/* * ==================================================== * Copyright 2004 Sun Microsystems, Inc. All Rights Reserved. * * Permission to use, copy, modify, and distribute this * software is freely granted, provided that this notice * is preserved. * ====================================================
*/
/* INDENT OFF */ //#include <sys/cdefs.h> //__FBSDID("$FreeBSD$");
/* __kernel_tan( x, y, k ) * kernel tan function on ~[-pi/4, pi/4] (except on -0), pi/4 ~ 0.7854 * Input x is assumed to be bounded by ~pi/4 in magnitude. * Input y is the tail of x. * Input k indicates whether tan (if k = 1) or -1/tan (if k = -1) is returned. * * Algorithm * 1. Since tan(-x) = -tan(x), we need only to consider positive x. * 2. Callers must return tan(-0) = -0 without calling here since our * odd polynomial is not evaluated in a way that preserves -0. * Callers may do the optimization tan(x) ~ x for tiny x. * 3. tan(x) is approximated by a odd polynomial of degree 27 on * [0,0.67434] * 3 27 * tan(x) ~ x + T1*x + ... + T13*x * where * * |tan(x) 2 4 26 | -59.2 * |----- - (1+T1*x +T2*x +.... +T13*x )| <= 2 * | x | * * Note: tan(x+y) = tan(x) + tan'(x)*y * ~ tan(x) + (1+x*x)*y * Therefore, for better accuracy in computing tan(x+y), let * 3 2 2 2 2 * r = x *(T2+x *(T3+x *(...+x *(T12+x *T13)))) * then * 3 2 * tan(x+y) = x + (T1*x + (x *(r+y)+y)) * * 4. For x in [0.67434,pi/4], let y = pi/4 - x, then * tan(x) = tan(pi/4-y) = (1-tan(y))/(1+tan(y)) * = 1 - 2*(tan(y) - (tan(y)^2)/(1+tan(y)))
*/
double
__kernel_tan(double x, double y, int iy) { double z, r, v, w, s;
int32_t ix, hx;
GET_HIGH_WORD(hx,x);
ix = hx & 0x7fffffff; /* high word of |x| */ if (ix >= 0x3FE59428) { /* |x| >= 0.6744 */ if (hx < 0) {
x = -x;
y = -y;
}
z = pio4 - x;
w = pio4lo - y;
x = z + w;
y = 0.0;
}
z = x * x;
w = z * z; /* * Break x^5*(T[1]+x^2*T[2]+...) into * x^5(T[1]+x^4*T[3]+...+x^20*T[11]) + * x^5(x^2*(T[2]+x^4*T[4]+...+x^22*[T12]))
*/
r = T[1] + w * (T[3] + w * (T[5] + w * (T[7] + w * (T[9] +
w * T[11]))));
v = z * (T[2] + w * (T[4] + w * (T[6] + w * (T[8] + w * (T[10] +
w * T[12])))));
s = z * x;
r = y + z * (s * (r + v) + y);
r += T[0] * s;
w = x + r; if (ix >= 0x3FE59428) {
v = (double) iy; return (double) (1 - ((hx >> 30) & 2)) *
(v - 2.0 * (x - (w * w / (w + v) - r)));
} if (iy == 1) return w; else { /* * if allow error up to 2 ulp, simply return * -1.0 / (x+r) here
*/ /* compute -1.0 / (x+r) accurately */ double a, t;
z = w;
SET_LOW_WORD(z,0);
v = r - (z - x); /* z+v = r+x */
t = a = -1.0 / w; /* a = -1.0/w */
SET_LOW_WORD(t,0);
s = 1.0 + t * z; return t + a * (s + t * v);
}
}
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