/* The fast (but not very precise, but tolerable) atan2f implementation. It is based (but not heavily) on GNU Radio's fast_atan2f.cc (which is C++, although not very) https://github.com/gnuradio/gnuradio/blob/b2c9623cbd548bd86250759007b80b61bd4a2a06/gnuradio-runtime/lib/math/fast_atan2f.cc * Copyright 2005,2013 Free Software Foundation, Inc. * * This file is part of GNU Radio * * SPDX-License-Identifier: GPL-3.0-or-later The code is greatly re-written, in many cases from scratch */ #define PI 3.14159265358979323846f #define PIHALF 1.57079632679489661923f #define TAN_MAP_RES 0.003921569f /* (smallest non-zero value in table) */ /* arctangents from 0 to pi/4 */ static float fast_atan_table[257] = { 0.000000e+00, 3.921549e-03, 7.842976e-03, 1.176416e-02, 1.568499e-02, 1.960533e-02, 2.352507e-02, 2.744409e-02, 3.136226e-02, 3.527947e-02, 3.919560e-02, 4.311053e-02, 4.702413e-02, 5.093629e-02, 5.484690e-02, 5.875582e-02, 6.266295e-02, 6.656816e-02, 7.047134e-02, 7.437238e-02, 7.827114e-02, 8.216752e-02, 8.606141e-02, 8.995267e-02, 9.384121e-02, 9.772691e-02, 1.016096e-01, 1.054893e-01, 1.093658e-01, 1.132390e-01, 1.171087e-01, 1.209750e-01, 1.248376e-01, 1.286965e-01, 1.325515e-01, 1.364026e-01, 1.402496e-01, 1.440924e-01, 1.479310e-01, 1.517652e-01, 1.555948e-01, 1.594199e-01, 1.632403e-01, 1.670559e-01, 1.708665e-01, 1.746722e-01, 1.784728e-01, 1.822681e-01, 1.860582e-01, 1.898428e-01, 1.936220e-01, 1.973956e-01, 2.011634e-01, 2.049255e-01, 2.086818e-01, 2.124320e-01, 2.161762e-01, 2.199143e-01, 2.236461e-01, 2.273716e-01, 2.310907e-01, 2.348033e-01, 2.385093e-01, 2.422086e-01, 2.459012e-01, 2.495869e-01, 2.532658e-01, 2.569376e-01, 2.606024e-01, 2.642600e-01, 2.679104e-01, 2.715535e-01, 2.751892e-01, 2.788175e-01, 2.824383e-01, 2.860514e-01, 2.896569e-01, 2.932547e-01, 2.968447e-01, 3.004268e-01, 3.040009e-01, 3.075671e-01, 3.111252e-01, 3.146752e-01, 3.182170e-01, 3.217506e-01, 3.252758e-01, 3.287927e-01, 3.323012e-01, 3.358012e-01, 3.392926e-01, 3.427755e-01, 3.462497e-01, 3.497153e-01, 3.531721e-01, 3.566201e-01, 3.600593e-01, 3.634896e-01, 3.669110e-01, 3.703234e-01, 3.737268e-01, 3.771211e-01, 3.805064e-01, 3.838825e-01, 3.872494e-01, 3.906070e-01, 3.939555e-01, 3.972946e-01, 4.006244e-01, 4.039448e-01, 4.072558e-01, 4.105574e-01, 4.138496e-01, 4.171322e-01, 4.204054e-01, 4.236689e-01, 4.269229e-01, 4.301673e-01, 4.334021e-01, 4.366272e-01, 4.398426e-01, 4.430483e-01, 4.462443e-01, 4.494306e-01, 4.526070e-01, 4.557738e-01, 4.589307e-01, 4.620778e-01, 4.652150e-01, 4.683424e-01, 4.714600e-01, 4.745676e-01, 4.776654e-01, 4.807532e-01, 4.838312e-01, 4.868992e-01, 4.899573e-01, 4.930055e-01, 4.960437e-01, 4.990719e-01, 5.020902e-01, 5.050985e-01, 5.080968e-01, 5.110852e-01, 5.140636e-01, 5.170320e-01, 5.199904e-01, 5.229388e-01, 5.258772e-01, 5.288056e-01, 5.317241e-01, 5.346325e-01, 5.375310e-01, 5.404195e-01, 5.432980e-01, 5.461666e-01, 5.490251e-01, 5.518738e-01, 5.547124e-01, 5.575411e-01, 5.603599e-01, 5.631687e-01, 5.659676e-01, 5.687566e-01, 5.715357e-01, 5.743048e-01, 5.770641e-01, 5.798135e-01, 5.825531e-01, 5.852828e-01, 5.880026e-01, 5.907126e-01, 5.934128e-01, 5.961032e-01, 5.987839e-01, 6.014547e-01, 6.041158e-01, 6.067672e-01, 6.094088e-01, 6.120407e-01, 6.146630e-01, 6.172755e-01, 6.198784e-01, 6.224717e-01, 6.250554e-01, 6.276294e-01, 6.301939e-01, 6.327488e-01, 6.352942e-01, 6.378301e-01, 6.403565e-01, 6.428734e-01, 6.453808e-01, 6.478788e-01, 6.503674e-01, 6.528466e-01, 6.553165e-01, 6.577770e-01, 6.602282e-01, 6.626701e-01, 6.651027e-01, 6.675261e-01, 6.699402e-01, 6.723452e-01, 6.747409e-01, 6.771276e-01, 6.795051e-01, 6.818735e-01, 6.842328e-01, 6.865831e-01, 6.889244e-01, 6.912567e-01, 6.935800e-01, 6.958943e-01, 6.981998e-01, 7.004964e-01, 7.027841e-01, 7.050630e-01, 7.073330e-01, 7.095943e-01, 7.118469e-01, 7.140907e-01, 7.163258e-01, 7.185523e-01, 7.207701e-01, 7.229794e-01, 7.251800e-01, 7.273721e-01, 7.295557e-01, 7.317307e-01, 7.338974e-01, 7.360555e-01, 7.382053e-01, 7.403467e-01, 7.424797e-01, 7.446045e-01, 7.467209e-01, 7.488291e-01, 7.509291e-01, 7.530208e-01, 7.551044e-01, 7.571798e-01, 7.592472e-01, 7.613064e-01, 7.633576e-01, 7.654008e-01, 7.674360e-01, 7.694633e-01, 7.714826e-01, 7.734940e-01, 7.754975e-01, 7.774932e-01, 7.794811e-01, 7.814612e-01, 7.834335e-01, 7.853982e-01, 7.853982e-01 }; /* The interface is deliberately copied from atan2f of the standard library: man atan2f, but with a simplification. float atan2f(float y, float x); Calculate the principal value of the arc tangent of y/x, using the signs of the two arguments to determine the quadrant of the result. The return value is in radians and within [-pi,pi] The simplification is the assumption of -ffast-math, in particular, that the arguments are NORMAL floats (no NaNs, no infinites, and no -0.0) This function uses table lookup and linear interpolation */ static inline float fast_atan2f(float const y, float const x) { // could have used signbit, but it doesn't seem to improve the code if (y==0.0f) return (x < 0.0f ? PI : 0.0f); // assuming y is not zero: already checked if (x==0.0f) return copysignf(PIHALF,y); /* We hereby assume that neither x nor y are zero */ float const y_abs = copysignf(y,1.0f); float const x_abs = copysignf(x,1.0f); /* 0 <= z <= 1 */ bool const ylessx = y_abs < x_abs; float const z = (ylessx ? (y_abs / x_abs) : (x_abs / y_abs)); /* tan z = z when z is sufficiently small */ float angle = z; if (z> TAN_MAP_RES) { /* find index and interpolation value */ /* XXX use modf? No, it is emitted as call to modff */ float const alpha = z * 255.0f; int const index = (int)alpha; /* assert (index>= 0); assert (index <= 255); */ /* linear interpolation */ angle = fast_atan_table[index]; angle += (fast_atan_table[index + 1] - angle) * (alpha - (float)index); } if(!ylessx) angle = PIHALF - angle; // could have used signbit, but it doesn't seem to improve the code if(x < 0.0f) angle = PI - angle; // II or III quandrant return copysignf(angle,y); } /* Max abs error 1.43051e-06 detected at x=-0.8125, y=-0.5625 Max rel error 3.96193e-06 detected at x=0.625, y=0.0625 Average error 5.2932e-07 over 1089 samples */

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