notes:ieee_754-1985

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notes:ieee_754-1985 [2013/02/20 16:19] andy |
notes:ieee_754-1985 [2013/02/24 00:11] andy [Normalised Values] |
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Briefly, floating point numbers are a way to represent a wide numeric range in a limited set of bits by storing a limited number of significant digits. As the absolute size of the number increases, the absolute precision decreases. | Briefly, floating point numbers are a way to represent a wide numeric range in a limited set of bits by storing a limited number of significant digits. As the absolute size of the number increases, the absolute precision decreases. | ||

- | The principle is similar to exponential notation for numbers (e.g. **3.523 x 10<sup>6</sup>**) except that IEEE floating point values use powers of 2 instead of 10. | + | The principle is similar to exponential notation for numbers (e.g. $3.523 \times 10^6$) except that IEEE floating point values use powers of 2 instead of 10. |

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The most common form of IEEE floating point numbers is the normalised form --- this is where the exponent has a value in the valid range (once the bias has been subtracted from the unsigned value stored). As explained above, the significand stores only the digits after the leading **1**, which is implicit. | The most common form of IEEE floating point numbers is the normalised form --- this is where the exponent has a value in the valid range (once the bias has been subtracted from the unsigned value stored). As explained above, the significand stores only the digits after the leading **1**, which is implicit. | ||

- | If the standard C library is available, the ''[[man>frexp|frexp()]]'' function normalises a floating point value such that the fractional part will be in the range **0.5 ≤ x < 1.0**. Multiplying this value by **2** and reducing the exponent by **1** yields a value in the desired range **1.0 ≤ x < 2.0**. At this point the leading digit can then be discarded as the implicit leading **1** (see the [[#Significand]] section for details). | + | If the standard C library is available, the ''[[man>frexp|frexp()]]'' function normalises a floating point value such that the fractional part will be in the range $0.5 \le \times < 1.0$. Multiplying this value by **2** and reducing the exponent by **1** yields a value in the desired range $1.0 \le \times < 2.0$. At this point the leading digit can then be discarded as the implicit leading **1** (see the [[#Significand]] section for details). |

If ''frexp()'' is available then it should be used, as it will likely use the underlying hardware representation to avoid expensive loops. However, a naive implementation can quite simply mimic its functionality --- the following version demonstrates the principle, but a production version would also need to check for special values (zero, NaN, infinities) as well as catching under- and overflows: | If ''frexp()'' is available then it should be used, as it will likely use the underlying hardware representation to avoid expensive loops. However, a naive implementation can quite simply mimic its functionality --- the following version demonstrates the principle, but a production version would also need to check for special values (zero, NaN, infinities) as well as catching under- and overflows: |

notes/ieee_754-1985.txt · Last modified: 2013/02/24 00:18 by andy