Add ellipke for simultaneous K(m) and E(m) computation

[mgyoo86]

Depends on: #509 (perf/elliptic-optimization) — this PR builds on the refactored elliptic integral infrastructure.

Motivation

In many physics and engineering applications, both complete elliptic integrals K(m) and E(m) are needed together—for example, when calculating magnetic fields from coils (as in VacuumFields.jl).

Currently, users have to call ellipk and ellipe separately, which duplicates overhead for input validation and index calculation. MATLAB provides ellipke for exactly this reason, and it makes sense for SpecialFunctions.jl to offer a similar optimized path.

Implementation

This implementation computes both integrals in a single pass. The speedup comes from two sources:

1. Shared indexing: The polynomial table index (idx = trunc(m * 10)) is computed once and reused for both K and E.
2. Shared logarithm: For m ≥ 0.9 (where computation is heaviest), the expensive log term is computed once and shared between K and E.

Benchmark

Environment

julia> versioninfo()
Julia Version 1.12.3
Commit 966d0af0fdf (2025-12-15 11:20 UTC)
Build Info:
  Official https://julialang.org release
Platform Info:
  OS: macOS (arm64-apple-darwin24.0.0)
  CPU: 10 × Apple M1 Pro
  WORD_SIZE: 64
  LLVM: libLLVM-18.1.7 (ORCJIT, apple-m1)
  GC: Built with stock GC
Threads: 1 default, 1 interactive, 1 GC (on 8 virtual cores)

(SpecialFunctions) pkg> st
Project SpecialFunctions v2.7.0
Status `~/.julia/dev/SpecialFunctions/Project.toml`
  [92d709cd] IrrationalConstants v0.2.6
  [2ab3a3ac] LogExpFunctions v0.3.29
  [efe28fd5] OpenSpecFun_jll v0.5.6+0
  [05823500] OpenLibm_jll v0.8.7+0

Calling ellipke(m) is consistently faster than (ellipk(m), ellipe(m)). The gain is most significant (~2.3x) near m = 1 due to shared log calculation.

m	ellipk + ellipe (ns)	ellipke (ns)	speedup
-19.0	43.5	21.9	1.99x
-4.7	13.1	10.0	1.32x
-0.33	13.1	7.2	1.82x
0.05	11.5	6.2	1.85x
0.55	11.8	9.0	1.31x
0.875	12.3	9.6	1.29x
0.95	42.1	18.5	2.27x
Avg	15.9	9.8	1.62x

Note: The (ellipk + ellipe) baseline already includes the optimizations from PR #509.

Usage

The function returns a tuple, behaving exactly like the standalone counterparts:

K, E = ellipke(0.5)
# K == ellipk(0.5), E == ellipe(0.5)  (bit-wise equal)

# Missing propagation follows Base.sincos convention
ellipke(missing)  # (missing, missing)

Tests

• Bit-wise equality against separate ellipk/ellipe calls across all polynomial branches
• Edge cases: 0, 1, -Inf, NaN, and negative values
• DomainError for m > 1

[codecov]

Codecov Report

✅ All modified and coverable lines are covered by tests.
✅ Project coverage is 94.16%. Comparing base (b846391) to head (ea1673d).

Additional details and impacted files

@@            Coverage Diff             @@
##           master     #510      +/-   ##
==========================================
- Coverage   94.17%   94.16%   -0.02%     
==========================================
  Files          14       14              
  Lines        2969     2997      +28     
==========================================
+ Hits         2796     2822      +26     
- Misses        173      175       +2     

Flag	Coverage Δ
unittests	94.16% <100.00%> (-0.02%)	⬇️

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