Average Error: 11.1 → 0.4
Time: 21.2s
Precision: 64
\[\frac{e^{x \cdot \log \left(\frac{x}{x + y}\right)}}{x}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -4.120156288663160099262452246870941235987 \cdot 10^{50} \lor \neg \left(x \le 1.115898472191414701666743120923343257633 \cdot 10^{-9}\right):\\ \;\;\;\;\frac{1}{e^{y} \cdot x}\\ \mathbf{else}:\\ \;\;\;\;\frac{e^{x \cdot \left(2 \cdot \log \left(\frac{\sqrt[3]{x}}{\sqrt[3]{x + y}}\right) + \log \left(\frac{\sqrt[3]{x}}{\sqrt[3]{x + y}}\right)\right)}}{x}\\ \end{array}\]
\frac{e^{x \cdot \log \left(\frac{x}{x + y}\right)}}{x}
↓
\begin{array}{l}
\mathbf{if}\;x \le -4.120156288663160099262452246870941235987 \cdot 10^{50} \lor \neg \left(x \le 1.115898472191414701666743120923343257633 \cdot 10^{-9}\right):\\
\;\;\;\;\frac{1}{e^{y} \cdot x}\\

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

\end{array}
double f(double x, double y) {
        double r274295 = x;
        double r274296 = y;
        double r274297 = r274295 + r274296;
        double r274298 = r274295 / r274297;
        double r274299 = log(r274298);
        double r274300 = r274295 * r274299;
        double r274301 = exp(r274300);
        double r274302 = r274301 / r274295;
        return r274302;
}
↓
double f(double x, double y) {
        double r274303 = x;
        double r274304 = -4.12015628866316e+50;
        bool r274305 = r274303 <= r274304;
        double r274306 = 1.1158984721914147e-09;
        bool r274307 = r274303 <= r274306;
        double r274308 = !r274307;
        bool r274309 = r274305 || r274308;
        double r274310 = 1.0;
        double r274311 = y;
        double r274312 = exp(r274311);
        double r274313 = r274312 * r274303;
        double r274314 = r274310 / r274313;
        double r274315 = 2.0;
        double r274316 = cbrt(r274303);
        double r274317 = r274303 + r274311;
        double r274318 = cbrt(r274317);
        double r274319 = r274316 / r274318;
        double r274320 = log(r274319);
        double r274321 = r274315 * r274320;
        double r274322 = r274321 + r274320;
        double r274323 = r274303 * r274322;
        double r274324 = exp(r274323);
        double r274325 = r274324 / r274303;
        double r274326 = r274309 ? r274314 : r274325;
        return r274326;
}

Error

Bits error versus x

Bits error versus y

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original11.1
Target8.0
Herbie0.4
\[\begin{array}{l} \mathbf{if}\;y \lt -3.73118442066479561492798134439269393419 \cdot 10^{94}:\\ \;\;\;\;\frac{e^{\frac{-1}{y}}}{x}\\ \mathbf{elif}\;y \lt 28179592427282878868860376020282245120:\\ \;\;\;\;\frac{{\left(\frac{x}{y + x}\right)}^{x}}{x}\\ \mathbf{elif}\;y \lt 2.347387415166997963747840232163110922613 \cdot 10^{178}:\\ \;\;\;\;\log \left(e^{\frac{{\left(\frac{x}{y + x}\right)}^{x}}{x}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{e^{\frac{-1}{y}}}{x}\\ \end{array}\]

Derivation

  1. Split input into 2 regimes

  2. if x < -4.12015628866316e+50 or 1.1158984721914147e-09 < x

    1. Initial program 11.7

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

    2. Taylor expanded around inf 0.5

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

    3. Simplified0.5

      \[\leadsto \color{blue}{\frac{e^{-y}}{x}}\]
    4. Using strategy rm

    5. Applied clear-num0.6

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

    6. Simplified0.5

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

    if -4.12015628866316e+50 < x < 1.1158984721914147e-09

    1. Initial program 10.6

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

    3. Applied add-cube-cbrt13.5

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

    4. Applied add-cube-cbrt10.6

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

    5. Applied times-frac10.6

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

    6. Applied log-prod2.0

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

    7. Simplified0.2

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

  4. Final simplification0.4

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -4.120156288663160099262452246870941235987 \cdot 10^{50} \lor \neg \left(x \le 1.115898472191414701666743120923343257633 \cdot 10^{-9}\right):\\ \;\;\;\;\frac{1}{e^{y} \cdot x}\\ \mathbf{else}:\\ \;\;\;\;\frac{e^{x \cdot \left(2 \cdot \log \left(\frac{\sqrt[3]{x}}{\sqrt[3]{x + y}}\right) + \log \left(\frac{\sqrt[3]{x}}{\sqrt[3]{x + y}}\right)\right)}}{x}\\ \end{array}\]

Reproduce

herbie shell --seed 2019303 
(FPCore (x y)
  :name "Numeric.SpecFunctions:invIncompleteBetaWorker from math-functions-0.1.5.2, F"
  :precision binary64

  :herbie-target
  (if (< y -3.73118442066479561e94) (/ (exp (/ -1 y)) x) (if (< y 2.81795924272828789e37) (/ (pow (/ x (+ y x)) x) x) (if (< y 2.347387415166998e178) (log (exp (/ (pow (/ x (+ y x)) x) x))) (/ (exp (/ -1 y)) x))))

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