fact stringlengths 12 30.2k | type stringclasses 14
values | library stringclasses 2
values | imports listlengths 1 5 | filename stringclasses 43
values | symbolic_name stringlengths 1 38 | docstring stringclasses 1
value |
|---|---|---|---|---|---|---|
Rmin_Rmax_overflow : forall x y z M, Rabs x <= M -> Rabs y <= M -> Rmin x y <= z <= Rmax x y -> Rabs z <= M. Proof. intros x y z M Hx Hy H. case (Rle_or_lt 0 z); intros Hz. rewrite Rabs_right. apply Rle_trans with (1:=proj2 H). generalize (proj2 H). apply Rmax_case_strong. intros; apply Rle_trans with (2:=Hx). apply RR... | Lemma | examples | [
"From Coq Require Import ZArith Reals Psatz.",
"From Flocq Require Import Core Plus_error."
] | examples/Average.v | Rmin_Rmax_overflow | |
radix2 := Build_radix 2 (refl_equal true). | Definition | examples | [
"From Coq Require Import ZArith Reals Psatz.",
"From Flocq Require Import Core Plus_error."
] | examples/Average.v | radix2 | |
bpow e := (bpow radix2 e). Variable emin prec : Z. Context { prec_gt_0_ : Prec_gt_0 prec }. | Notation | examples | [
"From Coq Require Import ZArith Reals Psatz.",
"From Flocq Require Import Core Plus_error."
] | examples/Average.v | bpow | |
format := (generic_format radix2 (FLT_exp emin prec)). | Notation | examples | [
"From Coq Require Import ZArith Reals Psatz.",
"From Flocq Require Import Core Plus_error."
] | examples/Average.v | format | |
round_flt :=(round radix2 (FLT_exp emin prec) ZnearestE). | Notation | examples | [
"From Coq Require Import ZArith Reals Psatz.",
"From Flocq Require Import Core Plus_error."
] | examples/Average.v | round_flt | |
ulp_flt :=(ulp radix2 (FLT_exp emin prec)). | Notation | examples | [
"From Coq Require Import ZArith Reals Psatz.",
"From Flocq Require Import Core Plus_error."
] | examples/Average.v | ulp_flt | |
cexp := (cexp radix2 (FLT_exp emin prec)). | Notation | examples | [
"From Coq Require Import ZArith Reals Psatz.",
"From Flocq Require Import Core Plus_error."
] | examples/Average.v | cexp | |
pred_flt := (pred radix2 (FLT_exp emin prec)). | Notation | examples | [
"From Coq Require Import ZArith Reals Psatz.",
"From Flocq Require Import Core Plus_error."
] | examples/Average.v | pred_flt | |
FLT_format_double : forall u, format u -> format (2*u). Proof with auto with typeclass_instances. intros u Fu. apply generic_format_FLT. apply FLT_format_generic in Fu... destruct Fu as [uf H1 H2 H3]. exists (Float radix2 (Fnum uf) (Fexp uf+1)). rewrite H1; unfold F2R; simpl. rewrite bpow_plus, bpow_1. simpl;ring. easy... | Lemma | examples | [
"From Coq Require Import ZArith Reals Psatz.",
"From Flocq Require Import Core Plus_error."
] | examples/Average.v | FLT_format_double | |
FLT_format_half : forall u, format u -> bpow (prec+emin) <= Rabs u -> format (u/2). Proof with auto with typeclass_instances. intros u Fu H. apply FLT_format_generic in Fu... destruct Fu as [[n e] H1 H2 H3]. simpl in H1, H2, H3. apply generic_format_FLT. exists (Float radix2 n (e-1)). rewrite H1; unfold F2R; simpl. unf... | Lemma | examples | [
"From Coq Require Import ZArith Reals Psatz.",
"From Flocq Require Import Core Plus_error."
] | examples/Average.v | FLT_format_half | |
FLT_round_half : forall z, bpow (prec+emin) <= Rabs z -> round_flt (z/2)= round_flt z /2. Proof with auto with typeclass_instances. intros z Hz. apply Rmult_eq_reg_l with 2. 2: apply sym_not_eq; auto with real. apply trans_eq with (round_flt z). 2: field. assert (z <> 0)%R. intros K; contradict Hz. rewrite K, Rabs_R0; ... | Lemma | examples | [
"From Coq Require Import ZArith Reals Psatz.",
"From Flocq Require Import Core Plus_error."
] | examples/Average.v | FLT_round_half | |
FLT_ulp_le_id : forall u, bpow emin <= u -> ulp_flt u <= u. Proof with auto with typeclass_instances. intros u H. rewrite ulp_neq_0. 2: apply Rgt_not_eq, Rlt_le_trans with (2:=H), bpow_gt_0. case (Rle_or_lt (bpow (emin+prec-1)) u); intros Hu. unfold ulp; rewrite cexp_FLT_FLX. unfold cexp, FLX_exp. destruct (mag radix2 ... | Lemma | examples | [
"From Coq Require Import ZArith Reals Psatz.",
"From Flocq Require Import Core Plus_error."
] | examples/Average.v | FLT_ulp_le_id | |
FLT_ulp_double : forall u, ulp_flt (2*u) <= 2*ulp_flt(u). Proof. intros u. case (Req_bool_spec u 0); intros Hu'. rewrite Hu', Rmult_0_r. rewrite <- (Rmult_1_l (ulp_flt 0)) at 1. apply Rmult_le_compat_r. apply ulp_ge_0. left; apply Rlt_plus_1. rewrite 2!ulp_neq_0; trivial. 2: lra. change 2 with (bpow 1) at 2. rewrite <-... | Lemma | examples | [
"From Coq Require Import ZArith Reals Psatz.",
"From Flocq Require Import Core Plus_error."
] | examples/Average.v | FLT_ulp_double | |
round_plus_small_id_aux : forall f h, format f -> (bpow (prec+emin) <= f) -> 0 < f -> Rabs h <= /4* ulp_flt f -> round_flt (f+h) = f. Proof with auto with typeclass_instances. intros f h Ff H1 H2 Hh. case (Rle_or_lt 0 h); intros H3;[destruct H3|idtac]. (* 0 < h *) rewrite Rabs_right in Hh. 2: now apply Rle_ge, Rlt_le. ... | Lemma | examples | [
"From Coq Require Import ZArith Reals Psatz.",
"From Flocq Require Import Core Plus_error."
] | examples/Average.v | round_plus_small_id_aux | |
round_plus_small_id : forall f h, format f -> (bpow (prec+emin) <= Rabs f) -> Rabs h <= /4* ulp_flt f -> round_flt (f+h) = f. intros f h Ff H1 H2. case (Rle_or_lt 0 f); intros V. case V; clear V; intros V. apply round_plus_small_id_aux; try assumption. rewrite Rabs_right in H1; try assumption. apply Rle_ge; now left. c... | Lemma | examples | [
"From Coq Require Import ZArith Reals Psatz.",
"From Flocq Require Import Core Plus_error."
] | examples/Average.v | round_plus_small_id | |
avg_naive (x y : R) :=round_flt(round_flt(x+y)/2). Variables x y:R. Hypothesis Fx: format x. Hypothesis Fy: format y. | Definition | examples | [
"From Coq Require Import ZArith Reals Psatz.",
"From Flocq Require Import Core Plus_error."
] | examples/Average.v | avg_naive | |
a :=(x+y)/2. | Let | examples | [
"From Coq Require Import ZArith Reals Psatz.",
"From Flocq Require Import Core Plus_error."
] | examples/Average.v | a | |
av :=avg_naive x y. | Let | examples | [
"From Coq Require Import ZArith Reals Psatz.",
"From Flocq Require Import Core Plus_error."
] | examples/Average.v | av | |
avg_naive_correct : av = round_flt a. Proof with auto with typeclass_instances. case (Rle_or_lt (bpow (prec + emin)) (Rabs (x+y))). (* normal case: division by 2 is exact *) intros H. unfold av,a,avg_naive. rewrite round_generic... now apply sym_eq, FLT_round_half. apply FLT_format_half. apply generic_format_round... a... | Lemma | examples | [
"From Coq Require Import ZArith Reals Psatz.",
"From Flocq Require Import Core Plus_error."
] | examples/Average.v | avg_naive_correct | |
avg_naive_symmetry : forall u v, avg_naive u v = avg_naive v u. Proof. intros u v; unfold avg_naive. rewrite Rplus_comm; reflexivity. Qed. | Lemma | examples | [
"From Coq Require Import ZArith Reals Psatz.",
"From Flocq Require Import Core Plus_error."
] | examples/Average.v | avg_naive_symmetry | |
avg_naive_symmetry_Ropp : forall u v, avg_naive (-u) (-v) = - avg_naive u v. Proof. intros u v; unfold avg_naive. replace (-u+-v) with (-(u+v)) by ring. rewrite round_NE_opp. replace (- round_flt (u + v) / 2) with (- (round_flt (u + v) / 2)) by (unfold Rdiv; ring). now rewrite round_NE_opp. Qed. | Lemma | examples | [
"From Coq Require Import ZArith Reals Psatz.",
"From Flocq Require Import Core Plus_error."
] | examples/Average.v | avg_naive_symmetry_Ropp | |
avg_naive_same_sign_1 : 0 <= a -> 0 <= av. Proof with auto with typeclass_instances. intros H. rewrite avg_naive_correct. apply round_ge_generic... apply generic_format_0. Qed. | Lemma | examples | [
"From Coq Require Import ZArith Reals Psatz.",
"From Flocq Require Import Core Plus_error."
] | examples/Average.v | avg_naive_same_sign_1 | |
avg_naive_same_sign_2 : a <= 0-> av <= 0. Proof with auto with typeclass_instances. intros H. rewrite avg_naive_correct. apply round_le_generic... apply generic_format_0. Qed. | Lemma | examples | [
"From Coq Require Import ZArith Reals Psatz.",
"From Flocq Require Import Core Plus_error."
] | examples/Average.v | avg_naive_same_sign_2 | |
avg_naive_between : Rmin x y <= av <= Rmax x y. Proof with auto with typeclass_instances. rewrite avg_naive_correct. split. apply round_ge_generic... now apply P_Rmin. apply Rmult_le_reg_l with (1 := Rlt_0_2). replace (2 * Rmin x y) with (Rmin x y + Rmin x y) by ring. replace (2 * a) with (x + y) by (unfold a; field). ... | Lemma | examples | [
"From Coq Require Import ZArith Reals Psatz.",
"From Flocq Require Import Core Plus_error."
] | examples/Average.v | avg_naive_between | |
avg_naive_zero : a = 0 -> av = 0. Proof with auto with typeclass_instances. intros H1; rewrite avg_naive_correct, H1. rewrite round_0... Qed. | Lemma | examples | [
"From Coq Require Import ZArith Reals Psatz.",
"From Flocq Require Import Core Plus_error."
] | examples/Average.v | avg_naive_zero | |
avg_naive_no_underflow : (bpow emin) <= Rabs a -> av <> 0. Proof with auto with typeclass_instances. intros H. (* *) cut (bpow emin <= Rabs av). intros H1 H2. rewrite H2 in H1; rewrite Rabs_R0 in H1. contradict H1. apply Rlt_not_le. apply bpow_gt_0. (* *) rewrite avg_naive_correct. apply abs_round_ge_generic... apply g... | Lemma | examples | [
"From Coq Require Import ZArith Reals Psatz.",
"From Flocq Require Import Core Plus_error."
] | examples/Average.v | avg_naive_no_underflow | |
avg_naive_correct_weak1 : Rabs (av -a) <= /2*ulp_flt a. Proof with auto with typeclass_instances. rewrite avg_naive_correct. apply error_le_half_ulp... Qed. | Lemma | examples | [
"From Coq Require Import ZArith Reals Psatz.",
"From Flocq Require Import Core Plus_error."
] | examples/Average.v | avg_naive_correct_weak1 | |
avg_naive_correct_weak2 : Rabs (av -a) <= 3/2*ulp_flt a. Proof with auto with typeclass_instances. apply Rle_trans with (1:=avg_naive_correct_weak1). apply Rmult_le_compat_r. unfold ulp; apply ulp_ge_0. lra. Qed. (* Hypothesis diff_sign: (0 <= x /\ y <= 0) \/ (x <= 0 /\ 0 <= y). is useless for properties: only useful f... | Lemma | examples | [
"From Coq Require Import ZArith Reals Psatz.",
"From Flocq Require Import Core Plus_error."
] | examples/Average.v | avg_naive_correct_weak2 | |
avg_sum_half (x y : R) :=round_flt(round_flt(x/2) + round_flt(y/2)). | Definition | examples | [
"From Coq Require Import ZArith Reals Psatz.",
"From Flocq Require Import Core Plus_error."
] | examples/Average.v | avg_sum_half | |
av2 :=avg_sum_half x y. | Let | examples | [
"From Coq Require Import ZArith Reals Psatz.",
"From Flocq Require Import Core Plus_error."
] | examples/Average.v | av2 | |
avg_sum_half_correct : bpow (emin +prec+prec+1) <= Rabs x -> av2 = round_flt a. Proof with auto with typeclass_instances. intros Hx. assert (G:(0 < prec)%Z). unfold Prec_gt_0 in prec_gt_0_; assumption. unfold av2, avg_sum_half. replace (round_flt (x/2)) with (x/2). 2: apply sym_eq, round_generic... 2: apply FLT_format_... | Lemma | examples | [
"From Coq Require Import ZArith Reals Psatz.",
"From Flocq Require Import Core Plus_error."
] | examples/Average.v | avg_sum_half_correct | |
bpow e := (bpow radix2 e). Variable emin prec : Z. Context { prec_gt_0_ : Prec_gt_0 prec }. | Notation | examples | [
"From Coq Require Import ZArith Reals Psatz.",
"From Flocq Require Import Core Plus_error."
] | examples/Average.v | bpow | |
format := (generic_format radix2 (FLT_exp emin prec)). | Notation | examples | [
"From Coq Require Import ZArith Reals Psatz.",
"From Flocq Require Import Core Plus_error."
] | examples/Average.v | format | |
round_flt :=(round radix2 (FLT_exp emin prec) ZnearestE). | Notation | examples | [
"From Coq Require Import ZArith Reals Psatz.",
"From Flocq Require Import Core Plus_error."
] | examples/Average.v | round_flt | |
ulp_flt :=(ulp radix2 (FLT_exp emin prec)). | Notation | examples | [
"From Coq Require Import ZArith Reals Psatz.",
"From Flocq Require Import Core Plus_error."
] | examples/Average.v | ulp_flt | |
cexp := (cexp radix2 (FLT_exp emin prec)). | Notation | examples | [
"From Coq Require Import ZArith Reals Psatz.",
"From Flocq Require Import Core Plus_error."
] | examples/Average.v | cexp | |
avg_half_sub (x y : R) :=round_flt(x+round_flt(round_flt(y-x)/2)). Variables x y:R. Hypothesis Fx: format x. Hypothesis Fy: format y. | Definition | examples | [
"From Coq Require Import ZArith Reals Psatz.",
"From Flocq Require Import Core Plus_error."
] | examples/Average.v | avg_half_sub | |
a :=(x+y)/2. | Let | examples | [
"From Coq Require Import ZArith Reals Psatz.",
"From Flocq Require Import Core Plus_error."
] | examples/Average.v | a | |
av :=avg_half_sub x y. | Let | examples | [
"From Coq Require Import ZArith Reals Psatz.",
"From Flocq Require Import Core Plus_error."
] | examples/Average.v | av | |
avg_half_sub_symmetry_Ropp : forall u v, avg_half_sub (-u) (-v) = - avg_half_sub u v. intros u v; unfold avg_half_sub. replace (-v--u) with (-(v-u)) by ring. rewrite round_NE_opp. replace (- round_flt (v-u) / 2) with (- (round_flt (v-u) / 2)) by (unfold Rdiv; ring). rewrite round_NE_opp. replace (- u + - round_flt (rou... | Lemma | examples | [
"From Coq Require Import ZArith Reals Psatz.",
"From Flocq Require Import Core Plus_error."
] | examples/Average.v | avg_half_sub_symmetry_Ropp | |
avg_half_sub_same_sign_1 : 0 <= a -> 0 <= av. Proof with auto with typeclass_instances. intros H. apply round_ge_generic... apply generic_format_0. apply Rplus_le_reg_l with (-x). ring_simplify. apply round_ge_generic... now apply generic_format_opp. apply Rmult_le_reg_l with (1 := Rlt_0_2). apply Rle_trans with (-(2*x... | Lemma | examples | [
"From Coq Require Import ZArith Reals Psatz.",
"From Flocq Require Import Core Plus_error."
] | examples/Average.v | avg_half_sub_same_sign_1 | |
avg_half_sub_same_sign_2 : a <= 0-> av <= 0. Proof with auto with typeclass_instances. intros H. apply round_le_generic... apply generic_format_0. apply Rplus_le_reg_l with (-x). ring_simplify. apply round_le_generic... now apply generic_format_opp. apply Rmult_le_reg_l with (1 := Rlt_0_2). apply Rle_trans with (-(2*x)... | Lemma | examples | [
"From Coq Require Import ZArith Reals Psatz.",
"From Flocq Require Import Core Plus_error."
] | examples/Average.v | avg_half_sub_same_sign_2 | |
avg_half_sub_between_aux : forall u v, format u -> format v -> u <= v -> u <= avg_half_sub u v <= v. Proof with auto with typeclass_instances. clear Fx Fy a av x y. intros x y Fx Fy M. split. (* . *) apply round_ge_generic... apply Rplus_le_reg_l with (-x). ring_simplify. apply round_ge_generic... apply generic_format_... | Lemma | examples | [
"From Coq Require Import ZArith Reals Psatz.",
"From Flocq Require Import Core Plus_error."
] | examples/Average.v | avg_half_sub_between_aux | |
avg_half_sub_between : Rmin x y <= av <= Rmax x y. Proof with auto with typeclass_instances. case (Rle_or_lt x y); intros M. (* x <= y *) rewrite Rmin_left; try exact M. rewrite Rmax_right; try exact M. now apply avg_half_sub_between_aux. (* y < x *) rewrite Rmin_right; try now left. rewrite Rmax_left; try now left. un... | Lemma | examples | [
"From Coq Require Import ZArith Reals Psatz.",
"From Flocq Require Import Core Plus_error."
] | examples/Average.v | avg_half_sub_between | |
avg_half_sub_zero : a = 0 -> av = 0. Proof with auto with typeclass_instances. intros H. assert (y=-x). apply Rplus_eq_reg_l with x. apply Rmult_eq_reg_r with (/2). apply trans_eq with a. reflexivity. rewrite H; ring. lra. unfold av, avg_half_sub. rewrite H0. replace (-x-x) with (-(2*x)) by ring. rewrite round_generic ... | Lemma | examples | [
"From Coq Require Import ZArith Reals Psatz.",
"From Flocq Require Import Core Plus_error."
] | examples/Average.v | avg_half_sub_zero | |
avg_half_sub_no_underflow_aux_aux : forall z:Z, (0 < z)%Z -> (ZnearestE (IZR z / 2) < z)%Z. Proof. intros z H1. case (Zle_lt_or_eq 1 z); [lia|intros H2|intros H2]. apply lt_IZR. apply Rplus_lt_reg_r with (- ((IZR z)/2)). apply Rle_lt_trans with (-(((IZR z) /2) - IZR (ZnearestE (IZR z / 2)))). right; ring. apply Rle_lt_... | Lemma | examples | [
"From Coq Require Import ZArith Reals Psatz.",
"From Flocq Require Import Core Plus_error."
] | examples/Average.v | avg_half_sub_no_underflow_aux_aux | |
avg_half_sub_no_underflow_aux1 : forall f, format f -> 0 < f -> f <= round_flt (f/2) -> False. Proof with auto with typeclass_instances. intros f Ff Hf1 Hf2. apply FLT_format_generic in Ff... destruct Ff as [g H1 H2 H3]. case (Zle_lt_or_eq emin (Fexp g)); try exact H3; intros H4. contradict Hf2. apply Rlt_not_le. rewri... | Lemma | examples | [
"From Coq Require Import ZArith Reals Psatz.",
"From Flocq Require Import Core Plus_error."
] | examples/Average.v | avg_half_sub_no_underflow_aux1 | |
avg_half_sub_no_underflow_aux2 : forall u v, format u -> format v -> (0 <= u /\ 0 <= v) \/ (u <= 0 /\ v <= 0) -> u <= v -> (bpow emin) <= Rabs ((u+v)/2) -> avg_half_sub u v <> 0. Proof with auto with typeclass_instances. clear Fx Fy a av x y; intros x y Fx Fy same_sign xLey H; unfold avg_half_sub. apply round_plus_neq_... | Lemma | examples | [
"From Coq Require Import ZArith Reals Psatz.",
"From Flocq Require Import Core Plus_error."
] | examples/Average.v | avg_half_sub_no_underflow_aux2 | |
avg_half_sub_no_underflow_aux3 : forall u v, format u -> format v -> (0 <= u /\ 0 <= v) \/ (u <= 0 /\ v <= 0) -> (bpow emin) <= Rabs ((u+v)/2) -> avg_half_sub u v <> 0. Proof with auto with typeclass_instances. clear Fx Fy a av x y; intros x y Fx Fy. intros same_sign H. case (Rle_or_lt x y); intros H1. now apply avg_ha... | Lemma | examples | [
"From Coq Require Import ZArith Reals Psatz.",
"From Flocq Require Import Core Plus_error."
] | examples/Average.v | avg_half_sub_no_underflow_aux3 | |
avg_half_sub_no_underflow : (0 <= x /\ 0 <= y) \/ (x <= 0 /\ y <= 0) -> (bpow emin) <= Rabs a -> av <> 0. Proof with auto with typeclass_instances. intros; now apply avg_half_sub_no_underflow_aux3. Qed. | Lemma | examples | [
"From Coq Require Import ZArith Reals Psatz.",
"From Flocq Require Import Core Plus_error."
] | examples/Average.v | avg_half_sub_no_underflow | |
avg_half_sub_correct_aux : forall u v, format u -> format v -> u <= v -> (0 <= u /\ 0 <= v) \/ (u <= 0 /\ v <= 0) -> 0 < Rabs ((u+v)/2) < bpow emin -> Rabs (avg_half_sub u v -((u+v)/2)) <= 3/2 * ulp_flt ((u+v)/2). Proof with auto with typeclass_instances. clear Fx Fy x y a av. intros u v Fu Fv uLev same_sign. pose (b:=... | Lemma | examples | [
"From Coq Require Import ZArith Reals Psatz.",
"From Flocq Require Import Core Plus_error."
] | examples/Average.v | avg_half_sub_correct_aux | |
avg_half_sub_correct_aux2 : forall u v, format u -> format v -> u <= v -> (0 <= u /\ 0 <= v) \/ (u <= 0 /\ v <= 0) -> Rabs (avg_half_sub u v -((u+v)/2)) <= 3/2 * ulp_flt ((u+v)/2). Proof with auto with typeclass_instances. clear Fx Fy a av x y. intros u v Fu Fv uLev same_sign. pose (b:=(u+v)/2); fold b. assert (T: fora... | Lemma | examples | [
"From Coq Require Import ZArith Reals Psatz.",
"From Flocq Require Import Core Plus_error."
] | examples/Average.v | avg_half_sub_correct_aux2 | |
avg_half_sub_correct : (0 <= x /\ 0 <= y) \/ (x <= 0 /\ y <= 0) -> Rabs (av-a) <= 3/2 * ulp_flt a. Proof with auto with typeclass_instances. intros same_sign; case (Rle_or_lt x y); intros H. now apply avg_half_sub_correct_aux2. unfold av, a. rewrite <- (Ropp_involutive x), <- (Ropp_involutive y). rewrite avg_half_sub_s... | Lemma | examples | [
"From Coq Require Import ZArith Reals Psatz.",
"From Flocq Require Import Core Plus_error."
] | examples/Average.v | avg_half_sub_correct | |
bpow e := (bpow radix2 e). Variable emin prec : Z. Context { prec_gt_0_ : Prec_gt_0 prec }. | Notation | examples | [
"From Coq Require Import ZArith Reals Psatz.",
"From Flocq Require Import Core Plus_error."
] | examples/Average.v | bpow | |
format := (generic_format radix2 (FLT_exp emin prec)). | Notation | examples | [
"From Coq Require Import ZArith Reals Psatz.",
"From Flocq Require Import Core Plus_error."
] | examples/Average.v | format | |
round_flt :=(round radix2 (FLT_exp emin prec) ZnearestE). | Notation | examples | [
"From Coq Require Import ZArith Reals Psatz.",
"From Flocq Require Import Core Plus_error."
] | examples/Average.v | round_flt | |
ulp_flt :=(ulp radix2 (FLT_exp emin prec)). | Notation | examples | [
"From Coq Require Import ZArith Reals Psatz.",
"From Flocq Require Import Core Plus_error."
] | examples/Average.v | ulp_flt | |
cexp := (cexp radix2 (FLT_exp emin prec)). | Notation | examples | [
"From Coq Require Import ZArith Reals Psatz.",
"From Flocq Require Import Core Plus_error."
] | examples/Average.v | cexp | |
average (x y : R) := let samesign := match (Rle_bool 0 x), (Rle_bool 0 y) with true , true => true | false , false => true | _,_ => false end in if samesign then match (Rle_bool (Rabs x) (Rabs y)) with true => avg_half_sub emin prec x y | false => avg_half_sub emin prec y x end else avg_naive emin prec x y. Variables x... | Definition | examples | [
"From Coq Require Import ZArith Reals Psatz.",
"From Flocq Require Import Core Plus_error."
] | examples/Average.v | average | |
a :=(x+y)/2. | Let | examples | [
"From Coq Require Import ZArith Reals Psatz.",
"From Flocq Require Import Core Plus_error."
] | examples/Average.v | a | |
av :=average x y. | Let | examples | [
"From Coq Require Import ZArith Reals Psatz.",
"From Flocq Require Import Core Plus_error."
] | examples/Average.v | av | |
average_symmetry : forall u v, average u v = average v u. Proof. intros u v; unfold average. case (Rle_bool_spec 0 u); case (Rle_bool_spec 0 v); intros. rewrite 2!Rabs_right; try apply Rle_ge; try assumption. case (Rle_bool_spec u v); case (Rle_bool_spec v u); trivial. intros; replace u with v; trivial; auto with real.... | Lemma | examples | [
"From Coq Require Import ZArith Reals Psatz.",
"From Flocq Require Import Core Plus_error."
] | examples/Average.v | average_symmetry | |
average_symmetry_Ropp : forall u v, format u -> format v -> average (-u) (-v) = - average u v. Proof with auto with typeclass_instances. (* first: nonnegative u *) assert (forall u v, 0 <= u -> format u -> format v -> average (-u) (-v) = - average u v). intros u v Hu Fu Fv; unfold average. rewrite 2!Rabs_Ropp. destruct... | Lemma | examples | [
"From Coq Require Import ZArith Reals Psatz.",
"From Flocq Require Import Core Plus_error."
] | examples/Average.v | average_symmetry_Ropp | |
average_same_sign_1 : 0 <= a -> 0 <= av. Proof with auto with typeclass_instances. intros H; unfold av, average. case (Rle_bool_spec 0 x); case (Rle_bool_spec 0 y); intros. case (Rle_bool_spec (Rabs x) (Rabs y)); intros. apply avg_half_sub_same_sign_1... apply avg_half_sub_same_sign_1... now rewrite Rplus_comm. apply a... | Lemma | examples | [
"From Coq Require Import ZArith Reals Psatz.",
"From Flocq Require Import Core Plus_error."
] | examples/Average.v | average_same_sign_1 | |
average_same_sign_2 : a <= 0-> av <= 0. Proof with auto with typeclass_instances. intros H; unfold av, average. case (Rle_bool_spec 0 x); case (Rle_bool_spec 0 y); intros. case (Rle_bool_spec (Rabs x) (Rabs y)); intros. apply avg_half_sub_same_sign_2... apply avg_half_sub_same_sign_2... now rewrite Rplus_comm. apply av... | Lemma | examples | [
"From Coq Require Import ZArith Reals Psatz.",
"From Flocq Require Import Core Plus_error."
] | examples/Average.v | average_same_sign_2 | |
average_correct : Rabs (av -a) <= 3/2 * ulp_flt a. Proof with auto with typeclass_instances. unfold av,a,average. case (Rle_bool_spec 0 x); case (Rle_bool_spec 0 y); intros. case (Rle_bool_spec (Rabs x) (Rabs y)); intros. apply avg_half_sub_correct... rewrite Rplus_comm. apply avg_half_sub_correct... apply avg_naive_co... | Lemma | examples | [
"From Coq Require Import ZArith Reals Psatz.",
"From Flocq Require Import Core Plus_error."
] | examples/Average.v | average_correct | |
average_between : Rmin x y <= av <= Rmax x y. Proof with auto with typeclass_instances. unfold av,a,average. case (Rle_bool_spec 0 x); case (Rle_bool_spec 0 y); intros. case (Rle_bool_spec (Rabs x) (Rabs y)); intros. apply avg_half_sub_between... rewrite Rmin_comm, Rmax_comm. apply avg_half_sub_between... apply avg_nai... | Lemma | examples | [
"From Coq Require Import ZArith Reals Psatz.",
"From Flocq Require Import Core Plus_error."
] | examples/Average.v | average_between | |
average_zero : a = 0 -> av = 0. Proof with auto with typeclass_instances. unfold av,a,average. case (Rle_bool_spec 0 x); case (Rle_bool_spec 0 y); intros. case (Rle_bool_spec (Rabs x) (Rabs y)); intros. apply avg_half_sub_zero... apply avg_half_sub_zero... now rewrite Rplus_comm. apply avg_naive_zero... apply avg_naive... | Lemma | examples | [
"From Coq Require Import ZArith Reals Psatz.",
"From Flocq Require Import Core Plus_error."
] | examples/Average.v | average_zero | |
average_no_underflow : (bpow emin) <= Rabs a -> av <> 0. Proof with auto with typeclass_instances. unfold av,a,average. case (Rle_bool_spec 0 x); case (Rle_bool_spec 0 y); intros. case (Rle_bool_spec (Rabs x) (Rabs y)); intros. apply avg_half_sub_no_underflow... apply avg_half_sub_no_underflow... now rewrite Rplus_comm... | Lemma | examples | [
"From Coq Require Import ZArith Reals Psatz.",
"From Flocq Require Import Core Plus_error."
] | examples/Average.v | average_no_underflow | |
rel_helper : forall xa xe b : R, xe <> 0 -> (Rabs ((xa - xe) / xe) <= b <-> Rabs (xa - xe) <= b * Rabs xe). Proof. intros xa xe b Zx. unfold Rdiv. rewrite Rabs_mult, Rabs_Rinv by exact Zx. split ; intros H. - apply Rmult_le_reg_r with (/ Rabs xe). apply Rinv_0_lt_compat. now apply Rabs_pos_lt. rewrite Rmult_assoc, Rinv... | Lemma | examples | [
"From Coq Require Import Reals.",
"From Flocq Require Import Core.",
"From Gappa Require Import Gappa_tactic.",
"From Interval Require Import Tactic."
] | examples/Cody_Waite.v | rel_helper | |
Rdiv_compat_r : forall a b c : R, a <> 0 -> c <> 0 -> (a*b) / (a*c) = b/c. Proof. intros a b c Ha Hc. field. apply conj. apply Hc. apply Ha. Qed. | Lemma | examples | [
"From Coq Require Import Reals.",
"From Flocq Require Import Core.",
"From Gappa Require Import Gappa_tactic.",
"From Interval Require Import Tactic."
] | examples/Cody_Waite.v | Rdiv_compat_r | |
pow2 := (bpow radix2). | Notation | examples | [
"From Coq Require Import Reals.",
"From Flocq Require Import Core.",
"From Gappa Require Import Gappa_tactic.",
"From Interval Require Import Tactic."
] | examples/Cody_Waite.v | pow2 | |
rnd := (round radix2 (FLT_exp (-1074) 53) ZnearestE). | Notation | examples | [
"From Coq Require Import Reals.",
"From Flocq Require Import Core.",
"From Gappa Require Import Gappa_tactic.",
"From Interval Require Import Tactic."
] | examples/Cody_Waite.v | rnd | |
add x y := rnd (x + y). | Definition | examples | [
"From Coq Require Import Reals.",
"From Flocq Require Import Core.",
"From Gappa Require Import Gappa_tactic.",
"From Interval Require Import Tactic."
] | examples/Cody_Waite.v | add | |
sub x y := rnd (x - y). | Definition | examples | [
"From Coq Require Import Reals.",
"From Flocq Require Import Core.",
"From Gappa Require Import Gappa_tactic.",
"From Interval Require Import Tactic."
] | examples/Cody_Waite.v | sub | |
mul x y := rnd (x * y). | Definition | examples | [
"From Coq Require Import Reals.",
"From Flocq Require Import Core.",
"From Gappa Require Import Gappa_tactic.",
"From Interval Require Import Tactic."
] | examples/Cody_Waite.v | mul | |
div x y := rnd (x / y). | Definition | examples | [
"From Coq Require Import Reals.",
"From Flocq Require Import Core.",
"From Gappa Require Import Gappa_tactic.",
"From Interval Require Import Tactic."
] | examples/Cody_Waite.v | div | |
nearbyint x := round radix2 (FIX_exp 0) ZnearestE x. | Definition | examples | [
"From Coq Require Import Reals.",
"From Flocq Require Import Core.",
"From Gappa Require Import Gappa_tactic.",
"From Interval Require Import Tactic."
] | examples/Cody_Waite.v | nearbyint | |
Log2h := 3048493539143 * pow2 (-42). | Definition | examples | [
"From Coq Require Import Reals.",
"From Flocq Require Import Core.",
"From Gappa Require Import Gappa_tactic.",
"From Interval Require Import Tactic."
] | examples/Cody_Waite.v | Log2h | |
Log2l := 544487923021427 * pow2 (-93). | Definition | examples | [
"From Coq Require Import Reals.",
"From Flocq Require Import Core.",
"From Gappa Require Import Gappa_tactic.",
"From Interval Require Import Tactic."
] | examples/Cody_Waite.v | Log2l | |
InvLog2 := 3248660424278399 * pow2 (-51). | Definition | examples | [
"From Coq Require Import Reals.",
"From Flocq Require Import Core.",
"From Gappa Require Import Gappa_tactic.",
"From Interval Require Import Tactic."
] | examples/Cody_Waite.v | InvLog2 | |
p0 := 1 * pow2 (-2). | Definition | examples | [
"From Coq Require Import Reals.",
"From Flocq Require Import Core.",
"From Gappa Require Import Gappa_tactic.",
"From Interval Require Import Tactic."
] | examples/Cody_Waite.v | p0 | |
p1 := 4002712888408905 * pow2 (-59). | Definition | examples | [
"From Coq Require Import Reals.",
"From Flocq Require Import Core.",
"From Gappa Require Import Gappa_tactic.",
"From Interval Require Import Tactic."
] | examples/Cody_Waite.v | p1 | |
p2 := 1218985200072455 * pow2 (-66). | Definition | examples | [
"From Coq Require Import Reals.",
"From Flocq Require Import Core.",
"From Gappa Require Import Gappa_tactic.",
"From Interval Require Import Tactic."
] | examples/Cody_Waite.v | p2 | |
q0 := 1 * pow2 (-1). | Definition | examples | [
"From Coq Require Import Reals.",
"From Flocq Require Import Core.",
"From Gappa Require Import Gappa_tactic.",
"From Interval Require Import Tactic."
] | examples/Cody_Waite.v | q0 | |
q1 := 8006155947364787 * pow2 (-57). | Definition | examples | [
"From Coq Require Import Reals.",
"From Flocq Require Import Core.",
"From Gappa Require Import Gappa_tactic.",
"From Interval Require Import Tactic."
] | examples/Cody_Waite.v | q1 | |
q2 := 4573527866750985 * pow2 (-63). | Definition | examples | [
"From Coq Require Import Reals.",
"From Flocq Require Import Core.",
"From Gappa Require Import Gappa_tactic.",
"From Interval Require Import Tactic."
] | examples/Cody_Waite.v | q2 | |
cw_exp (x : R) := let k := nearbyint (mul x InvLog2) in let t := sub (sub x (mul k Log2h)) (mul k Log2l) in let t2 := mul t t in let p := add p0 (mul t2 (add p1 (mul t2 p2))) in let q := add q0 (mul t2 (add q1 (mul t2 q2))) in pow2 (Zfloor k + 1) * (add (div (mul t p) (sub q (mul t p))) (1/2)). | Definition | examples | [
"From Coq Require Import Reals.",
"From Flocq Require Import Core.",
"From Gappa Require Import Gappa_tactic.",
"From Interval Require Import Tactic."
] | examples/Cody_Waite.v | cw_exp | |
method_error : forall t : R, let t2 := t * t in let p := p0 + t2 * (p1 + t2 * p2) in let q := q0 + t2 * (q1 + t2 * q2) in let f := 2 * ((t * p) / (q - t * p) + 1/2) in Rabs t <= 355 / 1024 -> Rabs ((f - exp t) / exp t) <= 23 * pow2 (-62). Proof. intros t t2 p q f Ht. unfold f, q, p, t2, p0, p1, p2, q0, q1, q2 ; interva... | Lemma | examples | [
"From Coq Require Import Reals.",
"From Flocq Require Import Core.",
"From Gappa Require Import Gappa_tactic.",
"From Interval Require Import Tactic."
] | examples/Cody_Waite.v | method_error | |
argument_reduction : forall x : R, generic_format radix2 (FLT_exp (-1074) 53) x -> -746 <= x <= 710 -> let k := nearbyint (mul x InvLog2) in let tf := sub (sub x (mul k Log2h)) (mul k Log2l) in let te := x - k * ln 2 in Rabs tf <= 355 / 1024 /\ Rabs (tf - te) <= 65537 * pow2 (-71). Proof with auto with typeclass_instan... | Lemma | examples | [
"From Coq Require Import Reals.",
"From Flocq Require Import Core.",
"From Gappa Require Import Gappa_tactic.",
"From Interval Require Import Tactic."
] | examples/Cody_Waite.v | argument_reduction | |
exp_correct : forall x : R, generic_format radix2 (FLT_exp (-1074) 53) x -> -746 <= x <= 710 -> Rabs ((cw_exp x - exp x) / exp x) <= 1 * pow2 (-51). Proof. intros x Fx Bx. generalize (argument_reduction x Fx Bx). unfold cw_exp. set (k := nearbyint (mul x InvLog2)). set (t := sub (sub x (mul k Log2h)) (mul k Log2l)). se... | Theorem | examples | [
"From Coq Require Import Reals.",
"From Flocq Require Import Core.",
"From Gappa Require Import Gappa_tactic.",
"From Interval Require Import Tactic."
] | examples/Cody_Waite.v | exp_correct | |
Rsgn_F2R : forall m e, Rlt_bool (F2R (Float beta m e)) 0 = Zlt_bool m 0. Proof. intros m e. case Zlt_bool_spec ; intros H. apply Rlt_bool_true. now apply F2R_lt_0. apply Rlt_bool_false. now apply F2R_ge_0. Qed. | Lemma | examples | [
"From Coq Require Import ZArith Reals.",
"From Flocq Require Import Core Bracket Round Operations Div Sqrt."
] | examples/Compute.v | Rsgn_F2R | |
plus (x y : float beta) := let (m, e) := Fplus x y in let s := Zlt_bool m 0 in let '(m', e', l) := truncate beta fexp (Z.abs m, e, loc_Exact) in Float beta (cond_Zopp s (choice s m' l)) e'. | Definition | examples | [
"From Coq Require Import ZArith Reals.",
"From Flocq Require Import Core Bracket Round Operations Div Sqrt."
] | examples/Compute.v | plus | |
plus_correct : forall x y : float beta, round beta fexp rnd (F2R x + F2R y) = F2R (plus x y). Proof. intros x y. unfold plus. rewrite <- F2R_plus. destruct (Fplus x y) as [m e]. rewrite (round_trunc_sign_any_correct beta fexp rnd choice rnd_choice _ (Z.abs m) e loc_Exact). 3: now right. destruct truncate as [[m' e'] l'... | Theorem | examples | [
"From Coq Require Import ZArith Reals.",
"From Flocq Require Import Core Bracket Round Operations Div Sqrt."
] | examples/Compute.v | plus_correct | |
mult (x y : float beta) := let (m, e) := Fmult x y in let s := Zlt_bool m 0 in let '(m', e', l) := truncate beta fexp (Z.abs m, e, loc_Exact) in Float beta (cond_Zopp s (choice s m' l)) e'. | Definition | examples | [
"From Coq Require Import ZArith Reals.",
"From Flocq Require Import Core Bracket Round Operations Div Sqrt."
] | examples/Compute.v | mult | |
mult_correct : forall x y : float beta, round beta fexp rnd (F2R x * F2R y) = F2R (mult x y). Proof. intros x y. unfold mult. rewrite <- F2R_mult. destruct (Fmult x y) as [m e]. rewrite (round_trunc_sign_any_correct beta fexp rnd choice rnd_choice _ (Z.abs m) e loc_Exact). 3: now right. destruct truncate as [[m' e'] l'... | Theorem | examples | [
"From Coq Require Import ZArith Reals.",
"From Flocq Require Import Core Bracket Round Operations Div Sqrt."
] | examples/Compute.v | mult_correct | |
sqrt (x : float beta) := if Zlt_bool 0 (Fnum x) then let '(m', e', l) := truncate beta fexp (Fsqrt fexp x) in Float beta (choice false m' l) e' else Float beta 0 0. | Definition | examples | [
"From Coq Require Import ZArith Reals.",
"From Flocq Require Import Core Bracket Round Operations Div Sqrt."
] | examples/Compute.v | sqrt | |
sqrt_correct : forall x : float beta, round beta fexp rnd (R_sqrt.sqrt (F2R x)) = F2R (sqrt x). Proof. intros x. unfold sqrt. case Zlt_bool_spec ; intros Hm. generalize (Fsqrt_correct fexp x (F2R_gt_0 _ _ Hm)). destruct Fsqrt as [[m' e'] l]. intros [Hs1 Hs2]. rewrite (round_trunc_sign_any_correct' beta fexp rnd choice ... | Theorem | examples | [
"From Coq Require Import ZArith Reals.",
"From Flocq Require Import Core Bracket Round Operations Div Sqrt."
] | examples/Compute.v | sqrt_correct | |
Rsgn_div : forall x y : R, x <> 0%R -> y <> 0%R -> Rlt_bool (x / y) 0 = xorb (Rlt_bool x 0) (Rlt_bool y 0). Proof. intros x y Hx0 Hy0. unfold Rdiv. case (Rlt_bool_spec x) ; intros Hx ; case (Rlt_bool_spec y) ; intros Hy. apply Rlt_bool_false. rewrite <- Rmult_opp_opp. apply Rlt_le, Rmult_lt_0_compat. now apply Ropp_gt_... | Lemma | examples | [
"From Coq Require Import ZArith Reals.",
"From Flocq Require Import Core Bracket Round Operations Div Sqrt."
] | examples/Compute.v | Rsgn_div | |
div (x y : float beta) := if Zeq_bool (Fnum x) 0 then Float beta 0 0 else let '(m, e, l) := truncate beta fexp (Fdiv fexp (Fabs x) (Fabs y)) in let s := xorb (Zlt_bool (Fnum x) 0) (Zlt_bool (Fnum y) 0) in Float beta (cond_Zopp s (choice s m l)) e. | Definition | examples | [
"From Coq Require Import ZArith Reals.",
"From Flocq Require Import Core Bracket Round Operations Div Sqrt."
] | examples/Compute.v | div |
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