Average Error: 5.9 → 1.1
Time: 4.4s
Precision: 64
\[x + \frac{e^{y \cdot \log \left(\frac{y}{z + y}\right)}}{y}\]
↓
\[\begin{array}{l} \mathbf{if}\;y \le 7.3367565723070908 \cdot 10^{-63}:\\ \;\;\;\;x + \frac{{\left(e^{y}\right)}^{\left(\log \left(\frac{y}{z + y}\right)\right)}}{y}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{e^{-1 \cdot z}}{y}\\ \end{array}\]
x + \frac{e^{y \cdot \log \left(\frac{y}{z + y}\right)}}{y}
↓
\begin{array}{l}
\mathbf{if}\;y \le 7.3367565723070908 \cdot 10^{-63}:\\
\;\;\;\;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 r499983 = x;
        double r499984 = y;
        double r499985 = z;
        double r499986 = r499985 + r499984;
        double r499987 = r499984 / r499986;
        double r499988 = log(r499987);
        double r499989 = r499984 * r499988;
        double r499990 = exp(r499989);
        double r499991 = r499990 / r499984;
        double r499992 = r499983 + r499991;
        return r499992;
}
↓
double f(double x, double y, double z) {
        double r499993 = y;
        double r499994 = 7.336756572307091e-63;
        bool r499995 = r499993 <= r499994;
        double r499996 = x;
        double r499997 = exp(r499993);
        double r499998 = z;
        double r499999 = r499998 + r499993;
        double r500000 = r499993 / r499999;
        double r500001 = log(r500000);
        double r500002 = pow(r499997, r500001);
        double r500003 = r500002 / r499993;
        double r500004 = r499996 + r500003;
        double r500005 = -1.0;
        double r500006 = r500005 * r499998;
        double r500007 = exp(r500006);
        double r500008 = r500007 / r499993;
        double r500009 = r499996 + r500008;
        double r500010 = r499995 ? r500004 : r500009;
        return r500010;
}

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
Target1.1
Herbie1.1
\[\begin{array}{l} \mathbf{if}\;\frac{y}{z + y} \lt 7.1154157598 \cdot 10^{-315}:\\ \;\;\;\;x + \frac{e^{\frac{-1}{z}}}{y}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{e^{\log \left({\left(\frac{y}{y + z}\right)}^{y}\right)}}{y}\\ \end{array}\]

Derivation

  1. Split input into 2 regimes

  2. if y < 7.336756572307091e-63

    1. Initial program 8.1

      \[x + \frac{e^{y \cdot \log \left(\frac{y}{z + y}\right)}}{y}\]
    2. Using strategy rm

    3. Applied add-log-exp29.8

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

    4. Applied exp-to-pow0.9

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

    if 7.336756572307091e-63 < y

    1. Initial program 1.5

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

    2. Taylor expanded around inf 1.6

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

  4. Final simplification1.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;y \le 7.3367565723070908 \cdot 10^{-63}:\\ \;\;\;\;x + \frac{{\left(e^{y}\right)}^{\left(\log \left(\frac{y}{z + y}\right)\right)}}{y}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{e^{-1 \cdot z}}{y}\\ \end{array}\]

Reproduce

herbie shell --seed 2020036 
(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)))