Jmat.Real.lambertw, newton loop step

Average Error: 13.5 → 0.2

Time: 7.6s

Precision: binary64

Math

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

↓

\[\begin{array}{l} t_0 := wj \cdot e^{wj}\\ t_1 := \frac{x}{e^{wj} \cdot \left(wj + 1\right)}\\ \mathbf{if}\;wj - \frac{t_0 - x}{e^{wj} + t_0} \leq 6.460728134164632 \cdot 10^{-9}:\\ \;\;\;\;\mathsf{fma}\left(1, t_1, \left(\mathsf{fma}\left(wj, wj, {wj}^{4}\right) - {wj}^{5}\right) - {wj}^{3}\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(1, t_1, -\mathsf{fma}\left(wj, \frac{1}{wj + 1}, -wj\right)\right)\\ \end{array} \]

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 (/ x (* (exp wj) (+ wj 1.0)))))
   (if (<= (- wj (/ (- t_0 x) (+ (exp wj) t_0))) 6.460728134164632e-9)
     (fma 1.0 t_1 (- (- (fma wj wj (pow wj 4.0)) (pow wj 5.0)) (pow wj 3.0)))
     (fma 1.0 t_1 (- (fma wj (/ 1.0 (+ wj 1.0)) (- 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 = x / (exp(wj) * (wj + 1.0));
	double tmp;
	if ((wj - ((t_0 - x) / (exp(wj) + t_0))) <= 6.460728134164632e-9) {
		tmp = fma(1.0, t_1, ((fma(wj, wj, pow(wj, 4.0)) - pow(wj, 5.0)) - pow(wj, 3.0)));
	} else {
		tmp = fma(1.0, t_1, -fma(wj, (1.0 / (wj + 1.0)), -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 = Float64(x / Float64(exp(wj) * Float64(wj + 1.0)))
	tmp = 0.0
	if (Float64(wj - Float64(Float64(t_0 - x) / Float64(exp(wj) + t_0))) <= 6.460728134164632e-9)
		tmp = fma(1.0, t_1, Float64(Float64(fma(wj, wj, (wj ^ 4.0)) - (wj ^ 5.0)) - (wj ^ 3.0)));
	else
		tmp = fma(1.0, t_1, Float64(-fma(wj, Float64(1.0 / Float64(wj + 1.0)), Float64(-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[(x / N[(N[Exp[wj], $MachinePrecision] * N[(wj + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(wj - N[(N[(t$95$0 - x), $MachinePrecision] / N[(N[Exp[wj], $MachinePrecision] + t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 6.460728134164632e-9], N[(1.0 * t$95$1 + N[(N[(N[(wj * wj + N[Power[wj, 4.0], $MachinePrecision]), $MachinePrecision] - N[Power[wj, 5.0], $MachinePrecision]), $MachinePrecision] - N[Power[wj, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 * t$95$1 + (-N[(wj * N[(1.0 / N[(wj + 1.0), $MachinePrecision]), $MachinePrecision] + (-wj)), $MachinePrecision])), $MachinePrecision]]]]

Error

Bits error versus wj

Bits error versus x

Target

Original	13.5
Target	12.9
Herbie	0.2

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

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))))) < 6.4607281341646e-9

   1. Initial program 17.8

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

   2. Simplified 17.8

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

   3. Applied egg-rr 9.3

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

   4. Taylor expanded in wj around 0 0.2

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

   5. Simplified 0.2

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

   if 6.4607281341646e-9 < (-.f64 wj (/.f64 (-.f64 (*.f64 wj (exp.f64 wj)) x) (+.f64 (exp.f64 wj) (*.f64 wj (exp.f64 wj)))))

   1. Initial program 2.2

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

   2. Simplified 0.2

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

   3. Applied egg-rr 0.2

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

   4. Applied egg-rr 0.2

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

4. Final simplification 0.2

   \[\leadsto \begin{array}{l} \mathbf{if}\;wj - \frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}} \leq 6.460728134164632 \cdot 10^{-9}:\\ \;\;\;\;\mathsf{fma}\left(1, \frac{x}{e^{wj} \cdot \left(wj + 1\right)}, \left(\mathsf{fma}\left(wj, wj, {wj}^{4}\right) - {wj}^{5}\right) - {wj}^{3}\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(1, \frac{x}{e^{wj} \cdot \left(wj + 1\right)}, -\mathsf{fma}\left(wj, \frac{1}{wj + 1}, -wj\right)\right)\\ \end{array} \]

Reproduce

herbie shell --seed 2022150 
(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))))))