Average Error: 0.3 → 0.3
Time: 2.3m
Precision: 64
Internal Precision: 576
\[\left(\left(\log \left(x + y\right) + \log z\right) - t\right) + \left(a - 0.5\right) \cdot \log t\]
↓
\[\left(\left(\log \left(x + y\right) + (\left(a - 0.5\right) \cdot \left(\log t\right) + \left(\log z\right))_*\right) - t\right) + 0\]

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. Applied simplify0.3

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

  4. Applied add-cube-cbrt1.3

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

  5. Applied add-cube-cbrt1.4

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

  6. Applied prod-diff1.4

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

  7. Applied simplify0.3

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

  8. Applied simplify0.3

    \[\leadsto \left(\left(\log \left(x + y\right) + (\left(a - 0.5\right) \cdot \left(\log t\right) + \left(\log z\right))_*\right) - t\right) + \color{blue}{0}\]

Runtime

Time bar (total: 2.3m)Debug logProfile

herbie shell --seed 2018193 +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))))