Average Error: 0.3 → 0.3
Time: 51.0s
Precision: 64
Internal Precision: 128
\[\left(\left(\log \left(x + y\right) + \log z\right) - t\right) + \left(a - 0.5\right) \cdot \log t\]
↓
\[\left(\left(\log \left(\sqrt[3]{z}\right) - t\right) + \left(a - 0.5\right) \cdot \log t\right) + \left(\log \left(\sqrt[3]{z} \cdot \sqrt[3]{\sqrt[3]{z}}\right) + \left(\log \left(y + x\right) + \log \left(\sqrt[3]{\sqrt[3]{z}} \cdot \sqrt[3]{\sqrt[3]{z}}\right)\right)\right)\]

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. Using strategy rm

  3. Applied add-cube-cbrt0.3

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

  4. Applied log-prod0.3

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

  5. Applied associate-+r+0.3

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

  6. Applied associate--l+0.3

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

  7. Applied associate-+l+0.3

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

  9. Applied add-cube-cbrt0.3

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

  10. Applied associate-*l*0.3

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

  11. Applied log-prod0.3

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

  12. Applied associate-+r+0.3

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

  13. Final simplification0.3

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

Reproduce

herbie shell --seed 2019088 +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))))