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

Apple iOS 11.2.6 - The latest version of...
iOS 11 sets a new standard for what is already the world’s most advanced mobile operating system. It makes iPhone better than before. It makes iPad more capable than ever. And now it opens up both to... Read more
EtreCheck 4.0.3 - 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
Opera 51.0.2830.34 - High-performance We...
Opera is a fast and secure browser trusted by millions of users. With the intuitive interface, Speed Dial and visual bookmarks for organizing favorite sites, news feature with fresh, relevant content... Read more
ffWorks 1.0.6 - Convert multimedia files...
ffWorks (was iFFmpeg), focused on simplicity, brings a fresh approach to the use of FFmpeg, allowing you to create ultra-high-quality movies without the need to write a single line of code on the... Read more
Adobe Photoshop CC 2018 19.1.1 - Profess...
Photoshop CC 2018 is available as part of Adobe Creative Cloud for as little as $19.99/month (or $9.99/month if you're a previous Photoshop customer). Adobe Photoshop CC 2018, the industry standard... Read more
Google Chrome 64.0.3282.167 - Modern and...
Google Chrome is a Web browser by Google, created to be a modern platform for Web pages and applications. It utilizes very fast loading of Web pages and has a V8 engine, which is a custom built... Read more
Adobe Acrobat Reader 18.011.20035 - View...
Adobe Acrobat Reader allows users to view PDF documents. You may not know what a PDF file is, but you've probably come across one at some point. PDF files are used by companies and even the IRS to... Read more
Adobe Acrobat 18.011.20035 - Powerful PD...
Acrobat DC is available only as a part of Adobe Creative Cloud, and can only be installed and/or updated through Adobe's Creative Cloud app. Adobe Acrobat DC with Adobe Document Cloud services is... Read more
Yummy FTP Pro 2.0 - $29.99
Yummy FTP Pro is an advanced Mac file transfer app which provides a full-featured professional toolkit combined with blazing speeds and impeccable reliability, so whether you want to transfer a few... Read more
WebSnapperPro 2.0.5 - $20.00
WebSnapperPro lets you capture full web pages exactly as they appear in your browser, with a single mouse click, without the need to "stitch" or cut-and-paste. Save the page as an image file or as... Read more

Latest Forum Discussions

See All

Check out these 5 games that are a lot l...
So you're in love with Minecraft, but you're looking for something else to play as well? You've come to the right place then, because this list is all about games that are a bit like Minecraft. Some of them, more than others. [Read more] | Read more »
Our top 5 characters from casual RPG Cre...
Creature Quest definitely lives up to its name with a host of collectible creatures based on fantasy tales and world mythologies. To celebrate Creature Quest’s first birthday, we’re going to lay out what we think are the five best characters in the... | Read more »
Around the Empire: What have you missed...
Did you know that Steel Media has a whole swathe of other sites dedicated to all aspects of mobile gaming? Sure you'll get the very best iPhone news, reviews, and opinions right here at 148Apps, but we don't want you missing out on a single piece... | Read more »
All the best games on sale for iPhone an...
Oh hi there, and welcome to our round-up of the best games that are currently on sale for iPhone and iPad. You thought I didn't see you there, did you, skulking behind the bushes? Trust me though, the bushes aren't where the best deals are. The... | Read more »
The Battle of Polytopia Guide - How to H...
A new update just released for The Battle of Polytopia (formerly Super Tribes), which introduces online multiplayer. For all the fans of Midjiwan’s lite take on Civilization, this is certainly welcome news, but playing online isn’t as easy and... | Read more »
Here are the very best mobile games to p...
It's Valentine's Day! Did you get loads of cards and chocolates and other tacky, simple expressions of human affection? Did you send out tat because you find it almost impossible to express emotion unless there's a section dedicated to it at your... | Read more »
Florence (Games)
Florence 1.0 Device: iOS Universal Category: Games Price: $2.99, Version: 1.0 (iTunes) Description: Florence is an interactive storybook from the award-winning lead designer of Monument Valley about the heart-racing highs and... | Read more »
Purrfect Date (Games)
Purrfect Date 1.0 Device: iOS Universal Category: Games Price: $4.99, Version: 1.0 (iTunes) Description: Cats are a lil’ like marmite. Either you absolutely head-over-heels love’ em, or… nahhh, who are we kidding?! Everyone... | Read more »
More monsters to collect and evolve in C...
A laid-back mix of RPG and TCG, Creature Quest is all about building your deck, evolving your creatures and winning in battle. It’s the creation of VC Mobile, set up by Might and Magic producer Jon Van Caneghem. There are elements of that classic... | Read more »
Check out this awesome hands-on with the...
Well, PlayerUnknown's Battlegrounds has come out on mobile. This isn't a clone, this isn't a riff on the battleroyale mechanics of the game, it's the official mobile port by Tencent. But there's a little bit of a hitch. [Read more] | Read more »

Price Scanner via MacPrices.net

B&H offers $200 discount on Silver 15″ Ma...
B&H Photo has Silver 15″ Apple MacBook Pros on sale for $200 off MSRP. Shipping is free, and B&H charges sales tax for NY & NJ residents only: – 15″ 2.8GHz Touch Bar MacBook Pro Silver (... Read more
12″ Apple iPad Pro Sale of the Year! Models u...
B&H Photo has 12″ #iPad Pros on sale for up to $150 off MSRP. Shipping is free, and B&H charges sales tax in NY & NJ only: – 12″ 64GB WiFi iPad Pro: $719 $80 off MSRP – 12″ 256GB WiFi... Read more
Deals on 32GB 9″ iPads: Up to $50 off MSRP, s...
B&H Photo has 2017 9.7″ 32GB iPads on sale for $299 including free shipping plus NY & NJ sales tax only. Their price is $30 off MSRP, and it’s currently the lowest price available for these... Read more
15″ 2.2GHz Retina MacBook Pro available for o...
Apple has Certified Refurbished 15″ 2.2GHz Retina MacBook Pros available for $1699. That’s $300 off MSRP for this model, and it’s the lowest price available for a 15″ MacBook Pro currently offered by... Read more
13″ 3.1GHz/256GB Silver Touch Bar MacBook Pro...
Amazon has the Silver 13″ 3.1GHz/256GB Touch Bar MacBook Pro (MPXX2LL/A) on sale for $1649.99 including free shipping. Their price is $150 off MSRP, and it’s the lowest price available for a new 13″... Read more
Saturday Sale: Amazon offers 13″ 1.8GHz/256GB...
Amazon has the 13″ 1.8GHz/256B Apple MacBook Air on sale today for $250 off MSRP including free shipping: – 13″ 1.8GHz/256GB MacBook Air (MQD42LL/A): $949.99, $250 off MSRP Their price is the lowest... Read more
Roundup of Apple Certified Refurbished 12″ Ma...
Apple has Certified Refurbished 2017 12″ Retina MacBooks available for $200-$240 off the cost of new models. Apple will include a standard one-year warranty with each MacBook, and shipping is free.... Read more
Apple offers Certified Refurbished 10″ and 12...
Apple is now offering Certified Refurbished 2017 10″ and 12″ iPad Pros for $100-$190 off MSRP, depending on the model. An Apple one-year warranty is included with each model, and shipping is free: –... Read more
Apple Canada offers Certified Refurbished Mac...
 Canadian shoppers can save up to $560 on the purchase of a 2017 current-generation MacBook Pro, MacBook, or MacBook Air with Certified Refurbished models at Apple Canada. Apple’s refurbished prices... Read more
Sale! 13″ MacBook Airs for up to $180 off MSR...
B&H Photo has 13″ MacBook Airs on sale for $50-$120 off MSRP. Shipping is free, and B&H charges sales tax for NY & NJ residents only: – 13″ 1.8GHz/128GB MacBook Air (MQD32LL/A): $899, $... Read more

Jobs Board

*Apple* Store Leader - Retail District Manag...
Job Description:Job SummaryAs more and more people discover Apple , they visit our retail stores seeking ways to incorporate our products into their lives. It's your Read more
*Apple* Retail - Multiple Positions - Apple,...
Job Description:SalesSpecialist - Retail Customer Service and SalesTransform Apple Store visitors into loyal Apple customers. When customers enter the store, Read more
*Apple* Retail - Multiple Positions - Apple,...
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* Retail - Multiple Positions - Apple,...
Job Description:SalesSpecialist - Retail Customer Service and SalesTransform Apple Store visitors into loyal Apple customers. When customers enter the store, Read more
*Apple* Retail - Multiple Positions - Apple,...
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.