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"""
This module contains SymPy functions mathcin corresponding to special math functions in the
C standard library (since C99, also available in C++11).
The functions defined in this module allows the user to express functions such as ``expm1``
as a SymPy function for symbolic manipulation.
"""
from sympy.core.function import ArgumentIndexError, Function
from sympy.core.numbers import Rational
from sympy.core.power import Pow
from sympy.core.singleton import S
from sympy.functions.elementary.exponential import exp, log
from sympy.functions.elementary.miscellaneous import sqrt
def _expm1(x):
return exp(x) - S.One
class expm1(Function):
"""
Represents the exponential function minus one.
Explanation
===========
The benefit of using ``expm1(x)`` over ``exp(x) - 1``
is that the latter is prone to cancellation under finite precision
arithmetic when x is close to zero.
Examples
========
>>> from sympy.abc import x
>>> from sympy.codegen.cfunctions import expm1
>>> '%.0e' % expm1(1e-99).evalf()
'1e-99'
>>> from math import exp
>>> exp(1e-99) - 1
0.0
>>> expm1(x).diff(x)
exp(x)
See Also
========
log1p
"""
nargs = 1
def fdiff(self, argindex=1):
"""
Returns the first derivative of this function.
"""
if argindex == 1:
return exp(*self.args)
else:
raise ArgumentIndexError(self, argindex)
def _eval_expand_func(self, **hints):
return _expm1(*self.args)
def _eval_rewrite_as_exp(self, arg, **kwargs):
return exp(arg) - S.One
_eval_rewrite_as_tractable = _eval_rewrite_as_exp
@classmethod
def eval(cls, arg):
exp_arg = exp.eval(arg)
if exp_arg is not None:
return exp_arg - S.One
def _eval_is_real(self):
return self.args[0].is_real
def _eval_is_finite(self):
return self.args[0].is_finite
def _log1p(x):
return log(x + S.One)
class log1p(Function):
"""
Represents the natural logarithm of a number plus one.
Explanation
===========
The benefit of using ``log1p(x)`` over ``log(x + 1)``
is that the latter is prone to cancellation under finite precision
arithmetic when x is close to zero.
Examples
========
>>> from sympy.abc import x
>>> from sympy.codegen.cfunctions import log1p
>>> from sympy import expand_log
>>> '%.0e' % expand_log(log1p(1e-99)).evalf()
'1e-99'
>>> from math import log
>>> log(1 + 1e-99)
0.0
>>> log1p(x).diff(x)
1/(x + 1)
See Also
========
expm1
"""
nargs = 1
def fdiff(self, argindex=1):
"""
Returns the first derivative of this function.
"""
if argindex == 1:
return S.One/(self.args[0] + S.One)
else:
raise ArgumentIndexError(self, argindex)
def _eval_expand_func(self, **hints):
return _log1p(*self.args)
def _eval_rewrite_as_log(self, arg, **kwargs):
return _log1p(arg)
_eval_rewrite_as_tractable = _eval_rewrite_as_log
@classmethod
def eval(cls, arg):
if arg.is_Rational:
return log(arg + S.One)
elif not arg.is_Float: # not safe to add 1 to Float
return log.eval(arg + S.One)
elif arg.is_number:
return log(Rational(arg) + S.One)
def _eval_is_real(self):
return (self.args[0] + S.One).is_nonnegative
def _eval_is_finite(self):
if (self.args[0] + S.One).is_zero:
return False
return self.args[0].is_finite
def _eval_is_positive(self):
return self.args[0].is_positive
def _eval_is_zero(self):
return self.args[0].is_zero
def _eval_is_nonnegative(self):
return self.args[0].is_nonnegative
_Two = S(2)
def _exp2(x):
return Pow(_Two, x)
class exp2(Function):
"""
Represents the exponential function with base two.
Explanation
===========
The benefit of using ``exp2(x)`` over ``2**x``
is that the latter is not as efficient under finite precision
arithmetic.
Examples
========
>>> from sympy.abc import x
>>> from sympy.codegen.cfunctions import exp2
>>> exp2(2).evalf() == 4.0
True
>>> exp2(x).diff(x)
log(2)*exp2(x)
See Also
========
log2
"""
nargs = 1
def fdiff(self, argindex=1):
"""
Returns the first derivative of this function.
"""
if argindex == 1:
return self*log(_Two)
else:
raise ArgumentIndexError(self, argindex)
def _eval_rewrite_as_Pow(self, arg, **kwargs):
return _exp2(arg)
_eval_rewrite_as_tractable = _eval_rewrite_as_Pow
def _eval_expand_func(self, **hints):
return _exp2(*self.args)
@classmethod
def eval(cls, arg):
if arg.is_number:
return _exp2(arg)
def _log2(x):
return log(x)/log(_Two)
class log2(Function):
"""
Represents the logarithm function with base two.
Explanation
===========
The benefit of using ``log2(x)`` over ``log(x)/log(2)``
is that the latter is not as efficient under finite precision
arithmetic.
Examples
========
>>> from sympy.abc import x
>>> from sympy.codegen.cfunctions import log2
>>> log2(4).evalf() == 2.0
True
>>> log2(x).diff(x)
1/(x*log(2))
See Also
========
exp2
log10
"""
nargs = 1
def fdiff(self, argindex=1):
"""
Returns the first derivative of this function.
"""
if argindex == 1:
return S.One/(log(_Two)*self.args[0])
else:
raise ArgumentIndexError(self, argindex)
@classmethod
def eval(cls, arg):
if arg.is_number:
result = log.eval(arg, base=_Two)
if result.is_Atom:
return result
elif arg.is_Pow and arg.base == _Two:
return arg.exp
def _eval_evalf(self, *args, **kwargs):
return self.rewrite(log).evalf(*args, **kwargs)
def _eval_expand_func(self, **hints):
return _log2(*self.args)
def _eval_rewrite_as_log(self, arg, **kwargs):
return _log2(arg)
_eval_rewrite_as_tractable = _eval_rewrite_as_log
def _fma(x, y, z):
return x*y + z
class fma(Function):
"""
Represents "fused multiply add".
Explanation
===========
The benefit of using ``fma(x, y, z)`` over ``x*y + z``
is that, under finite precision arithmetic, the former is
supported by special instructions on some CPUs.
Examples
========
>>> from sympy.abc import x, y, z
>>> from sympy.codegen.cfunctions import fma
>>> fma(x, y, z).diff(x)
y
"""
nargs = 3
def fdiff(self, argindex=1):
"""
Returns the first derivative of this function.
"""
if argindex in (1, 2):
return self.args[2 - argindex]
elif argindex == 3:
return S.One
else:
raise ArgumentIndexError(self, argindex)
def _eval_expand_func(self, **hints):
return _fma(*self.args)
def _eval_rewrite_as_tractable(self, arg, limitvar=None, **kwargs):
return _fma(arg)
_Ten = S(10)
def _log10(x):
return log(x)/log(_Ten)
class log10(Function):
"""
Represents the logarithm function with base ten.
Examples
========
>>> from sympy.abc import x
>>> from sympy.codegen.cfunctions import log10
>>> log10(100).evalf() == 2.0
True
>>> log10(x).diff(x)
1/(x*log(10))
See Also
========
log2
"""
nargs = 1
def fdiff(self, argindex=1):
"""
Returns the first derivative of this function.
"""
if argindex == 1:
return S.One/(log(_Ten)*self.args[0])
else:
raise ArgumentIndexError(self, argindex)
@classmethod
def eval(cls, arg):
if arg.is_number:
result = log.eval(arg, base=_Ten)
if result.is_Atom:
return result
elif arg.is_Pow and arg.base == _Ten:
return arg.exp
def _eval_expand_func(self, **hints):
return _log10(*self.args)
def _eval_rewrite_as_log(self, arg, **kwargs):
return _log10(arg)
_eval_rewrite_as_tractable = _eval_rewrite_as_log
def _Sqrt(x):
return Pow(x, S.Half)
class Sqrt(Function): # 'sqrt' already defined in sympy.functions.elementary.miscellaneous
"""
Represents the square root function.
Explanation
===========
The reason why one would use ``Sqrt(x)`` over ``sqrt(x)``
is that the latter is internally represented as ``Pow(x, S.Half)`` which
may not be what one wants when doing code-generation.
Examples
========
>>> from sympy.abc import x
>>> from sympy.codegen.cfunctions import Sqrt
>>> Sqrt(x)
Sqrt(x)
>>> Sqrt(x).diff(x)
1/(2*sqrt(x))
See Also
========
Cbrt
"""
nargs = 1
def fdiff(self, argindex=1):
"""
Returns the first derivative of this function.
"""
if argindex == 1:
return Pow(self.args[0], Rational(-1, 2))/_Two
else:
raise ArgumentIndexError(self, argindex)
def _eval_expand_func(self, **hints):
return _Sqrt(*self.args)
def _eval_rewrite_as_Pow(self, arg, **kwargs):
return _Sqrt(arg)
_eval_rewrite_as_tractable = _eval_rewrite_as_Pow
def _Cbrt(x):
return Pow(x, Rational(1, 3))
class Cbrt(Function): # 'cbrt' already defined in sympy.functions.elementary.miscellaneous
"""
Represents the cube root function.
Explanation
===========
The reason why one would use ``Cbrt(x)`` over ``cbrt(x)``
is that the latter is internally represented as ``Pow(x, Rational(1, 3))`` which
may not be what one wants when doing code-generation.
Examples
========
>>> from sympy.abc import x
>>> from sympy.codegen.cfunctions import Cbrt
>>> Cbrt(x)
Cbrt(x)
>>> Cbrt(x).diff(x)
1/(3*x**(2/3))
See Also
========
Sqrt
"""
nargs = 1
def fdiff(self, argindex=1):
"""
Returns the first derivative of this function.
"""
if argindex == 1:
return Pow(self.args[0], Rational(-_Two/3))/3
else:
raise ArgumentIndexError(self, argindex)
def _eval_expand_func(self, **hints):
return _Cbrt(*self.args)
def _eval_rewrite_as_Pow(self, arg, **kwargs):
return _Cbrt(arg)
_eval_rewrite_as_tractable = _eval_rewrite_as_Pow
def _hypot(x, y):
return sqrt(Pow(x, 2) + Pow(y, 2))
class hypot(Function):
"""
Represents the hypotenuse function.
Explanation
===========
The hypotenuse function is provided by e.g. the math library
in the C99 standard, hence one may want to represent the function
symbolically when doing code-generation.
Examples
========
>>> from sympy.abc import x, y
>>> from sympy.codegen.cfunctions import hypot
>>> hypot(3, 4).evalf() == 5.0
True
>>> hypot(x, y)
hypot(x, y)
>>> hypot(x, y).diff(x)
x/hypot(x, y)
"""
nargs = 2
def fdiff(self, argindex=1):
"""
Returns the first derivative of this function.
"""
if argindex in (1, 2):
return 2*self.args[argindex-1]/(_Two*self.func(*self.args))
else:
raise ArgumentIndexError(self, argindex)
def _eval_expand_func(self, **hints):
return _hypot(*self.args)
def _eval_rewrite_as_Pow(self, arg, **kwargs):
return _hypot(arg)
_eval_rewrite_as_tractable = _eval_rewrite_as_Pow