Floating Point

FP - Intro:

1. Given a value fractional value: d_md_{m-1}...d_0 . d_0...d_n
2. We would like to be able to represent it in binary
3. Floating-Point representation is one way in which we can achieve this

FP - Format:

1. Floating point numbers allow us to represent decimal values of the form: V = (-1)^s \times M \times 10^e
   • V : value
   • M : Fractional (significand) part of value (e.g 1.123)
   • s : Sign
   • e : Exponent part

FP - Bit Layout:

1. Single Precision:
   • 32 bit width:
     • float in C
2. Double Precision:
   • 64 bit width
     • double in C

Preliminary Remark:

1. Floating point is a representation that facilitates the encoding of fractional decimal values that are in scientific notation form.
   • e.g 23.12 \times 10^1
2. All decimal values that are encoded, are encoded such that only 1 digit
   left of the decimal point.

FP - Encoding Cases:

1. The arrangement of bits in a bit representation, effect the way we treat floating point:
   • 3 different Cases:
     1. Normalized:
        • If the exponent field is not all zeros and not all ones, then the floating point is said to be normalized.
     2. Denormalized:
        • If the exponent field is all zeros, then the floating point value is said to be denormal.
     3. Special:
        • If the exponent field is all 1’s, then the floating point value is a special value (e.g infinity, or NaN).

1. Normalized:

• Notice this means normalized fractional binary values (e.g 1.1010) can all only ever start with a 1 or 0. †

• A floating point value is normalized when the binary value starts with 1.

  • If a value starts with a 0 (e.g 0.101) we can normalize a value by shifting the binary point left by multiplying by 2^{-1}.

• Since all values that are normalized must start with a one, we can simply not store it in the bit field and make it implied.

• A FP is said to normalized, when value it encodes has a single leading 1 before the decimal point.

  • Example:
    • 1.110 - normalized
    • 11.110 - not normalized †: unlike base-10 a fractional decimal value can take any value 0 through 9

2. Denormalized:

• When a floating point is denormal, there is no implied 1.
• Allows us to represent the values closest to zero and zero itself.

Multiplication

Basics (Assuming A and B are normalized values):

• (A × B):
  1. Multiply the significands including the implicit 1.
  2. Add the exponents.
  3. Normalize the product.
  4. Round to ensure the product fits the floating-point format.

Example:

• Suppose A and B are floating point values with a significand of ‘p’ bits long.
• The significands of A and B are multiplied:
  • The final product will initially be 2p bits in width due to the multiplication.
  • To fit the floating-point format, the product must be truncated back to ‘p’ bits.

Truncation in Multiplication:

1. Adjustment: Reduce the 2p-bit product to ‘p’ bits, deciding which bits to keep.
2. Rounding: Apply a rounding strategy (e.g., round to nearest, toward zero) to minimize truncation error.
3. Overflow and Underflow: Check and handle cases where the product is too large (overflow) or too small (underflow).
4. Normalization: Adjust the significand and exponent so the significand falls within the expected range.
5. Sign Bit: Determine the sign of the result using the XOR of the operands’ sign bits.