the-french-artist/wikipedia_20220301.simple_sentence_split huggingface dataset:
And the second time was from November 9, 1846 to December 8 of that year.
List of Latin American subnational entities by HDI
List of Mexican state demonyms
List of Mexican state governors
List of Mexican states by area
List of Mexican states by HDI
List of Mexican states by population
Mexican state name etymologies
Ranked list of Mexican states
State governments of Mexico
Territorial evolution of Mexico
Political Constitution of the United Mexican States; articles 2, and 42 through 48
Law of Linguistic Rights or "Ley de los Derechos Lingüísticos" approved in 2001.juihu b
Dennis Seidenberg (born July 18, 1981 in Villingen-Schwenningen, Germany) is an NHL ice hockey defenceman that currently plays for the Boston Bruins.
He was drafted by the Philadelphia Flyers with the 172nd overall pick in the 1999 NHL Entry Draft.
Seidenberg played hockey formerly with the Philadelphia Flyers for 3 seasons, the Phoenix Coyotes for 2 seasons, the Carolina Hurricanes for 3 seasons and the Florida Panthers for 1 season.
He played in the NHL YoungStars Game in 2002.
Stanley Cup with the Boston Bruins in 2011.
Christopher Kelly (born November 11, 1980) is a Canadian former professional ice hockey player and current assistant coach for the Boston Bruins of the NHL.
He was drafted by the Ottawa Senators with the 94th overall pick in the 1999 NHL Entry Draft.
Kelly played in the OHL for 4 seasons with the London Knights and 1 season with the Sudbury Wolves.
He also played hockey formerly with the Ottawa Senators for 7 seasons and for one season with the Anahiem Ducks.
He won the Stanley Cup with the Bruins on June 15, 2011.
Megaloptera is an order of insects.
It contains the alderflies, dobsonflies and fishflies.
There are about 300 known species.
Michael Wayne Glen Ryder (born March 31, 1980 in Bonavista, Newfoundland and Labrador) is a Canadian professional ice hockey right winger.
He played from 1996 to 2015.
He was drafted by the Montreal Canadiens with the 216th overall pick in the 1998 NHL Entry Draft.
Ryder played in the QMJHL for 3 seasons with the Hull Olympiques and 2 seasons in the ECHL with the Tallahassee Tiger Sharks and Mississippi Sea Wolves.
He also played hockey formerly with the Montreal Canadiens for 4 seasons.
Competitive eating is a term for a sport where people play to see who can eat the most food in a given time.
The sport is most commonly played in the United States and Japan.
There are many different foods that can be eaten competitively.
Common ones include hot dogs, hamburgers, hot wings, and pies.
The most famous competitive eating contest is Nathan's Hot Dog Eating Contest on Coney Island.
It and other events are part of a tour known as Major League Eating.
There are often competitive eating contests at American fairs and carnivals.
The precision of a numeric value describes the number of digits that are used to show that value.
In a scientific setting, this would be the total number of digits (sometimes called the significant figures or significant digits) or, less commonly, the number of fractional digits or decimal places (the number of digits following the decimal point).
This second definition is useful in financial and engineering applications where the count of digits in the fractional part has particular importance.
In both cases, the term "precision" can be used to describe the position at which an inexact result will be rounded.
For example, in floating point arithmetic, a result is rounded to a given or fixed precision, which is the length of the resulting significand.
In financial calculations, a number is often rounded to a given number of places (for example, to two places after the decimal separator for many world currencies).
As an example, the decimal quantity 12.345 can be expressed with various numbers of significant digits or decimal places.
If insufficient precision is available then the number is rounded in some manner to fit the available precision.
The following table shows the results for various total precisions and decimal places rounded to the nearest value using the round-to-even method.
Note that it is often not appropriate to display a figure with more digits than that which can be measured.
For instance, if a device measures to the nearest gram and gives a reading of 12.345 kg, it would create false precision if the measurement were expressed "12.34500 kg" with 2 extra zeroes ("00") at the end.
The representation of a positive number x to a precision of p significant digits has a numerical value that is given by the formula
round(10−n·x)·10n, where n = floor(log10 x) + 1 − p.
Future Shop was a Canadian electronic retailer.
It was headquartered in Burnaby, British Columbia.
The largest Future Shop in Canada was in Edmonton, Alberta.
It was purchased on November 4, 2001 for $580 million by Best Buy.
It was founded in 1982 by Hassan Khosrowshahi, an Iranian entrepreneur who settled in Vancouver, British Columbia after leaving Iran.
On March 28, 2015, it was announced by Best Buy that they had decided to retire the Future Shop brand and close 66 locations.
The former Future Shop locations were re-opened as Best Buy stores.
1982 establishments in Canada
Shoppers Drug Mart Corporation (TSX: SC ) is Canada's largest pharmacy chain.
Shoppers Drug Mart is headquartered in the Toronto, Ontario.
The chain is founded by Murray Koffler in 1962.
The store is also called Pharmaprix in Quebec and Hawkesbury, Ontario.
Shoppers Drug Mart's Official website
Bianca di Angelo is a character in the fictional Percy Jackson and the Olympians book series, written by Rick Riordan.
Bianca is a half-blood, meaning one parent was a god, and the other was a mortal.
Her father is Hades, the god of the Underworld, and her mother is an Italian mortal, named Maria Di Angelo.
In "Percy Jackson and the Titan's Curse", she dies because she sacrifices herself after trying to rescue her friends.
Her brother, Nico di Angelo, believes it was Percy Jackson's fault that his sister died, and plans to take revenge on Percy in the fourth book of the series, Percy Jackson and the battle of the Labyrinth.
Document Records (founded 1986 by Johnny Parth in Austria) is a now British record label.
2000 Parth sold the label to Gary Atkinson.
Originally based in Vienna the label is now in Newton Stewart, Scotland.
The label publishes early American blues, bluegrass, gospel, spirituals jazz, and other rural American genres (this genres are also called "roots" music)and made between 1900 and 1945.
The label is also owner of the rights for unrealesed Music made by the Edison Company between 1914 and 1929.
Artists (selection)
Francis "Scrapper" Blackwell
Victoria Spivey
Document Records Homepage
The word "rounding" for a numerical value means replacing it by another value that is approximately equal but has a shorter, simpler, or more explicit form.
For example, US$23.74 could be rounded to US$24, the fraction 312/937 could be rounded to 1/3, and the expression  could be rounded to 1.41.
Rounding is often done on purpose to obtain a value that is easier to write and handle than the original.
It may also be done to indicate the accuracy of a computed number; for example, a quantity that was computed as 123,456, but is known to be accurate only to within a few hundred units, is better stated as "about 123,500".
On the other hand, rounding can introduce some round-off error as a result.
Rounding is almost unavoidable in many computations, especially when dividing two numbers in integer or doing fixed-point arithmetic; when computing mathematical functions such as square roots, logarithms, and sines; or when using a floating point representation with a fixed number of significant digits.
In a sequence of calculations, these rounding errors generally accumulate, and in certain "ill-conditioned" cases, they may make the result meaningless.
Accurate rounding of transcendental mathematical functions 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 "the table-maker's dilemma" (below).
Types of rounding
Approximating an irrational number by a fraction.
For example, π by 22/7.
Approximating a fraction with periodic decimal expansion by a finite decimal fraction.
For example, 5/3 by 1.6667.
Replacing a rational number by a fraction with smaller numerator and denominator.
For example, 3122/9417 by 1/3.
Replacing a fractional decimal number by one with fewer digits.
For example, 2.1784 dollars by 2.18 dollars.
Replacing a decimal integer by an integer with more trailing zeros.
23,217 people by 23,200 people.
Replacing a value by a multiple of a specified amount.
27.2 seconds by 30 seconds (a multiple of 15).
Rounding to a specified increment
In general, rounding a number x to a multiple of some specified increment m entails the following steps:
Divide x by m, let the result be y;
Round y to an integer value, call it q;
Multiply q by m to obtain the rounded value z.
For example, rounding x = 2.1784 dollars to whole cents (that is, to a multiple of 0.01) entails computing y = x/m = 2.1784/0.01 = 217.84, then rounding y to the integer q = 218, and finally computing z = q×m = 218×0.01 = 2.18.
When rounding to a predetermined number of significant digits, the increment m depends on the magnitude of the number to be rounded (or of the rounded result).
The increment m is normally a finite fraction in whatever numeral system that is used to represent the numbers.
For display to humans, that usually means the decimal numeral system (that is, m is an integer times a power of 10, like 1/1000 or 25/100).
For intermediate values stored in digital computers, it often means the binary numeral system (m is an integer times a power of 2).
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 rounding to integer section.
The abstract two-argument "round()" function is formally defined here, but in many cases it is used with the implicit value m = 1 for the increment and then reduces to the equivalent abstract single-argument function, with also the same dozen distinct concrete definitions.
Rounding to integer
The most basic form of rounding is to replace an arbitrary number by an integer.
All the following rounding modes are concrete implementations of the abstract single-argument "round()" function presented and used in the previous sections.
There are many ways of rounding a number y to an integer q.  The most common ones are
Round down (or take the floor, or round towards minus infinity):  q is the largest integer that does not exceed y.
Round up (or take the ceiling,  or round towards plus infinity):  q is the smallest integer that is not less than y.
Round towards zero (or truncate, or round away from infinity): q is the integer part of y, without its fraction digits.
Round away from zero (or round towards infinity): if y is an integer, q is y; else q is the integer that is closest to 0 and is such that y is between 0 and q.
Round to nearest: q is the integer that is closest to y. This is sometimes written as  (see below for tie-breaking rules).
The first four methods are called directed rounding, as the displacements from the original number y to the rounded value q are all directed towards or away from the same limiting value (0, +∞, or −∞).
If y is positive, round-down is the same as round-towards-zero, and round-up is the same as round-away-from-zero.
If y is negative, round-down is the same as round-away-from-zero, and round-up is the same as round-towards-zero.
In any case, if y is integer, q is just y. The following table illustrates these rounding methods:
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 index set up by the Vancouver Stock Exchange in 1982.
It was initially set at 1000.000, and after 22 months had fallen to about 520 — whereas stock prices had generally increased in the period.
The problem was caused by the index being recalculated thousands of times daily, and always being rounded down to 3 decimal places, in such a way that the rounding errors accumulated.
Recalculating with better rounding gave an index value of 1098.892 at the end of the same period.
Tie-breaking
Rounding a number y to the nearest integer requires some tie-breaking rule for those cases when y is exactly half-way between two integers — that is, when the fraction part of y is exactly 0.5.
Round half up
The following  tie-breaking rule, called round half up (or round half towards plus infinity), is widely used in many disciplines.
That is, half-way values y are always rounded up.
If the fraction of y is exactly 0.5, then q = y + 0.5.
If it were not for the 0.5 fractions, the roundoff errors introduced by the round to nearest method would be quite symmetric: for every fraction that gets rounded up (such as 0.268), there is a complementary fraction (namely, 0.732) that gets rounded down, by the same amount.
When rounding a large set of numbers with random fractional parts, these rounding errors would statistically compensate each other, and the expected (average) value of the rounded numbers would be equal to the expected value of the original numbers.
However, the round half up tie-breaking rule is not symmetric, as the fractions that are exactly 0.5 always get rounded up.
This asymmetry introduces a positive bias in the roundoff errors.
For example, if the fraction of y consists of three random decimal digits, then the expected value of q will be 0.0005 higher than the expected value of y. For this reason, round-to-nearest with the round half up rule is also (ambiguously) known as asymmetric rounding.
One reason for rounding up at 0.5 is that only one digit need be examined.
When seeing 17.50000..., for example, the first three figures, 17.5, determines that the figure would be rounded up to 18.
If the opposite rule were used (round half down), then all the zero decimal places would need to be examined to determine if the value is exactly 17.5.
One may also use round half down (or round half towards minus infinity) as opposed to the more common round half up (the round half up method is a common convention, but is nothing more than a convention).
If the fraction of y is exactly 0.5, then q = y − 0.5.
The round half down tie-breaking rule is not symmetric, as the fractions that are exactly 0.5 always get rounded down.
This asymmetry introduces a negative bias in the roundoff errors.
For example, if the fraction of y consists of three random decimal digits, then the expected value of q will be 0.0005 lower than the expected value of y. For this reason, round-to-nearest with the round half down rule is also (ambiguously) known as asymmetric rounding.
Round half away from zero
If the fraction of y is exactly 0.5, then q = y + 0.5 if y is positive, and q = y − 0.5 if y is negative.
This method treats positive and negative values symmetrically, and therefore is free of overall bias if the original numbers are positive or negative with equal probability.
However, this rule will still introduce a positive bias for positive numbers, and a negative bias for the negative ones.
Round half towards zero
One may also round half towards zero (or round half away from infinity) as opposed to the more common round half away from zero (the round half away from zero method is a common convention, but is nothing more than a convention).
If the fraction of y is exactly 0.5, then q = y − 0.5 if y is positive, and q = y + 0.5 if y is negative.
For example, 23.5 gets rounded to 23, and −23.5  gets rounded to −23.
This method also treats positive and negative values symmetrically, and therefore is free of overall bias if the original numbers are positive or negative with equal probability.
However, this rule will still introduce a negative bias for positive numbers, and a positive bias for the negative ones.
Round half to even
If the fraction of y is 0.5, then q is the even integer nearest to y.
In addition, for most reasonable distributions of y values, the expected (average) value of the rounded numbers is essentially the same as that of the original numbers, even if the latter are all positive (or all negative).
However, this rule will still introduce a positive bias for even numbers (including zero), and a negative bias for the odd ones.
This variant of the round-to-nearest method is also called unbiased rounding (ambiguously, and a bit abusively), convergent rounding, statistician's rounding, Dutch rounding, Gaussian rounding, or bankers' rounding.
This is widely used in bookkeeping.
Round half to odd
If the fraction of y is 0.5, then q is the odd integer nearest to y.
However, this rule will still introduce a negative bias for even numbers (including zero), and a positive bias for the odd ones.
This variant is almost never used in most computations, except in situations where one wants to avoid rounding 0.5 or −0.5 to zero, or to avoid increasing the scale of numbers represented as floating point (with limited ranges for the scaling exponent), so that a non infinite number would round to infinite, or that a small denormal value would round to a normal non-zero value (these could occur with the round half to even mode).
Effectively, this mode prefers preserving the existing scale of tie numbers, avoiding out of range results when possible.
Stochastic rounding
If the fractional part of y is .5, choose q randomly among y + 0.5 and y − 0.5, with equal probability.
Like round-half-to-even, this rule is essentially free of overall bias; but it is also fair among even and odd q values.
On the other hand, it introduces a random component into the result; performing the same computation twice on the same data may yield two different results.
Also, it is open to unconscious bias if humans (rather than computers or devices of chance) are "randomly" deciding in which direction to round.
Alternating tie-breaking
If the fractional part is 0.5, alternate round up and round down: for the first occurrence of a 0.5 fractional part, round up; for the second occurrence, round down; so on so forth.
This suppresses the random component of the result, if occurrences of 0.5 fractional parts can be effectively numbered.
But it can still introduce a positive or negative bias according to the direction of rounding assigned to the first occurrence, if the total number of occurrences is odd.
Simple dithering
In some contexts, all the rounding methods above may be unsatisfactory.
For example, suppose that y is an accurate measurement of an audio signal, which is being rounded to an integer q in order to reduce the storage or transmission costs.
If y changes slowly with time, any of the rounding method above will result in q being completely constant for long intervals, separated by sudden jumps of ±1.
When the q signal is played back, these steps will be heard as a very disagreeable noise, and any variations of the original signal between two integer values will be completely lost.
One way to avoid this problem is to round each value y upwards with probability equal to its fraction, and round it downwards with the complement of that probability.
For example, the number 23.17 would be rounded up to 24 with probability 0.17, and down to 23 with probability 1 - 0.17 = 0.83. (
This is equivalent to rounding down y + s, where s is a random number uniformly distributed between 0 and 1.)
With this special rounding, known as dithering, the sudden steps get replaced by a less objectionable noise, and even small variations in the original signal will be preserved to some extent.
Like the stochastic approach to tie-breaking, dithering has no bias: if all fraction values are equally likely, rounding up by a certain amount is as likely as rounding down by that same amount; and the same is true for the sum of several rounded numbers.
On the other hand, dithering introduces a random component in the result, much greater than that of stochastic tie-breaking.
More precisely, the roundoff error for each dithered number will be a uniformly distributed random variable with mean value of zero, but with a standard deviation , which is better than the 1/2 standard deviation with the simple predictive methods, but slightly higher than with the simpler stochastic method.
However, the sum of n rounded numbers will be a random variable with expected error zero, but with standard deviation  (the total remaining noise) which diverges semi-quadratically and may become easily perceptible, even if the standard deviation of the roundoff error per sample will be  which slowly converges semi-quadratically to zero.
So, this random distribution may still be too high for some applications that are rounding a lot of data.
Multidimensional dithering
This variant of the simple dithering method still rounds values with probability equal to its fraction.
However, instead of using a random distribution for rounding isolated samples, the roundoff error occurring at each rounded sample is totalled for the next surrounding elements to sample or compute; this accumulated value is then added to the value of these next sampled or computed values to round, so that the modified values will take into account this difference using a predictive model (such as Floyd–Steinberg dithering).
The modified values are then rounded with any one of the above rounding methods, the best ones being with stochastic or dithering methods: in this last case, the sum of n rounded numbers will still be a random variable with expected error zero but with an excellent constant standard deviation of , instead of diverging semi-quadratically when dithering isolated samples; and the overall average roundoff error deviation per rounded sample will be  that will converge hyperbolically to zero, faster than with the semi-hyperbolic convergence when dithering isolated samples.
In practice, when rounding large sets of sampled data (such as audio, image and video rendering), the accumulation of roundoff errors is most frequently used with a simple predictive rounding of the modified values (such as rounding towards zero), because it will still preserve the hyperbolic convergence towards zero of the overall mean roundoff error bias and of its standard deviation.
This enhancement is frequently used in image and audio processing (notably for accurate rescaling and antialiasing operations, where the simple probabilistic dithering of isolated values may still produce perceptible noise, sometimes even worse than the moiré effects occurring with simple non-probabilistic rounding methods applied to isolated samples).
The effective propagation of accumulated roundoff errors may depend on the discrete dimension of the sampled data to round: when sampling bidimensional images, including colored images (that add the discrete dimension of color planes), or tridimensional videos (that add a discrete time dimension), or on polyphonic audio data (using time and channel discrete dimensions), it may still be preferable to propagate this error into a preferred direction, or equally into several orthogonal dimensions, such as vertically vs. horizontally for bidimensional images, or into parallel color channels at the same position and/or timestamp, and depending on other properties of these orthogonal discrete dimensions (according to a perception model).
In those cases, several roundoff error accumulators may be used (at least one for each discrete dimension), or a (n-1)-dimension vector (or matrix) of accumulators.
Rounding to simple fractions
In some contexts it is desirable to round a given number x to a "neat" fraction — that is, the nearest fraction z = m/n
whose numerator m and denominator n 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 m. This problem is related to Farey sequences, the Stern-Brocot tree, and continued fractions.
Scaled rounding
This type of rounding, which is also named rounding to a logarithmic scale, is a variant of Rounding to a specified increment but with an increment that is modified depending on the scale and magnitude of the result.
Concretely, the intent is to limit the number of significant digits, rounding the value so that non-significant digits will be dropped.
This type of rounding occurs implicitly with numbers computed with floating-point values with limited precision (such as IEEE-754 float and double types), but it may be used more generally to round any real values with any positive number of significant digits and any strictly positive real base.
This type of rounding is based on a logarithmic scale defined by a fixed non-zero real scaling factor s (in most frequent cases this factor is s=1) and a fixed positive base b>1 (not necessarily an integer and most often different from the scaling factor), and a fixed integer number n>0 of significant digits in that base (which will determine the value of the increment to use for rounding, along with the computed effective scale of the rounded number).
The primary argument number (as well as the resulting rounded number) is first represented in exponential notation x = s·a·m·bc, such that the sign s is either +1 or −1, the absolute mantissa a is restricted to the half-open positive interval [1/b,1), and the exponent c is any (positive or negative) integer.
In that representation, all significant digits are in the fractional part of the absolute mantissa whose integer part is always zero.
If the source number (or rounded number) is 0, the absolute mantisssa a is defined as 0, the exponent c is fixed to an arbitrary value (0 in most conventions, but some floating-point representations cannot use a null absolute mantissa but reserve a specific maximum negative value for the exponent c to represent the number 0 itself), and the sign s may be arbitrarily chosen between −1 or +1 (it is generally set to +1 for simple zero, or it is set to the same sign as the argument in the rounded value if the number representation allows to differentiate positive and negative zeroes, even if they finally represent the same numeric value 0).
A scaled exponential representation as x = a·s·bc may also be used equivalently, with a signed mantissa a either equal to zero or within one of the two half-open intervals (−1,−1/b] and [+1/b,+1), and this will be the case in the algorithm below.
if x equals zero, simply return x; otherwise:
convert x into the scaled exponential representation, with a signed mantissa:
let x’ be the unscaled value of x, by dividing it by the scaling factor s: ;
let the scaling exponent c be one plus the base-b logarithm of the absolute value of x’, rounded down to an integer (towards minus infinity): ;
let the signed mantissa a be product of x’ divided by b to the power c:
compute the rounded value in this representation:
let c’ be the initial scaling exponent c of x’:
let m be the increment for rounding the mantissa a according to the number of significant digits to keep:
let a’ be the signed mantissa a rounded according to this increment m and the selected rounding mode:
if the absolute value of a’ is not lower than b, then decrement n (multiply the increment m by b), increment the scaling exponent c’, divide the signed mantissa a by b, and restart the rounding of the new signed mantissa a into a’ with the same formula; this step may be avoided only if the abtract "round()" function is always rounding a towards 0 (i.e. when it is a simple truncation), but is necessary if it may be rounding a towards infinity, because the rounded mantissa may have a higher scaling exponent in this case, leaving an extra digit of precision.
return the rounded value: .
For the abstract "round()" function, this type of rounding can use any one of the rounding to integer modes described more completely in the next section, but it is most frequently the round to nearest mode (with tie-breaking rules also described more completely below).
the scaled rounding of 1.234 with scaling factor 1 in base 10 and 3 significant digits (maximum relative precision=1/1000), when using any round to nearest mode, will return 1.23;
similar scaled rounding of 1.236 will return 1.24;
similar scaled rounding of 21.236 will return 21.2;
similar scaled rounding of 321.236 will return 321;
the scaled rounding of 1.234 scaling factor 1 in base 10 and 3 significant digits (maximum relative precision=1/1000), when using the round down mode, will return 1.23;
similar scaled rounding of 1.236 will also return 1.23;
the scaled rounding of  with scaling factor  in base 2 and 3 significant digits (maximum relative precision=1/8), when using the round down mode, will return ;
similar scaled rounding of  will return ;
similar scaled rounding of  will return .
similar scaled rounding of  will also return .
Round to available value
Floating-point rounding
In floating-point arithmetic, rounding aims to turn a given value x into a value z with a specified number of significant digits.
In other words, z should be a multiple of a number m that depends on the magnitude of z. The number m is a power of the base (usually 2 or 10) of the floating-point form.
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 Scaled rounding section above, but with a constant scaling factor s=1, and an integer base b>1.
For results where the rounded result would overflow the result for a directed rounding is either the appropriate signed infinity, or the highest representable positive finite number (or the lowest representable negative finite number if x is negative), depending on the direction of rounding.
The result of an overflow for the usual case of round to even is always the appropriate infinity.
In addition, if the rounded result would underflow, i.e. if the exponent would exceed the lowest representable integer value, the effective result may be either zero (possibly signed if the representation can maintain a distinction of signs for zeroes), or the smallest representable positive finite number (or the highest representable negative finite number if x is negative), possibly a denormal positive or negative number (if the mantissa is storing all its significant digits, in which case the most significant digit may still be stored in a lower position by setting the highest stored digits to zero, and this stored mantissa does not drop the most significant digit, something that is possible when base b=2 because the most significant digit is always 1 in that base), depending on the direction of rounding.
The result of an underflow for the usual case of round to even is always the appropriate zero.
Double rounding
Rounding a number twice in succession to different precisions, 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 when rounding to integer using rounding half to even, but would give 9 when rounded to integer directly.
Some computer languages and the IEEE 754-2008 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 x87 floating point.
Exact computation with rounded arithmetic
It is possible to use rounded arithmetic to evaluate the exact value of a function with a discrete domain and range.
For example, if we know that an integer n is a perfect square, we can compute its square root by converting n to a floating-point value x, computing the approximate square root y of x with floating point, and then rounding y to the nearest integer q. If n is not too big, the floating-point roundoff error in y will be less than 0.5, so the rounded value q will be the exact square root of n. In most modern computers, this method may be much faster than computing the square root of n by an all-integer algorithm.
The table-maker's dilemma
"Nobody knows how much it would cost to compute  y^w  correctly rounded for every two floating-point arguments at which it does not over/underflow.
Instead, reputable math libraries compute elementary transcendental functions mostly within slightly more than half an ulp and almost always well within one ulp.
Why can't Y^W 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 correctly 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."
The IEEE floating point standard guarantees that add, subtract, multiply, divide, square root, and floating point remainder will give the correctly rounded result of the infinite precision operation.
However, no such guarantee is given for more complex functions and they are typically only accurate to within the last bit at best.
Using the Gelfond–Schneider theorem and Lindemann–Weierstrass theorem, many of the standard elementary functions can be proved to return transcendental results when given rational non-zero arguments; therefore it is always possible to correctly round such functions.
However determining a limit for a given precision on how accurate results needs to be computed before a correctly rounded result can be guaranteed may demand a lot of computation time.
There are some packages around now that offer full accuracy.
The MPFR package gives correctly rounded arbitrary precision results.
IBM has written a package for fast and accurate IEEE elementary functions and in the future the standard libraries may offer such precision.
It is possible to devise well-defined computable numbers which it may never be possible to correctly round no matter how many digits are calculated.
For instance, if Goldbach's conjecture is true but unprovable, then it is impossible to correctly round down 0.5 + 10-n where n is the first even number greater than 4 which is not the sum of two primes, or 0.5 if there is no such number.
This can however be approximated to any given precision even if the conjecture is unprovable.
The concept of rounding is very old, perhaps older even than the concept of division.
Some ancient clay tablets found in Mesopotamia contain tables with rounded values of reciprocals and square roots in base 60.
The Round-to-even method has served as the ASTM (E-29) standard since 1940.
The origin of the terms unbiased rounding and statistician's rounding are fairly self-explanatory.
In the 1906 4th edition of Probability and Theory of Errors  Robert Simpson Woodward called this "the computer's rule" indicating that it was then in common use by human computers who calculated mathematical tables.
Churchill Eisenhart's 1947 paper "Effects of Rounding or Grouping Data" (in Selected Techniques of Statistical Analysis, McGrawHill, 1947, Eisenhart, Hastay, and Wallis, editors) indicated that the practice was already "well established" in data analysis.
The origin of the term "bankers' rounding" 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 The Introduction of the Euro and the Rounding of Currency Amounts  suggests that there had previously been no standard approach to rounding in banking; and it specifies that "half-way" amounts should be rounded up.
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 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 interval arithmetic.
Rounding functions in programming languages
Most programming languages provide functions or special syntax to round fractional numbers in various ways.
The earliest numeric languages, such as FORTRAN and C, would provide only one method, usually truncation (towards zero).
This default method could be implied in certain contexts, such as when assigning a fractional number to an integer variable, or using a fractional number as an index of an array.
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.
In the last decades, however, the syntax and/or the standard libraries of most languages have commonly provided at least the four basic rounding functions (up/ceiling, down/floor, to nearest, and towards zero).
The tie-breaking method may vary depending the language and version, and/or may be selectable by the programmer.
Several languages follow the lead of the IEEE-754 floating-point standard, and define these functions as taking a double precision float argument and returning the result of the same type, which then may be converted to an integer if necessary.
Since the IEEE double precision format has 52 fraction bits, this approach may avoid spurious overflows in languages have 32-bit integers.
Some languages, such as PHP, 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 "printf" 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 precision).
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.
Other rounding standards
U.S. Weather Observations
In a guideline issued in mid-1966, the U.S. 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 "round half away from zero".
Negative zero in meteorology
Some meteorologists may write "-0" to indicate a temperature between 0.0 and -0.5 degrees (exclusive) that was rounded to 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 Celsius scale, where below zero indicates freezing.
An introduction to different rounding algorithms  that is accessible to a general audience but especially useful to those studying computer science and electronics.
How To Implement Custom Rounding Procedures by Microsoft
Wheatland is an American city in the state of Wyoming.
It is in Platte County where it is the county seat.
The city has about 3,548 people.
Steamboat, the bucking horse on the license plate for the state of Wyoming and the logo for the athletics team of the University of Wyoming, was stabled in a barn in Wheatland.
Official City of Wheatland website
http://www.winchelsea.net/visiting/winchelsea_history.htm
http://www.winchelsea.net/
Charing is a village and civil parish in Kent, England.
In 2001 there were 2,694 people living in Charing.
Lenham is a village and civil parish in Kent, England.
In 2001 there were 3,301 people living in Lenham.
Hitoshi Matsumoto (born September 8, 1963 in Amagasaki Hyōgo) is an Japanese comedian, actor, producer and director.
Matsumoto is probably best known for being one-half of the comedy duo, Downtown where he is the boke also with member Masatoshi Hamada who is the tsukkomi.
While apart of Downtown, Matsumoto and Hamada made two television shows, Downtown no Gaki no Tsukai ya Arahende!!
He directed, produced and starred in a movie called Big Man Japan.
The movie was shown at the 2007 Cannes Film Festival in the Directors' Fortnight section.
The movie was nominated in 2007 for "Best Film" at the Sitges - Catalonian International Film Festival and in 2008 for "Best Visual Effects" at the Asian Film Awards.
He is married to his wife Ihara Rin and they have a daughter that was born on October 6, 2009.
Hitoshi Matsumoto at the Internet Movie Database
Japanese movie directors
Sturry is a village and civil parish in Kent, England.
In 2001 there were 6,257 people living in Sturry.
Dalston is a village and civil parish in City of Carlisle, Cumbria, England.
In 2001 there were 2,643 people living in Dalston.
Salcombe is a village and civil parish in South Hams, Devon, England.
In 2001, there were 1,893 people living in Salcombe.
Haslemere is a town and civil parish in Waverley, Surrey, England.
In 2001 there were 15,612 people living in Haslemere.
Kočevski Rog massacre was a series of massacres in Slovenia in late May 1945.
This was shortly after World War II.
It happened near Kočevski Rog.
Thousands of members of the Slovene Home Guard and their families were executed by special units of the Yugoslav Partisans.
Several thousand (between 10,000 and 12,000, according to certain sources) war prisoners died in these executions.
They had been repatriated by the British military authorities from Austria, where they had fled.
The Russian British author Nikolai Tolstoy wrote an account of the events in his book The Minister and the Massacres.
British author John Corsellis, who served in Austria with the British Army, has written a historic book of these events, called Slovenia 1945: Memories of Death and Survival after World War II.
Boris Karapandzic writes that there were 12,000 Slovenian "home guards", 3,000 Serbian volunteer troops, 1,000 Montenegrin "chetniks", and 2,500 Croatian "home guards".
Karapandzic's report is confirmed in a subsequent book by a group of scholars.
History of Slovenia
St Dennis is a parish and village in Cornwall, United Kingdom.
The village is on the B3279 road between Newquay and St Austell.
St Dennis had a population of 2,696 in the 2001 census.
The village is famous for its church built on the site of an Iron Age hillfort.
The area is full of tips and clay pits of china clay mining but St Dennis village itself is designated an 'island settlement' which prevents encroachment by the china clay industry.
Much of the parish is up to 500 feet above sea level, and the countryside is moorland with small fields enclosed by 'Cornish hedges'.
The church is on top of a hill overlooking the village and has views to the sea at Newquay.
It stands on the site of an Iron Age fort.
The tower is the oldest part of the church, and the newer church has been repaired after being destroyed by fire in 1985.
The village also has a strong history of Methodism and once had three Methodist chapels, two of these are still in use.
Illogan (pronounced i'lluggan) is a village and parish in Cornwall, United Kingdom.
It is two miles (3 km) northwest of Redruth and had a population of 5585 in the 2001 census.
The church is dedicated to Saint Illogan, of whom little is known.
Illogan was the birthplace of engineer and inventor Richard Trevithick.
St Illogan Church Parish stretching from Piece and Carnkie, to Portreath on the North Coast; Brea and Tuckingmill along the River Connor to the West; and Porthtowan, Sparnon Gate and Redruth Coombe to the East.
The population is approximately 14,000; making it more populous than Redruth.
It was the third largest in Cornwall.
Before 1935, the church and civil parish boundaries were the same.
Illogan Civil Parish stretching from Bridge and Harris Mill in the East; Tolvaddon and Bell Lake in the West; and from the A30 to Reskajeage on the North Cliffs.
In 1985, civil parish status was given back to Illogan, but as the population was too large, Illogan was divided into the civil parishes of Carn Brea, Illogan and Portreath.
St Keverne is a parish and village on the Lizard Peninsula in Cornwall, United Kingdom.
The population of the parish was 2107 in the 2001 census.
The Cornish Rebellion of 1497 started in St Keverne.
The leader of the rebellion Michael An Gof was a blacksmith from St Keverne and is commemorated by a statue in the village.
The parish is a large one and includes some 10 miles of coast from Nare Point at the mouth of the Helford River to Kennack Sands, and the Manacles offshore.
Inland the parish includes the hamlets of Zoar, Laddenvean, Traboe, Trelan and Gwenter.
St Keverne was in the Middle Ages the site of an important monastery.
The church is dedicated to St Akeveranus.
The church is very large for a village church and in its present form is 15th century.
The Cornish Rebellion of 1497 was a popular uprising by the people of Cornwall in the far southwest of Britain.
Its primary cause was the people's response to the raising of war taxes by King Henry VII to pay for a campaign against Scotland.
Tin miners were angered as the scale of the taxes overturned previous rights granted by Edward I of England to the Cornish Stannary Parliament which exempted Cornwall from all taxes of 10ths or 15ths of income.
In reaction to King Henry's tax levy, Michael Joseph (An Gof), a blacksmith from St Keverne and Thomas Flamank a lawyer from Bodmin, persuaded many of the people of Cornwall to revolt against the King.
An army of 15,000 marched into Devon, attracting support in terms of provisions and recruits as they went.
Apart from one isolated incident at Taunton their march was not accompanied by any violence or looting.
From Taunton, they moved on to Wells, where they were joined by the seventh Baron Audley, an able soldier.
From Wells to Winchester and Kent
After issuing a declaration of grievances, the army left Wells and marched to Winchester via Bristol and Salisbury, remarkably unopposed as they progressed across the south of England.
At this point, having come so far, there seems to have been some questioning of what exactly should be done.
Flamank proposed that they should head for Kent, 'the classic soil of protests', the home of the Peasants' Revolt of 1381 and Jack Cade's rebellion, to rally the volatile men of Kent to their banner.
Having failed to attract support in Kent part of the army abandoned the march and returned home while the rest turned towards Surrey.
In Surrey
Moving back west, by Tuesday 13 June 1497 the Cornish army arrived at Guildford.
Although shocked by the scale of the revolt, Henry VII had not been idle.
The army of 8000 men assembled for Scotland was recalled.
The same day that the Cornish arrived at Guildford, Lord Daubeney and his army took up position upon Hounslow Heath and were cheered by the arrival of food and wine dispatched by the Lord Mayor of London.
The Crown decided to take the offensive and test the strength and resolve of the Cornish forces.
Lord Daubeney sent out a force of 500 mounted spearmen and they clashed with the Cornish at 'Gill Down' outside Guildford on Wednesday 14 June 1497.
The Cornish army left Guildford and moved via Banstead and Chussex Plain to Blackheath where they pitched their final camp, looking down from the hill onto the Thames and City of London.
Somehow An Gof held his army together, but faced with overwhelming odds, some Cornish deserted and by morning there remained only some 9-10,000 Cornish stalwarts left in arms.
Battle of Deptford Bridge
The Battle of Deptford Bridge (also known as Battle of Blackheath) took place on 17 June 1497 on a site in present-day Deptford south-east London, near the River Ravensbourne and was the climax of the Cornish Rebellion.
Henry VII had an army of some 25,000 men and the Cornish lacked the supporting cavalry and artillery arms essential to the professional forces of the time.
After carefully spreading rumours that he would attack on the following Monday, Henry moved against the Cornish at dawn on Saturday (17 June 1497).
The Royal forces were divided into three 'battles', two under Lords Oxford, Essex and Suffolk, to wheel round the right flank and rear of enemy whilst the third waited in reserve.
When the Cornish were duly surrounded, Lord Daubeney and the third 'battle' were ordered into frontal attack.
Cornish force at the bridge
At the bridge at Deptford Strand, the Cornish had placed a body of archers to block the passage of the river.