03. float

'Computer Systems: A Programmer's Perspective 3rd (Randal E. Bryant)'를 주교재로 한 시스템 소프트웨어 강의 수업자료를 복습하며 정리한 것

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Fractional Binary Numbers

• e.g) 111.11 -> 2^2 + 2^1 + 2^0 + 2^(-1) + 2^(-2)

 

IEEE Floating Point

• IEEE Standard 754
  • Uniform standart for floating point arithmetic
  • Nice standard for rounding, overflow, underflow
  • Hard to make fast in H/W

• Representation
  • Numerical Form (-1)^s * M * 2^E
  • Sign bit s(positive or negative)
  • Significand M(유효숫자, 가수부) - normally fractional value in range [1.0, 2.0)
  • Exponent E(지수부)

• Encoding
  • MSB bit s
  • exp field encodes E(not equal to E)
  • frac field encodes M(not equal to M)
  • e.g.) Single precision : 32bits(1 + 8 + 23)

• Normalized Values
  • exp != 000...0 and exp != 111...1
  • Exponent coded as biased value
  • E = exp - Bias(= 2^(k - 1) - 1) where k is # of exponent bits
  • Significand coded with implied leading 1
  • M = 1.xxx...x (xxx...x : bits of frac field)
  • e.g.) Single precision

• Denormalized Values
  • where exp = 000...0, E = 1 - Bias, M = 0.xxx...x
  • possible results : +0, -0, closest to 0.0

• Special Values
  • where exp = 111...1
  • possible results : positive infinity, negative infinity, overflows, NaN

 

Rounding, addition, multiplication

• Rounding
  • modes
    1. Towards zero
    2. Round down : 버림
    3. Round up : 올림
    4. Nearest Even(default) : 가까운 값으로 근사, .5의 경우 짝수가 되는 쪽으로
• Addition
  • (-1)^s1 * M1 * 2^E1 + (-1)^s2 * M2 * 2^E2 = (-1)^s * M * 2^E (assume E1 > E2)
  • M >= 2라면, M을 오른쪽으로 쉬프트하고, E 증가
  • M < 1이라면, M을 왼쪽으로 k만큼 쉬프트, E를 kakszma rkath
  • E가 범위를 벗어나면 overflow
• Multiplication
  • ((-1)^s1 * M1 * 2^E1) * ((-1)^s2 * M2 * 2^E2) = (-1)^s * M * 2^E
  • Sign s : s1 ^ s2
  • M : M1 * M2
  • E : E1 + E2
  • M >= 2라면, M을 오른쪽으로 쉬프트, E를 증가
  • E가 범위를 벗어나면 overflow