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IEEE 754-1985 is a historic industry standard for representing
floating-point In computing, floating-point arithmetic (FP) is arithmetic on subsets of real numbers formed by a ''significand'' (a Sign (mathematics), signed sequence of a fixed number of digits in some Radix, base) multiplied by an integer power of that ba ...
numbers in
computers A computer is a machine that can be programmed to automatically carry out sequences of arithmetic or logical operations ('' computation''). Modern digital electronic computers can perform generic sets of operations known as ''programs'', ...
, officially adopted in 1985 and superseded in 2008 by
IEEE 754-2008 The Institute of Electrical and Electronics Engineers (IEEE) is an American 501(c)(3) public charity professional organization for electrical engineering, electronics engineering, and other related disciplines. The IEEE has a corporate office ...
, and then again in 2019 by minor revision IEEE 754-2019. During its 23 years, it was the most widely used format for floating-point computation. It was implemented in software, in the form of floating-point
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, and in hardware, in the instructions of many CPUs and FPUs. The first
integrated circuit An integrated circuit (IC), also known as a microchip or simply chip, is a set of electronic circuits, consisting of various electronic components (such as transistors, resistors, and capacitors) and their interconnections. These components a ...
to implement the draft of what was to become IEEE 754-1985 was the
Intel 8087 The Intel 8087, announced in 1980, was the first floating-point coprocessor for the 8086 line of microprocessors. The purpose of the chip was to speed up floating-point arithmetic operations, such as addition, subtraction, multiplication, div ...
. IEEE 754-1985 represents numbers in binary, providing definitions for four levels of precision, of which the two most commonly used are: The standard also defines representations for positive and negative
infinity Infinity is something which is boundless, endless, or larger than any natural number. It is denoted by \infty, called the infinity symbol. From the time of the Ancient Greek mathematics, ancient Greeks, the Infinity (philosophy), philosophic ...
, a "
negative zero Signed zero is zero with an associated sign. In ordinary arithmetic, the number 0 does not have a sign, so that −0, +0 and 0 are equivalent. However, in computing, some number representations allow for the existence of two zeros, often denoted by ...
", five exceptions to handle invalid results like
division by zero In mathematics, division by zero, division (mathematics), division where the divisor (denominator) is 0, zero, is a unique and problematic special case. Using fraction notation, the general example can be written as \tfrac a0, where a is the di ...
, special values called NaNs for representing those exceptions, denormal numbers to represent numbers smaller than shown above, and four
rounding Rounding or rounding off is the process of adjusting a number to an approximate, more convenient value, often with a shorter or simpler representation. For example, replacing $ with $, the fraction 312/937 with 1/3, or the expression √2 with ...
modes.


Representation of numbers

Floating-point numbers in IEEE 754 format consist of three fields: a sign bit, a biased exponent, and a fraction. The following example illustrates the meaning of each. The decimal number 0.1562510 represented in binary is 0.001012 (that is, 1/8 + 1/32). (Subscripts indicate the number base.) Analogous to
scientific notation Scientific notation is a way of expressing numbers that are too large or too small to be conveniently written in decimal form, since to do so would require writing out an inconveniently long string of digits. It may be referred to as scientif ...
, where numbers are written to have a single non-zero digit to the left of the decimal point, we rewrite this number so it has a single 1 bit to the left of the "binary point". We simply multiply by the appropriate power of 2 to compensate for shifting the bits left by three positions: : 0.00101_2 = 1.01_2 \times 2^ Now we can read off the fraction and the exponent: the fraction is .012 and the exponent is −3. As illustrated in the pictures, the three fields in the IEEE 754 representation of this number are: : ''sign'' = 0, because the number is positive. (1 indicates negative.) : ''biased exponent'' = −3 + the "bias". In single precision, the bias is 127, so in this example the biased exponent is 124; in double precision, the bias is 1023, so the biased exponent in this example is 1020. : ''fraction'' = .01000…2. IEEE 754 adds a
bias Bias is a disproportionate weight ''in favor of'' or ''against'' an idea or thing, usually in a way that is inaccurate, closed-minded, prejudicial, or unfair. Biases can be innate or learned. People may develop biases for or against an individ ...
to the exponent so that numbers can in many cases be compared conveniently by the same hardware that compares signed 2's-complement integers. Using a biased exponent, the lesser of two positive floating-point numbers will come out "less than" the greater following the same ordering as for sign and magnitude integers. If two floating-point numbers have different signs, the sign-and-magnitude comparison also works with biased exponents. However, if both biased-exponent floating-point numbers are negative, then the ordering must be reversed. If the exponent were represented as, say, a 2's-complement number, comparison to see which of two numbers is greater would not be as convenient. The leading 1 bit is omitted since all numbers except zero start with a leading 1; the leading 1 is implicit and doesn't actually need to be stored which gives an extra bit of precision for "free."


Zero

The number zero is represented specially: : ''sign'' = 0 for positive zero, 1 for
negative zero Signed zero is zero with an associated sign. In ordinary arithmetic, the number 0 does not have a sign, so that −0, +0 and 0 are equivalent. However, in computing, some number representations allow for the existence of two zeros, often denoted by ...
. : ''biased exponent'' = 0. : ''fraction'' = 0.


Denormalized numbers

The number representations described above are called ''normalized,'' meaning that the implicit leading binary digit is a 1. To reduce the loss of precision when an underflow occurs, IEEE 754 includes the ability to represent fractions smaller than are possible in the normalized representation, by making the implicit leading digit a 0. Such numbers are called denormal. They don't include as many
significant digits Significant figures, also referred to as significant digits, are specific digits within a number that is written in positional notation that carry both reliability and necessity in conveying a particular quantity. When presenting the outcom ...
as a normalized number, but they enable a gradual loss of precision when the result of an operation is not exactly zero but is too close to zero to be represented by a normalized number. A denormal number is represented with a biased exponent of all 0 bits, which represents an exponent of −126 in single precision (not −127), or −1022 in double precision (not −1023). In contrast, the smallest biased exponent representing a normal number is 1 (see
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below).


Representation of non-numbers

The biased-exponent field is filled with all 1 bits to indicate either infinity or an invalid result of a computation.


Positive and negative infinity

Positive and negative infinity are represented thus: : ''sign'' = 0 for positive infinity, 1 for negative infinity. : ''biased exponent'' = all 1 bits. : ''fraction'' = all 0 bits.


NaN

Some operations of
floating-point arithmetic In computing, floating-point arithmetic (FP) is arithmetic on subsets of real numbers formed by a ''significand'' (a Sign (mathematics), signed sequence of a fixed number of digits in some Radix, base) multiplied by an integer power of that ba ...
are invalid, such as taking the square root of a negative number. The act of reaching an invalid result is called a floating-point ''exception.'' An exceptional result is represented by a special code called a NaN, for " Not a Number". All NaNs in IEEE 754-1985 have this format: : ''sign'' = either 0 or 1. : ''biased exponent'' = all 1 bits. : ''fraction'' = anything except all 0 bits (since all 0 bits represents infinity).


Range and precision

Precision is defined as the minimum difference between two successive mantissa representations; thus it is a function only in the mantissa; while the gap is defined as the difference between two successive numbers.


Single precision

Single-precision numbers occupy 32 bits. In single precision: * The positive and negative numbers closest to zero (represented by the denormalized value with all 0s in the exponent field and the binary value 1 in the fraction field) are *: ±2−23 × 2−126 ≈ ±1.40130 * The positive and negative normalized numbers closest to zero (represented with the binary value 1 in the exponent field and 0 in the fraction field) are *: ±1 × 2−126 ≈ ±1.17549 * The finite positive and finite negative numbers furthest from zero (represented by the value with 254 in the exponent field and all 1s in the fraction field) are *: ±(2−2−23) × 2127 ≈ ±3.40282 Some example range and gap values for given exponents in single precision: As an example, 16,777,217 cannot be encoded as a 32-bit float as it will be rounded to 16,777,216. However, all integers within the representable range that are a power of 2 can be stored in a 32-bit float without rounding.


Double precision

Double-precision Double-precision floating-point format (sometimes called FP64 or float64) is a floating-point number format, usually occupying 64 bits in computer memory; it represents a wide range of numeric values by using a floating radix point. Double prec ...
numbers occupy 64 bits. In double precision: * The positive and negative numbers closest to zero (represented by the denormalized value with all 0s in the Exp field and the binary value 1 in the Fraction field) are *: ±2−52 × 2−1022 ≈ ±4.94066 * The positive and negative normalized numbers closest to zero (represented with the binary value 1 in the Exp field and 0 in the fraction field) are *: ±1 × 2−1022 ≈ ±2.22507 * The finite positive and finite negative numbers furthest from zero (represented by the value with 2046 in the Exp field and all 1s in the fraction field) are *: ±(2−2−52) × 21023 ≈ ±1.79769 Some example range and gap values for given exponents in double precision:


Extended formats

The standard also recommends extended format(s) to be used to perform internal computations at a higher precision than that required for the final result, to minimise round-off errors: the standard only specifies minimum precision and exponent requirements for such formats. The x87 80-bit extended format is the most commonly implemented extended format that meets these requirements.


Examples

Here are some examples of single-precision IEEE 754 representations:


Comparing floating-point numbers

Every possible bit combination is either a NaN or a number with a unique value in the
affinely extended real number system In mathematics, the extended real number system is obtained from the real number system \R by adding two elements denoted +\infty and -\infty that are respectively greater and lower than every real number. This allows for treating the potential ...
with its associated order, except for the two combinations of bits for negative zero and positive zero, which sometimes require special attention (see below). The
binary representation A binary number is a number expressed in the base-2 numeral system or binary numeral system, a method for representing numbers that uses only two symbols for the natural numbers: typically "0" (zero) and "1" (one). A ''binary number'' may also ...
has the special property that, excluding NaNs, any two numbers can be compared as sign and magnitude integers (
endianness file:Gullivers_travels.jpg, ''Gulliver's Travels'' by Jonathan Swift, the novel from which the term was coined In computing, endianness is the order in which bytes within a word (data type), word of digital data are transmitted over a data comm ...
issues apply). When comparing as 2's-complement integers: If the sign bits differ, the negative number precedes the positive number, so 2's complement gives the correct result (except that negative zero and positive zero should be considered equal). If both values are positive, the 2's complement comparison again gives the correct result. Otherwise (two negative numbers), the correct FP ordering is the opposite of the 2's complement ordering. Rounding errors inherent to floating point calculations may limit the use of comparisons for checking the exact equality of results. Choosing an acceptable range is a complex topic. A common technique is to use a comparison epsilon value to perform approximate comparisons. Depending on how lenient the comparisons are, common values include 1e-6 or 1e-5 for single-precision, and 1e-14 for double-precision. Another common technique is ULP, which checks what the difference is in the last place digits, effectively checking how many steps away the two values are. Although negative zero and positive zero are generally considered equal for comparison purposes, some
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s and similar constructs treat them as distinct. According to the
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Language Specification, comparison and equality operators treat them as equal, but Math.min() and Math.max() distinguish them (officially starting with Java version 1.1 but actually with 1.1.1), as do the comparison methods equals(), compareTo() and even compare() of classes Float and Double.


Rounding floating-point numbers

The IEEE standard has four different rounding modes; the first is the default; the others are called '' directed roundings''. * Round to Nearest – rounds to the nearest value; if the number falls midway it is rounded to the nearest value with an even (zero) least significant bit, which means it is rounded up 50% of the time (in
IEEE 754-2008 The Institute of Electrical and Electronics Engineers (IEEE) is an American 501(c)(3) public charity professional organization for electrical engineering, electronics engineering, and other related disciplines. The IEEE has a corporate office ...
this mode is called ''roundTiesToEven'' to distinguish it from another round-to-nearest mode) * Round toward 0 – directed rounding towards zero * Round toward +∞ – directed rounding towards positive infinity * Round toward −∞ – directed rounding towards negative infinity.


Extending the real numbers

The IEEE standard employs (and extends) the
affinely extended real number system In mathematics, the extended real number system is obtained from the real number system \R by adding two elements denoted +\infty and -\infty that are respectively greater and lower than every real number. This allows for treating the potential ...
, with separate positive and negative infinities. During drafting, there was a proposal for the standard to incorporate the projectively extended real number system, with a single unsigned infinity, by providing programmers with a mode selection option. In the interest of reducing the complexity of the final standard, the projective mode was dropped, however. The
Intel 8087 The Intel 8087, announced in 1980, was the first floating-point coprocessor for the 8086 line of microprocessors. The purpose of the chip was to speed up floating-point arithmetic operations, such as addition, subtraction, multiplication, div ...
and Intel 80287 floating point co-processors both support this projective mode.


Functions and predicates


Standard operations

The following functions must be provided: * Add, subtract, multiply, divide *
Square root In mathematics, a square root of a number is a number such that y^2 = x; in other words, a number whose ''square'' (the result of multiplying the number by itself, or y \cdot y) is . For example, 4 and −4 are square roots of 16 because 4 ...
*Floating point remainder. This is not like a normal
modulo operation In computing and mathematics, the modulo operation returns the remainder or signed remainder of a Division (mathematics), division, after one number is divided by another, the latter being called the ''modular arithmetic, modulus'' of the operatio ...
, it can be negative for two positive numbers. It returns the exact value of . * Round to nearest integer. For undirected rounding when halfway between two integers the even integer is chosen. *Comparison operations. Besides the more obvious results, IEEE 754 defines that −∞ = −∞, +∞ = +∞ and x ≠ NaN for any x (including NaN).


Recommended functions and predicates

* copysign(x,y) returns x with the sign of y, so abs(x) equals copysign(x,1.0). This is one of the few operations which operates on a NaN in a way resembling arithmetic. The function copysign is new in the C99 standard. * −x returns x with the sign reversed. This is different from 0−x in some cases, notably when x is 0. So −(0) is −0, but the sign of 0−0 depends on the rounding mode. * scalb(y, N) * logb(x) * finite(x) a predicate for "x is a finite value", equivalent to −Inf < x < Inf * isnan(x) a predicate for "x is a NaN", equivalent to "x ≠ x" * x <> y (x .LG. y in Fortran), which turns out to have different behavior than NOT(x = y) (x .NE. y in Fortran, x != y in C) due to NaN. * unordered(x, y) is true when "x is unordered with y", i.e., either x or y is a NaN. * class(x) * nextafter(x,y) returns the next representable value from x in the direction towards y


History

In 1976,
Intel Intel Corporation is an American multinational corporation and technology company headquartered in Santa Clara, California, and Delaware General Corporation Law, incorporated in Delaware. Intel designs, manufactures, and sells computer compo ...
was starting the development of a floating-point
coprocessor A coprocessor is a computer processor used to supplement the functions of the primary processor (the CPU). Operations performed by the coprocessor may be floating-point arithmetic, graphics, signal processing, string processing, cryptography or ...
. Intel hoped to be able to sell a chip containing good implementations of all the operations found in the widely varying maths software libraries. John Palmer, who managed the project, believed the effort should be backed by a standard unifying floating point operations across disparate processors. He contacted
William Kahan William "Velvel" Morton Kahan (born June 5, 1933) is a Canadian mathematician and computer scientist, who is a professor emeritus at University of California, Berkeley. He received the Turing Award in 1989 for "his fundamental contributions to nu ...
of the
University of California The University of California (UC) is a public university, public Land-grant university, land-grant research university, research university system in the U.S. state of California. Headquartered in Oakland, California, Oakland, the system is co ...
, who had helped improve the accuracy of
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's calculators. Kahan suggested that Intel use the floating point of
Digital Equipment Corporation Digital Equipment Corporation (DEC ), using the trademark Digital, was a major American company in the computer industry from the 1960s to the 1990s. The company was co-founded by Ken Olsen and Harlan Anderson in 1957. Olsen was president until ...
's (DEC) VAX. The first VAX, the
VAX-11/780 The VAX-11 is a discontinued family of 32-bit superminicomputers, running the Virtual Address eXtension (VAX) instruction set architecture (ISA), developed and manufactured by Digital Equipment Corporation (DEC). Development began in 1976. In ad ...
had just come out in late 1977, and its floating point was highly regarded. However, seeking to market their chip to the broadest possible market, Intel wanted the best floating point possible, and Kahan went on to draw up specifications. Kahan initially recommended that the floating point base be decimal but the hardware design of the coprocessor was too far along to make that change. The work within Intel worried other vendors, who set up a standardization effort to ensure a "level playing field". Kahan attended the second IEEE 754 standards working group meeting, held in November 1977. He subsequently received permission from Intel to put forward a draft proposal based on his work for their coprocessor; he was allowed to explain details of the format and its rationale, but not anything related to Intel's implementation architecture. The draft was co-written with Jerome Coonen and Harold Stone, and was initially known as the "Kahan-Coonen-Stone proposal" or "K-C-S format". As an 8-bit exponent was not wide enough for some operations desired for double-precision numbers, e.g. to store the product of two 32-bit numbers, both Kahan's proposal and a counter-proposal by DEC therefore used 11 bits, like the time-tested 60-bit floating-point format of the
CDC 6600 The CDC 6600 was the flagship of the 6000 series of mainframe computer systems manufactured by Control Data Corporation. Generally considered to be the first successful supercomputer, it outperformed the industry's prior recordholder, the I ...
from 1965. Kahan's proposal also provided for infinities, which are useful when dealing with division-by-zero conditions; not-a-number values, which are useful when dealing with invalid operations; denormal numbers, which help mitigate problems caused by underflow; and a better balanced exponent bias, which can help avoid overflow and underflow when taking the reciprocal of a number. Even before it was approved, the draft standard had been implemented by a number of manufacturers. The Intel 8087, which was announced in 1980, was the first chip to implement the draft standard. In 1980, the
Intel 8087 The Intel 8087, announced in 1980, was the first floating-point coprocessor for the 8086 line of microprocessors. The purpose of the chip was to speed up floating-point arithmetic operations, such as addition, subtraction, multiplication, div ...
chip was already released, but DEC remained opposed, to denormal numbers in particular, because of performance concerns and since it would give DEC a competitive advantage to standardise on DEC's format. The arguments over gradual underflow lasted until 1981 when an expert hired by DEC to assess it sided against the dissenters. DEC had the study done in order to demonstrate that gradual underflow was a bad idea, but the study concluded the opposite, and DEC gave in. In 1985, the standard was ratified, but it had already become the de facto standard a year earlier, implemented by many manufacturers.


See also

*
IEEE 754 The IEEE Standard for Floating-Point Arithmetic (IEEE 754) is a technical standard for floating-point arithmetic originally established in 1985 by the Institute of Electrical and Electronics Engineers (IEEE). The standard #Design rationale, add ...
*
Minifloat In computing, minifloats are floating-point values represented with very few bits. This reduced precision makes them ill-suited for general-purpose numerical calculations, but they are useful for special purposes such as: * Computer graphics, w ...
for simple examples of properties of IEEE 754 floating point numbers *
Fixed-point arithmetic In computing, fixed-point is a method of representing fractional (non-integer) numbers by storing a fixed number of digits of their fractional part. Dollar amounts, for example, are often stored with exactly two fractional digits, represen ...


Notes


References


Further reading

* * * * : A compendium of non-intuitive behaviours of floating-point on popular architectures, with implications for program verification and testing.


External links


Comparing floats

Coprocessor.info: x87 FPU pictures, development and manufacturer information


— History and minutes

{{DEFAULTSORT:Ieee 754-1985 Computer arithmetic IEEE standards Floating point Computer-related introductions in 1985