The Shape of Code

The numbers system I am developing attempts to match numeric literals contained in a file against a database of interesting numbers. One of the things I did to quickly build a reasonably sized database of reliable values was to extract numeric literals from a few well known programs that I thought I could trust.

R is a widely used statistical package and Maxima is a computer algebra system with a long history. Both contain a great deal of functionality and are actively maintained.

To my surprise the source code of both packages contain a large variety of different literal values for , or to be exact the number of digits contained in the literals varied by more than I expected. In the following table the value to the left of the representation is the number of occurrences; values listed in increasing literal order:

     Maxima                              R
   2 3.14159
                                      14 3.141592
   1 3.1415926
   1 3.14159265                        2 3.14159265
   3 3.1415926535
   4 3.14159265358979
  14 3.141592653589793
   3 3.1415926535897932385             3 3.1415926535897932385
   9 3.14159265358979324
                                       1 3.14159265359
                                       1 3.1415927
                                       1 3.141593

The comments in the Maxima source led me to believe that some thought had gone into ensuring that the numerical routines were robust. Over 3/4 of the literal representations of have a precision comparable to at least that of 64-bit floating-point (I’m assuming an IEEE 754 representation in this post).

In the R source approximately 2/3 of the literal representations of have a precision comparable to that of 32-bit floating-point.

Closer examination of the source suggests one reason for this difference. Both packages make heavy use of existing code (translated from Fortran to Lisp for Maxima and from Fortran to C for R); using existing code makes good sense and because of its use in scientific and engineering applications many numerical libraries have been written in Fortran. Maxima has adapted the slatec library, whereas the R developers have used a variety of different libraries (e.g., specfun).

How important is variation in the representation of Pi?

• A calculation based on a literal that is only accurate to 32-bits is likely to be limited to that level of accuracy (unless errors cancel out somewhere).
• Inconsistencies in the value used to represent Pi are a source of error. These inconsistencies may be implicit, for instance literals used to denote a value derived from such as often seem to be be based on more precise values of Pi than appear in the code.

The obvious solution to this representation issue of creating a file containing definitions of all of the frequently used literal values has possible drawbacks. For instance, numerical accuracy is a strange beast and increasing the precision of one literal without doing the same for other literals appearing in a calculation can sometimes reduce the accuracy of the final result.

Pulling together existing libraries to build a package is often very cost effective, but numerical accuracy is a slippery beast and this inconsistent usage of literals suggests that developers from these two communities have not addressed the system level consequences of software reuse.

Developers and testers rarely put any thought into working out the likely distribution of numeric values (final or intermediate) computed during the execution of the code they write.

The likely value of a variable is useful to know in a number of situations, including optimizing code (should it prove to be necessary) for the common case and testing (what distribution of input values are needed to be confident that all paths through a program are exercised?)

The answer for the ’simple’ distributions is actually more complicated to work with than the more ‘complicated’ distributions. For instance, the sum of two independent values having a normal distributions is a normal distribution and the sum of two Poisson distributions is also a Poisson distribution.

What if the values are uniformly distributed? If two independent, randomly chosen, uniformly distributed, variables, are added what is the distribution of the result? For instance, if the values of X and Y are independent of each other and take on any value between 0 and 9, with equal likelihood, what is the most (and least) likely value of X+Y?

Warning: Information spoilers follow.

You are probably thinking that the result will also be uniformly distributed and indeed it would be if the range of values taken by X and Y did not overlap. When the possible range of values overlap exactly the answer is the triangular distribution, with the mostly likely result being 9 and the least likely results being 0 and 18.

The variance of the actual result distribution is approximately six times smaller than the original distribution, meaning that the common cases occupy a much narrower value range. This value range ‘narrowing’ goes someway towards helping to explain the surprising discovery that during program execution a small set of (integer and floating) values often occur with such regularity that it might be worth cpu arithmetic units remembering previous operands and their results (i.e., to save time by returning the result rather than recalculating it).