1 | |
---|
2 | /* |
---|
3 | * Cephes Math Library Release 2.2: June, 1992 |
---|
4 | * Copyright 1984, 1987, 1988, 1992 by Stephen L. Moshier |
---|
5 | * Direct inquiries to 30 Frost Street, Cambridge, MA 02140 |
---|
6 | */ |
---|
7 | /* |
---|
8 | * |
---|
9 | * Error function |
---|
10 | * |
---|
11 | * |
---|
12 | * |
---|
13 | * SYNOPSIS: |
---|
14 | * |
---|
15 | * double x, y, erf(); |
---|
16 | * |
---|
17 | * y = erf( x ); |
---|
18 | * |
---|
19 | * |
---|
20 | * |
---|
21 | * DESCRIPTION: |
---|
22 | * |
---|
23 | * The integral is |
---|
24 | * |
---|
25 | * x |
---|
26 | * - |
---|
27 | * 2 | | 2 |
---|
28 | * erf(x) = -------- | exp( - t ) dt. |
---|
29 | * sqrt(pi) | | |
---|
30 | * - |
---|
31 | * 0 |
---|
32 | * |
---|
33 | * For 0 <= |x| < 1, erf(x) = x * P4(x**2)/Q5(x**2); otherwise |
---|
34 | * erf(x) = 1 - erfc(x). |
---|
35 | * |
---|
36 | * |
---|
37 | * |
---|
38 | * ACCURACY: |
---|
39 | * |
---|
40 | * Relative error: |
---|
41 | * arithmetic domain # trials peak rms |
---|
42 | * IEEE 0,1 30000 3.7e-16 1.0e-16 |
---|
43 | * |
---|
44 | */ |
---|
45 | /* |
---|
46 | * |
---|
47 | * Complementary error function |
---|
48 | * |
---|
49 | * |
---|
50 | * |
---|
51 | * SYNOPSIS: |
---|
52 | * |
---|
53 | * double x, y, erfc(); |
---|
54 | * |
---|
55 | * y = erfc( x ); |
---|
56 | * |
---|
57 | * |
---|
58 | * |
---|
59 | * DESCRIPTION: |
---|
60 | * |
---|
61 | * |
---|
62 | * 1 - erf(x) = |
---|
63 | * |
---|
64 | * inf. |
---|
65 | * - |
---|
66 | * 2 | | 2 |
---|
67 | * erfc(x) = -------- | exp( - t ) dt |
---|
68 | * sqrt(pi) | | |
---|
69 | * - |
---|
70 | * x |
---|
71 | * |
---|
72 | * |
---|
73 | * For small x, erfc(x) = 1 - erf(x); otherwise rational |
---|
74 | * approximations are computed. |
---|
75 | * |
---|
76 | * |
---|
77 | * |
---|
78 | * ACCURACY: |
---|
79 | * |
---|
80 | * Relative error: |
---|
81 | * arithmetic domain # trials peak rms |
---|
82 | * IEEE 0,26.6417 30000 5.7e-14 1.5e-14 |
---|
83 | */ |
---|
84 | |
---|
85 | #ifdef NEED_ERF |
---|
86 | |
---|
87 | #if FLOAT_SIZE>4 // DOUBLE_PRECISION |
---|
88 | double erf(double x); |
---|
89 | double erfc(double a); |
---|
90 | |
---|
91 | constant double PD[] = { |
---|
92 | 2.46196981473530512524E-10, |
---|
93 | 5.64189564831068821977E-1, |
---|
94 | 7.46321056442269912687E0, |
---|
95 | 4.86371970985681366614E1, |
---|
96 | 1.96520832956077098242E2, |
---|
97 | 5.26445194995477358631E2, |
---|
98 | 9.34528527171957607540E2, |
---|
99 | 1.02755188689515710272E3, |
---|
100 | 5.57535335369399327526E2 |
---|
101 | }; |
---|
102 | |
---|
103 | constant double QD[] = { |
---|
104 | /* 1.00000000000000000000E0, */ |
---|
105 | 1.32281951154744992508E1, |
---|
106 | 8.67072140885989742329E1, |
---|
107 | 3.54937778887819891062E2, |
---|
108 | 9.75708501743205489753E2, |
---|
109 | 1.82390916687909736289E3, |
---|
110 | 2.24633760818710981792E3, |
---|
111 | 1.65666309194161350182E3, |
---|
112 | 5.57535340817727675546E2 |
---|
113 | }; |
---|
114 | |
---|
115 | constant double RD[] = { |
---|
116 | 5.64189583547755073984E-1, |
---|
117 | 1.27536670759978104416E0, |
---|
118 | 5.01905042251180477414E0, |
---|
119 | 6.16021097993053585195E0, |
---|
120 | 7.40974269950448939160E0, |
---|
121 | 2.97886665372100240670E0 |
---|
122 | }; |
---|
123 | |
---|
124 | constant double SD[] = { |
---|
125 | /* 1.00000000000000000000E0, */ |
---|
126 | 2.26052863220117276590E0, |
---|
127 | 9.39603524938001434673E0, |
---|
128 | 1.20489539808096656605E1, |
---|
129 | 1.70814450747565897222E1, |
---|
130 | 9.60896809063285878198E0, |
---|
131 | 3.36907645100081516050E0 |
---|
132 | }; |
---|
133 | |
---|
134 | constant double TD[] = { |
---|
135 | 9.60497373987051638749E0, |
---|
136 | 9.00260197203842689217E1, |
---|
137 | 2.23200534594684319226E3, |
---|
138 | 7.00332514112805075473E3, |
---|
139 | 5.55923013010394962768E4 |
---|
140 | }; |
---|
141 | |
---|
142 | constant double UD[] = { |
---|
143 | /* 1.00000000000000000000E0, */ |
---|
144 | 3.35617141647503099647E1, |
---|
145 | 5.21357949780152679795E2, |
---|
146 | 4.59432382970980127987E3, |
---|
147 | 2.26290000613890934246E4, |
---|
148 | 4.92673942608635921086E4 |
---|
149 | }; |
---|
150 | |
---|
151 | double erfc(double a) |
---|
152 | { |
---|
153 | double MAXLOG = 88.72283905206835; |
---|
154 | double p, q, x, y, z; |
---|
155 | |
---|
156 | |
---|
157 | /*if (a < 0.0) |
---|
158 | x = -a; |
---|
159 | else |
---|
160 | x = a;*/ |
---|
161 | |
---|
162 | x = fabs(a); |
---|
163 | |
---|
164 | |
---|
165 | if (x < 1.0) { |
---|
166 | //The line bellow is a troublemaker for GPU, so sas_erf function |
---|
167 | //is explicit here for the case < 1.0 |
---|
168 | //return (1.0 - sas_erf(a)); |
---|
169 | z = x * x; |
---|
170 | y = x * polevl(z, TD, 4) / p1evl(z, UD, 5); |
---|
171 | |
---|
172 | return y; |
---|
173 | } |
---|
174 | |
---|
175 | z = -a * a; |
---|
176 | |
---|
177 | if (z < -MAXLOG) { |
---|
178 | if (a < 0) |
---|
179 | return (2.0); |
---|
180 | else |
---|
181 | return (0.0); |
---|
182 | } |
---|
183 | |
---|
184 | z = exp(z); |
---|
185 | |
---|
186 | |
---|
187 | if (x < 8.0) { |
---|
188 | p = polevl(x, PD, 8); |
---|
189 | q = p1evl(x, QD, 8); |
---|
190 | } |
---|
191 | else { |
---|
192 | p = polevl(x, RD, 5); |
---|
193 | q = p1evl(x, SD, 6); |
---|
194 | } |
---|
195 | y = (z * p) / q; |
---|
196 | |
---|
197 | if (a < 0) |
---|
198 | y = 2.0 - y; |
---|
199 | |
---|
200 | if (y == 0.0) { |
---|
201 | if (a < 0) |
---|
202 | return (2.0); |
---|
203 | else |
---|
204 | return (0.0); |
---|
205 | } |
---|
206 | return y; |
---|
207 | } |
---|
208 | |
---|
209 | |
---|
210 | double erf(double x) |
---|
211 | { |
---|
212 | double y, z; |
---|
213 | |
---|
214 | if (fabs(x) > 1.0) |
---|
215 | return (1.0 - erfc(x)); |
---|
216 | |
---|
217 | z = x * x; |
---|
218 | #if FLOAT_SIZE>4 |
---|
219 | y = x * polevl(z, TD, 4) / p1evl(z, UD, 5); |
---|
220 | #else |
---|
221 | y = x * polevl( z, TF, 6 ); |
---|
222 | #endif |
---|
223 | |
---|
224 | return y; |
---|
225 | } |
---|
226 | |
---|
227 | #else // SINGLE PRECISION |
---|
228 | |
---|
229 | double erff(double x); |
---|
230 | double erfcf(double a); |
---|
231 | |
---|
232 | /* erfc(x) = exp(-x^2) P(1/x), 1 < x < 2 */ |
---|
233 | constant double PF[] = { |
---|
234 | 2.326819970068386E-002, |
---|
235 | -1.387039388740657E-001, |
---|
236 | 3.687424674597105E-001, |
---|
237 | -5.824733027278666E-001, |
---|
238 | 6.210004621745983E-001, |
---|
239 | -4.944515323274145E-001, |
---|
240 | 3.404879937665872E-001, |
---|
241 | -2.741127028184656E-001, |
---|
242 | 5.638259427386472E-001 |
---|
243 | }; |
---|
244 | |
---|
245 | /* erfc(x) = exp(-x^2) 1/x P(1/x^2), 2 < x < 14 */ |
---|
246 | constant double RF[] = { |
---|
247 | -1.047766399936249E+001, |
---|
248 | 1.297719955372516E+001, |
---|
249 | -7.495518717768503E+000, |
---|
250 | 2.921019019210786E+000, |
---|
251 | -1.015265279202700E+000, |
---|
252 | 4.218463358204948E-001, |
---|
253 | -2.820767439740514E-001, |
---|
254 | 5.641895067754075E-001 |
---|
255 | }; |
---|
256 | |
---|
257 | /* erf(x) = x P(x^2), 0 < x < 1 */ |
---|
258 | constant double TF[] = { |
---|
259 | 7.853861353153693E-005, |
---|
260 | -8.010193625184903E-004, |
---|
261 | 5.188327685732524E-003, |
---|
262 | -2.685381193529856E-002, |
---|
263 | 1.128358514861418E-001, |
---|
264 | -3.761262582423300E-001, |
---|
265 | 1.128379165726710E+000 |
---|
266 | }; |
---|
267 | |
---|
268 | |
---|
269 | float erfcf(float a) |
---|
270 | { |
---|
271 | float MAXLOG = 88.72283905206835; |
---|
272 | float p, q, x, y, z; |
---|
273 | |
---|
274 | |
---|
275 | /*if (a < 0.0) |
---|
276 | x = -a; |
---|
277 | else |
---|
278 | x = a;*/ |
---|
279 | // TODO: tinycc does not support fabsf |
---|
280 | x = fabs(a); |
---|
281 | |
---|
282 | |
---|
283 | if (x < 1.0) { |
---|
284 | //The line below is a troublemaker for GPU, so sas_erf function |
---|
285 | //is explicit here for the case < 1.0 |
---|
286 | //return (1.0 - sas_erf(a)); |
---|
287 | z = x * x; |
---|
288 | y = x * polevl( z, TF, 6 ); |
---|
289 | |
---|
290 | return y; |
---|
291 | } |
---|
292 | |
---|
293 | z = -a * a; |
---|
294 | |
---|
295 | if (z < -MAXLOG) { |
---|
296 | if (a < 0) |
---|
297 | return (2.0); |
---|
298 | else |
---|
299 | return (0.0); |
---|
300 | } |
---|
301 | z = expf(z); |
---|
302 | |
---|
303 | |
---|
304 | q=1.0/x; |
---|
305 | y=q*q; |
---|
306 | if( x < 2.0 ) { |
---|
307 | p = polevl( y, PF, 8 ); |
---|
308 | } else { |
---|
309 | p = polevl( y, RF, 7 ); |
---|
310 | } |
---|
311 | y = z * q * p; |
---|
312 | |
---|
313 | if (a < 0) |
---|
314 | y = 2.0 - y; |
---|
315 | |
---|
316 | if (y == 0.0) { |
---|
317 | if (a < 0) |
---|
318 | return (2.0); |
---|
319 | else |
---|
320 | return (0.0); |
---|
321 | } |
---|
322 | return y; |
---|
323 | } |
---|
324 | |
---|
325 | |
---|
326 | float erff(float x) |
---|
327 | { |
---|
328 | float y, z; |
---|
329 | |
---|
330 | // TODO: tinycc does not support fabsf |
---|
331 | if (fabs(x) > 1.0) |
---|
332 | return (1.0 - erfcf(x)); |
---|
333 | |
---|
334 | z = x * x; |
---|
335 | y = x * polevl( z, TF, 6 ); |
---|
336 | |
---|
337 | return y; |
---|
338 | } |
---|
339 | |
---|
340 | #endif // SINGLE_PRECISION |
---|
341 | #endif // NEED_ERF |
---|
342 | |
---|
343 | #if FLOAT_SIZE>4 |
---|
344 | #define sas_erf erf |
---|
345 | #define sas_erfc erfc |
---|
346 | #else |
---|
347 | #define sas_erf erff |
---|
348 | #define sas_erfc erfcf |
---|
349 | #endif |
---|