Trigonometric Function Optimization

[Labdabeta]

I have the following trig functions, but I am wondering if there is a faster algorithm that I could implement:

static const double SINMIN=0.0009999998333333;
    static const double COSMIN=0.9999995000000417;
    static const double TANMIN=0.0010000003333334;
    static const double E=2.718281828459045235360;
    static const double PI=3.14159265358979323846;
    static const double MINVAL=0.01;
    double sin(double ax)
{
    double x=aabs(ax);
    if (x==MINVAL)
    {
        switch (quadrant(ax))
        {
            case 1:
            case 2:
            return SINMIN;
            case 3:
            case 4:
            return -SINMIN;
        }
    }
    else
        return (SINMIN*cos(x-MINVAL)+COSMIN*sin(x-MINVAL));
}
    double cos(double x)
{
    if (x==MINVAL)
    {
        switch (quadrant(ax))
        {
            case 1:
            case 4:
            return COSMIN;
            case 2:
            case 3:
            return -COSMIN;
        }
    }
    else
        return (COSMIN*cos(x-MINVAL)+SINMIN*sin(x-MINVAL));
}
    double tan(double x)
{
    if (x==MINVAL)
    {
        switch (quadrant(ax))
        {
            case 1:
            case 3:
            return TANMIN;
            case 2:
            case 4:
            return -TANMIN;
        }
    }
    else
        return ((TANMIN+tan(x-MINVAL))/(1.0000000000000-(TANMIN*tan(x-MINVAL))));
}

[nezachem]

Sure.
If you insist on recursive approach, it is better to half the argument, rather than subtract a small decrement. For example, to calculate sin(0.5), your code goes down 50 levels of recursion, while halfing does only 6. Another speedup is achieved by sincos, which calculates both sin and cos simultaneously, and performs faster than any of them:

sincos(double x, double * sin, double * cos)
{
    if(fabs(x) < epsilon) {
        *sin = x;
        *cos = 1.0 - x*x/2.0;
        return;
    }
    double s, c;
    sincos(x/2, &s, &c);
    *sin = 2.0*s*c;
    *cos = c*c - s*s;
}

As a side note, in the base case for sin and tan you should return x instead of SIN/TANMIN (for cos it is 1 - x*x/2).

[mike_2000_17]

First of all, if(x==MINVAL) makes no sense, I think you meant to write if(x < MINVAL) .

Second, I'm pretty certain that you would be better off using a Tailor series expansion. All you should need is a Taylor series expansion around 0 degrees for both the sine and cosine, which is valid (within you desired precision) on an interval from -45 degree to 45 degrees (i.e. -Pi/4 to Pi/4). With simple trig. identities, you should be able to use those two expansions to calculate a sine, cosine or tangent of any angle. For example, for the sine of an angle of 80 degrees, you can compute the cosine of -10 degrees (which is in the Taylor series' range for the cosine), and similarly for other ranges of values.

Third, you should also remember that branchings (e.g. conditional statements) are surprisingly expensive and slow, you should reduce those to a minimum. Also, mark your free-functions with the keyword "inline" to allow the compiler to inline if it leads to faster code.

Finally, if you are using or can use C++0x, you should mark those functions with constexpr keyword to allow compile-time computation of the values whenever possible.

[Labdabeta]

Thanks, those ideas helped a lot!

[mrnutty]

First of all, if(x==MINVAL) makes no sense, I think you meant to write if(x < MINVAL) .

Second, I'm pretty certain that you would be better off using a Tailor series expansion. All you should need is a Taylor series expansion around 0 degrees for both the sine and cosine, which is valid (within you desired precision) on an interval from -45 degree to 45 degrees (i.e. -Pi/4 to Pi/4). With simple trig. identities, you should be able to use those two expansions to calculate a sine, cosine or tangent of any angle. For example, for the sine of an angle of 80 degrees, you can compute the cosine of -10 degrees (which is in the Taylor series' range for the cosine), and similarly for other ranges of values.

Third, you should also remember that branchings (e.g. conditional statements) are surprisingly expensive and slow, you should reduce those to a minimum. Also, mark your free-functions with the keyword "inline" to allow the compiler to inline if it leads to faster code.

Finally, if you are using or can use C++0x, you should mark those functions with constexpr keyword to allow compile-time computation of the values whenever possible.

I would suspect using trig function be just as fast if not faster than taylor expansion.

@OP: Unless you are in a really limited environment, there is no need for all this complication. You should just use a simple one liner function call.