Cordic algorithm again

[Edwa...@alum.mit.edu]

Some time ago there was discussion here regarding the Cordic (coordinate rotation) algorithm for sines and cosines. Experience with the its implementation on a PDP8 leaves a bit to be desired. For example, with a double word argument and using double word arithmetic, the sine and cosine of 45 degrees is off by six in the last octal digit.

Two word calculation
Given angle: 45
Angle used: 0o20000000 45.0
Result
Sine:
Calc: 0o26501177 0.707107424
True: 0o26501171 0.7071067
Cosine:
Calc: 0o26501163 0.707105994
True: 0o26501171 0.7071067

In this calculation angles are scaled by 4000 0000 (octal)/90 (decimal) and the results by 4000 0000 (octal).

The advantage of the Cordic algorithm is significant: both sine and cosine can be obtained with 2 pages of code with no requirement for an EAE. This, as compared to the DEC distributed routines that require 3 pages (one each for sine, cosine and multiplication).

So is it possible to implement the Cordic algorithm in two pages and get results accurate to the last octal digit of a two word result? The answer is yes: there was enough left in the two pages of the two word version to calculate with three words. Here is the result of that implementation:

Three word calculation
Given angle: 45
Angle used: 0o200000000000 45.0
Result
Sine:
Calc: 0o265011714707 0.7071067823271
True: 0o265011714637 0.707106781
Cosine:
Calc: 0o265011714566 0.7071067799697
True: 0o265011714637 0.707106781

In the attachment are the python prototypes for both types of calculation, together with the pal assembly code and listing for both.

Yul

[Ian Schofield]

Dear All,

 Seeing we are now in the area of digital implementations of functions without resorting to floats/doubles etc. This tiny function

draws a (not too bad) circle using the sin/cos recursion technique and only 12 bits. Here is the c version which is actually

used to draw circles for a tic-tac-toe app that runs on our favourite computer.

This is actually a quadrature signal generator.

int circle(ix,iy)  // Draw circle at coord x,y

int ix,iy;

{

        int re,im,tm,i;

        re=0;

        im=1000;

        for (i=0;i<19;i++) {

                dispxy(ix+re/20,iy+im/20);

                tm=re+im/3;

                im=im-tm/3;

                re=tm;

        }

}

More of a curiosity than something useful! 

BW, Ian.