Some straightforward square root fractions are not simplified

asked 2021-12-03 15:46:29 +0200

I'm a Sage beginner who's trying to apply it to a set of equations I'm working on. In that process, I came across an issue where SageMath 9.4 is not simplifying some very straightforward square root fractions in my expressions. Let me show you a minimal example:

x = var('x')
sqrt(1-x)/(1-x)

$$-\frac{\sqrt{-x + 1}}{x - 1}$$

What I was expecting to get, of course, is $1 / \sqrt{1 - x}$. Calling simplify() or full_simplify() on the expression doesn't make a difference; I still get the same thing out. I experimented with other square root fractions as well to see if this issue recurs:

sqrt(x)/x

$$1/\sqrt{x}$$

sqrt(1+x)/(1+x)

$$1/\sqrt{1+x}$$

sqrt(x-1)/(x-1)

$$1/\sqrt{x-1}$$

So in other words, Sage automatically simplifies all of the other expressions I tried in exactly the way that I would expect. To try out things a bit further, I tried to see if it helps to apply Sympy:

( sqrt(1-x)/(1-x) )._sympy_().simplify()

$$1/\sqrt{1-x}$$

That works. (The simplify() argument is essential here; otherwise, the fraction is not automatically simplified.)

So, is there some subtle finesse here that I'm not understanding, or did I stumble across a bug in Sage's simplification algorithms?

Comments

Welcome to Ask Sage! Thank you for your question.

slelievre ( 2021-12-03 18:29:36 +0200 )edit

add a comment

answered 2021-12-04 10:06:57 +0200

Emmanuel Charpentier
7057 ●6 ●46 ●137

updated 2021-12-04 10:34:40 +0200

Sagemath formatting routine is somewhat asinine in radical-involving fraction handling, due to internal representation of expressions. Contemplate :

sage: 1/sqrt(2)
1/2*sqrt(2)

It turns out that, in this case, Sage denotes a division by a multiplication :

sage: (1/sqrt(2)).operator()
<function mul_vararg at 0x7fd16a1949d0>
sage: (1/sqrt(2)).operands()
[sqrt(2), 1/2]

On the other hand, if :

sage: (sqrt(x+1)/(x+1)).operator()
<built-in function pow>
sage: (sqrt(x+1)/(x+1)).operands()
[x + 1, -1/2]

the internal representation is different in your initial case :

sage: (sqrt(1-x)/(1-x)).operator()
<function mul_vararg at 0x7fd16a1949d0>
sage: (sqrt(1-x)/(1-x)).operands()
[1/(x - 1), sqrt(-x + 1), -1]

And; BTW, Sympy's internal representation is different :

sage: (sqrt(1-x)/(1-x))._sympy_().simplify()
1/sqrt(1 - x)
sage: (sqrt(1-x)/(1-x))._sympy_().simplify().func
<class 'sympy.core.power.Pow'>
sage: (sqrt(1-x)/(1-x))._sympy_().simplify().args
(1 - x, -1/2)

but this internal representation is lost when reverting to Sage :

sage: (sqrt(1-x)/(1-x))._sympy_()._sage_()
-sqrt(-x + 1)/(x - 1)
sage: (sqrt(1-x)/(1-x))._sympy_()._sage_().operator()
<function mul_vararg at 0x7fd16a1949d0>
sage: (sqrt(1-x)/(1-x))._sympy_()._sage_().operands()
[1/(x - 1), sqrt(-x + 1), -1]

EDIT : FWIW, the internal representation used by Mathematica is yet another :

sage: mathematica("M=Sqrt[1-x]/(1-x)")
1/Sqrt[1 - x]
sage: mathematica("Table[M[[u]], {u, Range[0,(Length[M])]}]")
{Power, 1 - x, -1/2}

HTH,

edit flag offensive delete link

Comments

Thanks! The takeaway message, the way I see it as a Sage beginner, is that it can always be useful to check with Sympy if it can help with further simplification if we see Sage results that look overly complex.

Erlend M. Viggen ( 2021-12-06 11:28:26 +0200 )edit

add a comment

Your Answer

Please start posting anonymously - your entry will be published after you log in or create a new account.

Some straightforward square root fractions are not simplified

Comments

1 Answer

Comments

Your Answer

Question Tools

Stats

Related questions

Some straightforward square root fractions are not simplified edit

Comments

1 Answer

Comments

Your Answer

Question Tools

Stats

Related questions

Some straightforward square root fractions are not simplified