Sine/Cosine Approximator

[thornahawk]

I have a proposal: seeing that we have so many different brain teasers coming from so many different sources... might it be time to make a new TI-Basic subforum? ;)

thornahawk

[Recursive Acronym]

A new subforum would be okay... this topic is on the small side; but since we have several problems going on at once in this thread, a new forum might be nice. It would definitely have more activity than the 68k Basic forum, though .
My program does cosine, accepting angles from 0 to 90 degrees. It also asks for a number of iterations. It works by dividing breaking the angle input into parts, each part being .5 the size of the previous. 90 would be broken up into 45+22.5+11.25+5.625+..., which approaches 90 as the number of terms increases. 58 would be broken up as 45+11.25+1.40625+.17578125..., which approaches 58. A greater number of iterations will increase the number of terms in this sequence, the related accuracy, and also the processing time. It also calculates the correct answer at the beginning and displays it alongside program's calculated value for comparison purposes. Once again, it is a proof of concept, and isn't optimized for size or speed.

[thornahawk]

I see recognizable portions of CORDIC in RA's program that he posted above.

You know, of course, that you don't have to switch to Degree mode to do this, right? ;)

thornahawk

P.S. It's a common theme, as always: efficiency, accuracy, generality, and code size can become conflicting goals when writing a program such as this.

[Recursive Acronym]

The switch to Degree mode is so that the actual answer can be stored to A and displayed when execution is finished.

[Weregoose]

It might be useful to know that the degree and radians symbols usually override the mode settings (but not when using the inverse trig functions). It's a good way to save a byte or two while leaving the user settings alone.

[DarkerLine]

You can kind of get around the inverse trig function problem by dividing by the appropriate unit instead... e.g. sin^-1(1/2)/1^o will return 30 regardless of the mode setting.

[thornahawk]

The "45" in RA's program can be changed to "π/4", and the program would now work in Radian mode. ;)

thornahawk

[DarkerLine]

And if you write it 45^o it will work in both modes.

[thornahawk]

There, I've added to and edited the solution I've posted. That indeed was an embarrassing bug. :blush:

I'm posting another solution that uses an approximant with better convergence properties. Here it is, with explanatory comments:

[highlight below]
[font="courier new"]
1→Y
If abs(X)>e-6:Then \\ e-6 : 10^-6
X→Z:0→K
While abs(Z)>π/4 \\ scale X accordingly if needed
K+1→K:Z/2→Z
End
3→B:Z²/4→A:1→Y \\ evaluate the continued fraction for tan(Z/2)...
Y→C:0→D
Repeat abs(H–1)=0 \\ ...using Lentz's algorithm
(B–AD)ֿ¹→D
B–A/C→C
CD→H:HY→Y
B+2→B
End
(4Y²–Z²)/(4Y²+Z²)→Y \\ use the identity relating tan(Z/2) and cos(Z)
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