Average Error: 2.0 → 2.0
Time: 1.2m
Precision: 64
Internal Precision: 128
\[\frac{x \cdot e^{\left(y \cdot \log z + \left(t - 1.0\right) \cdot \log a\right) - b}}{y}\]
↓
\[\frac{\left(\left(\sqrt[3]{e^{\left(\log z \cdot y + \left(t - 1.0\right) \cdot \log a\right) - b}} \cdot \sqrt[3]{\sqrt[3]{{e}^{\left(\left(\log z \cdot y + \left(t - 1.0\right) \cdot \log a\right) - b\right)}} \cdot \left(\sqrt[3]{{e}^{\left(\left(\log z \cdot y + \left(t - 1.0\right) \cdot \log a\right) - b\right)}} \cdot \sqrt[3]{{e}^{\left(\left(\log z \cdot y + \left(t - 1.0\right) \cdot \log a\right) - b\right)}}\right)}\right) \cdot \sqrt[3]{{e}^{\left(\left(\log z \cdot y + \left(t - 1.0\right) \cdot \log a\right) - b\right)}}\right) \cdot x}{y}\]

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 2.0

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

  3. Applied add-cube-cbrt2.0

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

  5. Applied *-un-lft-identity2.0

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

  6. Applied exp-prod1.9

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

  7. Simplified1.9

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

  9. Applied *-un-lft-identity1.9

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

  10. Applied exp-prod2.0

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

  11. Simplified2.0

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

  13. Applied add-cube-cbrt2.0

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

  14. Final simplification2.0

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

Reproduce

herbie shell --seed 2019051 
(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))