Jmat.Real.lambertw, newton loop step

Average Error: 13.4 → 0.3

Time: 7.6s

Precision: binary64

Math

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

↓

\[\begin{array}{l} t_0 := wj \cdot e^{wj}\\ \mathbf{if}\;wj - \frac{t_0 - x}{e^{wj} + t_0} \leq 5 \cdot 10^{-10}:\\ \;\;\;\;\mathsf{fma}\left(x, e^{\left(-wj\right) - \mathsf{log1p}\left(wj\right)}, \left(\mathsf{fma}\left(wj, wj, {wj}^{4}\right) - {wj}^{3}\right) - {wj}^{5}\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{\frac{x}{e^{wj}} - wj}{\mathsf{fma}\left(wj, wj, -1\right)}, wj + -1, wj\right)\\ \end{array} \]

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))))
   (if (<= (- wj (/ (- t_0 x) (+ (exp wj) t_0))) 5e-10)
     (fma
      x
      (exp (- (- wj) (log1p wj)))
      (- (- (fma wj wj (pow wj 4.0)) (pow wj 3.0)) (pow wj 5.0)))
     (fma (/ (- (/ x (exp wj)) wj) (fma wj 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 tmp;
	if ((wj - ((t_0 - x) / (exp(wj) + t_0))) <= 5e-10) {
		tmp = fma(x, exp((-wj - log1p(wj))), ((fma(wj, wj, pow(wj, 4.0)) - pow(wj, 3.0)) - pow(wj, 5.0)));
	} else {
		tmp = fma((((x / exp(wj)) - wj) / fma(wj, 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))
	tmp = 0.0
	if (Float64(wj - Float64(Float64(t_0 - x) / Float64(exp(wj) + t_0))) <= 5e-10)
		tmp = fma(x, exp(Float64(Float64(-wj) - log1p(wj))), Float64(Float64(fma(wj, wj, (wj ^ 4.0)) - (wj ^ 3.0)) - (wj ^ 5.0)));
	else
		tmp = fma(Float64(Float64(Float64(x / exp(wj)) - wj) / fma(wj, wj, -1.0)), Float64(wj + -1.0), 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]}, If[LessEqual[N[(wj - N[(N[(t$95$0 - x), $MachinePrecision] / N[(N[Exp[wj], $MachinePrecision] + t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 5e-10], N[(x * N[Exp[N[((-wj) - N[Log[1 + wj], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] + N[(N[(N[(wj * wj + N[Power[wj, 4.0], $MachinePrecision]), $MachinePrecision] - N[Power[wj, 3.0], $MachinePrecision]), $MachinePrecision] - N[Power[wj, 5.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(x / N[Exp[wj], $MachinePrecision]), $MachinePrecision] - wj), $MachinePrecision] / N[(wj * wj + -1.0), $MachinePrecision]), $MachinePrecision] * N[(wj + -1.0), $MachinePrecision] + wj), $MachinePrecision]]]

Error

Bits error versus wj

Bits error versus x

Target

Original	13.4
Target	12.7
Herbie	0.3

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

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))))) < 5.00000000000000031e-10

   1. Initial program 17.5

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

   2. Simplified 17.5

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

   3. Applied egg-rr 8.8

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

   4. Taylor expanded in wj around 0 0.3

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

   5. Applied egg-rr 0.4

      \[\leadsto \color{blue}{{\left(\mathsf{fma}\left(x, e^{\left(-wj\right) - \mathsf{log1p}\left(wj\right)}, -\left({wj}^{5} + \left({wj}^{3} - \mathsf{fma}\left(wj, wj, {wj}^{4}\right)\right)\right)\right)\right)}^{1}} \]

   if 5.00000000000000031e-10 < (-.f64 wj (/.f64 (-.f64 (*.f64 wj (exp.f64 wj)) x) (+.f64 (exp.f64 wj) (*.f64 wj (exp.f64 wj)))))

   1. Initial program 2.7

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

   2. Simplified 0.2

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

   3. Applied egg-rr 0.2

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

4. Final simplification 0.3

   \[\leadsto \begin{array}{l} \mathbf{if}\;wj - \frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}} \leq 5 \cdot 10^{-10}:\\ \;\;\;\;\mathsf{fma}\left(x, e^{\left(-wj\right) - \mathsf{log1p}\left(wj\right)}, \left(\mathsf{fma}\left(wj, wj, {wj}^{4}\right) - {wj}^{3}\right) - {wj}^{5}\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{\frac{x}{e^{wj}} - wj}{\mathsf{fma}\left(wj, wj, -1\right)}, wj + -1, wj\right)\\ \end{array} \]

Reproduce

herbie shell --seed 2022160 
(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))))))