Calculating Absolute Error
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Questions and Answers

What is the value of N in the IEEE 754 double precision format?

  • 64 (correct)
  • 32
  • 23
  • 52

    • What is the purpose of the hidden bit in the IEEE 754 standard?

  • To store the mantissa
  • To indicate the sign of the number
  • To store the exponent
  • To reduce the storage requirement (correct)

    • What is the range of emin in the IEEE 754 single precision format?

  • -1022 to 1023
  • -126 to 127 (correct)
  • -16382 to 16383
  • -2 to 2

    • What is the value of p in the IEEE 754 quadruple precision format?

    <p>112</p> Signup and view all the answers

    What is the binary representation of the number π in the IEEE 754 double precision format?

    <p>0 10000000000 1.1001001000011111101101010100010001000010110100011000</p> Signup and view all the answers

    What is the relationship between the binary representation of a number and its positional notation?

    <p>They are two different representations of the same number</p> Signup and view all the answers

    What is the purpose of the exponent in the IEEE 754 standard?

    <p>To shift the mantissa to its correct position</p> Signup and view all the answers

    What is the range of emax in the IEEE 754 double precision format?

    <p>1022 to 1023</p> Signup and view all the answers

    How many bits are used to store the exponent in the IEEE 754 single precision format?

    <p>11 bits</p> Signup and view all the answers

    What is the purpose of the normalization condition in the IEEE 754 standard?

    <p>To ensure the correct representation of the number</p> Signup and view all the answers

    Study Notes

    Error Measures

    • The absolute error between x and e is denoted by ∥x - e∥ := √(xk - ek)^2, where k=0 to n-1
    • The relative error of x to x ≠ 0 is defined by ∥x - e∥ / ∥x∥
    • The number of significant digits in x̃ is the number of leading digits that are correct relative to the true value x
    • x̃ has m significant digits with respect to x if |x - x̃| ≤ 5 · 10^(-m-1)

    Machine Representation of Numbers

    • A base β ≥ 2 is a positive integer, and Zβ = {0, 1,..., β - 1} are the digits
    • The β-positional representation of x ∈ R is xβ = (−1)^s (xn xn-1 ... x1 x0.x-1 x-2 ... x-m)β
    • Examples of number systems include binary (β = 2, Z2 = {0, 1}), decimal (β = 10, Z10 = {0, 1,..., 9}), and hexadecimal (β = 16, Z16 = {0, 1,..., 9, A, B, C, D, E, F})

    Fixed-Point Representation

    • The set of real numbers with at most D total number of digits in β-positional representation is denoted by Rβ,D
    • Zβ,D is the set of integers in Rβ,D with at most D total number of digits
    • A fixed-point number system is defined by FI(β, N, k) := {(−1)^s (xN-2 ... xk.xk-1 ... x0)β : s ∈ {0, 1}, {xk}N-2 k=0 ⊂ Zβ}
    • Theorem 2 states that |FI(β, N, k)| = 2^β^(N-1) - 1

    Floating-Point Representation

    • The IEEE 754 single precision format is FL(β = 2, N = 32, p = 23, emin = -126, emax = 127)
    • The IEEE 754 double precision format is FL(β = 2, N = 64, p = 52, emin = -1022, emax = 1023)
    • The IEEE 754 quadruple precision format is FL(β = 2, N = 128, p = 112, emin = -16382, emax = 16383)
    • IEEE 754 representations of transcendental numbers π and e are shown in 64-bit format

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    Description

    This quiz is about calculating the absolute error between a value x and a exact value e. It involves understanding the mathematical formula and concepts.

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