Rearranging expressions to minimize negative signs

edit

You can't really control the expression in this way because Sage does some automatic simplifications.

However, you can control how the expression is output, e.g. as $\LaTeX$:

def latex_with_few_minuses(frac):
    numer, denom = frac.numerator_denominator()
    permutations = ((1, numer, denom), (-1, numer, -denom), (-1, -numer, denom), (1, -numer, -denom))
    total_minuses = lambda args: sum(str(latex(arg)).count('-') for arg in args)
    prefactor, numer, denom = min(permutations, key=total_minuses)
    return LatexExpr(('-' if prefactor == -1 else '') + r'\frac{' + latex(numer) + '}{' + latex(denom) + '}')

In your example:

f, s, s_i = var("f, s, s_i")
thin_lens_formula = 1/f == 1/s + 1/s_i
sol = solve(thin_lens_formula, s)[0].rhs()
show(latex_with_few_minuses(sol))

displays

$$\frac{ f s_{i} }{ -f + s_{i} }$$