Jmat.Real.lambertw, newton loop step

Percentage Accurate: 77.6% → 99.5%

Time: 8.1s

Alternatives: 10

Speedup: 313.0×

• could not determine a ground truth

  1. wj = -1.5922497274662162e+267
  2. x = -1.1598947854241795e+86

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: 77.6% 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: 99.5% accurate, 2.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;wj \leq -5 \cdot 10^{-9}:\\ \;\;\;\;wj + \frac{\frac{x}{e^{wj}} - wj}{wj + 1}\\ \mathbf{elif}\;wj \leq 5.4 \cdot 10^{-9}:\\ \;\;\;\;wj \cdot wj + \left(x - wj \cdot \left(x + x\right)\right)\\ \mathbf{else}:\\ \;\;\;\;wj + \frac{x \cdot e^{-wj} - wj}{wj + 1}\\ \end{array} \end{array} \]

FPCore

(FPCore (wj x)
 :precision binary64
 (if (<= wj -5e-9)
   (+ wj (/ (- (/ x (exp wj)) wj) (+ wj 1.0)))
   (if (<= wj 5.4e-9)
     (+ (* wj wj) (- x (* wj (+ x x))))
     (+ wj (/ (- (* x (exp (- wj))) wj) (+ wj 1.0))))))

C

double code(double wj, double x) {
	double tmp;
	if (wj <= -5e-9) {
		tmp = wj + (((x / exp(wj)) - wj) / (wj + 1.0));
	} else if (wj <= 5.4e-9) {
		tmp = (wj * wj) + (x - (wj * (x + x)));
	} 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 (wj <= (-5d-9)) then
        tmp = wj + (((x / exp(wj)) - wj) / (wj + 1.0d0))
    else if (wj <= 5.4d-9) then
        tmp = (wj * wj) + (x - (wj * (x + x)))
    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 (wj <= -5e-9) {
		tmp = wj + (((x / Math.exp(wj)) - wj) / (wj + 1.0));
	} else if (wj <= 5.4e-9) {
		tmp = (wj * wj) + (x - (wj * (x + x)));
	} else {
		tmp = wj + (((x * Math.exp(-wj)) - wj) / (wj + 1.0));
	}
	return tmp;
}

Python

def code(wj, x):
	tmp = 0
	if wj <= -5e-9:
		tmp = wj + (((x / math.exp(wj)) - wj) / (wj + 1.0))
	elif wj <= 5.4e-9:
		tmp = (wj * wj) + (x - (wj * (x + x)))
	else:
		tmp = wj + (((x * math.exp(-wj)) - wj) / (wj + 1.0))
	return tmp

Julia

function code(wj, x)
	tmp = 0.0
	if (wj <= -5e-9)
		tmp = Float64(wj + Float64(Float64(Float64(x / exp(wj)) - wj) / Float64(wj + 1.0)));
	elseif (wj <= 5.4e-9)
		tmp = Float64(Float64(wj * wj) + Float64(x - Float64(wj * Float64(x + x))));
	else
		tmp = Float64(wj + Float64(Float64(Float64(x * exp(Float64(-wj))) - wj) / Float64(wj + 1.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(wj, x)
	tmp = 0.0;
	if (wj <= -5e-9)
		tmp = wj + (((x / exp(wj)) - wj) / (wj + 1.0));
	elseif (wj <= 5.4e-9)
		tmp = (wj * wj) + (x - (wj * (x + x)));
	else
		tmp = wj + (((x * exp(-wj)) - wj) / (wj + 1.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[wj_, x_] := If[LessEqual[wj, -5e-9], N[(wj + N[(N[(N[(x / N[Exp[wj], $MachinePrecision]), $MachinePrecision] - wj), $MachinePrecision] / N[(wj + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[wj, 5.4e-9], N[(N[(wj * wj), $MachinePrecision] + N[(x - N[(wj * N[(x + x), $MachinePrecision]), $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 3 regimes

2. if wj < -5.0000000000000001e-9

   1. Initial program 65.6%

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

      1. sub-neg 65.6%

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

      2. div-sub 65.6%

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

      3. sub-neg 65.6%

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

      4. +-commutative 65.6%

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

      5. distribute-neg-in 65.6%

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

      6. remove-double-neg 65.6%

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

      7. sub-neg 65.6%

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

      8. div-sub 65.6%

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

      9. distribute-rgt1-in 94.2%

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

      10. associate-/l/ 94.4%

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

   3. Simplified 94.4%

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

   if -5.0000000000000001e-9 < wj < 5.4000000000000004e-9

   1. Initial program 74.5%

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

      1. sub-neg 74.5%

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

      2. div-sub 74.5%

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

      3. sub-neg 74.5%

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

      4. +-commutative 74.5%

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

      5. distribute-neg-in 74.5%

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

      6. remove-double-neg 74.5%

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

      7. sub-neg 74.5%

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

      8. div-sub 74.5%

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

      9. distribute-rgt1-in 74.5%

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

      10. associate-/l/ 74.5%

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

   3. Simplified 74.5%

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

   4. Taylor expanded in wj around 0 74.5%

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

   5. Taylor expanded in wj around 0 99.7%

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

   6. Taylor expanded in x around 0 99.7%

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

      1. unpow2 99.7%

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

   8. Simplified 99.7%

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

   if 5.4000000000000004e-9 < wj

   1. Initial program 71.2%

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

      1. sub-neg 71.2%

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

      2. div-sub 71.2%

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

      3. sub-neg 71.2%

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

      4. +-commutative 71.2%

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

      5. distribute-neg-in 71.2%

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

      6. remove-double-neg 71.2%

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

      7. sub-neg 71.2%

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

      8. div-sub 71.2%

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

      9. distribute-rgt1-in 71.2%

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

      10. associate-/l/ 71.0%

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

   3. Simplified 99.6%

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

      1. clear-num 99.8%

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

      2. associate-/r/ 99.8%

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

      3. rec-exp 100.0%

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

   5. Applied egg-rr 100.0%

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

4. Final simplification 99.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;wj \leq -5 \cdot 10^{-9}:\\ \;\;\;\;wj + \frac{\frac{x}{e^{wj}} - wj}{wj + 1}\\ \mathbf{elif}\;wj \leq 5.4 \cdot 10^{-9}:\\ \;\;\;\;wj \cdot wj + \left(x - wj \cdot \left(x + x\right)\right)\\ \mathbf{else}:\\ \;\;\;\;wj + \frac{x \cdot e^{-wj} - wj}{wj + 1}\\ \end{array} \]

Alternative 2: 98.7% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x \cdot -4 + x \cdot 1.5\\ t_1 := wj \cdot e^{wj}\\ \mathbf{if}\;wj + \frac{x - t_1}{e^{wj} + t_1} \leq 5 \cdot 10^{-16}:\\ \;\;\;\;{wj}^{3} \cdot \left(\left(\left(-1 - -2 \cdot t_0\right) - x \cdot -3\right) - x \cdot 0.6666666666666666\right) + \left(\left(1 - t_0\right) \cdot {wj}^{2} + \left(x + -2 \cdot \left(wj \cdot x\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;wj + \frac{\frac{x}{e^{wj}} - wj}{wj + 1}\\ \end{array} \end{array} \]

FPCore

(FPCore (wj x)
 :precision binary64
 (let* ((t_0 (+ (* x -4.0) (* x 1.5))) (t_1 (* wj (exp wj))))
   (if (<= (+ wj (/ (- x t_1) (+ (exp wj) t_1))) 5e-16)
     (+
      (*
       (pow wj 3.0)
       (- (- (- -1.0 (* -2.0 t_0)) (* x -3.0)) (* x 0.6666666666666666)))
      (+ (* (- 1.0 t_0) (pow wj 2.0)) (+ x (* -2.0 (* wj x)))))
     (+ wj (/ (- (/ x (exp wj)) wj) (+ wj 1.0))))))

C

double code(double wj, double x) {
	double t_0 = (x * -4.0) + (x * 1.5);
	double t_1 = wj * exp(wj);
	double tmp;
	if ((wj + ((x - t_1) / (exp(wj) + t_1))) <= 5e-16) {
		tmp = (pow(wj, 3.0) * (((-1.0 - (-2.0 * t_0)) - (x * -3.0)) - (x * 0.6666666666666666))) + (((1.0 - t_0) * pow(wj, 2.0)) + (x + (-2.0 * (wj * x))));
	} 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) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = (x * (-4.0d0)) + (x * 1.5d0)
    t_1 = wj * exp(wj)
    if ((wj + ((x - t_1) / (exp(wj) + t_1))) <= 5d-16) then
        tmp = ((wj ** 3.0d0) * ((((-1.0d0) - ((-2.0d0) * t_0)) - (x * (-3.0d0))) - (x * 0.6666666666666666d0))) + (((1.0d0 - t_0) * (wj ** 2.0d0)) + (x + ((-2.0d0) * (wj * x))))
    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 t_0 = (x * -4.0) + (x * 1.5);
	double t_1 = wj * Math.exp(wj);
	double tmp;
	if ((wj + ((x - t_1) / (Math.exp(wj) + t_1))) <= 5e-16) {
		tmp = (Math.pow(wj, 3.0) * (((-1.0 - (-2.0 * t_0)) - (x * -3.0)) - (x * 0.6666666666666666))) + (((1.0 - t_0) * Math.pow(wj, 2.0)) + (x + (-2.0 * (wj * x))));
	} else {
		tmp = wj + (((x / Math.exp(wj)) - wj) / (wj + 1.0));
	}
	return tmp;
}

Python

def code(wj, x):
	t_0 = (x * -4.0) + (x * 1.5)
	t_1 = wj * math.exp(wj)
	tmp = 0
	if (wj + ((x - t_1) / (math.exp(wj) + t_1))) <= 5e-16:
		tmp = (math.pow(wj, 3.0) * (((-1.0 - (-2.0 * t_0)) - (x * -3.0)) - (x * 0.6666666666666666))) + (((1.0 - t_0) * math.pow(wj, 2.0)) + (x + (-2.0 * (wj * x))))
	else:
		tmp = wj + (((x / math.exp(wj)) - wj) / (wj + 1.0))
	return tmp

Julia

function code(wj, x)
	t_0 = Float64(Float64(x * -4.0) + Float64(x * 1.5))
	t_1 = Float64(wj * exp(wj))
	tmp = 0.0
	if (Float64(wj + Float64(Float64(x - t_1) / Float64(exp(wj) + t_1))) <= 5e-16)
		tmp = Float64(Float64((wj ^ 3.0) * Float64(Float64(Float64(-1.0 - Float64(-2.0 * t_0)) - Float64(x * -3.0)) - Float64(x * 0.6666666666666666))) + Float64(Float64(Float64(1.0 - t_0) * (wj ^ 2.0)) + Float64(x + Float64(-2.0 * Float64(wj * x)))));
	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)
	t_0 = (x * -4.0) + (x * 1.5);
	t_1 = wj * exp(wj);
	tmp = 0.0;
	if ((wj + ((x - t_1) / (exp(wj) + t_1))) <= 5e-16)
		tmp = ((wj ^ 3.0) * (((-1.0 - (-2.0 * t_0)) - (x * -3.0)) - (x * 0.6666666666666666))) + (((1.0 - t_0) * (wj ^ 2.0)) + (x + (-2.0 * (wj * x))));
	else
		tmp = wj + (((x / exp(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]}, Block[{t$95$1 = N[(wj * N[Exp[wj], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(wj + N[(N[(x - t$95$1), $MachinePrecision] / N[(N[Exp[wj], $MachinePrecision] + t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 5e-16], N[(N[(N[Power[wj, 3.0], $MachinePrecision] * N[(N[(N[(-1.0 - N[(-2.0 * t$95$0), $MachinePrecision]), $MachinePrecision] - N[(x * -3.0), $MachinePrecision]), $MachinePrecision] - N[(x * 0.6666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(N[(1.0 - t$95$0), $MachinePrecision] * N[Power[wj, 2.0], $MachinePrecision]), $MachinePrecision] + N[(x + N[(-2.0 * N[(wj * x), $MachinePrecision]), $MachinePrecision]), $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 (-.f64 wj (/.f64 (-.f64 (*.f64 wj (exp.f64 wj)) x) (+.f64 (exp.f64 wj) (*.f64 wj (exp.f64 wj))))) < 5.0000000000000004e-16

   1. Initial program 68.3%

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

      1. sub-neg 68.3%

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

      2. div-sub 68.3%

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

      3. sub-neg 68.3%

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

      4. +-commutative 68.3%

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

      5. distribute-neg-in 68.3%

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

      6. remove-double-neg 68.3%

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

      7. sub-neg 68.3%

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

      8. div-sub 68.3%

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

      9. distribute-rgt1-in 68.3%

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

      10. associate-/l/ 68.3%

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

   3. Simplified 68.3%

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

   4. Taylor expanded in wj around 0 98.7%

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

   if 5.0000000000000004e-16 < (-.f64 wj (/.f64 (-.f64 (*.f64 wj (exp.f64 wj)) x) (+.f64 (exp.f64 wj) (*.f64 wj (exp.f64 wj)))))

   1. Initial program 93.3%

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

      1. sub-neg 93.3%

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

      2. div-sub 93.3%

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

      3. sub-neg 93.3%

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

      4. +-commutative 93.3%

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

      5. distribute-neg-in 93.3%

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

      6. remove-double-neg 93.3%

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

      7. sub-neg 93.3%

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

      8. div-sub 93.3%

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

      9. distribute-rgt1-in 96.7%

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

      10. associate-/l/ 96.6%

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

   3. Simplified 100.0%

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

4. Final simplification 99.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;wj + \frac{x - wj \cdot e^{wj}}{e^{wj} + wj \cdot e^{wj}} \leq 5 \cdot 10^{-16}:\\ \;\;\;\;{wj}^{3} \cdot \left(\left(\left(-1 - -2 \cdot \left(x \cdot -4 + x \cdot 1.5\right)\right) - x \cdot -3\right) - x \cdot 0.6666666666666666\right) + \left(\left(1 - \left(x \cdot -4 + x \cdot 1.5\right)\right) \cdot {wj}^{2} + \left(x + -2 \cdot \left(wj \cdot x\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;wj + \frac{\frac{x}{e^{wj}} - wj}{wj + 1}\\ \end{array} \]

Alternative 3: 97.6% accurate, 2.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;wj \leq 5.5 \cdot 10^{-9}:\\ \;\;\;\;\left(1 - \left(x \cdot -4 + x \cdot 1.5\right)\right) \cdot {wj}^{2} + \left(x + -2 \cdot \left(wj \cdot x\right)\right)\\ \mathbf{else}:\\ \;\;\;\;wj + \frac{x \cdot e^{-wj} - wj}{wj + 1}\\ \end{array} \end{array} \]

FPCore

(FPCore (wj x)
 :precision binary64
 (if (<= wj 5.5e-9)
   (+
    (* (- 1.0 (+ (* x -4.0) (* x 1.5))) (pow wj 2.0))
    (+ x (* -2.0 (* wj x))))
   (+ wj (/ (- (* x (exp (- wj))) wj) (+ wj 1.0)))))

C

double code(double wj, double x) {
	double tmp;
	if (wj <= 5.5e-9) {
		tmp = ((1.0 - ((x * -4.0) + (x * 1.5))) * pow(wj, 2.0)) + (x + (-2.0 * (wj * x)));
	} 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 (wj <= 5.5d-9) then
        tmp = ((1.0d0 - ((x * (-4.0d0)) + (x * 1.5d0))) * (wj ** 2.0d0)) + (x + ((-2.0d0) * (wj * x)))
    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 (wj <= 5.5e-9) {
		tmp = ((1.0 - ((x * -4.0) + (x * 1.5))) * Math.pow(wj, 2.0)) + (x + (-2.0 * (wj * x)));
	} else {
		tmp = wj + (((x * Math.exp(-wj)) - wj) / (wj + 1.0));
	}
	return tmp;
}

Python

def code(wj, x):
	tmp = 0
	if wj <= 5.5e-9:
		tmp = ((1.0 - ((x * -4.0) + (x * 1.5))) * math.pow(wj, 2.0)) + (x + (-2.0 * (wj * x)))
	else:
		tmp = wj + (((x * math.exp(-wj)) - wj) / (wj + 1.0))
	return tmp

Julia

function code(wj, x)
	tmp = 0.0
	if (wj <= 5.5e-9)
		tmp = Float64(Float64(Float64(1.0 - Float64(Float64(x * -4.0) + Float64(x * 1.5))) * (wj ^ 2.0)) + Float64(x + Float64(-2.0 * Float64(wj * x))));
	else
		tmp = Float64(wj + Float64(Float64(Float64(x * exp(Float64(-wj))) - wj) / Float64(wj + 1.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(wj, x)
	tmp = 0.0;
	if (wj <= 5.5e-9)
		tmp = ((1.0 - ((x * -4.0) + (x * 1.5))) * (wj ^ 2.0)) + (x + (-2.0 * (wj * x)));
	else
		tmp = wj + (((x * exp(-wj)) - wj) / (wj + 1.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[wj_, x_] := If[LessEqual[wj, 5.5e-9], N[(N[(N[(1.0 - N[(N[(x * -4.0), $MachinePrecision] + N[(x * 1.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Power[wj, 2.0], $MachinePrecision]), $MachinePrecision] + N[(x + N[(-2.0 * N[(wj * x), $MachinePrecision]), $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 wj < 5.4999999999999996e-9

   1. Initial program 74.2%

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

      1. sub-neg 74.2%

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

      2. div-sub 74.2%

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

      3. sub-neg 74.2%

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

      4. +-commutative 74.2%

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

      5. distribute-neg-in 74.2%

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

      6. remove-double-neg 74.2%

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

      7. sub-neg 74.2%

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

      8. div-sub 74.2%

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

      9. distribute-rgt1-in 75.0%

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

      10. associate-/l/ 75.0%

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

   3. Simplified 75.0%

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

   4. Taylor expanded in wj around 0 98.0%

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

   if 5.4999999999999996e-9 < wj

   1. Initial program 71.2%

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

      1. sub-neg 71.2%

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

      2. div-sub 71.2%

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

      3. sub-neg 71.2%

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

      4. +-commutative 71.2%

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

      5. distribute-neg-in 71.2%

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

      6. remove-double-neg 71.2%

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

      7. sub-neg 71.2%

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

      8. div-sub 71.2%

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

      9. distribute-rgt1-in 71.2%

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

      10. associate-/l/ 71.0%

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

   3. Simplified 99.6%

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

      1. clear-num 99.8%

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

      2. associate-/r/ 99.8%

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

      3. rec-exp 100.0%

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

   5. Applied egg-rr 100.0%

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

4. Final simplification 98.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;wj \leq 5.5 \cdot 10^{-9}:\\ \;\;\;\;\left(1 - \left(x \cdot -4 + x \cdot 1.5\right)\right) \cdot {wj}^{2} + \left(x + -2 \cdot \left(wj \cdot x\right)\right)\\ \mathbf{else}:\\ \;\;\;\;wj + \frac{x \cdot e^{-wj} - wj}{wj + 1}\\ \end{array} \]

Alternative 4: 99.5% accurate, 2.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;wj \leq -5.5 \cdot 10^{-9} \lor \neg \left(wj \leq 5.2 \cdot 10^{-9}\right):\\ \;\;\;\;wj + \frac{\frac{x}{e^{wj}} - wj}{wj + 1}\\ \mathbf{else}:\\ \;\;\;\;wj \cdot wj + \left(x - wj \cdot \left(x + x\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (wj x)
 :precision binary64
 (if (or (<= wj -5.5e-9) (not (<= wj 5.2e-9)))
   (+ wj (/ (- (/ x (exp wj)) wj) (+ wj 1.0)))
   (+ (* wj wj) (- x (* wj (+ x x))))))

C

double code(double wj, double x) {
	double tmp;
	if ((wj <= -5.5e-9) || !(wj <= 5.2e-9)) {
		tmp = wj + (((x / exp(wj)) - wj) / (wj + 1.0));
	} else {
		tmp = (wj * wj) + (x - (wj * (x + x)));
	}
	return tmp;
}

Fortran

real(8) function code(wj, x)
    real(8), intent (in) :: wj
    real(8), intent (in) :: x
    real(8) :: tmp
    if ((wj <= (-5.5d-9)) .or. (.not. (wj <= 5.2d-9))) then
        tmp = wj + (((x / exp(wj)) - wj) / (wj + 1.0d0))
    else
        tmp = (wj * wj) + (x - (wj * (x + x)))
    end if
    code = tmp
end function

Java

public static double code(double wj, double x) {
	double tmp;
	if ((wj <= -5.5e-9) || !(wj <= 5.2e-9)) {
		tmp = wj + (((x / Math.exp(wj)) - wj) / (wj + 1.0));
	} else {
		tmp = (wj * wj) + (x - (wj * (x + x)));
	}
	return tmp;
}

Python

def code(wj, x):
	tmp = 0
	if (wj <= -5.5e-9) or not (wj <= 5.2e-9):
		tmp = wj + (((x / math.exp(wj)) - wj) / (wj + 1.0))
	else:
		tmp = (wj * wj) + (x - (wj * (x + x)))
	return tmp

Julia

function code(wj, x)
	tmp = 0.0
	if ((wj <= -5.5e-9) || !(wj <= 5.2e-9))
		tmp = Float64(wj + Float64(Float64(Float64(x / exp(wj)) - wj) / Float64(wj + 1.0)));
	else
		tmp = Float64(Float64(wj * wj) + Float64(x - Float64(wj * Float64(x + x))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(wj, x)
	tmp = 0.0;
	if ((wj <= -5.5e-9) || ~((wj <= 5.2e-9)))
		tmp = wj + (((x / exp(wj)) - wj) / (wj + 1.0));
	else
		tmp = (wj * wj) + (x - (wj * (x + x)));
	end
	tmp_2 = tmp;
end

Wolfram

code[wj_, x_] := If[Or[LessEqual[wj, -5.5e-9], N[Not[LessEqual[wj, 5.2e-9]], $MachinePrecision]], N[(wj + N[(N[(N[(x / N[Exp[wj], $MachinePrecision]), $MachinePrecision] - wj), $MachinePrecision] / N[(wj + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(wj * wj), $MachinePrecision] + N[(x - N[(wj * N[(x + x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if wj < -5.4999999999999996e-9 or 5.2000000000000002e-9 < wj

   1. Initial program 68.4%

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

      1. sub-neg 68.4%

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

      2. div-sub 68.4%

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

      3. sub-neg 68.4%

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

      4. +-commutative 68.4%

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

      5. distribute-neg-in 68.4%

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

      6. remove-double-neg 68.4%

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

      7. sub-neg 68.4%

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

      8. div-sub 68.4%

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

      9. distribute-rgt1-in 82.7%

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

      10. associate-/l/ 82.7%

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

   3. Simplified 97.0%

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

   if -5.4999999999999996e-9 < wj < 5.2000000000000002e-9

   1. Initial program 74.5%

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

      1. sub-neg 74.5%

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

      2. div-sub 74.5%

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

      3. sub-neg 74.5%

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

      4. +-commutative 74.5%

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

      5. distribute-neg-in 74.5%

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

      6. remove-double-neg 74.5%

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

      7. sub-neg 74.5%

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

      8. div-sub 74.5%

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

      9. distribute-rgt1-in 74.5%

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

      10. associate-/l/ 74.5%

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

   3. Simplified 74.5%

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

   4. Taylor expanded in wj around 0 74.5%

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

   5. Taylor expanded in wj around 0 99.7%

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

   6. Taylor expanded in x around 0 99.7%

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

      1. unpow2 99.7%

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

   8. Simplified 99.7%

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

4. Final simplification 99.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;wj \leq -5.5 \cdot 10^{-9} \lor \neg \left(wj \leq 5.2 \cdot 10^{-9}\right):\\ \;\;\;\;wj + \frac{\frac{x}{e^{wj}} - wj}{wj + 1}\\ \mathbf{else}:\\ \;\;\;\;wj \cdot wj + \left(x - wj \cdot \left(x + x\right)\right)\\ \end{array} \]

Alternative 5: 97.1% accurate, 2.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;wj \leq -2.9 \cdot 10^{-6}:\\ \;\;\;\;\frac{x}{e^{wj} \cdot \left(wj + 1\right)}\\ \mathbf{else}:\\ \;\;\;\;wj \cdot wj + \left(x - wj \cdot \left(x + x\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (wj x)
 :precision binary64
 (if (<= wj -2.9e-6)
   (/ x (* (exp wj) (+ wj 1.0)))
   (+ (* wj wj) (- x (* wj (+ x x))))))

C

double code(double wj, double x) {
	double tmp;
	if (wj <= -2.9e-6) {
		tmp = x / (exp(wj) * (wj + 1.0));
	} else {
		tmp = (wj * wj) + (x - (wj * (x + x)));
	}
	return tmp;
}

Fortran

real(8) function code(wj, x)
    real(8), intent (in) :: wj
    real(8), intent (in) :: x
    real(8) :: tmp
    if (wj <= (-2.9d-6)) then
        tmp = x / (exp(wj) * (wj + 1.0d0))
    else
        tmp = (wj * wj) + (x - (wj * (x + x)))
    end if
    code = tmp
end function

Java

public static double code(double wj, double x) {
	double tmp;
	if (wj <= -2.9e-6) {
		tmp = x / (Math.exp(wj) * (wj + 1.0));
	} else {
		tmp = (wj * wj) + (x - (wj * (x + x)));
	}
	return tmp;
}

Python

def code(wj, x):
	tmp = 0
	if wj <= -2.9e-6:
		tmp = x / (math.exp(wj) * (wj + 1.0))
	else:
		tmp = (wj * wj) + (x - (wj * (x + x)))
	return tmp

Julia

function code(wj, x)
	tmp = 0.0
	if (wj <= -2.9e-6)
		tmp = Float64(x / Float64(exp(wj) * Float64(wj + 1.0)));
	else
		tmp = Float64(Float64(wj * wj) + Float64(x - Float64(wj * Float64(x + x))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(wj, x)
	tmp = 0.0;
	if (wj <= -2.9e-6)
		tmp = x / (exp(wj) * (wj + 1.0));
	else
		tmp = (wj * wj) + (x - (wj * (x + x)));
	end
	tmp_2 = tmp;
end

Wolfram

code[wj_, x_] := If[LessEqual[wj, -2.9e-6], N[(x / N[(N[Exp[wj], $MachinePrecision] * N[(wj + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(wj * wj), $MachinePrecision] + N[(x - N[(wj * N[(x + x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if wj < -2.9000000000000002e-6

   1. Initial program 65.9%

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

      1. sub-neg 65.9%

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

      2. div-sub 65.9%

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

      3. sub-neg 65.9%

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

      4. +-commutative 65.9%

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

      5. distribute-neg-in 65.9%

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

      6. remove-double-neg 65.9%

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

      7. sub-neg 65.9%

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

      8. div-sub 65.9%

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

      9. distribute-rgt1-in 99.2%

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

      10. associate-/l/ 99.5%

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

   3. Simplified 99.5%

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

   4. Taylor expanded in x around inf 99.2%

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

   if -2.9000000000000002e-6 < wj

   1. Initial program 74.3%

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

      1. sub-neg 74.3%

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

      2. div-sub 74.3%

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

      3. sub-neg 74.3%

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

      4. +-commutative 74.3%

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

      5. distribute-neg-in 74.3%

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

      6. remove-double-neg 74.3%

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

      7. sub-neg 74.3%

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

      8. div-sub 74.3%

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

      9. distribute-rgt1-in 74.3%

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

      10. associate-/l/ 74.3%

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

   3. Simplified 75.1%

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

   4. Taylor expanded in wj around 0 74.2%

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

   5. Taylor expanded in wj around 0 97.5%

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

   6. Taylor expanded in x around 0 97.5%

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

      1. unpow2 97.5%

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

   8. Simplified 97.5%

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

4. Final simplification 97.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;wj \leq -2.9 \cdot 10^{-6}:\\ \;\;\;\;\frac{x}{e^{wj} \cdot \left(wj + 1\right)}\\ \mathbf{else}:\\ \;\;\;\;wj \cdot wj + \left(x - wj \cdot \left(x + x\right)\right)\\ \end{array} \]

Alternative 6: 97.1% accurate, 2.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;wj \leq -6.4 \cdot 10^{-6}:\\ \;\;\;\;\frac{\frac{x}{e^{wj}}}{wj + 1}\\ \mathbf{else}:\\ \;\;\;\;wj \cdot wj + \left(x - wj \cdot \left(x + x\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (wj x)
 :precision binary64
 (if (<= wj -6.4e-6)
   (/ (/ x (exp wj)) (+ wj 1.0))
   (+ (* wj wj) (- x (* wj (+ x x))))))

C

double code(double wj, double x) {
	double tmp;
	if (wj <= -6.4e-6) {
		tmp = (x / exp(wj)) / (wj + 1.0);
	} else {
		tmp = (wj * wj) + (x - (wj * (x + x)));
	}
	return tmp;
}

Fortran

real(8) function code(wj, x)
    real(8), intent (in) :: wj
    real(8), intent (in) :: x
    real(8) :: tmp
    if (wj <= (-6.4d-6)) then
        tmp = (x / exp(wj)) / (wj + 1.0d0)
    else
        tmp = (wj * wj) + (x - (wj * (x + x)))
    end if
    code = tmp
end function

Java

public static double code(double wj, double x) {
	double tmp;
	if (wj <= -6.4e-6) {
		tmp = (x / Math.exp(wj)) / (wj + 1.0);
	} else {
		tmp = (wj * wj) + (x - (wj * (x + x)));
	}
	return tmp;
}

Python

def code(wj, x):
	tmp = 0
	if wj <= -6.4e-6:
		tmp = (x / math.exp(wj)) / (wj + 1.0)
	else:
		tmp = (wj * wj) + (x - (wj * (x + x)))
	return tmp

Julia

function code(wj, x)
	tmp = 0.0
	if (wj <= -6.4e-6)
		tmp = Float64(Float64(x / exp(wj)) / Float64(wj + 1.0));
	else
		tmp = Float64(Float64(wj * wj) + Float64(x - Float64(wj * Float64(x + x))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(wj, x)
	tmp = 0.0;
	if (wj <= -6.4e-6)
		tmp = (x / exp(wj)) / (wj + 1.0);
	else
		tmp = (wj * wj) + (x - (wj * (x + x)));
	end
	tmp_2 = tmp;
end

Wolfram

code[wj_, x_] := If[LessEqual[wj, -6.4e-6], N[(N[(x / N[Exp[wj], $MachinePrecision]), $MachinePrecision] / N[(wj + 1.0), $MachinePrecision]), $MachinePrecision], N[(N[(wj * wj), $MachinePrecision] + N[(x - N[(wj * N[(x + x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if wj < -6.3999999999999997e-6

   1. Initial program 65.9%

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

      1. sub-neg 65.9%

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

      2. div-sub 65.9%

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

      3. sub-neg 65.9%

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

      4. +-commutative 65.9%

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

      5. distribute-neg-in 65.9%

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

      6. remove-double-neg 65.9%

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

      7. sub-neg 65.9%

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

      8. div-sub 65.9%

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

      9. distribute-rgt1-in 99.2%

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

      10. associate-/l/ 99.5%

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

   3. Simplified 99.5%

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

      1. clear-num 99.5%

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

      2. associate-/r/ 99.2%

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

      3. rec-exp 99.2%

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

   5. Applied egg-rr 99.2%

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

   6. Taylor expanded in x around inf 99.2%

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

      1. associate-/l* 99.5%

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

      2. +-commutative 99.5%

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

   8. Simplified 99.5%

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

   9. Taylor expanded in x around 0 99.2%

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

       1. *-commutative 99.2%

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

       2. exp-neg 99.2%

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

       3. associate-*r/ 99.5%

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

       4. metadata-eval 99.5%

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

       5. distribute-rgt-neg-in 99.5%

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

       6. *-commutative 99.5%

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

       7. mul-1-neg 99.5%

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

       8. remove-double-neg 99.5%

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

       9. +-commutative 99.5%

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

   11. Simplified 99.5%

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

   if -6.3999999999999997e-6 < wj

   1. Initial program 74.3%

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

      1. sub-neg 74.3%

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

      2. div-sub 74.3%

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

      3. sub-neg 74.3%

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

      4. +-commutative 74.3%

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

      5. distribute-neg-in 74.3%

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

      6. remove-double-neg 74.3%

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

      7. sub-neg 74.3%

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

      8. div-sub 74.3%

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

      9. distribute-rgt1-in 74.3%

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

      10. associate-/l/ 74.3%

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

   3. Simplified 75.1%

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

   4. Taylor expanded in wj around 0 74.2%

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

   5. Taylor expanded in wj around 0 97.5%

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

   6. Taylor expanded in x around 0 97.5%

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

      1. unpow2 97.5%

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

   8. Simplified 97.5%

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

4. Final simplification 97.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;wj \leq -6.4 \cdot 10^{-6}:\\ \;\;\;\;\frac{\frac{x}{e^{wj}}}{wj + 1}\\ \mathbf{else}:\\ \;\;\;\;wj \cdot wj + \left(x - wj \cdot \left(x + x\right)\right)\\ \end{array} \]

Alternative 7: 95.8% accurate, 28.5× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 74.1%

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

   1. sub-neg 74.1%

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

   2. div-sub 74.1%

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

   3. sub-neg 74.1%

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

   4. +-commutative 74.1%

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

   5. distribute-neg-in 74.1%

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

   6. remove-double-neg 74.1%

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

   7. sub-neg 74.1%

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

   8. div-sub 74.1%

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

   9. distribute-rgt1-in 74.9%

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

   10. associate-/l/ 74.9%

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

3. Simplified 75.7%

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

4. Taylor expanded in wj around 0 73.1%

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

5. Taylor expanded in wj around 0 95.9%

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

6. Taylor expanded in x around 0 95.8%

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

   1. unpow2 95.8%

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

8. Simplified 95.8%

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

9. Final simplification 95.8%

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

Alternative 8: 95.3% accurate, 62.6× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 74.1%

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

   1. sub-neg 74.1%

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

   2. div-sub 74.1%

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

   3. sub-neg 74.1%

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

   4. +-commutative 74.1%

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

   5. distribute-neg-in 74.1%

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

   6. remove-double-neg 74.1%

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

   7. sub-neg 74.1%

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

   8. div-sub 74.1%

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

   9. distribute-rgt1-in 74.9%

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

   10. associate-/l/ 74.9%

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

3. Simplified 75.7%

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

4. Taylor expanded in wj around 0 73.1%

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

5. Taylor expanded in wj around 0 95.9%

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

6. Taylor expanded in x around 0 95.8%

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

   1. unpow2 95.8%

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

8. Simplified 95.8%

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

9. Taylor expanded in wj around 0 95.3%

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

10. Final simplification 95.3%

    \[\leadsto x + wj \cdot wj \]

Alternative 9: 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 74.1%

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

   1. sub-neg 74.1%

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

   2. div-sub 74.1%

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

   3. sub-neg 74.1%

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

   4. +-commutative 74.1%

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

   5. distribute-neg-in 74.1%

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

   6. remove-double-neg 74.1%

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

   7. sub-neg 74.1%

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

   8. div-sub 74.1%

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

   9. distribute-rgt1-in 74.9%

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

   10. associate-/l/ 74.9%

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

3. Simplified 75.7%

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

4. Taylor expanded in wj around inf 4.5%

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

5. Final simplification 4.5%

   \[\leadsto wj \]

Alternative 10: 83.9% 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 74.1%

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

   1. sub-neg 74.1%

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

   2. div-sub 74.1%

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

   3. sub-neg 74.1%

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

   4. +-commutative 74.1%

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

   5. distribute-neg-in 74.1%

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

   6. remove-double-neg 74.1%

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

   7. sub-neg 74.1%

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

   8. div-sub 74.1%

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

   9. distribute-rgt1-in 74.9%

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

   10. associate-/l/ 74.9%

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

3. Simplified 75.7%

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

4. Taylor expanded in wj around 0 83.2%

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

5. Final simplification 83.2%

   \[\leadsto x \]

Developer target: 78.5% 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 2023171 
(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))))))