Floating Point Numbers: Representation and Errors

This lecture discusses the representation of floating point numbers, focusing on ensuring a uniform relative error within a given domain. It explores examples using scientific notation with a fixed number of significant digits, highlighting the trade-off between precision and range. The instructor explains the concept of relative error and its implications, emphasizing the importance of maintaining consistency in error distribution. Additionally, the lecture covers the representation of floating point numbers in binary, detailing the structure of a floating point number in base 2. It also addresses the maximum relative error in binary representation and compares it to decimal representation. The discussion includes practical examples and visual aids to illustrate the concepts effectively.

About this result

This page is automatically generated and may contain information that is not correct, complete, up-to-date, or relevant to your search query. The same applies to every other page on this website. Please make sure to verify the information with EPFL's official sources.

Related lectures (34)

Related concepts (36)

CS-119(d): Information, Computation, Communication

L'objectif de ce cours est d'introduire les étudiants à la pensée algorithmique, de les familiariser avec les fondamentaux de l'Informatique et de développer une première compétence en programmation (

Floating-point arithmetic

In computing, floating-point arithmetic (FP) is arithmetic that represents subsets of real numbers using an integer with a fixed precision, called the significand, scaled by an integer exponent of a fixed base. Numbers of this form are called floating-point numbers. For example, 12.345 is a floating-point number in base ten with five digits of precision: However, unlike 12.345, 12.3456 is not a floating-point number in base ten with five digits of precision—it needs six digits of precision; the nearest floating-point number with only five digits is 12.

Decimal floating point

Decimal floating-point (DFP) arithmetic refers to both a representation and operations on decimal floating-point numbers. Working directly with decimal (base-10) fractions can avoid the rounding errors that otherwise typically occur when converting between decimal fractions (common in human-entered data, such as measurements or financial information) and binary (base-2) fractions. The advantage of decimal floating-point representation over decimal fixed-point and integer representation is that it supports a much wider range of values.

Half-precision floating-point format

In computing, half precision (sometimes called FP16 or float16) is a binary floating-point computer number format that occupies 16 bits (two bytes in modern computers) in computer memory. It is intended for storage of floating-point values in applications where higher precision is not essential, in particular and neural networks. Almost all modern uses follow the IEEE 754-2008 standard, where the 16-bit base-2 format is referred to as binary16, and the exponent uses 5 bits.

Double-precision floating-point format

Double-precision floating-point format (sometimes called FP64 or float64) is a floating-point number format, usually occupying 64 bits in computer memory; it represents a wide dynamic range of numeric values by using a floating radix point. Floating point is used to represent fractional values, or when a wider range is needed than is provided by fixed point (of the same bit width), even if at the cost of precision. Double precision may be chosen when the range or precision of single precision would be insufficient.

Quadruple-precision floating-point format

In computing, quadruple precision (or quad precision) is a binary floating point–based computer number format that occupies 16 bytes (128 bits) with precision at least twice the 53-bit double precision. This 128-bit quadruple precision is designed not only for applications requiring results in higher than double precision, but also, as a primary function, to allow the computation of double precision results more reliably and accurately by minimising overflow and round-off errors in intermediate calculations and scratch variables.

Floating Point Numbers: Representation and Errors CS-119(c): Information, Computation, Communication

Explores the representation of floating point numbers and the impact of errors in numerical computations.

Computer Arithmetic: Floating Point Operations CS-208: Computer architecture I

Covers the basics of computer arithmetic, focusing on floating point numbers and their operations.

Computer Arithmetic: Floating Point Numbers CS-208: Computer architecture I

Explores computer arithmetic, emphasizing fixed-point and floating-point numbers, IEEE 754 standard, dynamic range, and floating-point operations in MIPS architecture.

Floating Point Numbers: Representation and Errors CS-119(d): Information, Computation, Communication

Explores the representation of floating point numbers and the importance of error analysis.

Information Representation: Binary Representations CS-119(d): Information, Computation, Communication

Explores binary information representation, efficiency, byte usage, and integer representation.