Average Error: 13.9 → 0.7
Time: 5.8s
Precision: binary64
\[wj - \frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}} \]
↓
\[\begin{array}{l} \mathbf{if}\;wj \leq 9.835644204217935 \cdot 10^{-7}:\\ \;\;\;\;{wj}^{3} \cdot \left(x \cdot -0.6666666666666666 - \left(x \cdot -3 + \left(1 + -2 \cdot \left(x \cdot -4 + x \cdot 1.5\right)\right)\right)\right) + \left(\left(1 + \left(x \cdot 4 + x \cdot -1.5\right)\right) \cdot {wj}^{2} + \left(x + -2 \cdot \left(wj \cdot x\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;wj + \frac{x \cdot e^{-wj} - wj}{wj + 1}\\ \end{array} \]
(FPCore (wj x)
 :precision binary64
 (- wj (/ (- (* wj (exp wj)) x) (+ (exp wj) (* wj (exp wj))))))
↓
(FPCore (wj x)
 :precision binary64
 (if (<= wj 9.835644204217935e-7)
   (+
    (*
     (pow wj 3.0)
     (-
      (* x -0.6666666666666666)
      (+ (* x -3.0) (+ 1.0 (* -2.0 (+ (* x -4.0) (* x 1.5)))))))
    (+
     (* (+ 1.0 (+ (* x 4.0) (* x -1.5))) (pow wj 2.0))
     (+ x (* -2.0 (* wj x)))))
   (+ wj (/ (- (* x (exp (- wj))) wj) (+ wj 1.0)))))
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 <= 9.835644204217935e-7) {
		tmp = (pow(wj, 3.0) * ((x * -0.6666666666666666) - ((x * -3.0) + (1.0 + (-2.0 * ((x * -4.0) + (x * 1.5))))))) + (((1.0 + ((x * 4.0) + (x * -1.5))) * pow(wj, 2.0)) + (x + (-2.0 * (wj * x))));
	} else {
		tmp = wj + (((x * exp(-wj)) - wj) / (wj + 1.0));
	}
	return tmp;
}

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 <= 9.835644204217935d-7) then
        tmp = ((wj ** 3.0d0) * ((x * (-0.6666666666666666d0)) - ((x * (-3.0d0)) + (1.0d0 + ((-2.0d0) * ((x * (-4.0d0)) + (x * 1.5d0))))))) + (((1.0d0 + ((x * 4.0d0) + (x * (-1.5d0)))) * (wj ** 2.0d0)) + (x + ((-2.0d0) * (wj * x))))
    else
        tmp = wj + (((x * exp(-wj)) - wj) / (wj + 1.0d0))
    end if
    code = tmp
end function

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 <= 9.835644204217935e-7) {
		tmp = (Math.pow(wj, 3.0) * ((x * -0.6666666666666666) - ((x * -3.0) + (1.0 + (-2.0 * ((x * -4.0) + (x * 1.5))))))) + (((1.0 + ((x * 4.0) + (x * -1.5))) * Math.pow(wj, 2.0)) + (x + (-2.0 * (wj * x))));
	} else {
		tmp = wj + (((x * Math.exp(-wj)) - wj) / (wj + 1.0));
	}
	return tmp;
}

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 <= 9.835644204217935e-7:
		tmp = (math.pow(wj, 3.0) * ((x * -0.6666666666666666) - ((x * -3.0) + (1.0 + (-2.0 * ((x * -4.0) + (x * 1.5))))))) + (((1.0 + ((x * 4.0) + (x * -1.5))) * math.pow(wj, 2.0)) + (x + (-2.0 * (wj * x))))
	else:
		tmp = wj + (((x * math.exp(-wj)) - wj) / (wj + 1.0))
	return tmp

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 <= 9.835644204217935e-7)
		tmp = Float64(Float64((wj ^ 3.0) * Float64(Float64(x * -0.6666666666666666) - Float64(Float64(x * -3.0) + Float64(1.0 + Float64(-2.0 * Float64(Float64(x * -4.0) + Float64(x * 1.5))))))) + Float64(Float64(Float64(1.0 + Float64(Float64(x * 4.0) + Float64(x * -1.5))) * (wj ^ 2.0)) + Float64(x + Float64(-2.0 * Float64(wj * x)))));
	else
		tmp = Float64(wj + Float64(Float64(Float64(x * exp(Float64(-wj))) - wj) / Float64(wj + 1.0)));
	end
	return tmp
end

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 <= 9.835644204217935e-7)
		tmp = ((wj ^ 3.0) * ((x * -0.6666666666666666) - ((x * -3.0) + (1.0 + (-2.0 * ((x * -4.0) + (x * 1.5))))))) + (((1.0 + ((x * 4.0) + (x * -1.5))) * (wj ^ 2.0)) + (x + (-2.0 * (wj * x))));
	else
		tmp = wj + (((x * exp(-wj)) - wj) / (wj + 1.0));
	end
	tmp_2 = tmp;
end

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, 9.835644204217935e-7], N[(N[(N[Power[wj, 3.0], $MachinePrecision] * N[(N[(x * -0.6666666666666666), $MachinePrecision] - N[(N[(x * -3.0), $MachinePrecision] + N[(1.0 + N[(-2.0 * N[(N[(x * -4.0), $MachinePrecision] + N[(x * 1.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(N[(1.0 + N[(N[(x * 4.0), $MachinePrecision] + N[(x * -1.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Power[wj, 2.0], $MachinePrecision]), $MachinePrecision] + N[(x + N[(-2.0 * N[(wj * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(wj + N[(N[(N[(x * N[Exp[(-wj)], $MachinePrecision]), $MachinePrecision] - wj), $MachinePrecision] / N[(wj + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

wj - \frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}}
↓
\begin{array}{l}
\mathbf{if}\;wj \leq 9.835644204217935 \cdot 10^{-7}:\\
\;\;\;\;{wj}^{3} \cdot \left(x \cdot -0.6666666666666666 - \left(x \cdot -3 + \left(1 + -2 \cdot \left(x \cdot -4 + x \cdot 1.5\right)\right)\right)\right) + \left(\left(1 + \left(x \cdot 4 + x \cdot -1.5\right)\right) \cdot {wj}^{2} + \left(x + -2 \cdot \left(wj \cdot x\right)\right)\right)\\

\mathbf{else}:\\
\;\;\;\;wj + \frac{x \cdot e^{-wj} - wj}{wj + 1}\\


\end{array}

Error

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Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original13.9
Target13.4
Herbie0.7
\[wj - \left(\frac{wj}{wj + 1} - \frac{x}{e^{wj} + wj \cdot e^{wj}}\right) \]

Derivation

  1. Split input into 2 regimes

  2. if wj < 9.835644204217935e-7

    1. Initial program 13.6

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

    2. Simplified13.6

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

    3. Taylor expanded in wj around 0 0.6

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

    if 9.835644204217935e-7 < wj

    1. Initial program 26.4

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

    2. Simplified1.3

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

    3. Applied egg-rr1.3

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

  4. Final simplification0.7

    \[\leadsto \begin{array}{l} \mathbf{if}\;wj \leq 9.835644204217935 \cdot 10^{-7}:\\ \;\;\;\;{wj}^{3} \cdot \left(x \cdot -0.6666666666666666 - \left(x \cdot -3 + \left(1 + -2 \cdot \left(x \cdot -4 + x \cdot 1.5\right)\right)\right)\right) + \left(\left(1 + \left(x \cdot 4 + x \cdot -1.5\right)\right) \cdot {wj}^{2} + \left(x + -2 \cdot \left(wj \cdot x\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;wj + \frac{x \cdot e^{-wj} - wj}{wj + 1}\\ \end{array} \]

Reproduce

herbie shell --seed 2022202 
(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))))))