[数据结构与算法]计算机组成1 --Number Representation

Logical Operations

• Shift operations

  • Left Shift

    throw away extra bits on left, fill with 0’s on the right

  • Right Shift

    throw away extra bits on right

    logical shift: fill with 0’s on the left

    arithmetic shift: replicate most significant bit(x >> k gives$?x/2^k?$ towards negative infinity)

Floating Point Representation

Biased Representation

The actual value is the binary value plus a fixed bias

Definition

$(?1{)}^s\times (1+F)\times 2^E$

• S represents Sign
  • 1 for negative, 0 for positive
• Significand
  • implicit leading 1, signed-magnitude (not 2’s complement)
• y’s represent Exponent

  • in biased notation

    $2^{k?1}\to 2^k?1$(bias of -127 for 32bits, -1023 for 64bits)

Special Cases

Overflow and Underflow

• Overflow ($>2.0\times 10^{38}$or$<?2.0\times 10^{38}$)

• Underflow ($?2.0\times 10^{38}<x<2.0\times 10^{38}$ without 0)

0,$\infty$ and NAN

• 0:Bit pattern all 0s

• $\infty$($1÷0$)

  • Sign bit 0 or 1, largest exponent (all 1s), 0 in fraction

• NaN($\infty ?\infty$,$0÷0$)

  • Sign bit 0 or 1, largest exponent (all 1s), not zero in fraction

op(NaN, X) = NaN

Demorms

Denormalized number:

• no (implied) leading 1(just $F$),frac nonzero
• exponent all 0,value = 1 – Bias (instead of 0 – Bias)