Average Error: 5.8 → 1.4
Time: 5.7s
Precision: 64
\[x + \frac{e^{y \cdot \log \left(\frac{y}{z + y}\right)}}{y}\]
↓
\[\begin{array}{l} \mathbf{if}\;\frac{e^{y \cdot \log \left(\frac{y}{z + y}\right)}}{y} = -\infty \lor \neg \left(\frac{e^{y \cdot \log \left(\frac{y}{z + y}\right)}}{y} \le -6.1512673171776875 \cdot 10^{-275} \lor \neg \left(\frac{e^{y \cdot \log \left(\frac{y}{z + y}\right)}}{y} \le -0.0\right)\right):\\ \;\;\;\;x + \frac{e^{y \cdot \left(2 \cdot \log \left(\frac{\sqrt[3]{\sqrt[3]{y} \cdot \sqrt[3]{y}} \cdot \sqrt[3]{\sqrt[3]{y}}}{\sqrt[3]{z + y}}\right)\right) + y \cdot \log \left(\frac{\sqrt[3]{y}}{\sqrt[3]{z + y}}\right)}}{y}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{e^{y \cdot \log \left(\frac{y}{z + y}\right)}}{y}\\ \end{array}\]
x + \frac{e^{y \cdot \log \left(\frac{y}{z + y}\right)}}{y}
↓
\begin{array}{l}
\mathbf{if}\;\frac{e^{y \cdot \log \left(\frac{y}{z + y}\right)}}{y} = -\infty \lor \neg \left(\frac{e^{y \cdot \log \left(\frac{y}{z + y}\right)}}{y} \le -6.1512673171776875 \cdot 10^{-275} \lor \neg \left(\frac{e^{y \cdot \log \left(\frac{y}{z + y}\right)}}{y} \le -0.0\right)\right):\\
\;\;\;\;x + \frac{e^{y \cdot \left(2 \cdot \log \left(\frac{\sqrt[3]{\sqrt[3]{y} \cdot \sqrt[3]{y}} \cdot \sqrt[3]{\sqrt[3]{y}}}{\sqrt[3]{z + y}}\right)\right) + y \cdot \log \left(\frac{\sqrt[3]{y}}{\sqrt[3]{z + y}}\right)}}{y}\\

\mathbf{else}:\\
\;\;\;\;x + \frac{e^{y \cdot \log \left(\frac{y}{z + y}\right)}}{y}\\

\end{array}
double f(double x, double y, double z) {
        double r387981 = x;
        double r387982 = y;
        double r387983 = z;
        double r387984 = r387983 + r387982;
        double r387985 = r387982 / r387984;
        double r387986 = log(r387985);
        double r387987 = r387982 * r387986;
        double r387988 = exp(r387987);
        double r387989 = r387988 / r387982;
        double r387990 = r387981 + r387989;
        return r387990;
}
↓
double f(double x, double y, double z) {
        double r387991 = y;
        double r387992 = z;
        double r387993 = r387992 + r387991;
        double r387994 = r387991 / r387993;
        double r387995 = log(r387994);
        double r387996 = r387991 * r387995;
        double r387997 = exp(r387996);
        double r387998 = r387997 / r387991;
        double r387999 = -inf.0;
        bool r388000 = r387998 <= r387999;
        double r388001 = -6.151267317177687e-275;
        bool r388002 = r387998 <= r388001;
        double r388003 = -0.0;
        bool r388004 = r387998 <= r388003;
        double r388005 = !r388004;
        bool r388006 = r388002 || r388005;
        double r388007 = !r388006;
        bool r388008 = r388000 || r388007;
        double r388009 = x;
        double r388010 = 2.0;
        double r388011 = cbrt(r387991);
        double r388012 = r388011 * r388011;
        double r388013 = cbrt(r388012);
        double r388014 = cbrt(r388011);
        double r388015 = r388013 * r388014;
        double r388016 = cbrt(r387993);
        double r388017 = r388015 / r388016;
        double r388018 = log(r388017);
        double r388019 = r388010 * r388018;
        double r388020 = r387991 * r388019;
        double r388021 = r388011 / r388016;
        double r388022 = log(r388021);
        double r388023 = r387991 * r388022;
        double r388024 = r388020 + r388023;
        double r388025 = exp(r388024);
        double r388026 = r388025 / r387991;
        double r388027 = r388009 + r388026;
        double r388028 = r388009 + r387998;
        double r388029 = r388008 ? r388027 : r388028;
        return r388029;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original5.8
Target1.0
Herbie1.4
\[\begin{array}{l} \mathbf{if}\;\frac{y}{z + y} \lt 7.1154157598 \cdot 10^{-315}:\\ \;\;\;\;x + \frac{e^{\frac{-1}{z}}}{y}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{e^{\log \left({\left(\frac{y}{y + z}\right)}^{y}\right)}}{y}\\ \end{array}\]

Derivation

  1. Split input into 2 regimes

  2. if (/ (exp (* y (log (/ y (+ z y))))) y) < -inf.0 or -6.151267317177687e-275 < (/ (exp (* y (log (/ y (+ z y))))) y) < -0.0

    1. Initial program 28.1

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

    3. Applied add-cube-cbrt34.7

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

    4. Applied add-cube-cbrt28.2

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

    5. Applied times-frac28.2

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

    6. Applied log-prod6.1

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

    7. Applied distribute-lft-in6.1

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

    8. Simplified0.2

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

    10. Applied add-cube-cbrt0.9

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

    11. Applied cbrt-prod2.5

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

    if -inf.0 < (/ (exp (* y (log (/ y (+ z y))))) y) < -6.151267317177687e-275 or -0.0 < (/ (exp (* y (log (/ y (+ z y))))) y)

    1. Initial program 1.1

      \[x + \frac{e^{y \cdot \log \left(\frac{y}{z + y}\right)}}{y}\]
  3. Recombined 2 regimes into one program.

  4. Final simplification1.4

    \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{e^{y \cdot \log \left(\frac{y}{z + y}\right)}}{y} = -\infty \lor \neg \left(\frac{e^{y \cdot \log \left(\frac{y}{z + y}\right)}}{y} \le -6.1512673171776875 \cdot 10^{-275} \lor \neg \left(\frac{e^{y \cdot \log \left(\frac{y}{z + y}\right)}}{y} \le -0.0\right)\right):\\ \;\;\;\;x + \frac{e^{y \cdot \left(2 \cdot \log \left(\frac{\sqrt[3]{\sqrt[3]{y} \cdot \sqrt[3]{y}} \cdot \sqrt[3]{\sqrt[3]{y}}}{\sqrt[3]{z + y}}\right)\right) + y \cdot \log \left(\frac{\sqrt[3]{y}}{\sqrt[3]{z + y}}\right)}}{y}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{e^{y \cdot \log \left(\frac{y}{z + y}\right)}}{y}\\ \end{array}\]

Reproduce

herbie shell --seed 2020034 
(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)))