Ch 3. Arithmetic for computers - (4)

4. Floating point number arithmetic

Normalized Scientific Notation

• Scientific notation
  coefficient X (base number)^exponent
  • coefficient: 0~9까지의 한 자리 숫자
    • Ex) 7.15576 x 10^4, 0.314 x 10^1
• Normalized scientific notation
  1 ≤ coefficient ≤ 10
  • 0.314 x 10^1 (X)
  • Binary number의 경우
    • 1.yyyy… x 2^z
    • 소수점 앞 1만 올수 있음 (0은 못옴)
• Representation: Normalized scientific notation
  • sign, exponent, significand :
    (-1)^sign x significand x 2^exponent
  • sign
    • if sign = 0, Positive floating number
    • if sign = 1, Negative floating number
  • exponent
    • bit가 커지면 훨씬 더 큰 수를 표현 (range 증가)
  • significand
    • bit가 커지면 정확성이 올라감
• IEEE 754 floating point standard
  • single precision (32bits)
    • float
    • sign: 1 bit
    • exponent: 8 bit
    • fraction 23 bit
      • 10진수로 6자릿수
  • double precision (64bits)
    • double
    • sign: 1 bit
    • exponent: 11 bit
    • fraction 52 bit
      • 10진수로 16자릿수

IEEE 754 floating-point standard

Binary number의 경우

• fraction의 1 bit은 암시적으로 확정 (0은 올 수 없으니)
• Exponent: biased 사용
  • single precision: bias of 127
  • double precision: bias of 1023
  • all 0s: smallest exponent
    • exponent value = 0 - 127 = -127
  • all 1s: largest exponent
    • exponent value = 255 - 127 = 128
• (-1)^sign X (1 + fraction) X 2^(exponent - bias)

Precision Range

• Exponent: all 0s와 all 1s 는 다른 목적으로 사용
• Smallest value
  • Exponent: 000…001
    • 1 - 127(1023) = -126(-1022)
  • Fraction: 000..00
    • 1 + fraction(0) = 1.0
  • ±1.0 × 2^-126(-1022)(2) or ±1.2(2.2) × 10^-38(-308)(10)
• Largest value
  • Exponent: 111…110
    • 254(2046) - 127(1023) = 127(1023)
  • Fraction: 111..11
    • 1 + fraction(1) = 2.0
  • ±2.0 × 2^127(1023)(2) or ±3.4(1.8) × 10^38(308)(10)

Floating Point Complexities

• exponent: all 0s, fraction: all 0s, sign = 0
  • 0 (O) 1.0 x 2^-127 (X)
• exponent 범위를 벗어나게 되면 underflow 발생
  • 보다 정확성이 더 요구 되는 부분
  • 0.5 x 2^-129

Denormalized Numbers

• Exponent: all 0s, Fraction: NonZero
  • hidden bit = 0
  • x = (-1)^S x (0 + fraction) x 2^-Bias
  • Denormalized Numbers * normalized number 보다 더 작은 숫자를 표현 할 수 있음 underflow 내 숫자
• Exponent: all 0s, Fraction: all 0s
  • x = (-1)^S x (0 + 0) x 2^-Bias = ±0.0
  • if sign = 0 positive 0
  • if sign = 1 negative 0

Infinities and NaNs

• Exponent: all 1s, Fraction: all 0s
  • ± ∞
• Exponent: all 1s, Fraction: NonZero
  • Not-a-Number (NaN)
  • 0.0 / 0.0