Why the fun9 well is closed only partly (diagnostics only, method unchanged). The well itself IS closed: reaching the root's side takes 4 evaluations with contra/defect against Chandrupatla's 3 (13 and 33 before). The remaining +20 (75 vs 55) is after it: (1) the exponential flank on the widest brackets - our step moves ~1 per evaluation down exp(x) (f/f' = 1), 10 evaluations where Chandrupatla halves 3 times; the analogy entry comes late because exp is self-similar by TRANSLATION, not scaling - the pair's x-scale does not shrink, so m = ln(f ratio)/ln(x ratio) is undefined or negative; (2) overconfident near-zero steps - on a steep flank the three-point slope is huge, z ~ 0, S ~ 1, the model promises the root at a point that is not it: 8 wasted evaluations on fun9 (27 without the decisions).
Co-Authored-By: Claude Opus 5 <noreply@anthropic.com>
This commit is contained in:
Claude Opus 5
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"""Why the fun9 well is closed only partly (Valeri, 2026-09-21).
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fun9 = exp(x) - 2 - 0.01/x^2 + 2e-6/x^3, brackets [2e-4, 1 / 3 / 9 / 27 / 81], root 0.7032.
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Every evaluation is recorded for the library Chandrupatla and for our variants. A run is
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split at its first evaluation with f > 0 inside the bracket (the left end is negative, the
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right end positive): before it the method is still looking for the root's side of the well,
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after it it converges. Our stats give how many steps the decision/contra/regime refused
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(pre_refused) and how many the safeguard forced. Nothing in the method changes.
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python -u diag_fun9_well.py counts and phases
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python -u diag_fun9_well.py fun9.4 plus the point sequences for one problem
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"""
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from __future__ import annotations
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import sys
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from simroot.baselines import chandrupatla_lib
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from simroot.pure import pure_root
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from sweep_full import MAXFEV, XTOL, load_problems
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VARIANTS = ("none", "contra", "defect", "contra+flatanalogmempt", "defect+flatanalogmempt")
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def recorded(f):
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pts = []
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def g(x):
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v = f(x)
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pts.append((float(x), float(v)))
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return v
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return g, pts
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def phases(pts, a, b):
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inner = [i for i, (x, v) in enumerate(pts) if x not in (a, b)]
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first_pos = next((i for i in inner if pts[i][1] > 0.0), None)
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n = len(pts)
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return (first_pos + 1 if first_pos is not None else n), (n - first_pos - 1 if first_pos is not None else 0)
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def main(show=None):
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print(f"{'problem':<8} {'method':<26} {'n':>4} {'to first f>0':>13} {'after':>6} "
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f"{'refused':>8} {'forced':>7}")
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for s, pid, f, a, b, root in load_problems():
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if not pid.startswith("fun9"):
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continue
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g, pts = recorded(f)
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chandrupatla_lib(g, a, b, XTOL, MAXFEV)
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p1, p2 = phases(pts, a, b)
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print(f"{pid:<8} {'Chandrupatla':<26} {len(pts):4d} {p1:13d} {p2:6d}")
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if pid == show:
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print(" Chandrupatla: " + " ".join(f"{x:.4g}:{v:+.2g}" for x, v in pts))
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for pre in VARIANTS:
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g, pts = recorded(f)
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st = {}
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pure_root(g, a, b, XTOL, MAXFEV, similarity="ratio", scale="derived3",
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guard="project", prestep=pre, stats=st)
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p1, p2 = phases(pts, a, b)
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print(f"{'':<8} {pre:<26} {len(pts):4d} {p1:13d} {p2:6d} "
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f"{st.get('pre_refused', 0):8d} {st.get('forced', 0):7d}")
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if pid == show:
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print(f" {pre}: " + " ".join(f"{x:.4g}:{v:+.2g}" for x, v in pts))
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print("diag done")
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if __name__ == "__main__":
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main(*sys.argv[1:2])
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"""How common are near-zero steps - the model promising 'the root is right here' at a point
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that is not the root, so the next evaluation lands almost on the same x and f barely moves?
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Seen on fun9 (diag_fun9_well.py): 0.0002 -> 0.0002, 6.75 -> 6.734, 1.964 -> 1.941, 20.2502 ->
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20.2501. On a steep flank the three-point slope is dominated by a huge value, the scale
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sigma = |slope| D is huge, z = |f|/sigma ~ 0 and S(z) ~ 1.
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DIAGNOSTIC thresholds (reporting only, never in the method): a step is near-zero when
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|x_new - x| < 1% of the bracket, and wasted when after it |f| kept >= 90% of its value.
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Counted on the recorded evaluations of the proposed rule (projection) and of the best
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variant, per group. Nothing in the method changes.
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python -u diag_near_zero_steps.py
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"""
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from __future__ import annotations
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from collections import defaultdict
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from simroot.pure import pure_root
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from sweep_full import MAXFEV, XTOL, load_problems
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FLAT = ("fun3", "fun4", "fun5", "fun6")
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def main():
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for pre in ("none", "defect+flatanalogmempt"):
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agg = defaultdict(lambda: [0, 0, 0, 0]) # problems, evals, near-zero, wasted
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for s, pid, f, a, b, _root in load_problems():
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pts = []
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def g(x, f=f):
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v = f(x)
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pts.append((float(x), float(v)))
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return v
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pure_root(g, a, b, XTOL, MAXFEV, similarity="ratio", scale="derived3",
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guard="project", prestep=pre)
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fam = pid.split(".")[0]
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grp = "APS" if s == "APS" else ("CHA-flat" if fam in FLAT else f"CHA-{fam}")
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lo, hi = min(a, b), max(a, b)
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near = wasted = 0
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for (x0, f0), (x1, f1) in zip(pts[2:], pts[3:]):
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width = hi - lo
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if width > 0 and abs(x1 - x0) < 0.01 * width:
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near += 1
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if f0 != 0.0 and abs(f1) >= 0.9 * abs(f0):
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wasted += 1
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if f1 != 0.0 and lo < x1 < hi:
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# track the bracket as the method does (sign against the left end)
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if (f1 > 0) == (f(lo) > 0):
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lo = x1
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else:
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hi = x1
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ag = agg[grp]
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ag[0] += 1
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ag[1] += len(pts)
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ag[2] += near
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ag[3] += wasted
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print(f"\n== {pre}")
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print(f" {'group':<9} {'problems':>8} {'evals':>6} {'near-zero':>10} {'wasted':>7}")
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for grp in sorted(agg):
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n, e, nz, w = agg[grp]
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print(f" {grp:<9} {n:8d} {e:6d} {nz:10d} {w:7d}")
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print("diag done")
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if __name__ == "__main__":
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main()
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@@ -0,0 +1,53 @@
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problem method n to first f>0 after refused forced
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fun9.1 Chandrupatla 8 4 4
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none 8 4 4 0 0
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contra 8 4 4 0 0
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defect 8 4 4 0 0
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contra+flatanalogmempt 8 4 4 0 0
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defect+flatanalogmempt 8 4 4 0 0
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fun9.2 Chandrupatla 9 3 6
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none 11 4 7 0 0
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contra 11 4 7 1 0
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defect 11 4 7 2 0
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contra+flatanalogmempt 11 4 7 1 0
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defect+flatanalogmempt 11 4 7 2 0
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fun9.3 Chandrupatla 11 3 8
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none 21 13 8 0 0
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contra 15 4 11 1 0
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defect 14 4 10 2 0
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contra+flatanalogmempt 15 4 11 1 0
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defect+flatanalogmempt 14 4 10 2 0
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fun9.4 Chandrupatla 13 3 10
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none 39 33 6 0 0
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contra 21 4 17 1 0
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defect 17 4 13 2 0
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contra+flatanalogmempt 21 4 17 1 0
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defect+flatanalogmempt 17 4 13 2 0
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fun9.5 Chandrupatla 14 3 11
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none 26 3 23 0 0
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contra 26 3 23 0 0
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defect 26 3 23 1 0
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contra+flatanalogmempt 25 3 22 4 0
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defect+flatanalogmempt 25 3 22 5 0
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diag done
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== none
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group problems evals near-zero wasted
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APS 154 2634 425 30
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CHA-flat 20 1823 695 188
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CHA-fun1 5 99 44 3
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CHA-fun2 5 141 18 3
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CHA-fun7 5 137 99 12
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CHA-fun8 5 52 15 0
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CHA-fun9 5 105 56 27
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== defect+flatanalogmempt
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group problems evals near-zero wasted
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APS 154 2384 438 52
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CHA-flat 20 789 447 147
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CHA-fun1 5 99 19 4
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CHA-fun2 5 144 13 2
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CHA-fun7 5 134 91 41
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CHA-fun8 5 57 16 0
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CHA-fun9 5 75 22 8
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diag done
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