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glibc  2.9
e_j0f.c
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00001 /* e_j0f.c -- float version of e_j0.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_j0f.c,v 1.4 1995/05/10 20:45:25 jtc Exp $";
00018 #endif
00019 
00020 #include "math.h"
00021 #include "math_private.h"
00022 
00023 #ifdef __STDC__
00024 static float pzerof(float), qzerof(float);
00025 #else
00026 static float pzerof(), qzerof();
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, 2.00] */
00039 R02  =  1.5625000000e-02, /* 0x3c800000 */
00040 R03  = -1.8997929874e-04, /* 0xb947352e */
00041 R04  =  1.8295404516e-06, /* 0x35f58e88 */
00042 R05  = -4.6183270541e-09, /* 0xb19eaf3c */
00043 S01  =  1.5619102865e-02, /* 0x3c7fe744 */
00044 S02  =  1.1692678527e-04, /* 0x38f53697 */
00045 S03  =  5.1354652442e-07, /* 0x3509daa6 */
00046 S04  =  1.1661400734e-09; /* 0x30a045e8 */
00047 
00048 #ifdef __STDC__
00049 static const float zero = 0.0;
00050 #else
00051 static float zero = 0.0;
00052 #endif
00053 
00054 #ifdef __STDC__
00055        float __ieee754_j0f(float x)
00056 #else
00057        float __ieee754_j0f(x)
00058        float x;
00059 #endif
00060 {
00061        float z, s,c,ss,cc,r,u,v;
00062        int32_t hx,ix;
00063 
00064        GET_FLOAT_WORD(hx,x);
00065        ix = hx&0x7fffffff;
00066        if(ix>=0x7f800000) return one/(x*x);
00067        x = fabsf(x);
00068        if(ix >= 0x40000000) {      /* |x| >= 2.0 */
00069               __sincosf (x, &s, &c);
00070               ss = s-c;
00071               cc = s+c;
00072               if(ix<0x7f000000) {  /* make sure x+x not overflow */
00073                   z = -__cosf(x+x);
00074                   if ((s*c)<zero) cc = z/ss;
00075                   else          ss = z/cc;
00076               }
00077        /*
00078         * j0(x) = 1/sqrt(pi) * (P(0,x)*cc - Q(0,x)*ss) / sqrt(x)
00079         * y0(x) = 1/sqrt(pi) * (P(0,x)*ss + Q(0,x)*cc) / sqrt(x)
00080         */
00081               if(ix>0x48000000) z = (invsqrtpi*cc)/__ieee754_sqrtf(x);
00082               else {
00083                   u = pzerof(x); v = qzerof(x);
00084                   z = invsqrtpi*(u*cc-v*ss)/__ieee754_sqrtf(x);
00085               }
00086               return z;
00087        }
00088        if(ix<0x39000000) {  /* |x| < 2**-13 */
00089            if(huge+x>one) { /* raise inexact if x != 0 */
00090                if(ix<0x32000000) return one;     /* |x|<2**-27 */
00091                else        return one - (float)0.25*x*x;
00092            }
00093        }
00094        z = x*x;
00095        r =  z*(R02+z*(R03+z*(R04+z*R05)));
00096        s =  one+z*(S01+z*(S02+z*(S03+z*S04)));
00097        if(ix < 0x3F800000) {       /* |x| < 1.00 */
00098            return one + z*((float)-0.25+(r/s));
00099        } else {
00100            u = (float)0.5*x;
00101            return((one+u)*(one-u)+z*(r/s));
00102        }
00103 }
00104 
00105 #ifdef __STDC__
00106 static const float
00107 #else
00108 static float
00109 #endif
00110 u00  = -7.3804296553e-02, /* 0xbd9726b5 */
00111 u01  =  1.7666645348e-01, /* 0x3e34e80d */
00112 u02  = -1.3818567619e-02, /* 0xbc626746 */
00113 u03  =  3.4745343146e-04, /* 0x39b62a69 */
00114 u04  = -3.8140706238e-06, /* 0xb67ff53c */
00115 u05  =  1.9559013964e-08, /* 0x32a802ba */
00116 u06  = -3.9820518410e-11, /* 0xae2f21eb */
00117 v01  =  1.2730483897e-02, /* 0x3c509385 */
00118 v02  =  7.6006865129e-05, /* 0x389f65e0 */
00119 v03  =  2.5915085189e-07, /* 0x348b216c */
00120 v04  =  4.4111031494e-10; /* 0x2ff280c2 */
00121 
00122 #ifdef __STDC__
00123        float __ieee754_y0f(float x)
00124 #else
00125        float __ieee754_y0f(x)
00126        float x;
00127 #endif
00128 {
00129        float z, s,c,ss,cc,u,v;
00130        int32_t hx,ix;
00131 
00132        GET_FLOAT_WORD(hx,x);
00133         ix = 0x7fffffff&hx;
00134     /* Y0(NaN) is NaN, y0(-inf) is Nan, y0(inf) is 0, y0(0) is -inf.  */
00135        if(ix>=0x7f800000) return  one/(x+x*x);
00136         if(ix==0) return -HUGE_VALF+x;  /* -inf and overflow exception.  */
00137         if(hx<0) return zero/(zero*x);
00138         if(ix >= 0x40000000) {  /* |x| >= 2.0 */
00139         /* y0(x) = sqrt(2/(pi*x))*(p0(x)*sin(x0)+q0(x)*cos(x0))
00140          * where x0 = x-pi/4
00141          *      Better formula:
00142          *              cos(x0) = cos(x)cos(pi/4)+sin(x)sin(pi/4)
00143          *                      =  1/sqrt(2) * (sin(x) + cos(x))
00144          *              sin(x0) = sin(x)cos(3pi/4)-cos(x)sin(3pi/4)
00145          *                      =  1/sqrt(2) * (sin(x) - cos(x))
00146          * To avoid cancellation, use
00147          *              sin(x) +- cos(x) = -cos(2x)/(sin(x) -+ cos(x))
00148          * to compute the worse one.
00149          */
00150               __sincosf (x, &s, &c);
00151                 ss = s-c;
00152                 cc = s+c;
00153        /*
00154         * j0(x) = 1/sqrt(pi) * (P(0,x)*cc - Q(0,x)*ss) / sqrt(x)
00155         * y0(x) = 1/sqrt(pi) * (P(0,x)*ss + Q(0,x)*cc) / sqrt(x)
00156         */
00157                 if(ix<0x7f000000) {  /* make sure x+x not overflow */
00158                     z = -__cosf(x+x);
00159                     if ((s*c)<zero) cc = z/ss;
00160                     else            ss = z/cc;
00161                 }
00162                 if(ix>0x48000000) z = (invsqrtpi*ss)/__ieee754_sqrtf(x);
00163                 else {
00164                     u = pzerof(x); v = qzerof(x);
00165                     z = invsqrtpi*(u*ss+v*cc)/__ieee754_sqrtf(x);
00166                 }
00167                 return z;
00168        }
00169        if(ix<=0x32000000) { /* x < 2**-27 */
00170            return(u00 + tpi*__ieee754_logf(x));
00171        }
00172        z = x*x;
00173        u = u00+z*(u01+z*(u02+z*(u03+z*(u04+z*(u05+z*u06)))));
00174        v = one+z*(v01+z*(v02+z*(v03+z*v04)));
00175        return(u/v + tpi*(__ieee754_j0f(x)*__ieee754_logf(x)));
00176 }
00177 
00178 /* The asymptotic expansions of pzero is
00179  *     1 - 9/128 s^2 + 11025/98304 s^4 - ...,    where s = 1/x.
00180  * For x >= 2, We approximate pzero by
00181  *     pzero(x) = 1 + (R/S)
00182  * where  R = pR0 + pR1*s^2 + pR2*s^4 + ... + pR5*s^10
00183  *       S = 1 + pS0*s^2 + ... + pS4*s^10
00184  * and
00185  *     | pzero(x)-1-R/S | <= 2  ** ( -60.26)
00186  */
00187 #ifdef __STDC__
00188 static const float pR8[6] = { /* for x in [inf, 8]=1/[0,0.125] */
00189 #else
00190 static float pR8[6] = { /* for x in [inf, 8]=1/[0,0.125] */
00191 #endif
00192   0.0000000000e+00, /* 0x00000000 */
00193  -7.0312500000e-02, /* 0xbd900000 */
00194  -8.0816707611e+00, /* 0xc1014e86 */
00195  -2.5706311035e+02, /* 0xc3808814 */
00196  -2.4852163086e+03, /* 0xc51b5376 */
00197  -5.2530439453e+03, /* 0xc5a4285a */
00198 };
00199 #ifdef __STDC__
00200 static const float pS8[5] = {
00201 #else
00202 static float pS8[5] = {
00203 #endif
00204   1.1653436279e+02, /* 0x42e91198 */
00205   3.8337448730e+03, /* 0x456f9beb */
00206   4.0597855469e+04, /* 0x471e95db */
00207   1.1675296875e+05, /* 0x47e4087c */
00208   4.7627726562e+04, /* 0x473a0bba */
00209 };
00210 #ifdef __STDC__
00211 static const float pR5[6] = { /* for x in [8,4.5454]=1/[0.125,0.22001] */
00212 #else
00213 static float pR5[6] = { /* for x in [8,4.5454]=1/[0.125,0.22001] */
00214 #endif
00215  -1.1412546255e-11, /* 0xad48c58a */
00216  -7.0312492549e-02, /* 0xbd8fffff */
00217  -4.1596107483e+00, /* 0xc0851b88 */
00218  -6.7674766541e+01, /* 0xc287597b */
00219  -3.3123129272e+02, /* 0xc3a59d9b */
00220  -3.4643338013e+02, /* 0xc3ad3779 */
00221 };
00222 #ifdef __STDC__
00223 static const float pS5[5] = {
00224 #else
00225 static float pS5[5] = {
00226 #endif
00227   6.0753936768e+01, /* 0x42730408 */
00228   1.0512523193e+03, /* 0x44836813 */
00229   5.9789707031e+03, /* 0x45bad7c4 */
00230   9.6254453125e+03, /* 0x461665c8 */
00231   2.4060581055e+03, /* 0x451660ee */
00232 };
00233 
00234 #ifdef __STDC__
00235 static const float pR3[6] = {/* for x in [4.547,2.8571]=1/[0.2199,0.35001] */
00236 #else
00237 static float pR3[6] = {/* for x in [4.547,2.8571]=1/[0.2199,0.35001] */
00238 #endif
00239  -2.5470459075e-09, /* 0xb12f081b */
00240  -7.0311963558e-02, /* 0xbd8fffb8 */
00241  -2.4090321064e+00, /* 0xc01a2d95 */
00242  -2.1965976715e+01, /* 0xc1afba52 */
00243  -5.8079170227e+01, /* 0xc2685112 */
00244  -3.1447946548e+01, /* 0xc1fb9565 */
00245 };
00246 #ifdef __STDC__
00247 static const float pS3[5] = {
00248 #else
00249 static float pS3[5] = {
00250 #endif
00251   3.5856033325e+01, /* 0x420f6c94 */
00252   3.6151397705e+02, /* 0x43b4c1ca */
00253   1.1936077881e+03, /* 0x44953373 */
00254   1.1279968262e+03, /* 0x448cffe6 */
00255   1.7358093262e+02, /* 0x432d94b8 */
00256 };
00257 
00258 #ifdef __STDC__
00259 static const float pR2[6] = {/* for x in [2.8570,2]=1/[0.3499,0.5] */
00260 #else
00261 static float pR2[6] = {/* for x in [2.8570,2]=1/[0.3499,0.5] */
00262 #endif
00263  -8.8753431271e-08, /* 0xb3be98b7 */
00264  -7.0303097367e-02, /* 0xbd8ffb12 */
00265  -1.4507384300e+00, /* 0xbfb9b1cc */
00266  -7.6356959343e+00, /* 0xc0f4579f */
00267  -1.1193166733e+01, /* 0xc1331736 */
00268  -3.2336456776e+00, /* 0xc04ef40d */
00269 };
00270 #ifdef __STDC__
00271 static const float pS2[5] = {
00272 #else
00273 static float pS2[5] = {
00274 #endif
00275   2.2220300674e+01, /* 0x41b1c32d */
00276   1.3620678711e+02, /* 0x430834f0 */
00277   2.7047027588e+02, /* 0x43873c32 */
00278   1.5387539673e+02, /* 0x4319e01a */
00279   1.4657617569e+01, /* 0x416a859a */
00280 };
00281 
00282 #ifdef __STDC__
00283        static float pzerof(float x)
00284 #else
00285        static float pzerof(x)
00286        float x;
00287 #endif
00288 {
00289 #ifdef __STDC__
00290        const float *p,*q;
00291 #else
00292        float *p,*q;
00293 #endif
00294        float z,r,s;
00295        int32_t ix;
00296        GET_FLOAT_WORD(ix,x);
00297        ix &= 0x7fffffff;
00298        if(ix>=0x41000000)     {p = pR8; q= pS8;}
00299        else if(ix>=0x40f71c58){p = pR5; q= pS5;}
00300        else if(ix>=0x4036db68){p = pR3; q= pS3;}
00301        else if(ix>=0x40000000){p = pR2; q= pS2;}
00302        z = one/(x*x);
00303        r = p[0]+z*(p[1]+z*(p[2]+z*(p[3]+z*(p[4]+z*p[5]))));
00304        s = one+z*(q[0]+z*(q[1]+z*(q[2]+z*(q[3]+z*q[4]))));
00305        return one+ r/s;
00306 }
00307 
00308 
00309 /* For x >= 8, the asymptotic expansions of qzero is
00310  *     -1/8 s + 75/1024 s^3 - ..., where s = 1/x.
00311  * We approximate pzero by
00312  *     qzero(x) = s*(-1.25 + (R/S))
00313  * where  R = qR0 + qR1*s^2 + qR2*s^4 + ... + qR5*s^10
00314  *       S = 1 + qS0*s^2 + ... + qS5*s^12
00315  * and
00316  *     | qzero(x)/s +1.25-R/S | <= 2  ** ( -61.22)
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   7.3242187500e-02, /* 0x3d960000 */
00325   1.1768206596e+01, /* 0x413c4a93 */
00326   5.5767340088e+02, /* 0x440b6b19 */
00327   8.8591972656e+03, /* 0x460a6cca */
00328   3.7014625000e+04, /* 0x471096a0 */
00329 };
00330 #ifdef __STDC__
00331 static const float qS8[6] = {
00332 #else
00333 static float qS8[6] = {
00334 #endif
00335   1.6377603149e+02, /* 0x4323c6aa */
00336   8.0983447266e+03, /* 0x45fd12c2 */
00337   1.4253829688e+05, /* 0x480b3293 */
00338   8.0330925000e+05, /* 0x49441ed4 */
00339   8.4050156250e+05, /* 0x494d3359 */
00340  -3.4389928125e+05, /* 0xc8a7eb69 */
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   1.8408595828e-11, /* 0x2da1ec79 */
00349   7.3242180049e-02, /* 0x3d95ffff */
00350   5.8356351852e+00, /* 0x40babd86 */
00351   1.3511157227e+02, /* 0x43071c90 */
00352   1.0272437744e+03, /* 0x448067cd */
00353   1.9899779053e+03, /* 0x44f8bf4b */
00354 };
00355 #ifdef __STDC__
00356 static const float qS5[6] = {
00357 #else
00358 static float qS5[6] = {
00359 #endif
00360   8.2776611328e+01, /* 0x42a58da0 */
00361   2.0778142090e+03, /* 0x4501dd07 */
00362   1.8847289062e+04, /* 0x46933e94 */
00363   5.6751113281e+04, /* 0x475daf1d */
00364   3.5976753906e+04, /* 0x470c88c1 */
00365  -5.3543427734e+03, /* 0xc5a752be */
00366 };
00367 
00368 #ifdef __STDC__
00369 static const float qR3[6] = {/* for x in [4.547,2.8571]=1/[0.2199,0.35001] */
00370 #else
00371 static float qR3[6] = {/* for x in [4.547,2.8571]=1/[0.2199,0.35001] */
00372 #endif
00373   4.3774099900e-09, /* 0x3196681b */
00374   7.3241114616e-02, /* 0x3d95ff70 */
00375   3.3442313671e+00, /* 0x405607e3 */
00376   4.2621845245e+01, /* 0x422a7cc5 */
00377   1.7080809021e+02, /* 0x432acedf */
00378   1.6673394775e+02, /* 0x4326bbe4 */
00379 };
00380 #ifdef __STDC__
00381 static const float qS3[6] = {
00382 #else
00383 static float qS3[6] = {
00384 #endif
00385   4.8758872986e+01, /* 0x42430916 */
00386   7.0968920898e+02, /* 0x44316c1c */
00387   3.7041481934e+03, /* 0x4567825f */
00388   6.4604252930e+03, /* 0x45c9e367 */
00389   2.5163337402e+03, /* 0x451d4557 */
00390  -1.4924745178e+02, /* 0xc3153f59 */
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   1.5044444979e-07, /* 0x342189db */
00399   7.3223426938e-02, /* 0x3d95f62a */
00400   1.9981917143e+00, /* 0x3fffc4bf */
00401   1.4495602608e+01, /* 0x4167edfd */
00402   3.1666231155e+01, /* 0x41fd5471 */
00403   1.6252708435e+01, /* 0x4182058c */
00404 };
00405 #ifdef __STDC__
00406 static const float qS2[6] = {
00407 #else
00408 static float qS2[6] = {
00409 #endif
00410   3.0365585327e+01, /* 0x41f2ecb8 */
00411   2.6934811401e+02, /* 0x4386ac8f */
00412   8.4478375244e+02, /* 0x44533229 */
00413   8.8293585205e+02, /* 0x445cbbe5 */
00414   2.1266638184e+02, /* 0x4354aa98 */
00415  -5.3109550476e+00, /* 0xc0a9f358 */
00416 };
00417 
00418 #ifdef __STDC__
00419        static float qzerof(float x)
00420 #else
00421        static float qzerof(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>=0x41000000)     {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).125 + r/s)/x;
00442 }