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From: John Phillips (phillips_at_[hidden])
Date: 2007-03-26 09:59:03

   I'll get the ball rolling on the question of whether building a
compile time only library is a good choice. My current thinking is yes,
it is. I have three basic reasons for this, so I'll list them all below.

   If the library is going to include a runtime component, it would be
for unit conversions on the fly. Various implementation details for this
have been discussed in the past, but they always boil down to choosing
some annointed set of units that the program will really use, and
converting anything that is not in those units to the preferred system
(usually during I/O). If answers for users are desired in something
other than the preferred system, this conversion is done after the
calculations are finished.

   The goal of this approach is to provide automatic unit conversion,
and thereby simplify code, while minimizing the impact on program
performance. In some settings I think this is a very usable answer to
the problem. However, for my work in specific I do not think it is a
good choice.

   My first problem is with code simplification. In theoretic physics
(which is my background), unit systems are chosen based on the problem
at hand. The goal is always to simplify the notation and to make
reasoning about the problem more natural. If this is done successfully,
expressing the resulting equations in code is cleaner and more obvious.
For example, the units chosen for many electro-dynamic calculations lead
to equations that have no constant multipliers. Lack of these
multipliers means no magic numbers hiding in the code, so no way to get
confused by said magic numbers. In the part of the code that is the
actual calculation, this leads to simplicity and easier debugging, as
well as eliminating a few multiplications or divisions.

   My second problem is with numerical accuracy and stability. The
distribution of floating point representable numbers inside the set of
the reals is not uniform. Nor is the relative error due to round off
uniformly distributed. In most cases, the relative error is minimized
close to 1.0. (See references such as Accuracy and Stability of
Numerical Algorithms by Nicholas Higham for details.) Thus, it is almost
always advisable to use units where the quantities in the code are in
the range around 1.0 in numerical calculations. There is no silver
bullet for numerical stability any more than there is for any other
coding issue, but this is sually good advice. In some cases, a wise
choice of units can even make a problem cease to use floating point
arithmetic and become a purely integer arithmetic problem (An example
would be monte carlo simulation of spontaneous magnetization using the
Ising model. More generally, it happens in situations where the discrete
nature of quantum states is important.). Obviously, since integer
arithmetic is exact, this eliminates all concerns with round off issues.

   The third problem I see is one of efficiency. In a well chosen unit
system there are very few multiplicative constants in the equations that
describe the evolution and state of the system. (Systems that manage to
have few or no constants and where the measurements of interest are of
order 1.0 are sometimes called natural units for the problem.) In the
primary calculation of a tight loop, every multiplication or division I
can avoid is important. If all calculations actually happen in some
pre-determined, but not natural for my problem set of units these
multiplications and divisions cannot be avoided, and in some cases
become the dominant part of the calculation's use of resources. For
simulation work where doubling the operation count on the main loop can
be the difference between a project that can be done and one that can't,
this is unacceptable.

   This is my opinion, and the reasons for it. In my work I would not be
able to use a system that imposes conversions. Nor would I suggest such
a system to my students.

   I think a system with conversions might be built on top of the
library Matthias and Steven have provided (though I'm not sure, since I
haven't tried), but one with conversions would be very hard to use as a
platform for those of us who need to not have conversions. Thus I think
the chosen design is a good one.


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