Basic definitions: float and rounding property
Require Import Raux.
Section Def.
Definition of a floating-point number
Record float (
beta :
radix) :=
Float {
Fnum :
Z ;
Fexp :
Z }.
Arguments Fnum {
beta}.
Arguments Fexp {
beta}.
Variable beta :
radix.
Definition F2R (
f :
float beta) :=
(
IZR (
Fnum f) *
bpow beta (
Fexp f))%
R.
Requirements on a rounding mode
Definition round_pred_total (
P :
R ->
R ->
Prop) :=
forall x,
exists f,
P x f.
Definition round_pred_monotone (
P :
R ->
R ->
Prop) :=
forall x y f g,
P x f ->
P y g -> (
x <=
y)%
R -> (
f <=
g)%
R.
Definition round_pred (
P :
R ->
R ->
Prop) :=
round_pred_total P /\
round_pred_monotone P.
End Def.
Arguments Fnum {
beta}.
Arguments Fexp {
beta}.
Arguments F2R {
beta}.
Section RND.
property of being a round toward -inf
Definition Rnd_DN_pt (
F :
R ->
Prop) (
x f :
R) :=
F f /\ (
f <=
x)%
R /\
forall g :
R,
F g -> (
g <=
x)%
R -> (
g <=
f)%
R.
property of being a round toward +inf
Definition Rnd_UP_pt (
F :
R ->
Prop) (
x f :
R) :=
F f /\ (
x <=
f)%
R /\
forall g :
R,
F g -> (
x <=
g)%
R -> (
f <=
g)%
R.
property of being a round toward zero
Definition Rnd_ZR_pt (
F :
R ->
Prop) (
x f :
R) :=
( (0 <=
x)%
R ->
Rnd_DN_pt F x f ) /\
( (
x <= 0)%
R ->
Rnd_UP_pt F x f ).
property of being a round to nearest
Definition Rnd_N_pt (
F :
R ->
Prop) (
x f :
R) :=
F f /\
forall g :
R,
F g -> (
Rabs (
f -
x) <=
Rabs (
g -
x))%
R.
Definition Rnd_NG_pt (
F :
R ->
Prop) (
P :
R ->
R ->
Prop) (
x f :
R) :=
Rnd_N_pt F x f /\
(
P x f \/
forall f2 :
R,
Rnd_N_pt F x f2 ->
f2 =
f ).
Definition Rnd_NA_pt (
F :
R ->
Prop) (
x f :
R) :=
Rnd_N_pt F x f /\
forall f2 :
R,
Rnd_N_pt F x f2 -> (
Rabs f2 <=
Rabs f)%
R.
Definition Rnd_N0_pt (
F :
R ->
Prop) (
x f :
R) :=
Rnd_N_pt F x f /\
forall f2 :
R,
Rnd_N_pt F x f2 -> (
Rabs f <=
Rabs f2)%
R.
End RND.