Read this article on the representation of real numbers. Read up to the section labeled "Addition and subtraction."
In computing, floatingpoint arithmetic (FP) is arithmetic using formulaic representation of real numbers as an approximation to support a tradeoff between range and precision. For this reason, floatingpoint computation is often found in systems which include very small and very large real numbers, which require fast processing times. A number is, in general, represented approximately to a fixed number of significant digits (the significand) and scaled using an exponent in some fixed base; the base for the scaling is normally two, ten, or sixteen. A number that can be represented exactly is of the following form:
where significand is an integer, base is an integer greater than or equal to two, and exponent is also an integer. For example:
The term floating point refers to the fact that a number's radix point (decimal point, or, more commonly in computers, binary point) can "float"; that is, it can be placed anywhere relative to the significant digits of the number. This position is indicated as the exponent component, and thus the floatingpoint representation can be thought of as a kind of scientific notation.
An early electromechanical programmable computer, the Z3, included floatingpoint arithmetic (replica on display at Deutsches Museum in Munich).
A floatingpoint system can be used to represent, with a fixed number of digits, numbers of different orders of magnitude: e.g. the distance between galaxies or the diameter of an atomic nucleus can be expressed with the same unit of length. 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 the chosen scale.
Singleprecision floating point numbers: the green lines mark representable values.
Over the years, a variety of floatingpoint representations have been used in computers. In 1985, the IEEE 754 Standard for FloatingPoint Arithmetic was established, and since the 1990s, the most commonly encountered representations are those defined by the IEEE.
The speed of floatingpoint operations, commonly measured in terms of FLOPS, is an important characteristic of a computer system, especially for applications that involve intensive mathematical calculations.
A floatingpoint unit (FPU, colloquially a math coprocessor) is a part of a computer system specially designed to carry out operations on floatingpoint 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 righthand end of the string, next to the least significant digit. In fixedpoint systems, a position in the string is specified for the radix point. So a fixedpoint 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 152,853.5047 seconds, a value that would be represented in standardform scientific notation as 1.528535047×10^{5} seconds.
Floatingpoint representation is similar in concept to scientific notation. Logically, a floatingpoint number consists of:
To derive the value of the floatingpoint 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 base10 (the familiar decimal notation) as an example, the number 152,853.5047, which has ten decimal digits of precision, is represented as the significand 1,528,535,047 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 10^{5} to give 1.528535047×10 ^{5}, or 152,853.5047. 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:
where s is the significand (ignoring any implied decimal point), p is the precision (the number of digits in the significand), b is the base (in our example, this is the number ten), and e is the exponent.
Historically, several number bases have been used for representing floatingpoint 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^{[nb 1]}), base eight (octal floating point), base four (quaternary floating point), base three (balanced ternary floating point) and even base 256 and base 65,536.
A floatingpoint number is a rational number, because it can be represented as one integer divided by another; for example 1.45×10^{3} is (145/100)×1000 or 145,000/100. The base determines the fractions that can be represented; for instance, 1/5 cannot be represented exactly as a floatingpoint number using a binary base, but 1/5 can be represented exactly using a decimal base (0.2, or 2×10^{−1}). 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 implementationdependent. The common IEEE formats are described in detail later and elsewhere, but as an example, in the binary singleprecision (32bit) floatingpoint representation, , and so the significand is a string of 24 bits. For instance, the number π's first 33 bits are:
In this binary expansion, let us denote the positions from 0 (leftmost bit, or most significant bit) to 32 (rightmost bit). The 24bit significand will stop at position 23, shown as the underlined bit 0 above. The next bit, at position 24, is called the round bit or rounding bit. It is used to round the 33bit approximation to the nearest 24bit number (there are specific rules for halfway values, which is not the case here). This bit, which is 1 in this example, is added to the integer formed by the leftmost 24 bits, yielding:
When this is stored in memory using the IEEE 754 encoding, this becomes the significand s. 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 lefttoright as follows:
where p is the precision (24 in this example), n is the position of the bit of the significand from the left (starting at 0 and finishing at 23 here) and e is the exponent (1 in this example).
It can be required that the most significant digit of the significand of a nonzero number be nonzero (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 0 and 1), this nonzero digit is necessarily 1. Therefore, it does not need to be represented in memory; allowing the format to have one more bit of precision. This rule is variously called the leading bit convention, the implicit bit convention, the hidden bit convention, or the assumed bit convention.
The floatingpoint representation is by far the most common way of representing in computers an approximation to real numbers. However, there are alternatives:
long double
.In 1914, Leonardo Torres y Quevedo designed an electromechanical version of Charles Babbage's Analytical Engine, and included floatingpoint arithmetic. In 1938, Konrad Zuse of Berlin completed the Z1, the first binary, programmable mechanical computer; it uses a 24bit binary floatingpoint number representation with a 7bit signed exponent, a 17bit significand (including one implicit bit), and a sign bit. The more reliable relaybased Z3, completed in 1941, has representations for both positive and negative infinities; in particular, it implements defined operations with infinity, such as , and it stops on undefined operations, such as .
Konrad Zuse, architect of the Z3 computer, which uses a 22bit binary floatingpoint representation.
Zuse also proposed, but did not complete, carefully rounded floatingpoint arithmetic that includes and NaN representations, anticipating features of the IEEE Standard by four decades. In contrast, von Neumann recommended against floatingpoint numbers for the 1951 IAS machine, arguing that fixedpoint arithmetic is preferable.
The first commercial computer with floatingpoint hardware was Zuse's Z4 computer, designed in 1942–1945. In 1946, Bell Laboratories introduced the Mark V, which implemented decimal floatingpoint numbers.
The Pilot ACE has binary floatingpoint arithmetic, and it became operational in 1950 at National Physical Laboratory, UK. Thirtythree were later sold commercially as the English Electric DEUCE. The arithmetic is actually implemented in software, but with a one megahertz clock rate, the speed of floatingpoint and fixedpoint operations in this machine were initially faster than those of many competing computers.
The massproduced IBM 704 followed in 1954; it introduced the use of a biased exponent. For many decades after that, floatingpoint hardware was typically an optional feature, and computers that had it were said to be "scientific computers", or to have "scientific computation" (SC) capability (see also Extensions for Scientific Computation (XSC)). It was not until the launch of the Intel i486 in 1989 that generalpurpose personal computers had floatingpoint capability in hardware as a standard feature.
The UNIVAC 1100/2200 series, introduced in 1962, supported two floatingpoint representations:
The IBM 7094, also introduced in 1962, supports singleprecision and doubleprecision representations, but with no relation to the UNIVAC's representations. Indeed, in 1964, IBM introduced hexadecimal floatingpoint representations in its System/360 mainframes; these same representations are still available for use in modern z/Architecture systems. However, in 1998, IBM included IEEEcompatible binary floatingpoint arithmetic to its mainframes; in 2005, IBM also added IEEEcompatible decimal floatingpoint arithmetic.
Initially, computers used many different representations for floatingpoint numbers. The lack of standardization at the mainframe level was an ongoing problem by the early 1970s for those writing and maintaining higherlevel source code; these manufacturer floatingpoint standards differed in the word sizes, the representations, and the rounding behavior and general accuracy of operations. Floatingpoint compatibility across multiple computing systems was in desperate need of standardization by the early 1980s, leading to the creation of the IEEE 754 standard once the 32bit (or 64bit) word had become commonplace. This standard was significantly based on a proposal from Intel, which was designing the i8087 numerical coprocessor; Motorola, which was designing the 68000 around the same time, gave significant input as well.
In 1989, mathematician and computer scientist William Kahan was honored with the Turing Award for being the primary architect behind this proposal; he was aided by his student (Jerome Coonen) and a visiting professor (Harold Stone).
Among the x86 innovations are these:
A floatingpoint number consists of two fixedpoint components, whose range depends exclusively on the number of bits or digits in their representation. Whereas components linearly depend on their range, the floatingpoint range linearly depends on the significand range and exponentially on the range of exponent component, which attaches outstandingly wider range to the number.
On a typical computer system, a doubleprecision (64bit) binary floatingpoint number has a coefficient of 53 bits (including 1 implied bit), an exponent of 11 bits, and 1 sign bit. Since 2^{10} = 1024, the complete range of the positive normal floatingpoint numbers in this format is from 2^{−1022} ≈ 2 × 10^{−308} to approximately 2^{1024} ≈ 2 × 10^{308}.
The number of normalized floatingpoint numbers in a system (B, P, L, U) where
There is a smallest positive normalized floatingpoint number,
which has a 1 as the leading digit and 0 for the remaining digits of the significand, and the smallest possible value for the exponent.
There is a largest floatingpoint number,
which has B − 1 as the value for each digit of the significand and the largest possible value for the exponent.
In addition, there are representable values strictly between −UFL and UFL. Namely, positive and negative zeros, as well as denormalized numbers.
Floatingpoint formats 

IEEE 754 

Other 


The IEEE standardized the computer representation for binary floatingpoint numbers in IEEE 754 (a.k.a. IEC 60559) in 1985. This first standard is followed by almost all modern machines. It was revised in 2008. IBM mainframes support IBM's own hexadecimal floating point format and IEEE 7542008 decimal floating point in addition to the IEEE 754 binary format. The Cray T90 series had an IEEE version, but the SV1 still uses Cray floatingpoint format.
The standard provides for many closely related formats, differing in only a few details. Five of these formats are called basic formats and others are termed extended formats; three of these are especially widely used in computer hardware and languages:
Increasing the precision of the floating point representation generally reduces the amount of accumulated roundoff error caused by intermediate calculations. Less common IEEE formats include:
Any integer with absolute value less than 2^{24} can be exactly represented in the single precision format, and any integer with absolute value less than 2^{53} can be exactly represented in the double precision format. Furthermore, a wide range of powers of 2 times such a number can be represented. These properties are sometimes used for purely integer data, to get 53bit integers on platforms that have double precision floats but only 32bit integers.
The standard specifies some special values, and their representation: positive infinity (+∞), negative infinity (−∞), a negative zero (−0) distinct from ordinary ("positive") zero, and "not a number" values (NaNs).
Comparison of floatingpoint numbers, as defined by the IEEE standard, is a bit different from usual integer comparison. Negative and positive zero compare equal, and every NaN compares unequal to every value, including itself. All values except NaN are strictly smaller than +∞ and strictly greater than −∞. Finite floatingpoint numbers are ordered in the same way as their values (in the set of real numbers).
Floatingpoint numbers are typically packed into a computer datum as the sign bit, the exponent field, and the significand or mantissa, from left to right. For the IEEE 754 binary formats (basic and extended) which have extant hardware implementations, they are apportioned as follows:
Type  Sign  Exponent  Significand field  Total bits  Exponent bias  Bits precision  Number of decimal digits 

Half (IEEE 7542008)  1  5  10  16  15  11  ~3.3 
Single  1  8  23  32  127  24  ~7.2 
Double  1  11  52  64  1023  53  ~15.9 
x86 extended precision  1  15  64  80  16383  64  ~19.2 
Quad  1  15  112  128  16383  113  ~34.0 
While the exponent can be positive or negative, in binary formats it is stored as an unsigned number that has a fixed "bias" added to it. Values of all 0s in this field are reserved for the zeros and subnormal numbers; values of all 1s are reserved for the infinities and NaNs. The exponent range for normalized numbers is [−126, 127] for single precision, [−1022, 1023] for double, or [−16382, 16383] for quad. Normalized numbers exclude subnormal values, zeros, infinities, and NaNs.
In the IEEE binary interchange formats the leading 1 bit of a normalized significand is not actually stored in the computer datum. It is called the "hidden" or "implicit" bit. Because of this, single precision format actually has a significand with 24 bits of precision, double precision format has 53, and quad has 113.
For example, it was shown above that π, rounded to 24 bits of precision, has:
The sum of the exponent bias (127) and the exponent (1) is 128, so this is represented in single precision format as
An example of a layout for 32bit floating point is
and the 64 bit layout is similar.
Main article: Signed zero
In the IEEE 754 standard, zero is signed, meaning that there exist both a "positive zero" (+0) and a "negative zero" (−0). In most runtime environments , positive zero is usually printed as "0" and the negative zero as "0". The two values behave as equal in numerical comparisons, but some operations return different results for +0 and −0. For instance, 1/(−0) returns negative infinity, while 1/+0 returns positive infinity (so that the identity 1/(1/±∞) = ±∞ is maintained). Other common functions with a discontinuity at x=0 which might treat +0 and −0 differently include log(x), signum(x), and the principal square root of y + xi for any negative number y. As with any approximation scheme, operations involving "negative zero" can occasionally cause confusion. For example, in IEEE 754, x = y does not always imply 1/x = 1/y, as 0 = −0 but 1/0 ≠ 1/−0.
Main article: Subnormal numbers
Subnormal values fill the underflow gap with values where the absolute distance between them is the same as for adjacent values just outside the underflow gap. This is an improvement over the older practice to just have zero in the underflow gap, and where underflowing results were replaced by zero (flush to zero).
Modern floatingpoint hardware usually handles subnormal values (as well as normal values), and does not require software emulation for subnormals.
Further information on the concept of infinite: Infinity
The infinities of the extended real number line can be represented in IEEE floatingpoint datatypes, just like ordinary floatingpoint values like 1, 1.5, etc. They are not error values in any way, though they are often (but not always, as it depends on the rounding) used as replacement values when there is an overflow. Upon a dividebyzero exception, a positive or negative infinity is returned as an exact result. An infinity can also be introduced as a numeral (like C's "INFINITY" macro, or "∞" if the programming language allows that syntax).
IEEE 754 requires infinities to be handled in a reasonable way, such as
Main article: NaN
IEEE 754 specifies a special value called "Not a Number" (NaN) to be returned as the result of certain "invalid" operations, such as 0/0, ∞×0, or sqrt(−1). In general, NaNs will be propagated i.e. most operations involving a NaN will result in a NaN, although functions that would give some defined result for any given floatingpoint value will do so for NaNs as well, e.g. NaN ^ 0 = 1. There are two kinds of NaNs: the default quiet NaNs and, optionally, signaling NaNs. A signaling NaN in any arithmetic operation (including numerical comparisons) will cause an "invalid operation" exception to be signaled.
The representation of NaNs specified by the standard has some unspecified bits that could be used to encode the type or source of error; but there is no standard for that encoding. In theory, signaling NaNs could be used by a runtime system to flag uninitialized variables, or extend the floatingpoint numbers with other special values without slowing down the computations with ordinary values, although such extensions are not common.
William Kahan. A primary architect of the Intel 80x87 floatingpoint coprocessor and IEEE 754 floatingpoint standard.
It is a common misconception that the more esoteric features of the IEEE 754 standard discussed here, such as extended formats, NaN, infinities, subnormals etc., are only of interest to numerical analysts, or for advanced numerical applications; in fact the opposite is true: these features are designed to give safe robust defaults for numerically unsophisticated programmers, in addition to supporting sophisticated numerical libraries by experts. The key designer of IEEE 754, William Kahan notes that it is incorrect to "... [deem] features of IEEE Standard 754 for Binary FloatingPoint Arithmetic that ...[are] not appreciated to be features usable by none but numerical experts. The facts are quite the opposite. In 1977 those features were designed into the Intel 8087 to serve the widest possible market... Erroranalysis tells us how to design floatingpoint arithmetic, like IEEE Standard 754, moderately tolerant of wellmeaning ignorance among programmers".
In addition to the widely used IEEE 754 standard formats, other floatingpoint formats are used, or have been used, in certain domainspecific areas.
Bfloat16 and TensorFloat32 formats specifications, compared with IEEE 754 halfprecision and singleprecision standard formats
Type  Sign  Exponent  Significand field  Total bits 

Halfprecision  1  5  10  16 
Bfloat16  1  8  7  16 
TensorFloat32  1  8  10  19 
Singleprecision  1  8  23  32 
Type  Sign  Exponent  Significand field  Total bits 

Halfprecision  1  5  10  16 
Bfloat16  1  8  7  16 
TensorFloat32  1  8  10  19 
Singleprecision  1  8  23  32 
By their nature, all numbers expressed in floatingpoint format are rational numbers with a terminating expansion in the relevant base (for example, a terminating decimal expansion in base10, or a terminating binary expansion in base2). Irrational numbers, such as π or √2, or nonterminating rational numbers, must be approximated. The number of digits (or bits) of precision also limits the set of rational numbers that can be represented exactly. For example, the decimal number 123456789 cannot be exactly represented if only eight decimal digits of precision are available (would be rounded to 123456790 or 123456780 where the rightmost digit 0 is not explicitly stored).
When a number is represented in some format (such as a character string) which is not a native floatingpoint representation supported in a computer implementation, then it will require a conversion before it can be used in that implementation. If the number can be represented exactly in the floatingpoint format then the conversion is exact. If there is not an exact representation then the conversion requires a choice of which floatingpoint number to use to represent the original value. The representation chosen will have a different value from the original, and the value thus adjusted is called the rounded value.
Whether or not a rational number has a terminating expansion depends on the base. For example, in base10 the number 1/2 has a terminating expansion (0.5) while the number 1/3 does not (0.333...). In base2 only rationals with denominators that are powers of 2 (such as 1/2 or 3/16) are terminating. Any rational with a denominator that has a prime factor other than 2 will have an infinite binary expansion. This means that numbers which appear to be short and exact when written in decimal format may need to be approximated when converted to binary floatingpoint. For example, the decimal number 0.1 is not representable in binary floatingpoint of any finite precision; the exact binary representation would have a "1100" sequence continuing endlessly:
where, as previously, s is the significand and e is the exponent.
When rounded to 24 bits this becomes
which is actually 0.100000001490116119384765625 in decimal.
As a further example, the real number π, represented in binary as an infinite sequence of bits is
but is
when approximated by rounding to a precision of 24 bits.
In binary singleprecision floatingpoint, this is represented as s = 1.10010010000111111011011 with e = 1. This has a decimal value of
whereas a more accurate approximation of the true value of π is
The result of rounding differs from the true value by about 0.03 parts per million, and matches the decimal representation of π in the first 7 digits. The difference is the discretization error and is limited by the machine epsilon.
The arithmetical difference between two consecutive representable floatingpoint numbers which have the same exponent is called a unit in the last place (ULP). For example, if there is no representable number lying between the representable numbers 1.45a70c22_{hex} and 1.45a70c24_{hex}, the ULP is 2×16^{−8}, or 2^{−31}. For numbers with a base2 exponent part of 0, i.e. numbers with an absolute value higher than or equal to 1 but lower than 2, an ULP is exactly 2^{−23} or about 10^{−7} in single precision, and exactly 2^{−53} or about 10^{−16} in double precision. The mandated behavior of IEEEcompliant hardware is that the result be within onehalf of a ULP.
Rounding is used when the exact result of a floatingpoint operation (or a conversion to floatingpoint format) would need more digits than there are digits in the significand. IEEE 754 requires correct rounding: that is, the rounded result is as if infinitely precise arithmetic was used to compute the value and then rounded (although in implementation only three extra bits are needed to ensure this). There are several different rounding schemes (or rounding modes). Historically, truncation was the typical approach. Since the introduction of IEEE 754, the default method (round to nearest, ties to even, sometimes called Banker's Rounding) is more commonly used. This method rounds the ideal (infinitely precise) result of an arithmetic operation to the nearest representable value, and gives that representation as the result. In the case of a tie, the value that would make the significand end in an even digit is chosen. The IEEE 754 standard requires the same rounding to be applied to all fundamental algebraic operations, including square root and conversions, when there is a numeric (nonNaN) result. It means that the results of IEEE 754 operations are completely determined in all bits of the result, except for the representation of NaNs. ("Library" functions such as cosine and log are not mandated.)
Alternative rounding options are also available. IEEE 754 specifies the following rounding modes:
Alternative modes are useful when the amount of error being introduced must be bounded. Applications that require a bounded error are multiprecision floatingpoint, and interval arithmetic. The alternative rounding modes are also useful in diagnosing numerical instability: if the results of a subroutine vary substantially between rounding to + and − infinity then it is likely numerically unstable and affected by roundoff error.
For ease of presentation and understanding, decimal radix with 7 digit precision will be used in the examples, as in the IEEE 754 decimal32 format. The fundamental principles are the same in any radix or precision, except that normalization is optional (it does not affect the numerical value of the result). Here, s denotes the significand and e denotes the exponent.
A simple method to add floatingpoint numbers is to first represent them with the same exponent. In the example below, the second number is shifted right by three digits, and one then proceeds with the usual addition method:
123456.7 = 1.234567 × 10^5 101.7654 = 1.017654 × 10^2 = 0.001017654 × 10^5
Hence: 123456.7 + 101.7654 = (1.234567 × 10^5) + (1.017654 × 10^2) = (1.234567 × 10^5) + (0.001017654 × 10^5) = (1.234567 + 0.001017654) × 10^5 = 1.235584654 × 10^5
In detail:
e=5; s=1.234567 (123456.7) + e=2; s=1.017654 (101.7654)
e=5; s=1.234567 + e=5; s=0.001017654 (after shifting)  e=5; s=1.235584654 (true sum: 123558.4654)
This is the true result, the exact sum of the operands. It will be rounded to seven digits and then normalized if necessary. The final result is
e=5; s=1.235585 (final sum: 123558.5)
The lowest three digits of the second operand (654) are essentially lost. This is roundoff error. In extreme cases, the sum of two nonzero numbers may be equal to one of them:
e=5; s=1.234567 + e=−3; s=9.876543
e=5; s=1.234567 + e=5; s=0.00000009876543 (after shifting)  e=5; s=1.23456709876543 (true sum) e=5; s=1.234567 (after rounding and normalization)
In the above conceptual examples it would appear that a large number of extra digits would need to be provided by the adder to ensure correct rounding; however, for binary addition or subtraction using careful implementation techniques only two extra guard bits and one extra sticky bit need to be carried beyond the precision of the operands.
Another problem of loss of significance occurs when two nearly equal numbers are subtracted. In the following example e = 5; s = 1.234571 and e = 5; s = 1.234567 are representations of the rationals 123457.1467 and 123456.659.
e=5; s=1.234571 − e=5; s=1.234567  e=5; s=0.000004 e=−1; s=4.000000 (after rounding and normalization)
The best representation of this difference is e = −1; s = 4.877000, which differs more than 20% from e = −1; s = 4.000000. In extreme cases, all significant digits of precision can be lost (although gradual underflow ensures that the result will not be zero unless the two operands were equal). This cancellation illustrates the danger in assuming that all of the digits of a computed result are meaningful. Dealing with the consequences of these errors is a topic in numerical analysis; see also Accuracy problems.
To multiply, the significands are multiplied while the exponents are added, and the result is rounded and normalized.
e=3; s=4.734612 × e=5; s=5.417242  e=8; s=25.648538980104 (true product) e=8; s=25.64854 (after rounding) e=9; s=2.564854 (after normalization)
Similarly, division is accomplished by subtracting the divisor's exponent from the dividend's exponent, and dividing the dividend's significand by the divisor's significand.
There are no cancellation or absorption problems with multiplication or division, though small errors may accumulate as operations are performed in succession. In practice, the way these operations are carried out in digital logic can be quite complex (see Booth's multiplication algorithm and Division algorithm). For a fast, simple method, see the Horner method.
Floatingpoint computation in a computer can run into three kinds of problems:
Prior to the IEEE standard, such conditions usually caused the program to terminate, or triggered some kind of trap that the programmer might be able to catch. How this worked was systemdependent, meaning that floatingpoint programs were not portable. (The term "exception" as used in IEEE 754 is a general term meaning an exceptional condition, which is not necessarily an error, and is a different usage to that typically defined in programming languages such as a C++ or Java, in which an "exception" is an alternative flow of control, closer to what is termed a "trap" in IEEE 754 terminology).
Here, the required default method of handling exceptions according to IEEE 754 is discussed (the IEEE 754 optional trapping and other "alternate exception handling" modes are not discussed). Arithmetic exceptions are (by default) required to be recorded in "sticky" status flag bits. That they are "sticky" means that they are not reset by the next (arithmetic) operation, but stay set until explicitly reset. The use of "sticky" flags thus allows for testing of exceptional conditions to be delayed until after a full floatingpoint expression or subroutine: without them exceptional conditions that could not be otherwise ignored would require explicit testing immediately after every floatingpoint operation. By default, an operation always returns a result according to specification without interrupting computation. For instance, 1/0 returns +∞, while also setting the dividebyzero flag bit (this default of ∞ is designed to often return a finite result when used in subsequent operations and so be safely ignored).
The original IEEE 754 standard, however, failed to recommend operations to handle such sets of arithmetic exception flag bits. So while these were implemented in hardware, initially programming language implementations typically did not provide a means to access them (apart from assembler). Over time some programming language standards (e.g., C99/C11 and Fortran) have been updated to specify methods to access and change status flag bits. The 2008 version of the IEEE 754 standard now specifies a few operations for accessing and handling the arithmetic flag bits. The programming model is based on a single thread of execution and use of them by multiple threads has to be handled by a means outside of the standard (e.g. C11 specifies that the flags have threadlocal storage).
IEEE 754 specifies five arithmetic exceptions that are to be recorded in the status flags ("sticky bits"):
Fig. 1: resistances in parallel, with total resistance
The default return value for each of the exceptions is designed to give the correct result in the majority of cases such that the exceptions can be ignored in the majority of codes. inexact returns a correctly rounded result, and underflow returns a denormalized small value and so can almost always be ignored. dividebyzero returns infinity exactly, which will typically then divide a finite number and so give zero, or else will give an invalid exception subsequently if not, and so can also typically be ignored. For example, the effective resistance of n resistors in parallel (see fig. 1) is given by . If a shortcircuit develops with set to 0, will return +infinity which will give a final of 0, as expected (see the continued fraction example of IEEE 754 design rationale for another example).
Overflow and invalid exceptions can typically not be ignored, but do not necessarily represent errors: for example, a rootfinding routine, as part of its normal operation, may evaluate a passedin function at values outside of its domain, returning NaN and an invalid exception flag to be ignored until finding a useful start point.
The fact that floatingpoint numbers cannot precisely represent all real numbers, and that floatingpoint operations cannot precisely represent true arithmetic operations, leads to many surprising situations. This is related to the finite precision with which computers generally represent numbers.
For example, the nonrepresentability of 0.1 and 0.01 (in binary) means that the result of attempting to square 0.1 is neither 0.01 nor the representable number closest to it. In 24bit (single precision) representation, 0.1 (decimal) was given previously as e = −4; s = 110011001100110011001101, which is
Squaring this number gives
Squaring it with singleprecision floatingpoint hardware (with rounding) gives
But the representable number closest to 0.01 is
Also, the nonrepresentability of π (and π/2) means that an attempted computation of tan(π/2) will not yield a result of infinity, nor will it even overflow. It is simply not possible for standard floatingpoint hardware to attempt to compute tan(π/2), because π/2 cannot be represented exactly. This computation in C:
/* Enough digits to be sure we get the correct approximation. */ double pi = 3.1415926535897932384626433832795; double z = tan(pi/2.0);
will give a result of 16331239353195370.0. In single precision (using the tanf function), the result will be −22877332.0.
By the same token, an attempted computation of sin(π) will not yield zero. The result will be (approximately) 0.1225×10^{−15} in double precision, or −0.8742×10^{−7} in single precision.
While floatingpoint addition and multiplication are both commutative (a + b = b + a and a × b = b × a), they are not necessarily associative. That is, (a + b) + c is not necessarily equal to a + ( b + c). Using 7digit significand decimal arithmetic:
a = 1234.567, b = 45.67834, c = 0.0004
(a + b) + c: 1234.567 (a) + 45.67834 (b) ____________ 1280.24534 rounds to 1280.245
1280.245 (a + b) + 0.0004 (c) ____________ 1280.2454 rounds to 1280.245 ← (a + b) + c
a + (b + c): 45.67834 (b) + 0.0004 (c) ____________ 45.67874
1234.567 (a) + 45.67874 (b + c) ____________ 1280.24574 rounds to 1280.246 ← a + (b + c)
They are also not necessarily distributive. That is, (a + b) × c may not be the same as a × c + b × c:
1234.567 × 3.333333 = 4115.223 1.234567 × 3.333333 = 4.115223 4115.223 + 4.115223 = 4119.338 but 1234.567 + 1.234567 = 1235.802 1235.802 × 3.333333 = 4119.340
In addition to loss of significance, inability to represent numbers such as π and 0.1 exactly, and other slight inaccuracies, the following phenomena may occur:
Machine precision is a quantity that characterizes the accuracy of a floatingpoint system, and is used in backward error analysis of floatingpoint algorithms. It is also known as unit roundoff or machine epsilon. Usually denoted Ε_{mach}, its value depends on the particular rounding being used.
With rounding to zero,
whereas rounding to nearest,
This is important since it bounds the relative error in representing any nonzero real number x within the normalized range of a floatingpoint system:
Backward error analysis, the theory of which was developed and popularized by James H. Wilkinson, can be used to establish that an algorithm implementing a numerical function is numerically stable. The basic approach is to show that although the calculated result, due to roundoff errors, will not be exactly correct, it is the exact solution to a nearby problem with slightly perturbed input data. If the perturbation required is small, on the order of the uncertainty in the input data, then the results are in some sense as accurate as the data "deserves". The algorithm is then defined as backward stable. Stability is a measure of the sensitivity to rounding errors of a given numerical procedure; by contrast, the condition number of a function for a given problem indicates the inherent sensitivity of the function to small perturbations in its input and is independent of the implementation used to solve the problem.
As a trivial example, consider a simple expression giving the inner product of (length two) vectors and , then
and so
where
where
by definition, which is the sum of two slightly perturbed (on the order of Ε_{mach}) input data, and so is backward stable. For more realistic examples in numerical linear algebra, see Higham 2002 and other references below.
Although, as noted previously, individual arithmetic operations of IEEE 754 are guaranteed accurate to within half a ULP, more complicated formulae can suffer from larger errors due to roundoff. The loss of accuracy can be substantial if a problem or its data are illconditioned, meaning that the correct result is hypersensitive to tiny perturbations in its data. However, even functions that are wellconditioned can suffer from large loss of accuracy if an algorithm numerically unstable for that data is used: apparently equivalent formulations of expressions in a programming language can differ markedly in their numerical stability. One approach to remove the risk of such loss of accuracy is the design and analysis of numerically stable algorithms, which is an aim of the branch of mathematics known as numerical analysis. Another approach that can protect against the risk of numerical instabilities is the computation of intermediate (scratch) values in an algorithm at a higher precision than the final result requires, which can remove, or reduce by orders of magnitude, such risk: IEEE 754 quadruple precision and extended precision are designed for this purpose when computing at double precision.
For example, the following algorithm is a direct implementation to compute the function A(x) = (x−1) / (exp(x−1) − 1) which is wellconditioned at 1.0, however it can be shown to be numerically unstable and lose up to half the significant digits carried by the arithmetic when computed near 1.0.
1 doubleA(doubleX) 2 { 3 doubleY,Z;// [1] 4 Y=X1.0; 5 Z=exp(Y); 6 if(Z!=1.0)Z=Y/(Z1.0);// [2] 7 return(Z); 8 }
If, however, intermediate computations are all performed in extended precision (e.g. by setting line [1] to C99 long double), then up to full precision in the final double result can be maintained. Alternatively, a numerical analysis of the algorithm reveals that if the following nonobvious change to line [2] is made:
if(Z!=1.0)Z=log(Z)/(Z1.0);
then the algorithm becomes numerically stable and can compute to full double precision.
To maintain the properties of such carefully constructed numerically stable programs, careful handling by the compiler is required. Certain "optimizations" that compilers might make (for example, reordering operations) can work against the goals of wellbehaved software. There is some controversy about the failings of compilers and language designs in this area: C99 is an example of a language where such optimizations are carefully specified to maintain numerical precision. See the external references at the bottom of this article.
A detailed treatment of the techniques for writing highquality floatingpoint software is beyond the scope of this article, and the reader is referred to, and the other references at the bottom of this article. Kahan suggests several rules of thumb that can substantially decrease by orders of magnitude the risk of numerical anomalies, in addition to, or in lieu of, a more careful numerical analysis. These include: as noted above, computing all expressions and intermediate results in the highest precision supported in hardware (a common rule of thumb is to carry twice the precision of the desired result i.e. compute in double precision for a final single precision result, or in double extended or quad precision for up to double precision results); and rounding input data and results to only the precision required and supported by the input data (carrying excess precision in the final result beyond that required and supported by the input data can be misleading, increases storage cost and decreases speed, and the excess bits can affect convergence of numerical procedures: notably, the first form of the iterative example given below converges correctly when using this rule of thumb). Brief descriptions of several additional issues and techniques follow.
As decimal fractions can often not be exactly represented in binary floatingpoint, such arithmetic is at its best when it is simply being used to measure realworld quantities over a wide range of scales (such as the orbital period of a moon around Saturn or the mass of a proton), and at its worst when it is expected to model the interactions of quantities expressed as decimal strings that are expected to be exact. An example of the latter case is financial calculations. For this reason, financial software tends not to use a binary floatingpoint number representation. The "decimal" data type of the C# and Python programming languages, and the decimal formats of the IEEE 7542008 standard, are designed to avoid the problems of binary floatingpoint representations when applied to humanentered exact decimal values, and make the arithmetic always behave as expected when numbers are printed in decimal.
Expectations from mathematics may not be realized in the field of floatingpoint computation. For example, it is known that , and that , however these facts cannot be relied on when the quantities involved are the result of floatingpoint computation.
The use of the equality test (if (x==y) ...
) requires care when dealing with floatingpoint numbers. Even simple expressions like 0.6/0.23==0
will, on most computers, fail to be true (in IEEE 754 double precision,
for example, 0.6/0.23
is approximately equal to 4.44089209850063e16). Consequently, such tests are sometimes replaced with "fuzzy" comparisons (if (abs(xy) < epsilon) ...
, where epsilon is sufficiently small
and tailored to the application, such as 1.0E−13). The wisdom of doing this varies greatly, and can require numerical analysis to bound epsilon. Values derived from the primary data representation and their comparisons should be performed in
a wider, extended, precision to minimize the risk of such inconsistencies due to roundoff errors. It is often better to organize the code in such a way that such tests are unnecessary. For example, in computational geometry, exact tests
of whether a point lies off or on a line or plane defined by other points can be performed using adaptive precision or exact arithmetic methods.
Small errors in floatingpoint arithmetic can grow when mathematical algorithms perform operations an enormous number of times. A few examples are matrix inversion, eigenvector computation, and differential equation solving. These algorithms must be very carefully designed, using numerical approaches such as Iterative refinement, if they are to work well.
Summation of a vector of floatingpoint values is a basic algorithm in scientific computing, and so an awareness of when loss of significance can occur is essential. For example, if one is adding a very large number of numbers, the individual addends are very small compared with the sum. This can lead to loss of significance. A typical addition would then be something like
3253.671 + 3.141276  3256.812
The low 3 digits of the addends are effectively lost. Suppose, for example, that one needs to add many numbers, all approximately equal to 3. After 1000 of them have been added, the running sum is about 3000; the lost digits are not regained. The Kahan summation algorithm may be used to reduce the errors.
Roundoff error can affect the convergence and accuracy of iterative numerical procedures. As an example, Archimedes approximated π by calculating the perimeters of polygons inscribing and circumscribing a circle, starting with hexagons, and successively doubling the number of sides. As noted above, computations may be rearranged in a way that is mathematically equivalent but less prone to error (numerical analysis). Two forms of the recurrence formula for the circumscribed polygon are:
Here is a computation using IEEE "double" (a significand with 53 bits of precision) arithmetic:
i 6 × 2^{i} × t_{i}, first form 6 × 2^{i} × t_{i}, second form  0 3.4641016151377543863 3.4641016151377543863 1 3.2153903091734710173 3.2153903091734723496 2 3.1596599420974940120 3.1596599420975006733 3 3.1460862151314012979 3.1460862151314352708 4 3.1427145996453136334 3.1427145996453689225 5 3.1418730499801259536 3.1418730499798241950 6 3.1416627470548084133 3.1416627470568494473 7 3.1416101765997805905 3.1416101766046906629 8 3.1415970343230776862 3.1415970343215275928 9 3.1415937488171150615 3.1415937487713536668 10 3.1415929278733740748 3.1415929273850979885 11 3.1415927256228504127 3.1415927220386148377 12 3.1415926717412858693 3.1415926707019992125 13 3.1415926189011456060 3.1415926578678454728 14 3.1415926717412858693 3.1415926546593073709 15 3.1415919358822321783 3.1415926538571730119 16 3.1415926717412858693 3.1415926536566394222 17 3.1415810075796233302 3.1415926536065061913 18 3.1415926717412858693 3.1415926535939728836 19 3.1414061547378810956 3.1415926535908393901 20 3.1405434924008406305 3.1415926535900560168 21 3.1400068646912273617 3.1415926535898608396 22 3.1349453756585929919 3.1415926535898122118 23 3.1400068646912273617 3.1415926535897995552 24 3.2245152435345525443 3.1415926535897968907 25 3.1415926535897962246 26 3.1415926535897962246 27 3.1415926535897962246 28 3.1415926535897962246 The true value is 3.14159265358979323846264338327...
While the two forms of the recurrence formula are clearly mathematically equivalent, the first subtracts 1 from a number extremely close to 1, leading to an increasingly problematic loss of significant digits. As the recurrence is applied repeatedly, the accuracy improves at first, but then it deteriorates. It never gets better than about 8 digits, even though 53bit arithmetic should be capable of about 16 digits of precision. When the second form of the recurrence is used, the value converges to 15 digits of precision.
Source: Wikipedia, https://en.wikipedia.org/wiki/Floatingpoint_arithmetic
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