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.