Average Error: 6.2 → 1.1
Time: 11.1s
Precision: 64
\[x + \frac{e^{y \cdot \log \left(\frac{y}{z + y}\right)}}{y}\]
↓
\[\begin{array}{l} \mathbf{if}\;y \le 8.521901356571207806640367488333446054748 \cdot 10^{-44}:\\ \;\;\;\;x + \frac{e^{y \cdot 0}}{y}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{1}{y \cdot e^{z}}\\ \end{array}\]
x + \frac{e^{y \cdot \log \left(\frac{y}{z + y}\right)}}{y}
↓
\begin{array}{l}
\mathbf{if}\;y \le 8.521901356571207806640367488333446054748 \cdot 10^{-44}:\\
\;\;\;\;x + \frac{e^{y \cdot 0}}{y}\\

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

\end{array}
double f(double x, double y, double z) {
        double r253937 = x;
        double r253938 = y;
        double r253939 = z;
        double r253940 = r253939 + r253938;
        double r253941 = r253938 / r253940;
        double r253942 = log(r253941);
        double r253943 = r253938 * r253942;
        double r253944 = exp(r253943);
        double r253945 = r253944 / r253938;
        double r253946 = r253937 + r253945;
        return r253946;
}
↓
double f(double x, double y, double z) {
        double r253947 = y;
        double r253948 = 8.521901356571208e-44;
        bool r253949 = r253947 <= r253948;
        double r253950 = x;
        double r253951 = 0.0;
        double r253952 = r253947 * r253951;
        double r253953 = exp(r253952);
        double r253954 = r253953 / r253947;
        double r253955 = r253950 + r253954;
        double r253956 = 1.0;
        double r253957 = z;
        double r253958 = exp(r253957);
        double r253959 = r253947 * r253958;
        double r253960 = r253956 / r253959;
        double r253961 = r253950 + r253960;
        double r253962 = r253949 ? r253955 : r253961;
        return r253962;
}

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

Original6.2
Target1.2
Herbie1.1
\[\begin{array}{l} \mathbf{if}\;\frac{y}{z + y} \lt 7.115415759790762719541517221498726780517 \cdot 10^{-315}:\\ \;\;\;\;x + \frac{e^{\frac{-1}{z}}}{y}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{e^{\log \left({\left(\frac{y}{y + z}\right)}^{y}\right)}}{y}\\ \end{array}\]

Derivation

  1. Split input into 2 regimes

  2. if y < 8.521901356571208e-44

    1. Initial program 8.3

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

    2. Taylor expanded around inf 1.1

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

    if 8.521901356571208e-44 < y

    1. Initial program 1.9

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

    2. Taylor expanded around inf 1.2

      \[\leadsto x + \frac{\color{blue}{e^{-1 \cdot z}}}{y}\]
    3. Using strategy rm

    4. Applied clear-num1.2

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

    5. Simplified1.2

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

  4. Final simplification1.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;y \le 8.521901356571207806640367488333446054748 \cdot 10^{-44}:\\ \;\;\;\;x + \frac{e^{y \cdot 0}}{y}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{1}{y \cdot e^{z}}\\ \end{array}\]

Reproduce

herbie shell --seed 2019291 
(FPCore (x y z)
  :name "Numeric.SpecFunctions:invIncompleteBetaWorker from math-functions-0.1.5.2, G"
  :precision binary64

  :herbie-target
  (if (< (/ y (+ z y)) 7.1154157598e-315) (+ x (/ (exp (/ -1 z)) y)) (+ x (/ (exp (log (pow (/ y (+ y z)) y))) y)))

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