4-1 FLOATING-POINT NUMBERS - GENERAL VIEW
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The real number system
----------------------
Scientific and engineering calculations are performed in the REAL
NUMBER SYSTEM, a highly abstract mathematical construct.
A real number is by definition a special infinite set of rational
numbers (integer fractions) - the so called Dedkind Cuts or an
equivalent formulation. The arithmetical operations are defined
between such sets and is a natural extension of the arithmetic of
rational numbers.
The real numbers have wonderful properties:
1) There is no lower or upper bound, in simple language
they go from minus infinity to plus infinity.
2) Infinite density - there is a real number between
any two real numbers.
3) A lot of algebraic axioms are satisfied, e.g. the
'field axioms'.
4) Completeness - they contain all their 'limit points'
(the limit of every converging sequence is also 'real').
5) They are ordered.
Many of these properties are not satisfied by computer arithmetic,
see the chapter on errors in floating-point computations for a short
review on properties that stay true in floating-point arithmetic.
In order to crunch quickly a lot of numbers, computers need a fixed
size representation of real numbers, that way the hardware can
efficiently perform the arithmetical operations.
The problems arising from using a fixed size representation are the
subject of the following chapters.
Finite number systems are discrete
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If you use a fixed size representation, let's say N binary digits (BITS)
long, you have at most 2**N bit-patterns, and so at most 2**N
representable numbers.
Such a finite set will have to be bounded - have a largest number and a
smallest number. We have already one problem, our computations must not
exceed these bounds.
In every bounded segment, there are infinitely many real numbers, but we
have at most 2**N available bit-patterns, so many real numbers will have
to be represented by one bit-pattern.
Of course one bit-pattern can't represent many numbers equally well, it
will represent one of them exactly and the others will be misrepresented.
We call numbers that can be represented exactly, FLOATING-POINT NUMBERS
(FPN), the term 'real numbers' will be reserved for the mathematical
constructs.
Roundoff errors are unavoidable
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Before we begin to study actual representations of real numbers,
let us develop a little an idea mentioned in the previous section.
We said that in a finite number system, many real numbers will have
to be represented by one bit-pattern, and that bit-pattern will
represent exactly only one of them. In other words many real numbers
will be 'rounded off' to that one bit-pattern.
This 'rounding off' may occur whenever we will enter a real number to
the computer (except in the rare case we will enter an exactly
representable number).
The same 'rounding off' may occur whenever we perform an arithmetical
operation. The result of an arithmetical operation usually will have
more binary digits than its operands, and will have to be converted
to one of the 'allowed' bit-patterns.
To make this more concrete, let's have an example using base 10
real numbers, and suppose that only two digit mantissas are allowed
(the fractional parts may have only 2 decimal digits):
0.12E+02 + 0.34E+00 = 12.00E+00 + 0.34E+00 = 12.34E+00 ==> 0.12E+02
This example is a bit artificial and incompletely defined (in our fixed
representation, only the size of the fractional part was specified,
the exponents were left unspecified), but the idea is clear, we can
see that computer arithmetic has to replace almost every number and
temporary result by a rounded form.
Instead of computing:
X + Y
We will really compute:
round(round(X) + round(Y))
The function 'round' can't be specified in general, it depends on the
representation and the floating-point arithmetical algorithms we use,
see the chapter 'radix conversion and rounding' for more information.
A possible implementation of round() for decimal floating-point numbers
(represented in radix 10) is:
e = INT(LOG10(X) + 1.0) (number of decimal digits in X)
INT(X * (10**(p-e)) + 0.5)
round(X) = ----------------------
10**(p-e)
The parameter p is the number of decimal digits in the representation.
Note that multiplying and dividing by (10**n) are just shifts of the
decimal point, and not error generating arithmetic operations.
Such seemingly complicated formulas can be implemented efficiently
(in radix 2) in hardware or reduced to a very small micro-code program
executed by the CPU.
In the following sections we will see that roundoff errors are an endless
source of errors, some of them unexpectedly large.
By the way, the distinction between real and floating-point numbers can
be summarized symbolically in our new notation by:
FPN = round(REAL)
A little basic theory
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Every real number x can be written in the form:
x = f X (2 ** e)
Where 'e' is an integer called the EXPONENT, and 'f' is a binary
fraction called the MANTISSA. The mantissa may satisfy one of the
normalization conditions:
1 <= |f| < 2 (IEEE)
1/2 <= |f| < 1 (DEC)
The mantissa is then said to be a NORMALIZED.
The IEEE normalization condition is equivalent to the requirement
that the MOST SIGNIFICANT BIT (MSB) in the mantissa = 1.
The DEC condition requires the two most significant bits to be 0,1.
On IBM 360, IBM 370 and Nova (Data General) computers, the base of the
exponent was 16 (it gives a larger range at the cost of precision):
x = f X (16 ** e)
The normalization condition was that the first HEX digit of the fraction
was not equal to 0, i.e. not all first 4 binary digits were 0.
The advantages of normalizing floating-point numbers are:
1) The representation is unique, there is exactly one way to
write a real number in such a form.
2) It's easy to compare two normalized numbers, you separately
test the sign, exponent and mantissa.
3) In a normalized form, a fixed size mantissa will use all
the 'digit cells' to store significant digits.
4) The IEEE and DEC normalization conditions makes the
representation always start with a 1-bit, this bit can
be omitted, and its place used for data. The omitted
bit is called the "hidden bit".
The normalized representation is used in almost all floating point
implementations, 'denormalized numbers' are used only to minimize
accuracy loss due to underflow (see next chapter).
Just like with rounding, we will have to normalize after arithmetical
operations, the result wouldn't be normalized in general.
Floating Point numbers in practise
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In our finite machines, we can keep only a finite number of the binary
digits of 'f' and 'e', let's say 'm' and 'n' digits respectively.
The vendor predetermine a few combinations of 'm' and 'n', usually one
or two combinations that the hardware executes efficiently, and maybe
one more that gives better precision.
The following table compares some floats used in practice, the REAL*n
notation is a common extension to FORTRAN, 'n' is the number of bytes
used in the representation. The representation radix, size (in bits)
of the various parts composing the floating-point number, and the
exponent bias are given.
The number of bits in the fraction part is counted without the
"hidden bit", if normalized mantissas are used, so the sizes here
are "physical" rather than "logical".
Table of float types (incomplete)
=================================
Float name Radix Sign Exponent Fraction Bias
---------- ----- ---- -------- -------- -----
IBM 370:
* REAL*4 16 1 7 24 64 0.f * 16**(e-64)
* REAL*8 16 1 7 56 64
VAX:
* REAL*4 (F_FLOAT) 2 1 8 23 128 0.1f * 2**(e-128)
* REAL*8 (D_FLOAT) 2 1 8 55 128 0.1f * 2**(e-128)
* REAL*8 (G_FLOAT) 2 1 11 52 1024 0.1f * 2**(e-1024)
* REAL*16(H_FLOAT) 2 1 15 113 16384 0.1f * 2**(e-16384)
Cray:
Single precision 2 1 15 48 16384
Double precision 2 1 15 96 16384
IEEE
* REAL*4 2 1 8 23 127 1.f * 2**(e-127)
extended 2 1 11+ 31+
* REAL*8 2 1 11 52 1023 1.f * 2**(e-1023)
extended 2 1 15+ 63+
REAL*10 2 1 15 64 16383
Intel (IEEE):
* Short real 2 1 8 23 127 1.f * 2**(e-127)
* Long real 2 1 11 52 1023 1.f * 2**(e-1023)
Temp real 2 1 15 64 16383 0.f * 2**(e-16384)
MIL 1750A:
REAL*4 None 8 24 None f * 2**e
REAL*8 None ? ?? None
HP 21MX:
Varian:
Honeywell:
Remarks:
1) Formats that use a sign bit (all except MIL 1750A),
use the sign convention: 0 = +, 1 = -
MIL 1750A uses a 2's complement mantissa with a
2's complement exponent.
2) '#' at the first column means that normalized mantissas
are used. Note that on IBM 370 the first hexadecimal
digit of the fraction (4 bits), couldn't be zero.
An important note
-----------------
The next chapter will provide a detailed example that will make the
abstract concepts more clear. To simplify our discussion, we will
give an incomplete treatment of this highly technical subject, and
with no proofs.
Readers interested in a deeper treatment of these subjects are
referred to:
Goldberg, David
What Every Computer Scientist Should
Know about Floating-Point arithmetic
ACM Computing Surveys
Vol. 23 #1 March 1991, pp. 5-48
+---------------------------------------------------------------------+
| SUMMARY |
| ======= |
| 1) x = f X (2 ** e) 2 > |f| => 1 b is integer |
| 2) There are a lot of float types |
| 3) IEEE/REAL*4 = 1 Sign bit, 8 exponent bits, 23 mantissa bits |
+---------------------------------------------------------------------+
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. 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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. 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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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