Average Error: 6.2 → 0.7
Time: 8.6s
Precision: 64
\[x + \frac{e^{y \cdot \log \left(\frac{y}{z + y}\right)}}{y}\]
↓
\[\begin{array}{l} \mathbf{if}\;y \le -9.2285838323510326 \cdot 10^{106}:\\ \;\;\;\;x + \frac{e^{-1 \cdot z}}{y}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{e^{\sqrt[3]{{\left(2 \cdot \log \left(\frac{\sqrt[3]{y}}{\sqrt[3]{z + y}}\right)\right)}^{3}} \cdot y}}{\frac{y}{\frac{{\left(\frac{\sqrt[3]{y}}{\sqrt[3]{z + y}}\right)}^{\left(\frac{y}{2}\right)} \cdot {\left(\frac{\sqrt[3]{y}}{\sqrt[3]{z + y}}\right)}^{\left(\frac{y}{2}\right)}}{1}}}\\ \end{array}\]
x + \frac{e^{y \cdot \log \left(\frac{y}{z + y}\right)}}{y}
↓
\begin{array}{l}
\mathbf{if}\;y \le -9.2285838323510326 \cdot 10^{106}:\\
\;\;\;\;x + \frac{e^{-1 \cdot z}}{y}\\

\mathbf{else}:\\
\;\;\;\;x + \frac{e^{\sqrt[3]{{\left(2 \cdot \log \left(\frac{\sqrt[3]{y}}{\sqrt[3]{z + y}}\right)\right)}^{3}} \cdot y}}{\frac{y}{\frac{{\left(\frac{\sqrt[3]{y}}{\sqrt[3]{z + y}}\right)}^{\left(\frac{y}{2}\right)} \cdot {\left(\frac{\sqrt[3]{y}}{\sqrt[3]{z + y}}\right)}^{\left(\frac{y}{2}\right)}}{1}}}\\

\end{array}
double f(double x, double y, double z) {
        double r440170 = x;
        double r440171 = y;
        double r440172 = z;
        double r440173 = r440172 + r440171;
        double r440174 = r440171 / r440173;
        double r440175 = log(r440174);
        double r440176 = r440171 * r440175;
        double r440177 = exp(r440176);
        double r440178 = r440177 / r440171;
        double r440179 = r440170 + r440178;
        return r440179;
}
↓
double f(double x, double y, double z) {
        double r440180 = y;
        double r440181 = -9.228583832351033e+106;
        bool r440182 = r440180 <= r440181;
        double r440183 = x;
        double r440184 = -1.0;
        double r440185 = z;
        double r440186 = r440184 * r440185;
        double r440187 = exp(r440186);
        double r440188 = r440187 / r440180;
        double r440189 = r440183 + r440188;
        double r440190 = 2.0;
        double r440191 = cbrt(r440180);
        double r440192 = r440185 + r440180;
        double r440193 = cbrt(r440192);
        double r440194 = r440191 / r440193;
        double r440195 = log(r440194);
        double r440196 = r440190 * r440195;
        double r440197 = 3.0;
        double r440198 = pow(r440196, r440197);
        double r440199 = cbrt(r440198);
        double r440200 = r440199 * r440180;
        double r440201 = exp(r440200);
        double r440202 = r440180 / r440190;
        double r440203 = pow(r440194, r440202);
        double r440204 = r440203 * r440203;
        double r440205 = 1.0;
        double r440206 = r440204 / r440205;
        double r440207 = r440180 / r440206;
        double r440208 = r440201 / r440207;
        double r440209 = r440183 + r440208;
        double r440210 = r440182 ? r440189 : r440209;
        return r440210;
}

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

Original6.2
Target1.1
Herbie0.7
\[\begin{array}{l} \mathbf{if}\;\frac{y}{z + y} \lt 7.1154157598 \cdot 10^{-315}:\\ \;\;\;\;x + \frac{e^{\frac{-1}{z}}}{y}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{e^{\log \left({\left(\frac{y}{y + z}\right)}^{y}\right)}}{y}\\ \end{array}\]

Derivation

  1. Split input into 2 regimes

  2. if y < -9.228583832351033e+106

    1. Initial program 2.4

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

    2. Taylor expanded around inf 0.0

      \[\leadsto x + \color{blue}{\frac{e^{-1 \cdot z}}{y}}\]

    if -9.228583832351033e+106 < y

    1. Initial program 7.0

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

    3. Applied add-cube-cbrt16.8

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

    4. Applied add-cube-cbrt7.0

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

    5. Applied times-frac7.0

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

    6. Applied log-prod2.0

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

    7. Applied distribute-rgt-in2.0

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

    8. Applied exp-sum2.0

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

    9. Applied associate-/l*2.0

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

    10. Simplified2.0

      \[\leadsto x + \frac{e^{\log \left(\frac{\sqrt[3]{y} \cdot \sqrt[3]{y}}{\sqrt[3]{z + y} \cdot \sqrt[3]{z + y}}\right) \cdot y}}{\color{blue}{\frac{y}{\frac{{\left(\frac{\sqrt[3]{y}}{\sqrt[3]{z + y}}\right)}^{y}}{1}}}}\]
    11. Using strategy rm

    12. Applied add-cbrt-cube2.0

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

    13. Simplified0.8

      \[\leadsto x + \frac{e^{\sqrt[3]{\color{blue}{{\left(2 \cdot \log \left(\frac{\sqrt[3]{y}}{\sqrt[3]{z + y}}\right)\right)}^{3}}} \cdot y}}{\frac{y}{\frac{{\left(\frac{\sqrt[3]{y}}{\sqrt[3]{z + y}}\right)}^{y}}{1}}}\]
    14. Using strategy rm

    15. Applied sqr-pow0.8

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

  4. Final simplification0.7

    \[\leadsto \begin{array}{l} \mathbf{if}\;y \le -9.2285838323510326 \cdot 10^{106}:\\ \;\;\;\;x + \frac{e^{-1 \cdot z}}{y}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{e^{\sqrt[3]{{\left(2 \cdot \log \left(\frac{\sqrt[3]{y}}{\sqrt[3]{z + y}}\right)\right)}^{3}} \cdot y}}{\frac{y}{\frac{{\left(\frac{\sqrt[3]{y}}{\sqrt[3]{z + y}}\right)}^{\left(\frac{y}{2}\right)} \cdot {\left(\frac{\sqrt[3]{y}}{\sqrt[3]{z + y}}\right)}^{\left(\frac{y}{2}\right)}}{1}}}\\ \end{array}\]

Reproduce

herbie shell --seed 2020049 
(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)))