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

Average Error: 5.9 → 1.2

Time: 17.5s

Precision: 64

Math

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

↓

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

C

double f(double x, double y, double z) {
        double r311680 = x;
        double r311681 = y;
        double r311682 = z;
        double r311683 = r311682 + r311681;
        double r311684 = r311681 / r311683;
        double r311685 = log(r311684);
        double r311686 = r311681 * r311685;
        double r311687 = exp(r311686);
        double r311688 = r311687 / r311681;
        double r311689 = r311680 + r311688;
        return r311689;
}

↓

double f(double x, double y, double z) {
        double r311690 = 2.0;
        double r311691 = y;
        double r311692 = r311690 * r311691;
        double r311693 = cbrt(r311691);
        double r311694 = z;
        double r311695 = r311691 + r311694;
        double r311696 = cbrt(r311695);
        double r311697 = r311693 / r311696;
        double r311698 = log(r311697);
        double r311699 = r311692 * r311698;
        double r311700 = exp(r311699);
        double r311701 = pow(r311697, r311691);
        double r311702 = r311700 * r311701;
        double r311703 = r311702 / r311691;
        double r311704 = x;
        double r311705 = r311703 + r311704;
        return r311705;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Target

Original	5.9
Target	1.1
Herbie	1.2

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

Derivation

1. Initial program 5.9

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

2. Simplified 5.9

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

4. Applied add-cube-cbrt 19.1

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

5. Applied add-cube-cbrt 5.9

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

6. Applied times-frac 5.9

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

7. Applied unpow-prod-down 2.3

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

8. Simplified 2.3

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

10. Applied add-exp-log 39.4

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

11. Applied add-exp-log 31.2

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

12. Applied div-exp 31.2

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

13. Applied add-exp-log 39.4

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

14. Applied add-exp-log 31.2

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

15. Applied div-exp 31.2

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

16. Applied prod-exp 31.2

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

17. Applied pow-exp 30.7

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

18. Simplified 1.2

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

19. Final simplification 1.2

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

Reproduce

herbie shell --seed 2019194 
(FPCore (x y z)
  :name "Numeric.SpecFunctions:invIncompleteBetaWorker from math-functions-0.1.5.2, G"

  :herbie-target
  (if (< (/ y (+ z y)) 7.1154157597908e-315) (+ x (/ (exp (/ -1.0 z)) y)) (+ x (/ (exp (log (pow (/ y (+ y z)) y))) y)))

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