Average Error: 6.1 → 0.1
Time: 4.4s
Precision: 64
\[x + \frac{e^{y \cdot \log \left(\frac{y}{z + y}\right)}}{y}\]
↓
\[\begin{array}{l} \mathbf{if}\;y \le -1116111820951900757028957439299745235337000 \lor \neg \left(y \le 1.292740669483186222166182233195286244154\right):\\ \;\;\;\;x + \frac{1 \cdot e^{-1 \cdot z}}{y}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{{\left(e^{y}\right)}^{\left(\log \left(\frac{y}{z + y}\right)\right)}}{y}\\ \end{array}\]
x + \frac{e^{y \cdot \log \left(\frac{y}{z + y}\right)}}{y}
↓
\begin{array}{l}
\mathbf{if}\;y \le -1116111820951900757028957439299745235337000 \lor \neg \left(y \le 1.292740669483186222166182233195286244154\right):\\
\;\;\;\;x + \frac{1 \cdot e^{-1 \cdot z}}{y}\\

\mathbf{else}:\\
\;\;\;\;x + \frac{{\left(e^{y}\right)}^{\left(\log \left(\frac{y}{z + y}\right)\right)}}{y}\\

\end{array}
double f(double x, double y, double z) {
        double r428174 = x;
        double r428175 = y;
        double r428176 = z;
        double r428177 = r428176 + r428175;
        double r428178 = r428175 / r428177;
        double r428179 = log(r428178);
        double r428180 = r428175 * r428179;
        double r428181 = exp(r428180);
        double r428182 = r428181 / r428175;
        double r428183 = r428174 + r428182;
        return r428183;
}
↓
double f(double x, double y, double z) {
        double r428184 = y;
        double r428185 = -1.1161118209519008e+42;
        bool r428186 = r428184 <= r428185;
        double r428187 = 1.2927406694831862;
        bool r428188 = r428184 <= r428187;
        double r428189 = !r428188;
        bool r428190 = r428186 || r428189;
        double r428191 = x;
        double r428192 = 1.0;
        double r428193 = -1.0;
        double r428194 = z;
        double r428195 = r428193 * r428194;
        double r428196 = exp(r428195);
        double r428197 = r428192 * r428196;
        double r428198 = r428197 / r428184;
        double r428199 = r428191 + r428198;
        double r428200 = exp(r428184);
        double r428201 = r428194 + r428184;
        double r428202 = r428184 / r428201;
        double r428203 = log(r428202);
        double r428204 = pow(r428200, r428203);
        double r428205 = r428204 / r428184;
        double r428206 = r428191 + r428205;
        double r428207 = r428190 ? r428199 : r428206;
        return r428207;
}

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
Target1.3
Herbie0.1
\[\begin{array}{l} \mathbf{if}\;\frac{y}{z + y} \lt 7.115415759790762719541517221498726780517 \cdot 10^{-315}:\\ \;\;\;\;x + \frac{e^{\frac{-1}{z}}}{y}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{e^{\log \left({\left(\frac{y}{y + z}\right)}^{y}\right)}}{y}\\ \end{array}\]

Derivation

  1. Split input into 2 regimes

  2. if y < -1.1161118209519008e+42 or 1.2927406694831862 < y

    1. Initial program 2.4

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

    3. Applied *-un-lft-identity2.4

      \[\leadsto x + \frac{e^{y \cdot \log \left(\frac{y}{\color{blue}{1 \cdot \left(z + y\right)}}\right)}}{y}\]

    4. Applied *-un-lft-identity2.4

      \[\leadsto x + \frac{e^{y \cdot \log \left(\frac{\color{blue}{1 \cdot y}}{1 \cdot \left(z + y\right)}\right)}}{y}\]

    5. Applied times-frac2.4

      \[\leadsto x + \frac{e^{y \cdot \log \color{blue}{\left(\frac{1}{1} \cdot \frac{y}{z + y}\right)}}}{y}\]

    6. Applied log-prod2.4

      \[\leadsto x + \frac{e^{y \cdot \color{blue}{\left(\log \left(\frac{1}{1}\right) + \log \left(\frac{y}{z + y}\right)\right)}}}{y}\]

    7. Applied distribute-lft-in2.4

      \[\leadsto x + \frac{e^{\color{blue}{y \cdot \log \left(\frac{1}{1}\right) + y \cdot \log \left(\frac{y}{z + y}\right)}}}{y}\]

    8. Applied exp-sum2.4

      \[\leadsto x + \frac{\color{blue}{e^{y \cdot \log \left(\frac{1}{1}\right)} \cdot e^{y \cdot \log \left(\frac{y}{z + y}\right)}}}{y}\]

    9. Simplified2.4

      \[\leadsto x + \frac{\color{blue}{1} \cdot e^{y \cdot \log \left(\frac{y}{z + y}\right)}}{y}\]

    10. Simplified2.4

      \[\leadsto x + \frac{1 \cdot \color{blue}{{\left(\frac{y}{z + y}\right)}^{y}}}{y}\]

    11. Taylor expanded around inf 0.0

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

    12. Simplified0.0

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

    if -1.1161118209519008e+42 < y < 1.2927406694831862

    1. Initial program 9.8

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

    3. Applied add-log-exp14.1

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

    4. Applied exp-to-pow0.1

      \[\leadsto x + \frac{\color{blue}{{\left(e^{y}\right)}^{\left(\log \left(\frac{y}{z + y}\right)\right)}}}{y}\]
  3. Recombined 2 regimes into one program.

  4. Final simplification0.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;y \le -1116111820951900757028957439299745235337000 \lor \neg \left(y \le 1.292740669483186222166182233195286244154\right):\\ \;\;\;\;x + \frac{1 \cdot e^{-1 \cdot z}}{y}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{{\left(e^{y}\right)}^{\left(\log \left(\frac{y}{z + y}\right)\right)}}{y}\\ \end{array}\]

Reproduce

herbie shell --seed 2020001 +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)))