Average Error: 13.7 → 0.2
Time: 6.0s
Precision: binary64
\[wj - \frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}} \]
↓
\[\begin{array}{l} \mathbf{if}\;wj \leq -4.62645130249213 \cdot 10^{-6}:\\ \;\;\;\;wj + \frac{\mathsf{fma}\left(wj, wj, -1\right) \cdot \left(\frac{x}{e^{wj}} - wj\right)}{\mathsf{fma}\left(wj, wj, -1\right) \cdot \left(wj + 1\right)}\\ \mathbf{elif}\;wj \leq 9.727382200534923 \cdot 10^{-10}:\\ \;\;\;\;\left(\mathsf{fma}\left(wj, wj, x\right) + x \cdot \mathsf{fma}\left(wj, \mathsf{fma}\left(2.5, wj, -2\right), {wj}^{3} \cdot -2.6666666666666665\right)\right) - {wj}^{3}\\ \mathbf{else}:\\ \;\;\;\;wj + \left(\frac{x}{e^{wj} \cdot \left(wj + 1\right)} - \frac{wj}{wj + 1}\right)\\ \end{array} \]
wj - \frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}}
↓
\begin{array}{l}
\mathbf{if}\;wj \leq -4.62645130249213 \cdot 10^{-6}:\\
\;\;\;\;wj + \frac{\mathsf{fma}\left(wj, wj, -1\right) \cdot \left(\frac{x}{e^{wj}} - wj\right)}{\mathsf{fma}\left(wj, wj, -1\right) \cdot \left(wj + 1\right)}\\

\mathbf{elif}\;wj \leq 9.727382200534923 \cdot 10^{-10}:\\
\;\;\;\;\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}\\

\mathbf{else}:\\
\;\;\;\;wj + \left(\frac{x}{e^{wj} \cdot \left(wj + 1\right)} - \frac{wj}{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 (<= wj -4.62645130249213e-6)
   (+
    wj
    (/
     (* (fma wj wj -1.0) (- (/ x (exp wj)) wj))
     (* (fma wj wj -1.0) (+ wj 1.0))))
   (if (<= wj 9.727382200534923e-10)
     (-
      (+
       (fma wj wj x)
       (* x (fma wj (fma 2.5 wj -2.0) (* (pow wj 3.0) -2.6666666666666665))))
      (pow wj 3.0))
     (+ wj (- (/ x (* (exp wj) (+ wj 1.0))) (/ wj (+ 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 tmp;
	if (wj <= -4.62645130249213e-6) {
		tmp = wj + ((fma(wj, wj, -1.0) * ((x / exp(wj)) - wj)) / (fma(wj, wj, -1.0) * (wj + 1.0)));
	} else if (wj <= 9.727382200534923e-10) {
		tmp = (fma(wj, wj, x) + (x * fma(wj, fma(2.5, wj, -2.0), (pow(wj, 3.0) * -2.6666666666666665)))) - pow(wj, 3.0);
	} else {
		tmp = wj + ((x / (exp(wj) * (wj + 1.0))) - (wj / (wj + 1.0)));
	}
	return tmp;
}

Error

Bits error versus wj

Bits error versus x

Target

Original13.7
Target13.1
Herbie0.2
\[wj - \left(\frac{wj}{wj + 1} - \frac{x}{e^{wj} + wj \cdot e^{wj}}\right) \]

Derivation

  1. Split input into 3 regimes

  2. if wj < -4.6264513024921298e-6

    1. Initial program 2.2

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

    2. Simplified2.1

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

    3. Applied flip-+_binary642.5

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

    4. Applied associate-/r/_binary642.4

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

    5. Simplified2.4

      \[\leadsto wj + \color{blue}{\frac{\frac{x}{e^{wj}} - wj}{\mathsf{fma}\left(wj, wj, -1\right)}} \cdot \left(wj - 1\right) \]

    6. Applied flip--_binary642.5

      \[\leadsto wj + \frac{\frac{x}{e^{wj}} - wj}{\mathsf{fma}\left(wj, wj, -1\right)} \cdot \color{blue}{\frac{wj \cdot wj - 1 \cdot 1}{wj + 1}} \]

    7. Applied frac-times_binary642.3

      \[\leadsto wj + \color{blue}{\frac{\left(\frac{x}{e^{wj}} - wj\right) \cdot \left(wj \cdot wj - 1 \cdot 1\right)}{\mathsf{fma}\left(wj, wj, -1\right) \cdot \left(wj + 1\right)}} \]

    8. Simplified2.3

      \[\leadsto wj + \frac{\color{blue}{\mathsf{fma}\left(wj, wj, -1\right) \cdot \left(\frac{x}{e^{wj}} - wj\right)}}{\mathsf{fma}\left(wj, wj, -1\right) \cdot \left(wj + 1\right)} \]

    if -4.6264513024921298e-6 < wj < 9.72738220053492348e-10

    1. Initial program 13.5

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

    2. Simplified13.5

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

    3. Taylor expanded in wj around 0 0.0

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

    4. Simplified0.0

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

    5. Applied fma-udef_binary640.0

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

    6. Applied associate--r+_binary640.0

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

    7. Simplified0.0

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

    if 9.72738220053492348e-10 < wj

    1. Initial program 25.4

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

    2. Simplified3.2

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

    3. Applied div-sub_binary643.2

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

    4. Simplified3.4

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

  4. Final simplification0.2

    \[\leadsto \begin{array}{l} \mathbf{if}\;wj \leq -4.62645130249213 \cdot 10^{-6}:\\ \;\;\;\;wj + \frac{\mathsf{fma}\left(wj, wj, -1\right) \cdot \left(\frac{x}{e^{wj}} - wj\right)}{\mathsf{fma}\left(wj, wj, -1\right) \cdot \left(wj + 1\right)}\\ \mathbf{elif}\;wj \leq 9.727382200534923 \cdot 10^{-10}:\\ \;\;\;\;\left(\mathsf{fma}\left(wj, wj, x\right) + x \cdot \mathsf{fma}\left(wj, \mathsf{fma}\left(2.5, wj, -2\right), {wj}^{3} \cdot -2.6666666666666665\right)\right) - {wj}^{3}\\ \mathbf{else}:\\ \;\;\;\;wj + \left(\frac{x}{e^{wj} \cdot \left(wj + 1\right)} - \frac{wj}{wj + 1}\right)\\ \end{array} \]

Reproduce

herbie shell --seed 2022019 
(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))))))