/**
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* Approxiamtes atan over the range [0, 1]. Safe to flip output for negative input.
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*
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* Based on Michal Drobot's approximation from ShaderFastLibs, which in turn is based on
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* "Efficient approximations for the arctangent function," Rajan, S. Sichun Wang Inkol, R. Joyal, A., May 2006.
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* Adapted from ShaderFastLibs under MIT License.
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*
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* Chosen for the following characteristics over range [0, 1]:
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* - basically no error at 0 and 1, important for getting around range limit (naive atan2 via atan requires infinite range atan)
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* - no visible artifacts from first-derivative discontinuities, unlike latitude via range-reduced sqrt asin approximations (at equator)
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*
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* The original code is x * (-0.1784 * abs(x) - 0.0663 * x * x + 1.0301);
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* Removed the abs() in here because it isn't needed, the input range is guaranteed as [0, 1] by how we're approximating atan2.
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*
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* @name czm_fastApproximateAtan
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* @glslFunction
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*
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* @param {float} x Value between 0 and 1 inclusive.
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*
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* @returns {float} Approximation of atan(x)
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*/
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float czm_fastApproximateAtan(float x) {
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return x * (-0.1784 * x - 0.0663 * x * x + 1.0301);
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}
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/**
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* Approximation of atan2.
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*
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* Range reduction math based on nvidia's cg reference implementation for atan2: http://developer.download.nvidia.com/cg/atan2.html
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* However, we replaced their atan curve with Michael Drobot's (see above).
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*
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* @name czm_fastApproximateAtan
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* @glslFunction
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*
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* @param {float} x Value between -1 and 1 inclusive.
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* @param {float} y Value between -1 and 1 inclusive.
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*
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* @returns {float} Approximation of atan2(x, y)
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*/
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float czm_fastApproximateAtan(float x, float y) {
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// atan approximations are usually only reliable over [-1, 1], or, in our case, [0, 1] due to modifications.
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// So range-reduce using abs and by flipping whether x or y is on top.
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float t = abs(x); // t used as swap and atan result.
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float opposite = abs(y);
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float adjacent = max(t, opposite);
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opposite = min(t, opposite);
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t = czm_fastApproximateAtan(opposite / adjacent);
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// Undo range reduction
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t = czm_branchFreeTernary(abs(y) > abs(x), czm_piOverTwo - t, t);
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t = czm_branchFreeTernary(x < 0.0, czm_pi - t, t);
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t = czm_branchFreeTernary(y < 0.0, -t, t);
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return t;
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}
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