<P> The use of the equality test (if (x = = y)...) requires care when dealing with floating - point numbers . Even simple expressions like 0.6 / 0.2 - 3 = = 0 will, on most computers, fail to be true (in IEEE 754 double precision, for example, 0.6 / 0.2 - 3 is approximately equal to - 4.44089209850063 e-16). Consequently, such tests are sometimes replaced with "fuzzy" comparisons (if (abs (x-y) <epsilon)..., where epsilon is sufficiently small and tailored to the application, such as 1.0 E − 13). The wisdom of doing this varies greatly, and can require numerical analysis to bound epsilon . Values derived from the primary data representation and their comparisons should be performed in a wider, extended, precision to minimize the risk of such inconsistencies due to round - off errors . It is often better to organize the code in such a way that such tests are unnecessary . For example, in computational geometry, exact tests of whether a point lies off or on a line or plane defined by other points can be performed using adaptive precision or exact arithmetic methods . </P> <P> Small errors in floating - point arithmetic can grow when mathematical algorithms perform operations an enormous number of times . A few examples are matrix inversion, eigenvector computation, and differential equation solving . These algorithms must be very carefully designed, using numerical approaches such as Iterative refinement, if they are to work well . </P> <P> Summation of a vector of floating - point values is a basic algorithm in scientific computing, and so an awareness of when loss of significance can occur is essential . For example, if one is adding a very large number of numbers, the individual addends are very small compared with the sum . This can lead to loss of significance . A typical addition would then be something like </P> <P> The low 3 digits of the addends are effectively lost . Suppose, for example, that one needs to add many numbers, all approximately equal to 3 . After 1000 of them have been added, the running sum is about 3000; the lost digits are not regained . The Kahan summation algorithm may be used to reduce the errors . </P>

Discuss the need of exponent and mantissa in the number systems and list the types with examples