Chapter 1 Homework Multiplication Performed The Output Step Because The

Floating Point Number Systems 1

1.3 Floating Point Number Systems

1. Provide the ﬂoating point equivalent for each of the following numbers from the

ﬂoating point number system F(10,4,0,4). Consider both chopping and round-

ing. Compute the absolute and relative error in each ﬂoating point equivalent.

(a) π(b) e

(c) √2(d) 1/7

(e) cos 22◦(f) ln 10

(g) 3

√9

In the following table, δdenotes the absolute error and ǫthe relative error.

Chopping Rounding

yfl(y)error fl(y)error

cos 22◦0.9271 δ= 8.385 ×10−50.9272 δ= 1.615 ×10−5

2. Prove the bounds on the absolute and relative roundoff error associated with

rounding:

|flround(y)−y| ≤ 1

2βe−kand |flround(y)−y|

|y|≤1

2β1−k.

Consider the ﬂoating point system F(β, k, m, M)with rounding. Let ybe a real

number whose expansion is given by

2Section 1.3

2βe−k.

Provided y6= 0, given the restriction on d1,

3. Show that machine precision is the smallest ﬂoating point number, v, such that

fl(1 + v)>1.

First consider the ﬂoating point number system F(β, k, m, M)with chopping. The

number one is represented by the expansion

If we let

Floating Point Number Systems 3

4. (a) Construct an algorithm to determine machine precision and another algo-

rithm to determine the smallest positive number of a ﬂoating point number

system.

(b) Implement the algorithms from part (a) to determine machine precision

and the smallest positive number on your computing system. Consider

both single and double precision.

(c) Assuming that your computing system uses β= 2 and rounding, use the

results from part (b) to determine the values for kand m.

(a) Assuming the ﬂoating point system uses rounding, here is an algorithm to

Here is an algorithm to determine the smallest positive number, assuming that

underﬂow is handled by setting the value to zero.

GIVEN: base β

4Section 1.3

5. Determine machine precision, the smallest positive number and the largest pos-

itive number for the ﬂoating point number system used by your calculator.

Assuming the calculator uses β= 10, determine the values for k,mand M.

Answers will of course vary. On a Casio fx −300SA, machine precision is 5×

6. Determine the number of significant decimal digits and the number of significant

binary digits to which each of the following pairs of numbers agree.

(a) 355/113 and π

(b) 685/252 and e

(c) √10002 and √10001

(d) 103/280 and 1/e

(a) Because 

(b) Because 

685

252 −e

Floating Point Number Systems 5

(c) Because 

(d) Because

7. The ideal gas law states that P V =nRT , where Pis the pressure of the gas, V

is the volume, nis the number of moles, Tis the temperature and R= 0.08206

atm·m3/moles·K is the universal gas constant.

(a) Experimentally, it has been determined that P= 0.750 atm, V= 1.15 m3

and T= 294.1K. Assuming that all values have been rounded to the digits

shown, in what range of values does nfall?

(b) Experimentally, it has been determined that V= 0.331 m3,n= 0.00712

moles and T= 264.7K. Assuming that all values have been rounded to the

digits shown, in what range of values does Pfall?

6Section 1.3

(a) With

0.7495 atm < P < 0.7505 atm

(b) With

0.3305 m3< V < 0.3315 m3

8. In a physics laboratory, students measure the mass of a rectangular block to be

243.27 ±0.005 grams. The length, width and depth of the block are measured

to be 7.8±0.05 cm, 3.1±0.05 cm and 4.2±0.05 cm, respectively.

(a) In what range of values does the volume of the block fall?

(b) In what range of values does the density of the block fall? Density is mass

per unit volume.

(a) With

7.75 cm <length <7.85 cm

(b) Density is defined as mass per unit volume. It is given that

9. Students are using a pendulum to experimentally determine the acceleration

due to gravity, g. They measure the period, T, of the pendulum to be 2.2

seconds, and the length, l, of the pendulum to be 1.15 meters. Assuming that

all values are correct to the digits shown, in what range of values does gfall?

The variables in this problem are related by the formula T= 2πpl/g.

10. Determine machine precision, the smallest positive number and the largest posi-

tive number in the IEEE standard double precision system. Approximately how

many significant decimal digits does the double precision standard supply?

With β= 2 and k= 53, machine precision with rounding is

11. In addition to the standard single and double precision ﬂoating point systems,

Intel microprocessors also have an extended precision system F(2,64,−16381,16384).

Determine machine precision, the smallest positive number and the largest pos-

itive number for this extended precision system.

8Section 1.3

12. IBM System/390 mainframes provide three ﬂoating point number systems: short

precision F(16,6,−64,63), long precision F(16,14,−64,63) and extended preci-

sion F(16,28,−64,63). Compare machine precision, the smallest positive num-

ber and the largest positive number for each of these number systems.

In the short precision system F(16,6,−64,63), machine precision with rounding is

13. A common ﬂoating point number system used on modern calculators is

F(10,10,−98,100). Determine machine precision, the smallest positive num-

ber and the largest positive number for this extended precision system.

With β= 10 and k= 10, machine precision with rounding is

Floating Point Number Systems 9

14. (a) Show that the number of elements in the set F(β, k, m, M ) is given by

1 + 2(β−1)βk−1(M−m+ 1).

(b) How many elements are in the IEEE standard single precision number

system?

(c) How many elements are in the IEEE standard double precision number

system?

(a) Let’s start by counting the number of positive elements in F(β, k, m, M).

The only restriction on the mantissa is that the first digit cannot be zero;

15. Consider the function f(x) = x2−4x+ 4.

(a) What are the zeros of f?

10 Section 1.3

(b) Suppose we were to change the constant term to 4 −10−8. What are

the zeros of this new function? Relative to the size of the change in the

constant term, how big is the change in the zeros of the function?

(c) Now, suppose we were to change the constant term to 4 + 10−8. What are

the zeros of this new function? Relative to the size of the change in the

constant term, how big is the change in the zeros of the function?

(c) Finally, consider the function f(x) = x2−4x+ (4 + 10−8). By the quadratic

formula, the zeros of this new function are

16. Consider the linear, first-order differential equation

dx

dt +1

tx=sin t

t.

(a) Solve this equation subject to the initial condition x(π/2) = x0.

(b) Solve this equation subject to the perturbed initial condition x(π/2) =

x0+ǫ.

(c) By considering the difference between the solutions obtained in parts (a)

and (b), comment on the conditioning of this problem.

Floating Point Number Systems 11

(b) The general solution to the differential equation remains

(c) The difference between the solutions obtained in parts (a) and (b) is

17. Consider the linear, first-order differential equation

dx

dt −1

tx=tsin t.

(a) Solve this equation subject to the initial condition x(π/2) = x0.

(b) Solve this equation subject to the perturbed initial condition x(π/2) =

x0+ǫ.

12 Section 1.3

(a) Multiplying

where Cis a constant of integration. Using the initial condition x(π/2) = x0,

(b) The general solution to the differential equation remains

(c) The difference between the solutions obtained in parts (a) and (b) is

18. Consider the linear system of equations

1.1 2.1

2 3.8  x

y=b.

Floating Point Number Systems 13

(a) Solve the system for the right-hand side vector b=3.2 5.8T.

(b) Solve the system for the right-hand side vector b=3.21 5.79 T.

(c) Solve the system for the right-hand side vector b=3.1 5.7T.

(d) By considering the difference between the solutions obtained in parts (a),

(b), and (c), comment on the conditioning of this problem.

(a) The system of equations is