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Understanding IEEE 754 Floating Point Standard: Representation & Operations

The IEEE 754-1985 standard defines the representation of floating point numbers in computing, encompassing both single (32 bits) and double precision (64 bits). This guide explores the structure of floating point numbers, including significands, exponents, and rounding methods. It highlights how to perform arithmetic operations like multiplication and division using the IEEE 754 format while also addressing unique features such as denormal numbers and handling special values (e.g., NaN, infinity). Gain insights into the precision and range of representable numbers within this critical standard.

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Understanding IEEE 754 Floating Point Standard: Representation & Operations

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  1. Floating point Number system corresponding to the decimal notation 1,837 * 10 significand exponent a great number of corresponding binary standards exists there is one common standard: IEEE 754-1985 (IEC 559) 4

  2. IEEE 754-1985 • Number representation Single precision (32 bits) sign: 1 bit exponent: 8 bits fraction: 23 bits Double precision (64 bits) sign: 1 bit exponent: 11 bits fraction: 52 bits Single extended and double extended numbers exists inside the floating point hardware

  3. IEEE 754-1985 SinglePrecision: 1 8 23 S E sign M exponent: excess 127 binary integer mantissa: sign + magnitude, normalized binary significand w/ hidden integer bit: 1.M actual exponent is e = E - 127 0 < E < 255 E-127 N = (-1) 2 (1.M) 0 = 0 00000000 0 . . . 0 -1.5 = 1 01111111 10 . . . 0 Magnitude of numbers that can be represented is in the range: -126 127 23 ) 2 (1.0) (2 - 2 to 2 which is approximately: -38 38 to 3.40 x 10 1.8 x 10

  4. IEEE 754-1985 • Fraction part: 23 / 52 bits; 0 ≤ x <1 • Significand: 1 + fraction part “1” is not stored; “hidden bit” corresponds to 7 resp. 16 decimal digits • Exponent: 127 / 1023 added to the exponent; “biased exponent” corresponds to 10 - 10 resp. 10 - 10 -39 39 -308 308

  5. IEEE 754-1985 • Special features: Correct rounding of “halfway” result (to even number) Includes special values: NaN Not a number ∞ Infinity -∞ - Infinity Uses denormal number to represent numbers less than 2 Rounds to nearest by default; Three other rounding modes exists. Sophisticated exception handling Emin

  6. Multiplication (s1 * 2 ) * (s2 * 2 ) = s1*s2 *2 so, multiply significands and add exponents Problem: Significand coded in signed- magnitude - use unsigned multiplication and take care of sign Round 2n bits significand to n bits significand Compute new exponent with respect to bias e1 e2 e1+e2

  7. P A x0 x1 x2 x3 x4 x5 g r s s s s P A x0 x1 x2 x3 x4 x5 r s s s s s Rounding 1. Multiply the two significands to get the 2n-bits product: Case 1: x0 = 0, shift needed: Case 2: x0 = 1, increment exponent, set g=r; r=s or r These four bits OR:ed together (“sticky bit”) guard round bit bit P A x1 x2 x3 x4 x5 g r s s s s

  8. Rounding 2: For both cases: if r = 0, P is the correctly rounded product. if r = 1 and s = 1, then P + 1 is the correctly rounded product if r = 1 and s = 0, (the “halfway case”), then P is the correctly rounded product if x5 (or g) is 0 P+1 is the correctly rounded product if x5 (or g) is 1

  9. Add / Sub e1 e2 e3 (s1 * e ) + (s2 * e ) = (s3 * e ) 1: Shift summands so they have the same exponent. (eg. if e2 < e1: shift s2 right and increment e2 until e1 = e2) 2: Add significands 3: Normalize number (shift s3 left and decrement e3 until MSB = 1) 4: Round s3 correctly (under the common assumption that more than 23 / 52 bits is internally used for addition) Subtraction use the same method s3

  10. Division (s1 * 2 ) / (s2 * 2 ) = (s1 / s2) * 2 e1 e1 e1-e2 • so, divide significands and subtract exponents • Problem: • Significand coded in signed- magnitude - use unsigned division (different algoritms exists) and take care of sign • Round n + 2 (guard and round) bits significand to n bits significand • Compute new exponent with respect to bias

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