Jmat.Real.lambertw, newton loop step

Percentage Accurate: 78.1% → 97.4%

Time: 11.7s

Alternatives: 7

Speedup: 313.0×

• could not determine a ground truth

  1. wj = -1.4824913387310231e+168
  2. x = -1.1349331058501772e-157

Specification

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := wj \cdot e^{wj}\\ wj - \frac{t_0 - x}{e^{wj} + t_0} \end{array} \end{array} \]

FPCore

(FPCore (wj x)
 :precision binary64
 (let* ((t_0 (* wj (exp wj)))) (- wj (/ (- t_0 x) (+ (exp wj) t_0)))))

C

double code(double wj, double x) {
	double t_0 = wj * exp(wj);
	return wj - ((t_0 - x) / (exp(wj) + t_0));
}

Fortran

real(8) function code(wj, x)
    real(8), intent (in) :: wj
    real(8), intent (in) :: x
    real(8) :: t_0
    t_0 = wj * exp(wj)
    code = wj - ((t_0 - x) / (exp(wj) + t_0))
end function

Java

public static double code(double wj, double x) {
	double t_0 = wj * Math.exp(wj);
	return wj - ((t_0 - x) / (Math.exp(wj) + t_0));
}

Python

def code(wj, x):
	t_0 = wj * math.exp(wj)
	return wj - ((t_0 - x) / (math.exp(wj) + t_0))

Julia

function code(wj, x)
	t_0 = Float64(wj * exp(wj))
	return Float64(wj - Float64(Float64(t_0 - x) / Float64(exp(wj) + t_0)))
end

MATLAB

function tmp = code(wj, x)
	t_0 = wj * exp(wj);
	tmp = wj - ((t_0 - x) / (exp(wj) + t_0));
end

Wolfram

code[wj_, x_] := Block[{t$95$0 = N[(wj * N[Exp[wj], $MachinePrecision]), $MachinePrecision]}, N[(wj - N[(N[(t$95$0 - x), $MachinePrecision] / N[(N[Exp[wj], $MachinePrecision] + t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Initial Program: 78.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := wj \cdot e^{wj}\\ wj - \frac{t_0 - x}{e^{wj} + t_0} \end{array} \end{array} \]

FPCore

(FPCore (wj x)
 :precision binary64
 (let* ((t_0 (* wj (exp wj)))) (- wj (/ (- t_0 x) (+ (exp wj) t_0)))))

C

double code(double wj, double x) {
	double t_0 = wj * exp(wj);
	return wj - ((t_0 - x) / (exp(wj) + t_0));
}

Fortran

real(8) function code(wj, x)
    real(8), intent (in) :: wj
    real(8), intent (in) :: x
    real(8) :: t_0
    t_0 = wj * exp(wj)
    code = wj - ((t_0 - x) / (exp(wj) + t_0))
end function

Java

public static double code(double wj, double x) {
	double t_0 = wj * Math.exp(wj);
	return wj - ((t_0 - x) / (Math.exp(wj) + t_0));
}

Python

def code(wj, x):
	t_0 = wj * math.exp(wj)
	return wj - ((t_0 - x) / (math.exp(wj) + t_0))

Julia

function code(wj, x)
	t_0 = Float64(wj * exp(wj))
	return Float64(wj - Float64(Float64(t_0 - x) / Float64(exp(wj) + t_0)))
end

MATLAB

function tmp = code(wj, x)
	t_0 = wj * exp(wj);
	tmp = wj - ((t_0 - x) / (exp(wj) + t_0));
end

Wolfram

code[wj_, x_] := Block[{t$95$0 = N[(wj * N[Exp[wj], $MachinePrecision]), $MachinePrecision]}, N[(wj - N[(N[(t$95$0 - x), $MachinePrecision] / N[(N[Exp[wj], $MachinePrecision] + t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Alternative 1: 97.4% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := wj \cdot e^{wj}\\ \mathbf{if}\;wj + \frac{x - t_0}{e^{wj} + t_0} \leq -5:\\ \;\;\;\;wj + \frac{1}{wj + 1} \cdot \left(\frac{x}{e^{wj}} - wj\right)\\ \mathbf{else}:\\ \;\;\;\;x + \left(-2 \cdot \left(wj \cdot x\right) - \left({wj}^{3} + wj \cdot \left(wj \cdot \left(-1 + x \cdot -2.5\right)\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (wj x)
 :precision binary64
 (let* ((t_0 (* wj (exp wj))))
   (if (<= (+ wj (/ (- x t_0) (+ (exp wj) t_0))) -5.0)
     (+ wj (* (/ 1.0 (+ wj 1.0)) (- (/ x (exp wj)) wj)))
     (+
      x
      (-
       (* -2.0 (* wj x))
       (+ (pow wj 3.0) (* wj (* wj (+ -1.0 (* x -2.5))))))))))

C

double code(double wj, double x) {
	double t_0 = wj * exp(wj);
	double tmp;
	if ((wj + ((x - t_0) / (exp(wj) + t_0))) <= -5.0) {
		tmp = wj + ((1.0 / (wj + 1.0)) * ((x / exp(wj)) - wj));
	} else {
		tmp = x + ((-2.0 * (wj * x)) - (pow(wj, 3.0) + (wj * (wj * (-1.0 + (x * -2.5))))));
	}
	return tmp;
}

Fortran

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 = wj * exp(wj)
    if ((wj + ((x - t_0) / (exp(wj) + t_0))) <= (-5.0d0)) then
        tmp = wj + ((1.0d0 / (wj + 1.0d0)) * ((x / exp(wj)) - wj))
    else
        tmp = x + (((-2.0d0) * (wj * x)) - ((wj ** 3.0d0) + (wj * (wj * ((-1.0d0) + (x * (-2.5d0)))))))
    end if
    code = tmp
end function

Java

public static double code(double wj, double x) {
	double t_0 = wj * Math.exp(wj);
	double tmp;
	if ((wj + ((x - t_0) / (Math.exp(wj) + t_0))) <= -5.0) {
		tmp = wj + ((1.0 / (wj + 1.0)) * ((x / Math.exp(wj)) - wj));
	} else {
		tmp = x + ((-2.0 * (wj * x)) - (Math.pow(wj, 3.0) + (wj * (wj * (-1.0 + (x * -2.5))))));
	}
	return tmp;
}

Python

def code(wj, x):
	t_0 = wj * math.exp(wj)
	tmp = 0
	if (wj + ((x - t_0) / (math.exp(wj) + t_0))) <= -5.0:
		tmp = wj + ((1.0 / (wj + 1.0)) * ((x / math.exp(wj)) - wj))
	else:
		tmp = x + ((-2.0 * (wj * x)) - (math.pow(wj, 3.0) + (wj * (wj * (-1.0 + (x * -2.5))))))
	return tmp

Julia

function code(wj, x)
	t_0 = Float64(wj * exp(wj))
	tmp = 0.0
	if (Float64(wj + Float64(Float64(x - t_0) / Float64(exp(wj) + t_0))) <= -5.0)
		tmp = Float64(wj + Float64(Float64(1.0 / Float64(wj + 1.0)) * Float64(Float64(x / exp(wj)) - wj)));
	else
		tmp = Float64(x + Float64(Float64(-2.0 * Float64(wj * x)) - Float64((wj ^ 3.0) + Float64(wj * Float64(wj * Float64(-1.0 + Float64(x * -2.5)))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(wj, x)
	t_0 = wj * exp(wj);
	tmp = 0.0;
	if ((wj + ((x - t_0) / (exp(wj) + t_0))) <= -5.0)
		tmp = wj + ((1.0 / (wj + 1.0)) * ((x / exp(wj)) - wj));
	else
		tmp = x + ((-2.0 * (wj * x)) - ((wj ^ 3.0) + (wj * (wj * (-1.0 + (x * -2.5))))));
	end
	tmp_2 = tmp;
end

Wolfram

code[wj_, x_] := Block[{t$95$0 = N[(wj * N[Exp[wj], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(wj + N[(N[(x - t$95$0), $MachinePrecision] / N[(N[Exp[wj], $MachinePrecision] + t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], -5.0], N[(wj + N[(N[(1.0 / N[(wj + 1.0), $MachinePrecision]), $MachinePrecision] * N[(N[(x / N[Exp[wj], $MachinePrecision]), $MachinePrecision] - wj), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x + N[(N[(-2.0 * N[(wj * x), $MachinePrecision]), $MachinePrecision] - N[(N[Power[wj, 3.0], $MachinePrecision] + N[(wj * N[(wj * N[(-1.0 + N[(x * -2.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

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

   1. Initial program 99.9%

      \[wj - \frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}} \]
   2. Step-by-step derivation

      1. div-sub 99.9%

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

      2. distribute-rgt1-in 99.9%

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

      3. times-frac 99.9%

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

      4. *-inverses 99.9%

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

      5. associate-*l/ 99.9%

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

      6. *-rgt-identity 99.9%

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

      7. distribute-rgt1-in 99.9%

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

      8. associate-/l/ 99.9%

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

      9. div-sub 99.9%

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

   3. Simplified 99.9%

      \[\leadsto \color{blue}{wj - \frac{wj - \frac{x}{e^{wj}}}{wj + 1}} \]
   4. Step-by-step derivation

      1. clear-num 99.6%

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

      2. associate-/r/ 100.0%

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

   5. Applied egg-rr 100.0%

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

   if -5 < (-.f64 wj (/.f64 (-.f64 (*.f64 wj (exp.f64 wj)) x) (+.f64 (exp.f64 wj) (*.f64 wj (exp.f64 wj)))))

   1. Initial program 74.1%

      \[wj - \frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}} \]
   2. Step-by-step derivation

      1. div-sub 74.1%

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

      2. distribute-rgt1-in 74.1%

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

      3. times-frac 74.1%

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

      4. *-inverses 74.1%

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

      5. associate-*l/ 74.1%

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

      6. *-rgt-identity 74.1%

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

      7. distribute-rgt1-in 74.1%

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

      8. associate-/l/ 74.1%

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

      9. div-sub 74.1%

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

   3. Simplified 74.1%

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

   4. Taylor expanded in wj around 0 98.5%

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

   5. Taylor expanded in x around 0 98.5%

      \[\leadsto x + \left(-2 \cdot \left(wj \cdot x\right) + \left(-1 \cdot \color{blue}{{wj}^{3}} + {wj}^{2} \cdot \left(1 - \left(-4 \cdot x + 1.5 \cdot x\right)\right)\right)\right) \]
   6. Step-by-step derivation

      1. add-cube-cbrt 98.2%

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

      2. pow3 98.2%

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

      3. distribute-rgt-out 98.2%

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

      4. metadata-eval 98.2%

         \[\leadsto x + \left(-2 \cdot \left(wj \cdot x\right) + \left(-1 \cdot {wj}^{3} + {\left(\sqrt[3]{{wj}^{2} \cdot \left(1 - x \cdot \color{blue}{-2.5}\right)}\right)}^{3}\right)\right) \]

   7. Applied egg-rr 98.2%

      \[\leadsto x + \left(-2 \cdot \left(wj \cdot x\right) + \left(-1 \cdot {wj}^{3} + \color{blue}{{\left(\sqrt[3]{{wj}^{2} \cdot \left(1 - x \cdot -2.5\right)}\right)}^{3}}\right)\right) \]
   8. Step-by-step derivation

      1. rem-cube-cbrt 98.5%

         \[\leadsto x + \left(-2 \cdot \left(wj \cdot x\right) + \left(-1 \cdot {wj}^{3} + \color{blue}{{wj}^{2} \cdot \left(1 - x \cdot -2.5\right)}\right)\right) \]

      2. *-commutative 98.5%

         \[\leadsto x + \left(-2 \cdot \left(wj \cdot x\right) + \left(-1 \cdot {wj}^{3} + \color{blue}{\left(1 - x \cdot -2.5\right) \cdot {wj}^{2}}\right)\right) \]

      3. unpow2 98.5%

         \[\leadsto x + \left(-2 \cdot \left(wj \cdot x\right) + \left(-1 \cdot {wj}^{3} + \left(1 - x \cdot -2.5\right) \cdot \color{blue}{\left(wj \cdot wj\right)}\right)\right) \]

      4. associate-*r* 98.5%

         \[\leadsto x + \left(-2 \cdot \left(wj \cdot x\right) + \left(-1 \cdot {wj}^{3} + \color{blue}{\left(\left(1 - x \cdot -2.5\right) \cdot wj\right) \cdot wj}\right)\right) \]

   9. Applied egg-rr 98.5%

      \[\leadsto x + \left(-2 \cdot \left(wj \cdot x\right) + \left(-1 \cdot {wj}^{3} + \color{blue}{\left(\left(1 - x \cdot -2.5\right) \cdot wj\right) \cdot wj}\right)\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 98.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;wj + \frac{x - wj \cdot e^{wj}}{e^{wj} + wj \cdot e^{wj}} \leq -5:\\ \;\;\;\;wj + \frac{1}{wj + 1} \cdot \left(\frac{x}{e^{wj}} - wj\right)\\ \mathbf{else}:\\ \;\;\;\;x + \left(-2 \cdot \left(wj \cdot x\right) - \left({wj}^{3} + wj \cdot \left(wj \cdot \left(-1 + x \cdot -2.5\right)\right)\right)\right)\\ \end{array} \]

Alternative 2: 97.7% accurate, 2.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;wj \leq -8.5 \cdot 10^{-9}:\\ \;\;\;\;wj + \frac{-1}{\frac{wj + 1}{wj - \frac{x}{e^{wj}}}}\\ \mathbf{else}:\\ \;\;\;\;x + \left(-2 \cdot \left(wj \cdot x\right) + {wj}^{2} \cdot \left(1 - \left(x \cdot -4 + x \cdot 1.5\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (wj x)
 :precision binary64
 (if (<= wj -8.5e-9)
   (+ wj (/ -1.0 (/ (+ wj 1.0) (- wj (/ x (exp wj))))))
   (+
    x
    (+ (* -2.0 (* wj x)) (* (pow wj 2.0) (- 1.0 (+ (* x -4.0) (* x 1.5))))))))

C

double code(double wj, double x) {
	double tmp;
	if (wj <= -8.5e-9) {
		tmp = wj + (-1.0 / ((wj + 1.0) / (wj - (x / exp(wj)))));
	} else {
		tmp = x + ((-2.0 * (wj * x)) + (pow(wj, 2.0) * (1.0 - ((x * -4.0) + (x * 1.5)))));
	}
	return tmp;
}

Fortran

real(8) function code(wj, x)
    real(8), intent (in) :: wj
    real(8), intent (in) :: x
    real(8) :: tmp
    if (wj <= (-8.5d-9)) then
        tmp = wj + ((-1.0d0) / ((wj + 1.0d0) / (wj - (x / exp(wj)))))
    else
        tmp = x + (((-2.0d0) * (wj * x)) + ((wj ** 2.0d0) * (1.0d0 - ((x * (-4.0d0)) + (x * 1.5d0)))))
    end if
    code = tmp
end function

Java

public static double code(double wj, double x) {
	double tmp;
	if (wj <= -8.5e-9) {
		tmp = wj + (-1.0 / ((wj + 1.0) / (wj - (x / Math.exp(wj)))));
	} else {
		tmp = x + ((-2.0 * (wj * x)) + (Math.pow(wj, 2.0) * (1.0 - ((x * -4.0) + (x * 1.5)))));
	}
	return tmp;
}

Python

def code(wj, x):
	tmp = 0
	if wj <= -8.5e-9:
		tmp = wj + (-1.0 / ((wj + 1.0) / (wj - (x / math.exp(wj)))))
	else:
		tmp = x + ((-2.0 * (wj * x)) + (math.pow(wj, 2.0) * (1.0 - ((x * -4.0) + (x * 1.5)))))
	return tmp

Julia

function code(wj, x)
	tmp = 0.0
	if (wj <= -8.5e-9)
		tmp = Float64(wj + Float64(-1.0 / Float64(Float64(wj + 1.0) / Float64(wj - Float64(x / exp(wj))))));
	else
		tmp = Float64(x + Float64(Float64(-2.0 * Float64(wj * x)) + Float64((wj ^ 2.0) * Float64(1.0 - Float64(Float64(x * -4.0) + Float64(x * 1.5))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(wj, x)
	tmp = 0.0;
	if (wj <= -8.5e-9)
		tmp = wj + (-1.0 / ((wj + 1.0) / (wj - (x / exp(wj)))));
	else
		tmp = x + ((-2.0 * (wj * x)) + ((wj ^ 2.0) * (1.0 - ((x * -4.0) + (x * 1.5)))));
	end
	tmp_2 = tmp;
end

Wolfram

code[wj_, x_] := If[LessEqual[wj, -8.5e-9], N[(wj + N[(-1.0 / N[(N[(wj + 1.0), $MachinePrecision] / N[(wj - N[(x / N[Exp[wj], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x + N[(N[(-2.0 * N[(wj * x), $MachinePrecision]), $MachinePrecision] + N[(N[Power[wj, 2.0], $MachinePrecision] * N[(1.0 - N[(N[(x * -4.0), $MachinePrecision] + N[(x * 1.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if wj < -8.5e-9

   1. Initial program 94.7%

      \[wj - \frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}} \]
   2. Step-by-step derivation

      1. div-sub 94.7%

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

      2. distribute-rgt1-in 94.7%

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

      3. times-frac 94.7%

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

      4. *-inverses 94.7%

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

      5. associate-*l/ 94.7%

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

      6. *-rgt-identity 94.7%

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

      7. distribute-rgt1-in 94.9%

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

      8. associate-/l/ 94.7%

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

      9. div-sub 94.7%

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

   3. Simplified 94.7%

      \[\leadsto \color{blue}{wj - \frac{wj - \frac{x}{e^{wj}}}{wj + 1}} \]
   4. Step-by-step derivation

      1. clear-num 94.9%

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

      2. inv-pow 94.9%

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

   5. Applied egg-rr 94.9%

      \[\leadsto wj - \frac{wj - \color{blue}{{\left(\frac{e^{wj}}{x}\right)}^{-1}}}{wj + 1} \]
   6. Step-by-step derivation

      1. div-sub 94.9%

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

      2. unpow-1 94.9%

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

      3. clear-num 94.7%

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

      4. div-sub 94.7%

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

      5. clear-num 95.3%

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

      6. inv-pow 95.3%

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

   7. Applied egg-rr 95.3%

      \[\leadsto wj - \color{blue}{{\left(\frac{wj + 1}{wj - \frac{x}{e^{wj}}}\right)}^{-1}} \]
   8. Step-by-step derivation

      1. unpow-1 95.3%

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

   9. Simplified 95.3%

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

   if -8.5e-9 < wj

   1. Initial program 79.8%

      \[wj - \frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}} \]
   2. Step-by-step derivation

      1. div-sub 79.8%

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

      2. distribute-rgt1-in 79.8%

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

      3. times-frac 79.8%

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

      4. *-inverses 79.8%

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

      5. associate-*l/ 79.8%

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

      6. *-rgt-identity 79.8%

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

      7. distribute-rgt1-in 79.8%

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

      8. associate-/l/ 79.8%

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

      9. div-sub 79.8%

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

   3. Simplified 79.8%

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

   4. Taylor expanded in wj around 0 98.6%

      \[\leadsto \color{blue}{x + \left(-2 \cdot \left(wj \cdot x\right) + {wj}^{2} \cdot \left(1 - \left(-4 \cdot x + 1.5 \cdot x\right)\right)\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 98.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;wj \leq -8.5 \cdot 10^{-9}:\\ \;\;\;\;wj + \frac{-1}{\frac{wj + 1}{wj - \frac{x}{e^{wj}}}}\\ \mathbf{else}:\\ \;\;\;\;x + \left(-2 \cdot \left(wj \cdot x\right) + {wj}^{2} \cdot \left(1 - \left(x \cdot -4 + x \cdot 1.5\right)\right)\right)\\ \end{array} \]

Alternative 3: 86.4% accurate, 2.9× speedup

Math

\[\begin{array}{l} \\ \frac{x}{e^{wj} \cdot \left(wj + 1\right)} \end{array} \]

FPCore

(FPCore (wj x) :precision binary64 (/ x (* (exp wj) (+ wj 1.0))))

C

double code(double wj, double x) {
	return x / (exp(wj) * (wj + 1.0));
}

Fortran

real(8) function code(wj, x)
    real(8), intent (in) :: wj
    real(8), intent (in) :: x
    code = x / (exp(wj) * (wj + 1.0d0))
end function

Java

public static double code(double wj, double x) {
	return x / (Math.exp(wj) * (wj + 1.0));
}

Python

def code(wj, x):
	return x / (math.exp(wj) * (wj + 1.0))

Julia

function code(wj, x)
	return Float64(x / Float64(exp(wj) * Float64(wj + 1.0)))
end

MATLAB

function tmp = code(wj, x)
	tmp = x / (exp(wj) * (wj + 1.0));
end

Wolfram

code[wj_, x_] := N[(x / N[(N[Exp[wj], $MachinePrecision] * N[(wj + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 80.2%

   \[wj - \frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}} \]
2. Step-by-step derivation

   1. div-sub 80.2%

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

   2. distribute-rgt1-in 80.2%

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

   3. times-frac 80.3%

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

   4. *-inverses 80.3%

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

   5. associate-*l/ 80.3%

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

   6. *-rgt-identity 80.3%

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

   7. distribute-rgt1-in 80.3%

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

   8. associate-/l/ 80.2%

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

   9. div-sub 80.2%

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

3. Simplified 80.2%

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

4. Taylor expanded in x around inf 86.8%

   \[\leadsto \color{blue}{\frac{x}{e^{wj} \cdot \left(1 + wj\right)}} \]
5. Step-by-step derivation

   1. +-commutative 86.8%

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

6. Simplified 86.8%

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

7. Final simplification 86.8%

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

Alternative 4: 84.6% accurate, 44.7× speedup

Math

\[\begin{array}{l} \\ x + -2 \cdot \left(wj \cdot x\right) \end{array} \]

FPCore

(FPCore (wj x) :precision binary64 (+ x (* -2.0 (* wj x))))

C

double code(double wj, double x) {
	return x + (-2.0 * (wj * x));
}

Fortran

real(8) function code(wj, x)
    real(8), intent (in) :: wj
    real(8), intent (in) :: x
    code = x + ((-2.0d0) * (wj * x))
end function

Java

public static double code(double wj, double x) {
	return x + (-2.0 * (wj * x));
}

Python

def code(wj, x):
	return x + (-2.0 * (wj * x))

Julia

function code(wj, x)
	return Float64(x + Float64(-2.0 * Float64(wj * x)))
end

MATLAB

function tmp = code(wj, x)
	tmp = x + (-2.0 * (wj * x));
end

Wolfram

code[wj_, x_] := N[(x + N[(-2.0 * N[(wj * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 80.2%

   \[wj - \frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}} \]
2. Step-by-step derivation

   1. div-sub 80.2%

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

   2. distribute-rgt1-in 80.2%

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

   3. times-frac 80.3%

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

   4. *-inverses 80.3%

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

   5. associate-*l/ 80.3%

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

   6. *-rgt-identity 80.3%

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

   7. distribute-rgt1-in 80.3%

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

   8. associate-/l/ 80.2%

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

   9. div-sub 80.2%

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

3. Simplified 80.2%

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

4. Taylor expanded in wj around 0 84.6%

   \[\leadsto \color{blue}{x + -2 \cdot \left(wj \cdot x\right)} \]
5. Step-by-step derivation

   1. *-commutative 84.6%

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

6. Simplified 84.6%

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

7. Final simplification 84.6%

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

Alternative 5: 84.7% accurate, 44.7× speedup

Math

\[\begin{array}{l} \\ \frac{x}{1 + wj \cdot 2} \end{array} \]

FPCore

(FPCore (wj x) :precision binary64 (/ x (+ 1.0 (* wj 2.0))))

C

double code(double wj, double x) {
	return x / (1.0 + (wj * 2.0));
}

Fortran

real(8) function code(wj, x)
    real(8), intent (in) :: wj
    real(8), intent (in) :: x
    code = x / (1.0d0 + (wj * 2.0d0))
end function

Java

public static double code(double wj, double x) {
	return x / (1.0 + (wj * 2.0));
}

Python

def code(wj, x):
	return x / (1.0 + (wj * 2.0))

Julia

function code(wj, x)
	return Float64(x / Float64(1.0 + Float64(wj * 2.0)))
end

MATLAB

function tmp = code(wj, x)
	tmp = x / (1.0 + (wj * 2.0));
end

Wolfram

code[wj_, x_] := N[(x / N[(1.0 + N[(wj * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 80.2%

   \[wj - \frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}} \]
2. Step-by-step derivation

   1. div-sub 80.2%

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

   2. distribute-rgt1-in 80.2%

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

   3. times-frac 80.3%

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

   4. *-inverses 80.3%

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

   5. associate-*l/ 80.3%

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

   6. *-rgt-identity 80.3%

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

   7. distribute-rgt1-in 80.3%

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

   8. associate-/l/ 80.2%

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

   9. div-sub 80.2%

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

3. Simplified 80.2%

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

4. Taylor expanded in x around inf 86.8%

   \[\leadsto \color{blue}{\frac{x}{e^{wj} \cdot \left(1 + wj\right)}} \]
5. Step-by-step derivation

   1. +-commutative 86.8%

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

6. Simplified 86.8%

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

7. Taylor expanded in wj around 0 84.6%

   \[\leadsto \frac{x}{\color{blue}{1 + 2 \cdot wj}} \]
8. Step-by-step derivation

   1. *-commutative 84.6%

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

9. Simplified 84.6%

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

10. Final simplification 84.6%

    \[\leadsto \frac{x}{1 + wj \cdot 2} \]

Alternative 6: 4.4% accurate, 313.0× speedup

Math

\[\begin{array}{l} \\ wj \end{array} \]

FPCore

(FPCore (wj x) :precision binary64 wj)

C

double code(double wj, double x) {
	return wj;
}

Fortran

real(8) function code(wj, x)
    real(8), intent (in) :: wj
    real(8), intent (in) :: x
    code = wj
end function

Java

public static double code(double wj, double x) {
	return wj;
}

Python

def code(wj, x):
	return wj

Julia

function code(wj, x)
	return wj
end

MATLAB

function tmp = code(wj, x)
	tmp = wj;
end

Wolfram

code[wj_, x_] := wj

Derivation

1. Initial program 80.2%

   \[wj - \frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}} \]
2. Step-by-step derivation

   1. div-sub 80.2%

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

   2. distribute-rgt1-in 80.2%

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

   3. times-frac 80.3%

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

   4. *-inverses 80.3%

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

   5. associate-*l/ 80.3%

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

   6. *-rgt-identity 80.3%

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

   7. distribute-rgt1-in 80.3%

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

   8. associate-/l/ 80.2%

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

   9. div-sub 80.2%

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

3. Simplified 80.2%

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

4. Taylor expanded in wj around inf 3.9%

   \[\leadsto \color{blue}{wj} \]

5. Final simplification 3.9%

   \[\leadsto wj \]

Alternative 7: 84.1% accurate, 313.0× speedup

Math

\[\begin{array}{l} \\ x \end{array} \]

FPCore

(FPCore (wj x) :precision binary64 x)

C

double code(double wj, double x) {
	return x;
}

Fortran

real(8) function code(wj, x)
    real(8), intent (in) :: wj
    real(8), intent (in) :: x
    code = x
end function

Java

public static double code(double wj, double x) {
	return x;
}

Python

def code(wj, x):
	return x

Julia

function code(wj, x)
	return x
end

MATLAB

function tmp = code(wj, x)
	tmp = x;
end

Wolfram

code[wj_, x_] := x

Derivation

1. Initial program 80.2%

   \[wj - \frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}} \]
2. Step-by-step derivation

   1. div-sub 80.2%

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

   2. distribute-rgt1-in 80.2%

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

   3. times-frac 80.3%

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

   4. *-inverses 80.3%

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

   5. associate-*l/ 80.3%

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

   6. *-rgt-identity 80.3%

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

   7. distribute-rgt1-in 80.3%

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

   8. associate-/l/ 80.2%

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

   9. div-sub 80.2%

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

3. Simplified 80.2%

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

4. Taylor expanded in wj around 0 84.2%

   \[\leadsto \color{blue}{x} \]

5. Final simplification 84.2%

   \[\leadsto x \]

Developer target: 79.2% accurate, 1.5× speedup

Math

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

FPCore

(FPCore (wj x)
 :precision binary64
 (- wj (- (/ wj (+ wj 1.0)) (/ x (+ (exp wj) (* wj (exp wj)))))))

C

double code(double wj, double x) {
	return wj - ((wj / (wj + 1.0)) - (x / (exp(wj) + (wj * exp(wj)))));
}

Fortran

real(8) function code(wj, x)
    real(8), intent (in) :: wj
    real(8), intent (in) :: x
    code = wj - ((wj / (wj + 1.0d0)) - (x / (exp(wj) + (wj * exp(wj)))))
end function

Java

public static double code(double wj, double x) {
	return wj - ((wj / (wj + 1.0)) - (x / (Math.exp(wj) + (wj * Math.exp(wj)))));
}

Python

def code(wj, x):
	return wj - ((wj / (wj + 1.0)) - (x / (math.exp(wj) + (wj * math.exp(wj)))))

Julia

function code(wj, x)
	return Float64(wj - Float64(Float64(wj / Float64(wj + 1.0)) - Float64(x / Float64(exp(wj) + Float64(wj * exp(wj))))))
end

MATLAB

function tmp = code(wj, x)
	tmp = wj - ((wj / (wj + 1.0)) - (x / (exp(wj) + (wj * exp(wj)))));
end

Wolfram

code[wj_, x_] := N[(wj - N[(N[(wj / N[(wj + 1.0), $MachinePrecision]), $MachinePrecision] - N[(x / N[(N[Exp[wj], $MachinePrecision] + N[(wj * N[Exp[wj], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Reproduce

herbie shell --seed 2023320 
(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))))))