Jmat.Real.lambertw, newton loop step

Average Error: 13.6 → 1.1

Time: 10.1s

Precision: binary64

Math

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

↓

\[\begin{array}{l} t_0 := wj \cdot e^{wj}\\ t_1 := \sqrt{e^{wj}}\\ \mathbf{if}\;wj + \frac{x - t_0}{e^{wj} + t_0} \leq 5 \cdot 10^{-14}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(-2, wj, 1\right), x, wj \cdot \left(wj \cdot \mathsf{fma}\left(x, \mathsf{fma}\left(wj, -2.6666666666666665, 2.5\right), 1 - wj\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;wj - \frac{t_0 - x}{\mathsf{fma}\left(wj \cdot t_1, t_1, e^{wj}\right)}\\ \end{array} \]

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))) (t_1 (sqrt (exp wj))))
   (if (<= (+ wj (/ (- x t_0) (+ (exp wj) t_0))) 5e-14)
     (fma
      (fma -2.0 wj 1.0)
      x
      (* wj (* wj (fma x (fma wj -2.6666666666666665 2.5) (- 1.0 wj)))))
     (- wj (/ (- t_0 x) (fma (* wj t_1) t_1 (exp wj)))))))

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 t_1 = sqrt(exp(wj));
	double tmp;
	if ((wj + ((x - t_0) / (exp(wj) + t_0))) <= 5e-14) {
		tmp = fma(fma(-2.0, wj, 1.0), x, (wj * (wj * fma(x, fma(wj, -2.6666666666666665, 2.5), (1.0 - wj)))));
	} else {
		tmp = wj - ((t_0 - x) / fma((wj * t_1), t_1, exp(wj)));
	}
	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))
	t_1 = sqrt(exp(wj))
	tmp = 0.0
	if (Float64(wj + Float64(Float64(x - t_0) / Float64(exp(wj) + t_0))) <= 5e-14)
		tmp = fma(fma(-2.0, wj, 1.0), x, Float64(wj * Float64(wj * fma(x, fma(wj, -2.6666666666666665, 2.5), Float64(1.0 - wj)))));
	else
		tmp = Float64(wj - Float64(Float64(t_0 - x) / fma(Float64(wj * t_1), t_1, exp(wj))));
	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]}, Block[{t$95$1 = N[Sqrt[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], 5e-14], N[(N[(-2.0 * wj + 1.0), $MachinePrecision] * x + N[(wj * N[(wj * N[(x * N[(wj * -2.6666666666666665 + 2.5), $MachinePrecision] + N[(1.0 - wj), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(wj - N[(N[(t$95$0 - x), $MachinePrecision] / N[(N[(wj * t$95$1), $MachinePrecision] * t$95$1 + N[Exp[wj], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Target

Original	13.6
Target	13.0
Herbie	1.1

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

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))))) < 5.0000000000000002e-14

   1. Initial program 18.0

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

   2. Taylor expanded in wj around 0 0.5

      \[\leadsto \color{blue}{-1 \cdot \left(\left(0.6666666666666666 \cdot x + \left(-3 \cdot x + \left(1 + -2 \cdot \left(-4 \cdot x + 1.5 \cdot x\right)\right)\right)\right) \cdot {wj}^{3}\right) + \left(\left(1 - \left(-4 \cdot x + 1.5 \cdot x\right)\right) \cdot {wj}^{2} + \left(-2 \cdot \left(wj \cdot x\right) + x\right)\right)} \]

   3. Simplified 0.5

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

   4. Taylor expanded in wj around 0 0.5

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

   5. Simplified 0.5

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

   if 5.0000000000000002e-14 < (-.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. Applied egg-rr 2.6

      \[\leadsto wj - \frac{wj \cdot e^{wj} - x}{\color{blue}{\mathsf{fma}\left(wj \cdot \sqrt{e^{wj}}, \sqrt{e^{wj}}, e^{wj}\right)}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 1.1

   \[\leadsto \begin{array}{l} \mathbf{if}\;wj + \frac{x - wj \cdot e^{wj}}{e^{wj} + wj \cdot e^{wj}} \leq 5 \cdot 10^{-14}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(-2, wj, 1\right), x, wj \cdot \left(wj \cdot \mathsf{fma}\left(x, \mathsf{fma}\left(wj, -2.6666666666666665, 2.5\right), 1 - wj\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;wj - \frac{wj \cdot e^{wj} - x}{\mathsf{fma}\left(wj \cdot \sqrt{e^{wj}}, \sqrt{e^{wj}}, e^{wj}\right)}\\ \end{array} \]

Reproduce

herbie shell --seed 2022210 
(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))))))