4-2 FLOATING-POINT NUMBERS - CONCRETE EXAMPLE
**********************************************
IEEE/REAL*4
-----------
To make things more concrete let's look at a typical floating-point
representation for a REAL (SINGLE PRECISION) - the single-precision
unextended IEEE (ANSI/IEEE Std 754-1985) that became a de facto
standard on workstations. The '*4' is a non-standard notation that
says that 4 bytes are allocated for the representation.
A schematic description of the representation follows, the 4 bytes
contain 32 bits that are partitioned into 3 parts (the letter 'S'
in the left part is short for 'Sign')
+-+--------+-----------------------+
|S| exp | fraction |
+-+--------+-----------------------+ Direction of
^ ^ <--- increasing addresses
Bit31 Bit0 (See discussion below)
A formula that gives the value of this float is:
Value = (-1)**S X 1.fffffffffffffffffffffff X 2**(exp - 127)
The most significant bit (MSB) is the sign bit, it is 0 for a positive
number and 1 for a negative number.
The next 8 bits describe the exponent which is BIASED by 127 (see the
formula above), so the range of values is [-127, 128]
The remaining 23 bits are taken as the binary digits of a binary fraction
that has a "whole part" = 1 (see the formula above), this condition is just
the normalization condition.
An IEEE normalized mantissa always has a leading '1' bit, so it is really
redundant and can be always omitted (an old 'trick' attributed to David
Goldberg), it 'saves' one bit that can be used to improve the precision.
The following program may help you examine the structure of REAL on
your machine, it is based on the plausible assumption that integers are
represented in two's complement format.
Of course we could use the Z edit descriptor, but it is not standard
FORTRAN 77, and so may not be implemented by all compilers.
PROGRAM RELREP
C ------------------------------------------------------------------
REAL
* X
C ------------------------------------------------------------------
WRITE(*,*) ' Enter a REAL number: '
READ(*,*) X
CALL BINREP(X)
C ------------------------------------------------------------------
END
SUBROUTINE BINREP(INT)
C ------------------------------------------------------------------
INTEGER
* I,
* INT
C ------------------------------------------------------------------
CHARACTER
* B*32
C ------------------------------------------------------------------
IF (INT .GE. 0) THEN
B(1:1) = '0'
DO I = 32, 2, -1
IF (MOD(INT,2) .EQ. 0) THEN
B(I:I) = '0'
ELSE
B(I:I) = '1'
ENDIF
INT = INT / 2
ENDDO
ELSE
B(1:1) = '1'
INT = ABS(INT + 1)
DO I = 32, 2, -1
IF (MOD(INT,2) .EQ. 0) THEN
B(I:I) = '1'
ELSE
B(I:I) = '0'
ENDIF
INT = INT / 2
ENDDO
ENDIF
C ------------------------------------------------------------------
WRITE(*,*) ' ', B(1:8),' ', B(9:16),' ', B(17:24),' ', B(25:32)
WRITE(*,*) ' ........ ........ ........ ........ '
WRITE(*,*) ' 21098765 43210987 65432109 87654321 '
WRITE(*,*) ' 3 2 1 '
WRITE(*,*) ' '
C ------------------------------------------------------------------
RETURN
END
Special numbers
---------------
Using normalized mantissas raises a little problem, how to represent
zero when the mantissa is not allowed to have zero value?
The IEEE solution is to represent the number zero by a zero fraction
and exponent, but no condition is imposed on the SIGN BIT, so we
have two 'zeros' +0 and -0!
Remember that the exponent is biased by 127, so that a zero exponent
really means that the binary fraction is 'multiplied' by (2 ** (-127)),
in other words, the minimal exponent is reserved to represent zero.
There is also an internal representation for 'INFINITY', it consists
of the maximal exponent = 255 (128 after debiasing) and all fraction
bits = 0. So we have also two 'infinities' one positive and one negative.
An even stranger phenomenon is the class of bit patterns called NaNs,
a NaN has exponent = 255 (128 after debiasing) and fraction bits
which are not all 0. NaN is short for 'Not A Number'.
The special numbers (except zero) were invented in order to implement
NON-STOP ARITHMETIC, instead of aborting the program in the case an
intermediary calculation gives a bad result, the result is replaced
by the appropriate special number and computation continues.
IEEE arithmetic implements an extension of the real numbers system,
the quantities +INFINITY, -INFINITY and the NaNs are added to the
real numbers, and arithmetic operations involving them are defined
in a plausible way. Many users find this extension confusing and
not very useful.
The 'representation density' of IEEE/REAL*4
-------------------------------------------
What is the spacing between two consecutive floating-point numbers?
Positive FPN are the product of a 'normalized' binary fraction with
23 binary digits, and (2 ** e), where e is in [-126,127].
Remember that the exponents -127 and +128 are reserved to represent
zero and infinity respectively.
The 'normal' FPNs can be partitioned into 254 disjoint sets, one for
each possible exponent, each set containing (2 ** 23) numbers, one
for each possible binary fraction of length 32.
The spacing between consecutive numbers belonging to the same set,
is the same, and equals (2 ** (-23)) * (2 ** e) = 2 ** (e -32).
It is clear that the spacing increases when e (and the magnitude
of the number) increases.
The minimal positive FPN is (+1.0) * (2 ** (-126)) = 2 ** (-126),
the spacing at that region is (2 ** (-126 - 32)) = 2 ** (-158).
We see that the minimal positive FPN is MUCH LARGER than the
local spacing.
The number space of IEEE/REAL*4
-------------------------------
If we will translate the binary data from previous sections to
decimal, we will find the range of numbers that can be represented
by the IEEE REAL*4 is:
(-3.4 X 10**+38, +3.4 X 10**+38)
Because the minimal FPN is so much larger than the nearby spacing,
it is more instructive to look at that range as the union of three
discrete segments:
(-3.4 X 10**+38, -1.2 X 10**-38)
(0.0)
(+1.2 X 10**-38, +3.4 X 10**+38)
In this floating-point representation we have a finite number of numbers
filling the three ranges, two of them with variable 'density'.
+---------------------------------------------------------------------+
| SUMMARY |
| ======= |
| 1) IEEE/REAL*4 = 1 Sign bit, 8 exponent bits, 23 mantissa bits |
| 2) There are all kinds of 'strange numbers' |
| 3) The number space is discrete, made of three parts, and has |
| maximal 'density' near zero |
+---------------------------------------------------------------------+
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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