[web] Implement ulps for path ops (flutter#19711)

* Implement ulps for path ops
* Address review comments
* cache abs()
* dartfmt and update licenses

ci/licenses_golden/licenses_flutter

@@ -524,6 +524,7 @@ FILE: ../../../flutter/lib/web_ui/lib/src/engine/text_editing/autofill_hint.dart
 FILE: ../../../flutter/lib/web_ui/lib/src/engine/text_editing/input_type.dart
 FILE: ../../../flutter/lib/web_ui/lib/src/engine/text_editing/text_capitalization.dart
 FILE: ../../../flutter/lib/web_ui/lib/src/engine/text_editing/text_editing.dart
+FILE: ../../../flutter/lib/web_ui/lib/src/engine/ulps.dart
 FILE: ../../../flutter/lib/web_ui/lib/src/engine/util.dart
 FILE: ../../../flutter/lib/web_ui/lib/src/engine/validators.dart
 FILE: ../../../flutter/lib/web_ui/lib/src/engine/vector_math.dart

lib/web_ui/lib/src/engine.dart

@@ -131,6 +131,7 @@ part 'engine/text_editing/autofill_hint.dart';
 part 'engine/text_editing/input_type.dart';
 part 'engine/text_editing/text_capitalization.dart';
 part 'engine/text_editing/text_editing.dart';
+part 'engine/ulps.dart';
 part 'engine/util.dart';
 part 'engine/validators.dart';
 part 'engine/vector_math.dart';

lib/web_ui/lib/src/engine/ulps.dart

@@ -0,0 +1,214 @@
+// Copyright 2013 The Flutter Authors. All rights reserved.
+// Use of this source code is governed by a BSD-style license that can be
+// found in the LICENSE file.
+
+part of engine;
+
+// This is a small library to handle stability for floating point operations.
+//
+// Since we are representing an infinite number of real numbers in finite
+// number of bits, when we perform comparisons of coordinates for paths for
+// example, we want to make sure that line and curve sections that are too
+// close to each other (number of floating point numbers
+// representable in bits between two numbers) are handled correctly and
+// don't cause algorithms to fail when we perform operations such as
+// subtraction or between checks.
+//
+// Small introduction into floating point comparison:
+//
+// For some good articles on the topic, see
+// https://randomascii.wordpress.com/category/floating-point/page/2/
+// Port based on:
+// https://github.com/google/skia/blob/master/include/private/SkFloatBits.h
+//
+// Here is the 32 bit IEEE representation:
+//   uint32_t mantissa : 23;
+//   uint32_t exponent : 8;
+//   uint32_t sign : 1;
+// As you can see it was carefully designed to be reinterpreted as an integer.
+//
+// Ulps stands for unit in the last place. ulp(x) is the gap between two
+// floating point numbers nearest x.
+
+/// Converts a sign-bit int (float interpreted as int) into a 2s complement
+/// int. Also converts 0x80000000 to 0. Allows result to be compared using
+/// int comparison.
+int signBitTo2sCompliment(int x) =>
+    (x & 0x80000000) != 0 ? (-(x & 0x7fffffff)) : x;
+
+/// Convert a 2s complement int to a sign-bit (i.e. int interpreted as float).
+int twosComplimentToSignBit(int x) {
+  if ((x & 0x80000000) == 0) {
+    return x;
+  }
+  x = ~x + 1;
+  x |= 0x80000000;
+  return x;
+}
+
+class _FloatBitConverter {
+  final Float32List float32List;
+  final Int32List int32List;
+  _FloatBitConverter._(this.float32List, this.int32List);
+
+  factory _FloatBitConverter() {
+    Float32List float32List = Float32List(1);
+    return _FloatBitConverter._(
+        float32List, float32List.buffer.asInt32List(0, 1));
+  }
+
+  int toInt(Float32List source, int index) {
+    float32List[0] = source[index];
+    return int32List[0];
+  }
+
+  int toBits(double x) {
+    float32List[0] = x;
+    return int32List[0];
+  }
+
+  double toDouble(int bits) {
+    int32List[0] = bits;
+    return float32List[0];
+  }
+}
+
+// Singleton bit converter to prevent typed array allocations.
+final _FloatBitConverter _floatBitConverter = _FloatBitConverter();
+
+// Converts float to bits.
+int float2Bits(Float32List source, int index) {
+  return _floatBitConverter.toInt(source, index);
+}
+
+// Converts bits to float.
+double bitsToFloat(int bits) {
+  return _floatBitConverter.toDouble(bits);
+}
+
+const int floatBitsExponentMask = 0x7F800000;
+const int floatBitsMatissaMask = 0x007FFFFF;
+
+/// Returns a float as 2s complement int to be able to compare floats to each
+/// other.
+int floatFromListAs2sCompliment(Float32List source, int index) =>
+    signBitTo2sCompliment(float2Bits(source, index));
+
+int floatAs2sCompliment(double x) =>
+    signBitTo2sCompliment(_floatBitConverter.toBits(x));
+
+double twosComplimentAsFloat(int x) => bitsToFloat(twosComplimentToSignBit(x));
+
+bool _argumentsDenormalized(double a, double b, int epsilon) {
+  double denormalizedCheck = kFltEpsilon * epsilon / 2;
+  return a.abs() <= denormalizedCheck && b.abs() <= denormalizedCheck;
+}
+
+bool equalUlps(double a, double b, int epsilon, int depsilon) {
+  if (_argumentsDenormalized(a, b, depsilon)) {
+    return true;
+  }
+  int aBits = floatAs2sCompliment(a);
+  int bBits = floatAs2sCompliment(b);
+  // Find the difference in ULPs.
+  return aBits < bBits + epsilon && bBits < aBits + epsilon;
+}
+
+/// General equality check that covers between, product and division by using
+/// ulps epsilon 16.
+bool almostEqualUlps(double a, double b) {
+  const int kUlpsEpsilon = 16;
+  return equalUlps(a, b, kUlpsEpsilon, kUlpsEpsilon);
+}
+
+/// Equality using the same error term for between comparison.
+bool almostBequalUlps(double a, double b) {
+  const int kUlpsEpsilon = 2;
+  return equalUlps(a, b, kUlpsEpsilon, kUlpsEpsilon);
+}
+
+/// Equality check for product.
+bool almostPequalUlps(double a, double b) {
+  const int kUlpsEpsilon = 8;
+  return equalUlps(a, b, kUlpsEpsilon, kUlpsEpsilon);
+}
+
+/// Equality check for division.
+bool almostDequalUlps(double a, double b) {
+  const int kUlpsEpsilon = 16;
+  return equalUlps(a, b, kUlpsEpsilon, kUlpsEpsilon);
+}
+
+/// Checks if 2 points are roughly equal (ulp 256) to each other.
+bool approximatelyEqual(double ax, double ay, double bx, double by) {
+  if (approximatelyEqualT(ax, bx) && approximatelyEqualT(ay, by)) {
+    return true;
+  }
+  if (!roughlyEqualUlps(ax, bx) || !roughlyEqualUlps(ay, by)) {
+    return false;
+  }
+  final double dx = (ax - bx);
+  final double dy = (ay - by);
+  double dist = math.sqrt(dx * dx + dy * dy);
+  double tiniest = math.min(math.min(math.min(ax, bx), ay), by);
+  double largest = math.max(math.max(math.max(ax, bx), ay), by);
+  largest = math.max(largest, -tiniest);
+  return almostDequalUlps(largest, largest + dist);
+}
+
+/// Equality check for comparing curve T values in the range of 0 to 1.
+///
+/// For general numbers (larger and smaller) use
+/// AlmostEqualUlps instead.
+bool approximatelyEqualT(double t1, double t2) {
+  return approximatelyZero(t1 - t2);
+}
+
+bool approximatelyZero(double value) => value.abs() < kFltEpsilon;
+
+bool roughlyEqualUlps(double a, double b) {
+  const int kUlpsEpsilon = 256;
+  const int kDUlpsEpsilon = 1024;
+  return equalUlps(a, b, kUlpsEpsilon, kDUlpsEpsilon);
+}
+
+bool dEqualUlpsEpsilon(double a, double b, int epsilon) {
+  int aBits = floatAs2sCompliment(a);
+  int bBits = floatAs2sCompliment(b);
+  // Find the difference in ULPs.
+  return aBits < bBits + epsilon && bBits < aBits + epsilon;
+}
+
+// Checks equality for division.
+bool almostDequalUlpsDouble(double a, double b) {
+  final double absA = a.abs();
+  final double absB = b.abs();
+  if (absA < kScalarMax && absB < kScalarMax) {
+    return almostDequalUlps(a, b);
+  }
+  return (a - b).abs() / math.max(absA, absB) < kDblEpsilonSubdivideErr;
+}
+
+const double kFltEpsilon = 1.19209290E-07; // == 1 / (2 ^ 23)
+const double kDblEpsilon = 2.22045e-16;
+const double kFltEpsilonCubed = kFltEpsilon * kFltEpsilon * kFltEpsilon;
+const double kFltEpsilonHalf = kFltEpsilon / 2;
+const double kFltEpsilonDouble = kFltEpsilon * 2;
+// Epsilon to use when ordering vectors.
+const double kFltEpsilonOrderableErr = kFltEpsilon * 16;
+const double kFltEpsilonSquared = kFltEpsilon * kFltEpsilon;
+// Use a compile-time constant for FLT_EPSILON_SQRT to avoid initializers.
+// A 17 digit constant guarantees exact results.
+const double kFltEpsilonSqrt = 0.00034526697709225118; // sqrt(kFltEpsilon);
+const double kFltEpsilonInverse = 1 / kFltEpsilon;
+const double kDblEpsilonErr = kDblEpsilon * 4;
+const double kDblEpsilonSubdivideErr = kDblEpsilon * 16;
+const double kRoughEpsilon = kFltEpsilon * 64;
+const double kMoreRoughEpsilon = kFltEpsilon * 256;
+const double kWayRoughEpsilon = kFltEpsilon * 2048;
+const double kBumpEpsilon = kFltEpsilon * 4096;
+
+// Scalar max is based on 32 bit float since [PathRef] stores values in
+// Float32List.
+const double kScalarMax = 3.402823466e+38;
+const double kScalarMin = -kScalarMax;

lib/web_ui/test/engine/ulps_test.dart

@@ -0,0 +1,88 @@
+// Copyright 2013 The Flutter Authors. All rights reserved.
+// Use of this source code is governed by a BSD-style license that can be
+// found in the LICENSE file.
+
+import 'dart:typed_data';
+import 'package:test/test.dart';
+import 'package:ui/src/engine.dart';
+
+void main() {
+  group('Float Int conversions', (){
+    test('Should convert signbit to 2\'s compliment', () {
+      expect(signBitTo2sCompliment(0), 0);
+      expect(signBitTo2sCompliment(0x7fffffff).toUnsigned(32), 0x7fffffff);
+      expect(signBitTo2sCompliment(0x80000000), 0);
+      expect(signBitTo2sCompliment(0x8f000000).toUnsigned(32), 0xf1000000);
+      expect(signBitTo2sCompliment(0x8fffffff).toUnsigned(32), 0xf0000001);
+      expect(signBitTo2sCompliment(0xffffffff).toUnsigned(32), 0x80000001);
+      expect(signBitTo2sCompliment(0x8f000000), -251658240);
+      expect(signBitTo2sCompliment(0x8fffffff), -268435455);
+      expect(signBitTo2sCompliment(0xffffffff), -2147483647);
+    });
+
+    test('Should convert 2s compliment to signbit', () {
+      expect(twosComplimentToSignBit(0), 0);
+      expect(twosComplimentToSignBit(0x7fffffff), 0x7fffffff);
+      expect(twosComplimentToSignBit(0), 0);
+      expect(twosComplimentToSignBit(0xf1000000).toRadixString(16), 0x8f000000.toRadixString(16));
+      expect(twosComplimentToSignBit(0xf0000001), 0x8fffffff);
+      expect(twosComplimentToSignBit(0x80000001), 0xffffffff);
+      expect(twosComplimentToSignBit(0x81234561), 0xfedcba9f);
+      expect(twosComplimentToSignBit(-5), 0x80000005);
+    });
+
+    test('Should convert float to bits', () {
+      Float32List floatList = Float32List(1);
+      floatList[0] = 0;
+      expect(float2Bits(floatList, 0), 0);
+      floatList[0] = 0.1;
+      expect(float2Bits(floatList, 0).toUnsigned(32).toRadixString(16), 0x3dcccccd.toRadixString(16));
+      floatList[0] = 123456.0;
+      expect(float2Bits(floatList, 0).toUnsigned(32).toRadixString(16), 0x47f12000.toRadixString(16));
+      floatList[0] = -0.1;
+      expect(float2Bits(floatList, 0).toUnsigned(32).toRadixString(16), 0xbdcccccd.toRadixString(16));
+      floatList[0] = -123456.0;
+      expect(float2Bits(floatList, 0).toUnsigned(32).toRadixString(16), 0xc7f12000.toRadixString(16));
+    });
+  });
+  group('Comparison', () {
+    test('Should compare equality based on ulps', () {
+      // If number of floats between a=1.1 and b are below 16, equals should
+      // return true.
+      final double a = 1.1;
+      int aBits = floatAs2sCompliment(a);
+      double b = twosComplimentAsFloat(aBits + 1);
+      expect(almostEqualUlps(a, b), true);
+      b = twosComplimentAsFloat(aBits + 15);
+      expect(almostEqualUlps(a, b), true);
+      b = twosComplimentAsFloat(aBits + 16);
+      expect(almostEqualUlps(a, b), false);
+
+      // Test between variant of equalUlps.
+      b = twosComplimentAsFloat(aBits + 1);
+      expect(almostBequalUlps(a, b), true);
+      b = twosComplimentAsFloat(aBits + 1);
+      expect(almostBequalUlps(a, b), true);
+      b = twosComplimentAsFloat(aBits + 2);
+      expect(almostBequalUlps(a, b), false);
+    });
+
+    test('Should compare 2 coordinates based on ulps', () {
+      double a = 1.1;
+      int aBits = floatAs2sCompliment(a);
+      double b = twosComplimentAsFloat(aBits + 1);
+      expect(approximatelyEqual(5.0, a, 5.0, b), true);
+      b = twosComplimentAsFloat(aBits + 16);
+      expect(approximatelyEqual(5.0, a, 5.0, b), true);
+
+      // Increase magnitude which should start checking with ulps rather than
+      // fltEpsilon.
+      a = 3000000.1;
+      aBits = floatAs2sCompliment(a);
+      b = twosComplimentAsFloat(aBits + 1);
+      expect(approximatelyEqual(5.0, a, 5.0, b), true);
+      b = twosComplimentAsFloat(aBits + 16);
+      expect(approximatelyEqual(5.0, a, 5.0, b), false);
+    });
+  });
+}