How the chain rule applies to integral equations.

In Calculus, one of the most basic things that can be solved for, is that a principal function receives a parameter, multiplies it by a multiplier, and then passes the product to a nested function, of which either the derivative or the integral can subsequently be found. But what needs to be done over the multiplier, is opposite for integration, from what it was for differentiation. The following two work-sheets illustrate:

PDF File for Desktop Computers

EPUB File for Mobile Devices

Please pardon the poor typesetting of the EPUB File. It’s the result of some compatibility issues (with EPUB readers which do not support EPUB3 that uses MathML.)

Continue reading How the chain rule applies to integral equations.

How to compute the sine function, on a CPU with no FPU.

There exists a maxim in the publishing world, which is, ‘Publish or Perish.’ I guess it’s a good thing I’m not a publisher, then. In any case, it’s been a while since I posted anything, so I decided to share with the community some wisdom that existed in the early days of computing, and when I say that, it really means, ‘back in the early days’. This is something that might have been used on mini-computers, or, on the computers in certain special applications, before PCs as such existed.

A standard capability which should exist, is to compute a decently accurate sine function. And one of the most lame reasons could be, the fact that audio files have been encoded with an amplitude, but that a decoder, or speech synthesis chip, might only need to be able to play back a sine-wave, that has that encoded peak amplitude. However, it’s not always a given that any ‘CPU’ (“Central Processing Unit”) actually possesses an ‘FPU’ (a “Floating-Point Unit”). In such situations, programmers back-when devised a trick.

It’s already known, that a table of pre-computed sine functions could be made part of a program, numbering maybe 256, but that, if all a program did was, to look up sine values from such a table once, ridiculously poor accuracies would initially result. But it was also known that, as long as the interval of 1 sine-wave was from (zero) to (two-times-pi), the derivative of the sine function was the cosine function. So, the trick, really, was, to make not one lookup into the table, but at least two, one to fetch an approximate sine value, and the next, to fetch an approximate cosine value, the latter of which was supposedly the derivative of the sine value at the same point. What could be done was, that a fractional part of the parameter, between table entries, could be multiplied by this derivative, and the result also added to the sine value, thus yielding a closer approximation to the real sine value. (:3)

But, a question which readers might have about this next could be, ‘Why does Dirk not just look up two adjacent sine-values, subtract to get the delta, and then, multiply the fractional part by this delta?’ And the answer is, ‘Because one can not only apply the first derivative, but also the second derivative, by squaring the fractional part and halving it (:1), before multiplying the result from that, by the negative of the sine function!’ One obtains a section of a parabola, and results from a 256-element table, that are close to 16 bits accurate!

The source code can be found in my binaries folder, which is:

And, in that folder, the compressed files of interest would be, ‘IntSine.tar.gz’ and ‘’. They are written in C. The variance that I get, from established values, in (16-bit) integer units squared, is “0.811416” “0.580644” (:2). Any variance lower than (1.0) should be considered ‘good’, since (±1) is actually the smallest-possible, per-result error.

(Updated 12/04/2020, 11h50… )

Continue reading How to compute the sine function, on a CPU with no FPU.