Version 5.0 The LEDA User Manual

5.9 A Floating Point Filter ( floatf )
1. Definition
The type floatf provides a clean and efficient way to approximately compute with large
integers. Consider an expression E with integer operands and operators +, −, and ∗,
and suppose that we want to determine the sign of E. In general, the integer arithmetic
provided by our machines does not suffice to evaluate E since intermediate results might
overflow. Resorting to arbitrary precision integer arithmetic is a costly process. An
alternative is to evaluate the expression using floating point arithmetic, i.e., to convert the
operands to doubles and to use floating-point addition, subtraction, and multiplication.
Of course, only an approximation E ′ of the true value E is computed. However, E ′ might
still be able to tell us something about the sign of E. If E ′ is far away from zero (the
forward error analysis carried out in the next section gives a precise meaning to ”far
away”) then the signs of E ′ and E agree and if E ′ is zero then we may be able to conclude
under certain circumstances that E is zero. Again, forward error analysis can be used to
say what ‘certain circumstances’ are.
The type floatf encapsulates this kind of approximate integer arithmetic. Any integer (=
object of type integer) can be converted to a floatf; floatfs can be added, subtracted,
multiplied, and their sign can be computed: for any floatf x the function Sign(x) returns
either the sign of x (−1 if x < 0, 0 if x = 0, and +1 if x > 0) or the special value
NO IDEA. If x approximates X, i.e., X is the integer value obtained by an exact
computation, then Sign(x)! = NO IDEA implies that Sign(x) is actually the sign of X
if Sign(x) = NO IDEA then no claim is made about the sign of X.
#include < LEDA/numbers/floatf .h >
2. Creation
floatf x; introduces a variable x of type floatf and initializes it with zero.
floatf x(integer i); introduces a variable x of type floatf and initializes it with integer
i.
3. Operations
floatf
floatf
floatf
const floatf & a + const floatf & b
Addition.
const floatf & a − const floatf & b
Subtraction.
const floatf & a ∗ const floatf & b
Multiplication.