Numeric.SpecFunctions:invIncompleteBetaWorker from math-functions-0.1.5.2, G

Average Error: 6.3 → 0.9

Time: 20.0s

Precision: 64

Math

\[x + \frac{e^{y \cdot \log \left(\frac{y}{z + y}\right)}}{y}\]

↓

\[x + \frac{e^{\left(2 \cdot \log \left(\frac{\sqrt[3]{y}}{\sqrt[3]{z + y}}\right)\right) \cdot y} \cdot {\left(\frac{\sqrt[3]{y}}{\sqrt[3]{z + y}}\right)}^{y}}{y}\]

C

double f(double x, double y, double z) {
        double r259906 = x;
        double r259907 = y;
        double r259908 = z;
        double r259909 = r259908 + r259907;
        double r259910 = r259907 / r259909;
        double r259911 = log(r259910);
        double r259912 = r259907 * r259911;
        double r259913 = exp(r259912);
        double r259914 = r259913 / r259907;
        double r259915 = r259906 + r259914;
        return r259915;
}

↓

double f(double x, double y, double z) {
        double r259916 = x;
        double r259917 = 2.0;
        double r259918 = y;
        double r259919 = cbrt(r259918);
        double r259920 = z;
        double r259921 = r259920 + r259918;
        double r259922 = cbrt(r259921);
        double r259923 = r259919 / r259922;
        double r259924 = log(r259923);
        double r259925 = r259917 * r259924;
        double r259926 = r259925 * r259918;
        double r259927 = exp(r259926);
        double r259928 = pow(r259923, r259918);
        double r259929 = r259927 * r259928;
        double r259930 = r259929 / r259918;
        double r259931 = r259916 + r259930;
        return r259931;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Target

Original	6.3
Target	0.9
Herbie	0.9

\[\begin{array}{l} \mathbf{if}\;\frac{y}{z + y} \lt 7.115415759790762719541517221498726780517 \cdot 10^{-315}:\\ \;\;\;\;x + \frac{e^{\frac{-1}{z}}}{y}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{e^{\log \left({\left(\frac{y}{y + z}\right)}^{y}\right)}}{y}\\ \end{array}\]

Derivation

1. Initial program 6.3

   \[x + \frac{e^{y \cdot \log \left(\frac{y}{z + y}\right)}}{y}\]

2. Simplified 6.3

   \[\leadsto \color{blue}{x + \frac{{\left(\frac{y}{z + y}\right)}^{y}}{y}}\]
3. Using strategy rm

4. Applied add-cube-cbrt 19.6

   \[\leadsto x + \frac{{\left(\frac{y}{\color{blue}{\left(\sqrt[3]{z + y} \cdot \sqrt[3]{z + y}\right) \cdot \sqrt[3]{z + y}}}\right)}^{y}}{y}\]

5. Applied add-cube-cbrt 6.3

   \[\leadsto x + \frac{{\left(\frac{\color{blue}{\left(\sqrt[3]{y} \cdot \sqrt[3]{y}\right) \cdot \sqrt[3]{y}}}{\left(\sqrt[3]{z + y} \cdot \sqrt[3]{z + y}\right) \cdot \sqrt[3]{z + y}}\right)}^{y}}{y}\]

6. Applied times-frac 6.3

   \[\leadsto x + \frac{{\color{blue}{\left(\frac{\sqrt[3]{y} \cdot \sqrt[3]{y}}{\sqrt[3]{z + y} \cdot \sqrt[3]{z + y}} \cdot \frac{\sqrt[3]{y}}{\sqrt[3]{z + y}}\right)}}^{y}}{y}\]

7. Applied unpow-prod-down 2.2

   \[\leadsto x + \frac{\color{blue}{{\left(\frac{\sqrt[3]{y} \cdot \sqrt[3]{y}}{\sqrt[3]{z + y} \cdot \sqrt[3]{z + y}}\right)}^{y} \cdot {\left(\frac{\sqrt[3]{y}}{\sqrt[3]{z + y}}\right)}^{y}}}{y}\]
8. Using strategy rm

9. Applied add-exp-log 40.2

   \[\leadsto x + \frac{{\left(\frac{\sqrt[3]{y} \cdot \sqrt[3]{y}}{\sqrt[3]{z + y} \cdot \color{blue}{e^{\log \left(\sqrt[3]{z + y}\right)}}}\right)}^{y} \cdot {\left(\frac{\sqrt[3]{y}}{\sqrt[3]{z + y}}\right)}^{y}}{y}\]

10. Applied add-exp-log 40.2

    \[\leadsto x + \frac{{\left(\frac{\sqrt[3]{y} \cdot \sqrt[3]{y}}{\color{blue}{e^{\log \left(\sqrt[3]{z + y}\right)}} \cdot e^{\log \left(\sqrt[3]{z + y}\right)}}\right)}^{y} \cdot {\left(\frac{\sqrt[3]{y}}{\sqrt[3]{z + y}}\right)}^{y}}{y}\]

11. Applied prod-exp 40.3

    \[\leadsto x + \frac{{\left(\frac{\sqrt[3]{y} \cdot \sqrt[3]{y}}{\color{blue}{e^{\log \left(\sqrt[3]{z + y}\right) + \log \left(\sqrt[3]{z + y}\right)}}}\right)}^{y} \cdot {\left(\frac{\sqrt[3]{y}}{\sqrt[3]{z + y}}\right)}^{y}}{y}\]

12. Applied add-exp-log 40.2

    \[\leadsto x + \frac{{\left(\frac{\sqrt[3]{y} \cdot \color{blue}{e^{\log \left(\sqrt[3]{y}\right)}}}{e^{\log \left(\sqrt[3]{z + y}\right) + \log \left(\sqrt[3]{z + y}\right)}}\right)}^{y} \cdot {\left(\frac{\sqrt[3]{y}}{\sqrt[3]{z + y}}\right)}^{y}}{y}\]

13. Applied add-exp-log 36.3

    \[\leadsto x + \frac{{\left(\frac{\color{blue}{e^{\log \left(\sqrt[3]{y}\right)}} \cdot e^{\log \left(\sqrt[3]{y}\right)}}{e^{\log \left(\sqrt[3]{z + y}\right) + \log \left(\sqrt[3]{z + y}\right)}}\right)}^{y} \cdot {\left(\frac{\sqrt[3]{y}}{\sqrt[3]{z + y}}\right)}^{y}}{y}\]

14. Applied prod-exp 32.1

    \[\leadsto x + \frac{{\left(\frac{\color{blue}{e^{\log \left(\sqrt[3]{y}\right) + \log \left(\sqrt[3]{y}\right)}}}{e^{\log \left(\sqrt[3]{z + y}\right) + \log \left(\sqrt[3]{z + y}\right)}}\right)}^{y} \cdot {\left(\frac{\sqrt[3]{y}}{\sqrt[3]{z + y}}\right)}^{y}}{y}\]

15. Applied div-exp 32.1

    \[\leadsto x + \frac{{\color{blue}{\left(e^{\left(\log \left(\sqrt[3]{y}\right) + \log \left(\sqrt[3]{y}\right)\right) - \left(\log \left(\sqrt[3]{z + y}\right) + \log \left(\sqrt[3]{z + y}\right)\right)}\right)}}^{y} \cdot {\left(\frac{\sqrt[3]{y}}{\sqrt[3]{z + y}}\right)}^{y}}{y}\]

16. Applied pow-exp 31.4

    \[\leadsto x + \frac{\color{blue}{e^{\left(\left(\log \left(\sqrt[3]{y}\right) + \log \left(\sqrt[3]{y}\right)\right) - \left(\log \left(\sqrt[3]{z + y}\right) + \log \left(\sqrt[3]{z + y}\right)\right)\right) \cdot y}} \cdot {\left(\frac{\sqrt[3]{y}}{\sqrt[3]{z + y}}\right)}^{y}}{y}\]

17. Simplified 0.9

    \[\leadsto x + \frac{e^{\color{blue}{\left(2 \cdot \log \left(\frac{\sqrt[3]{y}}{\sqrt[3]{z + y}}\right)\right) \cdot y}} \cdot {\left(\frac{\sqrt[3]{y}}{\sqrt[3]{z + y}}\right)}^{y}}{y}\]

18. Final simplification 0.9

    \[\leadsto x + \frac{e^{\left(2 \cdot \log \left(\frac{\sqrt[3]{y}}{\sqrt[3]{z + y}}\right)\right) \cdot y} \cdot {\left(\frac{\sqrt[3]{y}}{\sqrt[3]{z + y}}\right)}^{y}}{y}\]

Reproduce

herbie shell --seed 2019305 +o rules:numerics
(FPCore (x y z)
  :name "Numeric.SpecFunctions:invIncompleteBetaWorker from math-functions-0.1.5.2, G"
  :precision binary64

  :herbie-target
  (if (< (/ y (+ z y)) 7.1154157598e-315) (+ x (/ (exp (/ -1 z)) y)) (+ x (/ (exp (log (pow (/ y (+ y z)) y))) y)))

  (+ x (/ (exp (* y (log (/ y (+ z y))))) y)))