e_j0.c 14 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 __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. static const double
  119. u00 = -7.38042951086872317523e-02, /* 0xBFB2E4D6, 0x99CBD01F */
  120. u01 = 1.76666452509181115538e-01, /* 0x3FC69D01, 0x9DE9E3FC */
  121. u02 = -1.38185671945596898896e-02, /* 0xBF8C4CE8, 0xB16CFA97 */
  122. u03 = 3.47453432093683650238e-04, /* 0x3F36C54D, 0x20B29B6B */
  123. u04 = -3.81407053724364161125e-06, /* 0xBECFFEA7, 0x73D25CAD */
  124. u05 = 1.95590137035022920206e-08, /* 0x3E550057, 0x3B4EABD4 */
  125. u06 = -3.98205194132103398453e-11, /* 0xBDC5E43D, 0x693FB3C8 */
  126. v01 = 1.27304834834123699328e-02, /* 0x3F8A1270, 0x91C9C71A */
  127. v02 = 7.60068627350353253702e-05, /* 0x3F13ECBB, 0xF578C6C1 */
  128. v03 = 2.59150851840457805467e-07, /* 0x3E91642D, 0x7FF202FD */
  129. v04 = 4.41110311332675467403e-10; /* 0x3DFE5018, 0x3BD6D9EF */
  130. double __ieee754_y0(double x)
  131. {
  132. double z, s,c,ss,cc,u,v;
  133. int32_t hx,ix,lx;
  134. EXTRACT_WORDS(hx,lx,x);
  135. ix = 0x7fffffff&hx;
  136. /* Y0(NaN) is NaN, y0(-inf) is Nan, y0(inf) is 0 */
  137. if(ix>=0x7ff00000) return one/(x+x*x);
  138. if((ix|lx)==0) return -one/zero;
  139. if(hx<0) return zero/zero;
  140. if(ix >= 0x40000000) { /* |x| >= 2.0 */
  141. /* y0(x) = sqrt(2/(pi*x))*(p0(x)*sin(x0)+q0(x)*cos(x0))
  142. * where x0 = x-pi/4
  143. * Better formula:
  144. * cos(x0) = cos(x)cos(pi/4)+sin(x)sin(pi/4)
  145. * = 1/sqrt(2) * (sin(x) + cos(x))
  146. * sin(x0) = sin(x)cos(3pi/4)-cos(x)sin(3pi/4)
  147. * = 1/sqrt(2) * (sin(x) - cos(x))
  148. * To avoid cancellation, use
  149. * sin(x) +- cos(x) = -cos(2x)/(sin(x) -+ cos(x))
  150. * to compute the worse one.
  151. */
  152. s = sin(x);
  153. c = cos(x);
  154. ss = s-c;
  155. cc = s+c;
  156. /*
  157. * j0(x) = 1/sqrt(pi) * (P(0,x)*cc - Q(0,x)*ss) / sqrt(x)
  158. * y0(x) = 1/sqrt(pi) * (P(0,x)*ss + Q(0,x)*cc) / sqrt(x)
  159. */
  160. if(ix<0x7fe00000) { /* make sure x+x not overflow */
  161. z = -cos(x+x);
  162. if ((s*c)<zero) cc = z/ss;
  163. else ss = z/cc;
  164. }
  165. if(ix>0x48000000) z = (invsqrtpi*ss)/sqrt(x);
  166. else {
  167. u = pzero(x); v = qzero(x);
  168. z = invsqrtpi*(u*ss+v*cc)/sqrt(x);
  169. }
  170. return z;
  171. }
  172. if(ix<=0x3e400000) { /* x < 2**-27 */
  173. return(u00 + tpi*__ieee754_log(x));
  174. }
  175. z = x*x;
  176. u = u00+z*(u01+z*(u02+z*(u03+z*(u04+z*(u05+z*u06)))));
  177. v = one+z*(v01+z*(v02+z*(v03+z*v04)));
  178. return(u/v + tpi*(__ieee754_j0(x)*__ieee754_log(x)));
  179. }
  180. /* The asymptotic expansions of pzero is
  181. * 1 - 9/128 s^2 + 11025/98304 s^4 - ..., where s = 1/x.
  182. * For x >= 2, We approximate pzero by
  183. * pzero(x) = 1 + (R/S)
  184. * where R = pR0 + pR1*s^2 + pR2*s^4 + ... + pR5*s^10
  185. * S = 1 + pS0*s^2 + ... + pS4*s^10
  186. * and
  187. * | pzero(x)-1-R/S | <= 2 ** ( -60.26)
  188. */
  189. static const double pR8[6] = { /* for x in [inf, 8]=1/[0,0.125] */
  190. 0.00000000000000000000e+00, /* 0x00000000, 0x00000000 */
  191. -7.03124999999900357484e-02, /* 0xBFB1FFFF, 0xFFFFFD32 */
  192. -8.08167041275349795626e+00, /* 0xC02029D0, 0xB44FA779 */
  193. -2.57063105679704847262e+02, /* 0xC0701102, 0x7B19E863 */
  194. -2.48521641009428822144e+03, /* 0xC0A36A6E, 0xCD4DCAFC */
  195. -5.25304380490729545272e+03, /* 0xC0B4850B, 0x36CC643D */
  196. };
  197. static const double pS8[5] = {
  198. 1.16534364619668181717e+02, /* 0x405D2233, 0x07A96751 */
  199. 3.83374475364121826715e+03, /* 0x40ADF37D, 0x50596938 */
  200. 4.05978572648472545552e+04, /* 0x40E3D2BB, 0x6EB6B05F */
  201. 1.16752972564375915681e+05, /* 0x40FC810F, 0x8F9FA9BD */
  202. 4.76277284146730962675e+04, /* 0x40E74177, 0x4F2C49DC */
  203. };
  204. static const double pR5[6] = { /* for x in [8,4.5454]=1/[0.125,0.22001] */
  205. -1.14125464691894502584e-11, /* 0xBDA918B1, 0x47E495CC */
  206. -7.03124940873599280078e-02, /* 0xBFB1FFFF, 0xE69AFBC6 */
  207. -4.15961064470587782438e+00, /* 0xC010A370, 0xF90C6BBF */
  208. -6.76747652265167261021e+01, /* 0xC050EB2F, 0x5A7D1783 */
  209. -3.31231299649172967747e+02, /* 0xC074B3B3, 0x6742CC63 */
  210. -3.46433388365604912451e+02, /* 0xC075A6EF, 0x28A38BD7 */
  211. };
  212. static const double pS5[5] = {
  213. 6.07539382692300335975e+01, /* 0x404E6081, 0x0C98C5DE */
  214. 1.05125230595704579173e+03, /* 0x40906D02, 0x5C7E2864 */
  215. 5.97897094333855784498e+03, /* 0x40B75AF8, 0x8FBE1D60 */
  216. 9.62544514357774460223e+03, /* 0x40C2CCB8, 0xFA76FA38 */
  217. 2.40605815922939109441e+03, /* 0x40A2CC1D, 0xC70BE864 */
  218. };
  219. static const double pR3[6] = {/* for x in [4.547,2.8571]=1/[0.2199,0.35001] */
  220. -2.54704601771951915620e-09, /* 0xBE25E103, 0x6FE1AA86 */
  221. -7.03119616381481654654e-02, /* 0xBFB1FFF6, 0xF7C0E24B */
  222. -2.40903221549529611423e+00, /* 0xC00345B2, 0xAEA48074 */
  223. -2.19659774734883086467e+01, /* 0xC035F74A, 0x4CB94E14 */
  224. -5.80791704701737572236e+01, /* 0xC04D0A22, 0x420A1A45 */
  225. -3.14479470594888503854e+01, /* 0xC03F72AC, 0xA892D80F */
  226. };
  227. static const double pS3[5] = {
  228. 3.58560338055209726349e+01, /* 0x4041ED92, 0x84077DD3 */
  229. 3.61513983050303863820e+02, /* 0x40769839, 0x464A7C0E */
  230. 1.19360783792111533330e+03, /* 0x4092A66E, 0x6D1061D6 */
  231. 1.12799679856907414432e+03, /* 0x40919FFC, 0xB8C39B7E */
  232. 1.73580930813335754692e+02, /* 0x4065B296, 0xFC379081 */
  233. };
  234. static const double pR2[6] = {/* for x in [2.8570,2]=1/[0.3499,0.5] */
  235. -8.87534333032526411254e-08, /* 0xBE77D316, 0xE927026D */
  236. -7.03030995483624743247e-02, /* 0xBFB1FF62, 0x495E1E42 */
  237. -1.45073846780952986357e+00, /* 0xBFF73639, 0x8A24A843 */
  238. -7.63569613823527770791e+00, /* 0xC01E8AF3, 0xEDAFA7F3 */
  239. -1.11931668860356747786e+01, /* 0xC02662E6, 0xC5246303 */
  240. -3.23364579351335335033e+00, /* 0xC009DE81, 0xAF8FE70F */
  241. };
  242. static const double pS2[5] = {
  243. 2.22202997532088808441e+01, /* 0x40363865, 0x908B5959 */
  244. 1.36206794218215208048e+02, /* 0x4061069E, 0x0EE8878F */
  245. 2.70470278658083486789e+02, /* 0x4070E786, 0x42EA079B */
  246. 1.53875394208320329881e+02, /* 0x40633C03, 0x3AB6FAFF */
  247. 1.46576176948256193810e+01, /* 0x402D50B3, 0x44391809 */
  248. };
  249. static double pzero(double x)
  250. {
  251. const double *p = 0,*q = 0;
  252. double z,r,s;
  253. int32_t ix;
  254. GET_HIGH_WORD(ix,x);
  255. ix &= 0x7fffffff;
  256. if(ix>=0x40200000) {p = pR8; q= pS8;}
  257. else if(ix>=0x40122E8B){p = pR5; q= pS5;}
  258. else if(ix>=0x4006DB6D){p = pR3; q= pS3;}
  259. else if(ix>=0x40000000){p = pR2; q= pS2;}
  260. z = one/(x*x);
  261. r = p[0]+z*(p[1]+z*(p[2]+z*(p[3]+z*(p[4]+z*p[5]))));
  262. s = one+z*(q[0]+z*(q[1]+z*(q[2]+z*(q[3]+z*q[4]))));
  263. return one+ r/s;
  264. }
  265. /* For x >= 8, the asymptotic expansions of qzero is
  266. * -1/8 s + 75/1024 s^3 - ..., where s = 1/x.
  267. * We approximate pzero by
  268. * qzero(x) = s*(-1.25 + (R/S))
  269. * where R = qR0 + qR1*s^2 + qR2*s^4 + ... + qR5*s^10
  270. * S = 1 + qS0*s^2 + ... + qS5*s^12
  271. * and
  272. * | qzero(x)/s +1.25-R/S | <= 2 ** ( -61.22)
  273. */
  274. static const double qR8[6] = { /* for x in [inf, 8]=1/[0,0.125] */
  275. 0.00000000000000000000e+00, /* 0x00000000, 0x00000000 */
  276. 7.32421874999935051953e-02, /* 0x3FB2BFFF, 0xFFFFFE2C */
  277. 1.17682064682252693899e+01, /* 0x40278952, 0x5BB334D6 */
  278. 5.57673380256401856059e+02, /* 0x40816D63, 0x15301825 */
  279. 8.85919720756468632317e+03, /* 0x40C14D99, 0x3E18F46D */
  280. 3.70146267776887834771e+04, /* 0x40E212D4, 0x0E901566 */
  281. };
  282. static const double qS8[6] = {
  283. 1.63776026895689824414e+02, /* 0x406478D5, 0x365B39BC */
  284. 8.09834494656449805916e+03, /* 0x40BFA258, 0x4E6B0563 */
  285. 1.42538291419120476348e+05, /* 0x41016652, 0x54D38C3F */
  286. 8.03309257119514397345e+05, /* 0x412883DA, 0x83A52B43 */
  287. 8.40501579819060512818e+05, /* 0x4129A66B, 0x28DE0B3D */
  288. -3.43899293537866615225e+05, /* 0xC114FD6D, 0x2C9530C5 */
  289. };
  290. static const double qR5[6] = { /* for x in [8,4.5454]=1/[0.125,0.22001] */
  291. 1.84085963594515531381e-11, /* 0x3DB43D8F, 0x29CC8CD9 */
  292. 7.32421766612684765896e-02, /* 0x3FB2BFFF, 0xD172B04C */
  293. 5.83563508962056953777e+00, /* 0x401757B0, 0xB9953DD3 */
  294. 1.35111577286449829671e+02, /* 0x4060E392, 0x0A8788E9 */
  295. 1.02724376596164097464e+03, /* 0x40900CF9, 0x9DC8C481 */
  296. 1.98997785864605384631e+03, /* 0x409F17E9, 0x53C6E3A6 */
  297. };
  298. static const double qS5[6] = {
  299. 8.27766102236537761883e+01, /* 0x4054B1B3, 0xFB5E1543 */
  300. 2.07781416421392987104e+03, /* 0x40A03BA0, 0xDA21C0CE */
  301. 1.88472887785718085070e+04, /* 0x40D267D2, 0x7B591E6D */
  302. 5.67511122894947329769e+04, /* 0x40EBB5E3, 0x97E02372 */
  303. 3.59767538425114471465e+04, /* 0x40E19118, 0x1F7A54A0 */
  304. -5.35434275601944773371e+03, /* 0xC0B4EA57, 0xBEDBC609 */
  305. };
  306. static const double qR3[6] = {/* for x in [4.547,2.8571]=1/[0.2199,0.35001] */
  307. 4.37741014089738620906e-09, /* 0x3E32CD03, 0x6ADECB82 */
  308. 7.32411180042911447163e-02, /* 0x3FB2BFEE, 0x0E8D0842 */
  309. 3.34423137516170720929e+00, /* 0x400AC0FC, 0x61149CF5 */
  310. 4.26218440745412650017e+01, /* 0x40454F98, 0x962DAEDD */
  311. 1.70808091340565596283e+02, /* 0x406559DB, 0xE25EFD1F */
  312. 1.66733948696651168575e+02, /* 0x4064D77C, 0x81FA21E0 */
  313. };
  314. static const double qS3[6] = {
  315. 4.87588729724587182091e+01, /* 0x40486122, 0xBFE343A6 */
  316. 7.09689221056606015736e+02, /* 0x40862D83, 0x86544EB3 */
  317. 3.70414822620111362994e+03, /* 0x40ACF04B, 0xE44DFC63 */
  318. 6.46042516752568917582e+03, /* 0x40B93C6C, 0xD7C76A28 */
  319. 2.51633368920368957333e+03, /* 0x40A3A8AA, 0xD94FB1C0 */
  320. -1.49247451836156386662e+02, /* 0xC062A7EB, 0x201CF40F */
  321. };
  322. static const double qR2[6] = {/* for x in [2.8570,2]=1/[0.3499,0.5] */
  323. 1.50444444886983272379e-07, /* 0x3E84313B, 0x54F76BDB */
  324. 7.32234265963079278272e-02, /* 0x3FB2BEC5, 0x3E883E34 */
  325. 1.99819174093815998816e+00, /* 0x3FFFF897, 0xE727779C */
  326. 1.44956029347885735348e+01, /* 0x402CFDBF, 0xAAF96FE5 */
  327. 3.16662317504781540833e+01, /* 0x403FAA8E, 0x29FBDC4A */
  328. 1.62527075710929267416e+01, /* 0x403040B1, 0x71814BB4 */
  329. };
  330. static const double qS2[6] = {
  331. 3.03655848355219184498e+01, /* 0x403E5D96, 0xF7C07AED */
  332. 2.69348118608049844624e+02, /* 0x4070D591, 0xE4D14B40 */
  333. 8.44783757595320139444e+02, /* 0x408A6645, 0x22B3BF22 */
  334. 8.82935845112488550512e+02, /* 0x408B977C, 0x9C5CC214 */
  335. 2.12666388511798828631e+02, /* 0x406A9553, 0x0E001365 */
  336. -5.31095493882666946917e+00, /* 0xC0153E6A, 0xF8B32931 */
  337. };
  338. static double qzero(double x)
  339. {
  340. const double *p=0,*q=0;
  341. double s,r,z;
  342. int32_t ix;
  343. GET_HIGH_WORD(ix,x);
  344. ix &= 0x7fffffff;
  345. if(ix>=0x40200000) {p = qR8; q= qS8;}
  346. else if(ix>=0x40122E8B){p = qR5; q= qS5;}
  347. else if(ix>=0x4006DB6D){p = qR3; q= qS3;}
  348. else if(ix>=0x40000000){p = qR2; q= qS2;}
  349. z = one/(x*x);
  350. r = p[0]+z*(p[1]+z*(p[2]+z*(p[3]+z*(p[4]+z*p[5]))));
  351. s = one+z*(q[0]+z*(q[1]+z*(q[2]+z*(q[3]+z*(q[4]+z*q[5])))));
  352. return (-.125 + r/s)/x;
  353. }