Jmat.Real.lambertw, newton loop step

Average Error: 13.3 → 0.5

Time: 32.9s

Precision: binary64

Cost: 27204

Math

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

↓

\[\begin{array}{l} t_0 := wj \cdot e^{wj}\\ \mathbf{if}\;wj - \frac{t_0 - x}{e^{wj} + t_0} \leq 10^{-14}:\\ \;\;\;\;-1 \cdot \left(\left(\left(x \cdot \left(-2.3333333333333335 + 5\right) + 1\right) \cdot \left(wj \cdot wj\right)\right) \cdot wj\right) + \left(wj \cdot \left(wj \cdot \left(2.5 \cdot x\right) + wj\right) + \left(-2 \cdot \left(wj \cdot x\right) + x\right)\right)\\ \mathbf{else}:\\ \;\;\;\;wj - \frac{wj - \frac{x}{e^{wj}}}{wj + 1}\\ \end{array} \]

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))) 1e-14)
     (+
      (* -1.0 (* (* (+ (* x (+ -2.3333333333333335 5.0)) 1.0) (* wj wj)) wj))
      (+ (* wj (+ (* wj (* 2.5 x)) wj)) (+ (* -2.0 (* wj x)) x)))
     (- wj (/ (- wj (/ x (exp 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 - ((t_0 - x) / (exp(wj) + t_0))) <= 1e-14) {
		tmp = (-1.0 * ((((x * (-2.3333333333333335 + 5.0)) + 1.0) * (wj * wj)) * wj)) + ((wj * ((wj * (2.5 * x)) + wj)) + ((-2.0 * (wj * x)) + x));
	} else {
		tmp = wj - ((wj - (x / exp(wj))) / (wj + 1.0));
	}
	return tmp;
}

Fortran

real(8) function code(wj, x)
    real(8), intent (in) :: wj
    real(8), intent (in) :: x
    code = wj - (((wj * exp(wj)) - x) / (exp(wj) + (wj * exp(wj))))
end function

↓

real(8) function code(wj, x)
    real(8), intent (in) :: wj
    real(8), intent (in) :: x
    real(8) :: t_0
    real(8) :: tmp
    t_0 = wj * exp(wj)
    if ((wj - ((t_0 - x) / (exp(wj) + t_0))) <= 1d-14) then
        tmp = ((-1.0d0) * ((((x * ((-2.3333333333333335d0) + 5.0d0)) + 1.0d0) * (wj * wj)) * wj)) + ((wj * ((wj * (2.5d0 * x)) + wj)) + (((-2.0d0) * (wj * x)) + x))
    else
        tmp = wj - ((wj - (x / exp(wj))) / (wj + 1.0d0))
    end if
    code = tmp
end function

Java

public static double code(double wj, double x) {
	return wj - (((wj * Math.exp(wj)) - x) / (Math.exp(wj) + (wj * Math.exp(wj))));
}

↓

public static double code(double wj, double x) {
	double t_0 = wj * Math.exp(wj);
	double tmp;
	if ((wj - ((t_0 - x) / (Math.exp(wj) + t_0))) <= 1e-14) {
		tmp = (-1.0 * ((((x * (-2.3333333333333335 + 5.0)) + 1.0) * (wj * wj)) * wj)) + ((wj * ((wj * (2.5 * x)) + wj)) + ((-2.0 * (wj * x)) + x));
	} else {
		tmp = wj - ((wj - (x / Math.exp(wj))) / (wj + 1.0));
	}
	return tmp;
}

Python

def code(wj, x):
	return wj - (((wj * math.exp(wj)) - x) / (math.exp(wj) + (wj * math.exp(wj))))

↓

def code(wj, x):
	t_0 = wj * math.exp(wj)
	tmp = 0
	if (wj - ((t_0 - x) / (math.exp(wj) + t_0))) <= 1e-14:
		tmp = (-1.0 * ((((x * (-2.3333333333333335 + 5.0)) + 1.0) * (wj * wj)) * wj)) + ((wj * ((wj * (2.5 * x)) + wj)) + ((-2.0 * (wj * x)) + x))
	else:
		tmp = wj - ((wj - (x / math.exp(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(t_0 - x) / Float64(exp(wj) + t_0))) <= 1e-14)
		tmp = Float64(Float64(-1.0 * Float64(Float64(Float64(Float64(x * Float64(-2.3333333333333335 + 5.0)) + 1.0) * Float64(wj * wj)) * wj)) + Float64(Float64(wj * Float64(Float64(wj * Float64(2.5 * x)) + wj)) + Float64(Float64(-2.0 * Float64(wj * x)) + x)));
	else
		tmp = Float64(wj - Float64(Float64(wj - Float64(x / exp(wj))) / Float64(wj + 1.0)));
	end
	return tmp
end

MATLAB

function tmp = code(wj, x)
	tmp = wj - (((wj * exp(wj)) - x) / (exp(wj) + (wj * exp(wj))));
end

↓

function tmp_2 = code(wj, x)
	t_0 = wj * exp(wj);
	tmp = 0.0;
	if ((wj - ((t_0 - x) / (exp(wj) + t_0))) <= 1e-14)
		tmp = (-1.0 * ((((x * (-2.3333333333333335 + 5.0)) + 1.0) * (wj * wj)) * wj)) + ((wj * ((wj * (2.5 * x)) + wj)) + ((-2.0 * (wj * x)) + x));
	else
		tmp = wj - ((wj - (x / exp(wj))) / (wj + 1.0));
	end
	tmp_2 = 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], 1e-14], N[(N[(-1.0 * N[(N[(N[(N[(x * N[(-2.3333333333333335 + 5.0), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision] * N[(wj * wj), $MachinePrecision]), $MachinePrecision] * wj), $MachinePrecision]), $MachinePrecision] + N[(N[(wj * N[(N[(wj * N[(2.5 * x), $MachinePrecision]), $MachinePrecision] + wj), $MachinePrecision]), $MachinePrecision] + N[(N[(-2.0 * N[(wj * x), $MachinePrecision]), $MachinePrecision] + x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(wj - N[(N[(wj - N[(x / N[Exp[wj], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(wj + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Target

Original	13.3
Target	12.7
Herbie	0.5

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

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))))) < 9.99999999999999999e-15

   1. Initial program 17.6

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

   2. Simplified 17.6

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

   3. Taylor expanded in wj around 0 0.5

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

   4. Applied egg-rr 0.7

      \[\leadsto -1 \cdot \left(\left(0.6666666666666666 \cdot x + \left(-3 \cdot x + \left(1 + -2 \cdot \left(-4 \cdot x + 1.5 \cdot x\right)\right)\right)\right) \cdot {wj}^{3}\right) + \left(\color{blue}{{\left(\sqrt[3]{-\mathsf{fma}\left(x, -2.5, -1\right) \cdot \left(wj \cdot wj\right)}\right)}^{3}} + \left(-2 \cdot \left(wj \cdot x\right) + x\right)\right) \]

   5. Applied egg-rr 0.7

      \[\leadsto -1 \cdot \color{blue}{\left(\left(\left(x \cdot \left(-2.3333333333333335 + 5\right) + 1\right) \cdot \left(wj \cdot wj\right)\right) \cdot wj\right)} + \left({\left(\sqrt[3]{-\mathsf{fma}\left(x, -2.5, -1\right) \cdot \left(wj \cdot wj\right)}\right)}^{3} + \left(-2 \cdot \left(wj \cdot x\right) + x\right)\right) \]

   6. Applied egg-rr 0.5

      \[\leadsto -1 \cdot \left(\left(\left(x \cdot \left(-2.3333333333333335 + 5\right) + 1\right) \cdot \left(wj \cdot wj\right)\right) \cdot wj\right) + \left(\color{blue}{\mathsf{fma}\left(wj \cdot wj, 2.5 \cdot x, wj \cdot wj\right)} + \left(-2 \cdot \left(wj \cdot x\right) + x\right)\right) \]

   7. Simplified 0.5

      \[\leadsto -1 \cdot \left(\left(\left(x \cdot \left(-2.3333333333333335 + 5\right) + 1\right) \cdot \left(wj \cdot wj\right)\right) \cdot wj\right) + \left(\color{blue}{wj \cdot \left(wj \cdot \left(2.5 \cdot x\right) + wj\right)} + \left(-2 \cdot \left(wj \cdot x\right) + x\right)\right) \]
      Proof

   if 9.99999999999999999e-15 < (-.f64 wj (/.f64 (-.f64 (*.f64 wj (exp.f64 wj)) x) (+.f64 (exp.f64 wj) (*.f64 wj (exp.f64 wj)))))

   1. Initial program 2.4

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

   2. Simplified 0.4

      \[\leadsto \color{blue}{wj - \frac{wj - \frac{x}{e^{wj}}}{wj + 1}} \]
      Proof
3. Recombined 2 regimes into one program.

Alternatives

Alternative 1
Error	1.6
Cost	2112

\[-1 \cdot \left(\left(\left(x \cdot \left(-2.3333333333333335 + 5\right) + 1\right) \cdot \left(wj \cdot wj\right)\right) \cdot wj\right) + \left(wj \cdot \left(wj \cdot \left(2.5 \cdot x\right) + wj\right) + \left(-2 \cdot \left(wj \cdot x\right) + x\right)\right) \]

Alternative 2
Error	9.2
Cost	708

\[\begin{array}{l} \mathbf{if}\;x \leq -2.3 \cdot 10^{-256}:\\ \;\;\;\;\frac{1 - wj}{1 + wj} \cdot x\\ \mathbf{elif}\;x \leq 6 \cdot 10^{-278}:\\ \;\;\;\;wj \cdot wj\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 3
Error	2.0
Cost	704

\[wj \cdot wj + \left(-2 \cdot \left(wj \cdot x\right) + x\right) \]

Alternative 4
Error	9.2
Cost	580

\[\begin{array}{l} \mathbf{if}\;x \leq -5.4 \cdot 10^{-258}:\\ \;\;\;\;\left(-2 \cdot wj + 1\right) \cdot x\\ \mathbf{elif}\;x \leq 1.25 \cdot 10^{-281}:\\ \;\;\;\;wj \cdot wj\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 5
Error	9.4
Cost	456

\[\begin{array}{l} \mathbf{if}\;x \leq -6.5 \cdot 10^{-258}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq 1.55 \cdot 10^{-280}:\\ \;\;\;\;wj \cdot wj\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 6
Error	61.2
Cost	64

\[wj \]

Alternative 7
Error	9.1
Cost	64

\[x \]

Reproduce

herbie shell --seed 2023010 
(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))))))