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Re: Quotation concerning rational arithmetic
- Subject: Re: Quotation concerning rational arithmetic
- From: Guy.Steele at CMU-10A
- Date: Tue ,30 Dec 80 12:12:00 EDT
I stumbled across this familiar article again quite coincidentally,
looking for something else: "Computer Symbolic Math and Education:
A Radical Proposal" by David R. Stoutemeyer (U. Hawaii). Excerpts
follow. All emphases (*'s indicate italics) are Stoutemeyer's.
"Now for the most damnatory indictments of commonly-taught languages:
10. *The numbers and their arithmetic do not correspond
to those generally taught in schools!*
11. *The numbers and their arithmetic do not correspond
to those used in everyday life!*
"The limited-precision integer arithmetic of these languages is
bad enough in these regards, even without its usual overflow asymmetry
induced by 2's complement arithmetic. For the floating-point
arithmetic of these languages, we can add the indictment:
12. *Few other than the very best numerical analysts
fully understand the implications!*
"In contrast, the arithmetic that students learn in elementary
1. indefinite-precision rational arithmetic.
2. rounded and exact indefinite-precision decimal-fraction
"True, in high-school chemistry or physics students may learn
scientific notation, which could be regarded as indefinite-magnitude,
arbitrary-but-fixed-precision, rounded-decimal arithmetic. In contrast,
the floating-point arithmetic of commonly-taught languages is finite
magnitude, with only 1 to 3 alternative precisions, usually chopped
nondecimal. All of these internal differences fom true scientific
notation have external manifestations which are baffling to most
"Admittedly, extended sequences of indefinite-precision arithmetic
operations on experimental data suffers an unjustifiable growth in digits,
but to that one can respond:
1. Render onto floating-point arithmetic that which one must.
2. Render onto more rational arithmetic all that one can.
"Perhaps if floating-point computation becomes a choice rather than
an imposition, users will regard floating-point with more of the caution
"Perhaps indefinite-precision rational and decimal-fraction arithmetic
are inevitably less efficient than their respective finite-precision
floating-point and integer counterparts. However:
1. The difference in efficiency would greatly decrease if the
indefinite-precision arithmetics were microcoded or hardwired
as are their finite-precision counterparts in most computers.
2. ... [remark on "floating-slash arithmetic"]
3. Even if the indefinite-precision arithmetic is substantially
slower, computing has become so inexpensive that for the
computational needs of most people, *the cost of indefinite-
precision computation is negligible compared to the labor of
assessing results done in an unnatural arithmetic*.
"How negligible do computing costs have to become before software
and hardware designers abandon this historical obsession with efficiency?
[It'll never happen! But anyway... --GLS] If a certain computation
costs 10 times as much in rational arithmetic as in floating-point,
and the latter method was deemed worthwhile a few years ago when
computer costs were more than 10 times as much, is it not worthwhile now
to do the computation in a more humane arithmetic?
"In the early days of computers, scientific computation was usually
done using binary fixed-point fractions having magnitudes restricted
to lie between 0 and 1. The wide-spread acceptance of floating-point
brought substantially greater convenience for a large loss in efficiency.
For most work, computer costs have now decreased enough to justify
another such step in favor of human understanding. Those who cling
to efficiency-worship should defend fixed-point fractions rather
So how about it, fixed-point fans??