Programming and Problem Solving with Java

(^596) | Multidimensional Arrays and Numeric Computation
Here is the result of adding xto the sum of yand z:
(y) 1325000  100
(z) 5424  100
1330424  100 = 1330  103 (truncated to four digits)
(yz) 1330  103
(x) –1324  103
6  103 = 6000  100 ←x+ (y+ z)
These two answers are the same in the thousands place but are different there-
after. This discrepancy results from a representational error.
Representational error makes it unwise to use a floating-point variable as a
loop control variable. Because precision may be lost in calculations involving float-
ing-point numbers, it is difficult to predict when (or even if) a loop control variable
of type float(or double) will equal the termination value. As a consequence, a count-
controlled loop with a floating-point control variable can behave unpredictably.
Also because of representational errors, you should never compare floating-
point numbers for exact equality. Rarely are two floating-point numbers exactly
equal, and thus you should compare them only for near equality. If the difference between the
two numbers is less than some acceptable small value, you can consider them equal for the pur-
poses of the given problem.

Implementation of Floating-Point Numbers in the Computer

All computers limit the precision of floating-point numbers, although modern machines

use binary rather than decimal arithmetic. In our representation, we used only five digits to

simplify the examples. In fact, some computers really are limited to only four or five digits

of precision. Other systems provide 6 significant digits, 15 significant digits, and 19 signifi-

cant digits, respectively, for three sizes of floating-point types. We have shown only a single-

digit exponent, but most systems allow two digits for the smaller floating-point type and up

to four-digit exponents for a longer type.

Some languages leave the range and precision of floating-point types to each individual

compiler. Java, however, states the range and precision in the language specification in the

following formula:

sm 2 e

where sis +1 or 1,mis a positive integer less than 2^24 , and eis between 126 and 127, in-

clusive, for values of type float. For values of type double,mis less than 2^53 and eis between

1,022 and 1,023. No, we don’t expect you to calculate this value. Each Java numeric class (such

as Integeror Double) provides two constants,MAX_VALUEand MIN_VALUEthat contain those

values..

When you declare a floating-point variable, part of the memory location contains the

exponent, and the number itself (called themantissa) is assumed to be in the balance of the

Representational error An
arithmetic error that occurs
when the precision of the true
result of an arithmetic operation
is greater than the precision of
the machine