Jmat.Real.lambertw, newton loop step

Percentage Accurate: 78.0% → 97.8%

Time: 13.8s

Alternatives: 14

Speedup: 313.0×

• could not determine a ground truth

  1. wj = -4.727185388024249e+276
  2. x = 2.2779237559313842e-10

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: 78.0% 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: 97.8% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x \cdot -4 + x \cdot 1.5\\ \mathbf{if}\;wj \leq -3.7 \cdot 10^{-6}:\\ \;\;\;\;wj + \frac{\frac{x}{e^{wj}} - wj}{wj + 1}\\ \mathbf{else}:\\ \;\;\;\;x + \left(-2 \cdot \left(wj \cdot x\right) + \left({wj}^{3} \cdot \left(-1 - \left(x \cdot -3 + \left(-2 \cdot t_0 + x \cdot 0.6666666666666666\right)\right)\right) + {wj}^{2} \cdot \left(1 - t_0\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (wj x)
 :precision binary64
 (let* ((t_0 (+ (* x -4.0) (* x 1.5))))
   (if (<= wj -3.7e-6)
     (+ wj (/ (- (/ x (exp wj)) wj) (+ wj 1.0)))
     (+
      x
      (+
       (* -2.0 (* wj x))
       (+
        (*
         (pow wj 3.0)
         (- -1.0 (+ (* x -3.0) (+ (* -2.0 t_0) (* x 0.6666666666666666)))))
        (* (pow wj 2.0) (- 1.0 t_0))))))))

C

double code(double wj, double x) {
	double t_0 = (x * -4.0) + (x * 1.5);
	double tmp;
	if (wj <= -3.7e-6) {
		tmp = wj + (((x / exp(wj)) - wj) / (wj + 1.0));
	} else {
		tmp = x + ((-2.0 * (wj * x)) + ((pow(wj, 3.0) * (-1.0 - ((x * -3.0) + ((-2.0 * t_0) + (x * 0.6666666666666666))))) + (pow(wj, 2.0) * (1.0 - t_0))));
	}
	return tmp;
}

Fortran

real(8) function code(wj, x)
    real(8), intent (in) :: wj
    real(8), intent (in) :: x
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (x * (-4.0d0)) + (x * 1.5d0)
    if (wj <= (-3.7d-6)) then
        tmp = wj + (((x / exp(wj)) - wj) / (wj + 1.0d0))
    else
        tmp = x + (((-2.0d0) * (wj * x)) + (((wj ** 3.0d0) * ((-1.0d0) - ((x * (-3.0d0)) + (((-2.0d0) * t_0) + (x * 0.6666666666666666d0))))) + ((wj ** 2.0d0) * (1.0d0 - t_0))))
    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 <= -3.7e-6) {
		tmp = wj + (((x / Math.exp(wj)) - wj) / (wj + 1.0));
	} else {
		tmp = x + ((-2.0 * (wj * x)) + ((Math.pow(wj, 3.0) * (-1.0 - ((x * -3.0) + ((-2.0 * t_0) + (x * 0.6666666666666666))))) + (Math.pow(wj, 2.0) * (1.0 - t_0))));
	}
	return tmp;
}

Python

def code(wj, x):
	t_0 = (x * -4.0) + (x * 1.5)
	tmp = 0
	if wj <= -3.7e-6:
		tmp = wj + (((x / math.exp(wj)) - wj) / (wj + 1.0))
	else:
		tmp = x + ((-2.0 * (wj * x)) + ((math.pow(wj, 3.0) * (-1.0 - ((x * -3.0) + ((-2.0 * t_0) + (x * 0.6666666666666666))))) + (math.pow(wj, 2.0) * (1.0 - t_0))))
	return tmp

Julia

function code(wj, x)
	t_0 = Float64(Float64(x * -4.0) + Float64(x * 1.5))
	tmp = 0.0
	if (wj <= -3.7e-6)
		tmp = Float64(wj + Float64(Float64(Float64(x / exp(wj)) - wj) / Float64(wj + 1.0)));
	else
		tmp = Float64(x + Float64(Float64(-2.0 * Float64(wj * x)) + Float64(Float64((wj ^ 3.0) * Float64(-1.0 - Float64(Float64(x * -3.0) + Float64(Float64(-2.0 * t_0) + Float64(x * 0.6666666666666666))))) + Float64((wj ^ 2.0) * Float64(1.0 - t_0)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(wj, x)
	t_0 = (x * -4.0) + (x * 1.5);
	tmp = 0.0;
	if (wj <= -3.7e-6)
		tmp = wj + (((x / exp(wj)) - wj) / (wj + 1.0));
	else
		tmp = x + ((-2.0 * (wj * x)) + (((wj ^ 3.0) * (-1.0 - ((x * -3.0) + ((-2.0 * t_0) + (x * 0.6666666666666666))))) + ((wj ^ 2.0) * (1.0 - t_0))));
	end
	tmp_2 = tmp;
end

Wolfram

code[wj_, x_] := Block[{t$95$0 = N[(N[(x * -4.0), $MachinePrecision] + N[(x * 1.5), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[wj, -3.7e-6], N[(wj + N[(N[(N[(x / N[Exp[wj], $MachinePrecision]), $MachinePrecision] - wj), $MachinePrecision] / N[(wj + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x + N[(N[(-2.0 * N[(wj * x), $MachinePrecision]), $MachinePrecision] + N[(N[(N[Power[wj, 3.0], $MachinePrecision] * 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] + N[(N[Power[wj, 2.0], $MachinePrecision] * N[(1.0 - t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if wj < -3.7000000000000002e-6

   1. Initial program 33.1%

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

      1. distribute-rgt1-in 99.7%

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

      2. associate-/l/ 100.0%

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

      3. div-sub 33.3%

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

      4. associate-/l* 33.3%

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

      5. *-inverses 100.0%

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

      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

   if -3.7000000000000002e-6 < wj

   1. Initial program 73.5%

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

      1. distribute-rgt1-in 73.5%

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

      2. associate-/l/ 73.5%

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

      3. div-sub 73.5%

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

      4. associate-/l* 73.5%

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

      5. *-inverses 74.3%

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

      6. /-rgt-identity 74.3%

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

   3. Simplified 74.3%

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

   5. Taylor expanded in wj around 0 98.9%

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

4. Final simplification 98.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;wj \leq -3.7 \cdot 10^{-6}:\\ \;\;\;\;wj + \frac{\frac{x}{e^{wj}} - wj}{wj + 1}\\ \mathbf{else}:\\ \;\;\;\;x + \left(-2 \cdot \left(wj \cdot x\right) + \left({wj}^{3} \cdot \left(-1 - \left(x \cdot -3 + \left(-2 \cdot \left(x \cdot -4 + x \cdot 1.5\right) + x \cdot 0.6666666666666666\right)\right)\right) + {wj}^{2} \cdot \left(1 - \left(x \cdot -4 + x \cdot 1.5\right)\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 97.8% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;wj \leq -3.05 \cdot 10^{-6}:\\ \;\;\;\;wj + \frac{\frac{x}{e^{wj}} - wj}{wj + 1}\\ \mathbf{else}:\\ \;\;\;\;x + \left(-2 \cdot \left(wj \cdot x\right) + \left({wj}^{2} \cdot \left(1 - \left(x \cdot -4 + x \cdot 1.5\right)\right) - {wj}^{3}\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (wj x)
 :precision binary64
 (if (<= wj -3.05e-6)
   (+ wj (/ (- (/ x (exp wj)) wj) (+ wj 1.0)))
   (+
    x
    (+
     (* -2.0 (* wj x))
     (- (* (pow wj 2.0) (- 1.0 (+ (* x -4.0) (* x 1.5)))) (pow wj 3.0))))))

C

double code(double wj, double x) {
	double tmp;
	if (wj <= -3.05e-6) {
		tmp = wj + (((x / exp(wj)) - wj) / (wj + 1.0));
	} else {
		tmp = x + ((-2.0 * (wj * x)) + ((pow(wj, 2.0) * (1.0 - ((x * -4.0) + (x * 1.5)))) - pow(wj, 3.0)));
	}
	return tmp;
}

Fortran

real(8) function code(wj, x)
    real(8), intent (in) :: wj
    real(8), intent (in) :: x
    real(8) :: tmp
    if (wj <= (-3.05d-6)) then
        tmp = wj + (((x / exp(wj)) - wj) / (wj + 1.0d0))
    else
        tmp = x + (((-2.0d0) * (wj * x)) + (((wj ** 2.0d0) * (1.0d0 - ((x * (-4.0d0)) + (x * 1.5d0)))) - (wj ** 3.0d0)))
    end if
    code = tmp
end function

Java

public static double code(double wj, double x) {
	double tmp;
	if (wj <= -3.05e-6) {
		tmp = wj + (((x / Math.exp(wj)) - wj) / (wj + 1.0));
	} else {
		tmp = x + ((-2.0 * (wj * x)) + ((Math.pow(wj, 2.0) * (1.0 - ((x * -4.0) + (x * 1.5)))) - Math.pow(wj, 3.0)));
	}
	return tmp;
}

Python

def code(wj, x):
	tmp = 0
	if wj <= -3.05e-6:
		tmp = wj + (((x / math.exp(wj)) - wj) / (wj + 1.0))
	else:
		tmp = x + ((-2.0 * (wj * x)) + ((math.pow(wj, 2.0) * (1.0 - ((x * -4.0) + (x * 1.5)))) - math.pow(wj, 3.0)))
	return tmp

Julia

function code(wj, x)
	tmp = 0.0
	if (wj <= -3.05e-6)
		tmp = Float64(wj + Float64(Float64(Float64(x / exp(wj)) - wj) / Float64(wj + 1.0)));
	else
		tmp = Float64(x + Float64(Float64(-2.0 * Float64(wj * x)) + Float64(Float64((wj ^ 2.0) * Float64(1.0 - Float64(Float64(x * -4.0) + Float64(x * 1.5)))) - (wj ^ 3.0))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(wj, x)
	tmp = 0.0;
	if (wj <= -3.05e-6)
		tmp = wj + (((x / exp(wj)) - wj) / (wj + 1.0));
	else
		tmp = x + ((-2.0 * (wj * x)) + (((wj ^ 2.0) * (1.0 - ((x * -4.0) + (x * 1.5)))) - (wj ^ 3.0)));
	end
	tmp_2 = tmp;
end

Wolfram

code[wj_, x_] := If[LessEqual[wj, -3.05e-6], N[(wj + N[(N[(N[(x / N[Exp[wj], $MachinePrecision]), $MachinePrecision] - wj), $MachinePrecision] / N[(wj + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x + N[(N[(-2.0 * N[(wj * x), $MachinePrecision]), $MachinePrecision] + N[(N[(N[Power[wj, 2.0], $MachinePrecision] * N[(1.0 - N[(N[(x * -4.0), $MachinePrecision] + N[(x * 1.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[Power[wj, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if wj < -3.05000000000000002e-6

   1. Initial program 33.1%

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

      1. distribute-rgt1-in 99.7%

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

      2. associate-/l/ 100.0%

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

      3. div-sub 33.3%

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

      4. associate-/l* 33.3%

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

      5. *-inverses 100.0%

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

      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

   if -3.05000000000000002e-6 < wj

   1. Initial program 73.5%

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

      1. distribute-rgt1-in 73.5%

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

      2. associate-/l/ 73.5%

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

      3. div-sub 73.5%

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

      4. associate-/l* 73.5%

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

      5. *-inverses 74.3%

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

      6. /-rgt-identity 74.3%

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

   3. Simplified 74.3%

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

   5. Taylor expanded in wj around 0 98.9%

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

   6. Taylor expanded in x around 0 98.9%

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

4. Final simplification 98.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;wj \leq -3.05 \cdot 10^{-6}:\\ \;\;\;\;wj + \frac{\frac{x}{e^{wj}} - wj}{wj + 1}\\ \mathbf{else}:\\ \;\;\;\;x + \left(-2 \cdot \left(wj \cdot x\right) + \left({wj}^{2} \cdot \left(1 - \left(x \cdot -4 + x \cdot 1.5\right)\right) - {wj}^{3}\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 97.2% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;wj \leq -5.2 \cdot 10^{-9}:\\ \;\;\;\;wj + \frac{\frac{x}{e^{wj}} - wj}{wj + 1}\\ \mathbf{else}:\\ \;\;\;\;x + \left({wj}^{2} - {wj}^{3}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (wj x)
 :precision binary64
 (if (<= wj -5.2e-9)
   (+ wj (/ (- (/ x (exp wj)) wj) (+ wj 1.0)))
   (+ x (- (pow wj 2.0) (pow wj 3.0)))))

C

double code(double wj, double x) {
	double tmp;
	if (wj <= -5.2e-9) {
		tmp = wj + (((x / exp(wj)) - wj) / (wj + 1.0));
	} else {
		tmp = x + (pow(wj, 2.0) - pow(wj, 3.0));
	}
	return tmp;
}

Fortran

real(8) function code(wj, x)
    real(8), intent (in) :: wj
    real(8), intent (in) :: x
    real(8) :: tmp
    if (wj <= (-5.2d-9)) then
        tmp = wj + (((x / exp(wj)) - wj) / (wj + 1.0d0))
    else
        tmp = x + ((wj ** 2.0d0) - (wj ** 3.0d0))
    end if
    code = tmp
end function

Java

public static double code(double wj, double x) {
	double tmp;
	if (wj <= -5.2e-9) {
		tmp = wj + (((x / Math.exp(wj)) - wj) / (wj + 1.0));
	} else {
		tmp = x + (Math.pow(wj, 2.0) - Math.pow(wj, 3.0));
	}
	return tmp;
}

Python

def code(wj, x):
	tmp = 0
	if wj <= -5.2e-9:
		tmp = wj + (((x / math.exp(wj)) - wj) / (wj + 1.0))
	else:
		tmp = x + (math.pow(wj, 2.0) - math.pow(wj, 3.0))
	return tmp

Julia

function code(wj, x)
	tmp = 0.0
	if (wj <= -5.2e-9)
		tmp = Float64(wj + Float64(Float64(Float64(x / exp(wj)) - wj) / Float64(wj + 1.0)));
	else
		tmp = Float64(x + Float64((wj ^ 2.0) - (wj ^ 3.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(wj, x)
	tmp = 0.0;
	if (wj <= -5.2e-9)
		tmp = wj + (((x / exp(wj)) - wj) / (wj + 1.0));
	else
		tmp = x + ((wj ^ 2.0) - (wj ^ 3.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[wj_, x_] := If[LessEqual[wj, -5.2e-9], N[(wj + N[(N[(N[(x / N[Exp[wj], $MachinePrecision]), $MachinePrecision] - wj), $MachinePrecision] / N[(wj + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x + N[(N[Power[wj, 2.0], $MachinePrecision] - N[Power[wj, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if wj < -5.2000000000000002e-9

   1. Initial program 56.8%

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

      1. distribute-rgt1-in 96.8%

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

      2. associate-/l/ 97.0%

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

      3. div-sub 57.0%

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

      4. associate-/l* 57.0%

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

      5. *-inverses 97.0%

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

      6. /-rgt-identity 97.0%

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

   3. Simplified 97.0%

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

   if -5.2000000000000002e-9 < wj

   1. Initial program 73.2%

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

      1. distribute-rgt1-in 73.2%

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

      2. associate-/l/ 73.2%

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

      3. div-sub 73.2%

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

      4. associate-/l* 73.2%

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

      5. *-inverses 74.0%

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

      6. /-rgt-identity 74.0%

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

   3. Simplified 74.0%

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

   5. Taylor expanded in wj around 0 98.9%

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

   6. Taylor expanded in x around 0 98.9%

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

   7. Taylor expanded in x around 0 98.7%

      \[\leadsto x + \color{blue}{\left(-1 \cdot {wj}^{3} + {wj}^{2}\right)} \]
   8. Step-by-step derivation

      1. +-commutative 98.7%

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

      2. mul-1-neg 98.7%

         \[\leadsto x + \left({wj}^{2} + \color{blue}{\left(-{wj}^{3}\right)}\right) \]

      3. unsub-neg 98.7%

         \[\leadsto x + \color{blue}{\left({wj}^{2} - {wj}^{3}\right)} \]

   9. Simplified 98.7%

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

4. Final simplification 98.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;wj \leq -5.2 \cdot 10^{-9}:\\ \;\;\;\;wj + \frac{\frac{x}{e^{wj}} - wj}{wj + 1}\\ \mathbf{else}:\\ \;\;\;\;x + \left({wj}^{2} - {wj}^{3}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 86.0% accurate, 2.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -2 \cdot 10^{-274} \lor \neg \left(x \leq 2.5 \cdot 10^{-287}\right):\\ \;\;\;\;\frac{x}{e^{wj} \cdot \left(wj + 1\right)}\\ \mathbf{else}:\\ \;\;\;\;wj \cdot \left(wj + x \cdot -2\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (wj x)
 :precision binary64
 (if (or (<= x -2e-274) (not (<= x 2.5e-287)))
   (/ x (* (exp wj) (+ wj 1.0)))
   (* wj (+ wj (* x -2.0)))))

C

double code(double wj, double x) {
	double tmp;
	if ((x <= -2e-274) || !(x <= 2.5e-287)) {
		tmp = x / (exp(wj) * (wj + 1.0));
	} else {
		tmp = wj * (wj + (x * -2.0));
	}
	return tmp;
}

Fortran

real(8) function code(wj, x)
    real(8), intent (in) :: wj
    real(8), intent (in) :: x
    real(8) :: tmp
    if ((x <= (-2d-274)) .or. (.not. (x <= 2.5d-287))) then
        tmp = x / (exp(wj) * (wj + 1.0d0))
    else
        tmp = wj * (wj + (x * (-2.0d0)))
    end if
    code = tmp
end function

Java

public static double code(double wj, double x) {
	double tmp;
	if ((x <= -2e-274) || !(x <= 2.5e-287)) {
		tmp = x / (Math.exp(wj) * (wj + 1.0));
	} else {
		tmp = wj * (wj + (x * -2.0));
	}
	return tmp;
}

Python

def code(wj, x):
	tmp = 0
	if (x <= -2e-274) or not (x <= 2.5e-287):
		tmp = x / (math.exp(wj) * (wj + 1.0))
	else:
		tmp = wj * (wj + (x * -2.0))
	return tmp

Julia

function code(wj, x)
	tmp = 0.0
	if ((x <= -2e-274) || !(x <= 2.5e-287))
		tmp = Float64(x / Float64(exp(wj) * Float64(wj + 1.0)));
	else
		tmp = Float64(wj * Float64(wj + Float64(x * -2.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(wj, x)
	tmp = 0.0;
	if ((x <= -2e-274) || ~((x <= 2.5e-287)))
		tmp = x / (exp(wj) * (wj + 1.0));
	else
		tmp = wj * (wj + (x * -2.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[wj_, x_] := If[Or[LessEqual[x, -2e-274], N[Not[LessEqual[x, 2.5e-287]], $MachinePrecision]], N[(x / N[(N[Exp[wj], $MachinePrecision] * N[(wj + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(wj * N[(wj + N[(x * -2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -1.99999999999999993e-274 or 2.50000000000000013e-287 < x

   1. Initial program 75.2%

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

      1. distribute-rgt1-in 76.8%

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

      2. associate-/l/ 76.8%

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

      3. div-sub 75.2%

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

      4. associate-/l* 75.2%

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

      5. *-inverses 77.7%

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

      6. /-rgt-identity 77.7%

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

   3. Simplified 77.7%

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

   5. Taylor expanded in x around inf 88.9%

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

      1. +-commutative 88.9%

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

   7. Simplified 88.9%

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

   if -1.99999999999999993e-274 < x < 2.50000000000000013e-287

   1. Initial program 8.0%

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

      1. distribute-rgt1-in 8.0%

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

      2. associate-/l/ 8.0%

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

      3. div-sub 8.0%

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

      4. associate-/l* 8.0%

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

      5. *-inverses 8.0%

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

      6. /-rgt-identity 8.0%

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

   3. Simplified 8.0%

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

   5. Taylor expanded in wj around 0 99.5%

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

   6. Taylor expanded in x around 0 99.5%

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

   7. Taylor expanded in wj around inf 90.7%

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

      1. +-commutative 90.7%

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

      2. unpow2 90.7%

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

      3. associate-*r* 90.7%

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

      4. *-commutative 90.7%

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

      5. associate-*r* 90.7%

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

      6. distribute-rgt-out 90.7%

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

   9. Simplified 90.7%

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

4. Final simplification 89.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2 \cdot 10^{-274} \lor \neg \left(x \leq 2.5 \cdot 10^{-287}\right):\\ \;\;\;\;\frac{x}{e^{wj} \cdot \left(wj + 1\right)}\\ \mathbf{else}:\\ \;\;\;\;wj \cdot \left(wj + x \cdot -2\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 97.4% accurate, 2.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;wj \leq -6.2 \cdot 10^{-9}:\\ \;\;\;\;wj + \frac{\frac{x}{e^{wj}} - wj}{wj + 1}\\ \mathbf{else}:\\ \;\;\;\;x + \mathsf{fma}\left(wj, wj, x \cdot \left(wj \cdot -2\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (wj x)
 :precision binary64
 (if (<= wj -6.2e-9)
   (+ wj (/ (- (/ x (exp wj)) wj) (+ wj 1.0)))
   (+ x (fma wj wj (* x (* wj -2.0))))))

C

double code(double wj, double x) {
	double tmp;
	if (wj <= -6.2e-9) {
		tmp = wj + (((x / exp(wj)) - wj) / (wj + 1.0));
	} else {
		tmp = x + fma(wj, wj, (x * (wj * -2.0)));
	}
	return tmp;
}

Julia

function code(wj, x)
	tmp = 0.0
	if (wj <= -6.2e-9)
		tmp = Float64(wj + Float64(Float64(Float64(x / exp(wj)) - wj) / Float64(wj + 1.0)));
	else
		tmp = Float64(x + fma(wj, wj, Float64(x * Float64(wj * -2.0))));
	end
	return tmp
end

Wolfram

code[wj_, x_] := If[LessEqual[wj, -6.2e-9], N[(wj + N[(N[(N[(x / N[Exp[wj], $MachinePrecision]), $MachinePrecision] - wj), $MachinePrecision] / N[(wj + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x + N[(wj * wj + N[(x * N[(wj * -2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if wj < -6.2000000000000001e-9

   1. Initial program 52.0%

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

      1. distribute-rgt1-in 96.5%

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

      2. associate-/l/ 96.6%

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

      3. div-sub 52.2%

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

      4. associate-/l* 52.2%

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

      5. *-inverses 96.6%

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

      6. /-rgt-identity 96.6%

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

   3. Simplified 96.6%

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

   if -6.2000000000000001e-9 < wj

   1. Initial program 73.3%

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

      1. distribute-rgt1-in 73.3%

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

      2. associate-/l/ 73.3%

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

      3. div-sub 73.3%

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

      4. associate-/l* 73.3%

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

      5. *-inverses 74.1%

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

      6. /-rgt-identity 74.1%

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

   3. Simplified 74.1%

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

   5. Taylor expanded in wj around 0 98.7%

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

   6. Taylor expanded in x around 0 98.7%

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

      1. +-commutative 98.7%

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

      2. unpow2 98.7%

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

      3. fma-def 98.7%

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

      4. associate-*r* 98.7%

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

      5. *-commutative 98.7%

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

   8. Applied egg-rr 98.7%

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

4. Final simplification 98.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;wj \leq -6.2 \cdot 10^{-9}:\\ \;\;\;\;wj + \frac{\frac{x}{e^{wj}} - wj}{wj + 1}\\ \mathbf{else}:\\ \;\;\;\;x + \mathsf{fma}\left(wj, wj, x \cdot \left(wj \cdot -2\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 96.7% accurate, 2.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;wj \leq -0.0001:\\ \;\;\;\;\frac{x}{e^{wj} \cdot \left(wj + 1\right)}\\ \mathbf{else}:\\ \;\;\;\;x + \mathsf{fma}\left(wj, wj, x \cdot \left(wj \cdot -2\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (wj x)
 :precision binary64
 (if (<= wj -0.0001)
   (/ x (* (exp wj) (+ wj 1.0)))
   (+ x (fma wj wj (* x (* wj -2.0))))))

C

double code(double wj, double x) {
	double tmp;
	if (wj <= -0.0001) {
		tmp = x / (exp(wj) * (wj + 1.0));
	} else {
		tmp = x + fma(wj, wj, (x * (wj * -2.0)));
	}
	return tmp;
}

Julia

function code(wj, x)
	tmp = 0.0
	if (wj <= -0.0001)
		tmp = Float64(x / Float64(exp(wj) * Float64(wj + 1.0)));
	else
		tmp = Float64(x + fma(wj, wj, Float64(x * Float64(wj * -2.0))));
	end
	return tmp
end

Wolfram

code[wj_, x_] := If[LessEqual[wj, -0.0001], N[(x / N[(N[Exp[wj], $MachinePrecision] * N[(wj + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x + N[(wj * wj + N[(x * N[(wj * -2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if wj < -1.00000000000000005e-4

   1. Initial program 33.1%

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

      1. distribute-rgt1-in 99.7%

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

      2. associate-/l/ 100.0%

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

      3. div-sub 33.3%

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

      4. associate-/l* 33.3%

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

      5. *-inverses 100.0%

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

      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 inf 98.3%

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

      1. +-commutative 98.3%

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

   7. Simplified 98.3%

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

   if -1.00000000000000005e-4 < wj

   1. Initial program 73.5%

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

      1. distribute-rgt1-in 73.5%

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

      2. associate-/l/ 73.5%

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

      3. div-sub 73.5%

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

      4. associate-/l* 73.5%

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

      5. *-inverses 74.3%

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

      6. /-rgt-identity 74.3%

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

   3. Simplified 74.3%

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

   5. Taylor expanded in wj around 0 98.5%

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

   6. Taylor expanded in x around 0 98.4%

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

      1. +-commutative 98.4%

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

      2. unpow2 98.4%

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

      3. fma-def 98.4%

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

      4. associate-*r* 98.4%

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

      5. *-commutative 98.4%

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

   8. Applied egg-rr 98.4%

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

4. Final simplification 98.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;wj \leq -0.0001:\\ \;\;\;\;\frac{x}{e^{wj} \cdot \left(wj + 1\right)}\\ \mathbf{else}:\\ \;\;\;\;x + \mathsf{fma}\left(wj, wj, x \cdot \left(wj \cdot -2\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 83.9% accurate, 13.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -4.6 \cdot 10^{-243}:\\ \;\;\;\;\frac{x \cdot \left(1 - wj\right)}{wj + 1}\\ \mathbf{elif}\;x \leq 3.7 \cdot 10^{-289}:\\ \;\;\;\;wj \cdot \left(wj + x \cdot -2\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(\frac{1}{wj + 1} - \frac{wj}{wj + 1}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (wj x)
 :precision binary64
 (if (<= x -4.6e-243)
   (/ (* x (- 1.0 wj)) (+ wj 1.0))
   (if (<= x 3.7e-289)
     (* wj (+ wj (* x -2.0)))
     (* x (- (/ 1.0 (+ wj 1.0)) (/ wj (+ wj 1.0)))))))

C

double code(double wj, double x) {
	double tmp;
	if (x <= -4.6e-243) {
		tmp = (x * (1.0 - wj)) / (wj + 1.0);
	} else if (x <= 3.7e-289) {
		tmp = wj * (wj + (x * -2.0));
	} else {
		tmp = x * ((1.0 / (wj + 1.0)) - (wj / (wj + 1.0)));
	}
	return tmp;
}

Fortran

real(8) function code(wj, x)
    real(8), intent (in) :: wj
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= (-4.6d-243)) then
        tmp = (x * (1.0d0 - wj)) / (wj + 1.0d0)
    else if (x <= 3.7d-289) then
        tmp = wj * (wj + (x * (-2.0d0)))
    else
        tmp = x * ((1.0d0 / (wj + 1.0d0)) - (wj / (wj + 1.0d0)))
    end if
    code = tmp
end function

Java

public static double code(double wj, double x) {
	double tmp;
	if (x <= -4.6e-243) {
		tmp = (x * (1.0 - wj)) / (wj + 1.0);
	} else if (x <= 3.7e-289) {
		tmp = wj * (wj + (x * -2.0));
	} else {
		tmp = x * ((1.0 / (wj + 1.0)) - (wj / (wj + 1.0)));
	}
	return tmp;
}

Python

def code(wj, x):
	tmp = 0
	if x <= -4.6e-243:
		tmp = (x * (1.0 - wj)) / (wj + 1.0)
	elif x <= 3.7e-289:
		tmp = wj * (wj + (x * -2.0))
	else:
		tmp = x * ((1.0 / (wj + 1.0)) - (wj / (wj + 1.0)))
	return tmp

Julia

function code(wj, x)
	tmp = 0.0
	if (x <= -4.6e-243)
		tmp = Float64(Float64(x * Float64(1.0 - wj)) / Float64(wj + 1.0));
	elseif (x <= 3.7e-289)
		tmp = Float64(wj * Float64(wj + Float64(x * -2.0)));
	else
		tmp = Float64(x * Float64(Float64(1.0 / Float64(wj + 1.0)) - Float64(wj / Float64(wj + 1.0))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(wj, x)
	tmp = 0.0;
	if (x <= -4.6e-243)
		tmp = (x * (1.0 - wj)) / (wj + 1.0);
	elseif (x <= 3.7e-289)
		tmp = wj * (wj + (x * -2.0));
	else
		tmp = x * ((1.0 / (wj + 1.0)) - (wj / (wj + 1.0)));
	end
	tmp_2 = tmp;
end

Wolfram

code[wj_, x_] := If[LessEqual[x, -4.6e-243], N[(N[(x * N[(1.0 - wj), $MachinePrecision]), $MachinePrecision] / N[(wj + 1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 3.7e-289], N[(wj * N[(wj + N[(x * -2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x * N[(N[(1.0 / N[(wj + 1.0), $MachinePrecision]), $MachinePrecision] - N[(wj / N[(wj + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -4.6e-243

   1. Initial program 71.1%

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

      1. distribute-rgt1-in 73.6%

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

      2. associate-/l/ 73.6%

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

      3. div-sub 71.1%

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

      4. associate-/l* 71.1%

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

      5. *-inverses 75.2%

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

      6. /-rgt-identity 75.2%

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

   3. Simplified 75.2%

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

   5. Taylor expanded in wj around 0 71.1%

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

      1. mul-1-neg 71.1%

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

      2. unsub-neg 71.1%

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

      3. *-commutative 71.1%

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

   7. Simplified 71.1%

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

   8. Taylor expanded in x around inf 84.4%

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

      1. *-commutative 84.4%

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

      2. sub-div 84.4%

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

      3. associate-*l/ 84.4%

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

      4. +-commutative 84.4%

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

   10. Applied egg-rr 84.4%

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

   if -4.6e-243 < x < 3.69999999999999989e-289

   1. Initial program 10.4%

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

      1. distribute-rgt1-in 16.6%

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

      2. associate-/l/ 16.6%

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

      3. div-sub 10.4%

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

      4. associate-/l* 10.4%

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

      5. *-inverses 16.6%

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

      6. /-rgt-identity 16.6%

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

   3. Simplified 16.6%

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

   5. Taylor expanded in wj around 0 93.5%

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

   6. Taylor expanded in x around 0 93.5%

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

   7. Taylor expanded in wj around inf 76.0%

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

      1. +-commutative 76.0%

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

      2. unpow2 76.0%

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

      3. associate-*r* 76.0%

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

      4. *-commutative 76.0%

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

      5. associate-*r* 76.0%

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

      6. distribute-rgt-out 76.0%

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

   9. Simplified 76.0%

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

   if 3.69999999999999989e-289 < x

   1. Initial program 82.4%

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

      1. distribute-rgt1-in 82.4%

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

      2. associate-/l/ 82.4%

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

      3. div-sub 82.4%

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

      4. associate-/l* 82.4%

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

      5. *-inverses 82.4%

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

      6. /-rgt-identity 82.4%

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

   3. Simplified 82.4%

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

   5. Taylor expanded in wj around 0 81.6%

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

      1. mul-1-neg 81.6%

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

      2. unsub-neg 81.6%

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

      3. *-commutative 81.6%

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

   7. Simplified 81.6%

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

   8. Taylor expanded in x around inf 91.5%

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

4. Final simplification 87.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -4.6 \cdot 10^{-243}:\\ \;\;\;\;\frac{x \cdot \left(1 - wj\right)}{wj + 1}\\ \mathbf{elif}\;x \leq 3.7 \cdot 10^{-289}:\\ \;\;\;\;wj \cdot \left(wj + x \cdot -2\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(\frac{1}{wj + 1} - \frac{wj}{wj + 1}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 83.8% accurate, 16.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -5.4 \cdot 10^{-244} \lor \neg \left(x \leq 2.3 \cdot 10^{-289}\right):\\ \;\;\;\;\frac{x}{\frac{wj + 1}{1 - wj}}\\ \mathbf{else}:\\ \;\;\;\;wj \cdot \left(wj + x \cdot -2\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (wj x)
 :precision binary64
 (if (or (<= x -5.4e-244) (not (<= x 2.3e-289)))
   (/ x (/ (+ wj 1.0) (- 1.0 wj)))
   (* wj (+ wj (* x -2.0)))))

C

double code(double wj, double x) {
	double tmp;
	if ((x <= -5.4e-244) || !(x <= 2.3e-289)) {
		tmp = x / ((wj + 1.0) / (1.0 - wj));
	} else {
		tmp = wj * (wj + (x * -2.0));
	}
	return tmp;
}

Fortran

real(8) function code(wj, x)
    real(8), intent (in) :: wj
    real(8), intent (in) :: x
    real(8) :: tmp
    if ((x <= (-5.4d-244)) .or. (.not. (x <= 2.3d-289))) then
        tmp = x / ((wj + 1.0d0) / (1.0d0 - wj))
    else
        tmp = wj * (wj + (x * (-2.0d0)))
    end if
    code = tmp
end function

Java

public static double code(double wj, double x) {
	double tmp;
	if ((x <= -5.4e-244) || !(x <= 2.3e-289)) {
		tmp = x / ((wj + 1.0) / (1.0 - wj));
	} else {
		tmp = wj * (wj + (x * -2.0));
	}
	return tmp;
}

Python

def code(wj, x):
	tmp = 0
	if (x <= -5.4e-244) or not (x <= 2.3e-289):
		tmp = x / ((wj + 1.0) / (1.0 - wj))
	else:
		tmp = wj * (wj + (x * -2.0))
	return tmp

Julia

function code(wj, x)
	tmp = 0.0
	if ((x <= -5.4e-244) || !(x <= 2.3e-289))
		tmp = Float64(x / Float64(Float64(wj + 1.0) / Float64(1.0 - wj)));
	else
		tmp = Float64(wj * Float64(wj + Float64(x * -2.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(wj, x)
	tmp = 0.0;
	if ((x <= -5.4e-244) || ~((x <= 2.3e-289)))
		tmp = x / ((wj + 1.0) / (1.0 - wj));
	else
		tmp = wj * (wj + (x * -2.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[wj_, x_] := If[Or[LessEqual[x, -5.4e-244], N[Not[LessEqual[x, 2.3e-289]], $MachinePrecision]], N[(x / N[(N[(wj + 1.0), $MachinePrecision] / N[(1.0 - wj), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(wj * N[(wj + N[(x * -2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -5.3999999999999999e-244 or 2.3000000000000002e-289 < x

   1. Initial program 76.7%

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

      1. distribute-rgt1-in 78.0%

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

      2. associate-/l/ 78.0%

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

      3. div-sub 76.7%

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

      4. associate-/l* 76.7%

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

      5. *-inverses 78.8%

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

      6. /-rgt-identity 78.8%

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

   3. Simplified 78.8%

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

   5. Taylor expanded in wj around 0 76.3%

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

      1. mul-1-neg 76.3%

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

      2. unsub-neg 76.3%

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

      3. *-commutative 76.3%

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

   7. Simplified 76.3%

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

   8. Taylor expanded in x around inf 87.9%

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

      1. expm1-log1p-u 61.8%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(x \cdot \left(\frac{1}{1 + wj} - \frac{wj}{1 + wj}\right)\right)\right)} \]

      2. expm1-udef 26.8%

         \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(x \cdot \left(\frac{1}{1 + wj} - \frac{wj}{1 + wj}\right)\right)} - 1} \]

      3. sub-div 26.8%

         \[\leadsto e^{\mathsf{log1p}\left(x \cdot \color{blue}{\frac{1 - wj}{1 + wj}}\right)} - 1 \]

      4. associate-*r/ 26.8%

         \[\leadsto e^{\mathsf{log1p}\left(\color{blue}{\frac{x \cdot \left(1 - wj\right)}{1 + wj}}\right)} - 1 \]

      5. +-commutative 26.8%

         \[\leadsto e^{\mathsf{log1p}\left(\frac{x \cdot \left(1 - wj\right)}{\color{blue}{wj + 1}}\right)} - 1 \]

   10. Applied egg-rr 26.8%

       \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{x \cdot \left(1 - wj\right)}{wj + 1}\right)} - 1} \]
   11. Step-by-step derivation

       1. expm1-def 61.8%

          \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{x \cdot \left(1 - wj\right)}{wj + 1}\right)\right)} \]

       2. expm1-log1p 87.9%

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

       3. associate-/l* 87.9%

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

   12. Simplified 87.9%

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

   if -5.3999999999999999e-244 < x < 2.3000000000000002e-289

   1. Initial program 10.4%

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

      1. distribute-rgt1-in 16.6%

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

      2. associate-/l/ 16.6%

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

      3. div-sub 10.4%

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

      4. associate-/l* 10.4%

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

      5. *-inverses 16.6%

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

      6. /-rgt-identity 16.6%

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

   3. Simplified 16.6%

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

   5. Taylor expanded in wj around 0 93.5%

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

   6. Taylor expanded in x around 0 93.5%

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

   7. Taylor expanded in wj around inf 76.0%

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

      1. +-commutative 76.0%

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

      2. unpow2 76.0%

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

      3. associate-*r* 76.0%

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

      4. *-commutative 76.0%

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

      5. associate-*r* 76.0%

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

      6. distribute-rgt-out 76.0%

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

   9. Simplified 76.0%

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

4. Final simplification 87.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -5.4 \cdot 10^{-244} \lor \neg \left(x \leq 2.3 \cdot 10^{-289}\right):\\ \;\;\;\;\frac{x}{\frac{wj + 1}{1 - wj}}\\ \mathbf{else}:\\ \;\;\;\;wj \cdot \left(wj + x \cdot -2\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 83.9% accurate, 16.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.1 \cdot 10^{-244}:\\ \;\;\;\;\frac{x \cdot \left(1 - wj\right)}{wj + 1}\\ \mathbf{elif}\;x \leq 1.5 \cdot 10^{-285}:\\ \;\;\;\;wj \cdot \left(wj + x \cdot -2\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\frac{wj + 1}{1 - wj}}\\ \end{array} \end{array} \]

FPCore

(FPCore (wj x)
 :precision binary64
 (if (<= x -1.1e-244)
   (/ (* x (- 1.0 wj)) (+ wj 1.0))
   (if (<= x 1.5e-285)
     (* wj (+ wj (* x -2.0)))
     (/ x (/ (+ wj 1.0) (- 1.0 wj))))))

C

double code(double wj, double x) {
	double tmp;
	if (x <= -1.1e-244) {
		tmp = (x * (1.0 - wj)) / (wj + 1.0);
	} else if (x <= 1.5e-285) {
		tmp = wj * (wj + (x * -2.0));
	} else {
		tmp = x / ((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 (x <= (-1.1d-244)) then
        tmp = (x * (1.0d0 - wj)) / (wj + 1.0d0)
    else if (x <= 1.5d-285) then
        tmp = wj * (wj + (x * (-2.0d0)))
    else
        tmp = x / ((wj + 1.0d0) / (1.0d0 - wj))
    end if
    code = tmp
end function

Java

public static double code(double wj, double x) {
	double tmp;
	if (x <= -1.1e-244) {
		tmp = (x * (1.0 - wj)) / (wj + 1.0);
	} else if (x <= 1.5e-285) {
		tmp = wj * (wj + (x * -2.0));
	} else {
		tmp = x / ((wj + 1.0) / (1.0 - wj));
	}
	return tmp;
}

Python

def code(wj, x):
	tmp = 0
	if x <= -1.1e-244:
		tmp = (x * (1.0 - wj)) / (wj + 1.0)
	elif x <= 1.5e-285:
		tmp = wj * (wj + (x * -2.0))
	else:
		tmp = x / ((wj + 1.0) / (1.0 - wj))
	return tmp

Julia

function code(wj, x)
	tmp = 0.0
	if (x <= -1.1e-244)
		tmp = Float64(Float64(x * Float64(1.0 - wj)) / Float64(wj + 1.0));
	elseif (x <= 1.5e-285)
		tmp = Float64(wj * Float64(wj + Float64(x * -2.0)));
	else
		tmp = Float64(x / Float64(Float64(wj + 1.0) / Float64(1.0 - wj)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(wj, x)
	tmp = 0.0;
	if (x <= -1.1e-244)
		tmp = (x * (1.0 - wj)) / (wj + 1.0);
	elseif (x <= 1.5e-285)
		tmp = wj * (wj + (x * -2.0));
	else
		tmp = x / ((wj + 1.0) / (1.0 - wj));
	end
	tmp_2 = tmp;
end

Wolfram

code[wj_, x_] := If[LessEqual[x, -1.1e-244], N[(N[(x * N[(1.0 - wj), $MachinePrecision]), $MachinePrecision] / N[(wj + 1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 1.5e-285], N[(wj * N[(wj + N[(x * -2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x / N[(N[(wj + 1.0), $MachinePrecision] / N[(1.0 - wj), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -1.09999999999999992e-244

   1. Initial program 71.1%

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

      1. distribute-rgt1-in 73.6%

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

      2. associate-/l/ 73.6%

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

      3. div-sub 71.1%

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

      4. associate-/l* 71.1%

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

      5. *-inverses 75.2%

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

      6. /-rgt-identity 75.2%

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

   3. Simplified 75.2%

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

   5. Taylor expanded in wj around 0 71.1%

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

      1. mul-1-neg 71.1%

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

      2. unsub-neg 71.1%

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

      3. *-commutative 71.1%

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

   7. Simplified 71.1%

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

   8. Taylor expanded in x around inf 84.4%

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

      1. *-commutative 84.4%

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

      2. sub-div 84.4%

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

      3. associate-*l/ 84.4%

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

      4. +-commutative 84.4%

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

   10. Applied egg-rr 84.4%

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

   if -1.09999999999999992e-244 < x < 1.50000000000000002e-285

   1. Initial program 10.4%

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

      1. distribute-rgt1-in 16.6%

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

      2. associate-/l/ 16.6%

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

      3. div-sub 10.4%

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

      4. associate-/l* 10.4%

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

      5. *-inverses 16.6%

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

      6. /-rgt-identity 16.6%

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

   3. Simplified 16.6%

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

   5. Taylor expanded in wj around 0 93.5%

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

   6. Taylor expanded in x around 0 93.5%

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

   7. Taylor expanded in wj around inf 76.0%

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

      1. +-commutative 76.0%

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

      2. unpow2 76.0%

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

      3. associate-*r* 76.0%

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

      4. *-commutative 76.0%

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

      5. associate-*r* 76.0%

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

      6. distribute-rgt-out 76.0%

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

   9. Simplified 76.0%

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

   if 1.50000000000000002e-285 < x

   1. Initial program 82.4%

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

      1. distribute-rgt1-in 82.4%

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

      2. associate-/l/ 82.4%

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

      3. div-sub 82.4%

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

      4. associate-/l* 82.4%

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

      5. *-inverses 82.4%

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

      6. /-rgt-identity 82.4%

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

   3. Simplified 82.4%

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

   5. Taylor expanded in wj around 0 81.6%

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

      1. mul-1-neg 81.6%

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

      2. unsub-neg 81.6%

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

      3. *-commutative 81.6%

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

   7. Simplified 81.6%

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

   8. Taylor expanded in x around inf 91.5%

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

      1. expm1-log1p-u 86.8%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(x \cdot \left(\frac{1}{1 + wj} - \frac{wj}{1 + wj}\right)\right)\right)} \]

      2. expm1-udef 49.4%

         \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(x \cdot \left(\frac{1}{1 + wj} - \frac{wj}{1 + wj}\right)\right)} - 1} \]

      3. sub-div 49.4%

         \[\leadsto e^{\mathsf{log1p}\left(x \cdot \color{blue}{\frac{1 - wj}{1 + wj}}\right)} - 1 \]

      4. associate-*r/ 49.4%

         \[\leadsto e^{\mathsf{log1p}\left(\color{blue}{\frac{x \cdot \left(1 - wj\right)}{1 + wj}}\right)} - 1 \]

      5. +-commutative 49.4%

         \[\leadsto e^{\mathsf{log1p}\left(\frac{x \cdot \left(1 - wj\right)}{\color{blue}{wj + 1}}\right)} - 1 \]

   10. Applied egg-rr 49.4%

       \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{x \cdot \left(1 - wj\right)}{wj + 1}\right)} - 1} \]
   11. Step-by-step derivation

       1. expm1-def 86.8%

          \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{x \cdot \left(1 - wj\right)}{wj + 1}\right)\right)} \]

       2. expm1-log1p 91.5%

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

       3. associate-/l* 91.5%

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

   12. Simplified 91.5%

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

4. Final simplification 87.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.1 \cdot 10^{-244}:\\ \;\;\;\;\frac{x \cdot \left(1 - wj\right)}{wj + 1}\\ \mathbf{elif}\;x \leq 1.5 \cdot 10^{-285}:\\ \;\;\;\;wj \cdot \left(wj + x \cdot -2\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\frac{wj + 1}{1 - wj}}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 83.7% accurate, 18.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -8 \cdot 10^{-245} \lor \neg \left(x \leq 9.2 \cdot 10^{-290}\right):\\ \;\;\;\;x + -2 \cdot \left(wj \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;wj \cdot \left(wj + x \cdot -2\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (wj x)
 :precision binary64
 (if (or (<= x -8e-245) (not (<= x 9.2e-290)))
   (+ x (* -2.0 (* wj x)))
   (* wj (+ wj (* x -2.0)))))

C

double code(double wj, double x) {
	double tmp;
	if ((x <= -8e-245) || !(x <= 9.2e-290)) {
		tmp = x + (-2.0 * (wj * x));
	} else {
		tmp = wj * (wj + (x * -2.0));
	}
	return tmp;
}

Fortran

real(8) function code(wj, x)
    real(8), intent (in) :: wj
    real(8), intent (in) :: x
    real(8) :: tmp
    if ((x <= (-8d-245)) .or. (.not. (x <= 9.2d-290))) then
        tmp = x + ((-2.0d0) * (wj * x))
    else
        tmp = wj * (wj + (x * (-2.0d0)))
    end if
    code = tmp
end function

Java

public static double code(double wj, double x) {
	double tmp;
	if ((x <= -8e-245) || !(x <= 9.2e-290)) {
		tmp = x + (-2.0 * (wj * x));
	} else {
		tmp = wj * (wj + (x * -2.0));
	}
	return tmp;
}

Python

def code(wj, x):
	tmp = 0
	if (x <= -8e-245) or not (x <= 9.2e-290):
		tmp = x + (-2.0 * (wj * x))
	else:
		tmp = wj * (wj + (x * -2.0))
	return tmp

Julia

function code(wj, x)
	tmp = 0.0
	if ((x <= -8e-245) || !(x <= 9.2e-290))
		tmp = Float64(x + Float64(-2.0 * Float64(wj * x)));
	else
		tmp = Float64(wj * Float64(wj + Float64(x * -2.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(wj, x)
	tmp = 0.0;
	if ((x <= -8e-245) || ~((x <= 9.2e-290)))
		tmp = x + (-2.0 * (wj * x));
	else
		tmp = wj * (wj + (x * -2.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[wj_, x_] := If[Or[LessEqual[x, -8e-245], N[Not[LessEqual[x, 9.2e-290]], $MachinePrecision]], N[(x + N[(-2.0 * N[(wj * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(wj * N[(wj + N[(x * -2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -7.9999999999999994e-245 or 9.2000000000000003e-290 < x

   1. Initial program 76.7%

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

      1. distribute-rgt1-in 78.0%

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

      2. associate-/l/ 78.0%

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

      3. div-sub 76.7%

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

      4. associate-/l* 76.7%

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

      5. *-inverses 78.8%

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

      6. /-rgt-identity 78.8%

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

   3. Simplified 78.8%

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

   5. Taylor expanded in wj around 0 87.8%

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

      1. *-commutative 87.8%

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

   7. Simplified 87.8%

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

   if -7.9999999999999994e-245 < x < 9.2000000000000003e-290

   1. Initial program 10.4%

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

      1. distribute-rgt1-in 16.6%

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

      2. associate-/l/ 16.6%

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

      3. div-sub 10.4%

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

      4. associate-/l* 10.4%

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

      5. *-inverses 16.6%

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

      6. /-rgt-identity 16.6%

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

   3. Simplified 16.6%

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

   5. Taylor expanded in wj around 0 93.5%

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

   6. Taylor expanded in x around 0 93.5%

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

   7. Taylor expanded in wj around inf 76.0%

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

      1. +-commutative 76.0%

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

      2. unpow2 76.0%

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

      3. associate-*r* 76.0%

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

      4. *-commutative 76.0%

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

      5. associate-*r* 76.0%

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

      6. distribute-rgt-out 76.0%

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

   9. Simplified 76.0%

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

4. Final simplification 87.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -8 \cdot 10^{-245} \lor \neg \left(x \leq 9.2 \cdot 10^{-290}\right):\\ \;\;\;\;x + -2 \cdot \left(wj \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;wj \cdot \left(wj + x \cdot -2\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 83.9% accurate, 18.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -8.5 \cdot 10^{-244} \lor \neg \left(x \leq 9.8 \cdot 10^{-288}\right):\\ \;\;\;\;\frac{x}{1 + wj \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;wj \cdot \left(wj + x \cdot -2\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (wj x)
 :precision binary64
 (if (or (<= x -8.5e-244) (not (<= x 9.8e-288)))
   (/ x (+ 1.0 (* wj 2.0)))
   (* wj (+ wj (* x -2.0)))))

C

double code(double wj, double x) {
	double tmp;
	if ((x <= -8.5e-244) || !(x <= 9.8e-288)) {
		tmp = x / (1.0 + (wj * 2.0));
	} else {
		tmp = wj * (wj + (x * -2.0));
	}
	return tmp;
}

Fortran

real(8) function code(wj, x)
    real(8), intent (in) :: wj
    real(8), intent (in) :: x
    real(8) :: tmp
    if ((x <= (-8.5d-244)) .or. (.not. (x <= 9.8d-288))) then
        tmp = x / (1.0d0 + (wj * 2.0d0))
    else
        tmp = wj * (wj + (x * (-2.0d0)))
    end if
    code = tmp
end function

Java

public static double code(double wj, double x) {
	double tmp;
	if ((x <= -8.5e-244) || !(x <= 9.8e-288)) {
		tmp = x / (1.0 + (wj * 2.0));
	} else {
		tmp = wj * (wj + (x * -2.0));
	}
	return tmp;
}

Python

def code(wj, x):
	tmp = 0
	if (x <= -8.5e-244) or not (x <= 9.8e-288):
		tmp = x / (1.0 + (wj * 2.0))
	else:
		tmp = wj * (wj + (x * -2.0))
	return tmp

Julia

function code(wj, x)
	tmp = 0.0
	if ((x <= -8.5e-244) || !(x <= 9.8e-288))
		tmp = Float64(x / Float64(1.0 + Float64(wj * 2.0)));
	else
		tmp = Float64(wj * Float64(wj + Float64(x * -2.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(wj, x)
	tmp = 0.0;
	if ((x <= -8.5e-244) || ~((x <= 9.8e-288)))
		tmp = x / (1.0 + (wj * 2.0));
	else
		tmp = wj * (wj + (x * -2.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[wj_, x_] := If[Or[LessEqual[x, -8.5e-244], N[Not[LessEqual[x, 9.8e-288]], $MachinePrecision]], N[(x / N[(1.0 + N[(wj * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(wj * N[(wj + N[(x * -2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -8.4999999999999999e-244 or 9.80000000000000053e-288 < x

   1. Initial program 76.7%

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

      1. distribute-rgt1-in 78.0%

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

      2. associate-/l/ 78.0%

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

      3. div-sub 76.7%

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

      4. associate-/l* 76.7%

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

      5. *-inverses 78.8%

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

      6. /-rgt-identity 78.8%

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

   3. Simplified 78.8%

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

   5. Taylor expanded in x around inf 89.8%

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

      1. +-commutative 89.8%

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

   7. Simplified 89.8%

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

   8. Taylor expanded in wj around 0 87.9%

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

      1. *-commutative 87.9%

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

   10. Simplified 87.9%

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

   if -8.4999999999999999e-244 < x < 9.80000000000000053e-288

   1. Initial program 10.4%

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

      1. distribute-rgt1-in 16.6%

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

      2. associate-/l/ 16.6%

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

      3. div-sub 10.4%

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

      4. associate-/l* 10.4%

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

      5. *-inverses 16.6%

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

      6. /-rgt-identity 16.6%

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

   3. Simplified 16.6%

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

   5. Taylor expanded in wj around 0 93.5%

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

   6. Taylor expanded in x around 0 93.5%

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

   7. Taylor expanded in wj around inf 76.0%

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

      1. +-commutative 76.0%

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

      2. unpow2 76.0%

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

      3. associate-*r* 76.0%

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

      4. *-commutative 76.0%

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

      5. associate-*r* 76.0%

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

      6. distribute-rgt-out 76.0%

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

   9. Simplified 76.0%

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

4. Final simplification 87.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -8.5 \cdot 10^{-244} \lor \neg \left(x \leq 9.8 \cdot 10^{-288}\right):\\ \;\;\;\;\frac{x}{1 + wj \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;wj \cdot \left(wj + x \cdot -2\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 83.3% accurate, 18.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -7.5 \cdot 10^{-244}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq 3.5 \cdot 10^{-291}:\\ \;\;\;\;wj \cdot \left(wj + x \cdot -2\right)\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \end{array} \]

FPCore

(FPCore (wj x)
 :precision binary64
 (if (<= x -7.5e-244) x (if (<= x 3.5e-291) (* wj (+ wj (* x -2.0))) x)))

C

double code(double wj, double x) {
	double tmp;
	if (x <= -7.5e-244) {
		tmp = x;
	} else if (x <= 3.5e-291) {
		tmp = wj * (wj + (x * -2.0));
	} else {
		tmp = x;
	}
	return tmp;
}

Fortran

real(8) function code(wj, x)
    real(8), intent (in) :: wj
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= (-7.5d-244)) then
        tmp = x
    else if (x <= 3.5d-291) then
        tmp = wj * (wj + (x * (-2.0d0)))
    else
        tmp = x
    end if
    code = tmp
end function

Java

public static double code(double wj, double x) {
	double tmp;
	if (x <= -7.5e-244) {
		tmp = x;
	} else if (x <= 3.5e-291) {
		tmp = wj * (wj + (x * -2.0));
	} else {
		tmp = x;
	}
	return tmp;
}

Python

def code(wj, x):
	tmp = 0
	if x <= -7.5e-244:
		tmp = x
	elif x <= 3.5e-291:
		tmp = wj * (wj + (x * -2.0))
	else:
		tmp = x
	return tmp

Julia

function code(wj, x)
	tmp = 0.0
	if (x <= -7.5e-244)
		tmp = x;
	elseif (x <= 3.5e-291)
		tmp = Float64(wj * Float64(wj + Float64(x * -2.0)));
	else
		tmp = x;
	end
	return tmp
end

MATLAB

function tmp_2 = code(wj, x)
	tmp = 0.0;
	if (x <= -7.5e-244)
		tmp = x;
	elseif (x <= 3.5e-291)
		tmp = wj * (wj + (x * -2.0));
	else
		tmp = x;
	end
	tmp_2 = tmp;
end

Wolfram

code[wj_, x_] := If[LessEqual[x, -7.5e-244], x, If[LessEqual[x, 3.5e-291], N[(wj * N[(wj + N[(x * -2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], x]]

Derivation

1. Split input into 2 regimes

2. if x < -7.5000000000000003e-244 or 3.49999999999999996e-291 < x

   1. Initial program 76.7%

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

      1. distribute-rgt1-in 78.0%

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

      2. associate-/l/ 78.0%

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

      3. div-sub 76.7%

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

      4. associate-/l* 76.7%

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

      5. *-inverses 78.8%

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

      6. /-rgt-identity 78.8%

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

   3. Simplified 78.8%

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

   5. Taylor expanded in wj around 0 87.1%

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

   if -7.5000000000000003e-244 < x < 3.49999999999999996e-291

   1. Initial program 10.4%

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

      1. distribute-rgt1-in 16.6%

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

      2. associate-/l/ 16.6%

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

      3. div-sub 10.4%

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

      4. associate-/l* 10.4%

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

      5. *-inverses 16.6%

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

      6. /-rgt-identity 16.6%

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

   3. Simplified 16.6%

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

   5. Taylor expanded in wj around 0 93.5%

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

   6. Taylor expanded in x around 0 93.5%

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

   7. Taylor expanded in wj around inf 76.0%

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

      1. +-commutative 76.0%

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

      2. unpow2 76.0%

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

      3. associate-*r* 76.0%

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

      4. *-commutative 76.0%

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

      5. associate-*r* 76.0%

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

      6. distribute-rgt-out 76.0%

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

   9. Simplified 76.0%

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

4. Final simplification 86.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -7.5 \cdot 10^{-244}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq 3.5 \cdot 10^{-291}:\\ \;\;\;\;wj \cdot \left(wj + x \cdot -2\right)\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 4.5% 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 72.6%

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

   1. distribute-rgt1-in 74.1%

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

   2. associate-/l/ 74.2%

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

   3. div-sub 72.6%

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

   4. associate-/l* 72.6%

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

   5. *-inverses 74.9%

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

   6. /-rgt-identity 74.9%

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

3. Simplified 74.9%

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

5. Taylor expanded in wj around inf 4.1%

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

6. Final simplification 4.1%

   \[\leadsto wj \]
7. Add Preprocessing

Alternative 14: 83.5% 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 72.6%

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

   1. distribute-rgt1-in 74.1%

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

   2. associate-/l/ 74.2%

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

   3. div-sub 72.6%

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

   4. associate-/l* 72.6%

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

   5. *-inverses 74.9%

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

   6. /-rgt-identity 74.9%

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

3. Simplified 74.9%

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

5. Taylor expanded in wj around 0 83.1%

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

6. Final simplification 83.1%

   \[\leadsto x \]
7. Add Preprocessing

Developer target: 79.1% accurate, 1.5× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Reproduce

herbie shell --seed 2024021 
(FPCore (wj x)
  :name "Jmat.Real.lambertw, newton loop step"
  :precision binary64

  :herbie-target
  (- wj (- (/ wj (+ wj 1.0)) (/ x (+ (exp wj) (* wj (exp wj))))))

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