/* * CDDL HEADER START * * The contents of this file are subject to the terms of the * Common Development and Distribution License (the "License"). * You may not use this file except in compliance with the License. * * You can obtain a copy of the license at usr/src/OPENSOLARIS.LICENSE * or http://www.opensolaris.org/os/licensing. * See the License for the specific language governing permissions * and limitations under the License. * * When distributing Covered Code, include this CDDL HEADER in each * file and include the License file at usr/src/OPENSOLARIS.LICENSE. * If applicable, add the following below this CDDL HEADER, with the * fields enclosed by brackets "[]" replaced with your own identifying * information: Portions Copyright [yyyy] [name of copyright owner] * * CDDL HEADER END */ /* * Copyright 2011 Nexenta Systems, Inc. All rights reserved. */ /* * Copyright 2005 Sun Microsystems, Inc. All rights reserved. * Use is subject to license terms. */ /* * int __rem_pio2m(x,y,e0,nx,prec,ipio2) * double x[],y[]; int e0,nx,prec; const int ipio2[]; * * __rem_pio2m return the last three digits of N with * y = x - N*pi/2 * so that |y| < pi/4. * * 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[]. * Here PI could as well be a machine value pi. * * 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 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[] ouput 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 * precsion, 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] * * nx dimension of x[] * * prec an interger indicating the precision: * 0 24 bits (single) * 1 53 bits (double) * 2 64 bits (extended) * 3 113 bits (quad) * * ipio2[] * integer array, contains the (24*i)-th to (24*i+23)-th * bit of 2/pi or 2/PI after binary point. The corresponding * floating value is * * ipio2[i] * 2^(-24(i+1)). * * 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 recommended value is 3,4,4, * 6 for single, double, extended,and quad. * * 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 indicats q[] is >= 0.5, hence * it also indicates the *sign* of the result. * */ #include <assert.h> #include "libm.h" #if defined(__i386) && !defined(__amd64) extern int __swapRP(int); #endif static const int init_jk[] = { 3, 4, 4, 6 }; /* initial value for jk */ static const double pio2[] = { 1.57079625129699707031e+00, 7.54978941586159635335e-08, 5.39030252995776476554e-15, 3.28200341580791294123e-22, 1.27065575308067607349e-29, 1.22933308981111328932e-36, 2.73370053816464559624e-44, 2.16741683877804819444e-51, }; static const double zero = 0.0, one = 1.0, half = 0.5, eight = 8.0, eighth = 0.125, two24 = 16777216.0, twon24 = 5.960464477539062500E-8; int __rem_pio2m(double *x, double *y, int e0, int nx, int prec, const int *ipio2) { int jz, jx, jv, jp, jk, carry, n, iq[20]; int i, j, k, m, q0, ih; double z, fw, f[20], fq[20], q[20]; #if defined(__i386) && !defined(__amd64) int rp; rp = __swapRP(fp_extended); #endif fq[0] = NAN; /* Make gcc happy */ /* initialize jk */ jp = jk = init_jk[prec]; /* determine jx,jv,q0, note that 3>q0 */ jx = nx - 1; jv = (e0 - 3) / 24; if (jv < 0) jv = 0; q0 = e0 - 24 * (jv + 1); /* set up f[0] to f[jx+jk] where f[jx+jk] = ipio2[jv+jk] */ j = jv - jx; m = jx + jk; for (i = 0; i <= m; i++, j++) f[i] = (j < 0)? zero : (double)ipio2[j]; /* compute q[0],q[1],...q[jk] */ for (i = 0; i <= jk; i++) { for (j = 0, fw = zero; j <= jx; j++) fw += x[j] * f[jx+i-j]; q[i] = fw; } jz = jk; recompute: /* distill q[] into iq[] reversingly */ for (i = 0, j = jz, z = q[jz]; j > 0; i++, j--) { fw = (double)((int)(twon24 * z)); iq[i] = (int)(z - two24 * fw); z = q[j-1] + fw; } /* compute n */ z = scalbn(z, q0); /* actual value of z */ z -= eight * floor(z * eighth); /* trim off integer >= 8 */ n = (int)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); } else if (q0 == 0) { ih = iq[jz-1] >> 23; } else if (z >= half) { ih = 2; } if (ih > 0) { /* q > 0.5 */ n += 1; carry = 0; for (i = 0; i < jz; i++) { /* compute 1-q */ j = iq[i]; if (carry == 0) { if (j != 0) { carry = 1; iq[i] = 0x1000000 - j; } } else { iq[i] = 0xffffff - j; } } if (q0 > 0) { /* rare case: chance is 1 in 12 */ switch (q0) { case 1: iq[jz-1] &= 0x7fffff; break; case 2: iq[jz-1] &= 0x3fffff; break; } } if (ih == 2) { z = one - z; if (carry != 0) z -= scalbn(one, q0); } } /* check if recomputation is needed */ if (z == zero) { j = 0; for (i = jz - 1; i >= jk; i--) j |= iq[i]; if (j == 0) { /* need recomputation */ /* set k to no. of terms needed */ for (k = 1; iq[jk-k] == 0; k++) ; /* add q[jz+1] to q[jz+k] */ for (i = jz + 1; i <= jz + k; i++) { f[jx+i] = (double)ipio2[jv+i]; for (j = 0, fw = zero; j <= jx; j++) fw += x[j] * f[jx+i-j]; q[i] = fw; } jz += k; goto recompute; } } /* cut out zero terms */ if (z == zero) { jz -= 1; q0 -= 24; while (iq[jz] == 0) { jz--; q0 -= 24; } } else { /* break z into 24-bit if neccessary */ z = scalbn(z, -q0); if (z >= two24) { fw = (double)((int)(twon24 * z)); iq[jz] = (int)(z - two24 * fw); jz += 1; q0 += 24; iq[jz] = (int)fw; } else { iq[jz] = (int)z; } } /* convert integer "bit" chunk to floating-point value */ fw = scalbn(one, q0); for (i = jz; i >= 0; i--) { q[i] = fw * (double)iq[i]; fw *= twon24; } /* compute pio2[0,...,jp]*q[jz,...,0] */ for (i = jz; i >= 0; i--) { for (fw = zero, k = 0; k <= jp && k <= jz - i; k++) fw += pio2[k] * q[i+k]; fq[jz-i] = fw; } /* compress fq[] into y[] */ switch (prec) { case 0: fw = zero; for (i = jz; i >= 0; i--) fw += fq[i]; y[0] = (ih == 0)? fw : -fw; break; case 1: case 2: fw = zero; for (i = jz; i >= 0; i--) fw += fq[i]; y[0] = (ih == 0)? fw : -fw; assert(!isnan(fq[0])); fw = fq[0] - fw; for (i = 1; i <= jz; i++) fw += fq[i]; y[1] = (ih == 0)? fw : -fw; break; default: for (i = jz; i > 0; i--) { fw = fq[i-1] + fq[i]; fq[i] += fq[i-1] - fw; fq[i-1] = fw; } for (i = jz; i > 1; i--) { fw = fq[i-1] + fq[i]; fq[i] += fq[i-1] - fw; fq[i-1] = fw; } for (fw = zero, i = jz; i >= 2; i--) fw += fq[i]; if (ih == 0) { y[0] = fq[0]; y[1] = fq[1]; y[2] = fw; } else { y[0] = -fq[0]; y[1] = -fq[1]; y[2] = -fw; } } #if defined(__i386) && !defined(__amd64) (void) __swapRP(rp); #endif return (n & 7); }
