Average Error: 13.6 → 2.1
Time: 6.1s
Precision: binary64
\[wj - \frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}} \]
↓
\[\mathsf{fma}\left(wj, wj \cdot \mathsf{fma}\left(x, 2.5, 1\right), x \cdot \mathsf{fma}\left(-2, wj, 1\right)\right) \]
(FPCore (wj x)
 :precision binary64
 (- wj (/ (- (* wj (exp wj)) x) (+ (exp wj) (* wj (exp wj))))))
↓
(FPCore (wj x)
 :precision binary64
 (fma wj (* wj (fma x 2.5 1.0)) (* x (fma -2.0 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 * fma(x, 2.5, 1.0)), (x * fma(-2.0, 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 fma(wj, Float64(wj * fma(x, 2.5, 1.0)), Float64(x * fma(-2.0, 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[(wj * N[(wj * N[(x * 2.5 + 1.0), $MachinePrecision]), $MachinePrecision] + N[(x * N[(-2.0 * wj + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

wj - \frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}}
↓
\mathsf{fma}\left(wj, wj \cdot \mathsf{fma}\left(x, 2.5, 1\right), x \cdot \mathsf{fma}\left(-2, wj, 1\right)\right)

Error

Target

Original13.6
Target13.0
Herbie2.1
\[wj - \left(\frac{wj}{wj + 1} - \frac{x}{e^{wj} + wj \cdot e^{wj}}\right) \]

Derivation

  1. Initial program 13.6

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

  2. Taylor expanded in wj around 0 2.1

    \[\leadsto \color{blue}{\left(1 - \left(-4 \cdot x + 1.5 \cdot x\right)\right) \cdot {wj}^{2} + \left(-2 \cdot \left(wj \cdot x\right) + x\right)} \]

  3. Simplified2.1

    \[\leadsto \color{blue}{\mathsf{fma}\left(wj, wj \cdot \mathsf{fma}\left(x, 2.5, 1\right), \mathsf{fma}\left(-2, wj, 1\right) \cdot x\right)} \]

  4. Final simplification2.1

    \[\leadsto \mathsf{fma}\left(wj, wj \cdot \mathsf{fma}\left(x, 2.5, 1\right), x \cdot \mathsf{fma}\left(-2, wj, 1\right)\right) \]

Reproduce

herbie shell --seed 2022192 
(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))))))