Jmat.Real.lambertw, newton loop step

Average Error: 13.8 → 0.3

Time: 15.5s

Precision: binary64

Cost: 7368

Math

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

↓

\[\begin{array}{l} t_0 := \frac{x}{e^{wj}}\\ \mathbf{if}\;wj \leq -7.6 \cdot 10^{-9}:\\ \;\;\;\;wj + \left(t_0 - wj\right) \cdot \frac{1}{wj + 1}\\ \mathbf{elif}\;wj \leq 6.3 \cdot 10^{-9}:\\ \;\;\;\;\left(x - 2 \cdot \left(wj \cdot x\right)\right) + wj \cdot wj\\ \mathbf{else}:\\ \;\;\;\;wj - \frac{wj - t_0}{wj + 1}\\ \end{array} \]

FPCore

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

↓

(FPCore (wj x)
 :precision binary64
 (let* ((t_0 (/ x (exp wj))))
   (if (<= wj -7.6e-9)
     (+ wj (* (- t_0 wj) (/ 1.0 (+ wj 1.0))))
     (if (<= wj 6.3e-9)
       (+ (- x (* 2.0 (* wj x))) (* wj wj))
       (- wj (/ (- wj t_0) (+ 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 = x / exp(wj);
	double tmp;
	if (wj <= -7.6e-9) {
		tmp = wj + ((t_0 - wj) * (1.0 / (wj + 1.0)));
	} else if (wj <= 6.3e-9) {
		tmp = (x - (2.0 * (wj * x))) + (wj * wj);
	} else {
		tmp = wj - ((wj - t_0) / (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 = x / exp(wj)
    if (wj <= (-7.6d-9)) then
        tmp = wj + ((t_0 - wj) * (1.0d0 / (wj + 1.0d0)))
    else if (wj <= 6.3d-9) then
        tmp = (x - (2.0d0 * (wj * x))) + (wj * wj)
    else
        tmp = wj - ((wj - t_0) / (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 = x / Math.exp(wj);
	double tmp;
	if (wj <= -7.6e-9) {
		tmp = wj + ((t_0 - wj) * (1.0 / (wj + 1.0)));
	} else if (wj <= 6.3e-9) {
		tmp = (x - (2.0 * (wj * x))) + (wj * wj);
	} else {
		tmp = wj - ((wj - t_0) / (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 = x / math.exp(wj)
	tmp = 0
	if wj <= -7.6e-9:
		tmp = wj + ((t_0 - wj) * (1.0 / (wj + 1.0)))
	elif wj <= 6.3e-9:
		tmp = (x - (2.0 * (wj * x))) + (wj * wj)
	else:
		tmp = wj - ((wj - t_0) / (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(x / exp(wj))
	tmp = 0.0
	if (wj <= -7.6e-9)
		tmp = Float64(wj + Float64(Float64(t_0 - wj) * Float64(1.0 / Float64(wj + 1.0))));
	elseif (wj <= 6.3e-9)
		tmp = Float64(Float64(x - Float64(2.0 * Float64(wj * x))) + Float64(wj * wj));
	else
		tmp = Float64(wj - Float64(Float64(wj - t_0) / 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 = x / exp(wj);
	tmp = 0.0;
	if (wj <= -7.6e-9)
		tmp = wj + ((t_0 - wj) * (1.0 / (wj + 1.0)));
	elseif (wj <= 6.3e-9)
		tmp = (x - (2.0 * (wj * x))) + (wj * wj);
	else
		tmp = wj - ((wj - t_0) / (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[(x / N[Exp[wj], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[wj, -7.6e-9], N[(wj + N[(N[(t$95$0 - wj), $MachinePrecision] * N[(1.0 / N[(wj + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[wj, 6.3e-9], N[(N[(x - N[(2.0 * N[(wj * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(wj * wj), $MachinePrecision]), $MachinePrecision], N[(wj - N[(N[(wj - t$95$0), $MachinePrecision] / N[(wj + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Target

Original	13.8
Target	13.1
Herbie	0.3

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

Derivation

1. Split input into 3 regimes

2. if wj < -7.60000000000000023e-9

   1. Initial program 5.4

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

   2. Simplified 5.3

      \[\leadsto \color{blue}{wj + \frac{\frac{x}{e^{wj}} - wj}{wj + 1}} \]
      Proof
      (+.f64 wj (/.f64 (-.f64 (/.f64 x (exp.f64 wj)) wj) (+.f64 wj 1))): 0 points increase in error, 0 points decrease in error
      (+.f64 wj (Rewrite=> div-sub_binary64 (-.f64 (/.f64 (/.f64 x (exp.f64 wj)) (+.f64 wj 1)) (/.f64 wj (+.f64 wj 1))))): 0 points increase in error, 20 points decrease in error
      (+.f64 wj (-.f64 (Rewrite=> associate-/l/_binary64 (/.f64 x (*.f64 (+.f64 wj 1) (exp.f64 wj)))) (/.f64 wj (+.f64 wj 1)))): 0 points increase in error, 0 points decrease in error
      (+.f64 wj (-.f64 (/.f64 x (Rewrite<= distribute-rgt1-in_binary64 (+.f64 (exp.f64 wj) (*.f64 wj (exp.f64 wj))))) (/.f64 wj (+.f64 wj 1)))): 0 points increase in error, 0 points decrease in error
      (+.f64 wj (-.f64 (/.f64 x (+.f64 (exp.f64 wj) (*.f64 wj (exp.f64 wj)))) (Rewrite<= *-lft-identity_binary64 (*.f64 1 (/.f64 wj (+.f64 wj 1)))))): 20 points increase in error, 0 points decrease in error
      (+.f64 wj (-.f64 (/.f64 x (+.f64 (exp.f64 wj) (*.f64 wj (exp.f64 wj)))) (*.f64 (Rewrite<= *-inverses_binary64 (/.f64 (exp.f64 wj) (exp.f64 wj))) (/.f64 wj (+.f64 wj 1))))): 0 points increase in error, 20 points decrease in error
      (+.f64 wj (-.f64 (/.f64 x (+.f64 (exp.f64 wj) (*.f64 wj (exp.f64 wj)))) (Rewrite<= *-commutative_binary64 (*.f64 (/.f64 wj (+.f64 wj 1)) (/.f64 (exp.f64 wj) (exp.f64 wj)))))): 0 points increase in error, 0 points decrease in error
      (+.f64 wj (-.f64 (/.f64 x (+.f64 (exp.f64 wj) (*.f64 wj (exp.f64 wj)))) (Rewrite<= times-frac_binary64 (/.f64 (*.f64 wj (exp.f64 wj)) (*.f64 (+.f64 wj 1) (exp.f64 wj)))))): 0 points increase in error, 0 points decrease in error
      (+.f64 wj (-.f64 (/.f64 x (+.f64 (exp.f64 wj) (*.f64 wj (exp.f64 wj)))) (/.f64 (*.f64 wj (exp.f64 wj)) (Rewrite<= distribute-rgt1-in_binary64 (+.f64 (exp.f64 wj) (*.f64 wj (exp.f64 wj))))))): 0 points increase in error, 0 points decrease in error
      (+.f64 wj (Rewrite=> sub-neg_binary64 (+.f64 (/.f64 x (+.f64 (exp.f64 wj) (*.f64 wj (exp.f64 wj)))) (neg.f64 (/.f64 (*.f64 wj (exp.f64 wj)) (+.f64 (exp.f64 wj) (*.f64 wj (exp.f64 wj)))))))): 0 points increase in error, 0 points decrease in error
      (+.f64 wj (+.f64 (Rewrite<= remove-double-neg_binary64 (neg.f64 (neg.f64 (/.f64 x (+.f64 (exp.f64 wj) (*.f64 wj (exp.f64 wj))))))) (neg.f64 (/.f64 (*.f64 wj (exp.f64 wj)) (+.f64 (exp.f64 wj) (*.f64 wj (exp.f64 wj))))))): 20 points increase in error, 0 points decrease in error
      (+.f64 wj (Rewrite<= distribute-neg-in_binary64 (neg.f64 (+.f64 (neg.f64 (/.f64 x (+.f64 (exp.f64 wj) (*.f64 wj (exp.f64 wj))))) (/.f64 (*.f64 wj (exp.f64 wj)) (+.f64 (exp.f64 wj) (*.f64 wj (exp.f64 wj)))))))): 0 points increase in error, 20 points decrease in error
      (+.f64 wj (neg.f64 (Rewrite<= +-commutative_binary64 (+.f64 (/.f64 (*.f64 wj (exp.f64 wj)) (+.f64 (exp.f64 wj) (*.f64 wj (exp.f64 wj)))) (neg.f64 (/.f64 x (+.f64 (exp.f64 wj) (*.f64 wj (exp.f64 wj))))))))): 0 points increase in error, 0 points decrease in error
      (+.f64 wj (neg.f64 (Rewrite<= sub-neg_binary64 (-.f64 (/.f64 (*.f64 wj (exp.f64 wj)) (+.f64 (exp.f64 wj) (*.f64 wj (exp.f64 wj)))) (/.f64 x (+.f64 (exp.f64 wj) (*.f64 wj (exp.f64 wj)))))))): 0 points increase in error, 0 points decrease in error
      (+.f64 wj (neg.f64 (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
      (+.f64 wj (Rewrite=> neg-mul-1_binary64 (*.f64 -1 (/.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
      (+.f64 wj (Rewrite=> *-commutative_binary64 (*.f64 (/.f64 (-.f64 (*.f64 wj (exp.f64 wj)) x) (+.f64 (exp.f64 wj) (*.f64 wj (exp.f64 wj)))) -1))): 0 points increase in error, 0 points decrease in error
      (+.f64 wj (Rewrite<= *-commutative_binary64 (*.f64 -1 (/.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
      (+.f64 wj (Rewrite<= neg-mul-1_binary64 (neg.f64 (/.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
      (Rewrite<= sub-neg_binary64 (-.f64 wj (/.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 5.5

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

   if -7.60000000000000023e-9 < wj < 6.3000000000000002e-9

   1. Initial program 13.6

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

   2. Simplified 13.6

      \[\leadsto \color{blue}{wj - \frac{wj \cdot e^{wj} - x}{\left(wj + 1\right) \cdot e^{wj}}} \]
      Proof
      (+.f64 wj (/.f64 (-.f64 (/.f64 x (exp.f64 wj)) wj) (+.f64 wj 1))): 0 points increase in error, 0 points decrease in error
      (+.f64 wj (Rewrite=> div-sub_binary64 (-.f64 (/.f64 (/.f64 x (exp.f64 wj)) (+.f64 wj 1)) (/.f64 wj (+.f64 wj 1))))): 0 points increase in error, 20 points decrease in error
      (+.f64 wj (-.f64 (Rewrite=> associate-/l/_binary64 (/.f64 x (*.f64 (+.f64 wj 1) (exp.f64 wj)))) (/.f64 wj (+.f64 wj 1)))): 0 points increase in error, 0 points decrease in error
      (+.f64 wj (-.f64 (/.f64 x (Rewrite<= distribute-rgt1-in_binary64 (+.f64 (exp.f64 wj) (*.f64 wj (exp.f64 wj))))) (/.f64 wj (+.f64 wj 1)))): 0 points increase in error, 0 points decrease in error
      (+.f64 wj (-.f64 (/.f64 x (+.f64 (exp.f64 wj) (*.f64 wj (exp.f64 wj)))) (Rewrite<= *-lft-identity_binary64 (*.f64 1 (/.f64 wj (+.f64 wj 1)))))): 20 points increase in error, 0 points decrease in error
      (+.f64 wj (-.f64 (/.f64 x (+.f64 (exp.f64 wj) (*.f64 wj (exp.f64 wj)))) (*.f64 (Rewrite<= *-inverses_binary64 (/.f64 (exp.f64 wj) (exp.f64 wj))) (/.f64 wj (+.f64 wj 1))))): 0 points increase in error, 20 points decrease in error
      (+.f64 wj (-.f64 (/.f64 x (+.f64 (exp.f64 wj) (*.f64 wj (exp.f64 wj)))) (Rewrite<= *-commutative_binary64 (*.f64 (/.f64 wj (+.f64 wj 1)) (/.f64 (exp.f64 wj) (exp.f64 wj)))))): 0 points increase in error, 0 points decrease in error
      (+.f64 wj (-.f64 (/.f64 x (+.f64 (exp.f64 wj) (*.f64 wj (exp.f64 wj)))) (Rewrite<= times-frac_binary64 (/.f64 (*.f64 wj (exp.f64 wj)) (*.f64 (+.f64 wj 1) (exp.f64 wj)))))): 0 points increase in error, 0 points decrease in error
      (+.f64 wj (-.f64 (/.f64 x (+.f64 (exp.f64 wj) (*.f64 wj (exp.f64 wj)))) (/.f64 (*.f64 wj (exp.f64 wj)) (Rewrite<= distribute-rgt1-in_binary64 (+.f64 (exp.f64 wj) (*.f64 wj (exp.f64 wj))))))): 0 points increase in error, 0 points decrease in error
      (+.f64 wj (Rewrite=> sub-neg_binary64 (+.f64 (/.f64 x (+.f64 (exp.f64 wj) (*.f64 wj (exp.f64 wj)))) (neg.f64 (/.f64 (*.f64 wj (exp.f64 wj)) (+.f64 (exp.f64 wj) (*.f64 wj (exp.f64 wj)))))))): 0 points increase in error, 0 points decrease in error
      (+.f64 wj (+.f64 (Rewrite<= remove-double-neg_binary64 (neg.f64 (neg.f64 (/.f64 x (+.f64 (exp.f64 wj) (*.f64 wj (exp.f64 wj))))))) (neg.f64 (/.f64 (*.f64 wj (exp.f64 wj)) (+.f64 (exp.f64 wj) (*.f64 wj (exp.f64 wj))))))): 20 points increase in error, 0 points decrease in error
      (+.f64 wj (Rewrite<= distribute-neg-in_binary64 (neg.f64 (+.f64 (neg.f64 (/.f64 x (+.f64 (exp.f64 wj) (*.f64 wj (exp.f64 wj))))) (/.f64 (*.f64 wj (exp.f64 wj)) (+.f64 (exp.f64 wj) (*.f64 wj (exp.f64 wj)))))))): 0 points increase in error, 20 points decrease in error
      (+.f64 wj (neg.f64 (Rewrite<= +-commutative_binary64 (+.f64 (/.f64 (*.f64 wj (exp.f64 wj)) (+.f64 (exp.f64 wj) (*.f64 wj (exp.f64 wj)))) (neg.f64 (/.f64 x (+.f64 (exp.f64 wj) (*.f64 wj (exp.f64 wj))))))))): 0 points increase in error, 0 points decrease in error
      (+.f64 wj (neg.f64 (Rewrite<= sub-neg_binary64 (-.f64 (/.f64 (*.f64 wj (exp.f64 wj)) (+.f64 (exp.f64 wj) (*.f64 wj (exp.f64 wj)))) (/.f64 x (+.f64 (exp.f64 wj) (*.f64 wj (exp.f64 wj)))))))): 0 points increase in error, 0 points decrease in error
      (+.f64 wj (neg.f64 (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
      (+.f64 wj (Rewrite=> neg-mul-1_binary64 (*.f64 -1 (/.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
      (+.f64 wj (Rewrite=> *-commutative_binary64 (*.f64 (/.f64 (-.f64 (*.f64 wj (exp.f64 wj)) x) (+.f64 (exp.f64 wj) (*.f64 wj (exp.f64 wj)))) -1))): 0 points increase in error, 0 points decrease in error
      (+.f64 wj (Rewrite<= *-commutative_binary64 (*.f64 -1 (/.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
      (+.f64 wj (Rewrite<= neg-mul-1_binary64 (neg.f64 (/.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
      (Rewrite<= sub-neg_binary64 (-.f64 wj (/.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 13.6

      \[\leadsto wj - \frac{wj \cdot e^{wj} - x}{\color{blue}{1 + 2 \cdot wj}} \]

   4. Simplified 13.6

      \[\leadsto wj - \frac{wj \cdot e^{wj} - x}{\color{blue}{wj + \left(wj + 1\right)}} \]
      Proof
      (-.f64 wj (/.f64 (-.f64 (*.f64 wj (exp.f64 wj)) x) (+.f64 wj (+.f64 wj 1)))): 0 points increase in error, 0 points decrease in error
      (-.f64 wj (/.f64 (-.f64 (*.f64 wj (exp.f64 wj)) x) (Rewrite<= associate-+l+_binary64 (+.f64 (+.f64 wj wj) 1)))): 0 points increase in error, 4 points decrease in error
      (-.f64 wj (/.f64 (-.f64 (*.f64 wj (exp.f64 wj)) x) (+.f64 (Rewrite=> count-2_binary64 (*.f64 2 wj)) 1))): 0 points increase in error, 0 points decrease in error
      (-.f64 wj (/.f64 (-.f64 (*.f64 wj (exp.f64 wj)) x) (Rewrite<= +-commutative_binary64 (+.f64 1 (*.f64 2 wj))))): 0 points increase in error, 0 points decrease in error

   5. Taylor expanded in wj around 0 0.2

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

   6. Taylor expanded in x around 0 0.1

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

   7. Simplified 0.1

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

   if 6.3000000000000002e-9 < wj

   1. Initial program 28.3

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

   2. Simplified 2.6

      \[\leadsto \color{blue}{wj + \frac{\frac{x}{e^{wj}} - wj}{wj + 1}} \]
      Proof
      (+.f64 wj (/.f64 (-.f64 (/.f64 x (exp.f64 wj)) wj) (+.f64 wj 1))): 0 points increase in error, 0 points decrease in error
      (+.f64 wj (Rewrite=> div-sub_binary64 (-.f64 (/.f64 (/.f64 x (exp.f64 wj)) (+.f64 wj 1)) (/.f64 wj (+.f64 wj 1))))): 0 points increase in error, 20 points decrease in error
      (+.f64 wj (-.f64 (Rewrite=> associate-/l/_binary64 (/.f64 x (*.f64 (+.f64 wj 1) (exp.f64 wj)))) (/.f64 wj (+.f64 wj 1)))): 0 points increase in error, 0 points decrease in error
      (+.f64 wj (-.f64 (/.f64 x (Rewrite<= distribute-rgt1-in_binary64 (+.f64 (exp.f64 wj) (*.f64 wj (exp.f64 wj))))) (/.f64 wj (+.f64 wj 1)))): 0 points increase in error, 0 points decrease in error
      (+.f64 wj (-.f64 (/.f64 x (+.f64 (exp.f64 wj) (*.f64 wj (exp.f64 wj)))) (Rewrite<= *-lft-identity_binary64 (*.f64 1 (/.f64 wj (+.f64 wj 1)))))): 20 points increase in error, 0 points decrease in error
      (+.f64 wj (-.f64 (/.f64 x (+.f64 (exp.f64 wj) (*.f64 wj (exp.f64 wj)))) (*.f64 (Rewrite<= *-inverses_binary64 (/.f64 (exp.f64 wj) (exp.f64 wj))) (/.f64 wj (+.f64 wj 1))))): 0 points increase in error, 20 points decrease in error
      (+.f64 wj (-.f64 (/.f64 x (+.f64 (exp.f64 wj) (*.f64 wj (exp.f64 wj)))) (Rewrite<= *-commutative_binary64 (*.f64 (/.f64 wj (+.f64 wj 1)) (/.f64 (exp.f64 wj) (exp.f64 wj)))))): 0 points increase in error, 0 points decrease in error
      (+.f64 wj (-.f64 (/.f64 x (+.f64 (exp.f64 wj) (*.f64 wj (exp.f64 wj)))) (Rewrite<= times-frac_binary64 (/.f64 (*.f64 wj (exp.f64 wj)) (*.f64 (+.f64 wj 1) (exp.f64 wj)))))): 0 points increase in error, 0 points decrease in error
      (+.f64 wj (-.f64 (/.f64 x (+.f64 (exp.f64 wj) (*.f64 wj (exp.f64 wj)))) (/.f64 (*.f64 wj (exp.f64 wj)) (Rewrite<= distribute-rgt1-in_binary64 (+.f64 (exp.f64 wj) (*.f64 wj (exp.f64 wj))))))): 0 points increase in error, 0 points decrease in error
      (+.f64 wj (Rewrite=> sub-neg_binary64 (+.f64 (/.f64 x (+.f64 (exp.f64 wj) (*.f64 wj (exp.f64 wj)))) (neg.f64 (/.f64 (*.f64 wj (exp.f64 wj)) (+.f64 (exp.f64 wj) (*.f64 wj (exp.f64 wj)))))))): 0 points increase in error, 0 points decrease in error
      (+.f64 wj (+.f64 (Rewrite<= remove-double-neg_binary64 (neg.f64 (neg.f64 (/.f64 x (+.f64 (exp.f64 wj) (*.f64 wj (exp.f64 wj))))))) (neg.f64 (/.f64 (*.f64 wj (exp.f64 wj)) (+.f64 (exp.f64 wj) (*.f64 wj (exp.f64 wj))))))): 20 points increase in error, 0 points decrease in error
      (+.f64 wj (Rewrite<= distribute-neg-in_binary64 (neg.f64 (+.f64 (neg.f64 (/.f64 x (+.f64 (exp.f64 wj) (*.f64 wj (exp.f64 wj))))) (/.f64 (*.f64 wj (exp.f64 wj)) (+.f64 (exp.f64 wj) (*.f64 wj (exp.f64 wj)))))))): 0 points increase in error, 20 points decrease in error
      (+.f64 wj (neg.f64 (Rewrite<= +-commutative_binary64 (+.f64 (/.f64 (*.f64 wj (exp.f64 wj)) (+.f64 (exp.f64 wj) (*.f64 wj (exp.f64 wj)))) (neg.f64 (/.f64 x (+.f64 (exp.f64 wj) (*.f64 wj (exp.f64 wj))))))))): 0 points increase in error, 0 points decrease in error
      (+.f64 wj (neg.f64 (Rewrite<= sub-neg_binary64 (-.f64 (/.f64 (*.f64 wj (exp.f64 wj)) (+.f64 (exp.f64 wj) (*.f64 wj (exp.f64 wj)))) (/.f64 x (+.f64 (exp.f64 wj) (*.f64 wj (exp.f64 wj)))))))): 0 points increase in error, 0 points decrease in error
      (+.f64 wj (neg.f64 (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
      (+.f64 wj (Rewrite=> neg-mul-1_binary64 (*.f64 -1 (/.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
      (+.f64 wj (Rewrite=> *-commutative_binary64 (*.f64 (/.f64 (-.f64 (*.f64 wj (exp.f64 wj)) x) (+.f64 (exp.f64 wj) (*.f64 wj (exp.f64 wj)))) -1))): 0 points increase in error, 0 points decrease in error
      (+.f64 wj (Rewrite<= *-commutative_binary64 (*.f64 -1 (/.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
      (+.f64 wj (Rewrite<= neg-mul-1_binary64 (neg.f64 (/.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
      (Rewrite<= sub-neg_binary64 (-.f64 wj (/.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. Recombined 3 regimes into one program.

4. Final simplification 0.3

   \[\leadsto \begin{array}{l} \mathbf{if}\;wj \leq -7.6 \cdot 10^{-9}:\\ \;\;\;\;wj + \left(\frac{x}{e^{wj}} - wj\right) \cdot \frac{1}{wj + 1}\\ \mathbf{elif}\;wj \leq 6.3 \cdot 10^{-9}:\\ \;\;\;\;\left(x - 2 \cdot \left(wj \cdot x\right)\right) + wj \cdot wj\\ \mathbf{else}:\\ \;\;\;\;wj - \frac{wj - \frac{x}{e^{wj}}}{wj + 1}\\ \end{array} \]

Alternatives

Alternative 1
Error	0.5
Cost	35652

\[\begin{array}{l} t_0 := wj \cdot e^{wj}\\ t_1 := x \cdot 4 + x \cdot -1.5\\ \mathbf{if}\;wj + \frac{x - t_0}{e^{wj} + t_0} \leq 10^{-13}:\\ \;\;\;\;{wj}^{3} \cdot \left(x \cdot -0.6666666666666666 + \left(x \cdot 3 + \left(-1 + -2 \cdot t_1\right)\right)\right) + \left(\left(1 + t_1\right) \cdot {wj}^{2} + \left(x - 2 \cdot \left(wj \cdot x\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{x}{e^{wj}} - wj, \frac{1}{wj + 1}, wj\right)\\ \end{array} \]

Alternative 2
Error	0.8
Cost	7428

\[\begin{array}{l} \mathbf{if}\;wj \leq 1.45 \cdot 10^{-8}:\\ \;\;\;\;\left(\left(x - 2 \cdot \left(wj \cdot x\right)\right) + wj \cdot wj\right) - {wj}^{3}\\ \mathbf{else}:\\ \;\;\;\;wj - \frac{wj - \frac{x}{e^{wj}}}{wj + 1}\\ \end{array} \]

Alternative 3
Error	0.3
Cost	7369

\[\begin{array}{l} \mathbf{if}\;wj \leq -4.8 \cdot 10^{-9} \lor \neg \left(wj \leq 6.3 \cdot 10^{-9}\right):\\ \;\;\;\;wj - \frac{wj - \frac{x}{e^{wj}}}{wj + 1}\\ \mathbf{else}:\\ \;\;\;\;\left(x - 2 \cdot \left(wj \cdot x\right)\right) + wj \cdot wj\\ \end{array} \]

Alternative 4
Error	1.4
Cost	836

\[\begin{array}{l} \mathbf{if}\;wj \leq 0.016:\\ \;\;\;\;\left(x - 2 \cdot \left(wj \cdot x\right)\right) + wj \cdot wj\\ \mathbf{else}:\\ \;\;\;\;wj - \frac{wj}{wj + 1}\\ \end{array} \]

Alternative 5
Error	8.6
Cost	580

\[\begin{array}{l} \mathbf{if}\;wj \leq 0.016:\\ \;\;\;\;x \cdot \left(1 + wj \cdot -2\right)\\ \mathbf{else}:\\ \;\;\;\;wj - \frac{wj}{wj + 1}\\ \end{array} \]

Alternative 6
Error	8.6
Cost	580

\[\begin{array}{l} \mathbf{if}\;wj \leq 0.016:\\ \;\;\;\;\frac{x}{1 - wj \cdot -2}\\ \mathbf{else}:\\ \;\;\;\;wj - \frac{wj}{wj + 1}\\ \end{array} \]

Alternative 7
Error	9.5
Cost	448

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

Alternative 8
Error	9.8
Cost	320

\[x - wj \cdot x \]

Alternative 9
Error	9.8
Cost	320

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

Alternative 10
Error	61.1
Cost	64

\[wj \]

Alternative 11
Error	9.8
Cost	64

\[x \]

Reproduce

herbie shell --seed 2022340 
(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))))))