Jmat.Real.lambertw, newton loop step

Average Error: 14.1 → 0.4

Time: 6.6s

Precision: binary64

Math

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

↓

\[\begin{array}{l} t_0 := wj \cdot e^{wj}\\ \mathbf{if}\;wj - \frac{t_0 - x}{e^{wj} + t_0} \leq 4.1564083010500665 \cdot 10^{-15}:\\ \;\;\;\;\left(\mathsf{fma}\left(wj, wj, x\right) + \left(wj \cdot x\right) \cdot \left(wj \cdot 2.5 - 2\right)\right) - {wj}^{3} \cdot \mathsf{fma}\left(x, 2.6666666666666665, 1\right)\\ \mathbf{else}:\\ \;\;\;\;wj + \frac{x \cdot e^{-wj} - wj}{wj + 1}\\ \end{array} \]

FPCore

(FPCore (wj x)
 :precision binary64
 (- wj (/ (- (* wj (exp wj)) x) (+ (exp wj) (* wj (exp wj))))))

↓

(FPCore (wj x)
 :precision binary64
 (let* ((t_0 (* wj (exp wj))))
   (if (<= (- wj (/ (- t_0 x) (+ (exp wj) t_0))) 4.1564083010500665e-15)
     (-
      (+ (fma wj wj x) (* (* wj x) (- (* wj 2.5) 2.0)))
      (* (pow wj 3.0) (fma x 2.6666666666666665 1.0)))
     (+ wj (/ (- (* x (exp (- wj))) wj) (+ wj 1.0))))))

C

double code(double wj, double x) {
	return wj - (((wj * exp(wj)) - x) / (exp(wj) + (wj * exp(wj))));
}

↓

double code(double wj, double x) {
	double t_0 = wj * exp(wj);
	double tmp;
	if ((wj - ((t_0 - x) / (exp(wj) + t_0))) <= 4.1564083010500665e-15) {
		tmp = (fma(wj, wj, x) + ((wj * x) * ((wj * 2.5) - 2.0))) - (pow(wj, 3.0) * fma(x, 2.6666666666666665, 1.0));
	} else {
		tmp = wj + (((x * exp(-wj)) - wj) / (wj + 1.0));
	}
	return tmp;
}

Julia

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)
	t_0 = Float64(wj * exp(wj))
	tmp = 0.0
	if (Float64(wj - Float64(Float64(t_0 - x) / Float64(exp(wj) + t_0))) <= 4.1564083010500665e-15)
		tmp = Float64(Float64(fma(wj, wj, x) + Float64(Float64(wj * x) * Float64(Float64(wj * 2.5) - 2.0))) - Float64((wj ^ 3.0) * fma(x, 2.6666666666666665, 1.0)));
	else
		tmp = Float64(wj + Float64(Float64(Float64(x * exp(Float64(-wj))) - wj) / Float64(wj + 1.0)));
	end
	return tmp
end

Wolfram

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_] := Block[{t$95$0 = N[(wj * N[Exp[wj], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(wj - N[(N[(t$95$0 - x), $MachinePrecision] / N[(N[Exp[wj], $MachinePrecision] + t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 4.1564083010500665e-15], N[(N[(N[(wj * wj + x), $MachinePrecision] + N[(N[(wj * x), $MachinePrecision] * N[(N[(wj * 2.5), $MachinePrecision] - 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(N[Power[wj, 3.0], $MachinePrecision] * N[(x * 2.6666666666666665 + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(wj + N[(N[(N[(x * N[Exp[(-wj)], $MachinePrecision]), $MachinePrecision] - wj), $MachinePrecision] / N[(wj + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Error

Bits error versus wj

Bits error versus x

Target

Original	14.1
Target	13.5
Herbie	0.4

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

Derivation

1. Split input into 2 regimes

2. if (-.f64 wj (/.f64 (-.f64 (*.f64 wj (exp.f64 wj)) x) (+.f64 (exp.f64 wj) (*.f64 wj (exp.f64 wj))))) < 4.156408301e-15

   1. Initial program 18.4

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

   2. Simplified 18.4

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

   3. Taylor expanded in wj around 0 0.4

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

   4. Simplified 0.4

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

   5. Taylor expanded in x around 0 0.4

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

   6. Simplified 0.4

      \[\leadsto \color{blue}{\left(\mathsf{fma}\left(wj, wj, x\right) + \left(wj \cdot x\right) \cdot \left(wj \cdot 2.5 - 2\right)\right) - {wj}^{3} \cdot \mathsf{fma}\left(x, 2.6666666666666665, 1\right)} \]

   if 4.156408301e-15 < (-.f64 wj (/.f64 (-.f64 (*.f64 wj (exp.f64 wj)) x) (+.f64 (exp.f64 wj) (*.f64 wj (exp.f64 wj)))))

   1. Initial program 2.6

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

   2. Simplified 0.4

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

   3. Applied div-inv_binary64 0.4

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

   4. Simplified 0.4

      \[\leadsto wj + \frac{x \cdot \color{blue}{e^{-wj}} - wj}{wj + 1} \]
3. Recombined 2 regimes into one program.

4. Final simplification 0.4

   \[\leadsto \begin{array}{l} \mathbf{if}\;wj - \frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}} \leq 4.1564083010500665 \cdot 10^{-15}:\\ \;\;\;\;\left(\mathsf{fma}\left(wj, wj, x\right) + \left(wj \cdot x\right) \cdot \left(wj \cdot 2.5 - 2\right)\right) - {wj}^{3} \cdot \mathsf{fma}\left(x, 2.6666666666666665, 1\right)\\ \mathbf{else}:\\ \;\;\;\;wj + \frac{x \cdot e^{-wj} - wj}{wj + 1}\\ \end{array} \]

Reproduce

herbie shell --seed 2022129 
(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))))))