Approximate equality

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Task
Approximate equality
You are encouraged to solve this task according to the task description, using any language you may know.

Sometimes, when testing whether the solution to a task (for example, here on Rosetta Code) is correct, the difference in floating point calculations between different language implementations becomes significant.

For example, a difference between 32 bit and 64 bit floating point calculations may appear by about the 8th significant digit in base 10 arithmetic.


Task

Create a function which returns true if two floating point numbers are approximately equal.


The function should allow for differences in the magnitude of numbers, so that, for example,
100000000000000.01   may be approximately equal to   100000000000000.011,
even though   100.01   is not approximately equal to   100.011.

If the language has such a feature in its standard library, this may be used instead of a custom function.

Show the function results with comparisons on the following pairs of values:

  1.     100000000000000.01,   100000000000000.011     (note: should return true)
  2.     100.01,   100.011                                                     (note: should return false)
  3.     10000000000000.001 / 10000.0,   1000000000.0000001000
  4.     0.001,   0.0010000001
  5.     0.000000000000000000000101,   0.0
  6.      sqrt(2) * sqrt(2),    2.0
  7.     -sqrt(2) * sqrt(2),   -2.0
  8.     3.14159265358979323846,   3.14159265358979324


Answers should be true for the first example and false in the second, so that just rounding the numbers to a fixed number of decimals should not be enough. Otherwise answers may vary and still be correct. See the Python code for one type of solution.

Ada

with Ada.Text_IO; use Ada.Text_IO;
with Ada.Numerics.Generic_Elementary_Functions;

procedure Main is
   type Real is digits 18;
   package Real_Funcs is new Ada.Numerics.Generic_Elementary_Functions(Real);
   use Real_Funcs;
   package Real_IO is new Ada.Text_IO.Float_IO(Real);
   use Real_IO;
   
   function Approx_Equal (Left : Real; Right : Real) return Boolean is
      
      -- Calculate an epsilon value based upon the magnitude of the
      -- maximum value of the two parameters
      eps : Real := Real'Max(Left, Right) * 1.0e-9;
   begin
      if left > Right then
         return Left - Right < eps;
      else
         return Right - Left < eps;
      end if;
   end Approx_Equal;
   
   Type Index is (Left, Right);
   type Pairs_List is array (Index) of Real;
   type Pairs_Table is array(1..8) of Pairs_List;
   Table : Pairs_Table;
                          
begin
   Table := ((100000000000000.01,   100000000000000.011),
             (100.01,   100.011),
             (10000000000000.001 / 10000.0,   1000000000.0000001000),
             (0.001,   0.0010000001),
             (0.000000000000000000000101,   0.0),
             (sqrt(2.0) * sqrt(2.0),    2.0),
             (-sqrt(2.0) * sqrt(2.0),   -2.0),
             (3.14159265358979323846,   3.14159265358979324));

   for Pair of Table loop
      Put(Item => Pair(Left), Exp => 0, Aft => 16, Fore => 6);
      Put("  ");
      Put(Item => Pair(Right), Exp => 0, Aft => 16, Fore => 6);
      Put_Line("  " & Boolean'Image(Approx_Equal(Pair(Left), Pair(Right))));
   end loop;
   
end Main;
Output:
100000000000000.0100000000000000  100000000000000.0110000000000000  TRUE
   100.0100000000000000     100.0110000000000000  FALSE
1000000000.0000001000000000  1000000000.0000001000000000  TRUE
     0.0010000000000000       0.0010000001000000  FALSE
     0.0000000000000000       0.0000000000000000  FALSE
     2.0000000000000000       2.0000000000000000  TRUE
    -2.0000000000000000      -2.0000000000000000  FALSE
     3.1415926535897932       3.1415926535897932  TRUE

ALGOL 68

Translation of: Kotlin
BEGIN # test REAL values for approximate equality #
    # returns TRUE if value is approximately equal to other, FALSE otherwide #
    PROC approx equals = ( REAL value, REAL other, REAL epsilon )BOOL: ABS ( value - other ) < epsilon;
    # shows the result of testing a for approximate equality with b #
    PROC test = ( REAL a, b )VOID:
         BEGIN
            REAL epsilon = 1e-18;
            print( ( a, ", ", b, " => ", IF approx equals( a, b, epsilon ) THEN "true" ELSE "false" FI, newline ) )
         END # test # ;
    # task test cases #
    test( 100000000000000.01, 100000000000000.011 );
    test( 100.01, 100.011 );
    test( 10000000000000.001 / 10000.0, 1000000000.0000001000);
    test( 0.001, 0.0010000001 );
    test( 0.000000000000000000000101, 0.0 );
    test(   sqrt( 2 ) * sqrt( 2 ),  2.0 );
    test( - sqrt( 2 ) * sqrt( 2 ), -2.0 );
    test( 3.14159265358979323846, 3.14159265358979324 )
END
Output:
+1.00000000000000e +14, +1.00000000000000e +14 => true
+1.00010000000000e  +2, +1.00011000000000e  +2 => false
+1.00000000000000e  +9, +1.00000000000000e  +9 => false
+1.00000000000000e  -3, +1.00000010000000e  -3 => false
+1.01000000000000e -22, +0.00000000000000e  +0 => true
+2.00000000000000e  +0, +2.00000000000000e  +0 => false
-2.00000000000000e  +0, -2.00000000000000e  +0 => false
+3.14159265358979e  +0, +3.14159265358979e  +0 => true

AWK

# syntax: GAWK -f APPROXIMATE_EQUALITY.AWK
# converted from C#
BEGIN {
    epsilon = 1
    while (1 + epsilon != 1) {
      epsilon /= 2
    }
    printf("epsilon = %18.16g\n\n",epsilon)
    main("100000000000000.01","100000000000000.011")
    main("100.01","100.011")
    main("10000000000000.001"/"10000.0","1000000000.0000001000")
    main("0.001","0.0010000001")
    main("0.000000000000000000000101","0.0")
    main(sqrt(2.0)*sqrt(2.0),"2.0")
    main(-sqrt(2.0)*sqrt(2.0),"-2.0")
    main("3.14159265358979323846","3.14159265358979324")
    exit(0)
}
function main(a,b,  tmp) {
    tmp = abs(a - b) < epsilon
    printf("%d %27s %s\n",tmp,a,b)
}
function abs(x) { if (x >= 0) { return x } else { return -x } }
Output:
epsilon = 1.110223024625157e-016

1          100000000000000.01 100000000000000.011
0                      100.01 100.011
0                      1e+009 1000000000.0000001000
0                       0.001 0.0010000001
1  0.000000000000000000000101 0.0
0                           2 2.0
0                          -2 -2.0
1      3.14159265358979323846 3.14159265358979324

C

Translation of: Java
#include <math.h>
#include <stdbool.h>
#include <stdio.h>

bool approxEquals(double value, double other, double epsilon) {
    return fabs(value - other) < epsilon;
}

void test(double a, double b) {
    double epsilon = 1e-18;
    printf("%f, %f => %d\n", a, b, approxEquals(a, b, epsilon));
}

int main() {
    test(100000000000000.01, 100000000000000.011);
    test(100.01, 100.011);
    test(10000000000000.001 / 10000.0, 1000000000.0000001000);
    test(0.001, 0.0010000001);
    test(0.000000000000000000000101, 0.0);
    test(sqrt(2.0) * sqrt(2.0), 2.0);
    test(-sqrt(2.0) * sqrt(2.0), -2.0);
    test(3.14159265358979323846, 3.14159265358979324);
    return 0;
}
Output:
100000000000000.015625, 100000000000000.015625 => 1
100.010000, 100.011000 => 0
1000000000.000000, 1000000000.000000 => 0
0.001000, 0.001000 => 0
0.000000, 0.000000 => 1
2.000000, 2.000000 => 0
-2.000000, -2.000000 => 0
3.141593, 3.141593 => 1

C#

using System;

public static class Program
{
    public static void Main() {
        Test(100000000000000.01, 100000000000000.011);
        Test(100.01, 100.011);
        Test(10000000000000.001 / 10000.0, 1000000000.0000001000);
        Test(0.001, 0.0010000001);
        Test(0.000000000000000000000101, 0.0);
        Test(Math.Sqrt(2) * Math.Sqrt(2), 2.0);
        Test(-Math.Sqrt(2) * Math.Sqrt(2), -2.0);
        Test(3.14159265358979323846, 3.14159265358979324);

        void Test(double a, double b) {
            const double epsilon = 1e-18;
            WriteLine($"{a}, {b} => {a.ApproxEquals(b, epsilon)}");
        }
    }

    public static bool ApproxEquals(this double value, double other, double epsilon) => Math.Abs(value - other) < epsilon;
}
Output:
100000000000000.02, 100000000000000.02 => True
100.01, 100.011 => False
1000000000.0000002, 1000000000.0000001 => False
0.001, 0.0010000001 => False
1.01E-22, 0 => True
2.0000000000000004, 2 => False
-2.0000000000000004, -2 => False
3.141592653589793, 3.141592653589793 => True

C++

Translation of: C
#include <iomanip>
#include <iostream>
#include <cmath>

bool approxEquals(double a, double b, double e) {
    return fabs(a - b) < e;
}

void test(double a, double b) {
    constexpr double epsilon = 1e-18;
    std::cout << std::setprecision(21) << a;
    std::cout << ", ";
    std::cout << std::setprecision(21) << b;
    std::cout << " => ";
    std::cout << approxEquals(a, b, epsilon) << '\n';
}

int main() {
    test(100000000000000.01, 100000000000000.011);
    test(100.01, 100.011);
    test(10000000000000.001 / 10000.0, 1000000000.0000001000);
    test(0.001, 0.0010000001);
    test(0.000000000000000000000101, 0.0);
    test(sqrt(2.0) * sqrt(2.0), 2.0);
    test(-sqrt(2.0) * sqrt(2.0), -2.0);
    test(3.14159265358979323846, 3.14159265358979324);
    return 0;
}
Output:
100000000000000.015625, 100000000000000.015625 => 1
100.010000000000005116, 100.01099999999999568 => 0
1000000000.00000023842, 1000000000.00000011921 => 0
0.00100000000000000002082, 0.00100000010000000005492 => 0
1.0099999999999999762e-22, 0 => 1
2.00000000000000044409, 2 => 0
-2.00000000000000044409, -2 => 0
3.141592653589793116, 3.141592653589793116 => 1

Common Lisp

This solution compares the normalized (i.e. between 0.5 and 1 on implementations which use binary floating point) significands of the floating point numbers, correcting each significand by half the difference in the exponents so that the corrected numbers used for comparison have the same difference in order of magnitude as the original numbers and are stable when the order of the arguments is changed. Unlike the metric of comparing the difference to some fraction of the numbers' size, this approach only requires two floating point operations (the subtraction and comparison at the end), and more directly maps to the fundamental issue which leads to the need for floating-point comparisons, i.e. the limited precision of the significand.

(defun approx-equal (float1 float2 &optional (threshold 0.000001))
  "Determine whether float1 and float2 are equal; THRESHOLD is the
maximum allowable difference between normalized significands of floats
with the same exponent. The significands are scaled appropriately
before comparison for floats with different exponents."
  (multiple-value-bind (sig1 exp1 sign1) (decode-float float1)
    (multiple-value-bind (sig2 exp2 sign2) (decode-float float2)
      (let ((cmp1 (float-sign sign1 (scale-float sig1 (floor (- exp1 exp2) 2))))
            (cmp2 (float-sign sign2 (scale-float sig2 (floor (- exp2 exp1) 2)))))
        (< (abs (- cmp1 cmp2)) threshold)))))

D

Translation of: C#
import std.math;
import std.stdio;

auto approxEquals = (double a, double b, double epsilon) => abs(a - b) < epsilon;

void main() {
    void test(double a, double b) {
        double epsilon = 1e-18;
        writefln("%.18f, %.18f => %s", a, b, a.approxEquals(b, epsilon));
    }

    test(100000000000000.01, 100000000000000.011);
    test(100.01, 100.011);
    test(10000000000000.001 / 10000.0, 1000000000.0000001000);
    test(0.001, 0.0010000001);
    test(0.000000000000000000000101, 0.0);
    test(sqrt(2.0) * sqrt(2.0), 2.0);
    test(-sqrt(2.0) * sqrt(2.0), -2.0);
    test(3.14159265358979323846, 3.14159265358979324);
}
Output:
100000000000000.015620000000000000, 100000000000000.015620000000000000 => true
100.010000000000005110, 100.010999999999995680 => false
1000000000.000000119100000000, 1000000000.000000119100000000 => true
0.001000000000000000, 0.001000000100000000 => false
0.000000000000000000, 0.000000000000000000 => true
2.000000000000000000, 2.000000000000000000 => true
-2.000000000000000000, -2.000000000000000000 => true
3.141592653589793116, 3.141592653589793116 => true

Delphi

Library: System.Math

The Delphi has a Math.SameValue function for compare, but all float operations use by default Extended (High precision), we need use double cast for every operation, like division, multiply and square tree.

program Approximate_Equality;

{$APPTYPE CONSOLE}

uses
  System.SysUtils,
  System.Math;

const
  EPSILON: Double = 1E-18;

procedure Test(a, b: Double; Expected: Boolean);
var
  result: Boolean;
const
  STATUS: array[Boolean] of string = ('FAIL', 'OK');
begin
  result := SameValue(a, b, EPSILON);
  Write(a, ' ', b, ' => ', result, ' '^I);
  writeln(Expected, ^I, STATUS[Expected = result]);
end;

begin
  Test(100000000000000.01, 100000000000000.011, True);
  Test(100.01, 100.011, False);
  Test(double(10000000000000.001) / double(10000.0), double(1000000000.0000001000),
    False);
  Test(0.001, 0.0010000001, False);
  Test(0.000000000000000000000101, 0.0, True);
  Test(double(Sqrt(2)) * double(Sqrt(2)), 2.0, False);
  Test(-double(Sqrt(2)) * double(Sqrt(2)), -2.0, false);
  Test(3.14159265358979323846, 3.14159265358979324, True);
  Readln;
end.
Output:
 1.00000000000000E+0014  1.00000000000000E+0014 => TRUE         TRUE    OK
 1.00010000000000E+0002  1.00011000000000E+0002 => FALSE        FALSE   OK
 1.00000000000000E+0009  1.00000000000000E+0009 => FALSE        FALSE   OK
 1.00000000000000E-0003  1.00000010000000E-0003 => FALSE        FALSE   OK
 1.01000000000000E-0022  0.00000000000000E+0000 => TRUE         TRUE    OK
 2.00000000000000E+0000  2.00000000000000E+0000 => FALSE        FALSE   OK
-2.00000000000000E+0000 -2.00000000000000E+0000 => FALSE        FALSE   OK
 3.14159265358979E+0000  3.14159265358979E+0000 => TRUE         TRUE    OK

Factor

The ~ word takes three arguments: the two values to be compared, and an epsilon value representing the allowed distance between the two values. A positive epsilon performs an absolute distance test, an epsilon of zero performs an exact comparison, and a negative epsilon performs a relative distance test (as required by this task).

Works with: Factor version 0.99 development version 2019-07-10
USING: formatting generalizations kernel math math.functions ;

100000000000000.01             100000000000000.011
100.01                         100.011
10000000000000.001 10000.0 /f  1000000000.0000001000
0.001                          0.0010000001
0.000000000000000000000101     0.0
2 sqrt dup *                   2.0
2 sqrt dup neg *              -2.0
3.14159265358979323846         3.14159265358979324

[ 2dup -1e-15 ~ "%+47.30f %+47.30f -1e-15 ~ : %u\n" printf ]
2 8 mnapply
Output:
+100000000000000.015625000000000000000000000000 +100000000000000.015625000000000000000000000000 -1e-15 ~ : t
            +100.010000000000005115907697472721             +100.010999999999995679900166578591 -1e-15 ~ : f
     +1000000000.000000238418579101562500000000      +1000000000.000000119209289550781250000000 -1e-15 ~ : t
              +0.001000000000000000020816681712               +0.001000000100000000054917270731 -1e-15 ~ : f
              +0.000000000000000000000101000000               +0.000000000000000000000000000000 -1e-15 ~ : f
              +2.000000000000000444089209850063               +2.000000000000000000000000000000 -1e-15 ~ : t
              -2.000000000000000444089209850063               -2.000000000000000000000000000000 -1e-15 ~ : t
              +3.141592653589793115997963468544               +3.141592653589793115997963468544 -1e-15 ~ : t

Forth

Works with: GForth version 0.7.9_20211014

Genuine Forth word : f~ ( r1 r2 r3 – flag ) float-ext “f-proximate” ANS Forth medley for comparing r1 and r2 for equality: r3>0: f~abs; r3=0: bitwise comparison; r3<0:

  
: test-f~   ( f1 f2 -- )  
  1e-18                  \ epsilon
  f~                     \ AproximateEqual
  if ." True" else ." False" then   
;
Output:
  100000000000000.01e                     100000000000000.011e   test-f~ True 
  100.01e                                 100.011e               test-f~ False 
  10000000000000.001e  10000.0e f/        1000000000.0000001000e test-f~ False 
  0.001e                                  0.0010000001e          test-f~ False 
  0.000000000000000000000101e             0.0e                   test-f~ True 
  2.0e fdup fsqrt fswap fsqrt  f*         2.0e                   test-f~ False 
  2.0e fdup fsqrt fnegate fswap fsqrt f* -2.0e                   test-f~ False 
  3.14159265358979323846e                 3.14159265358979324e   test-f~ True

Fortran

Compare against the Python function documented at https://www.python.org/dev/peps/pep-0485/#proposed-implementation, and with the discussion at https://stackoverflow.com/questions/5595425/what-is-the-best-way-to-compare-floats-for-almost-equality-in-python#

program main
  implicit none

  integer                       :: i
  double precision, allocatable :: vals(:)

  vals = [ 100000000000000.01d0,          100000000000000.011d0,   &
    &      100.01d0,                      100.011d0,               &
    &      10000000000000.001d0/10000d0,  1000000000.0000001000d0, &
    &      0.001d0,                       0.0010000001d0,          &
    &      0.000000000000000000000101d0,  0d0,                     &
    &      sqrt(2d0)*sqrt(2d0),           2d0,                     &
    &     -sqrt(2d0)*sqrt(2d0),          -2d0,                     &
    &      3.14159265358979323846d0,      3.14159265358979324d0    ]

  do i = 1, size(vals)/2
    print '(ES30.18, A, ES30.18, A, L)', vals(2*i-1), ' == ', vals(2*i), ' ? ', eq_approx(vals(2*i-1), vals(2*i))
  end do

contains

  logical function eq_approx(a, b, reltol, abstol)
    !! is a approximately equal b?

    double precision, intent(in)           :: a, b
      !! values to compare
    double precision, intent(in), optional :: reltol, abstol
      !! relative and absolute error thresholds.
      !! defaults: epsilon, smallest non-denormal number

    double precision :: rt, at

    rt = epsilon(1d0)
    at = tiny(1d0)
    if (present(reltol)) rt = reltol
    if (present(abstol)) at = abstol

    eq_approx = abs(a - b) .le. max(rt * max(abs(a), abs(b)), at)
    return
  end function

end program
Output:
      1.000000000000000156E+14 ==       1.000000000000000156E+14 ? T
      1.000100000000000051E+02 ==       1.000109999999999957E+02 ? F
      1.000000000000000238E+09 ==       1.000000000000000119E+09 ? T
      1.000000000000000021E-03 ==       1.000000100000000055E-03 ? F
      1.009999999999999976E-22 ==       0.000000000000000000E+00 ? F
      2.000000000000000444E+00 ==       2.000000000000000000E+00 ? T
     -2.000000000000000444E+00 ==      -2.000000000000000000E+00 ? T
      3.141592653589793116E+00 ==       3.141592653589793116E+00 ? T


FreeBASIC

Translation of: AWK
#include "string.bi"

Dim Shared As Double epsilon = 1

Sub eq_approx(a As Double,b As Double)
    Dim As Boolean tmp = Abs(a - b) < epsilon
    Print Using "& & &";tmp;a;b
End Sub

While (1 + epsilon <> 1)
    epsilon /= 2
Wend

Print "epsilon = "; Format(epsilon, "0.000000000000000e-00")
Print
eq_approx(100000000000000.01, 100000000000000.011)
eq_approx(100.01, 100.011)
eq_approx(10000000000000.001/10000.0, 1000000000.0000001000)
eq_approx(0.001, 0.0010000001)
eq_approx(0.000000000000000000000101, 0.0)
eq_approx(Sqr(2)*Sqr(2), 2.0)
eq_approx(-Sqr(2)*Sqr(2), -2.0)
eq_approx(3.14159265358979323846, 3.14159265358979324)

Sleep


Go

Go's float64 type is limited to 15 or 16 digits of precision. As there are some numbers in this task which have more digits than this I've used big.Float instead.

package main

import (
    "fmt"
    "log"
    "math/big"
)

func max(a, b *big.Float) *big.Float {
    if a.Cmp(b) > 0 {
        return a
    }
    return b
}

func isClose(a, b *big.Float) bool {
    relTol := big.NewFloat(1e-9) // same as default for Python's math.isclose() function
    t := new(big.Float)
    t.Sub(a, b)
    t.Abs(t)
    u, v, w := new(big.Float), new(big.Float), new(big.Float)
    u.Mul(relTol, max(v.Abs(a), w.Abs(b)))
    return t.Cmp(u) <= 0
}

func nbf(s string) *big.Float {
    n, ok := new(big.Float).SetString(s)
    if !ok {
        log.Fatal("invalid floating point number")
    }
    return n
}

func main() {
    root2 := big.NewFloat(2.0)
    root2.Sqrt(root2)
    pairs := [][2]*big.Float{
        {nbf("100000000000000.01"), nbf("100000000000000.011")},
        {nbf("100.01"), nbf("100.011")},
        {nbf("0").Quo(nbf("10000000000000.001"), nbf("10000.0")), nbf("1000000000.0000001000")},
        {nbf("0.001"), nbf("0.0010000001")},
        {nbf("0.000000000000000000000101"), nbf("0.0")},
        {nbf("0").Mul(root2, root2), nbf("2.0")},
        {nbf("0").Mul(nbf("0").Neg(root2), root2), nbf("-2.0")},
        {nbf("100000000000000003.0"), nbf("100000000000000004.0")},
        {nbf("3.14159265358979323846"), nbf("3.14159265358979324")},
    }
    for _, pair := range pairs {
        s := "≉"
        if isClose(pair[0], pair[1]) {
            s = "≈"
        }
        fmt.Printf("% 21.19g %s %- 21.19g\n", pair[0], s, pair[1])
    }
}
Output:
   100000000000000.01 ≈  100000000000000.011 
               100.01 ≉  100.011             
   1000000000.0000001 ≈  1000000000.0000001  
                0.001 ≉  0.0010000001        
             1.01e-22 ≉  0                   
 2.000000000000000273 ≈  2                   
-2.000000000000000273 ≈ -2                   
   100000000000000003 ≈  100000000000000004  
 3.141592653589793239 ≈  3.14159265358979324 

Groovy

Translation of: Java
class Approximate {
    private static boolean approxEquals(double value, double other, double epsilon) {
        return Math.abs(value - other) < epsilon
    }

    private static void test(double a, double b) {
        double epsilon = 1e-18
        System.out.printf("%f, %f => %s\n", a, b, approxEquals(a, b, epsilon))
    }

    static void main(String[] args) {
        test(100000000000000.01, 100000000000000.011)
        test(100.01, 100.011)
        test(10000000000000.001 / 10000.0, 1000000000.0000001000)
        test(0.001, 0.0010000001)
        test(0.000000000000000000000101, 0.0)
        test(Math.sqrt(2.0) * Math.sqrt(2.0), 2.0)
        test(-Math.sqrt(2.0) * Math.sqrt(2.0), -2.0)
        test(3.14159265358979323846, 3.14159265358979324)
    }
}
Output:
100000000000000.020000, 100000000000000.020000 => true
100.010000, 100.011000 => false
1000000000.000000, 1000000000.000000 => true
0.001000, 0.001000 => false
0.000000, 0.000000 => true
2.000000, 2.000000 => false
-2.000000, -2.000000 => false
3.141593, 3.141593 => true

Haskell

class (Num a, Ord a, Eq a) => AlmostEq a where
  eps :: a

infix 4 ~=
(~=) :: AlmostEq a => a -> a -> Bool
a ~= b = or [ a == b
            , abs (a - b) < eps * abs(a + b)
            , abs (a - b) < eps ]

instance AlmostEq Int where eps = 0
instance AlmostEq Integer where eps = 0
instance AlmostEq Double where eps = 1e-14
instance AlmostEq Float where eps = 1e-5

Examples

λ> 0.000001 == (0 :: Float)
False
λ> 0.000001 ~= (0 :: Float)
True
λ> 0.000001 ~= (0 :: Double)
False
λ> (\x -> sqrt x * sqrt x == x) $ (2 :: Float)
False
λ> (\x -> sqrt x * sqrt x ~= x) $ (2 :: Float)
True
λ> (\x -> sqrt x * sqrt x == x) $ (2 :: Double)
False
λ> (\x -> sqrt x * sqrt x ~= x) $ (2 :: Double)
True

Assignment

test :: [(Double, Double)]
test = [(100000000000000.01, 100000000000000.011)
       ,(100.01, 100.011)
       ,(10000000000000.001 / 10000.0, 1000000000.0000001000)
       ,(0.001, 0.0010000001)
       ,(0.000000000000000000000101, 0.0)
       ,(sqrt 2 * sqrt 2, 2.0)
       ,(-sqrt 2 * sqrt 2, -2.0)
       ,(3.141592653589793, 3.141592653589794)
       ,(3.141592653589, 3.141592653589794)]

-- requires import Text.Printf
main = mapM_ runTest test
  where
    runTest (a, b) = do
      printf "%f == %f %v\n" a b (show $ a==b) :: IO ()
      printf "%f ~= %f %v\n\n" a b (show $ a~=b)
λ> main
100000000000000.02 == 100000000000000.02 True
100000000000000.02 ~= 100000000000000.02 True

100.01 == 100.011 False
100.01 ~= 100.011 False

1000000000.0000002 == 1000000000.0000001 False
1000000000.0000002 ~= 1000000000.0000001 True

0.001 == 0.0010000001 False
0.001 ~= 0.0010000001 False

0.000000000000000000000101 == 0.0 False
0.000000000000000000000101 ~= 0.0 True

2.0000000000000004 == 2.0 False
2.0000000000000004 ~= 2.0 True

-2.0000000000000004 == -2.0 False
-2.0000000000000004 ~= -2.0 True

3.141592653589793 == 3.141592653589794 False
3.141592653589793 ~= 3.141592653589794 True

3.141592653589 == 3.141592653589794 False
3.141592653589 ~= 3.141592653589794 False


J

Attributed to Ken Iverson, inventor of APL and of course his final dialect, j, "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."

J includes a "customization" conjunction ( !. ) that delivers variants of some verbs. Comparisons are tolerant by default, and their tolerance can be customized to some level. Specifying =!.0 specifies "no tolerance". Specifying a tolerance of 1e_8 is a domain error because that's no longer math. Write your own verb if you need this.

   NB. default comparison tolerance matches the python result
   ".;._2]0 :0
      100000000000000.01 =   100000000000000.011
      100.01 =   100.011                        
      (10000000000000.001 % 10000.0) =   1000000000.0000001000
      0.001 =   0.0010000001
      0.000000000000000000000101 =   0.0
      (= ([: *~ %:)) 2                    NB. sqrt(2)*sqrt(2)
      ((= -)~ ([: (* -) %:)) 2            NB. -sqrt(2) * sqrt(2),   -2.0
      3.14159265358979323846 =   3.14159265358979324
)
1 0 1 0 0 1 1 1


   NB. tolerance of 1e_12 matches the python result
   ".;._2]0 :0[CT=:1e_12
      100000000000000.01 =!.CT   100000000000000.011
      100.01 =!.CT   100.011                        
      (10000000000000.001 % 10000.0) =!.CT   1000000000.0000001000
      0.001 =!.CT   0.0010000001
      0.000000000000000000000101 =!.CT   0.0
      (=!.CT ([: *~ %:)) 2                    NB. sqrt(2)*sqrt(2)
      ((=!.CT -)~ ([: (* -) %:)) 2            NB. -sqrt(2) * sqrt(2),   -2.0
      3.14159265358979323846 =!.CT   3.14159265358979324
)
1 0 1 0 0 1 1 1


   NB. tight tolerance
   ".;._2]0 :0[CT=:1e_18
      100000000000000.01 =!.CT   100000000000000.011
      100.01 =!.CT   100.011                        
      (10000000000000.001 % 10000.0) =!.CT   1000000000.0000001000
      0.001 =!.CT   0.0010000001
      0.000000000000000000000101 =!.CT   0.0
      (=!.CT ([: *~ %:)) 2                    NB. sqrt(2)*sqrt(2)
      ((=!.CT -)~ ([: (* -) %:)) 2            NB. -sqrt(2) * sqrt(2),   -2.0
      3.14159265358979323846 =!.CT   3.14159265358979324
)
1 0 0 0 0 0 0 1

   2 (=!.1e_8) 9
|domain error
|   2(=    !.1e_8)9

Java

Translation of: Kotlin
public class Approximate {
    private static boolean approxEquals(double value, double other, double epsilon) {
        return Math.abs(value - other) < epsilon;
    }

    private static void test(double a, double b) {
        double epsilon = 1e-18;
        System.out.printf("%f, %f => %s\n", a, b, approxEquals(a, b, epsilon));
    }

    public static void main(String[] args) {
        test(100000000000000.01, 100000000000000.011);
        test(100.01, 100.011);
        test(10000000000000.001 / 10000.0, 1000000000.0000001000);
        test(0.001, 0.0010000001);
        test(0.000000000000000000000101, 0.0);
        test(Math.sqrt(2.0) * Math.sqrt(2.0), 2.0);
        test(-Math.sqrt(2.0) * Math.sqrt(2.0), -2.0);
        test(3.14159265358979323846, 3.14159265358979324);
    }
}
Output:
100000000000000.020000, 100000000000000.020000 => true
100.010000, 100.011000 => false
1000000000.000000, 1000000000.000000 => false
0.001000, 0.001000 => false
0.000000, 0.000000 => true
2.000000, 2.000000 => false
-2.000000, -2.000000 => false
3.141593, 3.141593 => true

jq

Translation of: Lobster
# Return whether the two numbers `a` and `b` are close.
# Closeness is determined by the `epsilon` parameter - 
# the numbers are considered close if the difference between them
# is no more than epsilon * max(abs(a), abs(b)).
def isclose(a; b; epsilon):
   ((a - b) | fabs) <= (([(a|fabs), (b|fabs)] | max) * epsilon);

def lpad($len; $fill): tostring | ($len - length) as $l | ($fill * $l)[:$l] + .;

def lpad: lpad(20; " ");

# test values
def tv: [
  {x:  100000000000000.01,             y: 100000000000000.011 },
  {x:  100.01,                         y: 100.011 },                            
  {x:  (10000000000000.001 / 10000.0), y: 1000000000.0000001000 },
  {x:  0.001,                          y: 0.0010000001 },
  {x:  0.000000000000000000000101,     y: 0.0 },
  {x:  ((2|sqrt) * (2|sqrt)),          y: 2.0 },
  {x:  (-(2|sqrt) * (2|sqrt)),         y: -2.0 },
  {x:  3.14159265358979323846,         y: 3.14159265358979324 }
 ]
;

tv[] | "\(.x|lpad) \(if isclose(.x; .y; 1.0e-9) then " ≈ " else " ≉ " end) \(.y|lpad)"
Output:

Using jq 1.6:

  100000000000000.02  ≈    100000000000000.02
              100.01  ≉               100.011
  1000000000.0000002  ≈    1000000000.0000001
               0.001  ≉          0.0010000001
            1.01e-22  ≉                     0
  2.0000000000000004  ≈                     2
 -2.0000000000000004  ≈                    -2
   3.141592653589793  ≈     3.141592653589793

Julia

Julia has an infix operator, ≈, which corresponds to Julia's buitin isapprox() function.

Translation of: Python
testvalues = [[100000000000000.01,           100000000000000.011],
              [100.01,                       100.011],
              [10000000000000.001 / 10000.0, 1000000000.0000001000],
              [0.001,                        0.0010000001],
              [0.000000000000000000000101,   0.0],
              [sqrt(2) * sqrt(2),            2.0],
              [-sqrt(2) * sqrt(2),          -2.0],
              [3.14159265358979323846,       3.14159265358979324]]

for (x, y) in testvalues
    println(rpad(x, 21), " ≈ ", lpad(y, 22), ": ", x  y)
end
Output:
1.0000000000000002e14 ≈  1.0000000000000002e14: true
100.01                ≈                100.011: false
1.0000000000000002e9  ≈   1.0000000000000001e9: true
0.001                 ≈           0.0010000001: false
1.01e-22              ≈                    0.0: false
2.0000000000000004    ≈                    2.0: true
-2.0000000000000004   ≈                   -2.0: true
3.141592653589793     ≈      3.141592653589793: true

Kotlin

Translation of: C#
import kotlin.math.abs
import kotlin.math.sqrt

fun approxEquals(value: Double, other: Double, epsilon: Double): Boolean {
    return abs(value - other) < epsilon
}

fun test(a: Double, b: Double) {
    val epsilon = 1e-18
    println("$a, $b => ${approxEquals(a, b, epsilon)}")
}

fun main() {
    test(100000000000000.01, 100000000000000.011)
    test(100.01, 100.011)
    test(10000000000000.001 / 10000.0, 1000000000.0000001000)
    test(0.001, 0.0010000001)
    test(0.000000000000000000000101, 0.0)
    test(sqrt(2.0) * sqrt(2.0), 2.0)
    test(-sqrt(2.0) * sqrt(2.0), -2.0)
    test(3.14159265358979323846, 3.14159265358979324)
}
Output:
1.0000000000000002E14, 1.0000000000000002E14 => true
100.01, 100.011 => false
1.0000000000000002E9, 1.0000000000000001E9 => false
0.001, 0.0010000001 => false
1.01E-22, 0.0 => true
2.0000000000000004, 2.0 => false
-2.0000000000000004, -2.0 => false
3.141592653589793, 3.141592653589793 => true

Lobster

Translation of: Rust
// Return whether the two numbers `a` and `b` are close.
// Closeness is determined by the `epsilon` parameter - 
// the numbers are considered close if the difference between them
// is no more than epsilon * max(abs(a), abs(b)).
//
def isclose(a, b, epsilon):
    return abs(a - b) <= max(abs(a), abs(b)) * epsilon

let tv = [
    xy { 100000000000000.01, 100000000000000.011 },
    xy { 100.01, 100.011 },
    xy { 10000000000000.001 / 10000.0, 1000000000.0000001000 },
    xy { 0.001, 0.0010000001 },
    xy { 0.000000000000000000000101, 0.0 },
    xy { sqrt(2.0) * sqrt(2.0), 2.0 },
    xy { -sqrt(2.0) * sqrt(2.0), -2.0 },
    xy { 3.14159265358979323846, 3.14159265358979324 }
    ]

for(tv) t:
    print concat_string([string(t.x), if isclose(t.x, t.y, 1.0e-9): """ ≈ """ else: """ ≉ """, string(t.y)], "")
Output:
100000000000000.0 ≈ 100000000000000.0
100.01 ≉ 100.011
1000000000.0 ≈ 1000000000.0
0.001 ≉ 0.0010000001
0.0 ≉ 0.0
2.0 ≈ 2.0
-2.0 ≈ -2.0
3.14159265359 ≈ 3.14159265359

Lua

Translation of: C
function approxEquals(value, other, epsilon)
    return math.abs(value - other) < epsilon
end

function test(a, b)
    local epsilon = 1e-18
    print(string.format("%f, %f => %s", a, b, tostring(approxEquals(a, b, epsilon))))
end

function main()
    test(100000000000000.01, 100000000000000.011);
    test(100.01, 100.011)
    test(10000000000000.001 / 10000.0, 1000000000.0000001000)
    test(0.001, 0.0010000001)
    test(0.000000000000000000000101, 0.0)
    test(math.sqrt(2.0) * math.sqrt(2.0), 2.0)
    test(-math.sqrt(2.0) * math.sqrt(2.0), -2.0)
    test(3.14159265358979323846, 3.14159265358979324)
end

main()
Output:
100000000000000.020000, 100000000000000.020000 => true
100.010000, 100.011000 => false
1000000000.000000, 1000000000.000000 => false
0.001000, 0.001000 => false
0.000000, 0.000000 => true
2.000000, 2.000000 => false
-2.000000, -2.000000 => false
3.141593, 3.141593 => true

Mathematica/Wolfram Language

ClearAll[CloseEnough]
CloseEnough[a_, b_, tol_] := Chop[a - b, tol] == 0
numbers = {
   {100000000000000.01, 100000000000000.011},
   {100.01, 100.011},
   {10000000000000.001/10000.0, 1000000000.0000001000},
   {0.001, 0.0010000001},
   {0.000000000000000000000101, 0.0},
   {Sqrt[2.0] Sqrt[2.0], 2.0}, {-Sqrt[2.0] Sqrt[2.0], -2.0},
   {3.14159265358979323846, 3.14159265358979324}
   };
(*And@@Flatten[Map[MachineNumberQ,numbers,{2}]]*)
{#1, #2, CloseEnough[#1, #2, 10^-9]} & @@@ numbers // Grid
Output:
1.*10^14	1.0000000000000001*10^14	True
100.01	100.011	False
1.*10^9	1.000000000000000100*10^9	False
0.001	0.001	True
1.01*10^-22	0.	True
2.	2.	True
-2.	-2.	True
3.1415926535897932385	3.1415926535897932	True

Nim

To compare the floating point values, we use a relative tolerance.

In order to display the values “a” and “b” as provided, without any rounding, we transmit them as strings to a comparison procedure which compute the floating point values. If the first value “a” is provided as an operation, we use a comparison procedure which accepts the computed value of “a” as second parameter. Here, “b” is never provided as an operation and can always be transmitted as a string.

from math import sqrt
import strformat
import strutils

const Tolerance = 1e-10

proc `~=`(a, b: float): bool =
  ## Check if "a" and "b" are close.
  ## We use a relative tolerance to compare the values.
  result = abs(a - b) < max(abs(a), abs(b)) * Tolerance

proc compare(a, b: string) =
  ## Compare "a" and "b" transmitted as strings.
  ## Values are computed using "parseFloat".
  let r = a.parseFloat() ~= b.parseFloat()
  echo fmt"{a} ~= {b} is {r}"

proc compare(a: string; avalue: float; b: string) =
  ## Compare "a" and "b" transmitted as strings.
  ## The value of "a" is transmitted and not computed.
  let r = avalue ~= b.parseFloat()
  echo fmt"{a} ~= {b} is {r}"


compare("100000000000000.01", "100000000000000.011")
compare("100.01", "100.011")
compare("10000000000000.001 / 10000.0", 10000000000000.001 / 10000.0, "1000000000.0000001000")
compare("0.001", "0.0010000001")
compare("0.000000000000000000000101", "0.0")
compare("sqrt(2) * sqrt(2)", sqrt(2.0) * sqrt(2.0), "2.0")
compare("-sqrt(2) * sqrt(2)", -sqrt(2.0) * sqrt(2.0), "-2.0")
compare("3.14159265358979323846", "3.14159265358979324")
Output:
100000000000000.01 ~= 100000000000000.011 is true
100.01 ~= 100.011 is false
10000000000000.001 / 10000.0 ~= 1000000000.0000001000 is true
0.001 ~= 0.0010000001 is false
0.000000000000000000000101 ~= 0.0 is false
sqrt(2) * sqrt(2) ~= 2.0 is true
-sqrt(2) * sqrt(2) ~= -2.0 is true
3.14159265358979323846 ~= 3.14159265358979324 is true

OCaml

let approx_eq v1 v2 epsilon =
  Float.abs (v1 -. v2) < epsilon

let test a b =
  let epsilon = 1e-18 in
  Printf.printf "%g, %g => %b\n" a b (approx_eq a b epsilon)

let () =
  test 100000000000000.01 100000000000000.011;
  test 100.01 100.011;
  test (10000000000000.001 /. 10000.0) 1000000000.0000001000;
  test 0.001 0.0010000001;
  test 0.000000000000000000000101 0.0;
  test ((sqrt 2.0) *. (sqrt 2.0)) 2.0;
  test (-. (sqrt 2.0) *. (sqrt 2.0)) (-2.0);
  test 3.14159265358979323846 3.14159265358979324;
;;

Pascal

Works with: Extended Pascal

The constants minReal, maxReal and epsReal are defined by the ISO standard 10206 (“Extended Pascal”). However, their specific values are “implementation defined”, i. e. it is up to the compiler vendors to assign concrete values to them.

program approximateEqual(output);

{
	\brief determines whether two `real` values are approximately equal
	\param x a reference value
	\param y the value to compare with \param x
	\return true if \param x is equal or approximately equal to \param y
}
function equal(protected x, y: real): Boolean;
	function approximate: Boolean;
		function d(protected x: real): integer;
		begin
			d := trunc(ln(abs(x) + minReal) / ln(2)) + 1
		end;
	begin
		approximate := abs(x - y) <= epsReal * (maxReal / (d(x) + d(y)))
	end;
begin
	equal := (x = y) or_else (x * y >= 0.0) and_then approximate
end;

{ --- auxilliary routines ---------------------------------------------- }
procedure test(protected x, y: real);
const
	{ ANSI escape code for color coding }
	CSI = chr(8#33) + '[';
	totalMinimumWidth = 40;
	postRadixDigits = 24;
begin
	write(x:totalMinimumWidth:postRadixDigits, '':1, CSI, '1;3');
	
	if equal(x, y) then
	begin
		if x = y then
		begin
			write('2m≅')
		end
		else
		begin
			write('5m≆')
		end
	end
	else
	begin
		write('1m≇')
	end;
	
	writeLn(CSI, 'm', '':1, y:totalMinimumWidth:postRadixDigits)
end;

{ === MAIN ============================================================= }
var
	n: integer;
	x: real;
begin
	{ Variables were used to thwart compile-time evaluation done }
	{ by /some/ compilers potentially confounding the results. }
	n := 2;
	x := 100000000000000.01;
	
	test(x, 100000000000000.011);
	test(100.01, 100.011);
	test(x / 10000.0, 1000000000.0000001000);
	test(0.001, 0.0010000001);
	test(0.000000000000000000000101, 0.0);
	x := sqrt(n);
	test(sqr(x), 2.0);
	test((-x) * x, -2.0);
	test(3.14159265358979323846, 3.14159265358979324)
end.
Output:
100000000000000.015625000000000000000000 ≅ 100000000000000.015625000000000000000000
            100.010000000000005115907697 ≆             100.010999999999995679900167
    10000000000.000001907348632812500000 ≆      1000000000.000000119209289550781250
              0.001000000000000000020817 ≇               0.001000000100000000054917
              0.000000000000000000000101 ≇               0.000000000000000000000000
              2.000000000000000444089210 ≆               2.000000000000000000000000
             -2.000000000000000444089210 ≆              -2.000000000000000000000000
              3.141592653589793115997963 ≅               3.141592653589793115997963

The shown output was generated by a program compiled by the GPC (GNU Pascal Compiler). Due to technical limitations it was not possible to reproduce 100.01 ≆ 100.011 as requested by the task specification. The computer had an IEEE-754-compliant FPU with 80-bit precision. Note that Pascal’s write/writeLn/writeStr routines (the last one is only available in Extended Pascal) produce rounded representations.

Perl

Passes task tests, but use the module Test::Number::Delta for anything of real importance.

use strict;
use warnings;

sub is_close {
    my($a,$b,$eps) = @_;
    $eps //= 15;
    my $epse = $eps;
    $epse++ if sprintf("%.${eps}f",$a) =~ /\./;
    $epse++ if sprintf("%.${eps}f",$a) =~ /\-/;
    my $afmt = substr((sprintf "%.${eps}f", $a), 0, $epse);
    my $bfmt = substr((sprintf "%.${eps}f", $b), 0, $epse);
    printf "%-5s %s ≅ %s\n", ($afmt eq $bfmt ? 'True' : 'False'), $afmt, $bfmt;
}

for (
    [100000000000000.01, 100000000000000.011],
    [100.01, 100.011],
    [10000000000000.001 / 10000.0, 1000000000.0000001000],
    [0.001, 0.0010000001],
    [0.000000000000000000000101, 0.0],
    [sqrt(2) * sqrt(2), 2.0],
    [-sqrt(2) * sqrt(2), -2.0],
    [100000000000000003.0, 100000000000000004.0],
    [3.14159265358979323846, 3.14159265358979324]
    ) {
        my($a,$b) = @$_;
        is_close($a,$b);
}

print "\nTolerance may be adjusted.\n";
my $real_pi  = 2 * atan2(1, 0);
my $roman_pi = 22/7;
is_close($real_pi,$roman_pi,$_) for <10 3>;
Output:
True  100000000000000.0 ≅ 100000000000000.0
False 100.0100000000000 ≅ 100.0109999999999
True  1000000000.000000 ≅ 1000000000.000000
False 0.001000000000000 ≅ 0.001000000100000
True  0.000000000000000 ≅ 0.000000000000000
True  2.000000000000000 ≅ 2.000000000000000
True  -2.000000000000000 ≅ -2.000000000000000
True  10000000000000000 ≅ 10000000000000000
True  3.141592653589793 ≅ 3.141592653589793

Tolerance may be adjusted.
False 3.141592653 ≅ 3.142857142
True  3.14 ≅ 3.14

Phix

Traditionally I have always just used sprintf() to compare floating point atoms in phix.
This task (imo) is trying to make a general-purpose routine out of code which is best tailored for each and every specific task.
It proved much harder to get decent-looking output than perform the tests, hence I allowed both the compare (cfmt) and display (dfmt) formats to be overridden.
I got a different result for test 4 to everyone else, but simply setting the cfmt to "%.10f" got it the NOT.
Likewise something similar for the trickier/ambiguous test 5, for which "0.000000" is as good as anything I can do, and both now show how to get either a true or false result.

with javascript_semantics
procedure test(atom a,b, string dfmt="%g", cfmt="%g")
    string ca = sprintf(cfmt,a),
           cb = sprintf(cfmt,b),
           eqs = iff(ca=cb?"":"NOT "),
           da = sprintf(dfmt,a),
           db = sprintf(dfmt,b)
    printf(1,"%30s and\n%30s are %sapproximately equal\n",{da,db,eqs})
end procedure
 
test(100000000000000.01,100000000000000.011,"%.3f")
test(100.01,100.011,"%.3f")
test(10000000000000.001/10000.0,1000000000.0000001000,"%.10f")
test(0.001,0.0010000001,"%.10f")         -- both
test(0.001,0.0010000001,"%.10f","%.10f")  -- ways
test(0.000000000000000000000101,0.0,"%f")         -- both
test(0.000000000000000000000101,0.0,"%f","%6f")   -- ways
test(sqrt(2)*sqrt(2),2.0)
test(-sqrt(2)*sqrt(2),-2.0)
test(3.14159265358979323846,3.14159265358979324,"%.20f")
Output:

64 bit (implied by some of the accuracies specified for this task):

           100000000000000.010 and
           100000000000000.011 are approximately equal
                       100.010 and
                       100.011 are NOT approximately equal
         1000000000.0000001001 and
         1000000000.0000001000 are approximately equal
                  0.0010000000 and
                  0.0010000001 are approximately equal
                  0.0010000000 and
                  0.0010000001 are NOT approximately equal
                      0.000000 and
                      0.000000 are NOT approximately equal
                      0.000000 and
                      0.000000 are approximately equal
                             2 and
                             2 are approximately equal
                            -2 and
                            -2 are approximately equal
        3.14159265358979323851 and
        3.14159265358979324003 are approximately equal

32 bit (in fact a couple of them, the first and last pairs, are actually genuinely identical):

           100000000000000.016 and
           100000000000000.016 are approximately equal
                       100.010 and
                       100.011 are NOT approximately equal
         1000000000.0000002384 and
         1000000000.0000001192 are approximately equal
                  0.0010000000 and
                  0.0010000001 are approximately equal
                  0.0010000000 and
                  0.0010000001 are NOT approximately equal
                      0.000000 and
                      0.000000 are NOT approximately equal
                      0.000000 and
                      0.000000 are approximately equal
                             2 and
                             2 are approximately equal
                            -2 and
                            -2 are approximately equal
            3.1415926535897931 and
            3.1415926535897931 are approximately equal

Python

The Python source documentation states:

math.isclose -> bool
    a: double
    b: double
    *
    rel_tol: double = 1e-09
        maximum difference for being considered "close", relative to the
        magnitude of the input values
    abs_tol: double = 0.0
        maximum difference for being considered "close", regardless of the
        magnitude of the input values
Determine whether two floating point numbers are close in value.
Return True if a is close in value to b, and False otherwise.
For the values to be considered close, the difference between them
must be smaller than at least one of the tolerances.
-inf, inf and NaN behave similarly to the IEEE 754 Standard.  That
is, NaN is not close to anything, even itself.  inf and -inf are
only close to themselves.
from numpy import sqrt
from math import isclose

testvalues = [[100000000000000.01,           100000000000000.011],
              [100.01,                       100.011],
              [10000000000000.001 / 10000.0, 1000000000.0000001000],
              [0.001,                        0.0010000001],
              [0.000000000000000000000101,   0.0],
              [sqrt(2) * sqrt(2),            2.0],
              [-sqrt(2) * sqrt(2),          -2.0],
              [3.14159265358979323846,       3.14159265358979324]]

for (x, y) in testvalues:
    maybenot = "is" if isclose(x, y) else "is NOT"
    print(x, maybenot, "approximately equal to ", y)
Output:
100000000000000.02 is approximately equal to  100000000000000.02
100.01 is NOT approximately equal to  100.011
1000000000.0000002 is approximately equal to  1000000000.0000001
0.001 is NOT approximately equal to  0.0010000001
1.01e-22 is NOT approximately equal to  0.0
2.0 is approximately equal to  2.0
-2.0 is approximately equal to  -2.0
3.141592653589793 is approximately equal to  3.141592653589793

R

The base library has the function all.equal() for this task. However, when the numbers are not equal, rather than return FALSE, it tries to explain the difference. To fix this, we use isTRUE(all.equal(....)) instead.

approxEq <- function(...) isTRUE(all.equal(...))
tests <- rbind(c(100000000000000.01, 100000000000000.011),
             c(100.01, 100.011),
             c(10000000000000.001 / 10000.0, 1000000000.0000001000),
             c(0.001, 0.0010000001),
             c(0.000000000000000000000101, 0.0),
             c(sqrt(2) * sqrt(2), 2.0),
             c(-sqrt(2) * sqrt(2), -2.0),
             c(3.14159265358979323846, 3.14159265358979324))
results <- mapply(approxEq, tests[, 1], tests[, 2])
#All that remains is to print out our results in a presentable way:
printableTests <- format(tests, scientific = FALSE)
print(data.frame(x = printableTests[, 1], y = printableTests[, 2], Equal = results, row.names = paste0("Test ", 1:8, ": ")))
Output:
                                                x                                        y Equal
Test 1:  100000000000000.015625000000000000000000 100000000000000.015625000000000000000000  TRUE
Test 2:              100.010000000000005115907697             100.010999999999995679900167 FALSE
Test 3:       1000000000.000000238418579101562500      1000000000.000000119209289550781250  TRUE
Test 4:                0.001000000000000000020817               0.001000000100000000054917 FALSE
Test 5:                0.000000000000000000000101               0.000000000000000000000000  TRUE
Test 6:                2.000000000000000444089210               2.000000000000000000000000  TRUE
Test 7:               -2.000000000000000444089210              -2.000000000000000000000000  TRUE
Test 8:                3.141592653589793115997963               3.141592653589793115997963  TRUE

Racket

In Racket, a number literal with decimal point is considered a flonum, an inexact number which could be either 30 or 62 bits depending on machines. By prefixing the literal with #e, it is now considered an exact, rational number. In this task, we test the approximate equality on both variants:

#lang racket

(define ( a b [tolerance 1e-9])
  (<= (abs (/ (- a b) (max a b))) tolerance))

(define all-tests
  `(([100000000000000.01 100000000000000.011]
     [100.01 100.011]
     [,(/ 10000000000000.001 10000.0) 1000000000.0000001000]
     [0.001 0.0010000001]
     [0.000000000000000000000101 0.0]
     [,(* (sqrt 2) (sqrt 2)) 2.0]
     [,(* (- (sqrt 2)) (sqrt 2)) -2.0]
     [100000000000000003.0 100000000000000004.0]
     [3.14159265358979323846 3.14159265358979324])
    ([#e100000000000000.01 #e100000000000000.011]
     [#e100.01 #e100.011]
     [,(/ #e10000000000000.001 #e10000.0) #e1000000000.0000001000]
     [#e0.001 #e0.0010000001]
     [#e0.000000000000000000000101 #e0.0]
     [,(* (sqrt 2) (sqrt 2)) #e2.0]
     [,(* (- (sqrt 2)) (sqrt 2)) #e-2.0]
     [100000000000000003 100000000000000004]
     [#e3.14159265358979323846 #e3.14159265358979324])))

(define (format-num x)
  (~a (~r x #:precision 30) #:min-width 50 #:align 'right))

(for ([tests (in-list all-tests)] [name '("inexact" "exact")])
  (printf "~a:\n" name)
  (for ([test (in-list tests)])
    (match-define (list a b) test)
    (printf "~a ~a: ~a\n" (format-num a) (format-num b) ( a b)))
  (newline))
Output:
inexact:
    100000000000000.015625000000000000310697263104     100000000000000.015625000000000000310697263104: #t
                100.010000000000005116710235406336                 100.010999999999995680439855480832: #f
         1000000000.000000238418579101562504740864          1000000000.000000119209289550781252370432: #t
                  0.001000000000000000013287555072                   0.001000000100000000093229940736: #f
                        0.000000000000000000000101                                                  0: #f
                  2.000000000000000444089209850063                                                  2: #t
                 -2.000000000000000444089209850063                                                 -2: #t
                                100000000000000000                                 100000000000000000: #t
                  3.141592653589793121575456735232                   3.141592653589793121575456735232: #t

exact:
                                100000000000000.01                                100000000000000.011: #t
                                            100.01                                            100.011: #f
                                1000000000.0000001                                 1000000000.0000001: #t
                                             0.001                                       0.0010000001: #f
                        0.000000000000000000000101                                                  0: #f
                  2.000000000000000444089209850063                                                  2: #t
                 -2.000000000000000444089209850063                                                 -2: #t
                                100000000000000003                                 100000000000000004: #t
                            3.14159265358979323846                                3.14159265358979324: #t

Raku

(formerly Perl 6)

Works with: Rakudo version 2019.07.1

Is approximately equal to is a built-in operator in Raku. Unicode ≅, or the ASCII equivalent: =~=. By default it uses a tolerance of 1e-15 times the order of magnitude of the larger comparand, though that is adjustable by setting the dynamic variable $*TOLERANCE to the desired value. Probably a good idea to localize the changed $*TOLERANCE as it will affect all comparisons within its scope.

Most of the following tests are somewhat pointless in Raku. To a large extent, when dealing with Rational values, you don't really need to worry about "approximately equal to", and all of the test values below, with the exception of sqrt(2), are Rats by default, and exact. You would have to specifically coerce them to Nums (floating point) to lose the precision.

For example, in Raku, the sum of .1, .2, .3, & .4 is identically equal to 1.

say 0.1 + 0.2 + 0.3 + 0.4 === 1.0000000000000000000000000000000000000000000000000000000000000000000000000; # True

It's also approximately equal to 1 but... ¯\_(ツ)_/¯

for
    100000000000000.01, 100000000000000.011,
    100.01, 100.011,
    10000000000000.001 / 10000.0, 1000000000.0000001000,
    0.001, 0.0010000001,
    0.000000000000000000000101, 0.0,
    sqrt(2) * sqrt(2), 2.0,
    -sqrt(2) * sqrt(2), -2.0,
    100000000000000003.0, 100000000000000004.0,
    3.14159265358979323846, 3.14159265358979324

  -> $a, $b {
    say "$a ≅ $b: ", $a$b;
}

say "\nTolerance may be adjusted.";

say 22/7, " ≅ ", π, ": ", 22/7π;
{ # Localize the tolerance to only this block
  my $*TOLERANCE = .001;
  say 22/7, " ≅ ", π, ": ", 22/7π;
}
Output:
100000000000000.01 ≅ 100000000000000.011: True
100.01 ≅ 100.011: False
1000000000.0000001 ≅ 1000000000.0000001: True
0.001 ≅ 0.0010000001: False
0.000000000000000000000101 ≅ 0: True
2.0000000000000004 ≅ 2: True
-2.0000000000000004 ≅ -2: True
100000000000000003 ≅ 100000000000000004: True
3.141592653589793226752 ≅ 3.14159265358979324: True

Tolerance may be adjusted.
3.142857 ≅ 3.141592653589793: False
3.142857 ≅ 3.141592653589793: True

ReScript

let approx_eq = (v1, v2, epsilon) => {
  abs_float (v1 -. v2) < epsilon
}

let test = (a, b) => {
  let epsilon = 1e-18
  Printf.printf("%g, %g => %b\n", a, b, approx_eq(a, b, epsilon))
}

{
  test(100000000000000.01, 100000000000000.011)
  test(100.01, 100.011)
  test(10000000000000.001 /. 10000.0, 1000000000.0000001000)
  test(0.001, 0.0010000001)
  test(0.000000000000000000000101, 0.0)
  test(sqrt(2.0) *. sqrt(2.0), 2.0)
  test(-. sqrt(2.0) *. sqrt(2.0), (-2.0))
  test(3.14159265358979323846, 3.14159265358979324)
}

REXX

Since the REXX language uses decimal digits (characters) for floating point numbers (and integers),   it's just a matter of
choosing the   number   of decimal digits for the precision to be used for arithmetic   (in this case, fifteen decimal digits).

The choosing of the number of decimal digits is performed via the REXX statement:   numeric digits   nnn

/*REXX program mimics an  "approximately equal to"  for comparing floating point numbers*/
numeric digits 15                                /*what other FP hardware normally uses.*/
@.=                                              /*assign default for the   @   array.  */
parse arg @.1                                    /*obtain optional argument from the CL.*/
if @.1=''  |  @.1==","  then do;   @.1= 100000000000000.01           100000000000000.011
                                   @.2= 100.01                       100.011
                                   @.3= 10000000000000.001 / 10000   1000000000.0000001000
                                   @.4= 0.001                        0.0010000001
                                   @.5= 0.00000000000000000000101    0.0
                                   @.6=  sqrt(2) * sqrt(2)           2.0
                                   @.7= -sqrt(2) * sqrt(2)           '-2.0'
                                   @.8= 3.14159265358979323846       3.14159265358979324
          /* added ───► */         @.9= 100000000000000003.0         100000000000000004.0
                             end
     do j=1  while @.j\==''                      /*process CL argument or the array #s. */
     say
     say center(' processing pair ' j" ",71,'═') /*display a title for the pair of #s.  */
     parse value  @.j  with  a  b                /*extract two values from a pair of #s.*/
     say 'A='   a                                /*display the value of  A  to the term.*/
     say 'B='   b                                /*   "     "    "    "  B   "  "    "  */
     say right('A approximately equal to B?', 65)   word("false true", 1 + approxEQ(a,b) )
     end   /*j*/                                 /* [↑]  right─justify text & true/false*/
exit                                             /*stick a fork in it,  we're all done. */
/*──────────────────────────────────────────────────────────────────────────────────────*/
approxEQ: procedure; parse arg x,y;   return x=y /*floating point compare with 15 digits*/
/*──────────────────────────────────────────────────────────────────────────────────────*/
sqrt: procedure; parse arg x;  if x=0  then return 0;  d=digits();  numeric digits;  h=d+6
      numeric form; m.=9; parse value format(x,2,1,,0) 'E0' with g "E" _ .; g=g *.5'e'_ %2
        do j=0  while h>9;      m.j=h;               h=h%2+1;       end  /*j*/
        do k=j+5  to 0  by -1;  numeric digits m.k;  g=(g+x/g)*.5;  end  /*k*/; return g/1
output   when using the internal default inputs:
═════════════════════════ processing pair  1 ══════════════════════════
A= 100000000000000.01
B= 100000000000000.011
                                      A approximately equal to B? true

═════════════════════════ processing pair  2 ══════════════════════════
A= 100.01
B= 100.011
                                      A approximately equal to B? false

═════════════════════════ processing pair  3 ══════════════════════════
A= 1000000000
B= 1000000000.0000001000
                                      A approximately equal to B? true

═════════════════════════ processing pair  4 ══════════════════════════
A= 0.001
B= 0.0010000001
                                      A approximately equal to B? false

═════════════════════════ processing pair  5 ══════════════════════════
A= 0.00000000000000000000101
B= 0.0
                                      A approximately equal to B? false

═════════════════════════ processing pair  6 ══════════════════════════
A= 2.00000000000000
B= 2.0
                                      A approximately equal to B? true

═════════════════════════ processing pair  7 ══════════════════════════
A= -2.00000000000000
B= -2.0
                                      A approximately equal to B? true

═════════════════════════ processing pair  8 ══════════════════════════
A= 3.14159265358979323846
B= 3.14159265358979324
                                      A approximately equal to B? true

═════════════════════════ processing pair  9 ══════════════════════════
A= 100000000000000003.0
B= 100000000000000004.0
                                      A approximately equal to B? true

Ruby

Most work went into handling weird Float values like NaN and Infinity.

require "bigdecimal"

testvalues = [[100000000000000.01,           100000000000000.011],
              [100.01,                       100.011],
              [10000000000000.001 / 10000.0, 1000000000.0000001000],
              [0.001,                        0.0010000001],
              [0.000000000000000000000101,   0.0],
              [(2**0.5) * (2**0.5),            2.0],
              [-(2**0.5) * (2**0.5),          -2.0],
              [BigDecimal("3.14159265358979323846"),       3.14159265358979324],
              [Float::NAN, Float::NAN,],
              [Float::INFINITY, Float::INFINITY],
               ]

class  Numeric
  def close_to?(num, tol = Float::EPSILON)
    return true  if self == num
    return false if (self.to_f.nan? or num.to_f.nan?)        # NaN is not even close to itself
    return false if [self, num].count( Float::INFINITY) == 1 # Infinity is only close to itself
    return false if [self, num].count(-Float::INFINITY) == 1
    (self-num).abs <= tol * ([self.abs, num.abs].max)
  end
end

testvalues.each do |a,b|
  puts "#{a} #{a.close_to?(b) ? '≈' : '≉'} #{b}"
end
Output:
100000000000000.02 ≈ 100000000000000.02
100.01 ≉ 100.011
1000000000.0000002 ≈ 1000000000.0000001
0.001 ≉ 0.0010000001
0.101e-21 ≉ 0.0
2.0000000000000004 ≈ 2.0
-2.0000000000000004 ≈ -2.0
0.314159265358979323846e1 ≈ 3.141592653589793
NaN ≉ NaN
Infinity ≈ Infinity

Rust

/// Return whether the two numbers `a` and `b` are close.
/// Closeness is determined by the `epsilon` parameter - 
/// the numbers are considered close if the difference between them
/// is no more than epsilon * max(abs(a), abs(b)).
fn isclose(a: f64, b: f64, epsilon: f64) -> bool {
    (a - b).abs() <= a.abs().max(b.abs()) * epsilon
}

fn main() {
    fn sqrt(x: f64) -> f64 { x.sqrt() }
    macro_rules! test {
        ($a: expr, $b: expr) => {
            let operator = if isclose($a, $b, 1.0e-9) { '≈' } else { '≉' };
            println!("{:>28} {} {}", stringify!($a), operator, stringify!($b))
        }
    }

    test!(100000000000000.01, 100000000000000.011);
    test!(100.01, 100.011);
    test!(10000000000000.001/10000.0, 1000000000.0000001000);
    test!(0.001, 0.0010000001);
    test!(0.000000000000000000000101, 0.0);
    test!( sqrt(2.0) * sqrt(2.0), 2.0);
    test!(-sqrt(2.0) * sqrt(2.0), -2.0);
    test!(3.14159265358979323846, 3.14159265358979324);
}
Output:
          100000000000000.01 ≈ 100000000000000.011
                      100.01 ≉ 100.011
10000000000000.001 / 10000.0 ≈ 1000000000.0000001000
                       0.001 ≉ 0.0010000001
  0.000000000000000000000101 ≉ 0.0
       sqrt(2.0) * sqrt(2.0) ≈ 2.0
      -sqrt(2.0) * sqrt(2.0) ≈ -2.0
      3.14159265358979323846 ≈ 3.14159265358979324

Scala

Output:
Best seen running in your browser by Scastie (remote JVM).
object Approximate extends App {
  val (ok, notOk, ε) = ("👌", "❌", 1e-18d)

  private def approxEquals(value: Double, other: Double, epsilon: Double) =
    scala.math.abs(value - other) < epsilon

  private def test(a: BigDecimal, b: BigDecimal, expected: Boolean): Unit = {
    val result = approxEquals(a.toDouble, b.toDouble, ε)
    println(f"$a%40.24f ≅ $b%40.24f => $result%5s ${if (expected == result) ok else notOk}")
  }

  test(BigDecimal("100000000000000.010"), BigDecimal("100000000000000.011"), true)
  test(BigDecimal("100.01"), BigDecimal("100.011"), false)
  test(BigDecimal(10000000000000.001 / 10000.0), BigDecimal("1000000000.0000001000"), false)
  test(BigDecimal("0.001"), BigDecimal("0.0010000001"), false)
  test(BigDecimal("0.000000000000000000000101"), BigDecimal(0), true)
  test(BigDecimal(math.sqrt(2) * math.sqrt(2d)), BigDecimal(2.0), false)
  test(BigDecimal(-Math.sqrt(2) * Math.sqrt(2)), BigDecimal(-2.0), false)
  test(BigDecimal("3.14159265358979323846"), BigDecimal("3.14159265358979324"), true)
}

Sidef

Two values can be compared for approximate equality by using the built-in operator , available in ASCII as =~=, which does approximate comparison by rounding both operands at (PREC>>2)-1 decimals. However, by default, Sidef uses a floating-point precision of 192 bits.

[
    100000000000000.01, 100000000000000.011,
    100.01, 100.011,
    10000000000000.001 / 10000.0, 1000000000.0000001000,
    0.001, 0.0010000001,
    0.000000000000000000000101, 0.0,
    sqrt(2) * sqrt(2), 2.0,
    -sqrt(2) * sqrt(2), -2.0,
    sqrt(-2) * sqrt(-2), -2.0,
    cbrt(3)**3, 3,
    cbrt(-3)**3, -3,
    100000000000000003.0, 100000000000000004.0,
    3.14159265358979323846, 3.14159265358979324
].each_slice(2, {|a,b|
    say ("#{a}#{b}: ", a  b)
})
Output:
100000000000000.01 ≅ 100000000000000.011: false
100.01 ≅ 100.011: false
1000000000.0000001 ≅ 1000000000.0000001: true
0.001 ≅ 0.0010000001: false
0.000000000000000000000101 ≅ 0: false
2 ≅ 2: true
-2 ≅ -2: true
-2 ≅ -2: true
3 ≅ 3: true
-3-7.82914889268316957969274243345625157631318402415e-58i ≅ -3: true
100000000000000003 ≅ 100000000000000004: false
3.14159265358979323846 ≅ 3.14159265358979324: false

The Number n.round(-k) can be used for rounding the number n to k decimal places. A positive argument can be used for rounding before the decimal point.

var a = 100000000000000.01
var b = 100000000000000.011

# Rounding at 2 and 3 decimal places, respectively
say (round(a, -2) == round(b, -2))      # true
say (round(a, -3) == round(b, -3))      # false

There is also the built-in approx_cmp(a, b, k) method, which is equivalent with a.round(k) <=> b.round(k).

var a = 22/7
var b = Num.pi

say ("22/7 ≅ π at 2 decimals: ", approx_cmp(a, b, -2) == 0)
say ("22/7 ≅ π at 3 decimals: ", approx_cmp(a, b, -3) == 0)
Output:
22/7 ≅ π at 2 decimals: true
22/7 ≅ π at 3 decimals: false

Additionally, the rat_approx method can be used for computing a very good rational approximation to a given real value:

say (1.33333333.rat_approx == 4/3)   # true
say (zeta(-5).rat_approx == -1/252)  # true

Rational approximations illustrated for substrings of PI:

for k in (3..19) {
    var r = Str(Num.pi).first(k)
    say ("rat_approx(#{r}) = ", Num(r).rat_approx.as_frac)
}
Output:
rat_approx(3.1) = 31/10
rat_approx(3.14) = 22/7
rat_approx(3.141) = 245/78
rat_approx(3.1415) = 333/106
rat_approx(3.14159) = 355/113
rat_approx(3.141592) = 355/113
rat_approx(3.1415926) = 86953/27678
rat_approx(3.14159265) = 102928/32763
rat_approx(3.141592653) = 103993/33102
rat_approx(3.1415926535) = 1354394/431117
rat_approx(3.14159265358) = 833719/265381
rat_approx(3.141592653589) = 17925491/5705861
rat_approx(3.1415926535897) = 126312511/40206521
rat_approx(3.14159265358979) = 144029661/45846065
rat_approx(3.141592653589793) = 325994779/103767361
rat_approx(3.1415926535897932) = 903259831/287516534
rat_approx(3.14159265358979323) = 1726375805/549522486

Smalltalk

This compares numbers given the number of ULPs by which they may differ (see wikipedia). There are slight differences in how this is named in the various dialects ¹. If required, you have to add a forwarding alias method to Number.

Works with: Smalltalk/X
{ #(100000000000000.01 100000000000000.011) .
  #(100.01 100.011) .
  {10000000000000.001 / 10000.0 . 1000000000.0000001000} .
  #(0.001 0.0010000001) .
  #(0.000000000000000000000101 0.0) .
  { 2 sqrt * 2 sqrt . 2.0} .
  { 2 sqrt negated * 2 sqrt . -2.0} .
  #(3.14159265358979323846  3.14159265358979324)
} pairsDo:[:val1 :val2 |
   Stdout printCR: e'{val1} =~= {val2} -> {val1 isAlmostEqualTo:val2 nEpsilon:2}'
]

In CUIS, this method is called isWithin:floatsFrom:.

Output:
100.01 =~= 100.011 -> false
1000000000.0 =~= 1000000000.0 -> true
0.001 =~= 0.0010000001 -> false
1.01e-22 =~= 0.0 -> true
2.0 =~= 2.0 -> true
-2.0 =~= -2.0 -> true
3.14159265358979 =~= 3.14159265358979 -> true

By default, double precision IEEE floats are used.

Swift

Using the solution proposed as an addition to the Swift standard library in SE-0259. Currently this is not accepted, but is likely to be included in the Swift Numerics module.

import Foundation

extension FloatingPoint {
  @inlinable
  public func isAlmostEqual(
    to other: Self,
    tolerance: Self = Self.ulpOfOne.squareRoot()
  ) -> Bool {
    // tolerances outside of [.ulpOfOne,1) yield well-defined but useless results,
    // so this is enforced by an assert rathern than a precondition.
    assert(tolerance >= .ulpOfOne && tolerance < 1, "tolerance should be in [.ulpOfOne, 1).")

    // The simple computation below does not necessarily give sensible
    // results if one of self or other is infinite; we need to rescale
    // the computation in that case.
    guard self.isFinite && other.isFinite else {
      return rescaledAlmostEqual(to: other, tolerance: tolerance)
    }

    // This should eventually be rewritten to use a scaling facility to be
    // defined on FloatingPoint suitable for hypot and scaled sums, but the
    // following is good enough to be useful for now.
    let scale = max(abs(self), abs(other), .leastNormalMagnitude)
    return abs(self - other) < scale*tolerance
  }

  @usableFromInline
  internal func rescaledAlmostEqual(to other: Self, tolerance: Self) -> Bool {
    // NaN is considered to be not approximately equal to anything, not even
    // itself.
    if self.isNaN || other.isNaN { return false }
    if self.isInfinite {
      if other.isInfinite { return self == other }

      // Self is infinite and other is finite. Replace self with the binade
      // of the greatestFiniteMagnitude, and reduce the exponent of other by
      // one to compensate.
      let scaledSelf = Self(sign: self.sign,
        exponent: Self.greatestFiniteMagnitude.exponent,
        significand: 1)
      let scaledOther = Self(sign: .plus,
        exponent: -1,
        significand: other)

      // Now both values are finite, so re-run the naive comparison.
      return scaledSelf.isAlmostEqual(to: scaledOther, tolerance: tolerance)
    }

    // If self is finite and other is infinite, flip order and use scaling
    // defined above, since this relation is symmetric.
    return other.rescaledAlmostEqual(to: self, tolerance: tolerance)
  }
}

let testCases = [
  (100000000000000.01, 100000000000000.011),
  (100.01, 100.011),
  (10000000000000.001 / 10000.0, 1000000000.0000001000),
  (0.001, 0.0010000001),
  (0.000000000000000000000101, 0.0),
  (sqrt(2) * sqrt(2), 2.0),
  (-sqrt(2) * sqrt(2), -2.0),
  (3.14159265358979323846, 3.14159265358979324)
]

for testCase in testCases {
  print("\(testCase.0), \(testCase.1) => \(testCase.0.isAlmostEqual(to: testCase.1))")
}
Output:
100000000000000.02, 100000000000000.02 => true
100.01, 100.011 => false
1000000000.0000002, 1000000000.0000001 => true
0.001, 0.0010000001 => false
1.01e-22, 0.0 => false
2.0000000000000004, 2.0 => true
-2.0000000000000004, -2.0 => true
3.141592653589793, 3.141592653589793 => true

Tcl

Using decimal library

Uses tcllib's decimal library. Using a tolerance of 9 significant digits.

catch {namespace delete test_almost_equal_decimal} ;# Start with a clean namespace

namespace eval test_almost_equal_decimal {
    package require Tcl 8.5 ;# required by tcllib
    package require math::decimal ;# from tcllib
    namespace import ::math::decimal::* ;# for: setVariable, fromstr, and compare

    array set yesno {0 Yes -1 No 1 No} ;# For nice output

    # More info here: http://speleotrove.com/decimal/dax3274.html
    # This puts the library into "simplified" mode. Which
    # rounds the "decimal digits" in the coefficient to the
    # number of digits that "precision" is set to.
    setVariable extended 0
    setVariable precision 9

    set data {
        {100000000000000.01 100000000000000.011}
        {100.01 100.011}
        {[expr {10000000000000.001 / 10000.0}] 1000000000.0000001000}
        {0.001 0.0010000001}
        {0.000000000000000000000101 0.0}
        {[expr { sqrt(2) * sqrt(2)}] 2.0}
        {[expr {-sqrt(2) * sqrt(2)}] -2.0}
        {3.14159265358979323846 3.14159265358979324}
    }
    set data [subst $data] ;# resolves expressions in the list

    foreach {a b} [join $data] {
        set a_d [fromstr $a]
        set b_d [fromstr $b]

        puts [format "Is %26s ≈ %21s ? %4s." $a $b $yesno([compare $a_d $b_d])]
    }
}
Output:
Is         100000000000000.01 ≈   100000000000000.011 ?  Yes.
Is                     100.01 ≈               100.011 ?   No.
Is         1000000000.0000002 ≈ 1000000000.0000001000 ?  Yes.
Is                      0.001 ≈          0.0010000001 ?   No.
Is 0.000000000000000000000101 ≈                   0.0 ?   No.
Is         2.0000000000000004 ≈                   2.0 ?  Yes.
Is        -2.0000000000000004 ≈                  -2.0 ?  Yes.
Is     3.14159265358979323846 ≈   3.14159265358979324 ?  Yes.

Using string manipulation

catch {namespace delete test_almost_equal_string} ;# Start with a clean namespace

namespace eval test_almost_equal_string {
    package require Tcl 8.4 ;# ?Maybe earlier?
    array set yesno {1 Yes 0 No} ;# For nice output

    proc isClose {a b {prec 9}} {
        proc toCoeff {n prec} {
            set repr 40 ;# Chosen to be arbitrarily large to handle most cases
            set long [format %0.${repr}f $n] ;# Take out of scientific notation
            set map [string map {. {}} $long] ;# Remove decimal point
            set trim [string trimleft $map 0] ;# Remove leading zeros
            # restore string for comparison
            set len [string length $trim]
            if {$len < $prec} {
                set trim "${trim}[string repeat 0 [expr ($prec+1)-$len]]"
            }
            # Round last decimal place
            set rounded [format %0.f "[string range $trim 0 [expr {$prec-1}]].[string index $trim $prec]"]
            return $rounded
        }
        set a_coeff [toCoeff $a $prec]
        set b_coeff [toCoeff $b $prec]

        return [expr {$a_coeff == $b_coeff}]
    }

    set data {
        {100000000000000.01 100000000000000.011}
        {100.01 100.011}
        {[expr {10000000000000.001 / 10000.0}] 1000000000.0000001000}
        {0.001 0.0010000001}
        {0.000000000000000000000101 0.0}
        {[expr { sqrt(2) * sqrt(2)}] 2.0}
        {[expr {-sqrt(2) * sqrt(2)}] -2.0}
        {3.14159265358979323846 3.14159265358979324}
    }
    set data [subst $data] ;# resolves expressions in the list

    foreach {a b} [join $data] {
        puts [format "Is %26s ≈ %21s ? %4s." $a $b $yesno([isClose $a $b])]
    }
}
Output:
Is         100000000000000.01 ≈   100000000000000.011 ?  Yes.
Is                     100.01 ≈               100.011 ?   No.
Is         1000000000.0000002 ≈ 1000000000.0000001000 ?  Yes.
Is                      0.001 ≈          0.0010000001 ?   No.
Is 0.000000000000000000000101 ≈                   0.0 ?   No.
Is         2.0000000000000004 ≈                   2.0 ?  Yes.
Is        -2.0000000000000004 ≈                  -2.0 ?  Yes.
Is     3.14159265358979323846 ≈   3.14159265358979324 ?  Yes.

Visual Basic .NET

Translation of: C#
Imports System.Runtime.CompilerServices

Module Module1

    <Extension()>
    Function ApproxEquals(ByVal value As Double, other As Double, epsilon As Double)
        Return Math.Abs(value - other) < epsilon
    End Function

    Sub Test(a As Double, b As Double)
        Dim epsilon = 1.0E-18
        Console.WriteLine($"{a}, {b} => {a.ApproxEquals(b, epsilon)}")
    End Sub

    Sub Main()
        Test(100000000000000.02, 100000000000000.02)
        Test(100.01, 100.011)
        Test(10000000000000.002 / 10000.0, 1000000000.0000001)
        Test(0.001, 0.0010000001)
        Test(1.01E-22, 0.0)
        Test(Math.Sqrt(2) * Math.Sqrt(2), 2.0)
        Test(-Math.Sqrt(2) * Math.Sqrt(2), -2.0)
        Test(3.1415926535897931, 3.1415926535897931)
    End Sub

End Module
Output:
100000000000000, 100000000000000 => True
100.01, 100.011 => False
1000000000, 1000000000 => False
0.001, 0.0010000001 => False
1.01E-22, 0 => True
2, 2 => False
-2, -2 => False
3.14159265358979, 3.14159265358979 => True

Wren

var tol = 1e-16
var pairs = [
    [100000000000000.01, 100000000000000.011],
    [100.01, 100.011],
    [10000000000000.001 / 10000.0, 1000000000.0000001000],
    [0.001, 0.0010000001],
    [0.000000000000000000000101, 0.0],
    [2.sqrt * 2.sqrt, 2.0],
    [-2.sqrt * 2.sqrt, -2.0],
    [3.14159265358979323846, 3.14159265358979324]
]
System.print("Approximate equality of test cases for a tolerance of %(tol):")
var i = 0
for (pair in pairs) {
    i = i + 1
    System.print("  %(i) -> %((pair[0] - pair[1]).abs < tol)")
}
Output:
Approximate equality of test cases for a tolerance of 1e-16:
  1 -> true
  2 -> false
  3 -> false
  4 -> false
  5 -> true
  6 -> false
  7 -> false
  8 -> true

XPL0

func ApproxEqual(A, B);         \Return 'true' if approximately equal
real A, B;
real Epsilon;
[Epsilon:= abs(A) * 1E-15;
return abs(A-B) < Epsilon;
];

real Data;
int  I;
[Format(0, 16);
Data:=[ [100000000000000.01,   100000000000000.011],    \should return true
        [100.01,   100.011],                            \should return false
        [10000000000000.001 / 10000.0,   1000000000.0000001000],
        [0.001,   0.0010000001],
        [0.000000000000000000000101,   0.0],            \is undefined
        [sqrt(2.0) * sqrt(2.0),    2.0],
        [-1.0 * sqrt(2.0) * sqrt(2.0),   -2.0],         \-sqrt doesn't compile!
        [3.14159265358979323846,   3.14159265358979324] ];
for I:= 0 to 7 do
        [IntOut(0, I+1);  Text(0, ". ");
        RlOut(0, Data(I,0));  ChOut(0, ^ );  RlOut(0, Data(I,1));
        Text(0, if ApproxEqual(Data(I,0), Data(I,1)) then " true" else " false");
        CrLf(0);
        ];
]
Output:
1.  1.0000000000000000E+014  1.0000000000000000E+014 true
2.  1.0001000000000000E+002  1.0001100000000000E+002 false
3.  1.0000000000000000E+009  1.0000000000000000E+009 true
4.  1.0000000000000000E-003  1.0000001000000000E-003 false
5.  1.0100000000000000E-022  0.0000000000000000E+000 false
6.  2.0000000000000000E+000  2.0000000000000000E+000 true
7. -2.0000000000000000E+000 -2.0000000000000000E+000 true
8.  3.1415926535897900E+000  3.1415926535897900E+000 true

zkl

Floats are 64 bit and have the closeTo method, which takes a comparison value and tolerance. If the tolerance is >=0, comparison is absolute. If tolerance is <0 (and x!=0 and y!=0), the comparison is relative.

testValues:=T( 
   T(100000000000000.01,100000000000000.011),
   T(100.01, 100.011),
   T(10000000000000.001 / 10000.0, 1000000000.0000001),
   T(0.001, 0.0010000001),
   T(0.00000000000000000101, 0.0),
   T(  (2.0).sqrt()*(2.0).sqrt(),  2.0),
   T( -(2.0).sqrt()*(2.0).sqrt(), -2.0),
   T(100000000000000003.0, 100000000000000004.0),
   T(3.14159265358979323846, 3.14159265358979324) 
);
 
tolerance:=-1e-9;	// <0 for relative comparison
foreach x,y in (testValues){
   maybeNot:=( if(x.closeTo(y,tolerance)) " \u2248" else "!\u2248" );
   println("% 25.19g %s %- 25.19g  %g".fmt(x,maybeNot,y, (x-y).abs()));
}
Output:
     100000000000000.0156  ≈  100000000000000.0156      0
     100.0100000000000051 !≈  100.0109999999999957      0.001
     1000000000.000000238  ≈  1000000000.000000119      1.19209e-07
  0.001000000000000000021 !≈  0.001000000100000000055   1e-10
 1.010000000000000018e-18 !≈  0                         1.01e-18
     2.000000000000000444  ≈  2                         4.44089e-16
    -2.000000000000000444  ≈ -2                         4.44089e-16
       100000000000000000  ≈  100000000000000000        0
     3.141592653589793116  ≈  3.141592653589793116      0