calc/qtrans.c

/*
 * qtrans - transcendental functions for real numbers
 *
 * Copyright (C) 1999-2007,2021-2023  David I. Bell, Landon Curt Noll and Ernest Bowen
 *
 * Primary author:  David I. Bell
 *
 * Calc is open software; you can redistribute it and/or modify it under
 * the terms of the version 2.1 of the GNU Lesser General Public License
 * as published by the Free Software Foundation.
 *
 * Calc is distributed in the hope that it will be useful, but WITHOUT
 * ANY WARRANTY; without even the implied warranty of MERCHANTABILITY
 * or FITNESS FOR A PARTICULAR PURPOSE.	 See the GNU Lesser General
 * Public License for more details.
 *
 * A copy of version 2.1 of the GNU Lesser General Public License is
 * distributed with calc under the filename COPYING-LGPL.  You should have
 * received a copy with calc; if not, write to Free Software Foundation, Inc.
 * 51 Franklin Street, Fifth Floor, Boston, MA  02110-1301, USA.
 *
 * Under source code control:	1990/02/15 01:48:22
 * File existed as early as:	before 1990
 *
 * Share and enjoy!  :-)	http://www.isthe.com/chongo/tech/comp/calc/
 */

/*
 * Transcendental functions for real numbers.
 * These are sin, cos, exp, ln, power, cosh, sinh.
 */


#include "qmath.h"


#include "errtbl.h"
#include "banned.h"	/* include after system header <> includes */


HALF _qlgenum_[] = { 36744 };
HALF _qlgeden_[] = { 25469 };
NUMBER _qlge_ = { { _qlgenum_, 1, 0 }, { _qlgeden_, 1, 0 }, 1, NULL };

/*
 * cache the natural logarithm of 10 and 2
 */
STATIC NUMBER *ln_10 = NULL;
STATIC NUMBER *ln_10_epsilon = NULL;
STATIC NUMBER *ln_2 = NULL;
STATIC NUMBER *ln_2_epsilon = NULL;
STATIC NUMBER *ln_n = NULL;
STATIC NUMBER *ln_n_epsilon = NULL;

/*
 * cache pi
 *
 * pivalue[LAST_PI_EPSILON] - last epsilon used to calculate pi
 * pivalue[LAST_PI_VALUE] - last calculated pi
 *			      given pivalue[LAST_PI_EPSILON] epsilon
 * pivalue[LAST_PI_DIV_180_EPSILON] - last epsilon used to calculate pi/180
 * pivalue[LAST_PI_DIV_180_VALUE] - last calculated pi/180 given
 *				      pivalue[LAST_PI_DIV_180_EPSILON] epsilon
 * pivalue[LAST_PI_DIV_200_EPSILON] - last epsilon used to calculate pi/200
 * pivalue[LAST_PI_DIV_200_VALUE] - last calculated pi/200 given
 *				      pivalue[LAST_PI_DIV_200_EPSILON] epsilon
 */
enum pi_cache {
	LAST_PI_EPSILON = 0,
	LAST_PI_VALUE,
	LAST_PI_DIV_180_EPSILON,
	LAST_PI_DIV_180_VALUE,
	LAST_PI_DIV_200_EPSILON,
	LAST_PI_DIV_200_VALUE,
	PI_CACHE_LEN	/* must be last */
};
STATIC NUMBER *pivalue[PI_CACHE_LEN] = {
	NULL,	/* LAST_PI_EPSILON */
	NULL,	/* LAST_PI_VALUE */
	NULL,	/* LAST_PI_DIV_180_EPSILON */
	NULL,	/* LAST_PI_DIV_180_VALUE */
	NULL,	/* LAST_PI_DIV_200_EPSILON */
	NULL,	/* LAST_PI_DIV_200_VALUE */
};

/*
 * other static function declarations
 */
STATIC NUMBER *qexprel(NUMBER *q, long bitnum);

/*
 * Evaluate and store in specified locations the sin and cos of a given
 * number to accuracy corresponding to a specified number of binary digits.
 */
void
qsincos(NUMBER *q, long bitnum, NUMBER **vs, NUMBER **vc)
{
	long n, m, k, h, s, t, d;
	NUMBER *qtmp1, *qtmp2;
	ZVALUE X, cossum, sinsum, mul, ztmp1, ztmp2, ztmp3;

	qtmp1 = qqabs(q);
	h = qilog2(qtmp1);
	qfree(qtmp1);
	k = bitnum + h + 1;
	if (k < 0) {
		*vs = qlink(&_qzero_);
		*vc = qlink(&_qone_);
		return;
	}
	s = k;
	if (k) {
		do {
			t = s;
			s = (s + k/s)/2;
		}
		while (t > s);
	}		/* s is int(sqrt(k)) */
	s++;
	if (s < -h)
		s = -h;
	n = h + s;	/* n is number of squaring that will be required */
	m = bitnum + n;
	while (s > 0) {		/* increasing m by ilog2(s) */
		s >>= 1;
		m++;
	}			/* m is working number of bits */
	qtmp1 = qscale(q, m - n);
	zquo(qtmp1->num, qtmp1->den, &X, 24);
	qfree(qtmp1);
	if (ziszero(X)) {
		zfree(X);
		*vs = qlink(&_qzero_);
		*vc = qlink(&_qone_);
		return;
	}
	zbitvalue(m, &cossum);
	zcopy(X, &sinsum);
	zcopy(X, &mul);
	d = 1;
	for (;;) {
		X.sign = !X.sign;
		zmul(X, mul, &ztmp1);
		zfree(X);
		zshift(ztmp1, -m, &ztmp2);
		zfree(ztmp1);
		zdivi(ztmp2, ++d, &X);
		zfree(ztmp2);
		if (ziszero(X))
			break;
		zadd(cossum, X, &ztmp1);
		zfree(cossum);
		cossum = ztmp1;
		zmul(X, mul, &ztmp1);
		zfree(X);
		zshift(ztmp1, -m, &ztmp2);
		zfree(ztmp1);
		zdivi(ztmp2, ++d, &X);
		zfree(ztmp2);
		if (ziszero(X))
			break;
		zadd(sinsum, X, &ztmp1);
		zfree(sinsum);
		sinsum = ztmp1;
	}
	zfree(X);
	zfree(mul);
	while (n-- > 0) {
		zsquare(cossum, &ztmp1);
		zsquare(sinsum, &ztmp2);
		zsub(ztmp1, ztmp2, &ztmp3);
		zfree(ztmp1);
		zfree(ztmp2);
		zmul(cossum, sinsum, &ztmp1);
		zfree(cossum);
		zfree(sinsum);
		zshift(ztmp3, -m, &cossum);
		zfree(ztmp3);
		zshift(ztmp1, 1 - m, &sinsum);
		zfree(ztmp1);
	}
	h = zlowbit(cossum);
	qtmp1 = qalloc();
	if (m > h) {
		zshift(cossum, -h, &qtmp1->num);
		zbitvalue(m - h, &qtmp1->den);
	} else {
		zshift(cossum, - m, &qtmp1->num);
	}
	zfree(cossum);
	*vc = qtmp1;
	h = zlowbit(sinsum);
	qtmp2 = qalloc();
	if (m > h) {
		zshift(sinsum, -h, &qtmp2->num);
		zbitvalue(m - h, &qtmp2->den);
	} else {
		zshift(sinsum, -m, &qtmp2->num);
	}
	zfree(sinsum);
	*vs = qtmp2;
	return;
}

/*
 * Calculate the cosine of a number to a near multiple of epsilon.
 * This calls qsincos() and discards the value of sin.
 */
NUMBER *
qcos(NUMBER *q, NUMBER *epsilon)
{
	NUMBER *sin, *cos, *res;
	long n;

	if (qiszero(epsilon)) {
		math_error("Zero epsilon value for cosine");
		not_reached();
	}
	if (qiszero(q))
		return qlink(&_qone_);
	n = -qilog2(epsilon);
	if (n < 0)
		return qlink(&_qzero_);
	qsincos(q, n + 2, &sin, &cos);
	qfree(sin);
	res = qmappr(cos, epsilon, 24);
	qfree(cos);
	return res;
}


/*
 * This calls qsincos() and discards the value of cos.
 */
NUMBER *
qsin(NUMBER *q, NUMBER *epsilon)
{
	NUMBER *sin, *cos, *res;
	long n;

	if (qiszero(epsilon)) {
		math_error("Zero epsilon value for sine");
		not_reached();
	}
	n = -qilog2(epsilon);
	if (qiszero(q) || n < 0)
		return qlink(&_qzero_);
	qsincos(q, n + 2, &sin, &cos);
	qfree(cos);
	res = qmappr(sin, epsilon, 24);
	qfree(sin);
	return res;
}


/*
 * Calculate the tangent function.
 */
NUMBER *
qtan(NUMBER *q, NUMBER *epsilon)
{
	NUMBER *sin, *cos, *tan, *res;
	long n, k, m;

	if (qiszero(epsilon)) {
		math_error("Zero epsilon value for tangent");
		not_reached();
	}
	if (qiszero(q))
		return qlink(q);
	n = qilog2(epsilon);
	k = (n > 0) ? 4 + n/2 : 4;
	for (;;) {
		qsincos(q, 2 * k - n, &sin, &cos);
		if (qiszero(cos)) {
			qfree(sin);
			qfree(cos);
			k = 2 * k - n + 4;
			continue;
		}
		m = -qilog2(cos);
		if (m < k)
			 break;
		qfree(sin);
		qfree(cos);
		k = m + 1;
	}
	tan = qqdiv(sin, cos);
	qfree(sin);
	qfree(cos);
	res = qmappr(tan, epsilon, 24);
	qfree(tan);
	return res;
}


/*
 * Calculate the cotangent function.
 */
NUMBER *
qcot(NUMBER *q, NUMBER *epsilon)
{
	NUMBER *sin, *cos, *cot, *res;
	long n, k, m;

	if (qiszero(epsilon)) {
		math_error("Zero epsilon value for cotangent");
		not_reached();
	}
	if (qiszero(q)) {
		math_error("Zero argument for cotangent");
		not_reached();
	}
	k = -qilog2(q);
	n = qilog2(epsilon);
	if (k < 0)
		k = (n > 0) ? n/2 : 0;
	k += 4;
	for (;;) {
		qsincos(q, 2 * k - n, &sin, &cos);
		if (qiszero(sin)) {
			qfree(sin);
			qfree(cos);
			k = 2 * k - n + 4;
			continue;
		}
		m = -qilog2(sin);
		if (m < k)
			 break;
		qfree(sin);
		qfree(cos);
		k = m + 1;
	}
	cot = qqdiv(cos, sin);
	qfree(sin);
	qfree(cos);
	res = qmappr(cot, epsilon, 24);
	qfree(cot);
	return res;
}


/*
 * Calculate the secant function.
 */
NUMBER *
qsec(NUMBER *q, NUMBER *epsilon)
{
	NUMBER *sin, *cos, *sec, *res;
	long n, k, m;

	if (qiszero(epsilon)) {
		math_error("Zero epsilon value for secant");
		not_reached();
	}
	if (qiszero(q))
		return qlink(&_qone_);
	n = qilog2(epsilon);
	k = (n > 0) ? 4 + n/2 : 4;
	for (;;) {
		qsincos(q, 2 * k - n, &sin, &cos);
		qfree(sin);
		if (qiszero(cos)) {
			qfree(cos);
			k = 2 * k - n + 4;
			continue;
		}
		m = -qilog2(cos);
		if (m < k)
			 break;
		qfree(cos);
		k = m + 1;
	}
	sec = qinv(cos);
	qfree(cos);
	res = qmappr(sec, epsilon, 24);
	qfree(sec);
	return res;
}


/*
 * Calculate the cosecant function.
 */
NUMBER *
qcsc(NUMBER *q, NUMBER *epsilon)
{
	NUMBER *sin, *cos, *csc, *res;
	long n, k, m;

	if (qiszero(epsilon)) {
		math_error("Zero epsilon value for cosecant");
		not_reached();
	}
	if (qiszero(q)) {
		math_error("Zero argument for cosecant");
		not_reached();
	}
	k = -qilog2(q);
	n = qilog2(epsilon);
	if (k < 0)
		k = (n > 0) ? n/2 : 0;
	k += 4;
	for (;;) {
		qsincos(q, 2 * k - n, &sin, &cos);
		qfree(cos);
		if (qiszero(sin)) {
			qfree(sin);
			k = 2 * k - n + 4;
			continue;
		}
		m = -qilog2(sin);
		if (m < k)
			 break;
		qfree(sin);
		k = m + 1;
	}
	csc = qinv(sin);
	qfree(sin);
	res = qmappr(csc, epsilon, 24);
	qfree(csc);
	return res;
}


/*
 * Calculate the arcsine function.
 * The result is in the range -pi/2 to pi/2.
 */
NUMBER *
qasin(NUMBER *q, NUMBER *epsilon)
{
	NUMBER *qtmp1, *qtmp2, *epsilon1;
	ZVALUE ztmp;
	bool neg;
	FLAG r;

	if (qiszero(epsilon)) {
		math_error("Zero epsilon value for asin");
		not_reached();
	}
	if (qiszero(q)) {
		return qlink(&_qzero_);
	}
	ztmp = q->num;
	neg = ztmp.sign;
	ztmp.sign = 0;
	r = zrel(ztmp, q->den);
	if (r > 0) {
		return NULL;
	}
	if (r == 0) {
		epsilon1 = qscale(epsilon, 1L);
		qtmp2 = qpi(epsilon1);
		qtmp1 = qscale(qtmp2, -1L);
	} else {
		epsilon1 = qscale(epsilon, -2L);
		qtmp1 = qalloc();
		zsquare(q->num, &qtmp1->num);
		zsquare(q->den, &ztmp);
		zsub(ztmp, qtmp1->num, &qtmp1->den);
		zfree(ztmp);
		qtmp2 = qsqrt(qtmp1, epsilon1, 24);
		qfree(qtmp1);
		qtmp1 = qatan(qtmp2, epsilon);
	}
	qfree(qtmp2);
	qfree(epsilon1);
	if (neg) {
		qtmp2 = qneg(qtmp1);
		qfree(qtmp1);
		return(qtmp2);
	}
	return qtmp1;
}



/*
 * Calculate the acos function.
 * The result is in the range 0 to pi.
 */
NUMBER *
qacos(NUMBER *q, NUMBER *epsilon)
{
	NUMBER *q1, *q2, *epsilon1;
	ZVALUE z;

	if (qiszero(epsilon)) {
		math_error("Zero epsilon value for acos");
		not_reached();
	}
	if (qisone(q))
		return qlink(&_qzero_);
	if (qisnegone(q))
		return qpi(epsilon);

	z = q->num;
	z.sign = 0;
	if (zrel(z, q->den) > 0)
		return NULL;
	epsilon1 = qscale(epsilon, -3L);		/* ??? */
	q1 = qalloc();
	zsub(q->den, q->num, &q1->num);
	zadd(q->den, q->num, &q1->den);
	q2 = qsqrt(q1, epsilon1, 24L);
	qfree(q1);
	qfree(epsilon1);
	epsilon1 = qscale(epsilon, -1L);
	q1 = qatan(q2, epsilon1);
	qfree(epsilon1);
	qfree(q2);
	q2 = qscale(q1, 1L);
	qfree(q1)
	return q2;
}


/*
 * Calculate the arctangent function to the nearest or next to nearest
 * multiple of epsilon.	 Algorithm uses
 *	atan(x) = 2 * atan(x/(1 + sqrt(1+x^2)))
 * to reduce x to a small value and then
 *	atan(x) = x - x^3/3 + ...
 */
NUMBER *
qatan(NUMBER *q, NUMBER *epsilon)
{
	long m, k, i;
	FULL d;
	ZVALUE X, D, DD, sum, mul, term, ztmp1, ztmp2;
	NUMBER *qtmp, *res;
	bool sign;

	if (qiszero(epsilon)) {
		math_error("Zero epsilon value for arctangent");
		not_reached();
	}
	if (qiszero(q))
		return qlink(&_qzero_);
	m = 12 - qilog2(epsilon);
		/* 4 bits for 4 doublings; 8 for rounding */
	if (m < 8)
		m = 8;		/* m is number of working binary digits */
	qtmp = qscale(q, m);
	zquo(qtmp->num, qtmp->den, &X, 24);
	qfree(qtmp);
	zbitvalue(m, &D);	/* q has become X/D */
	zsquare(D, &DD);
	i = 4;			/* maybe this should be larger */
	while (i-- > 0 && !ziszero(X)) {
		zsquare(X, &ztmp1);
		zadd(ztmp1, DD, &ztmp2);
		zfree(ztmp1);
		zsqrt(ztmp2, &ztmp1, 24L);
		zfree(ztmp2);
		zadd(ztmp1, D, &ztmp2);
		zfree(ztmp1);
		zshift(X, m, &ztmp1);
		zfree(X);
		zquo(ztmp1, ztmp2, &X, 24L);
		zfree(ztmp1);
		zfree(ztmp2);
	}
	zfree(DD);
	zfree(D);
	if (ziszero(X)) {
		zfree(X);
		return qlink(&_qzero_);
	}
	zcopy(X, &sum);
	zsquare(X, &ztmp1);
	zshift(ztmp1, -m, &mul);
	zfree(ztmp1);
	d = 3;
	sign = !X.sign;
	for (;;) {
		if (d > BASE) {
			math_error("Too many terms required for atan");
			not_reached();
		}
		zmul(X, mul, &ztmp1);
		zfree(X);
		zshift(ztmp1, -m, &X);	/* X now (original X)^d */
		zfree(ztmp1);
		zdivi(X, d, &term);
		if (ziszero(term)) {
			zfree(term);
			break;
		}
		term.sign = sign;
		zadd(sum, term, &ztmp1);
		zfree(sum);
		zfree(term);
		sum = ztmp1;
		sign = !sign;
		d += 2;
	}
	zfree(mul);
	zfree(X);
	qtmp = qalloc();
	k = zlowbit(sum);
	if (k) {
		zshift(sum, -k, &qtmp->num);
		zfree(sum);
	} else {
		qtmp->num = sum;
	}
	zbitvalue(m - 4 - k, &qtmp->den);
	res = qmappr(qtmp, epsilon, 24L);
	qfree(qtmp);
	return res;
}

/*
 * Inverse secant function
 */
NUMBER *
qasec(NUMBER *q, NUMBER *epsilon)
{
	NUMBER *tmp, *res;

	tmp = qinv(q);
	res = qacos(tmp, epsilon);
	qfree(tmp);
	return res;
}


/*
 * Inverse cosecant function
 */
NUMBER *
qacsc(NUMBER *q, NUMBER *epsilon)
{
	NUMBER *tmp, *res;

	tmp = qinv(q);
	res = qasin(tmp, epsilon);
	qfree(tmp);
	return res;
}


/*
 * Inverse cotangent function
 */
NUMBER *
qacot(NUMBER *q, NUMBER *epsilon)
{
	NUMBER *tmp1, *tmp2, *tmp3, *epsilon1;

	if (qiszero(epsilon)) {
		math_error("Zero epsilon for acot");
		not_reached();
	}
	if (qiszero(q)) {
		epsilon1 = qscale(epsilon, 1L);
		tmp1 = qpi(epsilon1);
		qfree(epsilon1);
		tmp2 = qscale(tmp1, -1L);
		qfree(tmp1);
		return tmp2;
	}
	tmp1 = qinv(q);
	if (!qisneg(q)) {
		tmp2 = qatan(tmp1, epsilon);
		qfree(tmp1);
		return tmp2;
	}
	epsilon1 = qscale(epsilon, -2L);
	tmp2 = qatan(tmp1, epsilon1);
	qfree(tmp1);
	tmp1 = qpi(epsilon1);
	qfree(epsilon1);
	tmp3 = qqadd(tmp1, tmp2);
	qfree(tmp1);
	qfree(tmp2);
	tmp1 = qmappr(tmp3, epsilon, 24L);
	qfree(tmp3);
	return tmp1;
}


/*
 * Calculate the angle which is determined by the point (x,y).
 * This is the same as atan(y/x) for positive x, but is continuous
 * except for y = 0, x <= 0.  By convention, y is the first argument.
 * For all x, y, -pi < atan2 <= pi.  For example, qatan2(1, -1) = 3/4 * pi.
 */
NUMBER *
qatan2(NUMBER *qy, NUMBER *qx, NUMBER *epsilon)
{
	NUMBER *tmp1, *tmp2, *tmp3, *epsilon2;

	if (qiszero(epsilon)) {
		math_error("Zero epsilon value for atan2");
		not_reached();
	}
	if (qiszero(qy) && qiszero(qx)) {
		/* conform to 4.3BSD ANSI/IEEE 754-1985 math lib */
		return qlink(&_qzero_);
	}
	/*
	 * If the point is on the negative real axis, then the answer is pi.
	 */
	if (qiszero(qy) && qisneg(qx))
		return qpi(epsilon);
	/*
	 * If the point is in the right half plane, then use the normal atan.
	 */
	if (!qisneg(qx) && !qiszero(qx)) {
		if (qiszero(qy))
			return qlink(&_qzero_);
		tmp1 = qqdiv(qy, qx);
		tmp2 = qatan(tmp1, epsilon);
		qfree(tmp1);
		return tmp2;
	}
	/*
	 * The point is in the left half plane (x <= 0) with nonzero y.
	 * Calculate the angle by using the formula:
	 *	atan2(y,x) = 2 * atan(sgn(y) * sqrt((x/y)^2 + 1) - x/y).
	 */
	epsilon2 = qscale(epsilon, -4L);
	tmp1 = qqdiv(qx, qy);
	tmp2 = qsquare(tmp1);
	tmp3 = qqadd(tmp2, &_qone_);
	qfree(tmp2);
	tmp2 = qsqrt(tmp3, epsilon2, 24L | (qy->num.sign * 64));
	qfree(tmp3);
	tmp3 = qsub(tmp2, tmp1);
	qfree(tmp2);
	qfree(tmp1);
	qfree(epsilon2);
	epsilon2 = qscale(epsilon, -1L);
	tmp1 = qatan(tmp3, epsilon2);
	qfree(epsilon2);
	qfree(tmp3);
	tmp2 = qscale(tmp1, 1L);
	qfree(tmp1);
	return tmp2;
}


/*
 * Calculate the value of pi to within the required epsilon.
 * This uses the following formula which only needs integer calculations
 * except for the final operation:
 *	pi = 1 / SUMOF(comb(2 * N, N) ^ 3 * (42 * N + 5) / 2 ^ (12 * N + 4)),
 * where the summation runs from N=0.  This formula gives about 6 bits of
 * accuracy per term.  Since the denominator for each term is a power of two,
 * we can simply use shifts to sum the terms.  The combinatorial numbers
 * in the formula are calculated recursively using the formula:
 *	comb(2*(N+1), N+1) = 2 * comb(2 * N, N) * (2 * N + 1) / N.
 */
NUMBER *
qpi(NUMBER *epsilon)
{
	ZVALUE comb;			/* current combinatorial value */
	ZVALUE sum;			/* current sum */
	ZVALUE tmp1, tmp2;
	NUMBER *r, *t1, qtmp;
	long shift;			/* current shift of result */
	long N;				/* current term number */
	long bits;			/* needed number of bits of precision */
	long t;

	/* firewall */
	if (qiszero(epsilon)) {
		math_error("zero epsilon value for pi");
		not_reached();
	}

	/* use pi cache if epsilon marches, else flush if needed */
	if (pivalue[LAST_PI_EPSILON] != NULL &&
	    pivalue[LAST_PI_VALUE] != NULL &&
	    epsilon == pivalue[LAST_PI_EPSILON]) {
		return qlink(pivalue[LAST_PI_VALUE]);
	}
	if (pivalue[LAST_PI_EPSILON] != NULL) {
		qfree(pivalue[LAST_PI_EPSILON]);
	}
	if (pivalue[LAST_PI_VALUE] != NULL) {
		qfree(pivalue[LAST_PI_VALUE]);
	}

	bits = -qilog2(epsilon) + 4;
	if (bits < 4)
		bits = 4;
	comb = _one_;
	itoz(5L, &sum);
	N = 0;
	shift = 4;
	do {
		t = 1 + (++N & 0x1);
		(void) zdivi(comb, N / (3 - t), &tmp1);
		zfree(comb);
		zmuli(tmp1, t * (2 * N - 1), &comb);
		zfree(tmp1);
		zsquare(comb, &tmp1);
		zmul(comb, tmp1, &tmp2);
		zfree(tmp1);
		zmuli(tmp2, 42 * N + 5, &tmp1);
		zfree(tmp2);
		zshift(sum, 12L, &tmp2);
		zfree(sum);
		zadd(tmp1, tmp2, &sum);
		t = zhighbit(tmp1);
		zfree(tmp1);
		zfree(tmp2);
		shift += 12;
	} while ((shift - t) < bits);
	zfree(comb);
	qtmp.num = _one_;
	qtmp.den = sum;
	t1 = qscale(&qtmp, shift);
	zfree(sum);
	r = qmappr(t1, epsilon, 24L);
	qfree(t1);
	pivalue[LAST_PI_EPSILON] = qlink(epsilon);
	pivalue[LAST_PI_VALUE] = qlink(r);
	return r;
}


/*
 * qpidiv180 - calculate pi / 180
 *
 * This function returns pi/180 as used to covert between radians and degrees.
 */
NUMBER *
qpidiv180(NUMBER *epsilon)
{
	NUMBER *pi, *pidiv180;

	/* firewall */
	if (qiszero(epsilon)) {
		math_error("zero epsilon value for qpidiv180");
		not_reached();
	}

	/* use pi/180 cache if epsilon marches, else flush if needed */
	if (pivalue[LAST_PI_DIV_180_EPSILON] != NULL &&
	    pivalue[LAST_PI_DIV_180_VALUE] != NULL &&
	    epsilon == pivalue[LAST_PI_DIV_180_EPSILON]) {
		return qlink(pivalue[LAST_PI_DIV_180_VALUE]);
	}
	if (pivalue[LAST_PI_DIV_180_EPSILON] != NULL) {
		qfree(pivalue[LAST_PI_DIV_180_EPSILON]);
	}
	if (pivalue[LAST_PI_DIV_180_VALUE] != NULL) {
		qfree(pivalue[LAST_PI_DIV_180_VALUE]);
	}

	/* let qpi() returned cached pi or calculate new as needed */
	pi = qpi(epsilon);

	/* calculate pi/180 */
	pidiv180 = qdivi(pi, 180);

	/* cache epsilon and pi/180 */
	pivalue[LAST_PI_DIV_180_EPSILON] = qlink(epsilon);
	pivalue[LAST_PI_DIV_180_VALUE] = qlink(pidiv180);

	/* return pi/180 */
	return pidiv180;
}


/*
 * qpidiv200 - calculate pi / 200
 *
 * This function returns pi/200 as used to covert between radians and gradians.
 */
NUMBER *
qpidiv200(NUMBER *epsilon)
{
	NUMBER *pi, *pidiv200;

	/* firewall */
	if (qiszero(epsilon)) {
		math_error("zero epsilon value for qpidiv200");
		not_reached();
	}

	/* use pi/200 cache if epsilon marches, else flush if needed */
	if (pivalue[LAST_PI_DIV_200_EPSILON] != NULL &&
	    pivalue[LAST_PI_DIV_200_VALUE] != NULL &&
	    epsilon == pivalue[LAST_PI_DIV_200_EPSILON]) {
		return qlink(pivalue[LAST_PI_DIV_200_VALUE]);
	}
	if (pivalue[LAST_PI_DIV_200_EPSILON] != NULL) {
		qfree(pivalue[LAST_PI_DIV_200_EPSILON]);
	}
	if (pivalue[LAST_PI_DIV_200_VALUE] != NULL) {
		qfree(pivalue[LAST_PI_DIV_200_VALUE]);
	}

	/* let qpi() returned cached pi or calculate new as needed */
	pi = qpi(epsilon);

	/* calculate pi/200 */
	pidiv200 = qdivi(pi, 200);

	/* cache epsilon and pi/200 */
	pivalue[LAST_PI_DIV_200_EPSILON] = qlink(epsilon);
	pivalue[LAST_PI_DIV_200_VALUE] = qlink(pidiv200);

	/* return pi/200 */
	return pidiv200;
}


/*
 * Calculate the exponential function to the nearest or next to nearest
 * multiple of the positive number epsilon.
 */
NUMBER *
qexp(NUMBER *q, NUMBER *epsilon)
{
	long m, n;
	NUMBER *tmp1, *tmp2;

	if (qiszero(epsilon)) {
		math_error("Zero epsilon value for exp");
		not_reached();
	}
	if (qiszero(q))
		return qlink(&_qone_);
	tmp1 = qmul(q, &_qlge_);
	m = qtoi(tmp1);		/* exp(q) < 2^(m+1) or m == MAXLONG */
	qfree(tmp1);

	if (m > (1 << 30))
		return NULL;

	n = qilog2(epsilon);	/* 2^n <= epsilon < 2^(n+1) */
	if (m < n)
		return qlink(&_qzero_);
	tmp1 = qqabs(q);
	tmp2 = qexprel(tmp1, m - n + 1);
	qfree(tmp1);
	if (tmp2 == NULL)
		return NULL;
	if (qisneg(q)) {
		tmp1 = qinv(tmp2);
		qfree(tmp2);
		tmp2 = tmp1;
	}
	tmp1 = qmappr(tmp2, epsilon, 24L);
	qfree(tmp2);
	return tmp1;
}


/*
 * Calculate the exponential function with relative error corresponding
 * to a specified number of significant bits
 * Requires *q >= 0, bitnum >= 0.
 * This returns NULL if more than 2^30 working bits would be required.
 */
S_FUNC NUMBER *
qexprel(NUMBER *q, long bitnum)
{
	long n, m, k, h, s, t, d;
	NUMBER *qtmp1;
	ZVALUE X, B, sum, term, ztmp1, ztmp2;

	h = qilog2(q);
	k = bitnum + h + 1;
	if (k < 0)
		return qlink(&_qone_);
	s = k;
	if (k) {
		do {
			t = s;
			s = (s + k/s)/2;
		}
		while (t > s);
	}		/* s is int(sqrt(k)) */
	s++;
	if (s < -h)
		s = -h;
	n = h + s;	/* n is number of squarings that will be required */
	m = bitnum + n;
	if (m > (1 << 30))
		return NULL;
	while (s > 0) {		/* increasing m by ilog2(s) */
		s >>= 1;
		m++;
	}			/* m is working number of bits */
	qtmp1 = qscale(q, m - n);
	zquo(qtmp1->num, qtmp1->den, &X, 24);
	qfree(qtmp1);
	if (ziszero(X)) {
		zfree(X);
		return qlink(&_qone_);
	}
	zbitvalue(m, &sum);
	zcopy(X, &term);
	d = 1;
	do {
		zadd(sum, term, &ztmp1);
		zfree(sum);
		sum = ztmp1;
		zmul(term, X, &ztmp1);
		zfree(term);
		zshift(ztmp1, -m, &ztmp2);
		zfree(ztmp1);
		zdivi(ztmp2, ++d, &term);
		zfree(ztmp2);
	}
	while (!ziszero(term));
	zfree(term);
	zfree(X);
	k = 0;
	zbitvalue(2 * m + 1, &B);
	while (n-- > 0) {
		k *= 2;
		zsquare(sum, &ztmp1);
		zfree(sum);
		if (zrel(ztmp1, B) >= 0) {
			zshift(ztmp1, -m - 1, &sum);
			k++;
		} else {
			zshift(ztmp1, -m, &sum);
		}
		zfree(ztmp1);
	}
	zfree(B);
	h = zlowbit(sum);
	qtmp1 = qalloc();
	if (m > h + k) {
		zshift(sum, -h, &qtmp1->num);
		zbitvalue(m - h - k, &qtmp1->den);
	}
	else
		zshift(sum, k - m, &qtmp1->num);
	zfree(sum);
	return qtmp1;
}


/*
 * Calculate the natural logarithm of a number accurate to the specified
 * positive epsilon.
 */
NUMBER *
qln(NUMBER *q, NUMBER *epsilon)
{
	long m, n, k, h, d;
	ZVALUE term, sum, mul, pow, X, D, B, ztmp;
	NUMBER *qtmp, *res;
	bool neg;

	if (qiszero(q)) {
		math_error("logarithm of 0");
		not_reached();
	}
	if (qiszero(epsilon)) {
		math_error("Zero epsilon value for ln");
		not_reached();
	}
	if (qisunit(q))
		return qlink(&_qzero_);
	q = qqabs(q);			/* Ignore sign of q */
	neg = (zrel(q->num, q->den) < 0);
	if (neg) {
		qtmp = qinv(q);
		qfree(q);
		q = qtmp;
	}
	k = qilog2(q);
	m = -qilog2(epsilon);		/* m will be number of working bits */
	if (m < 0)
		m = 0;
	h = k;
	while (h > 0) {
		h /= 2;
		m++;			/* Add 1 for each sqrt until X < 2 */
	}
	m += 18;	/* 8 more sqrts, 8 for rounding, 2 for epsilon/4 */
	qtmp = qscale(q, m - k);
	zquo(qtmp->num, qtmp->den, &X, 24L);
	qfree(q);
	qfree(qtmp);

	zbitvalue(m, &D);		/* Now "q" = X/D */
	zbitvalue(m - 8, &ztmp);
	zadd(D, ztmp, &B);		/* Will take sqrts until X <= B */
	zfree(ztmp);

	n = 1;			/* n is to count 1 + number of sqrts */

	while (k > 0 || zrel(X, B) > 0) {
		n++;
		zshift(X, m + (k & 1), &ztmp);
		zfree(X);
		zsqrt(ztmp, &X, 24);
		zfree(ztmp)
		k /= 2;
	}
	zfree(B);
	zsub(X, D, &pow);	/* pow, mul used as tmps */
	zadd(X, D, &mul);
	zfree(X);
	zfree(D);
	zshift(pow, m, &ztmp);
	zfree(pow);
	zquo(ztmp, mul, &pow, 24);	/* pow now (X - D)/(X + D) */
	zfree(ztmp);
	zfree(mul);

	zcopy(pow, &sum);	/* pow is first term of sum */
	zsquare(pow, &ztmp);
	zshift(ztmp, -m, &mul); /* mul is now multiplier for powers */
	zfree(ztmp);

	d = 1;
	for (;;) {
		zmul(pow, mul, &ztmp);
		zfree(pow);
		zshift(ztmp, -m, &pow);
		zfree(ztmp);
		d += 2;
		zdivi(pow, d, &term);	/* Round down div should be round off */
		if (ziszero(term)) {
			zfree(term);
			break;
		}
		zadd(sum, term, &ztmp);
		zfree(term);
		zfree(sum);
		sum = ztmp;
	}
	zfree(pow);
	zfree(mul);
	qtmp = qalloc();	/* qtmp is to be 2^n * sum / 2^m */
	k = zlowbit(sum);
	sum.sign = neg;
	if (k + n >= m) {
		zshift(sum, n - m, &qtmp->num);
		zfree(sum);
	} else {
		if (k) {
			zshift(sum, -k, &qtmp->num);
			zfree(sum);
		} else {
			qtmp->num = sum;
		}
		zbitvalue(m - k - n, &qtmp->den);
	}
	res = qmappr(qtmp, epsilon, 24L);
	qfree(qtmp);
	return res;
}


/*
 * Calculate the base 10 logarithm
 *
 *	log(q) = ln(q) / ln(10)
 */
NUMBER *
qlog(NUMBER *q, NUMBER *epsilon)
{
	int need_new_ln_10 = true;	/* false => use cached ln_10 value */
	NUMBER *ln_q;			/* ln(x) */
	NUMBER *ret;			/* base 10 logarithm of x */

	/* firewall */
	if (qiszero(q)) {
		math_error("logarithm of 0");
		not_reached();
	}
	if (qiszero(epsilon)) {
		math_error("Zero epsilon value for log");
		not_reached();
	}

	/*
	 * shortcut for small integer powers of 10
	 */
	if (qisint(q) && qispos(q) && !zge8192b(q->num) && ziseven(q->num)) {
		bool is_10_power;	/* true ==> q is a power of 10 */
		long ilog_10;		/* integer log base 10 */

		/* try for a quick small power of 10 log */
		ilog_10 = zlog10(q->num, &is_10_power );
		if (is_10_power == true) {
			/* is small power of 10, return log */
			return itoq(ilog_10);
		}
		/* q is an even integer that is not a power of 10 */
	}

	/*
	 * compute ln(c) first
	 */
	ln_q = qln(q, epsilon);
	/* quick return for log(1) == 0 */
	if (qiszero(ln_q)) {
		return ln_q;
	}

	/*
	 * save epsilon for ln(10) if needed
	 */
	if (ln_10_epsilon == NULL) {
		/* first time call */
		ln_10_epsilon = qcopy(epsilon);
	} else if (qcmp(ln_10_epsilon, epsilon) == true) {
		/* replaced cached value with epsilon arg */
		qfree(ln_10_epsilon);
		ln_10_epsilon = qcopy(epsilon);
	} else if (ln_10 != NULL) {
		/* the previously computed ln(2) is OK to use */
		need_new_ln_10 = false;
	}

	/*
	 * compute ln(10) if needed
	 */
	if (need_new_ln_10 == true) {
		if (ln_10 != NULL) {
			qfree(ln_10);
		}
		ln_10 = qln(&_qten_, ln_10_epsilon);
	}

	/*
	 * return ln(q) / ln(10)
	 */
	ret = qqdiv(ln_q, ln_10);
	qfree(ln_q);
	return ret;
}


/*
 * Calculate the base 2 logarithm
 *
 *	log(q) = ln(q) / ln(2)
 */
NUMBER *
qlog2(NUMBER *q, NUMBER *epsilon)
{
	int need_new_ln_2 = true;	/* false => use cached ln_2 value */
	NUMBER *ln_q;			/* ln(x) */
	NUMBER *ret;			/* base 2 logarithm of x */

	/* firewall */
	if (qiszero(q)) {
		math_error("logarithm of 0");
		not_reached();
	}
	if (qiszero(epsilon)) {
		math_error("Zero epsilon value for log2");
		not_reached();
	}

	/*
	 * special case: q is integer power of 2
	 */
	ret = qalloc();
	if (qispowerof2(q, &ret)) {
		return ret;
	}
	qfree(ret);

	/*
	 * compute ln(c) first
	 */
	ln_q = qln(q, epsilon);
	/* quick return for ln(1) == 0 */
	if (qiszero(ln_q)) {
		return ln_q;
	}

	/*
	 * save epsilon for ln(2) if needed
	 */
	if (ln_2_epsilon == NULL) {
		/* first time call */
		ln_2_epsilon = qcopy(epsilon);
	} else if (qcmp(ln_2_epsilon, epsilon) == true) {
		/* replaced cached value with epsilon arg */
		qfree(ln_2_epsilon);
		ln_2_epsilon = qcopy(epsilon);
	} else if (ln_2 != NULL) {
		/* the previously computed ln(2) is OK to use */
		need_new_ln_2 = false;
	}

	/*
	 * compute ln(2) if needed
	 */
	if (need_new_ln_2 == true) {
		if (ln_2 != NULL) {
			qfree(ln_2);
		}
		ln_2 = qln(&_qtwo_, ln_2_epsilon);
	}

	/*
	 * return ln(q) / ln(2)
	 */
	ret = qqdiv(ln_q, ln_2);
	qfree(ln_q);
	return ret;
}


/*
 * Calculate the base n logarithm
 *
 *	logn(q, n) = ln(q) / ln(n)
 */
NUMBER *
qlogn(NUMBER *q, NUMBER *n, NUMBER *epsilon)
{
	int need_new_ln_n = true;	/* false => use cached ln_n value */
	NUMBER *ln_q;			/* ln(x) */
	NUMBER *ret;			/* base 2 logarithm of x */

	/* firewall */
	if (qiszero(q)) {
		math_error("logarithm of 0");
		not_reached();
	}
	if (qiszero(epsilon)) {
		math_error("Zero epsilon value for logn");
		not_reached();
	}
	if (qiszero(n)) {
		math_error("invalid logarithm base of 0 for logn");
		not_reached();
	}
	if (qisone(n)) {
		math_error("invalid logarithm base of 1 for logn");
		not_reached();
	}

	/*
	 * special case: q is integer power of 2
	 */
	ret = qalloc();
	if (qispowerof2(q, &ret)) {
		return ret;
	}
	/* XXX - deal with n is integer power of 2 - XXX */
	qfree(ret);

	/*
	 * compute ln(q) first
	 */
	ln_q = qln(q, epsilon);
	/* quick return for ln(1) == 0 */
	if (qiszero(ln_q)) {
		return ln_q;
	}

	/*
	 * save epsilon for ln(n) if needed
	 */
	if (ln_n_epsilon == NULL) {
		/* first time call */
		ln_n_epsilon = qcopy(epsilon);
	} else if (qcmp(ln_n_epsilon, epsilon) == true) {
		/* replaced cached value with epsilon arg */
		qfree(ln_n_epsilon);
		ln_n_epsilon = qcopy(epsilon);
	} else if (ln_n != NULL) {
		/* the previously computed ln(2) is OK to use */
		need_new_ln_n = false;
	}

	/*
	 * compute ln(n) if needed
	 */
	if (need_new_ln_n == true) {
		if (ln_n != NULL) {
			qfree(ln_n);
		}
		ln_n = qln(&_qtwo_, ln_n_epsilon);
	}

	/*
	 * return ln(q) / ln(2)
	 */
	ret = qqdiv(ln_q, ln_n);
	qfree(ln_q);
	return ret;
}


/*
 * Calculate the result of raising one number to the power of another.
 * The result is calculated to the nearest or next to nearest multiple of
 * epsilon.
 */
NUMBER *
qpower(NUMBER *q1, NUMBER *q2, NUMBER *epsilon)
{
	NUMBER *tmp1, *tmp2, *epsilon2;
	NUMBER *q1tmp, *q2tmp;
	long m, n;

	if (qiszero(epsilon)) {
		math_error("Zero epsilon for power");
		not_reached();
	}
	if (qiszero(q1) && qisneg(q2)) {
		math_error("Negative power of zero");
		not_reached();
	}
	if (qiszero(q2) || qisone(q1))
		return qlink(&_qone_);
	if (qiszero(q1))
		return qlink(&_qzero_);
	if (qisneg(q1)) {
		math_error("Negative base for qpower");
		not_reached();
	}
	if (qisone(q2))
		return qmappr(q1, epsilon, 24);
	if (zrel(q1->num, q1->den) < 0) {
		q1tmp = qinv(q1);
		q2tmp = qneg(q2);
	} else {
		q1tmp = qlink(q1);
		q2tmp = qlink(q2);
	}
	if (qisone(q2tmp)) {
		qfree(q2tmp);
		q2tmp = qmappr(q1tmp, epsilon, 24);
		qfree(q1tmp);
		return q2tmp;
	}
	m = qilog2(q1tmp);
	n = qilog2(epsilon);
	if (qisneg(q2tmp)) {
		if (m > 0) {
			tmp1 = itoq(m);
			tmp2 = qmul(tmp1, q2tmp);
			m = qtoi(tmp2);
		} else {
			tmp1 = qdec(q1tmp);
			tmp2 = qqdiv(tmp1, q1tmp);
			qfree(tmp1);
			tmp1 = qmul(tmp2, q2tmp);
			qfree(tmp2);
			tmp2 = qmul(tmp1, &_qlge_);
			m = qtoi(tmp2);
		}
	} else {
		if (m > 0) {
			tmp1 = itoq(m + 1);
			tmp2 = qmul(tmp1, q2tmp);
			m = qtoi(tmp2);
		} else {
			tmp1 = qdec(q1tmp);
			tmp2 = qmul(tmp1, q2tmp);
			qfree(tmp1);
			tmp1 = qmul(tmp2, &_qlge_);
			m = qtoi(tmp1);
		}
	}
	qfree(tmp1);
	qfree(tmp2);
	if (m > (1 << 30)) {
		qfree(q1tmp);
		qfree(q2tmp);
		return NULL;
	}
	m += 1;
	if (m < n) {
		qfree(q1tmp);
		qfree(q2tmp);
		return qlink(&_qzero_);
	}
	tmp1 = qqdiv(epsilon, q2tmp);
	tmp2 = qscale(tmp1, -m - 4);
	epsilon2 = qqabs(tmp2);
	qfree(tmp1);
	qfree(tmp2);
	tmp1 = qln(q1tmp, epsilon2);
	qfree(epsilon2);
	tmp2 = qmul(tmp1, q2tmp);
	qfree(tmp1);
	qfree(q1tmp);
	qfree(q2tmp);
	if (qisneg(tmp2)) {
		tmp1 = qneg(tmp2);
		qfree(tmp2);
		tmp2 = qexprel(tmp1, m - n + 3);
		if (tmp2 == NULL) {
			qfree(tmp1);
			return NULL;
		}
		qfree(tmp1);
		tmp1 = qinv(tmp2);
	} else {
		tmp1 = qexprel(tmp2, m - n + 3) ;
		if (tmp1 == NULL) {
			qfree(tmp2);
			return NULL;
		}
	}
	qfree(tmp2);
	tmp2 = qmappr(tmp1, epsilon, 24L);
	qfree(tmp1);
	return tmp2;
}


/*
 * Calculate the K-th root of a number to within the specified accuracy.
 */
NUMBER *
qroot(NUMBER *q1, NUMBER *q2, NUMBER *epsilon)
{
	NUMBER *tmp1, *tmp2;
	int neg;

	if (qiszero(epsilon)) {
		math_error("Zero epsilon for root");
		not_reached();
	}
	if (qisneg(q2) || qiszero(q2) || qisfrac(q2)) {
		math_error("Taking bad root of number");
		not_reached();
	}
	if (qiszero(q1) || qisone(q1) || qisone(q2))
		return qlink(q1);
	if (qistwo(q2))
		return qsqrt(q1, epsilon, 24L);
	neg = qisneg(q1);
	if (neg) {
		if (ziseven(q2->num)) {
			math_error("Taking even root of negative number");
			not_reached();
		}
		q1 = qqabs(q1);
	}
	tmp2 = qinv(q2);
	tmp1 = qpower(q1, tmp2, epsilon);
	qfree(tmp2);
	if (tmp1 == NULL)
		return NULL;
	if (neg) {
		tmp2 = qneg(tmp1);
		qfree(tmp1);
		tmp1 = tmp2;
	}
	return tmp1;
}


/* Calculate the hyperbolic cosine function to the nearest or next to
 * nearest multiple of epsilon.
 * This is calculated using cosh(x) = (exp(x) + 1/exp(x))/2;
 */
NUMBER *
qcosh(NUMBER *q, NUMBER *epsilon)
{
	NUMBER *tmp1, *tmp2, *tmp3, *epsilon1;

	epsilon1 = qscale(epsilon, -2);
	tmp1 = qqabs(q);
	tmp2 = qexp(tmp1, epsilon1);
	qfree(tmp1);
	qfree(epsilon1);
	if (tmp2 == NULL)
		return NULL;
	tmp1 = qinv(tmp2);
	tmp3 = qqadd(tmp1, tmp2);
	qfree(tmp1)
	qfree(tmp2)
	tmp1 = qscale(tmp3, -1);
	qfree(tmp3);
	tmp2 = qmappr(tmp1, epsilon, 24L);
	qfree(tmp1);
	return tmp2;
}


/*
 * Calculate the hyperbolic sine to the nearest or next to nearest
 * multiple of epsilon.
 * This is calculated using sinh(x) = (exp(x) - 1/exp(x))/2.
 */
NUMBER *
qsinh(NUMBER *q, NUMBER *epsilon)
{
	NUMBER *tmp1, *tmp2, *tmp3, *epsilon1;

	if (qiszero(q))
		return qlink(&_qzero_);
	epsilon1 = qscale(epsilon, -3);
	tmp1 = qqabs(q);
	tmp2 = qexp(tmp1, epsilon1);
	qfree(tmp1);
	qfree(epsilon1);
	if (tmp2 == NULL)
		return NULL;
	tmp1 = qinv(tmp2);
	tmp3 = qispos(q) ? qsub(tmp2, tmp1) : qsub(tmp1, tmp2);
	qfree(tmp1)
	qfree(tmp2)
	tmp1 = qscale(tmp3, -1);
	qfree(tmp3);
	tmp2 = qmappr(tmp1, epsilon, 24L);
	qfree(tmp1);
	return tmp2;
}


/*
 * Calculate the hyperbolic tangent to the nearest or next to nearest
 * multiple of epsilon.
 * This is calculated using the formulae:
 *	tanh(x) = 1 or -1 for very large abs(x)
 *	tanh(x) = (+ or -)(1 - 2 * exp(2 * x))	for large abx(x)
 *	tanh(x) = (exp(2*x) - 1)/(exp(2*x) + 1) otherwise
 */
NUMBER *
qtanh(NUMBER *q, NUMBER *epsilon)
{
	NUMBER *tmp1, *tmp2, *tmp3;
	long n, m;

	n = qilog2(epsilon);
	if (n > 0 || qiszero(q))
		return qlink(&_qzero_);
	n = -n;
	tmp1 = qqabs(q);
	tmp2 = qmul(tmp1, &_qlge_);
	m = qtoi(tmp2);		/* exp(|q|) < 2^(m+1) or m == MAXLONG */
	qfree(tmp2);
	if (m > 1 + n/2) {
		qfree(tmp1);
		return q->num.sign ? qlink(&_qnegone_) : qlink(&_qone_);
	}
	tmp2 = qscale(tmp1, 1);
	qfree(tmp1);
	tmp1 = qexprel(tmp2, 2 + n);
	qfree(tmp2);
	if (m > 1 + n/4) {
		tmp2 = qqdiv(&_qtwo_, tmp1);
		qfree(tmp1);
		tmp1 = qsub(&_qone_, tmp2);
		qfree(tmp2);
	} else {
		tmp2 = qdec(tmp1);
		tmp3 = qinc(tmp1);
		qfree(tmp1);
		tmp1 = qqdiv(tmp2, tmp3);
		qfree(tmp2);
		qfree(tmp3);
	}
	tmp2 = qmappr(tmp1, epsilon, 24L);
	qfree(tmp1);
	if (qisneg(q)) {
		tmp1 = qneg(tmp2);
		qfree(tmp2);
		return tmp1;
	}
	return tmp2;
}


/*
 * Hyperbolic cotangent.
 * Calculated using coth(x) = 1 + 2/(exp(2*x) - 1)
 */
NUMBER *
qcoth(NUMBER *q, NUMBER *epsilon)
{
	NUMBER *tmp1, *tmp2, *res;
	long n, k;

	if (qiszero(epsilon)) {
		math_error("Zero epsilon value for coth");
		not_reached();
	}
	if (qiszero(q)) {
		math_error("Zero argument for coth");
		not_reached();
	}
	tmp1 = qscale(q, 1);
	tmp2 = qqabs(tmp1);
	qfree(tmp1);
	k = qilog2(tmp2);
	n = qilog2(epsilon);
	if (k > 0) {
		tmp1 = qmul(&_qlge_, tmp2);
		k = qtoi(tmp1);
		qfree(tmp1);
	} else {
		k = 2 * k;
	}
	k = 4 - k - n;
	if (k < 4)
		k = 4;
	tmp1 = qexprel(tmp2,  k);
	qfree(tmp2);
	tmp2 = qdec(tmp1);
	qfree(tmp1);
	if (qiszero(tmp2)) {
		math_error("This should not happen ????");
		not_reached();
	}
	tmp1 = qinv(tmp2);
	qfree(tmp2);
	tmp2 = qscale(tmp1, 1);
	qfree(tmp1);
	tmp1 = qinc(tmp2);
	qfree(tmp2);
	if (qisneg(q)) {
		tmp2 = qneg(tmp1);
		qfree(tmp1);
		tmp1 = tmp2;
	}
	res = qmappr(tmp1, epsilon, 24L);
	qfree(tmp1);
	return res;
}


NUMBER *
qsech(NUMBER *q, NUMBER *epsilon)
{
	NUMBER *tmp1, *tmp2, *tmp3, *res;
	long n, k;

	if (qiszero(epsilon)) {
		math_error("Zero epsilon value for sech");
		not_reached();
	}
	if (qiszero(q))
		return qlink(&_qone_);

	tmp1 = qqabs(q);
	k = 0;
	if (zrel(tmp1->num, tmp1->den) >= 0) {
		tmp2 = qmul(&_qlge_, tmp1);
		k = qtoi(tmp2);
		qfree(tmp2);
	}
	n = qilog2(epsilon);
	if (k + n > 1) {
		qfree(tmp1);
		return qlink(&_qzero_);
	}
	tmp2 = qexprel(tmp1, 4 - k - n);
	qfree(tmp1);
	tmp1 = qinv(tmp2);
	tmp3 = qqadd(tmp1, tmp2);
	qfree(tmp1);
	qfree(tmp2);
	tmp1 = qinv(tmp3);
	qfree(tmp3);
	tmp2 = qscale(tmp1, 1);
	qfree(tmp1);
	res = qmappr(tmp2, epsilon, 24L);
	qfree(tmp2);
	return res;
}


NUMBER *
qcsch(NUMBER *q, NUMBER *epsilon)
{
	NUMBER *tmp1, *tmp2, *tmp3, *res;
	long n, k;

	if (qiszero(epsilon)) {
		math_error("Zero epsilon value for csch");
		not_reached();
	}
	if (qiszero(q)) {
		math_error("Zero argument for csch");
		not_reached();
	}

	n = qilog2(epsilon);
	tmp1 = qqabs(q);
	if (zrel(tmp1->num, tmp1->den) >= 0) {
		tmp2 = qmul(&_qlge_, tmp1);
		k = qtoi(tmp2);
		qfree(tmp2);
	} else {
		k = 2 * qilog2(tmp1);
	}
	if (k + n >= 1) {
		qfree(tmp1);
		return qlink(&_qzero_);
	}
	tmp2 = qexprel(tmp1, 4 - k - n);
	qfree(tmp1);
	tmp1 = qinv(tmp2);
	if (qisneg(q))
		tmp3 = qsub(tmp1, tmp2);
	else
		tmp3 = qsub(tmp2, tmp1);
	qfree(tmp1);
	qfree(tmp2);
	tmp1 = qinv(tmp3);
	qfree(tmp3)
	tmp2 = qscale(tmp1, 1);
	qfree(tmp1);
	res = qmappr(tmp2, epsilon, 24L);
	qfree(tmp2);
	return res;
}


/*
 * Compute the hyperbolic arccosine within the specified accuracy.
 * This is calculated using the formula:
 *	acosh(x) = ln(x + sqrt(x^2 - 1)).
 */
NUMBER *
qacosh(NUMBER *q, NUMBER *epsilon)
{
	NUMBER *tmp1, *tmp2, *epsilon1;
	long n;

	if (qiszero(epsilon)) {
		math_error("Zero epsilon value for acosh");
		not_reached();
	}
	if (qisone(q))
		return qlink(&_qzero_);
	if (zrel(q->num, q->den) < 0)
		return NULL;
	n = qilog2(epsilon);
	epsilon1 = qbitvalue(n - 3);
	tmp1 = qsquare(q);
	tmp2 = qdec(tmp1);
	qfree(tmp1);
	tmp1 = qsqrt(tmp2, epsilon1, 24L);
	qfree(tmp2);
	tmp2 = qqadd(tmp1, q);
	qfree(tmp1);
	tmp1 = qln(tmp2, epsilon1);
	qfree(tmp2);
	qfree(epsilon1);
	tmp2 = qmappr(tmp1, epsilon, 24L);
	qfree(tmp1);
	return tmp2;
}


/*
 * Compute the hyperbolic arcsine within the specified accuracy.
 * This is calculated using the formula:
 *	asinh(x) = ln(x + sqrt(x^2 + 1)).
 */
NUMBER *
qasinh(NUMBER *q, NUMBER *epsilon)
{
	NUMBER *tmp1, *tmp2, *epsilon1;
	long n;
	bool neg;

	if (qiszero(epsilon)) {
		math_error("Zero epsilon value for asinh");
		not_reached();
	}
	if (qiszero(q))
		return qlink(&_qzero_);
	neg = qisneg(q);
	q = qqabs(q);
	n = qilog2(epsilon);
	epsilon1 = qbitvalue(n - 3);
	tmp1 = qsquare(q);
	tmp2 = qinc(tmp1);
	qfree(tmp1);
	tmp1 = qsqrt(tmp2, epsilon1, 24L);
	qfree(tmp2);
	tmp2 = qqadd(tmp1, q);
	qfree(tmp1);
	tmp1 = qln(tmp2, epsilon1);
	qfree(tmp2);
	qfree(q);
	qfree(epsilon1);
	tmp2 = qmappr(tmp1, epsilon, 24L);
	if (neg) {
		tmp1 = qneg(tmp2);
		qfree(tmp2);
		return tmp1;
	}
	return tmp2;
}


/*
 * Compute the hyperbolic arctangent within the specified accuracy.
 * This is calculated using the formula:
 *	atanh(x) = ln((1 + x) / (1 - x)) / 2.
 */
NUMBER *
qatanh(NUMBER *q, NUMBER *epsilon)
{
	NUMBER *tmp1, *tmp2, *tmp3, *epsilon1;
	ZVALUE z;

	if (qiszero(epsilon)) {
		math_error("Zero epsilon value for atanh");
		not_reached();
	}
	if (qiszero(q))
		return qlink(&_qzero_);
	z = q->num;
	z.sign = 0;
	if (zrel(z, q->den) >= 0)
		return NULL;
	tmp1 = qinc(q);
	tmp2 = qsub(&_qone_, q);
	tmp3 = qqdiv(tmp1, tmp2);
	qfree(tmp1);
	qfree(tmp2);
	epsilon1 = qscale(epsilon, 1L);
	tmp1 = qln(tmp3, epsilon1);
	qfree(tmp3);
	tmp2 = qscale(tmp1, -1L);
	qfree(tmp1);
	qfree(epsilon1);
	return tmp2;
}


/*
 * Inverse hyperbolic secant function
 */
NUMBER *
qasech(NUMBER *q, NUMBER *epsilon)
{
	NUMBER *tmp, *res;

	tmp = qinv(q);
	res = qacosh(tmp, epsilon);
	qfree(tmp);
	return res;
}


/*
 * Inverse hyperbolic cosecant function
 */
NUMBER *
qacsch(NUMBER *q, NUMBER *epsilon)
{
	NUMBER *tmp, *res;

	tmp = qinv(q);
	res = qasinh(tmp, epsilon);
	qfree(tmp);
	return res;
}


/*
 * Inverse hyperbolic cotangent function
 */
NUMBER *
qacoth(NUMBER *q, NUMBER *epsilon)
{
	NUMBER *tmp, *res;

	tmp = qinv(q);
	res = qatanh(tmp, epsilon);
	qfree(tmp);
	return res;
}


/*
 * qversin - versed sine for NUMBER values
 *
 * This uses the formula:
 *
 *	versin(x) = 1 - cos(x)
 *
 * given:
 *	q	    real value to pass to the trig function
 *	epsilon	    error tolerance / precision for trig calculation
 *
 * returns:
 *	real value result of trig function on q with error epsilon
 */
NUMBER *
qversin(NUMBER *q, NUMBER *epsilon)
{
	NUMBER *res;
	NUMBER *cos;

	/*
	 * firewall
	 */
	if (q == NULL) {
		math_error("q is NULL for %s", __func__);
		not_reached();
	}
	if (check_epsilon(epsilon) == false) {
		math_error("Invalid epsilon arg for %s", __func__);
		not_reached();
	}

	/*
	 * calculate trig function value
	 */
	cos = qcos(q, epsilon);
	res = qsub(&_qone_, cos);
	qfree(cos);

	/*
	 * return 1 - cos(x)
	 */
	return res;
}


/*
 * qaversin_or_NULL - inverse versed sine for NUMBER values
 *
 * This uses the formula:
 *
 *	aversin(x) = acos(1 - x)
 *
 * given:
 *	q	    real value to pass to the trig function
 *	epsilon	    error tolerance / precision for trig calculation
 *
 * returns:
 *	!= NULL ==> real value result of trig function on q with error epsilon,
 *	NULL    ==> trig function value cannot be expressed as a NUMBER
 *
 * NOTE: If this function returns NULL, consider calling the equivalent
 *	 COMPLEX function from comfunc.c.  See the help file for the
 *	 related builtin for details.
 */
NUMBER *
qaversin_or_NULL(NUMBER *q, NUMBER *epsilon)
{
	NUMBER *res;	/* inverse trig value result */
	NUMBER *x;	/* argument to inverse trig function */

	/*
	 * firewall
	 */
	if (q == NULL) {
		math_error("q is NULL for %s", __func__);
		not_reached();
	}
	if (check_epsilon(epsilon) == false) {
		math_error("Invalid epsilon arg for %s", __func__);
		not_reached();
	}

	/*
	 * calculate inverse trig function value
	 */
	x = qsub(&_qone_, q);
	res = qacos(x, epsilon);
	qfree(x);
	if (res == NULL) {
		return NULL;
	}

	/*
	 * return acos(1 - x)
	 */
	return res;
}


/*
 * qaversin - inverse versed sine for NUMBER values
 *
 * This uses the formula:
 *
 *	aversin(x) = acos(1 - x)
 *
 * given:
 *	q	    real value to pass to the trig function
 *	epsilon	    error tolerance / precision for trig calculation
 *
 * returns:
 *	real value result of trig function on q with error epsilon
 */
NUMBER *
qaversin(NUMBER *q, NUMBER *epsilon)
{
	NUMBER *res;	/* inverse trig value result */

	/*
	 * firewall
	 */
	if (q == NULL) {
		math_error("q is NULL for %s", __func__);
		not_reached();
	}
	if (check_epsilon(epsilon) == false) {
		math_error("Invalid epsilon arg for %s", __func__);
		not_reached();
	}

	/*
	 * calculate inverse trig function value
	 */
	res = qaversin_or_NULL(q, epsilon);
	if (res == NULL) {
		math_error("cannot compute inverse cos for aversin");
		not_reached();
	}

	/*
	 * return acos(1 - x)
	 */
	return res;
}


/*
 * qcoversin - coversed sine for NUMBER values
 *
 * This uses the formula:
 *
 *	coversin((x) = 1 - sin(x)
 *
 * given:
 *	q	    real value to pass to the trig function
 *	epsilon	    error tolerance / precision for trig calculation
 *
 * returns:
 *	real value result of trig function on q with error epsilon
 */
NUMBER *
qcoversin(NUMBER *q, NUMBER *epsilon)
{
	NUMBER *sin, *res;

	/*
	 * firewall
	 */
	if (q == NULL) {
		math_error("q is NULL for %s", __func__);
		not_reached();
	}
	if (check_epsilon(epsilon) == false) {
		math_error("Invalid epsilon arg for %s", __func__);
		not_reached();
	}

	/*
	 * calculate trig function value
	 */
	sin = qsin(q, epsilon);
	res = qsub(&_qone_, sin);
	qfree(sin);

	/*
	 * return 1 - sin(x)
	 */
	return res;
}


/*
 * qacoversin_or_NULL - inverse coversed sine for NUMBER values
 *
 * This uses the formula:
 *
 *	acoversin(x) = asin(1 - x)
 *
 * given:
 *	q	    real value to pass to the trig function
 *	epsilon	    error tolerance / precision for trig calculation
 *
 * returns:
 *	real value result of trig function on q with error epsilon
 *
 * returns:
 *	!= NULL ==> real value result of trig function on q with error epsilon,
 *	NULL    ==> trig function value cannot be expressed as a NUMBER
 *
 * NOTE: If this function returns NULL, consider calling the equivalent
 *	 COMPLEX function from comfunc.c.  See the help file for the
 *	 related builtin for details.
 */
NUMBER *
qacoversin_or_NULL(NUMBER *q, NUMBER *epsilon)
{
	NUMBER *res;	/* inverse trig value result */
	NUMBER *x;	/* argument to inverse trig function */

	/*
	 * firewall
	 */
	if (q == NULL) {
		math_error("q is NULL for %s", __func__);
		not_reached();
	}
	if (check_epsilon(epsilon) == false) {
		math_error("Invalid epsilon arg for %s", __func__);
		not_reached();
	}

	/*
	 * calculate inverse trig function value
	 */
	x = qsub(&_qone_, q);
	res = qasin(x, epsilon);
	qfree(x);
	if (res == NULL) {
		return NULL;
	}

	/*
	 * return asin(1 - x)
	 */
	return res;
}


/*
 * qacoversin - inverse coversed sine for NUMBER values
 *
 * This uses the formula:
 *
 *	acoversin(x) = asin(1 - x)
 *
 * given:
 *	q	    real value to pass to the trig function
 *	epsilon	    error tolerance / precision for trig calculation
 *
 * returns:
 *	real value result of trig function on q with error epsilon
 */
NUMBER *
qacoversin(NUMBER *q, NUMBER *epsilon)
{
	NUMBER *res;	/* inverse trig value result */

	/*
	 * firewall
	 */
	if (q == NULL) {
		math_error("q is NULL for %s", __func__);
		not_reached();
	}
	if (check_epsilon(epsilon) == false) {
		math_error("Invalid epsilon arg for %s", __func__);
		not_reached();
	}

	/*
	 * calculate inverse trig function value
	 */
	res = qacoversin_or_NULL(q, epsilon);
	if (res == NULL) {
		math_error("cannot compute inverse sin for acoversin");
		not_reached();
	}

	/*
	 * return asin(1 - x)
	 */
	return res;
}


/*
 * qvercos - versed sine for NUMBER values
 *
 * This uses the formula:
 *
 *	vercos(x) = 1 + cos(x)
 *
 * given:
 *	q	    real value to pass to the trig function
 *	epsilon	    error tolerance / precision for trig calculation
 *
 * returns:
 *	real value result of trig function on q with error epsilon
 */
NUMBER *
qvercos(NUMBER *q, NUMBER *epsilon)
{
	NUMBER *cos, *res;

	/*
	 * firewall
	 */
	if (q == NULL) {
		math_error("q is NULL for %s", __func__);
		not_reached();
	}
	if (check_epsilon(epsilon) == false) {
		math_error("Invalid epsilon arg for %s", __func__);
		not_reached();
	}

	/*
	 * calculate trig function value
	 */
	cos = qcos(q, epsilon);
	res = qqadd(&_qone_, cos);
	qfree(cos);

	/*
	 * return 1 + cos(x)
	 */
	return res;
}


/*
 * qavercos_or_NULL - inverse versed sine for NUMBER values
 *
 * This uses the formula:
 *
 *	avercos(x) = acos(x - 1)
 *
 * given:
 *	q	    real value to pass to the trig function
 *	epsilon	    error tolerance / precision for trig calculation
 *
 * returns:
 *	!= NULL ==> real value result of trig function on q with error epsilon,
 *	NULL    ==> trig function value cannot be expressed as a NUMBER
 *
 * NOTE: If this function returns NULL, consider calling the equivalent
 *	 COMPLEX function from comfunc.c.  See the help file for the
 *	 related builtin for details.
 */
NUMBER *
qavercos_or_NULL(NUMBER *q, NUMBER *epsilon)
{
	NUMBER *res;	/* inverse trig value result */
	NUMBER *x;	/* argument to inverse trig function */

	/*
	 * firewall
	 */
	if (q == NULL) {
		math_error("q is NULL for %s", __func__);
		not_reached();
	}
	if (check_epsilon(epsilon) == false) {
		math_error("Invalid epsilon arg for %s", __func__);
		not_reached();
	}

	/*
	 * calculate inverse trig function value
	 */
	x = qsub(q, &_qone_);
	res = qacos(x, epsilon);
	qfree(x);
	if (res == NULL) {
		return NULL;
	}

	/*
	 * return acos(x - 1)
	 */
	return res;
}


/*
 * qavercos - inverse versed sine for NUMBER values
 *
 * This uses the formula:
 *
 *	avercos(x) = acos(x - 1)
 *
 * given:
 *	q	    real value to pass to the trig function
 *	epsilon	    error tolerance / precision for trig calculation
 *
 * returns:
 *	real value result of trig function on q with error epsilon
 */
NUMBER *
qavercos(NUMBER *q, NUMBER *epsilon)
{
	NUMBER *res;	/* inverse trig value result */

	/*
	 * firewall
	 */
	if (q == NULL) {
		math_error("q is NULL for %s", __func__);
		not_reached();
	}
	if (check_epsilon(epsilon) == false) {
		math_error("Invalid epsilon arg for %s", __func__);
		not_reached();
	}

	/*
	 * calculate inverse trig function value
	 */
	res = qavercos_or_NULL(q, epsilon);
	if (res == NULL) {
		math_error("cannot compute inverse cos for avercos");
		not_reached();
	}

	/*
	 * return acos(x - 1)
	 */
	return res;
}


/*
 * qcovercos - coversed sine for NUMBER values
 *
 * This uses the formula:
 *
 *	covercos((x) = 1 + sin(x)
 *
 * given:
 *	q	    real value to pass to the trig function
 *	epsilon	    error tolerance / precision for trig calculation
 *
 * returns:
 *	real value result of trig function on q with error epsilon
 */
NUMBER *
qcovercos(NUMBER *q, NUMBER *epsilon)
{
	NUMBER *sin, *res;

	/*
	 * firewall
	 */
	if (q == NULL) {
		math_error("q is NULL for %s", __func__);
		not_reached();
	}
	if (check_epsilon(epsilon) == false) {
		math_error("Invalid epsilon arg for %s", __func__);
		not_reached();
	}

	/*
	 * calculate trig function value
	 */
	sin = qsin(q, epsilon);
	res = qqadd(&_qone_, sin);
	qfree(sin);

	/*
	 * return 1 + sin(x)
	 */
	return res;
}


/*
 * qacovercos_or_NULL - inverse coversed sine for NUMBER values
 *
 * This uses the formula:
 *
 *	acovercos(x) = asin(x - 1)
 *
 * given:
 *	q	    real value to pass to the trig function
 *	epsilon	    error tolerance / precision for trig calculation
 *
 * returns:
 *	real value result of trig function on q with error epsilon
 *
 * returns:
 *	!= NULL ==> real value result of trig function on q with error epsilon,
 *	NULL    ==> trig function value cannot be expressed as a NUMBER
 *
 * NOTE: If this function returns NULL, consider calling the equivalent
 *	 COMPLEX function from comfunc.c.  See the help file for the
 *	 related builtin for details.
 */
NUMBER *
qacovercos_or_NULL(NUMBER *q, NUMBER *epsilon)
{
	NUMBER *res;	/* inverse trig value result */
	NUMBER *x;	/* argument to inverse trig function */

	/*
	 * firewall
	 */
	if (q == NULL) {
		math_error("q is NULL for %s", __func__);
		not_reached();
	}
	if (check_epsilon(epsilon) == false) {
		math_error("Invalid epsilon arg for %s", __func__);
		not_reached();
	}

	/*
	 * calculate inverse trig function value
	 */
	x = qsub(&_qone_, q);
	res = qasin(x, epsilon);
	qfree(x);
	if (res == NULL) {
		return NULL;
	}

	/*
	 * return asin(x - 1)
	 */
	return res;
}


/*
 * qacovercos - inverse coversed sine for NUMBER values
 *
 * This uses the formula:
 *
 *	acovercos(x) = asin(x - 1)
 *
 * given:
 *	q	    real value to pass to the trig function
 *	epsilon	    error tolerance / precision for trig calculation
 *
 * returns:
 *	real value result of trig function on q with error epsilon
 */
NUMBER *
qacovercos(NUMBER *q, NUMBER *epsilon)
{
	NUMBER *res;	/* inverse trig value result */

	/*
	 * firewall
	 */
	if (q == NULL) {
		math_error("q is NULL for %s", __func__);
		not_reached();
	}
	if (check_epsilon(epsilon) == false) {
		math_error("Invalid epsilon arg for %s", __func__);
		not_reached();
	}

	/*
	 * calculate inverse trig function value
	 */
	res = qacovercos_or_NULL(q, epsilon);
	if (res == NULL) {
		math_error("cannot compute inverse sin for acovercos");
		not_reached();
	}

	/*
	 * return asin(x - 1)
	 */
	return res;
}