e_j0.c 15 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. /* __ieee754_j0(x), __ieee754_y0(x)
  12. * Bessel function of the first and second kinds of order zero.
  13. * Method -- j0(x):
  14. * 1. For tiny x, we use j0(x) = 1 - x^2/4 + x^4/64 - ...
  15. * 2. Reduce x to |x| since j0(x)=j0(-x), and
  16. * for x in (0,2)
  17. * j0(x) = 1-z/4+ z^2*R0/S0, where z = x*x;
  18. * (precision: |j0-1+z/4-z^2R0/S0 |<2**-63.67 )
  19. * for x in (2,inf)
  20. * j0(x) = sqrt(2/(pi*x))*(p0(x)*cos(x0)-q0(x)*sin(x0))
  21. * where x0 = x-pi/4. It is better to compute sin(x0),cos(x0)
  22. * as follow:
  23. * cos(x0) = cos(x)cos(pi/4)+sin(x)sin(pi/4)
  24. * = 1/sqrt(2) * (cos(x) + sin(x))
  25. * sin(x0) = sin(x)cos(pi/4)-cos(x)sin(pi/4)
  26. * = 1/sqrt(2) * (sin(x) - cos(x))
  27. * (To avoid cancellation, use
  28. * sin(x) +- cos(x) = -cos(2x)/(sin(x) -+ cos(x))
  29. * to compute the worse one.)
  30. *
  31. * 3 Special cases
  32. * j0(nan)= nan
  33. * j0(0) = 1
  34. * j0(inf) = 0
  35. *
  36. * Method -- y0(x):
  37. * 1. For x<2.
  38. * Since
  39. * y0(x) = 2/pi*(j0(x)*(ln(x/2)+Euler) + x^2/4 - ...)
  40. * therefore y0(x)-2/pi*j0(x)*ln(x) is an even function.
  41. * We use the following function to approximate y0,
  42. * y0(x) = U(z)/V(z) + (2/pi)*(j0(x)*ln(x)), z= x^2
  43. * where
  44. * U(z) = u00 + u01*z + ... + u06*z^6
  45. * V(z) = 1 + v01*z + ... + v04*z^4
  46. * with absolute approximation error bounded by 2**-72.
  47. * Note: For tiny x, U/V = u0 and j0(x)~1, hence
  48. * y0(tiny) = u0 + (2/pi)*ln(tiny), (choose tiny<2**-27)
  49. * 2. For x>=2.
  50. * y0(x) = sqrt(2/(pi*x))*(p0(x)*cos(x0)+q0(x)*sin(x0))
  51. * where x0 = x-pi/4. It is better to compute sin(x0),cos(x0)
  52. * by the method mentioned above.
  53. * 3. Special cases: y0(0)=-inf, y0(x<0)=NaN, y0(inf)=0.
  54. */
  55. #include "math.h"
  56. #include "math_private.h"
  57. static double pzero(double), qzero(double);
  58. static const double
  59. huge = 1e300,
  60. one = 1.0,
  61. invsqrtpi= 5.64189583547756279280e-01, /* 0x3FE20DD7, 0x50429B6D */
  62. tpi = 6.36619772367581382433e-01, /* 0x3FE45F30, 0x6DC9C883 */
  63. /* R0/S0 on [0, 2.00] */
  64. R02 = 1.56249999999999947958e-02, /* 0x3F8FFFFF, 0xFFFFFFFD */
  65. R03 = -1.89979294238854721751e-04, /* 0xBF28E6A5, 0xB61AC6E9 */
  66. R04 = 1.82954049532700665670e-06, /* 0x3EBEB1D1, 0x0C503919 */
  67. R05 = -4.61832688532103189199e-09, /* 0xBE33D5E7, 0x73D63FCE */
  68. S01 = 1.56191029464890010492e-02, /* 0x3F8FFCE8, 0x82C8C2A4 */
  69. S02 = 1.16926784663337450260e-04, /* 0x3F1EA6D2, 0xDD57DBF4 */
  70. S03 = 5.13546550207318111446e-07, /* 0x3EA13B54, 0xCE84D5A9 */
  71. S04 = 1.16614003333790000205e-09; /* 0x3E1408BC, 0xF4745D8F */
  72. static const double zero = 0.0;
  73. double attribute_hidden __ieee754_j0(double x)
  74. {
  75. double z, s,c,ss,cc,r,u,v;
  76. int32_t hx,ix;
  77. GET_HIGH_WORD(hx,x);
  78. ix = hx&0x7fffffff;
  79. if(ix>=0x7ff00000) return one/(x*x);
  80. x = fabs(x);
  81. if(ix >= 0x40000000) { /* |x| >= 2.0 */
  82. s = sin(x);
  83. c = cos(x);
  84. ss = s-c;
  85. cc = s+c;
  86. if(ix<0x7fe00000) { /* make sure x+x not overflow */
  87. z = -cos(x+x);
  88. if ((s*c)<zero) cc = z/ss;
  89. else ss = z/cc;
  90. }
  91. /*
  92. * j0(x) = 1/sqrt(pi) * (P(0,x)*cc - Q(0,x)*ss) / sqrt(x)
  93. * y0(x) = 1/sqrt(pi) * (P(0,x)*ss + Q(0,x)*cc) / sqrt(x)
  94. */
  95. if(ix>0x48000000) z = (invsqrtpi*cc)/sqrt(x);
  96. else {
  97. u = pzero(x); v = qzero(x);
  98. z = invsqrtpi*(u*cc-v*ss)/sqrt(x);
  99. }
  100. return z;
  101. }
  102. if(ix<0x3f200000) { /* |x| < 2**-13 */
  103. if(huge+x>one) { /* raise inexact if x != 0 */
  104. if(ix<0x3e400000) return one; /* |x|<2**-27 */
  105. else return one - 0.25*x*x;
  106. }
  107. }
  108. z = x*x;
  109. r = z*(R02+z*(R03+z*(R04+z*R05)));
  110. s = one+z*(S01+z*(S02+z*(S03+z*S04)));
  111. if(ix < 0x3FF00000) { /* |x| < 1.00 */
  112. return one + z*(-0.25+(r/s));
  113. } else {
  114. u = 0.5*x;
  115. return((one+u)*(one-u)+z*(r/s));
  116. }
  117. }
  118. /*
  119. * wrapper j0(double x)
  120. */
  121. #ifndef _IEEE_LIBM
  122. double j0(double x)
  123. {
  124. double z = __ieee754_j0(x);
  125. if (_LIB_VERSION == _IEEE_ || isnan(x))
  126. return z;
  127. if (fabs(x) > X_TLOSS)
  128. return __kernel_standard(x, x, 34); /* j0(|x|>X_TLOSS) */
  129. return z;
  130. }
  131. #else
  132. strong_alias(__ieee754_j0, j0)
  133. #endif
  134. static const double
  135. u00 = -7.38042951086872317523e-02, /* 0xBFB2E4D6, 0x99CBD01F */
  136. u01 = 1.76666452509181115538e-01, /* 0x3FC69D01, 0x9DE9E3FC */
  137. u02 = -1.38185671945596898896e-02, /* 0xBF8C4CE8, 0xB16CFA97 */
  138. u03 = 3.47453432093683650238e-04, /* 0x3F36C54D, 0x20B29B6B */
  139. u04 = -3.81407053724364161125e-06, /* 0xBECFFEA7, 0x73D25CAD */
  140. u05 = 1.95590137035022920206e-08, /* 0x3E550057, 0x3B4EABD4 */
  141. u06 = -3.98205194132103398453e-11, /* 0xBDC5E43D, 0x693FB3C8 */
  142. v01 = 1.27304834834123699328e-02, /* 0x3F8A1270, 0x91C9C71A */
  143. v02 = 7.60068627350353253702e-05, /* 0x3F13ECBB, 0xF578C6C1 */
  144. v03 = 2.59150851840457805467e-07, /* 0x3E91642D, 0x7FF202FD */
  145. v04 = 4.41110311332675467403e-10; /* 0x3DFE5018, 0x3BD6D9EF */
  146. double attribute_hidden __ieee754_y0(double x)
  147. {
  148. double z, s,c,ss,cc,u,v;
  149. int32_t hx,ix,lx;
  150. EXTRACT_WORDS(hx,lx,x);
  151. ix = 0x7fffffff&hx;
  152. /* Y0(NaN) is NaN, y0(-inf) is Nan, y0(inf) is 0 */
  153. if(ix>=0x7ff00000) return one/(x+x*x);
  154. if((ix|lx)==0) return -one/zero;
  155. if(hx<0) return zero/zero;
  156. if(ix >= 0x40000000) { /* |x| >= 2.0 */
  157. /* y0(x) = sqrt(2/(pi*x))*(p0(x)*sin(x0)+q0(x)*cos(x0))
  158. * where x0 = x-pi/4
  159. * Better formula:
  160. * cos(x0) = cos(x)cos(pi/4)+sin(x)sin(pi/4)
  161. * = 1/sqrt(2) * (sin(x) + cos(x))
  162. * sin(x0) = sin(x)cos(3pi/4)-cos(x)sin(3pi/4)
  163. * = 1/sqrt(2) * (sin(x) - cos(x))
  164. * To avoid cancellation, use
  165. * sin(x) +- cos(x) = -cos(2x)/(sin(x) -+ cos(x))
  166. * to compute the worse one.
  167. */
  168. s = sin(x);
  169. c = cos(x);
  170. ss = s-c;
  171. cc = s+c;
  172. /*
  173. * j0(x) = 1/sqrt(pi) * (P(0,x)*cc - Q(0,x)*ss) / sqrt(x)
  174. * y0(x) = 1/sqrt(pi) * (P(0,x)*ss + Q(0,x)*cc) / sqrt(x)
  175. */
  176. if(ix<0x7fe00000) { /* make sure x+x not overflow */
  177. z = -cos(x+x);
  178. if ((s*c)<zero) cc = z/ss;
  179. else ss = z/cc;
  180. }
  181. if(ix>0x48000000) z = (invsqrtpi*ss)/sqrt(x);
  182. else {
  183. u = pzero(x); v = qzero(x);
  184. z = invsqrtpi*(u*ss+v*cc)/sqrt(x);
  185. }
  186. return z;
  187. }
  188. if(ix<=0x3e400000) { /* x < 2**-27 */
  189. return(u00 + tpi*__ieee754_log(x));
  190. }
  191. z = x*x;
  192. u = u00+z*(u01+z*(u02+z*(u03+z*(u04+z*(u05+z*u06)))));
  193. v = one+z*(v01+z*(v02+z*(v03+z*v04)));
  194. return(u/v + tpi*(__ieee754_j0(x)*__ieee754_log(x)));
  195. }
  196. /*
  197. * wrapper y0(double x)
  198. */
  199. #ifndef _IEEE_LIBM
  200. double y0(double x)
  201. {
  202. double z = __ieee754_y0(x);
  203. if (_LIB_VERSION == _IEEE_ || isnan(x))
  204. return z;
  205. if (x <= 0.0) {
  206. if (x == 0.0) /* d= -one/(x-x); */
  207. return __kernel_standard(x, x, 8);
  208. /* d = zero/(x-x); */
  209. return __kernel_standard(x, x, 9);
  210. }
  211. if (x > X_TLOSS)
  212. return __kernel_standard(x, x, 35); /* y0(x>X_TLOSS) */
  213. return z;
  214. }
  215. #else
  216. strong_alias(__ieee754_y0, y0)
  217. #endif
  218. /* The asymptotic expansions of pzero is
  219. * 1 - 9/128 s^2 + 11025/98304 s^4 - ..., where s = 1/x.
  220. * For x >= 2, We approximate pzero by
  221. * pzero(x) = 1 + (R/S)
  222. * where R = pR0 + pR1*s^2 + pR2*s^4 + ... + pR5*s^10
  223. * S = 1 + pS0*s^2 + ... + pS4*s^10
  224. * and
  225. * | pzero(x)-1-R/S | <= 2 ** ( -60.26)
  226. */
  227. static const double pR8[6] = { /* for x in [inf, 8]=1/[0,0.125] */
  228. 0.00000000000000000000e+00, /* 0x00000000, 0x00000000 */
  229. -7.03124999999900357484e-02, /* 0xBFB1FFFF, 0xFFFFFD32 */
  230. -8.08167041275349795626e+00, /* 0xC02029D0, 0xB44FA779 */
  231. -2.57063105679704847262e+02, /* 0xC0701102, 0x7B19E863 */
  232. -2.48521641009428822144e+03, /* 0xC0A36A6E, 0xCD4DCAFC */
  233. -5.25304380490729545272e+03, /* 0xC0B4850B, 0x36CC643D */
  234. };
  235. static const double pS8[5] = {
  236. 1.16534364619668181717e+02, /* 0x405D2233, 0x07A96751 */
  237. 3.83374475364121826715e+03, /* 0x40ADF37D, 0x50596938 */
  238. 4.05978572648472545552e+04, /* 0x40E3D2BB, 0x6EB6B05F */
  239. 1.16752972564375915681e+05, /* 0x40FC810F, 0x8F9FA9BD */
  240. 4.76277284146730962675e+04, /* 0x40E74177, 0x4F2C49DC */
  241. };
  242. static const double pR5[6] = { /* for x in [8,4.5454]=1/[0.125,0.22001] */
  243. -1.14125464691894502584e-11, /* 0xBDA918B1, 0x47E495CC */
  244. -7.03124940873599280078e-02, /* 0xBFB1FFFF, 0xE69AFBC6 */
  245. -4.15961064470587782438e+00, /* 0xC010A370, 0xF90C6BBF */
  246. -6.76747652265167261021e+01, /* 0xC050EB2F, 0x5A7D1783 */
  247. -3.31231299649172967747e+02, /* 0xC074B3B3, 0x6742CC63 */
  248. -3.46433388365604912451e+02, /* 0xC075A6EF, 0x28A38BD7 */
  249. };
  250. static const double pS5[5] = {
  251. 6.07539382692300335975e+01, /* 0x404E6081, 0x0C98C5DE */
  252. 1.05125230595704579173e+03, /* 0x40906D02, 0x5C7E2864 */
  253. 5.97897094333855784498e+03, /* 0x40B75AF8, 0x8FBE1D60 */
  254. 9.62544514357774460223e+03, /* 0x40C2CCB8, 0xFA76FA38 */
  255. 2.40605815922939109441e+03, /* 0x40A2CC1D, 0xC70BE864 */
  256. };
  257. static const double pR3[6] = {/* for x in [4.547,2.8571]=1/[0.2199,0.35001] */
  258. -2.54704601771951915620e-09, /* 0xBE25E103, 0x6FE1AA86 */
  259. -7.03119616381481654654e-02, /* 0xBFB1FFF6, 0xF7C0E24B */
  260. -2.40903221549529611423e+00, /* 0xC00345B2, 0xAEA48074 */
  261. -2.19659774734883086467e+01, /* 0xC035F74A, 0x4CB94E14 */
  262. -5.80791704701737572236e+01, /* 0xC04D0A22, 0x420A1A45 */
  263. -3.14479470594888503854e+01, /* 0xC03F72AC, 0xA892D80F */
  264. };
  265. static const double pS3[5] = {
  266. 3.58560338055209726349e+01, /* 0x4041ED92, 0x84077DD3 */
  267. 3.61513983050303863820e+02, /* 0x40769839, 0x464A7C0E */
  268. 1.19360783792111533330e+03, /* 0x4092A66E, 0x6D1061D6 */
  269. 1.12799679856907414432e+03, /* 0x40919FFC, 0xB8C39B7E */
  270. 1.73580930813335754692e+02, /* 0x4065B296, 0xFC379081 */
  271. };
  272. static const double pR2[6] = {/* for x in [2.8570,2]=1/[0.3499,0.5] */
  273. -8.87534333032526411254e-08, /* 0xBE77D316, 0xE927026D */
  274. -7.03030995483624743247e-02, /* 0xBFB1FF62, 0x495E1E42 */
  275. -1.45073846780952986357e+00, /* 0xBFF73639, 0x8A24A843 */
  276. -7.63569613823527770791e+00, /* 0xC01E8AF3, 0xEDAFA7F3 */
  277. -1.11931668860356747786e+01, /* 0xC02662E6, 0xC5246303 */
  278. -3.23364579351335335033e+00, /* 0xC009DE81, 0xAF8FE70F */
  279. };
  280. static const double pS2[5] = {
  281. 2.22202997532088808441e+01, /* 0x40363865, 0x908B5959 */
  282. 1.36206794218215208048e+02, /* 0x4061069E, 0x0EE8878F */
  283. 2.70470278658083486789e+02, /* 0x4070E786, 0x42EA079B */
  284. 1.53875394208320329881e+02, /* 0x40633C03, 0x3AB6FAFF */
  285. 1.46576176948256193810e+01, /* 0x402D50B3, 0x44391809 */
  286. };
  287. static double pzero(double x)
  288. {
  289. const double *p = 0,*q = 0;
  290. double z,r,s;
  291. int32_t ix;
  292. GET_HIGH_WORD(ix,x);
  293. ix &= 0x7fffffff;
  294. if(ix>=0x40200000) {p = pR8; q= pS8;}
  295. else if(ix>=0x40122E8B){p = pR5; q= pS5;}
  296. else if(ix>=0x4006DB6D){p = pR3; q= pS3;}
  297. else if(ix>=0x40000000){p = pR2; q= pS2;}
  298. z = one/(x*x);
  299. r = p[0]+z*(p[1]+z*(p[2]+z*(p[3]+z*(p[4]+z*p[5]))));
  300. s = one+z*(q[0]+z*(q[1]+z*(q[2]+z*(q[3]+z*q[4]))));
  301. return one+ r/s;
  302. }
  303. /* For x >= 8, the asymptotic expansions of qzero is
  304. * -1/8 s + 75/1024 s^3 - ..., where s = 1/x.
  305. * We approximate pzero by
  306. * qzero(x) = s*(-1.25 + (R/S))
  307. * where R = qR0 + qR1*s^2 + qR2*s^4 + ... + qR5*s^10
  308. * S = 1 + qS0*s^2 + ... + qS5*s^12
  309. * and
  310. * | qzero(x)/s +1.25-R/S | <= 2 ** ( -61.22)
  311. */
  312. static const double qR8[6] = { /* for x in [inf, 8]=1/[0,0.125] */
  313. 0.00000000000000000000e+00, /* 0x00000000, 0x00000000 */
  314. 7.32421874999935051953e-02, /* 0x3FB2BFFF, 0xFFFFFE2C */
  315. 1.17682064682252693899e+01, /* 0x40278952, 0x5BB334D6 */
  316. 5.57673380256401856059e+02, /* 0x40816D63, 0x15301825 */
  317. 8.85919720756468632317e+03, /* 0x40C14D99, 0x3E18F46D */
  318. 3.70146267776887834771e+04, /* 0x40E212D4, 0x0E901566 */
  319. };
  320. static const double qS8[6] = {
  321. 1.63776026895689824414e+02, /* 0x406478D5, 0x365B39BC */
  322. 8.09834494656449805916e+03, /* 0x40BFA258, 0x4E6B0563 */
  323. 1.42538291419120476348e+05, /* 0x41016652, 0x54D38C3F */
  324. 8.03309257119514397345e+05, /* 0x412883DA, 0x83A52B43 */
  325. 8.40501579819060512818e+05, /* 0x4129A66B, 0x28DE0B3D */
  326. -3.43899293537866615225e+05, /* 0xC114FD6D, 0x2C9530C5 */
  327. };
  328. static const double qR5[6] = { /* for x in [8,4.5454]=1/[0.125,0.22001] */
  329. 1.84085963594515531381e-11, /* 0x3DB43D8F, 0x29CC8CD9 */
  330. 7.32421766612684765896e-02, /* 0x3FB2BFFF, 0xD172B04C */
  331. 5.83563508962056953777e+00, /* 0x401757B0, 0xB9953DD3 */
  332. 1.35111577286449829671e+02, /* 0x4060E392, 0x0A8788E9 */
  333. 1.02724376596164097464e+03, /* 0x40900CF9, 0x9DC8C481 */
  334. 1.98997785864605384631e+03, /* 0x409F17E9, 0x53C6E3A6 */
  335. };
  336. static const double qS5[6] = {
  337. 8.27766102236537761883e+01, /* 0x4054B1B3, 0xFB5E1543 */
  338. 2.07781416421392987104e+03, /* 0x40A03BA0, 0xDA21C0CE */
  339. 1.88472887785718085070e+04, /* 0x40D267D2, 0x7B591E6D */
  340. 5.67511122894947329769e+04, /* 0x40EBB5E3, 0x97E02372 */
  341. 3.59767538425114471465e+04, /* 0x40E19118, 0x1F7A54A0 */
  342. -5.35434275601944773371e+03, /* 0xC0B4EA57, 0xBEDBC609 */
  343. };
  344. static const double qR3[6] = {/* for x in [4.547,2.8571]=1/[0.2199,0.35001] */
  345. 4.37741014089738620906e-09, /* 0x3E32CD03, 0x6ADECB82 */
  346. 7.32411180042911447163e-02, /* 0x3FB2BFEE, 0x0E8D0842 */
  347. 3.34423137516170720929e+00, /* 0x400AC0FC, 0x61149CF5 */
  348. 4.26218440745412650017e+01, /* 0x40454F98, 0x962DAEDD */
  349. 1.70808091340565596283e+02, /* 0x406559DB, 0xE25EFD1F */
  350. 1.66733948696651168575e+02, /* 0x4064D77C, 0x81FA21E0 */
  351. };
  352. static const double qS3[6] = {
  353. 4.87588729724587182091e+01, /* 0x40486122, 0xBFE343A6 */
  354. 7.09689221056606015736e+02, /* 0x40862D83, 0x86544EB3 */
  355. 3.70414822620111362994e+03, /* 0x40ACF04B, 0xE44DFC63 */
  356. 6.46042516752568917582e+03, /* 0x40B93C6C, 0xD7C76A28 */
  357. 2.51633368920368957333e+03, /* 0x40A3A8AA, 0xD94FB1C0 */
  358. -1.49247451836156386662e+02, /* 0xC062A7EB, 0x201CF40F */
  359. };
  360. static const double qR2[6] = {/* for x in [2.8570,2]=1/[0.3499,0.5] */
  361. 1.50444444886983272379e-07, /* 0x3E84313B, 0x54F76BDB */
  362. 7.32234265963079278272e-02, /* 0x3FB2BEC5, 0x3E883E34 */
  363. 1.99819174093815998816e+00, /* 0x3FFFF897, 0xE727779C */
  364. 1.44956029347885735348e+01, /* 0x402CFDBF, 0xAAF96FE5 */
  365. 3.16662317504781540833e+01, /* 0x403FAA8E, 0x29FBDC4A */
  366. 1.62527075710929267416e+01, /* 0x403040B1, 0x71814BB4 */
  367. };
  368. static const double qS2[6] = {
  369. 3.03655848355219184498e+01, /* 0x403E5D96, 0xF7C07AED */
  370. 2.69348118608049844624e+02, /* 0x4070D591, 0xE4D14B40 */
  371. 8.44783757595320139444e+02, /* 0x408A6645, 0x22B3BF22 */
  372. 8.82935845112488550512e+02, /* 0x408B977C, 0x9C5CC214 */
  373. 2.12666388511798828631e+02, /* 0x406A9553, 0x0E001365 */
  374. -5.31095493882666946917e+00, /* 0xC0153E6A, 0xF8B32931 */
  375. };
  376. static double qzero(double x)
  377. {
  378. const double *p=0,*q=0;
  379. double s,r,z;
  380. int32_t ix;
  381. GET_HIGH_WORD(ix,x);
  382. ix &= 0x7fffffff;
  383. if(ix>=0x40200000) {p = qR8; q= qS8;}
  384. else if(ix>=0x40122E8B){p = qR5; q= qS5;}
  385. else if(ix>=0x4006DB6D){p = qR3; q= qS3;}
  386. else if(ix>=0x40000000){p = qR2; q= qS2;}
  387. z = one/(x*x);
  388. r = p[0]+z*(p[1]+z*(p[2]+z*(p[3]+z*(p[4]+z*p[5]))));
  389. s = one+z*(q[0]+z*(q[1]+z*(q[2]+z*(q[3]+z*(q[4]+z*q[5])))));
  390. return (-.125 + r/s)/x;
  391. }