1 /* 2 * ==================================================== 3 * Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved. 4 * 5 * Developed at SunSoft, a Sun Microsystems, Inc. business. 6 * Permission to use, copy, modify, and distribute this 7 * software is freely granted, provided that this notice 8 * is preserved. 9 * ==================================================== 10 */ 11 12 /* j1(x), y1(x) 13 * Bessel function of the first and second kinds of order zero. 14 * Method -- j1(x): 15 * 1. For tiny x, we use j1(x) = x/2 - x^3/16 + x^5/384 - ... 16 * 2. Reduce x to |x| since j1(x)=-j1(-x), and 17 * for x in (0,2) 18 * j1(x) = x/2 + x*z*R0/S0, where z = x*x; 19 * (precision: |j1/x - 1/2 - R0/S0 |<2**-61.51 ) 20 * for x in (2,inf) 21 * j1(x) = sqrt(2/(pi*x))*(p1(x)*cos(x1)-q1(x)*sin(x1)) 22 * y1(x) = sqrt(2/(pi*x))*(p1(x)*sin(x1)+q1(x)*cos(x1)) 23 * where x1 = x-3*pi/4. It is better to compute sin(x1),cos(x1) 24 * as follow: 25 * cos(x1) = cos(x)cos(3pi/4)+sin(x)sin(3pi/4) 26 * = 1/sqrt(2) * (sin(x) - cos(x)) 27 * sin(x1) = sin(x)cos(3pi/4)-cos(x)sin(3pi/4) 28 * = -1/sqrt(2) * (sin(x) + cos(x)) 29 * (To avoid cancellation, use 30 * sin(x) +- cos(x) = -cos(2x)/(sin(x) -+ cos(x)) 31 * to compute the worse one.) 32 * 33 * 3 Special cases 34 * j1(nan)= nan 35 * j1(0) = 0 36 * j1(inf) = 0 37 * 38 * Method -- y1(x): 39 * 1. screen out x<=0 cases: y1(0)=-inf, y1(x<0)=NaN 40 * 2. For x<2. 41 * Since 42 * y1(x) = 2/pi*(j1(x)*(ln(x/2)+Euler)-1/x-x/2+5/64*x^3-...) 43 * therefore y1(x)-2/pi*j1(x)*ln(x)-1/x is an odd function. 44 * We use the following function to approximate y1, 45 * y1(x) = x*U(z)/V(z) + (2/pi)*(j1(x)*ln(x)-1/x), z= x^2 46 * where for x in [0,2] (abs err less than 2**-65.89) 47 * U(z) = U0[0] + U0[1]*z + ... + U0[4]*z^4 48 * V(z) = 1 + v0[0]*z + ... + v0[4]*z^5 49 * Note: For tiny x, 1/x dominate y1 and hence 50 * y1(tiny) = -2/pi/tiny, (choose tiny<2**-54) 51 * 3. For x>=2. 52 * y1(x) = sqrt(2/(pi*x))*(p1(x)*sin(x1)+q1(x)*cos(x1)) 53 * where x1 = x-3*pi/4. It is better to compute sin(x1),cos(x1) 54 * by method mentioned above. 55 */ 56 57 #include "math.h" 58 #include "math_private.h" 59 60 static __inline double pone(double), qone(double); 61 62 static const volatile double vone = 1, vzero = 0; 63 64 static const double 65 huge = 1e300, 66 one = 1.0, 67 invsqrtpi= 5.64189583547756279280e-01, /* 0x3FE20DD7, 0x50429B6D */ 68 tpi = 6.36619772367581382433e-01, /* 0x3FE45F30, 0x6DC9C883 */ 69 /* R0/S0 on [0,2] */ 70 r00 = -6.25000000000000000000e-02, /* 0xBFB00000, 0x00000000 */ 71 r01 = 1.40705666955189706048e-03, /* 0x3F570D9F, 0x98472C61 */ 72 r02 = -1.59955631084035597520e-05, /* 0xBEF0C5C6, 0xBA169668 */ 73 r03 = 4.96727999609584448412e-08, /* 0x3E6AAAFA, 0x46CA0BD9 */ 74 s01 = 1.91537599538363460805e-02, /* 0x3F939D0B, 0x12637E53 */ 75 s02 = 1.85946785588630915560e-04, /* 0x3F285F56, 0xB9CDF664 */ 76 s03 = 1.17718464042623683263e-06, /* 0x3EB3BFF8, 0x333F8498 */ 77 s04 = 5.04636257076217042715e-09, /* 0x3E35AC88, 0xC97DFF2C */ 78 s05 = 1.23542274426137913908e-11; /* 0x3DAB2ACF, 0xCFB97ED8 */ 79 80 static const double zero = 0.0; 81 82 double 83 j1(double x) 84 { 85 double z, s,c,ss,cc,r,u,v,y; 86 int32_t hx,ix; 87 88 GET_HIGH_WORD(hx,x); 89 ix = hx&0x7fffffff; 90 if(ix>=0x7ff00000) return one/x; 91 y = fabs(x); 92 if(ix >= 0x40000000) { /* |x| >= 2.0 */ 93 sincos(y, &s, &c); 94 ss = -s-c; 95 cc = s-c; 96 if(ix<0x7fe00000) { /* make sure y+y not overflow */ 97 z = cos(y+y); 98 if ((s*c)>zero) cc = z/ss; 99 else ss = z/cc; 100 } 101 /* 102 * j1(x) = 1/sqrt(pi) * (P(1,x)*cc - Q(1,x)*ss) / sqrt(x) 103 * y1(x) = 1/sqrt(pi) * (P(1,x)*ss + Q(1,x)*cc) / sqrt(x) 104 */ 105 if(ix>0x48000000) z = (invsqrtpi*cc)/sqrt(y); 106 else { 107 u = pone(y); v = qone(y); 108 z = invsqrtpi*(u*cc-v*ss)/sqrt(y); 109 } 110 if(hx<0) return -z; 111 else return z; 112 } 113 if(ix<0x3e400000) { /* |x|<2**-27 */ 114 if(huge+x>one) return 0.5*x;/* inexact if x!=0 necessary */ 115 } 116 z = x*x; 117 r = z*(r00+z*(r01+z*(r02+z*r03))); 118 s = one+z*(s01+z*(s02+z*(s03+z*(s04+z*s05)))); 119 r *= x; 120 return(x*0.5+r/s); 121 } 122 123 static const double U0[5] = { 124 -1.96057090646238940668e-01, /* 0xBFC91866, 0x143CBC8A */ 125 5.04438716639811282616e-02, /* 0x3FA9D3C7, 0x76292CD1 */ 126 -1.91256895875763547298e-03, /* 0xBF5F55E5, 0x4844F50F */ 127 2.35252600561610495928e-05, /* 0x3EF8AB03, 0x8FA6B88E */ 128 -9.19099158039878874504e-08, /* 0xBE78AC00, 0x569105B8 */ 129 }; 130 static const double V0[5] = { 131 1.99167318236649903973e-02, /* 0x3F94650D, 0x3F4DA9F0 */ 132 2.02552581025135171496e-04, /* 0x3F2A8C89, 0x6C257764 */ 133 1.35608801097516229404e-06, /* 0x3EB6C05A, 0x894E8CA6 */ 134 6.22741452364621501295e-09, /* 0x3E3ABF1D, 0x5BA69A86 */ 135 1.66559246207992079114e-11, /* 0x3DB25039, 0xDACA772A */ 136 }; 137 138 double 139 y1(double x) 140 { 141 double z, s,c,ss,cc,u,v; 142 int32_t hx,ix,lx; 143 144 EXTRACT_WORDS(hx,lx,x); 145 ix = 0x7fffffff&hx; 146 /* 147 * y1(NaN) = NaN. 148 * y1(Inf) = 0. 149 * y1(-Inf) = NaN and raise invalid exception. 150 */ 151 if(ix>=0x7ff00000) return vone/(x+x*x); 152 /* y1(+-0) = -inf and raise divide-by-zero exception. */ 153 if((ix|lx)==0) return -one/vzero; 154 /* y1(x<0) = NaN and raise invalid exception. */ 155 if(hx<0) return vzero/vzero; 156 if(ix >= 0x40000000) { /* |x| >= 2.0 */ 157 sincos(x, &s, &c); 158 ss = -s-c; 159 cc = s-c; 160 if(ix<0x7fe00000) { /* make sure x+x not overflow */ 161 z = cos(x+x); 162 if ((s*c)>zero) cc = z/ss; 163 else ss = z/cc; 164 } 165 /* y1(x) = sqrt(2/(pi*x))*(p1(x)*sin(x0)+q1(x)*cos(x0)) 166 * where x0 = x-3pi/4 167 * Better formula: 168 * cos(x0) = cos(x)cos(3pi/4)+sin(x)sin(3pi/4) 169 * = 1/sqrt(2) * (sin(x) - cos(x)) 170 * sin(x0) = sin(x)cos(3pi/4)-cos(x)sin(3pi/4) 171 * = -1/sqrt(2) * (cos(x) + sin(x)) 172 * To avoid cancellation, use 173 * sin(x) +- cos(x) = -cos(2x)/(sin(x) -+ cos(x)) 174 * to compute the worse one. 175 */ 176 if(ix>0x48000000) z = (invsqrtpi*ss)/sqrt(x); 177 else { 178 u = pone(x); v = qone(x); 179 z = invsqrtpi*(u*ss+v*cc)/sqrt(x); 180 } 181 return z; 182 } 183 if(ix<=0x3c900000) { /* x < 2**-54 */ 184 return(-tpi/x); 185 } 186 z = x*x; 187 u = U0[0]+z*(U0[1]+z*(U0[2]+z*(U0[3]+z*U0[4]))); 188 v = one+z*(V0[0]+z*(V0[1]+z*(V0[2]+z*(V0[3]+z*V0[4])))); 189 return(x*(u/v) + tpi*(j1(x)*log(x)-one/x)); 190 } 191 192 /* For x >= 8, the asymptotic expansions of pone is 193 * 1 + 15/128 s^2 - 4725/2^15 s^4 - ..., where s = 1/x. 194 * We approximate pone by 195 * pone(x) = 1 + (R/S) 196 * where R = pr0 + pr1*s^2 + pr2*s^4 + ... + pr5*s^10 197 * S = 1 + ps0*s^2 + ... + ps4*s^10 198 * and 199 * | pone(x)-1-R/S | <= 2 ** ( -60.06) 200 */ 201 202 static const double pr8[6] = { /* for x in [inf, 8]=1/[0,0.125] */ 203 0.00000000000000000000e+00, /* 0x00000000, 0x00000000 */ 204 1.17187499999988647970e-01, /* 0x3FBDFFFF, 0xFFFFFCCE */ 205 1.32394806593073575129e+01, /* 0x402A7A9D, 0x357F7FCE */ 206 4.12051854307378562225e+02, /* 0x4079C0D4, 0x652EA590 */ 207 3.87474538913960532227e+03, /* 0x40AE457D, 0xA3A532CC */ 208 7.91447954031891731574e+03, /* 0x40BEEA7A, 0xC32782DD */ 209 }; 210 static const double ps8[5] = { 211 1.14207370375678408436e+02, /* 0x405C8D45, 0x8E656CAC */ 212 3.65093083420853463394e+03, /* 0x40AC85DC, 0x964D274F */ 213 3.69562060269033463555e+04, /* 0x40E20B86, 0x97C5BB7F */ 214 9.76027935934950801311e+04, /* 0x40F7D42C, 0xB28F17BB */ 215 3.08042720627888811578e+04, /* 0x40DE1511, 0x697A0B2D */ 216 }; 217 218 static const double pr5[6] = { /* for x in [8,4.5454]=1/[0.125,0.22001] */ 219 1.31990519556243522749e-11, /* 0x3DAD0667, 0xDAE1CA7D */ 220 1.17187493190614097638e-01, /* 0x3FBDFFFF, 0xE2C10043 */ 221 6.80275127868432871736e+00, /* 0x401B3604, 0x6E6315E3 */ 222 1.08308182990189109773e+02, /* 0x405B13B9, 0x452602ED */ 223 5.17636139533199752805e+02, /* 0x40802D16, 0xD052D649 */ 224 5.28715201363337541807e+02, /* 0x408085B8, 0xBB7E0CB7 */ 225 }; 226 static const double ps5[5] = { 227 5.92805987221131331921e+01, /* 0x404DA3EA, 0xA8AF633D */ 228 9.91401418733614377743e+02, /* 0x408EFB36, 0x1B066701 */ 229 5.35326695291487976647e+03, /* 0x40B4E944, 0x5706B6FB */ 230 7.84469031749551231769e+03, /* 0x40BEA4B0, 0xB8A5BB15 */ 231 1.50404688810361062679e+03, /* 0x40978030, 0x036F5E51 */ 232 }; 233 234 static const double pr3[6] = { 235 3.02503916137373618024e-09, /* 0x3E29FC21, 0xA7AD9EDD */ 236 1.17186865567253592491e-01, /* 0x3FBDFFF5, 0x5B21D17B */ 237 3.93297750033315640650e+00, /* 0x400F76BC, 0xE85EAD8A */ 238 3.51194035591636932736e+01, /* 0x40418F48, 0x9DA6D129 */ 239 9.10550110750781271918e+01, /* 0x4056C385, 0x4D2C1837 */ 240 4.85590685197364919645e+01, /* 0x4048478F, 0x8EA83EE5 */ 241 }; 242 static const double ps3[5] = { 243 3.47913095001251519989e+01, /* 0x40416549, 0xA134069C */ 244 3.36762458747825746741e+02, /* 0x40750C33, 0x07F1A75F */ 245 1.04687139975775130551e+03, /* 0x40905B7C, 0x5037D523 */ 246 8.90811346398256432622e+02, /* 0x408BD67D, 0xA32E31E9 */ 247 1.03787932439639277504e+02, /* 0x4059F26D, 0x7C2EED53 */ 248 }; 249 250 static const double pr2[6] = {/* for x in [2.8570,2]=1/[0.3499,0.5] */ 251 1.07710830106873743082e-07, /* 0x3E7CE9D4, 0xF65544F4 */ 252 1.17176219462683348094e-01, /* 0x3FBDFF42, 0xBE760D83 */ 253 2.36851496667608785174e+00, /* 0x4002F2B7, 0xF98FAEC0 */ 254 1.22426109148261232917e+01, /* 0x40287C37, 0x7F71A964 */ 255 1.76939711271687727390e+01, /* 0x4031B1A8, 0x177F8EE2 */ 256 5.07352312588818499250e+00, /* 0x40144B49, 0xA574C1FE */ 257 }; 258 static const double ps2[5] = { 259 2.14364859363821409488e+01, /* 0x40356FBD, 0x8AD5ECDC */ 260 1.25290227168402751090e+02, /* 0x405F5293, 0x14F92CD5 */ 261 2.32276469057162813669e+02, /* 0x406D08D8, 0xD5A2DBD9 */ 262 1.17679373287147100768e+02, /* 0x405D6B7A, 0xDA1884A9 */ 263 8.36463893371618283368e+00, /* 0x4020BAB1, 0xF44E5192 */ 264 }; 265 266 static __inline double 267 pone(double x) 268 { 269 const double *p,*q; 270 double z,r,s; 271 int32_t ix; 272 GET_HIGH_WORD(ix,x); 273 ix &= 0x7fffffff; 274 if(ix>=0x40200000) {p = pr8; q= ps8;} 275 else if(ix>=0x40122E8B){p = pr5; q= ps5;} 276 else if(ix>=0x4006DB6D){p = pr3; q= ps3;} 277 else {p = pr2; q= ps2;} /* ix>=0x40000000 */ 278 z = one/(x*x); 279 r = p[0]+z*(p[1]+z*(p[2]+z*(p[3]+z*(p[4]+z*p[5])))); 280 s = one+z*(q[0]+z*(q[1]+z*(q[2]+z*(q[3]+z*q[4])))); 281 return one+ r/s; 282 } 283 284 285 /* For x >= 8, the asymptotic expansions of qone is 286 * 3/8 s - 105/1024 s^3 - ..., where s = 1/x. 287 * We approximate pone by 288 * qone(x) = s*(0.375 + (R/S)) 289 * where R = qr1*s^2 + qr2*s^4 + ... + qr5*s^10 290 * S = 1 + qs1*s^2 + ... + qs6*s^12 291 * and 292 * | qone(x)/s -0.375-R/S | <= 2 ** ( -61.13) 293 */ 294 295 static const double qr8[6] = { /* for x in [inf, 8]=1/[0,0.125] */ 296 0.00000000000000000000e+00, /* 0x00000000, 0x00000000 */ 297 -1.02539062499992714161e-01, /* 0xBFBA3FFF, 0xFFFFFDF3 */ 298 -1.62717534544589987888e+01, /* 0xC0304591, 0xA26779F7 */ 299 -7.59601722513950107896e+02, /* 0xC087BCD0, 0x53E4B576 */ 300 -1.18498066702429587167e+04, /* 0xC0C724E7, 0x40F87415 */ 301 -4.84385124285750353010e+04, /* 0xC0E7A6D0, 0x65D09C6A */ 302 }; 303 static const double qs8[6] = { 304 1.61395369700722909556e+02, /* 0x40642CA6, 0xDE5BCDE5 */ 305 7.82538599923348465381e+03, /* 0x40BE9162, 0xD0D88419 */ 306 1.33875336287249578163e+05, /* 0x4100579A, 0xB0B75E98 */ 307 7.19657723683240939863e+05, /* 0x4125F653, 0x72869C19 */ 308 6.66601232617776375264e+05, /* 0x412457D2, 0x7719AD5C */ 309 -2.94490264303834643215e+05, /* 0xC111F969, 0x0EA5AA18 */ 310 }; 311 312 static const double qr5[6] = { /* for x in [8,4.5454]=1/[0.125,0.22001] */ 313 -2.08979931141764104297e-11, /* 0xBDB6FA43, 0x1AA1A098 */ 314 -1.02539050241375426231e-01, /* 0xBFBA3FFF, 0xCB597FEF */ 315 -8.05644828123936029840e+00, /* 0xC0201CE6, 0xCA03AD4B */ 316 -1.83669607474888380239e+02, /* 0xC066F56D, 0x6CA7B9B0 */ 317 -1.37319376065508163265e+03, /* 0xC09574C6, 0x6931734F */ 318 -2.61244440453215656817e+03, /* 0xC0A468E3, 0x88FDA79D */ 319 }; 320 static const double qs5[6] = { 321 8.12765501384335777857e+01, /* 0x405451B2, 0xFF5A11B2 */ 322 1.99179873460485964642e+03, /* 0x409F1F31, 0xE77BF839 */ 323 1.74684851924908907677e+04, /* 0x40D10F1F, 0x0D64CE29 */ 324 4.98514270910352279316e+04, /* 0x40E8576D, 0xAABAD197 */ 325 2.79480751638918118260e+04, /* 0x40DB4B04, 0xCF7C364B */ 326 -4.71918354795128470869e+03, /* 0xC0B26F2E, 0xFCFFA004 */ 327 }; 328 329 static const double qr3[6] = { 330 -5.07831226461766561369e-09, /* 0xBE35CFA9, 0xD38FC84F */ 331 -1.02537829820837089745e-01, /* 0xBFBA3FEB, 0x51AEED54 */ 332 -4.61011581139473403113e+00, /* 0xC01270C2, 0x3302D9FF */ 333 -5.78472216562783643212e+01, /* 0xC04CEC71, 0xC25D16DA */ 334 -2.28244540737631695038e+02, /* 0xC06C87D3, 0x4718D55F */ 335 -2.19210128478909325622e+02, /* 0xC06B66B9, 0x5F5C1BF6 */ 336 }; 337 static const double qs3[6] = { 338 4.76651550323729509273e+01, /* 0x4047D523, 0xCCD367E4 */ 339 6.73865112676699709482e+02, /* 0x40850EEB, 0xC031EE3E */ 340 3.38015286679526343505e+03, /* 0x40AA684E, 0x448E7C9A */ 341 5.54772909720722782367e+03, /* 0x40B5ABBA, 0xA61D54A6 */ 342 1.90311919338810798763e+03, /* 0x409DBC7A, 0x0DD4DF4B */ 343 -1.35201191444307340817e+02, /* 0xC060E670, 0x290A311F */ 344 }; 345 346 static const double qr2[6] = {/* for x in [2.8570,2]=1/[0.3499,0.5] */ 347 -1.78381727510958865572e-07, /* 0xBE87F126, 0x44C626D2 */ 348 -1.02517042607985553460e-01, /* 0xBFBA3E8E, 0x9148B010 */ 349 -2.75220568278187460720e+00, /* 0xC0060484, 0x69BB4EDA */ 350 -1.96636162643703720221e+01, /* 0xC033A9E2, 0xC168907F */ 351 -4.23253133372830490089e+01, /* 0xC04529A3, 0xDE104AAA */ 352 -2.13719211703704061733e+01, /* 0xC0355F36, 0x39CF6E52 */ 353 }; 354 static const double qs2[6] = { 355 2.95333629060523854548e+01, /* 0x403D888A, 0x78AE64FF */ 356 2.52981549982190529136e+02, /* 0x406F9F68, 0xDB821CBA */ 357 7.57502834868645436472e+02, /* 0x4087AC05, 0xCE49A0F7 */ 358 7.39393205320467245656e+02, /* 0x40871B25, 0x48D4C029 */ 359 1.55949003336666123687e+02, /* 0x40637E5E, 0x3C3ED8D4 */ 360 -4.95949898822628210127e+00, /* 0xC013D686, 0xE71BE86B */ 361 }; 362 363 static __inline double 364 qone(double x) 365 { 366 const double *p,*q; 367 double s,r,z; 368 int32_t ix; 369 GET_HIGH_WORD(ix,x); 370 ix &= 0x7fffffff; 371 if(ix>=0x40200000) {p = qr8; q= qs8;} 372 else if(ix>=0x40122E8B){p = qr5; q= qs5;} 373 else if(ix>=0x4006DB6D){p = qr3; q= qs3;} 374 else {p = qr2; q= qs2;} /* ix>=0x40000000 */ 375 z = one/(x*x); 376 r = p[0]+z*(p[1]+z*(p[2]+z*(p[3]+z*(p[4]+z*p[5])))); 377 s = one+z*(q[0]+z*(q[1]+z*(q[2]+z*(q[3]+z*(q[4]+z*q[5]))))); 378 return (.375 + r/s)/x; 379 } 380