CSAPP Float Point

Float Point

Fractional Binary Numbers

• Representation
  • Bits to right of "binary point" represent fractional powers of (2)
  • Represents rational number:
    [sum_{k=-j}^i b_k imes 2^k ]

we can represent any fractional binary number

Fractional Binary Numbers: Examples

Observations

• Divide by 2 by shifting right (unsigned)
• Multiply by 2 by shifting left
• Numbers of form 0.111111…2 are just below 1.0
  • Use notation (1.0^{-varepsilon})
    (varepsilon) depends how many bits you have to the right of binary point

Representable Numbers

Limitation #1

Can only exactly represent numbers of the form
(frac{x}{2^k})
example:
(1/3) Representation: ({0.0101010101[01]…}_2 )
(1/5) Representation: ({0.001100110011[0011]…}_2 )

Limitation #2

Just one setting of binary point within the (w) bits
Limited range of numbers:

• binary point shift right ( ightarrow) range of numbers (uparrow)
• binary point shift left ( ightarrow) range of fractional binary numbers (uparrow)

IEEE Floating Point

Floating Point Representation

• Numerical Form: ((-1)^s M 2^E)
  • Sign bit (s) determines whether number is negative or positive
  • Significand (M)(mantissa) normally a fractional value in range ([1.0,2.0)).
  • Exponent (E) weights value by power of two
• Encoding
  • 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)

Precision options

• Single precision: 32 bits
  (s):1 bit
  (exp): 8 bit
  (frac): 23 bit
• Double precision: 64 bits
  (s): 1 bit
  (exp): 11 bit
  (frac): 52 bit

"Normalized" Values

• When: exp ( ot =) (000…0) and exp ( ot =) (111…1)
• Exponent coded as a biased value: E = Exp – Bias(7 unsigned numbers)
  • Exp: unsigned value of exp field(we can compare two float numbers using Exp directly because of the unsigned value)
  • Bias = (2^{k-1} - 1),where (k) is number of exponent bits
    Single precision: 127 (Exp: 1…254, E: -126…127)(don't have 000..0 or 111..1)
    Double precision: 1023 (Exp: 1…2046, E: -1022…1023)(don't have 000..0 or 111..1)
• Significand coded with implied leading 1: M = 1.xx..x2
• xxx…x: bits of frac field(1 is drop,because we want a bit for free)
• Minimum when frac=000…0 (M = 1.0)