CSC161 2010F Imperative Problem Solving
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Summary: In this reading, we consider mechanisms for representing real numbers with a fixed number of bits. We emphasize the IEEE floatingpoint representations.
No human system of numeration can give a unique representation to every real number; there are just too many of them. So it is conventional to use approximations. For instance, the assertion that pi is 3.14159 is, strictly speaking, false, since pi is actually slightly larger than 3.14159; but in practice we sometimes use 3.14159 in calculations involving pi because it is a good enough approximation of pi.
One approach to representing real numbers, then, is to specify some
tolerance epsilon
and to say that a real number x
can be approximated by any number in the range from x 
epsilon
to x + epsilon
. Then, if a system of
numeration can represent selected numbers that are never more than twice
epsilon
apart, every real number has a representable
approximation. For instance, in the United States, the prices of stocks
are given in dollars and eighths of a dollar, and rounded to the nearest
eighth of a dollar; this corresponds to a tolerance of onesixteenth of a
dollar. In retail commerce, however, the conventional tolerance is half a
cent; that is, prices are rounded to the nearest cent. In this case, we
can represent a sum of money as an whole number of cents, or equivalently
as a number of dollars that is specified to two decimal places.
Such a representation, in which each real number is represented by a numeral for an approximation to some fixed number of decimal places, is called a fixedpoint representation. However, fixedpoint representations are unsatisfactory for most applications involving real numbers, for two reasons
0.01
, even though the former is more than twice as large as
the latter. Moreover, both 4/1000 and 7/10000 would be represented as
0.00
, and they don't even have the same sign! If one is
counting dollars, these differences are probably irrelevant, but there are
a lot of scientific and technological applications (navigation systems, for
instance) in which they are critical. These two considerations really have the same underlying cause: When approximating some real number, what we usually know is some number of significant digits at the beginning of the number. A fixedpoint system of representation conceals some of what we know when the number is small, because some of these significant digits are too far to the right of the decimal point, and demands more than we know when the number is large, because the number of digits demanded by the representation system is greater than the number that we can provide.
Scientists and engineers long ago learned to cope with this problem by
using scientific notation, in which a number is expressed as
the product of a mantissa and some power of ten. The mantissa
is a signed number with an absolute value greater than or equal to one
and less than ten. So, for instance, the speed of light in vacuum is
2.99792458 x 10^{8}
meters per second, and one can
specify only the digits about which one is completely confident.
Using scientific notation, one can easily see both that 1.4
x 10^{2}
is more than twice as large as 6
x 10^{3}
, and that both are close to 1 x
10^{2}
; and one can easily distinguish 4
x 10^{3}
and 7 x 10^{4}
as small numbers of opposite sign. The rules for calculating with
scientificnotation numerals are a little more complicated, but the
benefits are enormous.
The three varying components of a number represented in scientific notation are
A system of numeration for real numbers that is adapted to computers will typically store the same three kinds of values (a sign, a mantissa, and an exponent) into an allocated region of storage. By contrast with fixedpoint representations, these computer analogues of scientific notation are described as floatingpoint representations.
The exponent does not always indicate a power of ten; sometimes powers
of sixteen are used instead, or, most commonly of all, powers of two.
The numerals will be somewhat different depending how this choice is made.
For instance, the real number 0.125 will be expressed
as 1.25 x 10^{1}
if powers of ten are used, or
as 2 x 16^{1}
if powers of sixteen are used,
or as 1 x 2^{3}
if powers of two are used.
The absolute value of the mantissa is, however, always greater than or
equal to 1 and less than the base of numeration.
The particular system used in many modern computers and languages (including Java) was formulated and recommended as a standard by the Institute of Electrical and Electronics Engineers (IEEE) and is the most commonly used numeration system for computer representation of real numbers. Actually, their standard includes several variants of the system, depending on how much storage is available for a real number. We'll discuss two of these variants, both of which use binary numeration and powers of 2: the IEEE singleprecision representation, which fits in thirtytwo bits, and the IEEE doubleprecision representation, which occupies sixtyfour bits. We'll begin with singleprecision numbers.
In the IEEE singleprecision representation of a real number, one bit is
reserved for the sign, and it is set to 0
for a positive
number and to 1
for a negative one. A representation of the
exponent is stored in the next eight bits, and the remaining twentythree
bits are occupied by a representation of the mantissa of the number.
The exponent, which is a signed integer in the range from 126 to
127, is represented neither as a signed magnitude nor as a
twoscomplement number, but as a biased value. The idea here is
that the integers in the desired range of exponents are first adjusted by
adding a fixed bias
to each one. The bias is chosen to be large enough
to convert every integer in the range into a positive integer, which is
then stored as a binary numeral. The IEEE singleprecision representation
uses a bias of 127.
For example, the exponent 5 is represented by the eightbit pattern
01111010
, which is the binary numeral for 122, since 5 + 127
= 122. The least allowable exponent, 126, is represented by
00000001
(since 126 + 127 = 1);
the greatest allowable exponent, 127, is represented by the
binary numeral for 254, 11111110
. Note that neither
00000000
nor 11111111
are allowed as
exponents.
Since the base of numeration is two, the mantissa is always a number
greater than or equal to one and less than two. Its fractional part is
represented, using binary numeration, as a sum of the negative powers of
two. For instance, the binary numeral for 21/16 is 1.0101  one, plus no
halves, plus one quarter, plus no eighths, plus one sixteenth. (The
decimal point
in this case is actually a binary point
, separating
the digit in the units place from the digits representing multiples of
negative powers of two.)
Somewhat surprisingly, this means that approximations are used for many real numbers that can be represented exactly in decimal numeration. For instance, 7/5 is 1.4 exactly in decimal numeration, but the .4 part cannot be expressed as a sum of powers of two; 7/5 has an infinite binary expansion 1.011001100110011001100..., just as in decimal numeration a fraction like 27/11 leads to an infinite decimal expansion 2.4545454545.... In a singleprecision representation, the expansion is rounded off at the twentythird digit after the binary point. Thus 7/5 is not actually stored as 7/5, but (in single precision) as the very close approximation 11744051/8388608, which can be expressed as a sum of powers of two.
Only the part of the mantissa that comes after the binary point is actually
stored, since the bit to the left of the binary point is completely
predictable (it's always 1, since the mantissa is always greater than or
equal to one and less than two). This suppressed digit at the beginning of
the mantissa is called the hidden bit
.
Here's an example. Consider the thirtytwobit word
11000011100101100000000000000000
The number represented by this sequence of bits is negative (because
the sign bit is 1
). It has an exponent of 8 (because
the exponent bits, 10000111
, form the binary numeral
for 135, and therefore represent the exponent 8 when the bias of
127 is removed). Its mantissa is expressed by the binary numeral
1.00101100000000000000000
, where the initial 1
is the hidden bit and the remaining digits are taken from the right end
of the word. This last binary numeral expresses the number 75/64 (one,
plus no halves, plus no quarters, plus one eighth, plus no sixteenths,
plus one thirtysecond, plus one sixtyfourth). So the complete number
is (75/64) x 2^{8}, which is 300.0.
Conversely, let us find the IEEE singleprecision representation of,
say, 5.75. The sign bit is 0
, since the
number is positive. 5.75 is 23/4; to express this
as the product of a power of two and a mantissa greater than or equal
to one and less than two, one must factor out 2^{2}: 23/4 =
(23/16) x 2^{2}, so the exponent is 2. It will be stored, with
a bias of 127, as 10000001
(the binary numeral for 129).
The complete mantissa, extended to twentythree digits after the binary
point, is 1.01110000000000000000000
. We line up the sign
bit, the biased representation of the exponent, and the digits following
the binary point in the mantissa and get
01000000101110000000000000000000
The greatest number that has an exact IEEE singleprecision representation is 340282346638528859811704183484516925440.0 (2^{128}  2^{104}), which is represented by
01111111011111111111111111111111
and the least is the negative of this number, which has the same
representation except that the sign bit is 1
.
The alert reader will have noticed that there is a serious gap in
this scheme, as so far described: What about 0.0?
It is not possible to represent 0.0 as the product of a
power of two and a mantissa greater than or equal to one, so none of the
representations described above will do. However, not all the possible
settings of thirtytwo switches have been used for numbers. Recall that
the exponents permitted in IEEE singleprecision reals range from 126
to +127, so that the binary numerals for the biased exponents range from
00000001
to 11111110
. We haven't yet used any
of the bit patterns in which the exponent bits are all zeroes or all ones.
In the IEEE system, the allzero exponent is used for
numbers that are very close to zero  closer than
2^{126}, which is the least of the positive reals that
can be represented in the part of the system described above (it is
00000000100000000000000000000000
). Such numbers are
expressed in a slightly different form of scientific notation: The
exponent is held fixed at 126, and the mantissa is a number greater than
or equal to zero and less than one. So, for instance,
the mantissa used for 3 x 2^{129} is 0.011
(3
x 2^{129} is a quarter plus an eighth of 2^{126}).
Once again, only the part of the mantissa that follows the binary point
is stored explicitly, so the representation of 3 x 2^{129} is
00000000001100000000000000000000
(sign positive,
exponent 126, mantissa 0.01100000000000000000000
).
Mantissas less than one are said to be unnormalized
(because
the normal form
is the one in which the mantissa is greater than
or equal to one and less than the base of numeration), so an allzero
exponent indicates an unnormalized number. Unnormalized numbers are
stored less accurately than normalized ones (since there are fewer
significant digits in the mantissa), but without this special convention
for the allzero exponent it would not be possible to represent them
at all, and the designers of the IEEE standard felt that a degraded
approximation is better than none.
This convention allows zero to be represented in two different ways. In
one, the sign bit is 0
, the exponent bits are
00000000
, and the visible bits of the mantissa are
00000000000000000000000
 yielding
0.00000000000000000000000 x 2^{126}, or 0. The other representation is
the same, except that the sign bit is 1
. (So, as in
signedmagnitude representations, there is a nonnegative
and a
nonpositive
zero.)
The least positive real number that can be represented exactly in this way is 2^{149}, which is stored as
00000000000000000000000000000001
When the exponent bits are all ones, the value represented is not a real number at all, but a conventional signal of a computation that has gone wrong, either by going above the greatest representable real or below the least, or by attempting some undefined arithmetic operation, such as dividing by zero or taking the logarithm of a negative number. For instance, the thirtytwo bits
01111111100000000000000000000000
represent positive infinity
, a pseudonumber that indicates that some
unrepresentably large quantity was generated by an arithmetic operation.
Changing the sign bit to 1
yields a representation of negative
infinity, an indication of a similar problem at the other end of the range.
If some of the bits of the mantissa are 1
s, the pseudonumber
is called a NaN (Not a Number
); trying to compute 0/0, for
instance, typically produces a NaN. It should be clear that positive
infinity, negative infinity, and NaN are not real numbers,
although some language implementations will try to do something sensible
if they appear in places that are normally occupied by real numbers. The
value of such attempts at recovery is questionable, however, since the
appearance of a pseudonumber is supposed to be a danger signal to the
programmer and usually results from a programming error.
IEEE doubleprecision representations are quite similar. A
doubleprecision real begins with a sign bit, with the same interpretation
as in a singleprecision representation. The next eleven bits are used
for the exponent, which is an integer in the range from 1022 to +1023;
a bias of 1023 is added to the exponent, and the result is stored in a
binary numeral (the smallest is 00000000001
, the largest
11111111110
). The remaining fiftytwo bits are used for
the mantissa, and as above only the digits following the binary point of
the mantissa are actually stored; the 1
that precedes the
binary point is once again a hidden bit
. As in singleprecision
representations, the allzero exponent is used for unnormalized numbers
and (with an allzero mantissa) for 0, and the allone exponent is
used for the pseudonumbers positive infinity, negative infinity,
and NaN. The greatest real number that can be represented exactly as
a doubleprecision real is 2^{1024}  2^{971}, and the
least positive real that can be so represented is 2^{1074}.
Thursday, 22 February 1996 [John Stone]
Wednesday, 18 September 1996 [John Stone]
http://www.cs.grinnell.edu/~stone/courses/fundamentals/IEEEreals.html
.
Tuesday, 14 October 1997 [Samuel A. Rebelsky]
http://www.cs.grinnell.edu/~rebelsky/Courses/CS152/1997F/Readings/IEEEreals.html
.
Sunday, 2 February 2003 [Samuel A. Rebelsky]
Friday, 7 February 2003 [Samuel A. Rebelsky]
http://www.cs.grinnell.edu/~rebelsky/Courses/CS195/2003S/Readings/ieeereals.html
.
Wednesday, 15 September 2010 [Samuel A. Rebelsky]
http://www.cs.grinnell.edu/~rebelsky/Courses/CS161/2010F/Readings/ieeereals.html
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Related Courses:
[CSC195 2003S (Rebelsky)]
[CSC161 2009F (Coahran)]
[CSC161 2010S (Walker)]
Misc:
[SamR]
[ISO]
[GNU Coding Standards]
Disclaimer:
I usually create these pages on the fly
, which means that I rarely
proofread them and they may contain bad grammar and incorrect details.
It also means that I tend to update them regularly (see the history for
more details). Feel free to contact me with any suggestions for changes.
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.