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Floating point round off error

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• floating point "round off" error
  • Most values can not be represented exactly using the IEEE 754 (floating point) representation

    I.e.:

    The floating point representation has a round off error (for most of the values)

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  • Example: find the IEEE 754 code for the value 0.2:

    0.2 = 0.001100110011001100110011... Because: 0.2 *2 ------- 0 0.4 *2 ------- 0 0.8 *2 ------- 1 1.6 --> 0.6 *2 ------- 1 1.2 --> 0.2 *2 ------- 0 0.4 *2 ------- 0 0.8 *2 ------- 1 1.6 --> 0.6 *2 ------- 1 1.2 --> 0.2 and so on (repeats !)

    In other words:

    You will need an infinite number of bits to represent 0.2 exactly in binary

    Because the IEEE 754 representation uses a finite number of bits, the representation is truncated which results in a round off error !!

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  • This phenomenon is also present in the decimal number system.

    To represent 1/3 in decimal, you will need an infinite number of (decimal) digits:

    1/3 = 0.3333333333333333...

    When you truncate the representation to a finite number of digits, you will have a round off error also.

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• Caveat when working with floating point numbers
  • Consider this Java code segment:

    // /home/cs255001/demo/java/FloatError.java float f1, f2; f1 = 0.2f + 0.2f + 0.2f + 0.2f + 0.2f + 0.2f + 0.2f + 0.2f + 0.2f + 0.2f; f2 = 2.0f; if (f1 == f2) { System.out.println ("f1 == f2"); } else { System.out.println ("f1 != f2"); }

    Notice that:

    f1 = 0.2 + 0.2 + .... + 0.2 (10 times) f2 = 2.0 They should be equal to each other !!!

    But due to round off errors, if you run this program, you will get:

    f1 != f2

    The reason is: when you print the values f1 and f2 out, you will get:

    f1 = 2.000002 f2 = 2.0

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• Programming advice
  • Law on working with floating point numbers:

    Never use == and != to compare floating point numbers !!! If you want to check if the floating point values x and y are equal to each other, use: if ( Math.abs(x - y) < EPSILON ) { .... } where EPSILON is a small fraction, e.g.: EPSILON = 0.0001

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  • Example Program: (Demo above code)                                                
    • Prog file: /home/cs255001/demo/java/FloatRoundErr.java

    How to run the program:

    To compile:   javac FloatRoundErr.java          To run:   java FloatRoundErr