perf: quotient-sized division for short-quotient shapes (38–75× on 2^k+1 pathology)#89
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…k+1 pathology) New algorithms/division/QuotientSizedDivision.h: for ratio < 4/3 at large b, the (delta+1)-limb quotient is determined to +-a few units by the operand tops. With t = delta+4, divide (a >> B^(nb-t)) by (b >> B^(nb-t)) — truncating b perturbs q by <= ~B^(2-GUARD), sub-ulp — then reconstruct the exact remainder with one delta x nb back-multiply plus +-few fixups (cap 8, fallback to full Newton). Cost scales with the QUOTIENT, not the divisor. Dispatch: b >= NEWTON_BALANCED_B, a >= b + QSIZED_MIN_DELTA (64), ratio < 4/3. The tops divide (ratio ~2) goes to Newton directly at t >= 6144 — post-#85-87 Newton beats BZ at ratio 2 from ~6k limbs (13 vs 22 ms at 12k, 40 vs 299 ms at 30k, 0.16 s vs 5.3 s at 65537). Recursion cannot re-enter the band (tops ratio ~2 > 4/3). Also lowers NEWTON_BALANCED_B 98304 -> 24576: the generic ratio-4/3 Newton/BZ crossover sits at ~24k limbs after the wraparound PRs. Measured (M1 Max, min of 3): - nb=131073 (2^17+1) ratios 1.05/1.10/1.25: 1.07 s / 2.16 s / 5.35 s -> 28 / 43 / 71 ms (38-75x) - nb=120000 same ratios: 37/56/116 -> 30/39/66 ms - nb=48000 ratio 1.25: 16 ms vs Newton 39, FastDivision 733 - nb=131072 (BZ pow2 best case) 1.25: 65 vs 88 ms - no regression band This closes the last known residual in the division dispatch (ratio in (1, 4/3) at large b routing to BZ's transform-doubling blowup). Tests: 246 unit tests, div_correctness, identity + FastDivision cross-checks across the band (incl. all-max/sparse patterns, 2^k+1 sizes, band edges 98304->24576), Pow10 suite, ToString round-trips. Co-Authored-By: Claude Fable 5 <noreply@anthropic.com>
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What
Second half of the "do both" pair (#88 lowered the balanced band to 4/3; this closes the remaining ratio ∈ (1, 4/3) residual — the last known hole in division dispatch).
New
algorithms/division/QuotientSizedDivision.h: for short-quotient shapes, the (Δ+1)-limb quotient is determined to ±a few units by the operand tops (t = Δ+4limbs of b,Δ+tof a; truncating b perturbs q by ≤ ~B^(2−GUARD), sub-ulp). Divide the tops, then reconstruct the exact remainder with one Δ×nb back-multiply + ±few fixups (cap 8, fallback full Newton). Cost scales with the quotient, not the divisor.Dispatch band:
b ≥ NEWTON_BALANCED_B,a ≥ b + 64, ratio < 4/3. Tops divide (ratio ~2) routes to Newton directly att ≥ 6144— post-#85–87 Newton beats BZ at ratio 2 from ~6k limbs (40 vs 299 ms at 30k; 0.16 s vs 5.3 s at 65537). Recursion can't re-enter the band (tops ratio ~2 > 4/3).Also lowers
NEWTON_BALANCED_B98304 → 24576 (the generic ratio-4/3 crossover sits at ~24k limbs now).Results (M1 Max, min of 3)
Testing
div_correctnessq·b + r == a,r < b+ full FastDivision cross-check across ratios 1.01–1.32, sizes 98304–200000, all-max/sparse limb patterns, 2^k+1 sizes🤖 Generated with Claude Code