j1f.c 8.4 KB

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  1. /* origin: FreeBSD /usr/src/lib/msun/src/e_j1f.c */
  2. /*
  3. * Conversion to float by Ian Lance Taylor, Cygnus Support, [email protected].
  4. */
  5. /*
  6. * ====================================================
  7. * Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved.
  8. *
  9. * Developed at SunPro, a Sun Microsystems, Inc. business.
  10. * Permission to use, copy, modify, and distribute this
  11. * software is freely granted, provided that this notice
  12. * is preserved.
  13. * ====================================================
  14. */
  15. #include "libm.h"
  16. static float ponef(float), qonef(float);
  17. static const float
  18. invsqrtpi = 5.6418961287e-01, /* 0x3f106ebb */
  19. tpi = 6.3661974669e-01; /* 0x3f22f983 */
  20. static float common(uint32_t ix, float x, int y1, int sign)
  21. {
  22. double z,s,c,ss,cc;
  23. s = sinf(x);
  24. if (y1)
  25. s = -s;
  26. c = cosf(x);
  27. cc = s-c;
  28. if (ix < 0x7f000000) {
  29. ss = -s-c;
  30. z = cosf(2*x);
  31. if (s*c > 0)
  32. cc = z/ss;
  33. else
  34. ss = z/cc;
  35. if (ix < 0x58800000) {
  36. if (y1)
  37. ss = -ss;
  38. cc = ponef(x)*cc-qonef(x)*ss;
  39. }
  40. }
  41. if (sign)
  42. cc = -cc;
  43. return invsqrtpi*cc/sqrtf(x);
  44. }
  45. /* R0/S0 on [0,2] */
  46. static const float
  47. r00 = -6.2500000000e-02, /* 0xbd800000 */
  48. r01 = 1.4070566976e-03, /* 0x3ab86cfd */
  49. r02 = -1.5995563444e-05, /* 0xb7862e36 */
  50. r03 = 4.9672799207e-08, /* 0x335557d2 */
  51. s01 = 1.9153760746e-02, /* 0x3c9ce859 */
  52. s02 = 1.8594678841e-04, /* 0x3942fab6 */
  53. s03 = 1.1771846857e-06, /* 0x359dffc2 */
  54. s04 = 5.0463624390e-09, /* 0x31ad6446 */
  55. s05 = 1.2354227016e-11; /* 0x2d59567e */
  56. float j1f(float x)
  57. {
  58. float z,r,s;
  59. uint32_t ix;
  60. int sign;
  61. GET_FLOAT_WORD(ix, x);
  62. sign = ix>>31;
  63. ix &= 0x7fffffff;
  64. if (ix >= 0x7f800000)
  65. return 1/(x*x);
  66. if (ix >= 0x40000000) /* |x| >= 2 */
  67. return common(ix, fabsf(x), 0, sign);
  68. if (ix >= 0x32000000) { /* |x| >= 2**-27 */
  69. z = x*x;
  70. r = z*(r00+z*(r01+z*(r02+z*r03)));
  71. s = 1+z*(s01+z*(s02+z*(s03+z*(s04+z*s05))));
  72. z = 0.5f + r/s;
  73. } else
  74. /* raise inexact if x!=0 */
  75. z = 0.5f + x;
  76. return z*x;
  77. }
  78. static const float U0[5] = {
  79. -1.9605709612e-01, /* 0xbe48c331 */
  80. 5.0443872809e-02, /* 0x3d4e9e3c */
  81. -1.9125689287e-03, /* 0xbafaaf2a */
  82. 2.3525259166e-05, /* 0x37c5581c */
  83. -9.1909917899e-08, /* 0xb3c56003 */
  84. };
  85. static const float V0[5] = {
  86. 1.9916731864e-02, /* 0x3ca3286a */
  87. 2.0255257550e-04, /* 0x3954644b */
  88. 1.3560879779e-06, /* 0x35b602d4 */
  89. 6.2274145840e-09, /* 0x31d5f8eb */
  90. 1.6655924903e-11, /* 0x2d9281cf */
  91. };
  92. float y1f(float x)
  93. {
  94. float z,u,v;
  95. uint32_t ix;
  96. GET_FLOAT_WORD(ix, x);
  97. if ((ix & 0x7fffffff) == 0)
  98. return -1/0.0f;
  99. if (ix>>31)
  100. return 0/0.0f;
  101. if (ix >= 0x7f800000)
  102. return 1/x;
  103. if (ix >= 0x40000000) /* |x| >= 2.0 */
  104. return common(ix,x,1,0);
  105. if (ix < 0x32000000) /* x < 2**-27 */
  106. return -tpi/x;
  107. z = x*x;
  108. u = U0[0]+z*(U0[1]+z*(U0[2]+z*(U0[3]+z*U0[4])));
  109. v = 1.0f+z*(V0[0]+z*(V0[1]+z*(V0[2]+z*(V0[3]+z*V0[4]))));
  110. return x*(u/v) + tpi*(j1f(x)*logf(x)-1.0f/x);
  111. }
  112. /* For x >= 8, the asymptotic expansions of pone is
  113. * 1 + 15/128 s^2 - 4725/2^15 s^4 - ..., where s = 1/x.
  114. * We approximate pone by
  115. * pone(x) = 1 + (R/S)
  116. * where R = pr0 + pr1*s^2 + pr2*s^4 + ... + pr5*s^10
  117. * S = 1 + ps0*s^2 + ... + ps4*s^10
  118. * and
  119. * | pone(x)-1-R/S | <= 2 ** ( -60.06)
  120. */
  121. static const float pr8[6] = { /* for x in [inf, 8]=1/[0,0.125] */
  122. 0.0000000000e+00, /* 0x00000000 */
  123. 1.1718750000e-01, /* 0x3df00000 */
  124. 1.3239480972e+01, /* 0x4153d4ea */
  125. 4.1205184937e+02, /* 0x43ce06a3 */
  126. 3.8747453613e+03, /* 0x45722bed */
  127. 7.9144794922e+03, /* 0x45f753d6 */
  128. };
  129. static const float ps8[5] = {
  130. 1.1420736694e+02, /* 0x42e46a2c */
  131. 3.6509309082e+03, /* 0x45642ee5 */
  132. 3.6956207031e+04, /* 0x47105c35 */
  133. 9.7602796875e+04, /* 0x47bea166 */
  134. 3.0804271484e+04, /* 0x46f0a88b */
  135. };
  136. static const float pr5[6] = { /* for x in [8,4.5454]=1/[0.125,0.22001] */
  137. 1.3199052094e-11, /* 0x2d68333f */
  138. 1.1718749255e-01, /* 0x3defffff */
  139. 6.8027510643e+00, /* 0x40d9b023 */
  140. 1.0830818176e+02, /* 0x42d89dca */
  141. 5.1763616943e+02, /* 0x440168b7 */
  142. 5.2871520996e+02, /* 0x44042dc6 */
  143. };
  144. static const float ps5[5] = {
  145. 5.9280597687e+01, /* 0x426d1f55 */
  146. 9.9140142822e+02, /* 0x4477d9b1 */
  147. 5.3532670898e+03, /* 0x45a74a23 */
  148. 7.8446904297e+03, /* 0x45f52586 */
  149. 1.5040468750e+03, /* 0x44bc0180 */
  150. };
  151. static const float pr3[6] = {
  152. 3.0250391081e-09, /* 0x314fe10d */
  153. 1.1718686670e-01, /* 0x3defffab */
  154. 3.9329774380e+00, /* 0x407bb5e7 */
  155. 3.5119403839e+01, /* 0x420c7a45 */
  156. 9.1055007935e+01, /* 0x42b61c2a */
  157. 4.8559066772e+01, /* 0x42423c7c */
  158. };
  159. static const float ps3[5] = {
  160. 3.4791309357e+01, /* 0x420b2a4d */
  161. 3.3676245117e+02, /* 0x43a86198 */
  162. 1.0468714600e+03, /* 0x4482dbe3 */
  163. 8.9081134033e+02, /* 0x445eb3ed */
  164. 1.0378793335e+02, /* 0x42cf936c */
  165. };
  166. static const float pr2[6] = {/* for x in [2.8570,2]=1/[0.3499,0.5] */
  167. 1.0771083225e-07, /* 0x33e74ea8 */
  168. 1.1717621982e-01, /* 0x3deffa16 */
  169. 2.3685150146e+00, /* 0x401795c0 */
  170. 1.2242610931e+01, /* 0x4143e1bc */
  171. 1.7693971634e+01, /* 0x418d8d41 */
  172. 5.0735230446e+00, /* 0x40a25a4d */
  173. };
  174. static const float ps2[5] = {
  175. 2.1436485291e+01, /* 0x41ab7dec */
  176. 1.2529022980e+02, /* 0x42fa9499 */
  177. 2.3227647400e+02, /* 0x436846c7 */
  178. 1.1767937469e+02, /* 0x42eb5bd7 */
  179. 8.3646392822e+00, /* 0x4105d590 */
  180. };
  181. static float ponef(float x)
  182. {
  183. const float *p,*q;
  184. float_t z,r,s;
  185. uint32_t ix;
  186. GET_FLOAT_WORD(ix, x);
  187. ix &= 0x7fffffff;
  188. if (ix >= 0x41000000){p = pr8; q = ps8;}
  189. else if (ix >= 0x40f71c58){p = pr5; q = ps5;}
  190. else if (ix >= 0x4036db68){p = pr3; q = ps3;}
  191. else /*ix >= 0x40000000*/ {p = pr2; q = ps2;}
  192. z = 1.0f/(x*x);
  193. r = p[0]+z*(p[1]+z*(p[2]+z*(p[3]+z*(p[4]+z*p[5]))));
  194. s = 1.0f+z*(q[0]+z*(q[1]+z*(q[2]+z*(q[3]+z*q[4]))));
  195. return 1.0f + r/s;
  196. }
  197. /* For x >= 8, the asymptotic expansions of qone is
  198. * 3/8 s - 105/1024 s^3 - ..., where s = 1/x.
  199. * We approximate pone by
  200. * qone(x) = s*(0.375 + (R/S))
  201. * where R = qr1*s^2 + qr2*s^4 + ... + qr5*s^10
  202. * S = 1 + qs1*s^2 + ... + qs6*s^12
  203. * and
  204. * | qone(x)/s -0.375-R/S | <= 2 ** ( -61.13)
  205. */
  206. static const float qr8[6] = { /* for x in [inf, 8]=1/[0,0.125] */
  207. 0.0000000000e+00, /* 0x00000000 */
  208. -1.0253906250e-01, /* 0xbdd20000 */
  209. -1.6271753311e+01, /* 0xc1822c8d */
  210. -7.5960174561e+02, /* 0xc43de683 */
  211. -1.1849806641e+04, /* 0xc639273a */
  212. -4.8438511719e+04, /* 0xc73d3683 */
  213. };
  214. static const float qs8[6] = {
  215. 1.6139537048e+02, /* 0x43216537 */
  216. 7.8253862305e+03, /* 0x45f48b17 */
  217. 1.3387534375e+05, /* 0x4802bcd6 */
  218. 7.1965775000e+05, /* 0x492fb29c */
  219. 6.6660125000e+05, /* 0x4922be94 */
  220. -2.9449025000e+05, /* 0xc88fcb48 */
  221. };
  222. static const float qr5[6] = { /* for x in [8,4.5454]=1/[0.125,0.22001] */
  223. -2.0897993405e-11, /* 0xadb7d219 */
  224. -1.0253904760e-01, /* 0xbdd1fffe */
  225. -8.0564479828e+00, /* 0xc100e736 */
  226. -1.8366960144e+02, /* 0xc337ab6b */
  227. -1.3731937256e+03, /* 0xc4aba633 */
  228. -2.6124443359e+03, /* 0xc523471c */
  229. };
  230. static const float qs5[6] = {
  231. 8.1276550293e+01, /* 0x42a28d98 */
  232. 1.9917987061e+03, /* 0x44f8f98f */
  233. 1.7468484375e+04, /* 0x468878f8 */
  234. 4.9851425781e+04, /* 0x4742bb6d */
  235. 2.7948074219e+04, /* 0x46da5826 */
  236. -4.7191835938e+03, /* 0xc5937978 */
  237. };
  238. static const float qr3[6] = {
  239. -5.0783124372e-09, /* 0xb1ae7d4f */
  240. -1.0253783315e-01, /* 0xbdd1ff5b */
  241. -4.6101160049e+00, /* 0xc0938612 */
  242. -5.7847221375e+01, /* 0xc267638e */
  243. -2.2824453735e+02, /* 0xc3643e9a */
  244. -2.1921012878e+02, /* 0xc35b35cb */
  245. };
  246. static const float qs3[6] = {
  247. 4.7665153503e+01, /* 0x423ea91e */
  248. 6.7386511230e+02, /* 0x4428775e */
  249. 3.3801528320e+03, /* 0x45534272 */
  250. 5.5477290039e+03, /* 0x45ad5dd5 */
  251. 1.9031191406e+03, /* 0x44ede3d0 */
  252. -1.3520118713e+02, /* 0xc3073381 */
  253. };
  254. static const float qr2[6] = {/* for x in [2.8570,2]=1/[0.3499,0.5] */
  255. -1.7838172539e-07, /* 0xb43f8932 */
  256. -1.0251704603e-01, /* 0xbdd1f475 */
  257. -2.7522056103e+00, /* 0xc0302423 */
  258. -1.9663616180e+01, /* 0xc19d4f16 */
  259. -4.2325313568e+01, /* 0xc2294d1f */
  260. -2.1371921539e+01, /* 0xc1aaf9b2 */
  261. };
  262. static const float qs2[6] = {
  263. 2.9533363342e+01, /* 0x41ec4454 */
  264. 2.5298155212e+02, /* 0x437cfb47 */
  265. 7.5750280762e+02, /* 0x443d602e */
  266. 7.3939318848e+02, /* 0x4438d92a */
  267. 1.5594900513e+02, /* 0x431bf2f2 */
  268. -4.9594988823e+00, /* 0xc09eb437 */
  269. };
  270. static float qonef(float x)
  271. {
  272. const float *p,*q;
  273. float_t s,r,z;
  274. uint32_t ix;
  275. GET_FLOAT_WORD(ix, x);
  276. ix &= 0x7fffffff;
  277. if (ix >= 0x40200000){p = qr8; q = qs8;}
  278. else if (ix >= 0x40f71c58){p = qr5; q = qs5;}
  279. else if (ix >= 0x4036db68){p = qr3; q = qs3;}
  280. else /*ix >= 0x40000000*/ {p = qr2; q = qs2;}
  281. z = 1.0f/(x*x);
  282. r = p[0]+z*(p[1]+z*(p[2]+z*(p[3]+z*(p[4]+z*p[5]))));
  283. s = 1.0f+z*(q[0]+z*(q[1]+z*(q[2]+z*(q[3]+z*(q[4]+z*q[5])))));
  284. return (.375f + r/s)/x;
  285. }