use new math routines

lang/basic/lib/atn.c

@@ -14,90 +14,55 @@ double
 _atn(x)
 	double x;
 {
-	/*	The interval [0, infinity) is treated as follows:
-		Define partition points Xi
-			X0 = 0
-			X1 = tan(pi/16)
-			X2 = tan(3pi/16)
-			X3 = tan(5pi/16)
-			X4 = tan(7pi/16)
-			X5 = infinity
-		and evaluation nodes xi
-			x2 = tan(2pi/16)
-			x3 = tan(4pi/16)
-			x4 = tan(6pi/16)
-			x5 = infinity
-		An argument x in [Xn-1, Xn] is now reduced to an argument
-		t in [-X1, X1] by the following formulas:
-			
-			t = 1/xn - (1/(xn*xn) + 1)/((1/xn) + x)
-
-			arctan(x) = arctan(xi) + arctan(t)
-
-		For the interval [0, p/16] an approximation is used:
-			arctan(x) = x * P(x*x)/Q(x*x)
+	/*	Algorithm and coefficients from:
+			"Software manual for the elementary functions"
+			by W.J. Cody and W. Waite, Prentice-Hall, 1980
 	*/
-	static struct precomputed {
-		double X;		/* partition point */
-		double arctan;		/* arctan of evaluation node */
-		double one_o_x;		/* 1 / xn */
-		double one_o_xsq_p_1;	/* 1 / (xn*xn) + 1 */
-	} prec[5] = {
-		{ 0.19891236737965800691159762264467622,
-		  0.0,
-		  0.0,		/* these don't matter */
-		  0.0 } ,
-		{ 0.66817863791929891999775768652308076, /* tan(3pi/16)	*/
-		  M_PI_8,
-		  2.41421356237309504880168872420969808,
-		  6.82842712474619009760337744841939616 },
-		{ 1.49660576266548901760113513494247691, /* tan(5pi/16) */
-		  M_PI_4,
-		  1.0,
-		  2.0 },
-		{ 5.02733949212584810451497507106407238, /* tan(7pi/16) */
-		  M_3PI_8,
-		  0.41421356237309504880168872420969808,
-		  1.17157287525380998659662255158060384 },
-		{ MAXDOUBLE,
-		  M_PI_2,
-		  0.0,
-		  1.0 }};
 
-	/*	Hart & Cheney # 5037 */
-
-	static double p[5] = {
-		0.7698297257888171026986294745e+03,
-		0.1557282793158363491416585283e+04,
-		0.1033384651675161628243434662e+04,
-		0.2485841954911840502660889866e+03,
-		0.1566564964979791769948970100e+02
+	static double p[] = {
+		-0.13688768894191926929e+2,
+		-0.20505855195861651981e+2,
+		-0.84946240351320683534e+1,
+		-0.83758299368150059274e+0
+	};
+	static double q[] = {
+		 0.41066306682575781263e+2,
+		 0.86157349597130242515e+2,
+		 0.59578436142597344465e+2,
+		 0.15024001160028576121e+2,
+		 1.0
+	};
+	static double a[] = {
+		0.0,
+		0.52359877559829887307710723554658381,	/* pi/6 */
+		M_PI_2,
+		1.04719755119659774615421446109316763	/* pi/3 */
 	};
 
-	static double q[6] = {
-		0.7698297257888171026986294911e+03,
-		0.1813892701754635858982709369e+04,
-		0.1484049607102276827437401170e+04,
-		0.4904645326203706217748848797e+03,
-		0.5593479839280348664778328000e+02,
-		0.1000000000000000000000000000e+01
-	};
+	int	neg = x < 0;
+	int	n;
+	double	g;
 
-	int negative = x < 0.0;
-	register struct precomputed *pr = prec;
-
-	if (negative) {
+	if (neg) {
 		x = -x;
 	}
-	while (x > pr->X) pr++;
-	if (pr != prec) {
-		x = pr->arctan +
-			_atn(pr->one_o_x - pr->one_o_xsq_p_1/(pr->one_o_x + x));
+	if (x > 1.0) {
+		x = 1.0/x;
+		n = 2;
 	}
-	else {
-		double xsq = x*x;
+	else	n = 0;
 
-		x = x * POLYNOM4(xsq, p)/POLYNOM5(xsq, q);
+	if (x > 0.26794919243112270647) {	/* 2-sqtr(3) */
+		n = n + 1;
+		x = (((0.73205080756887729353*x-0.5)-0.5)+x)/
+			(1.73205080756887729353+x);
 	}
-	return negative ? -x : x;
+
+	/* ??? avoid underflow ??? */
+
+	g = x * x;
+	x += x * g * POLYNOM3(g, p) / POLYNOM4(g, q);
+	if (n > 1) x = -x;
+	x += a[n];
+	return neg ? -x : x;
 }