Average Error: 6.1 → 1.0
Time: 11.1s
Precision: 64
\[x + \frac{e^{y \cdot \log \left(\frac{y}{z + y}\right)}}{y}\]
↓
\[x + \frac{e^{y \cdot \left(2 \cdot \log \left(\frac{\sqrt[3]{y}}{\sqrt[3]{z + y}}\right)\right) + \log \left(\frac{\sqrt[3]{y}}{\sqrt[3]{z + y}}\right) \cdot y}}{y}\]
x + \frac{e^{y \cdot \log \left(\frac{y}{z + y}\right)}}{y}
↓
x + \frac{e^{y \cdot \left(2 \cdot \log \left(\frac{\sqrt[3]{y}}{\sqrt[3]{z + y}}\right)\right) + \log \left(\frac{\sqrt[3]{y}}{\sqrt[3]{z + y}}\right) \cdot y}}{y}
double f(double x, double y, double z) {
        double r395464 = x;
        double r395465 = y;
        double r395466 = z;
        double r395467 = r395466 + r395465;
        double r395468 = r395465 / r395467;
        double r395469 = log(r395468);
        double r395470 = r395465 * r395469;
        double r395471 = exp(r395470);
        double r395472 = r395471 / r395465;
        double r395473 = r395464 + r395472;
        return r395473;
}
↓
double f(double x, double y, double z) {
        double r395474 = x;
        double r395475 = y;
        double r395476 = 2.0;
        double r395477 = cbrt(r395475);
        double r395478 = z;
        double r395479 = r395478 + r395475;
        double r395480 = cbrt(r395479);
        double r395481 = r395477 / r395480;
        double r395482 = log(r395481);
        double r395483 = r395476 * r395482;
        double r395484 = r395475 * r395483;
        double r395485 = r395482 * r395475;
        double r395486 = r395484 + r395485;
        double r395487 = exp(r395486);
        double r395488 = r395487 / r395475;
        double r395489 = r395474 + r395488;
        return r395489;
}

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
Target1.0
Herbie1.0
\[\begin{array}{l} \mathbf{if}\;\frac{y}{z + y} \lt 7.1154157598 \cdot 10^{-315}:\\ \;\;\;\;x + \frac{e^{\frac{-1}{z}}}{y}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{e^{\log \left({\left(\frac{y}{y + z}\right)}^{y}\right)}}{y}\\ \end{array}\]

Derivation

  1. Initial program 6.1

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

  3. Applied add-cube-cbrt19.2

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

  4. Applied add-cube-cbrt6.1

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

  5. Applied times-frac6.1

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

  6. Applied log-prod2.1

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

  7. Applied distribute-lft-in2.1

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

  8. Simplified1.0

    \[\leadsto x + \frac{e^{\color{blue}{y \cdot \left(2 \cdot \log \left(\frac{\sqrt[3]{y}}{\sqrt[3]{z + y}}\right)\right)} + y \cdot \log \left(\frac{\sqrt[3]{y}}{\sqrt[3]{z + y}}\right)}}{y}\]

  9. Simplified1.0

    \[\leadsto x + \frac{e^{y \cdot \left(2 \cdot \log \left(\frac{\sqrt[3]{y}}{\sqrt[3]{z + y}}\right)\right) + \color{blue}{\log \left(\frac{\sqrt[3]{y}}{\sqrt[3]{z + y}}\right) \cdot y}}}{y}\]

  10. Final simplification1.0

    \[\leadsto x + \frac{e^{y \cdot \left(2 \cdot \log \left(\frac{\sqrt[3]{y}}{\sqrt[3]{z + y}}\right)\right) + \log \left(\frac{\sqrt[3]{y}}{\sqrt[3]{z + y}}\right) \cdot y}}{y}\]

Reproduce

herbie shell --seed 2020045 +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)))