Optimal 5th Degree Polynomial which Approximates the Sine Function

2022-03-29

Describes a 5th-degree polynomial that was found to approximate the sine function precisely up to 4 decimal digits. Useful for resource-constrained devices, games, or any applications where the precision of the least significant digits is not important.

Our goal is to approximate the sine function sin between 0 and pi/2.

We found the best coefficients for a polynomial of degree 5 in the form f(x) = c5 x^5 + c3 x^3 + c1 x to be the following:

c1 =  0.99977140751539240
c3 = -0.16582704279148017
c5 =  0.0075742477643552034

According to Wolfram Alpha, the largest error in the domain between 0 and pi/2 is of magnitude ~0.00016 at x~=1.5708, so this function should be good to use for up to 4 decimal digits.

Here's the query:

maximize abs(0.0075742477643552034 x^5
	+ -0.16582704279148017 x^3
	+ 0.99977140751539240 x - sin(x))
with x between 0 and pi/2

The coefficients were found by searching in R^3 for the optimal vector of coefficients (c5, c3, c1) for the function f(x) = c5 x^5 + c3 x^3 + c1 x which minimize the RMSE (Root Mean Squared Error) between f(x) and sin(x), assuming x in interval between 0 and pi/2.