Average Error: 6.2 → 1.0
Time: 3.9s
Precision: binary64
\[x + \frac{e^{y \cdot \log \left(\frac{y}{z + y}\right)}}{y}\]
↓
\[x + \frac{e^{y \cdot \left(2 \cdot \log \left(\frac{\sqrt[3]{y}}{\sqrt[3]{y + z}}\right)\right)} \cdot {\left(\frac{\sqrt[3]{y}}{\sqrt[3]{y + z}}\right)}^{y}}{y}\]
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]{y + z}}\right)\right)} \cdot {\left(\frac{\sqrt[3]{y}}{\sqrt[3]{y + z}}\right)}^{y}}{y}
(FPCore (x y z)
 :precision binary64
 (+ x (/ (exp (* y (log (/ y (+ z y))))) y)))
↓
(FPCore (x y z)
 :precision binary64
 (+
  x
  (/
   (*
    (exp (* y (* 2.0 (log (/ (cbrt y) (cbrt (+ y z)))))))
    (pow (/ (cbrt y) (cbrt (+ y z))) y))
   y)))
double code(double x, double y, double z) {
	return x + (exp(y * log(y / (z + y))) / y);
}
↓
double code(double x, double y, double z) {
	return x + ((exp(y * (2.0 * log(cbrt(y) / cbrt(y + z)))) * pow((cbrt(y) / cbrt(y + z)), y)) / y);
}

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

Original6.2
Target1.0
Herbie1.0
\[\begin{array}{l} \mathbf{if}\;\frac{y}{z + y} < 7.1154157597908 \cdot 10^{-315}:\\ \;\;\;\;x + \frac{e^{\frac{-1}{z}}}{y}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{e^{\log \left({\left(\frac{y}{y + z}\right)}^{y}\right)}}{y}\\ \end{array}\]

Derivation

  1. Initial program 6.2

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

  2. Simplified6.2

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

  4. Applied add-cube-cbrt_binary6419.5

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

  5. Applied add-cube-cbrt_binary646.2

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

  6. Applied times-frac_binary646.2

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

  7. Applied unpow-prod-down_binary642.0

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

  9. Applied add-exp-log_binary6439.3

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

  10. Applied add-exp-log_binary6439.4

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

  11. Applied prod-exp_binary6439.4

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

  12. Applied add-exp-log_binary6439.4

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

  13. Applied add-exp-log_binary6434.8

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

  14. Applied prod-exp_binary6430.8

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

  15. Applied div-exp_binary6430.8

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

  16. Applied pow-exp_binary6430.3

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

  17. Simplified1.0

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

  18. Final simplification1.0

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

Reproduce

herbie shell --seed 2020224 
(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 (/ -1.0 z)) y)) (+ x (/ (exp (log (pow (/ y (+ y z)) y))) y)))

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