SIMD median selection networks (AVX2): Hybrid
SIMD+scalar selection networks for n = 8, 16, 32 are 23–58% faster than
scalar median_net. Accelerates robScale,
robLoc, MAD, and ADM median
dispatch.
Raised sort network threshold (16 → 56):
Branchless sorting networks (sort_net) beat
std::sort by 1.2–3× for n ≤ 60. Benefits Qn
exact path (n ≤ 40), Sn micro-path, GMD, and
IQR sort-then-index. Ensemble uses a separate lower
threshold (20) to avoid L1 I-cache pressure in the bootstrap hot
loop.
Raised median network threshold (16 → 36):
Scalar median_net (selection networks) beats Floyd-Rivest
through n ≈ 36 for both odd and even sample sizes.
Earlier ensemble parallelism: TBB
parallel_for now fires for all ensemble bootstrap calls
with ≥ 2 cores and ≥ 16 replicates (was gated on
n × n_boot ≥ 10000). Ensemble is 4–8× faster for n = 3–49
on multi-core machines.
ADM AVX2 auto-vectorization restored:
adm_core_avx2 now contains the vectorizable loop body
directly instead of delegating to adm_core_scalar via plain
inline. Eliminates a 6–9% regression at medium n
(1024–4096) caused by lost auto-vectorization across the
target-attribute boundary.
Dropped AVX-512 dispatch: Removed
ROBSCALE_TARGET_AVX512F,
ROBSCALE_HAS_AVX512_DISPATCH, and the 8-wide AVX-512 tanh
path. The tanh backend hierarchy is now: Apple Accelerate → glibc
libmvec (AVX2 4-wide) → SLEEF (AVX2 4-wide) → OpenMP SIMD →
scalar.
Runtime AVX2 dispatch for ADM:
adm_core now takes an explicit use_avx2 flag,
computed once at each entry point via RuntimeConfig::get().
Non-AVX2 machines take the scalar path safely.
L2 cache plausibility guard: Hardware detection
now rejects l2_per_core values below 64 KB as implausible
and falls back to 256 KB. Applied across Linux, macOS, and Windows
detection paths.
configure sed delimiter: Replaced
| delimiter with ! in the 11-expression
sed block to survive TBB rpath values containing
|.
RcppParallel.h include ordering: Added
#include <RcppParallel.h> before system TBB headers
in rob_scale.cpp and rob_loc.cpp so that
RcppParallel’s bundled TBB 2019 include guards fire first.
Explicit <cmath> include:
Added direct #include <cmath> to
rob_scale.cpp and rob_loc.cpp for
portability.
macOS TBB detection: The configure script now
checks for libtbb.dylib in addition to
libtbb.so when detecting RcppParallel’s bundled
TBB.
SystemRequirements: Removed TBB
from SystemRequirements in DESCRIPTION. TBB is
provided by RcppParallel or detected at configure time;
listing it as a system requirement was incorrect.
<immintrin.h> guard: Narrowed
the <immintrin.h> include in
qnsn_kernels.h from an architecture check to
ROBSCALE_HAS_AVX2_DISPATCH.
Clang 21 ABI compatibility: Added
target("avx2,fma") and target("avx512f")
attributes to extern "C" declarations of libmvec/SLEEF
vectorized tanh functions. Clang 21 (CRAN Debian) requires target
attributes on __m256d/__m512d function
signatures even at declaration scope.
TBB guard alignment: Unified all TBB
preprocessor guards across qn_estimator.cpp,
sn_estimator.cpp, ensemble.cpp, and
worker_compat.h to use the combined
(ROBSCALE_HAS_SYSTEM_TBB || USE_DIRECT_TBB) pattern.
Previously these files only checked USE_DIRECT_TBB, causing
system-oneTBB builds (CRAN Linux) to fall back to RcppParallel instead
of using direct TBB parallelism for Qn, Sn, and ensemble
bootstrap.
Unused variable warning: Eliminated
use_avx2 warning on system-oneTBB builds by expanding the
guard that consumes it.
scale_robust(x, method = "qn") with
n >= threshold now correctly returns the Qn estimator
instead of silently returning gmd(x). The
auto_switch mechanism now only applies to
method = "ensemble"; named methods are always dispatched as
requested.scale_robust(..., ci = TRUE) now supports bootstrap
confidence intervals for individual named methods via
boot_method = "bca", "percentile", or
"parametric". The new C++ function
cpp_single_estimator_ci_bounds runs the bootstrap on the
single requested estimator.
boot_method = "analytical" is now accepted
explicitly by scale_robust(). It selects the chi-squared
interval for sd and the ARE-based normal approximation for
all other named methods.
print.robscale_ci now shows the CI method in the
output header (e.g.,
95% CI (analytical): [0.6123, 1.1456]).
AVX2 detection guarded for non-GCC/Clang
compilers (WU-PORT-1):
__builtin_cpu_supports("avx2") is now wrapped in
#if defined(__GNUC__) || defined(__clang__). MSVC and ICC
builds previously hit an undefined-identifier error here; on unsupported
compilers, SIMD capability defaults conservatively to
None.
hardware_concurrency() zero return
clamped (WU-PORT-1):
std::thread::hardware_concurrency() returns 0 on platforms
that cannot determine the thread count. The result is clamped to
std::max(..., 1u), preventing a zero-divisor in the TBB
grain-size calculation on such platforms.
sysfs L2-cache parse hardened (WU-PORT-1): The
Linux sysfs integer read now wraps std::stoul in
try { } catch (...). Malformed or out-of-range values fall
back to L2_FALLBACK_KB * 1024 and
num_logical_cores = 1 instead of propagating an uncaught
exception to the R session.
mad.cpp micro-buffer tied to
ROBSCALE_MICRO_BUFFER_SIZE (WU-PORT-2): Both
mad_impl_auto and mad_impl_center previously
hard-coded buf_micro[64] and n <= 64. Both
now reference buf_micro[ROBSCALE_MICRO_BUFFER_SIZE] and
n <= static_cast<int>(ROBSCALE_MICRO_BUFFER_SIZE).
Changing the package-wide constant in robscale_config.h now
propagates to both MAD paths without manual adjustment.
Ensemble bootstrap singleton overhead eliminated
(WU-PERF-1): ensemble_one_replicate called
robscale::qnsn::RuntimeConfig::get().sort_boost_threshold
on every bootstrap replicate. The call is replaced with the compile-time
constant ROBSCALE_SORT_BOOST_THRESHOLD, removing the
static-local singleton access from the hot path. Gate results: ~59%
speedup at n = 1,000–5,000 (stable-zone ratio ≈ 0.41).
AVX2 diff-fill enabled in TBB Qn refinement path
(WU-PERF-2): The qn_fill_diffs_avx2 kernel previously ran
only in the brute-force path. It now dispatches inside the TBB
parallel_for lambda when
use_avx2 && len >= 4; the use_avx2
flag is hoisted before the parallel region to avoid repeated CPUID
lookups inside the lambda. Gate results: ~11% speedup at n =
1,000–10,000 (stable-zone ratio ≈ 0.89).
AVX-512 tanh dispatch: On glibc ≥ 2.35 systems
where libmvec provides _ZGVeN8v_tanh, the bulk tanh kernel
processes 8 doubles per iteration instead of 4. CPUID dispatch ensures
the 8-wide path runs only on AVX-512F hardware; other systems fall
through to AVX2 with zero overhead.
GMD AVX2 FMA kernel: The Gini mean difference
weighted sum uses a shared gmd_weighted_sum kernel with
_mm256_fmadd_pd for n ≥ 8 on AVX2 hardware, replacing three
duplicate scalar loops.
Estimator-major bootstrap layout:
boot_results is stored as 7 × n_boot (estimator-major) so
the mean/variance reduction pass reads each estimator’s replicates
contiguously (stride-1 instead of stride-7).
Dead tmp parameter removed from
rob_scale_compute (WU-CLEAN-1): The function
accepted an unused double* ROBSCALE_RESTRICT tmp pointer
left over from a planned optimization that was never implemented.
Removed from the definition, its forward declaration, and all three call
sites.
STACK_SIZE unified to
ROBSCALE_SN_STACK_THRESHOLD (WU-CLEAN-1): Four
independent constexpr int STACK_SIZE = 2048 definitions in
gmd.cpp, iqr.cpp, mad.cpp, and
sn_estimator.cpp are replaced with the package-wide
constant in robscale_config.h.
iqr_select_and_interp helper extracted; dead
buf2 removed (WU-CLEAN-2): The pdqselect +
max-scan + Type-7 interpolation block was duplicated inline across the
Q1 and Q3 selection paths in iqr(). Extracting it as
static inline iqr_select_and_interp in
estimators_internal.h eliminated the duplication and
enabled compiler CSE across both call sites. Side effect: ~75% speedup
at n = 1,000 (ratio ≈ 0.25). The now-unused buf2 parameter
is removed from iqr() and its call site in
ensemble.cpp.
sn_inner_serial template extracted; Sn
inner-loop triplicated no more (WU-CLEAN-3): The Sn inner loop
appeared identically in SnWorker::operator(), the Tier-1
serial fallback, and the Tier-2 large-\(n\) branch. All three now delegate to
sn_inner_serial<T>, tagged
__attribute__((always_inline)). The attribute is
load-bearing: without it, GCC 15 declines to inline across all three
call sites at -O2, producing a ~5× regression at n =
1,000.
ROBSCALE_SORT_MEDIAN_THRESHOLD renamed
ROBSCALE_SORT_NETWORK_THRESHOLD (WU-CLEAN-4): The
macro guards the sort-network dispatch path, not median calculation. The
old name was misleading. Renamed across all six files; no value
change.
Documentation additions (WU-CLEAN-4):
ensemble_one_replicate receives a full doc comment covering
its parameters, the XorShift32 PRNG bit-shifts, and the sorted-once
design pattern. discover_linux,
discover_macos, and discover_windows in
qnsn_hardware_info.h receive brief function-level comments.
The diag.cpp section labels are replaced with semantic
names (“M-scale iteration diagnostics”, “Median crossover benchmark
helpers”).
iqr_scaled() dead stack eliminated
(OPT-I1+I2): The entry frame previously allocated both a 32 KB
buf_stack[4096] and a 2 KB buf_micro[256]
unconditionally, even for n=2. buf_stack is now declared
only inside a ROBSCALE_NOINLINE large-n helper
(n > 256); buf_stack size reduced 4096 →
2048 (IQR needs one copy). Stack at small n reduced from ~34 KB to ~2
KB.
ROBSCALE_RESTRICT on
iqr_impl_large and interp_q7
(OPT-I5): Pointer parameters carry ROBSCALE_RESTRICT,
allowing the compiler to generate stronger SIMD code in the selection
and scan loops.
iqr_scaled() n≤16 sort fast path
(OPT-I6): For n≤16, two pdqselect calls plus two min-scan passes are
replaced by a single small_sort (branch-free sorting
network) followed by direct index reads. Q1/Q3 are O(1) after the
sort.
Symmetric Q1 selection for n>16 with
frac1>0 (OPT-I3): When the Q1 quantile requires
interpolation ((n-1) % 4 ≠ 0, ~75% of n values), the
lower-partition scan is reduced from O(0.75n) to O(0.25n). Instead of
finding the minimum of buf[lo1+1..n-1], pdqselect places
lo1+1 exactly and a max-scan over buf[0..lo1]
recovers Q1. Applies to both the micro path (17≤n≤256) and the large-n
NOINLINE helper (n>257). Gate results: n=100 −4.3%, n=1000
−3.3%.
ROBSCALE_RESTRICT on
estimators_internal::iqr() (OPT-I8): Pointer
parameters of the inline iqr() function (used in the
ensemble compute path) carry ROBSCALE_RESTRICT, allowing
alias-free code generation in the pdqselect path.
iqr_sorted() added (OPT-I7): New
robscale::internal::iqr_sorted(sorted_x, n) for pre-sorted
data — O(1) Type 7 quantile reads, no copy, no selection. Completes the
sorted-variant family alongside sn_sorted() and
qn_sorted().
mad_scaled() small-n stack frame reduced 16
KB → 512 B (OPT-M1): mad_impl_auto and
mad_impl_center now route n > 64 through
ROBSCALE_NOINLINE helpers. The 16 KB
buf_stack[2048] is declared only in those helpers, never in
the entry frame.
#pragma omp simd on all deviation
loops (OPT-M2): All absolute-deviation loops in
mad_impl_auto, mad_impl_center, and their
large-n helpers are SIMD-annotated via the established
ROBSCALE_HAS_OMP_SIMD guard pattern.
ROBSCALE_RESTRICT on NOINLINE helper
parameters (OPT-M3): mad_impl_auto_large and
mad_impl_center_large carry ROBSCALE_RESTRICT
on their input pointers, allowing the compiler to generate stronger SIMD
code.
bulk_abs_diff /
bulk_abs_diff_inplace shared SIMD kernels
(OPT-M4): Two inline SIMD kernels added to
src/robust_core.h. All six deviation loops in
mad.cpp are replaced with kernel calls. n=1000 shows
~2–3.5% improvement.
mad_from_data uses
bulk_abs_diff_inplace (OPT-M5): The deviation loop
in estimators_internal.h::mad_from_data (used by the
ensemble compute path) is replaced with the shared kernel.
Ensemble MAD block uses
bulk_abs_diff (OPT-M6): The per-replicate
deviation loop in ensemble_one_replicate is replaced with
bulk_abs_diff.
V-shaped deviation median in ensemble MAD block
(OPT-M7): Since resample is sorted, the deviation array is
V-shaped. vshaped_mad finds the median of the two sorted
halves via O(log n) binary search (kth_of_two_sorted),
replacing the O(n) median_select call and eliminating the
work2 write-read cycle.
gmd() small-n stack frame reduced 17 KB → 1
KB (OPT-G1): gmd_impl now routes
n ≤ 128 through an ROBSCALE_NOINLINE helper
that allocates only a 128-double arena. The large
buf_stack[2048] is declared only in the
n > 128 path, eliminating 16 KB of unnecessary stack
allocation on every small-sample call.
Ensemble bootstrap sort upgraded to
boost::float_sort (OPT-G4): The per-replicate sort
in ensemble_one_replicate now dispatches to
boost::sort::spreadsort::float_sort (serial radix sort) for
n above the runtime-configured
sort_boost_threshold. Expected speedup ~1.5–2× for
bootstrap samples of size ≥ 512. tbb::parallel_sort is not
used here to avoid nested-TBB oversubscription inside the outer
tbb::parallel_for.
gmd() internal sort path uses
optimized_sort (OPT-G3):
internal::gmd() in estimators_internal.h now
uses robscale::qnsn::optimized_sort (tiered
std::sort → boost::float_sort →
tbb::parallel_sort) instead of plain
std::sort. Call sites (compute_all_estimators
at lines 216 and 330 of ensemble.cpp) are serial, so the
optional TBB dispatch is safe.
GMD kernel scale factor precomputed (OPT-G5):
constant * 2 / (n*(n-1)) is now computed once before the
accumulation loop across all three call sites (gmd_sorted,
internal::gmd(), ensemble inline block), reducing
per-element work to a single multiply.
ROBSCALE_RESTRICT annotations on
gmd_sorted and gmd_core (OPT-G6):
No-aliasing hints allow the compiler to generate tighter SIMD code for
the accumulation loop.
robScale()): The previous guard rejected a valid
extrapolation whenever the Aitken jump exceeded the two-step pair
displacement (|cand - s2| < |s2 - s0|). For convergence
rates \(r \gtrsim 0.73\) this condition
is never satisfied, so the acceleration never fired and the M-scale
iteration required up to 80 rho_sum evaluations per call at
small \(n\). Removing the guard —
keeping only the minimal safety candidate > 0 — cuts
typical rho_evals from 50–80 down to 7–12 for the
previously-slow cases (n=5 seed=47/247, n=7 seed=49/149). The case n=7
seed=49, which previously hit maxit without converging, now
converges in fewer than 40 evaluations.Fused AVX2 kernel threshold lowered from n≥8 to
n≥4 (robScale()): The use_fused flag
in rob_scale_compute now activates at \(n \geq 4\) instead of \(n \geq 8\). Inputs of length 4, 5, 6, and 7
now use the single-pass fused AVX2 kernel rather than the three-pass
scalar fallback, eliminating the remaining per-call overhead for very
small samples.
Fused single-pass NR kernel for
robLoc() (OPT-L1):
rob_loc_nr_step_avx2 collapses the three-pass
Newton–Raphson loop (scale residuals, bulk_tanh,
accumulate) into a single AVX2 pass. Two accumulators advance in
lockstep: acc_psi via addpd and
acc_p2 via fmadd (\(p_i^2 = \tanh^2(u_i)\)). The derivative sum
follows from \(\sum \text{sech}^2(u_i) = n -
\sum \tanh^2(u_i)\), avoiding a second transcendental evaluation.
A degenerate-scale guard (sum_dpsi < DBL_MIN) prevents
NaN on near-constant inputs.
RuntimeConfig hoist for robLoc()
(OPT-L2): The SIMD dispatch flag was re-read from thread-local storage
on every Newton–Raphson iteration. CPUID features are invariant over
process lifetime; the flag is now hoisted once before the loop. Added
bulk_tanh_dispatched() to robust_core.h to
accept the pre-hoisted flag.
Warm-cache MAD for robLoc()
(OPT-L4): rob_loc_core copies input into buf[]
and selects the median in place. MAD is permutation-invariant, so
mad_select now reads from buf[]—warm in L1/L2
cache—rather than the cold input array.
TBB parallel NR for robLoc()
(OPT-L3): For \(n \geq \max(4096,
L_2/32)\), rob_loc_parallel_compute partitions the
array across TBB threads via parallel_reduce. Each chunk
calls rob_loc_nr_step_avx2; an NRAccum struct
accumulates partial sums with operator+=.
Combined speedup: These four optimizations yield
2.6–4.1x speedup over revss for
robLoc() at \(n = 100\) to
\(10{,}000\) on x86_64 with AVX2
(benchmark ratios: 0.347 at \(n =
100\), 0.243 at \(n =
10{,}000\)).
Sorting-network threshold corrected for
robScale() (OPT-A): Lowered
ROBSCALE_SORT_MEDIAN_THRESHOLD from 64 to 16. The
median-net comparator count grows as \(O(n^{1.5})\) (\(n
= 64\): 337 swaps; \(n = 16\):
46 swaps); Floyd–Rivest is \(O(n)\),
crossing over at \(n \approx 16\).
Using sorting networks up to \(n = 64\)
incurred ~3.5× unnecessary overhead per median call. This change
eliminates the \(n = 500\)–\(1{,}000\) regression in
robScale(): benchmarks show ratios 0.201 (\(n = 500\)) and 0.174 (\(n = 1{,}000\)), versus 1.126 and 1.122
before the fix (a 5–6× reversal).
PLT elimination for
median_net<T> (OPT-B): Added
__attribute__((visibility("hidden"))) to the
median_net<T> template definition in
sort_net.h. On Linux -fPIC, this eliminates
PLT indirection for all intra-DSO calls, replacing the W (weak) symbol
with a direct \(t\) (local)
binding.
adm() hot-buffer pass (OPT-A):
adm_impl_auto() now passes the already-copied working
buffer to adm_core() rather than the cold SEXP pointer. The
\(n\) doubles remain in L1/L2 cache
from the initial copy through the deviation sum, avoiding an additional
cold read.
adm() stack-frame split (OPT-B):
Extracted adm_large_n() as a ROBSCALE_NOINLINE
helper. This reduces the adm_impl_auto() stack frame from
8,256 to 624 bytes (objdump verified), preventing the 32 KB large-\(n\) buffer from penalizing small
calls.
Benchmark seed pool widened to 7 seeds
(benchmarks/run_benchmarks.R): BENCH_SEEDS
increased from 3 to 7 seeds (spaced 50 apart: 42, 92, 142, 192, 242,
292, 342). At small \(n\) (\(n \leq 16\)) the previous 3-seed pool had
high variance because 2 of 3 seeds could land on the previously-slow
Aitken path, biasing the pooled-median timing toward the slow
cluster.
Added rob_loc_scalar_impl,
rob_loc_has_parallel, and rob_loc_serial_impl
diagnostic exports for TDD correctness gates.
Added compile-time diagnostic helpers (src/diag.cpp)
for sorting-network threshold validation; not user-facing.
Fused median-then-MAD: mad_scaled()
now computes absolute deviations in-place on the median selection
buffer, eliminating the second scratch array. Memory drops from \(2n\) to \(n\) doubles. The same fusion applies to
robScale() and the ensemble’s internal MAD and M-scale
paths, reducing total ensemble workspace from \(3n\) to \(2n\) doubles. Benchmarks show 7–15% speedup
at medium \(n\) on x86_64, with no
regressions on ARM64 or at small \(n\).
Noinline small-\(n\)
dispatch for robScale() and robLoc():
extracted shared core logic (rob_scale_core,
rob_loc_core) and separated the small-\(n\) path (\(n
\leq 64\)) into ROBSCALE_NOINLINE helpers with a 1
KB stack frame. The large-\(n\) path
retains its 32 KB buffer independently. Benchmarks showed no measurable
timing improvement (the R-to-C++ .Call() boundary dominates
at small \(n\)), but the refactoring
clarifies the code structure and prevents the compiler from penalizing
small calls with the large-\(n\) stack
reservation.
ROBSCALE_NOINLINE portability macro to
robscale_config.h (__attribute__((noinline))
on GCC/Clang, __declspec(noinline) on MSVC).estimators_internal.h: mad_from_data() and
rob_scale() signatures reduced from three to two buffer
arguments.ensemble.cpp: workspace allocation reduced from \(3n\) to \(2n\) doubles.revss v3.0.0, which changed robScale() and
robLoc() defaults to use bias-corrected “AA” constants. The
adm() cross-check remains active for all revss
versions.robScale() and robLoc(), decoupled from
upstream revss. Catches regressions in our own algorithm
independent of revss version changes.qn() and
sn() kernels, achieving optimal vectorization across x86_64
(AVX2/FMA) and ARM64 (NEON).scale_robust() provides a unified
dispatcher that automatically selects between a variance-weighted
ensemble of 7 estimators (for small samples) and the Gini Mean
Difference (GMD) for larger samples (n >= 20).gmd(), iqr_scaled(),
mad_scaled(), and sd_c4() to the public
API.ci = TRUE to all scale
estimators, providing analytical intervals (GMD, MAD, Sn, Qn,
bias-corrected SD) or bootstrap-based intervals (ensemble).stats::mad() and
stats::IQR() with optimized C++ implementations
(mad_scaled(), iqr_scaled()) using O(n)
selection via the pdqselect algorithm.robustbase, Hmisc,
GiniDistance, and collapse.inst/CITATION and documentation.std::sort for n =
9-16 on both ARM64 (Apple Silicon) and x86_64 (AMD Zen 3).robScale() that was inadvertently removed, causing
incorrect results when MAD collapses for n < 4.#pragma GCC diagnostic ignored "-Wdeprecated-volatile" from
src/qn_estimator.cpp, src/sn_estimator.cpp,
and src/qnsn_sort_utils.h to resolve the “pragmas
suppressing diagnostics” NOTE requested by CRAN maintainers.configure.win and
cleanup.win to suppress spurious win-builder warnings about
missing Windows configuration.qnsn_kernels.h that Clang 17 on macOS promoted to a
compilation WARNING.RcppParallel::LdFlags() to the
canonical RcppParallel::RcppParallelLibs() in
Makevars.win and Makevars.in.<Accelerate/Accelerate.h> include and resolving a
symbol conflict with R’s COMPLEX type.CXX_STD = CXX17 in
Makevars (required by CRAN policy for packages using C++17
features).GNU make in
SystemRequirements ($(shell) is a GNU
extension used for RcppParallel linking).inst/WORDLIST to whitelist technical terms
from aspell.DESCRIPTION to satisfy
metadata checks.-ltbb flag from
Makevars which caused warnings on some macOS
configurations.qn() and
sn() (Rousseeuw & Croux, 1993) with specialized sorting
network kernels and cache-aware parallelization.robustbase to Suggests for reference
and benchmarking.configure script
auto-detects SIMD capabilities (AVX2/FMA on x86_64, NEON on ARM64)
without requiring any environment variables.adm() now raises an error when
na.rm = FALSE and NAs are present, matching
robLoc() and robScale() behavior.new[]/delete[] with
std::unique_ptr for RAII memory safety in all C++ estimator
functions.PSI_NORM_CONST, INV_PSI_NORM_CONST) from
robust_core.h.half_inv_s above the
Newton–Raphson loop in rob_loc.cpp.robLoc and
robScale to README.adm(),
maxit edge cases for robLoc() and
robScale(), sorting network verification for n = 2–8.cph role to Authors@R in
DESCRIPTION.Initial release.
adm(), robLoc(), and
robScale() implementing the robust location and scale
M-estimators of Rousseeuw & Verboven (2002) for very small
samples.revss
package.tanh(x/2) identity for the logistic psi
function with platform-vectorized bulk evaluation (Apple Accelerate on
macOS, OpenMP SIMD hints on Linux).revss verified across 5,400
systematic comparisons (tolerance: sqrt(.Machine$double.eps)).
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