3
* Normal distribution function
17
* Returns the area under the Gaussian probability density
18
* function, integrated from minus infinity to x:
23
* ndtr(x) = --------- | exp( - t /2 ) dt
28
* = ( 1 + erf(z) ) / 2
31
* where z = x/sqrt(2). Computation is via the functions
38
* arithmetic domain # trials peak rms
39
* DEC -13,0 8000 2.1e-15 4.8e-16
40
* IEEE -13,0 30000 3.4e-14 6.7e-15
45
* message condition value returned
46
* erfc underflow x > 37.519379347 0.0
70
* erf(x) = -------- | exp( - t ) dt.
75
* The magnitude of x is limited to 9.231948545 for DEC
76
* arithmetic; 1 or -1 is returned outside this range.
78
* For 0 <= |x| < 1, erf(x) = x * P4(x**2)/Q5(x**2); otherwise
79
* erf(x) = 1 - erfc(x).
86
* arithmetic domain # trials peak rms
87
* DEC 0,1 14000 4.7e-17 1.5e-17
88
* IEEE 0,1 30000 3.7e-16 1.0e-16
93
* Complementary error function
99
* double x, y, erfc();
113
* erfc(x) = -------- | exp( - t ) dt
119
* For small x, erfc(x) = 1 - erf(x); otherwise rational
120
* approximations are computed.
127
* arithmetic domain # trials peak rms
128
* DEC 0, 9.2319 12000 5.1e-16 1.2e-16
129
* IEEE 0,26.6417 30000 5.7e-14 1.5e-14
134
* message condition value returned
135
* erfc underflow x > 9.231948545 (DEC) 0.0
142
Cephes Math Library Release 2.2: June, 1992
143
Copyright 1984, 1987, 1988, 1992 by Stephen L. Moshier
144
Direct inquiries to 30 Frost Street, Cambridge, MA 02140
150
extern double SQRTH, NAN;
151
extern double MAXLOG;
154
static double P[] = {
155
2.46196981473530512524E-10,
156
5.64189564831068821977E-1,
157
7.46321056442269912687E0,
158
4.86371970985681366614E1,
159
1.96520832956077098242E2,
160
5.26445194995477358631E2,
161
9.34528527171957607540E2,
162
1.02755188689515710272E3,
163
5.57535335369399327526E2
165
static double Q[] = {
166
/* 1.00000000000000000000E0,*/
167
1.32281951154744992508E1,
168
8.67072140885989742329E1,
169
3.54937778887819891062E2,
170
9.75708501743205489753E2,
171
1.82390916687909736289E3,
172
2.24633760818710981792E3,
173
1.65666309194161350182E3,
174
5.57535340817727675546E2
176
static double R[] = {
177
5.64189583547755073984E-1,
178
1.27536670759978104416E0,
179
5.01905042251180477414E0,
180
6.16021097993053585195E0,
181
7.40974269950448939160E0,
182
2.97886665372100240670E0
184
static double S[] = {
185
/* 1.00000000000000000000E0,*/
186
2.26052863220117276590E0,
187
9.39603524938001434673E0,
188
1.20489539808096656605E1,
189
1.70814450747565897222E1,
190
9.60896809063285878198E0,
191
3.36907645100081516050E0
193
static double T[] = {
194
9.60497373987051638749E0,
195
9.00260197203842689217E1,
196
2.23200534594684319226E3,
197
7.00332514112805075473E3,
198
5.55923013010394962768E4
200
static double U[] = {
201
/* 1.00000000000000000000E0,*/
202
3.35617141647503099647E1,
203
5.21357949780152679795E2,
204
4.59432382970980127987E3,
205
2.26290000613890934246E4,
206
4.92673942608635921086E4
209
#define UTHRESH 37.519379347
213
static unsigned short P[] = {
214
0030207,0054445,0011173,0021706,
215
0040020,0067272,0030661,0122075,
216
0040756,0151236,0173053,0067042,
217
0041502,0106175,0062555,0151457,
218
0042104,0102525,0047401,0003667,
219
0042403,0116176,0011446,0075303,
220
0042551,0120723,0061641,0123275,
221
0042600,0070651,0007264,0134516,
222
0042413,0061102,0167507,0176625
224
static unsigned short Q[] = {
225
/*0040200,0000000,0000000,0000000,*/
226
0041123,0123257,0165741,0017142,
227
0041655,0065027,0173413,0115450,
228
0042261,0074011,0021573,0004150,
229
0042563,0166530,0013662,0007200,
230
0042743,0176427,0162443,0105214,
231
0043014,0062546,0153727,0123772,
232
0042717,0012470,0006227,0067424,
233
0042413,0061103,0003042,0013254
235
static unsigned short R[] = {
236
0040020,0067272,0101024,0155421,
237
0040243,0037467,0056706,0026462,
238
0040640,0116017,0120665,0034315,
239
0040705,0020162,0143350,0060137,
240
0040755,0016234,0134304,0130157,
241
0040476,0122700,0051070,0015473
243
static unsigned short S[] = {
244
/*0040200,0000000,0000000,0000000,*/
245
0040420,0126200,0044276,0070413,
246
0041026,0053051,0007302,0063746,
247
0041100,0144203,0174051,0061151,
248
0041210,0123314,0126343,0177646,
249
0041031,0137125,0051431,0033011,
250
0040527,0117362,0152661,0066201
252
static unsigned short T[] = {
253
0041031,0126770,0170672,0166101,
254
0041664,0006522,0072360,0031770,
255
0043013,0100025,0162641,0126671,
256
0043332,0155231,0161627,0076200,
257
0044131,0024115,0021020,0117343
259
static unsigned short U[] = {
260
/*0040200,0000000,0000000,0000000,*/
261
0041406,0037461,0177575,0032714,
262
0042402,0053350,0123061,0153557,
263
0043217,0111227,0032007,0164217,
264
0043660,0145000,0004013,0160114,
265
0044100,0071544,0167107,0125471
271
static unsigned short P[] = {
272
0x6479,0xa24f,0xeb24,0x3df0,
273
0x3488,0x4636,0x0dd7,0x3fe2,
274
0x6dc4,0xdec5,0xda53,0x401d,
275
0xba66,0xacad,0x518f,0x4048,
276
0x20f7,0xa9e0,0x90aa,0x4068,
277
0xcf58,0xc264,0x738f,0x4080,
278
0x34d8,0x6c74,0x343a,0x408d,
279
0x972a,0x21d6,0x0e35,0x4090,
280
0xffb3,0x5de8,0x6c48,0x4081
282
static unsigned short Q[] = {
283
/*0x0000,0x0000,0x0000,0x3ff0,*/
284
0x23cc,0xfd7c,0x74d5,0x402a,
285
0x7365,0xfee1,0xad42,0x4055,
286
0x610d,0x246f,0x2f01,0x4076,
287
0x41d0,0x02f6,0x7dab,0x408e,
288
0x7151,0xfca4,0x7fa2,0x409c,
289
0xf4ff,0xdafa,0x8cac,0x40a1,
290
0xede2,0x0192,0xe2a7,0x4099,
291
0x42d6,0x60c4,0x6c48,0x4081
293
static unsigned short R[] = {
294
0x9b62,0x5042,0x0dd7,0x3fe2,
295
0xc5a6,0xebb8,0x67e6,0x3ff4,
296
0xa71a,0xf436,0x1381,0x4014,
297
0x0c0c,0x58dd,0xa40e,0x4018,
298
0x960e,0x9718,0xa393,0x401d,
299
0x0367,0x0a47,0xd4b8,0x4007
301
static unsigned short S[] = {
302
/*0x0000,0x0000,0x0000,0x3ff0,*/
303
0xce21,0x0917,0x1590,0x4002,
304
0x4cfd,0x21d8,0xcac5,0x4022,
305
0x2c4d,0x7f05,0x1910,0x4028,
306
0x7ff5,0x959c,0x14d9,0x4031,
307
0x26c1,0xaa63,0x37ca,0x4023,
308
0x2d90,0x5ab6,0xf3de,0x400a
310
static unsigned short T[] = {
311
0x5d88,0x1e37,0x35bf,0x4023,
312
0x067f,0x4e9e,0x81aa,0x4056,
313
0x35b7,0xbcb4,0x7002,0x40a1,
314
0xef90,0x3c72,0x5b53,0x40bb,
315
0x13dc,0xa442,0x2509,0x40eb
317
static unsigned short U[] = {
318
/*0x0000,0x0000,0x0000,0x3ff0,*/
319
0xa6ba,0x3fef,0xc7e6,0x4040,
320
0x3aee,0x14c6,0x4add,0x4080,
321
0xfd12,0xe680,0xf252,0x40b1,
322
0x7c0a,0x0101,0x1940,0x40d6,
323
0xf567,0x9dc8,0x0e6c,0x40e8
325
#define UTHRESH 37.519379347
329
static unsigned short P[] = {
330
0x3df0,0xeb24,0xa24f,0x6479,
331
0x3fe2,0x0dd7,0x4636,0x3488,
332
0x401d,0xda53,0xdec5,0x6dc4,
333
0x4048,0x518f,0xacad,0xba66,
334
0x4068,0x90aa,0xa9e0,0x20f7,
335
0x4080,0x738f,0xc264,0xcf58,
336
0x408d,0x343a,0x6c74,0x34d8,
337
0x4090,0x0e35,0x21d6,0x972a,
338
0x4081,0x6c48,0x5de8,0xffb3
340
static unsigned short Q[] = {
341
0x402a,0x74d5,0xfd7c,0x23cc,
342
0x4055,0xad42,0xfee1,0x7365,
343
0x4076,0x2f01,0x246f,0x610d,
344
0x408e,0x7dab,0x02f6,0x41d0,
345
0x409c,0x7fa2,0xfca4,0x7151,
346
0x40a1,0x8cac,0xdafa,0xf4ff,
347
0x4099,0xe2a7,0x0192,0xede2,
348
0x4081,0x6c48,0x60c4,0x42d6
350
static unsigned short R[] = {
351
0x3fe2,0x0dd7,0x5042,0x9b62,
352
0x3ff4,0x67e6,0xebb8,0xc5a6,
353
0x4014,0x1381,0xf436,0xa71a,
354
0x4018,0xa40e,0x58dd,0x0c0c,
355
0x401d,0xa393,0x9718,0x960e,
356
0x4007,0xd4b8,0x0a47,0x0367
358
static unsigned short S[] = {
359
0x4002,0x1590,0x0917,0xce21,
360
0x4022,0xcac5,0x21d8,0x4cfd,
361
0x4028,0x1910,0x7f05,0x2c4d,
362
0x4031,0x14d9,0x959c,0x7ff5,
363
0x4023,0x37ca,0xaa63,0x26c1,
364
0x400a,0xf3de,0x5ab6,0x2d90
366
static unsigned short T[] = {
367
0x4023,0x35bf,0x1e37,0x5d88,
368
0x4056,0x81aa,0x4e9e,0x067f,
369
0x40a1,0x7002,0xbcb4,0x35b7,
370
0x40bb,0x5b53,0x3c72,0xef90,
371
0x40eb,0x2509,0xa442,0x13dc
373
static unsigned short U[] = {
374
0x4040,0xc7e6,0x3fef,0xa6ba,
375
0x4080,0x4add,0x14c6,0x3aee,
376
0x40b1,0xf252,0xe680,0xfd12,
377
0x40d6,0x1940,0x0101,0x7c0a,
378
0x40e8,0x0e6c,0x9dc8,0xf567
380
#define UTHRESH 37.519379347
384
double polevl(), p1evl(), exp(), log(), fabs();
385
double erf(), erfc();
388
double ndtr(double a)
393
mtherr("ndtr", DOMAIN);
401
y = 0.5 + 0.5 * erf(x);
415
double erfc(double a)
420
mtherr("erfc", DOMAIN);
430
return( 1.0 - erf(a) );
437
mtherr( "erfc", UNDERFLOW );
448
p = polevl( x, P, 8 );
449
q = p1evl( x, Q, 8 );
453
p = polevl( x, R, 5 );
454
q = p1evl( x, S, 6 );
474
mtherr("erf", DOMAIN);
479
return( 1.0 - erfc(x) );
482
y = x * polevl( z, T, 4 ) / p1evl( z, U, 5 );