Jmat.Real.lambertw, newton loop step

Average Error: 14.0 → 0.2

Time: 15.7s

Precision: binary64

Cost: 7560

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;wj \leq -5.853250649290735 \cdot 10^{-9}:\\ \;\;\;\;\frac{x}{e^{wj} \cdot \left(1 + wj\right)} + \left(wj - \frac{wj}{1 + wj}\right)\\ \mathbf{elif}\;wj \leq 1.6348081454728468 \cdot 10^{-9}:\\ \;\;\;\;\left(\left(x + -2 \cdot \left(x \cdot wj\right)\right) + wj \cdot wj\right) - {wj}^{3}\\ \mathbf{else}:\\ \;\;\;\;wj + \frac{x \cdot e^{-wj} - wj}{1 + wj}\\ \end{array} \]

FPCore

(FPCore (wj x)
 :precision binary64
 (- wj (/ (- (* wj (exp wj)) x) (+ (exp wj) (* wj (exp wj))))))

↓

(FPCore (wj x)
 :precision binary64
 (if (<= wj -5.853250649290735e-9)
   (+ (/ x (* (exp wj) (+ 1.0 wj))) (- wj (/ wj (+ 1.0 wj))))
   (if (<= wj 1.6348081454728468e-9)
     (- (+ (+ x (* -2.0 (* x wj))) (* wj wj)) (pow wj 3.0))
     (+ wj (/ (- (* x (exp (- wj))) 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 tmp;
	if (wj <= -5.853250649290735e-9) {
		tmp = (x / (exp(wj) * (1.0 + wj))) + (wj - (wj / (1.0 + wj)));
	} else if (wj <= 1.6348081454728468e-9) {
		tmp = ((x + (-2.0 * (x * wj))) + (wj * wj)) - pow(wj, 3.0);
	} else {
		tmp = wj + (((x * exp(-wj)) - wj) / (1.0 + wj));
	}
	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) :: tmp
    if (wj <= (-5.853250649290735d-9)) then
        tmp = (x / (exp(wj) * (1.0d0 + wj))) + (wj - (wj / (1.0d0 + wj)))
    else if (wj <= 1.6348081454728468d-9) then
        tmp = ((x + ((-2.0d0) * (x * wj))) + (wj * wj)) - (wj ** 3.0d0)
    else
        tmp = wj + (((x * exp(-wj)) - wj) / (1.0d0 + wj))
    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 tmp;
	if (wj <= -5.853250649290735e-9) {
		tmp = (x / (Math.exp(wj) * (1.0 + wj))) + (wj - (wj / (1.0 + wj)));
	} else if (wj <= 1.6348081454728468e-9) {
		tmp = ((x + (-2.0 * (x * wj))) + (wj * wj)) - Math.pow(wj, 3.0);
	} else {
		tmp = wj + (((x * Math.exp(-wj)) - wj) / (1.0 + wj));
	}
	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):
	tmp = 0
	if wj <= -5.853250649290735e-9:
		tmp = (x / (math.exp(wj) * (1.0 + wj))) + (wj - (wj / (1.0 + wj)))
	elif wj <= 1.6348081454728468e-9:
		tmp = ((x + (-2.0 * (x * wj))) + (wj * wj)) - math.pow(wj, 3.0)
	else:
		tmp = wj + (((x * math.exp(-wj)) - 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)
	tmp = 0.0
	if (wj <= -5.853250649290735e-9)
		tmp = Float64(Float64(x / Float64(exp(wj) * Float64(1.0 + wj))) + Float64(wj - Float64(wj / Float64(1.0 + wj))));
	elseif (wj <= 1.6348081454728468e-9)
		tmp = Float64(Float64(Float64(x + Float64(-2.0 * Float64(x * wj))) + Float64(wj * wj)) - (wj ^ 3.0));
	else
		tmp = Float64(wj + Float64(Float64(Float64(x * exp(Float64(-wj))) - wj) / Float64(1.0 + wj)));
	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)
	tmp = 0.0;
	if (wj <= -5.853250649290735e-9)
		tmp = (x / (exp(wj) * (1.0 + wj))) + (wj - (wj / (1.0 + wj)));
	elseif (wj <= 1.6348081454728468e-9)
		tmp = ((x + (-2.0 * (x * wj))) + (wj * wj)) - (wj ^ 3.0);
	else
		tmp = wj + (((x * exp(-wj)) - wj) / (1.0 + wj));
	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_] := If[LessEqual[wj, -5.853250649290735e-9], N[(N[(x / N[(N[Exp[wj], $MachinePrecision] * N[(1.0 + wj), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(wj - N[(wj / N[(1.0 + wj), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[wj, 1.6348081454728468e-9], N[(N[(N[(x + N[(-2.0 * N[(x * wj), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(wj * wj), $MachinePrecision]), $MachinePrecision] - N[Power[wj, 3.0], $MachinePrecision]), $MachinePrecision], N[(wj + N[(N[(N[(x * N[Exp[(-wj)], $MachinePrecision]), $MachinePrecision] - wj), $MachinePrecision] / N[(1.0 + wj), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Target

Original	14.0
Target	13.2
Herbie	0.2

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

Derivation

1. Split input into 3 regimes

2. if wj < -5.8532506492907348e-9

   1. Initial program 4.2

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

   2. Taylor expanded in x around 0 4.3

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

   3. Simplified 4.3

      \[\leadsto \color{blue}{\frac{x}{e^{wj} \cdot \left(wj + 1\right)} + \left(wj - \frac{wj}{wj + 1}\right)} \]
      Proof
      (+.f64 (/.f64 x (*.f64 (exp.f64 wj) (+.f64 wj 1))) (-.f64 wj (/.f64 wj (+.f64 wj 1)))): 0 points increase in error, 0 points decrease in error
      (+.f64 (/.f64 x (Rewrite<= *-commutative_binary64 (*.f64 (+.f64 wj 1) (exp.f64 wj)))) (-.f64 wj (/.f64 wj (+.f64 wj 1)))): 0 points increase in error, 0 points decrease in error
      (+.f64 (/.f64 x (Rewrite<= distribute-lft1-in_binary64 (+.f64 (*.f64 wj (exp.f64 wj)) (exp.f64 wj)))) (-.f64 wj (/.f64 wj (+.f64 wj 1)))): 1 points increase in error, 2 points decrease in error
      (+.f64 (/.f64 x (+.f64 (Rewrite=> *-commutative_binary64 (*.f64 (exp.f64 wj) wj)) (exp.f64 wj))) (-.f64 wj (/.f64 wj (+.f64 wj 1)))): 0 points increase in error, 0 points decrease in error
      (+.f64 (/.f64 x (+.f64 (*.f64 (exp.f64 wj) wj) (exp.f64 wj))) (-.f64 wj (/.f64 (Rewrite<= *-rgt-identity_binary64 (*.f64 wj 1)) (+.f64 wj 1)))): 0 points increase in error, 0 points decrease in error
      (+.f64 (/.f64 x (+.f64 (*.f64 (exp.f64 wj) wj) (exp.f64 wj))) (-.f64 wj (Rewrite<= associate-*l/_binary64 (*.f64 (/.f64 wj (+.f64 wj 1)) 1)))): 0 points increase in error, 0 points decrease in error
      (+.f64 (/.f64 x (+.f64 (*.f64 (exp.f64 wj) wj) (exp.f64 wj))) (-.f64 wj (*.f64 (/.f64 wj (+.f64 wj 1)) (Rewrite<= *-inverses_binary64 (/.f64 (exp.f64 wj) (exp.f64 wj)))))): 1 points increase in error, 0 points decrease in error
      (+.f64 (/.f64 x (+.f64 (*.f64 (exp.f64 wj) wj) (exp.f64 wj))) (-.f64 wj (Rewrite<= times-frac_binary64 (/.f64 (*.f64 wj (exp.f64 wj)) (*.f64 (+.f64 wj 1) (exp.f64 wj)))))): 3 points increase in error, 0 points decrease in error
      (+.f64 (/.f64 x (+.f64 (*.f64 (exp.f64 wj) wj) (exp.f64 wj))) (-.f64 wj (/.f64 (*.f64 wj (exp.f64 wj)) (Rewrite<= distribute-lft1-in_binary64 (+.f64 (*.f64 wj (exp.f64 wj)) (exp.f64 wj)))))): 0 points increase in error, 1 points decrease in error
      (+.f64 (/.f64 x (+.f64 (*.f64 (exp.f64 wj) wj) (exp.f64 wj))) (-.f64 wj (/.f64 (*.f64 wj (exp.f64 wj)) (+.f64 (Rewrite=> *-commutative_binary64 (*.f64 (exp.f64 wj) wj)) (exp.f64 wj))))): 0 points increase in error, 0 points decrease in error
      (+.f64 (/.f64 x (+.f64 (*.f64 (exp.f64 wj) wj) (exp.f64 wj))) (-.f64 wj (/.f64 (Rewrite=> *-commutative_binary64 (*.f64 (exp.f64 wj) wj)) (+.f64 (*.f64 (exp.f64 wj) wj) (exp.f64 wj))))): 0 points increase in error, 0 points decrease in error
      (+.f64 (/.f64 x (+.f64 (*.f64 (exp.f64 wj) wj) (exp.f64 wj))) (-.f64 wj (/.f64 (Rewrite<= *-commutative_binary64 (*.f64 wj (exp.f64 wj))) (+.f64 (*.f64 (exp.f64 wj) wj) (exp.f64 wj))))): 0 points increase in error, 0 points decrease in error
      (Rewrite<= associate--l+_binary64 (-.f64 (+.f64 (/.f64 x (+.f64 (*.f64 (exp.f64 wj) wj) (exp.f64 wj))) wj) (/.f64 (*.f64 wj (exp.f64 wj)) (+.f64 (*.f64 (exp.f64 wj) wj) (exp.f64 wj))))): 51 points increase in error, 15 points decrease in error

   if -5.8532506492907348e-9 < wj < 1.63480814547284681e-9

   1. Initial program 13.7

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

   2. Simplified 13.7

      \[\leadsto \color{blue}{wj - \frac{wj - \frac{x}{e^{wj}}}{wj + 1}} \]
      Proof
      (-.f64 wj (/.f64 (-.f64 wj (/.f64 x (exp.f64 wj))) (+.f64 wj 1))): 0 points increase in error, 0 points decrease in error
      (-.f64 wj (Rewrite=> div-sub_binary64 (-.f64 (/.f64 wj (+.f64 wj 1)) (/.f64 (/.f64 x (exp.f64 wj)) (+.f64 wj 1))))): 0 points increase in error, 0 points decrease in error
      (-.f64 wj (-.f64 (Rewrite<= *-rgt-identity_binary64 (*.f64 (/.f64 wj (+.f64 wj 1)) 1)) (/.f64 (/.f64 x (exp.f64 wj)) (+.f64 wj 1)))): 0 points increase in error, 0 points decrease in error
      (-.f64 wj (-.f64 (*.f64 (/.f64 wj (+.f64 wj 1)) (Rewrite<= *-inverses_binary64 (/.f64 (exp.f64 wj) (exp.f64 wj)))) (/.f64 (/.f64 x (exp.f64 wj)) (+.f64 wj 1)))): 1 points increase in error, 0 points decrease in error
      (-.f64 wj (-.f64 (Rewrite<= times-frac_binary64 (/.f64 (*.f64 wj (exp.f64 wj)) (*.f64 (+.f64 wj 1) (exp.f64 wj)))) (/.f64 (/.f64 x (exp.f64 wj)) (+.f64 wj 1)))): 3 points increase in error, 0 points decrease in error
      (-.f64 wj (-.f64 (/.f64 (*.f64 wj (exp.f64 wj)) (Rewrite<= distribute-rgt1-in_binary64 (+.f64 (exp.f64 wj) (*.f64 wj (exp.f64 wj))))) (/.f64 (/.f64 x (exp.f64 wj)) (+.f64 wj 1)))): 0 points increase in error, 1 points decrease in error
      (-.f64 wj (-.f64 (/.f64 (*.f64 wj (exp.f64 wj)) (+.f64 (exp.f64 wj) (*.f64 wj (exp.f64 wj)))) (Rewrite=> associate-/l/_binary64 (/.f64 x (*.f64 (+.f64 wj 1) (exp.f64 wj)))))): 2 points increase in error, 2 points decrease in error
      (-.f64 wj (-.f64 (/.f64 (*.f64 wj (exp.f64 wj)) (+.f64 (exp.f64 wj) (*.f64 wj (exp.f64 wj)))) (/.f64 x (Rewrite<= distribute-rgt1-in_binary64 (+.f64 (exp.f64 wj) (*.f64 wj (exp.f64 wj))))))): 1 points increase in error, 2 points decrease in error
      (-.f64 wj (Rewrite<= div-sub_binary64 (/.f64 (-.f64 (*.f64 wj (exp.f64 wj)) x) (+.f64 (exp.f64 wj) (*.f64 wj (exp.f64 wj)))))): 0 points increase in error, 0 points decrease in error

   3. Taylor expanded in wj around 0 0.1

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

   4. Taylor expanded in x around 0 0.1

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

   5. Simplified 0.1

      \[\leadsto -1 \cdot \left(\left(0.6666666666666666 \cdot x + \left(-3 \cdot x + \left(1 + -2 \cdot \left(-4 \cdot x + 1.5 \cdot x\right)\right)\right)\right) \cdot {wj}^{3}\right) + \left(\color{blue}{wj \cdot wj} + \left(-2 \cdot \left(wj \cdot x\right) + x\right)\right) \]
      Proof
      (*.f64 wj wj): 0 points increase in error, 0 points decrease in error
      (Rewrite<= unpow2_binary64 (pow.f64 wj 2)): 0 points increase in error, 0 points decrease in error

   6. Taylor expanded in x around 0 0.0

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

   if 1.63480814547284681e-9 < wj

   1. Initial program 29.8

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

   2. Simplified 3.2

      \[\leadsto \color{blue}{wj - \frac{wj - \frac{x}{e^{wj}}}{wj + 1}} \]
      Proof
      (-.f64 wj (/.f64 (-.f64 wj (/.f64 x (exp.f64 wj))) (+.f64 wj 1))): 0 points increase in error, 0 points decrease in error
      (-.f64 wj (Rewrite=> div-sub_binary64 (-.f64 (/.f64 wj (+.f64 wj 1)) (/.f64 (/.f64 x (exp.f64 wj)) (+.f64 wj 1))))): 0 points increase in error, 0 points decrease in error
      (-.f64 wj (-.f64 (Rewrite<= *-rgt-identity_binary64 (*.f64 (/.f64 wj (+.f64 wj 1)) 1)) (/.f64 (/.f64 x (exp.f64 wj)) (+.f64 wj 1)))): 0 points increase in error, 0 points decrease in error
      (-.f64 wj (-.f64 (*.f64 (/.f64 wj (+.f64 wj 1)) (Rewrite<= *-inverses_binary64 (/.f64 (exp.f64 wj) (exp.f64 wj)))) (/.f64 (/.f64 x (exp.f64 wj)) (+.f64 wj 1)))): 1 points increase in error, 0 points decrease in error
      (-.f64 wj (-.f64 (Rewrite<= times-frac_binary64 (/.f64 (*.f64 wj (exp.f64 wj)) (*.f64 (+.f64 wj 1) (exp.f64 wj)))) (/.f64 (/.f64 x (exp.f64 wj)) (+.f64 wj 1)))): 3 points increase in error, 0 points decrease in error
      (-.f64 wj (-.f64 (/.f64 (*.f64 wj (exp.f64 wj)) (Rewrite<= distribute-rgt1-in_binary64 (+.f64 (exp.f64 wj) (*.f64 wj (exp.f64 wj))))) (/.f64 (/.f64 x (exp.f64 wj)) (+.f64 wj 1)))): 0 points increase in error, 1 points decrease in error
      (-.f64 wj (-.f64 (/.f64 (*.f64 wj (exp.f64 wj)) (+.f64 (exp.f64 wj) (*.f64 wj (exp.f64 wj)))) (Rewrite=> associate-/l/_binary64 (/.f64 x (*.f64 (+.f64 wj 1) (exp.f64 wj)))))): 2 points increase in error, 2 points decrease in error
      (-.f64 wj (-.f64 (/.f64 (*.f64 wj (exp.f64 wj)) (+.f64 (exp.f64 wj) (*.f64 wj (exp.f64 wj)))) (/.f64 x (Rewrite<= distribute-rgt1-in_binary64 (+.f64 (exp.f64 wj) (*.f64 wj (exp.f64 wj))))))): 1 points increase in error, 2 points decrease in error
      (-.f64 wj (Rewrite<= div-sub_binary64 (/.f64 (-.f64 (*.f64 wj (exp.f64 wj)) x) (+.f64 (exp.f64 wj) (*.f64 wj (exp.f64 wj)))))): 0 points increase in error, 0 points decrease in error

   3. Applied egg-rr 3.1

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

4. Final simplification 0.2

   \[\leadsto \begin{array}{l} \mathbf{if}\;wj \leq -5.853250649290735 \cdot 10^{-9}:\\ \;\;\;\;\frac{x}{e^{wj} \cdot \left(1 + wj\right)} + \left(wj - \frac{wj}{1 + wj}\right)\\ \mathbf{elif}\;wj \leq 1.6348081454728468 \cdot 10^{-9}:\\ \;\;\;\;\left(\left(x + -2 \cdot \left(x \cdot wj\right)\right) + wj \cdot wj\right) - {wj}^{3}\\ \mathbf{else}:\\ \;\;\;\;wj + \frac{x \cdot e^{-wj} - wj}{1 + wj}\\ \end{array} \]

Alternatives

Alternative 1
Error	1.7
Cost	15552

\[\begin{array}{l} t_0 := x \cdot 4 + x \cdot -1.5\\ {wj}^{3} \cdot \left(x \cdot -0.6666666666666666 + \left(x \cdot 3 + \left(-1 + -2 \cdot t_0\right)\right)\right) + \left(\left(1 + t_0\right) \cdot {wj}^{2} + \left(x + -2 \cdot \left(x \cdot wj\right)\right)\right) \end{array} \]

Alternative 2
Error	1.5
Cost	13636

\[\begin{array}{l} \mathbf{if}\;wj \leq -9.459341274338091 \cdot 10^{-6}:\\ \;\;\;\;\mathsf{fma}\left(wj - \frac{x}{e^{wj}}, \frac{-1}{1 + wj}, wj\right)\\ \mathbf{else}:\\ \;\;\;\;\left(1 + \left(x \cdot 4 + x \cdot -1.5\right)\right) \cdot {wj}^{2} + \left(x + -2 \cdot \left(x \cdot wj\right)\right)\\ \end{array} \]

Alternative 3
Error	1.5
Cost	7812

\[\begin{array}{l} \mathbf{if}\;wj \leq -9.459341274338091 \cdot 10^{-6}:\\ \;\;\;\;\frac{x}{e^{wj} \cdot \left(1 + wj\right)} + \left(wj - \frac{wj}{1 + wj}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(1 + \left(x \cdot 4 + x \cdot -1.5\right)\right) \cdot {wj}^{2} + \left(x + -2 \cdot \left(x \cdot wj\right)\right)\\ \end{array} \]

Alternative 4
Error	0.3
Cost	7492

\[\begin{array}{l} \mathbf{if}\;wj \leq -5.853250649290735 \cdot 10^{-9}:\\ \;\;\;\;\frac{x}{e^{wj} \cdot \left(1 + wj\right)} + \left(wj - \frac{wj}{1 + wj}\right)\\ \mathbf{elif}\;wj \leq 1.6348081454728468 \cdot 10^{-9}:\\ \;\;\;\;\left(x + -2 \cdot \left(x \cdot wj\right)\right) + wj \cdot wj\\ \mathbf{else}:\\ \;\;\;\;wj + \frac{x \cdot e^{-wj} - wj}{1 + wj}\\ \end{array} \]

Alternative 5
Error	0.3
Cost	7432

\[\begin{array}{l} \mathbf{if}\;wj \leq -5.853250649290735 \cdot 10^{-9}:\\ \;\;\;\;wj + \frac{\frac{x}{e^{wj}} - wj}{1 + wj}\\ \mathbf{elif}\;wj \leq 1.6348081454728468 \cdot 10^{-9}:\\ \;\;\;\;\left(x + -2 \cdot \left(x \cdot wj\right)\right) + wj \cdot wj\\ \mathbf{else}:\\ \;\;\;\;wj + \frac{x \cdot e^{-wj} - wj}{1 + wj}\\ \end{array} \]

Alternative 6
Error	0.3
Cost	7368

\[\begin{array}{l} t_0 := wj + \frac{\frac{x}{e^{wj}} - wj}{1 + wj}\\ \mathbf{if}\;wj \leq -5.853250649290735 \cdot 10^{-9}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;wj \leq 1.6348081454728468 \cdot 10^{-9}:\\ \;\;\;\;\left(x + -2 \cdot \left(x \cdot wj\right)\right) + wj \cdot wj\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 7
Error	9.0
Cost	704

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

Alternative 8
Error	2.0
Cost	704

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

Alternative 9
Error	9.1
Cost	576

\[\frac{x - x \cdot wj}{1 + wj} \]

Alternative 10
Error	9.1
Cost	448

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

Alternative 11
Error	9.1
Cost	448

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

Alternative 12
Error	61.1
Cost	64

\[wj \]

Alternative 13
Error	9.5
Cost	64

\[x \]

Reproduce

herbie shell --seed 2022311 
(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))))))