\[\left(\left(\log \left(x + y\right) + \log z\right) - t\right) + \left(a - 0.5\right) \cdot \log t\]

↓

\[\log \left(x + y\right) + \left((\left(a - 0.5\right) \cdot \left(\log t\right) + \left(\log z\right))_* - t\right)\]

Error

The X axis uses an exponential scale — Bits error versus `x`

Derivation

Initial program 0.3
\[\left(\left(\log \left(x + y\right) + \log z\right) - t\right) + \left(a - 0.5\right) \cdot \log t\]
Initial simplification0.3
\[\leadsto \log \left(y + x\right) + \left((\left(a - 0.5\right) \cdot \left(\log t\right) + \left(\log z\right))_* - t\right)\]
Taylor expanded around 0 0.3
\[\leadsto \log \left(y + x\right) + \color{blue}{\left(\left(a \cdot \log t + \log z\right) - \left(t + 0.5 \cdot \log t\right)\right)}\]
Simplified0.3
\[\leadsto \log \left(y + x\right) + \color{blue}{\left((\left(a - 0.5\right) \cdot \left(\log t\right) + \left(\log z\right))_* - t\right)}\]
Final simplification0.3
\[\leadsto \log \left(x + y\right) + \left((\left(a - 0.5\right) \cdot \left(\log t\right) + \left(\log z\right))_* - t\right)\]

Runtime

	Baseline	Herbie	Oracle	Span	%
Regimes	0.3	0.3	0.1	0.2	0%

herbie shell --seed 2018340 +o rules:numerics
(FPCore (x y z t a)
  :name "Numeric.SpecFunctions:logGammaL from math-functions-0.1.5.2"
  (+ (- (+ (log (+ x y)) (log z)) t) (* (- a 0.5) (log t))))