SkPathOpsTypes.cpp revision a2bbc6e19d5332e81784e582c290cc060f40c4c7
107393cab57ce74a4aae89a31fae9aaa9780fc19dcaryclark@google.com/* 207393cab57ce74a4aae89a31fae9aaa9780fc19dcaryclark@google.com * Copyright 2012 Google Inc. 307393cab57ce74a4aae89a31fae9aaa9780fc19dcaryclark@google.com * 407393cab57ce74a4aae89a31fae9aaa9780fc19dcaryclark@google.com * Use of this source code is governed by a BSD-style license that can be 507393cab57ce74a4aae89a31fae9aaa9780fc19dcaryclark@google.com * found in the LICENSE file. 607393cab57ce74a4aae89a31fae9aaa9780fc19dcaryclark@google.com */ 7cffbcc3b9665f2c928544b6fc6b8a0e22a4210fbcaryclark@google.com#include "SkFloatBits.h" 807393cab57ce74a4aae89a31fae9aaa9780fc19dcaryclark@google.com#include "SkPathOpsTypes.h" 907393cab57ce74a4aae89a31fae9aaa9780fc19dcaryclark@google.com 107eaa53d8f7e48fd17d02b5e3bd91f90e9c1899efcaryclark@google.comstatic bool arguments_denormalized(float a, float b, int epsilon) { 11a2bbc6e19d5332e81784e582c290cc060f40c4c7caryclark@google.com float denormalizedCheck = FLT_EPSILON * epsilon / 2; 12a2bbc6e19d5332e81784e582c290cc060f40c4c7caryclark@google.com return fabsf(a) <= denormalizedCheck && fabsf(b) <= denormalizedCheck; 137eaa53d8f7e48fd17d02b5e3bd91f90e9c1899efcaryclark@google.com} 147eaa53d8f7e48fd17d02b5e3bd91f90e9c1899efcaryclark@google.com 1507393cab57ce74a4aae89a31fae9aaa9780fc19dcaryclark@google.com// from http://randomascii.wordpress.com/2012/02/25/comparing-floating-point-numbers-2012-edition/ 16cffbcc3b9665f2c928544b6fc6b8a0e22a4210fbcaryclark@google.com// FIXME: move to SkFloatBits.h 17a2bbc6e19d5332e81784e582c290cc060f40c4c7caryclark@google.comstatic bool equal_ulps(float a, float b, int epsilon, int depsilon) { 18570863f2e22b8ea7d7c504bd15e4f766af097df2caryclark@google.com if (!SkScalarIsFinite(a) || !SkScalarIsFinite(b)) { 19570863f2e22b8ea7d7c504bd15e4f766af097df2caryclark@google.com return false; 2007393cab57ce74a4aae89a31fae9aaa9780fc19dcaryclark@google.com } 21a2bbc6e19d5332e81784e582c290cc060f40c4c7caryclark@google.com if (arguments_denormalized(a, b, depsilon)) { 227eaa53d8f7e48fd17d02b5e3bd91f90e9c1899efcaryclark@google.com return true; 237eaa53d8f7e48fd17d02b5e3bd91f90e9c1899efcaryclark@google.com } 247eaa53d8f7e48fd17d02b5e3bd91f90e9c1899efcaryclark@google.com int aBits = SkFloatAs2sCompliment(a); 257eaa53d8f7e48fd17d02b5e3bd91f90e9c1899efcaryclark@google.com int bBits = SkFloatAs2sCompliment(b); 267eaa53d8f7e48fd17d02b5e3bd91f90e9c1899efcaryclark@google.com // Find the difference in ULPs. 277eaa53d8f7e48fd17d02b5e3bd91f90e9c1899efcaryclark@google.com return aBits < bBits + epsilon && bBits < aBits + epsilon; 287eaa53d8f7e48fd17d02b5e3bd91f90e9c1899efcaryclark@google.com} 297eaa53d8f7e48fd17d02b5e3bd91f90e9c1899efcaryclark@google.com 307eaa53d8f7e48fd17d02b5e3bd91f90e9c1899efcaryclark@google.comstatic bool d_equal_ulps(float a, float b, int epsilon) { 317eaa53d8f7e48fd17d02b5e3bd91f90e9c1899efcaryclark@google.com if (!SkScalarIsFinite(a) || !SkScalarIsFinite(b)) { 327eaa53d8f7e48fd17d02b5e3bd91f90e9c1899efcaryclark@google.com return false; 337eaa53d8f7e48fd17d02b5e3bd91f90e9c1899efcaryclark@google.com } 34570863f2e22b8ea7d7c504bd15e4f766af097df2caryclark@google.com int aBits = SkFloatAs2sCompliment(a); 35570863f2e22b8ea7d7c504bd15e4f766af097df2caryclark@google.com int bBits = SkFloatAs2sCompliment(b); 3607393cab57ce74a4aae89a31fae9aaa9780fc19dcaryclark@google.com // Find the difference in ULPs. 37570863f2e22b8ea7d7c504bd15e4f766af097df2caryclark@google.com return aBits < bBits + epsilon && bBits < aBits + epsilon; 38570863f2e22b8ea7d7c504bd15e4f766af097df2caryclark@google.com} 39570863f2e22b8ea7d7c504bd15e4f766af097df2caryclark@google.com 40570863f2e22b8ea7d7c504bd15e4f766af097df2caryclark@google.comstatic bool not_equal_ulps(float a, float b, int epsilon) { 41570863f2e22b8ea7d7c504bd15e4f766af097df2caryclark@google.com if (!SkScalarIsFinite(a) || !SkScalarIsFinite(b)) { 42570863f2e22b8ea7d7c504bd15e4f766af097df2caryclark@google.com return false; 43570863f2e22b8ea7d7c504bd15e4f766af097df2caryclark@google.com } 447eaa53d8f7e48fd17d02b5e3bd91f90e9c1899efcaryclark@google.com if (arguments_denormalized(a, b, epsilon)) { 457eaa53d8f7e48fd17d02b5e3bd91f90e9c1899efcaryclark@google.com return false; 467eaa53d8f7e48fd17d02b5e3bd91f90e9c1899efcaryclark@google.com } 477eaa53d8f7e48fd17d02b5e3bd91f90e9c1899efcaryclark@google.com int aBits = SkFloatAs2sCompliment(a); 487eaa53d8f7e48fd17d02b5e3bd91f90e9c1899efcaryclark@google.com int bBits = SkFloatAs2sCompliment(b); 497eaa53d8f7e48fd17d02b5e3bd91f90e9c1899efcaryclark@google.com // Find the difference in ULPs. 507eaa53d8f7e48fd17d02b5e3bd91f90e9c1899efcaryclark@google.com return aBits >= bBits + epsilon || bBits >= aBits + epsilon; 517eaa53d8f7e48fd17d02b5e3bd91f90e9c1899efcaryclark@google.com} 527eaa53d8f7e48fd17d02b5e3bd91f90e9c1899efcaryclark@google.com 537eaa53d8f7e48fd17d02b5e3bd91f90e9c1899efcaryclark@google.comstatic bool d_not_equal_ulps(float a, float b, int epsilon) { 547eaa53d8f7e48fd17d02b5e3bd91f90e9c1899efcaryclark@google.com if (!SkScalarIsFinite(a) || !SkScalarIsFinite(b)) { 557eaa53d8f7e48fd17d02b5e3bd91f90e9c1899efcaryclark@google.com return false; 567eaa53d8f7e48fd17d02b5e3bd91f90e9c1899efcaryclark@google.com } 57570863f2e22b8ea7d7c504bd15e4f766af097df2caryclark@google.com int aBits = SkFloatAs2sCompliment(a); 58570863f2e22b8ea7d7c504bd15e4f766af097df2caryclark@google.com int bBits = SkFloatAs2sCompliment(b); 59570863f2e22b8ea7d7c504bd15e4f766af097df2caryclark@google.com // Find the difference in ULPs. 60570863f2e22b8ea7d7c504bd15e4f766af097df2caryclark@google.com return aBits >= bBits + epsilon || bBits >= aBits + epsilon; 6107e97fccd2d85076cd22ef411b0773ab92a18abecaryclark@google.com} 6207e97fccd2d85076cd22ef411b0773ab92a18abecaryclark@google.com 634fdbb229649caf74e5c1b55a1823926df903af34caryclark@google.comstatic bool less_ulps(float a, float b, int epsilon) { 64570863f2e22b8ea7d7c504bd15e4f766af097df2caryclark@google.com if (!SkScalarIsFinite(a) || !SkScalarIsFinite(b)) { 65570863f2e22b8ea7d7c504bd15e4f766af097df2caryclark@google.com return false; 66fa2aeee27af27f2934ee52a9732148f66481fb03caryclark@google.com } 677eaa53d8f7e48fd17d02b5e3bd91f90e9c1899efcaryclark@google.com if (arguments_denormalized(a, b, epsilon)) { 687eaa53d8f7e48fd17d02b5e3bd91f90e9c1899efcaryclark@google.com return a <= b - FLT_EPSILON * epsilon; 697eaa53d8f7e48fd17d02b5e3bd91f90e9c1899efcaryclark@google.com } 70570863f2e22b8ea7d7c504bd15e4f766af097df2caryclark@google.com int aBits = SkFloatAs2sCompliment(a); 71570863f2e22b8ea7d7c504bd15e4f766af097df2caryclark@google.com int bBits = SkFloatAs2sCompliment(b); 72fa2aeee27af27f2934ee52a9732148f66481fb03caryclark@google.com // Find the difference in ULPs. 73570863f2e22b8ea7d7c504bd15e4f766af097df2caryclark@google.com return aBits <= bBits - epsilon; 74570863f2e22b8ea7d7c504bd15e4f766af097df2caryclark@google.com} 75570863f2e22b8ea7d7c504bd15e4f766af097df2caryclark@google.com 76570863f2e22b8ea7d7c504bd15e4f766af097df2caryclark@google.comstatic bool less_or_equal_ulps(float a, float b, int epsilon) { 77570863f2e22b8ea7d7c504bd15e4f766af097df2caryclark@google.com if (!SkScalarIsFinite(a) || !SkScalarIsFinite(b)) { 78570863f2e22b8ea7d7c504bd15e4f766af097df2caryclark@google.com return false; 79570863f2e22b8ea7d7c504bd15e4f766af097df2caryclark@google.com } 807eaa53d8f7e48fd17d02b5e3bd91f90e9c1899efcaryclark@google.com if (arguments_denormalized(a, b, epsilon)) { 817eaa53d8f7e48fd17d02b5e3bd91f90e9c1899efcaryclark@google.com return a < b + FLT_EPSILON * epsilon; 827eaa53d8f7e48fd17d02b5e3bd91f90e9c1899efcaryclark@google.com } 83570863f2e22b8ea7d7c504bd15e4f766af097df2caryclark@google.com int aBits = SkFloatAs2sCompliment(a); 84570863f2e22b8ea7d7c504bd15e4f766af097df2caryclark@google.com int bBits = SkFloatAs2sCompliment(b); 85570863f2e22b8ea7d7c504bd15e4f766af097df2caryclark@google.com // Find the difference in ULPs. 86570863f2e22b8ea7d7c504bd15e4f766af097df2caryclark@google.com return aBits < bBits + epsilon; 87570863f2e22b8ea7d7c504bd15e4f766af097df2caryclark@google.com} 88570863f2e22b8ea7d7c504bd15e4f766af097df2caryclark@google.com 89570863f2e22b8ea7d7c504bd15e4f766af097df2caryclark@google.com// equality using the same error term as between 90570863f2e22b8ea7d7c504bd15e4f766af097df2caryclark@google.combool AlmostBequalUlps(float a, float b) { 91570863f2e22b8ea7d7c504bd15e4f766af097df2caryclark@google.com const int UlpsEpsilon = 2; 92a2bbc6e19d5332e81784e582c290cc060f40c4c7caryclark@google.com return equal_ulps(a, b, UlpsEpsilon, UlpsEpsilon); 93a2bbc6e19d5332e81784e582c290cc060f40c4c7caryclark@google.com} 94a2bbc6e19d5332e81784e582c290cc060f40c4c7caryclark@google.com 95a2bbc6e19d5332e81784e582c290cc060f40c4c7caryclark@google.combool AlmostPequalUlps(float a, float b) { 96a2bbc6e19d5332e81784e582c290cc060f40c4c7caryclark@google.com const int UlpsEpsilon = 8; 97a2bbc6e19d5332e81784e582c290cc060f40c4c7caryclark@google.com return equal_ulps(a, b, UlpsEpsilon, UlpsEpsilon); 98fa2aeee27af27f2934ee52a9732148f66481fb03caryclark@google.com} 99fa2aeee27af27f2934ee52a9732148f66481fb03caryclark@google.com 1007eaa53d8f7e48fd17d02b5e3bd91f90e9c1899efcaryclark@google.combool AlmostDequalUlps(float a, float b) { 1017eaa53d8f7e48fd17d02b5e3bd91f90e9c1899efcaryclark@google.com const int UlpsEpsilon = 16; 1027eaa53d8f7e48fd17d02b5e3bd91f90e9c1899efcaryclark@google.com return d_equal_ulps(a, b, UlpsEpsilon); 1037eaa53d8f7e48fd17d02b5e3bd91f90e9c1899efcaryclark@google.com} 1047eaa53d8f7e48fd17d02b5e3bd91f90e9c1899efcaryclark@google.com 1054fdbb229649caf74e5c1b55a1823926df903af34caryclark@google.combool AlmostEqualUlps(float a, float b) { 10607e97fccd2d85076cd22ef411b0773ab92a18abecaryclark@google.com const int UlpsEpsilon = 16; 107a2bbc6e19d5332e81784e582c290cc060f40c4c7caryclark@google.com return equal_ulps(a, b, UlpsEpsilon, UlpsEpsilon); 10807e97fccd2d85076cd22ef411b0773ab92a18abecaryclark@google.com} 10907e97fccd2d85076cd22ef411b0773ab92a18abecaryclark@google.com 110570863f2e22b8ea7d7c504bd15e4f766af097df2caryclark@google.combool NotAlmostEqualUlps(float a, float b) { 111570863f2e22b8ea7d7c504bd15e4f766af097df2caryclark@google.com const int UlpsEpsilon = 16; 112570863f2e22b8ea7d7c504bd15e4f766af097df2caryclark@google.com return not_equal_ulps(a, b, UlpsEpsilon); 113570863f2e22b8ea7d7c504bd15e4f766af097df2caryclark@google.com} 114570863f2e22b8ea7d7c504bd15e4f766af097df2caryclark@google.com 1157eaa53d8f7e48fd17d02b5e3bd91f90e9c1899efcaryclark@google.combool NotAlmostDequalUlps(float a, float b) { 1167eaa53d8f7e48fd17d02b5e3bd91f90e9c1899efcaryclark@google.com const int UlpsEpsilon = 16; 1177eaa53d8f7e48fd17d02b5e3bd91f90e9c1899efcaryclark@google.com return d_not_equal_ulps(a, b, UlpsEpsilon); 1187eaa53d8f7e48fd17d02b5e3bd91f90e9c1899efcaryclark@google.com} 1197eaa53d8f7e48fd17d02b5e3bd91f90e9c1899efcaryclark@google.com 1204fdbb229649caf74e5c1b55a1823926df903af34caryclark@google.combool RoughlyEqualUlps(float a, float b) { 12107e97fccd2d85076cd22ef411b0773ab92a18abecaryclark@google.com const int UlpsEpsilon = 256; 122a2bbc6e19d5332e81784e582c290cc060f40c4c7caryclark@google.com const int DUlpsEpsilon = 1024; 123a2bbc6e19d5332e81784e582c290cc060f40c4c7caryclark@google.com return equal_ulps(a, b, UlpsEpsilon, DUlpsEpsilon); 12407393cab57ce74a4aae89a31fae9aaa9780fc19dcaryclark@google.com} 12507393cab57ce74a4aae89a31fae9aaa9780fc19dcaryclark@google.com 126fa2aeee27af27f2934ee52a9732148f66481fb03caryclark@google.combool AlmostBetweenUlps(float a, float b, float c) { 127570863f2e22b8ea7d7c504bd15e4f766af097df2caryclark@google.com const int UlpsEpsilon = 2; 128570863f2e22b8ea7d7c504bd15e4f766af097df2caryclark@google.com return a <= c ? less_or_equal_ulps(a, b, UlpsEpsilon) && less_or_equal_ulps(b, c, UlpsEpsilon) 129570863f2e22b8ea7d7c504bd15e4f766af097df2caryclark@google.com : less_or_equal_ulps(b, a, UlpsEpsilon) && less_or_equal_ulps(c, b, UlpsEpsilon); 130570863f2e22b8ea7d7c504bd15e4f766af097df2caryclark@google.com} 131570863f2e22b8ea7d7c504bd15e4f766af097df2caryclark@google.com 132570863f2e22b8ea7d7c504bd15e4f766af097df2caryclark@google.combool AlmostLessUlps(float a, float b) { 133570863f2e22b8ea7d7c504bd15e4f766af097df2caryclark@google.com const int UlpsEpsilon = 16; 134570863f2e22b8ea7d7c504bd15e4f766af097df2caryclark@google.com return less_ulps(a, b, UlpsEpsilon); 135570863f2e22b8ea7d7c504bd15e4f766af097df2caryclark@google.com} 136570863f2e22b8ea7d7c504bd15e4f766af097df2caryclark@google.com 137570863f2e22b8ea7d7c504bd15e4f766af097df2caryclark@google.combool AlmostLessOrEqualUlps(float a, float b) { 138570863f2e22b8ea7d7c504bd15e4f766af097df2caryclark@google.com const int UlpsEpsilon = 16; 139570863f2e22b8ea7d7c504bd15e4f766af097df2caryclark@google.com return less_or_equal_ulps(a, b, UlpsEpsilon); 140fa2aeee27af27f2934ee52a9732148f66481fb03caryclark@google.com} 141fa2aeee27af27f2934ee52a9732148f66481fb03caryclark@google.com 1424fdbb229649caf74e5c1b55a1823926df903af34caryclark@google.comint UlpsDistance(float a, float b) { 143570863f2e22b8ea7d7c504bd15e4f766af097df2caryclark@google.com if (!SkScalarIsFinite(a) || !SkScalarIsFinite(b)) { 144570863f2e22b8ea7d7c504bd15e4f766af097df2caryclark@google.com return SK_MaxS32; 145570863f2e22b8ea7d7c504bd15e4f766af097df2caryclark@google.com } 1464fdbb229649caf74e5c1b55a1823926df903af34caryclark@google.com SkFloatIntUnion floatIntA, floatIntB; 1474fdbb229649caf74e5c1b55a1823926df903af34caryclark@google.com floatIntA.fFloat = a; 1484fdbb229649caf74e5c1b55a1823926df903af34caryclark@google.com floatIntB.fFloat = b; 1494fdbb229649caf74e5c1b55a1823926df903af34caryclark@google.com // Different signs means they do not match. 1504fdbb229649caf74e5c1b55a1823926df903af34caryclark@google.com if ((floatIntA.fSignBitInt < 0) != (floatIntB.fSignBitInt < 0)) { 1514fdbb229649caf74e5c1b55a1823926df903af34caryclark@google.com // Check for equality to make sure +0 == -0 1524fdbb229649caf74e5c1b55a1823926df903af34caryclark@google.com return a == b ? 0 : SK_MaxS32; 1534fdbb229649caf74e5c1b55a1823926df903af34caryclark@google.com } 1544fdbb229649caf74e5c1b55a1823926df903af34caryclark@google.com // Find the difference in ULPs. 1554fdbb229649caf74e5c1b55a1823926df903af34caryclark@google.com return abs(floatIntA.fSignBitInt - floatIntB.fSignBitInt); 1564fdbb229649caf74e5c1b55a1823926df903af34caryclark@google.com} 1574fdbb229649caf74e5c1b55a1823926df903af34caryclark@google.com 15807393cab57ce74a4aae89a31fae9aaa9780fc19dcaryclark@google.com// cube root approximation using bit hack for 64-bit float 15907393cab57ce74a4aae89a31fae9aaa9780fc19dcaryclark@google.com// adapted from Kahan's cbrt 160cffbcc3b9665f2c928544b6fc6b8a0e22a4210fbcaryclark@google.comstatic double cbrt_5d(double d) { 16107393cab57ce74a4aae89a31fae9aaa9780fc19dcaryclark@google.com const unsigned int B1 = 715094163; 16207393cab57ce74a4aae89a31fae9aaa9780fc19dcaryclark@google.com double t = 0.0; 16307393cab57ce74a4aae89a31fae9aaa9780fc19dcaryclark@google.com unsigned int* pt = (unsigned int*) &t; 16407393cab57ce74a4aae89a31fae9aaa9780fc19dcaryclark@google.com unsigned int* px = (unsigned int*) &d; 16507393cab57ce74a4aae89a31fae9aaa9780fc19dcaryclark@google.com pt[1] = px[1] / 3 + B1; 16607393cab57ce74a4aae89a31fae9aaa9780fc19dcaryclark@google.com return t; 16707393cab57ce74a4aae89a31fae9aaa9780fc19dcaryclark@google.com} 16807393cab57ce74a4aae89a31fae9aaa9780fc19dcaryclark@google.com 16907393cab57ce74a4aae89a31fae9aaa9780fc19dcaryclark@google.com// iterative cube root approximation using Halley's method (double) 170cffbcc3b9665f2c928544b6fc6b8a0e22a4210fbcaryclark@google.comstatic double cbrta_halleyd(const double a, const double R) { 17107393cab57ce74a4aae89a31fae9aaa9780fc19dcaryclark@google.com const double a3 = a * a * a; 17207393cab57ce74a4aae89a31fae9aaa9780fc19dcaryclark@google.com const double b = a * (a3 + R + R) / (a3 + a3 + R); 17307393cab57ce74a4aae89a31fae9aaa9780fc19dcaryclark@google.com return b; 17407393cab57ce74a4aae89a31fae9aaa9780fc19dcaryclark@google.com} 17507393cab57ce74a4aae89a31fae9aaa9780fc19dcaryclark@google.com 17607393cab57ce74a4aae89a31fae9aaa9780fc19dcaryclark@google.com// cube root approximation using 3 iterations of Halley's method (double) 17707393cab57ce74a4aae89a31fae9aaa9780fc19dcaryclark@google.comstatic double halley_cbrt3d(double d) { 17807393cab57ce74a4aae89a31fae9aaa9780fc19dcaryclark@google.com double a = cbrt_5d(d); 17907393cab57ce74a4aae89a31fae9aaa9780fc19dcaryclark@google.com a = cbrta_halleyd(a, d); 18007393cab57ce74a4aae89a31fae9aaa9780fc19dcaryclark@google.com a = cbrta_halleyd(a, d); 18107393cab57ce74a4aae89a31fae9aaa9780fc19dcaryclark@google.com return cbrta_halleyd(a, d); 18207393cab57ce74a4aae89a31fae9aaa9780fc19dcaryclark@google.com} 18307393cab57ce74a4aae89a31fae9aaa9780fc19dcaryclark@google.com 18407393cab57ce74a4aae89a31fae9aaa9780fc19dcaryclark@google.comdouble SkDCubeRoot(double x) { 18507393cab57ce74a4aae89a31fae9aaa9780fc19dcaryclark@google.com if (approximately_zero_cubed(x)) { 18607393cab57ce74a4aae89a31fae9aaa9780fc19dcaryclark@google.com return 0; 18707393cab57ce74a4aae89a31fae9aaa9780fc19dcaryclark@google.com } 18807393cab57ce74a4aae89a31fae9aaa9780fc19dcaryclark@google.com double result = halley_cbrt3d(fabs(x)); 18907393cab57ce74a4aae89a31fae9aaa9780fc19dcaryclark@google.com if (x < 0) { 19007393cab57ce74a4aae89a31fae9aaa9780fc19dcaryclark@google.com result = -result; 19107393cab57ce74a4aae89a31fae9aaa9780fc19dcaryclark@google.com } 19207393cab57ce74a4aae89a31fae9aaa9780fc19dcaryclark@google.com return result; 19307393cab57ce74a4aae89a31fae9aaa9780fc19dcaryclark@google.com} 194