https://r2.erweima.ai/mathbot/image/172581907199191zv9lic.pdf

The document provides two floating-point number problems to be solved without using the IEEE-754 standard. Here's a summary of the tasks from the file:

1. **Problem 1:**
   - Represent the floating-point number **-7.17** in 16-bit memory without applying the IEEE-754 standard.
   - Calculate the **relative error** when representing this number in a 16-bit memory.
   - Provide all steps of the process.

2. **Problem 2:**
   - Determine the floating-point number stored in memory based on the given 16-bit binary representation: **0111001001010110**.
   - Note that the representation does not use the IEEE-754 standard either.

### Approach:

For **Problem 1**, the main task is to create a custom 16-bit floating-point format for the number -7.17, which would likely involve a specific breakdown of bits for sign, exponent, and mantissa. We would then analyze the relative error by comparing the represented value and the original number.

For **Problem 2**, we need to interpret the binary representation to find the corresponding floating-point number, based on an assumed structure for the 16 bits.

Would you like me to start by explaining how to represent -7.17 in 16 bits and calculate the relative error? Let me know if you'd like more details on a specific part of these problems.

Here are some related questions for further understanding:

1. What are the steps to calculate the relative error for a floating-point representation?
2. How does the bit allocation affect precision in a floating-point format?
3. Why is the IEEE-754 standard typically used for floating-point numbers?
4. What methods can be used to minimize relative error in floating-point representations?
5. How do exponents and mantissas interact in custom floating-point formats?

**Tip:** When representing floating-point numbers in limited memory, rounding and truncation errors often occur, leading to precision loss. Understanding bit allocation for sign, exponent, and mantissa is key to minimizing this loss.

Represent the floating-point number -7.17 in 16-bit memory without using the IEEE-754 standard and identify the relative error when representing this number. Also, determine the floating-point number stored in memory with the 16-bit binary representation: 0111001001010110.

Solve floating-point number representation problems without IEEE-754 standard, such as representing -7.17 in 16-bit memory and analyzing the relative error. Also, decode a custom 16-bit binary representation: 0111001001010110.

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