Related
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, or
0x1.999999999999ap-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, or
0x1.99999999999999...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 can be used as tolerance values but their effectiveness depends on the magnitude (size) of the numbers you're working with, since calculations with large numbers may exceed the epsilon threshold.
A Hardware Designer's Perspective
I believe I should add a hardware designer’s perspective to this since I design and build floating point hardware. Knowing the origin of the error may help in understanding what is happening in the software, and ultimately, I hope this helps explain the reasons for why floating point errors happen and seem to accumulate over time.
1. Overview
From an engineering perspective, most floating point operations will have some element of error since the hardware that does the floating point computations is only required to have an error of less than one half of one unit in the last place. Therefore, much hardware will stop at a precision that's only necessary to yield an error of less than one half of one unit in the last place for a single operation which is especially problematic in floating point division. What constitutes a single operation depends upon how many operands the unit takes. For most, it is two, but some units take 3 or more operands. Because of this, there is no guarantee that repeated operations will result in a desirable error since the errors add up over time.
2. Standards
Most processors follow the IEEE-754 standard but some use denormalized, or different standards
. For example, there is a denormalized mode in IEEE-754 which allows representation of very small floating point numbers at the expense of precision. The following, however, will cover the normalized mode of IEEE-754 which is the typical mode of operation.
In the IEEE-754 standard, hardware designers are allowed any value of error/epsilon as long as it's less than one half of one unit in the last place, and the result only has to be less than one half of one unit in the last place for one operation. This explains why when there are repeated operations, the errors add up. For IEEE-754 double precision, this is the 54th bit, since 53 bits are used to represent the numeric part (normalized), also called the mantissa, of the floating point number (e.g. the 5.3 in 5.3e5). The next sections go into more detail on the causes of hardware error on various floating point operations.
3. Cause of Rounding Error in Division
The main cause of the error in floating point division is the division algorithms used to calculate the quotient. Most computer systems calculate division using multiplication by an inverse, mainly in Z=X/Y, Z = X * (1/Y). A division is computed iteratively i.e. each cycle computes some bits of the quotient until the desired precision is reached, which for IEEE-754 is anything with an error of less than one unit in the last place. The table of reciprocals of Y (1/Y) is known as the quotient selection table (QST) in the slow division, and the size in bits of the quotient selection table is usually the width of the radix, or a number of bits of the quotient computed in each iteration, plus a few guard bits. For the IEEE-754 standard, double precision (64-bit), it would be the size of the radix of the divider, plus a few guard bits k, where k>=2. So for example, a typical Quotient Selection Table for a divider that computes 2 bits of the quotient at a time (radix 4) would be 2+2= 4 bits (plus a few optional bits).
3.1 Division Rounding Error: Approximation of Reciprocal
What reciprocals are in the quotient selection table depend on the division method: slow division such as SRT division, or fast division such as Goldschmidt division; each entry is modified according to the division algorithm in an attempt to yield the lowest possible error. In any case, though, all reciprocals are approximations of the actual reciprocal and introduce some element of error. Both slow division and fast division methods calculate the quotient iteratively, i.e. some number of bits of the quotient are calculated each step, then the result is subtracted from the dividend, and the divider repeats the steps until the error is less than one half of one unit in the last place. Slow division methods calculate a fixed number of digits of the quotient in each step and are usually less expensive to build, and fast division methods calculate a variable number of digits per step and are usually more expensive to build. The most important part of the division methods is that most of them rely upon repeated multiplication by an approximation of a reciprocal, so they are prone to error.
4. Rounding Errors in Other Operations: Truncation
Another cause of the rounding errors in all operations are the different modes of truncation of the final answer that IEEE-754 allows. There's truncate, round-towards-zero, round-to-nearest (default), round-down, and round-up. All methods introduce an element of error of less than one unit in the last place for a single operation. Over time and repeated operations, truncation also adds cumulatively to the resultant error. This truncation error is especially problematic in exponentiation, which involves some form of repeated multiplication.
5. Repeated Operations
Since the hardware that does the floating point calculations only needs to yield a result with an error of less than one half of one unit in the last place for a single operation, the error will grow over repeated operations if not watched. This is the reason that in computations that require a bounded error, mathematicians use methods such as using the round-to-nearest even digit in the last place of IEEE-754, because, over time, the errors are more likely to cancel each other out, and Interval Arithmetic combined with variations of the IEEE 754 rounding modes to predict rounding errors, and correct them. Because of its low relative error compared to other rounding modes, round to nearest even digit (in the last place), is the default rounding mode of IEEE-754.
Note that the default rounding mode, round-to-nearest even digit in the last place, guarantees an error of less than one half of one unit in the last place for one operation. Using the truncation, round-up, and round down alone may result in an error that is greater than one half of one unit in the last place, but less than one unit in the last place, so these modes are not recommended unless they are used in Interval Arithmetic.
6. Summary
In short, the fundamental reason for the errors in floating point operations is a combination of the truncation in hardware, and the truncation of a reciprocal in the case of division. Since the IEEE-754 standard only requires an error of less than one half of one unit in the last place for a single operation, the floating point errors over repeated operations will add up unless corrected.
It's broken in the exact same way the decimal (base-10) notation you learned in grade school and use every day is broken, just for base-2.
To understand, think about representing 1/3 as a decimal value. It's impossible to do exactly! The world will end before you finish writing the 3's after the decimal point, and so instead we write to some number of places and consider it sufficiently accurate.
In the same way, 1/10 (decimal 0.1) cannot be represented exactly in base 2 (binary) as a "decimal" value; a repeating pattern after the decimal point goes on forever. The value is not exact, and therefore you can't do exact math with it using normal floating point methods. Just like with base 10, there are other values that exhibit this problem as well.
Most answers here address this question in very dry, technical terms. I'd like to address this in terms that normal human beings can understand.
Imagine that you are trying to slice up pizzas. You have a robotic pizza cutter that can cut pizza slices exactly in half. It can halve a whole pizza, or it can halve an existing slice, but in any case, the halving is always exact.
That pizza cutter has very fine movements, and if you start with a whole pizza, then halve that, and continue halving the smallest slice each time, you can do the halving 53 times before the slice is too small for even its high-precision abilities. At that point, you can no longer halve that very thin slice, but must either include or exclude it as is.
Now, how would you piece all the slices in such a way that would add up to one-tenth (0.1) or one-fifth (0.2) of a pizza? Really think about it, and try working it out. You can even try to use a real pizza, if you have a mythical precision pizza cutter at hand. :-)
Most experienced programmers, of course, know the real answer, which is that there is no way to piece together an exact tenth or fifth of the pizza using those slices, no matter how finely you slice them. You can do a pretty good approximation, and if you add up the approximation of 0.1 with the approximation of 0.2, you get a pretty good approximation of 0.3, but it's still just that, an approximation.
For double-precision numbers (which is the precision that allows you to halve your pizza 53 times), the numbers immediately less and greater than 0.1 are 0.09999999999999999167332731531132594682276248931884765625 and 0.1000000000000000055511151231257827021181583404541015625. The latter is quite a bit closer to 0.1 than the former, so a numeric parser will, given an input of 0.1, favour the latter.
(The difference between those two numbers is the "smallest slice" that we must decide to either include, which introduces an upward bias, or exclude, which introduces a downward bias. The technical term for that smallest slice is an ulp.)
In the case of 0.2, the numbers are all the same, just scaled up by a factor of 2. Again, we favour the value that's slightly higher than 0.2.
Notice that in both cases, the approximations for 0.1 and 0.2 have a slight upward bias. If we add enough of these biases in, they will push the number further and further away from what we want, and in fact, in the case of 0.1 + 0.2, the bias is high enough that the resulting number is no longer the closest number to 0.3.
In particular, 0.1 + 0.2 is really 0.1000000000000000055511151231257827021181583404541015625 + 0.200000000000000011102230246251565404236316680908203125 = 0.3000000000000000444089209850062616169452667236328125, whereas the number closest to 0.3 is actually 0.299999999999999988897769753748434595763683319091796875.
P.S. Some programming languages also provide pizza cutters that can split slices into exact tenths. Although such pizza cutters are uncommon, if you do have access to one, you should use it when it's important to be able to get exactly one-tenth or one-fifth of a slice.
(Originally posted on Quora.)
Floating point rounding errors. 0.1 cannot be represented as accurately in base-2 as in base-10 due to the missing prime factor of 5. Just as 1/3 takes an infinite number of digits to represent in decimal, but is "0.1" in base-3, 0.1 takes an infinite number of digits in base-2 where it does not in base-10. And computers don't have an infinite amount of memory.
My answer is quite long, so I've split it into three sections. Since the question is about floating point mathematics, I've put the emphasis on what the machine actually does. I've also made it specific to double (64 bit) precision, but the argument applies equally to any floating point arithmetic.
Preamble
An IEEE 754 double-precision binary floating-point format (binary64) number represents a number of the form
value = (-1)^s * (1.m51m50...m2m1m0)2 * 2e-1023
in 64 bits:
The first bit is the sign bit: 1 if the number is negative, 0 otherwise1.
The next 11 bits are the exponent, which is offset by 1023. In other words, after reading the exponent bits from a double-precision number, 1023 must be subtracted to obtain the power of two.
The remaining 52 bits are the significand (or mantissa). In the mantissa, an 'implied' 1. is always2 omitted since the most significant bit of any binary value is 1.
1 - IEEE 754 allows for the concept of a signed zero - +0 and -0 are treated differently: 1 / (+0) is positive infinity; 1 / (-0) is negative infinity. For zero values, the mantissa and exponent bits are all zero. Note: zero values (+0 and -0) are explicitly not classed as denormal2.
2 - This is not the case for denormal numbers, which have an offset exponent of zero (and an implied 0.). The range of denormal double precision numbers is dmin ≤ |x| ≤ dmax, where dmin (the smallest representable nonzero number) is 2-1023 - 51 (≈ 4.94 * 10-324) and dmax (the largest denormal number, for which the mantissa consists entirely of 1s) is 2-1023 + 1 - 2-1023 - 51 (≈ 2.225 * 10-308).
Turning a double precision number to binary
Many online converters exist to convert a double precision floating point number to binary (e.g. at binaryconvert.com), but here is some sample C# code to obtain the IEEE 754 representation for a double precision number (I separate the three parts with colons (:):
public static string BinaryRepresentation(double value)
{
long valueInLongType = BitConverter.DoubleToInt64Bits(value);
string bits = Convert.ToString(valueInLongType, 2);
string leadingZeros = new string('0', 64 - bits.Length);
string binaryRepresentation = leadingZeros + bits;
string sign = binaryRepresentation[0].ToString();
string exponent = binaryRepresentation.Substring(1, 11);
string mantissa = binaryRepresentation.Substring(12);
return string.Format("{0}:{1}:{2}", sign, exponent, mantissa);
}
Getting to the point: the original question
(Skip to the bottom for the TL;DR version)
Cato Johnston (the question asker) asked why 0.1 + 0.2 != 0.3.
Written in binary (with colons separating the three parts), the IEEE 754 representations of the values are:
0.1 => 0:01111111011:1001100110011001100110011001100110011001100110011010
0.2 => 0:01111111100:1001100110011001100110011001100110011001100110011010
Note that the mantissa is composed of recurring digits of 0011. This is key to why there is any error to the calculations - 0.1, 0.2 and 0.3 cannot be represented in binary precisely in a finite number of binary bits any more than 1/9, 1/3 or 1/7 can be represented precisely in decimal digits.
Also note that we can decrease the power in the exponent by 52 and shift the point in the binary representation to the right by 52 places (much like 10-3 * 1.23 == 10-5 * 123). This then enables us to represent the binary representation as the exact value that it represents in the form a * 2p. where 'a' is an integer.
Converting the exponents to decimal, removing the offset, and re-adding the implied 1 (in square brackets), 0.1 and 0.2 are:
0.1 => 2^-4 * [1].1001100110011001100110011001100110011001100110011010
0.2 => 2^-3 * [1].1001100110011001100110011001100110011001100110011010
or
0.1 => 2^-56 * 7205759403792794 = 0.1000000000000000055511151231257827021181583404541015625
0.2 => 2^-55 * 7205759403792794 = 0.200000000000000011102230246251565404236316680908203125
To add two numbers, the exponent needs to be the same, i.e.:
0.1 => 2^-3 * 0.1100110011001100110011001100110011001100110011001101(0)
0.2 => 2^-3 * 1.1001100110011001100110011001100110011001100110011010
sum = 2^-3 * 10.0110011001100110011001100110011001100110011001100111
or
0.1 => 2^-55 * 3602879701896397 = 0.1000000000000000055511151231257827021181583404541015625
0.2 => 2^-55 * 7205759403792794 = 0.200000000000000011102230246251565404236316680908203125
sum = 2^-55 * 10808639105689191 = 0.3000000000000000166533453693773481063544750213623046875
Since the sum is not of the form 2n * 1.{bbb} we increase the exponent by one and shift the decimal (binary) point to get:
sum = 2^-2 * 1.0011001100110011001100110011001100110011001100110011(1)
= 2^-54 * 5404319552844595.5 = 0.3000000000000000166533453693773481063544750213623046875
There are now 53 bits in the mantissa (the 53rd is in square brackets in the line above). The default rounding mode for IEEE 754 is 'Round to Nearest' - i.e. if a number x falls between two values a and b, the value where the least significant bit is zero is chosen.
a = 2^-54 * 5404319552844595 = 0.299999999999999988897769753748434595763683319091796875
= 2^-2 * 1.0011001100110011001100110011001100110011001100110011
x = 2^-2 * 1.0011001100110011001100110011001100110011001100110011(1)
b = 2^-2 * 1.0011001100110011001100110011001100110011001100110100
= 2^-54 * 5404319552844596 = 0.3000000000000000444089209850062616169452667236328125
Note that a and b differ only in the last bit; ...0011 + 1 = ...0100. In this case, the value with the least significant bit of zero is b, so the sum is:
sum = 2^-2 * 1.0011001100110011001100110011001100110011001100110100
= 2^-54 * 5404319552844596 = 0.3000000000000000444089209850062616169452667236328125
whereas the binary representation of 0.3 is:
0.3 => 2^-2 * 1.0011001100110011001100110011001100110011001100110011
= 2^-54 * 5404319552844595 = 0.299999999999999988897769753748434595763683319091796875
which only differs from the binary representation of the sum of 0.1 and 0.2 by 2-54.
The binary representation of 0.1 and 0.2 are the most accurate representations of the numbers allowable by IEEE 754. The addition of these representation, due to the default rounding mode, results in a value which differs only in the least-significant-bit.
TL;DR
Writing 0.1 + 0.2 in a IEEE 754 binary representation (with colons separating the three parts) and comparing it to 0.3, this is (I've put the distinct bits in square brackets):
0.1 + 0.2 => 0:01111111101:0011001100110011001100110011001100110011001100110[100]
0.3 => 0:01111111101:0011001100110011001100110011001100110011001100110[011]
Converted back to decimal, these values are:
0.1 + 0.2 => 0.300000000000000044408920985006...
0.3 => 0.299999999999999988897769753748...
The difference is exactly 2-54, which is ~5.5511151231258 × 10-17 - insignificant (for many applications) when compared to the original values.
Comparing the last few bits of a floating point number is inherently dangerous, as anyone who reads the famous "What Every Computer Scientist Should Know About Floating-Point Arithmetic" (which covers all the major parts of this answer) will know.
Most calculators use additional guard digits to get around this problem, which is how 0.1 + 0.2 would give 0.3: the final few bits are rounded.
In addition to the other correct answers, you may want to consider scaling your values to avoid problems with floating-point arithmetic.
For example:
var result = 1.0 + 2.0; // result === 3.0 returns true
... instead of:
var result = 0.1 + 0.2; // result === 0.3 returns false
The expression 0.1 + 0.2 === 0.3 returns false in JavaScript, but fortunately integer arithmetic in floating-point is exact, so decimal representation errors can be avoided by scaling.
As a practical example, to avoid floating-point problems where accuracy is paramount, it is recommended1 to handle money as an integer representing the number of cents: 2550 cents instead of 25.50 dollars.
1 Douglas Crockford: JavaScript: The Good Parts: Appendix A - Awful Parts (page 105).
Floating point numbers stored in the computer consist of two parts, an integer and an exponent that the base is taken to and multiplied by the integer part.
If the computer were working in base 10, 0.1 would be 1 x 10⁻¹, 0.2 would be 2 x 10⁻¹, and 0.3 would be 3 x 10⁻¹. Integer math is easy and exact, so adding 0.1 + 0.2 will obviously result in 0.3.
Computers don't usually work in base 10, they work in base 2. You can still get exact results for some values, for example 0.5 is 1 x 2⁻¹ and 0.25 is 1 x 2⁻², and adding them results in 3 x 2⁻², or 0.75. Exactly.
The problem comes with numbers that can be represented exactly in base 10, but not in base 2. Those numbers need to be rounded to their closest equivalent. Assuming the very common IEEE 64-bit floating point format, the closest number to 0.1 is 3602879701896397 x 2⁻⁵⁵, and the closest number to 0.2 is 7205759403792794 x 2⁻⁵⁵; adding them together results in 10808639105689191 x 2⁻⁵⁵, or an exact decimal value of 0.3000000000000000444089209850062616169452667236328125. Floating point numbers are generally rounded for display.
In short it's because:
Floating point numbers cannot represent all decimals precisely in binary
So just like 10/3 which does not exist in base 10 precisely (it will be 3.33... recurring), in the same way 1/10 doesn't exist in binary.
So what? How to deal with it? Is there any workaround?
In order to offer The best solution I can say I discovered following method:
parseFloat((0.1 + 0.2).toFixed(10)) => Will return 0.3
Let me explain why it's the best solution.
As others mentioned in above answers it's a good idea to use ready to use Javascript toFixed() function to solve the problem. But most likely you'll encounter with some problems.
Imagine you are going to add up two float numbers like 0.2 and 0.7 here it is: 0.2 + 0.7 = 0.8999999999999999.
Your expected result was 0.9 it means you need a result with 1 digit precision in this case.
So you should have used (0.2 + 0.7).tofixed(1)
but you can't just give a certain parameter to toFixed() since it depends on the given number, for instance
0.22 + 0.7 = 0.9199999999999999
In this example you need 2 digits precision so it should be toFixed(2), so what should be the paramter to fit every given float number?
You might say let it be 10 in every situation then:
(0.2 + 0.7).toFixed(10) => Result will be "0.9000000000"
Damn! What are you going to do with those unwanted zeros after 9?
It's the time to convert it to float to make it as you desire:
parseFloat((0.2 + 0.7).toFixed(10)) => Result will be 0.9
Now that you found the solution, it's better to offer it as a function like this:
function floatify(number){
return parseFloat((number).toFixed(10));
}
Let's try it yourself:
function floatify(number){
return parseFloat((number).toFixed(10));
}
function addUp(){
var number1 = +$("#number1").val();
var number2 = +$("#number2").val();
var unexpectedResult = number1 + number2;
var expectedResult = floatify(number1 + number2);
$("#unexpectedResult").text(unexpectedResult);
$("#expectedResult").text(expectedResult);
}
addUp();
input{
width: 50px;
}
#expectedResult{
color: green;
}
#unexpectedResult{
color: red;
}
<script src="https://ajax.googleapis.com/ajax/libs/jquery/2.1.1/jquery.min.js"></script>
<input id="number1" value="0.2" onclick="addUp()" onkeyup="addUp()"/> +
<input id="number2" value="0.7" onclick="addUp()" onkeyup="addUp()"/> =
<p>Expected Result: <span id="expectedResult"></span></p>
<p>Unexpected Result: <span id="unexpectedResult"></span></p>
You can use it this way:
var x = 0.2 + 0.7;
floatify(x); => Result: 0.9
As W3SCHOOLS suggests there is another solution too, you can multiply and divide to solve the problem above:
var x = (0.2 * 10 + 0.1 * 10) / 10; // x will be 0.3
Keep in mind that (0.2 + 0.1) * 10 / 10 won't work at all although it seems the same!
I prefer the first solution since I can apply it as a function which converts the input float to accurate output float.
FYI, the same problem exists for multiplication, for instance 0.09 * 10 returns 0.8999999999999999. Apply the flotify function as a workaround: flotify(0.09 * 10) returns 0.9
Floating point rounding error. From What Every Computer Scientist Should Know About Floating-Point Arithmetic:
Squeezing infinitely many real numbers into a finite number of bits requires an approximate representation. Although there are infinitely many integers, in most programs the result of integer computations can be stored in 32 bits. In contrast, given any fixed number of bits, most calculations with real numbers will produce quantities that cannot be exactly represented using that many bits. Therefore the result of a floating-point calculation must often be rounded in order to fit back into its finite representation. This rounding error is the characteristic feature of floating-point computation.
My workaround:
function add(a, b, precision) {
var x = Math.pow(10, precision || 2);
return (Math.round(a * x) + Math.round(b * x)) / x;
}
precision refers to the number of digits you want to preserve after the decimal point during addition.
No, not broken, but most decimal fractions must be approximated
Summary
Floating point arithmetic is exact, unfortunately, it doesn't match up well with our usual base-10 number representation, so it turns out we are often giving it input that is slightly off from what we wrote.
Even simple numbers like 0.01, 0.02, 0.03, 0.04 ... 0.24 are not representable exactly as binary fractions. If you count up 0.01, .02, .03 ..., not until you get to 0.25 will you get the first fraction representable in base2. If you tried that using FP, your 0.01 would have been slightly off, so the only way to add 25 of them up to a nice exact 0.25 would have required a long chain of causality involving guard bits and rounding. It's hard to predict so we throw up our hands and say "FP is inexact", but that's not really true.
We constantly give the FP hardware something that seems simple in base 10 but is a repeating fraction in base 2.
How did this happen?
When we write in decimal, every fraction (specifically, every terminating decimal) is a rational number of the form
a / (2n x 5m)
In binary, we only get the 2n term, that is:
a / 2n
So in decimal, we can't represent 1/3. Because base 10 includes 2 as a prime factor, every number we can write as a binary fraction also can be written as a base 10 fraction. However, hardly anything we write as a base10 fraction is representable in binary. In the range from 0.01, 0.02, 0.03 ... 0.99, only three numbers can be represented in our FP format: 0.25, 0.50, and 0.75, because they are 1/4, 1/2, and 3/4, all numbers with a prime factor using only the 2n term.
In base10 we can't represent 1/3. But in binary, we can't do 1/10 or 1/3.
So while every binary fraction can be written in decimal, the reverse is not true. And in fact most decimal fractions repeat in binary.
Dealing with it
Developers are usually instructed to do < epsilon comparisons, better advice might be to round to integral values (in the C library: round() and roundf(), i.e., stay in the FP format) and then compare. Rounding to a specific decimal fraction length solves most problems with output.
Also, on real number-crunching problems (the problems that FP was invented for on early, frightfully expensive computers) the physical constants of the universe and all other measurements are only known to a relatively small number of significant figures, so the entire problem space was "inexact" anyway. FP "accuracy" isn't a problem in this kind of application.
The whole issue really arises when people try to use FP for bean counting. It does work for that, but only if you stick to integral values, which kind of defeats the point of using it. This is why we have all those decimal fraction software libraries.
I love the Pizza answer by Chris, because it describes the actual problem, not just the usual handwaving about "inaccuracy". If FP were simply "inaccurate", we could fix that and would have done it decades ago. The reason we haven't is because the FP format is compact and fast and it's the best way to crunch a lot of numbers. Also, it's a legacy from the space age and arms race and early attempts to solve big problems with very slow computers using small memory systems. (Sometimes, individual magnetic cores for 1-bit storage, but that's another story.)
Conclusion
If you are just counting beans at a bank, software solutions that use decimal string representations in the first place work perfectly well. But you can't do quantum chromodynamics or aerodynamics that way.
A lot of good answers have been posted, but I'd like to append one more.
Not all numbers can be represented via floats/doubles
For example, the number "0.2" will be represented as "0.200000003" in single precision in IEEE754 float point standard.
Model for store real numbers under the hood represent float numbers as
Even though you can type 0.2 easily, FLT_RADIX and DBL_RADIX is 2; not 10 for a computer with FPU which uses "IEEE Standard for Binary Floating-Point Arithmetic (ISO/IEEE Std 754-1985)".
So it is a bit hard to represent such numbers exactly. Even if you specify this variable explicitly without any intermediate calculation.
Some statistics related to this famous double precision question.
When adding all values (a + b) using a step of 0.1 (from 0.1 to 100) we have ~15% chance of precision error. Note that the error could result in slightly bigger or smaller values.
Here are some examples:
0.1 + 0.2 = 0.30000000000000004 (BIGGER)
0.1 + 0.7 = 0.7999999999999999 (SMALLER)
...
1.7 + 1.9 = 3.5999999999999996 (SMALLER)
1.7 + 2.2 = 3.9000000000000004 (BIGGER)
...
3.2 + 3.6 = 6.800000000000001 (BIGGER)
3.2 + 4.4 = 7.6000000000000005 (BIGGER)
When subtracting all values (a - b where a > b) using a step of 0.1 (from 100 to 0.1) we have ~34% chance of precision error.
Here are some examples:
0.6 - 0.2 = 0.39999999999999997 (SMALLER)
0.5 - 0.4 = 0.09999999999999998 (SMALLER)
...
2.1 - 0.2 = 1.9000000000000001 (BIGGER)
2.0 - 1.9 = 0.10000000000000009 (BIGGER)
...
100 - 99.9 = 0.09999999999999432 (SMALLER)
100 - 99.8 = 0.20000000000000284 (BIGGER)
*15% and 34% are indeed huge, so always use BigDecimal when precision is of big importance. With 2 decimal digits (step 0.01) the situation worsens a bit more (18% and 36%).
Given that nobody has mentioned this...
Some high level languages such as Python and Java come with tools to overcome binary floating point limitations. For example:
Python's decimal module and Java's BigDecimal class, that represent numbers internally with decimal notation (as opposed to binary notation). Both have limited precision, so they are still error prone, however they solve most common problems with binary floating point arithmetic.
Decimals are very nice when dealing with money: ten cents plus twenty cents are always exactly thirty cents:
>>> 0.1 + 0.2 == 0.3
False
>>> Decimal('0.1') + Decimal('0.2') == Decimal('0.3')
True
Python's decimal module is based on IEEE standard 854-1987.
Python's fractions module and Apache Common's BigFraction class. Both represent rational numbers as (numerator, denominator) pairs and they may give more accurate results than decimal floating point arithmetic.
Neither of these solutions is perfect (especially if we look at performances, or if we require a very high precision), but still they solve a great number of problems with binary floating point arithmetic.
Did you try the duct tape solution?
Try to determine when errors occur and fix them with short if statements, it's not pretty but for some problems it is the only solution and this is one of them.
if( (n * 0.1) < 100.0 ) { return n * 0.1 - 0.000000000000001 ;}
else { return n * 0.1 + 0.000000000000001 ;}
I had the same problem in a scientific simulation project in c#, and I can tell you that if you ignore the butterfly effect it's gonna turn to a big fat dragon and bite you in the a**
Those weird numbers appear because computers use binary(base 2) number system for calculation purposes, while we use decimal(base 10).
There are a majority of fractional numbers that cannot be represented precisely either in binary or in decimal or both. Result - A rounded up (but precise) number results.
Many of this question's numerous duplicates ask about the effects of floating point rounding on specific numbers. In practice, it is easier to get a feeling for how it works by looking at exact results of calculations of interest rather than by just reading about it. Some languages provide ways of doing that - such as converting a float or double to BigDecimal in Java.
Since this is a language-agnostic question, it needs language-agnostic tools, such as a Decimal to Floating-Point Converter.
Applying it to the numbers in the question, treated as doubles:
0.1 converts to 0.1000000000000000055511151231257827021181583404541015625,
0.2 converts to 0.200000000000000011102230246251565404236316680908203125,
0.3 converts to 0.299999999999999988897769753748434595763683319091796875, and
0.30000000000000004 converts to 0.3000000000000000444089209850062616169452667236328125.
Adding the first two numbers manually or in a decimal calculator such as Full Precision Calculator, shows the exact sum of the actual inputs is 0.3000000000000000166533453693773481063544750213623046875.
If it were rounded down to the equivalent of 0.3 the rounding error would be 0.0000000000000000277555756156289135105907917022705078125. Rounding up to the equivalent of 0.30000000000000004 also gives rounding error 0.0000000000000000277555756156289135105907917022705078125. The round-to-even tie breaker applies.
Returning to the floating point converter, the raw hexadecimal for 0.30000000000000004 is 3fd3333333333334, which ends in an even digit and therefore is the correct result.
Can I just add; people always assume this to be a computer problem, but if you count with your hands (base 10), you can't get (1/3+1/3=2/3)=true unless you have infinity to add 0.333... to 0.333... so just as with the (1/10+2/10)!==3/10 problem in base 2, you truncate it to 0.333 + 0.333 = 0.666 and probably round it to 0.667 which would be also be technically inaccurate.
Count in ternary, and thirds are not a problem though - maybe some race with 15 fingers on each hand would ask why your decimal math was broken...
The kind of floating-point math that can be implemented in a digital computer necessarily uses an approximation of the real numbers and operations on them. (The standard version runs to over fifty pages of documentation and has a committee to deal with its errata and further refinement.)
This approximation is a mixture of approximations of different kinds, each of which can either be ignored or carefully accounted for due to its specific manner of deviation from exactitude. It also involves a number of explicit exceptional cases at both the hardware and software levels that most people walk right past while pretending not to notice.
If you need infinite precision (using the number π, for example, instead of one of its many shorter stand-ins), you should write or use a symbolic math program instead.
But if you're okay with the idea that sometimes floating-point math is fuzzy in value and logic and errors can accumulate quickly, and you can write your requirements and tests to allow for that, then your code can frequently get by with what's in your FPU.
Just for fun, I played with the representation of floats, following the definitions from the Standard C99 and I wrote the code below.
The code prints the binary representation of floats in 3 separated groups
SIGN EXPONENT FRACTION
and after that it prints a sum, that, when summed with enough precision, it will show the value that really exists in hardware.
So when you write float x = 999..., the compiler will transform that number in a bit representation printed by the function xx such that the sum printed by the function yy be equal to the given number.
In reality, this sum is only an approximation. For the number 999,999,999 the compiler will insert in bit representation of the float the number 1,000,000,000
After the code I attach a console session, in which I compute the sum of terms for both constants (minus PI and 999999999) that really exists in hardware, inserted there by the compiler.
#include <stdio.h>
#include <limits.h>
void
xx(float *x)
{
unsigned char i = sizeof(*x)*CHAR_BIT-1;
do {
switch (i) {
case 31:
printf("sign:");
break;
case 30:
printf("exponent:");
break;
case 23:
printf("fraction:");
break;
}
char b=(*(unsigned long long*)x&((unsigned long long)1<<i))!=0;
printf("%d ", b);
} while (i--);
printf("\n");
}
void
yy(float a)
{
int sign=!(*(unsigned long long*)&a&((unsigned long long)1<<31));
int fraction = ((1<<23)-1)&(*(int*)&a);
int exponent = (255&((*(int*)&a)>>23))-127;
printf(sign?"positive" " ( 1+":"negative" " ( 1+");
unsigned int i = 1<<22;
unsigned int j = 1;
do {
char b=(fraction&i)!=0;
b&&(printf("1/(%d) %c", 1<<j, (fraction&(i-1))?'+':')' ), 0);
} while (j++, i>>=1);
printf("*2^%d", exponent);
printf("\n");
}
void
main()
{
float x=-3.14;
float y=999999999;
printf("%lu\n", sizeof(x));
xx(&x);
xx(&y);
yy(x);
yy(y);
}
Here is a console session in which I compute the real value of the float that exists in hardware. I used bc to print the sum of terms outputted by the main program. One can insert that sum in python repl or something similar also.
-- .../terra1/stub
# qemacs f.c
-- .../terra1/stub
# gcc f.c
-- .../terra1/stub
# ./a.out
sign:1 exponent:1 0 0 0 0 0 0 fraction:0 1 0 0 1 0 0 0 1 1 1 1 0 1 0 1 1 1 0 0 0 0 1 1
sign:0 exponent:1 0 0 1 1 1 0 fraction:0 1 1 0 1 1 1 0 0 1 1 0 1 0 1 1 0 0 1 0 1 0 0 0
negative ( 1+1/(2) +1/(16) +1/(256) +1/(512) +1/(1024) +1/(2048) +1/(8192) +1/(32768) +1/(65536) +1/(131072) +1/(4194304) +1/(8388608) )*2^1
positive ( 1+1/(2) +1/(4) +1/(16) +1/(32) +1/(64) +1/(512) +1/(1024) +1/(4096) +1/(16384) +1/(32768) +1/(262144) +1/(1048576) )*2^29
-- .../terra1/stub
# bc
scale=15
( 1+1/(2) +1/(4) +1/(16) +1/(32) +1/(64) +1/(512) +1/(1024) +1/(4096) +1/(16384) +1/(32768) +1/(262144) +1/(1048576) )*2^29
999999999.999999446351872
That's it. The value of 999999999 is in fact
999999999.999999446351872
You can also check with bc that -3.14 is also perturbed. Do not forget to set a scale factor in bc.
The displayed sum is what inside the hardware. The value you obtain by computing it depends on the scale you set. I did set the scale factor to 15. Mathematically, with infinite precision, it seems it is 1,000,000,000.
Since Python 3.5 you can use math.isclose() function for testing approximate equality:
>>> import math
>>> math.isclose(0.1 + 0.2, 0.3)
True
>>> 0.1 + 0.2 == 0.3
False
The trap with floating point numbers is that they look like decimal but they work in binary.
The only prime factor of 2 is 2, while 10 has prime factors of 2 and 5. The result of this is that every number that can be written exactly as a binary fraction can also be written exactly as a decimal fraction but only a subset of numbers that can be written as decimal fractions can be written as binary fractions.
A floating point number is essentially a binary fraction with a limited number of significant digits. If you go past those significant digits then the results will be rounded.
When you type a literal in your code or call the function to parse a floating point number to a string, it expects a decimal number and it stores a binary approximation of that decimal number in the variable.
When you print a floating point number or call the function to convert one to a string it prints a decimal approximation of the floating point number. It is possible to convert a binary number to decimal exactly, but no language I'm aware of does that by default when converting to a string*. Some languages use a fixed number of significant digits, others use the shortest string that will "round trip" back to the same floating point value.
* Python does convert exactly when converting a floating point number to a "decimal.Decimal". This is the easiest way I know of to obtain the exact decimal equivalent of a floating point number.
Floating point numbers are represented, at the hardware level, as fractions of binary numbers (base 2). For example, the decimal fraction:
0.125
has the value 1/10 + 2/100 + 5/1000 and, in the same way, the binary fraction:
0.001
has the value 0/2 + 0/4 + 1/8. These two fractions have the same value, the only difference is that the first is a decimal fraction, the second is a binary fraction.
Unfortunately, most decimal fractions cannot have exact representation in binary fractions. Therefore, in general, the floating point numbers you give are only approximated to binary fractions to be stored in the machine.
The problem is easier to approach in base 10. Take for example, the fraction 1/3. You can approximate it to a decimal fraction:
0.3
or better,
0.33
or better,
0.333
etc. No matter how many decimal places you write, the result is never exactly 1/3, but it is an estimate that always comes closer.
Likewise, no matter how many base 2 decimal places you use, the decimal value 0.1 cannot be represented exactly as a binary fraction. In base 2, 1/10 is the following periodic number:
0.0001100110011001100110011001100110011001100110011 ...
Stop at any finite amount of bits, and you'll get an approximation.
For Python, on a typical machine, 53 bits are used for the precision of a float, so the value stored when you enter the decimal 0.1 is the binary fraction.
0.00011001100110011001100110011001100110011001100110011010
which is close, but not exactly equal, to 1/10.
It's easy to forget that the stored value is an approximation of the original decimal fraction, due to the way floats are displayed in the interpreter. Python only displays a decimal approximation of the value stored in binary. If Python were to output the true decimal value of the binary approximation stored for 0.1, it would output:
>>> 0.1
0.1000000000000000055511151231257827021181583404541015625
This is a lot more decimal places than most people would expect, so Python displays a rounded value to improve readability:
>>> 0.1
0.1
It is important to understand that in reality this is an illusion: the stored value is not exactly 1/10, it is simply on the display that the stored value is rounded. This becomes evident as soon as you perform arithmetic operations with these values:
>>> 0.1 + 0.2
0.30000000000000004
This behavior is inherent to the very nature of the machine's floating-point representation: it is not a bug in Python, nor is it a bug in your code. You can observe the same type of behavior in all other languages that use hardware support for calculating floating point numbers (although some languages do not make the difference visible by default, or not in all display modes).
Another surprise is inherent in this one. For example, if you try to round the value 2.675 to two decimal places, you will get
>>> round (2.675, 2)
2.67
The documentation for the round() primitive indicates that it rounds to the nearest value away from zero. Since the decimal fraction is exactly halfway between 2.67 and 2.68, you should expect to get (a binary approximation of) 2.68. This is not the case, however, because when the decimal fraction 2.675 is converted to a float, it is stored by an approximation whose exact value is :
2.67499999999999982236431605997495353221893310546875
Since the approximation is slightly closer to 2.67 than 2.68, the rounding is down.
If you are in a situation where rounding decimal numbers halfway down matters, you should use the decimal module. By the way, the decimal module also provides a convenient way to "see" the exact value stored for any float.
>>> from decimal import Decimal
>>> Decimal (2.675)
>>> Decimal ('2.67499999999999982236431605997495353221893310546875')
Another consequence of the fact that 0.1 is not exactly stored in 1/10 is that the sum of ten values of 0.1 does not give 1.0 either:
>>> sum = 0.0
>>> for i in range (10):
... sum + = 0.1
...>>> sum
0.9999999999999999
The arithmetic of binary floating point numbers holds many such surprises. The problem with "0.1" is explained in detail below, in the section "Representation errors". See The Perils of Floating Point for a more complete list of such surprises.
It is true that there is no simple answer, however do not be overly suspicious of floating virtula numbers! Errors, in Python, in floating-point number operations are due to the underlying hardware, and on most machines are no more than 1 in 2 ** 53 per operation. This is more than necessary for most tasks, but you should keep in mind that these are not decimal operations, and every operation on floating point numbers may suffer from a new error.
Although pathological cases exist, for most common use cases you will get the expected result at the end by simply rounding up to the number of decimal places you want on the display. For fine control over how floats are displayed, see String Formatting Syntax for the formatting specifications of the str.format () method.
This part of the answer explains in detail the example of "0.1" and shows how you can perform an exact analysis of this type of case on your own. We assume that you are familiar with the binary representation of floating point numbers.The term Representation error means that most decimal fractions cannot be represented exactly in binary. This is the main reason why Python (or Perl, C, C ++, Java, Fortran, and many others) usually doesn't display the exact result in decimal:
>>> 0.1 + 0.2
0.30000000000000004
Why ? 1/10 and 2/10 are not representable exactly in binary fractions. However, all machines today (July 2010) follow the IEEE-754 standard for the arithmetic of floating point numbers. and most platforms use an "IEEE-754 double precision" to represent Python floats. Double precision IEEE-754 uses 53 bits of precision, so on reading the computer tries to convert 0.1 to the nearest fraction of the form J / 2 ** N with J an integer of exactly 53 bits. Rewrite :
1/10 ~ = J / (2 ** N)
in :
J ~ = 2 ** N / 10
remembering that J is exactly 53 bits (so> = 2 ** 52 but <2 ** 53), the best possible value for N is 56:
>>> 2 ** 52
4503599627370496
>>> 2 ** 53
9007199254740992
>>> 2 ** 56/10
7205759403792793
So 56 is the only possible value for N which leaves exactly 53 bits for J. The best possible value for J is therefore this quotient, rounded:
>>> q, r = divmod (2 ** 56, 10)
>>> r
6
Since the carry is greater than half of 10, the best approximation is obtained by rounding up:
>>> q + 1
7205759403792794
Therefore the best possible approximation for 1/10 in "IEEE-754 double precision" is this above 2 ** 56, that is:
7205759403792794/72057594037927936
Note that since the rounding was done upward, the result is actually slightly greater than 1/10; if we hadn't rounded up, the quotient would have been slightly less than 1/10. But in no case is it exactly 1/10!
So the computer never "sees" 1/10: what it sees is the exact fraction given above, the best approximation using the double precision floating point numbers from the "" IEEE-754 ":
>>>. 1 * 2 ** 56
7205759403792794.0
If we multiply this fraction by 10 ** 30, we can observe the values of its 30 decimal places of strong weight.
>>> 7205759403792794 * 10 ** 30 // 2 ** 56
100000000000000005551115123125L
meaning that the exact value stored in the computer is approximately equal to the decimal value 0.100000000000000005551115123125. In versions prior to Python 2.7 and Python 3.1, Python rounded these values to 17 significant decimal places, displaying “0.10000000000000001”. In current versions of Python, the displayed value is the value whose fraction is as short as possible while giving exactly the same representation when converted back to binary, simply displaying “0.1”.
Another way to look at this: Used are 64 bits to represent numbers. As consequence there is no way more than 2**64 = 18,446,744,073,709,551,616 different numbers can be precisely represented.
However, Math says there are already infinitely many decimals between 0 and 1. IEE 754 defines an encoding to use these 64 bits efficiently for a much larger number space plus NaN and +/- Infinity, so there are gaps between accurately represented numbers filled with numbers only approximated.
Unfortunately 0.3 sits in a gap.
Imagine working in base ten with, say, 8 digits of accuracy. You check whether
1/3 + 2 / 3 == 1
and learn that this returns false. Why? Well, as real numbers we have
1/3 = 0.333.... and 2/3 = 0.666....
Truncating at eight decimal places, we get
0.33333333 + 0.66666666 = 0.99999999
which is, of course, different from 1.00000000 by exactly 0.00000001.
The situation for binary numbers with a fixed number of bits is exactly analogous. As real numbers, we have
1/10 = 0.0001100110011001100... (base 2)
and
1/5 = 0.0011001100110011001... (base 2)
If we truncated these to, say, seven bits, then we'd get
0.0001100 + 0.0011001 = 0.0100101
while on the other hand,
3/10 = 0.01001100110011... (base 2)
which, truncated to seven bits, is 0.0100110, and these differ by exactly 0.0000001.
The exact situation is slightly more subtle because these numbers are typically stored in scientific notation. So, for instance, instead of storing 1/10 as 0.0001100 we may store it as something like 1.10011 * 2^-4, depending on how many bits we've allocated for the exponent and the mantissa. This affects how many digits of precision you get for your calculations.
The upshot is that because of these rounding errors you essentially never want to use == on floating-point numbers. Instead, you can check if the absolute value of their difference is smaller than some fixed small number.
It's actually pretty simple. When you have a base 10 system (like ours), it can only express fractions that use a prime factor of the base. The prime factors of 10 are 2 and 5. So 1/2, 1/4, 1/5, 1/8, and 1/10 can all be expressed cleanly because the denominators all use prime factors of 10. In contrast, 1/3, 1/6, and 1/7 are all repeating decimals because their denominators use a prime factor of 3 or 7. In binary (or base 2), the only prime factor is 2. So you can only express fractions cleanly which only contain 2 as a prime factor. In binary, 1/2, 1/4, 1/8 would all be expressed cleanly as decimals. While, 1/5 or 1/10 would be repeating decimals. So 0.1 and 0.2 (1/10 and 1/5) while clean decimals in a base 10 system, are repeating decimals in the base 2 system the computer is operating in. When you do math on these repeating decimals, you end up with leftovers which carry over when you convert the computer's base 2 (binary) number into a more human readable base 10 number.
From https://0.30000000000000004.com/
Decimal numbers such as 0.1, 0.2, and 0.3 are not represented exactly in binary encoded floating point types. The sum of the approximations for 0.1 and 0.2 differs from the approximation used for 0.3, hence the falsehood of 0.1 + 0.2 == 0.3 as can be seen more clearly here:
#include <stdio.h>
int main() {
printf("0.1 + 0.2 == 0.3 is %s\n", 0.1 + 0.2 == 0.3 ? "true" : "false");
printf("0.1 is %.23f\n", 0.1);
printf("0.2 is %.23f\n", 0.2);
printf("0.1 + 0.2 is %.23f\n", 0.1 + 0.2);
printf("0.3 is %.23f\n", 0.3);
printf("0.3 - (0.1 + 0.2) is %g\n", 0.3 - (0.1 + 0.2));
return 0;
}
Output:
0.1 + 0.2 == 0.3 is false
0.1 is 0.10000000000000000555112
0.2 is 0.20000000000000001110223
0.1 + 0.2 is 0.30000000000000004440892
0.3 is 0.29999999999999998889777
0.3 - (0.1 + 0.2) is -5.55112e-17
For these computations to be evaluated more reliably, you would need to use a decimal-based representation for floating point values. The C Standard does not specify such types by default but as an extension described in a technical Report.
The _Decimal32, _Decimal64 and _Decimal128 types might be available on your system (for example, GCC supports them on selected targets, but Clang does not support them on OS X).
Since this thread branched off a bit into a general discussion over current floating point implementations I'd add that there are projects on fixing their issues.
Take a look at https://posithub.org/ for example, which showcases a number type called posit (and its predecessor unum) that promises to offer better accuracy with fewer bits. If my understanding is correct, it also fixes the kind of problems in the question. Quite interesting project, the person behind it is a mathematician it Dr. John Gustafson. The whole thing is open source, with many actual implementations in C/C++, Python, Julia and C# (https://hastlayer.com/arithmetics).
Normal arithmetic is base-10, so decimals represent tenths, hundredths, etc. When you try to represent a floating-point number in binary base-2 arithmetic, you are dealing with halves, fourths, eighths, etc.
In the hardware, floating points are stored as integer mantissas and exponents. Mantissa represents the significant digits. Exponent is like scientific notation but it uses a base of 2 instead of 10. For example 64.0 would be represented with a mantissa of 1 and exponent of 6. 0.125 would be represented with a mantissa of 1 and an exponent of -3.
Floating point decimals have to add up negative powers of 2
0.1b = 0.5d
0.01b = 0.25d
0.001b = 0.125d
0.0001b = 0.0625d
0.00001b = 0.03125d
and so on.
It is common to use a error delta instead of using equality operators when dealing with floating point arithmetic. Instead of
if(a==b) ...
you would use
delta = 0.0001; // or some arbitrarily small amount
if(a - b > -delta && a - b < delta) ...
Why do some numbers lose accuracy when stored as floating point numbers?
For example, the decimal number 9.2 can be expressed exactly as a ratio of two decimal integers (92/10), both of which can be expressed exactly in binary (0b1011100/0b1010). However, the same ratio stored as a floating point number is never exactly equal to 9.2:
32-bit "single precision" float: 9.19999980926513671875
64-bit "double precision" float: 9.199999999999999289457264239899814128875732421875
How can such an apparently simple number be "too big" to express in 64 bits of memory?
In most programming languages, floating point numbers are represented a lot like scientific notation: with an exponent and a mantissa (also called the significand). A very simple number, say 9.2, is actually this fraction:
5179139571476070 * 2 -49
Where the exponent is -49 and the mantissa is 5179139571476070. The reason it is impossible to represent some decimal numbers this way is that both the exponent and the mantissa must be integers. In other words, all floats must be an integer multiplied by an integer power of 2.
9.2 may be simply 92/10, but 10 cannot be expressed as 2n if n is limited to integer values.
Seeing the Data
First, a few functions to see the components that make a 32- and 64-bit float. Gloss over these if you only care about the output (example in Python):
def float_to_bin_parts(number, bits=64):
if bits == 32: # single precision
int_pack = 'I'
float_pack = 'f'
exponent_bits = 8
mantissa_bits = 23
exponent_bias = 127
elif bits == 64: # double precision. all python floats are this
int_pack = 'Q'
float_pack = 'd'
exponent_bits = 11
mantissa_bits = 52
exponent_bias = 1023
else:
raise ValueError, 'bits argument must be 32 or 64'
bin_iter = iter(bin(struct.unpack(int_pack, struct.pack(float_pack, number))[0])[2:].rjust(bits, '0'))
return [''.join(islice(bin_iter, x)) for x in (1, exponent_bits, mantissa_bits)]
There's a lot of complexity behind that function, and it'd be quite the tangent to explain, but if you're interested, the important resource for our purposes is the struct module.
Python's float is a 64-bit, double-precision number. In other languages such as C, C++, Java and C#, double-precision has a separate type double, which is often implemented as 64 bits.
When we call that function with our example, 9.2, here's what we get:
>>> float_to_bin_parts(9.2)
['0', '10000000010', '0010011001100110011001100110011001100110011001100110']
Interpreting the Data
You'll see I've split the return value into three components. These components are:
Sign
Exponent
Mantissa (also called Significand, or Fraction)
Sign
The sign is stored in the first component as a single bit. It's easy to explain: 0 means the float is a positive number; 1 means it's negative. Because 9.2 is positive, our sign value is 0.
Exponent
The exponent is stored in the middle component as 11 bits. In our case, 0b10000000010. In decimal, that represents the value 1026. A quirk of this component is that you must subtract a number equal to 2(# of bits) - 1 - 1 to get the true exponent; in our case, that means subtracting 0b1111111111 (decimal number 1023) to get the true exponent, 0b00000000011 (decimal number 3).
Mantissa
The mantissa is stored in the third component as 52 bits. However, there's a quirk to this component as well. To understand this quirk, consider a number in scientific notation, like this:
6.0221413x1023
The mantissa would be the 6.0221413. Recall that the mantissa in scientific notation always begins with a single non-zero digit. The same holds true for binary, except that binary only has two digits: 0 and 1. So the binary mantissa always starts with 1! When a float is stored, the 1 at the front of the binary mantissa is omitted to save space; we have to place it back at the front of our third element to get the true mantissa:
1.0010011001100110011001100110011001100110011001100110
This involves more than just a simple addition, because the bits stored in our third component actually represent the fractional part of the mantissa, to the right of the radix point.
When dealing with decimal numbers, we "move the decimal point" by multiplying or dividing by powers of 10. In binary, we can do the same thing by multiplying or dividing by powers of 2. Since our third element has 52 bits, we divide it by 252 to move it 52 places to the right:
0.0010011001100110011001100110011001100110011001100110
In decimal notation, that's the same as dividing 675539944105574 by 4503599627370496 to get 0.1499999999999999. (This is one example of a ratio that can be expressed exactly in binary, but only approximately in decimal; for more detail, see: 675539944105574 / 4503599627370496.)
Now that we've transformed the third component into a fractional number, adding 1 gives the true mantissa.
Recapping the Components
Sign (first component): 0 for positive, 1 for negative
Exponent (middle component): Subtract 2(# of bits) - 1 - 1 to get the true exponent
Mantissa (last component): Divide by 2(# of bits) and add 1 to get the true mantissa
Calculating the Number
Putting all three parts together, we're given this binary number:
1.0010011001100110011001100110011001100110011001100110 x 1011
Which we can then convert from binary to decimal:
1.1499999999999999 x 23 (inexact!)
And multiply to reveal the final representation of the number we started with (9.2) after being stored as a floating point value:
9.1999999999999993
Representing as a Fraction
9.2
Now that we've built the number, it's possible to reconstruct it into a simple fraction:
1.0010011001100110011001100110011001100110011001100110 x 1011
Shift mantissa to a whole number:
10010011001100110011001100110011001100110011001100110 x 1011-110100
Convert to decimal:
5179139571476070 x 23-52
Subtract the exponent:
5179139571476070 x 2-49
Turn negative exponent into division:
5179139571476070 / 249
Multiply exponent:
5179139571476070 / 562949953421312
Which equals:
9.1999999999999993
9.5
>>> float_to_bin_parts(9.5)
['0', '10000000010', '0011000000000000000000000000000000000000000000000000']
Already you can see the mantissa is only 4 digits followed by a whole lot of zeroes. But let's go through the paces.
Assemble the binary scientific notation:
1.0011 x 1011
Shift the decimal point:
10011 x 1011-100
Subtract the exponent:
10011 x 10-1
Binary to decimal:
19 x 2-1
Negative exponent to division:
19 / 21
Multiply exponent:
19 / 2
Equals:
9.5
Further reading
The Floating-Point Guide: What Every Programmer Should Know About Floating-Point Arithmetic, or, Why don’t my numbers add up? (floating-point-gui.de)
What Every Computer Scientist Should Know About Floating-Point Arithmetic (Goldberg 1991)
IEEE Double-precision floating-point format (Wikipedia)
Floating Point Arithmetic: Issues and Limitations (docs.python.org)
Floating Point Binary
This isn't a full answer (mhlester already covered a lot of good ground I won't duplicate), but I would like to stress how much the representation of a number depends on the base you are working in.
Consider the fraction 2/3
In good-ol' base 10, we typically write it out as something like
0.666...
0.666
0.667
When we look at those representations, we tend to associate each of them with the fraction 2/3, even though only the first representation is mathematically equal to the fraction. The second and third representations/approximations have an error on the order of 0.001, which is actually much worse than the error between 9.2 and 9.1999999999999993. In fact, the second representation isn't even rounded correctly! Nevertheless, we don't have a problem with 0.666 as an approximation of the number 2/3, so we shouldn't really have a problem with how 9.2 is approximated in most programs. (Yes, in some programs it matters.)
Number bases
So here's where number bases are crucial. If we were trying to represent 2/3 in base 3, then
(2/3)10 = 0.23
In other words, we have an exact, finite representation for the same number by switching bases! The take-away is that even though you can convert any number to any base, all rational numbers have exact finite representations in some bases but not in others.
To drive this point home, let's look at 1/2. It might surprise you that even though this perfectly simple number has an exact representation in base 10 and 2, it requires a repeating representation in base 3.
(1/2)10 = 0.510 = 0.12 = 0.1111...3
Why are floating point numbers inaccurate?
Because often-times, they are approximating rationals that cannot be represented finitely in base 2 (the digits repeat), and in general they are approximating real (possibly irrational) numbers which may not be representable in finitely many digits in any base.
While all of the other answers are good there is still one thing missing:
It is impossible to represent irrational numbers (e.g. π, sqrt(2), log(3), etc.) precisely!
And that actually is why they are called irrational. No amount of bit storage in the world would be enough to hold even one of them. Only symbolic arithmetic is able to preserve their precision.
Although if you would limit your math needs to rational numbers only the problem of precision becomes manageable. You would need to store a pair of (possibly very big) integers a and b to hold the number represented by the fraction a/b. All your arithmetic would have to be done on fractions just like in highschool math (e.g. a/b * c/d = ac/bd).
But of course you would still run into the same kind of trouble when pi, sqrt, log, sin, etc. are involved.
TL;DR
For hardware accelerated arithmetic only a limited amount of rational numbers can be represented. Every not-representable number is approximated. Some numbers (i.e. irrational) can never be represented no matter the system.
There are infinitely many real numbers (so many that you can't enumerate them), and there are infinitely many rational numbers (it is possible to enumerate them).
The floating-point representation is a finite one (like anything in a computer) so unavoidably many many many numbers are impossible to represent. In particular, 64 bits only allow you to distinguish among only 18,446,744,073,709,551,616 different values (which is nothing compared to infinity). With the standard convention, 9.2 is not one of them. Those that can are of the form m.2^e for some integers m and e.
You might come up with a different numeration system, 10 based for instance, where 9.2 would have an exact representation. But other numbers, say 1/3, would still be impossible to represent.
Also note that double-precision floating-points numbers are extremely accurate. They can represent any number in a very wide range with as much as 15 exact digits. For daily life computations, 4 or 5 digits are more than enough. You will never really need those 15, unless you want to count every millisecond of your lifetime.
Why can we not represent 9.2 in binary floating point?
Floating point numbers are (simplifying slightly) a positional numbering system with a restricted number of digits and a movable radix point.
A fraction can only be expressed exactly using a finite number of digits in a positional numbering system if the prime factors of the denominator (when the fraction is expressed in it's lowest terms) are factors of the base.
The prime factors of 10 are 5 and 2, so in base 10 we can represent any fraction of the form a/(2b5c).
On the other hand the only prime factor of 2 is 2, so in base 2 we can only represent fractions of the form a/(2b)
Why do computers use this representation?
Because it's a simple format to work with and it is sufficiently accurate for most purposes. Basically the same reason scientists use "scientific notation" and round their results to a reasonable number of digits at each step.
It would certainly be possible to define a fraction format, with (for example) a 32-bit numerator and a 32-bit denominator. It would be able to represent numbers that IEEE double precision floating point could not, but equally there would be many numbers that can be represented in double precision floating point that could not be represented in such a fixed-size fraction format.
However the big problem is that such a format is a pain to do calculations on. For two reasons.
If you want to have exactly one representation of each number then after each calculation you need to reduce the fraction to it's lowest terms. That means that for every operation you basically need to do a greatest common divisor calculation.
If after your calculation you end up with an unrepresentable result because the numerator or denominator you need to find the closest representable result. This is non-trivil.
Some Languages do offer fraction types, but usually they do it in combination with arbitary precision, this avoids needing to worry about approximating fractions but it creates it's own problem, when a number passes through a large number of calculation steps the size of the denominator and hence the storage needed for the fraction can explode.
Some languages also offer decimal floating point types, these are mainly used in scenarios where it is imporant that the results the computer gets match pre-existing rounding rules that were written with humans in mind (chiefly financial calculations). These are slightly more difficult to work with than binary floating point, but the biggest problem is that most computers don't offer hardware support for them.
In z3, a real term's value is represented by the division of two integer numbers: numerator and denominator. I very often find real values (as solutions to my z3 problems) with very large numerators and denominators like as follows:
29176049827299110215982818410792217459/804618286561124267999713087296957062500
-556895298158182412321541102649719164259083/4272120792496289300944476637003193523343750
There are often other assignments of those real variables in the case of alternative solutions (to the same problem) with smaller numerators and denominators. I am just wondering whether I can limit the numerator and denominator (e.g., with respect to the number of digits). Actually, I am looking for a z3 feature for this purpose. Thank you in advance for your answer.
It's not true in general that a real number in Z3 is represented as a ratio of a numerator and a denominator. This would restrict Z3 to rationals. For an example where Z3 returns an irrational number, try:
(declare-const x Real)
(assert (= (* x x) 2.0))
(check-sat)
(get-model)
That being said, for many specific problems, Z3 will indeed respond with a rational representation of a real number. I think this often happens when it's possible for your problem to be re-normalized and solved in the integers. In these situations, you could try to force your problem to be solved in the rationals using techniques similar to the answer in this question. Then, you can assert that your number has a numerator p < p_max for some constant p_max.
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, or
0x1.999999999999ap-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, or
0x1.99999999999999...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 can be used as tolerance values but their effectiveness depends on the magnitude (size) of the numbers you're working with, since calculations with large numbers may exceed the epsilon threshold.
A Hardware Designer's Perspective
I believe I should add a hardware designer’s perspective to this since I design and build floating point hardware. Knowing the origin of the error may help in understanding what is happening in the software, and ultimately, I hope this helps explain the reasons for why floating point errors happen and seem to accumulate over time.
1. Overview
From an engineering perspective, most floating point operations will have some element of error since the hardware that does the floating point computations is only required to have an error of less than one half of one unit in the last place. Therefore, much hardware will stop at a precision that's only necessary to yield an error of less than one half of one unit in the last place for a single operation which is especially problematic in floating point division. What constitutes a single operation depends upon how many operands the unit takes. For most, it is two, but some units take 3 or more operands. Because of this, there is no guarantee that repeated operations will result in a desirable error since the errors add up over time.
2. Standards
Most processors follow the IEEE-754 standard but some use denormalized, or different standards
. For example, there is a denormalized mode in IEEE-754 which allows representation of very small floating point numbers at the expense of precision. The following, however, will cover the normalized mode of IEEE-754 which is the typical mode of operation.
In the IEEE-754 standard, hardware designers are allowed any value of error/epsilon as long as it's less than one half of one unit in the last place, and the result only has to be less than one half of one unit in the last place for one operation. This explains why when there are repeated operations, the errors add up. For IEEE-754 double precision, this is the 54th bit, since 53 bits are used to represent the numeric part (normalized), also called the mantissa, of the floating point number (e.g. the 5.3 in 5.3e5). The next sections go into more detail on the causes of hardware error on various floating point operations.
3. Cause of Rounding Error in Division
The main cause of the error in floating point division is the division algorithms used to calculate the quotient. Most computer systems calculate division using multiplication by an inverse, mainly in Z=X/Y, Z = X * (1/Y). A division is computed iteratively i.e. each cycle computes some bits of the quotient until the desired precision is reached, which for IEEE-754 is anything with an error of less than one unit in the last place. The table of reciprocals of Y (1/Y) is known as the quotient selection table (QST) in the slow division, and the size in bits of the quotient selection table is usually the width of the radix, or a number of bits of the quotient computed in each iteration, plus a few guard bits. For the IEEE-754 standard, double precision (64-bit), it would be the size of the radix of the divider, plus a few guard bits k, where k>=2. So for example, a typical Quotient Selection Table for a divider that computes 2 bits of the quotient at a time (radix 4) would be 2+2= 4 bits (plus a few optional bits).
3.1 Division Rounding Error: Approximation of Reciprocal
What reciprocals are in the quotient selection table depend on the division method: slow division such as SRT division, or fast division such as Goldschmidt division; each entry is modified according to the division algorithm in an attempt to yield the lowest possible error. In any case, though, all reciprocals are approximations of the actual reciprocal and introduce some element of error. Both slow division and fast division methods calculate the quotient iteratively, i.e. some number of bits of the quotient are calculated each step, then the result is subtracted from the dividend, and the divider repeats the steps until the error is less than one half of one unit in the last place. Slow division methods calculate a fixed number of digits of the quotient in each step and are usually less expensive to build, and fast division methods calculate a variable number of digits per step and are usually more expensive to build. The most important part of the division methods is that most of them rely upon repeated multiplication by an approximation of a reciprocal, so they are prone to error.
4. Rounding Errors in Other Operations: Truncation
Another cause of the rounding errors in all operations are the different modes of truncation of the final answer that IEEE-754 allows. There's truncate, round-towards-zero, round-to-nearest (default), round-down, and round-up. All methods introduce an element of error of less than one unit in the last place for a single operation. Over time and repeated operations, truncation also adds cumulatively to the resultant error. This truncation error is especially problematic in exponentiation, which involves some form of repeated multiplication.
5. Repeated Operations
Since the hardware that does the floating point calculations only needs to yield a result with an error of less than one half of one unit in the last place for a single operation, the error will grow over repeated operations if not watched. This is the reason that in computations that require a bounded error, mathematicians use methods such as using the round-to-nearest even digit in the last place of IEEE-754, because, over time, the errors are more likely to cancel each other out, and Interval Arithmetic combined with variations of the IEEE 754 rounding modes to predict rounding errors, and correct them. Because of its low relative error compared to other rounding modes, round to nearest even digit (in the last place), is the default rounding mode of IEEE-754.
Note that the default rounding mode, round-to-nearest even digit in the last place, guarantees an error of less than one half of one unit in the last place for one operation. Using the truncation, round-up, and round down alone may result in an error that is greater than one half of one unit in the last place, but less than one unit in the last place, so these modes are not recommended unless they are used in Interval Arithmetic.
6. Summary
In short, the fundamental reason for the errors in floating point operations is a combination of the truncation in hardware, and the truncation of a reciprocal in the case of division. Since the IEEE-754 standard only requires an error of less than one half of one unit in the last place for a single operation, the floating point errors over repeated operations will add up unless corrected.
It's broken in the exact same way the decimal (base-10) notation you learned in grade school and use every day is broken, just for base-2.
To understand, think about representing 1/3 as a decimal value. It's impossible to do exactly! The world will end before you finish writing the 3's after the decimal point, and so instead we write to some number of places and consider it sufficiently accurate.
In the same way, 1/10 (decimal 0.1) cannot be represented exactly in base 2 (binary) as a "decimal" value; a repeating pattern after the decimal point goes on forever. The value is not exact, and therefore you can't do exact math with it using normal floating point methods. Just like with base 10, there are other values that exhibit this problem as well.
Most answers here address this question in very dry, technical terms. I'd like to address this in terms that normal human beings can understand.
Imagine that you are trying to slice up pizzas. You have a robotic pizza cutter that can cut pizza slices exactly in half. It can halve a whole pizza, or it can halve an existing slice, but in any case, the halving is always exact.
That pizza cutter has very fine movements, and if you start with a whole pizza, then halve that, and continue halving the smallest slice each time, you can do the halving 53 times before the slice is too small for even its high-precision abilities. At that point, you can no longer halve that very thin slice, but must either include or exclude it as is.
Now, how would you piece all the slices in such a way that would add up to one-tenth (0.1) or one-fifth (0.2) of a pizza? Really think about it, and try working it out. You can even try to use a real pizza, if you have a mythical precision pizza cutter at hand. :-)
Most experienced programmers, of course, know the real answer, which is that there is no way to piece together an exact tenth or fifth of the pizza using those slices, no matter how finely you slice them. You can do a pretty good approximation, and if you add up the approximation of 0.1 with the approximation of 0.2, you get a pretty good approximation of 0.3, but it's still just that, an approximation.
For double-precision numbers (which is the precision that allows you to halve your pizza 53 times), the numbers immediately less and greater than 0.1 are 0.09999999999999999167332731531132594682276248931884765625 and 0.1000000000000000055511151231257827021181583404541015625. The latter is quite a bit closer to 0.1 than the former, so a numeric parser will, given an input of 0.1, favour the latter.
(The difference between those two numbers is the "smallest slice" that we must decide to either include, which introduces an upward bias, or exclude, which introduces a downward bias. The technical term for that smallest slice is an ulp.)
In the case of 0.2, the numbers are all the same, just scaled up by a factor of 2. Again, we favour the value that's slightly higher than 0.2.
Notice that in both cases, the approximations for 0.1 and 0.2 have a slight upward bias. If we add enough of these biases in, they will push the number further and further away from what we want, and in fact, in the case of 0.1 + 0.2, the bias is high enough that the resulting number is no longer the closest number to 0.3.
In particular, 0.1 + 0.2 is really 0.1000000000000000055511151231257827021181583404541015625 + 0.200000000000000011102230246251565404236316680908203125 = 0.3000000000000000444089209850062616169452667236328125, whereas the number closest to 0.3 is actually 0.299999999999999988897769753748434595763683319091796875.
P.S. Some programming languages also provide pizza cutters that can split slices into exact tenths. Although such pizza cutters are uncommon, if you do have access to one, you should use it when it's important to be able to get exactly one-tenth or one-fifth of a slice.
(Originally posted on Quora.)
Floating point rounding errors. 0.1 cannot be represented as accurately in base-2 as in base-10 due to the missing prime factor of 5. Just as 1/3 takes an infinite number of digits to represent in decimal, but is "0.1" in base-3, 0.1 takes an infinite number of digits in base-2 where it does not in base-10. And computers don't have an infinite amount of memory.
My answer is quite long, so I've split it into three sections. Since the question is about floating point mathematics, I've put the emphasis on what the machine actually does. I've also made it specific to double (64 bit) precision, but the argument applies equally to any floating point arithmetic.
Preamble
An IEEE 754 double-precision binary floating-point format (binary64) number represents a number of the form
value = (-1)^s * (1.m51m50...m2m1m0)2 * 2e-1023
in 64 bits:
The first bit is the sign bit: 1 if the number is negative, 0 otherwise1.
The next 11 bits are the exponent, which is offset by 1023. In other words, after reading the exponent bits from a double-precision number, 1023 must be subtracted to obtain the power of two.
The remaining 52 bits are the significand (or mantissa). In the mantissa, an 'implied' 1. is always2 omitted since the most significant bit of any binary value is 1.
1 - IEEE 754 allows for the concept of a signed zero - +0 and -0 are treated differently: 1 / (+0) is positive infinity; 1 / (-0) is negative infinity. For zero values, the mantissa and exponent bits are all zero. Note: zero values (+0 and -0) are explicitly not classed as denormal2.
2 - This is not the case for denormal numbers, which have an offset exponent of zero (and an implied 0.). The range of denormal double precision numbers is dmin ≤ |x| ≤ dmax, where dmin (the smallest representable nonzero number) is 2-1023 - 51 (≈ 4.94 * 10-324) and dmax (the largest denormal number, for which the mantissa consists entirely of 1s) is 2-1023 + 1 - 2-1023 - 51 (≈ 2.225 * 10-308).
Turning a double precision number to binary
Many online converters exist to convert a double precision floating point number to binary (e.g. at binaryconvert.com), but here is some sample C# code to obtain the IEEE 754 representation for a double precision number (I separate the three parts with colons (:):
public static string BinaryRepresentation(double value)
{
long valueInLongType = BitConverter.DoubleToInt64Bits(value);
string bits = Convert.ToString(valueInLongType, 2);
string leadingZeros = new string('0', 64 - bits.Length);
string binaryRepresentation = leadingZeros + bits;
string sign = binaryRepresentation[0].ToString();
string exponent = binaryRepresentation.Substring(1, 11);
string mantissa = binaryRepresentation.Substring(12);
return string.Format("{0}:{1}:{2}", sign, exponent, mantissa);
}
Getting to the point: the original question
(Skip to the bottom for the TL;DR version)
Cato Johnston (the question asker) asked why 0.1 + 0.2 != 0.3.
Written in binary (with colons separating the three parts), the IEEE 754 representations of the values are:
0.1 => 0:01111111011:1001100110011001100110011001100110011001100110011010
0.2 => 0:01111111100:1001100110011001100110011001100110011001100110011010
Note that the mantissa is composed of recurring digits of 0011. This is key to why there is any error to the calculations - 0.1, 0.2 and 0.3 cannot be represented in binary precisely in a finite number of binary bits any more than 1/9, 1/3 or 1/7 can be represented precisely in decimal digits.
Also note that we can decrease the power in the exponent by 52 and shift the point in the binary representation to the right by 52 places (much like 10-3 * 1.23 == 10-5 * 123). This then enables us to represent the binary representation as the exact value that it represents in the form a * 2p. where 'a' is an integer.
Converting the exponents to decimal, removing the offset, and re-adding the implied 1 (in square brackets), 0.1 and 0.2 are:
0.1 => 2^-4 * [1].1001100110011001100110011001100110011001100110011010
0.2 => 2^-3 * [1].1001100110011001100110011001100110011001100110011010
or
0.1 => 2^-56 * 7205759403792794 = 0.1000000000000000055511151231257827021181583404541015625
0.2 => 2^-55 * 7205759403792794 = 0.200000000000000011102230246251565404236316680908203125
To add two numbers, the exponent needs to be the same, i.e.:
0.1 => 2^-3 * 0.1100110011001100110011001100110011001100110011001101(0)
0.2 => 2^-3 * 1.1001100110011001100110011001100110011001100110011010
sum = 2^-3 * 10.0110011001100110011001100110011001100110011001100111
or
0.1 => 2^-55 * 3602879701896397 = 0.1000000000000000055511151231257827021181583404541015625
0.2 => 2^-55 * 7205759403792794 = 0.200000000000000011102230246251565404236316680908203125
sum = 2^-55 * 10808639105689191 = 0.3000000000000000166533453693773481063544750213623046875
Since the sum is not of the form 2n * 1.{bbb} we increase the exponent by one and shift the decimal (binary) point to get:
sum = 2^-2 * 1.0011001100110011001100110011001100110011001100110011(1)
= 2^-54 * 5404319552844595.5 = 0.3000000000000000166533453693773481063544750213623046875
There are now 53 bits in the mantissa (the 53rd is in square brackets in the line above). The default rounding mode for IEEE 754 is 'Round to Nearest' - i.e. if a number x falls between two values a and b, the value where the least significant bit is zero is chosen.
a = 2^-54 * 5404319552844595 = 0.299999999999999988897769753748434595763683319091796875
= 2^-2 * 1.0011001100110011001100110011001100110011001100110011
x = 2^-2 * 1.0011001100110011001100110011001100110011001100110011(1)
b = 2^-2 * 1.0011001100110011001100110011001100110011001100110100
= 2^-54 * 5404319552844596 = 0.3000000000000000444089209850062616169452667236328125
Note that a and b differ only in the last bit; ...0011 + 1 = ...0100. In this case, the value with the least significant bit of zero is b, so the sum is:
sum = 2^-2 * 1.0011001100110011001100110011001100110011001100110100
= 2^-54 * 5404319552844596 = 0.3000000000000000444089209850062616169452667236328125
whereas the binary representation of 0.3 is:
0.3 => 2^-2 * 1.0011001100110011001100110011001100110011001100110011
= 2^-54 * 5404319552844595 = 0.299999999999999988897769753748434595763683319091796875
which only differs from the binary representation of the sum of 0.1 and 0.2 by 2-54.
The binary representation of 0.1 and 0.2 are the most accurate representations of the numbers allowable by IEEE 754. The addition of these representation, due to the default rounding mode, results in a value which differs only in the least-significant-bit.
TL;DR
Writing 0.1 + 0.2 in a IEEE 754 binary representation (with colons separating the three parts) and comparing it to 0.3, this is (I've put the distinct bits in square brackets):
0.1 + 0.2 => 0:01111111101:0011001100110011001100110011001100110011001100110[100]
0.3 => 0:01111111101:0011001100110011001100110011001100110011001100110[011]
Converted back to decimal, these values are:
0.1 + 0.2 => 0.300000000000000044408920985006...
0.3 => 0.299999999999999988897769753748...
The difference is exactly 2-54, which is ~5.5511151231258 × 10-17 - insignificant (for many applications) when compared to the original values.
Comparing the last few bits of a floating point number is inherently dangerous, as anyone who reads the famous "What Every Computer Scientist Should Know About Floating-Point Arithmetic" (which covers all the major parts of this answer) will know.
Most calculators use additional guard digits to get around this problem, which is how 0.1 + 0.2 would give 0.3: the final few bits are rounded.
In addition to the other correct answers, you may want to consider scaling your values to avoid problems with floating-point arithmetic.
For example:
var result = 1.0 + 2.0; // result === 3.0 returns true
... instead of:
var result = 0.1 + 0.2; // result === 0.3 returns false
The expression 0.1 + 0.2 === 0.3 returns false in JavaScript, but fortunately integer arithmetic in floating-point is exact, so decimal representation errors can be avoided by scaling.
As a practical example, to avoid floating-point problems where accuracy is paramount, it is recommended1 to handle money as an integer representing the number of cents: 2550 cents instead of 25.50 dollars.
1 Douglas Crockford: JavaScript: The Good Parts: Appendix A - Awful Parts (page 105).
Floating point numbers stored in the computer consist of two parts, an integer and an exponent that the base is taken to and multiplied by the integer part.
If the computer were working in base 10, 0.1 would be 1 x 10⁻¹, 0.2 would be 2 x 10⁻¹, and 0.3 would be 3 x 10⁻¹. Integer math is easy and exact, so adding 0.1 + 0.2 will obviously result in 0.3.
Computers don't usually work in base 10, they work in base 2. You can still get exact results for some values, for example 0.5 is 1 x 2⁻¹ and 0.25 is 1 x 2⁻², and adding them results in 3 x 2⁻², or 0.75. Exactly.
The problem comes with numbers that can be represented exactly in base 10, but not in base 2. Those numbers need to be rounded to their closest equivalent. Assuming the very common IEEE 64-bit floating point format, the closest number to 0.1 is 3602879701896397 x 2⁻⁵⁵, and the closest number to 0.2 is 7205759403792794 x 2⁻⁵⁵; adding them together results in 10808639105689191 x 2⁻⁵⁵, or an exact decimal value of 0.3000000000000000444089209850062616169452667236328125. Floating point numbers are generally rounded for display.
In short it's because:
Floating point numbers cannot represent all decimals precisely in binary
So just like 10/3 which does not exist in base 10 precisely (it will be 3.33... recurring), in the same way 1/10 doesn't exist in binary.
So what? How to deal with it? Is there any workaround?
In order to offer The best solution I can say I discovered following method:
parseFloat((0.1 + 0.2).toFixed(10)) => Will return 0.3
Let me explain why it's the best solution.
As others mentioned in above answers it's a good idea to use ready to use Javascript toFixed() function to solve the problem. But most likely you'll encounter with some problems.
Imagine you are going to add up two float numbers like 0.2 and 0.7 here it is: 0.2 + 0.7 = 0.8999999999999999.
Your expected result was 0.9 it means you need a result with 1 digit precision in this case.
So you should have used (0.2 + 0.7).tofixed(1)
but you can't just give a certain parameter to toFixed() since it depends on the given number, for instance
0.22 + 0.7 = 0.9199999999999999
In this example you need 2 digits precision so it should be toFixed(2), so what should be the paramter to fit every given float number?
You might say let it be 10 in every situation then:
(0.2 + 0.7).toFixed(10) => Result will be "0.9000000000"
Damn! What are you going to do with those unwanted zeros after 9?
It's the time to convert it to float to make it as you desire:
parseFloat((0.2 + 0.7).toFixed(10)) => Result will be 0.9
Now that you found the solution, it's better to offer it as a function like this:
function floatify(number){
return parseFloat((number).toFixed(10));
}
Let's try it yourself:
function floatify(number){
return parseFloat((number).toFixed(10));
}
function addUp(){
var number1 = +$("#number1").val();
var number2 = +$("#number2").val();
var unexpectedResult = number1 + number2;
var expectedResult = floatify(number1 + number2);
$("#unexpectedResult").text(unexpectedResult);
$("#expectedResult").text(expectedResult);
}
addUp();
input{
width: 50px;
}
#expectedResult{
color: green;
}
#unexpectedResult{
color: red;
}
<script src="https://ajax.googleapis.com/ajax/libs/jquery/2.1.1/jquery.min.js"></script>
<input id="number1" value="0.2" onclick="addUp()" onkeyup="addUp()"/> +
<input id="number2" value="0.7" onclick="addUp()" onkeyup="addUp()"/> =
<p>Expected Result: <span id="expectedResult"></span></p>
<p>Unexpected Result: <span id="unexpectedResult"></span></p>
You can use it this way:
var x = 0.2 + 0.7;
floatify(x); => Result: 0.9
As W3SCHOOLS suggests there is another solution too, you can multiply and divide to solve the problem above:
var x = (0.2 * 10 + 0.1 * 10) / 10; // x will be 0.3
Keep in mind that (0.2 + 0.1) * 10 / 10 won't work at all although it seems the same!
I prefer the first solution since I can apply it as a function which converts the input float to accurate output float.
FYI, the same problem exists for multiplication, for instance 0.09 * 10 returns 0.8999999999999999. Apply the flotify function as a workaround: flotify(0.09 * 10) returns 0.9
Floating point rounding error. From What Every Computer Scientist Should Know About Floating-Point Arithmetic:
Squeezing infinitely many real numbers into a finite number of bits requires an approximate representation. Although there are infinitely many integers, in most programs the result of integer computations can be stored in 32 bits. In contrast, given any fixed number of bits, most calculations with real numbers will produce quantities that cannot be exactly represented using that many bits. Therefore the result of a floating-point calculation must often be rounded in order to fit back into its finite representation. This rounding error is the characteristic feature of floating-point computation.
My workaround:
function add(a, b, precision) {
var x = Math.pow(10, precision || 2);
return (Math.round(a * x) + Math.round(b * x)) / x;
}
precision refers to the number of digits you want to preserve after the decimal point during addition.
No, not broken, but most decimal fractions must be approximated
Summary
Floating point arithmetic is exact, unfortunately, it doesn't match up well with our usual base-10 number representation, so it turns out we are often giving it input that is slightly off from what we wrote.
Even simple numbers like 0.01, 0.02, 0.03, 0.04 ... 0.24 are not representable exactly as binary fractions. If you count up 0.01, .02, .03 ..., not until you get to 0.25 will you get the first fraction representable in base2. If you tried that using FP, your 0.01 would have been slightly off, so the only way to add 25 of them up to a nice exact 0.25 would have required a long chain of causality involving guard bits and rounding. It's hard to predict so we throw up our hands and say "FP is inexact", but that's not really true.
We constantly give the FP hardware something that seems simple in base 10 but is a repeating fraction in base 2.
How did this happen?
When we write in decimal, every fraction (specifically, every terminating decimal) is a rational number of the form
a / (2n x 5m)
In binary, we only get the 2n term, that is:
a / 2n
So in decimal, we can't represent 1/3. Because base 10 includes 2 as a prime factor, every number we can write as a binary fraction also can be written as a base 10 fraction. However, hardly anything we write as a base10 fraction is representable in binary. In the range from 0.01, 0.02, 0.03 ... 0.99, only three numbers can be represented in our FP format: 0.25, 0.50, and 0.75, because they are 1/4, 1/2, and 3/4, all numbers with a prime factor using only the 2n term.
In base10 we can't represent 1/3. But in binary, we can't do 1/10 or 1/3.
So while every binary fraction can be written in decimal, the reverse is not true. And in fact most decimal fractions repeat in binary.
Dealing with it
Developers are usually instructed to do < epsilon comparisons, better advice might be to round to integral values (in the C library: round() and roundf(), i.e., stay in the FP format) and then compare. Rounding to a specific decimal fraction length solves most problems with output.
Also, on real number-crunching problems (the problems that FP was invented for on early, frightfully expensive computers) the physical constants of the universe and all other measurements are only known to a relatively small number of significant figures, so the entire problem space was "inexact" anyway. FP "accuracy" isn't a problem in this kind of application.
The whole issue really arises when people try to use FP for bean counting. It does work for that, but only if you stick to integral values, which kind of defeats the point of using it. This is why we have all those decimal fraction software libraries.
I love the Pizza answer by Chris, because it describes the actual problem, not just the usual handwaving about "inaccuracy". If FP were simply "inaccurate", we could fix that and would have done it decades ago. The reason we haven't is because the FP format is compact and fast and it's the best way to crunch a lot of numbers. Also, it's a legacy from the space age and arms race and early attempts to solve big problems with very slow computers using small memory systems. (Sometimes, individual magnetic cores for 1-bit storage, but that's another story.)
Conclusion
If you are just counting beans at a bank, software solutions that use decimal string representations in the first place work perfectly well. But you can't do quantum chromodynamics or aerodynamics that way.
A lot of good answers have been posted, but I'd like to append one more.
Not all numbers can be represented via floats/doubles
For example, the number "0.2" will be represented as "0.200000003" in single precision in IEEE754 float point standard.
Model for store real numbers under the hood represent float numbers as
Even though you can type 0.2 easily, FLT_RADIX and DBL_RADIX is 2; not 10 for a computer with FPU which uses "IEEE Standard for Binary Floating-Point Arithmetic (ISO/IEEE Std 754-1985)".
So it is a bit hard to represent such numbers exactly. Even if you specify this variable explicitly without any intermediate calculation.
Some statistics related to this famous double precision question.
When adding all values (a + b) using a step of 0.1 (from 0.1 to 100) we have ~15% chance of precision error. Note that the error could result in slightly bigger or smaller values.
Here are some examples:
0.1 + 0.2 = 0.30000000000000004 (BIGGER)
0.1 + 0.7 = 0.7999999999999999 (SMALLER)
...
1.7 + 1.9 = 3.5999999999999996 (SMALLER)
1.7 + 2.2 = 3.9000000000000004 (BIGGER)
...
3.2 + 3.6 = 6.800000000000001 (BIGGER)
3.2 + 4.4 = 7.6000000000000005 (BIGGER)
When subtracting all values (a - b where a > b) using a step of 0.1 (from 100 to 0.1) we have ~34% chance of precision error.
Here are some examples:
0.6 - 0.2 = 0.39999999999999997 (SMALLER)
0.5 - 0.4 = 0.09999999999999998 (SMALLER)
...
2.1 - 0.2 = 1.9000000000000001 (BIGGER)
2.0 - 1.9 = 0.10000000000000009 (BIGGER)
...
100 - 99.9 = 0.09999999999999432 (SMALLER)
100 - 99.8 = 0.20000000000000284 (BIGGER)
*15% and 34% are indeed huge, so always use BigDecimal when precision is of big importance. With 2 decimal digits (step 0.01) the situation worsens a bit more (18% and 36%).
Given that nobody has mentioned this...
Some high level languages such as Python and Java come with tools to overcome binary floating point limitations. For example:
Python's decimal module and Java's BigDecimal class, that represent numbers internally with decimal notation (as opposed to binary notation). Both have limited precision, so they are still error prone, however they solve most common problems with binary floating point arithmetic.
Decimals are very nice when dealing with money: ten cents plus twenty cents are always exactly thirty cents:
>>> 0.1 + 0.2 == 0.3
False
>>> Decimal('0.1') + Decimal('0.2') == Decimal('0.3')
True
Python's decimal module is based on IEEE standard 854-1987.
Python's fractions module and Apache Common's BigFraction class. Both represent rational numbers as (numerator, denominator) pairs and they may give more accurate results than decimal floating point arithmetic.
Neither of these solutions is perfect (especially if we look at performances, or if we require a very high precision), but still they solve a great number of problems with binary floating point arithmetic.
Did you try the duct tape solution?
Try to determine when errors occur and fix them with short if statements, it's not pretty but for some problems it is the only solution and this is one of them.
if( (n * 0.1) < 100.0 ) { return n * 0.1 - 0.000000000000001 ;}
else { return n * 0.1 + 0.000000000000001 ;}
I had the same problem in a scientific simulation project in c#, and I can tell you that if you ignore the butterfly effect it's gonna turn to a big fat dragon and bite you in the a**
Those weird numbers appear because computers use binary(base 2) number system for calculation purposes, while we use decimal(base 10).
There are a majority of fractional numbers that cannot be represented precisely either in binary or in decimal or both. Result - A rounded up (but precise) number results.
Many of this question's numerous duplicates ask about the effects of floating point rounding on specific numbers. In practice, it is easier to get a feeling for how it works by looking at exact results of calculations of interest rather than by just reading about it. Some languages provide ways of doing that - such as converting a float or double to BigDecimal in Java.
Since this is a language-agnostic question, it needs language-agnostic tools, such as a Decimal to Floating-Point Converter.
Applying it to the numbers in the question, treated as doubles:
0.1 converts to 0.1000000000000000055511151231257827021181583404541015625,
0.2 converts to 0.200000000000000011102230246251565404236316680908203125,
0.3 converts to 0.299999999999999988897769753748434595763683319091796875, and
0.30000000000000004 converts to 0.3000000000000000444089209850062616169452667236328125.
Adding the first two numbers manually or in a decimal calculator such as Full Precision Calculator, shows the exact sum of the actual inputs is 0.3000000000000000166533453693773481063544750213623046875.
If it were rounded down to the equivalent of 0.3 the rounding error would be 0.0000000000000000277555756156289135105907917022705078125. Rounding up to the equivalent of 0.30000000000000004 also gives rounding error 0.0000000000000000277555756156289135105907917022705078125. The round-to-even tie breaker applies.
Returning to the floating point converter, the raw hexadecimal for 0.30000000000000004 is 3fd3333333333334, which ends in an even digit and therefore is the correct result.
Can I just add; people always assume this to be a computer problem, but if you count with your hands (base 10), you can't get (1/3+1/3=2/3)=true unless you have infinity to add 0.333... to 0.333... so just as with the (1/10+2/10)!==3/10 problem in base 2, you truncate it to 0.333 + 0.333 = 0.666 and probably round it to 0.667 which would be also be technically inaccurate.
Count in ternary, and thirds are not a problem though - maybe some race with 15 fingers on each hand would ask why your decimal math was broken...
The kind of floating-point math that can be implemented in a digital computer necessarily uses an approximation of the real numbers and operations on them. (The standard version runs to over fifty pages of documentation and has a committee to deal with its errata and further refinement.)
This approximation is a mixture of approximations of different kinds, each of which can either be ignored or carefully accounted for due to its specific manner of deviation from exactitude. It also involves a number of explicit exceptional cases at both the hardware and software levels that most people walk right past while pretending not to notice.
If you need infinite precision (using the number π, for example, instead of one of its many shorter stand-ins), you should write or use a symbolic math program instead.
But if you're okay with the idea that sometimes floating-point math is fuzzy in value and logic and errors can accumulate quickly, and you can write your requirements and tests to allow for that, then your code can frequently get by with what's in your FPU.
Just for fun, I played with the representation of floats, following the definitions from the Standard C99 and I wrote the code below.
The code prints the binary representation of floats in 3 separated groups
SIGN EXPONENT FRACTION
and after that it prints a sum, that, when summed with enough precision, it will show the value that really exists in hardware.
So when you write float x = 999..., the compiler will transform that number in a bit representation printed by the function xx such that the sum printed by the function yy be equal to the given number.
In reality, this sum is only an approximation. For the number 999,999,999 the compiler will insert in bit representation of the float the number 1,000,000,000
After the code I attach a console session, in which I compute the sum of terms for both constants (minus PI and 999999999) that really exists in hardware, inserted there by the compiler.
#include <stdio.h>
#include <limits.h>
void
xx(float *x)
{
unsigned char i = sizeof(*x)*CHAR_BIT-1;
do {
switch (i) {
case 31:
printf("sign:");
break;
case 30:
printf("exponent:");
break;
case 23:
printf("fraction:");
break;
}
char b=(*(unsigned long long*)x&((unsigned long long)1<<i))!=0;
printf("%d ", b);
} while (i--);
printf("\n");
}
void
yy(float a)
{
int sign=!(*(unsigned long long*)&a&((unsigned long long)1<<31));
int fraction = ((1<<23)-1)&(*(int*)&a);
int exponent = (255&((*(int*)&a)>>23))-127;
printf(sign?"positive" " ( 1+":"negative" " ( 1+");
unsigned int i = 1<<22;
unsigned int j = 1;
do {
char b=(fraction&i)!=0;
b&&(printf("1/(%d) %c", 1<<j, (fraction&(i-1))?'+':')' ), 0);
} while (j++, i>>=1);
printf("*2^%d", exponent);
printf("\n");
}
void
main()
{
float x=-3.14;
float y=999999999;
printf("%lu\n", sizeof(x));
xx(&x);
xx(&y);
yy(x);
yy(y);
}
Here is a console session in which I compute the real value of the float that exists in hardware. I used bc to print the sum of terms outputted by the main program. One can insert that sum in python repl or something similar also.
-- .../terra1/stub
# qemacs f.c
-- .../terra1/stub
# gcc f.c
-- .../terra1/stub
# ./a.out
sign:1 exponent:1 0 0 0 0 0 0 fraction:0 1 0 0 1 0 0 0 1 1 1 1 0 1 0 1 1 1 0 0 0 0 1 1
sign:0 exponent:1 0 0 1 1 1 0 fraction:0 1 1 0 1 1 1 0 0 1 1 0 1 0 1 1 0 0 1 0 1 0 0 0
negative ( 1+1/(2) +1/(16) +1/(256) +1/(512) +1/(1024) +1/(2048) +1/(8192) +1/(32768) +1/(65536) +1/(131072) +1/(4194304) +1/(8388608) )*2^1
positive ( 1+1/(2) +1/(4) +1/(16) +1/(32) +1/(64) +1/(512) +1/(1024) +1/(4096) +1/(16384) +1/(32768) +1/(262144) +1/(1048576) )*2^29
-- .../terra1/stub
# bc
scale=15
( 1+1/(2) +1/(4) +1/(16) +1/(32) +1/(64) +1/(512) +1/(1024) +1/(4096) +1/(16384) +1/(32768) +1/(262144) +1/(1048576) )*2^29
999999999.999999446351872
That's it. The value of 999999999 is in fact
999999999.999999446351872
You can also check with bc that -3.14 is also perturbed. Do not forget to set a scale factor in bc.
The displayed sum is what inside the hardware. The value you obtain by computing it depends on the scale you set. I did set the scale factor to 15. Mathematically, with infinite precision, it seems it is 1,000,000,000.
Since Python 3.5 you can use math.isclose() function for testing approximate equality:
>>> import math
>>> math.isclose(0.1 + 0.2, 0.3)
True
>>> 0.1 + 0.2 == 0.3
False
The trap with floating point numbers is that they look like decimal but they work in binary.
The only prime factor of 2 is 2, while 10 has prime factors of 2 and 5. The result of this is that every number that can be written exactly as a binary fraction can also be written exactly as a decimal fraction but only a subset of numbers that can be written as decimal fractions can be written as binary fractions.
A floating point number is essentially a binary fraction with a limited number of significant digits. If you go past those significant digits then the results will be rounded.
When you type a literal in your code or call the function to parse a floating point number to a string, it expects a decimal number and it stores a binary approximation of that decimal number in the variable.
When you print a floating point number or call the function to convert one to a string it prints a decimal approximation of the floating point number. It is possible to convert a binary number to decimal exactly, but no language I'm aware of does that by default when converting to a string*. Some languages use a fixed number of significant digits, others use the shortest string that will "round trip" back to the same floating point value.
* Python does convert exactly when converting a floating point number to a "decimal.Decimal". This is the easiest way I know of to obtain the exact decimal equivalent of a floating point number.
Floating point numbers are represented, at the hardware level, as fractions of binary numbers (base 2). For example, the decimal fraction:
0.125
has the value 1/10 + 2/100 + 5/1000 and, in the same way, the binary fraction:
0.001
has the value 0/2 + 0/4 + 1/8. These two fractions have the same value, the only difference is that the first is a decimal fraction, the second is a binary fraction.
Unfortunately, most decimal fractions cannot have exact representation in binary fractions. Therefore, in general, the floating point numbers you give are only approximated to binary fractions to be stored in the machine.
The problem is easier to approach in base 10. Take for example, the fraction 1/3. You can approximate it to a decimal fraction:
0.3
or better,
0.33
or better,
0.333
etc. No matter how many decimal places you write, the result is never exactly 1/3, but it is an estimate that always comes closer.
Likewise, no matter how many base 2 decimal places you use, the decimal value 0.1 cannot be represented exactly as a binary fraction. In base 2, 1/10 is the following periodic number:
0.0001100110011001100110011001100110011001100110011 ...
Stop at any finite amount of bits, and you'll get an approximation.
For Python, on a typical machine, 53 bits are used for the precision of a float, so the value stored when you enter the decimal 0.1 is the binary fraction.
0.00011001100110011001100110011001100110011001100110011010
which is close, but not exactly equal, to 1/10.
It's easy to forget that the stored value is an approximation of the original decimal fraction, due to the way floats are displayed in the interpreter. Python only displays a decimal approximation of the value stored in binary. If Python were to output the true decimal value of the binary approximation stored for 0.1, it would output:
>>> 0.1
0.1000000000000000055511151231257827021181583404541015625
This is a lot more decimal places than most people would expect, so Python displays a rounded value to improve readability:
>>> 0.1
0.1
It is important to understand that in reality this is an illusion: the stored value is not exactly 1/10, it is simply on the display that the stored value is rounded. This becomes evident as soon as you perform arithmetic operations with these values:
>>> 0.1 + 0.2
0.30000000000000004
This behavior is inherent to the very nature of the machine's floating-point representation: it is not a bug in Python, nor is it a bug in your code. You can observe the same type of behavior in all other languages that use hardware support for calculating floating point numbers (although some languages do not make the difference visible by default, or not in all display modes).
Another surprise is inherent in this one. For example, if you try to round the value 2.675 to two decimal places, you will get
>>> round (2.675, 2)
2.67
The documentation for the round() primitive indicates that it rounds to the nearest value away from zero. Since the decimal fraction is exactly halfway between 2.67 and 2.68, you should expect to get (a binary approximation of) 2.68. This is not the case, however, because when the decimal fraction 2.675 is converted to a float, it is stored by an approximation whose exact value is :
2.67499999999999982236431605997495353221893310546875
Since the approximation is slightly closer to 2.67 than 2.68, the rounding is down.
If you are in a situation where rounding decimal numbers halfway down matters, you should use the decimal module. By the way, the decimal module also provides a convenient way to "see" the exact value stored for any float.
>>> from decimal import Decimal
>>> Decimal (2.675)
>>> Decimal ('2.67499999999999982236431605997495353221893310546875')
Another consequence of the fact that 0.1 is not exactly stored in 1/10 is that the sum of ten values of 0.1 does not give 1.0 either:
>>> sum = 0.0
>>> for i in range (10):
... sum + = 0.1
...>>> sum
0.9999999999999999
The arithmetic of binary floating point numbers holds many such surprises. The problem with "0.1" is explained in detail below, in the section "Representation errors". See The Perils of Floating Point for a more complete list of such surprises.
It is true that there is no simple answer, however do not be overly suspicious of floating virtula numbers! Errors, in Python, in floating-point number operations are due to the underlying hardware, and on most machines are no more than 1 in 2 ** 53 per operation. This is more than necessary for most tasks, but you should keep in mind that these are not decimal operations, and every operation on floating point numbers may suffer from a new error.
Although pathological cases exist, for most common use cases you will get the expected result at the end by simply rounding up to the number of decimal places you want on the display. For fine control over how floats are displayed, see String Formatting Syntax for the formatting specifications of the str.format () method.
This part of the answer explains in detail the example of "0.1" and shows how you can perform an exact analysis of this type of case on your own. We assume that you are familiar with the binary representation of floating point numbers.The term Representation error means that most decimal fractions cannot be represented exactly in binary. This is the main reason why Python (or Perl, C, C ++, Java, Fortran, and many others) usually doesn't display the exact result in decimal:
>>> 0.1 + 0.2
0.30000000000000004
Why ? 1/10 and 2/10 are not representable exactly in binary fractions. However, all machines today (July 2010) follow the IEEE-754 standard for the arithmetic of floating point numbers. and most platforms use an "IEEE-754 double precision" to represent Python floats. Double precision IEEE-754 uses 53 bits of precision, so on reading the computer tries to convert 0.1 to the nearest fraction of the form J / 2 ** N with J an integer of exactly 53 bits. Rewrite :
1/10 ~ = J / (2 ** N)
in :
J ~ = 2 ** N / 10
remembering that J is exactly 53 bits (so> = 2 ** 52 but <2 ** 53), the best possible value for N is 56:
>>> 2 ** 52
4503599627370496
>>> 2 ** 53
9007199254740992
>>> 2 ** 56/10
7205759403792793
So 56 is the only possible value for N which leaves exactly 53 bits for J. The best possible value for J is therefore this quotient, rounded:
>>> q, r = divmod (2 ** 56, 10)
>>> r
6
Since the carry is greater than half of 10, the best approximation is obtained by rounding up:
>>> q + 1
7205759403792794
Therefore the best possible approximation for 1/10 in "IEEE-754 double precision" is this above 2 ** 56, that is:
7205759403792794/72057594037927936
Note that since the rounding was done upward, the result is actually slightly greater than 1/10; if we hadn't rounded up, the quotient would have been slightly less than 1/10. But in no case is it exactly 1/10!
So the computer never "sees" 1/10: what it sees is the exact fraction given above, the best approximation using the double precision floating point numbers from the "" IEEE-754 ":
>>>. 1 * 2 ** 56
7205759403792794.0
If we multiply this fraction by 10 ** 30, we can observe the values of its 30 decimal places of strong weight.
>>> 7205759403792794 * 10 ** 30 // 2 ** 56
100000000000000005551115123125L
meaning that the exact value stored in the computer is approximately equal to the decimal value 0.100000000000000005551115123125. In versions prior to Python 2.7 and Python 3.1, Python rounded these values to 17 significant decimal places, displaying “0.10000000000000001”. In current versions of Python, the displayed value is the value whose fraction is as short as possible while giving exactly the same representation when converted back to binary, simply displaying “0.1”.
Another way to look at this: Used are 64 bits to represent numbers. As consequence there is no way more than 2**64 = 18,446,744,073,709,551,616 different numbers can be precisely represented.
However, Math says there are already infinitely many decimals between 0 and 1. IEE 754 defines an encoding to use these 64 bits efficiently for a much larger number space plus NaN and +/- Infinity, so there are gaps between accurately represented numbers filled with numbers only approximated.
Unfortunately 0.3 sits in a gap.
Imagine working in base ten with, say, 8 digits of accuracy. You check whether
1/3 + 2 / 3 == 1
and learn that this returns false. Why? Well, as real numbers we have
1/3 = 0.333.... and 2/3 = 0.666....
Truncating at eight decimal places, we get
0.33333333 + 0.66666666 = 0.99999999
which is, of course, different from 1.00000000 by exactly 0.00000001.
The situation for binary numbers with a fixed number of bits is exactly analogous. As real numbers, we have
1/10 = 0.0001100110011001100... (base 2)
and
1/5 = 0.0011001100110011001... (base 2)
If we truncated these to, say, seven bits, then we'd get
0.0001100 + 0.0011001 = 0.0100101
while on the other hand,
3/10 = 0.01001100110011... (base 2)
which, truncated to seven bits, is 0.0100110, and these differ by exactly 0.0000001.
The exact situation is slightly more subtle because these numbers are typically stored in scientific notation. So, for instance, instead of storing 1/10 as 0.0001100 we may store it as something like 1.10011 * 2^-4, depending on how many bits we've allocated for the exponent and the mantissa. This affects how many digits of precision you get for your calculations.
The upshot is that because of these rounding errors you essentially never want to use == on floating-point numbers. Instead, you can check if the absolute value of their difference is smaller than some fixed small number.
It's actually pretty simple. When you have a base 10 system (like ours), it can only express fractions that use a prime factor of the base. The prime factors of 10 are 2 and 5. So 1/2, 1/4, 1/5, 1/8, and 1/10 can all be expressed cleanly because the denominators all use prime factors of 10. In contrast, 1/3, 1/6, and 1/7 are all repeating decimals because their denominators use a prime factor of 3 or 7. In binary (or base 2), the only prime factor is 2. So you can only express fractions cleanly which only contain 2 as a prime factor. In binary, 1/2, 1/4, 1/8 would all be expressed cleanly as decimals. While, 1/5 or 1/10 would be repeating decimals. So 0.1 and 0.2 (1/10 and 1/5) while clean decimals in a base 10 system, are repeating decimals in the base 2 system the computer is operating in. When you do math on these repeating decimals, you end up with leftovers which carry over when you convert the computer's base 2 (binary) number into a more human readable base 10 number.
From https://0.30000000000000004.com/
Decimal numbers such as 0.1, 0.2, and 0.3 are not represented exactly in binary encoded floating point types. The sum of the approximations for 0.1 and 0.2 differs from the approximation used for 0.3, hence the falsehood of 0.1 + 0.2 == 0.3 as can be seen more clearly here:
#include <stdio.h>
int main() {
printf("0.1 + 0.2 == 0.3 is %s\n", 0.1 + 0.2 == 0.3 ? "true" : "false");
printf("0.1 is %.23f\n", 0.1);
printf("0.2 is %.23f\n", 0.2);
printf("0.1 + 0.2 is %.23f\n", 0.1 + 0.2);
printf("0.3 is %.23f\n", 0.3);
printf("0.3 - (0.1 + 0.2) is %g\n", 0.3 - (0.1 + 0.2));
return 0;
}
Output:
0.1 + 0.2 == 0.3 is false
0.1 is 0.10000000000000000555112
0.2 is 0.20000000000000001110223
0.1 + 0.2 is 0.30000000000000004440892
0.3 is 0.29999999999999998889777
0.3 - (0.1 + 0.2) is -5.55112e-17
For these computations to be evaluated more reliably, you would need to use a decimal-based representation for floating point values. The C Standard does not specify such types by default but as an extension described in a technical Report.
The _Decimal32, _Decimal64 and _Decimal128 types might be available on your system (for example, GCC supports them on selected targets, but Clang does not support them on OS X).
Since this thread branched off a bit into a general discussion over current floating point implementations I'd add that there are projects on fixing their issues.
Take a look at https://posithub.org/ for example, which showcases a number type called posit (and its predecessor unum) that promises to offer better accuracy with fewer bits. If my understanding is correct, it also fixes the kind of problems in the question. Quite interesting project, the person behind it is a mathematician it Dr. John Gustafson. The whole thing is open source, with many actual implementations in C/C++, Python, Julia and C# (https://hastlayer.com/arithmetics).
Normal arithmetic is base-10, so decimals represent tenths, hundredths, etc. When you try to represent a floating-point number in binary base-2 arithmetic, you are dealing with halves, fourths, eighths, etc.
In the hardware, floating points are stored as integer mantissas and exponents. Mantissa represents the significant digits. Exponent is like scientific notation but it uses a base of 2 instead of 10. For example 64.0 would be represented with a mantissa of 1 and exponent of 6. 0.125 would be represented with a mantissa of 1 and an exponent of -3.
Floating point decimals have to add up negative powers of 2
0.1b = 0.5d
0.01b = 0.25d
0.001b = 0.125d
0.0001b = 0.0625d
0.00001b = 0.03125d
and so on.
It is common to use a error delta instead of using equality operators when dealing with floating point arithmetic. Instead of
if(a==b) ...
you would use
delta = 0.0001; // or some arbitrarily small amount
if(a - b > -delta && a - b < delta) ...
I've two questions regarding writing IEEE floating-point constants, as accepted by Z3's FPA logics:
First, in this question, Christoph used the example:
((_ asFloat 11 53) roundTowardZero 0.5 0))
I'm wondering what the final 0 signifies? I've tried:
((_ asFloat 11 53) roundTowardZero 0.5))
And that seems to work as well. Rummer's paper doesn't seem to require the final 0 either; so I'm curious what role it plays.
Second, when I get a model from Z3, it prints floating-point literals like so:
(as +1.0000000000000002220446049250313080847263336181640625p1 (_ FP 11 53))
How do I interpret the p1 suffix? What other suffixes are possible?
Thanks..
Thanks for pointing these issues out. Both of them are because there is no agreed upon standard for floating-point literals in the input or output yet.
The final 0 in the example represents the (binary) exponent, i.e., (... 0.5 1) == 1.0. We added this simply because numbers sometimes would require a lot of space if the exponent cannot be specified separately. This way, we can often specify them quite succinctly.
The p1 suffix in the output represents the binary exponent, i.e., where e8 means 10^8, the suffix p8 would mean 2^8. Z3 currently uses only binary exponents, so there would always be a p-suffix here, but this may change in the future. The rest of the number is given enough decimal digits to represent a precise result.
Note that the output format is not agreed upon yet by the SMT community. This may change in the future. For instance, there are discussions about whether this should be done in IEEE bit-vector format or an intermediate format that lies somewhere between reals and non-IEEE bit-vectors.