The IEEE Standard for Binary Floating-Point Arithmetic (IEEE 754) is the most widely-used standard for floating-point computation, and is followed by many CPU and FPU implementations. The standard defines formats for representing floating-point numbers (including negative zero and denormal numbers) and special values (infinities and NaNs) together with a set of floating-point operations that operate on these values. It also specifies four rounding modes and five exceptions (including when the exceptions occur, and what happens when they do occur). The Institute of Electrical and Electronics Engineers or IEEE (pronounced as eye-triple-ee) is an international non-profit, professional organization incorporated in the State of New York, United States. ... A floating-point number is a digital representation for a number in a certain subset of the rational numbers, and is often used to approximate an arbitrary real number on a computer. ... “CPU” redirects here. ... A floating point unit (FPU) is a part of a computer system specially designed to carry out operations on floating point numbers. ... −0 is the representation of negative zero or minus zero, a number that exists in computing, in some signed number representations for integers, and in most floating point number representations. ... In computer science, denormal numbers or denormalized numbers (now often called subnormal numbers) fill the gap around zero in floating point arithmetic: any non-zero number which is smaller than the smallest normal number is sub-normal. For example, if the smallest positive normal number is 1×β-n (where β is... The infinity symbol ∞ in several typefaces. ... In computing, NaN (Not a Number) is a value or symbol that is usually produced as the result of an operation on invalid input operands, especially in floating-point calculations. ...

IEEE 754 specifies four formats for representing floating-point values: single-precision (32-bit), double-precision (64-bit), single-extended precision (≥ 43-bit, not commonly used) and double-extended precision (≥ 79-bit, usually implemented with 80 bits). Only 32-bit values are required by the standard; the others are optional. Many languages specify that IEEE formats and arithmetic be implemented, although sometimes it is optional. For example, the C programming language, which pre-dated IEEE 754, now allows but does not require IEEE arithmetic (the C float typically is used for IEEE single-precision and double uses IEEE double-precision). C is a general-purpose, block structured, procedural, imperative computer programming language developed in 1972 by Dennis Ritchie at the Bell Telephone Laboratories for use with the Unix operating system. ...

The full title of the standard is IEEE Standard for Binary Floating-Point Arithmetic (ANSI/IEEE Std 754-1985), and it is also known as IEC 60559:1989, Binary floating-point arithmetic for microprocessor systems (originally the reference number was IEC 559:1989).[1] Later there was an IEEE 854-1987 for "radix independent floating point" as long as the radix is 2 or 10.

Anatomy of a floating-point number

Following is a description of the standards' format for floating-point numbers.

Bit conventions used in this article

Bits within a word of width W are indexed by integers in the range 0 to W−1 inclusive. The bit with index 0 is drawn on the right. The lowest indexed bit is usually the lsb (Least Significant Bit, the one that if changed would cause the smallest variation of the represented value). This article is about the unit of information. ... In computing, word is a term for the natural unit of data used by a particular computer design. ... The integers are commonly denoted by the above symbol. ... Look up Inclusive in Wiktionary, the free dictionary. ...

General layout

The three fields in an IEEE 754 float

Binary floating-point numbers are stored in a sign-magnitude form where the most significant bit is the sign bit, exponent is the biased exponent, and "fraction" is the significand minus the most significant bit. Image File history File links No higher resolution available. ... In mathematics, negative numbers in any base are represented in the usual way, by prefixing them with a − sign. ... The binary representation of decimal 149, with the MSB highlighted. ... In computer science the sign bit is the bit in a computer numbering format which indicates the sign of the number. ... In IEEE 754 floating point numbers the exponent is biased in the engineering sense of the word – the value stored is offset from the actual value by the exponent bias. ... The significand (also coefficient or mantissa) is the part of a floating-point number that contains its significant digits. ...

Exponent biasing

The exponent is biased by 2^e−1−1. See also Excess-N. Biasing is done because exponents have to be signed values in order to be able to represent both tiny and huge values, but two's complement, the usual representation for signed values, would make comparison harder. To solve this the exponent is biased before being stored, by adjusting its value to put it within an unsigned range suitable for comparison. In mathematics, negative numbers in any base are represented in the usual way, by prefixing them with a − sign. ... A negative number is a number that is less than zero, such as −3. ... The twos complement of a binary number is the value obtained by subtracting the number from a large power of two (specifically, from 2N for an N-bit twos complement). ...

For example, to represent a number which has exponent of 17, exponent is 17 + 2^e−1−1. Assuming e = 8, the exponent is equal to 17 + 128 − 1 = 144.

Cases

The most significant bit of the significand (not stored) is determined by the value of exponent. If 0 < exponent < 2^e − 1, the most significant bit of the significand is 1, and the number is said to be normalized. If exponent is 0, the most significant bit of the significand is 0 and the number is said to be de-normalized. Three special cases arise: The significand (also coefficient or mantissa) is the part of a floating-point number that contains its significant digits. ...

1. if exponent is 0 and fraction is 0, the number is ±0 (depending on the sign bit)
2. if exponent = 2^e − 1 and fraction is 0, the number is ±infinity (again depending on the sign bit), and
3. if exponent = 2^e − 1 and fraction is not 0, the number being represented is not a number (NaN).

This can be summarized as: The infinity symbol ∞ in several typefaces. ... In computing, NaN (Not a Number) is a value or symbol that is usually produced as the result of an operation on invalid input operands, especially in floating-point calculations. ...

Single-precision 32 bit

A single-precision binary floating-point number is stored in 32 bits. In computing, single precision is a computer numbering format that occupies one storage locations in computer memory at address. ... This article is about the unit of information. ...

The exponent is biased by 2^{8 − 1} − 1 = 127 in this case (Exponents in the range −126 to +127 are representable. See the above explanation to understand why biasing is done). An exponent of −127 would be biased to the value 0 but this is reserved to encode that the value is a denormalized number or zero. An exponent of 128 would be biased to the value 255 but this is reserved to encode an infinity or not a number (NaN). See the chart above. Image File history File links No higher resolution available. ...

For normalised numbers, the most common, exponent is the biased exponent and fraction is the significand minus the most significant bit. The significand (also coefficient or mantissa) is the part of a floating-point number that contains its significant digits. ...

The number has value v:

v = s × 2^e × m

Where

s = +1 (positive numbers) when the sign bit is 0

s = −1 (negative numbers) when the sign bit is 1

e = Exp − 127 (in other words the exponent is stored with 127 added to it, also called "biased with 127")

m = 1.fraction in binary (that is, the significand is the binary number 1 followed by the radix point followed by the binary bits of the fraction). Therefore, 1 ≤ m < 2.

In the example shown above, the sign is zero, the exponent is −3, and the significand is 1.01 (in binary, which is 1.25 in decimal). The represented number is therefore +1.25 × 2⁻³, which is +0.15625.

Notes:

1. Denormalized numbers are the same except that e = −126 and m is 0.fraction. (e is NOT −127 : The fraction has to be shifted to the right by one more bit, in order to include the leading bit, which is not always 1 in this case. This is balanced by incrementing the exponent to −126 for the calculation.)
2. −126 is the smallest exponent for a normalized number
3. There are two Zeroes, +0 (s is 0) and −0 (s is 1)
4. There are two Infinities +∞ (s is 0) and −∞ (s is 1)
5. NaNs may have a sign and a fraction, but these have no meaning other than for diagnostics; the first bit of the fraction is often used to distinguish signaling NaNs from quiet NaNs
6. NaNs and Infinities have all 1s in the Exp field.
7. 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

   ±2⁻¹⁴⁹ ≈ ±1.4012985×10⁻⁴⁵

8. 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

   ±2⁻¹²⁶ ≈ ±1.175494351×10⁻³⁸

9. The finite positive and finite negative numbers furthest from zero (represented by the value with 254 in the Exp field and all 1s in the fraction field) are

   ±((1-(1/2)²⁴)2¹²⁸) ^[1] ≈ ±3.4028235×10³⁸

Here is the summary table from the previous section with some example 32-bit single-precision examples: −0 is the representation of negative zero or minus zero, a number that exists in computing, in some signed number representations for integers, and in most floating point number representations. ...

A more complex example

Bit values for the IEEE 754 32bit float -118.625

Let us encode the decimal number −118.625 using the IEEE 754 system. Image File history File links No higher resolution available. ...

1. First we need to get the sign, the exponent and the fraction. Because it is a negative number, the sign is "1".
2. Now, we write the number (without the sign; i.e. unsigned, no two's complement) using binary notation. The result is 1110110.101. We get the 101 after the decimal like this:
   1. 0.625 x 2 = 1.25 which means we write 1 after decimal and move on
   2. 0.25 x 2 = 0.5 which means we write 0 after the decimal and move on
   3. 0.5 x 2 = 1.00 which means we write 1 after the decimal and we are also finished since we have no residuum left to work with
3. Next, let's move the radix point left, leaving only a 1 at its left: 1110110.101 = 1.110110101 × 2⁶. This is a normalized floating point number. The first 1 binary digit is dropped. The fraction is the part at the right of the radix point, filled with 0 on the right until we get all 23 bits. That is 11011010100000000000000.
4. The exponent is 6, but we need to convert it to binary and bias it (so the most negative exponent is 0, and all exponents are non-negative binary numbers). For the 32-bit IEEE 754 format, the bias is 127 and so 6 + 127 = 133. In binary, this is written as 10000101.

The binary numeral system, or base-2 number system, is a numeral system that represents numeric values using two symbols, usually 0 and 1. ...

Double-precision 64 bit

The three fields in a 64bit IEEE 754 float

Double precision is essentially the same except that the fields are wider: Image File history File links No higher resolution available. ... In computing, double precision is a computer numbering format that occupies two storage locations in computer memory at address and address+1. ...

The fraction part is much larger, while the exponent is only slightly larger. The standard creators believed precision is more important than range.

NaNs and Infinities are represented with Exp being all 1s (2047).

For Normalized numbers the exponent bias is +1023 (so e is exponent (− 1023)). For Denormalized numbers the exponent is (−1022) (the minimum exponent for a normalized number—it is not (−1023) because normalised numbers have a leading 1 digit before the binary point and denormalized numbers do not). As before, both infinity and zero are signed.

Notes:

1. 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

   ±2⁻¹⁰⁷⁴ ≈ ±5×10⁻³²⁴

2. 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

   ±2⁻¹⁰²² ≈ ±2.2250738585072020×10⁻³⁰⁸

3. 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

   ±((1-(1/2)⁵³)2¹⁰²⁴) ^[1] ≈ ±1.7976931348623157×10³⁰⁸

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 with its associated order, except for the two bit combinations negative zero and positive zero, which sometimes require special attention (see below). The binary representation has the special property that, excluding NaNs, any two numbers can be compared like sign and magnitude integers (although with modern computer processors this is no longer directly applicable): if the sign bit is different, the negative number precedes the positive number (except that negative zero and positive zero should be considered equal), otherwise, relative order is the same as lexicographical order but inverted for two negative numbers; endianness issues apply. In mathematics, the affinely extended real number system is obtained from the real number system R by adding two elements: +∞ and −∞ (pronounced plus infinity and minus infinity). These new elements are not real numbers. ... In mathematics, negative numbers in any base are represented in the usual way, by prefixing them with a − sign. ... In mathematics, the lexicographic or lexicographical order, (also known as dictionary order, alphabetic order or lexicographic(al) product), is a natural order structure of the Cartesian product of two ordered sets. ... In computing, endianness is the byte (and sometimes bit) ordering in memory used to represent some kind of data. ...

Floating-point arithmetic is subject to rounding that may affect the outcome of comparisons on the results of the computations.

Although negative zero and positive zero should be considered equal for comparison purposes, some programming language relational operators and similar constructs might or do treat them as distinct. According to the Java Language Specification[2], 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. For C++, the standard does not seem to have anything to say on the subject, so it is probably important to verify this (one environment tested treated them as equal when using a floating-point variable and treated them as distinct and with negative zero preceding positive zero when comparing floating-point literals). A programming language is an artificial language that can be used to control the behavior of a machine, particularly a computer. ... In computer programming languages, a relational operator symbol or a relational operator name is a lexical or syntactic unit that denotes a relation, for example, equality or greater than, among two or more domains, the members of which are typically denoted by further expressions. ... “Java language” redirects here. ... C++ (pronounced see plus plus, IPA: ) is a general-purpose programming language with high-level and low-level capabilities. ...

Rounding floating-point numbers

The IEEE standard has four different rounding modes; the first is the default; the others are called directed roundings. Directed rounding refers to rounding a number in a certain direction - toward a certain number or limit. ...

• 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 occurs 50% of the time (in IEEE 754r 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.

IEEE 754r is an ongoing revision to the IEEE 754 floating point standard. ...

Extending the real numbers

The IEEE standard employs (and extends) the affinely extended real number system, 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 and Intel 80287 floating point co-processors both support this projective mode.^[2][3][4] In mathematics, the affinely extended real number system is obtained from the real number system R by adding two elements: +∞ and −∞ (pronounced plus infinity and minus infinity). These new elements are not real numbers. ... In real analysis, the real projective line (also called the one-point compactification of the real line, or the projectively extended real numbers), is the set , which will here be denoted . The symbol here means an unsigned infinity, an infinite quantity which is neither positive nor negative (it is sometimes... Intel C8087 Math Coprocessor The 8087 was the first math coprocessor designed by Intel and it was built to be paired with the ass] microprocessors. ... The Intel 80287 (287) was the math coprocessor for the Intel 80286 series of microprocessors. ... A floating-point number is a digital representation for a number in a certain subset of the rational numbers, and is often used to approximate an arbitrary real number on a computer. ... A co-processor is a secondary processor in a computer that handles tasks that the general-purpose CPU either cannot implement, or does not implement for efficiency reasons. ...

Recommended functions and predicates

• Under some C compilers, 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 which turns out to have different exception behavior than NOT(x = y).
• 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

In mathematics, a predicate is either a relation or the boolean-valued function that amounts to the characteristic function or the indicator function of such a relation. ...

References

1. ^ ^{a b} Prof. W. Kahan. "Lecture Notes on the Status of IEEE 754" (PDF). October 1, 1997 3:36 am. Elect. Eng. & Computer Science University of California. Retrieved on 2007-04-12.
2. ^ John R. Hauser (March 1996). "Handling Floating-Point Exceptions in Numeric Programs" (PDF). ACM Transactions on Programming Languages and Systems 18 (2). 
3. ^ David Stevenson (March 1981). "IEEE Task P754: A proposed standard for binary floating-point arithmetic". Computer 14 (3): 51–62. 
4. ^ Kahan, W. and Palmer, J. (1979). "On a proposed floating-point standard". SIGNUM Newsletter 14 (Special): 13–21. 

• Floating Point Unit by Jidan Al-Eryani

William Velvel Kahan (born June 5, 1933, in Toronto, Ontario, Canada) is an eminent mathematician and computer scientist. ... PDF is an abbreviation with several meanings: Portable Document Format Post-doctoral fellowship Probability density function There also is an electronic design automation company named PDF Solutions. ... Year 2007 (MMVII) is the current year, a common year starting on Monday of the Gregorian calendar and the AD/CE era. ... is the 102nd day of the year (103rd in leap years) in the Gregorian calendar. ...

Revision of the standard

Note that the IEEE 754 standard is currently under revision. See: IEEE 754r 2006 is a common year starting on Sunday of the Gregorian calendar. ... IEEE 754r is an ongoing revision to the IEEE 754 floating point standard. ...

See also

• minifloat for simple examples of properties of IEEE 754 floating point numbers
• −0 (negative zero)
• IEEE 754r working group to revise IEEE 754-1985.
• Intel 8087 (early implementation effort)
• Q (number format) For constant resolution

This article or section is in need of attention from an expert on the subject. ... −0 is the representation of negative zero or minus zero, a number that exists in computing, in some signed number representations for integers, and in most floating point number representations. ... IEEE 754r is an ongoing revision to the IEEE 754 floating point standard. ... Intel C8087 Math Coprocessor The 8087 was the first math coprocessor designed by Intel and it was built to be paired with the ass] microprocessors. ... Q is a fixed point number format where the number of fractional bits (and optionally the number of integer bits) is specified. ...

External links

• IEEE 754 references
• Let's Get To The (Floating) Point by Chris Hecker
• What Every Computer Scientist Should Know About Floating-Point Arithmetic by David Goldberg - a good introduction and explanation.
• A compendium of non-intuitive behaviours of floating-point on popular architectures, with implications for program verification and testing
• IEEE 854-1987 History and minutes
• Web Based Converter
• Another Web Based Converter
• Java Applet Converter
• Converter as MS-Windows program
• An Interview with the Old Man of Floating-Point
• Coprocessor.info : x87 FPU pictures, development and manufacturer information
• Understanding IEEE 754 - "Try it yourself"
• Comparing floats