add cas and cis trigonometric functions

Added the following new trigonometric functions:

    cas(x [,eps])		trigonometric cosine plus sine
    cis(x [,eps])		Euler's formula

CHANGES

@@ -24,6 +24,8 @@ The following are the changes from calc version 2.14.3.5 to date:
 	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
+	cas(x [,eps])		trigonometric cosine plus sine
+	cis(x [,eps])		Euler's formula
 
     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

cal/regress.cal

@@ -4182,6 +4182,52 @@ define test_trig()
 	    strcat(str(tnum++),
 		   ': round(acrd(2 + 3i, 1e-10), 10) == 1.0471975512+2.6339157938i'));
 
+	/* test cosine plus sine */
+	vrfy(cas(0, 1e-10) == 1,
+				strcat(str(tnum++), ': cas(0, 1e-10) == 1'));
+	vrfy(round(cas(0.2, 1e-10), 10) == 1.1787359086,
+	    strcat(str(tnum++),
+		   ': round(cas(0.2, 1e-10), 10) == 1.1787359086'));
+	vrfy(round(cas(3/7, 1e-10), 10) == 1.3251322067,
+	    strcat(str(tnum++),
+		   ': round(cas(3/7, 1e-10), 10) == 1.3251322067'));
+	vrfy(round(cas(-31, 1e-10), 10) == 1.3187800031,
+	    strcat(str(tnum++),
+		   ': round(cas(-31, 1e-10), 10) == 1.3187800031'));
+	vrfy(cas(pi/2, 1e-10) == 1,
+				strcat(str(tnum++), ': cas(pi/2, 1e-10) == 1'));
+	vrfy(cas(pi, 1e-10) == -1,
+				strcat(str(tnum++), ': cas(pi, 1e-10) == -1'));
+	vrfy(cas(3*pi/2, 1e-10) == -1,
+				strcat(str(tnum++), ': cas(3*pi/2, 1e-10) == -1'));
+	vrfy(round(cas(1, 1e-10), 10) == 1.3817732907,
+	    strcat(str(tnum++),
+		   ': round(cas(1, 1e-10), 10) == 1.3817732907'));
+	vrfy(round(cas(2 + 3i, 1e-10), 10) == 4.9648734559-13.2781348538i,
+	    strcat(str(tnum++),
+		   ': round(cas(2 + 3i, 1e-10), 10) == 4.9648734559-13.2781348538i'));
+
+	/* test Euler's formula */
+	vrfy(cis(0, 1e-10) == 1,
+				strcat(str(tnum++), ': cis(0, 1e-10) == 1'));
+	vrfy(cis(pi/2, 1e-10) == 1i,
+				strcat(str(tnum++), ': cis(pi/2, 1e-10) == 1i'));
+	vrfy(round(cis(0.2, 1e-10), 10) == 0.9800665778+0.1986693308i,
+	    strcat(str(tnum++),
+		   ': round(cis(0.2, 1e-10), 10) == 0.9800665778+0.1986693308i'));
+	vrfy(round(cis(3/7, 1e-10), 10) == 0.9095603517+0.415571855i,
+	    strcat(str(tnum++),
+		   ': round(cis(3/7, 1e-10), 10) == 0.9095603517+0.415571855i'));
+	vrfy(round(cis(-31, 1e-10), 10) == 0.9147423578+0.4040376453i,
+	    strcat(str(tnum++),
+		   ': round(cis(-31, 1e-10), 10) == 0.9147423578+0.4040376453i'));
+	vrfy(round(cis(1, 1e-10), 10) == 0.5403023059+0.8414709848i,
+	    strcat(str(tnum++),
+		   ': round(cis(1, 1e-10), 10) == 0.5403023059+0.8414709848i'));
+	vrfy(round(cis(2 + 3i, 1e-10), 10) == -0.020718731+0.0452712532i,
+	    strcat(str(tnum++),
+		   ': round(cis(2 + 3i, 1e-10), 10) == -0.020718731+0.0452712532i'));
+
 	print strcat(str(tnum++), ': Ending test_trig');
 }
 print '051: parsed test_trig()';
@@ -9879,9 +9925,11 @@ 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');
+vrfy(verify_crd(9517) == 0,		'9517: verify_crd(9517) == 0');
+vrfy(verify_cas(9518) == 0,		'9518: verify_cas(9518) == 0');
+vrfy(verify_cis(9519) == 0,		'9519: verify_cis(9519) == 0');
 
-print '9518: Ending trigonometric identities test set';
+print '9520: Ending trigonometric identities test set';
 
 
 /*
@@ -10890,6 +10938,12 @@ 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");
+vrfy_errsym(10600, 10600, "E_CAS_1");
+vrfy_errsym(10601, 10601, "E_CAS_2");
+vrfy_errsym(10602, 10602, "E_CAS_3");
+vrfy_errsym(10603, 10603, "E_CIS_1");
+vrfy_errsym(10604, 10604, "E_CIS_2");
+vrfy_errsym(10605, 10605, "E_CIS_3");
 
 /* ************************************************************** */
 /* NOTE: Reserve thru test 10998 for calc computation error codes */

cal/test9500.trigeq.cal

@@ -1312,3 +1312,122 @@ define verify_crd(testnum)
 	}
 	return error_count;
 }
+
+
+/*
+ * verify_cas - verify cosine plus sine
+ *
+ * We use the following trigonometric identity:
+ *
+ *	cas(x) = cos(x) + sin(x)
+ *
+ * given:
+ *	testnum		regression test number being performed
+ *
+ * returns:
+ *	number of tests that failed
+ */
+define verify_cas(testnum)
+{
+	local tval_len;		/* current length of the tval[] array */
+	local ident_val;	/* computed trig value trigonometric identity */
+	local trig_val;		/* computed value from the trigonometric function */
+	local error_count;	/* number of compare errors detected */
+	local i;
+
+	/*
+	 * firewall
+	 */
+	if (size(sin_tval) <= 0) {
+		precompute_trig();
+	}
+	if (size(cos_tval) <= 0) {
+		precompute_trig();
+	}
+
+	/*
+	 * for each test value, verify the trigonometric identity within epsilon
+	 */
+	tval_len = size(tval);
+	for (i=0; i < tval_len; ++i) {
+
+		/* compute trigonometric identity */
+		ident_val = cos_tval[i] + sin_tval[i];
+
+		/* compute trigonometric function */
+		trig_val = cas(tval[i]);
+
+		/* compare trigonometric identity with trigonometric function value */
+		if (compare(ident_val, trig_val, "cas", i, testnum)) {
+			++error_count;
+		}
+	}
+
+	/*
+	 * report test results
+	 */
+	if (error_count != 0) {
+		print '**** test', testnum : ': cas test failure count:', error_count;
+	}
+	return error_count;
+}
+
+
+/*
+ * verify_cis - verify Euler's formula
+ *
+ * We use the following trigonometric identity:
+ *
+ *	cis(x) = cos(x) + i*sin(x)
+ *	cis(x) = exp(1i * x)
+ *
+ * given:
+ *	testnum		regression test number being performed
+ *
+ * returns:
+ *	number of tests that failed
+ */
+define verify_cis(testnum)
+{
+	local tval_len;		/* current length of the tval[] array */
+	local ident_val;	/* computed trig value trigonometric identity */
+	local trig_val;		/* computed value from the trigonometric function */
+	local error_count;	/* number of compare errors detected */
+	local i;
+
+	/*
+	 * firewall
+	 */
+	if (size(sin_tval) <= 0) {
+		precompute_trig();
+	}
+	if (size(cos_tval) <= 0) {
+		precompute_trig();
+	}
+
+	/*
+	 * for each test value, verify the trigonometric identity within epsilon
+	 */
+	tval_len = size(tval);
+	for (i=0; i < tval_len; ++i) {
+
+		/* compute trigonometric identity */
+		ident_val = cos_tval[i] + 1i*sin_tval[i];
+
+		/* compute trigonometric function */
+		trig_val = cis(tval[i]);
+
+		/* compare trigonometric identity with trigonometric function value */
+		if (compare(ident_val, trig_val, "cis", i, testnum)) {
+			++error_count;
+		}
+	}
+
+	/*
+	 * report test results
+	 */
+	if (error_count != 0) {
+		print '**** test', testnum : ': cis test failure count:', error_count;
+	}
+	return error_count;
+}

cmath.h

@@ -74,6 +74,7 @@ E_FUNC COMPLEX *c_square(COMPLEX *c);
 E_FUNC COMPLEX *c_conj(COMPLEX *c);
 E_FUNC COMPLEX *c_real(COMPLEX *c);
 E_FUNC NUMBER *c_to_q(COMPLEX *c, bool cfree);
+E_FUNC COMPLEX *q_to_c(NUMBER *q);
 E_FUNC COMPLEX *c_imag(COMPLEX *c);
 E_FUNC COMPLEX *c_neg(COMPLEX *c);
 E_FUNC COMPLEX *c_inv(COMPLEX *c);
@@ -150,6 +151,8 @@ 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);
+E_FUNC COMPLEX *c_cas(COMPLEX *c, NUMBER *epsilon);
+E_FUNC COMPLEX *c_cis(COMPLEX *c, NUMBER *epsilon);
 
 
 

comfunc.c

@@ -2719,3 +2719,112 @@ c_acrd(COMPLEX *c, NUMBER *epsilon)
 	 */
 	return r;
 }
+
+
+/*
+ * c_cas - COMPLEX valued cosine plus sine
+ *
+ * This uses the formula:
+ *
+ *	cas(x) = cos(x) + sin(x)
+ *
+ * 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_cas(COMPLEX *c, NUMBER *epsilon)
+{
+	COMPLEX *r;		/* return COMPLEX value */
+	COMPLEX *csin;		/* complex sin(c) */
+	COMPLEX *ccos;		/* 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
+	 */
+	csin = c_sin(c, epsilon);
+	if (csin == NULL) {
+		math_error("Failed to compute complex sine for complex cas");
+		not_reached();
+	}
+	ccos = c_cos(c, epsilon);
+	if (ccos == NULL) {
+		comfree(csin);
+		math_error("Failed to compute complex cosine for complex cas");
+		not_reached();
+	}
+	r = c_add(csin, ccos);
+	comfree(csin);
+	comfree(ccos);
+
+	/*
+	 * return trigonometric result
+	 */
+	return r;
+}
+
+
+/*
+ * c_cis - COMPLEX valued Euler's formula
+ *
+ * This uses the formula:
+ *
+ *	cis(x) = cos(x) + 1i*sin(x)
+ *	cis(x) = exp(1i * x)
+ *
+ * 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_cis(COMPLEX *c, NUMBER *epsilon)
+{
+	COMPLEX *r;		/* return COMPLEX value */
+	COMPLEX *ctmp;		/* 1i * 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_mul(c, &_conei_);
+	r = c_exp(ctmp, epsilon);
+	comfree(ctmp);
+	if (r == NULL) {
+		math_error("Failed to compute complex exp for complex cis");
+		not_reached();
+	}
+
+	/*
+	 * return trigonometric result
+	 */
+	return r;
+}

commath.c

@@ -526,6 +526,38 @@ c_to_q(COMPLEX *c, bool cfree)
 }
 
 
+/*
+ * q_to_c - convert a NUMBER into an allocated COMPLEX
+ *
+ * given:
+ *	q	NUMBER to be converted
+ *
+ * returns:
+ *	allocated COMPLEX number whose real part is NUMBER and imag part is 0
+ */
+COMPLEX *
+q_to_c(NUMBER *q)
+{
+	COMPLEX *res;	/* COMPLEX number to return */
+
+	/*
+	 * allocate complex number
+	 */
+	res = comalloc();
+
+	/*
+	 * assign NUMBER to real part
+	 */
+	qfree(res->real);
+	res->real = qlink(q);
+
+	/*
+	 * return the allocated equivalent COMPLEX
+	 */
+	return res;
+}
+
+
 /*
  * Return the imaginary part of a complex number as a real.
  */

errtbl.c

@@ -749,6 +749,12 @@ CONST struct errtbl error_table[] =
 	{ 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" },
+	{ 10600,	"E_CAS_1",		"Bad epsilon for cas" },
+	{ 10601,	"E_CAS_2",		"Bad argument for cas" },
+	{ 10602,	"E_CAS_3",		"Invalid complex argument for cas" },
+	{ 10603,	"E_CIS_1",		"Non-real epsilon for cis" },
+	{ 10604,	"E_CIS_2",		"Bad first argument for cis" },
+	{ 10605,	"E_CIS_3",		"Too-large re(argument) for cis" },
 	/* IMPORTANT NOTE: add new entries above here and be sure their errnum numeric value is consecutive! */
 
 	/* The next NULL entry must be last */

func.c

@@ -2661,7 +2661,7 @@ f_cos(int count, VALUE **vals)
 	}
 
 	/*
-	 * compute cosinr to a given error tolerance
+	 * compute cosine to a given error tolerance
 	 */
 	switch (vals[0]->v_type) {
 	case V_NUM:
@@ -10848,7 +10848,7 @@ f_versin(int count, VALUE **vals)
 	}
 
 	/*
-	 * compute trig function to a given error tolerance
+	 * compute versed trigonometric sine to a given error tolerance
 	 */
 	switch (vals[0]->v_type) {
 		case V_NUM:
@@ -10906,7 +10906,7 @@ f_aversin(int count, VALUE **vals)
 	}
 
 	/*
-	 * compute inverse trig function to a given error tolerance
+	 * compute inverse versed trigonometric sine to a given error tolerance
 	 */
 	arg1 = *vals[0];
 	if (arg1.v_type == V_NUM) {
@@ -10988,7 +10988,7 @@ f_coversin(int count, VALUE **vals)
 	}
 
 	/*
-	 * compute trig function to a given error tolerance
+	 * compute coversed trigonometric sine to a given error tolerance
 	 */
 	switch (vals[0]->v_type) {
 		case V_NUM:
@@ -11046,7 +11046,7 @@ f_acoversin(int count, VALUE **vals)
 	}
 
 	/*
-	 * compute inverse trig function to a given error tolerance
+	 * compute inverse coversed trigonometric sine to a given error tolerance
 	 */
 	arg1 = *vals[0];
 	if (arg1.v_type == V_NUM) {
@@ -11128,7 +11128,7 @@ f_vercos(int count, VALUE **vals)
 	}
 
 	/*
-	 * compute trig function to a given error tolerance
+	 * compute versed trigonometric cosine to a given error tolerance
 	 */
 	switch (vals[0]->v_type) {
 		case V_NUM:
@@ -11186,7 +11186,7 @@ f_avercos(int count, VALUE **vals)
 	}
 
 	/*
-	 * compute inverse trig function to a given error tolerance
+	 * compute inverse versed trigonometric cosine to a given error tolerance
 	 */
 	arg1 = *vals[0];
 	if (arg1.v_type == V_NUM) {
@@ -11268,7 +11268,7 @@ f_covercos(int count, VALUE **vals)
 	}
 
 	/*
-	 * compute trig function to a given error tolerance
+	 * compute coversed trigonometric cosine to a given error tolerance
 	 */
 	switch (vals[0]->v_type) {
 		case V_NUM:
@@ -11326,7 +11326,7 @@ f_acovercos(int count, VALUE **vals)
 	}
 
 	/*
-	 * compute inverse trig function to a given error tolerance
+	 * compute inverse coversed trigonometric cosine to a given error tolerance
 	 */
 	arg1 = *vals[0];
 	if (arg1.v_type == V_NUM) {
@@ -11408,7 +11408,7 @@ f_haversin(int count, VALUE **vals)
 	}
 
 	/*
-	 * compute trig function to a given error tolerance
+	 * compute half versed trigonometric sine to a given error tolerance
 	 */
 	switch (vals[0]->v_type) {
 		case V_NUM:
@@ -11466,7 +11466,7 @@ f_ahaversin(int count, VALUE **vals)
 	}
 
 	/*
-	 * compute inverse trig function to a given error tolerance
+	 * compute inverse half versed trigonometric sine to a given error tolerance
 	 */
 	arg1 = *vals[0];
 	if (arg1.v_type == V_NUM) {
@@ -11548,7 +11548,7 @@ f_hacoversin(int count, VALUE **vals)
 	}
 
 	/*
-	 * compute trig function to a given error tolerance
+	 * compute half coversed trigonometric sine to a given error tolerance
 	 */
 	switch (vals[0]->v_type) {
 		case V_NUM:
@@ -11606,7 +11606,7 @@ f_ahacoversin(int count, VALUE **vals)
 	}
 
 	/*
-	 * compute inverse trig function to a given error tolerance
+	 * compute inverse half coversed trigonometric sine to a given error tolerance
 	 */
 	arg1 = *vals[0];
 	if (arg1.v_type == V_NUM) {
@@ -11688,7 +11688,7 @@ f_havercos(int count, VALUE **vals)
 	}
 
 	/*
-	 * compute trig function to a given error tolerance
+	 * compute half versed trigonometric cosine to a given error tolerance
 	 */
 	switch (vals[0]->v_type) {
 		case V_NUM:
@@ -11746,7 +11746,7 @@ f_ahavercos(int count, VALUE **vals)
 	}
 
 	/*
-	 * compute inverse trig function to a given error tolerance
+	 * compute inverse half versed trigonometric cosine to a given error tolerance
 	 */
 	arg1 = *vals[0];
 	if (arg1.v_type == V_NUM) {
@@ -11828,7 +11828,7 @@ f_hacovercos(int count, VALUE **vals)
 	}
 
 	/*
-	 * compute trig function to a given error tolerance
+	 * compute half coversed trigonometric cosine to a given error tolerance
 	 */
 	switch (vals[0]->v_type) {
 		case V_NUM:
@@ -11886,7 +11886,7 @@ f_ahacovercos(int count, VALUE **vals)
 	}
 
 	/*
-	 * compute inverse trig function to a given error tolerance
+	 * compute inverse half coversed trigonometric cosine to a given error tolerance
 	 */
 	arg1 = *vals[0];
 	if (arg1.v_type == V_NUM) {
@@ -12022,7 +12022,7 @@ f_aexsec(int count, VALUE **vals)
 	}
 
 	/*
-	 * compute inverse trig function to a given error tolerance
+	 * compute inverse exterior trigonometric secant to a given error tolerance
 	 */
 	arg1 = *vals[0];
 	if (arg1.v_type == V_NUM) {
@@ -12104,7 +12104,7 @@ f_excsc(int count, VALUE **vals)
 	}
 
 	/*
-	 * compute cosecant to a given error tolerance
+	 * compute exterior trigonometric cosecant to a given error tolerance
 	 */
 	switch (vals[0]->v_type) {
 	case V_NUM:
@@ -12164,7 +12164,7 @@ f_aexcsc(int count, VALUE **vals)
 	}
 
 	/*
-	 * compute inverse trig function to a given error tolerance
+	 * compute inverse exterior trigonometric cosecant to a given error tolerance
 	 */
 	arg1 = *vals[0];
 	if (arg1.v_type == V_NUM) {
@@ -12220,7 +12220,7 @@ f_aexcsc(int count, VALUE **vals)
 
 
 /*
- * f_crd - exterior trigonometric cosecant
+ * f_crd - trigonometric chord of a unit circle
  */
 S_FUNC VALUE
 f_crd(int count, VALUE **vals)
@@ -12246,7 +12246,7 @@ f_crd(int count, VALUE **vals)
 	}
 
 	/*
-	 * compute cosecant to a given error tolerance
+	 * compute chord of a unit circle to a given error tolerance
 	 */
 	switch (vals[0]->v_type) {
 	case V_NUM:
@@ -12273,7 +12273,7 @@ f_crd(int count, VALUE **vals)
 
 
 /*
- * f_acrd - exterior trigonometric cosecant
+ * f_acrd - inverse trigonometric chord of a unit circle
  */
 S_FUNC VALUE
 f_acrd(int count, VALUE **vals)
@@ -12300,7 +12300,7 @@ f_acrd(int count, VALUE **vals)
 	}
 
 	/*
-	 * compute inverse trig function to a given error tolerance
+	 * compute inverse trigonometric chord of a unit circle to a given error tolerance
 	 */
 	arg1 = *vals[0];
 	if (arg1.v_type == V_NUM) {
@@ -12355,6 +12355,137 @@ f_acrd(int count, VALUE **vals)
 }
 
 
+/*
+ * f_cas - trigonometric chord of a unit circle
+ */
+S_FUNC VALUE
+f_cas(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_CAS_1);
+		}
+		err = vals[1]->v_num;
+	}
+
+	/*
+	 * compute chord of a unit circle to a given error tolerance
+	 */
+	switch (vals[0]->v_type) {
+	case V_NUM:
+		result.v_num = qcas(vals[0]->v_num, err);
+		result.v_type = V_NUM;
+		break;
+	case V_COM:
+		c = c_cas(vals[0]->v_com, err);
+		if (c == NULL) {
+			return error_value(E_CAS_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_CAS_2);
+	}
+	return result;
+}
+
+
+/*
+ * f_cis - Euler's formula
+ */
+S_FUNC VALUE
+f_cis(int count, VALUE **vals)
+{
+	VALUE result;
+	NUMBER *eps;
+	COMPLEX *c;
+	COMPLEX *ctmp;
+
+	/* 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_CIS_1);
+		}
+		eps = vals[1]->v_num;
+	}
+
+	/*
+	 * compute Euler's formula to a given error tolerance
+	 */
+	switch (vals[0]->v_type) {
+	case V_NUM:
+		/*
+		 * convert arg to COMPLEX
+		 */
+		ctmp = q_to_c(vals[0]->v_num);
+
+		/*
+		 * compute cis of argument
+		 */
+		c = c_cis(ctmp, eps);
+		comfree(ctmp);
+		if (c == NULL) {
+			return error_value(E_CIS_3);
+		}
+		break;
+	case V_COM:
+
+		/*
+		 * compute cis of argument
+		 */
+		c = c_cis(vals[0]->v_com, eps);
+		if (c == NULL) {
+			return error_value(E_CIS_3);
+		}
+		break;
+	default:
+		return error_value(E_CIS_2);
+	}
+
+	/*
+	 * case: return NUMBER value
+	 */
+	if (cisreal(c)) {
+		result.v_num = c_to_q(c, true);
+		result.v_type = V_NUM;
+		return result;
+	}
+
+	/*
+	 * case: return COMPLEX value
+	 */
+	result.v_com = c;
+	result.v_type = V_COM;
+	return result;
+}
+
+
 #endif /* !FUNCLIST */
 
 
@@ -12487,6 +12618,8 @@ STATIC CONST struct builtin builtins[] = {
 	 "current CALCPATH search path value"},
 	{"calc_tty", 0, 0, 0, OP_NOP, {.null = NULL}, {.valfunc_0 = f_calc_tty},
 	 "set tty for interactivity"},
+	{"cas", 1, 2, 0, OP_NOP, {.null = NULL}, {.valfunc_cnt = f_cas},
+	 "cosine plus sine, within accuracy b"},
 	{"catalan", 1, 1, 0, OP_NOP, {.null = NULL}, {.valfunc_1 = f_catalan},
 	 "catalan number for index a"},
 	{"ceil", 1, 1, 0, OP_NOP, {.null = NULL}, {.valfunc_1 = f_ceil},
@@ -12498,6 +12631,8 @@ STATIC CONST struct builtin builtins[] = {
 	 "simplify number using continued fractions"},
 	{"char", 1, 1, 0, OP_NOP, {.null = NULL}, {.valfunc_1 = f_char},
 	 "character corresponding to integer value"},
+	{"cis", 1, 2, 0, OP_NOP, {.null = NULL}, {.valfunc_cnt = f_cis},
+	 "Euler's formula, within accuracy b"},
 	{"cmdbuf", 0, 0, 0, OP_NOP, {.null = NULL}, {.valfunc_0 = f_cmdbuf},
 	 "command buffer"},
 	{"cmp", 2, 2, 0, OP_CMP, {.null = NULL}, {.null = NULL},

help/Makefile

@@ -204,9 +204,9 @@ DETAIL_HELP= abs access acos acosh acot acoth acovercos acoversin \
 	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 \
+	cas catalan ceil cfappr cfsim char cis cmdbuf cmp comb conj cos cosh \
-	count covercos coversin cp crd csc csch ctime d2dm d2dms d2g d2r \
+	cot coth count covercos coversin cp crd csc csch ctime d2dm d2dms d2g \
-	delete den dereference det digit digits display dms2d dp epsilon \
+	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 \

help/acos

@@ -55,6 +55,7 @@ SEE ALSO
     ahaversin, hacoversin, havercos, ahacovercos
     exsec, aexsec, excsc, aexcsc
     crd, acrd
+    cas, cis
     epsilon
 
 EXTERNAL RESOURCES

help/acot

@@ -55,6 +55,7 @@ SEE ALSO
     ahaversin, hacoversin, havercos, ahacovercos
     exsec, aexsec, excsc, aexcsc
     crd, acrd
+    cas, cis
     epsilon
 
 EXTERNAL RESOURCES

help/acovercos

@@ -54,6 +54,7 @@ SEE ALSO
     ahaversin, hacoversin, havercos, ahacovercos
     exsec, aexsec, excsc, aexcsc
     crd, acrd
+    cas, cis
     epsilon
 
 EXTERNAL RESOURCES

help/acoversin

@@ -54,6 +54,7 @@ SEE ALSO
     ahaversin, hacoversin, havercos, ahacovercos
     exsec, aexsec, excsc, aexcsc
     crd, acrd
+    cas, cis
     epsilon
 
 EXTERNAL RESOURCES

help/acrd

@@ -51,6 +51,7 @@ SEE ALSO
     ahaversin, hacoversin, havercos, ahacovercos
     exsec, aexsec, excsc, aexcsc
     crd
+    cas, cis
     epsilon
 
 EXTERNAL RESOURCES

help/acsc

@@ -58,6 +58,7 @@ SEE ALSO
     ahaversin, hacoversin, havercos, ahacovercos
     exsec, aexsec, excsc, aexcsc
     crd, acrd
+    cas, cis
     epsilon
 
 EXTERNAL RESOURCES

help/aexcsc

@@ -60,6 +60,7 @@ SEE ALSO
     ahaversin, hacoversin, havercos, ahacovercos
     exsec, aexsec, excsc
     crd, acrd
+    cas, cis
     epsilon
 
 EXTERNAL RESOURCES

help/aexsec

@@ -57,6 +57,7 @@ SEE ALSO
     ahaversin, hacoversin, havercos, ahacovercos
     exsec, excsc, aexcsc
     crd, acrd
+    cas, cis
     epsilon
 
 EXTERNAL RESOURCES

help/ahacovercos

@@ -57,6 +57,7 @@ SEE ALSO
     ahaversin, hacoversin, havercos
     exsec, aexsec, excsc, aexcsc
     crd, acrd
+    cas, cis
     epsilon
 
 EXTERNAL RESOURCES

help/ahacoversin

@@ -57,6 +57,7 @@ SEE ALSO
     ahaversin, havercos, ahacovercos
     exsec, aexsec, excsc, aexcsc
     crd, acrd
+    cas, cis
     epsilon
 
 EXTERNAL RESOURCES

help/ahavercos

@@ -58,6 +58,7 @@ SEE ALSO
     ahaversin, hacoversin, ahacovercos
     exsec, aexsec, excsc, aexcsc
     crd, acrd
+    cas, cis
     epsilon
 
 EXTERNAL RESOURCES

help/ahaversin

@@ -58,6 +58,7 @@ SEE ALSO
     hacoversin, havercos, ahacovercos
     exsec, aexsec, excsc, aexcsc
     crd, acrd
+    cas, cis
     epsilon
 
 EXTERNAL RESOURCES

help/asec

@@ -55,6 +55,7 @@ SEE ALSO
     ahaversin, hacoversin, havercos, ahacovercos
     exsec, aexsec, excsc, aexcsc
     crd, acrd
+    cas, cis
     epsilon
 
 EXTERNAL RESOURCES

help/asin

@@ -55,6 +55,7 @@ SEE ALSO
     ahaversin, hacoversin, havercos, ahacovercos
     exsec, aexsec, excsc, aexcsc
     crd, acrd
+    cas, cis
     epsilon
 
 EXTERNAL RESOURCES

help/atan

@@ -55,6 +55,7 @@ SEE ALSO
     ahaversin, hacoversin, havercos, ahacovercos
     exsec, aexsec, excsc, aexcsc
     crd, acrd
+    cas, cis
     epsilon
 
 EXTERNAL RESOURCES

help/avercos

@@ -55,6 +55,7 @@ SEE ALSO
     ahaversin, hacoversin, havercos, ahacovercos
     exsec, aexsec, excsc, aexcsc
     crd, acrd
+    cas, cis
     epsilon
 
 EXTERNAL RESOURCES

help/aversin

@@ -55,6 +55,7 @@ SEE ALSO
     ahaversin, hacoversin, havercos, ahacovercos
     exsec, aexsec, excsc, aexcsc
     crd, acrd
+    cas, cis
     epsilon
 
 EXTERNAL RESOURCES

help/cas

@@ -0,0 +1,86 @@
+NAME
+    cas - trigonometric cosine plus sine
+
+SYNOPSIS
+    cas(x [,eps])
+
+TYPES
+    x		number (real or complex)
+    eps		0 < real < 1, defaults to epsilon()
+
+    return	number
+
+DESCRIPTION
+    Calculate the trigonometric cosine plus sine of x to a multiple of eps with error less in
+    absolute value than .75 * eps.
+
+    This function is sometimes called cosine and sine, is equivalent to:
+
+	cas(x) = cos(x) + sin(x)
+
+EXAMPLE
+    ; print cas(1/2), cas(5/7), cas(42/7)
+    1.35700810049457571639 1.41063924387921517267 0.68075478845144014774
+
+    ; print cas(1, 1e-5), cas(1, 1e-10), cas(1, 1e-15), cas(1, 1e-20)
+    1.38177 1.3817732907 1.381773290676036 1.38177329067603622405
+
+    ; print cas(2 + 3i, 1e-5), cas(2 + 3i, 1e-10)
+    4.96487-13.27814i 4.9648734559-13.2781348538i
+
+    ; pi = pi(1e-20)
+    ; print cas(pi/6), cas(pi/2), cas(pi)
+    1.36602540378443864676 1 -1
+
+LIMITS
+    0 < eps < 1
+
+LINK LIBRARY
+    NUMBER *qcas(NUMBER *x, NUMBER *eps)
+    COMPLEX *c_cas(COMPLEX *x, NUMBER *eps)
+
+SEE ALSO
+    cos, tan, cot, sec, csc
+    asin, acos, atan, acot, asec, acsc
+    versin, coversin, vercos, covercos
+    aversin, acoversin, avercos, acovercos
+    haversin, hacoversin, havercos, hacovercos
+    ahaversin, hacoversin, havercos, ahacovercos
+    exsec, aexsec, excsc, aexcsc
+    crd, acrd
+    cis
+    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 23:11:18
+## 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/cis

@@ -0,0 +1,92 @@
+NAME
+    cis - Euler's formula
+
+SYNOPSIS
+    cis(x [,eps])
+
+TYPES
+    x		number (real or complex)
+    eps		0 < real < 1, defaults to epsilon()
+
+    return	number
+
+DESCRIPTION
+    Calculate the Euler's formula value of x to a multiple of eps with error less in
+    absolute value than .75 * eps.
+
+    This function is sometimes called cosine and sine, is equivalent to:
+
+	cis(x) = cos(x) + 1i*sin(x)
+	cis(x) = exp(1i * x)
+
+EXAMPLE
+    ; print cis(1/2), cis(5/7)
+    0.87758256189037271612+0.47942553860420300027i 0.75556134670069659847+0.6550778971785185742i
+
+    ; print cis(42/7)
+    0.96017028665036602055-0.27941549819892587281i
+
+    ; print cis(1, 1e-5), cis(1, 1e-10), cis(1, 1e-15)
+    0.5403+0.84147i 0.5403023059+0.8414709848i 0.54030230586814+0.841470984807896i
+
+    ; print cis(1, 1e-20)
+    0.5403023058681397174+0.84147098480789650665i
+
+    ; print cis(2 + 3i, 1e-5), cis(2 + 3i, 1e-10)
+    -0.02072+0.04527i -0.020718731+0.0452712531i
+
+    ; pi = pi(1e-20)
+    ; print cis(pi/6), cis(pi/2), cis(pi)
+    0.86602540378443864676+0.5i 1i -1
+
+LIMITS
+    0 < eps < 1
+
+LINK LIBRARY
+    COMPLEX *c_cis(COMPLEX *x, NUMBER *eps)
+
+SEE ALSO
+    cos, tan, cot, sec, csc
+    asin, acos, atan, acot, asec, acsc
+    versin, coversin, vercos, covercos
+    aversin, acoversin, avercos, acovercos
+    haversin, hacoversin, havercos, hacovercos
+    ahaversin, hacoversin, havercos, ahacovercos
+    exsec, aexsec, excsc, aexcsc
+    crd, acrd
+    cis
+    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 23:11:18
+## 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/cos

@@ -44,6 +44,7 @@ SEE ALSO
     ahaversin, hacoversin, havercos, ahacovercos
     exsec, aexsec, excsc, aexcsc
     crd, acrd
+    cas, cis
     epsilon
 
 EXTERNAL RESOURCES

help/cot

@@ -50,6 +50,7 @@ SEE ALSO
     ahaversin, hacoversin, havercos, ahacovercos
     exsec, aexsec, excsc, aexcsc
     crd, acrd
+    cas, cis
     epsilon
 
 EXTERNAL RESOURCES

help/covercos

@@ -48,6 +48,7 @@ SEE ALSO
     ahaversin, hacoversin, havercos, ahacovercos
     exsec, aexsec, excsc, aexcsc
     crd, acrd
+    cas, cis
     epsilon
 
 EXTERNAL RESOURCES

help/coversin

@@ -48,6 +48,7 @@ SEE ALSO
     ahaversin, hacoversin, havercos, ahacovercos
     exsec, aexsec, excsc, aexcsc
     crd, acrd
+    cas, cis
     epsilon
 
 EXTERNAL RESOURCES

help/crd

@@ -51,6 +51,7 @@ SEE ALSO
     ahaversin, hacoversin, havercos, ahacovercos
     exsec, aexsec, excsc, aexcsc
     acrd
+    cas, cis
     epsilon
 
 EXTERNAL RESOURCES

help/csc

@@ -50,6 +50,7 @@ SEE ALSO
     ahaversin, hacoversin, havercos, ahacovercos
     exsec, aexsec, excsc, aexcsc
     crd, acrd
+    cas, cis
     epsilon
 
 EXTERNAL RESOURCES

help/excsc

@@ -50,6 +50,7 @@ SEE ALSO
     ahaversin, hacoversin, havercos, ahacovercos
     exsec, aexsec, aexcsc
     crd, acrd
+    cas, cis
     epsilon
 
 EXTERNAL RESOURCES

help/exp

@@ -39,7 +39,7 @@ LINK LIBRARY
     COMPLEX *c_exp(COMPLEX *x, NUMBER *eps)
 
 SEE ALSO
-    ln, cosh, sinh, tanh
+    ln, log, log2, logn, cosh, sinh, tanh
 
 ## Copyright (C) 1999,2021,2023  Landon Curt Noll
 ##

help/exsec

@@ -50,6 +50,7 @@ SEE ALSO
     ahaversin, hacoversin, havercos, ahacovercos
     aexsec, excsc, aexcsc
     crd, acrd
+    cas, cis
     epsilon
 
 EXTERNAL RESOURCES

help/hacovercos

@@ -48,6 +48,7 @@ SEE ALSO
     ahaversin, hacoversin, havercos, ahacovercos
     exsec, aexsec, excsc, aexcsc
     crd, acrd
+    cas, cis
     epsilon
 
 EXTERNAL RESOURCES

help/hacoversin

@@ -48,6 +48,7 @@ SEE ALSO
     ahaversin, hacoversin, havercos, ahacovercos
     exsec, aexsec, excsc, aexcsc
     crd, acrd
+    cas, cis
     epsilon
 
 EXTERNAL RESOURCES

help/havercos

@@ -48,6 +48,7 @@ SEE ALSO
     ahaversin, hacoversin, havercos, ahacovercos
     exsec, aexsec, excsc, aexcsc
     crd, acrd
+    cas, cis
     epsilon
 
 EXTERNAL RESOURCES

help/haversin

@@ -48,6 +48,7 @@ SEE ALSO
     ahaversin, hacoversin, havercos, ahacovercos
     exsec, aexsec, excsc, aexcsc
     crd, acrd
+    cas, cis
     epsilon
 
 EXTERNAL RESOURCES

help/sec

@@ -50,6 +50,7 @@ SEE ALSO
     ahaversin, hacoversin, havercos, ahacovercos
     exsec, aexsec, excsc, aexcsc
     crd, acrd
+    cas, cis
     epsilon
 
 EXTERNAL RESOURCES

help/sin

@@ -44,6 +44,7 @@ SEE ALSO
     ahaversin, hacoversin, havercos, ahacovercos
     exsec, aexsec, excsc, aexcsc
     crd, acrd
+    cas, cis
     epsilon
 
 EXTERNAL RESOURCES

help/tan

@@ -50,6 +50,7 @@ SEE ALSO
     ahaversin, hacoversin, havercos, ahacovercos
     exsec, aexsec, excsc, aexcsc
     crd, acrd
+    cas, cis
     epsilon
 
 EXTERNAL RESOURCES

help/vercos

@@ -48,6 +48,7 @@ SEE ALSO
     ahaversin, hacoversin, havercos, ahacovercos
     exsec, aexsec, excsc, aexcsc
     crd, acrd
+    cas, cis
     epsilon
 
 EXTERNAL RESOURCES

help/versin

@@ -48,6 +48,7 @@ SEE ALSO
     ahaversin, hacoversin, havercos, ahacovercos
     exsec, aexsec, excsc, aexcsc
     crd, acrd
+    cas, cis
     epsilon
 
 EXTERNAL RESOURCES

qmath.h

@@ -259,13 +259,13 @@ 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);
+E_FUNC NUMBER *qcas(NUMBER *q, NUMBER *epsilon);
 
 /*
  * pseudo-seed generator
  */
 E_FUNC NUMBER *pseudo_seed(void);
 
-
 /*
  * external swap functions
  */

qtrans.c

@@ -202,6 +202,7 @@ qsincos(NUMBER *q, long bitnum, NUMBER **vs, NUMBER **vc)
 	return;
 }
 
+
 /*
  * Calculate the cosine of a number to a near multiple of epsilon.
  * This calls qsincos() and discards the value of sin.
@@ -3623,3 +3624,69 @@ qacrd(NUMBER *q, NUMBER *epsilon)
 	 */
 	return res;
 }
+
+
+/*
+ * qcas - trigonometric chord of a unit circle
+ *
+ * This uses the formula:
+ *
+ *	cas(x) = cos(x) + 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 *
+qcas(NUMBER *q, NUMBER *epsilon)
+{
+	NUMBER *sin;	/* sin(x) */
+	NUMBER *tsin;	/* sin(x) rounded to nearest epsilon multiple */
+	NUMBER *cos;	/* cos(x) */
+	NUMBER *tcos;	/* cos(x) rounded to nearest epsilon multiple */
+	NUMBER *res;
+	long n;
+
+	/*
+	 * firewall
+	 */
+	if (qiszero(epsilon)) {
+		math_error("Zero epsilon value for cosine");
+		not_reached();
+	}
+
+	/*
+	 * case 0: quick return 1
+	 */
+	if (qiszero(q)) {
+		return qlink(&_qone_);
+	}
+
+	/*
+	 * case epsilon > 1: quick return 0
+	 */
+	n = -qilog2(epsilon);
+	if (n < 0) {
+		return qlink(&_qzero_);
+	}
+
+	/*
+	 * compute cosine and sine
+	 */
+	qsincos(q, n + 2, &sin, &cos);
+	tcos = qmappr(cos, epsilon, 24);
+	qfree(cos);
+	tsin = qmappr(sin, epsilon, 24);
+	qfree(sin);
+	res = qqadd(tcos, tsin);
+	qfree(tcos);
+	qfree(tsin);
+
+	/*
+	 * return trigonometric result
+	 */
+	return res;
+}