Average Error: 6.1 → 0.7
Time: 3.8s
Precision: 64
\[x + \frac{e^{y \cdot \log \left(\frac{y}{z + y}\right)}}{y}\]
↓
\[\begin{array}{l} \mathbf{if}\;y \le 1.184166370745772777259892531677086866132 \cdot 10^{-17}:\\ \;\;\;\;x + \frac{{\left(e^{y}\right)}^{\left(\log \left(\frac{y}{z + y}\right)\right)}}{y}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{e^{-1 \cdot z}}{y}\\ \end{array}\]
x + \frac{e^{y \cdot \log \left(\frac{y}{z + y}\right)}}{y}
↓
\begin{array}{l}
\mathbf{if}\;y \le 1.184166370745772777259892531677086866132 \cdot 10^{-17}:\\
\;\;\;\;x + \frac{{\left(e^{y}\right)}^{\left(\log \left(\frac{y}{z + y}\right)\right)}}{y}\\

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

\end{array}
double f(double x, double y, double z) {
        double r365375 = x;
        double r365376 = y;
        double r365377 = z;
        double r365378 = r365377 + r365376;
        double r365379 = r365376 / r365378;
        double r365380 = log(r365379);
        double r365381 = r365376 * r365380;
        double r365382 = exp(r365381);
        double r365383 = r365382 / r365376;
        double r365384 = r365375 + r365383;
        return r365384;
}
↓
double f(double x, double y, double z) {
        double r365385 = y;
        double r365386 = 1.1841663707457728e-17;
        bool r365387 = r365385 <= r365386;
        double r365388 = x;
        double r365389 = exp(r365385);
        double r365390 = z;
        double r365391 = r365390 + r365385;
        double r365392 = r365385 / r365391;
        double r365393 = log(r365392);
        double r365394 = pow(r365389, r365393);
        double r365395 = r365394 / r365385;
        double r365396 = r365388 + r365395;
        double r365397 = -1.0;
        double r365398 = r365397 * r365390;
        double r365399 = exp(r365398);
        double r365400 = r365399 / r365385;
        double r365401 = r365388 + r365400;
        double r365402 = r365387 ? r365396 : r365401;
        return r365402;
}

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

Original6.1
Target1.0
Herbie0.7
\[\begin{array}{l} \mathbf{if}\;\frac{y}{z + y} \lt 7.115415759790762719541517221498726780517 \cdot 10^{-315}:\\ \;\;\;\;x + \frac{e^{\frac{-1}{z}}}{y}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{e^{\log \left({\left(\frac{y}{y + z}\right)}^{y}\right)}}{y}\\ \end{array}\]

Derivation

  1. Split input into 2 regimes

  2. if y < 1.1841663707457728e-17

    1. Initial program 8.0

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

    3. Applied add-log-exp29.2

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

    4. Applied exp-to-pow0.7

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

    if 1.1841663707457728e-17 < y

    1. Initial program 1.8

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

    2. Taylor expanded around inf 0.4

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

  4. Final simplification0.7

    \[\leadsto \begin{array}{l} \mathbf{if}\;y \le 1.184166370745772777259892531677086866132 \cdot 10^{-17}:\\ \;\;\;\;x + \frac{{\left(e^{y}\right)}^{\left(\log \left(\frac{y}{z + y}\right)\right)}}{y}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{e^{-1 \cdot z}}{y}\\ \end{array}\]

Reproduce

herbie shell --seed 2020002 
(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)))