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glibc  2.9
e_j1f.c
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00001 /* e_j1f.c -- float version of e_j1.c.
00002  * Conversion to float by Ian Lance Taylor, Cygnus Support, ian@cygnus.com.
00003  */
00004 
00005 /*
00006  * ====================================================
00007  * Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved.
00008  *
00009  * Developed at SunPro, a Sun Microsystems, Inc. business.
00010  * Permission to use, copy, modify, and distribute this
00011  * software is freely granted, provided that this notice
00012  * is preserved.
00013  * ====================================================
00014  */
00015 
00016 #if defined(LIBM_SCCS) && !defined(lint)
00017 static char rcsid[] = "$NetBSD: e_j1f.c,v 1.4 1995/05/10 20:45:31 jtc Exp $";
00018 #endif
00019 
00020 #include "math.h"
00021 #include "math_private.h"
00022 
00023 #ifdef __STDC__
00024 static float ponef(float), qonef(float);
00025 #else
00026 static float ponef(), qonef();
00027 #endif
00028 
00029 #ifdef __STDC__
00030 static const float
00031 #else
00032 static float
00033 #endif
00034 huge    = 1e30,
00035 one    = 1.0,
00036 invsqrtpi=  5.6418961287e-01, /* 0x3f106ebb */
00037 tpi      =  6.3661974669e-01, /* 0x3f22f983 */
00038        /* R0/S0 on [0,2] */
00039 r00  = -6.2500000000e-02, /* 0xbd800000 */
00040 r01  =  1.4070566976e-03, /* 0x3ab86cfd */
00041 r02  = -1.5995563444e-05, /* 0xb7862e36 */
00042 r03  =  4.9672799207e-08, /* 0x335557d2 */
00043 s01  =  1.9153760746e-02, /* 0x3c9ce859 */
00044 s02  =  1.8594678841e-04, /* 0x3942fab6 */
00045 s03  =  1.1771846857e-06, /* 0x359dffc2 */
00046 s04  =  5.0463624390e-09, /* 0x31ad6446 */
00047 s05  =  1.2354227016e-11; /* 0x2d59567e */
00048 
00049 #ifdef __STDC__
00050 static const float zero    = 0.0;
00051 #else
00052 static float zero    = 0.0;
00053 #endif
00054 
00055 #ifdef __STDC__
00056        float __ieee754_j1f(float x)
00057 #else
00058        float __ieee754_j1f(x)
00059        float x;
00060 #endif
00061 {
00062        float z, s,c,ss,cc,r,u,v,y;
00063        int32_t hx,ix;
00064 
00065        GET_FLOAT_WORD(hx,x);
00066        ix = hx&0x7fffffff;
00067        if(ix>=0x7f800000) return one/x;
00068        y = fabsf(x);
00069        if(ix >= 0x40000000) {      /* |x| >= 2.0 */
00070               __sincosf (y, &s, &c);
00071               ss = -s-c;
00072               cc = s-c;
00073               if(ix<0x7f000000) {  /* make sure y+y not overflow */
00074                   z = __cosf(y+y);
00075                   if ((s*c)>zero) cc = z/ss;
00076                   else          ss = z/cc;
00077               }
00078        /*
00079         * j1(x) = 1/sqrt(pi) * (P(1,x)*cc - Q(1,x)*ss) / sqrt(x)
00080         * y1(x) = 1/sqrt(pi) * (P(1,x)*ss + Q(1,x)*cc) / sqrt(x)
00081         */
00082               if(ix>0x48000000) z = (invsqrtpi*cc)/__ieee754_sqrtf(y);
00083               else {
00084                   u = ponef(y); v = qonef(y);
00085                   z = invsqrtpi*(u*cc-v*ss)/__ieee754_sqrtf(y);
00086               }
00087               if(hx<0) return -z;
00088               else    return  z;
00089        }
00090        if(ix<0x32000000) {  /* |x|<2**-27 */
00091            if(huge+x>one) return (float)0.5*x;/* inexact if x!=0 necessary */
00092        }
00093        z = x*x;
00094        r =  z*(r00+z*(r01+z*(r02+z*r03)));
00095        s =  one+z*(s01+z*(s02+z*(s03+z*(s04+z*s05))));
00096        r *= x;
00097        return(x*(float)0.5+r/s);
00098 }
00099 
00100 #ifdef __STDC__
00101 static const float U0[5] = {
00102 #else
00103 static float U0[5] = {
00104 #endif
00105  -1.9605709612e-01, /* 0xbe48c331 */
00106   5.0443872809e-02, /* 0x3d4e9e3c */
00107  -1.9125689287e-03, /* 0xbafaaf2a */
00108   2.3525259166e-05, /* 0x37c5581c */
00109  -9.1909917899e-08, /* 0xb3c56003 */
00110 };
00111 #ifdef __STDC__
00112 static const float V0[5] = {
00113 #else
00114 static float V0[5] = {
00115 #endif
00116   1.9916731864e-02, /* 0x3ca3286a */
00117   2.0255257550e-04, /* 0x3954644b */
00118   1.3560879779e-06, /* 0x35b602d4 */
00119   6.2274145840e-09, /* 0x31d5f8eb */
00120   1.6655924903e-11, /* 0x2d9281cf */
00121 };
00122 
00123 #ifdef __STDC__
00124        float __ieee754_y1f(float x)
00125 #else
00126        float __ieee754_y1f(x)
00127        float x;
00128 #endif
00129 {
00130        float z, s,c,ss,cc,u,v;
00131        int32_t hx,ix;
00132 
00133        GET_FLOAT_WORD(hx,x);
00134         ix = 0x7fffffff&hx;
00135     /* if Y1(NaN) is NaN, Y1(-inf) is NaN, Y1(inf) is 0 */
00136        if(ix>=0x7f800000) return  one/(x+x*x);
00137         if(ix==0) return -HUGE_VALF+x;  /* -inf and overflow exception.  */
00138         if(hx<0) return zero/(zero*x);
00139         if(ix >= 0x40000000) {  /* |x| >= 2.0 */
00140               __sincosf (x, &s, &c);
00141                 ss = -s-c;
00142                 cc = s-c;
00143                 if(ix<0x7f000000) {  /* make sure x+x not overflow */
00144                     z = __cosf(x+x);
00145                     if ((s*c)>zero) cc = z/ss;
00146                     else            ss = z/cc;
00147                 }
00148         /* y1(x) = sqrt(2/(pi*x))*(p1(x)*sin(x0)+q1(x)*cos(x0))
00149          * where x0 = x-3pi/4
00150          *      Better formula:
00151          *              cos(x0) = cos(x)cos(3pi/4)+sin(x)sin(3pi/4)
00152          *                      =  1/sqrt(2) * (sin(x) - cos(x))
00153          *              sin(x0) = sin(x)cos(3pi/4)-cos(x)sin(3pi/4)
00154          *                      = -1/sqrt(2) * (cos(x) + sin(x))
00155          * To avoid cancellation, use
00156          *              sin(x) +- cos(x) = -cos(2x)/(sin(x) -+ cos(x))
00157          * to compute the worse one.
00158          */
00159                 if(ix>0x48000000) z = (invsqrtpi*ss)/__ieee754_sqrtf(x);
00160                 else {
00161                     u = ponef(x); v = qonef(x);
00162                     z = invsqrtpi*(u*ss+v*cc)/__ieee754_sqrtf(x);
00163                 }
00164                 return z;
00165         }
00166         if(ix<=0x24800000) {    /* x < 2**-54 */
00167             return(-tpi/x);
00168         }
00169         z = x*x;
00170         u = U0[0]+z*(U0[1]+z*(U0[2]+z*(U0[3]+z*U0[4])));
00171         v = one+z*(V0[0]+z*(V0[1]+z*(V0[2]+z*(V0[3]+z*V0[4]))));
00172         return(x*(u/v) + tpi*(__ieee754_j1f(x)*__ieee754_logf(x)-one/x));
00173 }
00174 
00175 /* For x >= 8, the asymptotic expansions of pone is
00176  *     1 + 15/128 s^2 - 4725/2^15 s^4 - ...,     where s = 1/x.
00177  * We approximate pone by
00178  *     pone(x) = 1 + (R/S)
00179  * where  R = pr0 + pr1*s^2 + pr2*s^4 + ... + pr5*s^10
00180  *       S = 1 + ps0*s^2 + ... + ps4*s^10
00181  * and
00182  *     | pone(x)-1-R/S | <= 2  ** ( -60.06)
00183  */
00184 
00185 #ifdef __STDC__
00186 static const float pr8[6] = { /* for x in [inf, 8]=1/[0,0.125] */
00187 #else
00188 static float pr8[6] = { /* for x in [inf, 8]=1/[0,0.125] */
00189 #endif
00190   0.0000000000e+00, /* 0x00000000 */
00191   1.1718750000e-01, /* 0x3df00000 */
00192   1.3239480972e+01, /* 0x4153d4ea */
00193   4.1205184937e+02, /* 0x43ce06a3 */
00194   3.8747453613e+03, /* 0x45722bed */
00195   7.9144794922e+03, /* 0x45f753d6 */
00196 };
00197 #ifdef __STDC__
00198 static const float ps8[5] = {
00199 #else
00200 static float ps8[5] = {
00201 #endif
00202   1.1420736694e+02, /* 0x42e46a2c */
00203   3.6509309082e+03, /* 0x45642ee5 */
00204   3.6956207031e+04, /* 0x47105c35 */
00205   9.7602796875e+04, /* 0x47bea166 */
00206   3.0804271484e+04, /* 0x46f0a88b */
00207 };
00208 
00209 #ifdef __STDC__
00210 static const float pr5[6] = { /* for x in [8,4.5454]=1/[0.125,0.22001] */
00211 #else
00212 static float pr5[6] = { /* for x in [8,4.5454]=1/[0.125,0.22001] */
00213 #endif
00214   1.3199052094e-11, /* 0x2d68333f */
00215   1.1718749255e-01, /* 0x3defffff */
00216   6.8027510643e+00, /* 0x40d9b023 */
00217   1.0830818176e+02, /* 0x42d89dca */
00218   5.1763616943e+02, /* 0x440168b7 */
00219   5.2871520996e+02, /* 0x44042dc6 */
00220 };
00221 #ifdef __STDC__
00222 static const float ps5[5] = {
00223 #else
00224 static float ps5[5] = {
00225 #endif
00226   5.9280597687e+01, /* 0x426d1f55 */
00227   9.9140142822e+02, /* 0x4477d9b1 */
00228   5.3532670898e+03, /* 0x45a74a23 */
00229   7.8446904297e+03, /* 0x45f52586 */
00230   1.5040468750e+03, /* 0x44bc0180 */
00231 };
00232 
00233 #ifdef __STDC__
00234 static const float pr3[6] = {
00235 #else
00236 static float pr3[6] = {/* for x in [4.547,2.8571]=1/[0.2199,0.35001] */
00237 #endif
00238   3.0250391081e-09, /* 0x314fe10d */
00239   1.1718686670e-01, /* 0x3defffab */
00240   3.9329774380e+00, /* 0x407bb5e7 */
00241   3.5119403839e+01, /* 0x420c7a45 */
00242   9.1055007935e+01, /* 0x42b61c2a */
00243   4.8559066772e+01, /* 0x42423c7c */
00244 };
00245 #ifdef __STDC__
00246 static const float ps3[5] = {
00247 #else
00248 static float ps3[5] = {
00249 #endif
00250   3.4791309357e+01, /* 0x420b2a4d */
00251   3.3676245117e+02, /* 0x43a86198 */
00252   1.0468714600e+03, /* 0x4482dbe3 */
00253   8.9081134033e+02, /* 0x445eb3ed */
00254   1.0378793335e+02, /* 0x42cf936c */
00255 };
00256 
00257 #ifdef __STDC__
00258 static const float pr2[6] = {/* for x in [2.8570,2]=1/[0.3499,0.5] */
00259 #else
00260 static float pr2[6] = {/* for x in [2.8570,2]=1/[0.3499,0.5] */
00261 #endif
00262   1.0771083225e-07, /* 0x33e74ea8 */
00263   1.1717621982e-01, /* 0x3deffa16 */
00264   2.3685150146e+00, /* 0x401795c0 */
00265   1.2242610931e+01, /* 0x4143e1bc */
00266   1.7693971634e+01, /* 0x418d8d41 */
00267   5.0735230446e+00, /* 0x40a25a4d */
00268 };
00269 #ifdef __STDC__
00270 static const float ps2[5] = {
00271 #else
00272 static float ps2[5] = {
00273 #endif
00274   2.1436485291e+01, /* 0x41ab7dec */
00275   1.2529022980e+02, /* 0x42fa9499 */
00276   2.3227647400e+02, /* 0x436846c7 */
00277   1.1767937469e+02, /* 0x42eb5bd7 */
00278   8.3646392822e+00, /* 0x4105d590 */
00279 };
00280 
00281 #ifdef __STDC__
00282        static float ponef(float x)
00283 #else
00284        static float ponef(x)
00285        float x;
00286 #endif
00287 {
00288 #ifdef __STDC__
00289        const float *p,*q;
00290 #else
00291        float *p,*q;
00292 #endif
00293        float z,r,s;
00294         int32_t ix;
00295        GET_FLOAT_WORD(ix,x);
00296        ix &= 0x7fffffff;
00297         if(ix>=0x41000000)     {p = pr8; q= ps8;}
00298         else if(ix>=0x40f71c58){p = pr5; q= ps5;}
00299         else if(ix>=0x4036db68){p = pr3; q= ps3;}
00300         else if(ix>=0x40000000){p = pr2; q= ps2;}
00301         z = one/(x*x);
00302         r = p[0]+z*(p[1]+z*(p[2]+z*(p[3]+z*(p[4]+z*p[5]))));
00303         s = one+z*(q[0]+z*(q[1]+z*(q[2]+z*(q[3]+z*q[4]))));
00304         return one+ r/s;
00305 }
00306 
00307 
00308 /* For x >= 8, the asymptotic expansions of qone is
00309  *     3/8 s - 105/1024 s^3 - ..., where s = 1/x.
00310  * We approximate pone by
00311  *     qone(x) = s*(0.375 + (R/S))
00312  * where  R = qr1*s^2 + qr2*s^4 + ... + qr5*s^10
00313  *       S = 1 + qs1*s^2 + ... + qs6*s^12
00314  * and
00315  *     | qone(x)/s -0.375-R/S | <= 2  ** ( -61.13)
00316  */
00317 
00318 #ifdef __STDC__
00319 static const float qr8[6] = { /* for x in [inf, 8]=1/[0,0.125] */
00320 #else
00321 static float qr8[6] = { /* for x in [inf, 8]=1/[0,0.125] */
00322 #endif
00323   0.0000000000e+00, /* 0x00000000 */
00324  -1.0253906250e-01, /* 0xbdd20000 */
00325  -1.6271753311e+01, /* 0xc1822c8d */
00326  -7.5960174561e+02, /* 0xc43de683 */
00327  -1.1849806641e+04, /* 0xc639273a */
00328  -4.8438511719e+04, /* 0xc73d3683 */
00329 };
00330 #ifdef __STDC__
00331 static const float qs8[6] = {
00332 #else
00333 static float qs8[6] = {
00334 #endif
00335   1.6139537048e+02, /* 0x43216537 */
00336   7.8253862305e+03, /* 0x45f48b17 */
00337   1.3387534375e+05, /* 0x4802bcd6 */
00338   7.1965775000e+05, /* 0x492fb29c */
00339   6.6660125000e+05, /* 0x4922be94 */
00340  -2.9449025000e+05, /* 0xc88fcb48 */
00341 };
00342 
00343 #ifdef __STDC__
00344 static const float qr5[6] = { /* for x in [8,4.5454]=1/[0.125,0.22001] */
00345 #else
00346 static float qr5[6] = { /* for x in [8,4.5454]=1/[0.125,0.22001] */
00347 #endif
00348  -2.0897993405e-11, /* 0xadb7d219 */
00349  -1.0253904760e-01, /* 0xbdd1fffe */
00350  -8.0564479828e+00, /* 0xc100e736 */
00351  -1.8366960144e+02, /* 0xc337ab6b */
00352  -1.3731937256e+03, /* 0xc4aba633 */
00353  -2.6124443359e+03, /* 0xc523471c */
00354 };
00355 #ifdef __STDC__
00356 static const float qs5[6] = {
00357 #else
00358 static float qs5[6] = {
00359 #endif
00360   8.1276550293e+01, /* 0x42a28d98 */
00361   1.9917987061e+03, /* 0x44f8f98f */
00362   1.7468484375e+04, /* 0x468878f8 */
00363   4.9851425781e+04, /* 0x4742bb6d */
00364   2.7948074219e+04, /* 0x46da5826 */
00365  -4.7191835938e+03, /* 0xc5937978 */
00366 };
00367 
00368 #ifdef __STDC__
00369 static const float qr3[6] = {
00370 #else
00371 static float qr3[6] = {/* for x in [4.547,2.8571]=1/[0.2199,0.35001] */
00372 #endif
00373  -5.0783124372e-09, /* 0xb1ae7d4f */
00374  -1.0253783315e-01, /* 0xbdd1ff5b */
00375  -4.6101160049e+00, /* 0xc0938612 */
00376  -5.7847221375e+01, /* 0xc267638e */
00377  -2.2824453735e+02, /* 0xc3643e9a */
00378  -2.1921012878e+02, /* 0xc35b35cb */
00379 };
00380 #ifdef __STDC__
00381 static const float qs3[6] = {
00382 #else
00383 static float qs3[6] = {
00384 #endif
00385   4.7665153503e+01, /* 0x423ea91e */
00386   6.7386511230e+02, /* 0x4428775e */
00387   3.3801528320e+03, /* 0x45534272 */
00388   5.5477290039e+03, /* 0x45ad5dd5 */
00389   1.9031191406e+03, /* 0x44ede3d0 */
00390  -1.3520118713e+02, /* 0xc3073381 */
00391 };
00392 
00393 #ifdef __STDC__
00394 static const float qr2[6] = {/* for x in [2.8570,2]=1/[0.3499,0.5] */
00395 #else
00396 static float qr2[6] = {/* for x in [2.8570,2]=1/[0.3499,0.5] */
00397 #endif
00398  -1.7838172539e-07, /* 0xb43f8932 */
00399  -1.0251704603e-01, /* 0xbdd1f475 */
00400  -2.7522056103e+00, /* 0xc0302423 */
00401  -1.9663616180e+01, /* 0xc19d4f16 */
00402  -4.2325313568e+01, /* 0xc2294d1f */
00403  -2.1371921539e+01, /* 0xc1aaf9b2 */
00404 };
00405 #ifdef __STDC__
00406 static const float qs2[6] = {
00407 #else
00408 static float qs2[6] = {
00409 #endif
00410   2.9533363342e+01, /* 0x41ec4454 */
00411   2.5298155212e+02, /* 0x437cfb47 */
00412   7.5750280762e+02, /* 0x443d602e */
00413   7.3939318848e+02, /* 0x4438d92a */
00414   1.5594900513e+02, /* 0x431bf2f2 */
00415  -4.9594988823e+00, /* 0xc09eb437 */
00416 };
00417 
00418 #ifdef __STDC__
00419        static float qonef(float x)
00420 #else
00421        static float qonef(x)
00422        float x;
00423 #endif
00424 {
00425 #ifdef __STDC__
00426        const float *p,*q;
00427 #else
00428        float *p,*q;
00429 #endif
00430        float  s,r,z;
00431        int32_t ix;
00432        GET_FLOAT_WORD(ix,x);
00433        ix &= 0x7fffffff;
00434        if(ix>=0x40200000)     {p = qr8; q= qs8;}
00435        else if(ix>=0x40f71c58){p = qr5; q= qs5;}
00436        else if(ix>=0x4036db68){p = qr3; q= qs3;}
00437        else if(ix>=0x40000000){p = qr2; q= qs2;}
00438        z = one/(x*x);
00439        r = p[0]+z*(p[1]+z*(p[2]+z*(p[3]+z*(p[4]+z*p[5]))));
00440        s = one+z*(q[0]+z*(q[1]+z*(q[2]+z*(q[3]+z*(q[4]+z*q[5])))));
00441        return ((float).375 + r/s)/x;
00442 }