Average Error: 1.9 → 1.4
Time: 40.7s
Precision: 64
\[\frac{x \cdot e^{\left(y \cdot \log z + \left(t - 1.0\right) \cdot \log a\right) - b}}{y}\]
↓
\[\frac{\sqrt[3]{x} \cdot \sqrt[3]{x}}{\frac{1}{\sqrt[3]{e^{\mathsf{fma}\left(t - 1.0, \log a, \log z \cdot y\right) - b}} \cdot \sqrt[3]{e^{\mathsf{fma}\left(t - 1.0, \log a, \log z \cdot y\right) - b}}}} \cdot \frac{\sqrt[3]{\sqrt[3]{x} \cdot \sqrt[3]{x}} \cdot \left(\sqrt[3]{\sqrt[3]{\sqrt[3]{x}} \cdot \sqrt[3]{\sqrt[3]{x}}} \cdot \sqrt[3]{\sqrt[3]{\sqrt[3]{x}}}\right)}{\frac{y}{\sqrt[3]{e^{\mathsf{fma}\left(t - 1.0, \log a, \log z \cdot y\right) - b}}}}\]
\frac{x \cdot e^{\left(y \cdot \log z + \left(t - 1.0\right) \cdot \log a\right) - b}}{y}
↓
\frac{\sqrt[3]{x} \cdot \sqrt[3]{x}}{\frac{1}{\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]{\sqrt[3]{x} \cdot \sqrt[3]{x}} \cdot \left(\sqrt[3]{\sqrt[3]{\sqrt[3]{x}} \cdot \sqrt[3]{\sqrt[3]{x}}} \cdot \sqrt[3]{\sqrt[3]{\sqrt[3]{x}}}\right)}{\frac{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 r17314866 = x;
        double r17314867 = y;
        double r17314868 = z;
        double r17314869 = log(r17314868);
        double r17314870 = r17314867 * r17314869;
        double r17314871 = t;
        double r17314872 = 1.0;
        double r17314873 = r17314871 - r17314872;
        double r17314874 = a;
        double r17314875 = log(r17314874);
        double r17314876 = r17314873 * r17314875;
        double r17314877 = r17314870 + r17314876;
        double r17314878 = b;
        double r17314879 = r17314877 - r17314878;
        double r17314880 = exp(r17314879);
        double r17314881 = r17314866 * r17314880;
        double r17314882 = r17314881 / r17314867;
        return r17314882;
}
↓
double f(double x, double y, double z, double t, double a, double b) {
        double r17314883 = x;
        double r17314884 = cbrt(r17314883);
        double r17314885 = r17314884 * r17314884;
        double r17314886 = 1.0;
        double r17314887 = t;
        double r17314888 = 1.0;
        double r17314889 = r17314887 - r17314888;
        double r17314890 = a;
        double r17314891 = log(r17314890);
        double r17314892 = z;
        double r17314893 = log(r17314892);
        double r17314894 = y;
        double r17314895 = r17314893 * r17314894;
        double r17314896 = fma(r17314889, r17314891, r17314895);
        double r17314897 = b;
        double r17314898 = r17314896 - r17314897;
        double r17314899 = exp(r17314898);
        double r17314900 = cbrt(r17314899);
        double r17314901 = r17314900 * r17314900;
        double r17314902 = r17314886 / r17314901;
        double r17314903 = r17314885 / r17314902;
        double r17314904 = cbrt(r17314885);
        double r17314905 = cbrt(r17314884);
        double r17314906 = r17314905 * r17314905;
        double r17314907 = cbrt(r17314906);
        double r17314908 = cbrt(r17314905);
        double r17314909 = r17314907 * r17314908;
        double r17314910 = r17314904 * r17314909;
        double r17314911 = r17314894 / r17314900;
        double r17314912 = r17314910 / r17314911;
        double r17314913 = r17314903 * r17314912;
        return r17314913;
}

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

Original1.9
Target10.5
Herbie1.4
\[\begin{array}{l} \mathbf{if}\;t \lt -0.8845848504127471:\\ \;\;\;\;\frac{x \cdot \frac{{a}^{\left(t - 1.0\right)}}{y}}{\left(b + 1\right) - y \cdot \log z}\\ \mathbf{elif}\;t \lt 852031.2288374073:\\ \;\;\;\;\frac{\frac{x}{y} \cdot {a}^{\left(t - 1.0\right)}}{e^{b - \log z \cdot y}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot \frac{{a}^{\left(t - 1.0\right)}}{y}}{\left(b + 1\right) - y \cdot \log z}\\ \end{array}\]

Derivation

  1. Initial program 1.9

    \[\frac{x \cdot e^{\left(y \cdot \log z + \left(t - 1.0\right) \cdot \log a\right) - b}}{y}\]
  2. Using strategy rm

  3. Applied associate-/l*2.0

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

  4. Simplified2.0

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

  6. Applied add-cube-cbrt2.0

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

  7. Applied *-un-lft-identity2.0

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

  8. Applied times-frac2.0

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

  9. Applied add-cube-cbrt2.0

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

  10. Applied times-frac1.4

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

  12. Applied add-cube-cbrt1.4

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

  13. Applied cbrt-prod1.4

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

  15. Applied add-cube-cbrt1.4

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

  16. Applied cbrt-prod1.4

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

  17. Final simplification1.4

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

Reproduce

herbie shell --seed 2019158 +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) (* 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) (* y (log z))))))

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