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2013-11-15T04:36:34.490
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<p>There are none.</p> <p>The IEEE754 representation is <em>canonical</em>, which means that all numbers have precisely one representation if any, and no others.</p> <h3>Format of an IEEE754 single-precision floating-point number</h3> <pre><code>31 0 seeeeeeeemmmmmmmmmmmmmmmmmmmmmmm </code></pre> <p>Mapping to</p> <p>(-1)^s * 2^(eeeeeeee - 127) * 1.mmmmmmmmmmmmmmmmmmmmmmm (binary) </p> <h3>Proof of uniqueness of representation</h3> <ul> <li>The sign is encoded in the sign bit, <code>s</code>. Two putatively different representations of the same real number must have the same value of <code>s</code>, because only that field controls the sign.</li> <li>The location of the most-significant <code>1</code>-bit, in digit positions left from the decimal comma, is encoded in biased format in the eight <code>eeeeeeee</code> exponent bits. Two putatively different representations of the same real number must have the same value of <code>eeeeeeee</code>, because only that field controls the position of the most significant 1-bit.</li> <li>The 23 bits immediately below the most-significant <code>1</code>-bit are encoded in the mantissa (<code>mmmm...</code>). Two putatively different representations of the same real number would have to have the same mantissa value, because only that field controls the 23 bits immediately below the most-significant <code>1</code>-bit of the real number.</li> </ul> <p>The conclusion is a contradiction. Two putatively different representations of a same real number would have to have the same sign, exponent and mantissa; In other words they must be identical.</p> <p>This discussion ignores all the special values, like infinities, NaNs and denormals.</p>
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1. POAll real numbers that have more than 1 representation in IEEE-754 of single precision
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