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 r432056 = x;
        double r432057 = y;
        double r432058 = z;
        double r432059 = r432058 + r432057;
        double r432060 = r432057 / r432059;
        double r432061 = log(r432060);
        double r432062 = r432057 * r432061;
        double r432063 = exp(r432062);
        double r432064 = r432063 / r432057;
        double r432065 = r432056 + r432064;
        return r432065;
}
↓
double f(double x, double y, double z) {
        double r432066 = y;
        double r432067 = 1.1841663707457728e-17;
        bool r432068 = r432066 <= r432067;
        double r432069 = x;
        double r432070 = exp(r432066);
        double r432071 = z;
        double r432072 = r432071 + r432066;
        double r432073 = r432066 / r432072;
        double r432074 = log(r432073);
        double r432075 = pow(r432070, r432074);
        double r432076 = r432075 / r432066;
        double r432077 = r432069 + r432076;
        double r432078 = -1.0;
        double r432079 = r432078 * r432071;
        double r432080 = exp(r432079);
        double r432081 = r432080 / r432066;
        double r432082 = r432069 + r432081;
        double r432083 = r432068 ? r432077 : r432082;
        return r432083;
}

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 +o rules:numerics
(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)))