Update for the new float.__repr__()

rhettinger · rhettinger · commit 8bd1d4f52c0e · 2009-04-24T03:09:06.000Z
diff --git a/Doc/tutorial/floatingpoint.rst b/Doc/tutorial/floatingpoint.rst
@@ -48,32 +48,43 @@ decimal value 0.1 cannot be represented exactly as a base 2 fraction.  In base
 
    0.0001100110011001100110011001100110011001100110011...
 
-Stop at any finite number of bits, and you get an approximation.  This is why
-you see things like::
+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 following 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
-   0.10000000000000001
+   0.1000000000000000055511151231257827021181583404541015625
 
-On most machines today, that is what you'll see if you enter 0.1 at a Python
-prompt.  You may not, though, because the number of bits used by the hardware to
-store floating-point values can vary across machines, and 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 ::
+That is more digits than most people find useful, so Python keeps the number
+of digits manageable by displaying a rounded value instead ::
 
-   >>> 0.1
-   0.1000000000000000055511151231257827021181583404541015625
+   >>> 1 / 10
+   0.1
 
-instead!  The Python prompt uses the built-in :func:`repr` function to obtain a
-string version of everything it displays.  For floats, ``repr(float)`` rounds
-the true decimal value to 17 significant digits, giving ::
+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.
 
-   0.10000000000000001
+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 and
+while still preserving the invariant ``eval(repr(x)) == x``.
 
-``repr(float)`` produces 17 significant digits because it turns out that's
-enough (on most machines) so that ``eval(repr(x)) == x`` exactly for all finite
-floats *x*, but rounding to 16 digits is not enough to make that true.
+Historically, the Python prompt and built-in :func:`repr` function would chose
+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
@@ -85,23 +96,28 @@ Python's built-in :func:`str` function produces only 12 significant digits, and
 you may wish to use that instead.  It's unusual for ``eval(str(x))`` to
 reproduce *x*, but the output may be more pleasant to look at::
 
-   >>> print(str(0.1))
-   0.1
+   >>> str(math.pi)
+   '3.14159265359'
+
+   >>> repr(math.pi)
+   '3.141592653589793'
+
+   >>> format(math.pi, '.2f')
+   '3.14'
 
-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.
+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.
 
 Other surprises follow from this one.  For example, after seeing ::
 
-   >>> 0.1
-   0.10000000000000001
+   >>> format(0.1, '.17g')
+   '0.10000000000000001'
 
 you may be tempted to use the :func:`round` function to chop it back to the
 single digit you expect.  But that makes no difference::
 
-   >>> round(0.1, 1)
-   0.10000000000000001
+   >>> format(round(0.1, 1), '.17g')
+   '0.10000000000000001'
 
 The problem is that the binary floating-point value stored for "0.1" was already
 the best possible binary approximation to 1/10, so trying to round it again
@@ -115,7 +131,7 @@ Another consequence is that since 0.1 is not exactly 1/10, summing ten values of
    ...     sum += 0.1
    ...
    >>> sum
-   0.99999999999999989
+   0.9999999999999999
 
 Binary floating-point arithmetic holds many surprises like this.  The problem
 with "0.1" is explained in precise detail below, in the "Representation Error"
@@ -191,10 +207,7 @@ floating-point representation is assumed.
 :dfn:`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
-   0.10000000000000001
+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
@@ -237,26 +250,38 @@ that over 2\*\*56, or ::
 
    7205759403792794 / 72057594037927936
 
+Dividing both the numerator and denominator by two reduces the fraction to::
+
+   3602879701896397 / 36028797018963968
+
 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
+   >>> 0.1 * 2 ** 55
+   3602879701896397.0
 
-If we multiply that fraction by 10\*\*30, we can see the (truncated) value of
-its 30 most significant decimal digits::
+If we multiply that fraction by 10\*\*60, we can see the value of out to
+60 decimal digits::
 
-   >>> 7205759403792794 * 10**30 / 2**56
-   100000000000000005551115123125
+   >>> 3602879701896397 * 10 ** 60 // 2 ** 55
+   1000000000000000055511151231257827021181583404541015625
 
 meaning that the exact number stored in the computer is approximately equal to
 the decimal value 0.100000000000000005551115123125.  Rounding that to 17
 significant digits gives the 0.10000000000000001 that Python displays (well,
 will display on any 754-conforming platform that does best-possible input and
 output conversions in its C library --- yours may not!).
 
+The :mod:`fractions` and :mod:`decimal` modules make these calculations
+easy::
 
+   >>> from decimal import Decimal
+   >>> from fractions import Fraction
+   >>> print(Fraction.from_float(0.1))
+   3602879701896397/36028797018963968
+   >>> print(Decimal.from_float(0.1))
+   0.1000000000000000055511151231257827021181583404541015625
