add trigonometric chord of a unit circle functions

Improve builtin function strings, as printed by help builtin, that use an optional accuracy (epsilon) arg by adding a comma. Added the following new trigonometric functions: crd(x [,eps]) trigonometric chord of a unit circle acrd(x [,eps]) inverse trigonometric chord of a unit circle
2025-08-16 01:03:29 +03:00 · 2023-10-02 22:43:33 -07:00
parent c78a893862
commit 26fc394089
44 changed files with 1246 additions and 118 deletions
--- a/51
+++ b/51
@@ -1,5 +1,30 @@
 The following are the changes from calc version 2.14.3.5 to date:

+    Added the following new trigonometric functions:
+
+	versin(x [,eps])	versed trigonometric sine
+	coversin(x [,eps])	coversed trigonometric sine
+	vercos(x [,eps])	versed trigonometric cosine
+	covercos(x [,eps])	coversed trigonometric cosine
+	aversin(x [,eps])	inverse versed trigonometric sine
+	acoversin(x [,eps])	inverse coversed trigonometric sine
+	avercos(x [,eps])	inverse versed trigonometric cosine
+	acovercos(x [,eps])	inverse coversed trigonometric cosine
+	haversin(x [,eps])	half versed trigonometric sine
+	hacoversin(x [,eps])	half coversed trigonometric sine
+	havercos(x [,eps])	half versed trigonometric cosine
+	hacovercos(x [,eps])	half coversed trigonometric cosine
+	ahaversin(x [,eps])	inverse half versed trigonometric sine
+	ahacoversin(x [,eps])	inverse half coversed trigonometric sine
+	ahavercos(x [,eps])	inverse half versed trigonometric cosine
+	ahacovercos(x [,eps])	inverse half coversed trigonometric cosine
+	exsec(x [,eps])		exterior trigonometric secant
+	aexsec(x [,eps])	inverse exterior trigonometric secant
+	excsc(x [,eps])		exterior trigonometric cosecant
+	aexcsc(x [,eps])	inverse exterior trigonometric cosecant
+	crd(x [,eps])		trigonometric chord of a unit circle
+	acrd(x [,eps])		inverse trigonometric chord of a unit circle
+
    As Msys2 is a fork of Cygwin, if the OSNAME is Msys, the Cygwin
    target will be used.  Thanks to GitHub user @iahung2 for the
    pull request.
@@ -336,29 +361,6 @@ The following are the changes from calc version 2.14.3.5 to date:

    Fixed SEE ALSO typo in help randperm.

-    Added the following new trigonometric functions:
-
-	versin(x [,eps])	versed trigonometric sine
-	coversin(x [,eps])	coversed trigonometric sine
-	vercos(x [,eps])	versed trigonometric cosine
-	covercos(x [,eps])	coversed trigonometric cosine
-	aversin(x [,eps])	inverse versed trigonometric sine
-	acoversin(x [,eps])	inverse coversed trigonometric sine
-	avercos(x [,eps])	inverse versed trigonometric cosine
-	acovercos(x [,eps])	inverse coversed trigonometric cosine
-	haversin(x [,eps])	half versed trigonometric sine
-	hacoversin(x [,eps])	half coversed trigonometric sine
-	havercos(x [,eps])	half versed trigonometric cosine
-	hacovercos(x [,eps])	half coversed trigonometric cosine
-	ahaversin(x [,eps])	inverse half versed trigonometric sine
-	ahacoversin(x [,eps])	inverse half coversed trigonometric sine
-	ahavercos(x [,eps])	inverse half versed trigonometric cosine
-	ahacovercos(x [,eps])	inverse half coversed trigonometric cosine
-	exsec(x [,eps])		exterior trigonometric secant
-	aexsec(x [,eps])	inverse exterior trigonometric secant
-	excsc(x [,eps])		exterior trigonometric cosecant
-	aexcsc(x [,eps])	inverse exterior trigonometric cosecant
-
    Fixed calc regression test 42dd to set the display value back to 20.

    Added to test 95dd and test9500.trigeq.cal to the calc regression test
@@ -366,6 +368,9 @@ The following are the changes from calc version 2.14.3.5 to date:

    Added to test 34dd, some if the missing inverse trigonometric tests.

+    Improve builtin function strings, as printed by help builtin,
+    that use an optional accuracy (epsilon) arg by adding a comma.
+

 The following are the changes from calc version 2.14.3.4 to 2.14.3.5:

--- a/cal/regress.cal
+++ b/cal/regress.cal
@@ -4138,6 +4138,50 @@ define test_trig()
 	    strcat(str(tnum++),
 		   ': round(aexcsc(2 + 3i, 1e-10), 10) == 0.1650861985-0.1681700706i'));

+	/* test trigonometric chord of a unit circle */
+	vrfy(crd(0, 1e-10) == 0,
+				strcat(str(tnum++), ': crd(0, 1e-10) == 0'));
+	vrfy(round(crd(0.2, 1e-10), 10) == 0.1996668332,
+	    strcat(str(tnum++),
+		   ': round(crd(0.2, 1e-10), 10) == 0.1996668332'));
+	vrfy(round(crd(3/7, 1e-10), 10) == 0.4252990674,
+	    strcat(str(tnum++),
+		   ': round(crd(3/7, 1e-10), 10) == 0.4252990674'));
+	vrfy(round(crd(-31, 1e-10), 10) == -0.4129349638,
+	    strcat(str(tnum++),
+		   ': round(crd(-31, 1e-10), 10) == -0.4129349638'));
+	vrfy(crd(pi/3, 1e-10) == 1,
+				strcat(str(tnum++), ': crd(pi/3, 1e-10) == 1'));
+	vrfy(crd(pi, 1e-10) == 2,
+				strcat(str(tnum++), ': crd(pi, 1e-10) == 2'));
+	vrfy(crd(5*pi/3, 1e-10) == 1,
+				strcat(str(tnum++), ': crd(5*pi/3, 1e-10) == 1'));
+	vrfy(round(crd(1, 1e-10), 10) == 0.9588510772,
+	    strcat(str(tnum++),
+		   ': round(crd(1, 1e-10), 10) == 0.9588510772'));
+	vrfy(round(crd(2 + 3i, 1e-10), 10) == 3.9589688712+2.3009091988i,
+	    strcat(str(tnum++),
+		   ': round(crd(2 + 3i, 1e-10), 10) == 3.9589688712+2.3009091988i'));
+
+	/* test inverse trigonometric chord of a unit circle */
+	vrfy(acrd(0, 1e-10) == 0,
+				strcat(str(tnum++), ': acrd(0, 1e-10) == 0'));
+	vrfy(round(acrd(0.2, 1e-10), 10) == 0.2003348424,
+	    strcat(str(tnum++),
+		   ': round(acrd(0.2, 1e-10), 10) == 0.2003348424'));
+	vrfy(round(acrd(3/7, 1e-10), 10) == 0.4319209974,
+	    strcat(str(tnum++),
+		   ': round(acrd(3/7, 1e-10), 10) == 0.4319209974'));
+	vrfy(round(acrd(-31, 1e-10), 10) == -3.1415926536+6.8658899902i,
+	    strcat(str(tnum++),
+		   ': round(acrd(-31, 1e-10), 10) == -3.1415926536+6.8658899902i'));
+	vrfy(round(acrd(1, 1e-10), 10) == 1.0471975512,
+	    strcat(str(tnum++),
+		   ': round(acrd(1, 1e-10), 10) == 1.0471975512'));
+	vrfy(round(acrd(2 + 3i, 1e-10), 10) == 1.0471975512+2.6339157938i,
+	    strcat(str(tnum++),
+		   ': round(acrd(2 + 3i, 1e-10), 10) == 1.0471975512+2.6339157938i'));
+
 	print strcat(str(tnum++), ': Ending test_trig');
 }
 print '051: parsed test_trig()';
@@ -9835,8 +9879,9 @@ vrfy(verify_havercos(9513) == 0,	'9513: verify_havercos(9513) == 0');
 vrfy(verify_hacovercos(9514) == 0,	'9514: verify_hacovercos(9514) == 0');
 vrfy(verify_exsec(9515) == 0,		'9515: verify_exsec(9515) == 0');
 vrfy(verify_excsc(9516) == 0,		'9516: verify_excsc(9516) == 0');
+vrfy(verify_crd(9517) == 0,		'9516: verify_excsc(9517) == 0');

-print '9517: Ending trigonometric identities test set';
+print '9518: Ending trigonometric identities test set';


 /*
@@ -10839,6 +10884,12 @@ vrfy_errsym(10590, 10590, "E_EXCSC_4");
 vrfy_errsym(10591, 10591, "E_AEXCSC_1");
 vrfy_errsym(10592, 10592, "E_AEXCSC_2");
 vrfy_errsym(10593, 10593, "E_AEXCSC_3");
+vrfy_errsym(10594, 10594, "E_CRD_1");
+vrfy_errsym(10595, 10595, "E_CRD_2");
+vrfy_errsym(10596, 10596, "E_CRD_3");
+vrfy_errsym(10597, 10597, "E_ACRD_1");
+vrfy_errsym(10598, 10598, "E_ACRD_2");
+vrfy_errsym(10599, 10599, "E_ACRD_3");

 /* ************************************************************** */
 /* NOTE: Reserve thru test 10998 for calc computation error codes */
--- a/cal/test9500.trigeq.cal
+++ b/cal/test9500.trigeq.cal
@@ -364,6 +364,16 @@ 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
 	 */
@@ -1244,3 +1254,61 @@ define verify_excsc(testnum)
 	}
 	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;
+}
--- a/cmath.h
+++ b/cmath.h
@@ -148,6 +148,8 @@ E_FUNC COMPLEX *c_exsec(COMPLEX *c, NUMBER *epsilon);
 E_FUNC COMPLEX *c_aexsec(COMPLEX *c, NUMBER *epsilon);
 E_FUNC COMPLEX *c_excsc(COMPLEX *c, NUMBER *epsilon);
 E_FUNC COMPLEX *c_aexcsc(COMPLEX *c, NUMBER *epsilon);
+E_FUNC COMPLEX *c_crd(COMPLEX *c, NUMBER *epsilon);
+E_FUNC COMPLEX *c_acrd(COMPLEX *c, NUMBER *epsilon);



--- a/comfunc.c
+++ b/comfunc.c
@@ -1655,7 +1655,7 @@ c_ilog(COMPLEX *c, ZVALUE base)
 *	versin(x) = 1 - cos(x)
 *
 * given:
- *	q	    complex value to pass to the trig function
+ *	c	    complex value to pass to the trig function
 *	epsilon	    error tolerance / precision for trig calculation
 *
 * returns:
@@ -1705,7 +1705,7 @@ c_versin(COMPLEX *c, NUMBER *epsilon)
 *	aversin(x) = acos(1 - x)
 *
 * given:
- *	q	    complex value to pass to the trig function
+ *	c	    complex value to pass to the trig function
 *	epsilon	    error tolerance / precision for trig calculation
 *
 * returns:
@@ -1751,7 +1751,7 @@ c_aversin(COMPLEX *c, NUMBER *epsilon)
 *	coversin(x) = 1 - cos(x)
 *
 * given:
- *	q	    complex value to pass to the trig function
+ *	c	    complex value to pass to the trig function
 *	epsilon	    error tolerance / precision for trig calculation
 *
 * returns:
@@ -1801,7 +1801,7 @@ c_coversin(COMPLEX *c, NUMBER *epsilon)
 *	acoversin(x) = asin(1 - x)
 *
 * given:
- *	q	    complex value to pass to the trig function
+ *	c	    complex value to pass to the trig function
 *	epsilon	    error tolerance / precision for trig calculation
 *
 * returns:
@@ -1847,7 +1847,7 @@ c_acoversin(COMPLEX *c, NUMBER *epsilon)
 *	vercos(x) = 1 + cos(x)
 *
 * given:
- *	q	    complex value to pass to the trig function
+ *	c	    complex value to pass to the trig function
 *	epsilon	    error tolerance / precision for trig calculation
 *
 * returns:
@@ -1897,7 +1897,7 @@ c_vercos(COMPLEX *c, NUMBER *epsilon)
 *	avercos(x) = acos(x - 1)
 *
 * given:
- *	q	    complex value to pass to the trig function
+ *	c	    complex value to pass to the trig function
 *	epsilon	    error tolerance / precision for trig calculation
 *
 * returns:
@@ -1943,7 +1943,7 @@ c_avercos(COMPLEX *c, NUMBER *epsilon)
 *	covercos(x) = 1 + sin(x)
 *
 * given:
- *	q	    complex value to pass to the trig function
+ *	c	    complex value to pass to the trig function
 *	epsilon	    error tolerance / precision for trig calculation
 *
 * returns:
@@ -1993,7 +1993,7 @@ c_covercos(COMPLEX *c, NUMBER *epsilon)
 *	acovercos(x) = asin(x - 1)
 *
 * given:
- *	q	    complex value to pass to the trig function
+ *	c	    complex value to pass to the trig function
 *	epsilon	    error tolerance / precision for trig calculation
 *
 * returns:
@@ -2039,7 +2039,7 @@ c_acovercos(COMPLEX *c, NUMBER *epsilon)
 *	haversin(x) = versin(x) / 2
 *
 * given:
- *	q	    complex value to pass to the trig function
+ *	c	    complex value to pass to the trig function
 *	epsilon	    error tolerance / precision for trig calculation
 *
 * returns:
@@ -2089,7 +2089,7 @@ c_haversin(COMPLEX *c, NUMBER *epsilon)
 *	ahaversin(x) = acos(1 - 2*x)
 *
 * given:
- *	q	    complex value to pass to the trig function
+ *	c	    complex value to pass to the trig function
 *	epsilon	    error tolerance / precision for trig calculation
 *
 * returns:
@@ -2138,7 +2138,7 @@ c_ahaversin(COMPLEX *c, NUMBER *epsilon)
 *	hacoversin(x) = coversin(x) / 2
 *
 * given:
- *	q	    complex value to pass to the trig function
+ *	c	    complex value to pass to the trig function
 *	epsilon	    error tolerance / precision for trig calculation
 *
 * returns:
@@ -2188,7 +2188,7 @@ c_hacoversin(COMPLEX *c, NUMBER *epsilon)
 *	ahacoversin(x) = asin(1 - 2*x)
 *
 * given:
- *	q	    complex value to pass to the trig function
+ *	c	    complex value to pass to the trig function
 *	epsilon	    error tolerance / precision for trig calculation
 *
 * returns:
@@ -2237,7 +2237,7 @@ c_ahacoversin(COMPLEX *c, NUMBER *epsilon)
 *	havercos(x) = vercos(x) / 2
 *
 * given:
- *	q	    complex value to pass to the trig function
+ *	c	    complex value to pass to the trig function
 *	epsilon	    error tolerance / precision for trig calculation
 *
 * returns:
@@ -2287,7 +2287,7 @@ c_havercos(COMPLEX *c, NUMBER *epsilon)
 *	ahavercos(x) = acos(2*x - 1)
 *
 * given:
- *	q	    complex value to pass to the trig function
+ *	c	    complex value to pass to the trig function
 *	epsilon	    error tolerance / precision for trig calculation
 *
 * returns:
@@ -2336,7 +2336,7 @@ c_ahavercos(COMPLEX *c, NUMBER *epsilon)
 *	hacovercos(x) = covercos(x) / 2
 *
 * given:
- *	q	    complex value to pass to the trig function
+ *	c	    complex value to pass to the trig function
 *	epsilon	    error tolerance / precision for trig calculation
 *
 * returns:
@@ -2386,7 +2386,7 @@ c_hacovercos(COMPLEX *c, NUMBER *epsilon)
 *	ahacovercos(x) = asin(2*x - 1)
 *
 * given:
- *	q	    complex value to pass to the trig function
+ *	c	    complex value to pass to the trig function
 *	epsilon	    error tolerance / precision for trig calculation
 *
 * returns:
@@ -2435,7 +2435,7 @@ c_ahacovercos(COMPLEX *c, NUMBER *epsilon)
 *	exsec(x) = sec(x) - 1
 *
 * given:
- *	q	    complex value to pass to the trig function
+ *	c	    complex value to pass to the trig function
 *	epsilon	    error tolerance / precision for trig calculation
 *
 * returns:
@@ -2485,7 +2485,7 @@ c_exsec(COMPLEX *c, NUMBER *epsilon)
 *	aexsec(x) = asec(x + 1)
 *
 * given:
- *	q	    complex value to pass to the trig function
+ *	c	    complex value to pass to the trig function
 *	epsilon	    error tolerance / precision for trig calculation
 *
 * returns:
@@ -2531,7 +2531,7 @@ c_aexsec(COMPLEX *c, NUMBER *epsilon)
 *	excsc(x) = csc(x) - 1
 *
 * given:
- *	q	    complex value to pass to the trig function
+ *	c	    complex value to pass to the trig function
 *	epsilon	    error tolerance / precision for trig calculation
 *
 * returns:
@@ -2581,7 +2581,7 @@ c_excsc(COMPLEX *c, NUMBER *epsilon)
 *	aexcsc(x) = acsc(x + 1)
 *
 * given:
- *	q	    complex value to pass to the trig function
+ *	c	    complex value to pass to the trig function
 *	epsilon	    error tolerance / precision for trig calculation
 *
 * returns:
@@ -2617,3 +2617,105 @@ c_aexcsc(COMPLEX *c, NUMBER *epsilon)
 	 */
 	return r;
 }
+
+
+/*
+ * c_crd - COMPLEX valued trigonometric chord of a unit circle
+ *
+ * This uses the formula:
+ *
+ *	crd(x) = 2 * sin(x / 2)
+ *
+ * given:
+ *	c	    complex value to pass to the trig function
+ *	epsilon	    error tolerance / precision for trig calculation
+ *
+ * returns:
+ *	complex value result of trig function on q with error epsilon
+ */
+COMPLEX *
+c_crd(COMPLEX *c, NUMBER *epsilon)
+{
+	COMPLEX *r;		/* return COMPLEX value */
+	COMPLEX *cdiv2;		/* complex c/2 */
+	COMPLEX *ctmp;		/* complex sin(c/2) */
+
+	/*
+	 * firewall
+	 */
+	if (c == NULL) {
+		math_error("%s: c is NULL", __func__);
+		not_reached();
+	}
+	if (check_epsilon(epsilon) == false) {
+		math_error("Invalid epsilon arg for %s", __func__);
+		not_reached();
+	}
+
+	/*
+	 * calculate complex trig function value
+	 */
+	cdiv2 = c_divq(c, &_qtwo_);
+	ctmp = c_sin(cdiv2, epsilon);
+	comfree(cdiv2);
+	if (ctmp == NULL) {
+		math_error("Failed to compute complex sine for complex crd");
+		not_reached();
+	}
+	r = c_mulq(ctmp, &_qtwo_);
+	comfree(ctmp);
+
+	/*
+	 * return trigonometric result
+	 */
+	return r;
+}
+
+
+/*
+ * c_acrd - COMPLEX valued inverse trigonometric chord of a unit circle
+ *
+ * This uses the formula:
+ *
+ *	acrd(x) = 2 * asin(x / 2)
+ *
+ * given:
+ *	c	    complex value to pass to the trig function
+ *	epsilon	    error tolerance / precision for trig calculation
+ *
+ * returns:
+ *	complex value result of trig function on q with error epsilon
+ */
+COMPLEX *
+c_acrd(COMPLEX *c, NUMBER *epsilon)
+{
+	COMPLEX *r;		/* inverse trig value result */
+	COMPLEX *cdiv2;		/* complex c/2 */
+	COMPLEX *ctmp;		/* complex asin(c/2) */
+
+	/*
+	 * firewall
+	 */
+	if (c == NULL) {
+		math_error("%s: c is NULL", __func__);
+		not_reached();
+	}
+	if (check_epsilon(epsilon) == false) {
+		math_error("Invalid epsilon arg for %s", __func__);
+		not_reached();
+	}
+
+	/*
+	 * calculate complex inverse trig function value
+	 */
+	cdiv2 = c_divq(c, &_qtwo_);
+	ctmp = c_asin(cdiv2, epsilon);
+	comfree(cdiv2);
+	r = c_mulq(ctmp, &_qtwo_);
+	comfree(ctmp);
+
+	/*
+	 * return inverse trigonometric result
+	 */
+	return r;
+}
--- a/errtbl.c
+++ b/errtbl.c
@@ -145,7 +145,8 @@
 * Do not remove such computation error codes, nor change their errnum values,
 * not change their E_STRING errsym codes.
 */
-CONST struct errtbl error_table[] = {
+CONST struct errtbl error_table[] =
+{

 	/* The E__BASE entry below must start with 10000 and must be first!! */
 	{ 10000,	"E__BASE",		"No error" },
@@ -742,6 +743,12 @@ CONST struct errtbl error_table[] = {
 	{ 10591,	"E_AEXCSC_1",		"Bad epsilon for aexcsc" },
 	{ 10592,	"E_AEXCSC_2",		"Bad argument for aexcsc" },
 	{ 10593,	"E_AEXCSC_3",		"Invalid value for calculating aexcsc" },
+	{ 10594,	"E_CRD_1",		"Bad epsilon for crd" },
+	{ 10595,	"E_CRD_2",		"Bad argument for crd" },
+	{ 10596,	"E_CRD_3",		"Invalid complex argument for crd" },
+	{ 10597,	"E_ACRD_1",		"Bad epsilon for acrd" },
+	{ 10598,	"E_ACRD_2",		"Bad argument for acrd" },
+	{ 10599,	"E_ACRD_3",		"Invalid value for calculating acrd" },
 	/* IMPORTANT NOTE: add new entries above here and be sure their errnum numeric value is consecutive! */

 	/* The next NULL entry must be last */
--- a/func.c
+++ b/func.c
@@ -12219,6 +12219,142 @@ f_aexcsc(int count, VALUE **vals)
 }


+/*
+ * f_crd - exterior trigonometric cosecant
+ */
+S_FUNC VALUE
+f_crd(int count, VALUE **vals)
+{
+	VALUE result;
+	COMPLEX *c;
+	NUMBER *err;
+
+	/* initialize VALUEs */
+	result.v_subtype = V_NOSUBTYPE;
+
+	/*
+	 * set error tolerance for builtin function
+	 *
+	 * Use err VALUE arg if given and value is in a valid range.
+	 */
+	err = conf->epsilon;
+	if (count == 2) {
+		if (verify_eps(vals[1]) == false) {
+			return error_value(E_CRD_1);
+		}
+		err = vals[1]->v_num;
+	}
+
+	/*
+	 * compute cosecant to a given error tolerance
+	 */
+	switch (vals[0]->v_type) {
+	case V_NUM:
+		result.v_num = qcrd(vals[0]->v_num, err);
+		result.v_type = V_NUM;
+		break;
+	case V_COM:
+		c = c_crd(vals[0]->v_com, err);
+		if (c == NULL) {
+			return error_value(E_CRD_3);
+		}
+		result.v_com = c;
+		result.v_type = V_COM;
+		if (cisreal(c)) {
+			result.v_num = c_to_q(c, true);
+			result.v_type = V_NUM;
+		}
+		break;
+	default:
+		return error_value(E_CRD_2);
+	}
+	return result;
+}
+
+
+/*
+ * f_acrd - exterior trigonometric cosecant
+ */
+S_FUNC VALUE
+f_acrd(int count, VALUE **vals)
+{
+	VALUE arg1;		/* 1st arg if it is a COMPLEX value */
+	VALUE result;		/* value to return */
+	COMPLEX *c;		/* COMPLEX trig result */
+	NUMBER *eps;		/* epsilon error tolerance */
+
+	/* initialize VALUE */
+	result.v_subtype = V_NOSUBTYPE;
+
+	/*
+	 * set error tolerance for builtin function
+	 *
+	 * Use eps VALUE arg if given and value is in a valid range.
+	 */
+	eps = conf->epsilon;
+	if (count == 2) {
+		if (verify_eps(vals[1]) == false) {
+			return error_value(E_ACRD_1);
+		}
+		eps = vals[1]->v_num;
+	}
+
+	/*
+	 * compute inverse trig function to a given error tolerance
+	 */
+	arg1 = *vals[0];
+	if (arg1.v_type == V_NUM) {
+
+		/* try to compute result using real trig function */
+		result.v_num = qacrd_or_NULL(arg1.v_num, eps);
+
+		/*
+		 * case: trig function returned a NUMBER
+		 */
+		if (result.v_num != NULL) {
+			result.v_type = V_NUM;
+
+		/*
+		 * case: trig function returned NULL - need to try COMPLEX trig function
+		 */
+		} else {
+			/* convert NUMBER argument from NUMBER to COMPLEX */
+			arg1.v_com = qqtoc(arg1.v_num, &_qzero_);
+			arg1.v_type = V_COM;
+		}
+	}
+	if (arg1.v_type == V_COM) {
+
+		/*
+		 * case: argument was COMPLEX or
+		 *	 trig function returned NULL and argument was converted to COMPLEX
+		 */
+		c = c_acrd(arg1.v_com, eps);
+		if (c == NULL) {
+			return error_value(E_ACRD_3);
+		}
+		result.v_com = c;
+		result.v_type = V_COM;
+
+		/*
+		 * case: complex trig function returned real, convert result to NUMBER
+		 */
+		if (cisreal(c)) {
+			result.v_num = c_to_q(c, true);
+			result.v_type = V_NUM;
+		}
+	}
+	if (arg1.v_type != V_NUM && arg1.v_type != V_COM) {
+
+		/*
+		 * case: argument type is not valid for this function
+		 */
+		return error_value(E_ACRD_2);
+	}
+	return result;
+}
+
+
 #endif /* !FUNCLIST */


@@ -12260,39 +12396,41 @@ f_aexcsc(int count, VALUE **vals)
 */
 STATIC CONST struct builtin builtins[] = {
 	{"abs", 1, 2, 0, OP_ABS, {.null = NULL}, {.null = NULL},
-	 "absolute value within accuracy b"},
+	 "absolute value, within accuracy b"},
 	{"access", 1, 2, 0, OP_NOP, {.null = NULL}, {.valfunc_cnt = f_access},
 	 "determine accessibility of file a for mode b"},
 	{"acos", 1, 2, 0, OP_NOP, {.null = NULL}, {.valfunc_cnt = f_acos},
-	 "inverse cosine of a within accuracy b"},
+	 "inverse cosine of a, within accuracy b"},
 	{"acosh", 1, 2, 0, OP_NOP, {.null = NULL}, {.valfunc_cnt = f_acosh},
-	 "inverse hyperbolic cosine of a within accuracy b"},
+	 "inverse hyperbolic cosine of a, within accuracy b"},
 	{"acot", 1, 2, 0, OP_NOP, {.null = NULL}, {.valfunc_cnt = f_acot},
-	 "inverse cotangent of a within accuracy b"},
+	 "inverse cotangent of a, within accuracy b"},
 	{"acoth", 1, 2, 0, OP_NOP, {.null = NULL}, {.valfunc_cnt = f_acoth},
-	 "inverse hyperbolic cotangent of a within accuracy b"},
+	 "inverse hyperbolic cotangent of a, within accuracy b"},
 	{"acovercos", 1, 2, 0, OP_NOP, {.null = NULL}, {.valfunc_cnt = f_acovercos},
-	 "inverse coversed cosine of a within accuracy b"},
+	 "inverse coversed cosine of a, within accuracy b"},
 	{"acoversin", 1, 2, 0, OP_NOP, {.null = NULL}, {.valfunc_cnt = f_acoversin},
-	 "inverse coversed sine of a within accuracy b"},
+	 "inverse coversed sine of a, within accuracy b"},
+	{"acrd", 1, 2, 0, OP_NOP, {.null = NULL}, {.valfunc_cnt = f_acrd},
+	 "angle of unit circle chord with length a, within accuracy b"},
 	{"acsc", 1, 2, 0, OP_NOP, {.null = NULL}, {.valfunc_cnt = f_acsc},
-	 "inverse cosecant of a within accuracy b"},
+	 "inverse cosecant of a, within accuracy b"},
 	{"acsch", 1, 2, 0, OP_NOP, {.null = NULL}, {.valfunc_cnt = f_acsch},
-	 "inverse csch of a within accuracy b"},
+	 "inverse csch of a, within accuracy b"},
 	{"aexcsc", 1, 2, 0, OP_NOP, {.null = NULL}, {.valfunc_cnt = f_aexcsc},
-	 "inverse exterior cosecant of a within accuracy b"},
+	 "inverse exterior cosecant of a, within accuracy b"},
 	{"aexsec", 1, 2, 0, OP_NOP, {.null = NULL}, {.valfunc_cnt = f_aexsec},
-	 "inverse exterior secant of a within accuracy b"},
+	 "inverse exterior secant of a, within accuracy b"},
 	{"agd", 1, 2, 0, OP_NOP, {.null = NULL}, {.valfunc_cnt = f_agd},
 	 "inverse Gudermannian function"},
 	{"ahacovercos", 1, 2, 0, OP_NOP, {.null = NULL}, {.valfunc_cnt = f_ahacovercos},
-	 "inverse half coversed cosine of a within accuracy b"},
+	 "inverse half coversed cosine of a, within accuracy b"},
 	{"ahacoversin", 1, 2, 0, OP_NOP, {.null = NULL}, {.valfunc_cnt = f_ahacoversin},
-	 "inverse half coversed sine of a within accuracy b"},
+	 "inverse half coversed sine of a, within accuracy b"},
 	{"ahavercos", 1, 2, 0, OP_NOP, {.null = NULL}, {.valfunc_cnt = f_ahavercos},
-	 "inverse half versed cosine of a within accuracy b"},
+	 "inverse half versed cosine of a, within accuracy b"},
 	{"ahaversin", 1, 2, 0, OP_NOP, {.null = NULL}, {.valfunc_cnt = f_ahaversin},
-	 "inverse half versed sine of a within accuracy b"},
+	 "inverse half versed sine of a, within accuracy b"},
 	{"append", 1, IN, FA, OP_NOP, {.null = NULL}, {.valfunc_cnt = f_listappend},
 	 "append values to end of list"},
 	{"appr", 1, 3, 0, OP_NOP, {.null = NULL}, {.valfunc_cnt = f_appr},
@@ -12302,25 +12440,25 @@ STATIC CONST struct builtin builtins[] = {
 	{"argv", 0, 1, 0, OP_NOP, {.null = NULL}, {.valfunc_cnt = f_argv},
 	 "calc argc or argv string"},
 	{"asec", 1, 2, 0, OP_NOP, {.null = NULL}, {.valfunc_cnt = f_asec},
-	 "inverse secant of a within accuracy b"},
+	 "inverse secant of a, within accuracy b"},
 	{"asech", 1, 2, 0, OP_NOP, {.null = NULL}, {.valfunc_cnt = f_asech},
-	 "inverse hyperbolic secant of a within accuracy b"},
+	 "inverse hyperbolic secant of a, within accuracy b"},
 	{"asin", 1, 2, 0, OP_NOP, {.null = NULL}, {.valfunc_cnt = f_asin},
-	 "inverse sine of a within accuracy b"},
+	 "inverse sine of a, within accuracy b"},
 	{"asinh", 1, 2, 0, OP_NOP, {.null = NULL}, {.valfunc_cnt = f_asinh},
-	 "inverse hyperbolic sine of a within accuracy b"},
+	 "inverse hyperbolic sine of a, within accuracy b"},
 	{"assoc", 0, 0, 0, OP_NOP, {.null = NULL}, {.valfunc_cnt = f_assoc},
 	 "create new association array"},
 	{"atan", 1, 2, 0, OP_NOP, {.null = NULL}, {.valfunc_cnt = f_atan},
-	 "inverse tangent of a within accuracy b"},
+	 "inverse tangent of a, within accuracy b"},
 	{"atan2", 2, 3, FE, OP_NOP, {.numfunc_3 = qatan2}, {.null = NULL},
 	 "angle to point (b,a) within accuracy c"},
 	{"atanh", 1, 2, 0, OP_NOP, {.null = NULL}, {.valfunc_cnt = f_atanh},
-	 "inverse hyperbolic tangent of a within accuracy b"},
+	 "inverse hyperbolic tangent of a, within accuracy b"},
 	{"avercos", 1, 2, 0, OP_NOP, {.null = NULL}, {.valfunc_cnt = f_avercos},
-	 "inverse versed cosine of a within accuracy b"},
+	 "inverse versed cosine of a, within accuracy b"},
 	{"aversin", 1, 2, 0, OP_NOP, {.null = NULL}, {.valfunc_cnt = f_aversin},
-	 "inverse versed sine of a within accuracy b"},
+	 "inverse versed sine of a, within accuracy b"},
 	{"avg", 0, IN, 0, OP_NOP, {.null = NULL}, {.valfunc_cnt = f_avg},
 	 "arithmetic mean of values"},
 	{"base", 0, 1, 0, OP_NOP, {.numfunc_cnt = f_base}, {.null = NULL},
@@ -12354,7 +12492,7 @@ STATIC CONST struct builtin builtins[] = {
 	{"ceil", 1, 1, 0, OP_NOP, {.null = NULL}, {.valfunc_1 = f_ceil},
 	 "smallest integer greater than or equal to number"},
 	{"cfappr", 1, 3, 0, OP_NOP, {.numfunc_cnt = f_cfappr}, {.null = NULL},
-	 "approximate a within accuracy b using\n"
+	 "approximate a, within accuracy b using\n"
 	 "\t\t\tcontinued fractions"},
 	{"cfsim", 1, 2, 0, OP_NOP, {.numfunc_cnt = f_cfsim}, {.null = NULL},
 	 "simplify number using continued fractions"},
@@ -12373,25 +12511,27 @@ STATIC CONST struct builtin builtins[] = {
 	{"copy", 2, 5, 0, OP_NOP, {.null = NULL}, {.valfunc_cnt = f_copy},
 	 "copy value to/from a block: copy(s,d,len,si,di)"},
 	{"cos", 1, 2, 0, OP_NOP, {.null = NULL}, {.valfunc_cnt = f_cos},
-	 "cosine of value a within accuracy b"},
+	 "cosine of value a, within accuracy b"},
 	{"cosh", 1, 2, 0, OP_NOP, {.null = NULL}, {.valfunc_cnt = f_cosh},
-	 "hyperbolic cosine of a within accuracy b"},
+	 "hyperbolic cosine of a, within accuracy b"},
 	{"cot", 1, 2, 0, OP_NOP, {.null = NULL}, {.valfunc_cnt = f_cot},
-	 "cotangent of a within accuracy b"},
+	 "cotangent of a, within accuracy b"},
 	{"coth", 1, 2, 0, OP_NOP, {.null = NULL}, {.valfunc_cnt = f_coth},
-	 "hyperbolic cotangent of a within accuracy b"},
+	 "hyperbolic cotangent of a, within accuracy b"},
 	{"count", 2, 2, 0, OP_NOP, {.null = NULL}, {.valfunc_2 = f_count},
 	 "count listr/matrix elements satisfying some condition"},
 	{"covercos", 1, 2, 0, OP_NOP, {.null = NULL}, {.valfunc_cnt = f_covercos},
-	 "coversed cosine of value a within accuracy b"},
+	 "coversed cosine of value a, within accuracy b"},
 	{"coversin", 1, 2, 0, OP_NOP, {.null = NULL}, {.valfunc_cnt = f_coversin},
-	 "coversed sine of value a within accuracy b"},
+	 "coversed sine of value a, within accuracy b"},
 	{"cp", 2, 2, 0, OP_NOP, {.null = NULL}, {.valfunc_2 = f_cp},
 	 "cross product of two vectors"},
+	{"crd", 1, 2, 0, OP_NOP, {.null = NULL}, {.valfunc_cnt = f_crd},
+	 "length of unit circle chord with angle a, within accuracy b"},
 	{"csc", 1, 2, 0, OP_NOP, {.null = NULL}, {.valfunc_cnt = f_csc},
-	 "cosecant of a within accuracy b"},
+	 "cosecant of a, within accuracy b"},
 	{"csch", 1, 2, 0, OP_NOP, {.null = NULL}, {.valfunc_cnt = f_csch},
-	 "hyperbolic cosecant of a within accuracy b"},
+	 "hyperbolic cosecant of a, within accuracy b"},
 	{"ctime", 0, 0, 0, OP_NOP, {.null = NULL}, {.valfunc_0 = f_ctime},
 	 "date and time as string"},
 	{"custom", 0, IN, 0, OP_NOP, {.null = NULL}, {.valfunc_cnt = f_custom},
@@ -12441,11 +12581,11 @@ STATIC CONST struct builtin builtins[] = {
 	{"eval", 1, 1, 0, OP_NOP, {.null = NULL}, {.valfunc_1 = f_eval},
 	 "evaluate expression from string to value"},
 	{"excsc", 1, 2, 0, OP_NOP, {.null = NULL}, {.valfunc_cnt = f_excsc},
-	 "exterior cosecant of a within accuracy b"},
+	 "exterior cosecant of a, within accuracy b"},
 	{"exp", 1, 2, 0, OP_NOP, {.null = NULL}, {.valfunc_cnt = f_exp},
-	 "exponential of value a within accuracy b"},
+	 "exponential of value a, within accuracy b"},
 	{"exsec", 1, 2, 0, OP_NOP, {.null = NULL}, {.valfunc_cnt = f_exsec},
-	 "exterior secant of a within accuracy b"},
+	 "exterior secant of a, within accuracy b"},
 	{"fact", 1, 1, 0, OP_NOP, {.null = NULL}, {.valfunc_1 = f_fact},
 	 "factorial"},
 	{"factor", 1, 3, 0, OP_NOP, {.numfunc_cnt = f_factor}, {.null = NULL},
@@ -12549,16 +12689,16 @@ STATIC CONST struct builtin builtins[] = {
 	{"h2hms", 4, 5, FA, OP_NOP, {.null = NULL}, {.valfunc_cnt = f_h2hms},
 	 "convert a to b hours, c min, d sec, rounding type e\n"},
 	{"hacovercos", 1, 2, 0, OP_NOP, {.null = NULL}, {.valfunc_cnt = f_hacovercos},
-	 "half coversed cosine of value a within accuracy b"},
+	 "half coversed cosine of value a, within accuracy b"},
 	{"hacoversin", 1, 2, 0, OP_NOP, {.null = NULL}, {.valfunc_cnt = f_hacoversin},
-	 "half coversed sine of value a within accuracy b"},
+	 "half coversed sine of value a, within accuracy b"},
 	{"hash", 1, IN, 0, OP_NOP, {.null = NULL}, {.valfunc_cnt = f_hash},
 	 "return non-negative hash value for one or\n"
 	 "\t\t\tmore values"},
 	{"havercos", 1, 2, 0, OP_NOP, {.null = NULL}, {.valfunc_cnt = f_havercos},
-	 "half versed cosine of value a within accuracy b"},
+	 "half versed cosine of value a, within accuracy b"},
 	{"haversin", 1, 2, 0, OP_NOP, {.null = NULL}, {.valfunc_cnt = f_haversin},
-	 "half versed sine of value a within accuracy b"},
+	 "half versed sine of value a, within accuracy b"},
 	{"head", 2, 2, 0, OP_NOP, {.null = NULL}, {.valfunc_2 = f_head},
 	 "return list of specified number at head of a list"},
 	{"highbit", 1, 1, 0, OP_HIGHBIT, {.null = NULL}, {.null = NULL},
@@ -12572,7 +12712,7 @@ STATIC CONST struct builtin builtins[] = {
 	{"hnrmod", 4, 4, 0, OP_NOP, {.numfunc_4 = f_hnrmod}, {.null = NULL},
 	 "v mod h*2^n+r, h>0, n>0, r  =  -1, 0 or 1"},
 	{"hypot", 2, 3, FE, OP_NOP, {.numfunc_3 = qhypot}, {.null = NULL},
-	 "hypotenuse of right triangle within accuracy c"},
+	 "hypotenuse of right triangle, within accuracy c"},
 	{"ilog", 2, 2, 0, OP_NOP, {.null = NULL}, {.valfunc_2 = f_ilog},
 	 "integral log of a to integral base b"},
 	{"ilog10", 1, 1, 0, OP_NOP, {.null = NULL}, {.valfunc_1 = f_ilog10},
@@ -12695,13 +12835,13 @@ STATIC CONST struct builtin builtins[] = {
 	{"list", 0, IN, 0, OP_NOP, {.null = NULL}, {.valfunc_cnt = f_list},
 	 "create list of specified values"},
 	{"ln", 1, 2, 0, OP_NOP, {.null = NULL}, {.valfunc_cnt = f_ln},
-	 "natural logarithm of value a within accuracy b"},
+	 "natural logarithm of value a, within accuracy b"},
 	{"log", 1, 2, 0, OP_NOP, {.null = NULL}, {.valfunc_cnt = f_log},
-	 "base 10 logarithm of value a within accuracy b"},
+	 "base 10 logarithm of value a, within accuracy b"},
 	{"log2", 1, 2, 0, OP_NOP, {.null = NULL}, {.valfunc_cnt = f_log2},
-	 "base 2 logarithm of value a within accuracy b"},
+	 "base 2 logarithm of value a, within accuracy b"},
 	{"logn", 2, 3, 0, OP_NOP, {.null = NULL}, {.valfunc_cnt = f_logn},
-	 "base b logarithm of value a within accuracy c"},
+	 "base b logarithm of value a, within accuracy c"},
 	{"lowbit", 1, 1, 0, OP_LOWBIT, {.null = NULL}, {.null = NULL},
 	 "low bit number in base 2 representation"},
 	{"ltol", 1, 2, FE, OP_NOP, {.numfunc_2 = f_legtoleg}, {.null = NULL},
@@ -12766,7 +12906,7 @@ STATIC CONST struct builtin builtins[] = {
 	{"pfact", 1, 1, 0, OP_NOP, {.numfunc_1 = qpfact}, {.null = NULL},
 	 "product of primes up till number"},
 	{"pi", 0, 1, FE, OP_NOP, {.numfunc_1 = qpi}, {.null = NULL},
-	 "value of pi accurate to within epsilon"},
+	 "value of pi, within accuracy a"},
 	{"pix", 1, 2, 0, OP_NOP, {.numfunc_cnt = f_pix}, {.null = NULL},
 	 "number of primes < =  a < 2^32, return b if error"},
 	{"places", 1, 2, 0, OP_NOP, {.null = NULL}, {.valfunc_cnt = f_places},
@@ -12783,7 +12923,7 @@ STATIC CONST struct builtin builtins[] = {
 	{"popcnt", 1, 2, 0, OP_NOP, {.numfunc_cnt = f_popcnt}, {.null = NULL},
 	 "number of bits in a that match b (or 1)"},
 	{"power", 2, 3, 0, OP_NOP, {.null = NULL}, {.valfunc_cnt = f_power},
-	 "value a raised to the power b within accuracy c"},
+	 "value a raised to the power b, within accuracy c"},
 	{"prevcand", 1, 5, 0, OP_NOP, {.numfunc_cnt = f_prevcand}, {.null = NULL},
 	 "largest value  =  =  d mod e < a, ptest(a,b,c) true"},
 	{"prevprime", 1, 2, 0, OP_NOP, {.numfunc_cnt = f_pprime}, {.null = NULL},
@@ -12840,7 +12980,7 @@ STATIC CONST struct builtin builtins[] = {
 	{"rm", 1, IN, 0, OP_NOP, {.null = NULL}, {.valfunc_cnt = f_rm},
 	 "remove file(s), -f turns off no-such-file errors"},
 	{"root", 2, 3, 0, OP_NOP, {.null = NULL}, {.valfunc_cnt = f_root},
-	 "value a taken to the b'th root within accuracy c"},
+	 "value a taken to the b'th root, within accuracy c"},
 	{"round", 1, 3, 0, OP_NOP, {.null = NULL}, {.valfunc_cnt = f_round},
 	 "round value a to b number of decimal places"},
 	{"rsearch", 2, 4, 0, OP_NOP, {.null = NULL}, {.valfunc_cnt = f_rsearch},
@@ -12862,9 +13002,9 @@ STATIC CONST struct builtin builtins[] = {
 	 "search matrix or list for value b starting\n"
 	 "\t\t\tat index c"},
 	{"sec", 1, 2, 0, OP_NOP, {.null = NULL}, {.valfunc_cnt = f_sec},
-	 "secant of a within accuracy b"},
+	 "secant of a, within accuracy b"},
 	{"sech", 1, 2, 0, OP_NOP, {.null = NULL}, {.valfunc_cnt = f_sech},
-	 "hyperbolic secant of a within accuracy b"},
+	 "hyperbolic secant of a, within accuracy b"},
 	{"seed", 0, 0, 0, OP_NOP, {.numfunc_0 = f_seed}, {.null = NULL},
 	 "return a 64 bit seed for a pseudo-random generator"},
 	{"segment", 2, 3, 0, OP_NOP, {.null = NULL}, {.valfunc_cnt = f_segment},
@@ -12878,9 +13018,9 @@ STATIC CONST struct builtin builtins[] = {
 	{"sha1", 0, IN, 0, OP_NOP, {.null = NULL}, {.valfunc_cnt = f_sha1},
 	 "Secure Hash Algorithm (SHS-1 FIPS Pub 180-1)"},
 	{"sin", 1, 2, 0, OP_NOP, {.null = NULL}, {.valfunc_cnt = f_sin},
-	 "sine of value a within accuracy b"},
+	 "sine of value a, within accuracy b"},
 	{"sinh", 1, 2, 0, OP_NOP, {.null = NULL}, {.valfunc_cnt = f_sinh},
-	 "hyperbolic sine of a within accuracy b"},
+	 "hyperbolic sine of a, within accuracy b"},
 	{"size", 1, 1, 0, OP_NOP, {.null = NULL}, {.valfunc_1 = f_size},
 	 "total number of elements in value"},
 	{"sizeof", 1, 1, 0, OP_NOP, {.null = NULL}, {.valfunc_1 = f_sizeof},
@@ -12890,7 +13030,7 @@ STATIC CONST struct builtin builtins[] = {
 	{"sort", 1, 1, 0, OP_NOP, {.null = NULL}, {.valfunc_1 = f_sort},
 	 "sort a copy of a matrix or list"},
 	{"sqrt", 1, 3, 0, OP_NOP, {.null = NULL}, {.valfunc_cnt = f_sqrt},
-	 "square root of value a within accuracy b"},
+	 "square root of value a, within accuracy b"},
 	{"srand", 0, 1, 0, OP_NOP, {.null = NULL}, {.valfunc_cnt = f_srand},
 	 "seed the rand() function"},
 	{"srandom", 0, 4, 0, OP_NOP, {.null = NULL}, {.valfunc_cnt = f_srandom},
@@ -12944,9 +13084,9 @@ STATIC CONST struct builtin builtins[] = {
 	{"tail", 2, 2, 0, OP_NOP, {.null = NULL}, {.valfunc_2 = f_tail},
 	 "retain list of specified number at tail of list"},
 	{"tan", 1, 2, 0, OP_NOP, {.null = NULL}, {.valfunc_cnt = f_tan},
-	 "tangent of a within accuracy b"},
+	 "tangent of a, within accuracy b"},
 	{"tanh", 1, 2, 0, OP_NOP, {.null = NULL}, {.valfunc_cnt = f_tanh},
-	 "hyperbolic tangent of a within accuracy b"},
+	 "hyperbolic tangent of a, within accuracy b"},
 	{"test", 1, 1, 0, OP_TEST, {.null = NULL}, {.null = NULL},
 	 "test that value is nonzero"},
 	{"time", 0, 0, 0, OP_NOP, {.numfunc_0 = f_time}, {.null = NULL},
@@ -12958,9 +13098,9 @@ STATIC CONST struct builtin builtins[] = {
 	{"usertime", 0, 0, 0, OP_NOP, {.numfunc_0 = f_usertime}, {.null = NULL},
 	 "user mode CPU time in seconds"},
 	{"vercos", 1, 2, 0, OP_NOP, {.null = NULL}, {.valfunc_cnt = f_vercos},
-	 "versed cosine of value a within accuracy b"},
+	 "versed cosine of value a, within accuracy b"},
 	{"versin", 1, 2, 0, OP_NOP, {.null = NULL}, {.valfunc_cnt = f_versin},
-	 "versed sine of value a within accuracy b"},
+	 "versed sine of value a, within accuracy b"},
 	{"version", 0, 0, 0, OP_NOP, {.null = NULL}, {.valfunc_0 = f_version},
 	 "calc version string"},
 	{"xor", 1, IN, 0, OP_NOP, {.null = NULL}, {.valfunc_cnt = f_xor},
--- a/help/Makefile
+++ b/help/Makefile
@@ -200,23 +200,24 @@ BLT_HELP_FILES= ${BLT_HELP_FILES_3} ${BLT_HELP_FILES_5} \
 # to keep this list in nice sorted order.
 #
 DETAIL_HELP= abs access acos acosh acot acoth acovercos acoversin \
-	acsc acsch address agd ahacovercos ahacoversin ahavercos ahaversin \
-	append appr arg argv arrow asec asech asin asinh assign atan atan2 \
-	atanh avercos aversin avg base base2 bernoulli bit blk blkcpy blkfree \
-	blocks bround btrunc calc_tty calclevel calcpath catalan ceil cfappr \
-	cfsim char cmdbuf cmp comb conj cos cosh cot coth count covercos \
-	coversin cp csc csch ctime d2dm d2dms d2g d2r delete den dereference \
-	det digit digits display dms2d dp epsilon errcount errmax errno error \
-	errsym estr euler eval exp fact factor fclose fcnt feof ferror fflush \
-	fgetc fgetfield fgetfile fgetline fgets fgetstr fib files floor fopen \
-	forall fpathopen fprintf fputc fputs fputstr frac free freebernoulli \
-	freeeuler freeglobals freeredc freestatics frem freopen fscan fscanf \
-	fseek fsize ftell g2d g2gm g2gms g2r gcd gcdrem gd getenv gms2g h2hm \
-	h2hms hacovercos hacoversin hash havercos haversin head highbit hmean \
-	hms2h hnrmod hypot ilog ilog10 ilog2 im indices inputlevel insert int \
-	inverse iroot isalnum isalpha isassoc isatty isblk iscntrl isconfig \
-	isdefined isdigit iserror iseven isfile isgraph ishash isident isint \
-	islist islower ismat ismult isnull isnum isobj isobjtype isodd isprime \
+	acrd acsc acsch address aexcsc aexsec agd ahacovercos ahacoversin \
+	ahavercos ahaversin append appr arg argv arrow asec asech asin asinh \
+	assign atan atan2 atanh avercos aversin avg base base2 bernoulli bit \
+	blk blkcpy blkfree blocks bround btrunc calc_tty calclevel calcpath \
+	catalan ceil cfappr cfsim char cmdbuf cmp comb conj cos cosh cot coth \
+	count covercos coversin cp crd csc csch ctime d2dm d2dms d2g d2r \
+	delete den dereference det digit digits display dms2d dp epsilon \
+	errcount errmax errno error errsym estr euler eval excsc exp exsec \
+	fact factor fclose fcnt feof ferror fflush fgetc fgetfield fgetfile \
+	fgetline fgets fgetstr fib files floor fopen forall fpathopen fprintf \
+	fputc fputs fputstr frac free freebernoulli freeeuler freeglobals \
+	freeredc freestatics frem freopen fscan fscanf fseek fsize ftell g2d \
+	g2gm g2gms g2r gcd gcdrem gd getenv gms2g h2hm h2hms hacovercos \
+	hacoversin hash havercos haversin head highbit hmean hms2h hnrmod \
+	hypot ilog ilog10 ilog2 im indices inputlevel insert int inverse iroot \
+	isalnum isalpha isassoc isatty isblk iscntrl isconfig isdefined \
+	isdigit iserror iseven isfile isgraph ishash isident isint islist \
+	islower ismat ismult isnull isnum isobj isobjtype isodd isprime \
 	isprint isptr ispunct isqrt isrand israndom isreal isrel issimple \
 	isspace issq isstr istype isupper isxdigit jacobi join lcm lcmfact \
 	lfactor ln log log2 logn lowbit ltol makelist matdim matfill matmax \
--- a/help/acos
+++ b/help/acos
@@ -54,8 +54,21 @@ SEE ALSO
    haversin, hacoversin, havercos, hacovercos
    ahaversin, hacoversin, havercos, ahacovercos
    exsec, aexsec, excsc, aexcsc
+    crd, acrd
    epsilon

+EXTERNAL RESOURCES
+    For general information on trigonometric functions, see:
+
+	https://en.wikipedia.org/wiki/Trigonometric_functions#Unit-circle_definitions
+	https://en.wikipedia.org/wiki/Versine
+	https://en.wikipedia.org/wiki/Exsecant
+	https://en.wikipedia.org/wiki/Inverse_trigonometric_functions
+	https://en.wikipedia.org/wiki/Chord_(geometry)
+	https://en.wikipedia.org/wiki/Secant_line
+	https://en.wikipedia.org/wiki/Hartley_transform#cas
+	https://en.wikipedia.org/wiki/Cis_(mathematics)
+
 ## Copyright (C) 1999,2023  Landon Curt Noll
 ##
 ## Calc is open software; you can redistribute it and/or modify it under
--- a/help/acot
+++ b/help/acot
@@ -54,8 +54,21 @@ SEE ALSO
    haversin, hacoversin, havercos, hacovercos
    ahaversin, hacoversin, havercos, ahacovercos
    exsec, aexsec, excsc, aexcsc
+    crd, acrd
    epsilon

+EXTERNAL RESOURCES
+    For general information on trigonometric functions, see:
+
+	https://en.wikipedia.org/wiki/Trigonometric_functions#Unit-circle_definitions
+	https://en.wikipedia.org/wiki/Versine
+	https://en.wikipedia.org/wiki/Exsecant
+	https://en.wikipedia.org/wiki/Inverse_trigonometric_functions
+	https://en.wikipedia.org/wiki/Chord_(geometry)
+	https://en.wikipedia.org/wiki/Secant_line
+	https://en.wikipedia.org/wiki/Hartley_transform#cas
+	https://en.wikipedia.org/wiki/Cis_(mathematics)
+
 ## Copyright (C) 1999,2021,2023  Landon Curt Noll
 ##
 ## Calc is open software; you can redistribute it and/or modify it under
--- a/help/acovercos
+++ b/help/acovercos
@@ -53,8 +53,21 @@ SEE ALSO
    haversin, hacoversin, havercos, hacovercos
    ahaversin, hacoversin, havercos, ahacovercos
    exsec, aexsec, excsc, aexcsc
+    crd, acrd
    epsilon

+EXTERNAL RESOURCES
+    For general information on trigonometric functions, see:
+
+	https://en.wikipedia.org/wiki/Trigonometric_functions#Unit-circle_definitions
+	https://en.wikipedia.org/wiki/Versine
+	https://en.wikipedia.org/wiki/Exsecant
+	https://en.wikipedia.org/wiki/Inverse_trigonometric_functions
+	https://en.wikipedia.org/wiki/Chord_(geometry)
+	https://en.wikipedia.org/wiki/Secant_line
+	https://en.wikipedia.org/wiki/Hartley_transform#cas
+	https://en.wikipedia.org/wiki/Cis_(mathematics)
+
 ## Copyright (C) 2023  Landon Curt Noll
 ##
 ## Calc is open software; you can redistribute it and/or modify it under
--- a/help/acoversin
+++ b/help/acoversin
@@ -53,8 +53,21 @@ SEE ALSO
    haversin, hacoversin, havercos, hacovercos
    ahaversin, hacoversin, havercos, ahacovercos
    exsec, aexsec, excsc, aexcsc
+    crd, acrd
    epsilon

+EXTERNAL RESOURCES
+    For general information on trigonometric functions, see:
+
+	https://en.wikipedia.org/wiki/Trigonometric_functions#Unit-circle_definitions
+	https://en.wikipedia.org/wiki/Versine
+	https://en.wikipedia.org/wiki/Exsecant
+	https://en.wikipedia.org/wiki/Inverse_trigonometric_functions
+	https://en.wikipedia.org/wiki/Chord_(geometry)
+	https://en.wikipedia.org/wiki/Secant_line
+	https://en.wikipedia.org/wiki/Hartley_transform#cas
+	https://en.wikipedia.org/wiki/Cis_(mathematics)
+
 ## Copyright (C) 2023  Landon Curt Noll
 ##
 ## Calc is open software; you can redistribute it and/or modify it under
--- a/help/acrd
+++ b/help/acrd
@@ -0,0 +1,88 @@
+NAME
+    acrd - inverse trigonometric chord of a unit circle
+
+SYNOPSIS
+    acrd(x [,eps])
+
+TYPES
+    x		number (real or complex)
+    eps		0 < real < 1, defaults to epsilon()
+
+    return	number
+
+DESCRIPTION
+    Calculate angle of unit circle chord of length x to a multiple of eps with error less in
+    absolute value than .75 * eps.
+
+    A chord of a circle is a straight line segment whose endpoints lie on a arc of a circle.
+    A chord extended infinitely on both directions into a line would produce a secant line.
+
+    This function is sometimes called chord, is equivalent to:
+
+	acrd(x) = 2 * asin(x / 2)
+
+EXAMPLE
+    ; print acrd(1/2), acrd(5/7), acrd(42/7)
+    0.50536051028415730696 0.73041444258073348124 3.14159265358979323846-3.52549434807817210092i
+
+    ; print acrd(1, 1e-5), acrd(1, 1e-10), acrd(1, 1e-15), acrd(1, 1e-20)
+    1.0472 1.0471975512 1.047197551196598 1.04719755119659774616
+
+    ; print acrd(2 + 3i, 1e-5), acrd(2 + 3i, 1e-10)
+    1.0472+2.63392i 1.0471975512+2.6339157938i
+
+    ; pi = pi(1e-20)
+    ; print acrd(pi/6), acrd(pi/2), acrd(pi)
+    0.52977229404847186194 1.80667822153302569472 3.14159265358979323846-2.04645495709510115864i
+
+LIMITS
+    0 < eps < 1
+
+LINK LIBRARY
+    NUMBER *qacrd(NUMBER *x, NUMBER *eps)
+    COMPLEX *c_acrd(COMPLEX *x, NUMBER *eps)
+
+SEE ALSO
+    sin, cos, tan, cot, sec, csc
+    asin, acos, atan, acot, asec, acsc
+    coversin, vercos, covercos
+    aversin, acoversin, avercos, acovercos
+    haversin, hacoversin, havercos, hacovercos
+    ahaversin, hacoversin, havercos, ahacovercos
+    exsec, aexsec, excsc, aexcsc
+    crd
+    epsilon
+
+EXTERNAL RESOURCES
+    For general information on trigonometric functions, see:
+
+	https://en.wikipedia.org/wiki/Trigonometric_functions#Unit-circle_definitions
+	https://en.wikipedia.org/wiki/Versine
+	https://en.wikipedia.org/wiki/Exsecant
+	https://en.wikipedia.org/wiki/Inverse_trigonometric_functions
+	https://en.wikipedia.org/wiki/Chord_(geometry)
+	https://en.wikipedia.org/wiki/Secant_line
+	https://en.wikipedia.org/wiki/Hartley_transform#cas
+	https://en.wikipedia.org/wiki/Cis_(mathematics)
+
+## Copyright (C) 2023  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:	2023/10/02 16:49:42
+## File existed as early as:	2023
+##
+## chongo <was here> /\oo/\	http://www.isthe.com/chongo/
+## Share and enjoy!  :-)	http://www.isthe.com/chongo/tech/comp/calc/
--- a/help/acsc
+++ b/help/acsc
@@ -57,8 +57,21 @@ SEE ALSO
    haversin, hacoversin, havercos, hacovercos
    ahaversin, hacoversin, havercos, ahacovercos
    exsec, aexsec, excsc, aexcsc
+    crd, acrd
    epsilon

+EXTERNAL RESOURCES
+    For general information on trigonometric functions, see:
+
+	https://en.wikipedia.org/wiki/Trigonometric_functions#Unit-circle_definitions
+	https://en.wikipedia.org/wiki/Versine
+	https://en.wikipedia.org/wiki/Exsecant
+	https://en.wikipedia.org/wiki/Inverse_trigonometric_functions
+	https://en.wikipedia.org/wiki/Chord_(geometry)
+	https://en.wikipedia.org/wiki/Secant_line
+	https://en.wikipedia.org/wiki/Hartley_transform#cas
+	https://en.wikipedia.org/wiki/Cis_(mathematics)
+
 ## Copyright (C) 1999,2021,2023  Landon Curt Noll
 ##
 ## Calc is open software; you can redistribute it and/or modify it under
--- a/help/aexcsc
+++ b/help/aexcsc
@@ -59,8 +59,21 @@ SEE ALSO
    haversin, hacoversin, havercos, hacovercos
    ahaversin, hacoversin, havercos, ahacovercos
    exsec, aexsec, excsc
+    crd, acrd
    epsilon

+EXTERNAL RESOURCES
+    For general information on trigonometric functions, see:
+
+	https://en.wikipedia.org/wiki/Trigonometric_functions#Unit-circle_definitions
+	https://en.wikipedia.org/wiki/Versine
+	https://en.wikipedia.org/wiki/Exsecant
+	https://en.wikipedia.org/wiki/Inverse_trigonometric_functions
+	https://en.wikipedia.org/wiki/Chord_(geometry)
+	https://en.wikipedia.org/wiki/Secant_line
+	https://en.wikipedia.org/wiki/Hartley_transform#cas
+	https://en.wikipedia.org/wiki/Cis_(mathematics)
+
 ## Copyright (C) 2023  Landon Curt Noll
 ##
 ## Calc is open software; you can redistribute it and/or modify it under
--- a/help/aexsec
+++ b/help/aexsec
@@ -56,8 +56,21 @@ SEE ALSO
    haversin, hacoversin, havercos, hacovercos
    ahaversin, hacoversin, havercos, ahacovercos
    exsec, excsc, aexcsc
+    crd, acrd
    epsilon

+EXTERNAL RESOURCES
+    For general information on trigonometric functions, see:
+
+	https://en.wikipedia.org/wiki/Trigonometric_functions#Unit-circle_definitions
+	https://en.wikipedia.org/wiki/Versine
+	https://en.wikipedia.org/wiki/Exsecant
+	https://en.wikipedia.org/wiki/Inverse_trigonometric_functions
+	https://en.wikipedia.org/wiki/Chord_(geometry)
+	https://en.wikipedia.org/wiki/Secant_line
+	https://en.wikipedia.org/wiki/Hartley_transform#cas
+	https://en.wikipedia.org/wiki/Cis_(mathematics)
+
 ## Copyright (C) 2023  Landon Curt Noll
 ##
 ## Calc is open software; you can redistribute it and/or modify it under
--- a/help/ahacovercos
+++ b/help/ahacovercos
@@ -56,8 +56,21 @@ SEE ALSO
    haversin, hacoversin, havercos, hacovercos
    ahaversin, hacoversin, havercos
    exsec, aexsec, excsc, aexcsc
+    crd, acrd
    epsilon

+EXTERNAL RESOURCES
+    For general information on trigonometric functions, see:
+
+	https://en.wikipedia.org/wiki/Trigonometric_functions#Unit-circle_definitions
+	https://en.wikipedia.org/wiki/Versine
+	https://en.wikipedia.org/wiki/Exsecant
+	https://en.wikipedia.org/wiki/Inverse_trigonometric_functions
+	https://en.wikipedia.org/wiki/Chord_(geometry)
+	https://en.wikipedia.org/wiki/Secant_line
+	https://en.wikipedia.org/wiki/Hartley_transform#cas
+	https://en.wikipedia.org/wiki/Cis_(mathematics)
+
 ## Copyright (C) 2023  Landon Curt Noll
 ##
 ## Calc is open software; you can redistribute it and/or modify it under
--- a/help/ahacoversin
+++ b/help/ahacoversin
@@ -56,8 +56,21 @@ SEE ALSO
    haversin, hacoversin, havercos, hacovercos
    ahaversin, havercos, ahacovercos
    exsec, aexsec, excsc, aexcsc
+    crd, acrd
    epsilon

+EXTERNAL RESOURCES
+    For general information on trigonometric functions, see:
+
+	https://en.wikipedia.org/wiki/Trigonometric_functions#Unit-circle_definitions
+	https://en.wikipedia.org/wiki/Versine
+	https://en.wikipedia.org/wiki/Exsecant
+	https://en.wikipedia.org/wiki/Inverse_trigonometric_functions
+	https://en.wikipedia.org/wiki/Chord_(geometry)
+	https://en.wikipedia.org/wiki/Secant_line
+	https://en.wikipedia.org/wiki/Hartley_transform#cas
+	https://en.wikipedia.org/wiki/Cis_(mathematics)
+
 ## Copyright (C) 2023  Landon Curt Noll
 ##
 ## Calc is open software; you can redistribute it and/or modify it under
--- a/help/ahavercos
+++ b/help/ahavercos
@@ -57,8 +57,21 @@ SEE ALSO
    haversin, hacoversin, havercos, hacovercos
    ahaversin, hacoversin, ahacovercos
    exsec, aexsec, excsc, aexcsc
+    crd, acrd
    epsilon

+EXTERNAL RESOURCES
+    For general information on trigonometric functions, see:
+
+	https://en.wikipedia.org/wiki/Trigonometric_functions#Unit-circle_definitions
+	https://en.wikipedia.org/wiki/Versine
+	https://en.wikipedia.org/wiki/Exsecant
+	https://en.wikipedia.org/wiki/Inverse_trigonometric_functions
+	https://en.wikipedia.org/wiki/Chord_(geometry)
+	https://en.wikipedia.org/wiki/Secant_line
+	https://en.wikipedia.org/wiki/Hartley_transform#cas
+	https://en.wikipedia.org/wiki/Cis_(mathematics)
+
 ## Copyright (C) 2023  Landon Curt Noll
 ##
 ## Calc is open software; you can redistribute it and/or modify it under
--- a/help/ahaversin
+++ b/help/ahaversin
@@ -57,8 +57,21 @@ SEE ALSO
    haversin, hacoversin, havercos, hacovercos
    hacoversin, havercos, ahacovercos
    exsec, aexsec, excsc, aexcsc
+    crd, acrd
    epsilon

+EXTERNAL RESOURCES
+    For general information on trigonometric functions, see:
+
+	https://en.wikipedia.org/wiki/Trigonometric_functions#Unit-circle_definitions
+	https://en.wikipedia.org/wiki/Versine
+	https://en.wikipedia.org/wiki/Exsecant
+	https://en.wikipedia.org/wiki/Inverse_trigonometric_functions
+	https://en.wikipedia.org/wiki/Chord_(geometry)
+	https://en.wikipedia.org/wiki/Secant_line
+	https://en.wikipedia.org/wiki/Hartley_transform#cas
+	https://en.wikipedia.org/wiki/Cis_(mathematics)
+
 ## Copyright (C) 2023  Landon Curt Noll
 ##
 ## Calc is open software; you can redistribute it and/or modify it under
--- a/help/asec
+++ b/help/asec
@@ -54,8 +54,21 @@ SEE ALSO
    haversin, hacoversin, havercos, hacovercos
    ahaversin, hacoversin, havercos, ahacovercos
    exsec, aexsec, excsc, aexcsc
+    crd, acrd
    epsilon

+EXTERNAL RESOURCES
+    For general information on trigonometric functions, see:
+
+	https://en.wikipedia.org/wiki/Trigonometric_functions#Unit-circle_definitions
+	https://en.wikipedia.org/wiki/Versine
+	https://en.wikipedia.org/wiki/Exsecant
+	https://en.wikipedia.org/wiki/Inverse_trigonometric_functions
+	https://en.wikipedia.org/wiki/Chord_(geometry)
+	https://en.wikipedia.org/wiki/Secant_line
+	https://en.wikipedia.org/wiki/Hartley_transform#cas
+	https://en.wikipedia.org/wiki/Cis_(mathematics)
+
 ## Copyright (C) 1999,2021,2023  Landon Curt Noll
 ##
 ## Calc is open software; you can redistribute it and/or modify it under
--- a/help/asin
+++ b/help/asin
@@ -54,8 +54,21 @@ SEE ALSO
    haversin, hacoversin, havercos, hacovercos
    ahaversin, hacoversin, havercos, ahacovercos
    exsec, aexsec, excsc, aexcsc
+    crd, acrd
    epsilon

+EXTERNAL RESOURCES
+    For general information on trigonometric functions, see:
+
+	https://en.wikipedia.org/wiki/Trigonometric_functions#Unit-circle_definitions
+	https://en.wikipedia.org/wiki/Versine
+	https://en.wikipedia.org/wiki/Exsecant
+	https://en.wikipedia.org/wiki/Inverse_trigonometric_functions
+	https://en.wikipedia.org/wiki/Chord_(geometry)
+	https://en.wikipedia.org/wiki/Secant_line
+	https://en.wikipedia.org/wiki/Hartley_transform#cas
+	https://en.wikipedia.org/wiki/Cis_(mathematics)
+
 ## Copyright (C) 1999,2021,2023  Landon Curt Noll
 ##
 ## Calc is open software; you can redistribute it and/or modify it under
--- a/help/atan
+++ b/help/atan
@@ -54,8 +54,21 @@ SEE ALSO
    haversin, hacoversin, havercos, hacovercos
    ahaversin, hacoversin, havercos, ahacovercos
    exsec, aexsec, excsc, aexcsc
+    crd, acrd
    epsilon

+EXTERNAL RESOURCES
+    For general information on trigonometric functions, see:
+
+	https://en.wikipedia.org/wiki/Trigonometric_functions#Unit-circle_definitions
+	https://en.wikipedia.org/wiki/Versine
+	https://en.wikipedia.org/wiki/Exsecant
+	https://en.wikipedia.org/wiki/Inverse_trigonometric_functions
+	https://en.wikipedia.org/wiki/Chord_(geometry)
+	https://en.wikipedia.org/wiki/Secant_line
+	https://en.wikipedia.org/wiki/Hartley_transform#cas
+	https://en.wikipedia.org/wiki/Cis_(mathematics)
+
 ## Copyright (C) 1999,2023  Landon Curt Noll
 ##
 ## Calc is open software; you can redistribute it and/or modify it under
--- a/help/avercos
+++ b/help/avercos
@@ -54,8 +54,21 @@ SEE ALSO
    haversin, hacoversin, havercos, hacovercos
    ahaversin, hacoversin, havercos, ahacovercos
    exsec, aexsec, excsc, aexcsc
+    crd, acrd
    epsilon

+EXTERNAL RESOURCES
+    For general information on trigonometric functions, see:
+
+	https://en.wikipedia.org/wiki/Trigonometric_functions#Unit-circle_definitions
+	https://en.wikipedia.org/wiki/Versine
+	https://en.wikipedia.org/wiki/Exsecant
+	https://en.wikipedia.org/wiki/Inverse_trigonometric_functions
+	https://en.wikipedia.org/wiki/Chord_(geometry)
+	https://en.wikipedia.org/wiki/Secant_line
+	https://en.wikipedia.org/wiki/Hartley_transform#cas
+	https://en.wikipedia.org/wiki/Cis_(mathematics)
+
 ## Copyright (C) 2023  Landon Curt Noll
 ##
 ## Calc is open software; you can redistribute it and/or modify it under
--- a/help/aversin
+++ b/help/aversin
@@ -54,8 +54,21 @@ SEE ALSO
    haversin, hacoversin, havercos, hacovercos
    ahaversin, hacoversin, havercos, ahacovercos
    exsec, aexsec, excsc, aexcsc
+    crd, acrd
    epsilon

+EXTERNAL RESOURCES
+    For general information on trigonometric functions, see:
+
+	https://en.wikipedia.org/wiki/Trigonometric_functions#Unit-circle_definitions
+	https://en.wikipedia.org/wiki/Versine
+	https://en.wikipedia.org/wiki/Exsecant
+	https://en.wikipedia.org/wiki/Inverse_trigonometric_functions
+	https://en.wikipedia.org/wiki/Chord_(geometry)
+	https://en.wikipedia.org/wiki/Secant_line
+	https://en.wikipedia.org/wiki/Hartley_transform#cas
+	https://en.wikipedia.org/wiki/Cis_(mathematics)
+
 ## Copyright (C) 2023  Landon Curt Noll
 ##
 ## Calc is open software; you can redistribute it and/or modify it under
--- a/help/cos
+++ b/help/cos
@@ -43,8 +43,21 @@ SEE ALSO
    haversin, hacoversin, havercos, hacovercos
    ahaversin, hacoversin, havercos, ahacovercos
    exsec, aexsec, excsc, aexcsc
+    crd, acrd
    epsilon

+EXTERNAL RESOURCES
+    For general information on trigonometric functions, see:
+
+	https://en.wikipedia.org/wiki/Trigonometric_functions#Unit-circle_definitions
+	https://en.wikipedia.org/wiki/Versine
+	https://en.wikipedia.org/wiki/Exsecant
+	https://en.wikipedia.org/wiki/Inverse_trigonometric_functions
+	https://en.wikipedia.org/wiki/Chord_(geometry)
+	https://en.wikipedia.org/wiki/Secant_line
+	https://en.wikipedia.org/wiki/Hartley_transform#cas
+	https://en.wikipedia.org/wiki/Cis_(mathematics)
+
 ## Copyright (C) 1999,2021,2023  Landon Curt Noll
 ##
 ## Calc is open software; you can redistribute it and/or modify it under
--- a/help/cot
+++ b/help/cot
@@ -49,8 +49,21 @@ SEE ALSO
    haversin, hacoversin, havercos, hacovercos
    ahaversin, hacoversin, havercos, ahacovercos
    exsec, aexsec, excsc, aexcsc
+    crd, acrd
    epsilon

+EXTERNAL RESOURCES
+    For general information on trigonometric functions, see:
+
+	https://en.wikipedia.org/wiki/Trigonometric_functions#Unit-circle_definitions
+	https://en.wikipedia.org/wiki/Versine
+	https://en.wikipedia.org/wiki/Exsecant
+	https://en.wikipedia.org/wiki/Inverse_trigonometric_functions
+	https://en.wikipedia.org/wiki/Chord_(geometry)
+	https://en.wikipedia.org/wiki/Secant_line
+	https://en.wikipedia.org/wiki/Hartley_transform#cas
+	https://en.wikipedia.org/wiki/Cis_(mathematics)
+
 ## Copyright (C) 1999,2021,2023  Landon Curt Noll
 ##
 ## Calc is open software; you can redistribute it and/or modify it under
--- a/help/covercos
+++ b/help/covercos
@@ -47,8 +47,21 @@ SEE ALSO
    haversin, hacoversin, havercos, hacovercos
    ahaversin, hacoversin, havercos, ahacovercos
    exsec, aexsec, excsc, aexcsc
+    crd, acrd
    epsilon

+EXTERNAL RESOURCES
+    For general information on trigonometric functions, see:
+
+	https://en.wikipedia.org/wiki/Trigonometric_functions#Unit-circle_definitions
+	https://en.wikipedia.org/wiki/Versine
+	https://en.wikipedia.org/wiki/Exsecant
+	https://en.wikipedia.org/wiki/Inverse_trigonometric_functions
+	https://en.wikipedia.org/wiki/Chord_(geometry)
+	https://en.wikipedia.org/wiki/Secant_line
+	https://en.wikipedia.org/wiki/Hartley_transform#cas
+	https://en.wikipedia.org/wiki/Cis_(mathematics)
+
 ## Copyright (C) 2023  Landon Curt Noll
 ##
 ## Calc is open software; you can redistribute it and/or modify it under
--- a/help/coversin
+++ b/help/coversin
@@ -47,8 +47,21 @@ SEE ALSO
    haversin, hacoversin, havercos, hacovercos
    ahaversin, hacoversin, havercos, ahacovercos
    exsec, aexsec, excsc, aexcsc
+    crd, acrd
    epsilon

+EXTERNAL RESOURCES
+    For general information on trigonometric functions, see:
+
+	https://en.wikipedia.org/wiki/Trigonometric_functions#Unit-circle_definitions
+	https://en.wikipedia.org/wiki/Versine
+	https://en.wikipedia.org/wiki/Exsecant
+	https://en.wikipedia.org/wiki/Inverse_trigonometric_functions
+	https://en.wikipedia.org/wiki/Chord_(geometry)
+	https://en.wikipedia.org/wiki/Secant_line
+	https://en.wikipedia.org/wiki/Hartley_transform#cas
+	https://en.wikipedia.org/wiki/Cis_(mathematics)
+
 ## Copyright (C) 2023  Landon Curt Noll
 ##
 ## Calc is open software; you can redistribute it and/or modify it under
--- a/help/crd
+++ b/help/crd
@@ -0,0 +1,88 @@
+NAME
+    crd - trigonometric chord of a unit circle
+
+SYNOPSIS
+    crd(x [,eps])
+
+TYPES
+    x		number (real or complex)
+    eps		0 < real < 1, defaults to epsilon()
+
+    return	number
+
+DESCRIPTION
+    Calculate length of unit circle chord with angle x to a multiple of eps with error less in
+    absolute value than .75 * eps.
+
+    A chord of a circle is a straight line segment whose endpoints lie on a arc of a circle.
+    A chord extended infinitely on both directions into a line would produce a secant line.
+
+    This function is sometimes called chord, is equivalent to:
+
+	crd(x) = 2 * sin(x / 2)
+
+EXAMPLE
+    ; print crd(1/2), crd(5/7), crd(42/7)
+    0.4948079185090458592 0.6991976162706840944 0.2822400161197344442
+
+    ; print crd(1, 1e-5), crd(1, 1e-10), crd(1, 1e-15), crd(1, 1e-20)
+    0.95886 0.9588510772 0.958851077208406 0.95885107720840600054
+
+    ; print crd(2 + 3i, 1e-5), crd(2 + 3i, 1e-10)
+    3.95896+2.3009i 3.9589688712+2.3009091988i
+
+    ; pi = pi(1e-20)
+    ; print crd(pi/6), crd(pi/2), crd(pi)
+    0.5176380902050415247 1.4142135623730950488 2
+
+LIMITS
+    0 < eps < 1
+
+LINK LIBRARY
+    NUMBER *qcrd(NUMBER *x, NUMBER *eps)
+    COMPLEX *c_crd(COMPLEX *x, NUMBER *eps)
+
+SEE ALSO
+    sin, cos, tan, cot, sec, csc
+    asin, acos, atan, acot, asec, acsc
+    coversin, vercos, covercos
+    aversin, acoversin, avercos, acovercos
+    haversin, hacoversin, havercos, hacovercos
+    ahaversin, hacoversin, havercos, ahacovercos
+    exsec, aexsec, excsc, aexcsc
+    acrd
+    epsilon
+
+EXTERNAL RESOURCES
+    For general information on trigonometric functions, see:
+
+	https://en.wikipedia.org/wiki/Trigonometric_functions#Unit-circle_definitions
+	https://en.wikipedia.org/wiki/Versine
+	https://en.wikipedia.org/wiki/Exsecant
+	https://en.wikipedia.org/wiki/Inverse_trigonometric_functions
+	https://en.wikipedia.org/wiki/Chord_(geometry)
+	https://en.wikipedia.org/wiki/Secant_line
+	https://en.wikipedia.org/wiki/Hartley_transform#cas
+	https://en.wikipedia.org/wiki/Cis_(mathematics)
+
+## Copyright (C) 2023  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:	2023/10/02 16:49:42
+## File existed as early as:	2023
+##
+## chongo <was here> /\oo/\	http://www.isthe.com/chongo/
+## Share and enjoy!  :-)	http://www.isthe.com/chongo/tech/comp/calc/
--- a/help/csc
+++ b/help/csc
@@ -49,8 +49,21 @@ SEE ALSO
    haversin, hacoversin, havercos, hacovercos
    ahaversin, hacoversin, havercos, ahacovercos
    exsec, aexsec, excsc, aexcsc
+    crd, acrd
    epsilon

+EXTERNAL RESOURCES
+    For general information on trigonometric functions, see:
+
+	https://en.wikipedia.org/wiki/Trigonometric_functions#Unit-circle_definitions
+	https://en.wikipedia.org/wiki/Versine
+	https://en.wikipedia.org/wiki/Exsecant
+	https://en.wikipedia.org/wiki/Inverse_trigonometric_functions
+	https://en.wikipedia.org/wiki/Chord_(geometry)
+	https://en.wikipedia.org/wiki/Secant_line
+	https://en.wikipedia.org/wiki/Hartley_transform#cas
+	https://en.wikipedia.org/wiki/Cis_(mathematics)
+
 ## Copyright (C) 1999,2023  Landon Curt Noll
 ##
 ## Calc is open software; you can redistribute it and/or modify it under
--- a/help/excsc
+++ b/help/excsc
@@ -49,8 +49,21 @@ SEE ALSO
    haversin, hacoversin, havercos, hacovercos
    ahaversin, hacoversin, havercos, ahacovercos
    exsec, aexsec, aexcsc
+    crd, acrd
    epsilon

+EXTERNAL RESOURCES
+    For general information on trigonometric functions, see:
+
+	https://en.wikipedia.org/wiki/Trigonometric_functions#Unit-circle_definitions
+	https://en.wikipedia.org/wiki/Versine
+	https://en.wikipedia.org/wiki/Exsecant
+	https://en.wikipedia.org/wiki/Inverse_trigonometric_functions
+	https://en.wikipedia.org/wiki/Chord_(geometry)
+	https://en.wikipedia.org/wiki/Secant_line
+	https://en.wikipedia.org/wiki/Hartley_transform#cas
+	https://en.wikipedia.org/wiki/Cis_(mathematics)
+
 ## Copyright (C) 2023  Landon Curt Noll
 ##
 ## Calc is open software; you can redistribute it and/or modify it under
--- a/help/exsec
+++ b/help/exsec
@@ -49,8 +49,21 @@ SEE ALSO
    haversin, hacoversin, havercos, hacovercos
    ahaversin, hacoversin, havercos, ahacovercos
    aexsec, excsc, aexcsc
+    crd, acrd
    epsilon

+EXTERNAL RESOURCES
+    For general information on trigonometric functions, see:
+
+	https://en.wikipedia.org/wiki/Trigonometric_functions#Unit-circle_definitions
+	https://en.wikipedia.org/wiki/Versine
+	https://en.wikipedia.org/wiki/Exsecant
+	https://en.wikipedia.org/wiki/Inverse_trigonometric_functions
+	https://en.wikipedia.org/wiki/Chord_(geometry)
+	https://en.wikipedia.org/wiki/Secant_line
+	https://en.wikipedia.org/wiki/Hartley_transform#cas
+	https://en.wikipedia.org/wiki/Cis_(mathematics)
+
 ## Copyright (C) 2023  Landon Curt Noll
 ##
 ## Calc is open software; you can redistribute it and/or modify it under
--- a/help/hacovercos
+++ b/help/hacovercos
@@ -47,8 +47,21 @@ SEE ALSO
    haversin, hacoversin, havercos
    ahaversin, hacoversin, havercos, ahacovercos
    exsec, aexsec, excsc, aexcsc
+    crd, acrd
    epsilon

+EXTERNAL RESOURCES
+    For general information on trigonometric functions, see:
+
+	https://en.wikipedia.org/wiki/Trigonometric_functions#Unit-circle_definitions
+	https://en.wikipedia.org/wiki/Versine
+	https://en.wikipedia.org/wiki/Exsecant
+	https://en.wikipedia.org/wiki/Inverse_trigonometric_functions
+	https://en.wikipedia.org/wiki/Chord_(geometry)
+	https://en.wikipedia.org/wiki/Secant_line
+	https://en.wikipedia.org/wiki/Hartley_transform#cas
+	https://en.wikipedia.org/wiki/Cis_(mathematics)
+
 ## Copyright (C) 2023  Landon Curt Noll
 ##
 ## Calc is open software; you can redistribute it and/or modify it under
--- a/help/hacoversin
+++ b/help/hacoversin
@@ -47,8 +47,21 @@ SEE ALSO
    haversin, havercos, hacovercos
    ahaversin, hacoversin, havercos, ahacovercos
    exsec, aexsec, excsc, aexcsc
+    crd, acrd
    epsilon

+EXTERNAL RESOURCES
+    For general information on trigonometric functions, see:
+
+	https://en.wikipedia.org/wiki/Trigonometric_functions#Unit-circle_definitions
+	https://en.wikipedia.org/wiki/Versine
+	https://en.wikipedia.org/wiki/Exsecant
+	https://en.wikipedia.org/wiki/Inverse_trigonometric_functions
+	https://en.wikipedia.org/wiki/Chord_(geometry)
+	https://en.wikipedia.org/wiki/Secant_line
+	https://en.wikipedia.org/wiki/Hartley_transform#cas
+	https://en.wikipedia.org/wiki/Cis_(mathematics)
+
 ## Copyright (C) 2023  Landon Curt Noll
 ##
 ## Calc is open software; you can redistribute it and/or modify it under
--- a/help/havercos
+++ b/help/havercos
@@ -47,8 +47,21 @@ SEE ALSO
    haversin, hacoversin, hacovercos
    ahaversin, hacoversin, havercos, ahacovercos
    exsec, aexsec, excsc, aexcsc
+    crd, acrd
    epsilon

+EXTERNAL RESOURCES
+    For general information on trigonometric functions, see:
+
+	https://en.wikipedia.org/wiki/Trigonometric_functions#Unit-circle_definitions
+	https://en.wikipedia.org/wiki/Versine
+	https://en.wikipedia.org/wiki/Exsecant
+	https://en.wikipedia.org/wiki/Inverse_trigonometric_functions
+	https://en.wikipedia.org/wiki/Chord_(geometry)
+	https://en.wikipedia.org/wiki/Secant_line
+	https://en.wikipedia.org/wiki/Hartley_transform#cas
+	https://en.wikipedia.org/wiki/Cis_(mathematics)
+
 ## Copyright (C) 2023  Landon Curt Noll
 ##
 ## Calc is open software; you can redistribute it and/or modify it under
--- a/help/haversin
+++ b/help/haversin
@@ -47,8 +47,21 @@ SEE ALSO
    hacoversin, havercos, hacovercos
    ahaversin, hacoversin, havercos, ahacovercos
    exsec, aexsec, excsc, aexcsc
+    crd, acrd
    epsilon

+EXTERNAL RESOURCES
+    For general information on trigonometric functions, see:
+
+	https://en.wikipedia.org/wiki/Trigonometric_functions#Unit-circle_definitions
+	https://en.wikipedia.org/wiki/Versine
+	https://en.wikipedia.org/wiki/Exsecant
+	https://en.wikipedia.org/wiki/Inverse_trigonometric_functions
+	https://en.wikipedia.org/wiki/Chord_(geometry)
+	https://en.wikipedia.org/wiki/Secant_line
+	https://en.wikipedia.org/wiki/Hartley_transform#cas
+	https://en.wikipedia.org/wiki/Cis_(mathematics)
+
 ## Copyright (C) 2023  Landon Curt Noll
 ##
 ## Calc is open software; you can redistribute it and/or modify it under
--- a/help/sec
+++ b/help/sec
@@ -49,8 +49,21 @@ SEE ALSO
    haversin, hacoversin, havercos, hacovercos
    ahaversin, hacoversin, havercos, ahacovercos
    exsec, aexsec, excsc, aexcsc
+    crd, acrd
    epsilon

+EXTERNAL RESOURCES
+    For general information on trigonometric functions, see:
+
+	https://en.wikipedia.org/wiki/Trigonometric_functions#Unit-circle_definitions
+	https://en.wikipedia.org/wiki/Versine
+	https://en.wikipedia.org/wiki/Exsecant
+	https://en.wikipedia.org/wiki/Inverse_trigonometric_functions
+	https://en.wikipedia.org/wiki/Chord_(geometry)
+	https://en.wikipedia.org/wiki/Secant_line
+	https://en.wikipedia.org/wiki/Hartley_transform#cas
+	https://en.wikipedia.org/wiki/Cis_(mathematics)
+
 ## Copyright (C) 1999,2023  Landon Curt Noll
 ##
 ## Calc is open software; you can redistribute it and/or modify it under
--- a/help/sin
+++ b/help/sin
@@ -43,8 +43,21 @@ SEE ALSO
    haversin, hacoversin, havercos, hacovercos
    ahaversin, hacoversin, havercos, ahacovercos
    exsec, aexsec, excsc, aexcsc
+    crd, acrd
    epsilon

+EXTERNAL RESOURCES
+    For general information on trigonometric functions, see:
+
+	https://en.wikipedia.org/wiki/Trigonometric_functions#Unit-circle_definitions
+	https://en.wikipedia.org/wiki/Versine
+	https://en.wikipedia.org/wiki/Exsecant
+	https://en.wikipedia.org/wiki/Inverse_trigonometric_functions
+	https://en.wikipedia.org/wiki/Chord_(geometry)
+	https://en.wikipedia.org/wiki/Secant_line
+	https://en.wikipedia.org/wiki/Hartley_transform#cas
+	https://en.wikipedia.org/wiki/Cis_(mathematics)
+
 ## Copyright (C) 1999,2021,2023  Landon Curt Noll
 ##
 ## Calc is open software; you can redistribute it and/or modify it under
--- a/help/tan
+++ b/help/tan
@@ -49,8 +49,21 @@ SEE ALSO
    haversin, hacoversin, havercos, hacovercos
    ahaversin, hacoversin, havercos, ahacovercos
    exsec, aexsec, excsc, aexcsc
+    crd, acrd
    epsilon

+EXTERNAL RESOURCES
+    For general information on trigonometric functions, see:
+
+	https://en.wikipedia.org/wiki/Trigonometric_functions#Unit-circle_definitions
+	https://en.wikipedia.org/wiki/Versine
+	https://en.wikipedia.org/wiki/Exsecant
+	https://en.wikipedia.org/wiki/Inverse_trigonometric_functions
+	https://en.wikipedia.org/wiki/Chord_(geometry)
+	https://en.wikipedia.org/wiki/Secant_line
+	https://en.wikipedia.org/wiki/Hartley_transform#cas
+	https://en.wikipedia.org/wiki/Cis_(mathematics)
+
 ## Copyright (C) 1999,2023  Landon Curt Noll
 ##
 ## Calc is open software; you can redistribute it and/or modify it under
--- a/help/vercos
+++ b/help/vercos
@@ -47,8 +47,22 @@ SEE ALSO
    haversin, hacoversin, havercos, hacovercos
    ahaversin, hacoversin, havercos, ahacovercos
    exsec, aexsec, excsc, aexcsc
+    crd, acrd
    epsilon

+EXTERNAL RESOURCES
+
+    For general information on trigonometric functions, see:
+
+	https://en.wikipedia.org/wiki/Trigonometric_functions#Unit-circle_definitions
+	https://en.wikipedia.org/wiki/Versine
+	https://en.wikipedia.org/wiki/Exsecant
+	https://en.wikipedia.org/wiki/Inverse_trigonometric_functions
+	https://en.wikipedia.org/wiki/Chord_(geometry)
+	https://en.wikipedia.org/wiki/Secant_line
+	https://en.wikipedia.org/wiki/Hartley_transform#cas
+	https://en.wikipedia.org/wiki/Cis_(mathematics)
+
 ## Copyright (C) 2023  Landon Curt Noll
 ##
 ## Calc is open software; you can redistribute it and/or modify it under
--- a/help/versin
+++ b/help/versin
@@ -47,8 +47,21 @@ SEE ALSO
    haversin, hacoversin, havercos, hacovercos
    ahaversin, hacoversin, havercos, ahacovercos
    exsec, aexsec, excsc, aexcsc
+    crd, acrd
    epsilon

+EXTERNAL RESOURCES
+    For general information on trigonometric functions, see:
+
+	https://en.wikipedia.org/wiki/Trigonometric_functions#Unit-circle_definitions
+	https://en.wikipedia.org/wiki/Versine
+	https://en.wikipedia.org/wiki/Exsecant
+	https://en.wikipedia.org/wiki/Inverse_trigonometric_functions
+	https://en.wikipedia.org/wiki/Chord_(geometry)
+	https://en.wikipedia.org/wiki/Secant_line
+	https://en.wikipedia.org/wiki/Hartley_transform#cas
+	https://en.wikipedia.org/wiki/Cis_(mathematics)
+
 ## Copyright (C) 2023  Landon Curt Noll
 ##
 ## Calc is open software; you can redistribute it and/or modify it under
--- a/qmath.h
+++ b/qmath.h
@@ -256,6 +256,9 @@ E_FUNC NUMBER *qaexsec(NUMBER *q, NUMBER *epsilon);
 E_FUNC NUMBER *qexcsc(NUMBER *q, NUMBER *epsilon);
 E_FUNC NUMBER *qaexcsc_or_NULL(NUMBER *q, NUMBER *epsilon);
 E_FUNC NUMBER *qaexcsc(NUMBER *q, NUMBER *epsilon);
+E_FUNC NUMBER *qcrd(NUMBER *q, NUMBER *epsilon);
+E_FUNC NUMBER *qacrd_or_NULL(NUMBER *q, NUMBER *epsilon);
+E_FUNC NUMBER *qacrd(NUMBER *q, NUMBER *epsilon);

 /*
 * pseudo-seed generator
--- a/qtrans.c
+++ b/qtrans.c
@@ -3467,3 +3467,159 @@ qaexcsc(NUMBER *q, NUMBER *epsilon)
 	 */
 	return res;
 }
+
+
+/*
+ * qcrd - trigonometric chord of a unit circle
+ *
+ * This uses the formula:
+ *
+ *	crd(x) = 2 * sin(x / 2)
+ *
+ * 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 *
+qcrd(NUMBER *q, NUMBER *epsilon)
+{
+	NUMBER *res;	/* inverse trig value result */
+	NUMBER *qdiv2;	/* q/2 */
+	NUMBER *qtmp;	/* 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 trig function value
+	 */
+	qdiv2 = qdivi(q, 2);
+	qtmp = qsin(qdiv2, epsilon);
+	qfree(qdiv2);
+	res = qmuli(qtmp, 2);
+	qfree(qtmp);
+
+	/*
+	 * return trigonometric result
+	 */
+	return res;
+}
+
+
+/*
+ * qacrd_or_NULL - return NULL or non-complex inverse trigonometric chord of a unit circle
+ *
+ * This uses the formula:
+ *
+ *	acrd(x) = 2 * asin(x / 2)
+ *
+ * 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 *
+qacrd_or_NULL(NUMBER *q, NUMBER *epsilon)
+{
+	NUMBER *res;	/* inverse trig value result */
+	NUMBER *qdiv2;	/* q/2 */
+	NUMBER *qtmp;	/* 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
+	 */
+	qdiv2 = qdivi(q, 2);
+	qtmp = qasin(qdiv2, epsilon);
+	qfree(qdiv2);
+	if (qtmp == NULL) {
+		return NULL;
+	}
+	res = qmuli(qtmp, 2);
+	qfree(qtmp);
+
+	/*
+	 * return inverse trigonometric result
+	 */
+	return res;
+}
+
+
+/*
+ * qacrd - non-complex inverse trigonometric chord of a unit circle
+ *
+ * This uses the formula:
+ *
+ *	acrd(x) = 2 * asin(x / 2)
+ *
+ * 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 *
+qacrd(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 = qacrd_or_NULL(q, epsilon);
+	if (res == NULL) {
+		math_error("cannot compute inverse sine for acrd");
+		not_reached();
+	}
+
+	/*
+	 * return inverse trigonometric result
+	 */
+	return res;
+}