IEEE 754 Single Precision Format Quiz

IEEE 754 Single Precision Format

• The IEEE 754-2008 standard defines the 32-bit base 2 format, also known as Binary32.
• Older computer formats for 4-byte floating-point numbers existed before this standard.
• Fortran was one of the first programming languages to include single and double-precision floating-point data types.
• Single precision is denoted as 'float' in programming languages such as C, C++, C#, and Java.
• IEEE 754 single-precision format uses an 8-bit exponent and a 23-bit significand.

Terms to Know

• Range: The range of values representable by the format.
  • Approximately 1.2 × 10^-38 to 3.4 × 10³⁸.
  • Despite this wide range, infinitely many numbers fall outside it.
• Accuracy: How close a represented number is to its true value.
  • Example: 0.1 cannot be precisely represented; however, an approximation within a range close to 0.1 exists.
• Precision: The amount of detail or information used to represent a value.
  • Higher precision often allows a number to be more accurate. Example: 1.666 has 4 decimal digits of precision, and 1.6660 has 5, but this does not make the latter more accurate.

IEEE 754 Storage

• Floating-point numbers are stored on byte boundaries.
• The IEEE-754 single-precision standard can be represented with a sign bit (1 bit), an exponent (8 bits), and a mantissa (23 bits).
• This format allows for numbers with 6 to 9 significant decimal digits. Converting to and from this format should maintain this accuracy.

Converting Decimal to IEEE 754

• Step 1: Convert the decimal number to binary.
• Step 2: Normalize the binary fractional number. Shift the decimal point until there is a "1" to the left of the decimal.
• Step 3: Convert the exponent to 8-bit excess 127 notation.
• Step 4: Convert the significant/mantissa to a "hidden bit format." Remove the initial "1" to the left of the decimal point.
• Step 5: Determine the sign bit. Positive numbers have 0, negative ones 1.
• Step 6: Assemble the three components into the final format.
• Step 7: Often useful to convert to hexadecimal format for better readability.

Example #1

• Convert 8.75₁₀ to IEEE 754 Single-Precision format.
  • Binary: 1000.11₂
  • Normalized: 1.00011 x 2³
  • Exponent (excess 127): 130 (binary: 10000010)
  • Mantissa: 00011
  • Final IEEE 754 format: 0 10000010 00011000000000000000000
  • Hexadecimal equivalent: 410C0000₁₆

Example #2

• Convert -0.5₁₀ to IEEE 754 Single-Precision format.
  • Binary: 0.1₂
  • Normalized: 1.0 x 2^-1
  • Exponent (excess 127): 126 (binary: 01111110)
  • Mantissa: 0
  • Final IEEE 754 format: 1 01111110 00000000000000000000000
  • Hexadecimal equivalent: BF000000₁₆

Example #3

• Convert -1.0625₁₀ to IEEE 754 Single-Precision format.
  • Binary: 1.0001₂ x 2^-0
  • Normalized: 1.0001 x 2⁰
  • Exponent (excess 127): 127 (binary: 01111111)
  • Mantissa: 0001
  • Final IEEE 754 format: 1 01111111 00010000000000000000000
  • Hexadecimal equivalent: BF880000₁₆

Test your knowledge on the IEEE 754-2008 standard for single precision format. This quiz covers the basics of Binary32, its historical context, and key terms like range, accuracy, and precision. Prepare to demonstrate your understanding of floating-point representation in programming languages such as C and Java.