Average Error: 2.0 → 1.1
Time: 38.2s
Precision: 64
\[\frac{x \cdot e^{\left(y \cdot \log z + \left(t - 1\right) \cdot \log a\right) - b}}{y}\]
↓
\[\frac{x}{\frac{\sqrt[3]{\sqrt[3]{y} \cdot \sqrt[3]{y}} \cdot \sqrt[3]{\sqrt[3]{y}}}{\sqrt[3]{{e}^{\left(\mathsf{fma}\left(\log a, t - 1, \log z \cdot y - b\right)\right)}}}} \cdot \frac{1}{\frac{\sqrt[3]{y} \cdot \sqrt[3]{y}}{\sqrt[3]{e^{\mathsf{fma}\left(\log a, t - 1, \log z \cdot y - b\right)}} \cdot \sqrt[3]{e^{\mathsf{fma}\left(\log a, t - 1, \log z \cdot y - b\right)}}}}\]
\frac{x \cdot e^{\left(y \cdot \log z + \left(t - 1\right) \cdot \log a\right) - b}}{y}
↓
\frac{x}{\frac{\sqrt[3]{\sqrt[3]{y} \cdot \sqrt[3]{y}} \cdot \sqrt[3]{\sqrt[3]{y}}}{\sqrt[3]{{e}^{\left(\mathsf{fma}\left(\log a, t - 1, \log z \cdot y - b\right)\right)}}}} \cdot \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)}}}}
double f(double x, double y, double z, double t, double a, double b) {
        double r17395803 = x;
        double r17395804 = y;
        double r17395805 = z;
        double r17395806 = log(r17395805);
        double r17395807 = r17395804 * r17395806;
        double r17395808 = t;
        double r17395809 = 1.0;
        double r17395810 = r17395808 - r17395809;
        double r17395811 = a;
        double r17395812 = log(r17395811);
        double r17395813 = r17395810 * r17395812;
        double r17395814 = r17395807 + r17395813;
        double r17395815 = b;
        double r17395816 = r17395814 - r17395815;
        double r17395817 = exp(r17395816);
        double r17395818 = r17395803 * r17395817;
        double r17395819 = r17395818 / r17395804;
        return r17395819;
}
↓
double f(double x, double y, double z, double t, double a, double b) {
        double r17395820 = x;
        double r17395821 = y;
        double r17395822 = cbrt(r17395821);
        double r17395823 = r17395822 * r17395822;
        double r17395824 = cbrt(r17395823);
        double r17395825 = cbrt(r17395822);
        double r17395826 = r17395824 * r17395825;
        double r17395827 = exp(1.0);
        double r17395828 = a;
        double r17395829 = log(r17395828);
        double r17395830 = t;
        double r17395831 = 1.0;
        double r17395832 = r17395830 - r17395831;
        double r17395833 = z;
        double r17395834 = log(r17395833);
        double r17395835 = r17395834 * r17395821;
        double r17395836 = b;
        double r17395837 = r17395835 - r17395836;
        double r17395838 = fma(r17395829, r17395832, r17395837);
        double r17395839 = pow(r17395827, r17395838);
        double r17395840 = cbrt(r17395839);
        double r17395841 = r17395826 / r17395840;
        double r17395842 = r17395820 / r17395841;
        double r17395843 = 1.0;
        double r17395844 = exp(r17395838);
        double r17395845 = cbrt(r17395844);
        double r17395846 = r17395845 * r17395845;
        double r17395847 = r17395823 / r17395846;
        double r17395848 = r17395843 / r17395847;
        double r17395849 = r17395842 * r17395848;
        return r17395849;
}

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
Target11.5
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 *-un-lft-identity1.1

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

  13. Applied exp-prod1.1

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

  14. Simplified1.1

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

  16. Applied add-cube-cbrt1.1

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

  17. Applied cbrt-prod1.1

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

  18. Final simplification1.1

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

Reproduce

herbie shell --seed 2019171 +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))