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Lecture

Floating Point Representation in Computers

In course

MATH-251(a): Numerical analysis

This course presents numerical methods to solve mathematical problems such as systems of linear and nonlinear equations, function approximation, integration and differentiation, and differential equations. ...

Description

This lecture covers the representation of real numbers in a computer using floating point representation, including the mantissa, exponent, and sign. It explains the use of simple and double precision, rounding errors, aligning exponents, and the dangers of significant digit cancellation.

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Instructors (4)

Pablo Antolin Sanchez

Fabio Nobile

Espen Sande

Daniel Kressner

Official source

Ontological neighbourhood

Mathematics

Analysis: Numerical analysis

Related concepts (32)

Round-off error

In computing, a roundoff error, also called rounding error, is the difference between the result produced by a given algorithm using exact arithmetic and the result produced by the same algorithm using finite-precision, rounded arithmetic. Rounding errors are due to inexactness in the representation of real numbers and the arithmetic operations done with them. This is a form of quantization error.

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.

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.

Extended precision

Extended precision refers to floating-point number formats that provide greater precision than the basic floating-point formats. Extended precision formats support a basic format by minimizing roundoff and overflow errors in intermediate values of expressions on the base format. In contrast to extended precision, arbitrary-precision arithmetic refers to implementations of much larger numeric types (with a storage count that usually is not a power of two) using special software (or, rarely, hardware).

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.

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