exponential polynomial approximation

[ZvRzyan18]

This pull request introduces a new implementations of glm::fastPow, glm::fastExp, glm::fastExp2, glm::fastLog, glm::fastLog2. These functions are numerically stable, even for large values.

Btw. I made a pull request before #1332 with the same purpose, but I closed it because I was still trying to find a more efficient code.

ALGORITHM

In this approximation I used Horner's method for polynomial evaluation ((((x + C0) * x + C1) * x + C2) * x + C3)
where C0 - Cn are coefficents calculated using the Remez Algorithm.
exp2() interval [0, 1], one and two degree polynomial (exp2 works more efficiently in IEEE 754 float)
log() interval[1, 2], two degree polynomial
pow(), uses 4 degree polynomial for both exp and log

SPECIAL CASES

these are not implemented for performance reasons

• glm::fastPow

If y is negative

1 / glm::fastPow(x, abs(-y));

If x is less than 1 (e.g 0.4, 0.5...)

1 / glm::fastPow(1 / x, y);

• glm::fastExp, glm::fastExp2

If x is negative

1 / glm::fastExp(abs(-x));

• glm::fastLog, glm::fastLog2

If x is less than 1 (e.g 0.7, 0.8...)

-glm::fastLog(1 / x);

ERROR

• glm::fastPow

x = 6.7
y = 0.00, error : 0.24760962%
y = 1.00, error : 0.14139716%
y = 2.00, error : 0.03850524%
y = 3.00, error : 0.22298415%
y = 4.00, error : 0.09447852%
y = 5.00, error : 0.11734231%
y = 6.00, error : 0.08545813%
y = 7.00, error : 0.23790078%
y = 8.00, error : 0.07089398%
y = 9.00, error : 0.08944029%
y = 10.00, error : 0.13199883%
y = 11.00, error : 0.24625938%
y = 12.00, error : 0.05305298%
y = 13.00, error : 0.05824120%
y = 14.00, error : 0.17782241%
y = 15.00, error : 0.24689905%
y = 16.00, error : 0.04042700%
y = 17.00, error : 0.02405633%
y = 18.00, error : 0.22297528%
y = 19.00, error : 0.23925050%
y = 20.00, error : 0.03312265%

• glm::fastExp

x = 0.00, error : 4.30356860%
x = 1.00, error : 2.98118323%
x = 2.00, error : 0.26578372%
x = 3.00, error : 2.36613181%
x = 4.00, error : 1.26317261%
x = 5.00, error : 0.94539768%
x = 6.00, error : 2.36313024%
x = 7.00, error : 1.41008692%
x = 8.00, error : 2.95274340%
x = 9.00, error : 1.87398441%
x = 10.00, error : 2.94006676%
x = 11.00, error : 0.03307902%
x = 12.00, error : 2.22139533%
x = 13.00, error : 1.44142235%
x = 14.00, error : 0.68031238%
x = 15.00, error : 2.47628319%
x = 16.00, error : 1.81461304%
x = 17.00, error : 2.98891599%
x = 18.00, error : 1.60186663%
x = 19.00, error : 2.88561219%
x = 20.00, error : 0.19276333%

• glm::fastExp2

x = 0.00, error : 0.24760962%
x = 1.00, error : 0.24760962%
x = 2.00, error : 0.24760962%
x = 3.00, error : 0.24760962%
x = 4.00, error : 0.24760962%
x = 5.00, error : 0.24760962%
x = 6.00, error : 0.24760962%
x = 7.00, error : 0.24760962%
x = 8.00, error : 0.24760962%
x = 9.00, error : 0.24760962%
x = 10.00, error : 0.24760962%
x = 11.00, error : 0.24760962%
x = 12.00, error : 0.24760962%
x = 13.00, error : 0.24760962%
x = 14.00, error : 0.24760962%
x = 15.00, error : 0.24760962%
x = 16.00, error : 0.24760962%
x = 17.00, error : 0.24760962%
x = 18.00, error : 0.24760962%
x = 19.00, error : 0.24760962%
x = 20.00, error : 0.24760962%

• glm::fastLog

x = 0.00, error : nan
x = 1.00, error : inf
x = 2.00, error : 0.49396884%
x = 3.00, error : 0.07884442%
x = 4.00, error : 0.24698456%
x = 5.00, error : 0.20669479%
x = 6.00, error : 0.04834334%
x = 7.00, error : 0.17220324%
x = 8.00, error : 0.16465646%
x = 9.00, error : 0.11049288%
x = 10.00, error : 0.14447338%
x = 11.00, error : 0.06858298%
x = 12.00, error : 0.03485839%
x = 13.00, error : 0.11212541%
x = 14.00, error : 0.12697873%
x = 15.00, error : 0.05432294%
x = 16.00, error : 0.12349242%
x = 17.00, error : 0.01063724%
x = 18.00, error : 0.08399524%
x = 19.00, error : 0.11310391%
x = 20.00, error : 0.11104532%

• glm::fastLog2

x = 0.00, error : nan
x = 1.00, error : inf
x = 2.00, error : 0.49397945%
x = 3.00, error : 0.07884461%
x = 4.00, error : 0.24698973%
x = 5.00, error : 0.20669022%
x = 6.00, error : 0.04834335%
x = 7.00, error : 0.17220546%
x = 8.00, error : 0.16465982%
x = 9.00, error : 0.11048829%
x = 10.00, error : 0.14447026%
x = 11.00, error : 0.06859365%
x = 12.00, error : 0.03485831%
x = 13.00, error : 0.11212646%
x = 14.00, error : 0.12697578%
x = 15.00, error : 0.05433159%
x = 16.00, error : 0.12350082%
x = 17.00, error : 0.01062610%
x = 18.00, error : 0.08398610%
x = 19.00, error : 0.11310125%
x = 20.00, error : 0.11103747%

PERFORMANCE TEST

• aarch 64

time : 52690 std::pow
time : 42250 glm::fastPow

time : 26476 std::exp
time : 18128 glm::fastExp

time : 23330 std::exp2
time : 20809 glm::fastExp2

time : 25646 std::log
time : 19567 glm::fastLog

time : 23910 std::log2
time : 17782 glm::fastLog2

• x86_64

time : 509 std::pow
time : 110 glm::fastPow

time : 171 std::exp
time : 67 glm::fastExp

time : 125 std::exp2
time : 48 glm::fastExp2

time : 220 std::log
time : 39 glm::fastLog

time : 140 std::log2
time : 44 glm::fastLog2