Average Error: 1.9 → 1.9
Time: 1.8m
Precision: 64
\[\frac{x \cdot e^{\left(y \cdot \log z + \left(t - 1.0\right) \cdot \log a\right) - b}}{y}\]
↓
\[\frac{\frac{x \cdot \left(\sqrt[3]{{e}^{\left(\left(\log a \cdot \left(t - 1.0\right) + \log z \cdot y\right) - b\right)}} \cdot \left(\sqrt[3]{{e}^{\left(\left(\log a \cdot \left(t - 1.0\right) + \log z \cdot y\right) - b\right)}} \cdot \sqrt[3]{e^{\left(\log a \cdot \left(t - 1.0\right) + \log z \cdot y\right) - b}}\right)\right)}{\sqrt[3]{y} \cdot \sqrt[3]{y}}}{\sqrt[3]{y}}\]
double f(double x, double y, double z, double t, double a, double b) {
        double r11184711 = x;
        double r11184712 = y;
        double r11184713 = z;
        double r11184714 = log(r11184713);
        double r11184715 = r11184712 * r11184714;
        double r11184716 = t;
        double r11184717 = 1.0;
        double r11184718 = r11184716 - r11184717;
        double r11184719 = a;
        double r11184720 = log(r11184719);
        double r11184721 = r11184718 * r11184720;
        double r11184722 = r11184715 + r11184721;
        double r11184723 = b;
        double r11184724 = r11184722 - r11184723;
        double r11184725 = exp(r11184724);
        double r11184726 = r11184711 * r11184725;
        double r11184727 = r11184726 / r11184712;
        return r11184727;
}
↓
double f(double x, double y, double z, double t, double a, double b) {
        double r11184728 = x;
        double r11184729 = exp(1.0);
        double r11184730 = a;
        double r11184731 = log(r11184730);
        double r11184732 = t;
        double r11184733 = 1.0;
        double r11184734 = r11184732 - r11184733;
        double r11184735 = r11184731 * r11184734;
        double r11184736 = z;
        double r11184737 = log(r11184736);
        double r11184738 = y;
        double r11184739 = r11184737 * r11184738;
        double r11184740 = r11184735 + r11184739;
        double r11184741 = b;
        double r11184742 = r11184740 - r11184741;
        double r11184743 = pow(r11184729, r11184742);
        double r11184744 = cbrt(r11184743);
        double r11184745 = exp(r11184742);
        double r11184746 = cbrt(r11184745);
        double r11184747 = r11184744 * r11184746;
        double r11184748 = r11184744 * r11184747;
        double r11184749 = r11184728 * r11184748;
        double r11184750 = cbrt(r11184738);
        double r11184751 = r11184750 * r11184750;
        double r11184752 = r11184749 / r11184751;
        double r11184753 = r11184752 / r11184750;
        return r11184753;
}


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

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

  6. Applied associate-/r*1.9

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

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

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

  9. Applied exp-prod1.9

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

  10. Simplified1.9

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

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

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

  13. Applied exp-prod1.9

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

  14. Simplified1.9

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

  15. Final simplification1.9

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

Reproduce

herbie shell --seed 2019101 +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))