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POFind min/max of a float/double that has the same internal representation
 

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    <p>Refreshing on <a href="http://docs.sun.com/source/806-3568/ncg_goldberg.html" rel="nofollow noreferrer">floating points</a> (also <a href="http://www.validlab.com/goldberg/paper.pdf" rel="nofollow noreferrer">PDF</a>), IEEE-754 and taking part <a href="https://stackoverflow.com/questions/1421520/formatting-doubles-for-output-in-c">in this discussion on floating point rounding when converting to strings</a>, brought me to tinker: how can I get the maximum and minimum value for a given floating point number whose binary representations are equal.</p> <p><strong>Disclaimer</strong>: for this discussion, I like to stick to 32 bit and 64 bit floating point as described by IEEE-754. I'm not interested in extended floating point (80-bits) or quads (128 bits IEEE-754-2008) or any other standard (IEEE-854).</p> <p><strong>Background</strong>: Computers are bad at representing <code>0.1</code> in binary representation. In C#, a float represents this as <code>3DCCCCCD</code> internally (C# uses round-to-nearest) and a double as <code>3FB999999999999A</code>. The same bit patterns are used for decimal <code>0.100000005</code> (float) and <code>0.1000000000000000124</code> (double), but not for <code>0.1000000000000000144</code> (double). </p> <p>For convenience, the following C# code gives these internal representations:</p> <pre><code>string GetHex(float f) { return BitConverter.ToUInt32(BitConverter.GetBytes(f), 0).ToString("X"); } string GetHex(double d) { return BitConverter.ToUInt64(BitConverter.GetBytes(d), 0).ToString("X"); } // float Console.WriteLine(GetHex(0.1F)); // double Console.WriteLine(GetHex(0.1)); </code></pre> <p>In the case of <code>0.1</code>, there is no lower decimal number that is represented with the same bit pattern, any <code>0.99...99</code> will yield a different bit representation (i.e., float for <code>0.999999937</code> yields <code>3F7FFFFF</code> internally).</p> <p><strong>My question</strong> is simple: how can I find the lowest and highest decimal value for a given float (or double) that is internally stored in the same binary representation.</p> <p><strong>Why</strong>: (I know you'll ask) to find the error in rounding in .NET when it converts to a string and when it converts from a string, to find the internal exact value and to understand my own rounding errors better.</p> <p>My guess is something like: take the mantissa, remove the rest, get its exact value, get one (mantissa-bit) higher, and calculate the mean: anything below that will yield the same bit pattern. My main problem is: how to get the fractional part as integer (bit manipulation it not my strongest asset). <a href="http://www.yoda.arachsys.com/csharp/DoubleConverter.cs" rel="nofollow noreferrer">Jon Skeet's DoubleConverter</a> class may be helpful.</p>
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