Average Error: 6.0 → 1.1
Time: 5.2s
Precision: binary64
\[x + \frac{e^{y \cdot \log \left(\frac{y}{z + y}\right)}}{y}\]
↓
\[x + \frac{{\left(\frac{\sqrt[3]{y}}{\sqrt[3]{y + z}}\right)}^{\left(y \cdot 2\right)} \cdot {\left(\frac{\sqrt[3]{y}}{\sqrt[3]{y + z}}\right)}^{y}}{y}\]
x + \frac{e^{y \cdot \log \left(\frac{y}{z + y}\right)}}{y}
↓
x + \frac{{\left(\frac{\sqrt[3]{y}}{\sqrt[3]{y + z}}\right)}^{\left(y \cdot 2\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
  (/
   (*
    (pow (/ (cbrt y) (cbrt (+ y z))) (* y 2.0))
    (pow (/ (cbrt y) (cbrt (+ y z))) y))
   y)))
double code(double x, double y, double z) {
	return ((double) (x + (((double) exp(((double) (y * ((double) log((y / ((double) (z + y))))))))) / y)));
}
↓
double code(double x, double y, double z) {
	return ((double) (x + (((double) (((double) pow((((double) cbrt(y)) / ((double) cbrt(((double) (y + z))))), ((double) (y * 2.0)))) * ((double) pow((((double) cbrt(y)) / ((double) cbrt(((double) (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

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

Derivation

  1. Initial program Error: 6.0 bits

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

  2. SimplifiedError: 6.0 bits

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

  4. Applied add-cube-cbrtError: 19.8 bits

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

  5. Applied add-cube-cbrtError: 6.0 bits

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

  6. Applied times-fracError: 6.0 bits

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

  7. Applied unpow-prod-downError: 2.1 bits

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

  8. SimplifiedError: 2.1 bits

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

  10. Applied pow1Error: 2.1 bits

    \[\leadsto x + \frac{{\left(\frac{\sqrt[3]{y}}{\sqrt[3]{y + z}} \cdot \color{blue}{{\left(\frac{\sqrt[3]{y}}{\sqrt[3]{y + z}}\right)}^{1}}\right)}^{y} \cdot {\left(\frac{\sqrt[3]{y}}{\sqrt[3]{y + z}}\right)}^{y}}{y}\]

  11. Applied pow1Error: 2.1 bits

    \[\leadsto x + \frac{{\left(\color{blue}{{\left(\frac{\sqrt[3]{y}}{\sqrt[3]{y + z}}\right)}^{1}} \cdot {\left(\frac{\sqrt[3]{y}}{\sqrt[3]{y + z}}\right)}^{1}\right)}^{y} \cdot {\left(\frac{\sqrt[3]{y}}{\sqrt[3]{y + z}}\right)}^{y}}{y}\]

  12. Applied pow-prod-upError: 2.1 bits

    \[\leadsto x + \frac{{\color{blue}{\left({\left(\frac{\sqrt[3]{y}}{\sqrt[3]{y + z}}\right)}^{\left(1 + 1\right)}\right)}}^{y} \cdot {\left(\frac{\sqrt[3]{y}}{\sqrt[3]{y + z}}\right)}^{y}}{y}\]

  13. Applied pow-powError: 1.1 bits

    \[\leadsto x + \frac{\color{blue}{{\left(\frac{\sqrt[3]{y}}{\sqrt[3]{y + z}}\right)}^{\left(\left(1 + 1\right) \cdot y\right)}} \cdot {\left(\frac{\sqrt[3]{y}}{\sqrt[3]{y + z}}\right)}^{y}}{y}\]

  14. SimplifiedError: 1.1 bits

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

  15. Final simplificationError: 1.1 bits

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

Reproduce

herbie shell --seed 2020204 
(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)))