Add Angle::sinCosSnap to avoid small errors, e.g. with buffer operations

[mwtoews]

This is a rather pedantic PR to avoid small rounding errors from std::sin(M_PI) and std::cos(M_PI_2), which are small non-zero values.

Consider a unit buffer geosop -a "POINT(0 0)" buffer 1 here with GEOS 3.12.0:
_{POLYGON ((1 0, 0.9807852804032304 -0.1950903220161282, 0.9238795325112867 -0.3826834323650898, 0.8314696123025452 -0.5555702330196022, 0.7071067811865476 -0.7071067811865475, 0.5555702330196023 -0.8314696123025452, 0.3826834323650898 -0.9238795325112867, 0.1950903220161283 -0.9807852804032304, 0.0000000000000001 -1, -0.1950903220161282 -0.9807852804032304, -0.3826834323650897 -0.9238795325112867, -0.555570233019602 -0.8314696123025455, -0.7071067811865475 -0.7071067811865476, -0.8314696123025453 -0.5555702330196022, -0.9238795325112867 -0.3826834323650899, -0.9807852804032304 -0.1950903220161286, -1 -0.0000000000000001, -0.9807852804032304 0.1950903220161284, -0.9238795325112868 0.3826834323650897, -0.8314696123025455 0.555570233019602, -0.7071067811865477 0.7071067811865475, -0.5555702330196022 0.8314696123025452, -0.3826834323650903 0.9238795325112865, -0.1950903220161287 0.9807852804032303, -0.0000000000000002 1, 0.1950903220161283 0.9807852804032304, 0.38268343236509 0.9238795325112866, 0.5555702330196018 0.8314696123025455, 0.7071067811865474 0.7071067811865477, 0.8314696123025452 0.5555702330196022, 0.9238795325112865 0.3826834323650904, 0.9807852804032303 0.1950903220161287, 1 0))}

and since #973 for main the small values are formatted using scientific notation:
_{POLYGON ((1 0, 0.9807852804032304 -0.1950903220161282, 0.9238795325112867 -0.3826834323650898, 0.8314696123025452 -0.5555702330196022, 0.7071067811865476 -0.7071067811865475, 0.5555702330196023 -0.8314696123025452, 0.3826834323650898 -0.9238795325112867, 0.1950903220161283 -0.9807852804032304, 6.123233995736766e-17 -1, -0.1950903220161282 -0.9807852804032304, -0.3826834323650897 -0.9238795325112867, -0.555570233019602 -0.8314696123025455, -0.7071067811865475 -0.7071067811865476, -0.8314696123025453 -0.5555702330196022, -0.9238795325112867 -0.3826834323650899, -0.9807852804032304 -0.1950903220161286, -1 -1.2246467991473532e-16, -0.9807852804032304 0.1950903220161284, -0.9238795325112868 0.3826834323650897, -0.8314696123025455 0.555570233019602, -0.7071067811865477 0.7071067811865475, -0.5555702330196022 0.8314696123025452, -0.3826834323650903 0.9238795325112865, -0.1950903220161287 0.9807852804032303, -1.8369701987210297e-16 1, 0.1950903220161283 0.9807852804032304, 0.38268343236509 0.9238795325112866, 0.5555702330196018 0.8314696123025455, 0.7071067811865474 0.7071067811865477, 0.8314696123025452 0.5555702330196022, 0.9238795325112865 0.3826834323650904, 0.9807852804032303 0.1950903220161287, 1 0))}

this PR would use exact values if the angles are a multiple of 45 degrees (pi/4) on the unit circle:
_{POLYGON ((1 0, 0.9807852804032304 -0.1950903220161282, 0.9238795325112867 -0.3826834323650898, 0.8314696123025452 -0.5555702330196022, 0.7071067811865476 -0.7071067811865476, 0.5555702330196023 -0.8314696123025452, 0.3826834323650898 -0.9238795325112867, 0.1950903220161283 -0.9807852804032304, 0 -1, -0.1950903220161282 -0.9807852804032304, -0.3826834323650897 -0.9238795325112867, -0.555570233019602 -0.8314696123025455, -0.7071067811865476 -0.7071067811865476, -0.8314696123025453 -0.5555702330196022, -0.9238795325112867 -0.3826834323650899, -0.9807852804032304 -0.1950903220161286, -1 0, -0.9807852804032304 0.1950903220161284, -0.9238795325112868 0.3826834323650897, -0.8314696123025455 0.555570233019602, -0.7071067811865476 0.7071067811865476, -0.5555702330196022 0.8314696123025452, -0.3826834323650903 0.9238795325112865, -0.1950903220161287 0.9807852804032303, 0 1, 0.1950903220161283 0.9807852804032304, 0.38268343236509 0.9238795325112866, 0.5555702330196018 0.8314696123025455, 0.7071067811865476 0.7071067811865476, 0.8314696123025452 0.5555702330196022, 0.9238795325112865 0.3826834323650904, 0.9807852804032303 0.1950903220161287, 1 0))}

---

Other changes in this PR use angle constants a bit more consistently, an also share Angle::sinCos in other places, where both calculations are performed.

Note that an alternative proposal for a much simpler Angle::sinCos would only zap small values from std::sin and std::cos to zero, rather than a lookup approach on 45 degrees of a unit circle currently in this PR. (Update: that's actually the one that ultimately ended up in this PR.)

[pramsey]

Are these called infrequently? Is there any advantage to inlining?

[mwtoews]

A typical point buffer will call these trig functions 33 times. I'll investigate inlining... (I'm still novice with C++)

[mwtoews]

I've been checking the performance of this PR with mpgrid.txt (a 20x20 MultiPoint grid):

./bin/geosop -a mpgrid.txt buffer 0.1 -v -r 100 > /dev/null

and on my oldish laptop the results show a minor slowdown, e.g. 810,667 usec with main vs. 822,922 usec with this PR.

I'm unsure how to restructure this PR as an inline function to check the function overhead. Tips welcome!

[dbaston]

Hi @mwtoews , you can move the implementation of Angle::sinCos under its declaration in Angle.h to inline it.

[mwtoews]

Thanks @dbaston that was helpful! I've not pushed the inline version here, but can if needed.

Running the same multipoint grid example shown above, I'm seeing pretty much the same range of benchmarks for three groups: (1) from main, (2) this PR and (3) an inline version of this PR. There isn't any clear outlier. I'm inclined to stick with the current form. Further review or ideas are welcome.

[mwtoews]

Another idea is to cache 32 pairs of results. Running ctest and capturing the frequency of angles shows some prominent hits, Here is the number of counts for the top 200 hits:

Rank, count, angle, turns

1. count 1495 at -90.0° or -1/4 turns
2. count 1453 at 90.0° or 1/4 turns
3. count 1375 at -45.0° or -1/8 turns
4. count 1372 at -112.5° or -5/16 turns
5. count 1371 at -101.25° or -9/32 turns
6. count 1371 at -123.75° or -11/32 turns
7. count 1370 at -135.0° or -3/8 turns
8. count 1368 at -67.5° or -3/16 turns
9. count 1366 at -157.5° or -7/16 turns
10. count 1364 at -56.25° or -5/32 turns
11. count 1364 at -78.75° or -7/32 turns
12. count 1363 at -146.25° or -13/32 turns
13. count 1363 at -168.75° or -15/32 turns
14. count 1362 at -180.0° or -1/2 turns
15. count 1360 at -22.5° or -1/16 turns
16. count 1358 at -11.25° or -1/32 turns
17. count 1358 at -33.75° or -3/32 turns
18. count 1289 at 135.0° or 3/8 turns
19. count 1277 at 112.5° or 5/16 turns
20. count 1275 at 123.75° or 11/32 turns
21. count 1275 at 101.25° or 9/32 turns
22. count 1266 at 157.5° or 7/16 turns
23. count 1264 at 168.75° or 15/32 turns
24. count 1264 at 146.25° or 13/32 turns
25. count 1247 at 0.0° or 0 turns
26. count 1236 at 45.0° or 1/8 turns
27. count 1224 at 67.5° or 3/16 turns
28. count 1222 at 78.75° or 7/32 turns
29. count 1222 at 56.25° or 5/32 turns
30. count 1222 at 33.75° or 3/32 turns
31. count 1222 at 22.5° or 1/16 turns
32. count 1222 at 11.25° or 1/32 turns
33. count 68 at 180.0° or 1/2 turns
34. count 55 at 225.0° or 5/8 turns
35. count 52 at 270.0° or 3/4 turns
36. count 50 at 213.75° or 19/32 turns
37. count 50 at 202.5° or 9/16 turns
38. count 50 at 191.25° or 17/32 turns
39. count 45 at 258.75° or 23/32 turns
40. count 45 at 247.5° or 11/16 turns
41. count 45 at 236.25° or 21/32 turns
42. count 41 at -57.3808° or -5742669988849355/36028797018963968 turns
43. count 40 at 3.57633° or 5726705228570285/576460752303423488 turns
44. count 37 at 59.3493° or 1484919363137221/9007199254740992 turns
45. count 31 at 68.1986° or 3412657661635439/18014398509481984 turns
46. count 30 at -202.5° or -9/16 turns
47. count 29 at -191.25° or -17/32 turns
48. count 29 at -213.75° or -19/32 turns
49. count 28 at 60.2551° or 3015164954524133/18014398509481984 turns
50. count 27 at -68.2538° or -3415419869406893/18014398509481984 turns
51. count 27 at -79.1269° or -3959509748388695/18014398509481984 turns
52. count 27 at -120.651° or -1509343885116327/4503599627370496 turns
53. count 26 at -119.745° or -5992039304216445/18014398509481984 turns
54. count 25 at 123.69° or 1547361771970713/4503599627370496 turns
55. count 24 at 183.576° or 2296535569983795/4503599627370496 turns
56. count 22 at -114.407° or -5724925806317515/18014398509481984 turns
57. count 22 at 77.7965° or 7785873075793973/36028797018963968 turns
58. count 22 at 65.593° or 6564546896846955/36028797018963968 turns
59. count 22 at 53.3894° or 1335802677475191/9007199254740992 turns
60. count 22 at 7.12502° or 5704575540801303/288230376151711744 turns
61. count 21 at -270.0° or -3/4 turns
62. count 21 at -102.204° or -5114287736841935/18014398509481984 turns
63. count 21 at -126.611° or -3167806957894477/9007199254740992 turns
64. count 20 at 122.619° or 6135854252317143/18014398509481984 turns
65. count 20 at 145.491° or 3640184518809783/9007199254740992 turns
66. count 20 at -17.354° or -6947152705190007/144115188075855872 turns
67. count 20 at -72.2553° or -7231309825678743/36028797018963968 turns
68. count 20 at -136.848° or -6847873353404419/18014398509481984 turns
69. count 19 at -225.0° or -5/8 turns
70. count 19 at 112.834° or 5646212892830251/18014398509481984 turns
71. count 18 at -315.0° or -7/8 turns
72. count 18 at -247.5° or -11/16 turns
73. count 18 at 43.1524° or 8637383669339671/72057594037927936 turns
74. count 17 at -292.5° or -13/16 turns
75. count 17 at -337.5° or -15/16 turns
76. count 17 at -236.25° or -21/32 turns
77. count 17 at -258.75° or -23/32 turns
78. count 17 at -66.6148° or -6666808632385781/36028797018963968 turns
79. count 17 at -111.801° or -5594521577107209/18014398509481984 turns
80. count 17 at -138.814° or -6946251985264533/18014398509481984 turns
81. count 16 at -281.25° or -25/32 turns
82. count 16 at -303.75° or -27/32 turns
83. count 16 at -326.25° or -29/32 turns
84. count 16 at -348.75° or -31/32 turns
85. count 16 at 79.4642° or 1988194119496081/9007199254740992 turns
86. count 16 at 68.9283° or 6898343693228483/36028797018963968 turns
87. count 16 at 101.417° or 5074906260100373/18014398509481984 turns
88. count 16 at 58.3925° or 5843920916471815/36028797018963968 turns
89. count 16 at 144.293° or 3610210561289839/9007199254740992 turns
90. count 16 at -67.1663° or -6722002747818999/36028797018963968 turns
91. count 15 at 41.1859° or 8243769061907489/72057594037927936 turns
92. count 15 at 42.1844° or 8443628805371019/72057594037927936 turns
93. count 15 at 5.35583° or 8576182752803457/576460752303423488 turns
94. count 15 at -78.3074° or -3918501971781693/18014398509481984 turns
95. count 14 at -56.3099° or -5635494325717109/36028797018963968 turns
96. count 14 at 79.7831° or 7984691987343623/36028797018963968 turns
97. count 14 at 69.5662° or 6962184719946253/36028797018963968 turns
98. count 14 at 112.46° or 5627497934378733/18014398509481984 turns
99. count 14 at 101.23° or 5065548780874615/18014398509481984 turns
100. count 14 at -137.816° or -6896312069396581/18014398509481984 turns
101. count 14 at -81.1277° or -4059629772104727/18014398509481984 turns
102. count 13 at 26.5651° or 2658634988023555/36028797018963968 turns
103. count 13 at 10.6197° or 4251277952247685/144115188075855872 turns
104. count 13 at -100.217° or -5014858265068767/18014398509481984 turns
105. count 13 at -110.434° or -5526116902767037/18014398509481984 turns
106. count 12 at 50.2676° or 2515390495875657/18014398509481984 turns
107. count 12 at -169.38° or -8475774498711273/18014398509481984 turns
108. count 12 at -121.608° or -3042631908251507/9007199254740992 turns
109. count 12 at -78.5832° or -3932303002639791/18014398509481984 turns
110. count 11 at -132.322° or -3310696166071771/9007199254740992 turns
111. count 11 at 79.0993° or 7916257289005935/36028797018963968 turns
112. count 11 at 111.746° or 5591769377334927/18014398509481984 turns
113. count 11 at 100.873° or 630960562794089/2251799813685248 turns
114. count 11 at -129.732° or -6491788742866991/18014398509481984 turns
115. count 10 at 51.3402° or 5138126790869483/36028797018963968 turns
116. count 10 at 80.085° or 2003726534210923/9007199254740992 turns
117. count 10 at 70.1701° or 3511311513472783/18014398509481984 turns
118. count 10 at 132.436° or 6627096891671545/18014398509481984 turns
119. count 10 at 162.646° or 508675322912015/1125899906842624 turns
120. count 10 at 113.385° or 2836892465274465/9007199254740992 turns
121. count 10 at -35.7067° or -3573526240325113/36028797018963968 turns
122. count 10 at -67.54° or -3379701320362259/18014398509481984 turns
123. count 10 at -78.77° or -3941650473866377/18014398509481984 turns
124. count 10 at -100.536° or -2515410511874001/9007199254740992 turns
125. count 10 at -111.072° or -1389510605031377/4503599627370496 turns
126. count 10 at -127.972° or -400232396884623/1125899906842624 turns
127. count 10 at -100.901° or -2524542811118391/9007199254740992 turns
128. count 10 at -100.803° or -2522090851321267/9007199254740992 turns
129. count 10 at -111.606° or -5584763777914573/18014398509481984 turns
130. count 10 at -122.409° or -1531336463296653/4503599627370496 turns
131. count 10 at -133.212° or -3332963964229325/9007199254740992 turns
132. count 10 at -144.015° or -28150429702073/70368744177664 turns
133. count 10 at -154.818° or -968386509875341/2251799813685248 turns
134. count 10 at -165.621° or -4143837077137383/9007199254740992 turns
135. count 9 at 315.0° or 7/8 turns
136. count 9 at 79.197° or 7926035104196915/36028797018963968 turns
137. count 9 at 68.3941° or 6844880961652009/36028797018963968 turns
138. count 9 at 57.5911° or 1440929202776983/9007199254740992 turns
139. count 9 at 46.7882° or 1170640667140757/9007199254740992 turns
140. count 9 at 35.9852° or 7202797036037901/72057594037927936 turns
141. count 9 at 25.1823° or 5040488750948091/72057594037927936 turns
142. count 9 at 14.3793° or 5756320899719873/144115188075855872 turns
143. count 9 at 190.62° or 4769312005385355/9007199254740992 turns
144. count 9 at 143.13° or 7162224607394879/18014398509481984 turns
145. count 9 at -112.306° or -5619791775016343/18014398509481984 turns
146. count 9 at -99.915° or -2499873093159573/9007199254740992 turns
147. count 9 at -109.83° or -5495892745267795/18014398509481984 turns
148. count 9 at 187.125° or 2340933556310289/4503599627370496 turns
149. count 9 at -34.5085° or -6907220788493989/72057594037927936 turns
150. count 9 at -100.581° or -5033072823561687/18014398509481984 turns
151. count 9 at -111.161° or -5562495979757019/18014398509481984 turns
152. count 9 at -121.742° or -3045984587974105/9007199254740992 turns
153. count 9 at -142.903° or -7150865528334733/18014398509481984 turns
154. count 9 at -153.483° or -7680288684530065/18014398509481984 turns
155. count 9 at -164.064° or -1026220235090157/2251799813685248 turns
156. count 8 at 32.0054° or 100096880217947/1125899906842624 turns
157. count 8 at 89.8461° or 4495898472007693/18014398509481984 turns
158. count 8 at -29.7449° or -1488434672846363/18014398509481984 turns
159. count 8 at -40.7004° or -4073295694974003/36028797018963968 turns
160. count 8 at -51.6558° or -1292428009064027/9007199254740992 turns
161. count 8 at -62.6113° or -6266138385537385/36028797018963968 turns
162. count 8 at -73.5668° or -3681282367409331/18014398509481984 turns
163. count 8 at -84.5223° or -2114747771024985/9007199254740992 turns
164. count 8 at -95.4777° or -2388851856345511/9007199254740992 turns
165. count 8 at -106.433° or -5325906879332489/18014398509481984 turns
166. count 8 at -117.389° or -5874145073971057/18014398509481984 turns
167. count 8 at -128.344° or -3211166614306883/9007199254740992 turns
168. count 8 at -139.3° or -6970571423252335/18014398509481984 turns
169. count 8 at 209.745° or 5247819465793471/9007199254740992 turns
170. count 8 at 171.198° or 4283373605592079/9007199254740992 turns
171. count 8 at 159.948° or 4001898628881423/9007199254740992 turns
172. count 8 at 148.698° or 3720423652170767/9007199254740992 turns
173. count 8 at 137.448° or 3438948675460111/9007199254740992 turns
174. count 8 at 126.198° or 6314947397498909/18014398509481984 turns
175. count 8 at 114.948° or 5751997444077597/18014398509481984 turns
176. count 8 at 103.698° or 5189047490656285/18014398509481984 turns
177. count 8 at 92.4481° or 4626102541234559/18014398509481984 turns
178. count 8 at 81.1981° or 8126305175626495/36028797018963968 turns
179. count 8 at 69.9481° or 7000405268783871/36028797018963968 turns
180. count 8 at 58.6981° or 5874505361941247/36028797018963968 turns
181. count 8 at 47.4481° or 4748605455098623/36028797018963968 turns
182. count 8 at 36.1981° or 7245411096511997/72057594037927936 turns
183. count 8 at 24.9481° or 2496805641413375/36028797018963968 turns
184. count 8 at 13.6981° or 1370905734570751/36028797018963968 turns
185. count 8 at 2.44812° or 7840250538494761/1152921504606846976 turns
186. count 8 at -8.80188° or -3523568310058651/144115188075855872 turns
187. count 8 at 77.539° or 3880051238963121/18014398509481984 turns
188. count 8 at 65.078° or 1628251425277873/9007199254740992 turns
189. count 8 at 52.617° or 2632954462148371/18014398509481984 turns
190. count 8 at 77.3428° or 3870233391775453/18014398509481984 turns
191. count 8 at 64.6856° or 6473734312360821/36028797018963968 turns
192. count 8 at 140.906° or 881366957075371/2251799813685248 turns
193. count 8 at 78.8469° or 7890997099095973/36028797018963968 turns
194. count 8 at -15.7086° or -6288466231689971/144115188075855872 turns
195. count 8 at -8.12071° or -812720589555197/36028797018963968 turns
196. count 8 at -19.8177° or -3966710503792901/72057594037927936 turns
197. count 8 at -31.5148° or -6308001846073587/72057594037927936 turns
198. count 8 at -43.2118° or -1081159146544491/9007199254740992 turns
199. count 8 at -54.9089° or -1373820564329577/9007199254740992 turns
200. count 8 at -66.6059° or -6665917920459479/36028797018963968 turns

However, I'm not sure how to implement this caching/lookup. Or if it would make single point buffers slower. It might be more worthwhile adding more prepared trig values beyond 0, 45, 90 etc. already implemented, e.g. exact trigonometric values.

Looking at the results a bit more, it could make sense to try adding a static const lookup table to Angle.h (similar to (src/deps/ryu/d2s_full_table.h) with 32 values that are each 1/32 of a turn.

[pramsey]

Don't polish the 💩 too much, you've already shown that it's only a small performance change, hard to measure, I think know that it's a super critical path, so perhaps just go ahead and commit.

[mwtoews]

Ok, I think I'm done the pedanticness. I tried the look-up-of-32-angles approach and didn't see any performance gains, so I'm going with a much simpler inline approach to just clip trig values within 5e-16 to zero. The performance is the same.