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  1. /* ef_j1.c -- float version of e_j1.c.
  2.  * Conversion to float by Ian Lance Taylor, Cygnus Support, ian@cygnus.com.
  3.  */
  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.  
  16. #include "fdlibm.h"
  17.  
  18. #ifdef __STDC__
  19. static float ponef(float), qonef(float);
  20. #else
  21. static float ponef(), qonef();
  22. #endif
  23.  
  24. #ifdef __STDC__
  25. static const float
  26. #else
  27. static float
  28. #endif
  29. huge    = 1e30,
  30. one     = 1.0,
  31. invsqrtpi=  5.6418961287e-01, /* 0x3f106ebb */
  32. tpi      =  6.3661974669e-01, /* 0x3f22f983 */
  33.         /* R0/S0 on [0,2] */
  34. r00  = -6.2500000000e-02, /* 0xbd800000 */
  35. r01  =  1.4070566976e-03, /* 0x3ab86cfd */
  36. r02  = -1.5995563444e-05, /* 0xb7862e36 */
  37. r03  =  4.9672799207e-08, /* 0x335557d2 */
  38. s01  =  1.9153760746e-02, /* 0x3c9ce859 */
  39. s02  =  1.8594678841e-04, /* 0x3942fab6 */
  40. s03  =  1.1771846857e-06, /* 0x359dffc2 */
  41. s04  =  5.0463624390e-09, /* 0x31ad6446 */
  42. s05  =  1.2354227016e-11; /* 0x2d59567e */
  43.  
  44. #ifdef __STDC__
  45. static const float zero    = 0.0;
  46. #else
  47. static float zero    = 0.0;
  48. #endif
  49.  
  50. #ifdef __STDC__
  51.         float __ieee754_j1f(float x)
  52. #else
  53.         float __ieee754_j1f(x)
  54.         float x;
  55. #endif
  56. {
  57.         float z, s,c,ss,cc,r,u,v,y;
  58.         __int32_t hx,ix;
  59.  
  60.         GET_FLOAT_WORD(hx,x);
  61.         ix = hx&0x7fffffff;
  62.         if(!FLT_UWORD_IS_FINITE(ix)) return one/x;
  63.         y = fabsf(x);
  64.         if(ix >= 0x40000000) {  /* |x| >= 2.0 */
  65.                 s = sinf(y);
  66.                 c = cosf(y);
  67.                 ss = -s-c;
  68.                 cc = s-c;
  69.                 if(ix<=FLT_UWORD_HALF_MAX) {  /* make sure y+y not overflow */
  70.                     z = cosf(y+y);
  71.                     if ((s*c)>zero) cc = z/ss;
  72.                     else            ss = z/cc;
  73.                 }
  74.         /*
  75.          * j1(x) = 1/sqrt(pi) * (P(1,x)*cc - Q(1,x)*ss) / sqrt(x)
  76.          * y1(x) = 1/sqrt(pi) * (P(1,x)*ss + Q(1,x)*cc) / sqrt(x)
  77.          */
  78.                 if(ix>0x80000000) z = (invsqrtpi*cc)/__ieee754_sqrtf(y);
  79.                 else {
  80.                     u = ponef(y); v = qonef(y);
  81.                     z = invsqrtpi*(u*cc-v*ss)/__ieee754_sqrtf(y);
  82.                 }
  83.                 if(hx<0) return -z;
  84.                 else     return  z;
  85.         }
  86.         if(ix<0x32000000) {     /* |x|<2**-27 */
  87.             if(huge+x>one) return (float)0.5*x;/* inexact if x!=0 necessary */
  88.         }
  89.         z = x*x;
  90.         r =  z*(r00+z*(r01+z*(r02+z*r03)));
  91.         s =  one+z*(s01+z*(s02+z*(s03+z*(s04+z*s05))));
  92.         r *= x;
  93.         return(x*(float)0.5+r/s);
  94. }
  95.  
  96. #ifdef __STDC__
  97. static const float U0[5] = {
  98. #else
  99. static float U0[5] = {
  100. #endif
  101.  -1.9605709612e-01, /* 0xbe48c331 */
  102.   5.0443872809e-02, /* 0x3d4e9e3c */
  103.  -1.9125689287e-03, /* 0xbafaaf2a */
  104.   2.3525259166e-05, /* 0x37c5581c */
  105.  -9.1909917899e-08, /* 0xb3c56003 */
  106. };
  107. #ifdef __STDC__
  108. static const float V0[5] = {
  109. #else
  110. static float V0[5] = {
  111. #endif
  112.   1.9916731864e-02, /* 0x3ca3286a */
  113.   2.0255257550e-04, /* 0x3954644b */
  114.   1.3560879779e-06, /* 0x35b602d4 */
  115.   6.2274145840e-09, /* 0x31d5f8eb */
  116.   1.6655924903e-11, /* 0x2d9281cf */
  117. };
  118.  
  119. #ifdef __STDC__
  120.         float __ieee754_y1f(float x)
  121. #else
  122.         float __ieee754_y1f(x)
  123.         float x;
  124. #endif
  125. {
  126.         float z, s,c,ss,cc,u,v;
  127.         __int32_t hx,ix;
  128.  
  129.         GET_FLOAT_WORD(hx,x);
  130.         ix = 0x7fffffff&hx;
  131.     /* if Y1(NaN) is NaN, Y1(-inf) is NaN, Y1(inf) is 0 */
  132.         if(!FLT_UWORD_IS_FINITE(ix)) return one/(x+x*x);
  133.         if(FLT_UWORD_IS_ZERO(ix)) return -one/zero;
  134.         if(hx<0) return zero/zero;
  135.         if(ix >= 0x40000000) {  /* |x| >= 2.0 */
  136.                 s = sinf(x);
  137.                 c = cosf(x);
  138.                 ss = -s-c;
  139.                 cc = s-c;
  140.                 if(ix<=FLT_UWORD_HALF_MAX) {  /* make sure x+x not overflow */
  141.                     z = cosf(x+x);
  142.                     if ((s*c)>zero) cc = z/ss;
  143.                     else            ss = z/cc;
  144.                 }
  145.         /* y1(x) = sqrt(2/(pi*x))*(p1(x)*sin(x0)+q1(x)*cos(x0))
  146.          * where x0 = x-3pi/4
  147.          *      Better formula:
  148.          *              cos(x0) = cos(x)cos(3pi/4)+sin(x)sin(3pi/4)
  149.          *                      =  1/sqrt(2) * (sin(x) - cos(x))
  150.          *              sin(x0) = sin(x)cos(3pi/4)-cos(x)sin(3pi/4)
  151.          *                      = -1/sqrt(2) * (cos(x) + sin(x))
  152.          * To avoid cancellation, use
  153.          *              sin(x) +- cos(x) = -cos(2x)/(sin(x) -+ cos(x))
  154.          * to compute the worse one.
  155.          */
  156.                 if(ix>0x48000000) z = (invsqrtpi*ss)/__ieee754_sqrtf(x);
  157.                 else {
  158.                     u = ponef(x); v = qonef(x);
  159.                     z = invsqrtpi*(u*ss+v*cc)/__ieee754_sqrtf(x);
  160.                 }
  161.                 return z;
  162.         }
  163.         if(ix<=0x24800000) {    /* x < 2**-54 */
  164.             return(-tpi/x);
  165.         }
  166.         z = x*x;
  167.         u = U0[0]+z*(U0[1]+z*(U0[2]+z*(U0[3]+z*U0[4])));
  168.         v = one+z*(V0[0]+z*(V0[1]+z*(V0[2]+z*(V0[3]+z*V0[4]))));
  169.         return(x*(u/v) + tpi*(__ieee754_j1f(x)*__ieee754_logf(x)-one/x));
  170. }
  171.  
  172. /* For x >= 8, the asymptotic expansions of pone is
  173.  *      1 + 15/128 s^2 - 4725/2^15 s^4 - ...,   where s = 1/x.
  174.  * We approximate pone by
  175.  *      pone(x) = 1 + (R/S)
  176.  * where  R = pr0 + pr1*s^2 + pr2*s^4 + ... + pr5*s^10
  177.  *        S = 1 + ps0*s^2 + ... + ps4*s^10
  178.  * and
  179.  *      | pone(x)-1-R/S | <= 2  ** ( -60.06)
  180.  */
  181.  
  182. #ifdef __STDC__
  183. static const float pr8[6] = { /* for x in [inf, 8]=1/[0,0.125] */
  184. #else
  185. static float pr8[6] = { /* for x in [inf, 8]=1/[0,0.125] */
  186. #endif
  187.   0.0000000000e+00, /* 0x00000000 */
  188.   1.1718750000e-01, /* 0x3df00000 */
  189.   1.3239480972e+01, /* 0x4153d4ea */
  190.   4.1205184937e+02, /* 0x43ce06a3 */
  191.   3.8747453613e+03, /* 0x45722bed */
  192.   7.9144794922e+03, /* 0x45f753d6 */
  193. };
  194. #ifdef __STDC__
  195. static const float ps8[5] = {
  196. #else
  197. static float ps8[5] = {
  198. #endif
  199.   1.1420736694e+02, /* 0x42e46a2c */
  200.   3.6509309082e+03, /* 0x45642ee5 */
  201.   3.6956207031e+04, /* 0x47105c35 */
  202.   9.7602796875e+04, /* 0x47bea166 */
  203.   3.0804271484e+04, /* 0x46f0a88b */
  204. };
  205.  
  206. #ifdef __STDC__
  207. static const float pr5[6] = { /* for x in [8,4.5454]=1/[0.125,0.22001] */
  208. #else
  209. static float pr5[6] = { /* for x in [8,4.5454]=1/[0.125,0.22001] */
  210. #endif
  211.   1.3199052094e-11, /* 0x2d68333f */
  212.   1.1718749255e-01, /* 0x3defffff */
  213.   6.8027510643e+00, /* 0x40d9b023 */
  214.   1.0830818176e+02, /* 0x42d89dca */
  215.   5.1763616943e+02, /* 0x440168b7 */
  216.   5.2871520996e+02, /* 0x44042dc6 */
  217. };
  218. #ifdef __STDC__
  219. static const float ps5[5] = {
  220. #else
  221. static float ps5[5] = {
  222. #endif
  223.   5.9280597687e+01, /* 0x426d1f55 */
  224.   9.9140142822e+02, /* 0x4477d9b1 */
  225.   5.3532670898e+03, /* 0x45a74a23 */
  226.   7.8446904297e+03, /* 0x45f52586 */
  227.   1.5040468750e+03, /* 0x44bc0180 */
  228. };
  229.  
  230. #ifdef __STDC__
  231. static const float pr3[6] = {
  232. #else
  233. static float pr3[6] = {/* for x in [4.547,2.8571]=1/[0.2199,0.35001] */
  234. #endif
  235.   3.0250391081e-09, /* 0x314fe10d */
  236.   1.1718686670e-01, /* 0x3defffab */
  237.   3.9329774380e+00, /* 0x407bb5e7 */
  238.   3.5119403839e+01, /* 0x420c7a45 */
  239.   9.1055007935e+01, /* 0x42b61c2a */
  240.   4.8559066772e+01, /* 0x42423c7c */
  241. };
  242. #ifdef __STDC__
  243. static const float ps3[5] = {
  244. #else
  245. static float ps3[5] = {
  246. #endif
  247.   3.4791309357e+01, /* 0x420b2a4d */
  248.   3.3676245117e+02, /* 0x43a86198 */
  249.   1.0468714600e+03, /* 0x4482dbe3 */
  250.   8.9081134033e+02, /* 0x445eb3ed */
  251.   1.0378793335e+02, /* 0x42cf936c */
  252. };
  253.  
  254. #ifdef __STDC__
  255. static const float pr2[6] = {/* for x in [2.8570,2]=1/[0.3499,0.5] */
  256. #else
  257. static float pr2[6] = {/* for x in [2.8570,2]=1/[0.3499,0.5] */
  258. #endif
  259.   1.0771083225e-07, /* 0x33e74ea8 */
  260.   1.1717621982e-01, /* 0x3deffa16 */
  261.   2.3685150146e+00, /* 0x401795c0 */
  262.   1.2242610931e+01, /* 0x4143e1bc */
  263.   1.7693971634e+01, /* 0x418d8d41 */
  264.   5.0735230446e+00, /* 0x40a25a4d */
  265. };
  266. #ifdef __STDC__
  267. static const float ps2[5] = {
  268. #else
  269. static float ps2[5] = {
  270. #endif
  271.   2.1436485291e+01, /* 0x41ab7dec */
  272.   1.2529022980e+02, /* 0x42fa9499 */
  273.   2.3227647400e+02, /* 0x436846c7 */
  274.   1.1767937469e+02, /* 0x42eb5bd7 */
  275.   8.3646392822e+00, /* 0x4105d590 */
  276. };
  277.  
  278. #ifdef __STDC__
  279.         static float ponef(float x)
  280. #else
  281.         static float ponef(x)
  282.         float x;
  283. #endif
  284. {
  285. #ifdef __STDC__
  286.         const float *p,*q;
  287. #else
  288.         float *p,*q;
  289. #endif
  290.         float z,r,s;
  291.         __int32_t ix;
  292.         GET_FLOAT_WORD(ix,x);
  293.         ix &= 0x7fffffff;
  294.         if(ix>=0x41000000)     {p = pr8; q= ps8;}
  295.         else if(ix>=0x40f71c58){p = pr5; q= ps5;}
  296.         else if(ix>=0x4036db68){p = pr3; q= ps3;}
  297.         else {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.                
  304.  
  305. /* For x >= 8, the asymptotic expansions of qone is
  306.  *      3/8 s - 105/1024 s^3 - ..., where s = 1/x.
  307.  * We approximate qone by
  308.  *      qone(x) = s*(0.375 + (R/S))
  309.  * where  R = qr1*s^2 + qr2*s^4 + ... + qr5*s^10
  310.  *        S = 1 + qs1*s^2 + ... + qs6*s^12
  311.  * and
  312.  *      | qone(x)/s -0.375-R/S | <= 2  ** ( -61.13)
  313.  */
  314.  
  315. #ifdef __STDC__
  316. static const float qr8[6] = { /* for x in [inf, 8]=1/[0,0.125] */
  317. #else
  318. static float qr8[6] = { /* for x in [inf, 8]=1/[0,0.125] */
  319. #endif
  320.   0.0000000000e+00, /* 0x00000000 */
  321.  -1.0253906250e-01, /* 0xbdd20000 */
  322.  -1.6271753311e+01, /* 0xc1822c8d */
  323.  -7.5960174561e+02, /* 0xc43de683 */
  324.  -1.1849806641e+04, /* 0xc639273a */
  325.  -4.8438511719e+04, /* 0xc73d3683 */
  326. };
  327. #ifdef __STDC__
  328. static const float qs8[6] = {
  329. #else
  330. static float qs8[6] = {
  331. #endif
  332.   1.6139537048e+02, /* 0x43216537 */
  333.   7.8253862305e+03, /* 0x45f48b17 */
  334.   1.3387534375e+05, /* 0x4802bcd6 */
  335.   7.1965775000e+05, /* 0x492fb29c */
  336.   6.6660125000e+05, /* 0x4922be94 */
  337.  -2.9449025000e+05, /* 0xc88fcb48 */
  338. };
  339.  
  340. #ifdef __STDC__
  341. static const float qr5[6] = { /* for x in [8,4.5454]=1/[0.125,0.22001] */
  342. #else
  343. static float qr5[6] = { /* for x in [8,4.5454]=1/[0.125,0.22001] */
  344. #endif
  345.  -2.0897993405e-11, /* 0xadb7d219 */
  346.  -1.0253904760e-01, /* 0xbdd1fffe */
  347.  -8.0564479828e+00, /* 0xc100e736 */
  348.  -1.8366960144e+02, /* 0xc337ab6b */
  349.  -1.3731937256e+03, /* 0xc4aba633 */
  350.  -2.6124443359e+03, /* 0xc523471c */
  351. };
  352. #ifdef __STDC__
  353. static const float qs5[6] = {
  354. #else
  355. static float qs5[6] = {
  356. #endif
  357.   8.1276550293e+01, /* 0x42a28d98 */
  358.   1.9917987061e+03, /* 0x44f8f98f */
  359.   1.7468484375e+04, /* 0x468878f8 */
  360.   4.9851425781e+04, /* 0x4742bb6d */
  361.   2.7948074219e+04, /* 0x46da5826 */
  362.  -4.7191835938e+03, /* 0xc5937978 */
  363. };
  364.  
  365. #ifdef __STDC__
  366. static const float qr3[6] = {
  367. #else
  368. static float qr3[6] = {/* for x in [4.547,2.8571]=1/[0.2199,0.35001] */
  369. #endif
  370.  -5.0783124372e-09, /* 0xb1ae7d4f */
  371.  -1.0253783315e-01, /* 0xbdd1ff5b */
  372.  -4.6101160049e+00, /* 0xc0938612 */
  373.  -5.7847221375e+01, /* 0xc267638e */
  374.  -2.2824453735e+02, /* 0xc3643e9a */
  375.  -2.1921012878e+02, /* 0xc35b35cb */
  376. };
  377. #ifdef __STDC__
  378. static const float qs3[6] = {
  379. #else
  380. static float qs3[6] = {
  381. #endif
  382.   4.7665153503e+01, /* 0x423ea91e */
  383.   6.7386511230e+02, /* 0x4428775e */
  384.   3.3801528320e+03, /* 0x45534272 */
  385.   5.5477290039e+03, /* 0x45ad5dd5 */
  386.   1.9031191406e+03, /* 0x44ede3d0 */
  387.  -1.3520118713e+02, /* 0xc3073381 */
  388. };
  389.  
  390. #ifdef __STDC__
  391. static const float qr2[6] = {/* for x in [2.8570,2]=1/[0.3499,0.5] */
  392. #else
  393. static float qr2[6] = {/* for x in [2.8570,2]=1/[0.3499,0.5] */
  394. #endif
  395.  -1.7838172539e-07, /* 0xb43f8932 */
  396.  -1.0251704603e-01, /* 0xbdd1f475 */
  397.  -2.7522056103e+00, /* 0xc0302423 */
  398.  -1.9663616180e+01, /* 0xc19d4f16 */
  399.  -4.2325313568e+01, /* 0xc2294d1f */
  400.  -2.1371921539e+01, /* 0xc1aaf9b2 */
  401. };
  402. #ifdef __STDC__
  403. static const float qs2[6] = {
  404. #else
  405. static float qs2[6] = {
  406. #endif
  407.   2.9533363342e+01, /* 0x41ec4454 */
  408.   2.5298155212e+02, /* 0x437cfb47 */
  409.   7.5750280762e+02, /* 0x443d602e */
  410.   7.3939318848e+02, /* 0x4438d92a */
  411.   1.5594900513e+02, /* 0x431bf2f2 */
  412.  -4.9594988823e+00, /* 0xc09eb437 */
  413. };
  414.  
  415. #ifdef __STDC__
  416.         static float qonef(float x)
  417. #else
  418.         static float qonef(x)
  419.         float x;
  420. #endif
  421. {
  422. #ifdef __STDC__
  423.         const float *p,*q;
  424. #else
  425.         float *p,*q;
  426. #endif
  427.         float  s,r,z;
  428.         __int32_t ix;
  429.         GET_FLOAT_WORD(ix,x);
  430.         ix &= 0x7fffffff;
  431.         if(ix>=0x40200000)     {p = qr8; q= qs8;}
  432.         else if(ix>=0x40f71c58){p = qr5; q= qs5;}
  433.         else if(ix>=0x4036db68){p = qr3; q= qs3;}
  434.       else {p = qr2; q= qs2;}
  435.         z = one/(x*x);
  436.         r = p[0]+z*(p[1]+z*(p[2]+z*(p[3]+z*(p[4]+z*p[5]))));
  437.         s = one+z*(q[0]+z*(q[1]+z*(q[2]+z*(q[3]+z*(q[4]+z*q[5])))));
  438.         return ((float).375 + r/s)/x;
  439. }
  440.