Average Error: 5.8 → 0.9
Time: 10.5s
Precision: 64
\[x + \frac{e^{y \cdot \log \left(\frac{y}{z + y}\right)}}{y}\]
↓
\[x + \frac{\left({\left(\left|\frac{\sqrt[3]{y}}{\sqrt[3]{z + y}}\right|\right)}^{y} \cdot {\left(\left|\frac{\sqrt[3]{y}}{\sqrt[3]{z + y}}\right|\right)}^{y}\right) \cdot {\left(\frac{\sqrt[3]{y}}{\sqrt[3]{z + y}}\right)}^{y}}{y}\]
x + \frac{e^{y \cdot \log \left(\frac{y}{z + y}\right)}}{y}
↓
x + \frac{\left({\left(\left|\frac{\sqrt[3]{y}}{\sqrt[3]{z + y}}\right|\right)}^{y} \cdot {\left(\left|\frac{\sqrt[3]{y}}{\sqrt[3]{z + y}}\right|\right)}^{y}\right) \cdot {\left(\frac{\sqrt[3]{y}}{\sqrt[3]{z + y}}\right)}^{y}}{y}
double f(double x, double y, double z) {
        double r442186 = x;
        double r442187 = y;
        double r442188 = z;
        double r442189 = r442188 + r442187;
        double r442190 = r442187 / r442189;
        double r442191 = log(r442190);
        double r442192 = r442187 * r442191;
        double r442193 = exp(r442192);
        double r442194 = r442193 / r442187;
        double r442195 = r442186 + r442194;
        return r442195;
}
↓
double f(double x, double y, double z) {
        double r442196 = x;
        double r442197 = y;
        double r442198 = cbrt(r442197);
        double r442199 = z;
        double r442200 = r442199 + r442197;
        double r442201 = cbrt(r442200);
        double r442202 = r442198 / r442201;
        double r442203 = fabs(r442202);
        double r442204 = pow(r442203, r442197);
        double r442205 = r442204 * r442204;
        double r442206 = pow(r442202, r442197);
        double r442207 = r442205 * r442206;
        double r442208 = r442207 / r442197;
        double r442209 = r442196 + r442208;
        return r442209;
}

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
Target0.9
Herbie0.9
\[\begin{array}{l} \mathbf{if}\;\frac{y}{z + y} \lt 7.1154157598 \cdot 10^{-315}:\\ \;\;\;\;x + \frac{e^{\frac{-1}{z}}}{y}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{e^{\log \left({\left(\frac{y}{y + z}\right)}^{y}\right)}}{y}\\ \end{array}\]

Derivation

  1. Initial program 5.8

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

  2. Simplified5.8

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

  4. Applied add-cube-cbrt19.6

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

  5. Applied add-cube-cbrt5.8

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

  6. Applied times-frac5.8

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

  7. Applied unpow-prod-down1.9

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

  9. Applied add-sqr-sqrt1.9

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

  10. Applied unpow-prod-down1.9

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

  11. Simplified1.9

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

  12. Simplified0.9

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

  13. Final simplification0.9

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

Reproduce

herbie shell --seed 2020047 +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)))