Average Error: 2.0 → 1.1
Time: 2.6m
Precision: 64
Internal Precision: 128
\[\frac{x \cdot e^{\left(y \cdot \log z + \left(t - 1.0\right) \cdot \log a\right) - b}}{y}\]
↓
\[\frac{x}{\frac{\sqrt[3]{\sqrt[3]{y}} \cdot \sqrt[3]{\sqrt[3]{y} \cdot \sqrt[3]{y}}}{\sqrt[3]{e^{(\left(\log a\right) \cdot t + \left(\log z \cdot y - b\right))_* - \log a \cdot 1.0}}}} \cdot \frac{1}{\frac{\sqrt[3]{y} \cdot \sqrt[3]{y}}{\sqrt[3]{e^{(\left(\log a\right) \cdot t + \left(\log z \cdot y - b\right))_* - \log a \cdot 1.0}} \cdot \sqrt[3]{e^{(\left(\log a\right) \cdot t + \left(\log z \cdot y - b\right))_* - \log a \cdot 1.0}}}}\]

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. Taylor expanded around inf 2.0

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

  3. Simplified2.0

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

  5. Applied associate-/l*1.9

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

  7. Applied add-cube-cbrt1.9

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

  8. Applied add-cube-cbrt1.9

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

  9. Applied times-frac1.9

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

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

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

  11. Applied times-frac1.1

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

  13. Applied add-cube-cbrt1.1

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

  14. Applied cbrt-prod1.1

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

  15. Final simplification1.1

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

Reproduce

herbie shell --seed 2019090 +o rules:numerics
(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))