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abc.rst.txt	7219	0644	editdlrm
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aetools.rst.txt	3532	0644	editdlrm
aetypes.rst.txt	4257	0644	editdlrm
aifc.rst.txt	7086	0644	editdlrm
al.rst.txt	5309	0644	editdlrm
allos.rst.txt	695	0644	editdlrm
anydbm.rst.txt	4111	0644	editdlrm
archiving.rst.txt	424	0644	editdlrm
argparse.rst.txt	74685	0644	editdlrm
array.rst.txt	10524	0644	editdlrm
ast.rst.txt	10111	0644	editdlrm
asynchat.rst.txt	9206	0644	editdlrm
asyncore.rst.txt	12937	0644	editdlrm
atexit.rst.txt	3910	0644	editdlrm
audioop.rst.txt	10395	0644	editdlrm
autogil.rst.txt	1015	0644	editdlrm
base64.rst.txt	6252	0644	editdlrm
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calendar.rst.txt	11282	0644	editdlrm
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cd.rst.txt	11974	0644	editdlrm
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chunk.rst.txt	4955	0644	editdlrm
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code.rst.txt	7115	0644	editdlrm
codecs.rst.txt	66966	0644	editdlrm
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collections.rst.txt	41448	0644	editdlrm
colorpicker.rst.txt	913	0644	editdlrm
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commands.rst.txt	2595	0644	editdlrm
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compiler.rst.txt	37464	0644	editdlrm
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crypt.rst.txt	2292	0644	editdlrm
crypto.rst.txt	355	0644	editdlrm
csv.rst.txt	22757	0644	editdlrm
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curses.ascii.rst.txt	9046	0644	editdlrm
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curses.rst.txt	74877	0644	editdlrm
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datetime.rst.txt	74679	0644	editdlrm
dbhash.rst.txt	3865	0644	editdlrm
dbm.rst.txt	3117	0644	editdlrm
debug.rst.txt	446	0644	editdlrm
decimal.rst.txt	70926	0644	editdlrm
development.rst.txt	640	0644	editdlrm
difflib.rst.txt	30716	0644	editdlrm
dircache.rst.txt	1813	0644	editdlrm
dis.rst.txt	23248	0644	editdlrm
distribution.rst.txt	426	0644	editdlrm
distutils.rst.txt	1958	0644	editdlrm
dl.rst.txt	3392	0644	editdlrm
doctest.rst.txt	73985	0644	editdlrm
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dumbdbm.rst.txt	2841	0644	editdlrm
dummy_thread.rst.txt	1058	0644	editdlrm
dummy_threading.rst.txt	799	0644	editdlrm
easydialogs.rst.txt	10346	0644	editdlrm
email-examples.rst.txt	1271	0644	editdlrm
email.charset.rst.txt	9654	0644	editdlrm
email.encoders.rst.txt	2376	0644	editdlrm
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exceptions.rst.txt	18343	0644	editdlrm
fcntl.rst.txt	7361	0644	editdlrm
filecmp.rst.txt	5348	0644	editdlrm
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fileinput.rst.txt	7417	0644	editdlrm
filesys.rst.txt	806	0644	editdlrm
fl.rst.txt	17646	0644	editdlrm
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fnmatch.rst.txt	3104	0644	editdlrm
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fpectl.rst.txt	4171	0644	editdlrm
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fractions.rst.txt	5297	0644	editdlrm
framework.rst.txt	11444	0644	editdlrm
frameworks.rst.txt	378	0644	editdlrm
ftplib.rst.txt	15720	0644	editdlrm
functions.rst.txt	75554	0644	editdlrm
functools.rst.txt	7448	0644	editdlrm
future_builtins.rst.txt	2010	0644	editdlrm
gc.rst.txt	9014	0644	editdlrm
gdbm.rst.txt	4882	0644	editdlrm
gensuitemodule.rst.txt	3113	0644	editdlrm
getopt.rst.txt	6670	0644	editdlrm
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hmac.rst.txt	3073	0644	editdlrm
hotshot.rst.txt	4289	0644	editdlrm
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httplib.rst.txt	37454	0644	editdlrm
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idle.rst.txt	22182	0644	editdlrm
imageop.rst.txt	4001	0644	editdlrm
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imghdr.rst.txt	2635	0644	editdlrm
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importlib.rst.txt	1126	0644	editdlrm
imputil.rst.txt	7023	0644	editdlrm
index.rst.txt	2287	0644	editdlrm
inspect.rst.txt	28151	0644	editdlrm
internet.rst.txt	950	0644	editdlrm
intro.rst.txt	2803	0644	editdlrm
io.rst.txt	39015	0644	editdlrm
ipc.rst.txt	630	0644	editdlrm
itertools.rst.txt	36324	0644	editdlrm
jpeg.rst.txt	3858	0644	editdlrm
json.rst.txt	25545	0644	editdlrm
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logging.config.rst.txt	31560	0644	editdlrm
logging.handlers.rst.txt	28153	0644	editdlrm
logging.rst.txt	46882	0644	editdlrm
mac.rst.txt	791	0644	editdlrm
macos.rst.txt	3824	0644	editdlrm
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macostools.rst.txt	4017	0644	editdlrm
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mailbox.rst.txt	68129	0644	editdlrm
mailcap.rst.txt	3675	0644	editdlrm
markup.rst.txt	1249	0644	editdlrm
marshal.rst.txt	5689	0644	editdlrm
math.rst.txt	10920	0644	editdlrm
md5.rst.txt	2815	0644	editdlrm
mhlib.rst.txt	3966	0644	editdlrm
mimetools.rst.txt	4504	0644	editdlrm
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mimewriter.rst.txt	3360	0644	editdlrm
mimify.rst.txt	3519	0644	editdlrm
miniaeframe.rst.txt	2564	0644	editdlrm
misc.rst.txt	248	0644	editdlrm
mm.rst.txt	447	0644	editdlrm
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modulefinder.rst.txt	3380	0644	editdlrm
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msilib.rst.txt	19062	0644	editdlrm
msvcrt.rst.txt	4346	0644	editdlrm
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mutex.rst.txt	1939	0644	editdlrm
netdata.rst.txt	432	0644	editdlrm
netrc.rst.txt	3119	0644	editdlrm
new.rst.txt	2653	0644	editdlrm
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nntplib.rst.txt	14531	0644	editdlrm
numbers.rst.txt	8007	0644	editdlrm
numeric.rst.txt	751	0644	editdlrm
operator.rst.txt	22092	0644	editdlrm
optparse.rst.txt	77101	0644	editdlrm
os.path.rst.txt	13094	0644	editdlrm
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othergui.rst.txt	2721	0644	editdlrm
parser.rst.txt	15385	0644	editdlrm
pdb.rst.txt	16043	0644	editdlrm
persistence.rst.txt	826	0644	editdlrm
pickle.rst.txt	37262	0644	editdlrm
pickletools.rst.txt	1997	0644	editdlrm
pipes.rst.txt	3786	0644	editdlrm
pkgutil.rst.txt	7714	0644	editdlrm
platform.rst.txt	9554	0644	editdlrm
plistlib.rst.txt	4134	0644	editdlrm
popen2.rst.txt	7022	0644	editdlrm
poplib.rst.txt	6220	0644	editdlrm
posix.rst.txt	3630	0644	editdlrm
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pprint.rst.txt	9071	0644	editdlrm
profile.rst.txt	28666	0644	editdlrm
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pyclbr.rst.txt	3296	0644	editdlrm
pydoc.rst.txt	4083	0644	editdlrm
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python.rst.txt	514	0644	editdlrm
py_compile.rst.txt	2479	0644	editdlrm
queue.rst.txt	7034	0644	editdlrm
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random.rst.txt	13319	0644	editdlrm
re.rst.txt	55102	0644	editdlrm
readline.rst.txt	10513	0644	editdlrm
repr.rst.txt	4718	0644	editdlrm
resource.rst.txt	9595	0644	editdlrm
restricted.rst.txt	3327	0644	editdlrm
rexec.rst.txt	11742	0644	editdlrm
rfc822.rst.txt	14037	0644	editdlrm
rlcompleter.rst.txt	2494	0644	editdlrm
robotparser.rst.txt	2190	0644	editdlrm
runpy.rst.txt	6938	0644	editdlrm
sched.rst.txt	4644	0644	editdlrm
scrolledtext.rst.txt	1379	0644	editdlrm
select.rst.txt	20705	0644	editdlrm
sets.rst.txt	14973	0644	editdlrm
sgi.rst.txt	322	0644	editdlrm
sgmllib.rst.txt	10664	0644	editdlrm
sha.rst.txt	2807	0644	editdlrm
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shlex.rst.txt	11313	0644	editdlrm
shutil.rst.txt	13491	0644	editdlrm
signal.rst.txt	10719	0644	editdlrm
simplehttpserver.rst.txt	4559	0644	editdlrm
simplexmlrpcserver.rst.txt	10881	0644	editdlrm
site.rst.txt	7880	0644	editdlrm
smtpd.rst.txt	2464	0644	editdlrm
smtplib.rst.txt	14906	0644	editdlrm
sndhdr.rst.txt	1759	0644	editdlrm
socket.rst.txt	40765	0644	editdlrm
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string.rst.txt	44085	0644	editdlrm
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:mod:`math` --- Mathematical functions ====================================== .. module:: math :synopsis: Mathematical functions (sin() etc.). .. testsetup:: from math import fsum This module is always available. It provides access to the mathematical functions defined by the C standard. These functions cannot be used with complex numbers; use the functions of the same name from the :mod:`cmath` module if you require support for complex numbers. The distinction between functions which support complex numbers and those which don't is made since most users do not want to learn quite as much mathematics as required to understand complex numbers. Receiving an exception instead of a complex result allows earlier detection of the unexpected complex number used as a parameter, so that the programmer can determine how and why it was generated in the first place. The following functions are provided by this module. Except when explicitly noted otherwise, all return values are floats. Number-theoretic and representation functions --------------------------------------------- .. function:: ceil(x) Return the ceiling of *x* as a float, the smallest integer value greater than or equal to *x*. .. function:: copysign(x, y) Return *x* with the sign of *y*. On a platform that supports signed zeros, ``copysign(1.0, -0.0)`` returns *-1.0*. .. versionadded:: 2.6 .. function:: fabs(x) Return the absolute value of *x*. .. function:: factorial(x) Return *x* factorial. Raises :exc:`ValueError` if *x* is not integral or is negative. .. versionadded:: 2.6 .. function:: floor(x) Return the floor of *x* as a float, the largest integer value less than or equal to *x*. .. function:: fmod(x, y) Return ``fmod(x, y)``, as defined by the platform C library. Note that the Python expression ``x % y`` may not return the same result. The intent of the C standard is that ``fmod(x, y)`` be exactly (mathematically; to infinite precision) equal to ``x - n*y`` for some integer *n* such that the result has the same sign as *x* and magnitude less than ``abs(y)``. Python's ``x % y`` returns a result with the sign of *y* instead, and may not be exactly computable for float arguments. For example, ``fmod(-1e-100, 1e100)`` is ``-1e-100``, but the result of Python's ``-1e-100 % 1e100`` is ``1e100-1e-100``, which cannot be represented exactly as a float, and rounds to the surprising ``1e100``. For this reason, function :func:`fmod` is generally preferred when working with floats, while Python's ``x % y`` is preferred when working with integers. .. function:: frexp(x) Return the mantissa and exponent of *x* as the pair ``(m, e)``. *m* is a float and *e* is an integer such that ``x == m * 2**e`` exactly. If *x* is zero, returns ``(0.0, 0)``, otherwise ``0.5 <= abs(m) < 1``. This is used to "pick apart" the internal representation of a float in a portable way. .. function:: fsum(iterable) Return an accurate floating point sum of values in the iterable. Avoids loss of precision by tracking multiple intermediate partial sums:: >>> sum([.1, .1, .1, .1, .1, .1, .1, .1, .1, .1]) 0.9999999999999999 >>> fsum([.1, .1, .1, .1, .1, .1, .1, .1, .1, .1]) 1.0 The algorithm's accuracy depends on IEEE-754 arithmetic guarantees and the typical case where the rounding mode is half-even. On some non-Windows builds, the underlying C library uses extended precision addition and may occasionally double-round an intermediate sum causing it to be off in its least significant bit. For further discussion and two alternative approaches, see the `ASPN cookbook recipes for accurate floating point summation `_\. .. versionadded:: 2.6 .. function:: isinf(x) Check if the float *x* is positive or negative infinity. .. versionadded:: 2.6 .. function:: isnan(x) Check if the float *x* is a NaN (not a number). For more information on NaNs, see the IEEE 754 standards. .. versionadded:: 2.6 .. function:: ldexp(x, i) Return ``x * (2**i)``. This is essentially the inverse of function :func:`frexp`. .. function:: modf(x) Return the fractional and integer parts of *x*. Both results carry the sign of *x* and are floats. .. function:: trunc(x) Return the :class:`~numbers.Real` value *x* truncated to an :class:`~numbers.Integral` (usually a long integer). Uses the ``__trunc__`` method. .. versionadded:: 2.6 Note that :func:`frexp` and :func:`modf` have a different call/return pattern than their C equivalents: they take a single argument and return a pair of values, rather than returning their second return value through an 'output parameter' (there is no such thing in Python). For the :func:`ceil`, :func:`floor`, and :func:`modf` functions, note that *all* floating-point numbers of sufficiently large magnitude are exact integers. Python floats typically carry no more than 53 bits of precision (the same as the platform C double type), in which case any float *x* with ``abs(x) >= 2**52`` necessarily has no fractional bits. Power and logarithmic functions ------------------------------- .. function:: exp(x) Return ``e**x``. .. function:: expm1(x) Return ``e**x - 1``. For small floats *x*, the subtraction in ``exp(x) - 1`` can result in a significant loss of precision; the :func:`expm1` function provides a way to compute this quantity to full precision:: >>> from math import exp, expm1 >>> exp(1e-5) - 1 # gives result accurate to 11 places 1.0000050000069649e-05 >>> expm1(1e-5) # result accurate to full precision 1.0000050000166668e-05 .. versionadded:: 2.7 .. function:: log(x[, base]) With one argument, return the natural logarithm of *x* (to base *e*). With two arguments, return the logarithm of *x* to the given *base*, calculated as ``log(x)/log(base)``. .. versionchanged:: 2.3 *base* argument added. .. function:: log1p(x) Return the natural logarithm of *1+x* (base *e*). The result is calculated in a way which is accurate for *x* near zero. .. versionadded:: 2.6 .. function:: log10(x) Return the base-10 logarithm of *x*. This is usually more accurate than ``log(x, 10)``. .. function:: pow(x, y) Return ``x`` raised to the power ``y``. Exceptional cases follow Annex 'F' of the C99 standard as far as possible. In particular, ``pow(1.0, x)`` and ``pow(x, 0.0)`` always return ``1.0``, even when ``x`` is a zero or a NaN. If both ``x`` and ``y`` are finite, ``x`` is negative, and ``y`` is not an integer then ``pow(x, y)`` is undefined, and raises :exc:`ValueError`. Unlike the built-in ``**`` operator, :func:`math.pow` converts both its arguments to type :class:`float`. Use ``**`` or the built-in :func:`pow` function for computing exact integer powers. .. versionchanged:: 2.6 The outcome of ``1**nan`` and ``nan**0`` was undefined. .. function:: sqrt(x) Return the square root of *x*. Trigonometric functions ----------------------- .. function:: acos(x) Return the arc cosine of *x*, in radians. .. function:: asin(x) Return the arc sine of *x*, in radians. .. function:: atan(x) Return the arc tangent of *x*, in radians. .. function:: atan2(y, x) Return ``atan(y / x)``, in radians. The result is between ``-pi`` and ``pi``. The vector in the plane from the origin to point ``(x, y)`` makes this angle with the positive X axis. The point of :func:`atan2` is that the signs of both inputs are known to it, so it can compute the correct quadrant for the angle. For example, ``atan(1)`` and ``atan2(1, 1)`` are both ``pi/4``, but ``atan2(-1, -1)`` is ``-3*pi/4``. .. function:: cos(x) Return the cosine of *x* radians. .. function:: hypot(x, y) Return the Euclidean norm, ``sqrt(x*x + y*y)``. This is the length of the vector from the origin to point ``(x, y)``. .. function:: sin(x) Return the sine of *x* radians. .. function:: tan(x) Return the tangent of *x* radians. Angular conversion ------------------ .. function:: degrees(x) Convert angle *x* from radians to degrees. .. function:: radians(x) Convert angle *x* from degrees to radians. Hyperbolic functions -------------------- .. function:: acosh(x) Return the inverse hyperbolic cosine of *x*. .. versionadded:: 2.6 .. function:: asinh(x) Return the inverse hyperbolic sine of *x*. .. versionadded:: 2.6 .. function:: atanh(x) Return the inverse hyperbolic tangent of *x*. .. versionadded:: 2.6 .. function:: cosh(x) Return the hyperbolic cosine of *x*. .. function:: sinh(x) Return the hyperbolic sine of *x*. .. function:: tanh(x) Return the hyperbolic tangent of *x*. Special functions ----------------- .. function:: erf(x) Return the error function at *x*. .. versionadded:: 2.7 .. function:: erfc(x) Return the complementary error function at *x*. .. versionadded:: 2.7 .. function:: gamma(x) Return the Gamma function at *x*. .. versionadded:: 2.7 .. function:: lgamma(x) Return the natural logarithm of the absolute value of the Gamma function at *x*. .. versionadded:: 2.7 Constants --------- .. data:: pi The mathematical constant π = 3.141592..., to available precision. .. data:: e The mathematical constant e = 2.718281..., to available precision. .. impl-detail:: The :mod:`math` module consists mostly of thin wrappers around the platform C math library functions. Behavior in exceptional cases follows Annex F of the C99 standard where appropriate. The current implementation will raise :exc:`ValueError` for invalid operations like ``sqrt(-1.0)`` or ``log(0.0)`` (where C99 Annex F recommends signaling invalid operation or divide-by-zero), and :exc:`OverflowError` for results that overflow (for example, ``exp(1000.0)``). A NaN will not be returned from any of the functions above unless one or more of the input arguments was a NaN; in that case, most functions will return a NaN, but (again following C99 Annex F) there are some exceptions to this rule, for example ``pow(float('nan'), 0.0)`` or ``hypot(float('nan'), float('inf'))``. Note that Python makes no effort to distinguish signaling NaNs from quiet NaNs, and behavior for signaling NaNs remains unspecified. Typical behavior is to treat all NaNs as though they were quiet. .. versionchanged:: 2.6 Behavior in special cases now aims to follow C99 Annex F. In earlier versions of Python the behavior in special cases was loosely specified. .. seealso:: Module :mod:`cmath` Complex number versions of many of these functions.