lib/libm/src/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  * $NetBSD: k_rem_pio2.c,v 1.11 2003/01/04 23:43:03 wiz Exp $
  13  * $DragonFly: src/lib/libm/src/k_rem_pio2.c,v 1.1 2005/07/26 21:15:20 joerg Exp $
  14  */
  15
  16 /*
  17  * __kernel_rem_pio2(x,y,e0,nx,prec,ipio2)
  18  * double x[],y[]; int e0,nx,prec; int ipio2[];
  19  *
  20  * __kernel_rem_pio2 return the last three digits of N with
  21  *              y = x - N*pi/2
  22  * so that |y| < pi/2.
  23  *
  24  * The method is to compute the integer (mod 8) and fraction parts of
  25  * (2/pi)*x without doing the full multiplication. In general we
  26  * skip the part of the product that are known to be a huge integer (
  27  * more accurately, = 0 mod 8 ). Thus the number of operations are
  28  * independent of the exponent of the input.
  29  *
  30  * (2/pi) is represented by an array of 24-bit integers in ipio2[].
  31  *
  32  * Input parameters:
  33  *      x[]     The input value (must be positive) is broken into nx
  34  *              pieces of 24-bit integers in double precision format.
  35  *              x[i] will be the i-th 24 bit of x. The scaled exponent
  36  *              of x[0] is given in input parameter e0 (i.e., x[0]*2^e0
  37  *              match x's up to 24 bits.
  38  *
  39  *              Example of breaking a double positive z into x[0]+x[1]+x[2]:
  40  *                      e0 = ilogb(z)-23
  41  *                      z  = scalbn(z,-e0)
  42  *              for i = 0,1,2
  43  *                      x[i] = floor(z)
  44  *                      z    = (z-x[i])*2**24
  45  *
  46  *
  47  *      y[]     output result in an array of double precision numbers.
  48  *              The dimension of y[] is:
  49  *                      24-bit  precision       1
  50  *                      53-bit  precision       2
  51  *                      64-bit  precision       2
  52  *                      113-bit precision       3
  53  *              The actual value is the sum of them. Thus for 113-bit
  54  *              precison, one may have to do something like:
  55  *
  56  *              long double t,w,r_head, r_tail;
  57  *              t = (long double)y[2] + (long double)y[1];
  58  *              w = (long double)y[0];
  59  *              r_head = t+w;
  60  *              r_tail = w - (r_head - t);
  61  *
  62  *      e0      The exponent of x[0]
  63  *
  64  *      nx      dimension of x[]
  65  *
  66  *      prec    an integer indicating the precision:
  67  *                      0       24  bits (single)
  68  *                      1       53  bits (double)
  69  *                      2       64  bits (extended)
  70  *                      3       113 bits (quad)
  71  *
  72  *      ipio2[]
  73  *              integer array, contains the (24*i)-th to (24*i+23)-th
  74  *              bit of 2/pi after binary point. The corresponding
  75  *              floating value is
  76  *
  77  *                      ipio2[i] * 2^(-24(i+1)).
  78  *
  79  * External function:
  80  *      double scalbn(), floor();
  81  *
  82  *
  83  * Here is the description of some local variables:
  84  *
  85  *      jk      jk+1 is the initial number of terms of ipio2[] needed
  86  *              in the computation. The recommended value is 2,3,4,
  87  *              6 for single, double, extended,and quad.
  88  *
  89  *      jz      local integer variable indicating the number of
  90  *              terms of ipio2[] used.
  91  *
  92  *      jx      nx - 1
  93  *
  94  *      jv      index for pointing to the suitable ipio2[] for the
  95  *              computation. In general, we want
  96  *                      ( 2^e0*x[0] * ipio2[jv-1]*2^(-24jv) )/8
  97  *              is an integer. Thus
  98  *                      e0-3-24*jv >= 0 or (e0-3)/24 >= jv
  99  *              Hence jv = max(0,(e0-3)/24).
 100  *
 101  *      jp      jp+1 is the number of terms in PIo2[] needed, jp = jk.
 102  *
 103  *      q[]     double array with integral value, representing the
 104  *              24-bits chunk of the product of x and 2/pi.
 105  *
 106  *      q0      the corresponding exponent of q[0]. Note that the
 107  *              exponent for q[i] would be q0-24*i.
 108  *
 109  *      PIo2[]  double precision array, obtained by cutting pi/2
 110  *              into 24 bits chunks.
 111  *
 112  *      f[]     ipio2[] in floating point
 113  *
 114  *      iq[]    integer array by breaking up q[] in 24-bits chunk.
 115  *
 116  *      fq[]    final product of x*(2/pi) in fq[0],..,fq[jk]
 117  *
 118  *      ih      integer. If >0 it indicates q[] is >= 0.5, hence
 119  *              it also indicates the *sign* of the result.
 120  *
 121  */
 122
 123
 124 /*
 125  * Constants:
 126  * The hexadecimal values are the intended ones for the following
 127  * constants. The decimal values may be used, provided that the
 128  * compiler will convert from decimal to binary accurately enough
 129  * to produce the hexadecimal values shown.
 130  */
 131
 132 #include <math.h>
 133 #include "math_private.h"
 134
 135 static const int init_jk[] = {2,3,4,6}; /* initial value for jk */
 136
 137 static const double PIo2[] = {
 138   1.57079625129699707031e+00, /* 0x3FF921FB, 0x40000000 */
 139   7.54978941586159635335e-08, /* 0x3E74442D, 0x00000000 */
 140   5.39030252995776476554e-15, /* 0x3CF84698, 0x80000000 */
 141   3.28200341580791294123e-22, /* 0x3B78CC51, 0x60000000 */
 142   1.27065575308067607349e-29, /* 0x39F01B83, 0x80000000 */
 143   1.22933308981111328932e-36, /* 0x387A2520, 0x40000000 */
 144   2.73370053816464559624e-44, /* 0x36E38222, 0x80000000 */
 145   2.16741683877804819444e-51, /* 0x3569F31D, 0x00000000 */
 146 };
 147
 148 static const double
 149 zero   = 0.0,
 150 one    = 1.0,
 151 two24   =  1.67772160000000000000e+07, /* 0x41700000, 0x00000000 */
 152 twon24  =  5.96046447753906250000e-08; /* 0x3E700000, 0x00000000 */
 153
 154 int
 155 __kernel_rem_pio2(double *x, double *y, int e0, int nx, int prec, const int32_t *ipio2)
 156 {
 157         int32_t jz,jx,jv,jp,jk,carry,n,iq[20],i,j,k,m,q0,ih;
 158         double z,fw,f[20],fq[20],q[20];
 159
 160     /* initialize jk*/
 161         jk = init_jk[prec];
 162         jp = jk;
 163
 164     /* determine jx,jv,q0, note that 3>q0 */
 165         jx =  nx-1;
 166         jv = (e0-3)/24; if(jv<0) jv=0;
 167         q0 =  e0-24*(jv+1);
 168
 169     /* set up f[0] to f[jx+jk] where f[jx+jk] = ipio2[jv+jk] */
 170         j = jv-jx; m = jx+jk;
 171         for(i=0;i<=m;i++,j++) f[i] = (j<0)? zero : (double) ipio2[j];
 172
 173     /* compute q[0],q[1],...q[jk] */
 174         for (i=0;i<=jk;i++) {
 175             for(j=0,fw=0.0;j<=jx;j++) fw += x[j]*f[jx+i-j]; q[i] = fw;
 176         }
 177
 178         jz = jk;
 179 recompute:
 180     /* distill q[] into iq[] reversingly */
 181         for(i=0,j=jz,z=q[jz];j>0;i++,j--) {
 182             fw    =  (double)((int32_t)(twon24* z));
 183             iq[i] =  (int32_t)(z-two24*fw);
 184             z     =  q[j-1]+fw;
 185         }
 186
 187     /* compute n */
 188         z  = scalbn(z,q0);              /* actual value of z */
 189         z -= 8.0*floor(z*0.125);                /* trim off integer >= 8 */
 190         n  = (int32_t) z;
 191         z -= (double)n;
 192         ih = 0;
 193         if(q0>0) {      /* need iq[jz-1] to determine n */
 194             i  = (iq[jz-1]>>(24-q0)); n += i;
 195             iq[jz-1] -= i<<(24-q0);
 196             ih = iq[jz-1]>>(23-q0);
 197         }
 198         else if(q0==0) ih = iq[jz-1]>>23;
 199         else if(z>=0.5) ih=2;
 200
 201         if(ih>0) {      /* q > 0.5 */
 202             n += 1; carry = 0;
 203             for(i=0;i<jz ;i++) {        /* compute 1-q */
 204                 j = iq[i];
 205                 if(carry==0) {
 206                     if(j!=0) {
 207                         carry = 1; iq[i] = 0x1000000- j;
 208                     }
 209                 } else  iq[i] = 0xffffff - j;
 210             }
 211             if(q0>0) {          /* rare case: chance is 1 in 12 */
 212                 switch(q0) {
 213                 case 1:
 214                    iq[jz-1] &= 0x7fffff; break;
 215                 case 2:
 216                    iq[jz-1] &= 0x3fffff; break;
 217                 }
 218             }
 219             if(ih==2) {
 220                 z = one - z;
 221                 if(carry!=0) z -= scalbn(one,q0);
 222             }
 223         }
 224
 225     /* check if recomputation is needed */
 226         if(z==zero) {
 227             j = 0;
 228             for (i=jz-1;i>=jk;i--) j |= iq[i];
 229             if(j==0) { /* need recomputation */
 230                 for(k=1;iq[jk-k]==0;k++);   /* k = no. of terms needed */
 231
 232                 for(i=jz+1;i<=jz+k;i++) {   /* add q[jz+1] to q[jz+k] */
 233                     f[jx+i] = (double) ipio2[jv+i];
 234                     for(j=0,fw=0.0;j<=jx;j++) fw += x[j]*f[jx+i-j];
 235                     q[i] = fw;
 236                 }
 237                 jz += k;
 238                 goto recompute;
 239             }
 240         }
 241
 242     /* chop off zero terms */
 243         if(z==0.0) {
 244             jz -= 1; q0 -= 24;
 245             while(iq[jz]==0) { jz--; q0-=24;}
 246         } else { /* break z into 24-bit if necessary */
 247             z = scalbn(z,-q0);
 248             if(z>=two24) {
 249                 fw = (double)((int32_t)(twon24*z));
 250                 iq[jz] = (int32_t)(z-two24*fw);
 251                 jz += 1; q0 += 24;
 252                 iq[jz] = (int32_t) fw;
 253             } else iq[jz] = (int32_t) z ;
 254         }
 255
 256     /* convert integer "bit" chunk to floating-point value */
 257         fw = scalbn(one,q0);
 258         for(i=jz;i>=0;i--) {
 259             q[i] = fw*(double)iq[i]; fw*=twon24;
 260         }
 261
 262     /* compute PIo2[0,...,jp]*q[jz,...,0] */
 263         for(i=jz;i>=0;i--) {
 264             for(fw=0.0,k=0;k<=jp&&k<=jz-i;k++) fw += PIo2[k]*q[i+k];
 265             fq[jz-i] = fw;
 266         }
 267
 268     /* compress fq[] into y[] */
 269         switch(prec) {
 270             case 0:
 271                 fw = 0.0;
 272                 for (i=jz;i>=0;i--) fw += fq[i];
 273                 y[0] = (ih==0)? fw: -fw;
 274                 break;
 275             case 1:
 276             case 2:
 277                 fw = 0.0;
 278                 for (i=jz;i>=0;i--) fw += fq[i];
 279                 y[0] = (ih==0)? fw: -fw;
 280                 fw = fq[0]-fw;
 281                 for (i=1;i<=jz;i++) fw += fq[i];
 282                 y[1] = (ih==0)? fw: -fw;
 283                 break;
 284             case 3:     /* painful */
 285                 for (i=jz;i>0;i--) {
 286                     fw      = fq[i-1]+fq[i];
 287                     fq[i]  += fq[i-1]-fw;
 288                     fq[i-1] = fw;
 289                 }
 290                 for (i=jz;i>1;i--) {
 291                     fw      = fq[i-1]+fq[i];
 292                     fq[i]  += fq[i-1]-fw;
 293                     fq[i-1] = fw;
 294                 }
 295                 for (fw=0.0,i=jz;i>=2;i--) fw += fq[i];
 296                 if(ih==0) {
 297                     y[0] =  fq[0]; y[1] =  fq[1]; y[2] =  fw;
 298                 } else {
 299                     y[0] = -fq[0]; y[1] = -fq[1]; y[2] = -fw;
 300                 }
 301         }
 302         return n&7;
 303 }