Jmat.Real.lambertw, newton loop step

Average Error: 13.5 → 0.4

Time: 10.2s

Precision: binary64

Math

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

↓

\[\begin{array}{l} t_0 := wj \cdot e^{wj}\\ \mathbf{if}\;wj + \frac{x - t_0}{e^{wj} + t_0} \leq 5 \cdot 10^{-13}:\\ \;\;\;\;x + \mathsf{fma}\left(wj, wj, wj \cdot \left(x \cdot \mathsf{fma}\left(wj, \mathsf{fma}\left(-2.6666666666666665, wj, 2.5\right), -2\right)\right) - {wj}^{3}\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x, \frac{e^{-wj}}{wj + 1}, wj - \frac{wj}{wj + 1}\right)\\ \end{array} \]

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))) 5e-13)
     (+
      x
      (fma
       wj
       wj
       (-
        (* wj (* x (fma wj (fma -2.6666666666666665 wj 2.5) -2.0)))
        (pow wj 3.0))))
     (fma 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))) <= 5e-13) {
		tmp = x + fma(wj, wj, ((wj * (x * fma(wj, fma(-2.6666666666666665, wj, 2.5), -2.0))) - pow(wj, 3.0)));
	} else {
		tmp = fma(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))) <= 5e-13)
		tmp = Float64(x + fma(wj, wj, Float64(Float64(wj * Float64(x * fma(wj, fma(-2.6666666666666665, wj, 2.5), -2.0))) - (wj ^ 3.0))));
	else
		tmp = fma(x, Float64(exp(Float64(-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], 5e-13], N[(x + N[(wj * wj + N[(N[(wj * N[(x * N[(wj * N[(-2.6666666666666665 * wj + 2.5), $MachinePrecision] + -2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[Power[wj, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x * N[(N[Exp[(-wj)], $MachinePrecision] / N[(wj + 1.0), $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	13.5
Target	12.8
Herbie	0.4

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

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))))) < 4.9999999999999999e-13

   1. Initial program 17.8

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

   2. Simplified 17.7

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

   3. Taylor expanded in wj around 0 0.5

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

   4. Simplified 0.6

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

   5. Taylor expanded in wj around 0 0.5

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

   6. Simplified 0.5

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

   7. Taylor expanded in x around 0 0.5

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

   8. Simplified 0.5

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

   if 4.9999999999999999e-13 < (-.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 0.3

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

   3. Applied egg-rr 0.3

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

4. Final simplification 0.4

   \[\leadsto \begin{array}{l} \mathbf{if}\;wj + \frac{x - wj \cdot e^{wj}}{e^{wj} + wj \cdot e^{wj}} \leq 5 \cdot 10^{-13}:\\ \;\;\;\;x + \mathsf{fma}\left(wj, wj, wj \cdot \left(x \cdot \mathsf{fma}\left(wj, \mathsf{fma}\left(-2.6666666666666665, wj, 2.5\right), -2\right)\right) - {wj}^{3}\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x, \frac{e^{-wj}}{wj + 1}, wj - \frac{wj}{wj + 1}\right)\\ \end{array} \]

Reproduce

herbie shell --seed 2022166 
(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))))))