Average Error: 13.9 → 0.7
Time: 7.3s
Precision: binary64
\[wj - \frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}} \]
↓
\[\begin{array}{l} \mathbf{if}\;wj \leq 2.15 \cdot 10^{-10}:\\ \;\;\;\;\mathsf{fma}\left(x, wj \cdot \left(wj \cdot 2.5\right), \mathsf{fma}\left(wj, wj, x\right)\right) - \mathsf{fma}\left(x, \mathsf{fma}\left(2, wj, 2.6666666666666665 \cdot {wj}^{3}\right), {wj}^{3}\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{\frac{x}{e^{wj}} - wj}{{wj}^{3} + 1}, \mathsf{fma}\left(wj, wj, 1\right) - wj, wj\right)\\ \end{array} \]
(FPCore (wj x)
 :precision binary64
 (- wj (/ (- (* wj (exp wj)) x) (+ (exp wj) (* wj (exp wj))))))
↓
(FPCore (wj x)
 :precision binary64
 (if (<= wj 2.15e-10)
   (-
    (fma x (* wj (* wj 2.5)) (fma wj wj x))
    (fma x (fma 2.0 wj (* 2.6666666666666665 (pow wj 3.0))) (pow wj 3.0)))
   (fma
    (/ (- (/ x (exp wj)) wj) (+ (pow wj 3.0) 1.0))
    (- (fma wj wj 1.0) wj)
    wj)))
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 <= 2.15e-10) {
		tmp = fma(x, (wj * (wj * 2.5)), fma(wj, wj, x)) - fma(x, fma(2.0, wj, (2.6666666666666665 * pow(wj, 3.0))), pow(wj, 3.0));
	} else {
		tmp = fma((((x / exp(wj)) - wj) / (pow(wj, 3.0) + 1.0)), (fma(wj, wj, 1.0) - wj), wj);
	}
	return tmp;
}

function code(wj, x)
	return Float64(wj - Float64(Float64(Float64(wj * exp(wj)) - x) / Float64(exp(wj) + Float64(wj * exp(wj)))))
end
↓
function code(wj, x)
	tmp = 0.0
	if (wj <= 2.15e-10)
		tmp = Float64(fma(x, Float64(wj * Float64(wj * 2.5)), fma(wj, wj, x)) - fma(x, fma(2.0, wj, Float64(2.6666666666666665 * (wj ^ 3.0))), (wj ^ 3.0)));
	else
		tmp = fma(Float64(Float64(Float64(x / exp(wj)) - wj) / Float64((wj ^ 3.0) + 1.0)), Float64(fma(wj, wj, 1.0) - wj), wj);
	end
	return tmp
end

code[wj_, x_] := N[(wj - N[(N[(N[(wj * N[Exp[wj], $MachinePrecision]), $MachinePrecision] - x), $MachinePrecision] / N[(N[Exp[wj], $MachinePrecision] + N[(wj * N[Exp[wj], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]
↓
code[wj_, x_] := If[LessEqual[wj, 2.15e-10], N[(N[(x * N[(wj * N[(wj * 2.5), $MachinePrecision]), $MachinePrecision] + N[(wj * wj + x), $MachinePrecision]), $MachinePrecision] - N[(x * N[(2.0 * wj + N[(2.6666666666666665 * N[Power[wj, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[Power[wj, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(x / N[Exp[wj], $MachinePrecision]), $MachinePrecision] - wj), $MachinePrecision] / N[(N[Power[wj, 3.0], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] * N[(N[(wj * wj + 1.0), $MachinePrecision] - wj), $MachinePrecision] + wj), $MachinePrecision]]

wj - \frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}}
↓
\begin{array}{l}
\mathbf{if}\;wj \leq 2.15 \cdot 10^{-10}:\\
\;\;\;\;\mathsf{fma}\left(x, wj \cdot \left(wj \cdot 2.5\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)\\

\mathbf{else}:\\
\;\;\;\;\mathsf{fma}\left(\frac{\frac{x}{e^{wj}} - wj}{{wj}^{3} + 1}, \mathsf{fma}\left(wj, wj, 1\right) - wj, wj\right)\\


\end{array}

Error

Bits error versus wj

Bits error versus x

Target

Original13.9
Target13.3
Herbie0.7
\[wj - \left(\frac{wj}{wj + 1} - \frac{x}{e^{wj} + wj \cdot e^{wj}}\right) \]

Derivation

  1. Split input into 2 regimes

  2. if wj < 2.15000000000000007e-10

    1. Initial program 13.6

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

    2. Simplified13.6

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

    3. Taylor expanded in wj around 0 0.6

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

    4. Simplified0.6

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

    if 2.15000000000000007e-10 < wj

    1. Initial program 23.4

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

    2. Simplified3.4

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

    3. Applied egg-rr3.4

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

  4. Final simplification0.7

    \[\leadsto \begin{array}{l} \mathbf{if}\;wj \leq 2.15 \cdot 10^{-10}:\\ \;\;\;\;\mathsf{fma}\left(x, wj \cdot \left(wj \cdot 2.5\right), \mathsf{fma}\left(wj, wj, x\right)\right) - \mathsf{fma}\left(x, \mathsf{fma}\left(2, wj, 2.6666666666666665 \cdot {wj}^{3}\right), {wj}^{3}\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{\frac{x}{e^{wj}} - wj}{{wj}^{3} + 1}, \mathsf{fma}\left(wj, wj, 1\right) - wj, wj\right)\\ \end{array} \]

Reproduce

herbie shell --seed 2022155 
(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))))))