Mar 192019

file:///C:/ti/430eforth-ide/lessons/lesson12.html

( Example 12. Sines and Cosines ) \ Sines and cosines of angles are among the most often encountered \ transdential functions, useful in drawing circles and many other \ different applications. They are usually computed using floating \ numbers for accuracy and dynamic range. However, for graphics \ applications in digital systems, single integers in the range from \ -32768 to 32767 are sufficient for most purposes. We shall \ study the computation of sines and cosines using the single \ integers. \ The value of sine or cosine of an angle lies between -1.0 and +1.0. \ We choose to use the integer 10000 in decimal to represent 1.0 \ in the computation so that the sines and cosines can be represented \ with enough precision for most applications. Pi is therefore \ 31416, and 90 degree angle is represented by 15708. Angles \ are first reduced in to the range from -90 to +90 degrees, \ and then converted to radians in the ranges from -15708 to \ +15708. From the radians we compute the values of sine and \ cosine. \ The sines and cosines thus computed are accurate to 1 part in \ 10000. This algorithm was first published by John Bumgarner \ in Forth Dimensions, Volume IV, No. 1, p. 7. DECIMAL 31415 CONSTANT PI 10000 CONSTANT 10K VARIABLE XS ( square of scaled angle ) : KN ( n1 n2 -- n3, n3=10000-n1*x*x/n2 where x is the angle ) XS @ SWAP / ( x*x/n2 ) NEGATE 10K */ ( -n1*x*x/n2 ) 10K + ( 10000-n1*x*x/n2 ) ; : (SIN) ( x -- sine*10K, x in radian*10K ) DUP DUP 10K */ ( x*x scaled by 10K ) XS ! ( save it in XS ) 10K 72 KN ( last term ) 42 KN 20 KN 6 KN ( terms 3, 2, and 1 ) 10K */ ( times x ) ; : (COS) ( x -- cosine*10K, x in radian*10K ) DUP 10K */ XS ! ( compute and save x*x ) 10K 56 KN 30 KN 12 KN 2 KN ( serial expansion ) ; : SIN ( degree -- sine*10K ) PI 180 */ ( convert to radian ) (SIN) ( compute sine ) ; : COS ( degree -- cosine*10K ) PI 180 */ (COS) ; \ To test the routines, type: \ 90 SIN . 10000 \ 45 SIN . 7070 \ 30 SIN . 4999 \ 0 SIN . 0 \ 90 COS . 0 \ 45 COS . 7072 \ 0 COS . 10000 flush