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Why3
why3
Commits
cc918120
Commit
cc918120
authored
Nov 09, 2010
by
Claude Marche
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exemple of cosine
parent
d48b4a25
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.gitignore
.gitignore
+1
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examples/programs/my_cosine.mlw
examples/programs/my_cosine.mlw
+48
-0
examples/programs/my_cosine/my_cosine.mlw_Pgm_WP_my_cosine.v
examples/programs/my_cosine/my_cosine.mlw_Pgm_WP_my_cosine.v
+169
-0
examples/programs/my_cosine/project.db
examples/programs/my_cosine/project.db
+0
-0
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.gitignore
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cc918120
*~
*.o
*.svn
.*.swp
...
...
examples/programs/my_cosine.mlw
0 → 100644
View file @
cc918120
{
use import real.RealInfix
use import real.Abs
use import real.Trigonometry
use import floating_point.Rounding
use import floating_point.Single
}
parameter single_of_real_exact : x:real ->
{ }
single
{ value result = x }
parameter add : x:single -> y:single ->
{ no_overflow NearestTiesToEven ((value x) +. (value y)) }
single
{ value result = round NearestTiesToEven ((value x) +. (value y))}
parameter sub : x:single -> y:single ->
{ no_overflow NearestTiesToEven ((value x) -. (value y)) }
single
{ value result = round NearestTiesToEven ((value x) -. (value y))}
parameter mul : x:single -> y:single ->
{ no_overflow NearestTiesToEven ((value x) *. (value y)) }
single
{ value result = round NearestTiesToEven ((value x) *. (value y))}
let my_cosine (x:single) : single =
{ abs (value x) <=. 0x1p-5 }
assert {
abs (1.0 -. (value x) *. (value x) *. 0.5 -. cos (value x)) <=. 0x1p-24
};
sub (single_of_real_exact 1.0)
(mul (mul x x) (single_of_real_exact 0.5))
{ abs (value result -. cos (value x)) <=. 0x1p-23 }
(*
Local Variables:
compile-command: "../../bin/whydb.byte my_cosine.mlw"
End:
*)
examples/programs/my_cosine/my_cosine.mlw_Pgm_WP_my_cosine.v
0 → 100644
View file @
cc918120
(
*
Beware
!
Only
edit
allowed
sections
below
*
)
(
*
This
file
is
generated
by
Why3
'
s
Coq
driver
*
)
Require
Import
ZArith
.
Require
Import
Rbase
.
Require
Import
Rbasic_fun
.
Require
Import
R_sqrt
.
Require
Import
Rtrigo
.
Require
Import
AltSeries
.
Definition
unit
:=
unit
.
Parameter
ignore
:
forall
(
a
:
Type
),
a
->
unit
.
Implicit
Arguments
ignore
.
Parameter
arrow
:
forall
(
a
:
Type
)
(
b
:
Type
),
Type
.
Parameter
ref
:
forall
(
a
:
Type
),
Type
.
Parameter
prefix_ex
:
forall
(
a
:
Type
),
(
ref
a
)
->
a
.
Implicit
Arguments
prefix_ex
.
Parameter
label
:
Type
.
Parameter
at1
:
forall
(
a
:
Type
),
a
->
label
->
a
.
Implicit
Arguments
at1
.
Parameter
old
:
forall
(
a
:
Type
),
a
->
a
.
Implicit
Arguments
old
.
Parameter
exn
:
Type
.
Definition
lt_nat
(
x
:
Z
)
(
y
:
Z
)
:
Prop
:=
(
0
%
Z
<=
y
)
%
Z
/
\
(
x
<
y
)
%
Z
.
Axiom
Refl
:
forall
(
x
:
R
),
(
x
<=
x
)
%
R
.
Axiom
Trans
:
forall
(
x
:
R
)
(
y
:
R
)
(
z
:
R
),
(
x
<=
y
)
%
R
->
((
y
<=
z
)
%
R
->
(
x
<=
z
)
%
R
).
Axiom
Antisymm
:
forall
(
x
:
R
)
(
y
:
R
),
(
x
<=
y
)
%
R
->
((
y
<=
x
)
%
R
->
(
x
=
y
)).
Axiom
Total
:
forall
(
x
:
R
)
(
y
:
R
),
(
x
<=
y
)
%
R
\
/
(
y
<=
x
)
%
R
.
Axiom
Abs_le
:
forall
(
x
:
R
)
(
y
:
R
),
((
Rabs
x
)
<=
y
)
%
R
<->
(((
-
y
)
%
R
<=
x
)
%
R
/
\
(
x
<=
y
)
%
R
).
Axiom
Abs_pos
:
forall
(
x
:
R
),
((
0
)
%
R
<=
(
Rabs
x
))
%
R
.
Axiom
Sqrt_positive
:
forall
(
x
:
R
),
((
0
)
%
R
<=
x
)
%
R
->
((
0
)
%
R
<=
(
sqrt
x
))
%
R
.
Axiom
Sqrt_square
:
forall
(
x
:
R
),
((
0
)
%
R
<=
x
)
%
R
->
(((
sqrt
x
)
*
(
sqrt
x
))
%
R
=
x
).
Axiom
Square_sqrt
:
forall
(
x
:
R
),
((
0
)
%
R
<=
x
)
%
R
->
((
sqrt
(
x
*
x
)
%
R
)
=
x
).
Axiom
Pythagorean_identity
:
forall
(
x
:
R
),
((((
cos
x
)
*
(
cos
x
))
%
R
+
((
sin
x
)
*
(
sin
x
))
%
R
)
%
R
=
(
1
)
%
R
).
Axiom
Cos_le_one
:
forall
(
x
:
R
),
((
Rabs
(
cos
x
))
<=
(
1
)
%
R
)
%
R
.
Axiom
Sin_le_one
:
forall
(
x
:
R
),
((
Rabs
(
sin
x
))
<=
(
1
)
%
R
)
%
R
.
Axiom
Cos_0
:
((
cos
(
0
)
%
R
)
=
(
1
)
%
R
).
Axiom
Sin_0
:
((
sin
(
0
)
%
R
)
=
(
0
)
%
R
).
Axiom
Pi_interval
:
((
314159265358979323846264338327950288419716939937510582097494459230781640628620899862803482534211706798214808651328230664709384460955058223172535940812848111745028410270193852110555964462294895493038196
/
100000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000
)
%
R
<
PI
)
%
R
/
\
(
PI
<
(
314159265358979323846264338327950288419716939937510582097494459230781640628620899862803482534211706798214808651328230664709384460955058223172535940812848111745028410270193852110555964462294895493038197
/
100000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000
)
%
R
)
%
R
.
Axiom
Cos_pi
:
((
cos
PI
)
=
(
-
(
1
)
%
R
)
%
R
).
Axiom
Sin_pi
:
((
sin
PI
)
=
(
0
)
%
R
).
Axiom
Cos_pi2
:
((
cos
((
05
/
10
)
%
R
*
PI
)
%
R
)
=
(
0
)
%
R
).
Axiom
Sin_pi2
:
((
sin
((
05
/
10
)
%
R
*
PI
)
%
R
)
=
(
1
)
%
R
).
Axiom
Cos_plus_pi
:
forall
(
x
:
R
),
((
cos
(
x
+
PI
)
%
R
)
=
(
-
(
cos
x
))
%
R
).
Axiom
Sin_plus_pi
:
forall
(
x
:
R
),
((
sin
(
x
+
PI
)
%
R
)
=
(
-
(
sin
x
))
%
R
).
Axiom
Cos_plus_pi2
:
forall
(
x
:
R
),
((
cos
(
x
+
((
05
/
10
)
%
R
*
PI
)
%
R
)
%
R
)
=
(
-
(
sin
x
))
%
R
).
Axiom
Sin_plus_pi2
:
forall
(
x
:
R
),
((
sin
(
x
+
((
05
/
10
)
%
R
*
PI
)
%
R
)
%
R
)
=
(
cos
x
)).
Axiom
Cos_neg
:
forall
(
x
:
R
),
((
cos
(
-
x
)
%
R
)
=
(
cos
x
)).
Axiom
Sin_neg
:
forall
(
x
:
R
),
((
sin
(
-
x
)
%
R
)
=
(
-
(
sin
x
))
%
R
).
Axiom
Cos_sum
:
forall
(
x
:
R
)
(
y
:
R
),
((
cos
(
x
+
y
)
%
R
)
=
(((
cos
x
)
*
(
cos
y
))
%
R
-
((
sin
x
)
*
(
sin
y
))
%
R
)
%
R
).
Axiom
Sin_sum
:
forall
(
x
:
R
)
(
y
:
R
),
((
sin
(
x
+
y
)
%
R
)
=
(((
sin
x
)
*
(
cos
y
))
%
R
+
((
cos
x
)
*
(
sin
y
))
%
R
)
%
R
).
Parameter
atan
:
R
->
R
.
Axiom
Tan_atan
:
forall
(
x
:
R
),
((
tan
(
atan
x
))
=
x
).
Inductive
mode
:=
|
nearestTiesToEven
:
mode
|
toZero
:
mode
|
up
:
mode
|
down
:
mode
|
nearTiesToAway
:
mode
.
Parameter
single
:
Type
.
Axiom
Zero
:
((
IZR
0
%
Z
)
=
(
0
)
%
R
).
Axiom
One
:
((
IZR
1
%
Z
)
=
(
1
)
%
R
).
Axiom
Add
:
forall
(
x
:
Z
)
(
y
:
Z
),
((
IZR
(
x
+
y
)
%
Z
)
=
((
IZR
x
)
+
(
IZR
y
))
%
R
).
Axiom
Sub
:
forall
(
x
:
Z
)
(
y
:
Z
),
((
IZR
(
x
-
y
)
%
Z
)
=
((
IZR
x
)
-
(
IZR
y
))
%
R
).
Axiom
Mul
:
forall
(
x
:
Z
)
(
y
:
Z
),
((
IZR
(
x
*
y
)
%
Z
)
=
((
IZR
x
)
*
(
IZR
y
))
%
R
).
Axiom
Neg
:
forall
(
x
:
Z
)
(
y
:
Z
),
((
IZR
(
-
x
)
%
Z
)
=
(
-
(
IZR
x
))
%
R
).
Parameter
round
:
mode
->
R
->
R
.
Parameter
round_logic
:
mode
->
R
->
single
.
Parameter
value
:
single
->
R
.
Parameter
exact
:
single
->
R
.
Parameter
model
:
single
->
R
.
Definition
round_error
(
x
:
single
)
:
R
:=
(
Rabs
((
value
x
)
-
(
exact
x
))
%
R
).
Definition
total_error
(
x
:
single
)
:
R
:=
(
Rabs
((
value
x
)
-
(
model
x
))
%
R
).
Definition
no_overflow
(
m
:
mode
)
(
x
:
R
)
:
Prop
:=
((
Rabs
(
round
m
x
))
<=
(
33554430
*
10141204801825835211973625643008
)
%
R
)
%
R
.
Axiom
Bounded_real_no_overflow
:
forall
(
m
:
mode
)
(
x
:
R
),
((
Rabs
x
)
<=
(
33554430
*
10141204801825835211973625643008
)
%
R
)
%
R
->
(
no_overflow
m
x
).
Axiom
Round_monotonic
:
forall
(
m
:
mode
)
(
x
:
R
)
(
y
:
R
),
(
x
<=
y
)
%
R
->
((
round
m
x
)
<=
(
round
m
y
))
%
R
.
Axiom
Exact_rounding_for_integers
:
forall
(
m
:
mode
)
(
i
:
Z
),
(((
-
16777216
%
Z
)
%
Z
<=
i
)
%
Z
/
\
(
i
<=
16777216
%
Z
)
%
Z
)
->
((
round
m
(
IZR
i
))
=
(
IZR
i
)).
Axiom
Round_down_le
:
forall
(
x
:
R
),
((
round
(
down
)
x
)
<=
x
)
%
R
.
Axiom
Round_up_ge
:
forall
(
x
:
R
),
(
x
<=
(
round
(
up
)
x
))
%
R
.
Axiom
Round_down_neg
:
forall
(
x
:
R
),
((
round
(
down
)
(
-
x
)
%
R
)
=
(
-
(
round
(
up
)
x
))
%
R
).
Axiom
Round_up_neg
:
forall
(
x
:
R
),
((
round
(
up
)
(
-
x
)
%
R
)
=
(
-
(
round
(
down
)
x
))
%
R
).
Theorem
WP_my_cosine
:
forall
(
x
:
single
),
((
Rabs
(
value
x
))
<=
(
1
/
32
)
%
R
)
%
R
->
((
Rabs
(((
1
)
%
R
-
(((
value
x
)
*
(
value
x
))
%
R
*
(
05
/
10
)
%
R
)
%
R
)
%
R
-
(
cos
(
value
x
)))
%
R
)
<=
(
1
/
16777216
)
%
R
)
%
R
.
(
*
YOU
MAY
EDIT
THE
PROOF
BELOW
*
)
intros
x
Hx
.
Require
Import
Interval_tactic
.
interval
with
(
i_bisect_diff
(
value
x
)).
Qed
.
(
*
DO
NOT
EDIT
BELOW
*
)
examples/programs/my_cosine/project.db
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View file @
cc918120
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