check if expression contains symbol - sympy

I would like to find out programatically if a SymPy expression contains a symbol. E.g., for
import sympy
x = sympy.Symbol('x')
y = sympy.Symbol('y')
a = 4 + x**2 + y
b = 4 + y**2
a contains both x and y, b contains only y.

>>> x in a.free_symbols, y in a.free_symbols
(True, True)
>>> x in b.free_symbols, y in b.free_symbols
(False, True)

You can also use .atoms(Symbol) to check that. atoms(Symbol) differs from .free_symbols in some cases. free_symbols doesn't return dummy symbols, like integration variables.
it's usually what you want, since expressions don't mathematically depend on dummy symbols
example:
>>> Integral(f(x), (x, 0, 1)).atoms(Symbol)
set([x])
>>> Integral(f(x), (x, 0, 1)).free_symbols
set([])

Related

Sympy: Is there a function that gives all the factors of an expression but does not work sums?

I have an expression which is composed only of factors (e.g. (x**2+1)*(x**2)*(x+4). I want to delete the factor x**2 from it using the function .args with an if condition. However, if I have the following equation x**2+1+x+4, the .args thinks I have x**2 in the expression which is not true (I only have one factor). I have the code below.:
if q**2 in expr.args:
expr = expr.func(*[term for term in expr.args if term != q**2])
else:
expr = expr*2
By using Mul.make_args(expr) you will get a singleton if the expression is not a product, otherwise a tuple of the factors:
>>> from sympy.abc import x, y
>>> from sympy import Mul
>>> Mul.make_args(x + y)
(x + y,)
>>> Mul.make_args(x*y)
(x, y)

How to generate random math expression trees with sympy?

I am scouring the web but I cannot find how to generate random math expressions with sympy. Is it even possible?
I would like to build an expression tree by randomly selecting functions (product, sum, cosine...) and symbols from a set of predefined functions and symbols.
For instance, given the set [+,.] of sum and product and the symbols [x,y] I'd like to generate expressions such as x+y, (x+y).x, y+(x.x+y)+x etc, controlling parameters as the tree depth, width and the number of nodes.
Any hints?
Something like the following might help you get started:
from random import choice, randint
from sympy import FunctionClass, Add, Mul, cos, sin, binomial, arity, S
def args(n, atoms, funcs):
a = funcs+atoms
g = []
for _ in range(n):
ai = choice(a)
if isinstance(ai, FunctionClass):
g.append(ai(*args(arity(ai), atoms, funcs)))
else:
g.append(ai)
return g
def expr(ops, atoms, funcs=()):
types = [Add, Mul]
atoms = tuple(atoms)
while 1:
e = S.Zero
while e.count_ops() < ops:
_ = choice(types)(*args(randint(1,3), atoms, funcs))
e = choice(types)(e, _)
if e is S.NaN: break
else:
return e
>>> [expr(5, (-1,0,1,x,y)) for do in range(2)]
[(x - 1)*(2*x + y + 2), x + y*(x + 4*y - 2) + y]
>>> expr(5, (-1,0,1,x,y), (cos, binomial))
x*y**2 + x + cos(1)
>>> expr(5, (-1,0,1,x,y), (cos, binomial))
(y + zoo*binomial(y, x) - 2)*(y + cos(1) + 1)
To generate rational expressions you could change make the 2nd _ arg be _**choice((1,-1)).

Sympy Piecewise and refine

I try to simplify this Piecewise expression using refine without success. I use sympy version 1.8.
import sympy as sp
x,y = sp.symbols('x,y', real=True, positive=True)
expr = sp.Piecewise((1, x>=y),(0, True))
expr variable contain
⎧1 for x ≥ y
⎨
⎩0 otherwise
now I try to obtain 1 assuming that x>y
sp.refine(expr, sp.Q.gt(x,y))
but I obtain the same expression
⎧1 for x ≥ y
⎨
⎩0 otherwise
Any ideas to force this simplification ?
Thank you for your help
If you give expr a value for x that satisfies the requirement, it will evaluate:
>>> eps = Symbol('eps', positive=True)
>>> expr.subs(x, y + eps).simplify()
1

Sympy .replace() doesn't detect product

I'm trying to make some replacements, but .replace() is confusingly missing occurrences:
from sympy import *
x, y = symbols('x y')
f = Function('f')
expr = x*f(x)*f(y)
expr.replace(x*f(x), 1)
# Output: x*f(x)*f(y)
# Expected: f(y)
Is this a bug?
It does work when I use .subs() instead of .replace(), but I have no idea why this would be needed.
(also, I wanna use this in the context of a Sum expression where .subs() would not work).

Errors with my argument error handing [duplicate]

I'm trying to make a function that will compare multiple variables to an integer and output a string of three letters. I was wondering if there was a way to translate this into Python. So say:
x = 0
y = 1
z = 3
mylist = []
if x or y or z == 0:
mylist.append("c")
if x or y or z == 1:
mylist.append("d")
if x or y or z == 2:
mylist.append("e")
if x or y or z == 3:
mylist.append("f")
which would return a list of:
["c", "d", "f"]
You misunderstand how boolean expressions work; they don't work like an English sentence and guess that you are talking about the same comparison for all names here. You are looking for:
if x == 1 or y == 1 or z == 1:
x and y are otherwise evaluated on their own (False if 0, True otherwise).
You can shorten that using a containment test against a tuple:
if 1 in (x, y, z):
or better still:
if 1 in {x, y, z}:
using a set to take advantage of the constant-cost membership test (i.e. in takes a fixed amount of time whatever the left-hand operand is).
Explanation
When you use or, python sees each side of the operator as separate expressions. The expression x or y == 1 is treated as first a boolean test for x, then if that is False, the expression y == 1 is tested.
This is due to operator precedence. The or operator has a lower precedence than the == test, so the latter is evaluated first.
However, even if this were not the case, and the expression x or y or z == 1 was actually interpreted as (x or y or z) == 1 instead, this would still not do what you expect it to do.
x or y or z would evaluate to the first argument that is 'truthy', e.g. not False, numeric 0 or empty (see boolean expressions for details on what Python considers false in a boolean context).
So for the values x = 2; y = 1; z = 0, x or y or z would resolve to 2, because that is the first true-like value in the arguments. Then 2 == 1 would be False, even though y == 1 would be True.
The same would apply to the inverse; testing multiple values against a single variable; x == 1 or 2 or 3 would fail for the same reasons. Use x == 1 or x == 2 or x == 3 or x in {1, 2, 3}.
Your problem is more easily addressed with a dictionary structure like:
x = 0
y = 1
z = 3
d = {0: 'c', 1:'d', 2:'e', 3:'f'}
mylist = [d[k] for k in [x, y, z]]
As stated by Martijn Pieters, the correct, and fastest, format is:
if 1 in {x, y, z}:
Using his advice you would now have separate if-statements so that Python will read each statement whether the former were True or False. Such as:
if 0 in {x, y, z}:
mylist.append("c")
if 1 in {x, y, z}:
mylist.append("d")
if 2 in {x, y, z}:
mylist.append("e")
...
This will work, but if you are comfortable using dictionaries (see what I did there), you can clean this up by making an initial dictionary mapping the numbers to the letters you want, then just using a for-loop:
num_to_letters = {0: "c", 1: "d", 2: "e", 3: "f"}
for number in num_to_letters:
if number in {x, y, z}:
mylist.append(num_to_letters[number])
The direct way to write x or y or z == 0 is
if any(map((lambda value: value == 0), (x,y,z))):
pass # write your logic.
But I dont think, you like it. :)
And this way is ugly.
The other way (a better) is:
0 in (x, y, z)
BTW lots of ifs could be written as something like this
my_cases = {
0: Mylist.append("c"),
1: Mylist.append("d")
# ..
}
for key in my_cases:
if key in (x,y,z):
my_cases[key]()
break
If you ARE very very lazy, you can put the values inside an array. Such as
list = []
list.append(x)
list.append(y)
list.append(z)
nums = [add numbers here]
letters = [add corresponding letters here]
for index in range(len(nums)):
for obj in list:
if obj == num[index]:
MyList.append(letters[index])
break
You can also put the numbers and letters in a dictionary and do it, but this is probably a LOT more complicated than simply if statements. That's what you get for trying to be extra lazy :)
One more thing, your
if x or y or z == 0:
will compile, but not in the way you want it to. When you simply put a variable in an if statement (example)
if b
the program will check if the variable is not null. Another way to write the above statement (which makes more sense) is
if bool(b)
Bool is an inbuilt function in python which basically does the command of verifying a boolean statement (If you don't know what that is, it is what you are trying to make in your if statement right now :))
Another lazy way I found is :
if any([x==0, y==0, z==0])
To check if a value is contained within a set of variables you can use the inbuilt modules itertools and operator.
For example:
Imports:
from itertools import repeat
from operator import contains
Declare variables:
x = 0
y = 1
z = 3
Create mapping of values (in the order you want to check):
check_values = (0, 1, 3)
Use itertools to allow repetition of the variables:
check_vars = repeat((x, y, z))
Finally, use the map function to create an iterator:
checker = map(contains, check_vars, check_values)
Then, when checking for the values (in the original order), use next():
if next(checker) # Checks for 0
# Do something
pass
elif next(checker) # Checks for 1
# Do something
pass
etc...
This has an advantage over the lambda x: x in (variables) because operator is an inbuilt module and is faster and more efficient than using lambda which has to create a custom in-place function.
Another option for checking if there is a non-zero (or False) value in a list:
not (x and y and z)
Equivalent:
not all((x, y, z))
Set is the good approach here, because it orders the variables, what seems to be your goal here. {z,y,x} is {0,1,3} whatever the order of the parameters.
>>> ["cdef"[i] for i in {z,x,y}]
['c', 'd', 'f']
This way, the whole solution is O(n).
I think this will handle it better:
my_dict = {0: "c", 1: "d", 2: "e", 3: "f"}
def validate(x, y, z):
for ele in [x, y, z]:
if ele in my_dict.keys():
return my_dict[ele]
Output:
print validate(0, 8, 9)
c
print validate(9, 8, 9)
None
print validate(9, 8, 2)
e
If you want to use if, else statements following is another solution:
myList = []
aList = [0, 1, 3]
for l in aList:
if l==0: myList.append('c')
elif l==1: myList.append('d')
elif l==2: myList.append('e')
elif l==3: myList.append('f')
print(myList)
All of the excellent answers provided here concentrate on the specific requirement of the original poster and concentrate on the if 1 in {x,y,z} solution put forward by Martijn Pieters.
What they ignore is the broader implication of the question:
How do I test one variable against multiple values?
The solution provided will not work for partial hits if using strings for example:
Test if the string "Wild" is in multiple values
>>> x = "Wild things"
>>> y = "throttle it back"
>>> z = "in the beginning"
>>> if "Wild" in {x, y, z}: print (True)
...
or
>>> x = "Wild things"
>>> y = "throttle it back"
>>> z = "in the beginning"
>>> if "Wild" in [x, y, z]: print (True)
...
for this scenario it's easiest to convert to a string
>>> [x, y, z]
['Wild things', 'throttle it back', 'in the beginning']
>>> {x, y, z}
{'in the beginning', 'throttle it back', 'Wild things'}
>>>
>>> if "Wild" in str([x, y, z]): print (True)
...
True
>>> if "Wild" in str({x, y, z}): print (True)
...
True
It should be noted however, as mentioned by #codeforester, that word boundries are lost with this method, as in:
>>> x=['Wild things', 'throttle it back', 'in the beginning']
>>> if "rot" in str(x): print(True)
...
True
the 3 letters rot do exist in combination in the list but not as an individual word. Testing for " rot " would fail but if one of the list items were "rot in hell", that would fail as well.
The upshot being, be careful with your search criteria if using this method and be aware that it does have this limitation.
d = {0:'c', 1:'d', 2:'e', 3: 'f'}
x, y, z = (0, 1, 3)
print [v for (k,v) in d.items() if x==k or y==k or z==k]
This code may be helpful
L ={x, y, z}
T= ((0,"c"),(1,"d"),(2,"e"),(3,"f"),)
List2=[]
for t in T :
if t[0] in L :
List2.append(t[1])
break;
You can try the method shown below. In this method, you will have the freedom to specify/input the number of variables that you wish to enter.
mydict = {0:"c", 1:"d", 2:"e", 3:"f"}
mylist= []
num_var = int(raw_input("How many variables? ")) #Enter 3 when asked for input.
for i in range(num_var):
''' Enter 0 as first input, 1 as second input and 3 as third input.'''
globals()['var'+str('i').zfill(3)] = int(raw_input("Enter an integer between 0 and 3 "))
mylist += mydict[globals()['var'+str('i').zfill(3)]]
print mylist
>>> ['c', 'd', 'f']
One line solution:
mylist = [{0: 'c', 1: 'd', 2: 'e', 3: 'f'}[i] for i in [0, 1, 2, 3] if i in (x, y, z)]
Or:
mylist = ['cdef'[i] for i in range(4) if i in (x, y, z)]
Maybe you need direct formula for output bits set.
x=0 or y=0 or z=0 is equivalent to x*y*z = 0
x=1 or y=1 or z=1 is equivalent to (x-1)*(y-1)*(z-1)=0
x=2 or y=2 or z=2 is equivalent to (x-2)*(y-2)*(z-2)=0
Let's map to bits: 'c':1 'd':0xb10 'e':0xb100 'f':0xb1000
Relation of isc (is 'c'):
if xyz=0 then isc=1 else isc=0
Use math if formula https://youtu.be/KAdKCgBGK0k?list=PLnI9xbPdZUAmUL8htSl6vToPQRRN3hhFp&t=315
[c]: (xyz=0 and isc=1) or (((xyz=0 and isc=1) or (isc=0)) and (isc=0))
[d]: ((x-1)(y-1)(z-1)=0 and isc=2) or (((xyz=0 and isd=2) or (isc=0)) and (isc=0))
...
Connect these formulas by following logic:
logic and is the sum of squares of equations
logic or is the product of equations
and you'll have a total equation
express sum and you have total formula of sum
then sum&1 is c, sum&2 is d, sum&4 is e, sum&5 is f
After this you may form predefined array where index of string elements would correspond to ready string.
array[sum] gives you the string.
The most pythonic way of representing your pseudo-code in Python would be:
x = 0
y = 1
z = 3
mylist = []
if any(v == 0 for v in (x, y, z)):
mylist.append("c")
if any(v == 1 for v in (x, y, z)):
mylist.append("d")
if any(v == 2 for v in (x, y, z)):
mylist.append("e")
if any(v == 3 for v in (x, y, z)):
mylist.append("f")
It can be done easily as
for value in [var1,var2,var3]:
li.append("targetValue")
To test multiple variables with one single value: if 1 in {a,b,c}:
To test multiple values with one variable: if a in {1, 2, 3}:
Looks like you're building some kind of Caesar cipher.
A much more generalized approach is this:
input_values = (0, 1, 3)
origo = ord('c')
[chr(val + origo) for val in inputs]
outputs
['c', 'd', 'f']
Not sure if it's a desired side effect of your code, but the order of your output will always be sorted.
If this is what you want, the final line can be changed to:
sorted([chr(val + origo) for val in inputs])
You can use dictionary :
x = 0
y = 1
z = 3
list=[]
dict = {0: 'c', 1: 'd', 2: 'e', 3: 'f'}
if x in dict:
list.append(dict[x])
else:
pass
if y in dict:
list.append(dict[y])
else:
pass
if z in dict:
list.append(dict[z])
else:
pass
print list
Without dict, try this solution:
x, y, z = 0, 1, 3
offset = ord('c')
[chr(i + offset) for i in (x,y,z)]
and gives:
['c', 'd', 'f']
This will help you.
def test_fun(val):
x = 0
y = 1
z = 2
myList = []
if val in (x, y, z) and val == 0:
myList.append("C")
if val in (x, y, z) and val == 1:
myList.append("D")
if val in (x, y, z) and val == 2:
myList.append("E")
test_fun(2);
You can unite this
x = 0
y = 1
z = 3
in one variable.
In [1]: xyz = (0,1,3,)
In [2]: mylist = []
Change our conditions as:
In [3]: if 0 in xyz:
...: mylist.append("c")
...: if 1 in xyz:
...: mylist.append("d")
...: if 2 in xyz:
...: mylist.append("e")
...: if 3 in xyz:
...: mylist.append("f")
Output:
In [21]: mylist
Out[21]: ['c', 'd', 'f']
you can develop it through two ways
def compareVariables(x,y,z):
mylist = []
if x==0 or y==0 or z==0:
mylist.append('c')
if x==1 or y==1 or z==1:
mylist.append('d')
if x==2 or y==2 or z==2:
mylist.append('e')
if x==3 or y==3 or z==3:
mylist.append('f')
else:
print("wrong input value!")
print('first:',mylist)
compareVariables(1, 3, 2)
Or
def compareVariables(x,y,z):
mylist = []
if 0 in (x,y,z):
mylist.append('c')
if 1 in (x,y,z):
mylist.append('d')
if 2 in (x,y,z):
mylist.append('e')
if 3 in (x,y,z):
mylist.append('f')
else:
print("wrong input value!")
print('second:',mylist)
compareVariables(1, 3, 2)
The or does not work like that, as explained by this answer.
While the generic answer would be use
if 0 in (x, y, z):
...
this is not the best one for the specific problem. In your case you're doing repeated tests, therefore it is worthwhile to compose a set of these variables:
values = {x, y, z}
if 0 in values:
mylist.append("c")
if 1 in values:
mylist.append("d")
We can simplify this using a dictionary - this will result in the same values:
mappings = {0: "c", 1: "d", ...}
for k in mappings:
if k in values:
mylist.append(mappings[k])
Or if the ordering of the mylist is arbitrary, you can loop over the values instead and match them to the mappings:
mappings = {0: "c", 1: "d", ...}
for v in (x, y, z):
if v in mappings:
mylist.append(mappings[v])
Problem
While the pattern for testing multiple values
>>> 2 in {1, 2, 3}
True
>>> 5 in {1, 2, 3}
False
is very readable and is working in many situation, there is one pitfall:
>>> 0 in {True, False}
True
But we want to have
>>> (0 is True) or (0 is False)
False
Solution
One generalization of the previous expression is based on the answer from ytpillai:
>>> any([0 is True, 0 is False])
False
which can be written as
>>> any(0 is item for item in (True, False))
False
While this expression returns the right result it is not as readable as the first expression :-(
Here is one more way to do it:
x = 0
y = 1
z = 3
mylist = []
if any(i in [0] for i in[x,y,z]):
mylist.append("c")
if any(i in [1] for i in[x,y,z]):
mylist.append("d")
if any(i in [2] for i in[x,y,z]):
mylist.append("e")
if any(i in [3] for i in[x,y,z]):
mylist.append("f")
It is a mix of list comprehension and any keyword.
usage without if example:
x,y,z = 0,1,3
values = {0:"c",1:"d",2:"e",3:"f"} # => as if usage
my_list = [values[i] for i in (x,y,z)]
print(my_list)
FIRST, A CORRECTION TO THE OR CONDITIONAL:
You need to say:
if x == 0 or y == 0 or z == 0:
The reason is that "or" splits up the condition into separate logical parts. The way your original statement was written, those parts were:
x
y
z == 0 // or 1, 2, 3 depending on the if statement
The last part was fine --- checking to see if z == 0, for instance --- but the first two parts just said essentially if x and if y. Since integers always evaluate to True unless they're 0, that means the first part of your condition was always True when x or y didn't equal 0 (which in the case of y was always, since you had y = 1, causing your whole condition (because of how OR works) to always be True.
To avoid that, you need to make sure all parts of your condition (each side of the OR) make sense on their own (you can do that by pretending that the other side(s) of the OR statement doesn't exist). That's how you can confirm whether or not your OR condition is correctly defined.
You would write the statements individually like so:
if x == 0
if y == 0
if z == 0
which means the correct mergin with the OR keyword would be:
if x == 0 or y == 0 or z == 0
SECOND, HOW TO SOLVE THE PROBLEM:
You're basically wanting to check to see if any of the variables match a given integer and if so, assign it a letter that matches it in a one-to-one mapping. You want to do that for a certain list of integers so that the output is a list of letters. You'd do that like this:
def func(x, y, z):
result = []
for integer, letter in zip([0, 1, 2, 3], ['c', 'd', 'e', 'f']):
if x == integer or y == integer or z == integer:
result.append(letter)
return result
Similarly, you could use LIST COMPREHENSION to achieve the same result faster:
def func(x, y, z):
return [
letter
for integer, letter in zip([0, 1, 2, 3], ['c', 'd', 'e', 'f'])
if x == integer or y == integer or z == integer
]
To test multiple variables against a single value:
Wrap the variables in a set object, e.g. {a, b, c}.
Use the in operator to test if the value is stored in any of the variables.
The in operator will return True if the value is stored in at least one of the variables.
# ✅ test multiple variables against single value using tuple
if 'a' in (a, b, c):
print('value is stored in at least one of the variables')
# ---------------------------------------------------------
# ✅ test multiple variables against single value using tuple
if 'a' in {a, b, c}:
print('value is stored in at least one of the variables')
# ---------------------------------------------------------
# ✅ test multiple variables against single value (OR operator chaining)
if a == 'a' or b == 'a' or c == 'a':
print('value is stored in at least one of the variables')