Average Error: 13.2 → 1.1
Time: 4.2s
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 ((((double) (wj + ((double) (((double) (x - ((double) (wj * ((double) exp(wj)))))) / ((double) (((double) exp(wj)) + ((double) (wj * ((double) exp(wj)))))))))) <= 7.992060590681988e-17)) {
		VAR = ((double) (x + ((double) (wj * ((double) (wj + ((double) (x * -2.0))))))));
	} else {
		VAR = ((double) (wj + ((double) (((double) (((double) cbrt(((double) (((double) (x / ((double) exp(wj)))) - wj)))) * ((double) cbrt(((double) (((double) (x / ((double) exp(wj)))) - wj)))))) / ((double) (((double) (wj + 1.0)) / ((double) cbrt(((double) (((double) (x / ((double) exp(wj)))) - wj))))))))));
	}
	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
Original13.2
Target12.7
Herbie1.1
\[\]
Derivation
  1. Split input into 2 regimes
  2. if  (- wj (/ (- (* wj (exp wj)) x) (+ (exp wj) (* wj (exp wj)))))  < 7.992060591e-17
    1. Initial program 17.5
      \[\]
    2. Simplified17.5
      \[\leadsto \]
    3. Taylor expanded around 0 0.7
      \[\leadsto \]
    4. Simplified0.8
      \[\leadsto \]
    if 7.992060591e-17 <  (- wj (/ (- (* wj (exp wj)) x) (+ (exp wj) (* wj (exp wj))))) 
    1. Initial program 2.3
      \[\]
    2. Simplified0.5
      \[\leadsto \]
    3. Using strategy rm
    4. Applied add-cube-cbrt1.8
      \[\leadsto \]
    5. Applied associate-/l*1.8
      \[\leadsto \]
  3. Recombined 2 regimes into one program.
  4. Final simplification1.1
    \[\leadsto \]
Reproduce
herbie shell --seed 2020192 
(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))))))