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  1. /* Copyright (C) 1994 DJ Delorie, see COPYING.DJ for details */
  2. /* e_j1f.c -- float version of e_j1.c.
  3.  * Conversion to float by Ian Lance Taylor, Cygnus Support, ian@cygnus.com.
  4.  */
  5.  
  6. /*
  7.  * ====================================================
  8.  * Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved.
  9.  *
  10.  * Developed at SunPro, a Sun Microsystems, Inc. business.
  11.  * Permission to use, copy, modify, and distribute this
  12.  * software is freely granted, provided that this notice
  13.  * is preserved.
  14.  * ====================================================
  15.  */
  16.  
  17. #if defined(LIBM_SCCS) && !defined(lint)
  18. static char rcsid[] = "$Id: e_j1f.c,v 1.2 1994/08/18 23:05:35 jtc Exp $";
  19. #endif
  20.  
  21. #include "math.h"
  22. #include "math_private.h"
  23.  
  24. #ifdef __STDC__
  25. static float ponef(float), qonef(float);
  26. #else
  27. static float ponef(), qonef();
  28. #endif
  29.  
  30. #ifdef __STDC__
  31. static const float
  32. #else
  33. static float
  34. #endif
  35. huge    = 1e30,
  36. one     = 1.0,
  37. invsqrtpi=  5.6418961287e-01, /* 0x3f106ebb */
  38. tpi      =  6.3661974669e-01, /* 0x3f22f983 */
  39.         /* R0/S0 on [0,2] */
  40. r00  = -6.2500000000e-02, /* 0xbd800000 */
  41. r01  =  1.4070566976e-03, /* 0x3ab86cfd */
  42. r02  = -1.5995563444e-05, /* 0xb7862e36 */
  43. r03  =  4.9672799207e-08, /* 0x335557d2 */
  44. s01  =  1.9153760746e-02, /* 0x3c9ce859 */
  45. s02  =  1.8594678841e-04, /* 0x3942fab6 */
  46. s03  =  1.1771846857e-06, /* 0x359dffc2 */
  47. s04  =  5.0463624390e-09, /* 0x31ad6446 */
  48. s05  =  1.2354227016e-11; /* 0x2d59567e */
  49.  
  50. #ifdef __STDC__
  51. static const float zero    = 0.0;
  52. #else
  53. static float zero    = 0.0;
  54. #endif
  55.  
  56. #ifdef __STDC__
  57.         float __ieee754_j1f(float x)
  58. #else
  59.         float __ieee754_j1f(x)
  60.         float x;
  61. #endif
  62. {
  63.         float z, s,c,ss,cc,r,u,v,y;
  64.         int32_t hx,ix;
  65.  
  66.         GET_FLOAT_WORD(hx,x);
  67.         ix = hx&0x7fffffff;
  68.         if(ix>=0x7f800000) return one/x;
  69.         y = fabsf(x);
  70.         if(ix >= 0x40000000) {  /* |x| >= 2.0 */
  71.                 s = sinf(y);
  72.                 c = cosf(y);
  73.                 ss = -s-c;
  74.                 cc = s-c;
  75.                 if(ix<0x7f000000) {  /* make sure y+y not overflow */
  76.                     z = cosf(y+y);
  77.                     if ((s*c)>zero) cc = z/ss;
  78.                     else            ss = z/cc;
  79.                 }
  80.         /*
  81.          * j1(x) = 1/sqrt(pi) * (P(1,x)*cc - Q(1,x)*ss) / sqrt(x)
  82.          * y1(x) = 1/sqrt(pi) * (P(1,x)*ss + Q(1,x)*cc) / sqrt(x)
  83.          */
  84.                 if(ix>0x80000000) z = (invsqrtpi*cc)/sqrtf(y);
  85.                 else {
  86.                     u = ponef(y); v = qonef(y);
  87.                     z = invsqrtpi*(u*cc-v*ss)/sqrtf(y);
  88.                 }
  89.                 if(hx<0) return -z;
  90.                 else     return  z;
  91.         }
  92.         if(ix<0x32000000) {     /* |x|<2**-27 */
  93.             if(huge+x>one) return (float)0.5*x;/* inexact if x!=0 necessary */
  94.         }
  95.         z = x*x;
  96.         r =  z*(r00+z*(r01+z*(r02+z*r03)));
  97.         s =  one+z*(s01+z*(s02+z*(s03+z*(s04+z*s05))));
  98.         r *= x;
  99.         return(x*(float)0.5+r/s);
  100. }
  101.  
  102. #ifdef __STDC__
  103. static const float U0[5] = {
  104. #else
  105. static float U0[5] = {
  106. #endif
  107.  -1.9605709612e-01, /* 0xbe48c331 */
  108.   5.0443872809e-02, /* 0x3d4e9e3c */
  109.  -1.9125689287e-03, /* 0xbafaaf2a */
  110.   2.3525259166e-05, /* 0x37c5581c */
  111.  -9.1909917899e-08, /* 0xb3c56003 */
  112. };
  113. #ifdef __STDC__
  114. static const float V0[5] = {
  115. #else
  116. static float V0[5] = {
  117. #endif
  118.   1.9916731864e-02, /* 0x3ca3286a */
  119.   2.0255257550e-04, /* 0x3954644b */
  120.   1.3560879779e-06, /* 0x35b602d4 */
  121.   6.2274145840e-09, /* 0x31d5f8eb */
  122.   1.6655924903e-11, /* 0x2d9281cf */
  123. };
  124.  
  125. #ifdef __STDC__
  126.         float __ieee754_y1f(float x)
  127. #else
  128.         float __ieee754_y1f(x)
  129.         float x;
  130. #endif
  131. {
  132.         float z, s,c,ss,cc,u,v;
  133.         int32_t hx,ix;
  134.  
  135.         GET_FLOAT_WORD(hx,x);
  136.         ix = 0x7fffffff&hx;
  137.     /* if Y1(NaN) is NaN, Y1(-inf) is NaN, Y1(inf) is 0 */
  138.         if(ix>=0x7f800000) return  one/(x+x*x);
  139.         if(ix==0) return -one/zero;
  140.         if(hx<0) return zero/zero;
  141.         if(ix >= 0x40000000) {  /* |x| >= 2.0 */
  142.                 s = sinf(x);
  143.                 c = cosf(x);
  144.                 ss = -s-c;
  145.                 cc = s-c;
  146.                 if(ix<0x7f000000) {  /* make sure x+x not overflow */
  147.                     z = cosf(x+x);
  148.                     if ((s*c)>zero) cc = z/ss;
  149.                     else            ss = z/cc;
  150.                 }
  151.         /* y1(x) = sqrt(2/(pi*x))*(p1(x)*sin(x0)+q1(x)*cos(x0))
  152.          * where x0 = x-3pi/4
  153.          *      Better formula:
  154.          *              cos(x0) = cos(x)cos(3pi/4)+sin(x)sin(3pi/4)
  155.          *                      =  1/sqrt(2) * (sin(x) - cos(x))
  156.          *              sin(x0) = sin(x)cos(3pi/4)-cos(x)sin(3pi/4)
  157.          *                      = -1/sqrt(2) * (cos(x) + sin(x))
  158.          * To avoid cancellation, use
  159.          *              sin(x) +- cos(x) = -cos(2x)/(sin(x) -+ cos(x))
  160.          * to compute the worse one.
  161.          */
  162.                 if(ix>0x48000000) z = (invsqrtpi*ss)/sqrtf(x);
  163.                 else {
  164.                     u = ponef(x); v = qonef(x);
  165.                     z = invsqrtpi*(u*ss+v*cc)/sqrtf(x);
  166.                 }
  167.                 return z;
  168.         }
  169.         if(ix<=0x24800000) {    /* x < 2**-54 */
  170.             return(-tpi/x);
  171.         }
  172.         z = x*x;
  173.         u = U0[0]+z*(U0[1]+z*(U0[2]+z*(U0[3]+z*U0[4])));
  174.         v = one+z*(V0[0]+z*(V0[1]+z*(V0[2]+z*(V0[3]+z*V0[4]))));
  175.         return(x*(u/v) + tpi*(__ieee754_j1f(x)*__ieee754_logf(x)-one/x));
  176. }
  177.  
  178. /* For x >= 8, the asymptotic expansions of pone is
  179.  *      1 + 15/128 s^2 - 4725/2^15 s^4 - ...,   where s = 1/x.
  180.  * We approximate pone by
  181.  *      pone(x) = 1 + (R/S)
  182.  * where  R = pr0 + pr1*s^2 + pr2*s^4 + ... + pr5*s^10
  183.  *        S = 1 + ps0*s^2 + ... + ps4*s^10
  184.  * and
  185.  *      | pone(x)-1-R/S | <= 2  ** ( -60.06)
  186.  */
  187.  
  188. #ifdef __STDC__
  189. static const float pr8[6] = { /* for x in [inf, 8]=1/[0,0.125] */
  190. #else
  191. static float pr8[6] = { /* for x in [inf, 8]=1/[0,0.125] */
  192. #endif
  193.   0.0000000000e+00, /* 0x00000000 */
  194.   1.1718750000e-01, /* 0x3df00000 */
  195.   1.3239480972e+01, /* 0x4153d4ea */
  196.   4.1205184937e+02, /* 0x43ce06a3 */
  197.   3.8747453613e+03, /* 0x45722bed */
  198.   7.9144794922e+03, /* 0x45f753d6 */
  199. };
  200. #ifdef __STDC__
  201. static const float ps8[5] = {
  202. #else
  203. static float ps8[5] = {
  204. #endif
  205.   1.1420736694e+02, /* 0x42e46a2c */
  206.   3.6509309082e+03, /* 0x45642ee5 */
  207.   3.6956207031e+04, /* 0x47105c35 */
  208.   9.7602796875e+04, /* 0x47bea166 */
  209.   3.0804271484e+04, /* 0x46f0a88b */
  210. };
  211.  
  212. #ifdef __STDC__
  213. static const float pr5[6] = { /* for x in [8,4.5454]=1/[0.125,0.22001] */
  214. #else
  215. static float pr5[6] = { /* for x in [8,4.5454]=1/[0.125,0.22001] */
  216. #endif
  217.   1.3199052094e-11, /* 0x2d68333f */
  218.   1.1718749255e-01, /* 0x3defffff */
  219.   6.8027510643e+00, /* 0x40d9b023 */
  220.   1.0830818176e+02, /* 0x42d89dca */
  221.   5.1763616943e+02, /* 0x440168b7 */
  222.   5.2871520996e+02, /* 0x44042dc6 */
  223. };
  224. #ifdef __STDC__
  225. static const float ps5[5] = {
  226. #else
  227. static float ps5[5] = {
  228. #endif
  229.   5.9280597687e+01, /* 0x426d1f55 */
  230.   9.9140142822e+02, /* 0x4477d9b1 */
  231.   5.3532670898e+03, /* 0x45a74a23 */
  232.   7.8446904297e+03, /* 0x45f52586 */
  233.   1.5040468750e+03, /* 0x44bc0180 */
  234. };
  235.  
  236. #ifdef __STDC__
  237. static const float pr3[6] = {
  238. #else
  239. static float pr3[6] = {/* for x in [4.547,2.8571]=1/[0.2199,0.35001] */
  240. #endif
  241.   3.0250391081e-09, /* 0x314fe10d */
  242.   1.1718686670e-01, /* 0x3defffab */
  243.   3.9329774380e+00, /* 0x407bb5e7 */
  244.   3.5119403839e+01, /* 0x420c7a45 */
  245.   9.1055007935e+01, /* 0x42b61c2a */
  246.   4.8559066772e+01, /* 0x42423c7c */
  247. };
  248. #ifdef __STDC__
  249. static const float ps3[5] = {
  250. #else
  251. static float ps3[5] = {
  252. #endif
  253.   3.4791309357e+01, /* 0x420b2a4d */
  254.   3.3676245117e+02, /* 0x43a86198 */
  255.   1.0468714600e+03, /* 0x4482dbe3 */
  256.   8.9081134033e+02, /* 0x445eb3ed */
  257.   1.0378793335e+02, /* 0x42cf936c */
  258. };
  259.  
  260. #ifdef __STDC__
  261. static const float pr2[6] = {/* for x in [2.8570,2]=1/[0.3499,0.5] */
  262. #else
  263. static float pr2[6] = {/* for x in [2.8570,2]=1/[0.3499,0.5] */
  264. #endif
  265.   1.0771083225e-07, /* 0x33e74ea8 */
  266.   1.1717621982e-01, /* 0x3deffa16 */
  267.   2.3685150146e+00, /* 0x401795c0 */
  268.   1.2242610931e+01, /* 0x4143e1bc */
  269.   1.7693971634e+01, /* 0x418d8d41 */
  270.   5.0735230446e+00, /* 0x40a25a4d */
  271. };
  272. #ifdef __STDC__
  273. static const float ps2[5] = {
  274. #else
  275. static float ps2[5] = {
  276. #endif
  277.   2.1436485291e+01, /* 0x41ab7dec */
  278.   1.2529022980e+02, /* 0x42fa9499 */
  279.   2.3227647400e+02, /* 0x436846c7 */
  280.   1.1767937469e+02, /* 0x42eb5bd7 */
  281.   8.3646392822e+00, /* 0x4105d590 */
  282. };
  283.  
  284. #ifdef __STDC__
  285.         static float ponef(float x)
  286. #else
  287.         static float ponef(x)
  288.         float x;
  289. #endif
  290. {
  291. #ifdef __STDC__
  292.         const float *p,*q;
  293. #else
  294.         float *p,*q;
  295. #endif
  296.         float z,r,s;
  297.         int32_t ix;
  298.         GET_FLOAT_WORD(ix,x);
  299.         ix &= 0x7fffffff;
  300.         if(ix>=0x41000000)     {p = pr8; q= ps8;}
  301.         else if(ix>=0x40f71c58){p = pr5; q= ps5;}
  302.         else if(ix>=0x4036db68){p = pr3; q= ps3;}
  303.         else if(ix>=0x40000000){p = pr2; q= ps2;}
  304.         z = one/(x*x);
  305.         r = p[0]+z*(p[1]+z*(p[2]+z*(p[3]+z*(p[4]+z*p[5]))));
  306.         s = one+z*(q[0]+z*(q[1]+z*(q[2]+z*(q[3]+z*q[4]))));
  307.         return one+ r/s;
  308. }
  309.                
  310.  
  311. /* For x >= 8, the asymptotic expansions of qone is
  312.  *      3/8 s - 105/1024 s^3 - ..., where s = 1/x.
  313.  * We approximate pone by
  314.  *      qone(x) = s*(0.375 + (R/S))
  315.  * where  R = qr1*s^2 + qr2*s^4 + ... + qr5*s^10
  316.  *        S = 1 + qs1*s^2 + ... + qs6*s^12
  317.  * and
  318.  *      | qone(x)/s -0.375-R/S | <= 2  ** ( -61.13)
  319.  */
  320.  
  321. #ifdef __STDC__
  322. static const float qr8[6] = { /* for x in [inf, 8]=1/[0,0.125] */
  323. #else
  324. static float qr8[6] = { /* for x in [inf, 8]=1/[0,0.125] */
  325. #endif
  326.   0.0000000000e+00, /* 0x00000000 */
  327.  -1.0253906250e-01, /* 0xbdd20000 */
  328.  -1.6271753311e+01, /* 0xc1822c8d */
  329.  -7.5960174561e+02, /* 0xc43de683 */
  330.  -1.1849806641e+04, /* 0xc639273a */
  331.  -4.8438511719e+04, /* 0xc73d3683 */
  332. };
  333. #ifdef __STDC__
  334. static const float qs8[6] = {
  335. #else
  336. static float qs8[6] = {
  337. #endif
  338.   1.6139537048e+02, /* 0x43216537 */
  339.   7.8253862305e+03, /* 0x45f48b17 */
  340.   1.3387534375e+05, /* 0x4802bcd6 */
  341.   7.1965775000e+05, /* 0x492fb29c */
  342.   6.6660125000e+05, /* 0x4922be94 */
  343.  -2.9449025000e+05, /* 0xc88fcb48 */
  344. };
  345.  
  346. #ifdef __STDC__
  347. static const float qr5[6] = { /* for x in [8,4.5454]=1/[0.125,0.22001] */
  348. #else
  349. static float qr5[6] = { /* for x in [8,4.5454]=1/[0.125,0.22001] */
  350. #endif
  351.  -2.0897993405e-11, /* 0xadb7d219 */
  352.  -1.0253904760e-01, /* 0xbdd1fffe */
  353.  -8.0564479828e+00, /* 0xc100e736 */
  354.  -1.8366960144e+02, /* 0xc337ab6b */
  355.  -1.3731937256e+03, /* 0xc4aba633 */
  356.  -2.6124443359e+03, /* 0xc523471c */
  357. };
  358. #ifdef __STDC__
  359. static const float qs5[6] = {
  360. #else
  361. static float qs5[6] = {
  362. #endif
  363.   8.1276550293e+01, /* 0x42a28d98 */
  364.   1.9917987061e+03, /* 0x44f8f98f */
  365.   1.7468484375e+04, /* 0x468878f8 */
  366.   4.9851425781e+04, /* 0x4742bb6d */
  367.   2.7948074219e+04, /* 0x46da5826 */
  368.  -4.7191835938e+03, /* 0xc5937978 */
  369. };
  370.  
  371. #ifdef __STDC__
  372. static const float qr3[6] = {
  373. #else
  374. static float qr3[6] = {/* for x in [4.547,2.8571]=1/[0.2199,0.35001] */
  375. #endif
  376.  -5.0783124372e-09, /* 0xb1ae7d4f */
  377.  -1.0253783315e-01, /* 0xbdd1ff5b */
  378.  -4.6101160049e+00, /* 0xc0938612 */
  379.  -5.7847221375e+01, /* 0xc267638e */
  380.  -2.2824453735e+02, /* 0xc3643e9a */
  381.  -2.1921012878e+02, /* 0xc35b35cb */
  382. };
  383. #ifdef __STDC__
  384. static const float qs3[6] = {
  385. #else
  386. static float qs3[6] = {
  387. #endif
  388.   4.7665153503e+01, /* 0x423ea91e */
  389.   6.7386511230e+02, /* 0x4428775e */
  390.   3.3801528320e+03, /* 0x45534272 */
  391.   5.5477290039e+03, /* 0x45ad5dd5 */
  392.   1.9031191406e+03, /* 0x44ede3d0 */
  393.  -1.3520118713e+02, /* 0xc3073381 */
  394. };
  395.  
  396. #ifdef __STDC__
  397. static const float qr2[6] = {/* for x in [2.8570,2]=1/[0.3499,0.5] */
  398. #else
  399. static float qr2[6] = {/* for x in [2.8570,2]=1/[0.3499,0.5] */
  400. #endif
  401.  -1.7838172539e-07, /* 0xb43f8932 */
  402.  -1.0251704603e-01, /* 0xbdd1f475 */
  403.  -2.7522056103e+00, /* 0xc0302423 */
  404.  -1.9663616180e+01, /* 0xc19d4f16 */
  405.  -4.2325313568e+01, /* 0xc2294d1f */
  406.  -2.1371921539e+01, /* 0xc1aaf9b2 */
  407. };
  408. #ifdef __STDC__
  409. static const float qs2[6] = {
  410. #else
  411. static float qs2[6] = {
  412. #endif
  413.   2.9533363342e+01, /* 0x41ec4454 */
  414.   2.5298155212e+02, /* 0x437cfb47 */
  415.   7.5750280762e+02, /* 0x443d602e */
  416.   7.3939318848e+02, /* 0x4438d92a */
  417.   1.5594900513e+02, /* 0x431bf2f2 */
  418.  -4.9594988823e+00, /* 0xc09eb437 */
  419. };
  420.  
  421. #ifdef __STDC__
  422.         static float qonef(float x)
  423. #else
  424.         static float qonef(x)
  425.         float x;
  426. #endif
  427. {
  428. #ifdef __STDC__
  429.         const float *p,*q;
  430. #else
  431.         float *p,*q;
  432. #endif
  433.         float  s,r,z;
  434.         int32_t ix;
  435.         GET_FLOAT_WORD(ix,x);
  436.         ix &= 0x7fffffff;
  437.         if(ix>=0x40200000)     {p = qr8; q= qs8;}
  438.         else if(ix>=0x40f71c58){p = qr5; q= qs5;}
  439.         else if(ix>=0x4036db68){p = qr3; q= qs3;}
  440.         else if(ix>=0x40000000){p = qr2; q= qs2;}
  441.         z = one/(x*x);
  442.         r = p[0]+z*(p[1]+z*(p[2]+z*(p[3]+z*(p[4]+z*p[5]))));
  443.         s = one+z*(q[0]+z*(q[1]+z*(q[2]+z*(q[3]+z*(q[4]+z*q[5])))));
  444.         return ((float).375 + r/s)/x;
  445. }
  446.