14. 浮點數運算：問題與限制
**************************

在計算機架構中，浮點數透過二進位小數表示。例如說，在十進位小數中：

   0.125

可被分為 1/10 + 2/100 + 5/1000，同樣的道理，二進位小數 ：

   0.001

可被分為 0/2 + 0/4 + 1/8。這兩個小數有相同的數值，而唯一真正的不同在於
前者以十進位表示，後者以二進位表示。

不幸的是，大多數十進位小數無法精準地以二進位小數表示。一般的結果為，您
輸入的十進位浮點數由實際存在計算機中的二進位浮點數近似。

在十進位中，這個問題更容易被理解。以分數 1/3 為例，您可以將其近似為十
進位小數：

   0.3

或者，更好的近似：

   0.33

或者，更好的近似：

   0.333

依此類推，不論你使用多少位數表示小數，最後的結果都無法精準地表示 1/3，
但你還是能越來越精準地表示 1/3。

同樣的道理，不論你願意以多少位數表示二進位小數，十進位小數 0.1 都無法
被二進位小數精準地表達。在二進位小數中， 1/10 會是一個無限循環小數：

   0.0001100110011001100110011001100110011001100110011...

Stop at any finite number of bits, and you get an approximation.

On a typical machine running Python, there are 53 bits of precision
available for a Python float, so the value stored internally when you
enter the decimal number "0.1" is the binary fraction

   0.00011001100110011001100110011001100110011001100110011010

which is close to, but not exactly equal to, 1/10.

It’s easy to forget that the stored value is an approximation to the
original decimal fraction, because of the way that floats are
displayed at the interpreter prompt.  Python only prints a decimal
approximation to the true decimal value of the binary approximation
stored by the machine.  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

這比一般人感到有用的位數還多，所以 Python 將位數保持在可以接受的範圍，
只顯示捨入後的數值：

   >>> 0.1
   0.1

It’s important to realize that this is, in a real sense, an illusion:
the value in the machine is not exactly 1/10, you’re simply rounding
the *display* of the true machine value.  This fact becomes apparent
as soon as you try to do arithmetic with these values

   >>> 0.1 + 0.2
   0.30000000000000004

注意，這是二進位浮點數理所當然的特性，並不是 Python 的錯誤 (bug)，更不
是您程式碼的錯誤。只要有程式語言支持硬體的浮點數運算，您將會看到同樣的
事情出現在其中（雖然某些程式語言預設不*顯示*差異，或者預設全部輸出）。

Other surprises follow from this one.  For example, if you try to
round the value 2.675 to two decimal places, you get this

   >>> round(2.675, 2)
   2.67

The documentation for the built-in "round()" function says that it
rounds to the nearest value, rounding ties away from zero.  Since the
decimal fraction 2.675 is exactly halfway between 2.67 and 2.68, you
might expect the result here to be (a binary approximation to) 2.68.
It’s not, because when the decimal string "2.675" is converted to a
binary floating-point number, it’s again replaced with a binary
approximation, whose exact value is

   2.67499999999999982236431605997495353221893310546875

Since this approximation is slightly closer to 2.67 than to 2.68, it’s
rounded down.

If you’re in a situation where you care which way your decimal
halfway-cases are rounded, you should consider using the "decimal"
module. Incidentally, the "decimal" module also provides a nice way to
「see」 the exact value that’s stored in any particular Python float

   >>> from decimal import Decimal
   >>> Decimal(2.675)
   Decimal('2.67499999999999982236431605997495353221893310546875')

Another consequence is that since 0.1 is not exactly 1/10, summing ten
values of 0.1 may not yield exactly 1.0, either:

   >>> sum = 0.0
   >>> for i in range(10):
   ...     sum += 0.1
   ...
   >>> sum
   0.9999999999999999

二進位浮點數架構擁有很多這樣的驚喜。底下的「表示法錯誤」章節，詳細的解
釋了「0.1」的問題。如果想要其他常見驚喜更完整的描述，可以參考 The
Perils of Floating Point（浮點數的風險）。

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.  For fine control over how a float is
displayed see the "str.format()" method’s format specifiers in Format
String Syntax.


14.1. 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:

   >>> 0.1 + 0.2
   0.30000000000000004

Why is that?  1/10 and 2/10 are not exactly representable as a binary
fraction. Almost all machines today (July 2010) 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
   4503599627370496
   >>> 2**53
   9007199254740992
   >>> 2**56/10
   7205759403792793

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 that over 2**56, or

   7205759403792794 / 72057594037927936

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:

   >>> .1 * 2**56
   7205759403792794.0

If we multiply that fraction by 10**30, we can see the (truncated)
value of its 30 most significant decimal digits:

   >>> 7205759403792794 * 10**30 // 2**56
   100000000000000005551115123125L

meaning that the exact number 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 this value
to 17 significant digits, giving 『0.10000000000000001』.  In current
versions, Python displays a value based on the shortest decimal
fraction that rounds correctly back to the true binary value,
resulting simply in 『0.1』.
