1 /* e_j0f.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 <sys/cdefs.h> 17 /* 18 * See e_j0.c for complete comments. 19 */ 20 21 #include "math.h" 22 #include "math_private.h" 23 24 static __inline float pzerof(float), qzerof(float); 25 26 static const volatile float vone = 1, vzero = 0; 27 28 static const float 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 static const float zero = 0, qrtr = 0.25; 44 45 float 46 j0f(float x) 47 { 48 float z, s,c,ss,cc,r,u,v; 49 int32_t hx,ix; 50 51 GET_FLOAT_WORD(hx,x); 52 ix = hx&0x7fffffff; 53 if(ix>=0x7f800000) return one/(x*x); 54 x = fabsf(x); 55 if(ix >= 0x40000000) { /* |x| >= 2.0 */ 56 sincosf(x, &s, &c); 57 ss = s-c; 58 cc = s+c; 59 if(ix<0x7f000000) { /* Make sure x+x does not overflow. */ 60 z = -cosf(x+x); 61 if ((s*c)<zero) cc = z/ss; 62 else ss = z/cc; 63 } 64 /* 65 * j0(x) = 1/sqrt(pi) * (P(0,x)*cc - Q(0,x)*ss) / sqrt(x) 66 * y0(x) = 1/sqrt(pi) * (P(0,x)*ss + Q(0,x)*cc) / sqrt(x) 67 */ 68 if(ix>0x58000000) z = (invsqrtpi*cc)/sqrtf(x); /* |x|>2**49 */ 69 else { 70 u = pzerof(x); v = qzerof(x); 71 z = invsqrtpi*(u*cc-v*ss)/sqrtf(x); 72 } 73 return z; 74 } 75 if(ix<0x3b000000) { /* |x| < 2**-9 */ 76 if(huge+x>one) { /* raise inexact if x != 0 */ 77 if(ix<0x39800000) return one; /* |x|<2**-12 */ 78 else return one - x*x/4; 79 } 80 } 81 z = x*x; 82 r = z*(R02+z*(R03+z*(R04+z*R05))); 83 s = one+z*(S01+z*(S02+z*(S03+z*S04))); 84 if(ix < 0x3F800000) { /* |x| < 1.00 */ 85 return one + z*((r/s)-qrtr); 86 } else { 87 u = x/2; 88 return((one+u)*(one-u)+z*(r/s)); 89 } 90 } 91 92 static const float 93 u00 = -7.3804296553e-02, /* 0xbd9726b5 */ 94 u01 = 1.7666645348e-01, /* 0x3e34e80d */ 95 u02 = -1.3818567619e-02, /* 0xbc626746 */ 96 u03 = 3.4745343146e-04, /* 0x39b62a69 */ 97 u04 = -3.8140706238e-06, /* 0xb67ff53c */ 98 u05 = 1.9559013964e-08, /* 0x32a802ba */ 99 u06 = -3.9820518410e-11, /* 0xae2f21eb */ 100 v01 = 1.2730483897e-02, /* 0x3c509385 */ 101 v02 = 7.6006865129e-05, /* 0x389f65e0 */ 102 v03 = 2.5915085189e-07, /* 0x348b216c */ 103 v04 = 4.4111031494e-10; /* 0x2ff280c2 */ 104 105 float 106 y0f(float x) 107 { 108 float z, s,c,ss,cc,u,v; 109 int32_t hx,ix; 110 111 GET_FLOAT_WORD(hx,x); 112 ix = 0x7fffffff&hx; 113 if(ix>=0x7f800000) return vone/(x+x*x); 114 if(ix==0) return -one/vzero; 115 if(hx<0) return vzero/vzero; 116 if(ix >= 0x40000000) { /* |x| >= 2.0 */ 117 /* y0(x) = sqrt(2/(pi*x))*(p0(x)*sin(x0)+q0(x)*cos(x0)) 118 * where x0 = x-pi/4 119 * Better formula: 120 * cos(x0) = cos(x)cos(pi/4)+sin(x)sin(pi/4) 121 * = 1/sqrt(2) * (sin(x) + cos(x)) 122 * sin(x0) = sin(x)cos(3pi/4)-cos(x)sin(3pi/4) 123 * = 1/sqrt(2) * (sin(x) - cos(x)) 124 * To avoid cancellation, use 125 * sin(x) +- cos(x) = -cos(2x)/(sin(x) -+ cos(x)) 126 * to compute the worse one. 127 */ 128 sincosf(x, &s, &c); 129 ss = s-c; 130 cc = s+c; 131 /* 132 * j0(x) = 1/sqrt(pi) * (P(0,x)*cc - Q(0,x)*ss) / sqrt(x) 133 * y0(x) = 1/sqrt(pi) * (P(0,x)*ss + Q(0,x)*cc) / sqrt(x) 134 */ 135 if(ix<0x7f000000) { /* make sure x+x not overflow */ 136 z = -cosf(x+x); 137 if ((s*c)<zero) cc = z/ss; 138 else ss = z/cc; 139 } 140 if(ix>0x58000000) z = (invsqrtpi*ss)/sqrtf(x); /* |x|>2**49 */ 141 else { 142 u = pzerof(x); v = qzerof(x); 143 z = invsqrtpi*(u*ss+v*cc)/sqrtf(x); 144 } 145 return z; 146 } 147 if(ix<=0x39000000) { /* x < 2**-13 */ 148 return(u00 + tpi*logf(x)); 149 } 150 z = x*x; 151 u = u00+z*(u01+z*(u02+z*(u03+z*(u04+z*(u05+z*u06))))); 152 v = one+z*(v01+z*(v02+z*(v03+z*v04))); 153 return(u/v + tpi*(j0f(x)*logf(x))); 154 } 155 156 /* The asymptotic expansions of pzero is 157 * 1 - 9/128 s^2 + 11025/98304 s^4 - ..., where s = 1/x. 158 * For x >= 2, We approximate pzero by 159 * pzero(x) = 1 + (R/S) 160 * where R = pR0 + pR1*s^2 + pR2*s^4 + ... + pR5*s^10 161 * S = 1 + pS0*s^2 + ... + pS4*s^10 162 * and 163 * | pzero(x)-1-R/S | <= 2 ** ( -60.26) 164 */ 165 static const float pR8[6] = { /* for x in [inf, 8]=1/[0,0.125] */ 166 0.0000000000e+00, /* 0x00000000 */ 167 -7.0312500000e-02, /* 0xbd900000 */ 168 -8.0816707611e+00, /* 0xc1014e86 */ 169 -2.5706311035e+02, /* 0xc3808814 */ 170 -2.4852163086e+03, /* 0xc51b5376 */ 171 -5.2530439453e+03, /* 0xc5a4285a */ 172 }; 173 static const float pS8[5] = { 174 1.1653436279e+02, /* 0x42e91198 */ 175 3.8337448730e+03, /* 0x456f9beb */ 176 4.0597855469e+04, /* 0x471e95db */ 177 1.1675296875e+05, /* 0x47e4087c */ 178 4.7627726562e+04, /* 0x473a0bba */ 179 }; 180 static const float pR5[6] = { /* for x in [8,4.5454]=1/[0.125,0.22001] */ 181 -1.1412546255e-11, /* 0xad48c58a */ 182 -7.0312492549e-02, /* 0xbd8fffff */ 183 -4.1596107483e+00, /* 0xc0851b88 */ 184 -6.7674766541e+01, /* 0xc287597b */ 185 -3.3123129272e+02, /* 0xc3a59d9b */ 186 -3.4643338013e+02, /* 0xc3ad3779 */ 187 }; 188 static const float pS5[5] = { 189 6.0753936768e+01, /* 0x42730408 */ 190 1.0512523193e+03, /* 0x44836813 */ 191 5.9789707031e+03, /* 0x45bad7c4 */ 192 9.6254453125e+03, /* 0x461665c8 */ 193 2.4060581055e+03, /* 0x451660ee */ 194 }; 195 196 static const float pR3[6] = {/* for x in [4.547,2.8571]=1/[0.2199,0.35001] */ 197 -2.5470459075e-09, /* 0xb12f081b */ 198 -7.0311963558e-02, /* 0xbd8fffb8 */ 199 -2.4090321064e+00, /* 0xc01a2d95 */ 200 -2.1965976715e+01, /* 0xc1afba52 */ 201 -5.8079170227e+01, /* 0xc2685112 */ 202 -3.1447946548e+01, /* 0xc1fb9565 */ 203 }; 204 static const float pS3[5] = { 205 3.5856033325e+01, /* 0x420f6c94 */ 206 3.6151397705e+02, /* 0x43b4c1ca */ 207 1.1936077881e+03, /* 0x44953373 */ 208 1.1279968262e+03, /* 0x448cffe6 */ 209 1.7358093262e+02, /* 0x432d94b8 */ 210 }; 211 212 static const float pR2[6] = {/* for x in [2.8570,2]=1/[0.3499,0.5] */ 213 -8.8753431271e-08, /* 0xb3be98b7 */ 214 -7.0303097367e-02, /* 0xbd8ffb12 */ 215 -1.4507384300e+00, /* 0xbfb9b1cc */ 216 -7.6356959343e+00, /* 0xc0f4579f */ 217 -1.1193166733e+01, /* 0xc1331736 */ 218 -3.2336456776e+00, /* 0xc04ef40d */ 219 }; 220 static const float pS2[5] = { 221 2.2220300674e+01, /* 0x41b1c32d */ 222 1.3620678711e+02, /* 0x430834f0 */ 223 2.7047027588e+02, /* 0x43873c32 */ 224 1.5387539673e+02, /* 0x4319e01a */ 225 1.4657617569e+01, /* 0x416a859a */ 226 }; 227 228 static __inline float 229 pzerof(float x) 230 { 231 const float *p,*q; 232 float z,r,s; 233 int32_t ix; 234 GET_FLOAT_WORD(ix,x); 235 ix &= 0x7fffffff; 236 if(ix>=0x41000000) {p = pR8; q= pS8;} 237 else if(ix>=0x409173eb){p = pR5; q= pS5;} 238 else if(ix>=0x4036d917){p = pR3; q= pS3;} 239 else {p = pR2; q= pS2;} /* ix>=0x40000000 */ 240 z = one/(x*x); 241 r = p[0]+z*(p[1]+z*(p[2]+z*(p[3]+z*(p[4]+z*p[5])))); 242 s = one+z*(q[0]+z*(q[1]+z*(q[2]+z*(q[3]+z*q[4])))); 243 return one+ r/s; 244 } 245 246 247 /* For x >= 8, the asymptotic expansions of qzero is 248 * -1/8 s + 75/1024 s^3 - ..., where s = 1/x. 249 * We approximate pzero by 250 * qzero(x) = s*(-1.25 + (R/S)) 251 * where R = qR0 + qR1*s^2 + qR2*s^4 + ... + qR5*s^10 252 * S = 1 + qS0*s^2 + ... + qS5*s^12 253 * and 254 * | qzero(x)/s +1.25-R/S | <= 2 ** ( -61.22) 255 */ 256 static const float qR8[6] = { /* for x in [inf, 8]=1/[0,0.125] */ 257 0.0000000000e+00, /* 0x00000000 */ 258 7.3242187500e-02, /* 0x3d960000 */ 259 1.1768206596e+01, /* 0x413c4a93 */ 260 5.5767340088e+02, /* 0x440b6b19 */ 261 8.8591972656e+03, /* 0x460a6cca */ 262 3.7014625000e+04, /* 0x471096a0 */ 263 }; 264 static const float qS8[6] = { 265 1.6377603149e+02, /* 0x4323c6aa */ 266 8.0983447266e+03, /* 0x45fd12c2 */ 267 1.4253829688e+05, /* 0x480b3293 */ 268 8.0330925000e+05, /* 0x49441ed4 */ 269 8.4050156250e+05, /* 0x494d3359 */ 270 -3.4389928125e+05, /* 0xc8a7eb69 */ 271 }; 272 273 static const float qR5[6] = { /* for x in [8,4.5454]=1/[0.125,0.22001] */ 274 1.8408595828e-11, /* 0x2da1ec79 */ 275 7.3242180049e-02, /* 0x3d95ffff */ 276 5.8356351852e+00, /* 0x40babd86 */ 277 1.3511157227e+02, /* 0x43071c90 */ 278 1.0272437744e+03, /* 0x448067cd */ 279 1.9899779053e+03, /* 0x44f8bf4b */ 280 }; 281 static const float qS5[6] = { 282 8.2776611328e+01, /* 0x42a58da0 */ 283 2.0778142090e+03, /* 0x4501dd07 */ 284 1.8847289062e+04, /* 0x46933e94 */ 285 5.6751113281e+04, /* 0x475daf1d */ 286 3.5976753906e+04, /* 0x470c88c1 */ 287 -5.3543427734e+03, /* 0xc5a752be */ 288 }; 289 290 static const float qR3[6] = {/* for x in [4.547,2.8571]=1/[0.2199,0.35001] */ 291 4.3774099900e-09, /* 0x3196681b */ 292 7.3241114616e-02, /* 0x3d95ff70 */ 293 3.3442313671e+00, /* 0x405607e3 */ 294 4.2621845245e+01, /* 0x422a7cc5 */ 295 1.7080809021e+02, /* 0x432acedf */ 296 1.6673394775e+02, /* 0x4326bbe4 */ 297 }; 298 static const float qS3[6] = { 299 4.8758872986e+01, /* 0x42430916 */ 300 7.0968920898e+02, /* 0x44316c1c */ 301 3.7041481934e+03, /* 0x4567825f */ 302 6.4604252930e+03, /* 0x45c9e367 */ 303 2.5163337402e+03, /* 0x451d4557 */ 304 -1.4924745178e+02, /* 0xc3153f59 */ 305 }; 306 307 static const float qR2[6] = {/* for x in [2.8570,2]=1/[0.3499,0.5] */ 308 1.5044444979e-07, /* 0x342189db */ 309 7.3223426938e-02, /* 0x3d95f62a */ 310 1.9981917143e+00, /* 0x3fffc4bf */ 311 1.4495602608e+01, /* 0x4167edfd */ 312 3.1666231155e+01, /* 0x41fd5471 */ 313 1.6252708435e+01, /* 0x4182058c */ 314 }; 315 static const float qS2[6] = { 316 3.0365585327e+01, /* 0x41f2ecb8 */ 317 2.6934811401e+02, /* 0x4386ac8f */ 318 8.4478375244e+02, /* 0x44533229 */ 319 8.8293585205e+02, /* 0x445cbbe5 */ 320 2.1266638184e+02, /* 0x4354aa98 */ 321 -5.3109550476e+00, /* 0xc0a9f358 */ 322 }; 323 324 static __inline float 325 qzerof(float x) 326 { 327 static const float eighth = 0.125; 328 const float *p,*q; 329 float s,r,z; 330 int32_t ix; 331 GET_FLOAT_WORD(ix,x); 332 ix &= 0x7fffffff; 333 if(ix>=0x41000000) {p = qR8; q= qS8;} 334 else if(ix>=0x409173eb){p = qR5; q= qS5;} 335 else if(ix>=0x4036d917){p = qR3; q= qS3;} 336 else {p = qR2; q= qS2;} /* ix>=0x40000000 */ 337 z = one/(x*x); 338 r = p[0]+z*(p[1]+z*(p[2]+z*(p[3]+z*(p[4]+z*p[5])))); 339 s = one+z*(q[0]+z*(q[1]+z*(q[2]+z*(q[3]+z*(q[4]+z*q[5]))))); 340 return (r/s-eighth)/x; 341 } 342