Jmat.Real.lambertw, newton loop step

Percentage Accurate: 78.1% → 99.7%

Time: 12.7s

Alternatives: 17

Speedup: 313.0×

• could not determine a ground truth

  1. wj = -3.440911654753403e+21
  2. x = 4.4164015877880217e+123

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 78.1% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.7% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := wj \cdot e^{wj}\\ t_1 := x - t\_0\\ t_2 := wj + \frac{t\_1}{e^{wj} + t\_0}\\ \mathbf{if}\;t\_2 \leq -4:\\ \;\;\;\;wj + \frac{t\_1}{e^{wj} \cdot \left(wj + 1\right)}\\ \mathbf{elif}\;t\_2 \leq 5 \cdot 10^{-15}:\\ \;\;\;\;x - wj \cdot \left(x \cdot 2 + wj \cdot \left(wj + -1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;wj + \frac{wj - \frac{x}{e^{wj}}}{-1 - wj}\\ \end{array} \end{array} \]

FPCore

(FPCore (wj x)
 :precision binary64
 (let* ((t_0 (* wj (exp wj)))
        (t_1 (- x t_0))
        (t_2 (+ wj (/ t_1 (+ (exp wj) t_0)))))
   (if (<= t_2 -4.0)
     (+ wj (/ t_1 (* (exp wj) (+ wj 1.0))))
     (if (<= t_2 5e-15)
       (- x (* wj (+ (* x 2.0) (* wj (+ wj -1.0)))))
       (+ wj (/ (- wj (/ x (exp wj))) (- -1.0 wj)))))))

C

double code(double wj, double x) {
	double t_0 = wj * exp(wj);
	double t_1 = x - t_0;
	double t_2 = wj + (t_1 / (exp(wj) + t_0));
	double tmp;
	if (t_2 <= -4.0) {
		tmp = wj + (t_1 / (exp(wj) * (wj + 1.0)));
	} else if (t_2 <= 5e-15) {
		tmp = x - (wj * ((x * 2.0) + (wj * (wj + -1.0))));
	} else {
		tmp = wj + ((wj - (x / exp(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) :: t_0
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_0 = wj * exp(wj)
    t_1 = x - t_0
    t_2 = wj + (t_1 / (exp(wj) + t_0))
    if (t_2 <= (-4.0d0)) then
        tmp = wj + (t_1 / (exp(wj) * (wj + 1.0d0)))
    else if (t_2 <= 5d-15) then
        tmp = x - (wj * ((x * 2.0d0) + (wj * (wj + (-1.0d0)))))
    else
        tmp = wj + ((wj - (x / exp(wj))) / ((-1.0d0) - wj))
    end if
    code = tmp
end function

Java

public static double code(double wj, double x) {
	double t_0 = wj * Math.exp(wj);
	double t_1 = x - t_0;
	double t_2 = wj + (t_1 / (Math.exp(wj) + t_0));
	double tmp;
	if (t_2 <= -4.0) {
		tmp = wj + (t_1 / (Math.exp(wj) * (wj + 1.0)));
	} else if (t_2 <= 5e-15) {
		tmp = x - (wj * ((x * 2.0) + (wj * (wj + -1.0))));
	} else {
		tmp = wj + ((wj - (x / Math.exp(wj))) / (-1.0 - wj));
	}
	return tmp;
}

Python

def code(wj, x):
	t_0 = wj * math.exp(wj)
	t_1 = x - t_0
	t_2 = wj + (t_1 / (math.exp(wj) + t_0))
	tmp = 0
	if t_2 <= -4.0:
		tmp = wj + (t_1 / (math.exp(wj) * (wj + 1.0)))
	elif t_2 <= 5e-15:
		tmp = x - (wj * ((x * 2.0) + (wj * (wj + -1.0))))
	else:
		tmp = wj + ((wj - (x / math.exp(wj))) / (-1.0 - wj))
	return tmp

Julia

function code(wj, x)
	t_0 = Float64(wj * exp(wj))
	t_1 = Float64(x - t_0)
	t_2 = Float64(wj + Float64(t_1 / Float64(exp(wj) + t_0)))
	tmp = 0.0
	if (t_2 <= -4.0)
		tmp = Float64(wj + Float64(t_1 / Float64(exp(wj) * Float64(wj + 1.0))));
	elseif (t_2 <= 5e-15)
		tmp = Float64(x - Float64(wj * Float64(Float64(x * 2.0) + Float64(wj * Float64(wj + -1.0)))));
	else
		tmp = Float64(wj + Float64(Float64(wj - Float64(x / exp(wj))) / Float64(-1.0 - wj)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(wj, x)
	t_0 = wj * exp(wj);
	t_1 = x - t_0;
	t_2 = wj + (t_1 / (exp(wj) + t_0));
	tmp = 0.0;
	if (t_2 <= -4.0)
		tmp = wj + (t_1 / (exp(wj) * (wj + 1.0)));
	elseif (t_2 <= 5e-15)
		tmp = x - (wj * ((x * 2.0) + (wj * (wj + -1.0))));
	else
		tmp = wj + ((wj - (x / exp(wj))) / (-1.0 - wj));
	end
	tmp_2 = tmp;
end

Wolfram

code[wj_, x_] := Block[{t$95$0 = N[(wj * N[Exp[wj], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(x - t$95$0), $MachinePrecision]}, Block[{t$95$2 = N[(wj + N[(t$95$1 / N[(N[Exp[wj], $MachinePrecision] + t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$2, -4.0], N[(wj + N[(t$95$1 / N[(N[Exp[wj], $MachinePrecision] * N[(wj + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, 5e-15], N[(x - N[(wj * N[(N[(x * 2.0), $MachinePrecision] + N[(wj * N[(wj + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(wj + N[(N[(wj - N[(x / N[Exp[wj], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(-1.0 - wj), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if (-.f64 wj (/.f64 (-.f64 (*.f64 wj (exp.f64 wj)) x) (+.f64 (exp.f64 wj) (*.f64 wj (exp.f64 wj))))) < -4

   1. Initial program 95.8%

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

      1. distribute-rgt1-in 100.0%

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

      2. *-commutative 100.0%

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

   3. Simplified 100.0%

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

   if -4 < (-.f64 wj (/.f64 (-.f64 (*.f64 wj (exp.f64 wj)) x) (+.f64 (exp.f64 wj) (*.f64 wj (exp.f64 wj))))) < 4.99999999999999999e-15

   1. Initial program 58.0%

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

      1. distribute-rgt1-in 58.0%

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

      2. associate-/l/ 58.0%

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

      3. div-sub 58.0%

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

      4. associate-/l* 58.0%

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

      5. *-inverses 58.0%

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

      6. *-rgt-identity 58.0%

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

   3. Simplified 58.0%

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

   5. Taylor expanded in wj around 0 99.9%

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

   6. Taylor expanded in x around 0 99.9%

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

      1. mul-1-neg 99.9%

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

   8. Simplified 99.9%

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

   9. Taylor expanded in wj around 0 99.9%

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

       1. neg-mul-1 99.9%

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

       2. sub-neg 99.9%

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

       3. distribute-rgt-out 99.9%

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

       4. metadata-eval 99.9%

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

       5. *-commutative 99.9%

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

   11. Simplified 99.9%

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

   12. Taylor expanded in x around 0 99.9%

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

   if 4.99999999999999999e-15 < (-.f64 wj (/.f64 (-.f64 (*.f64 wj (exp.f64 wj)) x) (+.f64 (exp.f64 wj) (*.f64 wj (exp.f64 wj)))))

   1. Initial program 93.3%

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

      1. distribute-rgt1-in 94.9%

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

      2. associate-/l/ 94.9%

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

      3. div-sub 93.4%

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

      4. associate-/l* 93.4%

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

      5. *-inverses 99.5%

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

      6. *-rgt-identity 99.5%

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

   3. Simplified 99.5%

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

4. Final simplification 99.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;wj + \frac{x - wj \cdot e^{wj}}{e^{wj} + wj \cdot e^{wj}} \leq -4:\\ \;\;\;\;wj + \frac{x - wj \cdot e^{wj}}{e^{wj} \cdot \left(wj + 1\right)}\\ \mathbf{elif}\;wj + \frac{x - wj \cdot e^{wj}}{e^{wj} + wj \cdot e^{wj}} \leq 5 \cdot 10^{-15}:\\ \;\;\;\;x - wj \cdot \left(x \cdot 2 + wj \cdot \left(wj + -1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;wj + \frac{wj - \frac{x}{e^{wj}}}{-1 - wj}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 99.6% accurate, 2.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x \cdot -4 + x \cdot 1.5\\ \mathbf{if}\;wj \leq -6.5 \cdot 10^{-6} \lor \neg \left(wj \leq 1.3 \cdot 10^{-8}\right):\\ \;\;\;\;wj + \frac{wj - \frac{x}{e^{wj}}}{-1 - wj}\\ \mathbf{else}:\\ \;\;\;\;x + wj \cdot \left(wj \cdot \left(\left(1 + wj \cdot \left(-1 - \left(x \cdot -3 + \left(-2 \cdot t\_0 + x \cdot 0.6666666666666666\right)\right)\right)\right) - t\_0\right) - x \cdot 2\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (wj x)
 :precision binary64
 (let* ((t_0 (+ (* x -4.0) (* x 1.5))))
   (if (or (<= wj -6.5e-6) (not (<= wj 1.3e-8)))
     (+ wj (/ (- wj (/ x (exp wj))) (- -1.0 wj)))
     (+
      x
      (*
       wj
       (-
        (*
         wj
         (-
          (+
           1.0
           (*
            wj
            (- -1.0 (+ (* x -3.0) (+ (* -2.0 t_0) (* x 0.6666666666666666))))))
          t_0))
        (* x 2.0)))))))

C

double code(double wj, double x) {
	double t_0 = (x * -4.0) + (x * 1.5);
	double tmp;
	if ((wj <= -6.5e-6) || !(wj <= 1.3e-8)) {
		tmp = wj + ((wj - (x / exp(wj))) / (-1.0 - wj));
	} else {
		tmp = x + (wj * ((wj * ((1.0 + (wj * (-1.0 - ((x * -3.0) + ((-2.0 * t_0) + (x * 0.6666666666666666)))))) - t_0)) - (x * 2.0)));
	}
	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 <= (-6.5d-6)) .or. (.not. (wj <= 1.3d-8))) then
        tmp = wj + ((wj - (x / exp(wj))) / ((-1.0d0) - wj))
    else
        tmp = x + (wj * ((wj * ((1.0d0 + (wj * ((-1.0d0) - ((x * (-3.0d0)) + (((-2.0d0) * t_0) + (x * 0.6666666666666666d0)))))) - t_0)) - (x * 2.0d0)))
    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 <= -6.5e-6) || !(wj <= 1.3e-8)) {
		tmp = wj + ((wj - (x / Math.exp(wj))) / (-1.0 - wj));
	} else {
		tmp = x + (wj * ((wj * ((1.0 + (wj * (-1.0 - ((x * -3.0) + ((-2.0 * t_0) + (x * 0.6666666666666666)))))) - t_0)) - (x * 2.0)));
	}
	return tmp;
}

Python

def code(wj, x):
	t_0 = (x * -4.0) + (x * 1.5)
	tmp = 0
	if (wj <= -6.5e-6) or not (wj <= 1.3e-8):
		tmp = wj + ((wj - (x / math.exp(wj))) / (-1.0 - wj))
	else:
		tmp = x + (wj * ((wj * ((1.0 + (wj * (-1.0 - ((x * -3.0) + ((-2.0 * t_0) + (x * 0.6666666666666666)))))) - t_0)) - (x * 2.0)))
	return tmp

Julia

function code(wj, x)
	t_0 = Float64(Float64(x * -4.0) + Float64(x * 1.5))
	tmp = 0.0
	if ((wj <= -6.5e-6) || !(wj <= 1.3e-8))
		tmp = Float64(wj + Float64(Float64(wj - Float64(x / exp(wj))) / Float64(-1.0 - wj)));
	else
		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))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(wj, x)
	t_0 = (x * -4.0) + (x * 1.5);
	tmp = 0.0;
	if ((wj <= -6.5e-6) || ~((wj <= 1.3e-8)))
		tmp = wj + ((wj - (x / exp(wj))) / (-1.0 - wj));
	else
		tmp = x + (wj * ((wj * ((1.0 + (wj * (-1.0 - ((x * -3.0) + ((-2.0 * t_0) + (x * 0.6666666666666666)))))) - t_0)) - (x * 2.0)));
	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[Or[LessEqual[wj, -6.5e-6], N[Not[LessEqual[wj, 1.3e-8]], $MachinePrecision]], N[(wj + N[(N[(wj - N[(x / N[Exp[wj], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(-1.0 - wj), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x + N[(wj * N[(N[(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]]]

Derivation

1. Split input into 2 regimes

2. if wj < -6.4999999999999996e-6 or 1.3000000000000001e-8 < wj

   1. Initial program 50.7%

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

      1. distribute-rgt1-in 77.6%

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

      2. associate-/l/ 77.5%

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

      3. div-sub 50.9%

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

      4. associate-/l* 50.9%

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

      5. *-inverses 97.5%

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

      6. *-rgt-identity 97.5%

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

   3. Simplified 97.5%

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

   if -6.4999999999999996e-6 < wj < 1.3000000000000001e-8

   1. Initial program 79.4%

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

      1. distribute-rgt1-in 79.4%

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

      2. associate-/l/ 79.4%

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

      3. div-sub 79.4%

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

      4. associate-/l* 79.4%

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

      5. *-inverses 79.4%

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

      6. *-rgt-identity 79.4%

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

   3. Simplified 79.4%

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

   5. Taylor expanded in wj around 0 100.0%

      \[\leadsto \color{blue}{x + wj \cdot \left(wj \cdot \left(\left(1 + -1 \cdot \left(wj \cdot \left(1 + \left(-3 \cdot x + \left(-2 \cdot \left(-4 \cdot x + 1.5 \cdot x\right) + 0.6666666666666666 \cdot x\right)\right)\right)\right)\right) - \left(-4 \cdot x + 1.5 \cdot x\right)\right) - 2 \cdot x\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 99.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;wj \leq -6.5 \cdot 10^{-6} \lor \neg \left(wj \leq 1.3 \cdot 10^{-8}\right):\\ \;\;\;\;wj + \frac{wj - \frac{x}{e^{wj}}}{-1 - wj}\\ \mathbf{else}:\\ \;\;\;\;x + wj \cdot \left(wj \cdot \left(\left(1 + wj \cdot \left(-1 - \left(x \cdot -3 + \left(-2 \cdot \left(x \cdot -4 + x \cdot 1.5\right) + x \cdot 0.6666666666666666\right)\right)\right)\right) - \left(x \cdot -4 + x \cdot 1.5\right)\right) - x \cdot 2\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 97.7% accurate, 6.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x \cdot -4 + x \cdot 1.5\\ \mathbf{if}\;wj \leq -0.00037:\\ \;\;\;\;wj + \frac{wj \cdot \left(1 + \left(x + wj \cdot \left(\left(x \cdot 0.5 + wj \cdot \frac{\left(x \cdot 0.16666666666666666\right) \cdot \left(x \cdot 0.8333333333333334\right)}{x \cdot 0.8333333333333334}\right) - x\right)\right)\right) - x}{-1 - wj}\\ \mathbf{elif}\;wj \leq 1.3 \cdot 10^{-8}:\\ \;\;\;\;x + wj \cdot \left(wj \cdot \left(\left(1 + wj \cdot \left(-1 - \left(x \cdot -3 + \left(-2 \cdot t\_0 + x \cdot 0.6666666666666666\right)\right)\right)\right) - t\_0\right) - x \cdot 2\right)\\ \mathbf{else}:\\ \;\;\;\;wj - \frac{wj + \frac{x}{-1 + wj \cdot \left(-1 - wj \cdot \left(0.5 + wj \cdot 0.16666666666666666\right)\right)}}{wj + 1}\\ \end{array} \end{array} \]

FPCore

(FPCore (wj x)
 :precision binary64
 (let* ((t_0 (+ (* x -4.0) (* x 1.5))))
   (if (<= wj -0.00037)
     (+
      wj
      (/
       (-
        (*
         wj
         (+
          1.0
          (+
           x
           (*
            wj
            (-
             (+
              (* x 0.5)
              (*
               wj
               (/
                (* (* x 0.16666666666666666) (* x 0.8333333333333334))
                (* x 0.8333333333333334))))
             x)))))
        x)
       (- -1.0 wj)))
     (if (<= wj 1.3e-8)
       (+
        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
           (+ -1.0 (* wj (- -1.0 (* wj (+ 0.5 (* wj 0.16666666666666666))))))))
         (+ wj 1.0)))))))

C

double code(double wj, double x) {
	double t_0 = (x * -4.0) + (x * 1.5);
	double tmp;
	if (wj <= -0.00037) {
		tmp = wj + (((wj * (1.0 + (x + (wj * (((x * 0.5) + (wj * (((x * 0.16666666666666666) * (x * 0.8333333333333334)) / (x * 0.8333333333333334)))) - x))))) - x) / (-1.0 - wj));
	} else if (wj <= 1.3e-8) {
		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 / (-1.0 + (wj * (-1.0 - (wj * (0.5 + (wj * 0.16666666666666666)))))))) / (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 <= (-0.00037d0)) then
        tmp = wj + (((wj * (1.0d0 + (x + (wj * (((x * 0.5d0) + (wj * (((x * 0.16666666666666666d0) * (x * 0.8333333333333334d0)) / (x * 0.8333333333333334d0)))) - x))))) - x) / ((-1.0d0) - wj))
    else if (wj <= 1.3d-8) 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 / ((-1.0d0) + (wj * ((-1.0d0) - (wj * (0.5d0 + (wj * 0.16666666666666666d0)))))))) / (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 <= -0.00037) {
		tmp = wj + (((wj * (1.0 + (x + (wj * (((x * 0.5) + (wj * (((x * 0.16666666666666666) * (x * 0.8333333333333334)) / (x * 0.8333333333333334)))) - x))))) - x) / (-1.0 - wj));
	} else if (wj <= 1.3e-8) {
		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 / (-1.0 + (wj * (-1.0 - (wj * (0.5 + (wj * 0.16666666666666666)))))))) / (wj + 1.0));
	}
	return tmp;
}

Python

def code(wj, x):
	t_0 = (x * -4.0) + (x * 1.5)
	tmp = 0
	if wj <= -0.00037:
		tmp = wj + (((wj * (1.0 + (x + (wj * (((x * 0.5) + (wj * (((x * 0.16666666666666666) * (x * 0.8333333333333334)) / (x * 0.8333333333333334)))) - x))))) - x) / (-1.0 - wj))
	elif wj <= 1.3e-8:
		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 / (-1.0 + (wj * (-1.0 - (wj * (0.5 + (wj * 0.16666666666666666)))))))) / (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 <= -0.00037)
		tmp = Float64(wj + Float64(Float64(Float64(wj * Float64(1.0 + Float64(x + Float64(wj * Float64(Float64(Float64(x * 0.5) + Float64(wj * Float64(Float64(Float64(x * 0.16666666666666666) * Float64(x * 0.8333333333333334)) / Float64(x * 0.8333333333333334)))) - x))))) - x) / Float64(-1.0 - wj)));
	elseif (wj <= 1.3e-8)
		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 / Float64(-1.0 + Float64(wj * Float64(-1.0 - Float64(wj * Float64(0.5 + Float64(wj * 0.16666666666666666)))))))) / 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 <= -0.00037)
		tmp = wj + (((wj * (1.0 + (x + (wj * (((x * 0.5) + (wj * (((x * 0.16666666666666666) * (x * 0.8333333333333334)) / (x * 0.8333333333333334)))) - x))))) - x) / (-1.0 - wj));
	elseif (wj <= 1.3e-8)
		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 / (-1.0 + (wj * (-1.0 - (wj * (0.5 + (wj * 0.16666666666666666)))))))) / (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, -0.00037], N[(wj + N[(N[(N[(wj * N[(1.0 + N[(x + N[(wj * N[(N[(N[(x * 0.5), $MachinePrecision] + N[(wj * N[(N[(N[(x * 0.16666666666666666), $MachinePrecision] * N[(x * 0.8333333333333334), $MachinePrecision]), $MachinePrecision] / N[(x * 0.8333333333333334), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - x), $MachinePrecision] / N[(-1.0 - wj), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[wj, 1.3e-8], 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[(-1.0 + N[(wj * N[(-1.0 - N[(wj * N[(0.5 + N[(wj * 0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(wj + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if wj < -3.6999999999999999e-4

   1. Initial program 51.4%

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

      1. distribute-rgt1-in 96.1%

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

      2. associate-/l/ 96.1%

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

      3. div-sub 51.6%

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

      4. associate-/l* 51.6%

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

      5. *-inverses 96.1%

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

      6. *-rgt-identity 96.1%

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

   3. Simplified 96.1%

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

   5. Taylor expanded in wj around 0 51.7%

      \[\leadsto wj - \frac{\color{blue}{wj \cdot \left(1 + \left(x + wj \cdot \left(-1 \cdot x + \left(0.5 \cdot x + wj \cdot \left(-1 \cdot \left(-1 \cdot x + 0.5 \cdot x\right) + \left(-0.5 \cdot x + 0.16666666666666666 \cdot x\right)\right)\right)\right)\right)\right) - x}}{wj + 1} \]
   6. Step-by-step derivation

      1. flip-+ 50.5%

         \[\leadsto wj - \frac{wj \cdot \left(1 + \left(x + wj \cdot \left(-1 \cdot x + \left(0.5 \cdot x + wj \cdot \color{blue}{\frac{\left(-1 \cdot \left(-1 \cdot x + 0.5 \cdot x\right)\right) \cdot \left(-1 \cdot \left(-1 \cdot x + 0.5 \cdot x\right)\right) - \left(-0.5 \cdot x + 0.16666666666666666 \cdot x\right) \cdot \left(-0.5 \cdot x + 0.16666666666666666 \cdot x\right)}{-1 \cdot \left(-1 \cdot x + 0.5 \cdot x\right) - \left(-0.5 \cdot x + 0.16666666666666666 \cdot x\right)}}\right)\right)\right)\right) - x}{wj + 1} \]

      2. pow2 50.5%

         \[\leadsto wj - \frac{wj \cdot \left(1 + \left(x + wj \cdot \left(-1 \cdot x + \left(0.5 \cdot x + wj \cdot \frac{\color{blue}{{\left(-1 \cdot \left(-1 \cdot x + 0.5 \cdot x\right)\right)}^{2}} - \left(-0.5 \cdot x + 0.16666666666666666 \cdot x\right) \cdot \left(-0.5 \cdot x + 0.16666666666666666 \cdot x\right)}{-1 \cdot \left(-1 \cdot x + 0.5 \cdot x\right) - \left(-0.5 \cdot x + 0.16666666666666666 \cdot x\right)}\right)\right)\right)\right) - x}{wj + 1} \]

      3. mul-1-neg 50.5%

         \[\leadsto wj - \frac{wj \cdot \left(1 + \left(x + wj \cdot \left(-1 \cdot x + \left(0.5 \cdot x + wj \cdot \frac{{\color{blue}{\left(-\left(-1 \cdot x + 0.5 \cdot x\right)\right)}}^{2} - \left(-0.5 \cdot x + 0.16666666666666666 \cdot x\right) \cdot \left(-0.5 \cdot x + 0.16666666666666666 \cdot x\right)}{-1 \cdot \left(-1 \cdot x + 0.5 \cdot x\right) - \left(-0.5 \cdot x + 0.16666666666666666 \cdot x\right)}\right)\right)\right)\right) - x}{wj + 1} \]

      4. distribute-rgt-out 50.5%

         \[\leadsto wj - \frac{wj \cdot \left(1 + \left(x + wj \cdot \left(-1 \cdot x + \left(0.5 \cdot x + wj \cdot \frac{{\left(-\color{blue}{x \cdot \left(-1 + 0.5\right)}\right)}^{2} - \left(-0.5 \cdot x + 0.16666666666666666 \cdot x\right) \cdot \left(-0.5 \cdot x + 0.16666666666666666 \cdot x\right)}{-1 \cdot \left(-1 \cdot x + 0.5 \cdot x\right) - \left(-0.5 \cdot x + 0.16666666666666666 \cdot x\right)}\right)\right)\right)\right) - x}{wj + 1} \]

      5. metadata-eval 50.5%

         \[\leadsto wj - \frac{wj \cdot \left(1 + \left(x + wj \cdot \left(-1 \cdot x + \left(0.5 \cdot x + wj \cdot \frac{{\left(-x \cdot \color{blue}{-0.5}\right)}^{2} - \left(-0.5 \cdot x + 0.16666666666666666 \cdot x\right) \cdot \left(-0.5 \cdot x + 0.16666666666666666 \cdot x\right)}{-1 \cdot \left(-1 \cdot x + 0.5 \cdot x\right) - \left(-0.5 \cdot x + 0.16666666666666666 \cdot x\right)}\right)\right)\right)\right) - x}{wj + 1} \]

      6. pow2 50.5%

         \[\leadsto wj - \frac{wj \cdot \left(1 + \left(x + wj \cdot \left(-1 \cdot x + \left(0.5 \cdot x + wj \cdot \frac{{\left(-x \cdot -0.5\right)}^{2} - \color{blue}{{\left(-0.5 \cdot x + 0.16666666666666666 \cdot x\right)}^{2}}}{-1 \cdot \left(-1 \cdot x + 0.5 \cdot x\right) - \left(-0.5 \cdot x + 0.16666666666666666 \cdot x\right)}\right)\right)\right)\right) - x}{wj + 1} \]

      7. distribute-rgt-out 50.5%

         \[\leadsto wj - \frac{wj \cdot \left(1 + \left(x + wj \cdot \left(-1 \cdot x + \left(0.5 \cdot x + wj \cdot \frac{{\left(-x \cdot -0.5\right)}^{2} - {\color{blue}{\left(x \cdot \left(-0.5 + 0.16666666666666666\right)\right)}}^{2}}{-1 \cdot \left(-1 \cdot x + 0.5 \cdot x\right) - \left(-0.5 \cdot x + 0.16666666666666666 \cdot x\right)}\right)\right)\right)\right) - x}{wj + 1} \]

      8. metadata-eval 50.5%

         \[\leadsto wj - \frac{wj \cdot \left(1 + \left(x + wj \cdot \left(-1 \cdot x + \left(0.5 \cdot x + wj \cdot \frac{{\left(-x \cdot -0.5\right)}^{2} - {\left(x \cdot \color{blue}{-0.3333333333333333}\right)}^{2}}{-1 \cdot \left(-1 \cdot x + 0.5 \cdot x\right) - \left(-0.5 \cdot x + 0.16666666666666666 \cdot x\right)}\right)\right)\right)\right) - x}{wj + 1} \]

      9. mul-1-neg 50.5%

         \[\leadsto wj - \frac{wj \cdot \left(1 + \left(x + wj \cdot \left(-1 \cdot x + \left(0.5 \cdot x + wj \cdot \frac{{\left(-x \cdot -0.5\right)}^{2} - {\left(x \cdot -0.3333333333333333\right)}^{2}}{\color{blue}{\left(-\left(-1 \cdot x + 0.5 \cdot x\right)\right)} - \left(-0.5 \cdot x + 0.16666666666666666 \cdot x\right)}\right)\right)\right)\right) - x}{wj + 1} \]

      10. distribute-rgt-out 50.5%

          \[\leadsto wj - \frac{wj \cdot \left(1 + \left(x + wj \cdot \left(-1 \cdot x + \left(0.5 \cdot x + wj \cdot \frac{{\left(-x \cdot -0.5\right)}^{2} - {\left(x \cdot -0.3333333333333333\right)}^{2}}{\left(-\color{blue}{x \cdot \left(-1 + 0.5\right)}\right) - \left(-0.5 \cdot x + 0.16666666666666666 \cdot x\right)}\right)\right)\right)\right) - x}{wj + 1} \]

      11. metadata-eval 50.5%

          \[\leadsto wj - \frac{wj \cdot \left(1 + \left(x + wj \cdot \left(-1 \cdot x + \left(0.5 \cdot x + wj \cdot \frac{{\left(-x \cdot -0.5\right)}^{2} - {\left(x \cdot -0.3333333333333333\right)}^{2}}{\left(-x \cdot \color{blue}{-0.5}\right) - \left(-0.5 \cdot x + 0.16666666666666666 \cdot x\right)}\right)\right)\right)\right) - x}{wj + 1} \]

      12. distribute-rgt-out 50.5%

          \[\leadsto wj - \frac{wj \cdot \left(1 + \left(x + wj \cdot \left(-1 \cdot x + \left(0.5 \cdot x + wj \cdot \frac{{\left(-x \cdot -0.5\right)}^{2} - {\left(x \cdot -0.3333333333333333\right)}^{2}}{\left(-x \cdot -0.5\right) - \color{blue}{x \cdot \left(-0.5 + 0.16666666666666666\right)}}\right)\right)\right)\right) - x}{wj + 1} \]

      13. metadata-eval 50.5%

          \[\leadsto wj - \frac{wj \cdot \left(1 + \left(x + wj \cdot \left(-1 \cdot x + \left(0.5 \cdot x + wj \cdot \frac{{\left(-x \cdot -0.5\right)}^{2} - {\left(x \cdot -0.3333333333333333\right)}^{2}}{\left(-x \cdot -0.5\right) - x \cdot \color{blue}{-0.3333333333333333}}\right)\right)\right)\right) - x}{wj + 1} \]

   7. Applied egg-rr 50.5%

      \[\leadsto wj - \frac{wj \cdot \left(1 + \left(x + wj \cdot \left(-1 \cdot x + \left(0.5 \cdot x + wj \cdot \color{blue}{\frac{{\left(-x \cdot -0.5\right)}^{2} - {\left(x \cdot -0.3333333333333333\right)}^{2}}{\left(-x \cdot -0.5\right) - x \cdot -0.3333333333333333}}\right)\right)\right)\right) - x}{wj + 1} \]
   8. Step-by-step derivation

      1. unpow2 50.5%

         \[\leadsto wj - \frac{wj \cdot \left(1 + \left(x + wj \cdot \left(-1 \cdot x + \left(0.5 \cdot x + wj \cdot \frac{\color{blue}{\left(-x \cdot -0.5\right) \cdot \left(-x \cdot -0.5\right)} - {\left(x \cdot -0.3333333333333333\right)}^{2}}{\left(-x \cdot -0.5\right) - x \cdot -0.3333333333333333}\right)\right)\right)\right) - x}{wj + 1} \]

      2. unpow2 50.5%

         \[\leadsto wj - \frac{wj \cdot \left(1 + \left(x + wj \cdot \left(-1 \cdot x + \left(0.5 \cdot x + wj \cdot \frac{\left(-x \cdot -0.5\right) \cdot \left(-x \cdot -0.5\right) - \color{blue}{\left(x \cdot -0.3333333333333333\right) \cdot \left(x \cdot -0.3333333333333333\right)}}{\left(-x \cdot -0.5\right) - x \cdot -0.3333333333333333}\right)\right)\right)\right) - x}{wj + 1} \]

      3. difference-of-squares 72.8%

         \[\leadsto wj - \frac{wj \cdot \left(1 + \left(x + wj \cdot \left(-1 \cdot x + \left(0.5 \cdot x + wj \cdot \frac{\color{blue}{\left(\left(-x \cdot -0.5\right) + x \cdot -0.3333333333333333\right) \cdot \left(\left(-x \cdot -0.5\right) - x \cdot -0.3333333333333333\right)}}{\left(-x \cdot -0.5\right) - x \cdot -0.3333333333333333}\right)\right)\right)\right) - x}{wj + 1} \]

      4. distribute-rgt-neg-in 72.8%

         \[\leadsto wj - \frac{wj \cdot \left(1 + \left(x + wj \cdot \left(-1 \cdot x + \left(0.5 \cdot x + wj \cdot \frac{\left(\color{blue}{x \cdot \left(--0.5\right)} + x \cdot -0.3333333333333333\right) \cdot \left(\left(-x \cdot -0.5\right) - x \cdot -0.3333333333333333\right)}{\left(-x \cdot -0.5\right) - x \cdot -0.3333333333333333}\right)\right)\right)\right) - x}{wj + 1} \]

      5. metadata-eval 72.8%

         \[\leadsto wj - \frac{wj \cdot \left(1 + \left(x + wj \cdot \left(-1 \cdot x + \left(0.5 \cdot x + wj \cdot \frac{\left(x \cdot \color{blue}{0.5} + x \cdot -0.3333333333333333\right) \cdot \left(\left(-x \cdot -0.5\right) - x \cdot -0.3333333333333333\right)}{\left(-x \cdot -0.5\right) - x \cdot -0.3333333333333333}\right)\right)\right)\right) - x}{wj + 1} \]

      6. distribute-lft-out 72.8%

         \[\leadsto wj - \frac{wj \cdot \left(1 + \left(x + wj \cdot \left(-1 \cdot x + \left(0.5 \cdot x + wj \cdot \frac{\color{blue}{\left(x \cdot \left(0.5 + -0.3333333333333333\right)\right)} \cdot \left(\left(-x \cdot -0.5\right) - x \cdot -0.3333333333333333\right)}{\left(-x \cdot -0.5\right) - x \cdot -0.3333333333333333}\right)\right)\right)\right) - x}{wj + 1} \]

      7. metadata-eval 72.8%

         \[\leadsto wj - \frac{wj \cdot \left(1 + \left(x + wj \cdot \left(-1 \cdot x + \left(0.5 \cdot x + wj \cdot \frac{\left(x \cdot \color{blue}{0.16666666666666666}\right) \cdot \left(\left(-x \cdot -0.5\right) - x \cdot -0.3333333333333333\right)}{\left(-x \cdot -0.5\right) - x \cdot -0.3333333333333333}\right)\right)\right)\right) - x}{wj + 1} \]

      8. distribute-rgt-neg-in 72.8%

         \[\leadsto wj - \frac{wj \cdot \left(1 + \left(x + wj \cdot \left(-1 \cdot x + \left(0.5 \cdot x + wj \cdot \frac{\left(x \cdot 0.16666666666666666\right) \cdot \left(\color{blue}{x \cdot \left(--0.5\right)} - x \cdot -0.3333333333333333\right)}{\left(-x \cdot -0.5\right) - x \cdot -0.3333333333333333}\right)\right)\right)\right) - x}{wj + 1} \]

      9. metadata-eval 72.8%

         \[\leadsto wj - \frac{wj \cdot \left(1 + \left(x + wj \cdot \left(-1 \cdot x + \left(0.5 \cdot x + wj \cdot \frac{\left(x \cdot 0.16666666666666666\right) \cdot \left(x \cdot \color{blue}{0.5} - x \cdot -0.3333333333333333\right)}{\left(-x \cdot -0.5\right) - x \cdot -0.3333333333333333}\right)\right)\right)\right) - x}{wj + 1} \]

      10. distribute-lft-out-- 72.8%

          \[\leadsto wj - \frac{wj \cdot \left(1 + \left(x + wj \cdot \left(-1 \cdot x + \left(0.5 \cdot x + wj \cdot \frac{\left(x \cdot 0.16666666666666666\right) \cdot \color{blue}{\left(x \cdot \left(0.5 - -0.3333333333333333\right)\right)}}{\left(-x \cdot -0.5\right) - x \cdot -0.3333333333333333}\right)\right)\right)\right) - x}{wj + 1} \]

      11. metadata-eval 72.8%

          \[\leadsto wj - \frac{wj \cdot \left(1 + \left(x + wj \cdot \left(-1 \cdot x + \left(0.5 \cdot x + wj \cdot \frac{\left(x \cdot 0.16666666666666666\right) \cdot \left(x \cdot \color{blue}{0.8333333333333334}\right)}{\left(-x \cdot -0.5\right) - x \cdot -0.3333333333333333}\right)\right)\right)\right) - x}{wj + 1} \]

      12. distribute-rgt-neg-in 72.8%

          \[\leadsto wj - \frac{wj \cdot \left(1 + \left(x + wj \cdot \left(-1 \cdot x + \left(0.5 \cdot x + wj \cdot \frac{\left(x \cdot 0.16666666666666666\right) \cdot \left(x \cdot 0.8333333333333334\right)}{\color{blue}{x \cdot \left(--0.5\right)} - x \cdot -0.3333333333333333}\right)\right)\right)\right) - x}{wj + 1} \]

      13. metadata-eval 72.8%

          \[\leadsto wj - \frac{wj \cdot \left(1 + \left(x + wj \cdot \left(-1 \cdot x + \left(0.5 \cdot x + wj \cdot \frac{\left(x \cdot 0.16666666666666666\right) \cdot \left(x \cdot 0.8333333333333334\right)}{x \cdot \color{blue}{0.5} - x \cdot -0.3333333333333333}\right)\right)\right)\right) - x}{wj + 1} \]

      14. distribute-lft-out-- 72.8%

          \[\leadsto wj - \frac{wj \cdot \left(1 + \left(x + wj \cdot \left(-1 \cdot x + \left(0.5 \cdot x + wj \cdot \frac{\left(x \cdot 0.16666666666666666\right) \cdot \left(x \cdot 0.8333333333333334\right)}{\color{blue}{x \cdot \left(0.5 - -0.3333333333333333\right)}}\right)\right)\right)\right) - x}{wj + 1} \]

      15. metadata-eval 72.8%

          \[\leadsto wj - \frac{wj \cdot \left(1 + \left(x + wj \cdot \left(-1 \cdot x + \left(0.5 \cdot x + wj \cdot \frac{\left(x \cdot 0.16666666666666666\right) \cdot \left(x \cdot 0.8333333333333334\right)}{x \cdot \color{blue}{0.8333333333333334}}\right)\right)\right)\right) - x}{wj + 1} \]

   9. Simplified 72.8%

      \[\leadsto wj - \frac{wj \cdot \left(1 + \left(x + wj \cdot \left(-1 \cdot x + \left(0.5 \cdot x + wj \cdot \color{blue}{\frac{\left(x \cdot 0.16666666666666666\right) \cdot \left(x \cdot 0.8333333333333334\right)}{x \cdot 0.8333333333333334}}\right)\right)\right)\right) - x}{wj + 1} \]

   if -3.6999999999999999e-4 < wj < 1.3000000000000001e-8

   1. Initial program 79.4%

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

      1. distribute-rgt1-in 79.4%

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

      2. associate-/l/ 79.4%

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

      3. div-sub 79.4%

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

      4. associate-/l* 79.4%

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

      5. *-inverses 79.4%

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

      6. *-rgt-identity 79.4%

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

   3. Simplified 79.4%

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

   5. Taylor expanded in wj around 0 100.0%

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

   if 1.3000000000000001e-8 < 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 50.0%

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

      2. associate-/l/ 49.7%

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

      3. div-sub 49.7%

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

      4. associate-/l* 49.7%

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

      5. *-inverses 99.7%

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

      6. *-rgt-identity 99.7%

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

   3. Simplified 99.7%

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

   5. Taylor expanded in wj around 0 98.3%

      \[\leadsto wj - \frac{wj - \frac{x}{\color{blue}{1 + wj \cdot \left(1 + wj \cdot \left(0.5 + 0.16666666666666666 \cdot wj\right)\right)}}}{wj + 1} \]
   6. Step-by-step derivation

      1. *-commutative 98.3%

         \[\leadsto wj - \frac{wj - \frac{x}{1 + wj \cdot \left(1 + wj \cdot \left(0.5 + \color{blue}{wj \cdot 0.16666666666666666}\right)\right)}}{wj + 1} \]

   7. Simplified 98.3%

      \[\leadsto wj - \frac{wj - \frac{x}{\color{blue}{1 + wj \cdot \left(1 + wj \cdot \left(0.5 + wj \cdot 0.16666666666666666\right)\right)}}}{wj + 1} \]
3. Recombined 3 regimes into one program.

4. Final simplification 99.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;wj \leq -0.00037:\\ \;\;\;\;wj + \frac{wj \cdot \left(1 + \left(x + wj \cdot \left(\left(x \cdot 0.5 + wj \cdot \frac{\left(x \cdot 0.16666666666666666\right) \cdot \left(x \cdot 0.8333333333333334\right)}{x \cdot 0.8333333333333334}\right) - x\right)\right)\right) - x}{-1 - wj}\\ \mathbf{elif}\;wj \leq 1.3 \cdot 10^{-8}:\\ \;\;\;\;x + wj \cdot \left(wj \cdot \left(\left(1 + wj \cdot \left(-1 - \left(x \cdot -3 + \left(-2 \cdot \left(x \cdot -4 + x \cdot 1.5\right) + x \cdot 0.6666666666666666\right)\right)\right)\right) - \left(x \cdot -4 + x \cdot 1.5\right)\right) - x \cdot 2\right)\\ \mathbf{else}:\\ \;\;\;\;wj - \frac{wj + \frac{x}{-1 + wj \cdot \left(-1 - wj \cdot \left(0.5 + wj \cdot 0.16666666666666666\right)\right)}}{wj + 1}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 97.7% accurate, 7.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;wj \leq -6.2 \cdot 10^{-5}:\\ \;\;\;\;wj + \frac{wj \cdot \left(1 + \left(x + wj \cdot \left(\left(x \cdot 0.5 + wj \cdot \frac{\left(x \cdot 0.16666666666666666\right) \cdot \left(x \cdot 0.8333333333333334\right)}{x \cdot 0.8333333333333334}\right) - x\right)\right)\right) - x}{-1 - wj}\\ \mathbf{elif}\;wj \leq 1.3 \cdot 10^{-8}:\\ \;\;\;\;x + wj \cdot \left(wj \cdot \left(\left(1 - wj\right) - x \cdot -2.5\right) - x \cdot 2\right)\\ \mathbf{else}:\\ \;\;\;\;wj - \frac{wj + \frac{x}{-1 + wj \cdot \left(-1 - wj \cdot \left(0.5 + wj \cdot 0.16666666666666666\right)\right)}}{wj + 1}\\ \end{array} \end{array} \]

FPCore

(FPCore (wj x)
 :precision binary64
 (if (<= wj -6.2e-5)
   (+
    wj
    (/
     (-
      (*
       wj
       (+
        1.0
        (+
         x
         (*
          wj
          (-
           (+
            (* x 0.5)
            (*
             wj
             (/
              (* (* x 0.16666666666666666) (* x 0.8333333333333334))
              (* x 0.8333333333333334))))
           x)))))
      x)
     (- -1.0 wj)))
   (if (<= wj 1.3e-8)
     (+ x (* wj (- (* wj (- (- 1.0 wj) (* x -2.5))) (* x 2.0))))
     (-
      wj
      (/
       (+
        wj
        (/
         x
         (+ -1.0 (* wj (- -1.0 (* wj (+ 0.5 (* wj 0.16666666666666666))))))))
       (+ wj 1.0))))))

C

double code(double wj, double x) {
	double tmp;
	if (wj <= -6.2e-5) {
		tmp = wj + (((wj * (1.0 + (x + (wj * (((x * 0.5) + (wj * (((x * 0.16666666666666666) * (x * 0.8333333333333334)) / (x * 0.8333333333333334)))) - x))))) - x) / (-1.0 - wj));
	} else if (wj <= 1.3e-8) {
		tmp = x + (wj * ((wj * ((1.0 - wj) - (x * -2.5))) - (x * 2.0)));
	} else {
		tmp = wj - ((wj + (x / (-1.0 + (wj * (-1.0 - (wj * (0.5 + (wj * 0.16666666666666666)))))))) / (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 <= (-6.2d-5)) then
        tmp = wj + (((wj * (1.0d0 + (x + (wj * (((x * 0.5d0) + (wj * (((x * 0.16666666666666666d0) * (x * 0.8333333333333334d0)) / (x * 0.8333333333333334d0)))) - x))))) - x) / ((-1.0d0) - wj))
    else if (wj <= 1.3d-8) then
        tmp = x + (wj * ((wj * ((1.0d0 - wj) - (x * (-2.5d0)))) - (x * 2.0d0)))
    else
        tmp = wj - ((wj + (x / ((-1.0d0) + (wj * ((-1.0d0) - (wj * (0.5d0 + (wj * 0.16666666666666666d0)))))))) / (wj + 1.0d0))
    end if
    code = tmp
end function

Java

public static double code(double wj, double x) {
	double tmp;
	if (wj <= -6.2e-5) {
		tmp = wj + (((wj * (1.0 + (x + (wj * (((x * 0.5) + (wj * (((x * 0.16666666666666666) * (x * 0.8333333333333334)) / (x * 0.8333333333333334)))) - x))))) - x) / (-1.0 - wj));
	} else if (wj <= 1.3e-8) {
		tmp = x + (wj * ((wj * ((1.0 - wj) - (x * -2.5))) - (x * 2.0)));
	} else {
		tmp = wj - ((wj + (x / (-1.0 + (wj * (-1.0 - (wj * (0.5 + (wj * 0.16666666666666666)))))))) / (wj + 1.0));
	}
	return tmp;
}

Python

def code(wj, x):
	tmp = 0
	if wj <= -6.2e-5:
		tmp = wj + (((wj * (1.0 + (x + (wj * (((x * 0.5) + (wj * (((x * 0.16666666666666666) * (x * 0.8333333333333334)) / (x * 0.8333333333333334)))) - x))))) - x) / (-1.0 - wj))
	elif wj <= 1.3e-8:
		tmp = x + (wj * ((wj * ((1.0 - wj) - (x * -2.5))) - (x * 2.0)))
	else:
		tmp = wj - ((wj + (x / (-1.0 + (wj * (-1.0 - (wj * (0.5 + (wj * 0.16666666666666666)))))))) / (wj + 1.0))
	return tmp

Julia

function code(wj, x)
	tmp = 0.0
	if (wj <= -6.2e-5)
		tmp = Float64(wj + Float64(Float64(Float64(wj * Float64(1.0 + Float64(x + Float64(wj * Float64(Float64(Float64(x * 0.5) + Float64(wj * Float64(Float64(Float64(x * 0.16666666666666666) * Float64(x * 0.8333333333333334)) / Float64(x * 0.8333333333333334)))) - x))))) - x) / Float64(-1.0 - wj)));
	elseif (wj <= 1.3e-8)
		tmp = Float64(x + Float64(wj * Float64(Float64(wj * Float64(Float64(1.0 - wj) - Float64(x * -2.5))) - Float64(x * 2.0))));
	else
		tmp = Float64(wj - Float64(Float64(wj + Float64(x / Float64(-1.0 + Float64(wj * Float64(-1.0 - Float64(wj * Float64(0.5 + Float64(wj * 0.16666666666666666)))))))) / Float64(wj + 1.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(wj, x)
	tmp = 0.0;
	if (wj <= -6.2e-5)
		tmp = wj + (((wj * (1.0 + (x + (wj * (((x * 0.5) + (wj * (((x * 0.16666666666666666) * (x * 0.8333333333333334)) / (x * 0.8333333333333334)))) - x))))) - x) / (-1.0 - wj));
	elseif (wj <= 1.3e-8)
		tmp = x + (wj * ((wj * ((1.0 - wj) - (x * -2.5))) - (x * 2.0)));
	else
		tmp = wj - ((wj + (x / (-1.0 + (wj * (-1.0 - (wj * (0.5 + (wj * 0.16666666666666666)))))))) / (wj + 1.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[wj_, x_] := If[LessEqual[wj, -6.2e-5], N[(wj + N[(N[(N[(wj * N[(1.0 + N[(x + N[(wj * N[(N[(N[(x * 0.5), $MachinePrecision] + N[(wj * N[(N[(N[(x * 0.16666666666666666), $MachinePrecision] * N[(x * 0.8333333333333334), $MachinePrecision]), $MachinePrecision] / N[(x * 0.8333333333333334), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - x), $MachinePrecision] / N[(-1.0 - wj), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[wj, 1.3e-8], N[(x + N[(wj * N[(N[(wj * N[(N[(1.0 - wj), $MachinePrecision] - N[(x * -2.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(x * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(wj - N[(N[(wj + N[(x / N[(-1.0 + N[(wj * N[(-1.0 - N[(wj * N[(0.5 + N[(wj * 0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(wj + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if wj < -6.20000000000000027e-5

   1. Initial program 51.4%

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

      1. distribute-rgt1-in 96.1%

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

      2. associate-/l/ 96.1%

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

      3. div-sub 51.6%

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

      4. associate-/l* 51.6%

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

      5. *-inverses 96.1%

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

      6. *-rgt-identity 96.1%

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

   3. Simplified 96.1%

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

   5. Taylor expanded in wj around 0 51.7%

      \[\leadsto wj - \frac{\color{blue}{wj \cdot \left(1 + \left(x + wj \cdot \left(-1 \cdot x + \left(0.5 \cdot x + wj \cdot \left(-1 \cdot \left(-1 \cdot x + 0.5 \cdot x\right) + \left(-0.5 \cdot x + 0.16666666666666666 \cdot x\right)\right)\right)\right)\right)\right) - x}}{wj + 1} \]
   6. Step-by-step derivation

      1. flip-+ 50.5%

         \[\leadsto wj - \frac{wj \cdot \left(1 + \left(x + wj \cdot \left(-1 \cdot x + \left(0.5 \cdot x + wj \cdot \color{blue}{\frac{\left(-1 \cdot \left(-1 \cdot x + 0.5 \cdot x\right)\right) \cdot \left(-1 \cdot \left(-1 \cdot x + 0.5 \cdot x\right)\right) - \left(-0.5 \cdot x + 0.16666666666666666 \cdot x\right) \cdot \left(-0.5 \cdot x + 0.16666666666666666 \cdot x\right)}{-1 \cdot \left(-1 \cdot x + 0.5 \cdot x\right) - \left(-0.5 \cdot x + 0.16666666666666666 \cdot x\right)}}\right)\right)\right)\right) - x}{wj + 1} \]

      2. pow2 50.5%

         \[\leadsto wj - \frac{wj \cdot \left(1 + \left(x + wj \cdot \left(-1 \cdot x + \left(0.5 \cdot x + wj \cdot \frac{\color{blue}{{\left(-1 \cdot \left(-1 \cdot x + 0.5 \cdot x\right)\right)}^{2}} - \left(-0.5 \cdot x + 0.16666666666666666 \cdot x\right) \cdot \left(-0.5 \cdot x + 0.16666666666666666 \cdot x\right)}{-1 \cdot \left(-1 \cdot x + 0.5 \cdot x\right) - \left(-0.5 \cdot x + 0.16666666666666666 \cdot x\right)}\right)\right)\right)\right) - x}{wj + 1} \]

      3. mul-1-neg 50.5%

         \[\leadsto wj - \frac{wj \cdot \left(1 + \left(x + wj \cdot \left(-1 \cdot x + \left(0.5 \cdot x + wj \cdot \frac{{\color{blue}{\left(-\left(-1 \cdot x + 0.5 \cdot x\right)\right)}}^{2} - \left(-0.5 \cdot x + 0.16666666666666666 \cdot x\right) \cdot \left(-0.5 \cdot x + 0.16666666666666666 \cdot x\right)}{-1 \cdot \left(-1 \cdot x + 0.5 \cdot x\right) - \left(-0.5 \cdot x + 0.16666666666666666 \cdot x\right)}\right)\right)\right)\right) - x}{wj + 1} \]

      4. distribute-rgt-out 50.5%

         \[\leadsto wj - \frac{wj \cdot \left(1 + \left(x + wj \cdot \left(-1 \cdot x + \left(0.5 \cdot x + wj \cdot \frac{{\left(-\color{blue}{x \cdot \left(-1 + 0.5\right)}\right)}^{2} - \left(-0.5 \cdot x + 0.16666666666666666 \cdot x\right) \cdot \left(-0.5 \cdot x + 0.16666666666666666 \cdot x\right)}{-1 \cdot \left(-1 \cdot x + 0.5 \cdot x\right) - \left(-0.5 \cdot x + 0.16666666666666666 \cdot x\right)}\right)\right)\right)\right) - x}{wj + 1} \]

      5. metadata-eval 50.5%

         \[\leadsto wj - \frac{wj \cdot \left(1 + \left(x + wj \cdot \left(-1 \cdot x + \left(0.5 \cdot x + wj \cdot \frac{{\left(-x \cdot \color{blue}{-0.5}\right)}^{2} - \left(-0.5 \cdot x + 0.16666666666666666 \cdot x\right) \cdot \left(-0.5 \cdot x + 0.16666666666666666 \cdot x\right)}{-1 \cdot \left(-1 \cdot x + 0.5 \cdot x\right) - \left(-0.5 \cdot x + 0.16666666666666666 \cdot x\right)}\right)\right)\right)\right) - x}{wj + 1} \]

      6. pow2 50.5%

         \[\leadsto wj - \frac{wj \cdot \left(1 + \left(x + wj \cdot \left(-1 \cdot x + \left(0.5 \cdot x + wj \cdot \frac{{\left(-x \cdot -0.5\right)}^{2} - \color{blue}{{\left(-0.5 \cdot x + 0.16666666666666666 \cdot x\right)}^{2}}}{-1 \cdot \left(-1 \cdot x + 0.5 \cdot x\right) - \left(-0.5 \cdot x + 0.16666666666666666 \cdot x\right)}\right)\right)\right)\right) - x}{wj + 1} \]

      7. distribute-rgt-out 50.5%

         \[\leadsto wj - \frac{wj \cdot \left(1 + \left(x + wj \cdot \left(-1 \cdot x + \left(0.5 \cdot x + wj \cdot \frac{{\left(-x \cdot -0.5\right)}^{2} - {\color{blue}{\left(x \cdot \left(-0.5 + 0.16666666666666666\right)\right)}}^{2}}{-1 \cdot \left(-1 \cdot x + 0.5 \cdot x\right) - \left(-0.5 \cdot x + 0.16666666666666666 \cdot x\right)}\right)\right)\right)\right) - x}{wj + 1} \]

      8. metadata-eval 50.5%

         \[\leadsto wj - \frac{wj \cdot \left(1 + \left(x + wj \cdot \left(-1 \cdot x + \left(0.5 \cdot x + wj \cdot \frac{{\left(-x \cdot -0.5\right)}^{2} - {\left(x \cdot \color{blue}{-0.3333333333333333}\right)}^{2}}{-1 \cdot \left(-1 \cdot x + 0.5 \cdot x\right) - \left(-0.5 \cdot x + 0.16666666666666666 \cdot x\right)}\right)\right)\right)\right) - x}{wj + 1} \]

      9. mul-1-neg 50.5%

         \[\leadsto wj - \frac{wj \cdot \left(1 + \left(x + wj \cdot \left(-1 \cdot x + \left(0.5 \cdot x + wj \cdot \frac{{\left(-x \cdot -0.5\right)}^{2} - {\left(x \cdot -0.3333333333333333\right)}^{2}}{\color{blue}{\left(-\left(-1 \cdot x + 0.5 \cdot x\right)\right)} - \left(-0.5 \cdot x + 0.16666666666666666 \cdot x\right)}\right)\right)\right)\right) - x}{wj + 1} \]

      10. distribute-rgt-out 50.5%

          \[\leadsto wj - \frac{wj \cdot \left(1 + \left(x + wj \cdot \left(-1 \cdot x + \left(0.5 \cdot x + wj \cdot \frac{{\left(-x \cdot -0.5\right)}^{2} - {\left(x \cdot -0.3333333333333333\right)}^{2}}{\left(-\color{blue}{x \cdot \left(-1 + 0.5\right)}\right) - \left(-0.5 \cdot x + 0.16666666666666666 \cdot x\right)}\right)\right)\right)\right) - x}{wj + 1} \]

      11. metadata-eval 50.5%

          \[\leadsto wj - \frac{wj \cdot \left(1 + \left(x + wj \cdot \left(-1 \cdot x + \left(0.5 \cdot x + wj \cdot \frac{{\left(-x \cdot -0.5\right)}^{2} - {\left(x \cdot -0.3333333333333333\right)}^{2}}{\left(-x \cdot \color{blue}{-0.5}\right) - \left(-0.5 \cdot x + 0.16666666666666666 \cdot x\right)}\right)\right)\right)\right) - x}{wj + 1} \]

      12. distribute-rgt-out 50.5%

          \[\leadsto wj - \frac{wj \cdot \left(1 + \left(x + wj \cdot \left(-1 \cdot x + \left(0.5 \cdot x + wj \cdot \frac{{\left(-x \cdot -0.5\right)}^{2} - {\left(x \cdot -0.3333333333333333\right)}^{2}}{\left(-x \cdot -0.5\right) - \color{blue}{x \cdot \left(-0.5 + 0.16666666666666666\right)}}\right)\right)\right)\right) - x}{wj + 1} \]

      13. metadata-eval 50.5%

          \[\leadsto wj - \frac{wj \cdot \left(1 + \left(x + wj \cdot \left(-1 \cdot x + \left(0.5 \cdot x + wj \cdot \frac{{\left(-x \cdot -0.5\right)}^{2} - {\left(x \cdot -0.3333333333333333\right)}^{2}}{\left(-x \cdot -0.5\right) - x \cdot \color{blue}{-0.3333333333333333}}\right)\right)\right)\right) - x}{wj + 1} \]

   7. Applied egg-rr 50.5%

      \[\leadsto wj - \frac{wj \cdot \left(1 + \left(x + wj \cdot \left(-1 \cdot x + \left(0.5 \cdot x + wj \cdot \color{blue}{\frac{{\left(-x \cdot -0.5\right)}^{2} - {\left(x \cdot -0.3333333333333333\right)}^{2}}{\left(-x \cdot -0.5\right) - x \cdot -0.3333333333333333}}\right)\right)\right)\right) - x}{wj + 1} \]
   8. Step-by-step derivation

      1. unpow2 50.5%

         \[\leadsto wj - \frac{wj \cdot \left(1 + \left(x + wj \cdot \left(-1 \cdot x + \left(0.5 \cdot x + wj \cdot \frac{\color{blue}{\left(-x \cdot -0.5\right) \cdot \left(-x \cdot -0.5\right)} - {\left(x \cdot -0.3333333333333333\right)}^{2}}{\left(-x \cdot -0.5\right) - x \cdot -0.3333333333333333}\right)\right)\right)\right) - x}{wj + 1} \]

      2. unpow2 50.5%

         \[\leadsto wj - \frac{wj \cdot \left(1 + \left(x + wj \cdot \left(-1 \cdot x + \left(0.5 \cdot x + wj \cdot \frac{\left(-x \cdot -0.5\right) \cdot \left(-x \cdot -0.5\right) - \color{blue}{\left(x \cdot -0.3333333333333333\right) \cdot \left(x \cdot -0.3333333333333333\right)}}{\left(-x \cdot -0.5\right) - x \cdot -0.3333333333333333}\right)\right)\right)\right) - x}{wj + 1} \]

      3. difference-of-squares 72.8%

         \[\leadsto wj - \frac{wj \cdot \left(1 + \left(x + wj \cdot \left(-1 \cdot x + \left(0.5 \cdot x + wj \cdot \frac{\color{blue}{\left(\left(-x \cdot -0.5\right) + x \cdot -0.3333333333333333\right) \cdot \left(\left(-x \cdot -0.5\right) - x \cdot -0.3333333333333333\right)}}{\left(-x \cdot -0.5\right) - x \cdot -0.3333333333333333}\right)\right)\right)\right) - x}{wj + 1} \]

      4. distribute-rgt-neg-in 72.8%

         \[\leadsto wj - \frac{wj \cdot \left(1 + \left(x + wj \cdot \left(-1 \cdot x + \left(0.5 \cdot x + wj \cdot \frac{\left(\color{blue}{x \cdot \left(--0.5\right)} + x \cdot -0.3333333333333333\right) \cdot \left(\left(-x \cdot -0.5\right) - x \cdot -0.3333333333333333\right)}{\left(-x \cdot -0.5\right) - x \cdot -0.3333333333333333}\right)\right)\right)\right) - x}{wj + 1} \]

      5. metadata-eval 72.8%

         \[\leadsto wj - \frac{wj \cdot \left(1 + \left(x + wj \cdot \left(-1 \cdot x + \left(0.5 \cdot x + wj \cdot \frac{\left(x \cdot \color{blue}{0.5} + x \cdot -0.3333333333333333\right) \cdot \left(\left(-x \cdot -0.5\right) - x \cdot -0.3333333333333333\right)}{\left(-x \cdot -0.5\right) - x \cdot -0.3333333333333333}\right)\right)\right)\right) - x}{wj + 1} \]

      6. distribute-lft-out 72.8%

         \[\leadsto wj - \frac{wj \cdot \left(1 + \left(x + wj \cdot \left(-1 \cdot x + \left(0.5 \cdot x + wj \cdot \frac{\color{blue}{\left(x \cdot \left(0.5 + -0.3333333333333333\right)\right)} \cdot \left(\left(-x \cdot -0.5\right) - x \cdot -0.3333333333333333\right)}{\left(-x \cdot -0.5\right) - x \cdot -0.3333333333333333}\right)\right)\right)\right) - x}{wj + 1} \]

      7. metadata-eval 72.8%

         \[\leadsto wj - \frac{wj \cdot \left(1 + \left(x + wj \cdot \left(-1 \cdot x + \left(0.5 \cdot x + wj \cdot \frac{\left(x \cdot \color{blue}{0.16666666666666666}\right) \cdot \left(\left(-x \cdot -0.5\right) - x \cdot -0.3333333333333333\right)}{\left(-x \cdot -0.5\right) - x \cdot -0.3333333333333333}\right)\right)\right)\right) - x}{wj + 1} \]

      8. distribute-rgt-neg-in 72.8%

         \[\leadsto wj - \frac{wj \cdot \left(1 + \left(x + wj \cdot \left(-1 \cdot x + \left(0.5 \cdot x + wj \cdot \frac{\left(x \cdot 0.16666666666666666\right) \cdot \left(\color{blue}{x \cdot \left(--0.5\right)} - x \cdot -0.3333333333333333\right)}{\left(-x \cdot -0.5\right) - x \cdot -0.3333333333333333}\right)\right)\right)\right) - x}{wj + 1} \]

      9. metadata-eval 72.8%

         \[\leadsto wj - \frac{wj \cdot \left(1 + \left(x + wj \cdot \left(-1 \cdot x + \left(0.5 \cdot x + wj \cdot \frac{\left(x \cdot 0.16666666666666666\right) \cdot \left(x \cdot \color{blue}{0.5} - x \cdot -0.3333333333333333\right)}{\left(-x \cdot -0.5\right) - x \cdot -0.3333333333333333}\right)\right)\right)\right) - x}{wj + 1} \]

      10. distribute-lft-out-- 72.8%

          \[\leadsto wj - \frac{wj \cdot \left(1 + \left(x + wj \cdot \left(-1 \cdot x + \left(0.5 \cdot x + wj \cdot \frac{\left(x \cdot 0.16666666666666666\right) \cdot \color{blue}{\left(x \cdot \left(0.5 - -0.3333333333333333\right)\right)}}{\left(-x \cdot -0.5\right) - x \cdot -0.3333333333333333}\right)\right)\right)\right) - x}{wj + 1} \]

      11. metadata-eval 72.8%

          \[\leadsto wj - \frac{wj \cdot \left(1 + \left(x + wj \cdot \left(-1 \cdot x + \left(0.5 \cdot x + wj \cdot \frac{\left(x \cdot 0.16666666666666666\right) \cdot \left(x \cdot \color{blue}{0.8333333333333334}\right)}{\left(-x \cdot -0.5\right) - x \cdot -0.3333333333333333}\right)\right)\right)\right) - x}{wj + 1} \]

      12. distribute-rgt-neg-in 72.8%

          \[\leadsto wj - \frac{wj \cdot \left(1 + \left(x + wj \cdot \left(-1 \cdot x + \left(0.5 \cdot x + wj \cdot \frac{\left(x \cdot 0.16666666666666666\right) \cdot \left(x \cdot 0.8333333333333334\right)}{\color{blue}{x \cdot \left(--0.5\right)} - x \cdot -0.3333333333333333}\right)\right)\right)\right) - x}{wj + 1} \]

      13. metadata-eval 72.8%

          \[\leadsto wj - \frac{wj \cdot \left(1 + \left(x + wj \cdot \left(-1 \cdot x + \left(0.5 \cdot x + wj \cdot \frac{\left(x \cdot 0.16666666666666666\right) \cdot \left(x \cdot 0.8333333333333334\right)}{x \cdot \color{blue}{0.5} - x \cdot -0.3333333333333333}\right)\right)\right)\right) - x}{wj + 1} \]

      14. distribute-lft-out-- 72.8%

          \[\leadsto wj - \frac{wj \cdot \left(1 + \left(x + wj \cdot \left(-1 \cdot x + \left(0.5 \cdot x + wj \cdot \frac{\left(x \cdot 0.16666666666666666\right) \cdot \left(x \cdot 0.8333333333333334\right)}{\color{blue}{x \cdot \left(0.5 - -0.3333333333333333\right)}}\right)\right)\right)\right) - x}{wj + 1} \]

      15. metadata-eval 72.8%

          \[\leadsto wj - \frac{wj \cdot \left(1 + \left(x + wj \cdot \left(-1 \cdot x + \left(0.5 \cdot x + wj \cdot \frac{\left(x \cdot 0.16666666666666666\right) \cdot \left(x \cdot 0.8333333333333334\right)}{x \cdot \color{blue}{0.8333333333333334}}\right)\right)\right)\right) - x}{wj + 1} \]

   9. Simplified 72.8%

      \[\leadsto wj - \frac{wj \cdot \left(1 + \left(x + wj \cdot \left(-1 \cdot x + \left(0.5 \cdot x + wj \cdot \color{blue}{\frac{\left(x \cdot 0.16666666666666666\right) \cdot \left(x \cdot 0.8333333333333334\right)}{x \cdot 0.8333333333333334}}\right)\right)\right)\right) - x}{wj + 1} \]

   if -6.20000000000000027e-5 < wj < 1.3000000000000001e-8

   1. Initial program 79.4%

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

      1. distribute-rgt1-in 79.4%

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

      2. associate-/l/ 79.4%

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

      3. div-sub 79.4%

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

      4. associate-/l* 79.4%

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

      5. *-inverses 79.4%

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

      6. *-rgt-identity 79.4%

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

   3. Simplified 79.4%

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

   5. Taylor expanded in wj around 0 100.0%

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

   6. Taylor expanded in x around 0 99.9%

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

      1. mul-1-neg 99.9%

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

   8. Simplified 99.9%

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

   9. Taylor expanded in wj around 0 99.9%

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

       1. neg-mul-1 99.9%

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

       2. sub-neg 99.9%

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

       3. distribute-rgt-out 99.9%

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

       4. metadata-eval 99.9%

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

       5. *-commutative 99.9%

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

   11. Simplified 99.9%

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

   if 1.3000000000000001e-8 < 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 50.0%

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

      2. associate-/l/ 49.7%

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

      3. div-sub 49.7%

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

      4. associate-/l* 49.7%

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

      5. *-inverses 99.7%

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

      6. *-rgt-identity 99.7%

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

   3. Simplified 99.7%

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

   5. Taylor expanded in wj around 0 98.3%

      \[\leadsto wj - \frac{wj - \frac{x}{\color{blue}{1 + wj \cdot \left(1 + wj \cdot \left(0.5 + 0.16666666666666666 \cdot wj\right)\right)}}}{wj + 1} \]
   6. Step-by-step derivation

      1. *-commutative 98.3%

         \[\leadsto wj - \frac{wj - \frac{x}{1 + wj \cdot \left(1 + wj \cdot \left(0.5 + \color{blue}{wj \cdot 0.16666666666666666}\right)\right)}}{wj + 1} \]

   7. Simplified 98.3%

      \[\leadsto wj - \frac{wj - \frac{x}{\color{blue}{1 + wj \cdot \left(1 + wj \cdot \left(0.5 + wj \cdot 0.16666666666666666\right)\right)}}}{wj + 1} \]
3. Recombined 3 regimes into one program.

4. Final simplification 98.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;wj \leq -6.2 \cdot 10^{-5}:\\ \;\;\;\;wj + \frac{wj \cdot \left(1 + \left(x + wj \cdot \left(\left(x \cdot 0.5 + wj \cdot \frac{\left(x \cdot 0.16666666666666666\right) \cdot \left(x \cdot 0.8333333333333334\right)}{x \cdot 0.8333333333333334}\right) - x\right)\right)\right) - x}{-1 - wj}\\ \mathbf{elif}\;wj \leq 1.3 \cdot 10^{-8}:\\ \;\;\;\;x + wj \cdot \left(wj \cdot \left(\left(1 - wj\right) - x \cdot -2.5\right) - x \cdot 2\right)\\ \mathbf{else}:\\ \;\;\;\;wj - \frac{wj + \frac{x}{-1 + wj \cdot \left(-1 - wj \cdot \left(0.5 + wj \cdot 0.16666666666666666\right)\right)}}{wj + 1}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 97.1% accurate, 11.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;wj \leq 1.3 \cdot 10^{-8}:\\ \;\;\;\;x + wj \cdot \left(wj \cdot \left(\left(1 - wj\right) - x \cdot -2.5\right) - x \cdot 2\right)\\ \mathbf{else}:\\ \;\;\;\;wj - \frac{wj + \frac{x}{-1 + wj \cdot \left(-1 - wj \cdot \left(0.5 + wj \cdot 0.16666666666666666\right)\right)}}{wj + 1}\\ \end{array} \end{array} \]

FPCore

(FPCore (wj x)
 :precision binary64
 (if (<= wj 1.3e-8)
   (+ x (* wj (- (* wj (- (- 1.0 wj) (* x -2.5))) (* x 2.0))))
   (-
    wj
    (/
     (+
      wj
      (/ x (+ -1.0 (* wj (- -1.0 (* wj (+ 0.5 (* wj 0.16666666666666666))))))))
     (+ wj 1.0)))))

C

double code(double wj, double x) {
	double tmp;
	if (wj <= 1.3e-8) {
		tmp = x + (wj * ((wj * ((1.0 - wj) - (x * -2.5))) - (x * 2.0)));
	} else {
		tmp = wj - ((wj + (x / (-1.0 + (wj * (-1.0 - (wj * (0.5 + (wj * 0.16666666666666666)))))))) / (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 <= 1.3d-8) then
        tmp = x + (wj * ((wj * ((1.0d0 - wj) - (x * (-2.5d0)))) - (x * 2.0d0)))
    else
        tmp = wj - ((wj + (x / ((-1.0d0) + (wj * ((-1.0d0) - (wj * (0.5d0 + (wj * 0.16666666666666666d0)))))))) / (wj + 1.0d0))
    end if
    code = tmp
end function

Java

public static double code(double wj, double x) {
	double tmp;
	if (wj <= 1.3e-8) {
		tmp = x + (wj * ((wj * ((1.0 - wj) - (x * -2.5))) - (x * 2.0)));
	} else {
		tmp = wj - ((wj + (x / (-1.0 + (wj * (-1.0 - (wj * (0.5 + (wj * 0.16666666666666666)))))))) / (wj + 1.0));
	}
	return tmp;
}

Python

def code(wj, x):
	tmp = 0
	if wj <= 1.3e-8:
		tmp = x + (wj * ((wj * ((1.0 - wj) - (x * -2.5))) - (x * 2.0)))
	else:
		tmp = wj - ((wj + (x / (-1.0 + (wj * (-1.0 - (wj * (0.5 + (wj * 0.16666666666666666)))))))) / (wj + 1.0))
	return tmp

Julia

function code(wj, x)
	tmp = 0.0
	if (wj <= 1.3e-8)
		tmp = Float64(x + Float64(wj * Float64(Float64(wj * Float64(Float64(1.0 - wj) - Float64(x * -2.5))) - Float64(x * 2.0))));
	else
		tmp = Float64(wj - Float64(Float64(wj + Float64(x / Float64(-1.0 + Float64(wj * Float64(-1.0 - Float64(wj * Float64(0.5 + Float64(wj * 0.16666666666666666)))))))) / Float64(wj + 1.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(wj, x)
	tmp = 0.0;
	if (wj <= 1.3e-8)
		tmp = x + (wj * ((wj * ((1.0 - wj) - (x * -2.5))) - (x * 2.0)));
	else
		tmp = wj - ((wj + (x / (-1.0 + (wj * (-1.0 - (wj * (0.5 + (wj * 0.16666666666666666)))))))) / (wj + 1.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[wj_, x_] := If[LessEqual[wj, 1.3e-8], N[(x + N[(wj * N[(N[(wj * N[(N[(1.0 - wj), $MachinePrecision] - N[(x * -2.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(x * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(wj - N[(N[(wj + N[(x / N[(-1.0 + N[(wj * N[(-1.0 - N[(wj * N[(0.5 + N[(wj * 0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(wj + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if wj < 1.3000000000000001e-8

   1. Initial program 78.4%

      \[wj - \frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}} \]
   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/ 80.0%

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

      3. div-sub 78.4%

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

      4. associate-/l* 78.4%

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

      5. *-inverses 80.0%

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

      6. *-rgt-identity 80.0%

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

   3. Simplified 80.0%

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

   5. Taylor expanded in wj around 0 97.3%

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

   6. Taylor expanded in x around 0 97.2%

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

      1. mul-1-neg 97.2%

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

   8. Simplified 97.2%

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

   9. Taylor expanded in wj around 0 97.2%

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

       1. neg-mul-1 97.2%

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

       2. sub-neg 97.2%

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

       3. distribute-rgt-out 97.2%

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

       4. metadata-eval 97.2%

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

       5. *-commutative 97.2%

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

   11. Simplified 97.2%

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

   if 1.3000000000000001e-8 < 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 50.0%

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

      2. associate-/l/ 49.7%

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

      3. div-sub 49.7%

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

      4. associate-/l* 49.7%

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

      5. *-inverses 99.7%

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

      6. *-rgt-identity 99.7%

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

   3. Simplified 99.7%

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

   5. Taylor expanded in wj around 0 98.3%

      \[\leadsto wj - \frac{wj - \frac{x}{\color{blue}{1 + wj \cdot \left(1 + wj \cdot \left(0.5 + 0.16666666666666666 \cdot wj\right)\right)}}}{wj + 1} \]
   6. Step-by-step derivation

      1. *-commutative 98.3%

         \[\leadsto wj - \frac{wj - \frac{x}{1 + wj \cdot \left(1 + wj \cdot \left(0.5 + \color{blue}{wj \cdot 0.16666666666666666}\right)\right)}}{wj + 1} \]

   7. Simplified 98.3%

      \[\leadsto wj - \frac{wj - \frac{x}{\color{blue}{1 + wj \cdot \left(1 + wj \cdot \left(0.5 + wj \cdot 0.16666666666666666\right)\right)}}}{wj + 1} \]
3. Recombined 2 regimes into one program.

4. Final simplification 97.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;wj \leq 1.3 \cdot 10^{-8}:\\ \;\;\;\;x + wj \cdot \left(wj \cdot \left(\left(1 - wj\right) - x \cdot -2.5\right) - x \cdot 2\right)\\ \mathbf{else}:\\ \;\;\;\;wj - \frac{wj + \frac{x}{-1 + wj \cdot \left(-1 - wj \cdot \left(0.5 + wj \cdot 0.16666666666666666\right)\right)}}{wj + 1}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 97.1% accurate, 13.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;wj \leq 1.3 \cdot 10^{-8}:\\ \;\;\;\;x + wj \cdot \left(wj \cdot \left(\left(1 - wj\right) - x \cdot -2.5\right) - x \cdot 2\right)\\ \mathbf{else}:\\ \;\;\;\;wj + \frac{\frac{x}{1 + wj \cdot \left(1 + wj \cdot 0.5\right)} - wj}{wj + 1}\\ \end{array} \end{array} \]

FPCore

(FPCore (wj x)
 :precision binary64
 (if (<= wj 1.3e-8)
   (+ x (* wj (- (* wj (- (- 1.0 wj) (* x -2.5))) (* x 2.0))))
   (+ wj (/ (- (/ x (+ 1.0 (* wj (+ 1.0 (* wj 0.5))))) wj) (+ wj 1.0)))))

C

double code(double wj, double x) {
	double tmp;
	if (wj <= 1.3e-8) {
		tmp = x + (wj * ((wj * ((1.0 - wj) - (x * -2.5))) - (x * 2.0)));
	} else {
		tmp = wj + (((x / (1.0 + (wj * (1.0 + (wj * 0.5))))) - wj) / (wj + 1.0));
	}
	return tmp;
}

Fortran

real(8) function code(wj, x)
    real(8), intent (in) :: wj
    real(8), intent (in) :: x
    real(8) :: tmp
    if (wj <= 1.3d-8) then
        tmp = x + (wj * ((wj * ((1.0d0 - wj) - (x * (-2.5d0)))) - (x * 2.0d0)))
    else
        tmp = wj + (((x / (1.0d0 + (wj * (1.0d0 + (wj * 0.5d0))))) - wj) / (wj + 1.0d0))
    end if
    code = tmp
end function

Java

public static double code(double wj, double x) {
	double tmp;
	if (wj <= 1.3e-8) {
		tmp = x + (wj * ((wj * ((1.0 - wj) - (x * -2.5))) - (x * 2.0)));
	} else {
		tmp = wj + (((x / (1.0 + (wj * (1.0 + (wj * 0.5))))) - wj) / (wj + 1.0));
	}
	return tmp;
}

Python

def code(wj, x):
	tmp = 0
	if wj <= 1.3e-8:
		tmp = x + (wj * ((wj * ((1.0 - wj) - (x * -2.5))) - (x * 2.0)))
	else:
		tmp = wj + (((x / (1.0 + (wj * (1.0 + (wj * 0.5))))) - wj) / (wj + 1.0))
	return tmp

Julia

function code(wj, x)
	tmp = 0.0
	if (wj <= 1.3e-8)
		tmp = Float64(x + Float64(wj * Float64(Float64(wj * Float64(Float64(1.0 - wj) - Float64(x * -2.5))) - Float64(x * 2.0))));
	else
		tmp = Float64(wj + Float64(Float64(Float64(x / Float64(1.0 + Float64(wj * Float64(1.0 + Float64(wj * 0.5))))) - wj) / Float64(wj + 1.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(wj, x)
	tmp = 0.0;
	if (wj <= 1.3e-8)
		tmp = x + (wj * ((wj * ((1.0 - wj) - (x * -2.5))) - (x * 2.0)));
	else
		tmp = wj + (((x / (1.0 + (wj * (1.0 + (wj * 0.5))))) - wj) / (wj + 1.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[wj_, x_] := If[LessEqual[wj, 1.3e-8], N[(x + N[(wj * N[(N[(wj * N[(N[(1.0 - wj), $MachinePrecision] - N[(x * -2.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(x * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(wj + N[(N[(N[(x / N[(1.0 + N[(wj * N[(1.0 + N[(wj * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - wj), $MachinePrecision] / N[(wj + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if wj < 1.3000000000000001e-8

   1. Initial program 78.4%

      \[wj - \frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}} \]
   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/ 80.0%

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

      3. div-sub 78.4%

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

      4. associate-/l* 78.4%

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

      5. *-inverses 80.0%

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

      6. *-rgt-identity 80.0%

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

   3. Simplified 80.0%

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

   5. Taylor expanded in wj around 0 97.3%

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

   6. Taylor expanded in x around 0 97.2%

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

      1. mul-1-neg 97.2%

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

   8. Simplified 97.2%

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

   9. Taylor expanded in wj around 0 97.2%

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

       1. neg-mul-1 97.2%

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

       2. sub-neg 97.2%

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

       3. distribute-rgt-out 97.2%

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

       4. metadata-eval 97.2%

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

       5. *-commutative 97.2%

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

   11. Simplified 97.2%

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

   if 1.3000000000000001e-8 < 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 50.0%

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

      2. associate-/l/ 49.7%

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

      3. div-sub 49.7%

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

      4. associate-/l* 49.7%

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

      5. *-inverses 99.7%

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

      6. *-rgt-identity 99.7%

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

   3. Simplified 99.7%

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

   5. Taylor expanded in wj around 0 95.1%

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

      1. *-commutative 95.1%

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

   7. Simplified 95.1%

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

4. Final simplification 97.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;wj \leq 1.3 \cdot 10^{-8}:\\ \;\;\;\;x + wj \cdot \left(wj \cdot \left(\left(1 - wj\right) - x \cdot -2.5\right) - x \cdot 2\right)\\ \mathbf{else}:\\ \;\;\;\;wj + \frac{\frac{x}{1 + wj \cdot \left(1 + wj \cdot 0.5\right)} - wj}{wj + 1}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 96.9% accurate, 17.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;wj \leq 1.3 \cdot 10^{-8}:\\ \;\;\;\;x - wj \cdot \left(x \cdot 2 + wj \cdot \left(wj + -1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;wj + \frac{wj + \frac{x}{-1 - wj}}{-1 - wj}\\ \end{array} \end{array} \]

FPCore

(FPCore (wj x)
 :precision binary64
 (if (<= wj 1.3e-8)
   (- x (* wj (+ (* x 2.0) (* wj (+ wj -1.0)))))
   (+ wj (/ (+ wj (/ x (- -1.0 wj))) (- -1.0 wj)))))

C

double code(double wj, double x) {
	double tmp;
	if (wj <= 1.3e-8) {
		tmp = x - (wj * ((x * 2.0) + (wj * (wj + -1.0))));
	} else {
		tmp = wj + ((wj + (x / (-1.0 - 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 <= 1.3d-8) then
        tmp = x - (wj * ((x * 2.0d0) + (wj * (wj + (-1.0d0)))))
    else
        tmp = wj + ((wj + (x / ((-1.0d0) - wj))) / ((-1.0d0) - wj))
    end if
    code = tmp
end function

Java

public static double code(double wj, double x) {
	double tmp;
	if (wj <= 1.3e-8) {
		tmp = x - (wj * ((x * 2.0) + (wj * (wj + -1.0))));
	} else {
		tmp = wj + ((wj + (x / (-1.0 - wj))) / (-1.0 - wj));
	}
	return tmp;
}

Python

def code(wj, x):
	tmp = 0
	if wj <= 1.3e-8:
		tmp = x - (wj * ((x * 2.0) + (wj * (wj + -1.0))))
	else:
		tmp = wj + ((wj + (x / (-1.0 - wj))) / (-1.0 - wj))
	return tmp

Julia

function code(wj, x)
	tmp = 0.0
	if (wj <= 1.3e-8)
		tmp = Float64(x - Float64(wj * Float64(Float64(x * 2.0) + Float64(wj * Float64(wj + -1.0)))));
	else
		tmp = Float64(wj + Float64(Float64(wj + Float64(x / Float64(-1.0 - wj))) / Float64(-1.0 - wj)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(wj, x)
	tmp = 0.0;
	if (wj <= 1.3e-8)
		tmp = x - (wj * ((x * 2.0) + (wj * (wj + -1.0))));
	else
		tmp = wj + ((wj + (x / (-1.0 - wj))) / (-1.0 - wj));
	end
	tmp_2 = tmp;
end

Wolfram

code[wj_, x_] := If[LessEqual[wj, 1.3e-8], N[(x - N[(wj * N[(N[(x * 2.0), $MachinePrecision] + N[(wj * N[(wj + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(wj + N[(N[(wj + N[(x / N[(-1.0 - wj), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(-1.0 - wj), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if wj < 1.3000000000000001e-8

   1. Initial program 78.4%

      \[wj - \frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}} \]
   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/ 80.0%

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

      3. div-sub 78.4%

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

      4. associate-/l* 78.4%

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

      5. *-inverses 80.0%

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

      6. *-rgt-identity 80.0%

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

   3. Simplified 80.0%

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

   5. Taylor expanded in wj around 0 97.3%

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

   6. Taylor expanded in x around 0 97.2%

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

      1. mul-1-neg 97.2%

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

   8. Simplified 97.2%

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

   9. Taylor expanded in wj around 0 97.2%

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

       1. neg-mul-1 97.2%

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

       2. sub-neg 97.2%

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

       3. distribute-rgt-out 97.2%

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

       4. metadata-eval 97.2%

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

       5. *-commutative 97.2%

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

   11. Simplified 97.2%

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

   12. Taylor expanded in x around 0 97.0%

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

   if 1.3000000000000001e-8 < 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 50.0%

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

      2. associate-/l/ 49.7%

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

      3. div-sub 49.7%

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

      4. associate-/l* 49.7%

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

      5. *-inverses 99.7%

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

      6. *-rgt-identity 99.7%

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

   3. Simplified 99.7%

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

   5. Taylor expanded in wj around 0 92.0%

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

      1. +-commutative 92.0%

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

   7. Simplified 92.0%

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

4. Final simplification 96.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;wj \leq 1.3 \cdot 10^{-8}:\\ \;\;\;\;x - wj \cdot \left(x \cdot 2 + wj \cdot \left(wj + -1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;wj + \frac{wj + \frac{x}{-1 - wj}}{-1 - wj}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 96.1% accurate, 18.4× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 77.7%

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

   1. distribute-rgt1-in 79.3%

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

   2. associate-/l/ 79.3%

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

   3. div-sub 77.7%

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

   4. associate-/l* 77.7%

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

   5. *-inverses 80.5%

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

   6. *-rgt-identity 80.5%

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

3. Simplified 80.5%

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

5. Taylor expanded in wj around 0 95.7%

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

6. Taylor expanded in x around 0 95.6%

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

   1. mul-1-neg 95.6%

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

8. Simplified 95.6%

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

9. Taylor expanded in wj around 0 95.6%

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

    1. neg-mul-1 95.6%

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

    2. sub-neg 95.6%

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

    3. distribute-rgt-out 95.6%

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

    4. metadata-eval 95.6%

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

    5. *-commutative 95.6%

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

11. Simplified 95.6%

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

12. Final simplification 95.6%

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

Alternative 9: 95.9% accurate, 24.1× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 77.7%

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

   1. distribute-rgt1-in 79.3%

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

   2. associate-/l/ 79.3%

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

   3. div-sub 77.7%

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

   4. associate-/l* 77.7%

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

   5. *-inverses 80.5%

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

   6. *-rgt-identity 80.5%

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

3. Simplified 80.5%

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

5. Taylor expanded in wj around 0 95.7%

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

6. Taylor expanded in x around 0 95.6%

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

   1. mul-1-neg 95.6%

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

8. Simplified 95.6%

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

9. Taylor expanded in wj around 0 95.6%

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

    1. neg-mul-1 95.6%

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

    2. sub-neg 95.6%

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

    3. distribute-rgt-out 95.6%

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

    4. metadata-eval 95.6%

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

    5. *-commutative 95.6%

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

11. Simplified 95.6%

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

12. Taylor expanded in x around 0 95.3%

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

13. Final simplification 95.3%

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

Alternative 10: 84.1% accurate, 28.5× speedup

Math

\[\begin{array}{l} \\ \frac{x}{1 + wj \cdot \left(2 + wj \cdot 1.5\right)} \end{array} \]

FPCore

(FPCore (wj x) :precision binary64 (/ x (+ 1.0 (* wj (+ 2.0 (* wj 1.5))))))

C

double code(double wj, double x) {
	return x / (1.0 + (wj * (2.0 + (wj * 1.5))));
}

Fortran

real(8) function code(wj, x)
    real(8), intent (in) :: wj
    real(8), intent (in) :: x
    code = x / (1.0d0 + (wj * (2.0d0 + (wj * 1.5d0))))
end function

Java

public static double code(double wj, double x) {
	return x / (1.0 + (wj * (2.0 + (wj * 1.5))));
}

Python

def code(wj, x):
	return x / (1.0 + (wj * (2.0 + (wj * 1.5))))

Julia

function code(wj, x)
	return Float64(x / Float64(1.0 + Float64(wj * Float64(2.0 + Float64(wj * 1.5)))))
end

MATLAB

function tmp = code(wj, x)
	tmp = x / (1.0 + (wj * (2.0 + (wj * 1.5))));
end

Wolfram

code[wj_, x_] := N[(x / N[(1.0 + N[(wj * N[(2.0 + N[(wj * 1.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 77.7%

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

   1. distribute-rgt1-in 79.3%

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

   2. associate-/l/ 79.3%

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

   3. div-sub 77.7%

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

   4. associate-/l* 77.7%

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

   5. *-inverses 80.5%

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

   6. *-rgt-identity 80.5%

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

3. Simplified 80.5%

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

5. Taylor expanded in x around inf 88.0%

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

6. Taylor expanded in wj around 0 86.1%

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

   1. *-commutative 86.1%

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

8. Simplified 86.1%

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

Alternative 11: 84.1% accurate, 28.5× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 77.7%

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

   1. distribute-rgt1-in 79.3%

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

   2. associate-/l/ 79.3%

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

   3. div-sub 77.7%

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

   4. associate-/l* 77.7%

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

   5. *-inverses 80.5%

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

   6. *-rgt-identity 80.5%

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

3. Simplified 80.5%

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

5. Taylor expanded in wj around 0 95.1%

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

   1. cancel-sign-sub-inv 95.1%

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

   2. distribute-rgt-out 95.1%

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

   3. metadata-eval 95.1%

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

   4. metadata-eval 95.1%

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

   5. *-commutative 95.1%

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

7. Simplified 95.1%

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

8. Taylor expanded in x around inf 86.1%

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

9. Final simplification 86.1%

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

Alternative 12: 84.1% accurate, 28.5× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 77.7%

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

   1. distribute-rgt1-in 79.3%

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

   2. associate-/l/ 79.3%

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

   3. div-sub 77.7%

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

   4. associate-/l* 77.7%

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

   5. *-inverses 80.5%

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

   6. *-rgt-identity 80.5%

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

3. Simplified 80.5%

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

5. Taylor expanded in wj around 0 95.1%

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

   1. cancel-sign-sub-inv 95.1%

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

   2. distribute-rgt-out 95.1%

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

   3. metadata-eval 95.1%

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

   4. metadata-eval 95.1%

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

   5. *-commutative 95.1%

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

7. Simplified 95.1%

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

8. Taylor expanded in x around inf 86.1%

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

9. Final simplification 86.1%

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

Alternative 13: 84.0% accurate, 44.7× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 77.7%

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

   1. distribute-rgt1-in 79.3%

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

   2. associate-/l/ 79.3%

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

   3. div-sub 77.7%

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

   4. associate-/l* 77.7%

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

   5. *-inverses 80.5%

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

   6. *-rgt-identity 80.5%

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

3. Simplified 80.5%

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

5. Taylor expanded in x around inf 88.0%

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

6. Taylor expanded in wj around 0 85.9%

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

   1. *-commutative 85.9%

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

8. Simplified 85.9%

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

Alternative 14: 83.9% 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 77.7%

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

   1. distribute-rgt1-in 79.3%

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

   2. associate-/l/ 79.3%

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

   3. div-sub 77.7%

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

   4. associate-/l* 77.7%

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

   5. *-inverses 80.5%

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

   6. *-rgt-identity 80.5%

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

3. Simplified 80.5%

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

5. Taylor expanded in wj around 0 85.8%

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

   1. *-commutative 85.8%

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

7. Simplified 85.8%

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

8. Final simplification 85.8%

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

Alternative 15: 83.4% 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 77.7%

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

   1. distribute-rgt1-in 79.3%

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

   2. associate-/l/ 79.3%

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

   3. div-sub 77.7%

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

   4. associate-/l* 77.7%

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

   5. *-inverses 80.5%

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

   6. *-rgt-identity 80.5%

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

3. Simplified 80.5%

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

5. Taylor expanded in wj around 0 85.0%

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

Alternative 16: 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 77.7%

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

   1. distribute-rgt1-in 79.3%

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

   2. associate-/l/ 79.3%

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

   3. div-sub 77.7%

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

   4. associate-/l* 77.7%

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

   5. *-inverses 80.5%

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

   6. *-rgt-identity 80.5%

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

3. Simplified 80.5%

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

5. Taylor expanded in wj around inf 4.3%

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

Alternative 17: 3.3% accurate, 313.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

def code(wj, x):
	return -1.0

Julia

function code(wj, x)
	return -1.0
end

MATLAB

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

Wolfram

code[wj_, x_] := -1.0

Derivation

1. Initial program 77.7%

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

   1. distribute-rgt1-in 79.3%

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

   2. associate-/l/ 79.3%

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

   3. div-sub 77.7%

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

   4. associate-/l* 77.7%

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

   5. *-inverses 80.5%

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

   6. *-rgt-identity 80.5%

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

3. Simplified 80.5%

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

5. Taylor expanded in wj around inf 4.5%

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

6. Taylor expanded in wj around 0 3.5%

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

Developer Target 1: 79.2% accurate, 1.5× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Reproduce

herbie shell --seed 2024144 
(FPCore (wj x)
  :name "Jmat.Real.lambertw, newton loop step"
  :precision binary64

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

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