Jmat.Real.lambertw, newton loop step

Percentage Accurate: 77.7% → 99.8%

Time: 11.2s

Alternatives: 15

Speedup: 313.0×

• could not determine a ground truth

  1. wj = -4.0333533080116007e+301
  2. x = -6.825184699304142e-42

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 77.7% 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: 99.8% accurate, 2.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;wj \leq -4.3 \cdot 10^{-7}:\\ \;\;\;\;wj + \frac{x \cdot e^{-wj} - wj}{wj + 1}\\ \mathbf{elif}\;wj \leq 6.2 \cdot 10^{-7}:\\ \;\;\;\;x + \left(-2 \cdot \left(wj \cdot x\right) + \left(wj \cdot wj - {wj}^{3}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;wj - \frac{wj - \frac{x}{e^{wj}}}{wj + 1}\\ \end{array} \end{array} \]

FPCore

(FPCore (wj x)
 :precision binary64
 (if (<= wj -4.3e-7)
   (+ wj (/ (- (* x (exp (- wj))) wj) (+ wj 1.0)))
   (if (<= wj 6.2e-7)
     (+ x (+ (* -2.0 (* wj x)) (- (* wj wj) (pow wj 3.0))))
     (- wj (/ (- wj (/ x (exp wj))) (+ wj 1.0))))))

C

double code(double wj, double x) {
	double tmp;
	if (wj <= -4.3e-7) {
		tmp = wj + (((x * exp(-wj)) - wj) / (wj + 1.0));
	} else if (wj <= 6.2e-7) {
		tmp = x + ((-2.0 * (wj * x)) + ((wj * wj) - pow(wj, 3.0)));
	} else {
		tmp = wj - ((wj - (x / exp(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 (wj <= (-4.3d-7)) then
        tmp = wj + (((x * exp(-wj)) - wj) / (wj + 1.0d0))
    else if (wj <= 6.2d-7) then
        tmp = x + (((-2.0d0) * (wj * x)) + ((wj * wj) - (wj ** 3.0d0)))
    else
        tmp = wj - ((wj - (x / exp(wj))) / (wj + 1.0d0))
    end if
    code = tmp
end function

Java

public static double code(double wj, double x) {
	double tmp;
	if (wj <= -4.3e-7) {
		tmp = wj + (((x * Math.exp(-wj)) - wj) / (wj + 1.0));
	} else if (wj <= 6.2e-7) {
		tmp = x + ((-2.0 * (wj * x)) + ((wj * wj) - Math.pow(wj, 3.0)));
	} else {
		tmp = wj - ((wj - (x / Math.exp(wj))) / (wj + 1.0));
	}
	return tmp;
}

Python

def code(wj, x):
	tmp = 0
	if wj <= -4.3e-7:
		tmp = wj + (((x * math.exp(-wj)) - wj) / (wj + 1.0))
	elif wj <= 6.2e-7:
		tmp = x + ((-2.0 * (wj * x)) + ((wj * wj) - math.pow(wj, 3.0)))
	else:
		tmp = wj - ((wj - (x / math.exp(wj))) / (wj + 1.0))
	return tmp

Julia

function code(wj, x)
	tmp = 0.0
	if (wj <= -4.3e-7)
		tmp = Float64(wj + Float64(Float64(Float64(x * exp(Float64(-wj))) - wj) / Float64(wj + 1.0)));
	elseif (wj <= 6.2e-7)
		tmp = Float64(x + Float64(Float64(-2.0 * Float64(wj * x)) + Float64(Float64(wj * wj) - (wj ^ 3.0))));
	else
		tmp = Float64(wj - Float64(Float64(wj - Float64(x / exp(wj))) / Float64(wj + 1.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(wj, x)
	tmp = 0.0;
	if (wj <= -4.3e-7)
		tmp = wj + (((x * exp(-wj)) - wj) / (wj + 1.0));
	elseif (wj <= 6.2e-7)
		tmp = x + ((-2.0 * (wj * x)) + ((wj * wj) - (wj ^ 3.0)));
	else
		tmp = wj - ((wj - (x / exp(wj))) / (wj + 1.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[wj_, x_] := If[LessEqual[wj, -4.3e-7], N[(wj + N[(N[(N[(x * N[Exp[(-wj)], $MachinePrecision]), $MachinePrecision] - wj), $MachinePrecision] / N[(wj + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[wj, 6.2e-7], N[(x + N[(N[(-2.0 * N[(wj * x), $MachinePrecision]), $MachinePrecision] + N[(N[(wj * wj), $MachinePrecision] - N[Power[wj, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(wj - N[(N[(wj - N[(x / N[Exp[wj], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(wj + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if wj < -4.3000000000000001e-7

   1. Initial program 55.5%

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

      1. div-sub 55.6%

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

      2. associate-/l* 55.9%

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

      3. distribute-rgt1-in 55.9%

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

      4. associate-/l* 55.9%

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

      5. *-inverses 55.9%

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

      6. /-rgt-identity 55.9%

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

      7. distribute-rgt1-in 97.6%

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

      8. associate-/l/ 97.6%

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

      9. div-sub 97.6%

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

   3. Simplified 97.6%

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

      1. clear-num 97.7%

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

      2. associate-/r/ 97.7%

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

      3. rec-exp 97.8%

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

   5. Applied egg-rr 97.8%

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

   if -4.3000000000000001e-7 < wj < 6.1999999999999999e-7

   1. Initial program 78.8%

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

      1. div-sub 78.8%

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

      2. associate-/l* 78.8%

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

      3. distribute-rgt1-in 78.8%

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

      4. associate-/l* 78.8%

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

      5. *-inverses 78.8%

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

      6. /-rgt-identity 78.8%

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

      7. distribute-rgt1-in 78.8%

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

      8. associate-/l/ 78.8%

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

      9. div-sub 78.8%

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

   3. Simplified 78.8%

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

   4. Taylor expanded in wj around 0 99.9%

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

   5. Taylor expanded in x around 0 99.9%

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

      1. unpow2 99.9%

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

   7. Simplified 99.9%

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

   8. Taylor expanded in x around 0 99.9%

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

   9. Taylor expanded in wj around 0 99.9%

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

       1. neg-mul-1 99.9%

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

       2. +-commutative 99.9%

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

       3. unsub-neg 99.9%

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

       4. unpow2 99.9%

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

   11. Simplified 99.9%

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

   if 6.1999999999999999e-7 < wj

   1. Initial program 20.0%

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

      1. div-sub 20.0%

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

      2. associate-/l* 20.0%

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

      3. distribute-rgt1-in 20.0%

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

      4. associate-/l* 20.0%

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

      5. *-inverses 100.0%

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

      6. /-rgt-identity 100.0%

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

      7. distribute-rgt1-in 100.0%

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

      8. associate-/l/ 100.0%

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

      9. div-sub 100.0%

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

   3. Simplified 100.0%

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

4. Final simplification 99.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;wj \leq -4.3 \cdot 10^{-7}:\\ \;\;\;\;wj + \frac{x \cdot e^{-wj} - wj}{wj + 1}\\ \mathbf{elif}\;wj \leq 6.2 \cdot 10^{-7}:\\ \;\;\;\;x + \left(-2 \cdot \left(wj \cdot x\right) + \left(wj \cdot wj - {wj}^{3}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;wj - \frac{wj - \frac{x}{e^{wj}}}{wj + 1}\\ \end{array} \]

Alternative 2: 98.9% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := wj \cdot e^{wj}\\ t_1 := x \cdot -4 + x \cdot 1.5\\ \mathbf{if}\;wj + \frac{x - t_0}{e^{wj} + t_0} \leq 2 \cdot 10^{-13}:\\ \;\;\;\;x + \left(-2 \cdot \left(wj \cdot x\right) + \left({wj}^{3} \cdot \left(-1 - \left(x \cdot -3 + \left(-2 \cdot t_1 + x \cdot 0.6666666666666666\right)\right)\right) + {wj}^{2} \cdot \left(1 - t_1\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;wj + \frac{x \cdot e^{-wj} - wj}{wj + 1}\\ \end{array} \end{array} \]

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))) 2e-13)
     (+
      x
      (+
       (* -2.0 (* wj x))
       (+
        (*
         (pow wj 3.0)
         (- -1.0 (+ (* x -3.0) (+ (* -2.0 t_1) (* x 0.6666666666666666)))))
        (* (pow wj 2.0) (- 1.0 t_1)))))
     (+ wj (/ (- (* x (exp (- wj))) wj) (+ wj 1.0))))))

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))) <= 2e-13) {
		tmp = x + ((-2.0 * (wj * x)) + ((pow(wj, 3.0) * (-1.0 - ((x * -3.0) + ((-2.0 * t_1) + (x * 0.6666666666666666))))) + (pow(wj, 2.0) * (1.0 - t_1))));
	} else {
		tmp = wj + (((x * exp(-wj)) - 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) :: 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))) <= 2d-13) then
        tmp = x + (((-2.0d0) * (wj * x)) + (((wj ** 3.0d0) * ((-1.0d0) - ((x * (-3.0d0)) + (((-2.0d0) * t_1) + (x * 0.6666666666666666d0))))) + ((wj ** 2.0d0) * (1.0d0 - t_1))))
    else
        tmp = wj + (((x * exp(-wj)) - wj) / (wj + 1.0d0))
    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))) <= 2e-13) {
		tmp = x + ((-2.0 * (wj * x)) + ((Math.pow(wj, 3.0) * (-1.0 - ((x * -3.0) + ((-2.0 * t_1) + (x * 0.6666666666666666))))) + (Math.pow(wj, 2.0) * (1.0 - t_1))));
	} else {
		tmp = wj + (((x * Math.exp(-wj)) - wj) / (wj + 1.0));
	}
	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))) <= 2e-13:
		tmp = x + ((-2.0 * (wj * x)) + ((math.pow(wj, 3.0) * (-1.0 - ((x * -3.0) + ((-2.0 * t_1) + (x * 0.6666666666666666))))) + (math.pow(wj, 2.0) * (1.0 - t_1))))
	else:
		tmp = wj + (((x * math.exp(-wj)) - wj) / (wj + 1.0))
	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))) <= 2e-13)
		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_1) + Float64(x * 0.6666666666666666))))) + Float64((wj ^ 2.0) * Float64(1.0 - t_1)))));
	else
		tmp = Float64(wj + Float64(Float64(Float64(x * exp(Float64(-wj))) - wj) / Float64(wj + 1.0)));
	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))) <= 2e-13)
		tmp = x + ((-2.0 * (wj * x)) + (((wj ^ 3.0) * (-1.0 - ((x * -3.0) + ((-2.0 * t_1) + (x * 0.6666666666666666))))) + ((wj ^ 2.0) * (1.0 - t_1))));
	else
		tmp = wj + (((x * exp(-wj)) - wj) / (wj + 1.0));
	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], 2e-13], 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$1), $MachinePrecision] + N[(x * 0.6666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[Power[wj, 2.0], $MachinePrecision] * N[(1.0 - t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(wj + N[(N[(N[(x * N[Exp[(-wj)], $MachinePrecision]), $MachinePrecision] - wj), $MachinePrecision] / N[(wj + 1.0), $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))))) < 2.0000000000000001e-13

   1. Initial program 71.6%

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

      1. div-sub 71.6%

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

      2. associate-/l* 71.6%

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

      3. distribute-rgt1-in 71.6%

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

      4. associate-/l* 71.6%

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

      5. *-inverses 71.6%

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

      6. /-rgt-identity 71.6%

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

      7. distribute-rgt1-in 72.2%

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

      8. associate-/l/ 72.2%

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

      9. div-sub 72.2%

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

   3. Simplified 72.2%

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

   4. Taylor expanded in wj around 0 98.7%

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

   if 2.0000000000000001e-13 < (-.f64 wj (/.f64 (-.f64 (*.f64 wj (exp.f64 wj)) x) (+.f64 (exp.f64 wj) (*.f64 wj (exp.f64 wj)))))

   1. Initial program 88.7%

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

      1. div-sub 88.7%

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

      2. associate-/l* 88.8%

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

      3. distribute-rgt1-in 88.8%

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

      4. associate-/l* 88.8%

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

      5. *-inverses 94.2%

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

      6. /-rgt-identity 94.2%

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

      7. distribute-rgt1-in 99.6%

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

      8. associate-/l/ 99.5%

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

      9. div-sub 99.5%

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

   3. Simplified 99.5%

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

      1. clear-num 99.4%

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

      2. associate-/r/ 99.6%

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

      3. rec-exp 99.6%

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

   5. Applied egg-rr 99.6%

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

4. Final simplification 99.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;wj + \frac{x - wj \cdot e^{wj}}{e^{wj} + wj \cdot e^{wj}} \leq 2 \cdot 10^{-13}:\\ \;\;\;\;x + \left(-2 \cdot \left(wj \cdot x\right) + \left({wj}^{3} \cdot \left(-1 - \left(x \cdot -3 + \left(-2 \cdot \left(x \cdot -4 + x \cdot 1.5\right) + x \cdot 0.6666666666666666\right)\right)\right) + {wj}^{2} \cdot \left(1 - \left(x \cdot -4 + x \cdot 1.5\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;wj + \frac{x \cdot e^{-wj} - wj}{wj + 1}\\ \end{array} \]

Alternative 3: 99.5% accurate, 2.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;wj \leq -2.9 \cdot 10^{-6} \lor \neg \left(wj \leq 6.6 \cdot 10^{-9}\right):\\ \;\;\;\;wj + \frac{\frac{x}{e^{wj}} - wj}{wj + 1}\\ \mathbf{else}:\\ \;\;\;\;x + \left(-2 \cdot \left(wj \cdot x\right) + \left(wj \cdot wj\right) \cdot \left(1 + x \cdot 2.5\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (wj x)
 :precision binary64
 (if (or (<= wj -2.9e-6) (not (<= wj 6.6e-9)))
   (+ wj (/ (- (/ x (exp wj)) wj) (+ wj 1.0)))
   (+ x (+ (* -2.0 (* wj x)) (* (* wj wj) (+ 1.0 (* x 2.5)))))))

C

double code(double wj, double x) {
	double tmp;
	if ((wj <= -2.9e-6) || !(wj <= 6.6e-9)) {
		tmp = wj + (((x / exp(wj)) - wj) / (wj + 1.0));
	} else {
		tmp = x + ((-2.0 * (wj * x)) + ((wj * wj) * (1.0 + (x * 2.5))));
	}
	return tmp;
}

Fortran

real(8) function code(wj, x)
    real(8), intent (in) :: wj
    real(8), intent (in) :: x
    real(8) :: tmp
    if ((wj <= (-2.9d-6)) .or. (.not. (wj <= 6.6d-9))) then
        tmp = wj + (((x / exp(wj)) - wj) / (wj + 1.0d0))
    else
        tmp = x + (((-2.0d0) * (wj * x)) + ((wj * wj) * (1.0d0 + (x * 2.5d0))))
    end if
    code = tmp
end function

Java

public static double code(double wj, double x) {
	double tmp;
	if ((wj <= -2.9e-6) || !(wj <= 6.6e-9)) {
		tmp = wj + (((x / Math.exp(wj)) - wj) / (wj + 1.0));
	} else {
		tmp = x + ((-2.0 * (wj * x)) + ((wj * wj) * (1.0 + (x * 2.5))));
	}
	return tmp;
}

Python

def code(wj, x):
	tmp = 0
	if (wj <= -2.9e-6) or not (wj <= 6.6e-9):
		tmp = wj + (((x / math.exp(wj)) - wj) / (wj + 1.0))
	else:
		tmp = x + ((-2.0 * (wj * x)) + ((wj * wj) * (1.0 + (x * 2.5))))
	return tmp

Julia

function code(wj, x)
	tmp = 0.0
	if ((wj <= -2.9e-6) || !(wj <= 6.6e-9))
		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 * wj) * Float64(1.0 + Float64(x * 2.5)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(wj, x)
	tmp = 0.0;
	if ((wj <= -2.9e-6) || ~((wj <= 6.6e-9)))
		tmp = wj + (((x / exp(wj)) - wj) / (wj + 1.0));
	else
		tmp = x + ((-2.0 * (wj * x)) + ((wj * wj) * (1.0 + (x * 2.5))));
	end
	tmp_2 = tmp;
end

Wolfram

code[wj_, x_] := If[Or[LessEqual[wj, -2.9e-6], N[Not[LessEqual[wj, 6.6e-9]], $MachinePrecision]], 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[(wj * wj), $MachinePrecision] * N[(1.0 + N[(x * 2.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if wj < -2.9000000000000002e-6 or 6.60000000000000037e-9 < wj

   1. Initial program 46.5%

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

      1. div-sub 46.6%

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

      2. associate-/l* 46.8%

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

      3. distribute-rgt1-in 46.8%

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

      4. associate-/l* 46.8%

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

      5. *-inverses 69.0%

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

      6. /-rgt-identity 69.0%

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

      7. distribute-rgt1-in 96.8%

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

      8. associate-/l/ 96.8%

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

      9. div-sub 96.8%

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

   3. Simplified 96.8%

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

   if -2.9000000000000002e-6 < wj < 6.60000000000000037e-9

   1. Initial program 78.8%

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

      1. div-sub 78.8%

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

      2. associate-/l* 78.8%

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

      3. distribute-rgt1-in 78.8%

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

      4. associate-/l* 78.8%

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

      5. *-inverses 78.8%

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

      6. /-rgt-identity 78.8%

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

      7. distribute-rgt1-in 78.8%

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

      8. associate-/l/ 78.8%

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

      9. div-sub 78.8%

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

   3. Simplified 78.8%

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

      1. clear-num 78.7%

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

      2. associate-/r/ 78.8%

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

      3. rec-exp 78.8%

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

   5. Applied egg-rr 78.8%

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

   6. Taylor expanded in wj around 0 78.8%

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

      1. associate-*r* 78.8%

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

      2. neg-mul-1 78.8%

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

      3. associate-*r* 78.8%

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

      4. distribute-rgt-out 78.8%

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

      5. *-commutative 78.8%

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

      6. unpow2 78.8%

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

   8. Simplified 78.8%

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

   9. Taylor expanded in wj around 0 99.7%

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

       1. +-commutative 99.7%

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

       2. fma-def 99.7%

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

       3. unpow2 99.7%

          \[\leadsto x + \mathsf{fma}\left(\color{blue}{wj \cdot wj}, \left(1 + 0.5 \cdot x\right) - -2 \cdot x, wj \cdot \left(-1 \cdot x - x\right)\right) \]

       4. associate--l+ 99.7%

          \[\leadsto x + \mathsf{fma}\left(wj \cdot wj, \color{blue}{1 + \left(0.5 \cdot x - -2 \cdot x\right)}, wj \cdot \left(-1 \cdot x - x\right)\right) \]

       5. distribute-rgt-out-- 99.7%

          \[\leadsto x + \mathsf{fma}\left(wj \cdot wj, 1 + \color{blue}{x \cdot \left(0.5 - -2\right)}, wj \cdot \left(-1 \cdot x - x\right)\right) \]

       6. metadata-eval 99.7%

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

       7. sub-neg 99.7%

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

       8. mul-1-neg 99.7%

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

       9. distribute-rgt-out 99.7%

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

       10. metadata-eval 99.7%

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

       11. associate-*l* 99.7%

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

       12. *-commutative 99.7%

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

       13. associate-*l* 99.7%

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

   11. Simplified 99.7%

       \[\leadsto \color{blue}{x + \mathsf{fma}\left(wj \cdot wj, 1 + x \cdot 2.5, x \cdot \left(wj \cdot -2\right)\right)} \]
   12. Step-by-step derivation

       1. fma-udef 99.7%

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

       2. *-commutative 99.7%

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

       3. *-commutative 99.7%

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

       4. associate-*r* 99.7%

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

   13. Applied egg-rr 99.7%

       \[\leadsto x + \color{blue}{\left(\left(wj \cdot wj\right) \cdot \left(1 + x \cdot 2.5\right) + -2 \cdot \left(wj \cdot x\right)\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 99.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;wj \leq -2.9 \cdot 10^{-6} \lor \neg \left(wj \leq 6.6 \cdot 10^{-9}\right):\\ \;\;\;\;wj + \frac{\frac{x}{e^{wj}} - wj}{wj + 1}\\ \mathbf{else}:\\ \;\;\;\;x + \left(-2 \cdot \left(wj \cdot x\right) + \left(wj \cdot wj\right) \cdot \left(1 + x \cdot 2.5\right)\right)\\ \end{array} \]

Alternative 4: 99.5% accurate, 2.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;wj \leq -2.9 \cdot 10^{-6}:\\ \;\;\;\;wj + \frac{x \cdot e^{-wj} - wj}{wj + 1}\\ \mathbf{elif}\;wj \leq 8.5 \cdot 10^{-9}:\\ \;\;\;\;x + \left(-2 \cdot \left(wj \cdot x\right) + \left(wj \cdot wj\right) \cdot \left(1 + x \cdot 2.5\right)\right)\\ \mathbf{else}:\\ \;\;\;\;wj - \frac{wj - \frac{x}{e^{wj}}}{wj + 1}\\ \end{array} \end{array} \]

FPCore

(FPCore (wj x)
 :precision binary64
 (if (<= wj -2.9e-6)
   (+ wj (/ (- (* x (exp (- wj))) wj) (+ wj 1.0)))
   (if (<= wj 8.5e-9)
     (+ x (+ (* -2.0 (* wj x)) (* (* wj wj) (+ 1.0 (* x 2.5)))))
     (- wj (/ (- wj (/ x (exp wj))) (+ wj 1.0))))))

C

double code(double wj, double x) {
	double tmp;
	if (wj <= -2.9e-6) {
		tmp = wj + (((x * exp(-wj)) - wj) / (wj + 1.0));
	} else if (wj <= 8.5e-9) {
		tmp = x + ((-2.0 * (wj * x)) + ((wj * wj) * (1.0 + (x * 2.5))));
	} else {
		tmp = wj - ((wj - (x / exp(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 (wj <= (-2.9d-6)) then
        tmp = wj + (((x * exp(-wj)) - wj) / (wj + 1.0d0))
    else if (wj <= 8.5d-9) then
        tmp = x + (((-2.0d0) * (wj * x)) + ((wj * wj) * (1.0d0 + (x * 2.5d0))))
    else
        tmp = wj - ((wj - (x / exp(wj))) / (wj + 1.0d0))
    end if
    code = tmp
end function

Java

public static double code(double wj, double x) {
	double tmp;
	if (wj <= -2.9e-6) {
		tmp = wj + (((x * Math.exp(-wj)) - wj) / (wj + 1.0));
	} else if (wj <= 8.5e-9) {
		tmp = x + ((-2.0 * (wj * x)) + ((wj * wj) * (1.0 + (x * 2.5))));
	} else {
		tmp = wj - ((wj - (x / Math.exp(wj))) / (wj + 1.0));
	}
	return tmp;
}

Python

def code(wj, x):
	tmp = 0
	if wj <= -2.9e-6:
		tmp = wj + (((x * math.exp(-wj)) - wj) / (wj + 1.0))
	elif wj <= 8.5e-9:
		tmp = x + ((-2.0 * (wj * x)) + ((wj * wj) * (1.0 + (x * 2.5))))
	else:
		tmp = wj - ((wj - (x / math.exp(wj))) / (wj + 1.0))
	return tmp

Julia

function code(wj, x)
	tmp = 0.0
	if (wj <= -2.9e-6)
		tmp = Float64(wj + Float64(Float64(Float64(x * exp(Float64(-wj))) - wj) / Float64(wj + 1.0)));
	elseif (wj <= 8.5e-9)
		tmp = Float64(x + Float64(Float64(-2.0 * Float64(wj * x)) + Float64(Float64(wj * wj) * Float64(1.0 + Float64(x * 2.5)))));
	else
		tmp = Float64(wj - Float64(Float64(wj - Float64(x / exp(wj))) / Float64(wj + 1.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(wj, x)
	tmp = 0.0;
	if (wj <= -2.9e-6)
		tmp = wj + (((x * exp(-wj)) - wj) / (wj + 1.0));
	elseif (wj <= 8.5e-9)
		tmp = x + ((-2.0 * (wj * x)) + ((wj * wj) * (1.0 + (x * 2.5))));
	else
		tmp = wj - ((wj - (x / exp(wj))) / (wj + 1.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[wj_, x_] := If[LessEqual[wj, -2.9e-6], N[(wj + N[(N[(N[(x * N[Exp[(-wj)], $MachinePrecision]), $MachinePrecision] - wj), $MachinePrecision] / N[(wj + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[wj, 8.5e-9], N[(x + N[(N[(-2.0 * N[(wj * x), $MachinePrecision]), $MachinePrecision] + N[(N[(wj * wj), $MachinePrecision] * N[(1.0 + N[(x * 2.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(wj - N[(N[(wj - N[(x / N[Exp[wj], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(wj + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if wj < -2.9000000000000002e-6

   1. Initial program 51.6%

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

      1. div-sub 51.7%

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

      2. associate-/l* 52.0%

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

      3. distribute-rgt1-in 52.0%

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

      4. associate-/l* 52.0%

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

      5. *-inverses 52.0%

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

      6. /-rgt-identity 52.0%

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

      7. distribute-rgt1-in 97.5%

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

      8. associate-/l/ 97.5%

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

      9. div-sub 97.5%

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

   3. Simplified 97.5%

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

      1. clear-num 97.6%

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

      2. associate-/r/ 97.5%

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

      3. rec-exp 97.6%

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

   5. Applied egg-rr 97.6%

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

   if -2.9000000000000002e-6 < wj < 8.5e-9

   1. Initial program 78.8%

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

      1. div-sub 78.8%

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

      2. associate-/l* 78.8%

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

      3. distribute-rgt1-in 78.8%

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

      4. associate-/l* 78.8%

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

      5. *-inverses 78.8%

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

      6. /-rgt-identity 78.8%

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

      7. distribute-rgt1-in 78.8%

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

      8. associate-/l/ 78.8%

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

      9. div-sub 78.8%

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

   3. Simplified 78.8%

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

      1. clear-num 78.7%

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

      2. associate-/r/ 78.8%

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

      3. rec-exp 78.8%

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

   5. Applied egg-rr 78.8%

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

   6. Taylor expanded in wj around 0 78.8%

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

      1. associate-*r* 78.8%

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

      2. neg-mul-1 78.8%

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

      3. associate-*r* 78.8%

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

      4. distribute-rgt-out 78.8%

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

      5. *-commutative 78.8%

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

      6. unpow2 78.8%

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

   8. Simplified 78.8%

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

   9. Taylor expanded in wj around 0 99.7%

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

       1. +-commutative 99.7%

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

       2. fma-def 99.7%

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

       3. unpow2 99.7%

          \[\leadsto x + \mathsf{fma}\left(\color{blue}{wj \cdot wj}, \left(1 + 0.5 \cdot x\right) - -2 \cdot x, wj \cdot \left(-1 \cdot x - x\right)\right) \]

       4. associate--l+ 99.7%

          \[\leadsto x + \mathsf{fma}\left(wj \cdot wj, \color{blue}{1 + \left(0.5 \cdot x - -2 \cdot x\right)}, wj \cdot \left(-1 \cdot x - x\right)\right) \]

       5. distribute-rgt-out-- 99.7%

          \[\leadsto x + \mathsf{fma}\left(wj \cdot wj, 1 + \color{blue}{x \cdot \left(0.5 - -2\right)}, wj \cdot \left(-1 \cdot x - x\right)\right) \]

       6. metadata-eval 99.7%

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

       7. sub-neg 99.7%

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

       8. mul-1-neg 99.7%

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

       9. distribute-rgt-out 99.7%

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

       10. metadata-eval 99.7%

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

       11. associate-*l* 99.7%

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

       12. *-commutative 99.7%

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

       13. associate-*l* 99.7%

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

   11. Simplified 99.7%

       \[\leadsto \color{blue}{x + \mathsf{fma}\left(wj \cdot wj, 1 + x \cdot 2.5, x \cdot \left(wj \cdot -2\right)\right)} \]
   12. Step-by-step derivation

       1. fma-udef 99.7%

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

       2. *-commutative 99.7%

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

       3. *-commutative 99.7%

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

       4. associate-*r* 99.7%

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

   13. Applied egg-rr 99.7%

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

   if 8.5e-9 < wj

   1. Initial program 38.6%

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

      1. div-sub 38.6%

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

      2. associate-/l* 38.6%

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

      3. distribute-rgt1-in 38.6%

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

      4. associate-/l* 38.6%

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

      5. *-inverses 95.7%

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

      6. /-rgt-identity 95.7%

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

      7. distribute-rgt1-in 95.7%

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

      8. associate-/l/ 95.7%

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

      9. div-sub 95.7%

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

   3. Simplified 95.7%

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

4. Final simplification 99.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;wj \leq -2.9 \cdot 10^{-6}:\\ \;\;\;\;wj + \frac{x \cdot e^{-wj} - wj}{wj + 1}\\ \mathbf{elif}\;wj \leq 8.5 \cdot 10^{-9}:\\ \;\;\;\;x + \left(-2 \cdot \left(wj \cdot x\right) + \left(wj \cdot wj\right) \cdot \left(1 + x \cdot 2.5\right)\right)\\ \mathbf{else}:\\ \;\;\;\;wj - \frac{wj - \frac{x}{e^{wj}}}{wj + 1}\\ \end{array} \]

Alternative 5: 97.3% accurate, 2.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;wj \leq -0.00028:\\ \;\;\;\;\frac{x \cdot e^{-wj}}{wj + 1}\\ \mathbf{else}:\\ \;\;\;\;x + \left(-2 \cdot \left(wj \cdot x\right) + \left(wj \cdot wj\right) \cdot \left(1 + x \cdot 2.5\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (wj x)
 :precision binary64
 (if (<= wj -0.00028)
   (/ (* x (exp (- wj))) (+ wj 1.0))
   (+ x (+ (* -2.0 (* wj x)) (* (* wj wj) (+ 1.0 (* x 2.5)))))))

C

double code(double wj, double x) {
	double tmp;
	if (wj <= -0.00028) {
		tmp = (x * exp(-wj)) / (wj + 1.0);
	} else {
		tmp = x + ((-2.0 * (wj * x)) + ((wj * wj) * (1.0 + (x * 2.5))));
	}
	return tmp;
}

Fortran

real(8) function code(wj, x)
    real(8), intent (in) :: wj
    real(8), intent (in) :: x
    real(8) :: tmp
    if (wj <= (-0.00028d0)) then
        tmp = (x * exp(-wj)) / (wj + 1.0d0)
    else
        tmp = x + (((-2.0d0) * (wj * x)) + ((wj * wj) * (1.0d0 + (x * 2.5d0))))
    end if
    code = tmp
end function

Java

public static double code(double wj, double x) {
	double tmp;
	if (wj <= -0.00028) {
		tmp = (x * Math.exp(-wj)) / (wj + 1.0);
	} else {
		tmp = x + ((-2.0 * (wj * x)) + ((wj * wj) * (1.0 + (x * 2.5))));
	}
	return tmp;
}

Python

def code(wj, x):
	tmp = 0
	if wj <= -0.00028:
		tmp = (x * math.exp(-wj)) / (wj + 1.0)
	else:
		tmp = x + ((-2.0 * (wj * x)) + ((wj * wj) * (1.0 + (x * 2.5))))
	return tmp

Julia

function code(wj, x)
	tmp = 0.0
	if (wj <= -0.00028)
		tmp = Float64(Float64(x * exp(Float64(-wj))) / Float64(wj + 1.0));
	else
		tmp = Float64(x + Float64(Float64(-2.0 * Float64(wj * x)) + Float64(Float64(wj * wj) * Float64(1.0 + Float64(x * 2.5)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(wj, x)
	tmp = 0.0;
	if (wj <= -0.00028)
		tmp = (x * exp(-wj)) / (wj + 1.0);
	else
		tmp = x + ((-2.0 * (wj * x)) + ((wj * wj) * (1.0 + (x * 2.5))));
	end
	tmp_2 = tmp;
end

Wolfram

code[wj_, x_] := If[LessEqual[wj, -0.00028], N[(N[(x * N[Exp[(-wj)], $MachinePrecision]), $MachinePrecision] / N[(wj + 1.0), $MachinePrecision]), $MachinePrecision], N[(x + N[(N[(-2.0 * N[(wj * x), $MachinePrecision]), $MachinePrecision] + N[(N[(wj * wj), $MachinePrecision] * N[(1.0 + N[(x * 2.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if wj < -2.7999999999999998e-4

   1. Initial program 49.4%

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

      1. div-sub 49.5%

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

      2. associate-/l* 49.5%

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

      3. distribute-rgt1-in 49.5%

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

      4. associate-/l* 49.5%

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

      5. *-inverses 49.5%

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

      6. /-rgt-identity 49.5%

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

      7. distribute-rgt1-in 99.5%

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

      8. associate-/l/ 99.5%

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

      9. div-sub 99.5%

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

   3. Simplified 99.5%

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

      1. clear-num 99.7%

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

      2. associate-/r/ 99.5%

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

      3. rec-exp 99.7%

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

   5. Applied egg-rr 99.7%

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

   6. Taylor expanded in x around inf 87.7%

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

   if -2.7999999999999998e-4 < wj

   1. Initial program 77.7%

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

      1. div-sub 77.7%

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

      2. associate-/l* 77.7%

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

      3. distribute-rgt1-in 77.7%

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

      4. associate-/l* 77.7%

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

      5. *-inverses 79.3%

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

      6. /-rgt-identity 79.3%

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

      7. distribute-rgt1-in 79.3%

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

      8. associate-/l/ 79.3%

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

      9. div-sub 79.3%

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

   3. Simplified 79.3%

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

      1. clear-num 79.1%

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

      2. associate-/r/ 79.3%

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

      3. rec-exp 79.3%

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

   5. Applied egg-rr 79.3%

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

   6. Taylor expanded in wj around 0 79.0%

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

      1. associate-*r* 79.0%

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

      2. neg-mul-1 79.0%

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

      3. associate-*r* 79.0%

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

      4. distribute-rgt-out 79.0%

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

      5. *-commutative 79.0%

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

      6. unpow2 79.0%

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

   8. Simplified 79.0%

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

   9. Taylor expanded in wj around 0 97.5%

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

       1. +-commutative 97.5%

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

       2. fma-def 97.5%

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

       3. unpow2 97.5%

          \[\leadsto x + \mathsf{fma}\left(\color{blue}{wj \cdot wj}, \left(1 + 0.5 \cdot x\right) - -2 \cdot x, wj \cdot \left(-1 \cdot x - x\right)\right) \]

       4. associate--l+ 97.5%

          \[\leadsto x + \mathsf{fma}\left(wj \cdot wj, \color{blue}{1 + \left(0.5 \cdot x - -2 \cdot x\right)}, wj \cdot \left(-1 \cdot x - x\right)\right) \]

       5. distribute-rgt-out-- 97.5%

          \[\leadsto x + \mathsf{fma}\left(wj \cdot wj, 1 + \color{blue}{x \cdot \left(0.5 - -2\right)}, wj \cdot \left(-1 \cdot x - x\right)\right) \]

       6. metadata-eval 97.5%

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

       7. sub-neg 97.5%

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

       8. mul-1-neg 97.5%

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

       9. distribute-rgt-out 97.5%

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

       10. metadata-eval 97.5%

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

       11. associate-*l* 97.5%

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

       12. *-commutative 97.5%

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

       13. associate-*l* 97.5%

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

   11. Simplified 97.5%

       \[\leadsto \color{blue}{x + \mathsf{fma}\left(wj \cdot wj, 1 + x \cdot 2.5, x \cdot \left(wj \cdot -2\right)\right)} \]
   12. Step-by-step derivation

       1. fma-udef 97.5%

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

       2. *-commutative 97.5%

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

       3. *-commutative 97.5%

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

       4. associate-*r* 97.5%

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

   13. Applied egg-rr 97.5%

       \[\leadsto x + \color{blue}{\left(\left(wj \cdot wj\right) \cdot \left(1 + x \cdot 2.5\right) + -2 \cdot \left(wj \cdot x\right)\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 97.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;wj \leq -0.00028:\\ \;\;\;\;\frac{x \cdot e^{-wj}}{wj + 1}\\ \mathbf{else}:\\ \;\;\;\;x + \left(-2 \cdot \left(wj \cdot x\right) + \left(wj \cdot wj\right) \cdot \left(1 + x \cdot 2.5\right)\right)\\ \end{array} \]

Alternative 6: 97.3% accurate, 2.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;wj \leq -0.00044:\\ \;\;\;\;\frac{x}{e^{wj} \cdot \left(wj + 1\right)}\\ \mathbf{else}:\\ \;\;\;\;x + \left(-2 \cdot \left(wj \cdot x\right) + \left(wj \cdot wj\right) \cdot \left(1 + x \cdot 2.5\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (wj x)
 :precision binary64
 (if (<= wj -0.00044)
   (/ x (* (exp wj) (+ wj 1.0)))
   (+ x (+ (* -2.0 (* wj x)) (* (* wj wj) (+ 1.0 (* x 2.5)))))))

C

double code(double wj, double x) {
	double tmp;
	if (wj <= -0.00044) {
		tmp = x / (exp(wj) * (wj + 1.0));
	} else {
		tmp = x + ((-2.0 * (wj * x)) + ((wj * wj) * (1.0 + (x * 2.5))));
	}
	return tmp;
}

Fortran

real(8) function code(wj, x)
    real(8), intent (in) :: wj
    real(8), intent (in) :: x
    real(8) :: tmp
    if (wj <= (-0.00044d0)) then
        tmp = x / (exp(wj) * (wj + 1.0d0))
    else
        tmp = x + (((-2.0d0) * (wj * x)) + ((wj * wj) * (1.0d0 + (x * 2.5d0))))
    end if
    code = tmp
end function

Java

public static double code(double wj, double x) {
	double tmp;
	if (wj <= -0.00044) {
		tmp = x / (Math.exp(wj) * (wj + 1.0));
	} else {
		tmp = x + ((-2.0 * (wj * x)) + ((wj * wj) * (1.0 + (x * 2.5))));
	}
	return tmp;
}

Python

def code(wj, x):
	tmp = 0
	if wj <= -0.00044:
		tmp = x / (math.exp(wj) * (wj + 1.0))
	else:
		tmp = x + ((-2.0 * (wj * x)) + ((wj * wj) * (1.0 + (x * 2.5))))
	return tmp

Julia

function code(wj, x)
	tmp = 0.0
	if (wj <= -0.00044)
		tmp = Float64(x / Float64(exp(wj) * Float64(wj + 1.0)));
	else
		tmp = Float64(x + Float64(Float64(-2.0 * Float64(wj * x)) + Float64(Float64(wj * wj) * Float64(1.0 + Float64(x * 2.5)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(wj, x)
	tmp = 0.0;
	if (wj <= -0.00044)
		tmp = x / (exp(wj) * (wj + 1.0));
	else
		tmp = x + ((-2.0 * (wj * x)) + ((wj * wj) * (1.0 + (x * 2.5))));
	end
	tmp_2 = tmp;
end

Wolfram

code[wj_, x_] := If[LessEqual[wj, -0.00044], N[(x / N[(N[Exp[wj], $MachinePrecision] * N[(wj + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x + N[(N[(-2.0 * N[(wj * x), $MachinePrecision]), $MachinePrecision] + N[(N[(wj * wj), $MachinePrecision] * N[(1.0 + N[(x * 2.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if wj < -4.40000000000000016e-4

   1. Initial program 49.4%

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

      1. div-sub 49.5%

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

      2. associate-/l* 49.5%

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

      3. distribute-rgt1-in 49.5%

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

      4. associate-/l* 49.5%

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

      5. *-inverses 49.5%

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

      6. /-rgt-identity 49.5%

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

      7. distribute-rgt1-in 99.5%

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

      8. associate-/l/ 99.5%

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

      9. div-sub 99.5%

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

   3. Simplified 99.5%

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

   4. Taylor expanded in x around inf 87.5%

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

      1. +-commutative 87.5%

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

   6. Simplified 87.5%

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

   if -4.40000000000000016e-4 < wj

   1. Initial program 77.7%

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

      1. div-sub 77.7%

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

      2. associate-/l* 77.7%

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

      3. distribute-rgt1-in 77.7%

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

      4. associate-/l* 77.7%

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

      5. *-inverses 79.3%

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

      6. /-rgt-identity 79.3%

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

      7. distribute-rgt1-in 79.3%

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

      8. associate-/l/ 79.3%

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

      9. div-sub 79.3%

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

   3. Simplified 79.3%

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

      1. clear-num 79.1%

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

      2. associate-/r/ 79.3%

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

      3. rec-exp 79.3%

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

   5. Applied egg-rr 79.3%

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

   6. Taylor expanded in wj around 0 79.0%

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

      1. associate-*r* 79.0%

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

      2. neg-mul-1 79.0%

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

      3. associate-*r* 79.0%

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

      4. distribute-rgt-out 79.0%

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

      5. *-commutative 79.0%

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

      6. unpow2 79.0%

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

   8. Simplified 79.0%

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

   9. Taylor expanded in wj around 0 97.5%

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

       1. +-commutative 97.5%

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

       2. fma-def 97.5%

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

       3. unpow2 97.5%

          \[\leadsto x + \mathsf{fma}\left(\color{blue}{wj \cdot wj}, \left(1 + 0.5 \cdot x\right) - -2 \cdot x, wj \cdot \left(-1 \cdot x - x\right)\right) \]

       4. associate--l+ 97.5%

          \[\leadsto x + \mathsf{fma}\left(wj \cdot wj, \color{blue}{1 + \left(0.5 \cdot x - -2 \cdot x\right)}, wj \cdot \left(-1 \cdot x - x\right)\right) \]

       5. distribute-rgt-out-- 97.5%

          \[\leadsto x + \mathsf{fma}\left(wj \cdot wj, 1 + \color{blue}{x \cdot \left(0.5 - -2\right)}, wj \cdot \left(-1 \cdot x - x\right)\right) \]

       6. metadata-eval 97.5%

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

       7. sub-neg 97.5%

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

       8. mul-1-neg 97.5%

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

       9. distribute-rgt-out 97.5%

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

       10. metadata-eval 97.5%

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

       11. associate-*l* 97.5%

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

       12. *-commutative 97.5%

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

       13. associate-*l* 97.5%

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

   11. Simplified 97.5%

       \[\leadsto \color{blue}{x + \mathsf{fma}\left(wj \cdot wj, 1 + x \cdot 2.5, x \cdot \left(wj \cdot -2\right)\right)} \]
   12. Step-by-step derivation

       1. fma-udef 97.5%

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

       2. *-commutative 97.5%

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

       3. *-commutative 97.5%

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

       4. associate-*r* 97.5%

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

   13. Applied egg-rr 97.5%

       \[\leadsto x + \color{blue}{\left(\left(wj \cdot wj\right) \cdot \left(1 + x \cdot 2.5\right) + -2 \cdot \left(wj \cdot x\right)\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 97.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;wj \leq -0.00044:\\ \;\;\;\;\frac{x}{e^{wj} \cdot \left(wj + 1\right)}\\ \mathbf{else}:\\ \;\;\;\;x + \left(-2 \cdot \left(wj \cdot x\right) + \left(wj \cdot wj\right) \cdot \left(1 + x \cdot 2.5\right)\right)\\ \end{array} \]

Alternative 7: 97.1% accurate, 2.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if wj < -1

   1. Initial program 28.3%

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

      1. div-sub 28.3%

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

      2. associate-/l* 28.3%

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

      3. distribute-rgt1-in 28.3%

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

      4. associate-/l* 28.3%

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

      5. *-inverses 28.3%

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

      6. /-rgt-identity 28.3%

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

      7. distribute-rgt1-in 99.8%

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

      8. associate-/l/ 100.0%

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

      9. div-sub 100.0%

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

   3. Simplified 100.0%

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

      1. clear-num 100.0%

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

      2. associate-/r/ 99.8%

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

      3. rec-exp 100.0%

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

   5. Applied egg-rr 100.0%

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

   6. Taylor expanded in x around inf 100.0%

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

      1. associate-/l* 100.0%

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

      2. +-commutative 100.0%

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

      3. associate-/r/ 100.0%

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

      4. exp-neg 99.8%

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

      5. times-frac 99.8%

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

      6. *-commutative 99.8%

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

      7. *-lft-identity 99.8%

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

      8. associate-/l/ 100.0%

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

   8. Simplified 100.0%

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

   9. Taylor expanded in wj around inf 78.8%

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

   if -1 < wj

   1. Initial program 77.9%

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

      1. div-sub 77.9%

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

      2. associate-/l* 77.9%

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

      3. distribute-rgt1-in 77.9%

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

      4. associate-/l* 77.9%

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

      5. *-inverses 79.6%

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

      6. /-rgt-identity 79.6%

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

      7. distribute-rgt1-in 79.6%

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

      8. associate-/l/ 79.5%

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

      9. div-sub 79.5%

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

   3. Simplified 79.5%

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

      1. clear-num 79.4%

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

      2. associate-/r/ 79.5%

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

      3. rec-exp 79.6%

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

   5. Applied egg-rr 79.6%

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

   6. Taylor expanded in wj around 0 78.9%

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

      1. associate-*r* 78.9%

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

      2. neg-mul-1 78.9%

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

      3. associate-*r* 78.9%

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

      4. distribute-rgt-out 78.9%

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

      5. *-commutative 78.9%

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

      6. unpow2 78.9%

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

   8. Simplified 78.9%

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

   9. Taylor expanded in wj around 0 96.9%

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

       1. +-commutative 96.9%

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

       2. fma-def 96.9%

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

       3. unpow2 96.9%

          \[\leadsto x + \mathsf{fma}\left(\color{blue}{wj \cdot wj}, \left(1 + 0.5 \cdot x\right) - -2 \cdot x, wj \cdot \left(-1 \cdot x - x\right)\right) \]

       4. associate--l+ 96.9%

          \[\leadsto x + \mathsf{fma}\left(wj \cdot wj, \color{blue}{1 + \left(0.5 \cdot x - -2 \cdot x\right)}, wj \cdot \left(-1 \cdot x - x\right)\right) \]

       5. distribute-rgt-out-- 96.9%

          \[\leadsto x + \mathsf{fma}\left(wj \cdot wj, 1 + \color{blue}{x \cdot \left(0.5 - -2\right)}, wj \cdot \left(-1 \cdot x - x\right)\right) \]

       6. metadata-eval 96.9%

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

       7. sub-neg 96.9%

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

       8. mul-1-neg 96.9%

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

       9. distribute-rgt-out 96.9%

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

       10. metadata-eval 96.9%

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

       11. associate-*l* 96.9%

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

       12. *-commutative 96.9%

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

       13. associate-*l* 96.9%

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

   11. Simplified 96.9%

       \[\leadsto \color{blue}{x + \mathsf{fma}\left(wj \cdot wj, 1 + x \cdot 2.5, x \cdot \left(wj \cdot -2\right)\right)} \]
   12. Step-by-step derivation

       1. fma-udef 96.9%

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

       2. *-commutative 96.9%

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

       3. *-commutative 96.9%

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

       4. associate-*r* 96.9%

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

   13. Applied egg-rr 96.9%

       \[\leadsto x + \color{blue}{\left(\left(wj \cdot wj\right) \cdot \left(1 + x \cdot 2.5\right) + -2 \cdot \left(wj \cdot x\right)\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 96.4%

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

Alternative 8: 96.2% accurate, 18.4× speedup

Math

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

FPCore

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

C

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

Fortran

real(8) function code(wj, x)
    real(8), intent (in) :: wj
    real(8), intent (in) :: x
    code = x + (((-2.0d0) * (wj * x)) + ((wj * wj) * (1.0d0 + (x * 2.5d0))))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 76.6%

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

   1. div-sub 76.6%

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

   2. associate-/l* 76.6%

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

   3. distribute-rgt1-in 76.6%

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

   4. associate-/l* 76.6%

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

   5. *-inverses 78.2%

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

   6. /-rgt-identity 78.2%

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

   7. distribute-rgt1-in 80.1%

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

   8. associate-/l/ 80.1%

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

   9. div-sub 80.1%

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

3. Simplified 80.1%

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

   1. clear-num 79.9%

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

   2. associate-/r/ 80.1%

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

   3. rec-exp 80.1%

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

5. Applied egg-rr 80.1%

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

6. Taylor expanded in wj around 0 77.3%

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

   1. associate-*r* 77.3%

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

   2. neg-mul-1 77.3%

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

   3. associate-*r* 77.3%

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

   4. distribute-rgt-out 77.3%

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

   5. *-commutative 77.3%

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

   6. unpow2 77.3%

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

8. Simplified 77.3%

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

9. Taylor expanded in wj around 0 94.3%

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

    1. +-commutative 94.3%

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

    2. fma-def 94.3%

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

    3. unpow2 94.3%

       \[\leadsto x + \mathsf{fma}\left(\color{blue}{wj \cdot wj}, \left(1 + 0.5 \cdot x\right) - -2 \cdot x, wj \cdot \left(-1 \cdot x - x\right)\right) \]

    4. associate--l+ 94.3%

       \[\leadsto x + \mathsf{fma}\left(wj \cdot wj, \color{blue}{1 + \left(0.5 \cdot x - -2 \cdot x\right)}, wj \cdot \left(-1 \cdot x - x\right)\right) \]

    5. distribute-rgt-out-- 94.3%

       \[\leadsto x + \mathsf{fma}\left(wj \cdot wj, 1 + \color{blue}{x \cdot \left(0.5 - -2\right)}, wj \cdot \left(-1 \cdot x - x\right)\right) \]

    6. metadata-eval 94.3%

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

    7. sub-neg 94.3%

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

    8. mul-1-neg 94.3%

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

    9. distribute-rgt-out 94.3%

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

    10. metadata-eval 94.3%

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

    11. associate-*l* 94.3%

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

    12. *-commutative 94.3%

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

    13. associate-*l* 94.3%

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

11. Simplified 94.3%

    \[\leadsto \color{blue}{x + \mathsf{fma}\left(wj \cdot wj, 1 + x \cdot 2.5, x \cdot \left(wj \cdot -2\right)\right)} \]
12. Step-by-step derivation

    1. fma-udef 94.3%

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

    2. *-commutative 94.3%

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

    3. *-commutative 94.3%

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

    4. associate-*r* 94.3%

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

13. Applied egg-rr 94.3%

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

14. Final simplification 94.3%

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

Alternative 9: 84.5% accurate, 23.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -3 \cdot 10^{-240} \lor \neg \left(x \leq 8 \cdot 10^{-287}\right):\\ \;\;\;\;\frac{x - wj \cdot x}{wj + 1}\\ \mathbf{else}:\\ \;\;\;\;wj \cdot wj\\ \end{array} \end{array} \]

FPCore

(FPCore (wj x)
 :precision binary64
 (if (or (<= x -3e-240) (not (<= x 8e-287)))
   (/ (- x (* wj x)) (+ wj 1.0))
   (* wj wj)))

C

double code(double wj, double x) {
	double tmp;
	if ((x <= -3e-240) || !(x <= 8e-287)) {
		tmp = (x - (wj * x)) / (wj + 1.0);
	} else {
		tmp = wj * wj;
	}
	return tmp;
}

Fortran

real(8) function code(wj, x)
    real(8), intent (in) :: wj
    real(8), intent (in) :: x
    real(8) :: tmp
    if ((x <= (-3d-240)) .or. (.not. (x <= 8d-287))) then
        tmp = (x - (wj * x)) / (wj + 1.0d0)
    else
        tmp = wj * wj
    end if
    code = tmp
end function

Java

public static double code(double wj, double x) {
	double tmp;
	if ((x <= -3e-240) || !(x <= 8e-287)) {
		tmp = (x - (wj * x)) / (wj + 1.0);
	} else {
		tmp = wj * wj;
	}
	return tmp;
}

Python

def code(wj, x):
	tmp = 0
	if (x <= -3e-240) or not (x <= 8e-287):
		tmp = (x - (wj * x)) / (wj + 1.0)
	else:
		tmp = wj * wj
	return tmp

Julia

function code(wj, x)
	tmp = 0.0
	if ((x <= -3e-240) || !(x <= 8e-287))
		tmp = Float64(Float64(x - Float64(wj * x)) / Float64(wj + 1.0));
	else
		tmp = Float64(wj * wj);
	end
	return tmp
end

MATLAB

function tmp_2 = code(wj, x)
	tmp = 0.0;
	if ((x <= -3e-240) || ~((x <= 8e-287)))
		tmp = (x - (wj * x)) / (wj + 1.0);
	else
		tmp = wj * wj;
	end
	tmp_2 = tmp;
end

Wolfram

code[wj_, x_] := If[Or[LessEqual[x, -3e-240], N[Not[LessEqual[x, 8e-287]], $MachinePrecision]], N[(N[(x - N[(wj * x), $MachinePrecision]), $MachinePrecision] / N[(wj + 1.0), $MachinePrecision]), $MachinePrecision], N[(wj * wj), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -2.99999999999999991e-240 or 8.00000000000000017e-287 < x

   1. Initial program 80.7%

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

      1. div-sub 80.7%

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

      2. associate-/l* 80.7%

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

      3. distribute-rgt1-in 80.7%

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

      4. associate-/l* 80.7%

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

      5. *-inverses 82.4%

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

      6. /-rgt-identity 82.4%

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

      7. distribute-rgt1-in 84.1%

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

      8. associate-/l/ 84.0%

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

      9. div-sub 84.0%

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

   3. Simplified 84.0%

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

      1. clear-num 83.9%

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

      2. associate-/r/ 84.1%

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

      3. rec-exp 84.1%

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

   5. Applied egg-rr 84.1%

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

   6. Taylor expanded in x around inf 88.2%

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

      1. associate-/l* 88.1%

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

      2. +-commutative 88.1%

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

      3. associate-/r/ 88.2%

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

      4. exp-neg 88.1%

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

      5. times-frac 88.1%

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

      6. *-commutative 88.1%

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

      7. *-lft-identity 88.1%

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

      8. associate-/l/ 88.1%

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

   8. Simplified 88.1%

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

   9. Taylor expanded in wj around 0 85.0%

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

       1. mul-1-neg 85.0%

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

       2. *-commutative 85.0%

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

   11. Simplified 85.0%

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

   if -2.99999999999999991e-240 < x < 8.00000000000000017e-287

   1. Initial program 22.3%

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

      1. div-sub 22.3%

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

      2. associate-/l* 22.3%

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

      3. distribute-rgt1-in 22.3%

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

      4. associate-/l* 22.3%

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

      5. *-inverses 22.3%

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

      6. /-rgt-identity 22.3%

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

      7. distribute-rgt1-in 27.8%

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

      8. associate-/l/ 27.8%

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

      9. div-sub 27.8%

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

   3. Simplified 27.8%

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

   4. Taylor expanded in x around 0 12.4%

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

      1. +-commutative 12.4%

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

   6. Simplified 12.4%

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

      1. flip3-- 10.4%

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

   8. Applied egg-rr 10.4%

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

   9. Taylor expanded in wj around 0 64.4%

      \[\leadsto \color{blue}{{wj}^{2}} \]
   10. Step-by-step derivation

       1. unpow2 64.4%

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

   11. Simplified 64.4%

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

4. Final simplification 83.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -3 \cdot 10^{-240} \lor \neg \left(x \leq 8 \cdot 10^{-287}\right):\\ \;\;\;\;\frac{x - wj \cdot x}{wj + 1}\\ \mathbf{else}:\\ \;\;\;\;wj \cdot wj\\ \end{array} \]

Alternative 10: 84.5% accurate, 28.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -2.6 \cdot 10^{-243} \lor \neg \left(x \leq 9 \cdot 10^{-287}\right):\\ \;\;\;\;x + -2 \cdot \left(wj \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;wj \cdot wj\\ \end{array} \end{array} \]

FPCore

(FPCore (wj x)
 :precision binary64
 (if (or (<= x -2.6e-243) (not (<= x 9e-287)))
   (+ x (* -2.0 (* wj x)))
   (* wj wj)))

C

double code(double wj, double x) {
	double tmp;
	if ((x <= -2.6e-243) || !(x <= 9e-287)) {
		tmp = x + (-2.0 * (wj * x));
	} else {
		tmp = wj * wj;
	}
	return tmp;
}

Fortran

real(8) function code(wj, x)
    real(8), intent (in) :: wj
    real(8), intent (in) :: x
    real(8) :: tmp
    if ((x <= (-2.6d-243)) .or. (.not. (x <= 9d-287))) then
        tmp = x + ((-2.0d0) * (wj * x))
    else
        tmp = wj * wj
    end if
    code = tmp
end function

Java

public static double code(double wj, double x) {
	double tmp;
	if ((x <= -2.6e-243) || !(x <= 9e-287)) {
		tmp = x + (-2.0 * (wj * x));
	} else {
		tmp = wj * wj;
	}
	return tmp;
}

Python

def code(wj, x):
	tmp = 0
	if (x <= -2.6e-243) or not (x <= 9e-287):
		tmp = x + (-2.0 * (wj * x))
	else:
		tmp = wj * wj
	return tmp

Julia

function code(wj, x)
	tmp = 0.0
	if ((x <= -2.6e-243) || !(x <= 9e-287))
		tmp = Float64(x + Float64(-2.0 * Float64(wj * x)));
	else
		tmp = Float64(wj * wj);
	end
	return tmp
end

MATLAB

function tmp_2 = code(wj, x)
	tmp = 0.0;
	if ((x <= -2.6e-243) || ~((x <= 9e-287)))
		tmp = x + (-2.0 * (wj * x));
	else
		tmp = wj * wj;
	end
	tmp_2 = tmp;
end

Wolfram

code[wj_, x_] := If[Or[LessEqual[x, -2.6e-243], N[Not[LessEqual[x, 9e-287]], $MachinePrecision]], N[(x + N[(-2.0 * N[(wj * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(wj * wj), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -2.5999999999999998e-243 or 9.00000000000000034e-287 < x

   1. Initial program 80.7%

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

      1. div-sub 80.7%

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

      2. associate-/l* 80.7%

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

      3. distribute-rgt1-in 80.7%

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

      4. associate-/l* 80.7%

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

      5. *-inverses 82.4%

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

      6. /-rgt-identity 82.4%

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

      7. distribute-rgt1-in 84.1%

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

      8. associate-/l/ 84.0%

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

      9. div-sub 84.0%

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

   3. Simplified 84.0%

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

   4. Taylor expanded in wj around 0 84.8%

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

      1. *-commutative 84.8%

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

   6. Simplified 84.8%

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

   if -2.5999999999999998e-243 < x < 9.00000000000000034e-287

   1. Initial program 22.3%

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

      1. div-sub 22.3%

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

      2. associate-/l* 22.3%

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

      3. distribute-rgt1-in 22.3%

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

      4. associate-/l* 22.3%

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

      5. *-inverses 22.3%

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

      6. /-rgt-identity 22.3%

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

      7. distribute-rgt1-in 27.8%

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

      8. associate-/l/ 27.8%

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

      9. div-sub 27.8%

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

   3. Simplified 27.8%

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

   4. Taylor expanded in x around 0 12.4%

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

      1. +-commutative 12.4%

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

   6. Simplified 12.4%

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

      1. flip3-- 10.4%

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

   8. Applied egg-rr 10.4%

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

   9. Taylor expanded in wj around 0 64.4%

      \[\leadsto \color{blue}{{wj}^{2}} \]
   10. Step-by-step derivation

       1. unpow2 64.4%

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

   11. Simplified 64.4%

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

4. Final simplification 83.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2.6 \cdot 10^{-243} \lor \neg \left(x \leq 9 \cdot 10^{-287}\right):\\ \;\;\;\;x + -2 \cdot \left(wj \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;wj \cdot wj\\ \end{array} \]

Alternative 11: 96.1% accurate, 28.5× speedup

Math

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

FPCore

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

C

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

Fortran

real(8) function code(wj, x)
    real(8), intent (in) :: wj
    real(8), intent (in) :: x
    code = x + (((-2.0d0) * (wj * x)) + (wj * wj))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 76.6%

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

   1. div-sub 76.6%

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

   2. associate-/l* 76.6%

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

   3. distribute-rgt1-in 76.6%

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

   4. associate-/l* 76.6%

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

   5. *-inverses 78.2%

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

   6. /-rgt-identity 78.2%

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

   7. distribute-rgt1-in 80.1%

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

   8. associate-/l/ 80.1%

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

   9. div-sub 80.1%

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

3. Simplified 80.1%

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

4. Taylor expanded in wj around 0 94.7%

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

5. Taylor expanded in x around 0 94.4%

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

   1. unpow2 94.4%

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

7. Simplified 94.4%

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

8. Taylor expanded in x around 0 94.4%

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

9. Taylor expanded in wj around 0 94.0%

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

    1. unpow2 94.0%

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

11. Simplified 94.0%

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

12. Final simplification 94.0%

    \[\leadsto x + \left(-2 \cdot \left(wj \cdot x\right) + wj \cdot wj\right) \]

Alternative 12: 84.1% accurate, 43.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -4.1 \cdot 10^{-243}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq 4 \cdot 10^{-287}:\\ \;\;\;\;wj \cdot wj\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \end{array} \]

FPCore

(FPCore (wj x)
 :precision binary64
 (if (<= x -4.1e-243) x (if (<= x 4e-287) (* wj wj) x)))

C

double code(double wj, double x) {
	double tmp;
	if (x <= -4.1e-243) {
		tmp = x;
	} else if (x <= 4e-287) {
		tmp = wj * wj;
	} 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 <= (-4.1d-243)) then
        tmp = x
    else if (x <= 4d-287) then
        tmp = wj * wj
    else
        tmp = x
    end if
    code = tmp
end function

Java

public static double code(double wj, double x) {
	double tmp;
	if (x <= -4.1e-243) {
		tmp = x;
	} else if (x <= 4e-287) {
		tmp = wj * wj;
	} else {
		tmp = x;
	}
	return tmp;
}

Python

def code(wj, x):
	tmp = 0
	if x <= -4.1e-243:
		tmp = x
	elif x <= 4e-287:
		tmp = wj * wj
	else:
		tmp = x
	return tmp

Julia

function code(wj, x)
	tmp = 0.0
	if (x <= -4.1e-243)
		tmp = x;
	elseif (x <= 4e-287)
		tmp = Float64(wj * wj);
	else
		tmp = x;
	end
	return tmp
end

MATLAB

function tmp_2 = code(wj, x)
	tmp = 0.0;
	if (x <= -4.1e-243)
		tmp = x;
	elseif (x <= 4e-287)
		tmp = wj * wj;
	else
		tmp = x;
	end
	tmp_2 = tmp;
end

Wolfram

code[wj_, x_] := If[LessEqual[x, -4.1e-243], x, If[LessEqual[x, 4e-287], N[(wj * wj), $MachinePrecision], x]]

Derivation

1. Split input into 2 regimes

2. if x < -4.09999999999999981e-243 or 4.00000000000000009e-287 < x

   1. Initial program 80.7%

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

      1. div-sub 80.7%

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

      2. associate-/l* 80.7%

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

      3. distribute-rgt1-in 80.7%

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

      4. associate-/l* 80.7%

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

      5. *-inverses 82.4%

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

      6. /-rgt-identity 82.4%

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

      7. distribute-rgt1-in 84.1%

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

      8. associate-/l/ 84.0%

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

      9. div-sub 84.0%

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

   3. Simplified 84.0%

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

   4. Taylor expanded in wj around 0 83.8%

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

   if -4.09999999999999981e-243 < x < 4.00000000000000009e-287

   1. Initial program 22.3%

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

      1. div-sub 22.3%

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

      2. associate-/l* 22.3%

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

      3. distribute-rgt1-in 22.3%

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

      4. associate-/l* 22.3%

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

      5. *-inverses 22.3%

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

      6. /-rgt-identity 22.3%

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

      7. distribute-rgt1-in 27.8%

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

      8. associate-/l/ 27.8%

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

      9. div-sub 27.8%

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

   3. Simplified 27.8%

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

   4. Taylor expanded in x around 0 12.4%

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

      1. +-commutative 12.4%

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

   6. Simplified 12.4%

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

      1. flip3-- 10.4%

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

   8. Applied egg-rr 10.4%

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

   9. Taylor expanded in wj around 0 64.4%

      \[\leadsto \color{blue}{{wj}^{2}} \]
   10. Step-by-step derivation

       1. unpow2 64.4%

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

   11. Simplified 64.4%

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

4. Final simplification 82.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -4.1 \cdot 10^{-243}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq 4 \cdot 10^{-287}:\\ \;\;\;\;wj \cdot wj\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 13: 84.3% accurate, 43.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;wj \leq 8.2 \cdot 10^{-31}:\\ \;\;\;\;x\\ \mathbf{elif}\;wj \leq 1.55 \cdot 10^{-16}:\\ \;\;\;\;wj \cdot wj\\ \mathbf{else}:\\ \;\;\;\;wj + x\\ \end{array} \end{array} \]

FPCore

(FPCore (wj x)
 :precision binary64
 (if (<= wj 8.2e-31) x (if (<= wj 1.55e-16) (* wj wj) (+ wj x))))

C

double code(double wj, double x) {
	double tmp;
	if (wj <= 8.2e-31) {
		tmp = x;
	} else if (wj <= 1.55e-16) {
		tmp = wj * wj;
	} else {
		tmp = wj + x;
	}
	return tmp;
}

Fortran

real(8) function code(wj, x)
    real(8), intent (in) :: wj
    real(8), intent (in) :: x
    real(8) :: tmp
    if (wj <= 8.2d-31) then
        tmp = x
    else if (wj <= 1.55d-16) then
        tmp = wj * wj
    else
        tmp = wj + x
    end if
    code = tmp
end function

Java

public static double code(double wj, double x) {
	double tmp;
	if (wj <= 8.2e-31) {
		tmp = x;
	} else if (wj <= 1.55e-16) {
		tmp = wj * wj;
	} else {
		tmp = wj + x;
	}
	return tmp;
}

Python

def code(wj, x):
	tmp = 0
	if wj <= 8.2e-31:
		tmp = x
	elif wj <= 1.55e-16:
		tmp = wj * wj
	else:
		tmp = wj + x
	return tmp

Julia

function code(wj, x)
	tmp = 0.0
	if (wj <= 8.2e-31)
		tmp = x;
	elseif (wj <= 1.55e-16)
		tmp = Float64(wj * wj);
	else
		tmp = Float64(wj + x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(wj, x)
	tmp = 0.0;
	if (wj <= 8.2e-31)
		tmp = x;
	elseif (wj <= 1.55e-16)
		tmp = wj * wj;
	else
		tmp = wj + x;
	end
	tmp_2 = tmp;
end

Wolfram

code[wj_, x_] := If[LessEqual[wj, 8.2e-31], x, If[LessEqual[wj, 1.55e-16], N[(wj * wj), $MachinePrecision], N[(wj + x), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if wj < 8.1999999999999993e-31

   1. Initial program 79.5%

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

      1. div-sub 79.5%

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

      2. associate-/l* 79.5%

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

      3. distribute-rgt1-in 79.5%

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

      4. associate-/l* 79.5%

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

      5. *-inverses 79.5%

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

      6. /-rgt-identity 79.5%

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

      7. distribute-rgt1-in 81.6%

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

      8. associate-/l/ 81.6%

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

      9. div-sub 81.6%

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

   3. Simplified 81.6%

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

   4. Taylor expanded in wj around 0 84.5%

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

   if 8.1999999999999993e-31 < wj < 1.55e-16

   1. Initial program 17.6%

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

      1. div-sub 17.6%

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

      2. associate-/l* 17.6%

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

      3. distribute-rgt1-in 17.6%

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

      4. associate-/l* 17.6%

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

      5. *-inverses 17.6%

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

      6. /-rgt-identity 17.6%

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

      7. distribute-rgt1-in 17.6%

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

      8. associate-/l/ 17.6%

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

      9. div-sub 17.6%

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

   3. Simplified 17.6%

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

   4. Taylor expanded in x around 0 5.3%

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

      1. +-commutative 5.3%

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

   6. Simplified 5.3%

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

      1. flip3-- 5.2%

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

   8. Applied egg-rr 5.2%

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

   9. Taylor expanded in wj around 0 70.0%

      \[\leadsto \color{blue}{{wj}^{2}} \]
   10. Step-by-step derivation

       1. unpow2 70.0%

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

   11. Simplified 70.0%

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

   if 1.55e-16 < wj

   1. Initial program 59.0%

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

      1. div-sub 59.0%

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

      2. associate-/l* 58.9%

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

      3. distribute-rgt1-in 58.9%

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

      4. associate-/l* 58.9%

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

      5. *-inverses 92.3%

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

      6. /-rgt-identity 92.3%

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

      7. distribute-rgt1-in 92.3%

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

      8. associate-/l/ 92.1%

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

      9. div-sub 92.1%

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

   3. Simplified 92.1%

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

   4. Taylor expanded in wj around 0 52.6%

      \[\leadsto wj - \color{blue}{-1 \cdot x} \]
   5. Step-by-step derivation

      1. neg-mul-1 52.6%

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

   6. Simplified 52.6%

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

4. Final simplification 82.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;wj \leq 8.2 \cdot 10^{-31}:\\ \;\;\;\;x\\ \mathbf{elif}\;wj \leq 1.55 \cdot 10^{-16}:\\ \;\;\;\;wj \cdot wj\\ \mathbf{else}:\\ \;\;\;\;wj + x\\ \end{array} \]

Alternative 14: 4.4% accurate, 313.0× speedup

Math

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

FPCore

(FPCore (wj x) :precision binary64 wj)

C

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

Fortran

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

Java

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

Python

def code(wj, x):
	return wj

Julia

function code(wj, x)
	return wj
end

MATLAB

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

Wolfram

code[wj_, x_] := wj

Derivation

1. Initial program 76.6%

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

   1. div-sub 76.6%

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

   2. associate-/l* 76.6%

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

   3. distribute-rgt1-in 76.6%

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

   4. associate-/l* 76.6%

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

   5. *-inverses 78.2%

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

   6. /-rgt-identity 78.2%

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

   7. distribute-rgt1-in 80.1%

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

   8. associate-/l/ 80.1%

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

   9. div-sub 80.1%

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

3. Simplified 80.1%

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

4. Taylor expanded in wj around inf 4.7%

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

5. Final simplification 4.7%

   \[\leadsto wj \]

Alternative 15: 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 76.6%

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

   1. div-sub 76.6%

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

   2. associate-/l* 76.6%

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

   3. distribute-rgt1-in 76.6%

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

   4. associate-/l* 76.6%

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

   5. *-inverses 78.2%

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

   6. /-rgt-identity 78.2%

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

   7. distribute-rgt1-in 80.1%

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

   8. associate-/l/ 80.1%

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

   9. div-sub 80.1%

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

3. Simplified 80.1%

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

4. Taylor expanded in wj around 0 80.1%

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

5. Final simplification 80.1%

   \[\leadsto x \]

Developer target: 78.7% 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 2023271 
(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))))))