Jmat.Real.lambertw, newton loop step

Percentage Accurate: 77.8% → 98.6%

Time: 10.0s

Alternatives: 10

Speedup: 26.0×

• could not determine a ground truth

  1. wj = 2.3481411065296882e+194
  2. x = -6.057763210450894e+49

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: 77.8% 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

\[\begin{array}{l} \\ \begin{array}{l} t_0 := wj \cdot e^{wj}\\ \mathbf{if}\;wj + \frac{x - t\_0}{e^{wj} + t\_0} \leq 5 \cdot 10^{-23}:\\ \;\;\;\;x + wj \cdot \left(x \cdot -2 + wj \cdot \left(1 + \left(x \cdot 2.5 + wj \cdot \left(x \cdot -2.6666666666666665 + -1\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;wj + \left(x \cdot \frac{e^{-wj}}{wj + 1} + \frac{wj}{-1 - wj}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (wj x)
 :precision binary64
 (let* ((t_0 (* wj (exp wj))))
   (if (<= (+ wj (/ (- x t_0) (+ (exp wj) t_0))) 5e-23)
     (+
      x
      (*
       wj
       (+
        (* x -2.0)
        (*
         wj
         (+ 1.0 (+ (* x 2.5) (* wj (+ (* x -2.6666666666666665) -1.0))))))))
     (+ wj (+ (* x (/ (exp (- wj)) (+ wj 1.0))) (/ wj (- -1.0 wj)))))))

C

double code(double wj, double x) {
	double t_0 = wj * exp(wj);
	double tmp;
	if ((wj + ((x - t_0) / (exp(wj) + t_0))) <= 5e-23) {
		tmp = x + (wj * ((x * -2.0) + (wj * (1.0 + ((x * 2.5) + (wj * ((x * -2.6666666666666665) + -1.0)))))));
	} else {
		tmp = wj + ((x * (exp(-wj) / (wj + 1.0))) + (wj / (-1.0 - wj)));
	}
	return tmp;
}

Fortran

real(8) function code(wj, x)
    real(8), intent (in) :: wj
    real(8), intent (in) :: x
    real(8) :: t_0
    real(8) :: tmp
    t_0 = wj * exp(wj)
    if ((wj + ((x - t_0) / (exp(wj) + t_0))) <= 5d-23) then
        tmp = x + (wj * ((x * (-2.0d0)) + (wj * (1.0d0 + ((x * 2.5d0) + (wj * ((x * (-2.6666666666666665d0)) + (-1.0d0))))))))
    else
        tmp = wj + ((x * (exp(-wj) / (wj + 1.0d0))) + (wj / ((-1.0d0) - wj)))
    end if
    code = tmp
end function

Java

public static double code(double wj, double x) {
	double t_0 = wj * Math.exp(wj);
	double tmp;
	if ((wj + ((x - t_0) / (Math.exp(wj) + t_0))) <= 5e-23) {
		tmp = x + (wj * ((x * -2.0) + (wj * (1.0 + ((x * 2.5) + (wj * ((x * -2.6666666666666665) + -1.0)))))));
	} else {
		tmp = wj + ((x * (Math.exp(-wj) / (wj + 1.0))) + (wj / (-1.0 - wj)));
	}
	return tmp;
}

Python

def code(wj, x):
	t_0 = wj * math.exp(wj)
	tmp = 0
	if (wj + ((x - t_0) / (math.exp(wj) + t_0))) <= 5e-23:
		tmp = x + (wj * ((x * -2.0) + (wj * (1.0 + ((x * 2.5) + (wj * ((x * -2.6666666666666665) + -1.0)))))))
	else:
		tmp = wj + ((x * (math.exp(-wj) / (wj + 1.0))) + (wj / (-1.0 - wj)))
	return tmp

Julia

function code(wj, x)
	t_0 = Float64(wj * exp(wj))
	tmp = 0.0
	if (Float64(wj + Float64(Float64(x - t_0) / Float64(exp(wj) + t_0))) <= 5e-23)
		tmp = Float64(x + Float64(wj * Float64(Float64(x * -2.0) + Float64(wj * Float64(1.0 + Float64(Float64(x * 2.5) + Float64(wj * Float64(Float64(x * -2.6666666666666665) + -1.0))))))));
	else
		tmp = Float64(wj + Float64(Float64(x * Float64(exp(Float64(-wj)) / Float64(wj + 1.0))) + Float64(wj / Float64(-1.0 - wj))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(wj, x)
	t_0 = wj * exp(wj);
	tmp = 0.0;
	if ((wj + ((x - t_0) / (exp(wj) + t_0))) <= 5e-23)
		tmp = x + (wj * ((x * -2.0) + (wj * (1.0 + ((x * 2.5) + (wj * ((x * -2.6666666666666665) + -1.0)))))));
	else
		tmp = wj + ((x * (exp(-wj) / (wj + 1.0))) + (wj / (-1.0 - wj)));
	end
	tmp_2 = tmp;
end

Wolfram

code[wj_, x_] := Block[{t$95$0 = N[(wj * N[Exp[wj], $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-23], N[(x + N[(wj * N[(N[(x * -2.0), $MachinePrecision] + N[(wj * N[(1.0 + N[(N[(x * 2.5), $MachinePrecision] + N[(wj * N[(N[(x * -2.6666666666666665), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(wj + N[(N[(x * N[(N[Exp[(-wj)], $MachinePrecision] / N[(wj + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(wj / N[(-1.0 - wj), $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))))) < 5.0000000000000002e-23

   1. Initial program 70.7%

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

      1. distribute-rgt1-in 70.7%

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

      2. associate-/l/ 70.8%

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

      3. div-sub 70.8%

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

      4. associate-/l* 70.8%

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

      5. *-inverses 70.8%

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

      6. *-rgt-identity 70.8%

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

   3. Simplified 70.8%

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

   5. Taylor expanded in x around inf 70.7%

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

      1. +-commutative 70.7%

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

      2. associate-/r* 70.7%

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

      3. rec-exp 70.7%

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

      4. +-commutative 70.7%

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

   7. Simplified 70.7%

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

   8. Taylor expanded in wj around 0 70.6%

      \[\leadsto wj - x \cdot \left(\frac{wj}{x \cdot \left(wj + 1\right)} - \frac{\color{blue}{1 + wj \cdot \left(wj \cdot \left(0.5 + wj \cdot \left(0.041666666666666664 \cdot wj - 0.16666666666666666\right)\right) - 1\right)}}{wj + 1}\right) \]

   9. Taylor expanded in x around 0 70.7%

      \[\leadsto \color{blue}{\left(wj + x \cdot \left(\frac{1}{1 + wj} + \frac{wj \cdot \left(wj \cdot \left(0.5 + wj \cdot \left(0.041666666666666664 \cdot wj - 0.16666666666666666\right)\right) - 1\right)}{1 + wj}\right)\right) - \frac{wj}{1 + wj}} \]

   10. Taylor expanded in wj around 0 99.2%

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

   if 5.0000000000000002e-23 < (-.f64 wj (/.f64 (-.f64 (*.f64 wj (exp.f64 wj)) x) (+.f64 (exp.f64 wj) (*.f64 wj (exp.f64 wj)))))

   1. Initial program 90.2%

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

      1. distribute-rgt1-in 92.9%

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

      2. associate-/l/ 92.9%

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

      3. div-sub 90.2%

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

      4. associate-/l* 90.2%

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

      5. *-inverses 99.8%

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

      6. *-rgt-identity 99.8%

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

   3. Simplified 99.8%

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

   5. Taylor expanded in x around inf 99.8%

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

      1. +-commutative 99.8%

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

      2. associate-/r* 99.8%

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

      3. rec-exp 99.8%

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

      4. +-commutative 99.8%

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

   7. Simplified 99.8%

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

   8. Taylor expanded in x around 0 99.8%

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

      1. associate--l+ 99.8%

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

      2. associate-/l* 99.8%

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

      3. +-commutative 99.8%

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

      4. +-commutative 99.8%

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

   10. Simplified 99.8%

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

4. Final simplification 99.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;wj + \frac{x - wj \cdot e^{wj}}{e^{wj} + wj \cdot e^{wj}} \leq 5 \cdot 10^{-23}:\\ \;\;\;\;x + wj \cdot \left(x \cdot -2 + wj \cdot \left(1 + \left(x \cdot 2.5 + wj \cdot \left(x \cdot -2.6666666666666665 + -1\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;wj + \left(x \cdot \frac{e^{-wj}}{wj + 1} + \frac{wj}{-1 - wj}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 98.3% accurate, 2.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;wj \leq 7.3 \cdot 10^{-6}:\\ \;\;\;\;x + wj \cdot \left(x \cdot -2 + wj \cdot \left(1 + \left(x \cdot 2.5 + wj \cdot \left(x \cdot -2.6666666666666665 + -1\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;wj + \frac{wj - \frac{x}{e^{wj}}}{-1 - wj}\\ \end{array} \end{array} \]

FPCore

(FPCore (wj x)
 :precision binary64
 (if (<= wj 7.3e-6)
   (+
    x
    (*
     wj
     (+
      (* x -2.0)
      (* wj (+ 1.0 (+ (* x 2.5) (* wj (+ (* x -2.6666666666666665) -1.0))))))))
   (+ wj (/ (- wj (/ x (exp wj))) (- -1.0 wj)))))

C

double code(double wj, double x) {
	double tmp;
	if (wj <= 7.3e-6) {
		tmp = x + (wj * ((x * -2.0) + (wj * (1.0 + ((x * 2.5) + (wj * ((x * -2.6666666666666665) + -1.0)))))));
	} else {
		tmp = wj + ((wj - (x / exp(wj))) / (-1.0 - wj));
	}
	return tmp;
}

Fortran

real(8) function code(wj, x)
    real(8), intent (in) :: wj
    real(8), intent (in) :: x
    real(8) :: tmp
    if (wj <= 7.3d-6) then
        tmp = x + (wj * ((x * (-2.0d0)) + (wj * (1.0d0 + ((x * 2.5d0) + (wj * ((x * (-2.6666666666666665d0)) + (-1.0d0))))))))
    else
        tmp = wj + ((wj - (x / exp(wj))) / ((-1.0d0) - wj))
    end if
    code = tmp
end function

Java

public static double code(double wj, double x) {
	double tmp;
	if (wj <= 7.3e-6) {
		tmp = x + (wj * ((x * -2.0) + (wj * (1.0 + ((x * 2.5) + (wj * ((x * -2.6666666666666665) + -1.0)))))));
	} else {
		tmp = wj + ((wj - (x / Math.exp(wj))) / (-1.0 - wj));
	}
	return tmp;
}

Python

def code(wj, x):
	tmp = 0
	if wj <= 7.3e-6:
		tmp = x + (wj * ((x * -2.0) + (wj * (1.0 + ((x * 2.5) + (wj * ((x * -2.6666666666666665) + -1.0)))))))
	else:
		tmp = wj + ((wj - (x / math.exp(wj))) / (-1.0 - wj))
	return tmp

Julia

function code(wj, x)
	tmp = 0.0
	if (wj <= 7.3e-6)
		tmp = Float64(x + Float64(wj * Float64(Float64(x * -2.0) + Float64(wj * Float64(1.0 + Float64(Float64(x * 2.5) + Float64(wj * Float64(Float64(x * -2.6666666666666665) + -1.0))))))));
	else
		tmp = Float64(wj + Float64(Float64(wj - Float64(x / exp(wj))) / Float64(-1.0 - wj)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(wj, x)
	tmp = 0.0;
	if (wj <= 7.3e-6)
		tmp = x + (wj * ((x * -2.0) + (wj * (1.0 + ((x * 2.5) + (wj * ((x * -2.6666666666666665) + -1.0)))))));
	else
		tmp = wj + ((wj - (x / exp(wj))) / (-1.0 - wj));
	end
	tmp_2 = tmp;
end

Wolfram

code[wj_, x_] := If[LessEqual[wj, 7.3e-6], N[(x + N[(wj * N[(N[(x * -2.0), $MachinePrecision] + N[(wj * N[(1.0 + N[(N[(x * 2.5), $MachinePrecision] + N[(wj * N[(N[(x * -2.6666666666666665), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(wj + N[(N[(wj - N[(x / N[Exp[wj], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(-1.0 - wj), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if wj < 7.30000000000000041e-6

   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 78.2%

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

      2. associate-/l/ 78.2%

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

      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 78.2%

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

      6. *-rgt-identity 78.2%

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

   3. Simplified 78.2%

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

   5. Taylor expanded in x around inf 78.2%

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

      1. +-commutative 78.2%

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

      2. associate-/r* 78.2%

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

      3. rec-exp 78.2%

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

      4. +-commutative 78.2%

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

   7. Simplified 78.2%

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

   8. Taylor expanded in wj around 0 77.3%

      \[\leadsto wj - x \cdot \left(\frac{wj}{x \cdot \left(wj + 1\right)} - \frac{\color{blue}{1 + wj \cdot \left(wj \cdot \left(0.5 + wj \cdot \left(0.041666666666666664 \cdot wj - 0.16666666666666666\right)\right) - 1\right)}}{wj + 1}\right) \]

   9. Taylor expanded in x around 0 77.4%

      \[\leadsto \color{blue}{\left(wj + x \cdot \left(\frac{1}{1 + wj} + \frac{wj \cdot \left(wj \cdot \left(0.5 + wj \cdot \left(0.041666666666666664 \cdot wj - 0.16666666666666666\right)\right) - 1\right)}{1 + wj}\right)\right) - \frac{wj}{1 + wj}} \]

   10. Taylor expanded in wj around 0 98.6%

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

   if 7.30000000000000041e-6 < wj

   1. Initial program 48.3%

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

      1. distribute-rgt1-in 48.0%

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

      2. associate-/l/ 48.6%

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

      3. div-sub 48.6%

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

      4. associate-/l* 48.6%

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

      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
3. Recombined 2 regimes into one program.

4. Final simplification 98.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;wj \leq 7.3 \cdot 10^{-6}:\\ \;\;\;\;x + wj \cdot \left(x \cdot -2 + wj \cdot \left(1 + \left(x \cdot 2.5 + wj \cdot \left(x \cdot -2.6666666666666665 + -1\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;wj + \frac{wj - \frac{x}{e^{wj}}}{-1 - wj}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 97.7% accurate, 11.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;wj \leq 0.69:\\ \;\;\;\;x + wj \cdot \left(x \cdot -2 + wj \cdot \left(1 + \left(x \cdot 2.5 + wj \cdot \left(x \cdot -2.6666666666666665 + -1\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;wj + \left(-1 + \frac{1 + \frac{-1 - \frac{-1}{wj}}{wj}}{wj}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (wj x)
 :precision binary64
 (if (<= wj 0.69)
   (+
    x
    (*
     wj
     (+
      (* x -2.0)
      (* wj (+ 1.0 (+ (* x 2.5) (* wj (+ (* x -2.6666666666666665) -1.0))))))))
   (+ wj (+ -1.0 (/ (+ 1.0 (/ (- -1.0 (/ -1.0 wj)) wj)) wj)))))

C

double code(double wj, double x) {
	double tmp;
	if (wj <= 0.69) {
		tmp = x + (wj * ((x * -2.0) + (wj * (1.0 + ((x * 2.5) + (wj * ((x * -2.6666666666666665) + -1.0)))))));
	} else {
		tmp = wj + (-1.0 + ((1.0 + ((-1.0 - (-1.0 / wj)) / 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 <= 0.69d0) then
        tmp = x + (wj * ((x * (-2.0d0)) + (wj * (1.0d0 + ((x * 2.5d0) + (wj * ((x * (-2.6666666666666665d0)) + (-1.0d0))))))))
    else
        tmp = wj + ((-1.0d0) + ((1.0d0 + (((-1.0d0) - ((-1.0d0) / wj)) / wj)) / wj))
    end if
    code = tmp
end function

Java

public static double code(double wj, double x) {
	double tmp;
	if (wj <= 0.69) {
		tmp = x + (wj * ((x * -2.0) + (wj * (1.0 + ((x * 2.5) + (wj * ((x * -2.6666666666666665) + -1.0)))))));
	} else {
		tmp = wj + (-1.0 + ((1.0 + ((-1.0 - (-1.0 / wj)) / wj)) / wj));
	}
	return tmp;
}

Python

def code(wj, x):
	tmp = 0
	if wj <= 0.69:
		tmp = x + (wj * ((x * -2.0) + (wj * (1.0 + ((x * 2.5) + (wj * ((x * -2.6666666666666665) + -1.0)))))))
	else:
		tmp = wj + (-1.0 + ((1.0 + ((-1.0 - (-1.0 / wj)) / wj)) / wj))
	return tmp

Julia

function code(wj, x)
	tmp = 0.0
	if (wj <= 0.69)
		tmp = Float64(x + Float64(wj * Float64(Float64(x * -2.0) + Float64(wj * Float64(1.0 + Float64(Float64(x * 2.5) + Float64(wj * Float64(Float64(x * -2.6666666666666665) + -1.0))))))));
	else
		tmp = Float64(wj + Float64(-1.0 + Float64(Float64(1.0 + Float64(Float64(-1.0 - Float64(-1.0 / wj)) / wj)) / wj)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(wj, x)
	tmp = 0.0;
	if (wj <= 0.69)
		tmp = x + (wj * ((x * -2.0) + (wj * (1.0 + ((x * 2.5) + (wj * ((x * -2.6666666666666665) + -1.0)))))));
	else
		tmp = wj + (-1.0 + ((1.0 + ((-1.0 - (-1.0 / wj)) / wj)) / wj));
	end
	tmp_2 = tmp;
end

Wolfram

code[wj_, x_] := If[LessEqual[wj, 0.69], N[(x + N[(wj * N[(N[(x * -2.0), $MachinePrecision] + N[(wj * N[(1.0 + N[(N[(x * 2.5), $MachinePrecision] + N[(wj * N[(N[(x * -2.6666666666666665), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(wj + N[(-1.0 + N[(N[(1.0 + N[(N[(-1.0 - N[(-1.0 / wj), $MachinePrecision]), $MachinePrecision] / wj), $MachinePrecision]), $MachinePrecision] / wj), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if wj < 0.68999999999999995

   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 78.4%

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

      2. associate-/l/ 78.4%

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

      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 78.4%

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

      6. *-rgt-identity 78.4%

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

   3. Simplified 78.4%

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

   5. Taylor expanded in x around inf 78.4%

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

      1. +-commutative 78.4%

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

      2. associate-/r* 78.4%

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

      3. rec-exp 78.4%

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

      4. +-commutative 78.4%

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

   7. Simplified 78.4%

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

   8. Taylor expanded in wj around 0 77.5%

      \[\leadsto wj - x \cdot \left(\frac{wj}{x \cdot \left(wj + 1\right)} - \frac{\color{blue}{1 + wj \cdot \left(wj \cdot \left(0.5 + wj \cdot \left(0.041666666666666664 \cdot wj - 0.16666666666666666\right)\right) - 1\right)}}{wj + 1}\right) \]

   9. Taylor expanded in x around 0 77.6%

      \[\leadsto \color{blue}{\left(wj + x \cdot \left(\frac{1}{1 + wj} + \frac{wj \cdot \left(wj \cdot \left(0.5 + wj \cdot \left(0.041666666666666664 \cdot wj - 0.16666666666666666\right)\right) - 1\right)}{1 + wj}\right)\right) - \frac{wj}{1 + wj}} \]

   10. Taylor expanded in wj around 0 98.0%

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

   if 0.68999999999999995 < wj

   1. Initial program 28.6%

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

      1. distribute-rgt1-in 28.6%

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

      2. associate-/l/ 28.6%

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

      3. div-sub 28.6%

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

      4. associate-/l* 28.6%

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

      5. *-inverses 100.0%

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

      6. *-rgt-identity 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in wj around 0 58.5%

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

      1. *-commutative 58.5%

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

   7. Simplified 58.5%

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

   8. Taylor expanded in wj around -inf 87.7%

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

      1. mul-1-neg 87.7%

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

      2. unsub-neg 87.7%

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

      3. mul-1-neg 87.7%

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

      4. unsub-neg 87.7%

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

      5. sub-neg 87.7%

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

      6. distribute-neg-frac 87.7%

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

      7. metadata-eval 87.7%

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

   10. Simplified 87.7%

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

4. Final simplification 97.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;wj \leq 0.69:\\ \;\;\;\;x + wj \cdot \left(x \cdot -2 + wj \cdot \left(1 + \left(x \cdot 2.5 + wj \cdot \left(x \cdot -2.6666666666666665 + -1\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;wj + \left(-1 + \frac{1 + \frac{-1 - \frac{-1}{wj}}{wj}}{wj}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 97.1% accurate, 15.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;wj \leq 1.42:\\ \;\;\;\;x + wj \cdot \left(x \cdot -2 + wj \cdot \left(1 + x \cdot 2.5\right)\right)\\ \mathbf{else}:\\ \;\;\;\;wj + \left(-1 + \frac{1 + \frac{-1 - \frac{-1}{wj}}{wj}}{wj}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (wj x)
 :precision binary64
 (if (<= wj 1.42)
   (+ x (* wj (+ (* x -2.0) (* wj (+ 1.0 (* x 2.5))))))
   (+ wj (+ -1.0 (/ (+ 1.0 (/ (- -1.0 (/ -1.0 wj)) wj)) wj)))))

C

double code(double wj, double x) {
	double tmp;
	if (wj <= 1.42) {
		tmp = x + (wj * ((x * -2.0) + (wj * (1.0 + (x * 2.5)))));
	} else {
		tmp = wj + (-1.0 + ((1.0 + ((-1.0 - (-1.0 / wj)) / 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 <= 1.42d0) then
        tmp = x + (wj * ((x * (-2.0d0)) + (wj * (1.0d0 + (x * 2.5d0)))))
    else
        tmp = wj + ((-1.0d0) + ((1.0d0 + (((-1.0d0) - ((-1.0d0) / wj)) / wj)) / wj))
    end if
    code = tmp
end function

Java

public static double code(double wj, double x) {
	double tmp;
	if (wj <= 1.42) {
		tmp = x + (wj * ((x * -2.0) + (wj * (1.0 + (x * 2.5)))));
	} else {
		tmp = wj + (-1.0 + ((1.0 + ((-1.0 - (-1.0 / wj)) / wj)) / wj));
	}
	return tmp;
}

Python

def code(wj, x):
	tmp = 0
	if wj <= 1.42:
		tmp = x + (wj * ((x * -2.0) + (wj * (1.0 + (x * 2.5)))))
	else:
		tmp = wj + (-1.0 + ((1.0 + ((-1.0 - (-1.0 / wj)) / wj)) / wj))
	return tmp

Julia

function code(wj, x)
	tmp = 0.0
	if (wj <= 1.42)
		tmp = Float64(x + Float64(wj * Float64(Float64(x * -2.0) + Float64(wj * Float64(1.0 + Float64(x * 2.5))))));
	else
		tmp = Float64(wj + Float64(-1.0 + Float64(Float64(1.0 + Float64(Float64(-1.0 - Float64(-1.0 / wj)) / wj)) / wj)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(wj, x)
	tmp = 0.0;
	if (wj <= 1.42)
		tmp = x + (wj * ((x * -2.0) + (wj * (1.0 + (x * 2.5)))));
	else
		tmp = wj + (-1.0 + ((1.0 + ((-1.0 - (-1.0 / wj)) / wj)) / wj));
	end
	tmp_2 = tmp;
end

Wolfram

code[wj_, x_] := If[LessEqual[wj, 1.42], N[(x + N[(wj * N[(N[(x * -2.0), $MachinePrecision] + N[(wj * N[(1.0 + N[(x * 2.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(wj + N[(-1.0 + N[(N[(1.0 + N[(N[(-1.0 - N[(-1.0 / wj), $MachinePrecision]), $MachinePrecision] / wj), $MachinePrecision]), $MachinePrecision] / wj), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if wj < 1.4199999999999999

   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 78.4%

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

      2. associate-/l/ 78.4%

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

      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 78.4%

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

      6. *-rgt-identity 78.4%

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

   3. Simplified 78.4%

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

   5. Taylor expanded in x around inf 78.4%

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

      1. +-commutative 78.4%

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

      2. associate-/r* 78.4%

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

      3. rec-exp 78.4%

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

      4. +-commutative 78.4%

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

   7. Simplified 78.4%

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

   8. Taylor expanded in wj around 0 77.5%

      \[\leadsto wj - x \cdot \left(\frac{wj}{x \cdot \left(wj + 1\right)} - \frac{\color{blue}{1 + wj \cdot \left(wj \cdot \left(0.5 + wj \cdot \left(0.041666666666666664 \cdot wj - 0.16666666666666666\right)\right) - 1\right)}}{wj + 1}\right) \]

   9. Taylor expanded in x around 0 77.6%

      \[\leadsto \color{blue}{\left(wj + x \cdot \left(\frac{1}{1 + wj} + \frac{wj \cdot \left(wj \cdot \left(0.5 + wj \cdot \left(0.041666666666666664 \cdot wj - 0.16666666666666666\right)\right) - 1\right)}{1 + wj}\right)\right) - \frac{wj}{1 + wj}} \]

   10. Taylor expanded in wj around 0 97.6%

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

   if 1.4199999999999999 < wj

   1. Initial program 28.6%

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

      1. distribute-rgt1-in 28.6%

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

      2. associate-/l/ 28.6%

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

      3. div-sub 28.6%

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

      4. associate-/l* 28.6%

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

      5. *-inverses 100.0%

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

      6. *-rgt-identity 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in wj around 0 58.5%

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

      1. *-commutative 58.5%

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

   7. Simplified 58.5%

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

   8. Taylor expanded in wj around -inf 87.7%

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

      1. mul-1-neg 87.7%

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

      2. unsub-neg 87.7%

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

      3. mul-1-neg 87.7%

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

      4. unsub-neg 87.7%

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

      5. sub-neg 87.7%

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

      6. distribute-neg-frac 87.7%

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

      7. metadata-eval 87.7%

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

   10. Simplified 87.7%

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

4. Final simplification 97.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;wj \leq 1.42:\\ \;\;\;\;x + wj \cdot \left(x \cdot -2 + wj \cdot \left(1 + x \cdot 2.5\right)\right)\\ \mathbf{else}:\\ \;\;\;\;wj + \left(-1 + \frac{1 + \frac{-1 - \frac{-1}{wj}}{wj}}{wj}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 97.1% accurate, 15.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;wj \leq 3.55:\\ \;\;\;\;x + wj \cdot \left(x \cdot -2 + wj \cdot \left(1 + x \cdot 2.5\right)\right)\\ \mathbf{else}:\\ \;\;\;\;wj + \left(-1 + \frac{1 + \frac{-1}{wj}}{wj}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (wj x)
 :precision binary64
 (if (<= wj 3.55)
   (+ x (* wj (+ (* x -2.0) (* wj (+ 1.0 (* x 2.5))))))
   (+ wj (+ -1.0 (/ (+ 1.0 (/ -1.0 wj)) wj)))))

C

double code(double wj, double x) {
	double tmp;
	if (wj <= 3.55) {
		tmp = x + (wj * ((x * -2.0) + (wj * (1.0 + (x * 2.5)))));
	} else {
		tmp = wj + (-1.0 + ((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 <= 3.55d0) then
        tmp = x + (wj * ((x * (-2.0d0)) + (wj * (1.0d0 + (x * 2.5d0)))))
    else
        tmp = wj + ((-1.0d0) + ((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 <= 3.55) {
		tmp = x + (wj * ((x * -2.0) + (wj * (1.0 + (x * 2.5)))));
	} else {
		tmp = wj + (-1.0 + ((1.0 + (-1.0 / wj)) / wj));
	}
	return tmp;
}

Python

def code(wj, x):
	tmp = 0
	if wj <= 3.55:
		tmp = x + (wj * ((x * -2.0) + (wj * (1.0 + (x * 2.5)))))
	else:
		tmp = wj + (-1.0 + ((1.0 + (-1.0 / wj)) / wj))
	return tmp

Julia

function code(wj, x)
	tmp = 0.0
	if (wj <= 3.55)
		tmp = Float64(x + Float64(wj * Float64(Float64(x * -2.0) + Float64(wj * Float64(1.0 + Float64(x * 2.5))))));
	else
		tmp = Float64(wj + Float64(-1.0 + Float64(Float64(1.0 + Float64(-1.0 / wj)) / wj)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(wj, x)
	tmp = 0.0;
	if (wj <= 3.55)
		tmp = x + (wj * ((x * -2.0) + (wj * (1.0 + (x * 2.5)))));
	else
		tmp = wj + (-1.0 + ((1.0 + (-1.0 / wj)) / wj));
	end
	tmp_2 = tmp;
end

Wolfram

code[wj_, x_] := If[LessEqual[wj, 3.55], N[(x + N[(wj * N[(N[(x * -2.0), $MachinePrecision] + N[(wj * N[(1.0 + N[(x * 2.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(wj + N[(-1.0 + N[(N[(1.0 + N[(-1.0 / wj), $MachinePrecision]), $MachinePrecision] / wj), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if wj < 3.5499999999999998

   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 78.4%

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

      2. associate-/l/ 78.4%

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

      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 78.4%

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

      6. *-rgt-identity 78.4%

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

   3. Simplified 78.4%

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

   5. Taylor expanded in x around inf 78.4%

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

      1. +-commutative 78.4%

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

      2. associate-/r* 78.4%

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

      3. rec-exp 78.4%

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

      4. +-commutative 78.4%

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

   7. Simplified 78.4%

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

   8. Taylor expanded in wj around 0 77.5%

      \[\leadsto wj - x \cdot \left(\frac{wj}{x \cdot \left(wj + 1\right)} - \frac{\color{blue}{1 + wj \cdot \left(wj \cdot \left(0.5 + wj \cdot \left(0.041666666666666664 \cdot wj - 0.16666666666666666\right)\right) - 1\right)}}{wj + 1}\right) \]

   9. Taylor expanded in x around 0 77.6%

      \[\leadsto \color{blue}{\left(wj + x \cdot \left(\frac{1}{1 + wj} + \frac{wj \cdot \left(wj \cdot \left(0.5 + wj \cdot \left(0.041666666666666664 \cdot wj - 0.16666666666666666\right)\right) - 1\right)}{1 + wj}\right)\right) - \frac{wj}{1 + wj}} \]

   10. Taylor expanded in wj around 0 97.6%

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

   if 3.5499999999999998 < wj

   1. Initial program 28.6%

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

      1. distribute-rgt1-in 28.6%

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

      2. associate-/l/ 28.6%

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

      3. div-sub 28.6%

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

      4. associate-/l* 28.6%

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

      5. *-inverses 100.0%

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

      6. *-rgt-identity 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in wj around 0 58.5%

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

      1. *-commutative 58.5%

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

   7. Simplified 58.5%

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

   8. Taylor expanded in wj around -inf 83.2%

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

      1. mul-1-neg 83.2%

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

      2. unsub-neg 83.2%

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

      3. sub-neg 83.2%

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

      4. distribute-neg-frac 83.2%

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

      5. metadata-eval 83.2%

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

   10. Simplified 83.2%

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

4. Final simplification 97.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;wj \leq 3.55:\\ \;\;\;\;x + wj \cdot \left(x \cdot -2 + wj \cdot \left(1 + x \cdot 2.5\right)\right)\\ \mathbf{else}:\\ \;\;\;\;wj + \left(-1 + \frac{1 + \frac{-1}{wj}}{wj}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 96.9% accurate, 17.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;wj \leq 3.15:\\ \;\;\;\;x - wj \cdot \left(x \cdot \left(2 - wj \cdot 2\right) - wj\right)\\ \mathbf{else}:\\ \;\;\;\;wj + \left(-1 + \frac{1 + \frac{-1}{wj}}{wj}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (wj x)
 :precision binary64
 (if (<= wj 3.15)
   (- x (* wj (- (* x (- 2.0 (* wj 2.0))) wj)))
   (+ wj (+ -1.0 (/ (+ 1.0 (/ -1.0 wj)) wj)))))

C

double code(double wj, double x) {
	double tmp;
	if (wj <= 3.15) {
		tmp = x - (wj * ((x * (2.0 - (wj * 2.0))) - wj));
	} else {
		tmp = wj + (-1.0 + ((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 <= 3.15d0) then
        tmp = x - (wj * ((x * (2.0d0 - (wj * 2.0d0))) - wj))
    else
        tmp = wj + ((-1.0d0) + ((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 <= 3.15) {
		tmp = x - (wj * ((x * (2.0 - (wj * 2.0))) - wj));
	} else {
		tmp = wj + (-1.0 + ((1.0 + (-1.0 / wj)) / wj));
	}
	return tmp;
}

Python

def code(wj, x):
	tmp = 0
	if wj <= 3.15:
		tmp = x - (wj * ((x * (2.0 - (wj * 2.0))) - wj))
	else:
		tmp = wj + (-1.0 + ((1.0 + (-1.0 / wj)) / wj))
	return tmp

Julia

function code(wj, x)
	tmp = 0.0
	if (wj <= 3.15)
		tmp = Float64(x - Float64(wj * Float64(Float64(x * Float64(2.0 - Float64(wj * 2.0))) - wj)));
	else
		tmp = Float64(wj + Float64(-1.0 + Float64(Float64(1.0 + Float64(-1.0 / wj)) / wj)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(wj, x)
	tmp = 0.0;
	if (wj <= 3.15)
		tmp = x - (wj * ((x * (2.0 - (wj * 2.0))) - wj));
	else
		tmp = wj + (-1.0 + ((1.0 + (-1.0 / wj)) / wj));
	end
	tmp_2 = tmp;
end

Wolfram

code[wj_, x_] := If[LessEqual[wj, 3.15], N[(x - N[(wj * N[(N[(x * N[(2.0 - N[(wj * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - wj), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(wj + N[(-1.0 + N[(N[(1.0 + N[(-1.0 / wj), $MachinePrecision]), $MachinePrecision] / wj), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if wj < 3.14999999999999991

   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 78.4%

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

      2. associate-/l/ 78.4%

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

      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 78.4%

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

      6. *-rgt-identity 78.4%

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

   3. Simplified 78.4%

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

   5. Taylor expanded in wj around 0 77.1%

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

      1. +-commutative 77.1%

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

   7. Simplified 77.1%

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

   8. Taylor expanded in wj around 0 97.5%

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

   9. Taylor expanded in x around 0 97.5%

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

   if 3.14999999999999991 < wj

   1. Initial program 28.6%

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

      1. distribute-rgt1-in 28.6%

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

      2. associate-/l/ 28.6%

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

      3. div-sub 28.6%

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

      4. associate-/l* 28.6%

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

      5. *-inverses 100.0%

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

      6. *-rgt-identity 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in wj around 0 58.5%

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

      1. *-commutative 58.5%

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

   7. Simplified 58.5%

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

   8. Taylor expanded in wj around -inf 83.2%

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

      1. mul-1-neg 83.2%

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

      2. unsub-neg 83.2%

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

      3. sub-neg 83.2%

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

      4. distribute-neg-frac 83.2%

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

      5. metadata-eval 83.2%

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

   10. Simplified 83.2%

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

4. Final simplification 97.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;wj \leq 3.15:\\ \;\;\;\;x - wj \cdot \left(x \cdot \left(2 - wj \cdot 2\right) - wj\right)\\ \mathbf{else}:\\ \;\;\;\;wj + \left(-1 + \frac{1 + \frac{-1}{wj}}{wj}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 85.7% accurate, 19.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;wj \leq 0.75:\\ \;\;\;\;x + -2 \cdot \left(wj \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;wj + \left(-1 + \frac{1 + \frac{-1}{wj}}{wj}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (wj x)
 :precision binary64
 (if (<= wj 0.75)
   (+ x (* -2.0 (* wj x)))
   (+ wj (+ -1.0 (/ (+ 1.0 (/ -1.0 wj)) wj)))))

C

double code(double wj, double x) {
	double tmp;
	if (wj <= 0.75) {
		tmp = x + (-2.0 * (wj * x));
	} else {
		tmp = wj + (-1.0 + ((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 <= 0.75d0) then
        tmp = x + ((-2.0d0) * (wj * x))
    else
        tmp = wj + ((-1.0d0) + ((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 <= 0.75) {
		tmp = x + (-2.0 * (wj * x));
	} else {
		tmp = wj + (-1.0 + ((1.0 + (-1.0 / wj)) / wj));
	}
	return tmp;
}

Python

def code(wj, x):
	tmp = 0
	if wj <= 0.75:
		tmp = x + (-2.0 * (wj * x))
	else:
		tmp = wj + (-1.0 + ((1.0 + (-1.0 / wj)) / wj))
	return tmp

Julia

function code(wj, x)
	tmp = 0.0
	if (wj <= 0.75)
		tmp = Float64(x + Float64(-2.0 * Float64(wj * x)));
	else
		tmp = Float64(wj + Float64(-1.0 + Float64(Float64(1.0 + Float64(-1.0 / wj)) / wj)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(wj, x)
	tmp = 0.0;
	if (wj <= 0.75)
		tmp = x + (-2.0 * (wj * x));
	else
		tmp = wj + (-1.0 + ((1.0 + (-1.0 / wj)) / wj));
	end
	tmp_2 = tmp;
end

Wolfram

code[wj_, x_] := If[LessEqual[wj, 0.75], N[(x + N[(-2.0 * N[(wj * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(wj + N[(-1.0 + N[(N[(1.0 + N[(-1.0 / wj), $MachinePrecision]), $MachinePrecision] / wj), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if wj < 0.75

   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 78.4%

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

      2. associate-/l/ 78.4%

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

      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 78.4%

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

      6. *-rgt-identity 78.4%

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

   3. Simplified 78.4%

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

   5. Taylor expanded in wj around 0 88.4%

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

      1. *-commutative 88.4%

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

   7. Simplified 88.4%

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

   if 0.75 < wj

   1. Initial program 28.6%

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

      1. distribute-rgt1-in 28.6%

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

      2. associate-/l/ 28.6%

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

      3. div-sub 28.6%

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

      4. associate-/l* 28.6%

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

      5. *-inverses 100.0%

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

      6. *-rgt-identity 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in wj around 0 58.5%

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

      1. *-commutative 58.5%

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

   7. Simplified 58.5%

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

   8. Taylor expanded in wj around -inf 83.2%

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

      1. mul-1-neg 83.2%

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

      2. unsub-neg 83.2%

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

      3. sub-neg 83.2%

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

      4. distribute-neg-frac 83.2%

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

      5. metadata-eval 83.2%

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

   10. Simplified 83.2%

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

4. Final simplification 88.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;wj \leq 0.75:\\ \;\;\;\;x + -2 \cdot \left(wj \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;wj + \left(-1 + \frac{1 + \frac{-1}{wj}}{wj}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 85.6% accurate, 26.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;wj \leq 0.2:\\ \;\;\;\;x + -2 \cdot \left(wj \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;wj + \left(-1 - \frac{-1}{wj}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (wj x)
 :precision binary64
 (if (<= wj 0.2) (+ x (* -2.0 (* wj x))) (+ wj (- -1.0 (/ -1.0 wj)))))

C

double code(double wj, double x) {
	double tmp;
	if (wj <= 0.2) {
		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 <= 0.2d0) 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 <= 0.2) {
		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 <= 0.2:
		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 <= 0.2)
		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 <= 0.2)
		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, 0.2], 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 < 0.20000000000000001

   1. Initial program 77.5%

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

      1. distribute-rgt1-in 78.3%

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

      2. associate-/l/ 78.4%

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

      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 78.4%

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

      6. *-rgt-identity 78.4%

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

   3. Simplified 78.4%

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

   5. Taylor expanded in wj around 0 88.7%

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

      1. *-commutative 88.7%

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

   7. Simplified 88.7%

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

   if 0.20000000000000001 < wj

   1. Initial program 37.0%

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

      1. distribute-rgt1-in 37.0%

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

      2. associate-/l/ 37.5%

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

      3. div-sub 37.5%

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

      4. associate-/l* 37.5%

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

      5. *-inverses 100.0%

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

      6. *-rgt-identity 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in wj around 0 63.7%

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

      1. *-commutative 63.7%

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

   7. Simplified 63.7%

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

   8. Taylor expanded in wj around inf 70.0%

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

4. Final simplification 88.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;wj \leq 0.2:\\ \;\;\;\;x + -2 \cdot \left(wj \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;wj + \left(-1 - \frac{-1}{wj}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 85.4% accurate, 26.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;wj \leq 0.92:\\ \;\;\;\;x + -2 \cdot \left(wj \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;wj + -1\\ \end{array} \end{array} \]

FPCore

(FPCore (wj x)
 :precision binary64
 (if (<= wj 0.92) (+ x (* -2.0 (* wj x))) (+ wj -1.0)))

C

double code(double wj, double x) {
	double tmp;
	if (wj <= 0.92) {
		tmp = x + (-2.0 * (wj * x));
	} else {
		tmp = wj + -1.0;
	}
	return tmp;
}

Fortran

real(8) function code(wj, x)
    real(8), intent (in) :: wj
    real(8), intent (in) :: x
    real(8) :: tmp
    if (wj <= 0.92d0) then
        tmp = x + ((-2.0d0) * (wj * x))
    else
        tmp = wj + (-1.0d0)
    end if
    code = tmp
end function

Java

public static double code(double wj, double x) {
	double tmp;
	if (wj <= 0.92) {
		tmp = x + (-2.0 * (wj * x));
	} else {
		tmp = wj + -1.0;
	}
	return tmp;
}

Python

def code(wj, x):
	tmp = 0
	if wj <= 0.92:
		tmp = x + (-2.0 * (wj * x))
	else:
		tmp = wj + -1.0
	return tmp

Julia

function code(wj, x)
	tmp = 0.0
	if (wj <= 0.92)
		tmp = Float64(x + Float64(-2.0 * Float64(wj * x)));
	else
		tmp = Float64(wj + -1.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(wj, x)
	tmp = 0.0;
	if (wj <= 0.92)
		tmp = x + (-2.0 * (wj * x));
	else
		tmp = wj + -1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[wj_, x_] := If[LessEqual[wj, 0.92], N[(x + N[(-2.0 * N[(wj * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(wj + -1.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if wj < 0.92000000000000004

   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 78.4%

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

      2. associate-/l/ 78.4%

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

      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 78.4%

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

      6. *-rgt-identity 78.4%

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

   3. Simplified 78.4%

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

   5. Taylor expanded in wj around 0 88.4%

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

      1. *-commutative 88.4%

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

   7. Simplified 88.4%

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

   if 0.92000000000000004 < wj

   1. Initial program 28.6%

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

      1. distribute-rgt1-in 28.6%

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

      2. associate-/l/ 28.6%

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

      3. div-sub 28.6%

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

      4. associate-/l* 28.6%

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

      5. *-inverses 100.0%

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

      6. *-rgt-identity 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in wj around inf 63.4%

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

   6. Taylor expanded in wj around 0 63.4%

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

4. Final simplification 87.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;wj \leq 0.92:\\ \;\;\;\;x + -2 \cdot \left(wj \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;wj + -1\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 4.1% accurate, 104.3× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 76.3%

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

   1. distribute-rgt1-in 77.0%

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

   2. associate-/l/ 77.1%

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

   3. div-sub 76.3%

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

   4. associate-/l* 76.3%

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

   5. *-inverses 79.0%

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

   6. *-rgt-identity 79.0%

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

3. Simplified 79.0%

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

5. Taylor expanded in wj around inf 5.2%

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

6. Taylor expanded in wj around 0 5.2%

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

7. Final simplification 5.2%

   \[\leadsto wj + -1 \]
8. Add Preprocessing

Developer Target 1: 78.8% 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 2024179 
(FPCore (wj x)
  :name "Jmat.Real.lambertw, newton loop step"
  :precision binary64

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

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