sysdeps/libm-ieee754/k_rem_pio2.c

   1 /* @(#)k_rem_pio2.c 5.1 93/09/24 */
   2 /*
   3  * ====================================================
   4  * Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved.
   5  *
   6  * Developed at SunPro, a Sun Microsystems, Inc. business.
   7  * Permission to use, copy, modify, and distribute this
   8  * software is freely granted, provided that this notice
   9  * is preserved.
  10  * ====================================================
  11  */
  12
  13 #if defined(LIBM_SCCS) && !defined(lint)
  14 static char rcsid[] = "$NetBSD: k_rem_pio2.c,v 1.7 1995/05/10 20:46:25 jtc Exp $";
  15 #endif
  16
  17 /*
  18  * __kernel_rem_pio2(x,y,e0,nx,prec,ipio2)
  19  * double x[],y[]; int e0,nx,prec; int ipio2[];
  20  *
  21  * __kernel_rem_pio2 return the last three digits of N with
  22  *              y = x - N*pi/2
  23  * so that |y| < pi/2.
  24  *
  25  * The method is to compute the integer (mod 8) and fraction parts of
  26  * (2/pi)*x without doing the full multiplication. In general we
  27  * skip the part of the product that are known to be a huge integer (
  28  * more accurately, = 0 mod 8 ). Thus the number of operations are
  29  * independent of the exponent of the input.
  30  *
  31  * (2/pi) is represented by an array of 24-bit integers in ipio2[].
  32  *
  33  * Input parameters:
  34  *      x[]     The input value (must be positive) is broken into nx
  35  *              pieces of 24-bit integers in double precision format.
  36  *              x[i] will be the i-th 24 bit of x. The scaled exponent
  37  *              of x[0] is given in input parameter e0 (i.e., x[0]*2^e0
  38  *              match x's up to 24 bits.
  39  *
  40  *              Example of breaking a double positive z into x[0]+x[1]+x[2]:
  41  *                      e0 = ilogb(z)-23
  42  *                      z  = scalbn(z,-e0)
  43  *              for i = 0,1,2
  44  *                      x[i] = floor(z)
  45  *                      z    = (z-x[i])*2**24
  46  *
  47  *
  48  *      y[]     ouput result in an array of double precision numbers.
  49  *              The dimension of y[] is:
  50  *                      24-bit  precision       1
  51  *                      53-bit  precision       2
  52  *                      64-bit  precision       2
  53  *                      113-bit precision       3
  54  *              The actual value is the sum of them. Thus for 113-bit
  55  *              precison, one may have to do something like:
  56  *
  57  *              long double t,w,r_head, r_tail;
  58  *              t = (long double)y[2] + (long double)y[1];
  59  *              w = (long double)y[0];
  60  *              r_head = t+w;
  61  *              r_tail = w - (r_head - t);
  62  *
  63  *      e0      The exponent of x[0]
  64  *
  65  *      nx      dimension of x[]
  66  *
  67  *      prec    an integer indicating the precision:
  68  *                      0       24  bits (single)
  69  *                      1       53  bits (double)
  70  *                      2       64  bits (extended)
  71  *                      3       113 bits (quad)
  72  *
  73  *      ipio2[]
  74  *              integer array, contains the (24*i)-th to (24*i+23)-th
  75  *              bit of 2/pi after binary point. The corresponding
  76  *              floating value is
  77  *
  78  *                      ipio2[i] * 2^(-24(i+1)).
  79  *
  80  * External function:
  81  *      double scalbn(), floor();
  82  *
  83  *
  84  * Here is the description of some local variables:
  85  *
  86  *      jk      jk+1 is the initial number of terms of ipio2[] needed
  87  *              in the computation. The recommended value is 2,3,4,
  88  *              6 for single, double, extended,and quad.
  89  *
  90  *      jz      local integer variable indicating the number of
  91  *              terms of ipio2[] used.
  92  *
  93  *      jx      nx - 1
  94  *
  95  *      jv      index for pointing to the suitable ipio2[] for the
  96  *              computation. In general, we want
  97  *                      ( 2^e0*x[0] * ipio2[jv-1]*2^(-24jv) )/8
  98  *              is an integer. Thus
  99  *                      e0-3-24*jv >= 0 or (e0-3)/24 >= jv
 100  *              Hence jv = max(0,(e0-3)/24).
 101  *
 102  *      jp      jp+1 is the number of terms in PIo2[] needed, jp = jk.
 103  *
 104  *      q[]     double array with integral value, representing the
 105  *              24-bits chunk of the product of x and 2/pi.
 106  *
 107  *      q0      the corresponding exponent of q[0]. Note that the
 108  *              exponent for q[i] would be q0-24*i.
 109  *
 110  *      PIo2[]  double precision array, obtained by cutting pi/2
 111  *              into 24 bits chunks.
 112  *
 113  *      f[]     ipio2[] in floating point
 114  *
 115  *      iq[]    integer array by breaking up q[] in 24-bits chunk.
 116  *
 117  *      fq[]    final product of x*(2/pi) in fq[0],..,fq[jk]
 118  *
 119  *      ih      integer. If >0 it indicates q[] is >= 0.5, hence
 120  *              it also indicates the *sign* of the result.
 121  *
 122  */
 123
 124
 125 /*
 126  * Constants:
 127  * The hexadecimal values are the intended ones for the following
 128  * constants. The decimal values may be used, provided that the
 129  * compiler will convert from decimal to binary accurately enough
 130  * to produce the hexadecimal values shown.
 131  */
 132
 133 #include "math.h"
 134 #include "math_private.h"
 135
 136 #ifdef __STDC__
 137 static const int init_jk[] = {2,3,4,6}; /* initial value for jk */
 138 #else
 139 static int init_jk[] = {2,3,4,6};
 140 #endif
 141
 142 #ifdef __STDC__
 143 static const double PIo2[] = {
 144 #else
 145 static double PIo2[] = {
 146 #endif
 147   1.57079625129699707031e+00, /* 0x3FF921FB, 0x40000000 */
 148   7.54978941586159635335e-08, /* 0x3E74442D, 0x00000000 */
 149   5.39030252995776476554e-15, /* 0x3CF84698, 0x80000000 */
 150   3.28200341580791294123e-22, /* 0x3B78CC51, 0x60000000 */
 151   1.27065575308067607349e-29, /* 0x39F01B83, 0x80000000 */
 152   1.22933308981111328932e-36, /* 0x387A2520, 0x40000000 */
 153   2.73370053816464559624e-44, /* 0x36E38222, 0x80000000 */
 154   2.16741683877804819444e-51, /* 0x3569F31D, 0x00000000 */
 155 };
 156
 157 #ifdef __STDC__
 158 static const double
 159 #else
 160 static double
 161 #endif
 162 zero   = 0.0,
 163 one    = 1.0,
 164 two24   =  1.67772160000000000000e+07, /* 0x41700000, 0x00000000 */
 165 twon24  =  5.96046447753906250000e-08; /* 0x3E700000, 0x00000000 */
 166
 167 #ifdef __STDC__
 168         int __kernel_rem_pio2(double *x, double *y, int e0, int nx, int prec, const int32_t *ipio2)
 169 #else
 170         int __kernel_rem_pio2(x,y,e0,nx,prec,ipio2)
 171         double x[], y[]; int e0,nx,prec; int32_t ipio2[];
 172 #endif
 173 {
 174         int32_t jz,jx,jv,jp,jk,carry,n,iq[20],i,j,k,m,q0,ih;
 175         double z,fw,f[20],fq[20],q[20];
 176
 177     /* initialize jk*/
 178         jk = init_jk[prec];
 179         jp = jk;
 180
 181     /* determine jx,jv,q0, note that 3>q0 */
 182         jx =  nx-1;
 183         jv = (e0-3)/24; if(jv<0) jv=0;
 184         q0 =  e0-24*(jv+1);
 185
 186     /* set up f[0] to f[jx+jk] where f[jx+jk] = ipio2[jv+jk] */
 187         j = jv-jx; m = jx+jk;
 188         for(i=0;i<=m;i++,j++) f[i] = (j<0)? zero : (double) ipio2[j];
 189
 190     /* compute q[0],q[1],...q[jk] */
 191         for (i=0;i<=jk;i++) {
 192             for(j=0,fw=0.0;j<=jx;j++) fw += x[j]*f[jx+i-j]; q[i] = fw;
 193         }
 194
 195         jz = jk;
 196 recompute:
 197     /* distill q[] into iq[] reversingly */
 198         for(i=0,j=jz,z=q[jz];j>0;i++,j--) {
 199             fw    =  (double)((int32_t)(twon24* z));
 200             iq[i] =  (int32_t)(z-two24*fw);
 201             z     =  q[j-1]+fw;
 202         }
 203
 204     /* compute n */
 205         z  = __scalbn(z,q0);            /* actual value of z */
 206         z -= 8.0*__floor(z*0.125);              /* trim off integer >= 8 */
 207         n  = (int32_t) z;
 208         z -= (double)n;
 209         ih = 0;
 210         if(q0>0) {      /* need iq[jz-1] to determine n */
 211             i  = (iq[jz-1]>>(24-q0)); n += i;
 212             iq[jz-1] -= i<<(24-q0);
 213             ih = iq[jz-1]>>(23-q0);
 214         }
 215         else if(q0==0) ih = iq[jz-1]>>23;
 216         else if(z>=0.5) ih=2;
 217
 218         if(ih>0) {      /* q > 0.5 */
 219             n += 1; carry = 0;
 220             for(i=0;i<jz ;i++) {        /* compute 1-q */
 221                 j = iq[i];
 222                 if(carry==0) {
 223                     if(j!=0) {
 224                         carry = 1; iq[i] = 0x1000000- j;
 225                     }
 226                 } else  iq[i] = 0xffffff - j;
 227             }
 228             if(q0>0) {          /* rare case: chance is 1 in 12 */
 229                 switch(q0) {
 230                 case 1:
 231                    iq[jz-1] &= 0x7fffff; break;
 232                 case 2:
 233                    iq[jz-1] &= 0x3fffff; break;
 234                 }
 235             }
 236             if(ih==2) {
 237                 z = one - z;
 238                 if(carry!=0) z -= __scalbn(one,q0);
 239             }
 240         }
 241
 242     /* check if recomputation is needed */
 243         if(z==zero) {
 244             j = 0;
 245             for (i=jz-1;i>=jk;i--) j |= iq[i];
 246             if(j==0) { /* need recomputation */
 247                 for(k=1;iq[jk-k]==0;k++);   /* k = no. of terms needed */
 248
 249                 for(i=jz+1;i<=jz+k;i++) {   /* add q[jz+1] to q[jz+k] */
 250                     f[jx+i] = (double) ipio2[jv+i];
 251                     for(j=0,fw=0.0;j<=jx;j++) fw += x[j]*f[jx+i-j];
 252                     q[i] = fw;
 253                 }
 254                 jz += k;
 255                 goto recompute;
 256             }
 257         }
 258
 259     /* chop off zero terms */
 260         if(z==0.0) {
 261             jz -= 1; q0 -= 24;
 262             while(iq[jz]==0) { jz--; q0-=24;}
 263         } else { /* break z into 24-bit if necessary */
 264             z = __scalbn(z,-q0);
 265             if(z>=two24) {
 266                 fw = (double)((int32_t)(twon24*z));
 267                 iq[jz] = (int32_t)(z-two24*fw);
 268                 jz += 1; q0 += 24;
 269                 iq[jz] = (int32_t) fw;
 270             } else iq[jz] = (int32_t) z ;
 271         }
 272
 273     /* convert integer "bit" chunk to floating-point value */
 274         fw = __scalbn(one,q0);
 275         for(i=jz;i>=0;i--) {
 276             q[i] = fw*(double)iq[i]; fw*=twon24;
 277         }
 278
 279     /* compute PIo2[0,...,jp]*q[jz,...,0] */
 280         for(i=jz;i>=0;i--) {
 281             for(fw=0.0,k=0;k<=jp&&k<=jz-i;k++) fw += PIo2[k]*q[i+k];
 282             fq[jz-i] = fw;
 283         }
 284
 285     /* compress fq[] into y[] */
 286         switch(prec) {
 287             case 0:
 288                 fw = 0.0;
 289                 for (i=jz;i>=0;i--) fw += fq[i];
 290                 y[0] = (ih==0)? fw: -fw;
 291                 break;
 292             case 1:
 293             case 2:
 294                 fw = 0.0;
 295                 for (i=jz;i>=0;i--) fw += fq[i];
 296                 y[0] = (ih==0)? fw: -fw;
 297                 fw = fq[0]-fw;
 298                 for (i=1;i<=jz;i++) fw += fq[i];
 299                 y[1] = (ih==0)? fw: -fw;
 300                 break;
 301             case 3:     /* painful */
 302                 for (i=jz;i>0;i--) {
 303                     fw      = fq[i-1]+fq[i];
 304                     fq[i]  += fq[i-1]-fw;
 305                     fq[i-1] = fw;
 306                 }
 307                 for (i=jz;i>1;i--) {
 308                     fw      = fq[i-1]+fq[i];
 309                     fq[i]  += fq[i-1]-fw;
 310                     fq[i-1] = fw;
 311                 }
 312                 for (fw=0.0,i=jz;i>=2;i--) fw += fq[i];
 313                 if(ih==0) {
 314                     y[0] =  fq[0]; y[1] =  fq[1]; y[2] =  fw;
 315                 } else {
 316                     y[0] = -fq[0]; y[1] = -fq[1]; y[2] = -fw;
 317                 }
 318         }
 319         return n&7;
 320 }