/*
 * test9500.trigeq - test of trigonometric identities for test 95dd
 *
 * We test over a wide variety of real (NUMBER) and complex (COMPLEX)
 * values, that various trigonometric identities hold for all calc
 * builtin trigonometric functions.
 *
 * We assume that the value produced by sin() (trigonometric sine)
 * and cos() (trigonometric cosine) are correct.  We can reasonably
 * assume this because of the calc regression test suite:
 *
 *	(such as test 34dd and in particular cal/test3400.trig.cal and
 *	 such as test 89dd and in particular cal/test8900.special.cal)
 *
 * will test such those function beforehand.
 *
 * We use various trigonometric identity to verify the non-inverse
 * trigonometric functions other than sin() and cos().
 *
 * we verify that it is true within the error tolerance of the eps
 * (epsilon) argument.
 *
 * Copyright (C) 2023  Landon Curt Noll
 *
 * Primary author:  Landon Curt Noll
 *
 * 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:	2023/09/10 14:33:06
 * File existed as early as:	2023
 *
 * Share and enjoy!  :-)	http://www.isthe.com/chongo/tech/comp/calc/
 */


/*
 * base_tval - base set of test values
 *
 * Our 16 base test values must be > 0 and <= 1.
 * We use these fractions because, when multiplied
 * by pi() will give values that are commonly used.
 *
 * NOTE: There is no reason to sort these values other than ascetics. :-)
 */
static base_tval = list(
	1/12,	/* ~ 0.08333333333333333333 */
	1/8,	/*   0.12500000000000000000 */
	1/6,	/* ~ 0.16666666666666666667 */
	1/5,	/*   0.20000000000000000000 */
	1/4,	/*   0.25000000000000000000 */
	1/3,	/* ~ 0.33333333333333333333 */
	3/8,	/*   0.37500000000000000000 */
	5/12,	/* ~ 0.41666666666666666667 */
	1/2,	/*   0.50000000000000000000 */
	2/3,	/* ~ 0.66666666666666666667 */
	3/4,	/*   0.75000000000000000000 */
	4/5,	/*   0.80000000000000000000 */
	5/6,	/* ~ 0.83333333333333333333 */
	7/8,	/*   0.87500000000000000000 */
	11/12,	/* ~ 0.91666666666666666667 */
	1	/*   1.00000000000000000000 */
);


/*
 * tval - full test list
 *
 * This long list is formed via the form_tval() function from the base_tval list.
 *
 * When the form_tval() function forms the full test test values
 * we will have 1025 values of various real and complex
 * values that also include random values added or subtracted.
 *
 * See the form_tval() function for details.
 */
tval = list();


/*
 * Lists that hold precomputed trigonometric functions on the tval full test list
 *
 * The precompute_trig() function will fill out these lists.
 * They will be the same size as size as the full test list.
 */
sin_tval = list();	/* trigonometric sine of each test test value */
cos_tval = list();	/* trigonometric cosine of each test test value */


/*
 * seed_rand - seed the subtractive 100 shuffle pseudo-random number generator
 *
 * We want to seed the subtractive 100 shuffle pseudo-random number generator
 * with a constant so that the result of this test set will be deterministic.
 *
 * We seed the subtractive 100 shuffle pseudo-random number generator with
 * a 128-bit seed value.  The 128-bit seed value was produced when a
 * Blum-Blum-Shub pseudo-random number generator was seeded with the
 * following 160-bit SHA1 hash of 3 calls to seed():
 *
 *	sha1(sha1(seed(), seed()<<64, seed()<<128)) == 0x732fda50b1bf6e34b604b7c75b993e5c37a4ad97
 *
 * This produced the following 160-bit value that was used to seed
 * the Blum-Blum-Shub pseudo-random number:
 *
 *	srandom(0x732fda50b1bf6e34b604b7c75b993e5c37a4ad97)
 *
 * The seeded Blum-Blum-Shub pseudo-random number generator was then
 * used to generate a 128-bit value:
 *
 *	randombit(128) == 0x37301cf47204f7fababc2f32e39ab338
 *
 * We seed the subtractive 100 shuffle pseudo-random number generator
 * with the above mentioned 128-bit value:
 *
 *	srand(0x37301cf47204f7fababc2f32e39ab338)
 */
define seed_rand()
{
	return srand(0x37301cf47204f7fababc2f32e39ab338);
}


/*
 * epsilon_bits - number of bits in epsilon()
 */
static epsilon_bits = highbit(1/epsilon()) + 1;


/*
 * epsilon_norm - norm of a epsilon()
 */
static epsilon_norm = norm(epsilon());


/*
 * random_rational_value - compute a random rational value > 0 and < 1
 *
 * We compute a pseudo-random value in the range (0,1) with a prevision of epsilon().
 *
 * We compute two distinct non-zero pseudo-random integers (i.e., > 0), a and b,
 * that are epsilon_bits (see above) long and produced by the seeded
 * subtractive 100 shuffle pseudo-random number generator (see seed_rand()).
 *
 * Return the smaller of a,b divided by the larger of a,b.
 */
define random_rational_value()
{
	local a;	/* epsilon_bits pseudo-random integer > 0 */
	local b;	/* epsilon_bits pseudo-random integer > 0 */

	/*
	 * compute 1st non-zero pseudo-random epsilon_bits integer
	 */
	do {
		a = randbit(epsilon_bits);
	} while (a == 0);

	/*
	 * compute 2nd non-zero pseudo-random epsilon_bits integer also != a
	 */
	do {
		b = randbit(epsilon_bits);
	} while (b == 0 || a == b);

	/*
	 * return the min(a,b) / max(a,b): a value > 0 and < 1
	 */
	return min(a,b) / max(a,b);
}


/*
 * form_tval - form a full list of test values
 *
 * Given an initial list of 16 values, we form 1025 test values.
 *
 * From the test set starts with 16 values from the base_tval[] list:
 *
 *	tval = base_tval;
 *
 * We form a list of 32 values by appending all of the above with:
 *
 *	tval[i] + 1
 *
 * We form a list of 64 values by appending all of the above with:
 *
 *	-tval[i]
 *
 * We form a list of 128 values by appending all of the above with:
 *
 *	For values >= 0:	tval[i] + random_rational_value
 *	For values < 0:		tval[i] - random_rational_value
 *
 * We form a list of 256 values by appending all of the above with:
 *
 *	tval[i] * pi()
 *
 * We form a list of 512 values by appending all of the above with:
 *
 *	tval[i] * 1i
 *
 * We for a list of 1024 values by appending new complex values where the
 * real part is a randomly selected real value (from the first 256 list values), and the
 * imaginary part is a randomly selected complex value (from the last 256 list values).
 *
 * Finally we append 0 to the list to form the full 1025 value list.
 */
define form_tval()
{
	local tval_len;		/* current length of the tval[] array */
	local real_cnt;		/* number of real values in tval[] array */
	local imag_cnt;		/* number of imaginary values in tval[] array */
	local i;

	/*
	 * seed the subtractive 100 shuffle pseudo-random number generator
	 */
	seed_rand();

	/*
	 * load base_tval[] array into tval[] array
	 */
	tval = base_tval;

	/*
	 * expand all tval with tval+1
	 */
	tval_len = size(tval);
	for (i=0; i < tval_len; ++i) {
		append(tval, tval[i] + 1);
	}

	/*
	 * expand all tval with -tval
	 */
	tval_len = size(tval);
	for (i=0; i < tval_len; ++i) {
		append(tval, -tval[i]);
	}

	/*
	 * expand all tval with tval +/- random_rational_value
	 *
	 * For values we add random_rational_value if > 0
	 *   or we subtract crandom_rational_value if < 0.
	 */
	tval_len = size(tval);
	for (i=0; i < tval_len; ++i) {

		/* use the sign to determine if we add or subtract */
		if (tval[i] >= 0) {
			append(tval, tval[i] + random_rational_value());
		} else {
			append(tval, tval[i] - random_rational_value());
		}
	}

	/*
	 * expand all tval with tval * pi()
	 */
	tval_len = size(tval);
	for (i=0; i < tval_len; ++i) {
		append(tval, tval[i] * pi());
	}

	/*
	 * expand all tval with tval * 1i
	 */
	tval_len = size(tval);
	/* tval[0] thru tval[real_cnt-1] are real values */
	real_cnt = tval_len;
	for (i=0; i < tval_len; ++i) {
		append(tval, tval[i] * 1i);
	}

	/*
	 * expand tval with randomly selected real value
	 * plus randomly selected complex value.
	 */
	tval_len = size(tval);
	/* tval[real_cnt] thru tval[real_cnt+imag_cnt-1] are complex values */
	imag_cnt = tval_len - real_cnt;
	for (i=0; i < tval_len; ++i) {
		append(tval, tval[rand(0,real_cnt)] + tval[real_cnt + rand(0,imag_cnt)]);
	}

	/*
	 * last, append our final 0 value
	 */
	append(tval, 0);
}


/*
 * compute - form precomputed trigonometric value lists
 */
define precompute_trig()
{
	local tval_len;		/* current length of the tval[] array */
	local i;

	/*
	 * firewall - be sure tval list is setup
	 */
	if (size(tval) <= 0) {
		form_tval();
	}
	tval_len = size(tval);
	if (tval_len <= 0) {
		quit "FATAL: tval_len is empty";
	}

	/*
	 * compute trigonometric sine of the tval full test list
	 */
	for (i=0; i < tval_len; ++i) {
		append(sin_tval, sin(tval[i]));
	}
	if (size(tval) != size(sin_tval)) {
		print "****: size(tval):", size(tval), "!= size(sin_tval)", size(sin_tval);
		quit "FATAL: sin_tval not same size as tval_len";
	}

	/*
	 * compute trigonometric cosine of the tval full test list
	 */
	for (i=0; i < tval_len; ++i) {
		append(cos_tval, cos(tval[i]));
	}
	if (size(tval) != size(cos_tval)) {
		print "****: size(tval):", size(tval), "!= size(cos_tval)", size(cos_tval);
		quit "FATAL: cos_tval not same size as tval_len";
	}
}


/*
 * compare - compare the trigonometric identity and the trigonometric function value
 *
 * given:
 *	ident_val	computed trig value trigonometric identity
 *	trig_val	computed value from the trigonometric function
 *	name		trig function name
 *	index		test index value
 *	testnum		regression test number being performed
 *
 * returns:
 *	0 ==> trigonometric identity and the trigonometric function value match
 *	1 ==> trigonometric identity and the trigonometric function value do NOT match
 */
define compare(ident_val, trig_val, name, index, testnum)
{
	local abs_diff;		/* absolute difference between ident_val and trig_val */

	/*
	 * firewall
	 */
	if (!isnum(trig_val)) {
		printf("**** trig test %d-%d failed: %s(tval[%d]): ",
		       testnum, index, name, index);
		printf("%d returned a non-numeric value\n", trig_val);
		return 1;
	}

	/*
	 * compute absolute difference
	 */
	abs_diff = abs(ident_val - trig_val);

	/*
	 * determine if ident_val within epsilon of trig_val
	 */
	if (near(norm(abs_diff), epsilon_norm) > 0) {
		printf("**** trig test %d-%d failed: %s(tval[%d]): ",
		       testnum, index, name, index);
		printf("%d is not near trig identity value: %d\n",
		       trig_val, ident_val);
		return 1;
	}
	return 0;
}


/*
 * not_near_zero - determine if a value is within epsilon() of 0
 *
 * given:
 *	value	real or complex value
 *
 * returns:
 *	1 ==> value is farther from 0 than epsilon (not near zero)
 *	0 ==> value is within epsilon of 0 (near zero)
 */
define not_near_zero(value)
{
	if (near(norm(value), epsilon_norm) > 0) {
		/* value is farther from 0 than epsilon */
		return 1;
	}

	/* value is within epsilon of 0 */
	return 0;
}


/*
 * verify_tan - verify trigonometric tangent
 *
 * We use the following trigonometric identity:
 *
 *	tan(x) = sin(x) / cos(x)
 *
 * given:
 *	testnum		regression test number being performed
 *
 * returns:
 *	number of tests that failed
 */
define verify_tan(testnum)
{
	local tval_len;		/* current length of the tval[] array */
	local ident_val;	/* computed trig value trigonometric identity */
	local trig_val;		/* computed value from the trigonometric function */
	local error_count;	/* number of compare errors detected */
	local i;

	/*
	 * firewall
	 */
	if (size(sin_tval) <= 0 || size(cos_tval) <= 0) {
		precompute_trig();
	}

	/*
	 * for each test value, verify the trigonometric identity within epsilon
	 */
	error_count = 0;
	tval_len = size(tval);
	for (i=0; i < tval_len; ++i) {

		/* skip test when cos(x) within epsilon of 0 */
		if (not_near_zero(cos_tval[i])) {

			/* compute trigonometric identity */
			ident_val = sin_tval[i] / cos_tval[i];

			/* compute trigonometric function */
			trig_val = tan(tval[i]);

			/* compare trigonometric identity with trigonometric function value */
			if (compare(ident_val, trig_val, "tan", i, testnum)) {
				++error_count;
			}
		}
	}

	/*
	 * report test results
	 */
	if (error_count != 0) {
		print '**** test', testnum : ': tan test failure count:', error_count;
	}
	return error_count;
}


/*
 * verify_cot - verify trigonometric cotangent
 *
 * We use the following trigonometric identity:
 *
 *	cot(x) = cos(x) / sin(x)
 *
 * given:
 *	testnum		regression test number being performed
 *
 * returns:
 *	number of tests that failed
 */
define verify_cot(testnum)
{
	local tval_len;		/* current length of the tval[] array */
	local ident_val;	/* computed trig value trigonometric identity */
	local trig_val;		/* computed value from the trigonometric function */
	local error_count;	/* number of compare errors detected */
	local i;

	/*
	 * firewall
	 */
	if (size(sin_tval) <= 0 || size(cos_tval) <= 0) {
		precompute_trig();
	}

	/*
	 * for each test value, verify the trigonometric identity within epsilon
	 */
	tval_len = size(tval);
	for (i=0; i < tval_len; ++i) {

		/* skip test when sin(x) within epsilon of 0 */
		if (not_near_zero(sin_tval[i])) {

			/* compute trigonometric identity */
			ident_val = cos_tval[i] / sin_tval[i];

			/* compute trigonometric function */
			trig_val = cot(tval[i]);

			/* compare trigonometric identity with trigonometric function value */
			if (compare(ident_val, trig_val, "cot", i, testnum)) {
				++error_count;
			}
		}
	}

	/*
	 * report test results
	 */
	if (error_count != 0) {
		print '**** test', testnum : ': cot test failure count:', error_count;
	}
	return error_count;
}


/*
 * verify_sec - verify trigonometric secant
 *
 * We use the following trigonometric identity:
 *
 *	sec(x) = 1 / cos(x)
 *
 * given:
 *	testnum		regression test number being performed
 *
 * returns:
 *	number of tests that failed
 */
define verify_sec(testnum)
{
	local tval_len;		/* current length of the tval[] array */
	local ident_val;	/* computed trig value trigonometric identity */
	local trig_val;		/* computed value from the trigonometric function */
	local error_count;	/* number of compare errors detected */
	local i;

	/*
	 * firewall
	 */
	if (size(cos_tval) <= 0) {
		precompute_trig();
	}

	/*
	 * for each test value, verify the trigonometric identity within epsilon
	 */
	tval_len = size(tval);
	for (i=0; i < tval_len; ++i) {

		/* skip test when cos(x) within epsilon of 0 */
		if (not_near_zero(cos_tval[i])) {

			/* compute trigonometric identity */
			ident_val = 1 / cos_tval[i];

			/* compute trigonometric function */
			trig_val = sec(tval[i]);

			/* compare trigonometric identity with trigonometric function value */
			if (compare(ident_val, trig_val, "sec", i, testnum)) {
				++error_count;
			}
		}
	}

	/*
	 * report test results
	 */
	if (error_count != 0) {
		print '**** test', testnum : ': sec test failure count:', error_count;
	}
	return error_count;
}


/*
 * verify_csc - verify trigonometric cosecant
 *
 * We use the following trigonometric identity:
 *
 *	csc(x) = 1 / sin(x)
 *
 * given:
 *	testnum		regression test number being performed
 *
 * returns:
 *	number of tests that failed
 */
define verify_csc(testnum)
{
	local tval_len;		/* current length of the tval[] array */
	local ident_val;	/* computed trig value trigonometric identity */
	local trig_val;		/* computed value from the trigonometric function */
	local error_count;	/* number of compare errors detected */
	local i;

	/*
	 * firewall
	 */
	if (size(sin_tval) <= 0) {
		precompute_trig();
	}

	/*
	 * for each test value, verify the trigonometric identity within epsilon
	 */
	tval_len = size(tval);
	for (i=0; i < tval_len; ++i) {

		/* skip test when sin(x) within epsilon of 0 */
		if (not_near_zero(sin_tval[i])) {

			/* compute trigonometric identity */
			ident_val = 1 / sin_tval[i];

			/* compute trigonometric function */
			trig_val = csc(tval[i]);

			/* compare trigonometric identity with trigonometric function value */
			if (compare(ident_val, trig_val, "csc", i, testnum)) {
				++error_count;
			}
		}
	}

	/*
	 * report test results
	 */
	if (error_count != 0) {
		print '**** test', testnum : ': csc test failure count:', error_count;
	}
	return error_count;
}


/*
 * verify_versin - verify versed trigonometric sine
 *
 * We use the following trigonometric identity:
 *
 *	versin(x) = 1 - cos(x)
 *
 * given:
 *	testnum		regression test number being performed
 *
 * returns:
 *	number of tests that failed
 */
define verify_versin(testnum)
{
	local tval_len;		/* current length of the tval[] array */
	local ident_val;	/* computed trig value trigonometric identity */
	local trig_val;		/* computed value from the trigonometric function */
	local error_count;	/* number of compare errors detected */
	local i;

	/*
	 * firewall
	 */
	if (size(cos_tval) <= 0) {
		precompute_trig();
	}

	/*
	 * for each test value, verify the trigonometric identity within epsilon
	 */
	tval_len = size(tval);
	for (i=0; i < tval_len; ++i) {

		/* skip test when cos(x) within epsilon of 0 */
		if (not_near_zero(cos_tval[i])) {

			/* compute trigonometric identity */
			ident_val = 1 - cos_tval[i];

			/* compute trigonometric function */
			trig_val = versin(tval[i]);

			/* compare trigonometric identity with trigonometric function value */
			if (compare(ident_val, trig_val, "versin", i, testnum)) {
				++error_count;
			}
		}
	}

	/*
	 * report test results
	 */
	if (error_count != 0) {
		print '**** test', testnum : ': versin test failure count:', error_count;
	}
	return error_count;
}


/*
 * verify_coversin - verify coversed trigonometric sine
 *
 * We use the following trigonometric identity:
 *
 *	coversin(x) = 1 - sin(x)
 *
 * given:
 *	testnum		regression test number being performed
 *
 * returns:
 *	number of tests that failed
 */
define verify_coversin(testnum)
{
	local tval_len;		/* current length of the tval[] array */
	local ident_val;	/* computed trig value trigonometric identity */
	local trig_val;		/* computed value from the trigonometric function */
	local error_count;	/* number of compare errors detected */
	local i;

	/*
	 * firewall
	 */
	if (size(sin_tval) <= 0) {
		precompute_trig();
	}

	/*
	 * for each test value, verify the trigonometric identity within epsilon
	 */
	tval_len = size(tval);
	for (i=0; i < tval_len; ++i) {

		/* skip test when sin(x) within epsilon of 0 */
		if (not_near_zero(sin_tval[i])) {

			/* compute trigonometric identity */
			ident_val = 1 - sin_tval[i];

			/* compute trigonometric function */
			trig_val = coversin(tval[i]);

			/* compare trigonometric identity with trigonometric function value */
			if (compare(ident_val, trig_val, "coversin", i, testnum)) {
				++error_count;
			}
		}
	}

	/*
	 * report test results
	 */
	if (error_count != 0) {
		print '**** test', testnum : ': coversin test failure count:', error_count;
	}
	return error_count;
}


/*
 * verify_vercos - verify versed trigonometric cosine
 *
 * We use the following trigonometric identity:
 *
 *	vercos(x) = 1 + cos(x)
 *
 * given:
 *	testnum		regression test number being performed
 *
 * returns:
 *	number of tests that failed
 */
define verify_vercos(testnum)
{
	local tval_len;		/* current length of the tval[] array */
	local ident_val;	/* computed trig value trigonometric identity */
	local trig_val;		/* computed value from the trigonometric function */
	local error_count;	/* number of compare errors detected */
	local i;

	/*
	 * firewall
	 */
	if (size(cos_tval) <= 0) {
		precompute_trig();
	}

	/*
	 * for each test value, verify the trigonometric identity within epsilon
	 */
	tval_len = size(tval);
	for (i=0; i < tval_len; ++i) {

		/* skip test when cos(x) within epsilon of 0 */
		if (not_near_zero(cos_tval[i])) {

			/* compute trigonometric identity */
			ident_val = 1 + cos_tval[i];

			/* compute trigonometric function */
			trig_val = vercos(tval[i]);

			/* compare trigonometric identity with trigonometric function value */
			if (compare(ident_val, trig_val, "vercos", i, testnum)) {
				++error_count;
			}
		}
	}

	/*
	 * report test results
	 */
	if (error_count != 0) {
		print '**** test', testnum : ': vercos test failure count:', error_count;
	}
	return error_count;
}


/*
 * verify_covercos - verify coversed trigonometric cosine
 *
 * We use the following trigonometric identity:
 *
 *	covercos(x) = 1 + sin(x)
 *
 * given:
 *	testnum		regression test number being performed
 *
 * returns:
 *	number of tests that failed
 */
define verify_covercos(testnum)
{
	local tval_len;		/* current length of the tval[] array */
	local ident_val;	/* computed trig value trigonometric identity */
	local trig_val;		/* computed value from the trigonometric function */
	local error_count;	/* number of compare errors detected */
	local i;

	/*
	 * firewall
	 */
	if (size(sin_tval) <= 0) {
		precompute_trig();
	}

	/*
	 * for each test value, verify the trigonometric identity within epsilon
	 */
	tval_len = size(tval);
	for (i=0; i < tval_len; ++i) {

		/* skip test when sin(x) within epsilon of 0 */
		if (not_near_zero(sin_tval[i])) {

			/* compute trigonometric identity */
			ident_val = 1 + sin_tval[i];

			/* compute trigonometric function */
			trig_val = covercos(tval[i]);

			/* compare trigonometric identity with trigonometric function value */
			if (compare(ident_val, trig_val, "covercos", i, testnum)) {
				++error_count;
			}
		}
	}

	/*
	 * report test results
	 */
	if (error_count != 0) {
		print '**** test', testnum : ': covercos test failure count:', error_count;
	}
	return error_count;
}


/*
 * verify_haversin - verify half versed trigonometric sine
 *
 * We use the following trigonometric identity:
 *
 *	haversin(x) = versin(x) / 2 = (1 - cos(x)) / 2
 *
 * given:
 *	testnum		regression test number being performed
 *
 * returns:
 *	number of tests that failed
 */
define verify_haversin(testnum)
{
	local tval_len;		/* current length of the tval[] array */
	local ident_val;	/* computed trig value trigonometric identity */
	local trig_val;		/* computed value from the trigonometric function */
	local error_count;	/* number of compare errors detected */
	local i;

	/*
	 * firewall
	 */
	if (size(cos_tval) <= 0) {
		precompute_trig();
	}

	/*
	 * for each test value, verify the trigonometric identity within epsilon
	 */
	tval_len = size(tval);
	for (i=0; i < tval_len; ++i) {

		/* skip test when cos(x) within epsilon of 0 */
		if (not_near_zero(cos_tval[i])) {

			/* compute trigonometric identity */
			ident_val = (1 - cos_tval[i]) / 2;

			/* compute trigonometric function */
			trig_val = haversin(tval[i]);

			/* compare trigonometric identity with trigonometric function value */
			if (compare(ident_val, trig_val, "haversin", i, testnum)) {
				++error_count;
			}
		}
	}

	/*
	 * report test results
	 */
	if (error_count != 0) {
		print '**** test', testnum : ': haversin test failure count:', error_count;
	}
	return error_count;
}


/*
 * verify_hacoversin - verify half versed trigonometric cosine
 *
 * We use the following trigonometric identity:
 *
 *	hacoversin(x) = coversin(x) / 2 = (1 - sin(x)) / 2
 *
 * given:
 *	testnum		regression test number being performed
 *
 * returns:
 *	number of tests that failed
 */
define verify_hacoversin(testnum)
{
	local tval_len;		/* current length of the tval[] array */
	local ident_val;	/* computed trig value trigonometric identity */
	local trig_val;		/* computed value from the trigonometric function */
	local error_count;	/* number of compare errors detected */
	local i;

	/*
	 * firewall
	 */
	if (size(sin_tval) <= 0) {
		precompute_trig();
	}

	/*
	 * for each test value, verify the trigonometric identity within epsilon
	 */
	tval_len = size(tval);
	for (i=0; i < tval_len; ++i) {

		/* skip test when cos(x) within epsilon of 0 */
		if (not_near_zero(sin_tval[i])) {

			/* compute trigonometric identity */
			ident_val = (1 - sin_tval[i]) / 2;

			/* compute trigonometric function */
			trig_val = hacoversin(tval[i]);

			/* compare trigonometric identity with trigonometric function value */
			if (compare(ident_val, trig_val, "hacoversin", i, testnum)) {
				++error_count;
			}
		}
	}

	/*
	 * report test results
	 */
	if (error_count != 0) {
		print '**** test', testnum : ': hacoversin test failure count:', error_count;
	}
	return error_count;
}


/*
 * verify_havercos - verify half versed trigonometric cosine
 *
 * We use the following trigonometric identity:
 *
 *	havercos(x) = vercos(x) / 2 = (1 + cos(x)) / 2
 *
 * given:
 *	testnum		regression test number being performed
 *
 * returns:
 *	number of tests that failed
 */
define verify_havercos(testnum)
{
	local tval_len;		/* current length of the tval[] array */
	local ident_val;	/* computed trig value trigonometric identity */
	local trig_val;		/* computed value from the trigonometric function */
	local error_count;	/* number of compare errors detected */
	local i;

	/*
	 * firewall
	 */
	if (size(cos_tval) <= 0) {
		precompute_trig();
	}

	/*
	 * for each test value, verify the trigonometric identity within epsilon
	 */
	tval_len = size(tval);
	for (i=0; i < tval_len; ++i) {

		/* skip test when cos(x) within epsilon of 0 */
		if (not_near_zero(cos_tval[i])) {

			/* compute trigonometric identity */
			ident_val = (1 + cos_tval[i]) / 2;

			/* compute trigonometric function */
			trig_val = havercos(tval[i]);

			/* compare trigonometric identity with trigonometric function value */
			if (compare(ident_val, trig_val, "havercos", i, testnum)) {
				++error_count;
			}
		}
	}

	/*
	 * report test results
	 */
	if (error_count != 0) {
		print '**** test', testnum : ': havercos test failure count:', error_count;
	}
	return error_count;
}


/*
 * verify_hacovercos - verify half coversed trigonometric cosine
 *
 * We use the following trigonometric identity:
 *
 *	hacovercos(x) = covercos(x) / 2 = (1 + sin(x)) / 2
 *
 * given:
 *	testnum		regression test number being performed
 *
 * returns:
 *	number of tests that failed
 */
define verify_hacovercos(testnum)
{
	local tval_len;		/* current length of the tval[] array */
	local ident_val;	/* computed trig value trigonometric identity */
	local trig_val;		/* computed value from the trigonometric function */
	local error_count;	/* number of compare errors detected */
	local i;

	/*
	 * firewall
	 */
	if (size(sin_tval) <= 0) {
		precompute_trig();
	}

	/*
	 * for each test value, verify the trigonometric identity within epsilon
	 */
	tval_len = size(tval);
	for (i=0; i < tval_len; ++i) {

		/* skip test when sin(x) within epsilon of 0 */
		if (not_near_zero(sin_tval[i])) {

			/* compute trigonometric identity */
			ident_val = (1 + sin_tval[i]) / 2;

			/* compute trigonometric function */
			trig_val = hacovercos(tval[i]);

			/* compare trigonometric identity with trigonometric function value */
			if (compare(ident_val, trig_val, "hacovercos", i, testnum)) {
				++error_count;
			}
		}
	}

	/*
	 * report test results
	 */
	if (error_count != 0) {
		print '**** test', testnum : ': hacovercos test failure count:', error_count;
	}
	return error_count;
}


/*
 * verify_exsec - exterior trigonometric secant
 *
 * We use the following trigonometric identity:
 *
 *	exsec(x) = sec(x) - 1 = (1 / cos(x)) - 1
 *
 * given:
 *	testnum		regression test number being performed
 *
 * returns:
 *	number of tests that failed
 */
define verify_exsec(testnum)
{
	local tval_len;		/* current length of the tval[] array */
	local ident_val;	/* computed trig value trigonometric identity */
	local trig_val;		/* computed value from the trigonometric function */
	local error_count;	/* number of compare errors detected */
	local i;

	/*
	 * firewall
	 */
	if (size(cos_tval) <= 0) {
		precompute_trig();
	}

	/*
	 * for each test value, verify the trigonometric identity within epsilon
	 */
	tval_len = size(tval);
	for (i=0; i < tval_len; ++i) {

		/* skip test when cos(x) within epsilon of 0 */
		if (not_near_zero(cos_tval[i])) {

			/* compute trigonometric identity */
			ident_val = (1 / cos_tval[i]) - 1;

			/* compute trigonometric function */
			trig_val = exsec(tval[i]);

			/* compare trigonometric identity with trigonometric function value */
			if (compare(ident_val, trig_val, "exsec", i, testnum)) {
				++error_count;
			}
		}
	}

	/*
	 * report test results
	 */
	if (error_count != 0) {
		print '**** test', testnum : ': exsec test failure count:', error_count;
	}
	return error_count;
}


/*
 * verify_excsc - verify trigonometric cosecant
 *
 * We use the following trigonometric identity:
 *
 *	excsc(x) = csc(x) - 1 = (1 / sin(x)) - 1
 *
 * given:
 *	testnum		regression test number being performed
 *
 * returns:
 *	number of tests that failed
 */
define verify_excsc(testnum)
{
	local tval_len;		/* current length of the tval[] array */
	local ident_val;	/* computed trig value trigonometric identity */
	local trig_val;		/* computed value from the trigonometric function */
	local error_count;	/* number of compare errors detected */
	local i;

	/*
	 * firewall
	 */
	if (size(sin_tval) <= 0) {
		precompute_trig();
	}

	/*
	 * for each test value, verify the trigonometric identity within epsilon
	 */
	tval_len = size(tval);
	for (i=0; i < tval_len; ++i) {

		/* skip test when sin(x) within epsilon of 0 */
		if (not_near_zero(sin_tval[i])) {

			/* compute trigonometric identity */
			ident_val = (1 / sin_tval[i]) - 1;

			/* compute trigonometric function */
			trig_val = excsc(tval[i]);

			/* compare trigonometric identity with trigonometric function value */
			if (compare(ident_val, trig_val, "excsc", i, testnum)) {
				++error_count;
			}
		}
	}

	/*
	 * report test results
	 */
	if (error_count != 0) {
		print '**** test', testnum : ': excsc test failure count:', error_count;
	}
	return error_count;
}


/*
 * verify_crd - verify trigonometric chord of a unit circle
 *
 * We use the following trigonometric identity:
 *
 *	crd(x) = 2 * sin(x / 2)
 *
 * given:
 *	testnum		regression test number being performed
 *
 * returns:
 *	number of tests that failed
 */
define verify_crd(testnum)
{
	local tval_len;		/* current length of the tval[] array */
	local ident_val;	/* computed trig value trigonometric identity */
	local trig_val;		/* computed value from the trigonometric function */
	local error_count;	/* number of compare errors detected */
	local i;

	/*
	 * firewall
	 */
	if (size(sin_tval) <= 0) {
		precompute_trig();
	}

	/*
	 * for each test value, verify the trigonometric identity within epsilon
	 */
	tval_len = size(tval);
	for (i=0; i < tval_len; ++i) {

		/* NOTE: We actually check the identity: crd(x*2) = 2 * sin(x) */

		/* compute trigonometric identity */
		ident_val = 2 * sin_tval[i];

		/* compute trigonometric function */
		trig_val = crd(tval[i] * 2);

		/* compare trigonometric identity with trigonometric function value */
		if (compare(ident_val, trig_val, "crd", i, testnum)) {
			++error_count;
		}
	}

	/*
	 * report test results
	 */
	if (error_count != 0) {
		print '**** test', testnum : ': crd test failure count:', error_count;
	}
	return error_count;
}


/*
 * verify_cas - verify cosine plus sine
 *
 * We use the following trigonometric identity:
 *
 *	cas(x) = cos(x) + sin(x)
 *
 * given:
 *	testnum		regression test number being performed
 *
 * returns:
 *	number of tests that failed
 */
define verify_cas(testnum)
{
	local tval_len;		/* current length of the tval[] array */
	local ident_val;	/* computed trig value trigonometric identity */
	local trig_val;		/* computed value from the trigonometric function */
	local error_count;	/* number of compare errors detected */
	local i;

	/*
	 * firewall
	 */
	if (size(sin_tval) <= 0) {
		precompute_trig();
	}
	if (size(cos_tval) <= 0) {
		precompute_trig();
	}

	/*
	 * for each test value, verify the trigonometric identity within epsilon
	 */
	tval_len = size(tval);
	for (i=0; i < tval_len; ++i) {

		/* compute trigonometric identity */
		ident_val = cos_tval[i] + sin_tval[i];

		/* compute trigonometric function */
		trig_val = cas(tval[i]);

		/* compare trigonometric identity with trigonometric function value */
		if (compare(ident_val, trig_val, "cas", i, testnum)) {
			++error_count;
		}
	}

	/*
	 * report test results
	 */
	if (error_count != 0) {
		print '**** test', testnum : ': cas test failure count:', error_count;
	}
	return error_count;
}


/*
 * verify_cis - verify Euler's formula
 *
 * We use the following trigonometric identity:
 *
 *	cis(x) = cos(x) + i*sin(x)
 *	cis(x) = exp(1i * x)
 *
 * given:
 *	testnum		regression test number being performed
 *
 * returns:
 *	number of tests that failed
 */
define verify_cis(testnum)
{
	local tval_len;		/* current length of the tval[] array */
	local ident_val;	/* computed trig value trigonometric identity */
	local trig_val;		/* computed value from the trigonometric function */
	local error_count;	/* number of compare errors detected */
	local i;

	/*
	 * firewall
	 */
	if (size(sin_tval) <= 0) {
		precompute_trig();
	}
	if (size(cos_tval) <= 0) {
		precompute_trig();
	}

	/*
	 * for each test value, verify the trigonometric identity within epsilon
	 */
	tval_len = size(tval);
	for (i=0; i < tval_len; ++i) {

		/* compute trigonometric identity */
		ident_val = cos_tval[i] + 1i*sin_tval[i];

		/* compute trigonometric function */
		trig_val = cis(tval[i]);

		/* compare trigonometric identity with trigonometric function value */
		if (compare(ident_val, trig_val, "cis", i, testnum)) {
			++error_count;
		}
	}

	/*
	 * report test results
	 */
	if (error_count != 0) {
		print '**** test', testnum : ': cis test failure count:', error_count;
	}
	return error_count;
}