Average Error: 5.8 → 2.1
Time: 20.3s
Precision: 64
\[x + \frac{e^{y \cdot \log \left(\frac{y}{z + y}\right)}}{y}\]
↓
\[\frac{{\left(\frac{\sqrt[3]{y} \cdot \sqrt[3]{y}}{\sqrt[3]{y + z} \cdot \sqrt[3]{y + z}}\right)}^{y}}{\frac{y}{{\left(\frac{\sqrt[3]{y}}{\sqrt[3]{y + z}}\right)}^{y}}} + x\]
x + \frac{e^{y \cdot \log \left(\frac{y}{z + y}\right)}}{y}
↓
\frac{{\left(\frac{\sqrt[3]{y} \cdot \sqrt[3]{y}}{\sqrt[3]{y + z} \cdot \sqrt[3]{y + z}}\right)}^{y}}{\frac{y}{{\left(\frac{\sqrt[3]{y}}{\sqrt[3]{y + z}}\right)}^{y}}} + x
double f(double x, double y, double z) {
        double r907153 = x;
        double r907154 = y;
        double r907155 = z;
        double r907156 = r907155 + r907154;
        double r907157 = r907154 / r907156;
        double r907158 = log(r907157);
        double r907159 = r907154 * r907158;
        double r907160 = exp(r907159);
        double r907161 = r907160 / r907154;
        double r907162 = r907153 + r907161;
        return r907162;
}
↓
double f(double x, double y, double z) {
        double r907163 = y;
        double r907164 = cbrt(r907163);
        double r907165 = r907164 * r907164;
        double r907166 = z;
        double r907167 = r907163 + r907166;
        double r907168 = cbrt(r907167);
        double r907169 = r907168 * r907168;
        double r907170 = r907165 / r907169;
        double r907171 = pow(r907170, r907163);
        double r907172 = r907164 / r907168;
        double r907173 = pow(r907172, r907163);
        double r907174 = r907163 / r907173;
        double r907175 = r907171 / r907174;
        double r907176 = x;
        double r907177 = r907175 + r907176;
        return r907177;
}

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

Original5.8
Target1.0
Herbie2.1
\[\begin{array}{l} \mathbf{if}\;\frac{y}{z + y} \lt 7.115415759790762719541517221498726780517 \cdot 10^{-315}:\\ \;\;\;\;x + \frac{e^{\frac{-1}{z}}}{y}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{e^{\log \left({\left(\frac{y}{y + z}\right)}^{y}\right)}}{y}\\ \end{array}\]

Derivation

  1. Initial program 5.8

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

  2. Simplified5.8

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

  4. Applied add-cube-cbrt19.1

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

  5. Applied add-cube-cbrt5.8

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

  6. Applied times-frac5.8

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

  7. Applied unpow-prod-down2.1

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

  8. Applied associate-/l*2.1

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

  9. Final simplification2.1

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

Reproduce

herbie shell --seed 2019179 
(FPCore (x y z)
  :name "Numeric.SpecFunctions:invIncompleteBetaWorker from math-functions-0.1.5.2, G"

  :herbie-target
  (if (< (/ y (+ z y)) 7.1154157597908e-315) (+ x (/ (exp (/ -1.0 z)) y)) (+ x (/ (exp (log (pow (/ y (+ y z)) y))) y)))

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