Algorithms in a Nutshell

Floating-Point Computations | 49

Patterns and

Domains

As you can readily determine, these three points are collinear on the liney= 5 *x.

When computing floating-point calculations, however, the errors inherent in

floating-point arithmetic affect the computation. Using floats as the values, the

calculation results in 0.00048828125; using doubles, the computed value is actu-

ally a very small negative number! Now we could introduce a small valueδto

determine≅between two floating-point values. Under this scheme, if |a–b|<δ,

then we consideraandbto be equal. However, it may be the case thatx≅yand

y≅z, but it might not be the case thatx≅z. This breaks the principle of Transitivity

and makes it really challenging to write correct code.

Special Quantities

While all possible 64-bit values could represent valid floating-point numbers, the

IEEE standard defines several values that are interpreted as special numbers (and

are often not able to participate in the standard mathematical computations, such

as ADD or MULT), shown in Table 3-3. These values have been designed to make

it easier to recover from common errors, such as divide by zero, square root of a

negative number, overflow of computations, and underflow of computations.

Note that the values of positive zero and negative zero are also included in this

table, even though they can be used in computations.

These special quantities are the result of computations that go outside the accept-

able bounds. For example, the quantity positive infinity could result from the Java

computationdouble x=1/0.0. As an interesting aside, the Java virtual machine

would throwjava.lang.ArithmeticExceptionif the statement had instead read

double x=1/0, since this expression computes the integer division of two numbers.

Performance

It is commonly accepted that computations over integer values will be more effi-

cient than their floating-point counterparts. Table 3-4 lists the computation times

of 10,000,000 operations on our high-end performance platform. A 1996 Sparc

Ultra-2 machine generated the values in the third column. As you can see, the

performance of individual operations can vary significantly from one platform to

another.

Table 3-3. Special IEEE 754 quantities

Special quantity 64-bit IEEE 754 representation

Positive infinity 0x7ff0000000000000L

Negative infinity 0xfff0000000000000L

Not a number (NaN) 0x7ff0000000000001L through

0x7fffffffffffffffL and

0xfff0000000000001L through

0xffffffffffffffffL

Negative zero 0x8000000000000000

Positive zero 0x0000000000000000

Algorithms in a Nutshell
Algorithms in a Nutshell By Gary Pollice, George T. Heineman, Stanley Selkow ISBN:
9780596516246 Publisher: O'Reilly Media, Inc.

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