Sine/Cosine Approximator

[Recursive Acronym]

Wow, I started this topic and I haven't posted for ages...
Anyway, here is a small challenge:
Create a program that approximately calculates the cosine (or sine, it doesn't matter) of a number given in degrees or radians without using trigonometric functions.

[Weregoose]

I have 21 bytes (within a program, minus the title).

[simplethinker]

How exact do you want it?

[edit] I can only get 26 bytes...

[another edit] 7 Bytes (in mem menu prog named A is 17 bytes, so 17-9-1=7). The method itself is 4 tokens

[Weregoose]

After another trial, I believe I've matched simplethinker's (four tokens).

[simplethinker]

[Weregoose]

I read off the number of bytes as it appears in the [2nd] [+] [2] [7] menu, then subtract the number of characters in the title of the program (just one if it is prgmA).

[DarkerLine]

How approximately are we approximating the sine or cosine here? And is something like real(e^(iAns "approximating" or just cheating?

[Weregoose]

To me, that's less cheating than what I think simplethinker and I came up with (no offense):

cos(θ) = [font="courier new"]P►Rx(1,θ

[Recursive Acronym]

Sorry, I should have been more specific. I don't really care about the size for this challenge. I hadn't thought of the polar conversions, but I'll say now that they are off limits. Mine only does the first quadrant, but it doesn't matter because of the symmetries of the trig functions. It gives results with an error of about 10^(-6) and takes about 3 seconds. I think there are polynomials that approximate trig functions, so that would be one way, but I used a different method.

[thornahawk]

Would using inverse trigonometric functions or complex exponentials e^(i*something) be disallowed as well, just to clear things up? ;)

thornahawk

P.S. It must be added as a reminder for the respondents to this challenge to note: once you can compute sine, it merely takes some modifications for your program to compute cosine (and vice-versa).

[simplethinker]

I have a slight modification to this. Have a program that approximates sin(x) without the use of: any form of i or e (including logs), no trig function (including hyperbolic and inverse), not complex, angle or polar functions, and has to work in Real mode. I have 34 bytes

[alexrudd]

You should specify accuracy, too. My program below has ± 1 x 10⁰ accuracy calculating sine or cosine in any quadrant.

[DarkerLine]

[thornahawk]

I have a cosine routine here I wrote with more emphasis on efficiency and as wide a range as possible (works even for, say X>1000 with relatively good accuracy) rather than size... unfortunately I had to use a logarithm, but with no complex number trickery, it must be added.

[highlight below]
[font="courier new"]
1→Y:X→Z
If real(X)<0
-X→Z
If abs(Z)>e-6:Then \\ e-6 : 10^-6
max(0,int(log(abs(Z))/log(2)))→K \\ estimate the size (or do nothing if Z is small)...
Z/2^K→Z \\ ...then scale accordingly by halving K times
1→T:0→J
Repeat abs(Y–O)=0 \\ use the Maclaurin series on the scaled argument
Y→O:J+2→J
-Z²T/J/(J–1)→T
Y+T→Y
End
For(J,1,K)
2Y²–1→Y \\ apply the double-angle formula K times
End
End
If imag(Y)=0 \\ ensures real results are returned as real numbers
real(Y)→Y
Y

[Weregoose]

Couldn't one take the input (mod 2[font="times new roman"]π

[thornahawk]

@Goose: Here is a paper you might want to look at regarding your question.

thornahawk

[Weregoose]

Oh, so the very build of the number in the input is essential to the algorithm. :)

[EDIT]

Okay, while some may not agree with this...

[s]2[font="times new roman"]π/79fPart(.5X/[font="times new roman"]π

[simplethinker]

[thornahawk]

I'll be doing a bit of explaining with the cosine program I've written, so here it goes as hidden text:


First, as it was mentioned in the paper, you have two possible choices for range reduction: you can either repeatedly subtract multiples of the period from the argument (mod 2π reduction) or scale the argument by halving repeatedly (which is the method I used). The importance of this, again, is that Maclaurin series (or for those with more sophistication, Padé approximants or economized rational approximations) are only worthwhile near their point of expansion, which is commonly taken to be 0. The further away the argument is from 0, the worse these approximations become.

Now that we've gotten the importance of reduction out of the way, I now justify why I chose halving over subtraction:

1. Subtractive cancellation - at large enough arguments, trying to subtract a multiple of 2π of comparable magnitude will leave you with much less significant digits than original. Because of this, any trig function you will be computing with this now not-so-accurate argument will be inaccurate, too. Special tricks like those mentioned in the paper must then be employed.

2. Generality - mod 2π reduction will definitely not be helpful if your argument has large imaginary part (e.g. cos(100i) and cos(40π+100i) would of course have the same value), and since large absolute values are bad for the underlying approximants, scaling by halving will be a much simpler solution for handling arguments of large absolute value.

Besides, I again scoped in advance: apart from complex numbers, you can modify the algorithm easily and it will compute a matrix cosine.



simplethinker: One can still cheat to use common logarithms instead of natural ones; log(X)/log(e)=ln(X) is valid for all complex values. ;)

Weregoose, it seems, got that cute inequality from the Wolfram Functions Site, which also allows us to compute the cosine and a bunch of other functions to arbitrary accuracy, so we all now have a standard to check against. :D

If everybody wants to stress-test their trig routines, here are a few choice arguments to test against:

π/4 (customary )
i (should give, with some factors, the hyperbolic equivalent)
arguments with absolute value >10^3, including complex ones (e.g. 4+1000i)

thornahawk

[simplethinker]

About the log, natural log thing. I was hoping that no natural logs would rule out people's thoughts of using logs to get it, because it is a form of e/natual log, which I had said no forms of. I'm probably making this even more confusing, since I was hoping for people to figure out what I meant without having to go into detail of what I meant. So, after the corrections to what I used for restrictions, I'm even more confused of what I meant, so disregard everything I said.