January 1983

DTACK GROUNDED, The Journal of Simple 68000 Systems

Issue # 16 January 1983 Copyright Digital Acoustics, Inc.

Once upon a time two young men founded a software company to write a BASIC interpreter for a new invention called a personal computer. Because everyone stole their software, their software became the standard for the new industry. As the new industry grew, so did the software company so that the software company continued to be the standard for the industry.

For many years this company prospered mightily by providing software, especially BASIC interpreters, which ran on 8 bit microprocessors. One day a blue giant appeared out of the forest and requested software for the 8 bit 8088. Since the 8088 is source compatible with the 8080, the software company was easily able to respond favorably to the giant's request. And the treasury of the software firm grew more rapidly as royalties from the giant poured in.

Since the giant's machine used an 8 bit processor and ran software written for yet another 8 bit processor, the giant's machine ran at about the same speed as earlier 8 bit machines supported by the software companies' products. This greatly puzzled many people since an evil sorcerer had cast a spell on the land so that the people would believe that the 8 bit machine was really a 16 bit machine.

The people resolved this problem by asserting that the giant's machine was a 16 bit machine (although it was not) and that it therefore ran swiftly (although it did not). But the two young men had amassed enough wealth to purchase an antidote for the evil sorcerer's spell and were aware of the true facts.

Among the true facts they were aware of was the fact that 99.999% of their (substantial) income came from 8 bit machines running at 8 bit machine speeds.

This posed a dilemma one day when 99 pygmies and one large oaf turned up and asked the software house to provide a BASIC interpreter for THEIR machines, all of which were based on a real, honest 16 bit processor called the 68000. The dilemma was caused by the fact that the 68000, being a true new generation 16 bit processor, was ten to twenty times the speed of the machines from which their (substantial) income derived.

And if those 99 pygmies (and one large oaf) began manufacturing machines which could run ten or twenty times faster than those 8 bit machines, then the sales of the 8 bit machines would doubtless diminish, and their (substantial) income would ALSO diminish. And the blue giant who had been friendly might become less so.

But the two young men and their assistants came up with a plan: they would produce a BASIC interpreter for those 100 68000 based machines which would run at EXACTLY THE SAME SPEED as all the 8 bit machines. And then they realized that the problem would recur as even more real, honest high performance 16 bit processors became available. The two young men are no dummies and they came up with a generic solutions

They would write a version of their BASIC interpreter in a sort of 'p-code'. Not literally PASCAL p-code but that sort of thing. They would then write native code for each of the real, honest 16 bit processors to interpret the 'p-code'. Therefore, TWO levels of interpretations would be involved and the real, honest new-generation 16 bit machines would run as slowly as all those older 8 bit machines.

The plan was carried out and it has worked beautifully. The large oaf has a BASIC interpreter for his expensive 68000 machine and is puzzled and angry over its sloth (for a 16 bit machine). Interface Age has just reviewed one of the pygmies' 68000 machines and discovered that it benchmarked more slowly than some of the older 8 bit machines.

And the (substantial) income of the software house remains safe and the blue giant remains friendly.

End of fable.

On page 277 of Oct '82 BYTE magazine we find the following: "I saw a new machine using the 68000 chip and got into a discussion of it with Carl Helmers..."

"It uses the UCSD p-code as the OPERATING SYSTEM," I said. "That's got to be the most inefficient thing I ever heard of."

"So what?" Carl replied, "The chip is so fast you don't notice."

(That excerpt is from Jerry Pournelle's User Column.)

That attitude is common, widespread, and ABSOLUTELY DEAD WRONG! It also provides an alternate explanation to the use of the TWO levels of interpretation in the 68000 version of the 'industry standard' BASIC.

This attitude is common to those who feel that since the 8 bit personal computers make nice text editors and do a good job of running checkbook balancing programs in BASIC that these are the same tasks which will be assigned to the new generation, high-performance 16 bit personal microcomputers. That is absolutely the WRONG approach to selling 68000 based machines, especially high priced 68000 machines.

Look: the 68000 brings enough performance to the individual personal computer operator (single-user single-tasking division) that NEW application areas are opened up. What are these new applications? Why, we don't know all of them or even very many of them. When the Pet, Apple and Trash 80 I were new nobody envisioned the electronic spreadsheets, data base management systems and such that are now commonly available.

However, our customers have already identified several applications areas where the power of the 68000 can be usefully applied. Unfortunately, the programs are proprietary since these customers are using that power to make money. But me can tell you that the 68000 is being used in on-line real-time financial analysis ("We are not using the 68000 instead of our IBM 4341 Mod II because it is faster than the IBM in this application. We are using it because this is a real-time on-line application and our 4341, like all mainframes, is 'down' an appreciable percentage of the time. On the other hand, our Apple/DTACK combo has proven highly reliable. We don't really object to the speed of the 68000, of course.")

An obvious application of the 68000 is in the CAD/CAM area. In one of those expensive (hundreds of dollars per year subscription) newsletters we find the following report of a 68000-based CAD system which sells for about $30K (add 15K for a plotter). This newsletter states, "...we were impressed by the briskness of the system's interactive response. It is as fast as any system we've seen, regardless of price." You DO understand that CAD machines in the $300K (not $30K) area are TYPICAL of the CAD industry (e.g. CALCOMP and Evans & Sutherland), don't you? What company are we talking about? Gee, we forget. As they have forgotten who makes their 68000 stuff.

However, we CAN tell you that since that newsletter (dated Nov '82) was published, the interactive speed has been approximately DOUBLED due to the adoption of a superior line-drawing algorithm. Presumably, this system is now TWICE "as fast as any system we've seen, regardless of price."

Modesty forbids that we reveal the source of the superior line-drawing algorithm.

The superior speed of the 68000 can even be applied to games. See MICRO magazine, Dec '82, page 86. You will find an advertisement for a chess-playing program for the Apple/DTACK combo. It should be apparent that the 68000 can play a better game than an 8 bit machine due to sheer raw speed. We have ordered this program along with the debugger and will report on them in the future. We left the PASCAL stuff alone.

It just makes sense that the future applications of the 68000 will take advantage of its pure blinding speed or its large linearly addressable memory space. Writing checkbook balancing programs for the 68000 is ridiculous!

If we use TWO LEVELS OF INTERPRETATION it is damned unlikely that we will achieve "...interactive response... as fast as any system we've seen, regardless of price." The people writing operating systems in p-code are either idiots or they are DELIBERATELY trying to sabotage the sales of their own computers! (With notable success, we add.) Or do these people think they can compete at $5000 and up with the $169 VIC-20 (or the $899 Apple II) with hardware having the SAME PERFORMANCE as the VIC-20 (or the Apple II)?

Has ANYONE asked himself (herself) why the world does not seem to be beating a path to the door of SAGE, Fortune or any of the other 98 manufacturers of expensive 68000 systems?

So all right, do you have a solution to the problem, you may ask? As a matter of fact, there are SEVERAL solutions, we answer. As witness the fact that several DTACK owners have already managed to find useful and profitable applications for their boards. ALL of these solutions have a common ground: assembly language. Assembly language, by definition, has zero levels of interpretation. And since each level of interpretation slows a microcomputer down by an order of magnitude, those people running assembly language programs on their DTACK boards are running on the order of 100 times faster than those expensive 68000 systems with their 'industry standard' BASIC and its two levels of interpretation.

And you will recall that the Concorde supersonic transport cannot fly 100 times faster than a normally athletic 14 year old girl can run, right?

But for those unfortunates who can't program in assembly language, we need to find an affordable higher level language with little interpretive overhead. Oh, yes: it should be easy to learn this language. And it should be interactive. (Sounds a tiny bit like BASIC!)

You will recall our discussion of HALGOL in newsletter #12. At that time it could only be used as an ASSEMBLED 'high level' language. It DID have the virtue of essentially zero run-time interpretive overhead. But it is common knowledge that we need an interpretive and interactive language to be user-friendly enough to encourage most of us to do our own programming.

Well, we have figured out a way to make HALGOL interpretive AND interactive to the user but with zero run-time interpretive overhead. By rapidly compiling when the command to RUN is given? No, by slowly (by 68000 standards) INTERPRETING the threaded code during program editing.

We have accordingly decided to make a bonafide effort to turn HALGOL into a real programming language. One which, like a (good) BASIC, is highly useful as a problem-solving language. Let us emphasize that last bit: HALGOL is intended to be USEFUL.

To be useful in a problem-solving environment, we need a generally usable floating point and transcendental package. (We use the word 'generally' here in its formal, not informal, meaning.) To be generally usable we need twelve or more decimal digits of precision and we need an accurate and complete set of transcendentals.

We published the double precision (fourteen decimal digit) floating point package in our last issue; we are printing most of the transcendental package in this newsletter and, (the good Lord willing and the creek don't rise) in the next issue, the interactive kernel of HALGOL. Let us emphasize that me have this kernel written and working NOW but it is not yet adequately documented, nor have me yet written the needed 6502 code to turn the Apple II into a terminal for the 68000.

Once we have a usable mathematical problem-solving language we can begin to add string functions, etc. An ideal early application of HALGOL would be an electronic spreadsheet program. The 8 bit versions are very, very slow with a large number of rows and columns AND the maximum number of rows and columns is limited by available memory. These are not problems for the 68000.

As a change of pace, this is a good time to drop a gripe we have on you innocent and unsuspecting folks:

On Thursday, Nov. 18 our technician, Ray Spinnett, apparently made an appointment to deliver two Dtack boards to persons who were going to drive down from Los Angeles to take personal delivery of one of our boards. We say apparently because today is Nov. 19 and Ray did not come in to work today. Let us note that your FNE had the flu last week and two other employees have already been out with the flu this week.

A few minutes past noon, when your FNE was the only person in the building, we get a phone call from one of the two people who had made an appointment. What can we do for you? we asked. "I would like to come down, evaluate your operation, and decide whether I would like to buy one of your boards" the caller replied. "I made in appointment with Mr. Ray Spinnett yesterday in this regard."

Well, Ray is not in today and we do not know what arrangements he had planned. However, it is not our policy to spend lots of time with visitors. In particular, we have no interest in having you evaluate our operation.

"I just want to see the board working before I pick it up. That's all right, isn't it?" Sure. As long as you understand that you are not going to got a whole lot of personalized attention.

Our caller retorted angrily: "I have bought a lot of electronic equipment and I have been ripped off a lot. I certainly will not buy one of your boards until I am satisfied with your operation. After all, I might want to be a distributor for you in the San Francisco area!"

All of our sales are factory direct, we replied. We HAVE no distributors... (Let us cut this short, you can see the direction this is going.) The call concluded thusly:

"It sounds like you are a rinky-dink outfit to me! And I do not want to do any business with rinky-dink outfits!" Slam.

Sigh. We just are not set up to accommodate demanding types like that. So we have someone in the S.F. area who is willing to give us a reverse recommendation. Which is not surprising, we also have someone in the LOCAL area with a similar attitude. On the occasion of his fourth personal visit to our facility he demanded that we stop what we were doing and repair his power supply. Since we are not in the business of repairing power supplies, either for cash or for charity, we declined. The visitor refused to pay for the Dtack board that was packaged and waiting for him and demanded that we refund his down payment (we did).

In the future we say refuse to sell to anyone who does not live in Fargo, North Dakota. Persons who live in Fargo do not demand to personally evaluate our facility as a condition of considering the purchase of one of our boards and they take their busted power supplies to local repair shops, not to us.

You will recall that TWO persons had made appointments to pick up their boards that day. The other guy showed up and took delivery of HIS board, personally, without incident except that he showed up during lunch when everybody was gone. So there was just one incompetent on the premises to help him. Guess who?

(This is being written a week later.) We have decided that we may well have to eliminate the practice of permitting individuals to pick up their boards personally. Although we can understand the motivation to assure oneself that the company is real and to see the product run, it is highly disruptive to us as a company. Nowadays, that is.

A year ago it was not disruptive because we did not have that much business. Right now we have all we can handle with the available personnel. We now have a policy that Ray Spinnett and ONLY Ray Spinnett handles personal pick-ups. And we have given him the option of turning down a personal visit (it is HIS option).

You see, Ray is a very patient person who will not get upset when some yo-yo demands to 'evaluate our operation'. In contrast, your FNE has a short fuse. Your FNE does not claim to be a nice, polite personage and no one who knows him personally makes that claim either.

About 5% of our sales to date have come from personal pick-ups. You understand that when we say that we may not permit personal pick-ups in the future, WE understand that we may well lose a substantial part of that 5%. If YOU get turned down we don't mind if you refuse to purchase our board as a result. We just want you to understand that our reason for refusing the arrangement is that we are busy, on a FULL TIME basis, manufacturing boards for persons who live in Fargo and order their boards via UPS.

(Oh, the trials of having a business that is successful enough that there is no tine during business hours to stop and smell the flowers...)

You will never believe who the latest personal computer manufacturer is to do the Fascist/Attila bit over protecting their absurd memory prices. Yep, Big blue.

Like Apple before them, IBM wants an exorbitant and unjustified price to upgrade from 16K to 64K. The operative word is 'gouge'. People have begun to react as expected: they started buying 16Ks, upgraded them themselves to 64K, and started selling them at a discount, typically $400. This is a good deal for IBM's customers, so IBM should be pleased, right?

OF COURSE NOT! IBM has reacted by restricting its sales of 16K PCs to a minuscule percentage of total shipments (5%). If you are an IBM distributor you can buy any version of the PC on IBMs' price list as long as it has at least 64K RAM at IBM RAM prices.

Gosh, and we were only recently assured by Mini-Micro World that IBM had learned from the mistakes that Apple made earlier...

NOT INCIDENTALLY, Chuck Peddle has joined the enemy. A 384K upgrade for the Victor 9000 is priced at $2400. Boy, is THAT price tag going to attract competition if Victor ever manages to sell some 9000s in this country. You Apple types say not know that Chuck led the design team on the original Pet.

More competition for understatement of the year: "At $36,000 per megabyte, we were not particularly competitive", said Derek Gardner (spokesman for PRIME Computer). Nooo... That quote is taken from an article in EN, 18 Oct. The article continues, "Prime's delinquency in dropping memory prices may have contributed to the firm picking up new competition..." DOUBLE nooo...

We have just purchased a MAX-80 from Lobo (see BYTE, Dec '82, p.551). We ordered it with a 64K expansion. Ninety-five bucks. That's $570 per 384K or $1520 per megabyte. Just who is kidding whom, fellows?

(If anyone knows of 100nsec static RAM being sold at prices lower than ours, PLEASE send us a copy of the ad.)

But do you know how Lobo is selling the MAX-80? Factory direct! And they expect you to pay in advance or else send a deposit and pay the balance COD! Gadzooks! Don't they know a company can't operate like that?

(The next 1 1/2 pages are an advertisement for some Apple/DTACK software from an outfit in West Germany. The claims made are theirs, not ours.)

THE M000SE is a high performance chess playing program for DTACK boards.

Do you expect a chess program to have catchy color graphics and noise effects? In this case THE M000SE is not the right program for you. THE M000SE is designed for high level tournament chess and advanced playing.

The very first version of THE M000SE was implemented in 1979 on a microprogrammable INTERDATA M85 at the University of Hamburg. It was developed by a team of five students of Computer Science which were both chess and assembler specialists. They exposed their careers to danger by sacrificing not only their spare time.

To improve its strength, THE M000SE had participated at same local chess tournaments (via modem) at Hamburg (W-Germany). As the tournament rules of most microcomputer chess championships forbid the participation of mainframe computers, we have decided to adapt the same program to a single chip microprocessor in 1981.

Looking for a suitable processor, the MC68000 seemed to be the best one for the special task of running a chess program. So we built up a 68000 hardware of our own and implemented the chess program on it. At that time there was very little software available, so that we had to design our own operating system. As THE M000SE is a very large and sophisticated program, we also had to write a powerful debugging tool, which later developed to 68TICID.

Although we had to do without microprogramming, the MC68000 program was much faster than the old INTERDATA chess, because of some algorithmic improvements.

THE M000SE should have taken part at the World Computer Chess Championship list year at Travemuende (W-Germany), but according to "Hofstadter's Law" which says: "It always takes longer than you expect, even when you take into account Hofstadter's Law", the program was not fully debugged at tournament start, so we are still waiting for the next chance to prove the strength of THE M000SE.

The last adaption has taken place in summer 1982 to make the program available to you, fellows.

So THE M000SE was split up into two parts working concurrently while the MC68000 is doing the heavy crunching, we keep Apple busy performing library search and I/O handling. In conjunction with powerful chess playing algorithms this is the reason for the almost unbeatable strength of THE M000SE. Of course both parts are written in native assembly language to make efficient use of the processors.

THE M000SE represents a modern implementation of the basic type-A strategy described by Claude Shannon. Block heads eight call this ignorantly "brute-force", but there are several sophisticated pruning techniques used. The Shannon type-A method implies that all legal moves up to a fixed ply level are considered in full width. Beyond this limit THE M000SE searches capture and check moves up to nearly unlimited depth (as long as there are pieces to capture on the board).

Although tree searching is performed by the well-known alpha-beta procedure, the number of searched positions is much less than done by most other programs because of several sophisticated pruning features.

The move generator of THE M000SE takes all legal chess moves according to FIDE resp. ICCA into consideration, i.e. enpassant captures, promotions, 50 moves rule and drawn by repetition rule. When promoting, THE MOOSE does not take blindly - like other famous programs - a queen, but also considers to promote to a knight, rook or bishop.

Some of the unique and paramount features are :

- Both processors (68000 and 6502) are working concurrently.
- Switches for detailed diagnostics and examination of positions and tree searching.
- Possibility to disconnect your Apple from THE M000SE to do other calculations with your Apple, while THE M000SE is still working on your DTACK board. Later reconnection is possible without losing any information.
- Convenient methods allowing you to justify the response time.
- Advanced facility for adaption to any CPU clock rates (xx MHz).
- Displaying the principle variation at full length up to an unlimited amount of plies.
- Possibility of taking back any number of moves without the need to change the board.
- The opening library may be used optionally. It contains mainly gambits which are normally not familiar to chess computers.

The authors of THE M000SE make every effort to improve the strength of the program. Later releases will be made accessible to its owners at modest prices.

As mentioned above, 68TICID is a powerful debugging tool for 68000 assembly language programming. Special efforts have been invested in compacting its code size to 2.5 kByte, so that even users of small DTACK boards are able to use it. 68TICID features 24 exciting commands :

- Breakpoint handling
- GO-UNTIL feature
- Search and modify-operations in preset ranges
- Stepping with variable step size (default: single-stepping)
- Supports any user defined trace operations
- Advanced display of register and memory contents in both hex and ASCII
- Indirect and relative address access (long and short)

68TICID handles all processor exceptions and displays pregnant error messages. All character I/O and hexadecimal output routines are user accessible to allow easy linking to user programs.

MUXA68 is a cross assembler for the 68000 processor running under the UCSD P-System. The current version is made for Apple Pascal 1.1, a version for the Softech UCSD IV.0 will be available if desired. MUXA68 is wholly written in Pascal, thus making it portable to nearly all kinds of micro and mini computers.

MUXA68 features a fully Motorola compatible instruction syntax and full support for all instructions and addressing modes as well as for total addressing range. It also supports format independent assembler syntax and many features of the Motorola assembler.

MUXA68 is delivered together with a loader and software to transfer object programs to DOS 3.3 diskettes, thus allowing 68000 programs to run under the environment supplied by DTACK and other manufacturers.

PCON68 is a software interface to connect the DTACK 68000 board to the Apple UCSD Pascal system. It allows 68000 programs and subroutines to run under the control of Apple Pascal host Programs and manages communication between both.

All programs are delivered with English manuals and are available from

M000SE SYSTEMS

STEENBARGKOPPEL 21

D - 2000 HAMBURG 65

WEST GERMANY

Prices : THE M000SE : $67 ; 68TICID : $47 ; MUXA68 : $70 ; PCON68 : $30

All customers must add $10 for the first and $6 for each of the remaining items of each order for s&h. All orders must be prepaid in us-$.

THIS YOU MAY NOT BELIEVE, but you can now buy a software package called 'The Bridge' for a VAX. This package simulates in (VAX) software the operation of an 8080 running CP/M. Now, the VAX is a fast machine but not fast enough to simulate an 8080 at full 8080 speeds, so the VAX will run a bit slower than a real 8080 CP/M machine. But there is LOTS of CP/M software, at LOTS lower prices than VAX software. An example cited is a DEC spreadsheet package which costs $10,000 and has a manual that is 20 inches (!) thick.

But if too many users (VAX terminal operators) call up that simulation, the VAX slows down such too such. So the same company which sells the software package also sells a 'Z board' which contains 4 Z80s, each with its own 64K RAM, which plugs into the VAX so that the VAX can run 8080 code more efficiently...

Isn't that wonderful? (Source: Electronics/Oct 6, 82 pp52-3.)

A LITTLE FLU BUG caused us to put the last newsletter to bed two pages prematurely. As a result, we learned 3 hours too late (moments before we headed home to treat the bug with the assistance of Dr. Calvert Fitzgerald) that HP had not been overtaken by in attack of rationality and offered a high performance computer at a reasonable price. The price we quoted, while real, does not permit turning on the computer and running a program,

You see, HP's 68000 BASIC interpreter occupies over 200K so you need the 512K version to be able to run the machine. If you have been paying attention to this newsletter you might suspect that HP wants an extraordinarily high price for those 24 additional 64K DRAMS needed to upgrade from 128K to 512K. Well, your suspicions are correct. In fact, there are a bunch of other hidden goodies you will need to pay (and pay and pay and..) for. Business as HP usual, in other words.

The Model 16 is a nice high-priced quality product which will appeal to the usual affluent HP customer. HP does make good stuff, but the price-performance just ain't there. And the Model 16 is NOT going to shoot down LISA (can we claim that the bug affected our judgement?).

ANOTHER NEWSLETTER EDITOR called us a '68000 fanatic' in his rag. We wrote back and explained that our writings were reasonable and carefully thought out and that we were really moderate in our views. We also threatened to sic our twin attack teddy bears, DTACK and Grounded, on him if he called us a fanatic again. He wrote back: "Attila the Hun was a moderate in his own way, too!"

Tsk.

The 68000 that you released commercially two years ago is a very nice and very high performance part. As of Dec '82 that is also the perception of the general public. General public meaning that group of electrical engineers, computer/peripheral designers and software types involved in what we loosely refer to as 'this industry'. We believe that this perception may well be in danger soon.

What we mean is that, within the next two months, the 'general perception' of the 68000 may be that of a slow part. You and we know that this is not the case but future 'design wins' depend upon the perception of the general public.

Let us discuss how this situation has developed. You know and we know that the potential sales of the 68000 in the personal computer marketplace are a very small percentage of (potential or real) total 68000 sales. You know and we know that most 68000s will be designed into intelligent peripherals and controllers for IBM, Honeywell and such, and that they will be buried away out of sight. Most 68000 chips are sold for such 'invisible' applications.

All things being equal, it is certainly sensible to concentrate your sales effort where the bulk of the sales are. Only a dummy would do otherwise. The problem is, all things are NOT equal.

Although the personal computer marketplace accounts for a negligibly small portion of 68000 sales, either real or potential, it is a spectacularly VISIBLE marketplace. Those hidden away 68000s are essentially INVISIBLE.

The people who are the decision makers for 'design wins', such as circuit design engineers and the engineering managers and the business managers are about to be shocked by the poor performance of a certain 68000 machine which is about to be introduced by a major personal computer manufacturer.

How can a 68000-based machine have poor performance? Read the front page of this newsletter. The way to make a 68000-based machine run slowly is to sabotage the software. Who has sabotaged 68000 software? Well, the original culprit was an outfit which you may have heard of called Motorola.

You see, Motorola originally marketed the 68000 EXCLUSIVELY as a minicomputer substitute and therefore supported the development of high-priced minicomputer type software EXCLUSIVELY. As a result, when the 68000 became an affordable part to use in a cheap personal computer, as it is now, the only available software was the high priced minicomputer stuff. (It is too bad that no one at Motorola foresaw the inevitable result of advances in IC production capability. The inevitable result? Why, cheap 68000s of course!)

Back to that certain 68000 machine which is about to be released by a major personal computer manufacturer: That company is watched very closely by the Wall Street Journal for reasons which we need not discuss here. Shortly after it becomes evident that it is SLOW for a 68000 based machine there will be a story by the Journal on the front page of the second section. And the business managers in this industry are going to read that story.

This same information will also be published in those journals associated with our industry such as Electronics and BYTE (both McGraw-Hill periodicals) and Infoworld. This is where the working engineers will learn about it. What magazines the engineering managers read we don't know, but what we are really saying is that this is not going to be a secret.

How do we know this? There was a review of the Fortune 68000-based machine in a recent Interface Age magazine in which the 68000 machine proved slower than some 8 bit varieties. A very recent issue of Infoworld published a short interview with Fortune management in which the reasons for the slowness of the machine was questioned. The reply was, "Oh, we are fine-tuning the operating system for speed" which is much more polite than "Go away, boy, you're bothering me" which it really means.

You don't think the industry standard BASIC uses two levels of interpretation? Check that very same Infoworld report where the Fortune spokesman specifically asserted that "their" BASIC could be transported to other processors specifically including the iAPX 286. Of course it can be transported! It is written in a sort of 'p-code' for that very purpose. And that is the second level of interpretation. And that is why the 'industry standard' BASIC is so slowww...

The thing is, not that many people read Interface Age and not that many people are following the Fortune machine closely. But there are an ENORMOUS number of people following that major personal computer manufacturer and anxiously awaiting news of its 68000 machine.

So what suggestion do we have to cure the situation? We do not have one that would be acceptable to Motorola. We know; we made a suggestion that would work and had it rejected, naturally. But Motorola should recognize the high visibility of the personal computer marketplace and not repeat its mistakes when the 68020 comes out.

Best regards, honest! (signed) your FNE

We suggested that Motorola invest about $300K in a 68000 BASIC with zero run-time interpretive overhead but fully interactive with the user. We then suggested that Motorola place this BASIC completely in the public domain, INCLUDING THE SOURCE CODE! Yes, we know this sort of thing just isn't done. Let as explain to you why it SHOULD be done:

It should be done because Motorola would not only be able to directly recover its dollar investment but would also gain ENORMOUS profits in an indirect way. Let us explain how this would come about.

First, if the BASIC had high performance (it would HAVE to be a high performance package with no interpretive overhead) and if it were FREE, then every one of those 100 manufacturers of 68000 based small computers would make that BASIC available on their computers. Which would make the new BASIC the NEW 'industry standard', at least for the 68000 section of the industry.

There would be a large demand for a BOOK describing the BASIC, including explanations of how the routines work and including the complete source code as a large appendix in the book. Such a book would recover at least $10 per copy for the publisher (Motorola), so that only 30,000 copies need be sold to recover the up-front investment.

IF the BASIC is GIVEN AWAY and so becomes ubiquitous, the demand for the book would be much greater than a mere 30,000 copies. It would not be unreasonable to expect to eventually recoup 10 times the original investment DIRECTLY..

But the INDIRECT profits would be enormously greater. First, each of the small 68000 based machines such as the SAGE and Fortune would instantly become the FASTEST personal computer available! The 'general perception' of the 68000 would be the correct one: it is by far the fastest microprocessor available for a small personal computer. The word in the general marketplace would be: You want to run fast, you use a 68000. THE RESULTING 'DESIGN WINS' WOULD BE THE MOST PROFITABLE RESULT OF IMPLEMENTING THIS SUGGESTION.

In fact, the only persons who could lose would be Motorola's competitors.

Let us consider some of the possible arguments against this program:

- Placing the source code in the public domain would
just help Motorola's competitors
### Page 8, Column 2

Baloney! Motorola's only major competitor is Intel, and Intel's chips cannot address a large memory space directly and they do not have enough address pointers to implement the language. Such a BASIC will inevitably use threaded code during run time and use lots of pointers and 32 bit arithmetic during the interactive phase. It MIGHT be possible to 'port' the language to the National 16032 but so what? Does anyone seriously think the 68000 is threatened by the 16032?

- There is no assurance that the BASIC mould become popular. The
profits, both direct and indirect, depend on in assumed popularity.
If the BASIC is both FAST and FREE, there is absolutely NO WAY it could fail to be popular. The only way to block it would be for some brain-damaged traditionalist at Motorola to decide at the last minute that since the language has such high performance, it should be licensed. THAT would, indeed, not merely damage its popularity but kill it dead.

- It is not possible to write a language with the attributes
described/asserted.
It is TOO possible! We are (slowly) doing it ourselves and we have progressed far enough that we KNOW it can be done! But our resources are smaller than Motorola's (candidate for understatement of the year) and we cannot bring this off quickly enough to gain the maximum benefit for Motorola. Also, we would NOT place our work in the public domain. Instead, we would (will) publish it with copyright notices and distribute copyable disks containing the source code.

- Maybe Motorola doesn't know HOW to write such a language? It does
sound slightly unconventional.
We have already offered to donate, it no charge, the work we have done toward implementing that language to Motorola. We here repeat, in writing, that offer.

- This sort of thing just ISN'T DONE in this industry.
Times change. It SHOULD be done. It should be done because Motorola would reap ENORMOUS benefits.

- What is in this for Digital Acoustics?
We will sell a lot more boards to work with the Apple and the Pet if such a language were available and was an industry standard (68000 section). That would produce standard APPLICATION software for 68000 machines, something which is somewhat lacking right now. Anything which makes the 68000 look more desirable will enhance our sales. We don't mind at all that it would ALSO enhance the sales of SAGE, Fortune and the other 90 manufacturers of small 68000-based computers.

(It is clear in retrospect that that suggestion was phrased improperly. It must have cost Motorola at least $100 million so far to develop the 68000 product line. Prudent businesspersons would invest about 15% of that sum in promotion. Promotion is whit you do to convince the public that the 68000 runs faster than a ring-tailed wowser.)

(We believe it would have been prudent to invest 2% of that promotional budget, or $300,000, in demonstration software. Since the persons making the design decisions are mostly engineers, the demonstration software should be slanted toward engineers' needs. Engineers need to solve problems and BASIC is the language of choice for engineering problem-solving. The $300,000 is for purely promotional purposes and the question of a possible direct payback is a red herring which should never have been raised.)

WANG LABORATORIES now sells over $1 BILLION in computers per year and is a FORTUNE 500 company. Wang is the ONLY computer company, as far as we know, which grew to that size from an origin of selling one CPU to set in front of one user. We think our recent experience with them might be enlightening.

We are seriously considering purchasing a replacement for our 10 year old 2200C, which is getting old and rheumy-eyed. Last issue, we told you a little about its current equivalent, the 2200 SVP. Since last issue we went to see a demonstrator SVP. The demonstration room is about 600 to 800 square feet, nicely carpeted and paneled. We forget whether there were two or just one additional model besides the SVP since that was the model we were interested in.

The SVP is HUGE! Aside from a terminal about the same size as our CBM 8032, the rest of the computer is in an absolutely HUGE cabinet with 8 inch double density double sided disks, one on each side. When we opened it up, we found that most of that size was to assure adequate cooling. After all, a computer which runs cool is a RELIABLE computer. But that huge cabinet has to be at desk height and also nearby since that's where the floppies are. And that huge cabinet will dominate any office it is in. Maybe that's the idea?

Once we got over our shock at the size of the beast, we looked around for a programming manual (a principal reason for the visit was to check for software compatibility with our old machine, especially some microprogrammed I/O instructions called $GIO). There was no programming manual in the demo room. In fact, as we looked around we realized that there was NOT ONE SINGLE SCRAP OF ANY KIND OF PAPER IN THE ROOM! Not even a snuffed out cigarette (there were no ashtrays).

So we went to the salesperson's cubicle, the service department AND the literature room looking for a programing manual for the SVP. There proved to be NOT ONE SVP PROGRAMING MANUAL in the building! So, we ordered one from Wang for $25. They had to send to Tewksbury Mass. for it. We got it last week (actually the salesperson delivered it personally and at no charge) and sure enough, the SVP does support those $GIO instructions. Fortunately, we did not immediately place in order because an article in this Monday's EN states that Wang is dropping its 2200 series prices to counter corresponding reductions in IBM's System 34 line. The price cuts appear to be in the 25 - 30% range but they have not yet been officially announced, so we'll wait.

Sure wish the SVP had a screen editor like our CBM 8032 and HIRES graphics like our (several) Apple IIs.

We have an associate in the local area who is well connected in the Pet world. We had an argument recently over whether Commodore's production costs for the $595 Model 64 are more or less than the production costs for the VIC 20, which is selling for as low as $168 locally. Our opinion is that the Model 64 costs MUST be higher, those eight 64K DRAMs aren't all that cheap. It is true that the 64 LOOKS easier to build than the VIC 20. Open the two side by side and you'll see what we mean.

Anyhow, this associate insists the 64 costs $20 less to build than the VIC 20, 64K DRAMs and all. Which means Commodore could market that little baby at the $150 level (discounted). That means possible BIG TROUBLE ahead for the revised Apple II if it stays pricey...

THE FOLLOWING TRADEMARKS ARE ACKNOWLEDGED: Apple, II and soft: Apple Computer Co. Pet: Commodore Business Machines. VAX and PDP11: Digital Equipment Corp. CP/M: Digital Research of CA. DTACK GROUNDED and HALGOL: Digital Acoustics, Inc.

SUBSCRIPTIONS: You can subscribe by sending $15/6 issues U.S. and CANADA or $25/6 issues elsewhere. Tell us which issue number to begin with. Make the check out to DTACK GROUNDED. The address is:

DTACK GROUNDED

1415 E. McFadden, Ste. F

SANTA ANA CA 92705

THERE IS SOME IDIOT WHO, under various pseudonyms, is constantly being quoted in various publications that the availability of computers will cause students to become mathematical illiterates. If we ever catch that guy we are personally going to bash his head in with a five pound sledgehammer!

Don't miss page 15!

In this issue we are publishing in REDLANDS (???) the standard BASIC transcendentals to go with the double precision floating point package found in the last issue. This includes LOG, EXP, SQR, SIN, COS, TAN and ARCTAN.

OBJECTIVES: All of us would like to have a computational machine with infinite precision which executes in zero time and which requires no storage space. You will find such a machine sitting right next to your free lunch.

As a practical matter, we must compromise. Who decides what the compromises are? In this case, your FNE has the exclusive authority and responsibility. This takes the job both easier and harder. One is not restricted to a predetermined format, as was the case when the Microsoft compatible 9 digit package was being written. On the other hand, one likes to do a respectable job so that those DTACK subscribers and board owners who are much better mathematicians than we are will not be disgusted or excessively amused.

Accordingly, we have spent CONSIDERABLE time (to the detriment of our hardware effort) checking the accuracy of the package. This will be reported as a separate section later.

HERE ARE OUR GROUND RULES: with a 48 bit mantissa we have just over 14 decimal digits of precision. It is possible in principle to approach that precision in calculating the transcendental functions, but it is very difficult. A more realistic goal is to try for 13.5 decimal digit accuracy in general, and hope that at least 13 digits is the poorest precision obtained.

And please: let us not fight over the difference between 'accuracy' and 'precision'. We will lean heavily on the book COMPUTER APPROXIMATIONS, and in that book 'precision' is used where some of us would prefer 'accuracy'.

We must recognize that poorly conditioned numbers will reduce the obtainable precision. For instance, the sine of 0.1 can be calculated more accurately than the sine of (2PI + 0.1) since we will lose some bits of precision in the argument during range reduction.

Aha! New jargon! You will recall that transcendentals are monadic functions. (If you do NOT recall this, review page 9 of our newsletter #1.) To calculate the square root (for instance) we pass a number to the SQR subroutine by leaving it in the floating point accumulator, FPACC1. This number is called the 'argument' of the function.

RANGE REDUCTION: If the argument passed to the SIN subroutine has an absolute value greater than 360 degrees, we must reduce its range to less than 360 degrees. In fact, we will reduce the range down to 0 to 90 degrees before performing any significant amount of computations. This is what is meant by 'range reduction'. The range reduction process for our chosen ARCTAN algorithm is quite complex.

Except for the SQR function (which uses Newton's method) the algorithms and stored constants used in this transcendental package are mostly taken from the book COMPUTER APPROXIMATIONS by Hart et al, Wiley 1968. This book is one of the SIAM series in applied mathematics.

SHORTHAND NOTATION: The techniques used (except for SQR) involve range reduction and then calculating a Chebyshev approximation using some form of series calculation. COMPUTER APPROXIMATIONS uses a shorthand notation which might confuse the reader on first confrontation (we believe 'confrontation' is the correct term). For instance:

Range Precision N Index [0,1] 4.12 3 1040

means that the exponent function base 2 can be calculated, over the range 0 to 1, to a precision of 4.12 decimal digits using the series form P(x) with N = 3. That series is calculated as follows:

Fn = P0 + X * (P1 + X * (P2 + X * P3))

Note the nested form of the computation. This reduces error buildup due to rounding errors which might be caused using an otherwise equivalent form, say E(X) with N = 3 which would be calculated is follows:

Fn = E0 + X * E1 + X * X * E2 + X * X * X * E3 where E0 = P0 E1 = P1 E2 = P2/P1 E3 = P3/(P1*P2)

Finally, 'Index' refers to a particular table in the back of the book, table number 1040. This table contains the four constants, P0 thru P3. The number of constants in a given table is always one greater than N.

And if the argument is not in the range 0 to 1 then some form of range reduction must be performed to accomplish that, and the needed corrections made after the series approximation is calculated.

OTHER SERIES FORMS: To repeat, the nested form is used to reduce error buildup. There are other forms of series approximations used, especially X*P(Z) where Z is a function of X (often equal to X squared). The SIN function is calculated using this form, for instance. If N is equal to 3, the calculation is performed as follows:

Fn = X * (P0 + Z * (P1 + Z * (P2 + Z * P3)))

There are many other forms of series approximations, most of which are NOT used in this transcendental package. Perhaps we should mention the form used to calculate the logarithm function, which is P(Z * Z) or P(Z squared) where:

Z = (X - 1)/(X + 1)

ABSOLUTE AND RELATIVE ERROR: If T = the true value of the given transcendental function of the argument and Fn is the calculated approximation to the argument, then:

Absolute error = T - Fn Relative error = (T - Fn)/T

The following crude drawings illustrate a Chebyshev approximation of the sine function (range reduced to 0 to 90 degrees). The error is somewhat exaggerated (we hope).

ABSOLUTE ERROR

RELATIVE ERROR

The relative error is NOT always the appropriate error criterion. Following are the appropriate criteria for each of the transcendental functions in this package:

RELATIVE: SQR, EXP, SIN, COS, TAN. ABSOLUTE: LOG, ARCTAN.

Note that even though the LOG function passes through zero, as does the SIN function, absolute error is the appropriate error criterion.

We will now describe the calculation of the EXP and ARCTAN functions in some detail. These descriptions will be followed by 'short form' descriptions of the remaining functions and then a description of the error testing we have performed on this package and the results of that testing. Oh, yes: then we will print the source code for the transcendental package.

To calculate the exponent function to the base of natural logarithms, e, we simply multiply the argument by one over the natural logarithm of two and THEN calculate the exponent of the result to the base 2.

Similarly, we can provide an exponent function to the base 10 by simply multiplying the argument by one over the logarithm to the base 10 of two and then calculating the exponent of the result to the base 2.

This means that we can easily and conveniently provide HALGOL with routines for calculating the exponent to any of the three bases 2, e and 10. And so we do!

For any given X along with a specified base a unique result exists. This result cannot be less than one for positive X. However, the result may be too large to be representable in the floating point format in use. In that case, an overflow message must be reported.

IDENTITIES (from HART et al): eqn 6.2.19: EXP(X+Y) = EXP(X)EXP(Y) eqn 6.2.24: EXPe(X) = EXP2(X/ln2) or, EXP10(X) = EXP2(X/log2) where EXPE(X) = exponent, base e EXP2(X) = exponent, base 2 EXP10(X) = exponent, base 10 ln2 = natural logarithm of 2 log2 = logarithm of 2 to the base 10 eqn 6.2.25 EXP(-X) = 1/EXP(X)

It is in general impractical for binary machines to calculate the exponent function (efficiently) in any other base than 2. To calculate the exponent of e, we use (6.2.24) and divide X by ln2 (actually we multiply by 1/ln2) and then proceed to calculate EXP2(X') where X' = X/ln2. Note that ln2 (or 1/ln2) is simply a stored constant whose value is published with great precision in HART.

We now assume that the problem is to calculate EXP2(X), referred to henceforth is simply EXP(X).

First we store the sign and set the sign of the argument positive (line 531). We also test for a zero argument (line 535). If X = 0, the result is one.

Next we reduce the range over which the value must be calculated so that a Chebyshev approximation can be applied. First we test whether the argument is ALREADY less than 1/16 (line 537). If so, no range reduction is necessary.

Next we test whether the exponent is greater than $100D. If so the argument is too large IF the sign was positive. So, if the sip is positive, we report an overflow message. In BASIC this is a fatal error and program execution stops. If the sign was negative, the result is zero and there is no error (line 511). (One over a number too large to represent is zero.)

We have chosen to reduce the range to 1/16 > R >= 0. We use R to indicate the Residue of the range reduction process. This range reduction can be accomplished by setting X equal to the sum of three parameters

X = INT(X) + C + R

So that, by 6.2.19:

EXP(X) = EXP(INT(X)) * EXP(C) * EXP(R)

Here, INT(X) has its usual meaning and C is equal one of sixteen values: zero, 1/16, ..., 15/16. R is the Residue, defined above.

Unfortunately, Hart does not contain an appropriate series approximation, so we have had to devise our own (see CONCLUDING DISCUSSION). We use N = 6 and the series form P(X). Let us suppose that EXP(R) has been calculated and continue.

The exponent of the integer of X is calculated very simply by adding the integer to the exponent portion of the floating point representation of EXP(R). The result is equivalent to multiplying EXP(INT(X)) * EXP(R). We must next determine which EXP(C) (if any) to multiply times this last result.

If C = 0, EXP(C) = 1 and there is little to gain by multiplying by one (line xxx). Only if C <> 0 do we multiply by one of 15 stored constants. Having multiplied by EXP(C) if necessary the result is the EXP(ABS(X)). We now test whether X was originally negative (line 585), and invert the result if so (6.2.25).

The exponent function calculation is now completed.

We need to store a 17 bit number which is the integer value of 16 times the argument of EXP2(X). The most significant 13 of these bits is the integer value of the argument and the least significant 4 bits determine which of the 15 stored constants will be used (or, if all 4 bits are zero, no EXP(C) correction is performed). Because the number of bits is more than 16, we use a 'long word' (32 bit) storage area called EXPADD to store this datum.

Note that its clear (.L, 32 bits) EXPADD in line 532 in case the argument is already less than 1/16 so that no range reduction is performed. In lines 539-40 we calculate the difference between the exponent and $100D. In line 546, we fetch the highest order 32 bits of the mantissa. Since we are going to store the 17 bit integer in a 32 bit space, we need to add #15 to this difference to determine the number of right shifts to be performed. We do this in line 547. The next line shifts the 32 bits in D2 right by the count in D1. This is a very rapid method of calculating the integer of 16 times the argument.

After performing the series calculation, at line 568 we recall the 17 bit stored value, shift it four bits right to obtain the integer of the argument (rather than 16 times the argument) and add the integer value to the exponent of the series calculation result.

Then in line 571 we load the lower word (note the MOVE.W and EXPADD+2) and mask the upper 4 bits. If the result is zero we can skip multiplying by EXP(C). Otherwise, in line 577 we multiply the number by 8 to determine the address of the desired stored constant after adding the starting address of the constant table, less 8. Simple, no?

Based on the series approximations of form P(X) that ARE published in Hart, we were almost certain that a series approximation of precision 15 (or so) could be found over the range 1/8 > R >= 0 with N = 6. See for instance the progression from Index 1043 to 1023 to 1003. However, we were unable to achieve a precision of better than 12.5 or so.

We even tried to obtain a series with N = 7 over that same range and failed. But when we used N = 6 and reduced the upper limit of the range to 1/16 we quickly found the constants for calculation to the needed precision. Unfortunately, this requires eight more stored constants (pre-calculated values of EXP(Cn)) than would otherwise have been needed.

We are almost certain that the failure to find the desired series approximation is due to our lack of expertise (translation: we don't know what the hell we're doing).

There is no 'good' way to calculate the ARCTAN (or ATN) function. The problem is to select the least undesirable method. The reason we say this is that this function is the most resistant to being easily approximated using a series approximation of reasonable length.

Although there are well known series functions which eventually converge on the desired value of the function, they converge very SLOWLY. Of those series approximations published in Hart, for a range of 0 to 1 (TAN(45 degrees)) the form

Fn = X * P(X * X)

has a precision of only 9.43 decimal digits for N = 10 (Index 4994). And each additional increment of N appears to add about 0.8 additional digit of precision. After our experience developing our own series approximation for EXP with N = 6 we are not about to try to develop our own series approximation for ATN with N = 17 or so!

This means we had to look for an alternate method. Hart provides such a method involving reducing the range to 0 to about 0.098 (TAN(5.625 degrees)). Unfortunately the method is NOT simple, at least not for us. And to make things worse there appears to be a discrepancy between what Hart does and what Hart says it does! We will explain this later.

There is a corresponding arc (or angle) for any possible argument. Therefore no error testing need be performed.

IDENTITIES (from Hart et al): eqn 6.5.3: ARCTAN(X) = ARCTAN(X) + kPI eqn 6.5.21 ARCTAN(-X) = - ARCTAN(X)

(the following formulae are said to be derived from the earlier formula 6.5.27).

T = 1/y - (1/(y * y) + 1)/(1/y + X) ARCTAN(X) = ARCTAN(y) + ARCTAN(T)

The two formulae above are given in Hart, section 6.5.e and are used in conjunction with a method of partitioning the argument so that the calculation can be made over the argument range 0 to 0.098 instead of 0 to 1. The text of Hart states that the argument, ranging from 0 to infinity, should be partitioned by eight partition points and that the appropriate constants are printed in Hart for those eight partition points.

The fact appears to be that the argument, ranging from 0 to 1, should be partitioned by four partition points and that appropriate constants are printed in Hart for these four partition points.

RANGE REDUCTION STRATEGY: If the sign is negative we set a sign flag and set the sign bit positive (take the absolute value of the argument). The argument is now in the range 0 to positive infinity. If the argument is greater than one we set an 'invert' flag and calculate the reciprocal of the argument. The range of the argument is now 0 to 1; we have two flags to test at the conclusion of the series approximation.

And NOW, boys and girls, the fun starts. We will use four partition points to partition the range 0 to 1 into five partitions. The partition points are:

X1 = TAN(PI/32), appx 0.098 X2 = TAN(3PI/32), " 0.303 X3 = TAN(5PI/32), " 0.535 X4 = TAN(7PI/32), " 0.821

For each of these four partition points, we need four stored constants. The first stored constant is the partition point itself, used to determine which partition the argument fits into. The second constant is the RECIPROCAL of y used in the formulae above (Hart uses lower case x with the subscript i instead of y). The four y values are:

y1 = TAN(PI/16), appx 0.199 y2 = TAN(PI/8), " 0.414 y3 = TAN(3PI/16), " 0.668 y4 = TAN(PI/4) = 1 exactly

The third stored constant is equal to the second constant squared, plus one. Note that this is equal to (1/(y*y)+1) in the first formula on this page. The fourth stored constant is equal to the arc tangent of y, so that the four constants are simply PI/16, PI/8, 3PI/16 and PI/4.

Although we have five partitions, if the argument falls between 0 and 0.098 (appx) no range reduction is necessary. This is similar to the exponent calculation, where only fifteen stored constants were needed for sixteen partitions.

In the table beginning on line 744, Xn is the partition point for the nth partition, An is 1/y, Bn is (1/(y*y)+1) and Cn is the ARCTAN(y).

LET US TALK THROUGH a typical calculation. Say we have an argument of -1.6. We set the sign flag and clear the most significant bit of S1, setting FPACC1 equal to 1.6. Next we test whether FPACC1 is greater than one. It is, so we set the invert flag and invert FPACC1 by dividing it into the number one. The result is 0.625. Now we have to determine which partition we are in.

Is FPACC1 less than X1? No. So we add #32 to our table pointer and check whether FPACC1 is less than X2. Nope. Not until we compare FPACC1 to X4 will we discover that we are less than the stored partition point. That means that the correct partition lies between X3 (0.535) and X4 (0.821). We will now use constants A3 and B3 to reduce FPACC1 to the range -0.098 to 0.098 via the formula at the top of the preceding column.

First we add FPACC1 (0.625) to A3 (1/0.668), result (1/y + X) or about 2.121. We divide this into B3 (1/(0.688*0.688)+1), result equal to the portion of the first formula on the preceding column to the right of the minus sign. FPACC1 is now about 1.527. Now we subtract #16 from our table pointer to point to A3 again and subtract FPACC1 from A3. This completes the calculation of T, which is about -0.0305.

Now we use the series approximation Hart Index 4903, N = 5, precision 16.52, range 0, TAN(PI/32) or 0 to about 0.098. BUT, folks, our argument is not in that range! Which is another mild gripe we have about Hart. You see, since the series approximation has the form X * P(X * X) AND ARCTAN(-X) * - ARCTAN(X), the range of the approximation is actually -TAN(PI/32) to TAN(PI/32). Anyhow, the series approximation gives us the right result, which is about -0.0304.

You will note that the result of the series approximation is almost the same as its argument (input). THAT is why the series approximation over this limited range of -0.098 to 0.098 is so efficient; it is easy to find approximations for functions whose output is nearly equal to their input. And the reason we went through that mumbo-jumbo range reduction was to gain access to the efficient approximation. Verstad?

Having calculated ARCTAN(T), we have to add ARCTAN(y) to obtain what we want, which is ARCTAN(X), right? WRONG! We test to see whether the original argument was in the range 0 to 0.098, in which case no range reduction occurred and we ALREADY HAVE ARCTAN(X) as the result. So we check the pointer to Cn. It is not zero, so we know that range reduction WAS performed. So we use the Cn pointer to find C3 and add 3PI/16 to -0.0304 to get 0.559, which is ARCTAN(0.625).

We are NOT done yet. Remember the sign flag and the invert flag? First we check the invert flag and it is set. So now we determine a new arc by subtracting FPACC1 from PI/2 (which is 90 degrees). The result is about 1.012. Next we check the sign flag and it is set too, so we set the sign of the result negative.

So the final result of our calculation is -1.012, which is the ARCTAN of -1.6. If that number doesn't look right to you, remember we are working in radians. Our result is about -57.99 degrees if you want to convert radians to degrees by multiplying by 180/PI.

See how simple it is to calculate the ARCTAN function?

Getting specific about the code: the ARCTAN subroutine starts on line 660. Instead of actual flags, we store the original sign and exponent in location SINSGN for later testing for the equivalent of those two flags. To test which segment we are in, we place #32 in D6 and the table address, minus #32, in D7.

Then we call a subroutine at line 671 which adds D6 to D7, places the result in pointer A0, and compares FPACC1 with Xn in the table. If the first call of the subroutine at line 694 results in an indication that FPACC1 is less than X1, a branch (in line 695) to X0 at line 716, bypassing the range reduction.

If we bypass the range reduction process, we have to know we did so. That's why the very first line of the ARCTAN function clears EXPADD, which is used as a pointer to the appropriate Cn. AFTER the series calculation, we check this pointer at line 722, and skip adding Cn if the pointer is zero. If the pointer is NOT zero, we move it to A0 and call subroutine FPADD to add FPACC1 to Cn.

(We seem to be going into overtime, so this subject will be concluded next issue. The reader is advised that the TAN, ARCTAN and EXP functions have been thoroughly tested but the SQR, LOG, SIN and COS functions have NOT yet been thoroughly tested. However, the algorithms used for them are essentially unchanged from our 9 digit Microsoft compatible package. No bugs are anticipated other than simple typos.)

IS IT TRUE that Charlie Winterble, Al Charpentier and an unnamed third engineer who left Commodore a couple of months back have come to roost at Atari? Charlie, besides being Commodore's former chief engineer, was the designer of the very fine sound chip in Commodore's Model 64 and doubtless played a major part in the design of the almost as good (for game-playing purposes) graphic controller chip also found in the Model 64. Can we expect to find EVEN BETTER sound and graphic controller chips in the Atari 1200, said to be ready for public unveiling in Feb '83??

IS IT TRUE that Al Charpentier purchased a new Mercedes two weeks after leaving Commodore? Did he pay for it with cash?

IS IT TRUE that the unnamed third member of Atari's Commodore Alumni Association is in fact the principal architect of Commodore's famous but unannounced 650,000 microprocessor (the number of zeros is said to be correct, anonymous sources indicate).

IS IT TRUE that, as a result of the loss of this principal architect the 650,000 project has been dropped at Commodore? Is it true that Atari has inadvertently done Commodore an ENORMOUS favor because the final design specification document for the 650,000, which is rumored to have 74 pages, reveals that the 650,000 is in fact a baby iAPX 432 we kid you not? Is it further true that this document was released on one of the first three days of Oct '81, a time when it was generally believed in the industry that the iAPX 432 would be a very high performance part?

IS IT TRUE that the final design document reportedly asserts on page 3, "(the 650,000 will have) ...capabilities somewhere between the 68000 and Intel's 432 micro-mainframe"? Is it further true that recent information which has come to light regarding the performance of the 432 indicates that that assertion would probably have been true had the project not been cancelled? Of course, the 8080 and 6502 ALSO have performance capabilities which lie between the 68000 and the iAPX 432...

According to an informed observer who declined to be named, "Commodore is on the verge of announcing an agreement with a manufacturer of 16 bit microprocessors", presumably to fill the void left by the (fortunately for Commodore) defunct 650,000 project. The manufacturer in question is neither Intel nor is it Motorola nor is it any of their second sources. We know who the manufacturer is but we promised not to tell...

IS IT TRUE that Commodore is experiencing severe financial difficulties, difficulties so severe that they recently had to sell an additional stock offering? Is it further true that the financial difficulties involve an inability to build factories fast enough to satisfy the public demand for their products? Is it true that a major competitor of Commodore located on the west coast has had NO DIFFICULTY WHATEVER in the past year building enough factories to keep up with public demand for their products? Why is Commodore unable to match the obvious management superiority of that west coast competitor?

IS IT TRUE that Tom Wolfe, the author of "The Right Stuff", a book about the beginnings of the astronaut program, is destined to write a book about a certain personal computer company which has in recent times shown an unwavering ability to select the worst possible decision at every available opportunity? Will that book be titled "The Wrong Stuff"??

IS IT TRUE that the absolutely totally confidential price list of a certain San Diego-based vendor of a personal computer operating system is printed on six different pieces of paper, one side only? Is it true that the quality of the paper stock rivals that used for the centerfold of certain girlie magazines? Is it true that the per copy royalty for small companies is MORE THAN NINE TIMES GREATER than the per copy royalty for large companies? Does this bear any relation to the fact that very few small companies have licensed that system for their 68000-based machines, even though the operating system is available for the 68000?

WE ARE ABLE TO REVEAL for the first time that Britain's Microcomputer Printout Magazine gossip columnist Inside Trader has moved to Newark, New Jersey. We have it on good authority that he prefers the living conditions in Armpit, er, Newark over those in Reading. However, he is reported to vigorously deny that the reception of several death threats from his fellow workers had anything to do with his decision to emigrate. We have faithfully promised the editor of Microcomputer Printout, Richard Paitson, not to reveal the source of this information...

IS IT TRUE that our Christmas card from the Apple Computer Co. is seriously overdue?

In a matter which is totally unrelated to any other item in this newsletter, IS IT TRUE that Zilog, which has been devoid of an ongoing domestic second source for its line of 16 bit microprocessors for exactly 13 months is about to sign an agreement which will fix that problem?

IT IS TRUE that subscriber Leroy L. has been assigned this month's D-TACK D-Merit Award for suggesting that portions of this newsletter are more suitable for the National Enquirer than for a technical publication!

1 OPT P=68000,BRS,FRS 2 ; NOTE: SEE BUG FIXES IN DG#17P22. 3 4 ; MOTOROLA 68000 TRANSCENDENTAL PACKAGE 5 ; 6 ; WORKS WITH THE DOUBLE PRECISION FLOATING POINT 7 ; MATH PACKAGE USING A 48 BIT MANTISSA. 8 ; 9 ; COPYRIGHT 1982 DIGITAL ACOUSTICS INC. 10 ; 002600 11 ORG $2600 ;TEMP FOR DEVEL 12 ; 13 ; FOLLOWING ARE I/O ASSIGNMENTS: 14 ; # 00000FFA 15 STATUS EQU $0FFA # 00000FF8 16 DATIN EQU $0FF8 # 00000FFA 17 DATOUT EQU $0FFA 18 ; 19 ; FOLLOWING ARE MEMORY ASSIGNMENTS 20 ; USED BY THE BOOTSTRAP ROM (MONITOR): 21 ; # 00000122 22 IDLE EQU $0122 # 0000109C 23 ADR EQU $109C # 000010A0 24 N EQU $10A0 # 000010A2 25 T EQU $10A2 26 ; 27 ; FOLLOWING ARE MEMORY ASSIGNMENTS FOR THIS CODE 28 ; # 000018F4 29 R EQU $18F4 # 00001902 30 S1 EQU R+14 # 00001904 31 M1 EQU S1+2 # 0000190A 32 S2 EQU S1+8 # 0000190C 33 M2 EQU S2+2 # 00001912 34 FPT EQU S2+8 # 0000191A 35 FPU EQU FPT+8 # 00001922 36 LOGX EQU FPU+8 # 00001922 37 SINSGN EQU LOGX # 00001924 38 SERCNT EQU LOGX+2 # 00001926 39 EXPADD EQU LOGX+4 ;EXP ADDER 40 ; # 000023A8 41 FPADD EQU $23A8 # 000023B0 42 FPADD1 EQU $23B0 # 000023A2 43 FPSUB EQU $23A2 # 0000239A 44 FPSUB1 EQU $239A # 00002220 45 FPMUL EQU $2220 # 00002228 46 FPMUL1 EQU $2228 # 000022FA 47 FPDIV EQU $22FA # 00002302 48 FPDIV1 EQU $2302 # 000021DA 49 ERROR EQU $21DA # 000021D8 50 OVFL EQU $21D8 # 000022F0 51 RZER EQU $22F0 # 00002514 52 NORM EQU $2514 # 00002570 53 SUBX EQU $2570 54 ; 002600 7E 03 55 ERR3 MOVEQ #3,D7 002602 66 02 56 BNE ERR4A 002604 7E 04 57 ERR4 MOVEQ #4,D7 002606 4EF8 21DA 58 ERR4A JMP ERROR 59 ; 60 61 ; COPYRIGHT 1982 DIGITAL ACOUSTICS, INC 62 ; 63 ; CALC LOG2 THEN MULT BY LOG10(2) RESULT LOG(10) 64 ; 00260A 61 0C 65 LOG10 BSR LOG2 ;CALC LOG BASE 2 00260C 5048 66 ADDQ.W #8,A0 ;PTR TO LOG10(2) 00260E 4EF8 2220 67 JMP FPMUL ;ACC = ACC * LOG(2) 68 ; 69 ; CALC LOG2(X) THEN MULT BY LOGE(2) RESULT LOGE(X) 70 ; 002612 61 04 71 LOGE BSR LOG2 ;CALC LOG BASE 2 002614 4EF8 2220 72 JMP FPMUL ;ACC = ACC * LN(2) 73 ; 74 ; CALCULATE THE LOGARITHM TO THE BASE 2 75 ; 002618 3038 1902 76 LOG2 MOVE.W S1,D0 00261C 6B E2 77 BMI ERR3 ;NO LOG IF NEG 00261E 0240 1FFF 78 ANDI.W #$1FFF,D0 ;MASK UPPER BITS 002622 67 E0 79 BEQ ERR4 ;NO LOG IF ZERO 80 ; 81 ; THE OPERAND IS LEGAL; PROCEED WITH LOG2 CALC 82 ; 002624 0440 1000 83 SUBI.W #$1000,D0 ;2'S COMP RESULT 002628 31C0 1922 84 MOVE.W D0,LOGX ;STORE INTGR PART 85 ; 86 ; SET THE OPERAND RANGE TO SQR(.5) =< X < SQR(2) 87 ; 00262C 31FC 1000 1902 88 MOVE.W #$1000,S1 002632 307C 26BC 89 MOVE.W #SQR2+2,A0 ;PTR TO SQR(2) 002636 4EB8 2B50 90 JSR CMPM1 ;COMPARE MANTS 00263A 67 0A 91 BEQ LOG2A ;OK IF SAME 00263C 6A 08 92 BPL LOG2A ;OK IF FPACC1 > 93 ; 00263E 5278 1902 94 ADDQ.W #1,S1 ;INCR EXPONENT 002642 5378 1922 95 SUBQ.W #1,LOGX ;DECR LOG EXP 96 ; 002646 4EB8 2756 97 LOG2A JSR FPZ ;FPACC1 TO FP5 00264A 307C 26C2 98 MOVE.W #LOGK,A0 00264E 4EB8 23A8 99 JSR FPADD ;CALC X-1 002652 4EB8 2764 100 JSR M1U2 ;(X-1) TO FPU 002656 21F8 1912 1902 101 MOVE.L FPT,S1 ;RECALL X 00265C 21F8 1916 1906 102 MOVE.L FPT+4,S1+4 002662 4EB8 23A8 103 JSR FPADD ;CALC X+1 002666 21F8 191A 190A 104 MOVE.L FPU,S2 ;(X-1) TO FPACC2 00266C 21F8 191E 190E 105 MOVE.L FPU+4,S2+4 002672 4EB8 2302 106 JSR FPDIV1 ;CALC (X-1)/(X+1) 107 ; 108 ; THE RANGE OF X IS NOW -0.172 TO +0.172. STORE 109 ; X IN FPU AND STORE Z= X*X IN FPT, FPACC1 110 ; 111 ; PERFORM SERIES EVALUATION, FORM IS Z*P(Z*Z) 112 ; WHERE Z = (X-1)/(X+1). 113 ; 002676 11FC 0006 1924 114 MOVE.B #6,SERCNT ;SET SERIES N = 7 00267C 4EB8 273C 115 JSR SERSQU ;DO SERIES EVAL 116 ; 117 118 ; COPYRIGHT 1982 DIGITAL ACOUSTICS, INC. 119 ; 120 ; THE FRACTIONAL PART OF THE LOG2 IS COMPLETED 121 ; 002680 42B8 190A 122 CLR.L S2 ;ZERO FPACC2 002684 42B8 190E 123 CLR.L M2+2 002688 31F8 1922 190C 124 MOVE.W LOGX,M2 00268E 67 1C 125 BEQ LGEXIT ;DONE IF LGEXP 0 002690 6B 1C 126 BMI LGNEG ;SKIP IF NEGATIVE 002692 3238 190C 127 MOVE.W M2,D1 ;MANT2 TO D1 002696 303C 1010 128 MOVE.W #$1010,D0 ;SET EXP = #16 00269A 5340 129 LOG2B SUBQ.W #1,D0 ;X2 = X2 - 1 00269C E341 130 ASL.W #1,D1 ;SHIFT MANT2 L 00269E 6A FA 131 BPL LOG2B ;REPT TILL D31= 1 0026A0 31C0 190A 132 MOVE.W D0,S2 ;STORE X2 IN FPACC2 0026A4 31C1 190C 133 MOVE.W D1,M2 ;STORE M2 IN FPACC2 0026A8 4EF8 23B0 134 JMP FPADD1 ;ADD INTEGER; EXIT 135 ; 0026AC 4E75 136 LGEXIT RTS ;LOG2 DONE 137 ; 138 ; THE INTEGER PORTION OF THE LOG IS NEGATIVE 139 ; 0026AE 303C 9010 140 LGNEG MOVE.W #$9010,D0 ;SET SIGN NEG 0026B2 4241 141 CLR.W D1 ;(EXP IS 2'S COMP) 0026B4 9278 190C 142 SUB.W M2,D1 ;ABS(EXPONENT) 0026B8 60 E0 143 BRA LOG2B ;ADD EXP TO LOG 144 ; 145 ; 146 ; CONSTANTS USED BY THE LOG ROUTINES 147 ; 0026BA 1001B504 148 SQR2 DC.L $1001B504 ;SQR(2) 0026BE F333F9DE 149 DC.L $F333F9DE 0026C2 90018000 150 LOGK DC.L $90018000 ;MINUS 1 0026C6 00000000 151 DC.L $00000000 0026CA 10018000 152 FPONE DC.L $10018000 ;ONE 0026CE 00000000 153 DC.L $00000000 0026D2 0FFEFA61 154 DC.L $0FFEFA61 ;P(6) 0026D6 18DC43A5 155 DC.L $18DC43A5 0026DA 0FFF85C8 156 DC.L $0FFF85C8 ;P(5) 0026DE 07A095A8 157 DC.L $07A095A8 0026E2 0FFFA428 158 DC.L $0FFFA428 ;P(4) 0026E6 9100003A 159 DC.L $9100003A 0026EA 0FFFD30B 160 DC.L $0FFFD30B ;P(3) 0026EE A7EE2159 161 DC.L $A7EE2159 0026F2 100093BB 162 DC.L $100093BB ;P(2) 0026F6 628EF5FA 163 DC.L $628EF5FA 0026FA 1000F638 164 DC.L $1000F638 ;P(1) 0026FE 4EE1CAE4 165 DC.L $4EE1CAE4 002702 1002B8AA 166 DC.L $1002B8AA ;P(0) 002706 3B395C18 167 DC.L $3B395C18 00270A 1000B172 168 DC.L $1000B172 ;LOGE(2) 00270E 17F7D1CF 169 DC.L $17F7D1CF 002712 0FFF9A20 170 DC.L $0FFF9A20 ;LOG10(2) 002716 9AA68151 171 DC.L $9AA68151 172 ; 173 174 ; COPYRIGHT 1982 DIGITAL ACOUSTICS, INC. 175 ; 176 ; PERFORM SERIES EVALUATION 177 ; 00271A 4EB8 2220 178 SERIES JSR FPMUL 00271E 60 10 179 BRA SER2 002720 21F8 1912 190A 180 SER1 MOVE.L FPT,S2 002726 21F8 1916 190E 181 MOVE.L FPT+4,S2+4 00272C 4EB8 2228 182 JSR FPMUL1 ;FPACC1 * FPT 002730 4EB8 23A8 183 SER2 JSR FPADD ;ADD NEXT P(I 002734 5338 1924 184 SUBQ.B #1,SERCNT ;DECR COUNT 002738 66 E6 185 BNE SER1 ;LOOP UNTIL ZERO 00273A 4E75 186 RTS 187 ; 00273C 61 12 188 SERSQU BSR FPSQU ;X SQUARED TO FPT 00273E 61 DA 189 BSR SERIES ;DO SERIES EVAL 190 ; 002740 21F8 191A 190A 191 MOVE.L FPU,S2 ;FPU TO FPACC2 002746 21F8 191E 190E 192 MOVE.L FPU+4,S2+4 00274C 4EF8 2228 193 JMP FPMUL1 ;SQUARE IT; DONE 194 ; 195 ; LEAVE A COPY OF FPACC1 IN FPU, THEN 196 ; SQUARE IT AND LEAVE A COPY IN FPT 197 ; 002750 61 12 198 FPSQU BSR M1U2 ;FPACC1 TO FPACC2 002752 4EB8 2228 199 JSR FPMUL1 ;SQUARE IT 002756 21F8 1902 1912 200 FPZ MOVE.L S1,FPT 00275C 21F8 1906 1916 201 MOVE.L S1+4,FPT+4 ;LEAVE Z IN FPT 002762 4E75 202 RTS 203 ; 204 ; COPY FPACC1 TO FPU AND FPACC2 205 ; 002764 21F8 1902 191A 206 M1U2 MOVE.L S1,FPU ;FPACC1 TO FPU 00276A 21F8 1906 191E 207 MOVE.L S1+4,FPU+4 208 ; 209 ; MOVE FPACC1 TO FPACC2 210 ; 002770 21F8 1902 190A 211 MOVE12 MOVE.L S1,S2 ;FPACC1 TO FPACC2 002776 21F8 1906 190E 212 MOVE.L S1+4,S2+4 00277C 4E75 213 RTS ;DONE 214 ; 215 216 ; COPYRIGHT 1982 DIGITAL ACOUSTICS, INC. 217 ; 218 ; CALCULATE THE COSINE OF THE 68000 FPACC1 219 ; 00277E 08B8 0007 1902 220 SCOS BCLR #7,S1 ;SET FPACC1 POS 002784 307C 2D0C 221 MOVE.W #COSK,A0 ;POINT A0 AT COSK 002788 4EB8 23A2 222 JSR FPSUB ;90 DEG - ACCUM 223 ; 224 ; 225 ; CALCULATE THE SINE OF THE 68000 FPACC#1 226 ; 00278C 11F8 1902 1922 227 SSIN MOVE.B S1,SINSGN ;STORE THE SIGN 002792 08B8 0007 1902 228 BCLR #7,S1 ;SET FPACC1 POS 002798 307C 2D14 229 MOVE.W #SINK,A0 ;POINT A0 AT SINK 00279C 4EB8 2220 230 JSR FPMUL ;MULT BY 1/2#PI 0027A0 61 52 231 BSR FRAC ;RESULT UND 360 DEG 0027A2 5478 1902 232 ADDQ.W #2,S1 ;MULT FPACC1 BY 4 0027A6 0C78 1002 1902 233 CMPI.W #$1002,S1 ;FPACC1 < 2 ? 0027AC 65 16 234 BLO UNDTWO ;SKIP IF UNDER TWO 235 ; 236 ; TOGGLE THE SIGN AND SUBTRACT TWO 237 ; 0027AE 0878 0007 1922 238 BCHG #7,SINSGN ;TOGGLE SIGN 0027B4 21FC 90028000 239 MOVE.L #$90028000,S2 190A 0027BC 42B8 190E 240 CLR.L S2+4 ;-2 TO FPACC2 0027C0 4EB8 23B0 241 JSR FPADD1 242 ; 243 ; THE OPERAND IS NOW IN THE RANGE 0 TO 1.999999 244 ; IF LESS THAN ONE, PROCEED WITH CALCULATION, 245 ; ELSE SUBTRACT OPERAND FROM ONE AND PROCEED 246 ; 0027C4 0C78 1001 1902 247 UNDTWO CMPI.W #$1001,S1 0027CA 65 0E 248 BLO UND1 ;SKIP IF < 1 249 ; 250 ; SET X = 90 DEGREES MINUS X 251 ; 0027CC 21F8 26CA 190A 252 MOVE.L FPONE,S2 0027D2 42B8 190E 253 CLR.L S2+4 ;ONE TO FPACC2 0027D6 4EB8 239A 254 JSR FPSUB1 ;SUBTRACT 255 ; 256 ; THE RANGE OF X IS NOW 0 TO 0.99999999. 257 ; NOW PERFORM SERIES EVALUATION, N = 7 258 ; 0027DA 11FC 0007 1924 259 UND1 MOVE.B #7,SERCNT ;SET SERIES N = 7 0027E0 4EB8 273C 260 JSR SERSQU ;FORM X*P(X*X) 0027E4 11F8 1922 1902 261 MOVE.B SINSGN,S1 ;RESTORE SIGN 0027EA 6A 06 262 BPL SINX ;EXIT IF SIGN POS 263 ; 0027EC 08F8 0007 1902 264 BSET #7,S1 ;SET SIGN NEG 0027F2 4E75 265 SINX RTS ;SINE CALC DONE 266 ; 267 268 ; COPYRIGHT 1982 DIGITAL ACOUSTICS, INC. 269 ; 270 ; CALCULATE THE FRACTIONAL PART OF FPACC1 271 ; 0027F4 3438 1902 272 FRAC MOVE.W S1,D2 ;FETCH THE EXP 0027F8 3202 273 MOVE.W D2,D1 ;SAVE SIGN 0027FA 0242 1FFF 274 ANDI.W #$1FFF,D2 ;MASK THE SIGN BIT 0027FE 0C42 1000 275 CMPI.W #$1000,D2 ;BR IF EQ OR LESS 002802 63 66 276 BLS FRACX ;DONE IF UNDER ONE 002804 0C42 1027 277 CMPI.W #$1027,D2 ;NO FRACTION? 002808 6200 FAE6 278 BHI RZER ;RESULT IS ZERO 00280C 0442 1000 279 SUBI.W #$1000,D2 ;CALC EXP- $1000 002810 0C42 0010 280 CMPI.W #$10,D2 ;< 16 SHIFTS? 002814 63 26 281 BLS UND16 ;SKIP IF LESS 282 ; 002816 0C02 0020 283 CMPI.B #$20,D2 ; < 32 SHIFTS? 00281A 63 12 284 BLS UND32 ;SKIP IF LESS 285 ; 286 ; WE HAVE 32 OR MORE SHIFTS TO PERFORM 287 ; 00281C 0402 0020 288 SUBI.B #$20,D2 002820 4245 289 CLR.W D5 002822 3838 1908 290 MOVE.W M1+4,D4 002826 E56C 291 LSL.W D2,D4 002828 4844 292 SWAP D4 00282A 4244 293 CLR.W D4 00282C 60 1E 294 BRA FRO ;NORMALIZE 295 ; 00282E 0402 0010 296 UND32 SUBI.B #16,D2 002832 4245 297 CLR.W D5 002834 2838 1906 298 MOVE.L M1+2,D4 002838 E5AC 299 LSL.L D2,D4 00283A 60 10 300 BRA FRO ;NORMALIZE 301 ; 00283C 2838 1904 302 UND16 MOVE.L M1,D4 002840 3A38 1908 303 MOVE.W M1+4,D5 002844 E34D 304 UND16A LSL.W #1,D5 ;-- SHIFT MANT 002846 E394 305 ROXL.L #1,D4 ; 1 BIT LEFT -- 002848 5342 306 SUBQ.W #1,D2 ;DECR SHIFT COUNT 00284A 66 F8 307 BNE UND16A ;LOOP UNTIL DONE 308 ; 00284C 31FC 1000 1902 309 FRO MOVE.W #$1000,S1 002852 3001 310 MOVE.W D1,D0 ;TEST SIGN 002854 6A 06 311 BPL FSPOS ;SKIP IF POS 002856 08F8 0007 1902 312 BSET #7,S1 ;SET SIGN NEG 313 ; 314 ; NORMALIZE; S1, X1 = MEM, M1 D4.L, D5.W 315 ; 00285C 2604 316 FSPOS MOVE.L D4,D3 ;NORMALIZED 00285E 6B 06 317 BMI FR2 ;DONE OF NORMAL'D 318 ; 002860 4241 319 CLR.W D1 002862 4EF8 2514 320 JMP NORM ;ELSE NORMALIZE 002866 4EF8 2570 321 FR2 JMP SUBX ;STORE RESULT, EXIT 00286A 4E75 322 FRACX RTS 323 ; 324 325 ; COPYRIGHT 1982 DIGITAL ACOUSTICS, INC. 326 ; 327 ; CALC THE SQUARE ROOT OF THE 68000 FPACC1 328 ; 00286C 3438 1902 329 SSQR MOVE.W S1,D2 ;CHK SIGN 002870 6B00 FD8E 330 BMI ERR3 ;ERROR IF NEG 331 ; 332 ; THE OPERAND IS LEGAL; PROCEED WITH SQR CALC 333 ; 002874 0242 1FFF 334 ANDI.W #$1FFF,D2 ;IS EXP ZERO? 002878 6700 FA76 335 BEQ RZER ;EXIT IF ZERO 00287C 2E38 1904 336 MOVE.L M1,D7 002880 31FC 1000 1902 337 MOVE.W #$1000,S1 ;1/2 =< X < 1 002886 3602 338 MOVE.W D2,D3 002888 E24B 339 LSR.W #1,D3 ;IS EXP EVEN? 00288A 64 06 340 BCC EVNEXP ;SKIP IF EVEN 00288C 5378 1902 341 SUBQ.W #1,S1 ;DECR EXP 002890 E28F 342 LSR.L #1,D7 ;MANT = MANT/2 343 ; 344 ; CALCULATE AND STORE THE EXP OF THE RESULT 345 ; 002892 363C 0800 346 EVNEXP MOVE.W #$0800,D3 002896 E24A 347 LSR.W #1,D2 002898 D543 348 ADDX.W D3,D2 00289A 31C2 1922 349 MOVE.W D2,LOGX 350 ; 351 ; SET Y0 = $FFFF (1ST GUESS DELIBERATELY HIGH) 352 ; 00289E 3E3C FFFF 353 MOVE.W #$FFFF,D7 354 ; 355 ; PERFORM Y(N+1) = (Y(N) + X/Y(N))/2 4 TIMES 356 ; 0028A2 303C 0003 357 MOVE.W #3,D0 ;SET FOR 4 LOOPS 0028A6 3A07 358 MOVE.W D7,D5 ;Y0 = $FFFF 0028A8 2C07 359 SQL1 MOVE.L D7,D6 ;MOVE X TO D6 0028AA 8CC5 360 DIVU D5,D6 ;D6 = X/Y 0028AC DA46 361 ADD.W D6,D5 ;D5 = Y + X/Y 0028AE E255 362 ROXR.W #1,D5 ;DIV BY 2 0028B0 51C8 FFF6 363 DBF D0,SQL1 ;LOOP 4 TIMES 364 ; 365 ; MOVE X TO TEMP FP REG U AND TO FPACC2 366 ; 0028B4 4EB8 2764 367 JSR M1U2 368 ; 369 ; FLOAT THE 16 BIT APPROXIMATION 370 ; 0028B8 42B8 1906 371 CLR.L M1+2 0028BC 31FC 1000 1902 372 MOVE.W #$1000,S1 0028C2 31C5 1904 373 MOVE.W D5,M1 0028C6 61 0A 374 BSR SQITER ;ONE ITERATION 0028C8 61 08 375 BSR SQITER ;ANOTHER ITER'TN 376 ; 377 ; RESTORE THE EXPONENT OF THE RESULT 378 ; 0028CA 31F8 1922 1902 379 MOVE.W LOGX,S1 ;(SIGN IS POS) 0028D0 4E75 380 RTS 381 ; 382 383 ; COPYRIGHT 1982 DIGITAL ACOUSTICS, INC. 384 ; 385 ; PERFORM ONE ITERATION OF Y1 = (Y0 + X/Y0)/2 386 ; FIRST MOVE Y0 TO TEMP FP REGISTER T 387 ; 0028D2 21F8 1902 1912 388 SQITER MOVE.L S1,FPT ;Y0 TO FPT 0028D8 21F8 1906 1916 389 MOVE.L S1+4,FPT+4 0028DE 21F8 191A 190A 390 MOVE.L FPU,S2 ;X TO FPACC2 0028E4 21F8 191E 190E 391 MOVE.L FPU+4,S2+4 0028EA 4EB8 2302 392 JSR FPDIV1 ;CALC X/Y0 0028EE 307C 1912 393 MOVE.W #FPT,A0 ;POINT AT Y0 0028F2 4EB8 23A8 394 JSR FPADD ;ADD Y0 TO X/Y0 0028F6 5378 1902 395 SUBQ.W #1,S1 ;DIVIDE BY 2 0028FA 4E75 396 RTS ;ITERATION DONE 397 ; 398 ; 399 ; CALCULATE THE TANGENT FUNCTION IN RADIANS 400 ; FIRST REDUCE THE RANGE TO -180 DEG < X < 180 DEG 401 ; 0028FC 307C 29B6 402 STAN MOVE.W #TANK,A0 ;PTR TO 1/PI 002900 4EB8 2220 403 JSR FPMUL ;(#1 = 180 DEG) 002904 4EB8 27F4 404 JSR FRAC ;RESULT < 180 DEG 002908 307C 0000 405 MOVE.W #0,A0 ;CLEAR FLAGS 00290C 31C8 1922 406 MOVE.W A0,SINSGN ;CLR SIGN FLAG 407 ; 408 ; REDUCE THE RANGE TO 0 =< X < 180 DEG 409 ; 002910 3238 1902 410 MOVE.W S1,D1 ;S1, X1 TO D1 002914 6A 0C 411 BPL TANPOS ;SKIP IF POSITIVE 002916 08F8 0007 1922 412 BSET #7,SINSGN ;SET SIGN FLAG 00291C 08B8 0007 1902 413 BCLR #7,S1 ;SET ARG POSITIVE 414 ; 415 ; REDUCE THE RANGE TO 0 =< X < 90 DEG 416 ; 002922 0C78 1000 1902 417 TANPOS CMPI.W #$1000,S1 ;FPACC < 1/2 ? 002928 65 14 418 BLO TUH ;SKIP IF < 1/2 00292A 0878 0007 1922 419 BCHG #7,SINSGN ;FLIP SIGN FLAG 002930 21F8 26CA 190A 420 MOVE.L FPONE,S2 ;ONE TO FPACC2 002936 42B8 190E 421 CLR.L S2+4 00293A 4EB8 239A 422 JSR FPSUB1 ;180 DEG - X 423 ; 424 ; IF THE ARGUMENT IS 45 DEGREES OR MORE, SET 425 ; THE INVERT FLAG AND LET X = 90 DEG - X 426 ; 00293E 0C78 0FFF 1902 427 TUH CMPI.W #$0FFF,S1 ;FPACC < 1/4 ? 002944 65 12 428 BLO TUQ ;SKIP IF < 1/4 002946 5448 429 ADDQ.W #2,A0 ;B1 IS INVERT FLAG 002948 21FC 10008000 430 MOVE.L #$10008000,S2 ;1/2 TO FPACC2 190A 002950 42B8 190E 431 CLR.L S2+4 002954 4EB8 239A 432 JSR FPSUB1 ;90 DEG - X 433 ; 434 ; ARGUMENT IS NOW LESS THAN 45 DEGREES; REDUCE 435 ; THE RANGE TO 22.5 DEGREES AND SET THE REDUCTION 436 ; (RED.) FLAG IF 22.5 DEG <= ARGUMENT. 437 ; 002958 0C78 0FFE 1902 438 TUQ CMPI.W #$0FFE,S1 ;FPACC < 1/8 ? 00295E 65 06 439 BLO TUE ;SKIP IF < 1/8 002960 5248 440 ADDQ.W #1,A0 ;B0 IS RED. FLAG 002962 5378 1902 441 SUBQ.W #1,S1 ;DIVIDE FPACC BY 2 442 ; 443 444 ; COPYRIGHT 1982 DIGITAL ACOUSTICS, INC. 445 ; 446 ; THE ARGUMENT IS NOW LESS THAN 22.5 DEGREES; 447 ; PERFORM SERIES EVALUATION HART 4187 FORM X*P(X*X) 448 ; 002966 5678 1902 449 TUE ADDQ.W #3,S1 ;RANGE 0 TO 1 00296A 31C8 1926 450 MOVE.W A0,EXPADD ;RESTORE FLAGS 00296E 307C 29BE 451 MOVE.W #TANK+8,A0 ;PTR TO SERIES K'S 002972 11FC 0008 1924 452 MOVE.B #8,SERCNT ;SET SERIES N = 8 002978 4EB8 273C 453 JSR SERSQU ;DO SERIES EVAL 454 ; 455 ; ADJUST FOR RANGE REDUCTION IF NEEDED 456 ; 00297C E2F8 1926 457 LSR EXPADD ;TEST RED. FLAG 002980 64 18 458 BCC NOTRNG ;SKIP OF NOT SET 002982 4EB8 2750 459 JSR FPSQU ;X TO FPU; SQUARE 002986 307C 26CA 460 MOVE.W #FPONE,A0 ;PTR TO 1 00298A 4EB8 23A2 461 JSR FPSUB ;1 - X*X 00298E 307C 191A 462 MOVE.W #FPU,A0 ;PTR TO X 002992 4EB8 22FA 463 JSR FPDIV ;X/(1 - X*X) 002996 5278 1902 464 ADDQ.W #1,S1 ;2X/(1 - X*X) 465 ; 466 ; INVERT IF THE INVERT FLAG IS SET 467 ; 00299A E2F8 1926 468 NOTRNG LSR EXPADD ;TEST INV. FLAG 00299E 64 08 469 BCC NOTINV ;SKIP IF NOT SET 0029A0 307C 26CA 470 MOVE.W #FPONE,A0 ;PTR TO 1 0029A4 4EB8 22FA 471 JSR FPDIV ;INVERT FPACC1 472 ; 473 ; CHANGE THE SIGN IF THE SIGN FLAG IS SET 474 ; 0029A8 3038 1922 475 NOTINV MOVE.W SINSGN,D0 ;TEST SIGN FLAG 0029AC 6A 06 476 BPL NOTNEG ;SKIP IF NOT SET 0029AE 08F8 0007 1902 477 BSET #7,S1 ;SET SIGN NEGATIVE 0029B4 4E75 478 NOTNEG RTS ;TANGENT CALC DONE 479 ; 480 ; CONSTANTS USED BY THE TANGENT ROUTINE 481 ; 0029B6 0FFFA2F9 482 TANK DC.L $0FFFA2F9 ;1/PI 0029BA 836E4E44 483 DC.L $836E4E44 0029BE 0FDFD967 484 DC.L $0FDFD967 ;P8 0029C2 E1E426B5 485 DC.L $E1E426B5 0029C6 0FE39C3D 486 DC.L $0FE39C3D ;P7 0029CA CBC63642 487 DC.L $CBC63642 0029CE 0FE7A369 488 DC.L $0FE7A369 ;P6 0029D2 61F9A940 489 DC.L $61F9A940 0029D6 0FEBA2F5 490 DC.L $0FEBA2F5 ;P5 0029DA 68557947 491 DC.L $68557947 0029DE 0FEFA2FA 492 DC.L $0FEFA2FA ;P4 0029E2 50DA798D 493 DC.L $50DA798D 0029E6 0FF3A2FF 494 DC.L $0FF3A2FF ;P3 0029EA FC90F626 495 DC.L $FC90F626 0029EE 0FF7A335 496 DC.L $0FF7A335 ;P2 0029F2 E33C201E 497 DC.L $E33C201E 0029F6 0FFBA55D 498 DC.L $0FFBA55D ;P1 0029FA E7312DAE 499 DC.L $E7312DAE 0029FE 0FFFC90F 500 DC.L $0FFFC90F ;P0 002A02 DAA22169 501 DC.L $DAA22169 502 ; 503 504 ; COPYRIGHT 1982 DIGITAL ACOUSTICS, INC. 505 ; 002A06 21F8 26CA 1902 506 XRONE MOVE.L FPONE,S1 ;RESULT IS ONE 002A0C 42B8 1906 507 CLR.L S1+4 002A10 4E75 508 EXP2X RTS ;RETURN 509 ; 002A12 1038 1922 510 EXOVFL MOVE.B SINSGN,D0 ;SIGN NEGATIVE? 002A16 6B00 F8D8 511 BMI RZER ;RESULT ZERO IF SO 002A1A 70 01 512 MOVEQ #1,D0 ;#1 = OVFL 002A1C 4EF8 21DA 513 JMP ERROR ;REPORT OVERFLOW 514 ; 515 ; CALCULATE THE EXPONENT FUNCTION, BASE 10 516 ; FIRST DIVIDE BY LOG10(2) AND THEN CALC EXP2 517 ; 002A20 307C 2CC4 518 XP10 MOVE.W #NVLOG2,A0 ;PTR TO 1/LOG10(2) 002A24 4EB8 2220 519 JSR FPMUL 002A28 60 08 520 BRA XP2 ;CALC EXP BASE 2 521 ; 522 ; CALCULATE THE EXPONENT FUNCTION, BASE E 523 ; FIRST DIVIDE BY LOG10(2) AND THEN CALC EXP2 524 ; 002A2A 307C 2CC4 525 XPE MOVE.W #NVLOG2,A0 ;POINT TO 1/LOGE(2) 002A2E 4EB8 2220 526 JSR FPMUL 527 ; 528 ; CALCULATE THE EXPONENT FUNCTION, BASE 2 529 ; 002A32 11F8 1902 1922 530 XP2 MOVE.B S1,SINSGN ;STORE THE SIGN 002A38 08B8 0007 1902 531 BCLR #7,S1 ;SET SIGN POSITIVE 002A3E 42B8 1926 532 CLR.L EXPADD ;CLR EXP ADDER 002A42 3038 1902 533 MOVE.W S1,D0 ;EXP TO D0 002A46 0240 1FFF 534 ANDI.W #$1FFF,D0 ;MASK SIGN BIT 002A4A 67 BA 535 BEQ XRONE ;RESULT ONE IF ZERO 002A4C 0C40 0FFD 536 CMPI.W #$0FFD,D0 ;LESS THAN 1/16? 002A50 65 24 537 BCS EXP2A ;SKIP OF UNDER 1/16 538 ; 002A52 323C 100D 539 MOVE.W #$100D,D1 ;-- TEST FOR ARG 002A56 9240 540 SUB.W D0,D1 ; OUT OF RANGE -- 002A58 6B B8 541 BMI EXOVFL ;OVFL IF X1>$100D 002A5A 67 B6 542 BEQ EXOVFL ;OVFL IF X1=$100D 543 ; 544 ; STORE THE INTEGER OF 16*FPACC1 IN EXPADD 545 ; 002A5C 2438 1904 546 MOVE.L M1,D2 002A60 0641 000F 547 ADDI.W #15,D1 ;CORRECT THE COUNT 002A64 E2AA 548 LSR.L D1,D2 ;SHIFT COUNT IN D1 002A66 21C2 1926 549 MOVE.L D2,EXPADD ;STORE EXP ADDER 550 ; 002A6A 5878 1902 551 ADDQ.W #4,S1 ;MULT FPACC1 * 16 002A6E 4EB8 27F4 552 JSR FRAC ;FRACTIONAL PART 002A72 5978 1902 553 SUBQ.W #4,S1 ;DIV FPACC1 BY 16 554 ; 555 ; FPACC RANGE IS NOW 0 TO ALMOST 1/16 556 ; PERFORM SERIES EVALUATION FORM P(X) 557 ; 002A76 307C 2CD4 558 EXP2A MOVE.W #EXPK2,A0 ;PTR TO P(X) 002A7A 11FC 0006 1924 559 MOVE.B #6,SERCNT ;SET N = 6 002A80 4EB8 2756 560 JSR FPZ ;X TO FPT 002A84 4EB8 271A 561 JSR SERIES ;SERIES EVAL 562 ; 563 564 ; COPYRIGHT 1982 DIGITAL ACOUSTICS, INC. 565 ; 566 ; NOW ADD THE INTEGER VALUE OF X TO THE EXP 567 ; 002A88 2038 1926 568 MOVE.L EXPADD,D0 ;FETCH ADDER 002A8C E888 569 LSR.L #4,D0 ;LEAST BIT VAL = 1 002A8E D178 1902 570 ADD.W D0,S1 ;ADD TO X1 002A92 3038 1928 571 MOVE.W EXPADD+2,D0 ;LOW 4 BITS = K 002A96 0240 000F 572 ANDI.W #$000F,D0 ;MASK ALL BUT 4 002A9A 67 0C 573 BEQ NOEXPK ;SKIP IF ZERO 574 ; 575 ; MULTIPLY FPACC1 BY ONE OF FIFTEEN CONSTANTS 576 ; 002A9C E748 577 LSL.W #3,D0 ;MULTIPLY BY 8 002A9E 0640 2AB6 578 ADDI.W #EXPK-8,D0 ;ADDR OF CONSTANT 002AA2 3040 579 MOVE.W D0,A0 ;SET PTR A0 002AA4 4EB8 2220 580 JSR FPMUL ;MULTIPLY BY K(N) 581 ; 582 ; INVERT FPACC1 IF THE SIGN IS NEGATIVE 583 ; 002AA8 1038 1922 584 NOEXPK MOVE.B SINSGN,D0 ;TEST SIGN OF ARG 002AAC 6A00 FF62 585 BPL EXP2X ;DONE OF POS 002AB0 21F8 26CA 190A 586 INV MOVE.L FPONE,S2 ;#1 TO FPACC2 002AB6 42B8 190E 587 CLR.L S2+4 002ABA 4EF8 2302 588 JMP FPDIV1 ;EXIT VIA DIVIDE 589 ; 590 ; CONSTANTS USED TO CORRECT FOR RANGE REDUCTION 591 ; 002ABE 100185AA 592 EXPK DC.L $100185AA ;EXP2(1/16) 002AC2 C367CC48 593 DC.L $C367CC48 ;(7B14C1) 002AC6 10018B95 594 DC.L $10018B95 ;EXP2(1/8) 002ACA C1E3EA8C 595 DC.L $C1E3EA8C 002ACE 100191C3 596 DC.L $100191C3 ;EXP2(3/16) 002AD2 D373AB12 597 DC.L $D373AB12 ;(C3360C) 002AD6 10019837 598 DC.L $10019837 ;EXP2(1/4) 002ADA F0518DB9 599 DC.L $F0518DB9 002ADE 10019EF5 600 DC.L $10019EF5 ;EXP2(5/16) 002AE2 326091A1 601 DC.L $326091A1 ;(11AD98) 002AE6 1001A5FE 602 DC.L $1001A5FE ;EXP2(3/8) 002AEA D6A9B151 603 DC.L $D6A9B151 002AEE 1001AD58 604 DC.L $1001AD58 ;EXP2(7/16) 002AF2 3EEA42A1 605 DC.L $3EEA42A1 ;(4AC644) 002AF6 1001B504 606 DC.L $1001B504 ;EXP2(1/2) 002AFA F333F9DE 607 DC.L $F333F9DE 002AFE 1001BD08 608 DC.L $1001BD08 ;EXP2(9/16) 002B02 A39F580C 609 DC.L $A39F580C ;(36BEA3) 002B06 1001C567 610 DC.L $1001C567 ;EXP2(5/8) 002B0A 2A115507 611 DC.L $2A115507 002B0E 1001CE24 612 DC.L $1001CE24 ;EXP2(11/16) 002B12 8C151F85 613 DC.L $8C151F85 ;(80E3DC) 002B16 1001D744 614 DC.L $1001D744 ;EXP2(3/4) 002B1A FCCAD69D 615 DC.L $FCCAD69D 002B1E 1001E0CC 616 DC.L $1001E0CC ;EXP2(13/16) 002B22 DEEC2A95 617 DC.L $DEEC2A95 ;(E110FF) 002B26 1001EAC0 618 DC.L $1001EAC0 ;EXP2(7/8) 002B2A C6E7DD24 619 DC.L $C6E7DD24 002B2E 1001F525 620 DC.L $1001F525 ;EXP2(15/16) 002B32 7D152487 621 DC.L $7D152487 ;(CC2C75) 622 ; 623 624 ; COPYRIGHT 1992 DIGITAL ACOUSTICS, INC. 625 ; 626 ; COMPARE FPACC WITH FP# IN MEMORY 627 ; 628 ; IF FPACC IS GREATER, N=0 (BPL FPACC>MEM) 629 ; IF FPACC IS LESS, N=1 (BMI FPACC<MEM) 630 ; IF FPACC IS SAME, Z=1 631 ; 002B36 70 80 632 CPFP1M MOVEQ #$80,D0 002B38 72 80 633 MOVEQ #$80,D1 002B3A C010 634 AND.B (A0),D0 ;MEMSIGN TO D0 002B3C C238 1902 635 AND.B S1,D1 ;FPACC SIGN TO D1 002B40 B001 636 CMP.B D1,D0 ;COMPARE SIGNS 002B42 66 1A 637 BNE NOTEQL ;DONE IF NOT SAME 638 ; 002B44 1001 639 MOVE.B D1,D0 ;BOTH SIGNS NEG? 002B46 6B 18 640 BMI NEGCMP ;BRANCH IF NEG 641 ; 642 ; COMPARE TWO POSITIVE NUMBERS 643 ; 002B48 3038 1902 644 CPFPOS MOVE.W S1,D0 002B4C B058 645 CMP.W (A0)+,D0 ;COMPARE EXP'S 002B4E 66 0E 646 BNE NOTEQL ;DONE IF NOT SAME 647 ; 648 ; COMPARE THE MANTISSA TO MEMORY 649 ; 002B50 2038 1904 650 CMPM1 MOVE.L M1,D0 002B54 B098 651 CMP.L (A0)+,D0 ;COMPARE 32 BITS 002B56 66 06 652 BNE NOTEQL ;DONE IF NOT SAME 653 ; 002B58 3038 1908 654 MOVE.W M1+4,D0 002B5C B090 655 CMP.L (A0),D0 ;COMPARE LEAST 16 656 ; 002B5E 4E75 657 NOTEQL RTS ;DONE 658 ; 002B60 3018 659 NEGCMP MOVE.W (A0)+,D0 002B62 B078 1902 660 CMP.W S1,D0 ;COMPARE EXP'S 002B66 66 F6 661 BNE NOTEQL ;DONE IF NOT EQUAL 662 ; 002B68 2018 663 MOVE.L (A0)+,D0 002B6A B0B8 1904 664 CMP.L M1,D0 ;COMPARE 32 BITS 002B6E 66 EE 665 BNE NOTEQL ;DONE IF NOT EQUAL 666 ; 002B70 3010 667 MOVE.W (A0),D0 002B72 B078 1908 668 CMP.W M1+4,D0 002B76 4E75 669 RTS ;DONE 670 ; 002B78 DE46 671 CPXN ADD.W D6,D7 ;ADD #32 TO D7 002B7A 3047 672 MOVE.W D7,A0 ;SET PTR TO XN 002B7C 60 CA 673 BRA CPFPOS ;CMP FPACC, XN 674 ; 675 676 ; COPYRIGHT 1982 DIGITAL ACOUSTICS, INC. 677 ; 678 ; CALCULATE THE ARC TANGENT FUNCTION IN RADIANS 679 ; 002B7E 4278 1926 680 SATN CLR.W EXPADD ;CLR PTR 002B82 3038 1902 681 MOVE.W S1,D0 ;S1,X1 TO D0 002B86 31C0 1922 682 MOVE.W D0,SINSGN ;STORE IN SINSGN 002B8A 08B8 0007 1902 683 BCLR #7,S1 ;SET SIGN POS 002B90 0240 1FFF 684 ANDI.W #$1FFF,D0 ;TEST IF EXP = 0 002B94 6700 F75A 685 BEQ RZER ;RESULT IS ZERO 002B98 0C40 1001 686 CMPI.W #$1001,D0 ;ONE OR GREATER? 002B9C 65 04 687 BCS UNDONE ;SKIP IF NOT 002B9E 4EB8 2AB0 688 JSR INV ;CALC RECIPROCAL 689 ; 690 ; TEST WHETHER RANGE REDUCTION IS NECESSARY 691 ; 002BA2 7C 20 692 UNDONE MOVEQ #32,D6 ;D6 = #32 002BA4 3E3C 2C24 693 MOVE.W #X1-32,D7 002BA8 61 CE 694 BSR CPXN ;COMPARE X1 002BAA 6B 2A 695 BMI X0 ;NO REDUCTION 002BAC 61 CA 696 BSR CPXN ;COMPARE X2 002BAE 6B 0A 697 BMI XN ;REDUCE ON X1-X2 002BB0 61 C6 698 BSR CPXN ;COMPARE X3 002BB2 6B 06 699 BMI XN ;REDUCTE ON X2-X3 002BB4 61 C2 700 BSR CPXN ;REDUCT ON X4 002BB6 6B 02 701 BMI XN ;REDUCE ON X3-X4 002BB8 DE46 702 ADD.W D6,D7 ;REDUCE ON X4-PI/4 703 ; 002BBA 5147 704 XN SUBQ.W #8,D7 ;POINT TO CN 002BBC 31C7 1926 705 MOVE.W D7,EXPADD ;STORE CN ADDR 002BC0 0447 0010 706 SUBI.W #16,D7 ;POINT TO AN 002BC4 3047 707 MOVE.W D7,A0 002BC6 4EB8 23A8 708 JSR FPADD ;X + AN 002BCA 4EB8 22FA 709 JSR FPDIV ;BN/(X+AN) 002BCE 90FC 0010 710 SUB.W #16,A0 ;PTR TO AN 002BD2 4EB8 23A2 711 JSR FPSUB ;AN-BN/(X+AN) 712 ; 713 ; RANGE REDUCTION TO -X1 TO X1 ACCOMPLISHED 714 ; PERFORM SERIES EVALUATION; FORM IS X*P(X*X) 715 ; 002BD6 11FC 0005 1924 716 X0 MOVE.B #5,SERCNT ;SET N = 5 002BDC 307C 2C14 717 MOVE.W #ATNK,A0 ;PTR TO ATN CONSTS 002BE0 4EB8 273C 718 JSR SERSQU ;DO SERIES EVAL 719 ; 720 ; ADD CN IF RANGE REDUCTION WAS DONE 721 ; 002BE4 3E38 1926 722 MOVE.W EXPADD,D7 ;FETCH CN PTR 002BE8 67 06 723 BEQ NOPTR ;SKIP IF ZERO 002BEA 3047 724 MOVE.W D7,A0 ;PTR TO CN 002BEC 4EB8 23A8 725 JSR FPADD ;ADD CN 726 ; 727 ; CORRECT FOR PREVIOUS INVERSION IF NEEDED 728 ; 002BF0 3038 1922 729 NOPTR MOVE.W SINSGN,D0 ;RECALL S1, X1 002BF4 0240 1FFF 730 ANDI.W #$1FFF,D0 ;MASK SIGN BIT 002BF8 0C40 1001 731 CMPI.W #$1001,D0 ;ONE OR GREATER? 002BFC 65 08 732 BCS NOINV ;SKIP IF NOT 002BFE 307C 2D0C 733 MOVE.W #COSK,A0 ;PTR TO PI/2 002C02 4EB8 23A2 734 JSR FPSUB ;90 DEG - FPACC1 735 ; 736 737 ; COPYRIGHT 1982 DIGITAL ACOUSTICS, INC. 738 ; 739 ; CHANGE SIGN IF ARGUMENT WAS NEGATIVE 740 ; 002C06 3038 1922 741 NOINV MOVE.W SINSGN,D0 ;RECALL S1 002C0A 6A 06 742 BPL ATNX ;DONE IF POS 002C0C 0878 0007 1902 743 BCHG #7,S1 ;ELSE CHANGE SIGN 002C12 4E75 744 ATNX RTS ;ARC TAN DONE 745 ; 746 ; CONSTANTS USED TO EVALUATE ARC TANGENT 747 ; 002C14 8FFDB5AB 748 ATNK DC.L $8FFDB5AB ;P5 002C18 6364E40E 749 DC.L $6364E40E 002C1C 0FFDE381 750 DC.L $0FFDE381 ;P4 002C20 AEE4C3E4 751 DC.L $AEE4C3E4 002C24 8FFE9249 752 DC.L $8FFE9249 ;P3 002C28 1C8532D1 753 DC.L $1C8532D1 002C2C 0FFECCCC 754 DC.L $0FFECCCC ;P2 002C30 CCC81F2E 755 DC.L $CCC81F2E 002C34 8FFFAAAA 756 DC.L $8FFFAAAA ;P1 002C38 AAAAAA2B 757 DC.L $AAAAAA2B 002C3C 10018000 758 DC.L $10018000 ;P0 002C40 00000000 759 DC.L $00000000 760 ; 761 ; CONSTANTS USED TO ADJUST RANGE OF ATN CALC 762 ; 002C44 0FFDC9B5 763 X1 DC.L $0FFDC9B5 ;X1 002C48 DC62D96D 764 DC.L $DC62D96D 002C4C 1003A0DF 765 DC.L $1003A0DF ;A1 002C50 F712123C 766 DC.L $F712123C 002C54 1005D231 767 DC.L $1005D231 ;B1 002C58 718DED74 768 DC.L $718DED74 002C5C 0FFEC90F 769 DC.L $0FFEC90F ;C1 = PI/16 002C60 DAA22169 770 DC.L $DAA22169 002C64 0FFF9B50 771 X2 DC.L $0FFF9B50 ;X2 002C68 41AAE31F 772 DC.L $41AAE31F 002C6C 10029A82 773 DC.L $10029A82 ;A2 002C70 7999FCEF 774 DC.L $7999FCEF 002C74 1003DA82 775 DC.L $1003DA82 ;B2 002C78 7999FCEF 776 DC.L $7999FCEF 002C7C 0FFFC90F 777 DC.L $0FFFC90F ;C2 = 2PI/16 002C80 DAA22169 778 DC.L $DAA22169 002C84 100088D5 779 X3 DC.L $100088D5 ;X3 002C88 B8C841A7 780 DC.L $B8C841A7 002C8C 1001BF90 781 DC.L $1001BF90 ;A3 002C90 C712D3A3 782 DC.L $C712D3A3 002C94 1002CF59 783 DC.L $1002CF59 ;B3 002C98 5AEEA7CA 784 DC.L $5AEEA7CA 002C9C 100096CB 785 DC.L $100096CB ;C3 = 3PI/16 002CA0 E3F9990F 786 DC.L $E3F9990F 002CA4 1000D218 787 X4 DC.L $1000D218 ;X4 002CA8 01572142 788 DC.L $01572142 002CAC 10018000 789 DC.L $10018000 ;A4 002CB0 00000000 790 DC.L $00000000 002CB4 10028000 791 DC.L $10028000 ;B4 002CB8 00000000 792 DC.L $00000000 002CBC 1000C90F 793 DC.L $1000C90F ;C4 = 4PI/16 002CC0 DAA22169 794 DC.L $DAA22169 795 ; 796 797 ; COPYRIGHT 1982 DIGITAL ACOUSTICS, INC. 798 ; 002CC4 1002D49A 799 NVLOG2 DC.L $1002D49A ;1/LOG10(2) 002CC8 784BCD1C 800 DC.L $784BCD1C 002CCC 1001B8AA 801 INVLN2 DC.L $1001B8AA ;1/LOGE(2) 002CD0 3B295C18 802 DC.L $3B295C18 803 ; 804 ; P(X); N= 6 805 ; 002CD4 0FF58B70 806 EXPK2 DC.L $0FF58B70 ;P6 002CD8 00000000 807 DC.L $00000000 002CDC 0FF7ABBF 808 DC.L $0FF7ABBF ;P5 002CE0 80000000 809 DC.L $80000000 002CE4 0FFA9D9C 810 DC.L $0FFA9D9C ;P4 002CE8 CC200000 811 DC.L $CC200000 002CEC 0FFCE358 812 DC.L $0FFCE358 ;P3 002CF0 36210000 813 DC.L $36210000 002CF4 0FFEF5FD 814 DC.L $0FFEF5FD ;P2 002CF8 F00C0800 815 DC.L $F00C0800 002CFC 1000B172 816 DC.L $1000B172 ;P1 002D00 17F7CD00 817 DC.L $17F7CD00 002D04 10018000 818 DC.L $10018000 ;P0 002D08 00000000 819 DC.L $00000000 820 ; 821 ; CONSTANTS USED BY THE COSINE AND SINE CALCS 822 ; 002D0C 1001C90F 823 COSK DC.L $1001C90F ;#PI/2 002D10 DAA22169 824 DC.L $DAA22169 002D14 0FFEA2F9 825 SINK DC.L $0FFEA2F9 ;1/(2#PI) 002D18 836E4E44 826 DC.L $836E4E44 827 ; 828 ; FORMULA 3344 HART, FORM X*P(X*X) 829 ; 002D1C 8FE2B131 830 SINK1 DC.L $8FE2B131 ;P7 002D20 3233A218 831 DC.L $3233A218 002D24 0FE8F44E 832 DC.L $0FE8F44E ;P6 002D28 7501852C 833 DC.L $7501852C 002D2C 8FEEF183 834 DC.L $8FEEF183 ;P5 002D30 11E19C26 835 DC.L $11E19C26 002D34 0FF4A83C 836 DC.L $0FF4A83C ;P4 002D38 1924E79B 837 DC.L $1924E79B 002D3C 8FF99969 838 DC.L $8FF99969 ;P3 002D40 6670BE99 839 DC.L $6670BE99 002D44 0FFDA335 840 DC.L $0FFDA335 ;P2 002D48 E33BA883 841 DC.L $E33BA883 002D4C 9000A55D 842 DC.L $9000A55D ;P1 002D50 E7312DEB 843 DC.L $E7312DEB 002D54 1001C90F 844 DC.L $1001C90F ;P0 002D58 DAA22169 845 DC.L $DAA22169 846 ;