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 How to Work Around Floating-Point Accuracy/Comparison - Q69333
 
 To reliably test whether two floating-point variables or expressions
 are equal (using IEEE format), you must subtract the two variables being
 compared and test whether their difference is less than a value chosen
 at the limits of significance for single or double precision. NO OTHER
 TEST FOR EQUALITY WILL BE RELIABLE. The following formulas reliably test
 whether X and Y are equal:
 
 1. For single precision, you must test whether the difference of X and
    Y is less than the value 7 significant digits smaller than X or Y.
    Divide X or Y by 10^7 to find the comparison value. For example:
 
    To try these examples in VBDOS.EXE:
 
    1. From the File menu, choose New Project.
    2. Copy the code example to the Code window.
    3. Press F5 to run the program.
 
       IF ABS(X! - Y!) <= (X! / 10^7) THEN
           PRINT "Equal within 7 digits"
       END IF
 
 2. For double precision, you must test whether the difference of X and
    Y is less than the value 15 significant digits smaller than X or Y.
    Divide X or Y by 10^15 to find the comparison value. For example:
 
       IF ABS(X# - Y#) <= (X# / 10^15) THEN
           PRINT "Equal within 15 digits"
       END IF
 
 More Information:
 
    NOTE: Significant digits in a calculated number can be lost due to
    the following: multiple calculations, especially addition of
    numbers far apart in value, or subtraction of numbers similar in
    value. When a number results from multiple calculations, you may
    need to change your test for equality to use fewer significant
    digits to reflect the mathematical loss of significant digits. If
    your test of significance uses too many significant digits, you may
    fail to discover that numbers compared for equality are actually
    equal within the possible limit of accuracy.
 
 In Visual Basic for MS-DOS, intermediate calculations are performed
 in an internal 64-bit temporary register, which has more bits of
 accuracy than are stored in single- or double-precision variables.
 This often results in an IF statement saying that the intermediate
 calculation is not equal to the expression being compared, as in the
 following example:
 
    X = 25
    Y = 60.1
    IF 1502.5 = (X * Y) THEN PRINT "equal"
 
 Running the above code will NOT print "equal". In contrast, the
 following method using a placeholder variable will print "equal", but
 is still NOT a reliable technique as a test for equality:
 
    Z = 25 * 60.1
    IF 1502.5 = Z THEN PRINT "equal"
 
 Note that explicit numeric type casts (! for single precision, # for
 double precision) will affect the precision in which calculations are
 stored and printed. Whichever type casting you perform, you may still
 see unexpected rounding results:
 
    PRINT 69.82! + 1    ' Single precision, prints 70.82.
    PRINT 69.82# + 1    ' Double precision, prints 70.81999999999999.
 
 For an exact decimal (base 10) numeric representation, such as for
 calculations of dollars and cents, you should use the CURRENCY (@)
 data type found in BASIC PDS 7.00/7.10. The CURRENCY data type exactly
 stores up to 19 digits, with 4 digits after the decimal place.
 
 Reference:
 
 The IEEE standard attempts to balance accuracy and precision with
 numeric range and speed. Accuracy measures how many significant bits of
 precision are not lost in calculations. Precision refers to the number
 of bits in the mantissa, which determines how many decimal digits can be
 represented.
 
 The IEEE format stores numbers of the form 1.x to the power of
 y (where x and y are base 2 numbers; x is the mantissa, and y is the
 exponent).
 
 IEEE single precision has 24 bits of mantissa, and double precision
 has 53 bits of mantissa. However, all single- and double-precision
 IEEE calculations in Visual Basic for MS-DOS version 1.0 are performed
 in a 64-bit temporary register for greater accuracy.
 
 Most numbers in decimal (base 10) notation do NOT have an exact
 representation in the binary (base 2) floating-point storage format
 used in single- and double-precision data types. IEEE format cannot
 exactly represent (and must round off) all numbers that are not of the
 form 1.x to the power of y (where x and y are base 2 numbers). The
 numbers that can be exactly represented are spread out over a very wide
 range. A high density of representable numbers is near 1.0 and
 -1.0, but fewer and fewer representable numbers occur as the numbers go
 towards 0 or infinity.
 
 The above limitations often cause BASIC to return floating-point
 results different than you might expect.