Average Error: 6.2 → 1.0
Time: 10.3s
Precision: 64
\[x + \frac{e^{y \cdot \log \left(\frac{y}{z + y}\right)}}{y}\]
↓
\[\begin{array}{l} \mathbf{if}\;y \le 1.4594586256060078 \cdot 10^{-19}:\\ \;\;\;\;x + \frac{1}{y}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{e^{-z}}{y}\\ \end{array}\]
x + \frac{e^{y \cdot \log \left(\frac{y}{z + y}\right)}}{y}
↓
\begin{array}{l}
\mathbf{if}\;y \le 1.4594586256060078 \cdot 10^{-19}:\\
\;\;\;\;x + \frac{1}{y}\\

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

\end{array}
double f(double x, double y, double z) {
        double r8092427 = x;
        double r8092428 = y;
        double r8092429 = z;
        double r8092430 = r8092429 + r8092428;
        double r8092431 = r8092428 / r8092430;
        double r8092432 = log(r8092431);
        double r8092433 = r8092428 * r8092432;
        double r8092434 = exp(r8092433);
        double r8092435 = r8092434 / r8092428;
        double r8092436 = r8092427 + r8092435;
        return r8092436;
}
↓
double f(double x, double y, double z) {
        double r8092437 = y;
        double r8092438 = 1.4594586256060078e-19;
        bool r8092439 = r8092437 <= r8092438;
        double r8092440 = x;
        double r8092441 = 1.0;
        double r8092442 = r8092441 / r8092437;
        double r8092443 = r8092440 + r8092442;
        double r8092444 = z;
        double r8092445 = -r8092444;
        double r8092446 = exp(r8092445);
        double r8092447 = r8092446 / r8092437;
        double r8092448 = r8092440 + r8092447;
        double r8092449 = r8092439 ? r8092443 : r8092448;
        return r8092449;
}

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

Original6.2
Target1.3
Herbie1.0
\[\begin{array}{l} \mathbf{if}\;\frac{y}{z + y} \lt 7.1154157597908 \cdot 10^{-315}:\\ \;\;\;\;x + \frac{e^{\frac{-1}{z}}}{y}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{e^{\log \left({\left(\frac{y}{y + z}\right)}^{y}\right)}}{y}\\ \end{array}\]

Derivation

  1. Split input into 2 regimes

  2. if y < 1.4594586256060078e-19

    1. Initial program 8.1

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

    2. Taylor expanded around inf 1.2

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

    if 1.4594586256060078e-19 < y

    1. Initial program 2.0

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

    2. Taylor expanded around inf 0.6

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

    3. Simplified0.6

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

  4. Final simplification1.0

    \[\leadsto \begin{array}{l} \mathbf{if}\;y \le 1.4594586256060078 \cdot 10^{-19}:\\ \;\;\;\;x + \frac{1}{y}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{e^{-z}}{y}\\ \end{array}\]

Reproduce

herbie shell --seed 2019156 
(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)))