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  1. /* Copyright (C) 1994 DJ Delorie, see COPYING.DJ for details */
  2. /* e_j0f.c -- float version of e_j0.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_j0f.c,v 1.2 1994/08/18 23:05:32 jtc Exp $";
  19. #endif
  20.  
  21. #include "math.h"
  22. #include "math_private.h"
  23.  
  24. #ifdef __STDC__
  25. static float pzerof(float), qzerof(float);
  26. #else
  27. static float pzerof(), qzerof();
  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, 2.00] */
  40. R02  =  1.5625000000e-02, /* 0x3c800000 */
  41. R03  = -1.8997929874e-04, /* 0xb947352e */
  42. R04  =  1.8295404516e-06, /* 0x35f58e88 */
  43. R05  = -4.6183270541e-09, /* 0xb19eaf3c */
  44. S01  =  1.5619102865e-02, /* 0x3c7fe744 */
  45. S02  =  1.1692678527e-04, /* 0x38f53697 */
  46. S03  =  5.1354652442e-07, /* 0x3509daa6 */
  47. S04  =  1.1661400734e-09; /* 0x30a045e8 */
  48.  
  49. #ifdef __STDC__
  50. static const float zero = 0.0;
  51. #else
  52. static float zero = 0.0;
  53. #endif
  54.  
  55. #ifdef __STDC__
  56.         float __ieee754_j0f(float x)
  57. #else
  58.         float __ieee754_j0f(x)
  59.         float x;
  60. #endif
  61. {
  62.         float z, s,c,ss,cc,r,u,v;
  63.         int32_t hx,ix;
  64.  
  65.         GET_FLOAT_WORD(hx,x);
  66.         ix = hx&0x7fffffff;
  67.         if(ix>=0x7f800000) return one/(x*x);
  68.         x = fabsf(x);
  69.         if(ix >= 0x40000000) {  /* |x| >= 2.0 */
  70.                 s = sinf(x);
  71.                 c = cosf(x);
  72.                 ss = s-c;
  73.                 cc = s+c;
  74.                 if(ix<0x7f000000) {  /* make sure x+x not overflow */
  75.                     z = -cosf(x+x);
  76.                     if ((s*c)<zero) cc = z/ss;
  77.                     else            ss = z/cc;
  78.                 }
  79.         /*
  80.          * j0(x) = 1/sqrt(pi) * (P(0,x)*cc - Q(0,x)*ss) / sqrt(x)
  81.          * y0(x) = 1/sqrt(pi) * (P(0,x)*ss + Q(0,x)*cc) / sqrt(x)
  82.          */
  83.                 if(ix>0x80000000) z = (invsqrtpi*cc)/sqrtf(x);
  84.                 else {
  85.                     u = pzerof(x); v = qzerof(x);
  86.                     z = invsqrtpi*(u*cc-v*ss)/sqrtf(x);
  87.                 }
  88.                 return z;
  89.         }
  90.         if(ix<0x39000000) {     /* |x| < 2**-13 */
  91.             if(huge+x>one) {    /* raise inexact if x != 0 */
  92.                 if(ix<0x32000000) return one;   /* |x|<2**-27 */
  93.                 else          return one - (float)0.25*x*x;
  94.             }
  95.         }
  96.         z = x*x;
  97.         r =  z*(R02+z*(R03+z*(R04+z*R05)));
  98.         s =  one+z*(S01+z*(S02+z*(S03+z*S04)));
  99.         if(ix < 0x3F800000) {   /* |x| < 1.00 */
  100.             return one + z*((float)-0.25+(r/s));
  101.         } else {
  102.             u = (float)0.5*x;
  103.             return((one+u)*(one-u)+z*(r/s));
  104.         }
  105. }
  106.  
  107. #ifdef __STDC__
  108. static const float
  109. #else
  110. static float
  111. #endif
  112. u00  = -7.3804296553e-02, /* 0xbd9726b5 */
  113. u01  =  1.7666645348e-01, /* 0x3e34e80d */
  114. u02  = -1.3818567619e-02, /* 0xbc626746 */
  115. u03  =  3.4745343146e-04, /* 0x39b62a69 */
  116. u04  = -3.8140706238e-06, /* 0xb67ff53c */
  117. u05  =  1.9559013964e-08, /* 0x32a802ba */
  118. u06  = -3.9820518410e-11, /* 0xae2f21eb */
  119. v01  =  1.2730483897e-02, /* 0x3c509385 */
  120. v02  =  7.6006865129e-05, /* 0x389f65e0 */
  121. v03  =  2.5915085189e-07, /* 0x348b216c */
  122. v04  =  4.4111031494e-10; /* 0x2ff280c2 */
  123.  
  124. #ifdef __STDC__
  125.         float __ieee754_y0f(float x)
  126. #else
  127.         float __ieee754_y0f(x)
  128.         float x;
  129. #endif
  130. {
  131.         float z, s,c,ss,cc,u,v;
  132.         int32_t hx,ix;
  133.  
  134.         GET_FLOAT_WORD(hx,x);
  135.         ix = 0x7fffffff&hx;
  136.     /* Y0(NaN) is NaN, y0(-inf) is Nan, y0(inf) is 0  */
  137.         if(ix>=0x7f800000) return  one/(x+x*x);
  138.         if(ix==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 = sinf(x);
  153.                 c = cosf(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<0x7f000000) {  /* make sure x+x not overflow */
  161.                     z = -cosf(x+x);
  162.                     if ((s*c)<zero) cc = z/ss;
  163.                     else            ss = z/cc;
  164.                 }
  165.                 if(ix>0x80000000) z = (invsqrtpi*ss)/sqrtf(x);
  166.                 else {
  167.                     u = pzerof(x); v = qzerof(x);
  168.                     z = invsqrtpi*(u*ss+v*cc)/sqrtf(x);
  169.                 }
  170.                 return z;
  171.         }
  172.         if(ix<=0x32000000) {    /* x < 2**-27 */
  173.             return(u00 + tpi*__ieee754_logf(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_j0f(x)*__ieee754_logf(x)));
  179. }
  180.  
  181. /* The asymptotic expansions of pzero is
  182.  *      1 - 9/128 s^2 + 11025/98304 s^4 - ...,  where s = 1/x.
  183.  * For x >= 2, We approximate pzero by
  184.  *      pzero(x) = 1 + (R/S)
  185.  * where  R = pR0 + pR1*s^2 + pR2*s^4 + ... + pR5*s^10
  186.  *        S = 1 + pS0*s^2 + ... + pS4*s^10
  187.  * and
  188.  *      | pzero(x)-1-R/S | <= 2  ** ( -60.26)
  189.  */
  190. #ifdef __STDC__
  191. static const float pR8[6] = { /* for x in [inf, 8]=1/[0,0.125] */
  192. #else
  193. static float pR8[6] = { /* for x in [inf, 8]=1/[0,0.125] */
  194. #endif
  195.   0.0000000000e+00, /* 0x00000000 */
  196.  -7.0312500000e-02, /* 0xbd900000 */
  197.  -8.0816707611e+00, /* 0xc1014e86 */
  198.  -2.5706311035e+02, /* 0xc3808814 */
  199.  -2.4852163086e+03, /* 0xc51b5376 */
  200.  -5.2530439453e+03, /* 0xc5a4285a */
  201. };
  202. #ifdef __STDC__
  203. static const float pS8[5] = {
  204. #else
  205. static float pS8[5] = {
  206. #endif
  207.   1.1653436279e+02, /* 0x42e91198 */
  208.   3.8337448730e+03, /* 0x456f9beb */
  209.   4.0597855469e+04, /* 0x471e95db */
  210.   1.1675296875e+05, /* 0x47e4087c */
  211.   4.7627726562e+04, /* 0x473a0bba */
  212. };
  213. #ifdef __STDC__
  214. static const float pR5[6] = { /* for x in [8,4.5454]=1/[0.125,0.22001] */
  215. #else
  216. static float pR5[6] = { /* for x in [8,4.5454]=1/[0.125,0.22001] */
  217. #endif
  218.  -1.1412546255e-11, /* 0xad48c58a */
  219.  -7.0312492549e-02, /* 0xbd8fffff */
  220.  -4.1596107483e+00, /* 0xc0851b88 */
  221.  -6.7674766541e+01, /* 0xc287597b */
  222.  -3.3123129272e+02, /* 0xc3a59d9b */
  223.  -3.4643338013e+02, /* 0xc3ad3779 */
  224. };
  225. #ifdef __STDC__
  226. static const float pS5[5] = {
  227. #else
  228. static float pS5[5] = {
  229. #endif
  230.   6.0753936768e+01, /* 0x42730408 */
  231.   1.0512523193e+03, /* 0x44836813 */
  232.   5.9789707031e+03, /* 0x45bad7c4 */
  233.   9.6254453125e+03, /* 0x461665c8 */
  234.   2.4060581055e+03, /* 0x451660ee */
  235. };
  236.  
  237. #ifdef __STDC__
  238. static const float pR3[6] = {/* for x in [4.547,2.8571]=1/[0.2199,0.35001] */
  239. #else
  240. static float pR3[6] = {/* for x in [4.547,2.8571]=1/[0.2199,0.35001] */
  241. #endif
  242.  -2.5470459075e-09, /* 0xb12f081b */
  243.  -7.0311963558e-02, /* 0xbd8fffb8 */
  244.  -2.4090321064e+00, /* 0xc01a2d95 */
  245.  -2.1965976715e+01, /* 0xc1afba52 */
  246.  -5.8079170227e+01, /* 0xc2685112 */
  247.  -3.1447946548e+01, /* 0xc1fb9565 */
  248. };
  249. #ifdef __STDC__
  250. static const float pS3[5] = {
  251. #else
  252. static float pS3[5] = {
  253. #endif
  254.   3.5856033325e+01, /* 0x420f6c94 */
  255.   3.6151397705e+02, /* 0x43b4c1ca */
  256.   1.1936077881e+03, /* 0x44953373 */
  257.   1.1279968262e+03, /* 0x448cffe6 */
  258.   1.7358093262e+02, /* 0x432d94b8 */
  259. };
  260.  
  261. #ifdef __STDC__
  262. static const float pR2[6] = {/* for x in [2.8570,2]=1/[0.3499,0.5] */
  263. #else
  264. static float pR2[6] = {/* for x in [2.8570,2]=1/[0.3499,0.5] */
  265. #endif
  266.  -8.8753431271e-08, /* 0xb3be98b7 */
  267.  -7.0303097367e-02, /* 0xbd8ffb12 */
  268.  -1.4507384300e+00, /* 0xbfb9b1cc */
  269.  -7.6356959343e+00, /* 0xc0f4579f */
  270.  -1.1193166733e+01, /* 0xc1331736 */
  271.  -3.2336456776e+00, /* 0xc04ef40d */
  272. };
  273. #ifdef __STDC__
  274. static const float pS2[5] = {
  275. #else
  276. static float pS2[5] = {
  277. #endif
  278.   2.2220300674e+01, /* 0x41b1c32d */
  279.   1.3620678711e+02, /* 0x430834f0 */
  280.   2.7047027588e+02, /* 0x43873c32 */
  281.   1.5387539673e+02, /* 0x4319e01a */
  282.   1.4657617569e+01, /* 0x416a859a */
  283. };
  284.  
  285. #ifdef __STDC__
  286.         static float pzerof(float x)
  287. #else
  288.         static float pzerof(x)
  289.         float x;
  290. #endif
  291. {
  292. #ifdef __STDC__
  293.         const float *p,*q;
  294. #else
  295.         float *p,*q;
  296. #endif
  297.         float z,r,s;
  298.         int32_t ix;
  299.         GET_FLOAT_WORD(ix,x);
  300.         ix &= 0x7fffffff;
  301.         if(ix>=0x41000000)     {p = pR8; q= pS8;}
  302.         else if(ix>=0x40f71c58){p = pR5; q= pS5;}
  303.         else if(ix>=0x4036db68){p = pR3; q= pS3;}
  304.         else if(ix>=0x40000000){p = pR2; q= pS2;}
  305.         z = one/(x*x);
  306.         r = p[0]+z*(p[1]+z*(p[2]+z*(p[3]+z*(p[4]+z*p[5]))));
  307.         s = one+z*(q[0]+z*(q[1]+z*(q[2]+z*(q[3]+z*q[4]))));
  308.         return one+ r/s;
  309. }
  310.                
  311.  
  312. /* For x >= 8, the asymptotic expansions of qzero is
  313.  *      -1/8 s + 75/1024 s^3 - ..., where s = 1/x.
  314.  * We approximate pzero by
  315.  *      qzero(x) = s*(-1.25 + (R/S))
  316.  * where  R = qR0 + qR1*s^2 + qR2*s^4 + ... + qR5*s^10
  317.  *        S = 1 + qS0*s^2 + ... + qS5*s^12
  318.  * and
  319.  *      | qzero(x)/s +1.25-R/S | <= 2  ** ( -61.22)
  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.   7.3242187500e-02, /* 0x3d960000 */
  328.   1.1768206596e+01, /* 0x413c4a93 */
  329.   5.5767340088e+02, /* 0x440b6b19 */
  330.   8.8591972656e+03, /* 0x460a6cca */
  331.   3.7014625000e+04, /* 0x471096a0 */
  332. };
  333. #ifdef __STDC__
  334. static const float qS8[6] = {
  335. #else
  336. static float qS8[6] = {
  337. #endif
  338.   1.6377603149e+02, /* 0x4323c6aa */
  339.   8.0983447266e+03, /* 0x45fd12c2 */
  340.   1.4253829688e+05, /* 0x480b3293 */
  341.   8.0330925000e+05, /* 0x49441ed4 */
  342.   8.4050156250e+05, /* 0x494d3359 */
  343.  -3.4389928125e+05, /* 0xc8a7eb69 */
  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.   1.8408595828e-11, /* 0x2da1ec79 */
  352.   7.3242180049e-02, /* 0x3d95ffff */
  353.   5.8356351852e+00, /* 0x40babd86 */
  354.   1.3511157227e+02, /* 0x43071c90 */
  355.   1.0272437744e+03, /* 0x448067cd */
  356.   1.9899779053e+03, /* 0x44f8bf4b */
  357. };
  358. #ifdef __STDC__
  359. static const float qS5[6] = {
  360. #else
  361. static float qS5[6] = {
  362. #endif
  363.   8.2776611328e+01, /* 0x42a58da0 */
  364.   2.0778142090e+03, /* 0x4501dd07 */
  365.   1.8847289062e+04, /* 0x46933e94 */
  366.   5.6751113281e+04, /* 0x475daf1d */
  367.   3.5976753906e+04, /* 0x470c88c1 */
  368.  -5.3543427734e+03, /* 0xc5a752be */
  369. };
  370.  
  371. #ifdef __STDC__
  372. static const float qR3[6] = {/* for x in [4.547,2.8571]=1/[0.2199,0.35001] */
  373. #else
  374. static float qR3[6] = {/* for x in [4.547,2.8571]=1/[0.2199,0.35001] */
  375. #endif
  376.   4.3774099900e-09, /* 0x3196681b */
  377.   7.3241114616e-02, /* 0x3d95ff70 */
  378.   3.3442313671e+00, /* 0x405607e3 */
  379.   4.2621845245e+01, /* 0x422a7cc5 */
  380.   1.7080809021e+02, /* 0x432acedf */
  381.   1.6673394775e+02, /* 0x4326bbe4 */
  382. };
  383. #ifdef __STDC__
  384. static const float qS3[6] = {
  385. #else
  386. static float qS3[6] = {
  387. #endif
  388.   4.8758872986e+01, /* 0x42430916 */
  389.   7.0968920898e+02, /* 0x44316c1c */
  390.   3.7041481934e+03, /* 0x4567825f */
  391.   6.4604252930e+03, /* 0x45c9e367 */
  392.   2.5163337402e+03, /* 0x451d4557 */
  393.  -1.4924745178e+02, /* 0xc3153f59 */
  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.   1.5044444979e-07, /* 0x342189db */
  402.   7.3223426938e-02, /* 0x3d95f62a */
  403.   1.9981917143e+00, /* 0x3fffc4bf */
  404.   1.4495602608e+01, /* 0x4167edfd */
  405.   3.1666231155e+01, /* 0x41fd5471 */
  406.   1.6252708435e+01, /* 0x4182058c */
  407. };
  408. #ifdef __STDC__
  409. static const float qS2[6] = {
  410. #else
  411. static float qS2[6] = {
  412. #endif
  413.   3.0365585327e+01, /* 0x41f2ecb8 */
  414.   2.6934811401e+02, /* 0x4386ac8f */
  415.   8.4478375244e+02, /* 0x44533229 */
  416.   8.8293585205e+02, /* 0x445cbbe5 */
  417.   2.1266638184e+02, /* 0x4354aa98 */
  418.  -5.3109550476e+00, /* 0xc0a9f358 */
  419. };
  420.  
  421. #ifdef __STDC__
  422.         static float qzerof(float x)
  423. #else
  424.         static float qzerof(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>=0x41000000)     {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).125 + r/s)/x;
  445. }
  446.