Average Error: 10.6 → 3.2
Time: 23.3s
Precision: 64
\[\frac{e^{x \cdot \log \left(\frac{x}{x + y}\right)}}{x}\]
↓
\[\begin{array}{l} \mathbf{if}\;y \le 74347.0824335901:\\ \;\;\;\;\frac{e^{\left(\log \left(\frac{\sqrt[3]{x}}{\sqrt[3]{x + y}}\right) + \left(\log \left(\frac{\sqrt[3]{x}}{\sqrt[3]{x + y}}\right) + \log \left(\frac{\sqrt[3]{x}}{\sqrt[3]{x + y}}\right)\right)\right) \cdot x}}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{e^{x \cdot \left(\left(\log \left(\frac{\sqrt[3]{x}}{\sqrt[3]{x + y}}\right) + \log \left(\frac{\sqrt[3]{x}}{\sqrt[3]{x + y}}\right)\right) + \log \left(\frac{\sqrt[3]{\sqrt[3]{x}} \cdot e^{\log \left(\sqrt[3]{\sqrt[3]{x} \cdot \sqrt[3]{x}}\right)}}{\sqrt[3]{x + y}}\right)\right)}}{x}\\ \end{array}\]
\frac{e^{x \cdot \log \left(\frac{x}{x + y}\right)}}{x}
↓
\begin{array}{l}
\mathbf{if}\;y \le 74347.0824335901:\\
\;\;\;\;\frac{e^{\left(\log \left(\frac{\sqrt[3]{x}}{\sqrt[3]{x + y}}\right) + \left(\log \left(\frac{\sqrt[3]{x}}{\sqrt[3]{x + y}}\right) + \log \left(\frac{\sqrt[3]{x}}{\sqrt[3]{x + y}}\right)\right)\right) \cdot x}}{x}\\

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

\end{array}
double f(double x, double y) {
        double r22691252 = x;
        double r22691253 = y;
        double r22691254 = r22691252 + r22691253;
        double r22691255 = r22691252 / r22691254;
        double r22691256 = log(r22691255);
        double r22691257 = r22691252 * r22691256;
        double r22691258 = exp(r22691257);
        double r22691259 = r22691258 / r22691252;
        return r22691259;
}
↓
double f(double x, double y) {
        double r22691260 = y;
        double r22691261 = 74347.0824335901;
        bool r22691262 = r22691260 <= r22691261;
        double r22691263 = x;
        double r22691264 = cbrt(r22691263);
        double r22691265 = r22691263 + r22691260;
        double r22691266 = cbrt(r22691265);
        double r22691267 = r22691264 / r22691266;
        double r22691268 = log(r22691267);
        double r22691269 = r22691268 + r22691268;
        double r22691270 = r22691268 + r22691269;
        double r22691271 = r22691270 * r22691263;
        double r22691272 = exp(r22691271);
        double r22691273 = r22691272 / r22691263;
        double r22691274 = cbrt(r22691264);
        double r22691275 = r22691264 * r22691264;
        double r22691276 = cbrt(r22691275);
        double r22691277 = log(r22691276);
        double r22691278 = exp(r22691277);
        double r22691279 = r22691274 * r22691278;
        double r22691280 = r22691279 / r22691266;
        double r22691281 = log(r22691280);
        double r22691282 = r22691269 + r22691281;
        double r22691283 = r22691263 * r22691282;
        double r22691284 = exp(r22691283);
        double r22691285 = r22691284 / r22691263;
        double r22691286 = r22691262 ? r22691273 : r22691285;
        return r22691286;
}

Error

Bits error versus x

Bits error versus y

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original10.6
Target7.7
Herbie3.2
\[\begin{array}{l} \mathbf{if}\;y \lt -3.7311844206647956 \cdot 10^{+94}:\\ \;\;\;\;\frac{e^{\frac{-1}{y}}}{x}\\ \mathbf{elif}\;y \lt 2.817959242728288 \cdot 10^{+37}:\\ \;\;\;\;\frac{{\left(\frac{x}{y + x}\right)}^{x}}{x}\\ \mathbf{elif}\;y \lt 2.347387415166998 \cdot 10^{+178}:\\ \;\;\;\;\log \left(e^{\frac{{\left(\frac{x}{y + x}\right)}^{x}}{x}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{e^{\frac{-1}{y}}}{x}\\ \end{array}\]

Derivation

  1. Split input into 2 regimes

  2. if y < 74347.0824335901

    1. Initial program 4.1

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

    3. Applied add-cube-cbrt28.5

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

    4. Applied add-cube-cbrt4.1

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

    5. Applied times-frac4.1

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

    6. Applied log-prod1.5

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

    7. Simplified0.9

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

    if 74347.0824335901 < y

    1. Initial program 32.8

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

    3. Applied add-cube-cbrt24.9

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

    4. Applied add-cube-cbrt32.8

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

    5. Applied times-frac32.8

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

    6. Applied log-prod24.0

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

    7. Simplified21.2

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

    9. Applied add-cube-cbrt17.8

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

    10. Applied cbrt-prod14.1

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

    12. Applied add-exp-log11.3

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

  4. Final simplification3.2

    \[\leadsto \begin{array}{l} \mathbf{if}\;y \le 74347.0824335901:\\ \;\;\;\;\frac{e^{\left(\log \left(\frac{\sqrt[3]{x}}{\sqrt[3]{x + y}}\right) + \left(\log \left(\frac{\sqrt[3]{x}}{\sqrt[3]{x + y}}\right) + \log \left(\frac{\sqrt[3]{x}}{\sqrt[3]{x + y}}\right)\right)\right) \cdot x}}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{e^{x \cdot \left(\left(\log \left(\frac{\sqrt[3]{x}}{\sqrt[3]{x + y}}\right) + \log \left(\frac{\sqrt[3]{x}}{\sqrt[3]{x + y}}\right)\right) + \log \left(\frac{\sqrt[3]{\sqrt[3]{x}} \cdot e^{\log \left(\sqrt[3]{\sqrt[3]{x} \cdot \sqrt[3]{x}}\right)}}{\sqrt[3]{x + y}}\right)\right)}}{x}\\ \end{array}\]

Reproduce

herbie shell --seed 2019164 
(FPCore (x y)
  :name "Numeric.SpecFunctions:invIncompleteBetaWorker from math-functions-0.1.5.2, F"

  :herbie-target
  (if (< y -3.7311844206647956e+94) (/ (exp (/ -1 y)) x) (if (< y 2.817959242728288e+37) (/ (pow (/ x (+ y x)) x) x) (if (< y 2.347387415166998e+178) (log (exp (/ (pow (/ x (+ y x)) x) x))) (/ (exp (/ -1 y)) x))))

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