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Lecture 4

# Xxxx2 xxxx the n bits of frac minimum when 0000 m 10

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Unformatted text preview: biased value: E = exp -­‐ Bias § exp is an unsigned value ranging from 1 to 2k-­‐2 (k == # bits in exp) § Bias = 2k-­‐1 -­‐ 1 §  Single precision: 127 (so exp: 1…254, E: -­‐126…127) §  Double precision: 1023 (so exp: 1…2046, E: -­‐1022…1023) § These enable nega=ve values for E, for represen=ng very small values ¢  Signiﬁcand coded with implied leading 1: M = 1.xxx…x2 §  xxx…x: the n bits of frac §  Minimum when 000…0 (M = 1.0) §  Maximum when 111…1 (M = 2.0 – ε) §  Get extra leading bit for “free” IEEE Floa-ng Point Standard University of Washington Normalized Encoding Example s E V = (–1) * M * 2 s exp frac n k ¢  Value: float f = 12345.0; §  1234510 = 110000001110012 = 1.10000001110012 x 213 (normalized form) ¢  Signiﬁcand: M = frac = ¢  1.10000001110012 100000011100100000000002 Exponent: E = exp -­‐ Bias, so exp = E + Bias E = Bias = exp = ¢  13 127 140 = 100011002 Result: 0 10001100 10000001110010000000000 s exp frac IEEE Floa-ng Point Standard University of Washington Integer & Floa-ng Point Numbers ¢  ¢  ¢  ¢  ¢  ¢  ¢  ¢  Representa-on of integers: unsigned and signed Unsigned and signed integers in C Arithme-c and shiBing Sign extension Background: frac-onal binary numbers IEEE ﬂoa-ng-­‐point standard Floa-ng-­‐point opera-ons and rounding Floa-ng-­‐point in C Floa-ng Point Opera-ons University of Washington How do we do opera-ons? ¢  Unlike the representa-on for integers, the representa-on for ﬂoa-ng-­‐point numbers is not exact Floa-ng Point Opera-ons University of Washington Floa-ng Point Opera-ons: Basic Idea s E V = (–1) * M * 2 s exp frac k ¢  x +f y = Round(x + y) ¢  x *f y = Round(x * y) ¢  n Basic idea for ﬂoa-ng point opera-ons: §  First, compute the exact result §  Then, round the result to...
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