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  1. /* ef_j0.c -- float version of e_j0.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 pzerof(float), qzerof(float);
  20. #else
  21. static float pzerof(), qzerof();
  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, 2.00] */
  34. R02  =  1.5625000000e-02, /* 0x3c800000 */
  35. R03  = -1.8997929874e-04, /* 0xb947352e */
  36. R04  =  1.8295404516e-06, /* 0x35f58e88 */
  37. R05  = -4.6183270541e-09, /* 0xb19eaf3c */
  38. S01  =  1.5619102865e-02, /* 0x3c7fe744 */
  39. S02  =  1.1692678527e-04, /* 0x38f53697 */
  40. S03  =  5.1354652442e-07, /* 0x3509daa6 */
  41. S04  =  1.1661400734e-09; /* 0x30a045e8 */
  42.  
  43. #ifdef __STDC__
  44. static const float zero = 0.0;
  45. #else
  46. static float zero = 0.0;
  47. #endif
  48.  
  49. #ifdef __STDC__
  50.         float __ieee754_j0f(float x)
  51. #else
  52.         float __ieee754_j0f(x)
  53.         float x;
  54. #endif
  55. {
  56.         float z, s,c,ss,cc,r,u,v;
  57.         __int32_t hx,ix;
  58.  
  59.         GET_FLOAT_WORD(hx,x);
  60.         ix = hx&0x7fffffff;
  61.         if(!FLT_UWORD_IS_FINITE(ix)) return one/(x*x);
  62.         x = fabsf(x);
  63.         if(ix >= 0x40000000) {  /* |x| >= 2.0 */
  64.                 s = sinf(x);
  65.                 c = cosf(x);
  66.                 ss = s-c;
  67.                 cc = s+c;
  68.                 if(ix<=FLT_UWORD_HALF_MAX) {  /* make sure x+x not overflow */
  69.                     z = -cosf(x+x);
  70.                     if ((s*c)<zero) cc = z/ss;
  71.                     else            ss = z/cc;
  72.                 }
  73.         /*
  74.          * j0(x) = 1/sqrt(pi) * (P(0,x)*cc - Q(0,x)*ss) / sqrt(x)
  75.          * y0(x) = 1/sqrt(pi) * (P(0,x)*ss + Q(0,x)*cc) / sqrt(x)
  76.          */
  77.                 if(ix>0x80000000) z = (invsqrtpi*cc)/__ieee754_sqrtf(x);
  78.                 else {
  79.                     u = pzerof(x); v = qzerof(x);
  80.                     z = invsqrtpi*(u*cc-v*ss)/__ieee754_sqrtf(x);
  81.                 }
  82.                 return z;
  83.         }
  84.         if(ix<0x39000000) {     /* |x| < 2**-13 */
  85.             if(huge+x>one) {    /* raise inexact if x != 0 */
  86.                 if(ix<0x32000000) return one;   /* |x|<2**-27 */
  87.                 else          return one - (float)0.25*x*x;
  88.             }
  89.         }
  90.         z = x*x;
  91.         r =  z*(R02+z*(R03+z*(R04+z*R05)));
  92.         s =  one+z*(S01+z*(S02+z*(S03+z*S04)));
  93.         if(ix < 0x3F800000) {   /* |x| < 1.00 */
  94.             return one + z*((float)-0.25+(r/s));
  95.         } else {
  96.             u = (float)0.5*x;
  97.             return((one+u)*(one-u)+z*(r/s));
  98.         }
  99. }
  100.  
  101. #ifdef __STDC__
  102. static const float
  103. #else
  104. static float
  105. #endif
  106. u00  = -7.3804296553e-02, /* 0xbd9726b5 */
  107. u01  =  1.7666645348e-01, /* 0x3e34e80d */
  108. u02  = -1.3818567619e-02, /* 0xbc626746 */
  109. u03  =  3.4745343146e-04, /* 0x39b62a69 */
  110. u04  = -3.8140706238e-06, /* 0xb67ff53c */
  111. u05  =  1.9559013964e-08, /* 0x32a802ba */
  112. u06  = -3.9820518410e-11, /* 0xae2f21eb */
  113. v01  =  1.2730483897e-02, /* 0x3c509385 */
  114. v02  =  7.6006865129e-05, /* 0x389f65e0 */
  115. v03  =  2.5915085189e-07, /* 0x348b216c */
  116. v04  =  4.4111031494e-10; /* 0x2ff280c2 */
  117.  
  118. #ifdef __STDC__
  119.         float __ieee754_y0f(float x)
  120. #else
  121.         float __ieee754_y0f(x)
  122.         float x;
  123. #endif
  124. {
  125.         float z, s,c,ss,cc,u,v;
  126.         __int32_t hx,ix;
  127.  
  128.         GET_FLOAT_WORD(hx,x);
  129.         ix = 0x7fffffff&hx;
  130.     /* Y0(NaN) is NaN, y0(-inf) is Nan, y0(inf) is 0  */
  131.         if(!FLT_UWORD_IS_FINITE(ix)) return  one/(x+x*x);
  132.         if(FLT_UWORD_IS_ZERO(ix)) return -one/zero;
  133.         if(hx<0) return zero/zero;
  134.         if(ix >= 0x40000000) {  /* |x| >= 2.0 */
  135.         /* y0(x) = sqrt(2/(pi*x))*(p0(x)*sin(x0)+q0(x)*cos(x0))
  136.          * where x0 = x-pi/4
  137.          *      Better formula:
  138.          *              cos(x0) = cos(x)cos(pi/4)+sin(x)sin(pi/4)
  139.          *                      =  1/sqrt(2) * (sin(x) + cos(x))
  140.          *              sin(x0) = sin(x)cos(3pi/4)-cos(x)sin(3pi/4)
  141.          *                      =  1/sqrt(2) * (sin(x) - cos(x))
  142.          * To avoid cancellation, use
  143.          *              sin(x) +- cos(x) = -cos(2x)/(sin(x) -+ cos(x))
  144.          * to compute the worse one.
  145.          */
  146.                 s = sinf(x);
  147.                 c = cosf(x);
  148.                 ss = s-c;
  149.                 cc = s+c;
  150.         /*
  151.          * j0(x) = 1/sqrt(pi) * (P(0,x)*cc - Q(0,x)*ss) / sqrt(x)
  152.          * y0(x) = 1/sqrt(pi) * (P(0,x)*ss + Q(0,x)*cc) / sqrt(x)
  153.          */
  154.                 if(ix<=FLT_UWORD_HALF_MAX) {  /* make sure x+x not overflow */
  155.                     z = -cosf(x+x);
  156.                     if ((s*c)<zero) cc = z/ss;
  157.                     else            ss = z/cc;
  158.                 }
  159.                 if(ix>0x80000000) z = (invsqrtpi*ss)/__ieee754_sqrtf(x);
  160.                 else {
  161.                     u = pzerof(x); v = qzerof(x);
  162.                     z = invsqrtpi*(u*ss+v*cc)/__ieee754_sqrtf(x);
  163.                 }
  164.                 return z;
  165.         }
  166.         if(ix<=0x32000000) {    /* x < 2**-27 */
  167.             return(u00 + tpi*__ieee754_logf(x));
  168.         }
  169.         z = x*x;
  170.         u = u00+z*(u01+z*(u02+z*(u03+z*(u04+z*(u05+z*u06)))));
  171.         v = one+z*(v01+z*(v02+z*(v03+z*v04)));
  172.         return(u/v + tpi*(__ieee754_j0f(x)*__ieee754_logf(x)));
  173. }
  174.  
  175. /* The asymptotic expansions of pzero is
  176.  *      1 - 9/128 s^2 + 11025/98304 s^4 - ...,  where s = 1/x.
  177.  * For x >= 2, We approximate pzero by
  178.  *      pzero(x) = 1 + (R/S)
  179.  * where  R = pR0 + pR1*s^2 + pR2*s^4 + ... + pR5*s^10
  180.  *        S = 1 + pS0*s^2 + ... + pS4*s^10
  181.  * and
  182.  *      | pzero(x)-1-R/S | <= 2  ** ( -60.26)
  183.  */
  184. #ifdef __STDC__
  185. static const float pR8[6] = { /* for x in [inf, 8]=1/[0,0.125] */
  186. #else
  187. static float pR8[6] = { /* for x in [inf, 8]=1/[0,0.125] */
  188. #endif
  189.   0.0000000000e+00, /* 0x00000000 */
  190.  -7.0312500000e-02, /* 0xbd900000 */
  191.  -8.0816707611e+00, /* 0xc1014e86 */
  192.  -2.5706311035e+02, /* 0xc3808814 */
  193.  -2.4852163086e+03, /* 0xc51b5376 */
  194.  -5.2530439453e+03, /* 0xc5a4285a */
  195. };
  196. #ifdef __STDC__
  197. static const float pS8[5] = {
  198. #else
  199. static float pS8[5] = {
  200. #endif
  201.   1.1653436279e+02, /* 0x42e91198 */
  202.   3.8337448730e+03, /* 0x456f9beb */
  203.   4.0597855469e+04, /* 0x471e95db */
  204.   1.1675296875e+05, /* 0x47e4087c */
  205.   4.7627726562e+04, /* 0x473a0bba */
  206. };
  207. #ifdef __STDC__
  208. static const float pR5[6] = { /* for x in [8,4.5454]=1/[0.125,0.22001] */
  209. #else
  210. static float pR5[6] = { /* for x in [8,4.5454]=1/[0.125,0.22001] */
  211. #endif
  212.  -1.1412546255e-11, /* 0xad48c58a */
  213.  -7.0312492549e-02, /* 0xbd8fffff */
  214.  -4.1596107483e+00, /* 0xc0851b88 */
  215.  -6.7674766541e+01, /* 0xc287597b */
  216.  -3.3123129272e+02, /* 0xc3a59d9b */
  217.  -3.4643338013e+02, /* 0xc3ad3779 */
  218. };
  219. #ifdef __STDC__
  220. static const float pS5[5] = {
  221. #else
  222. static float pS5[5] = {
  223. #endif
  224.   6.0753936768e+01, /* 0x42730408 */
  225.   1.0512523193e+03, /* 0x44836813 */
  226.   5.9789707031e+03, /* 0x45bad7c4 */
  227.   9.6254453125e+03, /* 0x461665c8 */
  228.   2.4060581055e+03, /* 0x451660ee */
  229. };
  230.  
  231. #ifdef __STDC__
  232. static const float pR3[6] = {/* for x in [4.547,2.8571]=1/[0.2199,0.35001] */
  233. #else
  234. static float pR3[6] = {/* for x in [4.547,2.8571]=1/[0.2199,0.35001] */
  235. #endif
  236.  -2.5470459075e-09, /* 0xb12f081b */
  237.  -7.0311963558e-02, /* 0xbd8fffb8 */
  238.  -2.4090321064e+00, /* 0xc01a2d95 */
  239.  -2.1965976715e+01, /* 0xc1afba52 */
  240.  -5.8079170227e+01, /* 0xc2685112 */
  241.  -3.1447946548e+01, /* 0xc1fb9565 */
  242. };
  243. #ifdef __STDC__
  244. static const float pS3[5] = {
  245. #else
  246. static float pS3[5] = {
  247. #endif
  248.   3.5856033325e+01, /* 0x420f6c94 */
  249.   3.6151397705e+02, /* 0x43b4c1ca */
  250.   1.1936077881e+03, /* 0x44953373 */
  251.   1.1279968262e+03, /* 0x448cffe6 */
  252.   1.7358093262e+02, /* 0x432d94b8 */
  253. };
  254.  
  255. #ifdef __STDC__
  256. static const float pR2[6] = {/* for x in [2.8570,2]=1/[0.3499,0.5] */
  257. #else
  258. static float pR2[6] = {/* for x in [2.8570,2]=1/[0.3499,0.5] */
  259. #endif
  260.  -8.8753431271e-08, /* 0xb3be98b7 */
  261.  -7.0303097367e-02, /* 0xbd8ffb12 */
  262.  -1.4507384300e+00, /* 0xbfb9b1cc */
  263.  -7.6356959343e+00, /* 0xc0f4579f */
  264.  -1.1193166733e+01, /* 0xc1331736 */
  265.  -3.2336456776e+00, /* 0xc04ef40d */
  266. };
  267. #ifdef __STDC__
  268. static const float pS2[5] = {
  269. #else
  270. static float pS2[5] = {
  271. #endif
  272.   2.2220300674e+01, /* 0x41b1c32d */
  273.   1.3620678711e+02, /* 0x430834f0 */
  274.   2.7047027588e+02, /* 0x43873c32 */
  275.   1.5387539673e+02, /* 0x4319e01a */
  276.   1.4657617569e+01, /* 0x416a859a */
  277. };
  278.  
  279. #ifdef __STDC__
  280.         static float pzerof(float x)
  281. #else
  282.         static float pzerof(x)
  283.         float x;
  284. #endif
  285. {
  286. #ifdef __STDC__
  287.         const float *p,*q;
  288. #else
  289.         float *p,*q;
  290. #endif
  291.         float z,r,s;
  292.         __int32_t ix;
  293.         GET_FLOAT_WORD(ix,x);
  294.         ix &= 0x7fffffff;
  295.         if(ix>=0x41000000)     {p = pR8; q= pS8;}
  296.         else if(ix>=0x40f71c58){p = pR5; q= pS5;}
  297.         else if(ix>=0x4036db68){p = pR3; q= pS3;}
  298.       else {p = pR2; q= pS2;}
  299.         z = one/(x*x);
  300.         r = p[0]+z*(p[1]+z*(p[2]+z*(p[3]+z*(p[4]+z*p[5]))));
  301.         s = one+z*(q[0]+z*(q[1]+z*(q[2]+z*(q[3]+z*q[4]))));
  302.         return one+ r/s;
  303. }
  304.                
  305.  
  306. /* For x >= 8, the asymptotic expansions of qzero is
  307.  *      -1/8 s + 75/1024 s^3 - ..., where s = 1/x.
  308.  * We approximate qzero by
  309.  *      qzero(x) = s*(-1.25 + (R/S))
  310.  * where  R = qR0 + qR1*s^2 + qR2*s^4 + ... + qR5*s^10
  311.  *        S = 1 + qS0*s^2 + ... + qS5*s^12
  312.  * and
  313.  *      | qzero(x)/s +1.25-R/S | <= 2  ** ( -61.22)
  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.   7.3242187500e-02, /* 0x3d960000 */
  322.   1.1768206596e+01, /* 0x413c4a93 */
  323.   5.5767340088e+02, /* 0x440b6b19 */
  324.   8.8591972656e+03, /* 0x460a6cca */
  325.   3.7014625000e+04, /* 0x471096a0 */
  326. };
  327. #ifdef __STDC__
  328. static const float qS8[6] = {
  329. #else
  330. static float qS8[6] = {
  331. #endif
  332.   1.6377603149e+02, /* 0x4323c6aa */
  333.   8.0983447266e+03, /* 0x45fd12c2 */
  334.   1.4253829688e+05, /* 0x480b3293 */
  335.   8.0330925000e+05, /* 0x49441ed4 */
  336.   8.4050156250e+05, /* 0x494d3359 */
  337.  -3.4389928125e+05, /* 0xc8a7eb69 */
  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.   1.8408595828e-11, /* 0x2da1ec79 */
  346.   7.3242180049e-02, /* 0x3d95ffff */
  347.   5.8356351852e+00, /* 0x40babd86 */
  348.   1.3511157227e+02, /* 0x43071c90 */
  349.   1.0272437744e+03, /* 0x448067cd */
  350.   1.9899779053e+03, /* 0x44f8bf4b */
  351. };
  352. #ifdef __STDC__
  353. static const float qS5[6] = {
  354. #else
  355. static float qS5[6] = {
  356. #endif
  357.   8.2776611328e+01, /* 0x42a58da0 */
  358.   2.0778142090e+03, /* 0x4501dd07 */
  359.   1.8847289062e+04, /* 0x46933e94 */
  360.   5.6751113281e+04, /* 0x475daf1d */
  361.   3.5976753906e+04, /* 0x470c88c1 */
  362.  -5.3543427734e+03, /* 0xc5a752be */
  363. };
  364.  
  365. #ifdef __STDC__
  366. static const float qR3[6] = {/* for x in [4.547,2.8571]=1/[0.2199,0.35001] */
  367. #else
  368. static float qR3[6] = {/* for x in [4.547,2.8571]=1/[0.2199,0.35001] */
  369. #endif
  370.   4.3774099900e-09, /* 0x3196681b */
  371.   7.3241114616e-02, /* 0x3d95ff70 */
  372.   3.3442313671e+00, /* 0x405607e3 */
  373.   4.2621845245e+01, /* 0x422a7cc5 */
  374.   1.7080809021e+02, /* 0x432acedf */
  375.   1.6673394775e+02, /* 0x4326bbe4 */
  376. };
  377. #ifdef __STDC__
  378. static const float qS3[6] = {
  379. #else
  380. static float qS3[6] = {
  381. #endif
  382.   4.8758872986e+01, /* 0x42430916 */
  383.   7.0968920898e+02, /* 0x44316c1c */
  384.   3.7041481934e+03, /* 0x4567825f */
  385.   6.4604252930e+03, /* 0x45c9e367 */
  386.   2.5163337402e+03, /* 0x451d4557 */
  387.  -1.4924745178e+02, /* 0xc3153f59 */
  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.   1.5044444979e-07, /* 0x342189db */
  396.   7.3223426938e-02, /* 0x3d95f62a */
  397.   1.9981917143e+00, /* 0x3fffc4bf */
  398.   1.4495602608e+01, /* 0x4167edfd */
  399.   3.1666231155e+01, /* 0x41fd5471 */
  400.   1.6252708435e+01, /* 0x4182058c */
  401. };
  402. #ifdef __STDC__
  403. static const float qS2[6] = {
  404. #else
  405. static float qS2[6] = {
  406. #endif
  407.   3.0365585327e+01, /* 0x41f2ecb8 */
  408.   2.6934811401e+02, /* 0x4386ac8f */
  409.   8.4478375244e+02, /* 0x44533229 */
  410.   8.8293585205e+02, /* 0x445cbbe5 */
  411.   2.1266638184e+02, /* 0x4354aa98 */
  412.  -5.3109550476e+00, /* 0xc0a9f358 */
  413. };
  414.  
  415. #ifdef __STDC__
  416.         static float qzerof(float x)
  417. #else
  418.         static float qzerof(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>=0x41000000)     {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).125 + r/s)/x;
  439. }
  440.