IEEE Floating-Point Representations of Real Numbers

Summary: In this reading, we consider mechanisms for representing real numbers with a fixed number of bits. We emphasize the IEEE floating-point representations.

• The Problem: Representing Reals
• A First Approach: Fixed Point Numbers
• Detour: Scientific Notation
• A Second Solution: Floating Point Representations
• The IEEE Standard
  • An Example: From Binary to Decimal
  • An Example: From Decimal to Binary
  • Bounds
  • Representing Zero
  • Some Special Values
• Double-Precision Representation

The Problem: Representing Reals

No human system of numeration can give a unique representation to every real number; there are just too many of them. So it is conventional to use approximations. For instance, the assertion that pi is 3.14159 is, strictly speaking, false, since pi is actually slightly larger than 3.14159; but in practice we sometimes use 3.14159 in calculations involving pi because it is a good enough approximation of pi.

A First Approach: Fixed Point Numbers

One approach to representing real numbers, then, is to specify some tolerance epsilon and to say that a real number x can be approximated by any number in the range from x - epsilon to x + epsilon. Then, if a system of numeration can represent selected numbers that are never more than twice epsilon apart, every real number has a representable approximation. For instance, in the United States, the prices of stocks are given in dollars and eighths of a dollar, and rounded to the nearest eighth of a dollar; this corresponds to a tolerance of one-sixteenth of a dollar. In retail commerce, however, the conventional tolerance is half a cent; that is, prices are rounded to the nearest cent. In this case, we can represent a sum of money as an whole number of cents, or equivalently as a number of dollars that is specified to two decimal places.

Such a representation, in which each real number is represented by a numeral for an approximation to some fixed number of decimal places, is called a fixed-point representation. However, fixed-point representations are unsatisfactory for most applications involving real numbers, for two reasons

1. Real numbers that are very small are not clearly distinguished. For instance, in a fixed-point representation using two decimal places, 14/1000 and 6/1000 would both be represented as 0.01, even though the former is more than twice as large as the latter. Moreover, both 4/1000 and -7/10000 would be represented as 0.00, and they don't even have the same sign! If one is counting dollars, these differences are probably irrelevant, but there are a lot of scientific and technological applications (navigation systems, for instance) in which they are critical.
2. In many cases, real numbers that are very large are not known accurately enough for a fixed-point representation to be selected. For instance, the speed of light in a vacuum is roughly 299792458 meters per second, but no one knows whether it is best approximated by 299792458.63 or 299792458.16 or 299792457.88, and there is no reason to prefer one of these fixed-point representations to the others.

These two considerations really have the same underlying cause: When approximating some real number, what we usually know is some number of significant digits at the beginning of the number. A fixed-point system of representation conceals some of what we know when the number is small, because some of these significant digits are too far to the right of the decimal point, and demands more than we know when the number is large, because the number of digits demanded by the representation system is greater than the number that we can provide.

Detour: Scientific Notation

Scientists and engineers long ago learned to cope with this problem by using scientific notation, in which a number is expressed as the product of a mantissa and some power of ten. The mantissa is a signed number with an absolute value greater than or equal to one and less than ten. So, for instance, the speed of light in vacuum is 2.99792458 x 108 meters per second, and one can specify only the digits about which one is completely confident.

Using scientific notation, one can easily see both that 1.4 x 10-2 is more than twice as large as 6 x 10-3, and that both are close to 1 x 10-2; and one can easily distinguish 4 x 10-3 and -7 x 10-4 as small numbers of opposite sign. The rules for calculating with scientific-notation numerals are a little more complicated, but the benefits are enormous.

A Second Solution: Floating Point Representations

The three varying components of a number represented in scientific notation are

1. the sign of the number (positive or negative),
2. the absolute value of the mantissa,
3. the exponent on the power of ten.

A system of numeration for real numbers that is adapted to computers will typically store the same three kinds of values (a sign, a mantissa, and an exponent) into an allocated region of storage. By contrast with fixed-point representations, these computer analogues of scientific notation are described as floating-point representations.

The exponent does not always indicate a power of ten; sometimes powers of sixteen are used instead, or, most commonly of all, powers of two. The numerals will be somewhat different depending how this choice is made. For instance, the real number -0.125 will be expressed as -1.25 x 10-1 if powers of ten are used, or as -2 x 16-1 if powers of sixteen are used, or as -1 x 2-3 if powers of two are used. The absolute value of the mantissa is, however, always greater than or equal to 1 and less than the base of numeration.

The IEEE Standard

The particular system used in many modern computers and languages (including Java) was formulated and recommended as a standard by the Institute of Electrical and Electronics Engineers (IEEE) and is the most commonly used numeration system for computer representation of real numbers. Actually, their standard includes several variants of the system, depending on how much storage is available for a real number. We'll discuss two of these variants, both of which use binary numeration and powers of 2: the IEEE single-precision representation, which fits in thirty-two bits, and the IEEE double-precision representation, which occupies sixty-four bits. We'll begin with single-precision numbers.

In the IEEE single-precision representation of a real number, one bit is reserved for the sign, and it is set to 0 for a positive number and to 1 for a negative one. A representation of the exponent is stored in the next eight bits, and the remaining twenty-three bits are occupied by a representation of the mantissa of the number.

The exponent, which is a signed integer in the range from -126 to 127, is represented neither as a signed magnitude nor as a twos-complement number, but as a biased value. The idea here is that the integers in the desired range of exponents are first adjusted by adding a fixed bias to each one. The bias is chosen to be large enough to convert every integer in the range into a positive integer, which is then stored as a binary numeral. The IEEE single-precision representation uses a bias of 127.

For example, the exponent -5 is represented by the eight-bit pattern 01111010, which is the binary numeral for 122, since -5 + 127 = 122. The least allowable exponent, -126, is represented by 00000001 (since -126 + 127 = 1); the greatest allowable exponent, 127, is represented by the binary numeral for 254, 11111110. Note that neither 00000000 nor 11111111 are allowed as exponents.

Since the base of numeration is two, the mantissa is always a number greater than or equal to one and less than two. Its fractional part is represented, using binary numeration, as a sum of the negative powers of two. For instance, the binary numeral for 21/16 is 1.0101 -- one, plus no halves, plus one quarter, plus no eighths, plus one sixteenth. (The decimal point in this case is actually a binary point, separating the digit in the units place from the digits representing multiples of negative powers of two.)

Somewhat surprisingly, this means that approximations are used for many real numbers that can be represented exactly in decimal numeration. For instance, 7/5 is 1.4 exactly in decimal numeration, but the .4 part cannot be expressed as a sum of powers of two; 7/5 has an infinite binary expansion 1.011001100110011001100..., just as in decimal numeration a fraction like 27/11 leads to an infinite decimal expansion 2.4545454545.... In a single-precision representation, the expansion is rounded off at the twenty-third digit after the binary point. Thus 7/5 is not actually stored as 7/5, but (in single precision) as the very close approximation 11744051/8388608, which can be expressed as a sum of powers of two.

Only the part of the mantissa that comes after the binary point is actually stored, since the bit to the left of the binary point is completely predictable (it's always 1, since the mantissa is always greater than or equal to one and less than two). This suppressed digit at the beginning of the mantissa is called the hidden bit.

An Example: From Binary to Decimal

Here's an example. Consider the thirty-two-bit word

11000011100101100000000000000000

The number represented by this sequence of bits is negative (because the sign bit is 1). It has an exponent of 8 (because the exponent bits, 10000111, form the binary numeral for 135, and therefore represent the exponent 8 when the bias of 127 is removed). Its mantissa is expressed by the binary numeral 1.00101100000000000000000, where the initial 1 is the hidden bit and the remaining digits are taken from the right end of the word. This last binary numeral expresses the number 75/64 (one, plus no halves, plus no quarters, plus one eighth, plus no sixteenths, plus one thirty-second, plus one sixty-fourth). So the complete number is -(75/64) x 2⁸, which is -300.0.

An Example: From Decimal to Binary

Conversely, let us find the IEEE single-precision representation of, say, 5.75. The sign bit is 0, since the number is positive. 5.75 is 23/4; to express this as the product of a power of two and a mantissa greater than or equal to one and less than two, one must factor out 2²: 23/4 = (23/16) x 2², so the exponent is 2. It will be stored, with a bias of 127, as 10000001 (the binary numeral for 129). The complete mantissa, extended to twenty-three digits after the binary point, is 1.01110000000000000000000. We line up the sign bit, the biased representation of the exponent, and the digits following the binary point in the mantissa and get

01000000101110000000000000000000

Bounds

The greatest number that has an exact IEEE single-precision representation is 340282346638528859811704183484516925440.0 (2¹²⁸ - 2¹⁰⁴), which is represented by

01111111011111111111111111111111

and the least is the negative of this number, which has the same representation except that the sign bit is 1.

Representing Zero

The alert reader will have noticed that there is a serious gap in this scheme, as so far described: What about 0.0? It is not possible to represent 0.0 as the product of a power of two and a mantissa greater than or equal to one, so none of the representations described above will do. However, not all the possible settings of thirty-two switches have been used for numbers. Recall that the exponents permitted in IEEE single-precision reals range from -126 to +127, so that the binary numerals for the biased exponents range from 00000001 to 11111110. We haven't yet used any of the bit patterns in which the exponent bits are all zeroes or all ones.

In the IEEE system, the all-zero exponent is used for numbers that are very close to zero -- closer than 2^-126, which is the least of the positive reals that can be represented in the part of the system described above (it is 00000000100000000000000000000000). Such numbers are expressed in a slightly different form of scientific notation: The exponent is held fixed at -126, and the mantissa is a number greater than or equal to zero and less than one. So, for instance, the mantissa used for 3 x 2^-129 is 0.011 (3 x 2^-129 is a quarter plus an eighth of 2^-126). Once again, only the part of the mantissa that follows the binary point is stored explicitly, so the representation of 3 x 2^-129 is

 00000000001100000000000000000000 

(sign positive, exponent -126, mantissa 0.01100000000000000000000). Mantissas less than one are said to be unnormalized (because the normal form is the one in which the mantissa is greater than or equal to one and less than the base of numeration), so an all-zero exponent indicates an unnormalized number. Unnormalized numbers are stored less accurately than normalized ones (since there are fewer significant digits in the mantissa), but without this special convention for the all-zero exponent it would not be possible to represent them at all, and the designers of the IEEE standard felt that a degraded approximation is better than none.

This convention allows zero to be represented in two different ways. In one, the sign bit is 0, the exponent bits are 00000000, and the visible bits of the mantissa are 00000000000000000000000 -- yielding 0.00000000000000000000000 x 2^-126, or 0. The other representation is the same, except that the sign bit is 1. (So, as in signed-magnitude representations, there is a non-negative and a non-positive zero.)

Some Special Values

The least positive real number that can be represented exactly in this way is 2^-149, which is stored as

00000000000000000000000000000001

When the exponent bits are all ones, the value represented is not a real number at all, but a conventional signal of a computation that has gone wrong, either by going above the greatest representable real or below the least, or by attempting some undefined arithmetic operation, such as dividing by zero or taking the logarithm of a negative number. For instance, the thirty-two bits

01111111100000000000000000000000

represent positive infinity, a pseudo-number that indicates that some unrepresentably large quantity was generated by an arithmetic operation. Changing the sign bit to 1 yields a representation of negative infinity, an indication of a similar problem at the other end of the range.

If some of the bits of the mantissa are 1s, the pseudo-number is called a NaN (Not a Number); trying to compute 0/0, for instance, typically produces a NaN. It should be clear that positive infinity, negative infinity, and NaN are not real numbers, although some language implementations will try to do something sensible if they appear in places that are normally occupied by real numbers. The value of such attempts at recovery is questionable, however, since the appearance of a pseudo-number is supposed to be a danger signal to the programmer and usually results from a programming error.

Double-Precision Representation

IEEE double-precision representations are quite similar. A double-precision real begins with a sign bit, with the same interpretation as in a single-precision representation. The next eleven bits are used for the exponent, which is an integer in the range from -1022 to +1023; a bias of 1023 is added to the exponent, and the result is stored in a binary numeral (the smallest is 00000000001, the largest 11111111110). The remaining fifty-two bits are used for the mantissa, and as above only the digits following the binary point of the mantissa are actually stored; the 1 that precedes the binary point is once again a hidden bit. As in single-precision representations, the all-zero exponent is used for unnormalized numbers and (with an all-zero mantissa) for 0, and the all-one exponent is used for the pseudo-numbers positive infinity, negative infinity, and NaN. The greatest real number that can be represented exactly as a double-precision real is 2¹⁰²⁴ - 2⁹⁷¹, and the least positive real that can be so represented is 2^-1074.