Average Error: 6.0 → 2.9
Time: 10.6s
Precision: 64
\[x + \frac{e^{y \cdot \log \left(\frac{y}{z + y}\right)}}{y}\]
↓
\[\begin{array}{l} \mathbf{if}\;y \le -7.547170526470774 \lor \neg \left(y \le 3.711281227157836 \cdot 10^{-137}\right):\\ \;\;\;\;x + \frac{e^{-z}}{y}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{{\left(\frac{1}{\sqrt[3]{z + y} \cdot \sqrt[3]{z + y}}\right)}^{y} \cdot {\left(\frac{y}{\sqrt[3]{z + y}}\right)}^{y}}{y}\\ \end{array}\]
x + \frac{e^{y \cdot \log \left(\frac{y}{z + y}\right)}}{y}
↓
\begin{array}{l}
\mathbf{if}\;y \le -7.547170526470774 \lor \neg \left(y \le 3.711281227157836 \cdot 10^{-137}\right):\\
\;\;\;\;x + \frac{e^{-z}}{y}\\

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

\end{array}
double f(double x, double y, double z) {
        double r444530 = x;
        double r444531 = y;
        double r444532 = z;
        double r444533 = r444532 + r444531;
        double r444534 = r444531 / r444533;
        double r444535 = log(r444534);
        double r444536 = r444531 * r444535;
        double r444537 = exp(r444536);
        double r444538 = r444537 / r444531;
        double r444539 = r444530 + r444538;
        return r444539;
}
↓
double f(double x, double y, double z) {
        double r444540 = y;
        double r444541 = -7.547170526470774;
        bool r444542 = r444540 <= r444541;
        double r444543 = 3.711281227157836e-137;
        bool r444544 = r444540 <= r444543;
        double r444545 = !r444544;
        bool r444546 = r444542 || r444545;
        double r444547 = x;
        double r444548 = z;
        double r444549 = -r444548;
        double r444550 = exp(r444549);
        double r444551 = r444550 / r444540;
        double r444552 = r444547 + r444551;
        double r444553 = 1.0;
        double r444554 = r444548 + r444540;
        double r444555 = cbrt(r444554);
        double r444556 = r444555 * r444555;
        double r444557 = r444553 / r444556;
        double r444558 = pow(r444557, r444540);
        double r444559 = r444540 / r444555;
        double r444560 = pow(r444559, r444540);
        double r444561 = r444558 * r444560;
        double r444562 = r444561 / r444540;
        double r444563 = r444547 + r444562;
        double r444564 = r444546 ? r444552 : r444563;
        return r444564;
}

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

Original6.0
Target1.1
Herbie2.9
\[\begin{array}{l} \mathbf{if}\;\frac{y}{z + y} \lt 7.1154157598 \cdot 10^{-315}:\\ \;\;\;\;x + \frac{e^{\frac{-1}{z}}}{y}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{e^{\log \left({\left(\frac{y}{y + z}\right)}^{y}\right)}}{y}\\ \end{array}\]

Derivation

  1. Split input into 2 regimes

  2. if y < -7.547170526470774 or 3.711281227157836e-137 < y

    1. Initial program 2.1

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

    2. Simplified2.1

      \[\leadsto \color{blue}{x + \frac{{\left(\frac{y}{z + y}\right)}^{y}}{y}}\]

    3. Taylor expanded around inf 2.2

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

    if -7.547170526470774 < y < 3.711281227157836e-137

    1. Initial program 13.0

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

    2. Simplified13.0

      \[\leadsto \color{blue}{x + \frac{{\left(\frac{y}{z + y}\right)}^{y}}{y}}\]
    3. Using strategy rm

    4. Applied add-cube-cbrt13.0

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

    5. Applied *-un-lft-identity13.0

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

    6. Applied times-frac13.0

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

    7. Applied unpow-prod-down4.1

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

  4. Final simplification2.9

    \[\leadsto \begin{array}{l} \mathbf{if}\;y \le -7.547170526470774 \lor \neg \left(y \le 3.711281227157836 \cdot 10^{-137}\right):\\ \;\;\;\;x + \frac{e^{-z}}{y}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{{\left(\frac{1}{\sqrt[3]{z + y} \cdot \sqrt[3]{z + y}}\right)}^{y} \cdot {\left(\frac{y}{\sqrt[3]{z + y}}\right)}^{y}}{y}\\ \end{array}\]

Reproduce

herbie shell --seed 2020043 
(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)))