TweetFollow Us on Twitter

SANE Normalized
Volume Number:6
Issue Number:1
Column Tag:XCMD Corner

SANE Normalized

By Donald Koscheka, Ernst & Young, MacTutor Contributing Editor

Note: Source code files accompanying article are located on MacTech CD-ROM or source code disks.

Ten years makes a big difference. When I started engineering school in 1973, I wanted to learn everything I could about the nascent microcomputer technology. For several years after graduation, however, I had trouble securing meaningful employment as a microcomputer engineer. With the exception of Silicon Valley and an instrumental little hotspot in Texas, there wasn’t much calling for people who knew about microcomputers. Employers were intrigued by my background, but they regarded the microcomputer as not much more than an interesting toy. They argued that micros didn’t have enough memory to do real work, they couldn’t do real number crunching, and so on.

I found myself apologizing for these shortcomings. Realizing that I would probably have to beef up my computer experience, I enrolled in graduate school in 1980 at the University of Illinois. At the time, the school used yet another derivative of the IBM 370. You can imagine my reaction when I discovered that this behemoth was running an operating system called CMS (Conversational Mode System or some such). The entire goal of this operating system was to transform this monolithic hunk of iron in hundreds of “virtual personal computers”. The same people that were pooh-poohing my microcomputer training were spending vast amounts of effort trying to make their mainframes look like personal computers!

Ten years later, I think it’s safe to say that the personal computer cum micro has come of age. Aside from the obvious advances in user interface design, tremendous progress has been made in the personal computer technology.

Nowhere is this more evident than in the floating point support that ships at no extra cost with each and every Macintosh: the Standard Apple Numerics Environment (SANE). Numerics on the Macintosh are as good as or better than numerics on many mainframes. The implications of this quiet little revolution are profound, you can trust your Macintosh to do real number crunching accurately and reliably!

SANE guarantees well-behaved results and you don’t have to be an expert in floating point arithmetic to use the Macintosh numerics package. If you do need to get into the details, SANE is beautifully documented in the Apple Numerics Manual, Second Addition by Addison Wesley. This is one of the best technical publications I have ever read; it is a paragon of simplicity and clarity.

I recently implemented a business graphics package as a set of XCMDs that accepts numbers from Hypercard and plots them into a windoid. I wanted to scale the picture so that it exactly fits within the dimensions of the windoid. I also didn’t want to limit the input data to the domain of integers; a floating point implementation was indicated.

When using SANE, the old adage that knowledge is power is a statement of fact (if one’s definition of power is the number of floating point operations per second). To exploit SANE, you should understand how floating point numbers work in the binary world.

HyperTalk’s callback mechnanism supports conversions between strings and extendeds (80 bit floating point numbers). While this is a good starting place, it only begins to untap the magic of SANE.

Floating point numbers come in a variety of flavors. You must consider factors such as range, precision, speed and space before settling on a format for your program. Simple applications might make do with the 32-bit single precision type, float. Most applications will be adequately served with the 64-bit double precision floating point type (double in “C” or Pascal). Understanding the internals of SANE can help you make a more informed decision.

For example, knowing that all SANE internal operations are performed on the extended type allows you to make an important design decision: if speed is important, you might want to consider doing all of your arithmetic with 80-bit extended numbers so that you can spare your code the overhead of automatic type conversions. If you know a priori that your product will have co-processor support, then the 96-bit extended type may better suit your needs.

For bean counters, there’s even a computational type that allows you to manipulate very large signed integers (64 bits).

The extended data type is the essential SANE type but it is implementation dependent. You should store your numbers in some other language specific format. If you intend to massage the data heavily, you might consider declaring your variables as extendeds so that no intermediate conversions will be made yielding speed for the potential loss of portability. This assumes that you have some worthwhile machine that you want to port to in the first place.

As I studied these floating point formats, I discovered some interesting properties of floating point representations in the binary world. I debug in TMON, so I need to be able to disassemble floating point numbers with the same ease that I disassemble integers. I needed to learn how to read floating point numbers from hex dumps. This is an illuminating exercise so I hope you won’t mind if I share it with you.

A decimal number can be broken down into the product of three numbers (if you ever learned how to use a slide rule, you’ll appreciate the value of this representation):

 -100110 = -1 * 1.001 * 103

Let’s call 1.001 the significand and 3 the exponent (the power of 10 that the significand is raised to). Any decimal number can be represented as the product of a sign, a significand and an exponent. It turns out that this is not just a property of decimal numbers. Binary numbers can be represented in the same fashion:

 1.001 * 23 

is equal to 9 base ten. Demonstrating this provides us with some insights into floating point numbers.

SANE stores numbers in either normalized or denormalized forms. Normalization maximizes precision for a given number of bits (can you prove this to yourself?) Unfortunately, very small numbers cannot be represented in this normalized format; how small the number has to be depends on the number of bits used to represent the number. Unless your idea of a fun afternoon is exploring the Mandelbrot set, you probably won’t need to concern yourself with the difference between normalized and denormalized numbers; suffice it to say that denormalized numbers are very small and characteristically hover around the origin.

SANE uses the format in figure 1 to store 80-bit normalized extended numbers.

Figure 1. Format of extended numbers in SANE.

The most significant bit is the sign bit, just as in signed integers. The next fifteen bits represent the exponent using the formula:

 2(e-16383)

This representation allows for numbers whose orders of magnitude range from 2-16385 to 2+16385. The next field in the number (the i-bit) is set if this is a normalized number, cleared otherwise. The “f” field represents the fractional portion of the significand. If the i-bit is set, then the significand is assembled as 1.f otherwise, the significand is assembled as 0.f. The exponent determines the range of the numbers while the significand determines the resolution of the numbers.

The complete representation for the extended type becomes the product of its components (for normalized numbers):

 (-1)s * 2(e-16383) * 1.f

To test this format, I wrote the following “C” program:

/* 1 */

main(){
 extended x;

 x = 9;
}

to determine the extended representation of 9 decimal. On debugging this number, I noticed that the integer 9 is first converted to a SANE extended which pops out as:

 $4002 9000 0000 0000 0000

To see if this is truly the extended representation of the number, let’s dissect it. The most significant bit is turned off so we know this is a positive number. The next 15 bits represent the exponent, in this case $4002 (hex) which is equal to 16386 decimal. Putting the sign and exponent together reveals the order of magnitude of the number:

 (-1)0 * 2(16386-16383) = 1 * 23 = 8

The rest of the number is the significand. The i-bit is set so this is a normalized number:

 1.0012

The significand is a binary fraction (the word decimal doesn’t quite seem to fit here).

When you see the decimal numbers 0.1, 0.01, 0.001 , you interpret them as 1/10, 1/100 and 1/1000 respectively. The binary numbers 0.12, 0.012 and 0.0012 have identical representations: 1/10, 1/100 and 1/1000 respectively, albeit in a different number system. To determine the value of a binary fraction, you need to know the decimal equivalent of these numbers. That’s simple: (1/10)2 is equivalent to (1/2)10. In the same fashion (1/100)2 = 1/4 and (1/1000)2 = 1/8. By now you should have inferred that these binary fractions are the negative powers of 2.

Armed with this knowledge, we can now determine that 1.0012 is equal to 1 + 1/8 or (1.125)10. We can now finish converting our extended number:

 1 * 23 * 1.125 = 8 * 1.125 = 9

If the significand raised to its exponent yields an integer (no fractional part) you can very quickly determine that value of the number:

  (-1)0 * 23 * 1.001 = 1 * 8 * 1.0012 = 910

In other words, just slide the significand to the right by the number of “decimal” places in the exponent. This is a simple trick that any student of the metric system understands but tends to be forgotten when we change to a non-decimal number system.

Try some of these problems on your own. You might want to consider exercises like finding the largest positive and negative numbers that a given format can represent. Equally interesting, is finding smallest number that can be represented in this format. What does 0 look like (watch this, it’s a trick question)?

Listing 1 contains a grab bag of SANE glue routines which I’ve provided as illustrations of how to interface with SANE. You may never need to use these conversions but knowing how this mechanism works will surely help you to debug code that references SANE.

SANE operations get dispatched via the trap _FP68K which most likely stands for “Floating Point, 68000” (SANE has been implemented on ALL Apple platforms since the mid-80s).

The conversions typically take an input parameter, an output parameter and an opword. The opwords are mnemonic, FX2D stands for extended to double and FL2X stands for long to extended. The conversion utilities in SANE give you a lot of control over how you want to represent your data and how you want to present it to the user. If you’re serious about these conversions, you might want to write a general purpose converter that can convert between any two formats.

If you want to explore SANE further, get a copy of the Apple Numerics Manual. The next time you run into one of those old hacks who believe that, “it ain’t a real computer unless it’s water cooled”, don’t get upset. They need all that power to compensate for the fact that some of those monoliths can’t even add as well as the Macintosh!

/* 2 */

void  ExtToDouble( ext, dbl )
 extended *ext;
 double *dbl;
/******************************
* given the extended IEEE number
* passed in, return its double
* representation
*
******************************/
{

asm{
 move.l 8(A6),-(sp); address of the extended
 move.l 12(A6),-(sp) ; address of the double
 move.w #FX2D,-(sp); push the appropriate opword
 _FP68K
 }
}

void  DoubleToExt( dbl, ext )
 double *dbl;
 extended *ext;
/******************************
* given the double number
* passed in, return its extended
* representation
*
******************************/
{

asm{
 move.l 8(A6),-(sp); address of the double
 move.l 12(A6),-(sp) ; address of the extended
 move.w #FD2X,-(sp); push the appropriate opword
 _FP68K
 }
}

void  LongToExt( lg, ext )
 long   *lg;
 extended *ext;
/******************************
* given the long  number
* passed in, return its extended
* representation
*
******************************/
{

asm{
 move.l 8(A6),-(sp); address of the long
 move.l 12(A6),-(sp) ; address of the extended
 move.w #FL2X,-(sp); push the appropriate opword
 _FP68K
 }
}

void  ExtToLong( ext, theint )
 extended *ext;
 long   *theint;
/******************************
* given the extended IEEE number
* passed in, return its long word
* representation
*
******************************/
{
asm{
 move.l 8(A6),-(sp); pointer to the extended
 move.l 12(A6),-(sp) ; address of the long
 move.w #FX2L,-(sp); push the appropriate opword
 _FP68K
 }
}

void  DoubleToLong( dbl, theint )
 double *dbl;
 long   *theint;
/******************************
* A simple conversion utility that might be useful
* for debugging at the TMON and MACSBUG level.
******************************/
{
 extended temp;
 
 DoubleToExt( dbl, &temp);
 ExtToLong( &temp, theint );
}

void  ExtendedToStr( ext, theStr )
 extended *ext;
 char   *theStr;
/*******************************
* convert an extended to a string
* 
* First convert the number to 
* a decimal record and then convert
* the decimal record to a string.
*
* The Hypercard callback “ExtToStr” does
* this for you.  I’ve added it here for those
* cases where you can’t make a callback
*
* The conversions uses the decimal record 
* structure that’s documented in Apple Numerics
* manual.
*******************************/
{
 decformdecrec;
 decimaldecnum;
 
 /*** convert the extended to a decimal ***/

 decrec.style = FIXEDDECIMAL;
 decrec.digits= 0;
 num2str( &decrec, *ext, theStr );
}

LISTING 1. Some Interesting SANE conversion utilities

 

Community Search:
MacTech Search:

Software Updates via MacUpdate

Latest Forum Discussions

See All

Six fantastic ways to spend National Vid...
As if anyone needed an excuse to play games today, I am about to give you one: it is National Video Games Day. A day for us to play games, like we no doubt do every day. Let’s not look a gift horse in the mouth. Instead, feast your eyes on this... | Read more »
Old School RuneScape players turn out in...
The sheer leap in technological advancements in our lifetime has been mind-blowing. We went from Commodore 64s to VR glasses in what feels like a heartbeat, but more importantly, the internet. It can be a dark mess, but it also brought hundreds of... | Read more »
Today's Best Mobile Game Discounts...
Every day, we pick out a curated list of the best mobile discounts on the App Store and post them here. This list won't be comprehensive, but it every game on it is recommended. Feel free to check out the coverage we did on them in the links below... | Read more »
Nintendo and The Pokémon Company's...
Unless you have been living under a rock, you know that Nintendo has been locked in an epic battle with Pocketpair, creator of the obvious Pokémon rip-off Palworld. Nintendo often resorts to legal retaliation at the drop of a hat, but it seems this... | Read more »
Apple exclusive mobile games don’t make...
If you are a gamer on phones, no doubt you have been as distressed as I am on one huge sticking point: exclusivity. For years, Xbox and PlayStation have done battle, and before this was the Sega Genesis and the Nintendo NES. On console, it makes... | Read more »
Regionally exclusive events make no sens...
Last week, over on our sister site AppSpy, I babbled excitedly about the Pokémon GO Safari Days event. You can get nine Eevees with an explorer hat per day. Or, can you? Specifically, you, reader. Do you have the time or funds to possibly fly for... | Read more »
As Jon Bellamy defends his choice to can...
Back in March, Jagex announced the appointment of a new CEO, Jon Bellamy. Mr Bellamy then decided to almost immediately paint a huge target on his back by cancelling the Runescapes Pride event. This led to widespread condemnation about his perceived... | Read more »
Marvel Contest of Champions adds two mor...
When I saw the latest two Marvel Contest of Champions characters, I scoffed. Mr Knight and Silver Samurai, thought I, they are running out of good choices. Then I realised no, I was being far too cynical. This is one of the things that games do best... | Read more »
Grass is green, and water is wet: Pokémo...
It must be a day that ends in Y, because Pokémon Trading Card Game Pocket has kicked off its Zoroark Drop Event. Here you can get a promo version of another card, and look forward to the next Wonder Pick Event and the next Mass Outbreak that will be... | Read more »
Enter the Gungeon review
It took me a minute to get around to reviewing this game for a couple of very good reasons. The first is that Enter the Gungeon's style of roguelike bullet-hell action is teetering on the edge of being straight-up malicious, which made getting... | Read more »

Price Scanner via MacPrices.net

Take $150 off every Apple 11-inch M3 iPad Air
Amazon is offering a $150 discount on 11-inch M3 WiFi iPad Airs right now. Shipping is free: – 11″ 128GB M3 WiFi iPad Air: $449, $150 off – 11″ 256GB M3 WiFi iPad Air: $549, $150 off – 11″ 512GB M3... Read more
Apple iPad minis back on sale for $100 off MS...
Amazon is offering $100 discounts (up to 20% off) on Apple’s newest 2024 WiFi iPad minis, each with free shipping. These are the lowest prices available for new minis among the Apple retailers we... Read more
Apple’s 16-inch M4 Max MacBook Pros are on sa...
Amazon has 16-inch M4 Max MacBook Pros (Silver and Black colors) on sale for up to $410 off Apple’s MSRP right now. Shipping is free. Be sure to select Amazon as the seller, rather than a third-party... Read more
Red Pocket Mobile is offering a $150 rebate o...
Red Pocket Mobile has new Apple iPhone 17’s on sale for $150 off MSRP when you switch and open up a new line of service. Red Pocket Mobile is a nationwide MVNO using all the major wireless carrier... Read more
Switch to Verizon, and get any iPhone 16 for...
With yesterday’s introduction of the new iPhone 17 models, Verizon responded by running “on us” promos across much of the iPhone 16 lineup: iPhone 16 and 16 Plus show as $0/mo for 36 months with bill... Read more
Here is a summary of the new features in Appl...
Apple’s September 2025 event introduced major updates across its most popular product lines, focusing on health, performance, and design breakthroughs. The AirPods Pro 3 now feature best-in-class... Read more
Apple’s Smartphone Lineup Could Use A Touch o...
COMMENTARY – Whatever happened to the old adage, “less is more”? Apple’s smartphone lineup. — which is due for its annual refresh either this month or next (possibly at an Apple Event on September 9... Read more
Take $50 off every 11th-generation A16 WiFi i...
Amazon has Apple’s 11th-generation A16 WiFi iPads in stock on sale for $50 off MSRP right now. Shipping is free: – 11″ 11th-generation 128GB WiFi iPads: $299 $50 off MSRP – 11″ 11th-generation 256GB... Read more
Sunday Sale: 14-inch M4 MacBook Pros for up t...
Don’t pay full price! Amazon has Apple’s 14-inch M4 MacBook Pros (Silver and Black colors) on sale for up to $220 off MSRP right now. Shipping is free. Be sure to select Amazon as the seller, rather... Read more
Mac mini with M4 Pro CPU back on sale for $12...
B&H Photo has Apple’s Mac mini with the M4 Pro CPU back on sale for $1259, $140 off MSRP. B&H offers free 1-2 day shipping to most US addresses: – Mac mini M4 Pro CPU (24GB/512GB): $1259, $... Read more

Jobs Board

All contents are Copyright 1984-2011 by Xplain Corporation. All rights reserved. Theme designed by Icreon.