Jmat.Real.lambertw, newton loop step

Average Error: 13.8 → 0.6

Time: 8.3s

Precision: binary64

Math

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

↓

\[\begin{array}{l} t_0 := wj \cdot e^{wj}\\ \mathbf{if}\;wj + \frac{x - t_0}{e^{wj} + t_0} \leq 1.9941353552350148 \cdot 10^{-22}:\\ \;\;\;\;\mathsf{fma}\left(2.5, x \cdot \left(wj \cdot wj\right), \mathsf{fma}\left(wj, wj, x\right)\right) - \mathsf{fma}\left(2, wj \cdot x, \mathsf{fma}\left(2.6666666666666665, x \cdot {wj}^{3}, {wj}^{3}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{e^{wj} \cdot \left(wj + 1\right)} + \left(wj - \frac{wj}{wj + 1}\right)\\ \end{array} \]

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 (/ (- x t_0) (+ (exp wj) t_0))) 1.9941353552350148e-22)
     (-
      (fma 2.5 (* x (* wj wj)) (fma wj wj x))
      (fma
       2.0
       (* wj x)
       (fma 2.6666666666666665 (* x (pow wj 3.0)) (pow wj 3.0))))
     (+ (/ x (* (exp wj) (+ wj 1.0))) (- 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 + ((x - t_0) / (exp(wj) + t_0))) <= 1.9941353552350148e-22) {
		tmp = fma(2.5, (x * (wj * wj)), fma(wj, wj, x)) - fma(2.0, (wj * x), fma(2.6666666666666665, (x * pow(wj, 3.0)), pow(wj, 3.0)));
	} else {
		tmp = (x / (exp(wj) * (wj + 1.0))) + (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(x - t_0) / Float64(exp(wj) + t_0))) <= 1.9941353552350148e-22)
		tmp = Float64(fma(2.5, Float64(x * Float64(wj * wj)), fma(wj, wj, x)) - fma(2.0, Float64(wj * x), fma(2.6666666666666665, Float64(x * (wj ^ 3.0)), (wj ^ 3.0))));
	else
		tmp = Float64(Float64(x / Float64(exp(wj) * Float64(wj + 1.0))) + Float64(wj - Float64(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[(x - t$95$0), $MachinePrecision] / N[(N[Exp[wj], $MachinePrecision] + t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 1.9941353552350148e-22], N[(N[(2.5 * N[(x * N[(wj * wj), $MachinePrecision]), $MachinePrecision] + N[(wj * wj + x), $MachinePrecision]), $MachinePrecision] - N[(2.0 * N[(wj * x), $MachinePrecision] + N[(2.6666666666666665 * N[(x * N[Power[wj, 3.0], $MachinePrecision]), $MachinePrecision] + N[Power[wj, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x / N[(N[Exp[wj], $MachinePrecision] * N[(wj + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(wj - N[(wj / N[(wj + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Error

Bits error versus wj

Bits error versus x

Target

Original	13.8
Target	13.1
Herbie	0.6

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

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))))) < 1.9941354e-22

   1. Initial program 18.3

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

   2. Simplified 18.3

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

   3. Applied egg-rr 35.5

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

   4. Taylor expanded in wj around 0 0.5

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

   5. Simplified 0.5

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

   if 1.9941354e-22 < (-.f64 wj (/.f64 (-.f64 (*.f64 wj (exp.f64 wj)) x) (+.f64 (exp.f64 wj) (*.f64 wj (exp.f64 wj)))))

   1. Initial program 3.5

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

   2. Simplified 1.0

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

   3. Applied egg-rr 1.0

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

   4. Applied egg-rr 0.9

      \[\leadsto \color{blue}{\left(\frac{x}{e^{wj} \cdot \left(wj + 1\right)} + 0\right) - \left(\frac{wj}{wj + 1} - wj\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 0.6

   \[\leadsto \begin{array}{l} \mathbf{if}\;wj + \frac{x - wj \cdot e^{wj}}{e^{wj} + wj \cdot e^{wj}} \leq 1.9941353552350148 \cdot 10^{-22}:\\ \;\;\;\;\mathsf{fma}\left(2.5, x \cdot \left(wj \cdot wj\right), \mathsf{fma}\left(wj, wj, x\right)\right) - \mathsf{fma}\left(2, wj \cdot x, \mathsf{fma}\left(2.6666666666666665, x \cdot {wj}^{3}, {wj}^{3}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{e^{wj} \cdot \left(wj + 1\right)} + \left(wj - \frac{wj}{wj + 1}\right)\\ \end{array} \]

Reproduce

herbie shell --seed 2022153 
(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))))))