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[mirror_edk2.git] / StdLib / LibC / Math / k_rem_pio2.c

/* @(#)k_rem_pio2.c 5.1 93/09/24 */\r
/*\r
 * ====================================================\r
 * Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved.\r
 *\r
 * Developed at SunPro, a Sun Microsystems, Inc. business.\r
 * Permission to use, copy, modify, and distribute this\r
 * software is freely granted, provided that this notice\r
 * is preserved.\r
 * ====================================================\r
 */\r
#include  <LibConfig.h>\r
#include  <sys/EfiCdefs.h>\r
#if defined(LIBM_SCCS) && !defined(lint)\r
__RCSID("$NetBSD: k_rem_pio2.c,v 1.11 2003/01/04 23:43:03 wiz Exp $");\r
#endif\r
\r
/*\r
 * __kernel_rem_pio2(x,y,e0,nx,prec,ipio2)\r
 * double x[],y[]; int e0,nx,prec; int ipio2[];\r
 *\r
 * __kernel_rem_pio2 return the last three digits of N with\r
 *    y = x - N*pi/2\r
 * so that |y| < pi/2.\r
 *\r
 * The method is to compute the integer (mod 8) and fraction parts of\r
 * (2/pi)*x without doing the full multiplication. In general we\r
 * skip the part of the product that are known to be a huge integer (\r
 * more accurately, = 0 mod 8 ). Thus the number of operations are\r
 * independent of the exponent of the input.\r
 *\r
 * (2/pi) is represented by an array of 24-bit integers in ipio2[].\r
 *\r
 * Input parameters:\r
 *  x[] The input value (must be positive) is broken into nx\r
 *    pieces of 24-bit integers in double precision format.\r
 *    x[i] will be the i-th 24 bit of x. The scaled exponent\r
 *    of x[0] is given in input parameter e0 (i.e., x[0]*2^e0\r
 *    match x's up to 24 bits.\r
 *\r
 *    Example of breaking a double positive z into x[0]+x[1]+x[2]:\r
 *      e0 = ilogb(z)-23\r
 *      z  = scalbn(z,-e0)\r
 *    for i = 0,1,2\r
 *      x[i] = floor(z)\r
 *      z    = (z-x[i])*2**24\r
 *\r
 *\r
 *  y[] output result in an array of double precision numbers.\r
 *    The dimension of y[] is:\r
 *      24-bit  precision 1\r
 *      53-bit  precision 2\r
 *      64-bit  precision 2\r
 *      113-bit precision 3\r
 *    The actual value is the sum of them. Thus for 113-bit\r
 *    precison, one may have to do something like:\r
 *\r
 *    long double t,w,r_head, r_tail;\r
 *    t = (long double)y[2] + (long double)y[1];\r
 *    w = (long double)y[0];\r
 *    r_head = t+w;\r
 *    r_tail = w - (r_head - t);\r
 *\r
 *  e0  The exponent of x[0]\r
 *\r
 *  nx  dimension of x[]\r
 *\r
 *    prec  an integer indicating the precision:\r
 *      0 24  bits (single)\r
 *      1 53  bits (double)\r
 *      2 64  bits (extended)\r
 *      3 113 bits (quad)\r
 *\r
 *  ipio2[]\r
 *    integer array, contains the (24*i)-th to (24*i+23)-th\r
 *    bit of 2/pi after binary point. The corresponding\r
 *    floating value is\r
 *\r
 *      ipio2[i] * 2^(-24(i+1)).\r
 *\r
 * External function:\r
 *  double scalbn(), floor();\r
 *\r
 *\r
 * Here is the description of some local variables:\r
 *\r
 *  jk  jk+1 is the initial number of terms of ipio2[] needed\r
 *    in the computation. The recommended value is 2,3,4,\r
 *    6 for single, double, extended,and quad.\r
 *\r
 *  jz  local integer variable indicating the number of\r
 *    terms of ipio2[] used.\r
 *\r
 *  jx  nx - 1\r
 *\r
 *  jv  index for pointing to the suitable ipio2[] for the\r
 *    computation. In general, we want\r
 *      ( 2^e0*x[0] * ipio2[jv-1]*2^(-24jv) )/8\r
 *    is an integer. Thus\r
 *      e0-3-24*jv >= 0 or (e0-3)/24 >= jv\r
 *    Hence jv = max(0,(e0-3)/24).\r
 *\r
 *  jp  jp+1 is the number of terms in PIo2[] needed, jp = jk.\r
 *\r
 *  q[] double array with integral value, representing the\r
 *    24-bits chunk of the product of x and 2/pi.\r
 *\r
 *  q0  the corresponding exponent of q[0]. Note that the\r
 *    exponent for q[i] would be q0-24*i.\r
 *\r
 *  PIo2[]  double precision array, obtained by cutting pi/2\r
 *    into 24 bits chunks.\r
 *\r
 *  f[] ipio2[] in floating point\r
 *\r
 *  iq[]  integer array by breaking up q[] in 24-bits chunk.\r
 *\r
 *  fq[]  final product of x*(2/pi) in fq[0],..,fq[jk]\r
 *\r
 *  ih  integer. If >0 it indicates q[] is >= 0.5, hence\r
 *    it also indicates the *sign* of the result.\r
 *\r
 */\r
\r
\r
/*\r
 * Constants:\r
 * The hexadecimal values are the intended ones for the following\r
 * constants. The decimal values may be used, provided that the\r
 * compiler will convert from decimal to binary accurately enough\r
 * to produce the hexadecimal values shown.\r
 */\r
\r
#include "math.h"\r
#include "math_private.h"\r
\r
static const int init_jk[] = {2,3,4,6}; /* initial value for jk */\r
\r
static const double PIo2[] = {\r
  1.57079625129699707031e+00, /* 0x3FF921FB, 0x40000000 */\r
  7.54978941586159635335e-08, /* 0x3E74442D, 0x00000000 */\r
  5.39030252995776476554e-15, /* 0x3CF84698, 0x80000000 */\r
  3.28200341580791294123e-22, /* 0x3B78CC51, 0x60000000 */\r
  1.27065575308067607349e-29, /* 0x39F01B83, 0x80000000 */\r
  1.22933308981111328932e-36, /* 0x387A2520, 0x40000000 */\r
  2.73370053816464559624e-44, /* 0x36E38222, 0x80000000 */\r
  2.16741683877804819444e-51, /* 0x3569F31D, 0x00000000 */\r
};\r
\r
static const double\r
zero   = 0.0,\r
one    = 1.0,\r
two24   =  1.67772160000000000000e+07, /* 0x41700000, 0x00000000 */\r
twon24  =  5.96046447753906250000e-08; /* 0x3E700000, 0x00000000 */\r
\r
int\r
__kernel_rem_pio2(double *x, double *y, int e0, int nx, int prec, const int32_t *ipio2)\r
{\r
  int32_t jz,jx,jv,jp,jk,carry,n,iq[20],i,j,k,m,q0,ih;\r
  double z,fw,f[20],fq[20],q[20];\r
\r
    /* initialize jk*/\r
  jk = init_jk[prec];\r
  jp = jk;\r
\r
    /* determine jx,jv,q0, note that 3>q0 */\r
  jx =  nx-1;\r
  jv = (e0-3)/24; if(jv<0) jv=0;\r
  q0 =  e0-24*(jv+1);\r
\r
    /* set up f[0] to f[jx+jk] where f[jx+jk] = ipio2[jv+jk] */\r
  j = jv-jx; m = jx+jk;\r
  for(i=0;i<=m;i++,j++) f[i] = (j<0)? zero : (double) ipio2[j];\r
\r
    /* compute q[0],q[1],...q[jk] */\r
  for (i=0;i<=jk;i++) {\r
      for(j=0,fw=0.0;j<=jx;j++) fw += x[j]*f[jx+i-j]; q[i] = fw;\r
  }\r
\r
  jz = jk;\r
recompute:\r
    /* distill q[] into iq[] reversingly */\r
  for(i=0,j=jz,z=q[jz];j>0;i++,j--) {\r
      fw    =  (double)((int32_t)(twon24* z));\r
      iq[i] =  (int32_t)(z-two24*fw);\r
      z     =  q[j-1]+fw;\r
  }\r
\r
    /* compute n */\r
  z  = scalbn(z,q0);    /* actual value of z */\r
  z -= 8.0*floor(z*0.125);    /* trim off integer >= 8 */\r
  n  = (int32_t) z;\r
  z -= (double)n;\r
  ih = 0;\r
  if(q0>0) {  /* need iq[jz-1] to determine n */\r
      i  = (iq[jz-1]>>(24-q0)); n += i;\r
      iq[jz-1] -= i<<(24-q0);\r
      ih = iq[jz-1]>>(23-q0);\r
  }\r
  else if(q0==0) ih = iq[jz-1]>>23;\r
  else if(z>=0.5) ih=2;\r
\r
  if(ih>0) {  /* q > 0.5 */\r
      n += 1; carry = 0;\r
      for(i=0;i<jz ;i++) {  /* compute 1-q */\r
    j = iq[i];\r
    if(carry==0) {\r
        if(j!=0) {\r
      carry = 1; iq[i] = 0x1000000- j;\r
        }\r
    } else  iq[i] = 0xffffff - j;\r
      }\r
      if(q0>0) {    /* rare case: chance is 1 in 12 */\r
          switch(q0) {\r
          case 1:\r
           iq[jz-1] &= 0x7fffff; break;\r
        case 2:\r
           iq[jz-1] &= 0x3fffff; break;\r
          }\r
      }\r
      if(ih==2) {\r
    z = one - z;\r
    if(carry!=0) z -= scalbn(one,q0);\r
      }\r
  }\r
\r
    /* check if recomputation is needed */\r
  if(z==zero) {\r
      j = 0;\r
      for (i=jz-1;i>=jk;i--) j |= iq[i];\r
      if(j==0) { /* need recomputation */\r
    for(k=1;iq[jk-k]==0;k++);   /* k = no. of terms needed */\r
\r
    for(i=jz+1;i<=jz+k;i++) {   /* add q[jz+1] to q[jz+k] */\r
        f[jx+i] = (double) ipio2[jv+i];\r
        for(j=0,fw=0.0;j<=jx;j++) fw += x[j]*f[jx+i-j];\r
        q[i] = fw;\r
    }\r
    jz += k;\r
    goto recompute;\r
      }\r
  }\r
\r
    /* chop off zero terms */\r
  if(z==0.0) {\r
      jz -= 1; q0 -= 24;\r
      while(iq[jz]==0) { jz--; q0-=24;}\r
  } else { /* break z into 24-bit if necessary */\r
      z = scalbn(z,-q0);\r
      if(z>=two24) {\r
    fw = (double)((int32_t)(twon24*z));\r
    iq[jz] = (int32_t)(z-two24*fw);\r
    jz += 1; q0 += 24;\r
    iq[jz] = (int32_t) fw;\r
      } else iq[jz] = (int32_t) z ;\r
  }\r
\r
    /* convert integer "bit" chunk to floating-point value */\r
  fw = scalbn(one,q0);\r
  for(i=jz;i>=0;i--) {\r
      q[i] = fw*(double)iq[i]; fw*=twon24;\r
  }\r
\r
    /* compute PIo2[0,...,jp]*q[jz,...,0] */\r
  for(i=jz;i>=0;i--) {\r
      for(fw=0.0,k=0;k<=jp&&k<=jz-i;k++) fw += PIo2[k]*q[i+k];\r
      fq[jz-i] = fw;\r
  }\r
\r
    /* compress fq[] into y[] */\r
  switch(prec) {\r
      case 0:\r
    fw = 0.0;\r
    for (i=jz;i>=0;i--) fw += fq[i];\r
    y[0] = (ih==0)? fw: -fw;\r
    break;\r
      case 1:\r
      case 2:\r
    fw = 0.0;\r
    for (i=jz;i>=0;i--) fw += fq[i];\r
    y[0] = (ih==0)? fw: -fw;\r
    fw = fq[0]-fw;\r
    for (i=1;i<=jz;i++) fw += fq[i];\r
    y[1] = (ih==0)? fw: -fw;\r
    break;\r
      case 3: /* painful */\r
    for (i=jz;i>0;i--) {\r
        fw      = fq[i-1]+fq[i];\r
        fq[i]  += fq[i-1]-fw;\r
        fq[i-1] = fw;\r
    }\r
    for (i=jz;i>1;i--) {\r
        fw      = fq[i-1]+fq[i];\r
        fq[i]  += fq[i-1]-fw;\r
        fq[i-1] = fw;\r
    }\r
    for (fw=0.0,i=jz;i>=2;i--) fw += fq[i];\r
    if(ih==0) {\r
        y[0] =  fq[0]; y[1] =  fq[1]; y[2] =  fw;\r
    } else {\r
        y[0] = -fq[0]; y[1] = -fq[1]; y[2] = -fw;\r
    }\r
  }\r
  return n&7;\r
}\r