Library Flocq.Prop.Fprop_relative
This file is part of the Flocq formalization of floating-point
arithmetic in Coq: http://flocq.gforge.inria.fr/
Copyright (C) 2010-2011 Sylvie Boldo
Copyright (C) 2010-2011 Guillaume Melquiond
This library is free software; you can redistribute it and/or modify it under the terms of the GNU Lesser General Public License as published by the Free Software Foundation; either version 3 of the License, or (at your option) any later version.
This library is distributed in the hope that it will be useful, but WITHOUT ANY WARRANTY; without even the implied warranty of MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the COPYING file for more details.
Copyright (C) 2010-2011 Sylvie Boldo
Copyright (C) 2010-2011 Guillaume Melquiond
This library is free software; you can redistribute it and/or modify it under the terms of the GNU Lesser General Public License as published by the Free Software Foundation; either version 3 of the License, or (at your option) any later version.
This library is distributed in the hope that it will be useful, but WITHOUT ANY WARRANTY; without even the implied warranty of MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the COPYING file for more details.
Require Import Fcore.
Section Fprop_relative.
Variable beta : radix.
Notation bpow e := (bpow beta e).
Section Fprop_relative_generic.
Variable fexp : Z -> Z.
Context { prop_exp : Valid_exp fexp }.
Section relative_error_conversion.
Variable rnd : R -> Z.
Context { valid_rnd : Valid_rnd rnd }.
Lemma relative_error_lt_conversion :
forall x b, (0 < b)%R ->
(Rabs (round beta fexp rnd x - x) < b * Rabs x)%R ->
exists eps,
(Rabs eps < b)%R /\ round beta fexp rnd x = (x * (1 + eps))%R.
Lemma relative_error_le_conversion :
forall x b, (0 <= b)%R ->
(Rabs (round beta fexp rnd x - x) <= b * Rabs x)%R ->
exists eps,
(Rabs eps <= b)%R /\ round beta fexp rnd x = (x * (1 + eps))%R.
End relative_error_conversion.
Variable emin p : Z.
Hypothesis Hmin : forall k, (emin < k)%Z -> (p <= k - fexp k)%Z.
Variable rnd : R -> Z.
Context { valid_rnd : Valid_rnd rnd }.
Theorem relative_error :
forall x,
(bpow emin <= Rabs x)%R ->
(Rabs (round beta fexp rnd x - x) < bpow (-p + 1) * Rabs x)%R.
Section Fprop_relative.
Variable beta : radix.
Notation bpow e := (bpow beta e).
Section Fprop_relative_generic.
Variable fexp : Z -> Z.
Context { prop_exp : Valid_exp fexp }.
Section relative_error_conversion.
Variable rnd : R -> Z.
Context { valid_rnd : Valid_rnd rnd }.
Lemma relative_error_lt_conversion :
forall x b, (0 < b)%R ->
(Rabs (round beta fexp rnd x - x) < b * Rabs x)%R ->
exists eps,
(Rabs eps < b)%R /\ round beta fexp rnd x = (x * (1 + eps))%R.
Lemma relative_error_le_conversion :
forall x b, (0 <= b)%R ->
(Rabs (round beta fexp rnd x - x) <= b * Rabs x)%R ->
exists eps,
(Rabs eps <= b)%R /\ round beta fexp rnd x = (x * (1 + eps))%R.
End relative_error_conversion.
Variable emin p : Z.
Hypothesis Hmin : forall k, (emin < k)%Z -> (p <= k - fexp k)%Z.
Variable rnd : R -> Z.
Context { valid_rnd : Valid_rnd rnd }.
Theorem relative_error :
forall x,
(bpow emin <= Rabs x)%R ->
(Rabs (round beta fexp rnd x - x) < bpow (-p + 1) * Rabs x)%R.
1+ε property in any rounding
Theorem relative_error_ex :
forall x,
(bpow emin <= Rabs x)%R ->
exists eps,
(Rabs eps < bpow (-p + 1))%R /\ round beta fexp rnd x = (x * (1 + eps))%R.
Theorem relative_error_F2R_emin :
forall m, let x := F2R (Float beta m emin) in
(x <> 0)%R ->
(Rabs (round beta fexp rnd x - x) < bpow (-p + 1) * Rabs x)%R.
Theorem relative_error_F2R_emin_ex :
forall m, let x := F2R (Float beta m emin) in
(x <> 0)%R ->
exists eps,
(Rabs eps < bpow (-p + 1))%R /\ round beta fexp rnd x = (x * (1 + eps))%R.
Theorem relative_error_round :
(0 < p)%Z ->
forall x,
(bpow emin <= Rabs x)%R ->
(Rabs (round beta fexp rnd x - x) < bpow (-p + 1) * Rabs (round beta fexp rnd x))%R.
Theorem relative_error_round_F2R_emin :
(0 < p)%Z ->
forall m, let x := F2R (Float beta m emin) in
(x <> 0)%R ->
(Rabs (round beta fexp rnd x - x) < bpow (-p + 1) * Rabs (round beta fexp rnd x))%R.
Variable choice : Z -> bool.
Theorem relative_error_N :
forall x,
(bpow emin <= Rabs x)%R ->
(Rabs (round beta fexp (Znearest choice) x - x) <= /2 * bpow (-p + 1) * Rabs x)%R.
forall x,
(bpow emin <= Rabs x)%R ->
exists eps,
(Rabs eps < bpow (-p + 1))%R /\ round beta fexp rnd x = (x * (1 + eps))%R.
Theorem relative_error_F2R_emin :
forall m, let x := F2R (Float beta m emin) in
(x <> 0)%R ->
(Rabs (round beta fexp rnd x - x) < bpow (-p + 1) * Rabs x)%R.
Theorem relative_error_F2R_emin_ex :
forall m, let x := F2R (Float beta m emin) in
(x <> 0)%R ->
exists eps,
(Rabs eps < bpow (-p + 1))%R /\ round beta fexp rnd x = (x * (1 + eps))%R.
Theorem relative_error_round :
(0 < p)%Z ->
forall x,
(bpow emin <= Rabs x)%R ->
(Rabs (round beta fexp rnd x - x) < bpow (-p + 1) * Rabs (round beta fexp rnd x))%R.
Theorem relative_error_round_F2R_emin :
(0 < p)%Z ->
forall m, let x := F2R (Float beta m emin) in
(x <> 0)%R ->
(Rabs (round beta fexp rnd x - x) < bpow (-p + 1) * Rabs (round beta fexp rnd x))%R.
Variable choice : Z -> bool.
Theorem relative_error_N :
forall x,
(bpow emin <= Rabs x)%R ->
(Rabs (round beta fexp (Znearest choice) x - x) <= /2 * bpow (-p + 1) * Rabs x)%R.
1+ε property in rounding to nearest
Theorem relative_error_N_ex :
forall x,
(bpow emin <= Rabs x)%R ->
exists eps,
(Rabs eps <= /2 * bpow (-p + 1))%R /\ round beta fexp (Znearest choice) x = (x * (1 + eps))%R.
Theorem relative_error_N_F2R_emin :
forall m, let x := F2R (Float beta m emin) in
(Rabs (round beta fexp (Znearest choice) x - x) <= /2 * bpow (-p + 1) * Rabs x)%R.
Theorem relative_error_N_F2R_emin_ex :
forall m, let x := F2R (Float beta m emin) in
exists eps,
(Rabs eps <= /2 * bpow (-p + 1))%R /\ round beta fexp (Znearest choice) x = (x * (1 + eps))%R.
Theorem relative_error_N_round :
(0 < p)%Z ->
forall x,
(bpow emin <= Rabs x)%R ->
(Rabs (round beta fexp (Znearest choice) x - x) <= /2 * bpow (-p + 1) * Rabs (round beta fexp (Znearest choice) x))%R.
Theorem relative_error_N_round_F2R_emin :
(0 < p)%Z ->
forall m, let x := F2R (Float beta m emin) in
(Rabs (round beta fexp (Znearest choice) x - x) <= /2 * bpow (-p + 1) * Rabs (round beta fexp (Znearest choice) x))%R.
End Fprop_relative_generic.
Section Fprop_relative_FLT.
Variable emin prec : Z.
Variable Hp : Zlt 0 prec.
Lemma relative_error_FLT_aux :
forall k, (emin + prec - 1 < k)%Z -> (prec <= k - FLT_exp emin prec k)%Z.
Variable rnd : R -> Z.
Context { valid_rnd : Valid_rnd rnd }.
Theorem relative_error_FLT_F2R_emin :
forall m, let x := F2R (Float beta m (emin + prec - 1)) in
(x <> 0)%R ->
(Rabs (round beta (FLT_exp emin prec) rnd x - x) < bpow (-prec + 1) * Rabs x)%R.
Theorem relative_error_FLT :
forall x,
(bpow (emin + prec - 1) <= Rabs x)%R ->
(Rabs (round beta (FLT_exp emin prec) rnd x - x) < bpow (-prec + 1) * Rabs x)%R.
Theorem relative_error_FLT_F2R_emin_ex :
forall m, let x := F2R (Float beta m (emin + prec - 1)) in
(x <> 0)%R ->
exists eps,
(Rabs eps < bpow (-prec + 1))%R /\ round beta (FLT_exp emin prec) rnd x = (x * (1 + eps))%R.
forall x,
(bpow emin <= Rabs x)%R ->
exists eps,
(Rabs eps <= /2 * bpow (-p + 1))%R /\ round beta fexp (Znearest choice) x = (x * (1 + eps))%R.
Theorem relative_error_N_F2R_emin :
forall m, let x := F2R (Float beta m emin) in
(Rabs (round beta fexp (Znearest choice) x - x) <= /2 * bpow (-p + 1) * Rabs x)%R.
Theorem relative_error_N_F2R_emin_ex :
forall m, let x := F2R (Float beta m emin) in
exists eps,
(Rabs eps <= /2 * bpow (-p + 1))%R /\ round beta fexp (Znearest choice) x = (x * (1 + eps))%R.
Theorem relative_error_N_round :
(0 < p)%Z ->
forall x,
(bpow emin <= Rabs x)%R ->
(Rabs (round beta fexp (Znearest choice) x - x) <= /2 * bpow (-p + 1) * Rabs (round beta fexp (Znearest choice) x))%R.
Theorem relative_error_N_round_F2R_emin :
(0 < p)%Z ->
forall m, let x := F2R (Float beta m emin) in
(Rabs (round beta fexp (Znearest choice) x - x) <= /2 * bpow (-p + 1) * Rabs (round beta fexp (Znearest choice) x))%R.
End Fprop_relative_generic.
Section Fprop_relative_FLT.
Variable emin prec : Z.
Variable Hp : Zlt 0 prec.
Lemma relative_error_FLT_aux :
forall k, (emin + prec - 1 < k)%Z -> (prec <= k - FLT_exp emin prec k)%Z.
Variable rnd : R -> Z.
Context { valid_rnd : Valid_rnd rnd }.
Theorem relative_error_FLT_F2R_emin :
forall m, let x := F2R (Float beta m (emin + prec - 1)) in
(x <> 0)%R ->
(Rabs (round beta (FLT_exp emin prec) rnd x - x) < bpow (-prec + 1) * Rabs x)%R.
Theorem relative_error_FLT :
forall x,
(bpow (emin + prec - 1) <= Rabs x)%R ->
(Rabs (round beta (FLT_exp emin prec) rnd x - x) < bpow (-prec + 1) * Rabs x)%R.
Theorem relative_error_FLT_F2R_emin_ex :
forall m, let x := F2R (Float beta m (emin + prec - 1)) in
(x <> 0)%R ->
exists eps,
(Rabs eps < bpow (-prec + 1))%R /\ round beta (FLT_exp emin prec) rnd x = (x * (1 + eps))%R.
1+ε property in any rounding in FLT
Theorem relative_error_FLT_ex :
forall x,
(bpow (emin + prec - 1) <= Rabs x)%R ->
exists eps,
(Rabs eps < bpow (-prec + 1))%R /\ round beta (FLT_exp emin prec) rnd x = (x * (1 + eps))%R.
Variable choice : Z -> bool.
Theorem relative_error_N_FLT :
forall x,
(bpow (emin + prec - 1) <= Rabs x)%R ->
(Rabs (round beta (FLT_exp emin prec) (Znearest choice) x - x) <= /2 * bpow (-prec + 1) * Rabs x)%R.
forall x,
(bpow (emin + prec - 1) <= Rabs x)%R ->
exists eps,
(Rabs eps < bpow (-prec + 1))%R /\ round beta (FLT_exp emin prec) rnd x = (x * (1 + eps))%R.
Variable choice : Z -> bool.
Theorem relative_error_N_FLT :
forall x,
(bpow (emin + prec - 1) <= Rabs x)%R ->
(Rabs (round beta (FLT_exp emin prec) (Znearest choice) x - x) <= /2 * bpow (-prec + 1) * Rabs x)%R.
1+ε property in rounding to nearest in FLT
Theorem relative_error_N_FLT_ex :
forall x,
(bpow (emin + prec - 1) <= Rabs x)%R ->
exists eps,
(Rabs eps <= /2 * bpow (-prec + 1))%R /\ round beta (FLT_exp emin prec) (Znearest choice) x = (x * (1 + eps))%R.
Theorem relative_error_N_FLT_round :
forall x,
(bpow (emin + prec - 1) <= Rabs x)%R ->
(Rabs (round beta (FLT_exp emin prec) (Znearest choice) x - x) <= /2 * bpow (-prec + 1) * Rabs (round beta (FLT_exp emin prec) (Znearest choice) x))%R.
Theorem relative_error_N_FLT_F2R_emin :
forall m, let x := F2R (Float beta m (emin + prec - 1)) in
(Rabs (round beta (FLT_exp emin prec) (Znearest choice) x - x) <= /2 * bpow (-prec + 1) * Rabs x)%R.
Theorem relative_error_N_FLT_F2R_emin_ex :
forall m, let x := F2R (Float beta m (emin + prec - 1)) in
exists eps,
(Rabs eps <= /2 * bpow (-prec + 1))%R /\ round beta (FLT_exp emin prec) (Znearest choice) x = (x * (1 + eps))%R.
Theorem relative_error_N_FLT_round_F2R_emin :
forall m, let x := F2R (Float beta m (emin + prec - 1)) in
(Rabs (round beta (FLT_exp emin prec) (Znearest choice) x - x) <= /2 * bpow (-prec + 1) * Rabs (round beta (FLT_exp emin prec) (Znearest choice) x))%R.
Lemma error_N_FLT_aux :
forall x,
(0 < x)%R ->
exists eps, exists eta,
(Rabs eps <= /2 * bpow (-prec + 1))%R /\
(Rabs eta <= /2 * bpow (emin))%R /\
(eps*eta=0)%R /\
round beta (FLT_exp emin prec) (Znearest choice) x = (x * (1 + eps) + eta)%R.
End Fprop_relative_FLT.
Section Fprop_relative_FLX.
Variable prec : Z.
Variable Hp : Zlt 0 prec.
Lemma relative_error_FLX_aux :
forall k, (prec <= k - FLX_exp prec k)%Z.
Variable rnd : R -> Z.
Context { valid_rnd : Valid_rnd rnd }.
Theorem relative_error_FLX :
forall x,
(x <> 0)%R ->
(Rabs (round beta (FLX_exp prec) rnd x - x) < bpow (-prec + 1) * Rabs x)%R.
forall x,
(bpow (emin + prec - 1) <= Rabs x)%R ->
exists eps,
(Rabs eps <= /2 * bpow (-prec + 1))%R /\ round beta (FLT_exp emin prec) (Znearest choice) x = (x * (1 + eps))%R.
Theorem relative_error_N_FLT_round :
forall x,
(bpow (emin + prec - 1) <= Rabs x)%R ->
(Rabs (round beta (FLT_exp emin prec) (Znearest choice) x - x) <= /2 * bpow (-prec + 1) * Rabs (round beta (FLT_exp emin prec) (Znearest choice) x))%R.
Theorem relative_error_N_FLT_F2R_emin :
forall m, let x := F2R (Float beta m (emin + prec - 1)) in
(Rabs (round beta (FLT_exp emin prec) (Znearest choice) x - x) <= /2 * bpow (-prec + 1) * Rabs x)%R.
Theorem relative_error_N_FLT_F2R_emin_ex :
forall m, let x := F2R (Float beta m (emin + prec - 1)) in
exists eps,
(Rabs eps <= /2 * bpow (-prec + 1))%R /\ round beta (FLT_exp emin prec) (Znearest choice) x = (x * (1 + eps))%R.
Theorem relative_error_N_FLT_round_F2R_emin :
forall m, let x := F2R (Float beta m (emin + prec - 1)) in
(Rabs (round beta (FLT_exp emin prec) (Znearest choice) x - x) <= /2 * bpow (-prec + 1) * Rabs (round beta (FLT_exp emin prec) (Znearest choice) x))%R.
Lemma error_N_FLT_aux :
forall x,
(0 < x)%R ->
exists eps, exists eta,
(Rabs eps <= /2 * bpow (-prec + 1))%R /\
(Rabs eta <= /2 * bpow (emin))%R /\
(eps*eta=0)%R /\
round beta (FLT_exp emin prec) (Znearest choice) x = (x * (1 + eps) + eta)%R.
End Fprop_relative_FLT.
Section Fprop_relative_FLX.
Variable prec : Z.
Variable Hp : Zlt 0 prec.
Lemma relative_error_FLX_aux :
forall k, (prec <= k - FLX_exp prec k)%Z.
Variable rnd : R -> Z.
Context { valid_rnd : Valid_rnd rnd }.
Theorem relative_error_FLX :
forall x,
(x <> 0)%R ->
(Rabs (round beta (FLX_exp prec) rnd x - x) < bpow (-prec + 1) * Rabs x)%R.
1+ε property in any rounding in FLX
Theorem relative_error_FLX_ex :
forall x,
(x <> 0)%R ->
exists eps,
(Rabs eps < bpow (-prec + 1))%R /\ round beta (FLX_exp prec) rnd x = (x * (1 + eps))%R.
Theorem relative_error_FLX_round :
forall x,
(x <> 0)%R ->
(Rabs (round beta (FLX_exp prec) rnd x - x) < bpow (-prec + 1) * Rabs (round beta (FLX_exp prec) rnd x))%R.
Variable choice : Z -> bool.
Theorem relative_error_N_FLX :
forall x,
(Rabs (round beta (FLX_exp prec) (Znearest choice) x - x) <= /2 * bpow (-prec + 1) * Rabs x)%R.
forall x,
(x <> 0)%R ->
exists eps,
(Rabs eps < bpow (-prec + 1))%R /\ round beta (FLX_exp prec) rnd x = (x * (1 + eps))%R.
Theorem relative_error_FLX_round :
forall x,
(x <> 0)%R ->
(Rabs (round beta (FLX_exp prec) rnd x - x) < bpow (-prec + 1) * Rabs (round beta (FLX_exp prec) rnd x))%R.
Variable choice : Z -> bool.
Theorem relative_error_N_FLX :
forall x,
(Rabs (round beta (FLX_exp prec) (Znearest choice) x - x) <= /2 * bpow (-prec + 1) * Rabs x)%R.
1+ε property in rounding to nearest in FLX
Theorem relative_error_N_FLX_ex :
forall x,
exists eps,
(Rabs eps <= /2 * bpow (-prec + 1))%R /\ round beta (FLX_exp prec) (Znearest choice) x = (x * (1 + eps))%R.
Theorem relative_error_N_FLX_round :
forall x,
(Rabs (round beta (FLX_exp prec) (Znearest choice) x - x) <= /2 * bpow (-prec + 1) * Rabs (round beta (FLX_exp prec) (Znearest choice) x))%R.
End Fprop_relative_FLX.
End Fprop_relative.
forall x,
exists eps,
(Rabs eps <= /2 * bpow (-prec + 1))%R /\ round beta (FLX_exp prec) (Znearest choice) x = (x * (1 + eps))%R.
Theorem relative_error_N_FLX_round :
forall x,
(Rabs (round beta (FLX_exp prec) (Znearest choice) x - x) <= /2 * bpow (-prec + 1) * Rabs (round beta (FLX_exp prec) (Znearest choice) x))%R.
End Fprop_relative_FLX.
End Fprop_relative.