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  1. /* Copyright (C) 1994 DJ Delorie, see COPYING.DJ for details */
  2. /* @(#)s_erf.c 5.1 93/09/24 */
  3. /*
  4.  * ====================================================
  5.  * Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved.
  6.  *
  7.  * Developed at SunPro, a Sun Microsystems, Inc. business.
  8.  * Permission to use, copy, modify, and distribute this
  9.  * software is freely granted, provided that this notice
  10.  * is preserved.
  11.  * ====================================================
  12.  */
  13.  
  14. #if defined(LIBM_SCCS) && !defined(lint)
  15. static char rcsid[] = "$Id: s_erf.c,v 1.6 1994/08/18 23:06:36 jtc Exp $";
  16. #endif
  17.  
  18. /* double erf(double x)
  19.  * double erfc(double x)
  20.  *                           x
  21.  *                    2      |\
  22.  *     erf(x)  =  ---------  | exp(-t*t)dt
  23.  *                 sqrt(pi) \|
  24.  *                           0
  25.  *
  26.  *     erfc(x) =  1-erf(x)
  27.  *  Note that
  28.  *              erf(-x) = -erf(x)
  29.  *              erfc(-x) = 2 - erfc(x)
  30.  *
  31.  * Method:
  32.  *      1. For |x| in [0, 0.84375]
  33.  *          erf(x)  = x + x*R(x^2)
  34.  *          erfc(x) = 1 - erf(x)           if x in [-.84375,0.25]
  35.  *                  = 0.5 + ((0.5-x)-x*R)  if x in [0.25,0.84375]
  36.  *         where R = P/Q where P is an odd poly of degree 8 and
  37.  *         Q is an odd poly of degree 10.
  38.  *                                               -57.90
  39.  *                      | R - (erf(x)-x)/x | <= 2
  40.  *     
  41.  *
  42.  *         Remark. The formula is derived by noting
  43.  *          erf(x) = (2/sqrt(pi))*(x - x^3/3 + x^5/10 - x^7/42 + ....)
  44.  *         and that
  45.  *          2/sqrt(pi) = 1.128379167095512573896158903121545171688
  46.  *         is close to one. The interval is chosen because the fix
  47.  *         point of erf(x) is near 0.6174 (i.e., erf(x)=x when x is
  48.  *         near 0.6174), and by some experiment, 0.84375 is chosen to
  49.  *         guarantee the error is less than one ulp for erf.
  50.  *
  51.  *      2. For |x| in [0.84375,1.25], let s = |x| - 1, and
  52.  *         c = 0.84506291151 rounded to single (24 bits)
  53.  *              erf(x)  = sign(x) * (c  + P1(s)/Q1(s))
  54.  *              erfc(x) = (1-c)  - P1(s)/Q1(s) if x > 0
  55.  *                        1+(c+P1(s)/Q1(s))    if x < 0
  56.  *              |P1/Q1 - (erf(|x|)-c)| <= 2**-59.06
  57.  *         Remark: here we use the taylor series expansion at x=1.
  58.  *              erf(1+s) = erf(1) + s*Poly(s)
  59.  *                       = 0.845.. + P1(s)/Q1(s)
  60.  *         That is, we use rational approximation to approximate
  61.  *                      erf(1+s) - (c = (single)0.84506291151)
  62.  *         Note that |P1/Q1|< 0.078 for x in [0.84375,1.25]
  63.  *         where
  64.  *              P1(s) = degree 6 poly in s
  65.  *              Q1(s) = degree 6 poly in s
  66.  *
  67.  *      3. For x in [1.25,1/0.35(~2.857143)],
  68.  *              erfc(x) = (1/x)*exp(-x*x-0.5625+R1/S1)
  69.  *              erf(x)  = 1 - erfc(x)
  70.  *         where
  71.  *              R1(z) = degree 7 poly in z, (z=1/x^2)
  72.  *              S1(z) = degree 8 poly in z
  73.  *
  74.  *      4. For x in [1/0.35,28]
  75.  *              erfc(x) = (1/x)*exp(-x*x-0.5625+R2/S2) if x > 0
  76.  *                      = 2.0 - (1/x)*exp(-x*x-0.5625+R2/S2) if -6<x<0
  77.  *                      = 2.0 - tiny            (if x <= -6)
  78.  *              erf(x)  = sign(x)*(1.0 - erfc(x)) if x < 6, else
  79.  *              erf(x)  = sign(x)*(1.0 - tiny)
  80.  *         where
  81.  *              R2(z) = degree 6 poly in z, (z=1/x^2)
  82.  *              S2(z) = degree 7 poly in z
  83.  *
  84.  *      Note1:
  85.  *         To compute exp(-x*x-0.5625+R/S), let s be a single
  86.  *         precision number and s := x; then
  87.  *              -x*x = -s*s + (s-x)*(s+x)
  88.  *              exp(-x*x-0.5626+R/S) =
  89.  *                      exp(-s*s-0.5625)*exp((s-x)*(s+x)+R/S);
  90.  *      Note2:
  91.  *         Here 4 and 5 make use of the asymptotic series
  92.  *                        exp(-x*x)
  93.  *              erfc(x) ~ ---------- * ( 1 + Poly(1/x^2) )
  94.  *                        x*sqrt(pi)
  95.  *         We use rational approximation to approximate
  96.  *              g(s)=f(1/x^2) = log(erfc(x)*x) - x*x + 0.5625
  97.  *         Here is the error bound for R1/S1 and R2/S2
  98.  *              |R1/S1 - f(x)|  < 2**(-62.57)
  99.  *              |R2/S2 - f(x)|  < 2**(-61.52)
  100.  *
  101.  *      5. For inf > x >= 28
  102.  *              erf(x)  = sign(x) *(1 - tiny)  (raise inexact)
  103.  *              erfc(x) = tiny*tiny (raise underflow) if x > 0
  104.  *                      = 2 - tiny if x<0
  105.  *
  106.  *      7. Special case:
  107.  *              erf(0)  = 0, erf(inf)  = 1, erf(-inf) = -1,
  108.  *              erfc(0) = 1, erfc(inf) = 0, erfc(-inf) = 2,
  109.  *              erfc/erf(NaN) is NaN
  110.  */
  111.  
  112.  
  113. #include "math.h"
  114. #include "math_private.h"
  115.  
  116. #ifdef __STDC__
  117. static const double
  118. #else
  119. static double
  120. #endif
  121. tiny        = 1e-300,
  122. half=  5.00000000000000000000e-01, /* 0x3FE00000, 0x00000000 */
  123. one =  1.00000000000000000000e+00, /* 0x3FF00000, 0x00000000 */
  124. two =  2.00000000000000000000e+00, /* 0x40000000, 0x00000000 */
  125.         /* c = (float)0.84506291151 */
  126. erx =  8.45062911510467529297e-01, /* 0x3FEB0AC1, 0x60000000 */
  127. /*
  128.  * Coefficients for approximation to  erf on [0,0.84375]
  129.  */
  130. efx =  1.28379167095512586316e-01, /* 0x3FC06EBA, 0x8214DB69 */
  131. efx8=  1.02703333676410069053e+00, /* 0x3FF06EBA, 0x8214DB69 */
  132. pp0  =  1.28379167095512558561e-01, /* 0x3FC06EBA, 0x8214DB68 */
  133. pp1  = -3.25042107247001499370e-01, /* 0xBFD4CD7D, 0x691CB913 */
  134. pp2  = -2.84817495755985104766e-02, /* 0xBF9D2A51, 0xDBD7194F */
  135. pp3  = -5.77027029648944159157e-03, /* 0xBF77A291, 0x236668E4 */
  136. pp4  = -2.37630166566501626084e-05, /* 0xBEF8EAD6, 0x120016AC */
  137. qq1  =  3.97917223959155352819e-01, /* 0x3FD97779, 0xCDDADC09 */
  138. qq2  =  6.50222499887672944485e-02, /* 0x3FB0A54C, 0x5536CEBA */
  139. qq3  =  5.08130628187576562776e-03, /* 0x3F74D022, 0xC4D36B0F */
  140. qq4  =  1.32494738004321644526e-04, /* 0x3F215DC9, 0x221C1A10 */
  141. qq5  = -3.96022827877536812320e-06, /* 0xBED09C43, 0x42A26120 */
  142. /*
  143.  * Coefficients for approximation to  erf  in [0.84375,1.25]
  144.  */
  145. pa0  = -2.36211856075265944077e-03, /* 0xBF6359B8, 0xBEF77538 */
  146. pa1  =  4.14856118683748331666e-01, /* 0x3FDA8D00, 0xAD92B34D */
  147. pa2  = -3.72207876035701323847e-01, /* 0xBFD7D240, 0xFBB8C3F1 */
  148. pa3  =  3.18346619901161753674e-01, /* 0x3FD45FCA, 0x805120E4 */
  149. pa4  = -1.10894694282396677476e-01, /* 0xBFBC6398, 0x3D3E28EC */
  150. pa5  =  3.54783043256182359371e-02, /* 0x3FA22A36, 0x599795EB */
  151. pa6  = -2.16637559486879084300e-03, /* 0xBF61BF38, 0x0A96073F */
  152. qa1  =  1.06420880400844228286e-01, /* 0x3FBB3E66, 0x18EEE323 */
  153. qa2  =  5.40397917702171048937e-01, /* 0x3FE14AF0, 0x92EB6F33 */
  154. qa3  =  7.18286544141962662868e-02, /* 0x3FB2635C, 0xD99FE9A7 */
  155. qa4  =  1.26171219808761642112e-01, /* 0x3FC02660, 0xE763351F */
  156. qa5  =  1.36370839120290507362e-02, /* 0x3F8BEDC2, 0x6B51DD1C */
  157. qa6  =  1.19844998467991074170e-02, /* 0x3F888B54, 0x5735151D */
  158. /*
  159.  * Coefficients for approximation to  erfc in [1.25,1/0.35]
  160.  */
  161. ra0  = -9.86494403484714822705e-03, /* 0xBF843412, 0x600D6435 */
  162. ra1  = -6.93858572707181764372e-01, /* 0xBFE63416, 0xE4BA7360 */
  163. ra2  = -1.05586262253232909814e+01, /* 0xC0251E04, 0x41B0E726 */
  164. ra3  = -6.23753324503260060396e+01, /* 0xC04F300A, 0xE4CBA38D */
  165. ra4  = -1.62396669462573470355e+02, /* 0xC0644CB1, 0x84282266 */
  166. ra5  = -1.84605092906711035994e+02, /* 0xC067135C, 0xEBCCABB2 */
  167. ra6  = -8.12874355063065934246e+01, /* 0xC0545265, 0x57E4D2F2 */
  168. ra7  = -9.81432934416914548592e+00, /* 0xC023A0EF, 0xC69AC25C */
  169. sa1  =  1.96512716674392571292e+01, /* 0x4033A6B9, 0xBD707687 */
  170. sa2  =  1.37657754143519042600e+02, /* 0x4061350C, 0x526AE721 */
  171. sa3  =  4.34565877475229228821e+02, /* 0x407B290D, 0xD58A1A71 */
  172. sa4  =  6.45387271733267880336e+02, /* 0x40842B19, 0x21EC2868 */
  173. sa5  =  4.29008140027567833386e+02, /* 0x407AD021, 0x57700314 */
  174. sa6  =  1.08635005541779435134e+02, /* 0x405B28A3, 0xEE48AE2C */
  175. sa7  =  6.57024977031928170135e+00, /* 0x401A47EF, 0x8E484A93 */
  176. sa8  = -6.04244152148580987438e-02, /* 0xBFAEEFF2, 0xEE749A62 */
  177. /*
  178.  * Coefficients for approximation to  erfc in [1/.35,28]
  179.  */
  180. rb0  = -9.86494292470009928597e-03, /* 0xBF843412, 0x39E86F4A */
  181. rb1  = -7.99283237680523006574e-01, /* 0xBFE993BA, 0x70C285DE */
  182. rb2  = -1.77579549177547519889e+01, /* 0xC031C209, 0x555F995A */
  183. rb3  = -1.60636384855821916062e+02, /* 0xC064145D, 0x43C5ED98 */
  184. rb4  = -6.37566443368389627722e+02, /* 0xC083EC88, 0x1375F228 */
  185. rb5  = -1.02509513161107724954e+03, /* 0xC0900461, 0x6A2E5992 */
  186. rb6  = -4.83519191608651397019e+02, /* 0xC07E384E, 0x9BDC383F */
  187. sb1  =  3.03380607434824582924e+01, /* 0x403E568B, 0x261D5190 */
  188. sb2  =  3.25792512996573918826e+02, /* 0x40745CAE, 0x221B9F0A */
  189. sb3  =  1.53672958608443695994e+03, /* 0x409802EB, 0x189D5118 */
  190. sb4  =  3.19985821950859553908e+03, /* 0x40A8FFB7, 0x688C246A */
  191. sb5  =  2.55305040643316442583e+03, /* 0x40A3F219, 0xCEDF3BE6 */
  192. sb6  =  4.74528541206955367215e+02, /* 0x407DA874, 0xE79FE763 */
  193. sb7  = -2.24409524465858183362e+01; /* 0xC03670E2, 0x42712D62 */
  194.  
  195. #ifdef __STDC__
  196.         double erf(double x)
  197. #else
  198.         double erf(x)
  199.         double x;
  200. #endif
  201. {
  202.         int32_t hx,ix,i;
  203.         double R,S,P,Q,s,y,z,r;
  204.         GET_HIGH_WORD(hx,x);
  205.         ix = hx&0x7fffffff;
  206.         if(ix>=0x7ff00000) {            /* erf(nan)=nan */
  207.             i = ((u_int32_t)hx>>31)<<1;
  208.             return (double)(1-i)+one/x; /* erf(+-inf)=+-1 */
  209.         }
  210.  
  211.         if(ix < 0x3feb0000) {           /* |x|<0.84375 */
  212.             if(ix < 0x3e300000) {       /* |x|<2**-28 */
  213.                 if (ix < 0x00800000)
  214.                     return 0.125*(8.0*x+efx8*x);  /*avoid underflow */
  215.                 return x + efx*x;
  216.             }
  217.             z = x*x;
  218.             r = pp0+z*(pp1+z*(pp2+z*(pp3+z*pp4)));
  219.             s = one+z*(qq1+z*(qq2+z*(qq3+z*(qq4+z*qq5))));
  220.             y = r/s;
  221.             return x + x*y;
  222.         }
  223.         if(ix < 0x3ff40000) {           /* 0.84375 <= |x| < 1.25 */
  224.             s = fabs(x)-one;
  225.             P = pa0+s*(pa1+s*(pa2+s*(pa3+s*(pa4+s*(pa5+s*pa6)))));
  226.             Q = one+s*(qa1+s*(qa2+s*(qa3+s*(qa4+s*(qa5+s*qa6)))));
  227.             if(hx>=0) return erx + P/Q; else return -erx - P/Q;
  228.         }
  229.         if (ix >= 0x40180000) {         /* inf>|x|>=6 */
  230.             if(hx>=0) return one-tiny; else return tiny-one;
  231.         }
  232.         x = fabs(x);
  233.         s = one/(x*x);
  234.         if(ix< 0x4006DB6E) {    /* |x| < 1/0.35 */
  235.             R=ra0+s*(ra1+s*(ra2+s*(ra3+s*(ra4+s*(
  236.                                 ra5+s*(ra6+s*ra7))))));
  237.             S=one+s*(sa1+s*(sa2+s*(sa3+s*(sa4+s*(
  238.                                 sa5+s*(sa6+s*(sa7+s*sa8)))))));
  239.         } else {        /* |x| >= 1/0.35 */
  240.             R=rb0+s*(rb1+s*(rb2+s*(rb3+s*(rb4+s*(
  241.                                 rb5+s*rb6)))));
  242.             S=one+s*(sb1+s*(sb2+s*(sb3+s*(sb4+s*(
  243.                                 sb5+s*(sb6+s*sb7))))));
  244.         }
  245.         z  = x;  
  246.         SET_LOW_WORD(z,0);
  247.         r  =  __ieee754_exp(-z*z-0.5625)*__ieee754_exp((z-x)*(z+x)+R/S);
  248.         if(hx>=0) return one-r/x; else return  r/x-one;
  249. }
  250.  
  251. #ifdef __STDC__
  252.         double erfc(double x)
  253. #else
  254.         double erfc(x)
  255.         double x;
  256. #endif
  257. {
  258.         int32_t hx,ix;
  259.         double R,S,P,Q,s,y,z,r;
  260.         GET_HIGH_WORD(hx,x);
  261.         ix = hx&0x7fffffff;
  262.         if(ix>=0x7ff00000) {                    /* erfc(nan)=nan */
  263.                                                 /* erfc(+-inf)=0,2 */
  264.             return (double)(((u_int32_t)hx>>31)<<1)+one/x;
  265.         }
  266.  
  267.         if(ix < 0x3feb0000) {           /* |x|<0.84375 */
  268.             if(ix < 0x3c700000)         /* |x|<2**-56 */
  269.                 return one-x;
  270.             z = x*x;
  271.             r = pp0+z*(pp1+z*(pp2+z*(pp3+z*pp4)));
  272.             s = one+z*(qq1+z*(qq2+z*(qq3+z*(qq4+z*qq5))));
  273.             y = r/s;
  274.             if(hx < 0x3fd00000) {       /* x<1/4 */
  275.                 return one-(x+x*y);
  276.             } else {
  277.                 r = x*y;
  278.                 r += (x-half);
  279.                 return half - r ;
  280.             }
  281.         }
  282.         if(ix < 0x3ff40000) {           /* 0.84375 <= |x| < 1.25 */
  283.             s = fabs(x)-one;
  284.             P = pa0+s*(pa1+s*(pa2+s*(pa3+s*(pa4+s*(pa5+s*pa6)))));
  285.             Q = one+s*(qa1+s*(qa2+s*(qa3+s*(qa4+s*(qa5+s*qa6)))));
  286.             if(hx>=0) {
  287.                 z  = one-erx; return z - P/Q;
  288.             } else {
  289.                 z = erx+P/Q; return one+z;
  290.             }
  291.         }
  292.         if (ix < 0x403c0000) {          /* |x|<28 */
  293.             x = fabs(x);
  294.             s = one/(x*x);
  295.             if(ix< 0x4006DB6D) {        /* |x| < 1/.35 ~ 2.857143*/
  296.                 R=ra0+s*(ra1+s*(ra2+s*(ra3+s*(ra4+s*(
  297.                                 ra5+s*(ra6+s*ra7))))));
  298.                 S=one+s*(sa1+s*(sa2+s*(sa3+s*(sa4+s*(
  299.                                 sa5+s*(sa6+s*(sa7+s*sa8)))))));
  300.             } else {                    /* |x| >= 1/.35 ~ 2.857143 */
  301.                 if(hx<0&&ix>=0x40180000) return two-tiny;/* x < -6 */
  302.                 R=rb0+s*(rb1+s*(rb2+s*(rb3+s*(rb4+s*(
  303.                                 rb5+s*rb6)))));
  304.                 S=one+s*(sb1+s*(sb2+s*(sb3+s*(sb4+s*(
  305.                                 sb5+s*(sb6+s*sb7))))));
  306.             }
  307.             z  = x;
  308.             SET_LOW_WORD(z,0);
  309.             r  =  __ieee754_exp(-z*z-0.5625)*
  310.                         __ieee754_exp((z-x)*(z+x)+R/S);
  311.             if(hx>0) return r/x; else return two-r/x;
  312.         } else {
  313.             if(hx>0) return tiny*tiny; else return two-tiny;
  314.         }
  315. }
  316.