Average Error: 6.1 → 0.8
Time: 17.5s
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 r25036546 = x;
        double r25036547 = y;
        double r25036548 = z;
        double r25036549 = r25036548 + r25036547;
        double r25036550 = r25036547 / r25036549;
        double r25036551 = log(r25036550);
        double r25036552 = r25036547 * r25036551;
        double r25036553 = exp(r25036552);
        double r25036554 = r25036553 / r25036547;
        double r25036555 = r25036546 + r25036554;
        return r25036555;
}
↓
double f(double x, double y, double z) {
        double r25036556 = y;
        double r25036557 = 0.028513151261711524;
        bool r25036558 = r25036556 <= r25036557;
        double r25036559 = x;
        double r25036560 = 1.0;
        double r25036561 = r25036560 / r25036556;
        double r25036562 = r25036559 + r25036561;
        double r25036563 = z;
        double r25036564 = -r25036563;
        double r25036565 = exp(r25036564);
        double r25036566 = r25036565 / r25036556;
        double r25036567 = r25036559 + r25036566;
        double r25036568 = r25036558 ? r25036562 : r25036567;
        return r25036568;
}

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 
(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)))