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# Trigonometric functions

Our research notes about implementation of fast trigonometric functions.

## Minimax polynomial for sin(x)

### Absolute error

Suppose we want to approximate sin(x) on [-π/2; π/2] with a polynomial P(x) such that the **absolute** error is never more than E:

We know sin(x) is an odd function, so instead we look for a polynomial Q(x) such that P(x) = xQ(x²):

Substitute y for x² and reduce the range to positive values:

Divide through by √y:

If we want to force the asymptotic behaviour in x=0, we substitute Q(y) with 1+yR(y):

Divide through by y:

We then use the following code:

static real myfun(real const &x) { real y = sqrt(x); return (sin(y) - y) / (x * y); } static real myerr(real const &x) { return re(x * sqrt(x)); } RemezSolver<6> solver; solver.Run(real::R_1 >> 400, real::R_PI_2 * real::R_PI_2, myfun, myerr, 15);

These are the resulting R, Q and P: