How Basic Operations Affect Application Results

To understand why floating-point calculations can yield different results, you need to know how a system performs the most basic operations: add, subtract, multiply, divide. Recall from “Inexact Result (Rounding)” that these operations always round their results to the nearest representable floating-point value using an algorithm specified by the rounding mode, and that this rounding introduces a rounding error into the result of the operation.

For example, suppose you run the following program. Two operations that are mathematically equivalent may produce different results:

Example 3-1 Sample Program: rounderr.f

PROGRAM ROUNDERR REAL A, B, C, D, E, F PRINT *, 'Enter 4 reals:' READ *, A, B, C, D E = (A + B) * (C + D) F = (A * C) + (A * D) + (B * C) + (B * D) IF (E .EQ. F) THEN WRITE (*, 20) E, ' equals ', F ELSE WRITE (*, 20) E, ' not equal to ', F WRITE (*, *) 'Math error!' ENDIF 20 FORMAT(F, A, F) END

Depending on the values read for A, B, C, and D, the program will indeed print "Math error!", although, of course, no error has actually occurred— only legitimate rounding errors:

$ f90 rounderr.f 
rounderr.f
 program ROUNDERR
 
21 Lines Compiled
$ ./a.out 
 Enter 4 reals: 
1.1 2.2 3.3 4.4 
     25.4100018 not equal to      25.4099998
 Math error!

This example also demonstrates a guiding principle of floating-point programming:

Be very careful about testing two floating-point values or expressions for exact equality or inequality.

This principle derives from the fact that two given floating-point values are almost never equal, even when the programmer might expect them to be equal from a purely mathematical standpoint. The inequality occurs because of the rounding errors incurred during the calculation of the two values being compared.

For further discussion of this issue, see “Testing Floating-Point Values for Equality”.