[builtins] Introduce proper Float64Atan and Float64Atan2 operators.

src/base/ieee754.cc

Index: src/base/ieee754.cc
diff --git a/src/base/ieee754.cc b/src/base/ieee754.cc
index cd074ee2c9ee1914f2ac10ad079f02457c90e4c4..2c20ce035206fb9a7395585c1375d76ea6eb8102 100644
--- a/src/base/ieee754.cc
+++ b/src/base/ieee754.cc
@@ -15,6 +15,7 @@
 
 #include "src/base/ieee754.h"
 
+#include <cmath>
 #include <limits>
 
 #include "src/base/build_config.h"
@@ -169,6 +170,228 @@ typedef union {
 
 }  // 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)...)
+ *      [7/16,11/16]  atan(x) = atan(1/2) + atan( (t-0.5)/(1+t/2) )
+ *      [11/16.19/16] atan(x) = atan( 1 ) + atan( (t-1)/(1+t) )
+ *      [19/16,39/16] atan(x) = atan(3/2) + atan( (t-1.5)/(1+1.5t) )
+ *      [39/16,INF]   atan(x) = atan(INF) + atan( -1/t )
+ *
+ * Constants:
+ * The hexadecimal values are the intended ones for the following
+ * constants. The decimal values may be used, provided that the
+ * compiler will convert from decimal to binary accurately enough
+ * to produce the hexadecimal values shown.
+ */
+double atan(double x) {
+  static const double atanhi[] = {
+      4.63647609000806093515e-01, /* atan(0.5)hi 0x3FDDAC67, 0x0561BB4F */
+      7.85398163397448278999e-01, /* atan(1.0)hi 0x3FE921FB, 0x54442D18 */
+      9.82793723247329054082e-01, /* atan(1.5)hi 0x3FEF730B, 0xD281F69B */
+      1.57079632679489655800e+00, /* atan(inf)hi 0x3FF921FB, 0x54442D18 */
+  };
+
+  static const double atanlo[] = {
+      2.26987774529616870924e-17, /* atan(0.5)lo 0x3C7A2B7F, 0x222F65E2 */
+      3.06161699786838301793e-17, /* atan(1.0)lo 0x3C81A626, 0x33145C07 */
+      1.39033110312309984516e-17, /* atan(1.5)lo 0x3C700788, 0x7AF0CBBD */
+      6.12323399573676603587e-17, /* atan(inf)lo 0x3C91A626, 0x33145C07 */
+  };
+
+  static const double aT[] = {
+      3.33333333333329318027e-01,  /* 0x3FD55555, 0x5555550D */
+      -1.99999999998764832476e-01, /* 0xBFC99999, 0x9998EBC4 */
+      1.42857142725034663711e-01,  /* 0x3FC24924, 0x920083FF */
+      -1.11111104054623557880e-01, /* 0xBFBC71C6, 0xFE231671 */
+      9.09088713343650656196e-02,  /* 0x3FB745CD, 0xC54C206E */
+      -7.69187620504482999495e-02, /* 0xBFB3B0F2, 0xAF749A6D */
+      6.66107313738753120669e-02,  /* 0x3FB10D66, 0xA0D03D51 */
+      -5.83357013379057348645e-02, /* 0xBFADDE2D, 0x52DEFD9A */
+      4.97687799461593236017e-02,  /* 0x3FA97B4B, 0x24760DEB */
+      -3.65315727442169155270e-02, /* 0xBFA2B444, 0x2C6A6C2F */
+      1.62858201153657823623e-02,  /* 0x3F90AD3A, 0xE322DA11 */
+  };
+
+  static const double one = 1.0, huge = 1.0e300;
+
+  double w, s1, s2, z;
+  int32_t ix, hx, id;
+
+  GET_HIGH_WORD(hx, x);
+  ix = hx & 0x7fffffff;
+  if (ix >= 0x44100000) { /* if |x| >= 2^66 */
+    u_int32_t low;
+    GET_LOW_WORD(low, x);
+    if (ix > 0x7ff00000 || (ix == 0x7ff00000 && (low != 0)))
+      return x + x; /* NaN */
+    if (hx > 0)
+      return atanhi[3] + *(volatile double *)&atanlo[3];
+    else
+      return -atanhi[3] - *(volatile double *)&atanlo[3];
+  }
+  if (ix < 0x3fdc0000) {            /* |x| < 0.4375 */
+    if (ix < 0x3e400000) {          /* |x| < 2^-27 */
+      if (huge + x > one) return x; /* raise inexact */
+    }
+    id = -1;
+  } else {
+    x = fabs(x);
+    if (ix < 0x3ff30000) {   /* |x| < 1.1875 */
+      if (ix < 0x3fe60000) { /* 7/16 <=|x|<11/16 */
+        id = 0;
+        x = (2.0 * x - one) / (2.0 + x);
+      } else { /* 11/16<=|x|< 19/16 */
+        id = 1;
+        x = (x - one) / (x + one);
+      }
+    } else {
+      if (ix < 0x40038000) { /* |x| < 2.4375 */
+        id = 2;
+        x = (x - 1.5) / (one + 1.5 * x);
+      } else { /* 2.4375 <= |x| < 2^66 */
+        id = 3;
+        x = -1.0 / x;
+      }
+    }
+  }
+  /* end of argument reduction */
+  z = x * x;
+  w = z * z;
+  /* break sum from i=0 to 10 aT[i]z**(i+1) into odd and even poly */
+  s1 = z * (aT[0] +
+            w * (aT[2] + w * (aT[4] + w * (aT[6] + w * (aT[8] + w * aT[10])))));
+  s2 = w * (aT[1] + w * (aT[3] + w * (aT[5] + w * (aT[7] + w * aT[9]))));
+  if (id < 0) {
+    return x - x * (s1 + s2);
+  } else {
+    z = atanhi[id] - ((x * (s1 + s2) - atanlo[id]) - x);
+    return (hx < 0) ? -z : z;
+  }
+}
+
+/* atan2(y,x)
+ * Method :
+ *  1. Reduce y to positive by atan2(y,x)=-atan2(-y,x).
+ *  2. Reduce x to positive by (if x and y are unexceptional):
+ *    ARG (x+iy) = arctan(y/x)       ... if x > 0,
+ *    ARG (x+iy) = pi - arctan[y/(-x)]   ... if x < 0,
+ *
+ * Special cases:
+ *
+ *  ATAN2((anything), NaN ) is NaN;
+ *  ATAN2(NAN , (anything) ) is NaN;
+ *  ATAN2(+-0, +(anything but NaN)) is +-0  ;
+ *  ATAN2(+-0, -(anything but NaN)) is +-pi ;
+ *  ATAN2(+-(anything but 0 and NaN), 0) is +-pi/2;
+ *  ATAN2(+-(anything but INF and NaN), +INF) is +-0 ;
+ *  ATAN2(+-(anything but INF and NaN), -INF) is +-pi;
+ *  ATAN2(+-INF,+INF ) is +-pi/4 ;
+ *  ATAN2(+-INF,-INF ) is +-3pi/4;
+ *  ATAN2(+-INF, (anything but,0,NaN, and INF)) is +-pi/2;
+ *
+ * Constants:
+ * The hexadecimal values are the intended ones for the following
+ * constants. The decimal values may be used, provided that the
+ * compiler will convert from decimal to binary accurately enough
+ * to produce the hexadecimal values shown.
+ */
+double atan2(double y, double x) {
+  static volatile double tiny = 1.0e-300;
+  static const double
+      zero = 0.0,
+      pi_o_4 = 7.8539816339744827900E-01, /* 0x3FE921FB, 0x54442D18 */
+      pi_o_2 = 1.5707963267948965580E+00, /* 0x3FF921FB, 0x54442D18 */
+      pi = 3.1415926535897931160E+00;     /* 0x400921FB, 0x54442D18 */
+  static volatile double pi_lo =
+      1.2246467991473531772E-16; /* 0x3CA1A626, 0x33145C07 */
+
+  double z;
+  int32_t k, m, hx, hy, ix, iy;
+  u_int32_t lx, ly;
+
+  EXTRACT_WORDS(hx, lx, x);
+  ix = hx & 0x7fffffff;
+  EXTRACT_WORDS(hy, ly, y);
+  iy = hy & 0x7fffffff;
+  if (((ix | ((lx | -static_cast<int32_t>(lx)) >> 31)) > 0x7ff00000) ||
+      ((iy | ((ly | -static_cast<int32_t>(ly)) >> 31)) > 0x7ff00000)) {
+    return x + y; /* x or y is NaN */
+  }
+  if (((hx - 0x3ff00000) | lx) == 0) return atan(y); /* x=1.0 */
+  m = ((hy >> 31) & 1) | ((hx >> 30) & 2);           /* 2*sign(x)+sign(y) */
+
+  /* when y = 0 */
+  if ((iy | ly) == 0) {
+    switch (m) {
+      case 0:
+      case 1:
+        return y; /* atan(+-0,+anything)=+-0 */
+      case 2:
+        return pi + tiny; /* atan(+0,-anything) = pi */
+      case 3:
+        return -pi - tiny; /* atan(-0,-anything) =-pi */
+    }
+  }
+  /* when x = 0 */
+  if ((ix | lx) == 0) return (hy < 0) ? -pi_o_2 - tiny : pi_o_2 + tiny;
+
+  /* when x is INF */
+  if (ix == 0x7ff00000) {
+    if (iy == 0x7ff00000) {
+      switch (m) {
+        case 0:
+          return pi_o_4 + tiny; /* atan(+INF,+INF) */
+        case 1:
+          return -pi_o_4 - tiny; /* atan(-INF,+INF) */
+        case 2:
+          return 3.0 * pi_o_4 + tiny; /*atan(+INF,-INF)*/
+        case 3:
+          return -3.0 * pi_o_4 - tiny; /*atan(-INF,-INF)*/
+      }
+    } else {
+      switch (m) {
+        case 0:
+          return zero; /* atan(+...,+INF) */
+        case 1:
+          return -zero; /* atan(-...,+INF) */
+        case 2:
+          return pi + tiny; /* atan(+...,-INF) */
+        case 3:
+          return -pi - tiny; /* atan(-...,-INF) */
+      }
+    }
+  }
+  /* when y is INF */
+  if (iy == 0x7ff00000) return (hy < 0) ? -pi_o_2 - tiny : pi_o_2 + tiny;
+
+  /* compute y/x */
+  k = (iy - ix) >> 20;
+  if (k > 60) { /* |y/x| >  2**60 */
+    z = pi_o_2 + 0.5 * pi_lo;
+    m &= 1;
+  } else if (hx < 0 && k < -60) {
+    z = 0.0; /* 0 > |y|/x > -2**-60 */
+  } else {
+    z = atan(fabs(y / x)); /* safe to do y/x */
+  }
+  switch (m) {
+    case 0:
+      return z; /* atan(+,+) */
+    case 1:
+      return -z; /* atan(-,+) */
+    case 2:
+      return pi - (z - pi_lo); /* atan(+,-) */
+    default:                   /* case 3 */
+      return (z - pi_lo) - pi; /* atan(-,-) */
+  }
+}
+
 /* log(x)
  * Return the logrithm of x
  *