[builtins] Introduce proper Float64Tan operator.

src/base/ieee754.cc

 // The following is adapted from fdlibm (http://www.netlib.org/fdlibm).
 //
 // ====================================================
 // Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved.
 //
 // Developed at SunSoft, a Sun Microsystems, Inc. business.
 // Permission to use, copy, modify, and distribute this
 // software is freely granted, provided that this notice
 // is preserved.
 // ====================================================
@@ skipping 743 matching lines @@
   z = x * x;
   v = z * x;
   r = S2 + z * (S3 + z * (S4 + z * (S5 + z * S6)));
   if (iy == 0) {
     return x + v * (S1 + z * r);
   } else {
     return x - ((z * (half * y - v * r) - y) - v * S1);
   }
 }
 
+/* __kernel_tan( x, y, k )
+ * 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
+ * -1/tan (if k= -1) is returned.
+ *
+ * Algorithm
+ *      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                                    |
+ *
+ *         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
+ *                   3      2      2       2       2
+ *              r = x *(T2+x *(T3+x *(...+x *(T12+x *T13))))
+ *         then
+ *                                  3    2
+ *              tan(x+y) = x + (T1*x + (x *(r+y)+y))
+ *
+ *      4. For x in [0.67434,pi/4],  let y = pi/4 - x, then
+ *              tan(x) = tan(pi/4-y) = (1-tan(y))/(1+tan(y))
+ *                     = 1 - 2*(tan(y) - (tan(y)^2)/(1+tan(y)))
+ */
+double __kernel_tan(double x, double y, int iy) {
+  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 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 (static_cast<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 = iy;
+    return (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);
+  }
+
+#undef one
+#undef pio4
+#undef pio4lo
+#undef T
+}
+
 }  // namespace
 
 /* atan(x)
  * Method
  *   1. Reduce x to positive by atan(x) = -atan(-x).
  *   2. According to the integer k=4t+0.25 chopped, t=x, the argument
  *      is further reduced to one of the following intervals and the
  *      arctangent of t is evaluated by the corresponding formula:
  *
  *      [0,7/16]      atan(x) = t-t^3*(a1+t^2*(a2+...(a10+t^2*a11)...)
@@ skipping 1342 matching lines @@
       case 1:
         return __kernel_cos(y[0], y[1]);
       case 2:
         return -__kernel_sin(y[0], y[1], 1);
       default:
         return -__kernel_cos(y[0], y[1]);
     }
   }
 }
 
+/* tan(x)
+ * Return tangent function of x.
+ *
+ * kernel function:
+ *      __kernel_tan            ... tangent function on [-pi/4,pi/4]
+ *      __ieee754_rem_pio2      ... argument reduction routine
+ *
+ * Method.
+ *      Let S,C and T denote the sin, cos and tan respectively on
+ *      [-PI/4, +PI/4]. Reduce the argument x to y1+y2 = x-k*pi/2
+ *      in [-pi/4 , +pi/4], and let n = k mod 4.
+ *      We have
+ *
+ *          n        sin(x)      cos(x)        tan(x)
+ *     ----------------------------------------------------------
+ *          0          S           C             T
+ *          1          C          -S            -1/T
+ *          2         -S          -C             T
+ *          3         -C           S            -1/T
+ *     ----------------------------------------------------------
+ *
+ * Special cases:
+ *      Let trig be any of sin, cos, or tan.
+ *      trig(+-INF)  is NaN, with signals;
+ *      trig(NaN)    is that NaN;
+ *
+ * Accuracy:
+ *      TRIG(x) returns trig(x) nearly rounded
+ */
+double tan(double x) {
+  double y[2], z = 0.0;
+  int32_t n, ix;
+
+  /* High word of x. */
+  GET_HIGH_WORD(ix, x);
+
+  /* |x| ~< pi/4 */
+  ix &= 0x7fffffff;
+  if (ix <= 0x3fe921fb) {
+    return __kernel_tan(x, z, 1);
+  } else if (ix >= 0x7ff00000) {
+    /* tan(Inf or NaN) is NaN */
+    return x - x; /* NaN */
+  } else {
+    /* argument reduction needed */
+    n = __ieee754_rem_pio2(x, y);
+    /* 1 -> n even, -1 -> n odd */
+    return __kernel_tan(y[0], y[1], 1 - ((n & 1) << 1));
+  }
+}
+
 }  // namespace ieee754
 }  // namespace base
 }  // namespace v8