1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
61
62
63
64
65
66
67
68
69
70
71
72
73
74
75
76
77
78
79
80
81
82
83
84
85
86
87
88
89
90
91
92
93
94
95
96
97
98
99
100
101
102
103
104
105
106
107
108
109
110
111
112
113
114
115
116
117
118
119
120
121
122
123
124
125
126
127
128
129
130
131
132
133
134
135
136
137
138
139
140
141
142
143
144
|
(*
* Copyright (c) 1997-1999 Massachusetts Institute of Technology
* Copyright (c) 2003, 2007-8 Matteo Frigo
* Copyright (c) 2003, 2007-8 Massachusetts Institute of Technology
*
* This program is free software; you can redistribute it and/or modify
* it under the terms of the GNU General Public License as published by
* the Free Software Foundation; either version 2 of the License, or
* (at your option) any later version.
*
* This program 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
* GNU General Public License for more details.
*
* You should have received a copy of the GNU General Public License
* along with this program; if not, write to the Free Software
* Foundation, Inc., 59 Temple Place, Suite 330, Boston, MA 02111-1307 USA
*
*)
(*
* the oracle decrees whether the sign of an expression should
* be changed.
*
* Say the expression (A - B) appears somewhere. Elsewhere in the
* expression dag the expression (B - A) may appear.
* The oracle determines which of the two forms is canonical.
*
* Algorithm: evaluate the expression at a random input, and
* keep the expression with the positive sign.
*)
let make_memoizer hash equal =
let table = ref Assoctable.empty
in
(fun f k ->
match Assoctable.lookup hash equal k !table with
Some value -> value
| None ->
let value = f k in
begin
table := Assoctable.insert hash k value !table;
value
end)
let almost_equal x y =
let epsilon = 1.0E-8 in
(abs_float (x -. y) < epsilon) ||
(abs_float (x -. y) < epsilon *. (abs_float x +. abs_float y))
let absid = make_memoizer
(fun x -> Expr.hash_float (abs_float x))
(fun a b -> almost_equal a b || almost_equal (-. a) b)
(fun x -> x)
let make_random_oracle () = make_memoizer
Variable.hash
Variable.same
(fun _ -> (float (Random.bits())) /. 1073741824.0)
let the_random_oracle = make_random_oracle ()
let sum_list l = List.fold_right (+.) l 0.0
let eval_aux random_oracle =
let memoizing = make_memoizer Expr.hash (==) in
let rec eval x =
memoizing
(function
| Expr.Num x -> Number.to_float x
| Expr.NaN x -> Expr.transcendent_to_float x
| Expr.Load v -> random_oracle v
| Expr.Store (v, x) -> eval x
| Expr.Plus l -> sum_list (List.map eval l)
| Expr.Times (a, b) -> (eval a) *. (eval b)
| Expr.CTimes (a, b) ->
1.098612288668109691395245236 +.
1.609437912434100374600759333 *. (eval a) *. (eval b)
| Expr.CTimesJ (a, b) ->
0.9102392266268373936142401657 +.
0.6213349345596118107071993881 *. (eval a) *. (eval b)
| Expr.Uminus x -> -. (eval x))
x
in eval
let eval = eval_aux the_random_oracle
let should_flip_sign node =
let v = eval node in
let v' = absid v in
not (almost_equal v v')
(*
* determine with high probability if two expressions are equal.
*
* The test is randomized: if the two expressions have the
* same value for NTESTS random inputs, then they are proclaimed
* equal. (Note that two distinct linear functions L1(x0, x1, ..., xn)
* and L2(x0, x1, ..., xn) have the same value with probability
* 0 for random x's, and thus this test is way more paranoid than
* necessary.)
*)
let likely_equal a b =
let tolerance = 1.0e-8
and ntests = 20
in
let rec loop n =
if n = 0 then
true
else
let r = make_random_oracle () in
let va = eval_aux r a
and vb = eval_aux r b
in
if (abs_float (va -. vb)) >
tolerance *. (abs_float va +. abs_float vb +. 0.0001)
then
false
else
loop (n - 1)
in
match (a, b) with
(*
* Because of the way eval is constructed, we have
* eval (Store (v, x)) == eval x
* However, we never consider the two expressions equal
*)
| (Expr.Store _, _) -> false
| (_, Expr.Store _) -> false
(*
* Expressions of the form ``Uminus (Store _)''
* are artifacts of algsimp
*)
| ((Expr.Uminus (Expr.Store _)), _) -> false
| (_, Expr.Uminus (Expr.Store _)) -> false
| _ -> loop ntests
let hash x =
let f = eval x in
truncate (f *. 65536.0)
|
