Average Error: 13.4 → 0.7
Time: 10.8s
Precision: binary64
Cost: 35652
\[wj - \frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}} \]
↓
\[\begin{array}{l} t_0 := \frac{wj}{wj + 1}\\ t_1 := wj \cdot e^{wj}\\ t_2 := x \cdot -4 + x \cdot 1.5\\ \mathbf{if}\;wj + \frac{x - t_1}{e^{wj} + t_1} \leq 5 \cdot 10^{-26}:\\ \;\;\;\;{wj}^{3} \cdot \left(x \cdot -0.6666666666666666 - \left(x \cdot -3 + \left(1 + -2 \cdot t_2\right)\right)\right) + \left(\left(1 - t_2\right) \cdot {wj}^{2} + \left(x - 2 \cdot \left(wj \cdot x\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\left(wj + 1\right) \cdot e^{wj}} + \frac{wj \cdot wj - t_0 \cdot t_0}{wj + t_0}\\ \end{array} \]
(FPCore (wj x)
 :precision binary64
 (- wj (/ (- (* wj (exp wj)) x) (+ (exp wj) (* wj (exp wj))))))
↓
(FPCore (wj x)
 :precision binary64
 (let* ((t_0 (/ wj (+ wj 1.0)))
        (t_1 (* wj (exp wj)))
        (t_2 (+ (* x -4.0) (* x 1.5))))
   (if (<= (+ wj (/ (- x t_1) (+ (exp wj) t_1))) 5e-26)
     (+
      (*
       (pow wj 3.0)
       (- (* x -0.6666666666666666) (+ (* x -3.0) (+ 1.0 (* -2.0 t_2)))))
      (+ (* (- 1.0 t_2) (pow wj 2.0)) (- x (* 2.0 (* wj x)))))
     (+
      (/ x (* (+ wj 1.0) (exp wj)))
      (/ (- (* wj wj) (* t_0 t_0)) (+ wj t_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 t_0 = wj / (wj + 1.0);
	double t_1 = wj * exp(wj);
	double t_2 = (x * -4.0) + (x * 1.5);
	double tmp;
	if ((wj + ((x - t_1) / (exp(wj) + t_1))) <= 5e-26) {
		tmp = (pow(wj, 3.0) * ((x * -0.6666666666666666) - ((x * -3.0) + (1.0 + (-2.0 * t_2))))) + (((1.0 - t_2) * pow(wj, 2.0)) + (x - (2.0 * (wj * x))));
	} else {
		tmp = (x / ((wj + 1.0) * exp(wj))) + (((wj * wj) - (t_0 * t_0)) / (wj + t_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) :: t_0
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_0 = wj / (wj + 1.0d0)
    t_1 = wj * exp(wj)
    t_2 = (x * (-4.0d0)) + (x * 1.5d0)
    if ((wj + ((x - t_1) / (exp(wj) + t_1))) <= 5d-26) then
        tmp = ((wj ** 3.0d0) * ((x * (-0.6666666666666666d0)) - ((x * (-3.0d0)) + (1.0d0 + ((-2.0d0) * t_2))))) + (((1.0d0 - t_2) * (wj ** 2.0d0)) + (x - (2.0d0 * (wj * x))))
    else
        tmp = (x / ((wj + 1.0d0) * exp(wj))) + (((wj * wj) - (t_0 * t_0)) / (wj + t_0))
    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 t_0 = wj / (wj + 1.0);
	double t_1 = wj * Math.exp(wj);
	double t_2 = (x * -4.0) + (x * 1.5);
	double tmp;
	if ((wj + ((x - t_1) / (Math.exp(wj) + t_1))) <= 5e-26) {
		tmp = (Math.pow(wj, 3.0) * ((x * -0.6666666666666666) - ((x * -3.0) + (1.0 + (-2.0 * t_2))))) + (((1.0 - t_2) * Math.pow(wj, 2.0)) + (x - (2.0 * (wj * x))));
	} else {
		tmp = (x / ((wj + 1.0) * Math.exp(wj))) + (((wj * wj) - (t_0 * t_0)) / (wj + t_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):
	t_0 = wj / (wj + 1.0)
	t_1 = wj * math.exp(wj)
	t_2 = (x * -4.0) + (x * 1.5)
	tmp = 0
	if (wj + ((x - t_1) / (math.exp(wj) + t_1))) <= 5e-26:
		tmp = (math.pow(wj, 3.0) * ((x * -0.6666666666666666) - ((x * -3.0) + (1.0 + (-2.0 * t_2))))) + (((1.0 - t_2) * math.pow(wj, 2.0)) + (x - (2.0 * (wj * x))))
	else:
		tmp = (x / ((wj + 1.0) * math.exp(wj))) + (((wj * wj) - (t_0 * t_0)) / (wj + t_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)
	t_0 = Float64(wj / Float64(wj + 1.0))
	t_1 = Float64(wj * exp(wj))
	t_2 = Float64(Float64(x * -4.0) + Float64(x * 1.5))
	tmp = 0.0
	if (Float64(wj + Float64(Float64(x - t_1) / Float64(exp(wj) + t_1))) <= 5e-26)
		tmp = Float64(Float64((wj ^ 3.0) * Float64(Float64(x * -0.6666666666666666) - Float64(Float64(x * -3.0) + Float64(1.0 + Float64(-2.0 * t_2))))) + Float64(Float64(Float64(1.0 - t_2) * (wj ^ 2.0)) + Float64(x - Float64(2.0 * Float64(wj * x)))));
	else
		tmp = Float64(Float64(x / Float64(Float64(wj + 1.0) * exp(wj))) + Float64(Float64(Float64(wj * wj) - Float64(t_0 * t_0)) / Float64(wj + t_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)
	t_0 = wj / (wj + 1.0);
	t_1 = wj * exp(wj);
	t_2 = (x * -4.0) + (x * 1.5);
	tmp = 0.0;
	if ((wj + ((x - t_1) / (exp(wj) + t_1))) <= 5e-26)
		tmp = ((wj ^ 3.0) * ((x * -0.6666666666666666) - ((x * -3.0) + (1.0 + (-2.0 * t_2))))) + (((1.0 - t_2) * (wj ^ 2.0)) + (x - (2.0 * (wj * x))));
	else
		tmp = (x / ((wj + 1.0) * exp(wj))) + (((wj * wj) - (t_0 * t_0)) / (wj + t_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_] := Block[{t$95$0 = N[(wj / N[(wj + 1.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(wj * N[Exp[wj], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(x * -4.0), $MachinePrecision] + N[(x * 1.5), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(wj + N[(N[(x - t$95$1), $MachinePrecision] / N[(N[Exp[wj], $MachinePrecision] + t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 5e-26], 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$2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(N[(1.0 - t$95$2), $MachinePrecision] * N[Power[wj, 2.0], $MachinePrecision]), $MachinePrecision] + N[(x - N[(2.0 * N[(wj * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x / N[(N[(wj + 1.0), $MachinePrecision] * N[Exp[wj], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(N[(wj * wj), $MachinePrecision] - N[(t$95$0 * t$95$0), $MachinePrecision]), $MachinePrecision] / N[(wj + t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

wj - \frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}}
↓
\begin{array}{l}
t_0 := \frac{wj}{wj + 1}\\
t_1 := wj \cdot e^{wj}\\
t_2 := x \cdot -4 + x \cdot 1.5\\
\mathbf{if}\;wj + \frac{x - t_1}{e^{wj} + t_1} \leq 5 \cdot 10^{-26}:\\
\;\;\;\;{wj}^{3} \cdot \left(x \cdot -0.6666666666666666 - \left(x \cdot -3 + \left(1 + -2 \cdot t_2\right)\right)\right) + \left(\left(1 - t_2\right) \cdot {wj}^{2} + \left(x - 2 \cdot \left(wj \cdot x\right)\right)\right)\\

\mathbf{else}:\\
\;\;\;\;\frac{x}{\left(wj + 1\right) \cdot e^{wj}} + \frac{wj \cdot wj - t_0 \cdot t_0}{wj + t_0}\\


\end{array}

Error

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original13.4
Target12.7
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 (-.f64 wj (/.f64 (-.f64 (*.f64 wj (exp.f64 wj)) x) (+.f64 (exp.f64 wj) (*.f64 wj (exp.f64 wj))))) < 5.00000000000000019e-26

    1. Initial program 17.9

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

    2. Simplified17.9

      \[\leadsto \color{blue}{wj - \frac{wj - \frac{x}{e^{wj}}}{wj + 1}} \]
      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)))): 3 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)))): 0 points increase in error, 2 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)))): 1 points increase in error, 0 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)))))): 1 points increase in error, 3 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, 0 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 0.5

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

    if 5.00000000000000019e-26 < (-.f64 wj (/.f64 (-.f64 (*.f64 wj (exp.f64 wj)) x) (+.f64 (exp.f64 wj) (*.f64 wj (exp.f64 wj)))))

    1. Initial program 3.3

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

    2. Simplified1.2

      \[\leadsto \color{blue}{wj - \frac{wj - \frac{x}{e^{wj}}}{wj + 1}} \]
      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)))): 3 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)))): 0 points increase in error, 2 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)))): 1 points increase in error, 0 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)))))): 1 points increase in error, 3 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, 0 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-rr1.1

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

    4. Applied egg-rr1.2

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

  4. Final simplification0.7

    \[\leadsto \begin{array}{l} \mathbf{if}\;wj + \frac{x - wj \cdot e^{wj}}{e^{wj} + wj \cdot e^{wj}} \leq 5 \cdot 10^{-26}:\\ \;\;\;\;{wj}^{3} \cdot \left(x \cdot -0.6666666666666666 - \left(x \cdot -3 + \left(1 + -2 \cdot \left(x \cdot -4 + x \cdot 1.5\right)\right)\right)\right) + \left(\left(1 - \left(x \cdot -4 + x \cdot 1.5\right)\right) \cdot {wj}^{2} + \left(x - 2 \cdot \left(wj \cdot x\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\left(wj + 1\right) \cdot e^{wj}} + \frac{wj \cdot wj - \frac{wj}{wj + 1} \cdot \frac{wj}{wj + 1}}{wj + \frac{wj}{wj + 1}}\\ \end{array} \]

Alternatives

Alternative 1
Error1.1
Cost26560
\[\mathsf{fma}\left(wj, wj, {wj}^{4} - {wj}^{3}\right) + \frac{x}{\left(wj + 1\right) \cdot e^{wj}} \]
Alternative 2
Error2.1
Cost7680
\[\left(1 - \left(x \cdot -4 + x \cdot 1.5\right)\right) \cdot {wj}^{2} + \left(x - 2 \cdot \left(wj \cdot x\right)\right) \]
Alternative 3
Error0.9
Cost7492
\[\begin{array}{l} \mathbf{if}\;wj \leq 6.5 \cdot 10^{-13}:\\ \;\;\;\;\left(x - 2 \cdot \left(wj \cdot x\right)\right) + wj \cdot wj\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\left(wj + 1\right) \cdot e^{wj}} + \left(wj - \frac{wj}{wj + 1}\right)\\ \end{array} \]
Alternative 4
Error0.9
Cost7236
\[\begin{array}{l} \mathbf{if}\;wj \leq 6.8 \cdot 10^{-9}:\\ \;\;\;\;\left(x - 2 \cdot \left(wj \cdot x\right)\right) + wj \cdot wj\\ \mathbf{else}:\\ \;\;\;\;wj + \frac{\frac{x}{e^{wj}} - wj}{wj + 1}\\ \end{array} \]
Alternative 5
Error2.1
Cost704
\[\left(x - 2 \cdot \left(wj \cdot x\right)\right) + wj \cdot wj \]
Alternative 6
Error10.1
Cost324
\[\begin{array}{l} \mathbf{if}\;wj \leq -7.1 \cdot 10^{-47}:\\ \;\;\;\;wj \cdot wj\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]
Alternative 7
Error2.5
Cost320
\[x + wj \cdot wj \]
Alternative 8
Error61.2
Cost64
\[wj \]
Alternative 9
Error9.4
Cost64
\[x \]

Error

Reproduce

herbie shell --seed 2022330 
(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))))))