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

Average Error: 6.3 → 2.0

Time: 24.4s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;z \le 2798330066458407314915328 \lor \neg \left(z \le 4.76689686693070055018888846744602636955 \cdot 10^{118}\right):\\ \;\;\;\;x + \frac{e^{\left(2 \cdot \log \left(\frac{\sqrt[3]{y}}{\sqrt[3]{z + y}}\right)\right) \cdot y} \cdot {\left(\frac{\sqrt[3]{y}}{\sqrt[3]{z + y}}\right)}^{y}}{y}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{{\left(\frac{\sqrt[3]{y} \cdot \sqrt[3]{y}}{\sqrt[3]{z + y} \cdot \sqrt[3]{z + y}}\right)}^{y} \cdot e^{\frac{-1}{3} \cdot z}}{y}\\ \end{array}\]

C

double f(double x, double y, double z) {
        double r252593 = x;
        double r252594 = y;
        double r252595 = z;
        double r252596 = r252595 + r252594;
        double r252597 = r252594 / r252596;
        double r252598 = log(r252597);
        double r252599 = r252594 * r252598;
        double r252600 = exp(r252599);
        double r252601 = r252600 / r252594;
        double r252602 = r252593 + r252601;
        return r252602;
}

↓

double f(double x, double y, double z) {
        double r252603 = z;
        double r252604 = 2.7983300664584073e+24;
        bool r252605 = r252603 <= r252604;
        double r252606 = 4.7668968669307006e+118;
        bool r252607 = r252603 <= r252606;
        double r252608 = !r252607;
        bool r252609 = r252605 || r252608;
        double r252610 = x;
        double r252611 = 2.0;
        double r252612 = y;
        double r252613 = cbrt(r252612);
        double r252614 = r252603 + r252612;
        double r252615 = cbrt(r252614);
        double r252616 = r252613 / r252615;
        double r252617 = log(r252616);
        double r252618 = r252611 * r252617;
        double r252619 = r252618 * r252612;
        double r252620 = exp(r252619);
        double r252621 = pow(r252616, r252612);
        double r252622 = r252620 * r252621;
        double r252623 = r252622 / r252612;
        double r252624 = r252610 + r252623;
        double r252625 = r252613 * r252613;
        double r252626 = r252615 * r252615;
        double r252627 = r252625 / r252626;
        double r252628 = pow(r252627, r252612);
        double r252629 = -0.3333333333333333;
        double r252630 = r252629 * r252603;
        double r252631 = exp(r252630);
        double r252632 = r252628 * r252631;
        double r252633 = r252632 / r252612;
        double r252634 = r252610 + r252633;
        double r252635 = r252609 ? r252624 : r252634;
        return r252635;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Target

Original	6.3
Target	1.1
Herbie	2.0

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

Derivation

1. Split input into 2 regimes

2. if z < 2.7983300664584073e+24 or 4.7668968669307006e+118 < z

   1. Initial program 5.9

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

   2. Simplified 5.9

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

   4. Applied add-cube-cbrt 19.3

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

   5. Applied add-cube-cbrt 5.9

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

   6. Applied times-frac 5.9

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

   7. Applied unpow-prod-down 1.8

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

   9. Applied add-exp-log 40.1

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

   10. Applied add-exp-log 40.1

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

   11. Applied prod-exp 40.1

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

   12. Applied add-exp-log 40.0

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

   13. Applied add-exp-log 36.2

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

   14. Applied prod-exp 32.0

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

   15. Applied div-exp 32.0

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

   16. Applied pow-exp 31.5

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

   17. Simplified 0.7

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

   if 2.7983300664584073e+24 < z < 4.7668968669307006e+118

   1. Initial program 10.9

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

   2. Simplified 10.9

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

   4. Applied add-cube-cbrt 17.9

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

   5. Applied add-cube-cbrt 10.9

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

   6. Applied times-frac 10.9

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

   7. Applied unpow-prod-down 6.3

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

   8. Taylor expanded around inf 18.4

      \[\leadsto x + \frac{{\left(\frac{\sqrt[3]{y} \cdot \sqrt[3]{y}}{\sqrt[3]{z + y} \cdot \sqrt[3]{z + y}}\right)}^{y} \cdot \color{blue}{e^{-\frac{1}{3} \cdot z}}}{y}\]

   9. Simplified 18.4

      \[\leadsto x + \frac{{\left(\frac{\sqrt[3]{y} \cdot \sqrt[3]{y}}{\sqrt[3]{z + y} \cdot \sqrt[3]{z + y}}\right)}^{y} \cdot \color{blue}{e^{\frac{-1}{3} \cdot z}}}{y}\]
3. Recombined 2 regimes into one program.

4. Final simplification 2.0

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \le 2798330066458407314915328 \lor \neg \left(z \le 4.76689686693070055018888846744602636955 \cdot 10^{118}\right):\\ \;\;\;\;x + \frac{e^{\left(2 \cdot \log \left(\frac{\sqrt[3]{y}}{\sqrt[3]{z + y}}\right)\right) \cdot y} \cdot {\left(\frac{\sqrt[3]{y}}{\sqrt[3]{z + y}}\right)}^{y}}{y}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{{\left(\frac{\sqrt[3]{y} \cdot \sqrt[3]{y}}{\sqrt[3]{z + y} \cdot \sqrt[3]{z + y}}\right)}^{y} \cdot e^{\frac{-1}{3} \cdot z}}{y}\\ \end{array}\]

Reproduce

herbie shell --seed 2019323 +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)) 7.1154157597908e-315) (+ x (/ (exp (/ -1 z)) y)) (+ x (/ (exp (log (pow (/ y (+ y z)) y))) y)))

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