# 15. Floating Point Arithmetic: Issues and Limitations¶

Floating-point numbers are represented in computer hardware as base 2 (binary) fractions. For example, the decimal fraction `0.125` has value 1/10 + 2/100 + 5/1000, and in the same way the binary fraction `0.001` has value 0/2 + 0/4 + 1/8. These two fractions have identical values, the only real difference being that the first is written in base 10 fractional notation, and the second in base 2.

```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。

```0.0001100110011001100110011001100110011001100110011...
```

どこか有限の桁で止めると、近似値を得ることになります。近年の殆どのコンピュータでは float 型は、最上位ビットから数えて最初の 53 ビットを分子、2 の冪乗を分母とした、二進小数で近似されます。1/10 の場合は、二進小数は `3602879701896397 / 2 ** 55` となります。これは、1/10 に近いですが、厳密に同じ値ではありません。

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
```

That is more digits than most people find useful, so Python keeps the number of digits manageable by displaying a rounded value instead

```>>> 1 / 10
0.1
```

Note that this is in the very nature of binary floating-point: this is not a bug in Python, and it is not a bug in your code either. You'll see the same kind of thing in all languages that support your hardware's floating-point arithmetic (although some languages may not display the difference by default, or in all output modes).

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'
```

これが、実際のコンピューター上の値の 表示 を丸めているだけの、いわば錯覚だということを認識しておいてください。

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
```

Binary floating-point arithmetic holds many surprises like this. The problem with "0.1" is explained in precise detail below, in the "Representation Error" section. See Examples of Floating Point Problems for a pleasant summary of how binary floating-point works and the kinds of problems commonly encountered in practice. Also see The Perils of Floating Point for a more complete account of other common surprises.

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.

あなたが浮動小数点演算のヘビーユーザーなら、SciPy プロジェクトが提供している NumPy パッケージやその他の数学用パッケージを調べてみるべきです。 <https://scipy.org> を参照してください。

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
```

この16進数表現は正確なので、値を (プラットフォームにも依存せず) バージョンの異なるPython 間でやり取りしたり、他のこのフォーマットをサポートした言語 (Java や C99 など) と正確にやり取りするのに利用することができます。

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。

なぜそうなるのでしょう？ 1/10 は２進法の小数で厳密に表現できません。少なくとも2000年以降、ほぼすべてのマシンは IEEE 754 2進数の浮動小数点演算を用いており、ほぼすべてのプラットフォームでは Python の浮動小数点を IEEE 754 binary64 "倍精度 (double precision)" 値に対応付けます。 IEEE 754 binary64 値は 53 ビットの精度を持つため、計算機に入力を行おうとすると、可能な限り 0.1 を最も近い値の分数に変換し、J/2**N の形式にしようと努力します。J はちょうど 53 ビットの精度の整数です。

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

を書き直すと

```J ~= 2**N / 10
```

and recalling that J has exactly 53 bits (is `>= 2**52` but `< 2**53`), the best value for N is 56:

```>>> 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
```

Since the remainder is more than half of 10, the best approximation is obtained by rounding up:

```>>> q+1
7205759403792794
```

```7205759403792794 / 2 ** 56
```

```3602879701896397 / 2 ** 55
```

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、IEEE 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'
```