Average Error: 6.3 → 1.2
Time: 6.3s
Precision: 64
\[x + \frac{e^{y \cdot \log \left(\frac{y}{z + y}\right)}}{y}\]
↓
\[x + \frac{{\left(e^{\sqrt[3]{y} \cdot \sqrt[3]{y}}\right)}^{\left(\sqrt[3]{y} \cdot \log \left(\frac{y}{z + y}\right)\right)}}{y}\]
x + \frac{e^{y \cdot \log \left(\frac{y}{z + y}\right)}}{y}
↓
x + \frac{{\left(e^{\sqrt[3]{y} \cdot \sqrt[3]{y}}\right)}^{\left(\sqrt[3]{y} \cdot \log \left(\frac{y}{z + y}\right)\right)}}{y}
double f(double x, double y, double z) {
        double r342267 = x;
        double r342268 = y;
        double r342269 = z;
        double r342270 = r342269 + r342268;
        double r342271 = r342268 / r342270;
        double r342272 = log(r342271);
        double r342273 = r342268 * r342272;
        double r342274 = exp(r342273);
        double r342275 = r342274 / r342268;
        double r342276 = r342267 + r342275;
        return r342276;
}
↓
double f(double x, double y, double z) {
        double r342277 = x;
        double r342278 = y;
        double r342279 = cbrt(r342278);
        double r342280 = r342279 * r342279;
        double r342281 = exp(r342280);
        double r342282 = z;
        double r342283 = r342282 + r342278;
        double r342284 = r342278 / r342283;
        double r342285 = log(r342284);
        double r342286 = r342279 * r342285;
        double r342287 = pow(r342281, r342286);
        double r342288 = r342287 / r342278;
        double r342289 = r342277 + r342288;
        return r342289;
}

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

Original6.3
Target1.2
Herbie1.2
\[\begin{array}{l} \mathbf{if}\;\frac{y}{z + y} \lt 7.1154157598 \cdot 10^{-315}:\\ \;\;\;\;x + \frac{e^{\frac{-1}{z}}}{y}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{e^{\log \left({\left(\frac{y}{y + z}\right)}^{y}\right)}}{y}\\ \end{array}\]

Derivation

  1. Initial program 6.3

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

  3. Applied add-log-exp35.1

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

  4. Applied exp-to-pow1.2

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

  6. Applied add-cube-cbrt1.2

    \[\leadsto x + \frac{{\left(e^{\color{blue}{\left(\sqrt[3]{y} \cdot \sqrt[3]{y}\right) \cdot \sqrt[3]{y}}}\right)}^{\left(\log \left(\frac{y}{z + y}\right)\right)}}{y}\]

  7. Applied exp-prod1.2

    \[\leadsto x + \frac{{\color{blue}{\left({\left(e^{\sqrt[3]{y} \cdot \sqrt[3]{y}}\right)}^{\left(\sqrt[3]{y}\right)}\right)}}^{\left(\log \left(\frac{y}{z + y}\right)\right)}}{y}\]

  8. Applied pow-pow1.2

    \[\leadsto x + \frac{\color{blue}{{\left(e^{\sqrt[3]{y} \cdot \sqrt[3]{y}}\right)}^{\left(\sqrt[3]{y} \cdot \log \left(\frac{y}{z + y}\right)\right)}}}{y}\]

  9. Final simplification1.2

    \[\leadsto x + \frac{{\left(e^{\sqrt[3]{y} \cdot \sqrt[3]{y}}\right)}^{\left(\sqrt[3]{y} \cdot \log \left(\frac{y}{z + y}\right)\right)}}{y}\]

Reproduce

herbie shell --seed 2020035 +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)))