25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis * CDDL HEADER START
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis * The contents of this file are subject to the terms of the
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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.
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis * See the License for the specific language governing permissions
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25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis * CDDL HEADER END
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis * Copyright 2011 Nexenta Systems, Inc. All rights reserved.
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis * Copyright 2006 Sun Microsystems, Inc. All rights reserved.
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis * Use is subject to license terms.
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis/* INDENT OFF */
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis * __k_tan( double x; double y; int k )
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis * kernel tan/cotan function on [-pi/4, pi/4], pi/4 ~ 0.785398164
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis * Input x is assumed to be bounded by ~pi/4 in magnitude.
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis * Input y is the tail of x.
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis * Input k indicate -- tan if k=0; else -1/tan
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis * Table look up algorithm
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis * 1. by tan(-x) = -tan(x), need only to consider positive x
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis * 2. if x < 5/32 = [0x3fc40000, 0] = 0.15625 , then
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis * if x < 2^-27 (hx < 0x3e400000 0), set w=x with inexact if x != 0
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis * w = x + (y+(x*z)*(t1+z*(t2+z*(t3+z*(t4+z*(t5+z*t6))))))
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis * return (k == 0)? w: 1/w;
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis * ht = (hx + 0x4000)&0x7fff8000 (round x to a break point t)
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis * i = (hy-0x3fc40000)>>15; (i<=64)
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis * x' = (x - t)+y (|x'| ~<= 2^-7)
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis * = (tan(t)+tan(x'))/(1-tan(x')tan(t))
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis * sin(x')+tan(t)*(tan(t)*sin(x'))
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis * = tan(t) + ------------------------------- for k=0
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis * cos(x') - tan(t)*sin(x')
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis * cos(x') - tan(t)*sin(x')
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis * = - -------------------------------------- for k=1
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis * tan(t) + tan(t)*(cos(x')-1) + sin(x')
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis * where tan(t) is from the table,
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis * sin(x') = x + pp1*x^3 + pp2*x^5
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis * cos(x') = 1 + qq1*x^2 + qq2*x^4
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtisextern const double _TBL_tan_hi[], _TBL_tan_lo[];
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtisstatic const double q[] = {
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis * |sin(x) - pp1*x*(pp2+x *(pp3+x )| <= 2 for |x|<1/64
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis/* pp1 = */ 8.33326120969096230395312119298978359438478946686e-0003,
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis/* pp2 = */ 1.20001038589438965215025680596868692381425944526e+0002,
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis/* pp3 = */ -2.00001730975089451192161504877731204032897949219e+0001,
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis * |cos(x) - (1+qq1*x (qq2+x ))| <= 2 for |x|<=1/128
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis/* qq1 = */ 4.16665486385721928197511942926212213933467864990e-0002,
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis/* qq2 = */ -1.20000339921340035687080671777948737144470214844e+0001,
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis * |tan(x) - PF(x)|
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis * |--------------| <= 2^-58.57 for |x|<0.15625
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis * where (let z = x*x)
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis * PF(x) = x + (t1*x*z)(t2 + z(t3 + z))(t4 + z)(t5 + z(t6 + z))
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis/* t1 = */ 3.71923358986516816929168705030406272271648049355e-0003,
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis/* t2 = */ 6.02645120354857866118436504621058702468872070312e+0000,
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis/* t3 = */ 2.42627327587398156083509093150496482849121093750e+0000,
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis/* t4 = */ 2.44968983934252770851003333518747240304946899414e+0000,
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis/* t5 = */ 6.07089252571767978849948121933266520500183105469e+0000,
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis/* t6 = */ -2.49403756995593761658369658107403665781021118164e+0000,
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis/* INDENT ON */
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis double a, t, z, w = 0.0L, s, c, r, rh, xh, xl;
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis if ((i = (int) x) == 0) /* generate inexact */
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis t = y + (((t1 * x) * z) * (t2 + z * (t3 + z))) *
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis * Compute -1/(x+T) with great care
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis * Let r = -1/(x+T), rh = r chopped to 20 bits.
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis * Also let xh = x+T chopped to 20 bits, xl = (x-xh)+T. Then
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis * -1/(x+T) = rh + (-1/(x+T)-rh) = rh + r*(1+rh*(x+T))
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis * = rh + r*((1+rh*xh)+rh*xl).
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis return (rh + r * ((one + rh * xh) + rh * xl));
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis ((int *) &t)[HIWORD] = j;
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis x = y - (t - x);
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis x = -y - (t + x);
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis s = (pp1 * x) * (pp2 + z * (pp3 + z)); /* sin(x) */
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis t = (qq1 * z) * (qq2 + z); /* cos(x) - 1 */
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis t = _TBL_tan_lo[i] + (s + a * w) / (one - (w - t));
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis return (hx < 0 ? -a - t : a + t);
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis w = s + a * t;
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis t = a * s - t;
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis * Now try to compute [(1-T)/(a+c)] accurately
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis * Let r = 1/(a+c), rh = (1-T)*r chopped to 20 bits.
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis * Also let xh = a+c chopped to 20 bits, xl = (a-xh)+c. Then
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis * (1-T)/(a+c) = rh + ((1-T)/(a+c)-rh)
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis * = rh + r*(1-T-rh*(a+c))
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis * = rh + r*((1-T-rh*xh)-rh*xl)
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis * = rh + r*(((1-rh*xh)-T)-rh*xl)
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis r = one / (a + c);
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis z = rh + r * (((one - rh * xh) - t) - rh * xl);
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis return (hx >= 0 ? -z : z);