poly-fast: add reusable interpolation plans - #9612
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The schoolbook path has quadratic coefficient work. Balanced multiplication and specialized squaring satisfy the three-recursion Karatsuba recurrence; the fixed 64-block skew dispatcher preserves that asymptotic bound.
Cost model: long division is quadratic because each leading-coefficient elimination updates a linear suffix. Cold Newton division costs O(M(n)): reciprocal doubling is a geometric series and the quotient/remainder use a constant number of clipped products.
The cached point/remainder tree evaluates bounded inputs in O(M(n) log n); the explicit Horner comparator performs Θ(n²) coefficient operations.
Reusable distinct-point plans cache the derivative weights and balanced point products. Direct Lagrange interpolation is cubic because it rebuilds n growing n-factor numerators; planned bottom-up interpolation costs O(M(n) log n) over the cached balanced tree.
# Conflicts: # .github/workflows/ci.yml # scripts/bench/proof_only_runtime_exemptions.json
# Conflicts: # .github/workflows/ci.yml # scripts/bench/proof_only_runtime_exemptions.json
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Depends on #9610.
Implements the interpolation portion of the hex-poly-fast SPEC:
Validation: