# Floating point math issues

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## Floating-point is an approximation to the real number system

One important thing to remember when writing computer programs is that floating-point mathematics is never exact but is only an approximation. 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

Note that you cannot represent numbers to -Infinity or +Infinity. Each number type is composed of a finite number of bytes, and can only represent numbers within a given range, as listed in the above table.

Here are some excellent references about floating-point math and ways to avoid pitfalls:

## 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,
Julio.

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

### Exact-value testing

#### Integers

If you are working with integer-valued numbers, then you can use the equality operator

• In Fortran: "==", ".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 +/- 0, 1, 2, 3, ...

#### Real numbers

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
```

#### Avoiding zero

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.

### Epsilon testing

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.

#### 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 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
```

which gives

```> X is within +/- EPSILON of 20.
```

--Bmy 12:59, 11 April 2008 (EDT)