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

↓

\[(\left(a - 0.5\right) \cdot \left(\log t\right) + \left((\left(\sqrt[3]{\log \left(y + x\right)} \cdot \sqrt[3]{\log \left(y + x\right)}\right) \cdot \left(\sqrt[3]{\log \left(y + x\right)}\right) + \left(\log z - t\right))_*\right))_*\]

Error

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

Derivation

Initial program 0.3
\[\left(\left(\log \left(x + y\right) + \log z\right) - t\right) + \left(a - 0.5\right) \cdot \log t\]
Applied simplify0.2
\[\leadsto \color{blue}{(\left(a - 0.5\right) \cdot \left(\log t\right) + \left(\log \left(y + x\right) + \left(\log z - t\right)\right))_*}\]
Using strategy rm
Applied add-cube-cbrt0.4
\[\leadsto (\left(a - 0.5\right) \cdot \left(\log t\right) + \left(\color{blue}{\left(\sqrt[3]{\log \left(y + x\right)} \cdot \sqrt[3]{\log \left(y + x\right)}\right) \cdot \sqrt[3]{\log \left(y + x\right)}} + \left(\log z - t\right)\right))_*\]
Applied fma-def0.4
\[\leadsto (\left(a - 0.5\right) \cdot \left(\log t\right) + \color{blue}{\left((\left(\sqrt[3]{\log \left(y + x\right)} \cdot \sqrt[3]{\log \left(y + x\right)}\right) \cdot \left(\sqrt[3]{\log \left(y + x\right)}\right) + \left(\log z - t\right))_*\right)})_*\]

Runtime

herbie shell --seed '#(1070100504 930361288 1279167582 284574201 1450237281 2578255382)' +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))))