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?

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

Answer 1

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;
}