Documentation for infix =~= assembled from the following pages:

Language documentation: Operators §

From Operators

(Operators) infix =~= §

multi sub infix:<=~=>(Any, Any)
multi sub infix:<=~=>(Int:D, Int:D)
multi sub infix:<=~=>(Num:D, Num:D)
multi sub infix:<=~=>(Rational:D, Rational:D)
multi sub infix:<=~=>(Real:D, Real:D)
multi sub infix:<=~=>(Complex:D, Complex:D)
multi sub infix:<=~=>(Numeric:D, Numeric:D)

The approximately-equal operator ≅, whose ASCII variant is =~=, calculates the relative difference between the left-hand and right-hand sides and returns True if the difference is less than $*TOLERANCE (which defaults to 1e-15). However, if either side is zero then it checks that the absolute difference between the sides is less than $*TOLERANCE. Note that this operator is not arithmetically symmetrical (doesn't do ± Δ):

my $x = 1;
say ($x + $*TOLERANCE) =~= $x;   # OUTPUT: «False␤» 
say ($x - $*TOLERANCE) =~= $x;   # OUTPUT: «True␤»

The tolerance is supposed to be modifiable via an adverb:

my ($x, $y) = 42, 42.1;
say $x =~= $y :tolerance(.1);

However, this is not yet implemented. The same effect can be achieved by assigning to $*TOLERANCE.

{
    my $*TOLERANCE = .1;
    say 11 =~= 10;        # OUTPUT: «True␤» 
}

Note that setting $*TOLERANCE = 0 will cause all comparisons to fail.

{
    my $*TOLERANCE = 0;
    say 1 =~= 1;          # OUTPUT: «False␤» 
}