The IEEE 754 standard defines the format for representing floating-point numbers in computer systems. Here’s an overview of the IEEE 754 standard for floating-point numbers:
Format:
- Sign bit: 1 bit, indicating the sign of the number (positive or negative).
- Exponent: A fixed number of bits representing the exponent of the number, typically biased to allow for both positive and negative exponents.
- Fraction (Significand or Mantissa): A fixed number of bits representing the significant digits of the number, including the fractional part.
Single Precision (32 bits):
- Sign bit: 1 bit
- Exponent: 8 bits
- Fraction: 23 bits
- Total: 32 bits
Double Precision (64 bits):
- Sign bit: 1 bit
- Exponent: 11 bits
- Fraction: 52 bits
- Total: 64 bits
Special Values:
- Zero: Exponent and fraction bits are all zero.
- Denormalized numbers: Exponent is all zeros, allowing for subnormal numbers with a reduced precision.
- Infinity: Exponent is all ones, and the fraction is all zeros.
- NaN (Not a Number): Exponent is all ones, and the fraction is non-zero. NaNs are used to represent undefined or invalid operations, such as the result of dividing zero by zero.
Normalization:
- The leading bit of the significand is typically assumed to be 1 (implicit leading bit), except for denormalized numbers, where the leading bit is explicitly set to zero.
Rounding:
- IEEE 754 specifies different rounding modes for arithmetic operations, including round to nearest, round towards zero, round towards positive infinity, and round towards negative infinity.
Accuracy:
- IEEE 754 ensures that floating-point arithmetic operations are correctly rounded to the nearest representable value within the specified precision.
Operations:
- The standard defines arithmetic operations (addition, subtraction, multiplication, division), comparison operations, and special functions (sqrt, log, exp, etc.) for floating-point numbers.
The IEEE 754 standard provides a widely accepted and standardized representation for floating-point numbers, ensuring interoperability and consistency across different computer architectures and programming languages. It is commonly implemented in hardware and software systems supporting floating-point arithmetic.