Optimize gnomonic projections in faceijk.c#1174
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Replace the transcendental-heavy inverse and forward gnomonic projections with algebraically exact identities. _hex2dToVec3 (inverse): drop atan2 by taking cos/sin of theta directly from the input vector; replace atan + final cos(r)/sin(r) with a single sqrt via cos(atan r) = 1/sqrt(1+r^2), sin(atan r) = r/sqrt(1+r^2); fold the per-face azimuth and Class III rotation into cos/sin via angle-addition against precomputed tables. ~6 transcendentals -> 1 sqrt. _vec3ToHex2d (forward): replace acos + tan with one sqrt via tan(acos x) = sqrt(1-x^2)/x (radicand factored to avoid cancellation near face centers); replace atan2 + two normalizations with direct cos/sin of the azimuth from un-normalized tangent-plane components (the common magnitude factor cancels); fold the face-axis offset and Class III rotation via angle-subtraction. The face-center guard is re-expressed on sqd (sqd < EPSILON*EPSILON) since acos is gone. ~5 transcendentals + 2 sqrt -> 2 sqrt. Removes the now-unused _vec3AzimuthRads. Adds precomputed cos/sin companion tables for faceAxesAzRadsCII[face][0], shared by both functions. All 317 C tests pass. Benchmarks (Release -O3, 40x interleaved runs, min microseconds/iter): benchmark before after delta latLngToCell 0.1789 0.1345 -24.8% cellToLatLng 0.0908 0.0696 -23.3% cellToBoundary 0.3887 0.2685 -30.9% cellArea (allCellsAtRes) 28247 19743 -30.1% directedEdgeToBoundary 1.2262 0.9250 -24.6% polygonToCellsSF 448.21 379.74 -15.3% polygonToCellsAlameda 689.77 607.62 -11.9% polygonToCellsSouthernExp 21252 19320 -9.1% cellToVertexes 0.6904 0.6897 -0.1% Co-Authored-By: Claude Opus 4.8 (1M context) <noreply@anthropic.com>
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Replace the transcendental-heavy inverse and forward gnomonic projections with algebraically exact identities.
_hex2dToVec3 (inverse): drop atan2 by taking cos/sin of theta directly from the input vector; replace atan + final cos(r)/sin(r) with a single sqrt via cos(atan r) = 1/sqrt(1+r^2), sin(atan r) = r/sqrt(1+r^2); fold the per-face azimuth and Class III rotation into cos/sin via angle-addition against precomputed tables. ~6 transcendentals -> 1 sqrt.
_vec3ToHex2d (forward): replace acos + tan with one sqrt via tan(acos x) = sqrt(1-x^2)/x (radicand factored to avoid cancellation near face centers); replace atan2 + two normalizations with direct cos/sin of the azimuth from un-normalized tangent-plane components (the common magnitude factor cancels); fold the face-axis offset and Class III rotation via angle-subtraction. The face-center guard is re-expressed on sqd (sqd < EPSILON*EPSILON) since acos is gone. ~5 transcendentals + 2 sqrt -> 2 sqrt. Removes the now-unused _vec3AzimuthRads.
Adds precomputed cos/sin companion tables for faceAxesAzRadsCII[face][0], shared by both functions.
All 317 C tests pass.
Benchmarks (Release -O3, 40x interleaved runs, min microseconds/iter):