(** 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. *)
(** * Helper function and theorem for computing the rounded quotient of two floating-point numbers. *)
Section Fcalc_div.
Variable beta : radix.
Notation bpow e := (bpow beta e).
(** Computes a mantissa of precision p, the corresponding exponent, and the position with respect to the real quotient of the input floating-point numbers.  The algorithm performs the following steps: - Shift dividend mantissa so that it has at least p2 + p digits. - Perform the Euclidean division. - Compute the position according to the division remainder.  Complexity is fine as long as p1 <= 2p and p2 <= p. *)
Definition Fdiv_core prec m1 e1 m2 e2 :=
let d1 := Zdigits beta m1 in
let d2 := Zdigits beta m2 in
let e := (e1 - e2)%Z in
let (m, e') :=
match (d2 + prec - d1)%Z with
| Zpos p => (m1 * Zpower_pos beta p, e + Zneg p)%Z
| _ => (m1, e)
end in
let '(q, r) :=  Zdiv_eucl m m2 in
(q, e', new_location m2 r loc_Exact).

Theorem Fdiv_core_correct :
forall prec m1 e1 m2 e2,
(0 < prec)%Z ->
(0 < m1)%Z -> (0 < m2)%Z ->
let '(m, e, l) := Fdiv_core prec m1 e1 m2 e2 in
(prec <= Zdigits beta m)%Z /\
inbetween_float beta m e (F2R (Float beta m1 e1) / F2R (Float beta m2 e2)) l.