Average Error: 1.8 → 1.5
Time: 3.6m
Precision: 64
Internal Precision: 128
\[\frac{x \cdot e^{\left(y \cdot \log z + \left(t - 1.0\right) \cdot \log a\right) - b}}{y}\]
↓
\[\left(\frac{\sqrt{e^{\left(\log a \cdot \left(t - 1.0\right) + y \cdot \log z\right) - b}} \cdot \sqrt[3]{x}}{\sqrt[3]{y} \cdot \sqrt[3]{y}} \cdot \frac{\sqrt{e^{\left(\log a \cdot \left(t - 1.0\right) + y \cdot \log z\right) - b}}}{\sqrt[3]{y}}\right) \cdot \left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right)\]

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 1.8

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

  3. Applied *-un-lft-identity1.8

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

  4. Applied add-cube-cbrt1.8

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

  5. Applied associate-*l*1.8

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

  6. Applied times-frac1.7

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

  7. Simplified1.7

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

  9. Applied add-cube-cbrt1.7

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

  10. Applied add-sqr-sqrt1.7

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

  11. Applied associate-*r*1.7

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

  12. Applied times-frac1.5

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

  13. Final simplification1.5

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

Reproduce

herbie shell --seed 2019088 
(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))