Average Error: 5.9 → 0.3
Time: 7.3s
Precision: 64
\[x + \frac{e^{y \cdot \log \left(\frac{y}{z + y}\right)}}{y}\]
↓
\[\begin{array}{l} \mathbf{if}\;y \le -3.86604521535774063 \cdot 10^{122} \lor \neg \left(y \le 0.0042390669371441449\right):\\ \;\;\;\;x + \frac{e^{-1 \cdot z}}{y}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{e^{y \cdot 0}}{y}\\ \end{array}\]
x + \frac{e^{y \cdot \log \left(\frac{y}{z + y}\right)}}{y}
↓
\begin{array}{l}
\mathbf{if}\;y \le -3.86604521535774063 \cdot 10^{122} \lor \neg \left(y \le 0.0042390669371441449\right):\\
\;\;\;\;x + \frac{e^{-1 \cdot z}}{y}\\

\mathbf{else}:\\
\;\;\;\;x + \frac{e^{y \cdot 0}}{y}\\

\end{array}
double f(double x, double y, double z) {
        double r408160 = x;
        double r408161 = y;
        double r408162 = z;
        double r408163 = r408162 + r408161;
        double r408164 = r408161 / r408163;
        double r408165 = log(r408164);
        double r408166 = r408161 * r408165;
        double r408167 = exp(r408166);
        double r408168 = r408167 / r408161;
        double r408169 = r408160 + r408168;
        return r408169;
}
↓
double f(double x, double y, double z) {
        double r408170 = y;
        double r408171 = -3.8660452153577406e+122;
        bool r408172 = r408170 <= r408171;
        double r408173 = 0.004239066937144145;
        bool r408174 = r408170 <= r408173;
        double r408175 = !r408174;
        bool r408176 = r408172 || r408175;
        double r408177 = x;
        double r408178 = -1.0;
        double r408179 = z;
        double r408180 = r408178 * r408179;
        double r408181 = exp(r408180);
        double r408182 = r408181 / r408170;
        double r408183 = r408177 + r408182;
        double r408184 = 0.0;
        double r408185 = r408170 * r408184;
        double r408186 = exp(r408185);
        double r408187 = r408186 / r408170;
        double r408188 = r408177 + r408187;
        double r408189 = r408176 ? r408183 : r408188;
        return r408189;
}

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

Original5.9
Target1.0
Herbie0.3
\[\begin{array}{l} \mathbf{if}\;\frac{y}{z + y} \lt 7.1154157598 \cdot 10^{-315}:\\ \;\;\;\;x + \frac{e^{\frac{-1}{z}}}{y}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{e^{\log \left({\left(\frac{y}{y + z}\right)}^{y}\right)}}{y}\\ \end{array}\]

Derivation

  1. Split input into 2 regimes

  2. if y < -3.8660452153577406e+122 or 0.004239066937144145 < y

    1. Initial program 1.9

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

    2. Taylor expanded around inf 0.1

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

    if -3.8660452153577406e+122 < y < 0.004239066937144145

    1. Initial program 9.2

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

    2. Taylor expanded around inf 0.6

      \[\leadsto x + \frac{e^{y \cdot \color{blue}{0}}}{y}\]
  3. Recombined 2 regimes into one program.

  4. Final simplification0.3

    \[\leadsto \begin{array}{l} \mathbf{if}\;y \le -3.86604521535774063 \cdot 10^{122} \lor \neg \left(y \le 0.0042390669371441449\right):\\ \;\;\;\;x + \frac{e^{-1 \cdot z}}{y}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{e^{y \cdot 0}}{y}\\ \end{array}\]

Reproduce

herbie shell --seed 2020039 
(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)))