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| [[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]]).]]
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| [[File:Float mantissa exponent.png|thumb|200px|A diagram showing a representation of a decimal floating-point number using a [[significand|mantissa]] and an [[exponent]].]]
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| In [[computing]], '''floating point''' describes a method of representing an approximation of a [[real number]] in a way that can support a wide range of values. The numbers are, in general, represented approximately to a fixed number of [[Significant figures|significant digits]] (the mantissa) and scaled using an [[Exponentiation|exponent]]. The base for the scaling is normally 2, 10 or 16. The typical number that can be represented exactly is of the form:
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| :''Significant digits'' × ''base''<sup>''exponent''</sup>
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| The idea of floating-point representation over intrinsically [[Integer (computer science)|integer]] [[fixed-point arithmetic|fixed-point]] numbers, which consist purely of [[significand]], is that expanding it with the exponent component achieves greater range. For instance, to represent large values, e.g. distances between galaxies, there is no need to keep all 39 decimal places down to [[femtometre]]-resolution (employed in particle physics). Assuming that the best resolution is in [[light years]], only the 9 most significant decimal digits matter, whereas the remaining 30 digits carry pure noise, and thus can be safely dropped. This represents a savings of 100 [[bit]]s of [[computer data storage]]. Instead of these 100 bits, much fewer are used to represent the scale (the exponent), e.g. 8 bits or 2 decimal digits. Given that one number can encode both astronomic and subatomic distances with the same nine digits of accuracy, but because a 9-digit number is 100 times less accurate than the 11 digits reserved for scale, this is considered a [[trade-off]] exchanging range for [[accuracy and precision|precision]]. The example of using scaling to extend the dynamic range reveals another contrast with fixed-point numbers: Floating-point values are not uniformly spaced. Small values, close to zero, can be represented with much higher resolution (e.g. one femtometre) than large ones because a greater scale (e.g. light years) must be selected for encoding significantly larger values.<ref>{{cite book |last=W.Smith |first= Steven |title=The Scientist and Engineer's Guide to Digital Signal Processing|url=http://www.dspguide.com/ch28/4.htm |accessdate=December 31, 2012 |year= 1997|publisher= California Technical Pub|location= |isbn=0966017633 |page=514 |chapter=Chapter 28, Fixed versus Floating Point }}</ref> That is, floating-point numbers cannot represent point coordinates with atomic accuracy at galactic distances, only close to the origin.
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| 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 in the internal representation, and floating point can thus be thought of as a computer realization of [[scientific notation]].
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| Over the years, a variety of floating-point representations have been used in computers. However, since the 1990s, the most commonly encountered representation is that defined by the [[IEEE 754]] Standard.
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| The speed of floating-point operations, commonly referred to in performance measurements as [[FLOPS]], is an important characteristic of a computer system, especially in [[software]] that performs large-scale mathematical calculations.
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| ==Overview==
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| A number representation (called a [[numeral system]] in mathematics) specifies some way of storing a number that may be encoded as a string of digits. The arithmetic is defined as a set of actions on the representation that simulate classical arithmetic operations.
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| 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 [[Decimal separator|"point" character]] (dot or comma) there. If the radix point is not specified then it is implicitly assumed to lie at the right (least significant) end of the string (that is, the number is an [[integer]]). In [[fixed-point arithmetic|fixed-point]] systems, some specific assumption is made about where the radix point is located in the string. For example, the convention could be that the string consists of 8 decimal digits with the decimal point in the middle, so that "00012345" has a value of 1.2345.
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| In [[scientific notation]], the given number is scaled by a [[exponentiation|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 revolution period of [[Jupiter]]'s moon [[Io (moon)|Io]] is 152853.5047 seconds, a value that would be represented in standard-form scientific notation as 1.528535047{{e|5}} seconds.
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| Floating-point representation is similar in concept to scientific notation. Logically, a floating-point number consists of:
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| *A signed (meaning positive or negative) digit string of a given length in a given [[base (exponentiation)|base]] (or [[radix]]). This digit string is referred to as the [[significand]], [[coefficient]] or, less often, the mantissa (see below). The length of the significand determines the ''precision'' to which numbers can be represented. The radix point position is assumed to always 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 will generally follow the convention that the radix point is just after the most significant (leftmost) digit.
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| *A signed integer [[exponent]], also referred to as the characteristic or scale, which modifies the magnitude of the number.
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| To derive the value of the floating-point number, one must multiply the ''significand'' 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.
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| Using base-10 (the familiar [[Decimal representation|decimal]] notation) as an example, the number 152853.5047, which has ten decimal digits of precision, is represented as the significand 1.528535047 together with an exponent of 5 (if the implied position of the radix point is after the first most significant digit, here 1). To determine the actual value, a decimal point is placed after the first digit of the significand and the result is multiplied by 10<sup>5</sup> to give 1.528535047 × 10<sup>5</sup>, or 152853.5047. 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.
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| Symbolically, this final value is
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| :<math>s \times b^e</math>
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| where ''s'' is the value of the significand (after taking into account the implied radix point), ''b'' is the base, and ''e'' is the exponent.
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| Equivalently:
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| : <math>\frac{s}{b^{p-1}} \times b^e</math>
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| where ''s'' here means the integer value of the entire significand, ignoring any implied decimal point, and ''p'' is the precision—the number of digits in the significand.
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| Historically, several number bases have been used for representing floating-point numbers, with base 2 ([[Binary numeral system|binary]]) being the most common, followed by base 10 (decimal), and other less common varieties, such as base 16 ([[Hexadecimal|hexadecimal notation]]), as well as some exotic ones like 3 (see [[Setun]]).
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| Floating-point numbers are [[rational number]]s because they can be represented as one integer divided by another. For example 1.45×10<sup>3</sup> is (145/100)*1000 or 145000/100. The base however 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 can be represented exactly using a decimal base (0.2, or 2×10<sup>−1</sup>). However 1/3 cannot be represented exactly by either binary (0.010101...) nor decimal (0.333....), but in [[Ternary numeral system|base 3]] it is trivial (0.1 or 1×3<sup>−1</sup>) . The occasions on which infinite expansions occur depend on the base and its [[prime factors]], as described in the article on [[Positional_notation#Infinite_representations|Positional Notation]].
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| The way in which the significand, exponent and sign bits are internally stored on 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 [[bit]]s. For instance, the number [[Pi|π]]'s first 33 bits are 11001001 00001111 11011010 10100010 0. Given that the 24th bit is zero, rounding to 24 bits in binary mode means attributing the 24th bit the value of the 25th which yields 11001001 00001111 11011011. When this is stored using the IEEE 754 encoding, this becomes the significand ''s'' with ''e'' = 1 (where ''s'' is assumed to have a binary point to the right of the first bit) after a left-adjustment (or ''normalization'') during which leading or trailing zeros are truncated should there be any. Note that they do not matter anyway. Then since the first bit of a non-zero binary significand is always 1 it need not be stored, giving an extra bit of precision. To calculate π the formula is
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| :<math>\begin{align}
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| &\left( 1 + \sum_{n=1}^{p-1} \text{bit}_n\times 2^{-n} \right) \times 2^e\\
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| = &\left( 1 + 1\times 2^{-1} + 0\times 2^{-2} + 1\times 2^{-4} + 1\times2^{-7} + \dots + 1\times 2^{-23} \right) \times 2^1\\
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| = &\; 1.5707964\times 2
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| \end{align}</math>
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| where n is the normalized significand's n-th bit from the left. Normalization, which is reversed when 1 is being added above, can be thought of as a form of compression; it allows a binary significand to be compressed into a field one bit shorter than the maximum precision, at the expense of extra processing.
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| The word "mantissa" is often used as a synonym for significand. Use of mantissa in place of significand or coefficient is discouraged, as the mantissa is traditionally defined as the fractional part of a logarithm, while the ''characteristic'' is the integer part. This terminology comes from the manner in which [[common logarithm|logarithm]] tables were used before computers became commonplace. Log tables were actually tables of mantissas.
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| ===Some other computer representations for non-integral numbers===
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| Floating-point representation, in particular the standard IEEE format, is by far the most common way of representing an approximation to real numbers in computers because it is efficiently handled in most large computer processors. However, there are alternatives:
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| * [[Fixed-point arithmetic|Fixed-point]] representation uses integer hardware operations controlled by a software implementation of a specific convention about the location of the binary or decimal point, for example, 6 bits or digits from the right. The hardware to manipulate these representations is less costly than floating point and is also commonly used to perform integer operations. Binary fixed point is usually used in special-purpose applications on embedded processors that can only do integer arithmetic, but decimal fixed point is common in commercial applications.
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| * [[Binary-coded decimal]] (BCD) is an encoding for decimal numbers in which each digit is represented by its own binary sequence. It is possible to implement a floating-point system with BCD encoding.
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| * [[Logarithmic number system]]s represent a real number by the logarithm of its absolute value and a sign bit. The value distribution is similar to floating point, but the value-to-representation curve, i. e. the graph of the logarithm function, is smooth (except at 0). Contrary to floating-point arithmetic, in a logarithmic number system multiplication, division and exponentiation are easy to implement but addition and subtraction are difficult. The [[Symmetric level-index arithmetic|level index arithmetic]] of Clenshaw, Olver, and Turner is a scheme based on a generalised logarithm representation.
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| * Where greater precision is desired, floating-point arithmetic can be implemented (typically in software) with variable-length significands (and sometimes exponents) that are sized depending on actual need and depending on how the calculation proceeds. This is called [[arbitrary-precision arithmetic|arbitrary-precision]] floating-point arithmetic.
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| * Some numbers (''e.g.,'' 1/3 and 1/10) cannot be represented exactly in binary floating-point, no matter what the precision is. Software packages that perform [[fraction (mathematics)|rational arithmetic]] represent numbers as fractions with integral numerator and denominator, and can therefore represent any rational number exactly. Such packages generally need to use "[[bignum]]" arithmetic for the individual integers.
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| * [[Computer algebra system]]s such as [[Mathematica]] and [[Maxima (software)|Maxima]] can often handle irrational numbers like <math>\pi</math> or <math>\sqrt{3}</math> in a completely "formal" way, without dealing with a specific encoding of the significand. Such programs can evaluate expressions like "<math>\sin 3\pi</math>" exactly, because they "know" the underlying mathematics.
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| ==Range of floating-point numbers==
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| A floating-point number consists of two [[Fixed-point arithmetic|fixed-point]] components, whose range depends exclusively on the number of bits or digits in their representation. Whereas components linearly depend on their range, the floating-point range linearly depends on the significant range and exponentially on the range of exponent component, which attaches outstandingly wider range to the number.
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| On a typical computer system, a 'double precision' (64-bit) binary floating-point number has a coefficient of 53 bits (one of which is implied), an exponent of 11 bits, and one sign bit. Positive floating-point numbers in this format have an approximate range of 10<sup>−308</sup> to 10<sup>308</sup>, because the range of the exponent is [−1022,1023] and 308 is approximately log<sub>10</sub>(2<sup>1023</sup>). The complete range of the format is from about −10<sup>308</sup> through +10<sup>308</sup> (see [[IEEE 754]]).
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| The number of normalized floating-point numbers in a system F (''B'', ''P'', ''L'', ''U'') (where ''B'' is the base of the system, ''P'' is the precision of the system to ''P'' numbers, ''L'' is the smallest exponent representable in the system, and ''U'' is the largest exponent used in the system) is:
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| <math>2 (B - 1) (B^{P-1}) (U - L + 1) + 1</math>.
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| There is a smallest positive normalized floating-point number,
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| Underflow level = UFL = <math>B^L</math>
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| 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.
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| There is a largest floating-point number,
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| Overflow level = OFL = <math>(1 - B^{-P}) (B^{U + 1})</math> which has ''B'' − 1 as the value for each digit of the significand and the largest possible value for the exponent.
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| In addition there are representable values strictly between −UFL and UFL. Namely, [[Signed_zero|positive and negative zeros]], as well as [[denormal numbers|denormalized numbers]].
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| ==History==
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| [[File:Leonardo Torres Quevedo.jpg|thumb|left|180px|[[Leonardo Torres y Quevedo]], in 1914 published an analysis of floating point based on the [[analytic engine]].]]
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| [[Leonardo Torres y Quevedo]] in 1914 designed an electro-mechanical version of the [[Analytical Engine]] of [[Charles Babbage]] which included floating-point arithmetic.<ref>{{cite book|author=B. Randell |title=From analytical engine to electronic digital computer: the contributions of Ludgate, Torres, and Bush. IEEE Annals of the History of Computing, 04(4) | pages=327–341 | year=1982}}</ref>
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| In 1938, [[Konrad Zuse]] of Berlin completed the [[Z1 (computer)|Z1]], the first mechanical binary programmable computer, this was however unreliable in operation.<ref>{{cite journal|url=http://ed-thelen.org/comp-hist/Zuse_Z1_and_Z3.pdf |title=Konrad Zuse’s Legacy: The Architecture of the Z1 and Z3 |journal=IEEE Annals of the History of Computing |volume=19 |number=2 |year=1997 |pages=5–15}}</ref> It worked with 24-bit binary floating-point numbers having a 7-bit signed exponent, a 16-bit significand (including one implicit bit), and a sign bit. The memory used sliding metal parts to store 64 words of such numbers. The [[relay]]-based [[Z3 (computer)|Z3]], completed in 1941 had representations for plus and minus infinity. It implemented defined operations with infinity such as {{math|1/∞ {{=}} 0}} and stopped on undefined operations like {{math|0×∞}}. It also implemented the square root operation in hardware.
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| [[File:Konrad Zuse (1992).jpg|thumb|220px|right|[[Konrad Zuse]], architect of the [[Z3 (computer)|Z3]] computer, which used 22-bit binary floating point.]]
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| Zuse also proposed, but did not complete, carefully rounded floating–point arithmetic that would have included ±∞ and NaNs, anticipating features of IEEE Standard floating–point by four decades.<ref name=kahansiam>{{cite web|url=http://www.cs.berkeley.edu/~wkahan/SIAMjvnl.pdf|title=The Baleful Effect of Computer Languages and Benchmarks upon Applied Mathematics, Physics and Chemistry| author=William Kahan |date=15 July 1997}}</ref>
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| By contrast, [[John von Neumann|von Neumann]] recommended against floating point for the 1951 [[IAS machine]], arguing that fixed point arithmetic was preferable.<ref>{{cite web|url=http://www.cs.berkeley.edu/~wkahan/SIAMjvnl.pdf|title=The Baleful Effect of Computer Languages and Benchmarks upon Applied Mathematics, Physics and Chemistry. John von Neumann Lecture|date=16 July 1997|page=3}}</ref>
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| The first ''commercial'' computer with floating-point hardware was Zuse's [[Z4 (computer)|Z4]] computer designed in 1942–1945. The Bell Laboratories Mark V computer implemented decimal floating point in 1946.<ref>{{cite book |editor-last=Randell |editor-first=Brian |title=The Origins of Digital Computers: Selected Papers |edition=3rd |publisher=Springer-Verlag |year=1982|origyear=1973 |location=Berlin; New York |page=244|isbn=3-540-11319-3}}</ref>
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| The [[Pilot ACE]] had binary floating-point arithmetic which became operational at [[National Physical Laboratory, UK]] in 1950. A total of 33 were later sold commercially as the [[English Electric DEUCE]]. The arithmetic was actually implemented as subroutines, but with a one megahertz clock rate, the speed of floating-point operations and fixed point was initially faster than many competing computers, and since it was only software, all the DEUCE's had it.
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| The mass-produced [[vacuum tube]]-based [[IBM 704]] followed in 1954; it introduced the use of a [[Exponent bias|biased exponent]]. For many decades after that, floating-point hardware was typically an optional feature, and computers that had it were said to be "scientific computers", or to have "scientific computing" capability. It was not until the launch of the Intel i486 in 1989 that ''general-purpose'' personal computers had floating-point capability in hardware as standard.
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| The [[UNIVAC 1100/2200 series]], introduced in 1962, supported two floating-point formats.
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| * Single precision used 36 bits, organized into a 1-bit sign, an 8-bit exponent, and a 27-bit significand.
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| * Double precision used 72 bits organized as a 1-bit sign, an 11-bit exponent, and a 60-bit significand.
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| The [[IBM 7094]], also introduced in 1962 supported single and double precision but with no relation to the UNIVAC formats. IBM mainframes until the mid-1980s had no real floating-point standardization between them.
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| The lack of standardization in floating-point numbers at the mainframe level was an ongoing problem by the early 1970s those writing and maintaining higher level language source code. Floating-point compatibility across multiple computing systems was in desperate need of standardization by the early 1980s.
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| Prior to the [[IEEE-754]] standard, computers used many different forms of floating point.
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| These manufacturer floating-point standards differed in the word sizes, the format of the representations, and the rounding behavior of operations. These differing systems implemented different parts of the arithmetic in hardware and software, with varying accuracy.
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| The IEEE-754 standard was created in the early 1980s after word sizes of 32 bits (or 64) had been generally settled upon. The IEEE-754 standard was significantly based on a proposal from Intel who were designing the [[Intel 8087|i8087]] numerical coprocessor. Motorola that was designing the [[68000]] around the same time had significant input in the IEEE-754 standardization process as well.
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| Mathematician and computer scientist [[William Kahan|Prof. William Kahan]] at the [[University of California, Berkeley]] was the primary architect behind this proposal, along with his student Jerome Coonen at U.C. Berkeley and visiting Prof. Harold Stone, for which he was awarding the 1989 Turing award.<ref>{{cite web |title=An Interview with the Old Man of Floating-Point |url=http://www.eecs.berkeley.edu/~wkahan/ieee754status/754story.html |first=Charles |last=Severance |date=20 February 1998}}</ref>
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| Among the x86 innovations are these:
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| * A precisely specified encoding of the floating-point datums at the bit-string level, so that all compliant computers would interpret bit patterns the same way. This made it possible to transfer floating-point numbers from one computer to another after [[endianness]] conversion.
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| * A precisely specified behavior of the arithmetic operations: arithmetic operations were required to be correctly rounded, i.e. to give the same result as if infinitely precise arithmetic were used and then the result were rounded. This meant that a given program, with given data, would always produce the same result on any compliant computer, provided the same format were used internally and expressions were written in a non-ambiguous way. This helped reduce the almost mystical reputation that floating-point computation had for seemingly nondeterministic behavior.
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| * The ability of exceptional conditions (overflow, divide by zero, etc.) to propagate through a computation in a benign manner and be handled by the software in a controlled way.
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| ==IEEE 754: floating point in modern computers==
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| {{Main|IEEE floating point}}
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| {{Floating-point}}
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| The [[IEEE]] has standardized the computer representation for binary floating-point numbers in [[IEEE floating point|IEEE 754]] (aka. IEC 60559). This standard is followed by almost all modern machines. Notable exceptions include IBM mainframes, which support [[IBM Floating Point Architecture|IBM's own format]] (in addition to the IEEE 754 binary and decimal formats), and [[Cray]] vector machines, where the [[Cray T90|T90]] series had an IEEE version, but the [[Cray SV1|SV1]] still uses Cray floating-point format.
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| 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:
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| *[[Single precision]], usually used to represent the "float" type in the C language family (though this is [[C data types#Basic types|not guaranteed]]). This is a binary format that occupies 32 bits (4 bytes) and its significand has a precision of 24 bits (about 7 decimal digits).
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| *[[Double precision]], usually used to represent the "double" type in the C language family (though this is [[C data types#Basic types|not guaranteed]]). This is a binary format that occupies 64 bits (8 bytes) and its significand has a precision of 53 bits (about 16 decimal digits).
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| *[[Extended precision|Double extended]], also called "extended precision" format. This is a binary format that occupies at least 79 bits (80 if the hidden/implicit bit rule is not used) and its significand has a precision of at least 64 bits (about 19 decimal digits). A format satisfying the minimal requirements (64-bit precision, 15-bit exponent, thus fitting on 80 bits) is provided by the [[x86 architecture]]. In general on such processors, this format can be used with "[[long double]]" in the C language family (the [[C99]] and [[C11 (C standard revision)|C11]] standards "IEC 60559 floating-point arithmetic extension- Annex F" recommend the 80-bit extended format to be provided as "long double" when available). On other processors, "long double" may be a synonym for "double" if any form of extended precision is not available, or may stand for a larger format, such as quadruple precision.
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| Increasing the precision of the floating point representation generally reduces the amount of accumulated [[round-off error]] caused by intermediate calculations.<ref>{{cite web |url=http://www.cs.berkeley.edu/~wkahan/Qdrtcs.pdf |title=W. Kahan. "On the Cost of Floating-Point Computation Without Extra-Precise Arithmetic" |date=20 November 2004 |format=PDF |accessdate=19 February 2012 }}</ref>
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| Less common IEEE formats include:
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| *[[Quadruple precision]] (binary128). This is a binary format that occupies 128 bits (16 bytes) and its significand has a precision of 113 bits (about 34 decimal digits).
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| *[[Decimal64 floating-point format|Double precision]] (decimal64) and [[Decimal128 floating-point format|quadruple precision]] (decimal128) decimal floating-point formats. These formats, along with the [[Decimal32 floating-point format|single precision]] (decimal32) format, are intended for performing decimal rounding correctly.
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| * [[Half precision|Half]], also called float16, a 16-bit floating-point value.
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| <!--In addition, some platforms use the non-IEEE "double-double" format, where the number is represented as unevaluated sum of two double precision numbers. It can have some strange properties unlike other formats. http://aggregate.org/NPAR/iccs2006.pdf-->
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| Any integer with absolute value less than 2<sup>24</sup> can be exactly represented in the single precision format, and any integer with absolute value less than 2<sup>53</sup> 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 53-bit integers on platforms that have double precision floats but only 32-bit integers.
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| 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 ([[NaN]]s).
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| Comparison of floating-point 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 floating-point numbers are ordered in the same way as their values (in the set of real numbers).
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| To a rough approximation, the bit representation of an IEEE binary floating-point number is proportional to its base 2 logarithm, with an average error of about 3%. (This is because the exponent field is in the more significant part of the datum.) This can be exploited in some applications, such as volume ramping in digital sound processing.
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| A project for revising the IEEE 754 standard was started in 2000 (see [[IEEE 754 revision]]); it was completed and approved in June 2008. It includes decimal floating-point formats and a 16-bit floating-point format ("binary16"). binary16 has the same structure and rules as the older formats, with 1 sign bit, 5 exponent bits and 10 trailing significand bits. It is being used in the NVIDIA [[Cg (programming language)|Cg]] graphics language, and in the openEXR standard.<ref>{{cite web|url=http://www.openexr.com/about.html |title=openEXR |publisher=openEXR |date= |accessdate=25 April 2012}}</ref>
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| ===Internal representation===
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| Floating-point numbers are typically packed into a computer datum as the sign bit, the exponent field, and the significand (mantissa), from left to right. For the IEEE 754 binary formats (basic and extended) which have extant hardware implementations, they are apportioned as follows:
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| {| class="wikitable" style="text-align: right;"
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| |-
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| !Type
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| !Sign
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| !Exponent
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| !Significand
| |
| !Total bits
| |
| !
| |
| !Exponent bias
| |
| !Bits precision
| |
| !Number of decimal digits
| |
| |-
| |
| |[[Half precision|Half]] ([[IEEE floating point|IEEE 754-2008]])
| |
| |1
| |
| |5
| |
| |10
| |
| |16
| |
| |
| |
| |15
| |
| |11
| |
| |~3.3
| |
| |-
| |
| |[[Single precision|Single]]
| |
| |1
| |
| |8
| |
| |23
| |
| |32
| |
| |
| |
| |127
| |
| |24
| |
| |~7.2
| |
| |-
| |
| |[[Double precision|Double]]
| |
| |1
| |
| |11
| |
| |52
| |
| |64
| |
| |
| |
| |1023
| |
| |53
| |
| |~15.9
| |
| |-
| |
| |[[Extended precision|Double extended (80-bit)]]
| |
| |1
| |
| |15
| |
| |64
| |
| |80
| |
| |
| |
| |16383
| |
| |64
| |
| |~19.2
| |
| |-
| |
| |[[Quad precision|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. Normalised 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:
| |
| * sign = 0 ; ''e'' = 1 ; ''s'' = 110010010000111111011011 (including the hidden bit)
| |
| The sum of the exponent bias (127) and the exponent (1) is 128, so this is represented in single precision format as
| |
| * 0 10000000 10010010000111111011011 (excluding the hidden bit) = 40490FDB<ref>http://babbage.cs.qc.edu/IEEE-754/32bit.html</ref> as a [[hexadecimal]] number.
| |
| | |
| ===Special values===
| |
| | |
| ====Signed zero====
| |
| {{Main|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 [[run-time environment]]s, positive zero is usually printed as "0", while negative zero may be printed 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 (exactly), while 1/+0 returns positive infinity (exactly) (so that the identity 1/(1/±∞) = ±∞ is maintained). A sign symmetric arccot operation will give different results for +0 and −0 without any exception. The difference between +0 and −0 is mostly noticeable for complex operations at so-called [[branch cut]]s.
| |
| | |
| ====Subnormal numbers====
| |
| {{Main|Subnormal numbers}}
| |
| Subnormal values fill the [[arithmetic underflow|underflow]] gap with values
| |
| where the absolute distance between them are the same as for
| |
| adjacent values just outside of 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 floating-point hardware usually handles subnormal values (as well as normal values),
| |
| and does not require software emulation for subnormals.
| |
| | |
| ====Infinities====
| |
| | |
| {{details|Infinity| the concept of infinite}}
| |
| | |
| The infinities of the [[extended real number line]] can be represented in IEEE floating-point datatypes,
| |
| just like ordinary floating-point 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 [[arithmetic overflow|overflow]]. Upon a divide-by-zero 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
| |
| | |
| * (+∞) + (+7) = (+∞)
| |
| * (+∞) × (−2) = (−∞)
| |
| * (+∞) × 0 = NaN – there is no meaningful thing to do
| |
| | |
| ====NaNs====
| |
| {{Main|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 floating-point 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" [[exception (computing)|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 uninitialised variables, or extend the floating-point numbers with other special values without slowing down the computations with ordinary values, although such extensions are not common.
| |
| | |
| ====IEEE 754 design rationale====
| |
| [[File:William Kahan.jpg|thumb|500p|right|[[William Kahan]]. A primary architect of the Intel [[80x87]] floating-point coprocessor and [[IEEE 754]] floating-point 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 analysis|numerical analyst]]s, 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|Prof. W. Kahan]] notes that it is incorrect to "... [deem] features of IEEE Standard 754 for Binary Floating- Point 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... . Error-analysis tells us how to design floating-point arithmetic, like IEEE Standard 754, moderately tolerant of well-meaning ignorance among programmers".<ref name=JavaHurt>{{cite web|url=http://www.cs.berkeley.edu/~wkahan/JAVAhurt.pdf|title=How JAVA's Floating-Point Hurts Everyone Everywhere| author=William Kahan |date=1 March 1998}}</ref>
| |
| | |
| *The special values such as infinity and NaN ensure that the floating-point arithmetic is algebraically completed, such that every floating-point operation produces a well-defined result and will not by default throw a machine interrupt or trap. Moreover, the choices of special values returned in exceptional cases were designed to give the correct answer in many cases, e.g. continued fractions such as R(z) := 7 − 3/(z − 2 − 1/(z − 7 + 10/(z − 2 − 2/(z − 3)))) will give the correct answer in all inputs under IEEE-754 arithmetic as the potential divide by zero in e.g. R(3)=4.6 is correctly handled as +infinity and so can be safely ignored.<ref name=whyieee>{{cite web|url=http://www.cs.berkeley.edu/~wkahan/ieee754status/why-ieee.pdf|title=Why do we need a floating-point arithmetic standard?| author=William Kahan |date=12 February 1981}}</ref> As noted by Kahan, the unhandled floating point overflow exception that caused the [[Cluster (spacecraft)|loss of an Ariane 5]] rocket would not have happened under IEEE 754 floating point.<ref name=JavaHurt />
| |
| *Subnormal numbers ensure that for ''finite'' floating-point numbers x and y, x - y == 0 if and only if x == y, as expected, but which did not hold under earlier floating-point representations.<ref>{{cite web|url=http://www.eecs.berkeley.edu/~wkahan/ieee754status/754story.html|title=An Interview with the Old Man of Floating-Point| author=[[Charles Severance]] |date=20 February 1998}}</ref>
| |
| *On the design rationale of the x87 [[Extended precision|80-bit format]], Prof. Kahan notes: "This Extended format is designed to be used, with negligible loss of speed, for all but the simplest arithmetic with float and double operands. For example, it should be used for scratch variables in loops that implement recurrences like polynomial evaluation, scalar products, partial and continued fractions. It often averts premature Over/Underflow or severe local cancellation that can spoil simple algorithms.<ref name=Baleful>{{cite web |url=http://www.cs.berkeley.edu/~wkahan/ieee754status/baleful.pdf |title=The Baleful Effect of Computer Benchmarks upon Applied Mathematics, Physics and Chemistry| author=William Kahan |date=11 June 1996}}</ref> Computing intermediate results in an extended format with high precision and extended exponent has precedents in the historical practice of scientific [[Significant figures#Arithmetic|calculation]] and in the design of scientific calculators e.g. Hewlett- Packard’s financial calculators performed arithmetic and financial functions to three more significant decimals than they stored or displayed.<ref name=Baleful /> The implementation of extended precision enabled standard elementary function libraries to be readily developed that normally gave double precision results within one [[unit in the last place]] (ULP) at high speed.
| |
| *Correct rounding of values to the nearest representable value avoids systematic biases in calculations and slows the growth of errors. Rounding ties to even removes the statistical bias that can occur in adding similar figures.
| |
| *Directed rounding was intended as an aid with checking error bounds, for instance in interval arithmetic. It is also used in the implementation of some functions.
| |
| *The mathematical basis of the operations enabled high precision multiword arithmetic subroutines to be built relatively easily.
| |
| *The single and double precision formats were designed to be easy to sort without using floating-point hardware.
| |
| | |
| ==Representable numbers, conversion and rounding==
| |
| By their nature, all numbers expressed in floating-point format are [[rational number]]s with a terminating expansion in the relevant base (for example, a terminating decimal expansion in base-10, or a terminating binary expansion in base-2). Irrational numbers, such as [[Pi|π]] or √2, or non-terminating 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 number 123456789 cannot be exactly represented if only eight decimal digits of precision are available.
| |
| | |
| When a number is represented in some format (such as a character string) which is not a native floating-point 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 floating-point format then the conversion is exact. If there is not an exact representation then the conversion requires a choice of which floating-point number to use to represent the original value. The representation chosen will have a different value to 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 base-10 the number 1/2 has a terminating expansion (0.5) while the number 1/3 does not (0.333...). In base-2 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 floating-point. For example, the decimal number 0.1 is not representable in binary floating-point of any finite precision; the exact binary representation would have a "1100" sequence continuing endlessly:
| |
| :''e'' = −4; ''s'' = 1100110011001100110011001100110011...,
| |
| where, as previously, ''s'' is the significand and ''e'' is the exponent.
| |
| | |
| When rounded to 24 bits this becomes
| |
| :''e'' = −4; ''s'' = 110011001100110011001101,
| |
| which is actually 0.100000001490116119384765625 in decimal.
| |
| <!-- Edit/rearrange this if you want, but please leave the 0.1 example in. (My previous reference to pi being more "sophisticated" than 0.1 was admittedly not artful.) I have known professional software engineers (who should have known better!) who believed that numbers with short decimal representations could always be represented exactly. Putting the many f.p. fallacies/superstitions to rest is important. -->
| |
| | |
| As a further example, the real number [[Pi|π]], represented in binary as an infinite series of bits is
| |
| :11.0010010000111111011010101000100010000101101000110000100011010011...
| |
| but is
| |
| :11.0010010000111111011011
| |
| when approximated by [[rounding]] to a precision of 24 bits.
| |
| | |
| In binary single-precision floating-point, this is represented as ''s'' = 1.10010010000111111011011 with ''e'' = 1.
| |
| This has a decimal value of
| |
| :'''3.141592'''7410125732421875,
| |
| whereas a more accurate approximation of the true value of π is
| |
| :'''3.14159265358979323846264338327950'''...
| |
| <!-- Before changing the above numbers, please discuss on talk page. Giving the actual value 10 more digits than the single-precision floating-point value is plenty - more digits do not help the reader -->
| |
| 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 floating-point 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<sub>hex</sub> and 1.45a70c24<sub>hex</sub>, the ULP is 2×16<sup>−8</sup>, or 2<sup>−31</sup>. For numbers with a base-2 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<sup>−23</sup> or about 10<sup>−7</sup> in single precision, and exactly 2<sup>−53</sup> or about 10<sup>−16</sup> in double precision. The mandated behavior of IEEE-compliant hardware is that the result be within one-half of a ULP.
| |
| | |
| ===Rounding modes===
| |
| Rounding is used when the exact result of a floating-point operation (or a conversion to floating-point 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|rounding schemes]] (or ''rounding modes''). Historically, [[truncation]] was the typical approach. Since the introduction of IEEE 754, the default method (''[[rounding|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.<ref>Computer hardware doesn't necessarily compute the exact value; it simply has to produce the equivalent rounded result as though it had computed the infinitely precise result.</ref> 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 (non-NaN) 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:
| |
| * round to nearest, where ties round to the nearest even digit in the required position (the default and by far the most common mode)
| |
| * round to nearest, where ties round away from zero (optional for binary floating-point and commonly used in decimal)
| |
| * round up (toward +∞; negative results thus round toward zero)
| |
| * round down (toward −∞; negative results thus round away from zero)
| |
| * round toward zero (truncation; it is similar to the common behavior of float-to-integer conversions, which convert −3.9 to −3 and 3.9 to 3)
| |
| | |
| Alternative modes are useful when the amount of error being introduced must be bounded. Applications that require a bounded error are multi-precision floating-point, 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 round-off error.<ref name>{{cite web|url=http://www.cs.berkeley.edu/~wkahan/Mindless.pdf | title=How Futile are Mindless Assessments of Roundoff in Floating-Point Computation ?| author=William Kahan |date=11 January 2006}}</ref>
| |
| A further use of rounding is when a number is explicitly rounded to a certain number of decimal (or binary) places, as when rounding a result to euros and cents (two decimal places).
| |
| | |
| ==Floating-point arithmetic operations==
| |
| 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.
| |
| | |
| ===Addition and subtraction===
| |
| A simple method to add floating-point numbers is to first represent them with the same exponent. In the example below, the second number is shifted right by three digits, and we then proceed 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)
| |
| | |
| Note that the low three digits of the second operand (654) are essentially lost. This is [[round-off error]]. In extreme cases, the sum of two non-zero 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/normalization)
| |
| | |
| Note that 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: in fact 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.<ref>{{cite web|url=http://www.validlab.com/goldberg/paper.pdf|title=What every computer scientist should know about floating-point arithmetic. ACM Computing Surveys, volume 23, issue 1|author=David Goldberg|date=March 1991 |page=195}}</ref>
| |
| | |
| Another problem of loss of significance occurs when two close 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/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 ''[[loss of significance|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|Accuracy problems]].
| |
| | |
| ===Multiplication and division===
| |
| 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.<ref name=goldberg>{{cite journal|
| |
| first = David| last = Goldberg|
| |
| title = What Every Computer Scientist Should Know About Floating-Point Arithmetic|
| |
| journal = ACM Computing Surveys|
| |
| year = 1991|
| |
| volume = 23|
| |
| pages = 5–48|
| |
| accessdate = 2 September 2010|
| |
| doi = 10.1145/103162.103163|
| |
| url = http://docs.sun.com/source/806-3568/ncg_goldberg.html
| |
| }}</ref> In practice, the way these operations are carried out in digital logic can be quite complex (see [[Booth's multiplication algorithm]] and [[Division algorithm]]).<ref>The enormous complexity of modern division algorithms once led to a famous error. An early version of the Intel Pentium chip was shipped with a division instruction that, on rare occasions, gave slightly incorrect results. Many computers had been shipped before the error was discovered. Until the defective computers were replaced, patched versions of compilers were developed that could avoid the failing cases. See ''[[Pentium FDIV bug]]''.</ref>
| |
| For a fast, simple method, see the [[Horner scheme#Floating point multiplication and division|Horner method]].
| |
| | |
| ==Dealing with exceptional cases== <!-- linked from a couple of places within the article -->
| |
| Floating-point computation in a computer can run into three kinds of problems:
| |
| * An operation can be mathematically undefined, such as ∞/∞, or division by zero.
| |
| * An operation can be legal in principle, but not supported by the specific format, for example, calculating the square root of −1 or the inverse sine of 2 (both of which result in [[complex number]]s).
| |
| * An operation can be legal in principle, but the result can be impossible to represent in the specified format, because the exponent is too large or too small to encode in the exponent field. Such an event is called an [[arithmetic overflow|overflow]] (exponent too large), [[arithmetic underflow|underflow]] (exponent too small) or [[Denormal number|denormalization]] (precision loss).
| |
| | |
| Prior to the IEEE standard, such conditions usually caused the program to terminate, or triggered some kind
| |
| of [[trap (computing)|trap]] that the programmer might be able to catch. How this worked was system-dependent,
| |
| meaning that floating-point programs were not [[porting|portable]]. (Note that 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 handling|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 floating-point expression or subroutine: without them exceptional conditions that could not be otherwise ignored would require explicit testing immediately after every floating-point operation. By default, an operation always returns a result according to specification without interrupting computation. For instance, 1/0 returns +∞, while also setting the divide-by-zero flag bit (this default of ∞ is designed so as 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 [[Concurrency (computer science)|means]] outside of the standard (e.g. [[C11 (C standard revision)|C11]] specifies that the flags have [[thread-local storage]]).
| |
| | |
| IEEE 754 specifies five arithmetic exceptions that are to be recorded in the status flags ("sticky bits"):
| |
| * '''inexact''', set if the rounded (and returned) value is different from the mathematically exact result of the operation.
| |
| * '''underflow''', set if the rounded value is tiny (as specified in IEEE 754) ''and'' inexact (or maybe limited to if it has denormalisation loss, as per the 1984 version of IEEE 754), returning a subnormal value including the zeros.
| |
| * '''overflow''', set if the absolute value of the rounded value is too large to be represented. An infinity or maximal finite value is returned, depending on which rounding is used.
| |
| * '''divide-by-zero''', set if the result is infinite given finite operands, returning an infinity, either +∞ or −∞.
| |
| * '''invalid''', set if a real-valued result cannot be returned e.g. sqrt(−1) or 0/0, returning a quiet NaN.
| |
| [[File:Resistors in Parallel.svg|thumb|left|200px|Fig. 1: resistances in parallel, with total resistance <math>R_{tot}</math>]]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 denormalised small value and so can almost always be ignored.<ref>{{cite web|url=http://www.cs.berkeley.edu/~wkahan/ieee754status/IEEE754.PDF|title=Lecture Notes on the Status of IEEE Standard 754 for Binary Floating-Point Arithmetic|author=William Kahan|date=1 October 1997}}</ref> ''divide-by-zero'' 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 <math>R_{tot}=1/(1/R_1+1/R_2+...+1/R_n)</math>. If a short-circuit develops with <math>R_1</math> set to 0, <math>1/R_1</math> will return +infinity which will give a final <math>R_{tot}</math> of 0, as expected<ref>{{cite web|url=http://www.intel.com/content/www/us/en/processors/architectures-software-developer-manuals.html|title=Intel® 64 and IA-32 Architectures Software Developers' Manuals. Volume 1. section D.3.2.1}}</ref> (see the continued fraction example of [[Floating point#IEEE 754: floating point in modern computers|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 [[Zero of a function|root-finding]] routine, as part of its normal operation, may evaluate a passed-in function at values outside of its domain, returning NaN and an ''invalid'' exception flag to be ignored until finding a useful start point.<ref>{{cite web|url=http://www.cs.berkeley.edu/~wkahan/ieee754status/IEEE754.PDF|title=Lecture Notes on the Status of IEEE Standard 754 for Binary Floating-Point Arithmetic (page 9)|author=William Kahan|date=1 October 1997}}</ref>
| |
| | |
| | |
| | |
| ==Accuracy problems== <!-- internally linked -->
| |
| | |
| The fact that floating-point numbers cannot precisely represent all real numbers, and that floating-point operations cannot precisely represent true arithmetic operations, leads to many surprising situations. This is related to the finite [[Precision (computer science)|precision]] with which computers generally represent numbers.
| |
| | |
| For example, the non-representability 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 24-bit (single precision) representation, 0.1 (decimal) was given previously as ''e'' = −4; ''s'' = 110011001100110011001101, which is
| |
| :0.100000001490116119384765625 exactly.
| |
| Squaring this number gives
| |
| :0.010000000298023226097399174250313080847263336181640625 exactly.
| |
| Squaring it with single-precision floating-point hardware (with rounding) gives
| |
| :0.010000000707805156707763671875 exactly.
| |
| But the representable number closest to 0.01 is
| |
| :0.009999999776482582092285156250 exactly.
| |
| | |
| Also, the non-representability 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 floating-point hardware to attempt to compute tan(π/2), because π/2 cannot be represented exactly. This computation in C:
| |
| <source lang="c">
| |
| /* Enough digits to be sure we get the correct approximation. */
| |
| double pi = 3.1415926535897932384626433832795;
| |
| double z = tan(pi/2.0);
| |
| </source>
| |
| 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{{e|-15}} in double precision, or −0.8742{{e|-7}} in single precision.<ref>But an attempted computation of cos(π) yields −1 exactly. Since the derivative is nearly zero near π, the effect of the inaccuracy in the argument is far smaller than the spacing of the floating-point numbers around −1, and the rounded result is exact.</ref>
| |
| | |
| While floating-point 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 7-digit mantissa 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:
| |
| * [[Loss of significance|Cancellation]]: subtraction of nearly equal operands may cause extreme loss of accuracy.<ref>{{Cite journal
| |
| | title = You're Going To Have To Think!
| |
| | author = Richard Harris
| |
| | journal = [[Overload (magazine)|Overload]]
| |
| | issue = [http://accu.org/var/uploads/journals/overload99.pdf 99]
| |
| |date=October 2010
| |
| | issn = 1354-3172
| |
| | pages = 5–10
| |
| | url = http://accu.org/index.php/journals/1702
| |
| | accessdate =24 September 2011
| |
| | quote = Far more worrying is cancellation error which can yield catastrophic loss of precision.}}</ref> When we subtract two almost equal numbers we set the most significant digits to zero, leaving ourselves with just the insignificant, and most erroneous, digits. For example, when determining a [[derivative]] of a function the following formula is used:
| |
| :<math>Q(h) = \frac{f(a + h) - f(a)}{h}.</math>
| |
| :Intuitively one would want an ''h'' very close to zero, however when using floating-point operations, the smallest number won't give the best approximation of a derivative. As ''h'' grows smaller the difference between f (a + h) and f(a) grows smaller, cancelling out the most significant and least erroneous digits and making the most erroneous digits more important. As a result the smallest number of ''h'' possible will give a more erroneous approximation of a derivative than a somewhat larger number. This is perhaps the most common and serious accuracy problem.
| |
| * Conversions to integer are not intuitive: converting (63.0/9.0) to integer yields 7, but converting (0.63/0.09) may yield 6. This is because conversions generally truncate rather than round. [[Floor and ceiling functions]] may produce answers which are off by one from the intuitively expected value.
| |
| * Limited exponent range: results might overflow yielding infinity, or underflow yielding a [[subnormal number]] or zero. In these cases precision will be lost.
| |
| * Testing for [[Division by zero#In computer arithmetic|safe division]] is problematic: Checking that the divisor is not zero does not guarantee that a division will not overflow.
| |
| * Testing for equality is problematic. Two computational sequences that are mathematically equal may well produce different floating-point values.
| |
| | |
| ===Incidents===
| |
| | |
| * On 25 February 1991, a [[loss of significance]] in a [[MIM-104 Patriot]] missile battery prevented it intercepting an incoming [[Al Hussein (missile)|Scud]] missile in [[Dhahran]], [[Saudi Arabia]], contributing to the death of 28 soldiers from the U.S. Army's [[14th Quartermaster Detachment]].<ref name="GAO report IMTEC 92-26">{{cite web|url=http://www.gao.gov/products/IMTEC-92-26|title=Patriot missile defense, Software problem led to system failure at Dharhan, Saudi Arabia; GAO report IMTEC 92-26|publisher=[[Government Accountability Office|US Government Accounting Office]]}}</ref> See also: [[MIM-104_Patriot#Failure_at_Dhahran|Failure at Dhahran]]
| |
| | |
| ===Machine precision and backward error analysis===
| |
| ''Machine precision'' is a quantity that characterizes the accuracy of a floating-point system, and is used in [[error analysis#Error analysis in numerical modeling|backward error analysis]] of floating-point algorithms. It is also known as unit roundoff or ''[[machine epsilon]]''. Usually denoted Ε<sub>mach</sub>, its value depends on the particular rounding being used.
| |
| | |
| With rounding to zero,
| |
| :<math>\Epsilon_\text{mach} = B^{1-P},\,</math>
| |
| whereas rounding to nearest,
| |
| :<math>\Epsilon_\text{mach} = \tfrac{1}{2} B^{1-P}.</math>
| |
| | |
| This is important since it bounds the ''[[relative error]]'' in representing any non-zero real number x within the normalised range of a floating-point system:
| |
| :<math>\left| \frac{fl(x) - x}{x} \right| \le \Epsilon_\text{mach}.</math>
| |
| | |
| 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.<ref name="RalstonReilly2003">{{cite book|author1=James H. Wilkinson |author2=Anthony Ralston(ed)|author3=Edwin D. Reilly(ed)|author4=David Hemmendinger(ed)|title="Error Analysis" in Encyclopedia of Computer Science. pp. 669-674|url=http://books.google.com/books?id=OLRwQgAACAAJ|accessdate=14 May 2013|date=8 September 2003|publisher=Wiley|isbn=978-0-470-86412-8}}</ref> 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 ''[[numerical stability#Forward, backward, and mixed stability|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.<ref name="Einarsson2005">{{cite book|author=Bo Einarsson|title=Accuracy and reliability in scientific computing|url=http://books.google.com/books?id=sh4orx_qB_QC&pg=PA50|accessdate=14 May 2013|year=2005|publisher=SIAM|isbn=978-0-89871-815-7|pages=50–}}</ref>
| |
| | |
| As a trivial example, consider a simple expression giving the inner product of (length two) vectors <math>x</math> and <math>y</math>, then
| |
| :<math>fl(x \cdot y)=fl(fl(x_1*y_1)+fl(x_2*y_2))</math> where <math>fl()</math> indicates correctly rounded floating-point arithmetic
| |
| :::<math>= fl((x_1*y_1)(1+\delta_1)+(x_2*y_2)(1+\delta_2))</math> where <math>\delta_n \leq \Epsilon_\text{mach}</math>, from above
| |
| :::<math>= ((x_1*y_1)(1+\delta_1)+(x_2*y_2)(1+\delta_2))(1+\delta_3)</math>
| |
| :::<math>= (x_1*y_1)(1+\delta_1)(1+\delta_3)+(x_2*y_2)(1+\delta_2)(1+\delta_3)</math>
| |
| and so
| |
| :<math>fl(x \cdot y)=\hat{x} \cdot \hat{y}</math> where
| |
| :<math>\hat{x}_1 = x_1(1+\delta_1)</math>; <math>\hat{x}_2=x_2(1+\delta_2)</math>;
| |
| :<math>\hat{y}_1 = y_1(1+\delta_3)</math>; <math>\hat{y}_2 = y_2(1+\delta_3)</math>
| |
| :where <math>\delta_n \leq \Epsilon_\text{mach}</math>, by definition
| |
| which is the sum of two slightly perturbed (on the order of Ε<sub>mach</sub>) input data, and so is backward stable. For more realistic examples in [[numerical linear algebra]] see Higham 2002 and other references below).
| |
| | |
| ===Minimizing the effect of accuracy problems===
| |
| 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 round-off. The loss of accuracy can be substantial if a problem or its data are [[Condition number|ill-conditioned]], meaning that the correct result is hypersensitive to tiny perturbations in its data. However, even functions that are well-conditioned can suffer from large loss of accuracy if an algorithm [[Numerical stability|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,<ref>Kahan estimates that the incidence of excessively inaccurate results near singularities is reduced by a factor of approx. 1/2000 uisng the 11 extra bits of precision of [[extended precision|double extended]]- {{cite web|author=[[William Kahan]]|title="Floating-Point Arithmetic Besieged by "Business Decisions". Keynote Address, IEEE-Sponsored ARITH 17 Symposium on Computer Arithmetic,.p. 18|url=http://www.cs.berkeley.edu/~wkahan/ARITH_17.pdf|date=15 July 2005|accessdate=23 May 2013 }}</ref> such risk: [[Quadruple-precision floating-point format|IEEE 754 quadruple precision]] and [[extended precision]] are designed for this purpose when computing at double precision.<ref name=debug>{{cite web|url=http://www.eecs.berkeley.edu/~wkahan/Boulder.pdf|title=Desperately Needed Remedies for the Undebuggability of Large Floating-Point Computations in Science and Engineering. IFIP/SIAM/NIST Working Conference on Uncertainty Quantification in Scientific Computing. Boulder CO.|page=33| author=William Kahan |date=3 August 2011}}</ref><ref>Kahan notes: "Except in extremely uncommon situations, extra-precise arithmetic generally attenuates risks due to roundoff at far less cost than the price of a competent error-analyst."</ref>
| |
| | |
| For example, the following algorithm is a direct implementation to compute the function A(x) = (x–1)/( exp(x–1) – 1) which is well-conditioned at 1.0,<ref>note: the Taylor expansion of this function demonstrates that it is well-conditioned near 1: A(x) = 1 – (x–1)/2 + (x–1)^2/12 – (x–1)^4/720 + (x–1)^6/30240 – (x–1)^8/1209600 + ... for |x–1| < π</ref> 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.<ref name=JavaHurt />
| |
| <source lang="c">
| |
| double A(double X)
| |
| {
| |
| double Y, Z; // [1]
| |
| Y = X - 1.0;
| |
| Z = exp(Y);
| |
| if (Z != 1.0) Z = Y/(Z - 1.0); // [2]
| |
| return(Z);
| |
| }
| |
| </source>
| |
| | |
| 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.<ref>if long double is IEEE quad precision then full double precision is retained; if long double is IEEE double extended precision then additional, but not full, precision is retained</ref> Alternatively, a numerical analysis of the algorithm reveals that if the following non-obvious change to line [2] is made:
| |
| <source lang="c">
| |
| if (Z != 1.0) Z = log(Z)/(Z - 1.0);
| |
| </source>
| |
| 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 well-behaved 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 optimisations are carefully specified so as to maintain numerical precision. See the external references at the bottom of this article.
| |
| | |
| A detailed treatment of the techniques for writing high-quality floating-point software is beyond the scope of this article, and the reader is referred to,<ref>{{cite book|first=Nicholas | last=Higham |title="Designing stable algorithms" in Accuracy and Stability of Numerical Algorithms (2 ed)| publisher=SIAM|year=2002 | pages=27–28 }}</ref><ref name=kahan_maths>{{cite web|url=http://www.cs.berkeley.edu/~wkahan/MktgMath.pdf |title="Four Rules of Thumb for Best Use of Modern Floating-point Hardware" in Marketing versus Mathematics |author=William Kahan|page=47}}</ref> and the other references at the bottom of this article. Kahan suggests several rules of thumb that can substantially decrease by orders of magnitude<ref name=kahan_maths>{{cite web|url=http://www.cs.berkeley.edu/~wkahan/MktgMath.pdf |title=Marketing versus Mathematics |author=William Kahan|page=35}}</ref> 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<ref>{{cite web|url=http://www.cs.berkeley.edu/~wkahan/ieee754status/why-ieee.pdf|title=Why do we need a floating-point arithmetic standard? (page 26)| author=William Kahan |date=12 February 1981}}</ref>); 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:<ref>{{cite web|url=http://www.cims.nyu.edu/~dbindel/class/cs279/notes-06-04.pdf|author=William Kahan (transcribed by David Bindel)|title=Lecture notes of System Support for Scientific Computation|date=4 June 2001}}</ref> 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 floating-point, such arithmetic is at its best when it is simply being used to measure real-world 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.<ref>{{cite web |url=http://www.cs.berkeley.edu/~wkahan/MktgMath.pdf |title=Marketing versus Mathematics (p 15)| author=Prof. W. Kahan |date=27 August 2000}}</ref><ref>{{cite web |url=http://www.cs.berkeley.edu/~wkahan/ARITH_17.pdf |title=Floating-Point Arithmetic Besieged by "Business Decisions": Keynote Address for the IEEE-Sponsored ARITH 17 Symposium on Computer Arithmetic | author=Prof. W. Kahan |date=5 July 2005|page=6}}</ref> An example of the latter case is financial calculations. For this reason, financial software tends not to use a binary floating-point number representation.<ref>{{cite web|url=http://speleotrove.com/decimal/ |title=General Decimal Arithmetic |publisher=Speleotrove.com |date= |accessdate=25 April 2012}}</ref> The "decimal" data type of the [[C Sharp (programming language)|C#]] and [[Python (programming language)|Python]] programming languages, and the [[IEEE 754-2008]] decimal floating-point standard, are designed to avoid the problems of binary floating-point representations when applied to human-entered exact decimal values, and make the arithmetic always behave as expected when numbers are printed in decimal.
| |
| | |
| Expectations from mathematics may not be realised in the field of floating-point computation. For example, it is known that <math>(x+y)(x-y) = x^2-y^2\,</math>, and that <math>\sin^2{\theta}+\cos^2{\theta} = 1\,</math>, however these facts cannot be relied on when the quantities involved are the result of floating-point computation.
| |
| | |
| The use of the equality test (<code>if (x==y) ...</code>) requires care when dealing with floating-point numbers. Even simple expressions like <code>0.6/0.2-3==0</code> will, on most computers, fail to be true<ref>{{cite web |url=http://perldoc.perl.org/5.8.8/perlfaq4.html#Why-is-int%28%29-broken? |title=perlfaq4 / Why is int() broken? |author=Tom Christiansen, Nathan Torkington, and others |year=2006 |publisher=perldoc.perl.org |accessdate=11 January 2011}}</ref> (in IEEE 754 double precision, for example, <code>0.6/0.2-3</code> is approximately equal to -4.44089209850063e-16). Consequently, such tests are sometimes replaced with "fuzzy" comparisons (<code>if (abs(x-y) < epsilon) ...</code>, where [[machine epsilon|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.<ref>{{cite book|first=Nicholas | last=Higham |title="Subtleties of floating point arithmetic" in Accuracy and Stability of Numerical Algorithms (2 ed)| publisher=SIAM|year=2002 | page=493 }}</ref> Values derived from the primary data representation and their comparisons should be performed in a wider, extended, precision to minimise the risk of such inconsistencies due to round-off errors.<ref name=kahan_maths /> 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.<ref>{{cite journal|url=http://www.cs.cmu.edu/~quake/robust.html|title=Adaptive Precision Floating-Point Arithmetic and Fast Robust Geometric Predicates, Discrete & Computational Geometry 18:305-363| author=Jonathan Richard Shewchuk |year=1997}}</ref>
| |
| | |
| Small errors in floating-point 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.<ref>{{cite web |url=http://www.cs.berkeley.edu/~wkahan/Cantilever.pdf |title=Roundoff Degrades an Idealized Cantilever| author=Prof. W. Kahan and Ms. Melody Y. Ivory |date=3 July 1997}}</ref>
| |
| | |
| Summation of a vector of floating-point values is a basic algorithm in [[Computational science|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.<ref>{{cite book|first=Nicholas | last=Higham |title=Summation in "Subtleties of floating point arithmetic" in Accuracy and Stability of Numerical Algorithms (2 ed)| publisher=SIAM|year=2002 | pages=110–123 }}</ref>
| |
| | |
| Round-off 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:
| |
| | |
| :<math>t_0 = \frac{1}{\sqrt{3}}</math>
| |
| | |
| :<math>\qquad\mathrm{first\ form:}\qquad t_{i+1} = \frac{\sqrt{t_i^2+1}-1}{t_i}\qquad\mathrm{second\ form:}\qquad t_{i+1} = \frac{t_i}{\sqrt{t_i^2+1}+1}</math>
| |
| | |
| :<math>\pi \sim 6 \times 2^i \times t_i,\qquad\mathrm{converging\ as\ i \rightarrow \infty}\,</math>
| |
| | |
| Here is a computation using IEEE "double" (a significand with 53 bits of precision) arithmetic:
| |
| | |
| i 6 × 2<sup>i</sup> × t<sub>i</sub>, first form 6 × 2<sup>i</sup> × t<sub>i</sub>, second form
| |
| | |
| 0 '''{{Fontcolor|purple|3}}'''.4641016151377543863 '''{{Fontcolor|purple|3}}'''.4641016151377543863
| |
| 1 '''{{Fontcolor|purple|3}}'''.2153903091734710173 '''{{Fontcolor|purple|3}}'''.2153903091734723496
| |
| 2 '''{{Fontcolor|purple|3.1}}'''596599420974940120 '''{{Fontcolor|purple|3.1}}'''596599420975006733
| |
| 3 '''{{Fontcolor|purple|3.14}}'''60862151314012979 '''{{Fontcolor|purple|3.14}}'''60862151314352708
| |
| 4 '''{{Fontcolor|purple|3.14}}'''27145996453136334 '''{{Fontcolor|purple|3.14}}'''27145996453689225
| |
| 5 '''{{Fontcolor|purple|3.141}}'''8730499801259536 '''{{Fontcolor|purple|3.141}}'''8730499798241950
| |
| 6 '''{{Fontcolor|purple|3.141}}'''6627470548084133 '''{{Fontcolor|purple|3.141}}'''6627470568494473
| |
| 7 '''{{Fontcolor|purple|3.141}}'''6101765997805905 '''{{Fontcolor|purple|3.141}}'''6101766046906629
| |
| 8 '''{{Fontcolor|purple|3.14159}}'''70343230776862 '''{{Fontcolor|purple|3.14159}}'''70343215275928
| |
| 9 '''{{Fontcolor|purple|3.14159}}'''37488171150615 '''{{Fontcolor|purple|3.14159}}'''37487713536668
| |
| 10 '''{{Fontcolor|purple|3.141592}}'''9278733740748 '''{{Fontcolor|purple|3.141592}}'''9273850979885
| |
| 11 '''{{Fontcolor|purple|3.141592}}'''7256228504127 '''{{Fontcolor|purple|3.141592}}'''7220386148377
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| 12 '''{{Fontcolor|purple|3.1415926}}'''717412858693 '''{{Fontcolor|purple|3.1415926}}'''707019992125
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| 13 '''{{Fontcolor|purple|3.1415926}}'''189011456060 '''{{Fontcolor|purple|3.14159265}}'''78678454728
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| 14 '''{{Fontcolor|purple|3.1415926}}'''717412858693 '''{{Fontcolor|purple|3.14159265}}'''46593073709
| |
| 15 '''{{Fontcolor|purple|3.14159}}'''19358822321783 '''{{Fontcolor|purple|3.141592653}}'''8571730119
| |
| 16 '''{{Fontcolor|purple|3.1415926}}'''717412858693 '''{{Fontcolor|purple|3.141592653}}'''6566394222
| |
| 17 '''{{Fontcolor|purple|3.1415}}'''810075796233302 '''{{Fontcolor|purple|3.141592653}}'''6065061913
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| 18 '''{{Fontcolor|purple|3.1415926}}'''717412858693 '''{{Fontcolor|purple|3.1415926535}}'''939728836
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| 19 '''{{Fontcolor|purple|3.141}}'''4061547378810956 '''{{Fontcolor|purple|3.1415926535}}'''908393901
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| 20 '''{{Fontcolor|purple|3.14}}'''05434924008406305 '''{{Fontcolor|purple|3.1415926535}}'''900560168
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| 21 '''{{Fontcolor|purple|3.14}}'''00068646912273617 '''{{Fontcolor|purple|3.141592653589}}'''8608396
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| 22 '''{{Fontcolor|purple|3.1}}'''349453756585929919 '''{{Fontcolor|purple|3.141592653589}}'''8122118
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| 23 '''{{Fontcolor|purple|3.14}}'''00068646912273617 '''{{Fontcolor|purple|3.14159265358979}}'''95552
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| 24 '''{{Fontcolor|purple|3}}'''.2245152435345525443 '''{{Fontcolor|purple|3.14159265358979}}'''68907
| |
| 25 '''{{Fontcolor|purple|3.14159265358979}}'''62246
| |
| 26 '''{{Fontcolor|purple|3.14159265358979}}'''62246
| |
| 27 '''{{Fontcolor|purple|3.14159265358979}}'''62246
| |
| 28 '''{{Fontcolor|purple|3.14159265358979}}'''62246
| |
| The true value is '''{{Fontcolor|purple|3.14159265358979323846264338327...}}'''
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| | |
| While the two forms of the recurrence formula are clearly mathematically equivalent,<ref>The equivalence of the two forms can be verified algebraically by noting that the [[denominator]] of the fraction in the second form is the [[conjugate (algebra)|conjugate]] of the [[numerator]] of the first. By multiplying the top and bottom of the first expression by this conjugate, one obtains the second expression.</ref> the first subtracts 1 from a number extremely close to 1, leading to an increasingly problematic loss of [[significant digit]]s. 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 53-bit 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.
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| | |
| ==See also==
| |
| {{Portal|Computer Science}}
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| <div style="column-count:3;-moz-column-count:3;-webkit-column-count:3">
| |
| * [[C99#IEEE 754 floating point support|C99]] for code examples demonstrating access and use of IEEE 754 features.
| |
| * [[Computable number]]
| |
| * [[Coprocessor]]
| |
| * [[Decimal floating point]]
| |
| * [[Double precision]]
| |
| * [[Experimental mathematics]]—utilises high precision floating-point computations
| |
| * [[Fixed-point arithmetic]]
| |
| * [[FLOPS]]
| |
| * [[Gal's accurate tables]]
| |
| * [[GNU Multi-Precision Library]]
| |
| * [[Half precision]]
| |
| * [[IEEE 754]] — Standard for Binary Floating-Point Arithmetic
| |
| * [[IBM Floating Point Architecture]]
| |
| * [[Kahan summation algorithm]]
| |
| * [[Microsoft Binary Format]]
| |
| * [[Minifloat]]
| |
| * [[Q (number format)]] for constant resolution
| |
| * [[Quad precision]]
| |
| * [[Significant digits]]
| |
| * [[Single precision]]
| |
| </div>
| |
| | |
| ==Notes and references==
| |
| {{Reflist|30em}}
| |
| | |
| ==Further reading==
| |
| * ''[http://download.oracle.com/docs/cd/E19422-01/819-3693/ncg_goldberg.html What Every Computer Scientist Should Know About Floating-Point Arithmetic]'', by David Goldberg, published in the March, 1991 issue of Computing Surveys.
| |
| * Nicholas Higham. ''Accuracy and Stability of Numerical Algorithms'', Second Edition. SIAM, 2002. ISBN 0-89871-355-2.
| |
| * Gene F. Golub and Charles F. van Loan. ''Matrix Computations'', Third Edition. Johns Hopkins University Press, 1986. ISBN 0-8018-5413.
| |
| * [[Donald Knuth]]. ''The Art of Computer Programming'', Volume 2: ''Seminumerical Algorithms'', Third Edition. Addison-Wesley, 1997. ISBN 0-201-89684-2. Section 4.2: Floating Point Arithmetic, pp. 214–264.
| |
| * Press et al. ''[[Numerical Recipes]] in [[C++]]. The Art of Scientific Computing'', ISBN 0-521-75033-4.
| |
| * [[James H. Wilkinson]]. ''Rounding errors in algebraic processes''. 1963. -- Classic influential treatises on floating point arithmetic.
| |
| * [[James H. Wilkinson]]. ''The Algebraic Eigenvalue Problem'', Clarendon Press, 1965.
| |
| * P.H. Sterbenz. ''Floating point computation''. 1974. -- Another classic book on floating point and [[error analysis#Error analysis in numerical modeling|error analysis]].
| |
| * Jean-Michel Muller, Nicolas Brisebarre, Florent de Dinechin, Claude-Pierre Jeannerod, Vincent Lefèvre, Guillaume Melquiond, Nathalie Revol, Damien Stehlé, and Serge Torres. ''Handbook of Floating-Point Arithmetic''. 2010. ISBN 978-0-8176-4705-6.
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| ==External links==
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| * Kahan, William and Darcy, Joseph (2001). [http://www.cs.berkeley.edu/~wkahan/JAVAhurt.pdf How Java's floating-point hurts everyone everywhere]. Retrieved 5 September 2003.
| |
| * [http://www.mrob.com/pub/math/floatformats.html Survey of Floating-Point Formats] This page gives a very brief summary of floating-point formats that have been used over the years.
| |
| * ''[http://hal.archives-ouvertes.fr/hal-00128124/en/ The pitfalls of verifying floating-point computations]'', by David Monniaux, also printed in ''[[Association for Computing Machinery|ACM]] Transactions on programming languages and systems (TOPLAS)'', May 2008: a compendium of non-intuitive behaviors of floating point on popular architectures, with implications for program verification and testing
| |
| * http://www.opencores.org The OpenCores website contains open source floating-point IP cores for the implementation of floating-point operators in FPGA or ASIC devices. The project, double_fpu, contains verilog source code of a double precision floating-point unit. The project, fpuvhdl, contains vhdl source code of a single precision floating-point unit.
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| * http://msdn.microsoft.com/en-us/library/aa289157(v=vs.71).aspx "Microsoft Visual C++ Floating-Point Optimization", by Eric Fleegal, MSDN, 2004
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| {{Data types}}
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| {{Use dmy dates|date=May 2012}}
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| {{DEFAULTSORT:Floating Point}}
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| [[Category:Data types]]
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| [[Category:Computer arithmetic]]
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| [[Category:Primitive types]]
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