Suppose that we use the floating-point format with 12 decimal digits, SEEEMMMMMMMM, to rep- resent a real number, where S is the digit to represent the sign of the mantissa (use 0 for pos- itive and 5 for negative), EEE are the 3 digits to represent the exponent in excess-500 format, MMMMMMMM are the 8 digits to represent the magnitude of the mantissa, and the decimal point of the mantissa is right to the left of MMMMMMMM (i.e., SEEEMMMMMMMM representing the real number ±0.MMMMMMMM × 10EEE-500.) When you use this format to represent a real number, in which scenario an overflow will occur and in which scenarios an underflow may occur?

Suppose that we use the floating-point format with 12 decimal digits, SEEEMMMMMMMM, to rep- resent a real number, where S is the digit to represent the sign of the mantissa (use 0 for pos- itive and 5 for negative), EEE are the 3 digits to represent the exponent in excess-500 format, MMMMMMMM are the 8 digits to represent the magnitude of the mantissa, and the decimal point of the mantissa is right to the left of MMMMMMMM (i.e., SEEEMMMMMMMM representing the real number ±0.MMMMMMMM × 10EEE-500.) When you use this format to represent a real number, in which scenario an overflow will occur and in which scenarios an underflow may occur?

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

See similar textbooks

Similar questions

uppose that we use the floating-point format with 12 decimal digits , SEEEMMMMMMMM , to rep- resent a real number, where S is the digit to represent the sign of the mantissa (use 0 for pos- itive and 5 for negative), EEE are the 3 digits to represent the exponent in excess-500 format, MMMMMMMM are the 8 digits to represent the magnitude of the mantissa, and the decimal point of the mantissa is right to the left of MMMMMMMM (i.e., SEEEMMMMMMMM representing the real number ± 0 . MMMMMMMM × 10^EEE − 500 .) Largest Positive Number = 0.99999999x 10^499 Second Largest Positive Number = 0.99999998 x 10^499 How much less is the second largest number compared to the largest number? Can you represent a number located somewhere between the largest number and the second largest number?
Suppose that we use the floating-point format with 12 decimal digits, SEEEMMMMMMMM, to rep- resent a real number, where S is the digit to represent the sign of the mantissa (use 0 for pos- itive and 5 for negative), EEE are the 3 digits to represent the exponent in excess-500 format, MMMMMMMM are the 8 digits to represent the magnitude of the mantissa, and the decimal point of the mantissa is right to the left of MMMMMMMM (i.e., SEEEMMMMMMMM representing the real number +0.MMMMMMMM × 10EEE-500) What are the smallest positive number and the largest positive number that can be represented in this format? Write them in exponential notation.
Any floating-point representation used in a computer can represent only certain real numbers exactly; all others must be approximated. If A' is the stored value approxi- mating the real value A, then the relative error, r, is expressed as A - A' A r = Represent the decimal quantity +0.4 in the following floating-point format: base exponent: biased, 4 bits; significand, 7 bits. What is the relative error? = 2;
4. The bfloat16 "brain floating point" format is a 16 bit format used in Google's machine learning and AI software. It is a binary floating point format which is very similar to the single precision IEEE-754 format: 1 bit is allocated for the sign, 8 bits for the exponent with a bias of 127, but only 7 bits are allocated for the fraction (the exponent is always chosen so that the first digit of the mantissa is 1, and then only the fraction is stored in memory). (a) What is the approximate decimal precision of a brain floating point? (b) If the bits are stored in the order: sign, exponent, fraction, and 0 corresponds to a positive sign, then calculate the decimal representation of the number stored as 1 00000110 0100010 (c) Given that the largest exponent actually used for numbers is 11111110, what is the largest number that can be expressed as a bfloat 16? (d) Supposing that numbers are truncated (rounded down), what is the maximum absolute rounding error in bfloat16?
In a given 32-bit floating-point representation of numbers, if the number of bits of the mantissa is reduced to accommodate an increase in the number of bits of the exponent, this will result in the following: (select the best answer) Increased range of the numbers at the expense of the precision Increased precision of the numbers at the expense of the range Both range and precision increase Both range and precision decrease
The following is a scheme for floating point number representations using 16 bits: s (1) e (6) m (9) Let s, e, and m be the numbers represented in binary in the sign, exponent, and mantissa fields respectively. Then the floating point number represented is (- 1)° (1 + m × 2 °) 2°-31, if the exponent # 111111. 0, otherwise What is the maximum difference between two successive real numbers representable in this system?
You are asked to design a 16-bit floating point number system to store the lengths of various man-made objects. This system should work in a similar way as the IEEE754 standard. Assume a value stored in the system denotes the length of an object in centimeters, assume also that the maximum length to be stored is 45845.0 centimeters (i.e. length of the biggest man-made oil-tanker, the “Seawise Giant”). Note: This representation has normalized, de-normalized and special cases as you have seen in IEEE754 standard. Answer the questions below: a) Is sign bit needed in this system? Why yes or why not. b) What is the minimum number of bits needed for the exponent? What is the value of the corresponding bias? Show your steps clearly. If you write the values directly without showing the steps, you will not get any point. c) What is the maximum length the system can represent? Please show your steps clearly, otherwise no point will be given.
To be completed after lecture on Monday or after watching the first Floating Point video: Take the values 12345 x 10-12 and -12345 x 1037 And encode them in a system that uses the format SEEMMMMM with the following biases: Bias of 47, 25, and 32. (you will have six values when you are done)
Suppose the characters 0,1,...,8,9, A,B,C,D,E,F are used to represent a hex adecimal (base-16) number. Here A = 10 B =11,..., F = 15 What is the largest base-10 integer that can be represented with a two-digit hexadecimal number, such as 14 or 3A? (A) 32 (B) 225 (C) 255 (D) 256 (E) 272
A barcode scanner for Universal Product Codes (UPCS) verifies the 12-digit code scanned by comparing the code's largest digit (called a check digit) to its own computation of the check digit from the first 11 digits as follows: 1. Calculate the sum of the digits in the odd-numbered positions (the first, third,... eleventh digits) and multiply this sum by 3. 2. Calculate the sum of the digits in the even-numbered position (the second, fourth...tenth digits) and add this to the previous result. 3. If the last digit of the result from step 2 is 0, then 0 is the check digit. Otherwise, subtract the last digit from 10 to calculate the check digit. 4. If the check digit matches the final digit of the 12-digit UPC, the UPC is assumed correct By using C++ write a program that prompts the user to enter the 12 digits of a barcode separated by space. The program should store the digits in an integer array, calculate the check digit, and compare it to the final barcode digit. If the digits match,…
Convert the decimal (base 10) number 63 to a binary (base 2) number that has only the digits '0' and '1'. Binary number: (For example, the decimal number 16 is the same as the binary number 10000; the decimal number 13 is the same as the binary number 1101.)
Suppose the characters 0, 1,...,8,9, A, B, C, D, E, F are used to represent a hex- adecimal (base-16) number. Here A = 10, B = 11,..., F = 15. What is the largest base-10 integer that can be represented with a two-digit hexadecimal number, such as 14 or 3A? (A) 32 (B) 225 (C) 255 (D) 256 (E) 272