ios4 - Value of sine 180 is coming out as 1.22465e-16

Question

I want to implement a Sine and Cosine calculator in ios4:

if([operation isEqual:@"sin"]){
    operand = (operand*M_PI/180.0);
    operand=sin(operand);
}

The code gives me correct answer for values from 0 through 90.

When I give value of 180, I get 1.22465e-16 as an answer. I expect zero.

Where does this small difference come from?

Answer 1

Just to be clear, your program is giving you the correct answer. That is to say, it is doing exactly what you told it to do in your code.

180*M_PI is correctly rounded (per IEEE-754), and gives the value:

565.4866776461627750904881395399570465087890625

dividing that by 180 is also correctly rounded, and gives the result:

3.141592653589793115997963468544185161590576171875

which is not exactly the mathematical value of π. In fact, it is:

π - 0.0000000000000001224646799147...

the first order term of the Taylor series for sin(x) around π is (π-x), so sin(π - x) is, for small x, nearly exactly -x. In fact, the result that you're getting is the correctly rounded result. The library couldn't possibly deliver a more accurate answer.

As Ben Voigt suggested, if this is actually a problem for you, you can work around it by reducing the argument into the range [-90, 90) before converting from degrees to radians. An even better suggestion is njuffa's to use a sinpi function that will do this work for you. iOS does not have such a function, but it does have vvsinpi, which implements sin(π*x) for vectors, and can be made to do what you want:

double result;
int vectorLength = 1;
vvsinpi(&result, &operand, &vectorLength);

Please also file a bug requesting that sinpi be added to the math library as an extension.