Lecture 4

# 020 exponent e weights value by a possibly negave

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Unformatted text preview: g Award to William Kahan (UC Berkeley) IEEE Floa-ng Point Standard University of Washington Floa-ng Point Representa-on ¢  Numerical form: V10 = (–1)s * M * 2E §  Sign bit s determines whether number is nega=ve or posi=ve §  Signiﬁcand (man=ssa) M normally a frac=onal value in range [1.0,2.0) §  Exponent E weights value by a (possibly nega=ve) power of two ¢  Representa-on in memory: §  MSB s is sign bit s §  exp ﬁeld encodes E (but is not equal to E) §  frac ﬁeld encodes M (but is not equal to M) s exp frac IEEE Floa-ng Point Standard University of Washington Precisions ¢  Single precision: 32 bits s exp 1 ¢  frac k=8 n=23 Double precision: 64 bits s exp 1 frac k=11 n=52 IEEE Floa-ng Point Standard University of Washington Normaliza-on and Special Values s E V = (–1) * M * 2 s exp frac k ¢  n “Normalized” means the man-ssa M has the form 1.xxxxx §  0.011 x 25 and 1.1 x 23 represent the same number, but the la[er makes be[er use of the available bits §  Since we know the man=ssa starts with a 1, we don't bother to store it! ¢  How do we represent 0.0? Or special / undeﬁned values like 1.0/0.0? IEEE Floa-ng Point Standard University of Washington Normaliza-on and Special Values s E V = (–1) * M * 2 s exp frac k ¢  n “Normalized” means the man-ssa M has the form 1.xxxxx §  0.011 x 25 and 1.1 x 23 represent the same number, but the la[er makes be[er use of the available bits §  Since we know the man=ssa starts with a 1, we don't bother to store it ¢  Special values: §  The bit pa[ern 00...0 represents zero §  If exp == 11...1 and frac == 00...0, it represents ∞ e.g. 1.0/0.0 = -1.0/-0.0 = +∞, 1.0/-0.0 = -1.0/0.0 = -∞ §  If exp == 11...1 and frac != 00...0, it represents NaN: “Not a Number” §  Results from opera=ons with undeﬁned result, e.g. sqrt(–1), ∞ - ∞, ∞ * 0 §  IEEE Floa-ng Point Standard University of Washington Normalized Values s E V = (–1) * M * 2 s exp frac k ¢  ¢  n Condi-on: exp ≠ 000…0 and exp ≠ 111…1 Exponent coded as...
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