xref: /freebsd/lib/msun/src/s_tanpi.c (revision dce5f3abed7181cc533ca5ed3de44517775e78dd)

/*-
 * Copyright (c) 2017 Steven G. Kargl
 * All rights reserved.
 *
 * Redistribution and use in source and binary forms, with or without
 * modification, are permitted provided that the following conditions
 * are met:
 * 1. Redistributions of source code must retain the above copyright
 *    notice unmodified, this list of conditions, and the following
 *    disclaimer.
 * 2. Redistributions in binary form must reproduce the above copyright
 *    notice, this list of conditions and the following disclaimer in the
 *    documentation and/or other materials provided with the distribution.
 *
 * THIS SOFTWARE IS PROVIDED BY THE AUTHOR ``AS IS'' AND ANY EXPRESS OR
 * IMPLIED WARRANTIES, INCLUDING, BUT NOT LIMITED TO, THE IMPLIED WARRANTIES
 * OF MERCHANTABILITY AND FITNESS FOR A PARTICULAR PURPOSE ARE DISCLAIMED.
 * IN NO EVENT SHALL THE AUTHOR BE LIABLE FOR ANY DIRECT, INDIRECT,
 * INCIDENTAL, SPECIAL, EXEMPLARY, OR CONSEQUENTIAL DAMAGES (INCLUDING, BUT
 * NOT LIMITED TO, PROCUREMENT OF SUBSTITUTE GOODS OR SERVICES; LOSS OF USE,
 * DATA, OR PROFITS; OR BUSINESS INTERRUPTION) HOWEVER CAUSED AND ON ANY
 * THEORY OF LIABILITY, WHETHER IN CONTRACT, STRICT LIABILITY, OR TORT
 * (INCLUDING NEGLIGENCE OR OTHERWISE) ARISING IN ANY WAY OUT OF THE USE OF
 * THIS SOFTWARE, EVEN IF ADVISED OF THE POSSIBILITY OF SUCH DAMAGE.
 */

/**
 * tanpi(x) computes tan(pi*x) without multiplication by pi (almost).  First,
 * note that tanpi(-x) = -tanpi(x), so the algorithm considers only |x| and
 * includes reflection symmetry by considering the sign of x on output.  The
 * method used depends on the magnitude of x.
 *
 * 1. For small |x|, tanpi(x) = pi * x where a sloppy threshold is used.  The
 *    threshold is |x| < 0x1pN with N = -(P/2+M).  P is the precision of the
 *    floating-point type and M = 2 to 4.  To achieve high accuracy, pi is
 *    decomposed into high and low parts with the high part containing a
 *    number of trailing zero bits.  x is also split into high and low parts.
 *
 * 2. For |x| < 1, argument reduction is not required and tanpi(x) is
 *    computed by a direct call to a kernel, which uses the kernel for
 *    tan(x).  See below.
 *
 * 3. For 1 <= |x| < 0x1p(P-1), argument reduction is required where
 *    |x| = j0 + r with j0 an integer and the remainder r satisfies
 *    0 <= r < 1.  With the given domain, a simplified inline floor(x)
 *    is used.  Also, note the following identity
 *
 *                                   tan(pi*j0) + tan(pi*r)
 *    tanpi(x) = tan(pi*(j0+r)) = ---------------------------- = tanpi(r)
 *                                 1 - tan(pi*j0) * tan(pi*r)
 *
 *    So, after argument reduction, the kernel is again invoked.
 *
 * 4. For |x| >= 0x1p(P-1), |x| is integral and tanpi(x) = copysign(0,x).
 *
 * 5. Special cases:
 *
 *    tanpi(+-0) = +-0
 *    tanpi(+-n) = +-0, for positive integers n.
 *    tanpi(+-n+1/4) = +-1, for positive integers n.
 *    tanpi(+-n+1/2) = NaN, for positive integers n.
 *    tanpi(+-inf) = NaN.  Raises the "invalid" floating-point exception.
 *    tanpi(nan) = NaN.  Raises the "invalid" floating-point exception.
 */

#include "math.h"
#include "math_private.h"

static const double
pi_hi =  3.1415926814079285e+00,	/* 0x400921fb 0x58000000 */
pi_lo = -2.7818135228334233e-08;	/* 0xbe5dde97 0x3dcb3b3a */

/*
 * The kernel for tanpi(x) multiplies x by an 80-bit approximation of
 * pi, where the hi and lo parts are used with with kernel for tan(x).
 */
static inline double
__kernel_tanpi(double x)
{
	double_t hi, lo, t;

	if (x < 0.25) {
		hi = (float)x;
		lo = x - hi;
		lo = lo * (pi_lo + pi_hi) + hi * pi_lo;
		hi *= pi_hi;
		_2sumF(hi, lo);
		t = __kernel_tan(hi, lo, 1);
	} else if (x > 0.25) {
		x = 0.5 - x;
		hi = (float)x;
		lo = x - hi;
		lo = lo * (pi_lo + pi_hi) + hi * pi_lo;
		hi *= pi_hi;
		_2sumF(hi, lo);
		t = - __kernel_tan(hi, lo, -1);
	} else
		t = 1;

	return (t);
}

volatile static const double vzero = 0;

double
tanpi(double x)
{
	double ax, hi, lo, t;
	uint32_t hx, ix, j0, lx;

	EXTRACT_WORDS(hx, lx, x);
	ix = hx & 0x7fffffff;
	INSERT_WORDS(ax, ix, lx);

	if (ix < 0x3ff00000) {			/* |x| < 1 */
		if (ix < 0x3fe00000) {		/* |x| < 0.5 */
			if (ix < 0x3e200000) {	/* |x| < 0x1p-29 */
				if (x == 0)
					return (x);
				/*
				 * To avoid issues with subnormal values,
				 * scale the computation and rescale on
				 * return.
				 */
				INSERT_WORDS(hi, hx, 0);
				hi *= 0x1p53;
				lo = x * 0x1p53 - hi;
				t = (pi_lo + pi_hi) * lo + pi_lo * hi +
				    pi_hi * hi;
				return (t * 0x1p-53);
			}
			t = __kernel_tanpi(ax);
		} else if (ax == 0.5)
			return ((ax - ax) / (ax - ax));
		else
			t = - __kernel_tanpi(1 - ax);
		return ((hx & 0x80000000) ? -t : t);
	}

	if (ix < 0x43300000) {		/* 1 <= |x| < 0x1p52 */
		/* Determine integer part of ax. */
		j0 = ((ix >> 20) & 0x7ff) - 0x3ff;
		if (j0 < 20) {
			ix &= ~(0x000fffff >> j0);
			lx = 0;
		} else {
			lx &= ~(((uint32_t)(0xffffffff)) >> (j0 - 20));
		}
		INSERT_WORDS(x,ix,lx);

		ax -= x;
		EXTRACT_WORDS(ix, lx, ax);

		if (ix < 0x3fe00000)		/* |x| < 0.5 */
			t = ax == 0 ? 0 : __kernel_tanpi(ax);
		else if (ax == 0.5)
			return ((ax - ax) / (ax - ax));
		else
			t = - __kernel_tanpi(1 - ax);

		return ((hx & 0x80000000) ? -t : t);
	}

	/* x = +-inf or nan. */
	if (ix >= 0x7f800000)
		return (vzero / vzero);

	/*
	 * |x| >= 0x1p52 is always an integer, so return +-0.
	 */
	return (copysign(0, x));
}

#if LDBL_MANT_DIG == 53
__weak_reference(tanpi, tanpil);
#endif