Comparison tolerance
⎕CT

Comparison tolerance (⎕CT
) is the parameter governing tolerant comparison, an inexact form of comparison used to mitigate the impact of floatingpoint rounding error on programs. Two numbers are considered equal when their relative difference is smaller than the comparison tolerance, which is accessed with the system variable ⎕CT
. In addition to the comparison functions, tolerance applies to Match and Not Match, Floor, Ceiling, and Modulus, and search functions defined in terms of Match (not Interval Index).
In an early talk Ken was explaining the advantages of tolerant comparison. A member of the audience asked incredulously, "Surely you don't mean that when A=B and B=C, A may not equal C?" Without skipping a beat, Ken replied, "Any carpenter knows that!" and went on to the next question.
—Paul Berry^{[1]}
The rule for tolerant comparison is that numbers a
and b
are equal when
(ab) ≤ ⎕CT×(a)⌈(b)
where ≤
is evaluated intolerantly. This means that the allowed difference between the numbers increases as they become larger in magnitude (a relative tolerance). Comparison with zero is intolerant: only zero can equal zero. A typical value for ⎕CT
is 1e¯14
, which is usually large enough to accomodate multiple iterations of doubleprecision rounding (which introduces error on the order of 1e¯16
) while being far smaller than typical precision of realworld measurements.
Comparison tolerance was available in some form since APL\360, where it was described as a "fuzz" (on the suggestion of Larry Breed^{[2]}) applied to some functions. The value ⎕CT
to control it was defined with the introduction of system variables in APL.SV. The formula now used for tolerant comparison was proposed by Dick Lathwell in 1976^{[3]} and later introduced in SHARP APL by Robert Bernecky and others^{[4]} and included in the APL standard ISO 8485:1989.^{[5]} However, tolerant comparison is not supported in many newer APLs such as ngn/apl, dzaima/APL, and Kap.
The application of nonzero comparison tolerance to search functions presents problems for standard hashbased search methods.^{[6]}^{[7]} Many algorithms can be sped up by setting ⎕CT←0
, thus disabling tolerant comparison.
External links
 Tolerated Comparison part 1 and part 2 by Marshall Lochbaum
Documentation
References
 ↑ Roger Hui. Ken Iverson Quotations and Anecdotes. 20050930.
 ↑ Adin Falkoff, and Ken Iverson. The Evolution of APL (web). ACM SIGPLAN Notices, Volume 13, Number 8. 197808.
 ↑ Dick Lathwell. APL comparison tolerance at APL76 (also reproduced in SATN23).
 ↑ Robert Bernecky. "Comparison Tolerance". SATN23. 19770610.
 ↑ Adin Falkoff and D. L. Orth. "Development of an APL standard" at APL79.
 ↑ Roger Hui. "Hashing for Tolerant IndexOf" at Dyalog '10.
 ↑ Roger Hui. "Tolerant Unique" (materials (1.5 MB), video (27 mins)) at Dyalog '17.