tango.math.ErrorFunction source code

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 =  &#915;
14  *  INTEGRAL = &#8747;
15  *  INTEGRATE = $(BIG &#8747;<sub>$(SMALL $1)</sub><sup>$2</sup>)
16  *  POWER = $1<sup>$2</sup>
17  *  BIGSUM = $(BIG &Sigma; <sup>$2</sup><sub>$(SMALL $1)</sub>)
18  *  CHOOSE = $(BIG &#40;) <sup>$(SMALL $1)</sup><sub>$(SMALL $2)</sub> $(BIG &#41;)
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 enum real SQRT2PI = 0x1.40d931ff62705966p+1L;    // 2.5066282746310005024
37 enum 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 __gshared immutable 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 __gshared immutable 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 __gshared immutable real [] R = [ 0x1.b9f6d8b78e22459ep-6, 0x1.1b84686b0a4ea43ap-1,
61    0x1.b8f6aebe96000c2ap+1, 0x1.cb1dbedac27c8ec2p+2, 0x1.cf885f8f572a4c14p+1
62 ];
63 
64 __gshared immutable 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 __gshared immutable 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 __gshared immutable 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)(&pi;)
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)(&pi;)
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     enum real erfc0_250 = 0.723663330078125 + 1.0279753638067014931732235184287934646022E-5;
205     enum real erfc0_375 = 0.5958709716796875 + 1.2118885490201676174914080878232469565953E-5;
206     enum real erfc0_500 = 0.4794921875 + 7.9346869534623172533461080354712635484242E-6;
207     enum real erfc0_625 = 0.3767547607421875 + 4.3570693945275513594941232097252997287766E-6;
208     enum real erfc0_750 = 0.2888336181640625 + 1.0748182422368401062165408589222625794046E-5;
209     enum real erfc0_875 = 0.215911865234375 + 1.3073705765341685464282101150637224028267E-5;
210     enum real erfc1_000 = 0.15728759765625 + 1.1609394035130658779364917390740703933002E-5;
211     enum real erfc1_125 = 0.111602783203125 + 8.9850951672359304215530728365232161564636E-6;
212 
213     enum 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     enum real M = 32768.0;
262     enum 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) &pi; $(INTEGRAL -$(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 __gshared immutable 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 __gshared immutable 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 __gshared immutable 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 __gshared immutable 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 __gshared immutable 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 __gshared immutable 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 __gshared immutable 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 __gshared immutable 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 }