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
145
146
147
148
149
150
151
152
153
154
155
156
157
158
159
160
161
162
163
164
165
166
167
168
169
170
171
172
173
174
175
176
177
178
179
180
181
182
183
184
185
186
187
188
189
190
191
192
193
194
195
196
197
198
199
200
201
202
203
204
205
206
207
208
209
210
211
212
213
214
215
216
217
218
219
220
221
222
223
224
225
226
227
228
229
230
231
232
233
234
235
236
237
238
239
240
241
242
243
244
245
246
247
248
249
250
251
252
253
254
255
256
257
258
259
260
261
262
263
264
265
266
267
268
269
270
271
272
273
274
275
276
277
278
279
280
281
282
283
284
285
286
287
288
289
290
291
292
293
294
295
296
297
298
299
300
301
302
303
304
305
306
307
308
309
310
311
312
313
314 | -- Testing the Beta special function
-- ** TO DO **:
-- - try gcov to see if all branches are covered
with Beta_function;
with Ada.Numerics.Generic_Elementary_Functions;
with Ada.Numerics.Float_Random;
with Ada.Text_IO; use Ada.Text_IO;
with System;
procedure Test_Beta is
type Real is digits System.Max_Digits;
-- 15 for Long_Float (IEEE double);
-- 18 max. for GNAT on x86
package RB is new Beta_function(Real);
package REF is new Ada.Numerics.Generic_Elementary_Functions(Real);
use RB, REF;
-- Check relative quasi-equality
--
function Very_Close (x, y, tol: Real) return Boolean is
begin
if abs y <= Real'Base'Model_Small then
return abs x <= Real'Base'Model_Small;
else
return abs (x / y - 1.0) <= tol;
end if;
end Very_Close;
Different_Beta_values: exception;
Different_Incomplete_Beta_values: exception;
Different_Inverse_Beta_values_1: exception;
Different_Inverse_Beta_values_2: exception;
Different_Regularized_Beta_values: exception;
Different_Symmetric_Regularized_Beta_values: exception;
Different_Inverse_Regularized_Beta_values: exception;
procedure Test_complete(a, b, pbab: Real; comment: String:= "") is
-- pbab is precomputed Beta(a,b), using another tool.
bab: constant Real:= Beta(a, b);
begin
Put_Line(
Real'Image(a) & "; " &
Real'Image(b) & "; " &
Real'Image(bab) & "; " & Real'Image(pbab) & "; " & comment
);
if not Very_Close(bab, pbab, 1.0E-13) then
raise Different_Beta_values;
end if;
end Test_complete;
procedure Test_incomplete_and_inverse(x, a, b, pBxab: Real; comment: String:= "") is
-- pBabx is precomputed Beta(x,a,b), using another tool.
Bxab: constant Real:= Beta(x, a, b);
x_again: constant Real:= Inverse_Beta (Bxab, a, b);
x_again_from_precomp: constant Real:= Inverse_Beta (pBxab, a, b);
begin
Put_Line(
Real'Image(x) & "; " &
Real'Image(a) & "; " &
Real'Image(b) & "; " &
Real'Image(Bxab) & "; " &
Real'Image(pBxab) & "; " & comment
);
if not Very_Close(Bxab, pBxab, 1.0E-13) then
raise Different_Incomplete_Beta_values;
end if;
if not Very_Close(x, x_again, 1.0E-8) then
raise Different_Inverse_Beta_values_1;
end if;
if not Very_Close(x, x_again_from_precomp, 1.0E-8) then
raise Different_Inverse_Beta_values_2;
end if;
end Test_incomplete_and_inverse;
-- Specific test when the "precomputed" value is I_x(a, b)
-- Typically, Excel, or some exact values
--
procedure Test_regularized(x, a, b, pBxab: Real; comment: String:= "") is
-- pBxab is precomputed Regularized_Beta(x,a,b), using another tool.
Bxab: constant Real:= Regularized_Beta(x, a, b);
y_sym: constant Real:= 1.0 - Regularized_Beta(1.0 - x, b, a);
begin
Put_Line(
Real'Image(x) & "; " &
Real'Image(a) & "; " &
Real'Image(b) & "; " &
Real'Image(Bxab) & "; " &
Real'Image(pBxab) & "; " & comment
);
if not Very_Close(Bxab, pBxab, 1.0E-13) then
raise Different_Regularized_Beta_values;
end if;
-- Check accuracy using symmetry property
--
if not Very_Close(Bxab, y_sym, 1.0E-13) then
raise Different_Symmetric_Regularized_Beta_values;
end if;
end Test_regularized;
procedure Test_regularized_on_analytic_values is
use Ada.Numerics.Float_Random;
gen: Generator;
x, y, a, b, diff_yb1, diff_ya1, max_diff_yb1, max_diff_ya1: Real;
iter : constant := 1_000_000;
begin
Put_Line(
"Random test with Regularized on cases with an analytic value; #iterations:" &
Integer'Image(iter)
);
Reset (gen, 1);
max_diff_ya1 := 0.0;
max_diff_yb1 := 0.0;
for i in 1 .. iter loop
-- I_x(a,1) = x^a
x := Real (Random (gen));
a := Real (Random (gen)) * 20.0;
y := Regularized_Beta (x, a, 1.0);
diff_yb1 := abs(y - x ** a);
max_diff_yb1 := Real'Max (max_diff_yb1, diff_yb1);
-- I_x(1,b) = 1 - (1-x)^b
x := Real (Random (gen));
b := Real (Random (gen)) * 20.0;
y := Regularized_Beta (x, 1.0, b);
diff_ya1 := abs(y-(1.0-(1.0-x)**b));
max_diff_ya1 := Real'Max (max_diff_ya1, diff_ya1);
end loop;
Put_Line ("Maximum difference for cases where a=1: " & Real'Image(max_diff_ya1));
Put_Line ("Maximum difference for cases where b=1: " & Real'Image(max_diff_yb1));
end Test_regularized_on_analytic_values;
procedure Test_inverse_regularized(y, a, b, pbiyab: Real; comment: String:= "") is
-- pbiyab is precomputed Inverse_Regularized_Beta(y,a,b), using another tool.
biyab: constant Real:= Inverse_Regularized_Beta(y, a, b);
diff: constant Real := abs(biyab - pbiyab);
begin
Put_Line(
Real'Image(y) & "; " &
Real'Image(a) & "; " &
Real'Image(b) & "; " &
Real'Image(biyab) & "; " &
Real'Image(pbiyab) & "; " &
Real'Image(diff) & "; " & comment
);
if not Very_Close(biyab, pbiyab, 1.0E-5) then
raise Different_Inverse_Regularized_Beta_values;
end if;
end Test_inverse_regularized;
procedure Test_regularized_and_inverse_regularized is
use Ada.Numerics.Float_Random;
gen: Generator;
x, y, x2, y2, a, b, diff_x, diff_y, max_diff_x, max_diff_y: Real;
iter : constant := 10_000_000;
bork_level: constant := 1.0e-4;
begin
Put_Line(
"Random test with Regularized then Inverse_Regularized, or vice versa; #iterations:" &
Integer'Image(iter)
);
Reset (gen, 1);
max_diff_x := 0.0;
max_diff_y := 0.0;
for i in 1 .. iter loop
x := Real (Random (gen)); -- Must be in [0;1], the domain of Regularized_Beta
-- We avoid cases of (a,b) where any approximation of Beta
-- or its inverse *must* become inaccurate
-- - discussed in Beta_function package spec.
a := 0.3 + Real (Random (gen)) * 3.0;
b := 0.3 + Real (Random (gen)) * 3.0;
y := Regularized_Beta (x, a, b);
x2 := Inverse_Regularized_Beta (y, a, b);
diff_x := abs(x-x2);
-- Put_Line (Real'Image(abs(x-x2)));
max_diff_x := Real'Max (max_diff_x, diff_x);
if diff_x > bork_level then
Put_Line ("Bork case!");
Put_Line(" x; a; b; " &
" Reg.Beta(x,a,b); I.R.Beta(R.Beta(x,a,b)); difference;");
Put_Line(
Real'Image(x) & "; " &
Real'Image(a) & "; " &
Real'Image(b) & "; " &
Real'Image(y) & "; " &
Real'Image(x2) & "; " &
Real'Image(diff_x)
);
end if;
-- Now the other way round
y := Real (Random (gen)); -- Must be in [0;1] (possible values of Regularized_Beta)
x := Inverse_Regularized_Beta (y, a, b);
y2 := Regularized_Beta (x, a, b);
diff_y := abs(y-y2);
max_diff_y := Real'Max (max_diff_y, diff_y);
if diff_y > bork_level then
Put_Line ("Bork case!");
Put_Line(" y; a; b; " &
" Inv.Reg.Beta(y,a,b); R.Beta(I.R.Beta(y,a,b)); difference;");
Put_Line(
Real'Image(y) & "; " &
Real'Image(a) & "; " &
Real'Image(b) & "; " &
Real'Image(x) & "; " &
Real'Image(y2) & "; " &
Real'Image(diff_y)
);
end if;
end loop;
Put_Line ("Maximum difference between x and IRB(RB(x)): " & Real'Image(max_diff_x));
Put_Line ("Maximum difference between y and RB(IRB(y)): " & Real'Image(max_diff_y));
end Test_regularized_and_inverse_regularized;
begin
Put_Line ("Real'Digits =" & Integer'Image (Real'Digits));
Put_Line ("Real'First =" & Real'Image(Real'First));
Put_Line ("Real'Last =" & Real'Image(Real'Last));
Put_Line("----");
Put_Line(" a; b; " &
" Beta(a,b); Precomputed Beta(a,b);");
--
-- https://www.wolframalpha.com/input/?i=Beta%5B5,+4%5D
--
Test_complete( 5.0, 4.0, 0.003571428571428571428571428571428571428571428571428571428);
Test_complete( 0.1, 3.5, 8.505924710949334225853324032128878460780976159294903434800);
Test_complete( 1.23, 4.56, 0.136801545778037926966061632922397598240588019774266210675);
--
Put_Line(" x; a; b; " &
" Beta(x,a,b); Precomputed Beta(x,a,b);");
-- Wolfram
Test_incomplete_and_inverse( 0.01, 5.0, 4.0, 1.95042732142857142857142857E-11, "a=5, b=4");
Test_incomplete_and_inverse( 0.1 , 5.0, 4.0, 1.541607142857142857142857E-6, "a=5, b=4");
Test_incomplete_and_inverse( 0.5 , 5.0, 4.0, 0.0012974330357142857142857142857, "a=5, b=4");
Test_incomplete_and_inverse( 0.9 , 5.0, 4.0, 0.0035534844642857142857142857, "a=5, b=4");
Test_incomplete_and_inverse( 0.99, 5.0, 4.0, 0.0035714261504342732142857142857, "a=5, b=4");
--
Test_incomplete_and_inverse( 0.01, 0.5, 0.5, 0.2003348423231195926910463589053, "a=0.5, b=0.5");
Test_incomplete_and_inverse( 0.1 , 0.5, 0.5, 0.6435011087932843868028092287173, "a=0.5, b=0.5");
Test_incomplete_and_inverse( 0.5 , 0.5, 0.5, 1.5707963267948966192313216916397, "a=0.5, b=0.5");
Test_incomplete_and_inverse( 0.9 , 0.5, 0.5, 2.4980915447965088516598341545621, "a=0.5, b=0.5");
Test_incomplete_and_inverse( 0.99, 0.5, 0.5, 2.9412578112666736457715970243741, "a=0.5, b=0.5");
--
Test_incomplete_and_inverse( 0.01, 5.0, 1.0, 2.0E-11, "a=5, b=1");
Test_incomplete_and_inverse( 0.1 , 5.0, 1.0, 2.0E-06, "a=5, b=1");
Test_incomplete_and_inverse( 0.5 , 5.0, 1.0, 0.00625, "a=5, b=1");
Test_incomplete_and_inverse( 0.9 , 5.0, 1.0, 0.118098, "a=5, b=1");
Test_incomplete_and_inverse( 0.99, 5.0, 1.0, 0.19019800998, "a=5, b=1");
--
Test_incomplete_and_inverse( 0.01, 1.0, 3.0, 0.00990033333333333333333333333333333, "a=1, b=3");
Test_incomplete_and_inverse( 0.1 , 1.0, 3.0, 0.09033333333333333333333333333333333, "a=1, b=3");
Test_incomplete_and_inverse( 0.5 , 1.0, 3.0, 0.29166666666666666666666666666666666, "a=1, b=3");
Test_incomplete_and_inverse( 0.9 , 1.0, 3.0, 0.333, "a=1, b=3");
Test_incomplete_and_inverse( 0.99, 1.0, 3.0, 0.333333, "a=1, b=3");
--
Test_incomplete_and_inverse( 0.01, 2.0, 2.0, 0.00004966666666666666666666666666666, "a=2, a=2");
Test_incomplete_and_inverse( 0.1 , 2.0, 2.0, 0.00466666666666666666666666666666666, "a=2, a=2");
Test_incomplete_and_inverse( 0.5 , 2.0, 2.0, 0.08333333333333333333333333333333333, "a=2, a=2");
Test_incomplete_and_inverse( 0.9 , 2.0, 2.0, 0.162, "a=2, a=2");
Test_incomplete_and_inverse( 0.99, 2.0, 2.0, 0.166617, "a=2, a=2");
--
Test_incomplete_and_inverse( 0.01, 2.0, 5.0, 0.00004868158683333333333333333333333, "a=2, a=5");
Test_incomplete_and_inverse( 0.1 , 2.0, 5.0, 0.00380883333333333333333333333333333, "a=2, a=5");
Test_incomplete_and_inverse( 0.5 , 2.0, 5.0, 0.0296875, "a=2, a=5");
Test_incomplete_and_inverse( 0.9 , 2.0, 5.0, 0.0333315, "a=2, a=5");
Test_incomplete_and_inverse( 0.99, 2.0, 5.0, 0.0333333333135, "a=2, a=5");
--
Put_Line(" x; a; b; " &
" Regularized_Beta(x,a,b); Precomp. Reg.Beta(x,a,b);");
Test_regularized( 0.1, 5.0, 4.0, 0.00043165,"Excel 2013");
Test_regularized( 0.2, 5.0, 4.0, 0.01040640,"Excel 2013");
Test_regularized( 0.3, 5.0, 4.0, 0.05796765,"Excel 2013");
Test_regularized( 0.4, 5.0, 4.0, 0.17367040,"Excel 2013");
Test_regularized( 0.5, 5.0, 4.0, 0.36328125,"Excel 2013");
Test_regularized( 0.6, 5.0, 4.0, 0.59408640,"Excel 2013");
Test_regularized( 0.7, 5.0, 4.0, 0.80589565,"Excel 2013");
Test_regularized( 0.8, 5.0, 4.0, 0.94371840,"Excel 2013");
Test_regularized( 0.9, 5.0, 4.0, 0.99497565,"Excel 2013");
Test_regularized( 0.9, 5.0, 4.0, 0.99497565,"Excel 2013");
--
-- Exact values
Test_regularized( 0.1234, 6.54321, 1.0, 0.1234 ** 6.54321, "I_x(a,1) = x^a");
Test_regularized( 0.7531, 1.0, 1.246, 1.0 - (1.0 - 0.7531) ** 1.246, "I_x(1,b) = 1 - (1-x)^b");
-- We do the same, randomized.
Test_regularized_on_analytic_values;
--
Put_Line(" y; a; b; " &
" Inv.Reg.Beta(y,a,b); Precomp. I.R.Beta(y,a,b); difference;");
Test_inverse_regularized( 0.01, 5.0, 4.0, 0.198202133178711, "Excel 2002");
Test_inverse_regularized( 0.1 , 5.0, 4.0, 0.344623088836670, "Excel 2002");
Test_inverse_regularized( 0.9 , 5.0, 4.0, 0.760338306427001, "Excel 2002");
Test_inverse_regularized( 0.99, 5.0, 4.0, 0.879049301147460, "Excel 2002");
--
Test_inverse_regularized( 0.01, 2.0, 5.0, 0.026762962341309, "Excel 2002");
Test_inverse_regularized( 0.1 , 2.0, 5.0, 0.092595100402832, "Excel 2002");
Test_inverse_regularized( 0.9 , 2.0, 5.0, 0.510316371917724, "Excel 2002");
Test_inverse_regularized( 0.99, 2.0, 5.0, 0.705684661865234, "Excel 2002");
--
-- Test_inverse_regularized(
-- 6.24895215034484863E-01,
-- 3.34854155778884888E+00,
-- 1.50668609421700239E-02,
-- 0.9999999999999999999999999999892137 , -- Wolfram [Note the 28 x '9' !...]
-- "Error with Excel 2002; would need floating-points with more than 28 digits"
-- );
--
-- x found by our ALGLIB version: 9.99999999999999500E-01
--
Test_regularized_and_inverse_regularized;
end Test_Beta;
|