Various methods of calculating cosine without math.h.
See the explanation and benchmarks at http://austinhenley.com/blog/cosine.html
For educational purposes only :)
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.
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.
- 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
- 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.