# Helper function and theorem for computing the rounded quotient of two floating-point numbers.

Require Import Fcore_Raux.
Require Import Fcore_defs.
Require Import Fcore_float_prop.
Require Import Fcore_digits.
Require Import Fcalc_bracket.
Require Import Fcalc_digits.

Section Fcalc_div.

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.
Proof.
intros prec m1 e1 m2 e2 Hprec Hm1 Hm2.
unfold Fdiv_core.
set (d1 := Zdigits beta m1).
set (d2 := Zdigits beta m2).
case_eq
(match (d2 + prec - d1)%Z with
| Zpos p => ((m1 * Zpower_pos beta p)%Z, (e1 - e2 + Zneg p)%Z)
| _ => (m1, (e1 - e2)%Z)
end).
intros m' e' Hme.
. the shifted mantissa m' has enough digits *)assert (Hs: F2R (Float beta m' (e' + e2)) = F2R (Float beta m1 e1) /\ (0 < m')%Z /\ (d2 + prec <= Zdigits beta m')%Z).
replace (d2 + prec)%Z with (d2 + prec - d1 + d1)%Z by ring.
destruct (d2 + prec - d1)%Z as [|p|p] ;
unfold d1 ;
inversion Hme.
ring_simplify (e1 - e2 + e2)%Z.
repeat split.
now rewrite <- H0.
apply Zle_refl.
replace (e1 - e2 + Zneg p + e2)%Z with (e1 - Zpos p)%Z by (unfold Zminus ; simpl ; ring).
fold (Zpower beta (Zpos p)).
split.
pattern (Zpos p) at 1 ; replace (Zpos p) with (e1 - (e1 - Zpos p))%Z by ring.
apply sym_eq.
apply F2R_change_exp.
assert (0 < Zpos p)%Z by easy.
omega.
split.
apply Zmult_lt_0_compat.
exact Hm1.
now apply Zpower_gt_0.
rewrite Zdigits_mult_Zpower.
rewrite Zplus_comm.
apply Zle_refl.
apply sym_not_eq.
now apply Zlt_not_eq.
easy.
split.
now ring_simplify (e1 - e2 + e2)%Z.
assert (Zneg p < 0)%Z by easy.
omega.
. *)destruct Hs as (Hs1, (Hs2, Hs3)).
rewrite <- Hs1.
generalize (Z_div_mod m' m2 (Zlt_gt _ _ Hm2)).
destruct (Zdiv_eucl m' m2) as (q, r).
intros (Hq, Hr).
split.
. the result mantissa q has enough digits *)cut (Zdigits beta m' <= d2 + Zdigits beta q)%Z. omega.
unfold d2.
rewrite Hq.
assert (Hq': (0 < q)%Z).
apply Zmult_lt_reg_r with (1 := Hm2).
assert (m2 < m')%Z.
apply lt_Zdigits with beta.
now apply Zlt_le_weak.
unfold d2 in Hs3.
clear -Hprec Hs3 ; omega.
cut (q * m2 = m' - r)%Z. clear -Hr H ; omega.
rewrite Hq.
ring.
apply Zle_trans with (Zdigits beta (m2 + q + m2 * q)).
apply Zdigits_le.
rewrite <- Hq.
now apply Zlt_le_weak.
clear -Hr Hq'. omega.
apply Zdigits_mult_strong ; apply Zlt_le_weak.
now apply Zle_lt_trans with r.
exact Hq'.
. the location is correctly computed *)unfold inbetween_float, F2R. simpl.
rewrite bpow_plus, Z2R_plus.
rewrite Hq, Z2R_plus, Z2R_mult.
replace ((Z2R m2 * Z2R q + Z2R r) * (bpow e' * bpow e2) / (Z2R m2 * bpow e2))%R
with ((Z2R q + Z2R r / Z2R m2) * bpow e')%R.
apply inbetween_mult_compat.
apply bpow_gt_0.
destruct (Z_lt_le_dec 1 m2) as [Hm2''|Hm2''].
replace (Z2R 1) with (Z2R m2 * /Z2R m2)%R.
apply new_location_correct ; try easy.
apply Rinv_0_lt_compat.
now apply (Z2R_lt 0).
now constructor.
apply Rinv_r.
apply Rgt_not_eq.
now apply (Z2R_lt 0).
assert (r = 0 /\ m2 = 1)%Z by (clear -Hr Hm2'' ; omega).
rewrite (proj1 H), (proj2 H).
unfold Rdiv.
rewrite Rmult_0_l, Rplus_0_r.
now constructor.
field.
split ; apply Rgt_not_eq.
apply bpow_gt_0.
now apply (Z2R_lt 0).
Qed.

End Fcalc_div.