|
132
|
1 /**
|
|
|
2 * Error Functions and Normal Distribution.
|
|
|
3 *
|
|
|
4 * Copyright: Copyright (C) 1984, 1995, 2000 Stephen L. Moshier
|
|
|
5 * Code taken from the Cephes Math Library Release 2.3: January, 1995
|
|
|
6 * License: BSD style: $(LICENSE)
|
|
|
7 * Authors: Stephen L. Moshier, ported to D by Don Clugston
|
|
|
8 */
|
|
|
9 /**
|
|
|
10 * Macros:
|
|
|
11 * NAN = $(RED NAN)
|
|
|
12 * SUP = <span style="vertical-align:super;font-size:smaller">$0</span>
|
|
|
13 * GAMMA = Γ
|
|
|
14 * INTEGRAL = ∫
|
|
|
15 * INTEGRATE = $(BIG ∫<sub>$(SMALL $1)</sub><sup>$2</sup>)
|
|
|
16 * POWER = $1<sup>$2</sup>
|
|
|
17 * BIGSUM = $(BIG Σ <sup>$2</sup><sub>$(SMALL $1)</sub>)
|
|
|
18 * CHOOSE = $(BIG () <sup>$(SMALL $1)</sup><sub>$(SMALL $2)</sub> $(BIG ))
|
|
|
19 * TABLE_SV = <table border=1 cellpadding=4 cellspacing=0>
|
|
|
20 * <caption>Special Values</caption>
|
|
|
21 * $0</table>
|
|
|
22 * SVH = $(TR $(TH $1) $(TH $2))
|
|
|
23 * SV = $(TR $(TD $1) $(TD $2))
|
|
|
24 */
|
|
|
25 module tango.math.ErrorFunction;
|
|
|
26
|
|
|
27 import tango.math.Math;
|
|
|
28 import tango.math.IEEE; // only required for unit tests
|
|
|
29
|
|
|
30 version(Windows) { // Some tests only pass on DMD Windows
|
|
|
31 version(DigitalMars) {
|
|
|
32 version = FailsOnLinux;
|
|
|
33 }
|
|
|
34 }
|
|
|
35
|
|
|
36 const real SQRT2PI = 0x1.40d931ff62705966p+1L; // 2.5066282746310005024
|
|
|
37 const real EXP_2 = 0.13533528323661269189L; /* exp(-2) */
|
|
|
38
|
|
|
39 private {
|
|
|
40
|
|
|
41 /* erfc(x) = exp(-x^2) P(1/x)/Q(1/x)
|
|
|
42 1/8 <= 1/x <= 1
|
|
|
43 Peak relative error 5.8e-21 */
|
|
|
44 const real [] P = [ -0x1.30dfa809b3cc6676p-17, 0x1.38637cd0913c0288p+18,
|
|
|
45 0x1.2f015e047b4476bp+22, 0x1.24726f46aa9ab08p+25, 0x1.64b13c6395dc9c26p+27,
|
|
|
46 0x1.294c93046ad55b5p+29, 0x1.5962a82f92576dap+30, 0x1.11a709299faba04ap+31,
|
|
|
47 0x1.11028065b087be46p+31, 0x1.0d8ef40735b097ep+30
|
|
|
48 ];
|
|
|
49
|
|
|
50 const real [] Q = [ 0x1.14d8e2a72dec49f4p+19, 0x1.0c880ff467626e1p+23,
|
|
|
51 0x1.04417ef060b58996p+26, 0x1.404e61ba86df4ebap+28, 0x1.0f81887bc82b873ap+30,
|
|
|
52 0x1.4552a5e39fb49322p+31, 0x1.11779a0ceb2a01cep+32, 0x1.3544dd691b5b1d5cp+32,
|
|
|
53 0x1.a91781f12251f02ep+31, 0x1.0d8ef3da605a1c86p+30, 1.0
|
|
|
54 ];
|
|
|
55
|
|
|
56
|
|
|
57 /* erfc(x) = exp(-x^2) 1/x R(1/x^2) / S(1/x^2)
|
|
|
58 1/128 <= 1/x < 1/8
|
|
|
59 Peak relative error 1.9e-21 */
|
|
|
60 const real [] R = [ 0x1.b9f6d8b78e22459ep-6, 0x1.1b84686b0a4ea43ap-1,
|
|
|
61 0x1.b8f6aebe96000c2ap+1, 0x1.cb1dbedac27c8ec2p+2, 0x1.cf885f8f572a4c14p+1
|
|
|
62 ];
|
|
|
63
|
|
|
64 const real [] S = [
|
|
|
65 0x1.87ae3cae5f65eb5ep-5, 0x1.01616f266f306d08p+0, 0x1.a4abe0411eed6c22p+2,
|
|
|
66 0x1.eac9ce3da600abaap+3, 0x1.5752a9ac2faebbccp+3, 1.0
|
|
|
67 ];
|
|
|
68
|
|
|
69 /* erf(x) = x P(x^2)/Q(x^2)
|
|
|
70 0 <= x <= 1
|
|
|
71 Peak relative error 7.6e-23 */
|
|
|
72 const real [] T = [ 0x1.0da01654d757888cp+20, 0x1.2eb7497bc8b4f4acp+17,
|
|
|
73 0x1.79078c19530f72a8p+15, 0x1.4eaf2126c0b2c23p+11, 0x1.1f2ea81c9d272a2ep+8,
|
|
|
74 0x1.59ca6e2d866e625p+2, 0x1.c188e0b67435faf4p-4
|
|
|
75 ];
|
|
|
76
|
|
|
77 const real [] U = [ 0x1.dde6025c395ae34ep+19, 0x1.c4bc8b6235df35aap+18,
|
|
|
78 0x1.8465900e88b6903ap+16, 0x1.855877093959ffdp+13, 0x1.e5c44395625ee358p+9,
|
|
|
79 0x1.6a0fed103f1c68a6p+5, 1.0
|
|
|
80 ];
|
|
|
81
|
|
|
82 }
|
|
|
83
|
|
|
84 /**
|
|
|
85 * Complementary error function
|
|
|
86 *
|
|
|
87 * erfc(x) = 1 - erf(x), and has high relative accuracy for
|
|
|
88 * values of x far from zero. (For values near zero, use erf(x)).
|
|
|
89 *
|
|
|
90 * 1 - erf(x) = 2/ $(SQRT)(π)
|
|
|
91 * $(INTEGRAL x, $(INFINITY)) exp( - $(POWER t, 2)) dt
|
|
|
92 *
|
|
|
93 *
|
|
|
94 * For small x, erfc(x) = 1 - erf(x); otherwise rational
|
|
|
95 * approximations are computed.
|
|
|
96 *
|
|
|
97 * A special function expx2(x) is used to suppress error amplification
|
|
|
98 * in computing exp(-x^2).
|
|
|
99 */
|
|
|
100 real erfc(real a)
|
|
|
101 {
|
|
|
102 if (a == real.infinity)
|
|
|
103 return 0.0;
|
|
|
104 if (a == -real.infinity)
|
|
|
105 return 2.0;
|
|
|
106
|
|
|
107 real x;
|
|
|
108
|
|
|
109 if (a < 0.0L )
|
|
|
110 x = -a;
|
|
|
111 else
|
|
|
112 x = a;
|
|
|
113 if (x < 1.0)
|
|
|
114 return 1.0 - erf(a);
|
|
|
115
|
|
|
116 real z = -a * a;
|
|
|
117
|
|
|
118 if (z < -MAXLOG){
|
|
|
119 // mtherr( "erfcl", UNDERFLOW );
|
|
|
120 if (a < 0) return 2.0;
|
|
|
121 else return 0.0;
|
|
|
122 }
|
|
|
123
|
|
|
124 /* Compute z = exp(z). */
|
|
|
125 z = expx2(a, -1);
|
|
|
126 real y = 1.0/x;
|
|
|
127
|
|
|
128 real p, q;
|
|
|
129
|
|
|
130 if( x < 8.0 ) y = z * rationalPoly(y, P, Q);
|
|
|
131 else y = z * y * rationalPoly(y * y, R, S);
|
|
|
132
|
|
|
133 if (a < 0.0L)
|
|
|
134 y = 2.0L - y;
|
|
|
135
|
|
|
136 if (y == 0.0) {
|
|
|
137 // mtherr( "erfcl", UNDERFLOW );
|
|
|
138 if (a < 0) return 2.0;
|
|
|
139 else return 0.0;
|
|
|
140 }
|
|
|
141
|
|
|
142 return y;
|
|
|
143 }
|
|
|
144
|
|
|
145
|
|
|
146 private {
|
|
|
147 /* Exponentially scaled erfc function
|
|
|
148 exp(x^2) erfc(x)
|
|
|
149 valid for x > 1.
|
|
|
150 Use with normalDistribution and expx2. */
|
|
|
151
|
|
|
152 real erfce(real x)
|
|
|
153 {
|
|
|
154 real p, q;
|
|
|
155
|
|
|
156 real y = 1.0/x;
|
|
|
157
|
|
|
158 if (x < 8.0) {
|
|
|
159 return rationalPoly( y, P, Q);
|
|
|
160 } else {
|
|
|
161 return y * rationalPoly(y*y, R, S);
|
|
|
162 }
|
|
|
163 }
|
|
|
164
|
|
|
165 }
|
|
|
166
|
|
|
167 /**
|
|
|
168 * Error function
|
|
|
169 *
|
|
|
170 * The integral is
|
|
|
171 *
|
|
|
172 * erf(x) = 2/ $(SQRT)(π)
|
|
|
173 * $(INTEGRAL 0, x) exp( - $(POWER t, 2)) dt
|
|
|
174 *
|
|
|
175 * The magnitude of x is limited to about 106.56 for IEEE 80-bit
|
|
|
176 * arithmetic; 1 or -1 is returned outside this range.
|
|
|
177 *
|
|
|
178 * For 0 <= |x| < 1, a rational polynomials are used; otherwise
|
|
|
179 * erf(x) = 1 - erfc(x).
|
|
|
180 *
|
|
|
181 * ACCURACY:
|
|
|
182 * Relative error:
|
|
|
183 * arithmetic domain # trials peak rms
|
|
|
184 * IEEE 0,1 50000 2.0e-19 5.7e-20
|
|
|
185 */
|
|
|
186 real erf(real x)
|
|
|
187 {
|
|
|
188 if (x == 0.0)
|
|
|
189 return x; // deal with negative zero
|
|
|
190 if (x == -real.infinity)
|
|
|
191 return -1.0;
|
|
|
192 if (x == real.infinity)
|
|
|
193 return 1.0;
|
|
|
194 if (abs(x) > 1.0L)
|
|
|
195 return 1.0L - erfc(x);
|
|
|
196
|
|
|
197 real z = x * x;
|
|
|
198 return x * rationalPoly(z, T, U);
|
|
|
199 }
|
|
|
200
|
|
|
201 debug(UnitTest) {
|
|
|
202 unittest {
|
|
|
203 // High resolution test points.
|
|
|
204 const real erfc0_250 = 0.723663330078125 + 1.0279753638067014931732235184287934646022E-5;
|
|
|
205 const real erfc0_375 = 0.5958709716796875 + 1.2118885490201676174914080878232469565953E-5;
|
|
|
206 const real erfc0_500 = 0.4794921875 + 7.9346869534623172533461080354712635484242E-6;
|
|
|
207 const real erfc0_625 = 0.3767547607421875 + 4.3570693945275513594941232097252997287766E-6;
|
|
|
208 const real erfc0_750 = 0.2888336181640625 + 1.0748182422368401062165408589222625794046E-5;
|
|
|
209 const real erfc0_875 = 0.215911865234375 + 1.3073705765341685464282101150637224028267E-5;
|
|
|
210 const real erfc1_000 = 0.15728759765625 + 1.1609394035130658779364917390740703933002E-5;
|
|
|
211 const real erfc1_125 = 0.111602783203125 + 8.9850951672359304215530728365232161564636E-6;
|
|
|
212
|
|
|
213 const real erf0_875 = (1-0.215911865234375) - 1.3073705765341685464282101150637224028267E-5;
|
|
|
214
|
|
|
215
|
|
|
216 assert(feqrel(erfc(0.250L), erfc0_250 )>=real.mant_dig-1);
|
|
|
217 assert(feqrel(erfc(0.375L), erfc0_375 )>=real.mant_dig-0);
|
|
|
218 assert(feqrel(erfc(0.500L), erfc0_500 )>=real.mant_dig-1);
|
|
|
219 assert(feqrel(erfc(0.625L), erfc0_625 )>=real.mant_dig-1);
|
|
|
220 assert(feqrel(erfc(0.750L), erfc0_750 )>=real.mant_dig-1);
|
|
|
221 assert(feqrel(erfc(0.875L), erfc0_875 )>=real.mant_dig-4);
|
|
|
222 version(FailsOnLinux) assert(feqrel(erfc(1.000L), erfc1_000 )>=real.mant_dig-0);
|
|
|
223 assert(feqrel(erfc(1.125L), erfc1_125 )>=real.mant_dig-2);
|
|
|
224 assert(feqrel(erf(0.875L), erf0_875 )>=real.mant_dig-1);
|
|
|
225 // The DMC implementation of erfc() fails this next test (just)
|
|
|
226 assert(feqrel(erfc(4.1L),0.67000276540848983727e-8L)>=real.mant_dig-4);
|
|
|
227
|
|
|
228 assert(isIdentical(erf(0.0),0.0));
|
|
|
229 assert(isIdentical(erf(-0.0),-0.0));
|
|
|
230 assert(erf(real.infinity) == 1.0);
|
|
|
231 assert(erf(-real.infinity) == -1.0);
|
|
|
232 assert(isIdentical(erf(NaN(0xDEF)),NaN(0xDEF)));
|
|
|
233 assert(isIdentical(erfc(NaN(0xDEF)),NaN(0xDEF)));
|
|
|
234 assert(isIdentical(erfc(real.infinity),0.0));
|
|
|
235 assert(erfc(-real.infinity) == 2.0);
|
|
|
236 assert(erfc(0) == 1.0);
|
|
|
237 }
|
|
|
238 }
|
|
|
239
|
|
|
240 /*
|
|
|
241 * Exponential of squared argument
|
|
|
242 *
|
|
|
243 * Computes y = exp(x*x) while suppressing error amplification
|
|
|
244 * that would ordinarily arise from the inexactness of the
|
|
|
245 * exponential argument x*x.
|
|
|
246 *
|
|
|
247 * If sign < 0, the result is inverted; i.e., y = exp(-x*x) .
|
|
|
248 *
|
|
|
249 * ACCURACY:
|
|
|
250 * Relative error:
|
|
|
251 * arithmetic domain # trials peak rms
|
|
|
252 * IEEE -106.566, 106.566 10^5 1.6e-19 4.4e-20
|
|
|
253 */
|
|
|
254
|
|
|
255 real expx2(real x, int sign)
|
|
|
256 {
|
|
|
257 /*
|
|
|
258 Cephes Math Library Release 2.9: June, 2000
|
|
|
259 Copyright 2000 by Stephen L. Moshier
|
|
|
260 */
|
|
|
261 const real M = 32768.0;
|
|
|
262 const real MINV = 3.0517578125e-5L;
|
|
|
263
|
|
|
264 x = abs(x);
|
|
|
265 if (sign < 0)
|
|
|
266 x = -x;
|
|
|
267
|
|
|
268 /* Represent x as an exact multiple of M plus a residual.
|
|
|
269 M is a power of 2 chosen so that exp(m * m) does not overflow
|
|
|
270 or underflow and so that |x - m| is small. */
|
|
|
271 real m = MINV * floor(M * x + 0.5L);
|
|
|
272 real f = x - m;
|
|
|
273
|
|
|
274 /* x^2 = m^2 + 2mf + f^2 */
|
|
|
275 real u = m * m;
|
|
|
276 real u1 = 2 * m * f + f * f;
|
|
|
277
|
|
|
278 if (sign < 0) {
|
|
|
279 u = -u;
|
|
|
280 u1 = -u1;
|
|
|
281 }
|
|
|
282
|
|
|
283 if ((u+u1) > MAXLOG)
|
|
|
284 return real.infinity;
|
|
|
285
|
|
|
286 /* u is exact, u1 is small. */
|
|
|
287 return exp(u) * exp(u1);
|
|
|
288 }
|
|
|
289
|
|
|
290
|
|
|
291 package {
|
|
|
292 /*
|
|
|
293 Computes the normal distribution function.
|
|
|
294
|
|
|
295 The normal (or Gaussian, or bell-shaped) distribution is
|
|
|
296 defined as:
|
|
|
297
|
|
|
298 normalDist(x) = 1/$(SQRT) π $(INTEGRAL -$(INFINITY), x) exp( - $(POWER t, 2)/2) dt
|
|
|
299 = 0.5 + 0.5 * erf(x/sqrt(2))
|
|
|
300 = 0.5 * erfc(- x/sqrt(2))
|
|
|
301
|
|
|
302 To maintain accuracy at high values of x, use
|
|
|
303 normalDistribution(x) = 1 - normalDistribution(-x).
|
|
|
304
|
|
|
305 Accuracy:
|
|
|
306 Within a few bits of machine resolution over the entire
|
|
|
307 range.
|
|
|
308
|
|
|
309 References:
|
|
|
310 $(LINK http://www.netlib.org/cephes/ldoubdoc.html),
|
|
|
311 G. Marsaglia, "Evaluating the Normal Distribution",
|
|
|
312 Journal of Statistical Software <b>11</b>, (July 2004).
|
|
|
313 */
|
|
|
314 real normalDistributionImpl(real a)
|
|
|
315 {
|
|
|
316 real x = a * SQRT1_2;
|
|
|
317 real z = abs(x);
|
|
|
318
|
|
|
319 if( z < 1.0 )
|
|
|
320 return 0.5L + 0.5L * erf(x);
|
|
|
321 else {
|
|
|
322 /* See below for erfce. */
|
|
|
323 real y = 0.5L * erfce(z);
|
|
|
324 /* Multiply by exp(-x^2 / 2) */
|
|
|
325 z = expx2(a, -1);
|
|
|
326 y = y * sqrt(z);
|
|
|
327 if( x > 0.0L )
|
|
|
328 y = 1.0L - y;
|
|
|
329 return y;
|
|
|
330 }
|
|
|
331 }
|
|
|
332
|
|
|
333 }
|
|
|
334
|
|
|
335 debug(UnitTest) {
|
|
|
336 unittest {
|
|
|
337 assert(fabs(normalDistributionImpl(1L) - (0.841344746068543))< 0.0000000000000005);
|
|
|
338 assert(isIdentical(normalDistributionImpl(NaN(0x325)), NaN(0x325)));
|
|
|
339 }
|
|
|
340 }
|
|
|
341
|
|
|
342 package {
|
|
|
343 /*
|
|
|
344 * Inverse of Normal distribution function
|
|
|
345 *
|
|
|
346 * Returns the argument, x, for which the area under the
|
|
|
347 * Normal probability density function (integrated from
|
|
|
348 * minus infinity to x) is equal to p.
|
|
|
349 *
|
|
|
350 * For small arguments 0 < p < exp(-2), the program computes
|
|
|
351 * z = sqrt( -2 log(p) ); then the approximation is
|
|
|
352 * x = z - log(z)/z - (1/z) P(1/z) / Q(1/z) .
|
|
|
353 * For larger arguments, x/sqrt(2 pi) = w + w^3 R(w^2)/S(w^2)) ,
|
|
|
354 * where w = p - 0.5 .
|
|
|
355 */
|
|
|
356 real normalDistributionInvImpl(real p)
|
|
|
357 in {
|
|
|
358 assert(p>=0.0L && p<=1.0L, "Domain error");
|
|
|
359 }
|
|
|
360 body
|
|
|
361 {
|
|
|
362 const real P0[] = [ -0x1.758f4d969484bfdcp-7, 0x1.53cee17a59259dd2p-3,
|
|
|
363 -0x1.ea01e4400a9427a2p-1, 0x1.61f7504a0105341ap+1, -0x1.09475a594d0399f6p+2,
|
|
|
364 0x1.7c59e7a0df99e3e2p+1, -0x1.87a81da52edcdf14p-1, 0x1.1fb149fd3f83600cp-7
|
|
|
365 ];
|
|
|
366
|
|
|
367 const real Q0[] = [ -0x1.64b92ae791e64bb2p-7, 0x1.7585c7d597298286p-3,
|
|
|
368 -0x1.40011be4f7591ce6p+0, 0x1.1fc067d8430a425ep+2, -0x1.21008ffb1e7ccdf2p+3,
|
|
|
369 0x1.3d1581cf9bc12fccp+3, -0x1.53723a89fd8f083cp+2, 1.0
|
|
|
370 ];
|
|
|
371
|
|
|
372 const real P1[] = [ 0x1.20ceea49ea142f12p-13, 0x1.cbe8a7267aea80bp-7,
|
|
|
373 0x1.79fea765aa787c48p-2, 0x1.d1f59faa1f4c4864p+1, 0x1.1c22e426a013bb96p+4,
|
|
|
374 0x1.a8675a0c51ef3202p+5, 0x1.75782c4f83614164p+6, 0x1.7a2f3d90948f1666p+6,
|
|
|
375 0x1.5cd116ee4c088c3ap+5, 0x1.1361e3eb6e3cc20ap+2
|
|
|
376 ];
|
|
|
377
|
|
|
378 const real Q1[] = [ 0x1.3a4ce1406cea98fap-13, 0x1.f45332623335cda2p-7,
|
|
|
379 0x1.98f28bbd4b98db1p-2, 0x1.ec3b24f9c698091cp+1, 0x1.1cc56ecda7cf58e4p+4,
|
|
|
380 0x1.92c6f7376bf8c058p+5, 0x1.4154c25aa47519b4p+6, 0x1.1b321d3b927849eap+6,
|
|
|
381 0x1.403a5f5a4ce7b202p+4, 1.0
|
|
|
382 ];
|
|
|
383
|
|
|
384 const real P2[] = [ 0x1.8c124a850116a6d8p-21, 0x1.534abda3c2fb90bap-13,
|
|
|
385 0x1.29a055ec93a4718cp-7, 0x1.6468e98aad6dd474p-3, 0x1.3dab2ef4c67a601cp+0,
|
|
|
386 0x1.e1fb3a1e70c67464p+1, 0x1.b6cce8035ff57b02p+2, 0x1.9f4c9e749ff35f62p+1
|
|
|
387 ];
|
|
|
388
|
|
|
389 const real Q2[] = [ 0x1.af03f4fc0655e006p-21, 0x1.713192048d11fb2p-13,
|
|
|
390 0x1.4357e5bbf5fef536p-7, 0x1.7fdac8749985d43cp-3, 0x1.4a080c813a2d8e84p+0,
|
|
|
391 0x1.c3a4b423cdb41bdap+1, 0x1.8160694e24b5557ap+2, 1.0
|
|
|
392 ];
|
|
|
393
|
|
|
394 const real P3[] = [ -0x1.55da447ae3806168p-34, -0x1.145635641f8778a6p-24,
|
|
|
395 -0x1.abf46d6b48040128p-17, -0x1.7da550945da790fcp-11, -0x1.aa0b2a31157775fap-8,
|
|
|
396 0x1.b11d97522eed26bcp-3, 0x1.1106d22f9ae89238p+1, 0x1.029a358e1e630f64p+1
|
|
|
397 ];
|
|
|
398
|
|
|
399 const real Q3[] = [ -0x1.74022dd5523e6f84p-34, -0x1.2cb60d61e29ee836p-24,
|
|
|
400 -0x1.d19e6ec03a85e556p-17, -0x1.9ea2a7b4422f6502p-11, -0x1.c54b1e852f107162p-8,
|
|
|
401 0x1.e05268dd3c07989ep-3, 0x1.239c6aff14afbf82p+1, 1.0
|
|
|
402 ];
|
|
|
403
|
|
|
404 if(p<=0.0L || p>=1.0L) {
|
|
|
405 if (p == 0.0L) {
|
|
|
406 return -real.infinity;
|
|
|
407 }
|
|
|
408 if( p == 1.0L ) {
|
|
|
409 return real.infinity;
|
|
|
410 }
|
|
|
411 return NaN(TANGO_NAN.NORMALDISTRIBUTION_INV_DOMAIN);
|
|
|
412 }
|
|
|
413 int code = 1;
|
|
|
414 real y = p;
|
|
|
415 if( y > (1.0L - EXP_2) ) {
|
|
|
416 y = 1.0L - y;
|
|
|
417 code = 0;
|
|
|
418 }
|
|
|
419
|
|
|
420 real x, z, y2, x0, x1;
|
|
|
421
|
|
|
422 if ( y > EXP_2 ) {
|
|
|
423 y = y - 0.5L;
|
|
|
424 y2 = y * y;
|
|
|
425 x = y + y * (y2 * rationalPoly( y2, P0, Q0));
|
|
|
426 return x * SQRT2PI;
|
|
|
427 }
|
|
|
428
|
|
|
429 x = sqrt( -2.0L * log(y) );
|
|
|
430 x0 = x - log(x)/x;
|
|
|
431 z = 1.0L/x;
|
|
|
432 if ( x < 8.0L ) {
|
|
|
433 x1 = z * rationalPoly( z, P1, Q1);
|
|
|
434 } else if( x < 32.0L ) {
|
|
|
435 x1 = z * rationalPoly( z, P2, Q2);
|
|
|
436 } else {
|
|
|
437 x1 = z * rationalPoly( z, P3, Q3);
|
|
|
438 }
|
|
|
439 x = x0 - x1;
|
|
|
440 if ( code != 0 ) {
|
|
|
441 x = -x;
|
|
|
442 }
|
|
|
443 return x;
|
|
|
444 }
|
|
|
445
|
|
|
446 }
|
|
|
447
|
|
|
448
|
|
|
449 debug(UnitTest) {
|
|
|
450 unittest {
|
|
|
451 // TODO: Use verified test points.
|
|
|
452 // The values below are from Excel 2003.
|
|
|
453 assert(fabs(normalDistributionInvImpl(0.001) - (-3.09023230616779))< 0.00000000000005);
|
|
|
454 assert(fabs(normalDistributionInvImpl(1e-50) - (-14.9333375347885))< 0.00000000000005);
|
|
|
455 assert(feqrel(normalDistributionInvImpl(0.999), -normalDistributionInvImpl(0.001))>real.mant_dig-6);
|
|
|
456
|
|
|
457 // Excel 2003 gets all the following values wrong!
|
|
|
458 assert(normalDistributionInvImpl(0.0)==-real.infinity);
|
|
|
459 assert(normalDistributionInvImpl(1.0)==real.infinity);
|
|
|
460 assert(normalDistributionInvImpl(0.5)==0);
|
|
|
461 // (Excel 2003 returns norminv(p) = -30 for all p < 1e-200).
|
|
|
462 // The value tested here is the one the function returned in Jan 2006.
|
|
|
463 real unknown1 = normalDistributionInvImpl(1e-250L);
|
|
|
464 assert( fabs(unknown1 -(-33.79958617269L) ) < 0.00000005);
|
|
|
465 }
|
|
|
466 } |
