Add base types for path ops

src/pathops/SkPathOpsTypes.cpp

+/*
+ * Copyright 2012 Google Inc.
+ *
+ * Use of this source code is governed by a BSD-style license that can be
+ * found in the LICENSE file.
+ */
+#include "SkPathOpsTypes.h"
+
+const int UlpsEpsilon = 16;
+
+// from http://randomascii.wordpress.com/2012/02/25/comparing-floating-point-numbers-2012-edition/
+union SkPathOpsUlpsFloat {
+    int32_t fInt;
+    float fFloat;
+
+    SkPathOpsUlpsFloat(float num = 0.0f) : fFloat(num) {}
+    bool negative() const { return (fInt >> 31) != 0; }

[whunt, 2013/03/22 18:16:06]

why not just compute (fInt < 0)?

[caryclark, 2013/03/22 19:38:51]

No reason. It's copied/pasted code. In general, I've done no performance
analysis on this code -- I've been concentrating only on completeness and
correctness.

+};
+
+bool AlmostEqualUlps(float A, float B) {
+    SkPathOpsUlpsFloat uA(A);
+    SkPathOpsUlpsFloat uB(B);
+    // Different signs means they do not match.
+    if (uA.negative() != uB.negative())
+    {
+        // Check for equality to make sure +0==-0

[whunt, 2013/03/22 18:16:06]

0 ==

[caryclark, 2013/03/22 19:38:51]

Done.

+        return A == B;
+    }
+    // Find the difference in ULPs.
+    int ulpsDiff = abs(uA.fInt - uB.fInt);
+    return ulpsDiff <= UlpsEpsilon;
+}
+
+// cube root approximation using bit hack for 64-bit float
+// adapted from Kahan's cbrt
+static  double cbrt_5d(double d) {
+    const unsigned int B1 = 715094163;
+    double t = 0.0;
+    unsigned int* pt = (unsigned int*) &t;
+    unsigned int* px = (unsigned int*) &d;
+    pt[1]=px[1]/3+B1;

[whunt, 2013/03/22 18:16:06]

spaces around the ops?

[caryclark, 2013/03/22 19:38:51]

Done.

+    return t;
+}
+
+// iterative cube root approximation using Halley's method (double)
+static  double cbrta_halleyd(const double a, const double R) {
+    const double a3 = a*a*a;
+    const double b= a * (a3 + R + R) / (a3 + a3 + R);

[whunt, 2013/03/22 18:16:06]

b =

[caryclark, 2013/03/22 19:38:51]

Done.

+    return b;
+}
+
+// cube root approximation using 3 iterations of Halley's method (double)
+static double halley_cbrt3d(double d) {
+    double a = cbrt_5d(d);
+    a = cbrta_halleyd(a, d);
+    a = cbrta_halleyd(a, d);
+    return cbrta_halleyd(a, d);
+}
+
+double SkDCubeRoot(double x) {
+    if (approximately_zero_cubed(x)) {
+        return 0;
+    }
+    double result = halley_cbrt3d(fabs(x));
+    if (x < 0) {
+        result = -result;
+    }
+    return result;
+}