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

↓

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

Error

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

Derivation

Initial program 0.3
\[\left(\left(\log \left(x + y\right) + \log z\right) - t\right) + \left(a - 0.5\right) \cdot \log t\]
Using strategy rm
Applied add-sqr-sqrt0.3
\[\leadsto \left(\left(\log \color{blue}{\left(\sqrt{x + y} \cdot \sqrt{x + y}\right)} + \log z\right) - t\right) + \left(a - 0.5\right) \cdot \log t\]
Applied log-prod0.3
\[\leadsto \left(\left(\color{blue}{\left(\log \left(\sqrt{x + y}\right) + \log \left(\sqrt{x + y}\right)\right)} + \log z\right) - t\right) + \left(a - 0.5\right) \cdot \log t\]
Applied associate-+l+0.3
\[\leadsto \left(\color{blue}{\left(\log \left(\sqrt{x + y}\right) + \left(\log \left(\sqrt{x + y}\right) + \log z\right)\right)} - t\right) + \left(a - 0.5\right) \cdot \log t\]

Runtime

herbie shell --seed '#(1070609872 3456127585 2380521889 2328837196 1765472538 734540918)' 
(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))))