Jmat.Real.lambertw, newton loop step

Percentage Accurate: 78.4% → 97.4%

Time: 9.3s

Alternatives: 13

Speedup: 313.0×

• could not determine a ground truth

  1. wj = -4.3907217176493484e+162
  2. x = 5.656483029217214e+202

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: 78.4% 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, 2.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -2 \cdot 10^{-23}:\\ \;\;\;\;wj + \frac{wj - \frac{x}{e^{wj}}}{-1 - wj}\\ \mathbf{else}:\\ \;\;\;\;x - wj \cdot \left(x \cdot 2 + x \cdot \left(\frac{wj \cdot \left(wj + -1\right)}{x} - wj \cdot \left(2.5 + wj \cdot -2.6666666666666665\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (wj x)
 :precision binary64
 (if (<= x -2e-23)
   (+ wj (/ (- wj (/ x (exp wj))) (- -1.0 wj)))
   (-
    x
    (*
     wj
     (+
      (* x 2.0)
      (*
       x
       (-
        (/ (* wj (+ wj -1.0)) x)
        (* wj (+ 2.5 (* wj -2.6666666666666665))))))))))

C

double code(double wj, double x) {
	double tmp;
	if (x <= -2e-23) {
		tmp = wj + ((wj - (x / exp(wj))) / (-1.0 - wj));
	} else {
		tmp = x - (wj * ((x * 2.0) + (x * (((wj * (wj + -1.0)) / x) - (wj * (2.5 + (wj * -2.6666666666666665)))))));
	}
	return tmp;
}

Fortran

real(8) function code(wj, x)
    real(8), intent (in) :: wj
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= (-2d-23)) then
        tmp = wj + ((wj - (x / exp(wj))) / ((-1.0d0) - wj))
    else
        tmp = x - (wj * ((x * 2.0d0) + (x * (((wj * (wj + (-1.0d0))) / x) - (wj * (2.5d0 + (wj * (-2.6666666666666665d0))))))))
    end if
    code = tmp
end function

Java

public static double code(double wj, double x) {
	double tmp;
	if (x <= -2e-23) {
		tmp = wj + ((wj - (x / Math.exp(wj))) / (-1.0 - wj));
	} else {
		tmp = x - (wj * ((x * 2.0) + (x * (((wj * (wj + -1.0)) / x) - (wj * (2.5 + (wj * -2.6666666666666665)))))));
	}
	return tmp;
}

Python

def code(wj, x):
	tmp = 0
	if x <= -2e-23:
		tmp = wj + ((wj - (x / math.exp(wj))) / (-1.0 - wj))
	else:
		tmp = x - (wj * ((x * 2.0) + (x * (((wj * (wj + -1.0)) / x) - (wj * (2.5 + (wj * -2.6666666666666665)))))))
	return tmp

Julia

function code(wj, x)
	tmp = 0.0
	if (x <= -2e-23)
		tmp = Float64(wj + Float64(Float64(wj - Float64(x / exp(wj))) / Float64(-1.0 - wj)));
	else
		tmp = Float64(x - Float64(wj * Float64(Float64(x * 2.0) + Float64(x * Float64(Float64(Float64(wj * Float64(wj + -1.0)) / x) - Float64(wj * Float64(2.5 + Float64(wj * -2.6666666666666665))))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(wj, x)
	tmp = 0.0;
	if (x <= -2e-23)
		tmp = wj + ((wj - (x / exp(wj))) / (-1.0 - wj));
	else
		tmp = x - (wj * ((x * 2.0) + (x * (((wj * (wj + -1.0)) / x) - (wj * (2.5 + (wj * -2.6666666666666665)))))));
	end
	tmp_2 = tmp;
end

Wolfram

code[wj_, x_] := If[LessEqual[x, -2e-23], N[(wj + N[(N[(wj - N[(x / N[Exp[wj], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(-1.0 - wj), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x - N[(wj * N[(N[(x * 2.0), $MachinePrecision] + N[(x * N[(N[(N[(wj * N[(wj + -1.0), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision] - N[(wj * N[(2.5 + N[(wj * -2.6666666666666665), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -1.99999999999999992e-23

   1. Initial program 92.4%

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

      1. distribute-rgt1-in 96.9%

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

      2. associate-/l/ 97.0%

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

      3. div-sub 92.4%

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

      4. associate-/l* 92.4%

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

      5. *-inverses 100.0%

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

      6. *-rgt-identity 100.0%

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

   3. Simplified 100.0%

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

   if -1.99999999999999992e-23 < x

   1. Initial program 69.8%

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

      1. distribute-rgt1-in 69.8%

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

      2. associate-/l/ 69.9%

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

      3. div-sub 69.9%

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

      4. associate-/l* 69.9%

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

      5. *-inverses 69.9%

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

      6. *-rgt-identity 69.9%

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

   3. Simplified 69.9%

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

   5. Taylor expanded in wj around 0 98.6%

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

   6. Taylor expanded in x around inf 98.6%

      \[\leadsto x + wj \cdot \left(\color{blue}{x \cdot \left(wj \cdot \left(2.5 + -2.6666666666666665 \cdot wj\right) + \frac{wj \cdot \left(1 + -1 \cdot wj\right)}{x}\right)} - 2 \cdot x\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 99.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2 \cdot 10^{-23}:\\ \;\;\;\;wj + \frac{wj - \frac{x}{e^{wj}}}{-1 - wj}\\ \mathbf{else}:\\ \;\;\;\;x - wj \cdot \left(x \cdot 2 + x \cdot \left(\frac{wj \cdot \left(wj + -1\right)}{x} - wj \cdot \left(2.5 + wj \cdot -2.6666666666666665\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 96.7% accurate, 12.5× speedup

Math

\[\begin{array}{l} \\ x - wj \cdot \left(x \cdot 2 + x \cdot \left(\frac{wj \cdot \left(wj + -1\right)}{x} - wj \cdot \left(2.5 + wj \cdot -2.6666666666666665\right)\right)\right) \end{array} \]

FPCore

(FPCore (wj x)
 :precision binary64
 (-
  x
  (*
   wj
   (+
    (* x 2.0)
    (*
     x
     (-
      (/ (* wj (+ wj -1.0)) x)
      (* wj (+ 2.5 (* wj -2.6666666666666665)))))))))

C

double code(double wj, double x) {
	return x - (wj * ((x * 2.0) + (x * (((wj * (wj + -1.0)) / x) - (wj * (2.5 + (wj * -2.6666666666666665)))))));
}

Fortran

real(8) function code(wj, x)
    real(8), intent (in) :: wj
    real(8), intent (in) :: x
    code = x - (wj * ((x * 2.0d0) + (x * (((wj * (wj + (-1.0d0))) / x) - (wj * (2.5d0 + (wj * (-2.6666666666666665d0))))))))
end function

Java

public static double code(double wj, double x) {
	return x - (wj * ((x * 2.0) + (x * (((wj * (wj + -1.0)) / x) - (wj * (2.5 + (wj * -2.6666666666666665)))))));
}

Python

def code(wj, x):
	return x - (wj * ((x * 2.0) + (x * (((wj * (wj + -1.0)) / x) - (wj * (2.5 + (wj * -2.6666666666666665)))))))

Julia

function code(wj, x)
	return Float64(x - Float64(wj * Float64(Float64(x * 2.0) + Float64(x * Float64(Float64(Float64(wj * Float64(wj + -1.0)) / x) - Float64(wj * Float64(2.5 + Float64(wj * -2.6666666666666665))))))))
end

MATLAB

function tmp = code(wj, x)
	tmp = x - (wj * ((x * 2.0) + (x * (((wj * (wj + -1.0)) / x) - (wj * (2.5 + (wj * -2.6666666666666665)))))));
end

Wolfram

code[wj_, x_] := N[(x - N[(wj * N[(N[(x * 2.0), $MachinePrecision] + N[(x * N[(N[(N[(wj * N[(wj + -1.0), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision] - N[(wj * N[(2.5 + N[(wj * -2.6666666666666665), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 75.6%

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

   1. distribute-rgt1-in 76.8%

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

   2. associate-/l/ 76.9%

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

   3. div-sub 75.7%

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

   4. associate-/l* 75.7%

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

   5. *-inverses 77.6%

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

   6. *-rgt-identity 77.6%

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

3. Simplified 77.6%

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

5. Taylor expanded in wj around 0 96.7%

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

6. Taylor expanded in x around inf 96.7%

   \[\leadsto x + wj \cdot \left(\color{blue}{x \cdot \left(wj \cdot \left(2.5 + -2.6666666666666665 \cdot wj\right) + \frac{wj \cdot \left(1 + -1 \cdot wj\right)}{x}\right)} - 2 \cdot x\right) \]

7. Final simplification 96.7%

   \[\leadsto x - wj \cdot \left(x \cdot 2 + x \cdot \left(\frac{wj \cdot \left(wj + -1\right)}{x} - wj \cdot \left(2.5 + wj \cdot -2.6666666666666665\right)\right)\right) \]
8. Add Preprocessing

Alternative 3: 96.7% accurate, 14.9× speedup

Math

\[\begin{array}{l} \\ x + wj \cdot \left(wj \cdot \left(\left(1 - wj\right) + x \cdot \left(2.5 + wj \cdot -2.6666666666666665\right)\right) - x \cdot 2\right) \end{array} \]

FPCore

(FPCore (wj x)
 :precision binary64
 (+
  x
  (*
   wj
   (-
    (* wj (+ (- 1.0 wj) (* x (+ 2.5 (* wj -2.6666666666666665)))))
    (* x 2.0)))))

C

double code(double wj, double x) {
	return x + (wj * ((wj * ((1.0 - wj) + (x * (2.5 + (wj * -2.6666666666666665))))) - (x * 2.0)));
}

Fortran

real(8) function code(wj, x)
    real(8), intent (in) :: wj
    real(8), intent (in) :: x
    code = x + (wj * ((wj * ((1.0d0 - wj) + (x * (2.5d0 + (wj * (-2.6666666666666665d0)))))) - (x * 2.0d0)))
end function

Java

public static double code(double wj, double x) {
	return x + (wj * ((wj * ((1.0 - wj) + (x * (2.5 + (wj * -2.6666666666666665))))) - (x * 2.0)));
}

Python

def code(wj, x):
	return x + (wj * ((wj * ((1.0 - wj) + (x * (2.5 + (wj * -2.6666666666666665))))) - (x * 2.0)))

Julia

function code(wj, x)
	return Float64(x + Float64(wj * Float64(Float64(wj * Float64(Float64(1.0 - wj) + Float64(x * Float64(2.5 + Float64(wj * -2.6666666666666665))))) - Float64(x * 2.0))))
end

MATLAB

function tmp = code(wj, x)
	tmp = x + (wj * ((wj * ((1.0 - wj) + (x * (2.5 + (wj * -2.6666666666666665))))) - (x * 2.0)));
end

Wolfram

code[wj_, x_] := N[(x + N[(wj * N[(N[(wj * N[(N[(1.0 - wj), $MachinePrecision] + N[(x * N[(2.5 + N[(wj * -2.6666666666666665), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(x * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 75.6%

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

   1. distribute-rgt1-in 76.8%

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

   2. associate-/l/ 76.9%

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

   3. div-sub 75.7%

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

   4. associate-/l* 75.7%

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

   5. *-inverses 77.6%

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

   6. *-rgt-identity 77.6%

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

3. Simplified 77.6%

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

5. Taylor expanded in wj around 0 96.7%

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

6. Taylor expanded in x around 0 96.7%

   \[\leadsto x + wj \cdot \left(\color{blue}{\left(wj \cdot \left(x \cdot \left(2.5 + -2.6666666666666665 \cdot wj\right)\right) + wj \cdot \left(1 + -1 \cdot wj\right)\right)} - 2 \cdot x\right) \]
7. Step-by-step derivation

   1. distribute-lft-out 96.7%

      \[\leadsto x + wj \cdot \left(\color{blue}{wj \cdot \left(x \cdot \left(2.5 + -2.6666666666666665 \cdot wj\right) + \left(1 + -1 \cdot wj\right)\right)} - 2 \cdot x\right) \]

   2. +-commutative 96.7%

      \[\leadsto x + wj \cdot \left(wj \cdot \left(x \cdot \color{blue}{\left(-2.6666666666666665 \cdot wj + 2.5\right)} + \left(1 + -1 \cdot wj\right)\right) - 2 \cdot x\right) \]

   3. *-commutative 96.7%

      \[\leadsto x + wj \cdot \left(wj \cdot \left(x \cdot \left(\color{blue}{wj \cdot -2.6666666666666665} + 2.5\right) + \left(1 + -1 \cdot wj\right)\right) - 2 \cdot x\right) \]

   4. mul-1-neg 96.7%

      \[\leadsto x + wj \cdot \left(wj \cdot \left(x \cdot \left(wj \cdot -2.6666666666666665 + 2.5\right) + \left(1 + \color{blue}{\left(-wj\right)}\right)\right) - 2 \cdot x\right) \]

   5. sub-neg 96.7%

      \[\leadsto x + wj \cdot \left(wj \cdot \left(x \cdot \left(wj \cdot -2.6666666666666665 + 2.5\right) + \color{blue}{\left(1 - wj\right)}\right) - 2 \cdot x\right) \]

8. Simplified 96.7%

   \[\leadsto x + wj \cdot \left(\color{blue}{wj \cdot \left(x \cdot \left(wj \cdot -2.6666666666666665 + 2.5\right) + \left(1 - wj\right)\right)} - 2 \cdot x\right) \]

9. Final simplification 96.7%

   \[\leadsto x + wj \cdot \left(wj \cdot \left(\left(1 - wj\right) + x \cdot \left(2.5 + wj \cdot -2.6666666666666665\right)\right) - x \cdot 2\right) \]
10. Add Preprocessing

Alternative 4: 96.2% accurate, 18.4× speedup

Math

\[\begin{array}{l} \\ x + wj \cdot \left(x \cdot \left(wj \cdot 2.5 + \frac{wj}{x}\right) + x \cdot -2\right) \end{array} \]

FPCore

(FPCore (wj x)
 :precision binary64
 (+ x (* wj (+ (* x (+ (* wj 2.5) (/ wj x))) (* x -2.0)))))

C

double code(double wj, double x) {
	return x + (wj * ((x * ((wj * 2.5) + (wj / x))) + (x * -2.0)));
}

Fortran

real(8) function code(wj, x)
    real(8), intent (in) :: wj
    real(8), intent (in) :: x
    code = x + (wj * ((x * ((wj * 2.5d0) + (wj / x))) + (x * (-2.0d0))))
end function

Java

public static double code(double wj, double x) {
	return x + (wj * ((x * ((wj * 2.5) + (wj / x))) + (x * -2.0)));
}

Python

def code(wj, x):
	return x + (wj * ((x * ((wj * 2.5) + (wj / x))) + (x * -2.0)))

Julia

function code(wj, x)
	return Float64(x + Float64(wj * Float64(Float64(x * Float64(Float64(wj * 2.5) + Float64(wj / x))) + Float64(x * -2.0))))
end

MATLAB

function tmp = code(wj, x)
	tmp = x + (wj * ((x * ((wj * 2.5) + (wj / x))) + (x * -2.0)));
end

Wolfram

code[wj_, x_] := N[(x + N[(wj * N[(N[(x * N[(N[(wj * 2.5), $MachinePrecision] + N[(wj / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(x * -2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 75.6%

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

   1. distribute-rgt1-in 76.8%

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

   2. associate-/l/ 76.9%

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

   3. div-sub 75.7%

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

   4. associate-/l* 75.7%

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

   5. *-inverses 77.6%

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

   6. *-rgt-identity 77.6%

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

3. Simplified 77.6%

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

5. Taylor expanded in wj around 0 96.3%

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

   1. cancel-sign-sub-inv 96.3%

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

   2. distribute-rgt-out 96.3%

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

   3. metadata-eval 96.3%

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

   4. metadata-eval 96.3%

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

   5. *-commutative 96.3%

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

7. Simplified 96.3%

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

8. Taylor expanded in x around inf 96.3%

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

9. Final simplification 96.3%

   \[\leadsto x + wj \cdot \left(x \cdot \left(wj \cdot 2.5 + \frac{wj}{x}\right) + x \cdot -2\right) \]
10. Add Preprocessing

Alternative 5: 84.4% accurate, 19.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;wj \leq -7 \cdot 10^{-49}:\\ \;\;\;\;\left(wj \cdot wj\right) \cdot \left(1 + wj \cdot \left(wj + -1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{1 + wj \cdot \left(2 + wj \cdot 1.5\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (wj x)
 :precision binary64
 (if (<= wj -7e-49)
   (* (* wj wj) (+ 1.0 (* wj (+ wj -1.0))))
   (/ x (+ 1.0 (* wj (+ 2.0 (* wj 1.5)))))))

C

double code(double wj, double x) {
	double tmp;
	if (wj <= -7e-49) {
		tmp = (wj * wj) * (1.0 + (wj * (wj + -1.0)));
	} else {
		tmp = x / (1.0 + (wj * (2.0 + (wj * 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 <= (-7d-49)) then
        tmp = (wj * wj) * (1.0d0 + (wj * (wj + (-1.0d0))))
    else
        tmp = x / (1.0d0 + (wj * (2.0d0 + (wj * 1.5d0))))
    end if
    code = tmp
end function

Java

public static double code(double wj, double x) {
	double tmp;
	if (wj <= -7e-49) {
		tmp = (wj * wj) * (1.0 + (wj * (wj + -1.0)));
	} else {
		tmp = x / (1.0 + (wj * (2.0 + (wj * 1.5))));
	}
	return tmp;
}

Python

def code(wj, x):
	tmp = 0
	if wj <= -7e-49:
		tmp = (wj * wj) * (1.0 + (wj * (wj + -1.0)))
	else:
		tmp = x / (1.0 + (wj * (2.0 + (wj * 1.5))))
	return tmp

Julia

function code(wj, x)
	tmp = 0.0
	if (wj <= -7e-49)
		tmp = Float64(Float64(wj * wj) * Float64(1.0 + Float64(wj * Float64(wj + -1.0))));
	else
		tmp = Float64(x / Float64(1.0 + Float64(wj * Float64(2.0 + Float64(wj * 1.5)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(wj, x)
	tmp = 0.0;
	if (wj <= -7e-49)
		tmp = (wj * wj) * (1.0 + (wj * (wj + -1.0)));
	else
		tmp = x / (1.0 + (wj * (2.0 + (wj * 1.5))));
	end
	tmp_2 = tmp;
end

Wolfram

code[wj_, x_] := If[LessEqual[wj, -7e-49], N[(N[(wj * wj), $MachinePrecision] * N[(1.0 + N[(wj * N[(wj + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x / N[(1.0 + N[(wj * N[(2.0 + N[(wj * 1.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if wj < -7.00000000000000012e-49

   1. Initial program 34.7%

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

      1. distribute-rgt1-in 47.8%

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

      2. associate-/l/ 48.3%

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

      3. div-sub 35.3%

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

      4. associate-/l* 35.3%

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

      5. *-inverses 48.3%

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

      6. *-rgt-identity 48.3%

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

   3. Simplified 48.3%

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

   5. Taylor expanded in x around 0 10.1%

      \[\leadsto \color{blue}{wj - \frac{wj}{1 + wj}} \]
   6. Step-by-step derivation

      1. +-commutative 10.1%

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

   7. Simplified 10.1%

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

   8. Taylor expanded in wj around 0 54.2%

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

      1. unpow2 54.2%

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

   10. Applied egg-rr 54.2%

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

   if -7.00000000000000012e-49 < wj

   1. Initial program 79.7%

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

      1. distribute-rgt1-in 79.7%

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

      2. associate-/l/ 79.7%

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

      3. div-sub 79.7%

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

      4. associate-/l* 79.7%

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

      5. *-inverses 80.5%

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

      6. *-rgt-identity 80.5%

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

   3. Simplified 80.5%

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

   5. Taylor expanded in x around inf 89.0%

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

      1. distribute-rgt-in 89.0%

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

      2. *-lft-identity 89.0%

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

      3. distribute-rgt1-in 89.0%

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

      4. *-commutative 89.0%

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

   7. Simplified 89.0%

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

   8. Taylor expanded in wj around 0 88.8%

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

      1. *-commutative 88.8%

         \[\leadsto \frac{x}{1 + wj \cdot \left(2 + \color{blue}{wj \cdot 1.5}\right)} \]

   10. Simplified 88.8%

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

4. Final simplification 85.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;wj \leq -7 \cdot 10^{-49}:\\ \;\;\;\;\left(wj \cdot wj\right) \cdot \left(1 + wj \cdot \left(wj + -1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{1 + wj \cdot \left(2 + wj \cdot 1.5\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 84.4% accurate, 19.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;wj \leq -4 \cdot 10^{-49}:\\ \;\;\;\;\left(wj \cdot wj\right) \cdot \left(1 + wj \cdot \left(wj + -1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(1 - wj \cdot \left(2 - wj \cdot 2.5\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (wj x)
 :precision binary64
 (if (<= wj -4e-49)
   (* (* wj wj) (+ 1.0 (* wj (+ wj -1.0))))
   (* x (- 1.0 (* wj (- 2.0 (* wj 2.5)))))))

C

double code(double wj, double x) {
	double tmp;
	if (wj <= -4e-49) {
		tmp = (wj * wj) * (1.0 + (wj * (wj + -1.0)));
	} else {
		tmp = x * (1.0 - (wj * (2.0 - (wj * 2.5))));
	}
	return tmp;
}

Fortran

real(8) function code(wj, x)
    real(8), intent (in) :: wj
    real(8), intent (in) :: x
    real(8) :: tmp
    if (wj <= (-4d-49)) then
        tmp = (wj * wj) * (1.0d0 + (wj * (wj + (-1.0d0))))
    else
        tmp = x * (1.0d0 - (wj * (2.0d0 - (wj * 2.5d0))))
    end if
    code = tmp
end function

Java

public static double code(double wj, double x) {
	double tmp;
	if (wj <= -4e-49) {
		tmp = (wj * wj) * (1.0 + (wj * (wj + -1.0)));
	} else {
		tmp = x * (1.0 - (wj * (2.0 - (wj * 2.5))));
	}
	return tmp;
}

Python

def code(wj, x):
	tmp = 0
	if wj <= -4e-49:
		tmp = (wj * wj) * (1.0 + (wj * (wj + -1.0)))
	else:
		tmp = x * (1.0 - (wj * (2.0 - (wj * 2.5))))
	return tmp

Julia

function code(wj, x)
	tmp = 0.0
	if (wj <= -4e-49)
		tmp = Float64(Float64(wj * wj) * Float64(1.0 + Float64(wj * Float64(wj + -1.0))));
	else
		tmp = Float64(x * Float64(1.0 - Float64(wj * Float64(2.0 - Float64(wj * 2.5)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(wj, x)
	tmp = 0.0;
	if (wj <= -4e-49)
		tmp = (wj * wj) * (1.0 + (wj * (wj + -1.0)));
	else
		tmp = x * (1.0 - (wj * (2.0 - (wj * 2.5))));
	end
	tmp_2 = tmp;
end

Wolfram

code[wj_, x_] := If[LessEqual[wj, -4e-49], N[(N[(wj * wj), $MachinePrecision] * N[(1.0 + N[(wj * N[(wj + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x * N[(1.0 - N[(wj * N[(2.0 - N[(wj * 2.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if wj < -3.99999999999999975e-49

   1. Initial program 34.7%

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

      1. distribute-rgt1-in 47.8%

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

      2. associate-/l/ 48.3%

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

      3. div-sub 35.3%

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

      4. associate-/l* 35.3%

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

      5. *-inverses 48.3%

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

      6. *-rgt-identity 48.3%

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

   3. Simplified 48.3%

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

   5. Taylor expanded in x around 0 10.1%

      \[\leadsto \color{blue}{wj - \frac{wj}{1 + wj}} \]
   6. Step-by-step derivation

      1. +-commutative 10.1%

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

   7. Simplified 10.1%

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

   8. Taylor expanded in wj around 0 54.2%

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

      1. unpow2 54.2%

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

   10. Applied egg-rr 54.2%

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

   if -3.99999999999999975e-49 < wj

   1. Initial program 79.7%

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

      1. distribute-rgt1-in 79.7%

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

      2. associate-/l/ 79.7%

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

      3. div-sub 79.7%

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

      4. associate-/l* 79.7%

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

      5. *-inverses 80.5%

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

      6. *-rgt-identity 80.5%

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

   3. Simplified 80.5%

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

   5. Taylor expanded in wj around 0 98.1%

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

      1. cancel-sign-sub-inv 98.1%

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

      2. distribute-rgt-out 98.1%

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

      3. metadata-eval 98.1%

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

      4. metadata-eval 98.1%

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

      5. *-commutative 98.1%

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

   7. Simplified 98.1%

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

   8. Taylor expanded in x around inf 88.7%

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

4. Final simplification 85.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;wj \leq -4 \cdot 10^{-49}:\\ \;\;\;\;\left(wj \cdot wj\right) \cdot \left(1 + wj \cdot \left(wj + -1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(1 - wj \cdot \left(2 - wj \cdot 2.5\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 96.1% accurate, 20.9× speedup

Math

\[\begin{array}{l} \\ x + wj \cdot \left(x \cdot -2 + wj \cdot \left(1 - x \cdot -2.5\right)\right) \end{array} \]

FPCore

(FPCore (wj x)
 :precision binary64
 (+ x (* wj (+ (* x -2.0) (* wj (- 1.0 (* x -2.5)))))))

C

double code(double wj, double x) {
	return x + (wj * ((x * -2.0) + (wj * (1.0 - (x * -2.5)))));
}

Fortran

real(8) function code(wj, x)
    real(8), intent (in) :: wj
    real(8), intent (in) :: x
    code = x + (wj * ((x * (-2.0d0)) + (wj * (1.0d0 - (x * (-2.5d0))))))
end function

Java

public static double code(double wj, double x) {
	return x + (wj * ((x * -2.0) + (wj * (1.0 - (x * -2.5)))));
}

Python

def code(wj, x):
	return x + (wj * ((x * -2.0) + (wj * (1.0 - (x * -2.5)))))

Julia

function code(wj, x)
	return Float64(x + Float64(wj * Float64(Float64(x * -2.0) + Float64(wj * Float64(1.0 - Float64(x * -2.5))))))
end

MATLAB

function tmp = code(wj, x)
	tmp = x + (wj * ((x * -2.0) + (wj * (1.0 - (x * -2.5)))));
end

Wolfram

code[wj_, x_] := N[(x + N[(wj * N[(N[(x * -2.0), $MachinePrecision] + N[(wj * N[(1.0 - N[(x * -2.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 75.6%

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

   1. distribute-rgt1-in 76.8%

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

   2. associate-/l/ 76.9%

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

   3. div-sub 75.7%

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

   4. associate-/l* 75.7%

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

   5. *-inverses 77.6%

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

   6. *-rgt-identity 77.6%

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

3. Simplified 77.6%

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

5. Taylor expanded in wj around 0 96.3%

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

   1. cancel-sign-sub-inv 96.3%

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

   2. distribute-rgt-out 96.3%

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

   3. metadata-eval 96.3%

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

   4. metadata-eval 96.3%

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

   5. *-commutative 96.3%

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

7. Simplified 96.3%

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

8. Final simplification 96.3%

   \[\leadsto x + wj \cdot \left(x \cdot -2 + wj \cdot \left(1 - x \cdot -2.5\right)\right) \]
9. Add Preprocessing

Alternative 8: 85.1% accurate, 28.5× speedup

Math

\[\begin{array}{l} \\ x \cdot \left(1 - wj \cdot \left(2 - wj \cdot 2.5\right)\right) \end{array} \]

FPCore

(FPCore (wj x) :precision binary64 (* x (- 1.0 (* wj (- 2.0 (* wj 2.5))))))

C

double code(double wj, double x) {
	return x * (1.0 - (wj * (2.0 - (wj * 2.5))));
}

Fortran

real(8) function code(wj, x)
    real(8), intent (in) :: wj
    real(8), intent (in) :: x
    code = x * (1.0d0 - (wj * (2.0d0 - (wj * 2.5d0))))
end function

Java

public static double code(double wj, double x) {
	return x * (1.0 - (wj * (2.0 - (wj * 2.5))));
}

Python

def code(wj, x):
	return x * (1.0 - (wj * (2.0 - (wj * 2.5))))

Julia

function code(wj, x)
	return Float64(x * Float64(1.0 - Float64(wj * Float64(2.0 - Float64(wj * 2.5)))))
end

MATLAB

function tmp = code(wj, x)
	tmp = x * (1.0 - (wj * (2.0 - (wj * 2.5))));
end

Wolfram

code[wj_, x_] := N[(x * N[(1.0 - N[(wj * N[(2.0 - N[(wj * 2.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 75.6%

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

   1. distribute-rgt1-in 76.8%

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

   2. associate-/l/ 76.9%

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

   3. div-sub 75.7%

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

   4. associate-/l* 75.7%

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

   5. *-inverses 77.6%

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

   6. *-rgt-identity 77.6%

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

3. Simplified 77.6%

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

5. Taylor expanded in wj around 0 96.3%

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

   1. cancel-sign-sub-inv 96.3%

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

   2. distribute-rgt-out 96.3%

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

   3. metadata-eval 96.3%

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

   4. metadata-eval 96.3%

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

   5. *-commutative 96.3%

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

7. Simplified 96.3%

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

8. Taylor expanded in x around inf 83.3%

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

9. Final simplification 83.3%

   \[\leadsto x \cdot \left(1 - wj \cdot \left(2 - wj \cdot 2.5\right)\right) \]
10. Add Preprocessing

Alternative 9: 85.0% 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 75.6%

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

   1. distribute-rgt1-in 76.8%

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

   2. associate-/l/ 76.9%

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

   3. div-sub 75.7%

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

   4. associate-/l* 75.7%

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

   5. *-inverses 77.6%

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

   6. *-rgt-identity 77.6%

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

3. Simplified 77.6%

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

5. Taylor expanded in x around inf 85.1%

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

   1. distribute-rgt-in 83.9%

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

   2. *-lft-identity 83.9%

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

   3. distribute-rgt1-in 85.1%

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

   4. *-commutative 85.1%

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

7. Simplified 85.1%

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

8. Taylor expanded in wj around 0 83.2%

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

   1. *-commutative 83.2%

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

10. Simplified 83.2%

    \[\leadsto \frac{x}{\color{blue}{1 + wj \cdot 2}} \]
11. Add Preprocessing

Alternative 10: 84.9% accurate, 44.7× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 75.6%

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

   1. distribute-rgt1-in 76.8%

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

   2. associate-/l/ 76.9%

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

   3. div-sub 75.7%

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

   4. associate-/l* 75.7%

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

   5. *-inverses 77.6%

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

   6. *-rgt-identity 77.6%

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

3. Simplified 77.6%

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

5. Taylor expanded in wj around 0 83.1%

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

   1. *-commutative 83.1%

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

7. Simplified 83.1%

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

Alternative 11: 84.3% 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 75.6%

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

   1. distribute-rgt1-in 76.8%

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

   2. associate-/l/ 76.9%

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

   3. div-sub 75.7%

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

   4. associate-/l* 75.7%

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

   5. *-inverses 77.6%

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

   6. *-rgt-identity 77.6%

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

3. Simplified 77.6%

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

5. Taylor expanded in wj around 0 82.3%

   \[\leadsto \color{blue}{x} \]
6. Add Preprocessing

Alternative 12: 4.3% 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 75.6%

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

   1. distribute-rgt1-in 76.8%

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

   2. associate-/l/ 76.9%

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

   3. div-sub 75.7%

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

   4. associate-/l* 75.7%

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

   5. *-inverses 77.6%

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

   6. *-rgt-identity 77.6%

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

3. Simplified 77.6%

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

5. Taylor expanded in wj around inf 4.4%

   \[\leadsto \color{blue}{wj} \]
6. Add Preprocessing

Alternative 13: 3.4% accurate, 313.0× speedup

Math

\[\begin{array}{l} \\ -1 \end{array} \]

FPCore

(FPCore (wj x) :precision binary64 -1.0)

C

double code(double wj, double x) {
	return -1.0;
}

Fortran

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

Java

public static double code(double wj, double x) {
	return -1.0;
}

Python

def code(wj, x):
	return -1.0

Julia

function code(wj, x)
	return -1.0
end

MATLAB

function tmp = code(wj, x)
	tmp = -1.0;
end

Wolfram

code[wj_, x_] := -1.0

Derivation

1. Initial program 75.6%

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

   1. distribute-rgt1-in 76.8%

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

   2. associate-/l/ 76.9%

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

   3. div-sub 75.7%

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

   4. associate-/l* 75.7%

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

   5. *-inverses 77.6%

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

   6. *-rgt-identity 77.6%

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

3. Simplified 77.6%

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

5. Taylor expanded in wj around inf 4.0%

   \[\leadsto wj - \color{blue}{1} \]

6. Taylor expanded in wj around 0 3.1%

   \[\leadsto \color{blue}{-1} \]
7. Add Preprocessing

Developer Target 1: 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 2024185 
(FPCore (wj x)
  :name "Jmat.Real.lambertw, newton loop step"
  :precision binary64

  :alt
  (! :herbie-platform default (let ((ew (exp wj))) (- wj (- (/ wj (+ wj 1)) (/ x (+ ew (* wj ew)))))))

  (- wj (/ (- (* wj (exp wj)) x) (+ (exp wj) (* wj (exp wj))))))