Jmat.Real.lambertw, newton loop step

Average Error: 13.8 → 0.6

Time: 7.1s

Precision: binary64

Math

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

↓

\[\begin{array}{l} t_0 := wj \cdot e^{wj}\\ t_1 := \sqrt{wj + 1}\\ \mathbf{if}\;wj - \frac{t_0 - x}{e^{wj} + t_0} \leq 5.148146744474318 \cdot 10^{-13}:\\ \;\;\;\;\left(wj \cdot x\right) \cdot \mathsf{fma}\left(wj, 2.5, -2\right) + \left(\mathsf{fma}\left(wj, wj, x\right) - {wj}^{3} \cdot \mathsf{fma}\left(x, 2.6666666666666665, 1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{1}{t_1}, \frac{\frac{x}{e^{wj}} - wj}{t_1}, wj\right)\\ \end{array} \]

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))) (t_1 (sqrt (+ wj 1.0))))
   (if (<= (- wj (/ (- t_0 x) (+ (exp wj) t_0))) 5.148146744474318e-13)
     (+
      (* (* wj x) (fma wj 2.5 -2.0))
      (- (fma wj wj x) (* (pow wj 3.0) (fma x 2.6666666666666665 1.0))))
     (fma (/ 1.0 t_1) (/ (- (/ x (exp wj)) wj) t_1) 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((wj + 1.0));
	double tmp;
	if ((wj - ((t_0 - x) / (exp(wj) + t_0))) <= 5.148146744474318e-13) {
		tmp = ((wj * x) * fma(wj, 2.5, -2.0)) + (fma(wj, wj, x) - (pow(wj, 3.0) * fma(x, 2.6666666666666665, 1.0)));
	} else {
		tmp = fma((1.0 / t_1), (((x / exp(wj)) - wj) / t_1), 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(Float64(wj + 1.0))
	tmp = 0.0
	if (Float64(wj - Float64(Float64(t_0 - x) / Float64(exp(wj) + t_0))) <= 5.148146744474318e-13)
		tmp = Float64(Float64(Float64(wj * x) * fma(wj, 2.5, -2.0)) + Float64(fma(wj, wj, x) - Float64((wj ^ 3.0) * fma(x, 2.6666666666666665, 1.0))));
	else
		tmp = fma(Float64(1.0 / t_1), Float64(Float64(Float64(x / exp(wj)) - wj) / t_1), 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[(wj + 1.0), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[N[(wj - N[(N[(t$95$0 - x), $MachinePrecision] / N[(N[Exp[wj], $MachinePrecision] + t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 5.148146744474318e-13], N[(N[(N[(wj * x), $MachinePrecision] * N[(wj * 2.5 + -2.0), $MachinePrecision]), $MachinePrecision] + N[(N[(wj * wj + x), $MachinePrecision] - N[(N[Power[wj, 3.0], $MachinePrecision] * N[(x * 2.6666666666666665 + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 / t$95$1), $MachinePrecision] * N[(N[(N[(x / N[Exp[wj], $MachinePrecision]), $MachinePrecision] - wj), $MachinePrecision] / t$95$1), $MachinePrecision] + wj), $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))))) < 5.14814674447e-13

   1. Initial program 18.1

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

   2. Simplified 18.1

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

   3. Taylor expanded in wj around 0 0.6

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

   4. Simplified 0.6

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

   5. Taylor expanded in x around 0 0.6

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

   6. Simplified 0.6

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

   if 5.14814674447e-13 < (-.f64 wj (/.f64 (-.f64 (*.f64 wj (exp.f64 wj)) x) (+.f64 (exp.f64 wj) (*.f64 wj (exp.f64 wj)))))

   1. Initial program 2.9

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

   2. Simplified 0.3

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

   3. Applied egg-rr 0.5

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

4. Final simplification 0.6

   \[\leadsto \begin{array}{l} \mathbf{if}\;wj - \frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}} \leq 5.148146744474318 \cdot 10^{-13}:\\ \;\;\;\;\left(wj \cdot x\right) \cdot \mathsf{fma}\left(wj, 2.5, -2\right) + \left(\mathsf{fma}\left(wj, wj, x\right) - {wj}^{3} \cdot \mathsf{fma}\left(x, 2.6666666666666665, 1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{1}{\sqrt{wj + 1}}, \frac{\frac{x}{e^{wj}} - wj}{\sqrt{wj + 1}}, wj\right)\\ \end{array} \]

Reproduce

herbie shell --seed 2022133 
(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))))))