A product is written letters before functions

[Rafael-SOWNet]

By hand, a product is written with the number first, then the letters and their powers, then the functions: y sin(x), x e^x, z Γ(z) (DLMF 5.5.1). The reason is that a function's argument can be written without parentheses. ISO 80000-2 allows sin nπ, so a letter written after a function reads as part of its argument. Simplify wrote sin(x) * y and e ^ x * x.

The answer of Simplify is now put in that order as its last step (Simplificator.InPrintOrder): numbers, then letters and their powers, then everything else, each kind keeping the order it had. e^x and 2^n count as functions, and x^n counts as a letter. A leading number goes into a quotient over a number, so the derivative of arcsin(2 * x) simplifies to 2 / sqrt(1 - (2 * x) ^ 2) rather than 2 * 1 / sqrt(…).

Only the answer is reordered. I first measured sorting letters first inside the search itself, and it lost four things:

• lim 1/x^2 - 1/sin(x)^2 at 0 became NaN, where it is -1/3, and its two siblings stopped answering.
• x * (x - 1)! stopped becoming x!.
• An integral stopped differentiating back.

The product rules are binary, and they meet 2 cos(x) sin(x) as neighbours only because the search's order sorts every function before every letter. With x between them, 2 x sin(x) cos(x) stopped becoming x sin(2x), and l'Hôpital's quotients grew past their guard. The numbers are on #1628. So the search keeps its order, and InnerSimplified and Differentiate return that order as before.

Simplify of	2.5.0	now
sin(x) * y	sin(x) * y	y * sin(x)
x * e ^ x	e ^ x * x	x * e ^ x
2 * sin(x) * y	2 * sin(x) * y	2 * y * sin(x)
a / sin(x)	csc(x) * a	a * csc(x)
2 * x * sin(x) * cos(x)	sin(2 * x) * x	x * sin(2 * x)
derivative(arcsin(2 * x), x)	1 / sqrt(1 - (2 * x) ^ 2) * 2	2 / sqrt(1 - (2 * x) ^ 2)

FactorOrderTest covers three things:

• the order itself, including exponentials and powers of letters;
• the round trip from printed form back to the same product;
• TheRulesStillMeetTheFactorsTheyCombine: 2 x sin(x) cos(x) → x sin(2x) and x (x - 1)! → x!. Both fail if the order is changed inside the search.

Twenty-one existing expectations are updated to the new spelling. None of them changes value.

Suite on net10.0, at 405b4d2a on master fac6ce16: 14202 passed, none failed. Every target framework builds.
Gate: allocation matches the baseline on all 19 gated benchmarks. The step reads a product's order before building anything, and most answers are in order already.

Closes #1628.

Part of #180.

🤖 Generated with Claude Code

https://claude.ai/code/session_012sonx8iAspMiwRwokT1Ura