(** 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. *)
(** * Relative error of the roundings *)
Section Fprop_relative.
Variable beta : radix.
Notation bpow e := (bpow beta e).
Section Fprop_relative_generic.
Variable fexp : Z -> Z.
Context { prop_exp : Valid_exp fexp }.
Section relative_error_conversion.
Variable rnd : R -> Z.
Context { valid_rnd : Valid_rnd rnd }.

(* Specifications: *)
Lemma relative_error_lt_conversion :
forall x b, (0 < b)%R ->
(x <> 0 -> Rabs (round beta fexp rnd x - x) < b * Rabs x)%R ->
exists eps,
(Rabs eps < b)%R /\ round beta fexp rnd x = (x * (1 + eps))%R.
Lemma relative_error_lt_conversion :
forall x b, (0 < b)%R ->
(x <> 0 -> Rabs (round beta fexp rnd x - x) < b * Rabs x)%R ->
exists eps,
(Rabs eps < b)%R /\ round beta fexp rnd x = (x * (1 + eps))%R.