add vercos(), avercos(), covercos(), acovercos()

Improved trig help files.

Added new vercos(x, [,eps]) for versed cosine and covercos(x, [,eps])
for inverse versed cosine.

Added new avercos(x, [,eps]) for inverse versed cosine and acovercos(x, [,eps])
for inverse coversed cosine.

CHANGES

@@ -128,7 +128,13 @@ The following are the changes from calc version 2.14.3.5 to date:
 
     Added c_to_q(COMPLEX *c, bool cfree) to make is easier to convert
     a COMPLEX value that is real (imag part is 0) into a NUMBER and
-    optionally free the COMPLEX value.
+    optionally free the COMPLEX value.  The func.c code now uses c_to_q().
+
+    Added new vercos(x, [,eps]) for versed cosine and covercos(x, [,eps])
+    for inverse versed cosine.
+
+    Added new avercos(x, [,eps]) for inverse versed cosine and acovercos(x, [,eps])
+    for inverse coversed cosine.
 
 
 The following are the changes from calc version 2.14.3.4 to 2.14.3.5:

cal/regress.cal

@@ -3506,9 +3506,9 @@ define test_trig()
 	/* test 3401-3407 */
 	tnum = test3400(1, 3401);
 	vrfy(tnum++ == 3407,	'3407: tnum == 3407');
+	pi = pi(1e-20);
 
 	/* test versed trigonometric sine */
-	pi = pi(1e-20);
 	vrfy(round(versin(0.2, 1e-10), 10) == 0.0199334222,
 	    strcat(str(tnum++),
 		   ': round(versin(0.2, 1e-10), 10) == 0.0199334222'));
@@ -3533,8 +3533,20 @@ define test_trig()
 	    strcat(str(tnum++),
 		   ': round(versin(2 + 3i, 1e-10), 10) == 5.189625691+9.1092278938i'));
 
+	/* test inverse versed trigonometric sine */
+	vrfy(round(aversin(0.5, 1e-10), 10) == 1.0471975512,
+	    strcat(str(tnum++),
+		   ': round(aversin(0.5, 1e-10), 10) == 1.0471975512'));
+	vrfy(aversin(0) == 0,
+		    strcat(str(tnum++), ': aversin(0) == 0'));
+	vrfy(round(aversin(-5, 1e-10), 10) == 2.4778887303i,
+	    strcat(str(tnum++),
+		   ': round(aversin(-5, 1e-10), 10) == 2.4778887303i'));
+	vrfy(round(aversin(2 + 3i, 1e-10), 10) == 1.8783999763+1.8641615439i,
+	    strcat(str(tnum++),
+		   ': round(aversin(2 + 3i, 1e-10), 10) == 1.8783999763+1.8641615439i'));
+
 	/* test coversed trigonometric sine */
-	pi = pi(1e-20);
 	vrfy(round(coversin(0.2, 1e-10), 10) == 0.8013306692,
 	    strcat(str(tnum++),
 		   ': round(coversin(0.2, 1e-10), 10) == 0.8013306692'));
@@ -3559,19 +3571,6 @@ define test_trig()
 	    strcat(str(tnum++),
 		   ': round(coversin(2 + 3i, 1e-10), 10) == -8.1544991469+4.16890696i'));
 
-	/* test inverse versed trigonometric sine */
-	vrfy(round(aversin(0.5, 1e-10), 10) == 1.0471975512,
-	    strcat(str(tnum++),
-		   ': round(aversin(0.5, 1e-10), 10) == 1.0471975512'));
-	vrfy(aversin(0) == 0,
-		    strcat(str(tnum++), ': aversin(0) == 0'));
-	vrfy(round(aversin(-5, 1e-10), 10) == 2.4778887303i,
-	    strcat(str(tnum++),
-		   ': round(aversin(-5, 1e-10), 10) == 2.4778887303i'));
-	vrfy(round(aversin(2 + 3i, 1e-10), 10) == 1.8783999763+1.8641615439i,
-	    strcat(str(tnum++),
-		   ': round(aversin(2 + 3i, 1e-10), 10) == 1.8783999763+1.8641615439i'));
-
 	/* test inverse coversed trigonometric sine */
 	vrfy(round(acoversin(0.5, 1e-10), 10) == 0.5235987756,
 	    strcat(str(tnum++),
@@ -3585,6 +3584,82 @@ define test_trig()
 	    strcat(str(tnum++),
 		   ': round(acoversin(2 + 3i, 1e-10), 10) == -0.3076036495-1.8641615442i'));
 
+	/* test versed trigonometric cosine */
+	vrfy(round(vercos(0.2, 1e-10), 10) == 1.9800665778,
+	    strcat(str(tnum++),
+		   ': round(vercos(0.2, 1e-10), 10) == 1.9800665778'));
+	vrfy(round(vercos(3/7, 1e-10), 10) == 1.9095603517,
+	    strcat(str(tnum++),
+		   ': round(vercos(3/7, 1e-10), 10) == 1.9095603517'));
+	vrfy(round(vercos(-31, 1e-10), 10) == 1.9147423578,
+	    strcat(str(tnum++),
+		   ': round(vercos(-31, 1e-10), 10) == 1.9147423578'));
+	vrfy(vercos(pi/3, 1e-10) == 1.5,
+				strcat(str(tnum++), ': vercos(pi/3, 1e-10) == 1.5'));
+	vrfy(vercos(pi/2, 1e-10) == 1,
+				strcat(str(tnum++), ': vercos(pi/2, 1e-10) == 1'));
+	vrfy(vercos(pi, 1e-10) == 0,
+				strcat(str(tnum++), ': vercos(pi, 1e-10) == 0'));
+	vrfy(vercos(3*pi/2, 1e-10) == 1,
+				strcat(str(tnum++), ': vercos(3*pi/2, 1e-10) == 1'));
+	vrfy(round(vercos(1, 1e-10), 10) == 1.5403023059,
+	    strcat(str(tnum++),
+		   ': round(vercos(1, 1e-10), 10) == 1.5403023059'));
+	vrfy(round(vercos(2 + 3i, 1e-10), 10) == -3.189625691-9.1092278938i,
+	    strcat(str(tnum++),
+		   ': round(vercos(2 + 3i, 1e-10), 10) == -3.189625691-9.1092278938i'));
+
+	/* test inverse versed trigonometric cosine */
+	vrfy(round(avercos(0.5, 1e-10), 10) == 2.0943951024,
+	    strcat(str(tnum++),
+		   ': round(avercos(0.5, 1e-10), 10) == 2.0943951024'));
+	vrfy(avercos(2) == 0,
+		    strcat(str(tnum++), ': avercos(2) == 0'));
+	vrfy(round(avercos(-5, 1e-10), 10) == 3.1415926536-2.4778887303i,
+	    strcat(str(tnum++),
+		   ': round(avercos(-5, 1e-10), 10) == 3.1415926536-2.4778887303i'));
+	vrfy(round(avercos(2 + 3i, 1e-10), 10) == 1.2631926773-1.8641615442i,
+	    strcat(str(tnum++),
+		   ': round(avercos(2 + 3i, 1e-10), 10) == 1.2631926773-1.8641615442i'));
+
+	/* test coversed trigonometric cosine */
+	vrfy(round(covercos(0.2, 1e-10), 10) == 1.1986693308,
+	    strcat(str(tnum++),
+		   ': round(covercos(0.2, 1e-10), 10) == 1.1986693308'));
+	vrfy(round(covercos(3/7, 1e-10), 10) == 1.415571855,
+	    strcat(str(tnum++),
+		   ': round(covercos(3/7, 1e-10), 10) == 1.415571855'));
+	vrfy(round(covercos(-31, 1e-10), 10) == 1.4040376453,
+	    strcat(str(tnum++),
+		   ': round(covercos(-31, 1e-10), 10) == 1.4040376453'));
+	vrfy(covercos(pi/6, 1e-10) == 1.5,
+				strcat(str(tnum++), ': covercos(pi/6, 1e-10) == 1.5'));
+	vrfy(covercos(pi/2, 1e-10) == 2,
+				strcat(str(tnum++), ': covercos(pi/2, 1e-10) == 2'));
+	vrfy(covercos(pi, 1e-10) == 1,
+				strcat(str(tnum++), ': covercos(pi, 1e-10) == 1'));
+	vrfy(covercos(3*pi/2, 1e-10) == 0,
+				strcat(str(tnum++), ': covercos(3*pi/2, 1e-10) == 0'));
+	vrfy(round(covercos(1, 1e-10), 10) == 1.8414709848,
+	    strcat(str(tnum++),
+		   ': round(covercos(1, 1e-10), 10) == 1.8414709848'));
+	vrfy(round(covercos(2 + 3i, 1e-10), 10) == 10.1544991469-4.16890696i,
+	    strcat(str(tnum++),
+		   ': round(covercos(2 + 3i, 1e-10), 10) == 10.1544991469-4.16890696i'));
+
+	/* test inverse coversed trigonometric cosine */
+	vrfy(round(acovercos(0.5, 1e-10), 10) == 0.5235987756,
+	    strcat(str(tnum++),
+		   ': round(acovercos(0.5, 1e-10), 10) == 0.5235987756'));
+	vrfy(acovercos(1) == 0,
+		    strcat(str(tnum++), ': acovercos(1) == 0'));
+	vrfy(round(acovercos(-5, 1e-10), 10) == -1.5707963268+2.4778887303i,
+	    strcat(str(tnum++),
+		   ': round(acovercos(-5, 1e-10), 10) == -1.5707963268+2.4778887303i'));
+	vrfy(round(acovercos(2 + 3i, 1e-10), 10) == 0.3076036495+1.8641615442i,
+	    strcat(str(tnum++),
+		   ': round(acovercos(2 + 3i, 1e-10), 10) == 0.3076036495+1.8641615442i'));
+
 	print strcat(str(tnum++), ': Ending test_trig');
 }
 print '051: parsed test_trig()';

calcerr.tbl

@@ -562,3 +562,15 @@ E_COVERSIN3	Too-large im(argument) for coversin
 E_ACOVERSIN1	Bad epsilon for acoversin
 E_ACOVERSIN2	Bad first argument for acoversin
 E_ACOVERSIN3	Too-large im(argument) for acoversin
+E_VERCOS1	Bad epsilon for vercos
+E_VERCOS2	Bad first argument for vercos
+E_VERCOS3	Too-large im(argument) for vercos
+E_AVERCOS1	Bad epsilon for avercos
+E_AVERCOS2	Bad first argument for avercos
+E_AVERCOS3	Too-large im(argument) for avercos
+E_COVERCOS1	Bad epsilon for covercos
+E_COVERCOS2	Bad first argument for covercos
+E_COVERCOS3	Too-large im(argument) for covercos
+E_ACOVERCOS1	Bad epsilon for acovercos
+E_ACOVERCOS2	Bad first argument for acovercos
+E_ACOVERCOS3	Too-large im(argument) for acovercos

cmath.h

@@ -127,6 +127,10 @@ E_FUNC COMPLEX *c_versin(COMPLEX *c, NUMBER *epsilon);
 E_FUNC COMPLEX *c_aversin(COMPLEX *c, NUMBER *epsilon);
 E_FUNC COMPLEX *c_coversin(COMPLEX *c, NUMBER *epsilon);
 E_FUNC COMPLEX *c_acoversin(COMPLEX *c, NUMBER *epsilon);
+E_FUNC COMPLEX *c_vercos(COMPLEX *c, NUMBER *epsilon);
+E_FUNC COMPLEX *c_avercos(COMPLEX *c, NUMBER *epsilon);
+E_FUNC COMPLEX *c_covercos(COMPLEX *c, NUMBER *epsilon);
+E_FUNC COMPLEX *c_acovercos(COMPLEX *c, NUMBER *epsilon);
 
 
 

comfunc.c

@@ -1428,7 +1428,7 @@ c_aversin(COMPLEX *c, NUMBER *epsilon)
  *
  * This uses the formula:
  *
- *	coversin(x) = 1 - sin(x)
+ *	coversin(x) = 1 - cos(x)
  *
  * given:
  *	q	    complex value to pass to the trig function
@@ -1517,3 +1517,195 @@ c_acoversin(COMPLEX *c, NUMBER *epsilon)
 	 */
 	return r;
 }
+
+
+/*
+ * c_vercos - versed sine for COMPLEX values
+ *
+ * This uses the formula:
+ *
+ *	vercos(x) = 1 + cos(x)
+ *
+ * given:
+ *	q	    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_vercos(COMPLEX *c, NUMBER *epsilon)
+{
+	COMPLEX *r;		/* return COMPLEX value */
+	COMPLEX *ctmp;		/* complex cos(c) */
+
+	/*
+	 * 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
+	 */
+	ctmp = c_cos(c, epsilon);
+	if (ctmp == NULL) {
+		math_error("Failed to compute complex cos for complex vercos");
+		not_reached();
+	}
+	r = c_add(&_cone_, ctmp);
+
+	/*
+	 * return complex 1 + cos(x)
+	 */
+	comfree(ctmp);
+	return r;
+}
+
+
+/*
+ * c_avercos - inverse versed sine for COMPLEX values
+ *
+ * This uses the formula:
+ *
+ *	avercos(x) = acos(x - 1)
+ *
+ * given:
+ *	q	    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_avercos(COMPLEX *c, NUMBER *epsilon)
+{
+	COMPLEX *r;	/* inverse trig value result */
+	COMPLEX *x;	/* argument to inverse trig function */
+
+	/*
+	 * 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
+	 */
+	x = c_sub(c, &_cone_);
+	r = c_acos(x, epsilon);
+	comfree(x);
+
+	/*
+	 * return complex acos(1 - x)
+	 */
+	return r;
+}
+
+
+/*
+ * c_covercos - coversed sine for COMPLEX values
+ *
+ * This uses the formula:
+ *
+ *	covercos(x) = 1 + sin(x)
+ *
+ * given:
+ *	q	    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_covercos(COMPLEX *c, NUMBER *epsilon)
+{
+	COMPLEX *r;		/* return COMPLEX value */
+	COMPLEX *ctmp;		/* complex sin(c) */
+
+	/*
+	 * 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
+	 */
+	ctmp = c_sin(c, epsilon);
+	if (ctmp == NULL) {
+		math_error("Failed to compute complex sin for complex covercos");
+		not_reached();
+	}
+	r = c_add(&_cone_, ctmp);
+
+	/*
+	 * return complex 1 + sin(x)
+	 */
+	comfree(ctmp);
+	return r;
+}
+
+
+/*
+ * c_acovercos - inverse versed sine for COMPLEX values
+ *
+ * This uses the formula:
+ *
+ *	acovercos(x) = asin(x - 1)
+ *
+ * given:
+ *	q	    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_acovercos(COMPLEX *c, NUMBER *epsilon)
+{
+	COMPLEX *r;	/* inverse trig value result */
+	COMPLEX *x;	/* argument to inverse trig function */
+
+	/*
+	 * 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
+	 */
+	x = c_sub(c, &_cone_);
+	r = c_asin(x, epsilon);
+	comfree(x);
+
+	/*
+	 * return complex asin(x - 1)
+	 */
+	return r;
+}

func.c

@@ -10841,6 +10841,286 @@ f_acoversin(int count, VALUE **vals)
 }
 
 
+/*
+ * f_vercos - versed cosine
+ */
+S_FUNC VALUE
+f_vercos(int count, VALUE **vals)
+{
+	VALUE result;
+	COMPLEX *c;
+	NUMBER *eps;
+
+	/* 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_VERCOS1);
+		}
+		eps = vals[1]->v_num;
+	}
+
+	/*
+	 * compute trig function to a given error tolerance
+	 */
+	switch (vals[0]->v_type) {
+		case V_NUM:
+			result.v_num = qvercos(vals[0]->v_num, eps);
+			result.v_type = V_NUM;
+			break;
+		case V_COM:
+			c = c_vercos(vals[0]->v_com, eps);
+			if (c == NULL) {
+				return error_value(E_VERCOS3);
+			}
+			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;
+			}
+			break;
+		default:
+			return error_value(E_VERCOS2);
+	}
+	return result;
+}
+
+
+/*
+ * f_avercos - inverse versed cosine
+ */
+S_FUNC VALUE
+f_avercos(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_AVERCOS1);
+		}
+		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 triv function */
+		result.v_num = qavercos_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_avercos(arg1.v_com, eps);
+		if (c == NULL) {
+			return error_value(E_AVERCOS3);
+		}
+		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_AVERCOS2);
+	}
+	return result;
+}
+
+
+/*
+ * f_covercos - coversed cosine
+ */
+S_FUNC VALUE
+f_covercos(int count, VALUE **vals)
+{
+	VALUE result;
+	COMPLEX *c;
+	NUMBER *eps;
+
+	/* 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_COVERCOS1);
+		}
+		eps = vals[1]->v_num;
+	}
+
+	/*
+	 * compute trig function to a given error tolerance
+	 */
+	switch (vals[0]->v_type) {
+		case V_NUM:
+			result.v_num = qcovercos(vals[0]->v_num, eps);
+			result.v_type = V_NUM;
+			break;
+		case V_COM:
+			c = c_covercos(vals[0]->v_com, eps);
+			if (c == NULL) {
+				return error_value(E_COVERCOS3);
+			}
+			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;
+			}
+			break;
+		default:
+			return error_value(E_COVERCOS2);
+	}
+	return result;
+}
+
+
+/*
+ * f_acovercos - inverse coversed cosine
+ */
+S_FUNC VALUE
+f_acovercos(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_ACOVERCOS1);
+		}
+		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 triv function */
+		result.v_num = qacovercos_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_acovercos(arg1.v_com, eps);
+		if (c == NULL) {
+			return error_value(E_ACOVERCOS3);
+		}
+		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_ACOVERCOS2);
+	}
+	return result;
+}
+
+
 #endif /* !FUNCLIST */
 
 
@@ -10893,6 +11173,8 @@ STATIC CONST struct builtin builtins[] = {
 	 "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"},
+	{"acovercos", 1, 2, 0, OP_NOP, {.null = NULL}, {.valfunc_cnt = f_acovercos},
+	 "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"},
 	{"acsc", 1, 2, 0, OP_NOP, {.null = NULL}, {.valfunc_cnt = f_acsc},
@@ -10925,6 +11207,8 @@ STATIC CONST struct builtin builtins[] = {
 	 "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"},
+	{"avercos", 1, 2, 0, OP_NOP, {.null = NULL}, {.valfunc_cnt = f_avercos},
+	 "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"},
 	{"avg", 0, IN, 0, OP_NOP, {.null = NULL}, {.valfunc_cnt = f_avg},
@@ -10988,6 +11272,8 @@ STATIC CONST struct builtin builtins[] = {
 	 "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"},
 	{"coversin", 1, 2, 0, OP_NOP, {.null = NULL}, {.valfunc_cnt = f_coversin},
 	 "coversed sine of value a within accuracy b"},
 	{"cp", 2, 2, 0, OP_NOP, {.null = NULL}, {.valfunc_2 = f_cp},
@@ -11547,6 +11833,8 @@ STATIC CONST struct builtin builtins[] = {
 	 "unget char read from file"},
 	{"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"},
 	{"versin", 1, 2, 0, OP_NOP, {.null = NULL}, {.valfunc_cnt = f_versin},
 	 "versed sine of value a within accuracy b"},
 	{"version", 0, 0, 0, OP_NOP, {.null = NULL}, {.valfunc_0 = f_version},

help/acos

@@ -36,8 +36,8 @@ LINK LIBRARY
 SEE ALSO
     sin, cos, tan, cot, sec, csc
     asin, atan, acot, asec, acsc
-    versin, coversin
+    versin, coversin, vercos, avercos
-    aversin. acoversin
+    aversin, acoversin, covercos, acovercos
     epsilon
 
 ## Copyright (C) 1999,2023  Landon Curt Noll

help/acot

@@ -34,8 +34,8 @@ LINK LIBRARY
 SEE ALSO
     sin, cos, tan, cot, sec, csc
     asin, acos, atan, asec, acsc
-    versin, coversin
+    versin, coversin, vercos, avercos
-    aversin. acoversin
+    aversin, acoversin, covercos, acovercos
     epsilon
 
 ## Copyright (C) 1999,2021,2023  Landon Curt Noll

help/acovercos

@@ -0,0 +1,62 @@
+NAME
+    acovercos - inverse coversed trigonometric cosine
+
+SYNOPSIS
+    acovercos(x [,eps])
+
+TYPES
+    x		number (real or complex)
+    eps		0 < real < 1, defaults to epsilon()
+
+    return	number
+
+DESCRIPTION
+    Calculate the inverse coversed trigonometric cosine of x to a multiple of eps with error less in
+    absolute value than .75 * eps.
+
+    This function is sometimes called acvc, or arccovercos, is equivalent to:
+
+	acovercos(x) = asin(x - 1)
+
+EXAMPLE
+    ; print acovercos(.5, 1e-5), acovercos(.5, 1e-10), acovercos(.5, 1e-15), acovercos(.5, 1e-20)
+    0.5236 0.5235987756 0.523598775598299 0.52359877559829887308
+
+    ; print acovercos(1), acovercos(-5), acovercos(2 + 3i)
+    0 -1.57079632679489661923+2.47788873028847500485i 0.30760364953071124992+1.86416154415788242834i
+
+LIMITS
+    0 < eps < 1
+
+LINK LIBRARY
+    NUMBER *qacovercos(NUMBER *x, NUMBER *eps)
+    COMPLEX *c_acovercos(COMPLEX *x, NUMBER *eps)
+
+SEE ALSO
+    sin, cos, tan, cot, sec, csc
+    asin, acos, atan, acot, asec, acsc
+    versin, coversin, vercos, avercos
+    aversin, acoversin, covercos, acovercos
+    epsilon
+
+## 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:	2023/09/05 23:30:08
+## 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/

help/acoversin

@@ -11,11 +11,10 @@ TYPES
     return	number
 
 DESCRIPTION
-    Calculate the inverse coversed sine of x to a multiple of eps with error less in
+    Calculate the inverse coversed trigonometric sine of x to a multiple of eps with error less in
     absolute value than .75 * eps.
 
-    The coversed sine function is sometimes called covers, or acosiv, or acvs,
+    This function is sometimes called acovers, or acvs, or arccoversin, is equivalent to:
-    may be defined as:
 
 	acoversin(x) = asin(1 - x)
 
@@ -36,8 +35,8 @@ LINK LIBRARY
 SEE ALSO
     sin, cos, tan, cot, sec, csc
     asin, acos, atan, acot, asec, acsc
-    versin, coversin
+    versin, coversin, vercos, avercos
-    aversin
+    aversin, acoversin, covercos, acovercos
     epsilon
 
 ## Copyright (C) 2023  Landon Curt Noll

help/acsc

@@ -34,8 +34,8 @@ LINK LIBRARY
 SEE ALSO
     sin, cos, tan, cot, sec, csc
     asin, acos, atan, acot, asec
-    versin, coversin
+    versin, coversin, vercos, avercos
-    aversin. acoversin
+    aversin, acoversin, covercos, acovercos
     epsilon
 
 ## Copyright (C) 1999,2021,2023  Landon Curt Noll

help/asec

@@ -34,8 +34,8 @@ LINK LIBRARY
 SEE ALSO
     sin, cos, tan, cot, sec, csc
     asin, acos, atan, acot, acsc
-    versin, coversin
+    versin, coversin, vercos, avercos
-    aversin. acoversin
+    aversin, acoversin, covercos, acovercos
     epsilon
 
 ## Copyright (C) 1999,2021,2023  Landon Curt Noll

help/asin

@@ -36,8 +36,8 @@ LINK LIBRARY
 SEE ALSO
     sin, cos, tan, cot, sec, csc
     acos, atan, acot, asec, acsc
-    versin, coversin
+    versin, coversin, vercos, avercos
-    aversin. acoversin
+    aversin, acoversin, covercos, acovercos
     epsilon
 
 ## Copyright (C) 1999,2021,2023  Landon Curt Noll

help/atan

@@ -34,8 +34,8 @@ LINK LIBRARY
 SEE ALSO
     sin, cos, tan, cot, sec, csc
     asin, acos, acot, asec, acsc
-    versin, coversin
+    versin, coversin, vercos, avercos
-    aversin. acoversin
+    aversin, acoversin, covercos, acovercos
     epsilon
 
 ## Copyright (C) 1999,2023  Landon Curt Noll

help/avercos

@@ -0,0 +1,63 @@
+NAME
+    avercos - inverse versed trigonometric cosine
+
+SYNOPSIS
+    avercos(x [,eps])
+
+TYPES
+    x		number (real or complex)
+    eps		0 < real < 1, defaults to epsilon()
+
+    return	real
+
+DESCRIPTION
+    Returns the inverse versed trigonometric cosine of x to a multiple of eps with error less in
+    absolute value than .75 * eps.
+
+    This function is sometimes called averc, or arcvercos, is equivalent to:
+
+	avercos(x) = acos(x - 1)
+
+EXAMPLE
+    ; print avercos(.5, 1e-5), avercos(.5, 1e-10), avercos(.5, 1e-15), avercos(.5, 1e-20)
+    2.0944 2.0943951024 2.094395102393195 2.09439510239319549231
+
+    ; print avercos(2), avercos(-5), avercos(2 + 3i)
+    0 3.14159265358979323846-2.47788873028847500481i 1.26319267726418536931-1.86416154415788242834i
+
+LIMITS
+    0 < eps < 1
+
+LINK LIBRARY
+    NUMBER *qavercos_or_NULL(NUMBER *q, NUMBER *epsilon);
+    NUMBER *qavercos(NUMBER *q, NUMBER *epsilon);
+    COMPLEX *c_avercos(COMPLEX *c, NUMBER *epsilon);
+
+SEE ALSO
+    sin, cos, tan, cot, sec, csc
+    asin, acos, atan, acot, asec, acsc
+    versin, coversin, vercos, avercos
+    aversin, acoversin, covercos, acovercos
+    epsilon
+
+## 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:	2023/09/05 23:40:24
+## 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/

help/aversin

@@ -11,11 +11,10 @@ TYPES
     return	real
 
 DESCRIPTION
-    Returns the inverse versed sine of x to a multiple of eps with error less in
+    Returns the inverse versed trigonometric sine of x to a multiple of eps with error less in
     absolute value than .75 * eps.
 
-    The inverse versed sine function is sometimes called avers, sometimes called aver,
+    This function is sometimes called avers, or arcversin, is equivalent to:
-    may be defined as:
 
 	aversin(x) = acos(1 - x)
 

help/cos

@@ -35,8 +35,8 @@ LINK LIBRARY
 SEE ALSO
     sin, tan, cot, sec, csc
     asin, acos, atan, acot, asec, acsc
-    versin, coversin
+    versin, coversin, vercos, avercos
-    aversin. acoversin
+    aversin, acoversin, covercos, acovercos
     epsilon
 
 ## Copyright (C) 1999,2021,2023  Landon Curt Noll

help/cot

@@ -28,8 +28,8 @@ LINK LIBRARY
 SEE ALSO
     sin, cos, tan, sec, csc
     asin, acos, atan, acot, asec, acsc
-    versin, coversin
+    versin, coversin, vercos, avercos
-    aversin. acoversin
+    aversin, acoversin, covercos, acovercos
     epsilon
 
 ## Copyright (C) 1999,2021,2023  Landon Curt Noll

help/covercos

@@ -0,0 +1,69 @@
+NAME
+    covercos - coversed trigonometric cosine
+
+SYNOPSIS
+    covercos(x [,eps])
+
+TYPES
+    x		number (real or complex)
+    eps		0 < real < 1, defaults to epsilon()
+
+    return	number
+
+DESCRIPTION
+    Calculate the coversed trigonometric cosine of x to a multiple of eps with error less in
+    absolute value than .75 * eps.
+
+    This function is sometimes called cvc, is equivalent to:
+
+	covercos(x) = 1 + sin(x)
+
+EXAMPLE
+    ; print covercos(0.2), covercos(3/7), covercos(-31)
+    1.19866933079506121546 1.41557185499305200807 1.40403764532306500605
+
+    ; print covercos(1, 1e-5), covercos(1, 1e-10), covercos(1, 1e-15), covercos(1, 1e-20)
+    1.84147 1.8414709848 1.841470984807896 1.84147098480789650665
+
+    ; print covercos(2 + 3i, 1e-5), covercos(2 + 3i, 1e-10)
+    10.1545-4.16891i 10.1544991469-4.16890696i
+
+    ; pi = pi(1e-20)
+    ; print covercos(pi/6, 1e-10), covercos(pi/2, 1e-10), covercos(pi, 1e-10), covercos(3*pi/2, 1e-10)
+    1.5 2 1 0
+
+LIMITS
+    0 < eps < 1
+
+LINK LIBRARY
+    NUMBER *qcovercos(NUMBER *x, NUMBER *eps)
+    COMPLEX *c_covercos(COMPLEX *x, NUMBER *eps)
+
+SEE ALSO
+    sin, cos, tan, cot, sec, csc
+    asin, acos, atan, acot, asec, acsc
+    versin, coversin, vercos, avercos
+    aversin, acoversin, covercos, acovercos
+    epsilon
+
+## 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:	2023/09/05 23:31:08
+## 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/

help/coversin

@@ -11,11 +11,10 @@ TYPES
     return	number
 
 DESCRIPTION
-    Calculate the coversed sine of x to a multiple of eps with error less in
+    Calculate the coversed trigonometric sine of x to a multiple of eps with error less in
     absolute value than .75 * eps.
 
-    The coversed sine function is sometimes called covers, or cosiv, or cvs,
+    This function is sometimes called covers, or cvs, is equivalent to:
-    may be defined as:
 
 	coversin(x) = 1 - sin(x)
 
@@ -43,8 +42,8 @@ LINK LIBRARY
 SEE ALSO
     sin, cos, tan, cot, sec, csc
     asin, acos, atan, acot, asec, acsc
-    versin
+    versin, coversin, vercos, avercos
-    aversin. acoversin
+    aversin, acoversin, covercos, acovercos
     epsilon
 
 ## Copyright (C) 2023  Landon Curt Noll

help/csc

@@ -27,8 +27,8 @@ LINK LIBRARY
 SEE ALSO
     sin, cos, tan, cot, sec
     asin, acos, atan, acot, asec, acsc
-    versin, coversin
+    versin, coversin, vercos, avercos
-    aversin. acoversin
+    aversin, acoversin, covercos, acovercos
     epsilon
 
 ## Copyright (C) 1999,2023  Landon Curt Noll

help/sec

@@ -28,8 +28,8 @@ LINK LIBRARY
 SEE ALSO
     sin, cos, tan, cot, csc
     asin, acos, atan, acot, asec, acsc
-    versin, coversin
+    versin, coversin, vercos, avercos
-    aversin. acoversin
+    aversin, acoversin, covercos, acovercos
     epsilon
 
 ## Copyright (C) 1999,2023  Landon Curt Noll

help/sin

@@ -35,8 +35,8 @@ LINK LIBRARY
 SEE ALSO
     cos, tan, cot, sec, csc
     asin, acos, atan, acot, asec, acsc
-    versin, coversin
+    versin, coversin, vercos, avercos
-    aversin. acoversin
+    aversin, acoversin, covercos, acovercos
     epsilon
 
 ## Copyright (C) 1999,2021,2023  Landon Curt Noll

help/tan

@@ -29,8 +29,8 @@ LINK LIBRARY
 SEE ALSO
     sin, cos, cot, sec, csc
     asin, acos, atan, acot, asec, acsc
-    versin, coversin
+    versin, coversin, vercos, avercos
-    aversin. acoversin
+    aversin, acoversin, covercos, acovercos
     epsilon
 
 ## Copyright (C) 1999,2023  Landon Curt Noll

help/vercos

@@ -0,0 +1,69 @@
+NAME
+    vercos - versed trigonometric cosine
+
+SYNOPSIS
+    vercos(x [,eps])
+
+TYPES
+    x		number (real or complex)
+    eps		0 < real < 1, defaults to epsilon()
+
+    return	number
+
+DESCRIPTION
+    Calculate the versed trigonometric cosine of x to a multiple of eps with error less in
+    absolute value than .75 * eps.
+
+    This function is sometimes called verc, is equivalent to:
+
+	vercos(x) = 1 + cos(x)
+
+EXAMPLE
+    ; print vercos(0.2), vercos(3/7), vercos(-31)
+    1.98006657784124163112 1.90956035167416667403 1.91474235780453127896
+
+    ; print vercos(1, 1e-5), vercos(1, 1e-10), vercos(1, 1e-15), vercos(1, 1e-20)
+    1.5403 1.5403023059 1.54030230586814 1.5403023058681397174
+
+    ; print vercos(2 + 3i, 1e-5), vercos(2 + 3i, 1e-10)
+    -3.18963-9.10923i -3.189625691-9.1092278938i
+
+    ; pi = pi(1e-20)
+    ; print vercos(pi/3, 1e-10), vercos(pi/2, 1e-10), vercos(pi, 1e-10), vercos(3*pi/2, 1e-10)
+    1.5 1 0 1
+
+LIMITS
+    0 < eps < 1
+
+LINK LIBRARY
+    NUMBER *qvercos(NUMBER *x, NUMBER *eps)
+    COMPLEX *c_vercos(COMPLEX *x, NUMBER *eps)
+
+SEE ALSO
+    sin, cos, tan, cot, sec, csc
+    asin, acos, atan, acot, asec, acsc
+    versin, coversin, vercos, avercos
+    aversin, acoversin, covercos, acovercos
+    epsilon
+
+## 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:	2023/09/05 23:25:28
+## 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/

help/versin

@@ -11,11 +11,10 @@ TYPES
     return	number
 
 DESCRIPTION
-    Calculate the versed sine of x to a multiple of eps with error less in
+    Calculate the versed trigonometric sine of x to a multiple of eps with error less in
     absolute value than .75 * eps.
 
-    The versed sine function is sometimes called vers, sometimes called ver,
+    This function is sometimes called vers, is equivalent to:
-    may be defined as:
 
 	versin(x) = 1 - cos(x)
 
@@ -43,8 +42,8 @@ LINK LIBRARY
 SEE ALSO
     sin, cos, tan, cot, sec, csc
     asin, acos, atan, acot, asec, acsc
-    coversin
+    versin, coversin, vercos, avercos
-    aversin. acoversin
+    aversin, acoversin, covercos, acovercos
     epsilon
 
 ## Copyright (C) 2023  Landon Curt Noll

qmath.h

@@ -232,6 +232,12 @@ E_FUNC NUMBER *qaversin(NUMBER *q, NUMBER *epsilon);
 E_FUNC NUMBER *qcoversin(NUMBER *q, NUMBER *epsilon);
 E_FUNC NUMBER *qacoversin_or_NULL(NUMBER *q, NUMBER *epsilon);
 E_FUNC NUMBER *qacoversin(NUMBER *q, NUMBER *epsilon);
+E_FUNC NUMBER *qvercos(NUMBER *q, NUMBER *epsilon);
+E_FUNC NUMBER *qavercos_or_NULL(NUMBER *q, NUMBER *epsilon);
+E_FUNC NUMBER *qavercos(NUMBER *q, NUMBER *epsilon);
+E_FUNC NUMBER *qcovercos(NUMBER *q, NUMBER *epsilon);
+E_FUNC NUMBER *qacovercos_or_NULL(NUMBER *q, NUMBER *epsilon);
+E_FUNC NUMBER *qacovercos(NUMBER *q, NUMBER *epsilon);
 
 
 /*

qtrans.c

@@ -2293,3 +2293,320 @@ qacoversin(NUMBER *q, NUMBER *epsilon)
 	 */
 	return res;
 }
+
+
+/*
+ * qvercos - versed sine for NUMBER values
+ *
+ * This uses the formula:
+ *
+ *	vercos(x) = 1 + cos(x)
+ *
+ * given:
+ *	q	    real value to pass to the trig function
+ *	epsilon	    error tolerance / precision for trig calculation
+ *
+ * returns:
+ *	real value result of trig function on q with error epsilon
+ */
+NUMBER *
+qvercos(NUMBER *q, NUMBER *epsilon)
+{
+	NUMBER *sin, *cos, *res;
+	NUMBER *vercos;
+	long n;
+
+	/*
+	 * 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
+	 */
+	n = -qilog2(epsilon);
+	qsincos(q, n + 2, &sin, &cos);
+	qfree(sin);
+	vercos = qqadd(&_qone_, cos);
+	qfree(cos);
+
+	/*
+	 * round value to nearest epsilon
+	 */
+	res = qmappr(vercos, epsilon, 24);
+	qfree(vercos);
+
+	/*
+	 * return 1 + cos(x)
+	 */
+	return res;
+}
+
+
+/*
+ * qavercos_or_NULL - inverse versed sine for NUMBER values
+ *
+ * This uses the formula:
+ *
+ *	avercos(x) = acos(x - 1)
+ *
+ * given:
+ *	q	    real value to pass to the trig function
+ *	epsilon	    error tolerance / precision for trig calculation
+ *
+ * returns:
+ *	!= NULL ==> real value result of trig function on q with error epsilon,
+ *	NULL    ==> trig function value cannot be expressed as a NUMBER
+ *
+ * NOTE: If this function returns NULL, consider calling the equivalent
+ *	 COMPLEX function from comfunc.c.  See the help file for the
+ *	 related builtin for details.
+ */
+NUMBER *
+qavercos_or_NULL(NUMBER *q, NUMBER *epsilon)
+{
+	NUMBER *res;	/* inverse trig value result */
+	NUMBER *x;	/* argument to inverse trig function */
+
+	/*
+	 * firewall
+	 */
+	if (q == NULL) {
+		math_error("q is NULL for %s", __func__);
+		not_reached();
+	}
+	if (check_epsilon(epsilon) == false) {
+		math_error("Invalid epsilon arg for %s", __func__);
+		not_reached();
+	}
+
+	/*
+	 * calculate inverse trig function value
+	 */
+	x = qsub(q, &_qone_);
+	res = qacos(x, epsilon);
+	qfree(x);
+	if (res == NULL) {
+		return NULL;
+	}
+
+	/*
+	 * return acos(x - 1)
+	 */
+	return res;
+}
+
+
+/*
+ * qavercos - inverse versed sine for NUMBER values
+ *
+ * This uses the formula:
+ *
+ *	avercos(x) = acos(x - 1)
+ *
+ * given:
+ *	q	    real value to pass to the trig function
+ *	epsilon	    error tolerance / precision for trig calculation
+ *
+ * returns:
+ *	real value result of trig function on q with error epsilon
+ */
+NUMBER *
+qavercos(NUMBER *q, NUMBER *epsilon)
+{
+	NUMBER *res;	/* inverse trig value result */
+
+	/*
+	 * firewall
+	 */
+	if (q == NULL) {
+		math_error("q is NULL for %s", __func__);
+		not_reached();
+	}
+	if (check_epsilon(epsilon) == false) {
+		math_error("Invalid epsilon arg for %s", __func__);
+		not_reached();
+	}
+
+	/*
+	 * calculate inverse trig function value
+	 */
+	res = qavercos_or_NULL(q, epsilon);
+	if (res == NULL) {
+		math_error("cannot compute inverse cos for avercos");
+		not_reached();
+	}
+
+	/*
+	 * return acos(x - 1)
+	 */
+	return res;
+}
+
+
+/*
+ * qcovercos - coversed sine for NUMBER values
+ *
+ * This uses the formula:
+ *
+ *	covercos((x) = 1 + sin(x)
+ *
+ * given:
+ *	q	    real value to pass to the trig function
+ *	epsilon	    error tolerance / precision for trig calculation
+ *
+ * returns:
+ *	real value result of trig function on q with error epsilon
+ */
+NUMBER *
+qcovercos(NUMBER *q, NUMBER *epsilon)
+{
+	NUMBER *sin, *cos, *res;
+	NUMBER *covercos;
+	long n;
+
+	/*
+	 * 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
+	 */
+	n = -qilog2(epsilon);
+	if (qiszero(q) || n < 0)
+		return qlink(&_qzero_);
+	qsincos(q, n + 2, &sin, &cos);
+	qfree(cos);
+	covercos = qqadd(&_qone_, sin);
+	qfree(sin);
+
+	/*
+	 * round value to nearest epsilon
+	 */
+	res = qmappr(covercos, epsilon, 24);
+	qfree(covercos);
+
+	/*
+	 * return 1 + sin(x)
+	 */
+	return res;
+}
+
+
+/*
+ * qacovercos_or_NULL - inverse coversed sine for NUMBER values
+ *
+ * This uses the formula:
+ *
+ *	acovercos(x) = asin(x - 1)
+ *
+ * given:
+ *	q	    real value to pass to the trig function
+ *	epsilon	    error tolerance / precision for trig calculation
+ *
+ * returns:
+ *	real value result of trig function on q with error epsilon
+ *
+ * returns:
+ *	!= NULL ==> real value result of trig function on q with error epsilon,
+ *	NULL    ==> trig function value cannot be expressed as a NUMBER
+ *
+ * NOTE: If this function returns NULL, consider calling the equivalent
+ *	 COMPLEX function from comfunc.c.  See the help file for the
+ *	 related builtin for details.
+ */
+NUMBER *
+qacovercos_or_NULL(NUMBER *q, NUMBER *epsilon)
+{
+	NUMBER *res;	/* inverse trig value result */
+	NUMBER *x;	/* argument to inverse trig function */
+
+	/*
+	 * firewall
+	 */
+	if (q == NULL) {
+		math_error("q is NULL for %s", __func__);
+		not_reached();
+	}
+	if (check_epsilon(epsilon) == false) {
+		math_error("Invalid epsilon arg for %s", __func__);
+		not_reached();
+	}
+
+	/*
+	 * calculate inverse trig function value
+	 */
+	x = qsub(&_qone_, q);
+	res = qasin(x, epsilon);
+	qfree(x);
+	if (res == NULL) {
+		return NULL;
+	}
+
+	/*
+	 * return asin(x - 1)
+	 */
+	return res;
+}
+
+
+/*
+ * qacovercos - inverse coversed sine for NUMBER values
+ *
+ * This uses the formula:
+ *
+ *	acovercos(x) = asin(x - 1)
+ *
+ * given:
+ *	q	    real value to pass to the trig function
+ *	epsilon	    error tolerance / precision for trig calculation
+ *
+ * returns:
+ *	real value result of trig function on q with error epsilon
+ */
+NUMBER *
+qacovercos(NUMBER *q, NUMBER *epsilon)
+{
+	NUMBER *res;	/* inverse trig value result */
+
+	/*
+	 * firewall
+	 */
+	if (q == NULL) {
+		math_error("q is NULL for %s", __func__);
+		not_reached();
+	}
+	if (check_epsilon(epsilon) == false) {
+		math_error("Invalid epsilon arg for %s", __func__);
+		not_reached();
+	}
+
+	/*
+	 * calculate inverse trig function value
+	 */
+	res = qacovercos_or_NULL(q, epsilon);
+	if (res == NULL) {
+		math_error("cannot compute inverse sin for acovercos");
+		not_reached();
+	}
+
+	/*
+	 * return asin(x - 1)
+	 */
+	return res;
+}