I have an application where I accumulate decimal values (both adding and subtracting.) I use the decimal type rather than double in order to avoid accumulation errors. However, I've run into a case where the behavior is not quite what I'd expect.
I have x = a + b, where a = 487.5M and b = 433.33333333333333333333333335M.
Computing the addition, I get x = 920.8333333333333333333333334M.
I then have y = 967.8750000000000000000000001M.
I want to assert that y - x = y - a - b. However,
y - x = 47.0416666666666666666666667
y - a - b = 47.04166666666666666666666675
I thought this kind of error was exactly what the decimal type was intended to avoid, so what's happening here?
Here is code that reproduces the issue:
static void Main()
{
decimal a = 487.5M;
decimal b = 433.33333333333333333333333335M;
decimal x = a + b;
decimal y = 967.8750000000000000000000001M;
Console.WriteLine(y - x);
Console.WriteLine(y - a - b);
if (y - x != y - a - b)
Console.WriteLine("x - y != y - a - b");
Console.ReadKey();
}
There was some discussion in comments as to why these high precisions are necessary, so I thought I'd address in summary here. For display purposes, I certainly round the results of these operations, but I use decimal for all internal representations. Some of the computations take fractions along the way, which results in numbers that are beyond the precision of the decimal type.
I take care, however, to try and keep everything stable for accumulation. So, for instance, if I split up a quantity into three thirds, I take x/3, x/3 and then (x - x/3 - x/3). This is a system that is accounting for physical quantities that are often divided up like this, so I don't want to introduce biases by rounding too soon. For instance, if I rounded the above for x=1 to three decimals, I would wind up with 0.333, 0.333, 0.334 as the three portions of the operation.
There are real physical limitations to the precision of what the system can do, but the logical accounting of what it's trying to do should ideally stay as precise as it can. The main critical requirement is that the sum total quantity of the system should not change as a result of these various operations. In the above case, I'm finding that decimal can violate this assumption, so I want to understand better why this is happening and how I can fix it.
The C# type Decimal is not like the decimal types used in COBOL, which actually store the numbers one decimal digit per nibble, and uses mathematical methods similar to doing decimal math by hand. Rather, it is a floating point type that simply assumes quantities will not get so large, so it uses fewer bits for exponents, and uses the remaining the bits of 128 rather than 64 for double to allow for greatly increased accuracy.
But being a floating point representation, even very simply fractional values are not represented exactly: 0.1, for example, requires a binary repeating fraction and may not be stored as an exact value. (It is not, for a double; Decimal may handle that particular value differently, but this is true in general.)
Therefore comparisons still need to be made using typical floating point math procedures, in which values are compared, added, subtracted, etc., by accepting them only to a certain point. Since there are approximately 23 decimal places of accuracy, select 16 as your standard, for example, and ignore those at the end.
For a good reference, read What Every Computer Scientist Should Know About Floating Point Precision.
The Decimal type is a floating-point type which has more bits of precision than any of the other types that have been built into .NET from the beginning, and whose values are all concisely representable in base-10 format. It is, however, bulky and slow, and because it is a floating-point type it is no more able to satisfy axioms typical of "precise" types (e.g. for any X and Y, (X+Y-Y)==X should either return true or throw an overflow exception). I would guess that it was made a floating-point type rather than fixed-point because of indecision regarding the number of digits that should be to the right of the decimal. In practice, it might would have been faster, and just as useful, to have a 128-bit fixed-point format, but the Decimal type is what it is.
Incidentally, languages like PL/I work well with fixed-point types because they recognize that precision is a function of a storage location rather than a value. Unfortunately, .NET does not provide any nice means via which a variable could be defined as holding a Fixed(6,3) and automatically scale and shift a Fixed(5,2) which is stored into it. Having the precision be part of the value means that storing a value into a variable will change the number of digits that variables represents to the right of the decimal place.
Related
In C# standard is said:
Conversions from int, uint, long, or ulong to float and from long or
ulong to double may cause a loss of precision, but will never cause a
loss of magnitude
can anyone explain me what does it means magnitude? and given
a number how I can calculate its magnitude? (i.e. to a long or an int).
for example:
var a = Int64.MaxValue; // a = 9223372036854775807L
var b = (float)a; // b = 9.223372037e+18
a and b have the same order of magnitude, they are very close to each other, but they are not equal...
Basically long (Int64) can represent whole numbers to an exact precision within a certain range. Floating point types sacrifice precision so they can represent numbers of a much larger range, and also fractions. So all the integral types in .net will fit into the floating points in the sense of range, but you may loos precision (digits after a certain decimal place might be incorrect. But the "scale" (order of magnitude) of the number and some of its more significant digits will be preserved...
https://en.wikipedia.org/wiki/Order_of_magnitude
It seems that it's unclear what the spec you quoted is referring to by "magnitude", and it's probably actually should be "order of magnitude".
Since Dictionary.com defines "magnitude"`as:
a number characteristic of a quantity and forming a basis for
comparison with similar quantities, as length.
and Wikipedia defines it says:
The magnitude of any number is usually called its "absolute value" or
"modulus", denoted by |x|.
you may conclude that the spec is saying that it's the quantity or actual value represented by the variable. However, as others have duly pointed out here, that is not the case.
That fact is made obvious by running a simple test (as again others here have done):
long x = 8223372036854775807; // arbitrary long number
double y = x; // implicit conversion to double
long z = Convert.ToInt64(y); // convert back to int64 (a.k.a. long)
System.Diagnostics.Debug.Print(x.ToString());
System.Diagnostics.Debug.Print(z.ToString());
This produces the output:
8223372036854775807
8223372036854775808
So, from this, you can see that the specification, while vague and imprecise, does not mean the definition of "magnitude" as defined by the dictionary or Wikipedia, but more closely resembles the definition of "order of magnitude". Specifically:
Orders of magnitude are written in powers of 10
and
Orders of magnitude are used to make approximate comparisons.
and
Two numbers of the same order of magnitude have roughly the same scale.
Which comports with the C# spec in question and also to the results we've seen from tests.
What is the maximum double value that can be represented\converted to a decimal?
How can this value be derived - example please.
Update
Given a maximum value for a double that can be converted to a decimal, I would expect to be able to round-trip the double to a decimal, and then back again. However, given a figure such as (2^52)-1 as in #Jirka's answer, this does not work. For example:
Test]
public void round_trip_double_to_decimal()
{
double maxDecimalAsDouble = (Math.Pow(2, 52) - 1);
decimal toDecimal = Convert.ToDecimal(maxDecimalAsDouble);
double toDouble = Convert.ToDouble(toDecimal);
//Fails.
Assert.That(toDouble, Is.EqualTo(maxDecimalAsDouble));
}
All integers between -9,007,199,254,740,992 and 9,007,199,254,740,991 can be exactly represented in a double. (Keep reading, though.)
The upper bound is derived as 2^53 - 1. The internal representation of it is something like (0x1.fffffffffffff * 2^52) if you pardon my hexadecimal syntax.
Outside of this range, many integers can be still exactly represented if they are a multiple of a power of two.
The highest integer whatsoever that can be accurately represented would therefore be 9,007,199,254,740,991 * (2 ^ 1023), which is even higher than Decimal.MaxValue but this is a pretty meaningless fact, given that the value does not bother to change, for example, when you subtract 1 in double arithmetic.
Based on the comments and further research, I am adding info on .NET and Mono implementations of C# that relativizes most conclusions you and I might want to make.
Math.Pow does not seem to guarantee any particular accuracy and it seems to deliver a bit or two fewer than what a double can represent. This is not too surprising with a floating point function. The Intel floating point hardware does not have an instruction for exponentiation and I expect that the computation involves logarithm and multiplication instructions, where intermediate results lose some precision. One would use BigInteger.Pow if integral accuracy was desired.
However, even (decimal)(double)9007199254740991M results in a round trip violation. This time it is, however, a known bug, a direct violation of Section 6.2.1 of the C# spec. Interestingly I see the same bug even in Mono 2.8. (The referenced source shows that this conversion bug can hit even with much lower values.)
Double literals are less rounded, but still a little: 9007199254740991D prints out as 9007199254740990D. This is an artifact of internal multiplication by 10 when parsing the string literal (before the upper and lower bound converge to the same double value based on the "first zero after the decimal point"). This again violates the C# spec, this time Section 9.4.4.3.
Unlike C, C# has no hexadecimal floating point literals, so we cannot avoid that multiplication by 10 by any other syntax, except perhaps by going through Decimal or BigInteger, if these only provided accurate conversion operators. I have not tested BigInteger.
The above could almost make you wonder whether C# does not invent its own unique floating point format with reduced precision. No, Section 11.1.6 references 64bit IEC 60559 representation. So the above are indeed bugs.
So, to conclude, you should be able to fit even 9007199254740991M in a double precisely, but it's quite a challenge to get the value in place!
The moral of the story is that the traditional belief that "Arithmetic should be barely more precise than the data and the desired result" is wrong, as this famous article demonstrates (page 36), albeit in the context of a different programming language.
Don't store integers in floating point variables unless you have to.
MSDN Double data type
Decimal vs double
The value of Decimal.MaxValue is positive 79,228,162,514,264,337,593,543,950,335.
Refreshing on floating points (also PDF), IEEE-754 and taking part in this discussion on floating point rounding when converting to strings, brought me to tinker: how can I get the maximum and minimum value for a given floating point number whose binary representations are equal.
Disclaimer: for this discussion, I like to stick to 32 bit and 64 bit floating point as described by IEEE-754. I'm not interested in extended floating point (80-bits) or quads (128 bits IEEE-754-2008) or any other standard (IEEE-854).
Background: Computers are bad at representing 0.1 in binary representation. In C#, a float represents this as 3DCCCCCD internally (C# uses round-to-nearest) and a double as 3FB999999999999A. The same bit patterns are used for decimal 0.100000005 (float) and 0.1000000000000000124 (double), but not for 0.1000000000000000144 (double).
For convenience, the following C# code gives these internal representations:
string GetHex(float f)
{
return BitConverter.ToUInt32(BitConverter.GetBytes(f), 0).ToString("X");
}
string GetHex(double d)
{
return BitConverter.ToUInt64(BitConverter.GetBytes(d), 0).ToString("X");
}
// float
Console.WriteLine(GetHex(0.1F));
// double
Console.WriteLine(GetHex(0.1));
In the case of 0.1, there is no lower decimal number that is represented with the same bit pattern, any 0.99...99 will yield a different bit representation (i.e., float for 0.999999937 yields 3F7FFFFF internally).
My question is simple: how can I find the lowest and highest decimal value for a given float (or double) that is internally stored in the same binary representation.
Why: (I know you'll ask) to find the error in rounding in .NET when it converts to a string and when it converts from a string, to find the internal exact value and to understand my own rounding errors better.
My guess is something like: take the mantissa, remove the rest, get its exact value, get one (mantissa-bit) higher, and calculate the mean: anything below that will yield the same bit pattern. My main problem is: how to get the fractional part as integer (bit manipulation it not my strongest asset). Jon Skeet's DoubleConverter class may be helpful.
One way to get at your question is to find the size of an ULP, or Unit in the Last Place, of your floating-point number. Simplifying a little bit, this is the distance between a given floating-point number and the next larger number. Again, simplifying a little bit, given a representable floating-point value x, any decimal string whose value is between (x - 1/2 ulp) and (x + 1/2 ulp) will be rounded to x when converted to a floating-point value.
The trick is that (x +/- 1/2 ulp) is not a representable floating-point number, so actually calculating its value requires that you use a wider floating-point type (if one is available) or an arbitrary width big decimal or similar type to do the computation.
How do you find the size of an ulp? One relatively easy way is roughly what you suggested, written here is C-ish pseudocode because I don't know C#:
float absX = absoluteValue(x);
uint32_t bitPattern = getRepresentationOfFloat(absx);
bitPattern++;
float nextFloatNumber = getFloatFromRepresentation(bitPattern);
float ulpOfX = (nextFloatNumber - absX);
This works because adding one to the bit pattern of x exactly corresponds to adding one ulp to the value of x. No floating-point rounding occurs in the subtraction because the values involved are so close (in particular, there is a theorem of ieee-754 floating-point arithmetic that if two numbers x and y satisfy y/2 <= x <= 2y, then x - y is computed exactly). The only caveats here are:
if x happens to be the largest finite floating point number, this won't work (it will return inf, which is clearly wrong).
if your platform does not correctly support gradual underflow (say an embedded device running in flush-to-zero mode), this won't work for very small values of x.
It sounds like you're not likely to be in either of those situations, so this should work just fine for your purposes.
Now that you know what an ulp of x is, you can find the interval of values that rounds to x. You can compute ulp(x)/2 exactly in floating-point, because floating-point division by 2 is exact (again, barring underflow). Then you need only compute the value of x +/- ulp(x)/2 suitable larger floating-point type (double will work if you're interested in float) or in a Big Decimal type, and you have your interval.
I made a few simplifying assumptions through this explanation. If you need this to really be spelled out exactly, leave a comment and I'll expand on the sections that are a bit fuzzy when I get the chance.
One other note the following statement in your question:
In the case of 0.1, there is no lower
decimal number that is represented
with the same bit pattern
is incorrect. You just happened to be looking at the wrong values (0.999999... instead of 0.099999... -- an easy typo to make).
Python 3.1 just implemented something like this: see the changelog (scroll down a bit), bug report.
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.
I always tell in c# a variable of type double is not suitable for money. All weird things could happen. But I can't seem to create an example to demonstrate some of these issues. Can anyone provide such an example?
(edit; this post was originally tagged C#; some replies refer to specific details of decimal, which therefore means System.Decimal).
(edit 2: I was specific asking for some c# code, so I don't think this is language agnostic only)
Very, very unsuitable. Use decimal.
double x = 3.65, y = 0.05, z = 3.7;
Console.WriteLine((x + y) == z); // false
(example from Jon's page here - recommended reading ;-p)
You will get odd errors effectively caused by rounding. In addition, comparisons with exact values are extremely tricky - you usually need to apply some sort of epsilon to check for the actual value being "near" a particular one.
Here's a concrete example:
using System;
class Test
{
static void Main()
{
double x = 0.1;
double y = x + x + x;
Console.WriteLine(y == 0.3); // Prints False
}
}
Yes it's unsuitable.
If I remember correctly double has about 17 significant numbers, so normally rounding errors will take place far behind the decimal point. Most financial software uses 4 decimals behind the decimal point, that leaves 13 decimals to work with so the maximum number you can work with for single operations is still very much higher than the USA national debt. But rounding errors will add up over time. If your software runs for a long time you'll eventually start losing cents. Certain operations will make this worse. For example adding large amounts to small amounts will cause a significant loss of precision.
You need fixed point datatypes for money operations, most people don't mind if you lose a cent here and there but accountants aren't like most people..
edit
According to this site http://msdn.microsoft.com/en-us/library/678hzkk9.aspx Doubles actually have 15 to 16 significant digits instead of 17.
#Jon Skeet decimal is more suitable than double because of its higher precision, 28 or 29 significant decimals. That means less chance of accumulated rounding errors becoming significant. Fixed point datatypes (ie integers that represent cents or 100th of a cent like I've seen used) like Boojum mentions are actually better suited.
Since decimal uses a scaling factor of multiples of 10, numbers like 0.1 can be represented exactly. In essence, the decimal type represents this as 1 / 10 ^ 1, whereas a double would represent this as 104857 / 2 ^ 20 (in reality it would be more like really-big-number / 2 ^ 1023).
A decimal can exactly represent any base 10 value with up to 28/29 significant digits (like 0.1). A double can't.
My understanding is that most financial systems express currency using integers -- i.e., counting everything in cents.
IEEE double precision actually can represent all integers exactly in the range -2^53 through +2^53. (Hacker's Delight, pg. 262) If you use only addition, subtraction and multiplication, and keep everything to integers within this range then you should see no loss of precision. I'd be very wary of division or more complex operations, however.
Using double when you don't know what you are doing is unsuitable.
"double" can represent an amount of a trillion dollars with an error of 1/90th of a cent. So you will get highly precise results. Want to calculate how much it costs to put a man on Mars and get him back alive? double will do just fine.
But with money there are often very specific rules saying that a certain calculation must give a certain result and no other. If you calculate an amount that is very very very close to $98.135 then there will often be a rule that determines whether the result should be $98.14 or $98.13 and you must follow that rule and get the result that is required.
Depending on where you live, using 64 bit integers to represent cents or pennies or kopeks or whatever is the smallest unit in your country will usually work just fine. For example, 64 bit signed integers representing cents can represent values up to 92,223 trillion dollars. 32 bit integers are usually unsuitable.
No a double will always have rounding errors, use "decimal" if you're on .Net...
Actually floating-point double is perfectly well suited to representing amounts of money as long as you pick a suitable unit.
See http://www.idinews.com/moneyRep.html
So is fixed-point long. Either consumes 8 bytes, surely preferable to the 16 consumed by a decimal item.
Whether or not something works (i.e. yields the expected and correct result) is not a matter of either voting or individual preference. A technique either works or it doesn't.