Rounding - Knowledge (XXG)

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correctly rounded result can be guaranteed, may demand a lot of computation time or may be out of reach. In practice, when this limit is not known (or only a very large bound is known), some decision has to be made in the implementation (see below); but according to a probabilistic model, correct rounding can be satisfied with a very high probability when using an intermediate accuracy of up to twice the number of digits of the target format plus some small constant (after taking special cases into account).

4221:, square root, and floating-point remainder will give the correctly rounded result of the infinite-precision operation. No such guarantee was given in the 1985 standard for more complex functions and they are typically only accurate to within the last bit at best. However, the 2008 standard guarantees that conforming implementations will give correctly rounded results which respect the active rounding mode; implementation of the functions, however, is optional. 1193: 1399: 32: 2707: 1012: 1236: 2200: 2006: 1799: 1614: 4495:

manufacturers. The standard allows the user to choose among several rounding modes, and in each case specifies precisely how the results should be rounded. These features made numerical computations more predictable and machine-independent, and made possible the efficient and consistent implementation of

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rather than the odd number that is equally near. The reason for this procedure is that in a series of several measurements of the same quantity it will be as apt to make a record too large as it will to make one too small, and so in the average of several such values will cause but a slight error, if

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Error diffusion tries to ensure the error, on average, is minimized. When dealing with a gentle slope from one to zero, the output would be zero for the first few terms until the sum of the error and the current value becomes greater than 0.5, in which case a 1 is output and the difference subtracted

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One method, more obscure than most, is to alternate direction when rounding a number with 0.5 fractional part. All others are rounded to the closest integer. Whenever the fractional part is 0.5, alternate rounding up or down: for the first occurrence of a 0.5 fractional part, round up, for the second

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With decimal arithmetic, final digits of 0 and 5 are avoided; if there is a choice between numbers with the least significant digit 0 or 1, 4 or 5, 5 or 6, 9 or 0, then the digit different from 0 or 5 shall be selected; otherwise, the choice is arbitrary. IBM defines that, in the latter case, a digit

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in 1982. It was initially set at 1000.000 (three decimal places of accuracy), and after 22 months had fallen to about 520, although the market appeared to be rising. The problem was caused by the index being recalculated thousands of times daily, and always being truncated (rounded down) to 3 decimal

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even is that its sign is arbitrary, or is not fixed by the computation as is the case with all the other errors. However, the computer's rule, which makes the last rounded figure of an interpolated value even when half a unit is to be disposed of, will, in the long run, make this error as often plus

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Rounding a number twice in succession to different levels of precision, with the latter precision being coarser, is not guaranteed to give the same result as rounding once to the final precision except in the case of directed rounding. For instance rounding 9.46 to one decimal gives 9.5, and then 10

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This method is also free from positive/negative bias and bias toward/away from zero, provided the numbers to be rounded are neither mostly even nor mostly odd. It also shares the round half to even property of distorting the original distribution, as it increases the probability of odds relative to

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In some algorithms, an intermediate result is computed in a larger precision, then must be rounded to the final precision. Double rounding can be avoided by choosing an adequate rounding for the intermediate computation. This consists in avoiding to round to midpoints for the final rounding (except

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can never be. For example, suppose one started with 0 and added 0.3 to that one hundred times while rounding the running total between every addition. The result would be 0 with regular rounding, but with stochastic rounding, the expected result would be 30, which is the same value obtained without

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where the rounding is randomly up or down. Stochastic rounding can be used for Monte Carlo arithmetic, but in general, just rounding up or down with equal probability is more often used. Repeated runs will give a random distribution of results which can indicate the stability of the computation.

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Office of the Federal Coordinator for Meteorology determined that weather data should be rounded to the nearest round number, with the "round half up" tie-breaking rule. For example, 1.5 rounded to integer should become 2, and −1.5 should become −1. Prior to that date, the tie-breaking rule was

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It is often used for currency conversions and price roundings (when the amount is first converted into the smallest significant subdivision of the currency, such as cents of a euro) as it is easy to explain by just considering the first fractional digit, independently of supplementary precision

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Until the 1980s, the rounding method used in floating-point computer arithmetic was usually fixed by the hardware, poorly documented, inconsistent, and different for each brand and model of computer. This situation changed after the IEEE 754 floating-point standard was adopted by most computer

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results, except on some well-known arguments; therefore, from a theoretical point of view, it is always possible to correctly round such functions. However, for an implementation of such a function, determining a limit for a given precision on how accurate results need to be computed, before a

4604:, and do not discriminate between integers and floating-point values; however, the implementations of these languages will typically convert these numbers into IEEE 754 double-precision floating-point values before exposing the computed digits with a limited precision (notably within standard 4319:'s libmcr of 2004, in the 4 rounding modes. For the difficult cases, this library also uses multiple precision, and the number of words is increased by 2 each time the Table-maker's dilemma occurs (with undefined behavior in the very unlikely event that some limit of the machine is reached). 4560:

of most languages have commonly provided at least the four basic rounding functions (up, down, to nearest, and toward zero). The tie-breaking method can vary depending on the language and version or might be selectable by the programmer. Several languages follow the lead of the IEEE 754

2255:. Thus, for example, 23.5 becomes 24, as does 24.5; however, −23.5 becomes −24, as does −24.5. This function minimizes the expected error when summing over rounded figures, even when the inputs are mostly positive or mostly negative, provided they are neither mostly even nor mostly odd. 1188:{\displaystyle y=\operatorname {truncate} (x)=\operatorname {sgn}(x)\left\lfloor \left|x\right|\right\rfloor =-\operatorname {sgn}(x)\left\lceil -\left|x\right|\right\rceil ={\begin{cases}\left\lfloor x\right\rfloor &x\geq 0\\\left\lceil x\right\rceil &x<0\end{cases}}} 1431:

If it were not for the 0.5 fractional parts, the round-off errors introduced by the round to nearest method would be symmetric: for every fraction that gets rounded down (such as 0.268), there is a complementary fraction (namely, 0.732) that gets rounded up by the same amount.

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The most basic form of rounding is to replace an arbitrary number by an integer. All the following rounding modes are concrete implementations of an abstract single-argument "round()" procedure. These are true functions (with the exception of those that use randomness).

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standard dictate that in straightforward calculations the result should not be rounded twice. This has been a particular problem with Java as it is designed to be run identically on different machines, special programming tricks have had to be used to achieve this with

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may write "−0" to indicate a temperature between 0.0 and −0.5 degrees (exclusive) that was rounded to an integer. This notation is used when the negative sign is considered important, no matter how small is the magnitude; for example, when rounding temperatures in the

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when rounding to integer using rounding half to even, but would give 9 when rounded to integer directly. Borman and Chatfield discuss the implications of double rounding when comparing data rounded to one decimal place to specification limits expressed using integers.

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The most common type of rounding is to round to an integer; or, more generally, to an integer multiple of some increment – such as rounding to whole tenths of seconds, hundredths of a dollar, to whole multiples of 1/2 or 1/8 inch, to whole dozens or thousands, etc.

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Where the rounded result would overflow the result for a directed rounding is either the appropriate signed infinity when "rounding away from zero", or the highest representable positive finite number (or the lowest representable negative finite number if

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with the smaller magnitude shall be selected. RPSP can be applied with the step between two consequent roundings as small as a single digit (for example, rounding to 1/10 can be applied after rounding to 1/100). For example, when rounding to integer,

1394:{\displaystyle y=\operatorname {sgn}(x)\left\lceil \left|x\right|\right\rceil =-\operatorname {sgn}(x)\left\lfloor -\left|x\right|\right\rfloor ={\begin{cases}\left\lceil x\right\rceil &x\geq 0\\\left\lfloor x\right\rfloor &x<0\end{cases}}} 3892:

Many design procedures describe how to calculate an approximate value, and then "round" to some standard size using phrases such as "round down to nearest standard value", "round up to nearest standard value", or "round to nearest standard value".

2071: 1877: 4129:, litigated between 1995 and 1997, the insurance companies argued that double rounding premiums was permissible and in fact required. The US courts ruled against the insurance companies and ordered them to adopt rules to ensure single rounding. 959: 3660: 2014:

This method treats positive and negative values symmetrically, and therefore is free of overall positive/negative bias if the original numbers are positive or negative with equal probability. It does, however, still have bias toward zero.

849: 2702:{\displaystyle \operatorname {Round} (x)={\begin{cases}\lfloor x\rfloor &{\text{ with probability }}1-(x-\lfloor x\rfloor )=\lfloor x\rfloor -x+1\\\lfloor x\rfloor +1&{\text{ with probability }}{x-\lfloor x\rfloor }\end{cases}}} 1679: 1494: 2223:, treats positive and negative values symmetrically, and therefore is free of overall positive/negative bias if the original numbers are positive or negative with equal probability. It does, however, still have bias away from zero. 4204:

be rounded within half an ulp like SQRT? Because nobody knows how much computation it would cost... No general way exists to predict how many extra digits will have to be carried to compute a transcendental expression and round it

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If occurrences of 0.5 fractional parts occur significantly more than a restart of the occurrence "counting", then it is effectively bias free. With guaranteed zero bias, it is useful if the numbers are to be summed or averaged.

4313:), first distributed in 2003. It supports the 4 rounding modes and is proved, using the knowledge of the hardest-to-round cases. More efficient than IBM MathLib. Succeeded by Metalibm (2014), which automates the formal proofs. 714:

places, in such a way that the rounding errors accumulated. Recalculating the index for the same period using rounding to the nearest thousandth rather than truncation corrected the index value from 524.811 up to 1098.892.

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A fraction perceptibly less than a half should be discarded and more than a half should always be considered as one more unit, but when it is uncertain which figure is the nearer one the universally adopted rule is to

4153:. Equivalently, it consists in returning the intermediate result when it is exactly representable, and the nearest floating-point number with an odd significand otherwise; this is why it is also known as 2344:

value would round to a normal non-zero value. Effectively, this mode prefers preserving the existing scale of tie numbers, avoiding out-of-range results when possible for numeral systems of even

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when the midpoint is exact). In binary arithmetic, the idea is to round the result toward zero, and set the least significant bit to 1 if the rounded result is inexact; this rounding is called

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remains more obscure. If this rounding method was ever a standard in banking, the evidence has proved extremely difficult to find. To the contrary, section 2 of the European Commission report

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This variant is almost never used in computations, except in situations where one wants to avoid increasing the scale of floating-point numbers, which have a limited exponent range. With

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However, this rule distorts the distribution by increasing the probability of evens relative to odds. Typically this is less important than the biases that are eliminated by this method.

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representation for the values to be rounded, because only the first omitted digit needs to be considered to determine if it rounds up or down. This is one method used when rounding to

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Rounding as follows to one of the closest integer toward negative infinity and the closest integer toward positive infinity, with a probability dependent on the proximity is called

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Bankers' rounding is used when truncating real numbers that end with .5; that is, odd numbers are rounded up to an even integer, even numbers are rounded down to an even integer.

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Apart from this detail, all the variants of rounding discussed above apply to the rounding of floating-point numbers as well. The algorithm for such rounding is presented in the

2195:{\displaystyle y=\operatorname {sgn}(x)\left\lfloor \left|x\right|+{\tfrac {1}{2}}\right\rfloor =-\operatorname {sgn}(x)\left\lceil -\left|x\right|-{\tfrac {1}{2}}\right\rceil } 2001:{\displaystyle y=\operatorname {sgn}(x)\left\lceil \left|x\right|-{\tfrac {1}{2}}\right\rceil =-\operatorname {sgn}(x)\left\lfloor -\left|x\right|+{\tfrac {1}{2}}\right\rfloor } 4340:

for which a rounded value can never be determined no matter how many digits are calculated. Specific instances cannot be given but this follows from the undecidability of the

4542:, would provide only one method, usually truncation (toward zero). This default method could be implied in certain contexts, such as when assigning a fractional number to an 4553:. Other kinds of rounding had to be programmed explicitly; for example, rounding a positive number to the nearest integer could be implemented by adding 0.5 and truncating. 4302:(abbreviated as APMathLib or just MathLib), also called libultim, in rounding to nearest only. This library uses up to 768 bits of working precision. It was included in the 3861:

do not exceed a given maximum. This problem is fairly distinct from that of rounding a value to a fixed number of decimal or binary digits, or to a multiple of a given unit

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fractional parts, the rounding errors by all values, with the omission of those having 0.5 fractional part, would statistically compensate each other. This means that the

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on a logarithmic scale. In particular, for resistors with a 10% accuracy, they are supplied with nominal values 100, 120, 150, 180, 220, etc. rounded to multiples of 10 (

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The abstract single-argument "round()" function that returns an integer from an arbitrary real value has at least a dozen distinct concrete definitions presented in the

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is difficult because the number of extra digits that need to be calculated to resolve whether to round up or down cannot be known in advance. This problem is known as "

1794:{\displaystyle y=\left\lceil x-{\tfrac {1}{2}}\right\rceil =-\left\lfloor -x+{\tfrac {1}{2}}\right\rfloor =\left\lfloor {\tfrac {1}{2}}\lceil 2x\rceil \right\rfloor } 1609:{\displaystyle y=\left\lfloor x+{\tfrac {1}{2}}\right\rfloor =-\left\lceil -x-{\tfrac {1}{2}}\right\rceil =\left\lceil {\tfrac {1}{2}}\lfloor 2x\rfloor \right\rceil } 769: 3945:, such as sound waves, the overall effect of a number of measurements is more important than the accuracy of each individual measurement. In these circumstances, 4581:). On the other hand, truncation (round to zero) is still the default rounding method used by many languages, especially for the division of two integer values. 4573:, provide functions that round a value to a specified number of decimal digits (e.g., from 4321.5678 to 4321.57 or 4300). In addition, many languages provide a 705:

Where many calculations are done in sequence, the choice of rounding method can have a very significant effect on the result. A famous instance involved a new

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or similar string formatting function, which allows one to convert a fractional number to a string, rounded to a user-specified number of decimal places (the

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were provided. According to its documentation, this library uses a first step with an accuracy a bit larger than double precision, a second step based on

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rounded to more than four digits. In contrast, truncation does not suffer from this problem; for example, a simple string search for "3.1415", which is

3704:. The logarithm of 165 is closer to the logarithm of 180 therefore a 180 ohm resistor would be the first choice if there are no other considerations. 5988: 5054:

Gupta, Suyog; Angrawl, Ankur; Gopalakrishnan, Kailash; Narayanan, Pritish (2016-02-09). "Deep Learning with Limited Numerical Precision". p. 3.

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values. An advantage over alternate tie-breaking is that the last direction of rounding on the 0.5 fractional part does not have to be "remembered".

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suggests that there had previously been no standard approach to rounding in banking; and it specifies that "half-way" amounts should be rounded up.

3681:. Rounding on a logarithmic scale is accomplished by taking the log of the amount and doing normal rounding to the nearest value on the log scale. 3282: 2360:. This can be achieved if all roundings except the final one are done using RPSP, and only the final rounding uses the externally requested mode. 4209:

to some preassigned number of digits. Even the fact (if true) that a finite number of extra digits will ultimately suffice may be a deep theorem.

2493:. "Up" and "down" can be any two rounding methods that oppose each other - toward and away from positive infinity or toward and away from zero. 2296:, will give a rounded result with an error that tends to grow in proportion to the square root of the number of operations rather than linearly. 5603:

de Dinechin, Florent; Lauter, Christoph; Muller, Jean-Michel (January–March 2007). "Fast and correctly rounded logarithms in double-precision".

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for the increment and then reduces to the equivalent abstract single-argument function, with also the same dozen distinct concrete definitions.

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section. The abstract two-argument "roundToMultiple()" function is formally defined here, but in many cases it is used with the implicit value

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of the rounded numbers is equal to the expected value of the original numbers when numbers with fractional part 0.5 from the set are removed.

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Approximating each of a finite set of real numbers by an integer so that the sum of the rounded numbers equals the rounded sum of the numbers

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argument and returning the result of the same type, which then may be converted to an integer if necessary. This approach may avoid spurious

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It is possible to use rounded arithmetic to evaluate the exact value of a function with integer domain and range. For example, if an integer

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Rounding is often done to obtain a value that is easier to report and communicate than the original. Rounding can also be important to avoid

3496:; for example, it is common in computing to need to round a number to a whole power of 2. The steps, in general, to round a positive number 4973: 4376:

exists and 1 otherwise. The value before rounding can however be approximated to any given precision even if the conjecture is unprovable.

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A rounding method should have utility in computer science or human arithmetic where finite precision is used, and speed is a consideration.

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package gives correctly rounded arbitrary precision results. Some other libraries implement elementary functions with correct rounding in

376:; i.e., once a number has been rounded, rounding it again to the same precision will not change its value. Rounding functions are also 4850: 3924:, which use notions of distance other than the simple difference – for example, a sequence may round to the integer with the smallest 277: 4229: 2530:

Like round-half-to-even and round-half-to-odd, this rule is essentially free of overall bias, but it is also fair among even and odd

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Muller, Jean-Michel; Brisebarre, Nicolas; de Dinechin, Florent; Jeannerod, Claude-Pierre; Lefèvre, Vincent; Melquiond, Guillaume;

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in the reported result. Rounding is almost unavoidable when reporting many computations – especially when dividing two numbers in

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A case where double rounding always leads to the same value as directly rounding to the final precision is when the radix is odd.

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For correct results, each rounding step must remove at least 2 binary digits, otherwise, wrong results may appear. For example,

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and Moshe Olshansky in 1999, correctly rounded to nearest only. This library was claimed to be portable, but only binaries for

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Selected Techniques of Statistical Analysis for Scientific and Industrial Research, and Production and Management Engineering

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Because it is not usually possible for a method to satisfy all ideal characteristics, many different rounding methods exist.

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floating point. The Java language was changed to allow different results where the difference does not matter and require a

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where the training may use low precision arithmetic iteratively. Stochastic rounding is also a way to achieve 1-dimensional

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is negative, round-down is the same as round-away-from-zero, and round-up is the same as round-toward-zero. In any case, if

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qualifier to be used when the results have to conform accurately; strict floating point has been restored in Java 17.

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that already exist between the domain and range. With finite precision (or a discrete domain), this translates to removing

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provide functions or special syntax to round fractional numbers in various ways. The earliest numeric languages, such as

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rounded to four digits is "3.1416" but a simple search for this string will not discover "3.14159" or any other value of

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two floating-point arguments at which it does not over/underflow. Instead, reputable math libraries compute elementary

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is used to achieve analog type output from an inertial device by rapidly pulsing the power with a variable duty cycle.

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As a result of (1) and (2), the output from rounding should be close to its input, often as close as possible by some

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Rounding to the nearest even number is also called 'bankers rounding' because the banks use this technique as well.

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that is accessible to a general audience but especially useful to those studying computer science and electronics.

5530: 42: 5981: 4697: 3759: 1451: 672: 284: 239: 49: 20: 6088: 5297:"A mechanically checked proof of the correctness of the kernel of the AMD5K86 floating-point division algorithm" 4592:

do not define any specific maximum precision for numbers and measurements, which they treat and expose in their

4295:, and a third step with a 768-bit precision based on arrays of IEEE 754 double-precision floating-point numbers. 2489:

occurrence, round down, and so on. Alternatively, the first 0.5 fractional part rounding can be determined by a

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Here we have a case in which the half of an odd number is required. A good rule to adopt in such a case is to

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is positive, round-down is the same as round-toward-zero, and round-up is the same as round-away-from-zero. If

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If the erroneous middle step is removed, the final rounding to integer rounds 3.25 to the correct value of 3.

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numbers are typically used, which have even more computational nuances because they are not equally spaced.

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Logarithmic and Other Mathematical Tables with Examples of their Use and Hints on the Art of Computation

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Standard Practice for Using Significant Digits in Test Data to Determine Conformance with Specifications

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The standard arose from a committee of the ASA working to standardize inch–millimeter conversion. See:

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of 10, like 1/1000 or 25/100). For intermediate values stored in digital computers, it often means the

299:) is sometimes used to indicate rounding of exact numbers, e.g. 9.98 ≈ 10. This sign was introduced by 195:

reporting of a computed number, measurement, or estimate; for example, a quantity that was computed as

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digits or sign of the amount (for strict equivalence between the paying and recipient of the amount).

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The concept of rounding is very old, perhaps older than the concept of division itself. Some ancient

338: 321: 64: 5617: 3889:, writing paper, capacitors, and many other products are usually sold in only a few standard sizes. 954:{\displaystyle y=\operatorname {ceil} (x)=\left\lceil x\right\rceil =-\left\lfloor -x\right\rfloor } 6149: 5316: 4687: 4557: 4496: 4477: 4322:

The CORE-MATH project (2022) provides some correctly rounded functions in the 4 rounding modes for

4157:. A concrete implementation of this approach, for binary and decimal arithmetic, is implemented as 3655:{\displaystyle \mathrm {roundToPower} (x,b)=b^{\mathrm {round} (\log _{b}x)},x>0,b>0,b\neq 1} 3450: 3423: 2213: 1636: 676: 521: 467: 330: 243: 134: 6127: 5900: 5875: 5257: 6046: 5930: 5630: 5372: 5238: 5055: 4894: 4827: 4778: 4049: 3874: 2865: 2855: 2845: 2311:, a similar tie-breaking rule to round half to even. In this approach, if the fractional part of 377: 5842: 4960: 4511: 3665:

Many of the caveats applicable to rounding to a multiple are applicable to rounding to a power.

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More general rounding rules can separate values at arbitrary break points, used for example in

844:{\displaystyle y=\mathrm {floor} (x)=\left\lfloor x\right\rfloor =-\left\lceil -x\right\rceil } 6110: 5473: 5463: 5195: 5191: 5164: 5117: 4995: 4865: 4807: 4790: 4337: 3942: 3921: 2394:

With binary arithmetic, this rounding is also called "round to odd" (not to be confused with "

2391:" section, rounding 9.46 to one decimal gives 9.4, which rounding to integer in turn gives 9. 359: 273: 4569:

because floating-point types have a larger range than integer types. Some languages, such as

4024:. The first of these and the differences of adjacent values give the desired rounded values: 172:

value that has a shorter, simpler, or more explicit representation. For example, replacing $

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in many studies, to evaluate the numeracy level of ancient populations. He came up with the

4316: 4169: 3685: 2721: 2323:. Thus, for example, 23.5 becomes 23, as does 22.5; while −23.5 becomes −23, as does −22.5. 1470:), a tie-breaking rule that is widely used in many disciplines. That is, half-way values of 5258:"Efficiently producing default orthogonal IEEE double results using extended IEEE hardware" 3823:

is negative), when "rounding toward zero". The result of an overflow for the usual case of

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For example, 1.6 would be rounded to 1 with probability 0.4 and to 2 with probability 0.6.

2031:), a tie-breaking rule that is commonly taught and used, namely: If the fractional part of 314:. This way, when the same input is rounded in different instances, the output is unchanged. 6092: 6013: 5116:

Bruce Trump, Christine Schneider. "Excel Formula Calculates Standard 1%-Resistor Values".

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in 2001, but the "slow paths" (providing correct rounding) were removed from 2018 to 2021.

3950: 3901: 3897: 2860: 2850: 2840: 435: 251: 211: 192: 5396: 5418: 5350:"Emulation of a FMA and correctly-rounded sums: proved algorithms using rounding to odd" 5147:

Borman, Phil; Chatfield, Marion (2015-11-10). "Avoid the perils of using rounded data".

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By eliminating bias, repeated addition or subtraction of independent numbers, as in a

625:{\displaystyle {\bigl \{}{\tfrac {3}{12}},{\tfrac {4}{12}},{\tfrac {5}{12}}{\bigr \}}} 89: 6143: 5956: 5451: 5184: 5134:"Monte Carlo Arithmetic: a framework for the statistical analysis of roundoff errors" 5133: 4692: 4665: 4648: 4629: 4503: 4480:

indicated the practice was already "well established" in data analysis by the 1940s.

4465: 4303: 3454: 726: 169: 6050: 5242: 4831: 380:; i.e., rounding two numbers to the same absolute precision will not exchange their 5634: 5491: 5376: 5345: 5105: 4890: 4671: 4620:

Some disciplines or institutions have issued standards or directives for rounding.

3917: 342: 5880:. Mathematical Monographs. Vol. 7. New York: J. Wiley & Son. p. 42. 5722: 2243:

without bias toward/away from zero. By this convention, if the fractional part of

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using different methods. For clarity, the graphs are shown displaced from integer

6014:"Quantifying Quantitative Literacy: Age Heaping and the History of Human Capital" 3430:

depends on the magnitude of the number to be rounded (or of the rounded result).

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Some programming languages (such as Java and Python) use "half down" to refer to

384:(but may give the same value). In the general case of a discrete range, they are 4861: 4682: 4642: 4507: 4460:

called this "the computer's rule", indicating that it was then in common use by

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is used to represent the numbers. For display to humans, that usually means the

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This method only requires checking one digit to determine rounding direction in

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Calculations done with rounding should be close to those done without rounding.

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is the first even number greater than 4 which is not the sum of two primes, or

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is known to be a perfect square, its square root can be computed by converting

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Some programming languages (such as Java and Python) use "half up" to refer to

6113: 6032: 5459: 4713: 4705:, an application of rounding to integers that has been thoroughly investigated 4609: 4605: 4349: 4288: 4106: 3689: 2543: 1424:

is exactly half-way between two integers – that is, when the fraction part of

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Currently, much research tends to round to multiples of 5 or 2. For example,

4426:, the length of the year, and the length of the month are also ancient – see 4172:

coined the term "The Table-Maker's Dilemma" for the unknown cost of rounding

3394:{\displaystyle \mathrm {roundToMultiple} (x,m)=\mathrm {round} (x/m)\times m} 2356:

This rounding mode is used to avoid getting a potentially wrong result after

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called it a "universally adopted rule" for recording physical measurements.

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to the nearest integer requires some tie-breaking rule for those cases when

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among regions possible without any historical sources where the population

5770: 5652: 5626: 5368: 5233: 5216: 6133: 4601: 4519: 4515: 4384:

Rounding can adversely affect a string search for a number. For example,

4310: 4241: 4217:

floating-point standard guarantees that add, subtract, multiply, divide,

4214: 4142: 2527:, with equal probability. All others are rounded to the closest integer. 2340: 2286: 181: 5691: 5547: 4372:=1 if there is no such number. The rounded result is 2 if such a number 5753: 5717:. Mathematical Software – ICMS 2014. Vol. 8592. pp. 713–717. 5534: 4898: 4653: 4535: 4464:

who calculated mathematical tables. For example, it was recommended in

4332:

libc provides some correctly rounded functions in the 4 rounding modes.

4272: 4268: 3442: 3411:

dollars to whole cents (i.e., to a multiple of 0.01) entails computing

2472: 2468: 517: 495: 347: 215: 5325: 5011:

Microsoft Pascal Compiler for the MS-DOS Operating System User's Guide

3692:). If a calculation indicates a resistor of 165 ohms is required then 203:

only to within a few hundred units is usually better stated as "about

6085: 6041: 5860: 5757: 5669: 5665: 5551: 5443: 4877:

Nievergelt, Yves (2000). "Rounding Errors to Knock Your Stocks Off".

4574: 4326:

processors. Proved using the knowledge of the hardest-to-round cases.

4323: 3886: 2205:

For example, 23.5 gets rounded to 24, and −23.5 gets rounded to −24.

2011:

For example, 23.5 gets rounded to 23, and −23.5 gets rounded to −23.

1804:

For example, 23.5 gets rounded to 23, and −23.5 gets rounded to −24.

1619:

For example, 23.5 gets rounded to 24, and −23.5 gets rounded to −23.

1404:

For example, 23.2 gets rounded to 24, and −23.2 gets rounded to −24.

1198:

For example, 23.7 gets rounded to 23, and −23.7 gets rounded to −23.

964:

For example, 23.2 gets rounded to 24, and −23.7 gets rounded to −23.

854:

For example, 23.7 gets rounded to 23, and −23.2 gets rounded to −24.

334: 165: 149:

values. In the SVG file, hover over a method to highlight it and, in

5712: 4232:, many of the standard elementary functions can be proved to return 290: 5982:"The Introduction of the Euro and the Rounding of Currency Amounts" 5565:"libultim – ultimate correctly-rounded elementary-function library" 5275: 5060: 2371:

20.01, 20.1, 20.9, 20.99, 21, 21.01, 21.9, 21.99 are rounded to 21;

5477: 4276: 3791: 2345: 4906:"Ever had problems rounding off figures? This stock exchange has" 4489:

The Introduction of the Euro and the Rounding of Currency Amounts

671:

are all directed toward or away from the same limiting value (0,

246:. In a sequence of calculations, these rounding errors generally 210:

On the other hand, rounding of exact numbers will introduce some

4561:

floating-point standard, and define these functions as taking a

4442: 4329: 235: 4810:(July 1977). "Mathematical foundation of computer arithmetic". 3965:

is a popular error diffusion procedure when digitizing images.

3900:

is equally spaced on a logarithmic scale, choosing the closest

3742:. The value 165 rounds to 180 in the resistors example because 5684:

Sibidanov, Alexei; Zimmermann, Paul; Glondu, Stéphane (2022).

5132:

Parker, D. Stott; Eggert, Paul R.; Pierce, Brad (2000-03-28).

4570: 4284: 4252: 4158: 4138: 25: 5935:. Philadelphia: Jefferson Laboratory of Physics. p. 29. 5790: 5186:

Class Action Dilemmas: Pursuing Public Goals for Private Gain

4961:"decimal – Decimal fixed point and floating point arithmetic" 2715:

Stochastic rounding can be accurate in a way that a rounding

2338:, a non-infinite number would round to infinity, and a small 5492:"NA Digest Sunday, April 18, 1999 Volume 99 : Issue 16" 5836:

Industrial Standardization and Commercial Standards Monthly

4988:

Postcards 4 Language Booster: Workbook with Grammar Builder

4453:

are fairly self-explanatory. In the 1906 fourth edition of

4421: 2695: 2258:

This variant of the round-to-nearest method is also called

1387: 1181: 419: 4787:

Linear Algebra as an Introduction to Abstract Mathematics.

4556:

In the last decades, however, the syntax and the standard

3984:

occur in order and each is to be rounded to a multiple of

541:

Approximating a value by a multiple of a specified amount

5276:"JEP 306: Restore Always-Strict Floating-Point Semantics" 4643:

Signed zero § In rounded values such as temperatures

3835:

In some contexts it is desirable to round a given number

2289:

operations for results in binary floating-point formats.

1218:

is the integer that is closest to 0 (or equivalently, to

5737:"libmcr – correctly-rounded elementary-function library" 3794:(usually 2 or 10) of the floating-point representation. 2398:"). For example, when rounding to 1/4 (0.01 in binary), 5692:

29th IEEE Symposium on Computer Arithmetic (ARITH 2022)

5295:

Moore, J. Strother; Lynch, Tom; Kaufmann, Matt (1996).

4668:, dealing with the absence of extremely low-value coins 4420:

and square roots in base 60. Rounded approximations to

2327:

evens. It was the method used for bank balances in the

639:

Sum of rounded elements equals rounded sum of elements

4986:

Abbs, Brian; Barker, Chris; Freebairn, Ingrid (2003).

4240:

Some programming packages offer correct rounding. The

4093:

is not too big, the floating-point round-off error in

2546:

rounding and will give an unbiased result on average.

2176: 2116: 1982: 1922: 1763: 1738: 1701: 1578: 1553: 1516: 604: 589: 574: 3839:

to a "neat" fraction – that is, the nearest fraction

3513: 3285: 2555: 2239:, a tie-breaking rule without positive/negative bias 2074: 1880: 1682: 1497: 1239: 1015: 893: 772: 565: 5840:

The standard was also more concisely advertised in:

5882:An important fact with regard to the error 1/2 for 4352:, then the result of rounding the following value, 3415:, then rounding that to 218, and finally computing 2732:

Comparison of approaches for rounding to an integer

306:Ideal characteristics of rounding methods include: 56:. Unsourced material may be challenged and removed. 5849:. Vol. 84, no. 11. Nov 1940. p. 93. 5679: 5677: 5183: 4445:standard E-29 since 1940. The origin of the terms 4396:truncated to four digits, will discover values of 3968:As a one-dimensional example, suppose the numbers 3908:. Such rounded values can be directly calculated. 3801:section above, but with a constant scaling factor 3678: 3654: 3393: 2701: 2194: 2000: 1793: 1608: 1393: 1187: 953: 843: 624: 153:-enabled browsers, click to select or deselect it. 5714:Metalibm: A Mathematical Functions Code Generator 5649:"CRlibm – Correctly Rounded mathematical library" 5149:Journal of Pharmaceutical and Biomedical Analysis 4549:, or using a fractional number as an index of an 4198:and almost always well within one ulp. Why can't 2460:3.5 round-half-to-even to 1 ⇒ result is 4 (wrong) 2208:This can be more efficient on computers that use 1624: 1474:are always rounded up. If the fractional part of 679:and is often required in financial calculations. 416:Approximating an irrational number by a fraction 6128:An introduction to different rounding algorithms 5808:Duncan J. Melville. "YBC 7289 clay tablet". 2006 4920:"Vancouver stock index has right number at last" 3953:, are normally used. A related technique called 663:, as the displacements from the original number 5444:"Chapter 12: Solving the Table Maker's Dilemma" 4309:CRlibm, written in the old Arénaire team (LIP, 4180:Nobody knows how much it would cost to compute 4178: 3492:is very different from rounding to a specified 3272:to a multiple of some specified positive value 5136:. IEEE Computation in Science and Engineering. 4852:Accuracy and stability of numerical algorithms 3920:. A related mathematically formalized tool is 1809: 880:is the smallest integer that is not less than 262: 5838:. Vol. 11, no. 9. pp. 230–233. 5711:Kupriianova, Olga; Lauter, Christoph (2014). 5049: 5047: 617: 568: 8: 6065:"ECMA-262 ECMAScript Language Specification" 4656:scale, where below zero indicates freezing. 4097:will be less than 0.5, so the rounded value 3904:to any given value can be seen as a form of 2688: 2682: 2659: 2653: 2634: 2628: 2619: 2613: 2588: 2582: 1783: 1774: 1598: 1589: 759:is the largest integer that does not exceed 254:cases they may make the result meaningless. 184:312/937 with 1/3, or the expression √2 with 6134:How To Implement Custom Rounding Procedures 5030:Schedule 1 of the Decimal Currency Act 1969 4526:Rounding functions in programming languages 3673:This type of rounding, which is also named 3422:When rounding to a predetermined number of 5040:IBM z/Architecture Principles of Operation 4002:2.9423 = 0.9677 + 0.9204 + 0.7451 + 0.3091 3437:is normally a finite fraction in whatever 2377:24.0, 24.1, 24.9, 24.99 are rounded to 24; 2374:22.0, 22.1, 22.9, 22.99 are rounded to 22; 2285:This is the default rounding mode used in 6040: 5963:. New York: McGraw-Hill. pp. 187–223 5616: 5315: 5232: 5059: 4470:Logarithmic and Other Mathematical Tables 4194:mostly within slightly more than half an 4159:Rounding to prepare for shorter precision 4057:Exact computation with rounded arithmetic 4048:Monte Carlo arithmetic is a technique in 3736:is greater than or less than the product 3684:For example, resistors are supplied with 3599: 3575: 3574: 3514: 3512: 3374: 3351: 3286: 3284: 2675: 2670: 2593: 2574: 2554: 2395: 2352:Rounding to prepare for shorter precision 2175: 2115: 2073: 1981: 1921: 1879: 1762: 1737: 1700: 1681: 1577: 1552: 1515: 1496: 1328: 1238: 1122: 1014: 892: 779: 771: 616: 615: 603: 588: 573: 567: 566: 564: 452:Approximating a fraction by a fractional 116:Learn how and when to remove this message 5905:. New York: Henry Holt. pp. 14–15. 5531:"Accurate Portable Mathematical Library" 5442:; Stehlé, Damien; Torres, Serge (2010). 4955: 4953: 4073:, computing the approximate square root 3465: 2735: 398: 129: 5959:. In Eisenhart; Hastay; Wallis (eds.). 5820:Rules for Rounding Off Numerical Values 4770: 4725: 4628:In a guideline issued in mid-1966, the 4081:with floating point, and then rounding 3730:depends upon whether the squared value 2756:Round to prepare for shorter precision 2467:RPSP is implemented in hardware in IBM 675:, or −∞). Directed rounding is used in 498:by an integer with more trailing zeros 438:by a fraction with smaller denominator 16:Replacing a number with a simpler value 5957:"Effects of Rounding or Grouping Data" 4938: 4936: 4703:Party-list proportional representation 4514:, which enables the comparison of the 4416:contain tables with rounded values of 4300:Accurate portable mathematical library 3762:, rounding aims to turn a given value 268:Rounding has many similarities to the 242:representation with a fixed number of 6086:Federal Meteorological Handbook No. 1 5791:"Math Functions — The LLVM C Library" 5448:Handbook of Floating-Point Arithmetic 5397:"21718 – real.c rounding not perfect" 4974:Engineering Drafting Standards Manual 345:. A classical range is the integers, 259:transcendental mathematical functions 7: 5217:"When is double rounding innocuous?" 4600:interface as strings as if they had 4400:truncated to more than four digits. 4109:could be used for exact arithmetic. 4004:, are each rounded to a multiple of 3988:. In this case the cumulative sums, 3905: 3827:is always the appropriate infinity. 3798: 2388: 2357: 54:adding citations to reliable sources 3500:to a power of some positive number 3461:is an integer times a power of 2). 1815:round half toward negative infinity 1653:round half toward negative infinity 1630:round half toward positive infinity 1468:round half toward positive infinity 396:Typical rounding problems include: 4744:, and for distributing the total 3588: 3585: 3582: 3579: 3576: 3548: 3545: 3542: 3539: 3536: 3533: 3530: 3527: 3524: 3521: 3518: 3515: 3364: 3361: 3358: 3355: 3352: 3329: 3326: 3323: 3320: 3317: 3314: 3311: 3308: 3305: 3302: 3299: 3296: 3293: 3290: 3287: 2454:3.125 RPSP to 1/4 ⇒ result is 3.25 2437:⇒ result is 2.75 (10.11 in binary) 2418:⇒ result is 2.25 (10.01 in binary) 2331:when it decimalized its currency. 990:is the integer that is closest to 792: 789: 786: 783: 780: 14: 4789:World Scientific, Singapur 2016, 4101:will be the exact square root of 3998:2.6332 = 0.9677 + 0.9204 + 0.7451 3782:that depends on the magnitude of 3778:should be a multiple of a number 2479:Randomized rounding to an integer 2427:⇒ result is 2.5 (10.10 in binary) 1833:) as opposed to the conventional 5877:Probability and theory of errors 5419:"A Logarithm Too Clever by Half" 5215:Samuel A. Figueroa (July 1995). 5076:"Zener Diode Voltage Regulators" 4455:Probability and Theory of Errors 4380:Interaction with string searches 4357: 4132:Some computer languages and the 3260:Rounding to a specified multiple 2720:rounding. This can be useful in 2457:3.25 RPSP to 1/2 ⇒ result is 3.5 1655:) as opposed to the more common 729:applied to the original number, 30: 6160:Statistical data transformation 5994:from the original on 2010-10-09 5348:; Melquiond, Guillaume (2008). 5256:Roger Golliver (October 1998). 5088:from the original on 2011-07-13 5013:. Microsoft Corporation. 1985. 3675:rounding to a logarithmic scale 2446:⇒ result is 3 (11.00 in binary) 2408:⇒ result is 2 (10.00 in binary) 2348:(such as binary and decimal).. 2251:is the even integer nearest to 1414:rounding to the nearest integer 1408:Rounding to the nearest integer 1006:, without its fraction digits. 661:directed rounding to an integer 655:Directed rounding to an integer 372:As a general rule, rounding is 329:To be considered rounding, the 41:needs additional citations for 5940:record the nearest even number 5824:American Standards Association 5357:IEEE Transactions on Computers 5304:IEEE Transactions on Computers 4891:10.1080/0025570X.2000.11996800 4812:IEEE Transactions on Computers 3881:Rounding to an available value 3611: 3592: 3564: 3552: 3382: 3368: 3345: 3333: 3268:In general, rounding a number 2622: 2604: 2568: 2562: 2383:25.01, 25.1 are rounded to 26. 2319:is the odd integer nearest to 2150: 2144: 2093: 2087: 1956: 1950: 1899: 1893: 1298: 1292: 1258: 1252: 1092: 1086: 1052: 1046: 1034: 1028: 994:such that it is between 0 and 912: 906: 874:round toward positive infinity 802: 796: 753:round toward negative infinity 659:These four methods are called 516:Approximating a large decimal 276:must be encoded by numbers or 1: 5832:"Man's Love Of Round Numbers" 5723:10.1007/978-3-662-44199-2_106 4963:. Python Software Foundation. 4732:This is needed e.g. for the 4633:"round half away from zero". 4360:cannot be determined: either 4261:Mathematical Library for Java 4230:Lindemann–Weierstrass theorem 3937:Dithering and error diffusion 3865:. This problem is related to 3831:Rounding to a simple fraction 3679:rounding to a specified power 3484:Rounding to a specified power 3276:entails the following steps: 1831:round half away from infinity 1639:and similar representations. 1435:When rounding a large set of 1412:These six methods are called 310:Rounding should be done by a 5955:Churchill Eisenhart (1947). 5874:Woodward, Robert S. (1906). 4918:Lilley, Wayne (1983-11-29). 4856:(2nd ed.). p. 54. 4742:Mathematics of apportionment 4637:Negative zero in meteorology 2672: with probability 2595: with probability 2019:Rounding half away from zero 1837:. If the fractional part of 1659:. If the fractional part of 386:piecewise constant functions 6021:Journal of Economic History 5182:Deborah R. Hensler (2000). 4904:Quinn, Kevin (1983-11-08). 4862:10.1137/1.9780898718027.ch2 4748:of an invoice to its items) 4709:Signed-digit representation 3949:, and a related technique, 3770:with a specified number of 2294:one-dimensional random walk 2219:This method, also known as 474:Approximating a fractional 6183: 6095:, Washington, DC., 104 pp. 5932:The Theory of Measurements 5161:10.1016/j.jpba.2015.07.021 4736:, implemented e.g. by the 4640: 4105:. This is essentially why 4069:to a floating-point value 3932:Rounding in other contexts 2741: 2505:If the fractional part of 2029:round half toward infinity 18: 6033:10.1017/S0022050709001120 5830:Agnew, P. G. (Sep 1940). 5460:10.1007/978-0-8176-4705-6 4698:Kahan summation algorithm 4246:IEEE 754 double precision 4226:Gelfond–Schneider theorem 3963:Floyd–Steinberg dithering 3760:floating-point arithmetic 3197: 3194: 3191: 3177: 3174: 3171: 3168: 3154: 3151: 3148: 3145: 3142: 3139: 3136: 3110: 3107: 3104: 3090: 3087: 3084: 3081: 3067: 3064: 3061: 3058: 3055: 3052: 3049: 3023: 3020: 3017: 3003: 3000: 2997: 2994: 2980: 2977: 2974: 2971: 2968: 2965: 2962: 2936: 2933: 2930: 2904: 2898: 2895: 2889: 2883: 2880: 2877: 2874: 2764: 2761: 2758: 2755: 2752: 2749: 2744: 2738: 2025:round half away from zero 1835:round half away from zero 1821:Rounding half toward zero 1625:round half away from zero 478:by one with fewer digits 354:Rounding should preserve 263:the table-maker's dilemma 21:Rounding (disambiguation) 5583:"Git - glibc.git/commit" 4976:(NASA), X-673-64-1F, p90 4944:"java.math.RoundingMode" 4738:largest remainder method 4616:Other rounding standards 4293:double-double arithmetic 4192:transcendental functions 4174:transcendental functions 3994:1.8881 = 0.9677 + 0.9204 3774:digits. In other words, 3488:Rounding to a specified 3255:Rounding to other values 2484:Alternating tie-breaking 1445:expected (average) value 984:round away from infinity 717:For the examples below, 711:Vancouver Stock Exchange 494:Approximating a decimal 5929:Tuttle, Lucius (1916). 5899:Newcomb, Simon (1882). 5843:"Rounding Off Decimals" 5771:"The CORE-MATH project" 5513:"Math Library for Java" 5106:"Build a Mirror Tester" 4824:10.1109/TC.1977.1674893 4624:US weather observations 4483:The origin of the term 4472:. Lucius Tuttle's 1916 4458:Robert Simpson Woodward 4451:statistician's rounding 4164: 4085:to the nearest integer 3961:from the error so far. 3754:Floating-point rounding 2264:statistician's rounding 2216:due to its simplicity. 1202:Rounding away from zero 1002:is the integer part of 301:Alfred George Greenhill 6072:ecma-international.org 5605:RAIRO-Theor. Inf. Appl 4734:apportionment of seats 4563:double-precision float 4474:Theory of Measurements 4358:up to the next integer 4211: 4186:correctly rounded for 4044:Monte Carlo arithmetic 3955:pulse-width modulation 3808:, and an integer base 3656: 3449:is an integer times a 3443:decimal numeral system 3413:2.1784 / 0.01 = 217.84 3404:For example, rounding 3395: 2759:Alternating tie-break 2703: 2380:25.0 is rounded to 25; 2368:20.0 is rounded to 20; 2196: 2002: 1827:round half toward zero 1810:round half toward zero 1795: 1610: 1395: 1189: 955: 845: 626: 536:3 significant figures 511:3 significant figures 297:approximately equal to 224:mathematical functions 220:fixed-point arithmetic 154: 6165:Theory of computation 6136:by Microsoft (broken) 5687:The CORE-MATH Project 5415:Kahan, William Morton 5369:10.1109/TC.2007.70819 5234:10.1145/221332.221334 5221:ACM SIGNUM Newsletter 4990:. Pearson Education. 4846:Higham, Nicholas John 4678:Gal's accurate tables 4674:, a similar operation 4612:interface bindings). 4532:programming languages 4437:method has served as 4346:Goldbach's conjecture 4165:Table-maker's dilemma 3657: 3455:binary numeral system 3396: 2704: 2387:In the example from " 2231:Rounding half to even 2197: 2035:is exactly 0.5, then 2003: 1841:is exactly 0.5, then 1796: 1663:is exactly 0.5, then 1611: 1478:is exactly 0.5, then 1441:uniformly distributed 1396: 1212:round toward infinity 1190: 956: 846: 667:to the rounded value 627: 257:Accurate rounding of 133: 6012:Baten, Jörg (2009). 4879:Mathematics Magazine 4123:Martinez v. Allstate 3511: 3479:Logarithmic rounding 3283: 2553: 2303:Rounding half to odd 2072: 1878: 1680: 1495: 1416:. Rounding a number 1237: 1208:round away from zero 1013: 968:Rounding toward zero 891: 770: 563: 550:Multiple of 15 447:1-digit-denominator 429:1-digit-denominator 193:misleadingly precise 50:improve this article 19:For other uses, see 6155:Computer arithmetic 5826:. 1940. Z25.1-1940. 5627:10.1051/ita:2007003 4913:Wall Street Journal 4688:Interval arithmetic 4497:interval arithmetic 4478:Churchill Eisenhart 4344:. For instance, if 4259:, which stands for 4050:Monte Carlo methods 3875:continued fractions 3504:other than 1, are: 3466:rounding to integer 2745:Randomized methods 2742:Functional methods 2538:Stochastic rounding 2501:Random tie-breaking 2260:convergent rounding 2221:commercial rounding 2214:significant figures 677:interval arithmetic 646:Rounding to integer 522:scientific notation 411:Rounding criterion 274:physical quantities 199:but is known to be 6111:Weisstein, Eric W. 6091:1999-04-20 at the 4808:Kulisch, Ulrich W. 4779:Bruno Nachtergaele 4602:infinite precision 4435:round-half-to-even 4338:computable numbers 4219:fused multiply–add 4127:Sendejo v. Farmers 4038:0.31 = 2.94 − 2.63 4034:0.74 = 2.63 − 1.89 4030:0.92 = 1.89 − 0.97 3943:continuous signals 3922:signpost sequences 3790:is a power of the 3677:, is a variant of 3652: 3424:significant digits 3391: 2750:Directed rounding 2699: 2694: 2358:multiple roundings 2336:round half to even 2237:round half to even 2192: 2185: 2125: 1998: 1991: 1931: 1791: 1772: 1747: 1710: 1643:Rounding half down 1606: 1587: 1562: 1525: 1391: 1386: 1185: 1180: 951: 841: 622: 613: 598: 583: 244:significant digits 238:; or when using a 164:means replacing a 155: 5909:write the nearest 5469:978-0-8176-4704-9 5326:10.1109/12.713311 5274:Darcy, Joseph D. 5227:(3). ACM: 21–25. 5190:. RAND. pp. 5118:Electronic Design 4871:978-0-89871-521-7 4795:978-981-4730-35-8 4485:bankers' rounding 4447:unbiased rounding 4439:American Standard 3928:(percent) error. 3871:Stern–Brocot tree 3748:150 × 180 = 27000 3686:preferred numbers 3417:218 × 0.01 = 2.18 3252: 3251: 2831:Half Away From 0 2825: 2812: 2793: 2780: 2762:Random tie-break 2753:Round to nearest 2673: 2596: 2513:randomly between 2396:round half to odd 2309:round half to odd 2280:bankers' rounding 2276:odd–even rounding 2272:Gaussian rounding 2184: 2124: 2050:is positive, and 1990: 1930: 1856:is positive, and 1771: 1746: 1709: 1586: 1561: 1524: 1226:is between 0 and 998:(included); i.e. 974:round toward zero 643: 642: 612: 597: 582: 489:2 decimal places 402:Rounding problem 392:Types of rounding 272:that occurs when 250:, and in certain 222:; when computing 126: 125: 118: 100: 6172: 6124: 6123: 6096: 6082: 6076: 6075: 6069: 6061: 6055: 6054: 6044: 6018: 6009: 6003: 6002: 6000: 5999: 5993: 5986: 5978: 5972: 5971: 5969: 5968: 5952: 5946: 5945: 5926: 5920: 5919: 5896: 5890: 5889: 5885: 5871: 5865: 5864: 5861:10.1520/E0029-13 5850: 5839: 5827: 5816: 5810: 5805: 5799: 5798: 5787: 5781: 5780: 5778: 5777: 5767: 5761: 5751: 5745: 5744: 5739:. Archived from 5733: 5727: 5726: 5708: 5702: 5701: 5699: 5698: 5681: 5672: 5663: 5657: 5656: 5651:. Archived from 5645: 5639: 5638: 5620: 5600: 5594: 5593: 5591: 5590: 5585:. Sourceware.org 5579: 5573: 5572: 5567:. Archived from 5561: 5555: 5545: 5539: 5538: 5533:. Archived from 5527: 5521: 5520: 5515:. Archived from 5509: 5503: 5502: 5500: 5499: 5488: 5482: 5481: 5435: 5429: 5428: 5426: 5425: 5411: 5405: 5404: 5393: 5387: 5386: 5384: 5383: 5354: 5342: 5336: 5335: 5333: 5332: 5319: 5301: 5292: 5286: 5285: 5283: 5282: 5271: 5265: 5264: 5262: 5253: 5247: 5246: 5236: 5212: 5206: 5205: 5189: 5179: 5173: 5172: 5144: 5138: 5137: 5129: 5123: 5114: 5108: 5103: 5097: 5096: 5094: 5093: 5087: 5080: 5072: 5066: 5065: 5063: 5051: 5042: 5037: 5031: 5028: 5022: 5021: 5008: 4983: 4977: 4971: 4965: 4964: 4957: 4948: 4947: 4940: 4931: 4930: 4927:The Toronto Star 4924: 4916: 4910: 4902: 4875: 4855: 4842: 4836: 4835: 4804: 4798: 4777:Isaiah Lankham, 4775: 4758: 4755: 4749: 4730: 4468:'s c. 1882 book 4428:base 60 examples 4424: 4399: 4395: 4391: 4387: 4375: 4371: 4367: 4363: 4355: 4317:Sun Microsystems 4203: 4185: 4170:William M. Kahan 4104: 4100: 4096: 4092: 4088: 4084: 4080: 4076: 4072: 4068: 4064: 4039: 4035: 4031: 4027: 4023: 4019: 4015: 4011: 4007: 4003: 3999: 3995: 3991: 3987: 3983: 3979: 3975: 3971: 3941:When digitizing 3898:preferred values 3864: 3860: 3857:and denominator 3856: 3853:whose numerator 3852: 3838: 3825:round to nearest 3822: 3814: 3807: 3789: 3785: 3781: 3777: 3769: 3765: 3749: 3746:is greater than 3745: 3741: 3735: 3729: 3725: 3721: 3707:Whether a value 3703: 3702:log(180) = 2.255 3699: 3698:log(165) = 2.217 3695: 3694:log(150) = 2.176 3661: 3659: 3658: 3653: 3615: 3614: 3604: 3603: 3591: 3551: 3503: 3499: 3474: 3460: 3448: 3436: 3429: 3426:, the increment 3418: 3414: 3410: 3400: 3398: 3397: 3392: 3378: 3367: 3332: 3275: 3271: 2817: 2804: 2785: 2772: 2736: 2722:machine learning 2708: 2706: 2705: 2700: 2698: 2697: 2691: 2674: 2671: 2597: 2594: 2533: 2526: 2519: 2512: 2508: 2445: 2436: 2426: 2417: 2407: 2322: 2318: 2314: 2254: 2250: 2246: 2201: 2199: 2198: 2193: 2191: 2187: 2186: 2177: 2171: 2131: 2127: 2126: 2117: 2111: 2064: 2060: 2049: 2045: 2034: 2007: 2005: 2004: 1999: 1997: 1993: 1992: 1983: 1977: 1937: 1933: 1932: 1923: 1917: 1870: 1866: 1855: 1851: 1840: 1800: 1798: 1797: 1792: 1790: 1786: 1773: 1764: 1753: 1749: 1748: 1739: 1716: 1712: 1711: 1702: 1673: 1662: 1637:two's complement 1615: 1613: 1612: 1607: 1605: 1601: 1588: 1579: 1568: 1564: 1563: 1554: 1531: 1527: 1526: 1517: 1488: 1477: 1473: 1458:Rounding half up 1428:is exactly 0.5. 1427: 1423: 1419: 1400: 1398: 1397: 1392: 1390: 1389: 1372: 1346: 1324: 1320: 1319: 1279: 1275: 1229: 1225: 1221: 1217: 1194: 1192: 1191: 1186: 1184: 1183: 1166: 1140: 1118: 1114: 1113: 1073: 1069: 1005: 1001: 997: 993: 989: 960: 958: 957: 952: 950: 946: 928: 883: 879: 850: 848: 847: 842: 840: 836: 818: 795: 762: 758: 732: 724: 701: 697: 693: 689: 685: 670: 666: 633: 631: 629: 628: 623: 621: 620: 614: 605: 599: 590: 584: 575: 572: 571: 533: 528: 508: 503: 483: 463: 434:Approximating a 422: 399: 285:wavy equals sign 206: 198: 187: 179: 175: 148: 144: 140: 121: 114: 110: 107: 101: 99: 58: 34: 26: 6182: 6181: 6175: 6174: 6173: 6171: 6170: 6169: 6140: 6139: 6109: 6108: 6105: 6100: 6099: 6093:Wayback Machine 6083: 6079: 6067: 6063: 6062: 6058: 6016: 6011: 6010: 6006: 5997: 5995: 5991: 5984: 5980: 5979: 5975: 5966: 5964: 5954: 5953: 5949: 5928: 5927: 5923: 5898: 5897: 5893: 5883: 5873: 5872: 5868: 5855:. ASTM. 2013 . 5851: 5841: 5829: 5818: 5817: 5813: 5806: 5802: 5789: 5788: 5784: 5775: 5773: 5769: 5768: 5764: 5752: 5748: 5735: 5734: 5730: 5710: 5709: 5705: 5696: 5694: 5683: 5682: 5675: 5664: 5660: 5647: 5646: 5642: 5635:ensl-00000007v2 5618:10.1.1.106.6652 5602: 5601: 5597: 5588: 5586: 5581: 5580: 5576: 5563: 5562: 5558: 5546: 5542: 5529: 5528: 5524: 5511: 5510: 5506: 5497: 5495: 5490: 5489: 5485: 5470: 5440:Revol, Nathalie 5437: 5436: 5432: 5423: 5421: 5413: 5412: 5408: 5395: 5394: 5390: 5381: 5379: 5352: 5344: 5343: 5339: 5330: 5328: 5299: 5294: 5293: 5289: 5280: 5278: 5273: 5272: 5268: 5260: 5255: 5254: 5250: 5214: 5213: 5209: 5202: 5181: 5180: 5176: 5146: 5145: 5141: 5131: 5130: 5126: 5115: 5111: 5104: 5100: 5091: 5089: 5085: 5078: 5074: 5073: 5069: 5053: 5052: 5045: 5038: 5034: 5029: 5025: 5009: 5002: 4985: 4984: 4980: 4972: 4968: 4959: 4958: 4951: 4942: 4941: 4934: 4922: 4917: 4908: 4903: 4876: 4872: 4844: 4843: 4839: 4806: 4805: 4801: 4776: 4772: 4767: 4762: 4761: 4756: 4752: 4731: 4727: 4722: 4662: 4645: 4639: 4626: 4618: 4528: 4462:human computers 4422: 4406: 4397: 4393: 4389: 4385: 4382: 4373: 4369: 4365: 4361: 4353: 4342:halting problem 4199: 4181: 4167: 4155:rounding to odd 4151:sticky rounding 4115: 4113:Double rounding 4102: 4098: 4094: 4090: 4086: 4082: 4078: 4074: 4070: 4066: 4062: 4059: 4046: 4037: 4033: 4029: 4025: 4021: 4017: 4013: 4009: 4005: 4001: 3997: 3993: 3989: 3985: 3981: 3977: 3973: 3969: 3951:error diffusion 3939: 3934: 3914: 3906:scaled rounding 3902:preferred value 3883: 3867:Farey sequences 3862: 3858: 3854: 3840: 3836: 3833: 3820: 3809: 3802: 3799:Scaled rounding 3787: 3783: 3779: 3775: 3767: 3763: 3756: 3747: 3743: 3737: 3731: 3727: 3723: 3708: 3701: 3697: 3693: 3671: 3669:Scaled rounding 3595: 3570: 3509: 3508: 3501: 3497: 3486: 3481: 3469: 3458: 3446: 3434: 3427: 3416: 3412: 3405: 3281: 3280: 3273: 3269: 3262: 3257: 2823: 2822: 2816: 2810: 2809: 2803: 2791: 2790: 2784: 2778: 2777: 2771: 2734: 2693: 2692: 2668: 2650: 2649: 2591: 2575: 2551: 2550: 2540: 2531: 2521: 2514: 2510: 2509:is 0.5, choose 2506: 2503: 2486: 2481: 2440: 2430: 2421: 2411: 2402: 2389:Double rounding 2354: 2320: 2316: 2312: 2305: 2252: 2248: 2244: 2233: 2161: 2157: 2153: 2101: 2100: 2096: 2070: 2069: 2062: 2051: 2047: 2036: 2032: 2021: 1967: 1963: 1959: 1907: 1906: 1902: 1876: 1875: 1868: 1857: 1853: 1842: 1838: 1823: 1761: 1757: 1727: 1723: 1693: 1689: 1678: 1677: 1664: 1660: 1649:round half down 1645: 1576: 1572: 1542: 1538: 1508: 1504: 1493: 1492: 1479: 1475: 1471: 1460: 1425: 1421: 1417: 1410: 1385: 1384: 1373: 1362: 1359: 1358: 1347: 1336: 1329: 1309: 1305: 1301: 1265: 1261: 1235: 1234: 1227: 1223: 1219: 1215: 1204: 1179: 1178: 1167: 1156: 1153: 1152: 1141: 1130: 1123: 1103: 1099: 1095: 1059: 1055: 1011: 1010: 1003: 999: 995: 991: 987: 970: 939: 935: 918: 889: 888: 881: 877: 860: 829: 825: 808: 768: 767: 760: 756: 739: 730: 718: 699: 695: 694:is an integer, 691: 687: 683: 668: 664: 657: 648: 561: 560: 558: 531: 526: 506: 501: 481: 461: 436:rational number 420: 394: 294: 278:digital signals 252:ill-conditioned 212:round-off error 204: 196: 185: 177: 173: 146: 142: 138: 137:of the result, 122: 111: 105: 102: 59: 57: 47: 35: 24: 17: 12: 11: 5: 6180: 6179: 6176: 6168: 6167: 6162: 6157: 6152: 6142: 6141: 6138: 6137: 6131: 6125: 6104: 6103:External links 6101: 6098: 6097: 6077: 6056: 6027:(3): 783–808. 6004: 5973: 5947: 5921: 5891: 5866: 5811: 5800: 5782: 5762: 5746: 5743:on 2021-02-25. 5728: 5703: 5673: 5658: 5655:on 2016-10-27. 5640: 5595: 5574: 5571:on 2021-03-01. 5556: 5540: 5537:on 2005-02-07. 5522: 5519:on 1999-05-08. 5504: 5483: 5468: 5450:(1 ed.). 5430: 5406: 5388: 5363:(4): 462–471. 5337: 5317:10.1.1.43.3309 5287: 5266: 5248: 5207: 5200: 5174: 5139: 5124: 5120:, 2002-01-21. 5109: 5098: 5067: 5043: 5032: 5023: 5000: 4978: 4966: 4949: 4932: 4870: 4837: 4818:(7): 610–621. 4799: 4783:Anne Schilling 4769: 4768: 4766: 4763: 4760: 4759: 4750: 4724: 4723: 4721: 4718: 4717: 4716: 4711: 4706: 4700: 4695: 4690: 4685: 4680: 4675: 4669: 4661: 4658: 4649:meteorologists 4641:Main article: 4638: 4635: 4625: 4622: 4617: 4614: 4527: 4524: 4522:was measured. 4405: 4402: 4381: 4378: 4334: 4333: 4327: 4320: 4314: 4307: 4296: 4234:transcendental 4208: 4189: 4166: 4163: 4114: 4111: 4058: 4055: 4045: 4042: 3938: 3935: 3933: 3930: 3913: 3912:Arbitrary bins 3910: 3896:When a set of 3882: 3879: 3832: 3829: 3773: 3755: 3752: 3670: 3667: 3663: 3662: 3651: 3648: 3645: 3642: 3639: 3636: 3633: 3630: 3627: 3624: 3621: 3618: 3613: 3610: 3607: 3602: 3598: 3594: 3590: 3587: 3584: 3581: 3578: 3573: 3569: 3566: 3563: 3560: 3557: 3554: 3550: 3547: 3544: 3541: 3538: 3535: 3532: 3529: 3526: 3523: 3520: 3517: 3485: 3482: 3480: 3477: 3439:numeral system 3433:The increment 3402: 3401: 3390: 3387: 3384: 3381: 3377: 3373: 3370: 3366: 3363: 3360: 3357: 3354: 3350: 3347: 3344: 3341: 3338: 3335: 3331: 3328: 3325: 3322: 3319: 3316: 3313: 3310: 3307: 3304: 3301: 3298: 3295: 3292: 3289: 3261: 3258: 3256: 3253: 3250: 3249: 3246: 3243: 3240: 3237: 3234: 3231: 3228: 3225: 3222: 3218: 3217: 3214: 3211: 3208: 3205: 3202: 3199: 3196: 3193: 3190: 3186: 3185: 3182: 3179: 3176: 3173: 3170: 3167: 3163: 3162: 3159: 3156: 3153: 3150: 3147: 3144: 3141: 3138: 3135: 3131: 3130: 3127: 3124: 3121: 3118: 3115: 3112: 3109: 3106: 3103: 3099: 3098: 3095: 3092: 3089: 3086: 3083: 3080: 3076: 3075: 3072: 3069: 3066: 3063: 3060: 3057: 3054: 3051: 3048: 3044: 3043: 3040: 3037: 3034: 3031: 3028: 3025: 3022: 3019: 3016: 3012: 3011: 3008: 3005: 3002: 2999: 2996: 2993: 2989: 2988: 2985: 2982: 2979: 2976: 2973: 2970: 2967: 2964: 2961: 2957: 2956: 2953: 2950: 2947: 2944: 2941: 2938: 2935: 2932: 2929: 2925: 2924: 2921: 2918: 2915: 2912: 2909: 2906: 2903: 2900: 2897: 2894: 2891: 2888: 2885: 2882: 2879: 2876: 2873: 2869: 2868: 2863: 2858: 2853: 2848: 2843: 2838: 2835: 2832: 2829: 2828:Half Toward 0 2826: 2820: 2819: 2813: 2807: 2806: 2800: 2797: 2794: 2788: 2787: 2781: 2775: 2774: 2767: 2766: 2763: 2760: 2757: 2754: 2751: 2747: 2746: 2743: 2740: 2733: 2730: 2710: 2709: 2696: 2690: 2687: 2684: 2681: 2678: 2669: 2667: 2664: 2661: 2658: 2655: 2652: 2651: 2648: 2645: 2642: 2639: 2636: 2633: 2630: 2627: 2624: 2621: 2618: 2615: 2612: 2609: 2606: 2603: 2600: 2592: 2590: 2587: 2584: 2581: 2580: 2578: 2573: 2570: 2567: 2564: 2561: 2558: 2539: 2536: 2502: 2499: 2485: 2482: 2480: 2477: 2462: 2461: 2458: 2455: 2448: 2447: 2438: 2428: 2419: 2409: 2385: 2384: 2381: 2378: 2375: 2372: 2369: 2353: 2350: 2343: 2329:United Kingdom 2304: 2301: 2268:Dutch rounding 2232: 2229: 2210:sign-magnitude 2203: 2202: 2190: 2183: 2180: 2174: 2170: 2167: 2164: 2160: 2156: 2152: 2149: 2146: 2143: 2140: 2137: 2134: 2130: 2123: 2120: 2114: 2110: 2107: 2104: 2099: 2095: 2092: 2089: 2086: 2083: 2080: 2077: 2020: 2017: 2009: 2008: 1996: 1989: 1986: 1980: 1976: 1973: 1970: 1966: 1962: 1958: 1955: 1952: 1949: 1946: 1943: 1940: 1936: 1929: 1926: 1920: 1916: 1913: 1910: 1905: 1901: 1898: 1895: 1892: 1889: 1886: 1883: 1822: 1819: 1802: 1801: 1789: 1785: 1782: 1779: 1776: 1770: 1767: 1760: 1756: 1752: 1745: 1742: 1736: 1733: 1730: 1726: 1722: 1719: 1715: 1708: 1705: 1699: 1696: 1692: 1688: 1685: 1644: 1641: 1617: 1616: 1604: 1600: 1597: 1594: 1591: 1585: 1582: 1575: 1571: 1567: 1560: 1557: 1551: 1548: 1545: 1541: 1537: 1534: 1530: 1523: 1520: 1514: 1511: 1507: 1503: 1500: 1459: 1456: 1452:floating-point 1409: 1406: 1402: 1401: 1388: 1383: 1380: 1377: 1374: 1371: 1368: 1365: 1361: 1360: 1357: 1354: 1351: 1348: 1345: 1342: 1339: 1335: 1334: 1332: 1327: 1323: 1318: 1315: 1312: 1308: 1304: 1300: 1297: 1294: 1291: 1288: 1285: 1282: 1278: 1274: 1271: 1268: 1264: 1260: 1257: 1254: 1251: 1248: 1245: 1242: 1203: 1200: 1196: 1195: 1182: 1177: 1174: 1171: 1168: 1165: 1162: 1159: 1155: 1154: 1151: 1148: 1145: 1142: 1139: 1136: 1133: 1129: 1128: 1126: 1121: 1117: 1112: 1109: 1106: 1102: 1098: 1094: 1091: 1088: 1085: 1082: 1079: 1076: 1072: 1068: 1065: 1062: 1058: 1054: 1051: 1048: 1045: 1042: 1039: 1036: 1033: 1030: 1027: 1024: 1021: 1018: 969: 966: 962: 961: 949: 945: 942: 938: 934: 931: 927: 924: 921: 917: 914: 911: 908: 905: 902: 899: 896: 859: 856: 852: 851: 839: 835: 832: 828: 824: 821: 817: 814: 811: 807: 804: 801: 798: 794: 791: 788: 785: 782: 778: 775: 738: 735: 725:refers to the 709:set up by the 656: 653: 647: 644: 641: 640: 637: 634: 619: 611: 608: 602: 596: 593: 587: 581: 578: 570: 556: 552: 551: 548: 545: 542: 538: 537: 534: 529: 524: 513: 512: 509: 504: 499: 491: 490: 487: 484: 479: 476:decimal number 471: 470: 468:decimal places 464: 459: 456: 454:decimal number 449: 448: 445: 442: 439: 431: 430: 427: 424: 417: 413: 412: 409: 406: 405:Example input 403: 393: 390: 367: 366: 363: 352: 327: 326: 325: 315: 288: 240:floating-point 141:, of rounding 124: 123: 38: 36: 29: 15: 13: 10: 9: 6: 4: 3: 2: 6178: 6177: 6166: 6163: 6161: 6158: 6156: 6153: 6151: 6148: 6147: 6145: 6135: 6132: 6129: 6126: 6121: 6120: 6115: 6112: 6107: 6106: 6102: 6094: 6090: 6087: 6081: 6078: 6073: 6066: 6060: 6057: 6052: 6048: 6043: 6038: 6034: 6030: 6026: 6022: 6015: 6008: 6005: 5990: 5983: 5977: 5974: 5962: 5958: 5951: 5948: 5944: 5941: 5934: 5933: 5925: 5922: 5918: 5916: 5913: 5910: 5904: 5903: 5895: 5892: 5888: 5879: 5878: 5870: 5867: 5862: 5858: 5854: 5848: 5844: 5837: 5833: 5825: 5821: 5815: 5812: 5809: 5804: 5801: 5796: 5795:libc.llvm.org 5792: 5786: 5783: 5772: 5766: 5763: 5759: 5755: 5750: 5747: 5742: 5738: 5732: 5729: 5724: 5720: 5716: 5715: 5707: 5704: 5693: 5689: 5688: 5680: 5678: 5674: 5671: 5667: 5662: 5659: 5654: 5650: 5644: 5641: 5636: 5632: 5628: 5624: 5619: 5614: 5611:(1): 85–102. 5610: 5606: 5599: 5596: 5584: 5578: 5575: 5570: 5566: 5560: 5557: 5553: 5549: 5544: 5541: 5536: 5532: 5526: 5523: 5518: 5514: 5508: 5505: 5493: 5487: 5484: 5479: 5475: 5471: 5465: 5461: 5457: 5453: 5449: 5445: 5441: 5434: 5431: 5420: 5416: 5410: 5407: 5402: 5398: 5392: 5389: 5378: 5374: 5370: 5366: 5362: 5358: 5351: 5347: 5346:Boldo, Sylvie 5341: 5338: 5327: 5323: 5318: 5313: 5309: 5305: 5298: 5291: 5288: 5277: 5270: 5267: 5259: 5252: 5249: 5244: 5240: 5235: 5230: 5226: 5222: 5218: 5211: 5208: 5203: 5201:0-8330-2601-1 5197: 5193: 5188: 5187: 5178: 5175: 5170: 5166: 5162: 5158: 5154: 5150: 5143: 5140: 5135: 5128: 5125: 5122: 5119: 5113: 5110: 5107: 5102: 5099: 5084: 5077: 5071: 5068: 5062: 5057: 5050: 5048: 5044: 5041: 5036: 5033: 5027: 5024: 5020: 5016: 5012: 5007: 5003: 5001:0-13-093904-8 4997: 4993: 4989: 4982: 4979: 4975: 4970: 4967: 4962: 4956: 4954: 4950: 4945: 4939: 4937: 4933: 4928: 4921: 4914: 4907: 4900: 4896: 4892: 4888: 4884: 4880: 4873: 4867: 4863: 4859: 4854: 4853: 4847: 4841: 4838: 4833: 4829: 4825: 4821: 4817: 4813: 4809: 4803: 4800: 4796: 4792: 4788: 4784: 4780: 4774: 4771: 4764: 4754: 4751: 4747: 4743: 4739: 4735: 4729: 4726: 4719: 4715: 4712: 4710: 4707: 4704: 4701: 4699: 4696: 4694: 4693:ISO/IEC 80000 4691: 4689: 4686: 4684: 4681: 4679: 4676: 4673: 4670: 4667: 4666:Cash rounding 4664: 4663: 4659: 4657: 4655: 4650: 4644: 4636: 4634: 4631: 4623: 4621: 4615: 4613: 4611: 4607: 4603: 4599: 4596:and in their 4595: 4591: 4587: 4584:In contrast, 4582: 4580: 4576: 4572: 4568: 4564: 4559: 4554: 4552: 4548: 4545: 4541: 4537: 4533: 4525: 4523: 4521: 4517: 4513: 4509: 4505: 4500: 4498: 4492: 4490: 4486: 4481: 4479: 4475: 4471: 4467: 4466:Simon Newcomb 4463: 4459: 4456: 4452: 4448: 4444: 4440: 4436: 4431: 4429: 4425: 4419: 4415: 4411: 4403: 4401: 4379: 4377: 4359: 4351: 4347: 4343: 4339: 4331: 4328: 4325: 4321: 4318: 4315: 4312: 4308: 4305: 4304:GNU C Library 4301: 4297: 4294: 4290: 4286: 4282: 4278: 4274: 4270: 4266: 4263:, written by 4262: 4258: 4254: 4251: 4250: 4249: 4247: 4243: 4238: 4235: 4231: 4227: 4222: 4220: 4216: 4210: 4206: 4202: 4197: 4193: 4187: 4184: 4177: 4175: 4171: 4162: 4160: 4156: 4152: 4146: 4144: 4140: 4135: 4134:IEEE 754-2008 4130: 4128: 4124: 4119: 4112: 4110: 4108: 4056: 4054: 4051: 4043: 4041: 3966: 3964: 3958: 3956: 3952: 3948: 3944: 3936: 3931: 3929: 3927: 3923: 3919: 3911: 3909: 3907: 3903: 3899: 3894: 3890: 3888: 3880: 3878: 3876: 3872: 3868: 3851: 3847: 3843: 3830: 3828: 3826: 3816: 3812: 3805: 3800: 3795: 3793: 3786:. 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