"how numbers are stored and used in computers"
Floating point formats are a type of binary format that represent real numbers in a computer system. They are used to store and manipulate decimal numbers with a fixed number of digits after the decimal point.
The IEEE 754 standard is a widely adopted standard for representing floating point numbers in binary systems. It defines two main formats for single precision (32-bit) and double precision (64-bit).
A 64-bit format offering very high precision and a wide dynamic range, commonly used in scientific computing.
FP64
A 32-bit format balancing precision and performance, widely used in general-purpose computing and graphics.
FP32
Optimized for memory efficiency and speed with lower precision and dynamic range, used in AI and graphics.
FP16
Same exponent size as FP32 but reduced mantissa, enabling faster computation while retaining dynamic range.
BF16
A low precision format for maximum efficiency, mainly used in deep learning models for training and inference.
FP8
An extremely compact format used in specialized machine learning and graphics applications.
FP4
A floating-point number in IEEE 754 format is composed of three main components: the sign bit, the exponent, and the significand (or mantissa). The sign bit determines the positivity or negativity of the number. The exponent, which is stored in a biased form, allows the representation of both very large and very small numbers. The significand, which includes an implicit leading bit, provides the precision of the number.
IEEE 754 defines several levels of precision, with single precision (32-bit) and double precision (64-bit) being the most common. The standard also specifies different rounding modes to handle the precision limitations inherent in floating-point arithmetic. These rounding modes include round to nearest, round toward zero, round toward positive infinity, and round toward negative infinity. The choice of rounding mode can significantly affect the outcome of numerical computations.
The standard introduces special values to handle exceptional cases in arithmetic operations. These include positive and negative infinity, which result from operations like division by zero, and NaN (Not a Number), which represents undefined or unrepresentable values, such as the result of 0/0.
IEEE 754 specifies how arithmetic operations should be performed to ensure consistent results. This includes rules for addition, subtraction, multiplication, division, and square root operations. The standard also addresses issues like overflow, underflow, and the handling of denormalized numbers, which are used to represent values closer to zero than the smallest normalized number.
The operations defined by the IEEE 754 standard are designed to be efficient. The time complexity for basic arithmetic operations (addition, subtraction, multiplication, division) is generally
By adhering to the IEEE 754 standard, developers and engineers can ensure that their applications perform numerical computations reliably and consistently, regardless of the underlying hardware or software environment.