Average Error: 14.0 → 0.6
Time: 10.5s
Precision: binary64
\[wj - \frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}} \]
↓
\[\begin{array}{l} \mathbf{if}\;\begin{array}{l} t_0 := wj \cdot e^{wj}\\ wj - \frac{t_0 - x}{e^{wj} + t_0} \leq 1.1510916157317222 \cdot 10^{-19} \end{array}:\\ \;\;\;\;\left(x + \left(x \cdot 2.5 + 1\right) \cdot \left(wj \cdot wj\right)\right) + \left(x \cdot \left(wj \cdot -2\right) - \left(1 + x \cdot 2.6666666666666665\right) \cdot {wj}^{3}\right)\\ \mathbf{else}:\\ \;\;\;\;wj + \frac{\frac{x}{e^{wj}} - wj}{wj \cdot wj - 1} \cdot \left(wj - 1\right)\\ \end{array} \]
wj - \frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}}
↓
\begin{array}{l}
\mathbf{if}\;\begin{array}{l}
t_0 := wj \cdot e^{wj}\\
wj - \frac{t_0 - x}{e^{wj} + t_0} \leq 1.1510916157317222 \cdot 10^{-19}
\end{array}:\\
\;\;\;\;\left(x + \left(x \cdot 2.5 + 1\right) \cdot \left(wj \cdot wj\right)\right) + \left(x \cdot \left(wj \cdot -2\right) - \left(1 + x \cdot 2.6666666666666665\right) \cdot {wj}^{3}\right)\\

\mathbf{else}:\\
\;\;\;\;wj + \frac{\frac{x}{e^{wj}} - wj}{wj \cdot wj - 1} \cdot \left(wj - 1\right)\\


\end{array}

(FPCore (wj x)
 :precision binary64
 (- wj (/ (- (* wj (exp wj)) x) (+ (exp wj) (* wj (exp wj))))))
↓
(FPCore (wj x)
 :precision binary64
 (if (let* ((t_0 (* wj (exp wj))))
       (<= (- wj (/ (- t_0 x) (+ (exp wj) t_0))) 1.1510916157317222e-19))
   (+
    (+ x (* (+ (* x 2.5) 1.0) (* wj wj)))
    (- (* x (* wj -2.0)) (* (+ 1.0 (* x 2.6666666666666665)) (pow wj 3.0))))
   (+ wj (* (/ (- (/ x (exp wj)) wj) (- (* wj wj) 1.0)) (- wj 1.0)))))
double code(double wj, double x) {
	return wj - (((wj * exp(wj)) - x) / (exp(wj) + (wj * exp(wj))));
}
↓
double code(double wj, double x) {
	double t_0 = wj * exp(wj);
	double tmp;
	if ((wj - ((t_0 - x) / (exp(wj) + t_0))) <= 1.1510916157317222e-19) {
		tmp = (x + (((x * 2.5) + 1.0) * (wj * wj))) + ((x * (wj * -2.0)) - ((1.0 + (x * 2.6666666666666665)) * pow(wj, 3.0)));
	} else {
		tmp = wj + ((((x / exp(wj)) - wj) / ((wj * wj) - 1.0)) * (wj - 1.0));
	}
	return tmp;
}

Error

Bits error versus wj

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original14.0
Target13.2
Herbie0.6
\[wj - \left(\frac{wj}{wj + 1} - \frac{x}{e^{wj} + wj \cdot e^{wj}}\right) \]

Derivation

  1. Split input into 2 regimes

  2. if (-.f64 wj (/.f64 (-.f64 (*.f64 wj (exp.f64 wj)) x) (+.f64 (exp.f64 wj) (*.f64 wj (exp.f64 wj))))) < 1.15109162e-19

    1. Initial program 18.2

      \[wj - \frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}} \]

    2. Simplified18.2

      \[\leadsto \color{blue}{wj + \frac{\frac{x}{e^{wj}} - wj}{wj + 1}} \]

    3. Taylor expanded around 0 0.4

      \[\leadsto \color{blue}{\left(2.5 \cdot \left({wj}^{2} \cdot x\right) + \left({wj}^{2} + x\right)\right) - \left(2 \cdot \left(wj \cdot x\right) + \left(2.6666666666666665 \cdot \left({wj}^{3} \cdot x\right) + {wj}^{3}\right)\right)} \]

    4. Simplified0.5

      \[\leadsto \color{blue}{\left(x + \left(2.5 \cdot x + 1\right) \cdot \left(wj \cdot wj\right)\right) + \left(x \cdot \left(wj \cdot -2\right) - \left(2.6666666666666665 \cdot x + 1\right) \cdot {wj}^{3}\right)} \]

    if 1.15109162e-19 < (-.f64 wj (/.f64 (-.f64 (*.f64 wj (exp.f64 wj)) x) (+.f64 (exp.f64 wj) (*.f64 wj (exp.f64 wj)))))

    1. Initial program 3.3

      \[wj - \frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}} \]

    2. Simplified0.7

      \[\leadsto \color{blue}{wj + \frac{\frac{x}{e^{wj}} - wj}{wj + 1}} \]
    3. Using strategy rm

    4. Applied flip-+_binary640.7

      \[\leadsto wj + \frac{\frac{x}{e^{wj}} - wj}{\color{blue}{\frac{wj \cdot wj - 1 \cdot 1}{wj - 1}}} \]

    5. Applied associate-/r/_binary640.7

      \[\leadsto wj + \color{blue}{\frac{\frac{x}{e^{wj}} - wj}{wj \cdot wj - 1 \cdot 1} \cdot \left(wj - 1\right)} \]
  3. Recombined 2 regimes into one program.

  4. Final simplification0.6

    \[\leadsto \begin{array}{l} \mathbf{if}\;wj - \frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}} \leq 1.1510916157317222 \cdot 10^{-19}:\\ \;\;\;\;\left(x + \left(x \cdot 2.5 + 1\right) \cdot \left(wj \cdot wj\right)\right) + \left(x \cdot \left(wj \cdot -2\right) - \left(1 + x \cdot 2.6666666666666665\right) \cdot {wj}^{3}\right)\\ \mathbf{else}:\\ \;\;\;\;wj + \frac{\frac{x}{e^{wj}} - wj}{wj \cdot wj - 1} \cdot \left(wj - 1\right)\\ \end{array} \]

Reproduce

herbie shell --seed 2021198 
(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))))))