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@

diff --git a/lib/libm/src/k_tan.c b/lib/libm/src/k_tan.c
index 91146c2..33d63a1 100644 (file)
--- 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);
        }
 }