Average Error: 13.6 → 1.7
Time: 17.1s
Precision: binary64
Cost: 15552
\[wj - \frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}} \]
↓
\[\begin{array}{l} t_0 := x \cdot 4 + x \cdot -1.5\\ {wj}^{3} \cdot \left(x \cdot -0.6666666666666666 + \left(x \cdot 3 + \left(-1 + -2 \cdot t_0\right)\right)\right) + \left(\left(1 + t_0\right) \cdot {wj}^{2} + \left(x + -2 \cdot \left(x \cdot wj\right)\right)\right) \end{array} \]
(FPCore (wj x)
 :precision binary64
 (- wj (/ (- (* wj (exp wj)) x) (+ (exp wj) (* wj (exp wj))))))
↓
(FPCore (wj x)
 :precision binary64
 (let* ((t_0 (+ (* x 4.0) (* x -1.5))))
   (+
    (*
     (pow wj 3.0)
     (+ (* x -0.6666666666666666) (+ (* x 3.0) (+ -1.0 (* -2.0 t_0)))))
    (+ (* (+ 1.0 t_0) (pow wj 2.0)) (+ x (* -2.0 (* x wj)))))))
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 * 4.0) + (x * -1.5);
	return (pow(wj, 3.0) * ((x * -0.6666666666666666) + ((x * 3.0) + (-1.0 + (-2.0 * t_0))))) + (((1.0 + t_0) * pow(wj, 2.0)) + (x + (-2.0 * (x * wj))));
}

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
    t_0 = (x * 4.0d0) + (x * (-1.5d0))
    code = ((wj ** 3.0d0) * ((x * (-0.6666666666666666d0)) + ((x * 3.0d0) + ((-1.0d0) + ((-2.0d0) * t_0))))) + (((1.0d0 + t_0) * (wj ** 2.0d0)) + (x + ((-2.0d0) * (x * wj))))
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 t_0 = (x * 4.0) + (x * -1.5);
	return (Math.pow(wj, 3.0) * ((x * -0.6666666666666666) + ((x * 3.0) + (-1.0 + (-2.0 * t_0))))) + (((1.0 + t_0) * Math.pow(wj, 2.0)) + (x + (-2.0 * (x * wj))));
}

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 * 4.0) + (x * -1.5)
	return (math.pow(wj, 3.0) * ((x * -0.6666666666666666) + ((x * 3.0) + (-1.0 + (-2.0 * t_0))))) + (((1.0 + t_0) * math.pow(wj, 2.0)) + (x + (-2.0 * (x * wj))))

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(Float64(x * 4.0) + Float64(x * -1.5))
	return Float64(Float64((wj ^ 3.0) * Float64(Float64(x * -0.6666666666666666) + Float64(Float64(x * 3.0) + Float64(-1.0 + Float64(-2.0 * t_0))))) + Float64(Float64(Float64(1.0 + t_0) * (wj ^ 2.0)) + Float64(x + Float64(-2.0 * Float64(x * wj)))))
end

function tmp = code(wj, x)
	tmp = wj - (((wj * exp(wj)) - x) / (exp(wj) + (wj * exp(wj))));
end
↓
function tmp = code(wj, x)
	t_0 = (x * 4.0) + (x * -1.5);
	tmp = ((wj ^ 3.0) * ((x * -0.6666666666666666) + ((x * 3.0) + (-1.0 + (-2.0 * t_0))))) + (((1.0 + t_0) * (wj ^ 2.0)) + (x + (-2.0 * (x * wj))));
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_] := Block[{t$95$0 = N[(N[(x * 4.0), $MachinePrecision] + N[(x * -1.5), $MachinePrecision]), $MachinePrecision]}, 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 * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(N[(1.0 + t$95$0), $MachinePrecision] * N[Power[wj, 2.0], $MachinePrecision]), $MachinePrecision] + N[(x + N[(-2.0 * N[(x * wj), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

wj - \frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}}
↓
\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}

Error

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original13.6
Target13.1
Herbie1.7
\[wj - \left(\frac{wj}{wj + 1} - \frac{x}{e^{wj} + wj \cdot e^{wj}}\right) \]

Derivation

  1. Initial program 13.6

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

  2. Simplified13.1

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

    [Start]13.6

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

    sub-neg [=>]13.6

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

    neg-mul-1 [=>]13.6

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

    *-commutative [=>]13.6

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

    *-commutative [<=]13.6

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

    neg-mul-1 [<=]13.6

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

    neg-sub0 [=>]13.6

    \[ wj + \color{blue}{\left(0 - \frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}}\right)} \]

    div-sub [=>]13.6

    \[ wj + \left(0 - \color{blue}{\left(\frac{wj \cdot e^{wj}}{e^{wj} + wj \cdot e^{wj}} - \frac{x}{e^{wj} + wj \cdot e^{wj}}\right)}\right) \]

    associate--r- [=>]13.6

    \[ wj + \color{blue}{\left(\left(0 - \frac{wj \cdot e^{wj}}{e^{wj} + wj \cdot e^{wj}}\right) + \frac{x}{e^{wj} + wj \cdot e^{wj}}\right)} \]

    +-commutative [=>]13.6

    \[ wj + \color{blue}{\left(\frac{x}{e^{wj} + wj \cdot e^{wj}} + \left(0 - \frac{wj \cdot e^{wj}}{e^{wj} + wj \cdot e^{wj}}\right)\right)} \]

    sub0-neg [=>]13.6

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

    sub-neg [<=]13.6

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

    distribute-rgt1-in [=>]13.6

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

    associate-/l/ [<=]13.6

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

    distribute-rgt1-in [=>]13.6

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

    times-frac [=>]13.6

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

    *-commutative [=>]13.6

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

    *-inverses [=>]13.1

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

    *-lft-identity [=>]13.1

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

    div-sub [<=]13.1

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

  3. Taylor expanded in wj around 0 1.7

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

  4. Final simplification1.7

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

Alternatives

Alternative 1
Error9.0
Cost7496
\[\begin{array}{l} \mathbf{if}\;wj \leq -1.65 \cdot 10^{-66}:\\ \;\;\;\;wj + \frac{\frac{x}{e^{wj}} - wj}{1 + wj}\\ \mathbf{elif}\;wj \leq -1.1 \cdot 10^{-80}:\\ \;\;\;\;wj \cdot wj\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{1 - wj \cdot wj} \cdot \left(1 - wj\right)}{e^{wj}}\\ \end{array} \]
Alternative 2
Error9.0
Cost7496
\[\begin{array}{l} \mathbf{if}\;wj \leq -5.6 \cdot 10^{-66}:\\ \;\;\;\;\frac{wj \cdot wj}{wj} + \frac{\frac{x}{e^{wj}} - wj}{1 + wj}\\ \mathbf{elif}\;wj \leq -1.1 \cdot 10^{-80}:\\ \;\;\;\;wj \cdot wj\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{1 - wj \cdot wj} \cdot \left(1 - wj\right)}{e^{wj}}\\ \end{array} \]
Alternative 3
Error1.8
Cost7296
\[\left(\left(x + -2 \cdot \left(x \cdot wj\right)\right) + wj \cdot wj\right) - {wj}^{3} \]
Alternative 4
Error9.0
Cost7236
\[\begin{array}{l} t_0 := \frac{x}{e^{wj}}\\ \mathbf{if}\;wj \leq -1.65 \cdot 10^{-66}:\\ \;\;\;\;wj + \frac{t_0 - wj}{1 + wj}\\ \mathbf{elif}\;wj \leq -1.1 \cdot 10^{-80}:\\ \;\;\;\;wj \cdot wj\\ \mathbf{else}:\\ \;\;\;\;\frac{t_0}{1 + wj}\\ \end{array} \]
Alternative 5
Error9.3
Cost7112
\[\begin{array}{l} \mathbf{if}\;wj \leq -1.65 \cdot 10^{-66}:\\ \;\;\;\;wj + \frac{\left(\left(x - x \cdot wj\right) + \left(wj \cdot wj\right) \cdot \left(x \cdot 0.5\right)\right) - wj}{1 + wj}\\ \mathbf{elif}\;wj \leq -1.1 \cdot 10^{-80}:\\ \;\;\;\;wj \cdot wj\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{e^{wj}}}{1 + wj}\\ \end{array} \]
Alternative 6
Error9.5
Cost1476
\[\begin{array}{l} t_0 := \left(x - x \cdot wj\right) + \left(wj \cdot wj\right) \cdot \left(x \cdot 0.5\right)\\ \mathbf{if}\;wj \leq -1.65 \cdot 10^{-66}:\\ \;\;\;\;wj + \frac{t_0 - wj}{1 + wj}\\ \mathbf{elif}\;wj \leq -1.1 \cdot 10^{-80}:\\ \;\;\;\;wj \cdot wj\\ \mathbf{else}:\\ \;\;\;\;\frac{t_0}{1 + wj}\\ \end{array} \]
Alternative 7
Error9.5
Cost1352
\[\begin{array}{l} t_0 := x - x \cdot wj\\ \mathbf{if}\;wj \leq -1.65 \cdot 10^{-66}:\\ \;\;\;\;wj + \frac{t_0 - wj}{1 + wj}\\ \mathbf{elif}\;wj \leq -1.1 \cdot 10^{-80}:\\ \;\;\;\;wj \cdot wj\\ \mathbf{else}:\\ \;\;\;\;\frac{t_0 + \left(wj \cdot wj\right) \cdot \left(x \cdot 0.5\right)}{1 + wj}\\ \end{array} \]
Alternative 8
Error9.6
Cost964
\[\begin{array}{l} \mathbf{if}\;wj \leq -1.65 \cdot 10^{-66}:\\ \;\;\;\;wj + \frac{\left(x - x \cdot wj\right) - wj}{1 + wj}\\ \mathbf{elif}\;wj \leq -1.1 \cdot 10^{-80}:\\ \;\;\;\;wj \cdot wj\\ \mathbf{else}:\\ \;\;\;\;\left(1 - wj\right) \cdot \frac{x}{1 + wj}\\ \end{array} \]
Alternative 9
Error9.9
Cost841
\[\begin{array}{l} \mathbf{if}\;wj \leq -1.65 \cdot 10^{-66} \lor \neg \left(wj \leq -1.1 \cdot 10^{-80}\right):\\ \;\;\;\;\left(1 - wj\right) \cdot \frac{x}{1 + wj}\\ \mathbf{else}:\\ \;\;\;\;wj \cdot wj\\ \end{array} \]
Alternative 10
Error10.0
Cost840
\[\begin{array}{l} \mathbf{if}\;wj \leq -1.65 \cdot 10^{-66}:\\ \;\;\;\;\frac{1 - wj}{\frac{1 + wj}{x}}\\ \mathbf{elif}\;wj \leq -1.1 \cdot 10^{-80}:\\ \;\;\;\;wj \cdot wj\\ \mathbf{else}:\\ \;\;\;\;\left(1 - wj\right) \cdot \frac{x}{1 + wj}\\ \end{array} \]
Alternative 11
Error9.9
Cost840
\[\begin{array}{l} \mathbf{if}\;wj \leq -1.65 \cdot 10^{-66}:\\ \;\;\;\;\frac{x - x \cdot wj}{1 + wj}\\ \mathbf{elif}\;wj \leq -1.1 \cdot 10^{-80}:\\ \;\;\;\;wj \cdot wj\\ \mathbf{else}:\\ \;\;\;\;\left(1 - wj\right) \cdot \frac{x}{1 + wj}\\ \end{array} \]
Alternative 12
Error10.0
Cost713
\[\begin{array}{l} \mathbf{if}\;wj \leq -1.65 \cdot 10^{-66} \lor \neg \left(wj \leq -1.1 \cdot 10^{-80}\right):\\ \;\;\;\;x + -2 \cdot \left(x \cdot wj\right)\\ \mathbf{else}:\\ \;\;\;\;wj \cdot wj\\ \end{array} \]
Alternative 13
Error10.0
Cost712
\[\begin{array}{l} \mathbf{if}\;wj \leq -1.65 \cdot 10^{-66}:\\ \;\;\;\;\frac{x}{1 + wj \cdot 2}\\ \mathbf{elif}\;wj \leq -1.1 \cdot 10^{-80}:\\ \;\;\;\;wj \cdot wj\\ \mathbf{else}:\\ \;\;\;\;x + -2 \cdot \left(x \cdot wj\right)\\ \end{array} \]
Alternative 14
Error10.0
Cost712
\[\begin{array}{l} \mathbf{if}\;wj \leq -1.65 \cdot 10^{-66}:\\ \;\;\;\;\frac{x}{wj + \left(1 + wj\right)}\\ \mathbf{elif}\;wj \leq -1.1 \cdot 10^{-80}:\\ \;\;\;\;wj \cdot wj\\ \mathbf{else}:\\ \;\;\;\;x + -2 \cdot \left(x \cdot wj\right)\\ \end{array} \]
Alternative 15
Error10.3
Cost456
\[\begin{array}{l} \mathbf{if}\;wj \leq -1.65 \cdot 10^{-66}:\\ \;\;\;\;x\\ \mathbf{elif}\;wj \leq -1.1 \cdot 10^{-80}:\\ \;\;\;\;wj \cdot wj\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]
Alternative 16
Error61.2
Cost64
\[wj \]
Alternative 17
Error9.5
Cost64
\[x \]

Error

Reproduce

herbie shell --seed 2022356 
(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))))))