(** 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 functions and theorems for computing the rounded square root of a floating-point number. *)
Section Fcalc_sqrt.
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 square root of the input floating-point number.  The algorithm performs the following steps: - Shift the mantissa so that it has at least 2p-1 digits; shift it one digit more if the new exponent is not even. - Compute the square root s (at least p digits) of the new mantissa, and its remainder r. - Compute the position according to the remainder: -- r == 0  =>  Eq, -- r <= s  =>  Lo, -- r >= s  =>  Up.  Complexity is fine as long as p1 <= 2p-1. *)
Definition Fsqrt_core prec m e :=
let d := Zdigits beta m in
let s := Zmax (2 * prec - d) 0 in
let e' := (e - s)%Z in
let (s', e'') := if Zeven e' then (s, e') else (s + 1, e' - 1)%Z in
let m' :=
match s' with
| Zpos p => (m * Zpower_pos beta p)%Z
| _ => m
end in
let (q, r) := Z.sqrtrem m' in
let l :=
if Zeq_bool r 0 then loc_Exact
else loc_Inexact (if Zle_bool r q then Lt else Gt) in
(q, Zdiv2 e'', l).

Theorem Fsqrt_core_correct :
forall prec m e,
(0 < m)%Z ->
let '(m', e', l) := Fsqrt_core prec m e in
(prec <= Zdigits beta m')%Z /\
inbetween_float beta m' e' (sqrt (F2R (Float beta m e))) l.