Inert-CompBench: Dual Benchmarking Frameworks

• Inert-CompBench is a dual-purpose benchmarking framework that evaluates both floating-point interval arithmetic libraries and compressed IDM parameter spaces in particle physics.
• It rigorously tests interval libraries on enclosure guarantees and cross-platform consistency using large-scale, precisely generated arithmetic expressions.
• For the compressed IDM, the framework defines benchmark points with finely tuned mass splittings essential for collider analyses and dark matter phenomenology.

Inert-CompBench refers to two unrelated but prominent benchmarking frameworks in computational science: (1) a cross-platform benchmark for the evaluation of floating-point interval arithmetic libraries, and (2) a collection of representative parameter points and limits for the compressed Inert Doublet Model (IDM) in particle physics. Both usages share the “CompBench” abbreviation, denoting “Compressed Benchmark” or “Computational Benchmark,” but arise in distinct research domains. Below, each usage is described in detail, with particular attention to benchmark methodology, scope, and technical findings.

1. Motivation and Conceptual Foundations

The first incarnation of Inert-CompBench is as a rigorous, experimental test suite for floating-point interval arithmetic libraries. Interval arithmetic is the only general methodology that guarantees a computed enclosure of the true mathematical result across a chain of floating-point operations, regardless of rounding errors, order permutations, or hardware-specific behaviors. In practice, minor differences in hardware-supported rounding modes, compiler optimizations, or even evaluation order can cause interval libraries to silently lose enclosure guarantees or fail to detect empty intervals.

The second variant, relevant to particle physics, refers to the definition of benchmark points and phenomenologically viable parameter ranges for a compressed Inert Doublet Model. The “compression” here denotes small mass splittings among the inert Higgs sector, which has significant implications for collider signals and dark matter phenomenology. Benchmark points (“Inert-CompBench points”) serve as standardized references for experimental searches.

2. Benchmark Suite Structure and Coverage

The interval arithmetic Inert-CompBench suite consists of a large, systematically generated set of tests, constructed as follows:

• Test Expressions: Four basic arithmetic operations (+, –, ×, ÷); four transcendental functions (√, exp, sin, cos); ten randomly generated composite arithmetic expressions; ten composites with both arithmetic and transcendental operations; and 104 real-world expressions from the FPBench repository (domains: finance, physics, geometry).
• Input Generation: For each expression, a large set of random rational inputs is generated.
• Reference Results: Each test + input is converted into an exact rational interval query, of the form

$$n_\ell/d_\ell \leq f(n_1/d_1, n_2/d_2, \ldots) \leq n_u/d_u$$

and the enclosure is validated using arbitrary-precision evaluations (Mathematica + GMP).

• Architectures and Compilers: Tests are run across x86-64 Windows, Intel macOS, Apple M1, and Linux, with MSVC, Clang, and GCC.
• Scale: The suite involves $O(10^3)$ expressions × $O(10^6)$ random inputs, yielding a robust but necessarily non-formal test of correctness, performance, and cross-platform consistency (Tang et al., 2021).

For the compressed IDM, Inert-CompBench comprises:

• Model Definitions: The most general $Z_2$-symmetric two-Higgs-doublet scalar potential, with explicit mass formulas depending on $\mu_1^2,\mu_2^2$ and the quartics $\lambda_1 \ldots \lambda_5$.
• Compression Realizations: Small mass splittings, defined via the quartic couplings,

$$\Delta^0 \equiv m_{A^0}-m_{H^0} \simeq -(\lambda_5 v^2)/m_{H^0}, \qquad \Delta^\pm\equiv m_{H^\pm}-m_{H^0} \simeq -[(\lambda_4+\lambda_5)v^2]/(2 m_{H^0})$$

with regimes set by global symmetry limits ($\lambda_5 \to 0$, $\lambda_4,\lambda_5 \to 0$).

• Benchmark Points: Four explicit $$(m_{H^0}, \Delta^0, \Delta^\pm, \lambda_L, \Omega h^2/\Omega_{\text{obs}}, \sigma_{\text{SI}})$$ tuples, representing parameter-space corners for collider and dark matter analyses (Blinov et al., 2015).

3. Methodology and Operational Definitions

Interval Arithmetic

• Operation Definitions: For two intervals $O(10^3)$0, elementary operations use outward rounding:
  • Addition: $O(10^3)$1
  • Multiplication: $O(10^3)$2
  • Transcendentals (e.g., $O(10^3)$3): computed via range reduction + Taylor expansions, rounding outward.
• Correctness: No library may produce an interval excluding the exact result, nor empty intervals (lower > upper). Measured are:
  • Interval width: $O(10^3)$4 as a percent of the true result.
  • Speed: Wall-clock time for $O(10^3)$5 random inputs × $O(10^3)$6 repeated evaluations per expression.
  • Result consistency: Bitwise identity of outputs across architectures and compilers.
  • Portability: Out-of-the-box buildability on all target platforms.

Compressed IDM

• Scalar Potential: The $O(10^3)$7-symmetric potential admits only one doublet ($O(10^3)$8) acquiring a VEV, such that the inert doublet remains “dark.”
• Split Control: Adjustable quartic couplings $O(10^3)$9 and $O(10^6)$0 allow arbitrarily small $O(10^6)$1, with technically natural suppression via global $O(10^6)$2 symmetry. RG running typically forces $O(10^6)$3.
• Collider Safety: $O(10^6)$4 (to avoid direct detection bounds), $O(10^6)$5 (for operationally soft but promptly decaying $O(10^6)$6).
• Observable Definitions: $O(10^6)$7 controls $O(10^6)$8–inert coupling; relic density $O(10^6)$9 calculated including coannihilation; direct detection cross section

$Z_2$0

4. Benchmark Results and Reference Comparison

Interval Arithmetic Library Results

Results across five libraries and multiple modes (see table):

Library	Arith.	Trans.	Composite	Size	Speed	Consis.	Portable
Boost.Interval	✓	✗	✗	1	6	✗	✓
filib	✓	✓	✓	2	3	✓	✓
filib++ native	✓	✓	✗	1	4	✗	✗
filib++ multiplicative	✓	✓	✓	3	1	✓	✗
filib++ pred-succ	✓	✓	✓	2	2	✓	✗
BIAS/PROFIL	✓	✓	✗	1	5	✗	✗

Key findings:

• Boost.Interval is the most portable and API-friendly but fails many transcendental and composite tests.
• Hardware rounding, as in Boost and filib++ native, achieves tighter intervals but often loses enclosure guarantees or portability.
• filib++ multiplicative mode gives superior speed and correctness on all tested expressions but lacks cross-platform portability.
• Only filib and filib++ software modes achieve bitwise-identical outputs across all tested systems.
• BIAS/PROFIL is unmaintained and performs poorly on modern systems (Tang et al., 2021).

Compressed IDM Benchmark Points

$Z_2$1 (GeV)	$Z_2$2 (MeV)	$Z_2$3 (GeV)	$Z_2$4	$Z_2$5	$Z_2$6 (cm$Z_2$7)
60	0.1	1	0.01	0.02	$Z_2$8
80	0.1	5	0.03	0.05	$Z_2$9
100	0.1	10	0.05	0.12	$\mu_1^2,\mu_2^2$0
535	negligible	10	0.00	1.00	$\mu_1^2,\mu_2^2$1

These points include the canonical low-mass, moderately split, and pure-gauge scenarios relevant to LHC, LEP, direct, and indirect dark matter searches (Blinov et al., 2015).

5. Guidelines, Limitations, and Recommendations

For interval computation, filib (bit-twiddling) is recommended for maximal correctness, reproducibility, and ease of build, with filib++ (multiplicative) as the fastest, still correct option if portability can be sacrificed. Boost.Interval should only be used for prototyping; BIAS is not recommended for new work. Limitations include finite coverage, lack of formal proof of correctness, and absence of higher-precision or non-C/C++ support.

For compressed IDM, current LEP and LHC bounds restrict $\mu_1^2,\mu_2^2$2 GeV, $\mu_1^2,\mu_2^2$3 GeV for substantial splits; softer splittings evade some LEP limits. Direct detection bounds from LUX restrict $\mu_1^2,\mu_2^2$4, and inelastic Z scattering requires $\mu_1^2,\mu_2^2$5 keV. The benchmarked models form the basis of current and future collider and dark matter experiment sensitivity projections (Blinov et al., 2015).

6. Future Directions

For interval arithmetic, extensions should add broader operation coverage (e.g., atan2, log, pow), higher-precision types (quad, double-double), and multi-language support (Julia, Rust), with automated integration in CI systems to capture library regressions and new hardware platforms (e.g., RISC-V, GPUs).

For compressed IDM benchmarks, natural evolutions include adding further symmetry-breaking deformations, extending to multi-inert models, and systematic scanning of cosmological and indirect detection constraints. As experiments increase sensitivity (HL-LHC, FCC-hh, ILC500, next-gen direct detection), these points will require updating.

Comprehensive code, data, and evaluation scripts for the Inert-CompBench suite are publicly available (Tang et al., 2021).