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Double Precision Reals

A B C D E F G H I J K L M N O P Q R S T U V W X Y Z overview

Double precision reals are stored as dual floating point numbers with a shifted exponent:

$\displaystyle x=\pm 1.m_2\cdots m_p\cdot 2^{n-1023}, $

occupying 8 bytes. According to the IEEE standard, 53 bits are used for sign and mantissa and 11 bits for the biased exponent $ n$:


0  1       11 12                                   63
$ \sigma$    $ n$    $ m_2$        $ \cdots$                $ m_{53}$

The extreme exponents $ \underline{n}=0$ and $ \overline{n}=2047$ are reserved for special purposes. Hence,

$\displaystyle x_{\min}=2^{-1022},\quad x_{\max}=(1-2^{-53})2^{1024} $

are the smallest and largest positive double precision reals.

The representation of exceptional cases is shown in the following table.


$ \sigma$ $ n$ $ m$
0 0 or 1 $ \underline{n}=0$ 0
underflow 0 or 1 $ \underline{n}$ $ m_1=0$ possible
overflow (Inf) 0 or 1 $ \overline{n}=2047$ $ m_2=m_3=\ldots =0$
NaN 0 or 1 $ \overline{n}$ $ m_2$ or $ m_3$ or $ \ldots\ne 0$

Underflow occurs if $ \vert x\vert<x_{\min}$. The storage convention permits the range of values

$\displaystyle 2^{-1022} > x\geq 0.0\cdots 01\cdot 2^{-1022} = 2^{-1074}. $

Overflow occurs if $ \vert x\vert>x_{\max}$ and is represented by the symbol Inf preceeded by a sign.

Finally, NaN stands for not-a-number indicating the result of an undefined operation, such as 0/0, Inf-Inf, etc.

Example:

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  automatically generated 6/11/2007