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# Floating-Point Complexities (2012)

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## Floating-level complexities

Binary floating-level math is complicated and subtle. I’ve silent here about a of my licensed oddball info about IEEE floating-level math, in step with the articles to this level in my floating-level series. The level of hobby in this list is on scurry collectively with the float nonetheless the identical ideas all observe to double.

These oddities don’t create floating-level math imperfect, and in lots of cases these oddities would possibly maybe also furthermore be neglected. However while you strive to simulate the limitless expanse of the precise-number line with 32-bit or 64-bit numbers then there will inevitably be locations where the abstraction breaks down, and it’s valid to be taught about them.

These type of information are functional, and some of them are exquisite. You rep to prefer which is which.

• Adjoining floats (of the identical sign) comprise adjacent integer representations, which makes producing the following (or all) floats trivial
• FLT_MIN is no longer the smallest certain scurry collectively with the float (FLT_MIN is the smallest certain normalized scurry collectively with the float)
• The smallest certain scurry collectively with the float – assuming denormals are supported, as they would possibly maybe also silent be – is 8,388,608 times smaller than FLT_MIN
• FLT_MAX is no longer the largest certain scurry collectively with the float (it’s the largest finite scurry collectively with the float, nonetheless the particular imprint infinity is bigger)
• 0.1 can’t be exactly represented in a scurry collectively with the float
• All floats would possibly maybe also furthermore be exactly represented in decimal
• Over a hundred decimal digits of mantissa are required to precisely level to the imprint of some floats
• 9 decimal digits of mantissa (plus sign and exponent) are enough to uniquely title any scurry collectively with the float
• The Visible C++ 2010 debugger shows floats with edifying 8 mantissa digits
• The integer representation of a scurry collectively with the float is a piecewise linear approximation of the base-2 logarithm of that scurry collectively with the float
• You would possibly per chance calculate the base-2 log of an integer by assigning it to a scurry collectively with the float
• Most scurry collectively with the float math offers inexact results attributable to rounding
• The elemental IEEE math operations guarantee edifying rounding
• Subtraction of floats with a connected values (f2 0.5

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## Knowasiak

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