Fappli_IEEE.v 51.5 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 32 33 34 35 36 37 38 39 40 41 42 43 44 45
Section AnyRadix.

Inductive full_float :=
  | F754_zero : bool -> full_float
  | F754_infinity : bool -> full_float
  | F754_nan : full_float
  | F754_finite : bool -> positive -> Z -> full_float.

Definition FF2R r x :=
  match x with
  | F754_finite s m e => F2R (Float r (cond_Zopp s (Zpos m)) e)
  | _ => R0
  end.

End AnyRadix.

46 47
Section Binary.

48
Variable prec emax : Z.
Guillaume Melquiond's avatar
Guillaume Melquiond committed
49
Hypothesis Hprec : (0 < prec)%Z.
50
Hypothesis Hmax : (prec < emax)%Z.
51

52
Let emin := (3 - emax - prec)%Z.
53
Let fexp := FLT_exp emin prec.
54 55
Let fexp_correct : valid_exp fexp := FLT_exp_correct _ _ Hprec.

56 57 58 59
Definition bounded_prec m e :=
  Zeq_bool (fexp (Z_of_nat (S (digits2_Pnat m)) + e)) e.

Definition bounded m e :=
60
  andb (bounded_prec m e) (Zle_bool e (emax - prec)).
61

62 63 64 65 66 67
Definition valid_binary x :=
  match x with
  | F754_finite _ m e => bounded m e
  | _ => true
  end.

68 69 70 71 72 73 74
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.

75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90
Definition FF2B x :=
  match x as x return valid_binary x = true -> binary_float with
  | F754_finite s m e => B754_finite s m e
  | F754_infinity s => fun _ => B754_infinity s
  | F754_zero s => fun _ => B754_zero s
  | F754_nan => fun _ => B754_nan
  end.

Definition B2FF x :=
  match x with
  | B754_finite s m e _ => F754_finite s m e
  | B754_infinity s => F754_infinity s
  | B754_zero s => F754_zero s
  | B754_nan => F754_nan
  end.

91 92 93 94
Definition radix2 := Build_radix 2 (refl_equal true).

Definition B2R f :=
  match f with
Guillaume Melquiond's avatar
Guillaume Melquiond committed
95
  | B754_finite s m e _ => F2R (Float radix2 (cond_Zopp s (Zpos m)) e)
96 97 98
  | _ => R0
  end.

99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119
Theorem FF2R_B2FF :
  forall x,
  FF2R radix2 (B2FF x) = B2R x.
Proof.
now intros [sx|sx| |sx mx ex Hx].
Qed.

Theorem B2FF_FF2B :
  forall x Hx,
  B2FF (FF2B x Hx) = x.
Proof.
now intros [sx|sx| |sx mx ex] Hx.
Qed.

Theorem B2R_FF2B :
  forall x Hx,
  B2R (FF2B x Hx) = FF2R radix2 x.
Proof.
now intros [sx|sx| |sx mx ex] Hx.
Qed.

120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137
Theorem match_FF2B :
  forall {T} fz fi fn ff x Hx,
  match FF2B x Hx return T with
  | B754_zero sx => fz sx
  | B754_infinity sx => fi sx
  | B754_nan => fn
  | B754_finite sx mx ex _ => ff sx mx ex
  end =
  match x with
  | F754_zero sx => fz sx
  | F754_infinity sx => fi sx
  | F754_nan => fn
  | F754_finite sx mx ex => ff sx mx ex
  end.
Proof.
now intros T fz fi fn ff [sx|sx| |sx mx ex] Hx.
Qed.

138 139 140
Theorem canonic_bounded_prec :
  forall (sx : bool) mx ex,
  bounded_prec mx ex = true ->
Guillaume Melquiond's avatar
Guillaume Melquiond committed
141
  canonic radix2 fexp (Float radix2 (cond_Zopp sx (Zpos mx)) ex).
142 143 144 145 146 147 148 149 150 151 152 153 154 155 156 157 158 159 160 161 162 163 164 165 166
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.

167 168 169 170 171 172 173 174 175 176 177 178 179 180 181 182 183 184 185 186 187 188 189
Theorem binary_unicity :
  forall x y : binary_float,
  B2FF x = B2FF y ->
  x = y.
Proof.
intros [sx|sx| |sx mx ex Hx] [sy|sy| |sy my ey Hy] ; try easy.
(* *)
intros H.
now inversion H.
(* *)
intros H.
now inversion H.
(* *)
intros H.
inversion H.
clear H.
revert Hx.
rewrite H2, H3.
intros Hx.
apply f_equal.
apply eqbool_irrelevance.
Qed.

190 191 192 193 194 195
Definition is_finite_strict f :=
  match f with
  | B754_finite _ _ _ _ => true
  | _ => false
  end.

196
Theorem finite_binary_unicity :
197 198 199 200 201 202 203 204 205 206 207 208 209 210 211 212 213 214 215 216 217 218 219 220 221 222 223 224 225 226 227 228 229 230 231 232 233 234 235 236 237 238 239 240 241 242
  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
243 244 245 246 247 248 249 250 251 252 253 254 255 256 257 258 259 260 261 262 263 264 265 266
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.

267 268 269
Theorem bounded_lt_emax :
  forall mx ex,
  bounded mx ex = true ->
270
  (F2R (Float radix2 (Zpos mx) ex) < bpow radix2 emax)%R.
271
Proof.
272
intros mx ex Hx.
273 274 275 276 277 278 279 280
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').
281 282
change (Zpos mx) with (Zabs (Zpos mx)).
rewrite <- abs_F2R.
283 284 285 286 287 288 289 290 291
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.
292 293 294
generalize (digits radix2 (Zpos mx)).
intros ; zify ; subst.
clear -H H2. clearbody emin.
295 296 297
omega.
Qed.

298 299
Theorem B2R_lt_emax :
  forall x,
300
  (Rabs (B2R x) < bpow radix2 emax)%R.
301 302 303 304 305 306
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.

307 308 309
Theorem bounded_canonic_lt_emax :
  forall mx ex,
  canonic radix2 fexp (Float radix2 (Zpos mx) ex) ->
310
  (F2R (Float radix2 (Zpos mx) ex) < bpow radix2 emax)%R ->
311 312
  bounded mx ex = true.
Proof.
313
intros mx ex Cx Bx.
314 315 316 317 318 319 320 321 322 323 324 325 326 327 328
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.
329 330
apply Zmax_lub.
cut (e' - 1 < emax)%Z. clear ; omega.
331 332 333 334 335 336 337
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.
338 339
unfold emin. clear -Hprec Hmax.
omega.
340 341
Qed.

342 343 344 345 346 347 348 349 350 351 352 353 354 355 356 357 358 359 360 361 362 363 364 365 366 367 368 369 370 371 372 373 374 375 376 377 378 379 380 381 382 383 384 385 386 387 388 389 390 391 392 393 394 395 396 397 398 399 400 401 402 403 404 405 406 407 408 409 410 411 412 413 414 415 416 417 418 419 420 421 422 423 424 425 426 427 428 429 430 431 432 433 434 435 436 437 438 439 440 441 442 443 444 445 446 447 448 449 450 451 452 453 454 455 456 457 458 459 460 461 462 463
Record shr_record := { shr_m : Z ; shr_r : bool ; shr_s : bool }.

Definition shr_1 mrs :=
  let '(Build_shr_record m r s) := mrs in
  let s := orb r s in
  match m with
  | Z0 => Build_shr_record Z0 false s
  | Zpos xH => Build_shr_record Z0 true s
  | Zpos (xO p) => Build_shr_record (Zpos p) false s
  | Zpos (xI p) => Build_shr_record (Zpos p) true s
  | Zneg xH => Build_shr_record Z0 true s
  | Zneg (xO p) => Build_shr_record (Zneg p) false s
  | Zneg (xI p) => Build_shr_record (Zneg p) true s
  end.

Definition loc_of_shr_record mrs :=
  match mrs with
  | Build_shr_record _ false false => loc_Exact
  | Build_shr_record _ false true => loc_Inexact Lt
  | Build_shr_record _ true false => loc_Inexact Eq
  | Build_shr_record _ true true => loc_Inexact Gt
  end.

Definition shr_record_of_loc m l :=
  match l with
  | loc_Exact => Build_shr_record m false false
  | loc_Inexact Lt => Build_shr_record m false true
  | loc_Inexact Eq => Build_shr_record m true false
  | loc_Inexact Gt => Build_shr_record m true true
  end.

Theorem shr_m_shr_record_of_loc :
  forall m l,
  shr_m (shr_record_of_loc m l) = m.
Proof.
now intros m [|[| |]].
Qed.

Theorem loc_of_shr_record_of_loc :
  forall m l,
  loc_of_shr_record (shr_record_of_loc m l) = l.
Proof.
now intros m [|[| |]].
Qed.

Definition shr mrs e n :=
  match n with
  | Zpos p => (iter_pos p _ shr_1 mrs, (e + n)%Z)
  | _ => (mrs, e)
  end.

Theorem inbetween_shr_1 :
  forall x mrs e,
  (0 <= shr_m mrs)%Z ->
  inbetween_float radix2 (shr_m mrs) e x (loc_of_shr_record mrs) ->
  inbetween_float radix2 (shr_m (shr_1 mrs)) (e + 1) x (loc_of_shr_record (shr_1 mrs)).
Proof.
intros x mrs e Hm Hl.
refine (_ (new_location_even_correct (F2R (Float radix2 (shr_m (shr_1 mrs)) (e + 1))) (bpow radix2 e) 2 _ _ _ x (if shr_r (shr_1 mrs) then 1 else 0) (loc_of_shr_record mrs) _ _)) ; try easy.
2: apply bpow_gt_0.
2: now case (shr_r (shr_1 mrs)) ; split.
change (Z2R 2) with (bpow radix2 1).
rewrite <- bpow_plus.
rewrite (Zplus_comm 1), <- (F2R_bpow radix2 (e + 1)).
unfold inbetween_float, F2R. simpl.
rewrite Z2R_plus, Rmult_plus_distr_r.
replace (new_location_even 2 (if shr_r (shr_1 mrs) then 1%Z else 0%Z) (loc_of_shr_record mrs)) with (loc_of_shr_record (shr_1 mrs)).
easy.
clear -Hm.
destruct mrs as (m, r, s).
now destruct m as [|[m|m|]|m] ; try (now elim Hm) ; destruct r as [|] ; destruct s as [|].
rewrite (F2R_change_exp radix2 e).
2: apply Zle_succ.
unfold F2R. simpl.
rewrite <- 2!Rmult_plus_distr_r, <- 2!Z2R_plus.
rewrite Zplus_assoc.
replace (shr_m (shr_1 mrs) * 2 ^ (e + 1 - e) + (if shr_r (shr_1 mrs) then 1%Z else 0%Z))%Z with (shr_m mrs).
exact Hl.
ring_simplify (e + 1 - e)%Z.
change (2^1)%Z with 2%Z.
rewrite Zmult_comm.
clear -Hm.
destruct mrs as (m, r, s).
now destruct m as [|[m|m|]|m] ; try (now elim Hm) ; destruct r as [|] ; destruct s as [|].
Qed.

Theorem inbetween_shr :
  forall x m e l n,
  (0 <= m)%Z ->
  inbetween_float radix2 m e x l ->
  let '(mrs, e') := shr (shr_record_of_loc m l) e n in
  inbetween_float radix2 (shr_m mrs) e' x (loc_of_shr_record mrs).
Proof.
intros x m e l n Hm Hl.
destruct n as [|n|n].
now destruct l as [|[| |]].
2: now destruct l as [|[| |]].
unfold shr.
rewrite iter_nat_of_P.
rewrite Zpos_eq_Z_of_nat_o_nat_of_P.
induction (nat_of_P n).
simpl.
rewrite Zplus_0_r.
now destruct l as [|[| |]].
simpl iter_nat.
rewrite inj_S.
unfold Zsucc.
rewrite  Zplus_assoc.
revert IHn0.
apply inbetween_shr_1.
clear -Hm.
induction n0.
now destruct l as [|[| |]].
simpl.
revert IHn0.
generalize (iter_nat n0 shr_record shr_1 (shr_record_of_loc m l)).
clear.
intros (m, r, s) Hm.
now destruct m as [|[m|m|]|m] ; try (now elim Hm) ; destruct r as [|] ; destruct s as [|].
Qed.

Definition shr_fexp m e l :=
464 465
  let d := match m with Z0 => m | Zpos p => Z_of_nat (S (digits2_Pnat p)) | Zneg p => Z_of_nat (S (digits2_Pnat p)) end in
  shr (shr_record_of_loc m l) e (fexp (d + e) - e).
466 467 468 469 470 471 472 473 474 475 476

Theorem shr_truncate :
  forall m e l,
  (0 <= m)%Z ->
  shr_fexp m e l =
  let '(m', e', l') := truncate radix2 fexp (m, e, l) in (shr_record_of_loc m' l', e').
Proof.
intros m e l Hm.
case_eq (truncate radix2 fexp (m, e, l)).
intros (m', e') l'.
unfold shr_fexp.
477 478 479
replace (match m with Z0 => m | Zpos p => Z_of_nat (S (digits2_Pnat p)) | Zneg p => Z_of_nat (S (digits2_Pnat p)) end)
  with (digits radix2 m).
2: now case m ; intros ; try rewrite Z_of_nat_S_digits2_Pnat.
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
case_eq (fexp (digits radix2 m + e) - e)%Z.
(* *)
intros He.
unfold truncate.
rewrite He.
simpl.
intros H.
now inversion H.
(* *)
intros p Hp.
assert (He: (e <= fexp (digits radix2 m + e))%Z).
clear -Hp ; zify ; omega.
destruct (inbetween_float_ex radix2 m e l) as (x, Hx).
generalize (inbetween_shr x m e l (fexp (digits radix2 m + e) - e) Hm Hx).
assert (Hx0 : (0 <= x)%R).
apply Rle_trans with (F2R (Float radix2 m e)).
now apply F2R_ge_0_compat.
exact (proj1 (inbetween_float_bounds _ _ _ _ _ Hx)).
case_eq (shr (shr_record_of_loc m l) e (fexp (digits radix2 m + e) - e)).
intros mrs e'' H3 H4 H1.
generalize (truncate_correct radix2 _ fexp_correct x m e l Hx0 Hx (or_introl _ He)).
rewrite H1.
intros (H2,_).
rewrite <- Hp, H3.
assert (e'' = e').
change (snd (mrs, e'') = snd (fst (m',e',l'))).
rewrite <- H1, <- H3.
unfold truncate.
now rewrite Hp.
rewrite H in H4 |- *.
apply (f_equal (fun v => (v, _))).
destruct (inbetween_float_unique _ _ _ _ _ _ _ H2 H4) as (H5, H6).
rewrite H5, H6.
case mrs.
now intros m0 [|] [|].
(* *)
intros p Hp.
unfold truncate.
rewrite Hp.
simpl.
intros H.
now inversion H.
Qed.

524 525 526 527 528 529 530 531 532 533
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.
534 535 536 537 538 539 540 541 542 543

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.

544 545 546 547 548 549 550 551 552 553 554 555 556
Definition overflow_to_inf m s :=
  match m with
  | mode_NE => true
  | mode_NA => true
  | mode_ZR => false
  | mode_UP => negb s
  | mode_DN => s
  end.

Definition binary_overflow m s :=
  if overflow_to_inf m s then F754_infinity s
  else F754_finite s (match (Zpower 2 prec - 1)%Z with Zpos p => p | _ => xH end) (emax - prec).

557
Definition binary_round_sign mode sx mx ex lx :=
558 559 560
  let '(mrs', e') := shr_fexp (Zpos mx) ex lx in
  let '(mrs'', e'') := shr_fexp (choice_mode mode sx (shr_m mrs') (loc_of_shr_record mrs')) e' loc_Exact in
  match shr_m mrs'' with
561
  | Z0 => F754_zero sx
562
  | Zpos m => if Zle_bool e'' (emax - prec) then F754_finite sx m e'' else binary_overflow mode sx
563
  | _ => F754_nan (* dummy *)
564 565 566 567 568 569
  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 ->
570
  valid_binary (binary_round_sign mode (Rlt_bool x 0) mx ex lx) = true /\
571
  if Rlt_bool (Rabs (round radix2 fexp (round_mode mode) x)) (bpow radix2 emax) then
572
    FF2R radix2 (binary_round_sign mode (Rlt_bool x 0) mx ex lx) = round radix2 fexp (round_mode mode) x
573
  else
574
    binary_round_sign mode (Rlt_bool x 0) mx ex lx = binary_overflow mode (Rlt_bool x 0).
575
Proof.
576
intros m x mx ex lx Bx Ex.
577
unfold binary_round_sign.
578
rewrite shr_truncate. 2: easy.
579 580 581
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).
582
rewrite loc_of_shr_record_of_loc, shr_m_shr_record_of_loc.
583 584 585 586 587 588 589 590 591 592 593 594 595 596 597 598 599 600 601 602 603 604 605 606 607 608 609
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 *)
610
rewrite shr_truncate. 2: apply Zle_refl.
611 612
generalize (truncate_0 radix2 fexp e1 loc_Exact).
destruct (truncate radix2 fexp (Z0, e1, loc_Exact)) as ((m2, e2), l2).
613
rewrite shr_m_shr_record_of_loc.
614
intros Hm2.
615 616 617
rewrite Hm2.
repeat split.
rewrite Rlt_bool_true.
618 619
apply sym_eq.
case Rlt_bool ; apply F2R_0.
620 621
rewrite abs_F2R, abs_cond_Zopp, F2R_0.
apply bpow_gt_0.
622 623 624 625 626 627 628 629 630 631 632 633 634 635 636 637 638 639 640 641
(* . 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.
642
rewrite shr_truncate. 2: easy.
643
destruct (truncate radix2 fexp (Zpos m1', e1, loc_Exact)) as ((m2, e2), l2).
644
rewrite shr_m_shr_record_of_loc.
645 646 647 648 649
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.
650 651
rewrite F2R_cond_Zopp, H3, abs_cond_Ropp, abs_F2R.
simpl Zabs.
652 653 654 655 656 657 658 659 660 661 662 663 664
case_eq (Zle_bool e2 (emax - prec)) ; intros He2.
assert (bounded m2 e2 = true).
apply andb_true_intro.
split.
unfold bounded_prec.
apply Zeq_bool_true.
rewrite Z_of_nat_S_digits2_Pnat.
rewrite <- ln_beta_F2R_digits.
apply sym_eq.
now rewrite H3 in H4.
discriminate.
exact He2.
apply (conj H).
665 666 667
rewrite Rlt_bool_true.
apply F2R_cond_Zopp.
now apply bounded_lt_emax.
668 669 670 671
rewrite (Rlt_bool_false _ (bpow radix2 emax)).
refine (conj _ (refl_equal _)).
unfold binary_overflow.
case overflow_to_inf.
672
apply refl_equal.
673 674 675 676 677 678 679 680 681 682 683 684 685 686 687 688 689 690 691 692 693 694 695 696 697 698 699 700 701 702 703 704
unfold valid_binary, bounded.
rewrite Zle_bool_refl.
rewrite Bool.andb_true_r.
apply Zeq_bool_true.
rewrite Z_of_nat_S_digits2_Pnat.
change Fcalc_digits.radix2 with radix2.
replace (digits radix2 (Zpos (match (Zpower 2 prec - 1)%Z with Zpos p => p | _ => xH end))) with prec.
unfold fexp, FLT_exp, emin.
clear -Hprec Hmax.
zify ; omega.
change 2%Z with (radix_val radix2).
case_eq (Zpower radix2 prec - 1)%Z.
simpl digits.
generalize (Zpower_gt_1 radix2 _ Hprec).
clear ; omega.
intros p Hp.
apply Zle_antisym.
cut (prec - 1 < digits radix2 (Zpos p))%Z. clear ; omega.
apply digits_gt_Zpower.
simpl Zabs. rewrite <- Hp.
cut (Zpower radix2 (prec - 1) < Zpower radix2 prec)%Z. clear ; omega.
apply lt_Z2R.
rewrite 2!Z2R_Zpower. 2: now apply Zlt_le_weak.
apply bpow_lt.
apply Zlt_pred.
now apply Zlt_0_le_0_pred.
apply digits_le_Zpower.
simpl Zabs. rewrite <- Hp.
apply Zlt_pred.
intros p Hp.
generalize (Zpower_gt_1 radix2 _ Hprec).
clear -Hp ; zify ; omega.
705 706
apply Rnot_lt_le.
intros Hx.
707 708 709 710
generalize (refl_equal (bounded m2 e2)).
unfold bounded at 2.
rewrite He2.
rewrite Bool.andb_false_r.
711 712
rewrite bounded_canonic_lt_emax with (2 := Hx).
discriminate.
713 714 715 716 717 718 719 720 721 722 723 724 725 726 727 728 729 730 731 732 733 734 735 736 737 738
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.
739

740 741 742 743 744 745 746 747
Definition Bsign x :=
  match x with
  | B754_nan => false
  | B754_zero s => s
  | B754_infinity s => s
  | B754_finite s _ _ _ => s
  end.

748 749 750 751 752 753 754 755
Lemma Bmult_correct_aux :
  forall m sx mx ex (Hx : bounded mx ex = true) sy my ey (Hy : bounded my ey = true),
  let x := F2R (Float radix2 (cond_Zopp sx (Zpos mx)) ex) in
  let y := F2R (Float radix2 (cond_Zopp sy (Zpos my)) ey) in
  let z := binary_round_sign m (xorb sx sy) (mx * my) (ex + ey) loc_Exact in
  valid_binary z = true /\
  if Rlt_bool (Rabs (round radix2 fexp (round_mode m) (x * y))) (bpow radix2 emax) then
    FF2R radix2 z = round radix2 fexp (round_mode m) (x * y)
756
  else
757
    z = binary_overflow m (xorb sx sy).
758
Proof.
759 760
intros m sx mx ex Hx sy my ey Hy x y.
unfold x, y.
761
rewrite <- mult_F2R.
762
simpl.
763 764 765 766 767 768 769 770 771 772 773 774 775 776 777
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).
778
clear x y sx sy Hx Hy H.
779 780 781 782
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)).
783 784
clear -Hprec Hmax.
unfold emin.
785 786 787 788 789 790 791 792 793 794 795 796 797 798
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.
799

800 801 802 803 804 805 806 807 808 809 810 811 812 813 814 815 816 817 818 819 820
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 =>
    FF2B _ (proj1 (Bmult_correct_aux m sx mx ex Hx sy my ey Hy))
  end.

Theorem Bmult_correct :
  forall m x y,
  if Rlt_bool (Rabs (round radix2 fexp (round_mode m) (B2R x * B2R y))) (bpow radix2 emax) then
    B2R (Bmult m x y) = round radix2 fexp (round_mode m) (B2R x * B2R y)
  else
821
    B2FF (Bmult m x y) = binary_overflow m (xorb (Bsign x) (Bsign y)).
822 823 824 825 826 827 828 829 830 831 832 833
Proof.
intros m [sx|sx| |sx mx ex Hx] [sy|sy| |sy my ey Hy] ;
  try ( rewrite ?Rmult_0_r, ?Rmult_0_l, round_0, Rabs_R0, Rlt_bool_true ; [ apply refl_equal | apply bpow_gt_0 ] ).
simpl.
case Bmult_correct_aux.
intros H1 H2.
revert H2.
case Rlt_bool ; intros H2.
now rewrite B2R_FF2B.
now rewrite B2FF_FF2B.
Qed.

834
Definition shl mx ex ex' :=
835 836 837 838 839
  match (ex' - ex)%Z with
  | Zneg d => (shift_pos d mx, ex')
  | _ => (mx, ex)
  end.

840 841 842 843 844
Theorem shl_correct :
  forall mx ex ex',
  let (mx', ex'') := shl mx ex ex' in
  F2R (Float radix2 (Zpos mx) ex) = F2R (Float radix2 (Zpos mx') ex'') /\
  (ex'' <= ex')%Z.
845
Proof.
846 847 848
intros mx ex ex'.
unfold shl.
case_eq (ex' - ex)%Z.
849 850 851 852 853 854 855 856
(* d = 0 *)
intros H.
repeat split.
rewrite Zminus_eq with (1 := H).
apply Zle_refl.
(* d > 0 *)
intros d Hd.
repeat split.
857
replace ex' with (ex' - ex + ex)%Z by ring.
858 859 860 861 862 863 864 865 866 867
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)).
change (Zpos d) with (Zopp (Zneg d)).
rewrite <- Hd.
split.
868
replace (- (ex' - ex))%Z with (ex - ex')%Z by ring.
869 870
apply F2R_change_exp.
apply Zle_0_minus_le.
871
replace (ex - ex')%Z with (- (ex' - ex))%Z by ring.
872 873 874
now rewrite Hd.
apply Zle_refl.
Qed.
875

876 877 878 879 880 881 882 883 884 885 886 887 888 889 890 891
Theorem snd_shl :
  forall mx ex ex',
  (ex' <= ex)%Z ->
  snd (shl mx ex ex') = ex'.
Proof.
intros mx ex ex' He.
unfold shl.
case_eq (ex' - ex)%Z ; simpl.
intros H.
now rewrite Zminus_eq with (1 := H).
intros p.
clear -He ; zify ; omega.
intros.
apply refl_equal.
Qed.

892 893 894 895 896 897 898 899 900 901 902 903 904 905 906 907 908 909 910 911 912 913 914 915
Definition shl_fexp mx ex :=
  shl mx ex (fexp (Z_of_nat (S (digits2_Pnat mx)) + ex)).

Theorem shl_fexp_correct :
  forall mx ex,
  let (mx', ex') := shl_fexp 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 shl_fexp.
generalize (shl_correct mx ex (fexp (Z_of_nat (S (digits2_Pnat mx)) + ex))).
rewrite Z_of_nat_S_digits2_Pnat.
case shl.
intros mx' ex' (H1, H2).
split.
exact H1.
rewrite <- ln_beta_F2R_digits. 2: easy.
rewrite <- H1.
now rewrite ln_beta_F2R_digits.
Qed.

Definition binary_round_sign_shl m sx mx ex :=
  let '(mz, ez) := shl_fexp mx ex in binary_round_sign m sx mz ez loc_Exact.
916

917
Theorem binary_round_sign_shl_correct :
918
  forall m sx mx ex,
919
  valid_binary (binary_round_sign_shl m sx mx ex) = true /\
920 921
  let x := F2R (Float radix2 (cond_Zopp sx (Zpos mx)) ex) in
  if Rlt_bool (Rabs (round radix2 fexp (round_mode m) x)) (bpow radix2 emax) then
922
    FF2R radix2 (binary_round_sign_shl m sx mx ex) = round radix2 fexp (round_mode m) x
923
  else
924
    binary_round_sign_shl m sx mx ex = binary_overflow m sx.
925 926
Proof.
intros m sx mx ex.
927 928 929
unfold binary_round_sign_shl.
generalize (shl_fexp_correct mx ex).
destruct (shl_fexp mx ex) as (mz, ez).
930 931 932 933 934 935 936 937 938 939 940 941 942 943 944 945 946
intros (H1, H2).
simpl.
set (x := F2R (Float radix2 (cond_Zopp sx (Zpos mx)) ex)).
replace sx with (Rlt_bool x 0).
apply binary_round_sign_correct.
constructor.
unfold x.
now rewrite abs_F2R, abs_cond_Zopp.
exact H2.
unfold x.
case sx.
apply Rlt_bool_true.
now apply F2R_lt_0_compat.
apply Rlt_bool_false.
now apply F2R_ge_0_compat.
Qed.

947 948 949 950 951 952 953 954 955 956 957 958 959 960
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 =>
961 962 963
    let ez := Zmin ex ey in
    match Zplus (cond_Zopp sx (Zpos (fst (shl mx ex ez)))) (cond_Zopp sy (Zpos (fst (shl my ey ez)))) with
    | Z0 =>
964
      match m with mode_DN => B754_zero true | _ => B754_zero false end
965
    | Zpos mz =>
966
      FF2B (binary_round_sign_shl m false mz ez) (proj1 (binary_round_sign_shl_correct _ _ _ _))
967
    | Zneg mz =>
968
      FF2B (binary_round_sign_shl m true mz ez) (proj1 (binary_round_sign_shl_correct _ _ _ _))
969
    end
970 971
  end.

972
Theorem Bplus_correct :
973 974 975
  forall m x y,
  is_finite x = true ->
  is_finite y = true ->
976
  if Rlt_bool (Rabs (round radix2 fexp (round_mode m) (B2R x + B2R y))) (bpow radix2 emax) then
977 978
    B2R (Bplus m x y) = round radix2 fexp (round_mode m) (B2R x + B2R y)
  else
979
    (B2FF (Bplus m x y) = binary_overflow m (Bsign x) /\ Bsign x = Bsign y).
980
Proof.
981
intros m [sx|sx| |sx mx ex Hx] [sy|sy| |sy my ey Hy] Fx Fy ; try easy.
982
(* *)
983
rewrite Rplus_0_r, round_0, Rabs_R0, Rlt_bool_true.
984 985 986
simpl.
case (Bool.eqb sx sy) ; try easy.
now case m.
987
apply bpow_gt_0.
988
(* *)
989 990 991
rewrite Rplus_0_l, round_generic, Rlt_bool_true.
apply refl_equal.
apply B2R_lt_emax.
992 993
apply generic_format_B2R.
(* *)
994 995 996
rewrite Rplus_0_r, round_generic, Rlt_bool_true.
apply refl_equal.
apply B2R_lt_emax.
997 998
apply generic_format_B2R.
(* *)
999 1000
clear Fx Fy.
simpl.
1001 1002 1003 1004 1005 1006 1007 1008 1009 1010 1011 1012 1013 1014 1015 1016 1017 1018 1019 1020 1021 1022
set (ez := Zmin ex ey).
set (mz := (cond_Zopp sx (Zpos (fst (shl mx ex ez))) + cond_Zopp sy (Zpos (fst (shl my ey ez))))%Z).
assert (Hp: (F2R (Float radix2 (cond_Zopp sx (Zpos mx)) ex) +
  F2R (Float radix2 (cond_Zopp sy (Zpos my)) ey))%R = F2R (Float radix2 mz ez)).
rewrite 2!F2R_cond_Zopp.
generalize (shl_correct mx ex ez).
generalize (shl_correct my ey ez).
generalize (snd_shl mx ex ez (Zle_min_l ex ey)).
generalize (snd_shl my ey ez (Zle_min_r ex ey)).
destruct (shl mx ex ez) as (mx', ex').
destruct (shl my ey ez) as (my', ey').
simpl.
intros H1 H2.
rewrite H1, H2.
clear H1 H2.
intros (H1, _) (H2, _).
rewrite H1, H2.
clear H1 H2.
rewrite <- 2!F2R_cond_Zopp.
unfold F2R. simpl.
now rewrite <- Rmult_plus_distr_r, <- Z2R_plus.
rewrite Hp.
1023
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).
1024
(* . *)
1025
rewrite <- Hp.
Guillaume Melquiond's avatar
Guillaume Melquiond committed
1026 1027
intros Bz.
destruct (Bool.bool_dec sx sy) as [Hs|Hs].
1028
(* .. *)
Guillaume Melquiond's avatar
Guillaume Melquiond committed
1029
refine (conj _ Hs).
1030 1031 1032 1033 1034 1035 1036 1037 1038 1039 1040 1041 1042
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
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
(* .. *)
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.
1095 1096 1097 1098 1099 1100
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 *)
1101
generalize (binary_round_sign_shl_correct m false mz ez).
1102 1103 1104 1105 1106 1107 1108 1109 1110 1111 1112 1113
simpl.
case Rlt_bool_spec.
intros _ (Vz, Rz).
now rewrite B2R_FF2B.
intros Hz' (Vz, Rz).
specialize (Sz Hz').
refine (conj _ (proj2 Sz)).
rewrite B2FF_FF2B.
rewrite (proj1 Sz).
rewrite Rlt_bool_false.
exact Rz.
now apply F2R_ge_0_compat.
1114
(* . mz < 0 *)
1115
generalize (binary_round_sign_shl_correct m true mz ez).
1116 1117 1118 1119 1120 1121 1122 1123 1124 1125 1126 1127
simpl.
case Rlt_bool_spec.
intros _ (Vz, Rz).
now rewrite B2R_FF2B.
intros Hz' (Vz, Rz).
specialize (Sz Hz').
refine (conj _ (proj2 Sz)).
rewrite B2FF_FF2B.
rewrite (proj1 Sz).
rewrite Rlt_bool_true.
exact Rz.
now apply F2R_lt_0_compat.
1128
Qed.
1129

1130 1131
Definition Bminus m x y := Bplus m x (Bopp y).

1132 1133 1134 1135 1136
Lemma Bdiv_correct_aux :
  forall m sx mx ex (Hx : bounded mx ex = true) sy my ey (Hy : bounded my ey = true),
  let x := F2R (Float radix2 (cond_Zopp sx (Zpos mx)) ex) in
  let y := F2R (Float radix2 (cond_Zopp sy (Zpos my)) ey) in
  let z :=
1137 1138 1139
    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
1140 1141 1142 1143 1144
    | _ => F754_nan (* dummy *)
    end in
  valid_binary z = true /\
  if Rlt_bool (Rabs (round radix2 fexp (round_mode m) (x / y))) (bpow radix2 emax) then
    FF2R radix2 z = round radix2 fexp (round_mode m) (x / y)
1145
  else
1146
    z = binary_overflow m (xorb sx sy).
1147
Proof.
1148
intros m sx mx ex Hx sy my ey Hy.
1149 1150 1151
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).
1152
simpl.
1153
replace (xorb sx sy) with (Rlt_bool (F2R (Float radix2 (cond_Zopp sx (Zpos mx)) ex) *
1154 1155
  / F2R (Float radix2 (cond_Zopp sy (Zpos my)) ey)) 0).
unfold Rdiv.
1156 1157 1158 1159 1160 1161 1162 1163
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.
1164 1165 1166 1167 1168
case sy.
apply Rlt_not_eq.
now apply F2R_lt_0_compat.
apply Rgt_not_eq.
now apply F2R_gt_0_compat.
1169 1170 1171 1172 1173 1174 1175
revert Pz.
generalize (digits radix2 (Zpos mz)).
unfold fexp, FLT_exp.
clear.
intros ; zify ; subst.
omega.
(* . mz < 0 *)
1176
elim Rlt_not_le with (1 := proj2 (inbetween_float_bounds _ _ _ _ _ Bz)).
1177 1178 1179 1180 1181 1182 1183 1184 1185 1186 1187 1188 1189 1190 1191 1192 1193 1194 1195 1196 1197 1198 1199 1200 1201 1202 1203 1204 1205 1206
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.
(* *)
case sy ; simpl.
change (Zneg my) with (Zopp (Zpos my)).
rewrite <- opp_F2R.
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.
1207 1208
apply Rgt_not_eq.
now apply F2R_gt_0_compat.
1209 1210 1211 1212 1213 1214 1215 1216 1217 1218 1219 1220 1221 1222 1223 1224 1225 1226
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.

1227 1228 1229 1230 1231 1232 1233 1234 1235 1236 1237 1238 1239 1240 1241 1242 1243 1244 1245 1246 1247 1248
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 =>
    FF2B _ (proj1 (Bdiv_correct_aux m sx mx ex Hx sy my ey Hy))
  end.

Theorem Bdiv_correct :
  forall m x y,
  B2R y <> R0 ->
  if Rlt_bool (Rabs (round radix2 fexp (round_mode m) (B2R x / B2R y))) (bpow radix2 emax) then
    B2R (Bdiv m x y) = round radix2 fexp (round_mode m) (B2R x / B2R y)
  else
1249
    B2FF (Bdiv m x y) = binary_overflow m (xorb (Bsign x) (Bsign y)).
1250 1251 1252 1253 1254 1255 1256 1257 1258 1259 1260 1261 1262 1263 1264 1265 1266 1267 1268 1269
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] ;
  try ( rewrite Rmult_0_l, round_0, Rabs_R0, Rlt_bool_true ; [ apply refl_equal | apply bpow_gt_0 ] ).
simpl.
case Bdiv_correct_aux.
intros H1 H2.
revert H2.
unfold Rdiv.
case Rlt_bool ; intros H2.
now rewrite B2R_FF2B.
now rewrite B2FF_FF2B.
Qed.

Lemma Bsqrt_correct_aux :
  forall m mx ex (Hx : bounded mx ex = true),
  let x := F2R (Float radix2 (Zpos mx) ex) in
  let z :=
1270 1271 1272
    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
1273 1274 1275 1276
    | _ => F754_nan (* dummy *)
    end in
  valid_binary z = true /\
  FF2R radix2 z = round radix2 fexp (round_mode m) (sqrt x).
1277
Proof.
1278
intros m mx ex Hx.
1279 1280 1281 1282 1283 1284 1285 1286 1287 1288
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).
destruct mz as [|mz|mz].
(* . mz = 0 *)
elim (Zlt_irrefl prec).
now apply Zle_lt_trans with Z0.
(* . mz > 0 *)
refine (_ (binary_round_sign_correct m (sqrt (F2R (Float radix2 (Zpos mx) ex))) mz ez lz _ _)).
1289
rewrite Rlt_bool_false. 2: apply sqrt_ge_0.
1290 1291
rewrite Rlt_bool_true.
easy.
Guillaume Melquiond's avatar
Guillaume Melquiond committed
1292
(* .. *)
1293
rewrite Rabs_pos_eq.
Guillaume Melquiond's avatar
Guillaume Melquiond committed
1294 1295 1296 1297 1298 1299 1300 1301 1302 1303 1304 1305 1306 1307 1308 1309 1310 1311 1312 1313 1314 1315 1316 1317 1318 1319 1320 1321 1322 1323 1324 1325 1326 1327 1328 1329 1330 1331 1332 1333 1334 1335 1336 1337 1338 1339 1340 1341 1342 1343 1344 1345 1346 1347 1348 1349 1350 1351 1352 1353 1354
refine (_ (relative_error_FLT_ex radix2 emin prec Hprec (round_mode m) (sqrt (F2R (Float radix2 (Zpos mx) ex))) _)).
fold fexp.
intros (eps, (Heps,