libm/e_j0.c

   1 /*
   2  * ====================================================
   3  * Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved.
   4  *
   5  * Developed at SunPro, 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 /* __ieee754_j0(x), __ieee754_y0(x)
  13  * Bessel function of the first and second kinds of order zero.
  14  * Method -- j0(x):
  15  *      1. For tiny x, we use j0(x) = 1 - x^2/4 + x^4/64 - ...
  16  *      2. Reduce x to |x| since j0(x)=j0(-x),  and
  17  *         for x in (0,2)
  18  *              j0(x) = 1-z/4+ z^2*R0/S0,  where z = x*x;
  19  *         (precision:  |j0-1+z/4-z^2R0/S0 |<2**-63.67 )
  20  *         for x in (2,inf)
  21  *              j0(x) = sqrt(2/(pi*x))*(p0(x)*cos(x0)-q0(x)*sin(x0))
  22  *         where x0 = x-pi/4. It is better to compute sin(x0),cos(x0)
  23  *         as follow:
  24  *              cos(x0) = cos(x)cos(pi/4)+sin(x)sin(pi/4)
  25  *                      = 1/sqrt(2) * (cos(x) + sin(x))
  26  *              sin(x0) = sin(x)cos(pi/4)-cos(x)sin(pi/4)
  27  *                      = 1/sqrt(2) * (sin(x) - cos(x))
  28  *         (To avoid cancellation, use
  29  *              sin(x) +- cos(x) = -cos(2x)/(sin(x) -+ cos(x))
  30  *          to compute the worse one.)
  31  *
  32  *      3 Special cases
  33  *              j0(nan)= nan
  34  *              j0(0) = 1
  35  *              j0(inf) = 0
  36  *
  37  * Method -- y0(x):
  38  *      1. For x<2.
  39  *         Since
  40  *              y0(x) = 2/pi*(j0(x)*(ln(x/2)+Euler) + x^2/4 - ...)
  41  *         therefore y0(x)-2/pi*j0(x)*ln(x) is an even function.
  42  *         We use the following function to approximate y0,
  43  *              y0(x) = U(z)/V(z) + (2/pi)*(j0(x)*ln(x)), z= x^2
  44  *         where
  45  *              U(z) = u00 + u01*z + ... + u06*z^6
  46  *              V(z) = 1  + v01*z + ... + v04*z^4
  47  *         with absolute approximation error bounded by 2**-72.
  48  *         Note: For tiny x, U/V = u0 and j0(x)~1, hence
  49  *              y0(tiny) = u0 + (2/pi)*ln(tiny), (choose tiny<2**-27)
  50  *      2. For x>=2.
  51  *              y0(x) = sqrt(2/(pi*x))*(p0(x)*cos(x0)+q0(x)*sin(x0))
  52  *         where x0 = x-pi/4. It is better to compute sin(x0),cos(x0)
  53  *         by the method mentioned above.
  54  *      3. Special cases: y0(0)=-inf, y0(x<0)=NaN, y0(inf)=0.
  55  */
  56
  57 #include "math.h"
  58 #include "math_private.h"
  59
  60 static double pzero(double), qzero(double);
  61
  62 static const double
  63 huge    = 1e300,
  64 one     = 1.0,
  65 invsqrtpi=  5.64189583547756279280e-01, /* 0x3FE20DD7, 0x50429B6D */
  66 tpi      =  6.36619772367581382433e-01, /* 0x3FE45F30, 0x6DC9C883 */
  67                 /* R0/S0 on [0, 2.00] */
  68 R02  =  1.56249999999999947958e-02, /* 0x3F8FFFFF, 0xFFFFFFFD */
  69 R03  = -1.89979294238854721751e-04, /* 0xBF28E6A5, 0xB61AC6E9 */
  70 R04  =  1.82954049532700665670e-06, /* 0x3EBEB1D1, 0x0C503919 */
  71 R05  = -4.61832688532103189199e-09, /* 0xBE33D5E7, 0x73D63FCE */
  72 S01  =  1.56191029464890010492e-02, /* 0x3F8FFCE8, 0x82C8C2A4 */
  73 S02  =  1.16926784663337450260e-04, /* 0x3F1EA6D2, 0xDD57DBF4 */
  74 S03  =  5.13546550207318111446e-07, /* 0x3EA13B54, 0xCE84D5A9 */
  75 S04  =  1.16614003333790000205e-09; /* 0x3E1408BC, 0xF4745D8F */
  76
  77 static const double zero = 0.0;
  78
  79 double attribute_hidden __ieee754_j0(double x)
  80 {
  81         double z, s,c,ss,cc,r,u,v;
  82         int32_t hx,ix;
  83
  84         GET_HIGH_WORD(hx,x);
  85         ix = hx&0x7fffffff;
  86         if(ix>=0x7ff00000) return one/(x*x);
  87         x = fabs(x);
  88         if(ix >= 0x40000000) {  /* |x| >= 2.0 */
  89                 s = sin(x);
  90                 c = cos(x);
  91                 ss = s-c;
  92                 cc = s+c;
  93                 if(ix<0x7fe00000) {  /* make sure x+x not overflow */
  94                     z = -cos(x+x);
  95                     if ((s*c)<zero) cc = z/ss;
  96                     else            ss = z/cc;
  97                 }
  98         /*
  99          * j0(x) = 1/sqrt(pi) * (P(0,x)*cc - Q(0,x)*ss) / sqrt(x)
 100          * y0(x) = 1/sqrt(pi) * (P(0,x)*ss + Q(0,x)*cc) / sqrt(x)
 101          */
 102                 if(ix>0x48000000) z = (invsqrtpi*cc)/sqrt(x);
 103                 else {
 104                     u = pzero(x); v = qzero(x);
 105                     z = invsqrtpi*(u*cc-v*ss)/sqrt(x);
 106                 }
 107                 return z;
 108         }
 109         if(ix<0x3f200000) {     /* |x| < 2**-13 */
 110             if(huge+x>one) {    /* raise inexact if x != 0 */
 111                 if(ix<0x3e400000) return one;   /* |x|<2**-27 */
 112                 else          return one - 0.25*x*x;
 113             }
 114         }
 115         z = x*x;
 116         r =  z*(R02+z*(R03+z*(R04+z*R05)));
 117         s =  one+z*(S01+z*(S02+z*(S03+z*S04)));
 118         if(ix < 0x3FF00000) {   /* |x| < 1.00 */
 119             return one + z*(-0.25+(r/s));
 120         } else {
 121             u = 0.5*x;
 122             return((one+u)*(one-u)+z*(r/s));
 123         }
 124 }
 125
 126 /*
 127  * wrapper j0(double x)
 128  */
 129 #ifndef _IEEE_LIBM
 130 double j0(double x)
 131 {
 132         double z = __ieee754_j0(x);
 133         if (_LIB_VERSION == _IEEE_ || isnan(x))
 134                 return z;
 135         if (fabs(x) > X_TLOSS)
 136                 return __kernel_standard(x, x, 34); /* j0(|x|>X_TLOSS) */
 137         return z;
 138 }
 139 #else
 140 strong_alias(__ieee754_j0, j0)
 141 #endif
 142
 143 static const double
 144 u00  = -7.38042951086872317523e-02, /* 0xBFB2E4D6, 0x99CBD01F */
 145 u01  =  1.76666452509181115538e-01, /* 0x3FC69D01, 0x9DE9E3FC */
 146 u02  = -1.38185671945596898896e-02, /* 0xBF8C4CE8, 0xB16CFA97 */
 147 u03  =  3.47453432093683650238e-04, /* 0x3F36C54D, 0x20B29B6B */
 148 u04  = -3.81407053724364161125e-06, /* 0xBECFFEA7, 0x73D25CAD */
 149 u05  =  1.95590137035022920206e-08, /* 0x3E550057, 0x3B4EABD4 */
 150 u06  = -3.98205194132103398453e-11, /* 0xBDC5E43D, 0x693FB3C8 */
 151 v01  =  1.27304834834123699328e-02, /* 0x3F8A1270, 0x91C9C71A */
 152 v02  =  7.60068627350353253702e-05, /* 0x3F13ECBB, 0xF578C6C1 */
 153 v03  =  2.59150851840457805467e-07, /* 0x3E91642D, 0x7FF202FD */
 154 v04  =  4.41110311332675467403e-10; /* 0x3DFE5018, 0x3BD6D9EF */
 155
 156 double attribute_hidden __ieee754_y0(double x)
 157 {
 158         double z, s,c,ss,cc,u,v;
 159         int32_t hx,ix,lx;
 160
 161         EXTRACT_WORDS(hx,lx,x);
 162         ix = 0x7fffffff&hx;
 163     /* Y0(NaN) is NaN, y0(-inf) is Nan, y0(inf) is 0  */
 164         if(ix>=0x7ff00000) return  one/(x+x*x);
 165         if((ix|lx)==0) return -one/zero;
 166         if(hx<0) return zero/zero;
 167         if(ix >= 0x40000000) {  /* |x| >= 2.0 */
 168         /* y0(x) = sqrt(2/(pi*x))*(p0(x)*sin(x0)+q0(x)*cos(x0))
 169          * where x0 = x-pi/4
 170          *      Better formula:
 171          *              cos(x0) = cos(x)cos(pi/4)+sin(x)sin(pi/4)
 172          *                      =  1/sqrt(2) * (sin(x) + cos(x))
 173          *              sin(x0) = sin(x)cos(3pi/4)-cos(x)sin(3pi/4)
 174          *                      =  1/sqrt(2) * (sin(x) - cos(x))
 175          * To avoid cancellation, use
 176          *              sin(x) +- cos(x) = -cos(2x)/(sin(x) -+ cos(x))
 177          * to compute the worse one.
 178          */
 179                 s = sin(x);
 180                 c = cos(x);
 181                 ss = s-c;
 182                 cc = s+c;
 183         /*
 184          * j0(x) = 1/sqrt(pi) * (P(0,x)*cc - Q(0,x)*ss) / sqrt(x)
 185          * y0(x) = 1/sqrt(pi) * (P(0,x)*ss + Q(0,x)*cc) / sqrt(x)
 186          */
 187                 if(ix<0x7fe00000) {  /* make sure x+x not overflow */
 188                     z = -cos(x+x);
 189                     if ((s*c)<zero) cc = z/ss;
 190                     else            ss = z/cc;
 191                 }
 192                 if(ix>0x48000000) z = (invsqrtpi*ss)/sqrt(x);
 193                 else {
 194                     u = pzero(x); v = qzero(x);
 195                     z = invsqrtpi*(u*ss+v*cc)/sqrt(x);
 196                 }
 197                 return z;
 198         }
 199         if(ix<=0x3e400000) {    /* x < 2**-27 */
 200             return(u00 + tpi*__ieee754_log(x));
 201         }
 202         z = x*x;
 203         u = u00+z*(u01+z*(u02+z*(u03+z*(u04+z*(u05+z*u06)))));
 204         v = one+z*(v01+z*(v02+z*(v03+z*v04)));
 205         return(u/v + tpi*(__ieee754_j0(x)*__ieee754_log(x)));
 206 }
 207
 208 /*
 209  * wrapper y0(double x)
 210  */
 211 #ifndef _IEEE_LIBM
 212 double y0(double x)
 213 {
 214         double z = __ieee754_y0(x);
 215         if (_LIB_VERSION == _IEEE_ || isnan(x))
 216                 return z;
 217         if (x <= 0.0) {
 218                 if (x == 0.0) /* d= -one/(x-x); */
 219                         return __kernel_standard(x, x, 8);
 220                 /* d = zero/(x-x); */
 221                 return __kernel_standard(x, x, 9);
 222         }
 223         if (x > X_TLOSS)
 224                 return __kernel_standard(x, x, 35); /* y0(x>X_TLOSS) */
 225         return z;
 226 }
 227 #else
 228 strong_alias(__ieee754_y0, y0)
 229 #endif
 230
 231
 232 /* The asymptotic expansions of pzero is
 233  *      1 - 9/128 s^2 + 11025/98304 s^4 - ...,  where s = 1/x.
 234  * For x >= 2, We approximate pzero by
 235  *      pzero(x) = 1 + (R/S)
 236  * where  R = pR0 + pR1*s^2 + pR2*s^4 + ... + pR5*s^10
 237  *        S = 1 + pS0*s^2 + ... + pS4*s^10
 238  * and
 239  *      | pzero(x)-1-R/S | <= 2  ** ( -60.26)
 240  */
 241 static const double pR8[6] = { /* for x in [inf, 8]=1/[0,0.125] */
 242   0.00000000000000000000e+00, /* 0x00000000, 0x00000000 */
 243  -7.03124999999900357484e-02, /* 0xBFB1FFFF, 0xFFFFFD32 */
 244  -8.08167041275349795626e+00, /* 0xC02029D0, 0xB44FA779 */
 245  -2.57063105679704847262e+02, /* 0xC0701102, 0x7B19E863 */
 246  -2.48521641009428822144e+03, /* 0xC0A36A6E, 0xCD4DCAFC */
 247  -5.25304380490729545272e+03, /* 0xC0B4850B, 0x36CC643D */
 248 };
 249 static const double pS8[5] = {
 250   1.16534364619668181717e+02, /* 0x405D2233, 0x07A96751 */
 251   3.83374475364121826715e+03, /* 0x40ADF37D, 0x50596938 */
 252   4.05978572648472545552e+04, /* 0x40E3D2BB, 0x6EB6B05F */
 253   1.16752972564375915681e+05, /* 0x40FC810F, 0x8F9FA9BD */
 254   4.76277284146730962675e+04, /* 0x40E74177, 0x4F2C49DC */
 255 };
 256
 257 static const double pR5[6] = { /* for x in [8,4.5454]=1/[0.125,0.22001] */
 258  -1.14125464691894502584e-11, /* 0xBDA918B1, 0x47E495CC */
 259  -7.03124940873599280078e-02, /* 0xBFB1FFFF, 0xE69AFBC6 */
 260  -4.15961064470587782438e+00, /* 0xC010A370, 0xF90C6BBF */
 261  -6.76747652265167261021e+01, /* 0xC050EB2F, 0x5A7D1783 */
 262  -3.31231299649172967747e+02, /* 0xC074B3B3, 0x6742CC63 */
 263  -3.46433388365604912451e+02, /* 0xC075A6EF, 0x28A38BD7 */
 264 };
 265 static const double pS5[5] = {
 266   6.07539382692300335975e+01, /* 0x404E6081, 0x0C98C5DE */
 267   1.05125230595704579173e+03, /* 0x40906D02, 0x5C7E2864 */
 268   5.97897094333855784498e+03, /* 0x40B75AF8, 0x8FBE1D60 */
 269   9.62544514357774460223e+03, /* 0x40C2CCB8, 0xFA76FA38 */
 270   2.40605815922939109441e+03, /* 0x40A2CC1D, 0xC70BE864 */
 271 };
 272
 273 static const double pR3[6] = {/* for x in [4.547,2.8571]=1/[0.2199,0.35001] */
 274  -2.54704601771951915620e-09, /* 0xBE25E103, 0x6FE1AA86 */
 275  -7.03119616381481654654e-02, /* 0xBFB1FFF6, 0xF7C0E24B */
 276  -2.40903221549529611423e+00, /* 0xC00345B2, 0xAEA48074 */
 277  -2.19659774734883086467e+01, /* 0xC035F74A, 0x4CB94E14 */
 278  -5.80791704701737572236e+01, /* 0xC04D0A22, 0x420A1A45 */
 279  -3.14479470594888503854e+01, /* 0xC03F72AC, 0xA892D80F */
 280 };
 281 static const double pS3[5] = {
 282   3.58560338055209726349e+01, /* 0x4041ED92, 0x84077DD3 */
 283   3.61513983050303863820e+02, /* 0x40769839, 0x464A7C0E */
 284   1.19360783792111533330e+03, /* 0x4092A66E, 0x6D1061D6 */
 285   1.12799679856907414432e+03, /* 0x40919FFC, 0xB8C39B7E */
 286   1.73580930813335754692e+02, /* 0x4065B296, 0xFC379081 */
 287 };
 288
 289 static const double pR2[6] = {/* for x in [2.8570,2]=1/[0.3499,0.5] */
 290  -8.87534333032526411254e-08, /* 0xBE77D316, 0xE927026D */
 291  -7.03030995483624743247e-02, /* 0xBFB1FF62, 0x495E1E42 */
 292  -1.45073846780952986357e+00, /* 0xBFF73639, 0x8A24A843 */
 293  -7.63569613823527770791e+00, /* 0xC01E8AF3, 0xEDAFA7F3 */
 294  -1.11931668860356747786e+01, /* 0xC02662E6, 0xC5246303 */
 295  -3.23364579351335335033e+00, /* 0xC009DE81, 0xAF8FE70F */
 296 };
 297 static const double pS2[5] = {
 298   2.22202997532088808441e+01, /* 0x40363865, 0x908B5959 */
 299   1.36206794218215208048e+02, /* 0x4061069E, 0x0EE8878F */
 300   2.70470278658083486789e+02, /* 0x4070E786, 0x42EA079B */
 301   1.53875394208320329881e+02, /* 0x40633C03, 0x3AB6FAFF */
 302   1.46576176948256193810e+01, /* 0x402D50B3, 0x44391809 */
 303 };
 304
 305 static double pzero(double x)
 306 {
 307         const double *p = 0,*q = 0;
 308         double z,r,s;
 309         int32_t ix;
 310         GET_HIGH_WORD(ix,x);
 311         ix &= 0x7fffffff;
 312         if(ix>=0x40200000)     {p = pR8; q= pS8;}
 313         else if(ix>=0x40122E8B){p = pR5; q= pS5;}
 314         else if(ix>=0x4006DB6D){p = pR3; q= pS3;}
 315         else if(ix>=0x40000000){p = pR2; q= pS2;}
 316         z = one/(x*x);
 317         r = p[0]+z*(p[1]+z*(p[2]+z*(p[3]+z*(p[4]+z*p[5]))));
 318         s = one+z*(q[0]+z*(q[1]+z*(q[2]+z*(q[3]+z*q[4]))));
 319         return one+ r/s;
 320 }
 321
 322
 323 /* For x >= 8, the asymptotic expansions of qzero is
 324  *      -1/8 s + 75/1024 s^3 - ..., where s = 1/x.
 325  * We approximate pzero by
 326  *      qzero(x) = s*(-1.25 + (R/S))
 327  * where  R = qR0 + qR1*s^2 + qR2*s^4 + ... + qR5*s^10
 328  *        S = 1 + qS0*s^2 + ... + qS5*s^12
 329  * and
 330  *      | qzero(x)/s +1.25-R/S | <= 2  ** ( -61.22)
 331  */
 332 static const double qR8[6] = { /* for x in [inf, 8]=1/[0,0.125] */
 333   0.00000000000000000000e+00, /* 0x00000000, 0x00000000 */
 334   7.32421874999935051953e-02, /* 0x3FB2BFFF, 0xFFFFFE2C */
 335   1.17682064682252693899e+01, /* 0x40278952, 0x5BB334D6 */
 336   5.57673380256401856059e+02, /* 0x40816D63, 0x15301825 */
 337   8.85919720756468632317e+03, /* 0x40C14D99, 0x3E18F46D */
 338   3.70146267776887834771e+04, /* 0x40E212D4, 0x0E901566 */
 339 };
 340 static const double qS8[6] = {
 341   1.63776026895689824414e+02, /* 0x406478D5, 0x365B39BC */
 342   8.09834494656449805916e+03, /* 0x40BFA258, 0x4E6B0563 */
 343   1.42538291419120476348e+05, /* 0x41016652, 0x54D38C3F */
 344   8.03309257119514397345e+05, /* 0x412883DA, 0x83A52B43 */
 345   8.40501579819060512818e+05, /* 0x4129A66B, 0x28DE0B3D */
 346  -3.43899293537866615225e+05, /* 0xC114FD6D, 0x2C9530C5 */
 347 };
 348
 349 static const double qR5[6] = { /* for x in [8,4.5454]=1/[0.125,0.22001] */
 350   1.84085963594515531381e-11, /* 0x3DB43D8F, 0x29CC8CD9 */
 351   7.32421766612684765896e-02, /* 0x3FB2BFFF, 0xD172B04C */
 352   5.83563508962056953777e+00, /* 0x401757B0, 0xB9953DD3 */
 353   1.35111577286449829671e+02, /* 0x4060E392, 0x0A8788E9 */
 354   1.02724376596164097464e+03, /* 0x40900CF9, 0x9DC8C481 */
 355   1.98997785864605384631e+03, /* 0x409F17E9, 0x53C6E3A6 */
 356 };
 357 static const double qS5[6] = {
 358   8.27766102236537761883e+01, /* 0x4054B1B3, 0xFB5E1543 */
 359   2.07781416421392987104e+03, /* 0x40A03BA0, 0xDA21C0CE */
 360   1.88472887785718085070e+04, /* 0x40D267D2, 0x7B591E6D */
 361   5.67511122894947329769e+04, /* 0x40EBB5E3, 0x97E02372 */
 362   3.59767538425114471465e+04, /* 0x40E19118, 0x1F7A54A0 */
 363  -5.35434275601944773371e+03, /* 0xC0B4EA57, 0xBEDBC609 */
 364 };
 365
 366 static const double qR3[6] = {/* for x in [4.547,2.8571]=1/[0.2199,0.35001] */
 367   4.37741014089738620906e-09, /* 0x3E32CD03, 0x6ADECB82 */
 368   7.32411180042911447163e-02, /* 0x3FB2BFEE, 0x0E8D0842 */
 369   3.34423137516170720929e+00, /* 0x400AC0FC, 0x61149CF5 */
 370   4.26218440745412650017e+01, /* 0x40454F98, 0x962DAEDD */
 371   1.70808091340565596283e+02, /* 0x406559DB, 0xE25EFD1F */
 372   1.66733948696651168575e+02, /* 0x4064D77C, 0x81FA21E0 */
 373 };
 374 static const double qS3[6] = {
 375   4.87588729724587182091e+01, /* 0x40486122, 0xBFE343A6 */
 376   7.09689221056606015736e+02, /* 0x40862D83, 0x86544EB3 */
 377   3.70414822620111362994e+03, /* 0x40ACF04B, 0xE44DFC63 */
 378   6.46042516752568917582e+03, /* 0x40B93C6C, 0xD7C76A28 */
 379   2.51633368920368957333e+03, /* 0x40A3A8AA, 0xD94FB1C0 */
 380  -1.49247451836156386662e+02, /* 0xC062A7EB, 0x201CF40F */
 381 };
 382
 383 static const double qR2[6] = {/* for x in [2.8570,2]=1/[0.3499,0.5] */
 384   1.50444444886983272379e-07, /* 0x3E84313B, 0x54F76BDB */
 385   7.32234265963079278272e-02, /* 0x3FB2BEC5, 0x3E883E34 */
 386   1.99819174093815998816e+00, /* 0x3FFFF897, 0xE727779C */
 387   1.44956029347885735348e+01, /* 0x402CFDBF, 0xAAF96FE5 */
 388   3.16662317504781540833e+01, /* 0x403FAA8E, 0x29FBDC4A */
 389   1.62527075710929267416e+01, /* 0x403040B1, 0x71814BB4 */
 390 };
 391 static const double qS2[6] = {
 392   3.03655848355219184498e+01, /* 0x403E5D96, 0xF7C07AED */
 393   2.69348118608049844624e+02, /* 0x4070D591, 0xE4D14B40 */
 394   8.44783757595320139444e+02, /* 0x408A6645, 0x22B3BF22 */
 395   8.82935845112488550512e+02, /* 0x408B977C, 0x9C5CC214 */
 396   2.12666388511798828631e+02, /* 0x406A9553, 0x0E001365 */
 397  -5.31095493882666946917e+00, /* 0xC0153E6A, 0xF8B32931 */
 398 };
 399
 400 static double qzero(double x)
 401 {
 402         const double *p=0,*q=0;
 403         double s,r,z;
 404         int32_t ix;
 405         GET_HIGH_WORD(ix,x);
 406         ix &= 0x7fffffff;
 407         if(ix>=0x40200000)     {p = qR8; q= qS8;}
 408         else if(ix>=0x40122E8B){p = qR5; q= qS5;}
 409         else if(ix>=0x4006DB6D){p = qR3; q= qS3;}
 410         else if(ix>=0x40000000){p = qR2; q= qS2;}
 411         z = one/(x*x);
 412         r = p[0]+z*(p[1]+z*(p[2]+z*(p[3]+z*(p[4]+z*p[5]))));
 413         s = one+z*(q[0]+z*(q[1]+z*(q[2]+z*(q[3]+z*(q[4]+z*q[5])))));
 414         return (-.125 + r/s)/x;
 415 }