Average Error: 0.3 → 0.3
Time: 41.8s
Precision: 64
Internal Precision: 128
\[\left(\left(\log \left(x + y\right) + \log z\right) - t\right) + \left(a - 0.5\right) \cdot \log t\]
↓
\[\log z + \left(\left(a - 0.5\right) \cdot \log t + \left(\log \left(x + y\right) - t\right)\right)\]

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. 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 fma-udef0.3

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

  5. Applied associate-+r+0.3

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

  7. Applied add-cube-cbrt0.7

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

  8. Applied associate-*r*0.7

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

  10. Applied pow1/331.7

    \[\leadsto \left(\left(\log \left(y + x\right) - t\right) + \left(\left(a - 0.5\right) \cdot \left(\sqrt[3]{\log t} \cdot \color{blue}{{\left(\log t\right)}^{\frac{1}{3}}}\right)\right) \cdot \sqrt[3]{\log t}\right) + \log z\]

  11. Applied pow1/331.7

    \[\leadsto \left(\left(\log \left(y + x\right) - t\right) + \left(\left(a - 0.5\right) \cdot \left(\color{blue}{{\left(\log t\right)}^{\frac{1}{3}}} \cdot {\left(\log t\right)}^{\frac{1}{3}}\right)\right) \cdot \sqrt[3]{\log t}\right) + \log z\]

  12. Applied pow-prod-down0.6

    \[\leadsto \left(\left(\log \left(y + x\right) - t\right) + \left(\left(a - 0.5\right) \cdot \color{blue}{{\left(\log t \cdot \log t\right)}^{\frac{1}{3}}}\right) \cdot \sqrt[3]{\log t}\right) + \log z\]

  13. Taylor expanded around 0 0.3

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

  14. Simplified0.3

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

  15. Final simplification0.3

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

Reproduce

herbie shell --seed 2019053 +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))))