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Rewrite J1's (4, 8] rational exactly to avoid cancellation - #1627
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The same approximation, expanded in t = (x^2 - 40) / 24 with its root near j(1,1) divided out. Double error on (4, 8] drops from 85 ulps to under 4. Closes #1626 Co-Authored-By: Claude Opus 5.5 <noreply@anthropic.com>
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The same approximation, expanded in t = (x^2 - 40) / 24 with its root near j(1,1) divided out.
Double error on (4, 8] drops from 85 ulps to under 4.
Closes #1626
The (4, 8] branch evaluated P2(x²), whose alternating coefficients cancel by up to 62x. This rewrites the same rational function exactly (no refit): P2 is divided by (y − r₀), r₀ its root near j(1,1)², and both polynomials are re-expanded about y = 40. The remaining cancellation is at most 5.8x. Evaluated at 80 digits, the new form agrees with the old one to 9e-20 on [4, 8], so the long double approximation is unchanged.
cyl_bessel_j(1, x)on (4, 8]Integer orders computed by recurrence from J₁ get the fix too: J₇(7.695) went from −58 to 0.7 eps and J₁₀(7.902) from −82 ulps to 0.7 eps.
cyl_bessel_j(1, x) on [4, 8], develop (orange) vs this PR (blue)
This PR alone
Near the x = 4 and x = 8 branch points
cyl_bessel_j(7, x) on [5, 8], where its recurrence uses J₁ on (4, 8]
Apple M-series, clang -O2; benchmark in
reporting/performance/cyl_bessel_j_performance.cpp. Y₁'s (4, 8] branch has the same kind of cancellation; that is for a separate PR.🤖 Generated with Claude Code