IEEE 754-1985 is a historic industry standard for representing floating-point numbers in computers, officially adopted in 1985 and superseded in 2008 by IEEE 754-2008, and then again in 2019 by minor revision IEEE 754-2019. During its 23 years, it was the most widely used format for floating-point computation. It was implemented in software, in the form of floating-point libraries, and in hardware, in the instructions of many CPUs and FPUs. The first integrated circuit to implement the draft of what was to become IEEE 754-1985 was the Intel 8087. IEEE 754-1985 represents numbers in binary, providing definitions for four levels of precision, of which the two most commonly used are:
The standard also defines representations for positive and negative infinity, a "negative zero", five exceptions to handle invalid results like division by zero, special values called NaNs for representing those exceptions, denormal numbers to represent numbers smaller than shown above, and four rounding modes.
Representation of numbers
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:
0.00101 2 = 1.01 2 × 2 − 3 {\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."
Zero The number zero is represented specially:
sign = 0 for positive zero, 1 for negative zero. biased exponent = 0. fraction = 0.
Denormalized numbers The number representations described above are called normalized, meaning that the implicit leading binary digit is a 1. To reduce the loss of precision when an underflow occurs, IEEE 754 includes the ability to represent fractions smaller than are possible in the normalized representation, by making the implicit leading digit a 0. Such numbers are called denormal. They don't include as many significant digits as a normalized number, but they enable a gradual loss of precision when the result of an operation is not exactly zero but is too close to zero to be represented by a normalized number. A denormal number is represented with a biased exponent of all 0 bits, which represents an exponent of −126 in single precision (not −127), or −1022 in double precision (not −1023). In contrast, the smallest biased exponent representing a normal number is 1 (see examples below).
Representation of non-numbers The biased-exponent field is filled with all 1 bits to indicate either infinity or an invalid result of a computation.
Positive and negative infinity Positive and negative infinity are represented thus:
sign = 0 for positive infinity, 1 for negative infinity. biased exponent = all 1 bits. fraction = all 0 bits.
NaN Some operations of floating-point arithmetic are invalid, such as taking the square root of a negative number. The act of reaching an invalid result is called a floating-point exception. An exceptional result is represented by a special code called a NaN, for "Not a Number". All NaNs in IEEE 754-1985 have this format:
sign = either 0 or 1. biased exponent = all 1 bits. fraction = anything except all 0 bits (since all 0 bits represents infinity).
Range and precision
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.
Single precision Single-precision numbers occupy 32 bits. In single precision:
The positive and negative numbers closest to zero (represented by the denormalized value with all 0s in the exponent field and the binary value 1 in the fraction field) are ±2−23 × 2−126 ≈ ±1.40130×10−45 The positive and negative normalized numbers closest to zero (represented with the binary value 1 in the exponent field and 0 in the fraction field) are ±1 × 2−126 ≈ ±1.17549×10−38 The finite positive and finite negative numbers furthest from zero (represented by the value with 254 in the exponent field and all 1s in the fraction field) are ±(2−2−23) × 2127 ≈ ±3.40282×1038 Some example range and gap values for given exponents in single precision:
As an example, 16,777,217 cannot be encoded as a 32-bit float as it will be rounded to 16,777,216. However, all integers within the representable range that are a power of 2 can be stored in a 32-bit float without rounding.
Double precision Double-precision numbers occupy 64 bits. In double precision:
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