Average Error: 2.0 → 1.1
Time: 37.3s
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 r18967839 = x;
        double r18967840 = y;
        double r18967841 = z;
        double r18967842 = log(r18967841);
        double r18967843 = r18967840 * r18967842;
        double r18967844 = t;
        double r18967845 = 1.0;
        double r18967846 = r18967844 - r18967845;
        double r18967847 = a;
        double r18967848 = log(r18967847);
        double r18967849 = r18967846 * r18967848;
        double r18967850 = r18967843 + r18967849;
        double r18967851 = b;
        double r18967852 = r18967850 - r18967851;
        double r18967853 = exp(r18967852);
        double r18967854 = r18967839 * r18967853;
        double r18967855 = r18967854 / r18967840;
        return r18967855;
}
↓
double f(double x, double y, double z, double t, double a, double b) {
        double r18967856 = 1.0;
        double r18967857 = y;
        double r18967858 = cbrt(r18967857);
        double r18967859 = r18967858 * r18967858;
        double r18967860 = a;
        double r18967861 = log(r18967860);
        double r18967862 = t;
        double r18967863 = 1.0;
        double r18967864 = r18967862 - r18967863;
        double r18967865 = z;
        double r18967866 = log(r18967865);
        double r18967867 = r18967866 * r18967857;
        double r18967868 = b;
        double r18967869 = r18967867 - r18967868;
        double r18967870 = fma(r18967861, r18967864, r18967869);
        double r18967871 = exp(r18967870);
        double r18967872 = cbrt(r18967871);
        double r18967873 = r18967872 * r18967872;
        double r18967874 = r18967859 / r18967873;
        double r18967875 = r18967856 / r18967874;
        double r18967876 = x;
        double r18967877 = cbrt(r18967858);
        double r18967878 = r18967877 * r18967877;
        double r18967879 = r18967877 * r18967878;
        double r18967880 = r18967879 / r18967872;
        double r18967881 = r18967876 / r18967880;
        double r18967882 = r18967875 * r18967881;
        return r18967882;
}

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

Target

Original2.0
Target10.8
Herbie1.1
\[\begin{array}{l} \mathbf{if}\;t \lt -0.8845848504127471478852839936735108494759:\\ \;\;\;\;\frac{x \cdot \frac{{a}^{\left(t - 1\right)}}{y}}{\left(b + 1\right) - y \cdot \log z}\\ \mathbf{elif}\;t \lt 852031.228837407310493290424346923828125:\\ \;\;\;\;\frac{\frac{x}{y} \cdot {a}^{\left(t - 1\right)}}{e^{b - \log z \cdot y}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot \frac{{a}^{\left(t - 1\right)}}{y}}{\left(b + 1\right) - y \cdot \log z}\\ \end{array}\]

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, A"

  :herbie-target
  (if (< t -0.8845848504127471) (/ (* x (/ (pow a (- t 1.0)) y)) (- (+ b 1.0) (* y (log z)))) (if (< t 852031.2288374073) (/ (* (/ x y) (pow a (- t 1.0))) (exp (- b (* (log z) y)))) (/ (* x (/ (pow a (- t 1.0)) y)) (- (+ b 1.0) (* y (log z))))))

  (/ (* x (exp (- (+ (* y (log z)) (* (- t 1.0) (log a))) b))) y))