Jmat.Real.lambertw, newton loop step

Percentage Accurate: 77.9% → 97.7%

Time: 15.5s

Alternatives: 16

Speedup: 313.0×

• could not determine a ground truth

  1. wj = -5.925334117508635e+219
  2. x = -1.8051098466275412e-286

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: 77.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := wj \cdot e^{wj}\\ wj - \frac{t\_0 - x}{e^{wj} + t\_0} \end{array} \end{array} \]

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 97.7% accurate, 2.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x \cdot -4 + x \cdot 1.5\\ \mathbf{if}\;wj \leq 5.4 \cdot 10^{-9}:\\ \;\;\;\;x + wj \cdot \left(wj \cdot \left(\left(1 + wj \cdot \left(-1 - \left(x \cdot -3 + \left(-2 \cdot t\_0 + x \cdot 0.6666666666666666\right)\right)\right)\right) - t\_0\right) - x \cdot 2\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(\frac{wj + \frac{wj}{-1 - wj}}{x} + \frac{e^{-wj}}{wj + 1}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (wj x)
 :precision binary64
 (let* ((t_0 (+ (* x -4.0) (* x 1.5))))
   (if (<= wj 5.4e-9)
     (+
      x
      (*
       wj
       (-
        (*
         wj
         (-
          (+
           1.0
           (*
            wj
            (- -1.0 (+ (* x -3.0) (+ (* -2.0 t_0) (* x 0.6666666666666666))))))
          t_0))
        (* x 2.0))))
     (* x (+ (/ (+ wj (/ wj (- -1.0 wj))) x) (/ (exp (- wj)) (+ wj 1.0)))))))

C

double code(double wj, double x) {
	double t_0 = (x * -4.0) + (x * 1.5);
	double tmp;
	if (wj <= 5.4e-9) {
		tmp = x + (wj * ((wj * ((1.0 + (wj * (-1.0 - ((x * -3.0) + ((-2.0 * t_0) + (x * 0.6666666666666666)))))) - t_0)) - (x * 2.0)));
	} else {
		tmp = x * (((wj + (wj / (-1.0 - wj))) / x) + (exp(-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 (wj <= 5.4d-9) then
        tmp = x + (wj * ((wj * ((1.0d0 + (wj * ((-1.0d0) - ((x * (-3.0d0)) + (((-2.0d0) * t_0) + (x * 0.6666666666666666d0)))))) - t_0)) - (x * 2.0d0)))
    else
        tmp = x * (((wj + (wj / ((-1.0d0) - wj))) / x) + (exp(-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 (wj <= 5.4e-9) {
		tmp = x + (wj * ((wj * ((1.0 + (wj * (-1.0 - ((x * -3.0) + ((-2.0 * t_0) + (x * 0.6666666666666666)))))) - t_0)) - (x * 2.0)));
	} else {
		tmp = x * (((wj + (wj / (-1.0 - wj))) / x) + (Math.exp(-wj) / (wj + 1.0)));
	}
	return tmp;
}

Python

def code(wj, x):
	t_0 = (x * -4.0) + (x * 1.5)
	tmp = 0
	if wj <= 5.4e-9:
		tmp = x + (wj * ((wj * ((1.0 + (wj * (-1.0 - ((x * -3.0) + ((-2.0 * t_0) + (x * 0.6666666666666666)))))) - t_0)) - (x * 2.0)))
	else:
		tmp = x * (((wj + (wj / (-1.0 - wj))) / x) + (math.exp(-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 (wj <= 5.4e-9)
		tmp = Float64(x + Float64(wj * Float64(Float64(wj * Float64(Float64(1.0 + Float64(wj * Float64(-1.0 - Float64(Float64(x * -3.0) + Float64(Float64(-2.0 * t_0) + Float64(x * 0.6666666666666666)))))) - t_0)) - Float64(x * 2.0))));
	else
		tmp = Float64(x * Float64(Float64(Float64(wj + Float64(wj / Float64(-1.0 - wj))) / x) + Float64(exp(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 (wj <= 5.4e-9)
		tmp = x + (wj * ((wj * ((1.0 + (wj * (-1.0 - ((x * -3.0) + ((-2.0 * t_0) + (x * 0.6666666666666666)))))) - t_0)) - (x * 2.0)));
	else
		tmp = x * (((wj + (wj / (-1.0 - wj))) / x) + (exp(-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[wj, 5.4e-9], N[(x + N[(wj * N[(N[(wj * N[(N[(1.0 + N[(wj * 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]), $MachinePrecision] - t$95$0), $MachinePrecision]), $MachinePrecision] - N[(x * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x * N[(N[(N[(wj + N[(wj / N[(-1.0 - wj), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision] + N[(N[Exp[(-wj)], $MachinePrecision] / N[(wj + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if wj < 5.4000000000000004e-9

   1. Initial program 80.5%

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

      1. distribute-rgt1-in 80.5%

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

      2. associate-/l/ 80.6%

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

      3. div-sub 80.6%

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

      4. associate-/l* 80.6%

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

      5. *-inverses 80.6%

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

      6. *-rgt-identity 80.6%

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

   3. Simplified 80.6%

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

   5. Taylor expanded in wj around 0 98.9%

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

   if 5.4000000000000004e-9 < wj

   1. Initial program 66.3%

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

      1. distribute-rgt1-in 66.3%

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

      2. associate-/l/ 66.5%

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

      3. div-sub 66.5%

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

      4. associate-/l* 66.5%

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

      5. *-inverses 99.8%

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

      6. *-rgt-identity 99.8%

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

   3. Simplified 99.8%

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

   5. Taylor expanded in x around -inf 99.7%

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

      1. associate-*r* 99.7%

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

      2. neg-mul-1 99.7%

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

      3. mul-1-neg 99.7%

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

      4. +-commutative 99.7%

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

      5. associate-/r* 100.0%

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

      6. exp-neg 100.0%

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

      7. +-commutative 100.0%

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

   7. Simplified 100.0%

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

4. Final simplification 98.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;wj \leq 5.4 \cdot 10^{-9}:\\ \;\;\;\;x + wj \cdot \left(wj \cdot \left(\left(1 + wj \cdot \left(-1 - \left(x \cdot -3 + \left(-2 \cdot \left(x \cdot -4 + x \cdot 1.5\right) + x \cdot 0.6666666666666666\right)\right)\right)\right) - \left(x \cdot -4 + x \cdot 1.5\right)\right) - x \cdot 2\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(\frac{wj + \frac{wj}{-1 - wj}}{x} + \frac{e^{-wj}}{wj + 1}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 97.8% accurate, 2.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x \cdot -4 + x \cdot 1.5\\ \mathbf{if}\;wj \leq 5.6 \cdot 10^{-6}:\\ \;\;\;\;x + wj \cdot \left(wj \cdot \left(\left(1 + wj \cdot \left(-1 - \left(x \cdot -3 + \left(-2 \cdot t\_0 + x \cdot 0.6666666666666666\right)\right)\right)\right) - t\_0\right) - x \cdot 2\right)\\ \mathbf{else}:\\ \;\;\;\;wj - \frac{wj - \frac{x}{e^{wj}}}{wj + 1}\\ \end{array} \end{array} \]

FPCore

(FPCore (wj x)
 :precision binary64
 (let* ((t_0 (+ (* x -4.0) (* x 1.5))))
   (if (<= wj 5.6e-6)
     (+
      x
      (*
       wj
       (-
        (*
         wj
         (-
          (+
           1.0
           (*
            wj
            (- -1.0 (+ (* x -3.0) (+ (* -2.0 t_0) (* x 0.6666666666666666))))))
          t_0))
        (* x 2.0))))
     (- wj (/ (- wj (/ x (exp wj))) (+ wj 1.0))))))

C

double code(double wj, double x) {
	double t_0 = (x * -4.0) + (x * 1.5);
	double tmp;
	if (wj <= 5.6e-6) {
		tmp = x + (wj * ((wj * ((1.0 + (wj * (-1.0 - ((x * -3.0) + ((-2.0 * t_0) + (x * 0.6666666666666666)))))) - t_0)) - (x * 2.0)));
	} else {
		tmp = wj - ((wj - (x / exp(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 (wj <= 5.6d-6) then
        tmp = x + (wj * ((wj * ((1.0d0 + (wj * ((-1.0d0) - ((x * (-3.0d0)) + (((-2.0d0) * t_0) + (x * 0.6666666666666666d0)))))) - t_0)) - (x * 2.0d0)))
    else
        tmp = wj - ((wj - (x / exp(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 (wj <= 5.6e-6) {
		tmp = x + (wj * ((wj * ((1.0 + (wj * (-1.0 - ((x * -3.0) + ((-2.0 * t_0) + (x * 0.6666666666666666)))))) - t_0)) - (x * 2.0)));
	} else {
		tmp = wj - ((wj - (x / Math.exp(wj))) / (wj + 1.0));
	}
	return tmp;
}

Python

def code(wj, x):
	t_0 = (x * -4.0) + (x * 1.5)
	tmp = 0
	if wj <= 5.6e-6:
		tmp = x + (wj * ((wj * ((1.0 + (wj * (-1.0 - ((x * -3.0) + ((-2.0 * t_0) + (x * 0.6666666666666666)))))) - t_0)) - (x * 2.0)))
	else:
		tmp = wj - ((wj - (x / math.exp(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 (wj <= 5.6e-6)
		tmp = Float64(x + Float64(wj * Float64(Float64(wj * Float64(Float64(1.0 + Float64(wj * Float64(-1.0 - Float64(Float64(x * -3.0) + Float64(Float64(-2.0 * t_0) + Float64(x * 0.6666666666666666)))))) - t_0)) - Float64(x * 2.0))));
	else
		tmp = Float64(wj - Float64(Float64(wj - Float64(x / exp(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 (wj <= 5.6e-6)
		tmp = x + (wj * ((wj * ((1.0 + (wj * (-1.0 - ((x * -3.0) + ((-2.0 * t_0) + (x * 0.6666666666666666)))))) - t_0)) - (x * 2.0)));
	else
		tmp = wj - ((wj - (x / exp(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[wj, 5.6e-6], N[(x + N[(wj * N[(N[(wj * N[(N[(1.0 + N[(wj * 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]), $MachinePrecision] - t$95$0), $MachinePrecision]), $MachinePrecision] - N[(x * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(wj - N[(N[(wj - N[(x / N[Exp[wj], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(wj + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if wj < 5.59999999999999975e-6

   1. Initial program 80.6%

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

      1. distribute-rgt1-in 80.6%

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

      2. associate-/l/ 80.7%

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

      3. div-sub 80.7%

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

      4. associate-/l* 80.7%

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

      5. *-inverses 80.7%

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

      6. *-rgt-identity 80.7%

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

   3. Simplified 80.7%

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

   5. Taylor expanded in wj around 0 98.9%

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

   if 5.59999999999999975e-6 < wj

   1. Initial program 62.1%

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

      1. distribute-rgt1-in 62.1%

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

      2. associate-/l/ 62.5%

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

      3. div-sub 62.5%

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

      4. associate-/l* 62.5%

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

      5. *-inverses 100.0%

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

      6. *-rgt-identity 100.0%

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

   3. Simplified 100.0%

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

4. Final simplification 98.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;wj \leq 5.6 \cdot 10^{-6}:\\ \;\;\;\;x + wj \cdot \left(wj \cdot \left(\left(1 + wj \cdot \left(-1 - \left(x \cdot -3 + \left(-2 \cdot \left(x \cdot -4 + x \cdot 1.5\right) + x \cdot 0.6666666666666666\right)\right)\right)\right) - \left(x \cdot -4 + x \cdot 1.5\right)\right) - x \cdot 2\right)\\ \mathbf{else}:\\ \;\;\;\;wj - \frac{wj - \frac{x}{e^{wj}}}{wj + 1}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 96.8% accurate, 6.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x \cdot -4 + x \cdot 1.5\\ \mathbf{if}\;wj \leq 0.01:\\ \;\;\;\;x + wj \cdot \left(wj \cdot \left(\left(1 + wj \cdot \left(-1 - \left(x \cdot -3 + \left(-2 \cdot t\_0 + x \cdot 0.6666666666666666\right)\right)\right)\right) - t\_0\right) - x \cdot 2\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(\frac{1}{wj + 1} + \left(\frac{wj + \frac{wj}{-1 - wj}}{x} - wj \cdot \left(\frac{1.5}{wj} + -0.5\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (wj x)
 :precision binary64
 (let* ((t_0 (+ (* x -4.0) (* x 1.5))))
   (if (<= wj 0.01)
     (+
      x
      (*
       wj
       (-
        (*
         wj
         (-
          (+
           1.0
           (*
            wj
            (- -1.0 (+ (* x -3.0) (+ (* -2.0 t_0) (* x 0.6666666666666666))))))
          t_0))
        (* x 2.0))))
     (*
      x
      (+
       (/ 1.0 (+ wj 1.0))
       (- (/ (+ wj (/ wj (- -1.0 wj))) x) (* wj (+ (/ 1.5 wj) -0.5))))))))

C

double code(double wj, double x) {
	double t_0 = (x * -4.0) + (x * 1.5);
	double tmp;
	if (wj <= 0.01) {
		tmp = x + (wj * ((wj * ((1.0 + (wj * (-1.0 - ((x * -3.0) + ((-2.0 * t_0) + (x * 0.6666666666666666)))))) - t_0)) - (x * 2.0)));
	} else {
		tmp = x * ((1.0 / (wj + 1.0)) + (((wj + (wj / (-1.0 - wj))) / x) - (wj * ((1.5 / wj) + -0.5))));
	}
	return tmp;
}

Fortran

real(8) function code(wj, x)
    real(8), intent (in) :: wj
    real(8), intent (in) :: x
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (x * (-4.0d0)) + (x * 1.5d0)
    if (wj <= 0.01d0) then
        tmp = x + (wj * ((wj * ((1.0d0 + (wj * ((-1.0d0) - ((x * (-3.0d0)) + (((-2.0d0) * t_0) + (x * 0.6666666666666666d0)))))) - t_0)) - (x * 2.0d0)))
    else
        tmp = x * ((1.0d0 / (wj + 1.0d0)) + (((wj + (wj / ((-1.0d0) - wj))) / x) - (wj * ((1.5d0 / wj) + (-0.5d0)))))
    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 (wj <= 0.01) {
		tmp = x + (wj * ((wj * ((1.0 + (wj * (-1.0 - ((x * -3.0) + ((-2.0 * t_0) + (x * 0.6666666666666666)))))) - t_0)) - (x * 2.0)));
	} else {
		tmp = x * ((1.0 / (wj + 1.0)) + (((wj + (wj / (-1.0 - wj))) / x) - (wj * ((1.5 / wj) + -0.5))));
	}
	return tmp;
}

Python

def code(wj, x):
	t_0 = (x * -4.0) + (x * 1.5)
	tmp = 0
	if wj <= 0.01:
		tmp = x + (wj * ((wj * ((1.0 + (wj * (-1.0 - ((x * -3.0) + ((-2.0 * t_0) + (x * 0.6666666666666666)))))) - t_0)) - (x * 2.0)))
	else:
		tmp = x * ((1.0 / (wj + 1.0)) + (((wj + (wj / (-1.0 - wj))) / x) - (wj * ((1.5 / wj) + -0.5))))
	return tmp

Julia

function code(wj, x)
	t_0 = Float64(Float64(x * -4.0) + Float64(x * 1.5))
	tmp = 0.0
	if (wj <= 0.01)
		tmp = Float64(x + Float64(wj * Float64(Float64(wj * Float64(Float64(1.0 + Float64(wj * Float64(-1.0 - Float64(Float64(x * -3.0) + Float64(Float64(-2.0 * t_0) + Float64(x * 0.6666666666666666)))))) - t_0)) - Float64(x * 2.0))));
	else
		tmp = Float64(x * Float64(Float64(1.0 / Float64(wj + 1.0)) + Float64(Float64(Float64(wj + Float64(wj / Float64(-1.0 - wj))) / x) - Float64(wj * Float64(Float64(1.5 / wj) + -0.5)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(wj, x)
	t_0 = (x * -4.0) + (x * 1.5);
	tmp = 0.0;
	if (wj <= 0.01)
		tmp = x + (wj * ((wj * ((1.0 + (wj * (-1.0 - ((x * -3.0) + ((-2.0 * t_0) + (x * 0.6666666666666666)))))) - t_0)) - (x * 2.0)));
	else
		tmp = x * ((1.0 / (wj + 1.0)) + (((wj + (wj / (-1.0 - wj))) / x) - (wj * ((1.5 / wj) + -0.5))));
	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[wj, 0.01], N[(x + N[(wj * N[(N[(wj * N[(N[(1.0 + N[(wj * 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]), $MachinePrecision] - t$95$0), $MachinePrecision]), $MachinePrecision] - N[(x * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x * N[(N[(1.0 / N[(wj + 1.0), $MachinePrecision]), $MachinePrecision] + N[(N[(N[(wj + N[(wj / N[(-1.0 - wj), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision] - N[(wj * N[(N[(1.5 / wj), $MachinePrecision] + -0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if wj < 0.0100000000000000002

   1. Initial program 80.7%

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

      1. distribute-rgt1-in 80.7%

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

      2. associate-/l/ 80.8%

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

      3. div-sub 80.8%

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

      4. associate-/l* 80.8%

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

      5. *-inverses 80.8%

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

      6. *-rgt-identity 80.8%

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

   3. Simplified 80.8%

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

   5. Taylor expanded in wj around 0 98.7%

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

   if 0.0100000000000000002 < wj

   1. Initial program 49.7%

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

      1. distribute-rgt1-in 49.7%

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

      2. associate-/l/ 50.0%

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

      3. div-sub 50.0%

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

      4. associate-/l* 50.0%

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

      5. *-inverses 100.0%

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

      6. *-rgt-identity 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in wj around 0 85.9%

      \[\leadsto wj - \frac{wj - \color{blue}{\left(x + wj \cdot \left(-1 \cdot \left(wj \cdot \left(-1 \cdot x + 0.5 \cdot x\right)\right) - x\right)\right)}}{wj + 1} \]
   6. Step-by-step derivation

      1. associate-*r* 85.9%

         \[\leadsto wj - \frac{wj - \left(x + wj \cdot \left(\color{blue}{\left(-1 \cdot wj\right) \cdot \left(-1 \cdot x + 0.5 \cdot x\right)} - x\right)\right)}{wj + 1} \]

      2. neg-mul-1 85.9%

         \[\leadsto wj - \frac{wj - \left(x + wj \cdot \left(\color{blue}{\left(-wj\right)} \cdot \left(-1 \cdot x + 0.5 \cdot x\right) - x\right)\right)}{wj + 1} \]

      3. distribute-rgt-out 85.9%

         \[\leadsto wj - \frac{wj - \left(x + wj \cdot \left(\left(-wj\right) \cdot \color{blue}{\left(x \cdot \left(-1 + 0.5\right)\right)} - x\right)\right)}{wj + 1} \]

      4. metadata-eval 85.9%

         \[\leadsto wj - \frac{wj - \left(x + wj \cdot \left(\left(-wj\right) \cdot \left(x \cdot \color{blue}{-0.5}\right) - x\right)\right)}{wj + 1} \]

   7. Simplified 85.9%

      \[\leadsto wj - \frac{wj - \color{blue}{\left(x + wj \cdot \left(\left(-wj\right) \cdot \left(x \cdot -0.5\right) - x\right)\right)}}{wj + 1} \]

   8. Taylor expanded in x around -inf 85.9%

      \[\leadsto \color{blue}{-1 \cdot \left(x \cdot \left(\left(-1 \cdot \frac{wj - \frac{wj}{1 + wj}}{x} + \frac{wj \cdot \left(1 + -0.5 \cdot wj\right)}{1 + wj}\right) - \frac{1}{1 + wj}\right)\right)} \]

   9. Taylor expanded in wj around inf 86.0%

      \[\leadsto -1 \cdot \left(x \cdot \left(\left(-1 \cdot \frac{wj - \frac{wj}{1 + wj}}{x} + \color{blue}{wj \cdot \left(1.5 \cdot \frac{1}{wj} - 0.5\right)}\right) - \frac{1}{1 + wj}\right)\right) \]
   10. Step-by-step derivation

       1. sub-neg 86.0%

          \[\leadsto -1 \cdot \left(x \cdot \left(\left(-1 \cdot \frac{wj - \frac{wj}{1 + wj}}{x} + wj \cdot \color{blue}{\left(1.5 \cdot \frac{1}{wj} + \left(-0.5\right)\right)}\right) - \frac{1}{1 + wj}\right)\right) \]

       2. associate-*r/ 86.0%

          \[\leadsto -1 \cdot \left(x \cdot \left(\left(-1 \cdot \frac{wj - \frac{wj}{1 + wj}}{x} + wj \cdot \left(\color{blue}{\frac{1.5 \cdot 1}{wj}} + \left(-0.5\right)\right)\right) - \frac{1}{1 + wj}\right)\right) \]

       3. metadata-eval 86.0%

          \[\leadsto -1 \cdot \left(x \cdot \left(\left(-1 \cdot \frac{wj - \frac{wj}{1 + wj}}{x} + wj \cdot \left(\frac{\color{blue}{1.5}}{wj} + \left(-0.5\right)\right)\right) - \frac{1}{1 + wj}\right)\right) \]

       4. metadata-eval 86.0%

          \[\leadsto -1 \cdot \left(x \cdot \left(\left(-1 \cdot \frac{wj - \frac{wj}{1 + wj}}{x} + wj \cdot \left(\frac{1.5}{wj} + \color{blue}{-0.5}\right)\right) - \frac{1}{1 + wj}\right)\right) \]

   11. Simplified 86.0%

       \[\leadsto -1 \cdot \left(x \cdot \left(\left(-1 \cdot \frac{wj - \frac{wj}{1 + wj}}{x} + \color{blue}{wj \cdot \left(\frac{1.5}{wj} + -0.5\right)}\right) - \frac{1}{1 + wj}\right)\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 98.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;wj \leq 0.01:\\ \;\;\;\;x + wj \cdot \left(wj \cdot \left(\left(1 + wj \cdot \left(-1 - \left(x \cdot -3 + \left(-2 \cdot \left(x \cdot -4 + x \cdot 1.5\right) + x \cdot 0.6666666666666666\right)\right)\right)\right) - \left(x \cdot -4 + x \cdot 1.5\right)\right) - x \cdot 2\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(\frac{1}{wj + 1} + \left(\frac{wj + \frac{wj}{-1 - wj}}{x} - wj \cdot \left(\frac{1.5}{wj} + -0.5\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 96.9% accurate, 9.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;wj \leq 0.042:\\ \;\;\;\;x + wj \cdot \left(x \cdot -2 + wj \cdot \left(x \cdot \left(\frac{1}{x} + 2.5\right) - wj \cdot \left(x \cdot \left(2.6666666666666665 + \frac{1}{x}\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(\frac{1}{wj + 1} + \left(\frac{wj + \frac{wj}{-1 - wj}}{x} - wj \cdot \left(\frac{1.5}{wj} + -0.5\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (wj x)
 :precision binary64
 (if (<= wj 0.042)
   (+
    x
    (*
     wj
     (+
      (* x -2.0)
      (*
       wj
       (-
        (* x (+ (/ 1.0 x) 2.5))
        (* wj (* x (+ 2.6666666666666665 (/ 1.0 x)))))))))
   (*
    x
    (+
     (/ 1.0 (+ wj 1.0))
     (- (/ (+ wj (/ wj (- -1.0 wj))) x) (* wj (+ (/ 1.5 wj) -0.5)))))))

C

double code(double wj, double x) {
	double tmp;
	if (wj <= 0.042) {
		tmp = x + (wj * ((x * -2.0) + (wj * ((x * ((1.0 / x) + 2.5)) - (wj * (x * (2.6666666666666665 + (1.0 / x))))))));
	} else {
		tmp = x * ((1.0 / (wj + 1.0)) + (((wj + (wj / (-1.0 - wj))) / x) - (wj * ((1.5 / wj) + -0.5))));
	}
	return tmp;
}

Fortran

real(8) function code(wj, x)
    real(8), intent (in) :: wj
    real(8), intent (in) :: x
    real(8) :: tmp
    if (wj <= 0.042d0) then
        tmp = x + (wj * ((x * (-2.0d0)) + (wj * ((x * ((1.0d0 / x) + 2.5d0)) - (wj * (x * (2.6666666666666665d0 + (1.0d0 / x))))))))
    else
        tmp = x * ((1.0d0 / (wj + 1.0d0)) + (((wj + (wj / ((-1.0d0) - wj))) / x) - (wj * ((1.5d0 / wj) + (-0.5d0)))))
    end if
    code = tmp
end function

Java

public static double code(double wj, double x) {
	double tmp;
	if (wj <= 0.042) {
		tmp = x + (wj * ((x * -2.0) + (wj * ((x * ((1.0 / x) + 2.5)) - (wj * (x * (2.6666666666666665 + (1.0 / x))))))));
	} else {
		tmp = x * ((1.0 / (wj + 1.0)) + (((wj + (wj / (-1.0 - wj))) / x) - (wj * ((1.5 / wj) + -0.5))));
	}
	return tmp;
}

Python

def code(wj, x):
	tmp = 0
	if wj <= 0.042:
		tmp = x + (wj * ((x * -2.0) + (wj * ((x * ((1.0 / x) + 2.5)) - (wj * (x * (2.6666666666666665 + (1.0 / x))))))))
	else:
		tmp = x * ((1.0 / (wj + 1.0)) + (((wj + (wj / (-1.0 - wj))) / x) - (wj * ((1.5 / wj) + -0.5))))
	return tmp

Julia

function code(wj, x)
	tmp = 0.0
	if (wj <= 0.042)
		tmp = Float64(x + Float64(wj * Float64(Float64(x * -2.0) + Float64(wj * Float64(Float64(x * Float64(Float64(1.0 / x) + 2.5)) - Float64(wj * Float64(x * Float64(2.6666666666666665 + Float64(1.0 / x)))))))));
	else
		tmp = Float64(x * Float64(Float64(1.0 / Float64(wj + 1.0)) + Float64(Float64(Float64(wj + Float64(wj / Float64(-1.0 - wj))) / x) - Float64(wj * Float64(Float64(1.5 / wj) + -0.5)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(wj, x)
	tmp = 0.0;
	if (wj <= 0.042)
		tmp = x + (wj * ((x * -2.0) + (wj * ((x * ((1.0 / x) + 2.5)) - (wj * (x * (2.6666666666666665 + (1.0 / x))))))));
	else
		tmp = x * ((1.0 / (wj + 1.0)) + (((wj + (wj / (-1.0 - wj))) / x) - (wj * ((1.5 / wj) + -0.5))));
	end
	tmp_2 = tmp;
end

Wolfram

code[wj_, x_] := If[LessEqual[wj, 0.042], N[(x + N[(wj * N[(N[(x * -2.0), $MachinePrecision] + N[(wj * N[(N[(x * N[(N[(1.0 / x), $MachinePrecision] + 2.5), $MachinePrecision]), $MachinePrecision] - N[(wj * N[(x * N[(2.6666666666666665 + N[(1.0 / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x * N[(N[(1.0 / N[(wj + 1.0), $MachinePrecision]), $MachinePrecision] + N[(N[(N[(wj + N[(wj / N[(-1.0 - wj), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision] - N[(wj * N[(N[(1.5 / wj), $MachinePrecision] + -0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if wj < 0.0420000000000000026

   1. Initial program 80.7%

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

      1. distribute-rgt1-in 80.7%

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

      2. associate-/l/ 80.8%

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

      3. div-sub 80.8%

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

      4. associate-/l* 80.8%

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

      5. *-inverses 80.8%

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

      6. *-rgt-identity 80.8%

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

   3. Simplified 80.8%

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

   5. Taylor expanded in x around -inf 90.7%

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

      1. associate-*r* 90.7%

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

      2. neg-mul-1 90.7%

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

      3. mul-1-neg 90.7%

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

      4. +-commutative 90.7%

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

      5. associate-/r* 90.7%

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

      6. exp-neg 90.7%

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

      7. +-commutative 90.7%

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

   7. Simplified 90.7%

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

   8. Taylor expanded in wj around 0 98.7%

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

   if 0.0420000000000000026 < wj

   1. Initial program 49.7%

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

      1. distribute-rgt1-in 49.7%

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

      2. associate-/l/ 50.0%

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

      3. div-sub 50.0%

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

      4. associate-/l* 50.0%

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

      5. *-inverses 100.0%

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

      6. *-rgt-identity 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in wj around 0 85.9%

      \[\leadsto wj - \frac{wj - \color{blue}{\left(x + wj \cdot \left(-1 \cdot \left(wj \cdot \left(-1 \cdot x + 0.5 \cdot x\right)\right) - x\right)\right)}}{wj + 1} \]
   6. Step-by-step derivation

      1. associate-*r* 85.9%

         \[\leadsto wj - \frac{wj - \left(x + wj \cdot \left(\color{blue}{\left(-1 \cdot wj\right) \cdot \left(-1 \cdot x + 0.5 \cdot x\right)} - x\right)\right)}{wj + 1} \]

      2. neg-mul-1 85.9%

         \[\leadsto wj - \frac{wj - \left(x + wj \cdot \left(\color{blue}{\left(-wj\right)} \cdot \left(-1 \cdot x + 0.5 \cdot x\right) - x\right)\right)}{wj + 1} \]

      3. distribute-rgt-out 85.9%

         \[\leadsto wj - \frac{wj - \left(x + wj \cdot \left(\left(-wj\right) \cdot \color{blue}{\left(x \cdot \left(-1 + 0.5\right)\right)} - x\right)\right)}{wj + 1} \]

      4. metadata-eval 85.9%

         \[\leadsto wj - \frac{wj - \left(x + wj \cdot \left(\left(-wj\right) \cdot \left(x \cdot \color{blue}{-0.5}\right) - x\right)\right)}{wj + 1} \]

   7. Simplified 85.9%

      \[\leadsto wj - \frac{wj - \color{blue}{\left(x + wj \cdot \left(\left(-wj\right) \cdot \left(x \cdot -0.5\right) - x\right)\right)}}{wj + 1} \]

   8. Taylor expanded in x around -inf 85.9%

      \[\leadsto \color{blue}{-1 \cdot \left(x \cdot \left(\left(-1 \cdot \frac{wj - \frac{wj}{1 + wj}}{x} + \frac{wj \cdot \left(1 + -0.5 \cdot wj\right)}{1 + wj}\right) - \frac{1}{1 + wj}\right)\right)} \]

   9. Taylor expanded in wj around inf 86.0%

      \[\leadsto -1 \cdot \left(x \cdot \left(\left(-1 \cdot \frac{wj - \frac{wj}{1 + wj}}{x} + \color{blue}{wj \cdot \left(1.5 \cdot \frac{1}{wj} - 0.5\right)}\right) - \frac{1}{1 + wj}\right)\right) \]
   10. Step-by-step derivation

       1. sub-neg 86.0%

          \[\leadsto -1 \cdot \left(x \cdot \left(\left(-1 \cdot \frac{wj - \frac{wj}{1 + wj}}{x} + wj \cdot \color{blue}{\left(1.5 \cdot \frac{1}{wj} + \left(-0.5\right)\right)}\right) - \frac{1}{1 + wj}\right)\right) \]

       2. associate-*r/ 86.0%

          \[\leadsto -1 \cdot \left(x \cdot \left(\left(-1 \cdot \frac{wj - \frac{wj}{1 + wj}}{x} + wj \cdot \left(\color{blue}{\frac{1.5 \cdot 1}{wj}} + \left(-0.5\right)\right)\right) - \frac{1}{1 + wj}\right)\right) \]

       3. metadata-eval 86.0%

          \[\leadsto -1 \cdot \left(x \cdot \left(\left(-1 \cdot \frac{wj - \frac{wj}{1 + wj}}{x} + wj \cdot \left(\frac{\color{blue}{1.5}}{wj} + \left(-0.5\right)\right)\right) - \frac{1}{1 + wj}\right)\right) \]

       4. metadata-eval 86.0%

          \[\leadsto -1 \cdot \left(x \cdot \left(\left(-1 \cdot \frac{wj - \frac{wj}{1 + wj}}{x} + wj \cdot \left(\frac{1.5}{wj} + \color{blue}{-0.5}\right)\right) - \frac{1}{1 + wj}\right)\right) \]

   11. Simplified 86.0%

       \[\leadsto -1 \cdot \left(x \cdot \left(\left(-1 \cdot \frac{wj - \frac{wj}{1 + wj}}{x} + \color{blue}{wj \cdot \left(\frac{1.5}{wj} + -0.5\right)}\right) - \frac{1}{1 + wj}\right)\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 98.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;wj \leq 0.042:\\ \;\;\;\;x + wj \cdot \left(x \cdot -2 + wj \cdot \left(x \cdot \left(\frac{1}{x} + 2.5\right) - wj \cdot \left(x \cdot \left(2.6666666666666665 + \frac{1}{x}\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(\frac{1}{wj + 1} + \left(\frac{wj + \frac{wj}{-1 - wj}}{x} - wj \cdot \left(\frac{1.5}{wj} + -0.5\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 96.9% accurate, 12.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;wj \leq 5.4 \cdot 10^{-9}:\\ \;\;\;\;x + wj \cdot \left(wj \cdot \left(\left(1 - wj\right) - \left(x \cdot -4 + x \cdot 1.5\right)\right) - x \cdot 2\right)\\ \mathbf{else}:\\ \;\;\;\;wj - \frac{wj + \left(wj \cdot \left(x - x \cdot \left(wj \cdot 0.5\right)\right) - x\right)}{wj + 1}\\ \end{array} \end{array} \]

FPCore

(FPCore (wj x)
 :precision binary64
 (if (<= wj 5.4e-9)
   (+ x (* wj (- (* wj (- (- 1.0 wj) (+ (* x -4.0) (* x 1.5)))) (* x 2.0))))
   (- wj (/ (+ wj (- (* wj (- x (* x (* wj 0.5)))) x)) (+ wj 1.0)))))

C

double code(double wj, double x) {
	double tmp;
	if (wj <= 5.4e-9) {
		tmp = x + (wj * ((wj * ((1.0 - wj) - ((x * -4.0) + (x * 1.5)))) - (x * 2.0)));
	} else {
		tmp = wj - ((wj + ((wj * (x - (x * (wj * 0.5)))) - x)) / (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 <= 5.4d-9) then
        tmp = x + (wj * ((wj * ((1.0d0 - wj) - ((x * (-4.0d0)) + (x * 1.5d0)))) - (x * 2.0d0)))
    else
        tmp = wj - ((wj + ((wj * (x - (x * (wj * 0.5d0)))) - x)) / (wj + 1.0d0))
    end if
    code = tmp
end function

Java

public static double code(double wj, double x) {
	double tmp;
	if (wj <= 5.4e-9) {
		tmp = x + (wj * ((wj * ((1.0 - wj) - ((x * -4.0) + (x * 1.5)))) - (x * 2.0)));
	} else {
		tmp = wj - ((wj + ((wj * (x - (x * (wj * 0.5)))) - x)) / (wj + 1.0));
	}
	return tmp;
}

Python

def code(wj, x):
	tmp = 0
	if wj <= 5.4e-9:
		tmp = x + (wj * ((wj * ((1.0 - wj) - ((x * -4.0) + (x * 1.5)))) - (x * 2.0)))
	else:
		tmp = wj - ((wj + ((wj * (x - (x * (wj * 0.5)))) - x)) / (wj + 1.0))
	return tmp

Julia

function code(wj, x)
	tmp = 0.0
	if (wj <= 5.4e-9)
		tmp = Float64(x + Float64(wj * Float64(Float64(wj * Float64(Float64(1.0 - wj) - Float64(Float64(x * -4.0) + Float64(x * 1.5)))) - Float64(x * 2.0))));
	else
		tmp = Float64(wj - Float64(Float64(wj + Float64(Float64(wj * Float64(x - Float64(x * Float64(wj * 0.5)))) - x)) / Float64(wj + 1.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(wj, x)
	tmp = 0.0;
	if (wj <= 5.4e-9)
		tmp = x + (wj * ((wj * ((1.0 - wj) - ((x * -4.0) + (x * 1.5)))) - (x * 2.0)));
	else
		tmp = wj - ((wj + ((wj * (x - (x * (wj * 0.5)))) - x)) / (wj + 1.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[wj_, x_] := If[LessEqual[wj, 5.4e-9], N[(x + N[(wj * N[(N[(wj * N[(N[(1.0 - wj), $MachinePrecision] - N[(N[(x * -4.0), $MachinePrecision] + N[(x * 1.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(x * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(wj - N[(N[(wj + N[(N[(wj * N[(x - N[(x * N[(wj * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - x), $MachinePrecision]), $MachinePrecision] / N[(wj + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if wj < 5.4000000000000004e-9

   1. Initial program 80.5%

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

      1. distribute-rgt1-in 80.5%

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

      2. associate-/l/ 80.6%

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

      3. div-sub 80.6%

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

      4. associate-/l* 80.6%

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

      5. *-inverses 80.6%

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

      6. *-rgt-identity 80.6%

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

   3. Simplified 80.6%

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

   5. Taylor expanded in wj around 0 98.9%

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

   6. Taylor expanded in x around 0 98.8%

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

   if 5.4000000000000004e-9 < wj

   1. Initial program 66.3%

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

      1. distribute-rgt1-in 66.3%

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

      2. associate-/l/ 66.5%

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

      3. div-sub 66.5%

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

      4. associate-/l* 66.5%

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

      5. *-inverses 99.8%

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

      6. *-rgt-identity 99.8%

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

   3. Simplified 99.8%

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

   5. Taylor expanded in wj around 0 83.7%

      \[\leadsto wj - \frac{wj - \color{blue}{\left(x + wj \cdot \left(-1 \cdot \left(wj \cdot \left(-1 \cdot x + 0.5 \cdot x\right)\right) - x\right)\right)}}{wj + 1} \]
   6. Step-by-step derivation

      1. associate-*r* 83.7%

         \[\leadsto wj - \frac{wj - \left(x + wj \cdot \left(\color{blue}{\left(-1 \cdot wj\right) \cdot \left(-1 \cdot x + 0.5 \cdot x\right)} - x\right)\right)}{wj + 1} \]

      2. neg-mul-1 83.7%

         \[\leadsto wj - \frac{wj - \left(x + wj \cdot \left(\color{blue}{\left(-wj\right)} \cdot \left(-1 \cdot x + 0.5 \cdot x\right) - x\right)\right)}{wj + 1} \]

      3. distribute-rgt-out 83.7%

         \[\leadsto wj - \frac{wj - \left(x + wj \cdot \left(\left(-wj\right) \cdot \color{blue}{\left(x \cdot \left(-1 + 0.5\right)\right)} - x\right)\right)}{wj + 1} \]

      4. metadata-eval 83.7%

         \[\leadsto wj - \frac{wj - \left(x + wj \cdot \left(\left(-wj\right) \cdot \left(x \cdot \color{blue}{-0.5}\right) - x\right)\right)}{wj + 1} \]

   7. Simplified 83.7%

      \[\leadsto wj - \frac{wj - \color{blue}{\left(x + wj \cdot \left(\left(-wj\right) \cdot \left(x \cdot -0.5\right) - x\right)\right)}}{wj + 1} \]

   8. Taylor expanded in wj around 0 83.7%

      \[\leadsto wj - \frac{wj - \left(x + \color{blue}{wj \cdot \left(-1 \cdot x + 0.5 \cdot \left(wj \cdot x\right)\right)}\right)}{wj + 1} \]
   9. Step-by-step derivation

      1. neg-mul-1 83.7%

         \[\leadsto wj - \frac{wj - \left(x + wj \cdot \left(\color{blue}{\left(-x\right)} + 0.5 \cdot \left(wj \cdot x\right)\right)\right)}{wj + 1} \]

      2. +-commutative 83.7%

         \[\leadsto wj - \frac{wj - \left(x + wj \cdot \color{blue}{\left(0.5 \cdot \left(wj \cdot x\right) + \left(-x\right)\right)}\right)}{wj + 1} \]

      3. sub-neg 83.7%

         \[\leadsto wj - \frac{wj - \left(x + wj \cdot \color{blue}{\left(0.5 \cdot \left(wj \cdot x\right) - x\right)}\right)}{wj + 1} \]

      4. associate-*r* 83.7%

         \[\leadsto wj - \frac{wj - \left(x + wj \cdot \left(\color{blue}{\left(0.5 \cdot wj\right) \cdot x} - x\right)\right)}{wj + 1} \]

      5. *-commutative 83.7%

         \[\leadsto wj - \frac{wj - \left(x + wj \cdot \left(\color{blue}{\left(wj \cdot 0.5\right)} \cdot x - x\right)\right)}{wj + 1} \]

   10. Simplified 83.7%

       \[\leadsto wj - \frac{wj - \left(x + \color{blue}{wj \cdot \left(\left(wj \cdot 0.5\right) \cdot x - x\right)}\right)}{wj + 1} \]
3. Recombined 2 regimes into one program.

4. Final simplification 98.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;wj \leq 5.4 \cdot 10^{-9}:\\ \;\;\;\;x + wj \cdot \left(wj \cdot \left(\left(1 - wj\right) - \left(x \cdot -4 + x \cdot 1.5\right)\right) - x \cdot 2\right)\\ \mathbf{else}:\\ \;\;\;\;wj - \frac{wj + \left(wj \cdot \left(x - x \cdot \left(wj \cdot 0.5\right)\right) - x\right)}{wj + 1}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 96.9% accurate, 13.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;wj \leq 5.4 \cdot 10^{-9}:\\ \;\;\;\;x + wj \cdot \left(x \cdot -2 + wj \cdot \left(1 - wj\right)\right)\\ \mathbf{else}:\\ \;\;\;\;wj - \frac{wj + \left(wj \cdot \left(x - x \cdot \left(wj \cdot 0.5\right)\right) - x\right)}{wj + 1}\\ \end{array} \end{array} \]

FPCore

(FPCore (wj x)
 :precision binary64
 (if (<= wj 5.4e-9)
   (+ x (* wj (+ (* x -2.0) (* wj (- 1.0 wj)))))
   (- wj (/ (+ wj (- (* wj (- x (* x (* wj 0.5)))) x)) (+ wj 1.0)))))

C

double code(double wj, double x) {
	double tmp;
	if (wj <= 5.4e-9) {
		tmp = x + (wj * ((x * -2.0) + (wj * (1.0 - wj))));
	} else {
		tmp = wj - ((wj + ((wj * (x - (x * (wj * 0.5)))) - x)) / (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 <= 5.4d-9) then
        tmp = x + (wj * ((x * (-2.0d0)) + (wj * (1.0d0 - wj))))
    else
        tmp = wj - ((wj + ((wj * (x - (x * (wj * 0.5d0)))) - x)) / (wj + 1.0d0))
    end if
    code = tmp
end function

Java

public static double code(double wj, double x) {
	double tmp;
	if (wj <= 5.4e-9) {
		tmp = x + (wj * ((x * -2.0) + (wj * (1.0 - wj))));
	} else {
		tmp = wj - ((wj + ((wj * (x - (x * (wj * 0.5)))) - x)) / (wj + 1.0));
	}
	return tmp;
}

Python

def code(wj, x):
	tmp = 0
	if wj <= 5.4e-9:
		tmp = x + (wj * ((x * -2.0) + (wj * (1.0 - wj))))
	else:
		tmp = wj - ((wj + ((wj * (x - (x * (wj * 0.5)))) - x)) / (wj + 1.0))
	return tmp

Julia

function code(wj, x)
	tmp = 0.0
	if (wj <= 5.4e-9)
		tmp = Float64(x + Float64(wj * Float64(Float64(x * -2.0) + Float64(wj * Float64(1.0 - wj)))));
	else
		tmp = Float64(wj - Float64(Float64(wj + Float64(Float64(wj * Float64(x - Float64(x * Float64(wj * 0.5)))) - x)) / Float64(wj + 1.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(wj, x)
	tmp = 0.0;
	if (wj <= 5.4e-9)
		tmp = x + (wj * ((x * -2.0) + (wj * (1.0 - wj))));
	else
		tmp = wj - ((wj + ((wj * (x - (x * (wj * 0.5)))) - x)) / (wj + 1.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[wj_, x_] := If[LessEqual[wj, 5.4e-9], N[(x + N[(wj * N[(N[(x * -2.0), $MachinePrecision] + N[(wj * N[(1.0 - wj), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(wj - N[(N[(wj + N[(N[(wj * N[(x - N[(x * N[(wj * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - x), $MachinePrecision]), $MachinePrecision] / N[(wj + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if wj < 5.4000000000000004e-9

   1. Initial program 80.5%

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

      1. distribute-rgt1-in 80.5%

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

      2. associate-/l/ 80.6%

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

      3. div-sub 80.6%

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

      4. associate-/l* 80.6%

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

      5. *-inverses 80.6%

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

      6. *-rgt-identity 80.6%

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

   3. Simplified 80.6%

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

   5. Taylor expanded in x around -inf 90.6%

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

      1. associate-*r* 90.6%

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

      2. neg-mul-1 90.6%

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

      3. mul-1-neg 90.6%

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

      4. +-commutative 90.6%

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

      5. associate-/r* 90.6%

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

      6. exp-neg 90.6%

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

      7. +-commutative 90.6%

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

   7. Simplified 90.6%

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

   8. Taylor expanded in wj around 0 98.8%

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

   9. Taylor expanded in x around 0 98.8%

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

       1. neg-mul-1 98.8%

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

       2. sub-neg 98.8%

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

   11. Simplified 98.8%

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

   if 5.4000000000000004e-9 < wj

   1. Initial program 66.3%

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

      1. distribute-rgt1-in 66.3%

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

      2. associate-/l/ 66.5%

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

      3. div-sub 66.5%

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

      4. associate-/l* 66.5%

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

      5. *-inverses 99.8%

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

      6. *-rgt-identity 99.8%

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

   3. Simplified 99.8%

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

   5. Taylor expanded in wj around 0 83.7%

      \[\leadsto wj - \frac{wj - \color{blue}{\left(x + wj \cdot \left(-1 \cdot \left(wj \cdot \left(-1 \cdot x + 0.5 \cdot x\right)\right) - x\right)\right)}}{wj + 1} \]
   6. Step-by-step derivation

      1. associate-*r* 83.7%

         \[\leadsto wj - \frac{wj - \left(x + wj \cdot \left(\color{blue}{\left(-1 \cdot wj\right) \cdot \left(-1 \cdot x + 0.5 \cdot x\right)} - x\right)\right)}{wj + 1} \]

      2. neg-mul-1 83.7%

         \[\leadsto wj - \frac{wj - \left(x + wj \cdot \left(\color{blue}{\left(-wj\right)} \cdot \left(-1 \cdot x + 0.5 \cdot x\right) - x\right)\right)}{wj + 1} \]

      3. distribute-rgt-out 83.7%

         \[\leadsto wj - \frac{wj - \left(x + wj \cdot \left(\left(-wj\right) \cdot \color{blue}{\left(x \cdot \left(-1 + 0.5\right)\right)} - x\right)\right)}{wj + 1} \]

      4. metadata-eval 83.7%

         \[\leadsto wj - \frac{wj - \left(x + wj \cdot \left(\left(-wj\right) \cdot \left(x \cdot \color{blue}{-0.5}\right) - x\right)\right)}{wj + 1} \]

   7. Simplified 83.7%

      \[\leadsto wj - \frac{wj - \color{blue}{\left(x + wj \cdot \left(\left(-wj\right) \cdot \left(x \cdot -0.5\right) - x\right)\right)}}{wj + 1} \]

   8. Taylor expanded in wj around 0 83.7%

      \[\leadsto wj - \frac{wj - \left(x + \color{blue}{wj \cdot \left(-1 \cdot x + 0.5 \cdot \left(wj \cdot x\right)\right)}\right)}{wj + 1} \]
   9. Step-by-step derivation

      1. neg-mul-1 83.7%

         \[\leadsto wj - \frac{wj - \left(x + wj \cdot \left(\color{blue}{\left(-x\right)} + 0.5 \cdot \left(wj \cdot x\right)\right)\right)}{wj + 1} \]

      2. +-commutative 83.7%

         \[\leadsto wj - \frac{wj - \left(x + wj \cdot \color{blue}{\left(0.5 \cdot \left(wj \cdot x\right) + \left(-x\right)\right)}\right)}{wj + 1} \]

      3. sub-neg 83.7%

         \[\leadsto wj - \frac{wj - \left(x + wj \cdot \color{blue}{\left(0.5 \cdot \left(wj \cdot x\right) - x\right)}\right)}{wj + 1} \]

      4. associate-*r* 83.7%

         \[\leadsto wj - \frac{wj - \left(x + wj \cdot \left(\color{blue}{\left(0.5 \cdot wj\right) \cdot x} - x\right)\right)}{wj + 1} \]

      5. *-commutative 83.7%

         \[\leadsto wj - \frac{wj - \left(x + wj \cdot \left(\color{blue}{\left(wj \cdot 0.5\right)} \cdot x - x\right)\right)}{wj + 1} \]

   10. Simplified 83.7%

       \[\leadsto wj - \frac{wj - \left(x + \color{blue}{wj \cdot \left(\left(wj \cdot 0.5\right) \cdot x - x\right)}\right)}{wj + 1} \]
3. Recombined 2 regimes into one program.

4. Final simplification 98.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;wj \leq 5.4 \cdot 10^{-9}:\\ \;\;\;\;x + wj \cdot \left(x \cdot -2 + wj \cdot \left(1 - wj\right)\right)\\ \mathbf{else}:\\ \;\;\;\;wj - \frac{wj + \left(wj \cdot \left(x - x \cdot \left(wj \cdot 0.5\right)\right) - x\right)}{wj + 1}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 97.0% accurate, 13.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;wj \leq 5.4 \cdot 10^{-9}:\\ \;\;\;\;x + wj \cdot \left(x \cdot -2 + wj \cdot \left(x \cdot \left(\frac{1}{x} + 2.5\right) - wj\right)\right)\\ \mathbf{else}:\\ \;\;\;\;wj - \frac{wj + \left(wj \cdot \left(x - x \cdot \left(wj \cdot 0.5\right)\right) - x\right)}{wj + 1}\\ \end{array} \end{array} \]

FPCore

(FPCore (wj x)
 :precision binary64
 (if (<= wj 5.4e-9)
   (+ x (* wj (+ (* x -2.0) (* wj (- (* x (+ (/ 1.0 x) 2.5)) wj)))))
   (- wj (/ (+ wj (- (* wj (- x (* x (* wj 0.5)))) x)) (+ wj 1.0)))))

C

double code(double wj, double x) {
	double tmp;
	if (wj <= 5.4e-9) {
		tmp = x + (wj * ((x * -2.0) + (wj * ((x * ((1.0 / x) + 2.5)) - wj))));
	} else {
		tmp = wj - ((wj + ((wj * (x - (x * (wj * 0.5)))) - x)) / (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 <= 5.4d-9) then
        tmp = x + (wj * ((x * (-2.0d0)) + (wj * ((x * ((1.0d0 / x) + 2.5d0)) - wj))))
    else
        tmp = wj - ((wj + ((wj * (x - (x * (wj * 0.5d0)))) - x)) / (wj + 1.0d0))
    end if
    code = tmp
end function

Java

public static double code(double wj, double x) {
	double tmp;
	if (wj <= 5.4e-9) {
		tmp = x + (wj * ((x * -2.0) + (wj * ((x * ((1.0 / x) + 2.5)) - wj))));
	} else {
		tmp = wj - ((wj + ((wj * (x - (x * (wj * 0.5)))) - x)) / (wj + 1.0));
	}
	return tmp;
}

Python

def code(wj, x):
	tmp = 0
	if wj <= 5.4e-9:
		tmp = x + (wj * ((x * -2.0) + (wj * ((x * ((1.0 / x) + 2.5)) - wj))))
	else:
		tmp = wj - ((wj + ((wj * (x - (x * (wj * 0.5)))) - x)) / (wj + 1.0))
	return tmp

Julia

function code(wj, x)
	tmp = 0.0
	if (wj <= 5.4e-9)
		tmp = Float64(x + Float64(wj * Float64(Float64(x * -2.0) + Float64(wj * Float64(Float64(x * Float64(Float64(1.0 / x) + 2.5)) - wj)))));
	else
		tmp = Float64(wj - Float64(Float64(wj + Float64(Float64(wj * Float64(x - Float64(x * Float64(wj * 0.5)))) - x)) / Float64(wj + 1.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(wj, x)
	tmp = 0.0;
	if (wj <= 5.4e-9)
		tmp = x + (wj * ((x * -2.0) + (wj * ((x * ((1.0 / x) + 2.5)) - wj))));
	else
		tmp = wj - ((wj + ((wj * (x - (x * (wj * 0.5)))) - x)) / (wj + 1.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[wj_, x_] := If[LessEqual[wj, 5.4e-9], N[(x + N[(wj * N[(N[(x * -2.0), $MachinePrecision] + N[(wj * N[(N[(x * N[(N[(1.0 / x), $MachinePrecision] + 2.5), $MachinePrecision]), $MachinePrecision] - wj), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(wj - N[(N[(wj + N[(N[(wj * N[(x - N[(x * N[(wj * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - x), $MachinePrecision]), $MachinePrecision] / N[(wj + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if wj < 5.4000000000000004e-9

   1. Initial program 80.5%

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

      1. distribute-rgt1-in 80.5%

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

      2. associate-/l/ 80.6%

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

      3. div-sub 80.6%

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

      4. associate-/l* 80.6%

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

      5. *-inverses 80.6%

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

      6. *-rgt-identity 80.6%

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

   3. Simplified 80.6%

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

   5. Taylor expanded in x around -inf 90.6%

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

      1. associate-*r* 90.6%

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

      2. neg-mul-1 90.6%

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

      3. mul-1-neg 90.6%

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

      4. +-commutative 90.6%

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

      5. associate-/r* 90.6%

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

      6. exp-neg 90.6%

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

      7. +-commutative 90.6%

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

   7. Simplified 90.6%

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

   8. Taylor expanded in wj around 0 98.8%

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

   9. Taylor expanded in x around 0 98.8%

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

   if 5.4000000000000004e-9 < wj

   1. Initial program 66.3%

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

      1. distribute-rgt1-in 66.3%

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

      2. associate-/l/ 66.5%

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

      3. div-sub 66.5%

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

      4. associate-/l* 66.5%

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

      5. *-inverses 99.8%

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

      6. *-rgt-identity 99.8%

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

   3. Simplified 99.8%

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

   5. Taylor expanded in wj around 0 83.7%

      \[\leadsto wj - \frac{wj - \color{blue}{\left(x + wj \cdot \left(-1 \cdot \left(wj \cdot \left(-1 \cdot x + 0.5 \cdot x\right)\right) - x\right)\right)}}{wj + 1} \]
   6. Step-by-step derivation

      1. associate-*r* 83.7%

         \[\leadsto wj - \frac{wj - \left(x + wj \cdot \left(\color{blue}{\left(-1 \cdot wj\right) \cdot \left(-1 \cdot x + 0.5 \cdot x\right)} - x\right)\right)}{wj + 1} \]

      2. neg-mul-1 83.7%

         \[\leadsto wj - \frac{wj - \left(x + wj \cdot \left(\color{blue}{\left(-wj\right)} \cdot \left(-1 \cdot x + 0.5 \cdot x\right) - x\right)\right)}{wj + 1} \]

      3. distribute-rgt-out 83.7%

         \[\leadsto wj - \frac{wj - \left(x + wj \cdot \left(\left(-wj\right) \cdot \color{blue}{\left(x \cdot \left(-1 + 0.5\right)\right)} - x\right)\right)}{wj + 1} \]

      4. metadata-eval 83.7%

         \[\leadsto wj - \frac{wj - \left(x + wj \cdot \left(\left(-wj\right) \cdot \left(x \cdot \color{blue}{-0.5}\right) - x\right)\right)}{wj + 1} \]

   7. Simplified 83.7%

      \[\leadsto wj - \frac{wj - \color{blue}{\left(x + wj \cdot \left(\left(-wj\right) \cdot \left(x \cdot -0.5\right) - x\right)\right)}}{wj + 1} \]

   8. Taylor expanded in wj around 0 83.7%

      \[\leadsto wj - \frac{wj - \left(x + \color{blue}{wj \cdot \left(-1 \cdot x + 0.5 \cdot \left(wj \cdot x\right)\right)}\right)}{wj + 1} \]
   9. Step-by-step derivation

      1. neg-mul-1 83.7%

         \[\leadsto wj - \frac{wj - \left(x + wj \cdot \left(\color{blue}{\left(-x\right)} + 0.5 \cdot \left(wj \cdot x\right)\right)\right)}{wj + 1} \]

      2. +-commutative 83.7%

         \[\leadsto wj - \frac{wj - \left(x + wj \cdot \color{blue}{\left(0.5 \cdot \left(wj \cdot x\right) + \left(-x\right)\right)}\right)}{wj + 1} \]

      3. sub-neg 83.7%

         \[\leadsto wj - \frac{wj - \left(x + wj \cdot \color{blue}{\left(0.5 \cdot \left(wj \cdot x\right) - x\right)}\right)}{wj + 1} \]

      4. associate-*r* 83.7%

         \[\leadsto wj - \frac{wj - \left(x + wj \cdot \left(\color{blue}{\left(0.5 \cdot wj\right) \cdot x} - x\right)\right)}{wj + 1} \]

      5. *-commutative 83.7%

         \[\leadsto wj - \frac{wj - \left(x + wj \cdot \left(\color{blue}{\left(wj \cdot 0.5\right)} \cdot x - x\right)\right)}{wj + 1} \]

   10. Simplified 83.7%

       \[\leadsto wj - \frac{wj - \left(x + \color{blue}{wj \cdot \left(\left(wj \cdot 0.5\right) \cdot x - x\right)}\right)}{wj + 1} \]
3. Recombined 2 regimes into one program.

4. Final simplification 98.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;wj \leq 5.4 \cdot 10^{-9}:\\ \;\;\;\;x + wj \cdot \left(x \cdot -2 + wj \cdot \left(x \cdot \left(\frac{1}{x} + 2.5\right) - wj\right)\right)\\ \mathbf{else}:\\ \;\;\;\;wj - \frac{wj + \left(wj \cdot \left(x - x \cdot \left(wj \cdot 0.5\right)\right) - x\right)}{wj + 1}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 96.7% accurate, 14.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;wj \leq 0.0073:\\ \;\;\;\;x + wj \cdot \left(x \cdot -2 + wj \cdot \left(1 - wj\right)\right)\\ \mathbf{else}:\\ \;\;\;\;wj - \frac{wj - \left(x + wj \cdot \left(x \cdot \left(wj \cdot 0.5\right)\right)\right)}{wj + 1}\\ \end{array} \end{array} \]

FPCore

(FPCore (wj x)
 :precision binary64
 (if (<= wj 0.0073)
   (+ x (* wj (+ (* x -2.0) (* wj (- 1.0 wj)))))
   (- wj (/ (- wj (+ x (* wj (* x (* wj 0.5))))) (+ wj 1.0)))))

C

double code(double wj, double x) {
	double tmp;
	if (wj <= 0.0073) {
		tmp = x + (wj * ((x * -2.0) + (wj * (1.0 - wj))));
	} else {
		tmp = wj - ((wj - (x + (wj * (x * (wj * 0.5))))) / (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 <= 0.0073d0) then
        tmp = x + (wj * ((x * (-2.0d0)) + (wj * (1.0d0 - wj))))
    else
        tmp = wj - ((wj - (x + (wj * (x * (wj * 0.5d0))))) / (wj + 1.0d0))
    end if
    code = tmp
end function

Java

public static double code(double wj, double x) {
	double tmp;
	if (wj <= 0.0073) {
		tmp = x + (wj * ((x * -2.0) + (wj * (1.0 - wj))));
	} else {
		tmp = wj - ((wj - (x + (wj * (x * (wj * 0.5))))) / (wj + 1.0));
	}
	return tmp;
}

Python

def code(wj, x):
	tmp = 0
	if wj <= 0.0073:
		tmp = x + (wj * ((x * -2.0) + (wj * (1.0 - wj))))
	else:
		tmp = wj - ((wj - (x + (wj * (x * (wj * 0.5))))) / (wj + 1.0))
	return tmp

Julia

function code(wj, x)
	tmp = 0.0
	if (wj <= 0.0073)
		tmp = Float64(x + Float64(wj * Float64(Float64(x * -2.0) + Float64(wj * Float64(1.0 - wj)))));
	else
		tmp = Float64(wj - Float64(Float64(wj - Float64(x + Float64(wj * Float64(x * Float64(wj * 0.5))))) / Float64(wj + 1.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(wj, x)
	tmp = 0.0;
	if (wj <= 0.0073)
		tmp = x + (wj * ((x * -2.0) + (wj * (1.0 - wj))));
	else
		tmp = wj - ((wj - (x + (wj * (x * (wj * 0.5))))) / (wj + 1.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[wj_, x_] := If[LessEqual[wj, 0.0073], N[(x + N[(wj * N[(N[(x * -2.0), $MachinePrecision] + N[(wj * N[(1.0 - wj), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(wj - N[(N[(wj - N[(x + N[(wj * N[(x * N[(wj * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(wj + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if wj < 0.00730000000000000007

   1. Initial program 80.7%

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

      1. distribute-rgt1-in 80.7%

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

      2. associate-/l/ 80.8%

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

      3. div-sub 80.8%

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

      4. associate-/l* 80.8%

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

      5. *-inverses 80.8%

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

      6. *-rgt-identity 80.8%

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

   3. Simplified 80.8%

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

   5. Taylor expanded in x around -inf 90.7%

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

      1. associate-*r* 90.7%

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

      2. neg-mul-1 90.7%

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

      3. mul-1-neg 90.7%

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

      4. +-commutative 90.7%

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

      5. associate-/r* 90.7%

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

      6. exp-neg 90.7%

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

      7. +-commutative 90.7%

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

   7. Simplified 90.7%

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

   8. Taylor expanded in wj around 0 98.7%

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

   9. Taylor expanded in x around 0 98.4%

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

       1. neg-mul-1 98.4%

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

       2. sub-neg 98.4%

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

   11. Simplified 98.4%

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

   if 0.00730000000000000007 < wj

   1. Initial program 49.7%

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

      1. distribute-rgt1-in 49.7%

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

      2. associate-/l/ 50.0%

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

      3. div-sub 50.0%

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

      4. associate-/l* 50.0%

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

      5. *-inverses 100.0%

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

      6. *-rgt-identity 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in wj around 0 85.9%

      \[\leadsto wj - \frac{wj - \color{blue}{\left(x + wj \cdot \left(-1 \cdot \left(wj \cdot \left(-1 \cdot x + 0.5 \cdot x\right)\right) - x\right)\right)}}{wj + 1} \]
   6. Step-by-step derivation

      1. associate-*r* 85.9%

         \[\leadsto wj - \frac{wj - \left(x + wj \cdot \left(\color{blue}{\left(-1 \cdot wj\right) \cdot \left(-1 \cdot x + 0.5 \cdot x\right)} - x\right)\right)}{wj + 1} \]

      2. neg-mul-1 85.9%

         \[\leadsto wj - \frac{wj - \left(x + wj \cdot \left(\color{blue}{\left(-wj\right)} \cdot \left(-1 \cdot x + 0.5 \cdot x\right) - x\right)\right)}{wj + 1} \]

      3. distribute-rgt-out 85.9%

         \[\leadsto wj - \frac{wj - \left(x + wj \cdot \left(\left(-wj\right) \cdot \color{blue}{\left(x \cdot \left(-1 + 0.5\right)\right)} - x\right)\right)}{wj + 1} \]

      4. metadata-eval 85.9%

         \[\leadsto wj - \frac{wj - \left(x + wj \cdot \left(\left(-wj\right) \cdot \left(x \cdot \color{blue}{-0.5}\right) - x\right)\right)}{wj + 1} \]

   7. Simplified 85.9%

      \[\leadsto wj - \frac{wj - \color{blue}{\left(x + wj \cdot \left(\left(-wj\right) \cdot \left(x \cdot -0.5\right) - x\right)\right)}}{wj + 1} \]

   8. Taylor expanded in wj around inf 85.8%

      \[\leadsto wj - \frac{wj - \left(x + wj \cdot \color{blue}{\left(0.5 \cdot \left(wj \cdot x\right)\right)}\right)}{wj + 1} \]
   9. Step-by-step derivation

      1. associate-*r* 85.8%

         \[\leadsto wj - \frac{wj - \left(x + wj \cdot \color{blue}{\left(\left(0.5 \cdot wj\right) \cdot x\right)}\right)}{wj + 1} \]

      2. *-commutative 85.8%

         \[\leadsto wj - \frac{wj - \left(x + wj \cdot \left(\color{blue}{\left(wj \cdot 0.5\right)} \cdot x\right)\right)}{wj + 1} \]

   10. Simplified 85.8%

       \[\leadsto wj - \frac{wj - \left(x + wj \cdot \color{blue}{\left(\left(wj \cdot 0.5\right) \cdot x\right)}\right)}{wj + 1} \]
3. Recombined 2 regimes into one program.

4. Final simplification 98.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;wj \leq 0.0073:\\ \;\;\;\;x + wj \cdot \left(x \cdot -2 + wj \cdot \left(1 - wj\right)\right)\\ \mathbf{else}:\\ \;\;\;\;wj - \frac{wj - \left(x + wj \cdot \left(x \cdot \left(wj \cdot 0.5\right)\right)\right)}{wj + 1}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 97.1% accurate, 17.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;wj \leq 0.0082:\\ \;\;\;\;x + wj \cdot \left(x \cdot -2 + wj \cdot \left(1 - wj\right)\right)\\ \mathbf{else}:\\ \;\;\;\;wj + \frac{wj}{-1 - wj}\\ \end{array} \end{array} \]

FPCore

(FPCore (wj x)
 :precision binary64
 (if (<= wj 0.0082)
   (+ x (* wj (+ (* x -2.0) (* wj (- 1.0 wj)))))
   (+ wj (/ wj (- -1.0 wj)))))

C

double code(double wj, double x) {
	double tmp;
	if (wj <= 0.0082) {
		tmp = x + (wj * ((x * -2.0) + (wj * (1.0 - wj))));
	} else {
		tmp = wj + (wj / (-1.0 - wj));
	}
	return tmp;
}

Fortran

real(8) function code(wj, x)
    real(8), intent (in) :: wj
    real(8), intent (in) :: x
    real(8) :: tmp
    if (wj <= 0.0082d0) then
        tmp = x + (wj * ((x * (-2.0d0)) + (wj * (1.0d0 - wj))))
    else
        tmp = wj + (wj / ((-1.0d0) - wj))
    end if
    code = tmp
end function

Java

public static double code(double wj, double x) {
	double tmp;
	if (wj <= 0.0082) {
		tmp = x + (wj * ((x * -2.0) + (wj * (1.0 - wj))));
	} else {
		tmp = wj + (wj / (-1.0 - wj));
	}
	return tmp;
}

Python

def code(wj, x):
	tmp = 0
	if wj <= 0.0082:
		tmp = x + (wj * ((x * -2.0) + (wj * (1.0 - wj))))
	else:
		tmp = wj + (wj / (-1.0 - wj))
	return tmp

Julia

function code(wj, x)
	tmp = 0.0
	if (wj <= 0.0082)
		tmp = Float64(x + Float64(wj * Float64(Float64(x * -2.0) + Float64(wj * Float64(1.0 - wj)))));
	else
		tmp = Float64(wj + Float64(wj / Float64(-1.0 - wj)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(wj, x)
	tmp = 0.0;
	if (wj <= 0.0082)
		tmp = x + (wj * ((x * -2.0) + (wj * (1.0 - wj))));
	else
		tmp = wj + (wj / (-1.0 - wj));
	end
	tmp_2 = tmp;
end

Wolfram

code[wj_, x_] := If[LessEqual[wj, 0.0082], N[(x + N[(wj * N[(N[(x * -2.0), $MachinePrecision] + N[(wj * N[(1.0 - wj), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(wj + N[(wj / N[(-1.0 - wj), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if wj < 0.00820000000000000069

   1. Initial program 80.7%

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

      1. distribute-rgt1-in 80.7%

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

      2. associate-/l/ 80.8%

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

      3. div-sub 80.8%

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

      4. associate-/l* 80.8%

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

      5. *-inverses 80.8%

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

      6. *-rgt-identity 80.8%

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

   3. Simplified 80.8%

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

   5. Taylor expanded in x around -inf 90.7%

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

      1. associate-*r* 90.7%

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

      2. neg-mul-1 90.7%

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

      3. mul-1-neg 90.7%

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

      4. +-commutative 90.7%

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

      5. associate-/r* 90.7%

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

      6. exp-neg 90.7%

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

      7. +-commutative 90.7%

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

   7. Simplified 90.7%

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

   8. Taylor expanded in wj around 0 98.7%

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

   9. Taylor expanded in x around 0 98.4%

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

       1. neg-mul-1 98.4%

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

       2. sub-neg 98.4%

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

   11. Simplified 98.4%

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

   if 0.00820000000000000069 < wj

   1. Initial program 49.7%

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

      1. distribute-rgt1-in 49.7%

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

      2. associate-/l/ 50.0%

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

      3. div-sub 50.0%

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

      4. associate-/l* 50.0%

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

      5. *-inverses 100.0%

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

      6. *-rgt-identity 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in x around 0 83.5%

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

      1. +-commutative 83.5%

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

   7. Simplified 83.5%

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

4. Final simplification 98.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;wj \leq 0.0082:\\ \;\;\;\;x + wj \cdot \left(x \cdot -2 + wj \cdot \left(1 - wj\right)\right)\\ \mathbf{else}:\\ \;\;\;\;wj + \frac{wj}{-1 - wj}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 96.8% accurate, 17.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;wj \leq 5.4 \cdot 10^{-9}:\\ \;\;\;\;x + wj \cdot \left(x \cdot -2 + wj \cdot \left(1 - wj\right)\right)\\ \mathbf{else}:\\ \;\;\;\;wj + \frac{wj + x \cdot \left(wj + -1\right)}{-1 - wj}\\ \end{array} \end{array} \]

FPCore

(FPCore (wj x)
 :precision binary64
 (if (<= wj 5.4e-9)
   (+ x (* wj (+ (* x -2.0) (* wj (- 1.0 wj)))))
   (+ wj (/ (+ wj (* x (+ wj -1.0))) (- -1.0 wj)))))

C

double code(double wj, double x) {
	double tmp;
	if (wj <= 5.4e-9) {
		tmp = x + (wj * ((x * -2.0) + (wj * (1.0 - wj))));
	} else {
		tmp = wj + ((wj + (x * (wj + -1.0))) / (-1.0 - wj));
	}
	return tmp;
}

Fortran

real(8) function code(wj, x)
    real(8), intent (in) :: wj
    real(8), intent (in) :: x
    real(8) :: tmp
    if (wj <= 5.4d-9) then
        tmp = x + (wj * ((x * (-2.0d0)) + (wj * (1.0d0 - wj))))
    else
        tmp = wj + ((wj + (x * (wj + (-1.0d0)))) / ((-1.0d0) - wj))
    end if
    code = tmp
end function

Java

public static double code(double wj, double x) {
	double tmp;
	if (wj <= 5.4e-9) {
		tmp = x + (wj * ((x * -2.0) + (wj * (1.0 - wj))));
	} else {
		tmp = wj + ((wj + (x * (wj + -1.0))) / (-1.0 - wj));
	}
	return tmp;
}

Python

def code(wj, x):
	tmp = 0
	if wj <= 5.4e-9:
		tmp = x + (wj * ((x * -2.0) + (wj * (1.0 - wj))))
	else:
		tmp = wj + ((wj + (x * (wj + -1.0))) / (-1.0 - wj))
	return tmp

Julia

function code(wj, x)
	tmp = 0.0
	if (wj <= 5.4e-9)
		tmp = Float64(x + Float64(wj * Float64(Float64(x * -2.0) + Float64(wj * Float64(1.0 - wj)))));
	else
		tmp = Float64(wj + Float64(Float64(wj + Float64(x * Float64(wj + -1.0))) / Float64(-1.0 - wj)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(wj, x)
	tmp = 0.0;
	if (wj <= 5.4e-9)
		tmp = x + (wj * ((x * -2.0) + (wj * (1.0 - wj))));
	else
		tmp = wj + ((wj + (x * (wj + -1.0))) / (-1.0 - wj));
	end
	tmp_2 = tmp;
end

Wolfram

code[wj_, x_] := If[LessEqual[wj, 5.4e-9], N[(x + N[(wj * N[(N[(x * -2.0), $MachinePrecision] + N[(wj * N[(1.0 - wj), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(wj + N[(N[(wj + N[(x * N[(wj + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(-1.0 - wj), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if wj < 5.4000000000000004e-9

   1. Initial program 80.5%

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

      1. distribute-rgt1-in 80.5%

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

      2. associate-/l/ 80.6%

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

      3. div-sub 80.6%

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

      4. associate-/l* 80.6%

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

      5. *-inverses 80.6%

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

      6. *-rgt-identity 80.6%

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

   3. Simplified 80.6%

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

   5. Taylor expanded in x around -inf 90.6%

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

      1. associate-*r* 90.6%

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

      2. neg-mul-1 90.6%

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

      3. mul-1-neg 90.6%

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

      4. +-commutative 90.6%

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

      5. associate-/r* 90.6%

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

      6. exp-neg 90.6%

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

      7. +-commutative 90.6%

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

   7. Simplified 90.6%

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

   8. Taylor expanded in wj around 0 98.8%

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

   9. Taylor expanded in x around 0 98.8%

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

       1. neg-mul-1 98.8%

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

       2. sub-neg 98.8%

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

   11. Simplified 98.8%

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

   if 5.4000000000000004e-9 < wj

   1. Initial program 66.3%

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

      1. distribute-rgt1-in 66.3%

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

      2. associate-/l/ 66.5%

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

      3. div-sub 66.5%

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

      4. associate-/l* 66.5%

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

      5. *-inverses 99.8%

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

      6. *-rgt-identity 99.8%

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

   3. Simplified 99.8%

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

   5. Taylor expanded in wj around 0 78.0%

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

      1. associate-*r* 78.0%

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

      2. neg-mul-1 78.0%

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

      3. distribute-rgt1-in 78.0%

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

      4. +-commutative 78.0%

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

      5. sub-neg 78.0%

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

   7. Simplified 78.0%

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

4. Final simplification 98.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;wj \leq 5.4 \cdot 10^{-9}:\\ \;\;\;\;x + wj \cdot \left(x \cdot -2 + wj \cdot \left(1 - wj\right)\right)\\ \mathbf{else}:\\ \;\;\;\;wj + \frac{wj + x \cdot \left(wj + -1\right)}{-1 - wj}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 85.9% accurate, 19.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;wj \leq 2.8:\\ \;\;\;\;x + wj \cdot \left(x \cdot \left(wj \cdot 2.5 - 2\right)\right)\\ \mathbf{else}:\\ \;\;\;\;wj + \frac{wj}{-1 - wj}\\ \end{array} \end{array} \]

FPCore

(FPCore (wj x)
 :precision binary64
 (if (<= wj 2.8)
   (+ x (* wj (* x (- (* wj 2.5) 2.0))))
   (+ wj (/ wj (- -1.0 wj)))))

C

double code(double wj, double x) {
	double tmp;
	if (wj <= 2.8) {
		tmp = x + (wj * (x * ((wj * 2.5) - 2.0)));
	} else {
		tmp = wj + (wj / (-1.0 - wj));
	}
	return tmp;
}

Fortran

real(8) function code(wj, x)
    real(8), intent (in) :: wj
    real(8), intent (in) :: x
    real(8) :: tmp
    if (wj <= 2.8d0) then
        tmp = x + (wj * (x * ((wj * 2.5d0) - 2.0d0)))
    else
        tmp = wj + (wj / ((-1.0d0) - wj))
    end if
    code = tmp
end function

Java

public static double code(double wj, double x) {
	double tmp;
	if (wj <= 2.8) {
		tmp = x + (wj * (x * ((wj * 2.5) - 2.0)));
	} else {
		tmp = wj + (wj / (-1.0 - wj));
	}
	return tmp;
}

Python

def code(wj, x):
	tmp = 0
	if wj <= 2.8:
		tmp = x + (wj * (x * ((wj * 2.5) - 2.0)))
	else:
		tmp = wj + (wj / (-1.0 - wj))
	return tmp

Julia

function code(wj, x)
	tmp = 0.0
	if (wj <= 2.8)
		tmp = Float64(x + Float64(wj * Float64(x * Float64(Float64(wj * 2.5) - 2.0))));
	else
		tmp = Float64(wj + Float64(wj / Float64(-1.0 - wj)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(wj, x)
	tmp = 0.0;
	if (wj <= 2.8)
		tmp = x + (wj * (x * ((wj * 2.5) - 2.0)));
	else
		tmp = wj + (wj / (-1.0 - wj));
	end
	tmp_2 = tmp;
end

Wolfram

code[wj_, x_] := If[LessEqual[wj, 2.8], N[(x + N[(wj * N[(x * N[(N[(wj * 2.5), $MachinePrecision] - 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(wj + N[(wj / N[(-1.0 - wj), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if wj < 2.7999999999999998

   1. Initial program 80.8%

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

      1. distribute-rgt1-in 80.8%

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

      2. associate-/l/ 80.9%

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

      3. div-sub 80.9%

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

      4. associate-/l* 80.9%

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

      5. *-inverses 80.9%

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

      6. *-rgt-identity 80.9%

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

   3. Simplified 80.9%

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

   5. Taylor expanded in x around -inf 90.7%

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

      1. associate-*r* 90.7%

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

      2. neg-mul-1 90.7%

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

      3. mul-1-neg 90.7%

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

      4. +-commutative 90.7%

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

      5. associate-/r* 90.7%

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

      6. exp-neg 90.7%

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

      7. +-commutative 90.7%

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

   7. Simplified 90.7%

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

   8. Taylor expanded in wj around 0 98.3%

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

   9. Taylor expanded in x around 0 98.2%

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

   10. Taylor expanded in x around inf 87.9%

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

   if 2.7999999999999998 < wj

   1. Initial program 40.0%

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

      1. distribute-rgt1-in 40.0%

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

      2. associate-/l/ 40.0%

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

      3. div-sub 40.0%

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

      4. associate-/l* 40.0%

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

      5. *-inverses 100.0%

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

      6. *-rgt-identity 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in x around 0 100.0%

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

      1. +-commutative 100.0%

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

   7. Simplified 100.0%

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

4. Final simplification 88.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;wj \leq 2.8:\\ \;\;\;\;x + wj \cdot \left(x \cdot \left(wj \cdot 2.5 - 2\right)\right)\\ \mathbf{else}:\\ \;\;\;\;wj + \frac{wj}{-1 - wj}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 96.7% accurate, 22.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;wj \leq 0.0076:\\ \;\;\;\;x + wj \cdot \left(wj + x \cdot -2\right)\\ \mathbf{else}:\\ \;\;\;\;wj + \frac{wj}{-1 - wj}\\ \end{array} \end{array} \]

FPCore

(FPCore (wj x)
 :precision binary64
 (if (<= wj 0.0076) (+ x (* wj (+ wj (* x -2.0)))) (+ wj (/ wj (- -1.0 wj)))))

C

double code(double wj, double x) {
	double tmp;
	if (wj <= 0.0076) {
		tmp = x + (wj * (wj + (x * -2.0)));
	} else {
		tmp = wj + (wj / (-1.0 - wj));
	}
	return tmp;
}

Fortran

real(8) function code(wj, x)
    real(8), intent (in) :: wj
    real(8), intent (in) :: x
    real(8) :: tmp
    if (wj <= 0.0076d0) then
        tmp = x + (wj * (wj + (x * (-2.0d0))))
    else
        tmp = wj + (wj / ((-1.0d0) - wj))
    end if
    code = tmp
end function

Java

public static double code(double wj, double x) {
	double tmp;
	if (wj <= 0.0076) {
		tmp = x + (wj * (wj + (x * -2.0)));
	} else {
		tmp = wj + (wj / (-1.0 - wj));
	}
	return tmp;
}

Python

def code(wj, x):
	tmp = 0
	if wj <= 0.0076:
		tmp = x + (wj * (wj + (x * -2.0)))
	else:
		tmp = wj + (wj / (-1.0 - wj))
	return tmp

Julia

function code(wj, x)
	tmp = 0.0
	if (wj <= 0.0076)
		tmp = Float64(x + Float64(wj * Float64(wj + Float64(x * -2.0))));
	else
		tmp = Float64(wj + Float64(wj / Float64(-1.0 - wj)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(wj, x)
	tmp = 0.0;
	if (wj <= 0.0076)
		tmp = x + (wj * (wj + (x * -2.0)));
	else
		tmp = wj + (wj / (-1.0 - wj));
	end
	tmp_2 = tmp;
end

Wolfram

code[wj_, x_] := If[LessEqual[wj, 0.0076], N[(x + N[(wj * N[(wj + N[(x * -2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(wj + N[(wj / N[(-1.0 - wj), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if wj < 0.00759999999999999998

   1. Initial program 80.7%

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

      1. distribute-rgt1-in 80.7%

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

      2. associate-/l/ 80.8%

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

      3. div-sub 80.8%

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

      4. associate-/l* 80.8%

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

      5. *-inverses 80.8%

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

      6. *-rgt-identity 80.8%

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

   3. Simplified 80.8%

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

   5. Taylor expanded in wj around 0 98.1%

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

      1. cancel-sign-sub-inv 98.1%

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

      2. metadata-eval 98.1%

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

      3. distribute-rgt-out 98.1%

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

      4. metadata-eval 98.1%

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

      5. *-commutative 98.1%

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

   7. Simplified 98.1%

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

   8. Taylor expanded in x around 0 97.9%

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

   if 0.00759999999999999998 < wj

   1. Initial program 49.7%

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

      1. distribute-rgt1-in 49.7%

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

      2. associate-/l/ 50.0%

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

      3. div-sub 50.0%

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

      4. associate-/l* 50.0%

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

      5. *-inverses 100.0%

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

      6. *-rgt-identity 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in x around 0 83.5%

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

      1. +-commutative 83.5%

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

   7. Simplified 83.5%

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

4. Final simplification 97.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;wj \leq 0.0076:\\ \;\;\;\;x + wj \cdot \left(wj + x \cdot -2\right)\\ \mathbf{else}:\\ \;\;\;\;wj + \frac{wj}{-1 - wj}\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 85.9% accurate, 26.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;wj \leq 0.0073:\\ \;\;\;\;x + -2 \cdot \left(wj \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;wj + \frac{wj}{-1 - wj}\\ \end{array} \end{array} \]

FPCore

(FPCore (wj x)
 :precision binary64
 (if (<= wj 0.0073) (+ x (* -2.0 (* wj x))) (+ wj (/ wj (- -1.0 wj)))))

C

double code(double wj, double x) {
	double tmp;
	if (wj <= 0.0073) {
		tmp = x + (-2.0 * (wj * x));
	} else {
		tmp = wj + (wj / (-1.0 - wj));
	}
	return tmp;
}

Fortran

real(8) function code(wj, x)
    real(8), intent (in) :: wj
    real(8), intent (in) :: x
    real(8) :: tmp
    if (wj <= 0.0073d0) then
        tmp = x + ((-2.0d0) * (wj * x))
    else
        tmp = wj + (wj / ((-1.0d0) - wj))
    end if
    code = tmp
end function

Java

public static double code(double wj, double x) {
	double tmp;
	if (wj <= 0.0073) {
		tmp = x + (-2.0 * (wj * x));
	} else {
		tmp = wj + (wj / (-1.0 - wj));
	}
	return tmp;
}

Python

def code(wj, x):
	tmp = 0
	if wj <= 0.0073:
		tmp = x + (-2.0 * (wj * x))
	else:
		tmp = wj + (wj / (-1.0 - wj))
	return tmp

Julia

function code(wj, x)
	tmp = 0.0
	if (wj <= 0.0073)
		tmp = Float64(x + Float64(-2.0 * Float64(wj * x)));
	else
		tmp = Float64(wj + Float64(wj / Float64(-1.0 - wj)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(wj, x)
	tmp = 0.0;
	if (wj <= 0.0073)
		tmp = x + (-2.0 * (wj * x));
	else
		tmp = wj + (wj / (-1.0 - wj));
	end
	tmp_2 = tmp;
end

Wolfram

code[wj_, x_] := If[LessEqual[wj, 0.0073], N[(x + N[(-2.0 * N[(wj * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(wj + N[(wj / N[(-1.0 - wj), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if wj < 0.00730000000000000007

   1. Initial program 80.7%

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

      1. distribute-rgt1-in 80.7%

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

      2. associate-/l/ 80.8%

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

      3. div-sub 80.8%

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

      4. associate-/l* 80.8%

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

      5. *-inverses 80.8%

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

      6. *-rgt-identity 80.8%

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

   3. Simplified 80.8%

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

   5. Taylor expanded in wj around 0 88.0%

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

      1. *-commutative 88.0%

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

   7. Simplified 88.0%

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

   if 0.00730000000000000007 < wj

   1. Initial program 49.7%

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

      1. distribute-rgt1-in 49.7%

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

      2. associate-/l/ 50.0%

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

      3. div-sub 50.0%

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

      4. associate-/l* 50.0%

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

      5. *-inverses 100.0%

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

      6. *-rgt-identity 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in x around 0 83.5%

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

      1. +-commutative 83.5%

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

   7. Simplified 83.5%

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

4. Final simplification 87.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;wj \leq 0.0073:\\ \;\;\;\;x + -2 \cdot \left(wj \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;wj + \frac{wj}{-1 - wj}\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 84.6% accurate, 44.7× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 80.0%

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

   1. distribute-rgt1-in 80.0%

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

   2. associate-/l/ 80.1%

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

   3. div-sub 80.1%

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

   4. associate-/l* 80.1%

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

   5. *-inverses 81.3%

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

   6. *-rgt-identity 81.3%

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

3. Simplified 81.3%

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

5. Taylor expanded in wj around 0 86.1%

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

   1. *-commutative 86.1%

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

7. Simplified 86.1%

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

8. Final simplification 86.1%

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

Alternative 15: 4.4% accurate, 313.0× speedup

Math

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

FPCore

(FPCore (wj x) :precision binary64 wj)

C

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

Fortran

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

Java

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

Python

def code(wj, x):
	return wj

Julia

function code(wj, x)
	return wj
end

MATLAB

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

Wolfram

code[wj_, x_] := wj

Derivation

1. Initial program 80.0%

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

   1. distribute-rgt1-in 80.0%

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

   2. associate-/l/ 80.1%

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

   3. div-sub 80.1%

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

   4. associate-/l* 80.1%

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

   5. *-inverses 81.3%

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

   6. *-rgt-identity 81.3%

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

3. Simplified 81.3%

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

5. Taylor expanded in wj around inf 4.3%

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

6. Final simplification 4.3%

   \[\leadsto wj \]
7. Add Preprocessing

Alternative 16: 84.2% accurate, 313.0× speedup

Math

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

FPCore

(FPCore (wj x) :precision binary64 x)

C

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

Fortran

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

Java

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

Python

def code(wj, x):
	return x

Julia

function code(wj, x)
	return x
end

MATLAB

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

Wolfram

code[wj_, x_] := x

Derivation

1. Initial program 80.0%

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

   1. distribute-rgt1-in 80.0%

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

   2. associate-/l/ 80.1%

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

   3. div-sub 80.1%

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

   4. associate-/l* 80.1%

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

   5. *-inverses 81.3%

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

   6. *-rgt-identity 81.3%

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

3. Simplified 81.3%

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

5. Taylor expanded in wj around 0 85.6%

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

6. Final simplification 85.6%

   \[\leadsto x \]
7. Add Preprocessing

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 2024110 
(FPCore (wj x)
  :name "Jmat.Real.lambertw, newton loop step"
  :precision binary64

  :alt
  (- wj (- (/ wj (+ wj 1.0)) (/ x (+ (exp wj) (* wj (exp wj))))))

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