s_erf.c 11 KB

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