Library Flocq.Prop.Fprop_plus_error

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

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.

Require Import Psatz.
Require Import Fcore_Raux.
Require Import Fcore_defs.
Require Import Fcore_float_prop.
Require Import Fcore_generic_fmt.
Require Import Fcore_FIX.
Require Import Fcore_FLX.
Require Import Fcore_FLT.
Require Import Fcore_ulp.
Require Import Fcalc_ops.

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 }.

Theorem 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,
  (canonic_exp beta fexp x ≤ canonic_exp 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_eq_zero_aux :
  ∀ x y,
  (canonic_exp beta fexp x ≤ canonic_exp beta fexp y)%Z →
  format x → format y →
  (0 ≤ x + y)%R →
  round beta fexp rnd (x + y) = 0%R →
  (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_eq_zero :
  ∀ 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).

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 := (canonic_exp beta fexp).

Lemma ex_shift :
  ∀ x e, format x → (e ≤ cexp x)%Z →
  ∃ m, (x = Z2R m × bpow e)%R.

Lemma ln_beta_minus1 :
  ∀ z, z ≠ 0%R →
  (ln_beta beta z - 1)%Z = ln_beta beta (z / Z2R beta).

Theorem round_plus_mult_ulp :
  ∀ x y, format x → format y → (x ≠ 0)%R →
  ∃ m, (round beta fexp rnd (x +y) = Z2R m × ulp beta fexp (x /Z2R beta))%R.

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 /Z2R 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_plus_ge_ulp_FLT : ∀ x y e,
  generic_format beta (FLT_exp emin prec) x → generic_format beta (FLT_exp emin prec) y →
  (bpow e ≤ Rabs x)%R →
  round beta (FLT_exp emin prec) rnd (x +y) = 0%R ∨
  (bpow (e - prec) ≤ Rabs (round beta (FLT_exp emin prec) rnd (x +y)))%R.

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

End Fprop_plus_ge_ulp.