Increase precision when finding the remainder after division by pi/2.

R=jkummerow@chromium.org

Review URL: https://codereview.chromium.org/66703005

git-svn-id: http://v8.googlecode.com/svn/branches/bleeding_edge@17933 ce2b1a6d-e550-0410-aec6-3dcde31c8c00

src/math.js

@@ -217,16 +217,19 @@ var InitTrigonometricFunctions;
 // Also define the initialization function that populates the lookup table
 // and then wires up the function definitions.
 function SetupTrigonometricFunctions() {
-  // TODO(yangguo): The following table size has been chosen to satisfy
-  // Sunspider's brittle result verification.  Reconsider relevance.
-  var samples = 4489;
-  var pi = 3.1415926535897932;
-  var pi_half = pi / 2;
-  var inverse_pi_half = 2 / pi;
-  var two_pi = 2 * pi;
-  var four_pi = 4 * pi;
-  var interval = pi_half / samples;
-  var inverse_interval = samples / pi_half;
+  var samples = 1800;   // Table size.  Do not change arbitrarily.
+  var inverse_pi_half      = 0.636619772367581343;      // 2 / pi
+  var inverse_pi_half_s_26 = 9.48637384723993156e-9;    // 2 / pi / (2^26)
+  var s_26 = 1 << 26;
+  var two_step_threshold   = 1 << 27;
+  var index_convert        = 1145.915590261646418;      // samples / (pi / 2)
+  // pi / 2 rounded up
+  var pi_half              = 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 pi_half_1            = 1.570796325802803040;      // 0x00000054fb21f93f
+  var pi_half_2            = 9.920935796805404252e-10;  // 0x3326a611460b113e
   var table_sin;
   var table_cos_interval;
 
@@ -234,6 +237,9 @@ function SetupTrigonometricFunctions() {
   // 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:
@@ -241,8 +247,30 @@ function SetupTrigonometricFunctions() {
   // 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.
-  var Interpolation = function(x) {
-    var double_index = x * inverse_interval;
+  var Interpolation = function(x, phase) {
+    if (x < 0 || x > pi_half) {
+      var multiple;
+      while (x < -two_step_threshold || x > two_step_threshold) {
+        // 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 * inverse_pi_half_s_26) * s_26;
+        x = x - multiple * pi_half_1 - multiple * pi_half_2;
+      }
+      multiple = MathFloor(x * inverse_pi_half);
+      x = x - multiple * pi_half_1 - multiple * pi_half_2;
+      phase += multiple;
+    }
+    var double_index = x * index_convert;
+    if (phase & 1) double_index = samples - double_index;
     var index = double_index | 0;
     var t1 = double_index - index;
     var t2 = 1 - t1;
@@ -251,26 +279,20 @@ function SetupTrigonometricFunctions() {
     var dy = y2 - y1;
     return (t2 * y1 + t1 * y2 +
                 t1 * t2 * ((table_cos_interval[index] - dy) * t2 +
-                           (dy - table_cos_interval[index + 1]) * t1));
+                           (dy - table_cos_interval[index + 1]) * t1))
+           * (1 - (phase & 2)) + 0;
   }
 
   var MathSinInterpolation = function(x) {
-    // This is to make Sunspider's result verification happy.
-    if (x > four_pi) x -= four_pi;
-    var multiple = MathFloor(x * inverse_pi_half);
-    if (%_IsMinusZero(multiple)) return multiple;
-    x = (multiple & 1) * pi_half +
-        (1 - ((multiple & 1) << 1)) * (x - multiple * pi_half);
-    return Interpolation(x) * (1 - (multiple & 2)) + 0;
+    x = x * 1;  // Convert to number and deal with -0.
+    if (%_IsMinusZero(x)) return x;
+    return Interpolation(x, 0);
   }
 
-  // Cosine is sine with a phase offset of pi/2.
+  // Cosine is sine with a phase offset.
   var MathCosInterpolation = function(x) {
-    var multiple = MathFloor(x * inverse_pi_half);
-    var phase = multiple + 1;
-    x = (phase & 1) * pi_half +
-        (1 - ((phase & 1) << 1)) * (x - multiple * pi_half);
-    return Interpolation(x) * (1 - (phase & 2)) + 0;
+    x = MathAbs(x);  // Convert to number and get rid of -0.
+    return Interpolation(x, 1);
   };
 
   %SetInlineBuiltinFlag(Interpolation);

test/mjsunit/mjsunit.js

@@ -54,6 +54,10 @@ var assertSame;
 // and the properties of non-Array objects).
 var assertEquals;
 
+
+// The difference between expected and found value is within certain tolerance.
+var assertEqualsDelta;
+
 // The found object is an Array with the same length and elements
 // as the expected object. The expected object doesn't need to be an Array,
 // as long as it's "array-ish".
@@ -247,6 +251,12 @@ var assertUnoptimized;
   };
 
 
+  assertEqualsDelta =
+      function assertEqualsDelta(expected, found, delta, name_opt) {
+    assertTrue(Math.abs(expected - found) <= delta, name_opt);
+  };
+
+
   assertArrayEquals = function assertArrayEquals(expected, found, name_opt) {
     var start = "";
     if (name_opt) {

test/mjsunit/sin-cos.js

@@ -27,6 +27,8 @@
 
 // Test Math.sin and Math.cos.
 
+// Flags: --allow-natives-syntax
+
 function sinTest() {
   assertEquals(0, Math.sin(0));
   assertEquals(1, Math.sin(Math.PI / 2));
@@ -97,7 +99,7 @@ function abs_error(fun, ref, x) {
 
 var test_inputs = [];
 for (var i = -10000; i < 10000; i += 177) test_inputs.push(i/1257);
-var epsilon = 0.000001;
+var epsilon = 0.0000001;
 
 test_inputs.push(0);
 test_inputs.push(0 + epsilon);
@@ -117,8 +119,8 @@ for (var i = 0; i < test_inputs.length; i++) {
   var x = test_inputs[i];
   var err_sin = abs_error(Math.sin, sin, x);
   var err_cos = abs_error(Math.cos, cos, x)
-  assertTrue(err_sin < 1E-13);
-  assertTrue(err_cos < 1E-13);
+  assertEqualsDelta(0, err_sin, 1E-13);
+  assertEqualsDelta(0, err_cos, 1E-13);
   squares.push(err_sin*err_sin + err_cos*err_cos);
 }
 
@@ -132,7 +134,7 @@ while (squares.length > 1) {
 }
 
 var err_rms = Math.sqrt(squares[0] / test_inputs.length / 2);
-assertTrue(err_rms < 1E-14);
+assertEqualsDelta(0, err_rms, 1E-14);
 
 assertEquals(-1, Math.cos({ valueOf: function() { return Math.PI; } }));
 assertEquals(0, Math.sin("0x00000"));
@@ -141,3 +143,40 @@ assertTrue(isNaN(Math.sin(Infinity)));
 assertTrue(isNaN(Math.cos("-Infinity")));
 assertEquals("Infinity", String(Math.tan(Math.PI/2)));
 assertEquals("-Infinity", String(Math.tan(-Math.PI/2)));
+assertEquals("-Infinity", String(1/Math.sin("-0")));
+
+// Assert that the remainder after division by pi is reasonably precise.
+function assertError(expected, x, epsilon) {
+  assertTrue(Math.abs(x - expected) < epsilon);
+}
+
+assertEqualsDelta(0.9367521275331447,  Math.cos(1e06),  1e-15);
+assertEqualsDelta(0.8731196226768560,  Math.cos(1e10),  1e-08);
+assertEqualsDelta(0.9367521275331447,  Math.cos(-1e06), 1e-15);
+assertEqualsDelta(0.8731196226768560,  Math.cos(-1e10), 1e-08);
+assertEqualsDelta(-0.3499935021712929, Math.sin(1e06),  1e-15);
+assertEqualsDelta(-0.4875060250875106, Math.sin(1e10),  1e-08);
+assertEqualsDelta(0.3499935021712929,  Math.sin(-1e06), 1e-15);
+assertEqualsDelta(0.4875060250875106,  Math.sin(-1e10), 1e-08);
+assertEqualsDelta(0.7796880066069787,  Math.sin(1e16),  1e-05);
+assertEqualsDelta(-0.6261681981330861, Math.cos(1e16),  1e-05);
+
+// Assert that remainder calculation terminates.
+for (var i = -1024; i < 1024; i++) {
+  assertFalse(isNaN(Math.sin(Math.pow(2, i))));
+}
+
+assertFalse(isNaN(Math.cos(1.57079632679489700)));
+assertFalse(isNaN(Math.cos(-1e-100)));
+assertFalse(isNaN(Math.cos(-1e-323)));
+
+
+function no_deopt_on_minus_zero(x) {
+  return Math.sin(x) + Math.cos(x) + Math.tan(x);
+}
+
+no_deopt_on_minus_zero(1);
+no_deopt_on_minus_zero(1);
+%OptimizeFunctionOnNextCall(no_deopt_on_minus_zero);
+no_deopt_on_minus_zero(-0);
+assertOptimized(no_deopt_on_minus_zero);

test/mozilla/mozilla.status

@@ -599,10 +599,6 @@
   # Negative hexadecimal literals are parsed as NaN. This test is outdated.
   'ecma/TypeConversion/9.3.1-3': [FAIL_OK],
 
-
-  # Math.tan expectations are more strict than the spec.
-  'ecma/Math/15.8.2.18': [FAIL_OK],
-
   ##################### FAILING TESTS #####################
 
   # This section is for tests that fail in V8 and pass in JSC.