Average Error: 13.6 → 0.0
Time: 5.7s
Precision: binary64
\[wj - \frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}} \]
↓
\[\mathsf{fma}\left(wj, wj, \frac{x}{e^{wj}}\right) \cdot \frac{1}{wj + 1} \]
(FPCore (wj x)
 :precision binary64
 (- wj (/ (- (* wj (exp wj)) x) (+ (exp wj) (* wj (exp wj))))))
↓
(FPCore (wj x)
 :precision binary64
 (* (fma wj wj (/ x (exp 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) {
	return fma(wj, wj, (x / exp(wj))) * (1.0 / (wj + 1.0));
}

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)
	return Float64(fma(wj, wj, Float64(x / exp(wj))) * Float64(1.0 / Float64(wj + 1.0)))
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_] := N[(N[(wj * wj + N[(x / N[Exp[wj], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(1.0 / N[(wj + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

wj - \frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}}
↓
\mathsf{fma}\left(wj, wj, \frac{x}{e^{wj}}\right) \cdot \frac{1}{wj + 1}

Error

Bits error versus wj

Bits error versus x

Target

Original13.6
Target12.9
Herbie0.0
\[wj - \left(\frac{wj}{wj + 1} - \frac{x}{e^{wj} + wj \cdot e^{wj}}\right) \]

Derivation

  1. Initial program 13.6

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

  2. Simplified0.0

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

  3. Applied egg-rr0.0

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

  4. Final simplification0.0

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

Reproduce

herbie shell --seed 2022131 
(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))))))