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double type Multiplication in C# giving me wrong values [duplicate]

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Is floating point math broken?
(31 answers)
Closed 7 years ago.
If I execute the following expression in C#:
double i = 10*0.69;
i is: 6.8999999999999995. Why?
I understand numbers such as 1/3 can be hard to represent in binary as it has infinite recurring decimal places but this is not the case for 0.69. And 0.69 can easily be represented in binary, one binary number for 69 and another to denote the position of the decimal place.
How do I work around this? Use the decimal type?

Because you've misunderstood floating point arithmetic and how data is stored.
In fact, your code isn't actually performing any arithmetic at execution time in this particular case - the compiler will have done it, then saved a constant in the generated executable. However, it can't store an exact value of 6.9, because that value cannot be precisely represented in floating point point format, just like 1/3 can't be precisely stored in a finite decimal representation.
See if this article helps you.

why doesn't the framework work around this and hide this problem from me and give me the
right answer,0.69!!!
Stop behaving like a dilbert manager, and accept that computers, though cool and awesome, have limits. In your specific case, it doesn't just "hide" the problem, because you have specifically told it not to. The language (the computer) provides alternatives to the format, that you didn't choose. You chose double, which has certain advantages over decimal, and certain downsides. Now, knowing the answer, you're upset that the downsides don't magically disappear.
As a programmer, you are responsible for hiding this downside from managers, and there are many ways to do that. However, the makers of C# have a responsibility to make floating point work correctly, and correct floating point will occasionally result in incorrect math.
So will every other number storage method, as we do not have infinite bits. Our job as programmers is to work with limited resources to make cool things happen. They got you 90% of the way there, just get the torch home.

And 0.69 can easily be represented in
binary, one binary number for 69 and
another to denote the position of the
decimal place.
I think this is a common mistake - you're thinking of floating point numbers as if they are base-10 (i.e decimal - hence my emphasis).
So - you're thinking that there are two whole-number parts to this double: 69 and divide by 100 to get the decimal place to move - which could also be expressed as:
69 x 10 to the power of -2.
However floats store the 'position of the point' as base-2.
Your float actually gets stored as:
68999999999999995 x 2 to the power of some big negative number
This isn't as much of a problem once you're used to it - most people know and expect that 1/3 can't be expressed accurately as a decimal or percentage. It's just that the fractions that can't be expressed in base-2 are different.

but why doesn't the framework work around this and hide this problem from me and give me the right answer,0.69!!!
Because you told it to use binary floating point, and the solution is to use decimal floating point, so you are suggesting that the framework should disregard the type you specified and use decimal instead, which is very much slower because it is not directly implemented in hardware.
A more efficient solution is to not output the full value of the representation and explicitly specify the accuracy required by your output. If you format the output to two decimal places, you will see the result you expect. However if this is a financial application decimal is precisely what you should use - you've seen Superman III (and Office Space) haven't you ;)
Note that it is all a finite approximation of an infinite range, it is merely that decimal and double use a different set of approximations. The advantage of decimal is it produces the same approximations that you would if you were performing the calculation yourself. For example if you calculated 1/3, you would eventually stop writing 3's when it was 'good enough'.

For the same reason that 1 / 3 in a decimal systems comes out as 0.3333333333333333333333333333333333333333333 and not the exact fraction, which is infinitely long.

To work around it (e.g. to display on screen) try this:
double i = (double) Decimal.Multiply(10, (Decimal) 0.69);
Everyone seems to have answered your first question, but ignored the second part.

C# - Possible to restore double precision from a text input?

I have a class that does some length calculations based on a height on a ticket. It's been in place for years and working quite well... Until we got a unique ticket size.
They are entered by sales people in inches and are normally nice numbers like 3, 4 or 3.5 and store in a database - This one is however 3.66666 recurring (or 11/3) But it is being entered as 3.666 and causing the calculation to fail due to lost precision.
I have thought of a bit of a hack to restore precision for certain numbers, but thought maybe someone knows of a better way of getting a 3.666 or a 93.1333 back to it's number + two thirds status?
Thanks,
Mick.

As you explained in comments I see your point now. I've checked the numbers:
168000 / 3.666 = 45826.5139
168000 / 3.666666 = 45818.1901488
168000 * 3 / 11 = 45818.1818182
It makes a difference of 8 tickets. I have a feeling that your issue can be solved in many ways. On the side of user input for example. Or on the side of database. But back to your question:
How do I convert 3.666 or a 93.1333 back to it's number + two thirds
status?
You are looking for converting decimal (or double) to fraction.
There is already a question on SO: Algorithm for simplifying decimal to fractions which has many answeres. I've tested some of them, and none of them were satisfying. Some of them don't even hanlde recurrence. Perhaps I've missed the correct one, you can look by yourself.
Anyway, I believe you don't need to fully implement a conversion from 1.666 to 3/2, since it's not easy and you have a real-world sizes. You've said, that most of the time numbers are aroung 3, 3.5, 4 etc. So I suggest you to take a look at a question I've linked above and search for an algorythm of detecting the recurrence number. It was also discussed here How to know the repeating decimal in a fraction?
After what just convert 1.666 to 1.666666, since 1/1000000 of inch won't mess your calculations, as numbers above show.

It would be difficult to get the accurate value of double as double is floating point.
The MSDN says:
Remember that a floating-point number
can only approximate a decimal number,
and that the precision of a
floating-point number determines how
accurately that number approximates a
decimal number. By default, a Double
value contains 15 decimal digits of
precision, although a maximum of 17
digits is maintained internally. The
precision of a floating-point number
has several consequences:
Two floating-point numbers that appear equal for a particular
precision might not compare equal
because their least significant digits
are different.
A mathematical or comparison operation that uses a floating-point
number might not yield the same result
if a decimal number is used because
the floating-point number might not
exactly approximate the decimal
number.

C# loss of precision when dividing doubles

I know this has been discussed time and time again, but I can't seem to get even the most simple example of a one-step division of doubles to result in the expected, unrounded outcome in C# - so I'm wondering if perhaps there's i.e. some compiler flag or something else strange I'm not thinking of. Consider this example:
double v1 = 0.7;
double v2 = 0.025;
double result = v1 / v2;
When I break after the last line and examine it in the VS debugger, the value of "result" is 27.999999999999996. I'm aware that I can resolve it by changing to "decimal," but that's not possible in the case of the surrounding program. Is it not strange that two low-precision doubles like this can't divide to the correct value of 28? Is the only solution really to Math.Round the result?

Is it not strange that two low-precision doubles like this can't divide to the correct value of 28?
No, not really. Neither 0.7 nor 0.025 can be exactly represented in the double type. The exact values involved are:
0.6999999999999999555910790149937383830547332763671875
0.025000000000000001387778780781445675529539585113525390625
Now are you surprised that the division doesn't give exactly 28? Garbage in, garbage out...
As you say, the right result to represent decimal numbers exactly is to use decimal. If the rest of your program is using the wrong type, that just means you need to work out which is higher: the cost of getting the wrong answer, or the cost of changing the whole program.

Precision is always a problem, in case you are dealing with float or double.
Its a known issue in Computer Science and every programming language is affected by it. To minimize these sort of errors, which are mostly related to rounding, a complete field of Numerical Analysis is dedicated to it.
For instance, let take the following code.
What would you expect?
You will expect the answer to be 1, but this is not the case, you will get 0.9999907.
float v = .001f;
float sum = 0;
for (int i = 0; i < 1000; i++ )
{
sum += v;
}

It has nothing to do with how 'simple' or 'small' the double numbers are. Strictly speaking, neither 0.7 or 0.025 may be stored as exactly those numbers in computer memory, so performing calculations on them may provide interesting results if you're after heavy precision.
So yes, use decimal or round.

To explain this by analogy:
Imagine that you are working in base 3. In base 3, 0.1 is (in decimal) 1/3, or 0.333333333'.
So you can EXACTLY represent 1/3 (decimal) in base 3, but you get rounding errors when trying to express it in decimal.
Well, you can get exactly the same thing with some decimal numbers: They can be exactly expressed in decimal, but they CAN'T be exactly expressed in binary; hence, you get rounding errors with them.

Short answer to your first question: No, it's not strange. Floating-point numbers are discrete approximations of the real numbers, which means that rounding errors will propagate and scale when you do arithmetic operations.
Theres' a whole field of mathematics called numerical analyis that basically deal with how to minimize the errors when working with such approximations.

It's the usual floating point imprecision. Not every number can be represented as a double, and those minor representation inaccuracies add up. It's also a reason why you should not compare doubles to exact numbers. I just tested it, and result.ToString() showed 28 (maybe some kind of rounding happens in double.ToString()?). result == 28 returned false though. And (int)result returned 27. So you'll just need to expect imprecisions like that.

Bitwise representation of division of floats - how division of floats works

A number can have multiple representations if we use a float, so the results of a division of floats may produce bitwise different floats. But what if the denominator is a power of 2?
AFAIK, dividing by a power of 2 would only shift the exponent, leaving the same mantissa, always producing bitwise identical floats. Is that right?
float a = xxx;
float result = n/1024f; // always the same result?
--- UPDATE ----------------------
Sorry for my lack of knowledge in the IEEE black magic for floating points :) , but I'm talking about those numbers Guvante mentioned: no representation for certain decimal numbers, 'inaccurate' floats. For the rest of this post I'll use 'accurate' and 'inaccurate' considering Guvante's definition of these words.
To simplify, let's say the numerator is always an 'accurate' number. Also, let's divide not by any power of 2, but always for 1024. Additionally, I'm doing the operation the same way every time (same method), so I'm talking about getting the same results in different executions (for the same inputs, sure).
I'm asking all this because I see different numbers coming from the same inputs, so I thought: well if I only use 'accurate' floats as numerators and divide by 1024 I will only shift the exponent, still having an 'accurate' float.
You asked for an example. The real problem is this: I have a simulator producing sometimes 0.02999994 and sometimes 0.03000000 for the same inputs. I thought I could multiply these numbers by 1024, round to get an 'integer' ('accurate' float) that would be the same for those two numbers, and then divide by 1024 to get an 'accurate' rounded float.
I was told (in my other question) that I could convert to decimal, round and cast to float, but I want to know if this way works.

A number can have multiple representations if we use a float
The question appears to be predicated on an incorrect premise; the only number that has multiple representations as a float is zero, which can be represented as either "positive zero" or "negative zero". Other than zero a given number only has one representation as a float, assuming that you are talking about the "double" or "float" types.
Or perhaps I misunderstand. Is the issue that you are referring to the fact that the compiler is permitted to do floating point operations in higher precision than the 32 or 64 bits available for storage? That can cause divisions and multiplications to produce different results in some cases.

Since people often don't fully grasp floating point numbers I will go over some of your points real quick. Each particular combination of bits in a floating point number represent a unique number. However because that number has a base 2 fractional component, there is no representation for certain decimal numbers. For instance 1.1. In those cases you take the closest number. IEEE 754-2008 specifies round to nearest, ties to even in these cases.
The real difficulty is when you combine two of these 'inaccurate' numbers. This can introduce problems as each intermediate step will involve rounding. If you calculate the same value using two different methods, you could come up with subtly different values. Typically this is handled with an epsilon when you want equality.
Now onto your real question, can you divide by a power of two and avoid introducing any additional 'inaccuracies'? Normally you can, however as with all floating point numbers, denormals and other odd cases have their own logic, and obviously if your mantissa overflows you will have difficulty. And again note, that no mathematical errors are introduced during any of this, it is simply math being done with limited percision, which involves intermittent rounding of results.
EDIT: In response to new question
What you are saying could work, but is pretty much equivalent to rounding. Additionally if you are just looking for equality, you should use an episilon as I mentioned earlier (a - b) < e for some small value e (0.0001 would work in your example). If you are looking to print out a pretty number, and the framework you are using isn't doing it to your liking, some rounding would be the most direct way of describing your solution, which is always a plus.

When should I use double instead of decimal?

I can name three advantages to using double (or float) instead of decimal:
Uses less memory.
Faster because floating point math operations are natively supported by processors.
Can represent a larger range of numbers.
But these advantages seem to apply only to calculation intensive operations, such as those found in modeling software. Of course, doubles should not be used when precision is required, such as financial calculations. So are there any practical reasons to ever choose double (or float) instead of decimal in "normal" applications?
Edited to add:
Thanks for all the great responses, I learned from them.
One further question: A few people made the point that doubles can more precisely represent real numbers. When declared I would think that they usually more accurately represent them as well. But is it a true statement that the accuracy may decrease (sometimes significantly) when floating point operations are performed?

I think you've summarised the advantages quite well. You are however missing one point. The decimal type is only more accurate at representing base 10 numbers (e.g. those used in currency/financial calculations). In general, the double type is going to offer at least as great precision (someone correct me if I'm wrong) and definitely greater speed for arbitrary real numbers. The simple conclusion is: when considering which to use, always use double unless you need the base 10 accuracy that decimal offers.
Edit:
Regarding your additional question about the decrease in accuracy of floating-point numbers after operations, this is a slightly more subtle issue. Indeed, precision (I use the term interchangeably for accuracy here) will steadily decrease after each operation is performed. This is due to two reasons:
the fact that certain numbers (most obviously decimals) can't be truly represented in floating point form
rounding errors occur, just as if you were doing the calculation by hand. It depends greatly on the context (how many operations you're performing) whether these errors are significant enough to warrant much thought however.
In all cases, if you want to compare two floating-point numbers that should in theory be equivalent (but were arrived at using different calculations), you need to allow a certain degree of tolerance (how much varies, but is typically very small).
For a more detailed overview of the particular cases where errors in accuracies can be introduced, see the Accuracy section of the Wikipedia article. Finally, if you want a seriously in-depth (and mathematical) discussion of floating-point numbers/operations at machine level, try reading the oft-quoted article What Every Computer Scientist Should Know About Floating-Point Arithmetic.

You seem spot on with the benefits of using a floating point type. I tend to design for decimals in all cases, and rely on a profiler to let me know if operations on decimal is causing bottlenecks or slow-downs. In those cases, I will "down cast" to double or float, but only do it internally, and carefully try to manage precision loss by limiting the number of significant digits in the mathematical operation being performed.
In general, if your value is transient (not reused), you're safe to use a floating point type. The real problem with floating point types is the following three scenarios.
You are aggregating floating point values (in which case the precision errors compound)
You build values based on the floating point value (for example in a recursive algorithm)
You are doing math with a very wide number of significant digits (for example, 123456789.1 * .000000000000000987654321)
EDIT
According to the reference documentation on C# decimals:
The decimal keyword denotes a
128-bit data type. Compared to
floating-point types, the decimal type
has a greater precision and a smaller
range, which makes it suitable for
financial and monetary calculations.
So to clarify my above statement:
I tend to design for decimals in all
cases, and rely on a profiler to let
me know if operations on decimal is
causing bottlenecks or slow-downs.
I have only ever worked in industries where decimals are favorable. If you're working on phsyics or graphics engines, it's probably much more beneficial to design for a floating point type (float or double).
Decimal is not infinitely precise (it is impossible to represent infinite precision for non-integral in a primitive data type), but it is far more precise than double:
decimal = 28-29 significant digits
double = 15-16 significant digits
float = 7 significant digits
EDIT 2
In response to Konrad Rudolph's comment, item # 1 (above) is definitely correct. Aggregation of imprecision does indeed compound. See the below code for an example:
private const float THREE_FIFTHS = 3f / 5f;
private const int ONE_MILLION = 1000000;
public static void Main(string[] args)
{
Console.WriteLine("Three Fifths: {0}", THREE_FIFTHS.ToString("F10"));
float asSingle = 0f;
double asDouble = 0d;
decimal asDecimal = 0M;
for (int i = 0; i < ONE_MILLION; i++)
{
asSingle += THREE_FIFTHS;
asDouble += THREE_FIFTHS;
asDecimal += (decimal) THREE_FIFTHS;
}
Console.WriteLine("Six Hundred Thousand: {0:F10}", THREE_FIFTHS * ONE_MILLION);
Console.WriteLine("Single: {0}", asSingle.ToString("F10"));
Console.WriteLine("Double: {0}", asDouble.ToString("F10"));
Console.WriteLine("Decimal: {0}", asDecimal.ToString("F10"));
Console.ReadLine();
}
This outputs the following:
Three Fifths: 0.6000000000
Six Hundred Thousand: 600000.0000000000
Single: 599093.4000000000
Double: 599999.9999886850
Decimal: 600000.0000000000
As you can see, even though we are adding from the same source constant, the results of the double is less precise (although probably will round correctly), and the float is far less precise, to the point where it has been reduced to only two significant digits.

Use decimal for base 10 values, e.g. financial calculations, as others have suggested.
But double is generally more accurate for arbitrary calculated values.
For example if you want to calculate the weight of each line in a portfolio, use double as the result will more nearly add up to 100%.
In the following example, doubleResult is closer to 1 than decimalResult:
// Add one third + one third + one third with decimal
decimal decimalValue = 1M / 3M;
decimal decimalResult = decimalValue + decimalValue + decimalValue;
// Add one third + one third + one third with double
double doubleValue = 1D / 3D;
double doubleResult = doubleValue + doubleValue + doubleValue;
So again taking the example of a portfolio:
The market value of each line in the portfolio is a monetary value and would probably be best represented as decimal.
The weight of each line in the portfolio (= Market Value / SUM(Market Value)) is usually better represented as double.

Use a double or a float when you don't need precision, for example, in a platformer game I wrote, I used a float to store the player velocities. Obviously I don't need super precision here because I eventually round to an Int for drawing on the screen.

In some Accounting, consider the possibility of using integral types instead or in conjunction. For example, let say that the rules you operate under require every calculation result carry forward with at least 6 decimal places and the final result will be rounded to the nearest penny.
A calculation of 1/6th of $100 yields $16.66666666666666..., so the value carried forth in a worksheet will be $16.666667. Both double and decimal should yield that result accurately to 6 decimal places. However, we can avoid any cumulative error by carrying the result forward as an integer 16666667. Each subsequent calculation can be made with the same precision and carried forward similarly. Continuing the example, I calculate Texas sales tax on that amount (16666667 * .0825 = 1375000). Adding the two (it's a short worksheet) 1666667 + 1375000 = 18041667. Moving the decimal point back in gives us 18.041667, or $18.04.
While this short example wouldn't yield a cumulative error using double or decimal, it's fairly easy to show cases where simply calculating the double or decimal and carrying forward would accumulate significant error. If the rules you operate under require a limited number of decimal places, storing each value as an integer by multiplying by 10^(required # of decimal place), and then dividing by 10^(required # of decimal places) to get the actual value will avoid any cumulative error.
In situations where fractions of pennies do not occur (for example, a vending machine), there is no reason to use non-integral types at all. Simply think of it as counting pennies, not dollars. I have seen code where every calculation involved only whole pennies, yet use of double led to errors! Integer only math removed the issue. So my unconventional answer is, when possible, forgo both double and decimal.

If you need to binary interrop with other languages or platforms, then you might need to use float or double, which are standardized.

Depends on what you need it for.
Because float and double are binary data types you have some diifculties and errrors in the way in rounds numbers, so for instance double would round 0.1 to 0.100000001490116, double would also round 1 / 3 to 0.33333334326441. Simply put not all real numbers have accurate representation in double types
Luckily C# also supports the so-called decimal floating-point arithmetic, where numbers are represented via the decimal numeric system rather than the binary system. Thus, the decimal floating point-arithmetic does not lose accuracy when storing and processing floating-point numbers. This makes it immensely suited to calculations where a high level of accuracy is needed.

Note: this post is based on information of the decimal type's capabilities from http://csharpindepth.com/Articles/General/Decimal.aspx and my own interpretation of what that means. I will assume Double is normal IEEE double precision.
Note2: smallest and largest in this post reffer to the magnitude of the number.
Pros of "decimal".
"decimal" can represent exactly numbers that can be written as (sufficiently short) decimal fractions, double cannot. This is important in financial ledgers and similar where it is important that the results exactly match what a human doing the calculations would give.
"decimal" has a much larger mantissa than "double". That means that for values within it's normalised range "decimal" will have a much higher precision than double.
Cons of decimal
It will be Much slower (I don't have benchmarks but I would guess at least an order of magnitude maybe more), decimal will not benefit from any hardware acceleration and arithmetic on it will require relatively expensive multiplication/division by powers of 10 (which is far more expensive than multiplication and dividion by powers of 2) to match the exponent before addition/subtraction and to bring the exponent back into range after multiplication/division.
decimal will overflow earlier tha double will. decimal can only represent numbers up to ±296-1 . By comparision double can represent numbers up to nearly ±21024
decimal will underflow earlier. The smallest numbers representable in decimal are ±10-28 . By comparision double can represent values down to 2-149 (approx 10-45) if subnromal numbers are supported and 2-126 (approx 10-38) if they are not.
decimal takes up twice as much memory as double.
My opinion is that you should default to using "decimal" for money work and other cases where matching human calculation exactly is important and that you should use use double as your default choice the rest of the time.

Use floating points if you value performance over correctness.

Choose the type in function of your application. If you need precision like in financial analysis, you have answered your question. But if your application can settle with an estimate your ok with double.
Is your application in need of a fast calculation or will he have all the time in the world to give you an answer? It really depends on the type of application.
Graphic hungry? float or double is enough. Financial data analysis, meteor striking a planet kind of precision ? Those would need a bit of precision :)

Decimal has wider bytes, double is natively supported by CPU. Decimal is base-10, so a decimal-to-double conversion is happening while a decimal is computed.
For accounting - decimal
For finance - double
For heavy computation - double
Keep in mind .NET CLR only supports Math.Pow(double,double). Decimal is not supported.
.NET Framework 4
[SecuritySafeCritical]
public static extern double Pow(double x, double y);

A double values will serialize to scientific notation by default if that notation is shorter than the decimal display. (e.g. .00000003 will be 3e-8) Decimal values will never serialize to scientific notation. When serializing for consumption by an external party, this may be a consideration.