Yet another fast trigonometry

Question

float sinx(float x)
{
    static const float a[] = {-.1666666664,.0083333315,-.0001984090,.0000027526,-.0000000239};
    float xsq = x*x;
    float temp = x*(1 + a[0]*xsq + a[1]*xsq*xsq + a[2]* xsq*xsq*xsq+a[3]*xsq*xsq*xsq*xsq+ a[4]*xsq*xsq*xsq*xsq*xsq);
    return temp;
}

How are those constants calculated? How to calculate cos and tan using this method? Can I extend this to get more precision? I guess I need to add more constants?

---

Plot of the error of the "fast" sine described above against a Taylor polynomial of equal degree.

Answer 1

Nearly all answers at the time of this writing refer to the Taylor expansion of function sin, but if the author of the function were serious, he would not use Taylor coefficients. Taylor coefficients tend to produce a polynomial approximation that's better than necessary near zero and increasingly bad away from zero. The goal is usually to obtain an approximation that is uniformly good on a range such as -π/2…π/2. For a polynomial approximation, this can be obtained by applying the Remez algorithm. A down-to-earth explanation is this post.

The polynomial coefficients obtained by that method are close to the Taylor coefficients, since both polynomials are trying to approximate the same function, but the polynomial may be more precise for the same number of operations, or involve fewer operations for the same (uniform) quality of approximation.

I cannot tell just by looking at them if the coefficients in your question are exactly Taylor coefficients or the slightly different coefficients obtained by the Remez algorithm, but it is probably what should have been used even if it wasn't.

Lastly, whoever wrote (1 + a[0]*xsq + a[1]*xsq*xsq + a[2]* xsq*xsq*xsq+a[3]*xsq*xsq*xsq*xsq+ a[4]*xsq*xsq*xsq*xsq*xsq) needs to read up about better polynomial evaluation schemes, for instance Horner's:

1 + xsq*(a[0]+ xsq*(a[1] + xsq*(a[2] + xsq*(a[3] + xsq*a[4])))) uses N multiplications instead of N²/2.

Answer 2

They are -1/6, 1/120, -1/5040 .. and so on.

Or rather: -1/3!, 1/5!, -1/7!, 1/9!... etc

Look at the taylor series for sin x in here:

It has cos x right below it:

For cos x, as seen from the picture above, the constants are -1/2!, 1/4!, -1/6!, 1/8!...

tan x is slightly different:

So to adjust this for cosx:

float cosx(float x)
{
    static const float a[] = {-.5, .0416666667,-.0013888889,.0000248016,-.0000002756};
    float xsq = x*x;
    float temp = (1 + a[0]*xsq + a[1]*xsq*xsq + a[2]* xsq*xsq*xsq+a[3]*xsq*xsq*xsq*xsq+ a[4]*xsq*xsq*xsq*xsq*xsq);
    return temp;
}