A polynomial may stand beside a root of a ± a cos(y) - #1474
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`x^3 sqrt(a + a cos(c + d x))` was left as written. The half angle at which `a ± a cos(y)` is a square answers the root alone, and it substitutes the angle -- so anything else that mentioned the variable stopped it, a polynomial among them, and the rule declined rather than handing on a question it had already made simpler. What is left of the variable is the variable itself. With `u` half the angle -- a quarter turn further for the sine's identity -- `x` is `(2u - offset)/rate`, so a polynomial in `x` is a polynomial in `u`, and what is handed on is one times a cosine of `u`: the shape the closed rules answer. Family 4 of the Rubi suite: 287 -> 290 of 422, 0 wrong; family 6 377/417 and the 1774-problem suite 1707 unchanged. Suite 12649 passed; allocation gate passed on all 19 gated benchmarks. Each row differentiates back with `a` positive and negative, where the sign the rule writes is what changes. Part of #718. Co-Authored-By: Claude Opus 5 (1M context) <noreply@anthropic.com> Claude-Session: https://claude.ai/code/session_012sonx8iAspMiwRwokT1Ura
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x^3 sqrt(a + a cos(c + d x))was left as written, whilesqrt(a + a cos(x))came out in four milliseconds.SolveByTheHalfAngleWhereOnePlusASineIsASquarerewritesa ± a cos(y)as a square of the half angle and then substitutes the angle — so anything else that mentioned the variable madeinU.ContainsNode(x)true and the rule declined, a polynomial among them.What is left of the variable is the variable itself. With
uhalf the angle (a quarter turn further for the sine's identity,u = y/2 + pi/4),xis(2u - offset)/rate, so a polynomial inxis a polynomial inu— and what is handed on is one times a cosine ofu, which the closed rule for a polynomial times a trigonometric answers.Measured (final build):
familycap.sh 4 6): 287 → 290/422, 0 wrong, timeouts 32 → 30 — 4.2.10:188/202 and 4.3.4.2:114; nothing lost.bench.sh: PASSED on all 19 gated benchmarks.apositive andanegative — the sign the rule writes (sgn(cos(y/2))orsgn(sin(y/2))) is what changes between them — and each was checked at five points, worst relative deviation 0.Part of #718.
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https://claude.ai/code/session_012sonx8iAspMiwRwokT1Ura