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 #include <sys/cdefs.h> 13 /* j1(x), y1(x) 14 * Bessel function of the first and second kinds of order zero. 15 * Method -- j1(x): 16 * 1. For tiny x, we use j1(x) = x/2 - x^3/16 + x^5/384 - ... 17 * 2. Reduce x to |x| since j1(x)=-j1(-x), and 18 * for x in (0,2) 19 * j1(x) = x/2 + x*z*R0/S0, where z = x*x; 20 * (precision: |j1/x - 1/2 - R0/S0 |<2**-61.51 ) 21 * for x in (2,inf) 22 * j1(x) = sqrt(2/(pi*x))*(p1(x)*cos(x1)-q1(x)*sin(x1)) 23 * y1(x) = sqrt(2/(pi*x))*(p1(x)*sin(x1)+q1(x)*cos(x1)) 24 * where x1 = x-3*pi/4. It is better to compute sin(x1),cos(x1) 25 * as follow: 26 * cos(x1) = cos(x)cos(3pi/4)+sin(x)sin(3pi/4) 27 * = 1/sqrt(2) * (sin(x) - cos(x)) 28 * sin(x1) = sin(x)cos(3pi/4)-cos(x)sin(3pi/4) 29 * = -1/sqrt(2) * (sin(x) + cos(x)) 30 * (To avoid cancellation, use 31 * sin(x) +- cos(x) = -cos(2x)/(sin(x) -+ cos(x)) 32 * to compute the worse one.) 33 * 34 * 3 Special cases 35 * j1(nan)= nan 36 * j1(0) = 0 37 * j1(inf) = 0 38 * 39 * Method -- y1(x): 40 * 1. screen out x<=0 cases: y1(0)=-inf, y1(x<0)=NaN 41 * 2. For x<2. 42 * Since 43 * y1(x) = 2/pi*(j1(x)*(ln(x/2)+Euler)-1/x-x/2+5/64*x^3-...) 44 * therefore y1(x)-2/pi*j1(x)*ln(x)-1/x is an odd function. 45 * We use the following function to approximate y1, 46 * y1(x) = x*U(z)/V(z) + (2/pi)*(j1(x)*ln(x)-1/x), z= x^2 47 * where for x in [0,2] (abs err less than 2**-65.89) 48 * U(z) = U0[0] + U0[1]*z + ... + U0[4]*z^4 49 * V(z) = 1 + v0[0]*z + ... + v0[4]*z^5 50 * Note: For tiny x, 1/x dominate y1 and hence 51 * y1(tiny) = -2/pi/tiny, (choose tiny<2**-54) 52 * 3. For x>=2. 53 * y1(x) = sqrt(2/(pi*x))*(p1(x)*sin(x1)+q1(x)*cos(x1)) 54 * where x1 = x-3*pi/4. It is better to compute sin(x1),cos(x1) 55 * by method mentioned above. 56 */ 57 58 #include "math.h" 59 #include "math_private.h" 60 61 static __inline double pone(double), qone(double); 62 63 static const volatile double vone = 1, vzero = 0; 64 65 static const double 66 huge = 1e300, 67 one = 1.0, 68 invsqrtpi= 5.64189583547756279280e-01, /* 0x3FE20DD7, 0x50429B6D */ 69 tpi = 6.36619772367581382433e-01, /* 0x3FE45F30, 0x6DC9C883 */ 70 /* R0/S0 on [0,2] */ 71 r00 = -6.25000000000000000000e-02, /* 0xBFB00000, 0x00000000 */ 72 r01 = 1.40705666955189706048e-03, /* 0x3F570D9F, 0x98472C61 */ 73 r02 = -1.59955631084035597520e-05, /* 0xBEF0C5C6, 0xBA169668 */ 74 r03 = 4.96727999609584448412e-08, /* 0x3E6AAAFA, 0x46CA0BD9 */ 75 s01 = 1.91537599538363460805e-02, /* 0x3F939D0B, 0x12637E53 */ 76 s02 = 1.85946785588630915560e-04, /* 0x3F285F56, 0xB9CDF664 */ 77 s03 = 1.17718464042623683263e-06, /* 0x3EB3BFF8, 0x333F8498 */ 78 s04 = 5.04636257076217042715e-09, /* 0x3E35AC88, 0xC97DFF2C */ 79 s05 = 1.23542274426137913908e-11; /* 0x3DAB2ACF, 0xCFB97ED8 */ 80 81 static const double zero = 0.0; 82 83 double 84 j1(double x) 85 { 86 double z, s,c,ss,cc,r,u,v,y; 87 int32_t hx,ix; 88 89 GET_HIGH_WORD(hx,x); 90 ix = hx&0x7fffffff; 91 if(ix>=0x7ff00000) return one/x; 92 y = fabs(x); 93 if(ix >= 0x40000000) { /* |x| >= 2.0 */ 94 sincos(y, &s, &c); 95 ss = -s-c; 96 cc = s-c; 97 if(ix<0x7fe00000) { /* make sure y+y not overflow */ 98 z = cos(y+y); 99 if ((s*c)>zero) cc = z/ss; 100 else ss = z/cc; 101 } 102 /* 103 * j1(x) = 1/sqrt(pi) * (P(1,x)*cc - Q(1,x)*ss) / sqrt(x) 104 * y1(x) = 1/sqrt(pi) * (P(1,x)*ss + Q(1,x)*cc) / sqrt(x) 105 */ 106 if(ix>0x48000000) z = (invsqrtpi*cc)/sqrt(y); 107 else { 108 u = pone(y); v = qone(y); 109 z = invsqrtpi*(u*cc-v*ss)/sqrt(y); 110 } 111 if(hx<0) return -z; 112 else return z; 113 } 114 if(ix<0x3e400000) { /* |x|<2**-27 */ 115 if(huge+x>one) return 0.5*x;/* inexact if x!=0 necessary */ 116 } 117 z = x*x; 118 r = z*(r00+z*(r01+z*(r02+z*r03))); 119 s = one+z*(s01+z*(s02+z*(s03+z*(s04+z*s05)))); 120 r *= x; 121 return(x*0.5+r/s); 122 } 123 124 static const double U0[5] = { 125 -1.96057090646238940668e-01, /* 0xBFC91866, 0x143CBC8A */ 126 5.04438716639811282616e-02, /* 0x3FA9D3C7, 0x76292CD1 */ 127 -1.91256895875763547298e-03, /* 0xBF5F55E5, 0x4844F50F */ 128 2.35252600561610495928e-05, /* 0x3EF8AB03, 0x8FA6B88E */ 129 -9.19099158039878874504e-08, /* 0xBE78AC00, 0x569105B8 */ 130 }; 131 static const double V0[5] = { 132 1.99167318236649903973e-02, /* 0x3F94650D, 0x3F4DA9F0 */ 133 2.02552581025135171496e-04, /* 0x3F2A8C89, 0x6C257764 */ 134 1.35608801097516229404e-06, /* 0x3EB6C05A, 0x894E8CA6 */ 135 6.22741452364621501295e-09, /* 0x3E3ABF1D, 0x5BA69A86 */ 136 1.66559246207992079114e-11, /* 0x3DB25039, 0xDACA772A */ 137 }; 138 139 double 140 y1(double x) 141 { 142 double z, s,c,ss,cc,u,v; 143 int32_t hx,ix,lx; 144 145 EXTRACT_WORDS(hx,lx,x); 146 ix = 0x7fffffff&hx; 147 /* 148 * y1(NaN) = NaN. 149 * y1(Inf) = 0. 150 * y1(-Inf) = NaN and raise invalid exception. 151 */ 152 if(ix>=0x7ff00000) return vone/(x+x*x); 153 /* y1(+-0) = -inf and raise divide-by-zero exception. */ 154 if((ix|lx)==0) return -one/vzero; 155 /* y1(x<0) = NaN and raise invalid exception. */ 156 if(hx<0) return vzero/vzero; 157 if(ix >= 0x40000000) { /* |x| >= 2.0 */ 158 sincos(x, &s, &c); 159 ss = -s-c; 160 cc = s-c; 161 if(ix<0x7fe00000) { /* make sure x+x not overflow */ 162 z = cos(x+x); 163 if ((s*c)>zero) cc = z/ss; 164 else ss = z/cc; 165 } 166 /* y1(x) = sqrt(2/(pi*x))*(p1(x)*sin(x0)+q1(x)*cos(x0)) 167 * where x0 = x-3pi/4 168 * Better formula: 169 * cos(x0) = cos(x)cos(3pi/4)+sin(x)sin(3pi/4) 170 * = 1/sqrt(2) * (sin(x) - cos(x)) 171 * sin(x0) = sin(x)cos(3pi/4)-cos(x)sin(3pi/4) 172 * = -1/sqrt(2) * (cos(x) + sin(x)) 173 * To avoid cancellation, use 174 * sin(x) +- cos(x) = -cos(2x)/(sin(x) -+ cos(x)) 175 * to compute the worse one. 176 */ 177 if(ix>0x48000000) z = (invsqrtpi*ss)/sqrt(x); 178 else { 179 u = pone(x); v = qone(x); 180 z = invsqrtpi*(u*ss+v*cc)/sqrt(x); 181 } 182 return z; 183 } 184 if(ix<=0x3c900000) { /* x < 2**-54 */ 185 return(-tpi/x); 186 } 187 z = x*x; 188 u = U0[0]+z*(U0[1]+z*(U0[2]+z*(U0[3]+z*U0[4]))); 189 v = one+z*(V0[0]+z*(V0[1]+z*(V0[2]+z*(V0[3]+z*V0[4])))); 190 return(x*(u/v) + tpi*(j1(x)*log(x)-one/x)); 191 } 192 193 /* For x >= 8, the asymptotic expansions of pone is 194 * 1 + 15/128 s^2 - 4725/2^15 s^4 - ..., where s = 1/x. 195 * We approximate pone by 196 * pone(x) = 1 + (R/S) 197 * where R = pr0 + pr1*s^2 + pr2*s^4 + ... + pr5*s^10 198 * S = 1 + ps0*s^2 + ... + ps4*s^10 199 * and 200 * | pone(x)-1-R/S | <= 2 ** ( -60.06) 201 */ 202 203 static const double pr8[6] = { /* for x in [inf, 8]=1/[0,0.125] */ 204 0.00000000000000000000e+00, /* 0x00000000, 0x00000000 */ 205 1.17187499999988647970e-01, /* 0x3FBDFFFF, 0xFFFFFCCE */ 206 1.32394806593073575129e+01, /* 0x402A7A9D, 0x357F7FCE */ 207 4.12051854307378562225e+02, /* 0x4079C0D4, 0x652EA590 */ 208 3.87474538913960532227e+03, /* 0x40AE457D, 0xA3A532CC */ 209 7.91447954031891731574e+03, /* 0x40BEEA7A, 0xC32782DD */ 210 }; 211 static const double ps8[5] = { 212 1.14207370375678408436e+02, /* 0x405C8D45, 0x8E656CAC */ 213 3.65093083420853463394e+03, /* 0x40AC85DC, 0x964D274F */ 214 3.69562060269033463555e+04, /* 0x40E20B86, 0x97C5BB7F */ 215 9.76027935934950801311e+04, /* 0x40F7D42C, 0xB28F17BB */ 216 3.08042720627888811578e+04, /* 0x40DE1511, 0x697A0B2D */ 217 }; 218 219 static const double pr5[6] = { /* for x in [8,4.5454]=1/[0.125,0.22001] */ 220 1.31990519556243522749e-11, /* 0x3DAD0667, 0xDAE1CA7D */ 221 1.17187493190614097638e-01, /* 0x3FBDFFFF, 0xE2C10043 */ 222 6.80275127868432871736e+00, /* 0x401B3604, 0x6E6315E3 */ 223 1.08308182990189109773e+02, /* 0x405B13B9, 0x452602ED */ 224 5.17636139533199752805e+02, /* 0x40802D16, 0xD052D649 */ 225 5.28715201363337541807e+02, /* 0x408085B8, 0xBB7E0CB7 */ 226 }; 227 static const double ps5[5] = { 228 5.92805987221131331921e+01, /* 0x404DA3EA, 0xA8AF633D */ 229 9.91401418733614377743e+02, /* 0x408EFB36, 0x1B066701 */ 230 5.35326695291487976647e+03, /* 0x40B4E944, 0x5706B6FB */ 231 7.84469031749551231769e+03, /* 0x40BEA4B0, 0xB8A5BB15 */ 232 1.50404688810361062679e+03, /* 0x40978030, 0x036F5E51 */ 233 }; 234 235 static const double pr3[6] = { 236 3.02503916137373618024e-09, /* 0x3E29FC21, 0xA7AD9EDD */ 237 1.17186865567253592491e-01, /* 0x3FBDFFF5, 0x5B21D17B */ 238 3.93297750033315640650e+00, /* 0x400F76BC, 0xE85EAD8A */ 239 3.51194035591636932736e+01, /* 0x40418F48, 0x9DA6D129 */ 240 9.10550110750781271918e+01, /* 0x4056C385, 0x4D2C1837 */ 241 4.85590685197364919645e+01, /* 0x4048478F, 0x8EA83EE5 */ 242 }; 243 static const double ps3[5] = { 244 3.47913095001251519989e+01, /* 0x40416549, 0xA134069C */ 245 3.36762458747825746741e+02, /* 0x40750C33, 0x07F1A75F */ 246 1.04687139975775130551e+03, /* 0x40905B7C, 0x5037D523 */ 247 8.90811346398256432622e+02, /* 0x408BD67D, 0xA32E31E9 */ 248 1.03787932439639277504e+02, /* 0x4059F26D, 0x7C2EED53 */ 249 }; 250 251 static const double pr2[6] = {/* for x in [2.8570,2]=1/[0.3499,0.5] */ 252 1.07710830106873743082e-07, /* 0x3E7CE9D4, 0xF65544F4 */ 253 1.17176219462683348094e-01, /* 0x3FBDFF42, 0xBE760D83 */ 254 2.36851496667608785174e+00, /* 0x4002F2B7, 0xF98FAEC0 */ 255 1.22426109148261232917e+01, /* 0x40287C37, 0x7F71A964 */ 256 1.76939711271687727390e+01, /* 0x4031B1A8, 0x177F8EE2 */ 257 5.07352312588818499250e+00, /* 0x40144B49, 0xA574C1FE */ 258 }; 259 static const double ps2[5] = { 260 2.14364859363821409488e+01, /* 0x40356FBD, 0x8AD5ECDC */ 261 1.25290227168402751090e+02, /* 0x405F5293, 0x14F92CD5 */ 262 2.32276469057162813669e+02, /* 0x406D08D8, 0xD5A2DBD9 */ 263 1.17679373287147100768e+02, /* 0x405D6B7A, 0xDA1884A9 */ 264 8.36463893371618283368e+00, /* 0x4020BAB1, 0xF44E5192 */ 265 }; 266 267 static __inline double 268 pone(double x) 269 { 270 const double *p,*q; 271 double z,r,s; 272 int32_t ix; 273 GET_HIGH_WORD(ix,x); 274 ix &= 0x7fffffff; 275 if(ix>=0x40200000) {p = pr8; q= ps8;} 276 else if(ix>=0x40122E8B){p = pr5; q= ps5;} 277 else if(ix>=0x4006DB6D){p = pr3; q= ps3;} 278 else {p = pr2; q= ps2;} /* ix>=0x40000000 */ 279 z = one/(x*x); 280 r = p[0]+z*(p[1]+z*(p[2]+z*(p[3]+z*(p[4]+z*p[5])))); 281 s = one+z*(q[0]+z*(q[1]+z*(q[2]+z*(q[3]+z*q[4])))); 282 return one+ r/s; 283 } 284 285 286 /* For x >= 8, the asymptotic expansions of qone is 287 * 3/8 s - 105/1024 s^3 - ..., where s = 1/x. 288 * We approximate pone by 289 * qone(x) = s*(0.375 + (R/S)) 290 * where R = qr1*s^2 + qr2*s^4 + ... + qr5*s^10 291 * S = 1 + qs1*s^2 + ... + qs6*s^12 292 * and 293 * | qone(x)/s -0.375-R/S | <= 2 ** ( -61.13) 294 */ 295 296 static const double qr8[6] = { /* for x in [inf, 8]=1/[0,0.125] */ 297 0.00000000000000000000e+00, /* 0x00000000, 0x00000000 */ 298 -1.02539062499992714161e-01, /* 0xBFBA3FFF, 0xFFFFFDF3 */ 299 -1.62717534544589987888e+01, /* 0xC0304591, 0xA26779F7 */ 300 -7.59601722513950107896e+02, /* 0xC087BCD0, 0x53E4B576 */ 301 -1.18498066702429587167e+04, /* 0xC0C724E7, 0x40F87415 */ 302 -4.84385124285750353010e+04, /* 0xC0E7A6D0, 0x65D09C6A */ 303 }; 304 static const double qs8[6] = { 305 1.61395369700722909556e+02, /* 0x40642CA6, 0xDE5BCDE5 */ 306 7.82538599923348465381e+03, /* 0x40BE9162, 0xD0D88419 */ 307 1.33875336287249578163e+05, /* 0x4100579A, 0xB0B75E98 */ 308 7.19657723683240939863e+05, /* 0x4125F653, 0x72869C19 */ 309 6.66601232617776375264e+05, /* 0x412457D2, 0x7719AD5C */ 310 -2.94490264303834643215e+05, /* 0xC111F969, 0x0EA5AA18 */ 311 }; 312 313 static const double qr5[6] = { /* for x in [8,4.5454]=1/[0.125,0.22001] */ 314 -2.08979931141764104297e-11, /* 0xBDB6FA43, 0x1AA1A098 */ 315 -1.02539050241375426231e-01, /* 0xBFBA3FFF, 0xCB597FEF */ 316 -8.05644828123936029840e+00, /* 0xC0201CE6, 0xCA03AD4B */ 317 -1.83669607474888380239e+02, /* 0xC066F56D, 0x6CA7B9B0 */ 318 -1.37319376065508163265e+03, /* 0xC09574C6, 0x6931734F */ 319 -2.61244440453215656817e+03, /* 0xC0A468E3, 0x88FDA79D */ 320 }; 321 static const double qs5[6] = { 322 8.12765501384335777857e+01, /* 0x405451B2, 0xFF5A11B2 */ 323 1.99179873460485964642e+03, /* 0x409F1F31, 0xE77BF839 */ 324 1.74684851924908907677e+04, /* 0x40D10F1F, 0x0D64CE29 */ 325 4.98514270910352279316e+04, /* 0x40E8576D, 0xAABAD197 */ 326 2.79480751638918118260e+04, /* 0x40DB4B04, 0xCF7C364B */ 327 -4.71918354795128470869e+03, /* 0xC0B26F2E, 0xFCFFA004 */ 328 }; 329 330 static const double qr3[6] = { 331 -5.07831226461766561369e-09, /* 0xBE35CFA9, 0xD38FC84F */ 332 -1.02537829820837089745e-01, /* 0xBFBA3FEB, 0x51AEED54 */ 333 -4.61011581139473403113e+00, /* 0xC01270C2, 0x3302D9FF */ 334 -5.78472216562783643212e+01, /* 0xC04CEC71, 0xC25D16DA */ 335 -2.28244540737631695038e+02, /* 0xC06C87D3, 0x4718D55F */ 336 -2.19210128478909325622e+02, /* 0xC06B66B9, 0x5F5C1BF6 */ 337 }; 338 static const double qs3[6] = { 339 4.76651550323729509273e+01, /* 0x4047D523, 0xCCD367E4 */ 340 6.73865112676699709482e+02, /* 0x40850EEB, 0xC031EE3E */ 341 3.38015286679526343505e+03, /* 0x40AA684E, 0x448E7C9A */ 342 5.54772909720722782367e+03, /* 0x40B5ABBA, 0xA61D54A6 */ 343 1.90311919338810798763e+03, /* 0x409DBC7A, 0x0DD4DF4B */ 344 -1.35201191444307340817e+02, /* 0xC060E670, 0x290A311F */ 345 }; 346 347 static const double qr2[6] = {/* for x in [2.8570,2]=1/[0.3499,0.5] */ 348 -1.78381727510958865572e-07, /* 0xBE87F126, 0x44C626D2 */ 349 -1.02517042607985553460e-01, /* 0xBFBA3E8E, 0x9148B010 */ 350 -2.75220568278187460720e+00, /* 0xC0060484, 0x69BB4EDA */ 351 -1.96636162643703720221e+01, /* 0xC033A9E2, 0xC168907F */ 352 -4.23253133372830490089e+01, /* 0xC04529A3, 0xDE104AAA */ 353 -2.13719211703704061733e+01, /* 0xC0355F36, 0x39CF6E52 */ 354 }; 355 static const double qs2[6] = { 356 2.95333629060523854548e+01, /* 0x403D888A, 0x78AE64FF */ 357 2.52981549982190529136e+02, /* 0x406F9F68, 0xDB821CBA */ 358 7.57502834868645436472e+02, /* 0x4087AC05, 0xCE49A0F7 */ 359 7.39393205320467245656e+02, /* 0x40871B25, 0x48D4C029 */ 360 1.55949003336666123687e+02, /* 0x40637E5E, 0x3C3ED8D4 */ 361 -4.95949898822628210127e+00, /* 0xC013D686, 0xE71BE86B */ 362 }; 363 364 static __inline double 365 qone(double x) 366 { 367 const double *p,*q; 368 double s,r,z; 369 int32_t ix; 370 GET_HIGH_WORD(ix,x); 371 ix &= 0x7fffffff; 372 if(ix>=0x40200000) {p = qr8; q= qs8;} 373 else if(ix>=0x40122E8B){p = qr5; q= qs5;} 374 else if(ix>=0x4006DB6D){p = qr3; q= qs3;} 375 else {p = qr2; q= qs2;} /* ix>=0x40000000 */ 376 z = one/(x*x); 377 r = p[0]+z*(p[1]+z*(p[2]+z*(p[3]+z*(p[4]+z*p[5])))); 378 s = one+z*(q[0]+z*(q[1]+z*(q[2]+z*(q[3]+z*(q[4]+z*q[5]))))); 379 return (.375 + r/s)/x; 380 } 381