Library Flocq.Prop.Plus_error

This file is part of the Flocq formalization of floating-point arithmetic in Coq: https://flocq.gitlabpages.inria.fr/

Copyright (C) 2010-2018 Sylvie Boldo
Copyright (C) 2010-2018 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.

Error of the rounded-to-nearest addition is representable.

From Coq Require Import ZArith Reals Psatz.

Require Import Core Operations Relative.

Section Fprop_plus_error.

Variable beta : radix.
Notation bpow e := (bpow beta e).

Variable fexp : Z → Z.
Context { valid_exp : Valid_exp fexp }.

Section round_repr_same_exp.

Variable rnd : R → Z.
Context { valid_rnd : Valid_rnd rnd }.

Lemma round_repr_same_exp :
  ∀ m e,
  ∃ m',
  round beta fexp rnd (F2R (Float beta m e)) = F2R (Float beta m' e).

End round_repr_same_exp.

Context { monotone_exp : Monotone_exp fexp }.
Notation format := (generic_format beta fexp).

Variable choice : Z → bool.

Lemma plus_error_aux :
  ∀ x y,
  (cexp beta fexp x ≤ cexp beta fexp y)%Z →
  format x → format y →
  format (round beta fexp (Znearest choice) (x + y) - (x + y ))%R.

Error of the addition

Theorem plus_error :
  ∀ x y,
  format x → format y →
  format (round beta fexp (Znearest choice) (x + y) - (x + y ))%R.

End Fprop_plus_error.

Section Fprop_plus_zero.

Variable beta : radix.
Notation bpow e := (bpow beta e).

Variable fexp : Z → Z.
Context { valid_exp : Valid_exp fexp }.
Context { exp_not_FTZ : Exp_not_FTZ fexp }.
Notation format := (generic_format beta fexp).

Section round_plus_eq_zero_aux.

Variable rnd : R → Z.
Context { valid_rnd : Valid_rnd rnd }.

Lemma round_plus_neq_0_aux :
  ∀ x y,
  (cexp beta fexp x ≤ cexp beta fexp y)%Z →
  format x → format y →
  (0 < x + y)%R →
  round beta fexp rnd (x + y) ≠ 0%R.

End round_plus_eq_zero_aux.

Variable rnd : R → Z.
Context { valid_rnd : Valid_rnd rnd }.

rnd(x+y)=0 -> x+y = 0 provided this is not a FTZ format

Theorem round_plus_neq_0 :
  ∀ x y,
  format x → format y →
  (x + y ≠ 0)%R →
  round beta fexp rnd (x + y) ≠ 0%R.

Theorem round_plus_eq_0 :
  ∀ x y,
  format x → format y →
  round beta fexp rnd (x + y) = 0%R →
  (x + y = 0)%R.

End Fprop_plus_zero.

Section Fprop_plus_FLT.
Variable beta : radix.

Notation bpow e := (bpow beta e).

Variable emin prec : Z.
Context { prec_gt_0_ : Prec_gt_0 prec }.

Theorem FLT_format_plus_small: ∀ x y,
  generic_format beta (FLT_exp emin prec) x →
  generic_format beta (FLT_exp emin prec) y →
   (Rabs (x +y) ≤ bpow (prec +emin))%R →
    generic_format beta (FLT_exp emin prec) (x +y).

Variable choice : Z → bool.

Lemma FLT_plus_error_N_ex : ∀ x y,
  generic_format beta (FLT_exp emin prec) x →
  generic_format beta (FLT_exp emin prec) y →
  ∃ eps ,
  (Rabs eps ≤ u_ro beta prec / (1 + u_ro beta prec ))%R ∧
  round beta (FLT_exp emin prec) (Znearest choice) (x + y)
  = ((x + y ) × (1 + eps ))%R.

Lemma FLT_plus_error_N_round_ex : ∀ x y,
  generic_format beta (FLT_exp emin prec) x →
  generic_format beta (FLT_exp emin prec) y →
  ∃ eps ,
  (Rabs eps ≤ u_ro beta prec)%R ∧
  (x + y
   = round beta (FLT_exp emin prec) (Znearest choice) (x + y) × (1 + eps ))%R.

End Fprop_plus_FLT.

Section Fprop_plus_mult_ulp.

Variable beta : radix.
Notation bpow e := (bpow beta e).

Variable fexp : Z → Z.
Context { valid_exp : Valid_exp fexp }.
Context { monotone_exp : Monotone_exp fexp }.
Variable rnd : R → Z.
Context { valid_rnd : Valid_rnd rnd }.

Notation format := (generic_format beta fexp).
Notation cexp := (cexp beta fexp).

Lemma ex_shift :
  ∀ x e, format x → (e ≤ cexp x)%Z →
  ∃ m , (x = IZR m × bpow e)%R.

Lemma mag_minus1 :
  ∀ z, z ≠ 0%R →
  (mag beta z - 1)%Z = mag beta (z / IZR beta).

Theorem round_plus_F2R :
  ∀ x y, format x → format y → (x ≠ 0)%R →
  ∃ m ,
  round beta fexp rnd (x +y) = F2R (Float beta m (cexp (x / IZR beta))).

Context {exp_not_FTZ : Exp_not_FTZ fexp}.

Theorem round_plus_ge_ulp :
  ∀ x y, format x → format y →
  round beta fexp rnd (x +y) ≠ 0%R →
  (ulp beta fexp (x /IZR beta) ≤ Rabs (round beta fexp rnd (x +y)))%R.

End Fprop_plus_mult_ulp.

Section Fprop_plus_ge_ulp.

Variable beta : radix.
Notation bpow e := (bpow beta e).

Variable rnd : R → Z.
Context { valid_rnd : Valid_rnd rnd }.
Variable emin prec : Z.
Context { prec_gt_0_ : Prec_gt_0 prec }.

Theorem round_FLT_plus_ge :
  ∀ x y e,
  generic_format beta (FLT_exp emin prec) x → generic_format beta (FLT_exp emin prec) y →
  (bpow (e + prec) ≤ Rabs x)%R →
  round beta (FLT_exp emin prec) rnd (x + y) ≠ 0%R →
  (bpow e ≤ Rabs (round beta (FLT_exp emin prec) rnd (x + y)))%R.

Lemma round_FLT_plus_ge' :
  ∀ x y e,
  generic_format beta (FLT_exp emin prec) x → generic_format beta (FLT_exp emin prec) y →
  (x ≠ 0%R → (bpow (e +prec) ≤ Rabs x)%R) →
  (x = 0%R → y ≠ 0%R → (bpow e ≤ Rabs y)%R) →
  round beta (FLT_exp emin prec) rnd (x +y) ≠ 0%R →
  (bpow e ≤ Rabs (round beta (FLT_exp emin prec) rnd (x +y)))%R.

Theorem round_FLX_plus_ge :
  ∀ x y e,
  generic_format beta (FLX_exp prec) x → generic_format beta (FLX_exp prec) y →
  (bpow (e +prec) ≤ Rabs x)%R →
  (round beta (FLX_exp prec) rnd (x +y) ≠ 0)%R →
  (bpow e ≤ Rabs (round beta (FLX_exp prec) rnd (x +y)))%R.

End Fprop_plus_ge_ulp.

Section Fprop_plus_le_ops.

Variable beta : radix.
Variable fexp : Z → Z.
Context { valid_exp : Valid_exp fexp }.
Variable choice : Z → bool.

Lemma plus_error_le_l :
  ∀ x y,
  generic_format beta fexp x → generic_format beta fexp y →
  (Rabs (round beta fexp (Znearest choice) (x + y) - (x + y )) ≤ Rabs x)%R.

Lemma plus_error_le_r :
  ∀ x y,
  generic_format beta fexp x → generic_format beta fexp y →
  (Rabs (round beta fexp (Znearest choice) (x + y) - (x + y )) ≤ Rabs y)%R.

End Fprop_plus_le_ops.