CSAPP Float Point

Float Point

Fractional Binary Numbers

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

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 \(\rightarrow\) range of numbers \(\uparrow\)
• binary point shift left \(\rightarrow\) 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 \(\not =\) \(000…0\) and exp \(\not =\) \(111…1\)
• 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)