About Standard Numerics (was Re: 4 hundred quadrillonth?)

Lawrence D'Oliveiro ldo at geek-central.gen.new_zealand
Sun May 24 21:39:02 EDT 2009

In message <9MWdnTfMPPrjQoTXnZ2dnUVZ_vadnZ2d at giganews.com>, Erik Max Francis 

> Lawrence D'Oliveiro wrote:
>> In message <mailman.525.1242941777.8015.python-list at python.org>,
>> Christian Heimes wrote:
>>> Welcome to IEEE 754 floating point land! :)
>> It used to be worse in the days before IEEE 754 became widespread.
>> Anybody remember a certain Prof William Kahan from Berkeley, and the
>> foreword he wrote to the Apple Numerics Manual, 2nd Edition, published in
>> 1988? It's such a classic piece that I think it should be posted
>> somewhere...
> I only see used versions of it available for purchase.  Care to hum a
> few bars?

Part I of this book is mainly for people who perform scientific, 
statistical, or engineering computations on Apple® computers. The rest is 
mainly for producers of software, especially of language processors, that 
people will use on Apple computers to perform computations in those fields 
and in finance and business too. Moreover, if the first edition was any 
indication, people who have nothing to do with Apple computers may well buy 
this book just to learn a little about an arcane subject, floating-point 
arithmetic on computers, and will wish they had an Apple.

Computer arithmetic has two properties that add to its mystery:

* What you see is often not what you get, and
* What you get is sometimes not what you wanted.

Floating-point arithmetic, the kind computers use for protracted work with 
approximate data, is intrinsically approximate because the alternative, 
exact arithmetic, could take longer than most people are willing to wait--
perhaps forever. Approximate results are customarily displayed or printed to 
show only as many of their leading digits as matter instead of all digits; 
what you see need not be exactly what you've got. To complicate matters, 
whatever digits you see are /decimal/ digits, the kind you saw first in 
school and the kind used in hand-held calculators. Nowadays almost no 
computers perform their arithmetic with decimal digits; most of them use 
/binary/, which is mathematically better than decimal where they differ, but 
different nonetheless. So, unless you have a small integer, what you see is 
rarely just what you have.

In the mid 1960's, computer architects discovered shortcuts that made 
arithmetic run faster at the cost of what they reckoned to be a slight 
increase in the level of rounding error; they thought you could not object 
to slight alterations in the rightmost digits of numbers since you could not 
see those digits anyway. They had the best intentions, but they accomplished 
the opposite of what they intended. Computer throughputs were not improved 
perceptibly by those shortcuts, but a few programs that had previously been 
trusted unreservedly turned treacherous, failing in mysterious ways on 
extremely rare occasions.

For instance, a very Important Bunch of Machines launched in 1964 were found 
to have two anomalies in their double-precision arithmetic (though not in 
single): First, multiplying a number /Z/ by 1.0 would lop off /Z/'s last 
digit. Second, the difference between two nearly equal numbers, whose digits 
mostly canceled, could be computed wrong by a factor almost as big as 16 
instead of being computed exactly as is normal. The anomalies introduced a 
kind of noise in the feedback loops by which some programs had compensated 
for their own rounding errors, so those programs lost their high accuracies. 
These anomalies were not "bugs"; they were "features" designed into the 
arithmetic by designers who thought nobody would care. Customers did care; 
the arithmetic was redesigned and repairs were retrofitted in 1967.

Not all Capriciously Designed Computer arithmetics have been repaired. One 
family of computers has enjoyed notoriety for two decades by allowing 
programs to generate tiny "partially underflowed" numbers. When one of these 
creatures turns up as the value of /T/ in an otherwise innocuous statement 

if T = 0.0 then Q := 0.0 else Q := 702345.6 / (T + 0.00189 / T);

it causes the computer to stop execution and emit a message alleging 
"Division by Zero". The machine's schizophrenic attitude toward zero comes 
about because the test for T = 0.0 is carried out by the adder, which 
examines at least 13 of /T/'s leading digits, whereas the divider and 
multiplier examine only 12 to recognize zero. Doing so saved less than a 
dollar's worth of transistors and maybe a picosecond of time, but at the 
cost of some disagreement about whether a very tiny number /T/ is zero or 
not. Fortunately, the divider agrees with the multiplier about whether /T/ 
is zero, so programmers could prevent spurious divisions by zero by slightly 
altering the foregoing statement as follows:

if 1.0 * T = 0.0 then Q := 0.0 else Q := 702345.6 / (T + 0.00189 / T);

Unfortunately, the Same Computer designer responsible for "partial 
underflow" designed another machine that can generate "partially 
underflowed" numbers /T/ for which this statement malfunctions. On that 
machine, /Q/ would be computed unexceptionably except that the product 1.0 * 
T causes the machine to stop and emit a message alleging "Overflow". How 
should a programmer rewrite that innocuous statement so that it will work 
correctly on both machines? We should be thankful that such a task is not 
encountered every day.

Anomalies related to roundoff are extremely difficult to diagnose. For 
instance, the machine on which 1.0 * T can overflow also divides in a 
peculiar way that causes quotients like 240.0 / 80.0, which ought to produce 
small integers, sometimes to produce nonintegers instead, sometimes slightly 
too big, sometimes slightly too small. The same machine multiplies in a 
peculiar way, and it subtracts in a peculiar way that can get the difference 
wrong by almost a factor of 2 when it ought to be exact because of 

Another peculiar kind of subtraction, but different, afflicts the machines 
that are schizophrenic about zero. Sets of three values /X/, /Y/ and /Z/ 
abound for which the statement

if (X = Y) and ((X - Z) > (Y - Z)) then writeln('Strange!');

will print "Strange!" on those machines. And many machines will print 
"Strange!" for unlucky values /X/ and /Y/ in the statement

if (X - Y = 0.0) and (X > Y) then writeln('Strange!');

because of underflow.

/These strange things cannot happen on current Apple computers./

I do not wish to suggest that all but Apple computers have had quirky 
arithmetics. A few other computer companies, some Highly Prestigious, have 
Demonstrated Exemplary Concern for arithmetic integrity over many years. Had 
their concern been shared more widely, numerical computation would now be 
easier to understand. Instead, because so many computers in the 1960's and 
1970's possessed so many different arithmetic anomalies, computational lore 
has become encumbered with a vast body of superstition purporting to cope 
with them. One such superstitious rule is "/Never/ ask whether floating-
point numbers are exactly equal".

Presumably the reasonable thing to do instead is to ask whether the numbers 
differ by less than some tolerance; and this /is/ truly reasonable provided 
you know what tolerance to choose. But the word /never/ is what turns the 
rule from reasonable into mere superstition. Even if every floating-point 
comparison in your program involved a tolerance, you would wish to predict 
which path execution would follow from various input data, and whether the 
different comparisons were mutually consistent. For instance, the predicates 
X < Y - TOL and Y - TOL > X seem equivalent to the naked eye, but computers 
exist (/not/ made by Apple!) on which one can be true and the other false 
for certain values of the variables. To ask "Which?" violates the 
superstitious rule.

There have been several attempts to avoid superstition by devising 
mathematical rules called /axioms/ that would be valid for all commercially 
significant computers and from which a programmer might hope to be able to 
deduce whether his program will function correctly on all those computers. 
Unfortunately, such attempts cannot succeed without failing! The paradox 
arises because any such rules, to be valid universally, have to encompass so 
wide a range of anomalies as to constitute the specifications for a 
hypothetical computer far worse arithmetically than any ever actually built. 
In consequence, many computations provably impossible on that hypothetical 
computer would be quite feasible on almost every actual computer. For 
instance, the axioms must imply limits to the accuracy with which 
differential equations can be solved, integrals evaluated, infinite series 
summed, and areas of triangles calculated; but these limits are routinely 
surpassed nowadays by programs that run on most commercially significant 
computers, although some computers may require programs that are so special 
that they would be useless on any other machine.

Arithmetic anarchy is where we seemed headed until a decade ago when work 
began upon IEEE Standard 754 for binary floating-point arithmetic. Apple's 
mathematicians and engineers helped from the very beginning. The resulting 
family of coherent designs for computer arithmetic has been adopted more 
widely, and by more computer manufacturers, than any other single design. 
Besides the undoubted benefits that flow from any standard, the principal  
benefit derived from the IEEE standard in particular is this:

/Program importability:/ Almost any application of floating-point 
arithmetic, designed to work on a few different families of computers in 
existence before the IEEE Standard and programmed in a higher-level 
language, will, after recompilation, work at least about as well on an Apple 
computer or on any other machine that conforms to IEEE Standard 754 as on 
any nonconforming computer with comparable capacity (memory, speed, and word 

The Standard Apple Numerics Environment (SANE) is the most thorough 
implementation of IEEE Standard 754 to date. The fanatical attention to 
detail that permeates SANE's implementation largely relieves Apple computer 
users from having to know any more about those details than they like. If 
you come to an Apple computer from some other computer that you were fond 
of, you will find the Apple computer's arithmetic at least about as good, 
and quite likely rather better. An Apple computer can be set up to mimic the 
worthwhile characteristics of almost any reasonable past computer 
arithmetic, so existing libraries of numerical software do not have to be 
discarded if they can be recompiled. SANE also offers features that are 
unique to the IEEE Standard, new capabilities that previous generations of 
computer users could only yearn for; but to learn what they are, you will 
have to read this book.

As one of the designers of IEEE Standard 754, I can only stand in awe of the 
efforts that Apple has expended to implement that standard faithfully both 
in hardware and in software, including language processors, so that users of 
Apple computers will actually reap tangible benefits from the Standard. And 
I thank Apple for letting me explain in this foreword why we needed that 

                                Professor W. Kahan
                                Mathematics Department and
                                Electrical Engineering and
                                Computer Science Department
                                University of California at Berkeley
                                December 16, 1987

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