25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis * CDDL HEADER START
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25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis * You may not use this file except in compliance with the License.
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis * You can obtain a copy of the license at usr/src/OPENSOLARIS.LICENSE
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis * or http://www.opensolaris.org/os/licensing.
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25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis * Copyright 2011 Nexenta Systems, Inc. All rights reserved.
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis * Copyright 2005 Sun Microsystems, Inc. All rights reserved.
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis * Use is subject to license terms.
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis * int __rem_pio2m(x,y,e0,nx,prec,ipio2)
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis * double x[],y[]; int e0,nx,prec; const int ipio2[];
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis * __rem_pio2m return the last three digits of N with
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis * y = x - N*pi/2
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis * so that |y| < pi/4.
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis * The method is to compute the integer (mod 8) and fraction parts of
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis * (2/pi)*x without doing the full multiplication. In general we
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis * skip the part of the product that are known to be a huge integer (
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis * more accurately, = 0 mod 8 ). Thus the number of operations are
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis * independent of the exponent of the input.
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis * (2/PI) is represented by an array of 24-bit integers in ipio2[].
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis * Here PI could as well be a machine value pi.
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis * Input parameters:
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis * x[] The input value (must be positive) is broken into nx
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis * pieces of 24-bit integers in double precision format.
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis * x[i] will be the i-th 24 bit of x. The scaled exponent
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis * of x[0] is given in input parameter e0 (i.e., x[0]*2^e0
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis * match x's up to 24 bits.
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis * Example of breaking a double z into x[0]+x[1]+x[2]:
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis * e0 = ilogb(z)-23
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis * z = scalbn(z,-e0)
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis * for i = 0,1,2
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis * x[i] = floor(z)
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis * z = (z-x[i])*2**24
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis * y[] ouput result in an array of double precision numbers.
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis * The dimension of y[] is:
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis * 24-bit precision 1
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis * 53-bit precision 2
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis * 64-bit precision 2
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis * 113-bit precision 3
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis * The actual value is the sum of them. Thus for 113-bit
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis * precsion, one may have to do something like:
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis * long double t,w,r_head, r_tail;
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis * t = (long double)y[2] + (long double)y[1];
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis * w = (long double)y[0];
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis * r_head = t+w;
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis * r_tail = w - (r_head - t);
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis * e0 The exponent of x[0]
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis * nx dimension of x[]
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis * prec an interger indicating the precision:
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis * 0 24 bits (single)
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis * 1 53 bits (double)
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis * 2 64 bits (extended)
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis * 3 113 bits (quad)
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis * integer array, contains the (24*i)-th to (24*i+23)-th
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis * bit of 2/pi or 2/PI after binary point. The corresponding
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis * floating value is
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis * ipio2[i] * 2^(-24(i+1)).
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis * External function:
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis * double scalbn( ), floor( );
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis * Here is the description of some local variables:
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis * jk jk+1 is the initial number of terms of ipio2[] needed
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis * in the computation. The recommended value is 3,4,4,
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis * 6 for single, double, extended,and quad.
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis * jz local integer variable indicating the number of
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis * terms of ipio2[] used.
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis * jv index for pointing to the suitable ipio2[] for the
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis * computation. In general, we want
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis * ( 2^e0*x[0] * ipio2[jv-1]*2^(-24jv) )/8
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis * is an integer. Thus
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis * e0-3-24*jv >= 0 or (e0-3)/24 >= jv
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis * Hence jv = max(0,(e0-3)/24).
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis * jp jp+1 is the number of terms in pio2[] needed, jp = jk.
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis * q[] double array with integral value, representing the
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis * 24-bits chunk of the product of x and 2/pi.
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis * q0 the corresponding exponent of q[0]. Note that the
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis * exponent for q[i] would be q0-24*i.
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis * pio2[] double precision array, obtained by cutting pi/2
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis * into 24 bits chunks.
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis * f[] ipio2[] in floating point
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis * iq[] integer array by breaking up q[] in 24-bits chunk.
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis * fq[] final product of x*(2/pi) in fq[0],..,fq[jk]
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis * ih integer. If >0 it indicats q[] is >= 0.5, hence
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis * it also indicates the *sign* of the result.
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtisextern int __swapRP(int);
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtisstatic const int init_jk[] = { 3, 4, 4, 6 }; /* initial value for jk */
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis 1.57079625129699707031e+00,
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis 7.54978941586159635335e-08,
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis 5.39030252995776476554e-15,
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis 3.28200341580791294123e-22,
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis 1.27065575308067607349e-29,
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis 1.22933308981111328932e-36,
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis 2.73370053816464559624e-44,
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis 2.16741683877804819444e-51,
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtisstatic const double
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis__rem_pio2m(double *x, double *y, int e0, int nx, int prec, const int *ipio2)
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis /* initialize jk */
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis /* determine jx,jv,q0, note that 3>q0 */
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis /* set up f[0] to f[jx+jk] where f[jx+jk] = ipio2[jv+jk] */
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis for (i = 0; i <= m; i++, j++)
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis /* compute q[0],q[1],...q[jk] */
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis for (i = 0; i <= jk; i++) {
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis /* distill q[] into iq[] reversingly */
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis for (i = 0, j = jz, z = q[jz]; j > 0; i++, j--) {
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis /* compute n */
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis z = scalbn(z, q0); /* actual value of z */
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis z -= eight * floor(z * eighth); /* trim off integer >= 8 */
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis z -= (double)n;
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis if (q0 > 0) { /* need iq[jz-1] to determine n */
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis } else if (q0 == 0) {
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis } else if (z >= half) {
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis for (i = 0; i < jz; i++) { /* compute 1-q */
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis if (q0 > 0) { /* rare case: chance is 1 in 12 */
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis /* check if recomputation is needed */
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis if (j == 0) { /* need recomputation */
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis /* set k to no. of terms needed */
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis /* add q[jz+1] to q[jz+k] */
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis /* cut out zero terms */
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis } else { /* break z into 24-bit if neccessary */
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis /* convert integer "bit" chunk to floating-point value */
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis for (i = jz; i >= 0; i--) {
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis /* compute pio2[0,...,jp]*q[jz,...,0] */
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis for (i = jz; i >= 0; i--) {
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis for (fw = zero, k = 0; k <= jp && k <= jz - i; k++)
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis /* compress fq[] into y[] */
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis for (i = jz; i >= 0; i--)
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis for (i = jz; i >= 0; i--)
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis for (i = jz; i > 0; i--) {
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis return (n & 7);