In IEEE-754 floating-point representation, the right-hand side neighbor of 1 is $1 + \epsilon_m$, where $\epsilon_m$ is the machine epsilon, which represents the smallest positive number that can be added to 1 to produce a result greater than 1 in the given floating-point precision.

To find the left-hand side neighbor of 1, we need to understand that the floating-point numbers in the system are spaced logarithmically. The left-hand side neighbor is the largest number smaller than 1 that can be represented in the same format. This number is given by:

$1 - \epsilon_m$

where $\epsilon_m$ is still the machine epsilon.

Explanation:

• Machine epsilon ($\epsilon_m$) is the smallest positive number such that $1 + \epsilon_m$ is distinct from 1 in the floating-point system. This value depends on the precision of the floating-point format (e.g., single precision or double precision).

• The right-hand side neighbor of 1 is $1 + \epsilon_m$, as this is the smallest number greater than 1 that can be represented.

• The left-hand side neighbor of 1 is $1 - \epsilon_m$, as this is the largest number smaller than 1 that can still be represented without underflowing to 0.

Example:

In IEEE-754 single precision (32-bit), the value of $\epsilon_m$ is approximately $2^{-23}$. Therefore:

• Right-hand neighbor of 1: $1 + \epsilon_m \approx 1 + 2^{-23}$
• Left-hand neighbor of 1: $1 - \epsilon_m \approx 1 - 2^{-23}$

Would you like further details on how machine epsilon works in different precisions or any other related concept?

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Related Questions:

1. How is the machine epsilon ($\epsilon_m$) calculated for different IEEE-754 formats?
2. What is the impact of machine epsilon on floating-point arithmetic accuracy?
3. How does floating-point precision affect numerical methods like root finding or integration?
4. What happens when we try to represent numbers smaller than the machine epsilon in IEEE-754?
5. Can we calculate the relative error in floating-point arithmetic based on $\epsilon_m$?

Tip:

For precise floating-point calculations, understanding and accounting for machine epsilon is crucial, especially when comparing values close to 1 or dealing with operations like subtraction of nearly equal numbers.