Step by Step solution about How to resolve the algorithm Approximate equality step by step in the Ruby programming language

Explanation of the Provided Ruby Code:

This Ruby code defines a method close_to? on the Numeric class, which checks if two numeric values are close to each other within a specified tolerance. The code also includes a series of test cases to demonstrate the usage of this method.

Implementation Details:

• The Numeric class is extended to add the close_to? method.

• The close_to? method takes two numeric arguments, num and an optional tol (tolerance) parameter, which defaults to Float::EPSILON.

• The method first checks if the two values are exactly equal, returning true if they are.

• If the values are not equal, it checks if either of them is either NaN (Not a Number) or Float::INFINITY. These values are considered not close to anything, so false is returned.

• If neither value is NaN or Float::INFINITY, the absolute difference between the two values is calculated and compared to the tolerance. The tolerance is multiplied by the maximum absolute value of the two numbers to account for different scales.

• If the absolute difference is less than or equal to the tolerance, the method returns true, indicating that the two values are close; otherwise, it returns false.

Test Cases:

The testvalues array contains several pairs of numeric values to be tested, including values with precision errors, NaN, infinity, and square root of 2.

Output:

After running the code, the following output is displayed:

1.0e+15 ≈ 1.0000000000000001e+15
100.01 ≈ 100.011
1.0e+11 ≈ 1000000000.0000001
0.001 ≈ 0.0010000001
1.01e-22 ≈ 0.0
2.0 ≈ 2.0
-2.0 ≈ -2.0
3.141592653589793 ≈ 3.141592653589793
nan ≈ nan
inf ≈ inf

Conclusion:

This code demonstrates a useful close_to? method for comparing numeric values while allowing for a specified tolerance. The test cases illustrate its behavior with various types of numeric values, including those with precision errors and special 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