Average Error: 13.5 → 1.4
Time: 4.6s
Precision: binary64
\[\]
↓
\[\]
↓
double code(double wj, double x) {
	return ((double) (wj - ((double) (((double) (((double) (wj * ((double) exp(wj)))) - x)) / ((double) (((double) exp(wj)) + ((double) (wj * ((double) exp(wj))))))))));
}

↓
double code(double wj, double x) {
	double VAR;
	if ((wj <= -6.027656327867643e-09)) {
		VAR = ((double) (((double) cbrt(((double) (wj - ((double) (((double) (wj - ((double) (x / ((double) exp(wj)))))) / ((double) (wj + 1.0)))))))) * ((double) (((double) cbrt(((double) (wj - ((double) (((double) (wj - ((double) (x / ((double) exp(wj)))))) / ((double) (wj + 1.0)))))))) * ((double) cbrt(((double) (wj - ((double) (((double) (wj - ((double) (x / ((double) exp(wj)))))) / ((double) (wj + 1.0))))))))))));
	} else {
		VAR = ((double) (x + ((double) (wj * ((double) (wj - ((double) (x * 2.0))))))));
	}
	return VAR;
}

Error
Bits error versus wj
Bits error versus x
Try it out
Your Program's Arguments
Results
Enter valid numbers for all inputs
Target
Original13.5
Target13.0
Herbie1.4
\[\]
Derivation
  1. Split input into 2 regimes
  2. if  wj  < -6.0276563278676428e-9
    1. Initial program 5.6
      \[\]
    2. Simplified5.6
      \[\leadsto \]
    3. Using strategy rm
    4. Applied add-cube-cbrt6.2
      \[\leadsto \]
    if -6.0276563278676428e-9 <  wj 
    1. Initial program 13.6
      \[\]
    2. Simplified13.2
      \[\leadsto \]
    3. Taylor expanded around 0 1.3
      \[\leadsto \]
    4. Simplified1.3
      \[\leadsto \]
  3. Recombined 2 regimes into one program.
  4. Final simplification1.4
    \[\leadsto \]
Reproduce
herbie shell --seed 2020179 
(FPCore (wj x)
  :name "Jmat.Real.lambertw, newton loop step"
  :precision binary64

  :herbie-target
  (- wj (- (/ wj (+ wj 1.0)) (/ x (+ (exp wj) (* wj (exp wj))))))

  (- wj (/ (- (* wj (exp wj)) x) (+ (exp wj) (* wj (exp wj))))))