Average Error: 13.8 → 1.3
Time: 5.6s
Precision: binary64
\[wj - \frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}} \]
↓
\[\begin{array}{l} \mathbf{if}\;x \leq -6.143922385183966 \cdot 10^{+41}:\\ \;\;\;\;wj + \frac{\frac{x}{e^{wj}} - wj}{wj + 1}\\ \mathbf{else}:\\ \;\;\;\;\left(\mathsf{fma}\left(wj, wj, x\right) + x \cdot \mathsf{fma}\left(wj, \mathsf{fma}\left(2.5, wj, -2\right), {wj}^{3} \cdot -2.6666666666666665\right)\right) - {wj}^{3}\\ \end{array} \]
wj - \frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}}
↓
\begin{array}{l}
\mathbf{if}\;x \leq -6.143922385183966 \cdot 10^{+41}:\\
\;\;\;\;wj + \frac{\frac{x}{e^{wj}} - wj}{wj + 1}\\

\mathbf{else}:\\
\;\;\;\;\left(\mathsf{fma}\left(wj, wj, x\right) + x \cdot \mathsf{fma}\left(wj, \mathsf{fma}\left(2.5, wj, -2\right), {wj}^{3} \cdot -2.6666666666666665\right)\right) - {wj}^{3}\\


\end{array}

(FPCore (wj x)
 :precision binary64
 (- wj (/ (- (* wj (exp wj)) x) (+ (exp wj) (* wj (exp wj))))))
↓
(FPCore (wj x)
 :precision binary64
 (if (<= x -6.143922385183966e+41)
   (+ wj (/ (- (/ x (exp wj)) wj) (+ wj 1.0)))
   (-
    (+
     (fma wj wj x)
     (* x (fma wj (fma 2.5 wj -2.0) (* (pow wj 3.0) -2.6666666666666665))))
    (pow wj 3.0))))
double code(double wj, double x) {
	return wj - (((wj * exp(wj)) - x) / (exp(wj) + (wj * exp(wj))));
}
↓
double code(double wj, double x) {
	double tmp;
	if (x <= -6.143922385183966e+41) {
		tmp = wj + (((x / exp(wj)) - wj) / (wj + 1.0));
	} else {
		tmp = (fma(wj, wj, x) + (x * fma(wj, fma(2.5, wj, -2.0), (pow(wj, 3.0) * -2.6666666666666665)))) - pow(wj, 3.0);
	}
	return tmp;
}

Error

Bits error versus wj

Bits error versus x

Target

Original13.8
Target13.1
Herbie1.3
\[wj - \left(\frac{wj}{wj + 1} - \frac{x}{e^{wj} + wj \cdot e^{wj}}\right) \]

Derivation

  1. Split input into 2 regimes

  2. if x < -6.1439223851839656e41

    1. Initial program 0.7

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

    2. Simplified0.0

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

    if -6.1439223851839656e41 < x

    1. Initial program 17.5

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

    2. Simplified16.8

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

    3. Taylor expanded in wj around 0 1.7

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

    4. Simplified1.7

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

    5. Applied fma-udef_binary641.7

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

    6. Applied associate--r+_binary641.7

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

    7. Simplified1.7

      \[\leadsto \color{blue}{\left(\mathsf{fma}\left(wj, wj, x\right) + x \cdot \mathsf{fma}\left(wj, \mathsf{fma}\left(2.5, wj, -2\right), {wj}^{3} \cdot -2.6666666666666665\right)\right)} - {wj}^{3} \]
  3. Recombined 2 regimes into one program.

  4. Final simplification1.3

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -6.143922385183966 \cdot 10^{+41}:\\ \;\;\;\;wj + \frac{\frac{x}{e^{wj}} - wj}{wj + 1}\\ \mathbf{else}:\\ \;\;\;\;\left(\mathsf{fma}\left(wj, wj, x\right) + x \cdot \mathsf{fma}\left(wj, \mathsf{fma}\left(2.5, wj, -2\right), {wj}^{3} \cdot -2.6666666666666665\right)\right) - {wj}^{3}\\ \end{array} \]

Reproduce

herbie shell --seed 2021224 
(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))))))