This section describes the IEEE 754 specification.
What Is IEEE Arithmetic?
- Two basic floating-point formats: single
and
double.
- The IEEE single format has a significand precision of 24 bits and occupies 32 bits overall. The IEEE double format has a significand precision of 53 bits and occupies 64 bits overall.
- Two classes of extended floating-point formats: single extended
and double extended.
- The standard does not prescribe the exact precision and size of these formats, but it does specify the minimum precision and size. For example, an IEEE double extended format must have a significand precision of at least 64 bits and occupy at least 79 bits overall.
- Accuracy requirements on floating-point operations: add, subtract, multiply, divide, square root, remainder, round numbers
in floating-point format to integer values, convert between different floating-point formats, convert between floating-point
and integer formats, and compare.
- The remainder and compare operations must be exact. Each of the other operations must deliver to its destination the exact result, unless there is no such result or that result does not fit in the destination's format. In the latter case, the operation must minimally modify the exact result according to the rules of prescribed rounding modes, presented below, and deliver the result so modified to the operation's destination.
- Accuracy, monotonicity and identity requirements for conversions
between decimal strings and binary floating-point numbers in either of the basic floating-point formats.
- For operands lying within specified ranges, these conversions must produce exact results, if possible, or minimally modify such exact results in accordance with the rules of the prescribed rounding modes. For operands not lying within the specified ranges, these conversions must produce results that differ from the exact result by no more than a specified tolerance that depends on the rounding mode.
- Five types of IEEE floating-point exceptions, and the conditions
for indicating to the user the occurrence of exceptions of these types.
- The five types of floating-point exceptions are invalid operation, division by zero, overflow, underflow, and inexact.
- Four rounding directions: toward the nearest representable value, with "even" values preferred whenever there are two nearest representable values; toward negative infinity (down); toward positive infinity (up); and toward 0 (chop).
- Rounding precision; for example, if a system delivers results in double extended format, the user should be able to specify that such results are to be rounded to the precision of either the single or double format.
The IEEE standard also recommends support for user handling of exceptions.
The features required by the IEEE standard make it possible to support interval arithmetic, the retrospective diagnosis of anomalies, efficient implementations of standard elementary functions like exp and cos, multiple precision arithmetic, and many other tools that are useful in numerical computation.
IEEE 754 floating-point arithmetic offers users greater control over computation than does any other kind of floating-point arithmetic. The IEEE standard simplifies the task of writing numerically sophisticated, portable programs not only by imposing rigorous requirements on conforming implementations, but also by allowing such implementations to provide refinements and enhancements to the standard itself.
IEEE Formats
This section describes how floating-point data is stored in memory. It summarizes the precisions and ranges of the different IEEE storage formats.
Storage Formats
A floating-point format is a data structure specifying the fields that comprise a floating-point numeral, the layout of those fields, and their arithmetic interpretation. A floating-point storage format specifies how a floating-point format is stored in memory. The IEEE standard defines the formats, but it leaves to implementors the choice of storage formats.
Assembly language software sometimes relies on using the storage formats, but higher level languages usually deal only with the linguistic notions of floating-point data types. These types have different names in different high-level languages, and correspond to the IEEE formats as shown in TABLE 2-1.
| single | float | REAL or REAL*4 |
| double | double | DOUBLE PRECISION or REAL*8 |
| double extended | long double | REAL*16 [SPARC only] |
IEEE 754 specifies exactly the single
and double floating-point formats, and it defines a class of extended formats for each of these two basic formats.
The long double
and REAL*16
types shown in TABLE 2-1 refer to one of the class of double extended formats defined by the
IEEE standard.
The following sections describe in detail each of the storage formats used for the IEEE floating-point formats on SPARC and x86 platforms.
Single Format
The IEEE single format consists of three
fields: a 23-bit fraction, f; an 8-bit biased exponent, e; and a 1-bit sign, s. These fields are stored contiguously in one 32-bit word, as shown
in FIGURE 2-1. Bits 0:22 contain the 23-bit fraction, f, with bit 0 being the least significant bit of the fraction and bit
22 being the most significant; bits 23:30 contain the 8-bit biased exponent, e, with bit 23 being the least significant bit of the biased exponent
and bit 30 being the most significant; and the highest-order bit 31 contains the sign bit, s.
FIGURE 2-1 Single-Storage Format
TABLE 2-2 shows the correspondence between the values of the three constituent
fields s,
e
and f,
on the one hand, and the value represented by the single- format bit pattern on the other; u means don't
care, that is, the value of the indicated field
is irrelevant to the determination of the value of the particular bit patterns in single format.
0 Notice that when e < 255, the value assigned to the single format bit pattern is
formed by inserting the binary radix point immediately to the left of the fraction's most significant bit, and
inserting an implicit bit immediately to the left of the binary point, thus representing in binary positional notation
a mixed number (whole number plus fraction, wherein 0 <= fraction < 1).
The mixed number thus formed is called the single-format significand. The implicit bit is so named because its value is not explicitly given in the single- format bit pattern, but is implied by the value of the biased exponent field.
For the single format, the difference between a normal number and a subnormal number is that the leading bit of the significand (the bit to left of the binary point) of a normal number is 1, whereas the leading bit of the significand of a subnormal number is 0. Single-format subnormal numbers were called single-format denormalized numbers in IEEE Standard 754.
The 23-bit fraction combined with the implicit leading significand bit provides 24 bits of precision in single-format normal numbers.
Examples of important bit patterns in the single-storage format are shown in TABLE 2-3. The maximum positive normal number is the largest finite number representable in IEEE single format. The minimum positive subnormal number is the smallest positive number representable in IEEE single format. The minimum positive normal number is often referred to as the underflow threshold. (The decimal values for the maximum and minimum normal and subnormal numbers are approximate; they are correct to the number of figures shown.)
A NaN (Not a Number) can be represented with any of the many bit patterns that satisfy the definition of a NaN. The hex value of the NaN shown in TABLE 2-3 is just one of the many bit patterns that can be used to represent a NaN.
Double Format
The IEEE double format consists of three
fields: a 52-bit fraction, f; an 11-bit biased exponent, e; and a 1-bit sign, s. These fields are stored contiguously in two successively addressed
32-bit words, as shown in FIGURE 2-2.
In the SPARC architecture, the higher address 32-bit word contains the least significant 32 bits of the fraction, while in the x86 architecture the lower address 32-bit word contains the least significant 32 bits of the fraction.
If we denote f[31:0] the least significant 32 bits of the fraction, then bit 0 is
the least significant bit of the entire fraction and bit 31 is the most significant of the 32 least significant
fraction bits.
In the other 32-bit word, bits 0:19 contain
the 20 most significant bits of the fraction, f[51:32], with bit 0 being the least significant of these 20 most significant
fraction bits, and bit 19 being the most significant bit of the entire fraction; bits 20:30 contain the 11-bit
biased exponent, e,
with bit 20 being the least significant bit of the biased exponent and bit 30 being the most significant; and the
highest-order bit 31 contains the sign bit, s.
FIGURE 2-2 numbers the bits as though the two contiguous 32-bit words were one
64-bit word in which bits 0:51 store the 52-bit fraction, f; bits 52:62 store the 11-bit biased exponent, e;
and bit 63 stores the sign bit, s.
FIGURE 2-2 Double-Storage Format
The values of the bit patterns in these three fields determine the value represented by the overall bit pattern.
TABLE 2-4 shows the correspondence between the values of the bits in the three constituent fields, on the one hand, and the value represented by the double-format bit pattern on the other; u means don't care, because the value of the indicated field is irrelevant to the determination of value for the particular bit pattern in double format.
Notice that when e < 2047, the value assigned to the double-format bit pattern is
formed by inserting the binary radix point immediately to the left of the fraction's most significant bit, and
inserting an implicit bit immediately to the left of the binary point. The number thus formed is called the significand.
The implicit bit is so named because its value is not explicitly given in the double-format bit pattern, but is
implied by the value of the biased exponent field.
For the double format, the difference between a normal number and a subnormal number is that the leading bit of the significand (the bit to the left of the binary point) of a normal number is 1, whereas the leading bit of the significand of a subnormal number is 0. Double-format subnormal numbers were called double-format denormalized numbers in IEEE Standard 754.
The 52-bit fraction combined with the implicit leading significand bit provides 53 bits of precision in double-format normal numbers.
Examples of important bit patterns in the double-storage format are shown in TABLE 2-5. The bit patterns in the second column appear as two 8-digit hexadecimal numbers. For the SPARC architecture, the left one is the value of the lower addressed 32-bit word, and the right one is the value of the higher addressed 32-bit word, while for the x86 architecture, the left one is the higher addressed word, and the right one is the lower addressed word. The maximum positive normal number is the largest finite number representable in the IEEE double format. The minimum positive subnormal number is the smallest positive number representable in IEEE double format. The minimum positive normal number is often referred to as the underflow threshold. (The decimal values for the maximum and minimum normal and subnormal numbers are approximate; they are correct to the number of figures shown.)
A NaN (Not a Number) can be represented by any of the many bit patterns that satisfy the definition of NaN. The hex value of the NaN shown in TABLE 2-5 is just one of the many bit patterns that can be used to represent a NaN.
Ranges and Precisions in Decimal Representation
This section covers the notions of range and precision for a given storage format. It includes the ranges and precisions corresponding to the IEEE single and double formats and to the implementations of IEEE double-extended format on SPARC and x86 architectures. For concreteness, in defining the notions of range and precision we refer to the IEEE single format.
The IEEE standard specifies that 32 bits be used to represent a floating point number in single format. Because there are only finitely many combinations of 32 zeroes and ones, only finitely many numbers can be represented by 32 bits.
- What are the decimal representations of the largest and smallest positive numbers that can be represented in this particular format?
Rephrase the question and introduce the notion of range:
- What is the range, in decimal notation, of numbers that can be represented by the IEEE single format?
Taking into account the precise definition of IEEE single format, one can prove that the range of floating-point numbers that can be represented in IEEE single format (if restricted to positive normalized numbers) is as follows:
- 1.175... × (10-38) to 3.402... × (10+38)
A second question refers to the precision (not to be confused with the accuracy or the number of significant digits) of the numbers represented in a given format. These notions are explained by looking at some pictures and examples.
The IEEE standard for binary floating-point arithmetic specifies the set of numerical values representable in the single format. Remember that this set of numerical values is described as a set of binary floating-point numbers. The significand of the IEEE single format has 23 bits, which together with the implicit leading bit, yield 24 digits (bits) of (binary) precision.
One obtains a different set of numerical values by marking the numbers:
- x = (x1.x2 x3...xq) × (10n)
(representable by q decimal digits in the significand) on the number line.
FIGURE 2-5 exemplifies this situation:
FIGURE 2-5 Comparison of a Set of Numbers Defined by Digital
and Binary Representation
Notice that the two sets are different. Therefore, estimating the number of significant decimal digits corresponding to 24 significant binary digits, requires reformulating the problem.
Reformulate the problem in terms of converting floating-point numbers between binary representations (the internal format used by the computer) and the decimal format (the format users are usually interested in). In fact, you may want to convert from decimal to binary and back to decimal, as well as convert from binary to decimal and back to binary.
It is important to notice that because the sets of numbers are different, conversions are in general inexact. If done correctly, converting a number from one set to a number in the other set results in choosing one of the two neighboring numbers from the second set (which one specifically is a question related to rounding).
Consider some examples. Suppose one is trying to represent a number with the following decimal representation in IEEE single format:
- x = x1.x2 x3... × 10n
Because there are only finitely many real numbers that can be represented exactly in IEEE single format, and not all numbers of the above form are among them, in general it will be impossible to represent such numbers exactly. For example, let
- y = 838861.2, z = 1.3
and run the following Fortran program:
REAL Y, Z Y = 838861.2 Z = 1.3 WRITE(*,40) Y 40 FORMAT("y: ",1PE18.11)
WRITE(*,50) Z 50 FORMAT("z: ",1PE18.11)
|
The output from this program should be similar to:
y: 8.38861187500E+05 z: 1.29999995232E+00 |
The difference between the value 8.388612 × 105 assigned to y and the value printed out is 0.000000125, which is seven decimal orders of magnitude smaller than y. The accuracy of representing y in IEEE single format is about 6 to 7 significant digits, or that y has about six significant digits if it is to be represented in IEEE single format.
Similarly, the difference between the value 1.3 assigned to z and the value printed out is 0.00000004768, which is eight decimal orders of magnitude smaller than z. The accuracy of representing z in IEEE single format is about 7 to 8 significant digits, or that z has about seven significant digits if it is to be represented in IEEE single format.
- Assume you convert a decimal floating point number a to its IEEE single format binary representation b, and then translate b back to a decimal number c; how many orders of magnitude are between a and a - c?
- What is the number of significant decimal digits of a in the IEEE single format representation, or how many decimal digits are to be trusted as accurate when one represents x in IEEE single format?
The number of significant decimal digits is always between 6 and 9, that is, at least 6 digits, but not more than 9 digits are accurate (with the exception of cases when the conversions are exact, when infinitely many digits could be accurate).
Conversely, if you convert a binary number in IEEE single format to a decimal number, and then convert it back to binary, generally, you need to use at least 9 decimal digits to ensure that after these two conversions you obtain the number you started from.
The complete picture is given in TABLE 2-10:
Underflow
Underflow occurs, roughly speaking, when the result of an arithmetic operation is so small that it cannot be stored in its intended destination format without suffering a rounding error that is larger than usual.
Underflow Thresholds
TABLE 2-11 shows the underflow thresholds for single, double.
| single | smallest normal number largest subnormal number |
1.17549435e-38 1.17549421e-38 |
| double | smallest normal number largest subnormal number |
2.2250738585072014e-308 2.2250738585072009e-308 |
The positive subnormal numbers are those numbers between the smallest normal number and zero. Subtracting two (positive) tiny numbers that are near the smallest normal number might produce a subnormal number. Or, dividing the smallest positive normal number by two produces a subnormal result.
The presence of subnormal numbers provides greater precision to floating-point calculations that involve small numbers, although the subnormal numbers themselves have fewer bits of precision than normal numbers. Producing subnormal numbers (rather than returning the answer zero) when the mathematically correct result has magnitude less than the smallest positive normal number is known as gradual underflow.
There are several other ways to deal with such underflow results. One way, common in the past, was to flush those results to zero. This method is known as Store 0 and was the default on most mainframes before the advent of the IEEE Standard.
The mathematicians and computer designers who drafted IEEE Standard 754 considered several alternatives while balancing the desire for a mathematically robust solution with the need to create a standard that could be implemented efficiently.