Floating point Numbers
- 1. Computer Architecture & organization ALU S.CIYAMALA KUSHBU ASSISTANT PROFESSOR/ECE
- 2. REVIEW OF FIXED POINT NUMBERS Limited in the range of numbers or the fractional number with fixed digits after radix point. Eg: 125.25,12345.89,10101.1010 Susceptible to problems of overflow. In a fixed-point processor, numbers are represented in integer format.
- 3. FLOATING POINT NUMBER Larger in the range of numbers or the fractional number with varying digits after radix point. Eg:1234565633.33333, 987891.23456789 etc .. Represented in scientific notation format. General syntax for floating point number can given as M x BE M -value of significand or Mantissa(takes +ve or –ve sign) B- Base E-Exponent(takes +ve or –ve sign) (i.e)The above example can be represented as 1.23*10^9 and 98.80*10^4. Representation of floating point is not unique. Eg:66.55 can be represented as 6.655*10^1 or 0.665*10^2or 0.06655*10^3.
- 4. Normalizing Floating Point Data Exactly one non-zero digit should appear before the radix point In a decimal number, this digit can be from 1 to 9. In a binary number, this digit should be 1. Normalized FP Numbers: Binary FP Normalization: Since the most significant bit is always 1 in binary, we can assume that it is implied and that we do not actually have to represent it.
- 5. There are three standard formats for representing floating-point numbers: • 32-bit format (single-precision) • 64-bit format (double-precision) • 80-bit format (extended precision) Single precision : • it is a binary format that occupies 32 bits • its significand has a precision of 24 bits Double precision : • it is a binary format that occupies 64 bits • its significand has a precision of 53 bits General syntax is (-1)S x 1.F x 2E
- 13. Floating-Point Addition EG: Consider a 4-digit decimal example 9.999 × 101 + 1.610 × 10–1 1. Align decimal points Shift number with smaller exponent 9.999 × 101 + 0.016 × 101 2. Add significands 9.999 × 101 + 0.016 × 101 = 10.015 × 101 3. Normalize result & check for over/underflow 1.0015 × 102 4. Round and renormalize if necessary 1.002 × 102
- 14. Floating-Point Binary Addition EG: Now consider a 4-digit binary example 0.5 + –0.4375 0.5)ten = 0.1)two x 20 = 1.000)two x 2-1 adding bias gives: 1.000 x 2126 –0.4375 )ten = -0.0111 )two x20 = -1.110 )two x 2-2 adding bias gives: -1.110 x 2125 1. Align binary points -1.110 x 2125 = -0.111x2126 2. Add significands 1.0 x2126 + (-0.111x2126 ) = 0.001 x 2126 3. Normalize result & check for over/underflow 1.0002 × 2–123 (with no over/underflow since biased exponent is between 0 and 255) 4. Round and renormalize if necessary No need to, it fits in 4 bits. Final answer with bias removed is: 1.000 x 2(123-127) = 1.000 x 2-4 = 0.0625)ten
- 16. Flowchart
- 17. Rounding Guard, Round and Sticky digit/bit Guard digit: first extra digit/bit to the right of mantissa -- used for rounding addition results Round digit: second extra digit/bit to the right of mantissa -- used for rounding multiplication results Sticky bit: third extra digit/bit to the right of mantissa – used for resolving ties such as 0.guatda.com/cmx.p50...00 vs. 0.guatda.com/cmx.p50...01
- 18. Rounding examples • An example without guard and round digits – Add 9.76 x 1025 and 2.59 x 1024 assuming 3 digit mantissa • Shift mantissa of the smaller number to the right: 0.25 x 1025 • Add mantissas: 10.01x 1025 • Check and normalize mantissa if necessary: 1.00x 1026 • An example with guard and round digits – Add 9.76 x 1025 and 2.59 x 1024 assuming 3 digit mantissa • Internal registers have extra two digits: 9.7600 x 1025 and 2.5900 x 1024 • Shift mantissa of the smaller number to the right: 0.2590 x 1025 • Add mantissas: 10.0190 x 1025 • Check and normalize mantissa if necessary: 1.0019 x 1026 • Round the result: 1.00 x 1026
- 19. Rounding examples • An example without guard and round digits – Add 9.78 x 1025 and 8.79 x 1024 assuming 3 digit mantissa • Shift mantissa of the smaller number to the right: 0.87 x 1025 • Add mantissas: 10.65 x 1025 • Normalize mantissa if necessary: 1.06 x 1026 • An example with guard and round digits – Add 9.78 x 1025 and 8.79 x 1024 assuming 3 digit mantissa • Internal registers have extra two digits: 9.7800 x 1025 and 8.7900 x 1024 • Shift mantissa of the smaller number to the right: 0.8790 x 1025 • Add mantissas (note extra digit on the left): 10.6590 x 1025 • Check and normalize mantissa if necessary: 1.0659 x 1026 • Round the result: 1.07 x 1026
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