Commit f89097cf by Guillaume Melquiond

### Proved that the error of multiplication is an FP number in FLX.

parent cc1cc5e3
 ... ... @@ -18,6 +18,7 @@ FILES = \ Calc/Fcalc_round.v \ Calc/Fcalc_round_FIX.v \ Calc/Fcalc_sqrt.v \ Prop/Fprop_mult_error.v \ Prop/Fprop_nearest.v \ Prop/Fprop_Sterbenz.v ... ...
 Require Import Fcore. Require Import Fcalc_ops. Section Fprop_mult_error. Variable beta : radix. Notation bpow e := (bpow beta e). Variable prec : Z. Variable Hp : Zlt 0 prec. Notation format := (generic_format beta (FLX_exp prec)). Notation cexp := (canonic_exponent beta (FLX_exp prec)). Theorem mult_error_FLX : forall rnd, forall x y, format x -> format y -> format (rounding beta (FLX_exp prec) rnd (x * y) - (x * y))%R. Proof. intros rnd x y Hx Hy. set (f := (rounding beta (FLX_exp prec) rnd (x * y))). destruct (Req_dec (f - x * y) 0) as [Hr0|Hr0]. rewrite Hr0. apply generic_format_0. destruct (Req_dec (x * y) 0) as [Hxy0|Hxy0]. unfold f. rewrite Hxy0. rewrite rounding_0. ring_simplify (0 - 0)%R. apply generic_format_0. destruct (ln_beta beta (x * y)) as (exy, Hexy). specialize (Hexy Hxy0). destruct (ln_beta beta (f - x * y)) as (er, Her). specialize (Her Hr0). destruct (ln_beta beta x) as (ex, Hex). assert (Hx0: (x <> 0)%R). contradict Hxy0. now rewrite Hxy0, Rmult_0_l. specialize (Hex Hx0). destruct (ln_beta 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 canonic_exponent, FLX_exp. rewrite ln_beta_unique with (1 := Hex). rewrite ln_beta_unique with (1 := Hey). rewrite ln_beta_unique with (1 := Hexy). cut (exy - 1 < ex + ey)%Z. omega. apply <- (bpow_lt beta). apply Rle_lt_trans with (1 := proj1 Hexy). rewrite Rabs_mult. rewrite bpow_add. 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 canonic_exponent, FLX_exp. rewrite ln_beta_unique with (1 := Hex). rewrite ln_beta_unique with (1 := Hey). rewrite ln_beta_unique with (1 := Hexy). cut ((ex - 1) + (ey - 1) < exy)%Z. omega. apply <- (bpow_lt beta). apply Rle_lt_trans with (2 := proj2 Hexy). rewrite Rabs_mult. rewrite bpow_add. 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)) + Zrnd rnd (scaled_mantissa beta (FLX_exp prec) (x * y)) * radix_val 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 <- mult_F2R. simpl. unfold f, rounding, Rminus. rewrite opp_F2R, Rplus_comm, <- plus_F2R. unfold Fplus. simpl. now rewrite Zle_imp_le_bool with (1 := Hc2). (* *) rewrite <- Hr. apply generic_format_canonic_exponent. rewrite Hr. clear Hr. apply Zle_trans with (cexp (x * y)%R - prec)%Z. unfold canonic_exponent, FLX_exp. apply Zplus_le_compat_r. rewrite ln_beta_unique with (1 := Her). rewrite ln_beta_unique with (1 := Hexy). apply (bpow_lt_bpow beta). apply Rle_lt_trans with (1 := proj1 Her). apply Rlt_le_trans with (ulp beta (FLX_exp prec) (x * y)). apply ulp_error. now apply FLX_exp_correct. unfold ulp. apply -> bpow_le. unfold canonic_exponent, FLX_exp. rewrite ln_beta_unique with (1 := Hexy). apply Zle_refl. apply Hc1. Qed. End Fprop_mult_error. \ No newline at end of file
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