Library Flocq.IEEE754.Binary

This file is part of the Flocq formalization of floating-point arithmetic in Coq: http://flocq.gforge.inria.fr/
Copyright (C) 2010-2018 Sylvie Boldo
Copyright (C) 2010-2018 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


From Coq Require Import ZArith Reals Psatz SpecFloat.

Require Import Core Round Bracket Operations Div Sqrt Relative BinarySingleNaN.

Arguments BinarySingleNaN.B754_zero {prec emax}.
Arguments BinarySingleNaN.B754_infinity {prec emax}.
Arguments BinarySingleNaN.B754_nan {prec emax}.
Arguments BinarySingleNaN.B754_finite {prec emax}.

Section AnyRadix.

Inductive full_float :=
  | F754_zero (s : bool)
  | F754_infinity (s : bool)
  | F754_nan (s : bool) (m : positive)
  | F754_finite (s : bool) (m : positive) (e : Z).

Definition FF2SF x :=
  match x with
  | F754_zero sS754_zero s
  | F754_infinity sS754_infinity s
  | F754_nan _ _S754_nan
  | F754_finite s m eS754_finite s m e
  end.

Definition FF2R beta x :=
  match x with
  | F754_finite s m eF2R (Float beta (cond_Zopp s (Zpos m)) e)
  | _ ⇒ 0%R
  end.

Lemma SF2R_FF2SF :
   beta x,
  SF2R beta (FF2SF x) = FF2R beta x.

Definition SF2FF x :=
  match x with
  | S754_zero sF754_zero s
  | S754_infinity sF754_infinity s
  | S754_nanF754_nan false xH
  | S754_finite s m eF754_finite s m e
  end.

Lemma FF2SF_SF2FF :
   x,
  FF2SF (SF2FF x) = x.

Lemma FF2R_SF2FF :
   beta x,
  FF2R beta (SF2FF x) = SF2R beta x.

Definition is_nan_FF f :=
  match f with
  | F754_nan _ _true
  | _false
  end.

Lemma is_nan_SF2FF :
   x,
  is_nan_FF (SF2FF x) = is_nan_SF x.

Lemma is_nan_FF2SF :
   x,
  is_nan_SF (FF2SF x) = is_nan_FF x.

Lemma SF2FF_FF2SF :
   x,
  is_nan_FF x = false
  SF2FF (FF2SF x) = x.

Definition sign_FF x :=
  match x with
  | F754_nan s _s
  | F754_zero ss
  | F754_infinity ss
  | F754_finite s _ _s
  end.

Definition is_finite_FF f :=
  match f with
  | F754_finite _ _ _true
  | F754_zero _true
  | _false
  end.

Lemma is_finite_SF2FF :
   x,
  is_finite_FF (SF2FF x) = is_finite_SF x.

Lemma sign_SF2FF :
   x,
  sign_FF (SF2FF x) = sign_SF x.

End AnyRadix.

Section Binary.

Arguments exist {A} {P}.

prec is the number of bits of the mantissa including the implicit one; emax is the exponent of the infinities. For instance, binary32 is defined by prec = 24 and emax = 128.
Variable prec emax : Z.
Context (prec_gt_0_ : Prec_gt_0 prec).
Context (prec_lt_emax_ : Prec_lt_emax prec emax).

Notation emin := (emin prec emax) (only parsing).
Notation fexp := (fexp prec emax) (only parsing).
Instance fexp_correct : Valid_exp fexp := FLT_exp_valid emin prec.
Instance fexp_monotone : Monotone_exp fexp := FLT_exp_monotone emin prec.

Notation canonical_mantissa := (canonical_mantissa prec emax).

Notation bounded := (SpecFloat.bounded prec emax).

Definition nan_pl pl :=
  Zlt_bool (Zpos (digits2_pos pl)) prec.

Notation valid_binary_SF := (valid_binary prec emax).

Definition valid_binary x :=
  match x with
  | F754_finite _ m ebounded m e
  | F754_nan _ plnan_pl pl
  | _true
  end.

Lemma valid_binary_SF2FF :
   x,
  is_nan_SF x = false
  valid_binary (SF2FF x) = valid_binary_SF x.

Basic type used for representing binary FP numbers. Note that there is exactly one such object per FP datum.

Inductive binary_float :=
  | B754_zero (s : bool)
  | B754_infinity (s : bool)
  | B754_nan (s : bool) (pl : positive) :
    nan_pl pl = true binary_float
  | B754_finite (s : bool) (m : positive) (e : Z) :
    bounded m e = true binary_float.

Definition B2BSN (x : binary_float) : BinarySingleNaN.binary_float prec emax :=
  match x with
  | B754_zero sBinarySingleNaN.B754_zero s
  | B754_infinity sBinarySingleNaN.B754_infinity s
  | B754_nan _ _ _BinarySingleNaN.B754_nan
  | B754_finite s m e HBinarySingleNaN.B754_finite s m e H
  end.

Definition FF2B x :=
  match x as x return valid_binary x = true binary_float with
  | F754_finite s m eB754_finite s m e
  | F754_infinity sfun _B754_infinity s
  | F754_zero sfun _B754_zero s
  | F754_nan b plfun HB754_nan b pl H
  end.

Definition B2FF x :=
  match x with
  | B754_finite s m e _F754_finite s m e
  | B754_infinity sF754_infinity s
  | B754_zero sF754_zero s
  | B754_nan b pl _F754_nan b pl
  end.

Definition B2R f :=
  match f with
  | B754_finite s m e _F2R (Float radix2 (cond_Zopp s (Zpos m)) e)
  | _ ⇒ 0%R
  end.

Definition B2SF (x : binary_float) :=
  match x with
  | B754_finite s m e _S754_finite s m e
  | B754_infinity sS754_infinity s
  | B754_zero sS754_zero s
  | B754_nan _ _ _S754_nan
  end.

Lemma B2SF_B2BSN :
   x,
  BinarySingleNaN.B2SF (B2BSN x) = B2SF x.

Lemma B2SF_FF2B :
   x Bx,
  B2SF (FF2B x Bx) = FF2SF x.

Lemma B2R_B2BSN :
   x, BinarySingleNaN.B2R (B2BSN x) = B2R x.

Lemma FF2SF_B2FF :
   x,
  FF2SF (B2FF x) = B2SF x.

Theorem FF2R_B2FF :
   x,
  FF2R radix2 (B2FF x) = B2R x.

Theorem B2FF_FF2B :
   x Hx,
  B2FF (FF2B x Hx) = x.

Theorem valid_binary_B2FF :
   x,
  valid_binary (B2FF x) = true.

Theorem FF2B_B2FF :
   x H,
  FF2B (B2FF x) H = x.

Theorem FF2B_B2FF_valid :
   x,
  FF2B (B2FF x) (valid_binary_B2FF x) = x.

Theorem B2R_FF2B :
   x Hx,
  B2R (FF2B x Hx) = FF2R radix2 x.

Theorem match_FF2B :
   {T} fz fi fn ff x Hx,
  match FF2B x Hx return T with
  | B754_zero sxfz sx
  | B754_infinity sxfi sx
  | B754_nan b p _fn b p
  | B754_finite sx mx ex _ff sx mx ex
  end =
  match x with
  | F754_zero sxfz sx
  | F754_infinity sxfi sx
  | F754_nan b pfn b p
  | F754_finite sx mx exff sx mx ex
  end.

Theorem canonical_canonical_mantissa :
   (sx : bool) mx ex,
  canonical_mantissa mx ex = true
  canonical radix2 fexp (Float radix2 (cond_Zopp sx (Zpos mx)) ex).

Theorem generic_format_B2R :
   x,
  generic_format radix2 fexp (B2R x).

Theorem FLT_format_B2R :
   x,
  FLT_format radix2 emin prec (B2R x).

Theorem B2FF_inj :
   x y : binary_float,
  B2FF x = B2FF y
  x = y.

Definition is_finite_strict f :=
  match f with
  | B754_finite _ _ _ _true
  | _false
  end.

Lemma is_finite_strict_B2BSN :
   x, BinarySingleNaN.is_finite_strict (B2BSN x) = is_finite_strict x.

Theorem B2R_inj:
   x y : binary_float,
  is_finite_strict x = true
  is_finite_strict y = true
  B2R x = B2R y
  x = y.

Definition Bsign x :=
  match x with
  | B754_nan s _ _s
  | B754_zero ss
  | B754_infinity ss
  | B754_finite s _ _ _s
  end.

Theorem Bsign_FF2B :
   x H,
  Bsign (FF2B x H) = sign_FF x.

Definition is_finite f :=
  match f with
  | B754_finite _ _ _ _true
  | B754_zero _true
  | _false
  end.

Lemma is_finite_B2BSN :
   x, BinarySingleNaN.is_finite (B2BSN x) = is_finite x.

Theorem is_finite_FF2B :
   x Hx,
  is_finite (FF2B x Hx) = is_finite_FF x.

Theorem is_finite_B2FF :
   x,
  is_finite_FF (B2FF x) = is_finite x.

Theorem B2R_Bsign_inj:
   x y : binary_float,
    is_finite x = true
    is_finite y = true
    B2R x = B2R y
    Bsign x = Bsign y
    x = y.

Definition is_nan f :=
  match f with
  | B754_nan _ _ _true
  | _false
  end.

Lemma is_nan_B2BSN :
   x,
  BinarySingleNaN.is_nan (B2BSN x) = is_nan x.

Theorem is_nan_FF2B :
   x Hx,
  is_nan (FF2B x Hx) = is_nan_FF x.

Theorem is_nan_B2FF :
   x,
  is_nan_FF (B2FF x) = is_nan x.

Definition get_nan_pl (x : binary_float) : positive :=
  match x with B754_nan _ pl _pl | _xH end.

Definition build_nan (x : { x | is_nan x = true }) : binary_float.

Theorem build_nan_correct :
   x : { x | is_nan x = true },
  build_nan x = proj1_sig x.

Theorem B2R_build_nan :
   x, B2R (build_nan x) = 0%R.

Theorem is_finite_build_nan :
   x, is_finite (build_nan x) = false.

Theorem is_nan_build_nan :
   x, is_nan (build_nan x) = true.

Definition BSN2B (nan : {x : binary_float | is_nan x = true }) (x : BinarySingleNaN.binary_float prec emax) : binary_float :=
  match x with
  | BinarySingleNaN.B754_nanbuild_nan nan
  | BinarySingleNaN.B754_zero sB754_zero s
  | BinarySingleNaN.B754_infinity sB754_infinity s
  | BinarySingleNaN.B754_finite s m e HB754_finite s m e H
  end.

Lemma B2BSN_BSN2B :
   nan x,
  B2BSN (BSN2B nan x) = x.

Lemma B2R_BSN2B :
   nan x, B2R (BSN2B nan x) = BinarySingleNaN.B2R x.

Lemma is_finite_BSN2B :
   nan x, is_finite (BSN2B nan x) = BinarySingleNaN.is_finite x.

Lemma is_nan_BSN2B :
   nan x, is_nan (BSN2B nan x) = BinarySingleNaN.is_nan x.

Lemma Bsign_B2BSN :
   x, is_nan x = false BinarySingleNaN.Bsign (B2BSN x) = Bsign x.

Lemma Bsign_BSN2B :
   nan x, BinarySingleNaN.is_nan x = false
  Bsign (BSN2B nan x) = BinarySingleNaN.Bsign x.

Definition BSN2B' (x : BinarySingleNaN.binary_float prec emax) : BinarySingleNaN.is_nan x = false binary_float.

Lemma B2BSN_BSN2B' :
   x Nx,
  B2BSN (BSN2B' x Nx) = x.

Lemma B2R_BSN2B' :
   x Nx,
  B2R (BSN2B' x Nx) = BinarySingleNaN.B2R x.

Lemma B2FF_BSN2B' :
   x Nx,
  B2FF (BSN2B' x Nx) = SF2FF (BinarySingleNaN.B2SF x).

Lemma Bsign_BSN2B' :
   x Nx,
  Bsign (BSN2B' x Nx) = BinarySingleNaN.Bsign x.

Lemma is_finite_BSN2B' :
   x Nx,
  is_finite (BSN2B' x Nx) = BinarySingleNaN.is_finite x.

Lemma is_nan_BSN2B' :
   x Nx,
  is_nan (BSN2B' x Nx) = false.

Definition erase (x : binary_float) : binary_float.

Theorem erase_correct :
   x, erase x = x.

Opposite

Definition Bopp opp_nan x :=
  match x with
  | B754_nan _ _ _build_nan (opp_nan x)
  | B754_infinity sxB754_infinity (negb sx)
  | B754_finite sx mx ex HxB754_finite (negb sx) mx ex Hx
  | B754_zero sxB754_zero (negb sx)
  end.

Theorem Bopp_involutive :
   opp_nan x,
  is_nan x = false
  Bopp opp_nan (Bopp opp_nan x) = x.

Theorem B2R_Bopp :
   opp_nan x,
  B2R (Bopp opp_nan x) = (- B2R x)%R.

Theorem is_finite_Bopp :
   opp_nan x,
  is_finite (Bopp opp_nan x) = is_finite x.

Lemma Bsign_Bopp :
   opp_nan x, is_nan x = false Bsign (Bopp opp_nan x) = negb (Bsign x).

Absolute value

Definition Babs abs_nan (x : binary_float) : binary_float :=
  match x with
  | B754_nan _ _ _build_nan (abs_nan x)
  | B754_infinity sxB754_infinity false
  | B754_finite sx mx ex HxB754_finite false mx ex Hx
  | B754_zero sxB754_zero false
  end.

Theorem B2R_Babs :
   abs_nan x,
  B2R (Babs abs_nan x) = Rabs (B2R x).

Theorem is_finite_Babs :
   abs_nan x,
  is_finite (Babs abs_nan x) = is_finite x.

Theorem Bsign_Babs :
   abs_nan x,
  is_nan x = false
  Bsign (Babs abs_nan x) = false.

Theorem Babs_idempotent :
   abs_nan (x: binary_float),
  is_nan x = false
  Babs abs_nan (Babs abs_nan x) = Babs abs_nan x.

Theorem Babs_Bopp :
   abs_nan opp_nan x,
  is_nan x = false
  Babs abs_nan (Bopp opp_nan x) = Babs abs_nan x.

Comparison
Some c means ordered as per c; None means unordered.

Definition Bcompare (f1 f2 : binary_float) : option comparison :=
  BinarySingleNaN.Bcompare (B2BSN f1) (B2BSN f2).

Theorem Bcompare_correct :
   f1 f2,
  is_finite f1 = true is_finite f2 = true
  Bcompare f1 f2 = Some (Rcompare (B2R f1) (B2R f2)).

Theorem Bcompare_swap :
   x y,
  Bcompare y x = match Bcompare x y with Some cSome (CompOpp c) | NoneNone end.

Theorem bounded_le_emax_minus_prec :
   mx ex,
  bounded mx ex = true
  (F2R (Float radix2 (Zpos mx) ex)
    bpow radix2 emax - bpow radix2 (emax - prec))%R.

Theorem bounded_lt_emax :
   mx ex,
  bounded mx ex = true
  (F2R (Float radix2 (Zpos mx) ex) < bpow radix2 emax)%R.

Theorem bounded_ge_emin :
   mx ex,
  bounded mx ex = true
  (bpow radix2 emin F2R (Float radix2 (Zpos mx) ex))%R.

Theorem abs_B2R_le_emax_minus_prec :
   x,
  (Rabs (B2R x) bpow radix2 emax - bpow radix2 (emax - prec))%R.

Theorem abs_B2R_lt_emax :
   x,
  (Rabs (B2R x) < bpow radix2 emax)%R.

Theorem abs_B2R_ge_emin :
   x,
  is_finite_strict x = true
  (bpow radix2 emin Rabs (B2R x))%R.

Theorem bounded_canonical_lt_emax :
   mx ex,
  canonical radix2 fexp (Float radix2 (Zpos mx) ex)
  (F2R (Float radix2 (Zpos mx) ex) < bpow radix2 emax)%R
  bounded mx ex = true.

Truncation

Notation shr_fexp := (shr_fexp prec emax) (only parsing).

Theorem shr_fexp_truncate :
   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').

Rounding modes
Multiplication

Definition Bmult mult_nan m x y :=
  BSN2B (mult_nan x y) (Bmult m (B2BSN x) (B2BSN y)).

Theorem Bmult_correct :
   mult_nan m x y,
  if Rlt_bool (Rabs (round radix2 fexp (round_mode m) (B2R x × B2R y))) (bpow radix2 emax) then
    B2R (Bmult mult_nan m x y) = round radix2 fexp (round_mode m) (B2R x × B2R y)
    is_finite (Bmult mult_nan m x y) = andb (is_finite x) (is_finite y)
    (is_nan (Bmult mult_nan m x y) = false
      Bsign (Bmult mult_nan m x y) = xorb (Bsign x) (Bsign y))
  else
    B2FF (Bmult mult_nan m x y) = binary_overflow m (xorb (Bsign x) (Bsign y)).

Normalization and rounding

Definition shl_align_fexp mx ex :=
  shl_align mx ex (fexp (Zpos (digits2_pos mx) + ex)).

Lemma shl_align_fexp_correct :
   mx ex,
  let (mx', ex') := shl_align_fexp mx ex in
  F2R (Float radix2 (Zpos mx) ex) = F2R (Float radix2 (Zpos mx') ex')
  (ex' fexp (Zdigits radix2 (Zpos mx') + ex'))%Z.

Definition binary_round m sx mx ex :=
  SF2FF (binary_round prec emax m sx mx ex).

Theorem binary_round_correct :
   m sx mx ex,
  let z := binary_round m sx mx ex in
  valid_binary z = true
  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
    FF2R radix2 z = round radix2 fexp (round_mode m) x
    is_finite_FF z = true
    sign_FF z = sx
  else
    z = binary_overflow m sx.

Definition binary_normalize mode m e szero :=
  BSN2B' _ (is_nan_binary_normalize prec emax _ _ mode m e szero).

Theorem binary_normalize_correct :
   m mx ex szero,
  if Rlt_bool (Rabs (round radix2 fexp (round_mode m) (F2R (Float radix2 mx ex)))) (bpow radix2 emax) then
    B2R (binary_normalize m mx ex szero) = round radix2 fexp (round_mode m) (F2R (Float radix2 mx ex))
    is_finite (binary_normalize m mx ex szero) = true
    Bsign (binary_normalize m mx ex szero) =
      match Rcompare (F2R (Float radix2 mx ex)) 0 with
        | Eqszero
        | Lttrue
        | Gtfalse
      end
  else
    B2FF (binary_normalize m mx ex szero) = binary_overflow m (Rlt_bool (F2R (Float radix2 mx ex)) 0).

Addition

Definition Bplus plus_nan m x y :=
  BSN2B (plus_nan x y) (Bplus m (B2BSN x) (B2BSN y)).

Theorem Bplus_correct :
   plus_nan m x y,
  is_finite x = true
  is_finite y = true
  if Rlt_bool (Rabs (round radix2 fexp (round_mode m) (B2R x + B2R y))) (bpow radix2 emax) then
    B2R (Bplus plus_nan m x y) = round radix2 fexp (round_mode m) (B2R x + B2R y)
    is_finite (Bplus plus_nan m x y) = true
    Bsign (Bplus plus_nan m x y) =
      match Rcompare (B2R x + B2R y) 0 with
        | Eqmatch m with mode_DNorb (Bsign x) (Bsign y)
                                 | _andb (Bsign x) (Bsign y) end
        | Lttrue
        | Gtfalse
      end
  else
    (B2FF (Bplus plus_nan m x y) = binary_overflow m (Bsign x) Bsign x = Bsign y).

Subtraction

Definition Bminus minus_nan m x y :=
  BSN2B (minus_nan x y) (Bminus m (B2BSN x) (B2BSN y)).

Theorem Bminus_correct :
   minus_nan m x y,
  is_finite x = true
  is_finite y = true
  if Rlt_bool (Rabs (round radix2 fexp (round_mode m) (B2R x - B2R y))) (bpow radix2 emax) then
    B2R (Bminus minus_nan m x y) = round radix2 fexp (round_mode m) (B2R x - B2R y)
    is_finite (Bminus minus_nan m x y) = true
    Bsign (Bminus minus_nan m x y) =
      match Rcompare (B2R x - B2R y) 0 with
        | Eqmatch m with mode_DNorb (Bsign x) (negb (Bsign y))
                                 | _andb (Bsign x) (negb (Bsign y)) end
        | Lttrue
        | Gtfalse
      end
  else
    (B2FF (Bminus minus_nan m x y) = binary_overflow m (Bsign x) Bsign x = negb (Bsign y)).

Fused Multiply-Add

Definition Bfma_szero m (x y z : binary_float) :=
  Bfma_szero prec emax m (B2BSN x) (B2BSN y) (B2BSN z).

Definition Bfma fma_nan m (x y z: binary_float) :=
  BSN2B (fma_nan x y z) (Bfma m (B2BSN x) (B2BSN y) (B2BSN z)).

Theorem Bfma_correct:
   fma_nan m x y z,
  is_finite x = true
  is_finite y = true
  is_finite z = true
  let res := (B2R x × B2R y + B2R z)%R in
  if Rlt_bool (Rabs (round radix2 fexp (round_mode m) res)) (bpow radix2 emax) then
    B2R (Bfma fma_nan m x y z) = round radix2 fexp (round_mode m) res
    is_finite (Bfma fma_nan m x y z) = true
    Bsign (Bfma fma_nan m x y z) =
      match Rcompare res 0 with
        | EqBfma_szero m x y z
        | Lttrue
        | Gtfalse
      end
  else
    B2FF (Bfma fma_nan m x y z) = binary_overflow m (Rlt_bool res 0).

Division

Definition Bdiv div_nan m x y :=
  BSN2B (div_nan x y) (Bdiv m (B2BSN x) (B2BSN y)).

Theorem Bdiv_correct :
   div_nan m x y,
  B2R y 0%R
  if Rlt_bool (Rabs (round radix2 fexp (round_mode m) (B2R x / B2R y))) (bpow radix2 emax) then
    B2R (Bdiv div_nan m x y) = round radix2 fexp (round_mode m) (B2R x / B2R y)
    is_finite (Bdiv div_nan m x y) = is_finite x
    (is_nan (Bdiv div_nan m x y) = false
      Bsign (Bdiv div_nan m x y) = xorb (Bsign x) (Bsign y))
  else
    B2FF (Bdiv div_nan m x y) = binary_overflow m (xorb (Bsign x) (Bsign y)).

Square root

Definition Bsqrt sqrt_nan m x :=
  BSN2B (sqrt_nan x) (Bsqrt m (B2BSN x)).

Theorem Bsqrt_correct :
   sqrt_nan m x,
  B2R (Bsqrt sqrt_nan m x) = round radix2 fexp (round_mode m) (sqrt (B2R x))
  is_finite (Bsqrt sqrt_nan m x) = match x with B754_zero _true | B754_finite false _ _ _true | _false end
  (is_nan (Bsqrt sqrt_nan m x) = false Bsign (Bsqrt sqrt_nan m x) = Bsign x).

NearbyInt and Trunc
A few values

Definition Bone :=
  BSN2B' _ (@is_nan_Bone prec emax _ _).

Theorem Bone_correct : B2R Bone = 1%R.

Lemma is_finite_Bone : is_finite Bone = true.

Lemma Bsign_Bone : Bsign Bone = false.

Definition Bmax_float :=
  BSN2B' Bmax_float eq_refl.

Extraction/modification of mantissa/exponent

Definition Bnormfr_mantissa x :=
  Bnormfr_mantissa (B2BSN x).

Definition lift x y (Ny : @BinarySingleNaN.is_nan prec emax y = is_nan x) : binary_float.

Lemma B2BSN_lift :
   x y Ny,
  B2BSN (lift x y Ny) = y.

Definition Bldexp (mode : mode) (x : binary_float) (e : Z) : binary_float.

Theorem Bldexp_correct :
   m (f : binary_float) e,
  if Rlt_bool
       (Rabs (round radix2 fexp (round_mode m) (B2R f × bpow radix2 e)))
       (bpow radix2 emax) then
    (B2R (Bldexp m f e)
     = round radix2 fexp (round_mode m) (B2R f × bpow radix2 e))%R
    is_finite (Bldexp m f e) = is_finite f
    Bsign (Bldexp m f e) = Bsign f
  else
    B2FF (Bldexp m f e) = binary_overflow m (Bsign f).

Section Bfrexp.

This hypothesis is needed to implement Bfrexp (otherwise, we have emin > - prec and Bfrexp cannot fit the mantissa in interval [0.5, 1))
Hypothesis Hemax : (2 < emax)%Z.

Definition Bfrexp (x : binary_float) : binary_float × Z.

Theorem Bfrexp_correct :
   f,
  is_finite_strict f = true
  let x := B2R f in
  let z := fst (Bfrexp f) in
  let e := snd (Bfrexp f) in
  (/2 Rabs (B2R z) < 1)%R
  (x = B2R z × bpow radix2 e)%R
  e = mag radix2 x.

End Bfrexp.

Ulp

Definition Bulp (x : binary_float) : binary_float.

Theorem Bulp_correct :
   x,
  is_finite x = true
  B2R (Bulp x) = ulp radix2 fexp (B2R x)
  is_finite (Bulp x) = true
  Bsign (Bulp x) = false.

Successor (and predecessor)

Definition Bsucc (x : binary_float) : binary_float.

Lemma Bsucc_correct :
   x,
  is_finite x = true
  if Rlt_bool (succ radix2 fexp (B2R x)) (bpow radix2 emax) then
    B2R (Bsucc x) = succ radix2 fexp (B2R x)
    is_finite (Bsucc x) = true
    (Bsign (Bsucc x) = Bsign x && is_finite_strict x)%bool
  else
    B2FF (Bsucc x) = F754_infinity false.

Definition Bpred (x : binary_float) : binary_float.

Lemma Bpred_correct :
   x,
  is_finite x = true
  if Rlt_bool (- bpow radix2 emax) (pred radix2 fexp (B2R x)) then
    B2R (Bpred x) = pred radix2 fexp (B2R x)
    is_finite (Bpred x) = true
    (Bsign (Bpred x) = Bsign x || negb (is_finite_strict x))%bool
  else
    B2FF (Bpred x) = F754_infinity true.

End Binary.