Average Error: 5.9 → 1.1
Time: 5.6s
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 r422229 = x;
        double r422230 = y;
        double r422231 = z;
        double r422232 = r422231 + r422230;
        double r422233 = r422230 / r422232;
        double r422234 = log(r422233);
        double r422235 = r422230 * r422234;
        double r422236 = exp(r422235);
        double r422237 = r422236 / r422230;
        double r422238 = r422229 + r422237;
        return r422238;
}
↓
double f(double x, double y, double z) {
        double r422239 = y;
        double r422240 = 7.336756572307091e-63;
        bool r422241 = r422239 <= r422240;
        double r422242 = x;
        double r422243 = exp(r422239);
        double r422244 = z;
        double r422245 = r422244 + r422239;
        double r422246 = r422239 / r422245;
        double r422247 = log(r422246);
        double r422248 = pow(r422243, r422247);
        double r422249 = r422248 / r422239;
        double r422250 = r422242 + r422249;
        double r422251 = -1.0;
        double r422252 = r422251 * r422244;
        double r422253 = exp(r422252);
        double r422254 = r422253 / r422239;
        double r422255 = r422242 + r422254;
        double r422256 = r422241 ? r422250 : r422255;
        return r422256;
}

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 +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)))