Tolerant Comparison Tolerant comparison ameliorates the ugly realities of finiteprecision representation and arithmetic. For example, 7 is not exactly equal to 100×0.07 , nor 2 exactly equal to the square of the square root of 2 : 7  100×0.07 ¯8.88178e¯16 2  (2*0.5)*2 ¯4.44089e¯16 The Dyalog APL Language Reference defines equality with a tolerance ⎕ct thus: Two numbers x and y are equal if (xy)≤⎕ct×(x)⌈(y) where ≤ is applied without tolerance. 7 = 100 × 0.07 1 2 = (2*0.5)*2 1 The tolerance ⎕ct may be assigned any value in the range
from 0 to 2*¯32 .
Historically, the upper bound 2*¯32 was chosen
so that comparisons involving Model of Tolerant Equal Tolerant comparison can be modelled by implementing the description above as a dynamic operator. The model is useful for exploring different tolerances, even those that are illegal settings for ⎕ct . teq←{⎕ct←0 ⋄ (⍺⍵)≤⍺⍺×(⍺)⌈(⍵)} 1 (0.1 teq) 0.899 0 1 (0.1 teq) 0.9 1 1 (0.1 teq) 1.1 1 1 (0.1 teq) 1.2 0 1 (0.1 teq) 0.899 0.9 1.1 1.12 0 1 1 0 Region of Tolerant Equality Adapted from R.K.W. Hui,
Hashing for Tolerant Index Of,
APL2010, Berlin, The definition of tolerant equality implies that the closed interval of numbers tolerantly equal to a real number x has endpoints x×1⎕ct and x÷1⎕ct . As the diagram (exaggeratingly) illustrates, the interval is not symmetric around x . The diagram is reflected about the origin if x is negative. The region of numbers tolerantly equal to a complex number z is a nearcircle with radius r←⎕ct×z . More precisely, it is the union of two circular regions: The smaller (in black) is centered on z with radius r . The larger derives as follows. A point u on the boundary with magnitude ≥ z satisfies (uz)=⎕ct×u . That is, the ratio between uz and u is ⎕ct , a constant; therefore, that boundary is a circle of Apollonius with foci z and the origin. The larger circle (in red) circumscribes the two points p and q on the smaller circle with magnitude z (in green) and the point s with magnitude (z)÷1⎕ct collinear with z and the origin.
Tolerance Less Than 1 A consequence of the definition is that tolerances greater than or equal to 1 do not give meaningful results: 1 (0.99 teq) 100 1 1 (0.99 teq) 100.1 0 1 (0.999 teq) 1000 1 1 (0.999 teq) 1000.1 0 1 (1 teq) 1e9 1 ∘.(1 teq)⍨ ?10⍴2e9 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 ∘.(1 teq)⍨ ?10⍴0 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 (That is, at tolerance ≥ 1 a number is equal to any other number.) Comparisons with 0 A further consequence is that comparisons with 0 are exact. By definition, x=y with tolerance t if (xy) ≤ t × (x)⌈(y) . If x is 0 , we get: (xy) ≤ t × (x)⌈(y) (0y) ≤ t × 0⌈y (y) ≤ t × y Since t is less than 1, y and hence y itself must be exactly 0 for the propositions to hold. This property can be exploited to effect exact comparison between x and y even when ⎕ct is nonzero: write 0=xy instead of x=y . Tolerant IndexOf If indexof is applied to floatingpoint numbers and it is known that tolerant comparison is not required, then {⎕ct←0 ⋄ ⍺⍳⍵} instead of ⍳ , exact indexof instead of tolerant indexof, is faster. Models The following models of tolerant comparisons are as presented in Robert Bernecky, Comparison Tolerance, SHARP APL Technical Notes 23, 19770610; Revision 1, 19780715. teq ← {⎕ct←0 ⋄ (⍺⍵)≤⍺⍺×(⍺)⌈(⍵)} ⍝ = tne ← {~⍺ (⍺⍺ teq) ⍵} ⍝ ≠ tlt ← {⎕ct←0 ⋄ (⍺<⍵)∧⍺(⍺⍺ tne)⍵} ⍝ < tle ← {⎕ct←0 ⋄ (⍺≤⍵)∨⍺(⍺⍺ teq)⍵} ⍝ ≤ tge ← {⎕ct←0 ⋄ (⍺≥⍵)∨⍺(⍺⍺ teq)⍵} ⍝ ≥ tgt ← {⎕ct←0 ⋄ (⍺>⍵)∧⍺(⍺⍺ tne)⍵} ⍝ > tfloor ← {⎕ct←0 ⋄ c⍵(⍺⍺ tlt)c←⌊0.5+⍵} ⍝ ⌊ tceiling ← {⎕ct←0 ⋄ c+⍵(⍺⍺ tgt)c←⌊0.5+⍵} ⍝ ⌈For example: ⎕←y←94+⍳13 94 95 96 97 98 99 100 101 102 103 104 105 106 100 (0.05 teq) y 0 1 1 1 1 1 1 1 1 1 1 1 0 100 (0.05 tne) y 1 0 0 0 0 0 0 0 0 0 0 0 1 100 (0.05 tlt) y 0 0 0 0 0 0 0 0 0 0 0 0 1 100 (0.05 tle) y 0 1 1 1 1 1 1 1 1 1 1 1 1 100 (0.05 tge) y 1 1 1 1 1 1 1 1 1 1 1 1 0 100 (0.05 tgt) y 1 0 0 0 0 0 0 0 0 0 0 0 0 (0.05 tfloor ) y÷100 0 0 1 1 1 1 1 1 1 1 1 1 1 (0.05 tceiling) y÷100 1 1 1 1 1 1 1 1 1 1 1 1 2 Alternatively, the computations can be modelled as a single dyadic dynamic operator, with the left operand being a function and a right operand the tolerance. tol←{ ⎕ct←0 f←⊃⎕cr'f'⊣0(f←⍺⍺)0 f='⌊' :c⍵(< tol ⍵⍵)c←⌊0.5+⍵ f='⌈' :c+⍵(> tol ⍵⍵)c←⌊0.5+⍵ e←(⍺⍵)≤⍵⍵×(⍺)⌈(⍵) f='=' : e f='≠' :~e f∊'≤≥':e∨⍺ ⍺⍺ ⍵ f∊'<>':e<⍺ ⍺⍺ ⍵ } 100 (= tol 0.05) y 0 1 1 1 1 1 1 1 1 1 1 1 0 100 (≠ tol 0.05) y 1 0 0 0 0 0 0 0 0 0 0 0 1 ⍝ etc. Previously appeared as the J Wiki Essay Tolerant Comparison, 20060127.
