/* * 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 2006 Sun Microsystems, Inc. All rights reserved. * Use is subject to license terms. */ /* INDENT OFF */ /* * __k_tan( double x; double y; int k ) * kernel tan/cotan function on [-pi/4, pi/4], pi/4 ~ 0.785398164 * Input x is assumed to be bounded by ~pi/4 in magnitude. * Input y is the tail of x. * Input k indicate -- tan if k=0; else -1/tan * * Table look up algorithm * 1. by tan(-x) = -tan(x), need only to consider positive x * 2. if x < 5/32 = [0x3fc40000, 0] = 0.15625 , then * if x < 2^-27 (hx < 0x3e400000 0), set w=x with inexact if x != 0 * else * z = x*x; * w = x + (y+(x*z)*(t1+z*(t2+z*(t3+z*(t4+z*(t5+z*t6)))))) * return (k == 0)? w: 1/w; * 3. else * ht = (hx + 0x4000)&0x7fff8000 (round x to a break point t) * lt = 0 * i = (hy-0x3fc40000)>>15; (i<=64) * x' = (x - t)+y (|x'| ~<= 2^-7) * By * tan(t+x') * = (tan(t)+tan(x'))/(1-tan(x')tan(t)) * We have * sin(x')+tan(t)*(tan(t)*sin(x')) * = tan(t) + ------------------------------- for k=0 * cos(x') - tan(t)*sin(x') * * cos(x') - tan(t)*sin(x') * = - -------------------------------------- for k=1 * tan(t) + tan(t)*(cos(x')-1) + sin(x') * * * where tan(t) is from the table, * sin(x') = x + pp1*x^3 + pp2*x^5 * cos(x') = 1 + qq1*x^2 + qq2*x^4 */ #include "libm.h" extern const double _TBL_tan_hi[], _TBL_tan_lo[]; static const double q[] = { /* one = */ 1.0, /* * 2 2 -59.56 * |sin(x) - pp1*x*(pp2+x *(pp3+x )| <= 2 for |x|<1/64 */ /* pp1 = */ 8.33326120969096230395312119298978359438478946686e-0003, /* pp2 = */ 1.20001038589438965215025680596868692381425944526e+0002, /* pp3 = */ -2.00001730975089451192161504877731204032897949219e+0001, /* * 2 2 -56.19 * |cos(x) - (1+qq1*x (qq2+x ))| <= 2 for |x|<=1/128 */ /* qq1 = */ 4.16665486385721928197511942926212213933467864990e-0002, /* qq2 = */ -1.20000339921340035687080671777948737144470214844e+0001, /* * |tan(x) - PF(x)| * |--------------| <= 2^-58.57 for |x|<0.15625 * | x | * * where (let z = x*x) * PF(x) = x + (t1*x*z)(t2 + z(t3 + z))(t4 + z)(t5 + z(t6 + z)) */ /* t1 = */ 3.71923358986516816929168705030406272271648049355e-0003, /* t2 = */ 6.02645120354857866118436504621058702468872070312e+0000, /* t3 = */ 2.42627327587398156083509093150496482849121093750e+0000, /* t4 = */ 2.44968983934252770851003333518747240304946899414e+0000, /* t5 = */ 6.07089252571767978849948121933266520500183105469e+0000, /* t6 = */ -2.49403756995593761658369658107403665781021118164e+0000, }; #define one q[0] #define pp1 q[1] #define pp2 q[2] #define pp3 q[3] #define qq1 q[4] #define qq2 q[5] #define t1 q[6] #define t2 q[7] #define t3 q[8] #define t4 q[9] #define t5 q[10] #define t6 q[11] /* INDENT ON */ double __k_tan(double x, double y, int k) { double a, t, z, w = 0.0L, s, c, r, rh, xh, xl; int i, j, hx, ix; t = one; hx = ((int *) &x)[HIWORD]; ix = hx & 0x7fffffff; if (ix < 0x3fc40000) { /* 0.15625 */ if (ix < 0x3e400000) { /* 2^-27 */ if ((i = (int) x) == 0) /* generate inexact */ w = x; t = y; } else { z = x * x; t = y + (((t1 * x) * z) * (t2 + z * (t3 + z))) * ((t4 + z) * (t5 + z * (t6 + z))); w = x + t; } if (k == 0) return (w); /* * Compute -1/(x+T) with great care * Let r = -1/(x+T), rh = r chopped to 20 bits. * Also let xh = x+T chopped to 20 bits, xl = (x-xh)+T. Then * -1/(x+T) = rh + (-1/(x+T)-rh) = rh + r*(1+rh*(x+T)) * = rh + r*((1+rh*xh)+rh*xl). */ rh = r = -one / w; ((int *) &rh)[LOWORD] = 0; xh = w; ((int *) &xh)[LOWORD] = 0; xl = (x - xh) + t; return (rh + r * ((one + rh * xh) + rh * xl)); } j = (ix + 0x4000) & 0x7fff8000; i = (j - 0x3fc40000) >> 15; ((int *) &t)[HIWORD] = j; if (hx > 0) x = y - (t - x); else x = -y - (t + x); a = _TBL_tan_hi[i]; z = x * x; s = (pp1 * x) * (pp2 + z * (pp3 + z)); /* sin(x) */ t = (qq1 * z) * (qq2 + z); /* cos(x) - 1 */ if (k == 0) { w = a * s; t = _TBL_tan_lo[i] + (s + a * w) / (one - (w - t)); return (hx < 0 ? -a - t : a + t); } else { w = s + a * t; c = w + _TBL_tan_lo[i]; t = a * s - t; /* * Now try to compute [(1-T)/(a+c)] accurately * * Let r = 1/(a+c), rh = (1-T)*r chopped to 20 bits. * Also let xh = a+c chopped to 20 bits, xl = (a-xh)+c. Then * (1-T)/(a+c) = rh + ((1-T)/(a+c)-rh) * = rh + r*(1-T-rh*(a+c)) * = rh + r*((1-T-rh*xh)-rh*xl) * = rh + r*(((1-rh*xh)-T)-rh*xl) */ r = one / (a + c); rh = (one - t) * r; ((int *) &rh)[LOWORD] = 0; xh = a + c; ((int *) &xh)[LOWORD] = 0; xl = (a - xh) + c; z = rh + r * (((one - rh * xh) - t) - rh * xl); return (hx >= 0 ? -z : z); } }
