4-5 FLOATING-POINT EXCEPTIONS ****************************** (Thanks to Sergio Gelato for the good comments on this chapter) IEEE exceptions --------------- While the CPU (or the FPU - Floating Point Unit) crunches your floating point numbers, the hardware may check the result of every individual arithmetical operation and take some action. IEEE arithmetic requires checking for the following conditions and automatic modifying the result to conform with the extended IEEE arithmetic (see below): Default result Exception name Generating conditions with no traps ----------------- ---------------------------------------- -------------- OVERFLOW Result larger than the maximum possible +/- infinity, or: +/- Xmax UNDERFLOW Result smaller than the minimum possible 0, +/- Xmin, Denormalized DIVIDE BY ZERO A division by zero was attempted +/- infinity INVALID OPERANDS Addition: +infinity + (-infinity) NaN Multiplication: 0 * infinity " Division: 0/0, infinity/infinity " Reminder: mod(X,0), mod(infinity,y) " Square root: sqrt(X) when X .lt. 0 " INEXACT OPERATION Result was rounded off (quite normal!) Rounded number Xmax - maximal representable number Xmin - minimal representable number Another available option in IEEE arithmetic is to establish a "trap" that will take some specified action when the condition is met. IEEE extended non-stop arithmetic --------------------------------- IEEE arithmetic extends the real number system by the two infinities. This procedure is known in mathematics as "Compactification of the real line", and serves to ....... To make the new system closed under arithmetic operations, another type of symbols has to be added: the NaNs (Not A Number), which are the results of INVALID OPERAND operations. Yet other beasts in the IEEE zoo are the signed zeros. In the extended IEEE arithmetic some result is defined for every arithmetic operation, and there is never an arithmetical need to abort a calculation. For every operation some result - either an ordinary real number, or one of the extended quantities is produced, and the calculation can proceed. The philosophy behind IEEE non-stop arithmetic maintains that the extended real system simplifies programming in some cases, and is useful when doing calculations that involve singular points. The first case may be illustrated by computing the relative error of two numbers: program test1 real x, y, ZERO parameter (ZERO = 0.0E0) write (*,*) 'Enter two real numbers: ' read (*,*) x, y if (y .ne. ZERO) then write (*,*) 'Relative error is: ', x / y else write (*,*) 'Relative error cannot be computed ' endif end With IEEE arithmetic we don't have to write the conditional statement, if "y" is zero then the result of "x/y" will be an infinity, and the "write" statement will take care of it: program test2 real x, y, ZERO parameter (ZERO = 0.0E0) write (*,*) 'Enter two real numbers: ' read (*,*) x, y write (*,*) 'Relative error is: ', x / y end Running such a program may give the following: Enter two real numbers: 1.0 0.0 Relative error is: Inf The second case may be illustrated by trying values of some function in order to find some special point, e.g. a point where the value of the function is zero. A possible example may be: 1 y = ----- - 0.5 x - 1 While trying we may stumble upon a point (namely "1") where computing the value leads to division by zero. In non-stop arithmetic nothing bad happens, we get an extended result that tells us what happeed, and can go on to try another point. With non-non-stop arithmetic the program is aborted, and we cannot continue the calculation. Non-stop arithmetic seems a nice improvement, but many users find the extended arithmetic confusing, and prefer to have calculations aborted with an appropriate error message when extended real results are produced. Exceptions in unextended arithmetic ----------------------------------- In a "normal" (i.e. done with unextended arithmetic) computation, none of the exceptions (except INEXACT) may occur, and their occurrence signifies one (or more) of the following: 1) There is a bug in the program, some intermediary calculation is done in the wrong way. 2) The input data to the program is bad. 3) A bad algorithm was used, or the problem was improperly analyzed before the program was written. 4) The problem/algorithm requires larger type of floating-point numbers with larger range and 'density' Having the operating system report these conditions is an invaluable tool for the programmer, helping him to locate problems that are otherwise hard to trace. Many users don't know that current IEEE-based workstations often don't trap *any* FP exceptions by default. It's important that users of these systems (Sun, IBM RS/6000, HP 9000/700 and HP 9000/800, probably others) will know how to trap overflows, invalid operands, and divisions by zero, if they need. To enable trapping of all exceptions: FORTRAN/CHECK=(UNDERFLOW,OVERFLOW) (VMS) f77 -fnonstd (Sun) xlf -qflttrap=inv:ov:zero:en:imp (IBM) f77 +FPVZOuiD (at link time) (HPUX) (system call or environment variable) (IRIX) f77 -check underflow overflow (DUNIX) f77 -check underflow overflow (ULTRIX) (UNICOS) The default behaviour of the IEEE standard of floating-point arithmetic, now implemented in most computers is to deliver a 'result' and continue in the computation. Underflow exceptions -------------------- Underflow occurs when the result (in absolute value) is less than the float type can represent, remember that there are gaps around zero in the three-segment representation of the number-space. It is clear that if we got an underflow condition the 'true' result must be very small - lesser than the smallest float, so it seems reasonable to handle that condition by assigning the value zero to the result. However, 'assign zero' underflow handling can create unexpectedly large errors (see the section errors of floating points), so a better possibility may be to abort the program. In any case the programmer (at least at the program development stage) must get an error message alerting him to that condition. Almost all machines let you choose between the two possibilities with compiler switches, other machines may require system calls A word would be useful on gradual vs. abrupt underflow. The IEEE default is gradual underflow (denormalized numbers). Abrupt underflow (set the result to zero right away on underflow) makes many algorithms converge faster, and is almost always appropriate. VS Fortran on IBM S/370 and ES/390 systems ("mainframes") running VM/CMS, MVS, AIX/370 or AIX/ESA traps underflow by default. Programs often run twice as fast if this trapping is disabled, which can be done by a CALL XUFLOW(0) from within Fortran, or at run-time by giving a special keyword (noxuflow, or -nospie under AIX) on the command line. Overflow exceptions ------------------- Invalid operand exceptions -------------------------- Division by zero exceptions ---------------------------Return to contents page