Average Error: 0.3 → 0.3
Time: 2.7m
Precision: 64
Internal Precision: 576
\[\left(\left(\log \left(x + y\right) + \log z\right) - t\right) + \left(a - 0.5\right) \cdot \log t\]
↓
\[\log t \cdot \left(a - 0.5\right) + \left(\left(\log \left({\left(x + y\right)}^{\left(\frac{1}{3} + \frac{1}{3}\right)}\right) + \left(\log z + \log \left(\sqrt[3]{x + y}\right)\right)\right) - t\right)\]

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Bits error versus a

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Your Program's Arguments

Results

Enter valid numbers for all inputs

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

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

  4. Applied log-prod0.3

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

  5. Applied associate-+l+0.3

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

  6. Simplified0.3

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

  7. Final simplification0.3

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

Runtime

Time bar (total: 2.7m)Debug logProfile

herbie shell --seed 2018214 +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))))