IEEE 754

Some numbers may have several possible exponential format representations. For instance, if b = 10, and p = 7, then −12.345 can be represented by −12345×10⁻³, −123450×10⁻⁴, and −1234500×10⁻⁵. However, for most operations, such as arithmetic operations, the result (value) does not depend on the representation of the inputs.

For the decimal formats, any representation is valid, and the set of these representations is called a cohort. When a result can have several representations, the standard specifies which member of the cohort is chosen.

For the binary formats, the representation is made unique by choosing the smallest representable exponent allowing the value to be represented exactly. Further, the exponent is not represented directly, but a bias is added so that the smallest representable exponent is represented as 1, with 0 used for subnormal numbers. For numbers with an exponent in the normal range (the exponent field being neither all ones nor all zeros), the leading bit of the significand will always be 1. Consequently, a leading 1 can be implied rather than explicitly present in the memory encoding, and under the standard the explicitly represented part of the significand will lie between 0 and 1. This rule is called leading bit convention, implicit bit convention, or hidden bit convention. This rule allows the binary format to have an extra bit of precision. The leading bit convention cannot be used for the subnormal numbers as they have an exponent outside the normal exponent range and scale by the smallest represented exponent as used for the smallest normal numbers.

Due to the possibility of multiple encodings (at least in formats called interchange formats), a NaN may carry other information: a sign bit (which has no meaning, but may be used by some operations) and a payload, which is intended for diagnostic information indicating the source of the NaN (but the payload may have other uses, such as NaN-boxing[10][11][12]).

The standard defines five basic formats that are named for their numeric base and the number of bits used in their interchange encoding. There are three binary floating-point basic formats (encoded with 32, 64 or 128 bits) and two decimal floating-point basic formats (encoded with 64 or 128 bits). The binary32 and binary64 formats are the single and double formats of IEEE 754-1985 respectively. A conforming implementation must fully implement at least one of the basic formats.

The standard also defines interchange formats, which generalize these basic formats.[13] For the binary formats, the leading bit convention is required. The following table summarizes some of the possible interchange formats (including the basic formats).

In the table above, integer values are exact where as values in decimal notation (e.g. 1.0) are rounded values. The minimum exponents listed are for normal numbers; the special subnormal number representation allows even smaller (in magnitude) numbers to be represented with some loss of precision. For example, the smallest positive number that can be represented in binary64 is 2⁻¹⁰⁷⁴; contributions to the −1074 figure include the emin value −1022 and all but one of the 53 significand bits (2^{−1022 − (53 − 1)} = 2⁻¹⁰⁷⁴). The decimal representation does not define subnormal numbers as such, but numbers with a mantissa with leading zero(s) can be interpreted as subnormal as they offer fewer digits available to express precision.

Decimal digits is the precision of the format expressed in terms of an equivalent number of decimal digits. It is computed as digits × log₁₀ base. E.g. binary128 has approximately the same precision as a 34 digit decimal number.

log₁₀ MAXVAL is a measure of the range of the encoding. Its integer part is the largest exponent shown on the output of a value in scientific notation with one leading digit in the significand before the decimal point (e.g. 1.698·10³⁸ is near the largest value in binary32, 9.999999·10⁹⁶ is the largest value in decimal32).

The binary32 (single) and binary64 (double) formats are two of the most common formats used today. The figure below shows the absolute precision for both formats over a range of values. This figure can be used to select an appropriate format given the expected value of a number and the required precision.

Precision of binary32 and binary64 in the range 10⁻¹² to 10¹²

An example of a layout for 32-bit floating point is

and the 64 bit layout is similar.

The standard specifies optional extended and extendable precision formats, which provide greater precision than the basic formats.[14] An extended precision format extends a basic format by using more precision and more exponent range. An extendable precision format allows the user to specify the precision and exponent range. An implementation may use whatever internal representation it chooses for such formats; all that needs to be defined are its parameters (b, p, and emax). These parameters uniquely describe the set of finite numbers (combinations of sign, significand, and exponent for the given radix) that it can represent.

The standard recommends that language standards provide a method of specifying p and emax for each supported base b.[15] The standard recommends that language standards and implementations support an extended format which has a greater precision than the largest basic format supported for each radix b.[16] For an extended format with a precision between two basic formats the exponent range must be as great as that of the next wider basic format. So for instance a 64-bit extended precision binary number must have an 'emax' of at least 16383. The x87 80-bit extended format meets this requirement.

Interchange formats are intended for the exchange of floating-point data using a bit string of fixed length for a given format.

For the exchange of binary floating-point numbers, interchange formats of length 16 bits, 32 bits, 64 bits, and any multiple of 32 bits ≥ 128[lower-alpha 5] are defined. The 16-bit format is intended for the exchange or storage of small numbers (e.g., for graphics).

The encoding scheme for these binary interchange formats is the same as that of IEEE 754-1985: a sign bit, followed by w exponent bits that describe the exponent offset by a bias, and p − 1 bits that describe the significand. The width of the exponent field for a k-bit format is computed as w = round(4 log₂(k)) − 13. The existing 64- and 128-bit formats follow this rule, but the 16- and 32-bit formats have more exponent bits (5 and 8 respectively) than this formula would provide (3 and 7 respectively).

As with IEEE 754-1985, the biased-exponent field is filled with all 1 bits to indicate either infinity (trailing significand field = 0) or a NaN (trailing significand field ≠ 0). For NaNs, quiet NaNs and signaling NaNs are distinguished by using the most significant bit of the trailing significand field exclusively,[lower-alpha 6] and the payload is carried in the remaining bits.

For the exchange of decimal floating-point numbers, interchange formats of any multiple of 32 bits are defined. As with binary interchange, the encoding scheme for the decimal interchange formats encodes the sign, exponent, and significand. Two different bit-level encodings are defined, and interchange is complicated by the fact that some external indicator of the encoding in use may be required.

The two options allow the significand to be encoded as a compressed sequence of decimal digits using densely packed decimal or, alternatively, as a binary integer. The former is more convenient for direct hardware implementation of the standard, while the latter is more suited to software emulation on a binary computer. In either case, the set of numbers (combinations of sign, significand, and exponent) that may be encoded is identical, and special values (±zero with the minimum exponent, ±infinity, quiet NaNs, and signaling NaNs) have identical encodings.

• Round to nearest, ties to even – rounds to the nearest value; if the number falls midway, it is rounded to the nearest value with an even least significant digit.
• Round to nearest, ties away from zero (or ties to away)  – rounds to the nearest value; if the number falls midway, it is rounded to the nearest value above (for positive numbers) or below (for negative numbers).

At the extremes, a value with a magnitude strictly less than ${\displaystyle k=b^{\text{emax}}\left(b-{\tfrac {1}{2}}b^{1-p}\right)}$ will be rounded to the minimum or maximum finite number (depending on the value's sign). Any numbers with exactly this magnitude are considered ties; this choice of tie may be conceptualized as the midpoint between ${\displaystyle \pm b^{\text{emax}}(b-b^{1-p})}$ and ${\displaystyle \pm b^{{\text{emax}}+1}}$, which, were the exponent not limited, would be the next representable floating-point numbers larger in magnitude. Numbers with a magnitude strictly larger than k are rounded to the corresponding infinity.[17]

"Round to nearest, ties to even" is the default for binary floating point and the recommended default for decimal. "Round to nearest, ties to away" is only required for decimal implementations.[18]

• Round toward 0 – directed rounding towards zero (also known as truncation).
• Round toward +∞ – directed rounding towards positive infinity (also known as rounding up or ceiling).
• Round toward −∞ – directed rounding towards negative infinity (also known as rounding down or floor).

Unless specified otherwise, the floating-point result of an operation is determined by applying the rounding function on the infinitely precise (mathematical) result. Such an operation is said to be correctly rounded. This requirement is called correct rounding.[19]

The standard provides comparison predicates to compare one floating-point datum to another in the supported arithmetic format.[31] Any comparison with a NaN is treated as unordered. −0 and +0 compare as equal.

The standard provides a predicate totalOrder, which defines a total ordering on canonical members of the supported arithmetic format.[32] The predicate agrees with the comparison predicates when one floating-point number is less than the other. The totalOrder predicate does not impose a total ordering on all encodings in a format. In particular, it does not distinguish among different encodings of the same floating-point representation, as when one or both encodings are non-canonical.[32] IEEE 754-2019 incorporates clarifications of totalOrder.

For the binary interchange formats whose encoding follows the IEEE 754-2008 recommendation on placement of the NaN signaling bit, the comparison is identical to one that type puns the floating-point numbers to a sign–magnitude integer (assuming a payload ordering consistent with this comparison), an old trick for FP comparison without an FPU.[33]

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 environments, positive zero is usually printed as "0" and the negative zero as "-0". The two values behave as equal in numerical comparisons, but some operations return different results for +0 and −0. For instance, 1/(−0) returns negative infinity, while 1/(+0) returns positive infinity (so that the identity 1/(1/±∞) = ±∞ is maintained). Other common functions with a discontinuity at x=0 which might treat +0 and −0 differently include log(x), signum(x), and the principal square root of y + xi for any negative number y. As with any approximation scheme, operations involving "negative zero" can occasionally cause confusion. For example, in IEEE 754, x = y does not always imply 1/x = 1/y, as 0 = −0 but 1/0 ≠ 1/(−0).[37]

Subnormal values fill the underflow gap with values where the absolute distance between them is the same as for adjacent values just outside the underflow gap. This is an improvement over the older practice to just have zero in the underflow gap, and where underflowing results were replaced by zero (flush to zero).[38]

Modern floating-point hardware usually handles subnormal values (as well as normal values), and does not require software emulation for subnormals.

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 (depends on the rounding) used as replacement values when there is an 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

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 operation" exception to be signaled.

The representation of NaNs specified by the standard has some unspecified bits that could be used to encode the type or source of error; but there is no standard for that encoding. In theory, signaling NaNs could be used by a runtime system to flag uninitialized variables, or extend the floating-point numbers with other special values without slowing down the computations with ordinary values, although such extensions are not common.

The standard recommends optional exception handling in various forms, including presubstitution of user-defined default values, and traps (exceptions that change the flow of control in some way) and other exception handling models that interrupt the flow, such as try/catch. The traps and other exception mechanisms remain optional, as they were in IEEE 754-1985.

Clause 9 in the standard recommends additional mathematical operations[43] that language standards should define.[44] None are required in order to conform to the standard.

The following are recommended arithmetic operations, which must round correctly:[45]

• ${\displaystyle e^{x}}$, ${\displaystyle 2^{x}}$, ${\displaystyle 10^{x}}$
• ${\displaystyle e^{x}-1}$, ${\displaystyle 2^{x}-1}$, ${\displaystyle 10^{x}-1}$
• ${\displaystyle \ln x}$, ${\displaystyle \log _{2}x}$, ${\displaystyle \log _{10}x}$
• ${\displaystyle \ln(1+x)}$, ${\displaystyle \log _{2}(1+x)}$, ${\displaystyle \log _{10}(1+x)}$
• ${\displaystyle {\sqrt {x^{2}+y^{2}}}}$
• ${\displaystyle {\sqrt {x}}}$
• ${\displaystyle (1+x)^{n}}$
• ${\displaystyle x^{\frac {1}{n}}}$
• ${\displaystyle x^{n}}$, ${\displaystyle x^{y}}$
• ${\displaystyle \sin x}$, ${\displaystyle \cos x}$, ${\displaystyle \tan x}$
• ${\displaystyle \arcsin x}$, ${\displaystyle \arccos x}$, ${\displaystyle \arctan x}$, ${\displaystyle \operatorname {atan2} (y,x)}$
• ${\displaystyle \operatorname {sinPi} x=\sin \pi x}$, ${\displaystyle \operatorname {cosPi} x=\cos \pi x}$, ${\displaystyle \operatorname {tanPi} x=\tan \pi x}$ (see also: Multiples of π)
• ${\displaystyle \operatorname {asinPi} x={\frac {\arcsin x}{\pi }}}$, ${\displaystyle \operatorname {acosPi} x={\frac {\arccos x}{\pi }}}$, ${\displaystyle \operatorname {atanPi} x={\frac {\arctan x}{\pi }}}$, ${\displaystyle \operatorname {atan2Pi} (y,x)={\frac {\operatorname {atan2} (y,x)}{\pi }}}$ (see also: Multiples of π)
• ${\displaystyle \sinh x}$, ${\displaystyle \cosh x}$, ${\displaystyle \tanh x}$
• ${\displaystyle \operatorname {arsinh} x}$, ${\displaystyle \operatorname {arcosh} x}$, ${\displaystyle \operatorname {artanh} x}$

The asinPi, acosPi and tanPi functions were not part of the IEEE 754-2008 standard because they were deemed less necessary.[46] asinPi, acosPi were mentioned, but this was regarded as an error.[5] All three were added in the 2019 revision.

The recommended operations also include setting and accessing dynamic mode rounding direction,[47] and implementation-defined vector reduction operations such as sum, scaled product, and dot product, whose accuracy is unspecified by the standard.[48]

As of 2019, augmented arithmetic operations[49] for the binary formats are also recommended. These operations, specified for addition, subtraction and multiplication, produce a pair of values consisting of a result correctly rounded to nearest in the format and the error term, which is representable exactly in the format. At the time of publication of the standard, no hardware implementations are known, but very similar operations were already implemented in software using well-known algorithms. The history and motivation for their standardization are explained in a background document.[50][51]

As of 2019, the formerly required minNum, maxNum, minNumMag, and maxNumMag in IEEE 754-2008 are now deprecated due to their non-associativity. Instead, two sets of new minimum and maximum operations are recommended.[52] The first set contains minimum, minimumNumber, maximum and maximumNumber. The second set contains minimumMagnitude, minimumMagnitudeNumber, maximumMagnitude and maximumMagnitudeNumber. The history and motivation for this change are explained in a background document.[53]

The standard recommends how language standards should specify the semantics of sequences of operations, and points out the subtleties of literal meanings and optimizations that change the value of a result. By contrast, the previous 1985 version of the standard left aspects of the language interface unspecified, which led to inconsistent behavior between compilers, or different optimization levels in an optimizing compiler.

Programming languages should allow a user to specify a minimum precision for intermediate calculations of expressions for each radix. This is referred to as preferredWidth in the standard, and it should be possible to set this on a per-block basis. Intermediate calculations within expressions should be calculated, and any temporaries saved, using the maximum of the width of the operands and the preferred width if set. Thus, for instance, a compiler targeting x87 floating-point hardware should have a means of specifying that intermediate calculations must use the double-extended format. The stored value of a variable must always be used when evaluating subsequent expressions, rather than any precursor from before rounding and assigning to the variable.

The IEEE 754-1985 version of the standard allowed many variations in implementations (such as the encoding of some values and the detection of certain exceptions). IEEE 754-2008 has reduced these allowances, but a few variations still remain (especially for binary formats). The reproducibility clause recommends that language standards should provide a means to write reproducible programs (i.e., programs that will produce the same result in all implementations of a language) and describes what needs to be done to achieve reproducible results.

The standard recommends providing conversions to and from external hexadecimal-significand character sequences, based on C99's hexadecimal floating point literals. Such a literal consists of an optional sign (+ or -), the indicator "0x", a hexadecimal number with or without a period, an exponent indicator "p", and a decimal exponent with optional sign. The syntax is not case-sensitive.[58] The decimal exponent scales by powers of 2, so for example 0x0.1p-4 is 1/256.[59]

• IEEE Standard for Binary Floating-Point Arithmetic. ANSI/IEEE STD 754-1985. IEEE. 1985-10-12. pp. 1–20. doi:10.1109/IEEESTD.1985.82928. ISBN 0-7381-1165-1.
• IEEE Computer Society (2008-08-29). IEEE Standard for Floating-Point Arithmetic. IEEE STD 754-2008. IEEE. pp. 1–70. doi:10.1109/IEEESTD.2008.4610935. ISBN 978-0-7381-5753-5. IEEE Std 754-2008.
• IEEE Computer Society (2019-07-22). IEEE Standard for Floating-Point Arithmetic. IEEE STD 754-2019. IEEE. pp. 1–84. doi:10.1109/IEEESTD.2019.8766229. ISBN 978-1-5044-5924-2. IEEE Std 754-2019.
• ISO/IEC JTC 1/SC 25 (June 2011). ISO/IEC/IEEE 60559:2011 — Information technology — Microprocessor Systems — Floating-Point arithmetic. ISO. pp. 1–58.
• ISO/IEC JTC 1/SC 25 (May 2020). ISO/IEC 60559:2020 — Information technology — Microprocessor Systems — Floating-Point arithmetic. ISO. pp. 1–74.

• Decimal floating-point arithmetic, FAQs, bibliography, and links
• Comparing binary floats
• IEEE 754 Reference Material
• IEEE 854-1987 – History and minutes
• Supplementary readings for IEEE 754. Includes historical perspectives.