Sun Studio 12: Fortran Programming Guide

Chapter 6 Floating-Point Arithmetic

This chapter considers floating-point arithmetic and suggests strategies for avoiding and detecting numerical computation errors.

For a detailed examination of floating-point computation on SPARC and x86 processors, see the Numerical Computation Guide.

6.1 Introduction

The Fortran 95 floating-point environment on SPARC processors implements the arithmetic model specified by the IEEE Standard 754 for Binary Floating Point Arithmetic. This environment enables you to develop robust, high-performance, portable numerical applications. It also provides tools to investigate any unusual behavior by a numerical program.

In numerical programs, there are many potential sources for computational error:

Finding the source of the errors in a numerical computation that has gone wrong can be extremely difficult. The chance of coding errors can be reduced by using commercially available and tested library packages whenever possible. Choice of algorithms is another critical issue. Using the appropriate computer arithmetic is another.

This chapter makes no attempt to teach or explain numerical error analysis. The material presented here is intended to introduce the IEEE floating-point model as implemented by Fortran 95.

6.2 IEEE Floating-Point Arithmetic

IEEE arithmetic is a relatively new way of dealing with arithmetic operations that result in such problems as invalid operand, division by zero, overflow, underflow, or inexact result. The differences are in rounding, handling numbers near zero, and handling numbers near the machine maximum.

The IEEE standard supports user handling of exceptions, rounding, and precision. Consequently, the standard supports interval arithmetic and diagnosis of anomalies. IEEE Standard 754 makes it possible to standardize elementary functions like exp and cos, to create high precision arithmetic, and to couple numerical and symbolic algebraic computation.

IEEE arithmetic offers users greater control over computation than does any other kind of floating-point arithmetic. The standard simplifies the task of writing numerically sophisticated, portable programs. Many questions about floating-point arithmetic concern elementary operations on numbers. For example:

Another class of questions concerns floating-point exceptions and exception handling. What happens if you:

In older arithmetic models, the first class of questions might not have the expected answers, while the exceptional cases in the second class might all have the same result: the program aborts on the spot or proceeds with garbage results.

The standard ensures that operations yield the mathematically expected results with the expected properties. It also ensures that exceptional cases yield specified results, unless the user specifically makes other choices.

For example, the exceptional values +Inf, -Inf, and NaN are introduced intuitively:

big*big = +Inf Positive infinity

big*(-big) = -Inf Negative infinity

num/0.0 = +Inf Where num > 0.0

num/0.0 = -Inf Where num < 0.0

0.0/0.0 = NaN Not a Number

Also, five types of floating-point exception are identified:

The implementation of the IEEE standard is described in the Numerical Computation Guide.

6.2.1 –ftrap=mode Compiler Options

The -ftrap=mode option enables trapping for floating-point exceptions. If no signal handler has been established by an ieee_handler() call, the exception terminates the program with a memory dump core file. See the Fortran User’s Guide for details on this compiler option. For example, to enable trapping for overflow, division by zero, and invalid operations, compile with -ftrap=common. (This is the f95 default.)

Note –

You must compile the application’s main program with -ftrap= for trapping to be enabled.

6.2.2 Floating-Point Exceptions

f95 programs do not automatically report on exceptions. An explicit call to ieee_retrospective(3M) is required to display a list of accrued floating-point exceptions on program termination. In general, a message results if any one of the invalid, division-by-zero, or overflow exceptions have occurred. Inexact exceptions do not generate messages because they occur so frequently in real programs. Retrospective Summary

The ieee_retrospective function queries the floating-point status registers to find out which exceptions have accrued and a message is printed to standard error to inform you which exceptions were raised but not cleared. The message typically looks like this; the format may vary with each compiler release:

Note: IEEE floating-point exception flags raised:
    Division by Zero;
IEEE floating-point exception traps enabled:
    inexact;  underflow;  overflow;  invalid operation;
See the Numerical Computation Guide, ieee_flags(3M),

A Fortran 95 program would need to call ieee_retrospective explicitly and compile with -xlang=f77 to link with the f77 compatibility library.

Compiling with the -f77 compatibility flag will enable the Fortran 77 convention of automatically calling ieee_retrospective at program termination.

You can turn off any or all of these messages with ieee_flags() by clearing exception status flags before the call to ieee_retrospective.

6.2.3 Handling Exceptions

Exception handling according to the IEEE standard is the default on SPARC and x86 processors. However, there is a difference between detecting a floating-point exception and generating a signal for a floating-point exception (SIGFPE).

Following the IEEE standard, two things happen when an untrapped exception occurs during a floating-point operation:

6.2.4 Trapping a Floating-Point Exception

f95 differs significantly from the earlier f77 compiler in the way it handles floating-point exceptions.

The default with f95 is to automatically trap on division by zero, overflow, and invalid operation. With f77, the default was not to automatically generate a signal to interrupt the running program for a floating-point exception. The assumption was that trapping would degrade performance while most exceptions were insignificant as long as expected values are returned.

The f95 command-line option -ftrap can be used to change the default. The default for f95 is -ftrap=common. To follow the earlier f77 default, compile the main program with -ftrap=%none.

6.2.5 Nonstandard Arithmetic

One aspect of standard IEEE arithmetic, called gradual underflow, can be manually disabled. When disabled, the program is considered to be running with nonstandard arithmetic.

The IEEE standard for arithmetic specifies a way of handling underflowed results gradually by dynamically adjusting the radix point of the significand. In IEEE floating-point format, the radix point occurs before the significand, and there is an implicit leading bit of 1. Gradual underflow allows the implicit leading bit to be cleared to 0 and shifts the radix point into the significand when the result of a floating-point computation would otherwise underflow. With a SPARC processor this result is not accomplished in hardware but in software. If your program generates many underflows (perhaps a sign of a problem with your algorithm), you may experience a performance loss.

Gradual underflow can be disabled either by compiling with the -fns option or by calling the library routine nonstandard_arithmetic() from within the program to turn it off. Call standard_arithmetic() to turn gradual underflow back on.

Note –

To be effective, the application’s main program must be compiled with -fns. See the Fortran User’s Guide.

For legacy applications, take note that:

6.3 IEEE Routines

The following interfaces help people use IEEE arithmetic and are described in man pages. These are mostly in the math library libsunmath and in several .h files.

6.3.1 Flags and ieee_flags()

The ieee_flags function is used to query and clear exception status flags. It is part of the libsunmath library shipped with Sun compilers and performs the following tasks:

The general form of a call to ieee_flags is:

flags = ieee_flags( action, mode, in, out )

Each of the four arguments is a string. The input is action, mode, and in. The output is out and flags. ieee_flags is an integer-valued function. Useful information is returned in flags as a set of 1-bit flags. Refer to the man page for ieee_flags(3m) for complete details.

Possible parameter values are shown in the following table

Table 6–1 ieee_flags( action, mode, in, out ) Argument Values


Values Allowed  


get, set, clear, clearall


direction, exception

in, out

nearest, tozero, negative, positive, extended, double single, inexact, division, underflow, overflow, invalid all, common

Note that these are literal character strings, and the output parameter out must be at least CHARACTER*9. The meanings of the possible values for in and out depend on the action and mode they are used with. These are summarized in the following table:

Table 6–2 ieee_flags in, out Argument Meanings

Value of in and out

Refers to  

nearest, tozero, negative, positive

Rounding direction 

extended, double, single

Rounding precision 

inexact, division, underflow, overflow, invalid



All five exceptions 


Common exceptions: invalid, division, overflow 

For example, to determine what is the highest priority exception that has a flag raised, pass the input argument in as the null string:

      CHARACTER *9, out
      ieeer = ieee_flags( ’get’, ’exception’, ’’, out )
      PRINT *, out, ’ flag raised’

Also, to determine if the overflow exception flag is raised, set the input argument in to overflow. On return, if out equals overflow, then the overflow exception flag is raised; otherwise it is not raised.

      ieeer = ieee_flags( ’get’, ’exception’, ’overflow’, out )
      IF ( out.eq. ’overflow’) PRINT *,’overflow flag raised’

Example: Clear the invalid exception:

      ieeer = ieee_flags( ’clear’, ’exception’, ’invalid’, out )

Example: Clear all exceptions:

      ieeer = ieee_flags( ’clear’, ’exception’, ’all’, out )

Example: Set rounding direction to zero:

      ieeer = ieee_flags( ’set’, ’direction’, ’tozero’, out )

Example: Set rounding precision to double:

      ieeer = ieee_flags( ’set’, ’precision’, ’double’, out ) Turning Off All Warning Messages With ieee_flags

Calling ieee_flags with an action of clear, as shown in the following example, resets any uncleared exceptions. Put this call before the program exits to suppress system warning messages about floating-point exceptions at program termination.

Example: Clear all accrued exceptions with ieee_flags():

      i = ieee_flags(’clear’, ’exception’, ’all’, out ) Detecting an Exception With ieee_flags

The following example demonstrates how to determine which floating-point exceptions have been raised by earlier computations. Bit masks defined in the system include file floatingpoint.h are applied to the value returned by ieee_flags.

In this example, DetExcFlg.F, the include file is introduced using the #include preprocessor directive, which requires us to name the source file with a .F suffix. Underflow is caused by dividing the smallest double-precision number by 2.

Example: Detect an exception using ieee_flags and decode it:

#include "floatingpoint.h"
         CHARACTER*16 out
         DOUBLE PRECISION d_max_subnormal, x
         INTEGER div, flgs, inv, inx, over, under

       x = d_max_subnormal() / 2.0                ! Cause underflow

       flgs=ieee_flags(’get’,’exception’,’’,out)  ! Which are raised?

       inx   = and(rshift(flgs, fp_inexact)  , 1)  ! Decode
       div   = and(rshift(flgs, fp_division) , 1)   ! the value
       under = and(rshift(flgs, fp_underflow), 1)    ! returned
       over  = and(rshift(flgs, fp_overflow) , 1)     ! by
       inv   = and(rshift(flgs, fp_invalid)  , 1)      ! ieee_flags

       PRINT *, "Highest priority exception is: ", out
       PRINT *, ’ invalid  divide  overflo underflo inexact’
       PRINT ’(5i8)’, inv, div, over, under, inx
       PRINT *, ’(1 = exception is raised; 0 = it is not)’
       i = ieee_flags(’clear’, ’exception’, ’all’, out)    ! Clear all

Example: Compile and run the preceding example (DetExcFlg.F):

demo% f95 DetExcFlg.F
demo% a.out
 Highest priority exception is: underflow
  invalid  divide  overflo underflo inexact
       0       0       0       1       1
 (1 = exception is raised; 0 = it is not)

6.3.2 IEEE Extreme Value Functions

The compilers provide a set of functions that can be called to return a special IEEE extreme value. These values, such as infinity or minimum normal, can be used directly in an application program.

Example: A convergence test based on the smallest number supported by the hardware would look like:

      IF ( delta .LE. r_min_normal() ) RETURN

The values available are listed in the following table:

Table 6–3 Functions Returning IEEE Values

IEEE Value  

Double Precision  

Single Precision  




quiet NaN



signaling NaN



min normal



min subnormal



max subnormal



max normal



The two NaN values (quiet and signaling) are unordered and should not be used in comparisons such as IF( To determine whether some value is a NaN, use the function ir_isnan(r) or id_isnan(d).

The Fortran names for these functions are listed in these man pages:

Also see:

6.3.3 Exception Handlers and ieee_handler()

Typical concerns about IEEE exceptions are:

Exception trapping to a user routine begins with the system generating a signal on a floating-point exception. The standard UNIX name for signal: floating-point exception is SIGFPE. The default situation on SPARC platforms is not to generate a SIGFPE when an exception occurs. For the system to generate a SIGFPE, exception trapping must first be enabled, usually by a call to ieee_handler(). Establishing an Exception Handler Function

To establish a function as an exception handler, pass the name of the function to ieee_handler(), together with the name of the exception to watch for and the action to take. Once you establish a handler, a SIGFPE signal is generated whenever the particular floating-point exception occurs, and the specified function is called.

The form for invoking ieee_handler() is shown in the following table:

Table 6–4 Arguments for ieee_handler( action , exception , handler)



Possible Values  



get, set, or clear



invalid, division, overflow, underflow, or inexact


Function name 

The name of the user handler function or SIGFPE_DEFAULT, SIGFPE_IGNORE, or SIGFPE_ABORT

Return value 


0 =OK

A Fortran 95 routine compiled with f95 that calls ieee_handler() should also declare:

#include ’floatingpoint.h’

The special arguments SIGFPE_DEFAULT, SIGFPE_IGNORE, and SIGFPE_ABORT are defined in these include files and can be used to change the behavior of the program for a specific exception:


No action taken when the specified exception occurs. 


Program aborts, possibly with dump file, on exception. Writing User Exception Handler Functions

The actions your exception handler takes are up to you. However, the routine must be an integer function with three arguments specified as shown:

handler_name( sig, sip, uap )

Example: An exception handler function:

      INTEGER FUNCTION hand( sig, sip, uap )
      INTEGER sig, location
      STRUCTURE /fault/
           INTEGER address
           INTEGER trapno
      STRUCTURE /siginfo/
           INTEGER si_signo
           INTEGER si_code
           INTEGER si_errno
           RECORD /fault/ fault
      RECORD /siginfo/ sip
      location = sip.fault.address
      ... actions you take ...

This example would have to be modified to run on 64 bit SPARC architectures by replacing all INTEGER declarations within each STRUCTURE with INTEGER*8.

If the handler routine enabled by ieee_handler() is in Fortran as shown in the example, the routine should not make any reference to its first argument (sig). This first argument is passed by value to the routine and can only be referenced as loc(sig). The value is the signal number.

Detecting an Exception by Handler

The following examples show how to create handler routines to detect floating-point exceptions.

Example: Detect exception and abort:

demo% cat DetExcHan.f
      EXTERNAL myhandler
      REAL :: r = 14.2 , s = 0.0
      i = ieee_handler (’set’, ’division’, myhandler )
      t = r/s

      INTEGER FUNCTION myhandler(sig,code,context)
      INTEGER sig, code, context(5)
      CALL abort()
demo% f95 DetExcHan.f
demo% a.out

SIGFPE is generated whenever that floating-point exception occurs. When the SIGFPE is detected, control passes to the myhandler function, which immediately aborts. Compile with -g and use dbx to find the location of the exception.

Locating an Exception by Handler

Example: Locate an exception (print address) and abort:

demo% cat LocExcHan.F
#include "floatingpoint.h"
      EXTERNAL Exhandler
      INTEGER Exhandler, i, ieee_handler
      REAL:: r = 14.2 ,  s = 0.0 , t
C  Detect division by zero
      i = ieee_handler( ’set’, ’division’, Exhandler )
      t = r/s

      INTEGER FUNCTION Exhandler( sig, sip, uap)
      INTEGER sig
      STRUCTURE /fault/
                  INTEGER address
      STRUCTURE /siginfo/
                  INTEGER si_signo
                  INTEGER si_code
                  INTEGER si_errno
                  RECORD /fault/ fault
      RECORD /siginfo/ sip
      WRITE (*,10)  sip.si_signo, sip.si_code, sip.fault.address
10        FORMAT(’Signal ’,i4,’ code ’,i4,’ at hex address ’, Z8 )
      CALL abort()
demo%  f95 -g LocExcHan.F
demo%  a.out
Signal   8 code    3 at hex address    11230

In 64–bit SPARC environments, replace the INTEGER declarations within each STRUCTURE with INTEGER*8, and the i4 formats with i8. (Note that this example relies on extensions to the f95 compiler to accept VAX Fortran STRUCTURE statements.)

In most cases, knowing the actual address of the exception is of little use, except with dbx:

demo% dbx a.out
(dbx) stopi at 0x11230     Set breakpoint at address
(2) stopi at &MAIN+0x68
(dbx) run               Run program
Running: a.out
(process id 18803)
stopped in MAIN at 0x11230
MAIN+0x68:      fdivs   %f3, %f2, %f2
(dbx) where           Shows the line number of the exception
=>[1] MAIN(), line 7 in "LocExcHan.F"
(dbx) list 7          Displays the source code line
    7         t = r/s
(dbx) cont            Continue after breakpoint, enter handler routine
Signal    8 code    3 at hex address    11230
abort: called
signal ABRT (Abort) in _kill at 0xef6e18a4
_kill+0x8:      bgeu    _kill+0x30
Current function is exhandler
   24         CALL abort()
(dbx) quit

Of course, there are easier ways to determine the source line that caused the error. However, this example does serve to show the basics of exception handling.

6.4 Debugging IEEE Exceptions

Locating where the exception occurred requires exception trapping be enabled. This can be done by either compiling with the -ftrap=common option (the default when compiling with f95) or by establishing an exception handler routine with ieee_handler(). With exception trapping enabled, run the program from dbx, using the dbx catch FPE command to see where the error occurs.

The advantage of compiling with -ftrap=common is that the source code need not be modified to trap the exceptions. However, by calling ieee_handler() you can be more selective as to which exceptions to look at.

Example: Compiling for and using dbx:

demo% f95 -g myprogram.f
demo% dbx a.out
Reading symbolic information for a.out
(dbx) catch FPE
(dbx) run
Running: a.out
(process id 19739)
signal FPE (floating point divide by zero) in MAIN at line 212 in file "myprogram.f"
  212               Z = X/Y
(dbx)  print Y
y = 0.0

If you find that the program terminates with overflow and other exceptions, you can locate the first overflow specifically by calling ieee_handler() to trap just overflows. This requires modifying the source code of at least the main program, as shown in the following example.

Example: Locate an overflow when other exceptions occur:

demo% cat myprog.F
#include “floatingpoint.h”
        program myprogram
      ier = ieee_handler(”set’,’overflow’,SIGFPE_ABORT)
demo% f95 -g myprog.F
demo% dbx a.out
Reading symbolic information for a.out
(dbx) catch FPE
(dbx) run
Running: a.out
(process id 19793)
signal FPE (floating point overflow) in MAIN at line 55 in file "myprog.F"
   55               w = rmax * 200.                     ! Cause of the overflow
(dbx) cont                                   ! Continue execution to completion
execution completed, exit code is 0

To be selective, the example introduces the #include, which required renaming the source file with a .F suffix and calling ieee_handler(). You could go further and create your own handler function to be invoked on the overflow exception to do some application-specific analysis and print intermediary or debug results before aborting.

6.5 Further Numerical Adventures

This section addresses some real world problems that involve arithmetic operations that may unwittingly generate invalid, division by zero, overflow, underflow, or inexact exceptions.

For instance, prior to the IEEE standard, if you multiplied two very small numbers on a computer, you could get zero. Most mainframes and minicomputers behaved that way. With IEEE arithmetic, gradual underflow expands the dynamic range of computations.

For example, consider a 32-bit processor with 1.0E-38 as the machine’s epsilon, the smallest representable value on the machine. Multiply two small numbers:

      a = 1.0E-30
      b = 1.0E-15
      x = a * b

In older arithmetic, you would get 0.0, but with IEEE arithmetic and the same word length, you get 1.40130E-45. Underflow tells you that you have an answer smaller than the machine naturally represents. This result is accomplished by “stealing” some bits from the mantissa and shifting them over to the exponent. The result, a denormalized number, is less precise in some sense, but more precise in another. The deep implications are beyond this discussion. If you are interested, consult Computer, January 1980, Volume 13, Number 1, particularly J. Coonen’s article, “Underflow and the Denormalized Numbers.”

Most scientific programs have sections of code that are sensitive to roundoff, often in an equation solution or matrix factorization. Without gradual underflow, programmers are left to implement their own methods of detecting the approach of an inaccuracy threshold. Otherwise they must abandon the quest for a robust, stable implementation of their algorithm.

For more details on these topics, see the Numerical Computation Guide.

6.5.1 Avoiding Simple Underflow

Some applications actually do a lot of computation very near zero. This is common in algorithms computing residuals or differential corrections. For maximum numerically safe performance, perform the key computations in extended precision arithmetic. If the application is a single-precision application, you can perform key computations in double precision.

Example: A simple dot product computation in single precision:

      sum = 0
      DO i = 1, n
         sum = sum + a(i) * b(i)
      END DO

If a(i) and b(i) are very small, many underflows occur. By forcing the computation to double precision, you compute the dot product with greater accuracy and do not suffer underflows:

      DO i = 1, n
         sum = sum + dble(a(i)) * dble(b(i))
      END DO
      result = sum

You can force a SPARC processor to behave like an older system with respect to underflow (Store Zero) by adding a call to the library routine nonstandard_arithmetic() or by compiling the application’s main program with the -fns option.

6.5.2 Continuing With the Wrong Answer

You might wonder why you would continue a computation if the answer is clearly wrong. IEEE arithmetic allows you to make distinctions about what kind of wrong answers can be ignored, such as NaN or Inf. Then decisions can be made based on such distinctions.

For an example, consider a circuit simulation. The only variable of interest (for the sake of argument) from a particular 50-line computation is the voltage. Further, assume that the only values that are possible are +5v, 0, -5v.

It is possible to carefully arrange each part of the calculation to coerce each sub-result to the correct range:

Furthermore, since Inf is not an allowed value, you need special logic to ensure that big numbers are not multiplied.

IEEE arithmetic allows the logic to be much simpler. The computation can be written in the obvious fashion, and only the final result need be coerced to the correct value—since Inf can occur and can be easily tested.

Furthermore, the special case of 0/0 can be detected and dealt with as you wish. The result is easier to read and faster in executing, since you don’t do unneeded comparisons.

6.5.3 Excessive Underflow

If two very small numbers are multiplied, the result underflows.

If you know in advance that the operands in a multiplication (or subtraction) may be small and underflow is likely, run the calculation in double precision and convert the result to single precision later.

For example, a dot product loop like this:

  real sum, a(maxn), b(maxn)
  do i =1, n
      sum = sum + a(i)*b(i)

where the a(*) and b(*) are known to have small elements, should be run in double precision to preserve numeric accuracy:

  real a(maxn), b(maxn)
  double sum
  do i =1, n
      sum = sum + a(i)*dble(b(i))

Doing so may also improve performance due to the software resolution of excessive underflows caused by the original loop. However, there is no hard and fast rule here; experiment with your intensely computational code to determine the most profitable solutions.

6.6 Interval Arithmetic

Note: Interval arithmetic is only available on SPARC platforms, currently.

The Fortran 95 compiler f95 supports intervals as an intrinsic data type. An interval is the closed compact set: [a, b] ={z | a≤ zb} defined by a pair of numbers, ab. Intervals can be used to:

By introducing intervals as an intrinsic data type to Fortran 95, all of the applicable syntax and semantics of Fortran 95 become immediately available to the developer. Besides the INTERVAL data types, f95 includes the following interval extensions to Fortran 95:

The f95 command-line option -xinterval enables the interval arithmetic features of the compiler. See the Fortran User’s Guide.

For detailed information on interval arithmetic in Fortran 95, see the Fortran 95 Interval Arithmetic Programming Reference.