Library Flocq.Prop.Plus_error

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


Require Import Psatz.
Require Import Raux Defs Float_prop Generic_fmt.
Require Import FIX FLX FLT Ulp Operations.

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).

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.