Average Error: 13.7 → 1.0
Time: 6.4s
Precision: binary64
\[wj - \frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}} \]
↓
\[\mathsf{fma}\left(\frac{\frac{x}{e^{wj}}}{\mathsf{fma}\left(wj, wj, -1\right)}, wj + -1, wj \cdot wj - \left({wj}^{5} + \left({wj}^{3} - {wj}^{4}\right)\right)\right) \]
(FPCore (wj x)
 :precision binary64
 (- wj (/ (- (* wj (exp wj)) x) (+ (exp wj) (* wj (exp wj))))))
↓
(FPCore (wj x)
 :precision binary64
 (fma
  (/ (/ x (exp wj)) (fma wj wj -1.0))
  (+ wj -1.0)
  (- (* wj wj) (+ (pow wj 5.0) (- (pow wj 3.0) (pow wj 4.0))))))
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(((x / exp(wj)) / fma(wj, wj, -1.0)), (wj + -1.0), ((wj * wj) - (pow(wj, 5.0) + (pow(wj, 3.0) - pow(wj, 4.0)))));
}

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 fma(Float64(Float64(x / exp(wj)) / fma(wj, wj, -1.0)), Float64(wj + -1.0), Float64(Float64(wj * wj) - Float64((wj ^ 5.0) + Float64((wj ^ 3.0) - (wj ^ 4.0)))))
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[(N[(x / N[Exp[wj], $MachinePrecision]), $MachinePrecision] / N[(wj * wj + -1.0), $MachinePrecision]), $MachinePrecision] * N[(wj + -1.0), $MachinePrecision] + N[(N[(wj * wj), $MachinePrecision] - N[(N[Power[wj, 5.0], $MachinePrecision] + N[(N[Power[wj, 3.0], $MachinePrecision] - N[Power[wj, 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

wj - \frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}}
↓
\mathsf{fma}\left(\frac{\frac{x}{e^{wj}}}{\mathsf{fma}\left(wj, wj, -1\right)}, wj + -1, wj \cdot wj - \left({wj}^{5} + \left({wj}^{3} - {wj}^{4}\right)\right)\right)

Error

Bits error versus wj

Bits error versus x

Target

Original13.7
Target13.3
Herbie1.0
\[wj - \left(\frac{wj}{wj + 1} - \frac{x}{e^{wj} + wj \cdot e^{wj}}\right) \]

Derivation

  1. Initial program 13.7

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

  2. Simplified13.3

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

  3. Applied egg-rr7.1

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

  4. Taylor expanded in wj around 0 1.0

    \[\leadsto \mathsf{fma}\left(\frac{\frac{x}{e^{wj}}}{\mathsf{fma}\left(wj, wj, -1\right)}, wj + -1, -\color{blue}{\left(\left({wj}^{5} + {wj}^{3}\right) - \left({wj}^{4} + {wj}^{2}\right)\right)}\right) \]

  5. Applied egg-rr1.0

    \[\leadsto \mathsf{fma}\left(\frac{\frac{x}{e^{wj}}}{\mathsf{fma}\left(wj, wj, -1\right)}, wj + -1, -\color{blue}{\left(\left({wj}^{5} + \left({wj}^{3} - {wj}^{4}\right)\right) + \left(-wj \cdot wj\right)\right)}\right) \]

  6. Final simplification1.0

    \[\leadsto \mathsf{fma}\left(\frac{\frac{x}{e^{wj}}}{\mathsf{fma}\left(wj, wj, -1\right)}, wj + -1, wj \cdot wj - \left({wj}^{5} + \left({wj}^{3} - {wj}^{4}\right)\right)\right) \]

Reproduce

herbie shell --seed 2022162 
(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))))))