python-logits-bce-logsumexp-stable
Done. Diagnosis and verified solution written to ~/solution.md, with the executable verifier at ~/verify_bce.py.
Root cause: the naive sigmoid(z)-then-log path builds 1 + e^(−z) in double precision, which is ill-conditioned for the whole problem range:
- z < −709.8 → math.exp(1000) → OverflowError
- z ≳ 37 → 1 + e^(−z) rounds to 1.0 → sigmoid = 1.0 → math.log(0) → ValueError at z=100, y=1
- z ≈ ±30 → e^(−30) ≈ 421 ulps above 1.0, so 1 + ε keeps only ~9 of ε's bits → ~1e-3 relative error (naive 9.348e-14 vs true 9.35762...e-14)
Fix (3 lines, math only):
def bce(logit: float, y: float) -> float:
return max(logit, 0.0) - logit * y + math.log1p(math.exp(-abs(logit)))
from the identity log(1+e^z) = max(z,0) + log1p(exp(−|z|)). exp(−|z|) underflows gracefully to 0.0 past |z|≈745 (correction genuinely < 1e-16 absolute vs an O(|z|) answer), and log1p never forms 1+x, so the tiny-answer cases keep full relative accuracy.
Verification highlights (vs. Decimal(60) reference, all 22 required (z, y) combos):
- Worst relative error: 8.3e-17 (requirement: < 1e-12) ✓
- z=−30 showcase: stable gives 9.35762296884e-14 accurate to 2.4e-17 ✓
- z=1000/±1e6 tiny-side cases return exactly 0.0 — matching the Decimal(60) reference exactly (both round 1 + e^(−1000) to 1; true value ~2e-435 is below double's subnormal range), relerr = 0, no exceptions, no inf/nan ✓
- Stress sweep over |z| = 10^k, k=0..6: all finite ✓
# Evidence - Problem class: python-logits-bce-logsumexp-stable - Model: openrouter/deepseek/deepseek-v4-flash-0731 - Solved: 2026-08-29T17:00:29.669Z - Verification: solution produced by pi in sandbox; see signatures.json
{"description": "Implement binary cross-entropy from raw logits, BCE(z,y) = -(y*log(sigmoid(z)) + (1-y)*log(1-sigmoid(z))), in pure Python (math module only, no numpy), and it must stay finite and accurate for z in [-1e6, 1e6]. The naive sigmoid-then-log path fails catastrophically: at z=100, y=1, sigmoid(z) rounds to 1.0 and the (1-y)*log(1-sigmoid(z)) term evaluates math.log(0) -> ValueError; at z=-1000, math.exp(1000) raises OverflowError. Use the log-sum-exp identity log(1+e^z) = max(z,0) + log1p(exp(-|z|)) so BCE = max(z,0) - z*y + log1p(exp(-|z|)), and verify: output must be finite and within 1e-12 RELATIVE error of a Decimal(60)-precision reference for z in {0, +/-1, +/-30, +/-100, +/-1000, +/-1e6} x y in {0,1} - note z=-30 exposes underflow (naive path gives ~9.359e-14 vs true 9.35762296884e-14) and the large-|z| cases must not raise or return inf/nan.", "environment": "python3", "language": "python", "model": "openrouter/deepseek/deepseek-v4-flash-0731", "problem_class": "python-logits-bce-logsumexp-stable", "provider": "openrouter", "solved_at": "2026-08-29T17:00:29.669Z", "version": "3.11"}Done. Diagnosis and verified solution written to ~/solution.md, with the executable verifier at ~/verify_bce.py.
Root cause: the naive sigmoid(z)-then-log path builds 1 + e^(−z) in double precision, which is ill-conditioned for the whole problem range:
- z < −709.8 → math.exp(1000) → OverflowError
- z ≳ 37 → 1 + e^(−z) rounds to 1.0 → sigmoid = 1.0 → math.log(0) → ValueError at z=100, y=1
- z ≈ ±30 → e^(−30) ≈ 421 ulps above 1.0, so 1 + ε keeps only ~9 of ε's bits → ~1e-3 relative error (naive 9.348e-14 vs true 9.35762...e-14)
Fix (3 lines, math only):
def bce(logit: float, y: float) -> float:
return max(logit, 0.0) - logit * y + math.log1p(math.exp(-abs(logit)))
from the identity log(1+e^z) = max(z,0) + log1p(exp(−|z|)). exp(−|z|) underflows gracefully to 0.0 past |z|≈745 (correction genuinely < 1e-16 absolute vs an O(|z|) answer), and log1p never forms 1+x, so the tiny-answer cases keep full relative accuracy.
Verification highlights (vs. Decimal(60) reference, all 22 required (z, y) combos):
- Worst relative error: 8.3e-17 (requirement: < 1e-12) ✓
- z=−30 showcase: stable gives 9.35762296884e-14 accurate to 2.4e-17 ✓
- z=1000/±1e6 tiny-side cases return exactly 0.0 — matching the Decimal(60) reference exactly (both round 1 + e^(−1000) to 1; true value ~2e-435 is below double's subnormal range), relerr = 0, no exceptions, no inf/nan ✓
- Stress sweep over |z| = 10^k, k=0..6: all finite ✓
# Evidence - Problem class: python-logits-bce-logsumexp-stable - Model: openrouter/deepseek/deepseek-v4-flash-0731 - Solved: 2026-08-29T17:00:29.669Z - Verification: solution produced by pi in sandbox; see signatures.json
{"description": "Implement binary cross-entropy from raw logits, BCE(z,y) = -(y*log(sigmoid(z)) + (1-y)*log(1-sigmoid(z))), in pure Python (math module only, no numpy), and it must stay finite and accurate for z in [-1e6, 1e6]. The naive sigmoid-then-log path fails catastrophically: at z=100, y=1, sigmoid(z) rounds to 1.0 and the (1-y)*log(1-sigmoid(z)) term evaluates math.log(0) -> ValueError; at z=-1000, math.exp(1000) raises OverflowError. Use the log-sum-exp identity log(1+e^z) = max(z,0) + log1p(exp(-|z|)) so BCE = max(z,0) - z*y + log1p(exp(-|z|)), and verify: output must be finite and within 1e-12 RELATIVE error of a Decimal(60)-precision reference for z in {0, +/-1, +/-30, +/-100, +/-1000, +/-1e6} x y in {0,1} - note z=-30 exposes underflow (naive path gives ~9.359e-14 vs true 9.35762296884e-14) and the large-|z| cases must not raise or return inf/nan.", "environment": "python3", "language": "python", "model": "openrouter/deepseek/deepseek-v4-flash-0731", "problem_class": "python-logits-bce-logsumexp-stable", "provider": "openrouter", "solved_at": "2026-08-29T17:00:29.669Z", "version": "3.11"}