4fd606d1f5abe38e1f42c38de1d2e895166bd0f4vboxsync/* @(#)k_rem_pio2.c 5.1 93/09/24 */
4fd606d1f5abe38e1f42c38de1d2e895166bd0f4vboxsync * ====================================================
4fd606d1f5abe38e1f42c38de1d2e895166bd0f4vboxsync * Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved.
4fd606d1f5abe38e1f42c38de1d2e895166bd0f4vboxsync * Developed at SunPro, a Sun Microsystems, Inc. business.
4fd606d1f5abe38e1f42c38de1d2e895166bd0f4vboxsync * Permission to use, copy, modify, and distribute this
4fd606d1f5abe38e1f42c38de1d2e895166bd0f4vboxsync * software is freely granted, provided that this notice
4fd606d1f5abe38e1f42c38de1d2e895166bd0f4vboxsync * is preserved.
4fd606d1f5abe38e1f42c38de1d2e895166bd0f4vboxsync * ====================================================
4fd606d1f5abe38e1f42c38de1d2e895166bd0f4vboxsync__RCSID("$NetBSD: k_rem_pio2.c,v 1.11 2003/01/04 23:43:03 wiz Exp $");
4fd606d1f5abe38e1f42c38de1d2e895166bd0f4vboxsync * __kernel_rem_pio2(x,y,e0,nx,prec,ipio2)
4fd606d1f5abe38e1f42c38de1d2e895166bd0f4vboxsync * double x[],y[]; int e0,nx,prec; int ipio2[];
4fd606d1f5abe38e1f42c38de1d2e895166bd0f4vboxsync * __kernel_rem_pio2 return the last three digits of N with
4fd606d1f5abe38e1f42c38de1d2e895166bd0f4vboxsync * y = x - N*pi/2
4fd606d1f5abe38e1f42c38de1d2e895166bd0f4vboxsync * so that |y| < pi/2.
4fd606d1f5abe38e1f42c38de1d2e895166bd0f4vboxsync * The method is to compute the integer (mod 8) and fraction parts of
4fd606d1f5abe38e1f42c38de1d2e895166bd0f4vboxsync * (2/pi)*x without doing the full multiplication. In general we
4fd606d1f5abe38e1f42c38de1d2e895166bd0f4vboxsync * skip the part of the product that are known to be a huge integer (
4fd606d1f5abe38e1f42c38de1d2e895166bd0f4vboxsync * more accurately, = 0 mod 8 ). Thus the number of operations are
4fd606d1f5abe38e1f42c38de1d2e895166bd0f4vboxsync * independent of the exponent of the input.
4fd606d1f5abe38e1f42c38de1d2e895166bd0f4vboxsync * (2/pi) is represented by an array of 24-bit integers in ipio2[].
4fd606d1f5abe38e1f42c38de1d2e895166bd0f4vboxsync * Input parameters:
4fd606d1f5abe38e1f42c38de1d2e895166bd0f4vboxsync * x[] The input value (must be positive) is broken into nx
4fd606d1f5abe38e1f42c38de1d2e895166bd0f4vboxsync * pieces of 24-bit integers in double precision format.
4fd606d1f5abe38e1f42c38de1d2e895166bd0f4vboxsync * x[i] will be the i-th 24 bit of x. The scaled exponent
4fd606d1f5abe38e1f42c38de1d2e895166bd0f4vboxsync * of x[0] is given in input parameter e0 (i.e., x[0]*2^e0
4fd606d1f5abe38e1f42c38de1d2e895166bd0f4vboxsync * match x's up to 24 bits.
4fd606d1f5abe38e1f42c38de1d2e895166bd0f4vboxsync * Example of breaking a double positive z into x[0]+x[1]+x[2]:
4fd606d1f5abe38e1f42c38de1d2e895166bd0f4vboxsync * e0 = ilogb(z)-23
4fd606d1f5abe38e1f42c38de1d2e895166bd0f4vboxsync * z = scalbn(z,-e0)
4fd606d1f5abe38e1f42c38de1d2e895166bd0f4vboxsync * for i = 0,1,2
4fd606d1f5abe38e1f42c38de1d2e895166bd0f4vboxsync * x[i] = floor(z)
4fd606d1f5abe38e1f42c38de1d2e895166bd0f4vboxsync * z = (z-x[i])*2**24
4fd606d1f5abe38e1f42c38de1d2e895166bd0f4vboxsync * y[] output result in an array of double precision numbers.
4fd606d1f5abe38e1f42c38de1d2e895166bd0f4vboxsync * The dimension of y[] is:
4fd606d1f5abe38e1f42c38de1d2e895166bd0f4vboxsync * 24-bit precision 1
4fd606d1f5abe38e1f42c38de1d2e895166bd0f4vboxsync * 53-bit precision 2
4fd606d1f5abe38e1f42c38de1d2e895166bd0f4vboxsync * 64-bit precision 2
4fd606d1f5abe38e1f42c38de1d2e895166bd0f4vboxsync * 113-bit precision 3
4fd606d1f5abe38e1f42c38de1d2e895166bd0f4vboxsync * The actual value is the sum of them. Thus for 113-bit
4fd606d1f5abe38e1f42c38de1d2e895166bd0f4vboxsync * precison, one may have to do something like:
4fd606d1f5abe38e1f42c38de1d2e895166bd0f4vboxsync * long double t,w,r_head, r_tail;
4fd606d1f5abe38e1f42c38de1d2e895166bd0f4vboxsync * t = (long double)y[2] + (long double)y[1];
4fd606d1f5abe38e1f42c38de1d2e895166bd0f4vboxsync * w = (long double)y[0];
4fd606d1f5abe38e1f42c38de1d2e895166bd0f4vboxsync * r_head = t+w;
4fd606d1f5abe38e1f42c38de1d2e895166bd0f4vboxsync * r_tail = w - (r_head - t);
4fd606d1f5abe38e1f42c38de1d2e895166bd0f4vboxsync * e0 The exponent of x[0]
4fd606d1f5abe38e1f42c38de1d2e895166bd0f4vboxsync * nx dimension of x[]
4fd606d1f5abe38e1f42c38de1d2e895166bd0f4vboxsync * prec an integer indicating the precision:
4fd606d1f5abe38e1f42c38de1d2e895166bd0f4vboxsync * 0 24 bits (single)
4fd606d1f5abe38e1f42c38de1d2e895166bd0f4vboxsync * 1 53 bits (double)
4fd606d1f5abe38e1f42c38de1d2e895166bd0f4vboxsync * 2 64 bits (extended)
4fd606d1f5abe38e1f42c38de1d2e895166bd0f4vboxsync * 3 113 bits (quad)
4fd606d1f5abe38e1f42c38de1d2e895166bd0f4vboxsync * integer array, contains the (24*i)-th to (24*i+23)-th
4fd606d1f5abe38e1f42c38de1d2e895166bd0f4vboxsync * bit of 2/pi after binary point. The corresponding
4fd606d1f5abe38e1f42c38de1d2e895166bd0f4vboxsync * floating value is
4fd606d1f5abe38e1f42c38de1d2e895166bd0f4vboxsync * ipio2[i] * 2^(-24(i+1)).
4fd606d1f5abe38e1f42c38de1d2e895166bd0f4vboxsync * External function:
4fd606d1f5abe38e1f42c38de1d2e895166bd0f4vboxsync * double scalbn(), floor();
4fd606d1f5abe38e1f42c38de1d2e895166bd0f4vboxsync * Here is the description of some local variables:
4fd606d1f5abe38e1f42c38de1d2e895166bd0f4vboxsync * jk jk+1 is the initial number of terms of ipio2[] needed
4fd606d1f5abe38e1f42c38de1d2e895166bd0f4vboxsync * in the computation. The recommended value is 2,3,4,
4fd606d1f5abe38e1f42c38de1d2e895166bd0f4vboxsync * 6 for single, double, extended,and quad.
4fd606d1f5abe38e1f42c38de1d2e895166bd0f4vboxsync * jz local integer variable indicating the number of
4fd606d1f5abe38e1f42c38de1d2e895166bd0f4vboxsync * terms of ipio2[] used.
4fd606d1f5abe38e1f42c38de1d2e895166bd0f4vboxsync * jx nx - 1
4fd606d1f5abe38e1f42c38de1d2e895166bd0f4vboxsync * jv index for pointing to the suitable ipio2[] for the
4fd606d1f5abe38e1f42c38de1d2e895166bd0f4vboxsync * computation. In general, we want
4fd606d1f5abe38e1f42c38de1d2e895166bd0f4vboxsync * ( 2^e0*x[0] * ipio2[jv-1]*2^(-24jv) )/8
4fd606d1f5abe38e1f42c38de1d2e895166bd0f4vboxsync * is an integer. Thus
4fd606d1f5abe38e1f42c38de1d2e895166bd0f4vboxsync * e0-3-24*jv >= 0 or (e0-3)/24 >= jv
4fd606d1f5abe38e1f42c38de1d2e895166bd0f4vboxsync * Hence jv = max(0,(e0-3)/24).
4fd606d1f5abe38e1f42c38de1d2e895166bd0f4vboxsync * jp jp+1 is the number of terms in PIo2[] needed, jp = jk.
4fd606d1f5abe38e1f42c38de1d2e895166bd0f4vboxsync * q[] double array with integral value, representing the
4fd606d1f5abe38e1f42c38de1d2e895166bd0f4vboxsync * 24-bits chunk of the product of x and 2/pi.
4fd606d1f5abe38e1f42c38de1d2e895166bd0f4vboxsync * q0 the corresponding exponent of q[0]. Note that the
4fd606d1f5abe38e1f42c38de1d2e895166bd0f4vboxsync * exponent for q[i] would be q0-24*i.
4fd606d1f5abe38e1f42c38de1d2e895166bd0f4vboxsync * PIo2[] double precision array, obtained by cutting pi/2
4fd606d1f5abe38e1f42c38de1d2e895166bd0f4vboxsync * into 24 bits chunks.
4fd606d1f5abe38e1f42c38de1d2e895166bd0f4vboxsync * f[] ipio2[] in floating point
4fd606d1f5abe38e1f42c38de1d2e895166bd0f4vboxsync * iq[] integer array by breaking up q[] in 24-bits chunk.
4fd606d1f5abe38e1f42c38de1d2e895166bd0f4vboxsync * fq[] final product of x*(2/pi) in fq[0],..,fq[jk]
4fd606d1f5abe38e1f42c38de1d2e895166bd0f4vboxsync * ih integer. If >0 it indicates q[] is >= 0.5, hence
4fd606d1f5abe38e1f42c38de1d2e895166bd0f4vboxsync * it also indicates the *sign* of the result.
4fd606d1f5abe38e1f42c38de1d2e895166bd0f4vboxsync * Constants:
4fd606d1f5abe38e1f42c38de1d2e895166bd0f4vboxsync * The hexadecimal values are the intended ones for the following
4fd606d1f5abe38e1f42c38de1d2e895166bd0f4vboxsync * constants. The decimal values may be used, provided that the
4fd606d1f5abe38e1f42c38de1d2e895166bd0f4vboxsync * compiler will convert from decimal to binary accurately enough
4fd606d1f5abe38e1f42c38de1d2e895166bd0f4vboxsync * to produce the hexadecimal values shown.
4fd606d1f5abe38e1f42c38de1d2e895166bd0f4vboxsyncstatic const int init_jk[] = {2,3,4,6}; /* initial value for jk */
4fd606d1f5abe38e1f42c38de1d2e895166bd0f4vboxsync 1.57079625129699707031e+00, /* 0x3FF921FB, 0x40000000 */
4fd606d1f5abe38e1f42c38de1d2e895166bd0f4vboxsync 7.54978941586159635335e-08, /* 0x3E74442D, 0x00000000 */
4fd606d1f5abe38e1f42c38de1d2e895166bd0f4vboxsync 5.39030252995776476554e-15, /* 0x3CF84698, 0x80000000 */
4fd606d1f5abe38e1f42c38de1d2e895166bd0f4vboxsync 3.28200341580791294123e-22, /* 0x3B78CC51, 0x60000000 */
4fd606d1f5abe38e1f42c38de1d2e895166bd0f4vboxsync 1.27065575308067607349e-29, /* 0x39F01B83, 0x80000000 */
4fd606d1f5abe38e1f42c38de1d2e895166bd0f4vboxsync 1.22933308981111328932e-36, /* 0x387A2520, 0x40000000 */
4fd606d1f5abe38e1f42c38de1d2e895166bd0f4vboxsync 2.73370053816464559624e-44, /* 0x36E38222, 0x80000000 */
4fd606d1f5abe38e1f42c38de1d2e895166bd0f4vboxsync 2.16741683877804819444e-51, /* 0x3569F31D, 0x00000000 */
4fd606d1f5abe38e1f42c38de1d2e895166bd0f4vboxsyncstatic const double
4fd606d1f5abe38e1f42c38de1d2e895166bd0f4vboxsynctwo24 = 1.67772160000000000000e+07, /* 0x41700000, 0x00000000 */
4fd606d1f5abe38e1f42c38de1d2e895166bd0f4vboxsynctwon24 = 5.96046447753906250000e-08; /* 0x3E700000, 0x00000000 */
4fd606d1f5abe38e1f42c38de1d2e895166bd0f4vboxsync__kernel_rem_pio2(double *x, double *y, int e0, int nx, int prec, const int32_t *ipio2)
4fd606d1f5abe38e1f42c38de1d2e895166bd0f4vboxsync int32_t jz,jx,jv,jp,jk,carry,n,iq[20],i,j,k,m,q0,ih;
4fd606d1f5abe38e1f42c38de1d2e895166bd0f4vboxsync /* initialize jk*/
4fd606d1f5abe38e1f42c38de1d2e895166bd0f4vboxsync /* determine jx,jv,q0, note that 3>q0 */
4fd606d1f5abe38e1f42c38de1d2e895166bd0f4vboxsync /* set up f[0] to f[jx+jk] where f[jx+jk] = ipio2[jv+jk] */
4fd606d1f5abe38e1f42c38de1d2e895166bd0f4vboxsync for(i=0;i<=m;i++,j++) f[i] = (j<0)? zero : (double) ipio2[j];
4fd606d1f5abe38e1f42c38de1d2e895166bd0f4vboxsync /* compute q[0],q[1],...q[jk] */
4fd606d1f5abe38e1f42c38de1d2e895166bd0f4vboxsync for (i=0;i<=jk;i++) {
4fd606d1f5abe38e1f42c38de1d2e895166bd0f4vboxsync for(j=0,fw=0.0;j<=jx;j++) fw += x[j]*f[jx+i-j]; q[i] = fw;
4fd606d1f5abe38e1f42c38de1d2e895166bd0f4vboxsync /* distill q[] into iq[] reversingly */
4fd606d1f5abe38e1f42c38de1d2e895166bd0f4vboxsync /* compute n */
4fd606d1f5abe38e1f42c38de1d2e895166bd0f4vboxsync z -= 8.0*floor(z*0.125); /* trim off integer >= 8 */
4fd606d1f5abe38e1f42c38de1d2e895166bd0f4vboxsync z -= (double)n;
4fd606d1f5abe38e1f42c38de1d2e895166bd0f4vboxsync /* check if recomputation is needed */
4fd606d1f5abe38e1f42c38de1d2e895166bd0f4vboxsync if(j==0) { /* need recomputation */
4fd606d1f5abe38e1f42c38de1d2e895166bd0f4vboxsync for(k=1;iq[jk-k]==0;k++); /* k = no. of terms needed */
4fd606d1f5abe38e1f42c38de1d2e895166bd0f4vboxsync for(i=jz+1;i<=jz+k;i++) { /* add q[jz+1] to q[jz+k] */
4fd606d1f5abe38e1f42c38de1d2e895166bd0f4vboxsync /* chop off zero terms */
4fd606d1f5abe38e1f42c38de1d2e895166bd0f4vboxsync if(z==0.0) {
4fd606d1f5abe38e1f42c38de1d2e895166bd0f4vboxsync } else { /* break z into 24-bit if necessary */
4fd606d1f5abe38e1f42c38de1d2e895166bd0f4vboxsync /* convert integer "bit" chunk to floating-point value */
4fd606d1f5abe38e1f42c38de1d2e895166bd0f4vboxsync for(i=jz;i>=0;i--) {
4fd606d1f5abe38e1f42c38de1d2e895166bd0f4vboxsync /* compute PIo2[0,...,jp]*q[jz,...,0] */
4fd606d1f5abe38e1f42c38de1d2e895166bd0f4vboxsync for(i=jz;i>=0;i--) {
4fd606d1f5abe38e1f42c38de1d2e895166bd0f4vboxsync for(fw=0.0,k=0;k<=jp&&k<=jz-i;k++) fw += PIo2[k]*q[i+k];
4fd606d1f5abe38e1f42c38de1d2e895166bd0f4vboxsync /* compress fq[] into y[] */
4fd606d1f5abe38e1f42c38de1d2e895166bd0f4vboxsync for (i=jz;i>0;i--) {
4fd606d1f5abe38e1f42c38de1d2e895166bd0f4vboxsync return n&7;