin < k)%Z -> (p <= k - fexp k)%Z
Hp:(0 < p)%Z
ex:Z
He':(emin < ex)%Z
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x:R
Hx:(bpow (ex - 1) <= x < bpow ex)%R
generic_format beta fexp (bpow (ex - 1))
apply generic_format_bpow.beta:radix
fexp:Z -> Z
prop_exp:Valid_exp fexp
emin, p:Z
Hmin:forall k : Z,
(emin < k)%Z -> (p <= k - fexp k)%Z
Hp:(0 < p)%Z
ex:Z
He':(emin < ex)%Z
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x:R
Hx:(bpow (ex - 1) <= x < bpow ex)%R
(fexp (ex - 1 + 1) <= ex - 1)%Z
ring_simplify (ex - 1 + 1)%Z.beta:radix
fexp:Z -> Z
prop_exp:Valid_exp fexp
emin, p:Z
Hmin:forall k : Z,
(emin < k)%Z -> (p <= k - fexp k)%Z
Hp:(0 < p)%Z
ex:Z
He':(emin < ex)%Z
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x:R
Hx:(bpow (ex - 1) <= x < bpow ex)%R
(fexp ex <= ex - 1)%Z
generalize (Hmin ex).beta:radix
fexp:Z -> Z
prop_exp:Valid_exp fexp
emin, p:Z
Hmin:forall k : Z,
(emin < k)%Z -> (p <= k - fexp k)%Z
Hp:(0 < p)%Z
ex:Z
He':(emin < ex)%Z
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x:R
Hx:(bpow (ex - 1) <= x < bpow ex)%R
((emin < ex)%Z -> (p <= ex - fexp ex)%Z) ->
(fexp ex <= ex - 1)%Z
omega.
Qed.

Theorem relative_error_round_F2R_emin :
  (0 < p)%Z ->
  forall m, let x := F2R (Float beta m emin) in
  (x <> 0)%R ->
  (Rabs (round beta fexp rnd x - x) < bpow (-p + 1) * Rabs (round beta fexp rnd x))%R.beta:radix
fexp:Z -> Z
prop_exp:Valid_exp fexp
emin, p:Z
Hmin:forall k : Z,
(emin < k)%Z -> (p <= k - fexp k)%Z
rnd:R -> Z
valid_rnd:Valid_rnd rnd
(0 < p)%Z ->
forall m : Z,
let x := F2R {| Fnum := m; Fexp := emin |} in
x <> 0%R ->
(Rabs (round beta fexp rnd x - x) <
 bpow (- p + 1) * Rabs (round beta fexp rnd x))%R
Proof.beta:radix
fexp:Z -> Z
prop_exp:Valid_exp fexp
emin, p:Z
Hmin:forall k : Z,
(emin < k)%Z -> (p <= k - fexp k)%Z
rnd:R -> Z
valid_rnd:Valid_rnd rnd
(0 < p)%Z ->
forall m : Z,
let x := F2R {| Fnum := m; Fexp := emin |} in
x <> 0%R ->
(Rabs (round beta fexp rnd x - x) <
 bpow (- p + 1) * Rabs (round beta fexp rnd x))%R
intros Hp m x Hx.beta:radix
fexp:Z -> Z
prop_exp:Valid_exp fexp
emin, p:Z
Hmin:forall k : Z,
(emin < k)%Z -> (p <= k - fexp k)%Z
rnd:R -> Z
valid_rnd:Valid_rnd rnd
Hp:(0 < p)%Z
m:Z
x:=F2R {| Fnum := m; Fexp := emin |}:R
Hx:x <> 0%R
(Rabs (round beta fexp rnd x - x) <
 bpow (- p + 1) * Rabs (round beta fexp rnd x))%R
apply relative_error_round.beta:radix
fexp:Z -> Z
prop_exp:Valid_exp fexp
emin, p:Z
Hmin:forall k : Z,
(emin < k)%Z -> (p <= k - fexp k)%Z
rnd:R -> Z
valid_rnd:Valid_rnd rnd
Hp:(0 < p)%Z
m:Z
x:=F2R {| Fnum := m; Fexp := emin |}:R
Hx:x <> 0%R
(0 < p)%Z
beta:radix
fexp:Z -> Z
prop_exp:Valid_exp fexp
emin, p:Z
Hmin:forall k : Z,
(emin < k)%Z -> (p <= k - fexp k)%Z
rnd:R -> Z
valid_rnd:Valid_rnd rnd
Hp:(0 < p)%Z
m:Z
x:=F2R {| Fnum := m; Fexp := emin |}:R
Hx:x <> 0%R
(bpow emin <= Rabs x)%R
exact Hp.beta:radix
fexp:Z -> Z
prop_exp:Valid_exp fexp
emin, p:Z
Hmin:forall k : Z,
(emin < k)%Z -> (p <= k - fexp k)%Z
rnd:R -> Z
valid_rnd:Valid_rnd rnd
Hp:(0 < p)%Z
m:Z
x:=F2R {| Fnum := m; Fexp := emin |}:R
Hx:x <> 0%R
(bpow emin <= Rabs x)%R
unfold x.beta:radix
fexp:Z -> Z
prop_exp:Valid_exp fexp
emin, p:Z
Hmin:forall k : Z,
(emin < k)%Z -> (p <= k - fexp k)%Z
rnd:R -> Z
valid_rnd:Valid_rnd rnd
Hp:(0 < p)%Z
m:Z
x:=F2R {| Fnum := m; Fexp := emin |}:R
Hx:x <> 0%R
(bpow emin <= Rabs (F2R {| Fnum := m; Fexp := emin |}))%R
rewrite <- F2R_Zabs.beta:radix
fexp:Z -> Z
prop_exp:Valid_exp fexp
emin, p:Z
Hmin:forall k : Z,
(emin < k)%Z -> (p <= k - fexp k)%Z
rnd:R -> Z
valid_rnd:Valid_rnd rnd
Hp:(0 < p)%Z
m:Z
x:=F2R {| Fnum := m; Fexp := emin |}:R
Hx:x <> 0%R
(bpow emin <= F2R {| Fnum := Z.abs m; Fexp := emin |})%R
apply bpow_le_F2R.beta:radix
fexp:Z -> Z
prop_exp:Valid_exp fexp
emin, p:Z
Hmin:forall k : Z,
(emin < k)%Z -> (p <= k - fexp k)%Z
rnd:R -> Z
valid_rnd:Valid_rnd rnd
Hp:(0 < p)%Z
m:Z
x:=F2R {| Fnum := m; Fexp := emin |}:R
Hx:x <> 0%R
(0 < Z.abs m)%Z
apply lt_F2R with beta emin.beta:radix
fexp:Z -> Z
prop_exp:Valid_exp fexp
emin, p:Z
Hmin:forall k : Z,
(emin < k)%Z -> (p <= k - fexp k)%Z
rnd:R -> Z
valid_rnd:Valid_rnd rnd
Hp:(0 < p)%Z
m:Z
x:=F2R {| Fnum := m; Fexp := emin |}:R
Hx:x <> 0%R
(F2R {| Fnum := 0; Fexp := emin |} <
 F2R {| Fnum := Z.abs m; Fexp := emin |})%R
rewrite F2R_0, F2R_Zabs.beta:radix
fexp:Z -> Z
prop_exp:Valid_exp fexp
emin, p:Z
Hmin:forall k : Z,
(emin < k)%Z -> (p <= k - fexp k)%Z
rnd:R -> Z
valid_rnd:Valid_rnd rnd
Hp:(0 < p)%Z
m:Z
x:=F2R {| Fnum := m; Fexp := emin |}:R
Hx:x <> 0%R
(0 < Rabs (F2R {| Fnum := m; Fexp := emin |}))%R
now apply Rabs_pos_lt.
Qed.

Variable choice : Z -> bool.

Theorem relative_error_N :
  forall x,
  (bpow emin <= Rabs x)%R ->
  (Rabs (round beta fexp (Znearest choice) x - x) <= /2 * bpow (-p + 1) * Rabs x)%R.beta:radix
fexp:Z -> Z
prop_exp:Valid_exp fexp
emin, p:Z
Hmin:forall k : Z,
(emin < k)%Z -> (p <= k - fexp k)%Z
rnd:R -> Z
valid_rnd:Valid_rnd rnd
choice:Z -> bool
forall x : R,
(bpow emin <= Rabs x)%R ->
(Rabs (round beta fexp (Znearest choice) x - x) <=
 / 2 * bpow (- p + 1) * Rabs x)%R
Proof.beta:radix
fexp:Z -> Z
prop_exp:Valid_exp fexp
emin, p:Z
Hmin:forall k : Z,
(emin < k)%Z -> (p <= k - fexp k)%Z
rnd:R -> Z
valid_rnd:Valid_rnd rnd
choice:Z -> bool
forall x : R,
(bpow emin <= Rabs x)%R ->
(Rabs (round beta fexp (Znearest choice) x - x) <=
 / 2 * bpow (- p + 1) * Rabs x)%R
intros x Hx.beta:radix
fexp:Z -> Z
prop_exp:Valid_exp fexp
emin, p:Z
Hmin:forall k : Z,
(emin < k)%Z -> (p <= k - fexp k)%Z
rnd:R -> Z
valid_rnd:Valid_rnd rnd
choice:Z -> bool
x:R
Hx:(bpow emin <= Rabs x)%R
(Rabs (round beta fexp (Znearest choice) x - x) <=
 / 2 * bpow (- p + 1) * Rabs x)%R
apply Rle_trans with (/2 * ulp beta fexp x)%R.beta:radix
fexp:Z -> Z
prop_exp:Valid_exp fexp
emin, p:Z
Hmin:forall k : Z,
(emin < k)%Z -> (p <= k - fexp k)%Z
rnd:R -> Z
valid_rnd:Valid_rnd rnd
choice:Z -> bool
x:R
Hx:(bpow emin <= Rabs x)%R
(Rabs (round beta fexp (Znearest choice) x - x) <=
 / 2 * ulp beta fexp x)%R
beta:radix
fexp:Z -> Z
prop_exp:Valid_exp fexp
emin, p:Z
Hmin:forall k : Z,
(emin < k)%Z -> (p <= k - fexp k)%Z
rnd:R -> Z
valid_rnd:Valid_rnd rnd
choice:Z -> bool
x:R
Hx:(bpow emin <= Rabs x)%R
(/ 2 * ulp beta fexp x <=
 / 2 * bpow (- p + 1) * Rabs x)%R
now apply error_le_half_ulp.beta:radix
fexp:Z -> Z
prop_exp:Valid_exp fexp
emin, p:Z
Hmin:forall k : Z,
(emin < k)%Z -> (p <= k - fexp k)%Z
rnd:R -> Z
valid_rnd:Valid_rnd rnd
choice:Z -> bool
x:R
Hx:(bpow emin <= Rabs x)%R
(/ 2 * ulp beta fexp x <=
 / 2 * bpow (- p + 1) * Rabs x)%R
rewrite Rmult_assoc.beta:radix
fexp:Z -> Z
prop_exp:Valid_exp fexp
emin, p:Z
Hmin:forall k : Z,
(emin < k)%Z -> (p <= k - fexp k)%Z
rnd:R -> Z
valid_rnd:Valid_rnd rnd
choice:Z -> bool
x:R
Hx:(bpow emin <= Rabs x)%R
(/ 2 * ulp beta fexp x <=
 / 2 * (bpow (- p + 1) * Rabs x))%R
apply Rmult_le_compat_l.beta:radix
fexp:Z -> Z
prop_exp:Valid_exp fexp
emin, p:Z
Hmin:forall k : Z,
(emin < k)%Z -> (p <= k - fexp k)%Z
rnd:R -> Z
valid_rnd:Valid_rnd rnd
choice:Z -> bool
x:R
Hx:(bpow emin <= Rabs x)%R
(0 <= / 2)%R
beta:radix
fexp:Z -> Z
prop_exp:Valid_exp fexp
emin, p:Z
Hmin:forall k : Z,
(emin < k)%Z -> (p <= k - fexp k)%Z
rnd:R -> Z
valid_rnd:Valid_rnd rnd
choice:Z -> bool
x:R
Hx:(bpow emin <= Rabs x)%R
(ulp beta fexp x <= bpow (- p + 1) * Rabs x)%R
apply Rlt_le.beta:radix
fexp:Z -> Z
prop_exp:Valid_exp fexp
emin, p:Z
Hmin:forall k : Z,
(emin < k)%Z -> (p <= k - fexp k)%Z
rnd:R -> Z
valid_rnd:Valid_rnd rnd
choice:Z -> bool
x:R
Hx:(bpow emin <= Rabs x)%R
(0 < / 2)%R
beta:radix
fexp:Z -> Z
prop_exp:Valid_exp fexp
emin, p:Z
Hmin:forall k : Z,
(emin < k)%Z -> (p <= k - fexp k)%Z
rnd:R -> Z
valid_rnd:Valid_rnd rnd
choice:Z -> bool
x:R
Hx:(bpow emin <= Rabs x)%R
(ulp beta fexp x <= bpow (- p + 1) * Rabs x)%R
apply Rinv_0_lt_compat.beta:radix
fexp:Z -> Z
prop_exp:Valid_exp fexp
emin, p:Z
Hmin:forall k : Z,
(emin < k)%Z -> (p <= k - fexp k)%Z
rnd:R -> Z
valid_rnd:Valid_rnd rnd
choice:Z -> bool
x:R
Hx:(bpow emin <= Rabs x)%R
(0 < 2)%R
beta:radix
fexp:Z -> Z
prop_exp:Valid_exp fexp
emin, p:Z
Hmin:forall k : Z,
(emin < k)%Z -> (p <= k - fexp k)%Z
rnd:R -> Z
valid_rnd:Valid_rnd rnd
choice:Z -> bool
x:R
Hx:(bpow emin <= Rabs x)%R
(ulp beta fexp x <= bpow (- p + 1) * Rabs x)%R
now apply IZR_lt.beta:radix
fexp:Z -> Z
prop_exp:Valid_exp fexp
emin, p:Z
Hmin:forall k : Z,
(emin < k)%Z -> (p <= k - fexp k)%Z
rnd:R -> Z
valid_rnd:Valid_rnd rnd
choice:Z -> bool
x:R
Hx:(bpow emin <= Rabs x)%R
(ulp beta fexp x <= bpow (- p + 1) * Rabs x)%R
assert (Hx': (x <> 0)%R).beta:radix
fexp:Z -> Z
prop_exp:Valid_exp fexp
emin, p:Z
Hmin:forall k : Z,
(emin < k)%Z -> (p <= k - fexp k)%Z
rnd:R -> Z
valid_rnd:Valid_rnd rnd
choice:Z -> bool
x:R
Hx:(bpow emin <= Rabs x)%R
x <> 0%R
beta:radix
fexp:Z -> Z
prop_exp:Valid_exp fexp
emin, p:Z
Hmin:forall k : Z,
(emin < k)%Z -> (p <= k - fexp k)%Z
rnd:R -> Z
valid_rnd:Valid_rnd rnd
choice:Z -> bool
x:R
Hx:(bpow emin <= Rabs x)%R
Hx':x <> 0%R
(ulp beta fexp x <= bpow (- p + 1) * Rabs x)%R
intros H.beta:radix
fexp:Z -> Z
prop_exp:Valid_exp fexp
emin, p:Z
Hmin:forall k : Z,
(emin < k)%Z -> (p <= k - fexp k)%Z
rnd:R -> Z
valid_rnd:Valid_rnd rnd
choice:Z -> bool
x:R
Hx:(bpow emin <= Rabs x)%R
H:x = 0%R
False
beta:radix
fexp:Z -> Z
prop_exp:Valid_exp fexp
emin, p:Z
Hmin:forall k : Z,
(emin < k)%Z -> (p <= k - fexp k)%Z
rnd:R -> Z
valid_rnd:Valid_rnd rnd
choice:Z -> bool
x:R
Hx:(bpow emin <= Rabs x)%R
Hx':x <> 0%R
(ulp beta fexp x <= bpow (- p + 1) * Rabs x)%R
apply Rlt_not_le with (2 := Hx).beta:radix
fexp:Z -> Z
prop_exp:Valid_exp fexp
emin, p:Z
Hmin:forall k : Z,
(emin < k)%Z -> (p <= k - fexp k)%Z
rnd:R -> Z
valid_rnd:Valid_rnd rnd
choice:Z -> bool
x:R
Hx:(bpow emin <= Rabs x)%R
H:x = 0%R
(Rabs x < bpow emin)%R
beta:radix
fexp:Z -> Z
prop_exp:Valid_exp fexp
emin, p:Z
Hmin:forall k : Z,
(emin < k)%Z -> (p <= k - fexp k)%Z
rnd:R -> Z
valid_rnd:Valid_rnd rnd
choice:Z -> bool
x:R
Hx:(bpow emin <= Rabs x)%R
Hx':x <> 0%R
(ulp beta fexp x <= bpow (- p + 1) * Rabs x)%R
rewrite H, Rabs_R0.beta:radix
fexp:Z -> Z
prop_exp:Valid_exp fexp
emin, p:Z
Hmin:forall k : Z,
(emin < k)%Z -> (p <= k - fexp k)%Z
rnd:R -> Z
valid_rnd:Valid_rnd rnd
choice:Z -> bool
x:R
Hx:(bpow emin <= Rabs x)%R
H:x = 0%R
(0 < bpow emin)%R
beta:radix
fexp:Z -> Z
prop_exp:Valid_exp fexp
emin, p:Z
Hmin:forall k : Z,
(emin < k)%Z -> (p <= k - fexp k)%Z
rnd:R -> Z
valid_rnd:Valid_rnd rnd
choice:Z -> bool
x:R
Hx:(bpow emin <= Rabs x)%R
Hx':x <> 0%R
(ulp beta fexp x <= bpow (- p + 1) * Rabs x)%R
apply bpow_gt_0.beta:radix
fexp:Z -> Z
prop_exp:Valid_exp fexp
emin, p:Z
Hmin:forall k : Z,
(emin < k)%Z -> (p <= k - fexp k)%Z
rnd:R -> Z
valid_rnd:Valid_rnd rnd
choice:Z -> bool
x:R
Hx:(bpow emin <= Rabs x)%R
Hx':x <> 0%R
(ulp beta fexp x <= bpow (- p + 1) * Rabs x)%R
rewrite ulp_neq_0; trivial.beta:radix
fexp:Z -> Z
prop_exp:Valid_exp fexp
emin, p:Z
Hmin:forall k : Z,
(emin < k)%Z -> (p <= k - fexp k)%Z
rnd:R -> Z
valid_rnd:Valid_rnd rnd
choice:Z -> bool
x:R
Hx:(bpow emin <= Rabs x)%R
Hx':x <> 0%R
(bpow (cexp beta fexp x) <= bpow (- p + 1) * Rabs x)%R
unfold cexp.beta:radix
fexp:Z -> Z
prop_exp:Valid_exp fexp
emin, p:Z
Hmin:forall k : Z,
(emin < k)%Z -> (p <= k - fexp k)%Z
rnd:R -> Z
valid_rnd:Valid_rnd rnd
choice:Z -> bool
x:R
Hx:(bpow emin <= Rabs x)%R
Hx':x <> 0%R
(bpow (fexp (mag beta x)) <= bpow (- p + 1) * Rabs x)%R
destruct (mag beta x) as (ex, He).beta:radix
fexp:Z -> Z
prop_exp:Valid_exp fexp
emin, p:Z
Hmin:forall k : Z,
(emin < k)%Z -> (p <= k - fexp k)%Z
rnd:R -> Z
valid_rnd:Valid_rnd rnd
choice:Z -> bool
x:R
Hx:(bpow emin <= Rabs x)%R
Hx':x <> 0%R
ex:Z
He:x <> 0%R -> (bpow (ex - 1) <= Rabs x < bpow ex)%R
(bpow (fexp (Build_mag_prop beta x ex He)) <=
 bpow (- p + 1) * Rabs x)%R
simpl.beta:radix
fexp:Z -> Z
prop_exp:Valid_exp fexp
emin, p:Z
Hmin:forall k : Z,
(emin < k)%Z -> (p <= k - fexp k)%Z
rnd:R -> Z
valid_rnd:Valid_rnd rnd
choice:Z -> bool
x:R
Hx:(bpow emin <= Rabs x)%R
Hx':x <> 0%R
ex:Z
He:x <> 0%R -> (bpow (ex - 1) <= Rabs x < bpow ex)%R
(bpow (fexp ex) <= bpow (- p + 1) * Rabs x)%R
specialize (He Hx').beta:radix
fexp:Z -> Z
prop_exp:Valid_exp fexp
emin, p:Z
Hmin:forall k : Z,
(emin < k)%Z -> (p <= k - fexp k)%Z
rnd:R -> Z
valid_rnd:Valid_rnd rnd
choice:Z -> bool
x:R
Hx:(bpow emin <= Rabs x)%R
Hx':x <> 0%R
ex:Z
He:(bpow (ex - 1) <= Rabs x < bpow ex)%R
(bpow (fexp ex) <= bpow (- p + 1) * Rabs x)%R
apply Rle_trans with (bpow (-p + 1) * bpow (ex - 1))%R.beta:radix
fexp:Z -> Z
prop_exp:Valid_exp fexp
emin, p:Z
Hmin:forall k : Z,
(emin < k)%Z -> (p <= k - fexp k)%Z
rnd:R -> Z
valid_rnd:Valid_rnd rnd
choice:Z -> bool
x:R
Hx:(bpow emin <= Rabs x)%R
Hx':x <> 0%R
ex:Z
He:(bpow (ex - 1) <= Rabs x < bpow ex)%R
(bpow (fexp ex) <= bpow (- p + 1) * bpow (ex - 1))%R
beta:radix
fexp:Z -> Z
prop_exp:Valid_exp fexp
emin, p:Z
Hmin:forall k : Z,
(emin < k)%Z -> (p <= k - fexp k)%Z
rnd:R -> Z
valid_rnd:Valid_rnd rnd
choice:Z -> bool
x:R
Hx:(bpow emin <= Rabs x)%R
Hx':x <> 0%R
ex:Z
He:(bpow (ex - 1) <= Rabs x < bpow ex)%R
(bpow (- p + 1) * bpow (ex - 1) <=
 bpow (- p + 1) * Rabs x)%R
rewrite <- bpow_plus.beta:radix
fexp:Z -> Z
prop_exp:Valid_exp fexp
emin, p:Z
Hmin:forall k : Z,
(emin < k)%Z -> (p <= k - fexp k)%Z
rnd:R -> Z
valid_rnd:Valid_rnd rnd
choice:Z -> bool
x:R
Hx:(bpow emin <= Rabs x)%R
Hx':x <> 0%R
ex:Z
He:(bpow (ex - 1) <= Rabs x < bpow ex)%R
(bpow (fexp ex) <= bpow (- p + 1 + (ex - 1)))%R
beta:radix
fexp:Z -> Z
prop_exp:Valid_exp fexp
emin, p:Z
Hmin:forall k : Z,
(emin < k)%Z -> (p <= k - fexp k)%Z
rnd:R -> Z
valid_rnd:Valid_rnd rnd
choice:Z -> bool
x:R
Hx:(bpow emin <= Rabs x)%R
Hx':x <> 0%R
ex:Z
He:(bpow (ex - 1) <= Rabs x < bpow ex)%R
(bpow (- p + 1) * bpow (ex - 1) <=
 bpow (- p + 1) * Rabs x)%R
apply bpow_le.beta:radix
fexp:Z -> Z
prop_exp:Valid_exp fexp
emin, p:Z
Hmin:forall k : Z,
(emin < k)%Z -> (p <= k - fexp k)%Z
rnd:R -> Z
valid_rnd:Valid_rnd rnd
choice:Z -> bool
x:R
Hx:(bpow emin <= Rabs x)%R
Hx':x <> 0%R
ex:Z
He:(bpow (ex - 1) <= Rabs x < bpow ex)%R
(fexp ex <= - p + 1 + (ex - 1))%Z
beta:radix
fexp:Z -> Z
prop_exp:Valid_exp fexp
emin, p:Z
Hmin:forall k : Z,
(emin < k)%Z -> (p <= k - fexp k)%Z
rnd:R -> Z
valid_rnd:Valid_rnd rnd
choice:Z -> bool
x:R
Hx:(bpow emin <= Rabs x)%R
Hx':x <> 0%R
ex:Z
He:(bpow (ex - 1) <= Rabs x < bpow ex)%R
(bpow (- p + 1) * bpow (ex - 1) <=
 bpow (- p + 1) * Rabs x)%R
assert (emin < ex)%Z.beta:radix
fexp:Z -> Z
prop_exp:Valid_exp fexp
emin, p:Z
Hmin:forall k : Z,
(emin < k)%Z -> (p <= k - fexp k)%Z
rnd:R -> Z
valid_rnd:Valid_rnd rnd
choice:Z -> bool
x:R
Hx:(bpow emin <= Rabs x)%R
Hx':x <> 0%R
ex:Z
He:(bpow (ex - 1) <= Rabs x < bpow ex)%R
(emin < ex)%Z
beta:radix
fexp:Z -> Z
prop_exp:Valid_exp fexp
emin, p:Z
Hmin:forall k : Z,
(emin < k)%Z -> (p <= k - fexp k)%Z
rnd:R -> Z
valid_rnd:Valid_rnd rnd
choice:Z -> bool
x:R
Hx:(bpow emin <= Rabs x)%R
Hx':x <> 0%R
ex:Z
He:(bpow (ex - 1) <= Rabs x < bpow ex)%R
H:(emin < ex)%Z
(fexp ex <= - p + 1 + (ex - 1))%Z
beta:radix
fexp:Z -> Z
prop_exp:Valid_exp fexp
emin, p:Z
Hmin:forall k : Z,
(emin < k)%Z -> (p <= k - fexp k)%Z
rnd:R -> Z
valid_rnd:Valid_rnd rnd
choice:Z -> bool
x:R
Hx:(bpow emin <= Rabs x)%R
Hx':x <> 0%R
ex:Z
He:(bpow (ex - 1) <= Rabs x < bpow ex)%R
(bpow (- p + 1) * bpow (ex - 1) <=
 bpow (- p + 1) * Rabs x)%R
apply (lt_bpow beta).beta:radix
fexp:Z -> Z
prop_exp:Valid_exp fexp
emin, p:Z
Hmin:forall k : Z,
(emin < k)%Z -> (p <= k - fexp k)%Z
rnd:R -> Z
valid_rnd:Valid_rnd rnd
choice:Z -> bool
x:R
Hx:(bpow emin <= Rabs x)%R
Hx':x <> 0%R
ex:Z
He:(bpow (ex - 1) <= Rabs x < bpow ex)%R
(bpow emin < bpow ex)%R
beta:radix
fexp:Z -> Z
prop_exp:Valid_exp fexp
emin, p:Z
Hmin:forall k : Z,
(emin < k)%Z -> (p <= k - fexp k)%Z
rnd:R -> Z
valid_rnd:Valid_rnd rnd
choice:Z -> bool
x:R
Hx:(bpow emin <= Rabs x)%R
Hx':x <> 0%R
ex:Z
He:(bpow (ex - 1) <= Rabs x < bpow ex)%R
H:(emin < ex)%Z
(fexp ex <= - p + 1 + (ex - 1))%Z
beta:radix
fexp:Z -> Z
prop_exp:Valid_exp fexp
emin, p:Z
Hmin:forall k : Z,
(emin < k)%Z -> (p <= k - fexp k)%Z
rnd:R -> Z
valid_rnd:Valid_rnd rnd
choice:Z -> bool
x:R
Hx:(bpow emin <= Rabs x)%R
Hx':x <> 0%R
ex:Z
He:(bpow (ex - 1) <= Rabs x < bpow ex)%R
(bpow (- p + 1) * bpow (ex - 1) <=
 bpow (- p + 1) * Rabs x)%R
apply Rle_lt_trans with (2 := proj2 He).beta:radix
fexp:Z -> Z
prop_exp:Valid_exp fexp
emin, p:Z
Hmin:forall k : Z,
(emin < k)%Z -> (p <= k - fexp k)%Z
rnd:R -> Z
valid_rnd:Valid_rnd rnd
choice:Z -> bool
x:R
Hx:(bpow emin <= Rabs x)%R
Hx':x <> 0%R
ex:Z
He:(bpow (ex - 1) <= Rabs x < bpow ex)%R
(bpow emin <= Rabs x)%R
beta:radix
fexp:Z -> Z
prop_exp:Valid_exp fexp
emin, p:Z
Hmin:forall k : Z,
(emin < k)%Z -> (p <= k - fexp k)%Z
rnd:R -> Z
valid_rnd:Valid_rnd rnd
choice:Z -> bool
x:R
Hx:(bpow emin <= Rabs x)%R
Hx':x <> 0%R
ex:Z
He:(bpow (ex - 1) <= Rabs x < bpow ex)%R
H:(emin < ex)%Z
(fexp ex <= - p + 1 + (ex - 1))%Z
beta:radix
fexp:Z -> Z
prop_exp:Valid_exp fexp
emin, p:Z
Hmin:forall k : Z,
(emin < k)%Z -> (p <= k - fexp k)%Z
rnd:R -> Z
valid_rnd:Valid_rnd rnd
choice:Z -> bool
x:R
Hx:(bpow emin <= Rabs x)%R
Hx':x <> 0%R
ex:Z
He:(bpow (ex - 1) <= Rabs x < bpow ex)%R
(bpow (- p + 1) * bpow (ex - 1) <=
 bpow (- p + 1) * Rabs x)%R
exact Hx.beta:radix
fexp:Z -> Z
prop_exp:Valid_exp fexp
emin, p:Z
Hmin:forall k : Z,
(emin < k)%Z -> (p <= k - fexp k)%Z
rnd:R -> Z
valid_rnd:Valid_rnd rnd
choice:Z -> bool
x:R
Hx:(bpow emin <= Rabs x)%R
Hx':x <> 0%R
ex:Z
He:(bpow (ex - 1) <= Rabs x < bpow ex)%R
H:(emin < ex)%Z
(fexp ex <= - p + 1 + (ex - 1))%Z
beta:radix
fexp:Z -> Z
prop_exp:Valid_exp fexp
emin, p:Z
Hmin:forall k : Z,
(emin < k)%Z -> (p <= k - fexp k)%Z
rnd:R -> Z
valid_rnd:Valid_rnd rnd
choice:Z -> bool
x:R
Hx:(bpow emin <= Rabs x)%R
Hx':x <> 0%R
ex:Z
He:(bpow (ex - 1) <= Rabs x < bpow ex)%R
(bpow (- p + 1) * bpow (ex - 1) <=
 bpow (- p + 1) * Rabs x)%R
generalize (Hmin ex).beta:radix
fexp:Z -> Z
prop_exp:Valid_exp fexp
emin, p:Z
Hmin:forall k : Z,
(emin < k)%Z -> (p <= k - fexp k)%Z
rnd:R -> Z
valid_rnd:Valid_rnd rnd
choice:Z -> bool
x:R
Hx:(bpow emin <= Rabs x)%R
Hx':x <> 0%R
ex:Z
He:(bpow (ex - 1) <= Rabs x < bpow ex)%R
H:(emin < ex)%Z
((emin < ex)%Z -> (p <= ex - fexp ex)%Z) ->
(fexp ex <= - p + 1 + (ex - 1))%Z
beta:radix
fexp:Z -> Z
prop_exp:Valid_exp fexp
emin, p:Z
Hmin:forall k : Z,
(emin < k)%Z -> (p <= k - fexp k)%Z
rnd:R -> Z
valid_rnd:Valid_rnd rnd
choice:Z -> bool
x:R
Hx:(bpow emin <= Rabs x)%R
Hx':x <> 0%R
ex:Z
He:(bpow (ex - 1) <= Rabs x < bpow ex)%R
(bpow (- p + 1) * bpow (ex - 1) <=
 bpow (- p + 1) * Rabs x)%R
omega.beta:radix
fexp:Z -> Z
prop_exp:Valid_exp fexp
emin, p:Z
Hmin:forall k : Z,
(emin < k)%Z -> (p <= k - fexp k)%Z
rnd:R -> Z
valid_rnd:Valid_rnd rnd
choice:Z -> bool
x:R
Hx:(bpow emin <= Rabs x)%R
Hx':x <> 0%R
ex:Z
He:(bpow (ex - 1) <= Rabs x < bpow ex)%R
(bpow (- p + 1) * bpow (ex - 1) <=
 bpow (- p + 1) * Rabs x)%R
apply Rmult_le_compat_l.beta:radix
fexp:Z -> Z
prop_exp:Valid_exp fexp
emin, p:Z
Hmin:forall k : Z,
(emin < k)%Z -> (p <= k - fexp k)%Z
rnd:R -> Z
valid_rnd:Valid_rnd rnd
choice:Z -> bool
x:R
Hx:(bpow emin <= Rabs x)%R
Hx':x <> 0%R
ex:Z
He:(bpow (ex - 1) <= Rabs x < bpow ex)%R
(0 <= bpow (- p + 1))%R
beta:radix
fexp:Z -> Z
prop_exp:Valid_exp fexp
emin, p:Z
Hmin:forall k : Z,
(emin < k)%Z -> (p <= k - fexp k)%Z
rnd:R -> Z
valid_rnd:Valid_rnd rnd
choice:Z -> bool
x:R
Hx:(bpow emin <= Rabs x)%R
Hx':x <> 0%R
ex:Z
He:(bpow (ex - 1) <= Rabs x < bpow ex)%R
(bpow (ex - 1) <= Rabs x)%R
apply bpow_ge_0.beta:radix
fexp:Z -> Z
prop_exp:Valid_exp fexp
emin, p:Z
Hmin:forall k : Z,
(emin < k)%Z -> (p <= k - fexp k)%Z
rnd:R -> Z
valid_rnd:Valid_rnd rnd
choice:Z -> bool
x:R
Hx:(bpow emin <= Rabs x)%R
Hx':x <> 0%R
ex:Z
He:(bpow (ex - 1) <= Rabs x < bpow ex)%R
(bpow (ex - 1) <= Rabs x)%R
apply He.
Qed.

1+ε property in rounding to nearest

Theorem relative_error_N_ex :
  forall x,
  (bpow emin <= Rabs x)%R ->
  exists eps,
  (Rabs eps <= /2 * bpow (-p + 1))%R /\ round beta fexp (Znearest choice) x = (x * (1 + eps))%R.beta:radix
fexp:Z -> Z
prop_exp:Valid_exp fexp
emin, p:Z
Hmin:forall k : Z,
(emin < k)%Z -> (p <= k - fexp k)%Z
rnd:R -> Z
valid_rnd:Valid_rnd rnd
choice:Z -> bool
forall x : R,
(bpow emin <= Rabs x)%R ->
exists eps : R,
  (Rabs eps <= / 2 * bpow (- p + 1))%R /\
  round beta fexp (Znearest choice) x =
  (x * (1 + eps))%R
Proof with auto with typeclass_instances.beta:radix
fexp:Z -> Z
prop_exp:Valid_exp fexp
emin, p:Z
Hmin:forall k : Z,
(emin < k)%Z -> (p <= k - fexp k)%Z
rnd:R -> Z
valid_rnd:Valid_rnd rnd
choice:Z -> bool
forall x : R,
(bpow emin <= Rabs x)%R ->
exists eps : R,
  (Rabs eps <= / 2 * bpow (- p + 1))%R /\
  round beta fexp (Znearest choice) x =
  (x * (1 + eps))%R
intros x Hx.beta:radix
fexp:Z -> Z
prop_exp:Valid_exp fexp
emin, p:Z
Hmin:forall k : Z,
(emin < k)%Z -> (p <= k - fexp k)%Z
rnd:R -> Z
valid_rnd:Valid_rnd rnd
choice:Z -> bool
x:R
Hx:(bpow emin <= Rabs x)%R
exists eps : R,
  (Rabs eps <= / 2 * bpow (- p + 1))%R /\
  round beta fexp (Znearest choice) x =
  (x * (1 + eps))%R
apply relative_error_le_conversion...beta:radix
fexp:Z -> Z
prop_exp:Valid_exp fexp
emin, p:Z
Hmin:forall k : Z,
(emin < k)%Z -> (p <= k - fexp k)%Z
rnd:R -> Z
valid_rnd:Valid_rnd rnd
choice:Z -> bool
x:R
Hx:(bpow emin <= Rabs x)%R
(0 <= / 2 * bpow (- p + 1))%R
beta:radix
fexp:Z -> Z
prop_exp:Valid_exp fexp
emin, p:Z
Hmin:forall k : Z,
(emin < k)%Z -> (p <= k - fexp k)%Z
rnd:R -> Z
valid_rnd:Valid_rnd rnd
choice:Z -> bool
x:R
Hx:(bpow emin <= Rabs x)%R
(Rabs (round beta fexp (Znearest choice) x - x) <=
 / 2 * bpow (- p + 1) * Rabs x)%R
apply Rlt_le.beta:radix
fexp:Z -> Z
prop_exp:Valid_exp fexp
emin, p:Z
Hmin:forall k : Z,
(emin < k)%Z -> (p <= k - fexp k)%Z
rnd:R -> Z
valid_rnd:Valid_rnd rnd
choice:Z -> bool
x:R
Hx:(bpow emin <= Rabs x)%R
(0 < / 2 * bpow (- p + 1))%R
beta:radix
fexp:Z -> Z
prop_exp:Valid_exp fexp
emin, p:Z
Hmin:forall k : Z,
(emin < k)%Z -> (p <= k - fexp k)%Z
rnd:R -> Z
valid_rnd:Valid_rnd rnd
choice:Z -> bool
x:R
Hx:(bpow emin <= Rabs x)%R
(Rabs (round beta fexp (Znearest choice) x - x) <=
 / 2 * bpow (- p + 1) * Rabs x)%R
apply Rmult_lt_0_compat.beta:radix
fexp:Z -> Z
prop_exp:Valid_exp fexp
emin, p:Z
Hmin:forall k : Z,
(emin < k)%Z -> (p <= k - fexp k)%Z
rnd:R -> Z
valid_rnd:Valid_rnd rnd
choice:Z -> bool
x:R
Hx:(bpow emin <= Rabs x)%R
(0 < / 2)%R
beta:radix
fexp:Z -> Z
prop_exp:Valid_exp fexp
emin, p:Z
Hmin:forall k : Z,
(emin < k)%Z -> (p <= k - fexp k)%Z
rnd:R -> Z
valid_rnd:Valid_rnd rnd
choice:Z -> bool
x:R
Hx:(bpow emin <= Rabs x)%R
(0 < bpow (- p + 1))%R
beta:radix
fexp:Z -> Z
prop_exp:Valid_exp fexp
emin, p:Z
Hmin:forall k : Z,
(emin < k)%Z -> (p <= k - fexp k)%Z
rnd:R -> Z
valid_rnd:Valid_rnd rnd
choice:Z -> bool
x:R
Hx:(bpow emin <= Rabs x)%R
(Rabs (round beta fexp (Znearest choice) x - x) <=
 / 2 * bpow (- p + 1) * Rabs x)%R
apply Rinv_0_lt_compat.beta:radix
fexp:Z -> Z
prop_exp:Valid_exp fexp
emin, p:Z
Hmin:forall k : Z,
(emin < k)%Z -> (p <= k - fexp k)%Z
rnd:R -> Z
valid_rnd:Valid_rnd rnd
choice:Z -> bool
x:R
Hx:(bpow emin <= Rabs x)%R
(0 < 2)%R
beta:radix
fexp:Z -> Z
prop_exp:Valid_exp fexp
emin, p:Z
Hmin:forall k : Z,
(emin < k)%Z -> (p <= k - fexp k)%Z
rnd:R -> Z
valid_rnd:Valid_rnd rnd
choice:Z -> bool
x:R
Hx:(bpow emin <= Rabs x)%R
(0 < bpow (- p + 1))%R
beta:radix
fexp:Z -> Z
prop_exp:Valid_exp fexp
emin, p:Z
Hmin:forall k : Z,
(emin < k)%Z -> (p <= k - fexp k)%Z
rnd:R -> Z
valid_rnd:Valid_rnd rnd
choice:Z -> bool
x:R
Hx:(bpow emin <= Rabs x)%R
(Rabs (round beta fexp (Znearest choice) x - x) <=
 / 2 * bpow (- p + 1) * Rabs x)%R
now apply IZR_lt.beta:radix
fexp:Z -> Z
prop_exp:Valid_exp fexp
emin, p:Z
Hmin:forall k : Z,
(emin < k)%Z -> (p <= k - fexp k)%Z
rnd:R -> Z
valid_rnd:Valid_rnd rnd
choice:Z -> bool
x:R
Hx:(bpow emin <= Rabs x)%R
(0 < bpow (- p + 1))%R
beta:radix
fexp:Z -> Z
prop_exp:Valid_exp fexp
emin, p:Z
Hmin:forall k : Z,
(emin < k)%Z -> (p <= k - fexp k)%Z
rnd:R -> Z
valid_rnd:Valid_rnd rnd
choice:Z -> bool
x:R
Hx:(bpow emin <= Rabs x)%R
(Rabs (round beta fexp (Znearest choice) x - x) <=
 / 2 * bpow (- p + 1) * Rabs x)%R
apply bpow_gt_0.beta:radix
fexp:Z -> Z
prop_exp:Valid_exp fexp
emin, p:Z
Hmin:forall k : Z,
(emin < k)%Z -> (p <= k - fexp k)%Z
rnd:R -> Z
valid_rnd:Valid_rnd rnd
choice:Z -> bool
x:R
Hx:(bpow emin <= Rabs x)%R
(Rabs (round beta fexp (Znearest choice) x - x) <=
 / 2 * bpow (- p + 1) * Rabs x)%R
now apply relative_error_N.
Qed.

Theorem relative_error_N_F2R_emin :
  forall m, let x := F2R (Float beta m emin) in
  (Rabs (round beta fexp (Znearest choice) x - x) <= /2 * bpow (-p + 1) * Rabs x)%R.beta:radix
fexp:Z -> Z
prop_exp:Valid_exp fexp
emin, p:Z
Hmin:forall k : Z,
(emin < k)%Z -> (p <= k - fexp k)%Z
rnd:R -> Z
valid_rnd:Valid_rnd rnd
choice:Z -> bool
forall m : Z,
let x := F2R {| Fnum := m; Fexp := emin |} in
(Rabs (round beta fexp (Znearest choice) x - x) <=
 / 2 * bpow (- p + 1) * Rabs x)%R
Proof with auto with typeclass_instances.beta:radix
fexp:Z -> Z
prop_exp:Valid_exp fexp
emin, p:Z
Hmin:forall k : Z,
(emin < k)%Z -> (p <= k - fexp k)%Z
rnd:R -> Z
valid_rnd:Valid_rnd rnd
choice:Z -> bool
forall m : Z,
let x := F2R {| Fnum := m; Fexp := emin |} in
(Rabs (round beta fexp (Znearest choice) x - x) <=
 / 2 * bpow (- p + 1) * Rabs x)%R
intros m x.beta:radix
fexp:Z -> Z
prop_exp:Valid_exp fexp
emin, p:Z
Hmin:forall k : Z,
(emin < k)%Z -> (p <= k - fexp k)%Z
rnd:R -> Z
valid_rnd:Valid_rnd rnd
choice:Z -> bool
m:Z
x:=F2R {| Fnum := m; Fexp := emin |}:R
(Rabs (round beta fexp (Znearest choice) x - x) <=
 / 2 * bpow (- p + 1) * Rabs x)%R
destruct (Req_dec x 0) as [Hx|Hx].beta:radix
fexp:Z -> Z
prop_exp:Valid_exp fexp
emin, p:Z
Hmin:forall k : Z,
(emin < k)%Z -> (p <= k - fexp k)%Z
rnd:R -> Z
valid_rnd:Valid_rnd rnd
choice:Z -> bool
m:Z
x:=F2R {| Fnum := m; Fexp := emin |}:R
Hx:x = 0%R
(Rabs (round beta fexp (Znearest choice) x - x) <=
 / 2 * bpow (- p + 1) * Rabs x)%R
beta:radix
fexp:Z -> Z
prop_exp:Valid_exp fexp
emin, p:Z
Hmin:forall k : Z,
(emin < k)%Z -> (p <= k - fexp k)%Z
rnd:R -> Z
valid_rnd:Valid_rnd rnd
choice:Z -> bool
m:Z
x:=F2R {| Fnum := m; Fexp := emin |}:R
Hx:x <> 0%R
(Rabs (round beta fexp (Znearest choice) x - x) <=
 / 2 * bpow (- p + 1) * Rabs x)%R
(* . *)
rewrite Hx, round_0...beta:radix
fexp:Z -> Z
prop_exp:Valid_exp fexp
emin, p:Z
Hmin:forall k : Z,
(emin < k)%Z -> (p <= k - fexp k)%Z
rnd:R -> Z
valid_rnd:Valid_rnd rnd
choice:Z -> bool
m:Z
x:=F2R {| Fnum := m; Fexp := emin |}:R
Hx:x = 0%R
(Rabs (0 - 0) <= / 2 * bpow (- p + 1) * Rabs 0)%R
beta:radix
fexp:Z -> Z
prop_exp:Valid_exp fexp
emin, p:Z
Hmin:forall k : Z,
(emin < k)%Z -> (p <= k - fexp k)%Z
rnd:R -> Z
valid_rnd:Valid_rnd rnd
choice:Z -> bool
m:Z
x:=F2R {| Fnum := m; Fexp := emin |}:R
Hx:x <> 0%R
(Rabs (round beta fexp (Znearest choice) x - x) <=
 / 2 * bpow (- p + 1) * Rabs x)%R
unfold Rminus.beta:radix
fexp:Z -> Z
prop_exp:Valid_exp fexp
emin, p:Z
Hmin:forall k : Z,
(emin < k)%Z -> (p <= k - fexp k)%Z
rnd:R -> Z
valid_rnd:Valid_rnd rnd
choice:Z -> bool
m:Z
x:=F2R {| Fnum := m; Fexp := emin |}:R
Hx:x = 0%R
(Rabs (0 + - 0) <= / 2 * bpow (- p + 1) * Rabs 0)%R
beta:radix
fexp:Z -> Z
prop_exp:Valid_exp fexp
emin, p:Z
Hmin:forall k : Z,
(emin < k)%Z -> (p <= k - fexp k)%Z
rnd:R -> Z
valid_rnd:Valid_rnd rnd
choice:Z -> bool
m:Z
x:=F2R {| Fnum := m; Fexp := emin |}:R
Hx:x <> 0%R
(Rabs (round beta fexp (Znearest choice) x - x) <=
 / 2 * bpow (- p + 1) * Rabs x)%R
rewrite Rplus_0_l, Rabs_Ropp, Rabs_R0.beta:radix
fexp:Z -> Z
prop_exp:Valid_exp fexp
emin, p:Z
Hmin:forall k : Z,
(emin < k)%Z -> (p <= k - fexp k)%Z
rnd:R -> Z
valid_rnd:Valid_rnd rnd
choice:Z -> bool
m:Z
x:=F2R {| Fnum := m; Fexp := emin |}:R
Hx:x = 0%R
(0 <= / 2 * bpow (- p + 1) * 0)%R
beta:radix
fexp:Z -> Z
prop_exp:Valid_exp fexp
emin, p:Z
Hmin:forall k : Z,
(emin < k)%Z -> (p <= k - fexp k)%Z
rnd:R -> Z
valid_rnd:Valid_rnd rnd
choice:Z -> bool
m:Z
x:=F2R {| Fnum := m; Fexp := emin |}:R
Hx:x <> 0%R
(Rabs (round beta fexp (Znearest choice) x - x) <=
 / 2 * bpow (- p + 1) * Rabs x)%R
rewrite Rmult_0_r.beta:radix
fexp:Z -> Z
prop_exp:Valid_exp fexp
emin, p:Z
Hmin:forall k : Z,
(emin < k)%Z -> (p <= k - fexp k)%Z
rnd:R -> Z
valid_rnd:Valid_rnd rnd
choice:Z -> bool
m:Z
x:=F2R {| Fnum := m; Fexp := emin |}:R
Hx:x = 0%R
(0 <= 0)%R
beta:radix
fexp:Z -> Z
prop_exp:Valid_exp fexp
emin, p:Z
Hmin:forall k : Z,
(emin < k)%Z -> (p <= k - fexp k)%Z
rnd:R -> Z
valid_rnd:Valid_rnd rnd
choice:Z -> bool
m:Z
x:=F2R {| Fnum := m; Fexp := emin |}:R
Hx:x <> 0%R
(Rabs (round beta fexp (Znearest choice) x - x) <=
 / 2 * bpow (- p + 1) * Rabs x)%R
apply Rle_refl.beta:radix
fexp:Z -> Z
prop_exp:Valid_exp fexp
emin, p:Z
Hmin:forall k : Z,
(emin < k)%Z -> (p <= k - fexp k)%Z
rnd:R -> Z
valid_rnd:Valid_rnd rnd
choice:Z -> bool
m:Z
x:=F2R {| Fnum := m; Fexp := emin |}:R
Hx:x <> 0%R
(Rabs (round beta fexp (Znearest choice) x - x) <=
 / 2 * bpow (- p + 1) * Rabs x)%R
(* . *)
apply relative_error_N.beta:radix
fexp:Z -> Z
prop_exp:Valid_exp fexp
emin, p:Z
Hmin:forall k : Z,
(emin < k)%Z -> (p <= k - fexp k)%Z
rnd:R -> Z
valid_rnd:Valid_rnd rnd
choice:Z -> bool
m:Z
x:=F2R {| Fnum := m; Fexp := emin |}:R
Hx:x <> 0%R
(bpow emin <= Rabs x)%R
unfold x.beta:radix
fexp:Z -> Z
prop_exp:Valid_exp fexp
emin, p:Z
Hmin:forall k : Z,
(emin < k)%Z -> (p <= k - fexp k)%Z
rnd:R -> Z
valid_rnd:Valid_rnd rnd
choice:Z -> bool
m:Z
x:=F2R {| Fnum := m; Fexp := emin |}:R
Hx:x <> 0%R
(bpow emin <= Rabs (F2R {| Fnum := m; Fexp := emin |}))%R
rewrite <- F2R_Zabs.beta:radix
fexp:Z -> Z
prop_exp:Valid_exp fexp
emin, p:Z
Hmin:forall k : Z,
(emin < k)%Z -> (p <= k - fexp k)%Z
rnd:R -> Z
valid_rnd:Valid_rnd rnd
choice:Z -> bool
m:Z
x:=F2R {| Fnum := m; Fexp := emin |}:R
Hx:x <> 0%R
(bpow emin <= F2R {| Fnum := Z.abs m; Fexp := emin |})%R
apply bpow_le_F2R.beta:radix
fexp:Z -> Z
prop_exp:Valid_exp fexp
emin, p:Z
Hmin:forall k : Z,
(emin < k)%Z -> (p <= k - fexp k)%Z
rnd:R -> Z
valid_rnd:Valid_rnd rnd
choice:Z -> bool
m:Z
x:=F2R {| Fnum := m; Fexp := emin |}:R
Hx:x <> 0%R
(0 < Z.abs m)%Z
apply lt_F2R with beta emin.beta:radix
fexp:Z -> Z
prop_exp:Valid_exp fexp
emin, p:Z
Hmin:forall k : Z,
(emin < k)%Z -> (p <= k - fexp k)%Z
rnd:R -> Z
valid_rnd:Valid_rnd rnd
choice:Z -> bool
m:Z
x:=F2R {| Fnum := m; Fexp := emin |}:R
Hx:x <> 0%R
(F2R {| Fnum := 0; Fexp := emin |} <
 F2R {| Fnum := Z.abs m; Fexp := emin |})%R
rewrite F2R_0, F2R_Zabs.beta:radix
fexp:Z -> Z
prop_exp:Valid_exp fexp
emin, p:Z
Hmin:forall k : Z,
(emin < k)%Z -> (p <= k - fexp k)%Z
rnd:R -> Z
valid_rnd:Valid_rnd rnd
choice:Z -> bool
m:Z
x:=F2R {| Fnum := m; Fexp := emin |}:R
Hx:x <> 0%R
(0 < Rabs (F2R {| Fnum := m; Fexp := emin |}))%R
now apply Rabs_pos_lt.
Qed.

Theorem relative_error_N_F2R_emin_ex :
  forall m, let x := F2R (Float beta m emin) in
  exists eps,
  (Rabs eps <= /2 * bpow (-p + 1))%R /\ round beta fexp (Znearest choice) x = (x * (1 + eps))%R.beta:radix
fexp:Z -> Z
prop_exp:Valid_exp fexp
emin, p:Z
Hmin:forall k : Z,
(emin < k)%Z -> (p <= k - fexp k)%Z
rnd:R -> Z
valid_rnd:Valid_rnd rnd
choice:Z -> bool
forall m : Z,
let x := F2R {| Fnum := m; Fexp := emin |} in
exists eps : R,
  (Rabs eps <= / 2 * bpow (- p + 1))%R /\
  round beta fexp (Znearest choice) x =
  (x * (1 + eps))%R
Proof with auto with typeclass_instances.beta:radix
fexp:Z -> Z
prop_exp:Valid_exp fexp
emin, p:Z
Hmin:forall k : Z,
(emin < k)%Z -> (p <= k - fexp k)%Z
rnd:R -> Z
valid_rnd:Valid_rnd rnd
choice:Z -> bool
forall m : Z,
let x := F2R {| Fnum := m; Fexp := emin |} in
exists eps : R,
  (Rabs eps <= / 2 * bpow (- p + 1))%R /\
  round beta fexp (Znearest choice) x =
  (x * (1 + eps))%R
intros m x.beta:radix
fexp:Z -> Z
prop_exp:Valid_exp fexp
emin, p:Z
Hmin:forall k : Z,
(emin < k)%Z -> (p <= k - fexp k)%Z
rnd:R -> Z
valid_rnd:Valid_rnd rnd
choice:Z -> bool
m:Z
x:=F2R {| Fnum := m; Fexp := emin |}:R
exists eps : R,
  (Rabs eps <= / 2 * bpow (- p + 1))%R /\
  round beta fexp (Znearest choice) x =
  (x * (1 + eps))%R
apply relative_error_le_conversion...beta:radix
fexp:Z -> Z
prop_exp:Valid_exp fexp
emin, p:Z
Hmin:forall k : Z,
(emin < k)%Z -> (p <= k - fexp k)%Z
rnd:R -> Z
valid_rnd:Valid_rnd rnd
choice:Z -> bool
m:Z
x:=F2R {| Fnum := m; Fexp := emin |}:R
(0 <= / 2 * bpow (- p + 1))%R
beta:radix
fexp:Z -> Z
prop_exp:Valid_exp fexp
emin, p:Z
Hmin:forall k : Z,
(emin < k)%Z -> (p <= k - fexp k)%Z
rnd:R -> Z
valid_rnd:Valid_rnd rnd
choice:Z -> bool
m:Z
x:=F2R {| Fnum := m; Fexp := emin |}:R
(Rabs (round beta fexp (Znearest choice) x - x) <=
 / 2 * bpow (- p + 1) * Rabs x)%R
apply Rlt_le.beta:radix
fexp:Z -> Z
prop_exp:Valid_exp fexp
emin, p:Z
Hmin:forall k : Z,
(emin < k)%Z -> (p <= k - fexp k)%Z
rnd:R -> Z
valid_rnd:Valid_rnd rnd
choice:Z -> bool
m:Z
x:=F2R {| Fnum := m; Fexp := emin |}:R
(0 < / 2 * bpow (- p + 1))%R
beta:radix
fexp:Z -> Z
prop_exp:Valid_exp fexp
emin, p:Z
Hmin:forall k : Z,
(emin < k)%Z -> (p <= k - fexp k)%Z
rnd:R -> Z
valid_rnd:Valid_rnd rnd
choice:Z -> bool
m:Z
x:=F2R {| Fnum := m; Fexp := emin |}:R
(Rabs (round beta fexp (Znearest choice) x - x) <=
 / 2 * bpow (- p + 1) * Rabs x)%R
apply Rmult_lt_0_compat.beta:radix
fexp:Z -> Z
prop_exp:Valid_exp fexp
emin, p:Z
Hmin:forall k : Z,
(emin < k)%Z -> (p <= k - fexp k)%Z
rnd:R -> Z
valid_rnd:Valid_rnd rnd
choice:Z -> bool
m:Z
x:=F2R {| Fnum := m; Fexp := emin |}:R
(0 < / 2)%R
beta:radix
fexp:Z -> Z
prop_exp:Valid_exp fexp
emin, p:Z
Hmin:forall k : Z,
(emin < k)%Z -> (p <= k - fexp k)%Z
rnd:R -> Z
valid_rnd:Valid_rnd rnd
choice:Z -> bool
m:Z
x:=F2R {| Fnum := m; Fexp := emin |}:R
(0 < bpow (- p + 1))%R
beta:radix
fexp:Z -> Z
prop_exp:Valid_exp fexp
emin, p:Z
Hmin:forall k : Z,
(emin < k)%Z -> (p <= k - fexp k)%Z
rnd:R -> Z
valid_rnd:Valid_rnd rnd
choice:Z -> bool
m:Z
x:=F2R {| Fnum := m; Fexp := emin |}:R
(Rabs (round beta fexp (Znearest choice) x - x) <=
 / 2 * bpow (- p + 1) * Rabs x)%R
apply Rinv_0_lt_compat.beta:radix
fexp:Z -> Z
prop_exp:Valid_exp fexp
emin, p:Z
Hmin:forall k : Z,
(emin < k)%Z -> (p <= k - fexp k)%Z
rnd:R -> Z
valid_rnd:Valid_rnd rnd
choice:Z -> bool
m:Z
x:=F2R {| Fnum := m; Fexp := emin |}:R
(0 < 2)%R
beta:radix
fexp:Z -> Z
prop_exp:Valid_exp fexp
emin, p:Z
Hmin:forall k : Z,
(emin < k)%Z -> (p <= k - fexp k)%Z
rnd:R -> Z
valid_rnd:Valid_rnd rnd
choice:Z -> bool
m:Z
x:=F2R {| Fnum := m; Fexp := emin |}:R
(0 < bpow (- p + 1))%R
beta:radix
fexp:Z -> Z
prop_exp:Valid_exp fexp
emin, p:Z
Hmin:forall k : Z,
(emin < k)%Z -> (p <= k - fexp k)%Z
rnd:R -> Z
valid_rnd:Valid_rnd rnd
choice:Z -> bool
m:Z
x:=F2R {| Fnum := m; Fexp := emin |}:R
(Rabs (round beta fexp (Znearest choice) x - x) <=
 / 2 * bpow (- p + 1) * Rabs x)%R
now apply IZR_lt.beta:radix
fexp:Z -> Z
prop_exp:Valid_exp fexp
emin, p:Z
Hmin:forall k : Z,
(emin < k)%Z -> (p <= k - fexp k)%Z
rnd:R -> Z
valid_rnd:Valid_rnd rnd
choice:Z -> bool
m:Z
x:=F2R {| Fnum := m; Fexp := emin |}:R
(0 < bpow (- p + 1))%R
beta:radix
fexp:Z -> Z
prop_exp:Valid_exp fexp
emin, p:Z
Hmin:forall k : Z,
(emin < k)%Z -> (p <= k - fexp k)%Z
rnd:R -> Z
valid_rnd:Valid_rnd rnd
choice:Z -> bool
m:Z
x:=F2R {| Fnum := m; Fexp := emin |}:R
(Rabs (round beta fexp (Znearest choice) x - x) <=
 / 2 * bpow (- p + 1) * Rabs x)%R
apply bpow_gt_0.beta:radix
fexp:Z -> Z
prop_exp:Valid_exp fexp
emin, p:Z
Hmin:forall k : Z,
(emin < k)%Z -> (p <= k - fexp k)%Z
rnd:R -> Z
valid_rnd:Valid_rnd rnd
choice:Z -> bool
m:Z
x:=F2R {| Fnum := m; Fexp := emin |}:R
(Rabs (round beta fexp (Znearest choice) x - x) <=
 / 2 * bpow (- p + 1) * Rabs x)%R
now apply relative_error_N_F2R_emin.
Qed.

Theorem relative_error_N_round :
  (0 < p)%Z ->
  forall x,
  (bpow emin <= Rabs x)%R ->
  (Rabs (round beta fexp (Znearest choice) x - x) <= /2 * bpow (-p + 1) * Rabs (round beta fexp (Znearest choice) x))%R.beta:radix
fexp:Z -> Z
prop_exp:Valid_exp fexp
emin, p:Z
Hmin:forall k : Z,
(emin < k)%Z -> (p <= k - fexp k)%Z
rnd:R -> Z
valid_rnd:Valid_rnd rnd
choice:Z -> bool
(0 < p)%Z ->
forall x : R,
(bpow emin <= Rabs x)%R ->
(Rabs (round beta fexp (Znearest choice) x - x) <=
 / 2 * bpow (- p + 1) *
 Rabs (round beta fexp (Znearest choice) x))%R
Proof with auto with typeclass_instances.beta:radix
fexp:Z -> Z
prop_exp:Valid_exp fexp
emin, p:Z
Hmin:forall k : Z,
(emin < k)%Z -> (p <= k - fexp k)%Z
rnd:R -> Z
valid_rnd:Valid_rnd rnd
choice:Z -> bool
(0 < p)%Z ->
forall x : R,
(bpow emin <= Rabs x)%R ->
(Rabs (round beta fexp (Znearest choice) x - x) <=
 / 2 * bpow (- p + 1) *
 Rabs (round beta fexp (Znearest choice) x))%R
intros Hp x Hx.beta:radix
fexp:Z -> Z
prop_exp:Valid_exp fexp
emin, p:Z
Hmin:forall k : Z,
(emin < k)%Z -> (p <= k - fexp k)%Z
rnd:R -> Z
valid_rnd:Valid_rnd rnd
choice:Z -> bool
Hp:(0 < p)%Z
x:R
Hx:(bpow emin <= Rabs x)%R
(Rabs (round beta fexp (Znearest choice) x - x) <=
 / 2 * bpow (- p + 1) *
 Rabs (round beta fexp (Znearest choice) x))%R
apply Rle_trans with (/2 * ulp beta fexp x)%R.beta:radix
fexp:Z -> Z
prop_exp:Valid_exp fexp
emin, p:Z
Hmin:forall k : Z,
(emin < k)%Z -> (p <= k - fexp k)%Z
rnd:R -> Z
valid_rnd:Valid_rnd rnd
choice:Z -> bool
Hp:(0 < p)%Z
x:R
Hx:(bpow emin <= Rabs x)%R
(Rabs (round beta fexp (Znearest choice) x - x) <=
 / 2 * ulp beta fexp x)%R
beta:radix
fexp:Z -> Z
prop_exp:Valid_exp fexp
emin, p:Z
Hmin:forall k : Z,
(emin < k)%Z -> (p <= k - fexp k)%Z
rnd:R -> Z
valid_rnd:Valid_rnd rnd
choice:Z -> bool
Hp:(0 < p)%Z
x:R
Hx:(bpow emin <= Rabs x)%R
(/ 2 * ulp beta fexp x <=
 / 2 * bpow (- p + 1) *
 Rabs (round beta fexp (Znearest choice) x))%R
now apply error_le_half_ulp.beta:radix
fexp:Z -> Z
prop_exp:Valid_exp fexp
emin, p:Z
Hmin:forall k : Z,
(emin < k)%Z -> (p <= k - fexp k)%Z
rnd:R -> Z
valid_rnd:Valid_rnd rnd
choice:Z -> bool
Hp:(0 < p)%Z
x:R
Hx:(bpow emin <= Rabs x)%R
(/ 2 * ulp beta fexp x <=
 / 2 * bpow (- p + 1) *
 Rabs (round beta fexp (Znearest choice) x))%R
rewrite Rmult_assoc.beta:radix
fexp:Z -> Z
prop_exp:Valid_exp fexp
emin, p:Z
Hmin:forall k : Z,
(emin < k)%Z -> (p <= k - fexp k)%Z
rnd:R -> Z
valid_rnd:Valid_rnd rnd
choice:Z -> bool
Hp:(0 < p)%Z
x:R
Hx:(bpow emin <= Rabs x)%R
(/ 2 * ulp beta fexp x <=
 / 2 *
 (bpow (- p + 1) *
  Rabs (round beta fexp (Znearest choice) x)))%R
apply Rmult_le_compat_l.beta:radix
fexp:Z -> Z
prop_exp:Valid_exp fexp
emin, p:Z
Hmin:forall k : Z,
(emin < k)%Z -> (p <= k - fexp k)%Z
rnd:R -> Z
valid_rnd:Valid_rnd rnd
choice:Z -> bool
Hp:(0 < p)%Z
x:R
Hx:(bpow emin <= Rabs x)%R
(0 <= / 2)%R
beta:radix
fexp:Z -> Z
prop_exp:Valid_exp fexp
emin, p:Z
Hmin:forall k : Z,
(emin < k)%Z -> (p <= k - fexp k)%Z
rnd:R -> Z
valid_rnd:Valid_rnd rnd
choice:Z -> bool
Hp:(0 < p)%Z
x:R
Hx:(bpow emin <= Rabs x)%R
(ulp beta fexp x <=
 bpow (- p + 1) *
 Rabs (round beta fexp (Znearest choice) x))%R
apply Rlt_le.beta:radix
fexp:Z -> Z
prop_exp:Valid_exp fexp
emin, p:Z
Hmin:forall k : Z,
(emin < k)%Z -> (p <= k - fexp k)%Z
rnd:R -> Z
valid_rnd:Valid_rnd rnd
choice:Z -> bool
Hp:(0 < p)%Z
x:R
Hx:(bpow emin <= Rabs x)%R
(0 < / 2)%R
beta:radix
fexp:Z -> Z
prop_exp:Valid_exp fexp
emin, p:Z
Hmin:forall k : Z,
(emin < k)%Z -> (p <= k - fexp k)%Z
rnd:R -> Z
valid_rnd:Valid_rnd rnd
choice:Z -> bool
Hp:(0 < p)%Z
x:R
Hx:(bpow emin <= Rabs x)%R
(ulp beta fexp x <=
 bpow (- p + 1) *
 Rabs (round beta fexp (Znearest choice) x))%R
apply Rinv_0_lt_compat.beta:radix
fexp:Z -> Z
prop_exp:Valid_exp fexp
emin, p:Z
Hmin:forall k : Z,
(emin < k)%Z -> (p <= k - fexp k)%Z
rnd:R -> Z
valid_rnd:Valid_rnd rnd
choice:Z -> bool
Hp:(0 < p)%Z
x:R
Hx:(bpow emin <= Rabs x)%R
(0 < 2)%R
beta:radix
fexp:Z -> Z
prop_exp:Valid_exp fexp
emin, p:Z
Hmin:forall k : Z,
(emin < k)%Z -> (p <= k - fexp k)%Z
rnd:R -> Z
valid_rnd:Valid_rnd rnd
choice:Z -> bool
Hp:(0 < p)%Z
x:R
Hx:(bpow emin <= Rabs x)%R
(ulp beta fexp x <=
 bpow (- p + 1) *
 Rabs (round beta fexp (Znearest choice) x))%R
now apply IZR_lt.beta:radix
fexp:Z -> Z
prop_exp:Valid_exp fexp
emin, p:Z
Hmin:forall k : Z,
(emin < k)%Z -> (p <= k - fexp k)%Z
rnd:R -> Z
valid_rnd:Valid_rnd rnd
choice:Z -> bool
Hp:(0 < p)%Z
x:R
Hx:(bpow emin <= Rabs x)%R
(ulp beta fexp x <=
 bpow (- p + 1) *
 Rabs (round beta fexp (Znearest choice) x))%R
assert (Hx': (x <> 0)%R).beta:radix
fexp:Z -> Z
prop_exp:Valid_exp fexp
emin, p:Z
Hmin:forall k : Z,
(emin < k)%Z -> (p <= k - fexp k)%Z
rnd:R -> Z
valid_rnd:Valid_rnd rnd
choice:Z -> bool
Hp:(0 < p)%Z
x:R
Hx:(bpow emin <= Rabs x)%R
x <> 0%R
beta:radix
fexp:Z -> Z
prop_exp:Valid_exp fexp
emin, p:Z
Hmin:forall k : Z,
(emin < k)%Z -> (p <= k - fexp k)%Z
rnd:R -> Z
valid_rnd:Valid_rnd rnd
choice:Z -> bool
Hp:(0 < p)%Z
x:R
Hx:(bpow emin <= Rabs x)%R
Hx':x <> 0%R
(ulp beta fexp x <=
 bpow (- p + 1) *
 Rabs (round beta fexp (Znearest choice) x))%R
intros H.beta:radix
fexp:Z -> Z
prop_exp:Valid_exp fexp
emin, p:Z
Hmin:forall k : Z,
(emin < k)%Z -> (p <= k - fexp k)%Z
rnd:R -> Z
valid_rnd:Valid_rnd rnd
choice:Z -> bool
Hp:(0 < p)%Z
x:R
Hx:(bpow emin <= Rabs x)%R
H:x = 0%R
False
beta:radix
fexp:Z -> Z
prop_exp:Valid_exp fexp
emin, p:Z
Hmin:forall k : Z,
(emin < k)%Z -> (p <= k - fexp k)%Z
rnd:R -> Z
valid_rnd:Valid_rnd rnd
choice:Z -> bool
Hp:(0 < p)%Z
x:R
Hx:(bpow emin <= Rabs x)%R
Hx':x <> 0%R
(ulp beta fexp x <=
 bpow (- p + 1) *
 Rabs (round beta fexp (Znearest choice) x))%R
apply Rlt_not_le with (2 := Hx).beta:radix
fexp:Z -> Z
prop_exp:Valid_exp fexp
emin, p:Z
Hmin:forall k : Z,
(emin < k)%Z -> (p <= k - fexp k)%Z
rnd:R -> Z
valid_rnd:Valid_rnd rnd
choice:Z -> bool
Hp:(0 < p)%Z
x:R
Hx:(bpow emin <= Rabs x)%R
H:x = 0%R
(Rabs x < bpow emin)%R
beta:radix
fexp:Z -> Z
prop_exp:Valid_exp fexp
emin, p:Z
Hmin:forall k : Z,
(emin < k)%Z -> (p <= k - fexp k)%Z
rnd:R -> Z
valid_rnd:Valid_rnd rnd
choice:Z -> bool
Hp:(0 < p)%Z
x:R
Hx:(bpow emin <= Rabs x)%R
Hx':x <> 0%R
(ulp beta fexp x <=
 bpow (- p + 1) *
 Rabs (round beta fexp (Znearest choice) x))%R
rewrite H, Rabs_R0.beta:radix
fexp:Z -> Z
prop_exp:Valid_exp fexp
emin, p:Z
Hmin:forall k : Z,
(emin < k)%Z -> (p <= k - fexp k)%Z
rnd:R -> Z
valid_rnd:Valid_rnd rnd
choice:Z -> bool
Hp:(0 < p)%Z
x:R
Hx:(bpow emin <= Rabs x)%R
H:x = 0%R
(0 < bpow emin)%R
beta:radix
fexp:Z -> Z
prop_exp:Valid_exp fexp
emin, p:Z
Hmin:forall k : Z,
(emin < k)%Z -> (p <= k - fexp k)%Z
rnd:R -> Z
valid_rnd:Valid_rnd rnd
choice:Z -> bool
Hp:(0 < p)%Z
x:R
Hx:(bpow emin <= Rabs x)%R
Hx':x <> 0%R
(ulp beta fexp x <=
 bpow (- p + 1) *
 Rabs (round beta fexp (Znearest choice) x))%R
apply bpow_gt_0.beta:radix
fexp:Z -> Z
prop_exp:Valid_exp fexp
emin, p:Z
Hmin:forall k : Z,
(emin < k)%Z -> (p <= k - fexp k)%Z
rnd:R -> Z
valid_rnd:Valid_rnd rnd
choice:Z -> bool
Hp:(0 < p)%Z
x:R
Hx:(bpow emin <= Rabs x)%R
Hx':x <> 0%R
(ulp beta fexp x <=
 bpow (- p + 1) *
 Rabs (round beta fexp (Znearest choice) x))%R
rewrite ulp_neq_0; trivial.beta:radix
fexp:Z -> Z
prop_exp:Valid_exp fexp
emin, p:Z
Hmin:forall k : Z,
(emin < k)%Z -> (p <= k - fexp k)%Z
rnd:R -> Z
valid_rnd:Valid_rnd rnd
choice:Z -> bool
Hp:(0 < p)%Z
x:R
Hx:(bpow emin <= Rabs x)%R
Hx':x <> 0%R
(bpow (cexp beta fexp x) <=
 bpow (- p + 1) *
 Rabs (round beta fexp (Znearest choice) x))%R
unfold cexp.beta:radix
fexp:Z -> Z
prop_exp:Valid_exp fexp
emin, p:Z
Hmin:forall k : Z,
(emin < k)%Z -> (p <= k - fexp k)%Z
rnd:R -> Z
valid_rnd:Valid_rnd rnd
choice:Z -> bool
Hp:(0 < p)%Z
x:R
Hx:(bpow emin <= Rabs x)%R
Hx':x <> 0%R
(bpow (fexp (mag beta x)) <=
 bpow (- p + 1) *
 Rabs (round beta fexp (Znearest choice) x))%R
destruct (mag beta x) as (ex, He).beta:radix
fexp:Z -> Z
prop_exp:Valid_exp fexp
emin, p:Z
Hmin:forall k : Z,
(emin < k)%Z -> (p <= k - fexp k)%Z
rnd:R -> Z
valid_rnd:Valid_rnd rnd
choice:Z -> bool
Hp:(0 < p)%Z
x:R
Hx:(bpow emin <= Rabs x)%R
Hx':x <> 0%R
ex:Z
He:x <> 0%R -> (bpow (ex - 1) <= Rabs x < bpow ex)%R
(bpow (fexp (Build_mag_prop beta x ex He)) <=
 bpow (- p + 1) *
 Rabs (round beta fexp (Znearest choice) x))%R
simpl.beta:radix
fexp:Z -> Z
prop_exp:Valid_exp fexp
emin, p:Z
Hmin:forall k : Z,
(emin < k)%Z -> (p <= k - fexp k)%Z
rnd:R -> Z
valid_rnd:Valid_rnd rnd
choice:Z -> bool
Hp:(0 < p)%Z
x:R
Hx:(bpow emin <= Rabs x)%R
Hx':x <> 0%R
ex:Z
He:x <> 0%R -> (bpow (ex - 1) <= Rabs x < bpow ex)%R
(bpow (fexp ex) <=
 bpow (- p + 1) *
 Rabs (round beta fexp (Znearest choice) x))%R
specialize (He Hx').beta:radix
fexp:Z -> Z
prop_exp:Valid_exp fexp
emin, p:Z
Hmin:forall k : Z,
(emin < k)%Z -> (p <= k - fexp k)%Z
rnd:R -> Z
valid_rnd:Valid_rnd rnd
choice:Z -> bool
Hp:(0 < p)%Z
x:R
Hx:(bpow emin <= Rabs x)%R
Hx':x <> 0%R
ex:Z
He:(bpow (ex - 1) <= Rabs x < bpow ex)%R
(bpow (fexp ex) <=
 bpow (- p + 1) *
 Rabs (round beta fexp (Znearest choice) x))%R
assert (He': (emin < ex)%Z).beta:radix
fexp:Z -> Z
prop_exp:Valid_exp fexp
emin, p:Z
Hmin:forall k : Z,
(emin < k)%Z -> (p <= k - fexp k)%Z
rnd:R -> Z
valid_rnd:Valid_rnd rnd
choice:Z -> bool
Hp:(0 < p)%Z
x:R
Hx:(bpow emin <= Rabs x)%R
Hx':x <> 0%R
ex:Z
He:(bpow (ex - 1) <= Rabs x < bpow ex)%R
(emin < ex)%Z
beta:radix
fexp:Z -> Z
prop_exp:Valid_exp fexp
emin, p:Z
Hmin:forall k : Z,
(emin < k)%Z -> (p <= k - fexp k)%Z
rnd:R -> Z
valid_rnd:Valid_rnd rnd
choice:Z -> bool
Hp:(0 < p)%Z
x:R
Hx:(bpow emin <= Rabs x)%R
Hx':x <> 0%R
ex:Z
He:(bpow (ex - 1) <= Rabs x < bpow ex)%R
He':(emin < ex)%Z
(bpow (fexp ex) <=
 bpow (- p + 1) *
 Rabs (round beta fexp (Znearest choice) x))%R
apply (lt_bpow beta).beta:radix
fexp:Z -> Z
prop_exp:Valid_exp fexp
emin, p:Z
Hmin:forall k : Z,
(emin < k)%Z -> (p <= k - fexp k)%Z
rnd:R -> Z
valid_rnd:Valid_rnd rnd
choice:Z -> bool
Hp:(0 < p)%Z
x:R
Hx:(bpow emin <= Rabs x)%R
Hx':x <> 0%R
ex:Z
He:(bpow (ex - 1) <= Rabs x < bpow ex)%R
(bpow emin < bpow ex)%R
beta:radix
fexp:Z -> Z
prop_exp:Valid_exp fexp
emin, p:Z
Hmin:forall k : Z,
(emin < k)%Z -> (p <= k - fexp k)%Z
rnd:R -> Z
valid_rnd:Valid_rnd rnd
choice:Z -> bool
Hp:(0 < p)%Z
x:R
Hx:(bpow emin <= Rabs x)%R
Hx':x <> 0%R
ex:Z
He:(bpow (ex - 1) <= Rabs x < bpow ex)%R
He':(emin < ex)%Z
(bpow (fexp ex) <=
 bpow (- p + 1) *
 Rabs (round beta fexp (Znearest choice) x))%R
apply Rle_lt_trans with (2 := proj2 He).beta:radix
fexp:Z -> Z
prop_exp:Valid_exp fexp
emin, p:Z
Hmin:forall k : Z,
(emin < k)%Z -> (p <= k - fexp k)%Z
rnd:R -> Z
valid_rnd:Valid_rnd rnd
choice:Z -> bool
Hp:(0 < p)%Z
x:R
Hx:(bpow emin <= Rabs x)%R
Hx':x <> 0%R
ex:Z
He:(bpow (ex - 1) <= Rabs x < bpow ex)%R
(bpow emin <= Rabs x)%R
beta:radix
fexp:Z -> Z
prop_exp:Valid_exp fexp
emin, p:Z
Hmin:forall k : Z,
(emin < k)%Z -> (p <= k - fexp k)%Z
rnd:R -> Z
valid_rnd:Valid_rnd rnd
choice:Z -> bool
Hp:(0 < p)%Z
x:R
Hx:(bpow emin <= Rabs x)%R
Hx':x <> 0%R
ex:Z
He:(bpow (ex - 1) <= Rabs x < bpow ex)%R
He':(emin < ex)%Z
(bpow (fexp ex) <=
 bpow (- p + 1) *
 Rabs (round beta fexp (Znearest choice) x))%R
exact Hx.beta:radix
fexp:Z -> Z
prop_exp:Valid_exp fexp
emin, p:Z
Hmin:forall k : Z,
(emin < k)%Z -> (p <= k - fexp k)%Z
rnd:R -> Z
valid_rnd:Valid_rnd rnd
choice:Z -> bool
Hp:(0 < p)%Z
x:R
Hx:(bpow emin <= Rabs x)%R
Hx':x <> 0%R
ex:Z
He:(bpow (ex - 1) <= Rabs x < bpow ex)%R
He':(emin < ex)%Z
(bpow (fexp ex) <=
 bpow (- p + 1) *
 Rabs (round beta fexp (Znearest choice) x))%R
apply Rle_trans with (bpow (-p + 1) * bpow (ex - 1))%R.beta:radix
fexp:Z -> Z
prop_exp:Valid_exp fexp
emin, p:Z
Hmin:forall k : Z,
(emin < k)%Z -> (p <= k - fexp k)%Z
rnd:R -> Z
valid_rnd:Valid_rnd rnd
choice:Z -> bool
Hp:(0 < p)%Z
x:R
Hx:(bpow emin <= Rabs x)%R
Hx':x <> 0%R
ex:Z
He:(bpow (ex - 1) <= Rabs x < bpow ex)%R
He':(emin < ex)%Z
(bpow (fexp ex) <= bpow (- p + 1) * bpow (ex - 1))%R
beta:radix
fexp:Z -> Z
prop_exp:Valid_exp fexp
emin, p:Z
Hmin:forall k : Z,
(emin < k)%Z -> (p <= k - fexp k)%Z
rnd:R -> Z
valid_rnd:Valid_rnd rnd
choice:Z -> bool
Hp:(0 < p)%Z
x:R
Hx:(bpow emin <= Rabs x)%R
Hx':x <> 0%R
ex:Z
He:(bpow (ex - 1) <= Rabs x < bpow ex)%R
He':(emin < ex)%Z
(bpow (- p + 1) * bpow (ex - 1) <=
 bpow (- p + 1) *
 Rabs (round beta fexp (Znearest choice) x))%R
rewrite <- bpow_plus.beta:radix
fexp:Z -> Z
prop_exp:Valid_exp fexp
emin, p:Z
Hmin:forall k : Z,
(emin < k)%Z -> (p <= k - fexp k)%Z
rnd:R -> Z
valid_rnd:Valid_rnd rnd
choice:Z -> bool
Hp:(0 < p)%Z
x:R
Hx:(bpow emin <= Rabs x)%R
Hx':x <> 0%R
ex:Z
He:(bpow (ex - 1) <= Rabs x < bpow ex)%R
He':(emin < ex)%Z
(bpow (fexp ex) <= bpow (- p + 1 + (ex - 1)))%R
beta:radix
fexp:Z -> Z
prop_exp:Valid_exp fexp
emin, p:Z
Hmin:forall k : Z,
(emin < k)%Z -> (p <= k - fexp k)%Z
rnd:R -> Z
valid_rnd:Valid_rnd rnd
choice:Z -> bool
Hp:(0 < p)%Z
x:R
Hx:(bpow emin <= Rabs x)%R
Hx':x <> 0%R
ex:Z
He:(bpow (ex - 1) <= Rabs x < bpow ex)%R
He':(emin < ex)%Z
(bpow (- p + 1) * bpow (ex - 1) <=
 bpow (- p + 1) *
 Rabs (round beta fexp (Znearest choice) x))%R
apply bpow_le.beta:radix
fexp:Z -> Z
prop_exp:Valid_exp fexp
emin, p:Z
Hmin:forall k : Z,
(emin < k)%Z -> (p <= k - fexp k)%Z
rnd:R -> Z
valid_rnd:Valid_rnd rnd
choice:Z -> bool
Hp:(0 < p)%Z
x:R
Hx:(bpow emin <= Rabs x)%R
Hx':x <> 0%R
ex:Z
He:(bpow (ex - 1) <= Rabs x < bpow ex)%R
He':(emin < ex)%Z
(fexp ex <= - p + 1 + (ex - 1))%Z
beta:radix
fexp:Z -> Z
prop_exp:Valid_exp fexp
emin, p:Z
Hmin:forall k : Z,
(emin < k)%Z -> (p <= k - fexp k)%Z
rnd:R -> Z
valid_rnd:Valid_rnd rnd
choice:Z -> bool
Hp:(0 < p)%Z
x:R
Hx:(bpow emin <= Rabs x)%R
Hx':x <> 0%R
ex:Z
He:(bpow (ex - 1) <= Rabs x < bpow ex)%R
He':(emin < ex)%Z
(bpow (- p + 1) * bpow (ex - 1) <=
 bpow (- p + 1) *
 Rabs (round beta fexp (Znearest choice) x))%R
generalize (Hmin ex).beta:radix
fexp:Z -> Z
prop_exp:Valid_exp fexp
emin, p:Z
Hmin:forall k : Z,
(emin < k)%Z -> (p <= k - fexp k)%Z
rnd:R -> Z
valid_rnd:Valid_rnd rnd
choice:Z -> bool
Hp:(0 < p)%Z
x:R
Hx:(bpow emin <= Rabs x)%R
Hx':x <> 0%R
ex:Z
He:(bpow (ex - 1) <= Rabs x < bpow ex)%R
He':(emin < ex)%Z
((emin < ex)%Z -> (p <= ex - fexp ex)%Z) ->
(fexp ex <= - p + 1 + (ex - 1))%Z
beta:radix
fexp:Z -> Z
prop_exp:Valid_exp fexp
emin, p:Z
Hmin:forall k : Z,
(emin < k)%Z -> (p <= k - fexp k)%Z
rnd:R -> Z
valid_rnd:Valid_rnd rnd
choice:Z -> bool
Hp:(0 < p)%Z
x:R
Hx:(bpow emin <= Rabs x)%R
Hx':x <> 0%R
ex:Z
He:(bpow (ex - 1) <= Rabs x < bpow ex)%R
He':(emin < ex)%Z
(bpow (- p + 1) * bpow (ex - 1) <=
 bpow (- p + 1) *
 Rabs (round beta fexp (Znearest choice) x))%R
omega.beta:radix
fexp:Z -> Z
prop_exp:Valid_exp fexp
emin, p:Z
Hmin:forall k : Z,
(emin < k)%Z -> (p <= k - fexp k)%Z
rnd:R -> Z
valid_rnd:Valid_rnd rnd
choice:Z -> bool
Hp:(0 < p)%Z
x:R
Hx:(bpow emin <= Rabs x)%R
Hx':x <> 0%R
ex:Z
He:(bpow (ex - 1) <= Rabs x < bpow ex)%R
He':(emin < ex)%Z
(bpow (- p + 1) * bpow (ex - 1) <=
 bpow (- p + 1) *
 Rabs (round beta fexp (Znearest choice) x))%R
apply Rmult_le_compat_l.beta:radix
fexp:Z -> Z
prop_exp:Valid_exp fexp
emin, p:Z
Hmin:forall k : Z,
(emin < k)%Z -> (p <= k - fexp k)%Z
rnd:R -> Z
valid_rnd:Valid_rnd rnd
choice:Z -> bool
Hp:(0 < p)%Z
x:R
Hx:(bpow emin <= Rabs x)%R
Hx':x <> 0%R
ex:Z
He:(bpow (ex - 1) <= Rabs x < bpow ex)%R
He':(emin < ex)%Z
(0 <= bpow (- p + 1))%R
beta:radix
fexp:Z -> Z
prop_exp:Valid_exp fexp
emin, p:Z
Hmin:forall k : Z,
(emin < k)%Z -> (p <= k - fexp k)%Z
rnd:R -> Z
valid_rnd:Valid_rnd rnd
choice:Z -> bool
Hp:(0 < p)%Z
x:R
Hx:(bpow emin <= Rabs x)%R
Hx':x <> 0%R
ex:Z
He:(bpow (ex - 1) <= Rabs x < bpow ex)%R
He':(emin < ex)%Z
(bpow (ex - 1) <=
 Rabs (round beta fexp (Znearest choice) x))%R
apply bpow_ge_0.beta:radix
fexp:Z -> Z
prop_exp:Valid_exp fexp
emin, p:Z
Hmin:forall k : Z,
(emin < k)%Z -> (p <= k - fexp k)%Z
rnd:R -> Z
valid_rnd:Valid_rnd rnd
choice:Z -> bool
Hp:(0 < p)%Z
x:R
Hx:(bpow emin <= Rabs x)%R
Hx':x <> 0%R
ex:Z
He:(bpow (ex - 1) <= Rabs x < bpow ex)%R
He':(emin < ex)%Z
(bpow (ex - 1) <=
 Rabs (round beta fexp (Znearest choice) x))%R
generalize He.beta:radix
fexp:Z -> Z
prop_exp:Valid_exp fexp
emin, p:Z
Hmin:forall k : Z,
(emin < k)%Z -> (p <= k - fexp k)%Z
rnd:R -> Z
valid_rnd:Valid_rnd rnd
choice:Z -> bool
Hp:(0 < p)%Z
x:R
Hx:(bpow emin <= Rabs x)%R
Hx':x <> 0%R
ex:Z
He:(bpow (ex - 1) <= Rabs x < bpow ex)%R
He':(emin < ex)%Z
(bpow (ex - 1) <= Rabs x < bpow ex)%R ->
(bpow (ex - 1) <=
 Rabs (round beta fexp (Znearest choice) x))%R
apply round_abs_abs...beta:radix
fexp:Z -> Z
prop_exp:Valid_exp fexp
emin, p:Z
Hmin:forall k : Z,
(emin < k)%Z -> (p <= k - fexp k)%Z
rnd:R -> Z
valid_rnd:Valid_rnd rnd
choice:Z -> bool
Hp:(0 < p)%Z
x:R
Hx:(bpow emin <= Rabs x)%R
Hx':x <> 0%R
ex:Z
He:(bpow (ex - 1) <= Rabs x < bpow ex)%R
He':(emin < ex)%Z
forall rnd0 : R -> Z,
Valid_rnd rnd0 ->
forall x0 : R,
(0 <= x0)%R ->
(bpow (ex - 1) <= x0 < bpow ex)%R ->
(bpow (ex - 1) <= round beta fexp rnd0 x0)%R
clear rnd valid_rnd x Hx Hx' He.beta:radix
fexp:Z -> Z
prop_exp:Valid_exp fexp
emin, p:Z
Hmin:forall k : Z,
(emin < k)%Z -> (p <= k - fexp k)%Z
choice:Z -> bool
Hp:(0 < p)%Z
ex:Z
He':(emin < ex)%Z
forall rnd : R -> Z,
Valid_rnd rnd ->
forall x : R,
(0 <= x)%R ->
(bpow (ex - 1) <= x < bpow ex)%R ->
(bpow (ex - 1) <= round beta fexp rnd x)%R
intros rnd valid_rnd x _ Hx.beta:radix
fexp:Z -> Z
prop_exp:Valid_exp fexp
emin, p:Z
Hmin:forall k : Z,
(emin < k)%Z -> (p <= k - fexp k)%Z
choice:Z -> bool
Hp:(0 < p)%Z
ex:Z
He':(emin < ex)%Z
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x:R
Hx:(bpow (ex - 1) <= x < bpow ex)%R
(bpow (ex - 1) <= round beta fexp rnd x)%R
rewrite <- (round_generic beta fexp rnd (bpow (ex - 1))).beta:radix
fexp:Z -> Z
prop_exp:Valid_exp fexp
emin, p:Z
Hmin:forall k : Z,
(emin < k)%Z -> (p <= k - fexp k)%Z
choice:Z -> bool
Hp:(0 < p)%Z
ex:Z
He':(emin < ex)%Z
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x:R
Hx:(bpow (ex - 1) <= x < bpow ex)%R
(round beta fexp rnd (bpow (ex - 1)) <=
 round beta fexp rnd x)%R
beta:radix
fexp:Z -> Z
prop_exp:Valid_exp fexp
emin, p:Z
Hmin:forall k : Z,
(emin < k)%Z -> (p <= k - fexp k)%Z
choice:Z -> bool
Hp:(0 < p)%Z
ex:Z
He':(emin < ex)%Z
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x:R
Hx:(bpow (ex - 1) <= x < bpow ex)%R
generic_format beta fexp (bpow (ex - 1))
now apply round_le.beta:radix
fexp:Z -> Z
prop_exp:Valid_exp fexp
emin, p:Z
Hmin:forall k : Z,
(emin < k)%Z -> (p <= k - fexp k)%Z
choice:Z -> bool
Hp:(0 < p)%Z
ex:Z
He':(emin < ex)%Z
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x:R
Hx:(bpow (ex - 1) <= x < bpow ex)%R
generic_format beta fexp (bpow (ex - 1))
apply generic_format_bpow.beta:radix
fexp:Z -> Z
prop_exp:Valid_exp fexp
emin, p:Z
Hmin:forall k : Z,
(emin < k)%Z -> (p <= k - fexp k)%Z
choice:Z -> bool
Hp:(0 < p)%Z
ex:Z
He':(emin < ex)%Z
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x:R
Hx:(bpow (ex - 1) <= x < bpow ex)%R
(fexp (ex - 1 + 1) <= ex - 1)%Z
ring_simplify (ex - 1 + 1)%Z.beta:radix
fexp:Z -> Z
prop_exp:Valid_exp fexp
emin, p:Z
Hmin:forall k : Z,
(emin < k)%Z -> (p <= k - fexp k)%Z
choice:Z -> bool
Hp:(0 < p)%Z
ex:Z
He':(emin < ex)%Z
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x:R
Hx:(bpow (ex - 1) <= x < bpow ex)%R
(fexp ex <= ex - 1)%Z
generalize (Hmin ex).beta:radix
fexp:Z -> Z
prop_exp:Valid_exp fexp
emin, p:Z
Hmin:forall k : Z,
(emin < k)%Z -> (p <= k - fexp k)%Z
choice:Z -> bool
Hp:(0 < p)%Z
ex:Z
He':(emin < ex)%Z
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x:R
Hx:(bpow (ex - 1) <= x < bpow ex)%R
((emin < ex)%Z -> (p <= ex - fexp ex)%Z) ->
(fexp ex <= ex - 1)%Z
omega.
Qed.

Theorem relative_error_N_round_F2R_emin :
  (0 < p)%Z ->
  forall m, let x := F2R (Float beta m emin) in
  (Rabs (round beta fexp (Znearest choice) x - x) <= /2 * bpow (-p + 1) * Rabs (round beta fexp (Znearest choice) x))%R.beta:radix
fexp:Z -> Z
prop_exp:Valid_exp fexp
emin, p:Z
Hmin:forall k : Z,
(emin < k)%Z -> (p <= k - fexp k)%Z
rnd:R -> Z
valid_rnd:Valid_rnd rnd
choice:Z -> bool
(0 < p)%Z ->
forall m : Z,
let x := F2R {| Fnum := m; Fexp := emin |} in
(Rabs (round beta fexp (Znearest choice) x - x) <=
 / 2 * bpow (- p + 1) *
 Rabs (round beta fexp (Znearest choice) x))%R
Proof with auto with typeclass_instances.beta:radix
fexp:Z -> Z
prop_exp:Valid_exp fexp
emin, p:Z
Hmin:forall k : Z,
(emin < k)%Z -> (p <= k - fexp k)%Z
rnd:R -> Z
valid_rnd:Valid_rnd rnd
choice:Z -> bool
(0 < p)%Z ->
forall m : Z,
let x := F2R {| Fnum := m; Fexp := emin |} in
(Rabs (round beta fexp (Znearest choice) x - x) <=
 / 2 * bpow (- p + 1) *
 Rabs (round beta fexp (Znearest choice) x))%R
intros Hp m x.beta:radix
fexp:Z -> Z
prop_exp:Valid_exp fexp
emin, p:Z
Hmin:forall k : Z,
(emin < k)%Z -> (p <= k - fexp k)%Z
rnd:R -> Z
valid_rnd:Valid_rnd rnd
choice:Z -> bool
Hp:(0 < p)%Z
m:Z
x:=F2R {| Fnum := m; Fexp := emin |}:R
(Rabs (round beta fexp (Znearest choice) x - x) <=
 / 2 * bpow (- p + 1) *
 Rabs (round beta fexp (Znearest choice) x))%R
destruct (Req_dec x 0) as [Hx|Hx].beta:radix
fexp:Z -> Z
prop_exp:Valid_exp fexp
emin, p:Z
Hmin:forall k : Z,
(emin < k)%Z -> (p <= k - fexp k)%Z
rnd:R -> Z
valid_rnd:Valid_rnd rnd
choice:Z -> bool
Hp:(0 < p)%Z
m:Z
x:=F2R {| Fnum := m; Fexp := emin |}:R
Hx:x = 0%R
(Rabs (round beta fexp (Znearest choice) x - x) <=
 / 2 * bpow (- p + 1) *
 Rabs (round beta fexp (Znearest choice) x))%R
beta:radix
fexp:Z -> Z
prop_exp:Valid_exp fexp
emin, p:Z
Hmin:forall k : Z,
(emin < k)%Z -> (p <= k - fexp k)%Z
rnd:R -> Z
valid_rnd:Valid_rnd rnd
choice:Z -> bool
Hp:(0 < p)%Z
m:Z
x:=F2R {| Fnum := m; Fexp := emin |}:R
Hx:x <> 0%R
(Rabs (round beta fexp (Znearest choice) x - x) <=
 / 2 * bpow (- p + 1) *
 Rabs (round beta fexp (Znearest choice) x))%R
(* . *)
rewrite Hx, round_0...beta:radix
fexp:Z -> Z
prop_exp:Valid_exp fexp
emin, p:Z
Hmin:forall k : Z,
(emin < k)%Z -> (p <= k - fexp k)%Z
rnd:R -> Z
valid_rnd:Valid_rnd rnd
choice:Z -> bool
Hp:(0 < p)%Z
m:Z
x:=F2R {| Fnum := m; Fexp := emin |}:R
Hx:x = 0%R
(Rabs (0 - 0) <= / 2 * bpow (- p + 1) * Rabs 0)%R
beta:radix
fexp:Z -> Z
prop_exp:Valid_exp fexp
emin, p:Z
Hmin:forall k : Z,
(emin < k)%Z -> (p <= k - fexp k)%Z
rnd:R -> Z
valid_rnd:Valid_rnd rnd
choice:Z -> bool
Hp:(0 < p)%Z
m:Z
x:=F2R {| Fnum := m; Fexp := emin |}:R
Hx:x <> 0%R
(Rabs (round beta fexp (Znearest choice) x - x) <=
 / 2 * bpow (- p + 1) *
 Rabs (round beta fexp (Znearest choice) x))%R
unfold Rminus.beta:radix
fexp:Z -> Z
prop_exp:Valid_exp fexp
emin, p:Z
Hmin:forall k : Z,
(emin < k)%Z -> (p <= k - fexp k)%Z
rnd:R -> Z
valid_rnd:Valid_rnd rnd
choice:Z -> bool
Hp:(0 < p)%Z
m:Z
x:=F2R {| Fnum := m; Fexp := emin |}:R
Hx:x = 0%R
(Rabs (0 + - 0) <= / 2 * bpow (- p + 1) * Rabs 0)%R
beta:radix
fexp:Z -> Z
prop_exp:Valid_exp fexp
emin, p:Z
Hmin:forall k : Z,
(emin < k)%Z -> (p <= k - fexp k)%Z
rnd:R -> Z
valid_rnd:Valid_rnd rnd
choice:Z -> bool
Hp:(0 < p)%Z
m:Z
x:=F2R {| Fnum := m; Fexp := emin |}:R
Hx:x <> 0%R
(Rabs (round beta fexp (Znearest choice) x - x) <=
 / 2 * bpow (- p + 1) *
 Rabs (round beta fexp (Znearest choice) x))%R
rewrite Rplus_0_l, Rabs_Ropp, Rabs_R0.beta:radix
fexp:Z -> Z
prop_exp:Valid_exp fexp
emin, p:Z
Hmin:forall k : Z,
(emin < k)%Z -> (p <= k - fexp k)%Z
rnd:R -> Z
valid_rnd:Valid_rnd rnd
choice:Z -> bool
Hp:(0 < p)%Z
m:Z
x:=F2R {| Fnum := m; Fexp := emin |}:R
Hx:x = 0%R
(0 <= / 2 * bpow (- p + 1) * 0)%R
beta:radix
fexp:Z -> Z
prop_exp:Valid_exp fexp
emin, p:Z
Hmin:forall k : Z,
(emin < k)%Z -> (p <= k - fexp k)%Z
rnd:R -> Z
valid_rnd:Valid_rnd rnd
choice:Z -> bool
Hp:(0 < p)%Z
m:Z
x:=F2R {| Fnum := m; Fexp := emin |}:R
Hx:x <> 0%R
(Rabs (round beta fexp (Znearest choice) x - x) <=
 / 2 * bpow (- p + 1) *
 Rabs (round beta fexp (Znearest choice) x))%R
rewrite Rmult_0_r.beta:radix
fexp:Z -> Z
prop_exp:Valid_exp fexp
emin, p:Z
Hmin:forall k : Z,
(emin < k)%Z -> (p <= k - fexp k)%Z
rnd:R -> Z
valid_rnd:Valid_rnd rnd
choice:Z -> bool
Hp:(0 < p)%Z
m:Z
x:=F2R {| Fnum := m; Fexp := emin |}:R
Hx:x = 0%R
(0 <= 0)%R
beta:radix
fexp:Z -> Z
prop_exp:Valid_exp fexp
emin, p:Z
Hmin:forall k : Z,
(emin < k)%Z -> (p <= k - fexp k)%Z
rnd:R -> Z
valid_rnd:Valid_rnd rnd
choice:Z -> bool
Hp:(0 < p)%Z
m:Z
x:=F2R {| Fnum := m; Fexp := emin |}:R
Hx:x <> 0%R
(Rabs (round beta fexp (Znearest choice) x - x) <=
 / 2 * bpow (- p + 1) *
 Rabs (round beta fexp (Znearest choice) x))%R
apply Rle_refl.beta:radix
fexp:Z -> Z
prop_exp:Valid_exp fexp
emin, p:Z
Hmin:forall k : Z,
(emin < k)%Z -> (p <= k - fexp k)%Z
rnd:R -> Z
valid_rnd:Valid_rnd rnd
choice:Z -> bool
Hp:(0 < p)%Z
m:Z
x:=F2R {| Fnum := m; Fexp := emin |}:R
Hx:x <> 0%R
(Rabs (round beta fexp (Znearest choice) x - x) <=
 / 2 * bpow (- p + 1) *
 Rabs (round beta fexp (Znearest choice) x))%R
(* . *)
apply relative_error_N_round with (1 := Hp).beta:radix
fexp:Z -> Z
prop_exp:Valid_exp fexp
emin, p:Z
Hmin:forall k : Z,
(emin < k)%Z -> (p <= k - fexp k)%Z
rnd:R -> Z
valid_rnd:Valid_rnd rnd
choice:Z -> bool
Hp:(0 < p)%Z
m:Z
x:=F2R {| Fnum := m; Fexp := emin |}:R
Hx:x <> 0%R
(bpow emin <= Rabs x)%R
unfold x.beta:radix
fexp:Z -> Z
prop_exp:Valid_exp fexp
emin, p:Z
Hmin:forall k : Z,
(emin < k)%Z -> (p <= k - fexp k)%Z
rnd:R -> Z
valid_rnd:Valid_rnd rnd
choice:Z -> bool
Hp:(0 < p)%Z
m:Z
x:=F2R {| Fnum := m; Fexp := emin |}:R
Hx:x <> 0%R
(bpow emin <= Rabs (F2R {| Fnum := m; Fexp := emin |}))%R
rewrite <- F2R_Zabs.beta:radix
fexp:Z -> Z
prop_exp:Valid_exp fexp
emin, p:Z
Hmin:forall k : Z,
(emin < k)%Z -> (p <= k - fexp k)%Z
rnd:R -> Z
valid_rnd:Valid_rnd rnd
choice:Z -> bool
Hp:(0 < p)%Z
m:Z
x:=F2R {| Fnum := m; Fexp := emin |}:R
Hx:x <> 0%R
(bpow emin <= F2R {| Fnum := Z.abs m; Fexp := emin |})%R
apply bpow_le_F2R.beta:radix
fexp:Z -> Z
prop_exp:Valid_exp fexp
emin, p:Z
Hmin:forall k : Z,
(emin < k)%Z -> (p <= k - fexp k)%Z
rnd:R -> Z
valid_rnd:Valid_rnd rnd
choice:Z -> bool
Hp:(0 < p)%Z
m:Z
x:=F2R {| Fnum := m; Fexp := emin |}:R
Hx:x <> 0%R
(0 < Z.abs m)%Z
apply lt_F2R with beta emin.beta:radix
fexp:Z -> Z
prop_exp:Valid_exp fexp
emin, p:Z
Hmin:forall k : Z,
(emin < k)%Z -> (p <= k - fexp k)%Z
rnd:R -> Z
valid_rnd:Valid_rnd rnd
choice:Z -> bool
Hp:(0 < p)%Z
m:Z
x:=F2R {| Fnum := m; Fexp := emin |}:R
Hx:x <> 0%R
(F2R {| Fnum := 0; Fexp := emin |} <
 F2R {| Fnum := Z.abs m; Fexp := emin |})%R
rewrite F2R_0, F2R_Zabs.beta:radix
fexp:Z -> Z
prop_exp:Valid_exp fexp
emin, p:Z
Hmin:forall k : Z,
(emin < k)%Z -> (p <= k - fexp k)%Z
rnd:R -> Z
valid_rnd:Valid_rnd rnd
choice:Z -> bool
Hp:(0 < p)%Z
m:Z
x:=F2R {| Fnum := m; Fexp := emin |}:R
Hx:x <> 0%R
(0 < Rabs (F2R {| Fnum := m; Fexp := emin |}))%R
now apply Rabs_pos_lt.
Qed.

End Fprop_relative_generic.

Section Fprop_relative_FLX.

Variable prec : Z.
Variable Hp : Z.lt 0 prec.

Lemma relative_error_FLX_aux :
  forall k, (prec <= k - FLX_exp prec k)%Z.beta:radix
prec:Z
Hp:(0 < prec)%Z
forall k : Z, (prec <= k - FLX_exp prec k)%Z
Proof.beta:radix
prec:Z
Hp:(0 < prec)%Z
forall k : Z, (prec <= k - FLX_exp prec k)%Z
intros k.beta:radix
prec:Z
Hp:(0 < prec)%Z
k:Z
(prec <= k - FLX_exp prec k)%Z
unfold FLX_exp.beta:radix
prec:Z
Hp:(0 < prec)%Z
k:Z
(prec <= k - (k - prec))%Z
omega.
Qed.

Variable rnd : R -> Z.
Context { valid_rnd : Valid_rnd rnd }.

Theorem relative_error_FLX :
  forall x,
  (x <> 0)%R ->
  (Rabs (round beta (FLX_exp prec) rnd x - x) < bpow (-prec + 1) * Rabs x)%R.beta:radix
prec:Z
Hp:(0 < prec)%Z
rnd:R -> Z
valid_rnd:Valid_rnd rnd
forall x : R,
x <> 0%R ->
(Rabs (round beta (FLX_exp prec) rnd x - x) <
 bpow (- prec + 1) * Rabs x)%R
Proof with auto with typeclass_instances.beta:radix
prec:Z
Hp:(0 < prec)%Z
rnd:R -> Z
valid_rnd:Valid_rnd rnd
forall x : R,
x <> 0%R ->
(Rabs (round beta (FLX_exp prec) rnd x - x) <
 bpow (- prec + 1) * Rabs x)%R
intros x Hx.beta:radix
prec:Z
Hp:(0 < prec)%Z
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x:R
Hx:x <> 0%R
(Rabs (round beta (FLX_exp prec) rnd x - x) <
 bpow (- prec + 1) * Rabs x)%R
destruct (mag beta x) as (ex, He).beta:radix
prec:Z
Hp:(0 < prec)%Z
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x:R
Hx:x <> 0%R
ex:Z
He:x <> 0%R -> (bpow (ex - 1) <= Rabs x < bpow ex)%R
(Rabs (round beta (FLX_exp prec) rnd x - x) <
 bpow (- prec + 1) * Rabs x)%R
specialize (He Hx).beta:radix
prec:Z
Hp:(0 < prec)%Z
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x:R
Hx:x <> 0%R
ex:Z
He:(bpow (ex - 1) <= Rabs x < bpow ex)%R
(Rabs (round beta (FLX_exp prec) rnd x - x) <
 bpow (- prec + 1) * Rabs x)%R
apply relative_error with (ex - 1)%Z...beta:radix
prec:Z
Hp:(0 < prec)%Z
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x:R
Hx:x <> 0%R
ex:Z
He:(bpow (ex - 1) <= Rabs x < bpow ex)%R
forall k : Z,
(ex - 1 < k)%Z -> (prec <= k - FLX_exp prec k)%Z
beta:radix
prec:Z
Hp:(0 < prec)%Z
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x:R
Hx:x <> 0%R
ex:Z
He:(bpow (ex - 1) <= Rabs x < bpow ex)%R
(bpow (ex - 1) <= Rabs x)%R
intros k _.beta:radix
prec:Z
Hp:(0 < prec)%Z
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x:R
Hx:x <> 0%R
ex:Z
He:(bpow (ex - 1) <= Rabs x < bpow ex)%R
k:Z
(prec <= k - FLX_exp prec k)%Z
beta:radix
prec:Z
Hp:(0 < prec)%Z
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x:R
Hx:x <> 0%R
ex:Z
He:(bpow (ex - 1) <= Rabs x < bpow ex)%R
(bpow (ex - 1) <= Rabs x)%R
apply relative_error_FLX_aux.beta:radix
prec:Z
Hp:(0 < prec)%Z
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x:R
Hx:x <> 0%R
ex:Z
He:(bpow (ex - 1) <= Rabs x < bpow ex)%R
(bpow (ex - 1) <= Rabs x)%R
apply He.
Qed.

1+ε property in any rounding in FLX

Theorem relative_error_FLX_ex :
  forall x,
  exists eps,
  (Rabs eps < bpow (-prec + 1))%R /\ round beta (FLX_exp prec) rnd x = (x * (1 + eps))%R.beta:radix
prec:Z
Hp:(0 < prec)%Z
rnd:R -> Z
valid_rnd:Valid_rnd rnd
forall x : R,
exists eps : R,
  (Rabs eps < bpow (- prec + 1))%R /\
  round beta (FLX_exp prec) rnd x = (x * (1 + eps))%R
Proof with auto with typeclass_instances.beta:radix
prec:Z
Hp:(0 < prec)%Z
rnd:R -> Z
valid_rnd:Valid_rnd rnd
forall x : R,
exists eps : R,
  (Rabs eps < bpow (- prec + 1))%R /\
  round beta (FLX_exp prec) rnd x = (x * (1 + eps))%R
intros x.beta:radix
prec:Z
Hp:(0 < prec)%Z
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x:R
exists eps : R,
  (Rabs eps < bpow (- prec + 1))%R /\
  round beta (FLX_exp prec) rnd x = (x * (1 + eps))%R
apply relative_error_lt_conversion...beta:radix
prec:Z
Hp:(0 < prec)%Z
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x:R
(0 < bpow (- prec + 1))%R
beta:radix
prec:Z
Hp:(0 < prec)%Z
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x:R
x <> 0%R ->
(Rabs (round beta (FLX_exp prec) rnd x - x) <
 bpow (- prec + 1) * Rabs x)%R
apply bpow_gt_0.beta:radix
prec:Z
Hp:(0 < prec)%Z
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x:R
x <> 0%R ->
(Rabs (round beta (FLX_exp prec) rnd x - x) <
 bpow (- prec + 1) * Rabs x)%R
now apply relative_error_FLX.
Qed.

Theorem relative_error_FLX_round :
  forall x,
  (x <> 0)%R ->
  (Rabs (round beta (FLX_exp prec) rnd x - x) < bpow (-prec + 1) * Rabs (round beta (FLX_exp prec) rnd x))%R.beta:radix
prec:Z
Hp:(0 < prec)%Z
rnd:R -> Z
valid_rnd:Valid_rnd rnd
forall x : R,
x <> 0%R ->
(Rabs (round beta (FLX_exp prec) rnd x - x) <
 bpow (- prec + 1) *
 Rabs (round beta (FLX_exp prec) rnd x))%R
Proof with auto with typeclass_instances.beta:radix
prec:Z
Hp:(0 < prec)%Z
rnd:R -> Z
valid_rnd:Valid_rnd rnd
forall x : R,
x <> 0%R ->
(Rabs (round beta (FLX_exp prec) rnd x - x) <
 bpow (- prec + 1) *
 Rabs (round beta (FLX_exp prec) rnd x))%R
intros x Hx.beta:radix
prec:Z
Hp:(0 < prec)%Z
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x:R
Hx:x <> 0%R
(Rabs (round beta (FLX_exp prec) rnd x - x) <
 bpow (- prec + 1) *
 Rabs (round beta (FLX_exp prec) rnd x))%R
destruct (mag beta x) as (ex, He).beta:radix
prec:Z
Hp:(0 < prec)%Z
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x:R
Hx:x <> 0%R
ex:Z
He:x <> 0%R -> (bpow (ex - 1) <= Rabs x < bpow ex)%R
(Rabs (round beta (FLX_exp prec) rnd x - x) <
 bpow (- prec + 1) *
 Rabs (round beta (FLX_exp prec) rnd x))%R
specialize (He Hx).beta:radix
prec:Z
Hp:(0 < prec)%Z
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x:R
Hx:x <> 0%R
ex:Z
He:(bpow (ex - 1) <= Rabs x < bpow ex)%R
(Rabs (round beta (FLX_exp prec) rnd x - x) <
 bpow (- prec + 1) *
 Rabs (round beta (FLX_exp prec) rnd x))%R
apply relative_error_round with (ex - 1)%Z...beta:radix
prec:Z
Hp:(0 < prec)%Z
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x:R
Hx:x <> 0%R
ex:Z
He:(bpow (ex - 1) <= Rabs x < bpow ex)%R
forall k : Z,
(ex - 1 < k)%Z -> (prec <= k - FLX_exp prec k)%Z
beta:radix
prec:Z
Hp:(0 < prec)%Z
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x:R
Hx:x <> 0%R
ex:Z
He:(bpow (ex - 1) <= Rabs x < bpow ex)%R
(bpow (ex - 1) <= Rabs x)%R
intros k _.beta:radix
prec:Z
Hp:(0 < prec)%Z
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x:R
Hx:x <> 0%R
ex:Z
He:(bpow (ex - 1) <= Rabs x < bpow ex)%R
k:Z
(prec <= k - FLX_exp prec k)%Z
beta:radix
prec:Z
Hp:(0 < prec)%Z
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x:R
Hx:x <> 0%R
ex:Z
He:(bpow (ex - 1) <= Rabs x < bpow ex)%R
(bpow (ex - 1) <= Rabs x)%R
apply relative_error_FLX_aux.beta:radix
prec:Z
Hp:(0 < prec)%Z
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x:R
Hx:x <> 0%R
ex:Z
He:(bpow (ex - 1) <= Rabs x < bpow ex)%R
(bpow (ex - 1) <= Rabs x)%R
apply He.
Qed.

Variable choice : Z -> bool.

Theorem relative_error_N_FLX :
  forall x,
  (Rabs (round beta (FLX_exp prec) (Znearest choice) x - x) <= /2 * bpow (-prec + 1) * Rabs x)%R.beta:radix
prec:Z
Hp:(0 < prec)%Z
rnd:R -> Z
valid_rnd:Valid_rnd rnd
choice:Z -> bool
forall x : R,
(Rabs
   (round beta (FLX_exp prec) (Znearest choice) x - x) <=
 / 2 * bpow (- prec + 1) * Rabs x)%R
Proof with auto with typeclass_instances.beta:radix
prec:Z
Hp:(0 < prec)%Z
rnd:R -> Z
valid_rnd:Valid_rnd rnd
choice:Z -> bool
forall x : R,
(Rabs
   (round beta (FLX_exp prec) (Znearest choice) x - x) <=
 / 2 * bpow (- prec + 1) * Rabs x)%R
intros x.beta:radix
prec:Z
Hp:(0 < prec)%Z
rnd:R -> Z
valid_rnd:Valid_rnd rnd
choice:Z -> bool
x:R
(Rabs
   (round beta (FLX_exp prec) (Znearest choice) x - x) <=
 / 2 * bpow (- prec + 1) * Rabs x)%R
destruct (Req_dec x 0) as [Hx|Hx].beta:radix
prec:Z
Hp:(0 < prec)%Z
rnd:R -> Z
valid_rnd:Valid_rnd rnd
choice:Z -> bool
x:R
Hx:x = 0%R
(Rabs
   (round beta (FLX_exp prec) (Znearest choice) x - x) <=
 / 2 * bpow (- prec + 1) * Rabs x)%R
beta:radix
prec:Z
Hp:(0 < prec)%Z
rnd:R -> Z
valid_rnd:Valid_rnd rnd
choice:Z -> bool
x:R
Hx:x <> 0%R
(Rabs
   (round beta (FLX_exp prec) (Znearest choice) x - x) <=
 / 2 * bpow (- prec + 1) * Rabs x)%R
(* . *)
rewrite Hx, round_0...beta:radix
prec:Z
Hp:(0 < prec)%Z
rnd:R -> Z
valid_rnd:Valid_rnd rnd
choice:Z -> bool
x:R
Hx:x = 0%R
(Rabs (0 - 0) <= / 2 * bpow (- prec + 1) * Rabs 0)%R
beta:radix
prec:Z
Hp:(0 < prec)%Z
rnd:R -> Z
valid_rnd:Valid_rnd rnd
choice:Z -> bool
x:R
Hx:x <> 0%R
(Rabs
   (round beta (FLX_exp prec) (Znearest choice) x - x) <=
 / 2 * bpow (- prec + 1) * Rabs x)%R
unfold Rminus.beta:radix
prec:Z
Hp:(0 < prec)%Z
rnd:R -> Z
valid_rnd:Valid_rnd rnd
choice:Z -> bool
x:R
Hx:x = 0%R
(Rabs (0 + - 0) <= / 2 * bpow (- prec + 1) * Rabs 0)%R
beta:radix
prec:Z
Hp:(0 < prec)%Z
rnd:R -> Z
valid_rnd:Valid_rnd rnd
choice:Z -> bool
x:R
Hx:x <> 0%R
(Rabs
   (round beta (FLX_exp prec) (Znearest choice) x - x) <=
 / 2 * bpow (- prec + 1) * Rabs x)%R
rewrite Rplus_0_l, Rabs_Ropp, Rabs_R0.beta:radix
prec:Z
Hp:(0 < prec)%Z
rnd:R -> Z
valid_rnd:Valid_rnd rnd
choice:Z -> bool
x:R
Hx:x = 0%R
(0 <= / 2 * bpow (- prec + 1) * 0)%R
beta:radix
prec:Z
Hp:(0 < prec)%Z
rnd:R -> Z
valid_rnd:Valid_rnd rnd
choice:Z -> bool
x:R
Hx:x <> 0%R
(Rabs
   (round beta (FLX_exp prec) (Znearest choice) x - x) <=
 / 2 * bpow (- prec + 1) * Rabs x)%R
rewrite Rmult_0_r.beta:radix
prec:Z
Hp:(0 < prec)%Z
rnd:R -> Z
valid_rnd:Valid_rnd rnd
choice:Z -> bool
x:R
Hx:x = 0%R
(0 <= 0)%R
beta:radix
prec:Z
Hp:(0 < prec)%Z
rnd:R -> Z
valid_rnd:Valid_rnd rnd
choice:Z -> bool
x:R
Hx:x <> 0%R
(Rabs
   (round beta (FLX_exp prec) (Znearest choice) x - x) <=
 / 2 * bpow (- prec + 1) * Rabs x)%R
apply Rle_refl.beta:radix
prec:Z
Hp:(0 < prec)%Z
rnd:R -> Z
valid_rnd:Valid_rnd rnd
choice:Z -> bool
x:R
Hx:x <> 0%R
(Rabs
   (round beta (FLX_exp prec) (Znearest choice) x - x) <=
 / 2 * bpow (- prec + 1) * Rabs x)%R
(* . *)
destruct (mag beta x) as (ex, He).beta:radix
prec:Z
Hp:(0 < prec)%Z
rnd:R -> Z
valid_rnd:Valid_rnd rnd
choice:Z -> bool
x:R
Hx:x <> 0%R
ex:Z
He:x <> 0%R -> (bpow (ex - 1) <= Rabs x < bpow ex)%R
(Rabs
   (round beta (FLX_exp prec) (Znearest choice) x - x) <=
 / 2 * bpow (- prec + 1) * Rabs x)%R
specialize (He Hx).beta:radix
prec:Z
Hp:(0 < prec)%Z
rnd:R -> Z
valid_rnd:Valid_rnd rnd
choice:Z -> bool
x:R
Hx:x <> 0%R
ex:Z
He:(bpow (ex - 1) <= Rabs x < bpow ex)%R
(Rabs
   (round beta (FLX_exp prec) (Znearest choice) x - x) <=
 / 2 * bpow (- prec + 1) * Rabs x)%R
apply relative_error_N with (ex - 1)%Z...beta:radix
prec:Z
Hp:(0 < prec)%Z
rnd:R -> Z
valid_rnd:Valid_rnd rnd
choice:Z -> bool
x:R
Hx:x <> 0%R
ex:Z
He:(bpow (ex - 1) <= Rabs x < bpow ex)%R
forall k : Z,
(ex - 1 < k)%Z -> (prec <= k - FLX_exp prec k)%Z
beta:radix
prec:Z
Hp:(0 < prec)%Z
rnd:R -> Z
valid_rnd:Valid_rnd rnd
choice:Z -> bool
x:R
Hx:x <> 0%R
ex:Z
He:(bpow (ex - 1) <= Rabs x < bpow ex)%R
(bpow (ex - 1) <= Rabs x)%R
intros k _.beta:radix
prec:Z
Hp:(0 < prec)%Z
rnd:R -> Z
valid_rnd:Valid_rnd rnd
choice:Z -> bool
x:R
Hx:x <> 0%R
ex:Z
He:(bpow (ex - 1) <= Rabs x < bpow ex)%R
k:Z
(prec <= k - FLX_exp prec k)%Z
beta:radix
prec:Z
Hp:(0 < prec)%Z
rnd:R -> Z
valid_rnd:Valid_rnd rnd
choice:Z -> bool
x:R
Hx:x <> 0%R
ex:Z
He:(bpow (ex - 1) <= Rabs x < bpow ex)%R
(bpow (ex - 1) <= Rabs x)%R
apply relative_error_FLX_aux.beta:radix
prec:Z
Hp:(0 < prec)%Z
rnd:R -> Z
valid_rnd:Valid_rnd rnd
choice:Z -> bool
x:R
Hx:x <> 0%R
ex:Z
He:(bpow (ex - 1) <= Rabs x < bpow ex)%R
(bpow (ex - 1) <= Rabs x)%R
apply He.
Qed.

unit roundoff

Definition u_ro := (/2 * bpow (-prec + 1))%R.

Lemma u_ro_pos : (0 <= u_ro)%R.beta:radix
prec:Z
Hp:(0 < prec)%Z
rnd:R -> Z
valid_rnd:Valid_rnd rnd
choice:Z -> bool
(0 <= u_ro)%R
Proof.beta:radix
prec:Z
Hp:(0 < prec)%Z
rnd:R -> Z
valid_rnd:Valid_rnd rnd
choice:Z -> bool
(0 <= u_ro)%R apply Rmult_le_pos; [lra|apply bpow_ge_0]. Qed.

Lemma u_ro_lt_1 : (u_ro < 1)%R.beta:radix
prec:Z
Hp:(0 < prec)%Z
rnd:R -> Z
valid_rnd:Valid_rnd rnd
choice:Z -> bool
(u_ro < 1)%R
Proof.beta:radix
prec:Z
Hp:(0 < prec)%Z
rnd:R -> Z
valid_rnd:Valid_rnd rnd
choice:Z -> bool
(u_ro < 1)%R
unfold u_ro; apply (Rmult_lt_reg_l 2); [lra|].beta:radix
prec:Z
Hp:(0 < prec)%Z
rnd:R -> Z
valid_rnd:Valid_rnd rnd
choice:Z -> bool
(2 * (/ 2 * bpow (- prec + 1)) < 2 * 1)%R
rewrite <-Rmult_assoc, Rinv_r, Rmult_1_l, Rmult_1_r; [|lra].beta:radix
prec:Z
Hp:(0 < prec)%Z
rnd:R -> Z
valid_rnd:Valid_rnd rnd
choice:Z -> bool
(bpow (- prec + 1) < 2)%R
apply (Rle_lt_trans _ (bpow 0));
  [apply bpow_le; omega|simpl; lra].
Qed.

Lemma u_rod1pu_ro_pos : (0 <= u_ro / (1 + u_ro))%R.beta:radix
prec:Z
Hp:(0 < prec)%Z
rnd:R -> Z
valid_rnd:Valid_rnd rnd
choice:Z -> bool
(0 <= u_ro / (1 + u_ro))%R
Proof.beta:radix
prec:Z
Hp:(0 < prec)%Z
rnd:R -> Z
valid_rnd:Valid_rnd rnd
choice:Z -> bool
(0 <= u_ro / (1 + u_ro))%R
apply Rmult_le_pos; [|apply Rlt_le, Rinv_0_lt_compat];
assert (H := u_ro_pos); lra.
Qed.

Lemma u_rod1pu_ro_le_u_ro : (u_ro / (1 + u_ro) <= u_ro)%R.beta:radix
prec:Z
Hp:(0 < prec)%Z
rnd:R -> Z
valid_rnd:Valid_rnd rnd
choice:Z -> bool
(u_ro / (1 + u_ro) <= u_ro)%R
Proof.beta:radix
prec:Z
Hp:(0 < prec)%Z
rnd:R -> Z
valid_rnd:Valid_rnd rnd
choice:Z -> bool
(u_ro / (1 + u_ro) <= u_ro)%R
assert (Pu_ro := u_ro_pos).beta:radix
prec:Z
Hp:(0 < prec)%Z
rnd:R -> Z
valid_rnd:Valid_rnd rnd
choice:Z -> bool
Pu_ro:(0 <= u_ro)%R
(u_ro / (1 + u_ro) <= u_ro)%R
apply (Rmult_le_reg_r (1 + u_ro)); [lra|].beta:radix
prec:Z
Hp:(0 < prec)%Z
rnd:R -> Z
valid_rnd:Valid_rnd rnd
choice:Z -> bool
Pu_ro:(0 <= u_ro)%R
(u_ro / (1 + u_ro) * (1 + u_ro) <= u_ro * (1 + u_ro))%R
unfold Rdiv; rewrite Rmult_assoc, Rinv_l; [|lra].beta:radix
prec:Z
Hp:(0 < prec)%Z
rnd:R -> Z
valid_rnd:Valid_rnd rnd
choice:Z -> bool
Pu_ro:(0 <= u_ro)%R
(u_ro * 1 <= u_ro * (1 + u_ro))%R
assert (0 <= u_ro * u_ro)%R; [apply Rmult_le_pos|]; lra.
Qed.

Theorem relative_error_N_FLX' :
  forall x,
  (Rabs (round beta (FLX_exp prec) (Znearest choice) x - x)
   <= u_ro / (1 + u_ro) * Rabs x)%R.beta:radix
prec:Z
Hp:(0 < prec)%Z
rnd:R -> Z
valid_rnd:Valid_rnd rnd
choice:Z -> bool
forall x : R,
(Rabs
   (round beta (FLX_exp prec) (Znearest choice) x - x) <=
 u_ro / (1 + u_ro) * Rabs x)%R
Proof with auto with typeclass_instances.beta:radix
prec:Z
Hp:(0 < prec)%Z
rnd:R -> Z
valid_rnd:Valid_rnd rnd
choice:Z -> bool
forall x : R,
(Rabs
   (round beta (FLX_exp prec) (Znearest choice) x - x) <=
 u_ro / (1 + u_ro) * Rabs x)%R
intro x.beta:radix
prec:Z
Hp:(0 < prec)%Z
rnd:R -> Z
valid_rnd:Valid_rnd rnd
choice:Z -> bool
x:R
(Rabs
   (round beta (FLX_exp prec) (Znearest choice) x - x) <=
 u_ro / (1 + u_ro) * Rabs x)%R
assert (Pu_ro : (0 <= u_ro)%R).beta:radix
prec:Z
Hp:(0 < prec)%Z
rnd:R -> Z
valid_rnd:Valid_rnd rnd
choice:Z -> bool
x:R
(0 <= u_ro)%R
beta:radix
prec:Z
Hp:(0 < prec)%Z
rnd:R -> Z
valid_rnd:Valid_rnd rnd
choice:Z -> bool
x:R
Pu_ro:(0 <= u_ro)%R
(Rabs
   (round beta (FLX_exp prec) (Znearest choice) x - x) <=
 u_ro / (1 + u_ro) * Rabs x)%R
{beta:radix
prec:Z
Hp:(0 < prec)%Z
rnd:R -> Z
valid_rnd:Valid_rnd rnd
choice:Z -> bool
x:R
(0 <= u_ro)%R apply Rmult_le_pos; [lra|apply bpow_ge_0]. }beta:radix
prec:Z
Hp:(0 < prec)%Z
rnd:R -> Z
valid_rnd:Valid_rnd rnd
choice:Z -> bool
x:R
Pu_ro:(0 <= u_ro)%R
(Rabs
   (round beta (FLX_exp prec) (Znearest choice) x - x) <=
 u_ro / (1 + u_ro) * Rabs x)%R
destruct (Req_dec x 0) as [Zx|Nzx].beta:radix
prec:Z
Hp:(0 < prec)%Z
rnd:R -> Z
valid_rnd:Valid_rnd rnd
choice:Z -> bool
x:R
Pu_ro:(0 <= u_ro)%R
Zx:x = 0%R
(Rabs
   (round beta (FLX_exp prec) (Znearest choice) x - x) <=
 u_ro / (1 + u_ro) * Rabs x)%R
beta:radix
prec:Z
Hp:(0 < prec)%Z
rnd:R -> Z
valid_rnd:Valid_rnd rnd
choice:Z -> bool
x:R
Pu_ro:(0 <= u_ro)%R
Nzx:x <> 0%R
(Rabs
   (round beta (FLX_exp prec) (Znearest choice) x - x) <=
 u_ro / (1 + u_ro) * Rabs x)%R
{beta:radix
prec:Z
Hp:(0 < prec)%Z
rnd:R -> Z
valid_rnd:Valid_rnd rnd
choice:Z -> bool
x:R
Pu_ro:(0 <= u_ro)%R
Zx:x = 0%R
(Rabs
   (round beta (FLX_exp prec) (Znearest choice) x - x) <=
 u_ro / (1 + u_ro) * Rabs x)%R rewrite Zx, Rabs_R0, Rmult_0_r, round_0...beta:radix
prec:Z
Hp:(0 < prec)%Z
rnd:R -> Z
valid_rnd:Valid_rnd rnd
choice:Z -> bool
x:R
Pu_ro:(0 <= u_ro)%R
Zx:x = 0%R
(Rabs (0 - 0) <= 0)%R
  now unfold Rminus; rewrite Rplus_0_l, Rabs_Ropp, Rabs_R0; right. }beta:radix
prec:Z
Hp:(0 < prec)%Z
rnd:R -> Z
valid_rnd:Valid_rnd rnd
choice:Z -> bool
x:R
Pu_ro:(0 <= u_ro)%R
Nzx:x <> 0%R
(Rabs
   (round beta (FLX_exp prec) (Znearest choice) x - x) <=
 u_ro / (1 + u_ro) * Rabs x)%R
set (ufpx := bpow (mag beta x - 1)%Z).beta:radix
prec:Z
Hp:(0 < prec)%Z
rnd:R -> Z
valid_rnd:Valid_rnd rnd
choice:Z -> bool
x:R
Pu_ro:(0 <= u_ro)%R
Nzx:x <> 0%R
ufpx:=bpow (mag beta x - 1):R
(Rabs
   (round beta (FLX_exp prec) (Znearest choice) x - x) <=
 u_ro / (1 + u_ro) * Rabs x)%R
set (rx := round _ _ _ _).beta:radix
prec:Z
Hp:(0 < prec)%Z
rnd:R -> Z
valid_rnd:Valid_rnd rnd
choice:Z -> bool
x:R
Pu_ro:(0 <= u_ro)%R
Nzx:x <> 0%R
ufpx:=bpow (mag beta x - 1):R
rx:=round beta (FLX_exp prec) (Znearest choice) x:R
(Rabs (rx - x) <= u_ro / (1 + u_ro) * Rabs x)%R
assert (Pufpx : (0 <= ufpx)%R); [now apply bpow_ge_0|].beta:radix
prec:Z
Hp:(0 < prec)%Z
rnd:R -> Z
valid_rnd:Valid_rnd rnd
choice:Z -> bool
x:R
Pu_ro:(0 <= u_ro)%R
Nzx:x <> 0%R
ufpx:=bpow (mag beta x - 1):R
rx:=round beta (FLX_exp prec) (Znearest choice) x:R
Pufpx:(0 <= ufpx)%R
(Rabs (rx - x) <= u_ro / (1 + u_ro) * Rabs x)%R
assert (H_2_1 : (Rabs (rx - x) <= u_ro * ufpx)%R).beta:radix
prec:Z
Hp:(0 < prec)%Z
rnd:R -> Z
valid_rnd:Valid_rnd rnd
choice:Z -> bool
x:R
Pu_ro:(0 <= u_ro)%R
Nzx:x <> 0%R
ufpx:=bpow (mag beta x - 1):R
rx:=round beta (FLX_exp prec) (Znearest choice) x:R
Pufpx:(0 <= ufpx)%R
(Rabs (rx - x) <= u_ro * ufpx)%R
beta:radix
prec:Z
Hp:(0 < prec)%Z
rnd:R -> Z
valid_rnd:Valid_rnd rnd
choice:Z -> bool
x:R
Pu_ro:(0 <= u_ro)%R
Nzx:x <> 0%R
ufpx:=bpow (mag beta x - 1):R
rx:=round beta (FLX_exp prec) (Znearest choice) x:R
Pufpx:(0 <= ufpx)%R
H_2_1:(Rabs (rx - x) <= u_ro * ufpx)%R
(Rabs (rx - x) <= u_ro / (1 + u_ro) * Rabs x)%R
{beta:radix
prec:Z
Hp:(0 < prec)%Z
rnd:R -> Z
valid_rnd:Valid_rnd rnd
choice:Z -> bool
x:R
Pu_ro:(0 <= u_ro)%R
Nzx:x <> 0%R
ufpx:=bpow (mag beta x - 1):R
rx:=round beta (FLX_exp prec) (Znearest choice) x:R
Pufpx:(0 <= ufpx)%R
(Rabs (rx - x) <= u_ro * ufpx)%R refine (Rle_trans _ _ _ (error_le_half_ulp _ _ _ _) _);
    [now apply FLX_exp_valid|right].beta:radix
prec:Z
Hp:(0 < prec)%Z
rnd:R -> Z
valid_rnd:Valid_rnd rnd
choice:Z -> bool
x:R
Pu_ro:(0 <= u_ro)%R
Nzx:x <> 0%R
ufpx:=bpow (mag beta x - 1):R
rx:=round beta (FLX_exp prec) (Znearest choice) x:R
Pufpx:(0 <= ufpx)%R
(/ 2 * ulp beta (FLX_exp prec) x)%R = (u_ro * ufpx)%R
  unfold ulp, cexp, FLX_exp, u_ro, ufpx; rewrite (Req_bool_false _ _ Nzx).beta:radix
prec:Z
Hp:(0 < prec)%Z
rnd:R -> Z
valid_rnd:Valid_rnd rnd
choice:Z -> bool
x:R
Pu_ro:(0 <= u_ro)%R
Nzx:x <> 0%R
ufpx:=bpow (mag beta x - 1):R
rx:=round beta (FLX_exp prec) (Znearest choice) x:R
Pufpx:(0 <= ufpx)%R
(/ 2 * bpow (mag beta x - prec))%R =
(/ 2 * bpow (- prec + 1) * bpow (mag beta x - 1))%R
  rewrite Rmult_assoc, <-bpow_plus; do 2 f_equal; ring. }beta:radix
prec:Z
Hp:(0 < prec)%Z
rnd:R -> Z
valid_rnd:Valid_rnd rnd
choice:Z -> bool
x:R
Pu_ro:(0 <= u_ro)%R
Nzx:x <> 0%R
ufpx:=bpow (mag beta x - 1):R
rx:=round beta (FLX_exp prec) (Znearest choice) x:R
Pufpx:(0 <= ufpx)%R
H_2_1:(Rabs (rx - x) <= u_ro * ufpx)%R
(Rabs (rx - x) <= u_ro / (1 + u_ro) * Rabs x)%R
assert (H_2_3 : (ufpx + Rabs (rx - x) <= Rabs x)%R).beta:radix
prec:Z
Hp:(0 < prec)%Z
rnd:R -> Z
valid_rnd:Valid_rnd rnd
choice:Z -> bool
x:R
Pu_ro:(0 <= u_ro)%R
Nzx:x <> 0%R
ufpx:=bpow (mag beta x - 1):R
rx:=round beta (FLX_exp prec) (Znearest choice) x:R
Pufpx:(0 <= ufpx)%R
H_2_1:(Rabs (rx - x) <= u_ro * ufpx)%R
(ufpx + Rabs (rx - x) <= Rabs x)%R
beta:radix
prec:Z
Hp:(0 < prec)%Z
rnd:R -> Z
valid_rnd:Valid_rnd rnd
choice:Z -> bool
x:R
Pu_ro:(0 <= u_ro)%R
Nzx:x <> 0%R
ufpx:=bpow (mag beta x - 1):R
rx:=round beta (FLX_exp prec) (Znearest choice) x:R
Pufpx:(0 <= ufpx)%R
H_2_1:(Rabs (rx - x) <= u_ro * ufpx)%R
H_2_3:(ufpx + Rabs (rx - x) <= Rabs x)%R
(Rabs (rx - x) <= u_ro / (1 + u_ro) * Rabs x)%R
{beta:radix
prec:Z
Hp:(0 < prec)%Z
rnd:R -> Z
valid_rnd:Valid_rnd rnd
choice:Z -> bool
x:R
Pu_ro:(0 <= u_ro)%R
Nzx:x <> 0%R
ufpx:=bpow (mag beta x - 1):R
rx:=round beta (FLX_exp prec) (Znearest choice) x:R
Pufpx:(0 <= ufpx)%R
H_2_1:(Rabs (rx - x) <= u_ro * ufpx)%R
(ufpx + Rabs (rx - x) <= Rabs x)%R apply (Rplus_le_reg_r (- ufpx)); ring_simplify.beta:radix
prec:Z
Hp:(0 < prec)%Z
rnd:R -> Z
valid_rnd:Valid_rnd rnd
choice:Z -> bool
x:R
Pu_ro:(0 <= u_ro)%R
Nzx:x <> 0%R
ufpx:=bpow (mag beta x - 1):R
rx:=round beta (FLX_exp prec) (Znearest choice) x:R
Pufpx:(0 <= ufpx)%R
H_2_1:(Rabs (rx - x) <= u_ro * ufpx)%R
(Rabs (rx - x) <= - ufpx + Rabs x)%R
  destruct (Rle_or_lt 0 x) as [Sx|Sx].beta:radix
prec:Z
Hp:(0 < prec)%Z
rnd:R -> Z
valid_rnd:Valid_rnd rnd
choice:Z -> bool
x:R
Pu_ro:(0 <= u_ro)%R
Nzx:x <> 0%R
ufpx:=bpow (mag beta x - 1):R
rx:=round beta (FLX_exp prec) (Znearest choice) x:R
Pufpx:(0 <= ufpx)%R
H_2_1:(Rabs (rx - x) <= u_ro * ufpx)%R
Sx:(0 <= x)%R
(Rabs (rx - x) <= - ufpx + Rabs x)%R
beta:radix
prec:Z
Hp:(0 < prec)%Z
rnd:R -> Z
valid_rnd:Valid_rnd rnd
choice:Z -> bool
x:R
Pu_ro:(0 <= u_ro)%R
Nzx:x <> 0%R
ufpx:=bpow (mag beta x - 1):R
rx:=round beta (FLX_exp prec) (Znearest choice) x:R
Pufpx:(0 <= ufpx)%R
H_2_1:(Rabs (rx - x) <= u_ro * ufpx)%R
Sx:(x < 0)%R
(Rabs (rx - x) <= - ufpx + Rabs x)%R
  {beta:radix
prec:Z
Hp:(0 < prec)%Z
rnd:R -> Z
valid_rnd:Valid_rnd rnd
choice:Z -> bool
x:R
Pu_ro:(0 <= u_ro)%R
Nzx:x <> 0%R
ufpx:=bpow (mag beta x - 1):R
rx:=round beta (FLX_exp prec) (Znearest choice) x:R
Pufpx:(0 <= ufpx)%R
H_2_1:(Rabs (rx - x) <= u_ro * ufpx)%R
Sx:(0 <= x)%R
(Rabs (rx - x) <= - ufpx + Rabs x)%R apply (Rle_trans _ (Rabs (ufpx - x))).beta:radix
prec:Z
Hp:(0 < prec)%Z
rnd:R -> Z
valid_rnd:Valid_rnd rnd
choice:Z -> bool
x:R
Pu_ro:(0 <= u_ro)%R
Nzx:x <> 0%R
ufpx:=bpow (mag beta x - 1):R
rx:=round beta (FLX_exp prec) (Znearest choice) x:R
Pufpx:(0 <= ufpx)%R
H_2_1:(Rabs (rx - x) <= u_ro * ufpx)%R
Sx:(0 <= x)%R
(Rabs (rx - x) <= Rabs (ufpx - x))%R
beta:radix
prec:Z
Hp:(0 < prec)%Z
rnd:R -> Z
valid_rnd:Valid_rnd rnd
choice:Z -> bool
x:R
Pu_ro:(0 <= u_ro)%R
Nzx:x <> 0%R
ufpx:=bpow (mag beta x - 1):R
rx:=round beta (FLX_exp prec) (Znearest choice) x:R
Pufpx:(0 <= ufpx)%R
H_2_1:(Rabs (rx - x) <= u_ro * ufpx)%R
Sx:(0 <= x)%R
(Rabs (ufpx - x) <= - ufpx + Rabs x)%R
    {beta:radix
prec:Z
Hp:(0 < prec)%Z
rnd:R -> Z
valid_rnd:Valid_rnd rnd
choice:Z -> bool
x:R
Pu_ro:(0 <= u_ro)%R
Nzx:x <> 0%R
ufpx:=bpow (mag beta x - 1):R
rx:=round beta (FLX_exp prec) (Znearest choice) x:R
Pufpx:(0 <= ufpx)%R
H_2_1:(Rabs (rx - x) <= u_ro * ufpx)%R
Sx:(0 <= x)%R
(Rabs (rx - x) <= Rabs (ufpx - x))%R apply round_N_pt; [now apply FLX_exp_valid|].beta:radix
prec:Z
Hp:(0 < prec)%Z
rnd:R -> Z
valid_rnd:Valid_rnd rnd
choice:Z -> bool
x:R
Pu_ro:(0 <= u_ro)%R
Nzx:x <> 0%R
ufpx:=bpow (mag beta x - 1):R
rx:=round beta (FLX_exp prec) (Znearest choice) x:R
Pufpx:(0 <= ufpx)%R
H_2_1:(Rabs (rx - x) <= u_ro * ufpx)%R
Sx:(0 <= x)%R
generic_format beta (FLX_exp prec) ufpx
      apply generic_format_bpow; unfold FLX_exp; lia. }beta:radix
prec:Z
Hp:(0 < prec)%Z
rnd:R -> Z
valid_rnd:Valid_rnd rnd
choice:Z -> bool
x:R
Pu_ro:(0 <= u_ro)%R
Nzx:x <> 0%R
ufpx:=bpow (mag beta x - 1):R
rx:=round beta (FLX_exp prec) (Znearest choice) x:R
Pufpx:(0 <= ufpx)%R
H_2_1:(Rabs (rx - x) <= u_ro * ufpx)%R
Sx:(0 <= x)%R
(Rabs (ufpx - x) <= - ufpx + Rabs x)%R
    rewrite Rabs_minus_sym, Rabs_pos_eq.beta:radix
prec:Z
Hp:(0 < prec)%Z
rnd:R -> Z
valid_rnd:Valid_rnd rnd
choice:Z -> bool
x:R
Pu_ro:(0 <= u_ro)%R
Nzx:x <> 0%R
ufpx:=bpow (mag beta x - 1):R
rx:=round beta (FLX_exp prec) (Znearest choice) x:R
Pufpx:(0 <= ufpx)%R
H_2_1:(Rabs (rx - x) <= u_ro * ufpx)%R
Sx:(0 <= x)%R
(x - ufpx <= - ufpx + Rabs x)%R
beta:radix
prec:Z
Hp:(0 < prec)%Z
rnd:R -> Z
valid_rnd:Valid_rnd rnd
choice:Z -> bool
x:R
Pu_ro:(0 <= u_ro)%R
Nzx:x <> 0%R
ufpx:=bpow (mag beta x - 1):R
rx:=round beta (FLX_exp prec) (Znearest choice) x:R
Pufpx:(0 <= ufpx)%R
H_2_1:(Rabs (rx - x) <= u_ro * ufpx)%R
Sx:(0 <= x)%R
(0 <= x - ufpx)%R
    {beta:radix
prec:Z
Hp:(0 < prec)%Z
rnd:R -> Z
valid_rnd:Valid_rnd rnd
choice:Z -> bool
x:R
Pu_ro:(0 <= u_ro)%R
Nzx:x <> 0%R
ufpx:=bpow (mag beta x - 1):R
rx:=round beta (FLX_exp prec) (Znearest choice) x:R
Pufpx:(0 <= ufpx)%R
H_2_1:(Rabs (rx - x) <= u_ro * ufpx)%R
Sx:(0 <= x)%R
(x - ufpx <= - ufpx + Rabs x)%R now rewrite Rabs_pos_eq; [right; ring|]. }beta:radix
prec:Z
Hp:(0 < prec)%Z
rnd:R -> Z
valid_rnd:Valid_rnd rnd
choice:Z -> bool
x:R
Pu_ro:(0 <= u_ro)%R
Nzx:x <> 0%R
ufpx:=bpow (mag beta x - 1):R
rx:=round beta (FLX_exp prec) (Znearest choice) x:R
Pufpx:(0 <= ufpx)%R
H_2_1:(Rabs (rx - x) <= u_ro * ufpx)%R
Sx:(0 <= x)%R
(0 <= x - ufpx)%R
    apply (Rplus_le_reg_r ufpx); ring_simplify.beta:radix
prec:Z
Hp:(0 < prec)%Z
rnd:R -> Z
valid_rnd:Valid_rnd rnd
choice:Z -> bool
x:R
Pu_ro:(0 <= u_ro)%R
Nzx:x <> 0%R
ufpx:=bpow (mag beta x - 1):R
rx:=round beta (FLX_exp prec) (Znearest choice) x:R
Pufpx:(0 <= ufpx)%R
H_2_1:(Rabs (rx - x) <= u_ro * ufpx)%R
Sx:(0 <= x)%R
(ufpx <= x)%R
    now rewrite <-(Rabs_pos_eq _ Sx); apply bpow_mag_le. }beta:radix
prec:Z
Hp:(0 < prec)%Z
rnd:R -> Z
valid_rnd:Valid_rnd rnd
choice:Z -> bool
x:R
Pu_ro:(0 <= u_ro)%R
Nzx:x <> 0%R
ufpx:=bpow (mag beta x - 1):R
rx:=round beta (FLX_exp prec) (Znearest choice) x:R
Pufpx:(0 <= ufpx)%R
H_2_1:(Rabs (rx - x) <= u_ro * ufpx)%R
Sx:(x < 0)%R
(Rabs (rx - x) <= - ufpx + Rabs x)%R
  apply (Rle_trans _ (Rabs (- ufpx - x))).beta:radix
prec:Z
Hp:(0 < prec)%Z
rnd:R -> Z
valid_rnd:Valid_rnd rnd
choice:Z -> bool
x:R
Pu_ro:(0 <= u_ro)%R
Nzx:x <> 0%R
ufpx:=bpow (mag beta x - 1):R
rx:=round beta (FLX_exp prec) (Znearest choice) x:R
Pufpx:(0 <= ufpx)%R
H_2_1:(Rabs (rx - x) <= u_ro * ufpx)%R
Sx:(x < 0)%R
(Rabs (rx - x) <= Rabs (- ufpx - x))%R
beta:radix
prec:Z
Hp:(0 < prec)%Z
rnd:R -> Z
valid_rnd:Valid_rnd rnd
choice:Z -> bool
x:R
Pu_ro:(0 <= u_ro)%R
Nzx:x <> 0%R
ufpx:=bpow (mag beta x - 1):R
rx:=round beta (FLX_exp prec) (Znearest choice) x:R
Pufpx:(0 <= ufpx)%R
H_2_1:(Rabs (rx - x) <= u_ro * ufpx)%R
Sx:(x < 0)%R
(Rabs (- ufpx - x) <= - ufpx + Rabs x)%R
  {beta:radix
prec:Z
Hp:(0 < prec)%Z
rnd:R -> Z
valid_rnd:Valid_rnd rnd
choice:Z -> bool
x:R
Pu_ro:(0 <= u_ro)%R
Nzx:x <> 0%R
ufpx:=bpow (mag beta x - 1):R
rx:=round beta (FLX_exp prec) (Znearest choice) x:R
Pufpx:(0 <= ufpx)%R
H_2_1:(Rabs (rx - x) <= u_ro * ufpx)%R
Sx:(x < 0)%R
(Rabs (rx - x) <= Rabs (- ufpx - x))%R apply round_N_pt; [now apply FLX_exp_valid|].beta:radix
prec:Z
Hp:(0 < prec)%Z
rnd:R -> Z
valid_rnd:Valid_rnd rnd
choice:Z -> bool
x:R
Pu_ro:(0 <= u_ro)%R
Nzx:x <> 0%R
ufpx:=bpow (mag beta x - 1):R
rx:=round beta (FLX_exp prec) (Znearest choice) x:R
Pufpx:(0 <= ufpx)%R
H_2_1:(Rabs (rx - x) <= u_ro * ufpx)%R
Sx:(x < 0)%R
generic_format beta (FLX_exp prec) (- ufpx)
    apply generic_format_opp, generic_format_bpow; unfold FLX_exp; lia. }beta:radix
prec:Z
Hp:(0 < prec)%Z
rnd:R -> Z
valid_rnd:Valid_rnd rnd
choice:Z -> bool
x:R
Pu_ro:(0 <= u_ro)%R
Nzx:x <> 0%R
ufpx:=bpow (mag beta x - 1):R
rx:=round beta (FLX_exp prec) (Znearest choice) x:R
Pufpx:(0 <= ufpx)%R
H_2_1:(Rabs (rx - x) <= u_ro * ufpx)%R
Sx:(x < 0)%R
(Rabs (- ufpx - x) <= - ufpx + Rabs x)%R
  rewrite Rabs_pos_eq; [now rewrite Rabs_left; [right|]|].beta:radix
prec:Z
Hp:(0 < prec)%Z
rnd:R -> Z
valid_rnd:Valid_rnd rnd
choice:Z -> bool
x:R
Pu_ro:(0 <= u_ro)%R
Nzx:x <> 0%R
ufpx:=bpow (mag beta x - 1):R
rx:=round beta (FLX_exp prec) (Znearest choice) x:R
Pufpx:(0 <= ufpx)%R
H_2_1:(Rabs (rx - x) <= u_ro * ufpx)%R
Sx:(x < 0)%R
(0 <= - ufpx - x)%R
  apply (Rplus_le_reg_r x); ring_simplify.beta:radix
prec:Z
Hp:(0 < prec)%Z
rnd:R -> Z
valid_rnd:Valid_rnd rnd
choice:Z -> bool
x:R
Pu_ro:(0 <= u_ro)%R
Nzx:x <> 0%R
ufpx:=bpow (mag beta x - 1):R
rx:=round beta (FLX_exp prec) (Znearest choice) x:R
Pufpx:(0 <= ufpx)%R
H_2_1:(Rabs (rx - x) <= u_ro * ufpx)%R
Sx:(x < 0)%R
(x <= - ufpx)%R
  rewrite <-(Ropp_involutive x); apply Ropp_le_contravar; unfold ufpx.beta:radix
prec:Z
Hp:(0 < prec)%Z
rnd:R -> Z
valid_rnd:Valid_rnd rnd
choice:Z -> bool
x:R
Pu_ro:(0 <= u_ro)%R
Nzx:x <> 0%R
ufpx:=bpow (mag beta x - 1):R
rx:=round beta (FLX_exp prec) (Znearest choice) x:R
Pufpx:(0 <= ufpx)%R
H_2_1:(Rabs (rx - x) <= u_ro * ufpx)%R
Sx:(x < 0)%R
(bpow (mag beta x - 1) <= - x)%R
  rewrite <-mag_opp, <-Rabs_pos_eq; [apply bpow_mag_le|]; lra. }beta:radix
prec:Z
Hp:(0 < prec)%Z
rnd:R -> Z
valid_rnd:Valid_rnd rnd
choice:Z -> bool
x:R
Pu_ro:(0 <= u_ro)%R
Nzx:x <> 0%R
ufpx:=bpow (mag beta x - 1):R
rx:=round beta (FLX_exp prec) (Znearest choice) x:R
Pufpx:(0 <= ufpx)%R
H_2_1:(Rabs (rx - x) <= u_ro * ufpx)%R
H_2_3:(ufpx + Rabs (rx - x) <= Rabs x)%R
(Rabs (rx - x) <= u_ro / (1 + u_ro) * Rabs x)%R
assert (H : (Rabs ((rx - x) / x) <= u_ro / (1 + u_ro))%R).beta:radix
prec:Z
Hp:(0 < prec)%Z
rnd:R -> Z
valid_rnd:Valid_rnd rnd
choice:Z -> bool
x:R
Pu_ro:(0 <= u_ro)%R
Nzx:x <> 0%R
ufpx:=bpow (mag beta x - 1):R
rx:=round beta (FLX_exp prec) (Znearest choice) x:R
Pufpx:(0 <= ufpx)%R
H_2_1:(Rabs (rx - x) <= u_ro * ufpx)%R
H_2_3:(ufpx + Rabs (rx - x) <= Rabs x)%R
(Rabs ((rx - x) / x) <= u_ro / (1 + u_ro))%R
beta:radix
prec:Z
Hp:(0 < prec)%Z
rnd:R -> Z
valid_rnd:Valid_rnd rnd
choice:Z -> bool
x:R
Pu_ro:(0 <= u_ro)%R
Nzx:x <> 0%R
ufpx:=bpow (mag beta x - 1):R
rx:=round beta (FLX_exp prec) (Znearest choice) x:R
Pufpx:(0 <= ufpx)%R
H_2_1:(Rabs (rx - x) <= u_ro * ufpx)%R
H_2_3:(ufpx + Rabs (rx - x) <= Rabs x)%R
H:(Rabs ((rx - x) / x) <= u_ro / (1 + u_ro))%R
(Rabs (rx - x) <= u_ro / (1 + u_ro) * Rabs x)%R
{beta:radix
prec:Z
Hp:(0 < prec)%Z
rnd:R -> Z
valid_rnd:Valid_rnd rnd
choice:Z -> bool
x:R
Pu_ro:(0 <= u_ro)%R
Nzx:x <> 0%R
ufpx:=bpow (mag beta x - 1):R
rx:=round beta (FLX_exp prec) (Znearest choice) x:R
Pufpx:(0 <= ufpx)%R
H_2_1:(Rabs (rx - x) <= u_ro * ufpx)%R
H_2_3:(ufpx + Rabs (rx - x) <= Rabs x)%R
(Rabs ((rx - x) / x) <= u_ro / (1 + u_ro))%R assert (H : (0 < ufpx + Rabs (rx - x))%R).beta:radix
prec:Z
Hp:(0 < prec)%Z
rnd:R -> Z
valid_rnd:Valid_rnd rnd
choice:Z -> bool
x:R
Pu_ro:(0 <= u_ro)%R
Nzx:x <> 0%R
ufpx:=bpow (mag beta x - 1):R
rx:=round beta (FLX_exp prec) (Znearest choice) x:R
Pufpx:(0 <= ufpx)%R
H_2_1:(Rabs (rx - x) <= u_ro * ufpx)%R
H_2_3:(ufpx + Rabs (rx - x) <= Rabs x)%R
(0 < ufpx + Rabs (rx - x))%R
beta:radix
prec:Z
Hp:(0 < prec)%Z
rnd:R -> Z
valid_rnd:Valid_rnd rnd
choice:Z -> bool
x:R
Pu_ro:(0 <= u_ro)%R
Nzx:x <> 0%R
ufpx:=bpow (mag beta x - 1):R
rx:=round beta (FLX_exp prec) (Znearest choice) x:R
Pufpx:(0 <= ufpx)%R
H_2_1:(Rabs (rx - x) <= u_ro * ufpx)%R
H_2_3:(ufpx + Rabs (rx - x) <= Rabs x)%R
H:(0 < ufpx + Rabs (rx - x))%R
(Rabs ((rx - x) / x) <= u_ro / (1 + u_ro))%R
  {beta:radix
prec:Z
Hp:(0 < prec)%Z
rnd:R -> Z
valid_rnd:Valid_rnd rnd
choice:Z -> bool
x:R
Pu_ro:(0 <= u_ro)%R
Nzx:x <> 0%R
ufpx:=bpow (mag beta x - 1):R
rx:=round beta (FLX_exp prec) (Znearest choice) x:R
Pufpx:(0 <= ufpx)%R
H_2_1:(Rabs (rx - x) <= u_ro * ufpx)%R
H_2_3:(ufpx + Rabs (rx - x) <= Rabs x)%R
(0 < ufpx + Rabs (rx - x))%R apply Rplus_lt_le_0_compat; [apply bpow_gt_0|apply Rabs_pos]. }beta:radix
prec:Z
Hp:(0 < prec)%Z
rnd:R -> Z
valid_rnd:Valid_rnd rnd
choice:Z -> bool
x:R
Pu_ro:(0 <= u_ro)%R
Nzx:x <> 0%R
ufpx:=bpow (mag beta x - 1):R
rx:=round beta (FLX_exp prec) (Znearest choice) x:R
Pufpx:(0 <= ufpx)%R
H_2_1:(Rabs (rx - x) <= u_ro * ufpx)%R
H_2_3:(ufpx + Rabs (rx - x) <= Rabs x)%R
H:(0 < ufpx + Rabs (rx - x))%R
(Rabs ((rx - x) / x) <= u_ro / (1 + u_ro))%R
  apply (Rle_trans _ (Rabs (rx - x) / (ufpx + Rabs (rx - x)))).beta:radix
prec:Z
Hp:(0 < prec)%Z
rnd:R -> Z
valid_rnd:Valid_rnd rnd
choice:Z -> bool
x:R
Pu_ro:(0 <= u_ro)%R
Nzx:x <> 0%R
ufpx:=bpow (mag beta x - 1):R
rx:=round beta (FLX_exp prec) (Znearest choice) x:R
Pufpx:(0 <= ufpx)%R
H_2_1:(Rabs (rx - x) <= u_ro * ufpx)%R
H_2_3:(ufpx + Rabs (rx - x) <= Rabs x)%R
H:(0 < ufpx + Rabs (rx - x))%R
(Rabs ((rx - x) / x) <=
 Rabs (rx - x) / (ufpx + Rabs (rx - x)))%R
beta:radix
prec:Z
Hp:(0 < prec)%Z
rnd:R -> Z
valid_rnd:Valid_rnd rnd
choice:Z -> bool
x:R
Pu_ro:(0 <= u_ro)%R
Nzx:x <> 0%R
ufpx:=bpow (mag beta x - 1):R
rx:=round beta (FLX_exp prec) (Znearest choice) x:R
Pufpx:(0 <= ufpx)%R
H_2_1:(Rabs (rx - x) <= u_ro * ufpx)%R
H_2_3:(ufpx + Rabs (rx - x) <= Rabs x)%R
H:(0 < ufpx + Rabs (rx - x))%R
(Rabs (rx - x) / (ufpx + Rabs (rx - x)) <=
 u_ro / (1 + u_ro))%R
  {beta:radix
prec:Z
Hp:(0 < prec)%Z
rnd:R -> Z
valid_rnd:Valid_rnd rnd
choice:Z -> bool
x:R
Pu_ro:(0 <= u_ro)%R
Nzx:x <> 0%R
ufpx:=bpow (mag beta x - 1):R
rx:=round beta (FLX_exp prec) (Znearest choice) x:R
Pufpx:(0 <= ufpx)%R
H_2_1:(Rabs (rx - x) <= u_ro * ufpx)%R
H_2_3:(ufpx + Rabs (rx - x) <= Rabs x)%R
H:(0 < ufpx + Rabs (rx - x))%R
(Rabs ((rx - x) / x) <=
 Rabs (rx - x) / (ufpx + Rabs (rx - x)))%R unfold Rdiv; rewrite Rabs_mult; apply Rmult_le_compat_l; [apply Rabs_pos|].beta:radix
prec:Z
Hp:(0 < prec)%Z
rnd:R -> Z
valid_rnd:Valid_rnd rnd
choice:Z -> bool
x:R
Pu_ro:(0 <= u_ro)%R
Nzx:x <> 0%R
ufpx:=bpow (mag beta x - 1):R
rx:=round beta (FLX_exp prec) (Znearest choice) x:R
Pufpx:(0 <= ufpx)%R
H_2_1:(Rabs (rx - x) <= u_ro * ufpx)%R
H_2_3:(ufpx + Rabs (rx - x) <= Rabs x)%R
H:(0 < ufpx + Rabs (rx - x))%R
(Rabs (/ x) <= / (ufpx + Rabs (rx - x)))%R
    now rewrite (Rabs_Rinv _ Nzx); apply Rinv_le. }beta:radix
prec:Z
Hp:(0 < prec)%Z
rnd:R -> Z
valid_rnd:Valid_rnd rnd
choice:Z -> bool
x:R
Pu_ro:(0 <= u_ro)%R
Nzx:x <> 0%R
ufpx:=bpow (mag beta x - 1):R
rx:=round beta (FLX_exp prec) (Znearest choice) x:R
Pufpx:(0 <= ufpx)%R
H_2_1:(Rabs (rx - x) <= u_ro * ufpx)%R
H_2_3:(ufpx + Rabs (rx - x) <= Rabs x)%R
H:(0 < ufpx + Rabs (rx - x))%R
(Rabs (rx - x) / (ufpx + Rabs (rx - x)) <=
 u_ro / (1 + u_ro))%R
  apply (Rmult_le_reg_r ((ufpx + Rabs (rx - x)) * (1 + u_ro))).beta:radix
prec:Z
Hp:(0 < prec)%Z
rnd:R -> Z
valid_rnd:Valid_rnd rnd
choice:Z -> bool
x:R
Pu_ro:(0 <= u_ro)%R
Nzx:x <> 0%R
ufpx:=bpow (mag beta x - 1):R
rx:=round beta (FLX_exp prec) (Znearest choice) x:R
Pufpx:(0 <= ufpx)%R
H_2_1:(Rabs (rx - x) <= u_ro * ufpx)%R
H_2_3:(ufpx + Rabs (rx - x) <= Rabs x)%R
H:(0 < ufpx + Rabs (rx - x))%R
(0 < (ufpx + Rabs (rx - x)) * (1 + u_ro))%R
beta:radix
prec:Z
Hp:(0 < prec)%Z
rnd:R -> Z
valid_rnd:Valid_rnd rnd
choice:Z -> bool
x:R
Pu_ro:(0 <= u_ro)%R
Nzx:x <> 0%R
ufpx:=bpow (mag beta x - 1):R
rx:=round beta (FLX_exp prec) (Znearest choice) x:R
Pufpx:(0 <= ufpx)%R
H_2_1:(Rabs (rx - x) <= u_ro * ufpx)%R
H_2_3:(ufpx + Rabs (rx - x) <= Rabs x)%R
H:(0 < ufpx + Rabs (rx - x))%R
(Rabs (rx - x) / (ufpx + Rabs (rx - x)) *
 ((ufpx + Rabs (rx - x)) * (1 + u_ro)) <=
 u_ro / (1 + u_ro) *
 ((ufpx + Rabs (rx - x)) * (1 + u_ro)))%R
  {beta:radix
prec:Z
Hp:(0 < prec)%Z
rnd:R -> Z
valid_rnd:Valid_rnd rnd
choice:Z -> bool
x:R
Pu_ro:(0 <= u_ro)%R
Nzx:x <> 0%R
ufpx:=bpow (mag beta x - 1):R
rx:=round beta (FLX_exp prec) (Znearest choice) x:R
Pufpx:(0 <= ufpx)%R
H_2_1:(Rabs (rx - x) <= u_ro * ufpx)%R
H_2_3:(ufpx + Rabs (rx - x) <= Rabs x)%R
H:(0 < ufpx + Rabs (rx - x))%R
(0 < (ufpx + Rabs (rx - x)) * (1 + u_ro))%R apply Rmult_lt_0_compat; lra. }beta:radix
prec:Z
Hp:(0 < prec)%Z
rnd:R -> Z
valid_rnd:Valid_rnd rnd
choice:Z -> bool
x:R
Pu_ro:(0 <= u_ro)%R
Nzx:x <> 0%R
ufpx:=bpow (mag beta x - 1):R
rx:=round beta (FLX_exp prec) (Znearest choice) x:R
Pufpx:(0 <= ufpx)%R
H_2_1:(Rabs (rx - x) <= u_ro * ufpx)%R
H_2_3:(ufpx + Rabs (rx - x) <= Rabs x)%R
H:(0 < ufpx + Rabs (rx - x))%R
(Rabs (rx - x) / (ufpx + Rabs (rx - x)) *
 ((ufpx + Rabs (rx - x)) * (1 + u_ro)) <=
 u_ro / (1 + u_ro) *
 ((ufpx + Rabs (rx - x)) * (1 + u_ro)))%R
  field_simplify; [try unfold Rdiv; rewrite ?Rinv_1, ?Rmult_1_r| |]; lra. }beta:radix
prec:Z
Hp:(0 < prec)%Z
rnd:R -> Z
valid_rnd:Valid_rnd rnd
choice:Z -> bool
x:R
Pu_ro:(0 <= u_ro)%R
Nzx:x <> 0%R
ufpx:=bpow (mag beta x - 1):R
rx:=round beta (FLX_exp prec) (Znearest choice) x:R
Pufpx:(0 <= ufpx)%R
H_2_1:(Rabs (rx - x) <= u_ro * ufpx)%R
H_2_3:(ufpx + Rabs (rx - x) <= Rabs x)%R
H:(Rabs ((rx - x) / x) <= u_ro / (1 + u_ro))%R
(Rabs (rx - x) <= u_ro / (1 + u_ro) * Rabs x)%R
revert H; unfold Rdiv; rewrite Rabs_mult, (Rabs_Rinv _ Nzx); intro H.beta:radix
prec:Z
Hp:(0 < prec)%Z
rnd:R -> Z
valid_rnd:Valid_rnd rnd
choice:Z -> bool
x:R
Pu_ro:(0 <= u_ro)%R
Nzx:x <> 0%R
ufpx:=bpow (mag beta x - 1):R
rx:=round beta (FLX_exp prec) (Znearest choice) x:R
Pufpx:(0 <= ufpx)%R
H_2_1:(Rabs (rx - x) <= u_ro * ufpx)%R
H_2_3:(ufpx + Rabs (rx - x) <= Rabs x)%R
H:(Rabs (rx - x) * / Rabs x <= u_ro * / (1 + u_ro))%R
(Rabs (rx - x) <= u_ro * / (1 + u_ro) * Rabs x)%R
apply (Rmult_le_reg_r (/ Rabs x)); [now apply Rinv_0_lt_compat, Rabs_pos_lt|].beta:radix
prec:Z
Hp:(0 < prec)%Z
rnd:R -> Z
valid_rnd:Valid_rnd rnd
choice:Z -> bool
x:R
Pu_ro:(0 <= u_ro)%R
Nzx:x <> 0%R
ufpx:=bpow (mag beta x - 1):R
rx:=round beta (FLX_exp prec) (Znearest choice) x:R
Pufpx:(0 <= ufpx)%R
H_2_1:(Rabs (rx - x) <= u_ro * ufpx)%R
H_2_3:(ufpx + Rabs (rx - x) <= Rabs x)%R
H:(Rabs (rx - x) * / Rabs x <= u_ro * / (1 + u_ro))%R
(Rabs (rx - x) * / Rabs x <=
 u_ro * / (1 + u_ro) * Rabs x * / Rabs x)%R
now apply (Rle_trans _ _ _ H); right; field; split; [apply Rabs_no_R0|lra].
Qed.

1+ε property in rounding to nearest in FLX

Theorem relative_error_N_FLX_ex :
  forall x,
  exists eps,
  (Rabs eps <= /2 * bpow (-prec + 1))%R /\ round beta (FLX_exp prec) (Znearest choice) x = (x * (1 + eps))%R.beta:radix
prec:Z
Hp:(0 < prec)%Z
rnd:R -> Z
valid_rnd:Valid_rnd rnd
choice:Z -> bool
forall x : R,
exists eps : R,
  (Rabs eps <= / 2 * bpow (- prec + 1))%R /\
  round beta (FLX_exp prec) (Znearest choice) x =
  (x * (1 + eps))%R
Proof with auto with typeclass_instances.beta:radix
prec:Z
Hp:(0 < prec)%Z
rnd:R -> Z
valid_rnd:Valid_rnd rnd
choice:Z -> bool
forall x : R,
exists eps : R,
  (Rabs eps <= / 2 * bpow (- prec + 1))%R /\
  round beta (FLX_exp prec) (Znearest choice) x =
  (x * (1 + eps))%R
intros x.beta:radix
prec:Z
Hp:(0 < prec)%Z
rnd:R -> Z
valid_rnd:Valid_rnd rnd
choice:Z -> bool
x:R
exists eps : R,
  (Rabs eps <= / 2 * bpow (- prec + 1))%R /\
  round beta (FLX_exp prec) (Znearest choice) x =
  (x * (1 + eps))%R
apply relative_error_le_conversion...beta:radix
prec:Z
Hp:(0 < prec)%Z
rnd:R -> Z
valid_rnd:Valid_rnd rnd
choice:Z -> bool
x:R
(0 <= / 2 * bpow (- prec + 1))%R
beta:radix
prec:Z
Hp:(0 < prec)%Z
rnd:R -> Z
valid_rnd:Valid_rnd rnd
choice:Z -> bool
x:R
(Rabs
   (round beta (FLX_exp prec) (Znearest choice) x - x) <=
 / 2 * bpow (- prec + 1) * Rabs x)%R
apply Rlt_le.beta:radix
prec:Z
Hp:(0 < prec)%Z
rnd:R -> Z
valid_rnd:Valid_rnd rnd
choice:Z -> bool
x:R
(0 < / 2 * bpow (- prec + 1))%R
beta:radix
prec:Z
Hp:(0 < prec)%Z
rnd:R -> Z
valid_rnd:Valid_rnd rnd
choice:Z -> bool
x:R
(Rabs
   (round beta (FLX_exp prec) (Znearest choice) x - x) <=
 / 2 * bpow (- prec + 1) * Rabs x)%R
apply Rmult_lt_0_compat.beta:radix
prec:Z
Hp:(0 < prec)%Z
rnd:R -> Z
valid_rnd:Valid_rnd rnd
choice:Z -> bool
x:R
(0 < / 2)%R
beta:radix
prec:Z
Hp:(0 < prec)%Z
rnd:R -> Z
valid_rnd:Valid_rnd rnd
choice:Z -> bool
x:R
(0 < bpow (- prec + 1))%R
beta:radix
prec:Z
Hp:(0 < prec)%Z
rnd:R -> Z
valid_rnd:Valid_rnd rnd
choice:Z -> bool
x:R
(Rabs
   (round beta (FLX_exp prec) (Znearest choice) x - x) <=
 / 2 * bpow (- prec + 1) * Rabs x)%R
apply Rinv_0_lt_compat.beta:radix
prec:Z
Hp:(0 < prec)%Z
rnd:R -> Z
valid_rnd:Valid_rnd rnd
choice:Z -> bool
x:R
(0 < 2)%R
beta:radix
prec:Z
Hp:(0 < prec)%Z
rnd:R -> Z
valid_rnd:Valid_rnd rnd
choice:Z -> bool
x:R
(0 < bpow (- prec + 1))%R
beta:radix
prec:Z
Hp:(0 < prec)%Z
rnd:R -> Z
valid_rnd:Valid_rnd rnd
choice:Z -> bool
x:R
(Rabs
   (round beta (FLX_exp prec) (Znearest choice) x - x) <=
 / 2 * bpow (- prec + 1) * Rabs x)%R
now apply IZR_lt.beta:radix
prec:Z
Hp:(0 < prec)%Z
rnd:R -> Z
valid_rnd:Valid_rnd rnd
choice:Z -> bool
x:R
(0 < bpow (- prec + 1))%R
beta:radix
prec:Z
Hp:(0 < prec)%Z
rnd:R -> Z
valid_rnd:Valid_rnd rnd
choice:Z -> bool
x:R
(Rabs
   (round beta (FLX_exp prec) (Znearest choice) x - x) <=
 / 2 * bpow (- prec + 1) * Rabs x)%R
apply bpow_gt_0.beta:radix
prec:Z
Hp:(0 < prec)%Z
rnd:R -> Z
valid_rnd:Valid_rnd rnd
choice:Z -> bool
x:R
(Rabs
   (round beta (FLX_exp prec) (Znearest choice) x - x) <=
 / 2 * bpow (- prec + 1) * Rabs x)%R
now apply relative_error_N_FLX.
Qed.

Theorem relative_error_N_FLX'_ex :
  forall x,
  exists eps,
  (Rabs eps <= u_ro / (1 + u_ro))%R /\
  round beta (FLX_exp prec) (Znearest choice) x = (x * (1 + eps))%R.beta:radix
prec:Z
Hp:(0 < prec)%Z
rnd:R -> Z
valid_rnd:Valid_rnd rnd
choice:Z -> bool
forall x : R,
exists eps : R,
  (Rabs eps <= u_ro / (1 + u_ro))%R /\
  round beta (FLX_exp prec) (Znearest choice) x =
  (x * (1 + eps))%R
Proof with auto with typeclass_instances.beta:radix
prec:Z
Hp:(0 < prec)%Z
rnd:R -> Z
valid_rnd:Valid_rnd rnd
choice:Z -> bool
forall x : R,
exists eps : R,
  (Rabs eps <= u_ro / (1 + u_ro))%R /\
  round beta (FLX_exp prec) (Znearest choice) x =
  (x * (1 + eps))%R
intros x.beta:radix
prec:Z
Hp:(0 < prec)%Z
rnd:R -> Z
valid_rnd:Valid_rnd rnd
choice:Z -> bool
x:R
exists eps : R,
  (Rabs eps <= u_ro / (1 + u_ro))%R /\
  round beta (FLX_exp prec) (Znearest choice) x =
  (x * (1 + eps))%R
apply relative_error_le_conversion...beta:radix
prec:Z
Hp:(0 < prec)%Z
rnd:R -> Z
valid_rnd:Valid_rnd rnd
choice:Z -> bool
x:R
(0 <= u_ro / (1 + u_ro))%R
beta:radix
prec:Z
Hp:(0 < prec)%Z
rnd:R -> Z
valid_rnd:Valid_rnd rnd
choice:Z -> bool
x:R
(Rabs
   (round beta (FLX_exp prec) (Znearest choice) x - x) <=
 u_ro / (1 + u_ro) * Rabs x)%R
{beta:radix
prec:Z
Hp:(0 < prec)%Z
rnd:R -> Z
valid_rnd:Valid_rnd rnd
choice:Z -> bool
x:R
(0 <= u_ro / (1 + u_ro))%R apply u_rod1pu_ro_pos. }beta:radix
prec:Z
Hp:(0 < prec)%Z
rnd:R -> Z
valid_rnd:Valid_rnd rnd
choice:Z -> bool
x:R
(Rabs
   (round beta (FLX_exp prec) (Znearest choice) x - x) <=
 u_ro / (1 + u_ro) * Rabs x)%R
now apply relative_error_N_FLX'.
Qed.

Lemma relative_error_N_round_ex_derive :
  forall x rx,
  (exists eps, (Rabs eps <= u_ro / (1 + u_ro))%R /\ rx = (x * (1 + eps))%R) ->
  exists eps, (Rabs eps <= u_ro)%R /\ x = (rx * (1 + eps))%R.beta:radix
prec:Z
Hp:(0 < prec)%Z
rnd:R -> Z
valid_rnd:Valid_rnd rnd
choice:Z -> bool
forall x rx : R,
(exists eps : R,
   (Rabs eps <= u_ro / (1 + u_ro))%R /\
   rx = (x * (1 + eps))%R) ->
exists eps : R,
  (Rabs eps <= u_ro)%R /\ x = (rx * (1 + eps))%R
Proof.beta:radix
prec:Z
Hp:(0 < prec)%Z
rnd:R -> Z
valid_rnd:Valid_rnd rnd
choice:Z -> bool
forall x rx : R,
(exists eps : R,
   (Rabs eps <= u_ro / (1 + u_ro))%R /\
   rx = (x * (1 + eps))%R) ->
exists eps : R,
  (Rabs eps <= u_ro)%R /\ x = (rx * (1 + eps))%R
intros x rx (d, (Bd, Hd)).beta:radix
prec:Z
Hp:(0 < prec)%Z
rnd:R -> Z
valid_rnd:Valid_rnd rnd
choice:Z -> bool
x, rx, d:R
Bd:(Rabs d <= u_ro / (1 + u_ro))%R
Hd:rx = (x * (1 + d))%R
exists eps : R,
  (Rabs eps <= u_ro)%R /\ x = (rx * (1 + eps))%R
assert (Pu_ro := u_ro_pos).beta:radix
prec:Z
Hp:(0 < prec)%Z
rnd:R -> Z
valid_rnd:Valid_rnd rnd
choice:Z -> bool
x, rx, d:R
Bd:(Rabs d <= u_ro / (1 + u_ro))%R
Hd:rx = (x * (1 + d))%R
Pu_ro:(0 <= u_ro)%R
exists eps : R,
  (Rabs eps <= u_ro)%R /\ x = (rx * (1 + eps))%R
assert (H := Rabs_le_inv _ _ Bd).beta:radix
prec:Z
Hp:(0 < prec)%Z
rnd:R -> Z
valid_rnd:Valid_rnd rnd
choice:Z -> bool
x, rx, d:R
Bd:(Rabs d <= u_ro / (1 + u_ro))%R
Hd:rx = (x * (1 + d))%R
Pu_ro:(0 <= u_ro)%R
H:(- (u_ro / (1 + u_ro)) <= d <= u_ro / (1 + u_ro))%R
exists eps : R,
  (Rabs eps <= u_ro)%R /\ x = (rx * (1 + eps))%R
assert (H' := u_rod1pu_ro_le_u_ro); assert (H'' := u_ro_lt_1).beta:radix
prec:Z
Hp:(0 < prec)%Z
rnd:R -> Z
valid_rnd:Valid_rnd rnd
choice:Z -> bool
x, rx, d:R
Bd:(Rabs d <= u_ro / (1 + u_ro))%R
Hd:rx = (x * (1 + d))%R
Pu_ro:(0 <= u_ro)%R
H:(- (u_ro / (1 + u_ro)) <= d <= u_ro / (1 + u_ro))%R
H':(u_ro / (1 + u_ro) <= u_ro)%R
H'':(u_ro < 1)%R
exists eps : R,
  (Rabs eps <= u_ro)%R /\ x = (rx * (1 + eps))%R
destruct (Req_dec rx 0) as [Zfx|Nzfx].beta:radix
prec:Z
Hp:(0 < prec)%Z
rnd:R -> Z
valid_rnd:Valid_rnd rnd
choice:Z -> bool
x, rx, d:R
Bd:(Rabs d <= u_ro / (1 + u_ro))%R
Hd:rx = (x * (1 + d))%R
Pu_ro:(0 <= u_ro)%R
H:(- (u_ro / (1 + u_ro)) <= d <= u_ro / (1 + u_ro))%R
H':(u_ro / (1 + u_ro) <= u_ro)%R
H'':(u_ro < 1)%R
Zfx:rx = 0%R
exists eps : R,
  (Rabs eps <= u_ro)%R /\ x = (rx * (1 + eps))%R
beta:radix
prec:Z
Hp:(0 < prec)%Z
rnd:R -> Z
valid_rnd:Valid_rnd rnd
choice:Z -> bool
x, rx, d:R
Bd:(Rabs d <= u_ro / (1 + u_ro))%R
Hd:rx = (x * (1 + d))%R
Pu_ro:(0 <= u_ro)%R
H:(- (u_ro / (1 + u_ro)) <= d <= u_ro / (1 + u_ro))%R
H':(u_ro / (1 + u_ro) <= u_ro)%R
H'':(u_ro < 1)%R
Nzfx:rx <> 0%R
exists eps : R,
  (Rabs eps <= u_ro)%R /\ x = (rx * (1 + eps))%R
{beta:radix
prec:Z
Hp:(0 < prec)%Z
rnd:R -> Z
valid_rnd:Valid_rnd rnd
choice:Z -> bool
x, rx, d:R
Bd:(Rabs d <= u_ro / (1 + u_ro))%R
Hd:rx = (x * (1 + d))%R
Pu_ro:(0 <= u_ro)%R
H:(- (u_ro / (1 + u_ro)) <= d <= u_ro / (1 + u_ro))%R
H':(u_ro / (1 + u_ro) <= u_ro)%R
H'':(u_ro < 1)%R
Zfx:rx = 0%R
exists eps : R,
  (Rabs eps <= u_ro)%R /\ x = (rx * (1 + eps))%R exists 0%R; split; [now rewrite Rabs_R0|].beta:radix
prec:Z
Hp:(0 < prec)%Z
rnd:R -> Z
valid_rnd:Valid_rnd rnd
choice:Z -> bool
x, rx, d:R
Bd:(Rabs d <= u_ro / (1 + u_ro))%R
Hd:rx = (x * (1 + d))%R
Pu_ro:(0 <= u_ro)%R
H:(- (u_ro / (1 + u_ro)) <= d <= u_ro / (1 + u_ro))%R
H':(u_ro / (1 + u_ro) <= u_ro)%R
H'':(u_ro < 1)%R
Zfx:rx = 0%R
x = (rx * (1 + 0))%R
  rewrite Rplus_0_r, Rmult_1_r, Zfx.beta:radix
prec:Z
Hp:(0 < prec)%Z
rnd:R -> Z
valid_rnd:Valid_rnd rnd
choice:Z -> bool
x, rx, d:R
Bd:(Rabs d <= u_ro / (1 + u_ro))%R
Hd:rx = (x * (1 + d))%R
Pu_ro:(0 <= u_ro)%R
H:(- (u_ro / (1 + u_ro)) <= d <= u_ro / (1 + u_ro))%R
H':(u_ro / (1 + u_ro) <= u_ro)%R
H'':(u_ro < 1)%R
Zfx:rx = 0%R
x = 0%R
  now rewrite Zfx in Hd; destruct (Rmult_integral _ _ (sym_eq Hd)); [|lra]. }beta:radix
prec:Z
Hp:(0 < prec)%Z
rnd:R -> Z
valid_rnd:Valid_rnd rnd
choice:Z -> bool
x, rx, d:R
Bd:(Rabs d <= u_ro / (1 + u_ro))%R
Hd:rx = (x * (1 + d))%R
Pu_ro:(0 <= u_ro)%R
H:(- (u_ro / (1 + u_ro)) <= d <= u_ro / (1 + u_ro))%R
H':(u_ro / (1 + u_ro) <= u_ro)%R
H'':(u_ro < 1)%R
Nzfx:rx <> 0%R
exists eps : R,
  (Rabs eps <= u_ro)%R /\ x = (rx * (1 + eps))%R
destruct (Req_dec x 0) as [Zx|Nzx].beta:radix
prec:Z
Hp:(0 < prec)%Z
rnd:R -> Z
valid_rnd:Valid_rnd rnd
choice:Z -> bool
x, rx, d:R
Bd:(Rabs d <= u_ro / (1 + u_ro))%R
Hd:rx = (x * (1 + d))%R
Pu_ro:(0 <= u_ro)%R
H:(- (u_ro / (1 + u_ro)) <= d <= u_ro / (1 + u_ro))%R
H':(u_ro / (1 + u_ro) <= u_ro)%R
H'':(u_ro < 1)%R
Nzfx:rx <> 0%R
Zx:x = 0%R
exists eps : R,
  (Rabs eps <= u_ro)%R /\ x = (rx * (1 + eps))%R
beta:radix
prec:Z
Hp:(0 < prec)%Z
rnd:R -> Z
valid_rnd:Valid_rnd rnd
choice:Z -> bool
x, rx, d:R
Bd:(Rabs d <= u_ro / (1 + u_ro))%R
Hd:rx = (x * (1 + d))%R
Pu_ro:(0 <= u_ro)%R
H:(- (u_ro / (1 + u_ro)) <= d <= u_ro / (1 + u_ro))%R
H':(u_ro / (1 + u_ro) <= u_ro)%R
H'':(u_ro < 1)%R
Nzfx:rx <> 0%R
Nzx:x <> 0%R
exists eps : R,
  (Rabs eps <= u_ro)%R /\ x = (rx * (1 + eps))%R
{beta:radix
prec:Z
Hp:(0 < prec)%Z
rnd:R -> Z
valid_rnd:Valid_rnd rnd
choice:Z -> bool
x, rx, d:R
Bd:(Rabs d <= u_ro / (1 + u_ro))%R
Hd:rx = (x * (1 + d))%R
Pu_ro:(0 <= u_ro)%R
H:(- (u_ro / (1 + u_ro)) <= d <= u_ro / (1 + u_ro))%R
H':(u_ro / (1 + u_ro) <= u_ro)%R
H'':(u_ro < 1)%R
Nzfx:rx <> 0%R
Zx:x = 0%R
exists eps : R,
  (Rabs eps <= u_ro)%R /\ x = (rx * (1 + eps))%R now exfalso; revert Hd; rewrite Zx, Rmult_0_l. }beta:radix
prec:Z
Hp:(0 < prec)%Z
rnd:R -> Z
valid_rnd:Valid_rnd rnd
choice:Z -> bool
x, rx, d:R
Bd:(Rabs d <= u_ro / (1 + u_ro))%R
Hd:rx = (x * (1 + d))%R
Pu_ro:(0 <= u_ro)%R
H:(- (u_ro / (1 + u_ro)) <= d <= u_ro / (1 + u_ro))%R
H':(u_ro / (1 + u_ro) <= u_ro)%R
H'':(u_ro < 1)%R
Nzfx:rx <> 0%R
Nzx:x <> 0%R
exists eps : R,
  (Rabs eps <= u_ro)%R /\ x = (rx * (1 + eps))%R
set (d' := ((x - rx) / rx)%R).beta:radix
prec:Z
Hp:(0 < prec)%Z
rnd:R -> Z
valid_rnd:Valid_rnd rnd
choice:Z -> bool
x, rx, d:R
Bd:(Rabs d <= u_ro / (1 + u_ro))%R
Hd:rx = (x * (1 + d))%R
Pu_ro:(0 <= u_ro)%R
H:(- (u_ro / (1 + u_ro)) <= d <= u_ro / (1 + u_ro))%R
H':(u_ro / (1 + u_ro) <= u_ro)%R
H'':(u_ro < 1)%R
Nzfx:rx <> 0%R
Nzx:x <> 0%R
d':=((x - rx) / rx)%R:R
exists eps : R,
  (Rabs eps <= u_ro)%R /\ x = (rx * (1 + eps))%R
assert (Hd' : (Rabs d' <= u_ro)%R).beta:radix
prec:Z
Hp:(0 < prec)%Z
rnd:R -> Z
valid_rnd:Valid_rnd rnd
choice:Z -> bool
x, rx, d:R
Bd:(Rabs d <= u_ro / (1 + u_ro))%R
Hd:rx = (x * (1 + d))%R
Pu_ro:(0 <= u_ro)%R
H:(- (u_ro / (1 + u_ro)) <= d <= u_ro / (1 + u_ro))%R
H':(u_ro / (1 + u_ro) <= u_ro)%R
H'':(u_ro < 1)%R
Nzfx:rx <> 0%R
Nzx:x <> 0%R
d':=((x - rx) / rx)%R:R
(Rabs d' <= u_ro)%R
beta:radix
prec:Z
Hp:(0 < prec)%Z
rnd:R -> Z
valid_rnd:Valid_rnd rnd
choice:Z -> bool
x, rx, d:R
Bd:(Rabs d <= u_ro / (1 + u_ro))%R
Hd:rx = (x * (1 + d))%R
Pu_ro:(0 <= u_ro)%R
H:(- (u_ro / (1 + u_ro)) <= d <= u_ro / (1 + u_ro))%R
H':(u_ro / (1 + u_ro) <= u_ro)%R
H'':(u_ro < 1)%R
Nzfx:rx <> 0%R
Nzx:x <> 0%R
d':=((x - rx) / rx)%R:R
Hd':(Rabs d' <= u_ro)%R
exists eps : R,
  (Rabs eps <= u_ro)%R /\ x = (rx * (1 + eps))%R
{beta:radix
prec:Z
Hp:(0 < prec)%Z
rnd:R -> Z
valid_rnd:Valid_rnd rnd
choice:Z -> bool
x, rx, d:R
Bd:(Rabs d <= u_ro / (1 + u_ro))%R
Hd:rx = (x * (1 + d))%R
Pu_ro:(0 <= u_ro)%R
H:(- (u_ro / (1 + u_ro)) <= d <= u_ro / (1 + u_ro))%R
H':(u_ro / (1 + u_ro) <= u_ro)%R
H'':(u_ro < 1)%R
Nzfx:rx <> 0%R
Nzx:x <> 0%R
d':=((x - rx) / rx)%R:R
(Rabs d' <= u_ro)%R unfold d'; rewrite Hd.beta:radix
prec:Z
Hp:(0 < prec)%Z
rnd:R -> Z
valid_rnd:Valid_rnd rnd
choice:Z -> bool
x, rx, d:R
Bd:(Rabs d <= u_ro / (1 + u_ro))%R
Hd:rx = (x * (1 + d))%R
Pu_ro:(0 <= u_ro)%R
H:(- (u_ro / (1 + u_ro)) <= d <= u_ro / (1 + u_ro))%R
H':(u_ro / (1 + u_ro) <= u_ro)%R
H'':(u_ro < 1)%R
Nzfx:rx <> 0%R
Nzx:x <> 0%R
d':=((x - rx) / rx)%R:R
(Rabs ((x - x * (1 + d)) / (x * (1 + d))) <= u_ro)%R
  replace (_ / _)%R with (- d / (1 + d))%R; [|now field; split; lra].beta:radix
prec:Z
Hp:(0 < prec)%Z
rnd:R -> Z
valid_rnd:Valid_rnd rnd
choice:Z -> bool
x, rx, d:R
Bd:(Rabs d <= u_ro / (1 + u_ro))%R
Hd:rx = (x * (1 + d))%R
Pu_ro:(0 <= u_ro)%R
H:(- (u_ro / (1 + u_ro)) <= d <= u_ro / (1 + u_ro))%R
H':(u_ro / (1 + u_ro) <= u_ro)%R
H'':(u_ro < 1)%R
Nzfx:rx <> 0%R
Nzx:x <> 0%R
d':=((x - rx) / rx)%R:R
(Rabs (- d / (1 + d)) <= u_ro)%R
  unfold Rdiv; rewrite Rabs_mult, Rabs_Ropp.beta:radix
prec:Z
Hp:(0 < prec)%Z
rnd:R -> Z
valid_rnd:Valid_rnd rnd
choice:Z -> bool
x, rx, d:R
Bd:(Rabs d <= u_ro / (1 + u_ro))%R
Hd:rx = (x * (1 + d))%R
Pu_ro:(0 <= u_ro)%R
H:(- (u_ro / (1 + u_ro)) <= d <= u_ro / (1 + u_ro))%R
H':(u_ro / (1 + u_ro) <= u_ro)%R
H'':(u_ro < 1)%R
Nzfx:rx <> 0%R
Nzx:x <> 0%R
d':=((x - rx) / rx)%R:R
(Rabs d * Rabs (/ (1 + d)) <= u_ro)%R
  rewrite (Rabs_pos_eq (/ _)); [|apply Rlt_le, Rinv_0_lt_compat; lra].beta:radix
prec:Z
Hp:(0 < prec)%Z
rnd:R -> Z
valid_rnd:Valid_rnd rnd
choice:Z -> bool
x, rx, d:R
Bd:(Rabs d <= u_ro / (1 + u_ro))%R
Hd:rx = (x * (1 + d))%R
Pu_ro:(0 <= u_ro)%R
H:(- (u_ro / (1 + u_ro)) <= d <= u_ro / (1 + u_ro))%R
H':(u_ro / (1 + u_ro) <= u_ro)%R
H'':(u_ro < 1)%R
Nzfx:rx <> 0%R
Nzx:x <> 0%R
d':=((x - rx) / rx)%R:R
(Rabs d * / (1 + d) <= u_ro)%R
  apply (Rmult_le_reg_r (1 + d)); [lra|].beta:radix
prec:Z
Hp:(0 < prec)%Z
rnd:R -> Z
valid_rnd:Valid_rnd rnd
choice:Z -> bool
x, rx, d:R
Bd:(Rabs d <= u_ro / (1 + u_ro))%R
Hd:rx = (x * (1 + d))%R
Pu_ro:(0 <= u_ro)%R
H:(- (u_ro / (1 + u_ro)) <= d <= u_ro / (1 + u_ro))%R
H':(u_ro / (1 + u_ro) <= u_ro)%R
H'':(u_ro < 1)%R
Nzfx:rx <> 0%R
Nzx:x <> 0%R
d':=((x - rx) / rx)%R:R
(Rabs d * / (1 + d) * (1 + d) <= u_ro * (1 + d))%R
  rewrite Rmult_assoc, Rinv_l, Rmult_1_r; [|lra].beta:radix
prec:Z
Hp:(0 < prec)%Z
rnd:R -> Z
valid_rnd:Valid_rnd rnd
choice:Z -> bool
x, rx, d:R
Bd:(Rabs d <= u_ro / (1 + u_ro))%R
Hd:rx = (x * (1 + d))%R
Pu_ro:(0 <= u_ro)%R
H:(- (u_ro / (1 + u_ro)) <= d <= u_ro / (1 + u_ro))%R
H':(u_ro / (1 + u_ro) <= u_ro)%R
H'':(u_ro < 1)%R
Nzfx:rx <> 0%R
Nzx:x <> 0%R
d':=((x - rx) / rx)%R:R
(Rabs d <= u_ro * (1 + d))%R
  apply (Rle_trans _ _ _ Bd).beta:radix
prec:Z
Hp:(0 < prec)%Z
rnd:R -> Z
valid_rnd:Valid_rnd rnd
choice:Z -> bool
x, rx, d:R
Bd:(Rabs d <= u_ro / (1 + u_ro))%R
Hd:rx = (x * (1 + d))%R
Pu_ro:(0 <= u_ro)%R
H:(- (u_ro / (1 + u_ro)) <= d <= u_ro / (1 + u_ro))%R
H':(u_ro / (1 + u_ro) <= u_ro)%R
H'':(u_ro < 1)%R
Nzfx:rx <> 0%R
Nzx:x <> 0%R
d':=((x - rx) / rx)%R:R
(u_ro / (1 + u_ro) <= u_ro * (1 + d))%R
  unfold Rdiv; apply Rmult_le_compat_l; [now apply u_ro_pos|].beta:radix
prec:Z
Hp:(0 < prec)%Z
rnd:R -> Z
valid_rnd:Valid_rnd rnd
choice:Z -> bool
x, rx, d:R
Bd:(Rabs d <= u_ro / (1 + u_ro))%R
Hd:rx = (x * (1 + d))%R
Pu_ro:(0 <= u_ro)%R
H:(- (u_ro / (1 + u_ro)) <= d <= u_ro / (1 + u_ro))%R
H':(u_ro / (1 + u_ro) <= u_ro)%R
H'':(u_ro < 1)%R
Nzfx:rx <> 0%R
Nzx:x <> 0%R
d':=((x - rx) / rx)%R:R
(/ (1 + u_ro) <= 1 + d)%R
  apply (Rle_trans _ (1 - u_ro / (1 + u_ro))); [right; field|]; lra. }beta:radix
prec:Z
Hp:(0 < prec)%Z
rnd:R -> Z
valid_rnd:Valid_rnd rnd
choice:Z -> bool
x, rx, d:R
Bd:(Rabs d <= u_ro / (1 + u_ro))%R
Hd:rx = (x * (1 + d))%R
Pu_ro:(0 <= u_ro)%R
H:(- (u_ro / (1 + u_ro)) <= d <= u_ro / (1 + u_ro))%R
H':(u_ro / (1 + u_ro) <= u_ro)%R
H'':(u_ro < 1)%R
Nzfx:rx <> 0%R
Nzx:x <> 0%R
d':=((x - rx) / rx)%R:R
Hd':(Rabs d' <= u_ro)%R
exists eps : R,
  (Rabs eps <= u_ro)%R /\ x = (rx * (1 + eps))%R
now exists d'; split; [|unfold d'; field].
Qed.

Theorem relative_error_N_FLX_round_ex :
  forall x,
  exists eps,
  (Rabs eps <= u_ro)%R /\
  x = (round beta (FLX_exp prec) (Znearest choice) x * (1 + eps))%R.beta:radix
prec:Z
Hp:(0 < prec)%Z
rnd:R -> Z
valid_rnd:Valid_rnd rnd
choice:Z -> bool
forall x : R,
exists eps : R,
  (Rabs eps <= u_ro)%R /\
  x =
  (round beta (FLX_exp prec) (Znearest choice) x *
   (1 + eps))%R
Proof.beta:radix
prec:Z
Hp:(0 < prec)%Z
rnd:R -> Z
valid_rnd:Valid_rnd rnd
choice:Z -> bool
forall x : R,
exists eps : R,
  (Rabs eps <= u_ro)%R /\
  x =
  (round beta (FLX_exp prec) (Znearest choice) x *
   (1 + eps))%R
intro x; apply relative_error_N_round_ex_derive, relative_error_N_FLX'_ex.
Qed.

Theorem relative_error_N_FLX_round :
  forall x,
  (Rabs (round beta (FLX_exp prec) (Znearest choice) x - x) <= /2 * bpow (-prec + 1) * Rabs(round beta (FLX_exp prec) (Znearest choice) x))%R.beta:radix
prec:Z
Hp:(0 < prec)%Z
rnd:R -> Z
valid_rnd:Valid_rnd rnd
choice:Z -> bool
forall x : R,
(Rabs
   (round beta (FLX_exp prec) (Znearest choice) x - x) <=
 / 2 * bpow (- prec + 1) *
 Rabs (round beta (FLX_exp prec) (Znearest choice) x))%R
Proof.beta:radix
prec:Z
Hp:(0 < prec)%Z
rnd:R -> Z
valid_rnd:Valid_rnd rnd
choice:Z -> bool
forall x : R,
(Rabs
   (round beta (FLX_exp prec) (Znearest choice) x - x) <=
 / 2 * bpow (- prec + 1) *
 Rabs (round beta (FLX_exp prec) (Znearest choice) x))%R
intro x.beta:radix
prec:Z
Hp:(0 < prec)%Z
rnd:R -> Z
valid_rnd:Valid_rnd rnd
choice:Z -> bool
x:R
(Rabs
   (round beta (FLX_exp prec) (Znearest choice) x - x) <=
 / 2 * bpow (- prec + 1) *
 Rabs (round beta (FLX_exp prec) (Znearest choice) x))%R
apply relative_error_le_conversion_round_inv, relative_error_N_FLX_round_ex.
Qed.

End Fprop_relative_FLX.

Section Fprop_relative_FLT.

Variable emin prec : Z.
Variable Hp : Z.lt 0 prec.

Lemma relative_error_FLT_aux :
  forall k, (emin + prec - 1 < k)%Z -> (prec <= k - FLT_exp emin prec k)%Z.beta:radix
emin, prec:Z
Hp:(0 < prec)%Z
forall k : Z,
(emin + prec - 1 < k)%Z ->
(prec <= k - FLT_exp emin prec k)%Z
Proof.beta:radix
emin, prec:Z
Hp:(0 < prec)%Z
forall k : Z,
(emin + prec - 1 < k)%Z ->
(prec <= k - FLT_exp emin prec k)%Z
intros k Hk.beta:radix
emin, prec:Z
Hp:(0 < prec)%Z
k:Z
Hk:(emin + prec - 1 < k)%Z
(prec <= k - FLT_exp emin prec k)%Z
unfold FLT_exp.beta:radix
emin, prec:Z
Hp:(0 < prec)%Z
k:Z
Hk:(emin + prec - 1 < k)%Z
(prec <= k - Z.max (k - prec) emin)%Z
generalize (Zmax_spec (k - prec) emin).beta:radix
emin, prec:Z
Hp:(0 < prec)%Z
k:Z
Hk:(emin + prec - 1 < k)%Z
(k - prec >= emin)%Z /\
Z.max (k - prec) emin = (k - prec)%Z \/
(k - prec < emin)%Z /\ Z.max (k - prec) emin = emin ->
(prec <= k - Z.max (k - prec) emin)%Z
omega.
Qed.

Variable rnd : R -> Z.
Context { valid_rnd : Valid_rnd rnd }.

Theorem relative_error_FLT :
  forall x,
  (bpow (emin + prec - 1) <= Rabs x)%R ->
  (Rabs (round beta (FLT_exp emin prec) rnd x - x) < bpow (-prec + 1) * Rabs x)%R.beta:radix
emin, prec:Z
Hp:(0 < prec)%Z
rnd:R -> Z
valid_rnd:Valid_rnd rnd
forall x : R,
(bpow (emin + prec - 1) <= Rabs x)%R ->
(Rabs (round beta (FLT_exp emin prec) rnd x - x) <
 bpow (- prec + 1) * Rabs x)%R
Proof with auto with typeclass_instances.beta:radix
emin, prec:Z
Hp:(0 < prec)%Z
rnd:R -> Z
valid_rnd:Valid_rnd rnd
forall x : R,
(bpow (emin + prec - 1) <= Rabs x)%R ->
(Rabs (round beta (FLT_exp emin prec) rnd x - x) <
 bpow (- prec + 1) * Rabs x)%R
intros x Hx.beta:radix
emin, prec:Z
Hp:(0 < prec)%Z
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x:R
Hx:(bpow (emin + prec - 1) <= Rabs x)%R
(Rabs (round beta (FLT_exp emin prec) rnd x - x) <
 bpow (- prec + 1) * Rabs x)%R
apply relative_error with (emin + prec - 1)%Z...beta:radix
emin, prec:Z
Hp:(0 < prec)%Z
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x:R
Hx:(bpow (emin + prec - 1) <= Rabs x)%R
forall k : Z,
(emin + prec - 1 < k)%Z ->
(prec <= k - FLT_exp emin prec k)%Z
apply relative_error_FLT_aux.
Qed.

Theorem relative_error_FLT_F2R_emin :
  forall m, let x := F2R (Float beta m emin) in
  (x <> 0)%R ->
  (Rabs (round beta (FLT_exp emin prec) rnd x - x) < bpow (-prec + 1) * Rabs x)%R.beta:radix
emin, prec:Z
Hp:(0 < prec)%Z
rnd:R -> Z
valid_rnd:Valid_rnd rnd
forall m : Z,
let x := F2R {| Fnum := m; Fexp := emin |} in
x <> 0%R ->
(Rabs (round beta (FLT_exp emin prec) rnd x - x) <
 bpow (- prec + 1) * Rabs x)%R
Proof with auto with typeclass_instances.beta:radix
emin, prec:Z
Hp:(0 < prec)%Z
rnd:R -> Z
valid_rnd:Valid_rnd rnd
forall m : Z,
let x := F2R {| Fnum := m; Fexp := emin |} in
x <> 0%R ->
(Rabs (round beta (FLT_exp emin prec) rnd x - x) <
 bpow (- prec + 1) * Rabs x)%R
intros m x Zx.beta:radix
emin, prec:Z
Hp:(0 < prec)%Z
rnd:R -> Z
valid_rnd:Valid_rnd rnd
m:Z
x:=F2R {| Fnum := m; Fexp := emin |}:R
Zx:x <> 0%R
(Rabs (round beta (FLT_exp emin prec) rnd x - x) <
 bpow (- prec + 1) * Rabs x)%R
destruct (Rlt_or_le (Rabs x) (bpow (emin + prec - 1))) as [Hx|Hx].beta:radix
emin, prec:Z
Hp:(0 < prec)%Z
rnd:R -> Z
valid_rnd:Valid_rnd rnd
m:Z
x:=F2R {| Fnum := m; Fexp := emin |}:R
Zx:x <> 0%R
Hx:(Rabs x < bpow (emin + prec - 1))%R
(Rabs (round beta (FLT_exp emin prec) rnd x - x) <
 bpow (- prec + 1) * Rabs x)%R
beta:radix
emin, prec:Z
Hp:(0 < prec)%Z
rnd:R -> Z
valid_rnd:Valid_rnd rnd
m:Z
x:=F2R {| Fnum := m; Fexp := emin |}:R
Zx:x <> 0%R
Hx:(bpow (emin + prec - 1) <= Rabs x)%R
(Rabs (round beta (FLT_exp emin prec) rnd x - x) <
 bpow (- prec + 1) * Rabs x)%R
rewrite round_generic...beta:radix
emin, prec:Z
Hp:(0 < prec)%Z
rnd:R -> Z
valid_rnd:Valid_rnd rnd
m:Z
x:=F2R {| Fnum := m; Fexp := emin |}:R
Zx:x <> 0%R
Hx:(Rabs x < bpow (emin + prec - 1))%R
(Rabs (x - x) < bpow (- prec + 1) * Rabs x)%R
beta:radix
emin, prec:Z
Hp:(0 < prec)%Z
rnd:R -> Z
valid_rnd:Valid_rnd rnd
m:Z
x:=F2R {| Fnum := m; Fexp := emin |}:R
Zx:x <> 0%R
Hx:(Rabs x < bpow (emin + prec - 1))%R
generic_format beta (FLT_exp emin prec) x
beta:radix
emin, prec:Z
Hp:(0 < prec)%Z
rnd:R -> Z
valid_rnd:Valid_rnd rnd
m:Z
x:=F2R {| Fnum := m; Fexp := emin |}:R
Zx:x <> 0%R
Hx:(bpow (emin + prec - 1) <= Rabs x)%R
(Rabs (round beta (FLT_exp emin prec) rnd x - x) <
 bpow (- prec + 1) * Rabs x)%R
unfold Rminus.beta:radix
emin, prec:Z
Hp:(0 < prec)%Z
rnd:R -> Z
valid_rnd:Valid_rnd rnd
m:Z
x:=F2R {| Fnum := m; Fexp := emin |}:R
Zx:x <> 0%R
Hx:(Rabs x < bpow (emin + prec - 1))%R
(Rabs (x + - x) < bpow (- prec + 1) * Rabs x)%R
beta:radix
emin, prec:Z
Hp:(0 < prec)%Z
rnd:R -> Z
valid_rnd:Valid_rnd rnd
m:Z
x:=F2R {| Fnum := m; Fexp := emin |}:R
Zx:x <> 0%R
Hx:(Rabs x < bpow (emin + prec - 1))%R
generic_format beta (FLT_exp emin prec) x
beta:radix
emin, prec:Z
Hp:(0 < prec)%Z
rnd:R -> Z
valid_rnd:Valid_rnd rnd
m:Z
x:=F2R {| Fnum := m; Fexp := emin |}:R
Zx:x <> 0%R
Hx:(bpow (emin + prec - 1) <= Rabs x)%R
(Rabs (round beta (FLT_exp emin prec) rnd x - x) <
 bpow (- prec + 1) * Rabs x)%R
rewrite Rplus_opp_r, Rabs_R0.beta:radix
emin, prec:Z
Hp:(0 < prec)%Z
rnd:R -> Z
valid_rnd:Valid_rnd rnd
m:Z
x:=F2R {| Fnum := m; Fexp := emin |}:R
Zx:x <> 0%R
Hx:(Rabs x < bpow (emin + prec - 1))%R
(0 < bpow (- prec + 1) * Rabs x)%R
beta:radix
emin, prec:Z
Hp:(0 < prec)%Z
rnd:R -> Z
valid_rnd:Valid_rnd rnd
m:Z
x:=F2R {| Fnum := m; Fexp := emin |}:R
Zx:x <> 0%R
Hx:(Rabs x < bpow (emin + prec - 1))%R
generic_format beta (FLT_exp emin prec) x
beta:radix
emin, prec:Z
Hp:(0 < prec)%Z
rnd:R -> Z
valid_rnd:Valid_rnd rnd
m:Z
x:=F2R {| Fnum := m; Fexp := emin |}:R
Zx:x <> 0%R
Hx:(bpow (emin + prec - 1) <= Rabs x)%R
(Rabs (round beta (FLT_exp emin prec) rnd x - x) <
 bpow (- prec + 1) * Rabs x)%R
apply Rmult_lt_0_compat.beta:radix
emin, prec:Z
Hp:(0 < prec)%Z
rnd:R -> Z
valid_rnd:Valid_rnd rnd
m:Z
x:=F2R {| Fnum := m; Fexp := emin |}:R
Zx:x <> 0%R
Hx:(Rabs x < bpow (emin + prec - 1))%R
(0 < bpow (- prec + 1))%R
beta:radix
emin, prec:Z
Hp:(0 < prec)%Z
rnd:R -> Z
valid_rnd:Valid_rnd rnd
m:Z
x:=F2R {| Fnum := m; Fexp := emin |}:R
Zx:x <> 0%R
Hx:(Rabs x < bpow (emin + prec - 1))%R
(0 < Rabs x)%R
beta:radix
emin, prec:Z
Hp:(0 < prec)%Z
rnd:R -> Z
valid_rnd:Valid_rnd rnd
m:Z
x:=F2R {| Fnum := m; Fexp := emin |}:R
Zx:x <> 0%R
Hx:(Rabs x < bpow (emin + prec - 1))%R
generic_format beta (FLT_exp emin prec) x
beta:radix
emin, prec:Z
Hp:(0 < prec)%Z
rnd:R -> Z
valid_rnd:Valid_rnd rnd
m:Z
x:=F2R {| Fnum := m; Fexp := emin |}:R
Zx:x <> 0%R
Hx:(bpow (emin + prec - 1) <= Rabs x)%R
(Rabs (round beta (FLT_exp emin prec) rnd x - x) <
 bpow (- prec + 1) * Rabs x)%R
apply bpow_gt_0.beta:radix
emin, prec:Z
Hp:(0 < prec)%Z
rnd:R -> Z
valid_rnd:Valid_rnd rnd
m:Z
x:=F2R {| Fnum := m; Fexp := emin |}:R
Zx:x <> 0%R
Hx:(Rabs x < bpow (emin + prec - 1))%R
(0 < Rabs x)%R
beta:radix
emin, prec:Z
Hp:(0 < prec)%Z
rnd:R -> Z
valid_rnd:Valid_rnd rnd
m:Z
x:=F2R {| Fnum := m; Fexp := emin |}:R
Zx:x <> 0%R
Hx:(Rabs x < bpow (emin + prec - 1))%R
generic_format beta (FLT_exp emin prec) x
beta:radix
emin, prec:Z
Hp:(0 < prec)%Z
rnd:R -> Z
valid_rnd:Valid_rnd rnd
m:Z
x:=F2R {| Fnum := m; Fexp := emin |}:R
Zx:x <> 0%R
Hx:(bpow (emin + prec - 1) <= Rabs x)%R
(Rabs (round beta (FLT_exp emin prec) rnd x - x) <
 bpow (- prec + 1) * Rabs x)%R
now apply Rabs_pos_lt.beta:radix
emin, prec:Z
Hp:(0 < prec)%Z
rnd:R -> Z
valid_rnd:Valid_rnd rnd
m:Z
x:=F2R {| Fnum := m; Fexp := emin |}:R
Zx:x <> 0%R
Hx:(Rabs x < bpow (emin + prec - 1))%R
generic_format beta (FLT_exp emin prec) x
beta:radix
emin, prec:Z
Hp:(0 < prec)%Z
rnd:R -> Z
valid_rnd:Valid_rnd rnd
m:Z
x:=F2R {| Fnum := m; Fexp := emin |}:R
Zx:x <> 0%R
Hx:(bpow (emin + prec - 1) <= Rabs x)%R
(Rabs (round beta (FLT_exp emin prec) rnd x - x) <
 bpow (- prec + 1) * Rabs x)%R
apply generic_format_FLT_FIX...beta:radix
emin, prec:Z
Hp:(0 < prec)%Z
rnd:R -> Z
valid_rnd:Valid_rnd rnd
m:Z
x:=F2R {| Fnum := m; Fexp := emin |}:R
Zx:x <> 0%R
Hx:(Rabs x < bpow (emin + prec - 1))%R
(Rabs x <= bpow (emin + prec))%R
beta:radix
emin, prec:Z
Hp:(0 < prec)%Z
rnd:R -> Z
valid_rnd:Valid_rnd rnd
m:Z
x:=F2R {| Fnum := m; Fexp := emin |}:R
Zx:x <> 0%R
Hx:(Rabs x < bpow (emin + prec - 1))%R
generic_format beta (FIX_exp emin) x
beta:radix
emin, prec:Z
Hp:(0 < prec)%Z
rnd:R -> Z
valid_rnd:Valid_rnd rnd
m:Z
x:=F2R {| Fnum := m; Fexp := emin |}:R
Zx:x <> 0%R
Hx:(bpow (emin + prec - 1) <= Rabs x)%R
(Rabs (round beta (FLT_exp emin prec) rnd x - x) <
 bpow (- prec + 1) * Rabs x)%R
apply Rlt_le.beta:radix
emin, prec:Z
Hp:(0 < prec)%Z
rnd:R -> Z
valid_rnd:Valid_rnd rnd
m:Z
x:=F2R {| Fnum := m; Fexp := emin |}:R
Zx:x <> 0%R
Hx:(Rabs x < bpow (emin + prec - 1))%R
(Rabs x < bpow (emin + prec))%R
beta:radix
emin, prec:Z
Hp:(0 < prec)%Z
rnd:R -> Z
valid_rnd:Valid_rnd rnd
m:Z
x:=F2R {| Fnum := m; Fexp := emin |}:R
Zx:x <> 0%R
Hx:(Rabs x < bpow (emin + prec - 1))%R
generic_format beta (FIX_exp emin) x
beta:radix
emin, prec:Z
Hp:(0 < prec)%Z
rnd:R -> Z
valid_rnd:Valid_rnd rnd
m:Z
x:=F2R {| Fnum := m; Fexp := emin |}:R
Zx:x <> 0%R
Hx:(bpow (emin + prec - 1) <= Rabs x)%R
(Rabs (round beta (FLT_exp emin prec) rnd x - x) <
 bpow (- prec + 1) * Rabs x)%R
apply Rlt_le_trans with (1 := Hx).beta:radix
emin, prec:Z
Hp:(0 < prec)%Z
rnd:R -> Z
valid_rnd:Valid_rnd rnd
m:Z
x:=F2R {| Fnum := m; Fexp := emin |}:R
Zx:x <> 0%R
Hx:(Rabs x < bpow (emin + prec - 1))%R
(bpow (emin + prec - 1) <= bpow (emin + prec))%R
beta:radix
emin, prec:Z
Hp:(0 < prec)%Z
rnd:R -> Z
valid_rnd:Valid_rnd rnd
m:Z
x:=F2R {| Fnum := m; Fexp := emin |}:R
Zx:x <> 0%R
Hx:(Rabs x < bpow (emin + prec - 1))%R
generic_format beta (FIX_exp emin) x
beta:radix
emin, prec:Z
Hp:(0 < prec)%Z
rnd:R -> Z
valid_rnd:Valid_rnd rnd
m:Z
x:=F2R {| Fnum := m; Fexp := emin |}:R
Zx:x <> 0%R
Hx:(bpow (emin + prec - 1) <= Rabs x)%R
(Rabs (round beta (FLT_exp emin prec) rnd x - x) <
 bpow (- prec + 1) * Rabs x)%R
apply bpow_le.beta:radix
emin, prec:Z
Hp:(0 < prec)%Z
rnd:R -> Z
valid_rnd:Valid_rnd rnd
m:Z
x:=F2R {| Fnum := m; Fexp := emin |}:R
Zx:x <> 0%R
Hx:(Rabs x < bpow (emin + prec - 1))%R
(emin + prec - 1 <= emin + prec)%Z
beta:radix
emin, prec:Z
Hp:(0 < prec)%Z
rnd:R -> Z
valid_rnd:Valid_rnd rnd
m:Z
x:=F2R {| Fnum := m; Fexp := emin |}:R
Zx:x <> 0%R
Hx:(Rabs x < bpow (emin + prec - 1))%R
generic_format beta (FIX_exp emin) x
beta:radix
emin, prec:Z
Hp:(0 < prec)%Z
rnd:R -> Z
valid_rnd:Valid_rnd rnd
m:Z
x:=F2R {| Fnum := m; Fexp := emin |}:R
Zx:x <> 0%R
Hx:(bpow (emin + prec - 1) <= Rabs x)%R
(Rabs (round beta (FLT_exp emin prec) rnd x - x) <
 bpow (- prec + 1) * Rabs x)%R
apply Zle_pred.beta:radix
emin, prec:Z
Hp:(0 < prec)%Z
rnd:R -> Z
valid_rnd:Valid_rnd rnd
m:Z
x:=F2R {| Fnum := m; Fexp := emin |}:R
Zx:x <> 0%R
Hx:(Rabs x < bpow (emin + prec - 1))%R
generic_format beta (FIX_exp emin) x
beta:radix
emin, prec:Z
Hp:(0 < prec)%Z
rnd:R -> Z
valid_rnd:Valid_rnd rnd
m:Z
x:=F2R {| Fnum := m; Fexp := emin |}:R
Zx:x <> 0%R
Hx:(bpow (emin + prec - 1) <= Rabs x)%R
(Rabs (round beta (FLT_exp emin prec) rnd x - x) <
 bpow (- prec + 1) * Rabs x)%R
apply generic_format_FIX.beta:radix
emin, prec:Z
Hp:(0 < prec)%Z
rnd:R -> Z
valid_rnd:Valid_rnd rnd
m:Z
x:=F2R {| Fnum := m; Fexp := emin |}:R
Zx:x <> 0%R
Hx:(Rabs x < bpow (emin + prec - 1))%R
FIX_format beta emin x
beta:radix
emin, prec:Z
Hp:(0 < prec)%Z
rnd:R -> Z
valid_rnd:Valid_rnd rnd
m:Z
x:=F2R {| Fnum := m; Fexp := emin |}:R
Zx:x <> 0%R
Hx:(bpow (emin + prec - 1) <= Rabs x)%R
(Rabs (round beta (FLT_exp emin prec) rnd x - x) <
 bpow (- prec + 1) * Rabs x)%R
now exists (Float beta m emin).beta:radix
emin, prec:Z
Hp:(0 < prec)%Z
rnd:R -> Z
valid_rnd:Valid_rnd rnd
m:Z
x:=F2R {| Fnum := m; Fexp := emin |}:R
Zx:x <> 0%R
Hx:(bpow (emin + prec - 1) <= Rabs x)%R
(Rabs (round beta (FLT_exp emin prec) rnd x - x) <
 bpow (- prec + 1) * Rabs x)%R
now apply relative_error_FLT.
Qed.

Theorem relative_error_FLT_F2R_emin_ex :
  forall m, let x := F2R (Float beta m emin) in
  exists eps,
  (Rabs eps < bpow (-prec + 1))%R /\ round beta (FLT_exp emin prec) rnd x = (x * (1 + eps))%R.beta:radix
emin, prec:Z
Hp:(0 < prec)%Z
rnd:R -> Z
valid_rnd:Valid_rnd rnd
forall m : Z,
let x := F2R {| Fnum := m; Fexp := emin |} in
exists eps : R,
  (Rabs eps < bpow (- prec + 1))%R /\
  round beta (FLT_exp emin prec) rnd x =
  (x * (1 + eps))%R
Proof with auto with typeclass_instances.beta:radix
emin, prec:Z
Hp:(0 < prec)%Z
rnd:R -> Z
valid_rnd:Valid_rnd rnd
forall m : Z,
let x := F2R {| Fnum := m; Fexp := emin |} in
exists eps : R,
  (Rabs eps < bpow (- prec + 1))%R /\
  round beta (FLT_exp emin prec) rnd x =
  (x * (1 + eps))%R
intros m x.beta:radix
emin, prec:Z
Hp:(0 < prec)%Z
rnd:R -> Z
valid_rnd:Valid_rnd rnd
m:Z
x:=F2R {| Fnum := m; Fexp := emin |}:R
exists eps : R,
  (Rabs eps < bpow (- prec + 1))%R /\
  round beta (FLT_exp emin prec) rnd x =
  (x * (1 + eps))%R
apply relative_error_lt_conversion...beta:radix
emin, prec:Z
Hp:(0 < prec)%Z
rnd:R -> Z
valid_rnd:Valid_rnd rnd
m:Z
x:=F2R {| Fnum := m; Fexp := emin |}:R
(0 < bpow (- prec + 1))%R
beta:radix
emin, prec:Z
Hp:(0 < prec)%Z
rnd:R -> Z
valid_rnd:Valid_rnd rnd
m:Z
x:=F2R {| Fnum := m; Fexp := emin |}:R
x <> 0%R ->
(Rabs (round beta (FLT_exp emin prec) rnd x - x) <
 bpow (- prec + 1) * Rabs x)%R
apply bpow_gt_0.beta:radix
emin, prec:Z
Hp:(0 < prec)%Z
rnd:R -> Z
valid_rnd:Valid_rnd rnd
m:Z
x:=F2R {| Fnum := m; Fexp := emin |}:R
x <> 0%R ->
(Rabs (round beta (FLT_exp emin prec) rnd x - x) <
 bpow (- prec + 1) * Rabs x)%R
now apply relative_error_FLT_F2R_emin.
Qed.

1+ε property in any rounding in FLT

Theorem relative_error_FLT_ex :
  forall x,
  (bpow (emin + prec - 1) <= Rabs x)%R ->
  exists eps,
  (Rabs eps < bpow (-prec + 1))%R /\ round beta (FLT_exp emin prec) rnd x = (x * (1 + eps))%R.beta:radix
emin, prec:Z
Hp:(0 < prec)%Z
rnd:R -> Z
valid_rnd:Valid_rnd rnd
forall x : R,
(bpow (emin + prec - 1) <= Rabs x)%R ->
exists eps : R,
  (Rabs eps < bpow (- prec + 1))%R /\
  round beta (FLT_exp emin prec) rnd x =
  (x * (1 + eps))%R
Proof with auto with typeclass_instances.beta:radix
emin, prec:Z
Hp:(0 < prec)%Z
rnd:R -> Z
valid_rnd:Valid_rnd rnd
forall x : R,
(bpow (emin + prec - 1) <= Rabs x)%R ->
exists eps : R,
  (Rabs eps < bpow (- prec + 1))%R /\
  round beta (FLT_exp emin prec) rnd x =
  (x * (1 + eps))%R
intros x Hx.beta:radix
emin, prec:Z
Hp:(0 < prec)%Z
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x:R
Hx:(bpow (emin + prec - 1) <= Rabs x)%R
exists eps : R,
  (Rabs eps < bpow (- prec + 1))%R /\
  round beta (FLT_exp emin prec) rnd x =
  (x * (1 + eps))%R
apply relative_error_lt_conversion...beta:radix
emin, prec:Z
Hp:(0 < prec)%Z
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x:R
Hx:(bpow (emin + prec - 1) <= Rabs x)%R
(0 < bpow (- prec + 1))%R
beta:radix
emin, prec:Z
Hp:(0 < prec)%Z
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x:R
Hx:(bpow (emin + prec - 1) <= Rabs x)%R
x <> 0%R ->
(Rabs (round beta (FLT_exp emin prec) rnd x - x) <
 bpow (- prec + 1) * Rabs x)%R
apply bpow_gt_0.beta:radix
emin, prec:Z
Hp:(0 < prec)%Z
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x:R
Hx:(bpow (emin + prec - 1) <= Rabs x)%R
x <> 0%R ->
(Rabs (round beta (FLT_exp emin prec) rnd x - x) <
 bpow (- prec + 1) * Rabs x)%R
intros _; now apply relative_error_FLT.
Qed.

Variable choice : Z -> bool.

Theorem relative_error_N_FLT :
  forall x,
  (bpow (emin + prec - 1) <= Rabs x)%R ->
  (Rabs (round beta (FLT_exp emin prec) (Znearest choice) x - x) <= /2 * bpow (-prec + 1) * Rabs x)%R.beta:radix
emin, prec:Z
Hp:(0 < prec)%Z
rnd:R -> Z
valid_rnd:Valid_rnd rnd
choice:Z -> bool
forall x : R,
(bpow (emin + prec - 1) <= Rabs x)%R ->
(Rabs
   (round beta (FLT_exp emin prec) (Znearest choice) x -
    x) <= / 2 * bpow (- prec + 1) * Rabs x)%R
Proof with auto with typeclass_instances.beta:radix
emin, prec:Z
Hp:(0 < prec)%Z
rnd:R -> Z
valid_rnd:Valid_rnd rnd
choice:Z -> bool
forall x : R,
(bpow (emin + prec - 1) <= Rabs x)%R ->
(Rabs
   (round beta (FLT_exp emin prec) (Znearest choice) x -
    x) <= / 2 * bpow (- prec + 1) * Rabs x)%R
intros x Hx.beta:radix
emin, prec:Z
Hp:(0 < prec)%Z
rnd:R -> Z
valid_rnd:Valid_rnd rnd
choice:Z -> bool
x:R
Hx:(bpow (emin + prec - 1) <= Rabs x)%R
(Rabs
   (round beta (FLT_exp emin prec) (Znearest choice) x -
    x) <= / 2 * bpow (- prec + 1) * Rabs x)%R
apply relative_error_N with (emin + prec - 1)%Z...beta:radix
emin, prec:Z
Hp:(0 < prec)%Z
rnd:R -> Z
valid_rnd:Valid_rnd rnd
choice:Z -> bool
x:R
Hx:(bpow (emin + prec - 1) <= Rabs x)%R
forall k : Z,
(emin + prec - 1 < k)%Z ->
(prec <= k - FLT_exp emin prec k)%Z
apply relative_error_FLT_aux.
Qed.

1+ε property in rounding to nearest in FLT

Theorem relative_error_N_FLT_ex :
  forall x,
  (bpow (emin + prec - 1) <= Rabs x)%R ->
  exists eps,
  (Rabs eps <= /2 * bpow (-prec + 1))%R /\ round beta (FLT_exp emin prec) (Znearest choice) x = (x * (1 + eps))%R.beta:radix
emin, prec:Z
Hp:(0 < prec)%Z
rnd:R -> Z
valid_rnd:Valid_rnd rnd
choice:Z -> bool
forall x : R,
(bpow (emin + prec - 1) <= Rabs x)%R ->
exists eps : R,
  (Rabs eps <= / 2 * bpow (- prec + 1))%R /\
  round beta (FLT_exp emin prec) (Znearest choice) x =
  (x * (1 + eps))%R
Proof with auto with typeclass_instances.beta:radix
emin, prec:Z
Hp:(0 < prec)%Z
rnd:R -> Z
valid_rnd:Valid_rnd rnd
choice:Z -> bool
forall x : R,
(bpow (emin + prec - 1) <= Rabs x)%R ->
exists eps : R,
  (Rabs eps <= / 2 * bpow (- prec + 1))%R /\
  round beta (FLT_exp emin prec) (Znearest choice) x =
  (x * (1 + eps))%R
intros x Hx.beta:radix
emin, prec:Z
Hp:(0 < prec)%Z
rnd:R -> Z
valid_rnd:Valid_rnd rnd
choice:Z -> bool
x:R
Hx:(bpow (emin + prec - 1) <= Rabs x)%R
exists eps : R,
  (Rabs eps <= / 2 * bpow (- prec + 1))%R /\
  round beta (FLT_exp emin prec) (Znearest choice) x =
  (x * (1 + eps))%R
apply relative_error_le_conversion...beta:radix
emin, prec:Z
Hp:(0 < prec)%Z
rnd:R -> Z
valid_rnd:Valid_rnd rnd
choice:Z -> bool
x:R
Hx:(bpow (emin + prec - 1) <= Rabs x)%R
(0 <= / 2 * bpow (- prec + 1))%R
beta:radix
emin, prec:Z
Hp:(0 < prec)%Z
rnd:R -> Z
valid_rnd:Valid_rnd rnd
choice:Z -> bool
x:R
Hx:(bpow (emin + prec - 1) <= Rabs x)%R
(Rabs
   (round beta (FLT_exp emin prec) (Znearest choice) x -
    x) <= / 2 * bpow (- prec + 1) * Rabs x)%R
apply Rlt_le.beta:radix
emin, prec:Z
Hp:(0 < prec)%Z
rnd:R -> Z
valid_rnd:Valid_rnd rnd
choice:Z -> bool
x:R
Hx:(bpow (emin + prec - 1) <= Rabs x)%R
(0 < / 2 * bpow (- prec + 1))%R
beta:radix
emin, prec:Z
Hp:(0 < prec)%Z
rnd:R -> Z
valid_rnd:Valid_rnd rnd
choice:Z -> bool
x:R
Hx:(bpow (emin + prec - 1) <= Rabs x)%R
(Rabs
   (round beta (FLT_exp emin prec) (Znearest choice) x -
    x) <= / 2 * bpow (- prec + 1) * Rabs x)%R
apply Rmult_lt_0_compat.beta:radix
emin, prec:Z
Hp:(0 < prec)%Z
rnd:R -> Z
valid_rnd:Valid_rnd rnd
choice:Z -> bool
x:R
Hx:(bpow (emin + prec - 1) <= Rabs x)%R
(0 < / 2)%R
beta:radix
emin, prec:Z
Hp:(0 < prec)%Z
rnd:R -> Z
valid_rnd:Valid_rnd rnd
choice:Z -> bool
x:R
Hx:(bpow (emin + prec - 1) <= Rabs x)%R
(0 < bpow (- prec + 1))%R
beta:radix
emin, prec:Z
Hp:(0 < prec)%Z
rnd:R -> Z
valid_rnd:Valid_rnd rnd
choice:Z -> bool
x:R
Hx:(bpow (emin + prec - 1) <= Rabs x)%R
(Rabs
   (round beta (FLT_exp emin prec) (Znearest choice) x -
    x) <= / 2 * bpow (- prec + 1) * Rabs x)%R
apply Rinv_0_lt_compat.beta:radix
emin, prec:Z
Hp:(0 < prec)%Z
rnd:R -> Z
valid_rnd:Valid_rnd rnd
choice:Z -> bool
x:R
Hx:(bpow (emin + prec - 1) <= Rabs x)%R
(0 < 2)%R
beta:radix
emin, prec:Z
Hp:(0 < prec)%Z
rnd:R -> Z
valid_rnd:Valid_rnd rnd
choice:Z -> bool
x:R
Hx:(bpow (emin + prec - 1) <= Rabs x)%R
(0 < bpow (- prec + 1))%R
beta:radix
emin, prec:Z
Hp:(0 < prec)%Z
rnd:R -> Z
valid_rnd:Valid_rnd rnd
choice:Z -> bool
x:R
Hx:(bpow (emin + prec - 1) <= Rabs x)%R
(Rabs
   (round beta (FLT_exp emin prec) (Znearest choice) x -
    x) <= / 2 * bpow (- prec + 1) * Rabs x)%R
now apply IZR_lt.beta:radix
emin, prec:Z
Hp:(0 < prec)%Z
rnd:R -> Z
valid_rnd:Valid_rnd rnd
choice:Z -> bool
x:R
Hx:(bpow (emin + prec - 1) <= Rabs x)%R
(0 < bpow (- prec + 1))%R
beta:radix
emin, prec:Z
Hp:(0 < prec)%Z
rnd:R -> Z
valid_rnd:Valid_rnd rnd
choice:Z -> bool
x:R
Hx:(bpow (emin + prec - 1) <= Rabs x)%R
(Rabs
   (round beta (FLT_exp emin prec) (Znearest choice) x -
    x) <= / 2 * bpow (- prec + 1) * Rabs x)%R
apply bpow_gt_0.beta:radix
emin, prec:Z
Hp:(0 < prec)%Z
rnd:R -> Z
valid_rnd:Valid_rnd rnd
choice:Z -> bool
x:R
Hx:(bpow (emin + prec - 1) <= Rabs x)%R
(Rabs
   (round beta (FLT_exp emin prec) (Znearest choice) x -
    x) <= / 2 * bpow (- prec + 1) * Rabs x)%R
now apply relative_error_N_FLT.
Qed.

Theorem relative_error_N_FLT_round :
  forall x,
  (bpow (emin + prec - 1) <= Rabs x)%R ->
  (Rabs (round beta (FLT_exp emin prec) (Znearest choice) x - x) <= /2 * bpow (-prec + 1) * Rabs (round beta (FLT_exp emin prec) (Znearest choice) x))%R.beta:radix
emin, prec:Z
Hp:(0 < prec)%Z
rnd:R -> Z
valid_rnd:Valid_rnd rnd
choice:Z -> bool
forall x : R,
(bpow (emin + prec - 1) <= Rabs x)%R ->
(Rabs
   (round beta (FLT_exp emin prec) (Znearest choice) x -
    x) <=
 / 2 * bpow (- prec + 1) *
 Rabs
   (round beta (FLT_exp emin prec) (Znearest choice) x))%R
Proof with auto with typeclass_instances.beta:radix
emin, prec:Z
Hp:(0 < prec)%Z
rnd:R -> Z
valid_rnd:Valid_rnd rnd
choice:Z -> bool
forall x : R,
(bpow (emin + prec - 1) <= Rabs x)%R ->
(Rabs
   (round beta (FLT_exp emin prec) (Znearest choice) x -
    x) <=
 / 2 * bpow (- prec + 1) *
 Rabs
   (round beta (FLT_exp emin prec) (Znearest choice) x))%R
intros x Hx.beta:radix
emin, prec:Z
Hp:(0 < prec)%Z
rnd:R -> Z
valid_rnd:Valid_rnd rnd
choice:Z -> bool
x:R
Hx:(bpow (emin + prec - 1) <= Rabs x)%R
(Rabs
   (round beta (FLT_exp emin prec) (Znearest choice) x -
    x) <=
 / 2 * bpow (- prec + 1) *
 Rabs
   (round beta (FLT_exp emin prec) (Znearest choice) x))%R
apply relative_error_N_round with (emin + prec - 1)%Z...beta:radix
emin, prec:Z
Hp:(0 < prec)%Z
rnd:R -> Z
valid_rnd:Valid_rnd rnd
choice:Z -> bool
x:R
Hx:(bpow (emin + prec - 1) <= Rabs x)%R
forall k : Z,
(emin + prec - 1 < k)%Z ->
(prec <= k - FLT_exp emin prec k)%Z
apply relative_error_FLT_aux.
Qed.

Theorem relative_error_N_FLT_F2R_emin :
  forall m, let x := F2R (Float beta m emin) in
  (Rabs (round beta (FLT_exp emin prec) (Znearest choice) x - x) <= /2 * bpow (-prec + 1) * Rabs x)%R.beta:radix
emin, prec:Z
Hp:(0 < prec)%Z
rnd:R -> Z
valid_rnd:Valid_rnd rnd
choice:Z -> bool
forall m : Z,
let x := F2R {| Fnum := m; Fexp := emin |} in
(Rabs
   (round beta (FLT_exp emin prec) (Znearest choice) x -
    x) <= / 2 * bpow (- prec + 1) * Rabs x)%R
Proof with auto with typeclass_instances.beta:radix
emin, prec:Z
Hp:(0 < prec)%Z
rnd:R -> Z
valid_rnd:Valid_rnd rnd
choice:Z -> bool
forall m : Z,
let x := F2R {| Fnum := m; Fexp := emin |} in
(Rabs
   (round beta (FLT_exp emin prec) (Znearest choice) x -
    x) <= / 2 * bpow (- prec + 1) * Rabs x)%R
intros m x.beta:radix
emin, prec:Z
Hp:(0 < prec)%Z
rnd:R -> Z
valid_rnd:Valid_rnd rnd
choice:Z -> bool
m:Z
x:=F2R {| Fnum := m; Fexp := emin |}:R
(Rabs
   (round beta (FLT_exp emin prec) (Znearest choice) x -
    x) <= / 2 * bpow (- prec + 1) * Rabs x)%R
destruct (Rlt_or_le (Rabs x) (bpow (emin + prec - 1))) as [Hx|Hx].beta:radix
emin, prec:Z
Hp:(0 < prec)%Z
rnd:R -> Z
valid_rnd:Valid_rnd rnd
choice:Z -> bool
m:Z
x:=F2R {| Fnum := m; Fexp := emin |}:R
Hx:(Rabs x < bpow (emin + prec - 1))%R
(Rabs
   (round beta (FLT_exp emin prec) (Znearest choice) x -
    x) <= / 2 * bpow (- prec + 1) * Rabs x)%R
beta:radix
emin, prec:Z
Hp:(0 < prec)%Z
rnd:R -> Z
valid_rnd:Valid_rnd rnd
choice:Z -> bool
m:Z
x:=F2R {| Fnum := m; Fexp := emin |}:R
Hx:(bpow (emin + prec - 1) <= Rabs x)%R
(Rabs
   (round beta (FLT_exp emin prec) (Znearest choice) x -
    x) <= / 2 * bpow (- prec + 1) * Rabs x)%R
rewrite round_generic...beta:radix
emin, prec:Z
Hp:(0 < prec)%Z
rnd:R -> Z
valid_rnd:Valid_rnd rnd
choice:Z -> bool
m:Z
x:=F2R {| Fnum := m; Fexp := emin |}:R
Hx:(Rabs x < bpow (emin + prec - 1))%R
(Rabs (x - x) <= / 2 * bpow (- prec + 1) * Rabs x)%R
beta:radix
emin, prec:Z
Hp:(0 < prec)%Z
rnd:R -> Z
valid_rnd:Valid_rnd rnd
choice:Z -> bool
m:Z
x:=F2R {| Fnum := m; Fexp := emin |}:R
Hx:(Rabs x < bpow (emin + prec - 1))%R
generic_format beta (FLT_exp emin prec) x
beta:radix
emin, prec:Z
Hp:(0 < prec)%Z
rnd:R -> Z
valid_rnd:Valid_rnd rnd
choice:Z -> bool
m:Z
x:=F2R {| Fnum := m; Fexp := emin |}:R
Hx:(bpow (emin + prec - 1) <= Rabs x)%R
(Rabs
   (round beta (FLT_exp emin prec) (Znearest choice) x -
    x) <= / 2 * bpow (- prec + 1) * Rabs x)%R
unfold Rminus.beta:radix
emin, prec:Z
Hp:(0 < prec)%Z
rnd:R -> Z
valid_rnd:Valid_rnd rnd
choice:Z -> bool
m:Z
x:=F2R {| Fnum := m; Fexp := emin |}:R
Hx:(Rabs x < bpow (emin + prec - 1))%R
(Rabs (x + - x) <= / 2 * bpow (- prec + 1) * Rabs x)%R
beta:radix
emin, prec:Z
Hp:(0 < prec)%Z
rnd:R -> Z
valid_rnd:Valid_rnd rnd
choice:Z -> bool
m:Z
x:=F2R {| Fnum := m; Fexp := emin |}:R
Hx:(Rabs x < bpow (emin + prec - 1))%R
generic_format beta (FLT_exp emin prec) x
beta:radix
emin, prec:Z
Hp:(0 < prec)%Z
rnd:R -> Z
valid_rnd:Valid_rnd rnd
choice:Z -> bool
m:Z
x:=F2R {| Fnum := m; Fexp := emin |}:R
Hx:(bpow (emin + prec - 1) <= Rabs x)%R
(Rabs
   (round beta (FLT_exp emin prec) (Znearest choice) x -
    x) <= / 2 * bpow (- prec + 1) * Rabs x)%R
rewrite Rplus_opp_r, Rabs_R0.beta:radix
emin, prec:Z
Hp:(0 < prec)%Z
rnd:R -> Z
valid_rnd:Valid_rnd rnd
choice:Z -> bool
m:Z
x:=F2R {| Fnum := m; Fexp := emin |}:R
Hx:(Rabs x < bpow (emin + prec - 1))%R
(0 <= / 2 * bpow (- prec + 1) * Rabs x)%R
beta:radix
emin, prec:Z
Hp:(0 < prec)%Z
rnd:R -> Z
valid_rnd:Valid_rnd rnd
choice:Z -> bool
m:Z
x:=F2R {| Fnum := m; Fexp := emin |}:R
Hx:(Rabs x < bpow (emin + prec - 1))%R
generic_format beta (FLT_exp emin prec) x
beta:radix
emin, prec:Z
Hp:(0 < prec)%Z
rnd:R -> Z
valid_rnd:Valid_rnd rnd
choice:Z -> bool
m:Z
x:=F2R {| Fnum := m; Fexp := emin |}:R
Hx:(bpow (emin + prec - 1) <= Rabs x)%R
(Rabs
   (round beta (FLT_exp emin prec) (Znearest choice) x -
    x) <= / 2 * bpow (- prec + 1) * Rabs x)%R
apply Rmult_le_pos.beta:radix
emin, prec:Z
Hp:(0 < prec)%Z
rnd:R -> Z
valid_rnd:Valid_rnd rnd
choice:Z -> bool
m:Z
x:=F2R {| Fnum := m; Fexp := emin |}:R
Hx:(Rabs x < bpow (emin + prec - 1))%R
(0 <= / 2 * bpow (- prec + 1))%R
beta:radix
emin, prec:Z
Hp:(0 < prec)%Z
rnd:R -> Z
valid_rnd:Valid_rnd rnd
choice:Z -> bool
m:Z
x:=F2R {| Fnum := m; Fexp := emin |}:R
Hx:(Rabs x < bpow (emin + prec - 1))%R
(0 <= Rabs x)%R
beta:radix
emin, prec:Z
Hp:(0 < prec)%Z
rnd:R -> Z
valid_rnd:Valid_rnd rnd
choice:Z -> bool
m:Z
x:=F2R {| Fnum := m; Fexp := emin |}:R
Hx:(Rabs x < bpow (emin + prec - 1))%R
generic_format beta (FLT_exp emin prec) x
beta:radix
emin, prec:Z
Hp:(0 < prec)%Z
rnd:R -> Z
valid_rnd:Valid_rnd rnd
choice:Z -> bool
m:Z
x:=F2R {| Fnum := m; Fexp := emin |}:R
Hx:(bpow (emin + prec - 1) <= Rabs x)%R
(Rabs
   (round beta (FLT_exp emin prec) (Znearest choice) x -
    x) <= / 2 * bpow (- prec + 1) * Rabs x)%R
apply Rmult_le_pos.beta:radix
emin, prec:Z
Hp:(0 < prec)%Z
rnd:R -> Z
valid_rnd:Valid_rnd rnd
choice:Z -> bool
m:Z
x:=F2R {| Fnum := m; Fexp := emin |}:R
Hx:(Rabs x < bpow (emin + prec - 1))%R
(0 <= / 2)%R
beta:radix
emin, prec:Z
Hp:(0 < prec)%Z
rnd:R -> Z
valid_rnd:Valid_rnd rnd
choice:Z -> bool
m:Z
x:=F2R {| Fnum := m; Fexp := emin |}:R
Hx:(Rabs x < bpow (emin + prec - 1))%R
(0 <= bpow (- prec + 1))%R
beta:radix
emin, prec:Z
Hp:(0 < prec)%Z
rnd:R -> Z
valid_rnd:Valid_rnd rnd
choice:Z -> bool
m:Z
x:=F2R {| Fnum := m; Fexp := emin |}:R
Hx:(Rabs x < bpow (emin + prec - 1))%R
(0 <= Rabs x)%R
beta:radix
emin, prec:Z
Hp:(0 < prec)%Z
rnd:R -> Z
valid_rnd:Valid_rnd rnd
choice:Z -> bool
m:Z
x:=F2R {| Fnum := m; Fexp := emin |}:R
Hx:(Rabs x < bpow (emin + prec - 1))%R
generic_format beta (FLT_exp emin prec) x
beta:radix
emin, prec:Z
Hp:(0 < prec)%Z
rnd:R -> Z
valid_rnd:Valid_rnd rnd
choice:Z -> bool
m:Z
x:=F2R {| Fnum := m; Fexp := emin |}:R
Hx:(bpow (emin + prec - 1) <= Rabs x)%R
(Rabs
   (round beta (FLT_exp emin prec) (Znearest choice) x -
    x) <= / 2 * bpow (- prec + 1) * Rabs x)%R
apply Rlt_le.beta:radix
emin, prec:Z
Hp:(0 < prec)%Z
rnd:R -> Z
valid_rnd:Valid_rnd rnd
choice:Z -> bool
m:Z
x:=F2R {| Fnum := m; Fexp := emin |}:R
Hx:(Rabs x < bpow (emin + prec - 1))%R
(0 < / 2)%R
beta:radix
emin, prec:Z
Hp:(0 < prec)%Z
rnd:R -> Z
valid_rnd:Valid_rnd rnd
choice:Z -> bool
m:Z
x:=F2R {| Fnum := m; Fexp := emin |}:R
Hx:(Rabs x < bpow (emin + prec - 1))%R
(0 <= bpow (- prec + 1))%R
beta:radix
emin, prec:Z
Hp:(0 < prec)%Z
rnd:R -> Z
valid_rnd:Valid_rnd rnd
choice:Z -> bool
m:Z
x:=F2R {| Fnum := m; Fexp := emin |}:R
Hx:(Rabs x < bpow (emin + prec - 1))%R
(0 <= Rabs x)%R
beta:radix
emin, prec:Z
Hp:(0 < prec)%Z
rnd:R -> Z
valid_rnd:Valid_rnd rnd
choice:Z -> bool
m:Z
x:=F2R {| Fnum := m; Fexp := emin |}:R
Hx:(Rabs x < bpow (emin + prec - 1))%R
generic_format beta (FLT_exp emin prec) x
beta:radix
emin, prec:Z
Hp:(0 < prec)%Z
rnd:R -> Z
valid_rnd:Valid_rnd rnd
choice:Z -> bool
m:Z
x:=F2R {| Fnum := m; Fexp := emin |}:R
Hx:(bpow (emin + prec - 1) <= Rabs x)%R
(Rabs
   (round beta (FLT_exp emin prec) (Znearest choice) x -
    x) <= / 2 * bpow (- prec + 1) * Rabs x)%R
apply (RinvN_pos 1).beta:radix
emin, prec:Z
Hp:(0 < prec)%Z
rnd:R -> Z
valid_rnd:Valid_rnd rnd
choice:Z -> bool
m:Z
x:=F2R {| Fnum := m; Fexp := emin |}:R
Hx:(Rabs x < bpow (emin + prec - 1))%R
(0 <= bpow (- prec + 1))%R
beta:radix
emin, prec:Z
Hp:(0 < prec)%Z
rnd:R -> Z
valid_rnd:Valid_rnd rnd
choice:Z -> bool
m:Z
x:=F2R {| Fnum := m; Fexp := emin |}:R
Hx:(Rabs x < bpow (emin + prec - 1))%R
(0 <= Rabs x)%R
beta:radix
emin, prec:Z
Hp:(0 < prec)%Z
rnd:R -> Z
valid_rnd:Valid_rnd rnd
choice:Z -> bool
m:Z
x:=F2R {| Fnum := m; Fexp := emin |}:R
Hx:(Rabs x < bpow (emin + prec - 1))%R
generic_format beta (FLT_exp emin prec) x
beta:radix
emin, prec:Z
Hp:(0 < prec)%Z
rnd:R -> Z
valid_rnd:Valid_rnd rnd
choice:Z -> bool
m:Z
x:=F2R {| Fnum := m; Fexp := emin |}:R
Hx:(bpow (emin + prec - 1) <= Rabs x)%R
(Rabs
   (round beta (FLT_exp emin prec) (Znearest choice) x -
    x) <= / 2 * bpow (- prec + 1) * Rabs x)%R
apply bpow_ge_0.beta:radix
emin, prec:Z
Hp:(0 < prec)%Z
rnd:R -> Z
valid_rnd:Valid_rnd rnd
choice:Z -> bool
m:Z
x:=F2R {| Fnum := m; Fexp := emin |}:R
Hx:(Rabs x < bpow (emin + prec - 1))%R
(0 <= Rabs x)%R
beta:radix
emin, prec:Z
Hp:(0 < prec)%Z
rnd:R -> Z
valid_rnd:Valid_rnd rnd
choice:Z -> bool
m:Z
x:=F2R {| Fnum := m; Fexp := emin |}:R
Hx:(Rabs x < bpow (emin + prec - 1))%R
generic_format beta (FLT_exp emin prec) x
beta:radix
emin, prec:Z
Hp:(0 < prec)%Z
rnd:R -> Z
valid_rnd:Valid_rnd rnd
choice:Z -> bool
m:Z
x:=F2R {| Fnum := m; Fexp := emin |}:R
Hx:(bpow (emin + prec - 1) <= Rabs x)%R
(Rabs
   (round beta (FLT_exp emin prec) (Znearest choice) x -
    x) <= / 2 * bpow (- prec + 1) * Rabs x)%R
apply Rabs_pos.beta:radix
emin, prec:Z
Hp:(0 < prec)%Z
rnd:R -> Z
valid_rnd:Valid_rnd rnd
choice:Z -> bool
m:Z
x:=F2R {| Fnum := m; Fexp := emin |}:R
Hx:(Rabs x < bpow (emin + prec - 1))%R
generic_format beta (FLT_exp emin prec) x
beta:radix
emin, prec:Z
Hp:(0 < prec)%Z
rnd:R -> Z
valid_rnd:Valid_rnd rnd
choice:Z -> bool
m:Z
x:=F2R {| Fnum := m; Fexp := emin |}:R
Hx:(bpow (emin + prec - 1) <= Rabs x)%R
(Rabs
   (round beta (FLT_exp emin prec) (Znearest choice) x -
    x) <= / 2 * bpow (- prec + 1) * Rabs x)%R
apply generic_format_FLT_FIX...beta:radix
emin, prec:Z
Hp:(0 < prec)%Z
rnd:R -> Z
valid_rnd:Valid_rnd rnd
choice:Z -> bool
m:Z
x:=F2R {| Fnum := m; Fexp := emin |}:R
Hx:(Rabs x < bpow (emin + prec - 1))%R
(Rabs x <= bpow (emin + prec))%R
beta:radix
emin, prec:Z
Hp:(0 < prec)%Z
rnd:R -> Z
valid_rnd:Valid_rnd rnd
choice:Z -> bool
m:Z
x:=F2R {| Fnum := m; Fexp := emin |}:R
Hx:(Rabs x < bpow (emin + prec - 1))%R
generic_format beta (FIX_exp emin) x
beta:radix
emin, prec:Z
Hp:(0 < prec)%Z
rnd:R -> Z
valid_rnd:Valid_rnd rnd
choice:Z -> bool
m:Z
x:=F2R {| Fnum := m; Fexp := emin |}:R
Hx:(bpow (emin + prec - 1) <= Rabs x)%R
(Rabs
   (round beta (FLT_exp emin prec) (Znearest choice) x -
    x) <= / 2 * bpow (- prec + 1) * Rabs x)%R
apply Rlt_le.beta:radix
emin, prec:Z
Hp:(0 < prec)%Z
rnd:R -> Z
valid_rnd:Valid_rnd rnd
choice:Z -> bool
m:Z
x:=F2R {| Fnum := m; Fexp := emin |}:R
Hx:(Rabs x < bpow (emin + prec - 1))%R
(Rabs x < bpow (emin + prec))%R
beta:radix
emin, prec:Z
Hp:(0 < prec)%Z
rnd:R -> Z
valid_rnd:Valid_rnd rnd
choice:Z -> bool
m:Z
x:=F2R {| Fnum := m; Fexp := emin |}:R
Hx:(Rabs x < bpow (emin + prec - 1))%R
generic_format beta (FIX_exp emin) x
beta:radix
emin, prec:Z
Hp:(0 < prec)%Z
rnd:R -> Z
valid_rnd:Valid_rnd rnd
choice:Z -> bool
m:Z
x:=F2R {| Fnum := m; Fexp := emin |}:R
Hx:(bpow (emin + prec - 1) <= Rabs x)%R
(Rabs
   (round beta (FLT_exp emin prec) (Znearest choice) x -
    x) <= / 2 * bpow (- prec + 1) * Rabs x)%R
apply Rlt_le_trans with (1 := Hx).beta:radix
emin, prec:Z
Hp:(0 < prec)%Z
rnd:R -> Z
valid_rnd:Valid_rnd rnd
choice:Z -> bool
m:Z
x:=F2R {| Fnum := m; Fexp := emin |}:R
Hx:(Rabs x < bpow (emin + prec - 1))%R
(bpow (emin + prec - 1) <= bpow (emin + prec))%R
beta:radix
emin, prec:Z
Hp:(0 < prec)%Z
rnd:R -> Z
valid_rnd:Valid_rnd rnd
choice:Z -> bool
m:Z
x:=F2R {| Fnum := m; Fexp := emin |}:R
Hx:(Rabs x < bpow (emin + prec - 1))%R
generic_format beta (FIX_exp emin) x
beta:radix
emin, prec:Z
Hp:(0 < prec)%Z
rnd:R -> Z
valid_rnd:Valid_rnd rnd
choice:Z -> bool
m:Z
x:=F2R {| Fnum := m; Fexp := emin |}:R
Hx:(bpow (emin + prec - 1) <= Rabs x)%R
(Rabs
   (round beta (FLT_exp emin prec) (Znearest choice) x -
    x) <= / 2 * bpow (- prec + 1) * Rabs x)%R
apply bpow_le.beta:radix
emin, prec:Z
Hp:(0 < prec)%Z
rnd:R -> Z
valid_rnd:Valid_rnd rnd
choice:Z -> bool
m:Z
x:=F2R {| Fnum := m; Fexp := emin |}:R
Hx:(Rabs x < bpow (emin + prec - 1))%R
(emin + prec - 1 <= emin + prec)%Z
beta:radix
emin, prec:Z
Hp:(0 < prec)%Z
rnd:R -> Z
valid_rnd:Valid_rnd rnd
choice:Z -> bool
m:Z
x:=F2R {| Fnum := m; Fexp := emin |}:R
Hx:(Rabs x < bpow (emin + prec - 1))%R
generic_format beta (FIX_exp emin) x
beta:radix
emin, prec:Z
Hp:(0 < prec)%Z
rnd:R -> Z
valid_rnd:Valid_rnd rnd
choice:Z -> bool
m:Z
x:=F2R {| Fnum := m; Fexp := emin |}:R
Hx:(bpow (emin + prec - 1) <= Rabs x)%R
(Rabs
   (round beta (FLT_exp emin prec) (Znearest choice) x -
    x) <= / 2 * bpow (- prec + 1) * Rabs x)%R
apply Zle_pred.beta:radix
emin, prec:Z
Hp:(0 < prec)%Z
rnd:R -> Z
valid_rnd:Valid_rnd rnd
choice:Z -> bool
m:Z
x:=F2R {| Fnum := m; Fexp := emin |}:R
Hx:(Rabs x < bpow (emin + prec - 1))%R
generic_format beta (FIX_exp emin) x
beta:radix
emin, prec:Z
Hp:(0 < prec)%Z
rnd:R -> Z
valid_rnd:Valid_rnd rnd
choice:Z -> bool
m:Z
x:=F2R {| Fnum := m; Fexp := emin |}:R
Hx:(bpow (emin + prec - 1) <= Rabs x)%R
(Rabs
   (round beta (FLT_exp emin prec) (Znearest choice) x -
    x) <= / 2 * bpow (- prec + 1) * Rabs x)%R
apply generic_format_FIX.beta:radix
emin, prec:Z
Hp:(0 < prec)%Z
rnd:R -> Z
valid_rnd:Valid_rnd rnd
choice:Z -> bool
m:Z
x:=F2R {| Fnum := m; Fexp := emin |}:R
Hx:(Rabs x < bpow (emin + prec - 1))%R
FIX_format beta emin x
beta:radix
emin, prec:Z
Hp:(0 < prec)%Z
rnd:R -> Z
valid_rnd:Valid_rnd rnd
choice:Z -> bool
m:Z
x:=F2R {| Fnum := m; Fexp := emin |}:R
Hx:(bpow (emin + prec - 1) <= Rabs x)%R
(Rabs
   (round beta (FLT_exp emin prec) (Znearest choice) x -
    x) <= / 2 * bpow (- prec + 1) * Rabs x)%R
now exists (Float beta m emin).beta:radix
emin, prec:Z
Hp:(0 < prec)%Z
rnd:R -> Z
valid_rnd:Valid_rnd rnd
choice:Z -> bool
m:Z
x:=F2R {| Fnum := m; Fexp := emin |}:R
Hx:(bpow (emin + prec - 1) <= Rabs x)%R
(Rabs
   (round beta (FLT_exp emin prec) (Znearest choice) x -
    x) <= / 2 * bpow (- prec + 1) * Rabs x)%R
now apply relative_error_N_FLT.
Qed.

Theorem relative_error_N_FLT_F2R_emin_ex :
  forall m, let x := F2R (Float beta m emin) in
  exists eps,
  (Rabs eps <= /2 * bpow (-prec + 1))%R /\ round beta (FLT_exp emin prec) (Znearest choice) x = (x * (1 + eps))%R.beta:radix
emin, prec:Z
Hp:(0 < prec)%Z
rnd:R -> Z
valid_rnd:Valid_rnd rnd
choice:Z -> bool
forall m : Z,
let x := F2R {| Fnum := m; Fexp := emin |} in
exists eps : R,
  (Rabs eps <= / 2 * bpow (- prec + 1))%R /\
  round beta (FLT_exp emin prec) (Znearest choice) x =
  (x * (1 + eps))%R
Proof with auto with typeclass_instances.beta:radix
emin, prec:Z
Hp:(0 < prec)%Z
rnd:R -> Z
valid_rnd:Valid_rnd rnd
choice:Z -> bool
forall m : Z,
let x := F2R {| Fnum := m; Fexp := emin |} in
exists eps : R,
  (Rabs eps <= / 2 * bpow (- prec + 1))%R /\
  round beta (FLT_exp emin prec) (Znearest choice) x =
  (x * (1 + eps))%R
intros m x.beta:radix
emin, prec:Z
Hp:(0 < prec)%Z
rnd:R -> Z
valid_rnd:Valid_rnd rnd
choice:Z -> bool
m:Z
x:=F2R {| Fnum := m; Fexp := emin |}:R
exists eps : R,
  (Rabs eps <= / 2 * bpow (- prec + 1))%R /\
  round beta (FLT_exp emin prec) (Znearest choice) x =
  (x * (1 + eps))%R
apply relative_error_le_conversion...beta:radix
emin, prec:Z
Hp:(0 < prec)%Z
rnd:R -> Z
valid_rnd:Valid_rnd rnd
choice:Z -> bool
m:Z
x:=F2R {| Fnum := m; Fexp := emin |}:R
(0 <= / 2 * bpow (- prec + 1))%R
beta:radix
emin, prec:Z
Hp:(0 < prec)%Z
rnd:R -> Z
valid_rnd:Valid_rnd rnd
choice:Z -> bool
m:Z
x:=F2R {| Fnum := m; Fexp := emin |}:R
(Rabs
   (round beta (FLT_exp emin prec) (Znearest choice) x -
    x) <= / 2 * bpow (- prec + 1) * Rabs x)%R
apply Rmult_le_pos.beta:radix
emin, prec:Z
Hp:(0 < prec)%Z
rnd:R -> Z
valid_rnd:Valid_rnd rnd
choice:Z -> bool
m:Z
x:=F2R {| Fnum := m; Fexp := emin |}:R
(0 <= / 2)%R
beta:radix
emin, prec:Z
Hp:(0 < prec)%Z
rnd:R -> Z
valid_rnd:Valid_rnd rnd
choice:Z -> bool
m:Z
x:=F2R {| Fnum := m; Fexp := emin |}:R
(0 <= bpow (- prec + 1))%R
beta:radix
emin, prec:Z
Hp:(0 < prec)%Z
rnd:R -> Z
valid_rnd:Valid_rnd rnd
choice:Z -> bool
m:Z
x:=F2R {| Fnum := m; Fexp := emin |}:R
(Rabs
   (round beta (FLT_exp emin prec) (Znearest choice) x -
    x) <= / 2 * bpow (- prec + 1) * Rabs x)%R
apply Rlt_le.beta:radix
emin, prec:Z
Hp:(0 < prec)%Z
rnd:R -> Z
valid_rnd:Valid_rnd rnd
choice:Z -> bool
m:Z
x:=F2R {| Fnum := m; Fexp := emin |}:R
(0 < / 2)%R
beta:radix
emin, prec:Z
Hp:(0 < prec)%Z
rnd:R -> Z
valid_rnd:Valid_rnd rnd
choice:Z -> bool
m:Z
x:=F2R {| Fnum := m; Fexp := emin |}:R
(0 <= bpow (- prec + 1))%R
beta:radix
emin, prec:Z
Hp:(0 < prec)%Z
rnd:R -> Z
valid_rnd:Valid_rnd rnd
choice:Z -> bool
m:Z
x:=F2R {| Fnum := m; Fexp := emin |}:R
(Rabs
   (round beta (FLT_exp emin prec) (Znearest choice) x -
    x) <= / 2 * bpow (- prec + 1) * Rabs x)%R
apply (RinvN_pos 1).beta:radix
emin, prec:Z
Hp:(0 < prec)%Z
rnd:R -> Z
valid_rnd:Valid_rnd rnd
choice:Z -> bool
m:Z
x:=F2R {| Fnum := m; Fexp := emin |}:R
(0 <= bpow (- prec + 1))%R
beta:radix
emin, prec:Z
Hp:(0 < prec)%Z
rnd:R -> Z
valid_rnd:Valid_rnd rnd
choice:Z -> bool
m:Z
x:=F2R {| Fnum := m; Fexp := emin |}:R
(Rabs
   (round beta (FLT_exp emin prec) (Znearest choice) x -
    x) <= / 2 * bpow (- prec + 1) * Rabs x)%R
apply bpow_ge_0.beta:radix
emin, prec:Z
Hp:(0 < prec)%Z
rnd:R -> Z
valid_rnd:Valid_rnd rnd
choice:Z -> bool
m:Z
x:=F2R {| Fnum := m; Fexp := emin |}:R
(Rabs
   (round beta (FLT_exp emin prec) (Znearest choice) x -
    x) <= / 2 * bpow (- prec + 1) * Rabs x)%R
now apply relative_error_N_FLT_F2R_emin.
Qed.


Theorem relative_error_N_FLT_round_F2R_emin :
  forall m, let x := F2R (Float beta m emin) in
  (Rabs (round beta (FLT_exp emin prec) (Znearest choice) x - x) <= /2 * bpow (-prec + 1) * Rabs (round beta (FLT_exp emin prec) (Znearest choice) x))%R.beta:radix
emin, prec:Z
Hp:(0 < prec)%Z
rnd:R -> Z
valid_rnd:Valid_rnd rnd
choice:Z -> bool
forall m : Z,
let x := F2R {| Fnum := m; Fexp := emin |} in
(Rabs
   (round beta (FLT_exp emin prec) (Znearest choice) x -
    x) <=
 / 2 * bpow (- prec + 1) *
 Rabs
   (round beta (FLT_exp emin prec) (Znearest choice) x))%R
Proof with auto with typeclass_instances.beta:radix
emin, prec:Z
Hp:(0 < prec)%Z
rnd:R -> Z
valid_rnd:Valid_rnd rnd
choice:Z -> bool
forall m : Z,
let x := F2R {| Fnum := m; Fexp := emin |} in
(Rabs
   (round beta (FLT_exp emin prec) (Znearest choice) x -
    x) <=
 / 2 * bpow (- prec + 1) *
 Rabs
   (round beta (FLT_exp emin prec) (Znearest choice) x))%R
intros m x.beta:radix
emin, prec:Z
Hp:(0 < prec)%Z
rnd:R -> Z
valid_rnd:Valid_rnd rnd
choice:Z -> bool
m:Z
x:=F2R {| Fnum := m; Fexp := emin |}:R
(Rabs
   (round beta (FLT_exp emin prec) (Znearest choice) x -
    x) <=
 / 2 * bpow (- prec + 1) *
 Rabs
   (round beta (FLT_exp emin prec) (Znearest choice) x))%R
destruct (Rlt_or_le (Rabs x) (bpow (emin + prec - 1))) as [Hx|Hx].beta:radix
emin, prec:Z
Hp:(0 < prec)%Z
rnd:R -> Z
valid_rnd:Valid_rnd rnd
choice:Z -> bool
m:Z
x:=F2R {| Fnum := m; Fexp := emin |}:R
Hx:(Rabs x < bpow (emin + prec - 1))%R
(Rabs
   (round beta (FLT_exp emin prec) (Znearest choice) x -
    x) <=
 / 2 * bpow (- prec + 1) *
 Rabs
   (round beta (FLT_exp emin prec) (Znearest choice) x))%R
beta:radix
emin, prec:Z
Hp:(0 < prec)%Z
rnd:R -> Z
valid_rnd:Valid_rnd rnd
choice:Z -> bool
m:Z
x:=F2R {| Fnum := m; Fexp := emin |}:R
Hx:(bpow (emin + prec - 1) <= Rabs x)%R
(Rabs
   (round beta (FLT_exp emin prec) (Znearest choice) x -
    x) <=
 / 2 * bpow (- prec + 1) *
 Rabs
   (round beta (FLT_exp emin prec) (Znearest choice) x))%R
rewrite round_generic...beta:radix
emin, prec:Z
Hp:(0 < prec)%Z
rnd:R -> Z
valid_rnd:Valid_rnd rnd
choice:Z -> bool
m:Z
x:=F2R {| Fnum := m; Fexp := emin |}:R
Hx:(Rabs x < bpow (emin + prec - 1))%R
(Rabs (x - x) <= / 2 * bpow (- prec + 1) * Rabs x)%R
beta:radix
emin, prec:Z
Hp:(0 < prec)%Z
rnd:R -> Z
valid_rnd:Valid_rnd rnd
choice:Z -> bool
m:Z
x:=F2R {| Fnum := m; Fexp := emin |}:R
Hx:(Rabs x < bpow (emin + prec - 1))%R
generic_format beta (FLT_exp emin prec) x
beta:radix
emin, prec:Z
Hp:(0 < prec)%Z
rnd:R -> Z
valid_rnd:Valid_rnd rnd
choice:Z -> bool
m:Z
x:=F2R {| Fnum := m; Fexp := emin |}:R
Hx:(bpow (emin + prec - 1) <= Rabs x)%R
(Rabs
   (round beta (FLT_exp emin prec) (Znearest choice) x -
    x) <=
 / 2 * bpow (- prec + 1) *
 Rabs
   (round beta (FLT_exp emin prec) (Znearest choice) x))%R
unfold Rminus.beta:radix
emin, prec:Z
Hp:(0 < prec)%Z
rnd:R -> Z
valid_rnd:Valid_rnd rnd
choice:Z -> bool
m:Z
x:=F2R {| Fnum := m; Fexp := emin |}:R
Hx:(Rabs x < bpow (emin + prec - 1))%R
(Rabs (x + - x) <= / 2 * bpow (- prec + 1) * Rabs x)%R
beta:radix
emin, prec:Z
Hp:(0 < prec)%Z
rnd:R -> Z
valid_rnd:Valid_rnd rnd
choice:Z -> bool
m:Z
x:=F2R {| Fnum := m; Fexp := emin |}:R
Hx:(Rabs x < bpow (emin + prec - 1))%R
generic_format beta (FLT_exp emin prec) x
beta:radix
emin, prec:Z
Hp:(0 < prec)%Z
rnd:R -> Z
valid_rnd:Valid_rnd rnd
choice:Z -> bool
m:Z
x:=F2R {| Fnum := m; Fexp := emin |}:R
Hx:(bpow (emin + prec - 1) <= Rabs x)%R
(Rabs
   (round beta (FLT_exp emin prec) (Znearest choice) x -
    x) <=
 / 2 * bpow (- prec + 1) *
 Rabs
   (round beta (FLT_exp emin prec) (Znearest choice) x))%R
rewrite Rplus_opp_r, Rabs_R0.beta:radix
emin, prec:Z
Hp:(0 < prec)%Z
rnd:R -> Z
valid_rnd:Valid_rnd rnd
choice:Z -> bool
m:Z
x:=F2R {| Fnum := m; Fexp := emin |}:R
Hx:(Rabs x < bpow (emin + prec - 1))%R
(0 <= / 2 * bpow (- prec + 1) * Rabs x)%R
beta:radix
emin, prec:Z
Hp:(0 < prec)%Z
rnd:R -> Z
valid_rnd:Valid_rnd rnd
choice:Z -> bool
m:Z
x:=F2R {| Fnum := m; Fexp := emin |}:R
Hx:(Rabs x < bpow (emin + prec - 1))%R
generic_format beta (FLT_exp emin prec) x
beta:radix
emin, prec:Z
Hp:(0 < prec)%Z
rnd:R -> Z
valid_rnd:Valid_rnd rnd
choice:Z -> bool
m:Z
x:=F2R {| Fnum := m; Fexp := emin |}:R
Hx:(bpow (emin + prec - 1) <= Rabs x)%R
(Rabs
   (round beta (FLT_exp emin prec) (Znearest choice) x -
    x) <=
 / 2 * bpow (- prec + 1) *
 Rabs
   (round beta (FLT_exp emin prec) (Znearest choice) x))%R
apply Rmult_le_pos.beta:radix
emin, prec:Z
Hp:(0 < prec)%Z
rnd:R -> Z
valid_rnd:Valid_rnd rnd
choice:Z -> bool
m:Z
x:=F2R {| Fnum := m; Fexp := emin |}:R
Hx:(Rabs x < bpow (emin + prec - 1))%R
(0 <= / 2 * bpow (- prec + 1))%R
beta:radix
emin, prec:Z
Hp:(0 < prec)%Z
rnd:R -> Z
valid_rnd:Valid_rnd rnd
choice:Z -> bool
m:Z
x:=F2R {| Fnum := m; Fexp := emin |}:R
Hx:(Rabs x < bpow (emin + prec - 1))%R
(0 <= Rabs x)%R
beta:radix
emin, prec:Z
Hp:(0 < prec)%Z
rnd:R -> Z
valid_rnd:Valid_rnd rnd
choice:Z -> bool
m:Z
x:=F2R {| Fnum := m; Fexp := emin |}:R
Hx:(Rabs x < bpow (emin + prec - 1))%R
generic_format beta (FLT_exp emin prec) x
beta:radix
emin, prec:Z
Hp:(0 < prec)%Z
rnd:R -> Z
valid_rnd:Valid_rnd rnd
choice:Z -> bool
m:Z
x:=F2R {| Fnum := m; Fexp := emin |}:R
Hx:(bpow (emin + prec - 1) <= Rabs x)%R
(Rabs
   (round beta (FLT_exp emin prec) (Znearest choice) x -
    x) <=
 / 2 * bpow (- prec + 1) *
 Rabs
   (round beta (FLT_exp emin prec) (Znearest choice) x))%R
apply Rmult_le_pos.beta:radix
emin, prec:Z
Hp:(0 < prec)%Z
rnd:R -> Z
valid_rnd:Valid_rnd rnd
choice:Z -> bool
m:Z
x:=F2R {| Fnum := m; Fexp := emin |}:R
Hx:(Rabs x < bpow (emin + prec - 1))%R
(0 <= / 2)%R
beta:radix
emin, prec:Z
Hp:(0 < prec)%Z
rnd:R -> Z
valid_rnd:Valid_rnd rnd
choice:Z -> bool
m:Z
x:=F2R {| Fnum := m; Fexp := emin |}:R
Hx:(Rabs x < bpow (emin + prec - 1))%R
(0 <= bpow (- prec + 1))%R
beta:radix
emin, prec:Z
Hp:(0 < prec)%Z
rnd:R -> Z
valid_rnd:Valid_rnd rnd
choice:Z -> bool
m:Z
x:=F2R {| Fnum := m; Fexp := emin |}:R
Hx:(Rabs x < bpow (emin + prec - 1))%R
(0 <= Rabs x)%R
beta:radix
emin, prec:Z
Hp:(0 < prec)%Z
rnd:R -> Z
valid_rnd:Valid_rnd rnd
choice:Z -> bool
m:Z
x:=F2R {| Fnum := m; Fexp := emin |}:R
Hx:(Rabs x < bpow (emin + prec - 1))%R
generic_format beta (FLT_exp emin prec) x
beta:radix
emin, prec:Z
Hp:(0 < prec)%Z
rnd:R -> Z
valid_rnd:Valid_rnd rnd
choice:Z -> bool
m:Z
x:=F2R {| Fnum := m; Fexp := emin |}:R
Hx:(bpow (emin + prec - 1) <= Rabs x)%R
(Rabs
   (round beta (FLT_exp emin prec) (Znearest choice) x -
    x) <=
 / 2 * bpow (- prec + 1) *
 Rabs
   (round beta (FLT_exp emin prec) (Znearest choice) x))%R
apply Rlt_le.beta:radix
emin, prec:Z
Hp:(0 < prec)%Z
rnd:R -> Z
valid_rnd:Valid_rnd rnd
choice:Z -> bool
m:Z
x:=F2R {| Fnum := m; Fexp := emin |}:R
Hx:(Rabs x < bpow (emin + prec - 1))%R
(0 < / 2)%R
beta:radix
emin, prec:Z
Hp:(0 < prec)%Z
rnd:R -> Z
valid_rnd:Valid_rnd rnd
choice:Z -> bool
m:Z
x:=F2R {| Fnum := m; Fexp := emin |}:R
Hx:(Rabs x < bpow (emin + prec - 1))%R
(0 <= bpow (- prec + 1))%R
beta:radix
emin, prec:Z
Hp:(0 < prec)%Z
rnd:R -> Z
valid_rnd:Valid_rnd rnd
choice:Z -> bool
m:Z
x:=F2R {| Fnum := m; Fexp := emin |}:R
Hx:(Rabs x < bpow (emin + prec - 1))%R
(0 <= Rabs x)%R
beta:radix
emin, prec:Z
Hp:(0 < prec)%Z
rnd:R -> Z
valid_rnd:Valid_rnd rnd
choice:Z -> bool
m:Z
x:=F2R {| Fnum := m; Fexp := emin |}:R
Hx:(Rabs x < bpow (emin + prec - 1))%R
generic_format beta (FLT_exp emin prec) x
beta:radix
emin, prec:Z
Hp:(0 < prec)%Z
rnd:R -> Z
valid_rnd:Valid_rnd rnd
choice:Z -> bool
m:Z
x:=F2R {| Fnum := m; Fexp := emin |}:R
Hx:(bpow (emin + prec - 1) <= Rabs x)%R
(Rabs
   (round beta (FLT_exp emin prec) (Znearest choice) x -
    x) <=
 / 2 * bpow (- prec + 1) *
 Rabs
   (round beta (FLT_exp emin prec) (Znearest choice) x))%R
apply (RinvN_pos 1).beta:radix
emin, prec:Z
Hp:(0 < prec)%Z
rnd:R -> Z
valid_rnd:Valid_rnd rnd
choice:Z -> bool
m:Z
x:=F2R {| Fnum := m; Fexp := emin |}:R
Hx:(Rabs x < bpow (emin + prec - 1))%R
(0 <= bpow (- prec + 1))%R
beta:radix
emin, prec:Z
Hp:(0 < prec)%Z
rnd:R -> Z
valid_rnd:Valid_rnd rnd
choice:Z -> bool
m:Z
x:=F2R {| Fnum := m; Fexp := emin |}:R
Hx:(Rabs x < bpow (emin + prec - 1))%R
(0 <= Rabs x)%R
beta:radix
emin, prec:Z
Hp:(0 < prec)%Z
rnd:R -> Z
valid_rnd:Valid_rnd rnd
choice:Z -> bool
m:Z
x:=F2R {| Fnum := m; Fexp := emin |}:R
Hx:(Rabs x < bpow (emin + prec - 1))%R
generic_format beta (FLT_exp emin prec) x
beta:radix
emin, prec:Z
Hp:(0 < prec)%Z
rnd:R -> Z
valid_rnd:Valid_rnd rnd
choice:Z -> bool
m:Z
x:=F2R {| Fnum := m; Fexp := emin |}:R
Hx:(bpow (emin + prec - 1) <= Rabs x)%R
(Rabs
   (round beta (FLT_exp emin prec) (Znearest choice) x -
    x) <=
 / 2 * bpow (- prec + 1) *
 Rabs
   (round beta (FLT_exp emin prec) (Znearest choice) x))%R
apply bpow_ge_0.beta:radix
emin, prec:Z
Hp:(0 < prec)%Z
rnd:R -> Z
valid_rnd:Valid_rnd rnd
choice:Z -> bool
m:Z
x:=F2R {| Fnum := m; Fexp := emin |}:R
Hx:(Rabs x < bpow (emin + prec - 1))%R
(0 <= Rabs x)%R
beta:radix
emin, prec:Z
Hp:(0 < prec)%Z
rnd:R -> Z
valid_rnd:Valid_rnd rnd
choice:Z -> bool
m:Z
x:=F2R {| Fnum := m; Fexp := emin |}:R
Hx:(Rabs x < bpow (emin + prec - 1))%R
generic_format beta (FLT_exp emin prec) x
beta:radix
emin, prec:Z
Hp:(0 < prec)%Z
rnd:R -> Z
valid_rnd:Valid_rnd rnd
choice:Z -> bool
m:Z
x:=F2R {| Fnum := m; Fexp := emin |}:R
Hx:(bpow (emin + prec - 1) <= Rabs x)%R
(Rabs
   (round beta (FLT_exp emin prec) (Znearest choice) x -
    x) <=
 / 2 * bpow (- prec + 1) *
 Rabs
   (round beta (FLT_exp emin prec) (Znearest choice) x))%R
apply Rabs_pos.beta:radix
emin, prec:Z
Hp:(0 < prec)%Z
rnd:R -> Z
valid_rnd:Valid_rnd rnd
choice:Z -> bool
m:Z
x:=F2R {| Fnum := m; Fexp := emin |}:R
Hx:(Rabs x < bpow (emin + prec - 1))%R
generic_format beta (FLT_exp emin prec) x
beta:radix
emin, prec:Z
Hp:(0 < prec)%Z
rnd:R -> Z
valid_rnd:Valid_rnd rnd
choice:Z -> bool
m:Z
x:=F2R {| Fnum := m; Fexp := emin |}:R
Hx:(bpow (emin + prec - 1) <= Rabs x)%R
(Rabs
   (round beta (FLT_exp emin prec) (Znearest choice) x -
    x) <=
 / 2 * bpow (- prec + 1) *
 Rabs
   (round beta (FLT_exp emin prec) (Znearest choice) x))%R
apply generic_format_FLT_FIX...beta:radix
emin, prec:Z
Hp:(0 < prec)%Z
rnd:R -> Z
valid_rnd:Valid_rnd rnd
choice:Z -> bool
m:Z
x:=F2R {| Fnum := m; Fexp := emin |}:R
Hx:(Rabs x < bpow (emin + prec - 1))%R
(Rabs x <= bpow (emin + prec))%R
beta:radix
emin, prec:Z
Hp:(0 < prec)%Z
rnd:R -> Z
valid_rnd:Valid_rnd rnd
choice:Z -> bool
m:Z
x:=F2R {| Fnum := m; Fexp := emin |}:R
Hx:(Rabs x < bpow (emin + prec - 1))%R
generic_format beta (FIX_exp emin) x
beta:radix
emin, prec:Z
Hp:(0 < prec)%Z
rnd:R -> Z
valid_rnd:Valid_rnd rnd
choice:Z -> bool
m:Z
x:=F2R {| Fnum := m; Fexp := emin |}:R
Hx:(bpow (emin + prec - 1) <= Rabs x)%R
(Rabs
   (round beta (FLT_exp emin prec) (Znearest choice) x -
    x) <=
 / 2 * bpow (- prec + 1) *
 Rabs
   (round beta (FLT_exp emin prec) (Znearest choice) x))%R
apply Rlt_le.beta:radix
emin, prec:Z
Hp:(0 < prec)%Z
rnd:R -> Z
valid_rnd:Valid_rnd rnd
choice:Z -> bool
m:Z
x:=F2R {| Fnum := m; Fexp := emin |}:R
Hx:(Rabs x < bpow (emin + prec - 1))%R
(Rabs x < bpow (emin + prec))%R
beta:radix
emin, prec:Z
Hp:(0 < prec)%Z
rnd:R -> Z
valid_rnd:Valid_rnd rnd
choice:Z -> bool
m:Z
x:=F2R {| Fnum := m; Fexp := emin |}:R
Hx:(Rabs x < bpow (emin + prec - 1))%R
generic_format beta (FIX_exp emin) x
beta:radix
emin, prec:Z
Hp:(0 < prec)%Z
rnd:R -> Z
valid_rnd:Valid_rnd rnd
choice:Z -> bool
m:Z
x:=F2R {| Fnum := m; Fexp := emin |}:R
Hx:(bpow (emin + prec - 1) <= Rabs x)%R
(Rabs
   (round beta (FLT_exp emin prec) (Znearest choice) x -
    x) <=
 / 2 * bpow (- prec + 1) *
 Rabs
   (round beta (FLT_exp emin prec) (Znearest choice) x))%R
apply Rlt_le_trans with (1 := Hx).beta:radix
emin, prec:Z
Hp:(0 < prec)%Z
rnd:R -> Z
valid_rnd:Valid_rnd rnd
choice:Z -> bool
m:Z
x:=F2R {| Fnum := m; Fexp := emin |}:R
Hx:(Rabs x < bpow (emin + prec - 1))%R
(bpow (emin + prec - 1) <= bpow (emin + prec))%R
beta:radix
emin, prec:Z
Hp:(0 < prec)%Z
rnd:R -> Z
valid_rnd:Valid_rnd rnd
choice:Z -> bool
m:Z
x:=F2R {| Fnum := m; Fexp := emin |}:R
Hx:(Rabs x < bpow (emin + prec - 1))%R
generic_format beta (FIX_exp emin) x
beta:radix
emin, prec:Z
Hp:(0 < prec)%Z
rnd:R -> Z
valid_rnd:Valid_rnd rnd
choice:Z -> bool
m:Z
x:=F2R {| Fnum := m; Fexp := emin |}:R
Hx:(bpow (emin + prec - 1) <= Rabs x)%R
(Rabs
   (round beta (FLT_exp emin prec) (Znearest choice) x -
    x) <=
 / 2 * bpow (- prec + 1) *
 Rabs
   (round beta (FLT_exp emin prec) (Znearest choice) x))%R
apply bpow_le.beta:radix
emin, prec:Z
Hp:(0 < prec)%Z
rnd:R -> Z
valid_rnd:Valid_rnd rnd
choice:Z -> bool
m:Z
x:=F2R {| Fnum := m; Fexp := emin |}:R
Hx:(Rabs x < bpow (emin + prec - 1))%R
(emin + prec - 1 <= emin + prec)%Z
beta:radix
emin, prec:Z
Hp:(0 < prec)%Z
rnd:R -> Z
valid_rnd:Valid_rnd rnd
choice:Z -> bool
m:Z
x:=F2R {| Fnum := m; Fexp := emin |}:R
Hx:(Rabs x < bpow (emin + prec - 1))%R
generic_format beta (FIX_exp emin) x
beta:radix
emin, prec:Z
Hp:(0 < prec)%Z
rnd:R -> Z
valid_rnd:Valid_rnd rnd
choice:Z -> bool
m:Z
x:=F2R {| Fnum := m; Fexp := emin |}:R
Hx:(bpow (emin + prec - 1) <= Rabs x)%R
(Rabs
   (round beta (FLT_exp emin prec) (Znearest choice) x -
    x) <=
 / 2 * bpow (- prec + 1) *
 Rabs
   (round beta (FLT_exp emin prec) (Znearest choice) x))%R
apply Zle_pred.beta:radix
emin, prec:Z
Hp:(0 < prec)%Z
rnd:R -> Z
valid_rnd:Valid_rnd rnd
choice:Z -> bool
m:Z
x:=F2R {| Fnum := m; Fexp := emin |}:R
Hx:(Rabs x < bpow (emin + prec - 1))%R
generic_format beta (FIX_exp emin) x
beta:radix
emin, prec:Z
Hp:(0 < prec)%Z
rnd:R -> Z
valid_rnd:Valid_rnd rnd
choice:Z -> bool
m:Z
x:=F2R {| Fnum := m; Fexp := emin |}:R
Hx:(bpow (emin + prec - 1) <= Rabs x)%R
(Rabs
   (round beta (FLT_exp emin prec) (Znearest choice) x -
    x) <=
 / 2 * bpow (- prec + 1) *
 Rabs
   (round beta (FLT_exp emin prec) (Znearest choice) x))%R
apply generic_format_FIX.beta:radix
emin, prec:Z
Hp:(0 < prec)%Z
rnd:R -> Z
valid_rnd:Valid_rnd rnd
choice:Z -> bool
m:Z
x:=F2R {| Fnum := m; Fexp := emin |}:R
Hx:(Rabs x < bpow (emin + prec - 1))%R
FIX_format beta emin x
beta:radix
emin, prec:Z
Hp:(0 < prec)%Z
rnd:R -> Z
valid_rnd:Valid_rnd rnd
choice:Z -> bool
m:Z
x:=F2R {| Fnum := m; Fexp := emin |}:R
Hx:(bpow (emin + prec - 1) <= Rabs x)%R
(Rabs
   (round beta (FLT_exp emin prec) (Znearest choice) x -
    x) <=
 / 2 * bpow (- prec + 1) *
 Rabs
   (round beta (FLT_exp emin prec) (Znearest choice) x))%R
now exists (Float beta m emin).beta:radix
emin, prec:Z
Hp:(0 < prec)%Z
rnd:R -> Z
valid_rnd:Valid_rnd rnd
choice:Z -> bool
m:Z
x:=F2R {| Fnum := m; Fexp := emin |}:R
Hx:(bpow (emin + prec - 1) <= Rabs x)%R
(Rabs
   (round beta (FLT_exp emin prec) (Znearest choice) x -
    x) <=
 / 2 * bpow (- prec + 1) *
 Rabs
   (round beta (FLT_exp emin prec) (Znearest choice) x))%R
apply relative_error_N_round with (emin := (emin + prec - 1)%Z)...beta:radix
emin, prec:Z
Hp:(0 < prec)%Z
rnd:R -> Z
valid_rnd:Valid_rnd rnd
choice:Z -> bool
m:Z
x:=F2R {| Fnum := m; Fexp := emin |}:R
Hx:(bpow (emin + prec - 1) <= Rabs x)%R
forall k : Z,
(emin + prec - 1 < k)%Z ->
(prec <= k - FLT_exp emin prec k)%Z
apply relative_error_FLT_aux.
Qed.

Lemma error_N_FLT_aux :
  forall x,
  (0 < x)%R ->
  exists eps, exists  eta,
  (Rabs eps <= /2 * bpow (-prec + 1))%R /\
  (Rabs eta <= /2 * bpow (emin))%R      /\
  (eps*eta=0)%R /\
  round beta (FLT_exp emin prec) (Znearest choice) x = (x * (1 + eps) + eta)%R.beta:radix
emin, prec:Z
Hp:(0 < prec)%Z
rnd:R -> Z
valid_rnd:Valid_rnd rnd
choice:Z -> bool
forall x : R,
(0 < x)%R ->
exists eps eta : R,
  (Rabs eps <= / 2 * bpow (- prec + 1))%R /\
  (Rabs eta <= / 2 * bpow emin)%R /\
  (eps * eta)%R = 0%R /\
  round beta (FLT_exp emin prec) (Znearest choice) x =
  (x * (1 + eps) + eta)%R
Proof.beta:radix
emin, prec:Z
Hp:(0 < prec)%Z
rnd:R -> Z
valid_rnd:Valid_rnd rnd
choice:Z -> bool
forall x : R,
(0 < x)%R ->
exists eps eta : R,
  (Rabs eps <= / 2 * bpow (- prec + 1))%R /\
  (Rabs eta <= / 2 * bpow emin)%R /\
  (eps * eta)%R = 0%R /\
  round beta (FLT_exp emin prec) (Znearest choice) x =
  (x * (1 + eps) + eta)%R
intros x Hx2.beta:radix
emin, prec:Z
Hp:(0 < prec)%Z
rnd:R -> Z
valid_rnd:Valid_rnd rnd
choice:Z -> bool
x:R
Hx2:(0 < x)%R
exists eps eta : R,
  (Rabs eps <= / 2 * bpow (- prec + 1))%R /\
  (Rabs eta <= / 2 * bpow emin)%R /\
  (eps * eta)%R = 0%R /\
  round beta (FLT_exp emin prec) (Znearest choice) x =
  (x * (1 + eps) + eta)%R
case (Rle_or_lt (bpow (emin+prec)) x); intros Hx.beta:radix
emin, prec:Z
Hp:(0 < prec)%Z
rnd:R -> Z
valid_rnd:Valid_rnd rnd
choice:Z -> bool
x:R
Hx2:(0 < x)%R
Hx:(bpow (emin + prec) <= x)%R
exists eps eta : R,
  (Rabs eps <= / 2 * bpow (- prec + 1))%R /\
  (Rabs eta <= / 2 * bpow emin)%R /\
  (eps * eta)%R = 0%R /\
  round beta (FLT_exp emin prec) (Znearest choice) x =
  (x * (1 + eps) + eta)%R
beta:radix
emin, prec:Z
Hp:(0 < prec)%Z
rnd:R -> Z
valid_rnd:Valid_rnd rnd
choice:Z -> bool
x:R
Hx2:(0 < x)%R
Hx:(x < bpow (emin + prec))%R
exists eps eta : R,
  (Rabs eps <= / 2 * bpow (- prec + 1))%R /\
  (Rabs eta <= / 2 * bpow emin)%R /\
  (eps * eta)%R = 0%R /\
  round beta (FLT_exp emin prec) (Znearest choice) x =
  (x * (1 + eps) + eta)%R
(* *)
destruct relative_error_N_ex with (FLT_exp emin prec) (emin+prec)%Z prec choice x
  as (eps,(Heps1,Heps2)).beta:radix
emin, prec:Z
Hp:(0 < prec)%Z
rnd:R -> Z
valid_rnd:Valid_rnd rnd
choice:Z -> bool
x:R
Hx2:(0 < x)%R
Hx:(bpow (emin + prec) <= x)%R
Valid_exp (FLT_exp emin prec)
beta:radix
emin, prec:Z
Hp:(0 < prec)%Z
rnd:R -> Z
valid_rnd:Valid_rnd rnd
choice:Z -> bool
x:R
Hx2:(0 < x)%R
Hx:(bpow (emin + prec) <= x)%R
forall k : Z,
(emin + prec < k)%Z ->
(prec <= k - FLT_exp emin prec k)%Z
beta:radix
emin, prec:Z
Hp:(0 < prec)%Z
rnd:R -> Z
valid_rnd:Valid_rnd rnd
choice:Z -> bool
x:R
Hx2:(0 < x)%R
Hx:(bpow (emin + prec) <= x)%R
(bpow (emin + prec) <= Rabs x)%R
beta:radix
emin, prec:Z
Hp:(0 < prec)%Z
rnd:R -> Z
valid_rnd:Valid_rnd rnd
choice:Z -> bool
x:R
Hx2:(0 < x)%R
Hx:(bpow (emin + prec) <= x)%R
eps:R
Heps1:(Rabs eps <= / 2 * bpow (- prec + 1))%R
Heps2:round beta (FLT_exp emin prec)
  (Znearest choice) x = (x * (1 + eps))%R
exists eps0 eta : R,
  (Rabs eps0 <= / 2 * bpow (- prec + 1))%R /\
  (Rabs eta <= / 2 * bpow emin)%R /\
  (eps0 * eta)%R = 0%R /\
  round beta (FLT_exp emin prec) (Znearest choice) x =
  (x * (1 + eps0) + eta)%R
beta:radix
emin, prec:Z
Hp:(0 < prec)%Z
rnd:R -> Z
valid_rnd:Valid_rnd rnd
choice:Z -> bool
x:R
Hx2:(0 < x)%R
Hx:(x < bpow (emin + prec))%R
exists eps eta : R,
  (Rabs eps <= / 2 * bpow (- prec + 1))%R /\
  (Rabs eta <= / 2 * bpow emin)%R /\
  (eps * eta)%R = 0%R /\
  round beta (FLT_exp emin prec) (Znearest choice) x =
  (x * (1 + eps) + eta)%R
now apply FLT_exp_valid.beta:radix
emin, prec:Z
Hp:(0 < prec)%Z
rnd:R -> Z
valid_rnd:Valid_rnd rnd
choice:Z -> bool
x:R
Hx2:(0 < x)%R
Hx:(bpow (emin + prec) <= x)%R
forall k : Z,
(emin + prec < k)%Z ->
(prec <= k - FLT_exp emin prec k)%Z
beta:radix
emin, prec:Z
Hp:(0 < prec)%Z
rnd:R -> Z
valid_rnd:Valid_rnd rnd
choice:Z -> bool
x:R
Hx2:(0 < x)%R
Hx:(bpow (emin + prec) <= x)%R
(bpow (emin + prec) <= Rabs x)%R
beta:radix
emin, prec:Z
Hp:(0 < prec)%Z
rnd:R -> Z
valid_rnd:Valid_rnd rnd
choice:Z -> bool
x:R
Hx2:(0 < x)%R
Hx:(bpow (emin + prec) <= x)%R
eps:R
Heps1:(Rabs eps <= / 2 * bpow (- prec + 1))%R
Heps2:round beta (FLT_exp emin prec)
  (Znearest choice) x = (x * (1 + eps))%R
exists eps0 eta : R,
  (Rabs eps0 <= / 2 * bpow (- prec + 1))%R /\
  (Rabs eta <= / 2 * bpow emin)%R /\
  (eps0 * eta)%R = 0%R /\
  round beta (FLT_exp emin prec) (Znearest choice) x =
  (x * (1 + eps0) + eta)%R
beta:radix
emin, prec:Z
Hp:(0 < prec)%Z
rnd:R -> Z
valid_rnd:Valid_rnd rnd
choice:Z -> bool
x:R
Hx2:(0 < x)%R
Hx:(x < bpow (emin + prec))%R
exists eps eta : R,
  (Rabs eps <= / 2 * bpow (- prec + 1))%R /\
  (Rabs eta <= / 2 * bpow emin)%R /\
  (eps * eta)%R = 0%R /\
  round beta (FLT_exp emin prec) (Znearest choice) x =
  (x * (1 + eps) + eta)%R
intros; unfold FLT_exp.beta:radix
emin, prec:Z
Hp:(0 < prec)%Z
rnd:R -> Z
valid_rnd:Valid_rnd rnd
choice:Z -> bool
x:R
Hx2:(0 < x)%R
Hx:(bpow (emin + prec) <= x)%R
k:Z
H:(emin + prec < k)%Z
(prec <= k - Z.max (k - prec) emin)%Z
beta:radix
emin, prec:Z
Hp:(0 < prec)%Z
rnd:R -> Z
valid_rnd:Valid_rnd rnd
choice:Z -> bool
x:R
Hx2:(0 < x)%R
Hx:(bpow (emin + prec) <= x)%R
(bpow (emin + prec) <= Rabs x)%R
beta:radix
emin, prec:Z
Hp:(0 < prec)%Z
rnd:R -> Z
valid_rnd:Valid_rnd rnd
choice:Z -> bool
x:R
Hx2:(0 < x)%R
Hx:(bpow (emin + prec) <= x)%R
eps:R
Heps1:(Rabs eps <= / 2 * bpow (- prec + 1))%R
Heps2:round beta (FLT_exp emin prec)
  (Znearest choice) x = (x * (1 + eps))%R
exists eps0 eta : R,
  (Rabs eps0 <= / 2 * bpow (- prec + 1))%R /\
  (Rabs eta <= / 2 * bpow emin)%R /\
  (eps0 * eta)%R = 0%R /\
  round beta (FLT_exp emin prec) (Znearest choice) x =
  (x * (1 + eps0) + eta)%R
beta:radix
emin, prec:Z
Hp:(0 < prec)%Z
rnd:R -> Z
valid_rnd:Valid_rnd rnd
choice:Z -> bool
x:R
Hx2:(0 < x)%R
Hx:(x < bpow (emin + prec))%R
exists eps eta : R,
  (Rabs eps <= / 2 * bpow (- prec + 1))%R /\
  (Rabs eta <= / 2 * bpow emin)%R /\
  (eps * eta)%R = 0%R /\
  round beta (FLT_exp emin prec) (Znearest choice) x =
  (x * (1 + eps) + eta)%R
rewrite Zmax_left; omega.beta:radix
emin, prec:Z
Hp:(0 < prec)%Z
rnd:R -> Z
valid_rnd:Valid_rnd rnd
choice:Z -> bool
x:R
Hx2:(0 < x)%R
Hx:(bpow (emin + prec) <= x)%R
(bpow (emin + prec) <= Rabs x)%R
beta:radix
emin, prec:Z
Hp:(0 < prec)%Z
rnd:R -> Z
valid_rnd:Valid_rnd rnd
choice:Z -> bool
x:R
Hx2:(0 < x)%R
Hx:(bpow (emin + prec) <= x)%R
eps:R
Heps1:(Rabs eps <= / 2 * bpow (- prec + 1))%R
Heps2:round beta (FLT_exp emin prec)
  (Znearest choice) x = (x * (1 + eps))%R
exists eps0 eta : R,
  (Rabs eps0 <= / 2 * bpow (- prec + 1))%R /\
  (Rabs eta <= / 2 * bpow emin)%R /\
  (eps0 * eta)%R = 0%R /\
  round beta (FLT_exp emin prec) (Znearest choice) x =
  (x * (1 + eps0) + eta)%R
beta:radix
emin, prec:Z
Hp:(0 < prec)%Z
rnd:R -> Z
valid_rnd:Valid_rnd rnd
choice:Z -> bool
x:R
Hx2:(0 < x)%R
Hx:(x < bpow (emin + prec))%R
exists eps eta : R,
  (Rabs eps <= / 2 * bpow (- prec + 1))%R /\
  (Rabs eta <= / 2 * bpow emin)%R /\
  (eps * eta)%R = 0%R /\
  round beta (FLT_exp emin prec) (Znearest choice) x =
  (x * (1 + eps) + eta)%R
rewrite Rabs_right;[assumption|apply Rle_ge; now left].beta:radix
emin, prec:Z
Hp:(0 < prec)%Z
rnd:R -> Z
valid_rnd:Valid_rnd rnd
choice:Z -> bool
x:R
Hx2:(0 < x)%R
Hx:(bpow (emin + prec) <= x)%R
eps:R
Heps1:(Rabs eps <= / 2 * bpow (- prec + 1))%R
Heps2:round beta (FLT_exp emin prec)
  (Znearest choice) x = (x * (1 + eps))%R
exists eps0 eta : R,
  (Rabs eps0 <= / 2 * bpow (- prec + 1))%R /\
  (Rabs eta <= / 2 * bpow emin)%R /\
  (eps0 * eta)%R = 0%R /\
  round beta (FLT_exp emin prec) (Znearest choice) x =
  (x * (1 + eps0) + eta)%R
beta:radix
emin, prec:Z
Hp:(0 < prec)%Z
rnd:R -> Z
valid_rnd:Valid_rnd rnd
choice:Z -> bool
x:R
Hx2:(0 < x)%R
Hx:(x < bpow (emin + prec))%R
exists eps eta : R,
  (Rabs eps <= / 2 * bpow (- prec + 1))%R /\
  (Rabs eta <= / 2 * bpow emin)%R /\
  (eps * eta)%R = 0%R /\
  round beta (FLT_exp emin prec) (Znearest choice) x =
  (x * (1 + eps) + eta)%R
exists eps; exists 0%R.beta:radix
emin, prec:Z
Hp:(0 < prec)%Z
rnd:R -> Z
valid_rnd:Valid_rnd rnd
choice:Z -> bool
x:R
Hx2:(0 < x)%R
Hx:(bpow (emin + prec) <= x)%R
eps:R
Heps1:(Rabs eps <= / 2 * bpow (- prec + 1))%R
Heps2:round beta (FLT_exp emin prec)
  (Znearest choice) x = (x * (1 + eps))%R
(Rabs eps <= / 2 * bpow (- prec + 1))%R /\
(Rabs 0 <= / 2 * bpow emin)%R /\
(eps * 0)%R = 0%R /\
round beta (FLT_exp emin prec) (Znearest choice) x =
(x * (1 + eps) + 0)%R
beta:radix
emin, prec:Z
Hp:(0 < prec)%Z
rnd:R -> Z
valid_rnd:Valid_rnd rnd
choice:Z -> bool
x:R
Hx2:(0 < x)%R
Hx:(x < bpow (emin + prec))%R
exists eps eta : R,
  (Rabs eps <= / 2 * bpow (- prec + 1))%R /\
  (Rabs eta <= / 2 * bpow emin)%R /\
  (eps * eta)%R = 0%R /\
  round beta (FLT_exp emin prec) (Znearest choice) x =
  (x * (1 + eps) + eta)%R
split;[assumption|split].beta:radix
emin, prec:Z
Hp:(0 < prec)%Z
rnd:R -> Z
valid_rnd:Valid_rnd rnd
choice:Z -> bool
x:R
Hx2:(0 < x)%R
Hx:(bpow (emin + prec) <= x)%R
eps:R
Heps1:(Rabs eps <= / 2 * bpow (- prec + 1))%R
Heps2:round beta (FLT_exp emin prec)
  (Znearest choice) x = (x * (1 + eps))%R
(Rabs 0 <= / 2 * bpow emin)%R
beta:radix
emin, prec:Z
Hp:(0 < prec)%Z
rnd:R -> Z
valid_rnd:Valid_rnd rnd
choice:Z -> bool
x:R
Hx2:(0 < x)%R
Hx:(bpow (emin + prec) <= x)%R
eps:R
Heps1:(Rabs eps <= / 2 * bpow (- prec + 1))%R
Heps2:round beta (FLT_exp emin prec)
  (Znearest choice) x = (x * (1 + eps))%R
(eps * 0)%R = 0%R /\
round beta (FLT_exp emin prec) (Znearest choice) x =
(x * (1 + eps) + 0)%R
beta:radix
emin, prec:Z
Hp:(0 < prec)%Z
rnd:R -> Z
valid_rnd:Valid_rnd rnd
choice:Z -> bool
x:R
Hx2:(0 < x)%R
Hx:(x < bpow (emin + prec))%R
exists eps eta : R,
  (Rabs eps <= / 2 * bpow (- prec + 1))%R /\
  (Rabs eta <= / 2 * bpow emin)%R /\
  (eps * eta)%R = 0%R /\
  round beta (FLT_exp emin prec) (Znearest choice) x =
  (x * (1 + eps) + eta)%R
rewrite Rabs_R0; apply Rmult_le_pos.beta:radix
emin, prec:Z
Hp:(0 < prec)%Z
rnd:R -> Z
valid_rnd:Valid_rnd rnd
choice:Z -> bool
x:R
Hx2:(0 < x)%R
Hx:(bpow (emin + prec) <= x)%R
eps:R
Heps1:(Rabs eps <= / 2 * bpow (- prec + 1))%R
Heps2:round beta (FLT_exp emin prec)
  (Znearest choice) x = (x * (1 + eps))%R
(0 <= / 2)%R
beta:radix
emin, prec:Z
Hp:(0 < prec)%Z
rnd:R -> Z
valid_rnd:Valid_rnd rnd
choice:Z -> bool
x:R
Hx2:(0 < x)%R
Hx:(bpow (emin + prec) <= x)%R
eps:R
Heps1:(Rabs eps <= / 2 * bpow (- prec + 1))%R
Heps2:round beta (FLT_exp emin prec)
  (Znearest choice) x = (x * (1 + eps))%R
(0 <= bpow emin)%R
beta:radix
emin, prec:Z
Hp:(0 < prec)%Z
rnd:R -> Z
valid_rnd:Valid_rnd rnd
choice:Z -> bool
x:R
Hx2:(0 < x)%R
Hx:(bpow (emin + prec) <= x)%R
eps:R
Heps1:(Rabs eps <= / 2 * bpow (- prec + 1))%R
Heps2:round beta (FLT_exp emin prec)
  (Znearest choice) x = (x * (1 + eps))%R
(eps * 0)%R = 0%R /\
round beta (FLT_exp emin prec) (Znearest choice) x =
(x * (1 + eps) + 0)%R
beta:radix
emin, prec:Z
Hp:(0 < prec)%Z
rnd:R -> Z
valid_rnd:Valid_rnd rnd
choice:Z -> bool
x:R
Hx2:(0 < x)%R
Hx:(x < bpow (emin + prec))%R
exists eps eta : R,
  (Rabs eps <= / 2 * bpow (- prec + 1))%R /\
  (Rabs eta <= / 2 * bpow emin)%R /\
  (eps * eta)%R = 0%R /\
  round beta (FLT_exp emin prec) (Znearest choice) x =
  (x * (1 + eps) + eta)%R
apply Rlt_le, pos_half_prf.beta:radix
emin, prec:Z
Hp:(0 < prec)%Z
rnd:R -> Z
valid_rnd:Valid_rnd rnd
choice:Z -> bool
x:R
Hx2:(0 < x)%R
Hx:(bpow (emin + prec) <= x)%R
eps:R
Heps1:(Rabs eps <= / 2 * bpow (- prec + 1))%R
Heps2:round beta (FLT_exp emin prec)
  (Znearest choice) x = (x * (1 + eps))%R
(0 <= bpow emin)%R
beta:radix
emin, prec:Z
Hp:(0 < prec)%Z
rnd:R -> Z
valid_rnd:Valid_rnd rnd
choice:Z -> bool
x:R
Hx2:(0 < x)%R
Hx:(bpow (emin + prec) <= x)%R
eps:R
Heps1:(Rabs eps <= / 2 * bpow (- prec + 1))%R
Heps2:round beta (FLT_exp emin prec)
  (Znearest choice) x = (x * (1 + eps))%R
(eps * 0)%R = 0%R /\
round beta (FLT_exp emin prec) (Znearest choice) x =
(x * (1 + eps) + 0)%R
beta:radix
emin, prec:Z
Hp:(0 < prec)%Z
rnd:R -> Z
valid_rnd:Valid_rnd rnd
choice:Z -> bool
x:R
Hx2:(0 < x)%R
Hx:(x < bpow (emin + prec))%R
exists eps eta : R,
  (Rabs eps <= / 2 * bpow (- prec + 1))%R /\
  (Rabs eta <= / 2 * bpow emin)%R /\
  (eps * eta)%R = 0%R /\
  round beta (FLT_exp emin prec) (Znearest choice) x =
  (x * (1 + eps) + eta)%R
apply bpow_ge_0.beta:radix
emin, prec:Z
Hp:(0 < prec)%Z
rnd:R -> Z
valid_rnd:Valid_rnd rnd
choice:Z -> bool
x:R
Hx2:(0 < x)%R
Hx:(bpow (emin + prec) <= x)%R
eps:R
Heps1:(Rabs eps <= / 2 * bpow (- prec + 1))%R
Heps2:round beta (FLT_exp emin prec)
  (Znearest choice) x = (x * (1 + eps))%R
(eps * 0)%R = 0%R /\
round beta (FLT_exp emin prec) (Znearest choice) x =
(x * (1 + eps) + 0)%R
beta:radix
emin, prec:Z
Hp:(0 < prec)%Z
rnd:R -> Z
valid_rnd:Valid_rnd rnd
choice:Z -> bool
x:R
Hx2:(0 < x)%R
Hx:(x < bpow (emin + prec))%R
exists eps eta : R,
  (Rabs eps <= / 2 * bpow (- prec + 1))%R /\
  (Rabs eta <= / 2 * bpow emin)%R /\
  (eps * eta)%R = 0%R /\
  round beta (FLT_exp emin prec) (Znearest choice) x =
  (x * (1 + eps) + eta)%R
split;[apply Rmult_0_r|idtac].beta:radix
emin, prec:Z
Hp:(0 < prec)%Z
rnd:R -> Z
valid_rnd:Valid_rnd rnd
choice:Z -> bool
x:R
Hx2:(0 < x)%R
Hx:(bpow (emin + prec) <= x)%R
eps:R
Heps1:(Rabs eps <= / 2 * bpow (- prec + 1))%R
Heps2:round beta (FLT_exp emin prec)
  (Znearest choice) x = (x * (1 + eps))%R
round beta (FLT_exp emin prec) (Znearest choice) x =
(x * (1 + eps) + 0)%R
beta:radix
emin, prec:Z
Hp:(0 < prec)%Z
rnd:R -> Z
valid_rnd:Valid_rnd rnd
choice:Z -> bool
x:R
Hx2:(0 < x)%R
Hx:(x < bpow (emin + prec))%R
exists eps eta : R,
  (Rabs eps <= / 2 * bpow (- prec + 1))%R /\
  (Rabs eta <= / 2 * bpow emin)%R /\
  (eps * eta)%R = 0%R /\
  round beta (FLT_exp emin prec) (Znearest choice) x =
  (x * (1 + eps) + eta)%R
now rewrite Rplus_0_r.beta:radix
emin, prec:Z
Hp:(0 < prec)%Z
rnd:R -> Z
valid_rnd:Valid_rnd rnd
choice:Z -> bool
x:R
Hx2:(0 < x)%R
Hx:(x < bpow (emin + prec))%R
exists eps eta : R,
  (Rabs eps <= / 2 * bpow (- prec + 1))%R /\
  (Rabs eta <= / 2 * bpow emin)%R /\
  (eps * eta)%R = 0%R /\
  round beta (FLT_exp emin prec) (Znearest choice) x =
  (x * (1 + eps) + eta)%R
(* *)
exists 0%R; exists (round beta (FLT_exp emin prec) (Znearest choice) x - x)%R.beta:radix
emin, prec:Z
Hp:(0 < prec)%Z
rnd:R -> Z
valid_rnd:Valid_rnd rnd
choice:Z -> bool
x:R
Hx2:(0 < x)%R
Hx:(x < bpow (emin + prec))%R
(Rabs 0 <= / 2 * bpow (- prec + 1))%R /\
(Rabs
   (round beta (FLT_exp emin prec) (Znearest choice) x -
    x) <= / 2 * bpow emin)%R /\
(0 *
 (round beta (FLT_exp emin prec) (Znearest choice) x -
  x))%R = 0%R /\
round beta (FLT_exp emin prec) (Znearest choice) x =
(x * (1 + 0) +
 (round beta (FLT_exp emin prec) (Znearest choice) x -
  x))%R
split.beta:radix
emin, prec:Z
Hp:(0 < prec)%Z
rnd:R -> Z
valid_rnd:Valid_rnd rnd
choice:Z -> bool
x:R
Hx2:(0 < x)%R
Hx:(x < bpow (emin + prec))%R
(Rabs 0 <= / 2 * bpow (- prec + 1))%R
beta:radix
emin, prec:Z
Hp:(0 < prec)%Z
rnd:R -> Z
valid_rnd:Valid_rnd rnd
choice:Z -> bool
x:R
Hx2:(0 < x)%R
Hx:(x < bpow (emin + prec))%R
(Rabs
   (round beta (FLT_exp emin prec) (Znearest choice) x -
    x) <= / 2 * bpow emin)%R /\
(0 *
 (round beta (FLT_exp emin prec) (Znearest choice) x -
  x))%R = 0%R /\
round beta (FLT_exp emin prec) (Znearest choice) x =
(x * (1 + 0) +
 (round beta (FLT_exp emin prec) (Znearest choice) x -
  x))%R
rewrite Rabs_R0; apply Rmult_le_pos.beta:radix
emin, prec:Z
Hp:(0 < prec)%Z
rnd:R -> Z
valid_rnd:Valid_rnd rnd
choice:Z -> bool
x:R
Hx2:(0 < x)%R
Hx:(x < bpow (emin + prec))%R
(0 <= / 2)%R
beta:radix
emin, prec:Z
Hp:(0 < prec)%Z
rnd:R -> Z
valid_rnd:Valid_rnd rnd
choice:Z -> bool
x:R
Hx2:(0 < x)%R
Hx:(x < bpow (emin + prec))%R
(0 <= bpow (- prec + 1))%R
beta:radix
emin, prec:Z
Hp:(0 < prec)%Z
rnd:R -> Z
valid_rnd:Valid_rnd rnd
choice:Z -> bool
x:R
Hx2:(0 < x)%R
Hx:(x < bpow (emin + prec))%R
(Rabs
   (round beta (FLT_exp emin prec) (Znearest choice) x -
    x) <= / 2 * bpow emin)%R /\
(0 *
 (round beta (FLT_exp emin prec) (Znearest choice) x -
  x))%R = 0%R /\
round beta (FLT_exp emin prec) (Znearest choice) x =
(x * (1 + 0) +
 (round beta (FLT_exp emin prec) (Znearest choice) x -
  x))%R
apply Rlt_le, pos_half_prf.beta:radix
emin, prec:Z
Hp:(0 < prec)%Z
rnd:R -> Z
valid_rnd:Valid_rnd rnd
choice:Z -> bool
x:R
Hx2:(0 < x)%R
Hx:(x < bpow (emin + prec))%R
(0 <= bpow (- prec + 1))%R
beta:radix
emin, prec:Z
Hp:(0 < prec)%Z
rnd:R -> Z
valid_rnd:Valid_rnd rnd
choice:Z -> bool
x:R
Hx2:(0 < x)%R
Hx:(x < bpow (emin + prec))%R
(Rabs
   (round beta (FLT_exp emin prec) (Znearest choice) x -
    x) <= / 2 * bpow emin)%R /\
(0 *
 (round beta (FLT_exp emin prec) (Znearest choice) x -
  x))%R = 0%R /\
round beta (FLT_exp emin prec) (Znearest choice) x =
(x * (1 + 0) +
 (round beta (FLT_exp emin prec) (Znearest choice) x -
  x))%R
apply bpow_ge_0.beta:radix
emin, prec:Z
Hp:(0 < prec)%Z
rnd:R -> Z
valid_rnd:Valid_rnd rnd
choice:Z -> bool
x:R
Hx2:(0 < x)%R
Hx:(x < bpow (emin + prec))%R
(Rabs
   (round beta (FLT_exp emin prec) (Znearest choice) x -
    x) <= / 2 * bpow emin)%R /\
(0 *
 (round beta (FLT_exp emin prec) (Znearest choice) x -
  x))%R = 0%R /\
round beta (FLT_exp emin prec) (Znearest choice) x =
(x * (1 + 0) +
 (round beta (FLT_exp emin prec) (Znearest choice) x -
  x))%R
split.beta:radix
emin, prec:Z
Hp:(0 < prec)%Z
rnd:R -> Z
valid_rnd:Valid_rnd rnd
choice:Z -> bool
x:R
Hx2:(0 < x)%R
Hx:(x < bpow (emin + prec))%R
(Rabs
   (round beta (FLT_exp emin prec) (Znearest choice) x -
    x) <= / 2 * bpow emin)%R
beta:radix
emin, prec:Z
Hp:(0 < prec)%Z
rnd:R -> Z
valid_rnd:Valid_rnd rnd
choice:Z -> bool
x:R
Hx2:(0 < x)%R
Hx:(x < bpow (emin + prec))%R
(0 *
 (round beta (FLT_exp emin prec) (Znearest choice) x -
  x))%R = 0%R /\
round beta (FLT_exp emin prec) (Znearest choice) x =
(x * (1 + 0) +
 (round beta (FLT_exp emin prec) (Znearest choice) x -
  x))%R
apply Rle_trans with (/2*ulp beta (FLT_exp emin prec) x)%R.beta:radix
emin, prec:Z
Hp:(0 < prec)%Z
rnd:R -> Z
valid_rnd:Valid_rnd rnd
choice:Z -> bool
x:R
Hx2:(0 < x)%R
Hx:(x < bpow (emin + prec))%R
(Rabs
   (round beta (FLT_exp emin prec) (Znearest choice) x -
    x) <= / 2 * ulp beta (FLT_exp emin prec) x)%R
beta:radix
emin, prec:Z
Hp:(0 < prec)%Z
rnd:R -> Z
valid_rnd:Valid_rnd rnd
choice:Z -> bool
x:R
Hx2:(0 < x)%R
Hx:(x < bpow (emin + prec))%R
(/ 2 * ulp beta (FLT_exp emin prec) x <=
 / 2 * bpow emin)%R
beta:radix
emin, prec:Z
Hp:(0 < prec)%Z
rnd:R -> Z
valid_rnd:Valid_rnd rnd
choice:Z -> bool
x:R
Hx2:(0 < x)%R
Hx:(x < bpow (emin + prec))%R
(0 *
 (round beta (FLT_exp emin prec) (Znearest choice) x -
  x))%R = 0%R /\
round beta (FLT_exp emin prec) (Znearest choice) x =
(x * (1 + 0) +
 (round beta (FLT_exp emin prec) (Znearest choice) x -
  x))%R
apply error_le_half_ulp.beta:radix
emin, prec:Z
Hp:(0 < prec)%Z
rnd:R -> Z
valid_rnd:Valid_rnd rnd
choice:Z -> bool
x:R
Hx2:(0 < x)%R
Hx:(x < bpow (emin + prec))%R
Valid_exp (FLT_exp emin prec)
beta:radix
emin, prec:Z
Hp:(0 < prec)%Z
rnd:R -> Z
valid_rnd:Valid_rnd rnd
choice:Z -> bool
x:R
Hx2:(0 < x)%R
Hx:(x < bpow (emin + prec))%R
(/ 2 * ulp beta (FLT_exp emin prec) x <=
 / 2 * bpow emin)%R
beta:radix
emin, prec:Z
Hp:(0 < prec)%Z
rnd:R -> Z
valid_rnd:Valid_rnd rnd
choice:Z -> bool
x:R
Hx2:(0 < x)%R
Hx:(x < bpow (emin + prec))%R
(0 *
 (round beta (FLT_exp emin prec) (Znearest choice) x -
  x))%R = 0%R /\
round beta (FLT_exp emin prec) (Znearest choice) x =
(x * (1 + 0) +
 (round beta (FLT_exp emin prec) (Znearest choice) x -
  x))%R
now apply FLT_exp_valid.beta:radix
emin, prec:Z
Hp:(0 < prec)%Z
rnd:R -> Z
valid_rnd:Valid_rnd rnd
choice:Z -> bool
x:R
Hx2:(0 < x)%R
Hx:(x < bpow (emin + prec))%R
(/ 2 * ulp beta (FLT_exp emin prec) x <=
 / 2 * bpow emin)%R
beta:radix
emin, prec:Z
Hp:(0 < prec)%Z
rnd:R -> Z
valid_rnd:Valid_rnd rnd
choice:Z -> bool
x:R
Hx2:(0 < x)%R
Hx:(x < bpow (emin + prec))%R
(0 *
 (round beta (FLT_exp emin prec) (Znearest choice) x -
  x))%R = 0%R /\
round beta (FLT_exp emin prec) (Znearest choice) x =
(x * (1 + 0) +
 (round beta (FLT_exp emin prec) (Znearest choice) x -
  x))%R
apply Rmult_le_compat_l.beta:radix
emin, prec:Z
Hp:(0 < prec)%Z
rnd:R -> Z
valid_rnd:Valid_rnd rnd
choice:Z -> bool
x:R
Hx2:(0 < x)%R
Hx:(x < bpow (emin + prec))%R
(0 <= / 2)%R
beta:radix
emin, prec:Z
Hp:(0 < prec)%Z
rnd:R -> Z
valid_rnd:Valid_rnd rnd
choice:Z -> bool
x:R
Hx2:(0 < x)%R
Hx:(x < bpow (emin + prec))%R
(ulp beta (FLT_exp emin prec) x <= bpow emin)%R
beta:radix
emin, prec:Z
Hp:(0 < prec)%Z
rnd:R -> Z
valid_rnd:Valid_rnd rnd
choice:Z -> bool
x:R
Hx2:(0 < x)%R
Hx:(x < bpow (emin + prec))%R
(0 *
 (round beta (FLT_exp emin prec) (Znearest choice) x -
  x))%R = 0%R /\
round beta (FLT_exp emin prec) (Znearest choice) x =
(x * (1 + 0) +
 (round beta (FLT_exp emin prec) (Znearest choice) x -
  x))%R
apply Rlt_le, pos_half_prf.beta:radix
emin, prec:Z
Hp:(0 < prec)%Z
rnd:R -> Z
valid_rnd:Valid_rnd rnd
choice:Z -> bool
x:R
Hx2:(0 < x)%R
Hx:(x < bpow (emin + prec))%R
(ulp beta (FLT_exp emin prec) x <= bpow emin)%R
beta:radix
emin, prec:Z
Hp:(0 < prec)%Z
rnd:R -> Z
valid_rnd:Valid_rnd rnd
choice:Z -> bool
x:R
Hx2:(0 < x)%R
Hx:(x < bpow (emin + prec))%R
(0 *
 (round beta (FLT_exp emin prec) (Znearest choice) x -
  x))%R = 0%R /\
round beta (FLT_exp emin prec) (Znearest choice) x =
(x * (1 + 0) +
 (round beta (FLT_exp emin prec) (Znearest choice) x -
  x))%R
rewrite ulp_neq_0.beta:radix
emin, prec:Z
Hp:(0 < prec)%Z
rnd:R -> Z
valid_rnd:Valid_rnd rnd
choice:Z -> bool
x:R
Hx2:(0 < x)%R
Hx:(x < bpow (emin + prec))%R
(bpow (cexp beta (FLT_exp emin prec) x) <= bpow emin)%R
beta:radix
emin, prec:Z
Hp:(0 < prec)%Z
rnd:R -> Z
valid_rnd:Valid_rnd rnd
choice:Z -> bool
x:R
Hx2:(0 < x)%R
Hx:(x < bpow (emin + prec))%R
x <> 0%R
beta:radix
emin, prec:Z
Hp:(0 < prec)%Z
rnd:R -> Z
valid_rnd:Valid_rnd rnd
choice:Z -> bool
x:R
Hx2:(0 < x)%R
Hx:(x < bpow (emin + prec))%R
(0 *
 (round beta (FLT_exp emin prec) (Znearest choice) x -
  x))%R = 0%R /\
round beta (FLT_exp emin prec) (Znearest choice) x =
(x * (1 + 0) +
 (round beta (FLT_exp emin prec) (Znearest choice) x -
  x))%R
2: now apply Rgt_not_eq.beta:radix
emin, prec:Z
Hp:(0 < prec)%Z
rnd:R -> Z
valid_rnd:Valid_rnd rnd
choice:Z -> bool
x:R
Hx2:(0 < x)%R
Hx:(x < bpow (emin + prec))%R
(bpow (cexp beta (FLT_exp emin prec) x) <= bpow emin)%R
beta:radix
emin, prec:Z
Hp:(0 < prec)%Z
rnd:R -> Z
valid_rnd:Valid_rnd rnd
choice:Z -> bool
x:R
Hx2:(0 < x)%R
Hx:(x < bpow (emin + prec))%R
(0 *
 (round beta (FLT_exp emin prec) (Znearest choice) x -
  x))%R = 0%R /\
round beta (FLT_exp emin prec) (Znearest choice) x =
(x * (1 + 0) +
 (round beta (FLT_exp emin prec) (Znearest choice) x -
  x))%R
apply bpow_le.beta:radix
emin, prec:Z
Hp:(0 < prec)%Z
rnd:R -> Z
valid_rnd:Valid_rnd rnd
choice:Z -> bool
x:R
Hx2:(0 < x)%R
Hx:(x < bpow (emin + prec))%R
(cexp beta (FLT_exp emin prec) x <= emin)%Z
beta:radix
emin, prec:Z
Hp:(0 < prec)%Z
rnd:R -> Z
valid_rnd:Valid_rnd rnd
choice:Z -> bool
x:R
Hx2:(0 < x)%R
Hx:(x < bpow (emin + prec))%R
(0 *
 (round beta (FLT_exp emin prec) (Znearest choice) x -
  x))%R = 0%R /\
round beta (FLT_exp emin prec) (Znearest choice) x =
(x * (1 + 0) +
 (round beta (FLT_exp emin prec) (Znearest choice) x -
  x))%R
unfold FLT_exp, cexp.beta:radix
emin, prec:Z
Hp:(0 < prec)%Z
rnd:R -> Z
valid_rnd:Valid_rnd rnd
choice:Z -> bool
x:R
Hx2:(0 < x)%R
Hx:(x < bpow (emin + prec))%R
(Z.max (mag beta x - prec) emin <= emin)%Z
beta:radix
emin, prec:Z
Hp:(0 < prec)%Z
rnd:R -> Z
valid_rnd:Valid_rnd rnd
choice:Z -> bool
x:R
Hx2:(0 < x)%R
Hx:(x < bpow (emin + prec))%R
(0 *
 (round beta (FLT_exp emin prec) (Znearest choice) x -
  x))%R = 0%R /\
round beta (FLT_exp emin prec) (Znearest choice) x =
(x * (1 + 0) +
 (round beta (FLT_exp emin prec) (Znearest choice) x -
  x))%R
rewrite Zmax_right.beta:radix
emin, prec:Z
Hp:(0 < prec)%Z
rnd:R -> Z
valid_rnd:Valid_rnd rnd
choice:Z -> bool
x:R
Hx2:(0 < x)%R
Hx:(x < bpow (emin + prec))%R
(emin <= emin)%Z
beta:radix
emin, prec:Z
Hp:(0 < prec)%Z
rnd:R -> Z
valid_rnd:Valid_rnd rnd
choice:Z -> bool
x:R
Hx2:(0 < x)%R
Hx:(x < bpow (emin + prec))%R
(mag beta x - prec <= emin)%Z
beta:radix
emin, prec:Z
Hp:(0 < prec)%Z
rnd:R -> Z
valid_rnd:Valid_rnd rnd
choice:Z -> bool
x:R
Hx2:(0 < x)%R
Hx:(x < bpow (emin + prec))%R
(0 *
 (round beta (FLT_exp emin prec) (Znearest choice) x -
  x))%R = 0%R /\
round beta (FLT_exp emin prec) (Znearest choice) x =
(x * (1 + 0) +
 (round beta (FLT_exp emin prec) (Znearest choice) x -
  x))%R
omega.beta:radix
emin, prec:Z
Hp:(0 < prec)%Z
rnd:R -> Z
valid_rnd:Valid_rnd rnd
choice:Z -> bool
x:R
Hx2:(0 < x)%R
Hx:(x < bpow (emin + prec))%R
(mag beta x - prec <= emin)%Z
beta:radix
emin, prec:Z
Hp:(0 < prec)%Z
rnd:R -> Z
valid_rnd:Valid_rnd rnd
choice:Z -> bool
x:R
Hx2:(0 < x)%R
Hx:(x < bpow (emin + prec))%R
(0 *
 (round beta (FLT_exp emin prec) (Znearest choice) x -
  x))%R = 0%R /\
round beta (FLT_exp emin prec) (Znearest choice) x =
(x * (1 + 0) +
 (round beta (FLT_exp emin prec) (Znearest choice) x -
  x))%R
destruct (mag beta x) as (e,He); simpl.beta:radix
emin, prec:Z
Hp:(0 < prec)%Z
rnd:R -> Z
valid_rnd:Valid_rnd rnd
choice:Z -> bool
x:R
Hx2:(0 < x)%R
Hx:(x < bpow (emin + prec))%R
e:Z
He:x <> 0%R -> (bpow (e - 1) <= Rabs x < bpow e)%R
(e - prec <= emin)%Z
beta:radix
emin, prec:Z
Hp:(0 < prec)%Z
rnd:R -> Z
valid_rnd:Valid_rnd rnd
choice:Z -> bool
x:R
Hx2:(0 < x)%R
Hx:(x < bpow (emin + prec))%R
(0 *
 (round beta (FLT_exp emin prec) (Znearest choice) x -
  x))%R = 0%R /\
round beta (FLT_exp emin prec) (Znearest choice) x =
(x * (1 + 0) +
 (round beta (FLT_exp emin prec) (Znearest choice) x -
  x))%R
assert (e-1 < emin+prec)%Z.beta:radix
emin, prec:Z
Hp:(0 < prec)%Z
rnd:R -> Z
valid_rnd:Valid_rnd rnd
choice:Z -> bool
x:R
Hx2:(0 < x)%R
Hx:(x < bpow (emin + prec))%R
e:Z
He:x <> 0%R -> (bpow (e - 1) <= Rabs x < bpow e)%R
(e - 1 < emin + prec)%Z
beta:radix
emin, prec:Z
Hp:(0 < prec)%Z
rnd:R -> Z
valid_rnd:Valid_rnd rnd
choice:Z -> bool
x:R
Hx2:(0 < x)%R
Hx:(x < bpow (emin + prec))%R
e:Z
He:x <> 0%R -> (bpow (e - 1) <= Rabs x < bpow e)%R
H:(e - 1 < emin + prec)%Z
(e - prec <= emin)%Z
beta:radix
emin, prec:Z
Hp:(0 < prec)%Z
rnd:R -> Z
valid_rnd:Valid_rnd rnd
choice:Z -> bool
x:R
Hx2:(0 < x)%R
Hx:(x < bpow (emin + prec))%R
(0 *
 (round beta (FLT_exp emin prec) (Znearest choice) x -
  x))%R = 0%R /\
round beta (FLT_exp emin prec) (Znearest choice) x =
(x * (1 + 0) +
 (round beta (FLT_exp emin prec) (Znearest choice) x -
  x))%R
apply (lt_bpow beta).beta:radix
emin, prec:Z
Hp:(0 < prec)%Z
rnd:R -> Z
valid_rnd:Valid_rnd rnd
choice:Z -> bool
x:R
Hx2:(0 < x)%R
Hx:(x < bpow (emin + prec))%R
e:Z
He:x <> 0%R -> (bpow (e - 1) <= Rabs x < bpow e)%R
(bpow (e - 1) < bpow (emin + prec))%R
beta:radix
emin, prec:Z
Hp:(0 < prec)%Z
rnd:R -> Z
valid_rnd:Valid_rnd rnd
choice:Z -> bool
x:R
Hx2:(0 < x)%R
Hx:(x < bpow (emin + prec))%R
e:Z
He:x <> 0%R -> (bpow (e - 1) <= Rabs x < bpow e)%R
H:(e - 1 < emin + prec)%Z
(e - prec <= emin)%Z
beta:radix
emin, prec:Z
Hp:(0 < prec)%Z
rnd:R -> Z
valid_rnd:Valid_rnd rnd
choice:Z -> bool
x:R
Hx2:(0 < x)%R
Hx:(x < bpow (emin + prec))%R
(0 *
 (round beta (FLT_exp emin prec) (Znearest choice) x -
  x))%R = 0%R /\
round beta (FLT_exp emin prec) (Znearest choice) x =
(x * (1 + 0) +
 (round beta (FLT_exp emin prec) (Znearest choice) x -
  x))%R
apply Rle_lt_trans with (2:=Hx).beta:radix
emin, prec:Z
Hp:(0 < prec)%Z
rnd:R -> Z
valid_rnd:Valid_rnd rnd
choice:Z -> bool
x:R
Hx2:(0 < x)%R
Hx:(x < bpow (emin + prec))%R
e:Z
He:x <> 0%R -> (bpow (e - 1) <= Rabs x < bpow e)%R
(bpow (e - 1) <= x)%R
beta:radix
emin, prec:Z
Hp:(0 < prec)%Z
rnd:R -> Z
valid_rnd:Valid_rnd rnd
choice:Z -> bool
x:R
Hx2:(0 < x)%R
Hx:(x < bpow (emin + prec))%R
e:Z
He:x <> 0%R -> (bpow (e - 1) <= Rabs x < bpow e)%R
H:(e - 1 < emin + prec)%Z
(e - prec <= emin)%Z
beta:radix
emin, prec:Z
Hp:(0 < prec)%Z
rnd:R -> Z
valid_rnd:Valid_rnd rnd
choice:Z -> bool
x:R
Hx2:(0 < x)%R
Hx:(x < bpow (emin + prec))%R
(0 *
 (round beta (FLT_exp emin prec) (Znearest choice) x -
  x))%R = 0%R /\
round beta (FLT_exp emin prec) (Znearest choice) x =
(x * (1 + 0) +
 (round beta (FLT_exp emin prec) (Znearest choice) x -
  x))%R
rewrite <- (Rabs_pos_eq x) by now apply Rlt_le.beta:radix
emin, prec:Z
Hp:(0 < prec)%Z
rnd:R -> Z
valid_rnd:Valid_rnd rnd
choice:Z -> bool
x:R
Hx2:(0 < x)%R
Hx:(x < bpow (emin + prec))%R
e:Z
He:x <> 0%R -> (bpow (e - 1) <= Rabs x < bpow e)%R
(bpow (e - 1) <= Rabs x)%R
beta:radix
emin, prec:Z
Hp:(0 < prec)%Z
rnd:R -> Z
valid_rnd:Valid_rnd rnd
choice:Z -> bool
x:R
Hx2:(0 < x)%R
Hx:(x < bpow (emin + prec))%R
e:Z
He:x <> 0%R -> (bpow (e - 1) <= Rabs x < bpow e)%R
H:(e - 1 < emin + prec)%Z
(e - prec <= emin)%Z
beta:radix
emin, prec:Z
Hp:(0 < prec)%Z
rnd:R -> Z
valid_rnd:Valid_rnd rnd
choice:Z -> bool
x:R
Hx2:(0 < x)%R
Hx:(x < bpow (emin + prec))%R
(0 *
 (round beta (FLT_exp emin prec) (Znearest choice) x -
  x))%R = 0%R /\
round beta (FLT_exp emin prec) (Znearest choice) x =
(x * (1 + 0) +
 (round beta (FLT_exp emin prec) (Znearest choice) x -
  x))%R
now apply He, Rgt_not_eq.beta:radix
emin, prec:Z
Hp:(0 < prec)%Z
rnd:R -> Z
valid_rnd:Valid_rnd rnd
choice:Z -> bool
x:R
Hx2:(0 < x)%R
Hx:(x < bpow (emin + prec))%R
e:Z
He:x <> 0%R -> (bpow (e - 1) <= Rabs x < bpow e)%R
H:(e - 1 < emin + prec)%Z
(e - prec <= emin)%Z
beta:radix
emin, prec:Z
Hp:(0 < prec)%Z
rnd:R -> Z
valid_rnd:Valid_rnd rnd
choice:Z -> bool
x:R
Hx2:(0 < x)%R
Hx:(x < bpow (emin + prec))%R
(0 *
 (round beta (FLT_exp emin prec) (Znearest choice) x -
  x))%R = 0%R /\
round beta (FLT_exp emin prec) (Znearest choice) x =
(x * (1 + 0) +
 (round beta (FLT_exp emin prec) (Znearest choice) x -
  x))%R
omega.beta:radix
emin, prec:Z
Hp:(0 < prec)%Z
rnd:R -> Z
valid_rnd:Valid_rnd rnd
choice:Z -> bool
x:R
Hx2:(0 < x)%R
Hx:(x < bpow (emin + prec))%R
(0 *
 (round beta (FLT_exp emin prec) (Znearest choice) x -
  x))%R = 0%R /\
round beta (FLT_exp emin prec) (Znearest choice) x =
(x * (1 + 0) +
 (round beta (FLT_exp emin prec) (Znearest choice) x -
  x))%R
split ; ring.
Qed.

Theorem relative_error_N_FLT'_ex :
  forall x,
  exists eps eta : R,
  (Rabs eps <= u_ro prec / (1 + u_ro prec))%R /\
  (Rabs eta <= /2 * bpow emin)%R /\
  (eps * eta = 0)%R /\
  round beta (FLT_exp emin prec) (Znearest choice) x
  = (x * (1 + eps) + eta)%R.beta:radix
emin, prec:Z
Hp:(0 < prec)%Z
rnd:R -> Z
valid_rnd:Valid_rnd rnd
choice:Z -> bool
forall x : R,
exists eps eta : R,
  (Rabs eps <= u_ro prec / (1 + u_ro prec))%R /\
  (Rabs eta <= / 2 * bpow emin)%R /\
  (eps * eta)%R = 0%R /\
  round beta (FLT_exp emin prec) (Znearest choice) x =
  (x * (1 + eps) + eta)%R
Proof.beta:radix
emin, prec:Z
Hp:(0 < prec)%Z
rnd:R -> Z
valid_rnd:Valid_rnd rnd
choice:Z -> bool
forall x : R,
exists eps eta : R,
  (Rabs eps <= u_ro prec / (1 + u_ro prec))%R /\
  (Rabs eta <= / 2 * bpow emin)%R /\
  (eps * eta)%R = 0%R /\
  round beta (FLT_exp emin prec) (Znearest choice) x =
  (x * (1 + eps) + eta)%R
intro x.beta:radix
emin, prec:Z
Hp:(0 < prec)%Z
rnd:R -> Z
valid_rnd:Valid_rnd rnd
choice:Z -> bool
x:R
exists eps eta : R,
  (Rabs eps <= u_ro prec / (1 + u_ro prec))%R /\
  (Rabs eta <= / 2 * bpow emin)%R /\
  (eps * eta)%R = 0%R /\
  round beta (FLT_exp emin prec) (Znearest choice) x =
  (x * (1 + eps) + eta)%R
set (rx := round _ _ _ x).beta:radix
emin, prec:Z
Hp:(0 < prec)%Z
rnd:R -> Z
valid_rnd:Valid_rnd rnd
choice:Z -> bool
x:R
rx:=round beta (FLT_exp emin prec) (Znearest choice) x:R
exists eps eta : R,
  (Rabs eps <= u_ro prec / (1 + u_ro prec))%R /\
  (Rabs eta <= / 2 * bpow emin)%R /\
  (eps * eta)%R = 0%R /\ rx = (x * (1 + eps) + eta)%R
assert (Pb := u_rod1pu_ro_pos prec).beta:radix
emin, prec:Z
Hp:(0 < prec)%Z
rnd:R -> Z
valid_rnd:Valid_rnd rnd
choice:Z -> bool
x:R
rx:=round beta (FLT_exp emin prec) (Znearest choice) x:R
Pb:(0 <= u_ro prec / (1 + u_ro prec))%R
exists eps eta : R,
  (Rabs eps <= u_ro prec / (1 + u_ro prec))%R /\
  (Rabs eta <= / 2 * bpow emin)%R /\
  (eps * eta)%R = 0%R /\ rx = (x * (1 + eps) + eta)%R
destruct (Rle_or_lt (bpow (emin + prec - 1)) (Rabs x)) as [MX|Mx].beta:radix
emin, prec:Z
Hp:(0 < prec)%Z
rnd:R -> Z
valid_rnd:Valid_rnd rnd
choice:Z -> bool
x:R
rx:=round beta (FLT_exp emin prec) (Znearest choice) x:R
Pb:(0 <= u_ro prec / (1 + u_ro prec))%R
MX:(bpow (emin + prec - 1) <= Rabs x)%R
exists eps eta : R,
  (Rabs eps <= u_ro prec / (1 + u_ro prec))%R /\
  (Rabs eta <= / 2 * bpow emin)%R /\
  (eps * eta)%R = 0%R /\ rx = (x * (1 + eps) + eta)%R
beta:radix
emin, prec:Z
Hp:(0 < prec)%Z
rnd:R -> Z
valid_rnd:Valid_rnd rnd
choice:Z -> bool
x:R
rx:=round beta (FLT_exp emin prec) (Znearest choice) x:R
Pb:(0 <= u_ro prec / (1 + u_ro prec))%R
Mx:(Rabs x < bpow (emin + prec - 1))%R
exists eps eta : R,
  (Rabs eps <= u_ro prec / (1 + u_ro prec))%R /\
  (Rabs eta <= / 2 * bpow emin)%R /\
  (eps * eta)%R = 0%R /\ rx = (x * (1 + eps) + eta)%R
{beta:radix
emin, prec:Z
Hp:(0 < prec)%Z
rnd:R -> Z
valid_rnd:Valid_rnd rnd
choice:Z -> bool
x:R
rx:=round beta (FLT_exp emin prec) (Znearest choice) x:R
Pb:(0 <= u_ro prec / (1 + u_ro prec))%R
MX:(bpow (emin + prec - 1) <= Rabs x)%R
exists eps eta : R,
  (Rabs eps <= u_ro prec / (1 + u_ro prec))%R /\
  (Rabs eta <= / 2 * bpow emin)%R /\
  (eps * eta)%R = 0%R /\ rx = (x * (1 + eps) + eta)%R destruct (relative_error_N_FLX'_ex prec Hp choice x) as (d, (Bd, Hd)).beta:radix
emin, prec:Z
Hp:(0 < prec)%Z
rnd:R -> Z
valid_rnd:Valid_rnd rnd
choice:Z -> bool
x:R
rx:=round beta (FLT_exp emin prec) (Znearest choice) x:R
Pb:(0 <= u_ro prec / (1 + u_ro prec))%R
MX:(bpow (emin + prec - 1) <= Rabs x)%R
d:R
Bd:(Rabs d <= u_ro prec / (1 + u_ro prec))%R
Hd:round beta (FLX_exp prec) (Znearest choice) x =
(x * (1 + d))%R
exists eps eta : R,
  (Rabs eps <= u_ro prec / (1 + u_ro prec))%R /\
  (Rabs eta <= / 2 * bpow emin)%R /\
  (eps * eta)%R = 0%R /\ rx = (x * (1 + eps) + eta)%R
  exists d, 0%R; split; [exact Bd|]; split.beta:radix
emin, prec:Z
Hp:(0 < prec)%Z
rnd:R -> Z
valid_rnd:Valid_rnd rnd
choice:Z -> bool
x:R
rx:=round beta (FLT_exp emin prec) (Znearest choice) x:R
Pb:(0 <= u_ro prec / (1 + u_ro prec))%R
MX:(bpow (emin + prec - 1) <= Rabs x)%R
d:R
Bd:(Rabs d <= u_ro prec / (1 + u_ro prec))%R
Hd:round beta (FLX_exp prec) (Znearest choice) x =
(x * (1 + d))%R
(Rabs 0 <= / 2 * bpow emin)%R
beta:radix
emin, prec:Z
Hp:(0 < prec)%Z
rnd:R -> Z
valid_rnd:Valid_rnd rnd
choice:Z -> bool
x:R
rx:=round beta (FLT_exp emin prec) (Znearest choice) x:R
Pb:(0 <= u_ro prec / (1 + u_ro prec))%R
MX:(bpow (emin + prec - 1) <= Rabs x)%R
d:R
Bd:(Rabs d <= u_ro prec / (1 + u_ro prec))%R
Hd:round beta (FLX_exp prec) (Znearest choice) x =
(x * (1 + d))%R
(d * 0)%R = 0%R /\ rx = (x * (1 + d) + 0)%R
  {beta:radix
emin, prec:Z
Hp:(0 < prec)%Z
rnd:R -> Z
valid_rnd:Valid_rnd rnd
choice:Z -> bool
x:R
rx:=round beta (FLT_exp emin prec) (Znearest choice) x:R
Pb:(0 <= u_ro prec / (1 + u_ro prec))%R
MX:(bpow (emin + prec - 1) <= Rabs x)%R
d:R
Bd:(Rabs d <= u_ro prec / (1 + u_ro prec))%R
Hd:round beta (FLX_exp prec) (Znearest choice) x =
(x * (1 + d))%R
(Rabs 0 <= / 2 * bpow emin)%R rewrite Rabs_R0; apply Rmult_le_pos; [lra|apply bpow_ge_0]. }beta:radix
emin, prec:Z
Hp:(0 < prec)%Z
rnd:R -> Z
valid_rnd:Valid_rnd rnd
choice:Z -> bool
x:R
rx:=round beta (FLT_exp emin prec) (Znearest choice) x:R
Pb:(0 <= u_ro prec / (1 + u_ro prec))%R
MX:(bpow (emin + prec - 1) <= Rabs x)%R
d:R
Bd:(Rabs d <= u_ro prec / (1 + u_ro prec))%R
Hd:round beta (FLX_exp prec) (Znearest choice) x =
(x * (1 + d))%R
(d * 0)%R = 0%R /\ rx = (x * (1 + d) + 0)%R
  rewrite Rplus_0_r, Rmult_0_r; split; [reflexivity|].beta:radix
emin, prec:Z
Hp:(0 < prec)%Z
rnd:R -> Z
valid_rnd:Valid_rnd rnd
choice:Z -> bool
x:R
rx:=round beta (FLT_exp emin prec) (Znearest choice) x:R
Pb:(0 <= u_ro prec / (1 + u_ro prec))%R
MX:(bpow (emin + prec - 1) <= Rabs x)%R
d:R
Bd:(Rabs d <= u_ro prec / (1 + u_ro prec))%R
Hd:round beta (FLX_exp prec) (Znearest choice) x =
(x * (1 + d))%R
rx = (x * (1 + d))%R
  now rewrite <- Hd; apply round_FLT_FLX. }beta:radix
emin, prec:Z
Hp:(0 < prec)%Z
rnd:R -> Z
valid_rnd:Valid_rnd rnd
choice:Z -> bool
x:R
rx:=round beta (FLT_exp emin prec) (Znearest choice) x:R
Pb:(0 <= u_ro prec / (1 + u_ro prec))%R
Mx:(Rabs x < bpow (emin + prec - 1))%R
exists eps eta : R,
  (Rabs eps <= u_ro prec / (1 + u_ro prec))%R /\
  (Rabs eta <= / 2 * bpow emin)%R /\
  (eps * eta)%R = 0%R /\ rx = (x * (1 + eps) + eta)%R
assert (H : (Rabs (rx - x) <= /2 * bpow emin)%R).beta:radix
emin, prec:Z
Hp:(0 < prec)%Z
rnd:R -> Z
valid_rnd:Valid_rnd rnd
choice:Z -> bool
x:R
rx:=round beta (FLT_exp emin prec) (Znearest choice) x:R
Pb:(0 <= u_ro prec / (1 + u_ro prec))%R
Mx:(Rabs x < bpow (emin + prec - 1))%R
(Rabs (rx - x) <= / 2 * bpow emin)%R
beta:radix
emin, prec:Z
Hp:(0 < prec)%Z
rnd:R -> Z
valid_rnd:Valid_rnd rnd
choice:Z -> bool
x:R
rx:=round beta (FLT_exp emin prec) (Znearest choice) x:R
Pb:(0 <= u_ro prec / (1 + u_ro prec))%R
Mx:(Rabs x < bpow (emin + prec - 1))%R
H:(Rabs (rx - x) <= / 2 * bpow emin)%R
exists eps eta : R,
  (Rabs eps <= u_ro prec / (1 + u_ro prec))%R /\
  (Rabs eta <= / 2 * bpow emin)%R /\
  (eps * eta)%R = 0%R /\ rx = (x * (1 + eps) + eta)%R
{beta:radix
emin, prec:Z
Hp:(0 < prec)%Z
rnd:R -> Z
valid_rnd:Valid_rnd rnd
choice:Z -> bool
x:R
rx:=round beta (FLT_exp emin prec) (Znearest choice) x:R
Pb:(0 <= u_ro prec / (1 + u_ro prec))%R
Mx:(Rabs x < bpow (emin + prec - 1))%R
(Rabs (rx - x) <= / 2 * bpow emin)%R refine (Rle_trans _ _ _ (error_le_half_ulp _ _ _ _) _);
    [now apply FLT_exp_valid|].beta:radix
emin, prec:Z
Hp:(0 < prec)%Z
rnd:R -> Z
valid_rnd:Valid_rnd rnd
choice:Z -> bool
x:R
rx:=round beta (FLT_exp emin prec) (Znearest choice) x:R
Pb:(0 <= u_ro prec / (1 + u_ro prec))%R
Mx:(Rabs x < bpow (emin + prec - 1))%R
(/ 2 * ulp beta (FLT_exp emin prec) x <=
 / 2 * bpow emin)%R
  rewrite ulp_FLT_small; [now right|now simpl|].beta:radix
emin, prec:Z
Hp:(0 < prec)%Z
rnd:R -> Z
valid_rnd:Valid_rnd rnd
choice:Z -> bool
x:R
rx:=round beta (FLT_exp emin prec) (Znearest choice) x:R
Pb:(0 <= u_ro prec / (1 + u_ro prec))%R
Mx:(Rabs x < bpow (emin + prec - 1))%R
(Rabs x < bpow (emin + prec))%R
  apply (Rlt_le_trans _ _ _ Mx), bpow_le; lia. }beta:radix
emin, prec:Z
Hp:(0 < prec)%Z
rnd:R -> Z
valid_rnd:Valid_rnd rnd
choice:Z -> bool
x:R
rx:=round beta (FLT_exp emin prec) (Znearest choice) x:R
Pb:(0 <= u_ro prec / (1 + u_ro prec))%R
Mx:(Rabs x < bpow (emin + prec - 1))%R
H:(Rabs (rx - x) <= / 2 * bpow emin)%R
exists eps eta : R,
  (Rabs eps <= u_ro prec / (1 + u_ro prec))%R /\
  (Rabs eta <= / 2 * bpow emin)%R /\
  (eps * eta)%R = 0%R /\ rx = (x * (1 + eps) + eta)%R
exists 0%R, (rx - x)%R; split; [now rewrite Rabs_R0|]; split; [exact H|].beta:radix
emin, prec:Z
Hp:(0 < prec)%Z
rnd:R -> Z
valid_rnd:Valid_rnd rnd
choice:Z -> bool
x:R
rx:=round beta (FLT_exp emin prec) (Znearest choice) x:R
Pb:(0 <= u_ro prec / (1 + u_ro prec))%R
Mx:(Rabs x < bpow (emin + prec - 1))%R
H:(Rabs (rx - x) <= / 2 * bpow emin)%R
(0 * (rx - x))%R = 0%R /\
rx = (x * (1 + 0) + (rx - x))%R
now rewrite Rmult_0_l, Rplus_0_r, Rmult_1_r; split; [|ring].
Qed.

Theorem relative_error_N_FLT'_ex_separate :
  forall x,
  exists x' : R,
  round beta (FLT_exp emin prec) (Znearest choice) x'
  = round beta (FLT_exp emin prec) (Znearest choice) x /\
  (exists eta, Rabs eta <= /2 * bpow emin /\ x' = x + eta)%R /\
  (exists eps, Rabs eps <= u_ro prec / (1 + u_ro prec) /\
               round beta (FLT_exp emin prec) (Znearest choice) x'
               = x' * (1 + eps))%R.beta:radix
emin, prec:Z
Hp:(0 < prec)%Z
rnd:R -> Z
valid_rnd:Valid_rnd rnd
choice:Z -> bool
forall x : R,
exists x' : R,
  round beta (FLT_exp emin prec) (Znearest choice) x' =
  round beta (FLT_exp emin prec) (Znearest choice) x /\
  (exists eta : R,
     (Rabs eta <= / 2 * bpow emin)%R /\
     x' = (x + eta)%R) /\
  (exists eps : R,
     (Rabs eps <= u_ro prec / (1 + u_ro prec))%R /\
     round beta (FLT_exp emin prec) (Znearest choice)
       x' = (x' * (1 + eps))%R)
Proof.beta:radix
emin, prec:Z
Hp:(0 < prec)%Z
rnd:R -> Z
valid_rnd:Valid_rnd rnd
choice:Z -> bool
forall x : R,
exists x' : R,
  round beta (FLT_exp emin prec) (Znearest choice) x' =
  round beta (FLT_exp emin prec) (Znearest choice) x /\
  (exists eta : R,
     (Rabs eta <= / 2 * bpow emin)%R /\
     x' = (x + eta)%R) /\
  (exists eps : R,
     (Rabs eps <= u_ro prec / (1 + u_ro prec))%R /\
     round beta (FLT_exp emin prec) (Znearest choice)
       x' = (x' * (1 + eps))%R)
intro x.beta:radix
emin, prec:Z
Hp:(0 < prec)%Z
rnd:R -> Z
valid_rnd:Valid_rnd rnd
choice:Z -> bool
x:R
exists x' : R,
  round beta (FLT_exp emin prec) (Znearest choice) x' =
  round beta (FLT_exp emin prec) (Znearest choice) x /\
  (exists eta : R,
     (Rabs eta <= / 2 * bpow emin)%R /\
     x' = (x + eta)%R) /\
  (exists eps : R,
     (Rabs eps <= u_ro prec / (1 + u_ro prec))%R /\
     round beta (FLT_exp emin prec) (Znearest choice)
       x' = (x' * (1 + eps))%R)
set (rx := round _ _ _ x).beta:radix
emin, prec:Z
Hp:(0 < prec)%Z
rnd:R -> Z
valid_rnd:Valid_rnd rnd
choice:Z -> bool
x:R
rx:=round beta (FLT_exp emin prec) (Znearest choice) x:R
exists x' : R,
  round beta (FLT_exp emin prec) (Znearest choice) x' =
  rx /\
  (exists eta : R,
     (Rabs eta <= / 2 * bpow emin)%R /\
     x' = (x + eta)%R) /\
  (exists eps : R,
     (Rabs eps <= u_ro prec / (1 + u_ro prec))%R /\
     round beta (FLT_exp emin prec) (Znearest choice)
       x' = (x' * (1 + eps))%R)
destruct (relative_error_N_FLT'_ex x) as (d, (e, (Bd, (Be, (Hde0, Hde))))).beta:radix
emin, prec:Z
Hp:(0 < prec)%Z
rnd:R -> Z
valid_rnd:Valid_rnd rnd
choice:Z -> bool
x:R
rx:=round beta (FLT_exp emin prec) (Znearest choice) x:R
d, e:R
Bd:(Rabs d <= u_ro prec / (1 + u_ro prec))%R
Be:(Rabs e <= / 2 * bpow emin)%R
Hde0:(d * e)%R = 0%R
Hde:round beta (FLT_exp emin prec) 
  (Znearest choice) x = (x * (1 + d) + e)%R
exists x' : R,
  round beta (FLT_exp emin prec) (Znearest choice) x' =
  rx /\
  (exists eta : R,
     (Rabs eta <= / 2 * bpow emin)%R /\
     x' = (x + eta)%R) /\
  (exists eps : R,
     (Rabs eps <= u_ro prec / (1 + u_ro prec))%R /\
     round beta (FLT_exp emin prec) (Znearest choice)
       x' = (x' * (1 + eps))%R)
destruct (Rlt_or_le (Rabs (d * x)) (Rabs e)) as [HdxLte|HeLedx].beta:radix
emin, prec:Z
Hp:(0 < prec)%Z
rnd:R -> Z
valid_rnd:Valid_rnd rnd
choice:Z -> bool
x:R
rx:=round beta (FLT_exp emin prec) (Znearest choice) x:R
d, e:R
Bd:(Rabs d <= u_ro prec / (1 + u_ro prec))%R
Be:(Rabs e <= / 2 * bpow emin)%R
Hde0:(d * e)%R = 0%R
Hde:round beta (FLT_exp emin prec) 
  (Znearest choice) x = (x * (1 + d) + e)%R
HdxLte:(Rabs (d * x) < Rabs e)%R
exists x' : R,
  round beta (FLT_exp emin prec) (Znearest choice) x' =
  rx /\
  (exists eta : R,
     (Rabs eta <= / 2 * bpow emin)%R /\
     x' = (x + eta)%R) /\
  (exists eps : R,
     (Rabs eps <= u_ro prec / (1 + u_ro prec))%R /\
     round beta (FLT_exp emin prec) (Znearest choice)
       x' = (x' * (1 + eps))%R)
beta:radix
emin, prec:Z
Hp:(0 < prec)%Z
rnd:R -> Z
valid_rnd:Valid_rnd rnd
choice:Z -> bool
x:R
rx:=round beta (FLT_exp emin prec) (Znearest choice) x:R
d, e:R
Bd:(Rabs d <= u_ro prec / (1 + u_ro prec))%R
Be:(Rabs e <= / 2 * bpow emin)%R
Hde0:(d * e)%R = 0%R
Hde:round beta (FLT_exp emin prec) 
  (Znearest choice) x = (x * (1 + d) + e)%R
HeLedx:(Rabs e <= Rabs (d * x))%R
exists x' : R,
  round beta (FLT_exp emin prec) (Znearest choice) x' =
  rx /\
  (exists eta : R,
     (Rabs eta <= / 2 * bpow emin)%R /\
     x' = (x + eta)%R) /\
  (exists eps : R,
     (Rabs eps <= u_ro prec / (1 + u_ro prec))%R /\
     round beta (FLT_exp emin prec) 
       (Znearest choice) x' = 
     (x' * (1 + eps))%R)
{beta:radix
emin, prec:Z
Hp:(0 < prec)%Z
rnd:R -> Z
valid_rnd:Valid_rnd rnd
choice:Z -> bool
x:R
rx:=round beta (FLT_exp emin prec) (Znearest choice) x:R
d, e:R
Bd:(Rabs d <= u_ro prec / (1 + u_ro prec))%R
Be:(Rabs e <= / 2 * bpow emin)%R
Hde0:(d * e)%R = 0%R
Hde:round beta (FLT_exp emin prec) 
  (Znearest choice) x = (x * (1 + d) + e)%R
HdxLte:(Rabs (d * x) < Rabs e)%R
exists x' : R,
  round beta (FLT_exp emin prec) (Znearest choice) x' =
  rx /\
  (exists eta : R,
     (Rabs eta <= / 2 * bpow emin)%R /\
     x' = (x + eta)%R) /\
  (exists eps : R,
     (Rabs eps <= u_ro prec / (1 + u_ro prec))%R /\
     round beta (FLT_exp emin prec) (Znearest choice)
       x' = (x' * (1 + eps))%R) exists rx; split; [|split].beta:radix
emin, prec:Z
Hp:(0 < prec)%Z
rnd:R -> Z
valid_rnd:Valid_rnd rnd
choice:Z -> bool
x:R
rx:=round beta (FLT_exp emin prec) (Znearest choice) x:R
d, e:R
Bd:(Rabs d <= u_ro prec / (1 + u_ro prec))%R
Be:(Rabs e <= / 2 * bpow emin)%R
Hde0:(d * e)%R = 0%R
Hde:round beta (FLT_exp emin prec) 
  (Znearest choice) x = (x * (1 + d) + e)%R
HdxLte:(Rabs (d * x) < Rabs e)%R
round beta (FLT_exp emin prec) (Znearest choice) rx =
rx
beta:radix
emin, prec:Z
Hp:(0 < prec)%Z
rnd:R -> Z
valid_rnd:Valid_rnd rnd
choice:Z -> bool
x:R
rx:=round beta (FLT_exp emin prec) (Znearest choice) x:R
d, e:R
Bd:(Rabs d <= u_ro prec / (1 + u_ro prec))%R
Be:(Rabs e <= / 2 * bpow emin)%R
Hde0:(d * e)%R = 0%R
Hde:round beta (FLT_exp emin prec) 
  (Znearest choice) x = (x * (1 + d) + e)%R
HdxLte:(Rabs (d * x) < Rabs e)%R
exists eta : R,
  (Rabs eta <= / 2 * bpow emin)%R /\ rx = (x + eta)%R
beta:radix
emin, prec:Z
Hp:(0 < prec)%Z
rnd:R -> Z
valid_rnd:Valid_rnd rnd
choice:Z -> bool
x:R
rx:=round beta (FLT_exp emin prec) (Znearest choice) x:R
d, e:R
Bd:(Rabs d <= u_ro prec / (1 + u_ro prec))%R
Be:(Rabs e <= / 2 * bpow emin)%R
Hde0:(d * e)%R = 0%R
Hde:round beta (FLT_exp emin prec) 
  (Znearest choice) x = (x * (1 + d) + e)%R
HdxLte:(Rabs (d * x) < Rabs e)%R
exists eps : R,
  (Rabs eps <= u_ro prec / (1 + u_ro prec))%R /\
  round beta (FLT_exp emin prec) (Znearest choice) rx =
  (rx * (1 + eps))%R
  {beta:radix
emin, prec:Z
Hp:(0 < prec)%Z
rnd:R -> Z
valid_rnd:Valid_rnd rnd
choice:Z -> bool
x:R
rx:=round beta (FLT_exp emin prec) (Znearest choice) x:R
d, e:R
Bd:(Rabs d <= u_ro prec / (1 + u_ro prec))%R
Be:(Rabs e <= / 2 * bpow emin)%R
Hde0:(d * e)%R = 0%R
Hde:round beta (FLT_exp emin prec) 
  (Znearest choice) x = (x * (1 + d) + e)%R
HdxLte:(Rabs (d * x) < Rabs e)%R
round beta (FLT_exp emin prec) (Znearest choice) rx =
rx apply round_generic; [now apply valid_rnd_N|].beta:radix
emin, prec:Z
Hp:(0 < prec)%Z
rnd:R -> Z
valid_rnd:Valid_rnd rnd
choice:Z -> bool
x:R
rx:=round beta (FLT_exp emin prec) (Znearest choice) x:R
d, e:R
Bd:(Rabs d <= u_ro prec / (1 + u_ro prec))%R
Be:(Rabs e <= / 2 * bpow emin)%R
Hde0:(d * e)%R = 0%R
Hde:round beta (FLT_exp emin prec) 
  (Znearest choice) x = (x * (1 + d) + e)%R
HdxLte:(Rabs (d * x) < Rabs e)%R
generic_format beta (FLT_exp emin prec) rx
    now apply generic_format_round; [apply FLT_exp_valid|apply valid_rnd_N]. }beta:radix
emin, prec:Z
Hp:(0 < prec)%Z
rnd:R -> Z
valid_rnd:Valid_rnd rnd
choice:Z -> bool
x:R
rx:=round beta (FLT_exp emin prec) (Znearest choice) x:R
d, e:R
Bd:(Rabs d <= u_ro prec / (1 + u_ro prec))%R
Be:(Rabs e <= / 2 * bpow emin)%R
Hde0:(d * e)%R = 0%R
Hde:round beta (FLT_exp emin prec) 
  (Znearest choice) x = (x * (1 + d) + e)%R
HdxLte:(Rabs (d * x) < Rabs e)%R
exists eta : R,
  (Rabs eta <= / 2 * bpow emin)%R /\ rx = (x + eta)%R
beta:radix
emin, prec:Z
Hp:(0 < prec)%Z
rnd:R -> Z
valid_rnd:Valid_rnd rnd
choice:Z -> bool
x:R
rx:=round beta (FLT_exp emin prec) (Znearest choice) x:R
d, e:R
Bd:(Rabs d <= u_ro prec / (1 + u_ro prec))%R
Be:(Rabs e <= / 2 * bpow emin)%R
Hde0:(d * e)%R = 0%R
Hde:round beta (FLT_exp emin prec) 
  (Znearest choice) x = (x * (1 + d) + e)%R
HdxLte:(Rabs (d * x) < Rabs e)%R
exists eps : R,
  (Rabs eps <= u_ro prec / (1 + u_ro prec))%R /\
  round beta (FLT_exp emin prec) (Znearest choice) rx =
  (rx * (1 + eps))%R
  {beta:radix
emin, prec:Z
Hp:(0 < prec)%Z
rnd:R -> Z
valid_rnd:Valid_rnd rnd
choice:Z -> bool
x:R
rx:=round beta (FLT_exp emin prec) (Znearest choice) x:R
d, e:R
Bd:(Rabs d <= u_ro prec / (1 + u_ro prec))%R
Be:(Rabs e <= / 2 * bpow emin)%R
Hde0:(d * e)%R = 0%R
Hde:round beta (FLT_exp emin prec) 
  (Znearest choice) x = (x * (1 + d) + e)%R
HdxLte:(Rabs (d * x) < Rabs e)%R
exists eta : R,
  (Rabs eta <= / 2 * bpow emin)%R /\ rx = (x + eta)%R exists e; split; [exact Be|].beta:radix
emin, prec:Z
Hp:(0 < prec)%Z
rnd:R -> Z
valid_rnd:Valid_rnd rnd
choice:Z -> bool
x:R
rx:=round beta (FLT_exp emin prec) (Znearest choice) x:R
d, e:R
Bd:(Rabs d <= u_ro prec / (1 + u_ro prec))%R
Be:(Rabs e <= / 2 * bpow emin)%R
Hde0:(d * e)%R = 0%R
Hde:round beta (FLT_exp emin prec) 
  (Znearest choice) x = (x * (1 + d) + e)%R
HdxLte:(Rabs (d * x) < Rabs e)%R
rx = (x + e)%R
    unfold rx; rewrite Hde; destruct (Rmult_integral _ _ Hde0) as [Zd|Ze].beta:radix
emin, prec:Z
Hp:(0 < prec)%Z
rnd:R -> Z
valid_rnd:Valid_rnd rnd
choice:Z -> bool
x:R
rx:=round beta (FLT_exp emin prec) (Znearest choice) x:R
d, e:R
Bd:(Rabs d <= u_ro prec / (1 + u_ro prec))%R
Be:(Rabs e <= / 2 * bpow emin)%R
Hde0:(d * e)%R = 0%R
Hde:round beta (FLT_exp emin prec) 
  (Znearest choice) x = (x * (1 + d) + e)%R
HdxLte:(Rabs (d * x) < Rabs e)%R
Zd:d = 0%R
(x * (1 + d) + e)%R = (x + e)%R
beta:radix
emin, prec:Z
Hp:(0 < prec)%Z
rnd:R -> Z
valid_rnd:Valid_rnd rnd
choice:Z -> bool
x:R
rx:=round beta (FLT_exp emin prec) (Znearest choice) x:R
d, e:R
Bd:(Rabs d <= u_ro prec / (1 + u_ro prec))%R
Be:(Rabs e <= / 2 * bpow emin)%R
Hde0:(d * e)%R = 0%R
Hde:round beta (FLT_exp emin prec) 
  (Znearest choice) x = (x * (1 + d) + e)%R
HdxLte:(Rabs (d * x) < Rabs e)%R
Ze:e = 0%R
(x * (1 + d) + e)%R = (x + e)%R
    {beta:radix
emin, prec:Z
Hp:(0 < prec)%Z
rnd:R -> Z
valid_rnd:Valid_rnd rnd
choice:Z -> bool
x:R
rx:=round beta (FLT_exp emin prec) (Znearest choice) x:R
d, e:R
Bd:(Rabs d <= u_ro prec / (1 + u_ro prec))%R
Be:(Rabs e <= / 2 * bpow emin)%R
Hde0:(d * e)%R = 0%R
Hde:round beta (FLT_exp emin prec) 
  (Znearest choice) x = (x * (1 + d) + e)%R
HdxLte:(Rabs (d * x) < Rabs e)%R
Zd:d = 0%R
(x * (1 + d) + e)%R = (x + e)%R now rewrite Zd, Rplus_0_r, Rmult_1_r. }beta:radix
emin, prec:Z
Hp:(0 < prec)%Z
rnd:R -> Z
valid_rnd:Valid_rnd rnd
choice:Z -> bool
x:R
rx:=round beta (FLT_exp emin prec) (Znearest choice) x:R
d, e:R
Bd:(Rabs d <= u_ro prec / (1 + u_ro prec))%R
Be:(Rabs e <= / 2 * bpow emin)%R
Hde0:(d * e)%R = 0%R
Hde:round beta (FLT_exp emin prec) 
  (Znearest choice) x = (x * (1 + d) + e)%R
HdxLte:(Rabs (d * x) < Rabs e)%R
Ze:e = 0%R
(x * (1 + d) + e)%R = (x + e)%R
    exfalso; revert HdxLte; rewrite Ze, Rabs_R0; apply Rle_not_lt, Rabs_pos. }beta:radix
emin, prec:Z
Hp:(0 < prec)%Z
rnd:R -> Z
valid_rnd:Valid_rnd rnd
choice:Z -> bool
x:R
rx:=round beta (FLT_exp emin prec) (Znearest choice) x:R
d, e:R
Bd:(Rabs d <= u_ro prec / (1 + u_ro prec))%R
Be:(Rabs e <= / 2 * bpow emin)%R
Hde0:(d * e)%R = 0%R
Hde:round beta (FLT_exp emin prec) 
  (Znearest choice) x = (x * (1 + d) + e)%R
HdxLte:(Rabs (d * x) < Rabs e)%R
exists eps : R,
  (Rabs eps <= u_ro prec / (1 + u_ro prec))%R /\
  round beta (FLT_exp emin prec) (Znearest choice) rx =
  (rx * (1 + eps))%R
  exists 0%R; split; [now rewrite Rabs_R0; apply u_rod1pu_ro_pos|].beta:radix
emin, prec:Z
Hp:(0 < prec)%Z
rnd:R -> Z
valid_rnd:Valid_rnd rnd
choice:Z -> bool
x:R
rx:=round beta (FLT_exp emin prec) (Znearest choice) x:R
d, e:R
Bd:(Rabs d <= u_ro prec / (1 + u_ro prec))%R
Be:(Rabs e <= / 2 * bpow emin)%R
Hde0:(d * e)%R = 0%R
Hde:round beta (FLT_exp emin prec) 
  (Znearest choice) x = (x * (1 + d) + e)%R
HdxLte:(Rabs (d * x) < Rabs e)%R
round beta (FLT_exp emin prec) (Znearest choice) rx =
(rx * (1 + 0))%R
  rewrite Rplus_0_r, Rmult_1_r; apply round_generic; [now apply valid_rnd_N|].beta:radix
emin, prec:Z
Hp:(0 < prec)%Z
rnd:R -> Z
valid_rnd:Valid_rnd rnd
choice:Z -> bool
x:R
rx:=round beta (FLT_exp emin prec) (Znearest choice) x:R
d, e:R
Bd:(Rabs d <= u_ro prec / (1 + u_ro prec))%R
Be:(Rabs e <= / 2 * bpow emin)%R
Hde0:(d * e)%R = 0%R
Hde:round beta (FLT_exp emin prec) 
  (Znearest choice) x = (x * (1 + d) + e)%R
HdxLte:(Rabs (d * x) < Rabs e)%R
generic_format beta (FLT_exp emin prec) rx
  now apply generic_format_round; [apply FLT_exp_valid|apply valid_rnd_N]. }beta:radix
emin, prec:Z
Hp:(0 < prec)%Z
rnd:R -> Z
valid_rnd:Valid_rnd rnd
choice:Z -> bool
x:R
rx:=round beta (FLT_exp emin prec) (Znearest choice) x:R
d, e:R
Bd:(Rabs d <= u_ro prec / (1 + u_ro prec))%R
Be:(Rabs e <= / 2 * bpow emin)%R
Hde0:(d * e)%R = 0%R
Hde:round beta (FLT_exp emin prec) 
  (Znearest choice) x = (x * (1 + d) + e)%R
HeLedx:(Rabs e <= Rabs (d * x))%R
exists x' : R,
  round beta (FLT_exp emin prec) (Znearest choice) x' =
  rx /\
  (exists eta : R,
     (Rabs eta <= / 2 * bpow emin)%R /\
     x' = (x + eta)%R) /\
  (exists eps : R,
     (Rabs eps <= u_ro prec / (1 + u_ro prec))%R /\
     round beta (FLT_exp emin prec) (Znearest choice)
       x' = (x' * (1 + eps))%R)
exists x; split; [now simpl|split].beta:radix
emin, prec:Z
Hp:(0 < prec)%Z
rnd:R -> Z
valid_rnd:Valid_rnd rnd
choice:Z -> bool
x:R
rx:=round beta (FLT_exp emin prec) (Znearest choice) x:R
d, e:R
Bd:(Rabs d <= u_ro prec / (1 + u_ro prec))%R
Be:(Rabs e <= / 2 * bpow emin)%R
Hde0:(d * e)%R = 0%R
Hde:round beta (FLT_exp emin prec) 
  (Znearest choice) x = (x * (1 + d) + e)%R
HeLedx:(Rabs e <= Rabs (d * x))%R
exists eta : R,
  (Rabs eta <= / 2 * bpow emin)%R /\ x = (x + eta)%R
beta:radix
emin, prec:Z
Hp:(0 < prec)%Z
rnd:R -> Z
valid_rnd:Valid_rnd rnd
choice:Z -> bool
x:R
rx:=round beta (FLT_exp emin prec) (Znearest choice) x:R
d, e:R
Bd:(Rabs d <= u_ro prec / (1 + u_ro prec))%R
Be:(Rabs e <= / 2 * bpow emin)%R
Hde0:(d * e)%R = 0%R
Hde:round beta (FLT_exp emin prec) 
  (Znearest choice) x = (x * (1 + d) + e)%R
HeLedx:(Rabs e <= Rabs (d * x))%R
exists eps : R,
  (Rabs eps <= u_ro prec / (1 + u_ro prec))%R /\
  round beta (FLT_exp emin prec) (Znearest choice) x =
  (x * (1 + eps))%R
{beta:radix
emin, prec:Z
Hp:(0 < prec)%Z
rnd:R -> Z
valid_rnd:Valid_rnd rnd
choice:Z -> bool
x:R
rx:=round beta (FLT_exp emin prec) (Znearest choice) x:R
d, e:R
Bd:(Rabs d <= u_ro prec / (1 + u_ro prec))%R
Be:(Rabs e <= / 2 * bpow emin)%R
Hde0:(d * e)%R = 0%R
Hde:round beta (FLT_exp emin prec) 
  (Znearest choice) x = (x * (1 + d) + e)%R
HeLedx:(Rabs e <= Rabs (d * x))%R
exists eta : R,
  (Rabs eta <= / 2 * bpow emin)%R /\ x = (x + eta)%R exists 0%R; split;
    [rewrite Rabs_R0; apply Rmult_le_pos; [lra|apply bpow_ge_0]|ring]. }beta:radix
emin, prec:Z
Hp:(0 < prec)%Z
rnd:R -> Z
valid_rnd:Valid_rnd rnd
choice:Z -> bool
x:R
rx:=round beta (FLT_exp emin prec) (Znearest choice) x:R
d, e:R
Bd:(Rabs d <= u_ro prec / (1 + u_ro prec))%R
Be:(Rabs e <= / 2 * bpow emin)%R
Hde0:(d * e)%R = 0%R
Hde:round beta (FLT_exp emin prec) 
  (Znearest choice) x = (x * (1 + d) + e)%R
HeLedx:(Rabs e <= Rabs (d * x))%R
exists eps : R,
  (Rabs eps <= u_ro prec / (1 + u_ro prec))%R /\
  round beta (FLT_exp emin prec) (Znearest choice) x =
  (x * (1 + eps))%R
exists d; rewrite Hde; destruct (Rmult_integral _ _ Hde0) as [Zd|Ze].beta:radix
emin, prec:Z
Hp:(0 < prec)%Z
rnd:R -> Z
valid_rnd:Valid_rnd rnd
choice:Z -> bool
x:R
rx:=round beta (FLT_exp emin prec) (Znearest choice) x:R
d, e:R
Bd:(Rabs d <= u_ro prec / (1 + u_ro prec))%R
Be:(Rabs e <= / 2 * bpow emin)%R
Hde0:(d * e)%R = 0%R
Hde:round beta (FLT_exp emin prec) 
  (Znearest choice) x = (x * (1 + d) + e)%R
HeLedx:(Rabs e <= Rabs (d * x))%R
Zd:d = 0%R
(Rabs d <= u_ro prec / (1 + u_ro prec))%R /\
(x * (1 + d) + e)%R = (x * (1 + d))%R
beta:radix
emin, prec:Z
Hp:(0 < prec)%Z
rnd:R -> Z
valid_rnd:Valid_rnd rnd
choice:Z -> bool
x:R
rx:=round beta (FLT_exp emin prec) (Znearest choice) x:R
d, e:R
Bd:(Rabs d <= u_ro prec / (1 + u_ro prec))%R
Be:(Rabs e <= / 2 * bpow emin)%R
Hde0:(d * e)%R = 0%R
Hde:round beta (FLT_exp emin prec) 
  (Znearest choice) x = (x * (1 + d) + e)%R
HeLedx:(Rabs e <= Rabs (d * x))%R
Ze:e = 0%R
(Rabs d <= u_ro prec / (1 + u_ro prec))%R /\
(x * (1 + d) + e)%R = (x * (1 + d))%R
{beta:radix
emin, prec:Z
Hp:(0 < prec)%Z
rnd:R -> Z
valid_rnd:Valid_rnd rnd
choice:Z -> bool
x:R
rx:=round beta (FLT_exp emin prec) (Znearest choice) x:R
d, e:R
Bd:(Rabs d <= u_ro prec / (1 + u_ro prec))%R
Be:(Rabs e <= / 2 * bpow emin)%R
Hde0:(d * e)%R = 0%R
Hde:round beta (FLT_exp emin prec) 
  (Znearest choice) x = (x * (1 + d) + e)%R
HeLedx:(Rabs e <= Rabs (d * x))%R
Zd:d = 0%R
(Rabs d <= u_ro prec / (1 + u_ro prec))%R /\
(x * (1 + d) + e)%R = (x * (1 + d))%R split; [exact Bd|].beta:radix
emin, prec:Z
Hp:(0 < prec)%Z
rnd:R -> Z
valid_rnd:Valid_rnd rnd
choice:Z -> bool
x:R
rx:=round beta (FLT_exp emin prec) (Znearest choice) x:R
d, e:R
Bd:(Rabs d <= u_ro prec / (1 + u_ro prec))%R
Be:(Rabs e <= / 2 * bpow emin)%R
Hde0:(d * e)%R = 0%R
Hde:round beta (FLT_exp emin prec) 
  (Znearest choice) x = (x * (1 + d) + e)%R
HeLedx:(Rabs e <= Rabs (d * x))%R
Zd:d = 0%R
(x * (1 + d) + e)%R = (x * (1 + d))%R
  assert (Ze : e = 0%R); [|now rewrite Ze, Rplus_0_r].beta:radix
emin, prec:Z
Hp:(0 < prec)%Z
rnd:R -> Z
valid_rnd:Valid_rnd rnd
choice:Z -> bool
x:R
rx:=round beta (FLT_exp emin prec) (Znearest choice) x:R
d, e:R
Bd:(Rabs d <= u_ro prec / (1 + u_ro prec))%R
Be:(Rabs e <= / 2 * bpow emin)%R
Hde0:(d * e)%R = 0%R
Hde:round beta (FLT_exp emin prec) 
  (Znearest choice) x = (x * (1 + d) + e)%R
HeLedx:(Rabs e <= Rabs (d * x))%R
Zd:d = 0%R
e = 0%R
  apply Rabs_eq_R0, Rle_antisym; [|now apply Rabs_pos].beta:radix
emin, prec:Z
Hp:(0 < prec)%Z
rnd:R -> Z
valid_rnd:Valid_rnd rnd
choice:Z -> bool
x:R
rx:=round beta (FLT_exp emin prec) (Znearest choice) x:R
d, e:R
Bd:(Rabs d <= u_ro prec / (1 + u_ro prec))%R
Be:(Rabs e <= / 2 * bpow emin)%R
Hde0:(d * e)%R = 0%R
Hde:round beta (FLT_exp emin prec) 
  (Znearest choice) x = (x * (1 + d) + e)%R
HeLedx:(Rabs e <= Rabs (d * x))%R
Zd:d = 0%R
(Rabs e <= 0)%R
  now revert HeLedx; rewrite Zd, Rmult_0_l, Rabs_R0. }beta:radix
emin, prec:Z
Hp:(0 < prec)%Z
rnd:R -> Z
valid_rnd:Valid_rnd rnd
choice:Z -> bool
x:R
rx:=round beta (FLT_exp emin prec) (Znearest choice) x:R
d, e:R
Bd:(Rabs d <= u_ro prec / (1 + u_ro prec))%R
Be:(Rabs e <= / 2 * bpow emin)%R
Hde0:(d * e)%R = 0%R
Hde:round beta (FLT_exp emin prec) 
  (Znearest choice) x = (x * (1 + d) + e)%R
HeLedx:(Rabs e <= Rabs (d * x))%R
Ze:e = 0%R
(Rabs d <= u_ro prec / (1 + u_ro prec))%R /\
(x * (1 + d) + e)%R = (x * (1 + d))%R
now rewrite Ze, Rplus_0_r.
Qed.

End Fprop_relative_FLT.

Theorem error_N_FLT :
  forall (emin prec : Z), (0 < prec)%Z ->
  forall (choice : Z -> bool),
  forall (x : R),
  exists eps eta : R,
  (Rabs eps <= /2 * bpow (-prec + 1))%R /\
  (Rabs eta <= /2 * bpow emin)%R /\
  (eps * eta = 0)%R /\
  (round beta (FLT_exp emin prec) (Znearest choice) x
   = x * (1 + eps) + eta)%R.beta:radix
forall emin prec : Z,
(0 < prec)%Z ->
forall (choice : Z -> bool) (x : R),
exists eps eta : R,
  (Rabs eps <= / 2 * bpow (- prec + 1))%R /\
  (Rabs eta <= / 2 * bpow emin)%R /\
  (eps * eta)%R = 0%R /\
  round beta (FLT_exp emin prec) (Znearest choice) x =
  (x * (1 + eps) + eta)%R
Proof.beta:radix
forall emin prec : Z,
(0 < prec)%Z ->
forall (choice : Z -> bool) (x : R),
exists eps eta : R,
  (Rabs eps <= / 2 * bpow (- prec + 1))%R /\
  (Rabs eta <= / 2 * bpow emin)%R /\
  (eps * eta)%R = 0%R /\
  round beta (FLT_exp emin prec) (Znearest choice) x =
  (x * (1 + eps) + eta)%R
intros emin prec Pprec choice x.beta:radix
emin, prec:Z
Pprec:(0 < prec)%Z
choice:Z -> bool
x:R
exists eps eta : R,
  (Rabs eps <= / 2 * bpow (- prec + 1))%R /\
  (Rabs eta <= / 2 * bpow emin)%R /\
  (eps * eta)%R = 0%R /\
  round beta (FLT_exp emin prec) (Znearest choice) x =
  (x * (1 + eps) + eta)%R
destruct (Rtotal_order x 0) as [Nx|[Zx|Px]].beta:radix
emin, prec:Z
Pprec:(0 < prec)%Z
choice:Z -> bool
x:R
Nx:(x < 0)%R
exists eps eta : R,
  (Rabs eps <= / 2 * bpow (- prec + 1))%R /\
  (Rabs eta <= / 2 * bpow emin)%R /\
  (eps * eta)%R = 0%R /\
  round beta (FLT_exp emin prec) (Znearest choice) x =
  (x * (1 + eps) + eta)%R
beta:radix
emin, prec:Z
Pprec:(0 < prec)%Z
choice:Z -> bool
x:R
Zx:x = 0%R
exists eps eta : R,
  (Rabs eps <= / 2 * bpow (- prec + 1))%R /\
  (Rabs eta <= / 2 * bpow emin)%R /\
  (eps * eta)%R = 0%R /\
  round beta (FLT_exp emin prec) (Znearest choice) x =
  (x * (1 + eps) + eta)%R
beta:radix
emin, prec:Z
Pprec:(0 < prec)%Z
choice:Z -> bool
x:R
Px:(x > 0)%R
exists eps eta : R,
  (Rabs eps <= / 2 * bpow (- prec + 1))%R /\
  (Rabs eta <= / 2 * bpow emin)%R /\
  (eps * eta)%R = 0%R /\
  round beta (FLT_exp emin prec) (Znearest choice) x =
  (x * (1 + eps) + eta)%R
{beta:radix
emin, prec:Z
Pprec:(0 < prec)%Z
choice:Z -> bool
x:R
Nx:(x < 0)%R
exists eps eta : R,
  (Rabs eps <= / 2 * bpow (- prec + 1))%R /\
  (Rabs eta <= / 2 * bpow emin)%R /\
  (eps * eta)%R = 0%R /\
  round beta (FLT_exp emin prec) (Znearest choice) x =
  (x * (1 + eps) + eta)%R assert (Pmx : (0 < - x)%R).beta:radix
emin, prec:Z
Pprec:(0 < prec)%Z
choice:Z -> bool
x:R
Nx:(x < 0)%R
(0 < - x)%R
beta:radix
emin, prec:Z
Pprec:(0 < prec)%Z
choice:Z -> bool
x:R
Nx:(x < 0)%R
Pmx:(0 < - x)%R
exists eps eta : R,
  (Rabs eps <= / 2 * bpow (- prec + 1))%R /\
  (Rabs eta <= / 2 * bpow emin)%R /\
  (eps * eta)%R = 0%R /\
  round beta (FLT_exp emin prec) (Znearest choice) x =
  (x * (1 + eps) + eta)%R
  {beta:radix
emin, prec:Z
Pprec:(0 < prec)%Z
choice:Z -> bool
x:R
Nx:(x < 0)%R
(0 < - x)%R now rewrite <- Ropp_0; apply Ropp_lt_contravar. }beta:radix
emin, prec:Z
Pprec:(0 < prec)%Z
choice:Z -> bool
x:R
Nx:(x < 0)%R
Pmx:(0 < - x)%R
exists eps eta : R,
  (Rabs eps <= / 2 * bpow (- prec + 1))%R /\
  (Rabs eta <= / 2 * bpow emin)%R /\
  (eps * eta)%R = 0%R /\
  round beta (FLT_exp emin prec) (Znearest choice) x =
  (x * (1 + eps) + eta)%R
  destruct (@error_N_FLT_aux emin prec Pprec
                             (fun t : Z => negb (choice (- (t + 1))%Z))
                             (- x)%R Pmx)
    as (d,(e,(Hd,(He,(Hde,Hr))))).beta:radix
emin, prec:Z
Pprec:(0 < prec)%Z
choice:Z -> bool
x:R
Nx:(x < 0)%R
Pmx:(0 < - x)%R
d, e:R
Hd:(Rabs d <= / 2 * bpow (- prec + 1))%R
He:(Rabs e <= / 2 * bpow emin)%R
Hde:(d * e)%R = 0%R
Hr:round beta (FLT_exp emin prec)
  (Znearest
     (fun t : Z => negb (choice (- (t + 1))%Z)))
  (- x) = (- x * (1 + d) + e)%R
exists eps eta : R,
  (Rabs eps <= / 2 * bpow (- prec + 1))%R /\
  (Rabs eta <= / 2 * bpow emin)%R /\
  (eps * eta)%R = 0%R /\
  round beta (FLT_exp emin prec) (Znearest choice) x =
  (x * (1 + eps) + eta)%R
  exists d; exists (- e)%R; split; [exact Hd|split; [|split]].beta:radix
emin, prec:Z
Pprec:(0 < prec)%Z
choice:Z -> bool
x:R
Nx:(x < 0)%R
Pmx:(0 < - x)%R
d, e:R
Hd:(Rabs d <= / 2 * bpow (- prec + 1))%R
He:(Rabs e <= / 2 * bpow emin)%R
Hde:(d * e)%R = 0%R
Hr:round beta (FLT_exp emin prec)
  (Znearest
     (fun t : Z => negb (choice (- (t + 1))%Z)))
  (- x) = (- x * (1 + d) + e)%R
(Rabs (- e) <= / 2 * bpow emin)%R
beta:radix
emin, prec:Z
Pprec:(0 < prec)%Z
choice:Z -> bool
x:R
Nx:(x < 0)%R
Pmx:(0 < - x)%R
d, e:R
Hd:(Rabs d <= / 2 * bpow (- prec + 1))%R
He:(Rabs e <= / 2 * bpow emin)%R
Hde:(d * e)%R = 0%R
Hr:round beta (FLT_exp emin prec)
  (Znearest
     (fun t : Z => negb (choice (- (t + 1))%Z)))
  (- x) = (- x * (1 + d) + e)%R
(d * - e)%R = 0%R
beta:radix
emin, prec:Z
Pprec:(0 < prec)%Z
choice:Z -> bool
x:R
Nx:(x < 0)%R
Pmx:(0 < - x)%R
d, e:R
Hd:(Rabs d <= / 2 * bpow (- prec + 1))%R
He:(Rabs e <= / 2 * bpow emin)%R
Hde:(d * e)%R = 0%R
Hr:round beta (FLT_exp emin prec)
  (Znearest
     (fun t : Z => negb (choice (- (t + 1))%Z)))
  (- x) = (- x * (1 + d) + e)%R
round beta (FLT_exp emin prec) (Znearest choice) x =
(x * (1 + d) + - e)%R
  {beta:radix
emin, prec:Z
Pprec:(0 < prec)%Z
choice:Z -> bool
x:R
Nx:(x < 0)%R
Pmx:(0 < - x)%R
d, e:R
Hd:(Rabs d <= / 2 * bpow (- prec + 1))%R
He:(Rabs e <= / 2 * bpow emin)%R
Hde:(d * e)%R = 0%R
Hr:round beta (FLT_exp emin prec)
  (Znearest
     (fun t : Z => negb (choice (- (t + 1))%Z)))
  (- x) = (- x * (1 + d) + e)%R
(Rabs (- e) <= / 2 * bpow emin)%R now rewrite Rabs_Ropp. }beta:radix
emin, prec:Z
Pprec:(0 < prec)%Z
choice:Z -> bool
x:R
Nx:(x < 0)%R
Pmx:(0 < - x)%R
d, e:R
Hd:(Rabs d <= / 2 * bpow (- prec + 1))%R
He:(Rabs e <= / 2 * bpow emin)%R
Hde:(d * e)%R = 0%R
Hr:round beta (FLT_exp emin prec)
  (Znearest
     (fun t : Z => negb (choice (- (t + 1))%Z)))
  (- x) = (- x * (1 + d) + e)%R
(d * - e)%R = 0%R
beta:radix
emin, prec:Z
Pprec:(0 < prec)%Z
choice:Z -> bool
x:R
Nx:(x < 0)%R
Pmx:(0 < - x)%R
d, e:R
Hd:(Rabs d <= / 2 * bpow (- prec + 1))%R
He:(Rabs e <= / 2 * bpow emin)%R
Hde:(d * e)%R = 0%R
Hr:round beta (FLT_exp emin prec)
  (Znearest
     (fun t : Z => negb (choice (- (t + 1))%Z)))
  (- x) = (- x * (1 + d) + e)%R
round beta (FLT_exp emin prec) (Znearest choice) x =
(x * (1 + d) + - e)%R
  {beta:radix
emin, prec:Z
Pprec:(0 < prec)%Z
choice:Z -> bool
x:R
Nx:(x < 0)%R
Pmx:(0 < - x)%R
d, e:R
Hd:(Rabs d <= / 2 * bpow (- prec + 1))%R
He:(Rabs e <= / 2 * bpow emin)%R
Hde:(d * e)%R = 0%R
Hr:round beta (FLT_exp emin prec)
  (Znearest
     (fun t : Z => negb (choice (- (t + 1))%Z)))
  (- x) = (- x * (1 + d) + e)%R
(d * - e)%R = 0%R now rewrite Ropp_mult_distr_r_reverse, <- Ropp_0; apply f_equal. }beta:radix
emin, prec:Z
Pprec:(0 < prec)%Z
choice:Z -> bool
x:R
Nx:(x < 0)%R
Pmx:(0 < - x)%R
d, e:R
Hd:(Rabs d <= / 2 * bpow (- prec + 1))%R
He:(Rabs e <= / 2 * bpow emin)%R
Hde:(d * e)%R = 0%R
Hr:round beta (FLT_exp emin prec)
  (Znearest
     (fun t : Z => negb (choice (- (t + 1))%Z)))
  (- x) = (- x * (1 + d) + e)%R
round beta (FLT_exp emin prec) (Znearest choice) x =
(x * (1 + d) + - e)%R
  rewrite <- (Ropp_involutive x), round_N_opp.beta:radix
emin, prec:Z
Pprec:(0 < prec)%Z
choice:Z -> bool
x:R
Nx:(x < 0)%R
Pmx:(0 < - x)%R
d, e:R
Hd:(Rabs d <= / 2 * bpow (- prec + 1))%R
He:(Rabs e <= / 2 * bpow emin)%R
Hde:(d * e)%R = 0%R
Hr:round beta (FLT_exp emin prec)
  (Znearest
     (fun t : Z => negb (choice (- (t + 1))%Z)))
  (- x) = (- x * (1 + d) + e)%R
(-
 round beta (FLT_exp emin prec)
   (Znearest
      (fun t : Z => negb (choice (- (t + 1))%Z)))
   (- x))%R = (- - x * (1 + d) + - e)%R
  now rewrite Ropp_mult_distr_l_reverse, <- Ropp_plus_distr; apply f_equal. }beta:radix
emin, prec:Z
Pprec:(0 < prec)%Z
choice:Z -> bool
x:R
Zx:x = 0%R
exists eps eta : R,
  (Rabs eps <= / 2 * bpow (- prec + 1))%R /\
  (Rabs eta <= / 2 * bpow emin)%R /\
  (eps * eta)%R = 0%R /\
  round beta (FLT_exp emin prec) (Znearest choice) x =
  (x * (1 + eps) + eta)%R
beta:radix
emin, prec:Z
Pprec:(0 < prec)%Z
choice:Z -> bool
x:R
Px:(x > 0)%R
exists eps eta : R,
  (Rabs eps <= / 2 * bpow (- prec + 1))%R /\
  (Rabs eta <= / 2 * bpow emin)%R /\
  (eps * eta)%R = 0%R /\
  round beta (FLT_exp emin prec) (Znearest choice) x =
  (x * (1 + eps) + eta)%R
{beta:radix
emin, prec:Z
Pprec:(0 < prec)%Z
choice:Z -> bool
x:R
Zx:x = 0%R
exists eps eta : R,
  (Rabs eps <= / 2 * bpow (- prec + 1))%R /\
  (Rabs eta <= / 2 * bpow emin)%R /\
  (eps * eta)%R = 0%R /\
  round beta (FLT_exp emin prec) (Znearest choice) x =
  (x * (1 + eps) + eta)%R assert (Ph2 : (0 <= / 2)%R).beta:radix
emin, prec:Z
Pprec:(0 < prec)%Z
choice:Z -> bool
x:R
Zx:x = 0%R
(0 <= / 2)%R
beta:radix
emin, prec:Z
Pprec:(0 < prec)%Z
choice:Z -> bool
x:R
Zx:x = 0%R
Ph2:(0 <= / 2)%R
exists eps eta : R,
  (Rabs eps <= / 2 * bpow (- prec + 1))%R /\
  (Rabs eta <= / 2 * bpow emin)%R /\
  (eps * eta)%R = 0%R /\
  round beta (FLT_exp emin prec) (Znearest choice) x =
  (x * (1 + eps) + eta)%R
  {beta:radix
emin, prec:Z
Pprec:(0 < prec)%Z
choice:Z -> bool
x:R
Zx:x = 0%R
(0 <= / 2)%R apply (Rmult_le_reg_l 2 _ _ Rlt_0_2).beta:radix
emin, prec:Z
Pprec:(0 < prec)%Z
choice:Z -> bool
x:R
Zx:x = 0%R
(2 * 0 <= 2 * / 2)%R
    rewrite Rmult_0_r, Rinv_r; [exact Rle_0_1|].beta:radix
emin, prec:Z
Pprec:(0 < prec)%Z
choice:Z -> bool
x:R
Zx:x = 0%R
2%R <> 0%R
    apply Rgt_not_eq, Rlt_gt, Rlt_0_2. }beta:radix
emin, prec:Z
Pprec:(0 < prec)%Z
choice:Z -> bool
x:R
Zx:x = 0%R
Ph2:(0 <= / 2)%R
exists eps eta : R,
  (Rabs eps <= / 2 * bpow (- prec + 1))%R /\
  (Rabs eta <= / 2 * bpow emin)%R /\
  (eps * eta)%R = 0%R /\
  round beta (FLT_exp emin prec) (Znearest choice) x =
  (x * (1 + eps) + eta)%R
  exists 0%R; exists 0%R; rewrite Zx; split; [|split; [|split]].beta:radix
emin, prec:Z
Pprec:(0 < prec)%Z
choice:Z -> bool
x:R
Zx:x = 0%R
Ph2:(0 <= / 2)%R
(Rabs 0 <= / 2 * bpow (- prec + 1))%R
beta:radix
emin, prec:Z
Pprec:(0 < prec)%Z
choice:Z -> bool
x:R
Zx:x = 0%R
Ph2:(0 <= / 2)%R
(Rabs 0 <= / 2 * bpow emin)%R
beta:radix
emin, prec:Z
Pprec:(0 < prec)%Z
choice:Z -> bool
x:R
Zx:x = 0%R
Ph2:(0 <= / 2)%R
(0 * 0)%R = 0%R
beta:radix
emin, prec:Z
Pprec:(0 < prec)%Z
choice:Z -> bool
x:R
Zx:x = 0%R
Ph2:(0 <= / 2)%R
round beta (FLT_exp emin prec) (Znearest choice) 0 =
(0 * (1 + 0) + 0)%R
  {beta:radix
emin, prec:Z
Pprec:(0 < prec)%Z
choice:Z -> bool
x:R
Zx:x = 0%R
Ph2:(0 <= / 2)%R
(Rabs 0 <= / 2 * bpow (- prec + 1))%R now rewrite Rabs_R0; apply Rmult_le_pos; [|apply bpow_ge_0]. }beta:radix
emin, prec:Z
Pprec:(0 < prec)%Z
choice:Z -> bool
x:R
Zx:x = 0%R
Ph2:(0 <= / 2)%R
(Rabs 0 <= / 2 * bpow emin)%R
beta:radix
emin, prec:Z
Pprec:(0 < prec)%Z
choice:Z -> bool
x:R
Zx:x = 0%R
Ph2:(0 <= / 2)%R
(0 * 0)%R = 0%R
beta:radix
emin, prec:Z
Pprec:(0 < prec)%Z
choice:Z -> bool
x:R
Zx:x = 0%R
Ph2:(0 <= / 2)%R
round beta (FLT_exp emin prec) (Znearest choice) 0 =
(0 * (1 + 0) + 0)%R
  {beta:radix
emin, prec:Z
Pprec:(0 < prec)%Z
choice:Z -> bool
x:R
Zx:x = 0%R
Ph2:(0 <= / 2)%R
(Rabs 0 <= / 2 * bpow emin)%R now rewrite Rabs_R0; apply Rmult_le_pos; [|apply bpow_ge_0]. }beta:radix
emin, prec:Z
Pprec:(0 < prec)%Z
choice:Z -> bool
x:R
Zx:x = 0%R
Ph2:(0 <= / 2)%R
(0 * 0)%R = 0%R
beta:radix
emin, prec:Z
Pprec:(0 < prec)%Z
choice:Z -> bool
x:R
Zx:x = 0%R
Ph2:(0 <= / 2)%R
round beta (FLT_exp emin prec) (Znearest choice) 0 =
(0 * (1 + 0) + 0)%R
  {beta:radix
emin, prec:Z
Pprec:(0 < prec)%Z
choice:Z -> bool
x:R
Zx:x = 0%R
Ph2:(0 <= / 2)%R
(0 * 0)%R = 0%R now rewrite Rmult_0_l. }beta:radix
emin, prec:Z
Pprec:(0 < prec)%Z
choice:Z -> bool
x:R
Zx:x = 0%R
Ph2:(0 <= / 2)%R
round beta (FLT_exp emin prec) (Znearest choice) 0 =
(0 * (1 + 0) + 0)%R
  now rewrite Rmult_0_l, Rplus_0_l, round_0; [|apply valid_rnd_N]. }beta:radix
emin, prec:Z
Pprec:(0 < prec)%Z
choice:Z -> bool
x:R
Px:(x > 0)%R
exists eps eta : R,
  (Rabs eps <= / 2 * bpow (- prec + 1))%R /\
  (Rabs eta <= / 2 * bpow emin)%R /\
  (eps * eta)%R = 0%R /\
  round beta (FLT_exp emin prec) (Znearest choice) x =
  (x * (1 + eps) + eta)%R
now apply error_N_FLT_aux.
Qed.

End Fprop_relative.