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Cosine

Various methods of calculating cosine without math.h.

See the explanation and benchmarks at http://austinhenley.com/blog/cosine.html

For educational purposes only :)

BUILDING

clang -O3 -mavx benchmarks.c -o benchmarks -lm
./benchmarks -h
Usage: ./benchmarks [-na] [-nt] [-t <testname>]
   -na - Don't run accuracy tests
   -nt - Don't run speed tests.
   -t <testname> - Run a particular test instead of all tests.
   -p - Print all test names.

TESTS

There are four types of tests:

  • Taylor series with literal values
  • Taylor series with running values
  • Table lookup with clamp to nearest
  • Table lookup with linear interpolation (LERP)

Table tests have the following properties

  • cos_table_1 - a cosine table with 1 value per radian
  • cos_table_0_1 - a cosine table with 10 values per radian
  • cos_table_0_01 - a cosine table with 100 values per radian
  • cos_table_0_001 - a cosine table with 1000 values per radian
  • cos_table_0_0001 - a cosine table with 10,000 values per radian
cos_taylor_literal_4terms_naive
cos_taylor_literal_6terms_naive
cos_taylor_literal_6terms_2pi
cos_taylor_literal_6terms_pi
cos_taylor_literal_6terms
cos_taylor_literal_10terms
cos_taylor_running_6terms
cos_taylor_running_8terms
cos_taylor_running_10terms
cos_taylor_running_16terms
cos_table_1
cos_table_0_1
cos_table_0_01
cos_table_0_001
cos_table_0_0001
cos_table_1_LERP
cos_table_0_1_LERP
cos_table_0_01_LERP
cos_table_0_001_LERP
cos_table_0_0001_LERP
cos_math_h
***NEW***
fast_cosine
fast_sine
fast_acc_cosine
fast_acc_sine 
***NEW***

You can run a particular test by using the -t switch:

./benchmarks -t cos_table_1_LERP
ACCURACY
cos_table_1_LERP                    0.1147496616359112

TIME
cos_table_1_LERP                    1.3534999999999999

Or run all tests with no arguments.

  1. macos arm64
clang -O3 -mavx benchmarks.c -o benchmarks -lm
./benchmarks 
Cosine benchmark

ACCURACY
cos_taylor_literal_4terms_naive     0.0000247372721736
cos_taylor_literal_6terms_naive     0.0000000063214679
cos_taylor_literal_6terms_2pi       0.0000000063214679
cos_taylor_literal_6terms_pi        0.0000000063214679
cos_taylor_literal_6terms           0.0000000063214679
cos_taylor_literal_10terms          0.0000000000000004
cos_taylor_running_6terms           0.0000000063214679
cos_taylor_running_8terms           0.0000000000005261
cos_taylor_running_10terms          0.0000000000000004
cos_taylor_running_16terms          0.0000000000000004
cos_table_1                         0.4868840381728474
cos_table_0_1                       0.0499943500331001
cos_table_0_01                      0.0049999675472431
cos_table_0_001                     0.0004999999109268
cos_table_0_0001                    0.0000499999164148
cos_table_1_LERP                    0.1076522572415404
cos_table_0_1_LERP                  0.0012481779730024
cos_table_0_01_LERP                 0.0000124998177090
cos_table_0_001_LERP                0.0000001249999818
cos_table_0_0001_LERP               0.0000000012500000
cos_math_h                          0.0000000000000000
fast_cosine                         0.0010902926026138
fast_sine                           0.0010902926026137
fast_acc_cosine                     0.0000233416851561
fast_acc_cosine_v2                  0.0000338483237610
fast_acc_sine                       0.0000233443382702

TIME
cos_taylor_literal_4terms_naive     0.1994540000000000
cos_taylor_literal_6terms_naive     0.3009390000000000
cos_taylor_literal_6terms_2pi       0.4623580000000000
cos_taylor_literal_6terms_pi        0.5693270000000000
cos_taylor_literal_6terms           0.4597660000000000
cos_taylor_literal_10terms          0.7167900000000000
cos_taylor_running_6terms           0.4068400000000000
cos_taylor_running_8terms           0.5226680000000000
cos_taylor_running_10terms          0.6387990000000000
cos_taylor_running_16terms          1.0606310000000001
cos_table_1                         0.1742500000000000
cos_table_0_1                       0.1843980000000000
cos_table_0_01                      0.1873230000000000
cos_table_0_001                     0.1868730000000000
cos_table_0_0001                    0.1880180000000000
cos_table_1_LERP                    0.2319910000000000
cos_table_0_1_LERP                  0.2450100000000000
cos_table_0_01_LERP                 0.2470970000000000
cos_table_0_001_LERP                0.2528150000000000
cos_table_0_0001_LERP               0.2469320000000000
cos_math_h                          0.3528410000000000
fast_cosine                         0.1374690000000000
fast_sine                           0.1332670000000000
fast_acc_cosine                     0.1389720000000000
fast_acc_cosine_v2                  0.1383920000000000
fast_acc_sine                       0.1289920000000000


Done
  1. Ubuntu18 g++-7
g++-7 -O3 -mavx benchmarks.c -o benchmarks -lm
ACCURACY
cos_taylor_literal_4terms_naive 0.0000247372721736
cos_taylor_literal_6terms_naive 0.0000000063214679
cos_taylor_literal_6terms_2pi 0.0000000063214679
cos_taylor_literal_6terms_pi 0.0000000063214679
cos_taylor_literal_6terms 0.0000000063214679
cos_taylor_literal_10terms 0.0000000000000004
cos_taylor_running_6terms 0.0000000063214679
cos_taylor_running_8terms 0.0000000000005261
cos_taylor_running_10terms 0.0000000000000004
cos_taylor_running_16terms 0.0000000000000004
cos_table_1 0.4868840381728474
cos_table_0_1 0.0499943500331001
cos_table_0_01 0.0049999675472431
cos_table_0_001 0.0004999999109268
cos_table_0_0001 0.0000499999164148
cos_table_1_LERP 0.1076522572415404
cos_table_0_1_LERP 0.0012481779730024
cos_table_0_01_LERP 0.0000124998177090
cos_table_0_001_LERP 0.0000001249999818
cos_table_0_0001_LERP 0.0000000012500000
cos_math_h 0.0000000000000000
fast_cosine 0.0010902926026138
fast_sine 0.0010902926026137
fast_acc_cosine 0.0000233416851561
fast_acc_sine 0.0000233443382702



TIME
cos_taylor_literal_4terms_naive 0.4052360000000000
cos_taylor_literal_6terms_naive 0.5750660000000000
cos_taylor_literal_6terms_2pi 1.2396400000000001
cos_taylor_literal_6terms_pi 1.4296800000000001
cos_taylor_literal_6terms 1.1930259999999999
cos_taylor_literal_10terms 1.5523950000000000
cos_taylor_running_6terms 0.9517200000000000
cos_taylor_running_8terms 1.1430320000000000
cos_taylor_running_10terms 1.3806490000000000
cos_taylor_running_16terms 2.0492330000000001
cos_table_1 0.3221170000000000
cos_table_0_1 0.3380720000000000
cos_table_0_01 0.3403360000000000
cos_table_0_001 0.3485470000000000
cos_table_0_0001 0.3479600000000000
cos_table_1_LERP 0.4599750000000000
cos_table_0_1_LERP 0.4492510000000000
cos_table_0_01_LERP 0.4504650000000000
cos_table_0_001_LERP 0.4588620000000000
cos_table_0_0001_LERP 0.4529580000000000
cos_math_h 0.9298620000000000
fast_cosine 0.2139590000000000
fast_sine 0.1922930000000000
fast_acc_cosine 0.2128010000000000
fast_acc_sine 0.1982350000000000

Your results will differ.