5973:
magnitude the risk of numerical anomalies, in addition to, or in lieu of, a more careful numerical analysis. These include: as noted above, computing all expressions and intermediate results in the highest precision supported in hardware (a common rule of thumb is to carry twice the precision of the desired result, i.e. compute in double precision for a final single-precision result, or in double extended or quad precision for up to double-precision results); and rounding input data and results to only the precision required and supported by the input data (carrying excess precision in the final result beyond that required and supported by the input data can be misleading, increases storage cost and decreases speed, and the excess bits can affect convergence of numerical procedures: notably, the first form of the iterative example given below converges correctly when using this rule of thumb). Brief descriptions of several additional issues and techniques follow.
246:
5300:
4639:
3515:"sticky" means that they are not reset by the next (arithmetic) operation, but stay set until explicitly reset. The use of "sticky" flags thus allows for testing of exceptional conditions to be delayed until after a full floating-point expression or subroutine: without them exceptional conditions that could not be otherwise ignored would require explicit testing immediately after every floating-point operation. By default, an operation always returns a result according to specification without interrupting computation. For instance, 1/0 returns +∞, while also setting the divide-by-zero flag bit (this default of ∞ is designed to often return a finite result when used in subsequent operations and so be safely ignored).
5295:{\displaystyle {\begin{aligned}\operatorname {fl} (x\cdot y)&=\operatorname {fl} {\big (}\operatorname {fl} (x_{1}\cdot y_{1})+\operatorname {fl} (x_{2}\cdot y_{2}){\big )},&&{\text{ where }}\operatorname {fl} (){\text{ indicates correctly rounded floating-point arithmetic}}\\&=\operatorname {fl} {\big (}(x_{1}\cdot y_{1})(1+\delta _{1})+(x_{2}\cdot y_{2})(1+\delta _{2}){\big )},&&{\text{ where }}\delta _{n}\leq \mathrm {E} _{\text{mach}},{\text{ from above}}\\&={\big (}(x_{1}\cdot y_{1})(1+\delta _{1})+(x_{2}\cdot y_{2})(1+\delta _{2}){\big )}(1+\delta _{3})\\&=(x_{1}\cdot y_{1})(1+\delta _{1})(1+\delta _{3})+(x_{2}\cdot y_{2})(1+\delta _{2})(1+\delta _{3}),\end{aligned}}}
3113:, sometimes called Banker's Rounding) is more commonly used. This method rounds the ideal (infinitely precise) result of an arithmetic operation to the nearest representable value, and gives that representation as the result. In the case of a tie, the value that would make the significand end in an even digit is chosen. The IEEE 754 standard requires the same rounding to be applied to all fundamental algebraic operations, including square root and conversions, when there is a numeric (non-NaN) result. It means that the results of IEEE 754 operations are completely determined in all bits of the result, except for the representation of NaNs. ("Library" functions such as cosine and log are not mandated.)
258:
8242:, while IEEE places the decimal point after the assumed bit. ieee_exp = msbin - 2; /* actually, msbin-1-128+127 */ _dmsbintoieee(double *src8, double *dest8) MS Binary Format byte order => m7 | m6 | m5 | m4 | m3 | m2 | m1 | exponent m1 is most significant byte => smmm|mmmm m7 is the least significant byte MBF is bias 128 and IEEE is bias 1023. MBF places the decimal point before the assumed bit, while IEEE places the decimal point after the assumed bit. ieee_exp = msbin - 128 - 1 + 1023;
4583:, can be used to establish that an algorithm implementing a numerical function is numerically stable. The basic approach is to show that although the calculated result, due to roundoff errors, will not be exactly correct, it is the exact solution to a nearby problem with slightly perturbed input data. If the perturbation required is small, on the order of the uncertainty in the input data, then the results are in some sense as accurate as the data "deserves". The algorithm is then defined as
6153:, where epsilon is sufficiently small and tailored to the application, such as 1.0E−13). The wisdom of doing this varies greatly, and can require numerical analysis to bound epsilon. Values derived from the primary data representation and their comparisons should be performed in a wider, extended, precision to minimize the risk of such inconsistencies due to round-off errors. It is often better to organize the code in such a way that such tests are unnecessary. For example, in
3578:
1423:
2959:
Any rational with a denominator that has a prime factor other than 2 will have an infinite binary expansion. This means that numbers that 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:
1073:
40:
1682:
2630:
1331:
771:
6870:. The "fast math" option on many compilers (ICC, GCC, Clang, MSVC...) turns on reassociation along with unsafe assumptions such as a lack of NaN and infinite numbers in IEEE 754. Some compilers also offer more granular options to only turn on reassociation. In either case, the programmer is exposed to many of the precision pitfalls mentioned above for the portion of the program using "fast" math.
5652:
3083:, the ULP is 2×16, or 2. For numbers with a base-2 exponent part of 0, i.e. numbers with an absolute value higher than or equal to 1 but lower than 2, an ULP is exactly 2 or about 10 in single precision, and exactly 2 or about 10 in double precision. The mandated behavior of IEEE-compliant hardware is that the result be within one-half of a ULP.
3523:/C11 and Fortran) have been updated to specify methods to access and change status flag bits. The 2008 version of the IEEE 754 standard now specifies a few operations for accessing and handling the arithmetic flag bits. The programming model is based on a single thread of execution and use of them by multiple threads has to be handled by a
5981:), and at its worst when it is expected to model the interactions of quantities expressed as decimal strings that are expected to be exact. An example of the latter case is financial calculations. For this reason, financial software tends not to use a binary floating-point number representation. The "decimal" data type of the
6846:. As the recurrence is applied repeatedly, the accuracy improves at first, but then it deteriorates. It never gets better than about 8 digits, even though 53-bit arithmetic should be capable of about 16 digits of precision. When the second form of the recurrence is used, the value converges to 15 digits of precision.
1758:(64-bit) binary floating-point number has a coefficient of 53 bits (including 1 implied bit), an exponent of 11 bits, and 1 sign bit. Since 2 = 1024, the complete range of the positive normal floating-point numbers in this format is from 2 ≈ 2 × 10 to approximately 2 ≈ 2 × 10.
2951:
floating-point format then the conversion is exact. If there is not an exact representation then the conversion requires a choice of which floating-point number to use to represent the original value. The representation chosen will have a different value from the original, and the value thus adjusted is called the
1068:{\displaystyle {\begin{aligned}&\left(\sum _{n=0}^{p-1}{\text{bit}}_{n}\times 2^{-n}\right)\times 2^{e}\\={}&\left(1\times 2^{-0}+1\times 2^{-1}+0\times 2^{-2}+0\times 2^{-3}+1\times 2^{-4}+\cdots +1\times 2^{-23}\right)\times 2^{1}\\\approx {}&1.5707964\times 2\\\approx {}&3.1415928\end{aligned}}}
1411:." The format he proposed shows the need for a fixed-sized significand as is presently used for floating-point data, fixing the location of the decimal point in the significand so that each representation was unique, and how to format such numbers by specifying a syntax to be used that could be entered through a
5380:
191:
3915:
Also, the non-representability of π (and π/2) means that an attempted computation of tan(π/2) will not yield a result of infinity, nor will it even overflow in the usual floating-point formats (assuming an accurate implementation of tan). It is simply not possible for standard floating-point hardware
3518:
The original IEEE 754 standard, however, failed to recommend operations to handle such sets of arithmetic exception flag bits. So while these were implemented in hardware, initially programming language implementations typically did not provide a means to access them (apart from assembler). Over time
2781:
based on the Nvidia Ampere architecture. The drawback of this format is its size, which is not a power of 2. However, according to Nvidia, this format should only be used internally by hardware to speed up computations, while inputs and outputs should be stored in the 32-bit single-precision IEEE 754
2749:
until
Microsoft adopted the IEEE-754 standard format in all its products starting in 1988 to their current releases. MBF consists of the MBF single-precision format (32 bits, "6-digit BASIC"), the MBF extended-precision format (40 bits, "9-digit BASIC"), and the MBF double-precision format (64 bits);
1748:
components, whose range depends exclusively on the number of bits or digits in their representation. Whereas components linearly depend on their range, the floating-point range linearly depends on the significand range and exponentially on the range of exponent component, which attaches outstandingly
1156:
representation uses integer hardware operations controlled by a software implementation of a specific convention about the location of the binary or decimal point, for example, 6 bits or digits from the right. The hardware to manipulate these representations is less costly than floating point, and it
396:
to which numbers can be represented. The radix point position is assumed always to be somewhere within the significand—often just after or just before the most significant digit, or to the right of the rightmost (least significant) digit. This article generally follows the convention that the radix
2938:
or √2, or non-terminating rational numbers, must be approximated. The number of digits (or bits) of precision also limits the set of rational numbers that can be represented exactly. For example, the decimal number 123456789 cannot be exactly represented if only eight decimal digits of precision are
5730:
for that data is used: apparently equivalent formulations of expressions in a programming language can differ markedly in their numerical stability. One approach to remove the risk of such loss of accuracy is the design and analysis of numerically stable algorithms, which is an aim of the branch of
2958:
Whether or not a rational number has a terminating expansion depends on the base. For example, in base-10 the number 1/2 has a terminating expansion (0.5) while the number 1/3 does not (0.333...). In base-2 only rationals with denominators that are powers of 2 (such as 1/2 or 3/16) are terminating.
3514:
Here, the required default method of handling exceptions according to IEEE 754 is discussed (the IEEE 754 optional trapping and other "alternate exception handling" modes are not discussed). Arithmetic exceptions are (by default) required to be recorded in "sticky" status flag bits. That they are
1723:
A precisely specified behavior for the arithmetic operations: A result is required to be produced as if infinitely precise arithmetic were used to yield a value that is then rounded according to specific rules. This means that a compliant computer program would always produce the same result when
1661:
Initially, computers used many different representations for floating-point numbers. The lack of standardization at the mainframe level was an ongoing problem by the early 1970s for those writing and maintaining higher-level source code; these manufacturer floating-point standards differed in the
8354:
Micikevicius, Paulius; Stosic, Dusan; Burgess, Neil; Cornea, Marius; Dubey, Pradeep; Grisenthwaite, Richard; Ha, Sangwon; Heinecke, Alexander; Judd, Patrick; Kamalu, John; Mellempudi, Naveen; Oberman, Stuart; Shoeybi, Mohammad; Siu, Michael; Wu, Hao (2022-09-12). "FP8 Formats for Deep
Learning".
2347:
Any integer with absolute value less than 2 can be exactly represented in the single-precision format, and any integer with absolute value less than 2 can be exactly represented in the double-precision format. Furthermore, a wide range of powers of 2 times such a number can be represented. These
207:
Floating-point arithmetic operations, such as addition and division, approximate the corresponding real number arithmetic operations by rounding any result that is not a floating-point number itself to a nearby floating-point number. For example, in a floating-point arithmetic with five base-ten
2590:
In the IEEE binary interchange formats the leading 1 bit of a normalized significand is not actually stored in the computer datum. It is called the "hidden" or "implicit" bit. Because of this, the single-precision format actually has a significand with 24 bits of precision, the double-precision
5972:
A detailed treatment of the techniques for writing high-quality floating-point software is beyond the scope of this article, and the reader is referred to, and the other references at the bottom of this article. Kahan suggests several rules of thumb that can substantially decrease by orders of
3876:
For example, the decimal numbers 0.1 and 0.01 cannot be represented exactly as binary floating-point numbers. In the IEEE 754 binary32 format with its 24-bit significand, the result of attempting to square the approximation to 0.1 is neither 0.01 nor the representable number closest to it. The
7719:
the Maniac's floating base, which is 2 = 65,536. The Maniac's large base permits a considerable increase in the speed of floating point arithmetic. Although such a large base implies the possibility of as many as 15 lead zeros, the large word size of 48 bits guarantees adequate significance.
2950:
When a number is represented in some format (such as a character string) which is not a native floating-point representation supported in a computer implementation, then it will require a conversion before it can be used in that implementation. If the number can be represented exactly in the
6896:
A common problem in "fast" math is that subexpressions may not be optimized identically from place to place, leading to unexpected differences. One interpretation of the issue is that "fast" math as implemented currently has a poorly defined semantics. One attempt at formalizing "fast" math
5968:
is required. Certain "optimizations" that compilers might make (for example, reordering operations) can work against the goals of well-behaved software. There is some controversy about the failings of compilers and language designs in this area: C99 is an example of a language where such
2374:
Comparison of floating-point numbers, as defined by the IEEE standard, is a bit different from usual integer comparison. Negative and positive zero compare equal, and every NaN compares unequal to every value, including itself. All finite floating-point numbers are strictly smaller than
2391:
Floating-point numbers are typically packed into a computer datum as the sign bit, the exponent field, and the significand or mantissa, from left to right. For the IEEE 754 binary formats (basic and extended) which have extant hardware implementations, they are apportioned as follows:
3148:
Converting a double-precision binary floating-point number to a decimal string is a common operation, but an algorithm producing results that are both accurate and minimal did not appear in print until 1990, with Steele and White's Dragon4. Some of the improvements since then include:
6179:, and so an awareness of when loss of significance can occur is essential. For example, if one is adding a very large number of numbers, the individual addends are very small compared with the sum. This can lead to loss of significance. A typical addition would then be something like
6197:
approximated π by calculating the perimeters of polygons inscribing and circumscribing a circle, starting with hexagons, and successively doubling the number of sides. As noted above, computations may be rearranged in a way that is mathematically equivalent but less prone to error
1715:
A precisely specified floating-point representation at the bit-string level, so that all compliant computers interpret bit patterns the same way. This makes it possible to accurately and efficiently transfer floating-point numbers from one computer to another (after accounting for
602:
The way in which the significand (including its sign) and exponent are stored in a computer is implementation-dependent. The common IEEE formats are described in detail later and elsewhere, but as an example, in the binary single-precision (32-bit) floating-point representation,
5976:
As decimal fractions can often not be exactly represented in binary floating-point, such arithmetic is at its best when it is simply being used to measure real-world quantities over a wide range of scales (such as the orbital period of a moon around Saturn or the mass of a
6888:
compilers, as allowed by the ISO/IEC 1539-1:2004 Fortran standard, reassociation is the default, with breakage largely prevented by the "protect parens" setting (also on by default). This setting stops the compiler from reassociating beyond the boundaries of parentheses.
8742:
4138:: subtraction of nearly equal operands may cause extreme loss of accuracy. When we subtract two almost equal numbers we set the most significant digits to zero, leaving ourselves with just the insignificant, and most erroneous, digits. For example, when determining a
343:, so that it lies within a specific range—typically between 1 and 10, with the radix point appearing immediately after the first digit. As a power of ten, the scaling factor is then indicated separately at the end of the number. For example, the orbital period of
5735:. Another approach that can protect against the risk of numerical instabilities is the computation of intermediate (scratch) values in an algorithm at a higher precision than the final result requires, which can remove, or reduce by orders of magnitude, such risk:
3358:
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
195:
However, unlike 12.345, 12.3456 is not a floating-point number in base ten with five digits of precision—it needs six digits of precision; the nearest floating-point number with only five digits is 12.346. In practice, most floating-point systems use
7291:
that, on rare occasions, gave slightly incorrect results. Many computers had been shipped before the error was discovered. Until the defective computers were replaced, patched versions of compilers were developed that could avoid the failing cases. See
1167:, the graph of the logarithm function) is smooth (except at 0). Conversely to floating-point arithmetic, in a logarithmic number system multiplication, division and exponentiation are simple to implement, but addition and subtraction are complex. The (
1662:
word sizes, the representations, and the rounding behavior and general accuracy of operations. Floating-point compatibility across multiple computing systems was in desperate need of standardization by the early 1980s, leading to the creation of the
3486:
An operation can be legal in principle, but the result can be impossible to represent in the specified format, because the exponent is too large or too small to encode in the exponent field. Such an event is called an overflow (exponent too large),
6185:
The low 3 digits of the addends are effectively lost. Suppose, for example, that one needs to add many numbers, all approximately equal to 3. After 1000 of them have been added, the running sum is about 3000; the lost digits are not regained. The
100:
7307:
But an attempted computation of cos(π) yields −1 exactly. Since the derivative is nearly zero near π, the effect of the inaccuracy in the argument is far smaller than the spacing of the floating-point numbers around −1, and the rounded result is
3185:
The problem of parsing a decimal string into a binary FP representation is complex, with an accurate parser not appearing until
Clinger's 1990 work (implemented in dtoa.c). Further work has likewise progressed in the direction of faster parsing.
2283:, also ambiguously called "extended precision" format. This is a binary format that occupies at least 79 bits (80 if the hidden/implicit bit rule is not used) and its significand has a precision of at least 64 bits (about 19 decimal digits). The
3140:. The alternative rounding modes are also useful in diagnosing numerical instability: if the results of a subroutine vary substantially between rounding to + and − infinity then it is likely numerically unstable and affected by round-off error.
3507:. (The term "exception" as used in IEEE 754 is a general term meaning an exceptional condition, which is not necessarily an error, and is a different usage to that typically defined in programming languages such as a C++ or Java, in which an "
3261:
In the above conceptual examples it would appear that a large number of extra digits would need to be provided by the adder to ensure correct rounding; however, for binary addition or subtraction using careful implementation techniques only a
1157:
can be used to perform normal integer operations, too. Binary fixed point is usually used in special-purpose applications on embedded processors that can only do integer arithmetic, but decimal fixed point is common in commercial applications.
4575:
3868:
The fact that floating-point numbers cannot accurately represent all real numbers, and that floating-point operations cannot accurately represent true arithmetic operations, leads to many surprising situations. This is related to the finite
2303:
purposes, many tools store this 80-bit value in a 96-bit or 128-bit space. On other processors, "long double" may stand for a larger format, such as quadruple precision, or just double precision, if any form of extended precision is not
3229:
Hence: 123456.7 + 101.7654 = (1.234567 × 10^5) + (1.017654 × 10^2) = (1.234567 × 10^5) + (0.001017654 × 10^5) = (1.234567 + 0.001017654) × 10^5 = 1.235584654 × 10^5
5993:
standard, are designed to avoid the problems of binary floating-point representations when applied to human-entered exact decimal values, and make the arithmetic always behave as expected when numbers are printed in decimal.
3222:
A simple method to add floating-point numbers is to first represent them with the same exponent. In the example below, the second number is shifted right by three digits, and one then proceeds with the usual addition method:
5647:{\displaystyle {\begin{aligned}{\hat {x}}_{1}&=x_{1}(1+\delta _{1});&{\hat {x}}_{2}&=x_{2}(1+\delta _{2});\\{\hat {y}}_{1}&=y_{1}(1+\delta _{3});&{\hat {y}}_{2}&=y_{2}(1+\delta _{3}),\\\end{aligned}}}
1595:. The arithmetic is actually implemented in software, but with a one megahertz clock rate, the speed of floating-point and fixed-point operations in this machine were initially faster than those of many competing computers.
3095:: that is, the rounded result is as if infinitely precise arithmetic was used to compute the value and then rounded (although in implementation only three extra bits are needed to ensure this). There are several different
2587:; values of all 1s are reserved for the infinities and NaNs. The exponent range for normal numbers is for single precision, for double, or for quad. Normal numbers exclude subnormal values, zeros, infinities, and NaNs.
8214:
579:/100. The base determines the fractions that can be represented; for instance, 1/5 cannot be represented exactly as a floating-point number using a binary base, but 1/5 can be represented exactly using a decimal base (
5371:
689:
331:
systems, a position in the string is specified for the radix point. So a fixed-point scheme might use a string of 8 decimal digits with the decimal point in the middle, whereby "00012345" would represent 0001.2345.
1896:
2772:
The TensorFloat-32 format combines the 8 bits of exponent of the Bfloat16 with the 10 bits of trailing significand field of half-precision formats, resulting in a size of 19 bits. This format was introduced by
4475:
3741:
428:, equivalent to shifting the radix point from its implied position by a number of places equal to the value of the exponent—to the right if the exponent is positive or to the left if the exponent is negative.
4126:
1234.567 × 3.333333 = 4115.223 1.234567 × 3.333333 = 4.115223 4115.223 + 4.115223 = 4119.338 but 1234.567 + 1.234567 = 1235.802 1235.802 × 3.333333 = 4119.340
1108:
It can be required that the most significant digit of the significand of a non-zero number be non-zero (except when the corresponding exponent would be smaller than the minimum one). This process is called
693:
In this binary expansion, let us denote the positions from 0 (leftmost bit, or most significant bit) to 32 (rightmost bit). The 24-bit significand will stop at position 23, shown as the underlined bit
2696:
prior to version 4.00. QuickBASIC version 4.00 and 4.50 switched to the IEEE 754-1985 format but can revert to the MBF format using the /MBF command option. MBF was designed and developed on a simulated
6415:
6330:
757:
2330:
5698:
6132:
2583:
While the exponent can be positive or negative, in binary formats it is stored as an unsigned number that has a fixed "bias" added to it. Values of all 0s in this field are reserved for the zeros and
2011:
4412:
5918:), then up to full precision in the final double result can be maintained. Alternatively, a numerical analysis of the algorithm reveals that if the following non-obvious change to line is made:
3091:
Rounding is used when the exact result of a floating-point operation (or a conversion to floating-point format) would need more digits than there are digits in the significand. IEEE 754 requires
5385:
4644:
776:
513:
6468:
5726:, meaning that the correct result is hypersensitive to tiny perturbations in its data. However, even functions that are well-conditioned can suffer from large loss of accuracy if an algorithm
3352:
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)
2717:
4-kilobytes memory. In
December 1975, the 8-kilobytes version added a double-precision (64 bits) format. A single-precision (40 bits) variant format was adopted for other CPU's, notably the
4268:
Conversions to integer are not intuitive: converting (63.0/9.0) to integer yields 7, but converting (0.63/0.09) may yield 6. This is because conversions generally truncate rather than round.
3173:
Schubfach, an always-succeeding algorithm that is based on a similar idea to Ryū, developed almost simultaneously and independently. Performs better than Ryū and Grisu3 in certain benchmarks.
6842:
While the two forms of the recurrence formula are clearly mathematically equivalent, the first subtracts 1 from a number extremely close to 1, leading to an increasingly problematic loss of
4593:
of a function for a given problem indicates the inherent sensitivity of the function to small perturbations in its input and is independent of the implementation used to solve the problem.
4218:
6245:
4507:
3258:
e=5; s=1.234567 + e=5; s=0.00000009876543 (after shifting) ---------------------- e=5; s=1.23456709876543 (true sum) e=5; s=1.234567 (after rounding and normalization)
6069:
3617:
The default return value for each of the exceptions is designed to give the correct result in the majority of cases such that the exceptions can be ignored in the majority of codes.
2934:
with a terminating expansion in the relevant base (for example, a terminating decimal expansion in base-10, or a terminating binary expansion in base-2). Irrational numbers, such as
219:
can "float" anywhere to the left, right, or between the significant digits of the number. This position is indicated by the exponent, so floating point can be considered a form of
8137:
Since the IEEE-754 floating-point specification does not define a 16-bit format, ILM created the "half" format. Half values have 1 sign bit, 5 exponent bits, and 10 mantissa bits.
7333:
of this function demonstrates that it is well-conditioned near 1: A(x) = 1 − (x−1)/2 + (x−1)^2/12 − (x−1)^4/720 + (x−1)^6/30240 − (x−1)^8/1209600 + ... for |x−1| < π.
7320:
notes: "Except in extremely uncommon situations, extra-precise arithmetic generally attenuates risks due to roundoff at far less cost than the price of a competent error-analyst."
6494:
8194:
5765:
which is well-conditioned at 1.0, however it can be shown to be numerically unstable and lose up to half the significant digits carried by the arithmetic when computed near 1.0.
1658:
systems. In 1998, IBM implemented IEEE-compatible binary floating-point arithmetic in its mainframes; in 2005, IBM also added IEEE-compatible decimal floating-point arithmetic.
1646:, also introduced in 1962, supported single-precision and double-precision representations, but with no relation to the UNIVAC's representations. Indeed, in 1964, IBM introduced
1449:; it uses a 24-bit binary floating-point number representation with a 7-bit signed exponent, a 17-bit significand (including one implicit bit), and a sign bit. The more reliable
1163:(LNSs) represent a real number by the logarithm of its absolute value and a sign bit. The value distribution is similar to floating point, but the value-to-representation curve (
7537:
315:
There are several mechanisms by which strings of digits can represent numbers. In standard mathematical notation, the digit string can be of any length, and the location of the
1499:
7552:
7270:
Computer hardware does not necessarily compute the exact value; it simply has to produce the equivalent rounded result as though it had computed the infinitely precise result.
443:
together with 5 as the exponent. To determine the actual value, a decimal point is placed after the first digit of the significand and the result is multiplied by 10 to give
9061:(NB. Kahan estimates that the incidence of excessively inaccurate results near singularities is reduced by a factor of approx. 1/2000 using the 11 extra bits of precision of
3355:
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.
1525:
3060:
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. The difference is the
1724:
given a particular input, thus mitigating the almost mystical reputation that floating-point computation had developed for its hitherto seemingly non-deterministic behavior.
8062:
3136:
Alternative modes are useful when the amount of error being introduced must be bounded. Applications that require a bounded error are multi-precision floating-point, and
1551:
1313:
1274:
1606:. For many decades after that, floating-point hardware was typically an optional feature, and computers that had it were said to be "scientific computers", or to have "
1204:
represent numbers as fractions with integral numerator and denominator, and can therefore represent any rational number exactly. Such packages generally need to use "
1196:, 1/3 and 1/10) cannot be represented exactly in binary floating point, no matter what the precision is. Using a different radix allows one to represent some of them (
8283:
4259:
grows smaller, cancelling out the most significant and least erroneous digits and making the most erroneous digits more important. As a result the smallest number of
3836:
3803:
3612:
3561:, set if the absolute value of the rounded value is too large to be represented. An infinity or maximal finite value is returned, depending on which rounding is used.
768:. The significand is assumed to have a binary point to the right of the leftmost bit. So, the binary representation of π is calculated from left-to-right as follows:
3768:
1930:
238:. For this reason, floating-point arithmetic is often used to allow very small and very large real numbers that require fast processing times. The result of this
1389:
1250:
627:
8656:
4130:
In addition to loss of significance, inability to represent numbers such as π and 0.1 exactly, and other slight inaccuracies, the following phenomena may occur:
186:{\displaystyle 12.345=\!\underbrace {12345} _{\text{significand}}\!\times \!\underbrace {10} _{\text{base}}\!\!\!\!\!\!\!\overbrace {{}^{-3}} ^{\text{exponent}}}
4634:
4614:
1149:
The floating-point representation is by far the most common way of representing in computers an approximation to real numbers. However, there are alternatives:
6157:, exact tests of whether a point lies off or on a line or plane defined by other points can be performed using adaptive precision or exact arithmetic methods.
5305:
3633:
exception subsequently if not, and so can also typically be ignored. For example, the effective resistance of n resistors in parallel (see fig. 1) is given by
1214:
allows one to represent numbers as intervals and obtain guaranteed bounds on results. It is generally based on other arithmetics, in particular floating point.
640:
242:
is that the numbers that can be represented are not uniformly spaced; the difference between two consecutive representable numbers varies with their exponent.
7746:
4265:
possible will give a more erroneous approximation of a derivative than a somewhat larger number. This is perhaps the most common and serious accuracy problem.
455:. In storing such a number, the base (10) need not be stored, since it will be the same for the entire range of supported numbers, and can thus be inferred.
8027:
6862:
cannot as effectively reorder arithmetic expressions as they could with integer and fixed-point arithmetic, presenting a roadblock in optimizations such as
3407:, when the decimal exponent is omitted, a decimal point is needed to differentiate them from integers. Other languages do not have an integer type (such as
245:
3333:
illustrates the danger in assuming that all of the digits of a computed result are meaningful. Dealing with the consequences of these errors is a topic in
2939:
available (it would be rounded to one of the two straddling representable values, 12345678 × 10 or 12345679 × 10), the same applies to
2789:
architecture GPUs provide two FP8 formats: one with the same numerical range as half-precision (E5M2) and one with higher precision, but less range (E4M3).
2765:
number. The tradeoff is a reduced precision, as the trailing significand field is reduced from 10 to 7 bits. This format is mainly used in the training of
1457:, completed in 1941, has representations for both positive and negative infinities; in particular, it implements defined operations with infinity, such as
1315:" exactly, because it is programmed to process the underlying mathematics directly, instead of using approximate values for each intermediate calculation.
273:
Standard for
Floating-Point Arithmetic was established, and since the 1990s, the most commonly encountered representations are those defined by the IEEE.
9816:. (NB. This website contains open source floating-point IP cores for the implementation of floating-point operators in FPGA or ASIC devices. The project
5722:, more complicated formulae can suffer from larger errors for a variety of reasons. The loss of accuracy can be substantial if a problem or its data are
4417:
2348:
properties are sometimes used for purely integer data, to get 53-bit integers on platforms that have double-precision floats but only 32-bit integers.
2214:
1125:. Therefore, it does not need to be represented in memory, allowing the format to have one more bit of precision. This rule is variously called the
6160:
Small errors in floating-point arithmetic can grow when mathematical algorithms perform operations an enormous number of times. A few examples are
3555:
inexact (or maybe limited to if it has denormalization loss, as per the 1985 version of IEEE 754), returning a subnormal value including the zeros.
2291:
standards of the C language family, in their annex F ("IEC 60559 floating-point arithmetic"), recommend such an extended format to be provided as "
8475:
717:
9810:(NB. A compendium of non-intuitive behaviors of floating point on popular architectures, with implications for program verification and testing.)
7178:
7174:
3242:
This is the true result, the exact sum of the operands. It will be rounded to seven digits and then normalized if necessary. The final result is
2333:(binary128). This is a binary format that occupies 128 bits (16 bytes) and its significand has a precision of 113 bits (about 34 decimal digits).
5661:
4341:
4289:
Testing for equality is problematic. Two computational sequences that are mathematically equal may well produce different floating-point values.
4227:
very close to zero; however, when using floating-point operations, the smallest number will not give the best approximation of a derivative. As
8971:
8924:
3167:
Errol3, an always-succeeding algorithm similar to, but slower than, Grisu3. Apparently not as good as an early-terminating Grisu with fallback.
9705:
Muller, Jean-Michel; Brunie, Nicolas; de
Dinechin, Florent; Jeannerod, Claude-Pierre; Joldes, Mioara; Lefèvre, Vincent; Melquiond, Guillaume;
8295:
7149:(1964), the Siemens 4004 (1965), 7.700 (1974), 7.800, 7.500 (1977) series mainframes and successors, the Unidata 7.000 series mainframes, the
2299:
architecture. Often on such processors, this format can be used with "long double", though extended precision is not available with MSVC. For
1813:
9829:
7985:
2295:". A format satisfying the minimal requirements (64-bit significand precision, 15-bit exponent, thus fitting on 80 bits) is provided by the
9186:
4365:
3132:
round toward zero (truncation; it is similar to the common behavior of float-to-integer conversions, which convert −3.9 to −3 and 3.9 to 3)
1361:
and described a way to store floating-point numbers in a consistent manner. He stated that numbers will be stored in exponential format as
9214:
8770:
2750:
each of them is represented with an 8-bit exponent, followed by a sign bit, followed by a significand of respectively 23, 31, and 55 bits.
7957:"Extended" is IEC 60559's double-extended data format. Extended refers to both the common 80-bit and quadruple 128-bit IEC 60559 formats.
2277:
family. This is a binary format that occupies 64 bits (8 bytes) and its significand has a precision of 53 bits (about 16 decimal digits).
461:
8694:
3636:
2267:
family. This is a binary format that occupies 32 bits (4 bytes) and its significand has a precision of 24 bits (about 7 decimal digits).
1936:
which has a 1 as the leading digit and 0 for the remaining digits of the significand, and the smallest possible value for the exponent.
7704:
7526:
9987:
8713:. The Morgan Kaufmann series in computer architecture and design (5th ed.). Waltham, Massachusetts, USA: Elsevier. p. 793.
7011:
6168:
computation, and differential equation solving. These algorithms must be very carefully designed, using numerical approaches such as
5736:
2096:
4146:
1391:, and offered three rules by which consistent manipulation of floating-point numbers by machines could be implemented. For Torres, "
7879:"The Baleful Effect of Computer Languages and Benchmarks upon Applied Mathematics, Physics and Chemistry. John von Neumann Lecture"
3120:
round to nearest, where ties round to the nearest even digit in the required position (the default and by far the most common mode)
2200:
8787:
7406:
Muller, Jean-Michel; Brisebarre, Nicolas; de
Dinechin, Florent; Jeannerod, Claude-Pierre; Lefèvre, Vincent; Melquiond, Guillaume;
6337:
6252:
5969:
optimizations are carefully specified to maintain numerical precision. See the external references at the bottom of this article.
3856:
routine, as part of its normal operation, may evaluate a passed-in function at values outside of its domain, returning NaN and an
3071:
The arithmetical difference between two consecutive representable floating-point numbers which have the same exponent is called a
10358:
9992:
7805:
2769:
models, where range is more valuable than precision. Many machine learning accelerators provide hardware support for this format.
2106:
8051:
6874:
4585:
3503:
that the programmer might be able to catch. How this worked was system-dependent, meaning that floating-point programs were not
9982:
9977:
9320:
7934:
7822:
7022:
6074:
2905:
2762:
2637:
2623:
2270:
2260:
2086:
2076:
1946:
1754:
9088:
7533:
3300:
e=5; s=1.234571 − e=5; s=1.234567 ---------------- e=5; s=0.000004 e=−1; s=4.000000 (after rounding and normalization)
2323:
floating-point formats. These formats (especially decimal128) are pervasive in financial transactions because, along with the
9732:
9641:
9608:
9568:
9545:
9158:
9116:
9046:
9010:
8979:
8932:
8895:
8387:
7918:
7885:
7592:
7500:
7433:
7375:
of the first. By multiplying the top and bottom of the first expression by this conjugate, one obtains the second expression.
3239:
e=5; s=1.234567 + e=5; s=0.001017654 (after shifting) -------------------- e=5; s=1.235584654 (true sum: 123558.4654)
1611:
1186:
6877:
at startup, affecting the floating-point behavior of not only the generated code, but also any program using such code as a
6149:
is approximately equal to -4.44089209850063e-16). Consequently, such tests are sometimes replaced with "fuzzy" comparisons (
5997:
Expectations from mathematics may not be realized in the field of floating-point computation. For example, it is known that
3839:
366:
Floating-point representation is similar in concept to scientific notation. Logically, a floating-point number consists of:
9965:
9866:
9280:
8718:
8461:
8097:
7670:
7576:
7157:(1982) computers, and in 360/370-compatible mainframe families made by Fujitsu, Amdahl and Hitachi. It is also used in the
6973:
2850:
2794:
Bfloat16, TensorFloat-32, and the two FP8 formats, compared with IEEE 754 half-precision and single-precision formats
2758:
2336:
2069:
231:
2244:
The standard provides for many closely related formats, differing in only a few details. Five of these formats are called
2153:
1735:, etc.) to propagate through a computation in a benign manner and then be handled by the software in a controlled fashion.
1704:
for being the primary architect behind this proposal; he was aided by his student Jerome Coonen and a visiting professor,
235:
9779:
3625:
returns a value less than or equal to the smallest positive normal number in magnitude and can almost always be ignored.
3360:
2610:
The sum of the exponent bias (127) and the exponent (1) is 128, so this is represented in the single-precision format as
1416:
7633:
6421:
3368:
7756:
9691:
3349:
To multiply, the significands are multiplied while the exponents are added, and the result is rounded and normalized.
2343:
graphics language, and in the openEXR standard (where it actually predates the introduction in the IEEE 754 standard).
10143:
9671:
9516:
9475:
7471:
6867:
6134:, however these facts cannot be relied on when the quantities involved are the result of floating-point computation.
3327: = 4.000000 of the approximations. In extreme cases, all significant digits of precision can be lost. This
2526:
9087:. IFIP/SIAM/NIST Working Conference on Uncertainty Quantification in Scientific Computing, Boulder, CO. p. 33.
3975:
By the same token, an attempted computation of sin(π) will not yield zero. The result will be (approximately) 0.1225
327:
and the unstated radix point would be off the right-hand end of the string, next to the least significant digit. In
9081:
Desperately Needed
Remedies for the Undebuggability of Large Floating-Point Computations in Science and Engineering
8740:, Huberto M Sierra, "Floating decimal point arithmetic control means for calculator", issued 1962-06-05
6863:
3123:
round to nearest, where ties round away from zero (optional for binary floating-point and commonly used in decimal)
2320:
2100:
1588:
1168:
3629:
returns infinity exactly, which will typically then divide a finite number and so give zero, or else will give an
3206:
or precision, except that normalization is optional (it does not affect the numerical value of the result). Here,
596:
10116:
9624:
8863:
8020:
7969:
7154:
6953:
2324:
2316:
2090:
2080:
553:
28:
7683:
Systems such as the DFS IV and DFS V were quaternary floating-point systems and used gain steps of 12 dB.
7554:(NB. This reference incorrectly gives the MANIAC II's floating point base as 256, whereas it actually is 65536.)
4589:. Stability is a measure of the sensitivity to rounding errors of a given numerical procedure; by contrast, the
1670:
had become commonplace. This standard was significantly based on a proposal from Intel, which was designing the
10233:
10038:
9970:
9932:
9655:
9560:
9498:
8121:
7351:
6984:
6911:
5982:
3524:
3499:
Prior to the IEEE standard, such conditions usually caused the program to terminate, or triggered some kind of
3475:
An operation can be legal in principle, but not supported by the specific format, for example, calculating the
3412:
2869:
2754:
2226:
2174:
2148:
2133:
1647:
1280:), without dealing with a specific encoding of the significand. Such a program can evaluate expressions like "
17:
8149:
7703:. Los Alamos, NM, USA: Los Alamos Scientific Laboratory of the University of California. p. 14. LA-2083.
7235:(1962) computer. It is also used in the Digital Field System DFS IV and V high-resolution site survey systems.
6208:
6000:
5964:
To maintain the properties of such carefully constructed numerically stable programs, careful handling by the
2652:
standard formats, other floating-point formats are used, or have been used, in certain domain-specific areas.
5986:
4325:
3545:, set if the rounded (and returned) value is different from the mathematically exact result of the operation.
262:
227:
8493:"Added Grisu3 algorithm support for double.ToString(). by mazong1123 · Pull Request #14646 · dotnet/coreclr"
8320:
8262:
8226:
_fmsbintoieee(float *src4, float *dest4) MS Binary Format byte order => m3 | m2 | m1 | exponent m1 is
4066:(a + b) + c: 1234.567 (a) + 45.67834 (b) ____________ 1280.24534 rounds to 1280.245
10414:
10133:
10063:
9911:
4286:
is problematic: Checking that the divisor is not zero does not guarantee that a division will not overflow.
3870:
3538:
IEEE 754 specifies five arithmetic exceptions that are to be recorded in the status flags ("sticky bits"):
2193:
1667:
591:). However, 1/3 cannot be represented exactly by either binary (0.010101...) or decimal (0.333...), but in
226:
A floating-point system can be used to represent, with a fixed number of digits, numbers of very different
6473:
6193:
Round-off error can affect the convergence and accuracy of iterative numerical procedures. As an example,
3075:(ULP). For example, if there is no representable number lying between the representable numbers 1.45a70c22
2308:
Increasing the precision of the floating-point representation generally reduces the amount of accumulated
540:
Historically, several number bases have been used for representing floating-point numbers, with base two (
10388:
10023:
9833:
9678:(1213 pages) (NB. This is a single-volume edition. This work was also available in a two-volume version.)
8340:
7655:"Chapter 2 - High resolution digital site survey systems - Chapter 2.1 - Digital field recording systems"
5911:
If, however, intermediate computations are all performed in extended precision (e.g. by setting line to
4269:
1403:
will be of order of tenths, the second of hundredths, etc, and one will write each quantity in the form:
9762:(NB. This page gives a very brief summary of floating-point formats that have been used over the years.)
8448:. PLDI '10: ACM SIGPLAN Conference on Programming Language Design and Implementation. pp. 233–243.
8438:
7461:
1200:, 1/10 in decimal floating point), but the possibilities remain limited. Software packages that perform
10011:
9600:
9002:
8737:
7938:
7573:
The
Mathematical-Function Computation Handbook - Programming Using the MathCW Portable Software Library
7162:
7134:
4283:
2786:
1460:
549:
295:) is a part of a computer system specially designed to carry out operations on floating-point numbers.
269:
Over the years, a variety of floating-point representations have been used in computers. In 1985, the
80:
9179:
2660:
was developed for the
Microsoft BASIC language products, including Microsoft's first ever product the
10321:
10273:
10185:
10163:
10158:
10086:
9906:
8763:
6989:
6187:
4135:
3528:
3329:
2713:. The initial release of July 1975 supported a single-precision (32 bits) format due to cost of the
2340:
2288:
1504:
1160:
8552:
7732:
4596:
As a trivial example, consider a simple expression giving the inner product of (length two) vectors
4309:
2021:− 1 as the value for each digit of the significand and the largest possible value for the exponent.
1530:
Zuse also proposed, but did not complete, carefully rounded floating-point arithmetic that includes
323:(dot or comma) there. If the radix point is not specified, then the string implicitly represents an
10195:
9859:
9536:. Prentice-Hall Series in Automatic Computation (1st ed.). Englewood Cliffs, New Jersey, USA:
8880:(2003-09-08). "Error Analysis". In Ralston, Anthony; Reilly, Edwin D.; Hemmendinger, David (eds.).
8417:
7077:
6963:
6942:
5707:
3404:
3392:
2778:
2300:
1176:
8697:
7982:
4570:{\displaystyle \left|{\frac {\operatorname {fl} (x)-x}{x}}\right|\leq \mathrm {E} _{\text{mach}}.}
3916:
to attempt to compute tan(π/2), because π/2 cannot be represented exactly. This computation in C:
3116:
Alternative rounding options are also available. IEEE 754 specifies the following rounding modes:
2221:(a.k.a. IEC 60559) in 1985. This first standard is followed by almost all modern machines. It was
1553:
and NaN representations, anticipating features of the IEEE Standard by four decades. In contrast,
10409:
10348:
10263:
9837:
9504:
6994:
3852:
exceptions can typically not be ignored, but do not necessarily represent errors: for example, a
3511:" is an alternative flow of control, closer to what is termed a "trap" in IEEE 754 terminology.)
2746:
2657:
2186:
2143:
1622:
1533:
1346:
1334:
1283:
1255:
1217:
9394:
8446:
Proceedings of the 31st ACM SIGPLAN Conference on Programming Language Design and Implementation
8234:
m = mantissa byte s = sign bit b = bit MBF is bias 128 and IEEE is bias 127. MBF places the
6499:
Here is a computation using IEEE "double" (a significand with 53 bits of precision) arithmetic:
10091:
9947:
9901:
9253:
9207:
8412:
8231:
6948:
6932:
6890:
6154:
5719:
5718:
Although individual arithmetic operations of IEEE 754 are guaranteed accurate to within half a
3072:
2230:
2222:
2052:
1745:
1607:
1592:
1577:
1172:
1153:
592:
545:
432:
328:
201:
3567:, set if the result is infinite given finite operands, returning an infinity, either +∞ or −∞.
10081:
10056:
9586:
9494:
9458:
8996:
8918:
8877:
8665:
8227:
7654:
7411:
6176:
4087:
3808:
3773:
3584:
3573:, set if a real-valued result cannot be returned e.g. sqrt(−1) or 0/0, returning a quiet NaN.
2742:
1180:
541:
371:
309:
7697:
95:. For example, 12.345 is a floating-point number in base ten with five digits of precision:
9883:
9485:
9309:
9203:
9175:
9147:
9105:
9075:
9029:
8759:
8376:
8086:
7874:
7317:
7166:
7053:
6855:
6169:
4313:
4301:
4027:
3746:
3532:
3488:
3061:
1908:
1277:
9313:
8860:"Patriot missile defense, Software problem led to system failure at Dharhan, Saudi Arabia"
4275:
Limited exponent range: results might overflow yielding infinity, or underflow yielding a
1368:
1235:
606:
8:
10353:
10331:
10258:
10111:
10103:
9852:
9578:
9468:
8811:
7368:
7347:
7158:
7017:
6916:
6878:
5727:
4340:
is a quantity that characterizes the accuracy of a floating-point system, and is used in
3388:
3137:
2441:
1446:
1211:
552:), base eight (octal floating point), base four (quaternary floating point), base three (
336:
288:
220:
9151:
9109:
9033:
8380:
7878:
7571:
Beebe, Nelson H. F. (2017-08-22). "Chapter H. Historical floating-point architectures".
5961:
then the algorithm becomes numerically stable and can compute to full double precision.
5746:
For example, the following algorithm is a direct implementation to compute the function
3400:
2761:, but allocates 8 bits to the exponent instead of 5, thus providing the same range as a
10336:
10316:
10268:
10243:
10028:
9997:
9801:
9783:
9651:
9619:
9412:
9079:
9062:
8831:
Far more worrying is cancellation error which can yield catastrophic loss of precision.
8764:"Lecture Notes on the Status of IEEE Standard 754 for Binary Floating-Point Arithmetic"
8684:
8633:
8615:
8533:
8467:
8356:
7851:
7606:
7280:
7232:
6199:
6141:) requires care when dealing with floating-point numbers. Even simple expressions like
5740:
5732:
4619:
4599:
4580:
3853:
3508:
3364:
3334:
2280:
2274:
2264:
2112:
706:. It is used to round the 33-bit approximation to the nearest 24-bit number (there are
8834:
8090:
3459:
3297: = 1.234567 are approximations to the rationals 123457.1467 and 123456.659.
2339:, also called binary16, a 16-bit floating-point value. It is being used in the NVIDIA
1728:
10223:
10153:
10128:
9942:
9937:
9805:
9738:
9728:
9667:
9637:
9604:
9594:
9590:
9582:
9564:
9541:
9512:
9471:
9376:
9006:
8975:
8928:
8891:
8887:
8816:
8714:
8688:
8637:
8537:
8457:
8411:(Technical report). NUMERICAL ANALYSIS MANUSCRIPT 90-10, AT&T BELL LABORATORIES.
7914:
7850:(2014-06-07). "The Z1: Architecture and Algorithms of Konrad Zuse's First Computer".
7783:, p. 545, Digital Computers: Origins, Encyclopedia of Computer Science, January 2003.
7780:
7752:
7666:
7598:
7588:
7496:
7467:
7439:
7429:
7245:
7005:
6922:
6843:
6145:
will, on most computers, fail to be true (in IEEE 754 double precision, for example,
3308:
2940:
2584:
2383:, and they are ordered in the same way as their values (in the set of real numbers).
1358:
1338:
1225:
320:
9820:
contains verilog source code of a double-precision floating-point unit. The project
9377:"55522 – -funsafe-math-optimizations is unexpectedly harmful, especially w/ -shared"
7797:
7610:
3303:
The floating-point difference is computed exactly because the numbers are close—the
1618:
personal computers had floating-point capability in hardware as a standard feature.
10368:
10253:
10051:
9793:
9720:
9435:
9289:
9276:"Adaptive Precision Floating-Point Arithmetic and Fast Robust Geometric Predicates"
8674:
8625:
8523:
8449:
7814:
7580:
7421:
7330:
7294:
6937:
6873:
In some compilers (GCC and Clang), turning on "fast" math may cause the program to
6161:
5723:
4590:
4276:
3500:
3492:
3469:
2766:
2714:
2710:
2498:
2470:
2029:
2024:
In addition, there are representable values strictly between −UFL and UFL. Namely,
1732:
1638:: 72 bits, organized as a 1-bit sign, an 11-bit exponent, and a 60-bit significand.
1554:
1229:
714:
in this example, is added to the integer formed by the leftmost 24 bits, yielding:
355:
seconds, a value that would be represented in standard-form scientific notation as
208:
digits of precision, the sum 12.345 + 1.0001 = 13.3451 might be rounded to 13.345.
48:
8471:
7363:
The equivalence of the two forms can be verified algebraically by noting that the
3248:
The lowest three digits of the second operand (654) are essentially lost. This is
3107:
was the typical approach. Since the introduction of IEEE 754, the default method (
1632:: 36 bits, organized as a 1-bit sign, an 8-bit exponent, and a 27-bit significand.
370:
A signed (meaning positive or negative) digit string of a given length in a given
257:
10373:
10238:
10190:
10123:
9757:
9710:
9508:
9481:
9462:
9440:
8965:
8961:
8881:
8202:
7989:
7910:
7219:
7215:
7211:
7207:
7109:
4346:
4305:
4079:
1234.567 (a) + 45.67874 (b + c) ____________ 1280.24574 rounds to
3252:. In extreme cases, the sum of two non-zero numbers may be equal to one of them:
3249:
3065:
2931:
2309:
1705:
1396:
1354:
564:
281:
8381:"How Futile are Mindless Assessments of Roundoff in Floating-Point Computation?"
4069:
1280.245 (a + b) + 0.0004 (c) ____________ 1280.2454 rounds to
707:
595:, it is trivial (0.1 or 1×3) . The occasions on which infinite expansions occur
284:, especially for applications that involve intensive mathematical calculations.
10326:
10148:
10138:
10046:
9706:
9663:
9633:
8492:
8002:
7407:
7203:
7170:
7150:
7146:
7142:
7138:
4494:
3480:
3464:
Floating-point computation in a computer can run into three kinds of problems:
3304:
2702:
2554:
2437:
2351:
The standard specifies some special values, and their representation: positive
2256:. Three formats are especially widely used in computer hardware and languages:
2138:
1655:
567:, because it can be represented as one integer divided by another; for example
439:, which has ten decimal digits of precision, is represented as the significand
340:
9724:
8586:
7584:
7425:
4579:
Backward error analysis, the theory of which was developed and popularized by
2614:
0 10000000 10010010000111111011011 (excluding the hidden bit) = 40490FDB as a
2594:
For example, it was shown above that π, rounded to 24 bits of precision, has:
2217:
standardized the computer representation for binary floating-point numbers in
10403:
10248:
9537:
8820:
8235:
7902:
7625:
7284:
7190:
6510:, second form --------------------------------------------------------- 0
5990:
4272:
may produce answers which are off by one from the intuitively expected value.
3226:
123456.7 = 1.234567 × 10^5 101.7654 = 1.017654 × 10^2 = 0.001017654 × 10^5
3143:
2738:
2734:
2726:
2364:
1697:
1685:
1654:
mainframes; these same representations are still available for use in modern
1603:
1569:
1454:
1442:
1430:
239:
197:
44:
9797:
9716:
9434:. CAV 2019: Computer Aided Verification. Vol. 11562. pp. 155–173.
8845:
8528:
8511:
8453:
8091:"On the Cost of Floating-Point Computation Without Extra-Precise Arithmetic"
7847:
7793:
7417:
5702:
by definition, which is the sum of two slightly perturbed (on the order of Ε
3379:
Literals for floating-point numbers depend on languages. They typically use
1353:, where he designed a special-purpose electromechanical calculator based on
761:
When this is stored in memory using the IEEE 754 encoding, this becomes the
10205:
10180:
9683:
8661:"What Every Computer Scientist Should Know About Floating-Point Arithmetic"
8170:
4321:
3170:
Ryū, an always-succeeding algorithm that is faster and simpler than Grisu3.
2661:
1701:
1587:
has binary floating-point arithmetic, and it became operational in 1950 at
1572:
computer, designed in 1942–1945. In 1946, Bell Laboratories introduced the
1325:
9430:
Becker, Heiko; Darulova, Eva; Myreen, Magnus O.; Tatlock, Zachary (2019).
8859:
8806:
8679:
8660:
6202:). Two forms of the recurrence formula for the circumscribed polygon are:
3968:
will give a result of 16331239353195370.0. In single precision (using the
2367:(−0) distinct from ordinary ("positive") zero, and "not a number" values (
10383:
10378:
10228:
10175:
10002:
9337:
8239:
7364:
7343:
7186:
7182:
7080:) is ambiguous, as it was historically also used to specify some form of
7044:
6927:
6165:
5366:{\displaystyle \operatorname {fl} (x\cdot y)={\hat {x}}\cdot {\hat {y}},}
3987:
3476:
2677:
2615:
2292:
2025:
1558:
1438:
1426:
1221:
1089:
is the position of the bit of the significand from the left (starting at
762:
684:{\displaystyle 11001001\ 00001111\ 1101101{\underline {0}}\ 10100010\ 0.}
380:
316:
292:
250:
216:
84:
76:
9782:(ACM) Transactions on programming languages and systems (TOPLAS): 1–41.
9358:
9346:
We support floating point reduction operations when -ffast-math is used.
8606:
Lemire, Daniel (2021-03-22). "Number parsing at a gigabyte per second".
5706:) input data, and so is backward stable. For more realistic examples in
3577:
3198:
with 7 digit precision will be used in the examples, as in the IEEE 754
312:
specifies some way of encoding a number, usually as a string of digits.
10288:
10283:
10200:
10168:
10073:
10016:
9767:
9294:
9275:
8566:
8321:"TensorFloat-32 in the A100 GPU Accelerates AI Training, HPC up to 20x"
6194:
6175:
Summation of a vector of floating-point values is a basic algorithm in
4139:
4076:
a + (b + c): 45.67834 (b) + 0.0004 (c) ____________ 45.67874
3408:
3319: = 4.877000, which differs more than 20% from the difference
3104:
2706:
2698:
2693:
1891:{\displaystyle 2\left(B-1\right)\left(B^{P-1}\right)\left(U-L+1\right)}
1717:
1671:
1651:
1412:
72:
9432:
Icing: Supporting Fast-Math Style Optimizations in a Verified Compiler
9045:, Symposium on Computer Arithmetic (Keynote Address). pp. 6, 18.
8439:"Printing floating-point numbers quickly and accurately with integers"
7818:
3311:
is supported. Despite this, the difference of the original numbers is
3011:
11.0010010000111111011010101000100010000101101000110000100011010011...
276:
The speed of floating-point operations, commonly measured in terms of
39:
10363:
10341:
10298:
10293:
9960:
9916:
9875:
9824:
contains vhdl source code of a single-precision floating-point unit.)
8629:
8291:
7662:
7488:
7372:
7258:
7057:
7000:
6859:
3281:
to two nearly equal numbers are subtracted. In the following example
2128:
1681:
1584:
1422:
348:
60:
8125:
6919:
for code examples demonstrating access and use of IEEE 754 features.
2930:
By their nature, all numbers expressed in floating-point format are
2777:, which provides hardware support for it in the Tensor Cores of its
2745:). All Microsoft language products from 1975 through 1987 used the
1625:, introduced in 1962, supported two floating-point representations:
10278:
9788:
9423:
9231:
9042:
8620:
8361:
8021:"Procedure Call Standard for the ARM 64-bit Architecture (AArch64)"
7095:
6978:
6968:
5965:
4344:
of floating-point algorithms. It is also known as unit roundoff or
3396:
3109:
3096:
3024:
2718:
2689:
2649:
2352:
2238:
2234:
2218:
2059:
2041:
1689:
1663:
1643:
1614:(XSC)). It was not until the launch of the Intel i486 in 1989 that
1201:
401:
270:
88:
9742:
8424:
7856:
7602:
7443:
5743:
are designed for this purpose when computing at double precision.
4470:{\displaystyle \mathrm {E} _{\text{mach}}={\tfrac {1}{2}}B^{1-P},}
3736:{\displaystyle R_{\text{tot}}=1/(1/R_{1}+1/R_{2}+\cdots +1/R_{n})}
3236:
e=5; s=1.234567 (123456.7) + e=2; s=1.017654 (101.7654)
3177:
Many modern language runtimes use Grisu3 with a Dragon4 fallback.
3157:, a practical open-source implementation of many ideas in Dragon4.
3129:
round down (toward −∞; negative results thus round away from zero)
3030:
In binary single-precision floating-point, this is represented as
2722:
1433:
computer, which uses a 22-bit binary floating-point representation
9490:(NB. Classic influential treatises on floating-point arithmetic.)
8711:
Computer Organization and Design, The Hardware/Software Interface
8693:(With the addendum "Differences Among IEEE 754 Implementations":
8258:
8210:
8206:
6885:
4317:
3900:
0.010000000298023226097399174250313080847263336181640625 exactly.
3860:
exception flag to be ignored until finding a useful start point.
3504:
2925:
2312:
caused by intermediate calculations. Other IEEE formats include:
1599:
1573:
344:
324:
9704:
9577:
7121:) is ambiguous, as it was historically also used to specify the
4332:
4324:, contributing to the death of 28 soldiers from the U.S. Army's
3921:/* Enough digits to be sure we get the correct approximation. */
3049:
whereas a more accurate approximation of the true value of π is
2629:
1337:, in 1914, published an analysis of floating point based on the
8591:
8571:
8409:
Correctly Rounded Binary-Decimal and Decimal-Binary Conversions
8353:
7493:
The Scientist and Engineer's Guide to Digital Signal Processing
5978:
3164:. Must be used with a fallback, as it fails for ~0.5% of cases.
3161:
2774:
2685:
2681:
2673:
2665:
2035:
1330:
1205:
52:
8284:"IEEE vs. Microsoft Binary Format; Rounding Issues (Complete)"
3551:, set if the rounded value is tiny (as specified in IEEE 754)
3468:
An operation can be mathematically undefined, such as ∞/∞, or
397:
point is set just after the most significant (leftmost) digit.
9844:
8792:
Intel 64 and IA-32 Architectures Software Developers' Manuals
7231:
Quaternary (base-4) floating-point arithmetic is used in the
6958:
6410:{\textstyle t_{i+1}={\frac {t_{i}}{{\sqrt {t_{i}^{2}+1}}+1}}}
6325:{\textstyle t_{i+1}={\frac {{\sqrt {t_{i}^{2}+1}}-1}{t_{i}}}}
5713:
3274:
bit need to be carried beyond the precision of the operands.
3203:
3195:
3126:
round up (toward +∞; negative results thus round toward zero)
2730:
2158:
1675:
1450:
375:
277:
9208:"Lecture notes of System Support for Scientific Computation"
7099:
of a floating-point number is sometimes also referred to as
7072:
by some authors is potentially misleading as well. The term
3000:
which is actually 0.100000001490116119384765625 in decimal.
752:{\displaystyle 11001001\ 00001111\ 1101101{\underline {1}}.}
525:
is the precision (the number of digits in the significand),
47:, included floating-point arithmetic (replica on display at
9896:
9429:
9252:
Christiansen, Tom; Torkington, Nathan; et al. (2006).
7761:
7735:, pp. 575–583, Revista de Obras Públicas, 19 November 1914.
7733:
Automática: Complemento de la Teoría de las Máquinas, (pdf)
7288:
4793: indicates correctly rounded floating-point arithmetic
4359:, its value depends on the particular rounding being used.
3007:, represented in binary as an infinite sequence of bits is
2669:
1901:
There is a smallest positive normal floating-point number,
9035:
Floating-Point Arithmetic Besieged by "Business Decisions"
7983:
Using the GNU Compiler Collection, i386 and x86-64 Options
7350:
then full double precision is retained; if long double is
5693:{\displaystyle \delta _{n}\leq \mathrm {E} _{\text{mach}}}
3986:
While floating-point addition and multiplication are both
3144:
Binary-to-decimal conversion with minimal number of digits
548:), and other less common varieties, such as base sixteen (
9891:
9813:
8052:"ARM Compiler toolchain Compiler Reference, Version 5.03"
7798:"Konrad Zuse's Legacy: The Architecture of the Z1 and Z3"
6127:{\displaystyle \sin ^{2}{\theta }+\cos ^{2}{\theta }=1\,}
5912:
3838:
of 0, as expected (see the continued fraction example of
3520:
2368:
2296:
2284:
2006:{\displaystyle \left(1-B^{-P}\right)\left(B^{U+1}\right)}
1761:
The number of normal floating-point numbers in a system (
1674:
numerical coprocessor; Motorola, which was designing the
1144:
630:
521:
is the significand (ignoring any implied decimal point),
8150:"Technical Introduction to OpenEXR – The half Data Type"
7202:
Octal (base-8) floating-point arithmetic is used in the
4504:
within the normalized range of a floating-point system:
3903:
Squaring it with rounding to the 24-bit precision gives
3479:
of −1 or the inverse sine of 2 (both of which result in
3403:
with a base-2 exponent instead of 10. In languages like
3160:
Grisu3, with a 4× speedup as it removes the use of
1557:
recommended against floating-point numbers for the 1951
698:
above. The next bit, at position 24, is called the
9768:"The pitfalls of verifying floating-point computations"
9142:
9140:
9138:
9136:
8702:
3581:
Fig. 1: resistances in parallel, with total resistance
3202:
format. The fundamental principles are the same in any
3004:
2935:
2643:
634:
9251:
9180:"Why do we need a floating-point arithmetic standard?"
8852:
8847:
PEP 485 -- A Function for testing approximate equality
8579:
8276:
8195:"Converting between Microsoft Binary and IEEE formats"
7466:. Englewood Cliffs, NJ, United States: Prentice-Hall.
6340:
6255:
6211:
5989:
programming languages, and the decimal formats of the
4437:
1678:
around the same time, gave significant input as well.
416:
To derive the value of the floating-point number, the
43:
An early electromechanical programmable computer, the
9772:
ACM Transactions on Programming Languages and Systems
9413:"Bug in zheevd · Issue #43 · Reference-LAPACK/lapack"
9302:
9110:"How Java's floating-point hurts everyone everywhere"
9024:
9022:
8259:"Create your own Version of Microsoft BASIC for 6502"
7145:(1970) as well as various newer IBM machines, in the
6945:– utilizes high precision floating-point computations
6849:
6476:
6424:
6077:
6003:
5664:
5383:
5308:
4642:
4622:
4602:
4510:
4420:
4407:{\displaystyle \mathrm {E} _{\text{mach}}=B^{1-P},\,}
4368:
4149:
3811:
3776:
3749:
3639:
3587:
2606:= 110010010000111111011011 (including the hidden bit)
1949:
1911:
1816:
1561:, arguing that fixed-point arithmetic is preferable.
1536:
1507:
1463:
1371:
1286:
1258:
1238:
774:
720:
643:
609:
464:
103:
27:"Floating point" redirects here. For other uses, see
9133:
8988:
8754:
8752:
7455:
7453:
7354:
then additional, but not full precision is retained.
7283:
once led to a famous error. An early version of the
7257:
Base-65536 floating-point arithmetic is used in the
3277:
Another problem of loss of significance occurs when
3194:
For ease of presentation and understanding, decimal
2757:
requires the same amount of memory (16 bits) as the
9596:
Numerical Recipes - The Art of Scientific Computing
9503:. Monographs on Numerical Analysis (1st ed.).
9467:(1st ed.). Englewood Cliffs, New Jersey, USA:
9098:
8369:
8188:
8186:
6858:of floating-point operations in general means that
2273:(binary64), usually used to represent the "double"
1591:. Thirty-three were later sold commercially as the
508:{\displaystyle {\frac {s}{b^{\,p-1}}}\times b^{e},}
9830:"Microsoft Visual C++ Floating-Point Optimization"
9622:(1997). "Section 4.2: Floating-Point Arithmetic".
9168:
9019:
8998:Writing Scientific Software: A Guide to Good Style
8252:
8250:
7869:
7867:
7244:Base-256 floating-point arithmetic is used in the
7052:by some authors—not to be confused with the
6488:
6463:{\displaystyle \pi \sim 6\times 2^{i}\times t_{i}}
6462:
6409:
6324:
6239:
6126:
6063:
5692:
5646:
5365:
5294:
4628:
4608:
4569:
4469:
4406:
4212:
3873:with which computers generally represent numbers.
3830:
3797:
3762:
3735:
3606:
3411:), or allow overloading of numeric types (such as
2263:(binary32), usually used to represent the "float"
2005:
1924:
1890:
1739:
1545:
1519:
1493:
1383:
1307:
1268:
1244:
1067:
751:
710:, which is not the case here). This bit, which is
683:
621:
507:
185:
9650:
9245:
8995:Oliveira, Suely; Stewart, David E. (2006-09-07).
8870:
8749:
8708:
8347:
8163:
8044:
7939:"An Interview with the Old Man of Floating-Point"
7907:The Origins of Digital Computers: Selected Papers
7450:
3453:
2947:to be rounded to either .55555555 or .55555556).
1568:computer with floating-point hardware was Zuse's
1113:. For binary formats (which uses only the digits
156:
155:
154:
153:
152:
151:
150:
132:
128:
110:
91:of a fixed base. Numbers of this form are called
10401:
9224:
9068:
8920:Accuracy and reliability in scientific computing
8651:
8649:
8647:
8183:
8013:
7527:"Rechnerarithmetik: Fest- und Gleitkommasysteme"
7520:
7518:
4060:. Using 7-digit significand decimal arithmetic:
3909:But the representable number closest to 0.01 is
3307:guarantees this, even in case of underflow when
1501:, and it stops on undefined operations, such as
529:is the base (in our example, this is the number
8994:
8709:Patterson, David A.; Hennessy, John L. (2014).
8247:
7864:
6981:– Standard for Binary Floating-Point Arithmetic
4279:or zero. In these cases precision will be lost.
3877:decimal number 0.1 is represented in binary as
3180:
2926:Representable numbers, conversion and rounding
2233:in addition to the IEEE 754 binary format. The
1189:, which does not appear to be used in practice.
544:) being the most common, followed by base ten (
392:. The length of the significand determines the
8972:Society for Industrial and Applied Mathematics
8967:Accuracy and Stability of Numerical Algorithms
8956:
8954:
8952:
8950:
8925:Society for Industrial and Applied Mathematics
8910:
8732:
8730:
8314:
8312:
7927:
7895:
7646:
5710:, see Higham 2002 and other references below.
4213:{\displaystyle Q(h)={\frac {f(a+h)-f(a)}{h}}.}
3805:will return +infinity which will give a final
3344:
1696:In 1989, mathematician and computer scientist
412:), which modifies the magnitude of the number.
9860:
9660:Computer Architecture: Concepts and Evolution
9555:Golub, Gene F.; van Loan, Charles F. (1986).
9554:
9395:"Code Gen Options (The GNU Fortran Compiler)"
9196:
8644:
8333:
8192:
7953:ISO/IEC 9899:1999 - Programming languages - C
7515:
7401:
7399:
7397:
7395:
7393:
7391:
7068:are also used by some. The usage of the term
7036:
5099:
4987:
4926:
4814:
4763:
4683:
4485:is the precision of the significand (in base
4333:Machine precision and backward error analysis
4142:of a function the following formula is used:
2194:
2036:IEEE 754: floating point in modern computers
1790:is the precision of the significand (in base
1602:followed in 1954; it introduced the use of a
249:Single-precision floating-point numbers on a
8430:
8142:
7781:Digital Computers, History of Origins, (pdf)
7689:
7566:
7564:
7562:
7560:
3189:
3038: = 1. This has a decimal value of
3034: = 1.10010010000111111011011 with
2327:format, they allow correct decimal rounding.
253:: the green lines mark representable values.
9330:
9314:"Roundoff Degrades an Idealized Cantilever"
8947:
8727:
8599:
8402:
8400:
8309:
8003:"long double (GCC specific) and __float128"
3972:function), the result will be −22877332.0.
3519:some programming language standards (e.g.,
3217:
2227:IBM's own hexadecimal floating point format
629:, and so the significand is a string of 24
378:). This digit string is referred to as the
9867:
9853:
9308:
9104:
8503:
8294:. 2006-11-21. Article ID KB35826, Q35826.
8079:
7840:
7786:
7617:
7388:
7367:of the fraction in the second form is the
7087:
7048:of a floating-point number is also called
6182:3253.671 + 3.141276 ----------- 3256.812
5714:Minimizing the effect of accuracy problems
3245:e=5; s=1.235585 (final sum: 123558.5)
2386:
2201:
2187:
1939:There is a largest floating-point number,
1648:hexadecimal floating-point representations
9787:
9493:
9457:
9439:
9293:
8916:
8876:
8678:
8619:
8544:
8527:
8416:
8360:
7995:
7933:
7855:
7652:
7575:(1st ed.). Salt Lake City, UT, USA:
7557:
7524:
7495:. California Technical Pub. p. 514.
7489:"Chapter 28, Fixed versus Floating Point"
7480:
7012:Quadruple-precision floating-point format
6240:{\textstyle t_{0}={\frac {1}{\sqrt {3}}}}
6123:
6060:
4498:in representing any non-zero real number
4403:
3912:0.009999999776482582092285156250 exactly.
3906:0.010000000707805156707763671875 exactly.
3422:Examples of floating-point literals are:
3415:). In these cases, digit strings such as
1232:can often handle irrational numbers like
1208:" arithmetic for the individual integers.
475:
303:
9765:
9531:
9273:
9267:
8655:
8397:
7459:
7410:; Stehlé, Damien; Torres, Serge (2010).
6064:{\displaystyle (x+y)(x-y)=x^{2}-y^{2}\,}
3621:returns a correctly rounded result, and
3576:
2970:= 1100110011001100110011001100110011...,
1744:A floating-point number consists of two
1680:
1421:
1329:
1179:and Peter Turner is a scheme based on a
597:depend on the base and its prime factors
256:
244:
38:
9827:
9369:
8436:
8114:
7901:
7806:IEEE Annals of the History of Computing
7767:
7695:
7405:
4063:a = 1234.567, b = 45.67834, c = 0.0004
2241:still uses Cray floating-point format.
1800:is the smallest exponent of the system,
34:Computer approximation for real numbers
14:
10402:
9681:
9615:(NB. Edition with source code CD-ROM.)
9464:Rounding Errors in Algebraic Processes
9405:
8974:(SIAM). pp. 27–28, 110–123, 493.
8960:
8838:
8804:
8798:
8605:
8512:"Ryū: fast float-to-string conversion"
8485:
8318:
7623:
7023:Single-precision floating-point format
6834:62246 The true value is
6828:62246 28
6822:62246 27
6816:62246 26
6810:68907 25
4492:This is important since it bounds the
4233:grows smaller, the difference between
3894:0.100000001490116119384765625 exactly.
3367:). For a fast, simple method, see the
3338:
3003:As a further example, the real number
1806:is the largest exponent of the system,
1145:Alternatives to floating-point numbers
1121:), this non-zero digit is necessarily
280:, is an important characteristic of a
9848:
9712:Handbook of Floating-Point Arithmetic
9618:
9281:Discrete & Computational Geometry
9202:
9174:
9146:
9074:
9028:
8780:
8758:
8559:
8553:"The Schubfach way to render doubles"
8550:
8509:
8375:
8341:"NVIDIA Hopper Architecture In-Depth"
8256:
8085:
7873:
7846:
7792:
7626:"The Decimal Floating-Point Standard"
7570:
7486:
7413:Handbook of Floating-Point Arithmetic
7251:
7238:
7225:
3889: = 110011001100110011001101
3419:may also be floating-point literals.
2985:When rounded to 24 bits this becomes
1711:Among the x86 innovations are these:
1666:standard once the 32-bit (or 64-bit)
1612:Extensions for Scientific Computation
1187:Tapered floating-point representation
261:Augmented version above showing both
215:refers to the fact that the number's
7970:"IEEE Floating-Point Representation"
7962:
7577:Springer International Publishing AG
7357:
7336:
7323:
7311:
7301:
7273:
7264:
7196:
7135:Hexadecimal (base-16) floating-point
6974:Half-precision floating-point format
6489:{\displaystyle i\rightarrow \infty }
3863:
3255:e=5; s=1.234567 + e=−3; s=9.876543
2644:Other notable floating-point formats
2237:series had an IEEE version, but the
435:notation) as an example, the number
319:is indicated by placing an explicit
9780:Association for Computing Machinery
9387:
9351:
9274:Shewchuk, Jonathan Richard (1997).
9206:(2001-06-04). Bindel, David (ed.).
8406:
7976:
7945:
7534:Friedrich-Schiller-Universität Jena
7525:Zehendner, Eberhard (Summer 2008).
7128:
7122:
3743:. If a short-circuit develops with
458:Symbolically, this final value is:
83:with a fixed precision, called the
24:
9758:"Survey of Floating-Point Formats"
9451:
7909:(3rd ed.). Berlin; New York:
7279:The enormous complexity of modern
6837:3.14159265358979323846264338327...
6483:
6190:may be used to reduce the errors.
5680:
4957:
4554:
4423:
4371:
3979:10 in double precision, or −0.8742
3460:IEEE 754 § Exception handling
3054:3.14159265358979323846264338327950
2628:
1540:
1514:
1480:
1395:will always be the same number of
1175:(LI and SLI) of Charles Clenshaw,
339:, the given number is scaled by a
25:
10426:
9750:
9312:; Ivory, Melody Y. (1997-07-03).
9254:"perlfaq4 / Why is int() broken?"
8608:Software: Practice and Experience
7696:Lazarus, Roger B. (1957-01-30) .
3933:3.1415926535897932384626433832795
3374:
3086:
2591:format has 53, and quad has 113.
1494:{\displaystyle ^{1}/_{\infty }=0}
1445:, the first binary, programmable
708:specific rules for halfway values
9500:The Algebraic Eigenvalue Problem
9326:from the original on 2003-12-05.
9220:from the original on 2013-05-17.
9192:from the original on 2004-12-04.
9164:from the original on 2003-08-15.
9094:from the original on 2013-06-20.
8883:Encyclopedia of Computer Science
8807:"You're Going To Have To Think!"
8805:Harris, Richard (October 2010).
8776:from the original on 2002-06-22.
8481:from the original on 2014-07-29.
8393:from the original on 2004-12-21.
7891:from the original on 2008-09-05.
7081:
7060:. Somewhat vague, terms such as
6864:common subexpression elimination
3361:Booth's multiplication algorithm
2709:, during spring of 1975 for the
1752:On a typical computer system, a
1589:National Physical Laboratory, UK
9694:from the original on 2018-07-03
9656:Brooks, Jr., Frederick Phillips
9625:The Art of Computer Programming
9122:from the original on 2000-08-16
9052:from the original on 2006-03-17
8864:US Government Accounting Office
8298:from the original on 2020-08-28
8265:from the original on 2016-05-30
8217:from the original on 2019-02-20
8103:from the original on 2006-05-25
8068:from the original on 2015-06-27
8033:from the original on 2013-07-31
7828:from the original on 2022-07-03
7773:
7738:
7725:
7710:from the original on 2018-08-07
7653:Parkinson, Roger (2000-12-07).
7636:from the original on 2018-07-03
7543:from the original on 2018-08-07
6985:IBM Floating Point Architecture
6954:Floating-point error mitigation
5915:
3454:Dealing with exceptional cases
2648:In addition to the widely used
2149:IBM floating-point architecture
1740:Range of floating-point numbers
1520:{\displaystyle 0\times \infty }
1399:(e.g. six), the first digit of
554:balanced ternary floating point
230:— such as the number of meters
29:Floating point (disambiguation)
9874:
9561:Johns Hopkins University Press
9152:"Marketing versus Mathematics"
8199:Technical Information Database
7532:(Lecture script) (in German).
7352:IEEE double extended precision
6912:Arbitrary-precision arithmetic
6480:
6151:if (abs(x-y) < epsilon) ...
6137:The use of the equality test (
6031:
6019:
6016:
6004:
5634:
5615:
5586:
5571:
5552:
5523:
5506:
5487:
5458:
5443:
5424:
5395:
5354:
5339:
5327:
5315:
5282:
5263:
5260:
5241:
5238:
5212:
5206:
5187:
5184:
5165:
5162:
5136:
5123:
5104:
5094:
5075:
5072:
5046:
5040:
5021:
5018:
4992:
4921:
4902:
4899:
4873:
4867:
4848:
4845:
4819:
4788:
4785:
4758:
4732:
4720:
4694:
4665:
4653:
4530:
4524:
4481:is the base of the system and
4310:prevented it from intercepting
4221:Intuitively one would want an
4198:
4192:
4183:
4171:
4159:
4153:
4086:They are also not necessarily
3730:
3661:
3531:specifies that the flags have
3527:outside of the standard (e.g.
3110:round to nearest, ties to even
2759:IEEE 754 half-precision format
1578:decimal floating-point numbers
1417:Electromechanical Arithmometer
1345:In 1914, the Spanish engineer
1302:
1293:
1276:in a completely "formal" way (
1192:Some simple rational numbers (
13:
1:
9933:Arbitrary-precision or bignum
8319:Kharya, Paresh (2020-05-14).
8257:Steil, Michael (2008-10-20).
8193:Borland staff (1998-07-02) .
7382:
4414:whereas rounding to nearest,
4326:14th Quartermaster Detachment
2658:Microsoft Binary Format (MBF)
1688:, principal architect of the
563:A floating-point number is a
556:) and even base 256 and base
9766:Monniaux, David (May 2008).
9682:Savard, John J. G. (2018) ,
9441:10.1007/978-3-030-25543-5_10
9338:"Auto-Vectorization in LLVM"
9232:"General Decimal Arithmetic"
7659:High Resolution Site Surveys
7624:Savard, John J. G. (2018) ,
6172:, if they are to work well.
5737:IEEE 754 quadruple precision
4294:
4045:is not necessarily equal to
4026:), they are not necessarily
3210:denotes the significand and
3181:Decimal-to-binary conversion
1610:" (SC) capability (see also
633:. For instance, the number
431:Using base-10 (the familiar
7:
9834:Microsoft Developer Network
8230:=> sbbb|bbbb m3 is the
7955:. Iso.org. §F.2, note 307.
7751:Springer, pp. 84–85, 2017.
6904:
6854:The aforementioned lack of
4270:Floor and ceiling functions
3897:Squaring this number gives
3345:Multiplication and division
3027:to a precision of 24 bits.
2996:= 110011001100110011001101,
2808:Trailing significand field
2622:An example of a layout for
2254:extendable precision format
2026:positive and negative zeros
1749:wider range to the number.
1546:{\displaystyle \pm \infty }
1308:{\displaystyle \sin(3\pi )}
1269:{\displaystyle {\sqrt {3}}}
424:raised to the power of the
298:
75:that represents subsets of
10:
10431:
9601:Cambridge University Press
9534:Floating-Point Computation
9003:Cambridge University Press
8587:"google/double-conversion"
7731:Torres Quevedo, Leonardo.
7463:Floating-Point Computation
7163:Data General Eclipse S/200
7137:arithmetic is used in the
7125:of floating-point numbers.
7084:of floating-point numbers.
6804:.2245152435345525443
6534:.2153903091734723496 2
6528:.2153903091734710173
6522:.4641016151377543863 1
6516:.4641016151377543863
3457:
3401:hexadecimal literal syntax
3289: = 1.234571 and
2379:and strictly greater than
2250:extended precision formats
2039:
1784:is the base of the system,
1323:
1319:
1161:Logarithmic number systems
550:hexadecimal floating point
291:(FPU, colloquially a math
236:between protons in an atom
26:
10307:
10274:Strongly typed identifier
10216:
10102:
10072:
10037:
9925:
9882:
9725:10.1007/978-3-319-76526-6
9709:; Torres, Serge (2018) .
9532:Sterbenz, Pat H. (1974).
9108:; Darcy, Joseph (2001) .
8866:. GAO report IMTEC 92-26.
8736:
8510:Adams, Ulf (2018-12-02).
8437:Loitsch, Florian (2010).
8124:. openEXR. Archived from
7585:10.1007/978-3-319-64110-2
7487:Smith, Steven W. (1997).
7460:Sterbenz, Pat H. (1974).
7426:10.1007/978-0-8176-4705-6
7014:(including double-double)
6990:Kahan summation algorithm
6897:optimizations is seen in
6506:, first form 6 × 2 × t
6188:Kahan summation algorithm
3840:IEEE 754 design rationale
3190:Floating-point operations
3019:11.0010010000111111011011
2763:IEEE 754 single-precision
2415:
2410:
2405:
2403:
2400:
2397:
2225:. IBM mainframes support
1415:, as was the case of his
404:(also referred to as the
65:floating-point arithmetic
9684:"Floating-Point Formats"
9630:Seminumerical Algorithms
7287:chip was shipped with a
7029:
6943:Experimental mathematics
6875:disable subnormal floats
6850:"Fast math" optimization
6780:349453756585929919
6546:596599420975006733 3
6540:596599420974940120
5920:
5767:
5708:numerical linear algebra
3983:10 in single precision.
3918:
3491:(exponent too small) or
3218:Addition and subtraction
2248:, and others are termed
1905:Underflow level = UFL =
1441:of Berlin completed the
1218:Computer algebra systems
10349:Parametric polymorphism
9798:10.1145/1353445.1353446
9505:Oxford University Press
9157:. pp. 15, 35, 47.
8529:10.1145/3296979.3192369
8454:10.1145/1806596.1806623
7119:excess n representation
7008:for constant resolution
6995:Microsoft Binary Format
6901:, a verified compiler.
6792:00068646912273617
6768:00068646912273617
6756:05434924008406305
6570:27145996453689225 5
6564:27145996453136334
6558:60862151314352708 4
6552:60862151314012979
4362:With rounding to zero,
4342:backward error analysis
4300:On 25 February 1991, a
4105:may not be the same as
3831:{\displaystyle R_{tot}}
3798:{\displaystyle 1/R_{1}}
3607:{\displaystyle R_{tot}}
3399:standard also define a
2978:is the significand and
2747:Microsoft Binary Format
2387:Internal representation
2144:Microsoft Binary Format
1943:Overflow level = OFL =
1692:floating-point standard
1623:UNIVAC 1100/2200 series
1347:Leonardo Torres Quevedo
1335:Leonardo Torres Quevedo
1326:IEEE 754 § History
1131:implicit bit convention
265:of representable values
87:, scaled by an integer
9828:Fleegal, Eric (2004).
9587:Vetterling, William T.
9495:Wilkinson, James Hardy
9459:Wilkinson, James Hardy
8927:(SIAM). pp. 50–.
8917:Einarsson, Bo (2005).
8878:Wilkinson, James Hardy
8738:US patent 3037701A
8407:Gay, David M. (1990).
8232:least significant byte
7248:computer (since 1958).
6949:Fixed-point arithmetic
6933:Decimal floating point
6893:is a notable outlier.
6891:Intel Fortran Compiler
6744:4061547378810956
6606:6101766046906629 8
6600:6101765997805905
6594:6627470568494473 7
6588:6627470548084133
6582:8730499798241950 6
6576:8730499801259536
6490:
6464:
6411:
6326:
6241:
6155:computational geometry
6128:
6065:
5694:
5648:
5367:
5296:
4630:
4610:
4571:
4471:
4408:
4214:
3842:for another example).
3832:
3799:
3764:
3737:
3614:
3608:
3393:C programming language
3214:denotes the exponent.
3073:unit in the last place
3064:and is limited by the
2974:where, as previously,
2941:non-terminating digits
2737:(MITS Altair 680) and
2633:
2527:x86 extended precision
2359:), negative infinity (
2275:type in the C language
2265:type in the C language
2231:decimal floating point
2007:
1926:
1892:
1729:exceptional conditions
1693:
1608:scientific computation
1593:English Electric DEUCE
1547:
1521:
1495:
1434:
1385:
1342:
1309:
1270:
1246:
1173:level-index arithmetic
1139:assumed bit convention
1127:leading bit convention
1069:
811:
753:
685:
637:'s first 33 bits are:
623:
546:decimal floating point
509:
304:Floating-point numbers
266:
254:
202:decimal floating point
187:
93:floating-point numbers
56:
9342:LLVM 13 documentation
9310:Kahan, William Morton
9204:Kahan, William Morton
9176:Kahan, William Morton
9148:Kahan, William Morton
9106:Kahan, William Morton
9076:Kahan, William Morton
9030:Kahan, William Morton
8962:Higham, Nicholas John
8760:Kahan, William Morton
8680:10.1145/103162.103163
8666:ACM Computing Surveys
8551:Giulietti, Rafaello.
8377:Kahan, William Morton
8228:most significant byte
8087:Kahan, William Morton
7875:Kahan, William Morton
7747:Numbers and Computers
6964:Gal's accurate tables
6720:810075796233302
6491:
6465:
6412:
6327:
6242:
6129:
6066:
5731:mathematics known as
5695:
5649:
5368:
5297:
4631:
4611:
4572:
4472:
4409:
4215:
3833:
3800:
3765:
3763:{\displaystyle R_{1}}
3738:
3609:
3580:
3458:Further information:
3023:when approximated by
2743:TRS-80 Color Computer
2632:
2624:32-bit floating point
2008:
1927:
1925:{\displaystyle B^{L}}
1893:
1700:was honored with the
1684:
1548:
1522:
1496:
1425:
1386:
1333:
1310:
1271:
1247:
1181:generalized logarithm
1135:hidden bit convention
1070:
785:
754:
686:
624:
575:is (145/100)×1000 or
510:
420:is multiplied by the
310:number representation
260:
248:
188:
42:
9636:. pp. 214–264.
9579:Press, William Henry
8890:. pp. 669–674.
8844:Christopher Barker:
8057:. 2013. Section 6.3
7770:, pp. 6, 11–13.
7289:division instruction
7167:Gould Powernode 9080
6696:19358822321783
6630:37487713536668 10
6624:37488171150615
6618:70343215275928 9
6612:70343230776862
6474:
6422:
6338:
6253:
6209:
6177:scientific computing
6170:iterative refinement
6075:
6001:
5728:numerically unstable
5662:
5381:
5306:
4640:
4620:
4600:
4508:
4418:
4366:
4302:loss of significance
4147:
3809:
3774:
3747:
3637:
3585:
3533:thread-local storage
3062:discretization error
1947:
1909:
1814:
1576:, which implemented
1534:
1505:
1461:
1384:{\displaystyle ^{m}}
1369:
1351:Essays on Automatics
1284:
1278:symbolic computation
1256:
1245:{\displaystyle \pi }
1236:
772:
718:
641:
622:{\displaystyle p=24}
607:
462:
101:
10415:Computer arithmetic
10354:Primitive data type
10259:Recursive data type
10112:Algebraic data type
9988:Quadruple precision
9652:Blaauw, Gerrit Anne
9620:Knuth, Donald Ervin
9557:Matrix Computations
9469:Prentice-Hall, Inc.
9359:"FloatingPointMath"
8567:"abolz/Drachennest"
8516:ACM SIGPLAN Notices
8171:"IEEE-754 Analysis"
7796:(April–June 1997).
7744:Ronald T. Kneusel.
7348:IEEE quad precision
7281:division algorithms
7159:Illinois ILLIAC III
7018:Significant figures
6654:7220386148377 12
6648:7256228504127
6642:9273850979885 11
6636:9278733740748
6389:
6294:
3389:scientific notation
3138:interval arithmetic
2795:
2640:layout is similar.
2331:Quadruple precision
2175:Arbitrary precision
1447:mechanical computer
1429:, architect of the
1212:Interval arithmetic
1202:rational arithmetic
337:scientific notation
289:floating-point unit
228:orders of magnitude
221:scientific notation
200:, though base ten (
10317:Abstract data type
9998:Extended precision
9957:Reduced precision
9591:Flannery, Brian P.
9583:Teukolsky, Saul A.
9295:10.1007/PL00009321
9256:. perldoc.perl.org
7988:2015-01-16 at the
7935:Severance, Charles
7233:Illinois ILLIAC II
6844:significant digits
6732:717412858693
6708:717412858693
6684:717412858693
6672:189011456060
6666:707019992125 13
6660:717412858693
6486:
6460:
6407:
6375:
6322:
6280:
6237:
6200:numerical analysis
6124:
6061:
5741:extended precision
5733:numerical analysis
5690:
5644:
5642:
5363:
5292:
5290:
4626:
4606:
4581:James H. Wilkinson
4567:
4467:
4446:
4404:
4350:. Usually denoted
4210:
3828:
3795:
3760:
3733:
3615:
3604:
3365:Division algorithm
3335:numerical analysis
3270:bit and one extra
2793:
2634:
2229:and IEEE 754-2008
2159:G.711 8-bit floats
2113:Extended precision
2003:
1922:
1888:
1694:
1598:The mass-produced
1543:
1517:
1491:
1435:
1381:
1343:
1305:
1266:
1242:
1105:in this example).
1085:in this example),
1081:is the precision (
1065:
1063:
749:
744:
681:
667:
619:
505:
267:
255:
204:) is also common.
183:
149:
142:
127:
120:
57:
10397:
10396:
10129:Associative array
9993:Octuple precision
9734:978-3-319-76525-9
9643:978-0-201-89684-8
9610:978-0-521-88407-5
9570:978-0-8018-5413-2
9547:978-0-13-322495-5
9234:. Speleotrove.com
9041:. IEEE-sponsored
9012:978-1-139-45862-7
8981:978-0-89871-521-7
8934:978-0-89871-815-7
8897:978-0-470-86412-8
8288:Microsoft Support
8261:. pagetable.com.
7920:978-3-540-11319-5
7819:10.1109/85.586067
7594:978-3-319-64109-6
7502:978-0-9660176-3-2
7435:978-0-8176-4704-9
7246:Rice Institute R1
7222:(1972) computers.
7076:(as used e.g. by
7006:Q (number format)
6923:Computable number
6690:46593073709 15
6678:78678454728 14
6405:
6396:
6320:
6301:
6235:
6234:
5687:
5589:
5526:
5461:
5398:
5357:
5342:
4973:
4964:
4940:
4939: where
4794:
4777:
4776: where
4629:{\displaystyle y}
4609:{\displaystyle x}
4561:
4543:
4445:
4430:
4378:
4338:Machine precision
4205:
3864:Accuracy problems
3647:
3495:(precision loss).
3449:in C and IEEE 754
3339:Accuracy problems
3323: = −1;
3315: = −1;
3309:gradual underflow
3103:). Historically,
3045:7410125732421875,
2982:is the exponent.
2923:
2922:
2638:64-bit ("double")
2585:subnormal numbers
2581:
2580:
2211:
2210:
2030:subnormal numbers
1359:analytical engine
1339:analytical engine
1264:
1101:is the exponent (
1093:and finishing at
816:
737:
732:
726:
677:
671:
660:
655:
649:
537:is the exponent.
487:
400:A signed integer
321:"point" character
182:
180:
175:
147:
135:
133:
125:
113:
111:
16:(Redirected from
10422:
10369:Type constructor
10254:Opaque data type
10186:Record or Struct
9983:Double precision
9978:Single precision
9869:
9862:
9855:
9846:
9845:
9841:
9836:. Archived from
9809:
9791:
9761:
9746:
9715:(2nd ed.).
9701:
9700:
9699:
9677:
9662:(1st ed.).
9647:
9632:(3rd ed.).
9614:
9599:(3rd ed.).
9574:
9559:(3rd ed.).
9551:
9528:
9526:
9525:
9489:
9446:
9445:
9443:
9427:
9421:
9420:
9409:
9403:
9402:
9391:
9385:
9384:
9373:
9367:
9366:
9355:
9349:
9348:
9334:
9328:
9327:
9325:
9318:
9306:
9300:
9299:
9297:
9271:
9265:
9264:
9262:
9261:
9249:
9243:
9242:
9240:
9239:
9228:
9222:
9221:
9219:
9212:
9200:
9194:
9193:
9191:
9184:
9172:
9166:
9165:
9163:
9156:
9144:
9131:
9130:
9128:
9127:
9121:
9114:
9102:
9096:
9095:
9093:
9086:
9072:
9066:
9060:
9058:
9057:
9051:
9040:
9026:
9017:
9016:
9005:. pp. 10–.
8992:
8986:
8985:
8984:. 0-89871-355-2.
8970:(2nd ed.).
8958:
8945:
8944:
8942:
8941:
8914:
8908:
8907:
8905:
8904:
8874:
8868:
8867:
8856:
8850:
8842:
8836:
8833:
8828:
8827:
8802:
8796:
8795:
8784:
8778:
8777:
8775:
8768:
8756:
8747:
8746:
8745:
8741:
8734:
8725:
8724:
8720:978-9-86605267-5
8706:
8700:
8692:
8682:
8653:
8642:
8641:
8630:10.1002/spe.2984
8623:
8614:(8): 1700–1727.
8603:
8597:
8596:
8583:
8577:
8576:
8563:
8557:
8556:
8548:
8542:
8541:
8531:
8507:
8501:
8500:
8489:
8483:
8482:
8480:
8463:978-1-45030019-3
8443:
8434:
8428:
8425:dtoa.c in netlab
8422:
8420:
8404:
8395:
8394:
8392:
8385:
8373:
8367:
8366:
8364:
8351:
8345:
8344:
8337:
8331:
8330:
8328:
8327:
8316:
8307:
8306:
8304:
8303:
8280:
8274:
8273:
8271:
8270:
8254:
8245:
8244:
8223:
8222:
8190:
8181:
8180:
8178:
8177:
8167:
8161:
8160:
8158:
8157:
8146:
8140:
8139:
8134:
8133:
8118:
8112:
8111:
8109:
8108:
8102:
8095:
8083:
8077:
8076:
8074:
8073:
8067:
8059:Basic data types
8056:
8048:
8042:
8041:
8039:
8038:
8032:
8025:
8017:
8011:
8010:
7999:
7993:
7980:
7974:
7973:
7966:
7960:
7959:
7949:
7943:
7942:
7931:
7925:
7924:
7899:
7893:
7892:
7890:
7883:
7871:
7862:
7861:
7859:
7844:
7838:
7836:
7834:
7833:
7827:
7802:
7790:
7784:
7779:Randell, Brian.
7777:
7771:
7765:
7759:
7742:
7736:
7729:
7723:
7722:
7716:
7715:
7709:
7702:
7693:
7687:
7685:
7680:
7679:
7672:978-0-20318604-6
7661:(1st ed.).
7650:
7644:
7643:
7642:
7641:
7621:
7615:
7614:
7568:
7555:
7551:
7549:
7548:
7542:
7531:
7522:
7513:
7512:
7510:
7509:
7484:
7478:
7477:
7457:
7448:
7447:
7416:(1st ed.).
7403:
7376:
7361:
7355:
7340:
7334:
7331:Taylor expansion
7327:
7321:
7315:
7309:
7305:
7299:
7295:Pentium FDIV bug
7277:
7271:
7268:
7262:
7261:(1956) computer.
7255:
7249:
7242:
7236:
7229:
7223:
7200:
7194:
7132:
7126:
7091:
7085:
7040:
6938:Double precision
6838:
6832:
6831:3.14159265358979
6826:
6825:3.14159265358979
6820:
6819:3.14159265358979
6814:
6813:3.14159265358979
6808:
6807:3.14159265358979
6802:
6796:
6795:3.14159265358979
6790:
6784:
6778:
6772:
6766:
6760:
6754:
6748:
6742:
6736:
6730:
6726:6065061913 18
6724:
6718:
6714:6566394222 17
6712:
6706:
6702:8571730119 16
6700:
6694:
6688:
6682:
6676:
6670:
6664:
6658:
6652:
6646:
6640:
6634:
6628:
6622:
6616:
6610:
6604:
6598:
6592:
6586:
6580:
6574:
6568:
6562:
6556:
6550:
6544:
6538:
6532:
6526:
6520:
6514:
6495:
6493:
6492:
6487:
6470:, converging as
6469:
6467:
6466:
6461:
6459:
6458:
6446:
6445:
6416:
6414:
6413:
6408:
6406:
6404:
6397:
6388:
6383:
6374:
6371:
6370:
6361:
6356:
6355:
6331:
6329:
6328:
6323:
6321:
6319:
6318:
6309:
6302:
6293:
6288:
6279:
6276:
6271:
6270:
6246:
6244:
6243:
6238:
6236:
6230:
6226:
6221:
6220:
6162:matrix inversion
6152:
6148:
6144:
6140:
6133:
6131:
6130:
6125:
6116:
6108:
6107:
6095:
6087:
6086:
6070:
6068:
6067:
6062:
6059:
6058:
6046:
6045:
5957:
5954:
5951:
5948:
5945:
5942:
5939:
5936:
5933:
5930:
5927:
5924:
5917:
5907:
5904:
5901:
5898:
5895:
5894:
5891:
5888:
5885:
5882:
5879:
5876:
5873:
5870:
5867:
5863:
5860:
5857:
5854:
5851:
5848:
5845:
5842:
5839:
5836:
5833:
5830:
5827:
5824:
5821:
5818:
5815:
5812:
5809:
5808:
5805:
5802:
5799:
5796:
5793:
5789:
5786:
5783:
5780:
5777:
5774:
5771:
5764:
5699:
5697:
5696:
5691:
5689:
5688:
5685:
5683:
5674:
5673:
5653:
5651:
5650:
5645:
5643:
5633:
5632:
5614:
5613:
5597:
5596:
5591:
5590:
5582:
5570:
5569:
5551:
5550:
5534:
5533:
5528:
5527:
5519:
5505:
5504:
5486:
5485:
5469:
5468:
5463:
5462:
5454:
5442:
5441:
5423:
5422:
5406:
5405:
5400:
5399:
5391:
5372:
5370:
5369:
5364:
5359:
5358:
5350:
5344:
5343:
5335:
5301:
5299:
5298:
5293:
5291:
5281:
5280:
5259:
5258:
5237:
5236:
5224:
5223:
5205:
5204:
5183:
5182:
5161:
5160:
5148:
5147:
5129:
5122:
5121:
5103:
5102:
5093:
5092:
5071:
5070:
5058:
5057:
5039:
5038:
5017:
5016:
5004:
5003:
4991:
4990:
4978:
4974:
4972: from above
4971:
4966:
4965:
4962:
4960:
4951:
4950:
4941:
4938:
4935:
4930:
4929:
4920:
4919:
4898:
4897:
4885:
4884:
4866:
4865:
4844:
4843:
4831:
4830:
4818:
4817:
4799:
4795:
4792:
4778:
4775:
4772:
4767:
4766:
4757:
4756:
4744:
4743:
4719:
4718:
4706:
4705:
4687:
4686:
4635:
4633:
4632:
4627:
4615:
4613:
4612:
4607:
4591:condition number
4576:
4574:
4573:
4568:
4563:
4562:
4559:
4557:
4548:
4544:
4539:
4516:
4503:
4476:
4474:
4473:
4468:
4463:
4462:
4447:
4438:
4432:
4431:
4428:
4426:
4413:
4411:
4410:
4405:
4399:
4398:
4380:
4379:
4376:
4374:
4358:
4308:missile battery
4277:subnormal number
4264:
4258:
4247:
4232:
4226:
4219:
4217:
4216:
4211:
4206:
4201:
4166:
4122:
4104:
4059:
4044:
4025:
4007:
3982:
3978:
3971:
3964:
3961:
3958:
3955:
3952:
3949:
3946:
3943:
3940:
3937:
3934:
3931:
3928:
3925:
3922:
3890:
3883:
3837:
3835:
3834:
3829:
3827:
3826:
3804:
3802:
3801:
3796:
3794:
3793:
3784:
3769:
3767:
3766:
3761:
3759:
3758:
3742:
3740:
3739:
3734:
3729:
3728:
3719:
3702:
3701:
3692:
3681:
3680:
3671:
3660:
3649:
3648:
3645:
3613:
3611:
3610:
3605:
3603:
3602:
3470:division by zero
3448:
3443:
3438:
3433:
3428:
3418:
3386:
3382:
3293: = 5;
3285: = 5;
3093:correct rounding
2946:
2932:rational numbers
2906:Single-precision
2796:
2792:
2767:machine learning
2715:MITS Altair 8800
2711:MITS Altair 8800
2705:, a dormmate of
2598:sign = 0 ;
2395:
2394:
2382:
2378:
2362:
2358:
2271:Double precision
2261:Single precision
2203:
2196:
2189:
2046:
2045:
2012:
2010:
2009:
2004:
2002:
1998:
1997:
1978:
1974:
1973:
1972:
1931:
1929:
1928:
1923:
1921:
1920:
1897:
1895:
1894:
1889:
1887:
1883:
1862:
1858:
1857:
1838:
1834:
1755:double-precision
1636:Double precision
1630:Single precision
1552:
1550:
1549:
1544:
1526:
1524:
1523:
1518:
1500:
1498:
1497:
1492:
1484:
1483:
1478:
1472:
1471:
1390:
1388:
1387:
1382:
1380:
1379:
1314:
1312:
1311:
1306:
1275:
1273:
1272:
1267:
1265:
1260:
1251:
1249:
1248:
1243:
1124:
1120:
1116:
1104:
1100:
1096:
1092:
1088:
1084:
1080:
1074:
1072:
1071:
1066:
1064:
1055:
1035:
1026:
1025:
1013:
1009:
1008:
1007:
980:
979:
958:
957:
936:
935:
914:
913:
892:
891:
866:
857:
856:
844:
840:
839:
838:
823:
822:
817:
814:
810:
799:
778:
767:
758:
756:
755:
750:
745:
730:
724:
713:
697:
690:
688:
687:
682:
675:
669:
668:
653:
647:
628:
626:
625:
620:
590:
588:
582:
578:
574:
572:
559:
536:
528:
524:
520:
514:
512:
511:
506:
501:
500:
488:
486:
485:
466:
454:
450:
448:
442:
438:
362:
360:
354:
232:between galaxies
192:
190:
189:
184:
181:
178:
176:
171:
170:
162:
159:
157:
148:
145:
143:
126:
123:
121:
49:Deutsches Museum
21:
10430:
10429:
10425:
10424:
10423:
10421:
10420:
10419:
10400:
10399:
10398:
10393:
10374:Type conversion
10309:
10303:
10239:Enumerated type
10212:
10098:
10092:null-terminated
10068:
10033:
9921:
9878:
9873:
9756:
9753:
9735:
9707:Revol, Nathalie
9697:
9695:
9674:
9644:
9611:
9571:
9548:
9523:
9521:
9519:
9509:Clarendon Press
9478:
9454:
9452:Further reading
9449:
9428:
9424:
9411:
9410:
9406:
9393:
9392:
9388:
9375:
9374:
9370:
9357:
9356:
9352:
9336:
9335:
9331:
9323:
9316:
9307:
9303:
9272:
9268:
9259:
9257:
9250:
9246:
9237:
9235:
9230:
9229:
9225:
9217:
9210:
9201:
9197:
9189:
9182:
9173:
9169:
9161:
9154:
9145:
9134:
9125:
9123:
9119:
9112:
9103:
9099:
9091:
9084:
9073:
9069:
9063:double extended
9055:
9053:
9049:
9038:
9027:
9020:
9013:
8993:
8989:
8982:
8959:
8948:
8939:
8937:
8935:
8915:
8911:
8902:
8900:
8898:
8875:
8871:
8858:
8857:
8853:
8843:
8839:
8825:
8823:
8803:
8799:
8786:
8785:
8781:
8773:
8766:
8757:
8750:
8743:
8735:
8728:
8721:
8707:
8703:
8657:Goldberg, David
8654:
8645:
8604:
8600:
8585:
8584:
8580:
8565:
8564:
8560:
8549:
8545:
8508:
8504:
8491:
8490:
8486:
8478:
8464:
8441:
8435:
8431:
8405:
8398:
8390:
8383:
8374:
8370:
8352:
8348:
8339:
8338:
8334:
8325:
8323:
8317:
8310:
8301:
8299:
8282:
8281:
8277:
8268:
8266:
8255:
8248:
8220:
8218:
8203:Embarcadero USA
8201:(TI1431C.txt).
8191:
8184:
8175:
8173:
8169:
8168:
8164:
8155:
8153:
8148:
8147:
8143:
8131:
8129:
8120:
8119:
8115:
8106:
8104:
8100:
8093:
8084:
8080:
8071:
8069:
8065:
8054:
8050:
8049:
8045:
8036:
8034:
8030:
8023:
8019:
8018:
8014:
8001:
8000:
7996:
7990:Wayback Machine
7981:
7977:
7968:
7967:
7963:
7951:
7950:
7946:
7932:
7928:
7921:
7913:. p. 244.
7911:Springer-Verlag
7905:, ed. (1982) .
7900:
7896:
7888:
7881:
7872:
7865:
7845:
7841:
7831:
7829:
7825:
7800:
7791:
7787:
7778:
7774:
7766:
7762:
7743:
7739:
7730:
7726:
7713:
7711:
7707:
7700:
7694:
7690:
7677:
7675:
7673:
7651:
7647:
7639:
7637:
7622:
7618:
7595:
7579:. p. 948.
7569:
7558:
7546:
7544:
7540:
7529:
7523:
7516:
7507:
7505:
7503:
7485:
7481:
7474:
7458:
7451:
7436:
7408:Revol, Nathalie
7404:
7389:
7385:
7380:
7379:
7362:
7358:
7341:
7337:
7328:
7324:
7316:
7312:
7306:
7302:
7278:
7274:
7269:
7265:
7256:
7252:
7243:
7239:
7230:
7226:
7220:Burroughs B7700
7216:Burroughs B6700
7212:Burroughs B5700
7208:Burroughs B5500
7201:
7197:
7181:as well as the
7133:
7129:
7110:biased exponent
7092:
7088:
7041:
7037:
7032:
7027:
6907:
6852:
6840:
6836:
6830:
6824:
6818:
6812:
6806:
6800:
6794:
6788:
6782:
6776:
6770:
6764:
6762:900560168 21
6758:
6752:
6750:908393901 20
6746:
6740:
6738:939728836 19
6734:
6728:
6722:
6716:
6710:
6704:
6698:
6692:
6686:
6680:
6674:
6668:
6662:
6656:
6650:
6644:
6638:
6632:
6626:
6620:
6614:
6608:
6602:
6596:
6590:
6584:
6578:
6572:
6566:
6560:
6554:
6548:
6542:
6536:
6530:
6524:
6518:
6512:
6509:
6505:
6475:
6472:
6471:
6454:
6450:
6441:
6437:
6423:
6420:
6419:
6384:
6379:
6373:
6372:
6366:
6362:
6360:
6345:
6341:
6339:
6336:
6335:
6314:
6310:
6289:
6284:
6278:
6277:
6275:
6260:
6256:
6254:
6251:
6250:
6225:
6216:
6212:
6210:
6207:
6206:
6183:
6150:
6146:
6142:
6138:
6112:
6103:
6099:
6091:
6082:
6078:
6076:
6073:
6072:
6054:
6050:
6041:
6037:
6002:
5999:
5998:
5959:
5958:
5955:
5952:
5949:
5946:
5943:
5940:
5937:
5934:
5931:
5928:
5925:
5922:
5909:
5908:
5905:
5902:
5899:
5896:
5892:
5889:
5886:
5883:
5880:
5877:
5874:
5871:
5868:
5865:
5864:
5861:
5858:
5855:
5852:
5849:
5846:
5843:
5840:
5837:
5834:
5831:
5828:
5825:
5822:
5819:
5816:
5813:
5810:
5806:
5803:
5800:
5797:
5794:
5791:
5790:
5787:
5784:
5781:
5778:
5775:
5772:
5769:
5762:
5758:
5754:
5750:
5747:
5724:ill-conditioned
5716:
5705:
5684:
5679:
5678:
5669:
5665:
5663:
5660:
5659:
5641:
5640:
5628:
5624:
5609:
5605:
5598:
5592:
5581:
5580:
5579:
5577:
5565:
5561:
5546:
5542:
5535:
5529:
5518:
5517:
5516:
5513:
5512:
5500:
5496:
5481:
5477:
5470:
5464:
5453:
5452:
5451:
5449:
5437:
5433:
5418:
5414:
5407:
5401:
5390:
5389:
5388:
5384:
5382:
5379:
5378:
5349:
5348:
5334:
5333:
5307:
5304:
5303:
5289:
5288:
5276:
5272:
5254:
5250:
5232:
5228:
5219:
5215:
5200:
5196:
5178:
5174:
5156:
5152:
5143:
5139:
5127:
5126:
5117:
5113:
5098:
5097:
5088:
5084:
5066:
5062:
5053:
5049:
5034:
5030:
5012:
5008:
4999:
4995:
4986:
4985:
4976:
4975:
4970:
4961:
4956:
4955:
4946:
4942:
4937:
4934:
4925:
4924:
4915:
4911:
4893:
4889:
4880:
4876:
4861:
4857:
4839:
4835:
4826:
4822:
4813:
4812:
4797:
4796:
4791:
4774:
4771:
4762:
4761:
4752:
4748:
4739:
4735:
4714:
4710:
4701:
4697:
4682:
4681:
4668:
4643:
4641:
4638:
4637:
4621:
4618:
4617:
4601:
4598:
4597:
4586:backward stable
4558:
4553:
4552:
4517:
4515:
4511:
4509:
4506:
4505:
4502:
4499:
4452:
4448:
4436:
4427:
4422:
4421:
4419:
4416:
4415:
4388:
4384:
4375:
4370:
4369:
4367:
4364:
4363:
4357:
4354:
4351:
4347:machine epsilon
4335:
4306:MIM-104 Patriot
4297:
4292:
4263:
4260:
4256:
4252:
4249:
4245:
4241:
4237:
4234:
4231:
4228:
4225:
4222:
4167:
4165:
4148:
4145:
4144:
4128:
4121:
4117:
4113:
4109:
4106:
4103:
4099:
4095:
4091:
4084:
4077:
4074:
4067:
4064:
4057:
4053:
4049:
4046:
4043:
4039:
4035:
4031:
4024:
4020:
4016:
4012:
4009:
4006:
4002:
3998:
3994:
3991:
3980:
3976:
3969:
3966:
3965:
3962:
3959:
3956:
3953:
3950:
3947:
3944:
3941:
3938:
3935:
3932:
3929:
3926:
3923:
3920:
3913:
3907:
3901:
3895:
3888:
3885:
3882: = −4
3881:
3878:
3866:
3816:
3812:
3810:
3807:
3806:
3789:
3785:
3780:
3775:
3772:
3771:
3754:
3750:
3748:
3745:
3744:
3724:
3720:
3715:
3697:
3693:
3688:
3676:
3672:
3667:
3656:
3644:
3640:
3638:
3635:
3634:
3592:
3588:
3586:
3583:
3582:
3493:denormalization
3481:complex numbers
3462:
3456:
3447:0x1.fffffep+127
3446:
3441:
3436:
3431:
3426:
3416:
3384:
3380:
3377:
3353:
3347:
3301:
3259:
3256:
3250:round-off error
3246:
3240:
3237:
3231:
3227:
3220:
3192:
3183:
3153:David M. Gay's
3146:
3089:
3082:
3078:
3066:machine epsilon
2944:
2928:
2888:TensorFloat-32
2755:Bfloat16 format
2666:TRS-80 LEVEL II
2646:
2418:decimal digits
2417:
2412:
2407:
2389:
2380:
2376:
2360:
2356:
2310:round-off error
2281:Double extended
2223:revised in 2008
2207:
2154:PMBus Linear-11
2044:
2038:
1987:
1983:
1979:
1965:
1961:
1954:
1950:
1948:
1945:
1944:
1916:
1912:
1910:
1907:
1906:
1867:
1863:
1847:
1843:
1839:
1824:
1820:
1815:
1812:
1811:
1742:
1727:The ability of
1616:general-purpose
1604:biased exponent
1535:
1532:
1531:
1506:
1503:
1502:
1479:
1474:
1473:
1467:
1464:
1462:
1459:
1458:
1375:
1372:
1370:
1367:
1366:
1355:Charles Babbage
1328:
1322:
1285:
1282:
1281:
1259:
1257:
1254:
1253:
1237:
1234:
1233:
1183:representation.
1147:
1122:
1118:
1114:
1102:
1098:
1094:
1090:
1086:
1082:
1078:
1062:
1061:
1056:
1054:
1048:
1047:
1036:
1034:
1028:
1027:
1021:
1017:
1000:
996:
972:
968:
950:
946:
928:
924:
906:
902:
884:
880:
873:
869:
867:
865:
859:
858:
852:
848:
831:
827:
818:
813:
812:
800:
789:
784:
780:
775:
773:
770:
769:
765:
736:
719:
716:
715:
711:
695:
659:
642:
639:
638:
608:
605:
604:
586:
584:
580:
576:
570:
568:
565:rational number
557:
534:
526:
522:
518:
496:
492:
474:
470:
465:
463:
460:
459:
452:
446:
444:
440:
436:
358:
356:
352:
306:
301:
282:computer system
177:
163:
161:
160:
158:
144:
134:
122:
112:
102:
99:
98:
35:
32:
23:
22:
15:
12:
11:
5:
10428:
10418:
10417:
10412:
10410:Floating point
10395:
10394:
10392:
10391:
10386:
10381:
10376:
10371:
10366:
10361:
10356:
10351:
10346:
10345:
10344:
10334:
10329:
10327:Data structure
10324:
10319:
10313:
10311:
10305:
10304:
10302:
10301:
10296:
10291:
10286:
10281:
10276:
10271:
10266:
10261:
10256:
10251:
10246:
10241:
10236:
10231:
10226:
10220:
10218:
10214:
10213:
10211:
10210:
10209:
10208:
10198:
10193:
10188:
10183:
10178:
10173:
10172:
10171:
10161:
10156:
10151:
10146:
10141:
10136:
10131:
10126:
10121:
10120:
10119:
10108:
10106:
10100:
10099:
10097:
10096:
10095:
10094:
10084:
10078:
10076:
10070:
10069:
10067:
10066:
10061:
10060:
10059:
10054:
10043:
10041:
10035:
10034:
10032:
10031:
10026:
10021:
10020:
10019:
10009:
10008:
10007:
10006:
10005:
9995:
9990:
9985:
9980:
9975:
9974:
9973:
9968:
9966:Half precision
9963:
9953:Floating point
9950:
9945:
9940:
9935:
9929:
9927:
9923:
9922:
9920:
9919:
9914:
9909:
9904:
9899:
9894:
9888:
9886:
9880:
9879:
9872:
9871:
9864:
9857:
9849:
9843:
9842:
9840:on 2017-07-06.
9825:
9811:
9763:
9752:
9751:External links
9749:
9748:
9747:
9733:
9702:
9679:
9672:
9664:Addison-Wesley
9648:
9642:
9634:Addison-Wesley
9616:
9609:
9575:
9569:
9552:
9546:
9529:
9517:
9491:
9476:
9453:
9450:
9448:
9447:
9422:
9404:
9386:
9368:
9350:
9329:
9301:
9288:(3): 305–363.
9266:
9244:
9223:
9195:
9185:. p. 26.
9178:(1981-02-12).
9167:
9150:(2000-08-27).
9132:
9097:
9078:(2011-08-03).
9067:
9032:(2005-07-15).
9018:
9011:
8987:
8980:
8946:
8933:
8909:
8896:
8869:
8851:
8837:
8797:
8794:. Vol. 1.
8779:
8762:(1997-10-01).
8748:
8726:
8719:
8701:
8659:(March 1991).
8643:
8598:
8578:
8558:
8543:
8522:(4): 270–282.
8502:
8484:
8462:
8429:
8418:10.1.1.31.4049
8396:
8379:(2006-01-11).
8368:
8346:
8332:
8308:
8275:
8246:
8182:
8162:
8141:
8113:
8089:(2004-11-20).
8078:
8043:
8026:. 2013-05-22.
8012:
7994:
7975:
7961:
7944:
7937:(1998-02-20).
7926:
7919:
7903:Randell, Brian
7894:
7877:(1997-07-15).
7863:
7839:
7785:
7772:
7760:
7757:978-3319505084
7737:
7724:
7688:
7671:
7665:. p. 24.
7645:
7616:
7593:
7556:
7514:
7501:
7479:
7472:
7449:
7434:
7386:
7384:
7381:
7378:
7377:
7356:
7335:
7322:
7310:
7300:
7272:
7263:
7250:
7237:
7224:
7204:Ferranti Atlas
7195:
7175:SEL Systems 85
7171:Interdata 8/32
7151:Manchester MU5
7147:RCA Spectra 70
7139:IBM System 360
7127:
7105:characteristic
7086:
7074:characteristic
7034:
7033:
7031:
7028:
7026:
7025:
7020:
7015:
7009:
7003:
6998:
6992:
6987:
6982:
6976:
6971:
6966:
6961:
6956:
6951:
6946:
6940:
6935:
6930:
6925:
6920:
6914:
6908:
6906:
6903:
6851:
6848:
6783:3.141592653589
6771:3.141592653589
6507:
6503:
6501:
6497:
6496:
6485:
6482:
6479:
6457:
6453:
6449:
6444:
6440:
6436:
6433:
6430:
6427:
6417:
6403:
6400:
6395:
6392:
6387:
6382:
6378:
6369:
6365:
6359:
6354:
6351:
6348:
6344:
6332:
6317:
6313:
6308:
6305:
6300:
6297:
6292:
6287:
6283:
6274:
6269:
6266:
6263:
6259:
6247:
6233:
6229:
6224:
6219:
6215:
6181:
6122:
6119:
6115:
6111:
6106:
6102:
6098:
6094:
6090:
6085:
6081:
6057:
6053:
6049:
6044:
6040:
6036:
6033:
6030:
6027:
6024:
6021:
6018:
6015:
6012:
6009:
6006:
5921:
5768:
5760:
5756:
5752:
5748:
5715:
5712:
5703:
5682:
5677:
5672:
5668:
5639:
5636:
5631:
5627:
5623:
5620:
5617:
5612:
5608:
5604:
5601:
5599:
5595:
5588:
5585:
5578:
5576:
5573:
5568:
5564:
5560:
5557:
5554:
5549:
5545:
5541:
5538:
5536:
5532:
5525:
5522:
5515:
5514:
5511:
5508:
5503:
5499:
5495:
5492:
5489:
5484:
5480:
5476:
5473:
5471:
5467:
5460:
5457:
5450:
5448:
5445:
5440:
5436:
5432:
5429:
5426:
5421:
5417:
5413:
5410:
5408:
5404:
5397:
5394:
5387:
5386:
5362:
5356:
5353:
5347:
5341:
5338:
5332:
5329:
5326:
5323:
5320:
5317:
5314:
5311:
5287:
5284:
5279:
5275:
5271:
5268:
5265:
5262:
5257:
5253:
5249:
5246:
5243:
5240:
5235:
5231:
5227:
5222:
5218:
5214:
5211:
5208:
5203:
5199:
5195:
5192:
5189:
5186:
5181:
5177:
5173:
5170:
5167:
5164:
5159:
5155:
5151:
5146:
5142:
5138:
5135:
5132:
5130:
5128:
5125:
5120:
5116:
5112:
5109:
5106:
5101:
5096:
5091:
5087:
5083:
5080:
5077:
5074:
5069:
5065:
5061:
5056:
5052:
5048:
5045:
5042:
5037:
5033:
5029:
5026:
5023:
5020:
5015:
5011:
5007:
5002:
4998:
4994:
4989:
4984:
4981:
4979:
4977:
4969:
4959:
4954:
4949:
4945:
4936:
4933:
4928:
4923:
4918:
4914:
4910:
4907:
4904:
4901:
4896:
4892:
4888:
4883:
4879:
4875:
4872:
4869:
4864:
4860:
4856:
4853:
4850:
4847:
4842:
4838:
4834:
4829:
4825:
4821:
4816:
4811:
4808:
4805:
4802:
4800:
4798:
4790:
4787:
4784:
4781:
4773:
4770:
4765:
4760:
4755:
4751:
4747:
4742:
4738:
4734:
4731:
4728:
4725:
4722:
4717:
4713:
4709:
4704:
4700:
4696:
4693:
4690:
4685:
4680:
4677:
4674:
4671:
4669:
4667:
4664:
4661:
4658:
4655:
4652:
4649:
4646:
4645:
4625:
4605:
4566:
4556:
4551:
4547:
4542:
4538:
4535:
4532:
4529:
4526:
4523:
4520:
4514:
4500:
4495:relative error
4466:
4461:
4458:
4455:
4451:
4444:
4441:
4435:
4425:
4402:
4397:
4394:
4391:
4387:
4383:
4373:
4355:
4352:
4334:
4331:
4330:
4329:
4296:
4293:
4291:
4290:
4287:
4280:
4273:
4266:
4261:
4254:
4250:
4243:
4239:
4235:
4229:
4223:
4209:
4204:
4200:
4197:
4194:
4191:
4188:
4185:
4182:
4179:
4176:
4173:
4170:
4164:
4161:
4158:
4155:
4152:
4132:
4125:
4119:
4115:
4111:
4107:
4101:
4097:
4093:
4083:← a + (b + c)
4078:
4075:
4073:← (a + b) + c
4068:
4065:
4062:
4055:
4051:
4047:
4041:
4037:
4033:
4022:
4018:
4014:
4010:
4004:
4000:
3996:
3992:
3919:
3911:
3905:
3899:
3893:
3886:
3879:
3865:
3862:
3825:
3822:
3819:
3815:
3792:
3788:
3783:
3779:
3757:
3753:
3732:
3727:
3723:
3718:
3714:
3711:
3708:
3705:
3700:
3696:
3691:
3687:
3684:
3679:
3675:
3670:
3666:
3663:
3659:
3655:
3652:
3643:
3627:divide-by-zero
3601:
3598:
3595:
3591:
3575:
3574:
3568:
3565:divide-by-zero
3562:
3556:
3546:
3497:
3496:
3484:
3473:
3455:
3452:
3451:
3450:
3444:
3439:
3434:
3429:
3376:
3375:Literal syntax
3373:
3351:
3346:
3343:
3305:Sterbenz lemma
3299:
3279:approximations
3257:
3254:
3244:
3238:
3235:
3228:
3225:
3219:
3216:
3191:
3188:
3182:
3179:
3175:
3174:
3171:
3168:
3165:
3158:
3145:
3142:
3134:
3133:
3130:
3127:
3124:
3121:
3101:rounding modes
3088:
3087:Rounding modes
3085:
3080:
3079:and 1.45a70c24
3076:
3058:
3057:
3047:
3046:
3021:
3020:
3013:
3012:
2998:
2997:
2972:
2971:
2927:
2924:
2921:
2920:
2917:
2914:
2911:
2908:
2902:
2901:
2898:
2895:
2892:
2889:
2885:
2884:
2881:
2878:
2875:
2872:
2866:
2865:
2862:
2859:
2856:
2853:
2851:Half-precision
2847:
2846:
2843:
2840:
2837:
2834:
2830:
2829:
2826:
2823:
2820:
2817:
2813:
2812:
2809:
2806:
2803:
2800:
2791:
2790:
2783:
2770:
2751:
2703:Monte Davidoff
2645:
2642:
2620:
2619:
2608:
2607:
2579:
2578:
2575:
2572:
2569:
2566:
2563:
2560:
2557:
2551:
2550:
2547:
2544:
2541:
2538:
2535:
2532:
2529:
2523:
2522:
2519:
2516:
2513:
2510:
2507:
2504:
2501:
2495:
2494:
2491:
2488:
2485:
2482:
2479:
2476:
2473:
2467:
2466:
2463:
2460:
2457:
2454:
2451:
2448:
2445:
2434:
2433:
2430:
2427:
2424:
2420:
2419:
2414:
2409:
2404:
2402:
2399:
2388:
2385:
2345:
2344:
2337:Half precision
2334:
2328:
2306:
2305:
2278:
2268:
2209:
2208:
2206:
2205:
2198:
2191:
2183:
2180:
2179:
2178:
2177:
2169:
2168:
2164:
2163:
2162:
2161:
2156:
2151:
2146:
2141:
2139:TensorFloat-32
2136:
2131:
2123:
2122:
2118:
2117:
2116:
2115:
2110:
2103:
2093:
2083:
2073:
2063:
2062:
2056:
2055:
2050:Floating-point
2040:Main article:
2037:
2034:
2015:
2014:
2001:
1996:
1993:
1990:
1986:
1982:
1977:
1971:
1968:
1964:
1960:
1957:
1953:
1934:
1933:
1919:
1915:
1886:
1882:
1879:
1876:
1873:
1870:
1866:
1861:
1856:
1853:
1850:
1846:
1842:
1837:
1833:
1830:
1827:
1823:
1819:
1808:
1807:
1801:
1795:
1785:
1741:
1738:
1737:
1736:
1733:divide by zero
1725:
1721:
1656:z/Architecture
1640:
1639:
1633:
1542:
1539:
1516:
1513:
1510:
1490:
1487:
1482:
1477:
1470:
1466:
1378:
1374:
1321:
1318:
1317:
1316:
1304:
1301:
1298:
1295:
1292:
1289:
1263:
1241:
1215:
1209:
1190:
1184:
1158:
1146:
1143:
1060:
1057:
1053:
1050:
1049:
1046:
1043:
1040:
1037:
1033:
1030:
1029:
1024:
1020:
1016:
1012:
1006:
1003:
999:
995:
992:
989:
986:
983:
978:
975:
971:
967:
964:
961:
956:
953:
949:
945:
942:
939:
934:
931:
927:
923:
920:
917:
912:
909:
905:
901:
898:
895:
890:
887:
883:
879:
876:
872:
868:
864:
861:
860:
855:
851:
847:
843:
837:
834:
830:
826:
821:
809:
806:
803:
798:
795:
792:
788:
783:
779:
777:
748:
743:
740:
735:
729:
723:
680:
674:
666:
663:
658:
652:
646:
618:
615:
612:
504:
499:
495:
491:
484:
481:
478:
473:
469:
414:
413:
406:characteristic
398:
305:
302:
300:
297:
213:floating point
174:
169:
166:
141:
138:
131:
119:
116:
109:
106:
33:
18:Floating-point
9:
6:
4:
3:
2:
10427:
10416:
10413:
10411:
10408:
10407:
10405:
10390:
10387:
10385:
10382:
10380:
10377:
10375:
10372:
10370:
10367:
10365:
10362:
10360:
10357:
10355:
10352:
10350:
10347:
10343:
10340:
10339:
10338:
10335:
10333:
10330:
10328:
10325:
10323:
10320:
10318:
10315:
10314:
10312:
10306:
10300:
10297:
10295:
10292:
10290:
10287:
10285:
10282:
10280:
10277:
10275:
10272:
10270:
10267:
10265:
10262:
10260:
10257:
10255:
10252:
10250:
10249:Function type
10247:
10245:
10242:
10240:
10237:
10235:
10232:
10230:
10227:
10225:
10222:
10221:
10219:
10215:
10207:
10204:
10203:
10202:
10199:
10197:
10194:
10192:
10189:
10187:
10184:
10182:
10179:
10177:
10174:
10170:
10167:
10166:
10165:
10162:
10160:
10157:
10155:
10152:
10150:
10147:
10145:
10142:
10140:
10137:
10135:
10132:
10130:
10127:
10125:
10122:
10118:
10115:
10114:
10113:
10110:
10109:
10107:
10105:
10101:
10093:
10090:
10089:
10088:
10085:
10083:
10080:
10079:
10077:
10075:
10071:
10065:
10062:
10058:
10055:
10053:
10050:
10049:
10048:
10045:
10044:
10042:
10040:
10036:
10030:
10027:
10025:
10022:
10018:
10015:
10014:
10013:
10010:
10004:
10001:
10000:
9999:
9996:
9994:
9991:
9989:
9986:
9984:
9981:
9979:
9976:
9972:
9969:
9967:
9964:
9962:
9959:
9958:
9956:
9955:
9954:
9951:
9949:
9946:
9944:
9941:
9939:
9936:
9934:
9931:
9930:
9928:
9924:
9918:
9915:
9913:
9910:
9908:
9905:
9903:
9900:
9898:
9895:
9893:
9890:
9889:
9887:
9885:
9884:Uninterpreted
9881:
9877:
9870:
9865:
9863:
9858:
9856:
9851:
9850:
9847:
9839:
9835:
9831:
9826:
9823:
9819:
9815:
9812:
9807:
9803:
9799:
9795:
9790:
9785:
9781:
9777:
9773:
9769:
9764:
9759:
9755:
9754:
9744:
9740:
9736:
9730:
9726:
9722:
9718:
9714:
9713:
9708:
9703:
9693:
9689:
9685:
9680:
9675:
9673:0-201-10557-8
9669:
9665:
9661:
9657:
9653:
9649:
9645:
9639:
9635:
9631:
9627:
9626:
9621:
9617:
9612:
9606:
9602:
9598:
9597:
9592:
9588:
9584:
9580:
9576:
9572:
9566:
9562:
9558:
9553:
9549:
9543:
9539:
9538:Prentice Hall
9535:
9530:
9520:
9518:9780198534037
9514:
9510:
9506:
9502:
9501:
9496:
9492:
9487:
9483:
9479:
9477:9780486679990
9473:
9470:
9466:
9465:
9460:
9456:
9455:
9442:
9437:
9433:
9426:
9418:
9414:
9408:
9400:
9396:
9390:
9382:
9378:
9372:
9364:
9360:
9354:
9347:
9343:
9339:
9333:
9322:
9315:
9311:
9305:
9296:
9291:
9287:
9283:
9282:
9277:
9270:
9255:
9248:
9233:
9227:
9216:
9209:
9205:
9199:
9188:
9181:
9177:
9171:
9160:
9153:
9149:
9143:
9141:
9139:
9137:
9118:
9111:
9107:
9101:
9090:
9083:
9082:
9077:
9071:
9064:
9048:
9044:
9037:
9036:
9031:
9025:
9023:
9014:
9008:
9004:
9000:
8999:
8991:
8983:
8977:
8973:
8969:
8968:
8963:
8957:
8955:
8953:
8951:
8936:
8930:
8926:
8922:
8921:
8913:
8899:
8893:
8889:
8885:
8884:
8879:
8873:
8865:
8861:
8855:
8849:
8848:
8841:
8835:
8832:
8822:
8818:
8814:
8813:
8808:
8801:
8793:
8789:
8783:
8772:
8769:. p. 9.
8765:
8761:
8755:
8753:
8739:
8733:
8731:
8722:
8716:
8712:
8705:
8698:
8695:
8690:
8686:
8681:
8676:
8672:
8668:
8667:
8662:
8658:
8652:
8650:
8648:
8639:
8635:
8631:
8627:
8622:
8617:
8613:
8609:
8602:
8595:. 2020-09-21.
8594:
8593:
8588:
8582:
8575:. 2022-11-10.
8574:
8573:
8568:
8562:
8554:
8547:
8539:
8535:
8530:
8525:
8521:
8517:
8513:
8506:
8498:
8494:
8488:
8477:
8473:
8469:
8465:
8459:
8455:
8451:
8447:
8440:
8433:
8426:
8419:
8414:
8410:
8403:
8401:
8389:
8382:
8378:
8372:
8363:
8358:
8350:
8343:. 2022-03-22.
8342:
8336:
8322:
8315:
8313:
8297:
8293:
8289:
8285:
8279:
8264:
8260:
8253:
8251:
8243:
8241:
8237:
8236:decimal point
8233:
8229:
8216:
8212:
8209:(originally:
8208:
8204:
8200:
8196:
8189:
8187:
8172:
8166:
8151:
8145:
8138:
8128:on 2013-05-08
8127:
8123:
8117:
8099:
8092:
8088:
8082:
8064:
8060:
8053:
8047:
8029:
8022:
8016:
8008:
8007:StackOverflow
8004:
7998:
7991:
7987:
7984:
7979:
7972:. 2021-08-03.
7971:
7965:
7958:
7954:
7948:
7940:
7936:
7930:
7922:
7916:
7912:
7908:
7904:
7898:
7887:
7884:. p. 3.
7880:
7876:
7870:
7868:
7858:
7853:
7849:
7843:
7824:
7820:
7816:
7812:
7808:
7807:
7799:
7795:
7789:
7782:
7776:
7769:
7764:
7758:
7754:
7750:
7748:
7741:
7734:
7728:
7721:
7706:
7699:
7692:
7684:
7674:
7668:
7664:
7660:
7656:
7649:
7635:
7631:
7627:
7620:
7612:
7608:
7604:
7600:
7596:
7590:
7586:
7582:
7578:
7574:
7567:
7565:
7563:
7561:
7553:
7539:
7536:. p. 2.
7535:
7528:
7521:
7519:
7504:
7498:
7494:
7490:
7483:
7475:
7473:0-13-322495-3
7469:
7465:
7464:
7456:
7454:
7445:
7441:
7437:
7431:
7427:
7423:
7419:
7415:
7414:
7409:
7402:
7400:
7398:
7396:
7394:
7392:
7387:
7374:
7370:
7366:
7360:
7353:
7349:
7345:
7339:
7332:
7326:
7319:
7318:William Kahan
7314:
7304:
7297:
7296:
7290:
7286:
7285:Intel Pentium
7282:
7276:
7267:
7260:
7254:
7247:
7241:
7234:
7228:
7221:
7217:
7213:
7209:
7205:
7199:
7192:
7191:Xerox Sigma 9
7188:
7184:
7180:
7176:
7173:(1970s), the
7172:
7168:
7164:
7160:
7156:
7152:
7148:
7144:
7140:
7136:
7131:
7124:
7120:
7116:
7115:exponent bias
7112:
7111:
7106:
7102:
7098:
7097:
7090:
7083:
7079:
7075:
7071:
7067:
7063:
7059:
7055:
7051:
7047:
7046:
7039:
7035:
7024:
7021:
7019:
7016:
7013:
7010:
7007:
7004:
7002:
6999:
6996:
6993:
6991:
6988:
6986:
6983:
6980:
6977:
6975:
6972:
6970:
6967:
6965:
6962:
6960:
6957:
6955:
6952:
6950:
6947:
6944:
6941:
6939:
6936:
6934:
6931:
6929:
6926:
6924:
6921:
6918:
6915:
6913:
6910:
6909:
6902:
6900:
6894:
6892:
6887:
6882:
6880:
6876:
6871:
6869:
6868:vectorization
6865:
6861:
6857:
6856:associativity
6847:
6845:
6839:
6833:
6827:
6821:
6815:
6809:
6803:
6797:
6791:
6786:8122118 23
6785:
6779:
6774:8608396 22
6773:
6767:
6761:
6755:
6749:
6743:
6737:
6731:
6725:
6719:
6713:
6707:
6701:
6695:
6689:
6683:
6677:
6671:
6665:
6659:
6653:
6647:
6641:
6635:
6629:
6623:
6617:
6611:
6605:
6599:
6593:
6587:
6581:
6575:
6569:
6563:
6557:
6551:
6545:
6539:
6533:
6527:
6521:
6515:
6502:i 6 × 2 × t
6500:
6477:
6455:
6451:
6447:
6442:
6438:
6434:
6431:
6428:
6425:
6418:
6401:
6398:
6393:
6390:
6385:
6380:
6376:
6367:
6363:
6357:
6352:
6349:
6346:
6342:
6334:second form:
6333:
6315:
6311:
6306:
6303:
6298:
6295:
6290:
6285:
6281:
6272:
6267:
6264:
6261:
6257:
6248:
6231:
6227:
6222:
6217:
6213:
6205:
6204:
6203:
6201:
6196:
6191:
6189:
6180:
6178:
6173:
6171:
6167:
6163:
6158:
6156:
6139:if (x==y) ...
6135:
6120:
6117:
6113:
6109:
6104:
6100:
6096:
6092:
6088:
6083:
6079:
6055:
6051:
6047:
6042:
6038:
6034:
6028:
6025:
6022:
6013:
6010:
6007:
5995:
5992:
5991:IEEE 754-2008
5988:
5984:
5980:
5974:
5970:
5967:
5962:
5919:
5914:
5766:
5744:
5742:
5738:
5734:
5729:
5725:
5721:
5711:
5709:
5700:
5675:
5670:
5666:
5657:
5654:
5637:
5629:
5625:
5621:
5618:
5610:
5606:
5602:
5600:
5593:
5583:
5574:
5566:
5562:
5558:
5555:
5547:
5543:
5539:
5537:
5530:
5520:
5509:
5501:
5497:
5493:
5490:
5482:
5478:
5474:
5472:
5465:
5455:
5446:
5438:
5434:
5430:
5427:
5419:
5415:
5411:
5409:
5402:
5392:
5376:
5373:
5360:
5351:
5345:
5336:
5330:
5324:
5321:
5318:
5312:
5309:
5285:
5277:
5273:
5269:
5266:
5255:
5251:
5247:
5244:
5233:
5229:
5225:
5220:
5216:
5209:
5201:
5197:
5193:
5190:
5179:
5175:
5171:
5168:
5157:
5153:
5149:
5144:
5140:
5133:
5131:
5118:
5114:
5110:
5107:
5089:
5085:
5081:
5078:
5067:
5063:
5059:
5054:
5050:
5043:
5035:
5031:
5027:
5024:
5013:
5009:
5005:
5000:
4996:
4982:
4980:
4967:
4952:
4947:
4943:
4931:
4916:
4912:
4908:
4905:
4894:
4890:
4886:
4881:
4877:
4870:
4862:
4858:
4854:
4851:
4840:
4836:
4832:
4827:
4823:
4809:
4806:
4803:
4801:
4782:
4779:
4768:
4753:
4749:
4745:
4740:
4736:
4729:
4726:
4723:
4715:
4711:
4707:
4702:
4698:
4691:
4688:
4678:
4675:
4672:
4670:
4662:
4659:
4656:
4650:
4647:
4623:
4603:
4594:
4592:
4588:
4587:
4582:
4577:
4564:
4549:
4545:
4540:
4536:
4533:
4527:
4521:
4518:
4512:
4497:
4496:
4490:
4488:
4484:
4480:
4464:
4459:
4456:
4453:
4449:
4442:
4439:
4433:
4400:
4395:
4392:
4389:
4385:
4381:
4360:
4349:
4348:
4343:
4339:
4327:
4323:
4319:
4315:
4311:
4307:
4303:
4299:
4298:
4288:
4285:
4284:safe division
4281:
4278:
4274:
4271:
4267:
4220:
4207:
4202:
4195:
4189:
4186:
4180:
4177:
4174:
4168:
4162:
4156:
4150:
4141:
4137:
4134:
4133:
4131:
4124:
4089:
4082:
4072:
4061:
4029:
3989:
3984:
3973:
3917:
3910:
3904:
3898:
3892:
3874:
3872:
3861:
3859:
3855:
3851:
3847:
3843:
3841:
3823:
3820:
3817:
3813:
3790:
3786:
3781:
3777:
3755:
3751:
3725:
3721:
3716:
3712:
3709:
3706:
3703:
3698:
3694:
3689:
3685:
3682:
3677:
3673:
3668:
3664:
3657:
3653:
3650:
3641:
3632:
3628:
3624:
3620:
3599:
3596:
3593:
3589:
3579:
3572:
3569:
3566:
3563:
3560:
3557:
3554:
3550:
3547:
3544:
3541:
3540:
3539:
3536:
3534:
3530:
3526:
3522:
3516:
3512:
3510:
3506:
3502:
3494:
3490:
3485:
3482:
3478:
3474:
3471:
3467:
3466:
3465:
3461:
3445:
3440:
3435:
3430:
3425:
3424:
3423:
3420:
3414:
3410:
3406:
3402:
3398:
3394:
3390:
3372:
3370:
3369:Horner method
3366:
3362:
3356:
3350:
3342:
3340:
3336:
3332:
3331:
3326:
3322:
3318:
3314:
3310:
3306:
3298:
3296:
3292:
3288:
3284:
3280:
3275:
3273:
3269:
3265:
3253:
3251:
3243:
3234:
3224:
3215:
3213:
3209:
3205:
3201:
3197:
3187:
3178:
3172:
3169:
3166:
3163:
3159:
3156:
3152:
3151:
3150:
3141:
3139:
3131:
3128:
3125:
3122:
3119:
3118:
3117:
3114:
3112:
3111:
3106:
3102:
3098:
3094:
3084:
3074:
3069:
3067:
3063:
3055:
3052:
3051:
3050:
3044:
3041:
3040:
3039:
3037:
3033:
3028:
3026:
3018:
3017:
3016:
3010:
3009:
3008:
3006:
3001:
2995:
2991:
2988:
2987:
2986:
2983:
2981:
2977:
2969:
2965:
2962:
2961:
2960:
2956:
2954:
2953:rounded value
2948:
2942:
2937:
2933:
2918:
2915:
2912:
2909:
2907:
2904:
2903:
2899:
2896:
2893:
2890:
2887:
2886:
2882:
2879:
2876:
2873:
2871:
2868:
2867:
2863:
2860:
2857:
2854:
2852:
2849:
2848:
2844:
2841:
2838:
2835:
2832:
2831:
2827:
2824:
2821:
2818:
2815:
2814:
2810:
2807:
2804:
2801:
2798:
2797:
2788:
2784:
2780:
2776:
2771:
2768:
2764:
2760:
2756:
2752:
2748:
2744:
2740:
2739:Motorola 6809
2736:
2735:Motorola 6800
2732:
2728:
2727:Commodore PET
2724:
2720:
2716:
2712:
2708:
2704:
2700:
2695:
2691:
2687:
2683:
2679:
2675:
2671:
2667:
2663:
2659:
2655:
2654:
2653:
2651:
2641:
2639:
2631:
2627:
2625:
2617:
2613:
2612:
2611:
2605:
2601:
2597:
2596:
2595:
2592:
2588:
2586:
2576:
2573:
2570:
2567:
2564:
2561:
2558:
2556:
2553:
2552:
2548:
2545:
2542:
2539:
2536:
2533:
2530:
2528:
2525:
2524:
2520:
2517:
2514:
2511:
2508:
2505:
2502:
2500:
2497:
2496:
2492:
2489:
2486:
2483:
2480:
2477:
2474:
2472:
2469:
2468:
2464:
2461:
2458:
2455:
2452:
2449:
2446:
2443:
2442:IEEE 754-2008
2439:
2436:
2435:
2431:
2428:
2425:
2422:
2421:
2396:
2393:
2384:
2372:
2370:
2366:
2365:negative zero
2354:
2349:
2342:
2338:
2335:
2332:
2329:
2326:
2322:
2318:
2315:
2314:
2313:
2311:
2302:
2298:
2294:
2290:
2286:
2282:
2279:
2276:
2272:
2269:
2266:
2262:
2259:
2258:
2257:
2255:
2251:
2247:
2246:basic formats
2242:
2240:
2236:
2232:
2228:
2224:
2220:
2216:
2204:
2199:
2197:
2192:
2190:
2185:
2184:
2182:
2181:
2176:
2173:
2172:
2171:
2170:
2166:
2165:
2160:
2157:
2155:
2152:
2150:
2147:
2145:
2142:
2140:
2137:
2135:
2132:
2130:
2127:
2126:
2125:
2124:
2120:
2119:
2114:
2111:
2108:
2104:
2102:
2099:(binary128),
2098:
2094:
2092:
2088:
2084:
2082:
2078:
2074:
2071:
2067:
2066:
2065:
2064:
2061:
2058:
2057:
2054:
2051:
2048:
2047:
2043:
2033:
2031:
2028:, as well as
2027:
2022:
2020:
1999:
1994:
1991:
1988:
1984:
1980:
1975:
1969:
1966:
1962:
1958:
1955:
1951:
1942:
1941:
1940:
1937:
1917:
1913:
1904:
1903:
1902:
1899:
1884:
1880:
1877:
1874:
1871:
1868:
1864:
1859:
1854:
1851:
1848:
1844:
1840:
1835:
1831:
1828:
1825:
1821:
1817:
1805:
1802:
1799:
1796:
1793:
1789:
1786:
1783:
1780:
1779:
1778:
1776:
1772:
1768:
1764:
1759:
1757:
1756:
1750:
1747:
1734:
1730:
1726:
1722:
1719:
1714:
1713:
1712:
1709:
1707:
1703:
1699:
1698:William Kahan
1691:
1687:
1686:William Kahan
1683:
1679:
1677:
1673:
1669:
1665:
1659:
1657:
1653:
1649:
1645:
1637:
1634:
1631:
1628:
1627:
1626:
1624:
1619:
1617:
1613:
1609:
1605:
1601:
1596:
1594:
1590:
1586:
1581:
1579:
1575:
1571:
1567:
1562:
1560:
1556:
1537:
1528:
1511:
1508:
1488:
1485:
1475:
1468:
1465:
1456:
1452:
1448:
1444:
1440:
1432:
1428:
1424:
1420:
1418:
1414:
1410:
1406:
1402:
1398:
1394:
1376:
1373:
1364:
1360:
1356:
1352:
1348:
1340:
1336:
1332:
1327:
1299:
1296:
1290:
1287:
1279:
1261:
1239:
1231:
1227:
1223:
1219:
1216:
1213:
1210:
1207:
1203:
1199:
1195:
1191:
1188:
1185:
1182:
1178:
1174:
1170:
1166:
1162:
1159:
1155:
1152:
1151:
1150:
1142:
1140:
1136:
1132:
1128:
1112:
1111:normalization
1106:
1075:
1058:
1051:
1044:
1041:
1038:
1031:
1022:
1018:
1014:
1010:
1004:
1001:
997:
993:
990:
987:
984:
981:
976:
973:
969:
965:
962:
959:
954:
951:
947:
943:
940:
937:
932:
929:
925:
921:
918:
915:
910:
907:
903:
899:
896:
893:
888:
885:
881:
877:
874:
870:
862:
853:
849:
845:
841:
835:
832:
828:
824:
819:
807:
804:
801:
796:
793:
790:
786:
781:
764:
759:
746:
741:
738:
733:
727:
721:
709:
705:
701:
691:
678:
672:
664:
661:
656:
650:
644:
636:
632:
616:
613:
610:
600:
598:
594:
566:
561:
555:
551:
547:
543:
538:
532:
515:
502:
497:
493:
489:
482:
479:
476:
471:
467:
456:
441:1,528,535,047
434:
429:
427:
423:
419:
411:
407:
403:
399:
395:
391:
387:
383:
382:
377:
373:
369:
368:
367:
364:
350:
346:
342:
338:
333:
330:
326:
322:
318:
313:
311:
296:
294:
290:
285:
283:
279:
274:
272:
264:
259:
252:
247:
243:
241:
240:dynamic range
237:
233:
229:
224:
222:
218:
214:
209:
205:
203:
199:
193:
172:
167:
164:
139:
136:
129:
117:
114:
107:
104:
96:
94:
90:
86:
82:
78:
74:
70:
66:
62:
54:
50:
46:
41:
37:
30:
19:
10154:Intersection
9952:
9838:the original
9821:
9817:
9775:
9771:
9711:
9696:, retrieved
9687:
9659:
9629:
9623:
9595:
9556:
9533:
9522:. Retrieved
9499:
9463:
9431:
9425:
9416:
9407:
9398:
9389:
9380:
9371:
9362:
9353:
9345:
9341:
9332:
9304:
9285:
9279:
9269:
9258:. Retrieved
9247:
9236:. Retrieved
9226:
9198:
9170:
9124:. Retrieved
9100:
9080:
9070:
9054:. Retrieved
9034:
8997:
8990:
8966:
8938:. Retrieved
8919:
8912:
8901:. Retrieved
8882:
8872:
8854:
8846:
8840:
8830:
8824:. Retrieved
8815:(99): 5–10.
8810:
8800:
8791:
8782:
8710:
8704:
8670:
8664:
8611:
8607:
8601:
8590:
8581:
8570:
8561:
8546:
8519:
8515:
8505:
8496:
8487:
8445:
8432:
8408:
8371:
8349:
8335:
8324:. Retrieved
8300:. Retrieved
8287:
8278:
8267:. Retrieved
8225:
8219:. Retrieved
8213:). ID 1400.
8198:
8174:. Retrieved
8165:
8154:. Retrieved
8144:
8136:
8130:. Retrieved
8126:the original
8116:
8105:. Retrieved
8081:
8070:. Retrieved
8058:
8046:
8035:. Retrieved
8015:
8006:
7997:
7978:
7964:
7956:
7952:
7947:
7929:
7906:
7897:
7842:
7830:. Retrieved
7810:
7804:
7788:
7775:
7768:Randell 1982
7763:
7745:
7740:
7727:
7718:
7712:. Retrieved
7691:
7682:
7676:. Retrieved
7658:
7648:
7638:, retrieved
7629:
7619:
7572:
7545:. Retrieved
7506:. Retrieved
7492:
7482:
7462:
7412:
7359:
7338:
7325:
7313:
7303:
7293:
7275:
7266:
7253:
7240:
7227:
7198:
7165:(ca. 1974),
7153:(1972), the
7130:
7118:
7114:
7108:
7104:
7100:
7094:
7089:
7073:
7069:
7065:
7061:
7049:
7043:
7038:
6898:
6895:
6883:
6872:
6853:
6841:
6835:
6829:
6823:
6817:
6811:
6805:
6799:
6793:
6787:
6781:
6775:
6769:
6763:
6759:3.1415926535
6757:
6751:
6747:3.1415926535
6745:
6739:
6735:3.1415926535
6733:
6727:
6721:
6715:
6709:
6703:
6697:
6691:
6685:
6679:
6673:
6667:
6661:
6655:
6649:
6643:
6637:
6631:
6625:
6619:
6613:
6607:
6601:
6595:
6589:
6583:
6577:
6571:
6565:
6559:
6553:
6547:
6541:
6535:
6529:
6523:
6517:
6511:
6498:
6249:First form:
6192:
6184:
6174:
6159:
6143:0.6/0.2-3==0
6136:
5996:
5975:
5971:
5963:
5960:
5910:
5745:
5717:
5701:
5658:
5655:
5377:
5374:
4595:
4584:
4578:
4493:
4491:
4486:
4482:
4478:
4361:
4345:
4337:
4336:
4322:Saudi Arabia
4312:an incoming
4282:Testing for
4143:
4136:Cancellation
4129:
4088:distributive
4085:
4080:
4070:
3985:
3974:
3967:
3914:
3908:
3902:
3896:
3875:
3867:
3857:
3854:root-finding
3849:
3845:
3844:
3630:
3626:
3622:
3618:
3616:
3570:
3564:
3558:
3552:
3548:
3542:
3537:
3517:
3513:
3498:
3463:
3421:
3378:
3357:
3354:
3348:
3330:cancellation
3328:
3324:
3320:
3316:
3312:
3302:
3294:
3290:
3286:
3282:
3278:
3276:
3271:
3267:
3263:
3260:
3247:
3241:
3232:
3221:
3211:
3207:
3199:
3193:
3184:
3176:
3154:
3147:
3135:
3115:
3108:
3100:
3099:schemes (or
3092:
3090:
3070:
3059:
3053:
3048:
3042:
3035:
3031:
3029:
3022:
3014:
3002:
2999:
2993:
2989:
2984:
2979:
2975:
2973:
2967:
2963:
2957:
2952:
2949:
2929:
2662:Altair BASIC
2647:
2635:
2621:
2609:
2603:
2599:
2593:
2589:
2582:
2429:Significand
2390:
2373:
2350:
2346:
2307:
2253:
2249:
2245:
2243:
2212:
2167:Alternatives
2089:(binary64),
2079:(binary32),
2049:
2023:
2018:
2016:
1938:
1935:
1900:
1809:
1803:
1797:
1791:
1787:
1781:
1774:
1770:
1766:
1762:
1760:
1753:
1751:
1743:
1710:
1706:Harold Stone
1702:Turing Award
1695:
1660:
1641:
1635:
1629:
1620:
1615:
1597:
1582:
1574:Model V
1565:
1563:
1529:
1436:
1408:
1404:
1400:
1392:
1362:
1350:
1344:
1197:
1193:
1164:
1148:
1138:
1134:
1130:
1126:
1110:
1107:
1076:
760:
704:rounding bit
703:
699:
692:
601:
562:
539:
530:
516:
457:
453:152,853.5047
437:152,853.5047
430:
425:
421:
417:
415:
409:
405:
393:
389:
385:
379:
365:
353:152,853.5047
334:
314:
307:
286:
275:
268:
225:
212:
210:
206:
194:
97:
92:
77:real numbers
68:
64:
58:
36:
10384:Type theory
10379:Type system
10229:Bottom type
10176:Option type
10117:generalized
10003:Long double
9948:Fixed point
9399:gcc.gnu.org
9381:gcc.gnu.org
8673:(1): 5–48.
8240:assumed bit
8238:before the
7848:Rojas, Raúl
7813:(2): 5–16.
7794:Rojas, Raúl
7698:"MANIAC II"
7686:(256 pages)
7365:denominator
7344:long double
7218:(1971) and
7189:(1966) and
7183:SDS Sigma 5
7141:(1964) and
7123:significand
7103:. The term
7062:coefficient
7045:significand
6928:Coprocessor
6798:95552 24
6723:3.141592653
6711:3.141592653
6699:3.141592653
6166:eigenvector
6147:0.6/0.2 - 3
6071:, and that
5916:long double
5759:−1) / (exp(
4316:missile in
4090:. That is,
4030:. That is,
4028:associative
3988:commutative
3891:, which is
3477:square root
3337:; see also
3233:In detail:
2833:FP8 (E5M2)
2816:FP8 (E4M3)
2811:Total bits
2678:IBM PC 5150
2616:hexadecimal
2602:= 1 ;
2293:long double
2109:(binary256)
1746:fixed-point
1731:(overflow,
1559:IAS machine
1555:von Neumann
1439:Konrad Zuse
1427:Konrad Zuse
1222:Mathematica
1177:Frank Olver
1154:Fixed-point
763:significand
445:1.528535047
418:significand
390:coefficient
381:significand
357:1.528535047
341:power of 10
329:fixed-point
317:radix point
293:coprocessor
251:number line
217:radix point
124:significand
85:significand
10404:Categories
10289:Empty type
10284:Type class
10234:Collection
10191:Refinement
10169:metaobject
10017:signedness
9876:Data types
9818:double_fpu
9789:cs/0701192
9743:2018935254
9717:Birkhäuser
9698:2018-07-16
9628:, Vol. 2:
9524:2016-02-11
9260:2011-01-11
9238:2012-04-25
9126:2003-09-05
9056:2013-05-23
8940:2013-05-14
8903:2013-05-14
8826:2011-09-24
8621:2101.11408
8362:2209.05433
8326:2020-05-16
8302:2010-02-24
8269:2016-05-30
8221:2016-05-30
8176:2024-08-29
8156:2024-04-16
8132:2012-04-25
8107:2012-02-19
8072:2019-11-08
8037:2019-09-22
7837:(12 pages)
7832:2022-07-03
7714:2018-08-07
7678:2019-08-18
7640:2018-07-16
7603:2017947446
7547:2018-08-07
7508:2012-12-31
7444:2009939668
7418:Birkhäuser
7383:References
6687:3.14159265
6675:3.14159265
6195:Archimedes
4140:derivative
3770:set to 0,
3409:JavaScript
3387:to denote
3105:truncation
2707:Bill Gates
2699:Intel 8080
2694:QuickBASIC
2413:precision
2321:decimal128
2304:available.
2101:decimal128
2072:(binary16)
2017:which has
1718:endianness
1652:System/360
1566:commercial
1564:The first
1419:in 1920.
1413:typewriter
1349:published
1324:See also:
1097:here) and
73:arithmetic
10364:Subtyping
10359:Interface
10342:metaclass
10294:Unit type
10264:Semaphore
10244:Exception
10149:Inductive
10139:Dependent
10104:Composite
10082:Character
10064:Reference
9961:Minifloat
9917:Bit array
9814:OpenCores
9806:218578808
9688:quadibloc
9593:(2007) .
8821:1354-3172
8788:"D.3.2.1"
8689:222008826
8638:231718830
8538:218472153
8413:CiteSeerX
8292:Microsoft
8152:. openEXR
8122:"openEXR"
7857:1406.1886
7663:CRC Press
7630:quadibloc
7373:numerator
7369:conjugate
7259:MANIAC II
7169:(1980s),
7058:logarithm
7001:Minifloat
6866:and auto-
6860:compilers
6729:3.1415926
6705:3.1415926
6681:3.1415926
6669:3.1415926
6663:3.1415926
6657:3.1415926
6484:∞
6481:→
6448:×
6435:×
6429:∼
6426:π
6304:−
6114:θ
6110:
6093:θ
6089:
6048:−
6026:−
5676:≤
5667:δ
5626:δ
5587:^
5563:δ
5524:^
5498:δ
5459:^
5435:δ
5396:^
5355:^
5346:⋅
5340:^
5322:⋅
5313:
5274:δ
5252:δ
5226:⋅
5198:δ
5176:δ
5150:⋅
5115:δ
5086:δ
5060:⋅
5032:δ
5006:⋅
4953:≤
4944:δ
4913:δ
4887:⋅
4859:δ
4833:⋅
4810:
4783:
4746:⋅
4730:
4708:⋅
4692:
4679:
4660:⋅
4651:
4550:≤
4534:−
4522:
4457:−
4393:−
4295:Incidents
4187:−
3871:precision
3707:⋯
3623:underflow
3549:underflow
3509:exception
3489:underflow
3200:decimal32
2805:Exponent
2426:Exponent
2416:Number of
2325:decimal32
2317:Decimal64
2301:alignment
2129:Minifloat
2105:256-bit:
2097:Quadruple
2095:128-bit:
2091:decimal64
2081:decimal32
1967:−
1959:−
1872:−
1852:−
1829:−
1585:Pilot ACE
1541:∞
1538:±
1515:∞
1512:×
1481:∞
1437:In 1938,
1300:π
1291:
1240:π
1169:symmetric
1137:, or the
1059:3.1415928
1052:≈
1042:×
1039:1.5707964
1032:≈
1015:×
1002:−
994:×
985:⋯
974:−
966:×
952:−
944:×
930:−
922:×
908:−
900:×
886:−
878:×
846:×
833:−
825:×
805:−
787:∑
742:_
700:round bit
665:_
490:×
480:−
394:precision
363:seconds.
211:The term
173:⏞
165:−
140:⏟
130:×
118:⏟
79:using an
61:computing
10389:Variable
10279:Top type
10144:Equality
10052:physical
10029:Rational
10024:Interval
9971:bfloat16
9692:archived
9658:(1997).
9497:(1965).
9461:(1963).
9363:GCC Wiki
9321:Archived
9215:Archived
9187:Archived
9159:Archived
9117:Archived
9089:Archived
9047:Archived
9043:ARITH 17
8964:(2002).
8812:Overload
8771:Archived
8476:Archived
8388:Archived
8296:Archived
8263:Archived
8215:Archived
8098:Archived
8063:Archived
8028:Archived
7986:Archived
7886:Archived
7823:Archived
7705:Archived
7634:archived
7611:30244721
7538:Archived
7214:(1971),
7210:(1964),
7206:(1962),
7185:(1967),
7161:(1966),
7096:exponent
7082:exponent
7070:fraction
7066:argument
7054:mantissa
7050:mantissa
6979:IEEE 754
6969:GNU MPFR
6905:See also
6884:In most
6651:3.141592
6645:3.141592
6639:3.141592
6633:3.141592
5966:compiler
5763:−1) − 1)
4081:1280.246
4071:1280.245
3846:Overflow
3559:overflow
3505:portable
3432:-5000.12
3397:IEEE 754
3395:and the
3268:rounding
3097:rounding
3043:3.141592
3025:rounding
2870:Bfloat16
2723:Apple //
2719:MOS 6502
2690:GW-BASIC
2664:(1975),
2650:IEEE 754
2636:and the
2406:Exponent
2353:infinity
2235:Cray T90
2219:IEEE 754
2134:bfloat16
2085:64-bit:
2075:32-bit:
2068:16-bit:
2060:IEEE 754
2042:IEEE 754
1777:) where
1690:IEEE 754
1664:IEEE 754
1644:IBM 7094
1220:such as
728:00001111
722:11001001
673:10100010
651:00001111
645:11001001
426:exponent
402:exponent
386:mantissa
347:'s moon
299:Overview
271:IEEE 754
198:base two
179:exponent
89:exponent
10332:Generic
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10224:Boolean
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10047:Address
10039:Pointer
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9938:Complex
9926:Numeric
9822:fpuvhdl
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7371:of the
7193:(1970).
6886:Fortran
6879:library
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6627:3.14159
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2782:format.
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2107:Octuple
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5987:Python
5979:proton
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5792:double
5779:double
5770:double
5656:where
5375:where
4477:where
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3924:double
3442:-3e-45
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3272:sticky
3155:dtoa.c
2992:= −4;
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2577:~34.0
2571:16383
2549:~19.2
2543:16383
2521:~15.9
2499:Double
2471:Single
2432:Total
2087:Double
2077:Single
1397:digits
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9784:arXiv
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9324:(PDF)
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7030:Notes
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6959:FLOPS
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6603:3.141
6597:3.141
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