Recovering the SIN() and COS() Functions
again, I had a chance to review the BALGOL 220 implementation of a function
versus Don Knuth's implementation of the same function for Clevite Corporation in
Cleveland. This time I was hoping to find a 30 word version of the SIN() function.
I first checked out the SIN() implementation in the BALGOL compiler for the Burroughs 220. Uh-oh, - 56 words versus the 31 words on the 205 that calculate both SIN() and COS().
the input by 1/Pi - effectively changing the radian input into "half cycles." For instance, 126.44910 radians (this is 20 x 2Pi
+Pi/4) becomes 40.249999. We throw away the digits before the decimal
point and are left with 0.249999 That is one quarter of a half-cycle or
- Now, double the value. The net effect to this point is to have multiplied by 2/Pi which converts our original input value in radians to a fraction of a quarter cycle. If that fraction is greater than 1, we subtract 2 from it. This always yields a decimal fraction between -1 and 1 corresponding to an angle between -90 and 90 degrees as input to the next step.
- Use a Power series of 5 terms to calculate a Sine value. 0.70710678 in this case.
SIN(x) = Ax+Bx^3+Cx^5+Dx^7+Ex^9
- Adjust the results if the original input was a value that required the result to be negative.
- Return to the calling program.
Enter Cecil J. Hastings, Jr., The Great Approximator.
Back to the 205 version of ALGOL 58
- Multiply by 1/360 - effectively, effectively changing the input into "full cycles." For instance, 7245 degrees
(this is 20 x 360 +45) becomes 20.125 We throw away the digits
before the decimal point and are left with 0.125 That is one eighth
of a full cycle or 45 degrees. No further adjustments are required.
- Use a power series of 9 terms to calculate a Sine value. 00.70710678 in this case.
- Return to the calling program,
How well does it work? I drew my own error curve and compared it to the BALGOL results. The manual only promises seven digits of accuracy and the routine comes close:
Using a MacLaurin series of equal power would also work very well out to 180 degrees but then definitely goes "off the chart." Where did the coefficients come from for the 205 version? I don't have an answer for that. But it is worth noting that the common method for finding a Sine or Cosine in the 1950s and 1960s, was to get out the Chemical Rubber Company's book of Standard Mathematical Tables and look them up.
This handy publication was produced in Cleveland with the assistance of considerable labor obtained at Case Institute. A commenter at the John Cook Blog left this description of how the labor was obtained:
- R E Van Valkenburgh
This also brings up the question of how the manual labor for number crunching has been acquired.
For those of you who remember the huge, “Handbook of Chemistry and Physics” (still?) published by the former Chemical Rubber Company (now CRC Press)… my dad used to tell us how the students at the then Case Institute of Technology (not Case-Western Reserve University) were conscripted to calculate the large number of large tables. Hey, it never hurt your grade if your professor asked you to volunteer.
Not only is it remarkable how some of these things were calculated; but even more remarkable how accurate they often were. (Case & the CRC had multiple independent calculations to help identify errors, according to dad.)
So there would have been plenty of expertise around the Case campus to support Sine approximations. Still, it might have been better to limit the input to range to just 180 degrees and adjust accordingly.
Of some interest is what happens as the initial input angle grows - for example, into the thousands or more, and we discard the first several digits - what happens to our accuracy? We could lose signifiance. Lop off the leading three or four digits of the angle, and we are left with only five or four digits as our input. The BALGOL manual addresses this - but only when we have an angle greater than ten million and lose all significance. The manual and the compiler note an error message: RESULT ILL-DEFINED FOR SIN On the 205, you are on your own to be aware of the problem.
Calculation of the COSINE
The Cosine function is a simple addition to the subroutine library. Since the Cosine function is equal to the Sine function offset by 90 degrees, this subroutine consists of a single addition instruction placed immediately before the Sine subroutine adding 90 degrees to the submitted angle.
Addendum for the Modern Era
is not the only concern when calculating Sines and Cosines on a
computer. Speed is also an issue. At about the same time as our two
Datatron compilers were being written, a new method for calculating
these functions was being developed within Convair Corporation's group
working on a navigation computer for the B-58. The method was not
published for a while but apparently became very popular among
calculators and computers that did not have a hardware multiplication
capability. There was a time when "multiplications per second" was a
key performance measure for computers. The new method is known as
CORDIC, involves no multiplications and is worth reading about either in Wikipedia or a serious paper if you have a math and hardware inclination. The always informative Ken Sherriff Blog has a post on CORDIC as found in the 8087 math coprocessor. He includes photos of the coefficients hard coded into the 8087.
Is speed a concern on either the BALGOL or Clevite implementation of SIN(x)? I don't think so; the Clevite routine runs faster than the square root subroutine on the 205. I will come back to the question, however, in a later blog post about benchmarking these old machines.
Additional Notes and Resources:
An exhaustive set of coefficients of polynomials for approximating SIN(x) carried out to 36 digits can be found under the title Fast Minimax Polynomial Approximations of Sine and Cosine; it can be found here. See the Summary in Appendix B for development details. This may be the best source of approximating polynomials on the Internet but I had no luck mapping them back to Knuth's set - probably because they were optimized on the interval (0, 90) rather than on (0, 360).
The original paper published in this area appears to be the Remez Exchange Algorithm published (in French) in 1934.
Numerical Methods for Scientists and Engineers, first published in 1962 by R. W. Hamming, is likely the definitive work on Approximation Theory from the early years. Hamming, then at Bell Labs, was also a presenter at the 1949 seminar in Endicott.
Modern treatment of the subject can be found in Approximation Theory and Approximation Practice by Lloyd N. Trefethen (SIAM, 2013)