Fappli_IEEE.v 3.71 KB
 Guillaume Melquiond committed Dec 02, 2010 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 151 152 153 154 155 156 157 158 159 160 161 162 163 164 165 166 ``````Require Import Fcore. Require Import Fcalc_digits. Section Binary. Variable prec emax : Z. Let fexp := FLT_exp (1 - (emax + prec)) prec. Definition bounded_prec m e := Zeq_bool (fexp (Z_of_nat (S (digits2_Pnat m)) + e)) e. Definition bounded m e := andb (bounded_prec m e) (Zle_bool e emax). Inductive binary_float := | B754_zero : bool -> binary_float | B754_infinity : bool -> binary_float | B754_nan : binary_float | B754_finite : bool -> forall (m : positive) (e : Z), bounded m e = true -> binary_float. Definition radix2 := Build_radix 2 (refl_equal true). Definition B2R f := match f with | B754_finite s m e _ => F2R (Float radix2 ((if s then Zneg else Zpos) m) e) | _ => R0 end. Theorem canonic_bounded_prec : forall (sx : bool) mx ex, bounded_prec mx ex = true -> canonic radix2 fexp (Float radix2 ((if sx then Zneg else Zpos) mx) ex). Proof. intros sx mx ex H. assert (Hx := Zeq_bool_eq _ _ H). clear H. apply sym_eq. simpl. pattern ex at 2 ; rewrite <- Hx. apply (f_equal fexp). rewrite ln_beta_F2R_digits. rewrite <- digits_abs. rewrite Z_of_nat_S_digits2_Pnat. now case sx. now case sx. Qed. Theorem generic_format_B2R : forall x, generic_format radix2 fexp (B2R x). Proof. intros [sx|sx| |sx mx ex Hx] ; try apply generic_format_0. simpl. apply generic_format_canonic. apply canonic_bounded_prec. now destruct (andb_prop _ _ Hx) as (H, _). Qed. Definition is_finite_strict f := match f with | B754_finite _ _ _ _ => true | _ => false end. Theorem binary_unicity : forall x y : binary_float, is_finite_strict x = true -> is_finite_strict y = true -> B2R x = B2R y -> x = y. Proof. intros [sx|sx| |sx mx ex Hx] [sy|sy| |sy my ey Hy] ; try easy. simpl. intros _ _ Heq. assert (Hs: sx = sy). (* *) revert Heq. clear. case sx ; case sy ; try easy ; intros Heq ; apply False_ind ; revert Heq. apply Rlt_not_eq. apply Rlt_trans with R0. now apply F2R_lt_0_compat. now apply F2R_gt_0_compat. apply Rgt_not_eq. apply Rgt_trans with R0. now apply F2R_gt_0_compat. now apply F2R_lt_0_compat. assert (mx = my /\ ex = ey). (* *) refine (_ (canonic_unicity _ fexp _ _ _ _ Heq)). rewrite Hs. now case sy ; intro H ; injection H ; split. apply canonic_bounded_prec. exact (proj1 (andb_prop _ _ Hx)). apply canonic_bounded_prec. exact (proj1 (andb_prop _ _ Hy)). (* *) revert Hx. rewrite Hs, (proj1 H), (proj2 H). intros Hx. apply f_equal. apply eqbool_irrelevance. Qed. Definition is_finite f := match f with | B754_finite _ _ _ _ => true | B754_zero _ => true | _ => false end. Inductive mode := mode_NE | mode_ZR | mode_DN | mode_UP | mode_NA. Definition round_mode m := match m with | mode_NE => rndNE | mode_ZR => rndZR | mode_DN => rndDN | mode_UP => rndUP | mode_NA => rndNA end. Definition Bplus m x y := match x, y with | B754_nan, _ => x | _, B754_nan => y | B754_infinity sx, B754_infinity sy => if Bool.eqb sx sy then x else B754_nan | B754_infinity _, _ => x | _, B754_infinity _ => y | B754_zero sx, B754_zero sy => if Bool.eqb sx sy then x else match m with mode_DN => B754_zero true | _ => B754_zero false end | B754_zero _, _ => y | _, B754_zero _ => x | B754_finite sx mx ex Hx, B754_finite sy my ey Hy => B754_nan end. Theorem Bplus_finite : forall m x y, is_finite x = true -> is_finite y = true -> B2R (Bplus m x y) = round radix2 fexp (round_mode m) (B2R x + B2R y)%R. Proof. intros m [sx|sx| |sx mx ex Hx] [sy|sy| |sy my ey Hy] ; try easy ; intros _ _. (* *) rewrite Rplus_0_r, round_0. simpl. case (Bool.eqb sx sy) ; try easy. now case m. (* *) rewrite Rplus_0_l. apply sym_eq. apply round_generic. apply generic_format_B2R. (* *) rewrite Rplus_0_r. apply sym_eq. apply round_generic. apply generic_format_B2R. (* *) Admitted. End Binary.``````