Average Error: 11.5 → 5.8
Time: 7.2s
Precision: 64
\[\frac{e^{x \cdot \log \left(\frac{x}{x + y}\right)}}{x}\]
↓
\[\begin{array}{l} \mathbf{if}\;y \le 193392245631.92206:\\ \;\;\;\;\frac{e^{x \cdot \left(2 \cdot \log \left(\frac{\sqrt[3]{x}}{\sqrt[3]{x + y}}\right)\right) + x \cdot \log \left(\frac{\sqrt[3]{x}}{\sqrt[3]{x + y}}\right)}}{x}\\ \mathbf{elif}\;y \le 3.17466847436457891 \cdot 10^{67}:\\ \;\;\;\;\frac{e^{x \cdot \log \left(\frac{\frac{x}{\sqrt[3]{x + y} \cdot \sqrt[3]{x + y}}}{\sqrt[3]{x + y}}\right)}}{x}\\ \mathbf{elif}\;y \le 3.9061854843159141 \cdot 10^{81}:\\ \;\;\;\;\frac{e^{x \cdot \left(2 \cdot \log \left(\frac{\sqrt[3]{x}}{\sqrt[3]{x + y}}\right)\right) + x \cdot \log \left(\frac{\sqrt[3]{x}}{\sqrt[3]{x + y}}\right)}}{x}\\ \mathbf{elif}\;y \le 6.7514169676994134 \cdot 10^{146}:\\ \;\;\;\;\frac{e^{x \cdot \log \left(\frac{x}{x \cdot x - y \cdot y} \cdot \left(x - y\right)\right)}}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{e^{x \cdot \left(2 \cdot \log \left(\frac{\sqrt[3]{x}}{\sqrt[3]{x + y}}\right)\right) + x \cdot \log \left(\frac{\sqrt[3]{x}}{\sqrt[3]{x + y}}\right)}}{x}\\ \end{array}\]
\frac{e^{x \cdot \log \left(\frac{x}{x + y}\right)}}{x}
↓
\begin{array}{l}
\mathbf{if}\;y \le 193392245631.92206:\\
\;\;\;\;\frac{e^{x \cdot \left(2 \cdot \log \left(\frac{\sqrt[3]{x}}{\sqrt[3]{x + y}}\right)\right) + x \cdot \log \left(\frac{\sqrt[3]{x}}{\sqrt[3]{x + y}}\right)}}{x}\\

\mathbf{elif}\;y \le 3.17466847436457891 \cdot 10^{67}:\\
\;\;\;\;\frac{e^{x \cdot \log \left(\frac{\frac{x}{\sqrt[3]{x + y} \cdot \sqrt[3]{x + y}}}{\sqrt[3]{x + y}}\right)}}{x}\\

\mathbf{elif}\;y \le 3.9061854843159141 \cdot 10^{81}:\\
\;\;\;\;\frac{e^{x \cdot \left(2 \cdot \log \left(\frac{\sqrt[3]{x}}{\sqrt[3]{x + y}}\right)\right) + x \cdot \log \left(\frac{\sqrt[3]{x}}{\sqrt[3]{x + y}}\right)}}{x}\\

\mathbf{elif}\;y \le 6.7514169676994134 \cdot 10^{146}:\\
\;\;\;\;\frac{e^{x \cdot \log \left(\frac{x}{x \cdot x - y \cdot y} \cdot \left(x - y\right)\right)}}{x}\\

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

\end{array}
double f(double x, double y) {
        double r428657 = x;
        double r428658 = y;
        double r428659 = r428657 + r428658;
        double r428660 = r428657 / r428659;
        double r428661 = log(r428660);
        double r428662 = r428657 * r428661;
        double r428663 = exp(r428662);
        double r428664 = r428663 / r428657;
        return r428664;
}
↓
double f(double x, double y) {
        double r428665 = y;
        double r428666 = 193392245631.92206;
        bool r428667 = r428665 <= r428666;
        double r428668 = x;
        double r428669 = 2.0;
        double r428670 = cbrt(r428668);
        double r428671 = r428668 + r428665;
        double r428672 = cbrt(r428671);
        double r428673 = r428670 / r428672;
        double r428674 = log(r428673);
        double r428675 = r428669 * r428674;
        double r428676 = r428668 * r428675;
        double r428677 = r428668 * r428674;
        double r428678 = r428676 + r428677;
        double r428679 = exp(r428678);
        double r428680 = r428679 / r428668;
        double r428681 = 3.174668474364579e+67;
        bool r428682 = r428665 <= r428681;
        double r428683 = r428672 * r428672;
        double r428684 = r428668 / r428683;
        double r428685 = r428684 / r428672;
        double r428686 = log(r428685);
        double r428687 = r428668 * r428686;
        double r428688 = exp(r428687);
        double r428689 = r428688 / r428668;
        double r428690 = 3.906185484315914e+81;
        bool r428691 = r428665 <= r428690;
        double r428692 = 6.751416967699413e+146;
        bool r428693 = r428665 <= r428692;
        double r428694 = r428668 * r428668;
        double r428695 = r428665 * r428665;
        double r428696 = r428694 - r428695;
        double r428697 = r428668 / r428696;
        double r428698 = r428668 - r428665;
        double r428699 = r428697 * r428698;
        double r428700 = log(r428699);
        double r428701 = r428668 * r428700;
        double r428702 = exp(r428701);
        double r428703 = r428702 / r428668;
        double r428704 = r428693 ? r428703 : r428680;
        double r428705 = r428691 ? r428680 : r428704;
        double r428706 = r428682 ? r428689 : r428705;
        double r428707 = r428667 ? r428680 : r428706;
        return r428707;
}

Error

Bits error versus x

Bits error versus y

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original11.5
Target8.3
Herbie5.8
\[\begin{array}{l} \mathbf{if}\;y \lt -3.73118442066479561 \cdot 10^{94}:\\ \;\;\;\;\frac{e^{\frac{-1}{y}}}{x}\\ \mathbf{elif}\;y \lt 2.81795924272828789 \cdot 10^{37}:\\ \;\;\;\;\frac{{\left(\frac{x}{y + x}\right)}^{x}}{x}\\ \mathbf{elif}\;y \lt 2.347387415166998 \cdot 10^{178}:\\ \;\;\;\;\log \left(e^{\frac{{\left(\frac{x}{y + x}\right)}^{x}}{x}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{e^{\frac{-1}{y}}}{x}\\ \end{array}\]

Derivation

  1. Split input into 3 regimes

  2. if y < 193392245631.92206 or 3.174668474364579e+67 < y < 3.906185484315914e+81 or 6.751416967699413e+146 < y

    1. Initial program 8.7

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

    3. Applied add-cube-cbrt28.4

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

    4. Applied add-cube-cbrt8.8

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

    5. Applied times-frac8.8

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

    6. Applied log-prod4.7

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

    7. Applied distribute-lft-in4.7

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

    8. Simplified3.4

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

    if 193392245631.92206 < y < 3.174668474364579e+67

    1. Initial program 39.1

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

    3. Applied add-cube-cbrt25.0

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

    4. Applied associate-/r*25.6

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

    if 3.906185484315914e+81 < y < 6.751416967699413e+146

    1. Initial program 32.7

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

    3. Applied flip-+21.1

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

    4. Applied associate-/r/28.3

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

  4. Final simplification5.8

    \[\leadsto \begin{array}{l} \mathbf{if}\;y \le 193392245631.92206:\\ \;\;\;\;\frac{e^{x \cdot \left(2 \cdot \log \left(\frac{\sqrt[3]{x}}{\sqrt[3]{x + y}}\right)\right) + x \cdot \log \left(\frac{\sqrt[3]{x}}{\sqrt[3]{x + y}}\right)}}{x}\\ \mathbf{elif}\;y \le 3.17466847436457891 \cdot 10^{67}:\\ \;\;\;\;\frac{e^{x \cdot \log \left(\frac{\frac{x}{\sqrt[3]{x + y} \cdot \sqrt[3]{x + y}}}{\sqrt[3]{x + y}}\right)}}{x}\\ \mathbf{elif}\;y \le 3.9061854843159141 \cdot 10^{81}:\\ \;\;\;\;\frac{e^{x \cdot \left(2 \cdot \log \left(\frac{\sqrt[3]{x}}{\sqrt[3]{x + y}}\right)\right) + x \cdot \log \left(\frac{\sqrt[3]{x}}{\sqrt[3]{x + y}}\right)}}{x}\\ \mathbf{elif}\;y \le 6.7514169676994134 \cdot 10^{146}:\\ \;\;\;\;\frac{e^{x \cdot \log \left(\frac{x}{x \cdot x - y \cdot y} \cdot \left(x - y\right)\right)}}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{e^{x \cdot \left(2 \cdot \log \left(\frac{\sqrt[3]{x}}{\sqrt[3]{x + y}}\right)\right) + x \cdot \log \left(\frac{\sqrt[3]{x}}{\sqrt[3]{x + y}}\right)}}{x}\\ \end{array}\]

Reproduce

herbie shell --seed 2020100 
(FPCore (x y)
  :name "Numeric.SpecFunctions:invIncompleteBetaWorker from math-functions-0.1.5.2, F"
  :precision binary64

  :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))