libm/k_rem_pio2.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 /*
  13  * __kernel_rem_pio2(x,y,e0,nx,prec,ipio2)
  14  * double x[],y[]; int e0,nx,prec; int ipio2[];
  15  *
  16  * __kernel_rem_pio2 return the last three digits of N with
  17  *              y = x - N*pi/2
  18  * so that |y| < pi/2.
  19  *
  20  * The method is to compute the integer (mod 8) and fraction parts of
  21  * (2/pi)*x without doing the full multiplication. In general we
  22  * skip the part of the product that are known to be a huge integer (
  23  * more accurately, = 0 mod 8 ). Thus the number of operations are
  24  * independent of the exponent of the input.
  25  *
  26  * (2/pi) is represented by an array of 24-bit integers in ipio2[].
  27  *
  28  * Input parameters:
  29  *      x[]     The input value (must be positive) is broken into nx
  30  *              pieces of 24-bit integers in double precision format.
  31  *              x[i] will be the i-th 24 bit of x. The scaled exponent
  32  *              of x[0] is given in input parameter e0 (i.e., x[0]*2^e0
  33  *              match x's up to 24 bits.
  34  *
  35  *              Example of breaking a double positive z into x[0]+x[1]+x[2]:
  36  *                      e0 = ilogb(z)-23
  37  *                      z  = scalbn(z,-e0)
  38  *              for i = 0,1,2
  39  *                      x[i] = floor(z)
  40  *                      z    = (z-x[i])*2**24
  41  *
  42  *
  43  *      y[]     ouput result in an array of double precision numbers.
  44  *              The dimension of y[] is:
  45  *                      24-bit  precision       1
  46  *                      53-bit  precision       2
  47  *                      64-bit  precision       2
  48  *                      113-bit precision       3
  49  *              The actual value is the sum of them. Thus for 113-bit
  50  *              precison, one may have to do something like:
  51  *
  52  *              long double t,w,r_head, r_tail;
  53  *              t = (long double)y[2] + (long double)y[1];
  54  *              w = (long double)y[0];
  55  *              r_head = t+w;
  56  *              r_tail = w - (r_head - t);
  57  *
  58  *      e0      The exponent of x[0]
  59  *
  60  *      nx      dimension of x[]
  61  *
  62  *      prec    an integer indicating the precision:
  63  *                      0       24  bits (single)
  64  *                      1       53  bits (double)
  65  *                      2       64  bits (extended)
  66  *                      3       113 bits (quad)
  67  *
  68  *      ipio2[]
  69  *              integer array, contains the (24*i)-th to (24*i+23)-th
  70  *              bit of 2/pi after binary point. The corresponding
  71  *              floating value is
  72  *
  73  *                      ipio2[i] * 2^(-24(i+1)).
  74  *
  75  * External function:
  76  *      double scalbn(), floor();
  77  *
  78  *
  79  * Here is the description of some local variables:
  80  *
  81  *      jk      jk+1 is the initial number of terms of ipio2[] needed
  82  *              in the computation. The recommended value is 2,3,4,
  83  *              6 for single, double, extended,and quad.
  84  *
  85  *      jz      local integer variable indicating the number of
  86  *              terms of ipio2[] used.
  87  *
  88  *      jx      nx - 1
  89  *
  90  *      jv      index for pointing to the suitable ipio2[] for the
  91  *              computation. In general, we want
  92  *                      ( 2^e0*x[0] * ipio2[jv-1]*2^(-24jv) )/8
  93  *              is an integer. Thus
  94  *                      e0-3-24*jv >= 0 or (e0-3)/24 >= jv
  95  *              Hence jv = max(0,(e0-3)/24).
  96  *
  97  *      jp      jp+1 is the number of terms in PIo2[] needed, jp = jk.
  98  *
  99  *      q[]     double array with integral value, representing the
 100  *              24-bits chunk of the product of x and 2/pi.
 101  *
 102  *      q0      the corresponding exponent of q[0]. Note that the
 103  *              exponent for q[i] would be q0-24*i.
 104  *
 105  *      PIo2[]  double precision array, obtained by cutting pi/2
 106  *              into 24 bits chunks.
 107  *
 108  *      f[]     ipio2[] in floating point
 109  *
 110  *      iq[]    integer array by breaking up q[] in 24-bits chunk.
 111  *
 112  *      fq[]    final product of x*(2/pi) in fq[0],..,fq[jk]
 113  *
 114  *      ih      integer. If >0 it indicates q[] is >= 0.5, hence
 115  *              it also indicates the *sign* of the result.
 116  *
 117  */
 118
 119
 120 /*
 121  * Constants:
 122  * The hexadecimal values are the intended ones for the following
 123  * constants. The decimal values may be used, provided that the
 124  * compiler will convert from decimal to binary accurately enough
 125  * to produce the hexadecimal values shown.
 126  */
 127
 128 #include "math.h"
 129 #include "math_private.h"
 130
 131 static const int init_jk[] = {2,3,4,6}; /* initial value for jk */
 132
 133 static const double PIo2[] = {
 134   1.57079625129699707031e+00, /* 0x3FF921FB, 0x40000000 */
 135   7.54978941586159635335e-08, /* 0x3E74442D, 0x00000000 */
 136   5.39030252995776476554e-15, /* 0x3CF84698, 0x80000000 */
 137   3.28200341580791294123e-22, /* 0x3B78CC51, 0x60000000 */
 138   1.27065575308067607349e-29, /* 0x39F01B83, 0x80000000 */
 139   1.22933308981111328932e-36, /* 0x387A2520, 0x40000000 */
 140   2.73370053816464559624e-44, /* 0x36E38222, 0x80000000 */
 141   2.16741683877804819444e-51, /* 0x3569F31D, 0x00000000 */
 142 };
 143
 144 static const double
 145 zero   = 0.0,
 146 one    = 1.0,
 147 two24   =  1.67772160000000000000e+07, /* 0x41700000, 0x00000000 */
 148 twon24  =  5.96046447753906250000e-08; /* 0x3E700000, 0x00000000 */
 149
 150 int __kernel_rem_pio2(double *x, double *y, int e0, int nx, int prec, const int32_t *ipio2)
 151 {
 152         int32_t jz,jx,jv,jp,jk,carry,n,iq[20],i,j,k,m,q0,ih;
 153         double z,fw,f[20],fq[20],q[20];
 154
 155     /* initialize jk*/
 156         jk = init_jk[prec];
 157         jp = jk;
 158
 159     /* determine jx,jv,q0, note that 3>q0 */
 160         jx =  nx-1;
 161         jv = (e0-3)/24; if(jv<0) jv=0;
 162         q0 =  e0-24*(jv+1);
 163
 164     /* set up f[0] to f[jx+jk] where f[jx+jk] = ipio2[jv+jk] */
 165         j = jv-jx; m = jx+jk;
 166         for(i=0;i<=m;i++,j++) f[i] = (j<0)? zero : (double) ipio2[j];
 167
 168     /* compute q[0],q[1],...q[jk] */
 169         for (i=0;i<=jk;i++) {
 170             for(j=0,fw=0.0;j<=jx;j++) fw += x[j]*f[jx+i-j];
 171             q[i] = fw;
 172         }
 173
 174         jz = jk;
 175 recompute:
 176     /* distill q[] into iq[] reversingly */
 177         for(i=0,j=jz,z=q[jz];j>0;i++,j--) {
 178             fw    =  (double)((int32_t)(twon24* z));
 179             iq[i] =  (int32_t)(z-two24*fw);
 180             z     =  q[j-1]+fw;
 181         }
 182
 183     /* compute n */
 184         z  = scalbn(z,q0);              /* actual value of z */
 185         z -= 8.0*floor(z*0.125);                /* trim off integer >= 8 */
 186         n  = (int32_t) z;
 187         z -= (double)n;
 188         ih = 0;
 189         if(q0>0) {      /* need iq[jz-1] to determine n */
 190             i  = (iq[jz-1]>>(24-q0)); n += i;
 191             iq[jz-1] -= i<<(24-q0);
 192             ih = iq[jz-1]>>(23-q0);
 193         }
 194         else if(q0==0) ih = iq[jz-1]>>23;
 195         else if(z>=0.5) ih=2;
 196
 197         if(ih>0) {      /* q > 0.5 */
 198             n += 1; carry = 0;
 199             for(i=0;i<jz ;i++) {        /* compute 1-q */
 200                 j = iq[i];
 201                 if(carry==0) {
 202                     if(j!=0) {
 203                         carry = 1; iq[i] = 0x1000000- j;
 204                     }
 205                 } else  iq[i] = 0xffffff - j;
 206             }
 207             if(q0>0) {          /* rare case: chance is 1 in 12 */
 208                 switch(q0) {
 209                 case 1:
 210                    iq[jz-1] &= 0x7fffff; break;
 211                 case 2:
 212                    iq[jz-1] &= 0x3fffff; break;
 213                 }
 214             }
 215             if(ih==2) {
 216                 z = one - z;
 217                 if(carry!=0) z -= scalbn(one,q0);
 218             }
 219         }
 220
 221     /* check if recomputation is needed */
 222         if(z==zero) {
 223             j = 0;
 224             for (i=jz-1;i>=jk;i--) j |= iq[i];
 225             if(j==0) { /* need recomputation */
 226                 for(k=1;iq[jk-k]==0;k++);   /* k = no. of terms needed */
 227
 228                 for(i=jz+1;i<=jz+k;i++) {   /* add q[jz+1] to q[jz+k] */
 229                     f[jx+i] = (double) ipio2[jv+i];
 230                     for(j=0,fw=0.0;j<=jx;j++) fw += x[j]*f[jx+i-j];
 231                     q[i] = fw;
 232                 }
 233                 jz += k;
 234                 goto recompute;
 235             }
 236         }
 237
 238     /* chop off zero terms */
 239         if(z==0.0) {
 240             jz -= 1; q0 -= 24;
 241             while(iq[jz]==0) { jz--; q0-=24;}
 242         } else { /* break z into 24-bit if necessary */
 243             z = scalbn(z,-q0);
 244             if(z>=two24) {
 245                 fw = (double)((int32_t)(twon24*z));
 246                 iq[jz] = (int32_t)(z-two24*fw);
 247                 jz += 1; q0 += 24;
 248                 iq[jz] = (int32_t) fw;
 249             } else iq[jz] = (int32_t) z ;
 250         }
 251
 252     /* convert integer "bit" chunk to floating-point value */
 253         fw = scalbn(one,q0);
 254         for(i=jz;i>=0;i--) {
 255             q[i] = fw*(double)iq[i]; fw*=twon24;
 256         }
 257
 258     /* compute PIo2[0,...,jp]*q[jz,...,0] */
 259         for(i=jz;i>=0;i--) {
 260             for(fw=0.0,k=0;k<=jp&&k<=jz-i;k++) fw += PIo2[k]*q[i+k];
 261             fq[jz-i] = fw;
 262         }
 263
 264     /* compress fq[] into y[] */
 265         switch(prec) {
 266             case 0:
 267                 fw = 0.0;
 268                 for (i=jz;i>=0;i--) fw += fq[i];
 269                 y[0] = (ih==0)? fw: -fw;
 270                 break;
 271             case 1:
 272             case 2:
 273                 fw = 0.0;
 274                 for (i=jz;i>=0;i--) fw += fq[i];
 275                 y[0] = (ih==0)? fw: -fw;
 276                 fw = fq[0]-fw;
 277                 for (i=1;i<=jz;i++) fw += fq[i];
 278                 y[1] = (ih==0)? fw: -fw;
 279                 break;
 280             case 3:     /* painful */
 281                 for (i=jz;i>0;i--) {
 282                     fw      = fq[i-1]+fq[i];
 283                     fq[i]  += fq[i-1]-fw;
 284                     fq[i-1] = fw;
 285                 }
 286                 for (i=jz;i>1;i--) {
 287                     fw      = fq[i-1]+fq[i];
 288                     fq[i]  += fq[i-1]-fw;
 289                     fq[i-1] = fw;
 290                 }
 291                 for (fw=0.0,i=jz;i>=2;i--) fw += fq[i];
 292                 if(ih==0) {
 293                     y[0] =  fq[0]; y[1] =  fq[1]; y[2] =  fw;
 294                 } else {
 295                     y[0] = -fq[0]; y[1] = -fq[1]; y[2] = -fw;
 296                 }
 297         }
 298         return n&7;
 299 }