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Again, keep in mind that even === has some tolerance (Internal`$SameQTolerance) when comparing machine precision numbers (though less than ==, Internal`$EqualTolerance) and 15/137. === Divide[15, 137.] …

You can lower the value of Internal`$EqualTolerance: Block[{Internal`$EqualTolerance = 0}, 0.999999999999988 >= 1.0 ] False This can lead to unexpected behaviors too: Block[{Internal`$EqualTolerance … NumericQ] := Block[{Internal`$EqualTolerance = x}, Boole[Not[0.999999999999988 >= 1.0] && (0.1 + 0.2 == 0.3)] ] Plot[correctEquals[x], {x, 0, Internal`$EqualTolerance}] …

There are two undocumented functions which control the tolerance of Equal and SameQ: Internal`$EqualTolerance and Internal`$SameQTolerance. …

So what is the fastest way to get rid of duplicates while allowing for a tolerance of comparisons, and preferably being able to control it (Internal`$EqualTolerance, Internal`$SameQTolerance)? …

The proper comparison is Block[{Internal`$EqualTolerance = -Infinity}, 1 == 1 + $MachineEpsilon] (* False *) Block[{Internal`$EqualTolerance = -Infinity}, 1 == 1 + $MachineEpsilon/2] (* True *) In … fact, the value of Internal`$EqualTolerance * Log2[10] is 7., meaning that it ignores the last seven bits, just as I discovered! …

For completeness: Internal`$EqualTolerance and Internal`$HashTolerance have no effect. The test is definitely SameQ. …

C definition of $MachineEpsilon Mathematica's view: floating-point numbers as distributions The nature of the distribution: interval arithmetic versus Gaussian error propagation $EqualTolerance; $SameQTolerance …

You can change the tolerance used by Equal with Internal`$EqualTolerance. …

(You might google for Internal`$EqualTolerance and Internal`$SameQTolerance if you wish to learn about these tolerances.) …

Internal`$EqualTolerance = N @ Log10[2^4]; a == b Internal`$EqualTolerance = N @ Log10[2^5]; a == b False True $SameQTolerance behaves the same only it applies to SameQ rather than Equal: Internal … ^13]; a == b Internal`$EqualTolerance = N@Log10[2^14]; a == b False True …

It must be said that this is still not as fast as Union, but at least it acknowledges Internal`$EqualTolerance while being faster than Split. … The value of Internal`$EqualTolerance is actually hard-coded into the bytecode on compilation, so it will be necessary to recompile if a different tolerance is required. …

There is a tolerance Internal`$EqualTolerance that is applied when testing Real numbers. If the numbers agree up to the last Internal`$EqualTolerance digits, then they are treated as equal. … Try this: eps = 1.0; p = 0; Block[{Internal`$EqualTolerance = 0.}, While[(1.0 + eps) > 1.0, eps = eps/2.0; p += 1]; ] eps p eps*2 (* 1.11022*10^-16 53 2.22045*10^-16 *) See How to make the computer …

Another workaround is to create a Floor that applies a machine-precision fudge factor, in the way that Equal has the fudge factor Internal`$EqualTolerance. … TrueQ[$in] := Block[{$in = True}, (* Villegas-Gayley *) Floor[x + 10^Internal`$EqualTolerance $MachineEpsilon Abs[x]]]; Protect@Floor; CoordinateBoundingBoxArray[v, rest] ]; Or even more …

In Mathematica similar (but relative) tolerances are built into SameQ and Equal, which are controlled by the the internal system parameters Internal`$SameQTolerance and Internal`$EqualTolerance respectively … out that the tolerance can be as small as 0.55, which is close to the value of Log10[4.] = 0.60.. that would be predicted by the observed rounding error): Block[{Internal`$SameQTolerance = Internal`$EqualTolerance …

Compare like Equal does (or in a similar way), with adjustable tolerance (see Internal`$EqualTolerance). It must work with at least machine precision numbers. …