TweetFollow Us on Twitter

FORTRAN Accuracy
Volume Number:8
Issue Number:5
Column Tag:Jörg's Folder

Accuracy & Speed in FORTRAN Compilers

MacFortran II compared with Language Systems Fortran on the Quadra.

By Jörg Langowski, MacTutor Regular Contributing Author

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

Absoft has recently announced MacFORTRAN II version 3.1, which is supposed to run very fast on Quadras. Since I reviewed Language Systems Fortran not long ago, I was curious how big the speed difference between these two compilers really was. So again, I went back to my benchmark programs. Two of them are (in)famous: the Whetstone benchmark, which tests primarily integer arithmetic and depends a lot on the efficiency of subroutine calling, because it works by calling lots of small routines repeatedly from a loop; and the Linpack, which tests the efficiency of floating point matrix operations using a package of linear algebra routines which are quite useful by themselves.

Furthermore, I found a third useful program between the demos included on the LS Fortran diskettes: it is called Paranoia (mentioned already in V8#1), and checks the consistency of the floating point arithmetic. Things are checked like whether 3*4 equals 4*3, 1+0 =1, 1*0 = 0 etc. These may seem trivial to you, but as you will see, they aren’t when you are dealing with Fortran compilers.

Numerical accuracy tests

I’d like to describe the Paranoia benchmark first, and its results on the two Fortran compilers. The program was original written in Basic and then ported to Fortran; the person who was responsible for the version I used is Richard Karpinski at the Computer Center of the University of California (San Francisco, CA 94143-0704). His description of the program follows:

===== PARANOIA =====

A test program that evaluates the quality of a numerics environment. Note that numerics quality depends on the compiler used as well as the underlying system hardware and software.

Paranoia is a rather large program, devised by Prof. Kahan of Berkeley, to explore the floating point system on your computer. These files are being redistributed as encouraged in the note copied below.

Paranoia.f single precision Fortran, 29 Jan. 1986

Dear Fellow Paranoid,

We ask:

1. Please distribute these programs as widely as you like. Be sure to include as much help as possible. Please include these requests as well.

2. Please let me know if you can provide it on other media. Other potential users may be saved from retyping 2500 lines of code if you can provide the programs on a Whizbang 200 disk/tape/notched-stick. It is reasonable to ask money for this.

3. PLEASE let me know about any source changes that you made to make Paranoia work on your system. The exact model/version/date of your system is quite important in order to understand the changes. If you send the new version, please indicate where the changes occur. Machine readable is preferred.

4. Please send me your results. Note which version of Paranoia gave the results and what machine/language model/version etc. they apply to.

5. Suggestions and comments are always welcome.

Thank you for participating in this unique investigation of contemporary floating-point arithmetic. Your help is vital to the project.

Yours truly,

Richard Karpinski

Modula Assured Quality Software, 6521 Raymond Street

Oakland, CA 94609

ps: Send a stamped, self addressed envelope to the above address to get a sheet on the current status of the project at any time.

The program is too long to print here (104 K, enclosed in compressed form on the source code disk). Just to give you a feeling for what it does, here are some typical lines of code:

 IF (0.0E0+0.0E0 .EQ. 0.0E0 .AND.
     1     1.0E0-1.0E0 .EQ. 0.0E0 .AND.
     1     1.0E0       .GT. 0.0E0 .AND.
     1     1.0E0+1.0E0 .EQ. 2.0E0) GOTO 930
920FORMAT(‘ FAILURE: violation of  0+0=0  or  1-1=0  or  
     1  1>0  or 1+1 = 2.’)

You wouldn’t believe that these expressions could be evaluated the wrong way, but there must be a reason why these tests are done

Less obvious are some other mistakes that are tested by Paranoia. For instance, the basic arithmetic operations should carry some ‘guard digits’, that is, some more digits that are the representation of the floating point number in memory. Also, they should be correctly rounded in a way that doesn’t introduce inconsistencies. It is not trivial to design floating point arithmetic in such a way that roundoff errors do not affect the consistency of the result. This is seen when the Paranoia benchmark is compiled with the Language Systems (v3.0) and Absoft (v3.1.2) compilers, using various settings of the optimizations and other options.

First, the code produced by the Language Systems compiler, at any optimization level, does not produce any error messages at all, so the arithmetic seems to behave in a consistent way. The rounding conforms to the IEEE p754 standard.

Compiling the benchmark with Absoft MacFortran 3.1.2, with and without the basic optimizations, produces a lot of error messages, like these:

(1-u1)-1/2 < 1/2 is false, so this program may malfunction.

Multiplication lacks a guard digit violating 1*x = x .

Division lacks a guard digit violating x/1 = x .

Computed value of 1/1.00...001 is not less than 1 .

The errors are typical of floating point implementations where roundoff is not handled correctly; it seems to be mainly due to the fact that Absoft tries to keep intermediate values in registers as long as possible. These errors are not as serious as they look, since they are mainly deviations of one unit in the last significant digit of the single-precision numbers used in the test. Some of the error messages are actually a consequence of the fact that the program computes the machine precision at the beginning, from the minimum distance of two floating point numbers close to 1.0. The Absoft-optimized code without any further options comes up with a much too small value for this distance, and therefore some later tests, where this machine precision is used, report errors as well.

All these considerations may seem quite academic to you, because we’re really dealing with very small differences between the computed and the ‘true’ result, of the order of 10-8 or so. But problems where the precise computation of a small difference between two large numbers is important are not that rare, and in those cases such errors will quickly blow up. It is for this reason that the IEEE standard for floating point arithmetic has been developed, and that Apple has implemented the SANE package which follows the IEEE proposal.

There is a remedy, however: one can force the Absoft compiler (with the -e option) to use extended precision in all subexpression calculations, and store values to memory from the registers and re-load them after every assignment. That seems to make the Paranoia benchmark work correctly, and - almost - no more errors are reported. Only now, there are some numbers x and y where x*y is not equal to y*x! (See listing - I reprinted the part of the Paranoia benchmark that reported the error). The only way around this error, and the way to make the benchmark run in a fully consistent fashion, is to compile the code non-optimized and with the -e option. In this case, however, Absoft MacFortran loses all its speed advantage over Language Systems Fortran.

Speed tests with Whetstone and Linpack

To compare the speed figures for the Whetstone and Linpack benchmarks, we should therefore use the Absoft compiler at least with the -e option, and maybe also drop the optimizations. Since now both LSF and ABF compilers exist in versions that support 68040 code, I’ve compared the benchmarks both on a Mac IIx and on a Quadra 700. LSF was always run at the highest optimization level, since the arithmetic seems to be OK at that level; for Absoft, I used the basic optimizations (-O) with and without the extended precision option (-e).

Linpack (single precision)

LSF 030/882 code (Mac IIx) 0.12 MFlops

(Quadra 700) 1.32 MFlops

LSF 040 code (Quadra 700) 1.37 MFlops

ABF 030/882 code -O (Quadra 700) 1.30 MFlops

ABF 030/882 code -O -e (Quadra 700) 1.27 MFlops

ABF 040 code -O (Quadra 700) 1.60 MFlops

ABF 040 code -O -e (Quadra 700) 1.58 MFlops

Whetstone (single precision)

LSF 030/882 code (Mac IIx) 974 K

(Quadra 700) 3976 K

LSF 040 code (Quadra 700) 3984 K

ABF 030/882 code -O (Mac IIx) 1215 K

ABF 030/882 code -O -e (Mac IIx) 955 K

ABF 030/882 code -O (Quadra 700) 4051 K

ABF 030/882 code -O -e (Quadra 700) 3829 K

ABF 040 code -O (Quadra 700) 5864 K

ABF 040 code -O -e (Quadra 700) 5381 K

The essence of these figures is that there is no significant execution speed difference between Absoft and Language Systems Fortran on a Mac IIx, if you make sure that both generate arithmetically consistent code. On the Quadra, however, the fact shows up that Language Systems really creates almost the same code for 68030 and 68040 systems; no additional speed is gained using the -68040 option, while Absoft gains 24% on the Linpack benchmark and 40% on the Whetstone benchmark when 68040 code generation is switched on. All in all, Absoft code on the Quadra is 15% faster than LSF code for the Linpack and 35% for the Whetstone benchmark.

If you are running Fortran on anything other than a Quadra, LSF is really the only choice, given its excellent support of the Macintosh user interface, System 7 features such as AppleEvents, 100% VAX compatibility going into such details as the syntax of I/O statements, and its documentation. For pure numeric performance on a Quadra, Absoft still features faster execution. Still, I’d like to see Absoft’s ‘threaded math library’ for the 68040 incorporated into LSF; that would be my dream system.

Forth news

For the Forth readers of this column, I recently got news from the creators of two of the best public domain Forth development systems for the Macintosh. You read in MacTutor, Vol. 8, No. 4, August 1992 issue about the update of Mops (2.1), the object-oriented Forth implemented by Michael Hore. He now sent me the tutorial that he is distributing together with Mops, a 340K Microsoft Word file.

I’m not going to print the Mops Tutorial here, obviously. But if you need it and don’t find it on any of the obvious sources (ftp from or, drop me some bytes at langowski@frembl.bitnet and I’ll be happy to mail the file to you.

Another message came from Chris Heilman, who created Pocket Forth:

From: N%”heilman@PC.bitnet”

To: angowski@FREMBL51.bitnet


Subj: Apple Events in Pocket Forth


Hello again. How have things been going for you? I read your latest article in MacTutor’s June (congrats by the way on getting it back together) about Apple Events in LS fortran. I’ve been working along exactly those lines.

Just two weeks ago I completed and released Pocket Forth 6. While release 5 was capable of handling high-level events, PF6 makes it easy and fun. Two new words, AE: and ;AE are used to define event handlers. Another word, ,S takes a 4 character token from the input stream and puts a 32 bit number on the stack. Use them like this:

  ,s misc ,s dosc AE: blah, blah blah, .... ;AE

This installs the dosc event handler into a list. The dictionary is then saved. The next time the program starts, it handles dosc events. Of course the four required events are handled automatically, but their actions can be changed at any time.

But wait, that’s not all! I’ve added SANE floating point for any numeric token with an ‘E’ or decimal point, new icons, drag&drop file loading and a rewritten manual in TeachText format.

I’d like to send you a copy, so expect a disk soon. I’d email a copy to you, but our system has been choking on loooong mail lately, and I’ve gotten real frustrated sending 200K files 3 or 4 times at 2400 bps.

Later, Chris Heilman

I’m looking forward to seeing the new version of Pocket Forth. Expect some lines in this column when I’ve reviewed it.

Example: check if multiplication commutes

Cvery short main program
 Implicit None

 REAL R9, X, X9, Y, Y9, Z, Z9

 FP0 = 0
 FP1 = 1
 FP2 = FP1+FP1
 FP3 = FP2+FP1
 HALF = FP1 / FP2
 ULPMIN = 5.96046440E-08
2920  WRITE(*,2921) NUMTRY
2921  FORMAT(/’ Does multiplication commute?’,
     1   ‘ Testing if  x*y = y*x  for’, I4,’ random pairs:’)
2930  R9 =  SQRT(FP3)
 I = NUMTRY + 1
 X9 = FP0 / FP3
2960  CALL RANDOM (X, Y, X9, R9)
 IF (I .GT. 0 .AND. Z9 .EQ. FP0) GOTO 2960
2970  IF (I .GT. 0) GOTO 3000
2980  X9=FP0+HALF/FP3
     1 -((ULPPLS+ULPMIN)+FP0)*(FP0+HALF/FP3)
 IF (Z9 .NE. FP0) GOTO 3000
2990  FORMAT(‘ No failure found in ‘,I4,’ randomly chosen pairs.’)
 WRITE(*, 3001) X9, Y9
 WRITE(*, 3002) Z, Y, Z9
 WRITE(*, 3003) NN
3001  FORMAT(‘ DEFECT:  x*y = y*x  violated at  x = ‘,E15.7,’, y = ‘,
     1 E15.7)
3002  FORMAT(‘  x*y =’,E15.7,’,  y*x =’,E15.7,’,  x*y-y*x =’,E15.7)
3003  FORMAT(‘    ... pair no.’, I4)
 REAL X, Y, X9, R9
2950  X=X9+R9


Community Search:
MacTech Search:

Software Updates via MacUpdate

Logic Pro X 10.1.1 - Music creation and...
Apple Logic Pro X is the most advanced version of Logic ever. Sophisticated new tools for professional songwriting, editing, and mixing are built around a modern interface that's designed to get... Read more
VLC Media Player 2.2.0 - Popular multime...
VLC Media Player is a highly portable multimedia player for various audio and video formats (MPEG-1, MPEG-2, MPEG-4, DivX, MP3, OGG, ...) as well as DVDs, VCDs, and various streaming protocols. It... Read more
Sound Studio 4.7.8 - Robust audio record...
Sound Studio lets you easily record and professionally edit audio on your Mac. Easily rip vinyls and digitize cassette tapes, or record lectures and voice memos. Prepare for live shows with live... Read more
LibreOffice - Free, open-source...
LibreOffice is an office suite (word processor, spreadsheet, presentations, drawing tool) compatible with other major office suites. The Document Foundation is coordinating development and... Read more
Freeway Pro 7.0.3 - Drag-and-drop Web de...
Freeway Pro lets you build websites with speed and precision... without writing a line of code! With its user-oriented drag-and-drop interface, Freeway Pro helps you piece together the website of... Read more
Cloud 3.3.0 - File sharing from your men...
Cloud is simple file sharing for the Mac. Drag a file from your Mac to the CloudApp icon in the menubar and we take care of the rest. A link to the file will automatically be copied to your clipboard... Read more
Cyberduck 4.6.5 - FTP and SFTP browser....
Cyberduck is a robust FTP/FTP-TLS/SFTP browser for the Mac whose lack of visual clutter and cleverly intuitive features make it easy to use. Support for external editors and system technologies such... Read more
Firefox 36.0 - Fast, safe Web browser. (...
Firefox for Mac offers a fast, safe Web browsing experience. Browse quickly, securely, and effortlessly. With its industry-leading features, Firefox is the choice of Web development professionals and... Read more
Thunderbird 31.5.0 - Email client from M...
As of July 2012, Thunderbird has transitioned to a new governance model, with new features being developed by the broader free software and open source community, and security fixes and improvements... Read more
VOX 2.4 - Music player that supports man...
VoxIt just sounds better! The beauty is in its simplicity, yet behind the minimal exterior lies a powerful music player with a ton of features & support for all audio formats you should ever need... Read more

Get The Whole Story – Lone Wolf Complete...
Get The Whole Story – Lone Wolf Complete is Now Available and On Sale Posted by Jessica Fisher on February 27th, 2015 [ permalink ] Universal App - Designed for iPhone and iPad | Read more »
Who Wore it Best? The Counting Dead vs....
Like it or not, the “clicker” genre, popularized by cute distractions like Candy Box and Cookie Clicker, seems like it’s here to stay. So Who Wore it Best? takes a look at two recent examples: The Counting Dead and AdVenture Capitalist. | Read more »
Card Crawl, the Mini Deck Building Game,...
Card Crawl, the Mini Deck Building Game, is Coming Soon Posted by Jessica Fisher on February 27th, 2015 [ permalink ] Tinytouchtales and Mexer have announced their new game, | Read more »
Witness an all new puzzle mechanic in Bl...
Well, BlastBall MAX is not one of those games and is bucking trends such as timers, elements of randomness, and tacked-on mechanics in favor of pure puzzle gameplay. When you first boot up the game you’ll see a grid made up of squares that are each... | Read more »
This Princess Has a Dragon and She isn’t...
This Princess Has a Dragon and She isn’t Afraid to Useit. | Read more »
Mecha Showdown Review
Mecha Showdown Review By Lee Hamlet on February 27th, 2015 Our Rating: :: IN A SPINUniversal App - Designed for iPhone and iPad Mecha Showdown replaces traditional buttons with a slot machine mechanic in this robot fighting game,... | Read more »
Reliance Games and Dreamworks Unveil Rea...
Reliance Games and Dreamworks Unveil Real Steel Champions Posted by Ellis Spice on February 27th, 2015 [ permalink ] Reliance Games and Dreamworks have announced that a third game in | Read more »
Sum Idea Review
Sum Idea Review By Jennifer Allen on February 27th, 2015 Our Rating: :: TAXING NUMBERSUniversal App - Designed for iPhone and iPad Sum Idea is a fairly charming but taxing puzzle game.   | Read more »
A New Badland Update Brings Daydream Lev...
A New Badland Update Brings Daydream Levels to Co-Op Posted by Ellis Spice on February 27th, 2015 [ permalink ] Universal App - Designed for iPhone and iPad | Read more »
Slashing Demons Review
Slashing Demons Review By Lee Hamlet on February 27th, 2015 Our Rating: :: IT'S A LONG WAY TO THE TOPUniversal App - Designed for iPhone and iPad Slashing Demons lacks the depth or scope to take it beyond the point of being just... | Read more »

Price Scanner via

Apple CEO Tim Cook to Deliver 2015 George Was...
Apple CEO Tim Cook will deliver the George Washington University’s Commencement address to GWU grads on May 17, at which time he will also be awarded an honorary doctorate of public service from the... Read more
Apple restocks refurbished Mac minis for up t...
The Apple Store has restocked Apple Certified Refurbished 2014 Mac minis, with models available starting at $419. Apple’s one-year warranty is included with each mini, and shipping is free: - 1.4GHz... Read more
Save up to $50 on iPad Air 2s, NY tax only, f...
 B&H Photo has iPad Air 2s on sale for $50 off MSRP including free shipping plus NY sales tax only: - 16GB iPad Air 2 WiFi: $469.99 $30 off - 64GB iPad Air 2 WiFi: $549 $50 off - 128GB iPad Air 2... Read more
16GB iPad Air 2 on sale for $447, save $52
Walmart has the 16GB iPad Air 2 WiFi on sale for $446.99 on their online store for a limited time. Choose free shipping or free local store pickup (if available). Sale price for online orders only,... Read more
iMacs on sale for up to $205 off MSRP
B&H Photo has 21″ and 27″ iMacs on sale for up to $205 off MSRP including free shipping plus NY sales tax only: - 21″ 1.4GHz iMac: $1029 $70 off - 21″ 2.7GHz iMac: $1199 $100 off - 21″ 2.9GHz... Read more
Apple Takes 89 Percent Share of Global Smartp...
According to the latest research from Strategy Analytics, global smartphone operating profit reached US$21 billion in Q4 2014. The Android operating system captured a record-low 11 percent global... Read more
New Travel Health App “My Travel Health” iOS...
Rochester, Minnesota based Travel Health and Wellness LLC has announced that its new iOS app help safeguard the user’s health when traveling abroad — “My Travel Health” is now available on the Apple... Read more
Sale! MacBook Airs for up to $115 off MSRP
B&H Photo has MacBook Airs on sale for up to $100 off MSRP. Shipping is free, and B&H charges NY sales tax only: - 11″ 128GB MacBook Air: $799 100 off MSRP - 11″ 256GB MacBook Air: $999 $100... Read more
15-inch 2.0GHz Retina MacBook Pro (refurbishe...
The Apple Store has Apple Certified Refurbished previous-generation 15″ 2.0GHz Retina MacBook Pros available for $1489 including free shipping plus Apple’s standard one-year warranty. Their price is... Read more
Wither The iPad mini? End Of The Road Imminen...
AppleDailyReport’s Dennis Sellers predicts that the iPad mini is going to be left to wither on the vine, as it were, and then just allowed to fade away — a casualty of the IPhone 6 Plus and other... Read more

Jobs Board

Sr. Technical Services Consultant, *Apple*...
**Job Summary** Apple Professional Services (APS) has an opening for a senior technical position that contributes to Apple 's efforts for strategic and transactional Read more
Event Director, *Apple* Retail Marketing -...
…This senior level position is responsible for leading and imagining the Apple Retail Team's global engagement strategy and team. Delivering an overarching brand Read more
*Apple* Pay - Site Reliability Engineer - Ap...
**Job Summary** Imagine what you could do here. At Apple , great ideas have a way of becoming great products, services, and customer experiences very quickly. Bring Read more
*Apple* Solutions Consultant - Retail Sales...
**Job Summary** The ASC is an Apple employee who serves as an Apple brand ambassador and influencer in a Reseller's store. The ASC's role is to grow Apple Read more
*Apple* Solutions Consultant - Retail Sales...
**Job Summary** As an Apple Solutions Consultant (ASC) you are the link between our customers and our products. Your role is to drive the Apple business in a retail Read more
All contents are Copyright 1984-2011 by Xplain Corporation. All rights reserved. Theme designed by Icreon.