How to substitute a subexpression with a function?

[Audrius-St]

Hello, In the following test code I am attempting to simplify the resulting expression

by replacing a subexpression with a function

import symengine as se

import sympy as sp

# Potential V

def V(x, y):

    v = (x*x + y*y)/2 + y*(x*x - (y*y)/3)

    return v

def main():

    # Symbols

    x, y, Ho = sp.symbols('x, y, Ho')

    expr = sp.Pow(Ho - V(x, y), -1)

    dexpr_dx = se.Derivative(expr, x, 1).xreplace({y*(x**2 + (-1/3)*y**2) + (-1/2)*x**2 + (-1/2)*y**2: V(x, y)})

    print("dexpr_dx =", dexpr_dx)

if __name__ == "__main__":

    main()

Taking in derivative w.r.t. x resulting in V(x, y) being replace the expression

dexpr_dx = -(-x - 2*x*y)/(Ho - y*(x**2 + (-1/3)*y**2) + (-1/2)*x**2 + (-1/2)*y**2)**2

which I would like to rewrite as -(-x - 2*x*y)/(Ho - V(x, y))**2 as I will be repeating the differentiation in the actual code.

In other words, I would like to reduce the output code bloat.

I've tried subs(), replace(), and xreplace() with no success. Any insight in how to do this would be appreciated.

[Chris Smith]

There are a few reasons this is not working.

1) if you have floats like (1/3) those will not match the rationals like Rational(1, 3)

2) xreplace will target full nodes, not sub-nodes. So when trying to match an Add that is a subexpression, it will fail.

3) you have a sign error in the key of the replacement dictionary: the first term should be `-y*...`

4) the replacement value, `V(x, y)` recreates the thing you are replacing.

Try the following:

>>> d = Derivative(expr, x, 1)
>>> d.subs(V(x,y), Symbol('V(x,y)'))
Derivative(1/(Ho - V(x,y)), x)

/c

[Hanspeter Schmid]

Regarding 1), if you write 1/3, then Python converts this into a float before it is handed over to sympy. To prevent it, you could import S (Singleton) from sympy and write S(1)/3.

dexpr_dx = se.Derivative(expr, x, 1).xreplace({y*(x**2 + (-S(1)/3)*y**2) + (-S(1)/2)*x**2 + (-S(1)/2)*y**2: V(x, y)})

[Audrius-St]

After a bit of experimentation, the following code fragment worked:

    dexpr_dx = se.Derivative(expr, x, 1).subs(

                    (Ho - y*(x**2 + sp.Rational(-1, 3)*y**2) +

                     sp.Rational(-1, 2)*x**2 +

                     sp.Rational(-1, 2)*y**2), sp.Symbol('(Ho - V(x, y))'))

yielding

dexpr_dx -(-x - 2*x*y)/(Ho - V(x, y))**2

Thank you both for your advice.