For example, a significand of is encoded as binary , with the leading 4 bits encoding 7; the first significand which requires a 24th bit (and thus the second encoding form) is 223 = . In the above cases, the value represented is: : (−1)sign × 10exponent−101 × significand Decimal64 and Decimal128 operate analogously, but with larger exponent continuation and significand fields. For Decimal128, the second encoding form is actually never used; the largest valid significand of 1034−1 = 1ED09BEAD87C0378D8E63FFFFFFFF16 can be represented in 113 bits.
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IEEE 754 defines the precision p to be the number of digits in the significand, including any implicit leading bit (e.g., p = 53 for the double-precision format), thus in a way independent from the encoding, and the term to express what is encoded (that is, the significand without its leading bit) is trailing significand field.
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As with IEEE 754-1985, the biased-exponent field is filled with all 1 bits to indicate either infinity (trailing significand field = 0) or a NaN (trailing significand field ≠ 0). For NaNs, quiet NaNs and signaling NaNs are distinguished by using the most significant bit of the trailing significand field exclusively, and the payload is carried in the remaining bits.
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IEEE 854 arithmetic was first commercially implemented in the HP-71B handheld computer, which used decimal floating point with 12 digits of significand, and an exponent range of ±499, with a 15 digit significand used for intermediate results.
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A binary floating number contains a sign bit, significant bits (known as the significand) and exponent bits (for simplicity, we don't consider base and combination field). The sign bits of each operand are XOR'd to get the sign of the answer. Then, the two exponents are added to get the exponent of the result. Finally, multiplication of each operand's significand will return the significand of the result.
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The number is represented as the following formula: (−1)sign × 0.significand × 16exponent−64.
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In this version, the significand is stored as a series of decimal digits. The leading digit is between 0 and 9 (3 or 4 binary bits), and the rest of the significand uses the densely packed decimal (DPD) encoding. The leading 2 bits of the exponent and the leading digit (3 or 4 bits) of the significand are combined into the five bits that follow the sign bit. These six bits after that are the exponent continuation field, providing the less-significant bits of the exponent. The last 20 bits are the significand continuation field, consisting of two 10-bit declets.
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In this version, the significand is stored as a series of decimal digits. The leading digit is between 0 and 9 (3 or 4 binary bits), and the rest of the significand uses the densely packed decimal (DPD) encoding. The leading 2 bits of the exponent and the leading digit (3 or 4 bits) of the significand are combined into the five bits that follow the sign bit. This twelve bits after that are the exponent continuation field, providing the less-significant bits of the exponent. The last 110 bits are the significand continuation field, consisting of eleven 10-bit declets.
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For a normalized number, the most significant digit is always non-zero. When working in binary, this constraint uniquely determines this digit to always be 1; as such, it does not need to be explicitly stored, being called the hidden bit. The significand is characterized by its width in (binary) digits, and depending on the context, the hidden bit may or may not be counted towards the width of the significand. For example, the same IEEE 754 double-precision format is commonly described as having either a 53-bit significand, including the hidden bit, or a 52-bit significand, excluding the hidden bit.
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The significand (also mantissa or coefficient, sometimes also argument, or ambiguously fraction or characteristic) is part of a number in scientific notation or a floating-point number, consisting of its significant digits. Depending on the interpretation of the exponent, the significand may represent an integer or a fraction.
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The leading bits of the significand field do not encode the most significant decimal digit; they are simply part of a larger pure-binary number. For example, a significand of is encoded as binary 2, with the leading 4 bits encoding 7; the first significand which requires a 54th bit is 253 = . The highest valid significant is whose binary encoding is 2 (with the 3 most significant bits (100) not stored but implicit as shown above; and the next bit is always zero in valid encodings).
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In this version, the significand is stored as a series of decimal digits. The leading digit is between 0 and 9 (3 or 4 binary bits), and the rest of the significand uses the densely packed decimal (DPD) encoding. The leading 2 bits of the exponent and the leading digit (3 or 4 bits) of the significand are combined into the five bits that follow the sign bit. This eight bits after that are the exponent continuation field, providing the less-significant bits of the exponent.
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The IEEE 754-2008 standard includes decimal floating-point number formats in which the significand and the exponent (and the payloads of NaNs) can be encoded in two ways, referred to as binary encoding and decimal encoding. Both formats break a number down into a sign bit s, an exponent q (between qmin and qmax), and a p-digit significand c (between 0 and 10p−1). The value encoded is (−1)s×10q×c. In both formats the range of possible values is identical, but they differ in how the significand c is represented.
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In a normal floating-point value, there are no leading zeros in the significand; rather, leading zeros are removed by adjusting the exponent (for example, the number 0.0123 would be written as ). Denormal numbers are numbers where this representation would result in an exponent that is below the smallest representable exponent (the exponent usually having a limited range). Such numbers are represented using leading zeros in the significand. The significand (or mantissa) of an IEEE floating- point number is the part of a floating-point number that represents the significant digits.
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Unums ("Universal Numbers") are an extension of variable length arithmetic proposed by John Gustafson. Unums have variable length fields for the exponent and significand lengths and error information is carried in a single bit, the ubit, representing possible error in the least significant bit of the significand (ULP). The efficacy of unums is questioned by William Kahan.
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If the 2 bits after the sign bit are "11", then the 10-bit exponent field is shifted 2 bits to the right (after both the sign bit and the "11" bits thereafter), and the represented significand is in the remaining 51 bits. In this case there is an implicit (that is, not stored) leading 3-bit sequence "100" for the most bits of the true significand (in the remaining lower bits ttt...ttt of the significand, not all possible values are used). s 1100eeeeeeee (100)t tttttttttt tttttttttt tttttttttt tttttttttt tttttttttt s 1101eeeeeeee (100)t tttttttttt tttttttttt tttttttttt tttttttttt tttttttttt s 1110eeeeeeee (100)t tttttttttt tttttttttt tttttttttt tttttttttt tttttttttt The 2-bit sequence "11" after the sign bit indicates that there is an implicit 3-bit prefix "100" to the significand. Compare having an implicit 1-bit prefix "1" in the significand of normal values for the binary formats. The 2-bit sequences "00", "01", or "10" after the sign bit are part of the exponent field.
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For base 2, this 1.xxxx form is also called a normalized significand. Finally, the value can be represented in the format given by the Language Independent Arithmetic standard and several programming language standards, including Ada, C, Fortran and Modula-2, as : 123.45 = 0.12345 × 10+3. Schmid called this representation with a significand ranging between 0.1 and 1.0 the true normalized form.
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This format uses a binary significand from 0 to 1016 − 1 = = 2386F26FC0FFFF16 = . The encoding, completely stored on 64 bits, can represent binary significands up to 10 × 250 − 1 = = 27FFFFFFFFFFFF16, but values larger than 1016 − 1 are illegal (and the standard requires implementations to treat them as 0, if encountered on input). As described above, the encoding varies depending on whether the most significant 4 bits of the significand are in the range 0 to 7 (00002 to 01112), or higher (10002 or 10012). If the 2 bits after the sign bit are "00", "01", or "10", then the exponent field consists of the 10 bits following the sign bit, and the significand is the remaining 53 bits, with an implicit leading 0 bit: s 00eeeeeeee (0)ttt tttttttttt tttttttttt tttttttttt tttttttttt tttttttttt s 01eeeeeeee (0)ttt tttttttttt tttttttttt tttttttttt tttttttttt tttttttttt s 10eeeeeeee (0)ttt tttttttttt tttttttttt tttttttttt tttttttttt tttttttttt This includes subnormal numbers where the leading significand digit is 0.
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The IBM 1130, sold in 1965, offered two floating-point formats: A 32-bit "standard precision" format and a 40-bit "extended precision" format. Standard precision format contains a 24-bit two's complement significand while extended precision utilizes a 32-bit two's complement significand. The latter format makes full use of the CPU's 32-bit integer operations. The characteristic in both formats is an 8-bit field containing the power of two biased by 128.
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In IEEE 754 decimal floating-point encoding, a negative zero is represented by an exponent being any valid exponent in the range for the encoding, the true significand being zero, and the sign bit being one.
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For this reason, the use of mantissa for significand is discouraged by some including the creator of the standard, William Kahan and prominent computer programmer and author of The Art of Computer Programming, Donald E. Knuth The confusion is because scientific notation and floating- point representation are log-linear, not logarithmic. To multiply two numbers, given their logarithms, one just adds the characteristic (integer part) and the mantissa (fractional part). By contrast, to multiply two floating-point numbers, one adds the exponent (which is logarithmic) and multiplies the significand (which is linear).
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These give decimal interchange formats with 7, 16, and 34-digit significands, which may be normalized or unnormalized. For maximum range and precision, the formats merge part of the exponent and significand into a combination field, and compress the remainder of the significand using either a decimal integer encoding (which uses Densely Packed Decimal, or DPD, a compressed form of BCD) encoding or conventional binary integer encoding. The basic formats are the two larger sizes, which have 64-bit and 128-bit encodings. Generalized formulae for some other interchange formats are also specified.
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For the exchange of decimal floating-point numbers, interchange formats of any multiple of 32 bits are defined. As with binary interchange, the encoding scheme for the decimal interchange formats encodes the sign, exponent, and significand. Two different bit-level encodings are defined, and interchange is complicated by the fact that some external indicator of the encoding in use may be required. The two options allow the significand to be encoded as a compressed sequence of decimal digits using densely packed decimal or, alternatively, as a binary integer.
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The number 123.45 can be represented as a decimal floating-point number with the integer 12345 as the significand and a 10−2 power term, also called characteristics, where −2 is the exponent (and 10 is the base). Its value is given by the following arithmetic: : 123.45 = 12345 × 10−2. This same value can also be represented in normalized form with 1.2345 as the fractional coefficient, and +2 as the exponent (and 10 as the base): : 123.45 = 1.2345 × 10+2. Schmid, however, called this representation with a significand ranging between 1.0 and 10 a modified normalized form.
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To multiply, the significands are multiplied while the exponents are added, and the result is rounded and normalized. e=3; s=4.734612 × e=5; s=5.417242 \----------------------- e=8; s=25.648538980104 (true product) e=8; s=25.64854 (after rounding) e=9; s=2.564854 (after normalization) Similarly, division is accomplished by subtracting the divisor's exponent from the dividend's exponent, and dividing the dividend's significand by the divisor's significand. There are no cancellation or absorption problems with multiplication or division, though small errors may accumulate as operations are performed in succession. In practice, the way these operations are carried out in digital logic can be quite complex (see Booth's multiplication algorithm and Division algorithm).
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Encodings of qNaN and sNaN are not specified in IEEE 754 and implemented differently on different processors. The x86 family and the ARM family processors use the most significant bit of the significand field to indicate a quiet NaN. The PA-RISC processors use the bit to indicate a signalling NaN.
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There are 27 numeric variables, `A` through `Z`, and `θ`. These can hold two types of values, real and complex. All numbers are stored in the RAM as floating-point numbers with 14-digit mantissa, or significand, and an exponent range of -128 to 127. Complex numbers are stored as two consecutive reals.
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In comparison to IEEE 754 floating-point, the IBM floating-point format has a longer significand, and a shorter exponent. All IBM floating-point formats have 7 bits of exponent with a bias of 64. The normalized range of representable numbers is from 16−65 to 1663 (approx. 5.39761 × 10−79 to 7.237005 × 1075).
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The IEEE 754 floating-point standard recommends that implementations provide extended precision formats. The standard specifies the minimum requirements for an extended format but does not specify an encoding. The encoding is the implementor's choice. The IA32, x86-64, and Itanium processors support an 80-bit "double extended" extended precision format with a 64-bit significand.
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In computing, decimal64 is a decimal floating-point computer numbering format that occupies 8 bytes (64 bits) in computer memory. It is intended for applications where it is necessary to emulate decimal rounding exactly, such as financial and tax computations. Decimal64 supports 16 decimal digits of significand and an exponent range of −383 to +384, i.e. to .
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In computing, decimal128 is a decimal floating-point computer numbering format that occupies 16 bytes (128 bits) in computer memory. It is intended for applications where it is necessary to emulate decimal rounding exactly, such as financial and tax computations. Decimal128 supports 34 decimal digits of significand and an exponent range of −6143 to +6144, i.e. to .
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For ease of presentation and understanding, decimal radix with 7 digit precision will be used in the examples, as in the IEEE 754 decimal32 format. The fundamental principles are the same in any radix or precision, except that normalization is optional (it does not affect the numerical value of the result). Here, s denotes the significand and e denotes the exponent.
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Nowadays, the word mantissa is generally used to describe the fractional part of a floating-point number on computers, though the recommended term is significand. Thus, log tables need only show the fractional part. Tables of common logarithms typically listed the mantissa, to four or five decimal places or more, of each number in a range, e.g., 1000 to 9999.
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Octuple precision is rarely implemented since usage of it is extremely rare. Apple Inc. had an implementation of addition, subtraction and multiplication of octuple-precision numbers with a 224-bit two's complement significand and a 32-bit exponent. One can use general arbitrary-precision arithmetic libraries to obtain octuple (or higher) precision, but specialized octuple-precision implementations may achieve higher performance.
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The former is more convenient for direct hardware implementation of the standard, while the latter is more suited to software emulation on a binary computer. In either case, the set of numbers (combinations of sign, significand, and exponent) that may be encoded is identical, and special values (±zero with the minimum exponent, ±infinity, quiet NaNs, and signaling NaNs) have identical encodings.
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An Elektronika B3-34 Elektronika B3-34 (Cyrillic: Электроника Б3-34) was a Soviet programmable calculator. It was released in 1980 and was sold for 85 rubles. B3-34 used reverse Polish notation and had 98 bytes of instruction memory, four stack user registers and 14 addressable registers. Each register could store up to 8 mantissa or Significand digits and two exponent digits in the range to .
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In computing, decimal32 is a decimal floating-point computer numbering format that occupies 4 bytes (32 bits) in computer memory. It is intended for applications where it is necessary to emulate decimal rounding exactly, such as financial and tax computations. Like the binary16 format, it is intended for memory saving storage. Decimal32 supports 7 decimal digits of significand and an exponent range of −95 to +96, i.e.
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While the machine epsilon is not to be confused with the underflow level (assuming subnormal numbers), it is closely related. The machine epsilon is dependent on the number of bits which make up the significand, whereas the underflow level depends on the number of digits which make up the exponent field. In most floating point systems, the underflow level is smaller than the machine epsilon.
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Just as in IEEE 754, NaN values are represented with either sign bit, all 8 exponent bits set (FFhex) and not all significand bits zero. Explicitly, val s_exponent_signcnd +NaN = 0_11111111_klmnopq -NaN = 1_11111111_klmonpq where at least one of k, l, m, n, o, p, or q is 1. As with IEEE 754, NaN values can be quiet or signaling, although there are no known uses of signaling bfloat16 NaNs as of September 2018.
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In this notation the significand is always meant to be hexadecimal, whereas the exponent is always meant to be decimal. This notation can be produced by implementations of the printf family of functions following the C99 specification and (Single Unix Specification) IEEE Std 1003.1 POSIX standard, when using the %a or %A conversion specifiers. Starting with C++11, C++ I/O functions could parse and print the P notation as well. Meanwhile, the notation has been fully adopted by the language standard since C++17.
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This consists in avoiding to round to midpoints for the final rounding (except when the midpoint is exact). In binary arithmetic, the idea is to round the result toward zero, and set the least significant bit to 1 if the rounded result is inexact; this rounding is called sticky rounding. Equivalently, it consists in returning the intermediate result when it is exactly representable, and the nearest floating-point number with an odd significand otherwise; this is why it is also known as rounding to odd.
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In floating-point calculations, NaN is not the same as infinity, although both are typically handled as special cases in floating-point representations of real numbers as well as in floating-point operations. An invalid operation is also not the same as an arithmetic overflow (which might return an infinity) or an arithmetic underflow (which would return the smallest normal number, a denormal number, or zero). IEEE 754 NaNs are encoded with the exponent field filled with ones (like infinity values), and some non-zero number in the significand field (to make them distinct from infinity values); this allows the definition of multiple distinct NaN values, depending on which bits are set in the significand field, but also on the value of the leading sign bit (but applications are not required to provide distinct semantics for those distinct NaN values). For example, a bit-wise IEEE floating-point standard single precision (32-bit) NaN would be :`s111 1111 1xxx xxxx xxxx xxxx xxxx xxxx` where s is the sign (most often ignored in applications) and the x sequence represents a non-zero number (the value zero encodes infinities).
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A decimal floating point number can be encoded in several ways, the different ways represent different precisions, for example 100.0 is encoded as 1000×10−1, while 100.00 is encoded as 10000×10−2. The set of possible encodings of the same numerical value is called a cohort in the standard. If the result of a calculation is inexact the largest amount of significant data is preserved by selecting the cohort member with the largest integer that can be stored in the significand along with the required exponent.
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The term significand was introduced by George Forsythe and Cleve Moler in 1967 and is the word used in the IEEE standard. However, in 1946 Arthur Burks used the terms mantissa and characteristic to describe the two parts of a floating-point number (Burks et al.) and that usage remains common among computer scientists today. Mantissa and characteristic have long described the two parts of the logarithm found on tables of common logarithms. While the two meanings of exponent are analogous, the two meanings of mantissa are not equivalent.
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However, these are uncommon formats, the most common formats including negative zero are the IEEE 754 floating-point formats, described below. Negative zero by IEEE 754 representation in binary32 In IEEE 754 binary floating-point numbers, zero values are represented by the biased exponent and significand both being zero. Negative zero has the sign bit set to one. One may obtain negative zero as the result of certain computations, for instance as the result of arithmetic underflow on a negative number, or `−1.0×0.0`, or simply as `−0.0`.
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Indeed, this is almost a form of fixed point arithmetic since the position of the radix point is implied. The Hertz and Chen–Ho encodings provide Boolean transformations for converting groups of three BCD-encoded digits to and from 10-bit values that can be efficiently encoded in hardware with only 2 or 3 gate delays. Densely packed decimal (DPD) is a similar scheme that is used for most of the significand, except the lead digit, for one of the two alternative decimal encodings specified in the IEEE 754-2008 floating-point standard.
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Consequently, a leading 1 can be implied rather than explicitly present in the memory encoding, and under the standard the explicitly represented part of the significand will lie between 0 and 1. This rule is called leading bit convention, implicit bit convention, or hidden bit convention. This rule allows the binary format to have an extra bit of precision. The leading bit convention cannot be used for the subnormal numbers as they have an exponent outside the normal exponent range and scale by the smallest represented exponent as used for the smallest normal numbers.
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The standard specifies optional extended and extendable precision formats, which provide greater precision than the basic formats.. An extended precision format extends a basic format by using more precision and more exponent range. An extendable precision format allows the user to specify the precision and exponent range. An implementation may use whatever internal representation it chooses for such formats; all that needs to be defined are its parameters (b, p, and emax). These parameters uniquely describe the set of finite numbers (combinations of sign, significand, and exponent for the given radix) that it can represent.
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Z3, included floating-point arithmetic (replica on display at Deutsches Museum in Munich). In computing, floating-point arithmetic (FP) is arithmetic using formulaic representation of real numbers as an approximation to support a trade-off between range and precision. For this reason, floating-point computation is often found in systems which include very small and very large real numbers, which require fast processing times. A number is, in general, represented approximately to a fixed number of significant digits (the significand) and scaled using an exponent in some fixed base; the base for the scaling is normally two, ten, or sixteen.
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Floating- point arithmetic is needed for very large or very small real numbers, or computations that require a large dynamic range. Floating-point representation is similar to scientific notation, except everything is carried out in base two, rather than base ten. The encoding scheme stores the sign, the exponent (in base two for Cray and VAX, base two or ten for IEEE floating point formats, and base 16 for IBM Floating Point Architecture) and the Significand (number after the radix point). While several similar formats are in use, the most common is ANSI/IEEE Std. 754-1985.
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The description of formats has been made more regular, with a distinction between arithmetic formats (in which arithmetic may be carried out) and interchange formats (which have a standard encoding). Conformance to the standard is now defined in these terms. The specification levels of a floating-point format have been enumerated, to clarify the distinction between: # the theoretical real numbers (an extended number line) # the entities which can be represented in the format (a finite set of numbers, together with −0, infinities, and NaN) # the particular representations of the entities: sign-exponent-significand, etc. # the bit- pattern (encoding) used.
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When a result can have several representations, the standard specifies which member of the cohort is chosen. For the binary formats, the representation is made unique by choosing the smallest representable exponent allowing the value to be represented exactly. Further, the exponent is not represented directly, but a bias is added so that the smallest representable exponent is represented as 1, with 0 used for subnormal numbers. For numbers with an exponent in the normal range (the exponent field being neither all ones nor all zeros), the leading bit of the significand will always be 1.
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These examples are given in bit representation, in hexadecimal and binary, of the floating-point value. This includes the sign, (biased) exponent, and significand. 3f80 = 0 01111111 0000000 = 1 c000 = 1 10000000 0000000 = −2 7f7f = 0 11111110 1111111 = (28 − 1) × 2−7 × 2127 ≈ 3.38953139 × 1038 (max finite positive value in bfloat16 precision) 0080 = 0 00000001 0000000 = 2−126 ≈ 1.175494351 × 10−38 (min normalized positive value in bfloat16 precision and single-precision floating point) The maximum positive finite value of a normal bfloat16 number is 3.38953139 × 1038, slightly below (224 − 1) × 2−23 × 2127 = 3.402823466 × 1038, the max finite positive value representable in single precision.
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Two hurricanes, the first in August, 1813, the second in October, 1815, destroyed more than two-hundred buildings, significand salt stores, and sank many vessels. By 1815, the United States, the primary client for Turks salt, had been at war with Britain (and hence Bermuda) for three years, and had established other sources of salt. With the destruction wrought by the storm, and the loss of market, many Bermudians abandoned the Turks, and those remaining were so distraught that they welcomed the visit of the Bahamian governor in 1819. The British government eventually assigned political control to the Bahamas, which the Turks and Caicos remained a part of until the 1840s.
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The sets of representable entities are then explained in detail, showing that they can be treated with the significand being considered either as a fraction or an integer. The particular sets known as basic formats are defined, and the encodings used for interchange of binary and decimal formats are explained. The binary interchange formats have the "half precision" (16-bit storage format) and "quad precision" (128-bit format) added, together with generalized formulae for some wider formats; the basic formats have 32-bit, 64-bit, and 128-bit encodings. Three new decimal formats are described, matching the lengths of the 32–128-bit binary formats.
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These examples are given in bit representation, in hexadecimal, of the floating-point value. This includes the sign, (biased) exponent, and significand. 0000 0000 0000 0000 0000 0000 0000 000116 = 2−16382 × 2−112 = 2−16494 ≈ 6.4751751194380251109244389582276465525 × 10−4966 (smallest positive subnormal number) 0000 ffff ffff ffff ffff ffff ffff ffff16 = 2−16382 × (1 − 2−112) ≈ 3.3621031431120935062626778173217519551 × 10−4932 (largest subnormal number) 0001 0000 0000 0000 0000 0000 0000 000016 = 2−16382 ≈ 3.3621031431120935062626778173217526026 × 10−4932 (smallest positive normal number) 7ffe ffff ffff ffff ffff ffff ffff ffff16 = 216383 × (2 − 2−112) ≈ 1.1897314953572317650857593266280070162 × 104932 (largest normal number) 3ffe ffff ffff ffff ffff ffff ffff ffff16 = 1 − 2−113 ≈ 0.9999999999999999999999999999999999037 (largest number less than one) 3fff 0000 0000 0000 0000 0000 0000 000016 = 1 (one) 3fff 0000 0000 0000 0000 0000 0000 000116 = 1 + 2−112 ≈ 1.0000000000000000000000000000000001926 (smallest number larger than one) c000 0000 0000 0000 0000 0000 0000 000016 = −2 0000 0000 0000 0000 0000 0000 0000 000016 = 0 8000 0000 0000 0000 0000 0000 0000 000016 = −0 7fff 0000 0000 0000 0000 0000 0000 000016 = infinity ffff 0000 0000 0000 0000 0000 0000 000016 = −infinity 4000 921f b544 42d1 8469 898c c517 01b816 ≈ π 3ffd 5555 5555 5555 5555 5555 5555 555516 ≈ 1/3 By default, 1/3 rounds down like double precision, because of the odd number of bits in the significand. So the bits beyond the rounding point are `0101...` which is less than 1/2 of a unit in the last place.
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These examples are given in bit representation, in hexadecimal and binary, of the floating-point value. This includes the sign, (biased) exponent, and significand. 0 00000000 000000000000000000000012 = 0000 000116 = 2−126 × 2−23 = 2−149 ≈ 1.4012984643 × 10−45 (smallest positive subnormal number) 0 00000000 111111111111111111111112 = 007f ffff16 = 2−126 × (1 − 2−23) ≈ 1.1754942107 ×10−38 (largest subnormal number) 0 00000001 000000000000000000000002 = 0080 000016 = 2−126 ≈ 1.1754943508 × 10−38 (smallest positive normal number) 0 11111110 111111111111111111111112 = 7f7f ffff16 = 2127 × (2 − 2−23) ≈ 3.4028234664 × 1038 (largest normal number) 0 01111110 111111111111111111111112 = 3f7f ffff16 = 1 − 2−24 ≈ 0.999999940395355225 (largest number less than one) 0 01111111 000000000000000000000002 = 3f80 000016 = 1 (one) 0 01111111 000000000000000000000012 = 3f80 000116 = 1 + 2−23 ≈ 1.00000011920928955 (smallest number larger than one) 1 10000000 000000000000000000000002 = c000 000016 = −2 0 00000000 000000000000000000000002 = 0000 000016 = 0 1 00000000 000000000000000000000002 = 8000 000016 = −0 0 11111111 000000000000000000000002 = 7f80 000016 = infinity 1 11111111 000000000000000000000002 = ff80 000016 = −infinity 0 10000000 100100100001111110110112 = 4049 0fdb16 ≈ 3.14159274101257324 ≈ π ( pi ) 0 01111101 010101010101010101010112 = 3eaa aaab16 ≈ 0.333333343267440796 ≈ 1/3 x 11111111 100000000000000000000012 = ffc0 000116 = qNaN (on x86 and ARM processors) x 11111111 000000000000000000000012 = ff80 000116 = sNaN (on x86 and ARM processors) By default, 1/3 rounds up, instead of down like double precision, because of the even number of bits in the significand. The bits of 1/3 beyond the rounding point are `1010...` which is more than 1/2 of a unit in the last place.
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The bfloat16 (Brain Floating Point) floating-point format is a computer number format occupying 16 bits in computer memory; it represents a wide dynamic range of numeric values by using a floating radix point. This format is a truncated (16-bit) version of the 32-bit IEEE 754 single-precision floating- point format (binary32) with the intent of accelerating machine learning and near-sensor computing. It preserves the approximate dynamic range of 32-bit floating-point numbers by retaining 8 exponent bits, but supports only an 8-bit precision rather than the 24-bit significand of the binary32 format. More so than single-precision 32-bit floating-point numbers, bfloat16 numbers are unsuitable for integer calculations, but this is not their intended use.
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Scale factors are also used in floating-point numbers, and most commonly are powers of two. For example, the double-precision format sets aside 11 bits for the scaling factor (a binary exponent) and 53 bits for the significand, allowing various degrees of precision for representing different ranges of numbers, and expanding the range of representable numbers beyond what could be represented using 64 explicit bits (though at the cost of precision). As an example of where precision is lost, a 16-bit unsigned integer (uint16) can only hold a value as large as 65,53510. If unsigned 16-bit integers are used to represent values from 0 to 131,07010, then a scale factor of would be introduced, such that the scaled values correspond exactly to the real-world even integers.
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These examples are given in bit representation, in hexadecimal, of the floating-point value. This includes the sign, (biased) exponent, and significand. 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 000016 = +0 8000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 000016 = −0 7fff f000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 000016 = +infinity ffff f000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 000016 = −infinity 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 000116 = 2−262142 × 2−236 = 2−262378 ≈ 2.24800708647703657297018614776265182597360918266100276294348974547709294462 × 10−78984 (smallest positive subnormal number) 0000 0fff ffff ffff ffff ffff ffff ffff ffff ffff ffff ffff ffff ffff ffff ffff16 = 2−262142 × (1 − 2−236) ≈ 2.4824279514643497882993282229138717236776877060796468692709532979137875392 × 10−78913 (largest subnormal number) 0000 1000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 000016 = 2−262142 ≈ 2.48242795146434978829932822291387172367768770607964686927095329791378756168 × 10−78913 (smallest positive normal number) 7fff efff ffff ffff ffff ffff ffff ffff ffff ffff ffff ffff ffff ffff ffff ffff16 = 2262143 × (2 − 2−236) ≈ 1.61132571748576047361957211845200501064402387454966951747637125049607182699 × 1078913 (largest normal number) 3fff efff ffff ffff ffff ffff ffff ffff ffff ffff ffff ffff ffff ffff ffff ffff16 = 1 − 2−237 ≈ 0.999999999999999999999999999999999999999999999999999999999999999999999995472 (largest number less than one) 3fff f000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 000016 = 1 (one) 3fff f000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 000116 = 1 + 2−236 ≈ 1.00000000000000000000000000000000000000000000000000000000000000000000000906 (smallest number larger than one) By default, 1/3 rounds down like double precision, because of the odd number of bits in the significand. So the bits beyond the rounding point are `0101...` which is less than 1/2 of a unit in the last place.
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Arbitrary-precision arithmetic is considerably slower than arithmetic using numbers that fit entirely within processor registers, since the latter are usually implemented in hardware arithmetic whereas the former must be implemented in software. Even if the computer lacks hardware for certain operations (such as integer division, or all floating-point operations) and software is provided instead, it will use number sizes closely related to the available hardware registers: one or two words only and definitely not N words. There are exceptions, as certain variable word length machines of the 1950s and 1960s, notably the IBM 1620, IBM 1401 and the Honeywell Liberator series, could manipulate numbers bound only by available storage, with an extra bit that delimited the value. Numbers can be stored in a fixed-point format, or in a floating-point format as a significand multiplied by an arbitrary exponent.
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This means that numbers which appear to be short and exact when written in decimal format may need to be approximated when converted to binary floating-point. For example, the decimal number 0.1 is not representable in binary floating-point of any finite precision; the exact binary representation would have a "1100" sequence continuing endlessly: : e = −4; s = 1100110011001100110011001100110011..., where, as previously, s is the significand and e is the exponent. When rounded to 24 bits this becomes : e = −4; s = 110011001100110011001101, which is actually 0.100000001490116119384765625 in decimal. As a further example, the real number π, represented in binary as an infinite sequence of bits is : 11.0010010000111111011010101000100010000101101000110000100011010011... but is : 11.0010010000111111011011 when approximated by rounding to a precision of 24 bits. In binary single-precision floating-point, this is represented as s = 1.10010010000111111011011 with e = 1\. This has a decimal value of : 3.1415927410125732421875, whereas a more accurate approximation of the true value of π is : 3.14159265358979323846264338327950... The result of rounding differs from the true value by about 0.03 parts per million, and matches the decimal representation of π in the first 7 digits.
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The last 50 bits are the significand continuation field, consisting of five 10-bit declets. Each declet encodes three decimal digits using the DPD encoding. If the first two bits after the sign bit are "00", "01", or "10", then those are the leading bits of the exponent, and the three bits "TTT" after that are interpreted as the leading decimal digit (0 to 7): s 00 TTT (00)eeeeeeee (0TTT)[tttttttttt][tttttttttt][tttttttttt][tttttttttt][tttttttttt] s 01 TTT (01)eeeeeeee (0TTT)[tttttttttt][tttttttttt][tttttttttt][tttttttttt][tttttttttt] s 10 TTT (10)eeeeeeee (0TTT)[tttttttttt][tttttttttt][tttttttttt][tttttttttt][tttttttttt] If the first two bits after the sign bit are "11", then the second 2-bits are the leading bits of the exponent, and the next bit "T" is prefixed with implicit bits "100" to form the leading decimal digit (8 or 9): s 1100 T (00)eeeeeeee (100T)[tttttttttt][tttttttttt][tttttttttt][tttttttttt][tttttttttt] s 1101 T (01)eeeeeeee (100T)[tttttttttt][tttttttttt][tttttttttt][tttttttttt][tttttttttt] s 1110 T (10)eeeeeeee (100T)[tttttttttt][tttttttttt][tttttttttt][tttttttttt][tttttttttt] The remaining two combinations (11 110 and 11 111) of the 5-bit field after the sign bit are used to represent ±infinity and NaNs, respectively. The DPD/3BCD transcoding for the declets is given by the following table. b9...b0 are the bits of the DPD, and d2...d0 are the three BCD digits.
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