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| <h3 class="section">20.3 Floating Point Numbers</h3> |
| |
| <p><a name="index-floating-point-2335"></a><a name="index-IEEE-754-2336"></a><a name="index-IEEE-floating-point-2337"></a> |
| Most computer hardware has support for two different kinds of numbers: |
| integers (<small class="dots">...</small>-3, -2, -1, 0, 1, 2, 3<small class="dots">...</small>) and |
| floating-point numbers. Floating-point numbers have three parts: the |
| <dfn>mantissa</dfn>, the <dfn>exponent</dfn>, and the <dfn>sign bit</dfn>. The real |
| number represented by a floating-point value is given by |
| (s ? -1 : 1) &middot; 2^e &middot; M |
| where s is the sign bit, e the exponent, and M |
| the mantissa. See <a href="Floating-Point-Concepts.html#Floating-Point-Concepts">Floating Point Concepts</a>, for details. (It is |
| possible to have a different <dfn>base</dfn> for the exponent, but all modern |
| hardware uses 2.) |
| |
| <p>Floating-point numbers can represent a finite subset of the real |
| numbers. While this subset is large enough for most purposes, it is |
| important to remember that the only reals that can be represented |
| exactly are rational numbers that have a terminating binary expansion |
| shorter than the width of the mantissa. Even simple fractions such as |
| 1/5 can only be approximated by floating point. |
| |
| <p>Mathematical operations and functions frequently need to produce values |
| that are not representable. Often these values can be approximated |
| closely enough for practical purposes, but sometimes they can't. |
| Historically there was no way to tell when the results of a calculation |
| were inaccurate. Modern computers implement the IEEE 754<!-- /@w --> standard |
| for numerical computations, which defines a framework for indicating to |
| the program when the results of calculation are not trustworthy. This |
| framework consists of a set of <dfn>exceptions</dfn> that indicate why a |
| result could not be represented, and the special values <dfn>infinity</dfn> |
| and <dfn>not a number</dfn> (NaN). |
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