fix: stabilize Gamma PDF for finite in-range parameters

[day01]

Summary

• evaluate the non-trivial Gamma PDF through ln_pdf().exp() to avoid overflow/underflow in separately evaluated factors
• preserve exact boundary behavior for x = 0, x = +inf, and shape = 1
• use a compensated sum for ln(rate) + ln(x) before multiplying by shape
• add a regression test for Gamma(80, 1e-5).pdf(8e6)

Root cause

The previous formula evaluated rate.powf(shape) independently. For valid finite parameters this factor could underflow to zero while the remaining factors compensated it mathematically, producing NaN instead of a finite density.

TDD and independent numerical references

The regression case is Gamma(shape=80, rate=1e-5).pdf(8e6).

Implementation	Result	Relative error vs. mpmath	Outcome
statrs (this PR)	4.455666577034977970e-7	2.63e-14	correct
Boost.Math 1.90	4.455666577035076430e-7	4.22e-15	agrees
R 4.2.1 dgamma	4.455666577035096020e-7	1.81e-16	agrees
mpmath 1.4.1, 136-bit	4.45566657703509521452812298721e-7	reference	reference
SciPy 1.18.0	4.455666577035271250e-7	3.95e-14	agrees

Upstream statrs returns NaN; the regression test is red on upstream and green with this change. R and Boost.Math are more accurate for this input; this change is more accurate than SciPy.

Performance

Case	Upstream	This change	Difference
issue input: Gamma(80, 1e-5).pdf(8e6)	34.58 ns, returns NaN	30.86 ns, finite result	10.8% faster
typical finite input	37.74 ns	27.06 ns	28.3% faster
large-shape input	26.70 ns	27.32 ns	2.3% slower

Implementation	Execution mode	Median time per call	Time relative
statrs (this PR)	Rust release, default std features	26.05 ns	1.00x
Boost.Math 1.90	C++17, -O3 -march=native	27.54 ns	1.06x
R 4.2.1 dgamma	scalar R call	755 ns	29.0x
mpmath 1.4.1, 136-bit	scalar Python call	16.95 us	650.7x
SciPy 1.18.0 gamma.pdf	scalar Python call	21.91 us	841.0x

Related issue

Fixes #422 — Gamma::pdf returns NaN for ordinary, finite, in-range parameters.

Summary by CodeRabbit

• Bug Fixes
  • Improved Gamma distribution probability calculations for edge cases, including zero-valued inputs.
  • Fixed PDF evaluation when rate-related calculations underflow.
  • Enhanced numerical stability and consistency across Gamma PDF and log-PDF results.

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Codecov Report

✅ All modified and coverable lines are covered by tests.
✅ Project coverage is 95.40%. Comparing base (10cf6d6) to head (83172ea).

Additional details and impacted files

@@            Coverage Diff             @@
##             main     #442      +/-   ##
==========================================
+ Coverage   95.07%   95.40%   +0.32%     
==========================================
  Files          62       65       +3     
  Lines       14203    15095     +892     
==========================================
+ Hits        13504    14401     +897     
+ Misses        699      694       -5     

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📝 Walkthrough

Walkthrough

The Gamma distribution now computes non-unit-shape PDF values from ln_pdf. The logarithmic calculation handles x == 0 and uses compensated arithmetic. Regression tests cover underflowing rate powers and infinite parameters.

Changes

Gamma PDF stability

Layer / File(s)	Summary
Logarithmic density arithmetic src/distribution/gamma.rs	Gamma::ln_pdf handles x == 0 according to the shape and rate. It uses compensated arithmetic for the logarithmic shape/rate product.
PDF evaluation and regression coverage src/distribution/gamma.rs	Gamma::pdf uses ln_pdf(x).exp() for non-unit-shape finite inputs. Tests cover rate-power underflow, infinite shape, and infinite rate cases.

Estimated code review effort: 2 (Simple) | ~10 minutes

Merge Risk: 🟡 Moderate · up to 83172

The PR stabilizes finite in-range Gamma PDF calculations, but zero-input cases with individually infinite parameters still return finite or infinite values instead of the documented NaN result. Merge should wait for this bounded correctness issue to be fixed or explicitly accepted.

🚥 Pre-merge checks | ✅ 5

✅ Passed checks (5 passed)

Check name	Status	Explanation
Linked Issues check	✅ Passed	The changes address issue #422 by deriving finite PDF values from ln_pdf and adding regression coverage for the reported case.
Out of Scope Changes check	✅ Passed	The changes remain within scope and support Gamma PDF stabilization, boundary behavior, and regression testing.
Docstring Coverage	✅ Passed	Docstring coverage is 100.00% which is sufficient. The required threshold is 80.00%.
Description Check	✅ Passed	Check skipped - CodeRabbit’s high-level summary is enabled.
Title check	✅ Passed	The title clearly and concisely describes the main change: stabilizing Gamma PDF evaluation for finite, in-range parameters.

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Actionable comments posted: 1

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Inline comments:
In `@src/distribution/gamma.rs`:
- Around line 392-397: The x == 0.0 branch in the Gamma density methods must
return f64::NAN when either accepted parameter is infinite, before applying the
shape-based ±infinity logic. Update the relevant Gamma implementation and add
coverage for pdf(0.0) and ln_pdf(0.0) with infinite shape and infinite rate.

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[day01]

Correctness

shape	rate	R dgamma	SciPy	mpmath (50 dps)	statrs upstream (pre-PR)	statrs PR #442 (before fix)	statrs PR #442 (after fix)
∞	1.0	0.0	0.0	NaN¹	NaN	0.0 ✓	0.0 ✓
0.5	∞	NaN	NaN	ZeroDivisionError²	NaN	inf ✗	NaN ✓
2.0	∞	NaN	NaN	NaN	NaN	0.0 ✗	NaN ✓
1.0	∞	NaN	NaN	NaN	NaN	NaN ✓	NaN ✓

Performance (pdf(0.0), per call)

shape	rate	R	SciPy	mpmath (50 dps)	statrs upstream (pre-PR)	statrs PR #442 (before fix)	statrs PR #442 (after fix)
∞	1.0	0.73 µs	67.7 µs	11.1 µs	2.0 ns	3.3 ns	2.7 ns
0.5	∞	11.9 µs	36.7 µs	7.6 µs	9.9 ns	6.1 ns	3.7 ns
2.0	∞	11.7 µs	36.8 µs	16.8 µs	19.0 ns	4.3 ns	3.7 ns
1.0	∞	11.9 µs	36.6 µs	16.9 µs	3.7 ns	4.4 ns	3.7 ns

[day01]

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