Library Flocq.Prop.Fprop_Sterbenz
This file is part of the Flocq formalization of floating-point
arithmetic in Coq: http://flocq.gforge.inria.fr/
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.
Copyright (C) 2010-2011 Guillaume Melquiond
Sterbenz conditions for exact subtraction
Require Import Fcore_Raux.
Require Import Fcore_defs.
Require Import Fcore_generic_fmt.
Require Import Fcalc_ops.
Section Fprop_Sterbenz.
Variable beta : radix.
Notation bpow e := (bpow beta e).
Variable fexp : Z -> Z.
Context { valid_exp : Valid_exp fexp }.
Context { monotone_exp : Monotone_exp fexp }.
Notation format := (generic_format beta fexp).
Theorem generic_format_plus :
forall x y,
format x -> format y ->
(Rabs (x + y) < bpow (Zmin (ln_beta beta x) (ln_beta beta y)))%R ->
format (x + y)%R.
Theorem generic_format_plus_weak :
forall x y,
format x -> format y ->
(Rabs (x + y) <= Rmin (Rabs x) (Rabs y))%R ->
format (x + y)%R.
Lemma sterbenz_aux :
forall x y, format x -> format y ->
(y <= x <= 2 * y)%R ->
format (x - y)%R.
Theorem sterbenz :
forall x y, format x -> format y ->
(y / 2 <= x <= 2 * y)%R ->
format (x - y)%R.
End Fprop_Sterbenz.