Jmat.Real.lambertw, newton loop step

Percentage Accurate: 77.2% → 98.6%

Time: 14.2s

Alternatives: 13

Speedup: 313.0×

• could not determine a ground truth

  1. wj = -1.3407821353004777e+149
  2. x = 1.9207596028713045e-213

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 77.2% 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: 98.6% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := wj \cdot e^{wj}\\ t_1 := x \cdot -4 + x \cdot 1.5\\ \mathbf{if}\;wj + \frac{x - t\_0}{e^{wj} + t\_0} \leq 5 \cdot 10^{-19}:\\ \;\;\;\;x + wj \cdot \left(wj \cdot \left(\left(1 + wj \cdot \left(-1 - \left(x \cdot -3 + \left(-2 \cdot t\_1 + x \cdot 0.6666666666666666\right)\right)\right)\right) - t\_1\right) - x \cdot 2\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(\frac{wj}{x \cdot \left(-1 - wj\right)} - \left(\frac{e^{-wj}}{-1 - wj} - \frac{wj}{x}\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (wj x)
 :precision binary64
 (let* ((t_0 (* wj (exp wj))) (t_1 (+ (* x -4.0) (* x 1.5))))
   (if (<= (+ wj (/ (- x t_0) (+ (exp wj) t_0))) 5e-19)
     (+
      x
      (*
       wj
       (-
        (*
         wj
         (-
          (+
           1.0
           (*
            wj
            (- -1.0 (+ (* x -3.0) (+ (* -2.0 t_1) (* x 0.6666666666666666))))))
          t_1))
        (* x 2.0))))
     (*
      x
      (-
       (/ wj (* x (- -1.0 wj)))
       (- (/ (exp (- wj)) (- -1.0 wj)) (/ wj x)))))))

C

double code(double wj, double x) {
	double t_0 = wj * exp(wj);
	double t_1 = (x * -4.0) + (x * 1.5);
	double tmp;
	if ((wj + ((x - t_0) / (exp(wj) + t_0))) <= 5e-19) {
		tmp = x + (wj * ((wj * ((1.0 + (wj * (-1.0 - ((x * -3.0) + ((-2.0 * t_1) + (x * 0.6666666666666666)))))) - t_1)) - (x * 2.0)));
	} else {
		tmp = x * ((wj / (x * (-1.0 - wj))) - ((exp(-wj) / (-1.0 - wj)) - (wj / x)));
	}
	return tmp;
}

Fortran

real(8) function code(wj, x)
    real(8), intent (in) :: wj
    real(8), intent (in) :: x
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = wj * exp(wj)
    t_1 = (x * (-4.0d0)) + (x * 1.5d0)
    if ((wj + ((x - t_0) / (exp(wj) + t_0))) <= 5d-19) then
        tmp = x + (wj * ((wj * ((1.0d0 + (wj * ((-1.0d0) - ((x * (-3.0d0)) + (((-2.0d0) * t_1) + (x * 0.6666666666666666d0)))))) - t_1)) - (x * 2.0d0)))
    else
        tmp = x * ((wj / (x * ((-1.0d0) - wj))) - ((exp(-wj) / ((-1.0d0) - wj)) - (wj / x)))
    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 * -4.0) + (x * 1.5);
	double tmp;
	if ((wj + ((x - t_0) / (Math.exp(wj) + t_0))) <= 5e-19) {
		tmp = x + (wj * ((wj * ((1.0 + (wj * (-1.0 - ((x * -3.0) + ((-2.0 * t_1) + (x * 0.6666666666666666)))))) - t_1)) - (x * 2.0)));
	} else {
		tmp = x * ((wj / (x * (-1.0 - wj))) - ((Math.exp(-wj) / (-1.0 - wj)) - (wj / x)));
	}
	return tmp;
}

Python

def code(wj, x):
	t_0 = wj * math.exp(wj)
	t_1 = (x * -4.0) + (x * 1.5)
	tmp = 0
	if (wj + ((x - t_0) / (math.exp(wj) + t_0))) <= 5e-19:
		tmp = x + (wj * ((wj * ((1.0 + (wj * (-1.0 - ((x * -3.0) + ((-2.0 * t_1) + (x * 0.6666666666666666)))))) - t_1)) - (x * 2.0)))
	else:
		tmp = x * ((wj / (x * (-1.0 - wj))) - ((math.exp(-wj) / (-1.0 - wj)) - (wj / x)))
	return tmp

Julia

function code(wj, x)
	t_0 = Float64(wj * exp(wj))
	t_1 = Float64(Float64(x * -4.0) + Float64(x * 1.5))
	tmp = 0.0
	if (Float64(wj + Float64(Float64(x - t_0) / Float64(exp(wj) + t_0))) <= 5e-19)
		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_1) + Float64(x * 0.6666666666666666)))))) - t_1)) - Float64(x * 2.0))));
	else
		tmp = Float64(x * Float64(Float64(wj / Float64(x * Float64(-1.0 - wj))) - Float64(Float64(exp(Float64(-wj)) / Float64(-1.0 - wj)) - Float64(wj / x))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(wj, x)
	t_0 = wj * exp(wj);
	t_1 = (x * -4.0) + (x * 1.5);
	tmp = 0.0;
	if ((wj + ((x - t_0) / (exp(wj) + t_0))) <= 5e-19)
		tmp = x + (wj * ((wj * ((1.0 + (wj * (-1.0 - ((x * -3.0) + ((-2.0 * t_1) + (x * 0.6666666666666666)))))) - t_1)) - (x * 2.0)));
	else
		tmp = x * ((wj / (x * (-1.0 - wj))) - ((exp(-wj) / (-1.0 - wj)) - (wj / x)));
	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[(N[(x * -4.0), $MachinePrecision] + N[(x * 1.5), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(wj + N[(N[(x - t$95$0), $MachinePrecision] / N[(N[Exp[wj], $MachinePrecision] + t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 5e-19], 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$1), $MachinePrecision] + N[(x * 0.6666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - t$95$1), $MachinePrecision]), $MachinePrecision] - N[(x * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x * N[(N[(wj / N[(x * N[(-1.0 - wj), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(N[(N[Exp[(-wj)], $MachinePrecision] / N[(-1.0 - wj), $MachinePrecision]), $MachinePrecision] - N[(wj / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 wj (/.f64 (-.f64 (*.f64 wj (exp.f64 wj)) x) (+.f64 (exp.f64 wj) (*.f64 wj (exp.f64 wj))))) < 5.0000000000000004e-19

   1. Initial program 69.7%

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

      1. distribute-rgt1-in 69.7%

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

      2. associate-/l/ 69.7%

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

      3. div-sub 69.7%

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

      4. associate-/l* 69.7%

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

      5. *-inverses 69.7%

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

      6. *-rgt-identity 69.7%

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

   3. Simplified 69.7%

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

   5. Taylor expanded in wj around 0 98.9%

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

   if 5.0000000000000004e-19 < (-.f64 wj (/.f64 (-.f64 (*.f64 wj (exp.f64 wj)) x) (+.f64 (exp.f64 wj) (*.f64 wj (exp.f64 wj)))))

   1. Initial program 92.8%

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

      1. distribute-rgt1-in 94.2%

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

      2. associate-/l/ 94.3%

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

      3. div-sub 93.0%

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

      4. associate-/l* 93.0%

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

      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

   5. Taylor expanded in x around inf 99.5%

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

      1. +-commutative 99.5%

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

      2. associate-/r* 99.5%

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

      3. exp-neg 99.5%

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

      4. +-commutative 99.5%

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

      5. +-commutative 99.5%

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

   7. Simplified 99.5%

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

4. Final simplification 99.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;wj + \frac{x - wj \cdot e^{wj}}{e^{wj} + wj \cdot e^{wj}} \leq 5 \cdot 10^{-19}:\\ \;\;\;\;x + wj \cdot \left(wj \cdot \left(\left(1 + wj \cdot \left(-1 - \left(x \cdot -3 + \left(-2 \cdot \left(x \cdot -4 + x \cdot 1.5\right) + x \cdot 0.6666666666666666\right)\right)\right)\right) - \left(x \cdot -4 + x \cdot 1.5\right)\right) - x \cdot 2\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(\frac{wj}{x \cdot \left(-1 - wj\right)} - \left(\frac{e^{-wj}}{-1 - wj} - \frac{wj}{x}\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 97.8% accurate, 2.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;wj \leq 2.8 \cdot 10^{-7}:\\ \;\;\;\;x + wj \cdot \left(wj \cdot \left(1 - wj\right) - x \cdot 2\right)\\ \mathbf{else}:\\ \;\;\;\;wj + \frac{wj - \frac{x}{e^{wj}}}{-1 - wj}\\ \end{array} \end{array} \]

FPCore

(FPCore (wj x)
 :precision binary64
 (if (<= wj 2.8e-7)
   (+ x (* wj (- (* wj (- 1.0 wj)) (* x 2.0))))
   (+ wj (/ (- wj (/ x (exp wj))) (- -1.0 wj)))))

C

double code(double wj, double x) {
	double tmp;
	if (wj <= 2.8e-7) {
		tmp = x + (wj * ((wj * (1.0 - wj)) - (x * 2.0)));
	} 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) :: tmp
    if (wj <= 2.8d-7) then
        tmp = x + (wj * ((wj * (1.0d0 - wj)) - (x * 2.0d0)))
    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 tmp;
	if (wj <= 2.8e-7) {
		tmp = x + (wj * ((wj * (1.0 - wj)) - (x * 2.0)));
	} else {
		tmp = wj + ((wj - (x / Math.exp(wj))) / (-1.0 - wj));
	}
	return tmp;
}

Python

def code(wj, x):
	tmp = 0
	if wj <= 2.8e-7:
		tmp = x + (wj * ((wj * (1.0 - wj)) - (x * 2.0)))
	else:
		tmp = wj + ((wj - (x / math.exp(wj))) / (-1.0 - wj))
	return tmp

Julia

function code(wj, x)
	tmp = 0.0
	if (wj <= 2.8e-7)
		tmp = Float64(x + Float64(wj * Float64(Float64(wj * Float64(1.0 - wj)) - Float64(x * 2.0))));
	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)
	tmp = 0.0;
	if (wj <= 2.8e-7)
		tmp = x + (wj * ((wj * (1.0 - wj)) - (x * 2.0)));
	else
		tmp = wj + ((wj - (x / exp(wj))) / (-1.0 - wj));
	end
	tmp_2 = tmp;
end

Wolfram

code[wj_, x_] := If[LessEqual[wj, 2.8e-7], N[(x + N[(wj * N[(N[(wj * N[(1.0 - wj), $MachinePrecision]), $MachinePrecision] - N[(x * 2.0), $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 2 regimes

2. if wj < 2.80000000000000019e-7

   1. Initial program 77.4%

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

      1. distribute-rgt1-in 77.9%

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

      2. associate-/l/ 77.9%

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

      3. div-sub 77.4%

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

      4. associate-/l* 77.4%

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

      5. *-inverses 77.9%

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

      6. *-rgt-identity 77.9%

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

   3. Simplified 77.9%

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

   5. Taylor expanded in wj around 0 98.3%

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

   6. Taylor expanded in x around 0 98.2%

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

   7. Taylor expanded in x around 0 98.4%

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

      1. neg-mul-1 98.4%

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

      2. unsub-neg 98.4%

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

   9. Simplified 98.4%

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

   if 2.80000000000000019e-7 < wj

   1. Initial program 59.1%

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

      1. distribute-rgt1-in 59.3%

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

      2. associate-/l/ 60.0%

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

      3. div-sub 60.0%

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

      4. associate-/l* 60.0%

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

      5. *-inverses 96.4%

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

      6. *-rgt-identity 96.4%

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

   3. Simplified 96.4%

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

4. Final simplification 98.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;wj \leq 2.8 \cdot 10^{-7}:\\ \;\;\;\;x + wj \cdot \left(wj \cdot \left(1 - wj\right) - x \cdot 2\right)\\ \mathbf{else}:\\ \;\;\;\;wj + \frac{wj - \frac{x}{e^{wj}}}{-1 - wj}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 97.0% accurate, 15.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;wj \leq 0.00023:\\ \;\;\;\;x + wj \cdot \left(wj \cdot \left(1 - x \cdot -2.5\right) + x \cdot -2\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(\frac{wj}{x} + \frac{wj}{x \cdot \left(-1 - wj\right)}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (wj x)
 :precision binary64
 (if (<= wj 0.00023)
   (+ x (* wj (+ (* wj (- 1.0 (* x -2.5))) (* x -2.0))))
   (* x (+ (/ wj x) (/ wj (* x (- -1.0 wj)))))))

C

double code(double wj, double x) {
	double tmp;
	if (wj <= 0.00023) {
		tmp = x + (wj * ((wj * (1.0 - (x * -2.5))) + (x * -2.0)));
	} else {
		tmp = x * ((wj / x) + (wj / (x * (-1.0 - wj))));
	}
	return tmp;
}

Fortran

real(8) function code(wj, x)
    real(8), intent (in) :: wj
    real(8), intent (in) :: x
    real(8) :: tmp
    if (wj <= 0.00023d0) then
        tmp = x + (wj * ((wj * (1.0d0 - (x * (-2.5d0)))) + (x * (-2.0d0))))
    else
        tmp = x * ((wj / x) + (wj / (x * ((-1.0d0) - wj))))
    end if
    code = tmp
end function

Java

public static double code(double wj, double x) {
	double tmp;
	if (wj <= 0.00023) {
		tmp = x + (wj * ((wj * (1.0 - (x * -2.5))) + (x * -2.0)));
	} else {
		tmp = x * ((wj / x) + (wj / (x * (-1.0 - wj))));
	}
	return tmp;
}

Python

def code(wj, x):
	tmp = 0
	if wj <= 0.00023:
		tmp = x + (wj * ((wj * (1.0 - (x * -2.5))) + (x * -2.0)))
	else:
		tmp = x * ((wj / x) + (wj / (x * (-1.0 - wj))))
	return tmp

Julia

function code(wj, x)
	tmp = 0.0
	if (wj <= 0.00023)
		tmp = Float64(x + Float64(wj * Float64(Float64(wj * Float64(1.0 - Float64(x * -2.5))) + Float64(x * -2.0))));
	else
		tmp = Float64(x * Float64(Float64(wj / x) + Float64(wj / Float64(x * Float64(-1.0 - wj)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(wj, x)
	tmp = 0.0;
	if (wj <= 0.00023)
		tmp = x + (wj * ((wj * (1.0 - (x * -2.5))) + (x * -2.0)));
	else
		tmp = x * ((wj / x) + (wj / (x * (-1.0 - wj))));
	end
	tmp_2 = tmp;
end

Wolfram

code[wj_, x_] := If[LessEqual[wj, 0.00023], N[(x + N[(wj * N[(N[(wj * N[(1.0 - N[(x * -2.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(x * -2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x * N[(N[(wj / x), $MachinePrecision] + N[(wj / N[(x * N[(-1.0 - wj), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if wj < 2.3000000000000001e-4

   1. Initial program 77.6%

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

      1. distribute-rgt1-in 78.0%

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

      2. associate-/l/ 78.0%

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

      3. div-sub 77.6%

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

      4. associate-/l* 77.6%

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

      5. *-inverses 78.0%

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

      6. *-rgt-identity 78.0%

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

   3. Simplified 78.0%

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

   5. Taylor expanded in wj around 0 98.0%

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

      1. cancel-sign-sub-inv 98.0%

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

      2. distribute-rgt-out 98.4%

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

      3. metadata-eval 98.4%

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

      4. metadata-eval 98.4%

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

      5. *-commutative 98.4%

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

   7. Simplified 98.4%

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

   if 2.3000000000000001e-4 < wj

   1. Initial program 50.1%

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

      1. distribute-rgt1-in 50.4%

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

      2. associate-/l/ 51.3%

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

      3. div-sub 51.3%

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

      4. associate-/l* 51.3%

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

      5. *-inverses 95.7%

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

      6. *-rgt-identity 95.7%

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

   3. Simplified 95.7%

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

   5. Taylor expanded in wj around 0 53.6%

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

      1. associate-*r* 53.6%

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

      2. neg-mul-1 53.6%

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

      3. distribute-rgt-out 53.6%

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

      4. metadata-eval 53.6%

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

   7. Simplified 53.6%

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

   8. Taylor expanded in x around inf 54.6%

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

   9. Taylor expanded in x around 0 91.4%

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

4. Final simplification 98.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;wj \leq 0.00023:\\ \;\;\;\;x + wj \cdot \left(wj \cdot \left(1 - x \cdot -2.5\right) + x \cdot -2\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(\frac{wj}{x} + \frac{wj}{x \cdot \left(-1 - wj\right)}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 96.8% accurate, 17.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;wj \leq 0.00035:\\ \;\;\;\;x + wj \cdot \left(wj + x \cdot -2\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(\frac{wj}{x} + \frac{wj}{x \cdot \left(-1 - wj\right)}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (wj x)
 :precision binary64
 (if (<= wj 0.00035)
   (+ x (* wj (+ wj (* x -2.0))))
   (* x (+ (/ wj x) (/ wj (* x (- -1.0 wj)))))))

C

double code(double wj, double x) {
	double tmp;
	if (wj <= 0.00035) {
		tmp = x + (wj * (wj + (x * -2.0)));
	} else {
		tmp = x * ((wj / x) + (wj / (x * (-1.0 - wj))));
	}
	return tmp;
}

Fortran

real(8) function code(wj, x)
    real(8), intent (in) :: wj
    real(8), intent (in) :: x
    real(8) :: tmp
    if (wj <= 0.00035d0) then
        tmp = x + (wj * (wj + (x * (-2.0d0))))
    else
        tmp = x * ((wj / x) + (wj / (x * ((-1.0d0) - wj))))
    end if
    code = tmp
end function

Java

public static double code(double wj, double x) {
	double tmp;
	if (wj <= 0.00035) {
		tmp = x + (wj * (wj + (x * -2.0)));
	} else {
		tmp = x * ((wj / x) + (wj / (x * (-1.0 - wj))));
	}
	return tmp;
}

Python

def code(wj, x):
	tmp = 0
	if wj <= 0.00035:
		tmp = x + (wj * (wj + (x * -2.0)))
	else:
		tmp = x * ((wj / x) + (wj / (x * (-1.0 - wj))))
	return tmp

Julia

function code(wj, x)
	tmp = 0.0
	if (wj <= 0.00035)
		tmp = Float64(x + Float64(wj * Float64(wj + Float64(x * -2.0))));
	else
		tmp = Float64(x * Float64(Float64(wj / x) + Float64(wj / Float64(x * Float64(-1.0 - wj)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(wj, x)
	tmp = 0.0;
	if (wj <= 0.00035)
		tmp = x + (wj * (wj + (x * -2.0)));
	else
		tmp = x * ((wj / x) + (wj / (x * (-1.0 - wj))));
	end
	tmp_2 = tmp;
end

Wolfram

code[wj_, x_] := If[LessEqual[wj, 0.00035], N[(x + N[(wj * N[(wj + N[(x * -2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x * N[(N[(wj / x), $MachinePrecision] + N[(wj / N[(x * N[(-1.0 - wj), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if wj < 3.49999999999999996e-4

   1. Initial program 77.6%

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

      1. distribute-rgt1-in 78.0%

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

      2. associate-/l/ 78.0%

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

      3. div-sub 77.6%

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

      4. associate-/l* 77.6%

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

      5. *-inverses 78.0%

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

      6. *-rgt-identity 78.0%

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

   3. Simplified 78.0%

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

   5. Taylor expanded in wj around 0 98.0%

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

      1. cancel-sign-sub-inv 98.0%

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

      2. distribute-rgt-out 98.4%

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

      3. metadata-eval 98.4%

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

      4. metadata-eval 98.4%

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

      5. *-commutative 98.4%

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

   7. Simplified 98.4%

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

   8. Taylor expanded in x around 0 98.1%

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

   if 3.49999999999999996e-4 < wj

   1. Initial program 50.1%

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

      1. distribute-rgt1-in 50.4%

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

      2. associate-/l/ 51.3%

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

      3. div-sub 51.3%

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

      4. associate-/l* 51.3%

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

      5. *-inverses 95.7%

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

      6. *-rgt-identity 95.7%

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

   3. Simplified 95.7%

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

   5. Taylor expanded in wj around 0 53.6%

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

      1. associate-*r* 53.6%

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

      2. neg-mul-1 53.6%

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

      3. distribute-rgt-out 53.6%

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

      4. metadata-eval 53.6%

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

   7. Simplified 53.6%

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

   8. Taylor expanded in x around inf 54.6%

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

   9. Taylor expanded in x around 0 91.4%

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

4. Final simplification 97.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;wj \leq 0.00035:\\ \;\;\;\;x + wj \cdot \left(wj + x \cdot -2\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(\frac{wj}{x} + \frac{wj}{x \cdot \left(-1 - wj\right)}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 97.2% accurate, 17.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;wj \leq 0.00039:\\ \;\;\;\;x + wj \cdot \left(wj \cdot \left(1 - wj\right) - x \cdot 2\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(\frac{wj}{x} + \frac{wj}{x \cdot \left(-1 - wj\right)}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (wj x)
 :precision binary64
 (if (<= wj 0.00039)
   (+ x (* wj (- (* wj (- 1.0 wj)) (* x 2.0))))
   (* x (+ (/ wj x) (/ wj (* x (- -1.0 wj)))))))

C

double code(double wj, double x) {
	double tmp;
	if (wj <= 0.00039) {
		tmp = x + (wj * ((wj * (1.0 - wj)) - (x * 2.0)));
	} else {
		tmp = x * ((wj / x) + (wj / (x * (-1.0 - wj))));
	}
	return tmp;
}

Fortran

real(8) function code(wj, x)
    real(8), intent (in) :: wj
    real(8), intent (in) :: x
    real(8) :: tmp
    if (wj <= 0.00039d0) then
        tmp = x + (wj * ((wj * (1.0d0 - wj)) - (x * 2.0d0)))
    else
        tmp = x * ((wj / x) + (wj / (x * ((-1.0d0) - wj))))
    end if
    code = tmp
end function

Java

public static double code(double wj, double x) {
	double tmp;
	if (wj <= 0.00039) {
		tmp = x + (wj * ((wj * (1.0 - wj)) - (x * 2.0)));
	} else {
		tmp = x * ((wj / x) + (wj / (x * (-1.0 - wj))));
	}
	return tmp;
}

Python

def code(wj, x):
	tmp = 0
	if wj <= 0.00039:
		tmp = x + (wj * ((wj * (1.0 - wj)) - (x * 2.0)))
	else:
		tmp = x * ((wj / x) + (wj / (x * (-1.0 - wj))))
	return tmp

Julia

function code(wj, x)
	tmp = 0.0
	if (wj <= 0.00039)
		tmp = Float64(x + Float64(wj * Float64(Float64(wj * Float64(1.0 - wj)) - Float64(x * 2.0))));
	else
		tmp = Float64(x * Float64(Float64(wj / x) + Float64(wj / Float64(x * Float64(-1.0 - wj)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(wj, x)
	tmp = 0.0;
	if (wj <= 0.00039)
		tmp = x + (wj * ((wj * (1.0 - wj)) - (x * 2.0)));
	else
		tmp = x * ((wj / x) + (wj / (x * (-1.0 - wj))));
	end
	tmp_2 = tmp;
end

Wolfram

code[wj_, x_] := If[LessEqual[wj, 0.00039], N[(x + N[(wj * N[(N[(wj * N[(1.0 - wj), $MachinePrecision]), $MachinePrecision] - N[(x * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x * N[(N[(wj / x), $MachinePrecision] + N[(wj / N[(x * N[(-1.0 - wj), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if wj < 3.89999999999999993e-4

   1. Initial program 77.6%

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

      1. distribute-rgt1-in 78.0%

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

      2. associate-/l/ 78.0%

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

      3. div-sub 77.6%

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

      4. associate-/l* 77.6%

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

      5. *-inverses 78.0%

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

      6. *-rgt-identity 78.0%

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

   3. Simplified 78.0%

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

   5. Taylor expanded in wj around 0 98.3%

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

   6. Taylor expanded in x around 0 98.2%

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

   7. Taylor expanded in x around 0 98.2%

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

      1. neg-mul-1 98.2%

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

      2. unsub-neg 98.2%

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

   9. Simplified 98.2%

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

   if 3.89999999999999993e-4 < wj

   1. Initial program 50.1%

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

      1. distribute-rgt1-in 50.4%

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

      2. associate-/l/ 51.3%

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

      3. div-sub 51.3%

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

      4. associate-/l* 51.3%

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

      5. *-inverses 95.7%

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

      6. *-rgt-identity 95.7%

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

   3. Simplified 95.7%

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

   5. Taylor expanded in wj around 0 53.6%

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

      1. associate-*r* 53.6%

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

      2. neg-mul-1 53.6%

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

      3. distribute-rgt-out 53.6%

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

      4. metadata-eval 53.6%

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

   7. Simplified 53.6%

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

   8. Taylor expanded in x around inf 54.6%

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

   9. Taylor expanded in x around 0 91.4%

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

4. Final simplification 98.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;wj \leq 0.00039:\\ \;\;\;\;x + wj \cdot \left(wj \cdot \left(1 - wj\right) - x \cdot 2\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(\frac{wj}{x} + \frac{wj}{x \cdot \left(-1 - wj\right)}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 96.9% accurate, 22.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;wj \leq 6.5 \cdot 10^{-5}:\\ \;\;\;\;x + wj \cdot \left(wj + x \cdot -2\right)\\ \mathbf{else}:\\ \;\;\;\;wj + \frac{wj}{-1 - wj}\\ \end{array} \end{array} \]

FPCore

(FPCore (wj x)
 :precision binary64
 (if (<= wj 6.5e-5) (+ x (* wj (+ wj (* x -2.0)))) (+ wj (/ wj (- -1.0 wj)))))

C

double code(double wj, double x) {
	double tmp;
	if (wj <= 6.5e-5) {
		tmp = x + (wj * (wj + (x * -2.0)));
	} else {
		tmp = wj + (wj / (-1.0 - wj));
	}
	return tmp;
}

Fortran

real(8) function code(wj, x)
    real(8), intent (in) :: wj
    real(8), intent (in) :: x
    real(8) :: tmp
    if (wj <= 6.5d-5) then
        tmp = x + (wj * (wj + (x * (-2.0d0))))
    else
        tmp = wj + (wj / ((-1.0d0) - wj))
    end if
    code = tmp
end function

Java

public static double code(double wj, double x) {
	double tmp;
	if (wj <= 6.5e-5) {
		tmp = x + (wj * (wj + (x * -2.0)));
	} else {
		tmp = wj + (wj / (-1.0 - wj));
	}
	return tmp;
}

Python

def code(wj, x):
	tmp = 0
	if wj <= 6.5e-5:
		tmp = x + (wj * (wj + (x * -2.0)))
	else:
		tmp = wj + (wj / (-1.0 - wj))
	return tmp

Julia

function code(wj, x)
	tmp = 0.0
	if (wj <= 6.5e-5)
		tmp = Float64(x + Float64(wj * Float64(wj + Float64(x * -2.0))));
	else
		tmp = Float64(wj + Float64(wj / Float64(-1.0 - wj)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(wj, x)
	tmp = 0.0;
	if (wj <= 6.5e-5)
		tmp = x + (wj * (wj + (x * -2.0)));
	else
		tmp = wj + (wj / (-1.0 - wj));
	end
	tmp_2 = tmp;
end

Wolfram

code[wj_, x_] := If[LessEqual[wj, 6.5e-5], N[(x + N[(wj * N[(wj + N[(x * -2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(wj + N[(wj / N[(-1.0 - wj), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if wj < 6.49999999999999943e-5

   1. Initial program 77.6%

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

      1. distribute-rgt1-in 78.0%

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

      2. associate-/l/ 78.0%

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

      3. div-sub 77.6%

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

      4. associate-/l* 77.6%

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

      5. *-inverses 78.0%

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

      6. *-rgt-identity 78.0%

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

   3. Simplified 78.0%

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

   5. Taylor expanded in wj around 0 98.0%

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

      1. cancel-sign-sub-inv 98.0%

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

      2. distribute-rgt-out 98.4%

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

      3. metadata-eval 98.4%

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

      4. metadata-eval 98.4%

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

      5. *-commutative 98.4%

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

   7. Simplified 98.4%

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

   8. Taylor expanded in x around 0 98.1%

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

   if 6.49999999999999943e-5 < wj

   1. Initial program 50.1%

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

      1. distribute-rgt1-in 50.4%

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

      2. associate-/l/ 51.3%

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

      3. div-sub 51.3%

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

      4. associate-/l* 51.3%

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

      5. *-inverses 95.7%

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

      6. *-rgt-identity 95.7%

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

   3. Simplified 95.7%

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

   5. Taylor expanded in x around 0 90.8%

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

      1. +-commutative 90.8%

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

   7. Simplified 90.8%

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

4. Final simplification 97.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;wj \leq 6.5 \cdot 10^{-5}:\\ \;\;\;\;x + wj \cdot \left(wj + x \cdot -2\right)\\ \mathbf{else}:\\ \;\;\;\;wj + \frac{wj}{-1 - wj}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 96.8% accurate, 22.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;wj \leq 0.00038:\\ \;\;\;\;x + wj \cdot \left(wj + x \cdot -2\right)\\ \mathbf{else}:\\ \;\;\;\;wj + \frac{1}{\frac{-1 - wj}{wj}}\\ \end{array} \end{array} \]

FPCore

(FPCore (wj x)
 :precision binary64
 (if (<= wj 0.00038)
   (+ x (* wj (+ wj (* x -2.0))))
   (+ wj (/ 1.0 (/ (- -1.0 wj) wj)))))

C

double code(double wj, double x) {
	double tmp;
	if (wj <= 0.00038) {
		tmp = x + (wj * (wj + (x * -2.0)));
	} else {
		tmp = wj + (1.0 / ((-1.0 - wj) / wj));
	}
	return tmp;
}

Fortran

real(8) function code(wj, x)
    real(8), intent (in) :: wj
    real(8), intent (in) :: x
    real(8) :: tmp
    if (wj <= 0.00038d0) then
        tmp = x + (wj * (wj + (x * (-2.0d0))))
    else
        tmp = wj + (1.0d0 / (((-1.0d0) - wj) / wj))
    end if
    code = tmp
end function

Java

public static double code(double wj, double x) {
	double tmp;
	if (wj <= 0.00038) {
		tmp = x + (wj * (wj + (x * -2.0)));
	} else {
		tmp = wj + (1.0 / ((-1.0 - wj) / wj));
	}
	return tmp;
}

Python

def code(wj, x):
	tmp = 0
	if wj <= 0.00038:
		tmp = x + (wj * (wj + (x * -2.0)))
	else:
		tmp = wj + (1.0 / ((-1.0 - wj) / wj))
	return tmp

Julia

function code(wj, x)
	tmp = 0.0
	if (wj <= 0.00038)
		tmp = Float64(x + Float64(wj * Float64(wj + Float64(x * -2.0))));
	else
		tmp = Float64(wj + Float64(1.0 / Float64(Float64(-1.0 - wj) / wj)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(wj, x)
	tmp = 0.0;
	if (wj <= 0.00038)
		tmp = x + (wj * (wj + (x * -2.0)));
	else
		tmp = wj + (1.0 / ((-1.0 - wj) / wj));
	end
	tmp_2 = tmp;
end

Wolfram

code[wj_, x_] := If[LessEqual[wj, 0.00038], N[(x + N[(wj * N[(wj + N[(x * -2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(wj + N[(1.0 / N[(N[(-1.0 - wj), $MachinePrecision] / wj), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if wj < 3.8000000000000002e-4

   1. Initial program 77.6%

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

      1. distribute-rgt1-in 78.0%

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

      2. associate-/l/ 78.0%

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

      3. div-sub 77.6%

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

      4. associate-/l* 77.6%

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

      5. *-inverses 78.0%

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

      6. *-rgt-identity 78.0%

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

   3. Simplified 78.0%

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

   5. Taylor expanded in wj around 0 98.0%

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

      1. cancel-sign-sub-inv 98.0%

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

      2. distribute-rgt-out 98.4%

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

      3. metadata-eval 98.4%

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

      4. metadata-eval 98.4%

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

      5. *-commutative 98.4%

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

   7. Simplified 98.4%

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

   8. Taylor expanded in x around 0 98.1%

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

   if 3.8000000000000002e-4 < wj

   1. Initial program 50.1%

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

      1. distribute-rgt1-in 50.4%

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

      2. associate-/l/ 51.3%

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

      3. div-sub 51.3%

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

      4. associate-/l* 51.3%

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

      5. *-inverses 95.7%

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

      6. *-rgt-identity 95.7%

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

   3. Simplified 95.7%

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

   5. Taylor expanded in x around 0 90.8%

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

      1. +-commutative 90.8%

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

   7. Simplified 90.8%

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

      1. clear-num 90.9%

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

      2. inv-pow 90.9%

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

   9. Applied egg-rr 90.9%

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

       1. unpow-1 90.9%

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

   11. Simplified 90.9%

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

4. Final simplification 97.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;wj \leq 0.00038:\\ \;\;\;\;x + wj \cdot \left(wj + x \cdot -2\right)\\ \mathbf{else}:\\ \;\;\;\;wj + \frac{1}{\frac{-1 - wj}{wj}}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 85.0% accurate, 26.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;wj \leq 0.00039:\\ \;\;\;\;x + -2 \cdot \left(wj \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;wj - \left(1 + \frac{-1}{wj}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (wj x)
 :precision binary64
 (if (<= wj 0.00039) (+ x (* -2.0 (* wj x))) (- wj (+ 1.0 (/ -1.0 wj)))))

C

double code(double wj, double x) {
	double tmp;
	if (wj <= 0.00039) {
		tmp = x + (-2.0 * (wj * x));
	} else {
		tmp = wj - (1.0 + (-1.0 / wj));
	}
	return tmp;
}

Fortran

real(8) function code(wj, x)
    real(8), intent (in) :: wj
    real(8), intent (in) :: x
    real(8) :: tmp
    if (wj <= 0.00039d0) then
        tmp = x + ((-2.0d0) * (wj * x))
    else
        tmp = wj - (1.0d0 + ((-1.0d0) / wj))
    end if
    code = tmp
end function

Java

public static double code(double wj, double x) {
	double tmp;
	if (wj <= 0.00039) {
		tmp = x + (-2.0 * (wj * x));
	} else {
		tmp = wj - (1.0 + (-1.0 / wj));
	}
	return tmp;
}

Python

def code(wj, x):
	tmp = 0
	if wj <= 0.00039:
		tmp = x + (-2.0 * (wj * x))
	else:
		tmp = wj - (1.0 + (-1.0 / wj))
	return tmp

Julia

function code(wj, x)
	tmp = 0.0
	if (wj <= 0.00039)
		tmp = Float64(x + Float64(-2.0 * Float64(wj * x)));
	else
		tmp = Float64(wj - Float64(1.0 + Float64(-1.0 / wj)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(wj, x)
	tmp = 0.0;
	if (wj <= 0.00039)
		tmp = x + (-2.0 * (wj * x));
	else
		tmp = wj - (1.0 + (-1.0 / wj));
	end
	tmp_2 = tmp;
end

Wolfram

code[wj_, x_] := If[LessEqual[wj, 0.00039], N[(x + N[(-2.0 * N[(wj * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(wj - N[(1.0 + N[(-1.0 / wj), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if wj < 3.89999999999999993e-4

   1. Initial program 77.6%

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

      1. distribute-rgt1-in 78.0%

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

      2. associate-/l/ 78.0%

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

      3. div-sub 77.6%

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

      4. associate-/l* 77.6%

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

      5. *-inverses 78.0%

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

      6. *-rgt-identity 78.0%

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

   3. Simplified 78.0%

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

   5. Taylor expanded in wj around 0 84.8%

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

      1. *-commutative 84.8%

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

   7. Simplified 84.8%

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

   if 3.89999999999999993e-4 < wj

   1. Initial program 50.1%

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

      1. distribute-rgt1-in 50.4%

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

      2. associate-/l/ 51.3%

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

      3. div-sub 51.3%

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

      4. associate-/l* 51.3%

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

      5. *-inverses 95.7%

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

      6. *-rgt-identity 95.7%

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

   3. Simplified 95.7%

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

   5. Taylor expanded in x around 0 90.8%

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

      1. +-commutative 90.8%

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

   7. Simplified 90.8%

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

   8. Taylor expanded in wj around inf 58.8%

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

4. Final simplification 83.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;wj \leq 0.00039:\\ \;\;\;\;x + -2 \cdot \left(wj \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;wj - \left(1 + \frac{-1}{wj}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 85.6% accurate, 26.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;wj \leq 8.5 \cdot 10^{-5}:\\ \;\;\;\;x + -2 \cdot \left(wj \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;wj + \frac{wj}{-1 - wj}\\ \end{array} \end{array} \]

FPCore

(FPCore (wj x)
 :precision binary64
 (if (<= wj 8.5e-5) (+ x (* -2.0 (* wj x))) (+ wj (/ wj (- -1.0 wj)))))

C

double code(double wj, double x) {
	double tmp;
	if (wj <= 8.5e-5) {
		tmp = x + (-2.0 * (wj * x));
	} else {
		tmp = wj + (wj / (-1.0 - wj));
	}
	return tmp;
}

Fortran

real(8) function code(wj, x)
    real(8), intent (in) :: wj
    real(8), intent (in) :: x
    real(8) :: tmp
    if (wj <= 8.5d-5) then
        tmp = x + ((-2.0d0) * (wj * x))
    else
        tmp = wj + (wj / ((-1.0d0) - wj))
    end if
    code = tmp
end function

Java

public static double code(double wj, double x) {
	double tmp;
	if (wj <= 8.5e-5) {
		tmp = x + (-2.0 * (wj * x));
	} else {
		tmp = wj + (wj / (-1.0 - wj));
	}
	return tmp;
}

Python

def code(wj, x):
	tmp = 0
	if wj <= 8.5e-5:
		tmp = x + (-2.0 * (wj * x))
	else:
		tmp = wj + (wj / (-1.0 - wj))
	return tmp

Julia

function code(wj, x)
	tmp = 0.0
	if (wj <= 8.5e-5)
		tmp = Float64(x + Float64(-2.0 * Float64(wj * x)));
	else
		tmp = Float64(wj + Float64(wj / Float64(-1.0 - wj)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(wj, x)
	tmp = 0.0;
	if (wj <= 8.5e-5)
		tmp = x + (-2.0 * (wj * x));
	else
		tmp = wj + (wj / (-1.0 - wj));
	end
	tmp_2 = tmp;
end

Wolfram

code[wj_, x_] := If[LessEqual[wj, 8.5e-5], N[(x + N[(-2.0 * N[(wj * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(wj + N[(wj / N[(-1.0 - wj), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if wj < 8.500000000000001e-5

   1. Initial program 77.6%

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

      1. distribute-rgt1-in 78.0%

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

      2. associate-/l/ 78.0%

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

      3. div-sub 77.6%

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

      4. associate-/l* 77.6%

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

      5. *-inverses 78.0%

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

      6. *-rgt-identity 78.0%

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

   3. Simplified 78.0%

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

   5. Taylor expanded in wj around 0 84.8%

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

      1. *-commutative 84.8%

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

   7. Simplified 84.8%

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

   if 8.500000000000001e-5 < wj

   1. Initial program 50.1%

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

      1. distribute-rgt1-in 50.4%

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

      2. associate-/l/ 51.3%

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

      3. div-sub 51.3%

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

      4. associate-/l* 51.3%

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

      5. *-inverses 95.7%

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

      6. *-rgt-identity 95.7%

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

   3. Simplified 95.7%

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

   5. Taylor expanded in x around 0 90.8%

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

      1. +-commutative 90.8%

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

   7. Simplified 90.8%

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

4. Final simplification 85.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;wj \leq 8.5 \cdot 10^{-5}:\\ \;\;\;\;x + -2 \cdot \left(wj \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;wj + \frac{wj}{-1 - wj}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 84.2% accurate, 44.7× speedup

Math

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

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 76.7%

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

   1. distribute-rgt1-in 77.1%

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

   2. associate-/l/ 77.1%

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

   3. div-sub 76.7%

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

   4. associate-/l* 76.7%

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

   5. *-inverses 78.6%

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

   6. *-rgt-identity 78.6%

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

3. Simplified 78.6%

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

5. Taylor expanded in wj around 0 76.3%

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

   1. associate-*r* 76.3%

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

   2. neg-mul-1 76.3%

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

   3. distribute-rgt-out 76.3%

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

   4. metadata-eval 76.3%

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

7. Simplified 76.3%

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

8. Taylor expanded in x around inf 76.7%

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

9. Taylor expanded in wj around 0 82.0%

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

    1. *-commutative 82.0%

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

11. Simplified 82.0%

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

12. Final simplification 82.0%

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

Alternative 11: 84.2% 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 76.7%

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

   1. distribute-rgt1-in 77.1%

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

   2. associate-/l/ 77.1%

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

   3. div-sub 76.7%

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

   4. associate-/l* 76.7%

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

   5. *-inverses 78.6%

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

   6. *-rgt-identity 78.6%

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

3. Simplified 78.6%

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

5. Taylor expanded in wj around 0 82.0%

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

   1. *-commutative 82.0%

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

7. Simplified 82.0%

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

8. Final simplification 82.0%

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

Alternative 12: 4.3% accurate, 313.0× speedup

Math

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

FPCore

(FPCore (wj x) :precision binary64 wj)

C

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

Fortran

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

Java

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

Python

def code(wj, x):
	return wj

Julia

function code(wj, x)
	return wj
end

MATLAB

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

Wolfram

code[wj_, x_] := wj

Derivation

1. Initial program 76.7%

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

   1. distribute-rgt1-in 77.1%

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

   2. associate-/l/ 77.1%

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

   3. div-sub 76.7%

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

   4. associate-/l* 76.7%

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

   5. *-inverses 78.6%

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

   6. *-rgt-identity 78.6%

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

3. Simplified 78.6%

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

5. Taylor expanded in wj around inf 5.1%

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

6. Final simplification 5.1%

   \[\leadsto wj \]
7. Add Preprocessing

Alternative 13: 83.7% 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 76.7%

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

   1. distribute-rgt1-in 77.1%

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

   2. associate-/l/ 77.1%

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

   3. div-sub 76.7%

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

   4. associate-/l* 76.7%

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

   5. *-inverses 78.6%

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

   6. *-rgt-identity 78.6%

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

3. Simplified 78.6%

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

5. Taylor expanded in wj around 0 81.4%

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

6. Final simplification 81.4%

   \[\leadsto x \]
7. Add Preprocessing

Developer target: 78.1% 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 2024071 
(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))))))