feat: add symbolic definite integration for piecewise functions

[KingArth0r]

Summary

Add symbolic definite integration of piecewise expressions over finite real intervals. Partition the interval at established switch points, select each expression on the resulting open subintervals, and delegate the smooth pieces to the existing Integrate machinery.

• Support If/Which with first-true branch ordering, Min/Max, Floor, Ceil, and Fract.
• Keep the affine fast path; reuse bounded-domain Solve for supported nonlinear crossings and periodic root families.
• Reuse CE's exact-constant sign resolver for irrational/transcendental boundaries instead of replacing them with floating-point roots or tolerance-based ties.
• Include poles and real-domain boundaries when classifying conditions. Preserve existing one-sided pole checks when integrating the selected pieces.
• Use derivatives and established critical points to bound nonlinear integer-level partitions.
• Preserve variable binding, reversed bounds, isolated exceptional point values, optional integration-provider dispatch, angular units, and user-defined library overrides.

Scope and safeguards

This is finite, real, symbolic definite integration, not a general inequality solver or an indefinite piecewise antiderivative implementation. Unsupported conditions, incomplete root sets, undecidable ordering/domain checks, and unresolved cell integrals remain unevaluated. Work is bounded by 128 interior cuts, 128 nonlinear solve calls, and a traversal budget. Infinite jump sets and general periodic/infinite-series reductions are not included. Nested combinations that cannot be classified completely may remain unevaluated.

Files

• src/compute-engine/symbolic/piecewise-integrate.ts: partitioning, classification, and integration.
• src/compute-engine/library/calculus.ts: Integrate dispatch (plus one formatter-required line wrap).
• test/compute-engine/integrate-piecewise.test.ts: 60 regression tests.
• test/compute-engine/compile-integrate.test.ts: expect exact compilation for the newly solvable example; retain numerical-integration coverage with an unsupported nonlinear trig condition.

Validation

• 1,044 tests across 13 integration, compiler, parser, contour, and bounded-solver suites passed; 221 snapshots passed.
• The final piecewise suite was rerun after the root-cache adjustment: all 60 tests passed.
• TypeScript no-emit check, targeted source ESLint, formatting checks, and git diff checks passed.
• Dependency check found no circular imports.
• The full repository test suite and corpus pipeline were not run.

Built on upstream main at 5130eec. Integral Benchmark changes are intentionally excluded.

[arnog]

Thanks, looks good, merged.

While reviewing it we found that EvaluateAt turned irrational bounds into floats (∫₂^√5 4 dx gave 0.944…), which made many piecewise results inexact. That is fixed on main now, so ∫₀³ ⌊x²⌋ dx gives 21 − 3√2 − √3 − √5 − √6 − √7.