# Floating point math issues

## The pitfalls of floating-point mathematics

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:

## 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.

Therefore, 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)
!
!
!  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