Function dF

scripts/rubi/benchmark.ts:164–194

Richardson-extrapolated central difference F′(x) at a (possibly * complex) point, differencing along the real axis: combines steps h and * h/2 for O(h⁴) truncation — high-order poles like x⁻¹⁰⁰ and degree-20 * polynomial antiderivatives are far outside plain central-difference * accuracy at REL_

(
  F: BoxedExpression,
  x: string,
  xv: number | C,
  h: number
)

  if (!Number.isFinite(re) || !Number.isFinite(im)) return null;
  return { re, im };
}

/** Richardson-extrapolated central difference F′(x) at a (possibly
 * complex) point, differencing along the real axis: combines steps h and
 * h/2 for O(h⁴) truncation — high-order poles like x⁻¹⁰⁰ and degree-20
 * polynomial antiderivatives are far outside plain central-difference
 * accuracy at REL_TOL. */
function dF(
  F: BoxedExpression,
  x: string,
  xv: number | C,
  h: number
): C | null {
  const at = (d: number): C | null =>
    complexAt(
      F,
      x,
      typeof xv === 'number' ? xv + d : { re: xv.re + d, im: xv.im }
    );
  const Fp = at(h);
  const Fm = at(-h);
  if (!Fp || !Fm) return null;
  // catastrophic-cancellation guard: when F(x±h) agree to ~9+ digits the
  // difference is double-precision noise (degree-20 expanded polynomial
  // antiderivatives evaluated near a root of the integrand) — no usable
  // derivative signal at this point
  const fMag = Math.max(
    Math.hypot(Fp.re, Fp.im),
    Math.hypot(Fm.re, Fm.im)
  );
  if (Math.hypot(Fp.re - Fm.re, Fp.im - Fm.im) < 1e-9 * fMag) return null;
  const d1 = { re: (Fp.re - Fm.re) / (2 * h), im: (Fp.im - Fm.im) / (2 * h) };
  const Fp2 = at(h / 2);
  const Fm2 = at(-h / 2);