StdLib/LibC/Math/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 #include  <LibConfig.h>
  13 #include  <sys/EfiCdefs.h>
  14 #if defined(LIBM_SCCS) && !defined(lint)
  15 __RCSID("$NetBSD: k_rem_pio2.c,v 1.11 2003/01/04 23:43:03 wiz 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[] output 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
 157 __kernel_rem_pio2(double *x, double *y, int e0, int nx, int prec, const int32_t *ipio2)
 158 {
 159   int32_t jz,jx,jv,jp,jk,carry,n,iq[20],i,j,k,m,q0,ih;
 160   double z,fw,f[20],fq[20],q[20];
 161
 162     /* initialize jk*/
 163   jk = init_jk[prec];
 164   jp = jk;
 165
 166     /* determine jx,jv,q0, note that 3>q0 */
 167   jx =  nx-1;
 168   jv = (e0-3)/24; if(jv<0) jv=0;
 169   q0 =  e0-24*(jv+1);
 170
 171     /* set up f[0] to f[jx+jk] where f[jx+jk] = ipio2[jv+jk] */
 172   j = jv-jx; m = jx+jk;
 173   for(i=0;i<=m;i++,j++) f[i] = (j<0)? zero : (double) ipio2[j];
 174
 175     /* compute q[0],q[1],...q[jk] */
 176   for (i=0;i<=jk;i++) {
 177       for(j=0,fw=0.0;j<=jx;j++) fw += x[j]*f[jx+i-j];
 178       q[i] = fw;
 179   }
 180
 181   jz = jk;
 182 recompute:
 183     /* distill q[] into iq[] reversingly */
 184   for(i=0,j=jz,z=q[jz];j>0;i++,j--) {
 185       fw    =  (double)((int32_t)(twon24* z));
 186       iq[i] =  (int32_t)(z-two24*fw);
 187       z     =  q[j-1]+fw;
 188   }
 189
 190     /* compute n */
 191   z  = scalbn(z,q0);    /* actual value of z */
 192   z -= 8.0*floor(z*0.125);    /* trim off integer >= 8 */
 193   n  = (int32_t) z;
 194   z -= (double)n;
 195   ih = 0;
 196   if(q0>0) {  /* need iq[jz-1] to determine n */
 197       i  = (iq[jz-1]>>(24-q0)); n += i;
 198       iq[jz-1] -= i<<(24-q0);
 199       ih = iq[jz-1]>>(23-q0);
 200   }
 201   else if(q0==0) ih = iq[jz-1]>>23;
 202   else if(z>=0.5) ih=2;
 203
 204   if(ih>0) {  /* q > 0.5 */
 205       n += 1; carry = 0;
 206       for(i=0;i<jz ;i++) {  /* compute 1-q */
 207     j = iq[i];
 208     if(carry==0) {
 209         if(j!=0) {
 210       carry = 1; iq[i] = 0x1000000- j;
 211         }
 212     } else  iq[i] = 0xffffff - j;
 213       }
 214       if(q0>0) {    /* rare case: chance is 1 in 12 */
 215           switch(q0) {
 216           case 1:
 217            iq[jz-1] &= 0x7fffff; break;
 218         case 2:
 219            iq[jz-1] &= 0x3fffff; break;
 220           }
 221       }
 222       if(ih==2) {
 223     z = one - z;
 224     if(carry!=0) z -= scalbn(one,q0);
 225       }
 226   }
 227
 228     /* check if recomputation is needed */
 229   if(z==zero) {
 230       j = 0;
 231       for (i=jz-1;i>=jk;i--) j |= iq[i];
 232       if(j==0) { /* need recomputation */
 233     for(k=1;iq[jk-k]==0;k++);   /* k = no. of terms needed */
 234
 235     for(i=jz+1;i<=jz+k;i++) {   /* add q[jz+1] to q[jz+k] */
 236         f[jx+i] = (double) ipio2[jv+i];
 237         for(j=0,fw=0.0;j<=jx;j++) fw += x[j]*f[jx+i-j];
 238         q[i] = fw;
 239     }
 240     jz += k;
 241     goto recompute;
 242       }
 243   }
 244
 245     /* chop off zero terms */
 246   if(z==0.0) {
 247       jz -= 1; q0 -= 24;
 248       while(iq[jz]==0) { jz--; q0-=24;}
 249   } else { /* break z into 24-bit if necessary */
 250       z = scalbn(z,-q0);
 251       if(z>=two24) {
 252     fw = (double)((int32_t)(twon24*z));
 253     iq[jz] = (int32_t)(z-two24*fw);
 254     jz += 1; q0 += 24;
 255     iq[jz] = (int32_t) fw;
 256       } else iq[jz] = (int32_t) z ;
 257   }
 258
 259     /* convert integer "bit" chunk to floating-point value */
 260   fw = scalbn(one,q0);
 261   for(i=jz;i>=0;i--) {
 262       q[i] = fw*(double)iq[i]; fw*=twon24;
 263   }
 264
 265     /* compute PIo2[0,...,jp]*q[jz,...,0] */
 266   for(i=jz;i>=0;i--) {
 267       for(fw=0.0,k=0;k<=jp&&k<=jz-i;k++) fw += PIo2[k]*q[i+k];
 268       fq[jz-i] = fw;
 269   }
 270
 271     /* compress fq[] into y[] */
 272   switch(prec) {
 273       case 0:
 274     fw = 0.0;
 275     for (i=jz;i>=0;i--) fw += fq[i];
 276     y[0] = (ih==0)? fw: -fw;
 277     break;
 278       case 1:
 279       case 2:
 280     fw = 0.0;
 281     for (i=jz;i>=0;i--) fw += fq[i];
 282     y[0] = (ih==0)? fw: -fw;
 283     fw = fq[0]-fw;
 284     for (i=1;i<=jz;i++) fw += fq[i];
 285     y[1] = (ih==0)? fw: -fw;
 286     break;
 287       case 3: /* painful */
 288     for (i=jz;i>0;i--) {
 289         fw      = fq[i-1]+fq[i];
 290         fq[i]  += fq[i-1]-fw;
 291         fq[i-1] = fw;
 292     }
 293     for (i=jz;i>1;i--) {
 294         fw      = fq[i-1]+fq[i];
 295         fq[i]  += fq[i-1]-fw;
 296         fq[i-1] = fw;
 297     }
 298     for (fw=0.0,i=jz;i>=2;i--) fw += fq[i];
 299     if(ih==0) {
 300         y[0] =  fq[0]; y[1] =  fq[1]; y[2] =  fw;
 301     } else {
 302         y[0] = -fq[0]; y[1] = -fq[1]; y[2] = -fw;
 303     }
 304   }
 305   return n&7;
 306 }