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Discrete Mathematical Structures

1 Floating-point numbers

In a digital computer, the set of real numbers R is approximated by a set of floating-point numbers.
Representing a quantity as a floating-point number is very similar to representing it using scientific
notation. In scientific notation, a number that has very large or very small magnitude is represented
as a number of moderate magnitude multiplied by some power of ten. For example, 0.000008119
can be expressed as 8.119 ×10-6. The decimal point moves, or floats, as the power of 10 changes.

A set of floating-point numbers F is defined by four integers.

β the base

p the precision

[L ,U] the exponent range

A floating point number x ∈F is written as follows.

where E is an integer such that L≤ E ≤U

The string of base-β digits is called the fraction. Notice that the value of the precision
p determines how many digits there are in the floating-point number. The number E is
called the exponent. A set of floating-point numbers is said to be normalized if the lead digit
unless the number represented is zero . In other words, in a normalized set of floating point numbers,
a number has a lead digit of 0 if and only if that number is 0. In virtually every modern
computer, floating point numbers are normalized and use base 2 (i.e. β= 2). Internally to the
computer, a floating point number is represented by three integers. the sign (1-bit), the fraction,
and the exponent.

A real number y ∈ R is not always exactly representable in a floating-point system F . In this
case, y is usually approximated by x ∈ F where x = fl(y) is the closest floating-point number to
y. This function fl is called round to nearest. In the case where two floating point numbers are the
same distance from x, the nearest floating-point number ending in an even digit is chosen.

2 IEEE Standard 754

The two sets of floating point numbers most commonly supported on modern computer systems
are specified by IEEE Standard 754. They are.

IEEE single-precision. a base-2 (i.e. β= 2) system with 1 sign-bit, 8-bit exponents, 23-bit fractions
and a bias of 127

IEEE double-precision. a base-2 (i.e. β= 2) system with 1 sign-bit, 11-bit exponents, 52-bit
fractions and a bias of 1023

We denote the exponent as E, the sign-bit as S, the fraction as F and the bias as b. The bias allows
the IEEE 754 standard to represent both positive and negative exponents . Assume we have have a
bias of b where b is a positive integer. If the value of the exponent bits, interpreted as an unsigned
integer, is E then the exponent of the floating-point number is E - b. The fraction, F, can be
interpreted in two ways. When representing a normalized value, the fraction represents the value
1.F where an implicit leading 1 is pre-pended followed by the radix point and then the bits making
up F. When representing unnormalized values, the fraction is regarded as representing 0.F . The
standard also includes special values used to represent and NaN which stands for “Not a
Number”. We can summarize the rules for single precision IEEE 754 numbers as follow.

1. If E = 0 and F = 0 then the value is 0 with S indicating positive or negative.

2. If E = 255 and F ≠ 0, then the value is NaN.

3. If E = 255 and F = 0 then the value is with S indicating positive or negative.

4. If 0 < E < 255 then the number represented is the normalized value
5. If E = 0 and F ≠ 0 then the number represented is the unnormalized value

Similar rules apply for double precision numbers, just replace E = 255 with E = 2047 and use
for unnormalized values and for normalized values.

3 Floating-Point Arithmetic

To add or subtract two floating-point numbers, the exponents must match before the fractions can
be added or subtracted. If they do not match, the fraction of one must must be shifted until the two
exponents match. As a result, the sum or difference of two floating point numbers will not necessarily
be equal to the true sum of the two real numbers they represent. In other words, since the
sum of two p-digit numbers can have more than p digits, the excess digits cannot be represented by
a p-digit floating point number and will be lost. While multiplication and division of floating -point
numbers does not require the exponents to match, these operations may also produce answers different
from the corresponding operation on real numbers.

Since floating point operations only approximate traditional arithmetic operations on real numbers,
they are often represented by different symbols : , represent floating point addition,
subtraction, multiplication, and division respectively.

The associative law does not hold for floating-point numbers. The following laws do.

if and only if v = -u

4 Problems

In answering the following questions, remember that in a binary number, binary digits after the
“decimal point” are multiplied by successively smaller powers of 2, e.g.
. It may be useful to review section 3.6 of textbook which discusses representing
integers in different bases.

1. Floating-Point Number Systems [10 points]

(a) A Small Set of Floating-Point Numbers

Suppose we create a simple normalized floating-point system F with

The smallest positive number we can represent in this system is

What is the largest positive number representable in the system?

How many numbers are in this system?

(b) Picturing Floating-Point Numbers

Using decimal notation, write out the entire set of numbers representable in the floatingpoint
system described above (e.g. we have seen 0.5 is one of the numbers represented
in the system so it will be in the list you write out).

Now, draw a number line representing the real numbers in the interval [-3, 3] and put
tick marks to show where the floating-point numbers in F fall on the line. Are the
floating-point numbers uniformly distributed on the line ?

2. IEEE Standard Floating-Point Systems [15 points]
Answer the following questions for both single precision numbers and double precision numbers.

(a) How many numbers are in the set?

(b) What is the smallest positive number that can be represented?

(c) What is the largest positive number that can be represented?

3. Round to Nearest [10 points]
Suppose we have a round to nearest function fl that maps real numbers to the small floating
point system F described in Problem 1. Is this function one-to-one? Is it onto? Justify your
answers. Hint. for one-to-one use a specific example, for onto you can write a simple direct
proof showing that any x ∈ F has a real number that maps to it .

4. Arithmetic Operations [15 Points]

(a) We will prove floating-point addition is not associative. Suppose we have a decimal (i.e.
β= 10) floating point number system with a precision of 8 digits that uses round to
nearest. Under these conditions, we would have the following example.

Show that a different answer is reached for

(b) Consider a 2-digit decimal number system in which uses round to nearest. In this system,
Consider the algebraic identity . Is this identity valid
in the number system we are using? To answer that question, try setting a = 1.8 and
b = 1.7 and calculate both the left hand side and the right-hand side of the identity using
2-digit decimal arithmetic, rounding if necessary after each operation. What answers do
you get?

(c) This problem involves fixed-point arithmetic but the lesson about the perils of finite
precision are relevant for floating-point numbers as well. In 1982 the Vancouver Stock
Exchange instituted a new index initialized to a decimal value of 1000.000. By design,
indexed was fixed to always have three digits after the decimal point. After each transaction,
a new index value was computed and truncated to three trailing decimal digits.
Twenty two months later, the value of the index should have been 1098.892. Instead, the
calculated value was 524.881. In one or two sentences, explain what you think caused
this inaccuracy.

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