Consider the following code:
0.1 + 0.2 == 0.3 -> false
0.1 + 0.2 -> 0.30000000000000004
Why do these inaccuracies happen?
Binary floating point math is like this. In most programming languages, it is based on the IEEE 754 standard. The crux of the problem is that numbers are represented in this format as a whole number times a power of two; rational numbers (such as 0.1
, which is 1/10
) whose denominator is not a power of two cannot be exactly represented.
For 0.1
in the standard binary64
format, the representation can be written exactly as
0.1000000000000000055511151231257827021181583404541015625
in decimal, or0x1.999999999999ap-4
in C99 hexfloat notation.In contrast, the rational number 0.1
, which is 1/10
, can be written exactly as
0.1
in decimal, or0x1.99999999999999...p-4
in an analogue of C99 hexfloat notation, where the ...
represents an unending sequence of 9's.The constants 0.2
and 0.3
in your program will also be approximations to their true values. It happens that the closest double
to 0.2
is larger than the rational number 0.2
but that the closest double
to 0.3
is smaller than the rational number 0.3
. The sum of 0.1
and 0.2
winds up being larger than the rational number 0.3
and hence disagreeing with the constant in your code.
A fairly comprehensive treatment of floating-point arithmetic issues is What Every Computer Scientist Should Know About Floating-Point Arithmetic. For an easier-to-digest explanation, see floating-point-gui.de.
Side Note: All positional (base-N) number systems share this problem with precision
Plain old decimal (base 10) numbers have the same issues, which is why numbers like 1/3 end up as 0.333333333...
You've just stumbled on a number (3/10) that happens to be easy to represent with the decimal system, but doesn't fit the binary system. It goes both ways (to some small degree) as well: 1/16 is an ugly number in decimal (0.0625), but in binary it looks as neat as a 10,000th does in decimal (0.0001)** - if we were in the habit of using a base-2 number system in our daily lives, you'd even look at that number and instinctively understand you could arrive there by halving something, halving it again, and again and again.
** Of course, that's not exactly how floating-point numbers are stored in memory (they use a form of scientific notation). However, it does illustrate the point that binary floating-point precision errors tend to crop up because the "real world" numbers we are usually interested in working with are so often powers of ten - but only because we use a decimal number system day-to-day. This is also why we'll say things like 71% instead of "5 out of every 7" (71% is an approximation, since 5/7 can't be represented exactly with any decimal number).
So no: binary floating point numbers are not broken, they just happen to be as imperfect as every other base-N number system :)
Side Side Note: Working with Floats in Programming
In practice, this problem of precision means you need to use rounding functions to round your floating point numbers off to however many decimal places you're interested in before you display them.
You also need to replace equality tests with comparisons that allow some amount of tolerance, which means:
Do not do if (x == y) { ... }
Instead do if (abs(x - y) < myToleranceValue) { ... }
.
where abs
is the absolute value. myToleranceValue
needs to be chosen for your particular application - and it will have a lot to do with how much "wiggle room" you are prepared to allow, and what the largest number you are going to be comparing may be (due to loss of precision issues). Beware of "epsilon" style constants in your language of choice. These are not to be used as tolerance values.
I think "some error constant" is more correct than "The Epsilon" because there is no "The Epsilon" which could be used in all cases. Different epsilons need to be used in different situations. And the machine epsilon is almost never a good constant to use.
It's not quite true that all floating-point math is based on the IEEE [754] standard. There are still some systems in use that have the old IBM hexadecimal FP, for example, and there are still graphics cards that don't support IEEE-754 arithmetic. It's true to a reasonably approximation, however.
Cray ditched IEEE-754 compliance for speed. Java loosened its adherence as an optimization as well.
I think you should add something to this answer about how computations on money should always, always be done with fixed-point arithmetic on integers, because money is quantized. (It may make sense to do internal accounting computations in tiny fractions of a cent, or whatever your smallest currency unit is - this often helps with e.g. reducing round-off error when converting "$29.99 a month" to a daily rate - but it should still be fixed-point arithmetic.)
Interesting fact: this very 0.1 not being exactly represented in binary floating-point caused an infamous Patriot missile software bug which resulted in 28 people killed during the first Iraq war.