perf: SIMD transcendental kernels, Tier-2 LogNormal/Pareto, dedup helpers, test realignment

.clang-tidy

@@ -211,7 +211,7 @@ CheckOptions:
     value: ''
   - key: readability-identifier-naming.NamespaceSuffix
     value: ''
-  
+
   # Performance and modernization options
   - key: modernize-use-auto.MinTypeNameLength
     value: '5'
@@ -223,13 +223,13 @@ CheckOptions:
     value: 'true'
   - key: performance-unnecessary-value-param.IncludeStyle
     value: 'llvm'
-    
+
   # Certificate and security options
   - key: cert-dcl16-c.NewSuffixes
     value: 'L;LL;LU;LLU'
   - key: cert-oop54-cpp.WarnOnlyIfThisHasSuspiciousField
     value: 'false'
-    
+
   # Core guidelines options
   - key: cppcoreguidelines-special-member-functions.AllowSoleDefaultDtor
     value: 'true'

.git-blame-ignore-revs

@@ -4,3 +4,6 @@
 
 # style: bulk reformat all source files with clang-format (2026-04-23)
 7221753
+
+# style: apply clang-format 19.1.7 to all source files (2026-05-03)
+662c172

.gitattributes

@@ -30,10 +30,10 @@ CMakeLists.txt text eol=lf
 # Scripts — always LF so they run correctly in bash/sh
 *.sh text eol=lf
 
-# Windows-only scripts stay CRLF
+# Windows batch/cmd scripts stay CRLF; PowerShell handles LF on all platforms
 *.bat text eol=crlf
 *.cmd text eol=crlf
-*.ps1 text eol=crlf
+*.ps1 text eol=lf
 
 # XML (HMM model files)
 *.xml text eol=lf

CHANGELOG.md

@@ -7,6 +7,54 @@ and this project adheres to [Semantic Versioning](https://semver.org/spec/v2.0.0
 
 ## [Unreleased]
 
+## [3.3.0] - 2026-05-03
+
+SIMD performance phase: explicit vector kernels for transcendental
+operations and two additional Tier-2 distributions. 37/37 tests pass.
+
+### Added
+
+- **SIMD transcendental kernels** (`src/performance/transcendental_kernels.cpp`):
+  five inner-loop kernels used by `ForwardBackwardCalculator` (FB max-reduce
+  recurrence) and `BaumWelchTrainer` (dense-xi accumulation) now have
+  AVX-512 / AVX / SSE2 / NEON backends. The vector `exp` helper uses a
+  13-term Horner polynomial with Cephes `ln2` range reduction and branch-free
+  underflow masking at `MIN_LOG_PROBABILITY`. AVX path stays AVX-1 compatible
+  for Ivy Bridge / Catalina. Benchmarks on Zen 4 / AVX-512 (T=1000):
+  FB max-reduce 5.7× faster at N=32; BW xi accumulation 1.03–1.15×.
+- **LogNormal and Pareto promoted to Tier 2** (`src/distributions/`): explicit
+  SIMD `getBatchLogProbabilities` via a vector `log` helper (IEEE-754 exponent
+  extraction, 7-term Horner, split-LN2 reconstruction, ≤5 ULP).
+- **`simd_kernels_internal.h`**: single source of truth for vector exp/log
+  primitives shared by all Tier-2 distribution TUs and the transcendental
+  kernels TU.
+- **FB recurrence crossover retuned** (`fb_recurrence_policy.h`): threshold
+  moved from N≥5 to N≥4 on x86 after profiling post-SIMD (MaxReduce is 1.7×
+  faster at N=4).
+- **New tests** (37 total, up from 33):
+  - `test_simd_platform`: compile-time ISA hierarchy invariants (`#error`) and
+    runtime contracts on `simd_platform.h` utility functions.
+  - `test_transcendental_kernels`: SIMD vs `std::exp` parity for all five
+    kernels across 11 sizes; 1e-12 rel / 1e-15 abs tolerance.
+  - `test_fb_mode_parity`: Pairwise vs MaxReduce FB log-likelihood agreement.
+  - `test_bw_parity`: BW determinism (bit-exact) and EM monotonicity.
+- **New tools**: `bw_hotspot` (BW E-step phase breakdown), `hotspot_breakdown`
+  (FB phase-level timings), `fb_crossover_sweep` (Pairwise vs MaxReduce
+  timing across N), `fb_contour_sweep` (2-D N×T timing heatmap data).
+
+### Changed
+
+- `fb_recurrence_policy.h` moved from `include/libhmm/calculators/` to
+  `include/libhmm/performance/` (cross-cutting primitive, not calculator-specific).
+- Test group labels in `tests/CMakeLists.txt` changed from numeric Level N
+  notation to semantic names; Performance Primitives group reordered before
+  Distributions to reflect dependency order.
+- `performance/PERFORMANCE_ARCHITECTURE.md` updated: Tier-2 coverage,
+  delivered recurrence-kernel SIMD, corrected `LIBHMM_SIMD_SOURCES` list.
+- `*.ps1` line-ending rule in `.gitattributes` changed from `eol=crlf` to
+  `eol=lf` (PowerShell handles LF on all platforms; avoids CI pre-commit
+  mixed-line-ending failures).
+
 ## [3.2.1] - 2026-05-02
 
 CI hygiene fix; no functional changes.

CMakeLists.txt

@@ -58,7 +58,7 @@ if(APPLE AND NOT CMAKE_CXX_COMPILER)
 endif()
 
 project(libhmm
-    VERSION 3.2.1
+    VERSION 3.3.0
     DESCRIPTION "Modern C++20 Hidden Markov Model Library"
     LANGUAGES CXX
 )
@@ -479,6 +479,15 @@ set(LIBHMM_SIMD_SOURCES
     src/distributions/weibull_distribution.cpp
 )
 
+# Additional TUs that include simd_kernels_internal.h or transcendental_kernels.h
+# and therefore need LIBHMM_BEST_SIMD_FLAGS to activate the #if LIBHMM_HAS_* cascade.
+# (log_normal and pareto are already in LIBHMM_SIMD_SOURCES above.)
+list(APPEND LIBHMM_SIMD_SOURCES
+    src/performance/transcendental_kernels.cpp
+    src/calculators/forward_backward_calculator.cpp
+    src/training/baum_welch_trainer.cpp
+)
+
 if(LIBHMM_BEST_SIMD_FLAGS)
     foreach(simd_src ${LIBHMM_SIMD_SOURCES})
         set_source_files_properties(
@@ -499,6 +508,7 @@ set(LIBHMM_SOURCES
     src/common/common.cpp
     src/common/string_tokenizer.cpp
     src/common/numerical_stability.cpp
+    src/performance/transcendental_kernels.cpp
     src/distributions/distribution_base.cpp
     src/distributions/discrete_distribution.cpp
     src/distributions/gaussian_distribution.cpp

WARP.md

@@ -6,8 +6,8 @@ This file provides guidance to Warp (warp.dev) when working in this repository.
 
 ## Current Status
 
-**Version**: v3.2.1 — latest tag and published release on `main`.
-**Tests**: 33/33 passing on all four CI platforms (Linux/GCC, Linux/Clang, macOS/AppleClang, Windows/MSVC).
+**Version**: v3.3.0 — latest tag and published release on `main`.
+**Tests**: 37/37 passing on all four CI platforms (Linux/GCC, Linux/Clang, macOS/AppleClang, Windows/MSVC).
 **Active phase**: Complete. All phases through Post-Phase 5 (CI/tooling, benchmarks) are done.
 
 ---
@@ -36,7 +36,7 @@ include/libhmm/
 │   └── segmental_kmeans_trainer.h     # Discrete-state initialisation
 └── io/             # XML I/O
 src/                # Implementation (mirrors include/)
-tests/              # GTest suite — levels 0–7 (see tests/CMakeLists.txt)
+tests/              # GTest suite — semantic groups (see tests/CMakeLists.txt)
 examples/           # 13 usage demonstrations (all canonical API)
 tools/              # Standalone diagnostic/benchmarking executables
 benchmarks/         # Comparative benchmarks
@@ -70,7 +70,7 @@ Both are always produced regardless of `BUILD_SHARED_LIBS`. Tests link against
 
 2. **Two canonical calculators** — `ForwardBackwardCalculator` (log-space, precomputed log-trans) and `ViterbiCalculator`. Both call `getBatchLogProbabilities()` per state per time step.
 
-3. **Compile-time SIMD dispatch** — source-distributed; each machine builds for its own CPU. GCC/Clang: `-march=native`. MSVC: `check_cxx_source_runs`-verified `/arch:AVX512`/`AVX2`/`AVX`. All 15 distribution TUs in `LIBHMM_SIMD_SOURCES`. Tier 2 explicit intrinsics: Gaussian + Exponential via `detail::` free functions (extractable to separate TU for future runtime dispatch).
+3. **Compile-time SIMD dispatch** — source-distributed; each machine builds for its own CPU. GCC/Clang: `-march=native`. MSVC: `check_cxx_source_runs`-verified `/arch:AVX512`/`AVX2`/`AVX`. All 15 distribution TUs plus transcendental kernels, FB calculator, and BW trainer in `LIBHMM_SIMD_SOURCES`. Tier 2 explicit intrinsics: Gaussian, Exponential, LogNormal, Pareto via `detail::` free functions; recurrence kernels (FB max-reduce, BW xi) via `TranscendentalKernels` in `src/performance/`. Shared vector exp/log helpers in `include/libhmm/performance/simd_kernels_internal.h`.
 
 4. **Thread-safe cache** — `std::atomic<bool> cacheValid_` in `DistributionBase`. Avoids mutex; safe for concurrent const reads if the library is invoked from multiple threads (calculators and trainers themselves run single-threaded — see `performance/PERFORMANCE_ARCHITECTURE.md`).
 
@@ -210,24 +210,26 @@ CRLF: `.gitattributes` enforces LF. CRLF warnings on `git add` are normal.
 
 - Always run `./scripts/configure_catalina.sh build` for the first configure.
 - The script sanitizes toolchain-related environment variables, pins AppleClang via `xcrun`, and sets `CMAKE_OSX_DEPLOYMENT_TARGET=10.15`.
+- **Build type:** the script defaults to `Release` (`-O3`). This is required for correctness: at `-O0`, AppleClang inserts `VZEROUPPER` in the prologue of large-frame AVX functions before saving the `__m256d` argument, silently zeroing `x[2]` and `x[3]`. For debuggable builds use `RelWithDebInfo` (`-O2 -g`) — SIMD helpers inline at `-O2` so the issue cannot occur: `./scripts/configure_catalina.sh build -DCMAKE_BUILD_TYPE=RelWithDebInfo`. Pure `Debug` (`-O0`) is unsafe for any code path that passes `__m256d` through a real call boundary.
 - Do not point Catalina builds at Homebrew LLVM/libc++ (`/usr/local/opt/llvm`, `Cellar/llvm*`, libc++ include paths). The root `CMakeLists.txt` guard fails configure when those hints are detected.
 - Use `-DLIBHMM_ALLOW_UNSUPPORTED_CATALINA_HOMEBREW_LIBCXX=ON` only for explicit troubleshooting; runtime stability is not guaranteed.
 
 ---
 
 ## Test Suite Structure
 
-Tests in `tests/CMakeLists.txt` use `add_hmm_test()` helper organized into 8 levels:
+Tests in `tests/CMakeLists.txt` use `add_hmm_test()` helper organized into semantic groups:
 
-| Level | Content |
+| Group | Content |
 |---|---|
-| 1 | Math & Numerics |
-| 2 | Linear Algebra |
-| 3 | Distributions (all 15 + traits/header/type_safety) |
-| 4 | Core HMM |
-| 5 | Calculators (canonical + continuous + edge cases) |
-| 6 | Trainers (canonical + training + edge cases + BW convergence) |
-| 7 | IO + Integration (stream IO + end-to-end casino) |
+| Platform Capabilities | No tests yet (placeholder) |
+| Math & Numerics | constants, numerical stability, common types |
+| Performance Primitives | transcendental kernels (SIMD parity vs `std::exp`) |
+| Distributions | all 15 + traits/header/type_safety |
+| Core HMM | HMM construction and state management |
+| Calculators | canonical + continuous + edge cases + FB mode parity |
+| Trainers | canonical + training + edge cases + BW convergence + BW parity |
+| IO & Integration | stream IO + end-to-end casino |
 
 Custom targets: `check` (correctness, parallel), `check_timing` (serial).
 Note: named `check` not `run_tests` to avoid cmake's built-in `RUN_TESTS` on Windows.

benchmark-analysis/fb_contour_sweep_adaptive_static_v1.csv

@@ -0,0 +1,23 @@
+mode,n,t,runs,warmup,recurrence_work,emission_work,transition_ms,obs_copy_ms,emission_ms,alloc_ms,forward_ms,backward_ms,reduction_ms,total_ms
+adaptive_static_v1,2,1000,5,2,3996,2000,0.0002,0.0006,0.0006,0.0005,0.0555,0.053,0.0001,0.1109
+adaptive_static_v1,2,10000,5,2,39996,20000,0.0007,0.0071,0.0045,0.043,0.3578,0.3551,0,0.7707
+adaptive_static_v1,2,100000,5,2,399996,200000,0.0026,0.1488,0.2834,0.508,3.8598,3.6578,0.0003,9.0083
+adaptive_static_v1,2,1000000,5,2,3999996,2000000,0.0031,2.0429,3.4685,3.7612,36.9812,36.2041,0.0002,82.1594
+adaptive_static_v1,4,1000,5,2,15984,4000,0.001,0.0007,0.0106,0.0154,0.2256,0.2209,0.0001,0.4701
+adaptive_static_v1,4,10000,5,2,159984,40000,0.0018,0.0104,0.014,0.0139,1.4938,1.5459,0.0005,3.0504
+adaptive_static_v1,4,100000,5,2,1599984,400000,0.0036,0.1141,0.58,0.9126,14.5554,14.3194,0.0007,30.568
+adaptive_static_v1,8,1000,5,2,63936,8000,0.0012,0.0024,0.0157,0.0294,0.3975,0.3908,0.0002,0.8399
+adaptive_static_v1,8,5000,5,2,319936,40000,0.0006,0.0022,0.007,0.0059,1.9524,1.9707,0.0002,3.9503
+adaptive_static_v1,8,10000,5,2,639936,80000,0.002,0.0087,0.019,0.2104,3.9859,4.0981,0.0006,8.434
+adaptive_static_v1,16,1000,5,2,255744,16000,0.0024,0.0036,0.0276,0.0427,1.4421,1.4556,0.0005,2.9893
+adaptive_static_v1,16,2000,5,2,511744,32000,0.0015,0.0017,0.0057,0.0056,2.8761,2.9113,0.0005,5.7923
+adaptive_static_v1,16,5000,5,2,1279744,80000,0.0029,0.005,0.0262,0.1948,7.2773,7.3363,0.0007,14.8745
+adaptive_static_v1,32,500,5,2,510976,16000,0.0102,0.0007,0.0276,0.0519,4.0494,4.2193,0.0008,8.3801
+adaptive_static_v1,32,1000,5,2,1022976,32000,0.0134,0.0031,0.044,0.0831,8.221,8.6986,0.001,17.1867
+adaptive_static_v1,32,2000,5,2,2046976,64000,0.0158,0.0056,0.0887,0.1513,16.2641,16.9673,0.001,33.4698
+adaptive_static_v1,64,200,5,2,815104,12800,0.0268,0.0006,0.0238,0.0412,8.7132,8.7867,0.0017,17.5748
+adaptive_static_v1,64,500,5,2,2043904,32000,0.0417,0.0027,0.0657,0.1169,36.6388,36.9101,0.0019,74.5554
+adaptive_static_v1,64,1000,5,2,4091904,64000,0.0355,0.0045,0.1179,0.1798,45.2402,47.7388,0.0015,93.3553
+adaptive_static_v1,128,100,5,2,1622016,12800,0.0678,0.0005,0.0268,0.0428,21.5884,25.9046,0.0023,50.4003
+adaptive_static_v1,128,250,5,2,4079616,32000,0.0685,0.001,0.0247,0.0602,54.7442,59.1274,0.0025,111.21
+adaptive_static_v1,128,500,5,2,8175616,64000,0.0821,0.0013,0.0333,0.032,115.191,122.896,0.0026,231.18

benchmark-analysis/fb_contour_sweep_max_reduce.csv

@@ -0,0 +1,23 @@
+mode,n,t,runs,warmup,recurrence_work,emission_work,transition_ms,obs_copy_ms,emission_ms,alloc_ms,forward_ms,backward_ms,reduction_ms,total_ms
+max_reduce,2,1000,5,2,3996,2000,0.0001,0.0003,0.0004,0.0003,0.0541,0.0557,0,0.1112
+max_reduce,2,10000,5,2,39996,20000,0.0003,0.0033,0.0036,0.0029,0.5451,0.5607,0.0001,1.1176
+max_reduce,2,100000,5,2,399996,200000,0.0024,0.1024,0.292,0.5074,5.9164,5.8783,0.0006,12.7317
+max_reduce,2,1000000,5,2,3999996,2000000,0.0019,1.5644,3.6518,4.0798,61.6187,65.8737,0.0008,138.632
+max_reduce,4,1000,5,2,15984,4000,0.0002,0.0003,0.0072,0.0148,0.1365,0.1401,0.0001,0.3002
+max_reduce,4,10000,5,2,159984,40000,0.0005,0.0036,0.0072,0.0061,1.3655,1.4421,0.0002,2.8389
+max_reduce,4,100000,5,2,1599984,400000,0.0039,0.1803,0.544,0.8251,14.3255,14.7261,0.0007,30.5996
+max_reduce,8,1000,5,2,63936,8000,0.0005,0.0024,0.015,0.0308,0.3906,0.4051,0.0002,0.8435
+max_reduce,8,5000,5,2,319936,40000,0.0015,0.0127,0.0492,0.094,1.9496,2.0359,0.0003,4.1927
+max_reduce,8,10000,5,2,639936,80000,0.0024,0.0097,0.0191,0.1943,3.9162,4.15,0.0005,8.2942
+max_reduce,16,1000,5,2,255744,16000,0.0012,0.0027,0.0325,0.045,1.4214,1.4575,0.0004,2.963
+max_reduce,16,2000,5,2,511744,32000,0.0018,0.0063,0.0454,0.0944,2.8557,2.9186,0.0006,6.0147
+max_reduce,16,5000,5,2,1279744,80000,0.0036,0.0147,0.1311,0.186,7.0892,7.4272,0.0006,15.147
+max_reduce,32,500,5,2,510976,16000,0.0045,0.0023,0.0257,0.0451,4.0341,4.1987,0.0008,8.3059
+max_reduce,32,1000,5,2,1022976,32000,0.0064,0.0067,0.0439,0.0748,8.1545,8.4885,0.0008,16.8164
+max_reduce,32,2000,5,2,2046976,64000,0.0069,0.0067,0.0793,0.151,16.8425,17.4785,0.0013,35.1039
+max_reduce,64,200,5,2,815104,12800,0.0297,0.0025,0.0322,0.0434,9.1157,9.1911,0.0018,18.3756
+max_reduce,64,500,5,2,2043904,32000,0.0483,0.0029,0.0804,0.1053,27.1055,28.3244,0.0024,55.0267
+max_reduce,64,1000,5,2,4091904,64000,0.0318,0.0042,0.1039,0.1689,62.8022,63.4727,0.0016,120.995
+max_reduce,128,100,5,2,1622016,12800,0.071,0.0007,0.0337,0.0426,21.6621,21.5886,0.0024,43.8249
+max_reduce,128,250,5,2,4079616,32000,0.0696,0.0008,0.0513,0.0852,77.0032,61.7649,0.0023,137.852
+max_reduce,128,500,5,2,8175616,64000,0.0756,0.0031,0.085,0.1356,128.719,119.591,0.0025,243.712

benchmark-analysis/fb_contour_sweep_pairwise.csv

@@ -0,0 +1,23 @@
+mode,n,t,runs,warmup,recurrence_work,emission_work,transition_ms,obs_copy_ms,emission_ms,alloc_ms,forward_ms,backward_ms,reduction_ms,total_ms
+pairwise,2,1000,5,1,3996,2000,0.0001,0.0003,0.0004,0.0003,0.0343,0.0336,0.0001,0.0693
+pairwise,2,10000,5,1,39996,20000,0.0001,0.0024,0.0047,0.0023,0.3434,0.3354,0,0.6895
+pairwise,2,100000,5,1,399996,200000,0.001,0.1048,0.2501,0.4206,3.461,3.3926,0.0001,7.6391
+pairwise,2,1000000,5,1,3999996,2000000,0.0049,1.5373,2.8471,3.7466,34.7657,34.3781,0.0004,78.5542
+pairwise,4,1000,5,1,15984,4000,0.0003,0.0004,0.0101,0.0187,0.2189,0.2153,0.0001,0.4634
+pairwise,4,10000,5,1,159984,40000,0.0019,0.0122,0.0167,0.0218,3.4942,3.2695,0.0002,6.8535
+pairwise,4,100000,5,1,1599984,400000,0.0033,0.1415,0.6652,1.1502,29.2175,26.0248,0.0002,58.7034
+pairwise,8,1000,5,1,63936,8000,0.0005,0.0034,0.0159,0.0316,1.166,1.1765,0.0002,2.3957
+pairwise,8,5000,5,1,319936,40000,0.0016,0.0156,0.052,0.1019,5.8452,5.8658,0.0002,11.8913
+pairwise,8,10000,5,1,639936,80000,0.0022,0.0079,0.0197,0.204,11.6961,11.7406,0.0002,23.715
+pairwise,16,1000,5,1,255744,16000,0.0019,0.0042,0.0326,0.0477,5.3054,5.3313,0.0004,10.7288
+pairwise,16,2000,5,1,511744,32000,0.0033,0.0073,0.0434,0.0883,10.6612,10.8194,0.0005,21.7072
+pairwise,16,5000,5,1,1279744,80000,0.0051,0.0149,0.0966,0.2077,26.5814,26.6937,0.0005,53.6173
+pairwise,32,500,5,1,510976,16000,0.0047,0.0028,0.029,0.044,9.7704,9.8929,0.0006,19.7958
+pairwise,32,1000,5,1,1022976,32000,0.0058,0.0047,0.0453,0.0761,19.5781,19.7934,0.0007,39.505
+pairwise,32,2000,5,1,2046976,64000,0.0064,0.0065,0.0791,0.1424,39.3132,40.2802,0.0008,80.4737
+pairwise,64,200,5,1,815104,12800,0.0311,0.0022,0.0302,0.0409,14.4688,14.2692,0.0014,28.7968
+pairwise,64,500,5,1,2043904,32000,0.0293,0.002,0.0509,0.0823,37.0369,38.7809,0.0014,76.2688
+pairwise,64,1000,5,1,4091904,64000,0.0298,0.0036,0.0765,0.1626,70.9994,71.0655,0.0013,142.836
+pairwise,128,100,5,1,1622016,12800,0.0658,0.0008,0.0361,0.044,27.5451,27.7767,0.002,55.5736
+pairwise,128,250,5,1,4079616,32000,0.0637,0.0008,0.0164,0.0593,66.9222,67.2184,0.002,134.272
+pairwise,128,500,5,1,8175616,64000,0.0677,0.001,0.0482,0.0731,133.704,135.611,0.0023,269.665

benchmark-analysis/focus-n2-8-catalina-ivybridge/focused_max_reduce_n2_8.csv

@@ -0,0 +1,43 @@
+mode,n,t,runs,warmup,fb_total_ms,forward_ms,backward_ms
+max_reduce,2,500,5,2,0.3,0.114,0.13
+max_reduce,2,1000,5,2,0.637,0.233,0.252
+max_reduce,2,2000,5,2,1.217,0.467,0.527
+max_reduce,2,5000,5,2,3.092,1.191,1.347
+max_reduce,2,10000,5,2,6.021,2.482,2.443
+max_reduce,2,100000,5,2,63.802,26.135,26.283
+max_reduce,3,500,5,2,0.589,0.234,0.258
+max_reduce,3,1000,5,2,1.107,0.455,0.501
+max_reduce,3,2000,5,2,2.289,0.94,1.034
+max_reduce,3,5000,5,2,5.686,2.326,2.592
+max_reduce,3,10000,5,2,12.027,4.796,5.664
+max_reduce,3,100000,5,2,120.989,49.523,55.446
+max_reduce,4,500,5,2,0.884,0.372,0.416
+max_reduce,4,1000,5,2,1.879,0.792,0.877
+max_reduce,4,2000,5,2,3.776,1.606,1.767
+max_reduce,4,5000,5,2,9.505,4.148,4.381
+max_reduce,4,10000,5,2,19.404,8.402,8.949
+max_reduce,4,100000,5,2,201.829,84.693,96.849
+max_reduce,5,500,5,2,1.317,0.568,0.632
+max_reduce,5,1000,5,2,2.775,1.196,1.337
+max_reduce,5,2000,5,2,5.672,2.391,2.801
+max_reduce,5,5000,5,2,13.83,5.923,6.682
+max_reduce,5,10000,5,2,29.043,12.056,14.445
+max_reduce,5,100000,5,2,291.988,124.124,142.458
+max_reduce,6,500,5,2,1.933,0.836,0.951
+max_reduce,6,1000,5,2,4.947,2.178,2.407
+max_reduce,6,2000,5,2,8.027,3.517,3.891
+max_reduce,6,5000,5,2,19.475,8.439,9.547
+max_reduce,6,10000,5,2,39.116,17.027,19.181
+max_reduce,6,100000,5,2,410.151,176.87,203.052
+max_reduce,7,500,5,2,2.623,1.146,1.304
+max_reduce,7,1000,5,2,5.839,2.317,3.179
+max_reduce,7,2000,5,2,10.765,4.824,5.204
+max_reduce,7,5000,5,2,25.732,11.46,12.566
+max_reduce,7,10000,5,2,53.622,23.214,27.048
+max_reduce,7,100000,5,2,548.109,240.248,271.739
+max_reduce,8,500,5,2,3.935,1.592,2.096
+max_reduce,8,1000,5,2,7.416,3.137,3.887
+max_reduce,8,2000,5,2,13.338,5.863,6.718
+max_reduce,8,5000,5,2,35.927,14.932,19.053
+max_reduce,8,10000,5,2,67.716,29.651,34.379
+max_reduce,8,100000,5,2,707.026,309.823,357.473