Section Fprop_mult_error.
Variable beta : radix.
Notation bpow e := (bpow beta e).
Variable prec : Z.
Context { prec_gt_0_ : Prec_gt_0 prec }.
Notation format := (generic_format beta (FLX_exp prec)).
Notation cexp := (cexp beta (FLX_exp prec)).
Variable rnd : R -> Z.
Context { valid_rnd : Valid_rnd rnd }.
Proof with auto with typeclass_instances.
intros x y Hx Hy Hz.
set (f := (round beta (FLX_exp prec) rnd (x * y))).
destruct (Req_dec (x * y) 0) as [Hxy0|Hxy0].
contradict Hz.
rewrite Hxy0.
rewrite round_0...
ring.
destruct (mag beta (x * y)) as (exy, Hexy).
specialize (Hexy Hxy0).
destruct (mag beta (f - x * y)) as (er, Her).
specialize (Her Hz).
destruct (mag beta x) as (ex, Hex).
assert (Hx0: (x <> 0)%R).
contradict Hxy0.
now rewrite Hxy0, Rmult_0_l.
specialize (Hex Hx0).
destruct (mag beta y) as (ey, Hey).
assert (Hy0: (y <> 0)%R).
contradict Hxy0.
now rewrite Hxy0, Rmult_0_r.
specialize (Hey Hy0).
(* *)
assert (Hc1: (cexp (x * y)%R - prec <= cexp x + cexp y)%Z).
unfold cexp, FLX_exp.
rewrite mag_unique with (1 := Hex).
rewrite mag_unique with (1 := Hey).
rewrite mag_unique with (1 := Hexy).
cut (exy - 1 < ex + ey)%Z. omega.
apply (lt_bpow beta).
apply Rle_lt_trans with (1 := proj1 Hexy).
rewrite Rabs_mult.
rewrite bpow_plus.
apply Rmult_le_0_lt_compat.
apply Rabs_pos.
apply Rabs_pos.
apply Hex.
apply Hey.
(* *)
assert (Hc2: (cexp x + cexp y <= cexp (x * y)%R)%Z).
unfold cexp, FLX_exp.
rewrite mag_unique with (1 := Hex).
rewrite mag_unique with (1 := Hey).
rewrite mag_unique with (1 := Hexy).
cut ((ex - 1) + (ey - 1) < exy)%Z.
generalize (prec_gt_0 prec).
clear ; omega.
apply (lt_bpow beta).
apply Rle_lt_trans with (2 := proj2 Hexy).
rewrite Rabs_mult.
rewrite bpow_plus.
apply Rmult_le_compat.
apply bpow_ge_0.
apply bpow_ge_0.
apply Hex.
apply Hey.
(* *)
assert (Hr: ((F2R (Float beta (- (Ztrunc (scaled_mantissa beta (FLX_exp prec) x) *
Ztrunc (scaled_mantissa beta (FLX_exp prec) y)) + rnd (scaled_mantissa beta (FLX_exp prec) (x * y)) *
beta ^ (cexp (x * y)%R - (cexp x + cexp y))) (cexp x + cexp y))) = f - x * y)%R).
rewrite Hx at 6.
rewrite Hy at 6.
rewrite <- F2R_mult.
simpl.
unfold f, round, Rminus.
rewrite <- F2R_opp, Rplus_comm, <- F2R_plus.
unfold Fplus. simpl.
now rewrite Zle_imp_le_bool with (1 := Hc2).
(* *)
exists (Float beta (- (Ztrunc (scaled_mantissa beta (FLX_exp prec) x) *
Ztrunc (scaled_mantissa beta (FLX_exp prec) y)) + rnd (scaled_mantissa beta (FLX_exp prec) (x * y)) *
beta ^ (cexp (x * y)%R - (cexp x + cexp y))) (cexp x + cexp y)).
split;[assumption|split].
rewrite Hr.
simpl.
clear Hr.
apply Z.le_trans with (cexp (x * y)%R - prec)%Z.
unfold cexp, FLX_exp.
apply Zplus_le_compat_r.
rewrite mag_unique with (1 := Hexy).
apply mag_le_bpow with (1 := Hz).
replace (bpow (exy - prec)) with (ulp beta (FLX_exp prec) (x * y)).
apply error_lt_ulp...
rewrite ulp_neq_0; trivial.
unfold cexp.
now rewrite mag_unique with (1 := Hexy).
apply Hc1.
reflexivity.
Qed.

(** Error of the multiplication in FLX *)
Theorem mult_error_FLX :
forall x y,
format x -> format y ->
format (round beta (FLX_exp prec) rnd (x * y) - (x * y))%R.