Library Flocq.Core.Fcore_FLX

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

Copyright (C) 2010-2011 Sylvie Boldo
Copyright (C) 2010-2011 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 without underflow

Require Import Fcore_Raux.
Require Import Fcore_defs.
Require Import Fcore_rnd.
Require Import Fcore_generic_fmt.
Require Import Fcore_float_prop.
Require Import Fcore_FIX.
Require Import Fcore_rnd_ne.

Section RND_FLX.

Variable beta : radix.

Notation bpow e := (bpow beta e).

Variable prec : Z.

Class Prec_gt_0 :=
  prec_gt_0 : (0 < prec)%Z.

Context { prec_gt_0_ : Prec_gt_0 }.

Definition FLX_format (x : R) :=
  exists f : float beta,
  x = F2R f /\ (Zabs (Fnum f) < Zpower beta prec)%Z.

Definition FLX_exp (e : Z) := (e - prec)%Z.

Properties of the FLX format

Global Instance FLX_exp_valid : Valid_exp FLX_exp.

Theorem FIX_format_FLX :
  forall x e,
  (bpow (e - 1) <= Rabs x <= bpow e)%R ->
  FLX_format x ->
  FIX_format beta (e - prec) x.

Theorem FLX_format_generic :
  forall x, generic_format beta FLX_exp x -> FLX_format x.

Theorem generic_format_FLX :
  forall x, FLX_format x -> generic_format beta FLX_exp x.

Theorem FLX_format_satisfies_any :
  satisfies_any FLX_format.

Theorem FLX_format_FIX :
  forall x e,
  (bpow (e - 1) <= Rabs x <= bpow e)%R ->
  FIX_format beta (e - prec) x ->
  FLX_format x.

unbounded floating-point format with normal mantissas
Definition FLXN_format (x : R) :=
  exists f : float beta,
  x = F2R f /\ (x <> R0 ->
  Zpower beta (prec - 1) <= Zabs (Fnum f) < Zpower beta prec)%Z.

Theorem generic_format_FLXN :
  forall x, FLXN_format x -> generic_format beta FLX_exp x.

Theorem FLXN_format_generic :
  forall x, generic_format beta FLX_exp x -> FLXN_format x.

Theorem FLXN_format_satisfies_any :
  satisfies_any FLXN_format.

FLX is a nice format: it has a monotone exponent...
and it allows a rounding to nearest, ties to even.
Hypothesis NE_prop : Zeven beta = false \/ (1 < prec)%Z.

Global Instance exists_NE_FLX : Exists_NE beta FLX_exp.