Let a numeric factor reach the coefficient when collecting like terms

[Rafael-SOWNet]

Closes #855.

The defect

CollectLikeTerms failed on the expression its own documentation cites as the reason it exists:

(x+1)^2 * (x+1)^2   ->   1 + 4x + 2x^2 + 4x^2 + 4x^3 + x^4

The two x^2 terms sit side by side instead of being added. Correct, but one term longer than it should be — and (x+1)^4, (x+1)*(x+1)*(x+1)*(x+1) and (x+1)^2*(x+2)^2 all collect properly, which is what made it look arbitrary.

The cause, from instrumenting the monomial keys

[collect] term=1              key=''              coeff=1
[collect] term=2 * x          key='x^1'           coeff=2
[collect] term=x ^ 2          key='x^2'           coeff=1
[collect] term=(2 * x) ^ 2    key='4 * x ^ 2^1'   coeff=1     <-- here

(2 * x)^2 reduces to 4 * x^2, which is a product, not a power, so it fell to the exponent-of-one branch and the 4 was baked into the monomial key instead of being extracted into the coefficient. Keyed on 4 * x ^ 2, it could never meet the plain x^2 terms.

The fix

A factor that reduces to a product is split again, and an integer power of a product is distributed over its factors — so the number reaches the coefficient in both shapes. (a*b)^n = a^n * b^n is used only for integer n, where it holds for every a and b.

Only for polynomial-shaped factors, and that restriction is the whole design, not a caveat.

What the restriction is for

Opening a product is what lets like terms meet. It is also what stops a product cancelling as a whole:

(1/2 * sin(2t))^2 * csc(t)^2     collapses to cos(t)^2 while intact
                                 does not, once the 1/2 is lifted out

DoubleAngleOverSineTest.TheReportersSecondExpressionIsAlsoZero exists to catch exactly that, and it caught it. Collection is a statement about polynomials, so it now opens polynomials and leaves anything carrying a function to the rules that know it.

Two other shapes tried and rejected — against the test, not against reasoning

1. Split only products carrying a number. Does not help: 1/2 is a number, so the trigonometric term is opened anyway.
2. Build both forms and keep the shorter by SimplifiedRate. Does not work either, and the reason is already written down in this repository — the comment on TheReportersSecondExpressionIsAlsoZero says the opened form "was offered to the complexity metric in the one shape where its payoff had not happened yet". A local comparison cannot see a payoff that only materialises after later passes. I tried it at the Simplificator level too, offering the uncollected expansion as a candidate; still no.

Recording these because each looks obviously right until it is run.

Measured

	before	after
(x+1)^2 * (x+1)^2	1 + 4x + 2x² + 4x² + 4x³ + x⁴	1 + 4x + 6x² + 4x³ + x⁴
(x+y)^2 * (x+y)^2	… + 2x²y² + (2xy)² + …	… + 6x²y² + …
(2x+1)^2 * (2x+1)^2	uncollected	1 + 8x + 24x² + 32x³ + 16x⁴
(3x)^2 + x^2	uncollected	10x²
(x*y+1)^2 * (x*y+1)^2	uncollected	1 + 4xy + 6x²y² + 4x³y³ + x⁴y⁴

All binomial coefficients 1, 4, 6, 4, 1 as they should be.

Passed! - Failed: 0, Passed: 6067, Skipped: 14, Total: 6081

Six new cases, including one that asserts the trigonometric term is not opened — so the restriction has a test of its own rather than only the pre-existing one it was derived from.

🤖 Generated with Claude Code