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.

Unfortunately, most decimal fractions cannot be represented exactly as
binary fractions.  A consequence is that, in general, the decimal
floating-point numbers you enter are only approximated by the binary
floating-point numbers actually stored in the machine.

The problem is easier to understand at first in base 10.  Consider the
fraction 1/3.  You can approximate that as a base 10 fraction:

   0.3

or, better,

   0.33

or, better,

   0.333

and so on.  No matter how many digits you're willing to write down,
the result will never be exactly 1/3, but will be an increasingly
better approximation of 1/3.

In the same way, no matter how many base 2 digits you're willing to
use, the decimal value 0.1 cannot be represented exactly as a base 2
fraction.  In base 2, 1/10 is the infinitely repeating fraction

   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

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
   0.1

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

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).

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

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


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'.
