Floating point math issues
- 1 Floating-point is an approximation to the real number system
- 2 Safe floating-point division
- 3 Testing for equality
- 4 Testing for non-representable values
Floating-point is an approximation to the real number system
The real number system may be thought of as a number line with the following characteristics:
- Zero is the primary reference point
- From zero, the line extends infinitely in both directions towards + Infinity and -Infinity
- Between any two real numbers, there are an infinite number of other real numbers
A pictorial representation of the real number system looks like this:
-Infinity <---------- 0 ---A---B----> +Infinity
where between points A and B there are an infinite number of real numbered-values.
It is important to realize that floating-point mathematics (as implemented in all modern computer systems) is never exact but is only an approximation to the real number system. In most programming languages, floating-point real numbers are composed of groups of 4 or 8 bytes. This means that floating-point numbers are not infinitely precise, but have a maximum precision. As a consequence, floating-point math operations (especially multiplication and division) can often lead to different results than one would normally anticipate.
Here are the common number types used in both IDL and Fortran:
|IDL Number Type||Fortran Equivalent||Number of bytes||Approx. range|
|byte||BYTE||1||0 to 255|
|fix||INTEGER*2||2||0 - 32767|
|long||INTEGER*4||4||0 - 2000000000|
|float||REAL*4||4||-1e-38 to 1e+38|
|double||REAL*8||8||-1e-312 to 1e+312|
A pictorial representation of the floating point mathematics system would look like this:
-Min Representable Value < . . . . . . 0 . . A . B . . > +Max Representable Value
Because each floating-point number type is composed of a finite number of bytes, we can only represent numbers within a given range (see table above). Therefore, it is not possible to represent numbers to -Infinity or +Infinity with floating-point mathematics. Also, in on the floating-point number line pictured above, there are a finite number of possible floating-point values between points A and B.
Here are some excellent references about floating-point math and ways to avoid pitfalls:
- The Perils of Floating Point by Bruce M. Bush
- Fun with Floating Point Arithmetic
- Fortran common pitfalls (good reference for some floating-point issues)
- Avoiding floating point errors in Fortran
Safe floating-point division
Due to the approximate nature of floating-point mathematics, you need to take special precautions when dividing by small numbers. Division by zero is of course not allowed. However, under certain circumstances, the quotient of the division might not fall in the range of representable numbers.
For example, if you divided a very big number by a very small number:
REAL*4 :: A, B A = 1e20 B = 1e-20 PRINT*, A/B
then this will result in +Infinity because the order of magnitude of the answer (1e40) is not representable with a REAL*4 variable (max value ~ 1e38).
Therefore it is adviseable to use "safe division" in certain critical computations where you might have very small denominators. The algorithm is described below:
- Thank you to all that offered their suggestions for a "safe division" routine to prevent overflow. For those that are curious about the solution to the problem, I found useful to adopt a subroutine along the lines suggested by William Long:
if(exponent(a) - exponent (b) >= maxexponent(a) .OR. b==0)Then q=altv else q=a/b endif
- The = in the >= is to take into account the case when the fractional part of a is 1.111... and that of b is 1.
- It works very well.
- Thank you again,
In other words, we test if the order of magnitude of the result is in the allowable range for the given number type. If it is not, then instead of doing the division, we return the alternate value as the result.
NOTE: The alternate value that you substitute depends on the type of computation that you are doing...there is no "one-size-fits-all" substitute value.
We have implemented the above algorithm into GEOS-Chem. See routine SAFE_DIV in error_mod.f:
FUNCTION SAFE_DIV( N, D, ALTV ) RESULT( Q ) ! !****************************************************************************** ! Subroutine SAFE_DIV performs "safe division", that is to prevent overflow, ! underflow, NaN, or infinity errors. An alternate value is returned if the ! division cannot be performed. (bmy, 2/26/08) ! ! For more information, see the discussion on: http://groups.google.com/group/comp.lang.fortran/browse_thread/thread/8b367f44c419fa1d/ ! ! Arguments as Input: ! ============================================================================ ! (1 ) N : Numerator for the division ! (2 ) D : Divisor for the division ! (3 ) ALTV : Alternate value to be returned if the division can't be done ! ! NOTES: !****************************************************************************** ! ! Arguments REAL*8, INTENT(IN) :: N, D, ALTV ! Function value REAL*8 :: Q !================================================================== ! SAFE_DIV begins here! !================================================================== IF ( EXPONENT(N) - EXPONENT(D) >= MAXEXPONENT(N) .or. D==0 ) THEN Q = ALTV ELSE Q = N / D ENDIF ! Return to calling program END FUNCTION SAFE_DIV
Testing for equality
If you are working with integer-valued numbers, then you can use the equality operator
- In Fortran: == or .eq.
- In IDL: eq
directly to test if a variable is equal to a given value, e.g.
INTEGER :: A A = 1 IF ( A == 1 ) THEN PRINT*, 'A is 1' ELSE PRINT*, 'A is not 1' ENDIF
Then you would get the result
> A is 1
This is because integers are always exactly ... -3, -2, -1, 0, 1, 2, 3 ...
However, if you are working with floating-point real numbers (i.e. REAL*4 or REAL*8), then you have to be a little bit more careful. Recall from the previous section that floating-point numbers are not represented with infinite precision. Therefore, if you try to do the following equality test:
REAL*8 :: A A = 1d0 / 3d0 IF ( A == 0.333d0 ) THEN PRINT*, 'A is 1/3' ELSE PRINT*, 'A is not 1/3 but is ', A ENDIF
You will get the result:
> A is not 1/3 but is 0.333333333333333
A is a repeating decimal 0.333333333333333 up to the limit of the precision of the machine (usually about 15 or 16 decimal places). However, we tested for 0.333 and not 0.333333333333333. This is why the equality test failed.
TIP: You should never use a direct equality test with floating-point real numbers!
A better way to test for equality is to use the greater-than or less-than operators
- In Fortran: >, .gt., <, .lt.
- In IDL: gt, lt
in the following manner:
REAL*8 :: A A = 1d0/3d0 IF ( A > 1d0/3d0 .or. A < 1d0/3d0 ) THEN PRINT*, 'A is not 1d0/3d0 but is ', A ELSE PRINT*, 'A is exactly 1d0/3d0' ENDIF
This will reject all other values except exactly 1d0/3d0. Running the above example gives:
> A is exactly 1d0/3d0
In many instances it is necessary to take appropriate action if a number is zero (for example, to avoid a division by zero error). A simple test is as follows:
REAL*8 :: A A = 0d0 IF ( ABS( A ) > 0d0 ) THEN PRINT*, 'A is not 0 but is ', A ELSE PRINT*, 'A is exactly 0' ENDIF
This example gives the result:
> A is exactly 0
Note that we have used the mathematical relation ABS( x ) > a, which is true if x > a or x < -a. This test will reject all numbers that are not exactly 0.
The above algorithms are useful when you need to test if a variable is EXACTLY EQUAL to a given value. However, it is often better to test whether the variable is within some tolerance of a given value. Here are some examples:
Avoiding division by zero
For example, let's say we have the following division:
REAL*8 :: X, Y, Q ! Make X artificially large, Y artificially small for demo purposes X = 1.0d+75 Y = 1.0d-250 ! Do the division if Y is not exactly zero IF ( ABS( Y ) > 0d0 ) THEN Q = X / Y ENDIF PRINT*, 'Q is : ', Q
and we are testing to prevent the denominator Y from being zero. However, when we compile and run this code, we get the result
> Q is : Infinity
Although this is a rather contrived example, it illustrates the point that the denominator of the expression does not have to be exactly zero to cause a floating point error. In this case the error check on Y failed because we were testing for Y strictly equal to zero.
In this case it is preferable to do the division only if the magnitude of Y is greater than some specified minimum value (called an epsilon value). A good choice for the epsilon is usually 1e-30 or 1e-32. We can rewrite the above code as follows:
REAL*8 :: X, Y, Q ! Epsilon value REAL*8, PARAMETER :: EPSILON = 1d-30 ! Make X artificially large, Y artificially small for demo purposes X = 1.0d+75 Y = 1.0d-250 ! Only do the division if Y does not lie within +/- EPSILON of zero ! Otherwise return a default value IF ( ABS( Y ) > EPSILON ) THEN Q = X / Y ELSE Q = -999d0 ENDIF PRINT*, 'Q is : ', Q
Which gives the result:
> Q is : -999.000000000000
In this case, notice that the "epsilon test" determined that Y was too small to be placed in the denominator of the division, and thus prevented the division from taking place. The alternate value of -999.0 was returned.
As described in the previous section, the mathematical identity ABS( X ) > EPSILON is true only if X > +EPSILON or X < -EPSILON.
(Of course, you could also use the routine SAFE_DIV in GEOS-Chem error_mod.f which will perform a safe division, as described above.)
Testing for values close to zero
If you would like to test whether a given variable lies within a specified tolerance of zero, then we can just reverse the logic. We'll test for ABS( X ) < EPSILON, as this identity is true only if -EPSILON < X < +EPSILON. For example:
REAL*8 :: X ! Epsilon value REAL*8, PARAMETER :: EPSILON = 1d-3 X = 1d-5 IF ( ABS( X ) < EPSILON ) THEN PRINT*, 'X is within +/- EPSILON of zero.' ELSE PRINT*, 'X is not within +/- EPSILON of zero.' ENDIF
which results in:
> X is within +/- EPSILON of zero.
Testing for values close to a non-zero number
The procedure for testing if values lie within a specfied tolerance of a non-zero number is similar to the above example, just that we have to test for ABS( X - VALUE ) < EPSILON.
REAL*8 :: X, VALUE ! Epsilon value REAL*8, PARAMETER :: EPSILON = 1d-3 X = 20.00001d0 ! Value of X VALUE = 20.d0 ! Target value IF ( ABS( X - VALUE ) < EPSILON ) THEN PRINT*, 'X is within +/- EPSILON of 20.' ELSE PRINT*, 'X is not within +/- EPSILON of 20.' ENDIF
> X is within +/- EPSILON of 20.
Testing for non-representable values
Recall the floating point number line from the above discussion. Each number type (e.g. INTEGER, REAL*4, REAL*8) has a minimum and maximum representable value. If we are not careful, the result of a mathematical computation may result in a value that cannot be represented for the given number type. There are two common non-representable values: Not-A-Number, (or NaN), and Infinity.
NaN (Not-A-Number) is the IEEE designation for a value that cannot be represented for a given number type. Some computations resulting in NaN are:
SQRT( x ) for all x < 0 LOG( x ) for all x <= 0 LOG10( x ) for all x <= 0 ACOS( x ) for all x > 1 and x < -1 ASIN( x ) for all x > 1 and x < -1
It is important to trap for NaN values, otherwise they will propagate throughout your code. For example:
X + NaN = NaN X - NaN = NaN X * NaN = NaN X / NaN = NaN
Each compiler supplies a function to trap for NaN values. Here are the
Section to be added
--Bmy 16:40, 14 April 2008 (EDT)