Hidden Bit

In computing, floating-point arithmetic (FP) is arithmetic that represents real numbers approximately, using an integer with a fixed precision, called the significand, scaled by an integer exponent of a fixed base. For example, 12.345 can be represented as a base-ten floating-point number:

$12.345 = \underbrace_\text \times \underbrace_\text\!\!\!\!\!\!^$

In practice, most floating-point systems use base two, though base ten ( decimal floating point) is also common.

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 — such as the number of meters between galaxies or 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.

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.

Overview

Floating-point numbers

A number representation specifies some way of encoding a number, usually as a string of digits.

There are several mechanisms by which strings of digits can represent numbers. In common mathematical notation, the digit string can be of any length, and the location of the radix point is indicated by placing an explicit "point" character (dot or comma) there. If the radix point is not specified, then the string implicitly represents an integer and the unstated radix point would be off the right-hand end of the string, next to the least significant digit. In fixed-point systems, a position in the string is specified for the radix point. So a fixed-point scheme might be to use a string of 8 decimal digits with the decimal point in the middle, whereby "00012345" would represent 0001.2345.

In scientific notation, the given number is scaled by a power of 10, so that it lies within a certain range—typically between 1 and 10, with the radix point appearing immediately after the first digit. The scaling factor, as a power of ten, is then indicated separately at the end of the number. For example, the orbital period of Jupiter's moon Io is seconds, a value that would be represented in standard-form scientific notation as seconds.

Floating-point representation is similar in concept to scientific notation. Logically, a floating-point number consists of:
* A signed (meaning positive or negative) digit string of a given length in a given base (or radix). This digit string is referred to as the ''significand'', ''mantissa'', or ''coefficient''. The length of the significand determines the ''precision'' to which numbers can be represented. The radix point position is assumed always to be somewhere within the significand—often just after or just before the most significant digit, or to the right of the rightmost (least significant) digit. This article generally follows the convention that the radix point is set just after the most significant (leftmost) digit.
* A signed integer exponent (also referred to as the ''characteristic'', or ''scale''), which modifies the magnitude of the number.

To derive the value of the floating-point number, the ''significand'' is multiplied by the ''base'' raised to the power of the ''exponent'', equivalent to shifting the radix point from its implied position by a number of places equal to the value of the exponent—to the right if the exponent is positive or to the left if the exponent is negative.

Using base-10 (the familiar decimal notation) as an example, the number , which has ten decimal digits of precision, is represented as the significand together with 5 as the exponent. To determine the actual value, a decimal point is placed after the first digit of the significand and the result is multiplied by to give , or . In storing such a number, the base (10) need not be stored, since it will be the same for the entire range of supported numbers, and can thus be inferred.

Symbolically, this final value is:
$\frac \times b^e,$

where is the significand (ignoring any implied decimal point), is the precision (the number of digits in the significand), is the base (in our example, this is the number ''ten''), and is the exponent.

Historically, several number bases have been used for representing floating-point numbers, with base two ( binary) being the most common, followed by base ten ( decimal floating point), and other less common varieties, such as base sixteen ( hexadecimal floating point), base eight (octal floating point), base four (quaternary floating point), base three ( balanced ternary floating point) and even base 256 and base .

A floating-point number is a rational number, because it can be represented as one integer divided by another; for example is (145/100)×1000 or /100. The base determines the fractions that can be represented; for instance, 1/5 cannot be represented exactly as a floating-point number using a binary base, but 1/5 can be represented exactly using a decimal base (, or ). However, 1/3 cannot be represented exactly by either binary (0.010101...) or decimal (0.333...), but in base 3, it is trivial (0.1 or 1×3⁻¹) . The occasions on which infinite expansions occur depend on the base and its prime factors.

The way in which the significand (including its sign) and exponent are stored in a computer is implementation-dependent. The common IEEE formats are described in detail later and elsewhere, but as an example, in the binary single-precision (32-bit) floating-point representation, $p = 24$, and so the significand is a string of 24 bits. For instance, the number π's first 33 bits are:
$11001001\ 00001111\ 1101101\underline\ 10100010\ 0.$

In this binary expansion, let us denote the positions from 0 (leftmost bit, or most significant bit) to 32 (rightmost bit). The 24-bit significand will stop at position 23, shown as the underlined bit above. The next bit, at position 24, is called the ''round bit'' or ''rounding bit''. It is used to round the 33-bit approximation to the nearest 24-bit number (there are specific rules for halfway values, which is not the case here). This bit, which is in this example, is added to the integer formed by the leftmost 24 bits, yielding:
$11001001\ 00001111\ 1101101\underline.$

When this is stored in memory using the IEEE 754 encoding, this becomes the significand . The significand is assumed to have a binary point to the right of the leftmost bit. So, the binary representation of π is calculated from left-to-right as follows:
$\begin &\left(\sum_^ \text_n \times 2^\right) \times 2^e \\ = &\left(1 \times 2^ + 1 \times 2^ + 0 \times 2^ + 0 \times 2^ + 1 \times2^ + \cdots + 1 \times 2^\right) \times 2^1 \\ \approx &1.5707964 \times 2 \\ \approx &3.1415928 \end$

where is the precision ( in this example), is the position of the bit of the significand from the left (starting at and finishing at here) and is the exponent ( in this example).

It can be required that the most significant digit of the significand of a non-zero number be non-zero (except when the corresponding exponent would be smaller than the minimum one). This process is called ''normalization''. For binary formats (which uses only the digits and ), this non-zero digit is necessarily . Therefore, it does not need to be represented in memory; allowing the format to have one