Average Error: 6.3 → 2.5

Time: 4.3s

Precision: 64

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

↓

\[x + \frac{e^{0}}{y}\]

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

↓

x + \frac{e^{0}}{y}

double code(double x, double y, double z) {
	return ((double) (x + ((double) (((double) exp(((double) (y * ((double) log(((double) (y / ((double) (z + y)))))))))) / y))));
}

↓

double code(double x, double y, double z) {
	return ((double) (x + ((double) (((double) exp(0.0)) / y))));
}

Error

The X axis uses an exponential scale — Bits error versus `x`

Try it out

In
Out

Enter valid numbers for all inputs

Target

Original	6.3
Target	1.1
Herbie	2.5

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

Derivation

Initial program 6.3
\[x + \frac{e^{y \cdot \log \left(\frac{y}{z + y}\right)}}{y}\]
Taylor expanded around inf 2.5
\[\leadsto x + \frac{e^{\color{blue}{0}}}{y}\]
Final simplification2.5
\[\leadsto x + \frac{e^{0}}{y}\]

Reproduce

herbie shell --seed 2020128 
(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)))