The Shape of Code

Numeric values are an integral to communication between people. What is the distribution of integer values in text/speech, and does the use of integer literals in source code have a similar distribution?

• The paper Numbers in Context: Cardinals, Ordinals, and Nominals in American English by Greg Woodin, and Bodo Winter studied the 9+ million numbers contained in the Corpus of Contemporary American English (7,744,038 integer values). The plot below shows the number of occurrences of the smaller integer values, and a fitted regression line for values in the range 1..50 (code+data):

  

  The frequency of integer values in this corpus is proportional to: .

• The paper Frequency of occurrence of numbers in the World Wide Web by Dorogovtsev, Mendes, and Gama Oliveira found that the number of web pages containing a given integer value declines as the value increases, with the decline for non-round numbers being roughly proportional to (round numbers are much more frequent than adjacent values and bias fitted models), and including all values gives (for values up to ).

Programs are an implementation of a sliver of the world in which people live, and it is to be expected that the frequency of numeric literal values in source code is highly correlated with real world frequency. Numeric values also appear in the algorithms and mathematical expressions used to create implementations. I am not aware of any studies looking at the frequency of use of numeric constants in algorithms and mathematics. As an aside, the frequency of occurrence of mathematical expressions containing a given number of operators is similar to that in C source

What are the usage characteristics of integer literals in source code (floating-point literal use is very rare outside of particular application domains)?

The plot below shows occurrences of decimal (green) and hexadecimal (blue) literals in C source (data from fig 825.1 from my C book) with a regression line fitted to values 1..50 of the decimal data (code+data):

The frequency of decimal literal values in C source is proportional to: . Adding the hexadecimal values to the model has little effect.

The paper What do developers consider magic literals? A smalltalk perspective by Anquetil, Delplanque, Ducasse, Zaitsev, Fuhrman, and Guéhéneuc studied the use of literals in Smalltalk. The plot below shows the number of occurrences of all kinds of integer literals and a fitted regression line (code+data):

The frequency of integer literal values in Smalltalk source is proportional to: .

The distribution of integer literals in both human communication and source code is well-fitted by a power law. Smalltalk appears to be the outlier, with an exponent of 1.7 vs 1.3-1.4. Perhaps it’s a sample size issue; 14,054 integer literals for Smalltalk and a million+ for the other datasets.

I had expected source code to contain a lot more zeroes/ones, relative to other values, than human communication. Zero/one are such common values that there are implicit short-cuts that people can use to express them; removing the effort/cost needed to explicitly specify them. Some programming languages specify default 0/1 values for common idioms, but C-like languages generally require explicit specification of values.