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 /* 17 * See e_j0.c for complete comments. 18 */ 19 20 #include "math.h" 21 #include "math_private.h" 22 23 static __inline float pzerof(float), qzerof(float); 24 25 static const volatile float vone = 1, vzero = 0; 26 27 static const float 28 huge = 1e30, 29 one = 1.0, 30 invsqrtpi= 5.6418961287e-01, /* 0x3f106ebb */ 31 tpi = 6.3661974669e-01, /* 0x3f22f983 */ 32 /* R0/S0 on [0, 2.00] */ 33 R02 = 1.5625000000e-02, /* 0x3c800000 */ 34 R03 = -1.8997929874e-04, /* 0xb947352e */ 35 R04 = 1.8295404516e-06, /* 0x35f58e88 */ 36 R05 = -4.6183270541e-09, /* 0xb19eaf3c */ 37 S01 = 1.5619102865e-02, /* 0x3c7fe744 */ 38 S02 = 1.1692678527e-04, /* 0x38f53697 */ 39 S03 = 5.1354652442e-07, /* 0x3509daa6 */ 40 S04 = 1.1661400734e-09; /* 0x30a045e8 */ 41 42 static const float zero = 0, qrtr = 0.25; 43 44 float 45 j0f(float x) 46 { 47 float z, s,c,ss,cc,r,u,v; 48 int32_t hx,ix; 49 50 GET_FLOAT_WORD(hx,x); 51 ix = hx&0x7fffffff; 52 if(ix>=0x7f800000) return one/(x*x); 53 x = fabsf(x); 54 if(ix >= 0x40000000) { /* |x| >= 2.0 */ 55 sincosf(x, &s, &c); 56 ss = s-c; 57 cc = s+c; 58 if(ix<0x7f000000) { /* Make sure x+x does not overflow. */ 59 z = -cosf(x+x); 60 if ((s*c)<zero) cc = z/ss; 61 else ss = z/cc; 62 } 63 /* 64 * j0(x) = 1/sqrt(pi) * (P(0,x)*cc - Q(0,x)*ss) / sqrt(x) 65 * y0(x) = 1/sqrt(pi) * (P(0,x)*ss + Q(0,x)*cc) / sqrt(x) 66 */ 67 if(ix>0x58000000) z = (invsqrtpi*cc)/sqrtf(x); /* |x|>2**49 */ 68 else { 69 u = pzerof(x); v = qzerof(x); 70 z = invsqrtpi*(u*cc-v*ss)/sqrtf(x); 71 } 72 return z; 73 } 74 if(ix<0x3b000000) { /* |x| < 2**-9 */ 75 if(huge+x>one) { /* raise inexact if x != 0 */ 76 if(ix<0x39800000) return one; /* |x|<2**-12 */ 77 else return one - x*x/4; 78 } 79 } 80 z = x*x; 81 r = z*(R02+z*(R03+z*(R04+z*R05))); 82 s = one+z*(S01+z*(S02+z*(S03+z*S04))); 83 if(ix < 0x3F800000) { /* |x| < 1.00 */ 84 return one + z*((r/s)-qrtr); 85 } else { 86 u = x/2; 87 return((one+u)*(one-u)+z*(r/s)); 88 } 89 } 90 91 static const float 92 u00 = -7.3804296553e-02, /* 0xbd9726b5 */ 93 u01 = 1.7666645348e-01, /* 0x3e34e80d */ 94 u02 = -1.3818567619e-02, /* 0xbc626746 */ 95 u03 = 3.4745343146e-04, /* 0x39b62a69 */ 96 u04 = -3.8140706238e-06, /* 0xb67ff53c */ 97 u05 = 1.9559013964e-08, /* 0x32a802ba */ 98 u06 = -3.9820518410e-11, /* 0xae2f21eb */ 99 v01 = 1.2730483897e-02, /* 0x3c509385 */ 100 v02 = 7.6006865129e-05, /* 0x389f65e0 */ 101 v03 = 2.5915085189e-07, /* 0x348b216c */ 102 v04 = 4.4111031494e-10; /* 0x2ff280c2 */ 103 104 float 105 y0f(float x) 106 { 107 float z, s,c,ss,cc,u,v; 108 int32_t hx,ix; 109 110 GET_FLOAT_WORD(hx,x); 111 ix = 0x7fffffff&hx; 112 if(ix>=0x7f800000) return vone/(x+x*x); 113 if(ix==0) return -one/vzero; 114 if(hx<0) return vzero/vzero; 115 if(ix >= 0x40000000) { /* |x| >= 2.0 */ 116 /* y0(x) = sqrt(2/(pi*x))*(p0(x)*sin(x0)+q0(x)*cos(x0)) 117 * where x0 = x-pi/4 118 * Better formula: 119 * cos(x0) = cos(x)cos(pi/4)+sin(x)sin(pi/4) 120 * = 1/sqrt(2) * (sin(x) + cos(x)) 121 * sin(x0) = sin(x)cos(3pi/4)-cos(x)sin(3pi/4) 122 * = 1/sqrt(2) * (sin(x) - cos(x)) 123 * To avoid cancellation, use 124 * sin(x) +- cos(x) = -cos(2x)/(sin(x) -+ cos(x)) 125 * to compute the worse one. 126 */ 127 sincosf(x, &s, &c); 128 ss = s-c; 129 cc = s+c; 130 /* 131 * j0(x) = 1/sqrt(pi) * (P(0,x)*cc - Q(0,x)*ss) / sqrt(x) 132 * y0(x) = 1/sqrt(pi) * (P(0,x)*ss + Q(0,x)*cc) / sqrt(x) 133 */ 134 if(ix<0x7f000000) { /* make sure x+x not overflow */ 135 z = -cosf(x+x); 136 if ((s*c)<zero) cc = z/ss; 137 else ss = z/cc; 138 } 139 if(ix>0x58000000) z = (invsqrtpi*ss)/sqrtf(x); /* |x|>2**49 */ 140 else { 141 u = pzerof(x); v = qzerof(x); 142 z = invsqrtpi*(u*ss+v*cc)/sqrtf(x); 143 } 144 return z; 145 } 146 if(ix<=0x39000000) { /* x < 2**-13 */ 147 return(u00 + tpi*logf(x)); 148 } 149 z = x*x; 150 u = u00+z*(u01+z*(u02+z*(u03+z*(u04+z*(u05+z*u06))))); 151 v = one+z*(v01+z*(v02+z*(v03+z*v04))); 152 return(u/v + tpi*(j0f(x)*logf(x))); 153 } 154 155 /* The asymptotic expansions of pzero is 156 * 1 - 9/128 s^2 + 11025/98304 s^4 - ..., where s = 1/x. 157 * For x >= 2, We approximate pzero by 158 * pzero(x) = 1 + (R/S) 159 * where R = pR0 + pR1*s^2 + pR2*s^4 + ... + pR5*s^10 160 * S = 1 + pS0*s^2 + ... + pS4*s^10 161 * and 162 * | pzero(x)-1-R/S | <= 2 ** ( -60.26) 163 */ 164 static const float pR8[6] = { /* for x in [inf, 8]=1/[0,0.125] */ 165 0.0000000000e+00, /* 0x00000000 */ 166 -7.0312500000e-02, /* 0xbd900000 */ 167 -8.0816707611e+00, /* 0xc1014e86 */ 168 -2.5706311035e+02, /* 0xc3808814 */ 169 -2.4852163086e+03, /* 0xc51b5376 */ 170 -5.2530439453e+03, /* 0xc5a4285a */ 171 }; 172 static const float pS8[5] = { 173 1.1653436279e+02, /* 0x42e91198 */ 174 3.8337448730e+03, /* 0x456f9beb */ 175 4.0597855469e+04, /* 0x471e95db */ 176 1.1675296875e+05, /* 0x47e4087c */ 177 4.7627726562e+04, /* 0x473a0bba */ 178 }; 179 static const float pR5[6] = { /* for x in [8,4.5454]=1/[0.125,0.22001] */ 180 -1.1412546255e-11, /* 0xad48c58a */ 181 -7.0312492549e-02, /* 0xbd8fffff */ 182 -4.1596107483e+00, /* 0xc0851b88 */ 183 -6.7674766541e+01, /* 0xc287597b */ 184 -3.3123129272e+02, /* 0xc3a59d9b */ 185 -3.4643338013e+02, /* 0xc3ad3779 */ 186 }; 187 static const float pS5[5] = { 188 6.0753936768e+01, /* 0x42730408 */ 189 1.0512523193e+03, /* 0x44836813 */ 190 5.9789707031e+03, /* 0x45bad7c4 */ 191 9.6254453125e+03, /* 0x461665c8 */ 192 2.4060581055e+03, /* 0x451660ee */ 193 }; 194 195 static const float pR3[6] = {/* for x in [4.547,2.8571]=1/[0.2199,0.35001] */ 196 -2.5470459075e-09, /* 0xb12f081b */ 197 -7.0311963558e-02, /* 0xbd8fffb8 */ 198 -2.4090321064e+00, /* 0xc01a2d95 */ 199 -2.1965976715e+01, /* 0xc1afba52 */ 200 -5.8079170227e+01, /* 0xc2685112 */ 201 -3.1447946548e+01, /* 0xc1fb9565 */ 202 }; 203 static const float pS3[5] = { 204 3.5856033325e+01, /* 0x420f6c94 */ 205 3.6151397705e+02, /* 0x43b4c1ca */ 206 1.1936077881e+03, /* 0x44953373 */ 207 1.1279968262e+03, /* 0x448cffe6 */ 208 1.7358093262e+02, /* 0x432d94b8 */ 209 }; 210 211 static const float pR2[6] = {/* for x in [2.8570,2]=1/[0.3499,0.5] */ 212 -8.8753431271e-08, /* 0xb3be98b7 */ 213 -7.0303097367e-02, /* 0xbd8ffb12 */ 214 -1.4507384300e+00, /* 0xbfb9b1cc */ 215 -7.6356959343e+00, /* 0xc0f4579f */ 216 -1.1193166733e+01, /* 0xc1331736 */ 217 -3.2336456776e+00, /* 0xc04ef40d */ 218 }; 219 static const float pS2[5] = { 220 2.2220300674e+01, /* 0x41b1c32d */ 221 1.3620678711e+02, /* 0x430834f0 */ 222 2.7047027588e+02, /* 0x43873c32 */ 223 1.5387539673e+02, /* 0x4319e01a */ 224 1.4657617569e+01, /* 0x416a859a */ 225 }; 226 227 static __inline float 228 pzerof(float x) 229 { 230 const float *p,*q; 231 float z,r,s; 232 int32_t ix; 233 GET_FLOAT_WORD(ix,x); 234 ix &= 0x7fffffff; 235 if(ix>=0x41000000) {p = pR8; q= pS8;} 236 else if(ix>=0x409173eb){p = pR5; q= pS5;} 237 else if(ix>=0x4036d917){p = pR3; q= pS3;} 238 else {p = pR2; q= pS2;} /* ix>=0x40000000 */ 239 z = one/(x*x); 240 r = p[0]+z*(p[1]+z*(p[2]+z*(p[3]+z*(p[4]+z*p[5])))); 241 s = one+z*(q[0]+z*(q[1]+z*(q[2]+z*(q[3]+z*q[4])))); 242 return one+ r/s; 243 } 244 245 246 /* For x >= 8, the asymptotic expansions of qzero is 247 * -1/8 s + 75/1024 s^3 - ..., where s = 1/x. 248 * We approximate pzero by 249 * qzero(x) = s*(-1.25 + (R/S)) 250 * where R = qR0 + qR1*s^2 + qR2*s^4 + ... + qR5*s^10 251 * S = 1 + qS0*s^2 + ... + qS5*s^12 252 * and 253 * | qzero(x)/s +1.25-R/S | <= 2 ** ( -61.22) 254 */ 255 static const float qR8[6] = { /* for x in [inf, 8]=1/[0,0.125] */ 256 0.0000000000e+00, /* 0x00000000 */ 257 7.3242187500e-02, /* 0x3d960000 */ 258 1.1768206596e+01, /* 0x413c4a93 */ 259 5.5767340088e+02, /* 0x440b6b19 */ 260 8.8591972656e+03, /* 0x460a6cca */ 261 3.7014625000e+04, /* 0x471096a0 */ 262 }; 263 static const float qS8[6] = { 264 1.6377603149e+02, /* 0x4323c6aa */ 265 8.0983447266e+03, /* 0x45fd12c2 */ 266 1.4253829688e+05, /* 0x480b3293 */ 267 8.0330925000e+05, /* 0x49441ed4 */ 268 8.4050156250e+05, /* 0x494d3359 */ 269 -3.4389928125e+05, /* 0xc8a7eb69 */ 270 }; 271 272 static const float qR5[6] = { /* for x in [8,4.5454]=1/[0.125,0.22001] */ 273 1.8408595828e-11, /* 0x2da1ec79 */ 274 7.3242180049e-02, /* 0x3d95ffff */ 275 5.8356351852e+00, /* 0x40babd86 */ 276 1.3511157227e+02, /* 0x43071c90 */ 277 1.0272437744e+03, /* 0x448067cd */ 278 1.9899779053e+03, /* 0x44f8bf4b */ 279 }; 280 static const float qS5[6] = { 281 8.2776611328e+01, /* 0x42a58da0 */ 282 2.0778142090e+03, /* 0x4501dd07 */ 283 1.8847289062e+04, /* 0x46933e94 */ 284 5.6751113281e+04, /* 0x475daf1d */ 285 3.5976753906e+04, /* 0x470c88c1 */ 286 -5.3543427734e+03, /* 0xc5a752be */ 287 }; 288 289 static const float qR3[6] = {/* for x in [4.547,2.8571]=1/[0.2199,0.35001] */ 290 4.3774099900e-09, /* 0x3196681b */ 291 7.3241114616e-02, /* 0x3d95ff70 */ 292 3.3442313671e+00, /* 0x405607e3 */ 293 4.2621845245e+01, /* 0x422a7cc5 */ 294 1.7080809021e+02, /* 0x432acedf */ 295 1.6673394775e+02, /* 0x4326bbe4 */ 296 }; 297 static const float qS3[6] = { 298 4.8758872986e+01, /* 0x42430916 */ 299 7.0968920898e+02, /* 0x44316c1c */ 300 3.7041481934e+03, /* 0x4567825f */ 301 6.4604252930e+03, /* 0x45c9e367 */ 302 2.5163337402e+03, /* 0x451d4557 */ 303 -1.4924745178e+02, /* 0xc3153f59 */ 304 }; 305 306 static const float qR2[6] = {/* for x in [2.8570,2]=1/[0.3499,0.5] */ 307 1.5044444979e-07, /* 0x342189db */ 308 7.3223426938e-02, /* 0x3d95f62a */ 309 1.9981917143e+00, /* 0x3fffc4bf */ 310 1.4495602608e+01, /* 0x4167edfd */ 311 3.1666231155e+01, /* 0x41fd5471 */ 312 1.6252708435e+01, /* 0x4182058c */ 313 }; 314 static const float qS2[6] = { 315 3.0365585327e+01, /* 0x41f2ecb8 */ 316 2.6934811401e+02, /* 0x4386ac8f */ 317 8.4478375244e+02, /* 0x44533229 */ 318 8.8293585205e+02, /* 0x445cbbe5 */ 319 2.1266638184e+02, /* 0x4354aa98 */ 320 -5.3109550476e+00, /* 0xc0a9f358 */ 321 }; 322 323 static __inline float 324 qzerof(float x) 325 { 326 static const float eighth = 0.125; 327 const float *p,*q; 328 float s,r,z; 329 int32_t ix; 330 GET_FLOAT_WORD(ix,x); 331 ix &= 0x7fffffff; 332 if(ix>=0x41000000) {p = qR8; q= qS8;} 333 else if(ix>=0x409173eb){p = qR5; q= qS5;} 334 else if(ix>=0x4036d917){p = qR3; q= qS3;} 335 else {p = qR2; q= qS2;} /* ix>=0x40000000 */ 336 z = one/(x*x); 337 r = p[0]+z*(p[1]+z*(p[2]+z*(p[3]+z*(p[4]+z*p[5])))); 338 s = one+z*(q[0]+z*(q[1]+z*(q[2]+z*(q[3]+z*(q[4]+z*q[5]))))); 339 return (r/s-eighth)/x; 340 } 341