Optimal 5th Degree Polynomial which Approximates the Sine Function
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 aproximate the sine function
sin between 0 and
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
x in interval between 0 and