libjava/classpath/native/fdlibm/k_rem_pio2.c

   1
   2 /* @(#)k_rem_pio2.c 1.3 95/01/18 */
   3 /*
   4  * ====================================================
   5  * Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved.
   6  *
   7  * Developed at SunSoft, a Sun Microsystems, Inc. business.
   8  * Permission to use, copy, modify, and distribute this
   9  * software is freely granted, provided that this notice
  10  * is preserved.
  11  * ====================================================
  12  */
  13
  14 /*
  15  * __kernel_rem_pio2(x,y,e0,nx,prec,ipio2)
  16  * double x[],y[]; int e0,nx,prec; int ipio2[];
  17  *
  18  * __kernel_rem_pio2 return the last three digits of N with
  19  *              y = x - N*pi/2
  20  * so that |y| < pi/2.
  21  *
  22  * The method is to compute the integer (mod 8) and fraction parts of
  23  * (2/pi)*x without doing the full multiplication. In general we
  24  * skip the part of the product that are known to be a huge integer (
  25  * more accurately, = 0 mod 8 ). Thus the number of operations are
  26  * independent of the exponent of the input.
  27  *
  28  * (2/pi) is represented by an array of 24-bit integers in ipio2[].
  29  *
  30  * Input parameters:
  31  *      x[]     The input value (must be positive) is broken into nx
  32  *              pieces of 24-bit integers in double precision format.
  33  *              x[i] will be the i-th 24 bit of x. The scaled exponent
  34  *              of x[0] is given in input parameter e0 (i.e., x[0]*2^e0
  35  *              match x's up to 24 bits.
  36  *
  37  *              Example of breaking a double positive z into x[0]+x[1]+x[2]:
  38  *                      e0 = ilogb(z)-23
  39  *                      z  = scalbn(z,-e0)
  40  *              for i = 0,1,2
  41  *                      x[i] = floor(z)
  42  *                      z    = (z-x[i])*2**24
  43  *
  44  *
  45  *      y[]     ouput result in an array of double precision numbers.
  46  *              The dimension of y[] is:
  47  *                      24-bit  precision       1
  48  *                      53-bit  precision       2
  49  *                      64-bit  precision       2
  50  *                      113-bit precision       3
  51  *              The actual value is the sum of them. Thus for 113-bit
  52  *              precison, one may have to do something like:
  53  *
  54  *              long double t,w,r_head, r_tail;
  55  *              t = (long double)y[2] + (long double)y[1];
  56  *              w = (long double)y[0];
  57  *              r_head = t+w;
  58  *              r_tail = w - (r_head - t);
  59  *
  60  *      e0      The exponent of x[0]
  61  *
  62  *      nx      dimension of x[]
  63  *
  64  *      prec    an integer indicating the precision:
  65  *                      0       24  bits (single)
  66  *                      1       53  bits (double)
  67  *                      2       64  bits (extended)
  68  *                      3       113 bits (quad)
  69  *
  70  *      ipio2[]
  71  *              integer array, contains the (24*i)-th to (24*i+23)-th
  72  *              bit of 2/pi after binary point. The corresponding
  73  *              floating value is
  74  *
  75  *                      ipio2[i] * 2^(-24(i+1)).
  76  *
  77  * External function:
  78  *      double scalbn(), floor();
  79  *
  80  *
  81  * Here is the description of some local variables:
  82  *
  83  *      jk      jk+1 is the initial number of terms of ipio2[] needed
  84  *              in the computation. The recommended value is 2,3,4,
  85  *              6 for single, double, extended,and quad.
  86  *
  87  *      jz      local integer variable indicating the number of
  88  *              terms of ipio2[] used.
  89  *
  90  *      jx      nx - 1
  91  *
  92  *      jv      index for pointing to the suitable ipio2[] for the
  93  *              computation. In general, we want
  94  *                      ( 2^e0*x[0] * ipio2[jv-1]*2^(-24jv) )/8
  95  *              is an integer. Thus
  96  *                      e0-3-24*jv >= 0 or (e0-3)/24 >= jv
  97  *              Hence jv = max(0,(e0-3)/24).
  98  *
  99  *      jp      jp+1 is the number of terms in PIo2[] needed, jp = jk.
 100  *
 101  *      q[]     double array with integral value, representing the
 102  *              24-bits chunk of the product of x and 2/pi.
 103  *
 104  *      q0      the corresponding exponent of q[0]. Note that the
 105  *              exponent for q[i] would be q0-24*i.
 106  *
 107  *      PIo2[]  double precision array, obtained by cutting pi/2
 108  *              into 24 bits chunks.
 109  *
 110  *      f[]     ipio2[] in floating point
 111  *
 112  *      iq[]    integer array by breaking up q[] in 24-bits chunk.
 113  *
 114  *      fq[]    final product of x*(2/pi) in fq[0],..,fq[jk]
 115  *
 116  *      ih      integer. If >0 it indicates q[] is >= 0.5, hence
 117  *              it also indicates the *sign* of the result.
 118  *
 119  */
 120
 121
 122 /*
 123  * Constants:
 124  * The hexadecimal values are the intended ones for the following
 125  * constants. The decimal values may be used, provided that the
 126  * compiler will convert from decimal to binary accurately enough
 127  * to produce the hexadecimal values shown.
 128  */
 129
 130 #include "fdlibm.h"
 131
 132 #ifdef __STDC__
 133 static const int init_jk[] = {2,3,4,6}; /* initial value for jk */
 134 #else
 135 static int init_jk[] = {2,3,4,6};
 136 #endif
 137
 138 #ifdef __STDC__
 139 static const double PIo2[] = {
 140 #else
 141 static double PIo2[] = {
 142 #endif
 143   1.57079625129699707031e+00, /* 0x3FF921FB, 0x40000000 */
 144   7.54978941586159635335e-08, /* 0x3E74442D, 0x00000000 */
 145   5.39030252995776476554e-15, /* 0x3CF84698, 0x80000000 */
 146   3.28200341580791294123e-22, /* 0x3B78CC51, 0x60000000 */
 147   1.27065575308067607349e-29, /* 0x39F01B83, 0x80000000 */
 148   1.22933308981111328932e-36, /* 0x387A2520, 0x40000000 */
 149   2.73370053816464559624e-44, /* 0x36E38222, 0x80000000 */
 150   2.16741683877804819444e-51, /* 0x3569F31D, 0x00000000 */
 151 };
 152
 153 #ifdef __STDC__
 154 static const double
 155 #else
 156 static double
 157 #endif
 158 zero   = 0.0,
 159 one    = 1.0,
 160 two24   =  1.67772160000000000000e+07, /* 0x41700000, 0x00000000 */
 161 twon24  =  5.96046447753906250000e-08; /* 0x3E700000, 0x00000000 */
 162
 163 #ifdef __STDC__
 164         int __kernel_rem_pio2(double *x, double *y, int e0, int nx, int prec, const int *ipio2)
 165 #else
 166         int __kernel_rem_pio2(x,y,e0,nx,prec,ipio2)
 167         double x[], y[]; int e0,nx,prec; int ipio2[];
 168 #endif
 169 {
 170         int jz,jx,jv,jp,jk,carry,n,iq[20],i,j,k,m,q0,ih;
 171         double z,fw,f[20],fq[20],q[20];
 172
 173     /* initialize jk*/
 174         jk = init_jk[prec];
 175         jp = jk;
 176
 177     /* determine jx,jv,q0, note that 3>q0 */
 178         jx =  nx-1;
 179         jv = (e0-3)/24; if(jv<0) jv=0;
 180         q0 =  e0-24*(jv+1);
 181
 182     /* set up f[0] to f[jx+jk] where f[jx+jk] = ipio2[jv+jk] */
 183         j = jv-jx; m = jx+jk;
 184         for(i=0;i<=m;i++,j++) f[i] = (j<0)? zero : (double) ipio2[j];
 185
 186     /* compute q[0],q[1],...q[jk] */
 187         for (i=0;i<=jk;i++) {
 188             for(j=0,fw=0.0;j<=jx;j++) fw += x[j]*f[jx+i-j]; q[i] = fw;
 189         }
 190
 191         jz = jk;
 192 recompute:
 193     /* distill q[] into iq[] reversingly */
 194         for(i=0,j=jz,z=q[jz];j>0;i++,j--) {
 195             fw    =  (double)((int)(twon24* z));
 196             iq[i] =  (int)(z-two24*fw);
 197             z     =  q[j-1]+fw;
 198         }
 199
 200     /* compute n */
 201         z  = scalbn(z,q0);              /* actual value of z */
 202         z -= 8.0*floor(z*0.125);                /* trim off integer >= 8 */
 203         n  = (int) z;
 204         z -= (double)n;
 205         ih = 0;
 206         if(q0>0) {      /* need iq[jz-1] to determine n */
 207             i  = (iq[jz-1]>>(24-q0)); n += i;
 208             iq[jz-1] -= i<<(24-q0);
 209             ih = iq[jz-1]>>(23-q0);
 210         }
 211         else if(q0==0) ih = iq[jz-1]>>23;
 212         else if(z>=0.5) ih=2;
 213
 214         if(ih>0) {      /* q > 0.5 */
 215             n += 1; carry = 0;
 216             for(i=0;i<jz ;i++) {        /* compute 1-q */
 217                 j = iq[i];
 218                 if(carry==0) {
 219                     if(j!=0) {
 220                         carry = 1; iq[i] = 0x1000000- j;
 221                     }
 222                 } else  iq[i] = 0xffffff - j;
 223             }
 224             if(q0>0) {          /* rare case: chance is 1 in 12 */
 225                 switch(q0) {
 226                 case 1:
 227                    iq[jz-1] &= 0x7fffff; break;
 228                 case 2:
 229                    iq[jz-1] &= 0x3fffff; break;
 230                 }
 231             }
 232             if(ih==2) {
 233                 z = one - z;
 234                 if(carry!=0) z -= scalbn(one,q0);
 235             }
 236         }
 237
 238     /* check if recomputation is needed */
 239         if(z==zero) {
 240             j = 0;
 241             for (i=jz-1;i>=jk;i--) j |= iq[i];
 242             if(j==0) { /* need recomputation */
 243                 for(k=1;iq[jk-k]==0;k++);   /* k = no. of terms needed */
 244
 245                 for(i=jz+1;i<=jz+k;i++) {   /* add q[jz+1] to q[jz+k] */
 246                     f[jx+i] = (double) ipio2[jv+i];
 247                     for(j=0,fw=0.0;j<=jx;j++) fw += x[j]*f[jx+i-j];
 248                     q[i] = fw;
 249                 }
 250                 jz += k;
 251                 goto recompute;
 252             }
 253         }
 254
 255     /* chop off zero terms */
 256         if(z==0.0) {
 257             jz -= 1; q0 -= 24;
 258             while(iq[jz]==0) { jz--; q0-=24;}
 259         } else { /* break z into 24-bit if necessary */
 260             z = scalbn(z,-q0);
 261             if(z>=two24) {
 262                 fw = (double)((int)(twon24*z));
 263                 iq[jz] = (int)(z-two24*fw);
 264                 jz += 1; q0 += 24;
 265                 iq[jz] = (int) fw;
 266             } else iq[jz] = (int) z ;
 267         }
 268
 269     /* convert integer "bit" chunk to floating-point value */
 270         fw = scalbn(one,q0);
 271         for(i=jz;i>=0;i--) {
 272             q[i] = fw*(double)iq[i]; fw*=twon24;
 273         }
 274
 275     /* compute PIo2[0,...,jp]*q[jz,...,0] */
 276         for(i=jz;i>=0;i--) {
 277             for(fw=0.0,k=0;k<=jp&&k<=jz-i;k++) fw += PIo2[k]*q[i+k];
 278             fq[jz-i] = fw;
 279         }
 280
 281     /* compress fq[] into y[] */
 282         switch(prec) {
 283             case 0:
 284                 fw = 0.0;
 285                 for (i=jz;i>=0;i--) fw += fq[i];
 286                 y[0] = (ih==0)? fw: -fw;
 287                 break;
 288             case 1:
 289             case 2:
 290                 fw = 0.0;
 291                 for (i=jz;i>=0;i--) fw += fq[i];
 292                 y[0] = (ih==0)? fw: -fw;
 293                 fw = fq[0]-fw;
 294                 for (i=1;i<=jz;i++) fw += fq[i];
 295                 y[1] = (ih==0)? fw: -fw;
 296                 break;
 297             case 3:     /* painful */
 298                 for (i=jz;i>0;i--) {
 299                     fw      = fq[i-1]+fq[i];
 300                     fq[i]  += fq[i-1]-fw;
 301                     fq[i-1] = fw;
 302                 }
 303                 for (i=jz;i>1;i--) {
 304                     fw      = fq[i-1]+fq[i];
 305                     fq[i]  += fq[i-1]-fw;
 306                     fq[i-1] = fw;
 307                 }
 308                 for (fw=0.0,i=jz;i>=2;i--) fw += fq[i];
 309                 if(ih==0) {
 310                     y[0] =  fq[0]; y[1] =  fq[1]; y[2] =  fw;
 311                 } else {
 312                     y[0] = -fq[0]; y[1] = -fq[1]; y[2] = -fw;
 313                 }
 314         }
 315         return n&7;
 316 }