Average Error: 2.0 → 1.0
Time: 45.8s
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 r1979657 = x;
        double r1979658 = y;
        double r1979659 = z;
        double r1979660 = log(r1979659);
        double r1979661 = r1979658 * r1979660;
        double r1979662 = t;
        double r1979663 = 1.0;
        double r1979664 = r1979662 - r1979663;
        double r1979665 = a;
        double r1979666 = log(r1979665);
        double r1979667 = r1979664 * r1979666;
        double r1979668 = r1979661 + r1979667;
        double r1979669 = b;
        double r1979670 = r1979668 - r1979669;
        double r1979671 = exp(r1979670);
        double r1979672 = r1979657 * r1979671;
        double r1979673 = r1979672 / r1979658;
        return r1979673;
}
↓
double f(double x, double y, double z, double t, double a, double b) {
        double r1979674 = 1.0;
        double r1979675 = y;
        double r1979676 = cbrt(r1979675);
        double r1979677 = r1979676 * r1979676;
        double r1979678 = t;
        double r1979679 = 1.0;
        double r1979680 = r1979678 - r1979679;
        double r1979681 = a;
        double r1979682 = log(r1979681);
        double r1979683 = z;
        double r1979684 = log(r1979683);
        double r1979685 = r1979684 * r1979675;
        double r1979686 = fma(r1979680, r1979682, r1979685);
        double r1979687 = b;
        double r1979688 = r1979686 - r1979687;
        double r1979689 = exp(r1979688);
        double r1979690 = cbrt(r1979689);
        double r1979691 = r1979690 * r1979690;
        double r1979692 = r1979677 / r1979691;
        double r1979693 = r1979674 / r1979692;
        double r1979694 = x;
        double r1979695 = r1979676 / r1979690;
        double r1979696 = r1979694 / r1979695;
        double r1979697 = r1979693 * r1979696;
        return r1979697;
}

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))