IEEE 754-1985

The number 0.15625 represented as a single-precision IEEE 754-1985 floating-point number. See text for explanation.

The three fields in a 64bit IEEE 754 float

Floating-point numbers in IEEE 754 format consist of three fields: a sign bit, a biased exponent, and a fraction. The following example illustrates the meaning of each.

The decimal number 0.15625₁₀ represented in binary is 0.00101₂ (that is, 1/8 + 1/32). (Subscripts indicate the number base.) Analogous to scientific notation, where numbers are written to have a single non-zero digit to the left of the decimal point, we rewrite this number so it has a single 1 bit to the left of the "binary point". We simply multiply by the appropriate power of 2 to compensate for shifting the bits left by three positions:

${\displaystyle 0.00101_{2}=1.01_{2}\times 2^{-3}}$

Now we can read off the fraction and the exponent: the fraction is .01₂ and the exponent is −3.

As illustrated in the pictures, the three fields in the IEEE 754 representation of this number are:

sign = 0, because the number is positive. (1 indicates negative.)

biased exponent = −3 + the "bias". In single precision, the bias is 127, so in this example the biased exponent is 124; in double precision, the bias is 1023, so the biased exponent in this example is 1020.

fraction = .01000…₂.

IEEE 754 adds a bias to the exponent so that numbers can in many cases be compared conveniently by the same hardware that compares signed 2's-complement integers. Using a biased exponent, the lesser of two positive floating-point numbers will come out "less than" the greater following the same ordering as for sign and magnitude integers. If two floating-point numbers have different signs, the sign-and-magnitude comparison also works with biased exponents. However, if both biased-exponent floating-point numbers are negative, then the ordering must be reversed. If the exponent were represented as, say, a 2's-complement number, comparison to see which of two numbers is greater would not be as convenient.

The leading 1 bit is omitted since all numbers except zero start with a leading 1; the leading 1 is implicit and doesn't actually need to be stored which gives an extra bit of precision for "free."

The biased-exponent field is filled with all 1 bits to indicate either infinity or an invalid result of a computation.

Relative precision of single (binary32) and double precision (binary64) numbers, compared with decimal representations using a fixed number of significant digits. Relative precision is defined here as ulp(x)/x, where ulp(x) is the unit in the last place in the representation of x, i.e. the gap between x and the next representable number.

Precision is defined as the minimum difference between two successive mantissa representations; thus it is a function only in the mantissa; while the gap is defined as the difference between two successive numbers.[4]

Here are some examples of single-precision IEEE 754 representations:

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 combinations of bits for 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 as sign and magnitude integers (endianness issues apply). When comparing as 2's-complement integers: If the sign bits differ, the negative number precedes the positive number, so 2's complement gives the correct result (except that negative zero and positive zero should be considered equal). If both values are positive, the 2's complement comparison again gives the correct result. Otherwise (two negative numbers), the correct FP ordering is the opposite of the 2's complement ordering.

Rounding errors inherent to floating point calculations may limit the use of comparisons for checking the exact equality of results. Choosing an acceptable range is a complex topic. A common technique is to use a comparison epsilon value to perform approximate comparisons.[6] Depending on how lenient the comparisons are, common values include 1e-6 or 1e-5 for single-precision, and 1e-14 for double-precision.[7][8] Another common technique is ULP, which checks what the difference is in the last place digits, effectively checking how many steps away the two values are.[9]

Although negative zero and positive zero are generally considered equal for comparison purposes, some programming language relational operators and similar constructs treat them as distinct. According to the Java Language Specification,[10] 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.

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 means it is rounded up 50% of the time (in IEEE 754-2008 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.

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.[11][12][13]

In 1976, Intel was starting the development of a floating-point coprocessor.[14][15] Intel hoped to be able to sell a chip containing good implementations of all the operations found in the widely varying maths software libraries.[14][16]

John Palmer, who managed the project, believed the effort should be backed by a standard unifying floating point operations across disparate processors. He contacted William Kahan of the University of California, who had helped improve the accuracy of Hewlett-Packard's calculators. Kahan suggested that Intel use the floating point of Digital Equipment Corporation's (DEC) VAX. The first VAX, the VAX-11/780 had just come out in late 1977, and its floating point was highly regarded. However, seeking to market their chip to the broadest possible market, Intel wanted the best floating point possible, and Kahan went on to draw up specifications.[14] Kahan initially recommended that the floating point base be decimal[17] but the hardware design of the coprocessor was too far along to make that change.

The work within Intel worried other vendors, who set up a standardization effort to ensure a "level playing field". Kahan attended the second IEEE 754 standards working group meeting, held in November 1977. He subsequently received permission from Intel to put forward a draft proposal based on his work for their coprocessor; he was allowed to explain details of the format and its rationale, but not anything related to Intel's implementation architecture. The draft was co-written with Jerome Coonen and Harold Stone, and was initially known as the "Kahan-Coonen-Stone proposal" or "K-C-S format".[14][15][16][18]

As an 8-bit exponent was not wide enough for some operations desired for double-precision numbers, e.g. to store the product of two 32-bit numbers,[19] both Kahan's proposal and a counter-proposal by DEC therefore used 11 bits, like the time-tested 60-bit floating-point format of the CDC 6600 from 1965.[15][18][20] Kahan's proposal also provided for infinities, which are useful when dealing with division-by-zero conditions; not-a-number values, which are useful when dealing with invalid operations; denormal numbers, which help mitigate problems caused by underflow;[18][21][22] and a better balanced exponent bias, which can help avoid overflow and underflow when taking the reciprocal of a number.[23][24]

Even before it was approved, the draft standard had been implemented by a number of manufacturers.[25][26] The Intel 8087, which was announced in 1980, was the first chip to implement the draft standard.

Intel 8087 floating-point coprocessor

In 1980, the Intel 8087 chip was already released,[27] but DEC remained opposed, to denormal numbers in particular, because of performance concerns and since it would give DEC a competitive advantage to standardise on DEC's format.

The arguments over gradual underflow lasted until 1981 when an expert hired by DEC to assess it sided against the dissenters. DEC had the study done in order to demonstrate that gradual underflow was a bad idea, but the study concluded the opposite, and DEC gave in. In 1985, the standard was ratified, but it had already become the de facto standard a year earlier, implemented by many manufacturers.[15][18][5]

• IEEE 754
• Minifloat for simple examples of properties of IEEE 754 floating point numbers
• Fixed-point arithmetic

• Charles Severance (March 1998). "IEEE 754: An Interview with William Kahan" (PDF). IEEE Computer. 31 (3): 114–115. doi:10.1109/MC.1998.660194. S2CID 33291145. Archived from the original (PDF) on 2009-08-23. Retrieved 2008-04-28.
• David Goldberg (March 1991). "What Every Computer Scientist Should Know About Floating-Point Arithmetic" (PDF). ACM Computing Surveys. 23 (1): 5–48. doi:10.1145/103162.103163. S2CID 222008826. Retrieved 2008-04-28.
• Chris Hecker (February 1996). "Let's Get To The (Floating) Point" (PDF). Game Developer Magazine: 19–24. ISSN 1073-922X. Archived from the original (PDF) on 2007-02-03.
• David Monniaux (May 2008). "The pitfalls of verifying floating-point computations". ACM Transactions on Programming Languages and Systems. 30 (3): 1–41. arXiv:cs/0701192. doi:10.1145/1353445.1353446. ISSN 0164-0925. S2CID 218578808.: A compendium of non-intuitive behaviours of floating-point on popular architectures, with implications for program verification and testing.

• Comparing floats
• Coprocessor.info: x87 FPU pictures, development and manufacturer information
• IEEE 854-1987 — History and minutes
• IEEE754 (Single and Double precision) Online Converter