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Flocq
flocq
Commits
0a6a2cd1
Commit
0a6a2cd1
authored
Apr 12, 2014
by
Pierre Roux
Browse files
Properties about innocuous double rounding of usual arithmetic operations.
parent
cf6e4536
Changes
5
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Remakefile.in
View file @
0a6a2cd1
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@@ -27,7 +27,8 @@ FILES = \
Prop/Fprop_Sterbenz.v \
Appli/Fappli_rnd_odd.v \
Appli/Fappli_IEEE.v \
Appli/Fappli_IEEE_bits.v
Appli/Fappli_IEEE_bits.v \
Appli/Fappli_double_round.v
OBJS = $(addprefix src/,$(addsuffix o,$(FILES)))
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examples/Double_round_beta_odd.v
0 → 100644
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0a6a2cd1
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src/Appli/Fappli_double_round.v
0 → 100644
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0a6a2cd1
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src/Core/Fcore_Raux.v
View file @
0a6a2cd1
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@@ -2095,6 +2095,58 @@ replace (ex - 1 - 1)%Z with (ex - 2)%Z by ring.
now
apply
Rabs_ge
;
right
.
Qed
.
Lemma
ln_beta_div
:
forall
x
y
:
R
,
(
0
<
x
)
%
R
->
(
0
<
y
)
%
R
->
(
ln_beta
x
-
ln_beta
y
<=
ln_beta
(
x
/
y
)
<=
ln_beta
x
-
ln_beta
y
+
1
)
%
Z
.
Proof
.
intros
x
y
Px
Py
.
destruct
(
ln_beta
x
)
as
(
ex
,
Hex
).
destruct
(
ln_beta
y
)
as
(
ey
,
Hey
).
simpl
.
unfold
Rdiv
.
rewrite
Rabs_right
in
Hex
;
[
|
now
apply
Rle_ge
;
apply
Rlt_le
].
rewrite
Rabs_right
in
Hey
;
[
|
now
apply
Rle_ge
;
apply
Rlt_le
].
assert
(
Heiy
:
(
bpow
(
-
ey
)
<
/
y
<=
bpow
(
-
ey
+
1
))
%
R
).
{
split
.
-
rewrite
bpow_opp
.
apply
Rinv_lt_contravar
.
+
apply
Rmult_lt_0_compat
;
[
exact
Py
|
].
now
apply
bpow_gt_0
.
+
apply
Hey
.
now
apply
Rgt_not_eq
.
-
replace
(
_
+
_
)
%
Z
with
(
-
(
ey
-
1
))
%
Z
by
ring
.
rewrite
bpow_opp
.
apply
Rinv_le
;
[
now
apply
bpow_gt_0
|
].
apply
Hey
.
now
apply
Rgt_not_eq
.
}
split
.
-
apply
ln_beta_ge_bpow
.
apply
Rabs_ge
;
right
.
replace
(
_
-
_
)
%
Z
with
(
ex
-
1
-
ey
)
%
Z
by
ring
.
unfold
Zminus
at
1
;
rewrite
bpow_plus
.
apply
Rmult_le_compat
.
+
now
apply
bpow_ge_0
.
+
now
apply
bpow_ge_0
.
+
apply
Hex
.
now
apply
Rgt_not_eq
.
+
apply
Rlt_le
;
apply
Heiy
.
-
assert
(
Pxy
:
(
0
<
x
*
/
y
)
%
R
).
{
apply
Rmult_lt_0_compat
;
[
exact
Px
|
].
now
apply
Rinv_0_lt_compat
.
}
apply
ln_beta_le_bpow
.
+
now
apply
Rgt_not_eq
.
+
rewrite
Rabs_right
;
[
|
now
apply
Rle_ge
;
apply
Rlt_le
].
replace
(
_
+
1
)
%
Z
with
(
ex
+
(
-
ey
+
1
))
%
Z
by
ring
.
rewrite
bpow_plus
.
apply
Rlt_le_trans
with
(
bpow
ex
*
/
y
)
%
R
.
*
apply
Rmult_lt_compat_r
;
[
now
apply
Rinv_0_lt_compat
|
].
apply
Hex
.
now
apply
Rgt_not_eq
.
*
apply
Rmult_le_compat_l
;
[
now
apply
bpow_ge_0
|
].
apply
Heiy
.
Qed
.
Lemma
ln_beta_sqrt
:
forall
x
,
(
0
<
x
)
%
R
->
...
...
src/Core/Fcore_generic_fmt.v
View file @
0a6a2cd1
...
...
@@ -2365,6 +2365,37 @@ End Generic.
Notation
ZnearestA
:=
(
Znearest
(
Zle_bool
0
)).
Section
rndNA_opp
.
Lemma
round_NA_opp
:
forall
beta
:
radix
,
forall
(
fexp
:
Z
->
Z
),
forall
x
,
(
round
beta
fexp
ZnearestA
(
-
x
)
=
-
round
beta
fexp
ZnearestA
x
)
%
R
.
Proof
.
intros
beta
fexp
x
.
rewrite
round_N_opp
.
apply
Ropp_eq_compat
.
apply
round_ext
.
clear
x
;
intro
x
.
unfold
Znearest
.
case_eq
(
Rcompare
(
x
-
Z2R
(
Zfloor
x
))
(
/
2
));
intro
C
;
[
|
reflexivity
|
reflexivity
].
apply
Rcompare_Eq_inv
in
C
.
assert
(
H
:
negb
(
0
<=?
-
(
Zfloor
x
+
1
))
%
Z
=
(
0
<=?
Zfloor
x
)
%
Z
);
[
|
now
rewrite
H
].
rewrite
negb_Zle_bool
.
case_eq
(
0
<=?
Zfloor
x
)
%
Z
;
intro
C
'
.
-
apply
Zle_bool_imp_le
in
C
'
.
apply
Zlt_bool_true
.
omega
.
-
rewrite
Z
.
leb_gt
in
C
'
.
apply
Zlt_bool_false
.
omega
.
Qed
.
End
rndNA_opp
.
(
**
Notations
for
backward
-
compatibility
with
Flocq
1.4
.
*
)
Notation
rndDN
:=
Zfloor
(
only
parsing
).
Notation
rndUP
:=
Zceil
(
only
parsing
).
...
...
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