111
both to the nearest built in type and do the operation there. For subtypes of
Integral
,
this means that
__add__()
and
__radd__()
should be defined as:
class MyIntegral(Integral):
def __add__(self, other):
if isinstance(other, MyIntegral):
return do_my_adding_stuff(self, other)
elif isinstance(other, OtherTypeIKnowAbout):
return do_my_other_adding_stuff(self, other)
else:
return NotImplemented
def __radd__(self, other):
if isinstance(other, MyIntegral):
return do_my_adding_stuff(other, self)
elif isinstance(other, OtherTypeIKnowAbout):
return do_my_other_adding_stuff(other, self)
elif isinstance(other, Integral):
return int(other) + int(self)
elif isinstance(other, Real):
return float(other) + float(self)
elif isinstance(other, Complex):
return complex(other) + complex(self)
else:
return NotImplemented
There are 5 different cases for a mixed-type operation on subclasses of
Complex
. I’ll refer
to all of the above code that doesn’t refer to
MyIntegral
and
OtherTypeIKnowAbout
as
“boilerplate”.
a
will be an instance of
A
, which is a subtype of
Complex
(
a : A <:
Complex
), and
b : B <: Complex
. I’ll consider
a + b
:
1. If
A
defines an
__add__()
which accepts
b
, all is well.
2. If
A
falls back to the boilerplate code, and it were to return a value from
__add__()
, we’d miss the possibility that
B
defines a more intelligent
__radd__()
,
so the boilerplate should return
NotImplemented
from
__add__()
. (Or
A
may not
implement
__add__()
at all.)
3. Then
B
‘s
__radd__()
gets a chance. If it accepts
a
, all is well.
4. If it falls back to the boilerplate, there are no more possible methods to try, so
this is where the default implementation should live.
5. If
B <: A
, Python tries
B.__radd__
before
A.__add__
. This is ok, because it was
implemented with knowledge of
A
, so it can handle those instances before
delegating to
Complex
.
If
A <: Complex
and
B <: Real
without sharing any other knowledge, then the
appropriate shared operation is the one involving the built in
complex
, and both
__radd__()
s land there, so
a+b == b+a
.
50
Because most of the operations on any given type will be very similar, it can be useful to
define a helper function which generates the forward and reverse instances of any given
operator. For example,
fractions.Fraction
uses:
def _operator_fallbacks(monomorphic_operator, fallback_operator):
def forward(a, b):
if isinstance(b, (int, long, Fraction)):
return monomorphic_operator(a, b)
elif isinstance(b, float):
return fallback_operator(float(a), b)
elif isinstance(b, complex):
return fallback_operator(complex(a), b)
else:
return NotImplemented
forward.__name__ = '__' + fallback_operator.__name__ + '__'
forward.__doc__ = monomorphic_operator.__doc__
def reverse(b, a):
if isinstance(a, Rational):
# Includes ints.
return monomorphic_operator(a, b)
elif isinstance(a, numbers.Real):
return fallback_operator(float(a), float(b))
elif isinstance(a, numbers.Complex):
return fallback_operator(complex(a), complex(b))
else:
return NotImplemented
reverse.__name__ = '__r' + fallback_operator.__name__ + '__'
reverse.__doc__ = monomorphic_operator.__doc__
return forward, reverse
def _add(a, b):
"""a + b"""
return Fraction(a.numerator * b.denominator +
b.numerator * a.denominator,
a.denominator * b.denominator)
__add__, __radd__ = _operator_fallbacks(_add, operator.add)
# ...
index
modules |
next |
previous |
Python » Python v2.7.6 documentation » The Python Standard Library » 9. Numeric and
Mathematical Modules »
© Copyright
1990-2013, Python Software Foundation.
The Python Software Foundation is a non-profit corporation. Please donate.
Last updated on Nov 10, 2013. Found a bug
?
Created using Sphinx
1.0.7.
Online Remove password from protected PDF file If we need a password from you, it will not be read or stored. To hlep protect your PDF document in C# project, XDoc.PDF provides some PDF security settings.
pdf password encryption; pdf secure signature
65
index
modules |
next |
previous |
9.2.
math
— Mathematical functions
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
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.
9.2.1. Number-theoretic and representation functions
math.
ceil
(
x
)
Return the ceiling of x as a float, the smallest integer value greater than or equal to x.
math.
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.
New in version 2.6.
math.
fabs
(
x
)
Return the absolute value of x.
math.
factorial
(
x
)
Return x factorial. Raises
ValueError
if x is not integral or is negative.
New in version 2.6.
math.
floor
(
x
)
Return the floor of x as a float, the largest integer value less than or equal to x.
math.
fmod
(
x, y
)
Python » Python v2.7.6 documentation » The Python Standard Library » 9. Numeric and
Mathematical Modules »
C# HTML5 Viewer: Deployment on AzureCloudService All. Create a New AzureCloudService Project in RasterEdge.XDoc.PDF.HTML5Editor.dll. validateIntegratedModeConfiguration="false"/> <security> <requestFiltering
change security settings pdf; create encrypted pdf C# HTML5 Viewer: Deployment on ASP.NET MVC Create a New ASP.NET MVC3 RasterEdge.XDoc.PDF.HTML5Editor.dll. validateIntegratedM odeConfiguration="false"/> <security> <requestFiltering allowDoubleEscaping
copy text from locked pdf; convert secure pdf to word
80
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
fmod()
is generally preferred when working with floats, while Python’s
x % y
is preferred when working with integers.
math.
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.
math.
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.
New in version 2.6.
math.
isinf
(
x
)
Check if the float x is positive or negative infinity.
New in version 2.6.
math.
isnan
(
x
)
76
Check if the float x is a NaN (not a number). For more information on NaNs, see the
IEEE 754 standards.
New in version 2.6.
math.
ldexp
(
x, i
)
Return
x * (2**i)
. This is essentially the inverse of function
frexp()
.
math.
modf
(
x
)
Return the fractional and integer parts of x. Both results carry the sign of x and are
floats.
math.
trunc
(
x
)
Return the
Real
value x truncated to an
Integral
(usually a long integer). Uses the
__trunc__
method.
New in version 2.6.
Note that
frexp()
and
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
ceil()
,
floor()
, and
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.
9.2.2. Power and logarithmic functions
math.
exp
(
x
)
Return
e**x
.
math.
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
expm1()
function provides a way to compute this
quantity to full precision:
97
>>> 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
New in version 2.7.
math.
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)
.
Changed in version 2.3: base argument added.
math.
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.
New in version 2.6.
math.
log10
(
x
)
Return the base-10 logarithm of x. This is usually more accurate than
log(x, 10)
.
math.
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
ValueError
.
Unlike the built-in
**
operator,
math.pow()
converts both its arguments to type
float
.
Use
**
or the built-in
pow()
function for computing exact integer powers.
Changed in version 2.6: The outcome of
1**nan
and
nan**0
was undefined.
math.
sqrt
(
x
)
Return the square root of x.
9.2.3. Trigonometric functions
math.
acos
(
x
)
Return the arc cosine of x, in radians.
91
math.
asin
(
x
)
Return the arc sine of x, in radians.
math.
atan
(
x
)
Return the arc tangent of x, in radians.
math.
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
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
.
math.
cos
(
x
)
Return the cosine of x radians.
math.
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)
.
math.
sin
(
x
)
Return the sine of x radians.
math.
tan
(
x
)
Return the tangent of x radians.
9.2.4. Angular conversion
math.
degrees
(
x
)
Converts angle x from radians to degrees.
math.
radians
(
x
)
Converts angle x from degrees to radians.
9.2.5. Hyperbolic functions
math.
acosh
(
x
)
Return the inverse hyperbolic cosine of x.
62
New in version 2.6.
math.
asinh
(
x
)
Return the inverse hyperbolic sine of x.
New in version 2.6.
math.
atanh
(
x
)
Return the inverse hyperbolic tangent of x.
New in version 2.6.
math.
cosh
(
x
)
Return the hyperbolic cosine of x.
math.
sinh
(
x
)
Return the hyperbolic sine of x.
math.
tanh
(
x
)
Return the hyperbolic tangent of x.
9.2.6. Special functions
math.
erf
(
x
)
Return the error function at x.
New in version 2.7.
math.
erfc
(
x
)
Return the complementary error function at x.
New in version 2.7.
math.
gamma
(
x
)
Return the Gamma function at x.
New in version 2.7.
math.
lgamma
(
x
)
Return the natural logarithm of the absolute value of the Gamma function at x.
New in version 2.7.
54
9.2.7. Constants
math.
pi
The mathematical constant π = 3.141592..., to available precision.
math.
e
The mathematical constant e = 2.718281..., to available precision.
CPython implementation detail: The
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
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
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.
Changed in version 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.
See also:
Module
cmath
Complex number versions of many of these functions.
index
modules |
next |
previous |
Python » Python v2.7.6 documentation » The Python Standard Library » 9. Numeric and
Mathematical Modules »
© Copyright
1990-2013, Python Software Foundation.
The Python Software Foundation is a non-profit corporation. Please donate.
Last updated on Nov 10, 2013. Found a bug
?
Created using Sphinx
1.0.7.
Documents you may be interested
Documents you may be interested
