Convert The Floating Point Number To IEEE Single Precision

Convert the floating point number to IEEE Single Precision Floating-Point Representation

The task involves converting the floating-point number -17 into its IEEE 754 single-precision floating-point format. Additionally, the assignment requires performing calculations on 8-bit data using two’s complement representation, specifically for given decimal numbers.

This assignment combines understanding of floating-point binary representations and two’s complement operations in binary, essential skills in computer architecture and low-level programming. To effectively complete this workload, an understanding of IEEE 754 standards, binary arithmetic, and two’s complement operations is necessary.

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IEEE 754 single-precision floating-point format is a widely used standard for representing real numbers in binary within computer systems. It uses a 32-bit structure comprised of three parts: a sign bit, an exponent, and a fraction (mantissa). To convert the decimal number -17 into this format, the process involves several steps: converting to binary, normalizing the binary number, determining the exponent, and then composing the final 32-bit pattern.

First, convert 17 to binary: 17 in binary is 10001. Since the original number is negative, the sign bit will be 1. Next, normalize the binary number: 17 as binary normalized form is 1.0001 × 2^4. The exponent in IEEE 754 is stored with a bias of 127, so the exponent field becomes 4 + 127 = 131, which in binary is 10000011. The mantissa consists of the fractional part of the normalized binary number after the leading 1, which are the bits 0001 followed by zeros to fill 23 bits.

Putting these together, the sign bit is 1, the exponent is 10000011, and the mantissa is 00010000000000000000000. Therefore, the 32-bit IEEE 754 representation of -17 is:

1 10000011 00010000000000000000000

or, in hexadecimal, this corresponds to 0xC1100000.

For the second part, the calculation involves performing arithmetic operations in 8-bit two’s complement binary representation. The operations include addition of positive and negative integers represented in two’s complement form.

Given the data: +25, -65, -25, 50, the operations are as follows:

  • + (25)10 (decimal) in binary is 00011001. In 8-bit two’s complement, positive numbers are represented directly, so 25 is 00011001.
  • + (-65)10: To represent -65 in two’s complement, first write 65 in binary (01000001), then invert bits (10111110) and add 1, giving 10111111.
  • + (-25)10: 25 in binary is 00011001; invert bits (11100110), add 1 (11100111), so -25 is 11100111.
  • + (50)10: 50 in binary is 00110010, which remains as is in two’s complement for positive numbers.

Next, perform the addition operations in binary with two’s complement, taking care of overflow and sign bits:

  • 25 + (-65): 00011001 + 10111111 = 11111000 (which is -8 in two’s complement).
  • -65 + (-25): 10111111 + 11100111 = hojz (resulting in negative sum with correct two’s complement notation).
  • 25 + 50: 00011001 + 00110010 = 01001011 (which is 75 in decimal).

These calculations demonstrate the use of two’s complement for handling signed integers in binary arithmetic, a fundamental concept for low-level data manipulation and digital logic design.

In conclusion, converting decimal numbers to IEEE 754 format and performing binary arithmetic in two’s complement are critical foundational skills in computer engineering and programming. Mastering these operations enhances understanding of how data is represented and manipulated at the hardware level, which is crucial for system programming, compiler development, and designing digital systems.

References

  • IEEE Standard for Floating-Point Arithmetic (IEEE 754-2008). (2008). IEEE.
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  • Roth, C. H., & Kinney, L. F. (2014). Fundamentals of Logic Design (7th ed.). Cengage Learning.
  • Bhasker, J. (2004). A VHDL Primer (3rd ed.). Prentice Hall.
  • Peterson, L. L., & Davie, B. S. (2011). Computer Networks: A Systems Approach (5th ed.). Morgan Kaufmann.
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