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
|
/* ndtrif.c
*
* Inverse of Normal distribution function
*
*
*
* SYNOPSIS:
*
* float x, y, ndtrif();
*
* x = ndtrif( y );
*
*
*
* DESCRIPTION:
*
* Returns the argument, x, for which the area under the
* Gaussian probability density function (integrated from
* minus infinity to x) is equal to y.
*
*
* For small arguments 0 < y < exp(-2), the program computes
* z = sqrt( -2.0 * log(y) ); then the approximation is
* x = z - log(z)/z - (1/z) P(1/z) / Q(1/z).
* There are two rational functions P/Q, one for 0 < y < exp(-32)
* and the other for y up to exp(-2). For larger arguments,
* w = y - 0.5, and x/sqrt(2pi) = w + w**3 R(w**2)/S(w**2)).
*
*
* ACCURACY:
*
* Relative error:
* arithmetic domain # trials peak rms
* IEEE 1e-38, 1 30000 3.6e-7 5.0e-8
*
*
* ERROR MESSAGES:
*
* message condition value returned
* ndtrif domain x <= 0 -MAXNUM
* ndtrif domain x >= 1 MAXNUM
*
*/
/*
Cephes Math Library Release 2.2: July, 1992
Copyright 1984, 1987, 1989, 1992 by Stephen L. Moshier
Direct inquiries to 30 Frost Street, Cambridge, MA 02140
*/
#include <math.h>
extern float MAXNUMF;
/* sqrt(2pi) */
static float s2pi = 2.50662827463100050242;
/* approximation for 0 <= |y - 0.5| <= 3/8 */
static float P0[5] = {
-5.99633501014107895267E1,
9.80010754185999661536E1,
-5.66762857469070293439E1,
1.39312609387279679503E1,
-1.23916583867381258016E0,
};
static float Q0[8] = {
/* 1.00000000000000000000E0,*/
1.95448858338141759834E0,
4.67627912898881538453E0,
8.63602421390890590575E1,
-2.25462687854119370527E2,
2.00260212380060660359E2,
-8.20372256168333339912E1,
1.59056225126211695515E1,
-1.18331621121330003142E0,
};
/* Approximation for interval z = sqrt(-2 log y ) between 2 and 8
* i.e., y between exp(-2) = .135 and exp(-32) = 1.27e-14.
*/
static float P1[9] = {
4.05544892305962419923E0,
3.15251094599893866154E1,
5.71628192246421288162E1,
4.40805073893200834700E1,
1.46849561928858024014E1,
2.18663306850790267539E0,
-1.40256079171354495875E-1,
-3.50424626827848203418E-2,
-8.57456785154685413611E-4,
};
static float Q1[8] = {
/* 1.00000000000000000000E0,*/
1.57799883256466749731E1,
4.53907635128879210584E1,
4.13172038254672030440E1,
1.50425385692907503408E1,
2.50464946208309415979E0,
-1.42182922854787788574E-1,
-3.80806407691578277194E-2,
-9.33259480895457427372E-4,
};
/* Approximation for interval z = sqrt(-2 log y ) between 8 and 64
* i.e., y between exp(-32) = 1.27e-14 and exp(-2048) = 3.67e-890.
*/
static float P2[9] = {
3.23774891776946035970E0,
6.91522889068984211695E0,
3.93881025292474443415E0,
1.33303460815807542389E0,
2.01485389549179081538E-1,
1.23716634817820021358E-2,
3.01581553508235416007E-4,
2.65806974686737550832E-6,
6.23974539184983293730E-9,
};
static float Q2[8] = {
/* 1.00000000000000000000E0,*/
6.02427039364742014255E0,
3.67983563856160859403E0,
1.37702099489081330271E0,
2.16236993594496635890E-1,
1.34204006088543189037E-2,
3.28014464682127739104E-4,
2.89247864745380683936E-6,
6.79019408009981274425E-9,
};
#ifdef ANSIC
float polevlf(float, float *, int);
float p1evlf(float, float *, int);
float logf(float), sqrtf(float);
#else
float polevlf(), p1evlf(), logf(), sqrtf();
#endif
float ndtrif(float yy0)
{
float y0, x, y, z, y2, x0, x1;
int code;
y0 = yy0;
if( y0 <= 0.0 )
{
mtherr( "ndtrif", DOMAIN );
return( -MAXNUMF );
}
if( y0 >= 1.0 )
{
mtherr( "ndtrif", DOMAIN );
return( MAXNUMF );
}
code = 1;
y = y0;
if( y > (1.0 - 0.13533528323661269189) ) /* 0.135... = exp(-2) */
{
y = 1.0 - y;
code = 0;
}
if( y > 0.13533528323661269189 )
{
y = y - 0.5;
y2 = y * y;
x = y + y * (y2 * polevlf( y2, P0, 4)/p1evlf( y2, Q0, 8 ));
x = x * s2pi;
return(x);
}
x = sqrtf( -2.0 * logf(y) );
x0 = x - logf(x)/x;
z = 1.0/x;
if( x < 8.0 ) /* y > exp(-32) = 1.2664165549e-14 */
x1 = z * polevlf( z, P1, 8 )/p1evlf( z, Q1, 8 );
else
x1 = z * polevlf( z, P2, 8 )/p1evlf( z, Q2, 8 );
x = x0 - x1;
if( code != 0 )
x = -x;
return( x );
}
|
