``` 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: '
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: '
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
---------------------------

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