Author: mark.dickinson Date: Fri Jul 30 14:58:44 2010 New Revision: 83276 Log: Update the floating-point section of the tutorial for the short float repr. Modified: python/branches/release27-maint/Doc/tutorial/floatingpoint.rst Modified: python/branches/release27-maint/Doc/tutorial/floatingpoint.rst ============================================================================== --- python/branches/release27-maint/Doc/tutorial/floatingpoint.rst (original) +++ python/branches/release27-maint/Doc/tutorial/floatingpoint.rst Fri Jul 30 14:58:44 2010 @@ -48,32 +48,37 @@ 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 a typical +machine, there are 53 bits of precision available, so the value stored +internally 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 - 0.10000000000000001 + 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 + 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 :: - - 0.10000000000000001 - -``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. +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 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 @@ -81,31 +86,32 @@ (although some languages may not *display* the difference by default, or in all output modes). -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 - -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. - -Other surprises follow from this one. For example, after seeing :: - - >>> 0.1 - 0.10000000000000001 +Other surprises follow from this one. For example, if you try to round the value +2.675 to two decimal places, you get this :: -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(2.675, 2) + 2.67 - >>> round(0.1, 1) - 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 -can't make it better: it was already as good as it gets. +The documentation for the built-in :func:`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 literal ``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 :mod:`decimal` module. +Incidentally, the :mod:`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:: @@ -150,15 +156,15 @@ 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 + >>> 0.1 + 0.2 + 0.30000000000000004 -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 :: +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) @@ -211,9 +217,8 @@ 100000000000000005551115123125L 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 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'.