Jmat.Real.lambertw, newton loop step

Percentage Accurate: 78.5% → 98.0%

Time: 17.6s

Alternatives: 13

Speedup: 313.0×

• could not determine a ground truth

  1. wj = -8.003486471052602e+147
  2. x = 4.253225023112373e-171

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: 78.5% 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: 98.0% accurate, 2.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;wj \leq -3.45 \cdot 10^{-6}:\\ \;\;\;\;wj + x \cdot \left(\frac{e^{-wj}}{wj + 1} - \frac{wj}{x \cdot \left(wj + 1\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;x + wj \cdot \left(wj \cdot \left(\left(1 - wj\right) - \left(x \cdot -4 + x \cdot 1.5\right)\right) - x \cdot 2\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (wj x)
 :precision binary64
 (if (<= wj -3.45e-6)
   (+ wj (* x (- (/ (exp (- wj)) (+ wj 1.0)) (/ wj (* x (+ wj 1.0))))))
   (+ x (* wj (- (* wj (- (- 1.0 wj) (+ (* x -4.0) (* x 1.5)))) (* x 2.0))))))

C

double code(double wj, double x) {
	double tmp;
	if (wj <= -3.45e-6) {
		tmp = wj + (x * ((exp(-wj) / (wj + 1.0)) - (wj / (x * (wj + 1.0)))));
	} else {
		tmp = x + (wj * ((wj * ((1.0 - wj) - ((x * -4.0) + (x * 1.5)))) - (x * 2.0)));
	}
	return tmp;
}

Fortran

real(8) function code(wj, x)
    real(8), intent (in) :: wj
    real(8), intent (in) :: x
    real(8) :: tmp
    if (wj <= (-3.45d-6)) then
        tmp = wj + (x * ((exp(-wj) / (wj + 1.0d0)) - (wj / (x * (wj + 1.0d0)))))
    else
        tmp = x + (wj * ((wj * ((1.0d0 - wj) - ((x * (-4.0d0)) + (x * 1.5d0)))) - (x * 2.0d0)))
    end if
    code = tmp
end function

Java

public static double code(double wj, double x) {
	double tmp;
	if (wj <= -3.45e-6) {
		tmp = wj + (x * ((Math.exp(-wj) / (wj + 1.0)) - (wj / (x * (wj + 1.0)))));
	} else {
		tmp = x + (wj * ((wj * ((1.0 - wj) - ((x * -4.0) + (x * 1.5)))) - (x * 2.0)));
	}
	return tmp;
}

Python

def code(wj, x):
	tmp = 0
	if wj <= -3.45e-6:
		tmp = wj + (x * ((math.exp(-wj) / (wj + 1.0)) - (wj / (x * (wj + 1.0)))))
	else:
		tmp = x + (wj * ((wj * ((1.0 - wj) - ((x * -4.0) + (x * 1.5)))) - (x * 2.0)))
	return tmp

Julia

function code(wj, x)
	tmp = 0.0
	if (wj <= -3.45e-6)
		tmp = Float64(wj + Float64(x * Float64(Float64(exp(Float64(-wj)) / Float64(wj + 1.0)) - Float64(wj / Float64(x * Float64(wj + 1.0))))));
	else
		tmp = Float64(x + Float64(wj * Float64(Float64(wj * Float64(Float64(1.0 - wj) - Float64(Float64(x * -4.0) + Float64(x * 1.5)))) - Float64(x * 2.0))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(wj, x)
	tmp = 0.0;
	if (wj <= -3.45e-6)
		tmp = wj + (x * ((exp(-wj) / (wj + 1.0)) - (wj / (x * (wj + 1.0)))));
	else
		tmp = x + (wj * ((wj * ((1.0 - wj) - ((x * -4.0) + (x * 1.5)))) - (x * 2.0)));
	end
	tmp_2 = tmp;
end

Wolfram

code[wj_, x_] := If[LessEqual[wj, -3.45e-6], N[(wj + N[(x * N[(N[(N[Exp[(-wj)], $MachinePrecision] / N[(wj + 1.0), $MachinePrecision]), $MachinePrecision] - N[(wj / N[(x * N[(wj + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x + N[(wj * N[(N[(wj * N[(N[(1.0 - wj), $MachinePrecision] - N[(N[(x * -4.0), $MachinePrecision] + N[(x * 1.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(x * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if wj < -3.45e-6

   1. Initial program 65.4%

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

      1. distribute-rgt1-in 98.9%

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

      2. associate-/l/ 98.6%

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

      3. div-sub 65.3%

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

      4. associate-/l* 65.3%

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

      5. *-inverses 98.6%

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

      6. *-rgt-identity 98.6%

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

   3. Simplified 98.6%

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

   5. Taylor expanded in x around inf 99.1%

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

      1. +-commutative 99.1%

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

      2. associate-/r* 98.9%

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

      3. exp-neg 98.9%

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

      4. +-commutative 98.9%

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

   7. Simplified 98.9%

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

   if -3.45e-6 < wj

   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/ 79.5%

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

      3. div-sub 79.5%

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

      4. associate-/l* 79.5%

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

      5. *-inverses 80.3%

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

      6. *-rgt-identity 80.3%

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

   3. Simplified 80.3%

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

   5. Taylor expanded in wj around 0 98.7%

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

   6. Taylor expanded in x around 0 98.7%

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

4. Final simplification 98.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;wj \leq -3.45 \cdot 10^{-6}:\\ \;\;\;\;wj + x \cdot \left(\frac{e^{-wj}}{wj + 1} - \frac{wj}{x \cdot \left(wj + 1\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;x + wj \cdot \left(wj \cdot \left(\left(1 - wj\right) - \left(x \cdot -4 + x \cdot 1.5\right)\right) - x \cdot 2\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 98.0% accurate, 2.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x \cdot -4 + x \cdot 1.5\\ \mathbf{if}\;wj \leq -1.16 \cdot 10^{-5}:\\ \;\;\;\;wj + \frac{wj - \frac{x}{e^{wj}}}{-1 - wj}\\ \mathbf{else}:\\ \;\;\;\;x + wj \cdot \left(wj \cdot \left(\left(1 + wj \cdot \left(-1 - \left(x \cdot -3 + \left(t\_0 \cdot -2 + x \cdot 0.6666666666666666\right)\right)\right)\right) - t\_0\right) - x \cdot 2\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (wj x)
 :precision binary64
 (let* ((t_0 (+ (* x -4.0) (* x 1.5))))
   (if (<= wj -1.16e-5)
     (+ wj (/ (- wj (/ x (exp wj))) (- -1.0 wj)))
     (+
      x
      (*
       wj
       (-
        (*
         wj
         (-
          (+
           1.0
           (*
            wj
            (- -1.0 (+ (* x -3.0) (+ (* t_0 -2.0) (* 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 <= -1.16e-5) {
		tmp = wj + ((wj - (x / exp(wj))) / (-1.0 - wj));
	} else {
		tmp = x + (wj * ((wj * ((1.0 + (wj * (-1.0 - ((x * -3.0) + ((t_0 * -2.0) + (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 <= (-1.16d-5)) then
        tmp = wj + ((wj - (x / exp(wj))) / ((-1.0d0) - wj))
    else
        tmp = x + (wj * ((wj * ((1.0d0 + (wj * ((-1.0d0) - ((x * (-3.0d0)) + ((t_0 * (-2.0d0)) + (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 <= -1.16e-5) {
		tmp = wj + ((wj - (x / Math.exp(wj))) / (-1.0 - wj));
	} else {
		tmp = x + (wj * ((wj * ((1.0 + (wj * (-1.0 - ((x * -3.0) + ((t_0 * -2.0) + (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 <= -1.16e-5:
		tmp = wj + ((wj - (x / math.exp(wj))) / (-1.0 - wj))
	else:
		tmp = x + (wj * ((wj * ((1.0 + (wj * (-1.0 - ((x * -3.0) + ((t_0 * -2.0) + (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 <= -1.16e-5)
		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(t_0 * -2.0) + 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 <= -1.16e-5)
		tmp = wj + ((wj - (x / exp(wj))) / (-1.0 - wj));
	else
		tmp = x + (wj * ((wj * ((1.0 + (wj * (-1.0 - ((x * -3.0) + ((t_0 * -2.0) + (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[LessEqual[wj, -1.16e-5], 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[(t$95$0 * -2.0), $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 < -1.1600000000000001e-5

   1. Initial program 61.0%

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

      1. distribute-rgt1-in 98.7%

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

      2. associate-/l/ 98.7%

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

      3. div-sub 61.2%

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

      4. associate-/l* 61.2%

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

      5. *-inverses 98.7%

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

      6. *-rgt-identity 98.7%

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

   3. Simplified 98.7%

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

   if -1.1600000000000001e-5 < wj

   1. Initial program 79.5%

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

      1. distribute-rgt1-in 79.5%

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

      2. associate-/l/ 79.6%

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

      3. div-sub 79.6%

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

      4. associate-/l* 79.6%

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

      5. *-inverses 80.4%

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

      6. *-rgt-identity 80.4%

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

   3. Simplified 80.4%

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

   5. Taylor expanded in wj around 0 98.7%

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

4. Final simplification 98.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;wj \leq -1.16 \cdot 10^{-5}:\\ \;\;\;\;wj + \frac{wj - \frac{x}{e^{wj}}}{-1 - wj}\\ \mathbf{else}:\\ \;\;\;\;x + wj \cdot \left(wj \cdot \left(\left(1 + wj \cdot \left(-1 - \left(x \cdot -3 + \left(\left(x \cdot -4 + x \cdot 1.5\right) \cdot -2 + x \cdot 0.6666666666666666\right)\right)\right)\right) - \left(x \cdot -4 + x \cdot 1.5\right)\right) - x \cdot 2\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 96.8% accurate, 6.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x \cdot -4 + x \cdot 1.5\\ \mathbf{if}\;wj \leq -0.00017:\\ \;\;\;\;wj - \frac{wj + \left(wj \cdot \left(x + wj \cdot \left(wj \cdot \frac{\left(x \cdot 0.16666666666666666\right) \cdot \left(x \cdot 0.8333333333333334\right)}{x \cdot 0.8333333333333334} + \left(x \cdot 0.5 - x\right)\right)\right) - x\right)}{wj + 1}\\ \mathbf{else}:\\ \;\;\;\;x + wj \cdot \left(wj \cdot \left(\left(1 + wj \cdot \left(-1 - \left(x \cdot -3 + \left(t\_0 \cdot -2 + x \cdot 0.6666666666666666\right)\right)\right)\right) - t\_0\right) - x \cdot 2\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (wj x)
 :precision binary64
 (let* ((t_0 (+ (* x -4.0) (* x 1.5))))
   (if (<= wj -0.00017)
     (-
      wj
      (/
       (+
        wj
        (-
         (*
          wj
          (+
           x
           (*
            wj
            (+
             (*
              wj
              (/
               (* (* x 0.16666666666666666) (* x 0.8333333333333334))
               (* x 0.8333333333333334)))
             (- (* x 0.5) x)))))
         x))
       (+ wj 1.0)))
     (+
      x
      (*
       wj
       (-
        (*
         wj
         (-
          (+
           1.0
           (*
            wj
            (- -1.0 (+ (* x -3.0) (+ (* t_0 -2.0) (* 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 <= -0.00017) {
		tmp = wj - ((wj + ((wj * (x + (wj * ((wj * (((x * 0.16666666666666666) * (x * 0.8333333333333334)) / (x * 0.8333333333333334))) + ((x * 0.5) - x))))) - x)) / (wj + 1.0));
	} else {
		tmp = x + (wj * ((wj * ((1.0 + (wj * (-1.0 - ((x * -3.0) + ((t_0 * -2.0) + (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 <= (-0.00017d0)) then
        tmp = wj - ((wj + ((wj * (x + (wj * ((wj * (((x * 0.16666666666666666d0) * (x * 0.8333333333333334d0)) / (x * 0.8333333333333334d0))) + ((x * 0.5d0) - x))))) - x)) / (wj + 1.0d0))
    else
        tmp = x + (wj * ((wj * ((1.0d0 + (wj * ((-1.0d0) - ((x * (-3.0d0)) + ((t_0 * (-2.0d0)) + (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 <= -0.00017) {
		tmp = wj - ((wj + ((wj * (x + (wj * ((wj * (((x * 0.16666666666666666) * (x * 0.8333333333333334)) / (x * 0.8333333333333334))) + ((x * 0.5) - x))))) - x)) / (wj + 1.0));
	} else {
		tmp = x + (wj * ((wj * ((1.0 + (wj * (-1.0 - ((x * -3.0) + ((t_0 * -2.0) + (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 <= -0.00017:
		tmp = wj - ((wj + ((wj * (x + (wj * ((wj * (((x * 0.16666666666666666) * (x * 0.8333333333333334)) / (x * 0.8333333333333334))) + ((x * 0.5) - x))))) - x)) / (wj + 1.0))
	else:
		tmp = x + (wj * ((wj * ((1.0 + (wj * (-1.0 - ((x * -3.0) + ((t_0 * -2.0) + (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 <= -0.00017)
		tmp = Float64(wj - Float64(Float64(wj + Float64(Float64(wj * Float64(x + Float64(wj * Float64(Float64(wj * Float64(Float64(Float64(x * 0.16666666666666666) * Float64(x * 0.8333333333333334)) / Float64(x * 0.8333333333333334))) + Float64(Float64(x * 0.5) - x))))) - x)) / Float64(wj + 1.0)));
	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(t_0 * -2.0) + 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 <= -0.00017)
		tmp = wj - ((wj + ((wj * (x + (wj * ((wj * (((x * 0.16666666666666666) * (x * 0.8333333333333334)) / (x * 0.8333333333333334))) + ((x * 0.5) - x))))) - x)) / (wj + 1.0));
	else
		tmp = x + (wj * ((wj * ((1.0 + (wj * (-1.0 - ((x * -3.0) + ((t_0 * -2.0) + (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[LessEqual[wj, -0.00017], N[(wj - N[(N[(wj + N[(N[(wj * N[(x + N[(wj * N[(N[(wj * N[(N[(N[(x * 0.16666666666666666), $MachinePrecision] * N[(x * 0.8333333333333334), $MachinePrecision]), $MachinePrecision] / N[(x * 0.8333333333333334), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(x * 0.5), $MachinePrecision] - x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - x), $MachinePrecision]), $MachinePrecision] / N[(wj + 1.0), $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[(t$95$0 * -2.0), $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 < -1.7e-4

   1. Initial program 61.0%

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

      1. distribute-rgt1-in 98.7%

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

      2. associate-/l/ 98.7%

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

      3. div-sub 61.2%

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

      4. associate-/l* 61.2%

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

      5. *-inverses 98.7%

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

      6. *-rgt-identity 98.7%

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

   3. Simplified 98.7%

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

   5. Taylor expanded in wj around 0 61.6%

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

      1. flip-+ 48.1%

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

      2. pow2 48.1%

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

      3. mul-1-neg 48.1%

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

      4. distribute-rgt-out 48.1%

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

      5. metadata-eval 48.1%

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

      6. pow2 48.1%

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

      7. distribute-rgt-out 48.1%

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

      8. metadata-eval 48.1%

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

      9. mul-1-neg 48.1%

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

      10. distribute-rgt-out 48.1%

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

      11. metadata-eval 48.1%

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

      12. distribute-rgt-out 48.1%

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

      13. metadata-eval 48.1%

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

   7. Applied egg-rr 48.1%

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

      1. unpow2 48.1%

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

      2. unpow2 48.1%

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

      3. difference-of-squares 73.1%

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

      4. distribute-rgt-neg-in 73.1%

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

      5. metadata-eval 73.1%

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

      6. distribute-lft-out 73.1%

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

      7. metadata-eval 73.1%

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

      8. distribute-rgt-neg-in 73.1%

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

      9. metadata-eval 73.1%

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

      10. distribute-lft-out-- 73.1%

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

      11. metadata-eval 73.1%

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

      12. distribute-rgt-neg-in 73.1%

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

      13. metadata-eval 73.1%

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

      14. distribute-lft-out-- 73.1%

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

      15. metadata-eval 73.1%

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

   9. Simplified 73.1%

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

   if -1.7e-4 < wj

   1. Initial program 79.5%

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

      1. distribute-rgt1-in 79.5%

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

      2. associate-/l/ 79.6%

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

      3. div-sub 79.6%

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

      4. associate-/l* 79.6%

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

      5. *-inverses 80.4%

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

      6. *-rgt-identity 80.4%

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

   3. Simplified 80.4%

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

   5. Taylor expanded in wj around 0 98.7%

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

4. Final simplification 97.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;wj \leq -0.00017:\\ \;\;\;\;wj - \frac{wj + \left(wj \cdot \left(x + wj \cdot \left(wj \cdot \frac{\left(x \cdot 0.16666666666666666\right) \cdot \left(x \cdot 0.8333333333333334\right)}{x \cdot 0.8333333333333334} + \left(x \cdot 0.5 - x\right)\right)\right) - x\right)}{wj + 1}\\ \mathbf{else}:\\ \;\;\;\;x + wj \cdot \left(wj \cdot \left(\left(1 + wj \cdot \left(-1 - \left(x \cdot -3 + \left(\left(x \cdot -4 + x \cdot 1.5\right) \cdot -2 + x \cdot 0.6666666666666666\right)\right)\right)\right) - \left(x \cdot -4 + x \cdot 1.5\right)\right) - x \cdot 2\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 96.7% accurate, 7.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;wj \leq -0.00028:\\ \;\;\;\;wj - \frac{wj + \left(wj \cdot \left(x + wj \cdot \left(wj \cdot \frac{\left(x \cdot 0.16666666666666666\right) \cdot \left(x \cdot 0.8333333333333334\right)}{x \cdot 0.8333333333333334} + \left(x \cdot 0.5 - x\right)\right)\right) - x\right)}{wj + 1}\\ \mathbf{else}:\\ \;\;\;\;x + wj \cdot \left(wj \cdot \left(\left(1 - wj\right) - \left(x \cdot -4 + x \cdot 1.5\right)\right) - x \cdot 2\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (wj x)
 :precision binary64
 (if (<= wj -0.00028)
   (-
    wj
    (/
     (+
      wj
      (-
       (*
        wj
        (+
         x
         (*
          wj
          (+
           (*
            wj
            (/
             (* (* x 0.16666666666666666) (* x 0.8333333333333334))
             (* x 0.8333333333333334)))
           (- (* x 0.5) x)))))
       x))
     (+ wj 1.0)))
   (+ x (* wj (- (* wj (- (- 1.0 wj) (+ (* x -4.0) (* x 1.5)))) (* x 2.0))))))

C

double code(double wj, double x) {
	double tmp;
	if (wj <= -0.00028) {
		tmp = wj - ((wj + ((wj * (x + (wj * ((wj * (((x * 0.16666666666666666) * (x * 0.8333333333333334)) / (x * 0.8333333333333334))) + ((x * 0.5) - x))))) - x)) / (wj + 1.0));
	} else {
		tmp = x + (wj * ((wj * ((1.0 - wj) - ((x * -4.0) + (x * 1.5)))) - (x * 2.0)));
	}
	return tmp;
}

Fortran

real(8) function code(wj, x)
    real(8), intent (in) :: wj
    real(8), intent (in) :: x
    real(8) :: tmp
    if (wj <= (-0.00028d0)) then
        tmp = wj - ((wj + ((wj * (x + (wj * ((wj * (((x * 0.16666666666666666d0) * (x * 0.8333333333333334d0)) / (x * 0.8333333333333334d0))) + ((x * 0.5d0) - x))))) - x)) / (wj + 1.0d0))
    else
        tmp = x + (wj * ((wj * ((1.0d0 - wj) - ((x * (-4.0d0)) + (x * 1.5d0)))) - (x * 2.0d0)))
    end if
    code = tmp
end function

Java

public static double code(double wj, double x) {
	double tmp;
	if (wj <= -0.00028) {
		tmp = wj - ((wj + ((wj * (x + (wj * ((wj * (((x * 0.16666666666666666) * (x * 0.8333333333333334)) / (x * 0.8333333333333334))) + ((x * 0.5) - x))))) - x)) / (wj + 1.0));
	} else {
		tmp = x + (wj * ((wj * ((1.0 - wj) - ((x * -4.0) + (x * 1.5)))) - (x * 2.0)));
	}
	return tmp;
}

Python

def code(wj, x):
	tmp = 0
	if wj <= -0.00028:
		tmp = wj - ((wj + ((wj * (x + (wj * ((wj * (((x * 0.16666666666666666) * (x * 0.8333333333333334)) / (x * 0.8333333333333334))) + ((x * 0.5) - x))))) - x)) / (wj + 1.0))
	else:
		tmp = x + (wj * ((wj * ((1.0 - wj) - ((x * -4.0) + (x * 1.5)))) - (x * 2.0)))
	return tmp

Julia

function code(wj, x)
	tmp = 0.0
	if (wj <= -0.00028)
		tmp = Float64(wj - Float64(Float64(wj + Float64(Float64(wj * Float64(x + Float64(wj * Float64(Float64(wj * Float64(Float64(Float64(x * 0.16666666666666666) * Float64(x * 0.8333333333333334)) / Float64(x * 0.8333333333333334))) + Float64(Float64(x * 0.5) - x))))) - x)) / Float64(wj + 1.0)));
	else
		tmp = Float64(x + Float64(wj * Float64(Float64(wj * Float64(Float64(1.0 - wj) - Float64(Float64(x * -4.0) + Float64(x * 1.5)))) - Float64(x * 2.0))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(wj, x)
	tmp = 0.0;
	if (wj <= -0.00028)
		tmp = wj - ((wj + ((wj * (x + (wj * ((wj * (((x * 0.16666666666666666) * (x * 0.8333333333333334)) / (x * 0.8333333333333334))) + ((x * 0.5) - x))))) - x)) / (wj + 1.0));
	else
		tmp = x + (wj * ((wj * ((1.0 - wj) - ((x * -4.0) + (x * 1.5)))) - (x * 2.0)));
	end
	tmp_2 = tmp;
end

Wolfram

code[wj_, x_] := If[LessEqual[wj, -0.00028], N[(wj - N[(N[(wj + N[(N[(wj * N[(x + N[(wj * N[(N[(wj * N[(N[(N[(x * 0.16666666666666666), $MachinePrecision] * N[(x * 0.8333333333333334), $MachinePrecision]), $MachinePrecision] / N[(x * 0.8333333333333334), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(x * 0.5), $MachinePrecision] - x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - x), $MachinePrecision]), $MachinePrecision] / N[(wj + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x + N[(wj * N[(N[(wj * N[(N[(1.0 - wj), $MachinePrecision] - N[(N[(x * -4.0), $MachinePrecision] + N[(x * 1.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(x * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if wj < -2.7999999999999998e-4

   1. Initial program 61.0%

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

      1. distribute-rgt1-in 98.7%

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

      2. associate-/l/ 98.7%

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

      3. div-sub 61.2%

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

      4. associate-/l* 61.2%

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

      5. *-inverses 98.7%

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

      6. *-rgt-identity 98.7%

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

   3. Simplified 98.7%

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

   5. Taylor expanded in wj around 0 61.6%

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

      1. flip-+ 48.1%

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

      2. pow2 48.1%

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

      3. mul-1-neg 48.1%

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

      4. distribute-rgt-out 48.1%

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

      5. metadata-eval 48.1%

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

      6. pow2 48.1%

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

      7. distribute-rgt-out 48.1%

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

      8. metadata-eval 48.1%

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

      9. mul-1-neg 48.1%

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

      10. distribute-rgt-out 48.1%

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

      11. metadata-eval 48.1%

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

      12. distribute-rgt-out 48.1%

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

      13. metadata-eval 48.1%

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

   7. Applied egg-rr 48.1%

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

      1. unpow2 48.1%

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

      2. unpow2 48.1%

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

      3. difference-of-squares 73.1%

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

      4. distribute-rgt-neg-in 73.1%

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

      5. metadata-eval 73.1%

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

      6. distribute-lft-out 73.1%

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

      7. metadata-eval 73.1%

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

      8. distribute-rgt-neg-in 73.1%

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

      9. metadata-eval 73.1%

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

      10. distribute-lft-out-- 73.1%

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

      11. metadata-eval 73.1%

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

      12. distribute-rgt-neg-in 73.1%

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

      13. metadata-eval 73.1%

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

      14. distribute-lft-out-- 73.1%

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

      15. metadata-eval 73.1%

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

   9. Simplified 73.1%

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

   if -2.7999999999999998e-4 < wj

   1. Initial program 79.5%

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

      1. distribute-rgt1-in 79.5%

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

      2. associate-/l/ 79.6%

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

      3. div-sub 79.6%

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

      4. associate-/l* 79.6%

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

      5. *-inverses 80.4%

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

      6. *-rgt-identity 80.4%

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

   3. Simplified 80.4%

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

   5. Taylor expanded in wj around 0 98.7%

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

   6. Taylor expanded in x around 0 98.6%

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

4. Final simplification 97.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;wj \leq -0.00028:\\ \;\;\;\;wj - \frac{wj + \left(wj \cdot \left(x + wj \cdot \left(wj \cdot \frac{\left(x \cdot 0.16666666666666666\right) \cdot \left(x \cdot 0.8333333333333334\right)}{x \cdot 0.8333333333333334} + \left(x \cdot 0.5 - x\right)\right)\right) - x\right)}{wj + 1}\\ \mathbf{else}:\\ \;\;\;\;x + wj \cdot \left(wj \cdot \left(\left(1 - wj\right) - \left(x \cdot -4 + x \cdot 1.5\right)\right) - x \cdot 2\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 96.3% accurate, 13.6× speedup

Math

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

FPCore

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

C

double code(double wj, double x) {
	return x * (1.0 + (wj * ((wj * (2.5 + ((1.0 / x) + (wj * ((-1.0 / x) - 2.6666666666666665))))) - 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 + ((1.0d0 / x) + (wj * (((-1.0d0) / x) - 2.6666666666666665d0))))) - 2.0d0)))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 79.0%

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

   1. distribute-rgt1-in 80.1%

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

   2. associate-/l/ 80.2%

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

   3. div-sub 79.0%

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

   4. associate-/l* 79.0%

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

   5. *-inverses 81.0%

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

   6. *-rgt-identity 81.0%

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

3. Simplified 81.0%

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

5. Taylor expanded in wj around 0 79.8%

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

6. Taylor expanded in x around inf 80.8%

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

7. Taylor expanded in wj around 0 96.2%

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

8. Final simplification 96.2%

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

Alternative 6: 96.3% accurate, 14.9× speedup

Math

\[\begin{array}{l} \\ x + wj \cdot \left(wj \cdot \left(\left(1 - wj\right) - \left(x \cdot -4 + x \cdot 1.5\right)\right) - x \cdot 2\right) \end{array} \]

FPCore

(FPCore (wj x)
 :precision binary64
 (+ x (* wj (- (* wj (- (- 1.0 wj) (+ (* x -4.0) (* x 1.5)))) (* x 2.0)))))

C

double code(double wj, double x) {
	return x + (wj * ((wj * ((1.0 - wj) - ((x * -4.0) + (x * 1.5)))) - (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 * (-4.0d0)) + (x * 1.5d0)))) - (x * 2.0d0)))
end function

Java

public static double code(double wj, double x) {
	return x + (wj * ((wj * ((1.0 - wj) - ((x * -4.0) + (x * 1.5)))) - (x * 2.0)));
}

Python

def code(wj, x):
	return x + (wj * ((wj * ((1.0 - wj) - ((x * -4.0) + (x * 1.5)))) - (x * 2.0)))

Julia

function code(wj, x)
	return Float64(x + Float64(wj * Float64(Float64(wj * Float64(Float64(1.0 - wj) - Float64(Float64(x * -4.0) + Float64(x * 1.5)))) - Float64(x * 2.0))))
end

MATLAB

function tmp = code(wj, x)
	tmp = x + (wj * ((wj * ((1.0 - wj) - ((x * -4.0) + (x * 1.5)))) - (x * 2.0)));
end

Wolfram

code[wj_, x_] := N[(x + N[(wj * N[(N[(wj * N[(N[(1.0 - wj), $MachinePrecision] - N[(N[(x * -4.0), $MachinePrecision] + N[(x * 1.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(x * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 79.0%

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

   1. distribute-rgt1-in 80.1%

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

   2. associate-/l/ 80.2%

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

   3. div-sub 79.0%

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

   4. associate-/l* 79.0%

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

   5. *-inverses 81.0%

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

   6. *-rgt-identity 81.0%

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

3. Simplified 81.0%

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

5. Taylor expanded in wj around 0 96.3%

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

6. Taylor expanded in x around 0 96.2%

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

7. Final simplification 96.2%

   \[\leadsto x + wj \cdot \left(wj \cdot \left(\left(1 - wj\right) - \left(x \cdot -4 + x \cdot 1.5\right)\right) - x \cdot 2\right) \]
8. Add Preprocessing

Alternative 7: 96.0% accurate, 24.1× speedup

Math

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

FPCore

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

C

double code(double wj, double x) {
	return x + (wj * ((wj * (1.0 - wj)) - (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 * 2.0d0)))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 79.0%

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

   1. distribute-rgt1-in 80.1%

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

   2. associate-/l/ 80.2%

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

   3. div-sub 79.0%

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

   4. associate-/l* 79.0%

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

   5. *-inverses 81.0%

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

   6. *-rgt-identity 81.0%

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

3. Simplified 81.0%

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

5. Taylor expanded in wj around 0 96.3%

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

6. Taylor expanded in x around 0 96.2%

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

7. Taylor expanded in x around 0 96.0%

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

   1. neg-mul-1 96.0%

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

   2. sub-neg 96.0%

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

9. Simplified 96.0%

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

10. Final simplification 96.0%

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

Alternative 8: 85.2% 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 79.0%

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

   1. distribute-rgt1-in 80.1%

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

   2. associate-/l/ 80.2%

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

   3. div-sub 79.0%

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

   4. associate-/l* 79.0%

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

   5. *-inverses 81.0%

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

   6. *-rgt-identity 81.0%

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

3. Simplified 81.0%

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

5. Taylor expanded in wj around 0 96.3%

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

6. Taylor expanded in x around 0 96.2%

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

7. Taylor expanded in x around inf 84.3%

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

8. Final simplification 84.3%

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

Alternative 9: 85.1% accurate, 28.5× speedup

Math

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

FPCore

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

C

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

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)))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 79.0%

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

   1. distribute-rgt1-in 80.1%

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

   2. associate-/l/ 80.2%

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

   3. div-sub 79.0%

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

   4. associate-/l* 79.0%

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

   5. *-inverses 81.0%

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

   6. *-rgt-identity 81.0%

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

3. Simplified 81.0%

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

5. Taylor expanded in wj around 0 96.3%

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

6. Taylor expanded in x around 0 96.2%

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

7. Taylor expanded in wj around inf 84.3%

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

   1. neg-mul-1 84.3%

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

9. Simplified 84.3%

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

10. Final simplification 84.3%

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

Alternative 10: 85.1% accurate, 34.8× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 79.0%

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

   1. distribute-rgt1-in 80.1%

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

   2. associate-/l/ 80.2%

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

   3. div-sub 79.0%

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

   4. associate-/l* 79.0%

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

   5. *-inverses 81.0%

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

   6. *-rgt-identity 81.0%

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

3. Simplified 81.0%

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

5. Taylor expanded in wj around 0 79.2%

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

   1. associate-*r* 79.2%

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

   2. neg-mul-1 79.2%

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

   3. distribute-rgt1-in 79.2%

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

   4. +-commutative 79.2%

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

   5. sub-neg 79.2%

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

7. Simplified 79.2%

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

8. Taylor expanded in x around inf 84.2%

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

   1. div-sub 84.2%

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

   2. +-commutative 84.2%

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

10. Simplified 84.2%

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

11. Final simplification 84.2%

    \[\leadsto x \cdot \frac{1 - wj}{wj + 1} \]
12. Add Preprocessing

Alternative 11: 85.0% 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 79.0%

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

   1. distribute-rgt1-in 80.1%

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

   2. associate-/l/ 80.2%

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

   3. div-sub 79.0%

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

   4. associate-/l* 79.0%

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

   5. *-inverses 81.0%

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

   6. *-rgt-identity 81.0%

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

3. Simplified 81.0%

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

5. Taylor expanded in wj around 0 84.1%

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

   1. *-commutative 84.1%

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

7. Simplified 84.1%

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

8. Final simplification 84.1%

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

Alternative 12: 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 79.0%

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

   1. distribute-rgt1-in 80.1%

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

   2. associate-/l/ 80.2%

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

   3. div-sub 79.0%

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

   4. associate-/l* 79.0%

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

   5. *-inverses 81.0%

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

   6. *-rgt-identity 81.0%

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

3. Simplified 81.0%

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

5. Taylor expanded in wj around inf 4.3%

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

6. Final simplification 4.3%

   \[\leadsto wj \]
7. Add Preprocessing

Alternative 13: 84.4% 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 79.0%

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

   1. distribute-rgt1-in 80.1%

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

   2. associate-/l/ 80.2%

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

   3. div-sub 79.0%

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

   4. associate-/l* 79.0%

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

   5. *-inverses 81.0%

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

   6. *-rgt-identity 81.0%

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

3. Simplified 81.0%

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

5. Taylor expanded in wj around 0 83.5%

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

6. Final simplification 83.5%

   \[\leadsto x \]
7. Add Preprocessing

Developer target: 79.5% accurate, 1.5× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Reproduce

herbie shell --seed 2024118 
(FPCore (wj x)
  :name "Jmat.Real.lambertw, newton loop step"
  :precision binary64

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

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