Simplify gathers a quotient of two powers with the same exponent into a single power — a^p / b^p becomes (a/b)^p — but it stops doing so as soon as one of the bases is itself a power.
"(y + 1) ^ x / y ^ x".Simplify() // (1 + 1/y) ^ x ✅ gathered
"(x ^ 2 + 1) ^ x / (x ^ 2 + 2) ^ x".Simplify() // (1 + (-1)/(x^2 + 2)) ^ x ✅ gathered
"a ^ x / b ^ x".Simplify() // (a / b) ^ x ✅ gathered
"(a ^ 2 + 1) ^ x / (a ^ 2) ^ x".Simplify() // unchanged ❌
"(x ^ 2 + 1) ^ x / (x ^ 2) ^ x".Simplify() // unchanged ❌
"(x ^ 3 + 1) ^ x / (x ^ 3) ^ x".Simplify() // unchanged ❌
It is the shape of the base, not the variable: a^2 fails exactly as x^2 does, while a plain variable or a sum gathers fine.
A second way in, with the same outcome by a different route — here the denominator is flattened first, so the two exponents stop matching and there is nothing left to gather on:
"(sqrt(x) + 1) ^ x / sqrt(x) ^ x".Simplify() // (sqrt(x) + 1) ^ x / x ^ (x/2)
Why it is worth fixing
Gathering is what makes these tractable downstream. The limit machinery reads a 1^oo off the single power and cannot see one in the quotient, so the same limit is answered or not depending only on how it was written:
"((x ^ 2 + 1) / x ^ 2) ^ x".Limit("x", "+oo") // 1 ✅ (20 ms)
"(x ^ 2 + 1) ^ x / (x ^ 2) ^ x".Limit("x", "+oo") // unevaluated ❌ (5.5 s)Both are the same function, and the second spends five seconds not answering it.
This is not a wrong answer — the quotient form declines rather than returning something false — so it is a gap rather than a defect.
Context
Found while fixing #738, where the sibling quotients were gathered and so were being answered wrongly rather than not at all. That fix re-reads the second remarkable limit after simplification, which covers everything the gathering reaches; this issue is exactly the remainder it cannot reach, because for these no single power is ever produced.
Measured on master at 318ac9f.
Simplifygathers a quotient of two powers with the same exponent into a single power —a^p / b^pbecomes(a/b)^p— but it stops doing so as soon as one of the bases is itself a power.It is the shape of the base, not the variable:
a^2fails exactly asx^2does, while a plain variable or a sum gathers fine.A second way in, with the same outcome by a different route — here the denominator is flattened first, so the two exponents stop matching and there is nothing left to gather on:
Why it is worth fixing
Gathering is what makes these tractable downstream. The limit machinery reads a
1^oooff the single power and cannot see one in the quotient, so the same limit is answered or not depending only on how it was written:Both are the same function, and the second spends five seconds not answering it.
This is not a wrong answer — the quotient form declines rather than returning something false — so it is a gap rather than a defect.
Context
Found while fixing #738, where the sibling quotients were gathered and so were being answered wrongly rather than not at all. That fix re-reads the second remarkable limit after simplification, which covers everything the gathering reaches; this issue is exactly the remainder it cannot reach, because for these no single power is ever produced.
Measured on
masterat 318ac9f.