Average Error: 2.0 → 1.1
Time: 42.4s
Precision: 64
\[\frac{x \cdot e^{\left(y \cdot \log z + \left(t - 1\right) \cdot \log a\right) - b}}{y}\]
↓
\[\frac{1}{\frac{\sqrt[3]{y} \cdot \sqrt[3]{y}}{\sqrt[3]{e^{\mathsf{fma}\left(\log a, t - 1, \log z \cdot y - b\right)}} \cdot \sqrt[3]{e^{\mathsf{fma}\left(\log a, t - 1, \log z \cdot y - b\right)}}}} \cdot \frac{x}{\frac{\sqrt[3]{\sqrt[3]{y}} \cdot \left(\sqrt[3]{\sqrt[3]{y}} \cdot \sqrt[3]{\sqrt[3]{y}}\right)}{\sqrt[3]{e^{\mathsf{fma}\left(\log a, t - 1, \log z \cdot y - b\right)}}}}\]
\frac{x \cdot e^{\left(y \cdot \log z + \left(t - 1\right) \cdot \log a\right) - b}}{y}
↓
\frac{1}{\frac{\sqrt[3]{y} \cdot \sqrt[3]{y}}{\sqrt[3]{e^{\mathsf{fma}\left(\log a, t - 1, \log z \cdot y - b\right)}} \cdot \sqrt[3]{e^{\mathsf{fma}\left(\log a, t - 1, \log z \cdot y - b\right)}}}} \cdot \frac{x}{\frac{\sqrt[3]{\sqrt[3]{y}} \cdot \left(\sqrt[3]{\sqrt[3]{y}} \cdot \sqrt[3]{\sqrt[3]{y}}\right)}{\sqrt[3]{e^{\mathsf{fma}\left(\log a, t - 1, \log z \cdot y - b\right)}}}}
double f(double x, double y, double z, double t, double a, double b) {
        double r2823764 = x;
        double r2823765 = y;
        double r2823766 = z;
        double r2823767 = log(r2823766);
        double r2823768 = r2823765 * r2823767;
        double r2823769 = t;
        double r2823770 = 1.0;
        double r2823771 = r2823769 - r2823770;
        double r2823772 = a;
        double r2823773 = log(r2823772);
        double r2823774 = r2823771 * r2823773;
        double r2823775 = r2823768 + r2823774;
        double r2823776 = b;
        double r2823777 = r2823775 - r2823776;
        double r2823778 = exp(r2823777);
        double r2823779 = r2823764 * r2823778;
        double r2823780 = r2823779 / r2823765;
        return r2823780;
}
↓
double f(double x, double y, double z, double t, double a, double b) {
        double r2823781 = 1.0;
        double r2823782 = y;
        double r2823783 = cbrt(r2823782);
        double r2823784 = r2823783 * r2823783;
        double r2823785 = a;
        double r2823786 = log(r2823785);
        double r2823787 = t;
        double r2823788 = 1.0;
        double r2823789 = r2823787 - r2823788;
        double r2823790 = z;
        double r2823791 = log(r2823790);
        double r2823792 = r2823791 * r2823782;
        double r2823793 = b;
        double r2823794 = r2823792 - r2823793;
        double r2823795 = fma(r2823786, r2823789, r2823794);
        double r2823796 = exp(r2823795);
        double r2823797 = cbrt(r2823796);
        double r2823798 = r2823797 * r2823797;
        double r2823799 = r2823784 / r2823798;
        double r2823800 = r2823781 / r2823799;
        double r2823801 = x;
        double r2823802 = cbrt(r2823783);
        double r2823803 = r2823802 * r2823802;
        double r2823804 = r2823802 * r2823803;
        double r2823805 = r2823804 / r2823797;
        double r2823806 = r2823801 / r2823805;
        double r2823807 = r2823800 * r2823806;
        return r2823807;
}

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

    \[\frac{x \cdot e^{\left(y \cdot \log z + \left(t - 1\right) \cdot \log a\right) - b}}{y}\]
  2. Using strategy rm

  3. Applied associate-/l*1.9

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

  4. Simplified1.9

    \[\leadsto \frac{x}{\color{blue}{\frac{y}{e^{\mathsf{fma}\left(\log a, t - 1, \log z \cdot y - b\right)}}}}\]
  5. Using strategy rm

  6. Applied add-cube-cbrt1.9

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

  7. Applied add-cube-cbrt1.9

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

  8. Applied times-frac1.9

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

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

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

  10. Applied times-frac1.1

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

  12. Applied add-cube-cbrt1.1

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

  13. Final simplification1.1

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

Reproduce

herbie shell --seed 2019174 +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))