Average Error: 2.0 → 1.0
Time: 39.1s
Precision: 64
\[\frac{x \cdot e^{\left(y \cdot \log z + \left(t - 1.0\right) \cdot \log a\right) - b}}{y}\]
↓
\[\frac{1}{\frac{\sqrt[3]{y} \cdot \sqrt[3]{y}}{\sqrt[3]{e^{\mathsf{fma}\left(t - 1.0, \log a, \log z \cdot y\right) - b}} \cdot \sqrt[3]{e^{\mathsf{fma}\left(t - 1.0, \log a, \log z \cdot y\right) - b}}}} \cdot \frac{x}{\frac{\sqrt[3]{y}}{\sqrt[3]{e^{\mathsf{fma}\left(t - 1.0, \log a, \log z \cdot y\right) - b}}}}\]
\frac{x \cdot e^{\left(y \cdot \log z + \left(t - 1.0\right) \cdot \log a\right) - b}}{y}
↓
\frac{1}{\frac{\sqrt[3]{y} \cdot \sqrt[3]{y}}{\sqrt[3]{e^{\mathsf{fma}\left(t - 1.0, \log a, \log z \cdot y\right) - b}} \cdot \sqrt[3]{e^{\mathsf{fma}\left(t - 1.0, \log a, \log z \cdot y\right) - b}}}} \cdot \frac{x}{\frac{\sqrt[3]{y}}{\sqrt[3]{e^{\mathsf{fma}\left(t - 1.0, \log a, \log z \cdot y\right) - b}}}}
double f(double x, double y, double z, double t, double a, double b) {
        double r1922769 = x;
        double r1922770 = y;
        double r1922771 = z;
        double r1922772 = log(r1922771);
        double r1922773 = r1922770 * r1922772;
        double r1922774 = t;
        double r1922775 = 1.0;
        double r1922776 = r1922774 - r1922775;
        double r1922777 = a;
        double r1922778 = log(r1922777);
        double r1922779 = r1922776 * r1922778;
        double r1922780 = r1922773 + r1922779;
        double r1922781 = b;
        double r1922782 = r1922780 - r1922781;
        double r1922783 = exp(r1922782);
        double r1922784 = r1922769 * r1922783;
        double r1922785 = r1922784 / r1922770;
        return r1922785;
}
↓
double f(double x, double y, double z, double t, double a, double b) {
        double r1922786 = 1.0;
        double r1922787 = y;
        double r1922788 = cbrt(r1922787);
        double r1922789 = r1922788 * r1922788;
        double r1922790 = t;
        double r1922791 = 1.0;
        double r1922792 = r1922790 - r1922791;
        double r1922793 = a;
        double r1922794 = log(r1922793);
        double r1922795 = z;
        double r1922796 = log(r1922795);
        double r1922797 = r1922796 * r1922787;
        double r1922798 = fma(r1922792, r1922794, r1922797);
        double r1922799 = b;
        double r1922800 = r1922798 - r1922799;
        double r1922801 = exp(r1922800);
        double r1922802 = cbrt(r1922801);
        double r1922803 = r1922802 * r1922802;
        double r1922804 = r1922789 / r1922803;
        double r1922805 = r1922786 / r1922804;
        double r1922806 = x;
        double r1922807 = r1922788 / r1922802;
        double r1922808 = r1922806 / r1922807;
        double r1922809 = r1922805 * r1922808;
        return r1922809;
}

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 associate-/l*1.9

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

  4. Simplified1.9

    \[\leadsto \frac{x}{\color{blue}{\frac{y}{e^{\mathsf{fma}\left(t - 1.0, \log a, \log z \cdot y\right) - b}}}}\]
  5. Using strategy rm

  6. Applied add-cube-cbrt1.9

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

  7. Applied add-cube-cbrt1.9

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

  8. Applied times-frac1.9

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

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

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

  10. Applied times-frac1.0

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

  11. Final simplification1.0

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

Reproduce

herbie shell --seed 2019153 +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))