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:

Maclaurin series are a type of Mathematic series expansion in which all terms are nonnegative real powers of the variable. The Maclaurin series expansion for sin(x) is given by the following formula that is valid for all real values of x such that x is in radians (Note that: radians(x) = x X 1/180): sin(x) = x - 3! 5! Implement a Java program to compute the value of Maclaurin series expansion for sin(x) where x is a nonnegative real value according to the following: a. Write a java method named Factorial that takes as an argument an integer value n and returns the mathematical factorial n! of n as a long value such that: n! = n x (n – 1) x (n – 2) ... × 2 x 1 b. In the main method: i. Ask the user to enter the value of the angle x to be calculate in the Maclaurin series expansion for sin(x) as given above. x Should be entered in degrees, i.e., 0° < x< 360° and then converted into radians using the formula: (radians(x) = x × 1/180). ii. Ask the user to enter the number of terms to be…

Write a code in sim8085 for the following problem: The pressure of two boilers is monitored and controlled by a microcomputer works based on microprocessor programming. A set of 6 readings of first boiler, recorded by six pressure sensors, which are stored in the memory location starting from 2050H. A corresponding set of 6 reading from the second boiler is stored at the memory location starting from 2060H. Each reading from the first set is expected to be higher than the corresponding position in the second set of readings. Write an 8085 sequence to check whether the first set of reading is higher than the second one or not. If all the readings of first set is higher than the second set, store 00 in the ‘D’ register. If any one of the readings is lower than the corresponding reading of second set, stop the process and store FF in the register ‘D’. Data (H): First set: 78, 89, 6A, 80, 90, 85 Second Set:71, 78, 65, 89, 56, 75

All numbers given in parantheses are IEEE-754 floating point numbers in hexadecimal representation. Please see the example below. Example Values for IEEE-754 Floating Point Numbers Base-10 Number 2 Hexadecimal Binary 0100000000000000 (4000) To find to root (zero) of the function: x(4200) – (4400)x = (4880) - starting with the initial value given below Xo = (3c00) write down the results from Newton-Raphson method for 15 iterations. Use IEEE-754 half precision floating point numbers in hexadecimal representation.

Assume you have a base 10 computer that stores floating-point numbers using a 5 digit normalized mantissa (x.xxxx), a 4 digit exponent, and a sign for each. (a) For this machine, what is machine epsilon? (b) What is the smallest positive normalized number that can be represented exactly in this machine?

VI. Floating point representation Consider a 12-bit variant of the IEEE floating point format as follows: • Sign bit 5-bit exponent with a bias of 15. • 6-bit significand All of the rules for IEEE 754 Standard apply. Fill in the numeric value represented by the following bit patterns. You must write your number in decimal form (e.g. 0.0146485375, -0.0146485375). Bit Pattern 010011101110 111011101011 100101001111 001010111010 Numerical Value

In a certain computer program, two positive integers are added together, resulting in an overflow error. Which of the following best explains why the error occurs? A) The program attempted to perform an operation that is considered an undecidable problem. B) The precision of the result is limited due to the constraints of using a floating-point representation. C) The program can only use a fixed number of bits to represent integers; the computed sum is greater than the maximum representable value. D) The program cannot represent integers; the integers are converted into decimal approximations, leading to rounding errors.

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.

2. Constrained by the size of the register in your old personal computer at home, youdecide to build a computer with a larger floating point register called R. Five octitscan be represented in R. You decide on the following format for the representation offloating point numbers:R: sign characteristic mantissaYou have available a 1 octit characteristic and a 3 octit mantissa.(b) Write the following decimal numbers as they would be represented in R: (i) 9.25 (ii) -3.5

Suppose there are less than 1,000,000,000,000,000,000,000 (or 10) stars in the universe. If you have to store that number (Any integer or round number less than 10) exactly in a floating-point number (Stored in binary), minimum how many bits must be in mantissa and exponent?