Green-sell.info

Новые технологии
0 просмотров
Рейтинг статьи
1 звезда2 звезды3 звезды4 звезды5 звезд
Загрузка...

Floating point c

1. Floating point basics

The core idea of floating-point representations (as opposed to fixed point representations as used by, say, ints), is that a number x is written as m*b e where m is a mantissa or fractional part, b is a base, and e is an exponent. On modern computers the base is almost always 2, and for most floating-point representations the mantissa will be scaled to be between 1 and b. This is done by adjusting the exponent, e.g.

The mantissa is usually represented in base b, as a binary fraction. So (in a very low-precision format), 1 would be 1.000*2 0 , 2 would be 1.000*2 1 , and 0.375 would be 1.100*2 -2 , where the first 1 after the decimal point counts as 1/2, the second as 1/4, etc. Note that for a properly-scaled (or normalized) floating-point number in base 2 the digit before the decimal point is always 1. For this reason it is usually dropped (although this requires a special representation for 0).

Negative values are typically handled by adding a sign bit that is 0 for positive numbers and 1 for negative numbers.

2. Floating-point constants

Any number that has a decimal point in it will be interpreted by the compiler as a floating-point number. Note that you have to put at least one digit after the decimal point: 2.0, 3.75, -12.6112. You can specific a floating point number in scientific notation using e for the exponent: 6.022e23.

3. Operators

Floating-point types in C support most of the same arithmetic and relational operators as integer types; x > y, x / y, x + y all make sense when x and y are floats. If you mix two different floating-point types together, the less-precise one will be extended to match the precision of the more-precise one; this also works if you mix integer and floating point types as in 2 / 3.0. Unlike integer division, floating-point division does not discard the fractional part (although it may produce round-off error: 2.0/3.0 gives 0.66666666666666663, which is not quite exact). Be careful about accidentally using integer division when you mean to use floating-point division: 2/3 is . Casts can be used to force floating-point division (see below).

Some operators that work on integers will not work on floating-point types. These are % (use modf from the math library if you really need to get a floating-point remainder) and all of the bitwise operators

4. Conversion to and from integer types

Mixed uses of floating-point and integer types will convert the integers to floating-point.

You can convert floating-point numbers to and from integer types explicitly using casts. A typical use might be:

If we didn’t put in the (double) to convert sum to a double, we’d end up doing integer division, which would truncate the fractional part of our average.

In the other direction, we can write:

to convert a float f to int i. This conversion loses information by throwing away the fractional part of f: if f was 3.2, i will end up being just 3.

5. The IEEE-754 floating-point standard

The IEEE-754 floating-point standard is a standard for representing and manipulating floating-point quantities that is followed by all modern computer systems. It defines several standard representations of floating-point numbers, all of which have the following basic pattern (the specific layout here is for 32-bit floats):

The bit numbers are counting from the least-significant bit. The first bit is the sign (0 for positive, 1 for negative). The following 8 bits are the exponent in excess-127 binary notation; this means that the binary pattern 01111111 = 127 represents an exponent of 0, 1000000 = 128, represents 1, 01111110 = 126 represents -1, and so forth. The mantissa fits in the remaining 24 bits, with its leading 1 stripped off as described above.

Certain numbers have a special representation. Because 0 cannot be represented in the standard form (there is no 1 before the decimal point), it is given the special representation 0 00000000 00000000000000000000000. (There is also a -0 = 1 00000000 00000000000000000000000, which looks equal to +0 but prints differently.) Numbers with exponents of 11111111 = 255 = 2 128 represent non-numeric quantities such as «not a number» (NaN), returned by operations like (0.0/0.0) and positive or negative infinity. A table of some typical floating-point numbers (generated by the program float.c) is given below:

What this means in practice is that a 32-bit floating-point value (e.g. a float) can represent any number between 1.17549435e-38 and 3.40282347e+38, where the e separates the (base 10) exponent. Operations that would create a smaller value will underflow to 0 (slowly—IEEE 754 allows «denormalized» floating point numbers with reduced precision for very small values) and operations that would create a larger value will produce inf or -inf instead.

For a 64-bit double, the size of both the exponent and mantissa are larger; this gives a range from 1.7976931348623157e+308 to 2.2250738585072014e-308, with similar behavior on underflow and overflow.

Intel processors internally use an even larger 80-bit floating-point format for all operations. Unless you declare your variables as long double, this should not be visible to you from C except that some operations that might otherwise produce overflow errors will not do so, provided all the variables involved sit in registers (typically the case only for local variables and function parameters).

6. Error

In general, floating-point numbers are not exact: they are likely to contain round-off error because of the truncation of the mantissa to a fixed number of bits. This is particularly noticeable for large values (e.g. 1e+12 in the table above), but can also be seen in fractions with values that aren’t powers of 2 in the denominator (e.g. 0.1). Round-off error is often invisible with the default float output formats, since they produce fewer digits than are stored internally, but can accumulate over time, particularly if you subtract floating-point quantities with values that are close (this wipes out the mantissa without wiping out the error, making the error much larger relative to the number that remains).

The easiest way to avoid accumulating error is to use high-precision floating-point numbers (this means using double instead of float). On modern CPUs there is little or no time penalty for doing so, although storing doubles instead of floats will take twice as much space in memory.

Note that a consequence of the internal structure of IEEE 754 floating-point numbers is that small integers and fractions with small numerators and power-of-2 denominators can be represented exactly—indeed, the IEEE 754 standard carefully defines floating-point operations so that arithmetic on such exact integers will give the same answers as integer arithmetic would (except, of course, for division that produces a remainder). This fact can sometimes be exploited to get higher precision on integer values than is available from the standard integer types; for example, a double can represent any integer between -2 53 and 2 53 exactly, which is a much wider range than the values from 2^-31^ to 2^31^-1 that fit in a 32-bit int or long. (A 64-bit long long does better.) So double` should be considered for applications where large precise integers are needed (such as calculating the net worth in pennies of a billionaire.)

One consequence of round-off error is that it is very difficult to test floating-point numbers for equality, unless you are sure you have an exact value as described above. It is generally not the case, for example, that (0.1+0.1+0.1) == 0.3 in C. This can produce odd results if you try writing something like for(f = 0.0; f to generate infinite quantities. The macros isinf and isnan can be used to detect such quantities if they occur.

9. The math library

Many mathematical functions on floating-point values are not linked into C programs by default, but can be obtained by linking in the math library. Examples would be the trigonometric functions sin, cos, and tan (plus more exotic ones), sqrt for taking square roots, pow for exponentiation, log and exp for base-e logs and exponents, and fmod for when you really want to write x%y but one or both variables is a double. The standard math library functions all take doubles as arguments and return double values; most implementations also provide some extra functions with similar names (e.g., sinf) that use floats instead, for applications where space or speed is more important than accuracy.

Читать еще:  Как добавить песню в презентацию powerpoint

There are two parts to using the math library. The first is to include the line

somewhere at the top of your source file. This tells the preprocessor to paste in the declarations of the math library functions found in /usr/include/math.h.

The second step is to link to the math library when you compile. This is done by passing the flag -lm to gcc after your C program source file(s). A typical command might be:

If you don’t do this, you will get errors from the compiler about missing functions. The reason is that the math library is not linked in by default, since for many system programs it’s not needed.

Floating points

Floating-point numbers are numbers that have fractional parts (usually expressed with a decimal point). You might wonder why there’s isn’t just a single data type for dealing with numbers (fractions or no fractions), but that’s because it’s a lot faster for the computer to deal with whole numbers than with numbers containing fractions. Therefore it makes sense to distinguish between them — if you know that you will only be dealing with whole numbers, pick an integer data type. Otherwise, use one of the floating point data types, as described in this article.

Using a floating point value is just as easy as using an integer, even though there are quite a few more concerns with floating point values, which we’ll discuss later. For now, let’s see what it looks like when declaring one of the most commonly used floating point data type: the double.

Just like an integer, you can of course assign a value to it at the same time as declaring it:

The same goes for the float and decimal types, which will discuss in just a second, but here, the notation is slightly different:

Notice the «f» and «m» after the numbers — it tells the compiler that we are assigning a float and a decimal value. Without it, C# will interpret the numbers as double, which can’t be automatically converted to either a float or decimal.

float, double or decimal?

Dealing with floating point values in programming has always caused a lot of questions and concerns. For instance, C# has at least three data types for dealing with non-whole/non-integer numbers:

  • float (an alias for System.Single)
  • double (an alias for System.Double)
  • decimal (an alias for System.Decimal)

The underlying difference might be a bit difficult to understand, unless you have a lot of knowledge about how a computer works internally, so let’s stick to the more practical stuff here.

In general, the difference between the float, double and decimal data types lies in the precision and therefore also in how much memory is used to hold them. The float is the least expensive one — it can represent a number with up to 7 digits. The double is more precise, with up to 16 digits, while the decimal is the most precise, with a whooping maximum of 29 digits.

You might wonder what you need all that precision for but the answer is «math stuff». The classic example to understand the difference is to divide 10 with 3. Most of us will do that in the head and say that the result is 3.33, but a lot of people also know that it’s not entirely accurate. The real answer is 3.33 followed by an amount of extra 3’s — how many, when doing this calculation with C#, is determined by the data type. Check out this example:

I do the exact same calculation, but with different data types. The result will look like this:

The difference is quite clear, but how much precision do you really need for most tasks?

How to choose

First of all, you to consider how many digits you need to store. A float can only contain 7 digits, so if you need a bigger number than that, you may want to go with a double or decimal instead.

Second of all, both the float and double will represent values as an approximation of the actual value — in other words, it might not be comparable down to the very last digit. That also means that once you start doing more and more calculations with these variables, they might not be as precise anymore, which basically means that two values which should check out as equals will suddenly not be equal anyway.

So, for situations where precision is the primary concern, you should go with the decimal type. A great example is for representing financial numbers (money) — you don’t want to add 10 amounts in your bookkeeping, just to find out that the result is not what you expected. On the other hand, if performance is more important, you should go with a float (for small numbers) or a double (for larger numbers). A decimal is, thanks to its extra precision, much slower than a float — some tests shows that its up to 20 times slower!

Summary

When dealing with floating point values, you should use a float or a double data type when precision is less important than performance. On the other hand, if you want the maximum amount of precision and you are willing to accept a lower level of performance, you should go with the decimal data type — especially when dealing with financial numbers.

If you want to know more about the underlying differences between these data types, on a much more detailed level, you should have a look at this very detailed article: What Every Computer Scientist Should Know About Floating-Point Arithmetic

Binary floating point and .NET

Lots of people are at first surprised when some of their arithmetic comes out «wrong» in .NET. This isn’t something specific to .NET in particular — most languages/platforms use something called «floating point» arithmetic for representing non-integer numbers. This is fine in itself, but you need to be a bit aware of what’s going on under the covers, otherwise you’ll be surprised at some of the results.

It’s worth noting that I am not an expert on this matter. Since writing this article, I’ve found another one — this time written by someone who really is an expert, Jeffrey Sax. I strongly recommend that you read his article on floating point concepts too.

What is floating point?

Computers always need some way of representing data, and ultimately those representations will always boil down to binary (0s and 1s). Integers are easy to represent (with appropriate conventions for negative numbers, and with well-specified ranges to know how big the representation is to start with) but non-integers are a bit more tricky. Whatever you come up with, there’ll be a problem with it. For instance, take our own normal way of writing numbers in decimal: that can’t (in itself) express a third. You end up with a recurring 3. Whatever base you come up with, you’ll have the same problem with some numbers — and in particular, «irrational» numbers (numbers which can’t be represented as fractions) like the mathematical constants pi and e are always going to give trouble.

Читать еще:  Lenovo z570 wifi не работает

You could store all the rational numbers exactly as two integers, with the number being the first number divided by the second — but the integers can grow quite large quite quickly even for «simple» operations, and things like square roots will tend to produce irrational numbers. There are various other schemes which also pose problems, but the one most systems use in one form or other is floating point. The idea of this is that basically you have one integer (the mantissa) which gives some scaled representation of the number, and another (the exponent) which says what the scale is, in terms of «where does the dot go». For instance, 34.5 could be represented in «decimal floating point» as mantissa 3.45 with an exponent of 1, whereas 3450 would have the same mantissa but an exponent of 3 (as 34.5 is 3.45×10 1 , and 3450 is 3.45×10 3 ). Now, that example is in decimal just for simplicity, but the most common formats of floating point are for binary. For instance, the binary mantissa 1.1 with an exponent of -1 would mean decimal 0.75 (binary 1.1==decimal 1.5, and the exponent of -1 means «divide by 2» in the same way that a decimal exponent of -1 means «divide by 10»).

It’s very important to understand that in the same way that you can’t represent a third exactly in a (finite) decimal expansion, there are lots of numbers which look simple in decimal, but which have long or infinite expansions in a binary expansion. This means that (for instance) a binary floating point variable can’t have the exact value of decimal 0.1. Instead, suppose you have some code like this:

The variable x will actually store the closest available double to that value. Once you can get your head round that, it becomes obvious why some calculations seem to be «wrong». If you were asked to add a third to a third, but could only represent the thirds using 3 decimal places, you’d get the «wrong» answer: the closest you could get to a third is 0.333, and adding two of those together gives 0.666, rather than 0.667 (which is closer to the exact value of two thirds). An example in binary floating point is that 3.65d+0.05d != 3.7d (although it may be displayed as 3.7 in some situations).

What floating point types are available in .NET?

The C# standard only lists double and float as floating points available (those being the C# shorthand for System.Double and System.Single ), but the decimal type (shorthand for System.Decimal ) is also a floating point type really — it’s just it’s decimal floating point, and the ranges of exponents are interesting. The decimal type is described in another article, so this one doesn’t go into it any further — we’re concentrating on double and float . Both of these are binary floating point types, conforming to IEEE 754 (a standard defining various floating point types). float is a 32 bit type (1 bit of sign, 23 bits of mantissa, and 8 bits of exponent), and double is a 64 bit type (1 bit of sign, 52 bits of mantissa and 11 bits of exponent).

Isn’t it bad that results aren’t what I’d expect?

Well, that depends on the situation. If you’re writing financial applications, you probably have very rigidly defined ways of treating errors, and the amounts are also intuitively represented as decimal — in which case the decimal type is more likely to be appropriate than float or double . If, however, you’re writing a scientific app, the link with the decimal representation is likely to be weaker, and you’re also likely to be dealing with less precise amounts to start with (a dollar is exactly a dollar, but if you’ve measured a length to be a metre, that’s likely to have some sort of inaccuracy in it to start with).

Comparing floating point numbers

One consequence of all of this is that you should very, very rarely be comparing binary floating point numbers for equality directly. It’s usually fine to compare in terms of greater-than or less-than, but when you’re interested in equality you should always consider whether what you actually want is near equality: is one number almost the same as another. One simple way of doing this is to subtract one from the other, use Math.Abs to find the absolute value of the difference, and then check whether this is lower than a certain tolerance level.

There are some cases which are particularly pathological though, and these are due to JIT optimisations. Look at the following code:

It should always print True , right? Wrong, unfortunately. When running under debug, where the JIT can’t make as many optimisations as normal, it will print True . When running normally, the JIT can store the result of the sum more accurately than a float can really represent — it can use the default x86 80-bit representation, for instance, for the sum itself, the return value, and the local variable. See the ECMA CLI spec, partition 1, section 12.1.3 for more details. Uncommenting one of the commented out lines in the above may make the JIT behave a bit more conservatively, leading to a result of True . However, this depends on the exact implementation, CLR version, processor etc — it’s not something you should rely on. (Indeed, in some environments only some of the commented-out lines will affect the results.) This is another reason to avoid equality comparisons even if you’re really sure that the results should be the same.

How does .NET format floating point numbers?

There’s no built-in way to see the exact decimal value of a floating point number in .NET, although you can do it with a bit of work. (See the bottom of this article for some code to do this.) By default, .NET formats a double to 15 decimal places, and a float to 7. (In some cases it will use scientific notation; see the MSDN page on standard numeric format strings for more information.) If you use the round-trip format specifier (» r «), it formats the number to the shortest form which, when parsed (to the same type), will get back to the original number. If you are storing floating point numbers as strings and the exact value is important to you, you should definitely use the round-trip specifier, as otherwise you are very likely to lose data.

What exactly does a floating point number look like in memory?

As it says above, a floating point number basically has a sign, an exponent and a mantissa. All of these are integers, and the combination of the three of them specifies exactly what number is being represented. There are various classes of floating point number: normalised, subnormal, infinity and not a number (NaN). Most numbers are normalised, which means that the first bit of the binary mantissa is assumed to be 1, which means you don’t actually need to store it. For instance, the binary number 1.01101 could be expressed as just .01101 — the leading 1 is assumed, as if it were 0 a different exponent would be used. That technique only works when the number is in the range where you can choose the exponent suitably. Numbers which don’t lie in that range (very, very small numbers) are called subnormal, and no leading bit is assumed. «Not a number» (NaN) values are for things like the result of dividing 0 by 0, etc. There are various different classes of NaN, and there’s some odd behaviour there as well. Subnormal numbers are also sometimes called denormalised numbers.

Читать еще:  Как установить powerpoint 2020

The actual representation of the sign, exponent and mantissa at the bit level is for each of them to be an unsigned integer, with the stored value being just the concatenation of the sign, then the exponent, then the mantissa. The «real» exponent is biased — for instance, in the case of a double , the exponent is biased by 1023, so a stored exponent value of 1026 really means 3 when you come to work out the actual value. The table below shows what each combination of sign, exponent and mantissa means, using double as an example. The same principles apply for float , just with slightly different values (such as the bias). Note that the exponent value given here is the stored exponent, before the bias is applied. (That’s why the bias is shown in the «value» column.)

Learn C — C Floating-Point Type

Floating-point numbers hold values that are written with a decimal point.

The following are examples of floating-point values:

The last constant is integral, but it will be stored as a floating-point value.

Floating-point numbers are often expressed as a decimal value multiplied by some power of 10, where the power of 10 is called the exponent.

The following table shows how to Express Floating-Point Numbers.

The code above generates the following result.

Floating-Point Variables

You have a choice of three types of floating-point variables, and these are shown in the following Table.

The following code defines two float point numbers.

To write a constant of type float, you append an f to the number to distinguish it from type double.

You could initialize the previous two variables when you declare them like this:

The variable radius has the initial value 2.5, and the variable biggest is initialized to the number that corresponds to 123 followed by 30 zeroes.

To specify a long double constant, you append an uppercase or lowercase letter L.

long double huge = 1234567.98765L;

The following code shows how to float type variable declaration and initializations.

The code above generates the following result.

Division Using Floating-Point Values

Here’s a simple example that divides one floating-point value by another and outputs the result:

The code above generates the following result.

Number of Decimal Places

You can specify the number of places that you want to see after the decimal point in the format specifier.

To obtain the output to two decimal places, you would write the format specifier as %.2f .

To get three decimal places, you would write %.3f .

The following code changes the printf() statement so that it will produce more suitable output:

The code above generates the following result.

Output Field Width

You can specify the field width and the number of decimal places.

The general form of the format specifier for floating-point values can be written like this:

The square brackets means optional.

You can omit width or .precision or modifier or any combination of these.

The width value is an integer specifying the total number of characters in the output including spaces. And it is the output field width.

The precision value is an integer specifying the number of decimal places that are after the decimal point.

The modifier part is L when the value you are outputting is type long double, otherwise you omit it.

The code above generates the following result.

precision

To create precision with floating-point numbers, adjust the conversion specifier using numbering schemes between the % sign and the f character conversion specifier.

The code above generates the following result.

Left align

When you specify the field width, the output value will be right aligned by default.

If you want the value to be left aligned in the field, just put a minus sign following the %.

You can specify a field width and the alignment in the field with a specification for outputting an integer value.

For example, %-15d specifies an integer value will be presented left aligned in a field width of 15 characters.

The code above generates the following result.

Example

The following code shows how to calculate the circumference and area of a circular table.

The code above generates the following result.

Example 2

The following code uses defined constants from limit.h and float.

The code above generates the following result.

Example 3

The following code displays float value in two ways.

The code above generates the following result.

Example 4

Read float type number for command line.

The code above generates the following result.

  • Next »
  • « Previous

Вопросы с тегом ‘floating-point’

Количество результатов: 3457

Читаем данные из столбцов Excel и сохранить его в Numpy массива с самостоятельными написано функциями. У меня есть проблема только с одним из них. Вот функция: impor.

log1pexp(x) инвентарь log(1 + exp(x)) в числовом стабильно. К сожалению, эта функция не существует в tensorflow.math модуль. Это на самом деле довольно просто реализо.

Мне нужно проверить, если один поплавок/двойной кратна другой поплавка/двойной. В целом числе легко $isMultiple = $x % $y == 0; но в поплавках/двойниках нет. Перва.

Я новичок в Java и написание класса для представления комплексных чисел. //imports public final class Complex extends Object implements Serializable< .

У меня есть эта функция, она округляет все числа до 8 знаков после запятой. exports.roundDown = function(number, decimals) < try < if (!exports.isNumeric(number.

Я пытаюсь сохранить список чисел с плавающей точкой в ​​динамическом массиве с использованием MIPS. Программа запрашивает у пользователя числа поплавков для входа, а з.

Я выполняю операцию простой суммы на числах с плавающей точкой с использованием math.fsum() функция в Python, но результат не соответствует тому, что я ожидал. Вы мо.

Я пишу фрагмент кода, где я хранить значения из одного файла в другой с Grep. Так в коде ниже, я читаю из ./stats.txt и хранения его двух различных файлов на основе то.

Моя программа C++ вычисляет некоторое число с плавающей точкой х и выводит его на экран: cout << x; Это очень важно для меня, что выход будет одинаковым во вс.

Как правильно добавить или вычесть с помощью поплавков? Например, как выполнить: 2.4e-07 — 1e-8 так что она возвращает 2.3e-7 вместо того 2.2999999999999997e-07. П.

В предыдущем проекте шахты (который я разместил вопрос здесь), я был очень раздражен осознанием того, что я хотел бы иметь дело с ошибками округления в числе с плавающ.

Если a и b положительные целые числа меньше, чем Number.MAX_SAFE_INTEGER и a делится на b, Является то, что гарантировано console.log(a / b) будет выводить целое число.

Я использую алгоритм пузырьковой сортировки для сортировки списка значений с плавающей точкой, полученных из текстового файла. Этот код пузырьковой сортировки работает.

Когда я делаю некоторые расчеты с гением я получаю ответы, как фракция. Например: genius> 1-((3/4)^4) = 175 — 256 или genius> 10/54 = 5/27 Как я могу по.

Для относительно простых поплавков, числовая точность достаточна, чтобы представлять их в точности. Так, например, 17,5, равна 17,5 Для более сложных поплавков, такие.

Console.WriteLine(» 0.0 float: 0x00000000, <0>«, (float)0x00000000); Console.WriteLine(«-0.0 float: 0x80000000, <0>«, (float)0x80000000); C# печатает 0,0 для первой.

Моя цель состоит в том, чтобы проверить, есть ли остаток от деления остались 2 поплавков, и если есть, дать, что остаточные обратно пользователю. Учитывая следующий к.

если пользователь Я вошел 5.123, как я могу сохранить его в памяти, как 5.1 в Java Примечание: мне не нужно, чтобы отобразить его Я хочу (магазин) его в одну цифру пос.

Я пытаюсь запрограммировать матричное умножение в C с использованием SIMD встроенных функций. Я был уверен в моей реализации, но когда я исполняю, я получаю некоторые .

Ссылка на основную публикацию
Adblock
detector