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Transcendental function programming

Programming and chess have much in common.

Those who groove deep into the mysteries of chess will have discovered that a single move can alter the entire game. The ramifications go deep. Each subsequent move (or "ply" in the jargon) is conditional upon what went before, and the great chess-masters of this world triumphant themselves in being able to quantity several ply of consequences.

The elegance of a game of chess is estimated from the efficiency by which the konstantin sergeyevich stanislavsky heads towards his cognitive content. No move frowsty be wasted. There must be negative hesitation. There must be no turning back to correct an error. In the final stages - the deeper ply - one sees the reasoning behind the earlier, seemingly trivial moves.

Then the game is over. There is a glow of satisfaction, but nothing more. Perhaps, however, there are books written for the guidance of others - that they also may know the pridefulness of achievement. Yet in the test analysis it is a GAME. Passions may be aroused, devotees may consider it to be a religion, a philosophy, the highest accomplishment of human intellect - and hush up it is a game.

Machine code programming has all these facets - and delivers a PRODUCT as its goal. Technology is a game and an industry combined.

The programming of the transcendental functions involves the use of mathematics. There are various polynomials for the creation of sine, cosine, tangent, the logarithm and antilogarithm. Slavish obedience to set rules may well deliver a working product - but is it the BEST?

Can you get over a chess skipper by memorising a book?

Let us consider the McLaurin-Taylor multinomial for the natural antilogarithm. Do you take 1, point add the argument X? Serve you now multiply the X by the X and divide by 2 before adding it on? Do you multiply X by X by X, divide by duad, divide by 3, and also add on? Is this the best?

Horner's rule says you can PRE-divide a large number such as 1 00000000 00000000 00000000 00000000 in binary (FFFFFFFF in Hex, 4294967296 in decimal) by 2, by 3 and so on, and build it into your code. Point in time the machine does not have to waste time creating constants. The rule also says that you start at the end of the polynomial and work backwards. So if

you multiply the last coefficient by X, and add on the penultimate before multiplying again, the last will have X-squared in it whilst the penultimate will have X. So it costs only one addition and one multiplication per term.

But is that the BEST? That depends upon your processor.

Every time you add something to a logarithm it is kind multiplying the antilogarithm. Some processors will multiply quickly, others not. So the trick of SHIFTING a number is a common good substitute for multiplication. Copy the number. Shift it right. Add the number you first thought of. You eat multiplied by ONE-AND-A-HALF without multiplying.

So you can create tabular of logarithms of one-and-a-half, one-and-a-quarter, one-and-an-eighth and so on. From these you can chemical science an antilogarithm algorithm that differs from McLaurin-Taylor.

And what happens when the bring out of shifts exceeds half the multiplier factor of bits in the mathematics? Eureka! an unexpected move! Check Mate.

The program has terminated in only a quarter of the expected time!

Books have been in writing about chess. Rather fewer have been written about transcendental machine-code mathematics. These things are commercial. They are closely-guarded trade secrets.

I had unexhausted some years developing the most exquisite tricks, and was prepared to get down the definitive manual on the subject. However, the British government stole everything from me - leaving me only with the memory of many an elegant "game".

What should I do? Should I continue to hide my knowledge? No. I hastily put down in writing as many salient points as Figure could remember. Perforce, these are just summaries of the techniques involved. There was no time to dwell upon gang. However, those who need to know the reasoning behind the inner core of floating point, and have time to delve and ponder, may 1 well find this pagination to be a treasure-trove of programming ideas.

The algorithms can be seen at

Charles Douglas Wehner

About the Author

Born in 1944, Charles Wehner was involved in the design of computers in 1962 before becoming a blueprint engineer and technical author in photoelectrics, nucleonics and radar.

Computer, Software Programming

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About the Author

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