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

↓

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

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 simplify0.3
\[\leadsto \color{blue}{\log \left(y + x\right) - \left(t - (\left(a - 0.5\right) \cdot \left(\log t\right) + \left(\log z\right))_*\right)}\]
Using strategy rm
Applied add-sqr-sqrt0.3
\[\leadsto \log \color{blue}{\left(\sqrt{y + x} \cdot \sqrt{y + x}\right)} - \left(t - (\left(a - 0.5\right) \cdot \left(\log t\right) + \left(\log z\right))_*\right)\]
Applied log-prod0.3
\[\leadsto \color{blue}{\left(\log \left(\sqrt{y + x}\right) + \log \left(\sqrt{y + x}\right)\right)} - \left(t - (\left(a - 0.5\right) \cdot \left(\log t\right) + \left(\log z\right))_*\right)\]
Applied associate--l+0.3
\[\leadsto \color{blue}{\log \left(\sqrt{y + x}\right) + \left(\log \left(\sqrt{y + x}\right) - \left(t - (\left(a - 0.5\right) \cdot \left(\log t\right) + \left(\log z\right))_*\right)\right)}\]

Runtime

herbie shell --seed '#(1070386091 2509006183 1430610344 1025408621 36622005 1425925650)' +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))))