Average Error: 1.8 → 1.1
Time: 35.5s
Precision: 64
\[\frac{x \cdot e^{\left(y \cdot \log z + \left(t - 1\right) \cdot \log a\right) - b}}{y}\]
↓
\[\left(x \cdot \left(\frac{\sqrt[3]{e^{\mathsf{fma}\left(t - 1, \log a, y \cdot \log z - b\right)}}}{\sqrt[3]{y}} \cdot \frac{\sqrt[3]{e^{\mathsf{fma}\left(t - 1, \log a, y \cdot \log z - b\right)}}}{\sqrt[3]{y}}\right)\right) \cdot \frac{\sqrt[3]{e^{\mathsf{fma}\left(t - 1, \log a, y \cdot \log z - b\right)}}}{\sqrt[3]{y}}\]
\frac{x \cdot e^{\left(y \cdot \log z + \left(t - 1\right) \cdot \log a\right) - b}}{y}
↓
\left(x \cdot \left(\frac{\sqrt[3]{e^{\mathsf{fma}\left(t - 1, \log a, y \cdot \log z - b\right)}}}{\sqrt[3]{y}} \cdot \frac{\sqrt[3]{e^{\mathsf{fma}\left(t - 1, \log a, y \cdot \log z - b\right)}}}{\sqrt[3]{y}}\right)\right) \cdot \frac{\sqrt[3]{e^{\mathsf{fma}\left(t - 1, \log a, y \cdot \log z - b\right)}}}{\sqrt[3]{y}}
double f(double x, double y, double z, double t, double a, double b) {
        double r19706843 = x;
        double r19706844 = y;
        double r19706845 = z;
        double r19706846 = log(r19706845);
        double r19706847 = r19706844 * r19706846;
        double r19706848 = t;
        double r19706849 = 1.0;
        double r19706850 = r19706848 - r19706849;
        double r19706851 = a;
        double r19706852 = log(r19706851);
        double r19706853 = r19706850 * r19706852;
        double r19706854 = r19706847 + r19706853;
        double r19706855 = b;
        double r19706856 = r19706854 - r19706855;
        double r19706857 = exp(r19706856);
        double r19706858 = r19706843 * r19706857;
        double r19706859 = r19706858 / r19706844;
        return r19706859;
}
↓
double f(double x, double y, double z, double t, double a, double b) {
        double r19706860 = x;
        double r19706861 = t;
        double r19706862 = 1.0;
        double r19706863 = r19706861 - r19706862;
        double r19706864 = a;
        double r19706865 = log(r19706864);
        double r19706866 = y;
        double r19706867 = z;
        double r19706868 = log(r19706867);
        double r19706869 = r19706866 * r19706868;
        double r19706870 = b;
        double r19706871 = r19706869 - r19706870;
        double r19706872 = fma(r19706863, r19706865, r19706871);
        double r19706873 = exp(r19706872);
        double r19706874 = cbrt(r19706873);
        double r19706875 = cbrt(r19706866);
        double r19706876 = r19706874 / r19706875;
        double r19706877 = r19706876 * r19706876;
        double r19706878 = r19706860 * r19706877;
        double r19706879 = r19706878 * r19706876;
        return r19706879;
}

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

Original1.8
Target11.1
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 1.8

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

  3. Applied *-un-lft-identity1.8

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

  4. Applied times-frac2.2

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

  5. Simplified2.2

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

  6. Simplified2.2

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

  8. Applied add-cube-cbrt2.2

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

  9. Applied add-cube-cbrt2.2

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

  10. Applied times-frac2.2

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

  11. Applied associate-*r*1.1

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

  12. Simplified1.1

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

  14. Applied *-un-lft-identity1.1

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

  15. Applied cbrt-prod1.1

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

  16. Simplified1.1

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

  17. Final simplification1.1

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

Reproduce

herbie shell --seed 2019170 +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))