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

Microsoft Office 2016 15.25 - Popular pr...
Microsoft Office 2016 - Unmistakably Office, designed for Mac. The new versions of Word, Excel, PowerPoint, Outlook and OneNote provide the best of both worlds for Mac users - the familiar Office... Read more
FileZilla 3.21.0 - Fast and reliable FTP...
FileZilla (ported from Windows) is a fast and reliable FTP client and server with lots of useful features and an intuitive interface. Version 3.21.0: Fixed Vulnerabilities Fixed a string format... Read more
Fantastical 2.2.5 - Create calendar even...
Fantastical 2 is the Mac calendar you'll actually enjoy using. Creating an event with Fantastical is quick, easy, and fun: Open Fantastical with a single click or keystroke Type in your event... Read more
The Hit List 1.1.26 - Advanced reminder...
The Hit List manages the daily chaos of your modern life. It's easy to learn - it's as easy as making lists. And it's powerful enough to let you plan, then forget, then act when the time is right.... Read more
Typinator 6.10 - Speedy and reliable tex...
Typinator turbo-charges your typing productivity. Type a little. Typinator does the rest. We've all faced projects that require repetitive typing tasks. With Typinator, you can store commonly used... Read more
EtreCheck 3.0.2 - For troubleshooting yo...
EtreCheck is an app that displays the important details of your system configuration and allow you to copy that information to the Clipboard. It is meant to be used with Apple Support Communities to... Read more
FileZilla 3.21.0 - Fast and reliable FTP...
FileZilla (ported from Windows) is a fast and reliable FTP client and server with lots of useful features and an intuitive interface. Version 3.21.0: Fixed Vulnerabilities Fixed a string format... Read more
EtreCheck 3.0.2 - For troubleshooting yo...
EtreCheck is an app that displays the important details of your system configuration and allow you to copy that information to the Clipboard. It is meant to be used with Apple Support Communities to... Read more
Microsoft Office 2016 15.25 - Popular pr...
Microsoft Office 2016 - Unmistakably Office, designed for Mac. The new versions of Word, Excel, PowerPoint, Outlook and OneNote provide the best of both worlds for Mac users - the familiar Office... Read more
Typinator 6.10 - Speedy and reliable tex...
Typinator turbo-charges your typing productivity. Type a little. Typinator does the rest. We've all faced projects that require repetitive typing tasks. With Typinator, you can store commonly used... Read more

Bowmasters tips, tricks and hints
At least for this writer, archery was one of the more pleasant surprises of the 2016 Rio Olympics. As opposed to target shooting with guns, which was dreadfully boring, watching people shoot arrows at targets was pretty darn cool. [Read more] | Read more »
Best apps for watching live TV
The Olympics have come and gone, leaving nearly everyone in a temporary state of "What the heck am I going to watch on TV right now?" Besides old reruns of Golden Girls, but that goes without saying. [Read more] | Read more »
What is Flip Diving, and why has it take...
Move over Pokemon GO. There's a new king in town, and it's "the world's #1 cliff diving game." [Read more] | Read more »
5 places where Pokemon GO is still numbe...
In the U.S., the bloom is off the Pokemon Go rose ever so slightly. It's still doing great, sitting atop the top grossing chart as it has for some time, but it's no longer among the top 10 free apps in downloads, possibly because darn near... | Read more »
Madden NFL Mobile: How defense has chang...
Saying that defense is not a priority in Madden NFL Mobile is a bit of an understatement. In asynchronous head-to-head play, you don't take control of your defenders at all, as the AI manages them while your opponent plays offense. When it's your... | Read more »
Feed Hawk (News)
Feed Hawk 1.0.1 Device: iOS Universal Category: News Price: $2.99, Version: 1.0.1 (iTunes) Description: Feed Hawk makes it easy to subscribe to the RSS feed of the website you are visiting. From within Safari, simply open a share... | Read more »
Reigns character guide: Who's who i...
Know your foes. Keep your friends close, but your enemies closer. And there are probably some other cliches that would apply to your perilous spot on the throne in Reigns as well. [Read more] | Read more »
Match 3 puzzler Small Lime is now availa...
Set to hit Android and IOS on the 17th August, Small Lime is the newest match 3 mobile game, and hopes to throw something a little different into the mix. If you love match 3 puzzles, but are tired of the same old ideas being re-hashed again and... | Read more »
Deus Ex GO tips, tricks, and hints
When Square Enix Montreal first hit us with Hitman GO,it was seen as a clever board game twist on a property that you wouldn't normally think would fit that kind of format. Lara Croft GOexpanded things even further while keeping some of the same... | Read more »
Leap of Fate (Games)
Leap of Fate 1.0 Device: iOS Universal Category: Games Price: $3.99, Version: 1.0 (iTunes) Description: *** Minimum hardware: iPad 4, iPad mini 2, iPhone 5s. *** | Read more »

Price Scanner via MacPrices.net

Typinator 6.10 comes with 50 improvements – G...
Ergonis Software today announced release of Typinator 6.10, a new version of their text expander utility for macOS. Typinator 6.10 comes with 50 improvements, including new features, compatibility... Read more
Taxi Sim 2016 Puts Users Behind the Wheel in...
Ovilex Soft today announces Taxi Sim 2016, an update to their ultra-realistic 3D driving simulator app for iOS and Android devices — literally a global event what with the company’s nearly 450,000... Read more
11-inch 1.6GHz/128GB MacBook Air on sale for...
Amazon has the current-generation 11″ 1.6GHz/128GB MacBook Air (sku MJVM2LL/A) on sale for $788 for a limited time. Their price is $111 off MSRP, and it’s the lowest price available for this model. Read more
Apple refurbished Mac minis available for up...
Apple has Certified Refurbished Mac minis available starting at $419. Apple’s one-year warranty is included with each mini, and shipping is free: - 1.4GHz Mac mini: $419 $80 off MSRP - 2.6GHz Mac... Read more
Apple refurbished 13-inch Retina MacBook Pros...
Apple has Certified Refurbished 13″ Retina MacBook Pros available for up to $270 off the cost of new models. An Apple one-year warranty is included with each model, and shipping is free: - 13″ 2.7GHz... Read more
12-inch 32GB and 128GB WiFi iPad Pros on sale...
B&H Photo has 12″ 32GB & 128GB WiFi Apple iPad Pros on sale for up to $70 off MSRP, each including free shipping. B&H charges sales tax in NY only: - 12″ Space Gray 32GB WiFi iPad Pro: $... Read more
Apple refurbished 11-inch MacBook Airs availa...
Apple has Certified Refurbished 11″ MacBook Airs (the latest models), available for up to $170 off the cost of new models. An Apple one-year warranty is included with each MacBook, and shipping is... Read more
WaterField Launches Kickstarter for Intrepid...
San Francisco based WaterField Design have announced their first Kickstarter campaign for the one-of-a-kind Intrepid iPhone Travel Wallet. The all-new design includes iPhone play-through capability... Read more
Five of Top 10 Worldwide Mobile Phone Vendors...
Global sales of smartphones to end users totaled 344 million units in the second quarter of 2016, a 4.3 percent increase over the same period in 2015, according to Gartner, Inc. Overall sales of... Read more
DriveSavers Offers $300 Off Data Recovery Ser...
DriveSavers, with more than 30 years of experience recovering photos, videos, contact lists, financial records and other important data that may have been kept on devices damaged or even destroyed by... Read more

Jobs Board

*Apple* Retail - Multiple Positions Germanto...
Job Description: Sales Specialist - Retail Customer Service and Sales Transform Apple Store visitors into loyal Apple customers. When customers enter the store, Read more
*Apple* Professional Learning Specialist - A...
# Apple Professional Learning Specialist Job Number: 51234379 Portland, Maine, Maine, United States Posted: Aug. 18, 2016 Weekly Hours: 40.00 **Job Summary** The Read more
Lead *Apple* Solutions Consultant - Apple (...
# Lead Apple Solutions Consultant Job Number: 51218465 Richmond, VA, Virginia, United States Posted: Aug. 18, 2016 Weekly Hours: 40.00 **Job Summary** The Lead ASC Read more
*Apple* Solutions Consultant - Apple (United...
# Apple Solutions Consultant Job Number: 51218534 Pleasant Hill, California, United States Posted: Aug. 18, 2016 Weekly Hours: 40.00 **Job Summary** As an Apple Read more
*Apple* Retail - Multiple Positions Chestnut...
Job Description: Sales Specialist - Retail Customer Service and Sales Transform Apple Store visitors into loyal Apple customers. When customers enter the store, Read more
All contents are Copyright 1984-2011 by Xplain Corporation. All rights reserved. Theme designed by Icreon.