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

Average Error: 5.8 → 0.9

Time: 10.2s

Precision: 64

• Falling back on racket egraph because egg-herbie package not installed

Math

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

↓

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

C

double f(double x, double y, double z) {
        double r480756 = x;
        double r480757 = y;
        double r480758 = z;
        double r480759 = r480758 + r480757;
        double r480760 = r480757 / r480759;
        double r480761 = log(r480760);
        double r480762 = r480757 * r480761;
        double r480763 = exp(r480762);
        double r480764 = r480763 / r480757;
        double r480765 = r480756 + r480764;
        return r480765;
}

↓

double f(double x, double y, double z) {
        double r480766 = x;
        double r480767 = y;
        double r480768 = 2.0;
        double r480769 = cbrt(r480767);
        double r480770 = z;
        double r480771 = r480770 + r480767;
        double r480772 = cbrt(r480771);
        double r480773 = r480769 / r480772;
        double r480774 = log(r480773);
        double r480775 = r480768 * r480774;
        double r480776 = r480767 * r480775;
        double r480777 = exp(r480776);
        double r480778 = pow(r480773, r480767);
        double r480779 = r480777 * r480778;
        double r480780 = r480779 / r480767;
        double r480781 = r480766 + r480780;
        return r480781;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Target

Original	5.8
Target	0.9
Herbie	0.9

\[\begin{array}{l} \mathbf{if}\;\frac{y}{z + y} \lt 7.1154157598 \cdot 10^{-315}:\\ \;\;\;\;x + \frac{e^{\frac{-1}{z}}}{y}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{e^{\log \left({\left(\frac{y}{y + z}\right)}^{y}\right)}}{y}\\ \end{array}\]

Derivation

1. Initial program 5.8

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

2. Simplified 5.8

   \[\leadsto \color{blue}{x + \frac{{\left(\frac{y}{z + y}\right)}^{y}}{y}}\]
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 5.8

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

6. Applied times-frac 5.8

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

7. Applied unpow-prod-down 1.9

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

9. Applied add-exp-log 39.5

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

10. Applied add-exp-log 39.5

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

11. Applied prod-exp 39.6

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

12. Applied add-exp-log 39.5

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

13. Applied add-exp-log 35.8

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

14. Applied prod-exp 31.6

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

15. Applied div-exp 31.6

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

16. Applied pow-exp 31.1

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

17. Simplified 0.9

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

18. Final simplification 0.9

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

Reproduce

herbie shell --seed 2020047 
(FPCore (x y z)
  :name "Numeric.SpecFunctions:invIncompleteBetaWorker from math-functions-0.1.5.2, G"
  :precision binary64

  :herbie-target
  (if (< (/ y (+ z y)) 7.1154157597908e-315) (+ x (/ (exp (/ -1 z)) y)) (+ x (/ (exp (log (pow (/ y (+ y z)) y))) y)))

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