Simplifications in calculus on manifolds with derivatives of symbolic functions

[egourgoulhon]

The function sage.manifolds.utilities.simplify_sqrt_real, which is involved in calculus on manifolds (see ​here), does not simplify sqrt's as soon as the expression contains some derivative of a symbolic function:

sage: from sage.manifolds.utilities import simplify_sqrt_real
sage: f = function('f')
sage: assume(x<1)
sage: simplify_sqrt_real(sqrt(x^2-2*x+1))  # OK
-x + 1
sage: simplify_sqrt_real(sqrt(x^2-2*x+1) * diff(f(x), x))  # nothing done
sqrt(x^2 - 2*x + 1)*diff(f(x), x)

The same limitation holds for derivatives denoted by D[...] instead of diff:

sage: y = var('y'); s = sqrt(x^2-2*x+1) * diff(f(x), x).subs(x=y^2); s
sqrt(x^2 - 2*x + 1)*D[0](f)(y^2)
sage: simplify_sqrt_real(s)  # nothing done
sqrt(x^2 - 2*x + 1)*D[0](f)(y^2)

Similarly, the function sage.manifolds.utilities.simplify_abs_trig does not perform any simplification when a derivative is present:

sage: from sage.manifolds.utilities import simplify_abs_trig
sage: assume(x>0); assumptions()
[x < 1, x > 0]
sage: simplify_abs_trig(abs(sin(x)))  # OK
sin(x)
sage: simplify_abs_trig(abs(sin(x)) * diff(f(x), x))  # nothing done
abs(sin(x))*diff(f(x), x)

The code introduced in this ticket repairs this. We have now, with the same assumptions as above:

sage: simplify_sqrt_real(sqrt(x^2-2*x+1) * diff(f(x), x))
-(x - 1)*diff(f(x), x)
sage: simplify_sqrt_real(s)
-(x - 1)*D[0](f)(y^2)
sage: simplify_abs_trig(abs(sin(x)) * diff(f(x), x))
sin(x)*diff(f(x), x)

In addition, simplify_abs_trig is now capable to treat abs(cos(...)):

sage: simplify_abs_trig(abs(cos(x))) # recall that 0 < x < 1 < pi/2
cos(x)
sage: simplify_abs_trig(abs(cos(x+2)))
-cos(x + 2)
sage: simplify_abs_trig(abs(cos(x+2)) * diff(f(x), x))
-cos(x + 2)*diff(f(x), x)

[egourgoulhon]

New commits:

​6eb6bde

Improve simplifications in calculus on manifolds when derivatives of symbolic functions are involved

[tscrim]

LGTM.

As a possible improvement for another ticket, instead of string manipulations, you might want to consider contacting, e.g., Ralf (rws) to see if there is a better way by directly manipulating the expression tree. Or if there is an easy way to fix some of these problems upstream.

[egourgoulhon]

Replying to tscrim:

As a possible improvement for another ticket, instead of string manipulations, you might want to consider contacting, e.g., Ralf (rws) to see if there is a better way by directly manipulating the expression tree. Or if there is an easy way to fix some of these problems upstream.

Thanks for the suggestion. I agree that using the expression tree would be an improvement.

And thank you for the review!

[egourgoulhon]

Replying to egourgoulhon:

Replying to tscrim:

As a possible improvement for another ticket, instead of string manipulations, you might want to consider contacting, e.g., Ralf (rws) to see if there is a better way by directly manipulating the expression tree. Or if there is an easy way to fix some of these problems upstream.

Thanks for the suggestion. I agree that using the expression tree would be an improvement.

Done in #24232

[egourgoulhon]

This is superseded by #24232.