Mise à jour terminée. Pour connaître les apports de la version 13.8.4 par rapport à notre ancienne version vous pouvez lire les "Release Notes" suivantes :
https://about.gitlab.com/releases/2021/02/11/security-release-gitlab-13-8-4-released/
https://about.gitlab.com/releases/2021/02/05/gitlab-13-8-3-released/

Fappli_IEEE.v 34.9 KB
Newer Older
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20
(**
This file is part of the Flocq formalization of floating-point
arithmetic in Coq: http://flocq.gforge.inria.fr/

Copyright (C) 2010 Sylvie Boldo
#<br />#
Copyright (C) 2010 Guillaume Melquiond

This library is free software; you can redistribute it and/or
modify it under the terms of the GNU Lesser General Public
License as published by the Free Software Foundation; either
version 3 of the License, or (at your option) any later version.

This library is distributed in the hope that it will be useful,
but WITHOUT ANY WARRANTY; without even the implied warranty of
MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the
COPYING file for more details.
*)

(** * IEEE-754 arithmetic *)
21 22
Require Import Fcore.
Require Import Fcalc_digits.
Guillaume Melquiond's avatar
Guillaume Melquiond committed
23 24 25
Require Import Fcalc_round.
Require Import Fcalc_bracket.
Require Import Fcalc_ops.
26
Require Import Fcalc_div.
27
Require Import Fcalc_sqrt.
Guillaume Melquiond's avatar
Guillaume Melquiond committed
28
Require Import Fprop_relative.
29 30 31

Section Binary.

32
Variable prec emax : Z.
Guillaume Melquiond's avatar
Guillaume Melquiond committed
33
Hypothesis Hprec : (0 < prec)%Z.
34
Hypothesis Hmax : (prec < emax)%Z.
35

36
Let emin := (3 - emax - prec)%Z.
37
Let fexp := FLT_exp emin prec.
38 39
Let fexp_correct : valid_exp fexp := FLT_exp_correct _ _ Hprec.

40 41 42 43
Definition bounded_prec m e :=
  Zeq_bool (fexp (Z_of_nat (S (digits2_Pnat m)) + e)) e.

Definition bounded m e :=
44
  andb (bounded_prec m e) (Zle_bool e (emax - prec)).
45 46 47 48 49 50 51 52 53 54 55 56

Inductive binary_float :=
  | B754_zero : bool -> binary_float
  | B754_infinity : bool -> binary_float
  | B754_nan : binary_float
  | B754_finite : bool ->
    forall (m : positive) (e : Z), bounded m e = true -> binary_float.

Definition radix2 := Build_radix 2 (refl_equal true).

Definition B2R f :=
  match f with
Guillaume Melquiond's avatar
Guillaume Melquiond committed
57
  | B754_finite s m e _ => F2R (Float radix2 (cond_Zopp s (Zpos m)) e)
58 59 60 61 62 63
  | _ => R0
  end.

Theorem canonic_bounded_prec :
  forall (sx : bool) mx ex,
  bounded_prec mx ex = true ->
Guillaume Melquiond's avatar
Guillaume Melquiond committed
64
  canonic radix2 fexp (Float radix2 (cond_Zopp sx (Zpos mx)) ex).
65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142
Proof.
intros sx mx ex H.
assert (Hx := Zeq_bool_eq _ _ H). clear H.
apply sym_eq.
simpl.
pattern ex at 2 ; rewrite <- Hx.
apply (f_equal fexp).
rewrite ln_beta_F2R_digits.
rewrite <- digits_abs.
rewrite Z_of_nat_S_digits2_Pnat.
now case sx.
now case sx.
Qed.

Theorem generic_format_B2R :
  forall x,
  generic_format radix2 fexp (B2R x).
Proof.
intros [sx|sx| |sx mx ex Hx] ; try apply generic_format_0.
simpl.
apply generic_format_canonic.
apply canonic_bounded_prec.
now destruct (andb_prop _ _ Hx) as (H, _).
Qed.

Definition is_finite_strict f :=
  match f with
  | B754_finite _ _ _ _ => true
  | _ => false
  end.

Theorem binary_unicity :
  forall x y : binary_float,
  is_finite_strict x = true ->
  is_finite_strict y = true ->
  B2R x = B2R y ->
  x = y.
Proof.
intros [sx|sx| |sx mx ex Hx] [sy|sy| |sy my ey Hy] ; try easy.
simpl.
intros _ _ Heq.
assert (Hs: sx = sy).
(* *)
revert Heq. clear.
case sx ; case sy ; try easy ;
  intros Heq ; apply False_ind ; revert Heq.
apply Rlt_not_eq.
apply Rlt_trans with R0.
now apply F2R_lt_0_compat.
now apply F2R_gt_0_compat.
apply Rgt_not_eq.
apply Rgt_trans with R0.
now apply F2R_gt_0_compat.
now apply F2R_lt_0_compat.
assert (mx = my /\ ex = ey).
(* *)
refine (_ (canonic_unicity _ fexp _ _ _ _ Heq)).
rewrite Hs.
now case sy ; intro H ; injection H ; split.
apply canonic_bounded_prec.
exact (proj1 (andb_prop _ _ Hx)).
apply canonic_bounded_prec.
exact (proj1 (andb_prop _ _ Hy)).
(* *)
revert Hx.
rewrite Hs, (proj1 H), (proj2 H).
intros Hx.
apply f_equal.
apply eqbool_irrelevance.
Qed.

Definition is_finite f :=
  match f with
  | B754_finite _ _ _ _ => true
  | B754_zero _ => true
  | _ => false
  end.

Guillaume Melquiond's avatar
Guillaume Melquiond committed
143 144 145 146 147 148 149 150 151 152 153 154 155 156 157 158 159 160 161 162 163 164 165 166
Definition Bopp x :=
  match x with
  | B754_nan => x
  | B754_infinity sx => B754_infinity (negb sx)
  | B754_finite sx mx ex Hx => B754_finite (negb sx) mx ex Hx
  | B754_zero sx => B754_zero (negb sx)
  end.

Theorem Bopp_involutive :
  forall x, Bopp (Bopp x) = x.
Proof.
now intros [sx|sx| |sx mx ex Hx] ; simpl ; try rewrite Bool.negb_involutive.
Qed.

Theorem B2R_Bopp :
  forall x,
  B2R (Bopp x) = (- B2R x)%R.
Proof.
intros [sx|sx| |sx mx ex Hx] ; apply sym_eq ; try apply Ropp_0.
simpl.
rewrite opp_F2R.
now case sx.
Qed.

167 168 169
Theorem bounded_lt_emax :
  forall mx ex,
  bounded mx ex = true ->
170
  (F2R (Float radix2 (Zpos mx) ex) < bpow radix2 emax)%R.
171
Proof.
172
intros mx ex Hx.
173 174 175 176 177 178 179 180
destruct (andb_prop _ _ Hx) as (H1,H2).
generalize (Zeq_bool_eq _ _ H1). clear H1. intro H1.
generalize (Zle_bool_imp_le _ _ H2). clear H2. intro H2.
generalize (ln_beta_F2R_digits radix2 (Zpos mx) ex).
destruct (ln_beta radix2 (F2R (Float radix2 (Zpos mx) ex))) as (e',Ex).
unfold ln_beta_val.
intros H.
apply Rlt_le_trans with (bpow radix2 e').
181 182
change (Zpos mx) with (Zabs (Zpos mx)).
rewrite <- abs_F2R.
183 184 185 186 187 188 189 190 191
apply Ex.
apply Rgt_not_eq.
now apply F2R_gt_0_compat.
apply bpow_le.
rewrite H. 2: discriminate.
revert H1. clear -H2.
rewrite Z_of_nat_S_digits2_Pnat.
change Fcalc_digits.radix2 with radix2.
unfold fexp, FLT_exp.
192 193 194
generalize (digits radix2 (Zpos mx)).
intros ; zify ; subst.
clear -H H2. clearbody emin.
195 196 197
omega.
Qed.

198 199
Theorem B2R_lt_emax :
  forall x,
200
  (Rabs (B2R x) < bpow radix2 emax)%R.
201 202 203 204 205 206
Proof.
intros [sx|sx| |sx mx ex Hx] ; simpl ; try ( rewrite Rabs_R0 ; apply bpow_gt_0 ).
rewrite abs_F2R, abs_cond_Zopp.
now apply bounded_lt_emax.
Qed.

207 208 209
Theorem bounded_canonic_lt_emax :
  forall mx ex,
  canonic radix2 fexp (Float radix2 (Zpos mx) ex) ->
210
  (F2R (Float radix2 (Zpos mx) ex) < bpow radix2 emax)%R ->
211 212
  bounded mx ex = true.
Proof.
213
intros mx ex Cx Bx.
214 215 216 217 218 219 220 221 222 223 224 225 226 227 228
apply andb_true_intro.
split.
unfold bounded_prec.
unfold canonic, Fexp in Cx.
rewrite Cx at 2.
rewrite Z_of_nat_S_digits2_Pnat.
change Fcalc_digits.radix2 with radix2.
unfold canonic_exponent.
rewrite ln_beta_F2R_digits. 2: discriminate.
now apply -> Zeq_is_eq_bool.
apply Zle_bool_true.
unfold canonic, Fexp in Cx.
rewrite Cx.
unfold canonic_exponent, fexp, FLT_exp.
destruct (ln_beta radix2 (F2R (Float radix2 (Zpos mx) ex))) as (e',Ex). simpl.
229 230
apply Zmax_lub.
cut (e' - 1 < emax)%Z. clear ; omega.
231 232 233 234 235 236 237
apply lt_bpow with radix2.
apply Rle_lt_trans with (2 := Bx).
change (Zpos mx) with (Zabs (Zpos mx)).
rewrite <- abs_F2R.
apply Ex.
apply Rgt_not_eq.
now apply F2R_gt_0_compat.
238 239
unfold emin. clear -Hprec Hmax.
omega.
240 241
Qed.

242 243 244 245 246 247 248 249 250 251
Inductive mode := mode_NE | mode_ZR | mode_DN | mode_UP | mode_NA.

Definition round_mode m :=
  match m with
  | mode_NE => rndNE
  | mode_ZR => rndZR
  | mode_DN => rndDN
  | mode_UP => rndUP
  | mode_NA => rndNA
  end.
252 253 254 255 256 257 258 259 260 261 262 263 264 265 266 267 268 269 270 271 272 273 274 275 276 277 278

Definition choice_mode m sx mx lx :=
  match m with
  | mode_NE => cond_incr (round_N (negb (Zeven mx)) lx) mx
  | mode_ZR => mx
  | mode_DN => cond_incr (round_sign_DN sx lx) mx
  | mode_UP => cond_incr (round_sign_UP sx lx) mx
  | mode_NA => cond_incr (round_N true lx) mx
  end.

Definition binary_round_sign mode sx mx ex lx :=
  let '(m', e', l') := truncate radix2 fexp (Zpos mx, ex, lx) in
  let '(m'', e'', l'') := truncate radix2 fexp (choice_mode mode sx m' l', e', loc_Exact) in
  match m'' with
  | Z0 => B754_zero sx
  | Zpos m =>
    match Sumbool.sumbool_of_bool (bounded m e'') with
    | left H => B754_finite sx m e'' H
    | right _ => B754_infinity sx
    end
  | _ => B754_nan (* dummy *)
  end.

Theorem binary_round_sign_correct :
  forall mode x mx ex lx,
  inbetween_float radix2 (Zpos mx) ex (Rabs x) lx ->
  (ex <= fexp (digits radix2 (Zpos mx) + ex))%Z ->
279
  if Rlt_bool (Rabs (round radix2 fexp (round_mode mode) x)) (bpow radix2 emax) then
280 281 282
    B2R (binary_round_sign mode (Rlt_bool x 0) mx ex lx) = round radix2 fexp (round_mode mode) x
  else
    binary_round_sign mode (Rlt_bool x 0) mx ex lx = B754_infinity (Rlt_bool x 0).
283
Proof.
284
intros m x mx ex lx Bx Ex.
285 286 287 288 289 290 291 292 293 294 295 296 297 298 299 300 301 302 303 304 305 306 307 308 309 310 311 312 313 314 315
unfold binary_round_sign.
refine (_ (round_trunc_sign_any_correct _ _ fexp_correct (round_mode m) (choice_mode m) _ x (Zpos mx) ex lx Bx (or_introl _ Ex))).
refine (_ (truncate_correct_partial _ _ fexp_correct _ _ _ _ _ Bx Ex)).
destruct (truncate radix2 fexp (Zpos mx, ex, lx)) as ((m1, e1), l1).
set (m1' := choice_mode m (Rlt_bool x 0) m1 l1).
intros (H1a,H1b) H1c.
rewrite H1c.
assert (Hm: (m1 <= m1')%Z).
(* . *)
unfold m1', choice_mode, cond_incr.
case m ;
  try apply Zle_refl ;
  match goal with |- (m1 <= if ?b then _ else _)%Z =>
    case b ; [ apply Zle_succ | apply Zle_refl ] end.
assert (Hr: Rabs (round radix2 fexp (round_mode m) x) = F2R (Float radix2 m1' e1)).
(* . *)
rewrite <- (Zabs_eq m1').
replace (Zabs m1') with (Zabs (cond_Zopp (Rlt_bool x 0) m1')).
rewrite <- abs_F2R.
now apply f_equal.
apply abs_cond_Zopp.
apply Zle_trans with (2 := Hm).
apply Zlt_succ_le.
apply F2R_gt_0_reg with radix2 e1.
apply Rle_lt_trans with (1 := Rabs_pos x).
exact (proj2 (inbetween_float_bounds _ _ _ _ _ H1a)).
(* . *)
assert (Br: inbetween_float radix2 m1' e1 (Rabs (round radix2 fexp (round_mode m) x)) loc_Exact).
now apply inbetween_Exact.
destruct m1' as [|m1'|m1'].
(* . m1' = 0 *)
316
rewrite Rlt_bool_true.
317 318 319 320 321 322
generalize (truncate_0 radix2 fexp e1 loc_Exact).
destruct (truncate radix2 fexp (Z0, e1, loc_Exact)) as ((m2, e2), l2).
intros Hm2.
rewrite Hm2. simpl.
apply sym_eq.
case Rlt_bool ; apply F2R_0.
323 324
rewrite abs_F2R, abs_cond_Zopp, F2R_0.
apply bpow_gt_0.
325 326 327 328 329 330 331 332 333 334 335 336 337 338 339 340 341 342 343 344 345 346 347 348 349 350
(* . 0 < m1' *)
assert (He: (e1 <= fexp (digits radix2 (Zpos m1') + e1))%Z).
rewrite <- ln_beta_F2R_digits, <- Hr, ln_beta_abs.
2: discriminate.
rewrite H1b.
rewrite canonic_exponent_abs.
fold (canonic_exponent radix2 fexp (round radix2 fexp (round_mode m) x)).
apply canonic_exponent_round.
apply fexp_correct.
apply FLT_exp_monotone.
rewrite H1c.
case (Rlt_bool x 0).
apply Rlt_not_eq.
now apply F2R_lt_0_compat.
apply Rgt_not_eq.
now apply F2R_gt_0_compat.
refine (_ (truncate_correct_partial _ _ fexp_correct _ _ _ _ _ Br He)).
2: now rewrite Hr ; apply F2R_gt_0_compat.
refine (_ (truncate_correct_format radix2 fexp (Zpos m1') e1 _ _ He)).
2: discriminate.
destruct (truncate radix2 fexp (Zpos m1', e1, loc_Exact)) as ((m2, e2), l2).
intros (H3,H4) (H2,_).
destruct m2 as [|m2|m2].
elim Rgt_not_eq with (2 := H3).
rewrite F2R_0.
now apply F2R_gt_0_compat.
351 352
rewrite F2R_cond_Zopp, H3, abs_cond_Ropp, abs_F2R.
simpl Zabs.
353
case (Sumbool.sumbool_of_bool (bounded m2 e2)) ; intros Hb.
354
rewrite Rlt_bool_true.
355
simpl.
356 357 358 359 360 361 362 363 364
apply F2R_cond_Zopp.
now apply bounded_lt_emax.
rewrite Rlt_bool_false.
apply refl_equal.
apply Rnot_lt_le.
intros Hx.
revert Hb.
rewrite bounded_canonic_lt_emax with (2 := Hx).
discriminate.
365 366 367 368 369 370 371 372 373 374 375 376 377 378 379 380 381 382 383 384 385 386 387 388 389 390
unfold canonic.
now rewrite <- H3.
elim Rgt_not_eq with (2 := H3).
apply Rlt_trans with R0.
now apply F2R_lt_0_compat.
now apply F2R_gt_0_compat.
rewrite <- Hr.
apply generic_format_abs.
apply generic_format_round.
apply fexp_correct.
(* . not m1' < 0 *)
elim Rgt_not_eq with (2 := Hr).
apply Rlt_le_trans with R0.
now apply F2R_lt_0_compat.
apply Rabs_pos.
(* *)
apply Rlt_le_trans with (2 := proj1 (inbetween_float_bounds _ _ _ _ _ Bx)).
now apply F2R_gt_0_compat.
(* all the modes are valid *)
clear. case m.
exact inbetween_int_NE_sign.
exact inbetween_int_ZR_sign.
exact inbetween_int_DN_sign.
exact inbetween_int_UP_sign.
exact inbetween_int_NA_sign.
Qed.
391

392 393 394 395 396 397 398 399
Definition Bsign x :=
  match x with
  | B754_nan => false
  | B754_zero s => s
  | B754_infinity s => s
  | B754_finite s _ _ _ => s
  end.

400 401 402 403 404 405 406 407 408 409 410 411 412 413 414 415
Definition Bmult m x y :=
  match x, y with
  | B754_nan, _ => x
  | _, B754_nan => y
  | B754_infinity sx, B754_infinity sy => B754_infinity (xorb sx sy)
  | B754_infinity sx, B754_finite sy _ _ _ => B754_infinity (xorb sx sy)
  | B754_finite sx _ _ _, B754_infinity sy => B754_infinity (xorb sx sy)
  | B754_infinity _, B754_zero _ => B754_nan
  | B754_zero _, B754_infinity _ => B754_nan
  | B754_finite sx _ _ _, B754_zero sy => B754_zero (xorb sx sy)
  | B754_zero sx, B754_finite sy _ _ _ => B754_zero (xorb sx sy)
  | B754_zero sx, B754_zero sy => B754_zero (xorb sx sy)
  | B754_finite sx mx ex Hx, B754_finite sy my ey Hy =>
    binary_round_sign m (xorb sx sy) (Pmult mx my) (ex + ey) loc_Exact
  end.

416
Theorem Bmult_correct :
417
  forall m x y,
418
  if Rlt_bool (Rabs (round radix2 fexp (round_mode m) (B2R x * B2R y))) (bpow radix2 emax) then
419 420 421
    B2R (Bmult m x y) = round radix2 fexp (round_mode m) (B2R x * B2R y)
  else
    Bmult m x y = B754_infinity (xorb (Bsign x) (Bsign y)).
422 423
Proof.
intros m [sx|sx| |sx mx ex Hx] [sy|sy| |sy my ey Hy] ;
424
  try ( rewrite ?Rmult_0_r, ?Rmult_0_l, round_0, Rabs_R0, Rlt_bool_true ; [ apply refl_equal | apply bpow_gt_0 ] ).
425 426 427 428 429 430 431 432 433 434 435 436 437 438 439 440 441 442 443 444 445 446 447
simpl.
rewrite <- mult_F2R.
simpl Fmult.
replace (xorb sx sy) with (Rlt_bool (F2R (Float radix2 (cond_Zopp sx (Zpos mx) * cond_Zopp sy (Zpos my)) (ex + ey))) 0).
apply binary_round_sign_correct.
constructor.
rewrite abs_F2R.
apply F2R_eq_compat.
rewrite Zabs_Zmult.
now rewrite 2!abs_cond_Zopp.
(* *)
change (Zpos (mx * my)) with (Zpos mx * Zpos my)%Z.
assert (forall m e, bounded m e = true -> fexp (digits radix2 (Zpos m) + e) = e)%Z.
clear. intros m e Hb.
destruct (andb_prop _ _ Hb) as (H,_).
apply Zeq_bool_eq.
now rewrite <- Z_of_nat_S_digits2_Pnat.
generalize (H _ _ Hx) (H _ _ Hy).
clear sx sy Hx Hy H.
unfold fexp, FLT_exp.
refine (_ (digits_mult_ge radix2 (Zpos mx) (Zpos my) _ _)) ; try discriminate.
refine (_ (digits_gt_0 radix2 (Zpos mx) _) (digits_gt_0 radix2 (Zpos my) _)) ; try discriminate.
generalize (digits radix2 (Zpos mx)) (digits radix2 (Zpos my)) (digits radix2 (Zpos mx * Zpos my)).
448 449
clear -Hprec Hmax.
unfold emin.
450 451 452 453 454 455 456 457 458 459 460 461 462 463
intros dx dy dxy Hx Hy Hxy.
zify ; intros ; subst.
omega.
(* *)
case sx ; case sy.
apply Rlt_bool_false.
now apply F2R_ge_0_compat.
apply Rlt_bool_true.
now apply F2R_lt_0_compat.
apply Rlt_bool_true.
now apply F2R_lt_0_compat.
apply Rlt_bool_false.
now apply F2R_ge_0_compat.
Qed.
464 465 466 467 468 469 470 471 472 473 474 475

Theorem Bmult_correct_finite :
  forall m x y,
  is_finite (Bmult m x y) = true ->
  B2R (Bmult m x y) = round radix2 fexp (round_mode m) (B2R x * B2R y)%R.
Proof.
intros m x y.
generalize (Bmult_correct m x y).
destruct (Bmult m x y) as [sz|sz| |sz mz ez Hz] ; try easy.
now case Rlt_bool.
now case Rlt_bool.
Qed.
476 477 478 479 480 481 482 483 484 485 486 487 488 489 490 491 492 493 494 495 496 497 498 499 500 501 502 503 504 505 506 507 508 509 510 511 512 513 514 515 516 517 518 519 520 521 522 523 524 525 526 527

Definition fexp_scale mx ex :=
  let ex' := fexp (Z_of_nat (S (digits2_Pnat mx)) + ex) in
  match (ex' - ex)%Z with
  | Zneg d => (shift_pos d mx, ex')
  | _ => (mx, ex)
  end.

Theorem fexp_scale_correct :
  forall mx ex,
  let (mx', ex') := fexp_scale mx ex in
  F2R (Float radix2 (Zpos mx) ex) = F2R (Float radix2 (Zpos mx') ex') /\
  (ex' <= fexp (digits radix2 (Zpos mx') + ex'))%Z.
Proof.
intros mx ex.
unfold fexp_scale.
rewrite Z_of_nat_S_digits2_Pnat.
change (Fcalc_digits.radix2) with radix2.
set (e' := fexp (digits radix2 (Zpos mx) + ex)).
pattern e' at 2 ; replace e' with (e' - ex + ex)%Z by ring.
case_eq (e' - ex)%Z ; fold e'.
(* d = 0 *)
intros H.
repeat split.
rewrite Zminus_eq with (1 := H).
apply Zle_refl.
(* d > 0 *)
intros d Hd.
repeat split.
replace e' with (e' - ex + ex)%Z by ring.
rewrite Hd.
pattern ex at 1 ; rewrite <- Zplus_0_l.
now apply Zplus_le_compat_r.
(* d < 0 *)
intros d Hd.
rewrite shift_pos_correct, Zmult_comm.
change (Zpower_pos 2 d) with (Zpower radix2 (Zpos d)).
rewrite digits_shift ; try easy.
change (Zpos d) with (Zopp (Zneg d)).
rewrite <- Hd.
split.
replace (- (e' - ex))%Z with (ex - e')%Z by ring.
replace (e' - ex + ex)%Z with e' by ring.
apply F2R_change_exp.
apply Zle_0_minus_le.
replace (ex - e')%Z with (-(e' - ex))%Z by ring.
now rewrite Hd.
ring_simplify (digits radix2 (Zpos mx) + - (e' - ex) + (e' - ex + ex))%Z.
fold e'.
ring_simplify.
apply Zle_refl.
Qed.
528 529 530 531 532 533 534 535 536 537 538 539 540 541 542

Definition Bplus m x y :=
  match x, y with
  | B754_nan, _ => x
  | _, B754_nan => y
  | B754_infinity sx, B754_infinity sy =>
    if Bool.eqb sx sy then x else B754_nan
  | B754_infinity _, _ => x
  | _, B754_infinity _ => y
  | B754_zero sx, B754_zero sy =>
    if Bool.eqb sx sy then x else
    match m with mode_DN => B754_zero true | _ => B754_zero false end
  | B754_zero _, _ => y
  | _, B754_zero _ => x
  | B754_finite sx mx ex Hx, B754_finite sy my ey Hy =>
543 544 545 546 547 548 549 550
    let fx := Float radix2 (cond_Zopp sx (Zpos mx)) ex in
    let fy := Float radix2 (cond_Zopp sy (Zpos my)) ey in
    match Fplus radix2 fx fy with
    | Float Z0 _ =>
      match m with mode_DN => B754_zero true | _ => B754_zero false end
    | Float (Zpos mz) ez => let '(m', e') := fexp_scale mz ez in binary_round_sign m false m' e' loc_Exact
    | Float (Zneg mz) ez => let '(m', e') := fexp_scale mz ez in binary_round_sign m true m' e' loc_Exact
    end
551 552
  end.

553
Theorem Bplus_correct :
554 555 556
  forall m x y,
  is_finite x = true ->
  is_finite y = true ->
557
  if Rlt_bool (Rabs (round radix2 fexp (round_mode m) (B2R x + B2R y))) (bpow radix2 emax) then
558 559 560
    B2R (Bplus m x y) = round radix2 fexp (round_mode m) (B2R x + B2R y)
  else
    (Bplus m x y = B754_infinity (Bsign x) /\ Bsign x = Bsign y).
561
Proof.
562
intros m [sx|sx| |sx mx ex Hx] [sy|sy| |sy my ey Hy] Fx Fy ; try easy.
563
(* *)
564
rewrite Rplus_0_r, round_0, Rabs_R0, Rlt_bool_true.
565 566 567
simpl.
case (Bool.eqb sx sy) ; try easy.
now case m.
568
apply bpow_gt_0.
569
(* *)
570 571 572
rewrite Rplus_0_l, round_generic, Rlt_bool_true.
apply refl_equal.
apply B2R_lt_emax.
573 574
apply generic_format_B2R.
(* *)
575 576 577
rewrite Rplus_0_r, round_generic, Rlt_bool_true.
apply refl_equal.
apply B2R_lt_emax.
578 579
apply generic_format_B2R.
(* *)
580 581 582 583 584 585
clear Fx Fy.
simpl.
rewrite <- plus_F2R.
case_eq (Fplus radix2 (Float radix2 (cond_Zopp sx (Zpos mx)) ex)
  (Float radix2 (cond_Zopp sy (Zpos my)) ey)).
intros mz ez Hz.
586
assert (Sz: (bpow radix2 emax <= Rabs (round radix2 fexp (round_mode m) (F2R (Float radix2 mz ez))))%R -> sx = Rlt_bool (F2R (Float radix2 mz ez)) 0 /\ sx = sy).
587 588 589
(* . *)
rewrite <- Hz.
rewrite plus_F2R.
Guillaume Melquiond's avatar
Guillaume Melquiond committed
590 591
intros Bz.
destruct (Bool.bool_dec sx sy) as [Hs|Hs].
592
(* .. *)
Guillaume Melquiond's avatar
Guillaume Melquiond committed
593
refine (conj _ Hs).
594 595 596 597 598 599 600 601 602 603 604 605 606
rewrite Hs.
apply sym_eq.
case sy.
apply Rlt_bool_true.
rewrite <- (Rplus_0_r 0).
apply Rplus_lt_compat.
now apply F2R_lt_0_compat.
now apply F2R_lt_0_compat.
apply Rlt_bool_false.
rewrite <- (Rplus_0_r 0).
apply Rplus_le_compat.
now apply F2R_ge_0_compat.
now apply F2R_ge_0_compat.
Guillaume Melquiond's avatar
Guillaume Melquiond committed
607 608 609 610 611 612 613 614 615 616 617 618 619 620 621 622 623 624 625 626 627 628 629 630 631 632 633 634 635 636 637 638 639 640 641 642 643 644 645 646 647 648 649 650 651 652 653 654 655 656 657 658
(* .. *)
elim Rle_not_lt with (1 := Bz).
generalize (bounded_lt_emax _ _ Hx) (bounded_lt_emax _ _ Hy) (andb_prop _ _ Hx) (andb_prop _ _ Hy).
intros Bx By (Hx',_) (Hy',_).
generalize (canonic_bounded_prec sx _ _ Hx') (canonic_bounded_prec sy _ _ Hy').
clear -Bx By Hs fexp_correct.
intros Cx Cy.
destruct sx.
(* ... *)
destruct sy.
now elim Hs.
clear Hs.
apply Rabs_lt.
split.
apply Rlt_le_trans with (F2R (Float radix2 (cond_Zopp true (Zpos mx)) ex)).
rewrite <- opp_F2R.
now apply Ropp_lt_contravar.
apply round_monotone_l.
apply fexp_correct.
now apply generic_format_canonic.
pattern (F2R (Float radix2 (cond_Zopp true (Zpos mx)) ex)) at 1 ; rewrite <- Rplus_0_r.
apply Rplus_le_compat_l.
now apply F2R_ge_0_compat.
apply Rle_lt_trans with (2 := By).
apply round_monotone_r.
apply fexp_correct.
now apply generic_format_canonic.
rewrite <- (Rplus_0_l (F2R (Float radix2 (Zpos my) ey))).
apply Rplus_le_compat_r.
now apply F2R_le_0_compat.
(* ... *)
destruct sy.
2: now elim Hs.
clear Hs.
apply Rabs_lt.
split.
apply Rlt_le_trans with (F2R (Float radix2 (cond_Zopp true (Zpos my)) ey)).
rewrite <- opp_F2R.
now apply Ropp_lt_contravar.
apply round_monotone_l.
apply fexp_correct.
now apply generic_format_canonic.
pattern (F2R (Float radix2 (cond_Zopp true (Zpos my)) ey)) at 1 ; rewrite <- Rplus_0_l.
apply Rplus_le_compat_r.
now apply F2R_ge_0_compat.
apply Rle_lt_trans with (2 := Bx).
apply round_monotone_r.
apply fexp_correct.
now apply generic_format_canonic.
rewrite <- (Rplus_0_r (F2R (Float radix2 (Zpos mx) ex))).
apply Rplus_le_compat_l.
now apply F2R_le_0_compat.
659 660 661 662 663 664 665 666 667 668 669 670 671 672 673 674 675 676 677 678 679 680 681 682 683 684 685 686 687 688 689 690 691 692 693 694 695 696 697 698 699 700
destruct mz as [|mz|mz].
(* . mz = 0 *)
rewrite F2R_0, round_0, Rabs_R0, Rlt_bool_true.
now case m.
apply bpow_gt_0.
(* . mz > 0 *)
generalize (fexp_scale_correct mz ez).
destruct (fexp_scale mz ez) as (m', e').
intros (Hz', Ez).
refine (_ (binary_round_sign_correct m (F2R (Float radix2 (Zpos mz) ez)) m' e' loc_Exact _ Ez)).
2: constructor ; now rewrite abs_F2R.
revert Sz.
rewrite (Rlt_bool_false _ 0).
2: now apply F2R_ge_0_compat.
intros Sz.
case Rlt_bool_spec ; intros Bz.
easy.
specialize (Sz Bz).
intros H.
split.
rewrite H.
now apply f_equal.
apply Sz.
(* . mz < 0 *)
generalize (fexp_scale_correct mz ez).
destruct (fexp_scale mz ez) as (m', e').
intros (Hz', Ez).
refine (_ (binary_round_sign_correct m (F2R (Float radix2 (Zneg mz) ez)) m' e' loc_Exact _ Ez)).
2: constructor ; now rewrite abs_F2R.
revert Sz.
rewrite (Rlt_bool_true _ 0).
intros Sz.
2: now apply F2R_lt_0_compat.
case Rlt_bool_spec ; intros Bz.
easy.
specialize (Sz Bz).
intros H.
split.
rewrite H.
now apply f_equal.
apply Sz.
Qed.
701

702 703 704 705 706 707 708 709 710 711 712 713 714 715 716 717 718 719 720 721 722 723 724 725 726
Definition Bminus m x y := Bplus m x (Bopp y).

Definition Bdiv m x y :=
  match x, y with
  | B754_nan, _ => x
  | _, B754_nan => y
  | B754_infinity sx, B754_infinity sy => B754_nan
  | B754_infinity sx, B754_finite sy _ _ _ => B754_infinity (xorb sx sy)
  | B754_finite sx _ _ _, B754_infinity sy => B754_infinity (xorb sx sy)
  | B754_infinity sx, B754_zero sy => B754_infinity (xorb sx sy)
  | B754_zero sx, B754_infinity sy => B754_zero (xorb sx sy)
  | B754_finite sx _ _ _, B754_zero sy => B754_infinity (xorb sx sy)
  | B754_zero sx, B754_finite sy _ _ _ => B754_zero (xorb sx sy)
  | B754_zero sx, B754_zero sy => B754_nan
  | B754_finite sx mx ex Hx, B754_finite sy my ey Hy =>
    let '(mz, ez, lz) := Fdiv_core radix2 prec (Zpos mx) ex (Zpos my) ey in
    match mz with
    | Zpos mz => binary_round_sign m (xorb sx sy) mz ez lz
    | _ => B754_nan (* dummy *)
    end
  end.

Theorem Bdiv_correct :
  forall m x y,
  B2R y <> R0 ->
727
  if Rlt_bool (Rabs (round radix2 fexp (round_mode m) (B2R x / B2R y))) (bpow radix2 emax) then
728 729 730 731 732 733 734 735
    B2R (Bdiv m x y) = round radix2 fexp (round_mode m) (B2R x / B2R y)
  else
    Bdiv m x y = B754_infinity (xorb (Bsign x) (Bsign y)).
Proof.
intros m x [sy|sy| |sy my ey Hy] Zy ; try now elim Zy.
revert x.
unfold Rdiv.
intros [sx|sx| |sx mx ex Hx] ;
736
  try ( rewrite Rmult_0_l, round_0, Rabs_R0, Rlt_bool_true ; [ apply refl_equal | apply bpow_gt_0 ] ).
737 738 739 740 741 742 743 744 745 746 747 748 749 750 751 752 753 754 755 756 757 758 759 760 761 762 763 764 765 766 767 768 769 770 771 772 773 774 775 776 777 778 779 780 781 782 783 784 785 786 787 788 789 790 791 792 793 794 795 796 797 798 799 800 801 802 803 804 805 806 807 808 809 810 811 812 813
simpl.
refine (_ (Fdiv_core_correct radix2 prec (Zpos mx) ex (Zpos my) ey _ _ _)) ; try easy.
destruct (Fdiv_core radix2 prec (Zpos mx) ex (Zpos my) ey) as ((mz, ez), lz).
intros (Pz, Bz).
replace (xorb sx sy) with (Rlt_bool (F2R (Float radix2 (cond_Zopp sx (Zpos mx)) ex) *
   / F2R (Float radix2 (cond_Zopp sy (Zpos my)) ey)) 0).
destruct mz as [|mz|mz].
(* . mz = 0 *)
elim (Zlt_irrefl prec).
now apply Zle_lt_trans with Z0.
(* . mz > 0 *)
apply binary_round_sign_correct.
rewrite Rabs_mult, Rabs_Rinv.
now rewrite 2!abs_F2R, 2!abs_cond_Zopp.
exact Zy.
revert Pz.
generalize (digits radix2 (Zpos mz)).
unfold fexp, FLT_exp.
clear.
intros ; zify ; subst.
omega.
(* . mz < 0 *)
elim Rlt_not_le  with (1 := proj2 (inbetween_float_bounds _ _ _ _ _ Bz)).
apply Rle_trans with R0.
apply F2R_le_0_compat.
now case mz.
apply Rmult_le_pos.
now apply F2R_ge_0_compat.
apply Rlt_le.
apply Rinv_0_lt_compat.
now apply F2R_gt_0_compat.
(* *)
revert Zy. clear.
case sy ; simpl.
change (Zneg my) with (Zopp (Zpos my)).
rewrite <- opp_F2R.
intros Zy.
rewrite <- Ropp_inv_permute.
rewrite Ropp_mult_distr_r_reverse.
case sx ; simpl.
apply Rlt_bool_false.
rewrite <- Ropp_mult_distr_l_reverse.
apply Rmult_le_pos.
rewrite opp_F2R.
now apply F2R_ge_0_compat.
apply Rlt_le.
apply Rinv_0_lt_compat.
now apply F2R_gt_0_compat.
apply Rlt_bool_true.
rewrite <- Ropp_0.
apply Ropp_lt_contravar.
apply Rmult_lt_0_compat.
now apply F2R_gt_0_compat.
apply Rinv_0_lt_compat.
now apply F2R_gt_0_compat.
contradict Zy.
rewrite Zy.
apply Ropp_0.
intros Zy.
case sx.
apply Rlt_bool_true.
rewrite <- opp_F2R.
rewrite Ropp_mult_distr_l_reverse.
rewrite <- Ropp_0.
apply Ropp_lt_contravar.
apply Rmult_lt_0_compat.
now apply F2R_gt_0_compat.
apply Rinv_0_lt_compat.
now apply F2R_gt_0_compat.
apply Rlt_bool_false.
apply Rmult_le_pos.
now apply F2R_ge_0_compat.
apply Rlt_le.
apply Rinv_0_lt_compat.
now apply F2R_gt_0_compat.
Qed.

814 815 816 817 818 819 820 821 822 823 824 825 826 827 828 829 830 831 832 833 834 835 836 837 838 839 840 841 842 843 844 845 846 847 848 849 850 851 852 853 854 855 856 857
Definition Bsqrt m x :=
  match x with
  | B754_nan => x
  | B754_infinity false => x
  | B754_infinity true => B754_nan
  | B754_finite true _ _ _ => B754_nan
  | B754_zero _ => x
  | B754_finite sx mx ex Hx =>
    let '(mz, ez, lz) := Fsqrt_core radix2 prec (Zpos mx) ex in
    match mz with
    | Zpos mz => binary_round_sign m false mz ez lz
    | _ => B754_nan (* dummy *)
    end
  end.

Theorem Bsqrt_correct :
  forall m x,
  B2R (Bsqrt m x) = round radix2 fexp (round_mode m) (sqrt (B2R x)).
Proof.
intros m [sx|[|]| |sx mx ex Hx] ; try ( now simpl ; rewrite sqrt_0, round_0 ).
simpl.
refine (_ (Fsqrt_core_correct radix2 prec (Zpos mx) ex _)) ; try easy.
destruct (Fsqrt_core radix2 prec (Zpos mx) ex) as ((mz, ez), lz).
intros (Pz, Bz).
case sx.
apply sym_eq.
unfold sqrt.
case Rcase_abs.
intros _.
apply round_0.
intros H.
elim Rge_not_lt with (1 := H).
now apply F2R_lt_0_compat.
destruct mz as [|mz|mz].
(* . mz = 0 *)
elim (Zlt_irrefl prec).
now apply Zle_lt_trans with Z0.
(* . mz > 0 *)
simpl.
refine (_ (binary_round_sign_correct m (sqrt (F2R (Float radix2 (Zpos mx) ex))) mz ez lz _ _)).
rewrite Rlt_bool_true.
rewrite Rlt_bool_false.
easy.
apply sqrt_ge_0.
Guillaume Melquiond's avatar
Guillaume Melquiond committed
858
(* .. *)
859
rewrite Rabs_pos_eq.
Guillaume Melquiond's avatar
Guillaume Melquiond committed
860 861 862 863 864 865 866 867 868 869 870 871 872 873 874 875 876 877 878 879 880 881 882 883 884 885 886 887 888 889 890 891 892 893 894 895 896 897 898 899 900 901 902 903 904 905 906 907 908 909 910 911 912 913 914 915 916 917 918 919 920
refine (_ (relative_error_FLT_ex radix2 emin prec Hprec (round_mode m) (sqrt (F2R (Float radix2 (Zpos mx) ex))) _)).
fold fexp.
intros (eps, (Heps, Hr)).
rewrite Hr.
assert (Heps': (Rabs eps < 1)%R).
apply Rlt_le_trans with (1 := Heps).
fold (bpow radix2 0).
apply bpow_le.
clear -Hprec. omega.
apply Rsqr_incrst_0.
3: apply bpow_ge_0.
rewrite Rsqr_mult.
rewrite Rsqr_sqrt.
2: now apply F2R_ge_0_compat.
unfold Rsqr.
apply Rmult_ge_0_gt_0_lt_compat.
apply Rle_ge.
apply Rle_0_sqr.
apply bpow_gt_0.
now apply bounded_lt_emax.
apply Rlt_le_trans with 4%R.
apply Rsqr_incrst_1.
apply Rplus_lt_compat_l.
apply (Rabs_lt_inv _ _ Heps').
rewrite <- (Rplus_opp_r 1).
apply Rplus_le_compat_l.
apply Rlt_le.
apply (Rabs_lt_inv _ _ Heps').
now apply (Z2R_le 0 2).
change 4%R with (bpow radix2 2).
apply bpow_le.
clear -Hprec Hmax.
omega.
apply Rmult_le_pos.
apply sqrt_ge_0.
rewrite <- (Rplus_opp_r 1).
apply Rplus_le_compat_l.
apply Rlt_le.
apply (Rabs_lt_inv _ _ Heps').
rewrite Rabs_pos_eq.
2: apply sqrt_ge_0.
apply Rsqr_incr_0.
2: apply bpow_ge_0.
2: apply sqrt_ge_0.
rewrite Rsqr_sqrt.
2: now apply F2R_ge_0_compat.
apply Rle_trans with (bpow radix2 emin).
unfold Rsqr.
rewrite <- bpow_plus.
apply bpow_le.
unfold emin.
clear -Hprec Hmax.
omega.
apply generic_format_ge_bpow with fexp.
intros.
apply Zle_max_r.
now apply F2R_gt_0_compat.
apply generic_format_canonic.
apply (canonic_bounded_prec false).
apply (andb_prop _ _ Hx).
(* .. *)
921 922 923 924 925 926 927 928 929 930 931 932 933 934 935 936 937 938 939 940 941
apply round_monotone_l.
apply fexp_correct.
apply generic_format_0.
apply sqrt_ge_0.
rewrite Rabs_pos_eq.
exact Bz.
apply sqrt_ge_0.
revert Pz.
generalize (digits radix2 (Zpos mz)).
unfold fexp, FLT_exp.
clear.
intros ; zify ; subst.
omega.
(* . mz < 0 *)
elim Rlt_not_le  with (1 := proj2 (inbetween_float_bounds _ _ _ _ _ Bz)).
apply Rle_trans with R0.
apply F2R_le_0_compat.
now case mz.
apply sqrt_ge_0.
Qed.

942
End Binary.
943 944 945 946 947 948 949 950 951 952 953 954 955 956 957 958 959 960 961 962 963 964 965 966 967 968 969 970 971 972 973 974 975 976 977 978 979 980 981 982 983 984 985 986 987 988 989 990 991 992 993 994 995 996 997 998 999 1000 1001 1002 1003 1004 1005 1006 1007 1008 1009 1010 1011 1012 1013 1014 1015 1016 1017 1018 1019 1020 1021 1022 1023 1024 1025 1026 1027 1028 1029 1030 1031 1032 1033 1034 1035 1036 1037 1038 1039 1040 1041 1042 1043 1044 1045 1046 1047 1048 1049 1050 1051 1052 1053 1054 1055 1056 1057 1058 1059 1060 1061 1062 1063 1064 1065 1066 1067 1068 1069 1070 1071 1072 1073 1074 1075 1076 1077 1078 1079 1080 1081 1082 1083 1084 1085 1086 1087 1088 1089 1090 1091 1092 1093 1094 1095 1096 1097 1098 1099 1100 1101 1102 1103 1104 1105 1106 1107 1108 1109 1110 1111 1112 1113 1114 1115 1116 1117 1118 1119 1120 1121 1122 1123 1124 1125 1126 1127 1128 1129 1130 1131 1132 1133 1134 1135 1136 1137 1138 1139 1140 1141 1142

Section Binary_Bits.

Variable mw ew : Z.
Hypothesis Hmw : (0 <= mw)%Z.
Hypothesis Hew : (0 < ew)%Z.

Let emax := Zpower 2 (ew - 1).
Let prec := (mw + 1)%Z.
Let emin := (3 - emax - prec)%Z.
Let binary_float := binary_float prec emax.

Let Hprec : (0 < prec)%Z.
unfold prec.
now apply Zle_lt_succ.
Qed.

Let Hm_gt_0 : (0 < 2^mw)%Z.
now apply (Zpower_gt_0 radix2).
Qed.

Let He_gt_0 : (0 < 2^ew)%Z.
apply (Zpower_gt_0 radix2).
now apply Zlt_le_weak.
Qed.

Hypothesis Hmax : (prec < emax)%Z.

Definition join_bits (s : bool) m e :=
  (((if s then Zpower 2 ew else 0) + e) * Zpower 2 mw + m)%Z.

Definition split_bits x :=
  let mm := Zpower 2 mw in
  let em := Zpower 2 ew in
  (Zle_bool (mm * em) x, Zmod x mm, Zmod (Zdiv x mm) em)%Z.

Theorem split_join_bits :
  forall s m e,
  (0 <= m < Zpower 2 mw)%Z ->
  (0 <= e < Zpower 2 ew)%Z ->
  split_bits (join_bits s m e) = (s, m, e).
Proof.
intros s m e Hm He.
unfold split_bits, join_bits.
apply f_equal2.
apply f_equal2.
(* *)
case s.
apply Zle_bool_true.
apply Zle_0_minus_le.
ring_simplify.
apply Zplus_le_0_compat.
apply Zmult_le_0_compat.
apply He.
now apply Zlt_le_weak.
apply Hm.
apply Zle_bool_false.
apply Zplus_lt_reg_l with (2^mw * (-e))%Z.
replace (2 ^ mw * - e + ((0 + e) * 2 ^ mw + m))%Z with (m * 1)%Z by ring.
rewrite <- Zmult_plus_distr_r.
apply Zlt_le_trans with (2^mw * 1)%Z.
now apply Zmult_lt_compat_r.
apply Zmult_le_compat_l.
clear -He. omega.
now apply Zlt_le_weak.
(* *)
rewrite Zplus_comm.
rewrite Z_mod_plus_full.
now apply Zmod_small.
(* *)
rewrite Z_div_plus_full_l.
rewrite Zdiv_small with (1 := Hm).
rewrite Zplus_0_r.
case s.
replace (2^ew + e)%Z with (e + 1 * 2^ew)%Z by ring.
rewrite Z_mod_plus_full.
now apply Zmod_small.
now apply Zmod_small.
now apply Zgt_not_eq.
Qed.

Theorem join_split_bits :
  forall x,
  (0 <= x < Zpower 2 (mw + ew + 1))%Z ->
  let '(s, m, e) := split_bits x in
  join_bits s m e = x.
Proof.
intros x Hx.
unfold split_bits, join_bits.
pattern x at 4 ; rewrite Z_div_mod_eq_full with x (2^mw)%Z.
apply (f_equal (fun v => (v + _)%Z)).
rewrite Zmult_comm.
apply f_equal.
pattern (x / (2^mw))%Z at 2 ; rewrite Z_div_mod_eq_full with (x / (2^mw))%Z (2^ew)%Z.
apply (f_equal (fun v => (v + _)%Z)).
replace (x / 2 ^ mw / 2 ^ ew)%Z with (if Zle_bool (2 ^ mw * 2 ^ ew) x then 1 else 0)%Z.
case Zle_bool.
now rewrite Zmult_1_r.
now rewrite Zmult_0_r.
rewrite Zdiv_Zdiv.
apply sym_eq.
case Zle_bool_spec ; intros Hs.
apply Zle_antisym.
cut (x / (2^mw * 2^ew) < 2)%Z. clear ; omega.
apply Zdiv_lt_upper_bound.
apply Hx.
now apply Zmult_lt_0_compat.
rewrite <- Zpower_exp.
change 2%Z at 1 with (Zpower 2 1).
rewrite <- Zpower_exp.
now rewrite Zplus_comm.
discriminate.
apply Zle_ge.
apply Zplus_le_0_compat with (1 := Hmw).
now apply Zlt_le_weak.
now apply Zle_ge.
apply Zle_ge.
now apply Zlt_le_weak.
apply Zdiv_le_lower_bound.
apply Hx.
now apply Zmult_lt_0_compat.
now rewrite Zmult_1_l.
apply Zdiv_small.
now split.
now apply Zlt_le_weak.
now apply Zlt_le_weak.
now apply Zgt_not_eq.
now apply Zgt_not_eq.
Qed.

Definition bits_of_binary_float (x : binary_float) :=
  match x with
  | B754_zero sx => join_bits sx 0 0
  | B754_infinity sx => join_bits sx 0 (Zpower 2 ew - 1)
  | B754_nan => join_bits false (Zpower 2 mw - 1) (Zpower 2 ew - 1)
  | B754_finite sx mx ex _ =>
    if Zle_bool (Zpower 2 mw) (Zpos mx) then
      join_bits sx (Zpos mx - Zpower 2 mw) (ex - emin + 1)
    else
      join_bits sx (Zpos mx) 0
  end.

Definition split_bits_of_binary_float (x : binary_float) :=
  match x with
  | B754_zero sx => (sx, 0, 0)%Z
  | B754_infinity sx => (sx, 0, Zpower 2 ew - 1)%Z
  | B754_nan => (false, Zpower 2 mw - 1, Zpower 2 ew - 1)%Z
  | B754_finite sx mx ex _ =>
    if Zle_bool (Zpower 2 mw) (Zpos mx) then
      (sx, Zpos mx - Zpower 2 mw, ex - emin + 1)%Z
    else
      (sx, Zpos mx, 0)%Z
  end.

Theorem split_bits_of_binary_float_correct :
  forall x,
  split_bits (bits_of_binary_float x) = split_bits_of_binary_float x.
Proof.
intros [sx|sx| |sx mx ex Hx] ;
  try ( simpl ; apply split_join_bits ; split ; try apply Zle_refl ; try apply Zlt_pred ; trivial ; omega ).
unfold bits_of_binary_float, split_bits_of_binary_float.
assert (Hf: (emin <= ex /\ digits radix2 (Zpos mx) <= prec)%Z).
destruct (andb_prop _ _ Hx) as (Hx', _).
unfold bounded_prec in Hx'.
rewrite Z_of_nat_S_digits2_Pnat in Hx'.
generalize (Zeq_bool_eq _ _ Hx').
unfold FLT_exp.
change (Fcalc_digits.radix2) with radix2.
unfold emin.
clear ; zify ; omega.
destruct (Zle_bool_spec (2^mw) (Zpos mx)) as [H|H] ;
  apply split_join_bits ; try now split.
(* *)
split.
clear -He_gt_0 H ; omega.
cut (Zpos mx < 2 * 2^mw)%Z. clear ; omega.
replace (2 * 2^mw)%Z with (2^prec)%Z.
apply (Zpower_gt_digits radix2 _ (Zpos mx)).
apply Hf.
unfold prec.
rewrite Zplus_comm.
now apply Zpower_exp ; apply Zle_ge.
(* *)
split.
generalize (proj1 Hf).
clear ; omega.
destruct (andb_prop _ _ Hx) as (_, Hx').
unfold emin.
replace (2^ew)%Z with (2 * emax)%Z.
generalize (Zle_bool_imp_le _ _ Hx').
clear ; omega.
apply sym_eq.
rewrite (Zsucc_pred ew).
unfold Zsucc.
rewrite Zplus_comm.
apply Zpower_exp ; apply Zle_ge.
discriminate.
now apply Zlt_0_le_0_pred.
Qed.

1143 1144 1145 1146 1147 1148 1149 1150
Lemma binary_float_of_bits_aux1 :
  forall x sx mx ex,
  split_bits x = (sx, Zpos mx, ex) ->
  bounded prec emax mx emin = true.
Proof.
intros x sx mx ex Hx.
injection Hx.
intros Hex Hmx _.
1151 1152 1153
assert (digits radix2 (Zpos mx) <= mw)%Z.
apply digits_le_Zpower.
simpl.
1154
rewrite <- Hmx.
1155 1156 1157 1158 1159 1160 1161 1162 1163 1164 1165 1166 1167 1168 1169 1170 1171 1172 1173 1174 1175 1176 1177 1178 1179
eapply Z_mod_lt.
apply Zlt_gt.
now apply (Zpower_gt_0 radix2).
apply bounded_canonic_lt_emax ; try assumption.
unfold canonic, canonic_exponent.
fold emin.
rewrite ln_beta_F2R_digits. 2: discriminate.
unfold Fexp, FLT_exp.
apply sym_eq.
apply Zmax_right.
clear -H Hprec.
unfold prec ; omega.
apply Rnot_le_lt.
intros H0.
refine (_ (ln_beta_monotone radix2 _ _ _ H0)).
rewrite ln_beta_bpow.
rewrite ln_beta_F2R_digits. 2: discriminate.
unfold emin, prec.
apply Zlt_not_le.
cut (0 < emax)%Z. clear -H Hew ; omega.
apply (Zpower_gt_0 radix2).
clear -Hew ; omega.
apply bpow_gt_0.
Qed.

1180 1181 1182 1183 1184 1185 1186 1187 1188 1189 1190 1191
Lemma binary_float_of_bits_aux2 :
  forall x sx mx ex px,
  split_bits x = (sx, mx, ex) ->
  Zeq_bool ex 0 = false ->
  Zeq_bool ex (Zpower 2 ew - 1) = false ->
  (mx + Zpower 2 mw)%Z = Zpos px ->
  bounded prec emax px (ex + emin - 1) = true.
Proof.
intros x sx mx ex px Hx Hex Hex' Hmx.
injection Hx.
intros Hex'' Hmx' _.
assert (prec = digits radix2 (Zpos px)).
1192 1193 1194 1195 1196 1197 1198 1199
rewrite digits_ln_beta. 2: discriminate.
apply sym_eq.
apply ln_beta_unique.
rewrite <- Z2R_abs.
unfold Zabs.
replace (prec - 1)%Z with mw by ( unfold prec ; ring ).
rewrite <- Z2R_Zpower with (1 := Hmw).
rewrite <- Z2R_Zpower. 2: now apply Zlt_le_weak.
1200 1201
rewrite <- Hmx.
rewrite <- Hmx'.
1202 1203 1204 1205 1206 1207 1208 1209 1210 1211 1212 1213 1214 1215 1216 1217 1218 1219 1220 1221 1222
split.
apply Z2R_le.
change (radix2^mw)%Z with (0 + 2^mw)%Z.
apply Zplus_le_compat_r.
eapply Z_mod_lt.
apply Zlt_gt.
now apply (Zpower_gt_0 radix2).
apply Z2R_lt.
unfold prec.
rewrite Zpower_exp. 2: now apply Zle_ge. 2: discriminate.
rewrite <- Zplus_diag_eq_mult_2.
apply Zplus_lt_compat_r.
eapply Z_mod_lt.
apply Zlt_gt.
now apply (Zpower_gt_0 radix2).
(* *)
apply bounded_canonic_lt_emax ; try assumption.
unfold canonic, canonic_exponent.
rewrite ln_beta_F2R_digits. 2: discriminate.
unfold Fexp, FLT_exp.
rewrite <- H.
1223
replace (prec + (ex + emin - 1) - prec)%Z with (ex + emin - 1)%Z by ring.
1224 1225 1226
apply sym_eq.
apply Zmax_left.
generalize (Zeq_bool_neq _ _ Hex).
1227
cut (0 <= ex)%Z.
1228 1229
unfold emin.
clear ; intros H1 H2 ; omega.
1230
rewrite <- Hex''.
1231 1232 1233 1234 1235 1236 1237 1238 1239 1240 1241 1242 1243 1244 1245
eapply Z_mod_lt.
apply Zlt_gt.
apply (Zpower_gt_0 radix2).
now apply Zlt_le_weak.
apply Rnot_le_lt.
intros H0.
refine (_ (ln_beta_monotone radix2 _ _ _ H0)).
rewrite ln_beta_bpow.
rewrite ln_beta_F2R_digits. 2: discriminate.
rewrite <- H.
apply Zlt_not_le.
unfold emin.
apply Zplus_lt_reg_r with (emax - 1)%Z.
ring_simplify.
generalize (Zeq_bool_neq _ _ Hex').
1246
cut (ex < 2^ew)%Z.
1247 1248 1249 1250 1251 1252 1253
replace (2^ew)%Z with (2 * emax)%Z.
clear ; intros H1 H2 ; omega.
replace ew with (1 + (ew - 1))%Z by ring.
rewrite Zpower_exp.
apply refl_equal.
discriminate.
clear -Hew ; omega.
1254
rewrite <- Hex''.
1255 1256 1257 1258 1259 1260 1261
eapply Z_mod_lt.
apply Zlt_gt.
apply (Zpower_gt_0 radix2).
now apply Zlt_le_weak.
apply bpow_gt_0.
Qed.

1262 1263 1264 1265 1266 1267 1268 1269 1270 1271 1272 1273 1274 1275 1276 1277 1278 1279 1280 1281 1282 1283 1284
Definition binary_float_of_bits (x : Z) :=
  match split_bits x as v1 return split_bits x = v1 -> binary_float with
  | (sx, mx, ex) =>
    match Zeq_bool ex 0 as v2 return _ = v2 -> _ -> binary_float with
    | true => fun _ =>
      match mx as v3 return split_bits x = (sx, v3, ex) -> binary_float with
      | Zpos px => fun H1 => B754_finite prec emax sx px emin (binary_float_of_bits_aux1 x sx px ex H1)
      | Z0 => fun _ => B754_zero prec emax sx
      | _ => fun _ => B754_nan prec emax (* dummy *)
      end
    | false => fun H2 =>
      match Zeq_bool ex (Zpower 2 ew - 1) as v3 return _ = v3 -> _ -> binary_float with
      | true => fun _ _ =>
        if Zeq_bool mx 0 then B754_infinity prec emax sx else B754_nan prec emax
      | false => fun H3 H1 =>
        match (mx + Zpower 2 mw)%Z as v4 return _ = v4 -> binary_float with
        | Zpos px => fun H4 => B754_finite prec emax sx px _ (binary_float_of_bits_aux2 x sx mx ex px H1 H2 H3 H4)
        | _ => fun _ => B754_nan prec emax (* dummy *)
        end (refl_equal _)
      end (refl_equal _)
    end (refl_equal _)
  end (refl_equal _).

1285
End Binary_Bits.