Introduction to Floating Point Representation | Digital Circuits - Electronics and Communication Engineering (ECE) PDF Download

Floating-point representation is a method used in computing to represent real numbers, including very large and very small values, with a high degree of precision. This guide explains how to convert between floating-point and decimal representations in a 32-bit format, along with the advantages and disadvantages of this system.

Converting Floating-Point to Decimal

In a 32-bit floating-point representation, a number is divided into three components:

• Sign: Indicates whether the number is positive or negative.
• Exponent: Represents the power of 2 that scales the number.
• Mantissa: Represents the significant digits of the number.

Components of a 32-bit Floating-Point Number

1. Sign Bit (1 bit)
   • The first bit of the 32-bit representation.
   • 0 = positive number, 1 = negative number.
   • Example: 11000001110100000000000000000000 → Negative (starts with 1).
2. Exponent (8 bits)
   • The next 8 bits after the sign bit.
   • A bias of 127 is added to the exponent value for 32-bit representation.
   • Bias formula: $2k-1-12^\{k-1\}\; -\; 1$2k−1−1, where $kk$kis the number of exponent bits.
     • For 8-bit representation: $23-1-1=32^\{3-1\}\; -\; 1\; =\; 3$23−1−1=3.
     • For 32-bit representation: $28-1-1=1272^\{8-1\}\; -\; 1\; =\; 127$28−1−1=127.
   • Example: 01000001110100000000000000000000
     • Exponent bits: 10000011 = $13110131\_\{10\}$13110​.
     • Actual exponent: $131-127=4131\; -\; 127\; =\; 4$131−127=4.
     • This represents $24=162^4\; =\; 16$24=16.
3. Mantissa (23 bits)
   • The remaining 23 bits, with an implied leading 1 (normalized form).
   • The fractional part is calculated as a sum of powers of 2:
     • Example: 10100000000000000000000
       • $1\cdot \left(1/2\right)+0\cdot \left(1/4\right)+1\cdot \left(1/8\right)+0\cdot \left(1/16\right)+\cdots =0.5+0+0.125+0+\cdots =0.6251\; \backslash cdot\; (1/2)\; +\; 0\; \backslash cdot\; (1/4)\; +\; 1\; \backslash cdot\; (1/8)\; +\; 0\; \backslash cdot\; (1/16)\; +\; \backslash dots\; =\; 0.5\; +\; 0\; +\; 0.125\; +\; 0\; +\; \backslash dots\; =\; 0.625$1⋅(1/2)+0⋅(1/4)+1⋅(1/8)+0⋅(1/16)+⋯=0.5+0+0.125+0+⋯=0.625.
       • Mantissa = $1+0.625=1.6251\; +\; 0.625\; =\; 1.625$1+0.625=1.625.

Final Calculation

• Formula: $(-1)sign\cdot 2exponent\cdot mantissa\; (-1)^\{\backslash text\{sign\}\}\; \backslash cdot\; 2^\{\backslash text\{exponent\}\}\; \backslash cdot\; \backslash text\{mantissa\}$(−1)sign⋅2exponent⋅mantissa.
• Example: 01000001110100000000000000000000
  • Sign = 0 (positive).
  • Exponent = $24=162^4\; =\; 16$24=16.
  • Mantissa = $1.6251.625$1.625.
  • Result: $(-1)0\cdot 16\cdot 1.625=26(-1)^0\; \backslash cdot\; 16\; \backslash cdot\; 1.625\; =\; 26$(−1)0⋅16⋅1.625=26.

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Converting Decimal to Floating-Point

To convert a decimal number (e.g., $-17-17$−17) into a 32-bit floating-point representation, follow these steps:

Components

1. Sign Bit (1 bit)
   • -17 is negative → Sign bit = 1.
2. Exponent (8 bits)
   • Find the nearest power of 2 less than or equal to the absolute value (17 → 16 = $242^4$24).
   • Exponent = 4.
   • Add bias (127 for 32-bit): $127+4=131127\; +\; 4\; =\; 131$127+4=131.
   • Convert to binary: $13110=100000112131\_\{10\}\; =\; 10000011\_2$13110​=100000112​.
3. Mantissa (23 bits)
   • $1710=10001217\_\{10\}\; =\; 10001\_2$1710​=100012​.
   • Normalize: Move the binary point to get $1.0001\cdot 241.0001\; \backslash cdot\; 2^4$1.0001⋅24.
   • Take the fractional part (0001) and pad with zeros to 23 bits:
     • 00010000000000000000000.

Final Representation

• -17 in 32-bit floating-point:
  • 1 10000011 00010000000000000000000.

Advantages of Floating-Point Representation

1. Wide Range of Values
   • Can represent very large and very small numbers efficiently.
2. High Precision
   • Ideal for scientific and engineering calculations requiring accuracy.
3. Compatibility
   • Widely supported across hardware and software systems.
4. Ease of Use
   • Built-in support in most programming languages simplifies implementation.

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Disadvantages of Floating-Point Representation

1. Complexity
   • Difficult to understand without knowledge of the underlying math.
2. Rounding Errors
   • Approximate representations can lead to small inaccuracies.
3. Speed
   • Slower than integer operations, especially on older hardware.
4. Limited Precision
   • Fixed number of significant digits limits accuracy in some cases.