复旦大学软件学院计算机系统课件 Floating Point.ppt

* * * * * * * * * * * * * * * * * * * * * * * * * * * Round-to-Even Default Rounding Mode Hard to get any other kind without dropping into assembly All others are statistically biased Sum of set of positive numbers will consistently be over- or under- estimated * Round-to-Even Applying to Other Decimal Places When exactly halfway between two possible values Round so that least significant digit is even E.g., round to nearest hundredth 1.2349999 1.23 (Less than half way) 1.2350001 1.24 (Greater than half way) 1.2350000 1.24 (Half way—round up) 1.2450000 1.24 (Half way—round down) * Rounding Binary Number “Even” when least significant bit is 0 Half way when bits to right of rounding position = 100…2 Value Binary Rounded Action Round Decimal 2 3/32 10.00011 10.00 Down 2 2 3/16 10.0011 10.01 Up 2 1/4 2 7/8 10.111 11.00 Up 3 2 5/8 10.101 10.10 Down 2 1/2 * Floating-Point Operations Conceptual View First compute exact result Make it fit into desired precision Possibly overflow if exponent too large Possibly round to fit into frac * FP Multiplication Operands (–1)s1 M1 2E1 (–1)s2 M2 2E2 Exact Result (–1)s M 2E Sign s : s1 ^?s2 Significand M : M1 *?M2 Exponent E : E1 +?E2 * FP Multiplication Fixing If M ≥ 2, shift M right, increment E If E out of range, overflow Round M to fit frac precision * FP Addition Operands (–1)s1 M1 2E1 (–1)s2 M2 2E2 Assume E1 E2 Exact Result (–1)s M 2E Sign s, significand M: Result of signed align add Exponent E : E1 * FP Addition (–1)s1 m1 (–1)s2 m2 E1–E2 + (–1)s m * FP Addition Fixing If M ≥ 2, shift M right, increment E if M 1, shift M left k positions, decrement E by k Overflow if E out of range Round M to fit frac precision * Floating Point in C C Guarantees Two Levels float single precision double double precision * Floating Point Puzzles int x = …; float f = …; double d = …; Assume neither d nor f is NAN or infinity * Floating Point Puzzles x == (int)(float) x x == (int)(double) x f == (float)(double) f d == (float