Library Flocq.Core.FTZ

This file is part of the Flocq formalization of floating-point arithmetic in Coq:
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.

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
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.