(** This file is part of the Flocq formalization of floating-point arithmetic in Coq: http://flocq.gforge.inria.fr/  Copyright (C) 2010-2013 Sylvie Boldo #<br /># Copyright (C) 2010-2013 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 gradual underflow *)
Section RND_FLT.
Variable beta : radix.
Notation bpow e := (bpow beta e).
Variable emin prec : Z.
Context { prec_gt_0_ : Prec_gt_0 prec }.
(* floating-point format with gradual underflow *)
Definition FLT_format (x : R) :=
exists f : float beta,
x = F2R f /\ (Zabs (Fnum f) < Zpower beta prec)%Z /\ (emin <= Fexp f)%Z.
Definition FLT_exp e := Zmax (e - prec) emin.
(** Properties of the FLT format *)
Global Instance FLT_exp_valid : Valid_exp FLT_exp.

(* Specifications: *)
Theorem generic_format_FLT :
forall x, FLT_format x -> generic_format beta FLT_exp x.
Theorem FLT_format_generic :
forall x, generic_format beta FLT_exp x -> FLT_format x.
Theorem FLT_format_bpow :
forall e, (emin <= e)%Z -> generic_format beta FLT_exp (bpow e).
Theorem FLT_format_satisfies_any :
satisfies_any FLT_format.
Theorem canonic_exp_FLT_FLX :
forall x,
(bpow (emin + prec - 1) <= Rabs x)%R ->
canonic_exp beta FLT_exp x = canonic_exp beta (FLX_exp prec) x.
(** Links between FLT and FLX *)
Theorem generic_format_FLT_FLX :
forall x : R,
(bpow (emin + prec - 1) <= Rabs x)%R ->
generic_format beta (FLX_exp prec) x ->
generic_format beta FLT_exp x.
Theorem generic_format_FLX_FLT :
forall x : R,
generic_format beta FLT_exp x -> generic_format beta (FLX_exp prec) x.
Theorem round_FLT_FLX : forall rnd x,
(bpow (emin + prec - 1) <= Rabs x)%R ->
round beta FLT_exp rnd x = round beta (FLX_exp prec) rnd x.
Theorem generic_format_FLT :
forall x, FLT_format x -> generic_format beta FLT_exp x.
Theorem FLT_format_generic :
forall x, generic_format beta FLT_exp x -> FLT_format x.
Theorem FLT_format_bpow :
forall e, (emin <= e)%Z -> generic_format beta FLT_exp (bpow e).
Theorem FLT_format_satisfies_any :
satisfies_any FLT_format.
Theorem canonic_exp_FLT_FLX :
forall x,
(bpow (emin + prec - 1) <= Rabs x)%R ->
canonic_exp beta FLT_exp x = canonic_exp beta (FLX_exp prec) x.
(** Links between FLT and FLX *)
Theorem generic_format_FLT_FLX :
forall x : R,
(bpow (emin + prec - 1) <= Rabs x)%R ->
generic_format beta (FLX_exp prec) x ->
generic_format beta FLT_exp x.
Theorem generic_format_FLX_FLT :
forall x : R,
generic_format beta FLT_exp x -> generic_format beta (FLX_exp prec) x.
Theorem round_FLT_FLX : forall rnd x,
(bpow (emin + prec - 1) <= Rabs x)%R ->
round beta FLT_exp rnd x = round beta (FLX_exp prec) rnd x.