The Coq Proof Assistant

# Library Coq.Floats.SpecFloat

Require Import ZArith FloatClass.

# Specification of floating-point arithmetic

This specification is mostly borrowed from the IEEE754.Binary module of the Flocq library (see http://flocq.gforge.inria.fr/)

## Inductive specification of floating-point numbers

Similar to Flocq.IEEE754.Binary.full_float, but with no NaN payload.
Variant spec_float :=
| S754_zero (s : bool)
| S754_infinity (s : bool)
| S754_nan
| S754_finite (s : bool) (m : positive) (e : Z).

## Parameterized definitions

prec is the number of bits of the mantissa including the implicit one; emax is the exponent of the infinities.
For instance, Binary64 is defined by prec = 53 and emax = 1024.
Section FloatOps.
Variable prec emax : Z.
Definition emin := (3-emax-prec)%Z.
Definition fexp e := Z.max (e - prec) emin.
Section Zdigits2.
Fixpoint digits2_pos (n : positive) : positive :=
match n with
| xH => xH
| xO p => Pos.succ (digits2_pos p)
| xI p => Pos.succ (digits2_pos p)
end.
Definition Zdigits2 n :=
match n with
| Z0 => n
| Zpos p => Zpos (digits2_pos p)
| Zneg p => Zpos (digits2_pos p)
end.
End Zdigits2.
Section ValidBinary.
Definition canonical_mantissa m e :=
Zeq_bool (fexp (Zpos (digits2_pos m) + e)) e.
Definition bounded m e :=
andb (canonical_mantissa m e) (Zle_bool e (emax - prec)).
Definition valid_binary x :=
match x with
| S754_finite _ m e => bounded m e
| _ => true
end.
End ValidBinary.
Section Iter.
Context {A : Type}.
Variable (f : A -> A).
Fixpoint iter_pos (n : positive) (x : A) {struct n} : A :=
match n with
| xI n' => iter_pos n' (iter_pos n' (f x))
| xO n' => iter_pos n' (iter_pos n' x)
| xH => f x
end.
End Iter.
Section Rounding.
Inductive location := loc_Exact | loc_Inexact : comparison -> location.
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.
Definition shr mrs e n :=
match n with
| Zpos p => (iter_pos shr_1 p mrs, (e + n)%Z)
| _ => (mrs, e)
end.
Definition shr_fexp m e l :=
shr (shr_record_of_loc m l) e (fexp (Zdigits2 m + e) - e).
Definition round_nearest_even mx lx :=
match lx with
| loc_Exact => mx
| loc_Inexact Lt => mx
| loc_Inexact Eq => if Z.even mx then mx else (mx + 1)%Z
| loc_Inexact Gt => (mx + 1)%Z
end.
Definition binary_round_aux sx mx ex lx :=
let '(mrs', e') := shr_fexp mx ex lx in
let '(mrs'', e'') := shr_fexp (round_nearest_even (shr_m mrs') (loc_of_shr_record mrs')) e' loc_Exact in
match shr_m mrs'' with
| Z0 => S754_zero sx
| Zpos m => if Zle_bool e'' (emax - prec) then S754_finite sx m e'' else S754_infinity sx
| _ => S754_nan
end.
Definition shl_align mx ex ex' :=
match (ex' - ex)%Z with
| Zneg d => (shift_pos d mx, ex')
| _ => (mx, ex)
end.
Definition binary_round sx mx ex :=
let '(mz, ez) := shl_align mx ex (fexp (Zpos (digits2_pos mx) + ex))in
binary_round_aux sx (Zpos mz) ez loc_Exact.
Definition binary_normalize m e szero :=
match m with
| Z0 => S754_zero szero
| Zpos m => binary_round false m e
| Zneg m => binary_round true m e
end.
End Rounding.

## Define operations

Definition SFopp x :=
match x with
| S754_nan => S754_nan
| S754_infinity sx => S754_infinity (negb sx)
| S754_finite sx mx ex => S754_finite (negb sx) mx ex
| S754_zero sx => S754_zero (negb sx)
end.
Definition SFabs x :=
match x with
| S754_nan => S754_nan
| S754_infinity sx => S754_infinity false
| S754_finite sx mx ex => S754_finite false mx ex
| S754_zero sx => S754_zero false
end.
Definition SFcompare f1 f2 :=
match f1, f2 with
| S754_nan , _ | _, S754_nan => None
| S754_infinity s1, S754_infinity s2 =>
Some match s1, s2 with
| true, true => Eq
| false, false => Eq
| true, false => Lt
| false, true => Gt
end
| S754_infinity s, _ => Some (if s then Lt else Gt)
| _, S754_infinity s => Some (if s then Gt else Lt)
| S754_finite s _ _, S754_zero _ => Some (if s then Lt else Gt)
| S754_zero _, S754_finite s _ _ => Some (if s then Gt else Lt)
| S754_zero _, S754_zero _ => Some Eq
| S754_finite s1 m1 e1, S754_finite s2 m2 e2 =>
Some match s1, s2 with
| true, false => Lt
| false, true => Gt
| false, false =>
match Z.compare e1 e2 with
| Lt => Lt
| Gt => Gt
| Eq => Pcompare m1 m2 Eq
end
| true, true =>
match Z.compare e1 e2 with
| Lt => Gt
| Gt => Lt
| Eq => CompOpp (Pcompare m1 m2 Eq)
end
end
end.
Definition SFeqb f1 f2 :=
match SFcompare f1 f2 with
| Some Eq => true
| _ => false
end.
Definition SFltb f1 f2 :=
match SFcompare f1 f2 with
| Some Lt => true
| _ => false
end.
Definition SFleb f1 f2 :=
match SFcompare f1 f2 with
| Some (Lt | Eq) => true
| _ => false
end.
Definition SFclassify f :=
match f with
| S754_nan => NaN
| S754_infinity false => PInf
| S754_infinity true => NInf
| S754_zero false => PZero
| S754_zero true => NZero
| S754_finite false m _ =>
if (digits2_pos m =? Z.to_pos prec)%positive then PNormal
else PSubn
| S754_finite true m _ =>
if (digits2_pos m =? Z.to_pos prec)%positive then NNormal
else NSubn
end.
Definition SFmul x y :=
match x, y with
| S754_nan, _ | _, S754_nan => S754_nan
| S754_infinity sx, S754_infinity sy => S754_infinity (xorb sx sy)
| S754_infinity sx, S754_finite sy _ _ => S754_infinity (xorb sx sy)
| S754_finite sx _ _, S754_infinity sy => S754_infinity (xorb sx sy)
| S754_infinity _, S754_zero _ => S754_nan
| S754_zero _, S754_infinity _ => S754_nan
| S754_finite sx _ _, S754_zero sy => S754_zero (xorb sx sy)
| S754_zero sx, S754_finite sy _ _ => S754_zero (xorb sx sy)
| S754_zero sx, S754_zero sy => S754_zero (xorb sx sy)
| S754_finite sx mx ex, S754_finite sy my ey =>
binary_round_aux (xorb sx sy) (Zpos (mx * my)) (ex + ey) loc_Exact
end.
Definition cond_Zopp (b : bool) m := if b then Z.opp m else m.
Definition SFadd x y :=
match x, y with
| S754_nan, _ | _, S754_nan => S754_nan
| S754_infinity sx, S754_infinity sy =>
if Bool.eqb sx sy then x else S754_nan
| S754_infinity _, _ => x
| _, S754_infinity _ => y
| S754_zero sx, S754_zero sy =>
if Bool.eqb sx sy then x else
S754_zero false
| S754_zero _, _ => y
| _, S754_zero _ => x
| S754_finite sx mx ex, S754_finite sy my ey =>
let ez := Z.min ex ey in
binary_normalize (Zplus (cond_Zopp sx (Zpos (fst (shl_align mx ex ez)))) (cond_Zopp sy (Zpos (fst (shl_align my ey ez)))))
ez false
end.
Definition SFsub x y :=
match x, y with
| S754_nan, _ | _, S754_nan => S754_nan
| S754_infinity sx, S754_infinity sy =>
if Bool.eqb sx (negb sy) then x else S754_nan
| S754_infinity _, _ => x
| _, S754_infinity sy => S754_infinity (negb sy)
| S754_zero sx, S754_zero sy =>
if Bool.eqb sx (negb sy) then x else
S754_zero false
| S754_zero _, S754_finite sy my ey => S754_finite (negb sy) my ey
| _, S754_zero _ => x
| S754_finite sx mx ex, S754_finite sy my ey =>
let ez := Z.min ex ey in
binary_normalize (Zminus (cond_Zopp sx (Zpos (fst (shl_align mx ex ez)))) (cond_Zopp sy (Zpos (fst (shl_align my ey ez)))))
ez false
end.
Definition new_location_even nb_steps k :=
if Zeq_bool k 0 then loc_Exact
else loc_Inexact (Z.compare (2 * k) nb_steps).
Definition new_location_odd nb_steps k :=
if Zeq_bool k 0 then loc_Exact
else
loc_Inexact
match Z.compare (2 * k + 1) nb_steps with
| Lt => Lt
| Eq => Lt
| Gt => Gt
end.
Definition new_location nb_steps :=
if Z.even nb_steps then new_location_even nb_steps else new_location_odd nb_steps.
Definition SFdiv_core_binary m1 e1 m2 e2 :=
let d1 := Zdigits2 m1 in
let d2 := Zdigits2 m2 in
let e' := Z.min (fexp (d1 + e1 - (d2 + e2))) (e1 - e2) in
let s := (e1 - e2 - e')%Z in
let m' :=
match s with
| Zpos _ => Z.shiftl m1 s
| Z0 => m1
| Zneg _ => Z0
end in
let '(q, r) := Z.div_eucl m' m2 in
(q, e', new_location m2 r).
Definition SFdiv x y :=
match x, y with
| S754_nan, _ | _, S754_nan => S754_nan
| S754_infinity sx, S754_infinity sy => S754_nan
| S754_infinity sx, S754_finite sy _ _ => S754_infinity (xorb sx sy)
| S754_finite sx _ _, S754_infinity sy => S754_zero (xorb sx sy)
| S754_infinity sx, S754_zero sy => S754_infinity (xorb sx sy)
| S754_zero sx, S754_infinity sy => S754_zero (xorb sx sy)
| S754_finite sx _ _, S754_zero sy => S754_infinity (xorb sx sy)
| S754_zero sx, S754_finite sy _ _ => S754_zero (xorb sx sy)
| S754_zero sx, S754_zero sy => S754_nan
| S754_finite sx mx ex, S754_finite sy my ey =>
let '(mz, ez, lz) := SFdiv_core_binary (Zpos mx) ex (Zpos my) ey in
binary_round_aux (xorb sx sy) mz ez lz
end.
Definition SFsqrt_core_binary m e :=
let d := Zdigits2 m in
let e' := Z.min (fexp (Z.div2 (d + e + 1))) (Z.div2 e) in
let s := (e - 2 * e')%Z in
let m' :=
match s with
| Zpos p => Z.shiftl m s
| Z0 => m
| Zneg _ => Z0
end in
let (q, r) := Z.sqrtrem m' in
let l :=
if Zeq_bool r 0 then loc_Exact
else loc_Inexact (if Zle_bool r q then Lt else Gt) in
(q, e', l).
Definition SFsqrt x :=
match x with
| S754_nan => S754_nan
| S754_infinity false => x
| S754_infinity true => S754_nan
| S754_finite true _ _ => S754_nan
| S754_zero _ => x
| S754_finite false mx ex =>
let '(mz, ez, lz) := SFsqrt_core_binary (Zpos mx) ex in
binary_round_aux false mz ez lz
end.
Definition SFnormfr_mantissa f :=
match f with
| S754_finite _ mx ex =>
if Z.eqb ex (-prec) then Npos mx else 0%N
| _ => 0%N
end.
Definition SFldexp f e :=
match f with
| S754_finite sx mx ex => binary_round sx mx (ex+e)
| _ => f
end.
Definition SFfrexp f :=
match f with
| S754_finite sx mx ex =>
if (Z.to_pos prec <=? digits2_pos mx)%positive then
(S754_finite sx mx (-prec), (ex+prec)%Z)
else
let d := (prec - Z.pos (digits2_pos mx))%Z in
(S754_finite sx (shift_pos (Z.to_pos d) mx) (-prec), (ex+prec-d)%Z)
| _ => (f, (-2*emax-prec)%Z)
end.
Definition SFone := binary_round false 1 0.
Definition SFulp x := SFldexp SFone (fexp (snd (SFfrexp x))).
Definition SFpred_pos x :=
match x with
| S754_finite _ mx _ =>
let d :=
if (mx~0 =? shift_pos (Z.to_pos prec) 1)%positive then
SFldexp SFone (fexp (snd (SFfrexp x) - 1))
else
SFulp x in
SFsub x d
| _ => x
end.
Definition SFmax_float :=
S754_finite false (shift_pos (Z.to_pos prec) 1 - 1) (emax - prec).
Definition SFsucc x :=
match x with
| S754_zero _ => SFldexp SFone emin
| S754_infinity false => x
| S754_infinity true => SFopp SFmax_float
| S754_nan => x
| S754_finite false _ _ => SFadd x (SFulp x)
| S754_finite true _ _ => SFopp (SFpred_pos (SFopp x))
end.
Definition SFpred f := SFopp (SFsucc (SFopp f)).
End FloatOps.