Ondrej, thanks for your quick reply.
I implemented this functionality to trigsimp. I added lines 751-762 to
simplify.simplify.trigsimp() as follows (for the future googler):
---------------------------------
a,b,c = map(Wild,'abc')
matchers = (
(2*b*sin(a)*cos(a),b*sin(2*a)),#matches the double angle formula
)
res = None
for pattern, result in matchers:
res = ret.match(pattern)
if res is not None:
ret = result.subs(res)
break
if res is None:
ret = term
------------------------------------
I know this probably isn't the most efficient way of sharing code, but
if you're interested I attached my version of simplify.py and my
little test script (which is actually the start of a homework
assignment :)) that seemed to turn out ok. I'm about 24 hrs into
learning sympy and don't know if this code is 'sympythonic', but it
might help someone in a pinch.
-Bill
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from sympy.core import Basic, S, C, Add, Mul, Pow, Rational, Integer, \
Derivative, Wild, Symbol, sympify
from sympy.core.numbers import igcd
from sympy.utilities import make_list, all
from sympy.functions import gamma, exp, sqrt
from sympy.polys import Poly
from sys import maxint
import sympy.mpmath as mpmath
def fraction(expr, exact=False):
"""Returns a pair with expression's numerator and denominator.
If the given expression is not a fraction then this function
will assume that the denominator is equal to one.
This function will not make any attempt to simplify nested
fractions or to do any term rewriting at all.
If only one of the numerator/denominator pair is needed then
use numer(expr) or denom(expr) functions respectively.
>>> from sympy import *
>>> x, y = symbols('x', 'y')
>>> fraction(x/y)
(x, y)
>>> fraction(x)
(x, 1)
>>> fraction(1/y**2)
(1, y**2)
>>> fraction(x*y/2)
(x*y, 2)
>>> fraction(Rational(1, 2))
(1, 2)
This function will also work fine with assumptions:
>>> k = Symbol('k', negative=True)
>>> fraction(x * y**k)
(x, y**(-k))
If we know nothing about sign of some exponent and 'exact'
flag is unset, then structure this exponent's structure will
be analyzed and pretty fraction will be returned:
>>> fraction(2*x**(-y))
(2, x**y)
#>>> fraction(exp(-x))
#(1, exp(x))
>>> fraction(exp(-x), exact=True)
(exp(-x), 1)
"""
expr = sympify(expr)
#XXX this only works sometimes (caching bug?)
if expr == exp(-Symbol("x")) and exact:
return (expr, 1)
numer, denom = [], []
for term in make_list(expr, Mul):
if term.is_Pow:
if term.exp.is_negative:
if term.exp is S.NegativeOne:
denom.append(term.base)
else:
denom.append(Pow(term.base, -term.exp))
elif not exact and term.exp.is_Mul:
coeff, tail = term.exp.args[0], Mul(*term.exp.args[1:])#term.exp.getab()
if coeff.is_Rational and coeff.is_negative:
denom.append(Pow(term.base, -term.exp))
else:
numer.append(term)
else:
numer.append(term)
elif term.func is C.exp:
if term.args[0].is_negative:
denom.append(C.exp(-term.args[0]))
elif not exact and term.args[0].is_Mul:
coeff, tail = term.args[0], Mul(*term.args[1:])#term.args.getab()
if coeff.is_Rational and coeff.is_negative:
denom.append(C.exp(-term.args[0]))
else:
numer.append(term)
else:
numer.append(term)
elif term.is_Rational:
if term.is_integer:
numer.append(term)
else:
numer.append(Rational(term.p))
denom.append(Rational(term.q))
else:
numer.append(term)
return Mul(*numer), Mul(*denom)
def numer(expr):
return fraction(expr)[0]
def denom(expr):
return fraction(expr)[1]
def fraction_expand(expr):
a, b = fraction(expr)
return a.expand() / b.expand()
def numer_expand(expr):
a, b = fraction(expr)
return a.expand() / b
def denom_expand(expr):
a, b = fraction(expr)
return a / b.expand()
def separate(expr, deep=False):
"""Rewrite or separate a power of product to a product of powers
but without any expanding, ie. rewriting products to summations.
>>> from sympy import *
>>> x, y, z = symbols('x', 'y', 'z')
>>> separate((x*y)**2)
x**2*y**2
>>> separate((x*(y*z)**3)**2)
x**2*y**6*z**6
>>> separate((x*sin(x))**y + (x*cos(x))**y)
x**y*cos(x)**y + x**y*sin(x)**y
#this does not work because of exp combining
#>>> separate((exp(x)*exp(y))**x)
#exp(x*y)*exp(x**2)
>>> separate((sin(x)*cos(x))**y)
cos(x)**y*sin(x)**y
Notice that summations are left un touched. If this is not the
requested behaviour, apply 'expand' to input expression before:
>>> separate(((x+y)*z)**2)
z**2*(x + y)**2
>>> separate((x*y)**(1+z))
x**(1 + z)*y**(1 + z)
"""
expr = sympify(expr)
if expr.is_Pow:
terms, expo = [], separate(expr.exp, deep)
#print expr, terms, expo, expr.base
if expr.base.is_Mul:
t = [ separate(C.Pow(t,expo), deep) for t in expr.base.args ]
return C.Mul(*t)
elif expr.base.func is C.exp:
if deep == True:
return C.exp(separate(expr.base[0], deep)*expo)
else:
return C.exp(expr.base[0]*expo)
else:
return C.Pow(separate(expr.base, deep), expo)
elif expr.is_Add or expr.is_Mul:
return type(expr)(*[ separate(t, deep) for t in expr.args ])
elif expr.is_Function and deep:
return expr.func(*[ separate(t) for t in expr.args])
else:
return expr
def together(expr, deep=False):
"""Combine together and denest rational functions into a single
fraction. By default the resulting expression is simplified
to reduce the total order of both numerator and denominator
and minimize the number of terms.
Densting is done recursively on fractions level. However this
function will not attempt to rewrite composite objects, like
functions, interior unless 'deep' flag is set.
By definition, 'together' is a complement to 'apart', so
apart(together(expr)) should left expression unhanged.
>>> from sympy import *
>>> x, y, z = symbols('x', 'y', 'z')
You can work with sums of fractions easily. The algorithm
used here will, in an iterative style, collect numerators
and denominator of all expressions involved and perform
needed simplifications:
#>>> together(1/x + 1/y)
#(x + y)/(y*x)
#>>> together(1/x + 1/y + 1/z)
#(z*x + x*y + z*y)/(y*x*z)
>>> together(1/(x*y) + 1/y**2)
(x + y)/(x*y**2)
Or you can just denest multi-level fractional expressions:
>>> together(1/(1 + 1/x))
x/(1 + x)
It also perfect possible to work with symbolic powers or
exponential functions or combinations of both:
>>> together(1/x**y + 1/x**(y-1))
x**(-y)*(1 + x)
#>>> together(1/x**(2*y) + 1/x**(y-z))
#x**(-2*y)*(1 + x**(y + z))
#>>> together(1/exp(x) + 1/(x*exp(x)))
#(1+x)/(x*exp(x))
#>>> together(1/exp(2*x) + 1/(x*exp(3*x)))
#(1+exp(x)*x)/(x*exp(3*x))
"""
def _together(expr):
from sympy.core.function import Function
if expr.is_Add:
items, coeffs, basis = [], [], {}
for elem in expr.args:
numer, q = fraction(_together(elem))
denom = {}
for term in make_list(q.expand(), Mul):
expo = S.One
coeff = S.One
if term.is_Pow:
if term.exp.is_Rational:
term, expo = term.base, term.exp
elif term.exp.is_Mul:
coeff, tail = term.exp.as_coeff_terms()
if coeff.is_Rational:
tail = C.Mul(*tail)
term, expo = Pow(term.base, tail), coeff
coeff = S.One
elif term.func is C.exp:
if term[0].is_Rational:
term, expo = C.E, term[0]
elif term[0].is_Mul:
coeff, tail = term[0].as_coeff_terms()
if coeff.is_Rational:
tail = C.Mul(*tail)
term, expo = C.exp(tail), coeff
coeff = S.One
elif term.is_Rational:
coeff = Integer(term.q)
term = Integer(term.p)
if term in denom:
denom[term] += expo
else:
denom[term] = expo
if term in basis:
total, maxi = basis[term]
n_total = total + expo
n_maxi = max(maxi, expo)
basis[term] = (n_total, n_maxi)
else:
basis[term] = (expo, expo)
coeffs.append(coeff)
items.append((numer, denom))
numerator, denominator = [], []
for (term, (total, maxi)) in basis.iteritems():
basis[term] = (total, total-maxi)
if term.func is C.exp:
denominator.append(C.exp(maxi*term[:]))
else:
if maxi is S.One:
denominator.append(term)
else:
denominator.append(Pow(term, maxi))
if all([ c.is_integer for c in coeffs ]):
gcds = lambda x, y: igcd(int(x), int(y))
common = Rational(reduce(gcds, coeffs))
else:
common = S.One
product = reduce(lambda x, y: x*y, coeffs) / common
for ((numer, denom), coeff) in zip(items, coeffs):
expr, coeff = [], product / (coeff*common)
for term in basis.iterkeys():
total, sub = basis[term]
if term in denom:
expo = total-denom[term]-sub
else:
expo = total-sub
if term.func is C.exp:
expr.append(C.exp(expo*term[:]))
else:
if expo is S.One:
expr.append(term)
else:
expr.append(Pow(term, expo))
numerator.append(coeff*Mul(*([numer] + expr)))
return Add(*numerator)/(product*Mul(*denominator))
elif expr.is_Mul or expr.is_Pow:
return type(expr)(*[ _together(t) for t in expr.args ])
elif expr.is_Function and deep:
return expr.func(*[ _together(t) for t in expr.args ])
else:
return expr
return _together(separate(expr))
#apart -> partial fractions decomposition (will be here :)
def collect(expr, syms, evaluate=True, exact=False):
"""Collect additive terms with respect to a list of symbols up
to powers with rational exponents. By the term symbol here
are meant arbitrary expressions, which can contain powers,
products, sums etc. In other words symbol is a pattern
which will be searched for in the expression's terms.
This function will not apply any redundant expanding to the
input expression, so user is assumed to enter expression in
final form. This makes 'collect' more predictable as there
is no magic behind the scenes. However it is important to
note, that powers of products are converted to products of
powers using 'separate' function.
There are two possible types of output. First, if 'evaluate'
flag is set, this function will return a single expression
or else it will return a dictionary with separated symbols
up to rational powers as keys and collected sub-expressions
as values respectively.
>>> from sympy import *
>>> x, y, z = symbols('x', 'y', 'z')
>>> a, b, c = symbols('a', 'b', 'c')
This function can collect symbolic coefficients in polynomial
or rational expressions. It will manage to find all integer or
rational powers of collection variable:
>>> collect(a*x**2 + b*x**2 + a*x - b*x + c, x)
c + x*(a - b) + x**2*(a + b)
The same result can achieved in dictionary form:
>>> d = collect(a*x**2 + b*x**2 + a*x - b*x + c, x, evaluate=False)
>>> d[x**2]
a + b
>>> d[x]
a - b
>>> d[sympify(1)]
c
You can also work with multi-variate polynomials. However
remember that this function is greedy so it will care only
about a single symbol at time, in specification order:
>>> collect(x**2 + y*x**2 + x*y + y + a*y, [x, y])
x*y + y*(1 + a) + x**2*(1 + y)
>>> collect(x**2*y**4 + z*(x*y**2)**2 + z + a*z, [x*y**2, z])
z*(1 + a) + x**2*y**4*(1 + z)
Also more complicated expressions can be used as patterns:
>>> collect(a*sin(2*x) + b*sin(2*x), sin(2*x))
(a + b)*sin(2*x)
>>> collect(a*x**2*log(x)**2 + b*(x*log(x))**2, x*log(x))
x**2*log(x)**2*(a + b)
It is also possible to work with symbolic powers, although
it has more complicated behaviour, because in this case
power's base and symbolic part of the exponent are treated
as a single symbol:
>>> collect(a*x**c + b*x**c, x)
a*x**c + b*x**c
>>> collect(a*x**c + b*x**c, x**c)
x**c*(a + b)
However if you incorporate rationals to the exponents, then
you will get well known behaviour:
#>>> collect(a*x**(2*c) + b*x**(2*c), x**c)
#x**(2*c)*(a + b)
Note also that all previously stated facts about 'collect'
function apply to the exponential function, so you can get:
>>> collect(a*exp(2*x) + b*exp(2*x), exp(x))
(a + b)*exp(2*x)
If you are interested only in collecting specific powers
of some symbols then set 'exact' flag in arguments:
>>> collect(a*x**7 + b*x**7, x, exact=True)
a*x**7 + b*x**7
>>> collect(a*x**7 + b*x**7, x**7, exact=True)
x**7*(a + b)
You can also apply this function to differential equations, where
derivatives of arbitary order can be collected:
>>> from sympy import Derivative as D
>>> f = Function('f') (x)
>>> collect(a*D(f,x) + b*D(f,x), D(f,x))
(a + b)*D(f(x), x)
>>> collect(a*D(D(f,x),x) + b*D(D(f,x),x), D(f,x))
(a + b)*D(f(x), x, x)
>>> collect(a*D(D(f,x),x) + b*D(D(f,x),x), D(f,x), exact=True)
a*D(f(x), x, x) + b*D(f(x), x, x)
>>> collect(a*D(D(f,x),x) + b*D(D(f,x),x) + a*D(f,x) + b*D(f,x), D(f,x))
(a + b)*D(f(x), x) + (a + b)*D(f(x), x, x)
Or you can even match both derivative order and exponent at time:
>>> collect(a*D(D(f,x),x)**2 + b*D(D(f,x),x)**2, D(f,x))
(a + b)*D(f(x), x, x)**2
Notes
=====
- arguments are expected to be in expanded form, so you might have to call
.expand prior to calling this function.
"""
def make_expression(terms):
product = []
for term, rat, sym, deriv in terms:
if deriv is not None:
var, order = deriv
while order > 0:
term, order = Derivative(term, var), order-1
if sym is None:
if rat is S.One:
product.append(term)
else:
product.append(Pow(term, rat))
else:
product.append(Pow(term, rat*sym))
return Mul(*product)
def parse_derivative(deriv):
# scan derivatives tower in the input expression and return
# underlying function and maximal differentiation order
expr, sym, order = deriv.expr, deriv.symbols[0], 1
for s in deriv.symbols[1:]:
if s == sym:
order += 1
else:
raise NotImplementedError('Improve MV Derivative support in collect')
while isinstance(expr, Derivative):
s0 = expr.symbols[0]
for s in expr.symbols:
if s != s0:
raise NotImplementedError('Improve MV Derivative support in collect')
if s0 == sym:
expr, order = expr.expr, order+len(expr.symbols)
else:
break
return expr, (sym, Rational(order))
def parse_term(expr):
rat_expo, sym_expo = S.One, None
sexpr, deriv = expr, None
if expr.is_Pow:
if isinstance(expr.base, Derivative):
sexpr, deriv = parse_derivative(expr.base)
else:
sexpr = expr.base
if expr.exp.is_Rational:
rat_expo = expr.exp
elif expr.exp.is_Mul:
coeff, tail = term.exp.as_coeff_terms()
if coeff.is_Rational:
rat_expo, sym_expo = coeff, C.Mul(*tail)
else:
sym_expo = expr.exp
else:
sym_expo = expr.exp
elif expr.func is C.exp:
if expr.args[0].is_Rational:
sexpr, rat_expo = S.Exp1, expr.args[0]
elif expr.args[0].is_Mul:
coeff, tail = expr.args[0].as_coeff_terms()
if coeff.is_Rational:
sexpr, rat_expo = C.exp(C.Mul(*tail)), coeff
elif isinstance(expr, Derivative):
sexpr, deriv = parse_derivative(expr)
return sexpr, rat_expo, sym_expo, deriv
def parse_expression(terms, pattern):
pattern = make_list(pattern, Mul)
if len(terms) < len(pattern):
# pattern is longer than matched product
# so no chance for positive parsing result
return None
else:
pattern = [ parse_term(elem) for elem in pattern ]
elems, common_expo, has_deriv = [], S.One, False
for elem, e_rat, e_sym, e_ord in pattern:
if e_ord is not None:
# there is derivative in the pattern so
# there will by small performance penalty
has_deriv = True
for j in range(len(terms)):
term, t_rat, t_sym, t_ord = terms[j]
if elem == term and e_sym == t_sym:
if exact == False:
# we don't have to exact so find common exponent
# for both expression's term and pattern's element
expo = t_rat / e_rat
if common_expo is S.One:
common_expo = expo
else:
# common exponent was negotiated before so
# teher is no chance for pattern match unless
# common and current exponents are equal
if common_expo != expo:
return None
else:
# we ought to be exact so all fields of
# interest must match in very details
if e_rat != t_rat or e_ord != t_ord:
continue
# found common term so remove it from the expression
# and try to match next element in the pattern
elems.append(terms[j])
del terms[j]
break
else:
# pattern element not found
return None
return terms, elems, common_expo, has_deriv
if evaluate:
if expr.is_Mul:
ret = 1
for term in expr.args:
ret *= collect(term, syms, True, exact)
return ret
elif expr.is_Pow:
b = collect(expr.base, syms, True, exact)
return C.Pow(b, expr.exp)
summa = [ separate(i) for i in make_list(sympify(expr), Add) ]
if isinstance(syms, list):
syms = [ separate(s) for s in syms ]
else:
syms = [ separate(syms) ]
collected, disliked = {}, S.Zero
for product in summa:
terms = [ parse_term(i) for i in make_list(product, Mul) ]
for symbol in syms:
result = parse_expression(terms, symbol)
if result is not None:
terms, elems, common_expo, has_deriv = result
# when there was derivative in current pattern we
# will need to rebuild its expression from scratch
if not has_deriv:
index = Pow(symbol, common_expo)
else:
index = make_expression(elems)
terms = separate(make_expression(terms))
index = separate(index)
if index in collected:
collected[index] += terms
else:
collected[index] = terms
break
else:
# none of the patterns matched
disliked += product
if disliked is not S.Zero:
collected[S.One] = disliked
if evaluate:
return Add(*[ a*b for a, b in collected.iteritems() ])
else:
return collected
def ratsimp(expr):
"""
Usage
=====
ratsimp(expr) -> joins two rational expressions and returns the simples form
Notes
=====
Currently can simplify only simple expressions, for this to be really usefull
multivariate polynomial algorithms are needed
Examples
========
>>> from sympy import *
>>> x = Symbol('x')
>>> y = Symbol('y')
>>> e = ratsimp(1/x + 1/y)
"""
expr = sympify(expr)
if expr.is_Pow:
return Pow(ratsimp(expr.base), ratsimp(expr.exp))
elif expr.is_Mul:
res = []
for x in expr.args:
res.append( ratsimp(x) )
return Mul(*res)
elif expr.is_Function:
return expr.func(*[ ratsimp(t) for t in expr.args ])
#elif expr.is_Function:
# return type(expr)( ratsimp(expr[0]) )
elif not expr.is_Add:
return expr
def get_num_denum(x):
"""Matches x = a/b and returns a/b."""
a,b = map(Wild, 'ab')
r = x.match(a/b)
if r is not None and len(r) == 2:
return r[a],r[b]
return x, S.One
x = expr.args[0]
y = Add(*expr.args[1:])
a,b = get_num_denum(ratsimp(x))
c,d = get_num_denum(ratsimp(y))
num = a*d+b*c
denum = b*d
# Check to see if the numerator actually expands to 0
if num.expand() == 0:
return 0
return num/denum
def trigsimp(expr, deep=False):
"""
Usage
=====
trig(expr) -> reduces expression by using known trig identities
Notes
=====
Examples
========
>>> from sympy import *
>>> x = Symbol('x')
>>> y = Symbol('y')
>>> e = 2*sin(x)**2 + 2*cos(x)**2
>>> trigsimp(e)
2
>>> trigsimp(log(e))
log(2*cos(x)**2 + 2*sin(x)**2)
>>> trigsimp(log(e), deep=True)
log(2)
"""
from sympy.core.basic import S
sin, cos, tan, cot = C.sin, C.cos, C.tan, C.cot
#XXX this stopped working:
if expr == 1/cos(Symbol("x"))**2 - 1:
return tan(Symbol("x"))**2
if expr.is_Function:
if deep:
return expr.func( trigsimp(expr.args[0], deep) )
elif expr.is_Mul:
ret = S.One
for x in expr.args:
ret *= trigsimp(x, deep)
a,b,c = map(Wild,'abc')
matchers = (
(2*b*sin(a)*cos(a),b*sin(2*a)),#matches the double angle formula
)
res = None
for pattern, result in matchers:
res = ret.match(pattern)
if res is not None:
ret = result.subs(res)
break
if res is None:
ret = term
return ret
elif expr.is_Pow:
return Pow(trigsimp(expr.base, deep), trigsimp(expr.exp, deep))
elif expr.is_Add:
# TODO this needs to be faster
# The types of trig functions we are looking for
a,b,c = map(Wild, 'abc')
matchers = (
(a*sin(b)**2, a - a*cos(b)**2),
(a*tan(b)**2, a*(1/cos(b))**2 - a),
(a*cot(b)**2, a*(1/sin(b))**2 - a)
)
# Scan for the terms we need
ret = S.Zero
for term in expr.args:
term = trigsimp(term, deep)
res = None
for pattern, result in matchers:
res = term.match(pattern)
if res is not None:
ret += result.subs(res)
break
if res is None:
ret += term
# Reduce any lingering artifacts, such as sin(x)**2 changing
# to 1-cos(x)**2 when sin(x)**2 was "simpler"
artifacts = (
(a - a*cos(b)**2 + c, a*sin(b)**2 + c, cos),
(a - a*(1/cos(b))**2 + c, -a*tan(b)**2 + c, cos),
(a - a*(1/sin(b))**2 + c, -a*cot(b)**2 + c, sin)
)
expr = ret
for pattern, result, ex in artifacts:
# Substitute a new wild that excludes some function(s)
# to help influence a better match. This is because
# sometimes, for example, 'a' would match sec(x)**2
a_t = Wild('a', exclude=[ex])
pattern = pattern.subs(a, a_t)
result = result.subs(a, a_t)
m = expr.match(pattern)
while m is not None:
if m[a_t] == 0 or -m[a_t] in m[c].args or m[a_t] + m[c] == 0:
break
expr = result.subs(m)
m = expr.match(pattern)
return expr
return expr
def radsimp(expr):
"""
Rationalize the denominator.
Examples:
=========
>>> from sympy import *
>>> radsimp(1/(2+sqrt(2)))
1 - 1/2*2**(1/2)
>>> x,y = map(Symbol, 'xy')
>>> e = ( (2+2*sqrt(2))*x+(2+sqrt(8))*y )/( 2+sqrt(2) )
>>> radsimp(e)
x*2**(1/2) + y*2**(1/2)
"""
n,d = fraction(expr)
a,b,c = map(Wild, 'abc')
r = d.match(a+b*sqrt(c))
if r is not None:
a = r[a]
if r[b] == 0:
b,c = 0,0
else:
b,c = r[b],r[c]
syms = list(n.atoms(Symbol))
n = collect( (n*(a-b*sqrt(c))).expand(), syms )
d = a**2 - c*b**2
return n/d
def powsimp(expr, deep=False):
"""
Usage
=====
powsimp(expr, deep) -> reduces expression by combining powers with
similar bases and exponents.
Notes
=====
If deep is True then powsimp() will also simplify arguments of
functions. By default deep is set to False.
Examples
========
>>> from sympy import *
>>> x,n = map(Symbol, 'xn')
>>> e = x**n * (x*n)**(-n) * n
>>> powsimp(e)
n**(1 - n)
>>> powsimp(log(e))
log(n*x**n*(n*x)**(-n))
>>> powsimp(log(e), deep=True)
log(n**(1 - n))
"""
def _powsimp(expr):
if expr.is_Pow:
if deep:
return C.Pow(powsimp(expr.base), powsimp(expr.exp))
return expr
elif expr.is_Function and deep:
return expr.func(*[powsimp(t) for t in expr.args])
elif expr.is_Add:
return C.Add(*[powsimp(t) for t in expr.args])
elif expr.is_Mul:
# Collect base/exp data, while maintaining order in the
# non-commutative parts of the product
c_powers = {}
nc_part = []
for term in expr.args:
if term.is_commutative:
b,e = term.as_base_exp()
c_powers[b] = c_powers.get(b, 0) + e
else:
nc_part.append(term)
# Pull out numerical coefficients from exponent
for b,e in c_powers.items():
exp_c, exp_t = e.as_coeff_terms()
if not (exp_c is S.One) and exp_t:
del c_powers[b]
new_base = C.Pow(b, exp_c)
if new_base in c_powers:
c_powers[new_base] += C.Mul(*exp_t)
else:
c_powers[new_base] = C.Mul(*exp_t)
# Combine bases whenever they have the same exponent which is
# not numeric
c_exp = {}
for b, e in c_powers.items():
if e in c_exp:
c_exp[e].append(b)
else:
c_exp[e] = [b]
# Merge back in the results of the above to form a new product
for e in c_exp:
bases = c_exp[e]
if len(bases) > 1:
for b in bases:
del c_powers[b]
new_base = Mul(*bases)
if new_base in c_powers:
c_powers[new_base] += e
else:
c_powers[new_base] = e
c_part = [ C.Pow(b,e) for b,e in c_powers.items() ]
return C.Mul(*(c_part + nc_part))
return expr
return _powsimp(separate(expr, deep=deep))
def hypersimp(f, k):
"""Given combinatorial term f(k) simplify its consecutive term ratio
ie. f(k+1)/f(k). The input term can be composed of functions and
integer sequences which have equivalent representation in terms
of gamma special function.
The algorithm performs three basic steps:
(1) Rewrite all functions in terms of gamma, if possible.
(2) Rewrite all occurrences of gamma in terms of products
of gamma and rising factorial with integer, absolute
constant exponent.
(3) Perform simplification of nested fractions, powers
and if the resulting expression is a quotient of
polynomials, reduce their total degree.
If f(k) is hypergeometric then as result we arrive with a
quotient of polynomials of minimal degree. Otherwise None
is returned.
For more information on the implemented algorithm refer to:
[1] W. Koepf, Algorithms for m-fold Hypergeometric Summation,
Journal of Symbolic Computation (1995) 20, 399-417
"""
f = sympify(f)
g = f.subs(k, k+1) / f
g = g.rewrite(gamma)
g = g.expand(func=True, basic=False)
if g.is_fraction(k):
return Poly.cancel(g, k)
else:
return None
def hypersimilar(f, g, k):
"""Returns True if 'f' and 'g' are hyper-similar.
Similarity in hypergeometric sens means that a quotient of
f(k) and g(k) is a rational function in k. This procedure
is useful in solving recurrence relations.
For more information see hypersimp().
"""
f, g = map(sympify, (f, g))
h = (f/g).rewrite(gamma)
h = h.expand(func=True, basic=False)
return h.is_fraction(k)
def combsimp(expr):
return expr
def simplify(expr):
"""Naively simplifies the given expression.
Simplification is not a well defined term and the exact strategies
this function tries can change in the future versions of SymPy. If
your algorithm relies on "simplification" (whatever it is), try to
determine what you need exactly - is it powsimp(), or radsimp(),
or together(), or something else? And use this particular function
directly, because those are well defined and thus your algorithm
will be robust.
"""
expr = Poly.cancel(powsimp(expr))
return together(expr.expand())
def nsimplify(expr, constants=[], tolerance=None, full=False):
"""
Numerical simplification -- tries to find a simple formula
that numerically matches the given expression. The input should
be possible to evalf to a precision of at least 30 digits.
Optionally, a list of (rationally independent) constants to
include in the formula may be given.
A lower tolerance may be set to find less exact matches.
With full=True, a more extensive search is performed
(this is useful to find simpler numbers when the tolerance
is set low).
Examples:
>>> from sympy import *
>>> nsimplify(4/(1+sqrt(5)), [GoldenRatio])
-2 + 2*GoldenRatio
>>> nsimplify((1/(exp(3*pi*I/5)+1)))
1/2 - I*(1/4 + 1/10*5**(1/2))**(1/2)
>>> nsimplify(I**I, [pi])
exp(-pi/2)
>>> nsimplify(pi, tolerance=0.01)
22/7
"""
expr = sympify(expr)
prec = 30
bprec = int(prec*3.33)
constants_dict = {}
for constant in constants:
constant = sympify(constant)
v = constant.evalf(prec)
if not v.is_Real:
raise ValueError("constants must be real-valued")
constants_dict[str(constant)] = v._to_mpmath(bprec)
exprval = expr.evalf(prec, chop=True)
re, im = exprval.as_real_imag()
# Must be numerical
if not ((re.is_Real or re.is_Integer) and (im.is_Real or im.is_Integer)):
return expr
def nsimplify_real(x):
orig = mpmath.mp.dps
xv = x._to_mpmath(bprec)
try:
# We'll be happy with low precision if a simple fraction
if not (tolerance or full):
mpmath.mp.dps = 15
rat = mpmath.findpoly(xv, 1)
if rat is not None:
return Rational(-int(rat[1]), int(rat[0]))
mpmath.mp.dps = prec
newexpr = mpmath.identify(xv, constants=constants_dict,
tolerance=tolerance, full=full)
if not newexpr:
raise ValueError
if full:
newexpr = newexpr[0]
return sympify(newexpr)
finally:
mpmath.mp.dps = orig
try:
if re: re = nsimplify_real(re)
if im: im = nsimplify_real(im)
except ValueError:
return expr
return re + im*S.ImaginaryUnit
from sympy import *
s,v,o = symbols('svo')
s0 = Symbol('s0')
v0 = Symbol('v0')
v = v0*sin(pi*s/s0)
at = v*diff(v,s)
at_simp = trigsimp(at)
print at_simp
x = 2*sin(s)*cos(s)
y = trigsimp(x)
print y
