Average Error: 6.1 → 0.8
Time: 16.9s
Precision: 64
\[x + \frac{e^{y \cdot \log \left(\frac{y}{z + y}\right)}}{y}\]
↓
\[\begin{array}{l} \mathbf{if}\;y \le 0.02851315126171152444789491653409641003236:\\ \;\;\;\;x + \frac{1}{y}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{e^{-z}}{y}\\ \end{array}\]
x + \frac{e^{y \cdot \log \left(\frac{y}{z + y}\right)}}{y}
↓
\begin{array}{l}
\mathbf{if}\;y \le 0.02851315126171152444789491653409641003236:\\
\;\;\;\;x + \frac{1}{y}\\

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

\end{array}
double f(double x, double y, double z) {
        double r14121440 = x;
        double r14121441 = y;
        double r14121442 = z;
        double r14121443 = r14121442 + r14121441;
        double r14121444 = r14121441 / r14121443;
        double r14121445 = log(r14121444);
        double r14121446 = r14121441 * r14121445;
        double r14121447 = exp(r14121446);
        double r14121448 = r14121447 / r14121441;
        double r14121449 = r14121440 + r14121448;
        return r14121449;
}
↓
double f(double x, double y, double z) {
        double r14121450 = y;
        double r14121451 = 0.028513151261711524;
        bool r14121452 = r14121450 <= r14121451;
        double r14121453 = x;
        double r14121454 = 1.0;
        double r14121455 = r14121454 / r14121450;
        double r14121456 = r14121453 + r14121455;
        double r14121457 = z;
        double r14121458 = -r14121457;
        double r14121459 = exp(r14121458);
        double r14121460 = r14121459 / r14121450;
        double r14121461 = r14121453 + r14121460;
        double r14121462 = r14121452 ? r14121456 : r14121461;
        return r14121462;
}

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
Herbie0.8
\[\begin{array}{l} \mathbf{if}\;\frac{y}{z + y} \lt 7.115415759790762719541517221498726780517 \cdot 10^{-315}:\\ \;\;\;\;x + \frac{e^{\frac{-1}{z}}}{y}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{e^{\log \left({\left(\frac{y}{y + z}\right)}^{y}\right)}}{y}\\ \end{array}\]

Derivation

  1. Split input into 2 regimes

  2. if y < 0.028513151261711524

    1. Initial program 7.9

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

    2. Taylor expanded around inf 1.1

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

    if 0.028513151261711524 < y

    1. Initial program 1.7

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

    2. Taylor expanded around inf 0.2

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

    3. Simplified0.2

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

  4. Final simplification0.8

    \[\leadsto \begin{array}{l} \mathbf{if}\;y \le 0.02851315126171152444789491653409641003236:\\ \;\;\;\;x + \frac{1}{y}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{e^{-z}}{y}\\ \end{array}\]

Reproduce

herbie shell --seed 2019171 +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 (/ -1.0 z)) y)) (+ x (/ (exp (log (pow (/ y (+ y z)) y))) y)))

  (+ x (/ (exp (* y (log (/ y (+ z y))))) y)))