Average Error: 1.9 → 1.9
Time: 54.3s
Precision: 64
Internal Precision: 128
\[\frac{x \cdot e^{\left(y \cdot \log z + \left(t - 1.0\right) \cdot \log a\right) - b}}{y}\]
↓
\[\sqrt[3]{\frac{x \cdot e^{\left(\log a \cdot \left(t - 1.0\right) + \log z \cdot y\right) - b}}{y}} \cdot \left(\sqrt[3]{\frac{x \cdot e^{\left(\log a \cdot \left(t - 1.0\right) + \log z \cdot y\right) - b}}{y}} \cdot \sqrt[3]{\frac{\left(\sqrt[3]{e^{\log z \cdot y + \left(\left(\log a \cdot t - b\right) - \log a \cdot 1.0\right)}} \cdot \left(\sqrt[3]{e^{\log z \cdot y + \left(\left(\log a \cdot t - b\right) - \log a \cdot 1.0\right)}} \cdot \sqrt[3]{e^{\log z \cdot y + \left(\left(\log a \cdot t - b\right) - \log a \cdot 1.0\right)}}\right)\right) \cdot x}{y}}\right)\]

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Bits error versus a

Bits error versus b

Derivation

  1. Initial program 1.9

    \[\frac{x \cdot e^{\left(y \cdot \log z + \left(t - 1.0\right) \cdot \log a\right) - b}}{y}\]
  2. Using strategy rm

  3. Applied add-cube-cbrt1.9

    \[\leadsto \color{blue}{\left(\sqrt[3]{\frac{x \cdot e^{\left(y \cdot \log z + \left(t - 1.0\right) \cdot \log a\right) - b}}{y}} \cdot \sqrt[3]{\frac{x \cdot e^{\left(y \cdot \log z + \left(t - 1.0\right) \cdot \log a\right) - b}}{y}}\right) \cdot \sqrt[3]{\frac{x \cdot e^{\left(y \cdot \log z + \left(t - 1.0\right) \cdot \log a\right) - b}}{y}}}\]

  4. Taylor expanded around inf 1.9

    \[\leadsto \left(\sqrt[3]{\frac{x \cdot e^{\left(y \cdot \log z + \left(t - 1.0\right) \cdot \log a\right) - b}}{y}} \cdot \sqrt[3]{\frac{x \cdot \color{blue}{e^{1.0 \cdot \log \left(\frac{1}{a}\right) - \left(\log \left(\frac{1}{z}\right) \cdot y + \left(b + t \cdot \log \left(\frac{1}{a}\right)\right)\right)}}}{y}}\right) \cdot \sqrt[3]{\frac{x \cdot e^{\left(y \cdot \log z + \left(t - 1.0\right) \cdot \log a\right) - b}}{y}}\]

  5. Simplified1.9

    \[\leadsto \left(\sqrt[3]{\frac{x \cdot e^{\left(y \cdot \log z + \left(t - 1.0\right) \cdot \log a\right) - b}}{y}} \cdot \sqrt[3]{\frac{x \cdot \color{blue}{e^{\log z \cdot y + \left(\left(t \cdot \log a - b\right) - 1.0 \cdot \log a\right)}}}{y}}\right) \cdot \sqrt[3]{\frac{x \cdot e^{\left(y \cdot \log z + \left(t - 1.0\right) \cdot \log a\right) - b}}{y}}\]
  6. Using strategy rm

  7. Applied add-cube-cbrt1.9

    \[\leadsto \left(\sqrt[3]{\frac{x \cdot e^{\left(y \cdot \log z + \left(t - 1.0\right) \cdot \log a\right) - b}}{y}} \cdot \sqrt[3]{\frac{x \cdot \color{blue}{\left(\left(\sqrt[3]{e^{\log z \cdot y + \left(\left(t \cdot \log a - b\right) - 1.0 \cdot \log a\right)}} \cdot \sqrt[3]{e^{\log z \cdot y + \left(\left(t \cdot \log a - b\right) - 1.0 \cdot \log a\right)}}\right) \cdot \sqrt[3]{e^{\log z \cdot y + \left(\left(t \cdot \log a - b\right) - 1.0 \cdot \log a\right)}}\right)}}{y}}\right) \cdot \sqrt[3]{\frac{x \cdot e^{\left(y \cdot \log z + \left(t - 1.0\right) \cdot \log a\right) - b}}{y}}\]

  8. Final simplification1.9

    \[\leadsto \sqrt[3]{\frac{x \cdot e^{\left(\log a \cdot \left(t - 1.0\right) + \log z \cdot y\right) - b}}{y}} \cdot \left(\sqrt[3]{\frac{x \cdot e^{\left(\log a \cdot \left(t - 1.0\right) + \log z \cdot y\right) - b}}{y}} \cdot \sqrt[3]{\frac{\left(\sqrt[3]{e^{\log z \cdot y + \left(\left(\log a \cdot t - b\right) - \log a \cdot 1.0\right)}} \cdot \left(\sqrt[3]{e^{\log z \cdot y + \left(\left(\log a \cdot t - b\right) - \log a \cdot 1.0\right)}} \cdot \sqrt[3]{e^{\log z \cdot y + \left(\left(\log a \cdot t - b\right) - \log a \cdot 1.0\right)}}\right)\right) \cdot x}{y}}\right)\]

Reproduce

herbie shell --seed 2019089 
(FPCore (x y z t a b)
  :name "Numeric.SpecFunctions:incompleteBetaWorker from math-functions-0.1.5.2"
  (/ (* x (exp (- (+ (* y (log z)) (* (- t 1.0) (log a))) b))) y))