- Generated on Wed Feb 19 2014 14:47:01 for POK by
1.7.6.1
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POK
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00001 /* 00002 * POK header 00003 * 00004 * The following file is a part of the POK project. Any modification should 00005 * made according to the POK licence. You CANNOT use this file or a part of 00006 * this file is this part of a file for your own project 00007 * 00008 * For more information on the POK licence, please see our LICENCE FILE 00009 * 00010 * Please follow the coding guidelines described in doc/CODING_GUIDELINES 00011 * 00012 * Copyright (c) 2007-2009 POK team 00013 * 00014 * Created by julien on Fri Jan 30 14:41:34 2009 00015 */ 00016 00017 /* @(#)s_erf.c 5.1 93/09/24 */ 00018 /* 00019 * ==================================================== 00020 * Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved. 00021 * 00022 * Developed at SunPro, a Sun Microsystems, Inc. business. 00023 * Permission to use, copy, modify, and distribute this 00024 * software is freely granted, provided that this notice 00025 * is preserved. 00026 * ==================================================== 00027 */ 00028 00029 /* double erf(double x) 00030 * double erfc(double x) 00031 * x 00032 * 2 |\ 00033 * erf(x) = --------- | exp(-t*t)dt 00034 * sqrt(pi) \| 00035 * 0 00036 * 00037 * erfc(x) = 1-erf(x) 00038 * Note that 00039 * erf(-x) = -erf(x) 00040 * erfc(-x) = 2 - erfc(x) 00041 * 00042 * Method: 00043 * 1. For |x| in [0, 0.84375] 00044 * erf(x) = x + x*R(x^2) 00045 * erfc(x) = 1 - erf(x) if x in [-.84375,0.25] 00046 * = 0.5 + ((0.5-x)-x*R) if x in [0.25,0.84375] 00047 * where R = P/Q where P is an odd poly of degree 8 and 00048 * Q is an odd poly of degree 10. 00049 * -57.90 00050 * | R - (erf(x)-x)/x | <= 2 00051 * 00052 * 00053 * Remark. The formula is derived by noting 00054 * erf(x) = (2/sqrt(pi))*(x - x^3/3 + x^5/10 - x^7/42 + ....) 00055 * and that 00056 * 2/sqrt(pi) = 1.128379167095512573896158903121545171688 00057 * is close to one. The interval is chosen because the fix 00058 * point of erf(x) is near 0.6174 (i.e., erf(x)=x when x is 00059 * near 0.6174), and by some experiment, 0.84375 is chosen to 00060 * guarantee the error is less than one ulp for erf. 00061 * 00062 * 2. For |x| in [0.84375,1.25], let s = |x| - 1, and 00063 * c = 0.84506291151 rounded to single (24 bits) 00064 * erf(x) = sign(x) * (c + P1(s)/Q1(s)) 00065 * erfc(x) = (1-c) - P1(s)/Q1(s) if x > 0 00066 * 1+(c+P1(s)/Q1(s)) if x < 0 00067 * |P1/Q1 - (erf(|x|)-c)| <= 2**-59.06 00068 * Remark: here we use the taylor series expansion at x=1. 00069 * erf(1+s) = erf(1) + s*Poly(s) 00070 * = 0.845.. + P1(s)/Q1(s) 00071 * That is, we use rational approximation to approximate 00072 * erf(1+s) - (c = (single)0.84506291151) 00073 * Note that |P1/Q1|< 0.078 for x in [0.84375,1.25] 00074 * where 00075 * P1(s) = degree 6 poly in s 00076 * Q1(s) = degree 6 poly in s 00077 * 00078 * 3. For x in [1.25,1/0.35(~2.857143)], 00079 * erfc(x) = (1/x)*exp(-x*x-0.5625+R1/S1) 00080 * erf(x) = 1 - erfc(x) 00081 * where 00082 * R1(z) = degree 7 poly in z, (z=1/x^2) 00083 * S1(z) = degree 8 poly in z 00084 * 00085 * 4. For x in [1/0.35,28] 00086 * erfc(x) = (1/x)*exp(-x*x-0.5625+R2/S2) if x > 0 00087 * = 2.0 - (1/x)*exp(-x*x-0.5625+R2/S2) if -6<x<0 00088 * = 2.0 - tiny (if x <= -6) 00089 * erf(x) = sign(x)*(1.0 - erfc(x)) if x < 6, else 00090 * erf(x) = sign(x)*(1.0 - tiny) 00091 * where 00092 * R2(z) = degree 6 poly in z, (z=1/x^2) 00093 * S2(z) = degree 7 poly in z 00094 * 00095 * Note1: 00096 * To compute exp(-x*x-0.5625+R/S), let s be a single 00097 * precision number and s := x; then 00098 * -x*x = -s*s + (s-x)*(s+x) 00099 * exp(-x*x-0.5626+R/S) = 00100 * exp(-s*s-0.5625)*exp((s-x)*(s+x)+R/S); 00101 * Note2: 00102 * Here 4 and 5 make use of the asymptotic series 00103 * exp(-x*x) 00104 * erfc(x) ~ ---------- * ( 1 + Poly(1/x^2) ) 00105 * x*sqrt(pi) 00106 * We use rational approximation to approximate 00107 * g(s)=f(1/x^2) = log(erfc(x)*x) - x*x + 0.5625 00108 * Here is the error bound for R1/S1 and R2/S2 00109 * |R1/S1 - f(x)| < 2**(-62.57) 00110 * |R2/S2 - f(x)| < 2**(-61.52) 00111 * 00112 * 5. For inf > x >= 28 00113 * erf(x) = sign(x) *(1 - tiny) (raise inexact) 00114 * erfc(x) = tiny*tiny (raise underflow) if x > 0 00115 * = 2 - tiny if x<0 00116 * 00117 * 7. Special case: 00118 * erf(0) = 0, erf(inf) = 1, erf(-inf) = -1, 00119 * erfc(0) = 1, erfc(inf) = 0, erfc(-inf) = 2, 00120 * erfc/erf(NaN) is NaN 00121 */ 00122 00123 #ifdef POK_NEEDS_LIBMATH 00124 #include <libm.h> 00125 #include "math_private.h" 00126 00127 static const double 00128 tiny = 1e-300, 00129 half= 5.00000000000000000000e-01, /* 0x3FE00000, 0x00000000 */ 00130 one = 1.00000000000000000000e+00, /* 0x3FF00000, 0x00000000 */ 00131 two = 2.00000000000000000000e+00, /* 0x40000000, 0x00000000 */ 00132 /* c = (float)0.84506291151 */ 00133 erx = 8.45062911510467529297e-01, /* 0x3FEB0AC1, 0x60000000 */ 00134 /* 00135 * Coefficients for approximation to erf on [0,0.84375] 00136 */ 00137 efx = 1.28379167095512586316e-01, /* 0x3FC06EBA, 0x8214DB69 */ 00138 efx8= 1.02703333676410069053e+00, /* 0x3FF06EBA, 0x8214DB69 */ 00139 pp0 = 1.28379167095512558561e-01, /* 0x3FC06EBA, 0x8214DB68 */ 00140 pp1 = -3.25042107247001499370e-01, /* 0xBFD4CD7D, 0x691CB913 */ 00141 pp2 = -2.84817495755985104766e-02, /* 0xBF9D2A51, 0xDBD7194F */ 00142 pp3 = -5.77027029648944159157e-03, /* 0xBF77A291, 0x236668E4 */ 00143 pp4 = -2.37630166566501626084e-05, /* 0xBEF8EAD6, 0x120016AC */ 00144 qq1 = 3.97917223959155352819e-01, /* 0x3FD97779, 0xCDDADC09 */ 00145 qq2 = 6.50222499887672944485e-02, /* 0x3FB0A54C, 0x5536CEBA */ 00146 qq3 = 5.08130628187576562776e-03, /* 0x3F74D022, 0xC4D36B0F */ 00147 qq4 = 1.32494738004321644526e-04, /* 0x3F215DC9, 0x221C1A10 */ 00148 qq5 = -3.96022827877536812320e-06, /* 0xBED09C43, 0x42A26120 */ 00149 /* 00150 * Coefficients for approximation to erf in [0.84375,1.25] 00151 */ 00152 pa0 = -2.36211856075265944077e-03, /* 0xBF6359B8, 0xBEF77538 */ 00153 pa1 = 4.14856118683748331666e-01, /* 0x3FDA8D00, 0xAD92B34D */ 00154 pa2 = -3.72207876035701323847e-01, /* 0xBFD7D240, 0xFBB8C3F1 */ 00155 pa3 = 3.18346619901161753674e-01, /* 0x3FD45FCA, 0x805120E4 */ 00156 pa4 = -1.10894694282396677476e-01, /* 0xBFBC6398, 0x3D3E28EC */ 00157 pa5 = 3.54783043256182359371e-02, /* 0x3FA22A36, 0x599795EB */ 00158 pa6 = -2.16637559486879084300e-03, /* 0xBF61BF38, 0x0A96073F */ 00159 qa1 = 1.06420880400844228286e-01, /* 0x3FBB3E66, 0x18EEE323 */ 00160 qa2 = 5.40397917702171048937e-01, /* 0x3FE14AF0, 0x92EB6F33 */ 00161 qa3 = 7.18286544141962662868e-02, /* 0x3FB2635C, 0xD99FE9A7 */ 00162 qa4 = 1.26171219808761642112e-01, /* 0x3FC02660, 0xE763351F */ 00163 qa5 = 1.36370839120290507362e-02, /* 0x3F8BEDC2, 0x6B51DD1C */ 00164 qa6 = 1.19844998467991074170e-02, /* 0x3F888B54, 0x5735151D */ 00165 /* 00166 * Coefficients for approximation to erfc in [1.25,1/0.35] 00167 */ 00168 ra0 = -9.86494403484714822705e-03, /* 0xBF843412, 0x600D6435 */ 00169 ra1 = -6.93858572707181764372e-01, /* 0xBFE63416, 0xE4BA7360 */ 00170 ra2 = -1.05586262253232909814e+01, /* 0xC0251E04, 0x41B0E726 */ 00171 ra3 = -6.23753324503260060396e+01, /* 0xC04F300A, 0xE4CBA38D */ 00172 ra4 = -1.62396669462573470355e+02, /* 0xC0644CB1, 0x84282266 */ 00173 ra5 = -1.84605092906711035994e+02, /* 0xC067135C, 0xEBCCABB2 */ 00174 ra6 = -8.12874355063065934246e+01, /* 0xC0545265, 0x57E4D2F2 */ 00175 ra7 = -9.81432934416914548592e+00, /* 0xC023A0EF, 0xC69AC25C */ 00176 sa1 = 1.96512716674392571292e+01, /* 0x4033A6B9, 0xBD707687 */ 00177 sa2 = 1.37657754143519042600e+02, /* 0x4061350C, 0x526AE721 */ 00178 sa3 = 4.34565877475229228821e+02, /* 0x407B290D, 0xD58A1A71 */ 00179 sa4 = 6.45387271733267880336e+02, /* 0x40842B19, 0x21EC2868 */ 00180 sa5 = 4.29008140027567833386e+02, /* 0x407AD021, 0x57700314 */ 00181 sa6 = 1.08635005541779435134e+02, /* 0x405B28A3, 0xEE48AE2C */ 00182 sa7 = 6.57024977031928170135e+00, /* 0x401A47EF, 0x8E484A93 */ 00183 sa8 = -6.04244152148580987438e-02, /* 0xBFAEEFF2, 0xEE749A62 */ 00184 /* 00185 * Coefficients for approximation to erfc in [1/.35,28] 00186 */ 00187 rb0 = -9.86494292470009928597e-03, /* 0xBF843412, 0x39E86F4A */ 00188 rb1 = -7.99283237680523006574e-01, /* 0xBFE993BA, 0x70C285DE */ 00189 rb2 = -1.77579549177547519889e+01, /* 0xC031C209, 0x555F995A */ 00190 rb3 = -1.60636384855821916062e+02, /* 0xC064145D, 0x43C5ED98 */ 00191 rb4 = -6.37566443368389627722e+02, /* 0xC083EC88, 0x1375F228 */ 00192 rb5 = -1.02509513161107724954e+03, /* 0xC0900461, 0x6A2E5992 */ 00193 rb6 = -4.83519191608651397019e+02, /* 0xC07E384E, 0x9BDC383F */ 00194 sb1 = 3.03380607434824582924e+01, /* 0x403E568B, 0x261D5190 */ 00195 sb2 = 3.25792512996573918826e+02, /* 0x40745CAE, 0x221B9F0A */ 00196 sb3 = 1.53672958608443695994e+03, /* 0x409802EB, 0x189D5118 */ 00197 sb4 = 3.19985821950859553908e+03, /* 0x40A8FFB7, 0x688C246A */ 00198 sb5 = 2.55305040643316442583e+03, /* 0x40A3F219, 0xCEDF3BE6 */ 00199 sb6 = 4.74528541206955367215e+02, /* 0x407DA874, 0xE79FE763 */ 00200 sb7 = -2.24409524465858183362e+01; /* 0xC03670E2, 0x42712D62 */ 00201 00202 double 00203 erf(double x) 00204 { 00205 int32_t hx,ix,i; 00206 double R,S,P,Q,s,y,z,r; 00207 GET_HIGH_WORD(hx,x); 00208 ix = hx&0x7fffffff; 00209 if(ix>=0x7ff00000) { /* erf(nan)=nan */ 00210 i = ((uint32_t)hx>>31)<<1; 00211 return (double)(1-i)+one/x; /* erf(+-inf)=+-1 */ 00212 } 00213 00214 if(ix < 0x3feb0000) { /* |x|<0.84375 */ 00215 if(ix < 0x3e300000) { /* |x|<2**-28 */ 00216 if (ix < 0x00800000) 00217 return 0.125*(8.0*x+efx8*x); /*avoid underflow */ 00218 return x + efx*x; 00219 } 00220 z = x*x; 00221 r = pp0+z*(pp1+z*(pp2+z*(pp3+z*pp4))); 00222 s = one+z*(qq1+z*(qq2+z*(qq3+z*(qq4+z*qq5)))); 00223 y = r/s; 00224 return x + x*y; 00225 } 00226 if(ix < 0x3ff40000) { /* 0.84375 <= |x| < 1.25 */ 00227 s = fabs(x)-one; 00228 P = pa0+s*(pa1+s*(pa2+s*(pa3+s*(pa4+s*(pa5+s*pa6))))); 00229 Q = one+s*(qa1+s*(qa2+s*(qa3+s*(qa4+s*(qa5+s*qa6))))); 00230 if(hx>=0) return erx + P/Q; else return -erx - P/Q; 00231 } 00232 if (ix >= 0x40180000) { /* inf>|x|>=6 */ 00233 if(hx>=0) return one-tiny; else return tiny-one; 00234 } 00235 x = fabs(x); 00236 s = one/(x*x); 00237 if(ix< 0x4006DB6E) { /* |x| < 1/0.35 */ 00238 R=ra0+s*(ra1+s*(ra2+s*(ra3+s*(ra4+s*( 00239 ra5+s*(ra6+s*ra7)))))); 00240 S=one+s*(sa1+s*(sa2+s*(sa3+s*(sa4+s*( 00241 sa5+s*(sa6+s*(sa7+s*sa8))))))); 00242 } else { /* |x| >= 1/0.35 */ 00243 R=rb0+s*(rb1+s*(rb2+s*(rb3+s*(rb4+s*( 00244 rb5+s*rb6))))); 00245 S=one+s*(sb1+s*(sb2+s*(sb3+s*(sb4+s*( 00246 sb5+s*(sb6+s*sb7)))))); 00247 } 00248 z = x; 00249 SET_LOW_WORD(z,0); 00250 r = __ieee754_exp(-z*z-0.5625)*__ieee754_exp((z-x)*(z+x)+R/S); 00251 if(hx>=0) return one-r/x; else return r/x-one; 00252 } 00253 00254 double 00255 erfc(double x) 00256 { 00257 int32_t hx,ix; 00258 double R,S,P,Q,s,y,z,r; 00259 GET_HIGH_WORD(hx,x); 00260 ix = hx&0x7fffffff; 00261 if(ix>=0x7ff00000) { /* erfc(nan)=nan */ 00262 /* erfc(+-inf)=0,2 */ 00263 return (double)(((uint32_t)hx>>31)<<1)+one/x; 00264 } 00265 00266 if(ix < 0x3feb0000) { /* |x|<0.84375 */ 00267 if(ix < 0x3c700000) /* |x|<2**-56 */ 00268 return one-x; 00269 z = x*x; 00270 r = pp0+z*(pp1+z*(pp2+z*(pp3+z*pp4))); 00271 s = one+z*(qq1+z*(qq2+z*(qq3+z*(qq4+z*qq5)))); 00272 y = r/s; 00273 if(hx < 0x3fd00000) { /* x<1/4 */ 00274 return one-(x+x*y); 00275 } else { 00276 r = x*y; 00277 r += (x-half); 00278 return half - r ; 00279 } 00280 } 00281 if(ix < 0x3ff40000) { /* 0.84375 <= |x| < 1.25 */ 00282 s = fabs(x)-one; 00283 P = pa0+s*(pa1+s*(pa2+s*(pa3+s*(pa4+s*(pa5+s*pa6))))); 00284 Q = one+s*(qa1+s*(qa2+s*(qa3+s*(qa4+s*(qa5+s*qa6))))); 00285 if(hx>=0) { 00286 z = one-erx; return z - P/Q; 00287 } else { 00288 z = erx+P/Q; return one+z; 00289 } 00290 } 00291 if (ix < 0x403c0000) { /* |x|<28 */ 00292 x = fabs(x); 00293 s = one/(x*x); 00294 if(ix< 0x4006DB6D) { /* |x| < 1/.35 ~ 2.857143*/ 00295 R=ra0+s*(ra1+s*(ra2+s*(ra3+s*(ra4+s*( 00296 ra5+s*(ra6+s*ra7)))))); 00297 S=one+s*(sa1+s*(sa2+s*(sa3+s*(sa4+s*( 00298 sa5+s*(sa6+s*(sa7+s*sa8))))))); 00299 } else { /* |x| >= 1/.35 ~ 2.857143 */ 00300 if(hx<0&&ix>=0x40180000) return two-tiny;/* x < -6 */ 00301 R=rb0+s*(rb1+s*(rb2+s*(rb3+s*(rb4+s*( 00302 rb5+s*rb6))))); 00303 S=one+s*(sb1+s*(sb2+s*(sb3+s*(sb4+s*( 00304 sb5+s*(sb6+s*sb7)))))); 00305 } 00306 z = x; 00307 SET_LOW_WORD(z,0); 00308 r = __ieee754_exp(-z*z-0.5625)* 00309 __ieee754_exp((z-x)*(z+x)+R/S); 00310 if(hx>0) return r/x; else return two-r/x; 00311 } else { 00312 if(hx>0) return tiny*tiny; else return two-tiny; 00313 } 00314 } 00315 #endif
1.7.6.1