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
---------------------------
A much more important factor in the social movement than those already mentioned was the ever-increasing influence of women. This probably stood at the lowest point to which it has ever fallen, during the classic age of Greek life and thought. In the history of Thucydides, so far as it forms a connected series of events, four times only during a period of nearly seventy years does a woman cross the scene. In each instance her apparition only lasts for a moment. In three of the four instances she is a queen or a princess, and belongs either to the half-barbarous kingdoms of northern Hellas or to wholly barbarous Thrace. In the one remaining instance208— that of the woman who helps some of the trapped Thebans to make their escape from Plataea—while her deed of mercy will live for ever, her name is for ever lost.319 But no sooner did philosophy abandon physics for ethics and religion than the importance of those subjects to women was perceived, first by Socrates, and after him by Xenophon and Plato. Women are said to have attended Plato’s lectures disguised as men. Women formed part of the circle which gathered round Epicurus in his suburban retreat. Others aspired not only to learn but to teach. Arêtê, the daughter of Aristippus, handed on the Cyrenaic doctrine to her son, the younger Aristippus. Hipparchia, the wife of Crates the Cynic, earned a place among the representatives of his school. But all these were exceptions; some of them belonged to the class of Hetaerae; and philosophy, although it might address itself to them, remained unaffected by their influence. The case was widely different in Rome, where women were far more highly honoured than in Greece;320 and even if the prominent part assigned to them in the legendary history of the city be a proof, among others, of its untrustworthiness, still that such stories should be thought worth inventing and preserving is an indirect proof of the extent to which feminine influence prevailed. With the loss of political liberty, their importance, as always happens at such a conjuncture, was considerably increased. Under a personal government there is far more scope for intrigue than where law is king; and as intriguers women are at least the209 equals of men. Moreover, they profited fully by the levelling tendencies of the age. One great service of the imperial jurisconsults was to remove some of the disabilities under which women formerly suffered. According to the old law, they were placed under male guardianship through their whole life, but this restraint was first reduced to a legal fiction by compelling the guardian to do what they wished, and at last it was entirely abolished. Their powers both of inheritance and bequest were extended; they frequently possessed immense wealth; and their wealth was sometimes expended for purposes of public munificence. Their social freedom seems to have been unlimited, and they formed combinations among themselves which probably served to increase their general influence.321 The old religions of Greece and Italy were essentially oracular. While inculcating the existence of supernatural beings, and prescribing the modes according to which such beings were to be worshipped, they paid most attention to the interpretation of the signs by which either future events in general, or the consequences of particular actions, were supposed to be divinely revealed. Of these intimations, some were given to the whole world, so that he who ran might read, others were reserved for certain favoured localities, and only communicated through the appointed ministers of the god. The Delphic oracle in particular enjoyed an enormous reputation both among Greeks and barbarians for guidance afforded under the latter conditions; and during a considerable period it may even be said to have directed the course of Hellenic civilisation. It was also under this form that supernatural religion suffered most injury from the great intellectual movement which followed the Persian wars. Men who had learned to study the constant sequences of Nature for themselves, and to shape their conduct according to fixed principles of prudence or of justice, either thought it irreverent to trouble the god about questions on which they were competent to form an opinion for themselves, or did not choose to place a well-considered scheme at the mercy of his possibly interested responses. That such a revolution occurred about the middle of the fifth century B.C., seems proved by the great change of tone in reference to this subject which one perceives on passing from Aeschylus to Sophocles. That anyone should question the veracity of an oracle is a supposition which never crosses the mind of the elder dramatist. A knowledge of augury counts among the greatest benefits222 conferred by Prometheus on mankind, and the Titan brings Zeus himself to terms by his acquaintance with the secrets of destiny. Sophocles, on the other hand, evidently has to deal with a sceptical generation, despising prophecies and needing to be warned of the fearful consequences brought about by neglecting their injunctions. The stranger had a pleasant, round face, with eyes that twinkled in spite of the creases around them that showed worry. No wonder he was worried, Sandy thought: having deserted the craft they had foiled in its attempt to get the gems, the man had returned from some short foray to discover his craft replaced by another. “Thanks,” Dick retorted, without smiling. When they reached him, in the dying glow of the flashlight Dick trained on a body lying in a heap, they identified the man who had been warned by his gypsy fortune teller to “look out for a hidden enemy.” He was lying at full length in the mould and leaves. "But that is sport," she answered carelessly. On the retirement of Townshend, Walpole reigned supreme and without a rival in the Cabinet. Henry Pelham was made Secretary at War; Compton Earl of Wilmington Privy Seal. He left foreign affairs chiefly to Stanhope, now Lord Harrington, and to the Duke of Newcastle, impressing on them by all means to avoid quarrels with foreign Powers, and maintain the blessings of peace. With all the faults of Walpole, this was the praise of his political system, which system, on the meeting of Parliament in the spring of 1731, was violently attacked by Wyndham and Pulteney, on the plea that we were making ruinous treaties, and sacrificing British interests, in order to benefit Hanover, the eternal millstone round the neck of England. Pulteney and Bolingbroke carried the same attack into the pages of The Craftsman, but they failed to move Walpole, or to shake his power. The English Government, instead of treating Wilkes with a dignified indifference, was weak enough to show how deeply it was touched by him, dismissed him from his commission of Colonel of the Buckinghamshire Militia, and treated Lord Temple as an abettor of his, by depriving him of the Lord-Lieutenancy of the same county, and striking his name from the list of Privy Councillors, giving the Lord-Lieutenancy to Dashwood, now Lord Le Despencer. "I tell you what I'll do," said the Deacon, after a little consideration. "I feel as if both Si and you kin stand a little more'n you had yesterday. I'll cook two to-day. We'll send a big cupful over to Capt. McGillicuddy. That'll leave us two for to-morrer. After that we'll have to trust to Providence." "Indeed you won't," said the Surgeon decisively. "You'll go straight home, and stay there until you are well. You won't be fit for duty for at least a month yet, if then. If you went out into camp now you would have a relapse, and be dead inside of a week. The country between here and Chattanooga is dotted with the graves of men who have been sent back to the front too soon." "Adone do wud that—though you sound more as if you wur in a black temper wud me than as if you pitied me." "Wot about this gal he's married?" "Don't come any further." "Davy, it 'ud be cruel of us to go and leave him." "Insolent priest!" interrupted De Boteler, "do you dare to justify what you have done? Now, by my faith, if you had with proper humility acknowledged your fault and sued for pardon—pardon you should have had. But now, you leave this castle instantly. I will teach you that De Boteler will yet be master of his own house, and his own vassals. And here I swear (and the baron of Sudley uttered an imprecation) that, for your meddling knavery, no priest or monk shall ever again abide here. If the varlets want to shrieve, they can go to the Abbey; and if they want to hear mass, a priest can come from Winchcombe. But never shall another of your meddling fraternity abide at Sudley while Roland de Boteler is its lord." "My lord," said Edith, in her defence, "this woman has sworn falsely. The medicine I gave was a sovereign remedy, if given as I ordered. Ten drops would have saved the child's life; but the contents of the phial destroyed it. The words I uttered were prayers for the life of the child. My children, and all who know me, can bear witness that I have a custom of asking His blessing upon all I take in hand. I raised my eyes towards heaven, and muttered words; but, my lord, they were words of prayer—and I looked up as I prayed, to the footstool of the Lord. But it is in vain to contend: the malice of the wicked will triumph, and Edith Holgrave, who even in thought never harmed one of God's creatures, must be sacrificed to cover the guilt, or hide the thoughtlessness of another." "Aye, Sir Treasurer, thou hast reason to sink thy head! Thy odious poll-tax has mingled vengeance—nay, blood—with the cry of the bond." HoME古一级毛片免费观看
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