Some straightforward square root fractions are not simplified

Erlend M. Viggen
23 ●1 ●1 ●5

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?

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Welcome to Ask Sage! Thank you for your question.

slelievre ( 3 years ago )

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answered 3 years ago

Emmanuel Charpentier
7804 ●8 ●52 ●155

updated 3 years ago

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,

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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 ( 3 years ago )

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