/* @(#)k_rem_pio2.c 1.3 95/01/18 */ /* * ==================================================== * Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved. * * Developed at SunSoft, a Sun Microsystems, Inc. business. * Permission to use, copy, modify, and distribute this * software is freely granted, provided that this notice * is preserved. * ====================================================
*/
//#include <sys/cdefs.h> //__FBSDID("$FreeBSD$");
/* * __kernel_rem_pio2(x,y,e0,nx,prec) * double x[],y[]; int e0,nx,prec; * * __kernel_rem_pio2 return the last three digits of N with * y = x - N*pi/2 * so that |y| < pi/2. * * The method is to compute the integer (mod 8) and fraction parts of * (2/pi)*x without doing the full multiplication. In general we * skip the part of the product that are known to be a huge integer ( * more accurately, = 0 mod 8 ). Thus the number of operations are * independent of the exponent of the input. * * (2/pi) is represented by an array of 24-bit integers in ipio2[]. * * Input parameters: * x[] The input value (must be positive) is broken into nx * pieces of 24-bit integers in double precision format. * x[i] will be the i-th 24 bit of x. The scaled exponent * of x[0] is given in input parameter e0 (i.e., x[0]*2^e0 * match x's up to 24 bits. * * Example of breaking a double positive z into x[0]+x[1]+x[2]: * e0 = ilogb(z)-23 * z = scalbn(z,-e0) * for i = 0,1,2 * x[i] = floor(z) * z = (z-x[i])*2**24 * * * y[] output result in an array of double precision numbers. * The dimension of y[] is: * 24-bit precision 1 * 53-bit precision 2 * 64-bit precision 2 * 113-bit precision 3 * The actual value is the sum of them. Thus for 113-bit * precision, one may have to do something like: * * long double t,w,r_head, r_tail; * t = (long double)y[2] + (long double)y[1]; * w = (long double)y[0]; * r_head = t+w; * r_tail = w - (r_head - t); * * e0 The exponent of x[0]. Must be <= 16360 or you need to * expand the ipio2 table. * * nx dimension of x[] * * prec an integer indicating the precision: * 0 24 bits (single) * 1 53 bits (double) * 2 64 bits (extended) * 3 113 bits (quad) * * External function: * double scalbn(), floor(); * * * Here is the description of some local variables: * * jk jk+1 is the initial number of terms of ipio2[] needed * in the computation. The minimum and recommended value * for jk is 3,4,4,6 for single, double, extended, and quad. * jk+1 must be 2 larger than you might expect so that our * recomputation test works. (Up to 24 bits in the integer * part (the 24 bits of it that we compute) and 23 bits in * the fraction part may be lost to cancellation before we * recompute.) * * jz local integer variable indicating the number of * terms of ipio2[] used. * * jx nx - 1 * * jv index for pointing to the suitable ipio2[] for the * computation. In general, we want * ( 2^e0*x[0] * ipio2[jv-1]*2^(-24jv) )/8 * is an integer. Thus * e0-3-24*jv >= 0 or (e0-3)/24 >= jv * Hence jv = max(0,(e0-3)/24). * * jp jp+1 is the number of terms in PIo2[] needed, jp = jk. * * q[] double array with integral value, representing the * 24-bits chunk of the product of x and 2/pi. * * q0 the corresponding exponent of q[0]. Note that the * exponent for q[i] would be q0-24*i. * * PIo2[] double precision array, obtained by cutting pi/2 * into 24 bits chunks. * * f[] ipio2[] in floating point * * iq[] integer array by breaking up q[] in 24-bits chunk. * * fq[] final product of x*(2/pi) in fq[0],..,fq[jk] * * ih integer. If >0 it indicates q[] is >= 0.5, hence * it also indicates the *sign* of the result. *
*/
/* * Constants: * The hexadecimal values are the intended ones for the following * constants. The decimal values may be used, provided that the * compiler will convert from decimal to binary accurately enough * to produce the hexadecimal values shown.
*/
#include <float.h>
#include"math_private.h"
staticconstint init_jk[] = {3,4,4,6}; /* initial value for jk */
/* * Table of constants for 2/pi, 396 Hex digits (476 decimal) of 2/pi * * integer array, contains the (24*i)-th to (24*i+23)-th * bit of 2/pi after binary point. The corresponding * floating value is * * ipio2[i] * 2^(-24(i+1)). * * NB: This table must have at least (e0-3)/24 + jk terms. * For quad precision (e0 <= 16360, jk = 6), this is 686.
*/ staticconst int32_t ipio2[] = {
0xA2F983, 0x6E4E44, 0x1529FC, 0x2757D1, 0xF534DD, 0xC0DB62,
0x95993C, 0x439041, 0xFE5163, 0xABDEBB, 0xC561B7, 0x246E3A,
0x424DD2, 0xE00649, 0x2EEA09, 0xD1921C, 0xFE1DEB, 0x1CB129,
0xA73EE8, 0x8235F5, 0x2EBB44, 0x84E99C, 0x7026B4, 0x5F7E41,
0x3991D6, 0x398353, 0x39F49C, 0x845F8B, 0xBDF928, 0x3B1FF8,
0x97FFDE, 0x05980F, 0xEF2F11, 0x8B5A0A, 0x6D1F6D, 0x367ECF,
0x27CB09, 0xB74F46, 0x3F669E, 0x5FEA2D, 0x7527BA, 0xC7EBE5,
0xF17B3D, 0x0739F7, 0x8A5292, 0xEA6BFB, 0x5FB11F, 0x8D5D08,
0x560330, 0x46FC7B, 0x6BABF0, 0xCFBC20, 0x9AF436, 0x1DA9E3,
0x91615E, 0xE61B08, 0x659985, 0x5F14A0, 0x68408D, 0xFFD880,
0x4D7327, 0x310606, 0x1556CA, 0x73A8C9, 0x60E27B, 0xC08C6B,
int
__kernel_rem_pio2(double *x, double *y, int e0, int nx, int prec)
{
int32_t jz,jx,jv,jp,jk,carry,n,iq[20],i,j,k,m,q0,ih; double z,fw,f[20],fq[20],q[20];
/* compute n */
z = scalbn(z,q0); /* actual value of z */
z -= 8.0*floor(z*0.125); /* trim off integer >= 8 */
n = (int32_t) z;
z -= (double)n;
ih = 0; if(q0>0) { /* need iq[jz-1] to determine n */
i = (iq[jz-1]>>(24-q0)); n += i;
iq[jz-1] -= i<<(24-q0);
ih = iq[jz-1]>>(23-q0);
} elseif(q0==0) ih = iq[jz-1]>>23; elseif(z>=0.5) ih=2;
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