Programming Trigonometric Functions

[Dan894]

What's the best way to do this. I've heard make a look up table (which I'm not entirely sure how to do anyway, I've never done arrays or array-like things in C), and Taylor polynomials. I've been trying the Taylor approach, but encounter this crude reality that most computers do NOT have infinite memory.

Is defining an integral exponential function like this (recursively) generally a bad idea?

float exp(float base, int exponent) {
if(exponent < 0)
return exp(base, -1*exponent)
else if(exponent == 0)
return 1;
else
return exp(base, exponent-1)
}

I did a similar thing with factorial...

unsigned long factorial (unsigned character n) {
if(n==0)
return 1;
else
return n*factorial(n-1);
}

I doubt I'd ever be using that for something bigger than 12! or so.

Of course, since I had some debugging to do, I wound up defining the trig functions using a much more brute force mentality:

float sin(float x) {
return x-(x*x*x)/6+(x*x*x*x*x)/120-(x*x*x*x*x*x*x)/5040+(x*x*x*x*x*x*x*x*x)/362880;
}
float cos(float x) {
return 1-(x*x)/2+(x*x*x*x)/24-(x*x*x*x*x*x)/720+(x*x*x*x*x*x*x*x)/40320;
}
Then I defined the other four using ratios... and the arctrig...

float arcsin(float x) {
return x + x*x*x/6 + 3*x*x*x*x*x/8 + 135*x*x*x*x*x*x*x/48 + 1215*x*x*x*x*x*x*x*x*x/384;
}

float arctan(float x) {
if(-1 < x && x < 1)
return x-x*x*x/3+x*x*x*x*x/5-x*x*x*x*x*x*x/7+x*x*x*x*x*x*x*x*x/9;
else if(x >= 1)
return pi/2-1/x+1/(3*x*x*x)-1/(5*x*x*x*x*x)+1/(7*x*x*x*x*x*x*x)-1/(9*x*x*x*x*x*x*x*x*x);
else
return -pi/2-1/x+1/(3*x*x*x)-1/(5*x*x*x*x*x)+1/(7*x*x*x*x*x*x*x)-1/(9*x*x*x*x*x*x*x*x*x);
}

And the remaining ones I defined using triangle identities.

Around the term x*x*x*x*x*x*x*x*x/362880, I start getting an error message like

Hence I just give up the accuracy and only get accurate to the 1.6^8/40320 or whichever term I gave up at. And I still don't trust that this code will, you know, actually work. (I also wound up defining
float pi = 3.141592653589793238462548838279; )

[Keith Watson]

What accuracy (how many decimal places) do you really need? For example, take an angles of 1.0, 1.1, 1.01, 1.001, etc and use those to calculate distances in the size of the playing field. What accuracy is good enough?

I forget which college math class it is but they teach how to stop the series based on how many decimal places of accuracy you want. Also consider how many bits is the variable you are using and what accuracy actually fits inside of that. Once these things are considered 20 decimals places for pi does not add to the accuracy.

If you choose to use the math library instead, those almost always have the accuracy and overflow issues sorted out.

[Dan894]

Am I going to have any problem with slow calculations using the Taylors?

There was an error too... Math section can not fit the section.

And in Calc II they cover Taylor's inequality, which for these trig functions, it gives my accuracy down to (pi/2)^n/n! where n is one bigger than the exponent in the last term.

[Kevin Watson]

...I've heard make a look up table (which I'm not entirely sure how to do anyway, I've never done arrays or array-like things in C)...From a couple of days ago: http://www.chiefdelphi.com/forums/showpost.php?p=449278&postcount=4. If you try the search function, you'll find many posts on this topic.

-Kevin

[esquared]

What's the best way to do this. I've heard make a look up table (which I'm not entirely sure how to do anyway, I've never done arrays or array-like things in C)...

We started with the command line version, and one of students developed a gui frontend, of the program in the following thread:

http://www.chiefdelphi.com/forums/showthread.php?t=44012

Does the lookup table generation for you.


-Eric