Variations in the literal representation of Pi
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.
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.
Update 6 April: After further rummaging around in the R source distribution I found that things are not as bad as they first appear. Only two of the single precision instances of listed above occur in the C or Fortran source code, the rest appear in support files (e.g., m4 scripts and R examples).