15. 浮点算术：争议和限制
************************

浮点数在计算机硬件中以二进制分数表示。举例而言，十进制分数

   0.125

等于 1/10 + 2/100 + 5/1000 ，同理，二进制分数

   0.001

等于0/2 + 0/4 + 1/8。这两个分数的值相同，唯一真正的区别是第一个用十进
制表示法写入，第二个用二进制表示。

不幸的是，大多数的十进制数都不能完全表示为二进制分数。这导致，大多数情
况下，你输入的十进制浮点数都只能近似地以二进制浮点数储存在计算机中。

用十进制来理解这个问题显得更加容易一些。考虑分数 1/3 。我们可以得到它
在十进制下的一个近似值

   0.3

或者，更近似的，:

   0.33

或者，更近似的，:

   0.333

以此类推。结果是无论你写下多少的数字，它都永远不会等于 1/3 ，只是更加
更加地接近 1/3 。

同样的道理，无论你写下多少的二进制数字，十进制分数 0.1 都无法恰好表示
为一个二进制分数。在二进制下， 1/10 是一个无限循环小数

   0.0001100110011001100110011001100110011001100110011...

在任何一个位置停下，你都只能得到一个近似值。因此，在今天的大部分架构上
，浮点数都只能近似地使用二进制分数表达，分子使用每 8 字节的前 53 位表
示，分母则表示为 2 的幂次。在 1/10 这个例子中，相应的二进制分数是
"3602879701896397 / 2 ** 55" ，它很接近 1/10 ，但并不是 1/10 。

大部分用户都不会意识到这个差异的存在，因为 Python 只会打印计算机中存储
的二进制值的十进制近似值。在大部分计算机中，如果 Python 想把 0.1 的二
进制对应的精确十进制打印出来，将会变成这样

   >>> 0.1
   0.1000000000000000055511151231257827021181583404541015625

这比大多数人认为有用的数字更多，因此Python通过显示舍入值来保持可管理的
位数

   >>> 1 / 10
   0.1

牢记，即使输出的结果看起来好像就是 1/10 的精确值，实际储存的值只是最接
近 1/10 的计算机可表示的二进制分数。

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".

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'

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

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 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.

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 Format
String Syntax.

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. Representation Error
==========================

This section explains the "0.1" example in detail, and shows how you
can perform an exact analysis of cases like this yourself.  Basic
familiarity with binary floating-point representation is assumed.

*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)

as

   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

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

Note that since we rounded up, this is actually a little bit larger
than 1/10; if we had not rounded up, the quotient would have been a
little bit smaller than 1/10.  But in no case can it be *exactly*
1/10!

So the computer never "sees" 1/10:  what it sees is the exact fraction
given above, the best 754 double approximation it can get:

   >>> 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'
