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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 &y) { real x = sqrt(y); return (sin(x) - x) / (x * y); } static real myerr(real const &y) { return re(y * sqrt(y)); } 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:

With the following coefficients:

a0 = -1.66666666666658080941942898789420724e-1; a1 = +8.33333333326271609442503773834687308e-3; a2 = -1.98412698200591143928364634696492885e-4; a3 = +2.75573160733868922065738227278330896e-6; a4 = -2.50518513021429359590028300127165228e-8; a5 = +1.60472959182597740337401201006549498e-10; a6 = -7.36458957326227991327065122848667046e-13;