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implement a fast atan approximation function (#1583)
implement a fast atan function using an approximate function for x in [-1, 1], for |x| > 1, the identity arctan(x) = sign(pi/2, x) - arctan(1/x) (Thanks to Eric) is used. Two versions of the approximate functions are included. 1. Efficient Approximations for the Arctangent Function by Rajan 2006: Equation 9: Max error ~0.0015 rad 2. https://stackoverflow.com/questions/42537957/fast-accurate-atan-arctan-approximation-algorithm: Max error ~0.00063 rad Generally I find the form from stackoverflow is more accurate and also faster. 1) is also discussed in 2). In general, from testing, atanf is roughly more than 2 times faster than std::atan.
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700 changes: 350 additions & 350 deletions
700
unit_test/test_ase/make_table/ci-benchmarks/ase_nse_net_make_table_unit_test.out
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| CEXE_headers += approx_math.H |
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| #ifndef APPROX_MATH_H | ||
| #define APPROX_MATH_H | ||
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| #include <AMReX_REAL.H> | ||
| #include <microphysics_math.H> | ||
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| using namespace amrex::literals; | ||
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| AMREX_GPU_HOST_DEVICE AMREX_INLINE | ||
| amrex::Real fast_atan_1(const amrex::Real x) { | ||
| /// | ||
| /// This calculates atan within [-1, 1] range. | ||
| /// | ||
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| /// | ||
| /// Approximation of atan for x in [-1, 1] | ||
| /// Max absolute error for this is ~0.0015 rad | ||
| /// See Ref: | ||
| /// Efficient Approximations for the Arctangent Function by Rajan 2006 | ||
| /// | ||
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| // constexpr amrex::Real PI_4 = GCEM_PI / 4.0_rt; | ||
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| // return PI_4*x - x*(std::abs(x) - 1.0_rt) * | ||
| // (0.2447_rt + 0.0663_rt*std::abs(x)); | ||
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| /// | ||
| /// Another approximation of atan for x in [-1, 1] | ||
| /// Max absolute error for this is ~0.00063 rad | ||
| /// See Ref: | ||
| /// https://stackoverflow.com/questions/42537957/fast-accurate-atan-arctan-approximation-algorithm | ||
| /// | ||
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| constexpr amrex::Real A = 0.0776509570923569_rt; | ||
| constexpr amrex::Real B = -0.287434475393028_rt; | ||
| constexpr amrex::Real C = GCEM_PI / 4.0_rt - A - B; | ||
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| amrex::Real x2 = x*x; | ||
| return ((A*x2 + B)*x2 + C)*x; | ||
| } | ||
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| AMREX_GPU_HOST_DEVICE AMREX_INLINE | ||
| amrex::Real fast_atan(const amrex::Real x) { | ||
| /// | ||
| /// Fast atan approximation calculations. | ||
| /// | ||
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| constexpr amrex::Real PI_2 = 0.5_rt * GCEM_PI; | ||
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| /// | ||
| /// If x < 0.113, then using arctan(x) ~ x | ||
| /// gives better answer than the approximation below. | ||
| /// And accuracy increase as x << 0.113. | ||
| /// Also significantly faster. | ||
| /// | ||
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| if (std::abs(x) < 0.113_rt) { | ||
| return x; | ||
| } | ||
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| // Check for large number, close to infinity. | ||
| // Error is ~1e-8 rad by not checking actual inf | ||
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| if (x > 1.e8_rt) { | ||
| return PI_2; | ||
| } | ||
| if (x < -1.e8_rt) { | ||
| return -PI_2; | ||
| } | ||
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| // Now calculate Atan(x) using approximations | ||
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| if (x > 1.0_rt) { | ||
| return PI_2 - fast_atan_1(1.0_rt / x); | ||
| } | ||
| if (x < -1.0_rt) { | ||
| return -PI_2 - fast_atan_1(1.0_rt / x); | ||
| } | ||
| return fast_atan_1(x); | ||
| } | ||
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| #endif |
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| PRECISION = DOUBLE | ||
| PROFILE = FALSE | ||
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| DEBUG = FALSE | ||
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| DIM = 3 | ||
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| COMP = gnu | ||
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| USE_MPI = FALSE | ||
| USE_OMP = FALSE | ||
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| USE_REACT = TRUE | ||
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| EBASE = main | ||
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| # define the location of the Microphysics top directory | ||
| MICROPHYSICS_HOME ?= ../../.. | ||
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| # This sets the EOS directory | ||
| EOS_DIR := helmholtz | ||
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| # This sets the network directory | ||
| NETWORK_DIR := aprox21 | ||
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| CONDUCTIVITY_DIR := stellar | ||
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| INTEGRATOR_DIR = VODE | ||
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| ifeq ($(USE_CUDA), TRUE) | ||
| INTEGRATOR_DIR := VODE | ||
| endif | ||
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| EXTERN_SEARCH += . | ||
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| Bpack := ../Make.package ./Make.package | ||
| Blocs := ../ . | ||
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| include $(MICROPHYSICS_HOME)/unit_test/Make.unit_test |
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| CEXE_sources += main.cpp |
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| #include <test_fast_atan.H> | ||
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| int main() { | ||
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| // Accuracy tests | ||
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| // Test values including edge cases and some typical values | ||
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| test_fast_atan_accuracy(0.0_rt); | ||
| test_fast_atan_accuracy(0.01_rt); | ||
| test_fast_atan_accuracy(0.08_rt); | ||
| test_fast_atan_accuracy(0.1_rt); | ||
| test_fast_atan_accuracy(0.5_rt); | ||
| test_fast_atan_accuracy(1.0_rt); | ||
| test_fast_atan_accuracy(5.0_rt); | ||
| test_fast_atan_accuracy(8.0_rt); | ||
| test_fast_atan_accuracy(10.0_rt); | ||
| test_fast_atan_accuracy(25.0_rt); | ||
| test_fast_atan_accuracy(100.0_rt); | ||
| test_fast_atan_accuracy(500.0_rt); | ||
| test_fast_atan_accuracy(5.0e8_rt); | ||
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| test_fast_atan_accuracy(-0.01_rt); | ||
| test_fast_atan_accuracy(-0.08_rt); | ||
| test_fast_atan_accuracy(-0.1_rt); | ||
| test_fast_atan_accuracy(-0.5_rt); | ||
| test_fast_atan_accuracy(-1.0_rt); | ||
| test_fast_atan_accuracy(-5.0_rt); | ||
| test_fast_atan_accuracy(-8.0_rt); | ||
| test_fast_atan_accuracy(-10.0_rt); | ||
| test_fast_atan_accuracy(-25.0_rt); | ||
| test_fast_atan_accuracy(-100.0_rt); | ||
| test_fast_atan_accuracy(-500.0_rt); | ||
| test_fast_atan_accuracy(-5.0e8_rt); | ||
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| // Inf Case | ||
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| test_fast_atan_accuracy(std::numeric_limits<amrex::Real>::infinity()); | ||
| test_fast_atan_accuracy(-std::numeric_limits<amrex::Real>::infinity()); | ||
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| std::cout << "Accuracy tests passed!" << std::endl; | ||
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| // Now performance test | ||
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| int iters = 5; | ||
| amrex::Real test_value = 160.0_rt; | ||
| test_fast_atan_speed(100, iters, test_value); | ||
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| iters = 10; | ||
| test_fast_atan_speed(100, iters, test_value); | ||
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| iters = 20; | ||
| test_fast_atan_speed(100, iters, test_value); | ||
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| iters = 30; | ||
| test_fast_atan_speed(100, iters, test_value); | ||
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| iters = 50; | ||
| test_fast_atan_speed(100, iters, test_value); | ||
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| iters = 70; | ||
| test_fast_atan_speed(100, iters, test_value); | ||
| } |
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