Average Error: 0.3 → 0.3
Time: 48.3s
Precision: 64
Internal Precision: 128
\[\left(\left(\log \left(x + y\right) + \log z\right) - t\right) + \left(a - 0.5\right) \cdot \log t\]
↓
\[\left(\left(\left(\log z - t\right) + \log t \cdot \left(a - 0.5\right)\right) + \log \left(\sqrt[3]{y + x}\right)\right) + \log \left({\left(y + x\right)}^{\frac{2}{3}}\right)\]

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 associate--l+0.3

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

  4. Applied associate-+l+0.2

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

  6. Applied add-cube-cbrt0.2

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

  7. Applied log-prod0.3

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

  8. Applied associate-+l+0.3

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

  10. Applied pow1/30.3

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

  11. Applied pow1/30.3

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

  12. Applied pow-prod-up0.3

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

  13. Simplified0.3

    \[\leadsto \log \left({\left(x + y\right)}^{\color{blue}{\frac{2}{3}}}\right) + \left(\log \left(\sqrt[3]{x + y}\right) + \left(\left(\log z - t\right) + \left(a - 0.5\right) \cdot \log t\right)\right)\]

  14. Final simplification0.3

    \[\leadsto \left(\left(\left(\log z - t\right) + \log t \cdot \left(a - 0.5\right)\right) + \log \left(\sqrt[3]{y + x}\right)\right) + \log \left({\left(y + x\right)}^{\frac{2}{3}}\right)\]

Reproduce

herbie shell --seed 2019050 
(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))))