Implement trigonometric functions using a fdlibm port.

src/math.js

Index: src/math.js
diff --git a/src/math.js b/src/math.js
index 9dc4b37d0ce2115ed9e9b5078f204fd060f926b9..6323451d1b94dc5c3ec59922640da024eacc44a5 100644
--- a/src/math.js
+++ b/src/math.js
@@ -56,12 +56,6 @@ function MathCeil(x) {
   return -MathFloor(-x);
 }
 
-// ECMA 262 - 15.8.2.7
-function MathCos(x) {
-  x = MathAbs(x);  // Convert to number and get rid of -0.
-  return TrigonometricInterpolation(x, 1);
-}
-
 // ECMA 262 - 15.8.2.8
 function MathExp(x) {
   return %MathExpRT(TO_NUMBER_INLINE(x));
@@ -164,94 +158,251 @@ function MathRound(x) {
   return %RoundNumber(TO_NUMBER_INLINE(x));
 }
 
-// ECMA 262 - 15.8.2.16
-function MathSin(x) {
-  x = x * 1;  // Convert to number and deal with -0.
-  if (%_IsMinusZero(x)) return x;
-  return TrigonometricInterpolation(x, 0);
-}
-
 // ECMA 262 - 15.8.2.17
 function MathSqrt(x) {
   return %_MathSqrtRT(TO_NUMBER_INLINE(x));
 }
 
-// ECMA 262 - 15.8.2.18
-function MathTan(x) {
-  return MathSin(x) / MathCos(x);
-}
-
 // Non-standard extension.
 function MathImul(x, y) {
   return %NumberImul(TO_NUMBER_INLINE(x), TO_NUMBER_INLINE(y));
 }
 
+// -------------------------------------------------------------------
+
+// A straightforward translation of fdlibm routines for sin, cos, and
+// tan, by Raymond Toy (rtoy@google.com).
+
+// ====================================================
+// 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.
+// ====================================================
+
+var kTrig;  // Initialized to a Float64Array during genesis.
+
+macro REMPIO2(X)
+  var n, y0, y1;
+  var hx = %_DoubleHi(X);
+  var ix = hx & 0x7fffffff;
+
+  if (ix < 0x4002d97c) {
+    // |X| ~< 3*pi/4, special case with n = +/- 1
+    if (hx > 0) {
+      var z = X - kTrig[1];
+      if (ix != 0x3ff921fb) {
+        // 33+53 bit pi is good enough
+        y0 = z - kTrig[2];
+        y1 = (z - y0) - kTrig[2];
+      } else {
+        // near pi/2, use 33+33+53 bit pi
+        z -= kTrig[3];
+        y0 = z - kTrig[4];
+        y1 = (z - y0) - kTrig[4];
+      }
+      n = 1;
+    } else {
+      // Negative X
+      var z = X + kTrig[1];
+      if (ix != 0x3ff921fb) {
+        // 33+53 bit pi is good enough
+        y0 = z + kTrig[2];
+        y1 = (z - y0) + kTrig[2];
+      } else {
+        // near pi/2, use 33+33+53 bit pi
+        z += kTrig[3];
+        y0 = z + kTrig[4];
+        y1 = (z - y0) + kTrig[4];
+      }
+      n = -1;
+    }
+  } else if (ix <= 0x413921fb) {
+    // |X| ~<= 2^19*(pi/2), medium size
+    var t = MathAbs(X);
+    n = (t * kTrig[0] + 0.5) | 0;
+    var r = t - n * kTrig[1];
+    var w = n * kTrig[2];
+    // First round good to 85 bit
+    y0 = r - w;
+    if (ix - (%_DoubleHi(y0) & 0x7ff00000) > 0x1000000) {
+      // 2nd iteration needed, good to 118
+      t = r;
+      w = n * kTrig[3];
+      r = t - w;
+      w = n * kTrig[4] - ((t - r) - w);
+      y0 = r - w;
+      if (ix - (%_DoubleHi(y0) & 0x7ff00000) > 0x3100000) {
+        // 3rd iteration needed. 151 bits accuracy
+        t = r;
+        w = n * kTrig[5];
+        r = t - w;
+        w = n * kTrig[6] - ((t - r) - w);
+        y0 = r - w;
+      }
+    }
+    y1 = (r - y0) - w;
+    if (hx < 0) {
+      n = -n;
+      y0 = -y0;
+      y1 = -y1;
+    }
+  } else {
+    // Need to do full Payne-Hanek reduction here!
+    var r = %RemPiO2(X);
+    n = r[0];
+    y0 = r[1];
+    y1 = r[2];
+  }
+endmacro
+
+// Sine for [-pi/4, pi/4], pi/4 ~ 0.7854
+macro RETURN_KERNELSIN(X0, X1, SIGN)
+  var z = X0 * X0;
+  var v = z * X0;
+  var r = kTrig[8] + z * (kTrig[9] + z * (kTrig[10] +
+                     z * (kTrig[11] + z * kTrig[12])));
+  return (X0 - ((z * (0.5 * X1 - v * r) - X1) - v * kTrig[7])) SIGN;
+endmacro
 
-var kInversePiHalf      = 0.636619772367581343;      // 2 / pi
-var kInversePiHalfS26   = 9.48637384723993156e-9;    // 2 / pi / (2^26)
-var kS26                = 1 << 26;
-var kTwoStepThreshold   = 1 << 27;
-// pi / 2 rounded up
-var kPiHalf             = 1.570796326794896780;      // 0x192d4454fb21f93f
-// We use two parts for pi/2 to emulate a higher precision.
-// pi_half_1 only has 26 significant bits for mantissa.
-// Note that pi_half > pi_half_1 + pi_half_2
-var kPiHalf1            = 1.570796325802803040;      // 0x00000054fb21f93f
-var kPiHalf2            = 9.920935796805404252e-10;  // 0x3326a611460b113e
-
-var kSamples;            // Initialized to a number during genesis.
-var kIndexConvert;       // Initialized to kSamples / (pi/2) during genesis.
-var kSinTable;           // Initialized to a Float64Array during genesis.
-var kCosXIntervalTable;  // Initialized to a Float64Array during genesis.
-
-// This implements sine using the following algorithm.
-// 1) Multiplication takes care of to-number conversion.
-// 2) Reduce x to the first quadrant [0, pi/2].
-//    Conveniently enough, in case of +/-Infinity, we get NaN.
-//    Note that we try to use only 26 instead of 52 significant bits for
-//    mantissa to avoid rounding errors when multiplying.  For very large
-//    input we therefore have additional steps.
-// 3) Replace x by (pi/2-x) if x was in the 2nd or 4th quadrant.
-// 4) Do a table lookup for the closest samples to the left and right of x.
-// 5) Find the derivatives at those sampling points by table lookup:
-//    dsin(x)/dx = cos(x) = sin(pi/2-x) for x in [0, pi/2].
-// 6) Use cubic spline interpolation to approximate sin(x).
-// 7) Negate the result if x was in the 3rd or 4th quadrant.
-// 8) Get rid of -0 by adding 0.
-function TrigonometricInterpolation(x, phase) {
-  if (x < 0 || x > kPiHalf) {
-    var multiple;
-    while (x < -kTwoStepThreshold || x > kTwoStepThreshold) {
-      // Let's assume this loop does not terminate.
-      // All numbers x in each loop forms a set S.
-      // (1) abs(x) > 2^27 for all x in S.
-      // (2) abs(multiple) != 0 since (2^27 * inverse_pi_half_s26) > 1
-      // (3) multiple is rounded down in 2^26 steps, so the rounding error is
-      //     at most max(ulp, 2^26).
-      // (4) so for x > 2^27, we subtract at most (1+pi/4)x and at least
-      //     (1-pi/4)x
-      // (5) The subtraction results in x' so that abs(x') <= abs(x)*pi/4.
-      //     Note that this difference cannot be simply rounded off.
-      // Set S cannot exist since (5) violates (1).  Loop must terminate.
-      multiple = MathFloor(x * kInversePiHalfS26) * kS26;
-      x = x - multiple * kPiHalf1 - multiple * kPiHalf2;
+// Cosine for [-pi/4, pi/4], pi/4 ~ 0.7854
+macro RETURN_KERNELCOS(X0, X1, SIGN)
+  var ix = %_DoubleHi(X0) & 0x7fffffff;
+  var z = X0 * X0;
+  var r = z * (kTrig[13] + z * (kTrig[14] + z * (kTrig[15] +
+          z * (kTrig[16] + z * (kTrig[17] + z * kTrig[18])))));
+  if (ix < 0x3fd33333) {
+    return (1 - (0.5 * z - (z * r - X0 * X1))) SIGN;
+  } else {
+    var qx;
+    if (ix > 0x3fe90000) {
+      qx = 0.28125;
+    } else {
+      qx = %_ConstructDouble(%_DoubleHi(0.25 * X0), 0);
     }
-    multiple = MathFloor(x * kInversePiHalf);
-    x = x - multiple * kPiHalf1 - multiple * kPiHalf2;
-    phase += multiple;
+    var hz = 0.5 * z - qx;
+    return (1 - qx - (hz - (z * r - X0 * X1))) SIGN;
+  }
+endmacro
+
+// Tangent for [-pi/4, pi/4], pi/4 ~ 0.7854
+function KernelTan(x, y, returnTan) {
+  var z;
+  var w;
+  var hx = %_DoubleHi(x);
+  var ix = hx & 0x7fffffff;
+
+  if (ix < 0x3e300000) {
+    // x < 2^-28
+    if (((ix | %_DoubleLo(x)) | (returnTan + 1)) == 0) {
+      return 1 / MathAbs(x);
+    } else {
+      if (returnTan == 1) {
+        return x;
+      } else {
+        // Compute -1/(x + y) carefully
+        var w = x + y;
+        var z = %_ConstructDouble(%_DoubleHi(w), 0);
+        var v = y - (z - x);
+        var a = -1 / w;
+        var t = %_ConstructDouble(%_DoubleHi(a), 0);
+        var s = 1 + t * z;
+        return t + a * (s + t * v);
+      }
+    }
+  }
+  if (ix >= 0x3fe59429) {
+    // |x| > .6744
+    if (x < 0) {
+      x = -x;
+      y = -y;
+    }
+    z = kTrig[32] - x;
+    w = kTrig[33] - y;
+    x = z + w;
+    y = 0;
+  }
+  z = x * x;
+  w = z * z;
+
+  var r = kTrig[20] + w * (kTrig[22] + w * (kTrig[24] +
+                      w * (kTrig[26] + w * (kTrig[28] + w * kTrig[30]))));
+  var v = z * (kTrig[21] + w * (kTrig[23] + w * (kTrig[25] +
+                           w * (kTrig[27] + w * (kTrig[29] + w * kTrig[31])))));
+  var s = z * x;
+  r = y + z * (s * (r + v) + y);
+  r = r + kTrig[19] * s;
+  w = x + r;
+  if (ix >= 0x3fe59428) {
+    return (1 - ((hx >> 30) & 2)) *
+      (returnTan - 2.0 * (x - (w * w / (w + returnTan) - r)));
+  }
+  if (returnTan == 1) {
+    return w;
+  } else {
+    z = %_ConstructDouble(%_DoubleHi(w), 0);
+    v = r - (z - x);
+    var a = -1 / w;
+    var t = %_ConstructDouble(%_DoubleHi(a), 0);
+    s = 1 + t * z;
+    return t + a * (s + t * v);
+  }
+}
+
+function MathSinSlow(x) {
+  REMPIO2(x);
+  var sign = 1 - (n & 2);
+  if (n & 1) {
+    RETURN_KERNELCOS(y0, y1, * sign);
+  } else {
+    RETURN_KERNELSIN(y0, y1, * sign);
+  }
+}
+
+function MathCosSlow(x) {
+  REMPIO2(x);
+  if (n & 1) {
+    var sign = (n & 2) - 1;
+    RETURN_KERNELSIN(y0, y1, * sign);
+  } else {
+    var sign = 1 - (n & 2);
+    RETURN_KERNELCOS(y0, y1, * sign);
+  }
+}
+
+// ECMA 262 - 15.8.2.16
+function MathSin(x) {
+  x = x * 1;  // Convert to number.
+  if ((%_DoubleHi(x) & 0x7fffffff) <= 0x3fe921fb) {
+    // |x| < pi/4, approximately.  No reduction needed.
+    if (%_IsMinusZero(x)) return x;
+    RETURN_KERNELSIN(x, 0, /* empty */);
+  }
+  return MathSinSlow(x);
+}
+
+// ECMA 262 - 15.8.2.7
+function MathCos(x) {
+  x = x * 1;  // Convert to number.
+  if ((%_DoubleHi(x) & 0x7fffffff) <= 0x3fe921fb) {
+    // |x| < pi/4, approximately.  No reduction needed.
+    RETURN_KERNELCOS(x, 0, /* empty */);
+  }
+  return MathCosSlow(x);
+}
+
+// ECMA 262 - 15.8.2.18
+function MathTan(x) {
+  x = x * 1;  // Convert to number.
+  if ((%_DoubleHi(x) & 0x7fffffff) <= 0x3fe921fb) {
+    // |x| < pi/4, approximately.  No reduction needed.
+    if (%_IsMinusZero(x)) return x;
+    return KernelTan(x, 0, 1);
   }
-  var double_index = x * kIndexConvert;
-  if (phase & 1) double_index = kSamples - double_index;
-  var index = double_index | 0;
-  var t1 = double_index - index;
-  var t2 = 1 - t1;
-  var y1 = kSinTable[index];
-  var y2 = kSinTable[index + 1];
-  var dy = y2 - y1;
-  return (t2 * y1 + t1 * y2 +
-              t1 * t2 * ((kCosXIntervalTable[index] - dy) * t2 +
-                         (dy - kCosXIntervalTable[index + 1]) * t1))
-         * (1 - (phase & 2)) + 0;
+  REMPIO2(x);
+  return KernelTan(y0, y1, (n & 1) ? -1 : 1);
 }
 
 
@@ -537,8 +688,6 @@ function SetUpMath() {
   %SetInlineBuiltinFlag(MathRandom);
   %SetInlineBuiltinFlag(MathSin);
   %SetInlineBuiltinFlag(MathCos);
-  %SetInlineBuiltinFlag(MathTan);
-  %SetInlineBuiltinFlag(TrigonometricInterpolation);
 }
 
 SetUpMath();