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 "math.h" 17 #include "math_private.h" 18 19 static float pzerof(float), qzerof(float); 20 21 static const float 22 huge = 1e30, 23 one = 1.0, 24 invsqrtpi= 5.6418961287e-01, /* 0x3f106ebb */ 25 tpi = 6.3661974669e-01, /* 0x3f22f983 */ 26 /* R0/S0 on [0, 2.00] */ 27 R02 = 1.5625000000e-02, /* 0x3c800000 */ 28 R03 = -1.8997929874e-04, /* 0xb947352e */ 29 R04 = 1.8295404516e-06, /* 0x35f58e88 */ 30 R05 = -4.6183270541e-09, /* 0xb19eaf3c */ 31 S01 = 1.5619102865e-02, /* 0x3c7fe744 */ 32 S02 = 1.1692678527e-04, /* 0x38f53697 */ 33 S03 = 5.1354652442e-07, /* 0x3509daa6 */ 34 S04 = 1.1661400734e-09; /* 0x30a045e8 */ 35 36 static const float zero = 0.0; 37 38 float 39 j0f(float x) 40 { 41 float z, s,c,ss,cc,r,u,v; 42 int32_t hx,ix; 43 44 GET_FLOAT_WORD(hx,x); 45 ix = hx&0x7fffffff; 46 if(ix>=0x7f800000) return one/(x*x); 47 x = fabsf(x); 48 if(ix >= 0x40000000) { /* |x| >= 2.0 */ 49 s = sinf(x); 50 c = cosf(x); 51 ss = s-c; 52 cc = s+c; 53 if(ix<0x7f000000) { /* make sure x+x not overflow */ 54 z = -cosf(x+x); 55 if ((s*c)<zero) cc = z/ss; 56 else ss = z/cc; 57 } 58 /* 59 * j0(x) = 1/sqrt(pi) * (P(0,x)*cc - Q(0,x)*ss) / sqrt(x) 60 * y0(x) = 1/sqrt(pi) * (P(0,x)*ss + Q(0,x)*cc) / sqrt(x) 61 */ 62 if((u_int32_t)ix>0x80000000U) z = (invsqrtpi*cc)/sqrtf(x); 63 else { 64 u = pzerof(x); v = qzerof(x); 65 z = invsqrtpi*(u*cc-v*ss)/sqrtf(x); 66 } 67 return z; 68 } 69 if(ix<0x39000000) { /* |x| < 2**-13 */ 70 if(huge+x>one) { /* raise inexact if x != 0 */ 71 if(ix<0x32000000) return one; /* |x|<2**-27 */ 72 else return one - (float)0.25*x*x; 73 } 74 } 75 z = x*x; 76 r = z*(R02+z*(R03+z*(R04+z*R05))); 77 s = one+z*(S01+z*(S02+z*(S03+z*S04))); 78 if(ix < 0x3F800000) { /* |x| < 1.00 */ 79 return one + z*((float)-0.25+(r/s)); 80 } else { 81 u = (float)0.5*x; 82 return((one+u)*(one-u)+z*(r/s)); 83 } 84 } 85 86 static const float 87 u00 = -7.3804296553e-02, /* 0xbd9726b5 */ 88 u01 = 1.7666645348e-01, /* 0x3e34e80d */ 89 u02 = -1.3818567619e-02, /* 0xbc626746 */ 90 u03 = 3.4745343146e-04, /* 0x39b62a69 */ 91 u04 = -3.8140706238e-06, /* 0xb67ff53c */ 92 u05 = 1.9559013964e-08, /* 0x32a802ba */ 93 u06 = -3.9820518410e-11, /* 0xae2f21eb */ 94 v01 = 1.2730483897e-02, /* 0x3c509385 */ 95 v02 = 7.6006865129e-05, /* 0x389f65e0 */ 96 v03 = 2.5915085189e-07, /* 0x348b216c */ 97 v04 = 4.4111031494e-10; /* 0x2ff280c2 */ 98 99 float 100 y0f(float x) 101 { 102 float z, s,c,ss,cc,u,v; 103 int32_t hx,ix; 104 105 GET_FLOAT_WORD(hx,x); 106 ix = 0x7fffffff&hx; 107 /* Y0(NaN) is NaN, y0(-inf) is Nan, y0(inf) is 0 */ 108 if(ix>=0x7f800000) return one/(x+x*x); 109 if(ix==0) return -one/zero; 110 if(hx<0) return zero/zero; 111 if(ix >= 0x40000000) { /* |x| >= 2.0 */ 112 /* y0(x) = sqrt(2/(pi*x))*(p0(x)*sin(x0)+q0(x)*cos(x0)) 113 * where x0 = x-pi/4 114 * Better formula: 115 * cos(x0) = cos(x)cos(pi/4)+sin(x)sin(pi/4) 116 * = 1/sqrt(2) * (sin(x) + cos(x)) 117 * sin(x0) = sin(x)cos(3pi/4)-cos(x)sin(3pi/4) 118 * = 1/sqrt(2) * (sin(x) - cos(x)) 119 * To avoid cancellation, use 120 * sin(x) +- cos(x) = -cos(2x)/(sin(x) -+ cos(x)) 121 * to compute the worse one. 122 */ 123 s = sinf(x); 124 c = cosf(x); 125 ss = s-c; 126 cc = s+c; 127 /* 128 * j0(x) = 1/sqrt(pi) * (P(0,x)*cc - Q(0,x)*ss) / sqrt(x) 129 * y0(x) = 1/sqrt(pi) * (P(0,x)*ss + Q(0,x)*cc) / sqrt(x) 130 */ 131 if(ix<0x7f000000) { /* make sure x+x not overflow */ 132 z = -cosf(x+x); 133 if ((s*c)<zero) cc = z/ss; 134 else ss = z/cc; 135 } 136 if((u_int32_t)ix>0x80000000U) z = (invsqrtpi*ss)/sqrtf(x); 137 else { 138 u = pzerof(x); v = qzerof(x); 139 z = invsqrtpi*(u*ss+v*cc)/sqrtf(x); 140 } 141 return z; 142 } 143 if(ix<=0x32000000) { /* x < 2**-27 */ 144 return(u00 + tpi*logf(x)); 145 } 146 z = x*x; 147 u = u00+z*(u01+z*(u02+z*(u03+z*(u04+z*(u05+z*u06))))); 148 v = one+z*(v01+z*(v02+z*(v03+z*v04))); 149 return(u/v + tpi*(j0f(x)*logf(x))); 150 } 151 152 /* The asymptotic expansions of pzero is 153 * 1 - 9/128 s^2 + 11025/98304 s^4 - ..., where s = 1/x. 154 * For x >= 2, We approximate pzero by 155 * pzero(x) = 1 + (R/S) 156 * where R = pR0 + pR1*s^2 + pR2*s^4 + ... + pR5*s^10 157 * S = 1 + pS0*s^2 + ... + pS4*s^10 158 * and 159 * | pzero(x)-1-R/S | <= 2 ** ( -60.26) 160 */ 161 static const float pR8[6] = { /* for x in [inf, 8]=1/[0,0.125] */ 162 0.0000000000e+00, /* 0x00000000 */ 163 -7.0312500000e-02, /* 0xbd900000 */ 164 -8.0816707611e+00, /* 0xc1014e86 */ 165 -2.5706311035e+02, /* 0xc3808814 */ 166 -2.4852163086e+03, /* 0xc51b5376 */ 167 -5.2530439453e+03, /* 0xc5a4285a */ 168 }; 169 static const float pS8[5] = { 170 1.1653436279e+02, /* 0x42e91198 */ 171 3.8337448730e+03, /* 0x456f9beb */ 172 4.0597855469e+04, /* 0x471e95db */ 173 1.1675296875e+05, /* 0x47e4087c */ 174 4.7627726562e+04, /* 0x473a0bba */ 175 }; 176 static const float pR5[6] = { /* for x in [8,4.5454]=1/[0.125,0.22001] */ 177 -1.1412546255e-11, /* 0xad48c58a */ 178 -7.0312492549e-02, /* 0xbd8fffff */ 179 -4.1596107483e+00, /* 0xc0851b88 */ 180 -6.7674766541e+01, /* 0xc287597b */ 181 -3.3123129272e+02, /* 0xc3a59d9b */ 182 -3.4643338013e+02, /* 0xc3ad3779 */ 183 }; 184 static const float pS5[5] = { 185 6.0753936768e+01, /* 0x42730408 */ 186 1.0512523193e+03, /* 0x44836813 */ 187 5.9789707031e+03, /* 0x45bad7c4 */ 188 9.6254453125e+03, /* 0x461665c8 */ 189 2.4060581055e+03, /* 0x451660ee */ 190 }; 191 192 static const float pR3[6] = {/* for x in [4.547,2.8571]=1/[0.2199,0.35001] */ 193 -2.5470459075e-09, /* 0xb12f081b */ 194 -7.0311963558e-02, /* 0xbd8fffb8 */ 195 -2.4090321064e+00, /* 0xc01a2d95 */ 196 -2.1965976715e+01, /* 0xc1afba52 */ 197 -5.8079170227e+01, /* 0xc2685112 */ 198 -3.1447946548e+01, /* 0xc1fb9565 */ 199 }; 200 static const float pS3[5] = { 201 3.5856033325e+01, /* 0x420f6c94 */ 202 3.6151397705e+02, /* 0x43b4c1ca */ 203 1.1936077881e+03, /* 0x44953373 */ 204 1.1279968262e+03, /* 0x448cffe6 */ 205 1.7358093262e+02, /* 0x432d94b8 */ 206 }; 207 208 static const float pR2[6] = {/* for x in [2.8570,2]=1/[0.3499,0.5] */ 209 -8.8753431271e-08, /* 0xb3be98b7 */ 210 -7.0303097367e-02, /* 0xbd8ffb12 */ 211 -1.4507384300e+00, /* 0xbfb9b1cc */ 212 -7.6356959343e+00, /* 0xc0f4579f */ 213 -1.1193166733e+01, /* 0xc1331736 */ 214 -3.2336456776e+00, /* 0xc04ef40d */ 215 }; 216 static const float pS2[5] = { 217 2.2220300674e+01, /* 0x41b1c32d */ 218 1.3620678711e+02, /* 0x430834f0 */ 219 2.7047027588e+02, /* 0x43873c32 */ 220 1.5387539673e+02, /* 0x4319e01a */ 221 1.4657617569e+01, /* 0x416a859a */ 222 }; 223 224 static float 225 pzerof(float x) 226 { 227 const float *p,*q; 228 float z,r,s; 229 int32_t ix; 230 GET_FLOAT_WORD(ix,x); 231 ix &= 0x7fffffff; 232 if(ix>=0x41000000) {p = pR8; q= pS8;} 233 else if(ix>=0x40f71c58){p = pR5; q= pS5;} 234 else if(ix>=0x4036db68){p = pR3; q= pS3;} 235 else /*if(ix>=0x40000000)*/ {p = pR2; q= pS2;} 236 z = one/(x*x); 237 r = p[0]+z*(p[1]+z*(p[2]+z*(p[3]+z*(p[4]+z*p[5])))); 238 s = one+z*(q[0]+z*(q[1]+z*(q[2]+z*(q[3]+z*q[4])))); 239 return one+ r/s; 240 } 241 242 243 /* For x >= 8, the asymptotic expansions of qzero is 244 * -1/8 s + 75/1024 s^3 - ..., where s = 1/x. 245 * We approximate pzero by 246 * qzero(x) = s*(-1.25 + (R/S)) 247 * where R = qR0 + qR1*s^2 + qR2*s^4 + ... + qR5*s^10 248 * S = 1 + qS0*s^2 + ... + qS5*s^12 249 * and 250 * | qzero(x)/s +1.25-R/S | <= 2 ** ( -61.22) 251 */ 252 static const float qR8[6] = { /* for x in [inf, 8]=1/[0,0.125] */ 253 0.0000000000e+00, /* 0x00000000 */ 254 7.3242187500e-02, /* 0x3d960000 */ 255 1.1768206596e+01, /* 0x413c4a93 */ 256 5.5767340088e+02, /* 0x440b6b19 */ 257 8.8591972656e+03, /* 0x460a6cca */ 258 3.7014625000e+04, /* 0x471096a0 */ 259 }; 260 static const float qS8[6] = { 261 1.6377603149e+02, /* 0x4323c6aa */ 262 8.0983447266e+03, /* 0x45fd12c2 */ 263 1.4253829688e+05, /* 0x480b3293 */ 264 8.0330925000e+05, /* 0x49441ed4 */ 265 8.4050156250e+05, /* 0x494d3359 */ 266 -3.4389928125e+05, /* 0xc8a7eb69 */ 267 }; 268 269 static const float qR5[6] = { /* for x in [8,4.5454]=1/[0.125,0.22001] */ 270 1.8408595828e-11, /* 0x2da1ec79 */ 271 7.3242180049e-02, /* 0x3d95ffff */ 272 5.8356351852e+00, /* 0x40babd86 */ 273 1.3511157227e+02, /* 0x43071c90 */ 274 1.0272437744e+03, /* 0x448067cd */ 275 1.9899779053e+03, /* 0x44f8bf4b */ 276 }; 277 static const float qS5[6] = { 278 8.2776611328e+01, /* 0x42a58da0 */ 279 2.0778142090e+03, /* 0x4501dd07 */ 280 1.8847289062e+04, /* 0x46933e94 */ 281 5.6751113281e+04, /* 0x475daf1d */ 282 3.5976753906e+04, /* 0x470c88c1 */ 283 -5.3543427734e+03, /* 0xc5a752be */ 284 }; 285 286 static const float qR3[6] = {/* for x in [4.547,2.8571]=1/[0.2199,0.35001] */ 287 4.3774099900e-09, /* 0x3196681b */ 288 7.3241114616e-02, /* 0x3d95ff70 */ 289 3.3442313671e+00, /* 0x405607e3 */ 290 4.2621845245e+01, /* 0x422a7cc5 */ 291 1.7080809021e+02, /* 0x432acedf */ 292 1.6673394775e+02, /* 0x4326bbe4 */ 293 }; 294 static const float qS3[6] = { 295 4.8758872986e+01, /* 0x42430916 */ 296 7.0968920898e+02, /* 0x44316c1c */ 297 3.7041481934e+03, /* 0x4567825f */ 298 6.4604252930e+03, /* 0x45c9e367 */ 299 2.5163337402e+03, /* 0x451d4557 */ 300 -1.4924745178e+02, /* 0xc3153f59 */ 301 }; 302 303 static const float qR2[6] = {/* for x in [2.8570,2]=1/[0.3499,0.5] */ 304 1.5044444979e-07, /* 0x342189db */ 305 7.3223426938e-02, /* 0x3d95f62a */ 306 1.9981917143e+00, /* 0x3fffc4bf */ 307 1.4495602608e+01, /* 0x4167edfd */ 308 3.1666231155e+01, /* 0x41fd5471 */ 309 1.6252708435e+01, /* 0x4182058c */ 310 }; 311 static const float qS2[6] = { 312 3.0365585327e+01, /* 0x41f2ecb8 */ 313 2.6934811401e+02, /* 0x4386ac8f */ 314 8.4478375244e+02, /* 0x44533229 */ 315 8.8293585205e+02, /* 0x445cbbe5 */ 316 2.1266638184e+02, /* 0x4354aa98 */ 317 -5.3109550476e+00, /* 0xc0a9f358 */ 318 }; 319 320 static float 321 qzerof(float x) 322 { 323 const float *p,*q; 324 float s,r,z; 325 int32_t ix; 326 GET_FLOAT_WORD(ix,x); 327 ix &= 0x7fffffff; 328 if(ix>=0x41000000) {p = qR8; q= qS8;} 329 else if(ix>=0x40f71c58){p = qR5; q= qS5;} 330 else if(ix>=0x4036db68){p = qR3; q= qS3;} 331 else /*if(ix>=0x40000000)*/ {p = qR2; q= qS2;} 332 z = one/(x*x); 333 r = p[0]+z*(p[1]+z*(p[2]+z*(p[3]+z*(p[4]+z*p[5])))); 334 s = one+z*(q[0]+z*(q[1]+z*(q[2]+z*(q[3]+z*(q[4]+z*q[5]))))); 335 return (-(float).125 + r/s)/x; 336 } 337