Library Flocq.Core.FTZ
This file is part of the Flocq formalization of floating-point
arithmetic in Coq: https://flocq.gitlabpages.inria.fr/
Copyright (C) 2009-2018 Sylvie Boldo
Copyright (C) 2009-2018 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.
Copyright (C) 2009-2018 Guillaume Melquiond
Floating-point format with abrupt underflow
From Coq Require Import ZArith Reals Lia.
Require Import Zaux Raux Defs Round_pred Generic_fmt Float_prop Ulp FLX.
Section RND_FTZ.
Variable beta : radix.
Notation bpow e := (bpow beta e).
Variable emin prec : Z.
Context { prec_gt_0_ : Prec_gt_0 prec }.
Inductive FTZ_format (x : R) : Prop :=
FTZ_spec (f : float beta) :
x = F2R f →
(x ≠ 0%R → Zpower beta (prec - 1) ≤ Z.abs (Fnum f) < Zpower beta prec)%Z →
(emin ≤ Fexp f)%Z →
FTZ_format x.
Definition FTZ_exp e := if Zlt_bool (e - prec) emin then (emin + prec - 1)%Z else (e - prec)%Z.
Properties of the FTZ format
Global Instance FTZ_exp_valid : Valid_exp FTZ_exp.
Theorem FLXN_format_FTZ :
∀ x, FTZ_format x → FLXN_format beta prec x.
Theorem generic_format_FTZ :
∀ x, FTZ_format x → generic_format beta FTZ_exp x.
Theorem FTZ_format_generic :
∀ x, generic_format beta FTZ_exp x → FTZ_format x.
Theorem FTZ_format_satisfies_any :
satisfies_any FTZ_format.
Theorem FTZ_format_FLXN :
∀ x : R,
(bpow (emin + prec - 1) ≤ Rabs x)%R →
FLXN_format beta prec x → FTZ_format x.
Theorem ulp_FTZ_0 :
ulp beta FTZ_exp 0 = bpow (emin+prec-1).
Section FTZ_round.
Theorem FLXN_format_FTZ :
∀ x, FTZ_format x → FLXN_format beta prec x.
Theorem generic_format_FTZ :
∀ x, FTZ_format x → generic_format beta FTZ_exp x.
Theorem FTZ_format_generic :
∀ x, generic_format beta FTZ_exp x → FTZ_format x.
Theorem FTZ_format_satisfies_any :
satisfies_any FTZ_format.
Theorem FTZ_format_FLXN :
∀ x : R,
(bpow (emin + prec - 1) ≤ Rabs x)%R →
FLXN_format beta prec x → FTZ_format x.
Theorem ulp_FTZ_0 :
ulp beta FTZ_exp 0 = bpow (emin+prec-1).
Section FTZ_round.
Rounding with FTZ
Variable rnd : R → Z.
Context { valid_rnd : Valid_rnd rnd }.
Definition Zrnd_FTZ x :=
if Rle_bool 1 (Rabs x) then rnd x else Z0.
Global Instance valid_rnd_FTZ : Valid_rnd Zrnd_FTZ.
Theorem round_FTZ_FLX :
∀ x : R,
(bpow (emin + prec - 1) ≤ Rabs x)%R →
round beta FTZ_exp Zrnd_FTZ x = round beta (FLX_exp prec) rnd x.
Theorem round_FTZ_small :
∀ x : R,
(Rabs x < bpow (emin + prec - 1))%R →
round beta FTZ_exp Zrnd_FTZ x = 0%R.
End FTZ_round.
End RND_FTZ.
Context { valid_rnd : Valid_rnd rnd }.
Definition Zrnd_FTZ x :=
if Rle_bool 1 (Rabs x) then rnd x else Z0.
Global Instance valid_rnd_FTZ : Valid_rnd Zrnd_FTZ.
Theorem round_FTZ_FLX :
∀ x : R,
(bpow (emin + prec - 1) ≤ Rabs x)%R →
round beta FTZ_exp Zrnd_FTZ x = round beta (FLX_exp prec) rnd x.
Theorem round_FTZ_small :
∀ x : R,
(Rabs x < bpow (emin + prec - 1))%R →
round beta FTZ_exp Zrnd_FTZ x = 0%R.
End FTZ_round.
End RND_FTZ.