Average Error: 2.0 → 1.6
Time: 2.7m
Precision: 64
\[\frac{x \cdot e^{\left(y \cdot \log z + \left(t - 1.0\right) \cdot \log a\right) - b}}{y}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -2.5605062893222377 \cdot 10^{-295}:\\ \;\;\;\;\frac{x \cdot \left(\sqrt[3]{\sqrt[3]{{\left(e^{\sqrt[3]{\left(\log a \cdot \left(t - 1.0\right) + y \cdot \log z\right) - b} \cdot \sqrt[3]{\left(\log a \cdot \left(t - 1.0\right) + y \cdot \log z\right) - b}}\right)}^{\left(\sqrt[3]{\left(\log a \cdot \left(t - 1.0\right) + y \cdot \log z\right) - b}\right)}} \cdot \left(\sqrt[3]{e^{\left(\log a \cdot \left(t - 1.0\right) + y \cdot \log z\right) - b}} \cdot \sqrt[3]{e^{\left(\log a \cdot \left(t - 1.0\right) + y \cdot \log z\right) - b}}\right)} \cdot \left(\sqrt[3]{e^{\left(\log a \cdot \left(t - 1.0\right) + y \cdot \log z\right) - b}} \cdot \sqrt[3]{e^{\left(\log a \cdot \left(t - 1.0\right) + y \cdot \log z\right) - b}}\right)\right)}{y}\\ \mathbf{elif}\;x \le 1.291514918677971 \cdot 10^{-42}:\\ \;\;\;\;\frac{x}{\sqrt[3]{y} \cdot \sqrt[3]{y}} \cdot \frac{e^{(\left(\log a\right) \cdot \left(t - 1.0\right) + \left(y \cdot \log z\right))_* - b}}{\sqrt[3]{y}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x \cdot \left(\sqrt[3]{\left(\sqrt[3]{e^{\left(\log a \cdot \left(t - 1.0\right) + y \cdot \log z\right) - b}} \cdot \sqrt[3]{e^{\left(\log a \cdot \left(t - 1.0\right) + y \cdot \log z\right) - b}}\right) \cdot \sqrt[3]{e^{\left(\log a \cdot \left(t - 1.0\right) + y \cdot \log z\right) - b}}} \cdot \left(\sqrt[3]{e^{\left(\log a \cdot \left(t - 1.0\right) + y \cdot \log z\right) - b}} \cdot \sqrt[3]{e^{\left(\log a \cdot \left(t - 1.0\right) + y \cdot \log z\right) - b}}\right)\right)}{\sqrt[3]{y} \cdot \sqrt[3]{y}}}{\sqrt[3]{y}}\\ \end{array}\]

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. Split input into 3 regimes

  2. if x < -2.5605062893222377e-295

    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

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

    5. Applied add-cube-cbrt1.9

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

    7. Applied add-cube-cbrt2.0

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

    8. Applied exp-prod2.0

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

    if -2.5605062893222377e-295 < x < 1.291514918677971e-42

    1. Initial program 3.4

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

    3. Applied add-cube-cbrt3.4

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

    5. Applied add-cube-cbrt3.4

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

    6. Applied times-frac1.7

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

    7. Simplified1.7

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

    if 1.291514918677971e-42 < x

    1. Initial program 1.0

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

    3. Applied add-cube-cbrt1.0

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

    5. Applied add-cube-cbrt1.0

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

    7. Applied add-cube-cbrt1.0

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

    8. Applied associate-/r*1.0

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

  4. Final simplification1.6

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -2.5605062893222377 \cdot 10^{-295}:\\ \;\;\;\;\frac{x \cdot \left(\sqrt[3]{\sqrt[3]{{\left(e^{\sqrt[3]{\left(\log a \cdot \left(t - 1.0\right) + y \cdot \log z\right) - b} \cdot \sqrt[3]{\left(\log a \cdot \left(t - 1.0\right) + y \cdot \log z\right) - b}}\right)}^{\left(\sqrt[3]{\left(\log a \cdot \left(t - 1.0\right) + y \cdot \log z\right) - b}\right)}} \cdot \left(\sqrt[3]{e^{\left(\log a \cdot \left(t - 1.0\right) + y \cdot \log z\right) - b}} \cdot \sqrt[3]{e^{\left(\log a \cdot \left(t - 1.0\right) + y \cdot \log z\right) - b}}\right)} \cdot \left(\sqrt[3]{e^{\left(\log a \cdot \left(t - 1.0\right) + y \cdot \log z\right) - b}} \cdot \sqrt[3]{e^{\left(\log a \cdot \left(t - 1.0\right) + y \cdot \log z\right) - b}}\right)\right)}{y}\\ \mathbf{elif}\;x \le 1.291514918677971 \cdot 10^{-42}:\\ \;\;\;\;\frac{x}{\sqrt[3]{y} \cdot \sqrt[3]{y}} \cdot \frac{e^{(\left(\log a\right) \cdot \left(t - 1.0\right) + \left(y \cdot \log z\right))_* - b}}{\sqrt[3]{y}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x \cdot \left(\sqrt[3]{\left(\sqrt[3]{e^{\left(\log a \cdot \left(t - 1.0\right) + y \cdot \log z\right) - b}} \cdot \sqrt[3]{e^{\left(\log a \cdot \left(t - 1.0\right) + y \cdot \log z\right) - b}}\right) \cdot \sqrt[3]{e^{\left(\log a \cdot \left(t - 1.0\right) + y \cdot \log z\right) - b}}} \cdot \left(\sqrt[3]{e^{\left(\log a \cdot \left(t - 1.0\right) + y \cdot \log z\right) - b}} \cdot \sqrt[3]{e^{\left(\log a \cdot \left(t - 1.0\right) + y \cdot \log z\right) - b}}\right)\right)}{\sqrt[3]{y} \cdot \sqrt[3]{y}}}{\sqrt[3]{y}}\\ \end{array}\]

Reproduce

herbie shell --seed 2019100 +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))