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) 2010-2013 Sylvie Boldo
Copyright (C) 2010-2013 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) 2010-2013 Guillaume Melquiond
Relative error of the roundings
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' :
∀ x b, (0 < b)%R →
(x ≠ 0 → Rabs (round beta fexp rnd x - x) < b × Rabs x)%R →
∃ eps,
(Rabs eps < b)%R ∧ round beta fexp rnd x = (x × (1 + eps))%R.
Lemma relative_error_lt_conversion :
∀ x b, (0 < b)%R →
(Rabs (round beta fexp rnd x - x) < b × Rabs x)%R →
∃ eps,
(Rabs eps < b)%R ∧ round beta fexp rnd x = (x × (1 + eps))%R.
Lemma relative_error_le_conversion :
∀ x b, (0 ≤ b)%R →
(Rabs (round beta fexp rnd x - x) ≤ b × Rabs x)%R →
∃ eps,
(Rabs eps ≤ b)%R ∧ round beta fexp rnd x = (x × (1 + eps))%R.
End relative_error_conversion.
Variable emin p : Z.
Hypothesis Hmin : ∀ k, (emin < k)%Z → (p ≤ k - fexp k)%Z.
Variable rnd : R → Z.
Context { valid_rnd : Valid_rnd rnd }.
Theorem relative_error :
∀ 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' :
∀ x b, (0 < b)%R →
(x ≠ 0 → Rabs (round beta fexp rnd x - x) < b × Rabs x)%R →
∃ eps,
(Rabs eps < b)%R ∧ round beta fexp rnd x = (x × (1 + eps))%R.
Lemma relative_error_lt_conversion :
∀ x b, (0 < b)%R →
(Rabs (round beta fexp rnd x - x) < b × Rabs x)%R →
∃ eps,
(Rabs eps < b)%R ∧ round beta fexp rnd x = (x × (1 + eps))%R.
Lemma relative_error_le_conversion :
∀ x b, (0 ≤ b)%R →
(Rabs (round beta fexp rnd x - x) ≤ b × Rabs x)%R →
∃ eps,
(Rabs eps ≤ b)%R ∧ round beta fexp rnd x = (x × (1 + eps))%R.
End relative_error_conversion.
Variable emin p : Z.
Hypothesis Hmin : ∀ k, (emin < k)%Z → (p ≤ k - fexp k)%Z.
Variable rnd : R → Z.
Context { valid_rnd : Valid_rnd rnd }.
Theorem relative_error :
∀ 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 :
∀ x,
(bpow emin ≤ Rabs x)%R →
∃ eps,
(Rabs eps < bpow (-p + 1))%R ∧ round beta fexp rnd x = (x × (1 + eps))%R.
Theorem relative_error_F2R_emin :
∀ 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' :
∀ m, let x := F2R (Float beta m emin) in
∃ eps,
(Rabs eps < bpow (-p + 1))%R ∧ round beta fexp rnd x = (x × (1 + eps))%R.
Theorem relative_error_F2R_emin_ex :
∀ m, let x := F2R (Float beta m emin) in
(x ≠ 0)%R →
∃ eps,
(Rabs eps < bpow (-p + 1))%R ∧ round beta fexp rnd x = (x × (1 + eps))%R.
Theorem relative_error_round :
(0 < p)%Z →
∀ 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 →
∀ 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 :
∀ x,
(bpow emin ≤ Rabs x)%R →
(Rabs (round beta fexp (Znearest choice) x - x) ≤ /2 × bpow (-p + 1) × Rabs x)%R.
∀ x,
(bpow emin ≤ Rabs x)%R →
∃ eps,
(Rabs eps < bpow (-p + 1))%R ∧ round beta fexp rnd x = (x × (1 + eps))%R.
Theorem relative_error_F2R_emin :
∀ 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' :
∀ m, let x := F2R (Float beta m emin) in
∃ eps,
(Rabs eps < bpow (-p + 1))%R ∧ round beta fexp rnd x = (x × (1 + eps))%R.
Theorem relative_error_F2R_emin_ex :
∀ m, let x := F2R (Float beta m emin) in
(x ≠ 0)%R →
∃ eps,
(Rabs eps < bpow (-p + 1))%R ∧ round beta fexp rnd x = (x × (1 + eps))%R.
Theorem relative_error_round :
(0 < p)%Z →
∀ 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 →
∀ 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 :
∀ 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 :
∀ x,
(bpow emin ≤ Rabs x)%R →
∃ eps,
(Rabs eps ≤ /2 × bpow (-p + 1))%R ∧ round beta fexp (Znearest choice) x = (x × (1 + eps))%R.
Theorem relative_error_N_F2R_emin :
∀ 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 :
∀ m, let x := F2R (Float beta m emin) in
∃ 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 →
∀ 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 →
∀ 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 :
∀ 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 :
∀ 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 :
∀ 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' :
∀ m, let x := F2R (Float beta m emin) in
(x ≠ 0)%R →
(Rabs (round beta (FLT_exp emin prec) rnd x - x) < bpow (-prec + 1) × Rabs x)%R.
Theorem relative_error_FLT_F2R_emin_ex' :
∀ m, let x := F2R (Float beta m emin) in
∃ eps,
(Rabs eps < bpow (-prec + 1))%R ∧ round beta (FLT_exp emin prec) rnd x = (x × (1 + eps))%R.
Theorem relative_error_FLT_F2R_emin_ex :
∀ m, let x := F2R (Float beta m (emin + prec - 1)) in
(x ≠ 0)%R →
∃ eps,
(Rabs eps < bpow (-prec + 1))%R ∧ round beta (FLT_exp emin prec) rnd x = (x × (1 + eps))%R.
∀ x,
(bpow emin ≤ Rabs x)%R →
∃ eps,
(Rabs eps ≤ /2 × bpow (-p + 1))%R ∧ round beta fexp (Znearest choice) x = (x × (1 + eps))%R.
Theorem relative_error_N_F2R_emin :
∀ 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 :
∀ m, let x := F2R (Float beta m emin) in
∃ 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 →
∀ 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 →
∀ 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 :
∀ 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 :
∀ 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 :
∀ 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' :
∀ m, let x := F2R (Float beta m emin) in
(x ≠ 0)%R →
(Rabs (round beta (FLT_exp emin prec) rnd x - x) < bpow (-prec + 1) × Rabs x)%R.
Theorem relative_error_FLT_F2R_emin_ex' :
∀ m, let x := F2R (Float beta m emin) in
∃ eps,
(Rabs eps < bpow (-prec + 1))%R ∧ round beta (FLT_exp emin prec) rnd x = (x × (1 + eps))%R.
Theorem relative_error_FLT_F2R_emin_ex :
∀ m, let x := F2R (Float beta m (emin + prec - 1)) in
(x ≠ 0)%R →
∃ 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 :
∀ x,
(bpow (emin + prec - 1) ≤ Rabs x)%R →
∃ 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 :
∀ 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.
∀ x,
(bpow (emin + prec - 1) ≤ Rabs x)%R →
∃ 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 :
∀ 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 :
∀ x,
(bpow (emin + prec - 1) ≤ Rabs x)%R →
∃ 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 :
∀ 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' :
∀ m, let x := F2R (Float beta m emin) 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 :
∀ 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' :
∀ m, let x := F2R (Float beta m emin) in
∃ 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_F2R_emin_ex :
∀ m, let x := F2R (Float beta m (emin + prec - 1)) in
∃ 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' :
∀ m, let x := F2R (Float beta m emin) 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.
Theorem relative_error_N_FLT_round_F2R_emin :
∀ 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 :
∀ x,
(0 < x)%R →
∃ eps, ∃ 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.
Lemma error_N_FLT :
∀ (emin prec : Z), (0 < prec)%Z →
∀ (choice : Z → bool),
∀ (x : R),
∃ eps eta : R,
(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.
Section Fprop_relative_FLX.
Variable prec : Z.
Variable Hp : Zlt 0 prec.
Lemma relative_error_FLX_aux :
∀ k, (prec ≤ k - FLX_exp prec k)%Z.
Variable rnd : R → Z.
Context { valid_rnd : Valid_rnd rnd }.
Theorem relative_error_FLX :
∀ x,
(x ≠ 0)%R →
(Rabs (round beta (FLX_exp prec) rnd x - x) < bpow (-prec + 1) × Rabs x)%R.
∀ x,
(bpow (emin + prec - 1) ≤ Rabs x)%R →
∃ 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 :
∀ 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' :
∀ m, let x := F2R (Float beta m emin) 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 :
∀ 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' :
∀ m, let x := F2R (Float beta m emin) in
∃ 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_F2R_emin_ex :
∀ m, let x := F2R (Float beta m (emin + prec - 1)) in
∃ 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' :
∀ m, let x := F2R (Float beta m emin) 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.
Theorem relative_error_N_FLT_round_F2R_emin :
∀ 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 :
∀ x,
(0 < x)%R →
∃ eps, ∃ 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.
Lemma error_N_FLT :
∀ (emin prec : Z), (0 < prec)%Z →
∀ (choice : Z → bool),
∀ (x : R),
∃ eps eta : R,
(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.
Section Fprop_relative_FLX.
Variable prec : Z.
Variable Hp : Zlt 0 prec.
Lemma relative_error_FLX_aux :
∀ k, (prec ≤ k - FLX_exp prec k)%Z.
Variable rnd : R → Z.
Context { valid_rnd : Valid_rnd rnd }.
Theorem relative_error_FLX :
∀ 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' :
∀ x,
∃ eps,
(Rabs eps < bpow (-prec + 1))%R ∧ round beta (FLX_exp prec) rnd x = (x × (1 + eps))%R.
Theorem relative_error_FLX_ex :
∀ x,
(x ≠ 0)%R →
∃ eps,
(Rabs eps < bpow (-prec + 1))%R ∧ round beta (FLX_exp prec) rnd x = (x × (1 + eps))%R.
Theorem relative_error_FLX_round :
∀ 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 :
∀ x,
(Rabs (round beta (FLX_exp prec) (Znearest choice) x - x) ≤ /2 × bpow (-prec + 1) × Rabs x)%R.
∀ x,
∃ eps,
(Rabs eps < bpow (-prec + 1))%R ∧ round beta (FLX_exp prec) rnd x = (x × (1 + eps))%R.
Theorem relative_error_FLX_ex :
∀ x,
(x ≠ 0)%R →
∃ eps,
(Rabs eps < bpow (-prec + 1))%R ∧ round beta (FLX_exp prec) rnd x = (x × (1 + eps))%R.
Theorem relative_error_FLX_round :
∀ 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 :
∀ 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 :
∀ x,
∃ 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 :
∀ 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.
∀ x,
∃ 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 :
∀ 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.