use more accurate algorithm for tan. from fdlibm 5.3:

>     2. k_tan.c error was > 1 ulp target for FDLIBM
>         5.2: Worst error at least 1.45 ulp at
>         tan(1.7765241907548024E+269) = 1.7733884462610958E+16
>         5.3: Worst error 0.96 ulp
ok millert@

1 files changed, 101 insertions, 69 deletions

diff --git a/lib/libm/src/k_tan.c b/lib/libm/src/k_tan.c
index 91146c2f767..33d63a1024d 100644
--- a/lib/libm/src/k_tan.c
+++ b/lib/libm/src/k_tan.c
@@ -5,7 +5,7 @@
  *
  * Developed at SunPro, a Sun Microsystems, Inc. business.
  * Permission to use, copy, modify, and distribute this
- * software is freely granted, provided that this notice 
+ * software is freely granted, provided that this notice
  * is preserved.
  * ====================================================
  */
@@ -18,25 +18,25 @@ static char rcsid[] = "$NetBSD: k_tan.c,v 1.8 1995/05/10 20:46:37 jtc Exp $";
  * kernel tan function on [-pi/4, pi/4], pi/4 ~ 0.7854
  * Input x is assumed to be bounded by ~pi/4 in magnitude.
  * Input y is the tail of x.
- * Input k indicates whether tan (if k=1) or 
+ * Input k indicates whether tan (if k=1) or
  * -1/tan (if k= -1) is returned.
  *
  * Algorithm
- *	1. Since tan(-x) = -tan(x), we need only to consider positive x. 
+ *	1. Since tan(-x) = -tan(x), we need only to consider positive x.
  *	2. if x < 2^-28 (hx<0x3e300000 0), return x with inexact if x!=0.
  *	3. tan(x) is approximated by a odd polynomial of degree 27 on
  *	   [0,0.67434]
  *		  	         3             27
  *	   	tan(x) ~ x + T1*x + ... + T13*x
  *	   where
- *	
+ *
  * 	        |tan(x)         2     4            26   |     -59.2
  * 	        |----- - (1+T1*x +T2*x +.... +T13*x    )| <= 2
- * 	        |  x 					| 
- * 
+ * 	        |  x 					|
+ *
  *	   Note: tan(x+y) = tan(x) + tan'(x)*y
  *		          ~ tan(x) + (1+x*x)*y
- *	   Therefore, for better accuracy in computing tan(x+y), let 
+ *	   Therefore, for better accuracy in computing tan(x+y), let
  *		     3      2      2       2       2
  *		r = x *(T2+x *(T3+x *(...+x *(T12+x *T13))))
  *	   then
@@ -50,74 +50,106 @@ static char rcsid[] = "$NetBSD: k_tan.c,v 1.8 1995/05/10 20:46:37 jtc Exp $";
 
 #include "math.h"
 #include "math_private.h"
-static const double 
-one   =  1.00000000000000000000e+00, /* 0x3FF00000, 0x00000000 */
-pio4  =  7.85398163397448278999e-01, /* 0x3FE921FB, 0x54442D18 */
-pio4lo=  3.06161699786838301793e-17, /* 0x3C81A626, 0x33145C07 */
-T[] =  {
-  3.33333333333334091986e-01, /* 0x3FD55555, 0x55555563 */
-  1.33333333333201242699e-01, /* 0x3FC11111, 0x1110FE7A */
-  5.39682539762260521377e-02, /* 0x3FABA1BA, 0x1BB341FE */
-  2.18694882948595424599e-02, /* 0x3F9664F4, 0x8406D637 */
-  8.86323982359930005737e-03, /* 0x3F8226E3, 0xE96E8493 */
-  3.59207910759131235356e-03, /* 0x3F6D6D22, 0xC9560328 */
-  1.45620945432529025516e-03, /* 0x3F57DBC8, 0xFEE08315 */
-  5.88041240820264096874e-04, /* 0x3F4344D8, 0xF2F26501 */
-  2.46463134818469906812e-04, /* 0x3F3026F7, 0x1A8D1068 */
-  7.81794442939557092300e-05, /* 0x3F147E88, 0xA03792A6 */
-  7.14072491382608190305e-05, /* 0x3F12B80F, 0x32F0A7E9 */
- -1.85586374855275456654e-05, /* 0xBEF375CB, 0xDB605373 */
-  2.59073051863633712884e-05, /* 0x3EFB2A70, 0x74BF7AD4 */
+
+static const double xxx[] = {
+		 3.33333333333334091986e-01,	/* 3FD55555, 55555563 */
+		 1.33333333333201242699e-01,	/* 3FC11111, 1110FE7A */
+		 5.39682539762260521377e-02,	/* 3FABA1BA, 1BB341FE */
+		 2.18694882948595424599e-02,	/* 3F9664F4, 8406D637 */
+		 8.86323982359930005737e-03,	/* 3F8226E3, E96E8493 */
+		 3.59207910759131235356e-03,	/* 3F6D6D22, C9560328 */
+		 1.45620945432529025516e-03,	/* 3F57DBC8, FEE08315 */
+		 5.88041240820264096874e-04,	/* 3F4344D8, F2F26501 */
+		 2.46463134818469906812e-04,	/* 3F3026F7, 1A8D1068 */
+		 7.81794442939557092300e-05,	/* 3F147E88, A03792A6 */
+		 7.14072491382608190305e-05,	/* 3F12B80F, 32F0A7E9 */
+		-1.85586374855275456654e-05,	/* BEF375CB, DB605373 */
+		 2.59073051863633712884e-05,	/* 3EFB2A70, 74BF7AD4 */
+/* one */	 1.00000000000000000000e+00,	/* 3FF00000, 00000000 */
+/* pio4 */	 7.85398163397448278999e-01,	/* 3FE921FB, 54442D18 */
+/* pio4lo */	 3.06161699786838301793e-17	/* 3C81A626, 33145C07 */
 };
+#define	one	xxx[13]
+#define	pio4	xxx[14]
+#define	pio4lo	xxx[15]
+#define	T	xxx
 
 double
 __kernel_tan(double x, double y, int iy)
 {
-	double z,r,v,w,s;
-	int32_t ix,hx;
-	GET_HIGH_WORD(hx,x);
-	ix = hx&0x7fffffff;	/* high word of |x| */
-	if(ix<0x3e300000)			/* x < 2**-28 */
-	    {if((int)x==0) {			/* generate inexact */
-	        u_int32_t low;
-		GET_LOW_WORD(low,x);
-		if(((ix|low)|(iy+1))==0) return one/fabs(x);
-		else return (iy==1)? x: -one/x;
-	    }
-	    }
-	if(ix>=0x3FE59428) { 			/* |x|>=0.6744 */
-	    if(hx<0) {x = -x; y = -y;}
-	    z = pio4-x;
-	    w = pio4lo-y;
-	    x = z+w; y = 0.0;
+	double z, r, v, w, s;
+	int32_t ix, hx;
+
+	GET_HIGH_WORD(hx, x);	/* high word of x */
+	ix = hx & 0x7fffffff;			/* high word of |x| */
+	if (ix < 0x3e300000) {			/* x < 2**-28 */
+		if ((int) x == 0) {		/* generate inexact */
+			u_int32_t low;
+			GET_LOW_WORD(low, x);
+			if(((ix | low) | (iy + 1)) == 0)
+				return one / fabs(x);
+			else {
+				if (iy == 1)
+					return x;
+				else {	/* compute -1 / (x+y) carefully */
+					double a, t;
+
+					z = w = x + y;
+					SET_LOW_WORD(z, 0);
+					v = y - (z - x);
+					t = a = -one / w;
+					SET_LOW_WORD(t, 0);
+					s = one + t * z;
+					return t + a * (s + t * v);
+				}
+			}
+		}
+	}
+	if (ix >= 0x3FE59428) {	/* |x| >= 0.6744 */
+		if (hx < 0) {
+			x = -x;
+			y = -y;
+		}
+		z = pio4 - x;
+		w = pio4lo - y;
+		x = z + w;
+		y = 0.0;
 	}
-	z	=  x*x;
-	w 	=  z*z;
-    /* Break x^5*(T[1]+x^2*T[2]+...) into
-     *	  x^5(T[1]+x^4*T[3]+...+x^20*T[11]) +
-     *	  x^5(x^2*(T[2]+x^4*T[4]+...+x^22*[T12]))
-     */
-	r = T[1]+w*(T[3]+w*(T[5]+w*(T[7]+w*(T[9]+w*T[11]))));
-	v = z*(T[2]+w*(T[4]+w*(T[6]+w*(T[8]+w*(T[10]+w*T[12])))));
-	s = z*x;
-	r = y + z*(s*(r+v)+y);
-	r += T[0]*s;
-	w = x+r;
-	if(ix>=0x3FE59428) {
-	    v = (double)iy;
-	    return (double)(1-((hx>>30)&2))*(v-2.0*(x-(w*w/(w+v)-r)));
+	z = x * x;
+	w = z * z;
+	/*
+	 * Break x^5*(T[1]+x^2*T[2]+...) into
+	 * x^5(T[1]+x^4*T[3]+...+x^20*T[11]) +
+	 * x^5(x^2*(T[2]+x^4*T[4]+...+x^22*[T12]))
+	 */
+	r = T[1] + w * (T[3] + w * (T[5] + w * (T[7] + w * (T[9] +
+		w * T[11]))));
+	v = z * (T[2] + w * (T[4] + w * (T[6] + w * (T[8] + w * (T[10] +
+		w * T[12])))));
+	s = z * x;
+	r = y + z * (s * (r + v) + y);
+	r += T[0] * s;
+	w = x + r;
+	if (ix >= 0x3FE59428) {
+		v = (double) iy;
+		return (double) (1 - ((hx >> 30) & 2)) *
+			(v - 2.0 * (x - (w * w / (w + v) - r)));
 	}
-	if(iy==1) return w;
-	else {		/* if allow error up to 2 ulp, 
-			   simply return -1.0/(x+r) here */
-     /*  compute -1.0/(x+r) accurately */
-	    double a,t;
-	    z  = w;
-	    SET_LOW_WORD(z,0);
-	    v  = r-(z - x); 	/* z+v = r+x */
-	    t = a  = -1.0/w;	/* a = -1.0/w */
-	    SET_LOW_WORD(t,0);
-	    s  = 1.0+t*z;
-	    return t+a*(s+t*v);
+	if (iy == 1)
+		return w;
+	else {
+		/*
+		 * if allow error up to 2 ulp, simply return
+		 * -1.0 / (x+r) here
+		 */
+		/* compute -1.0 / (x+r) accurately */
+		double a, t;
+		z = w;
+		SET_LOW_WORD(z, 0);
+		v = r - (z - x);	/* z+v = r+x */
+		t = a = -1.0 / w;	/* a = -1.0/w */
+		SET_LOW_WORD(t, 0);
+		s = 1.0 + t * z;
+		return t + a * (s + t * v);
 	}
 }