compiler/mlib/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 #ifndef lint
  15 static char rcsid[] = "$FreeBSD: src/lib/msun/src/k_rem_pio2.c,v 1.7 2005/02/04 18:26:06 das Exp $";
  16 #endif
  17
  18 /*
  19  * __kernel_rem_pio2(x,y,e0,nx,prec,ipio2)
  20  * double x[],y[]; int e0,nx,prec; int ipio2[];
  21  *
  22  * __kernel_rem_pio2 return the last three digits of N with
  23  *              y = x - N*pi/2
  24  * so that |y| < pi/2.
  25  *
  26  * The method is to compute the integer (mod 8) and fraction parts of
  27  * (2/pi)*x without doing the full multiplication. In general we
  28  * skip the part of the product that are known to be a huge integer (
  29  * more accurately, = 0 mod 8 ). Thus the number of operations are
  30  * independent of the exponent of the input.
  31  *
  32  * (2/pi) is represented by an array of 24-bit integers in ipio2[].
  33  *
  34  * Input parameters:
  35  *      x[]     The input value (must be positive) is broken into nx
  36  *              pieces of 24-bit integers in double precision format.
  37  *              x[i] will be the i-th 24 bit of x. The scaled exponent
  38  *              of x[0] is given in input parameter e0 (i.e., x[0]*2^e0
  39  *              match x's up to 24 bits.
  40  *
  41  *              Example of breaking a double positive z into x[0]+x[1]+x[2]:
  42  *                      e0 = ilogb(z)-23
  43  *                      z  = scalbn(z,-e0)
  44  *              for i = 0,1,2
  45  *                      x[i] = floor(z)
  46  *                      z    = (z-x[i])*2**24
  47  *
  48  *
  49  *      y[]     ouput result in an array of double precision numbers.
  50  *              The dimension of y[] is:
  51  *                      24-bit  precision       1
  52  *                      53-bit  precision       2
  53  *                      64-bit  precision       2
  54  *                      113-bit precision       3
  55  *              The actual value is the sum of them. Thus for 113-bit
  56  *              precison, one may have to do something like:
  57  *
  58  *              long double t,w,r_head, r_tail;
  59  *              t = (long double)y[2] + (long double)y[1];
  60  *              w = (long double)y[0];
  61  *              r_head = t+w;
  62  *              r_tail = w - (r_head - t);
  63  *
  64  *      e0      The exponent of x[0]
  65  *
  66  *      nx      dimension of x[]
  67  *
  68  *      prec    an integer indicating the precision:
  69  *                      0       24  bits (single)
  70  *                      1       53  bits (double)
  71  *                      2       64  bits (extended)
  72  *                      3       113 bits (quad)
  73  *
  74  *      ipio2[]
  75  *              integer array, contains the (24*i)-th to (24*i+23)-th
  76  *              bit of 2/pi after binary point. The corresponding
  77  *              floating value is
  78  *
  79  *                      ipio2[i] * 2^(-24(i+1)).
  80  *
  81  * External function:
  82  *      double scalbn(), floor();
  83  *
  84  *
  85  * Here is the description of some local variables:
  86  *
  87  *      jk      jk+1 is the initial number of terms of ipio2[] needed
  88  *              in the computation. The recommended value is 2,3,4,
  89  *              6 for single, double, extended,and quad.
  90  *
  91  *      jz      local integer variable indicating the number of
  92  *              terms of ipio2[] used.
  93  *
  94  *      jx      nx - 1
  95  *
  96  *      jv      index for pointing to the suitable ipio2[] for the
  97  *              computation. In general, we want
  98  *                      ( 2^e0*x[0] * ipio2[jv-1]*2^(-24jv) )/8
  99  *              is an integer. Thus
 100  *                      e0-3-24*jv >= 0 or (e0-3)/24 >= jv
 101  *              Hence jv = max(0,(e0-3)/24).
 102  *
 103  *      jp      jp+1 is the number of terms in PIo2[] needed, jp = jk.
 104  *
 105  *      q[]     double array with integral value, representing the
 106  *              24-bits chunk of the product of x and 2/pi.
 107  *
 108  *      q0      the corresponding exponent of q[0]. Note that the
 109  *              exponent for q[i] would be q0-24*i.
 110  *
 111  *      PIo2[]  double precision array, obtained by cutting pi/2
 112  *              into 24 bits chunks.
 113  *
 114  *      f[]     ipio2[] in floating point
 115  *
 116  *      iq[]    integer array by breaking up q[] in 24-bits chunk.
 117  *
 118  *      fq[]    final product of x*(2/pi) in fq[0],..,fq[jk]
 119  *
 120  *      ih      integer. If >0 it indicates q[] is >= 0.5, hence
 121  *              it also indicates the *sign* of the result.
 122  *
 123  */
 124
 125
 126 /*
 127  * Constants:
 128  * The hexadecimal values are the intended ones for the following
 129  * constants. The decimal values may be used, provided that the
 130  * compiler will convert from decimal to binary accurately enough
 131  * to produce the hexadecimal values shown.
 132  */
 133
 134 #include "math.h"
 135 #include "math_private.h"
 136
 137 static const int init_jk[] = {2,3,4,6}; /* initial value for jk */
 138
 139 static const double PIo2[] = {
 140   1.57079625129699707031e+00, /* 0x3FF921FB, 0x40000000 */
 141   7.54978941586159635335e-08, /* 0x3E74442D, 0x00000000 */
 142   5.39030252995776476554e-15, /* 0x3CF84698, 0x80000000 */
 143   3.28200341580791294123e-22, /* 0x3B78CC51, 0x60000000 */
 144   1.27065575308067607349e-29, /* 0x39F01B83, 0x80000000 */
 145   1.22933308981111328932e-36, /* 0x387A2520, 0x40000000 */
 146   2.73370053816464559624e-44, /* 0x36E38222, 0x80000000 */
 147   2.16741683877804819444e-51, /* 0x3569F31D, 0x00000000 */
 148 };
 149
 150 static const double
 151 zero   = 0.0,
 152 one    = 1.0,
 153 two24   =  1.67772160000000000000e+07, /* 0x41700000, 0x00000000 */
 154 twon24  =  5.96046447753906250000e-08; /* 0x3E700000, 0x00000000 */
 155
 156         int __kernel_rem_pio2(double *x, double *y, int e0, int nx, int prec, const int32_t *ipio2)
 157 {
 158         int32_t jz,jx,jv,jp,jk,carry,n,iq[20],i,j,k,m,q0,ih;
 159         double z,fw,f[20],fq[20],q[20];
 160
 161     /* initialize jk*/
 162         jk = init_jk[prec];
 163         jp = jk;
 164
 165     /* determine jx,jv,q0, note that 3>q0 */
 166         jx =  nx-1;
 167         jv = (e0-3)/24; if(jv<0) jv=0;
 168         q0 =  e0-24*(jv+1);
 169
 170     /* set up f[0] to f[jx+jk] where f[jx+jk] = ipio2[jv+jk] */
 171         j = jv-jx; m = jx+jk;
 172         for(i=0;i<=m;i++,j++) f[i] = (j<0)? zero : (double) ipio2[j];
 173
 174     /* compute q[0],q[1],...q[jk] */
 175         for (i=0;i<=jk;i++) {
 176             for(j=0,fw=0.0;j<=jx;j++) fw += x[j]*f[jx+i-j]; q[i] = fw;
 177         }
 178
 179         jz = jk;
 180 recompute:
 181     /* distill q[] into iq[] reversingly */
 182         for(i=0,j=jz,z=q[jz];j>0;i++,j--) {
 183             fw    =  (double)((int32_t)(twon24* z));
 184             iq[i] =  (int32_t)(z-two24*fw);
 185             z     =  q[j-1]+fw;
 186         }
 187
 188     /* compute n */
 189         z  = scalbn(z,q0);              /* actual value of z */
 190         z -= 8.0*floor(z*0.125);                /* trim off integer >= 8 */
 191         n  = (int32_t) z;
 192         z -= (double)n;
 193         ih = 0;
 194         if(q0>0) {      /* need iq[jz-1] to determine n */
 195             i  = (iq[jz-1]>>(24-q0)); n += i;
 196             iq[jz-1] -= i<<(24-q0);
 197             ih = iq[jz-1]>>(23-q0);
 198         }
 199         else if(q0==0) ih = iq[jz-1]>>23;
 200         else if(z>=0.5) ih=2;
 201
 202         if(ih>0) {      /* q > 0.5 */
 203             n += 1; carry = 0;
 204             for(i=0;i<jz ;i++) {        /* compute 1-q */
 205                 j = iq[i];
 206                 if(carry==0) {
 207                     if(j!=0) {
 208                         carry = 1; iq[i] = 0x1000000- j;
 209                     }
 210                 } else  iq[i] = 0xffffff - j;
 211             }
 212             if(q0>0) {          /* rare case: chance is 1 in 12 */
 213                 switch(q0) {
 214                 case 1:
 215                    iq[jz-1] &= 0x7fffff; break;
 216                 case 2:
 217                    iq[jz-1] &= 0x3fffff; break;
 218                 }
 219             }
 220             if(ih==2) {
 221                 z = one - z;
 222                 if(carry!=0) z -= scalbn(one,q0);
 223             }
 224         }
 225
 226     /* check if recomputation is needed */
 227         if(z==zero) {
 228             j = 0;
 229             for (i=jz-1;i>=jk;i--) j |= iq[i];
 230             if(j==0) { /* need recomputation */
 231                 for(k=1;iq[jk-k]==0;k++);   /* k = no. of terms needed */
 232
 233                 for(i=jz+1;i<=jz+k;i++) {   /* add q[jz+1] to q[jz+k] */
 234                     f[jx+i] = (double) ipio2[jv+i];
 235                     for(j=0,fw=0.0;j<=jx;j++) fw += x[j]*f[jx+i-j];
 236                     q[i] = fw;
 237                 }
 238                 jz += k;
 239                 goto recompute;
 240             }
 241         }
 242
 243     /* chop off zero terms */
 244         if(z==0.0) {
 245             jz -= 1; q0 -= 24;
 246             while(iq[jz]==0) { jz--; q0-=24;}
 247         } else { /* break z into 24-bit if necessary */
 248             z = scalbn(z,-q0);
 249             if(z>=two24) {
 250                 fw = (double)((int32_t)(twon24*z));
 251                 iq[jz] = (int32_t)(z-two24*fw);
 252                 jz += 1; q0 += 24;
 253                 iq[jz] = (int32_t) fw;
 254             } else iq[jz] = (int32_t) z ;
 255         }
 256
 257     /* convert integer "bit" chunk to floating-point value */
 258         fw = scalbn(one,q0);
 259         for(i=jz;i>=0;i--) {
 260             q[i] = fw*(double)iq[i]; fw*=twon24;
 261         }
 262
 263     /* compute PIo2[0,...,jp]*q[jz,...,0] */
 264         for(i=jz;i>=0;i--) {
 265             for(fw=0.0,k=0;k<=jp&&k<=jz-i;k++) fw += PIo2[k]*q[i+k];
 266             fq[jz-i] = fw;
 267         }
 268
 269     /* compress fq[] into y[] */
 270         switch(prec) {
 271             case 0:
 272                 fw = 0.0;
 273                 for (i=jz;i>=0;i--) fw += fq[i];
 274                 y[0] = (ih==0)? fw: -fw;
 275                 break;
 276             case 1:
 277             case 2:
 278                 fw = 0.0;
 279                 for (i=jz;i>=0;i--) fw += fq[i];
 280                 y[0] = (ih==0)? fw: -fw;
 281                 fw = fq[0]-fw;
 282                 for (i=1;i<=jz;i++) fw += fq[i];
 283                 y[1] = (ih==0)? fw: -fw;
 284                 break;
 285             case 3:     /* painful */
 286                 for (i=jz;i>0;i--) {
 287                     fw      = fq[i-1]+fq[i];
 288                     fq[i]  += fq[i-1]-fw;
 289                     fq[i-1] = fw;
 290                 }
 291                 for (i=jz;i>1;i--) {
 292                     fw      = fq[i-1]+fq[i];
 293                     fq[i]  += fq[i-1]-fw;
 294                     fq[i-1] = fw;
 295                 }
 296                 for (fw=0.0,i=jz;i>=2;i--) fw += fq[i];
 297                 if(ih==0) {
 298                     y[0] =  fq[0]; y[1] =  fq[1]; y[2] =  fw;
 299                 } else {
 300                     y[0] = -fq[0]; y[1] = -fq[1]; y[2] = -fw;
 301                 }
 302         }
 303         return n&7;
 304 }