Jmat.Real.lambertw, newton loop step

Percentage Accurate: 78.4% → 98.8%

Time: 16.7s

Alternatives: 10

Speedup: 313.0×

• could not determine a ground truth

  1. wj = -5.3495630325723885e+113
  2. x = 7.387659279266024e+102

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 78.4% 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: 98.8% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := wj \cdot e^{wj}\\ t_1 := x \cdot -4 + x \cdot 1.5\\ \mathbf{if}\;wj + \frac{x - t\_0}{e^{wj} + t\_0} \leq 10^{-11}:\\ \;\;\;\;x - wj \cdot \left(x \cdot 2 - wj \cdot \left(\left(1 + wj \cdot \left(-1 - \left(x \cdot -3 + \left(-2 \cdot t\_1 + x \cdot 0.6666666666666666\right)\right)\right)\right) - t\_1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(\frac{e^{-wj}}{wj + 1} + \left(\frac{wj}{x} + \frac{\frac{wj}{x}}{-1 - wj}\right)\right)\\ \end{array} \end{array} \]

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))) 1e-11)
     (-
      x
      (*
       wj
       (-
        (* x 2.0)
        (*
         wj
         (-
          (+
           1.0
           (*
            wj
            (- -1.0 (+ (* x -3.0) (+ (* -2.0 t_1) (* x 0.6666666666666666))))))
          t_1)))))
     (*
      x
      (+ (/ (exp (- wj)) (+ wj 1.0)) (+ (/ wj x) (/ (/ wj x) (- -1.0 wj))))))))

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))) <= 1e-11) {
		tmp = x - (wj * ((x * 2.0) - (wj * ((1.0 + (wj * (-1.0 - ((x * -3.0) + ((-2.0 * t_1) + (x * 0.6666666666666666)))))) - t_1))));
	} else {
		tmp = x * ((exp(-wj) / (wj + 1.0)) + ((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) :: 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))) <= 1d-11) then
        tmp = x - (wj * ((x * 2.0d0) - (wj * ((1.0d0 + (wj * ((-1.0d0) - ((x * (-3.0d0)) + (((-2.0d0) * t_1) + (x * 0.6666666666666666d0)))))) - t_1))))
    else
        tmp = x * ((exp(-wj) / (wj + 1.0d0)) + ((wj / x) + ((wj / x) / ((-1.0d0) - wj))))
    end if
    code = tmp
end function

Java

public static double code(double wj, double x) {
	double t_0 = wj * Math.exp(wj);
	double t_1 = (x * -4.0) + (x * 1.5);
	double tmp;
	if ((wj + ((x - t_0) / (Math.exp(wj) + t_0))) <= 1e-11) {
		tmp = x - (wj * ((x * 2.0) - (wj * ((1.0 + (wj * (-1.0 - ((x * -3.0) + ((-2.0 * t_1) + (x * 0.6666666666666666)))))) - t_1))));
	} else {
		tmp = x * ((Math.exp(-wj) / (wj + 1.0)) + ((wj / x) + ((wj / x) / (-1.0 - wj))));
	}
	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))) <= 1e-11:
		tmp = x - (wj * ((x * 2.0) - (wj * ((1.0 + (wj * (-1.0 - ((x * -3.0) + ((-2.0 * t_1) + (x * 0.6666666666666666)))))) - t_1))))
	else:
		tmp = x * ((math.exp(-wj) / (wj + 1.0)) + ((wj / x) + ((wj / x) / (-1.0 - wj))))
	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))) <= 1e-11)
		tmp = Float64(x - Float64(wj * Float64(Float64(x * 2.0) - 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)))));
	else
		tmp = Float64(x * Float64(Float64(exp(Float64(-wj)) / Float64(wj + 1.0)) + Float64(Float64(wj / x) + Float64(Float64(wj / x) / Float64(-1.0 - wj)))));
	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))) <= 1e-11)
		tmp = x - (wj * ((x * 2.0) - (wj * ((1.0 + (wj * (-1.0 - ((x * -3.0) + ((-2.0 * t_1) + (x * 0.6666666666666666)))))) - t_1))));
	else
		tmp = x * ((exp(-wj) / (wj + 1.0)) + ((wj / x) + ((wj / x) / (-1.0 - wj))));
	end
	tmp_2 = tmp;
end

Wolfram

code[wj_, x_] := Block[{t$95$0 = N[(wj * N[Exp[wj], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(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], 1e-11], N[(x - N[(wj * N[(N[(x * 2.0), $MachinePrecision] - 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]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x * N[(N[(N[Exp[(-wj)], $MachinePrecision] / N[(wj + 1.0), $MachinePrecision]), $MachinePrecision] + N[(N[(wj / x), $MachinePrecision] + N[(N[(wj / x), $MachinePrecision] / N[(-1.0 - wj), $MachinePrecision]), $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))))) < 9.99999999999999939e-12

   1. Initial program 72.5%

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

      1. distribute-rgt1-in 72.5%

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

      2. associate-/l/ 72.5%

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

      3. div-sub 72.5%

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

      4. associate-/l* 72.5%

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

      5. *-inverses 72.5%

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

      6. *-rgt-identity 72.5%

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

   3. Simplified 72.5%

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

   5. Taylor expanded in wj around 0 99.1%

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

   if 9.99999999999999939e-12 < (-.f64 wj (/.f64 (-.f64 (*.f64 wj (exp.f64 wj)) x) (+.f64 (exp.f64 wj) (*.f64 wj (exp.f64 wj)))))

   1. Initial program 93.9%

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

      1. distribute-rgt1-in 96.7%

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

      2. associate-/l/ 96.7%

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

      3. div-sub 93.9%

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

      4. associate-/l* 93.9%

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

      5. *-inverses 99.6%

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

      6. *-rgt-identity 99.6%

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

   3. Simplified 99.6%

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

      1. add-cbrt-cube 43.2%

         \[\leadsto \color{blue}{\sqrt[3]{\left(\left(wj - \frac{wj - \frac{x}{e^{wj}}}{wj + 1}\right) \cdot \left(wj - \frac{wj - \frac{x}{e^{wj}}}{wj + 1}\right)\right) \cdot \left(wj - \frac{wj - \frac{x}{e^{wj}}}{wj + 1}\right)}} \]

      2. pow1/3 38.4%

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

      3. pow3 38.4%

         \[\leadsto {\color{blue}{\left({\left(wj - \frac{wj - \frac{x}{e^{wj}}}{wj + 1}\right)}^{3}\right)}}^{0.3333333333333333} \]

   6. Applied egg-rr 38.4%

      \[\leadsto \color{blue}{{\left({\left(wj - \frac{wj - \frac{x}{e^{wj}}}{wj + 1}\right)}^{3}\right)}^{0.3333333333333333}} \]

   7. Taylor expanded in x around inf 99.5%

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

      1. associate--l+ 99.6%

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

      2. associate-/r* 99.6%

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

      3. rec-exp 99.6%

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

      4. +-commutative 99.6%

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

      5. associate-/r* 99.6%

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

      6. +-commutative 99.6%

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

   9. Simplified 99.6%

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

4. Final simplification 99.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;wj + \frac{x - wj \cdot e^{wj}}{e^{wj} + wj \cdot e^{wj}} \leq 10^{-11}:\\ \;\;\;\;x - wj \cdot \left(x \cdot 2 - wj \cdot \left(\left(1 + wj \cdot \left(-1 - \left(x \cdot -3 + \left(-2 \cdot \left(x \cdot -4 + x \cdot 1.5\right) + x \cdot 0.6666666666666666\right)\right)\right)\right) - \left(x \cdot -4 + x \cdot 1.5\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(\frac{e^{-wj}}{wj + 1} + \left(\frac{wj}{x} + \frac{\frac{wj}{x}}{-1 - wj}\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 97.9% accurate, 2.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;wj \leq 3.3 \cdot 10^{-7}:\\ \;\;\;\;x + wj \cdot \left(wj \cdot \left(1 - wj\right) - x \cdot 2\right)\\ \mathbf{else}:\\ \;\;\;\;wj + \frac{wj - \frac{x}{e^{wj}}}{-1 - wj}\\ \end{array} \end{array} \]

FPCore

(FPCore (wj x)
 :precision binary64
 (if (<= wj 3.3e-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 <= 3.3e-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 <= 3.3d-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 <= 3.3e-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 <= 3.3e-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 <= 3.3e-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 <= 3.3e-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, 3.3e-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 < 3.3000000000000002e-7

   1. Initial program 78.3%

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

      1. distribute-rgt1-in 79.1%

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

      2. associate-/l/ 79.1%

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

      3. div-sub 78.3%

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

      4. associate-/l* 78.3%

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

      5. *-inverses 79.1%

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

      6. *-rgt-identity 79.1%

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

   3. Simplified 79.1%

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

   5. Taylor expanded in wj around 0 99.1%

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

   6. Taylor expanded in x around 0 99.1%

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

      1. mul-1-neg 99.1%

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

      2. unsub-neg 99.1%

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

   8. Simplified 99.1%

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

   if 3.3000000000000002e-7 < wj

   1. Initial program 79.0%

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

      1. distribute-rgt1-in 79.1%

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

      2. associate-/l/ 79.3%

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

      3. div-sub 79.3%

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

      4. associate-/l* 79.3%

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

      5. *-inverses 97.4%

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

      6. *-rgt-identity 97.4%

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

   3. Simplified 97.4%

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

4. Final simplification 99.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;wj \leq 3.3 \cdot 10^{-7}:\\ \;\;\;\;x + wj \cdot \left(wj \cdot \left(1 - wj\right) - x \cdot 2\right)\\ \mathbf{else}:\\ \;\;\;\;wj + \frac{wj - \frac{x}{e^{wj}}}{-1 - wj}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 97.5% accurate, 6.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x \cdot -4 + x \cdot 1.5\\ \mathbf{if}\;wj \leq 0.43:\\ \;\;\;\;x - wj \cdot \left(x \cdot 2 - wj \cdot \left(\left(1 + wj \cdot \left(-1 - \left(x \cdot -3 + \left(-2 \cdot t\_0 + x \cdot 0.6666666666666666\right)\right)\right)\right) - t\_0\right)\right)\\ \mathbf{else}:\\ \;\;\;\;wj + \frac{wj}{-1 - wj}\\ \end{array} \end{array} \]

FPCore

(FPCore (wj x)
 :precision binary64
 (let* ((t_0 (+ (* x -4.0) (* x 1.5))))
   (if (<= wj 0.43)
     (-
      x
      (*
       wj
       (-
        (* x 2.0)
        (*
         wj
         (-
          (+
           1.0
           (*
            wj
            (- -1.0 (+ (* x -3.0) (+ (* -2.0 t_0) (* x 0.6666666666666666))))))
          t_0)))))
     (+ wj (/ wj (- -1.0 wj))))))

C

double code(double wj, double x) {
	double t_0 = (x * -4.0) + (x * 1.5);
	double tmp;
	if (wj <= 0.43) {
		tmp = x - (wj * ((x * 2.0) - (wj * ((1.0 + (wj * (-1.0 - ((x * -3.0) + ((-2.0 * t_0) + (x * 0.6666666666666666)))))) - t_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) :: t_0
    real(8) :: tmp
    t_0 = (x * (-4.0d0)) + (x * 1.5d0)
    if (wj <= 0.43d0) then
        tmp = x - (wj * ((x * 2.0d0) - (wj * ((1.0d0 + (wj * ((-1.0d0) - ((x * (-3.0d0)) + (((-2.0d0) * t_0) + (x * 0.6666666666666666d0)))))) - t_0))))
    else
        tmp = wj + (wj / ((-1.0d0) - wj))
    end if
    code = tmp
end function

Java

public static double code(double wj, double x) {
	double t_0 = (x * -4.0) + (x * 1.5);
	double tmp;
	if (wj <= 0.43) {
		tmp = x - (wj * ((x * 2.0) - (wj * ((1.0 + (wj * (-1.0 - ((x * -3.0) + ((-2.0 * t_0) + (x * 0.6666666666666666)))))) - t_0))));
	} else {
		tmp = wj + (wj / (-1.0 - wj));
	}
	return tmp;
}

Python

def code(wj, x):
	t_0 = (x * -4.0) + (x * 1.5)
	tmp = 0
	if wj <= 0.43:
		tmp = x - (wj * ((x * 2.0) - (wj * ((1.0 + (wj * (-1.0 - ((x * -3.0) + ((-2.0 * t_0) + (x * 0.6666666666666666)))))) - t_0))))
	else:
		tmp = wj + (wj / (-1.0 - wj))
	return tmp

Julia

function code(wj, x)
	t_0 = Float64(Float64(x * -4.0) + Float64(x * 1.5))
	tmp = 0.0
	if (wj <= 0.43)
		tmp = Float64(x - Float64(wj * Float64(Float64(x * 2.0) - Float64(wj * Float64(Float64(1.0 + Float64(wj * Float64(-1.0 - Float64(Float64(x * -3.0) + Float64(Float64(-2.0 * t_0) + Float64(x * 0.6666666666666666)))))) - t_0)))));
	else
		tmp = Float64(wj + Float64(wj / Float64(-1.0 - wj)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(wj, x)
	t_0 = (x * -4.0) + (x * 1.5);
	tmp = 0.0;
	if (wj <= 0.43)
		tmp = x - (wj * ((x * 2.0) - (wj * ((1.0 + (wj * (-1.0 - ((x * -3.0) + ((-2.0 * t_0) + (x * 0.6666666666666666)))))) - t_0))));
	else
		tmp = wj + (wj / (-1.0 - wj));
	end
	tmp_2 = tmp;
end

Wolfram

code[wj_, x_] := Block[{t$95$0 = N[(N[(x * -4.0), $MachinePrecision] + N[(x * 1.5), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[wj, 0.43], N[(x - N[(wj * N[(N[(x * 2.0), $MachinePrecision] - N[(wj * N[(N[(1.0 + N[(wj * N[(-1.0 - N[(N[(x * -3.0), $MachinePrecision] + N[(N[(-2.0 * t$95$0), $MachinePrecision] + N[(x * 0.6666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(wj + N[(wj / N[(-1.0 - wj), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if wj < 0.429999999999999993

   1. Initial program 78.6%

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

      1. distribute-rgt1-in 79.4%

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

      2. associate-/l/ 79.4%

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

      3. div-sub 78.6%

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

      4. associate-/l* 78.6%

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

      5. *-inverses 79.4%

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

      6. *-rgt-identity 79.4%

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

   3. Simplified 79.4%

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

   5. Taylor expanded in wj around 0 98.0%

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

   if 0.429999999999999993 < wj

   1. Initial program 66.4%

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

      1. distribute-rgt1-in 66.7%

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

      2. associate-/l/ 66.7%

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

      3. div-sub 66.7%

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

      4. associate-/l* 66.7%

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

      5. *-inverses 100.0%

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

      6. *-rgt-identity 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in x around 0 83.5%

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

      1. +-commutative 83.5%

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

   7. Simplified 83.5%

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

4. Final simplification 97.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;wj \leq 0.43:\\ \;\;\;\;x - wj \cdot \left(x \cdot 2 - wj \cdot \left(\left(1 + wj \cdot \left(-1 - \left(x \cdot -3 + \left(-2 \cdot \left(x \cdot -4 + x \cdot 1.5\right) + x \cdot 0.6666666666666666\right)\right)\right)\right) - \left(x \cdot -4 + x \cdot 1.5\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;wj + \frac{wj}{-1 - wj}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 97.3% accurate, 13.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;wj \leq 112:\\ \;\;\;\;x - wj \cdot \left(x \cdot 2 - x \cdot \left(wj \cdot \left(2.5 + \frac{1 - wj}{x}\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;wj + \frac{wj}{-1 - wj}\\ \end{array} \end{array} \]

FPCore

(FPCore (wj x)
 :precision binary64
 (if (<= wj 112.0)
   (- x (* wj (- (* x 2.0) (* x (* wj (+ 2.5 (/ (- 1.0 wj) x)))))))
   (+ wj (/ wj (- -1.0 wj)))))

C

double code(double wj, double x) {
	double tmp;
	if (wj <= 112.0) {
		tmp = x - (wj * ((x * 2.0) - (x * (wj * (2.5 + ((1.0 - 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 <= 112.0d0) then
        tmp = x - (wj * ((x * 2.0d0) - (x * (wj * (2.5d0 + ((1.0d0 - 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 <= 112.0) {
		tmp = x - (wj * ((x * 2.0) - (x * (wj * (2.5 + ((1.0 - wj) / x))))));
	} else {
		tmp = wj + (wj / (-1.0 - wj));
	}
	return tmp;
}

Python

def code(wj, x):
	tmp = 0
	if wj <= 112.0:
		tmp = x - (wj * ((x * 2.0) - (x * (wj * (2.5 + ((1.0 - wj) / x))))))
	else:
		tmp = wj + (wj / (-1.0 - wj))
	return tmp

Julia

function code(wj, x)
	tmp = 0.0
	if (wj <= 112.0)
		tmp = Float64(x - Float64(wj * Float64(Float64(x * 2.0) - Float64(x * Float64(wj * Float64(2.5 + Float64(Float64(1.0 - 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 <= 112.0)
		tmp = x - (wj * ((x * 2.0) - (x * (wj * (2.5 + ((1.0 - wj) / x))))));
	else
		tmp = wj + (wj / (-1.0 - wj));
	end
	tmp_2 = tmp;
end

Wolfram

code[wj_, x_] := If[LessEqual[wj, 112.0], N[(x - N[(wj * N[(N[(x * 2.0), $MachinePrecision] - N[(x * N[(wj * N[(2.5 + N[(N[(1.0 - wj), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(wj + N[(wj / N[(-1.0 - wj), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if wj < 112

   1. Initial program 78.7%

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

      1. distribute-rgt1-in 79.5%

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

      2. associate-/l/ 79.5%

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

      3. div-sub 78.7%

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

      4. associate-/l* 78.7%

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

      5. *-inverses 79.5%

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

      6. *-rgt-identity 79.5%

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

   3. Simplified 79.5%

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

   5. Taylor expanded in wj around 0 97.6%

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

   6. Taylor expanded in x around inf 97.6%

      \[\leadsto x + wj \cdot \left(\color{blue}{x \cdot \left(wj \cdot \left(2.5 + -2.6666666666666665 \cdot wj\right) + \frac{wj \cdot \left(1 + -1 \cdot wj\right)}{x}\right)} - 2 \cdot x\right) \]
   7. Step-by-step derivation

      1. associate-/l* 97.5%

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

      2. distribute-lft-out 97.5%

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

      3. +-commutative 97.5%

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

      4. *-commutative 97.5%

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

      5. fma-define 97.5%

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

      6. mul-1-neg 97.5%

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

      7. unsub-neg 97.5%

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

   8. Simplified 97.5%

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

   9. Taylor expanded in wj around 0 97.5%

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

   if 112 < wj

   1. Initial program 60.0%

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

      1. distribute-rgt1-in 60.0%

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

      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 100.0%

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

      6. *-rgt-identity 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in x around 0 100.0%

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

      1. +-commutative 100.0%

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

   7. Simplified 100.0%

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

4. Final simplification 97.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;wj \leq 112:\\ \;\;\;\;x - wj \cdot \left(x \cdot 2 - x \cdot \left(wj \cdot \left(2.5 + \frac{1 - wj}{x}\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;wj + \frac{wj}{-1 - wj}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 97.3% accurate, 17.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;wj \leq 0.43:\\ \;\;\;\;x + wj \cdot \left(wj \cdot \left(1 - wj\right) - x \cdot 2\right)\\ \mathbf{else}:\\ \;\;\;\;wj + \frac{wj}{-1 - wj}\\ \end{array} \end{array} \]

FPCore

(FPCore (wj x)
 :precision binary64
 (if (<= wj 0.43)
   (+ x (* wj (- (* wj (- 1.0 wj)) (* x 2.0))))
   (+ wj (/ wj (- -1.0 wj)))))

C

double code(double wj, double x) {
	double tmp;
	if (wj <= 0.43) {
		tmp = x + (wj * ((wj * (1.0 - wj)) - (x * 2.0)));
	} else {
		tmp = wj + (wj / (-1.0 - wj));
	}
	return tmp;
}

Fortran

real(8) function code(wj, x)
    real(8), intent (in) :: wj
    real(8), intent (in) :: x
    real(8) :: tmp
    if (wj <= 0.43d0) then
        tmp = x + (wj * ((wj * (1.0d0 - wj)) - (x * 2.0d0)))
    else
        tmp = wj + (wj / ((-1.0d0) - wj))
    end if
    code = tmp
end function

Java

public static double code(double wj, double x) {
	double tmp;
	if (wj <= 0.43) {
		tmp = x + (wj * ((wj * (1.0 - wj)) - (x * 2.0)));
	} else {
		tmp = wj + (wj / (-1.0 - wj));
	}
	return tmp;
}

Python

def code(wj, x):
	tmp = 0
	if wj <= 0.43:
		tmp = x + (wj * ((wj * (1.0 - wj)) - (x * 2.0)))
	else:
		tmp = wj + (wj / (-1.0 - wj))
	return tmp

Julia

function code(wj, x)
	tmp = 0.0
	if (wj <= 0.43)
		tmp = Float64(x + Float64(wj * Float64(Float64(wj * Float64(1.0 - wj)) - Float64(x * 2.0))));
	else
		tmp = Float64(wj + Float64(wj / Float64(-1.0 - wj)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(wj, x)
	tmp = 0.0;
	if (wj <= 0.43)
		tmp = x + (wj * ((wj * (1.0 - wj)) - (x * 2.0)));
	else
		tmp = wj + (wj / (-1.0 - wj));
	end
	tmp_2 = tmp;
end

Wolfram

code[wj_, x_] := If[LessEqual[wj, 0.43], N[(x + N[(wj * N[(N[(wj * N[(1.0 - wj), $MachinePrecision]), $MachinePrecision] - N[(x * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(wj + N[(wj / N[(-1.0 - wj), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if wj < 0.429999999999999993

   1. Initial program 78.6%

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

      1. distribute-rgt1-in 79.4%

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

      2. associate-/l/ 79.4%

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

      3. div-sub 78.6%

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

      4. associate-/l* 78.6%

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

      5. *-inverses 79.4%

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

      6. *-rgt-identity 79.4%

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

   3. Simplified 79.4%

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

   5. Taylor expanded in wj around 0 98.0%

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

   6. Taylor expanded in x around 0 97.8%

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

      1. mul-1-neg 97.8%

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

      2. unsub-neg 97.8%

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

   8. Simplified 97.8%

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

   if 0.429999999999999993 < wj

   1. Initial program 66.4%

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

      1. distribute-rgt1-in 66.7%

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

      2. associate-/l/ 66.7%

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

      3. div-sub 66.7%

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

      4. associate-/l* 66.7%

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

      5. *-inverses 100.0%

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

      6. *-rgt-identity 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in x around 0 83.5%

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

      1. +-commutative 83.5%

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

   7. Simplified 83.5%

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

4. Final simplification 97.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;wj \leq 0.43:\\ \;\;\;\;x + wj \cdot \left(wj \cdot \left(1 - wj\right) - x \cdot 2\right)\\ \mathbf{else}:\\ \;\;\;\;wj + \frac{wj}{-1 - wj}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 86.2% accurate, 19.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;wj \leq 112:\\ \;\;\;\;x + wj \cdot \left(x \cdot \left(-2 + wj \cdot 2.5\right)\right)\\ \mathbf{else}:\\ \;\;\;\;wj + \frac{wj}{-1 - wj}\\ \end{array} \end{array} \]

FPCore

(FPCore (wj x)
 :precision binary64
 (if (<= wj 112.0)
   (+ x (* wj (* x (+ -2.0 (* wj 2.5)))))
   (+ wj (/ wj (- -1.0 wj)))))

C

double code(double wj, double x) {
	double tmp;
	if (wj <= 112.0) {
		tmp = x + (wj * (x * (-2.0 + (wj * 2.5))));
	} 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 <= 112.0d0) then
        tmp = x + (wj * (x * ((-2.0d0) + (wj * 2.5d0))))
    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 <= 112.0) {
		tmp = x + (wj * (x * (-2.0 + (wj * 2.5))));
	} else {
		tmp = wj + (wj / (-1.0 - wj));
	}
	return tmp;
}

Python

def code(wj, x):
	tmp = 0
	if wj <= 112.0:
		tmp = x + (wj * (x * (-2.0 + (wj * 2.5))))
	else:
		tmp = wj + (wj / (-1.0 - wj))
	return tmp

Julia

function code(wj, x)
	tmp = 0.0
	if (wj <= 112.0)
		tmp = Float64(x + Float64(wj * Float64(x * Float64(-2.0 + Float64(wj * 2.5)))));
	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 <= 112.0)
		tmp = x + (wj * (x * (-2.0 + (wj * 2.5))));
	else
		tmp = wj + (wj / (-1.0 - wj));
	end
	tmp_2 = tmp;
end

Wolfram

code[wj_, x_] := If[LessEqual[wj, 112.0], N[(x + N[(wj * N[(x * N[(-2.0 + N[(wj * 2.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(wj + N[(wj / N[(-1.0 - wj), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if wj < 112

   1. Initial program 78.7%

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

      1. distribute-rgt1-in 79.5%

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

      2. associate-/l/ 79.5%

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

      3. div-sub 78.7%

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

      4. associate-/l* 78.7%

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

      5. *-inverses 79.5%

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

      6. *-rgt-identity 79.5%

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

   3. Simplified 79.5%

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

   5. Taylor expanded in wj around 0 97.6%

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

   6. Taylor expanded in x around inf 85.4%

      \[\leadsto x + wj \cdot \left(\color{blue}{wj \cdot \left(x \cdot \left(2.5 + -2.6666666666666665 \cdot wj\right)\right)} - 2 \cdot x\right) \]
   7. Step-by-step derivation

      1. *-commutative 85.4%

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

   8. Simplified 85.4%

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

   9. Taylor expanded in wj around 0 85.3%

      \[\leadsto x + \color{blue}{wj \cdot \left(-2 \cdot x + 2.5 \cdot \left(wj \cdot x\right)\right)} \]
   10. Step-by-step derivation

       1. +-commutative 85.3%

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

       2. associate-*r* 85.3%

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

       3. distribute-rgt-out 85.3%

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

       4. *-commutative 85.3%

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

   11. Simplified 85.3%

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

   if 112 < wj

   1. Initial program 60.0%

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

      1. distribute-rgt1-in 60.0%

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

      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 100.0%

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

      6. *-rgt-identity 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in x around 0 100.0%

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

      1. +-commutative 100.0%

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

   7. Simplified 100.0%

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

4. Final simplification 85.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;wj \leq 112:\\ \;\;\;\;x + wj \cdot \left(x \cdot \left(-2 + wj \cdot 2.5\right)\right)\\ \mathbf{else}:\\ \;\;\;\;wj + \frac{wj}{-1 - wj}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 86.4% accurate, 26.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;wj \leq 4.2 \cdot 10^{-6}:\\ \;\;\;\;x + -2 \cdot \left(wj \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;wj + \frac{wj}{-1 - wj}\\ \end{array} \end{array} \]

FPCore

(FPCore (wj x)
 :precision binary64
 (if (<= wj 4.2e-6) (+ x (* -2.0 (* wj x))) (+ wj (/ wj (- -1.0 wj)))))

C

double code(double wj, double x) {
	double tmp;
	if (wj <= 4.2e-6) {
		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 <= 4.2d-6) 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 <= 4.2e-6) {
		tmp = x + (-2.0 * (wj * x));
	} else {
		tmp = wj + (wj / (-1.0 - wj));
	}
	return tmp;
}

Python

def code(wj, x):
	tmp = 0
	if wj <= 4.2e-6:
		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 <= 4.2e-6)
		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 <= 4.2e-6)
		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, 4.2e-6], 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 < 4.1999999999999996e-6

   1. Initial program 78.3%

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

      1. distribute-rgt1-in 79.1%

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

      2. associate-/l/ 79.1%

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

      3. div-sub 78.3%

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

      4. associate-/l* 78.3%

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

      5. *-inverses 79.1%

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

      6. *-rgt-identity 79.1%

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

   3. Simplified 79.1%

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

   5. Taylor expanded in wj around 0 87.0%

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

      1. *-commutative 87.0%

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

   7. Simplified 87.0%

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

   if 4.1999999999999996e-6 < wj

   1. Initial program 79.0%

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

      1. distribute-rgt1-in 79.1%

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

      2. associate-/l/ 79.3%

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

      3. div-sub 79.3%

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

      4. associate-/l* 79.3%

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

      5. *-inverses 97.4%

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

      6. *-rgt-identity 97.4%

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

   3. Simplified 97.4%

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

   5. Taylor expanded in x around 0 55.7%

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

      1. +-commutative 55.7%

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

   7. Simplified 55.7%

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

4. Final simplification 85.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;wj \leq 4.2 \cdot 10^{-6}:\\ \;\;\;\;x + -2 \cdot \left(wj \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;wj + \frac{wj}{-1 - wj}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 85.2% 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 78.3%

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

   1. distribute-rgt1-in 79.1%

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

   2. associate-/l/ 79.1%

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

   3. div-sub 78.3%

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

   4. associate-/l* 78.3%

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

   5. *-inverses 79.9%

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

   6. *-rgt-identity 79.9%

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

3. Simplified 79.9%

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

5. Taylor expanded in wj around 0 83.7%

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

   1. *-commutative 83.7%

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

7. Simplified 83.7%

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

8. Final simplification 83.7%

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

Alternative 9: 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 78.3%

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

   1. distribute-rgt1-in 79.1%

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

   2. associate-/l/ 79.1%

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

   3. div-sub 78.3%

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

   4. associate-/l* 78.3%

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

   5. *-inverses 79.9%

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

   6. *-rgt-identity 79.9%

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

3. Simplified 79.9%

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

5. Taylor expanded in wj around inf 4.2%

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

6. Final simplification 4.2%

   \[\leadsto wj \]
7. Add Preprocessing

Alternative 10: 84.7% 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 78.3%

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

   1. distribute-rgt1-in 79.1%

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

   2. associate-/l/ 79.1%

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

   3. div-sub 78.3%

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

   4. associate-/l* 78.3%

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

   5. *-inverses 79.9%

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

   6. *-rgt-identity 79.9%

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

3. Simplified 79.9%

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

5. Taylor expanded in wj around 0 83.4%

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

6. Final simplification 83.4%

   \[\leadsto x \]
7. Add Preprocessing

Developer target: 79.2% accurate, 1.5× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Reproduce

herbie shell --seed 2024078 
(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))))))