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| <h5 class="subsubsection">A.5.3.1 Floating Point Representation Concepts</h5> |
| |
| <p>This section introduces the terminology for describing floating point |
| representations. |
| |
| <p>You are probably already familiar with most of these concepts in terms |
| of scientific or exponential notation for floating point numbers. For |
| example, the number <code>123456.0</code> could be expressed in exponential |
| notation as <code>1.23456e+05</code>, a shorthand notation indicating that the |
| mantissa <code>1.23456</code> is multiplied by the base <code>10</code> raised to |
| power <code>5</code>. |
| |
| <p>More formally, the internal representation of a floating point number |
| can be characterized in terms of the following parameters: |
| |
| <ul> |
| <li><a name="index-sign-_0028of-floating-point-number_0029-3770"></a>The <dfn>sign</dfn> is either <code>-1</code> or <code>1</code>. |
| |
| <li><a name="index-base-_0028of-floating-point-number_0029-3771"></a><a name="index-radix-_0028of-floating-point-number_0029-3772"></a>The <dfn>base</dfn> or <dfn>radix</dfn> for exponentiation, an integer greater |
| than <code>1</code>. This is a constant for a particular representation. |
| |
| <li><a name="index-exponent-_0028of-floating-point-number_0029-3773"></a>The <dfn>exponent</dfn> to which the base is raised. The upper and lower |
| bounds of the exponent value are constants for a particular |
| representation. |
| |
| <p><a name="index-bias-_0028of-floating-point-number-exponent_0029-3774"></a>Sometimes, in the actual bits representing the floating point number, |
| the exponent is <dfn>biased</dfn> by adding a constant to it, to make it |
| always be represented as an unsigned quantity. This is only important |
| if you have some reason to pick apart the bit fields making up the |
| floating point number by hand, which is something for which the GNU |
| library provides no support. So this is ignored in the discussion that |
| follows. |
| |
| <li><a name="index-mantissa-_0028of-floating-point-number_0029-3775"></a><a name="index-significand-_0028of-floating-point-number_0029-3776"></a>The <dfn>mantissa</dfn> or <dfn>significand</dfn> is an unsigned integer which is a |
| part of each floating point number. |
| |
| <li><a name="index-precision-_0028of-floating-point-number_0029-3777"></a>The <dfn>precision</dfn> of the mantissa. If the base of the representation |
| is <var>b</var>, then the precision is the number of base-<var>b</var> digits in |
| the mantissa. This is a constant for a particular representation. |
| |
| <p><a name="index-hidden-bit-_0028of-floating-point-number-mantissa_0029-3778"></a>Many floating point representations have an implicit <dfn>hidden bit</dfn> in |
| the mantissa. This is a bit which is present virtually in the mantissa, |
| but not stored in memory because its value is always 1 in a normalized |
| number. The precision figure (see above) includes any hidden bits. |
| |
| <p>Again, the GNU library provides no facilities for dealing with such |
| low-level aspects of the representation. |
| </ul> |
| |
| <p>The mantissa of a floating point number represents an implicit fraction |
| whose denominator is the base raised to the power of the precision. Since |
| the largest representable mantissa is one less than this denominator, the |
| value of the fraction is always strictly less than <code>1</code>. The |
| mathematical value of a floating point number is then the product of this |
| fraction, the sign, and the base raised to the exponent. |
| |
| <p><a name="index-normalized-floating-point-number-3779"></a>We say that the floating point number is <dfn>normalized</dfn> if the |
| fraction is at least <code>1/</code><var>b</var>, where <var>b</var> is the base. In |
| other words, the mantissa would be too large to fit if it were |
| multiplied by the base. Non-normalized numbers are sometimes called |
| <dfn>denormal</dfn>; they contain less precision than the representation |
| normally can hold. |
| |
| <p>If the number is not normalized, then you can subtract <code>1</code> from the |
| exponent while multiplying the mantissa by the base, and get another |
| floating point number with the same value. <dfn>Normalization</dfn> consists |
| of doing this repeatedly until the number is normalized. Two distinct |
| normalized floating point numbers cannot be equal in value. |
| |
| <p>(There is an exception to this rule: if the mantissa is zero, it is |
| considered normalized. Another exception happens on certain machines |
| where the exponent is as small as the representation can hold. Then |
| it is impossible to subtract <code>1</code> from the exponent, so a number |
| may be normalized even if its fraction is less than <code>1/</code><var>b</var>.) |
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