Jmat.Real.lambertw, newton loop step

Percentage Accurate: 78.0% → 96.6%

Time: 10.3s

Alternatives: 11

Speedup: 313.0×

• could not determine a ground truth

  1. wj = -1.0112838441356869e+89
  2. x = 2.1190396170990004e-296

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: 78.0% 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: 96.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 6.8 \cdot 10^{-149}:\\ \;\;\;\;\mathsf{fma}\left(-2, x \cdot wj, x\right) + {wj}^{2} \cdot \left(\mathsf{fma}\left(2.5, x, 1\right) - wj\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{e^{wj} \cdot \left(wj + 1\right)} + \left(wj - \frac{wj}{wj + 1}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (wj x)
 :precision binary64
 (if (<= x 6.8e-149)
   (+ (fma -2.0 (* x wj) x) (* (pow wj 2.0) (- (fma 2.5 x 1.0) wj)))
   (+ (/ x (* (exp wj) (+ wj 1.0))) (- wj (/ wj (+ wj 1.0))))))

C

double code(double wj, double x) {
	double tmp;
	if (x <= 6.8e-149) {
		tmp = fma(-2.0, (x * wj), x) + (pow(wj, 2.0) * (fma(2.5, x, 1.0) - wj));
	} else {
		tmp = (x / (exp(wj) * (wj + 1.0))) + (wj - (wj / (wj + 1.0)));
	}
	return tmp;
}

Julia

function code(wj, x)
	tmp = 0.0
	if (x <= 6.8e-149)
		tmp = Float64(fma(-2.0, Float64(x * wj), x) + Float64((wj ^ 2.0) * Float64(fma(2.5, x, 1.0) - wj)));
	else
		tmp = Float64(Float64(x / Float64(exp(wj) * Float64(wj + 1.0))) + Float64(wj - Float64(wj / Float64(wj + 1.0))));
	end
	return tmp
end

Wolfram

code[wj_, x_] := If[LessEqual[x, 6.8e-149], N[(N[(-2.0 * N[(x * wj), $MachinePrecision] + x), $MachinePrecision] + N[(N[Power[wj, 2.0], $MachinePrecision] * N[(N[(2.5 * x + 1.0), $MachinePrecision] - wj), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x / N[(N[Exp[wj], $MachinePrecision] * N[(wj + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(wj - N[(wj / N[(wj + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 6.7999999999999998e-149

   1. Initial program 72.4%

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

      1. div-sub 72.4%

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

      2. distribute-rgt1-in 72.4%

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

      3. times-frac 72.5%

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

      4. *-inverses 72.5%

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

      5. associate-*l/ 72.5%

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

      6. *-rgt-identity 72.5%

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

      7. distribute-rgt1-in 72.5%

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

      8. associate-/l/ 72.5%

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

      9. div-sub 72.5%

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

   3. Simplified 72.5%

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

   4. Taylor expanded in wj around 0 100.0%

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

      1. associate-+r+ 100.0%

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

      2. +-commutative 100.0%

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

      3. fma-def 100.0%

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

      4. *-commutative 100.0%

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

      5. +-commutative 100.0%

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

      6. mul-1-neg 100.0%

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

      7. unsub-neg 100.0%

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

   6. Simplified 100.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(-2, x \cdot wj, x\right) + \left({wj}^{2} \cdot \left(1 + x \cdot 2.5\right) - {wj}^{3} \cdot \left(1 + \mathsf{fma}\left(x, -3, \mathsf{fma}\left(x, 0.6666666666666666, x \cdot 5\right)\right)\right)\right)} \]

   7. Taylor expanded in x around 0 100.0%

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

   8. Taylor expanded in wj around 0 100.0%

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

      1. neg-mul-1 100.0%

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

      2. +-commutative 100.0%

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

      3. *-commutative 100.0%

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

      4. +-commutative 100.0%

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

      5. *-commutative 100.0%

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

      6. fma-udef 100.0%

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

      7. unsub-neg 100.0%

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

      8. cube-mult 100.0%

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

      9. unpow2 100.0%

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

      10. distribute-rgt-out-- 100.0%

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

      11. fma-udef 100.0%

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

      12. *-commutative 100.0%

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

      13. fma-udef 100.0%

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

   10. Simplified 100.0%

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

   if 6.7999999999999998e-149 < x

   1. Initial program 97.1%

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

      1. div-sub 97.1%

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

      2. distribute-rgt1-in 97.1%

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

      3. times-frac 97.0%

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

      4. *-inverses 99.3%

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

      5. associate-*l/ 99.3%

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

      6. *-rgt-identity 99.3%

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

      7. distribute-rgt1-in 99.3%

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

      8. associate-/l/ 99.2%

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

      9. div-sub 99.2%

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

   3. Simplified 99.2%

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

   4. Taylor expanded in x around 0 99.3%

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

      1. sub-neg 99.3%

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

      2. +-commutative 99.3%

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

      3. associate-+l+ 99.6%

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

      4. +-commutative 99.6%

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

      5. +-commutative 99.6%

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

      6. sub-neg 99.6%

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

   6. Simplified 99.6%

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

4. Final simplification 99.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 6.8 \cdot 10^{-149}:\\ \;\;\;\;\mathsf{fma}\left(-2, x \cdot wj, x\right) + {wj}^{2} \cdot \left(\mathsf{fma}\left(2.5, x, 1\right) - wj\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{e^{wj} \cdot \left(wj + 1\right)} + \left(wj - \frac{wj}{wj + 1}\right)\\ \end{array} \]

Alternative 2: 96.6% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x \cdot -4 + x \cdot 1.5\\ \mathbf{if}\;x \leq 6.8 \cdot 10^{-149}:\\ \;\;\;\;x + \left(-2 \cdot \left(x \cdot wj\right) + \left({wj}^{3} \cdot \left(-1 - \left(x \cdot -3 + \left(-2 \cdot t_0 + x \cdot 0.6666666666666666\right)\right)\right) + {wj}^{2} \cdot \left(1 - t_0\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{e^{wj} \cdot \left(wj + 1\right)} + \left(wj - \frac{wj}{wj + 1}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (wj x)
 :precision binary64
 (let* ((t_0 (+ (* x -4.0) (* x 1.5))))
   (if (<= x 6.8e-149)
     (+
      x
      (+
       (* -2.0 (* x wj))
       (+
        (*
         (pow wj 3.0)
         (- -1.0 (+ (* x -3.0) (+ (* -2.0 t_0) (* x 0.6666666666666666)))))
        (* (pow wj 2.0) (- 1.0 t_0)))))
     (+ (/ x (* (exp wj) (+ wj 1.0))) (- wj (/ wj (+ wj 1.0)))))))

C

double code(double wj, double x) {
	double t_0 = (x * -4.0) + (x * 1.5);
	double tmp;
	if (x <= 6.8e-149) {
		tmp = x + ((-2.0 * (x * wj)) + ((pow(wj, 3.0) * (-1.0 - ((x * -3.0) + ((-2.0 * t_0) + (x * 0.6666666666666666))))) + (pow(wj, 2.0) * (1.0 - t_0))));
	} else {
		tmp = (x / (exp(wj) * (wj + 1.0))) + (wj - (wj / (wj + 1.0)));
	}
	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 = (x * (-4.0d0)) + (x * 1.5d0)
    if (x <= 6.8d-149) then
        tmp = x + (((-2.0d0) * (x * wj)) + (((wj ** 3.0d0) * ((-1.0d0) - ((x * (-3.0d0)) + (((-2.0d0) * t_0) + (x * 0.6666666666666666d0))))) + ((wj ** 2.0d0) * (1.0d0 - t_0))))
    else
        tmp = (x / (exp(wj) * (wj + 1.0d0))) + (wj - (wj / (wj + 1.0d0)))
    end if
    code = tmp
end function

Java

public static double code(double wj, double x) {
	double t_0 = (x * -4.0) + (x * 1.5);
	double tmp;
	if (x <= 6.8e-149) {
		tmp = x + ((-2.0 * (x * wj)) + ((Math.pow(wj, 3.0) * (-1.0 - ((x * -3.0) + ((-2.0 * t_0) + (x * 0.6666666666666666))))) + (Math.pow(wj, 2.0) * (1.0 - t_0))));
	} else {
		tmp = (x / (Math.exp(wj) * (wj + 1.0))) + (wj - (wj / (wj + 1.0)));
	}
	return tmp;
}

Python

def code(wj, x):
	t_0 = (x * -4.0) + (x * 1.5)
	tmp = 0
	if x <= 6.8e-149:
		tmp = x + ((-2.0 * (x * wj)) + ((math.pow(wj, 3.0) * (-1.0 - ((x * -3.0) + ((-2.0 * t_0) + (x * 0.6666666666666666))))) + (math.pow(wj, 2.0) * (1.0 - t_0))))
	else:
		tmp = (x / (math.exp(wj) * (wj + 1.0))) + (wj - (wj / (wj + 1.0)))
	return tmp

Julia

function code(wj, x)
	t_0 = Float64(Float64(x * -4.0) + Float64(x * 1.5))
	tmp = 0.0
	if (x <= 6.8e-149)
		tmp = Float64(x + Float64(Float64(-2.0 * Float64(x * wj)) + Float64(Float64((wj ^ 3.0) * Float64(-1.0 - Float64(Float64(x * -3.0) + Float64(Float64(-2.0 * t_0) + Float64(x * 0.6666666666666666))))) + Float64((wj ^ 2.0) * Float64(1.0 - t_0)))));
	else
		tmp = Float64(Float64(x / Float64(exp(wj) * Float64(wj + 1.0))) + Float64(wj - Float64(wj / Float64(wj + 1.0))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(wj, x)
	t_0 = (x * -4.0) + (x * 1.5);
	tmp = 0.0;
	if (x <= 6.8e-149)
		tmp = x + ((-2.0 * (x * wj)) + (((wj ^ 3.0) * (-1.0 - ((x * -3.0) + ((-2.0 * t_0) + (x * 0.6666666666666666))))) + ((wj ^ 2.0) * (1.0 - t_0))));
	else
		tmp = (x / (exp(wj) * (wj + 1.0))) + (wj - (wj / (wj + 1.0)));
	end
	tmp_2 = tmp;
end

Wolfram

code[wj_, x_] := Block[{t$95$0 = N[(N[(x * -4.0), $MachinePrecision] + N[(x * 1.5), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, 6.8e-149], N[(x + N[(N[(-2.0 * N[(x * wj), $MachinePrecision]), $MachinePrecision] + N[(N[(N[Power[wj, 3.0], $MachinePrecision] * N[(-1.0 - N[(N[(x * -3.0), $MachinePrecision] + N[(N[(-2.0 * t$95$0), $MachinePrecision] + N[(x * 0.6666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[Power[wj, 2.0], $MachinePrecision] * N[(1.0 - t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x / N[(N[Exp[wj], $MachinePrecision] * N[(wj + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(wj - N[(wj / N[(wj + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if x < 6.7999999999999998e-149

   1. Initial program 72.4%

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

      1. div-sub 72.4%

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

      2. distribute-rgt1-in 72.4%

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

      3. times-frac 72.5%

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

      4. *-inverses 72.5%

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

      5. associate-*l/ 72.5%

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

      6. *-rgt-identity 72.5%

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

      7. distribute-rgt1-in 72.5%

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

      8. associate-/l/ 72.5%

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

      9. div-sub 72.5%

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

   3. Simplified 72.5%

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

   4. Taylor expanded in wj around 0 100.0%

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

   if 6.7999999999999998e-149 < x

   1. Initial program 97.1%

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

      1. div-sub 97.1%

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

      2. distribute-rgt1-in 97.1%

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

      3. times-frac 97.0%

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

      4. *-inverses 99.3%

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

      5. associate-*l/ 99.3%

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

      6. *-rgt-identity 99.3%

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

      7. distribute-rgt1-in 99.3%

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

      8. associate-/l/ 99.2%

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

      9. div-sub 99.2%

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

   3. Simplified 99.2%

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

   4. Taylor expanded in x around 0 99.3%

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

      1. sub-neg 99.3%

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

      2. +-commutative 99.3%

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

      3. associate-+l+ 99.6%

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

      4. +-commutative 99.6%

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

      5. +-commutative 99.6%

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

      6. sub-neg 99.6%

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

   6. Simplified 99.6%

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

4. Final simplification 99.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 6.8 \cdot 10^{-149}:\\ \;\;\;\;x + \left(-2 \cdot \left(x \cdot wj\right) + \left({wj}^{3} \cdot \left(-1 - \left(x \cdot -3 + \left(-2 \cdot \left(x \cdot -4 + x \cdot 1.5\right) + x \cdot 0.6666666666666666\right)\right)\right) + {wj}^{2} \cdot \left(1 - \left(x \cdot -4 + x \cdot 1.5\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{e^{wj} \cdot \left(wj + 1\right)} + \left(wj - \frac{wj}{wj + 1}\right)\\ \end{array} \]

Alternative 3: 96.2% accurate, 2.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 1.55 \cdot 10^{-149}:\\ \;\;\;\;x + \left(-2 \cdot \left(x \cdot wj\right) + {wj}^{2} \cdot \left(1 + x \cdot 2.5\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{e^{wj} \cdot \left(wj + 1\right)} + \left(wj - \frac{wj}{wj + 1}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (wj x)
 :precision binary64
 (if (<= x 1.55e-149)
   (+ x (+ (* -2.0 (* x wj)) (* (pow wj 2.0) (+ 1.0 (* x 2.5)))))
   (+ (/ x (* (exp wj) (+ wj 1.0))) (- wj (/ wj (+ wj 1.0))))))

C

double code(double wj, double x) {
	double tmp;
	if (x <= 1.55e-149) {
		tmp = x + ((-2.0 * (x * wj)) + (pow(wj, 2.0) * (1.0 + (x * 2.5))));
	} else {
		tmp = (x / (exp(wj) * (wj + 1.0))) + (wj - (wj / (wj + 1.0)));
	}
	return tmp;
}

Fortran

real(8) function code(wj, x)
    real(8), intent (in) :: wj
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= 1.55d-149) then
        tmp = x + (((-2.0d0) * (x * wj)) + ((wj ** 2.0d0) * (1.0d0 + (x * 2.5d0))))
    else
        tmp = (x / (exp(wj) * (wj + 1.0d0))) + (wj - (wj / (wj + 1.0d0)))
    end if
    code = tmp
end function

Java

public static double code(double wj, double x) {
	double tmp;
	if (x <= 1.55e-149) {
		tmp = x + ((-2.0 * (x * wj)) + (Math.pow(wj, 2.0) * (1.0 + (x * 2.5))));
	} else {
		tmp = (x / (Math.exp(wj) * (wj + 1.0))) + (wj - (wj / (wj + 1.0)));
	}
	return tmp;
}

Python

def code(wj, x):
	tmp = 0
	if x <= 1.55e-149:
		tmp = x + ((-2.0 * (x * wj)) + (math.pow(wj, 2.0) * (1.0 + (x * 2.5))))
	else:
		tmp = (x / (math.exp(wj) * (wj + 1.0))) + (wj - (wj / (wj + 1.0)))
	return tmp

Julia

function code(wj, x)
	tmp = 0.0
	if (x <= 1.55e-149)
		tmp = Float64(x + Float64(Float64(-2.0 * Float64(x * wj)) + Float64((wj ^ 2.0) * Float64(1.0 + Float64(x * 2.5)))));
	else
		tmp = Float64(Float64(x / Float64(exp(wj) * Float64(wj + 1.0))) + Float64(wj - Float64(wj / Float64(wj + 1.0))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(wj, x)
	tmp = 0.0;
	if (x <= 1.55e-149)
		tmp = x + ((-2.0 * (x * wj)) + ((wj ^ 2.0) * (1.0 + (x * 2.5))));
	else
		tmp = (x / (exp(wj) * (wj + 1.0))) + (wj - (wj / (wj + 1.0)));
	end
	tmp_2 = tmp;
end

Wolfram

code[wj_, x_] := If[LessEqual[x, 1.55e-149], N[(x + N[(N[(-2.0 * N[(x * wj), $MachinePrecision]), $MachinePrecision] + N[(N[Power[wj, 2.0], $MachinePrecision] * N[(1.0 + N[(x * 2.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x / N[(N[Exp[wj], $MachinePrecision] * N[(wj + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(wj - N[(wj / N[(wj + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 1.54999999999999994e-149

   1. Initial program 72.4%

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

      1. div-sub 72.4%

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

      2. distribute-rgt1-in 72.4%

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

      3. times-frac 72.5%

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

      4. *-inverses 72.5%

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

      5. associate-*l/ 72.5%

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

      6. *-rgt-identity 72.5%

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

      7. distribute-rgt1-in 72.5%

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

      8. associate-/l/ 72.5%

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

      9. div-sub 72.5%

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

   3. Simplified 72.5%

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

   4. Taylor expanded in wj around 0 100.0%

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

      1. associate-+r+ 100.0%

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

      2. +-commutative 100.0%

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

      3. fma-def 100.0%

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

      4. *-commutative 100.0%

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

      5. +-commutative 100.0%

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

      6. mul-1-neg 100.0%

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

      7. unsub-neg 100.0%

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

   6. Simplified 100.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(-2, x \cdot wj, x\right) + \left({wj}^{2} \cdot \left(1 + x \cdot 2.5\right) - {wj}^{3} \cdot \left(1 + \mathsf{fma}\left(x, -3, \mathsf{fma}\left(x, 0.6666666666666666, x \cdot 5\right)\right)\right)\right)} \]

   7. Taylor expanded in x around 0 100.0%

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

   8. Taylor expanded in wj around 0 100.0%

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

   if 1.54999999999999994e-149 < x

   1. Initial program 97.1%

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

      1. div-sub 97.1%

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

      2. distribute-rgt1-in 97.1%

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

      3. times-frac 97.0%

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

      4. *-inverses 99.3%

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

      5. associate-*l/ 99.3%

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

      6. *-rgt-identity 99.3%

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

      7. distribute-rgt1-in 99.3%

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

      8. associate-/l/ 99.2%

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

      9. div-sub 99.2%

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

   3. Simplified 99.2%

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

   4. Taylor expanded in x around 0 99.3%

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

      1. sub-neg 99.3%

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

      2. +-commutative 99.3%

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

      3. associate-+l+ 99.6%

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

      4. +-commutative 99.6%

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

      5. +-commutative 99.6%

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

      6. sub-neg 99.6%

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

   6. Simplified 99.6%

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

4. Final simplification 99.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 1.55 \cdot 10^{-149}:\\ \;\;\;\;x + \left(-2 \cdot \left(x \cdot wj\right) + {wj}^{2} \cdot \left(1 + x \cdot 2.5\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{e^{wj} \cdot \left(wj + 1\right)} + \left(wj - \frac{wj}{wj + 1}\right)\\ \end{array} \]

Alternative 4: 85.9% accurate, 2.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 2 \cdot 10^{-172}:\\ \;\;\;\;\frac{x}{e^{wj} \cdot \left(wj + 1\right)}\\ \mathbf{else}:\\ \;\;\;\;wj + \frac{1}{wj + 1} \cdot \left(\frac{x}{e^{wj}} - wj\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (wj x)
 :precision binary64
 (if (<= x 2e-172)
   (/ x (* (exp wj) (+ wj 1.0)))
   (+ wj (* (/ 1.0 (+ wj 1.0)) (- (/ x (exp wj)) wj)))))

C

double code(double wj, double x) {
	double tmp;
	if (x <= 2e-172) {
		tmp = x / (exp(wj) * (wj + 1.0));
	} else {
		tmp = wj + ((1.0 / (wj + 1.0)) * ((x / exp(wj)) - wj));
	}
	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-172) then
        tmp = x / (exp(wj) * (wj + 1.0d0))
    else
        tmp = wj + ((1.0d0 / (wj + 1.0d0)) * ((x / exp(wj)) - wj))
    end if
    code = tmp
end function

Java

public static double code(double wj, double x) {
	double tmp;
	if (x <= 2e-172) {
		tmp = x / (Math.exp(wj) * (wj + 1.0));
	} else {
		tmp = wj + ((1.0 / (wj + 1.0)) * ((x / Math.exp(wj)) - wj));
	}
	return tmp;
}

Python

def code(wj, x):
	tmp = 0
	if x <= 2e-172:
		tmp = x / (math.exp(wj) * (wj + 1.0))
	else:
		tmp = wj + ((1.0 / (wj + 1.0)) * ((x / math.exp(wj)) - wj))
	return tmp

Julia

function code(wj, x)
	tmp = 0.0
	if (x <= 2e-172)
		tmp = Float64(x / Float64(exp(wj) * Float64(wj + 1.0)));
	else
		tmp = Float64(wj + Float64(Float64(1.0 / Float64(wj + 1.0)) * Float64(Float64(x / exp(wj)) - wj)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(wj, x)
	tmp = 0.0;
	if (x <= 2e-172)
		tmp = x / (exp(wj) * (wj + 1.0));
	else
		tmp = wj + ((1.0 / (wj + 1.0)) * ((x / exp(wj)) - wj));
	end
	tmp_2 = tmp;
end

Wolfram

code[wj_, x_] := If[LessEqual[x, 2e-172], N[(x / N[(N[Exp[wj], $MachinePrecision] * N[(wj + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(wj + N[(N[(1.0 / N[(wj + 1.0), $MachinePrecision]), $MachinePrecision] * N[(N[(x / N[Exp[wj], $MachinePrecision]), $MachinePrecision] - wj), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 2.0000000000000001e-172

   1. Initial program 72.8%

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

      1. div-sub 72.8%

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

      2. distribute-rgt1-in 72.8%

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

      3. times-frac 72.8%

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

      4. *-inverses 72.8%

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

      5. associate-*l/ 72.8%

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

      6. *-rgt-identity 72.8%

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

      7. distribute-rgt1-in 72.8%

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

      8. associate-/l/ 72.8%

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

      9. div-sub 72.8%

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

   3. Simplified 72.8%

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

   4. Taylor expanded in x around inf 88.9%

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

      1. +-commutative 88.9%

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

   6. Simplified 88.9%

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

   if 2.0000000000000001e-172 < x

   1. Initial program 95.4%

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

      1. div-sub 95.4%

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

      2. distribute-rgt1-in 95.4%

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

      3. times-frac 95.3%

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

      4. *-inverses 97.5%

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

      5. associate-*l/ 97.5%

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

      6. *-rgt-identity 97.5%

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

      7. distribute-rgt1-in 97.5%

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

      8. associate-/l/ 97.4%

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

      9. div-sub 97.4%

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

   3. Simplified 97.4%

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

      1. clear-num 97.2%

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

      2. associate-/r/ 97.5%

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

   5. Applied egg-rr 97.5%

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

4. Final simplification 92.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 2 \cdot 10^{-172}:\\ \;\;\;\;\frac{x}{e^{wj} \cdot \left(wj + 1\right)}\\ \mathbf{else}:\\ \;\;\;\;wj + \frac{1}{wj + 1} \cdot \left(\frac{x}{e^{wj}} - wj\right)\\ \end{array} \]

Alternative 5: 89.1% accurate, 2.7× speedup

Math

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

FPCore

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

C

double code(double wj, double x) {
	return (x / (exp(wj) * (wj + 1.0))) + (wj - (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))) + (wj - (wj / (wj + 1.0d0)))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 81.0%

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

   1. div-sub 81.0%

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

   2. distribute-rgt1-in 81.0%

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

   3. times-frac 81.0%

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

   4. *-inverses 81.8%

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

   5. associate-*l/ 81.8%

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

   6. *-rgt-identity 81.8%

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

   7. distribute-rgt1-in 81.8%

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

   8. associate-/l/ 81.8%

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

   9. div-sub 81.8%

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

3. Simplified 81.8%

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

4. Taylor expanded in x around 0 81.8%

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

   1. sub-neg 81.8%

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

   2. +-commutative 81.8%

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

   3. associate-+l+ 92.3%

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

   4. +-commutative 92.3%

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

   5. +-commutative 92.3%

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

   6. sub-neg 92.3%

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

6. Simplified 92.3%

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

7. Final simplification 92.3%

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

Alternative 6: 85.9% accurate, 2.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 2 \cdot 10^{-172}:\\ \;\;\;\;\frac{x}{e^{wj} \cdot \left(wj + 1\right)}\\ \mathbf{else}:\\ \;\;\;\;wj + \frac{\frac{x}{e^{wj}} - wj}{wj + 1}\\ \end{array} \end{array} \]

FPCore

(FPCore (wj x)
 :precision binary64
 (if (<= x 2e-172)
   (/ x (* (exp wj) (+ wj 1.0)))
   (+ wj (/ (- (/ x (exp wj)) wj) (+ wj 1.0)))))

C

double code(double wj, double x) {
	double tmp;
	if (x <= 2e-172) {
		tmp = x / (exp(wj) * (wj + 1.0));
	} else {
		tmp = wj + (((x / exp(wj)) - wj) / (wj + 1.0));
	}
	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-172) then
        tmp = x / (exp(wj) * (wj + 1.0d0))
    else
        tmp = wj + (((x / exp(wj)) - wj) / (wj + 1.0d0))
    end if
    code = tmp
end function

Java

public static double code(double wj, double x) {
	double tmp;
	if (x <= 2e-172) {
		tmp = x / (Math.exp(wj) * (wj + 1.0));
	} else {
		tmp = wj + (((x / Math.exp(wj)) - wj) / (wj + 1.0));
	}
	return tmp;
}

Python

def code(wj, x):
	tmp = 0
	if x <= 2e-172:
		tmp = x / (math.exp(wj) * (wj + 1.0))
	else:
		tmp = wj + (((x / math.exp(wj)) - wj) / (wj + 1.0))
	return tmp

Julia

function code(wj, x)
	tmp = 0.0
	if (x <= 2e-172)
		tmp = Float64(x / Float64(exp(wj) * Float64(wj + 1.0)));
	else
		tmp = Float64(wj + Float64(Float64(Float64(x / exp(wj)) - wj) / Float64(wj + 1.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(wj, x)
	tmp = 0.0;
	if (x <= 2e-172)
		tmp = x / (exp(wj) * (wj + 1.0));
	else
		tmp = wj + (((x / exp(wj)) - wj) / (wj + 1.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[wj_, x_] := If[LessEqual[x, 2e-172], N[(x / N[(N[Exp[wj], $MachinePrecision] * N[(wj + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(wj + N[(N[(N[(x / N[Exp[wj], $MachinePrecision]), $MachinePrecision] - wj), $MachinePrecision] / N[(wj + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 2.0000000000000001e-172

   1. Initial program 72.8%

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

      1. div-sub 72.8%

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

      2. distribute-rgt1-in 72.8%

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

      3. times-frac 72.8%

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

      4. *-inverses 72.8%

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

      5. associate-*l/ 72.8%

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

      6. *-rgt-identity 72.8%

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

      7. distribute-rgt1-in 72.8%

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

      8. associate-/l/ 72.8%

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

      9. div-sub 72.8%

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

   3. Simplified 72.8%

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

   4. Taylor expanded in x around inf 88.9%

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

      1. +-commutative 88.9%

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

   6. Simplified 88.9%

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

   if 2.0000000000000001e-172 < x

   1. Initial program 95.4%

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

      1. div-sub 95.4%

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

      2. distribute-rgt1-in 95.4%

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

      3. times-frac 95.3%

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

      4. *-inverses 97.5%

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

      5. associate-*l/ 97.5%

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

      6. *-rgt-identity 97.5%

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

      7. distribute-rgt1-in 97.5%

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

      8. associate-/l/ 97.4%

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

      9. div-sub 97.4%

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

   3. Simplified 97.4%

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

4. Final simplification 92.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 2 \cdot 10^{-172}:\\ \;\;\;\;\frac{x}{e^{wj} \cdot \left(wj + 1\right)}\\ \mathbf{else}:\\ \;\;\;\;wj + \frac{\frac{x}{e^{wj}} - wj}{wj + 1}\\ \end{array} \]

Alternative 7: 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 81.0%

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

   1. div-sub 81.0%

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

   2. distribute-rgt1-in 81.0%

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

   3. times-frac 81.0%

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

   4. *-inverses 81.8%

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

   5. associate-*l/ 81.8%

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

   6. *-rgt-identity 81.8%

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

   7. distribute-rgt1-in 81.8%

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

   8. associate-/l/ 81.8%

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

   9. div-sub 81.8%

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

3. Simplified 81.8%

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

4. Taylor expanded in x around inf 90.4%

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

   1. +-commutative 90.4%

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

6. Simplified 90.4%

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

7. Final simplification 90.4%

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

Alternative 8: 84.5% 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 81.0%

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

   1. div-sub 81.0%

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

   2. distribute-rgt1-in 81.0%

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

   3. times-frac 81.0%

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

   4. *-inverses 81.8%

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

   5. associate-*l/ 81.8%

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

   6. *-rgt-identity 81.8%

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

   7. distribute-rgt1-in 81.8%

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

   8. associate-/l/ 81.8%

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

   9. div-sub 81.8%

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

3. Simplified 81.8%

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

4. Taylor expanded in wj around 0 89.7%

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

   1. *-commutative 89.7%

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

6. Simplified 89.7%

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

7. Final simplification 89.7%

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

Alternative 9: 84.5% 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 81.0%

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

   1. div-sub 81.0%

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

   2. distribute-rgt1-in 81.0%

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

   3. times-frac 81.0%

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

   4. *-inverses 81.8%

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

   5. associate-*l/ 81.8%

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

   6. *-rgt-identity 81.8%

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

   7. distribute-rgt1-in 81.8%

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

   8. associate-/l/ 81.8%

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

   9. div-sub 81.8%

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

3. Simplified 81.8%

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

   1. clear-num 81.6%

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

   2. inv-pow 81.6%

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

5. Applied egg-rr 81.6%

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

   1. unpow-1 81.6%

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

7. Simplified 81.6%

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

8. Taylor expanded in x around inf 90.4%

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

   1. *-commutative 90.4%

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

   2. +-commutative 90.4%

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

10. Simplified 90.4%

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

11. Taylor expanded in wj around 0 89.8%

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

    1. *-commutative 89.8%

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

13. Simplified 89.8%

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

14. Final simplification 89.8%

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

Alternative 10: 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 81.0%

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

   1. div-sub 81.0%

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

   2. distribute-rgt1-in 81.0%

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

   3. times-frac 81.0%

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

   4. *-inverses 81.8%

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

   5. associate-*l/ 81.8%

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

   6. *-rgt-identity 81.8%

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

   7. distribute-rgt1-in 81.8%

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

   8. associate-/l/ 81.8%

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

   9. div-sub 81.8%

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

3. Simplified 81.8%

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

4. Taylor expanded in wj around inf 4.4%

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

5. Final simplification 4.4%

   \[\leadsto wj \]

Alternative 11: 84.0% 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 81.0%

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

   1. div-sub 81.0%

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

   2. distribute-rgt1-in 81.0%

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

   3. times-frac 81.0%

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

   4. *-inverses 81.8%

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

   5. associate-*l/ 81.8%

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

   6. *-rgt-identity 81.8%

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

   7. distribute-rgt1-in 81.8%

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

   8. associate-/l/ 81.8%

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

   9. div-sub 81.8%

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

3. Simplified 81.8%

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

4. Taylor expanded in wj around 0 89.2%

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

5. Final simplification 89.2%

   \[\leadsto x \]

Developer target: 78.9% 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 2023314 
(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))))))