Average Error: 14.0 → 0.1
Time: 6.0s
Precision: binary64
\[wj - \frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}}\]
↓
\[\begin{array}{l} \mathbf{if}\;wj \leq -2.883523057983485 \cdot 10^{-07}:\\ \;\;\;\;wj + \frac{\frac{x}{e^{wj}} - wj}{wj + 1}\\ \mathbf{elif}\;wj \leq 9.233660653634418 \cdot 10^{-07}:\\ \;\;\;\;{wj}^{2} + \left(x - \left({wj}^{3} + x \cdot \left(\left(wj + wj\right) + {wj}^{3} \cdot 2.6666666666666665\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;wj - \left(\frac{wj}{wj + 1} - \frac{x}{e^{wj} \cdot \left(wj + 1\right)}\right)\\ \end{array}\]
wj - \frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}}
↓
\begin{array}{l}
\mathbf{if}\;wj \leq -2.883523057983485 \cdot 10^{-07}:\\
\;\;\;\;wj + \frac{\frac{x}{e^{wj}} - wj}{wj + 1}\\

\mathbf{elif}\;wj \leq 9.233660653634418 \cdot 10^{-07}:\\
\;\;\;\;{wj}^{2} + \left(x - \left({wj}^{3} + x \cdot \left(\left(wj + wj\right) + {wj}^{3} \cdot 2.6666666666666665\right)\right)\right)\\

\mathbf{else}:\\
\;\;\;\;wj - \left(\frac{wj}{wj + 1} - \frac{x}{e^{wj} \cdot \left(wj + 1\right)}\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 -2.883523057983485e-07)
   (+ wj (/ (- (/ x (exp wj)) wj) (+ wj 1.0)))
   (if (<= wj 9.233660653634418e-07)
     (+
      (pow wj 2.0)
      (-
       x
       (+
        (pow wj 3.0)
        (* x (+ (+ wj wj) (* (pow wj 3.0) 2.6666666666666665))))))
     (- wj (- (/ wj (+ wj 1.0)) (/ x (* (exp 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 <= -2.883523057983485e-07) {
		tmp = wj + (((x / exp(wj)) - wj) / (wj + 1.0));
	} else if (wj <= 9.233660653634418e-07) {
		tmp = pow(wj, 2.0) + (x - (pow(wj, 3.0) + (x * ((wj + wj) + (pow(wj, 3.0) * 2.6666666666666665)))));
	} else {
		tmp = wj - ((wj / (wj + 1.0)) - (x / (exp(wj) * (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
Target13.4
Herbie0.1
\[wj - \left(\frac{wj}{wj + 1} - \frac{x}{e^{wj} + wj \cdot e^{wj}}\right)\]

Derivation

  1. Split input into 3 regimes

  2. if wj < -2.8835230579834852e-7

    1. Initial program 3.8

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

    2. Simplified3.6

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

    4. Applied *-un-lft-identity_binary643.6

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

    if -2.8835230579834852e-7 < wj < 9.23366065363441782e-7

    1. Initial program 13.9

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

    2. Simplified13.9

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

    3. Taylor expanded around 0 0.0

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

    4. Simplified0.0

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

    5. Taylor expanded around 0 0.0

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

    if 9.23366065363441782e-7 < wj

    1. Initial program 26.4

      \[wj - \frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}}\]
    2. Using strategy rm

    3. Applied div-sub_binary6426.4

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

    4. Simplified1.8

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

    5. Simplified1.8

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

  4. Final simplification0.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;wj \leq -2.883523057983485 \cdot 10^{-07}:\\ \;\;\;\;wj + \frac{\frac{x}{e^{wj}} - wj}{wj + 1}\\ \mathbf{elif}\;wj \leq 9.233660653634418 \cdot 10^{-07}:\\ \;\;\;\;{wj}^{2} + \left(x - \left({wj}^{3} + x \cdot \left(\left(wj + wj\right) + {wj}^{3} \cdot 2.6666666666666665\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;wj - \left(\frac{wj}{wj + 1} - \frac{x}{e^{wj} \cdot \left(wj + 1\right)}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2021175 
(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))))))