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positive de finite

IEEE 754-1985

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).

Summary

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).

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). Later there was an IEEE 854-1987 for "radix independent floating point" as long as the radix is 2 or 10. In June 2008, a major revision to IEEE 754 and IEEE 854 was approved by the IEEE. See IEEE 754r.

Structure of a floating-point number

Following is a description of the standard's 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. Unless otherwise stated, the lowest indexed bit is the lsb (Least Significant Bit, the one that if changed would cause the smallest variation of the represented value).

General layout

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 without the most significant bit.

Exponent biasing

The exponent is biased by (2^{e-1})-1, where e is the number of bits used for the exponent field (e.g. if e=8, then (2^{8-1})-1 = 128 - 1 = 127 ). 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.

For example, to represent a number which has exponent of 17 in an exponent field 8 bits wide:
exponentfield=17 + (2^{8-1}) - 1 = 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:

  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:

Type Exponent Fraction
Zeroes 0 0
Denormalized numbers 0 non zero
Normalized numbers 1 to 2^e-2 any
Infinities 2^e-1 0
NaNs 2^e-1 non zero

Single-precision 32-bit

A single-precision binary floating-point number is stored in 32 bits.

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.

For normalized numbers, which are the most common, the exponent is the biased exponent and fraction is the significand without the most significant bit.

The number has value v:

v = s × 2e × m

or

v = (-1)^{sign} times 2^{exponent - exponent~bias} times (1 + fraction)

Where

s = +1 (positive numbers and +0) when the sign bit is 0
s = −1 (negative numbers and −0) when the sign bit is 1
e = exponent − 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 so s is +1, the exponent is 124 so e is −3, and the significand m 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
  8. : ±2−149 ≈ ±1.4012985
  9. 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
  10. : ±2−126 ≈ ±1.175494351
  11. 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
  12. : ±(1-2-24)×2128 ≈ ±3.4028235

Here is the summary table from the previous section with some 32-bit single-precision examples:

Type Sign Exp Exp+Bias Exponent Significand Value
Zero 0 -127 0 0000 0000 000 0000 0000 0000 0000 0000 0.0
One 0 0 127 0111 1111 000 0000 0000 0000 0000 0000 1.0
Minus One 1 0 127 0111 1111 000 0000 0000 0000 0000 0000 −1.0
Smallest denormalized number * -127 0 0000 0000 000 0000 0000 0000 0000 0001 ±2−23 × 2−126 = ±2−149 ≈ ±1.4
"Middle" denormalized number * -127 0 0000 0000 100 0000 0000 0000 0000 0000 ±2−1 × 2−126 = ±2−127 ≈ ±5.88
Largest denormalized number * -127 0 0000 0000 111 1111 1111 1111 1111 1111 ±(1−2−23) × 2−126 ≈ ±1.18
Smallest normalized number * -126 1 0000 0001 000 0000 0000 0000 0000 0000 ±2−126 ≈ ±1.18
Largest normalized number * 127 254 1111 1110 111 1111 1111 1111 1111 1111 ±(2−2−23) × 2127 ≈ ±3.4
Positive infinity 0 128 255 1111 1111 000 0000 0000 0000 0000 0000 +∞
Negative infinity 1 128 255 1111 1111 000 0000 0000 0000 0000 0000 −∞
Not a number * 128 255 1111 1111 non zero NaN
* Sign bit can be either 0 or 1 .

Range and Precision Table

Some example range and precision values for given exponents:

Exponent Minimum Maximum Precision
0 1 1.999999880791 1.19209289551e-7
1 2 3.99999976158 2.38418579102e-7
2 4 7.99999952316 4.76837158203e-7
10 1024 2047.99987793 1.220703125e-4
11 2048 4095.99975586 2.44140625e-4
23 8388608 16777215 1
24 16777216 33554430 2
127 1.7014e38 3.4028e38 2.02824096037e31

As an example, 16,777,217 can not be encoded as a 32-bit float as it will be rounded to 16,777,216. Which is a good example of why floating point arithmetic is unsuitable for accounting software. However, all integers within the representable range that are a power of 2 can be stored in a 32-bit float without rounding.

A more complex example

The decimal number −118.625 is encoded using the IEEE 754 system as follows:

  1. The sign, the exponent, and the fraction are extracted from the original number. First, because the number is negative, the sign bit is "1".
  2. Next, the number (without the sign; i.e., unsigned, no two's complement) is converted to binary notation, giving 1110110.101. The 101 after the binary point has the value 0.625 because it is the sum of:
    1. (2−1) × 1, from the first digit after the binary point
    2. (2−2) × 0, from the second digit
    3. (2−3) × 1, from the third digit.
  3. That binary number is then normalized; that is, the binary point is moved left, leaving only a 1 to its left. The number of places it is moved gives the (power of two) exponent: 1110110.101 becomes 1.110110101 × 26. After this process, the first binary digit is always a 1, so it need not be included in the encoding. The rest is the part to the right of the binary point, which is then padded with zeros on the right to make 23 bits in all, which becomes the significand bits in the encoding: That is, 11011010100000000000000.
  4. The exponent is 6. This is encoded by converting it to binary and biasing it (so the most negative encodable 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 encoded as 10000101.

Double-precision 64 bit

Double precision is essentially the same except that the fields are wider:

The fraction part is much larger, while the exponent is only slightly larger. NaNs and Infinities are represented with Exp being all 1s (2047). If the fraction part is all zero then it is Infinity, else it is NaN.

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. : ±2−1074 ≈ ±5
  3. 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
  4. : ±2−1022 ≈ ±2.2250738585072020
  5. 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
  6. : ±((1-(1/2)53)21024) ≈ ±1.7976931348623157

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.

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 are generally 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, 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 have anything to say on the subject, so it is 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).

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 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 +infty – directed rounding towards positive infinity
  • Round toward -infty – directed rounding towards negative infinity.

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.

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

See also

References

Further reading

External links

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