The TPK algorithm is a simple program introduced by Donald Knuth and Luis Trabb Pardo to illustrate the evolution of computer programming languages. In their 1977 work "The Early Development of Programming Languages", Trabb Pardo and Knuth introduced a small program that involved arrays, indexing, mathematical functions, subroutines, I/O, conditionals and iteration. They then wrote implementations of the algorithm in several early programming languages to show how such concepts were expressed.
To explain the name "TPK", the authors referred to Grimm's law (which concerns the consonants 't', 'p', and 'k'), the sounds in the word "typical", and their own initials (Trabb Pardo and Knuth). In a talk based on the paper, Knuth said:
Knuth describes it as follows:[1]
In pseudocode:
ask for 11 numbers to be read into a sequence S reverse sequence S for each item in sequence S call a function to do an operation if result overflows alert user else print result
The algorithm reads eleven numbers from an input device, stores them in an array, and then processes them in reverse order, applying a user-defined function to each value and reporting either the value of the function or a message to the effect that the value has exceeded some threshold.
In the original paper, which covered "roughly the first decade" of the development of high-level programming languages (from 1945 up to 1957), they gave the following example implementation "in a dialect of ALGOL 60", noting that ALGOL 60 was a later development than the languages actually discussed in the paper:
As many of the early high-level languages could not handle the TPK algorithm exactly, they allow the following modifications:
sqrt(x) means the largest integer not exceeding\sqrt{x}
'TOO LARGE', output the number 999.a0,a1,\ldots,a10
10,f(10),9,f(9),\ldots,0,f(0)
f(i)
f(a[i]) with an expression equivalent to\sqrt{|ai|}+5x3
With these modifications when necessary, the authors implement this algorithm in Konrad Zuse's Plankalkül, in Goldstine and von Neumann's flow diagrams, in Haskell Curry's proposed notation, in Short Code of John Mauchly and others, in the Intermediate Program Language of Arthur Burks, in the notation of Heinz Rutishauser, in the language and compiler by Corrado Böhm in 1951–52, in Autocode of Alick Glennie, in the A-2 system of Grace Hopper, in the Laning and Zierler system, in the earliest proposed Fortran (1954) of John Backus, in the Autocode for Mark 1 by Tony Brooker, in ПП-2 of Andrey Ershov, in BACAIC of Mandalay Grems and R. E. Porter, in Kompiler 2 of A. Kenton Elsworth and others, in ADES of E. K. Blum, the Internal Translator of Alan Perlis, in Fortran of John Backus, in ARITH-MATIC and MATH-MATIC from Grace Hopper's lab, in the system of Bauer and Samelson, and (in addenda in 2003 and 2009) PACT I and TRANSCODE. They then describe what kind of arithmetic was available, and provide a subjective rating of these languages on parameters of "implementation", "readability", "control structures", "data structures", "machine independence" and "impact", besides mentioning what each was the first to do.
This shows a C implementation equivalent to the above ALGOL 60.
double f(double t)
int main(void)
This shows a Python implementation.
def f(t): return sqrt(abs(t)) + 5 * t ** 3
a = [float(input) for _ in range(11)]for i, t in reversed(list(enumerate(a))): y = f(t) print(i, "TOO LARGE" if y > 400 else y)
This shows a Rust implementation.
fn f(t: f64) -> f64
fn main