Library Flocq.Prop.Sterbenz

This file is part of the Flocq formalization of floating-point arithmetic in Coq: https://flocq.gitlabpages.inria.fr/

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.

Sterbenz conditions for exact subtraction

From Coq Require Import ZArith Reals.

Require Import Zaux Raux Defs Generic_fmt Operations.

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 :
  ∀ x y,
  format x → format y →
  (Rabs (x + y) ≤ bpow (Z.min (mag beta x) (mag beta y)))%R →
  format (x + y)%R.

Theorem generic_format_plus_weak :
  ∀ x y,
  format x → format y →
  (Rabs (x + y) ≤ Rmin (Rabs x) (Rabs y))%R →
  format (x + y)%R.

Lemma sterbenz_aux :
  ∀ x y, format x → format y →
  (y ≤ x ≤ 2 × y)%R →
  format (x - y)%R.

Theorem sterbenz :
  ∀ x y, format x → format y →
  (y / 2 ≤ x ≤ 2 × y)%R →
  format (x - y)%R.

End Fprop_Sterbenz.