Average Error: 6.0 → 0.8
Time: 17.4s
Precision: 64
\[x + \frac{e^{y \cdot \log \left(\frac{y}{z + y}\right)}}{y}\]
↓
\[\begin{array}{l} \mathbf{if}\;y \le 0.06656101021950764:\\ \;\;\;\;x + \frac{1}{y}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{1}{y \cdot e^{z}}\\ \end{array}\]
x + \frac{e^{y \cdot \log \left(\frac{y}{z + y}\right)}}{y}
↓
\begin{array}{l}
\mathbf{if}\;y \le 0.06656101021950764:\\
\;\;\;\;x + \frac{1}{y}\\

\mathbf{else}:\\
\;\;\;\;x + \frac{1}{y \cdot e^{z}}\\

\end{array}
double f(double x, double y, double z) {
        double r17815510 = x;
        double r17815511 = y;
        double r17815512 = z;
        double r17815513 = r17815512 + r17815511;
        double r17815514 = r17815511 / r17815513;
        double r17815515 = log(r17815514);
        double r17815516 = r17815511 * r17815515;
        double r17815517 = exp(r17815516);
        double r17815518 = r17815517 / r17815511;
        double r17815519 = r17815510 + r17815518;
        return r17815519;
}
↓
double f(double x, double y, double z) {
        double r17815520 = y;
        double r17815521 = 0.06656101021950764;
        bool r17815522 = r17815520 <= r17815521;
        double r17815523 = x;
        double r17815524 = 1.0;
        double r17815525 = r17815524 / r17815520;
        double r17815526 = r17815523 + r17815525;
        double r17815527 = z;
        double r17815528 = exp(r17815527);
        double r17815529 = r17815520 * r17815528;
        double r17815530 = r17815524 / r17815529;
        double r17815531 = r17815523 + r17815530;
        double r17815532 = r17815522 ? r17815526 : r17815531;
        return r17815532;
}

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
Herbie0.8
\[\begin{array}{l} \mathbf{if}\;\frac{y}{z + y} \lt 7.1154157597908 \cdot 10^{-315}:\\ \;\;\;\;x + \frac{e^{\frac{-1}{z}}}{y}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{e^{\log \left({\left(\frac{y}{y + z}\right)}^{y}\right)}}{y}\\ \end{array}\]

Derivation

  1. Split input into 2 regimes

  2. if y < 0.06656101021950764

    1. Initial program 7.7

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

    2. Taylor expanded around inf 1.2

      \[\leadsto x + \frac{e^{\color{blue}{0}}}{y}\]

    if 0.06656101021950764 < y

    1. Initial program 2.0

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

    2. Taylor expanded around inf 0.0

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

    3. Simplified0.0

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

    5. Applied clear-num0.0

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

    6. Simplified0.0

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

  4. Final simplification0.8

    \[\leadsto \begin{array}{l} \mathbf{if}\;y \le 0.06656101021950764:\\ \;\;\;\;x + \frac{1}{y}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{1}{y \cdot e^{z}}\\ \end{array}\]

Reproduce

herbie shell --seed 2019163 +o rules:numerics
(FPCore (x y z)
  :name "Numeric.SpecFunctions:invIncompleteBetaWorker from math-functions-0.1.5.2, G"

  :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)))