Editing Floating-point arithmetic (section)

{{Redirect|Floating point}}
{{Short description|Computer approximation for real numbers}}
{{Use dmy dates|date=May 2019|cs1-dates=y}}
[[File:Z3 Deutsches Museum.JPG|thumb|200px|An early electromechanical programmable computer, the [[Z3 (computer)|Z3]], included floating-point arithmetic (replica on display at [[Deutsches Museum]] in [[Munich]]).]]
{{Floating-point}}

In [[computing]], '''floating-point arithmetic''' ('''FP''') is [[arithmetic]] on subsets of [[real number]]s formed by a ''[[significand]]'' (a [[Sign (mathematics)|signed]] sequence of a fixed number of digits in some [[Radix|base]]) multiplied by an [[integer power]] of that base.
Numbers of this form are called '''floating-point numbers'''.<ref name="Muller_2010"/>{{rp|3}}<ref name="sterbenz1974fpcomp">{{cite book
|last1=Sterbenz
|first1=Pat H.
|title=Floating-Point Computation
|date=1974
|publisher=Prentice-Hall
|location=Englewood Cliffs, NJ, United States
|isbn=0-13-322495-3
|url=https://archive.org/details/SterbenzFloatingPointComputation/mode/2up}}</ref>{{rp|10}}

For example, the number 2469/200 is a floating-point number in base ten with five digits:
<math display=block>2469/200 = 12.345 = \! \underbrace{12345}_\text{significand} \! \times \! \underbrace{10}_\text{base}\!\!\!\!\!\!\!\overbrace{{}^{-3}}^{\text{exponent}}</math>
However, 7716/625 = 12.3456 is not a floating-point number in base ten with five digits&mdash;it needs six digits.
The nearest floating-point number with only five digits is 12.346.
And 1/3 = 0.3333… is not a floating-point number in base ten with any finite number of digits.
In practice, most floating-point systems use [[Binary number|base two]], though base ten ([[decimal floating point]]) is also common.

Floating-point arithmetic operations, such as addition and division, approximate the corresponding real number arithmetic operations by [[rounding]] any result that is not a floating-point number itself to a nearby floating-point number.<ref name="Muller_2010"/>{{rp|22}}<ref name="sterbenz1974fpcomp"/>{{rp|10}}
For example, in a floating-point arithmetic with five base-ten digits, the sum 12.345 + 1.0001 = 13.3451 might be rounded to 13.345.

The term ''floating point'' refers to the fact that the number's [[radix point]] can "float" anywhere to the left, right, or between the [[significant digits]] of the number. This position is indicated by the exponent, so floating point can be considered a form of [[scientific notation]].

A floating-point system can be used to represent, with a fixed number of digits, numbers of very different [[Orders of magnitude (numbers)|orders of magnitude]] — such as the number of meters [[Orders of magnitude (length)#100_zettametres|between galaxies]] or [[Orders of magnitude (length)#10_femtometres|between protons in an atom]].  For this reason, floating-point arithmetic is often used to allow very small and very large real numbers that require fast processing times. The result of this [[dynamic range]] is that the numbers that can be represented are not uniformly spaced; the difference between two consecutive representable numbers varies with their exponent.<ref name="Smith_1997"/>

[[File:A number line representing single-precision floating point's numbers and numbers that it cannot display.png|500x500px|thumb|right|Single-precision floating-point numbers on a [[number line]]: the green lines mark representable values.]]
[[File:FloatingPointPrecisionAugmented.png|500x500px|thumb|right|Augmented version above showing both [[Signed number representations|signs]] of representable values]]

Over the years, a variety of floating-point representations have been used in computers. In 1985, the [[IEEE 754]] Standard for Floating-Point Arithmetic was established, and since the 1990s, the most commonly encountered representations are those defined by the IEEE.

The speed of floating-point operations, commonly measured in terms of [[FLOPS]], is an important characteristic of a [[computer system]], especially for applications that involve intensive mathematical calculations.

A [[floating-point unit]] (FPU, colloquially a math [[coprocessor]]) is a part of a computer system specially designed to carry out operations on floating-point numbers.