15. 浮動小数点演算、その問題と制限

浮動小数点数は、計算機ハードウェアの中では、基数を 2 とする (2進法の) 分数として表現されています。例えば、小数

0.125

は、 1/10 + 2/100 + 5/1000 という値を持ちますが、これと同様に、2 進法の分数

0.001

は 0/2 + 0/4 + 1/8 という値になります。これら二つの分数は同じ値を持っていますが、ただ一つ、最初の分数は基数 10 で記述されており、二番目の分数は基数 2 で記述されていることが違います。

残念なことに、ほとんどの小数は 2 進法の分数として正確に表わすことができません。その結果、一般に、入力した 10 進の浮動小数点数は、 2 進法の浮動小数点数で近似された後、実際にマシンに記憶されます。

最初は基数 10 を使うと問題を簡単に理解できます。分数 1/3 を考えてみましょう。分数 1/3 は、基数 10 の分数として、以下のように近似することができます:

0.3

さらに正確な近似は、

0.33

さらに正確な近似は、

0.333

となり、以後同様です。何個桁数を増やして書こうが、結果は決して厳密な 1/3 にはなりません。しかし、少しづつ正確な近似にはなっていくでしょう。

同様に、基数を 2 とした表現で何桁使おうとも、10 進数の 0.1 は基数を 2 とした小数で正確に表現することはできません。基数 2 では、1/10 は循環小数 (repeating fraction) となります

0.0001100110011001100110011001100110011001100110011...

Stop at any finite number of bits, and you get an approximation. On most machines today, floats are approximated using a binary fraction with the numerator using the first 53 bits starting with the most significant bit and with the denominator as a power of two. In the case of 1/10, the binary fraction is 3602879701896397 / 2 ** 55 which is close to but not exactly equal to the true value of 1/10.

Many users are not aware of the approximation because of the way values are displayed. Python only prints a decimal approximation to the true decimal value of the binary approximation stored by the machine. On most machines, if Python were to print the true decimal value of the binary approximation stored for 0.1, it would have to display

>>> 0.1
0.1000000000000000055511151231257827021181583404541015625

これは、ほとんどの人が必要と感じるよりも多すぎる桁数です。なので、Python は丸めた値を表示することで、桁数を扱いやすい範囲にとどめます

>>> 1 / 10
0.1

Just remember, even though the printed result looks like the exact value of 1/10, the actual stored value is the nearest representable binary fraction.

Interestingly, there are many different decimal numbers that share the same nearest approximate binary fraction. For example, the numbers 0.1 and 0.10000000000000001 and 0.1000000000000000055511151231257827021181583404541015625 are all approximated by 3602879701896397 / 2 ** 55. Since all of these decimal values share the same approximation, any one of them could be displayed while still preserving the invariant eval(repr(x)) == x.

Historically, the Python prompt and built-in repr() function would choose the one with 17 significant digits, 0.10000000000000001. Starting with Python 3.1, Python (on most systems) is now able to choose the shortest of these and simply display 0.1.

この動作は2進数の浮動小数点にとってはごく自然なものです。これは Python のバグではありませんし、あなたのコードのバグでもありません。ハードウェアの浮動小数点演算をサポートしている全ての言語で同じ種類の問題を見つけることができます (いくつかの言語ではデフォルトの、あるいはどの出力モードを選んでも、この差を 表示 しないかもしれませんが)。

For more pleasant output, you may wish to use string formatting to produce a limited number of significant digits:

>>> format(math.pi, '.12g')  # give 12 significant digits
'3.14159265359'

>>> format(math.pi, '.2f')   # give 2 digits after the point
'3.14'

>>> repr(math.pi)
'3.141592653589793'

It’s important to realize that this is, in a real sense, an illusion: you’re simply rounding the display of the true machine value.

One illusion may beget another. For example, since 0.1 is not exactly 1/10, summing three values of 0.1 may not yield exactly 0.3, either:

>>> .1 + .1 + .1 == .3
False

Also, since the 0.1 cannot get any closer to the exact value of 1/10 and 0.3 cannot get any closer to the exact value of 3/10, then pre-rounding with round() function cannot help:

>>> round(.1, 1) + round(.1, 1) + round(.1, 1) == round(.3, 1)
False

Though the numbers cannot be made closer to their intended exact values, the round() function can be useful for post-rounding so that results with inexact values become comparable to one another:

>>> round(.1 + .1 + .1, 10) == round(.3, 10)
True

2 進の浮動小数点数に対する算術演算は、このような意外性をたくさん持っています。 「0.1」 に関する問題は、以下の 「表現エラー」 の章で詳細に説明します。 2 進法の浮動小数点演算にともなうその他のよく知られた意外な事象に関しては The Perils of Floating Point を参照してください。

As that says near the end, 「there are no easy answers.」 Still, don’t be unduly wary of floating-point! The errors in Python float operations are inherited from the floating-point hardware, and on most machines are on the order of no more than 1 part in 2**53 per operation. That’s more than adequate for most tasks, but you do need to keep in mind that it’s not decimal arithmetic and that every float operation can suffer a new rounding error.

While pathological cases do exist, for most casual use of floating-point arithmetic you’ll see the result you expect in the end if you simply round the display of your final results to the number of decimal digits you expect. str() usually suffices, and for finer control see the str.format() method’s format specifiers in 書式指定文字列の文法.

For use cases which require exact decimal representation, try using the decimal module which implements decimal arithmetic suitable for accounting applications and high-precision applications.

Another form of exact arithmetic is supported by the fractions module which implements arithmetic based on rational numbers (so the numbers like 1/3 can be represented exactly).

If you are a heavy user of floating point operations you should take a look at the Numerical Python package and many other packages for mathematical and statistical operations supplied by the SciPy project. See <https://scipy.org>.

Python provides tools that may help on those rare occasions when you really do want to know the exact value of a float. The float.as_integer_ratio() method expresses the value of a float as a fraction:

>>> x = 3.14159
>>> x.as_integer_ratio()
(3537115888337719, 1125899906842624)

Since the ratio is exact, it can be used to losslessly recreate the original value:

>>> x == 3537115888337719 / 1125899906842624
True

The float.hex() method expresses a float in hexadecimal (base 16), again giving the exact value stored by your computer:

>>> x.hex()
'0x1.921f9f01b866ep+1'

This precise hexadecimal representation can be used to reconstruct the float value exactly:

>>> x == float.fromhex('0x1.921f9f01b866ep+1')
True

Since the representation is exact, it is useful for reliably porting values across different versions of Python (platform independence) and exchanging data with other languages that support the same format (such as Java and C99).

Another helpful tool is the math.fsum() function which helps mitigate loss-of-precision during summation. It tracks 「lost digits」 as values are added onto a running total. That can make a difference in overall accuracy so that the errors do not accumulate to the point where they affect the final total:

>>> sum([0.1] * 10) == 1.0
False
>>> math.fsum([0.1] * 10) == 1.0
True

15.1. 表現エラー

この章では、」0.1」 の例について詳細に説明し、このようなケースに対してどのようにすれば正確な分析を自分で行えるかを示します。ここでは、 2 進法表現の浮動小数点数についての基礎的な知識があるものとして話を進めます。

Representation error refers to the fact that some (most, actually) decimal fractions cannot be represented exactly as binary (base 2) fractions. This is the chief reason why Python (or Perl, C, C++, Java, Fortran, and many others) often won’t display the exact decimal number you expect.

Why is that? 1/10 is not exactly representable as a binary fraction. Almost all machines today (November 2000) use IEEE-754 floating point arithmetic, and almost all platforms map Python floats to IEEE-754 「double precision」. 754 doubles contain 53 bits of precision, so on input the computer strives to convert 0.1 to the closest fraction it can of the form J/2**N where J is an integer containing exactly 53 bits. Rewriting

1 / 10 ~= J / (2**N)

を書き直すと

J ~= 2**N / 10

となります。 J は厳密に 53 ビットの精度を持っている (>= 2**52 だが < 2**53 ) ことを思い出すと、 N として最適な値は 56 になります:

>>> 2**52 <=  2**56 // 10  < 2**53
True

That is, 56 is the only value for N that leaves J with exactly 53 bits. The best possible value for J is then that quotient rounded:

>>> q, r = divmod(2**56, 10)
>>> r
6

残りは 10 の半分以上なので、最良の近似は丸め値を一つ増やした (round up) ものになります:

>>> q+1
7205759403792794

Therefore the best possible approximation to 1/10 in 754 double precision is:

7205759403792794 / 2 ** 56

Dividing both the numerator and denominator by two reduces the fraction to:

3602879701896397 / 2 ** 55

となります。丸め値を 1 増やしたので、この値は実際には 1/10 より少し小さいことに注意してください; 丸め値を 1 増やさない場合、商は 1/10 よりもわずかに小さくなります。しかし、どちらにしろ 厳密に 1/10 ではありません!

つまり、計算機は 1/10 を 「理解する」 ことは決してありません。計算機が理解できるのは、上記のような厳密な分数であり、 754 の倍精度浮動小数点数で得られるもっともよい近似は以下になります:

>>> 0.1 * 2 ** 55
3602879701896397.0

If we multiply that fraction by 10**55, we can see the value out to 55 decimal digits:

>>> 3602879701896397 * 10 ** 55 // 2 ** 55
1000000000000000055511151231257827021181583404541015625

meaning that the exact number stored in the computer is equal to the decimal value 0.1000000000000000055511151231257827021181583404541015625. Instead of displaying the full decimal value, many languages (including older versions of Python), round the result to 17 significant digits:

>>> format(0.1, '.17f')
'0.10000000000000001'

The fractions and decimal modules make these calculations easy:

>>> from decimal import Decimal
>>> from fractions import Fraction

>>> Fraction.from_float(0.1)
Fraction(3602879701896397, 36028797018963968)

>>> (0.1).as_integer_ratio()
(3602879701896397, 36028797018963968)

>>> Decimal.from_float(0.1)
Decimal('0.1000000000000000055511151231257827021181583404541015625')

>>> format(Decimal.from_float(0.1), '.17')
'0.10000000000000001'