Mercurial > projects > ldc
diff tango/tango/math/ErrorFunction.d @ 132:1700239cab2e trunk
[svn r136] MAJOR UNSTABLE UPDATE!!!
Initial commit after moving to Tango instead of Phobos.
Lots of bugfixes...
This build is not suitable for most things.
author | lindquist |
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date | Fri, 11 Jan 2008 17:57:40 +0100 |
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--- /dev/null Thu Jan 01 00:00:00 1970 +0000 +++ b/tango/tango/math/ErrorFunction.d Fri Jan 11 17:57:40 2008 +0100 @@ -0,0 +1,466 @@ +/** + * Error Functions and Normal Distribution. + * + * Copyright: Copyright (C) 1984, 1995, 2000 Stephen L. Moshier + * Code taken from the Cephes Math Library Release 2.3: January, 1995 + * License: BSD style: $(LICENSE) + * Authors: Stephen L. Moshier, ported to D by Don Clugston + */ +/** + * Macros: + * NAN = $(RED NAN) + * SUP = <span style="vertical-align:super;font-size:smaller">$0</span> + * GAMMA = Γ + * INTEGRAL = ∫ + * INTEGRATE = $(BIG ∫<sub>$(SMALL $1)</sub><sup>$2</sup>) + * POWER = $1<sup>$2</sup> + * BIGSUM = $(BIG Σ <sup>$2</sup><sub>$(SMALL $1)</sub>) + * CHOOSE = $(BIG () <sup>$(SMALL $1)</sup><sub>$(SMALL $2)</sub> $(BIG )) + * TABLE_SV = <table border=1 cellpadding=4 cellspacing=0> + * <caption>Special Values</caption> + * $0</table> + * SVH = $(TR $(TH $1) $(TH $2)) + * SV = $(TR $(TD $1) $(TD $2)) + */ +module tango.math.ErrorFunction; + +import tango.math.Math; +import tango.math.IEEE; // only required for unit tests + +version(Windows) { // Some tests only pass on DMD Windows + version(DigitalMars) { + version = FailsOnLinux; +} +} + +const real SQRT2PI = 0x1.40d931ff62705966p+1L; // 2.5066282746310005024 +const real EXP_2 = 0.13533528323661269189L; /* exp(-2) */ + +private { + +/* erfc(x) = exp(-x^2) P(1/x)/Q(1/x) + 1/8 <= 1/x <= 1 + Peak relative error 5.8e-21 */ +const real [] P = [ -0x1.30dfa809b3cc6676p-17, 0x1.38637cd0913c0288p+18, + 0x1.2f015e047b4476bp+22, 0x1.24726f46aa9ab08p+25, 0x1.64b13c6395dc9c26p+27, + 0x1.294c93046ad55b5p+29, 0x1.5962a82f92576dap+30, 0x1.11a709299faba04ap+31, + 0x1.11028065b087be46p+31, 0x1.0d8ef40735b097ep+30 +]; + +const real [] Q = [ 0x1.14d8e2a72dec49f4p+19, 0x1.0c880ff467626e1p+23, + 0x1.04417ef060b58996p+26, 0x1.404e61ba86df4ebap+28, 0x1.0f81887bc82b873ap+30, + 0x1.4552a5e39fb49322p+31, 0x1.11779a0ceb2a01cep+32, 0x1.3544dd691b5b1d5cp+32, + 0x1.a91781f12251f02ep+31, 0x1.0d8ef3da605a1c86p+30, 1.0 +]; + + +/* erfc(x) = exp(-x^2) 1/x R(1/x^2) / S(1/x^2) + 1/128 <= 1/x < 1/8 + Peak relative error 1.9e-21 */ +const real [] R = [ 0x1.b9f6d8b78e22459ep-6, 0x1.1b84686b0a4ea43ap-1, + 0x1.b8f6aebe96000c2ap+1, 0x1.cb1dbedac27c8ec2p+2, 0x1.cf885f8f572a4c14p+1 +]; + +const real [] S = [ + 0x1.87ae3cae5f65eb5ep-5, 0x1.01616f266f306d08p+0, 0x1.a4abe0411eed6c22p+2, + 0x1.eac9ce3da600abaap+3, 0x1.5752a9ac2faebbccp+3, 1.0 +]; + +/* erf(x) = x P(x^2)/Q(x^2) + 0 <= x <= 1 + Peak relative error 7.6e-23 */ +const real [] T = [ 0x1.0da01654d757888cp+20, 0x1.2eb7497bc8b4f4acp+17, + 0x1.79078c19530f72a8p+15, 0x1.4eaf2126c0b2c23p+11, 0x1.1f2ea81c9d272a2ep+8, + 0x1.59ca6e2d866e625p+2, 0x1.c188e0b67435faf4p-4 +]; + +const real [] U = [ 0x1.dde6025c395ae34ep+19, 0x1.c4bc8b6235df35aap+18, + 0x1.8465900e88b6903ap+16, 0x1.855877093959ffdp+13, 0x1.e5c44395625ee358p+9, + 0x1.6a0fed103f1c68a6p+5, 1.0 +]; + +} + +/** + * Complementary error function + * + * erfc(x) = 1 - erf(x), and has high relative accuracy for + * values of x far from zero. (For values near zero, use erf(x)). + * + * 1 - erf(x) = 2/ $(SQRT)(π) + * $(INTEGRAL x, $(INFINITY)) exp( - $(POWER t, 2)) dt + * + * + * For small x, erfc(x) = 1 - erf(x); otherwise rational + * approximations are computed. + * + * A special function expx2(x) is used to suppress error amplification + * in computing exp(-x^2). + */ +real erfc(real a) +{ + if (a == real.infinity) + return 0.0; + if (a == -real.infinity) + return 2.0; + + real x; + + if (a < 0.0L ) + x = -a; + else + x = a; + if (x < 1.0) + return 1.0 - erf(a); + + real z = -a * a; + + if (z < -MAXLOG){ +// mtherr( "erfcl", UNDERFLOW ); + if (a < 0) return 2.0; + else return 0.0; + } + + /* Compute z = exp(z). */ + z = expx2(a, -1); + real y = 1.0/x; + + real p, q; + + if( x < 8.0 ) y = z * rationalPoly(y, P, Q); + else y = z * y * rationalPoly(y * y, R, S); + + if (a < 0.0L) + y = 2.0L - y; + + if (y == 0.0) { +// mtherr( "erfcl", UNDERFLOW ); + if (a < 0) return 2.0; + else return 0.0; + } + + return y; +} + + +private { +/* Exponentially scaled erfc function + exp(x^2) erfc(x) + valid for x > 1. + Use with normalDistribution and expx2. */ + +real erfce(real x) +{ + real p, q; + + real y = 1.0/x; + + if (x < 8.0) { + return rationalPoly( y, P, Q); + } else { + return y * rationalPoly(y*y, R, S); + } +} + +} + +/** + * Error function + * + * The integral is + * + * erf(x) = 2/ $(SQRT)(π) + * $(INTEGRAL 0, x) exp( - $(POWER t, 2)) dt + * + * The magnitude of x is limited to about 106.56 for IEEE 80-bit + * arithmetic; 1 or -1 is returned outside this range. + * + * For 0 <= |x| < 1, a rational polynomials are used; otherwise + * erf(x) = 1 - erfc(x). + * + * ACCURACY: + * Relative error: + * arithmetic domain # trials peak rms + * IEEE 0,1 50000 2.0e-19 5.7e-20 + */ +real erf(real x) +{ + if (x == 0.0) + return x; // deal with negative zero + if (x == -real.infinity) + return -1.0; + if (x == real.infinity) + return 1.0; + if (abs(x) > 1.0L) + return 1.0L - erfc(x); + + real z = x * x; + return x * rationalPoly(z, T, U); +} + +debug(UnitTest) { +unittest { + // High resolution test points. + const real erfc0_250 = 0.723663330078125 + 1.0279753638067014931732235184287934646022E-5; + const real erfc0_375 = 0.5958709716796875 + 1.2118885490201676174914080878232469565953E-5; + const real erfc0_500 = 0.4794921875 + 7.9346869534623172533461080354712635484242E-6; + const real erfc0_625 = 0.3767547607421875 + 4.3570693945275513594941232097252997287766E-6; + const real erfc0_750 = 0.2888336181640625 + 1.0748182422368401062165408589222625794046E-5; + const real erfc0_875 = 0.215911865234375 + 1.3073705765341685464282101150637224028267E-5; + const real erfc1_000 = 0.15728759765625 + 1.1609394035130658779364917390740703933002E-5; + const real erfc1_125 = 0.111602783203125 + 8.9850951672359304215530728365232161564636E-6; + + const real erf0_875 = (1-0.215911865234375) - 1.3073705765341685464282101150637224028267E-5; + + + assert(feqrel(erfc(0.250L), erfc0_250 )>=real.mant_dig-1); + assert(feqrel(erfc(0.375L), erfc0_375 )>=real.mant_dig-0); + assert(feqrel(erfc(0.500L), erfc0_500 )>=real.mant_dig-1); + assert(feqrel(erfc(0.625L), erfc0_625 )>=real.mant_dig-1); + assert(feqrel(erfc(0.750L), erfc0_750 )>=real.mant_dig-1); + assert(feqrel(erfc(0.875L), erfc0_875 )>=real.mant_dig-4); + version(FailsOnLinux) assert(feqrel(erfc(1.000L), erfc1_000 )>=real.mant_dig-0); + assert(feqrel(erfc(1.125L), erfc1_125 )>=real.mant_dig-2); + assert(feqrel(erf(0.875L), erf0_875 )>=real.mant_dig-1); + // The DMC implementation of erfc() fails this next test (just) + assert(feqrel(erfc(4.1L),0.67000276540848983727e-8L)>=real.mant_dig-4); + + assert(isIdentical(erf(0.0),0.0)); + assert(isIdentical(erf(-0.0),-0.0)); + assert(erf(real.infinity) == 1.0); + assert(erf(-real.infinity) == -1.0); + assert(isIdentical(erf(NaN(0xDEF)),NaN(0xDEF))); + assert(isIdentical(erfc(NaN(0xDEF)),NaN(0xDEF))); + assert(isIdentical(erfc(real.infinity),0.0)); + assert(erfc(-real.infinity) == 2.0); + assert(erfc(0) == 1.0); +} +} + +/* + * Exponential of squared argument + * + * Computes y = exp(x*x) while suppressing error amplification + * that would ordinarily arise from the inexactness of the + * exponential argument x*x. + * + * If sign < 0, the result is inverted; i.e., y = exp(-x*x) . + * + * ACCURACY: + * Relative error: + * arithmetic domain # trials peak rms + * IEEE -106.566, 106.566 10^5 1.6e-19 4.4e-20 + */ + +real expx2(real x, int sign) +{ + /* + Cephes Math Library Release 2.9: June, 2000 + Copyright 2000 by Stephen L. Moshier + */ + const real M = 32768.0; + const real MINV = 3.0517578125e-5L; + + x = abs(x); + if (sign < 0) + x = -x; + + /* Represent x as an exact multiple of M plus a residual. + M is a power of 2 chosen so that exp(m * m) does not overflow + or underflow and so that |x - m| is small. */ + real m = MINV * floor(M * x + 0.5L); + real f = x - m; + + /* x^2 = m^2 + 2mf + f^2 */ + real u = m * m; + real u1 = 2 * m * f + f * f; + + if (sign < 0) { + u = -u; + u1 = -u1; + } + + if ((u+u1) > MAXLOG) + return real.infinity; + + /* u is exact, u1 is small. */ + return exp(u) * exp(u1); +} + + +package { +/* +Computes the normal distribution function. + +The normal (or Gaussian, or bell-shaped) distribution is +defined as: + +normalDist(x) = 1/$(SQRT) π $(INTEGRAL -$(INFINITY), x) exp( - $(POWER t, 2)/2) dt + = 0.5 + 0.5 * erf(x/sqrt(2)) + = 0.5 * erfc(- x/sqrt(2)) + +To maintain accuracy at high values of x, use +normalDistribution(x) = 1 - normalDistribution(-x). + +Accuracy: +Within a few bits of machine resolution over the entire +range. + +References: +$(LINK http://www.netlib.org/cephes/ldoubdoc.html), +G. Marsaglia, "Evaluating the Normal Distribution", +Journal of Statistical Software <b>11</b>, (July 2004). +*/ +real normalDistributionImpl(real a) +{ + real x = a * SQRT1_2; + real z = abs(x); + + if( z < 1.0 ) + return 0.5L + 0.5L * erf(x); + else { + /* See below for erfce. */ + real y = 0.5L * erfce(z); + /* Multiply by exp(-x^2 / 2) */ + z = expx2(a, -1); + y = y * sqrt(z); + if( x > 0.0L ) + y = 1.0L - y; + return y; + } +} + +} + +debug(UnitTest) { +unittest { +assert(fabs(normalDistributionImpl(1L) - (0.841344746068543))< 0.0000000000000005); +assert(isIdentical(normalDistributionImpl(NaN(0x325)), NaN(0x325))); +} +} + +package { +/* + * Inverse of Normal distribution function + * + * Returns the argument, x, for which the area under the + * Normal probability density function (integrated from + * minus infinity to x) is equal to p. + * + * For small arguments 0 < p < exp(-2), the program computes + * z = sqrt( -2 log(p) ); then the approximation is + * x = z - log(z)/z - (1/z) P(1/z) / Q(1/z) . + * For larger arguments, x/sqrt(2 pi) = w + w^3 R(w^2)/S(w^2)) , + * where w = p - 0.5 . + */ +real normalDistributionInvImpl(real p) +in { + assert(p>=0.0L && p<=1.0L, "Domain error"); +} +body +{ +const real P0[] = [ -0x1.758f4d969484bfdcp-7, 0x1.53cee17a59259dd2p-3, + -0x1.ea01e4400a9427a2p-1, 0x1.61f7504a0105341ap+1, -0x1.09475a594d0399f6p+2, + 0x1.7c59e7a0df99e3e2p+1, -0x1.87a81da52edcdf14p-1, 0x1.1fb149fd3f83600cp-7 +]; + +const real Q0[] = [ -0x1.64b92ae791e64bb2p-7, 0x1.7585c7d597298286p-3, + -0x1.40011be4f7591ce6p+0, 0x1.1fc067d8430a425ep+2, -0x1.21008ffb1e7ccdf2p+3, + 0x1.3d1581cf9bc12fccp+3, -0x1.53723a89fd8f083cp+2, 1.0 +]; + +const real P1[] = [ 0x1.20ceea49ea142f12p-13, 0x1.cbe8a7267aea80bp-7, + 0x1.79fea765aa787c48p-2, 0x1.d1f59faa1f4c4864p+1, 0x1.1c22e426a013bb96p+4, + 0x1.a8675a0c51ef3202p+5, 0x1.75782c4f83614164p+6, 0x1.7a2f3d90948f1666p+6, + 0x1.5cd116ee4c088c3ap+5, 0x1.1361e3eb6e3cc20ap+2 +]; + +const real Q1[] = [ 0x1.3a4ce1406cea98fap-13, 0x1.f45332623335cda2p-7, + 0x1.98f28bbd4b98db1p-2, 0x1.ec3b24f9c698091cp+1, 0x1.1cc56ecda7cf58e4p+4, + 0x1.92c6f7376bf8c058p+5, 0x1.4154c25aa47519b4p+6, 0x1.1b321d3b927849eap+6, + 0x1.403a5f5a4ce7b202p+4, 1.0 +]; + +const real P2[] = [ 0x1.8c124a850116a6d8p-21, 0x1.534abda3c2fb90bap-13, + 0x1.29a055ec93a4718cp-7, 0x1.6468e98aad6dd474p-3, 0x1.3dab2ef4c67a601cp+0, + 0x1.e1fb3a1e70c67464p+1, 0x1.b6cce8035ff57b02p+2, 0x1.9f4c9e749ff35f62p+1 +]; + +const real Q2[] = [ 0x1.af03f4fc0655e006p-21, 0x1.713192048d11fb2p-13, + 0x1.4357e5bbf5fef536p-7, 0x1.7fdac8749985d43cp-3, 0x1.4a080c813a2d8e84p+0, + 0x1.c3a4b423cdb41bdap+1, 0x1.8160694e24b5557ap+2, 1.0 +]; + +const real P3[] = [ -0x1.55da447ae3806168p-34, -0x1.145635641f8778a6p-24, + -0x1.abf46d6b48040128p-17, -0x1.7da550945da790fcp-11, -0x1.aa0b2a31157775fap-8, + 0x1.b11d97522eed26bcp-3, 0x1.1106d22f9ae89238p+1, 0x1.029a358e1e630f64p+1 +]; + +const real Q3[] = [ -0x1.74022dd5523e6f84p-34, -0x1.2cb60d61e29ee836p-24, + -0x1.d19e6ec03a85e556p-17, -0x1.9ea2a7b4422f6502p-11, -0x1.c54b1e852f107162p-8, + 0x1.e05268dd3c07989ep-3, 0x1.239c6aff14afbf82p+1, 1.0 +]; + + if(p<=0.0L || p>=1.0L) { + if (p == 0.0L) { + return -real.infinity; + } + if( p == 1.0L ) { + return real.infinity; + } + return NaN(TANGO_NAN.NORMALDISTRIBUTION_INV_DOMAIN); + } + int code = 1; + real y = p; + if( y > (1.0L - EXP_2) ) { + y = 1.0L - y; + code = 0; + } + + real x, z, y2, x0, x1; + + if ( y > EXP_2 ) { + y = y - 0.5L; + y2 = y * y; + x = y + y * (y2 * rationalPoly( y2, P0, Q0)); + return x * SQRT2PI; + } + + x = sqrt( -2.0L * log(y) ); + x0 = x - log(x)/x; + z = 1.0L/x; + if ( x < 8.0L ) { + x1 = z * rationalPoly( z, P1, Q1); + } else if( x < 32.0L ) { + x1 = z * rationalPoly( z, P2, Q2); + } else { + x1 = z * rationalPoly( z, P3, Q3); + } + x = x0 - x1; + if ( code != 0 ) { + x = -x; + } + return x; +} + +} + + +debug(UnitTest) { +unittest { + // TODO: Use verified test points. + // The values below are from Excel 2003. +assert(fabs(normalDistributionInvImpl(0.001) - (-3.09023230616779))< 0.00000000000005); +assert(fabs(normalDistributionInvImpl(1e-50) - (-14.9333375347885))< 0.00000000000005); +assert(feqrel(normalDistributionInvImpl(0.999), -normalDistributionInvImpl(0.001))>real.mant_dig-6); + +// Excel 2003 gets all the following values wrong! +assert(normalDistributionInvImpl(0.0)==-real.infinity); +assert(normalDistributionInvImpl(1.0)==real.infinity); +assert(normalDistributionInvImpl(0.5)==0); +// (Excel 2003 returns norminv(p) = -30 for all p < 1e-200). +// The value tested here is the one the function returned in Jan 2006. +real unknown1 = normalDistributionInvImpl(1e-250L); +assert( fabs(unknown1 -(-33.79958617269L) ) < 0.00000005); +} +} \ No newline at end of file