Average Error: 0.3 → 0.3
Time: 43.6s
Precision: 64
Internal Precision: 128
\[\left(\left(\log \left(x + y\right) + \log z\right) - t\right) + \left(a - 0.5\right) \cdot \log t\]
↓
\[\left((\left(a - 0.5\right) \cdot \left(\log t\right) + \left(\log z\right))_* + (\left(-\sqrt[3]{t}\right) \cdot \left(\sqrt[3]{t} \cdot \sqrt[3]{t}\right) + \left(\sqrt[3]{t} \cdot \left(\sqrt[3]{t} \cdot \sqrt[3]{t}\right)\right))_*\right) + \left(\log \left(y + x\right) - t\right)\]

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Bits error versus a

Derivation

  1. Initial program 0.3

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

  2. Simplified0.3

    \[\leadsto \color{blue}{\left(\log \left(y + x\right) - t\right) + (\left(a - 0.5\right) \cdot \left(\log t\right) + \left(\log z\right))_*}\]
  3. Using strategy rm

  4. Applied add-cube-cbrt0.7

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

  5. Applied add-cube-cbrt0.9

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

  6. Applied prod-diff0.9

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

  7. Applied associate-+l+0.9

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

  8. Simplified0.3

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

  9. Final simplification0.3

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

Reproduce

herbie shell --seed 2019089 +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))))