Floating Point课件.ppt

– * – CS 105 Floating Point Topics IEEE Floating-Point Standard Rounding Floating-Point Operations Mathematical Properties CS 105“Tour of the Black Holes of Computing!” Floating-Point Puzzles For each of the following C expressions, either: Argue that it is true for all argument values Explain why not true x == (int)(float) x x == (int)(double) x f == (float)(double) f d == (float) d f == -(-f) 2/3 == 2/3.0 d 0.0 ??? ((d*2) 0.0) d f ??? -f -d d * d = 0.0 (d+f)-d == f int x = …; float f = …; double d = …; Assume neither d nor f is NaN IEEE Floating Point IEEE Standard 754 Established in 1985 as uniform standard for floating-point arithmetic Before that, many idiosyncratic formats Supported by all major CPUs Driven by numerical concerns Nice standards for rounding, overflow, underflow Hard to make go fast Numerical analysts predominated over hardware types in defining standard Fractional Binary Numbers Representation Bits to right of “binary point” represent fractional powers of 2 Represents rational number: bi bi–1 b2 b1 b0 b–1 b–2 b–3 b–j ? ? ? ? ? ? . 1 2 4 2i–1 2i ? ? ? ? ? ? 1/2 1/4 1/8 2–j Fractional Binary Number Examples Value Representation 5-3/4 101.112 2-7/8 10.1112 63/64 0.1111112 Observations Divide by 2 by shifting right Multiply by 2 by shifting left Numbers of form 0.111111…2 are just below 1.0 1/2 + 1/4 + 1/8 + … + 1/2i + … ? 1.0 Use notation 1.0 – ? Representable Numbers Limitation Can only exactly represent numbers of the form x/2k Other numbers have repeating bit representations Value Representation 1/3 0.0101010101[01]…2 1/5 0.001100110011[0011]…2 1/10 0.0001100110011[0011]…2 Numerical Form –1s M 2E Sign bit s determines whether number is negative or positive Significand M normally a fractional value in range [1.0,2.0). Exponent E weights value by power of two Encoding MSB is sign bit exp field encodes E frac field encodes M Floating-Point Representation s exp frac Encoding MSB is sign bit exp field encodes E frac field encodes M S