1. The natural exponent e = 2.71828182459 ... Answer the following questions: a. Assuming we follow the standard rules for representing floating point numbers, What is the exponent required to represent e as a floating point binary number? Report your answer in both the base-2 and base-10 representations
1. The natural exponent e = 2.71828182459 ... Answer the following questions: a. Assuming we follow the standard rules for representing floating point numbers, What is the exponent required to represent e as a floating point binary number? Report your answer in both the base-2 and base-10 representations
Database System Concepts
7th Edition
ISBN:9780078022159
Author:Abraham Silberschatz Professor, Henry F. Korth, S. Sudarshan
Publisher:Abraham Silberschatz Professor, Henry F. Korth, S. Sudarshan
Chapter1: Introduction
Section: Chapter Questions
Problem 1PE
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Question
1.1
![1. The natural exponent \( e = 2.71828182459 \ldots \)
Answer the following questions:
a. Assuming we follow the standard rules for representing floating point numbers, what is the exponent required to represent \( e \) as a floating point binary number? Report your answer in both the base-2 and base-10 representations.
b. How many bits of data are necessary to represent the decimal portion (i.e., mantissa) of \( \sqrt{e} \) to the fifth decimal place (1.64872)?
c. Suppose you have a calculator that stores floating point numbers using a six-bit mantissa. What is the machine precision, “eps” for this calculator?
d. If the calculator employs rounding, what is the largest possible relative round-off error introduced by a single calculation with the calculator from part ‘c’?
e. Suppose instead that this calculator uses the following number format:
\[ FL(x) = S|E|E|E|M|M|M \]
- **Bit 1:** Sign of the number.
- **Bits 2-4:** Biased exponent – subtract 4 to achieve the final exponent value.
- **Bits 5-8:** Mantissa.
Assuming that the calculator uses exact rounding, what is the expected relative error produced when performing the calculation \( \sqrt{e} \)?](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F79826512-1148-4736-b4a7-6e7f4ec13f28%2F100db944-0cba-468d-9b0a-1cbe064ca126%2Fq1l0lr_processed.png&w=3840&q=75)
Transcribed Image Text:1. The natural exponent \( e = 2.71828182459 \ldots \)
Answer the following questions:
a. Assuming we follow the standard rules for representing floating point numbers, what is the exponent required to represent \( e \) as a floating point binary number? Report your answer in both the base-2 and base-10 representations.
b. How many bits of data are necessary to represent the decimal portion (i.e., mantissa) of \( \sqrt{e} \) to the fifth decimal place (1.64872)?
c. Suppose you have a calculator that stores floating point numbers using a six-bit mantissa. What is the machine precision, “eps” for this calculator?
d. If the calculator employs rounding, what is the largest possible relative round-off error introduced by a single calculation with the calculator from part ‘c’?
e. Suppose instead that this calculator uses the following number format:
\[ FL(x) = S|E|E|E|M|M|M \]
- **Bit 1:** Sign of the number.
- **Bits 2-4:** Biased exponent – subtract 4 to achieve the final exponent value.
- **Bits 5-8:** Mantissa.
Assuming that the calculator uses exact rounding, what is the expected relative error produced when performing the calculation \( \sqrt{e} \)?
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