Jmat.Real.lambertw, newton loop step

Percentage Accurate: 77.4% → 99.3%

Time: 20.8s

Alternatives: 22

Speedup: 313.0×

• could not determine a ground truth

  1. wj = -4.2091038317165086e+155
  2. x = 3.508560567941444e+242

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: 77.4% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.3% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := wj - \frac{x}{e^{wj}}\\ t_1 := \frac{wj}{wj + 1}\\ t_2 := \sqrt{\frac{t\_0}{wj + 1}}\\ \mathbf{if}\;wj \leq -3.8 \cdot 10^{-6}:\\ \;\;\;\;\left(wj - t\_1\right) + \mathsf{fma}\left(\frac{-1}{wj + 1}, t\_0, t\_1\right)\\ \mathbf{elif}\;wj \leq -7.8 \cdot 10^{-32}:\\ \;\;\;\;x + wj \cdot \left({wj}^{2} \cdot \left(\left(\left(-1 - \left(x \cdot -3 + \left(-2 \cdot \left(x \cdot -4 + x \cdot 1.5\right) + x \cdot 0.6666666666666666\right)\right)\right) + \frac{1}{wj}\right) - \left(-4 \cdot \frac{x}{wj} + 1.5 \cdot \frac{x}{wj}\right)\right) - x \cdot 2\right)\\ \mathbf{elif}\;wj \leq 5 \cdot 10^{-83}:\\ \;\;\;\;x - wj \cdot \left(x \cdot 2 - wj\right)\\ \mathbf{elif}\;wj \leq 3.7 \cdot 10^{-7}:\\ \;\;\;\;x - wj \cdot \left(x \cdot 2 + wj \cdot \left(wj + x \cdot \left(\left(\frac{-1}{x} - wj \cdot -2.6666666666666665\right) - 2.5\right)\right)\right)\\ \mathbf{elif}\;wj \leq 4.1 \cdot 10^{-5}:\\ \;\;\;\;x - wj \cdot \left(x \cdot 2 + \left(0 - {\left(e^{3 \cdot \log \left(wj \cdot \left(\mathsf{fma}\left(-1, \mathsf{fma}\left(wj, \mathsf{fma}\left(-3, x, \mathsf{fma}\left(-2, x \cdot -2.5, x \cdot 0.6666666666666666\right)\right), wj\right), 1\right) - x \cdot -2.5\right)\right)}\right)}^{0.3333333333333333}\right)\right)\\ \mathbf{elif}\;wj \leq 0.00055:\\ \;\;\;\;x \cdot \frac{e^{-wj}}{wj + 1}\\ \mathbf{elif}\;wj \leq 0.014:\\ \;\;\;\;\left(t\_2 + \sqrt{wj}\right) \cdot \left(\sqrt{wj} - t\_2\right)\\ \mathbf{else}:\\ \;\;\;\;wj + \frac{t\_0}{-1 - wj}\\ \end{array} \end{array} \]

FPCore

(FPCore (wj x)
 :precision binary64
 (let* ((t_0 (- wj (/ x (exp wj))))
        (t_1 (/ wj (+ wj 1.0)))
        (t_2 (sqrt (/ t_0 (+ wj 1.0)))))
   (if (<= wj -3.8e-6)
     (+ (- wj t_1) (fma (/ -1.0 (+ wj 1.0)) t_0 t_1))
     (if (<= wj -7.8e-32)
       (+
        x
        (*
         wj
         (-
          (*
           (pow wj 2.0)
           (-
            (+
             (-
              -1.0
              (+
               (* x -3.0)
               (+ (* -2.0 (+ (* x -4.0) (* x 1.5))) (* x 0.6666666666666666))))
             (/ 1.0 wj))
            (+ (* -4.0 (/ x wj)) (* 1.5 (/ x wj)))))
          (* x 2.0))))
       (if (<= wj 5e-83)
         (- x (* wj (- (* x 2.0) wj)))
         (if (<= wj 3.7e-7)
           (-
            x
            (*
             wj
             (+
              (* x 2.0)
              (*
               wj
               (+
                wj
                (* x (- (- (/ -1.0 x) (* wj -2.6666666666666665)) 2.5)))))))
           (if (<= wj 4.1e-5)
             (-
              x
              (*
               wj
               (+
                (* x 2.0)
                (-
                 0.0
                 (pow
                  (exp
                   (*
                    3.0
                    (log
                     (*
                      wj
                      (-
                       (fma
                        -1.0
                        (fma
                         wj
                         (fma
                          -3.0
                          x
                          (fma -2.0 (* x -2.5) (* x 0.6666666666666666)))
                         wj)
                        1.0)
                       (* x -2.5))))))
                  0.3333333333333333)))))
             (if (<= wj 0.00055)
               (* x (/ (exp (- wj)) (+ wj 1.0)))
               (if (<= wj 0.014)
                 (* (+ t_2 (sqrt wj)) (- (sqrt wj) t_2))
                 (+ wj (/ t_0 (- -1.0 wj))))))))))))

C

double code(double wj, double x) {
	double t_0 = wj - (x / exp(wj));
	double t_1 = wj / (wj + 1.0);
	double t_2 = sqrt((t_0 / (wj + 1.0)));
	double tmp;
	if (wj <= -3.8e-6) {
		tmp = (wj - t_1) + fma((-1.0 / (wj + 1.0)), t_0, t_1);
	} else if (wj <= -7.8e-32) {
		tmp = x + (wj * ((pow(wj, 2.0) * (((-1.0 - ((x * -3.0) + ((-2.0 * ((x * -4.0) + (x * 1.5))) + (x * 0.6666666666666666)))) + (1.0 / wj)) - ((-4.0 * (x / wj)) + (1.5 * (x / wj))))) - (x * 2.0)));
	} else if (wj <= 5e-83) {
		tmp = x - (wj * ((x * 2.0) - wj));
	} else if (wj <= 3.7e-7) {
		tmp = x - (wj * ((x * 2.0) + (wj * (wj + (x * (((-1.0 / x) - (wj * -2.6666666666666665)) - 2.5))))));
	} else if (wj <= 4.1e-5) {
		tmp = x - (wj * ((x * 2.0) + (0.0 - pow(exp((3.0 * log((wj * (fma(-1.0, fma(wj, fma(-3.0, x, fma(-2.0, (x * -2.5), (x * 0.6666666666666666))), wj), 1.0) - (x * -2.5)))))), 0.3333333333333333))));
	} else if (wj <= 0.00055) {
		tmp = x * (exp(-wj) / (wj + 1.0));
	} else if (wj <= 0.014) {
		tmp = (t_2 + sqrt(wj)) * (sqrt(wj) - t_2);
	} else {
		tmp = wj + (t_0 / (-1.0 - wj));
	}
	return tmp;
}

Julia

function code(wj, x)
	t_0 = Float64(wj - Float64(x / exp(wj)))
	t_1 = Float64(wj / Float64(wj + 1.0))
	t_2 = sqrt(Float64(t_0 / Float64(wj + 1.0)))
	tmp = 0.0
	if (wj <= -3.8e-6)
		tmp = Float64(Float64(wj - t_1) + fma(Float64(-1.0 / Float64(wj + 1.0)), t_0, t_1));
	elseif (wj <= -7.8e-32)
		tmp = Float64(x + Float64(wj * Float64(Float64((wj ^ 2.0) * Float64(Float64(Float64(-1.0 - Float64(Float64(x * -3.0) + Float64(Float64(-2.0 * Float64(Float64(x * -4.0) + Float64(x * 1.5))) + Float64(x * 0.6666666666666666)))) + Float64(1.0 / wj)) - Float64(Float64(-4.0 * Float64(x / wj)) + Float64(1.5 * Float64(x / wj))))) - Float64(x * 2.0))));
	elseif (wj <= 5e-83)
		tmp = Float64(x - Float64(wj * Float64(Float64(x * 2.0) - wj)));
	elseif (wj <= 3.7e-7)
		tmp = Float64(x - Float64(wj * Float64(Float64(x * 2.0) + Float64(wj * Float64(wj + Float64(x * Float64(Float64(Float64(-1.0 / x) - Float64(wj * -2.6666666666666665)) - 2.5)))))));
	elseif (wj <= 4.1e-5)
		tmp = Float64(x - Float64(wj * Float64(Float64(x * 2.0) + Float64(0.0 - (exp(Float64(3.0 * log(Float64(wj * Float64(fma(-1.0, fma(wj, fma(-3.0, x, fma(-2.0, Float64(x * -2.5), Float64(x * 0.6666666666666666))), wj), 1.0) - Float64(x * -2.5)))))) ^ 0.3333333333333333)))));
	elseif (wj <= 0.00055)
		tmp = Float64(x * Float64(exp(Float64(-wj)) / Float64(wj + 1.0)));
	elseif (wj <= 0.014)
		tmp = Float64(Float64(t_2 + sqrt(wj)) * Float64(sqrt(wj) - t_2));
	else
		tmp = Float64(wj + Float64(t_0 / Float64(-1.0 - wj)));
	end
	return tmp
end

Wolfram

code[wj_, x_] := Block[{t$95$0 = N[(wj - N[(x / N[Exp[wj], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(wj / N[(wj + 1.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[Sqrt[N[(t$95$0 / N[(wj + 1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[wj, -3.8e-6], N[(N[(wj - t$95$1), $MachinePrecision] + N[(N[(-1.0 / N[(wj + 1.0), $MachinePrecision]), $MachinePrecision] * t$95$0 + t$95$1), $MachinePrecision]), $MachinePrecision], If[LessEqual[wj, -7.8e-32], N[(x + N[(wj * N[(N[(N[Power[wj, 2.0], $MachinePrecision] * N[(N[(N[(-1.0 - N[(N[(x * -3.0), $MachinePrecision] + N[(N[(-2.0 * N[(N[(x * -4.0), $MachinePrecision] + N[(x * 1.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(x * 0.6666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(1.0 / wj), $MachinePrecision]), $MachinePrecision] - N[(N[(-4.0 * N[(x / wj), $MachinePrecision]), $MachinePrecision] + N[(1.5 * N[(x / wj), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(x * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[wj, 5e-83], N[(x - N[(wj * N[(N[(x * 2.0), $MachinePrecision] - wj), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[wj, 3.7e-7], N[(x - N[(wj * N[(N[(x * 2.0), $MachinePrecision] + N[(wj * N[(wj + N[(x * N[(N[(N[(-1.0 / x), $MachinePrecision] - N[(wj * -2.6666666666666665), $MachinePrecision]), $MachinePrecision] - 2.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[wj, 4.1e-5], N[(x - N[(wj * N[(N[(x * 2.0), $MachinePrecision] + N[(0.0 - N[Power[N[Exp[N[(3.0 * N[Log[N[(wj * N[(N[(-1.0 * N[(wj * N[(-3.0 * x + N[(-2.0 * N[(x * -2.5), $MachinePrecision] + N[(x * 0.6666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + wj), $MachinePrecision] + 1.0), $MachinePrecision] - N[(x * -2.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision], 0.3333333333333333], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[wj, 0.00055], N[(x * N[(N[Exp[(-wj)], $MachinePrecision] / N[(wj + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[wj, 0.014], N[(N[(t$95$2 + N[Sqrt[wj], $MachinePrecision]), $MachinePrecision] * N[(N[Sqrt[wj], $MachinePrecision] - t$95$2), $MachinePrecision]), $MachinePrecision], N[(wj + N[(t$95$0 / N[(-1.0 - wj), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]]]]]

Derivation

1. Split input into 8 regimes

2. if wj < -3.8e-6

   1. Initial program 59.6%

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

      1. distribute-rgt1-in 97.1%

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

      2. *-commutative 97.1%

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

      3. associate-/r* 97.1%

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

      4. div-sub 59.6%

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

      5. associate-/l* 59.6%

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

      6. *-inverses 97.1%

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

      7. *-rgt-identity 97.1%

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

   3. Simplified 97.1%

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

      1. *-un-lft-identity 97.1%

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

      2. div-inv 97.2%

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

      3. prod-diff 59.8%

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

      4. associate-/r/ 59.5%

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

      5. clear-num 59.6%

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

      6. fma-neg 59.6%

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

      7. *-un-lft-identity 59.6%

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

      8. associate-/r/ 59.6%

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

      9. clear-num 59.8%

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

   6. Applied egg-rr 59.8%

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

      1. distribute-neg-frac 59.8%

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

      2. metadata-eval 59.8%

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

   8. Simplified 59.8%

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

   9. Taylor expanded in x around 0 66.8%

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

       1. +-commutative 66.8%

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

   11. Simplified 66.8%

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

   12. Taylor expanded in x around 0 97.1%

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

       1. +-commutative 97.1%

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

   14. Simplified 97.1%

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

   if -3.8e-6 < wj < -7.8000000000000003e-32

   1. Initial program 70.8%

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

      1. distribute-rgt1-in 70.7%

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

      2. *-commutative 70.7%

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

      3. associate-/r* 70.8%

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

      4. div-sub 70.8%

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

      5. associate-/l* 70.8%

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

      6. *-inverses 70.8%

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

      7. *-rgt-identity 70.8%

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

   3. Simplified 70.8%

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

   5. Taylor expanded in wj around 0 98.4%

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

   6. Taylor expanded in wj around inf 98.5%

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

   if -7.8000000000000003e-32 < wj < 5e-83

   1. Initial program 81.3%

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

      1. distribute-rgt1-in 81.3%

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

      2. *-commutative 81.3%

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

      3. associate-/r* 81.3%

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

      4. div-sub 81.3%

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

      5. associate-/l* 81.3%

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

      6. *-inverses 81.3%

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

      7. *-rgt-identity 81.3%

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

   3. Simplified 81.3%

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

   5. Taylor expanded in wj around 0 99.6%

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

   6. Taylor expanded in x around 0 100.0%

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

      1. mul-1-neg 100.0%

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

      2. unsub-neg 100.0%

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

   8. Simplified 100.0%

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

   9. Taylor expanded in wj around 0 100.0%

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

   if 5e-83 < wj < 3.70000000000000004e-7

   1. Initial program 51.6%

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

      1. distribute-rgt1-in 51.6%

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

      2. *-commutative 51.6%

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

      3. associate-/r* 51.6%

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

      4. div-sub 51.6%

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

      5. associate-/l* 51.6%

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

      6. *-inverses 51.6%

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

      7. *-rgt-identity 51.6%

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

   3. Simplified 51.6%

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

   5. Taylor expanded in wj around 0 99.9%

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

   6. Taylor expanded in x around 0 99.9%

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

      1. distribute-lft-out 99.9%

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

      2. *-commutative 99.9%

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

      3. mul-1-neg 99.9%

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

   8. Simplified 99.9%

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

   9. Taylor expanded in x around 0 99.9%

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

   10. Taylor expanded in x around inf 100.0%

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

   if 3.70000000000000004e-7 < wj < 4.10000000000000005e-5

   1. Initial program 68.1%

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

      1. distribute-rgt1-in 66.6%

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

      2. *-commutative 66.6%

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

      3. associate-/r* 69.7%

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

      4. div-sub 69.7%

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

      5. associate-/l* 69.7%

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

      6. *-inverses 69.7%

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

      7. *-rgt-identity 69.7%

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

   3. Simplified 69.7%

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

   5. Taylor expanded in wj around 0 85.0%

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

      1. add-cbrt-cube 85.0%

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

      2. pow1/3 85.0%

         \[\leadsto x + wj \cdot \left(\color{blue}{{\left(\left(\left(wj \cdot \left(\left(1 + -1 \cdot \left(wj \cdot \left(1 + \left(-3 \cdot x + \left(-2 \cdot \left(-4 \cdot x + 1.5 \cdot x\right) + 0.6666666666666666 \cdot x\right)\right)\right)\right)\right) - \left(-4 \cdot x + 1.5 \cdot x\right)\right)\right) \cdot \left(wj \cdot \left(\left(1 + -1 \cdot \left(wj \cdot \left(1 + \left(-3 \cdot x + \left(-2 \cdot \left(-4 \cdot x + 1.5 \cdot x\right) + 0.6666666666666666 \cdot x\right)\right)\right)\right)\right) - \left(-4 \cdot x + 1.5 \cdot x\right)\right)\right)\right) \cdot \left(wj \cdot \left(\left(1 + -1 \cdot \left(wj \cdot \left(1 + \left(-3 \cdot x + \left(-2 \cdot \left(-4 \cdot x + 1.5 \cdot x\right) + 0.6666666666666666 \cdot x\right)\right)\right)\right)\right) - \left(-4 \cdot x + 1.5 \cdot x\right)\right)\right)\right)}^{0.3333333333333333}} - 2 \cdot x\right) \]

   7. Applied egg-rr 85.0%

      \[\leadsto x + wj \cdot \left(\color{blue}{{\left({\left(wj \cdot \left(\mathsf{fma}\left(-1, wj \cdot \left(\mathsf{fma}\left(-3, x, \mathsf{fma}\left(-2, x \cdot -2.5, x \cdot 0.6666666666666666\right)\right) + 1\right), 1\right) - x \cdot -2.5\right)\right)}^{3}\right)}^{0.3333333333333333}} - 2 \cdot x\right) \]
   8. Step-by-step derivation

      1. add-exp-log 85.0%

         \[\leadsto x + wj \cdot \left({\color{blue}{\left(e^{\log \left({\left(wj \cdot \left(\mathsf{fma}\left(-1, wj \cdot \left(\mathsf{fma}\left(-3, x, \mathsf{fma}\left(-2, x \cdot -2.5, x \cdot 0.6666666666666666\right)\right) + 1\right), 1\right) - x \cdot -2.5\right)\right)}^{3}\right)}\right)}}^{0.3333333333333333} - 2 \cdot x\right) \]

      2. log-pow 85.0%

         \[\leadsto x + wj \cdot \left({\left(e^{\color{blue}{3 \cdot \log \left(wj \cdot \left(\mathsf{fma}\left(-1, wj \cdot \left(\mathsf{fma}\left(-3, x, \mathsf{fma}\left(-2, x \cdot -2.5, x \cdot 0.6666666666666666\right)\right) + 1\right), 1\right) - x \cdot -2.5\right)\right)}}\right)}^{0.3333333333333333} - 2 \cdot x\right) \]

   9. Applied egg-rr 85.0%

      \[\leadsto x + wj \cdot \left({\color{blue}{\left(e^{3 \cdot \log \left(wj \cdot \left(\mathsf{fma}\left(-1, \mathsf{fma}\left(wj, \mathsf{fma}\left(-3, x, \mathsf{fma}\left(-2, x \cdot -2.5, x \cdot 0.6666666666666666\right)\right), wj\right), 1\right) - x \cdot -2.5\right)\right)}\right)}}^{0.3333333333333333} - 2 \cdot x\right) \]

   if 4.10000000000000005e-5 < wj < 5.50000000000000033e-4

   1. Initial program 100.0%

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

      1. distribute-rgt1-in 98.4%

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

      2. *-commutative 98.4%

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

      3. associate-/r* 98.4%

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

      4. div-sub 98.4%

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

      5. associate-/l* 98.4%

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

      6. *-inverses 98.4%

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

      7. *-rgt-identity 98.4%

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

   3. Simplified 98.4%

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

   5. Taylor expanded in x around inf 98.4%

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

      1. +-commutative 98.4%

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

      2. associate-/r* 100.0%

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

      3. exp-neg 100.0%

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

      4. +-commutative 100.0%

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

   7. Simplified 100.0%

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

   8. Taylor expanded in x around inf 98.4%

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

      1. associate-/l* 100.0%

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

      2. +-commutative 100.0%

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

   10. Simplified 100.0%

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

   if 5.50000000000000033e-4 < wj < 0.0140000000000000003

   1. Initial program 94.8%

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

      1. distribute-rgt1-in 94.8%

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

      2. *-commutative 94.8%

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

      3. associate-/r* 90.9%

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

      4. div-sub 90.9%

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

      5. associate-/l* 90.9%

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

      6. *-inverses 90.9%

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

      7. *-rgt-identity 90.9%

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

   3. Simplified 90.9%

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

      1. add-sqr-sqrt 90.9%

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

      2. add-sqr-sqrt 94.8%

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

      3. difference-of-squares 100.0%

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

   6. Applied egg-rr 100.0%

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

      1. +-commutative 100.0%

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

   8. Simplified 100.0%

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

   if 0.0140000000000000003 < wj

   1. Initial program 74.6%

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

      1. distribute-rgt1-in 75.0%

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

      2. *-commutative 75.0%

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

      3. associate-/r* 75.0%

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

      4. div-sub 75.0%

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

      5. associate-/l* 75.0%

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

      6. *-inverses 100.0%

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

      7. *-rgt-identity 100.0%

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

   3. Simplified 100.0%

      \[\leadsto \color{blue}{wj - \frac{wj - \frac{x}{e^{wj}}}{wj + 1}} \]
   4. Add Preprocessing
3. Recombined 8 regimes into one program.

4. Final simplification 99.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;wj \leq -3.8 \cdot 10^{-6}:\\ \;\;\;\;\left(wj - \frac{wj}{wj + 1}\right) + \mathsf{fma}\left(\frac{-1}{wj + 1}, wj - \frac{x}{e^{wj}}, \frac{wj}{wj + 1}\right)\\ \mathbf{elif}\;wj \leq -7.8 \cdot 10^{-32}:\\ \;\;\;\;x + wj \cdot \left({wj}^{2} \cdot \left(\left(\left(-1 - \left(x \cdot -3 + \left(-2 \cdot \left(x \cdot -4 + x \cdot 1.5\right) + x \cdot 0.6666666666666666\right)\right)\right) + \frac{1}{wj}\right) - \left(-4 \cdot \frac{x}{wj} + 1.5 \cdot \frac{x}{wj}\right)\right) - x \cdot 2\right)\\ \mathbf{elif}\;wj \leq 5 \cdot 10^{-83}:\\ \;\;\;\;x - wj \cdot \left(x \cdot 2 - wj\right)\\ \mathbf{elif}\;wj \leq 3.7 \cdot 10^{-7}:\\ \;\;\;\;x - wj \cdot \left(x \cdot 2 + wj \cdot \left(wj + x \cdot \left(\left(\frac{-1}{x} - wj \cdot -2.6666666666666665\right) - 2.5\right)\right)\right)\\ \mathbf{elif}\;wj \leq 4.1 \cdot 10^{-5}:\\ \;\;\;\;x - wj \cdot \left(x \cdot 2 + \left(0 - {\left(e^{3 \cdot \log \left(wj \cdot \left(\mathsf{fma}\left(-1, \mathsf{fma}\left(wj, \mathsf{fma}\left(-3, x, \mathsf{fma}\left(-2, x \cdot -2.5, x \cdot 0.6666666666666666\right)\right), wj\right), 1\right) - x \cdot -2.5\right)\right)}\right)}^{0.3333333333333333}\right)\right)\\ \mathbf{elif}\;wj \leq 0.00055:\\ \;\;\;\;x \cdot \frac{e^{-wj}}{wj + 1}\\ \mathbf{elif}\;wj \leq 0.014:\\ \;\;\;\;\left(\sqrt{\frac{wj - \frac{x}{e^{wj}}}{wj + 1}} + \sqrt{wj}\right) \cdot \left(\sqrt{wj} - \sqrt{\frac{wj - \frac{x}{e^{wj}}}{wj + 1}}\right)\\ \mathbf{else}:\\ \;\;\;\;wj + \frac{wj - \frac{x}{e^{wj}}}{-1 - wj}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 99.5% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{wj}{wj + 1}\\ t_1 := wj - \frac{x}{e^{wj}}\\ t_2 := \sqrt{\frac{t\_1}{wj + 1}}\\ \mathbf{if}\;wj \leq -3.8 \cdot 10^{-6}:\\ \;\;\;\;\left(wj - t\_0\right) + \mathsf{fma}\left(\frac{-1}{wj + 1}, t\_1, t\_0\right)\\ \mathbf{elif}\;wj \leq -5 \cdot 10^{-32}:\\ \;\;\;\;x + wj \cdot \left({wj}^{2} \cdot \left(\left(\left(-1 - \left(x \cdot -3 + \left(-2 \cdot \left(x \cdot -4 + x \cdot 1.5\right) + x \cdot 0.6666666666666666\right)\right)\right) + \frac{1}{wj}\right) - \left(-4 \cdot \frac{x}{wj} + 1.5 \cdot \frac{x}{wj}\right)\right) - x \cdot 2\right)\\ \mathbf{elif}\;wj \leq 5 \cdot 10^{-83}:\\ \;\;\;\;x - wj \cdot \left(x \cdot 2 - wj\right)\\ \mathbf{elif}\;wj \leq 3.7 \cdot 10^{-7}:\\ \;\;\;\;x - wj \cdot \left(x \cdot 2 + wj \cdot \left(wj + x \cdot \left(\left(\frac{-1}{x} - wj \cdot -2.6666666666666665\right) - 2.5\right)\right)\right)\\ \mathbf{elif}\;wj \leq 0.000155:\\ \;\;\;\;x - \left(-wj\right) \cdot \left({\left({wj}^{3} \cdot {\left(1 - wj\right)}^{3}\right)}^{0.3333333333333333} + 2 \cdot \left(-x\right)\right)\\ \mathbf{elif}\;wj \leq 0.00058:\\ \;\;\;\;x \cdot \frac{e^{-wj}}{wj + 1}\\ \mathbf{elif}\;wj \leq 0.014:\\ \;\;\;\;\left(t\_2 + \sqrt{wj}\right) \cdot \left(\sqrt{wj} - t\_2\right)\\ \mathbf{else}:\\ \;\;\;\;wj + \frac{t\_1}{-1 - wj}\\ \end{array} \end{array} \]

FPCore

(FPCore (wj x)
 :precision binary64
 (let* ((t_0 (/ wj (+ wj 1.0)))
        (t_1 (- wj (/ x (exp wj))))
        (t_2 (sqrt (/ t_1 (+ wj 1.0)))))
   (if (<= wj -3.8e-6)
     (+ (- wj t_0) (fma (/ -1.0 (+ wj 1.0)) t_1 t_0))
     (if (<= wj -5e-32)
       (+
        x
        (*
         wj
         (-
          (*
           (pow wj 2.0)
           (-
            (+
             (-
              -1.0
              (+
               (* x -3.0)
               (+ (* -2.0 (+ (* x -4.0) (* x 1.5))) (* x 0.6666666666666666))))
             (/ 1.0 wj))
            (+ (* -4.0 (/ x wj)) (* 1.5 (/ x wj)))))
          (* x 2.0))))
       (if (<= wj 5e-83)
         (- x (* wj (- (* x 2.0) wj)))
         (if (<= wj 3.7e-7)
           (-
            x
            (*
             wj
             (+
              (* x 2.0)
              (*
               wj
               (+
                wj
                (* x (- (- (/ -1.0 x) (* wj -2.6666666666666665)) 2.5)))))))
           (if (<= wj 0.000155)
             (-
              x
              (*
               (- wj)
               (+
                (pow (* (pow wj 3.0) (pow (- 1.0 wj) 3.0)) 0.3333333333333333)
                (* 2.0 (- x)))))
             (if (<= wj 0.00058)
               (* x (/ (exp (- wj)) (+ wj 1.0)))
               (if (<= wj 0.014)
                 (* (+ t_2 (sqrt wj)) (- (sqrt wj) t_2))
                 (+ wj (/ t_1 (- -1.0 wj))))))))))))

C

double code(double wj, double x) {
	double t_0 = wj / (wj + 1.0);
	double t_1 = wj - (x / exp(wj));
	double t_2 = sqrt((t_1 / (wj + 1.0)));
	double tmp;
	if (wj <= -3.8e-6) {
		tmp = (wj - t_0) + fma((-1.0 / (wj + 1.0)), t_1, t_0);
	} else if (wj <= -5e-32) {
		tmp = x + (wj * ((pow(wj, 2.0) * (((-1.0 - ((x * -3.0) + ((-2.0 * ((x * -4.0) + (x * 1.5))) + (x * 0.6666666666666666)))) + (1.0 / wj)) - ((-4.0 * (x / wj)) + (1.5 * (x / wj))))) - (x * 2.0)));
	} else if (wj <= 5e-83) {
		tmp = x - (wj * ((x * 2.0) - wj));
	} else if (wj <= 3.7e-7) {
		tmp = x - (wj * ((x * 2.0) + (wj * (wj + (x * (((-1.0 / x) - (wj * -2.6666666666666665)) - 2.5))))));
	} else if (wj <= 0.000155) {
		tmp = x - (-wj * (pow((pow(wj, 3.0) * pow((1.0 - wj), 3.0)), 0.3333333333333333) + (2.0 * -x)));
	} else if (wj <= 0.00058) {
		tmp = x * (exp(-wj) / (wj + 1.0));
	} else if (wj <= 0.014) {
		tmp = (t_2 + sqrt(wj)) * (sqrt(wj) - t_2);
	} else {
		tmp = wj + (t_1 / (-1.0 - wj));
	}
	return tmp;
}

Julia

function code(wj, x)
	t_0 = Float64(wj / Float64(wj + 1.0))
	t_1 = Float64(wj - Float64(x / exp(wj)))
	t_2 = sqrt(Float64(t_1 / Float64(wj + 1.0)))
	tmp = 0.0
	if (wj <= -3.8e-6)
		tmp = Float64(Float64(wj - t_0) + fma(Float64(-1.0 / Float64(wj + 1.0)), t_1, t_0));
	elseif (wj <= -5e-32)
		tmp = Float64(x + Float64(wj * Float64(Float64((wj ^ 2.0) * Float64(Float64(Float64(-1.0 - Float64(Float64(x * -3.0) + Float64(Float64(-2.0 * Float64(Float64(x * -4.0) + Float64(x * 1.5))) + Float64(x * 0.6666666666666666)))) + Float64(1.0 / wj)) - Float64(Float64(-4.0 * Float64(x / wj)) + Float64(1.5 * Float64(x / wj))))) - Float64(x * 2.0))));
	elseif (wj <= 5e-83)
		tmp = Float64(x - Float64(wj * Float64(Float64(x * 2.0) - wj)));
	elseif (wj <= 3.7e-7)
		tmp = Float64(x - Float64(wj * Float64(Float64(x * 2.0) + Float64(wj * Float64(wj + Float64(x * Float64(Float64(Float64(-1.0 / x) - Float64(wj * -2.6666666666666665)) - 2.5)))))));
	elseif (wj <= 0.000155)
		tmp = Float64(x - Float64(Float64(-wj) * Float64((Float64((wj ^ 3.0) * (Float64(1.0 - wj) ^ 3.0)) ^ 0.3333333333333333) + Float64(2.0 * Float64(-x)))));
	elseif (wj <= 0.00058)
		tmp = Float64(x * Float64(exp(Float64(-wj)) / Float64(wj + 1.0)));
	elseif (wj <= 0.014)
		tmp = Float64(Float64(t_2 + sqrt(wj)) * Float64(sqrt(wj) - t_2));
	else
		tmp = Float64(wj + Float64(t_1 / Float64(-1.0 - wj)));
	end
	return tmp
end

Wolfram

code[wj_, x_] := Block[{t$95$0 = N[(wj / N[(wj + 1.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(wj - N[(x / N[Exp[wj], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[Sqrt[N[(t$95$1 / N[(wj + 1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[wj, -3.8e-6], N[(N[(wj - t$95$0), $MachinePrecision] + N[(N[(-1.0 / N[(wj + 1.0), $MachinePrecision]), $MachinePrecision] * t$95$1 + t$95$0), $MachinePrecision]), $MachinePrecision], If[LessEqual[wj, -5e-32], N[(x + N[(wj * N[(N[(N[Power[wj, 2.0], $MachinePrecision] * N[(N[(N[(-1.0 - N[(N[(x * -3.0), $MachinePrecision] + N[(N[(-2.0 * N[(N[(x * -4.0), $MachinePrecision] + N[(x * 1.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(x * 0.6666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(1.0 / wj), $MachinePrecision]), $MachinePrecision] - N[(N[(-4.0 * N[(x / wj), $MachinePrecision]), $MachinePrecision] + N[(1.5 * N[(x / wj), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(x * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[wj, 5e-83], N[(x - N[(wj * N[(N[(x * 2.0), $MachinePrecision] - wj), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[wj, 3.7e-7], N[(x - N[(wj * N[(N[(x * 2.0), $MachinePrecision] + N[(wj * N[(wj + N[(x * N[(N[(N[(-1.0 / x), $MachinePrecision] - N[(wj * -2.6666666666666665), $MachinePrecision]), $MachinePrecision] - 2.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[wj, 0.000155], N[(x - N[((-wj) * N[(N[Power[N[(N[Power[wj, 3.0], $MachinePrecision] * N[Power[N[(1.0 - wj), $MachinePrecision], 3.0], $MachinePrecision]), $MachinePrecision], 0.3333333333333333], $MachinePrecision] + N[(2.0 * (-x)), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[wj, 0.00058], N[(x * N[(N[Exp[(-wj)], $MachinePrecision] / N[(wj + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[wj, 0.014], N[(N[(t$95$2 + N[Sqrt[wj], $MachinePrecision]), $MachinePrecision] * N[(N[Sqrt[wj], $MachinePrecision] - t$95$2), $MachinePrecision]), $MachinePrecision], N[(wj + N[(t$95$1 / N[(-1.0 - wj), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]]]]]

Derivation

1. Split input into 8 regimes

2. if wj < -3.8e-6

   1. Initial program 59.6%

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

      1. distribute-rgt1-in 97.1%

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

      2. *-commutative 97.1%

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

      3. associate-/r* 97.1%

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

      4. div-sub 59.6%

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

      5. associate-/l* 59.6%

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

      6. *-inverses 97.1%

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

      7. *-rgt-identity 97.1%

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

   3. Simplified 97.1%

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

      1. *-un-lft-identity 97.1%

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

      2. div-inv 97.2%

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

      3. prod-diff 59.8%

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

      4. associate-/r/ 59.5%

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

      5. clear-num 59.6%

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

      6. fma-neg 59.6%

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

      7. *-un-lft-identity 59.6%

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

      8. associate-/r/ 59.6%

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

      9. clear-num 59.8%

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

   6. Applied egg-rr 59.8%

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

      1. distribute-neg-frac 59.8%

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

      2. metadata-eval 59.8%

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

   8. Simplified 59.8%

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

   9. Taylor expanded in x around 0 66.8%

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

       1. +-commutative 66.8%

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

   11. Simplified 66.8%

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

   12. Taylor expanded in x around 0 97.1%

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

       1. +-commutative 97.1%

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

   14. Simplified 97.1%

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

   if -3.8e-6 < wj < -5e-32

   1. Initial program 70.8%

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

      1. distribute-rgt1-in 70.7%

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

      2. *-commutative 70.7%

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

      3. associate-/r* 70.8%

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

      4. div-sub 70.8%

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

      5. associate-/l* 70.8%

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

      6. *-inverses 70.8%

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

      7. *-rgt-identity 70.8%

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

   3. Simplified 70.8%

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

   5. Taylor expanded in wj around 0 98.4%

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

   6. Taylor expanded in wj around inf 98.5%

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

   if -5e-32 < wj < 5e-83

   1. Initial program 81.3%

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

      1. distribute-rgt1-in 81.3%

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

      2. *-commutative 81.3%

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

      3. associate-/r* 81.3%

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

      4. div-sub 81.3%

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

      5. associate-/l* 81.3%

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

      6. *-inverses 81.3%

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

      7. *-rgt-identity 81.3%

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

   3. Simplified 81.3%

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

   5. Taylor expanded in wj around 0 99.6%

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

   6. Taylor expanded in x around 0 100.0%

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

      1. mul-1-neg 100.0%

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

      2. unsub-neg 100.0%

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

   8. Simplified 100.0%

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

   9. Taylor expanded in wj around 0 100.0%

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

   if 5e-83 < wj < 3.70000000000000004e-7

   1. Initial program 51.6%

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

      1. distribute-rgt1-in 51.6%

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

      2. *-commutative 51.6%

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

      3. associate-/r* 51.6%

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

      4. div-sub 51.6%

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

      5. associate-/l* 51.6%

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

      6. *-inverses 51.6%

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

      7. *-rgt-identity 51.6%

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

   3. Simplified 51.6%

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

   5. Taylor expanded in wj around 0 99.9%

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

   6. Taylor expanded in x around 0 99.9%

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

      1. distribute-lft-out 99.9%

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

      2. *-commutative 99.9%

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

      3. mul-1-neg 99.9%

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

   8. Simplified 99.9%

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

   9. Taylor expanded in x around 0 99.9%

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

   10. Taylor expanded in x around inf 100.0%

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

   if 3.70000000000000004e-7 < wj < 1.55e-4

   1. Initial program 68.1%

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

      1. distribute-rgt1-in 66.6%

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

      2. *-commutative 66.6%

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

      3. associate-/r* 69.7%

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

      4. div-sub 69.7%

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

      5. associate-/l* 69.7%

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

      6. *-inverses 69.7%

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

      7. *-rgt-identity 69.7%

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

   3. Simplified 69.7%

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

   5. Taylor expanded in wj around 0 85.0%

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

      1. add-cbrt-cube 85.0%

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

      2. pow1/3 85.0%

         \[\leadsto x + wj \cdot \left(\color{blue}{{\left(\left(\left(wj \cdot \left(\left(1 + -1 \cdot \left(wj \cdot \left(1 + \left(-3 \cdot x + \left(-2 \cdot \left(-4 \cdot x + 1.5 \cdot x\right) + 0.6666666666666666 \cdot x\right)\right)\right)\right)\right) - \left(-4 \cdot x + 1.5 \cdot x\right)\right)\right) \cdot \left(wj \cdot \left(\left(1 + -1 \cdot \left(wj \cdot \left(1 + \left(-3 \cdot x + \left(-2 \cdot \left(-4 \cdot x + 1.5 \cdot x\right) + 0.6666666666666666 \cdot x\right)\right)\right)\right)\right) - \left(-4 \cdot x + 1.5 \cdot x\right)\right)\right)\right) \cdot \left(wj \cdot \left(\left(1 + -1 \cdot \left(wj \cdot \left(1 + \left(-3 \cdot x + \left(-2 \cdot \left(-4 \cdot x + 1.5 \cdot x\right) + 0.6666666666666666 \cdot x\right)\right)\right)\right)\right) - \left(-4 \cdot x + 1.5 \cdot x\right)\right)\right)\right)}^{0.3333333333333333}} - 2 \cdot x\right) \]

   7. Applied egg-rr 85.0%

      \[\leadsto x + wj \cdot \left(\color{blue}{{\left({\left(wj \cdot \left(\mathsf{fma}\left(-1, wj \cdot \left(\mathsf{fma}\left(-3, x, \mathsf{fma}\left(-2, x \cdot -2.5, x \cdot 0.6666666666666666\right)\right) + 1\right), 1\right) - x \cdot -2.5\right)\right)}^{3}\right)}^{0.3333333333333333}} - 2 \cdot x\right) \]

   8. Taylor expanded in x around 0 85.0%

      \[\leadsto x + wj \cdot \left({\color{blue}{\left({wj}^{3} \cdot {\left(1 + -1 \cdot wj\right)}^{3}\right)}}^{0.3333333333333333} - 2 \cdot x\right) \]
   9. Step-by-step derivation

      1. neg-mul-1 85.0%

         \[\leadsto x + wj \cdot \left({\left({wj}^{3} \cdot {\left(1 + \color{blue}{\left(-wj\right)}\right)}^{3}\right)}^{0.3333333333333333} - 2 \cdot x\right) \]

      2. sub-neg 85.0%

         \[\leadsto x + wj \cdot \left({\left({wj}^{3} \cdot {\color{blue}{\left(1 - wj\right)}}^{3}\right)}^{0.3333333333333333} - 2 \cdot x\right) \]

   10. Simplified 85.0%

       \[\leadsto x + wj \cdot \left({\color{blue}{\left({wj}^{3} \cdot {\left(1 - wj\right)}^{3}\right)}}^{0.3333333333333333} - 2 \cdot x\right) \]

   if 1.55e-4 < wj < 5.8e-4

   1. Initial program 100.0%

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

      1. distribute-rgt1-in 98.4%

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

      2. *-commutative 98.4%

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

      3. associate-/r* 98.4%

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

      4. div-sub 98.4%

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

      5. associate-/l* 98.4%

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

      6. *-inverses 98.4%

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

      7. *-rgt-identity 98.4%

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

   3. Simplified 98.4%

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

   5. Taylor expanded in x around inf 98.4%

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

      1. +-commutative 98.4%

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

      2. associate-/r* 100.0%

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

      3. exp-neg 100.0%

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

      4. +-commutative 100.0%

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

   7. Simplified 100.0%

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

   8. Taylor expanded in x around inf 98.4%

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

      1. associate-/l* 100.0%

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

      2. +-commutative 100.0%

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

   10. Simplified 100.0%

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

   if 5.8e-4 < wj < 0.0140000000000000003

   1. Initial program 94.8%

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

      1. distribute-rgt1-in 94.8%

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

      2. *-commutative 94.8%

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

      3. associate-/r* 90.9%

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

      4. div-sub 90.9%

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

      5. associate-/l* 90.9%

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

      6. *-inverses 90.9%

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

      7. *-rgt-identity 90.9%

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

   3. Simplified 90.9%

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

      1. add-sqr-sqrt 90.9%

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

      2. add-sqr-sqrt 94.8%

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

      3. difference-of-squares 100.0%

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

   6. Applied egg-rr 100.0%

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

      1. +-commutative 100.0%

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

   8. Simplified 100.0%

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

   if 0.0140000000000000003 < wj

   1. Initial program 74.6%

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

      1. distribute-rgt1-in 75.0%

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

      2. *-commutative 75.0%

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

      3. associate-/r* 75.0%

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

      4. div-sub 75.0%

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

      5. associate-/l* 75.0%

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

      6. *-inverses 100.0%

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

      7. *-rgt-identity 100.0%

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

   3. Simplified 100.0%

      \[\leadsto \color{blue}{wj - \frac{wj - \frac{x}{e^{wj}}}{wj + 1}} \]
   4. Add Preprocessing
3. Recombined 8 regimes into one program.

4. Final simplification 99.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;wj \leq -3.8 \cdot 10^{-6}:\\ \;\;\;\;\left(wj - \frac{wj}{wj + 1}\right) + \mathsf{fma}\left(\frac{-1}{wj + 1}, wj - \frac{x}{e^{wj}}, \frac{wj}{wj + 1}\right)\\ \mathbf{elif}\;wj \leq -5 \cdot 10^{-32}:\\ \;\;\;\;x + wj \cdot \left({wj}^{2} \cdot \left(\left(\left(-1 - \left(x \cdot -3 + \left(-2 \cdot \left(x \cdot -4 + x \cdot 1.5\right) + x \cdot 0.6666666666666666\right)\right)\right) + \frac{1}{wj}\right) - \left(-4 \cdot \frac{x}{wj} + 1.5 \cdot \frac{x}{wj}\right)\right) - x \cdot 2\right)\\ \mathbf{elif}\;wj \leq 5 \cdot 10^{-83}:\\ \;\;\;\;x - wj \cdot \left(x \cdot 2 - wj\right)\\ \mathbf{elif}\;wj \leq 3.7 \cdot 10^{-7}:\\ \;\;\;\;x - wj \cdot \left(x \cdot 2 + wj \cdot \left(wj + x \cdot \left(\left(\frac{-1}{x} - wj \cdot -2.6666666666666665\right) - 2.5\right)\right)\right)\\ \mathbf{elif}\;wj \leq 0.000155:\\ \;\;\;\;x - \left(-wj\right) \cdot \left({\left({wj}^{3} \cdot {\left(1 - wj\right)}^{3}\right)}^{0.3333333333333333} + 2 \cdot \left(-x\right)\right)\\ \mathbf{elif}\;wj \leq 0.00058:\\ \;\;\;\;x \cdot \frac{e^{-wj}}{wj + 1}\\ \mathbf{elif}\;wj \leq 0.014:\\ \;\;\;\;\left(\sqrt{\frac{wj - \frac{x}{e^{wj}}}{wj + 1}} + \sqrt{wj}\right) \cdot \left(\sqrt{wj} - \sqrt{\frac{wj - \frac{x}{e^{wj}}}{wj + 1}}\right)\\ \mathbf{else}:\\ \;\;\;\;wj + \frac{wj - \frac{x}{e^{wj}}}{-1 - wj}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 99.5% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{wj}{wj + 1}\\ t_1 := \frac{-1}{wj + 1}\\ t_2 := wj - \frac{x}{e^{wj}}\\ \mathbf{if}\;wj \leq -3.8 \cdot 10^{-6}:\\ \;\;\;\;\left(wj - t\_0\right) + \mathsf{fma}\left(t\_1, t\_2, t\_0\right)\\ \mathbf{elif}\;wj \leq -5 \cdot 10^{-32}:\\ \;\;\;\;x + wj \cdot \left({wj}^{2} \cdot \left(\left(\left(-1 - \left(x \cdot -3 + \left(-2 \cdot \left(x \cdot -4 + x \cdot 1.5\right) + x \cdot 0.6666666666666666\right)\right)\right) + \frac{1}{wj}\right) - \left(-4 \cdot \frac{x}{wj} + 1.5 \cdot \frac{x}{wj}\right)\right) - x \cdot 2\right)\\ \mathbf{elif}\;wj \leq 5 \cdot 10^{-83}:\\ \;\;\;\;x - wj \cdot \left(x \cdot 2 - wj\right)\\ \mathbf{elif}\;wj \leq 3.7 \cdot 10^{-7}:\\ \;\;\;\;x - wj \cdot \left(x \cdot 2 + wj \cdot \left(wj + x \cdot \left(\left(\frac{-1}{x} - wj \cdot -2.6666666666666665\right) - 2.5\right)\right)\right)\\ \mathbf{elif}\;wj \leq 0.00029:\\ \;\;\;\;x - \left(-wj\right) \cdot \left({\left({wj}^{3} \cdot {\left(1 - wj\right)}^{3}\right)}^{0.3333333333333333} + 2 \cdot \left(-x\right)\right)\\ \mathbf{elif}\;wj \leq 0.00055:\\ \;\;\;\;x \cdot \frac{e^{-wj}}{wj + 1}\\ \mathbf{else}:\\ \;\;\;\;\left(wj - {\left(\frac{wj + 1}{t\_2}\right)}^{-1}\right) + \mathsf{fma}\left(t\_1, t\_2, \frac{t\_2}{wj + 1}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (wj x)
 :precision binary64
 (let* ((t_0 (/ wj (+ wj 1.0)))
        (t_1 (/ -1.0 (+ wj 1.0)))
        (t_2 (- wj (/ x (exp wj)))))
   (if (<= wj -3.8e-6)
     (+ (- wj t_0) (fma t_1 t_2 t_0))
     (if (<= wj -5e-32)
       (+
        x
        (*
         wj
         (-
          (*
           (pow wj 2.0)
           (-
            (+
             (-
              -1.0
              (+
               (* x -3.0)
               (+ (* -2.0 (+ (* x -4.0) (* x 1.5))) (* x 0.6666666666666666))))
             (/ 1.0 wj))
            (+ (* -4.0 (/ x wj)) (* 1.5 (/ x wj)))))
          (* x 2.0))))
       (if (<= wj 5e-83)
         (- x (* wj (- (* x 2.0) wj)))
         (if (<= wj 3.7e-7)
           (-
            x
            (*
             wj
             (+
              (* x 2.0)
              (*
               wj
               (+
                wj
                (* x (- (- (/ -1.0 x) (* wj -2.6666666666666665)) 2.5)))))))
           (if (<= wj 0.00029)
             (-
              x
              (*
               (- wj)
               (+
                (pow (* (pow wj 3.0) (pow (- 1.0 wj) 3.0)) 0.3333333333333333)
                (* 2.0 (- x)))))
             (if (<= wj 0.00055)
               (* x (/ (exp (- wj)) (+ wj 1.0)))
               (+
                (- wj (pow (/ (+ wj 1.0) t_2) -1.0))
                (fma t_1 t_2 (/ t_2 (+ wj 1.0))))))))))))

C

double code(double wj, double x) {
	double t_0 = wj / (wj + 1.0);
	double t_1 = -1.0 / (wj + 1.0);
	double t_2 = wj - (x / exp(wj));
	double tmp;
	if (wj <= -3.8e-6) {
		tmp = (wj - t_0) + fma(t_1, t_2, t_0);
	} else if (wj <= -5e-32) {
		tmp = x + (wj * ((pow(wj, 2.0) * (((-1.0 - ((x * -3.0) + ((-2.0 * ((x * -4.0) + (x * 1.5))) + (x * 0.6666666666666666)))) + (1.0 / wj)) - ((-4.0 * (x / wj)) + (1.5 * (x / wj))))) - (x * 2.0)));
	} else if (wj <= 5e-83) {
		tmp = x - (wj * ((x * 2.0) - wj));
	} else if (wj <= 3.7e-7) {
		tmp = x - (wj * ((x * 2.0) + (wj * (wj + (x * (((-1.0 / x) - (wj * -2.6666666666666665)) - 2.5))))));
	} else if (wj <= 0.00029) {
		tmp = x - (-wj * (pow((pow(wj, 3.0) * pow((1.0 - wj), 3.0)), 0.3333333333333333) + (2.0 * -x)));
	} else if (wj <= 0.00055) {
		tmp = x * (exp(-wj) / (wj + 1.0));
	} else {
		tmp = (wj - pow(((wj + 1.0) / t_2), -1.0)) + fma(t_1, t_2, (t_2 / (wj + 1.0)));
	}
	return tmp;
}

Julia

function code(wj, x)
	t_0 = Float64(wj / Float64(wj + 1.0))
	t_1 = Float64(-1.0 / Float64(wj + 1.0))
	t_2 = Float64(wj - Float64(x / exp(wj)))
	tmp = 0.0
	if (wj <= -3.8e-6)
		tmp = Float64(Float64(wj - t_0) + fma(t_1, t_2, t_0));
	elseif (wj <= -5e-32)
		tmp = Float64(x + Float64(wj * Float64(Float64((wj ^ 2.0) * Float64(Float64(Float64(-1.0 - Float64(Float64(x * -3.0) + Float64(Float64(-2.0 * Float64(Float64(x * -4.0) + Float64(x * 1.5))) + Float64(x * 0.6666666666666666)))) + Float64(1.0 / wj)) - Float64(Float64(-4.0 * Float64(x / wj)) + Float64(1.5 * Float64(x / wj))))) - Float64(x * 2.0))));
	elseif (wj <= 5e-83)
		tmp = Float64(x - Float64(wj * Float64(Float64(x * 2.0) - wj)));
	elseif (wj <= 3.7e-7)
		tmp = Float64(x - Float64(wj * Float64(Float64(x * 2.0) + Float64(wj * Float64(wj + Float64(x * Float64(Float64(Float64(-1.0 / x) - Float64(wj * -2.6666666666666665)) - 2.5)))))));
	elseif (wj <= 0.00029)
		tmp = Float64(x - Float64(Float64(-wj) * Float64((Float64((wj ^ 3.0) * (Float64(1.0 - wj) ^ 3.0)) ^ 0.3333333333333333) + Float64(2.0 * Float64(-x)))));
	elseif (wj <= 0.00055)
		tmp = Float64(x * Float64(exp(Float64(-wj)) / Float64(wj + 1.0)));
	else
		tmp = Float64(Float64(wj - (Float64(Float64(wj + 1.0) / t_2) ^ -1.0)) + fma(t_1, t_2, Float64(t_2 / Float64(wj + 1.0))));
	end
	return tmp
end

Wolfram

code[wj_, x_] := Block[{t$95$0 = N[(wj / N[(wj + 1.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(-1.0 / N[(wj + 1.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(wj - N[(x / N[Exp[wj], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[wj, -3.8e-6], N[(N[(wj - t$95$0), $MachinePrecision] + N[(t$95$1 * t$95$2 + t$95$0), $MachinePrecision]), $MachinePrecision], If[LessEqual[wj, -5e-32], N[(x + N[(wj * N[(N[(N[Power[wj, 2.0], $MachinePrecision] * N[(N[(N[(-1.0 - N[(N[(x * -3.0), $MachinePrecision] + N[(N[(-2.0 * N[(N[(x * -4.0), $MachinePrecision] + N[(x * 1.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(x * 0.6666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(1.0 / wj), $MachinePrecision]), $MachinePrecision] - N[(N[(-4.0 * N[(x / wj), $MachinePrecision]), $MachinePrecision] + N[(1.5 * N[(x / wj), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(x * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[wj, 5e-83], N[(x - N[(wj * N[(N[(x * 2.0), $MachinePrecision] - wj), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[wj, 3.7e-7], N[(x - N[(wj * N[(N[(x * 2.0), $MachinePrecision] + N[(wj * N[(wj + N[(x * N[(N[(N[(-1.0 / x), $MachinePrecision] - N[(wj * -2.6666666666666665), $MachinePrecision]), $MachinePrecision] - 2.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[wj, 0.00029], N[(x - N[((-wj) * N[(N[Power[N[(N[Power[wj, 3.0], $MachinePrecision] * N[Power[N[(1.0 - wj), $MachinePrecision], 3.0], $MachinePrecision]), $MachinePrecision], 0.3333333333333333], $MachinePrecision] + N[(2.0 * (-x)), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[wj, 0.00055], N[(x * N[(N[Exp[(-wj)], $MachinePrecision] / N[(wj + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(wj - N[Power[N[(N[(wj + 1.0), $MachinePrecision] / t$95$2), $MachinePrecision], -1.0], $MachinePrecision]), $MachinePrecision] + N[(t$95$1 * t$95$2 + N[(t$95$2 / N[(wj + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]]]]

Derivation

1. Split input into 7 regimes

2. if wj < -3.8e-6

   1. Initial program 59.6%

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

      1. distribute-rgt1-in 97.1%

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

      2. *-commutative 97.1%

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

      3. associate-/r* 97.1%

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

      4. div-sub 59.6%

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

      5. associate-/l* 59.6%

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

      6. *-inverses 97.1%

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

      7. *-rgt-identity 97.1%

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

   3. Simplified 97.1%

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

      1. *-un-lft-identity 97.1%

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

      2. div-inv 97.2%

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

      3. prod-diff 59.8%

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

      4. associate-/r/ 59.5%

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

      5. clear-num 59.6%

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

      6. fma-neg 59.6%

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

      7. *-un-lft-identity 59.6%

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

      8. associate-/r/ 59.6%

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

      9. clear-num 59.8%

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

   6. Applied egg-rr 59.8%

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

      1. distribute-neg-frac 59.8%

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

      2. metadata-eval 59.8%

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

   8. Simplified 59.8%

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

   9. Taylor expanded in x around 0 66.8%

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

       1. +-commutative 66.8%

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

   11. Simplified 66.8%

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

   12. Taylor expanded in x around 0 97.1%

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

       1. +-commutative 97.1%

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

   14. Simplified 97.1%

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

   if -3.8e-6 < wj < -5e-32

   1. Initial program 70.8%

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

      1. distribute-rgt1-in 70.7%

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

      2. *-commutative 70.7%

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

      3. associate-/r* 70.8%

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

      4. div-sub 70.8%

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

      5. associate-/l* 70.8%

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

      6. *-inverses 70.8%

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

      7. *-rgt-identity 70.8%

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

   3. Simplified 70.8%

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

   5. Taylor expanded in wj around 0 98.4%

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

   6. Taylor expanded in wj around inf 98.5%

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

   if -5e-32 < wj < 5e-83

   1. Initial program 81.3%

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

      1. distribute-rgt1-in 81.3%

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

      2. *-commutative 81.3%

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

      3. associate-/r* 81.3%

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

      4. div-sub 81.3%

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

      5. associate-/l* 81.3%

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

      6. *-inverses 81.3%

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

      7. *-rgt-identity 81.3%

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

   3. Simplified 81.3%

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

   5. Taylor expanded in wj around 0 99.6%

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

   6. Taylor expanded in x around 0 100.0%

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

      1. mul-1-neg 100.0%

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

      2. unsub-neg 100.0%

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

   8. Simplified 100.0%

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

   9. Taylor expanded in wj around 0 100.0%

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

   if 5e-83 < wj < 3.70000000000000004e-7

   1. Initial program 51.6%

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

      1. distribute-rgt1-in 51.6%

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

      2. *-commutative 51.6%

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

      3. associate-/r* 51.6%

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

      4. div-sub 51.6%

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

      5. associate-/l* 51.6%

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

      6. *-inverses 51.6%

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

      7. *-rgt-identity 51.6%

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

   3. Simplified 51.6%

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

   5. Taylor expanded in wj around 0 99.9%

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

   6. Taylor expanded in x around 0 99.9%

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

      1. distribute-lft-out 99.9%

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

      2. *-commutative 99.9%

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

      3. mul-1-neg 99.9%

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

   8. Simplified 99.9%

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

   9. Taylor expanded in x around 0 99.9%

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

   10. Taylor expanded in x around inf 100.0%

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

   if 3.70000000000000004e-7 < wj < 2.9e-4

   1. Initial program 68.1%

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

      1. distribute-rgt1-in 66.6%

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

      2. *-commutative 66.6%

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

      3. associate-/r* 69.7%

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

      4. div-sub 69.7%

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

      5. associate-/l* 69.7%

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

      6. *-inverses 69.7%

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

      7. *-rgt-identity 69.7%

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

   3. Simplified 69.7%

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

   5. Taylor expanded in wj around 0 85.0%

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

      1. add-cbrt-cube 85.0%

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

      2. pow1/3 85.0%

         \[\leadsto x + wj \cdot \left(\color{blue}{{\left(\left(\left(wj \cdot \left(\left(1 + -1 \cdot \left(wj \cdot \left(1 + \left(-3 \cdot x + \left(-2 \cdot \left(-4 \cdot x + 1.5 \cdot x\right) + 0.6666666666666666 \cdot x\right)\right)\right)\right)\right) - \left(-4 \cdot x + 1.5 \cdot x\right)\right)\right) \cdot \left(wj \cdot \left(\left(1 + -1 \cdot \left(wj \cdot \left(1 + \left(-3 \cdot x + \left(-2 \cdot \left(-4 \cdot x + 1.5 \cdot x\right) + 0.6666666666666666 \cdot x\right)\right)\right)\right)\right) - \left(-4 \cdot x + 1.5 \cdot x\right)\right)\right)\right) \cdot \left(wj \cdot \left(\left(1 + -1 \cdot \left(wj \cdot \left(1 + \left(-3 \cdot x + \left(-2 \cdot \left(-4 \cdot x + 1.5 \cdot x\right) + 0.6666666666666666 \cdot x\right)\right)\right)\right)\right) - \left(-4 \cdot x + 1.5 \cdot x\right)\right)\right)\right)}^{0.3333333333333333}} - 2 \cdot x\right) \]

   7. Applied egg-rr 85.0%

      \[\leadsto x + wj \cdot \left(\color{blue}{{\left({\left(wj \cdot \left(\mathsf{fma}\left(-1, wj \cdot \left(\mathsf{fma}\left(-3, x, \mathsf{fma}\left(-2, x \cdot -2.5, x \cdot 0.6666666666666666\right)\right) + 1\right), 1\right) - x \cdot -2.5\right)\right)}^{3}\right)}^{0.3333333333333333}} - 2 \cdot x\right) \]

   8. Taylor expanded in x around 0 85.0%

      \[\leadsto x + wj \cdot \left({\color{blue}{\left({wj}^{3} \cdot {\left(1 + -1 \cdot wj\right)}^{3}\right)}}^{0.3333333333333333} - 2 \cdot x\right) \]
   9. Step-by-step derivation

      1. neg-mul-1 85.0%

         \[\leadsto x + wj \cdot \left({\left({wj}^{3} \cdot {\left(1 + \color{blue}{\left(-wj\right)}\right)}^{3}\right)}^{0.3333333333333333} - 2 \cdot x\right) \]

      2. sub-neg 85.0%

         \[\leadsto x + wj \cdot \left({\left({wj}^{3} \cdot {\color{blue}{\left(1 - wj\right)}}^{3}\right)}^{0.3333333333333333} - 2 \cdot x\right) \]

   10. Simplified 85.0%

       \[\leadsto x + wj \cdot \left({\color{blue}{\left({wj}^{3} \cdot {\left(1 - wj\right)}^{3}\right)}}^{0.3333333333333333} - 2 \cdot x\right) \]

   if 2.9e-4 < wj < 5.50000000000000033e-4

   1. Initial program 100.0%

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

      1. distribute-rgt1-in 98.4%

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

      2. *-commutative 98.4%

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

      3. associate-/r* 98.4%

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

      4. div-sub 98.4%

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

      5. associate-/l* 98.4%

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

      6. *-inverses 98.4%

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

      7. *-rgt-identity 98.4%

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

   3. Simplified 98.4%

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

   5. Taylor expanded in x around inf 98.4%

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

      1. +-commutative 98.4%

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

      2. associate-/r* 100.0%

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

      3. exp-neg 100.0%

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

      4. +-commutative 100.0%

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

   7. Simplified 100.0%

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

   8. Taylor expanded in x around inf 98.4%

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

      1. associate-/l* 100.0%

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

      2. +-commutative 100.0%

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

   10. Simplified 100.0%

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

   if 5.50000000000000033e-4 < wj

   1. Initial program 78.6%

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

      1. distribute-rgt1-in 79.0%

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

      2. *-commutative 79.0%

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

      3. associate-/r* 78.2%

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

      4. div-sub 78.2%

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

      5. associate-/l* 78.2%

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

      6. *-inverses 98.2%

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

      7. *-rgt-identity 98.2%

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

   3. Simplified 98.2%

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

      1. *-un-lft-identity 98.2%

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

      2. div-inv 98.2%

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

      3. prod-diff 98.1%

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

      4. associate-/r/ 99.7%

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

      5. clear-num 98.1%

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

      6. fma-neg 98.1%

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

      7. *-un-lft-identity 98.1%

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

      8. associate-/r/ 97.8%

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

      9. clear-num 98.1%

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

   6. Applied egg-rr 98.1%

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

      1. distribute-neg-frac 98.1%

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

      2. metadata-eval 98.1%

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

   8. Simplified 98.1%

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

      1. clear-num 99.7%

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

      2. inv-pow 99.7%

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

   10. Applied egg-rr 99.7%

       \[\leadsto \left(wj - \color{blue}{{\left(\frac{wj + 1}{wj - \frac{x}{e^{wj}}}\right)}^{-1}}\right) + \mathsf{fma}\left(\frac{-1}{wj + 1}, wj - \frac{x}{e^{wj}}, \frac{wj - \frac{x}{e^{wj}}}{wj + 1}\right) \]
3. Recombined 7 regimes into one program.

4. Final simplification 99.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;wj \leq -3.8 \cdot 10^{-6}:\\ \;\;\;\;\left(wj - \frac{wj}{wj + 1}\right) + \mathsf{fma}\left(\frac{-1}{wj + 1}, wj - \frac{x}{e^{wj}}, \frac{wj}{wj + 1}\right)\\ \mathbf{elif}\;wj \leq -5 \cdot 10^{-32}:\\ \;\;\;\;x + wj \cdot \left({wj}^{2} \cdot \left(\left(\left(-1 - \left(x \cdot -3 + \left(-2 \cdot \left(x \cdot -4 + x \cdot 1.5\right) + x \cdot 0.6666666666666666\right)\right)\right) + \frac{1}{wj}\right) - \left(-4 \cdot \frac{x}{wj} + 1.5 \cdot \frac{x}{wj}\right)\right) - x \cdot 2\right)\\ \mathbf{elif}\;wj \leq 5 \cdot 10^{-83}:\\ \;\;\;\;x - wj \cdot \left(x \cdot 2 - wj\right)\\ \mathbf{elif}\;wj \leq 3.7 \cdot 10^{-7}:\\ \;\;\;\;x - wj \cdot \left(x \cdot 2 + wj \cdot \left(wj + x \cdot \left(\left(\frac{-1}{x} - wj \cdot -2.6666666666666665\right) - 2.5\right)\right)\right)\\ \mathbf{elif}\;wj \leq 0.00029:\\ \;\;\;\;x - \left(-wj\right) \cdot \left({\left({wj}^{3} \cdot {\left(1 - wj\right)}^{3}\right)}^{0.3333333333333333} + 2 \cdot \left(-x\right)\right)\\ \mathbf{elif}\;wj \leq 0.00055:\\ \;\;\;\;x \cdot \frac{e^{-wj}}{wj + 1}\\ \mathbf{else}:\\ \;\;\;\;\left(wj - {\left(\frac{wj + 1}{wj - \frac{x}{e^{wj}}}\right)}^{-1}\right) + \mathsf{fma}\left(\frac{-1}{wj + 1}, wj - \frac{x}{e^{wj}}, \frac{wj - \frac{x}{e^{wj}}}{wj + 1}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 99.5% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{wj}{wj + 1}\\ t_1 := wj - \frac{x}{e^{wj}}\\ t_2 := \mathsf{fma}\left(\frac{-1}{wj + 1}, t\_1, t\_0\right)\\ \mathbf{if}\;wj \leq -3.8 \cdot 10^{-6}:\\ \;\;\;\;\left(wj - t\_0\right) + t\_2\\ \mathbf{elif}\;wj \leq -5 \cdot 10^{-32}:\\ \;\;\;\;x + wj \cdot \left({wj}^{2} \cdot \left(\left(\left(-1 - \left(x \cdot -3 + \left(-2 \cdot \left(x \cdot -4 + x \cdot 1.5\right) + x \cdot 0.6666666666666666\right)\right)\right) + \frac{1}{wj}\right) - \left(-4 \cdot \frac{x}{wj} + 1.5 \cdot \frac{x}{wj}\right)\right) - x \cdot 2\right)\\ \mathbf{elif}\;wj \leq 5 \cdot 10^{-83}:\\ \;\;\;\;x - wj \cdot \left(x \cdot 2 - wj\right)\\ \mathbf{elif}\;wj \leq 3.7 \cdot 10^{-7}:\\ \;\;\;\;x - wj \cdot \left(x \cdot 2 + wj \cdot \left(wj + x \cdot \left(\left(\frac{-1}{x} - wj \cdot -2.6666666666666665\right) - 2.5\right)\right)\right)\\ \mathbf{elif}\;wj \leq 0.00043:\\ \;\;\;\;x - \left(-wj\right) \cdot \left({\left({wj}^{3} \cdot {\left(1 - wj\right)}^{3}\right)}^{0.3333333333333333} + 2 \cdot \left(-x\right)\right)\\ \mathbf{elif}\;wj \leq 0.00056:\\ \;\;\;\;x \cdot \frac{e^{-wj}}{wj + 1}\\ \mathbf{elif}\;wj \leq 0.014:\\ \;\;\;\;t\_2 + \left(wj - {\left(\frac{wj + 1}{t\_1}\right)}^{-1}\right)\\ \mathbf{else}:\\ \;\;\;\;wj + \frac{t\_1}{-1 - wj}\\ \end{array} \end{array} \]

FPCore

(FPCore (wj x)
 :precision binary64
 (let* ((t_0 (/ wj (+ wj 1.0)))
        (t_1 (- wj (/ x (exp wj))))
        (t_2 (fma (/ -1.0 (+ wj 1.0)) t_1 t_0)))
   (if (<= wj -3.8e-6)
     (+ (- wj t_0) t_2)
     (if (<= wj -5e-32)
       (+
        x
        (*
         wj
         (-
          (*
           (pow wj 2.0)
           (-
            (+
             (-
              -1.0
              (+
               (* x -3.0)
               (+ (* -2.0 (+ (* x -4.0) (* x 1.5))) (* x 0.6666666666666666))))
             (/ 1.0 wj))
            (+ (* -4.0 (/ x wj)) (* 1.5 (/ x wj)))))
          (* x 2.0))))
       (if (<= wj 5e-83)
         (- x (* wj (- (* x 2.0) wj)))
         (if (<= wj 3.7e-7)
           (-
            x
            (*
             wj
             (+
              (* x 2.0)
              (*
               wj
               (+
                wj
                (* x (- (- (/ -1.0 x) (* wj -2.6666666666666665)) 2.5)))))))
           (if (<= wj 0.00043)
             (-
              x
              (*
               (- wj)
               (+
                (pow (* (pow wj 3.0) (pow (- 1.0 wj) 3.0)) 0.3333333333333333)
                (* 2.0 (- x)))))
             (if (<= wj 0.00056)
               (* x (/ (exp (- wj)) (+ wj 1.0)))
               (if (<= wj 0.014)
                 (+ t_2 (- wj (pow (/ (+ wj 1.0) t_1) -1.0)))
                 (+ wj (/ t_1 (- -1.0 wj))))))))))))

C

double code(double wj, double x) {
	double t_0 = wj / (wj + 1.0);
	double t_1 = wj - (x / exp(wj));
	double t_2 = fma((-1.0 / (wj + 1.0)), t_1, t_0);
	double tmp;
	if (wj <= -3.8e-6) {
		tmp = (wj - t_0) + t_2;
	} else if (wj <= -5e-32) {
		tmp = x + (wj * ((pow(wj, 2.0) * (((-1.0 - ((x * -3.0) + ((-2.0 * ((x * -4.0) + (x * 1.5))) + (x * 0.6666666666666666)))) + (1.0 / wj)) - ((-4.0 * (x / wj)) + (1.5 * (x / wj))))) - (x * 2.0)));
	} else if (wj <= 5e-83) {
		tmp = x - (wj * ((x * 2.0) - wj));
	} else if (wj <= 3.7e-7) {
		tmp = x - (wj * ((x * 2.0) + (wj * (wj + (x * (((-1.0 / x) - (wj * -2.6666666666666665)) - 2.5))))));
	} else if (wj <= 0.00043) {
		tmp = x - (-wj * (pow((pow(wj, 3.0) * pow((1.0 - wj), 3.0)), 0.3333333333333333) + (2.0 * -x)));
	} else if (wj <= 0.00056) {
		tmp = x * (exp(-wj) / (wj + 1.0));
	} else if (wj <= 0.014) {
		tmp = t_2 + (wj - pow(((wj + 1.0) / t_1), -1.0));
	} else {
		tmp = wj + (t_1 / (-1.0 - wj));
	}
	return tmp;
}

Julia

function code(wj, x)
	t_0 = Float64(wj / Float64(wj + 1.0))
	t_1 = Float64(wj - Float64(x / exp(wj)))
	t_2 = fma(Float64(-1.0 / Float64(wj + 1.0)), t_1, t_0)
	tmp = 0.0
	if (wj <= -3.8e-6)
		tmp = Float64(Float64(wj - t_0) + t_2);
	elseif (wj <= -5e-32)
		tmp = Float64(x + Float64(wj * Float64(Float64((wj ^ 2.0) * Float64(Float64(Float64(-1.0 - Float64(Float64(x * -3.0) + Float64(Float64(-2.0 * Float64(Float64(x * -4.0) + Float64(x * 1.5))) + Float64(x * 0.6666666666666666)))) + Float64(1.0 / wj)) - Float64(Float64(-4.0 * Float64(x / wj)) + Float64(1.5 * Float64(x / wj))))) - Float64(x * 2.0))));
	elseif (wj <= 5e-83)
		tmp = Float64(x - Float64(wj * Float64(Float64(x * 2.0) - wj)));
	elseif (wj <= 3.7e-7)
		tmp = Float64(x - Float64(wj * Float64(Float64(x * 2.0) + Float64(wj * Float64(wj + Float64(x * Float64(Float64(Float64(-1.0 / x) - Float64(wj * -2.6666666666666665)) - 2.5)))))));
	elseif (wj <= 0.00043)
		tmp = Float64(x - Float64(Float64(-wj) * Float64((Float64((wj ^ 3.0) * (Float64(1.0 - wj) ^ 3.0)) ^ 0.3333333333333333) + Float64(2.0 * Float64(-x)))));
	elseif (wj <= 0.00056)
		tmp = Float64(x * Float64(exp(Float64(-wj)) / Float64(wj + 1.0)));
	elseif (wj <= 0.014)
		tmp = Float64(t_2 + Float64(wj - (Float64(Float64(wj + 1.0) / t_1) ^ -1.0)));
	else
		tmp = Float64(wj + Float64(t_1 / Float64(-1.0 - wj)));
	end
	return tmp
end

Wolfram

code[wj_, x_] := Block[{t$95$0 = N[(wj / N[(wj + 1.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(wj - N[(x / N[Exp[wj], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(-1.0 / N[(wj + 1.0), $MachinePrecision]), $MachinePrecision] * t$95$1 + t$95$0), $MachinePrecision]}, If[LessEqual[wj, -3.8e-6], N[(N[(wj - t$95$0), $MachinePrecision] + t$95$2), $MachinePrecision], If[LessEqual[wj, -5e-32], N[(x + N[(wj * N[(N[(N[Power[wj, 2.0], $MachinePrecision] * N[(N[(N[(-1.0 - N[(N[(x * -3.0), $MachinePrecision] + N[(N[(-2.0 * N[(N[(x * -4.0), $MachinePrecision] + N[(x * 1.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(x * 0.6666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(1.0 / wj), $MachinePrecision]), $MachinePrecision] - N[(N[(-4.0 * N[(x / wj), $MachinePrecision]), $MachinePrecision] + N[(1.5 * N[(x / wj), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(x * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[wj, 5e-83], N[(x - N[(wj * N[(N[(x * 2.0), $MachinePrecision] - wj), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[wj, 3.7e-7], N[(x - N[(wj * N[(N[(x * 2.0), $MachinePrecision] + N[(wj * N[(wj + N[(x * N[(N[(N[(-1.0 / x), $MachinePrecision] - N[(wj * -2.6666666666666665), $MachinePrecision]), $MachinePrecision] - 2.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[wj, 0.00043], N[(x - N[((-wj) * N[(N[Power[N[(N[Power[wj, 3.0], $MachinePrecision] * N[Power[N[(1.0 - wj), $MachinePrecision], 3.0], $MachinePrecision]), $MachinePrecision], 0.3333333333333333], $MachinePrecision] + N[(2.0 * (-x)), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[wj, 0.00056], N[(x * N[(N[Exp[(-wj)], $MachinePrecision] / N[(wj + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[wj, 0.014], N[(t$95$2 + N[(wj - N[Power[N[(N[(wj + 1.0), $MachinePrecision] / t$95$1), $MachinePrecision], -1.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(wj + N[(t$95$1 / N[(-1.0 - wj), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]]]]]

Derivation

1. Split input into 8 regimes

2. if wj < -3.8e-6

   1. Initial program 59.6%

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

      1. distribute-rgt1-in 97.1%

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

      2. *-commutative 97.1%

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

      3. associate-/r* 97.1%

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

      4. div-sub 59.6%

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

      5. associate-/l* 59.6%

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

      6. *-inverses 97.1%

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

      7. *-rgt-identity 97.1%

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

   3. Simplified 97.1%

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

      1. *-un-lft-identity 97.1%

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

      2. div-inv 97.2%

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

      3. prod-diff 59.8%

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

      4. associate-/r/ 59.5%

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

      5. clear-num 59.6%

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

      6. fma-neg 59.6%

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

      7. *-un-lft-identity 59.6%

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

      8. associate-/r/ 59.6%

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

      9. clear-num 59.8%

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

   6. Applied egg-rr 59.8%

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

      1. distribute-neg-frac 59.8%

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

      2. metadata-eval 59.8%

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

   8. Simplified 59.8%

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

   9. Taylor expanded in x around 0 66.8%

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

       1. +-commutative 66.8%

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

   11. Simplified 66.8%

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

   12. Taylor expanded in x around 0 97.1%

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

       1. +-commutative 97.1%

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

   14. Simplified 97.1%

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

   if -3.8e-6 < wj < -5e-32

   1. Initial program 70.8%

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

      1. distribute-rgt1-in 70.7%

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

      2. *-commutative 70.7%

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

      3. associate-/r* 70.8%

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

      4. div-sub 70.8%

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

      5. associate-/l* 70.8%

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

      6. *-inverses 70.8%

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

      7. *-rgt-identity 70.8%

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

   3. Simplified 70.8%

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

   5. Taylor expanded in wj around 0 98.4%

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

   6. Taylor expanded in wj around inf 98.5%

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

   if -5e-32 < wj < 5e-83

   1. Initial program 81.3%

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

      1. distribute-rgt1-in 81.3%

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

      2. *-commutative 81.3%

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

      3. associate-/r* 81.3%

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

      4. div-sub 81.3%

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

      5. associate-/l* 81.3%

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

      6. *-inverses 81.3%

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

      7. *-rgt-identity 81.3%

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

   3. Simplified 81.3%

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

   5. Taylor expanded in wj around 0 99.6%

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

   6. Taylor expanded in x around 0 100.0%

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

      1. mul-1-neg 100.0%

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

      2. unsub-neg 100.0%

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

   8. Simplified 100.0%

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

   9. Taylor expanded in wj around 0 100.0%

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

   if 5e-83 < wj < 3.70000000000000004e-7

   1. Initial program 51.6%

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

      1. distribute-rgt1-in 51.6%

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

      2. *-commutative 51.6%

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

      3. associate-/r* 51.6%

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

      4. div-sub 51.6%

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

      5. associate-/l* 51.6%

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

      6. *-inverses 51.6%

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

      7. *-rgt-identity 51.6%

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

   3. Simplified 51.6%

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

   5. Taylor expanded in wj around 0 99.9%

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

   6. Taylor expanded in x around 0 99.9%

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

      1. distribute-lft-out 99.9%

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

      2. *-commutative 99.9%

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

      3. mul-1-neg 99.9%

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

   8. Simplified 99.9%

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

   9. Taylor expanded in x around 0 99.9%

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

   10. Taylor expanded in x around inf 100.0%

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

   if 3.70000000000000004e-7 < wj < 4.29999999999999989e-4

   1. Initial program 68.1%

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

      1. distribute-rgt1-in 66.6%

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

      2. *-commutative 66.6%

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

      3. associate-/r* 69.7%

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

      4. div-sub 69.7%

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

      5. associate-/l* 69.7%

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

      6. *-inverses 69.7%

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

      7. *-rgt-identity 69.7%

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

   3. Simplified 69.7%

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

   5. Taylor expanded in wj around 0 85.0%

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

      1. add-cbrt-cube 85.0%

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

      2. pow1/3 85.0%

         \[\leadsto x + wj \cdot \left(\color{blue}{{\left(\left(\left(wj \cdot \left(\left(1 + -1 \cdot \left(wj \cdot \left(1 + \left(-3 \cdot x + \left(-2 \cdot \left(-4 \cdot x + 1.5 \cdot x\right) + 0.6666666666666666 \cdot x\right)\right)\right)\right)\right) - \left(-4 \cdot x + 1.5 \cdot x\right)\right)\right) \cdot \left(wj \cdot \left(\left(1 + -1 \cdot \left(wj \cdot \left(1 + \left(-3 \cdot x + \left(-2 \cdot \left(-4 \cdot x + 1.5 \cdot x\right) + 0.6666666666666666 \cdot x\right)\right)\right)\right)\right) - \left(-4 \cdot x + 1.5 \cdot x\right)\right)\right)\right) \cdot \left(wj \cdot \left(\left(1 + -1 \cdot \left(wj \cdot \left(1 + \left(-3 \cdot x + \left(-2 \cdot \left(-4 \cdot x + 1.5 \cdot x\right) + 0.6666666666666666 \cdot x\right)\right)\right)\right)\right) - \left(-4 \cdot x + 1.5 \cdot x\right)\right)\right)\right)}^{0.3333333333333333}} - 2 \cdot x\right) \]

   7. Applied egg-rr 85.0%

      \[\leadsto x + wj \cdot \left(\color{blue}{{\left({\left(wj \cdot \left(\mathsf{fma}\left(-1, wj \cdot \left(\mathsf{fma}\left(-3, x, \mathsf{fma}\left(-2, x \cdot -2.5, x \cdot 0.6666666666666666\right)\right) + 1\right), 1\right) - x \cdot -2.5\right)\right)}^{3}\right)}^{0.3333333333333333}} - 2 \cdot x\right) \]

   8. Taylor expanded in x around 0 85.0%

      \[\leadsto x + wj \cdot \left({\color{blue}{\left({wj}^{3} \cdot {\left(1 + -1 \cdot wj\right)}^{3}\right)}}^{0.3333333333333333} - 2 \cdot x\right) \]
   9. Step-by-step derivation

      1. neg-mul-1 85.0%

         \[\leadsto x + wj \cdot \left({\left({wj}^{3} \cdot {\left(1 + \color{blue}{\left(-wj\right)}\right)}^{3}\right)}^{0.3333333333333333} - 2 \cdot x\right) \]

      2. sub-neg 85.0%

         \[\leadsto x + wj \cdot \left({\left({wj}^{3} \cdot {\color{blue}{\left(1 - wj\right)}}^{3}\right)}^{0.3333333333333333} - 2 \cdot x\right) \]

   10. Simplified 85.0%

       \[\leadsto x + wj \cdot \left({\color{blue}{\left({wj}^{3} \cdot {\left(1 - wj\right)}^{3}\right)}}^{0.3333333333333333} - 2 \cdot x\right) \]

   if 4.29999999999999989e-4 < wj < 5.5999999999999995e-4

   1. Initial program 100.0%

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

      1. distribute-rgt1-in 98.4%

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

      2. *-commutative 98.4%

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

      3. associate-/r* 98.4%

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

      4. div-sub 98.4%

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

      5. associate-/l* 98.4%

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

      6. *-inverses 98.4%

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

      7. *-rgt-identity 98.4%

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

   3. Simplified 98.4%

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

   5. Taylor expanded in x around inf 98.4%

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

      1. +-commutative 98.4%

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

      2. associate-/r* 100.0%

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

      3. exp-neg 100.0%

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

      4. +-commutative 100.0%

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

   7. Simplified 100.0%

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

   8. Taylor expanded in x around inf 98.4%

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

      1. associate-/l* 100.0%

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

      2. +-commutative 100.0%

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

   10. Simplified 100.0%

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

   if 5.5999999999999995e-4 < wj < 0.0140000000000000003

   1. Initial program 94.8%

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

      1. distribute-rgt1-in 94.8%

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

      2. *-commutative 94.8%

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

      3. associate-/r* 90.9%

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

      4. div-sub 90.9%

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

      5. associate-/l* 90.9%

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

      6. *-inverses 90.9%

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

      7. *-rgt-identity 90.9%

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

   3. Simplified 90.9%

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

      1. *-un-lft-identity 90.9%

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

      2. div-inv 90.9%

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

      3. prod-diff 90.6%

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

      4. associate-/r/ 98.4%

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

      5. clear-num 90.6%

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

      6. fma-neg 90.6%

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

      7. *-un-lft-identity 90.6%

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

      8. associate-/r/ 89.0%

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

      9. clear-num 90.6%

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

   6. Applied egg-rr 90.6%

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

      1. distribute-neg-frac 90.6%

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

      2. metadata-eval 90.6%

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

   8. Simplified 90.6%

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

   9. Taylor expanded in x around 0 90.6%

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

       1. +-commutative 90.6%

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

   11. Simplified 90.6%

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

       1. clear-num 98.4%

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

       2. inv-pow 98.4%

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

   13. Applied egg-rr 98.4%

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

   if 0.0140000000000000003 < wj

   1. Initial program 74.6%

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

      1. distribute-rgt1-in 75.0%

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

      2. *-commutative 75.0%

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

      3. associate-/r* 75.0%

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

      4. div-sub 75.0%

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

      5. associate-/l* 75.0%

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

      6. *-inverses 100.0%

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

      7. *-rgt-identity 100.0%

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

   3. Simplified 100.0%

      \[\leadsto \color{blue}{wj - \frac{wj - \frac{x}{e^{wj}}}{wj + 1}} \]
   4. Add Preprocessing
3. Recombined 8 regimes into one program.

4. Final simplification 99.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;wj \leq -3.8 \cdot 10^{-6}:\\ \;\;\;\;\left(wj - \frac{wj}{wj + 1}\right) + \mathsf{fma}\left(\frac{-1}{wj + 1}, wj - \frac{x}{e^{wj}}, \frac{wj}{wj + 1}\right)\\ \mathbf{elif}\;wj \leq -5 \cdot 10^{-32}:\\ \;\;\;\;x + wj \cdot \left({wj}^{2} \cdot \left(\left(\left(-1 - \left(x \cdot -3 + \left(-2 \cdot \left(x \cdot -4 + x \cdot 1.5\right) + x \cdot 0.6666666666666666\right)\right)\right) + \frac{1}{wj}\right) - \left(-4 \cdot \frac{x}{wj} + 1.5 \cdot \frac{x}{wj}\right)\right) - x \cdot 2\right)\\ \mathbf{elif}\;wj \leq 5 \cdot 10^{-83}:\\ \;\;\;\;x - wj \cdot \left(x \cdot 2 - wj\right)\\ \mathbf{elif}\;wj \leq 3.7 \cdot 10^{-7}:\\ \;\;\;\;x - wj \cdot \left(x \cdot 2 + wj \cdot \left(wj + x \cdot \left(\left(\frac{-1}{x} - wj \cdot -2.6666666666666665\right) - 2.5\right)\right)\right)\\ \mathbf{elif}\;wj \leq 0.00043:\\ \;\;\;\;x - \left(-wj\right) \cdot \left({\left({wj}^{3} \cdot {\left(1 - wj\right)}^{3}\right)}^{0.3333333333333333} + 2 \cdot \left(-x\right)\right)\\ \mathbf{elif}\;wj \leq 0.00056:\\ \;\;\;\;x \cdot \frac{e^{-wj}}{wj + 1}\\ \mathbf{elif}\;wj \leq 0.014:\\ \;\;\;\;\mathsf{fma}\left(\frac{-1}{wj + 1}, wj - \frac{x}{e^{wj}}, \frac{wj}{wj + 1}\right) + \left(wj - {\left(\frac{wj + 1}{wj - \frac{x}{e^{wj}}}\right)}^{-1}\right)\\ \mathbf{else}:\\ \;\;\;\;wj + \frac{wj - \frac{x}{e^{wj}}}{-1 - wj}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 99.6% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{wj}{wj + 1}\\ t_1 := wj \cdot e^{wj}\\ t_2 := wj - \frac{x}{e^{wj}}\\ \mathbf{if}\;wj \leq -3.8 \cdot 10^{-6}:\\ \;\;\;\;\left(wj - t\_0\right) + \mathsf{fma}\left(\frac{-1}{wj + 1}, t\_2, t\_0\right)\\ \mathbf{elif}\;wj \leq -6 \cdot 10^{-32}:\\ \;\;\;\;x + wj \cdot \left({wj}^{2} \cdot \left(\left(\left(-1 - \left(x \cdot -3 + \left(-2 \cdot \left(x \cdot -4 + x \cdot 1.5\right) + x \cdot 0.6666666666666666\right)\right)\right) + \frac{1}{wj}\right) - \left(-4 \cdot \frac{x}{wj} + 1.5 \cdot \frac{x}{wj}\right)\right) - x \cdot 2\right)\\ \mathbf{elif}\;wj \leq 5 \cdot 10^{-83}:\\ \;\;\;\;x - wj \cdot \left(x \cdot 2 - wj\right)\\ \mathbf{elif}\;wj \leq 3.7 \cdot 10^{-7}:\\ \;\;\;\;x - wj \cdot \left(x \cdot 2 + wj \cdot \left(wj + x \cdot \left(\left(\frac{-1}{x} - wj \cdot -2.6666666666666665\right) - 2.5\right)\right)\right)\\ \mathbf{elif}\;wj \leq 3.9 \cdot 10^{-7}:\\ \;\;\;\;x - \left(-wj\right) \cdot \left({\left({wj}^{3} \cdot {\left(1 - wj\right)}^{3}\right)}^{0.3333333333333333} + 2 \cdot \left(-x\right)\right)\\ \mathbf{elif}\;wj \leq 0.68:\\ \;\;\;\;wj + \frac{x - t\_1}{e^{wj} + t\_1}\\ \mathbf{else}:\\ \;\;\;\;wj + \frac{t\_2}{-1 - wj}\\ \end{array} \end{array} \]

FPCore

(FPCore (wj x)
 :precision binary64
 (let* ((t_0 (/ wj (+ wj 1.0)))
        (t_1 (* wj (exp wj)))
        (t_2 (- wj (/ x (exp wj)))))
   (if (<= wj -3.8e-6)
     (+ (- wj t_0) (fma (/ -1.0 (+ wj 1.0)) t_2 t_0))
     (if (<= wj -6e-32)
       (+
        x
        (*
         wj
         (-
          (*
           (pow wj 2.0)
           (-
            (+
             (-
              -1.0
              (+
               (* x -3.0)
               (+ (* -2.0 (+ (* x -4.0) (* x 1.5))) (* x 0.6666666666666666))))
             (/ 1.0 wj))
            (+ (* -4.0 (/ x wj)) (* 1.5 (/ x wj)))))
          (* x 2.0))))
       (if (<= wj 5e-83)
         (- x (* wj (- (* x 2.0) wj)))
         (if (<= wj 3.7e-7)
           (-
            x
            (*
             wj
             (+
              (* x 2.0)
              (*
               wj
               (+
                wj
                (* x (- (- (/ -1.0 x) (* wj -2.6666666666666665)) 2.5)))))))
           (if (<= wj 3.9e-7)
             (-
              x
              (*
               (- wj)
               (+
                (pow (* (pow wj 3.0) (pow (- 1.0 wj) 3.0)) 0.3333333333333333)
                (* 2.0 (- x)))))
             (if (<= wj 0.68)
               (+ wj (/ (- x t_1) (+ (exp wj) t_1)))
               (+ wj (/ t_2 (- -1.0 wj)))))))))))

C

double code(double wj, double x) {
	double t_0 = wj / (wj + 1.0);
	double t_1 = wj * exp(wj);
	double t_2 = wj - (x / exp(wj));
	double tmp;
	if (wj <= -3.8e-6) {
		tmp = (wj - t_0) + fma((-1.0 / (wj + 1.0)), t_2, t_0);
	} else if (wj <= -6e-32) {
		tmp = x + (wj * ((pow(wj, 2.0) * (((-1.0 - ((x * -3.0) + ((-2.0 * ((x * -4.0) + (x * 1.5))) + (x * 0.6666666666666666)))) + (1.0 / wj)) - ((-4.0 * (x / wj)) + (1.5 * (x / wj))))) - (x * 2.0)));
	} else if (wj <= 5e-83) {
		tmp = x - (wj * ((x * 2.0) - wj));
	} else if (wj <= 3.7e-7) {
		tmp = x - (wj * ((x * 2.0) + (wj * (wj + (x * (((-1.0 / x) - (wj * -2.6666666666666665)) - 2.5))))));
	} else if (wj <= 3.9e-7) {
		tmp = x - (-wj * (pow((pow(wj, 3.0) * pow((1.0 - wj), 3.0)), 0.3333333333333333) + (2.0 * -x)));
	} else if (wj <= 0.68) {
		tmp = wj + ((x - t_1) / (exp(wj) + t_1));
	} else {
		tmp = wj + (t_2 / (-1.0 - wj));
	}
	return tmp;
}

Julia

function code(wj, x)
	t_0 = Float64(wj / Float64(wj + 1.0))
	t_1 = Float64(wj * exp(wj))
	t_2 = Float64(wj - Float64(x / exp(wj)))
	tmp = 0.0
	if (wj <= -3.8e-6)
		tmp = Float64(Float64(wj - t_0) + fma(Float64(-1.0 / Float64(wj + 1.0)), t_2, t_0));
	elseif (wj <= -6e-32)
		tmp = Float64(x + Float64(wj * Float64(Float64((wj ^ 2.0) * Float64(Float64(Float64(-1.0 - Float64(Float64(x * -3.0) + Float64(Float64(-2.0 * Float64(Float64(x * -4.0) + Float64(x * 1.5))) + Float64(x * 0.6666666666666666)))) + Float64(1.0 / wj)) - Float64(Float64(-4.0 * Float64(x / wj)) + Float64(1.5 * Float64(x / wj))))) - Float64(x * 2.0))));
	elseif (wj <= 5e-83)
		tmp = Float64(x - Float64(wj * Float64(Float64(x * 2.0) - wj)));
	elseif (wj <= 3.7e-7)
		tmp = Float64(x - Float64(wj * Float64(Float64(x * 2.0) + Float64(wj * Float64(wj + Float64(x * Float64(Float64(Float64(-1.0 / x) - Float64(wj * -2.6666666666666665)) - 2.5)))))));
	elseif (wj <= 3.9e-7)
		tmp = Float64(x - Float64(Float64(-wj) * Float64((Float64((wj ^ 3.0) * (Float64(1.0 - wj) ^ 3.0)) ^ 0.3333333333333333) + Float64(2.0 * Float64(-x)))));
	elseif (wj <= 0.68)
		tmp = Float64(wj + Float64(Float64(x - t_1) / Float64(exp(wj) + t_1)));
	else
		tmp = Float64(wj + Float64(t_2 / Float64(-1.0 - wj)));
	end
	return tmp
end

Wolfram

code[wj_, x_] := Block[{t$95$0 = N[(wj / N[(wj + 1.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(wj * N[Exp[wj], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(wj - N[(x / N[Exp[wj], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[wj, -3.8e-6], N[(N[(wj - t$95$0), $MachinePrecision] + N[(N[(-1.0 / N[(wj + 1.0), $MachinePrecision]), $MachinePrecision] * t$95$2 + t$95$0), $MachinePrecision]), $MachinePrecision], If[LessEqual[wj, -6e-32], N[(x + N[(wj * N[(N[(N[Power[wj, 2.0], $MachinePrecision] * N[(N[(N[(-1.0 - N[(N[(x * -3.0), $MachinePrecision] + N[(N[(-2.0 * N[(N[(x * -4.0), $MachinePrecision] + N[(x * 1.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(x * 0.6666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(1.0 / wj), $MachinePrecision]), $MachinePrecision] - N[(N[(-4.0 * N[(x / wj), $MachinePrecision]), $MachinePrecision] + N[(1.5 * N[(x / wj), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(x * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[wj, 5e-83], N[(x - N[(wj * N[(N[(x * 2.0), $MachinePrecision] - wj), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[wj, 3.7e-7], N[(x - N[(wj * N[(N[(x * 2.0), $MachinePrecision] + N[(wj * N[(wj + N[(x * N[(N[(N[(-1.0 / x), $MachinePrecision] - N[(wj * -2.6666666666666665), $MachinePrecision]), $MachinePrecision] - 2.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[wj, 3.9e-7], N[(x - N[((-wj) * N[(N[Power[N[(N[Power[wj, 3.0], $MachinePrecision] * N[Power[N[(1.0 - wj), $MachinePrecision], 3.0], $MachinePrecision]), $MachinePrecision], 0.3333333333333333], $MachinePrecision] + N[(2.0 * (-x)), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[wj, 0.68], N[(wj + N[(N[(x - t$95$1), $MachinePrecision] / N[(N[Exp[wj], $MachinePrecision] + t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(wj + N[(t$95$2 / N[(-1.0 - wj), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]]]]

Derivation

1. Split input into 7 regimes

2. if wj < -3.8e-6

   1. Initial program 59.6%

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

      1. distribute-rgt1-in 97.1%

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

      2. *-commutative 97.1%

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

      3. associate-/r* 97.1%

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

      4. div-sub 59.6%

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

      5. associate-/l* 59.6%

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

      6. *-inverses 97.1%

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

      7. *-rgt-identity 97.1%

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

   3. Simplified 97.1%

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

      1. *-un-lft-identity 97.1%

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

      2. div-inv 97.2%

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

      3. prod-diff 59.8%

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

      4. associate-/r/ 59.5%

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

      5. clear-num 59.6%

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

      6. fma-neg 59.6%

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

      7. *-un-lft-identity 59.6%

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

      8. associate-/r/ 59.6%

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

      9. clear-num 59.8%

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

   6. Applied egg-rr 59.8%

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

      1. distribute-neg-frac 59.8%

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

      2. metadata-eval 59.8%

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

   8. Simplified 59.8%

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

   9. Taylor expanded in x around 0 66.8%

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

       1. +-commutative 66.8%

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

   11. Simplified 66.8%

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

   12. Taylor expanded in x around 0 97.1%

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

       1. +-commutative 97.1%

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

   14. Simplified 97.1%

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

   if -3.8e-6 < wj < -6.0000000000000001e-32

   1. Initial program 70.8%

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

      1. distribute-rgt1-in 70.7%

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

      2. *-commutative 70.7%

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

      3. associate-/r* 70.8%

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

      4. div-sub 70.8%

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

      5. associate-/l* 70.8%

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

      6. *-inverses 70.8%

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

      7. *-rgt-identity 70.8%

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

   3. Simplified 70.8%

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

   5. Taylor expanded in wj around 0 98.4%

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

   6. Taylor expanded in wj around inf 98.5%

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

   if -6.0000000000000001e-32 < wj < 5e-83

   1. Initial program 81.3%

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

      1. distribute-rgt1-in 81.3%

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

      2. *-commutative 81.3%

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

      3. associate-/r* 81.3%

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

      4. div-sub 81.3%

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

      5. associate-/l* 81.3%

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

      6. *-inverses 81.3%

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

      7. *-rgt-identity 81.3%

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

   3. Simplified 81.3%

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

   5. Taylor expanded in wj around 0 99.6%

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

   6. Taylor expanded in x around 0 100.0%

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

      1. mul-1-neg 100.0%

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

      2. unsub-neg 100.0%

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

   8. Simplified 100.0%

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

   9. Taylor expanded in wj around 0 100.0%

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

   if 5e-83 < wj < 3.70000000000000004e-7

   1. Initial program 51.6%

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

      1. distribute-rgt1-in 51.6%

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

      2. *-commutative 51.6%

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

      3. associate-/r* 51.6%

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

      4. div-sub 51.6%

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

      5. associate-/l* 51.6%

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

      6. *-inverses 51.6%

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

      7. *-rgt-identity 51.6%

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

   3. Simplified 51.6%

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

   5. Taylor expanded in wj around 0 99.9%

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

   6. Taylor expanded in x around 0 99.9%

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

      1. distribute-lft-out 99.9%

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

      2. *-commutative 99.9%

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

      3. mul-1-neg 99.9%

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

   8. Simplified 99.9%

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

   9. Taylor expanded in x around 0 99.9%

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

   10. Taylor expanded in x around inf 100.0%

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

   if 3.70000000000000004e-7 < wj < 3.90000000000000025e-7

   1. Initial program 68.1%

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

      1. distribute-rgt1-in 66.6%

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

      2. *-commutative 66.6%

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

      3. associate-/r* 69.7%

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

      4. div-sub 69.7%

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

      5. associate-/l* 69.7%

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

      6. *-inverses 69.7%

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

      7. *-rgt-identity 69.7%

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

   3. Simplified 69.7%

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

   5. Taylor expanded in wj around 0 85.0%

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

      1. add-cbrt-cube 85.0%

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

      2. pow1/3 85.0%

         \[\leadsto x + wj \cdot \left(\color{blue}{{\left(\left(\left(wj \cdot \left(\left(1 + -1 \cdot \left(wj \cdot \left(1 + \left(-3 \cdot x + \left(-2 \cdot \left(-4 \cdot x + 1.5 \cdot x\right) + 0.6666666666666666 \cdot x\right)\right)\right)\right)\right) - \left(-4 \cdot x + 1.5 \cdot x\right)\right)\right) \cdot \left(wj \cdot \left(\left(1 + -1 \cdot \left(wj \cdot \left(1 + \left(-3 \cdot x + \left(-2 \cdot \left(-4 \cdot x + 1.5 \cdot x\right) + 0.6666666666666666 \cdot x\right)\right)\right)\right)\right) - \left(-4 \cdot x + 1.5 \cdot x\right)\right)\right)\right) \cdot \left(wj \cdot \left(\left(1 + -1 \cdot \left(wj \cdot \left(1 + \left(-3 \cdot x + \left(-2 \cdot \left(-4 \cdot x + 1.5 \cdot x\right) + 0.6666666666666666 \cdot x\right)\right)\right)\right)\right) - \left(-4 \cdot x + 1.5 \cdot x\right)\right)\right)\right)}^{0.3333333333333333}} - 2 \cdot x\right) \]

   7. Applied egg-rr 85.0%

      \[\leadsto x + wj \cdot \left(\color{blue}{{\left({\left(wj \cdot \left(\mathsf{fma}\left(-1, wj \cdot \left(\mathsf{fma}\left(-3, x, \mathsf{fma}\left(-2, x \cdot -2.5, x \cdot 0.6666666666666666\right)\right) + 1\right), 1\right) - x \cdot -2.5\right)\right)}^{3}\right)}^{0.3333333333333333}} - 2 \cdot x\right) \]

   8. Taylor expanded in x around 0 85.0%

      \[\leadsto x + wj \cdot \left({\color{blue}{\left({wj}^{3} \cdot {\left(1 + -1 \cdot wj\right)}^{3}\right)}}^{0.3333333333333333} - 2 \cdot x\right) \]
   9. Step-by-step derivation

      1. neg-mul-1 85.0%

         \[\leadsto x + wj \cdot \left({\left({wj}^{3} \cdot {\left(1 + \color{blue}{\left(-wj\right)}\right)}^{3}\right)}^{0.3333333333333333} - 2 \cdot x\right) \]

      2. sub-neg 85.0%

         \[\leadsto x + wj \cdot \left({\left({wj}^{3} \cdot {\color{blue}{\left(1 - wj\right)}}^{3}\right)}^{0.3333333333333333} - 2 \cdot x\right) \]

   10. Simplified 85.0%

       \[\leadsto x + wj \cdot \left({\color{blue}{\left({wj}^{3} \cdot {\left(1 - wj\right)}^{3}\right)}}^{0.3333333333333333} - 2 \cdot x\right) \]

   if 3.90000000000000025e-7 < wj < 0.680000000000000049

   1. Initial program 97.4%

      \[wj - \frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}} \]
   2. Add Preprocessing

   if 0.680000000000000049 < wj

   1. Initial program 74.6%

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

      1. distribute-rgt1-in 75.0%

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

      2. *-commutative 75.0%

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

      3. associate-/r* 75.0%

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

      4. div-sub 75.0%

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

      5. associate-/l* 75.0%

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

      6. *-inverses 100.0%

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

      7. *-rgt-identity 100.0%

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

   3. Simplified 100.0%

      \[\leadsto \color{blue}{wj - \frac{wj - \frac{x}{e^{wj}}}{wj + 1}} \]
   4. Add Preprocessing
3. Recombined 7 regimes into one program.

4. Final simplification 99.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;wj \leq -3.8 \cdot 10^{-6}:\\ \;\;\;\;\left(wj - \frac{wj}{wj + 1}\right) + \mathsf{fma}\left(\frac{-1}{wj + 1}, wj - \frac{x}{e^{wj}}, \frac{wj}{wj + 1}\right)\\ \mathbf{elif}\;wj \leq -6 \cdot 10^{-32}:\\ \;\;\;\;x + wj \cdot \left({wj}^{2} \cdot \left(\left(\left(-1 - \left(x \cdot -3 + \left(-2 \cdot \left(x \cdot -4 + x \cdot 1.5\right) + x \cdot 0.6666666666666666\right)\right)\right) + \frac{1}{wj}\right) - \left(-4 \cdot \frac{x}{wj} + 1.5 \cdot \frac{x}{wj}\right)\right) - x \cdot 2\right)\\ \mathbf{elif}\;wj \leq 5 \cdot 10^{-83}:\\ \;\;\;\;x - wj \cdot \left(x \cdot 2 - wj\right)\\ \mathbf{elif}\;wj \leq 3.7 \cdot 10^{-7}:\\ \;\;\;\;x - wj \cdot \left(x \cdot 2 + wj \cdot \left(wj + x \cdot \left(\left(\frac{-1}{x} - wj \cdot -2.6666666666666665\right) - 2.5\right)\right)\right)\\ \mathbf{elif}\;wj \leq 3.9 \cdot 10^{-7}:\\ \;\;\;\;x - \left(-wj\right) \cdot \left({\left({wj}^{3} \cdot {\left(1 - wj\right)}^{3}\right)}^{0.3333333333333333} + 2 \cdot \left(-x\right)\right)\\ \mathbf{elif}\;wj \leq 0.68:\\ \;\;\;\;wj + \frac{x - wj \cdot e^{wj}}{e^{wj} + wj \cdot e^{wj}}\\ \mathbf{else}:\\ \;\;\;\;wj + \frac{wj - \frac{x}{e^{wj}}}{-1 - wj}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 99.6% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{wj}{wj + 1}\\ t_1 := wj \cdot e^{wj}\\ t_2 := wj - \frac{x}{e^{wj}}\\ \mathbf{if}\;wj \leq -3.8 \cdot 10^{-6}:\\ \;\;\;\;\left(wj - t\_0\right) + \mathsf{fma}\left(\frac{-1}{wj + 1}, t\_2, t\_0\right)\\ \mathbf{elif}\;wj \leq -5 \cdot 10^{-32}:\\ \;\;\;\;x + wj \cdot \left({wj}^{2} \cdot \left(\left(\left(-1 - \left(x \cdot -3 + \left(-2 \cdot \left(x \cdot -4 + x \cdot 1.5\right) + x \cdot 0.6666666666666666\right)\right)\right) + \frac{1}{wj}\right) - \left(-4 \cdot \frac{x}{wj} + 1.5 \cdot \frac{x}{wj}\right)\right) - x \cdot 2\right)\\ \mathbf{elif}\;wj \leq 5 \cdot 10^{-83}:\\ \;\;\;\;x - wj \cdot \left(x \cdot 2 - wj\right)\\ \mathbf{elif}\;wj \leq 3.2 \cdot 10^{-6}:\\ \;\;\;\;x - wj \cdot \left(x \cdot 2 + wj \cdot \left(wj + x \cdot \left(\left(\frac{-1}{x} - wj \cdot -2.6666666666666665\right) - 2.5\right)\right)\right)\\ \mathbf{elif}\;wj \leq 0.5:\\ \;\;\;\;wj + \frac{x - t\_1}{e^{wj} + t\_1}\\ \mathbf{else}:\\ \;\;\;\;wj + \frac{t\_2}{-1 - wj}\\ \end{array} \end{array} \]

FPCore

(FPCore (wj x)
 :precision binary64
 (let* ((t_0 (/ wj (+ wj 1.0)))
        (t_1 (* wj (exp wj)))
        (t_2 (- wj (/ x (exp wj)))))
   (if (<= wj -3.8e-6)
     (+ (- wj t_0) (fma (/ -1.0 (+ wj 1.0)) t_2 t_0))
     (if (<= wj -5e-32)
       (+
        x
        (*
         wj
         (-
          (*
           (pow wj 2.0)
           (-
            (+
             (-
              -1.0
              (+
               (* x -3.0)
               (+ (* -2.0 (+ (* x -4.0) (* x 1.5))) (* x 0.6666666666666666))))
             (/ 1.0 wj))
            (+ (* -4.0 (/ x wj)) (* 1.5 (/ x wj)))))
          (* x 2.0))))
       (if (<= wj 5e-83)
         (- x (* wj (- (* x 2.0) wj)))
         (if (<= wj 3.2e-6)
           (-
            x
            (*
             wj
             (+
              (* x 2.0)
              (*
               wj
               (+
                wj
                (* x (- (- (/ -1.0 x) (* wj -2.6666666666666665)) 2.5)))))))
           (if (<= wj 0.5)
             (+ wj (/ (- x t_1) (+ (exp wj) t_1)))
             (+ wj (/ t_2 (- -1.0 wj))))))))))

C

double code(double wj, double x) {
	double t_0 = wj / (wj + 1.0);
	double t_1 = wj * exp(wj);
	double t_2 = wj - (x / exp(wj));
	double tmp;
	if (wj <= -3.8e-6) {
		tmp = (wj - t_0) + fma((-1.0 / (wj + 1.0)), t_2, t_0);
	} else if (wj <= -5e-32) {
		tmp = x + (wj * ((pow(wj, 2.0) * (((-1.0 - ((x * -3.0) + ((-2.0 * ((x * -4.0) + (x * 1.5))) + (x * 0.6666666666666666)))) + (1.0 / wj)) - ((-4.0 * (x / wj)) + (1.5 * (x / wj))))) - (x * 2.0)));
	} else if (wj <= 5e-83) {
		tmp = x - (wj * ((x * 2.0) - wj));
	} else if (wj <= 3.2e-6) {
		tmp = x - (wj * ((x * 2.0) + (wj * (wj + (x * (((-1.0 / x) - (wj * -2.6666666666666665)) - 2.5))))));
	} else if (wj <= 0.5) {
		tmp = wj + ((x - t_1) / (exp(wj) + t_1));
	} else {
		tmp = wj + (t_2 / (-1.0 - wj));
	}
	return tmp;
}

Julia

function code(wj, x)
	t_0 = Float64(wj / Float64(wj + 1.0))
	t_1 = Float64(wj * exp(wj))
	t_2 = Float64(wj - Float64(x / exp(wj)))
	tmp = 0.0
	if (wj <= -3.8e-6)
		tmp = Float64(Float64(wj - t_0) + fma(Float64(-1.0 / Float64(wj + 1.0)), t_2, t_0));
	elseif (wj <= -5e-32)
		tmp = Float64(x + Float64(wj * Float64(Float64((wj ^ 2.0) * Float64(Float64(Float64(-1.0 - Float64(Float64(x * -3.0) + Float64(Float64(-2.0 * Float64(Float64(x * -4.0) + Float64(x * 1.5))) + Float64(x * 0.6666666666666666)))) + Float64(1.0 / wj)) - Float64(Float64(-4.0 * Float64(x / wj)) + Float64(1.5 * Float64(x / wj))))) - Float64(x * 2.0))));
	elseif (wj <= 5e-83)
		tmp = Float64(x - Float64(wj * Float64(Float64(x * 2.0) - wj)));
	elseif (wj <= 3.2e-6)
		tmp = Float64(x - Float64(wj * Float64(Float64(x * 2.0) + Float64(wj * Float64(wj + Float64(x * Float64(Float64(Float64(-1.0 / x) - Float64(wj * -2.6666666666666665)) - 2.5)))))));
	elseif (wj <= 0.5)
		tmp = Float64(wj + Float64(Float64(x - t_1) / Float64(exp(wj) + t_1)));
	else
		tmp = Float64(wj + Float64(t_2 / Float64(-1.0 - wj)));
	end
	return tmp
end

Wolfram

code[wj_, x_] := Block[{t$95$0 = N[(wj / N[(wj + 1.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(wj * N[Exp[wj], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(wj - N[(x / N[Exp[wj], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[wj, -3.8e-6], N[(N[(wj - t$95$0), $MachinePrecision] + N[(N[(-1.0 / N[(wj + 1.0), $MachinePrecision]), $MachinePrecision] * t$95$2 + t$95$0), $MachinePrecision]), $MachinePrecision], If[LessEqual[wj, -5e-32], N[(x + N[(wj * N[(N[(N[Power[wj, 2.0], $MachinePrecision] * N[(N[(N[(-1.0 - N[(N[(x * -3.0), $MachinePrecision] + N[(N[(-2.0 * N[(N[(x * -4.0), $MachinePrecision] + N[(x * 1.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(x * 0.6666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(1.0 / wj), $MachinePrecision]), $MachinePrecision] - N[(N[(-4.0 * N[(x / wj), $MachinePrecision]), $MachinePrecision] + N[(1.5 * N[(x / wj), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(x * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[wj, 5e-83], N[(x - N[(wj * N[(N[(x * 2.0), $MachinePrecision] - wj), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[wj, 3.2e-6], N[(x - N[(wj * N[(N[(x * 2.0), $MachinePrecision] + N[(wj * N[(wj + N[(x * N[(N[(N[(-1.0 / x), $MachinePrecision] - N[(wj * -2.6666666666666665), $MachinePrecision]), $MachinePrecision] - 2.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[wj, 0.5], N[(wj + N[(N[(x - t$95$1), $MachinePrecision] / N[(N[Exp[wj], $MachinePrecision] + t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(wj + N[(t$95$2 / N[(-1.0 - wj), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]]]

Derivation

1. Split input into 6 regimes

2. if wj < -3.8e-6

   1. Initial program 59.6%

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

      1. distribute-rgt1-in 97.1%

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

      2. *-commutative 97.1%

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

      3. associate-/r* 97.1%

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

      4. div-sub 59.6%

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

      5. associate-/l* 59.6%

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

      6. *-inverses 97.1%

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

      7. *-rgt-identity 97.1%

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

   3. Simplified 97.1%

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

      1. *-un-lft-identity 97.1%

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

      2. div-inv 97.2%

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

      3. prod-diff 59.8%

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

      4. associate-/r/ 59.5%

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

      5. clear-num 59.6%

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

      6. fma-neg 59.6%

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

      7. *-un-lft-identity 59.6%

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

      8. associate-/r/ 59.6%

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

      9. clear-num 59.8%

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

   6. Applied egg-rr 59.8%

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

      1. distribute-neg-frac 59.8%

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

      2. metadata-eval 59.8%

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

   8. Simplified 59.8%

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

   9. Taylor expanded in x around 0 66.8%

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

       1. +-commutative 66.8%

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

   11. Simplified 66.8%

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

   12. Taylor expanded in x around 0 97.1%

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

       1. +-commutative 97.1%

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

   14. Simplified 97.1%

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

   if -3.8e-6 < wj < -5e-32

   1. Initial program 70.8%

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

      1. distribute-rgt1-in 70.7%

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

      2. *-commutative 70.7%

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

      3. associate-/r* 70.8%

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

      4. div-sub 70.8%

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

      5. associate-/l* 70.8%

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

      6. *-inverses 70.8%

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

      7. *-rgt-identity 70.8%

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

   3. Simplified 70.8%

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

   5. Taylor expanded in wj around 0 98.4%

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

   6. Taylor expanded in wj around inf 98.5%

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

   if -5e-32 < wj < 5e-83

   1. Initial program 81.3%

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

      1. distribute-rgt1-in 81.3%

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

      2. *-commutative 81.3%

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

      3. associate-/r* 81.3%

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

      4. div-sub 81.3%

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

      5. associate-/l* 81.3%

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

      6. *-inverses 81.3%

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

      7. *-rgt-identity 81.3%

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

   3. Simplified 81.3%

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

   5. Taylor expanded in wj around 0 99.6%

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

   6. Taylor expanded in x around 0 100.0%

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

      1. mul-1-neg 100.0%

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

      2. unsub-neg 100.0%

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

   8. Simplified 100.0%

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

   9. Taylor expanded in wj around 0 100.0%

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

   if 5e-83 < wj < 3.1999999999999999e-6

   1. Initial program 52.2%

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

      1. distribute-rgt1-in 52.1%

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

      2. *-commutative 52.1%

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

      3. associate-/r* 52.3%

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

      4. div-sub 52.3%

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

      5. associate-/l* 52.3%

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

      6. *-inverses 52.3%

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

      7. *-rgt-identity 52.3%

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

   3. Simplified 52.3%

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

   5. Taylor expanded in wj around 0 99.4%

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

   6. Taylor expanded in x around 0 99.4%

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

      1. distribute-lft-out 99.4%

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

      2. *-commutative 99.4%

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

      3. mul-1-neg 99.4%

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

   8. Simplified 99.4%

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

   9. Taylor expanded in x around 0 99.4%

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

   10. Taylor expanded in x around inf 99.5%

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

   if 3.1999999999999999e-6 < wj < 0.5

   1. Initial program 97.4%

      \[wj - \frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}} \]
   2. Add Preprocessing

   if 0.5 < wj

   1. Initial program 74.6%

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

      1. distribute-rgt1-in 75.0%

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

      2. *-commutative 75.0%

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

      3. associate-/r* 75.0%

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

      4. div-sub 75.0%

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

      5. associate-/l* 75.0%

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

      6. *-inverses 100.0%

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

      7. *-rgt-identity 100.0%

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

   3. Simplified 100.0%

      \[\leadsto \color{blue}{wj - \frac{wj - \frac{x}{e^{wj}}}{wj + 1}} \]
   4. Add Preprocessing
3. Recombined 6 regimes into one program.

4. Final simplification 99.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;wj \leq -3.8 \cdot 10^{-6}:\\ \;\;\;\;\left(wj - \frac{wj}{wj + 1}\right) + \mathsf{fma}\left(\frac{-1}{wj + 1}, wj - \frac{x}{e^{wj}}, \frac{wj}{wj + 1}\right)\\ \mathbf{elif}\;wj \leq -5 \cdot 10^{-32}:\\ \;\;\;\;x + wj \cdot \left({wj}^{2} \cdot \left(\left(\left(-1 - \left(x \cdot -3 + \left(-2 \cdot \left(x \cdot -4 + x \cdot 1.5\right) + x \cdot 0.6666666666666666\right)\right)\right) + \frac{1}{wj}\right) - \left(-4 \cdot \frac{x}{wj} + 1.5 \cdot \frac{x}{wj}\right)\right) - x \cdot 2\right)\\ \mathbf{elif}\;wj \leq 5 \cdot 10^{-83}:\\ \;\;\;\;x - wj \cdot \left(x \cdot 2 - wj\right)\\ \mathbf{elif}\;wj \leq 3.2 \cdot 10^{-6}:\\ \;\;\;\;x - wj \cdot \left(x \cdot 2 + wj \cdot \left(wj + x \cdot \left(\left(\frac{-1}{x} - wj \cdot -2.6666666666666665\right) - 2.5\right)\right)\right)\\ \mathbf{elif}\;wj \leq 0.5:\\ \;\;\;\;wj + \frac{x - wj \cdot e^{wj}}{e^{wj} + wj \cdot e^{wj}}\\ \mathbf{else}:\\ \;\;\;\;wj + \frac{wj - \frac{x}{e^{wj}}}{-1 - wj}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 99.6% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{wj}{wj + 1}\\ t_1 := wj - \frac{x}{e^{wj}}\\ \mathbf{if}\;wj \leq -3.8 \cdot 10^{-6}:\\ \;\;\;\;\left(wj - t\_0\right) + \mathsf{fma}\left(\frac{-1}{wj + 1}, t\_1, t\_0\right)\\ \mathbf{elif}\;wj \leq -5 \cdot 10^{-32}:\\ \;\;\;\;x + wj \cdot \left({wj}^{2} \cdot \left(\left(\left(-1 - \left(x \cdot -3 + \left(-2 \cdot \left(x \cdot -4 + x \cdot 1.5\right) + x \cdot 0.6666666666666666\right)\right)\right) + \frac{1}{wj}\right) - \left(-4 \cdot \frac{x}{wj} + 1.5 \cdot \frac{x}{wj}\right)\right) - x \cdot 2\right)\\ \mathbf{elif}\;wj \leq 5 \cdot 10^{-83}:\\ \;\;\;\;x - wj \cdot \left(x \cdot 2 - wj\right)\\ \mathbf{elif}\;wj \leq 0.00052:\\ \;\;\;\;x - wj \cdot \left(x \cdot 2 + wj \cdot \left(wj + x \cdot \left(\left(\frac{-1}{x} - wj \cdot -2.6666666666666665\right) - 2.5\right)\right)\right)\\ \mathbf{elif}\;wj \leq 0.00055:\\ \;\;\;\;x \cdot \frac{e^{-wj}}{wj + 1}\\ \mathbf{elif}\;wj \leq 0.105:\\ \;\;\;\;wj + \frac{wj \cdot e^{wj} - x}{e^{wj} \cdot \left(-1 - wj\right)}\\ \mathbf{else}:\\ \;\;\;\;wj + \frac{t\_1}{-1 - wj}\\ \end{array} \end{array} \]

FPCore

(FPCore (wj x)
 :precision binary64
 (let* ((t_0 (/ wj (+ wj 1.0))) (t_1 (- wj (/ x (exp wj)))))
   (if (<= wj -3.8e-6)
     (+ (- wj t_0) (fma (/ -1.0 (+ wj 1.0)) t_1 t_0))
     (if (<= wj -5e-32)
       (+
        x
        (*
         wj
         (-
          (*
           (pow wj 2.0)
           (-
            (+
             (-
              -1.0
              (+
               (* x -3.0)
               (+ (* -2.0 (+ (* x -4.0) (* x 1.5))) (* x 0.6666666666666666))))
             (/ 1.0 wj))
            (+ (* -4.0 (/ x wj)) (* 1.5 (/ x wj)))))
          (* x 2.0))))
       (if (<= wj 5e-83)
         (- x (* wj (- (* x 2.0) wj)))
         (if (<= wj 0.00052)
           (-
            x
            (*
             wj
             (+
              (* x 2.0)
              (*
               wj
               (+
                wj
                (* x (- (- (/ -1.0 x) (* wj -2.6666666666666665)) 2.5)))))))
           (if (<= wj 0.00055)
             (* x (/ (exp (- wj)) (+ wj 1.0)))
             (if (<= wj 0.105)
               (+ wj (/ (- (* wj (exp wj)) x) (* (exp wj) (- -1.0 wj))))
               (+ wj (/ t_1 (- -1.0 wj)))))))))))

C

double code(double wj, double x) {
	double t_0 = wj / (wj + 1.0);
	double t_1 = wj - (x / exp(wj));
	double tmp;
	if (wj <= -3.8e-6) {
		tmp = (wj - t_0) + fma((-1.0 / (wj + 1.0)), t_1, t_0);
	} else if (wj <= -5e-32) {
		tmp = x + (wj * ((pow(wj, 2.0) * (((-1.0 - ((x * -3.0) + ((-2.0 * ((x * -4.0) + (x * 1.5))) + (x * 0.6666666666666666)))) + (1.0 / wj)) - ((-4.0 * (x / wj)) + (1.5 * (x / wj))))) - (x * 2.0)));
	} else if (wj <= 5e-83) {
		tmp = x - (wj * ((x * 2.0) - wj));
	} else if (wj <= 0.00052) {
		tmp = x - (wj * ((x * 2.0) + (wj * (wj + (x * (((-1.0 / x) - (wj * -2.6666666666666665)) - 2.5))))));
	} else if (wj <= 0.00055) {
		tmp = x * (exp(-wj) / (wj + 1.0));
	} else if (wj <= 0.105) {
		tmp = wj + (((wj * exp(wj)) - x) / (exp(wj) * (-1.0 - wj)));
	} else {
		tmp = wj + (t_1 / (-1.0 - wj));
	}
	return tmp;
}

Julia

function code(wj, x)
	t_0 = Float64(wj / Float64(wj + 1.0))
	t_1 = Float64(wj - Float64(x / exp(wj)))
	tmp = 0.0
	if (wj <= -3.8e-6)
		tmp = Float64(Float64(wj - t_0) + fma(Float64(-1.0 / Float64(wj + 1.0)), t_1, t_0));
	elseif (wj <= -5e-32)
		tmp = Float64(x + Float64(wj * Float64(Float64((wj ^ 2.0) * Float64(Float64(Float64(-1.0 - Float64(Float64(x * -3.0) + Float64(Float64(-2.0 * Float64(Float64(x * -4.0) + Float64(x * 1.5))) + Float64(x * 0.6666666666666666)))) + Float64(1.0 / wj)) - Float64(Float64(-4.0 * Float64(x / wj)) + Float64(1.5 * Float64(x / wj))))) - Float64(x * 2.0))));
	elseif (wj <= 5e-83)
		tmp = Float64(x - Float64(wj * Float64(Float64(x * 2.0) - wj)));
	elseif (wj <= 0.00052)
		tmp = Float64(x - Float64(wj * Float64(Float64(x * 2.0) + Float64(wj * Float64(wj + Float64(x * Float64(Float64(Float64(-1.0 / x) - Float64(wj * -2.6666666666666665)) - 2.5)))))));
	elseif (wj <= 0.00055)
		tmp = Float64(x * Float64(exp(Float64(-wj)) / Float64(wj + 1.0)));
	elseif (wj <= 0.105)
		tmp = Float64(wj + Float64(Float64(Float64(wj * exp(wj)) - x) / Float64(exp(wj) * Float64(-1.0 - wj))));
	else
		tmp = Float64(wj + Float64(t_1 / Float64(-1.0 - wj)));
	end
	return tmp
end

Wolfram

code[wj_, x_] := Block[{t$95$0 = N[(wj / N[(wj + 1.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(wj - N[(x / N[Exp[wj], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[wj, -3.8e-6], N[(N[(wj - t$95$0), $MachinePrecision] + N[(N[(-1.0 / N[(wj + 1.0), $MachinePrecision]), $MachinePrecision] * t$95$1 + t$95$0), $MachinePrecision]), $MachinePrecision], If[LessEqual[wj, -5e-32], N[(x + N[(wj * N[(N[(N[Power[wj, 2.0], $MachinePrecision] * N[(N[(N[(-1.0 - N[(N[(x * -3.0), $MachinePrecision] + N[(N[(-2.0 * N[(N[(x * -4.0), $MachinePrecision] + N[(x * 1.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(x * 0.6666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(1.0 / wj), $MachinePrecision]), $MachinePrecision] - N[(N[(-4.0 * N[(x / wj), $MachinePrecision]), $MachinePrecision] + N[(1.5 * N[(x / wj), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(x * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[wj, 5e-83], N[(x - N[(wj * N[(N[(x * 2.0), $MachinePrecision] - wj), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[wj, 0.00052], N[(x - N[(wj * N[(N[(x * 2.0), $MachinePrecision] + N[(wj * N[(wj + N[(x * N[(N[(N[(-1.0 / x), $MachinePrecision] - N[(wj * -2.6666666666666665), $MachinePrecision]), $MachinePrecision] - 2.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[wj, 0.00055], N[(x * N[(N[Exp[(-wj)], $MachinePrecision] / N[(wj + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[wj, 0.105], N[(wj + N[(N[(N[(wj * N[Exp[wj], $MachinePrecision]), $MachinePrecision] - x), $MachinePrecision] / N[(N[Exp[wj], $MachinePrecision] * N[(-1.0 - wj), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(wj + N[(t$95$1 / N[(-1.0 - wj), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]]]

Derivation

1. Split input into 7 regimes

2. if wj < -3.8e-6

   1. Initial program 59.6%

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

      1. distribute-rgt1-in 97.1%

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

      2. *-commutative 97.1%

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

      3. associate-/r* 97.1%

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

      4. div-sub 59.6%

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

      5. associate-/l* 59.6%

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

      6. *-inverses 97.1%

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

      7. *-rgt-identity 97.1%

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

   3. Simplified 97.1%

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

      1. *-un-lft-identity 97.1%

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

      2. div-inv 97.2%

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

      3. prod-diff 59.8%

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

      4. associate-/r/ 59.5%

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

      5. clear-num 59.6%

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

      6. fma-neg 59.6%

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

      7. *-un-lft-identity 59.6%

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

      8. associate-/r/ 59.6%

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

      9. clear-num 59.8%

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

   6. Applied egg-rr 59.8%

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

      1. distribute-neg-frac 59.8%

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

      2. metadata-eval 59.8%

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

   8. Simplified 59.8%

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

   9. Taylor expanded in x around 0 66.8%

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

       1. +-commutative 66.8%

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

   11. Simplified 66.8%

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

   12. Taylor expanded in x around 0 97.1%

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

       1. +-commutative 97.1%

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

   14. Simplified 97.1%

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

   if -3.8e-6 < wj < -5e-32

   1. Initial program 70.8%

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

      1. distribute-rgt1-in 70.7%

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

      2. *-commutative 70.7%

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

      3. associate-/r* 70.8%

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

      4. div-sub 70.8%

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

      5. associate-/l* 70.8%

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

      6. *-inverses 70.8%

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

      7. *-rgt-identity 70.8%

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

   3. Simplified 70.8%

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

   5. Taylor expanded in wj around 0 98.4%

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

   6. Taylor expanded in wj around inf 98.5%

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

   if -5e-32 < wj < 5e-83

   1. Initial program 81.3%

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

      1. distribute-rgt1-in 81.3%

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

      2. *-commutative 81.3%

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

      3. associate-/r* 81.3%

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

      4. div-sub 81.3%

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

      5. associate-/l* 81.3%

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

      6. *-inverses 81.3%

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

      7. *-rgt-identity 81.3%

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

   3. Simplified 81.3%

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

   5. Taylor expanded in wj around 0 99.6%

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

   6. Taylor expanded in x around 0 100.0%

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

      1. mul-1-neg 100.0%

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

      2. unsub-neg 100.0%

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

   8. Simplified 100.0%

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

   9. Taylor expanded in wj around 0 100.0%

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

   if 5e-83 < wj < 5.19999999999999954e-4

   1. Initial program 52.2%

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

      1. distribute-rgt1-in 52.1%

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

      2. *-commutative 52.1%

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

      3. associate-/r* 52.3%

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

      4. div-sub 52.3%

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

      5. associate-/l* 52.3%

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

      6. *-inverses 52.3%

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

      7. *-rgt-identity 52.3%

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

   3. Simplified 52.3%

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

   5. Taylor expanded in wj around 0 99.4%

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

   6. Taylor expanded in x around 0 99.4%

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

      1. distribute-lft-out 99.4%

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

      2. *-commutative 99.4%

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

      3. mul-1-neg 99.4%

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

   8. Simplified 99.4%

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

   9. Taylor expanded in x around 0 99.4%

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

   10. Taylor expanded in x around inf 99.5%

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

   if 5.19999999999999954e-4 < wj < 5.50000000000000033e-4

   1. Initial program 100.0%

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

      1. distribute-rgt1-in 98.4%

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

      2. *-commutative 98.4%

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

      3. associate-/r* 98.4%

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

      4. div-sub 98.4%

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

      5. associate-/l* 98.4%

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

      6. *-inverses 98.4%

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

      7. *-rgt-identity 98.4%

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

   3. Simplified 98.4%

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

   5. Taylor expanded in x around inf 98.4%

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

      1. +-commutative 98.4%

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

      2. associate-/r* 100.0%

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

      3. exp-neg 100.0%

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

      4. +-commutative 100.0%

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

   7. Simplified 100.0%

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

   8. Taylor expanded in x around inf 98.4%

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

      1. associate-/l* 100.0%

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

      2. +-commutative 100.0%

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

   10. Simplified 100.0%

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

   if 5.50000000000000033e-4 < wj < 0.104999999999999996

   1. Initial program 94.8%

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

      1. distribute-rgt1-in 94.8%

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

      2. *-commutative 94.8%

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

   3. Simplified 94.8%

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

   if 0.104999999999999996 < wj

   1. Initial program 74.6%

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

      1. distribute-rgt1-in 75.0%

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

      2. *-commutative 75.0%

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

      3. associate-/r* 75.0%

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

      4. div-sub 75.0%

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

      5. associate-/l* 75.0%

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

      6. *-inverses 100.0%

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

      7. *-rgt-identity 100.0%

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

   3. Simplified 100.0%

      \[\leadsto \color{blue}{wj - \frac{wj - \frac{x}{e^{wj}}}{wj + 1}} \]
   4. Add Preprocessing
3. Recombined 7 regimes into one program.

4. Final simplification 99.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;wj \leq -3.8 \cdot 10^{-6}:\\ \;\;\;\;\left(wj - \frac{wj}{wj + 1}\right) + \mathsf{fma}\left(\frac{-1}{wj + 1}, wj - \frac{x}{e^{wj}}, \frac{wj}{wj + 1}\right)\\ \mathbf{elif}\;wj \leq -5 \cdot 10^{-32}:\\ \;\;\;\;x + wj \cdot \left({wj}^{2} \cdot \left(\left(\left(-1 - \left(x \cdot -3 + \left(-2 \cdot \left(x \cdot -4 + x \cdot 1.5\right) + x \cdot 0.6666666666666666\right)\right)\right) + \frac{1}{wj}\right) - \left(-4 \cdot \frac{x}{wj} + 1.5 \cdot \frac{x}{wj}\right)\right) - x \cdot 2\right)\\ \mathbf{elif}\;wj \leq 5 \cdot 10^{-83}:\\ \;\;\;\;x - wj \cdot \left(x \cdot 2 - wj\right)\\ \mathbf{elif}\;wj \leq 0.00052:\\ \;\;\;\;x - wj \cdot \left(x \cdot 2 + wj \cdot \left(wj + x \cdot \left(\left(\frac{-1}{x} - wj \cdot -2.6666666666666665\right) - 2.5\right)\right)\right)\\ \mathbf{elif}\;wj \leq 0.00055:\\ \;\;\;\;x \cdot \frac{e^{-wj}}{wj + 1}\\ \mathbf{elif}\;wj \leq 0.105:\\ \;\;\;\;wj + \frac{wj \cdot e^{wj} - x}{e^{wj} \cdot \left(-1 - wj\right)}\\ \mathbf{else}:\\ \;\;\;\;wj + \frac{wj - \frac{x}{e^{wj}}}{-1 - wj}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 99.6% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := wj + \frac{wj - \frac{x}{e^{wj}}}{-1 - wj}\\ \mathbf{if}\;wj \leq -4.5 \cdot 10^{-6}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;wj \leq -5 \cdot 10^{-32}:\\ \;\;\;\;x + wj \cdot \left({wj}^{2} \cdot \left(\left(\left(-1 - \left(x \cdot -3 + \left(-2 \cdot \left(x \cdot -4 + x \cdot 1.5\right) + x \cdot 0.6666666666666666\right)\right)\right) + \frac{1}{wj}\right) - \left(-4 \cdot \frac{x}{wj} + 1.5 \cdot \frac{x}{wj}\right)\right) - x \cdot 2\right)\\ \mathbf{elif}\;wj \leq 5 \cdot 10^{-83}:\\ \;\;\;\;x - wj \cdot \left(x \cdot 2 - wj\right)\\ \mathbf{elif}\;wj \leq 0.00052:\\ \;\;\;\;x - wj \cdot \left(x \cdot 2 + wj \cdot \left(wj + x \cdot \left(\left(\frac{-1}{x} - wj \cdot -2.6666666666666665\right) - 2.5\right)\right)\right)\\ \mathbf{elif}\;wj \leq 0.00055:\\ \;\;\;\;x \cdot \frac{e^{-wj}}{wj + 1}\\ \mathbf{elif}\;wj \leq 0.11:\\ \;\;\;\;wj + \frac{wj \cdot e^{wj} - x}{e^{wj} \cdot \left(-1 - wj\right)}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (wj x)
 :precision binary64
 (let* ((t_0 (+ wj (/ (- wj (/ x (exp wj))) (- -1.0 wj)))))
   (if (<= wj -4.5e-6)
     t_0
     (if (<= wj -5e-32)
       (+
        x
        (*
         wj
         (-
          (*
           (pow wj 2.0)
           (-
            (+
             (-
              -1.0
              (+
               (* x -3.0)
               (+ (* -2.0 (+ (* x -4.0) (* x 1.5))) (* x 0.6666666666666666))))
             (/ 1.0 wj))
            (+ (* -4.0 (/ x wj)) (* 1.5 (/ x wj)))))
          (* x 2.0))))
       (if (<= wj 5e-83)
         (- x (* wj (- (* x 2.0) wj)))
         (if (<= wj 0.00052)
           (-
            x
            (*
             wj
             (+
              (* x 2.0)
              (*
               wj
               (+
                wj
                (* x (- (- (/ -1.0 x) (* wj -2.6666666666666665)) 2.5)))))))
           (if (<= wj 0.00055)
             (* x (/ (exp (- wj)) (+ wj 1.0)))
             (if (<= wj 0.11)
               (+ wj (/ (- (* wj (exp wj)) x) (* (exp wj) (- -1.0 wj))))
               t_0))))))))

C

double code(double wj, double x) {
	double t_0 = wj + ((wj - (x / exp(wj))) / (-1.0 - wj));
	double tmp;
	if (wj <= -4.5e-6) {
		tmp = t_0;
	} else if (wj <= -5e-32) {
		tmp = x + (wj * ((pow(wj, 2.0) * (((-1.0 - ((x * -3.0) + ((-2.0 * ((x * -4.0) + (x * 1.5))) + (x * 0.6666666666666666)))) + (1.0 / wj)) - ((-4.0 * (x / wj)) + (1.5 * (x / wj))))) - (x * 2.0)));
	} else if (wj <= 5e-83) {
		tmp = x - (wj * ((x * 2.0) - wj));
	} else if (wj <= 0.00052) {
		tmp = x - (wj * ((x * 2.0) + (wj * (wj + (x * (((-1.0 / x) - (wj * -2.6666666666666665)) - 2.5))))));
	} else if (wj <= 0.00055) {
		tmp = x * (exp(-wj) / (wj + 1.0));
	} else if (wj <= 0.11) {
		tmp = wj + (((wj * exp(wj)) - x) / (exp(wj) * (-1.0 - wj)));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(wj, x)
    real(8), intent (in) :: wj
    real(8), intent (in) :: x
    real(8) :: t_0
    real(8) :: tmp
    t_0 = wj + ((wj - (x / exp(wj))) / ((-1.0d0) - wj))
    if (wj <= (-4.5d-6)) then
        tmp = t_0
    else if (wj <= (-5d-32)) then
        tmp = x + (wj * (((wj ** 2.0d0) * ((((-1.0d0) - ((x * (-3.0d0)) + (((-2.0d0) * ((x * (-4.0d0)) + (x * 1.5d0))) + (x * 0.6666666666666666d0)))) + (1.0d0 / wj)) - (((-4.0d0) * (x / wj)) + (1.5d0 * (x / wj))))) - (x * 2.0d0)))
    else if (wj <= 5d-83) then
        tmp = x - (wj * ((x * 2.0d0) - wj))
    else if (wj <= 0.00052d0) then
        tmp = x - (wj * ((x * 2.0d0) + (wj * (wj + (x * ((((-1.0d0) / x) - (wj * (-2.6666666666666665d0))) - 2.5d0))))))
    else if (wj <= 0.00055d0) then
        tmp = x * (exp(-wj) / (wj + 1.0d0))
    else if (wj <= 0.11d0) then
        tmp = wj + (((wj * exp(wj)) - x) / (exp(wj) * ((-1.0d0) - wj)))
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double wj, double x) {
	double t_0 = wj + ((wj - (x / Math.exp(wj))) / (-1.0 - wj));
	double tmp;
	if (wj <= -4.5e-6) {
		tmp = t_0;
	} else if (wj <= -5e-32) {
		tmp = x + (wj * ((Math.pow(wj, 2.0) * (((-1.0 - ((x * -3.0) + ((-2.0 * ((x * -4.0) + (x * 1.5))) + (x * 0.6666666666666666)))) + (1.0 / wj)) - ((-4.0 * (x / wj)) + (1.5 * (x / wj))))) - (x * 2.0)));
	} else if (wj <= 5e-83) {
		tmp = x - (wj * ((x * 2.0) - wj));
	} else if (wj <= 0.00052) {
		tmp = x - (wj * ((x * 2.0) + (wj * (wj + (x * (((-1.0 / x) - (wj * -2.6666666666666665)) - 2.5))))));
	} else if (wj <= 0.00055) {
		tmp = x * (Math.exp(-wj) / (wj + 1.0));
	} else if (wj <= 0.11) {
		tmp = wj + (((wj * Math.exp(wj)) - x) / (Math.exp(wj) * (-1.0 - wj)));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(wj, x):
	t_0 = wj + ((wj - (x / math.exp(wj))) / (-1.0 - wj))
	tmp = 0
	if wj <= -4.5e-6:
		tmp = t_0
	elif wj <= -5e-32:
		tmp = x + (wj * ((math.pow(wj, 2.0) * (((-1.0 - ((x * -3.0) + ((-2.0 * ((x * -4.0) + (x * 1.5))) + (x * 0.6666666666666666)))) + (1.0 / wj)) - ((-4.0 * (x / wj)) + (1.5 * (x / wj))))) - (x * 2.0)))
	elif wj <= 5e-83:
		tmp = x - (wj * ((x * 2.0) - wj))
	elif wj <= 0.00052:
		tmp = x - (wj * ((x * 2.0) + (wj * (wj + (x * (((-1.0 / x) - (wj * -2.6666666666666665)) - 2.5))))))
	elif wj <= 0.00055:
		tmp = x * (math.exp(-wj) / (wj + 1.0))
	elif wj <= 0.11:
		tmp = wj + (((wj * math.exp(wj)) - x) / (math.exp(wj) * (-1.0 - wj)))
	else:
		tmp = t_0
	return tmp

Julia

function code(wj, x)
	t_0 = Float64(wj + Float64(Float64(wj - Float64(x / exp(wj))) / Float64(-1.0 - wj)))
	tmp = 0.0
	if (wj <= -4.5e-6)
		tmp = t_0;
	elseif (wj <= -5e-32)
		tmp = Float64(x + Float64(wj * Float64(Float64((wj ^ 2.0) * Float64(Float64(Float64(-1.0 - Float64(Float64(x * -3.0) + Float64(Float64(-2.0 * Float64(Float64(x * -4.0) + Float64(x * 1.5))) + Float64(x * 0.6666666666666666)))) + Float64(1.0 / wj)) - Float64(Float64(-4.0 * Float64(x / wj)) + Float64(1.5 * Float64(x / wj))))) - Float64(x * 2.0))));
	elseif (wj <= 5e-83)
		tmp = Float64(x - Float64(wj * Float64(Float64(x * 2.0) - wj)));
	elseif (wj <= 0.00052)
		tmp = Float64(x - Float64(wj * Float64(Float64(x * 2.0) + Float64(wj * Float64(wj + Float64(x * Float64(Float64(Float64(-1.0 / x) - Float64(wj * -2.6666666666666665)) - 2.5)))))));
	elseif (wj <= 0.00055)
		tmp = Float64(x * Float64(exp(Float64(-wj)) / Float64(wj + 1.0)));
	elseif (wj <= 0.11)
		tmp = Float64(wj + Float64(Float64(Float64(wj * exp(wj)) - x) / Float64(exp(wj) * Float64(-1.0 - wj))));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(wj, x)
	t_0 = wj + ((wj - (x / exp(wj))) / (-1.0 - wj));
	tmp = 0.0;
	if (wj <= -4.5e-6)
		tmp = t_0;
	elseif (wj <= -5e-32)
		tmp = x + (wj * (((wj ^ 2.0) * (((-1.0 - ((x * -3.0) + ((-2.0 * ((x * -4.0) + (x * 1.5))) + (x * 0.6666666666666666)))) + (1.0 / wj)) - ((-4.0 * (x / wj)) + (1.5 * (x / wj))))) - (x * 2.0)));
	elseif (wj <= 5e-83)
		tmp = x - (wj * ((x * 2.0) - wj));
	elseif (wj <= 0.00052)
		tmp = x - (wj * ((x * 2.0) + (wj * (wj + (x * (((-1.0 / x) - (wj * -2.6666666666666665)) - 2.5))))));
	elseif (wj <= 0.00055)
		tmp = x * (exp(-wj) / (wj + 1.0));
	elseif (wj <= 0.11)
		tmp = wj + (((wj * exp(wj)) - x) / (exp(wj) * (-1.0 - wj)));
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[wj_, x_] := Block[{t$95$0 = N[(wj + N[(N[(wj - N[(x / N[Exp[wj], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(-1.0 - wj), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[wj, -4.5e-6], t$95$0, If[LessEqual[wj, -5e-32], N[(x + N[(wj * N[(N[(N[Power[wj, 2.0], $MachinePrecision] * N[(N[(N[(-1.0 - N[(N[(x * -3.0), $MachinePrecision] + N[(N[(-2.0 * N[(N[(x * -4.0), $MachinePrecision] + N[(x * 1.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(x * 0.6666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(1.0 / wj), $MachinePrecision]), $MachinePrecision] - N[(N[(-4.0 * N[(x / wj), $MachinePrecision]), $MachinePrecision] + N[(1.5 * N[(x / wj), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(x * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[wj, 5e-83], N[(x - N[(wj * N[(N[(x * 2.0), $MachinePrecision] - wj), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[wj, 0.00052], N[(x - N[(wj * N[(N[(x * 2.0), $MachinePrecision] + N[(wj * N[(wj + N[(x * N[(N[(N[(-1.0 / x), $MachinePrecision] - N[(wj * -2.6666666666666665), $MachinePrecision]), $MachinePrecision] - 2.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[wj, 0.00055], N[(x * N[(N[Exp[(-wj)], $MachinePrecision] / N[(wj + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[wj, 0.11], N[(wj + N[(N[(N[(wj * N[Exp[wj], $MachinePrecision]), $MachinePrecision] - x), $MachinePrecision] / N[(N[Exp[wj], $MachinePrecision] * N[(-1.0 - wj), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]]]]]]

Derivation

1. Split input into 6 regimes

2. if wj < -4.50000000000000011e-6 or 0.110000000000000001 < wj

   1. Initial program 64.6%

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

      1. distribute-rgt1-in 89.7%

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

      2. *-commutative 89.7%

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

      3. associate-/r* 89.7%

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

      4. div-sub 64.7%

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

      5. associate-/l* 64.7%

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

      6. *-inverses 98.0%

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

      7. *-rgt-identity 98.0%

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

   3. Simplified 98.0%

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

   if -4.50000000000000011e-6 < wj < -5e-32

   1. Initial program 70.8%

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

      1. distribute-rgt1-in 70.7%

         \[\leadsto wj - \frac{wj \cdot e^{wj} - x}{\color{blue}{\left(wj + 1\right) \cdot e^{wj}}} \]

      2. *-commutative 70.7%

         \[\leadsto wj - \frac{wj \cdot e^{wj} - x}{\color{blue}{e^{wj} \cdot \left(wj + 1\right)}} \]

      3. associate-/r* 70.8%

         \[\leadsto wj - \color{blue}{\frac{\frac{wj \cdot e^{wj} - x}{e^{wj}}}{wj + 1}} \]

      4. div-sub 70.8%

         \[\leadsto wj - \frac{\color{blue}{\frac{wj \cdot e^{wj}}{e^{wj}} - \frac{x}{e^{wj}}}}{wj + 1} \]

      5. associate-/l* 70.8%

         \[\leadsto wj - \frac{\color{blue}{wj \cdot \frac{e^{wj}}{e^{wj}}} - \frac{x}{e^{wj}}}{wj + 1} \]

      6. *-inverses 70.8%

         \[\leadsto wj - \frac{wj \cdot \color{blue}{1} - \frac{x}{e^{wj}}}{wj + 1} \]

      7. *-rgt-identity 70.8%

         \[\leadsto wj - \frac{\color{blue}{wj} - \frac{x}{e^{wj}}}{wj + 1} \]

   3. Simplified 70.8%

      \[\leadsto \color{blue}{wj - \frac{wj - \frac{x}{e^{wj}}}{wj + 1}} \]
   4. Add Preprocessing

   5. Taylor expanded in wj around 0 98.4%

      \[\leadsto \color{blue}{x + wj \cdot \left(wj \cdot \left(\left(1 + -1 \cdot \left(wj \cdot \left(1 + \left(-3 \cdot x + \left(-2 \cdot \left(-4 \cdot x + 1.5 \cdot x\right) + 0.6666666666666666 \cdot x\right)\right)\right)\right)\right) - \left(-4 \cdot x + 1.5 \cdot x\right)\right) - 2 \cdot x\right)} \]

   6. Taylor expanded in wj around inf 98.5%

      \[\leadsto x + wj \cdot \left(\color{blue}{{wj}^{2} \cdot \left(\left(-1 \cdot \left(1 + \left(-3 \cdot x + \left(-2 \cdot \left(-4 \cdot x + 1.5 \cdot x\right) + 0.6666666666666666 \cdot x\right)\right)\right) + \frac{1}{wj}\right) - \left(-4 \cdot \frac{x}{wj} + 1.5 \cdot \frac{x}{wj}\right)\right)} - 2 \cdot x\right) \]

   if -5e-32 < wj < 5e-83

   1. Initial program 81.3%

      \[wj - \frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}} \]
   2. Step-by-step derivation

      1. distribute-rgt1-in 81.3%

         \[\leadsto wj - \frac{wj \cdot e^{wj} - x}{\color{blue}{\left(wj + 1\right) \cdot e^{wj}}} \]

      2. *-commutative 81.3%

         \[\leadsto wj - \frac{wj \cdot e^{wj} - x}{\color{blue}{e^{wj} \cdot \left(wj + 1\right)}} \]

      3. associate-/r* 81.3%

         \[\leadsto wj - \color{blue}{\frac{\frac{wj \cdot e^{wj} - x}{e^{wj}}}{wj + 1}} \]

      4. div-sub 81.3%

         \[\leadsto wj - \frac{\color{blue}{\frac{wj \cdot e^{wj}}{e^{wj}} - \frac{x}{e^{wj}}}}{wj + 1} \]

      5. associate-/l* 81.3%

         \[\leadsto wj - \frac{\color{blue}{wj \cdot \frac{e^{wj}}{e^{wj}}} - \frac{x}{e^{wj}}}{wj + 1} \]

      6. *-inverses 81.3%

         \[\leadsto wj - \frac{wj \cdot \color{blue}{1} - \frac{x}{e^{wj}}}{wj + 1} \]

      7. *-rgt-identity 81.3%

         \[\leadsto wj - \frac{\color{blue}{wj} - \frac{x}{e^{wj}}}{wj + 1} \]

   3. Simplified 81.3%

      \[\leadsto \color{blue}{wj - \frac{wj - \frac{x}{e^{wj}}}{wj + 1}} \]
   4. Add Preprocessing

   5. Taylor expanded in wj around 0 99.6%

      \[\leadsto \color{blue}{x + wj \cdot \left(wj \cdot \left(\left(1 + -1 \cdot \left(wj \cdot \left(1 + \left(-3 \cdot x + \left(-2 \cdot \left(-4 \cdot x + 1.5 \cdot x\right) + 0.6666666666666666 \cdot x\right)\right)\right)\right)\right) - \left(-4 \cdot x + 1.5 \cdot x\right)\right) - 2 \cdot x\right)} \]

   6. Taylor expanded in x around 0 100.0%

      \[\leadsto x + wj \cdot \left(\color{blue}{wj \cdot \left(1 + -1 \cdot wj\right)} - 2 \cdot x\right) \]
   7. Step-by-step derivation

      1. mul-1-neg 100.0%

         \[\leadsto x + wj \cdot \left(wj \cdot \left(1 + \color{blue}{\left(-wj\right)}\right) - 2 \cdot x\right) \]

      2. unsub-neg 100.0%

         \[\leadsto x + wj \cdot \left(wj \cdot \color{blue}{\left(1 - wj\right)} - 2 \cdot x\right) \]

   8. Simplified 100.0%

      \[\leadsto x + wj \cdot \left(\color{blue}{wj \cdot \left(1 - wj\right)} - 2 \cdot x\right) \]

   9. Taylor expanded in wj around 0 100.0%

      \[\leadsto x + wj \cdot \left(\color{blue}{wj} - 2 \cdot x\right) \]

   if 5e-83 < wj < 5.19999999999999954e-4

   1. Initial program 52.2%

      \[wj - \frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}} \]
   2. Step-by-step derivation

      1. distribute-rgt1-in 52.1%

         \[\leadsto wj - \frac{wj \cdot e^{wj} - x}{\color{blue}{\left(wj + 1\right) \cdot e^{wj}}} \]

      2. *-commutative 52.1%

         \[\leadsto wj - \frac{wj \cdot e^{wj} - x}{\color{blue}{e^{wj} \cdot \left(wj + 1\right)}} \]

      3. associate-/r* 52.3%

         \[\leadsto wj - \color{blue}{\frac{\frac{wj \cdot e^{wj} - x}{e^{wj}}}{wj + 1}} \]

      4. div-sub 52.3%

         \[\leadsto wj - \frac{\color{blue}{\frac{wj \cdot e^{wj}}{e^{wj}} - \frac{x}{e^{wj}}}}{wj + 1} \]

      5. associate-/l* 52.3%

         \[\leadsto wj - \frac{\color{blue}{wj \cdot \frac{e^{wj}}{e^{wj}}} - \frac{x}{e^{wj}}}{wj + 1} \]

      6. *-inverses 52.3%

         \[\leadsto wj - \frac{wj \cdot \color{blue}{1} - \frac{x}{e^{wj}}}{wj + 1} \]

      7. *-rgt-identity 52.3%

         \[\leadsto wj - \frac{\color{blue}{wj} - \frac{x}{e^{wj}}}{wj + 1} \]

   3. Simplified 52.3%

      \[\leadsto \color{blue}{wj - \frac{wj - \frac{x}{e^{wj}}}{wj + 1}} \]
   4. Add Preprocessing

   5. Taylor expanded in wj around 0 99.4%

      \[\leadsto \color{blue}{x + wj \cdot \left(wj \cdot \left(\left(1 + -1 \cdot \left(wj \cdot \left(1 + \left(-3 \cdot x + \left(-2 \cdot \left(-4 \cdot x + 1.5 \cdot x\right) + 0.6666666666666666 \cdot x\right)\right)\right)\right)\right) - \left(-4 \cdot x + 1.5 \cdot x\right)\right) - 2 \cdot x\right)} \]

   6. Taylor expanded in x around 0 99.4%

      \[\leadsto x + wj \cdot \left(\color{blue}{\left(wj \cdot \left(x \cdot \left(2.5 + -2.6666666666666665 \cdot wj\right)\right) + wj \cdot \left(1 + -1 \cdot wj\right)\right)} - 2 \cdot x\right) \]
   7. Step-by-step derivation

      1. distribute-lft-out 99.4%

         \[\leadsto x + wj \cdot \left(\color{blue}{wj \cdot \left(x \cdot \left(2.5 + -2.6666666666666665 \cdot wj\right) + \left(1 + -1 \cdot wj\right)\right)} - 2 \cdot x\right) \]

      2. *-commutative 99.4%

         \[\leadsto x + wj \cdot \left(wj \cdot \left(x \cdot \left(2.5 + \color{blue}{wj \cdot -2.6666666666666665}\right) + \left(1 + -1 \cdot wj\right)\right) - 2 \cdot x\right) \]

      3. mul-1-neg 99.4%

         \[\leadsto x + wj \cdot \left(wj \cdot \left(x \cdot \left(2.5 + wj \cdot -2.6666666666666665\right) + \left(1 + \color{blue}{\left(-wj\right)}\right)\right) - 2 \cdot x\right) \]

   8. Simplified 99.4%

      \[\leadsto x + wj \cdot \left(\color{blue}{wj \cdot \left(x \cdot \left(2.5 + wj \cdot -2.6666666666666665\right) + \left(1 + \left(-wj\right)\right)\right)} - 2 \cdot x\right) \]

   9. Taylor expanded in x around 0 99.4%

      \[\leadsto x + wj \cdot \left(wj \cdot \color{blue}{\left(\left(1 + x \cdot \left(2.5 + -2.6666666666666665 \cdot wj\right)\right) - wj\right)} - 2 \cdot x\right) \]

   10. Taylor expanded in x around inf 99.5%

       \[\leadsto x + wj \cdot \left(wj \cdot \left(\color{blue}{x \cdot \left(2.5 + \left(-2.6666666666666665 \cdot wj + \frac{1}{x}\right)\right)} - wj\right) - 2 \cdot x\right) \]

   if 5.19999999999999954e-4 < wj < 5.50000000000000033e-4

   1. Initial program 100.0%

      \[wj - \frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}} \]
   2. Step-by-step derivation

      1. distribute-rgt1-in 98.4%

         \[\leadsto wj - \frac{wj \cdot e^{wj} - x}{\color{blue}{\left(wj + 1\right) \cdot e^{wj}}} \]

      2. *-commutative 98.4%

         \[\leadsto wj - \frac{wj \cdot e^{wj} - x}{\color{blue}{e^{wj} \cdot \left(wj + 1\right)}} \]

      3. associate-/r* 98.4%

         \[\leadsto wj - \color{blue}{\frac{\frac{wj \cdot e^{wj} - x}{e^{wj}}}{wj + 1}} \]

      4. div-sub 98.4%

         \[\leadsto wj - \frac{\color{blue}{\frac{wj \cdot e^{wj}}{e^{wj}} - \frac{x}{e^{wj}}}}{wj + 1} \]

      5. associate-/l* 98.4%

         \[\leadsto wj - \frac{\color{blue}{wj \cdot \frac{e^{wj}}{e^{wj}}} - \frac{x}{e^{wj}}}{wj + 1} \]

      6. *-inverses 98.4%

         \[\leadsto wj - \frac{wj \cdot \color{blue}{1} - \frac{x}{e^{wj}}}{wj + 1} \]

      7. *-rgt-identity 98.4%

         \[\leadsto wj - \frac{\color{blue}{wj} - \frac{x}{e^{wj}}}{wj + 1} \]

   3. Simplified 98.4%

      \[\leadsto \color{blue}{wj - \frac{wj - \frac{x}{e^{wj}}}{wj + 1}} \]
   4. Add Preprocessing

   5. Taylor expanded in x around inf 98.4%

      \[\leadsto wj - \color{blue}{x \cdot \left(\frac{wj}{x \cdot \left(1 + wj\right)} - \frac{1}{e^{wj} \cdot \left(1 + wj\right)}\right)} \]
   6. Step-by-step derivation

      1. +-commutative 98.4%

         \[\leadsto wj - x \cdot \left(\frac{wj}{x \cdot \color{blue}{\left(wj + 1\right)}} - \frac{1}{e^{wj} \cdot \left(1 + wj\right)}\right) \]

      2. associate-/r* 100.0%

         \[\leadsto wj - x \cdot \left(\frac{wj}{x \cdot \left(wj + 1\right)} - \color{blue}{\frac{\frac{1}{e^{wj}}}{1 + wj}}\right) \]

      3. exp-neg 100.0%

         \[\leadsto wj - x \cdot \left(\frac{wj}{x \cdot \left(wj + 1\right)} - \frac{\color{blue}{e^{-wj}}}{1 + wj}\right) \]

      4. +-commutative 100.0%

         \[\leadsto wj - x \cdot \left(\frac{wj}{x \cdot \left(wj + 1\right)} - \frac{e^{-wj}}{\color{blue}{wj + 1}}\right) \]

   7. Simplified 100.0%

      \[\leadsto wj - \color{blue}{x \cdot \left(\frac{wj}{x \cdot \left(wj + 1\right)} - \frac{e^{-wj}}{wj + 1}\right)} \]

   8. Taylor expanded in x around inf 98.4%

      \[\leadsto \color{blue}{\frac{x \cdot e^{-wj}}{1 + wj}} \]
   9. Step-by-step derivation

      1. associate-/l* 100.0%

         \[\leadsto \color{blue}{x \cdot \frac{e^{-wj}}{1 + wj}} \]

      2. +-commutative 100.0%

         \[\leadsto x \cdot \frac{e^{-wj}}{\color{blue}{wj + 1}} \]

   10. Simplified 100.0%

       \[\leadsto \color{blue}{x \cdot \frac{e^{-wj}}{wj + 1}} \]

   if 5.50000000000000033e-4 < wj < 0.110000000000000001

   1. Initial program 94.8%

      \[wj - \frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}} \]
   2. Step-by-step derivation

      1. distribute-rgt1-in 94.8%

         \[\leadsto wj - \frac{wj \cdot e^{wj} - x}{\color{blue}{\left(wj + 1\right) \cdot e^{wj}}} \]

      2. *-commutative 94.8%

         \[\leadsto wj - \frac{wj \cdot e^{wj} - x}{\color{blue}{e^{wj} \cdot \left(wj + 1\right)}} \]

   3. Simplified 94.8%

      \[\leadsto \color{blue}{wj - \frac{wj \cdot e^{wj} - x}{e^{wj} \cdot \left(wj + 1\right)}} \]
   4. Add Preprocessing
3. Recombined 6 regimes into one program.

4. Final simplification 99.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;wj \leq -4.5 \cdot 10^{-6}:\\ \;\;\;\;wj + \frac{wj - \frac{x}{e^{wj}}}{-1 - wj}\\ \mathbf{elif}\;wj \leq -5 \cdot 10^{-32}:\\ \;\;\;\;x + wj \cdot \left({wj}^{2} \cdot \left(\left(\left(-1 - \left(x \cdot -3 + \left(-2 \cdot \left(x \cdot -4 + x \cdot 1.5\right) + x \cdot 0.6666666666666666\right)\right)\right) + \frac{1}{wj}\right) - \left(-4 \cdot \frac{x}{wj} + 1.5 \cdot \frac{x}{wj}\right)\right) - x \cdot 2\right)\\ \mathbf{elif}\;wj \leq 5 \cdot 10^{-83}:\\ \;\;\;\;x - wj \cdot \left(x \cdot 2 - wj\right)\\ \mathbf{elif}\;wj \leq 0.00052:\\ \;\;\;\;x - wj \cdot \left(x \cdot 2 + wj \cdot \left(wj + x \cdot \left(\left(\frac{-1}{x} - wj \cdot -2.6666666666666665\right) - 2.5\right)\right)\right)\\ \mathbf{elif}\;wj \leq 0.00055:\\ \;\;\;\;x \cdot \frac{e^{-wj}}{wj + 1}\\ \mathbf{elif}\;wj \leq 0.11:\\ \;\;\;\;wj + \frac{wj \cdot e^{wj} - x}{e^{wj} \cdot \left(-1 - wj\right)}\\ \mathbf{else}:\\ \;\;\;\;wj + \frac{wj - \frac{x}{e^{wj}}}{-1 - wj}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 99.5% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := wj + \frac{wj - \frac{x}{e^{wj}}}{-1 - wj}\\ \mathbf{if}\;wj \leq -3.5 \cdot 10^{-6}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;wj \leq -5 \cdot 10^{-32}:\\ \;\;\;\;x + wj \cdot \left({wj}^{2} \cdot \left(\left(\left(-1 - \left(x \cdot -3 + \left(-2 \cdot \left(x \cdot -4 + x \cdot 1.5\right) + x \cdot 0.6666666666666666\right)\right)\right) + \frac{1}{wj}\right) - \left(-4 \cdot \frac{x}{wj} + 1.5 \cdot \frac{x}{wj}\right)\right) - x \cdot 2\right)\\ \mathbf{elif}\;wj \leq 5 \cdot 10^{-83}:\\ \;\;\;\;x - wj \cdot \left(x \cdot 2 - wj\right)\\ \mathbf{elif}\;wj \leq 0.00052:\\ \;\;\;\;x - wj \cdot \left(x \cdot 2 + wj \cdot \left(wj + x \cdot \left(\left(\frac{-1}{x} - wj \cdot -2.6666666666666665\right) - 2.5\right)\right)\right)\\ \mathbf{elif}\;wj \leq 0.00146:\\ \;\;\;\;x \cdot \frac{e^{-wj}}{wj + 1}\\ \mathbf{elif}\;wj \leq 0.014:\\ \;\;\;\;wj + wj \cdot \frac{e^{wj}}{e^{wj} \cdot \left(-1 - wj\right)}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (wj x)
 :precision binary64
 (let* ((t_0 (+ wj (/ (- wj (/ x (exp wj))) (- -1.0 wj)))))
   (if (<= wj -3.5e-6)
     t_0
     (if (<= wj -5e-32)
       (+
        x
        (*
         wj
         (-
          (*
           (pow wj 2.0)
           (-
            (+
             (-
              -1.0
              (+
               (* x -3.0)
               (+ (* -2.0 (+ (* x -4.0) (* x 1.5))) (* x 0.6666666666666666))))
             (/ 1.0 wj))
            (+ (* -4.0 (/ x wj)) (* 1.5 (/ x wj)))))
          (* x 2.0))))
       (if (<= wj 5e-83)
         (- x (* wj (- (* x 2.0) wj)))
         (if (<= wj 0.00052)
           (-
            x
            (*
             wj
             (+
              (* x 2.0)
              (*
               wj
               (+
                wj
                (* x (- (- (/ -1.0 x) (* wj -2.6666666666666665)) 2.5)))))))
           (if (<= wj 0.00146)
             (* x (/ (exp (- wj)) (+ wj 1.0)))
             (if (<= wj 0.014)
               (+ wj (* wj (/ (exp wj) (* (exp wj) (- -1.0 wj)))))
               t_0))))))))

C

double code(double wj, double x) {
	double t_0 = wj + ((wj - (x / exp(wj))) / (-1.0 - wj));
	double tmp;
	if (wj <= -3.5e-6) {
		tmp = t_0;
	} else if (wj <= -5e-32) {
		tmp = x + (wj * ((pow(wj, 2.0) * (((-1.0 - ((x * -3.0) + ((-2.0 * ((x * -4.0) + (x * 1.5))) + (x * 0.6666666666666666)))) + (1.0 / wj)) - ((-4.0 * (x / wj)) + (1.5 * (x / wj))))) - (x * 2.0)));
	} else if (wj <= 5e-83) {
		tmp = x - (wj * ((x * 2.0) - wj));
	} else if (wj <= 0.00052) {
		tmp = x - (wj * ((x * 2.0) + (wj * (wj + (x * (((-1.0 / x) - (wj * -2.6666666666666665)) - 2.5))))));
	} else if (wj <= 0.00146) {
		tmp = x * (exp(-wj) / (wj + 1.0));
	} else if (wj <= 0.014) {
		tmp = wj + (wj * (exp(wj) / (exp(wj) * (-1.0 - wj))));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(wj, x)
    real(8), intent (in) :: wj
    real(8), intent (in) :: x
    real(8) :: t_0
    real(8) :: tmp
    t_0 = wj + ((wj - (x / exp(wj))) / ((-1.0d0) - wj))
    if (wj <= (-3.5d-6)) then
        tmp = t_0
    else if (wj <= (-5d-32)) then
        tmp = x + (wj * (((wj ** 2.0d0) * ((((-1.0d0) - ((x * (-3.0d0)) + (((-2.0d0) * ((x * (-4.0d0)) + (x * 1.5d0))) + (x * 0.6666666666666666d0)))) + (1.0d0 / wj)) - (((-4.0d0) * (x / wj)) + (1.5d0 * (x / wj))))) - (x * 2.0d0)))
    else if (wj <= 5d-83) then
        tmp = x - (wj * ((x * 2.0d0) - wj))
    else if (wj <= 0.00052d0) then
        tmp = x - (wj * ((x * 2.0d0) + (wj * (wj + (x * ((((-1.0d0) / x) - (wj * (-2.6666666666666665d0))) - 2.5d0))))))
    else if (wj <= 0.00146d0) then
        tmp = x * (exp(-wj) / (wj + 1.0d0))
    else if (wj <= 0.014d0) then
        tmp = wj + (wj * (exp(wj) / (exp(wj) * ((-1.0d0) - wj))))
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double wj, double x) {
	double t_0 = wj + ((wj - (x / Math.exp(wj))) / (-1.0 - wj));
	double tmp;
	if (wj <= -3.5e-6) {
		tmp = t_0;
	} else if (wj <= -5e-32) {
		tmp = x + (wj * ((Math.pow(wj, 2.0) * (((-1.0 - ((x * -3.0) + ((-2.0 * ((x * -4.0) + (x * 1.5))) + (x * 0.6666666666666666)))) + (1.0 / wj)) - ((-4.0 * (x / wj)) + (1.5 * (x / wj))))) - (x * 2.0)));
	} else if (wj <= 5e-83) {
		tmp = x - (wj * ((x * 2.0) - wj));
	} else if (wj <= 0.00052) {
		tmp = x - (wj * ((x * 2.0) + (wj * (wj + (x * (((-1.0 / x) - (wj * -2.6666666666666665)) - 2.5))))));
	} else if (wj <= 0.00146) {
		tmp = x * (Math.exp(-wj) / (wj + 1.0));
	} else if (wj <= 0.014) {
		tmp = wj + (wj * (Math.exp(wj) / (Math.exp(wj) * (-1.0 - wj))));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(wj, x):
	t_0 = wj + ((wj - (x / math.exp(wj))) / (-1.0 - wj))
	tmp = 0
	if wj <= -3.5e-6:
		tmp = t_0
	elif wj <= -5e-32:
		tmp = x + (wj * ((math.pow(wj, 2.0) * (((-1.0 - ((x * -3.0) + ((-2.0 * ((x * -4.0) + (x * 1.5))) + (x * 0.6666666666666666)))) + (1.0 / wj)) - ((-4.0 * (x / wj)) + (1.5 * (x / wj))))) - (x * 2.0)))
	elif wj <= 5e-83:
		tmp = x - (wj * ((x * 2.0) - wj))
	elif wj <= 0.00052:
		tmp = x - (wj * ((x * 2.0) + (wj * (wj + (x * (((-1.0 / x) - (wj * -2.6666666666666665)) - 2.5))))))
	elif wj <= 0.00146:
		tmp = x * (math.exp(-wj) / (wj + 1.0))
	elif wj <= 0.014:
		tmp = wj + (wj * (math.exp(wj) / (math.exp(wj) * (-1.0 - wj))))
	else:
		tmp = t_0
	return tmp

Julia

function code(wj, x)
	t_0 = Float64(wj + Float64(Float64(wj - Float64(x / exp(wj))) / Float64(-1.0 - wj)))
	tmp = 0.0
	if (wj <= -3.5e-6)
		tmp = t_0;
	elseif (wj <= -5e-32)
		tmp = Float64(x + Float64(wj * Float64(Float64((wj ^ 2.0) * Float64(Float64(Float64(-1.0 - Float64(Float64(x * -3.0) + Float64(Float64(-2.0 * Float64(Float64(x * -4.0) + Float64(x * 1.5))) + Float64(x * 0.6666666666666666)))) + Float64(1.0 / wj)) - Float64(Float64(-4.0 * Float64(x / wj)) + Float64(1.5 * Float64(x / wj))))) - Float64(x * 2.0))));
	elseif (wj <= 5e-83)
		tmp = Float64(x - Float64(wj * Float64(Float64(x * 2.0) - wj)));
	elseif (wj <= 0.00052)
		tmp = Float64(x - Float64(wj * Float64(Float64(x * 2.0) + Float64(wj * Float64(wj + Float64(x * Float64(Float64(Float64(-1.0 / x) - Float64(wj * -2.6666666666666665)) - 2.5)))))));
	elseif (wj <= 0.00146)
		tmp = Float64(x * Float64(exp(Float64(-wj)) / Float64(wj + 1.0)));
	elseif (wj <= 0.014)
		tmp = Float64(wj + Float64(wj * Float64(exp(wj) / Float64(exp(wj) * Float64(-1.0 - wj)))));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(wj, x)
	t_0 = wj + ((wj - (x / exp(wj))) / (-1.0 - wj));
	tmp = 0.0;
	if (wj <= -3.5e-6)
		tmp = t_0;
	elseif (wj <= -5e-32)
		tmp = x + (wj * (((wj ^ 2.0) * (((-1.0 - ((x * -3.0) + ((-2.0 * ((x * -4.0) + (x * 1.5))) + (x * 0.6666666666666666)))) + (1.0 / wj)) - ((-4.0 * (x / wj)) + (1.5 * (x / wj))))) - (x * 2.0)));
	elseif (wj <= 5e-83)
		tmp = x - (wj * ((x * 2.0) - wj));
	elseif (wj <= 0.00052)
		tmp = x - (wj * ((x * 2.0) + (wj * (wj + (x * (((-1.0 / x) - (wj * -2.6666666666666665)) - 2.5))))));
	elseif (wj <= 0.00146)
		tmp = x * (exp(-wj) / (wj + 1.0));
	elseif (wj <= 0.014)
		tmp = wj + (wj * (exp(wj) / (exp(wj) * (-1.0 - wj))));
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[wj_, x_] := Block[{t$95$0 = N[(wj + N[(N[(wj - N[(x / N[Exp[wj], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(-1.0 - wj), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[wj, -3.5e-6], t$95$0, If[LessEqual[wj, -5e-32], N[(x + N[(wj * N[(N[(N[Power[wj, 2.0], $MachinePrecision] * N[(N[(N[(-1.0 - N[(N[(x * -3.0), $MachinePrecision] + N[(N[(-2.0 * N[(N[(x * -4.0), $MachinePrecision] + N[(x * 1.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(x * 0.6666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(1.0 / wj), $MachinePrecision]), $MachinePrecision] - N[(N[(-4.0 * N[(x / wj), $MachinePrecision]), $MachinePrecision] + N[(1.5 * N[(x / wj), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(x * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[wj, 5e-83], N[(x - N[(wj * N[(N[(x * 2.0), $MachinePrecision] - wj), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[wj, 0.00052], N[(x - N[(wj * N[(N[(x * 2.0), $MachinePrecision] + N[(wj * N[(wj + N[(x * N[(N[(N[(-1.0 / x), $MachinePrecision] - N[(wj * -2.6666666666666665), $MachinePrecision]), $MachinePrecision] - 2.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[wj, 0.00146], N[(x * N[(N[Exp[(-wj)], $MachinePrecision] / N[(wj + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[wj, 0.014], N[(wj + N[(wj * N[(N[Exp[wj], $MachinePrecision] / N[(N[Exp[wj], $MachinePrecision] * N[(-1.0 - wj), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]]]]]]

Derivation

1. Split input into 6 regimes

2. if wj < -3.49999999999999995e-6 or 0.0140000000000000003 < wj

   1. Initial program 64.6%

      \[wj - \frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}} \]
   2. Step-by-step derivation

      1. distribute-rgt1-in 89.7%

         \[\leadsto wj - \frac{wj \cdot e^{wj} - x}{\color{blue}{\left(wj + 1\right) \cdot e^{wj}}} \]

      2. *-commutative 89.7%

         \[\leadsto wj - \frac{wj \cdot e^{wj} - x}{\color{blue}{e^{wj} \cdot \left(wj + 1\right)}} \]

      3. associate-/r* 89.7%

         \[\leadsto wj - \color{blue}{\frac{\frac{wj \cdot e^{wj} - x}{e^{wj}}}{wj + 1}} \]

      4. div-sub 64.7%

         \[\leadsto wj - \frac{\color{blue}{\frac{wj \cdot e^{wj}}{e^{wj}} - \frac{x}{e^{wj}}}}{wj + 1} \]

      5. associate-/l* 64.7%

         \[\leadsto wj - \frac{\color{blue}{wj \cdot \frac{e^{wj}}{e^{wj}}} - \frac{x}{e^{wj}}}{wj + 1} \]

      6. *-inverses 98.0%

         \[\leadsto wj - \frac{wj \cdot \color{blue}{1} - \frac{x}{e^{wj}}}{wj + 1} \]

      7. *-rgt-identity 98.0%

         \[\leadsto wj - \frac{\color{blue}{wj} - \frac{x}{e^{wj}}}{wj + 1} \]

   3. Simplified 98.0%

      \[\leadsto \color{blue}{wj - \frac{wj - \frac{x}{e^{wj}}}{wj + 1}} \]
   4. Add Preprocessing

   if -3.49999999999999995e-6 < wj < -5e-32

   1. Initial program 70.8%

      \[wj - \frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}} \]
   2. Step-by-step derivation

      1. distribute-rgt1-in 70.7%

         \[\leadsto wj - \frac{wj \cdot e^{wj} - x}{\color{blue}{\left(wj + 1\right) \cdot e^{wj}}} \]

      2. *-commutative 70.7%

         \[\leadsto wj - \frac{wj \cdot e^{wj} - x}{\color{blue}{e^{wj} \cdot \left(wj + 1\right)}} \]

      3. associate-/r* 70.8%

         \[\leadsto wj - \color{blue}{\frac{\frac{wj \cdot e^{wj} - x}{e^{wj}}}{wj + 1}} \]

      4. div-sub 70.8%

         \[\leadsto wj - \frac{\color{blue}{\frac{wj \cdot e^{wj}}{e^{wj}} - \frac{x}{e^{wj}}}}{wj + 1} \]

      5. associate-/l* 70.8%

         \[\leadsto wj - \frac{\color{blue}{wj \cdot \frac{e^{wj}}{e^{wj}}} - \frac{x}{e^{wj}}}{wj + 1} \]

      6. *-inverses 70.8%

         \[\leadsto wj - \frac{wj \cdot \color{blue}{1} - \frac{x}{e^{wj}}}{wj + 1} \]

      7. *-rgt-identity 70.8%

         \[\leadsto wj - \frac{\color{blue}{wj} - \frac{x}{e^{wj}}}{wj + 1} \]

   3. Simplified 70.8%

      \[\leadsto \color{blue}{wj - \frac{wj - \frac{x}{e^{wj}}}{wj + 1}} \]
   4. Add Preprocessing

   5. Taylor expanded in wj around 0 98.4%

      \[\leadsto \color{blue}{x + wj \cdot \left(wj \cdot \left(\left(1 + -1 \cdot \left(wj \cdot \left(1 + \left(-3 \cdot x + \left(-2 \cdot \left(-4 \cdot x + 1.5 \cdot x\right) + 0.6666666666666666 \cdot x\right)\right)\right)\right)\right) - \left(-4 \cdot x + 1.5 \cdot x\right)\right) - 2 \cdot x\right)} \]

   6. Taylor expanded in wj around inf 98.5%

      \[\leadsto x + wj \cdot \left(\color{blue}{{wj}^{2} \cdot \left(\left(-1 \cdot \left(1 + \left(-3 \cdot x + \left(-2 \cdot \left(-4 \cdot x + 1.5 \cdot x\right) + 0.6666666666666666 \cdot x\right)\right)\right) + \frac{1}{wj}\right) - \left(-4 \cdot \frac{x}{wj} + 1.5 \cdot \frac{x}{wj}\right)\right)} - 2 \cdot x\right) \]

   if -5e-32 < wj < 5e-83

   1. Initial program 81.3%

      \[wj - \frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}} \]
   2. Step-by-step derivation

      1. distribute-rgt1-in 81.3%

         \[\leadsto wj - \frac{wj \cdot e^{wj} - x}{\color{blue}{\left(wj + 1\right) \cdot e^{wj}}} \]

      2. *-commutative 81.3%

         \[\leadsto wj - \frac{wj \cdot e^{wj} - x}{\color{blue}{e^{wj} \cdot \left(wj + 1\right)}} \]

      3. associate-/r* 81.3%

         \[\leadsto wj - \color{blue}{\frac{\frac{wj \cdot e^{wj} - x}{e^{wj}}}{wj + 1}} \]

      4. div-sub 81.3%

         \[\leadsto wj - \frac{\color{blue}{\frac{wj \cdot e^{wj}}{e^{wj}} - \frac{x}{e^{wj}}}}{wj + 1} \]

      5. associate-/l* 81.3%

         \[\leadsto wj - \frac{\color{blue}{wj \cdot \frac{e^{wj}}{e^{wj}}} - \frac{x}{e^{wj}}}{wj + 1} \]

      6. *-inverses 81.3%

         \[\leadsto wj - \frac{wj \cdot \color{blue}{1} - \frac{x}{e^{wj}}}{wj + 1} \]

      7. *-rgt-identity 81.3%

         \[\leadsto wj - \frac{\color{blue}{wj} - \frac{x}{e^{wj}}}{wj + 1} \]

   3. Simplified 81.3%

      \[\leadsto \color{blue}{wj - \frac{wj - \frac{x}{e^{wj}}}{wj + 1}} \]
   4. Add Preprocessing

   5. Taylor expanded in wj around 0 99.6%

      \[\leadsto \color{blue}{x + wj \cdot \left(wj \cdot \left(\left(1 + -1 \cdot \left(wj \cdot \left(1 + \left(-3 \cdot x + \left(-2 \cdot \left(-4 \cdot x + 1.5 \cdot x\right) + 0.6666666666666666 \cdot x\right)\right)\right)\right)\right) - \left(-4 \cdot x + 1.5 \cdot x\right)\right) - 2 \cdot x\right)} \]

   6. Taylor expanded in x around 0 100.0%

      \[\leadsto x + wj \cdot \left(\color{blue}{wj \cdot \left(1 + -1 \cdot wj\right)} - 2 \cdot x\right) \]
   7. Step-by-step derivation

      1. mul-1-neg 100.0%

         \[\leadsto x + wj \cdot \left(wj \cdot \left(1 + \color{blue}{\left(-wj\right)}\right) - 2 \cdot x\right) \]

      2. unsub-neg 100.0%

         \[\leadsto x + wj \cdot \left(wj \cdot \color{blue}{\left(1 - wj\right)} - 2 \cdot x\right) \]

   8. Simplified 100.0%

      \[\leadsto x + wj \cdot \left(\color{blue}{wj \cdot \left(1 - wj\right)} - 2 \cdot x\right) \]

   9. Taylor expanded in wj around 0 100.0%

      \[\leadsto x + wj \cdot \left(\color{blue}{wj} - 2 \cdot x\right) \]

   if 5e-83 < wj < 5.19999999999999954e-4

   1. Initial program 52.2%

      \[wj - \frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}} \]
   2. Step-by-step derivation

      1. distribute-rgt1-in 52.1%

         \[\leadsto wj - \frac{wj \cdot e^{wj} - x}{\color{blue}{\left(wj + 1\right) \cdot e^{wj}}} \]

      2. *-commutative 52.1%

         \[\leadsto wj - \frac{wj \cdot e^{wj} - x}{\color{blue}{e^{wj} \cdot \left(wj + 1\right)}} \]

      3. associate-/r* 52.3%

         \[\leadsto wj - \color{blue}{\frac{\frac{wj \cdot e^{wj} - x}{e^{wj}}}{wj + 1}} \]

      4. div-sub 52.3%

         \[\leadsto wj - \frac{\color{blue}{\frac{wj \cdot e^{wj}}{e^{wj}} - \frac{x}{e^{wj}}}}{wj + 1} \]

      5. associate-/l* 52.3%

         \[\leadsto wj - \frac{\color{blue}{wj \cdot \frac{e^{wj}}{e^{wj}}} - \frac{x}{e^{wj}}}{wj + 1} \]

      6. *-inverses 52.3%

         \[\leadsto wj - \frac{wj \cdot \color{blue}{1} - \frac{x}{e^{wj}}}{wj + 1} \]

      7. *-rgt-identity 52.3%

         \[\leadsto wj - \frac{\color{blue}{wj} - \frac{x}{e^{wj}}}{wj + 1} \]

   3. Simplified 52.3%

      \[\leadsto \color{blue}{wj - \frac{wj - \frac{x}{e^{wj}}}{wj + 1}} \]
   4. Add Preprocessing

   5. Taylor expanded in wj around 0 99.4%

      \[\leadsto \color{blue}{x + wj \cdot \left(wj \cdot \left(\left(1 + -1 \cdot \left(wj \cdot \left(1 + \left(-3 \cdot x + \left(-2 \cdot \left(-4 \cdot x + 1.5 \cdot x\right) + 0.6666666666666666 \cdot x\right)\right)\right)\right)\right) - \left(-4 \cdot x + 1.5 \cdot x\right)\right) - 2 \cdot x\right)} \]

   6. Taylor expanded in x around 0 99.4%

      \[\leadsto x + wj \cdot \left(\color{blue}{\left(wj \cdot \left(x \cdot \left(2.5 + -2.6666666666666665 \cdot wj\right)\right) + wj \cdot \left(1 + -1 \cdot wj\right)\right)} - 2 \cdot x\right) \]
   7. Step-by-step derivation

      1. distribute-lft-out 99.4%

         \[\leadsto x + wj \cdot \left(\color{blue}{wj \cdot \left(x \cdot \left(2.5 + -2.6666666666666665 \cdot wj\right) + \left(1 + -1 \cdot wj\right)\right)} - 2 \cdot x\right) \]

      2. *-commutative 99.4%

         \[\leadsto x + wj \cdot \left(wj \cdot \left(x \cdot \left(2.5 + \color{blue}{wj \cdot -2.6666666666666665}\right) + \left(1 + -1 \cdot wj\right)\right) - 2 \cdot x\right) \]

      3. mul-1-neg 99.4%

         \[\leadsto x + wj \cdot \left(wj \cdot \left(x \cdot \left(2.5 + wj \cdot -2.6666666666666665\right) + \left(1 + \color{blue}{\left(-wj\right)}\right)\right) - 2 \cdot x\right) \]

   8. Simplified 99.4%

      \[\leadsto x + wj \cdot \left(\color{blue}{wj \cdot \left(x \cdot \left(2.5 + wj \cdot -2.6666666666666665\right) + \left(1 + \left(-wj\right)\right)\right)} - 2 \cdot x\right) \]

   9. Taylor expanded in x around 0 99.4%

      \[\leadsto x + wj \cdot \left(wj \cdot \color{blue}{\left(\left(1 + x \cdot \left(2.5 + -2.6666666666666665 \cdot wj\right)\right) - wj\right)} - 2 \cdot x\right) \]

   10. Taylor expanded in x around inf 99.5%

       \[\leadsto x + wj \cdot \left(wj \cdot \left(\color{blue}{x \cdot \left(2.5 + \left(-2.6666666666666665 \cdot wj + \frac{1}{x}\right)\right)} - wj\right) - 2 \cdot x\right) \]

   if 5.19999999999999954e-4 < wj < 0.0014599999999999999

   1. Initial program 100.0%

      \[wj - \frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}} \]
   2. Step-by-step derivation

      1. distribute-rgt1-in 98.4%

         \[\leadsto wj - \frac{wj \cdot e^{wj} - x}{\color{blue}{\left(wj + 1\right) \cdot e^{wj}}} \]

      2. *-commutative 98.4%

         \[\leadsto wj - \frac{wj \cdot e^{wj} - x}{\color{blue}{e^{wj} \cdot \left(wj + 1\right)}} \]

      3. associate-/r* 98.4%

         \[\leadsto wj - \color{blue}{\frac{\frac{wj \cdot e^{wj} - x}{e^{wj}}}{wj + 1}} \]

      4. div-sub 98.4%

         \[\leadsto wj - \frac{\color{blue}{\frac{wj \cdot e^{wj}}{e^{wj}} - \frac{x}{e^{wj}}}}{wj + 1} \]

      5. associate-/l* 98.4%

         \[\leadsto wj - \frac{\color{blue}{wj \cdot \frac{e^{wj}}{e^{wj}}} - \frac{x}{e^{wj}}}{wj + 1} \]

      6. *-inverses 98.4%

         \[\leadsto wj - \frac{wj \cdot \color{blue}{1} - \frac{x}{e^{wj}}}{wj + 1} \]

      7. *-rgt-identity 98.4%

         \[\leadsto wj - \frac{\color{blue}{wj} - \frac{x}{e^{wj}}}{wj + 1} \]

   3. Simplified 98.4%

      \[\leadsto \color{blue}{wj - \frac{wj - \frac{x}{e^{wj}}}{wj + 1}} \]
   4. Add Preprocessing

   5. Taylor expanded in x around inf 98.4%

      \[\leadsto wj - \color{blue}{x \cdot \left(\frac{wj}{x \cdot \left(1 + wj\right)} - \frac{1}{e^{wj} \cdot \left(1 + wj\right)}\right)} \]
   6. Step-by-step derivation

      1. +-commutative 98.4%

         \[\leadsto wj - x \cdot \left(\frac{wj}{x \cdot \color{blue}{\left(wj + 1\right)}} - \frac{1}{e^{wj} \cdot \left(1 + wj\right)}\right) \]

      2. associate-/r* 100.0%

         \[\leadsto wj - x \cdot \left(\frac{wj}{x \cdot \left(wj + 1\right)} - \color{blue}{\frac{\frac{1}{e^{wj}}}{1 + wj}}\right) \]

      3. exp-neg 100.0%

         \[\leadsto wj - x \cdot \left(\frac{wj}{x \cdot \left(wj + 1\right)} - \frac{\color{blue}{e^{-wj}}}{1 + wj}\right) \]

      4. +-commutative 100.0%

         \[\leadsto wj - x \cdot \left(\frac{wj}{x \cdot \left(wj + 1\right)} - \frac{e^{-wj}}{\color{blue}{wj + 1}}\right) \]

   7. Simplified 100.0%

      \[\leadsto wj - \color{blue}{x \cdot \left(\frac{wj}{x \cdot \left(wj + 1\right)} - \frac{e^{-wj}}{wj + 1}\right)} \]

   8. Taylor expanded in x around inf 98.4%

      \[\leadsto \color{blue}{\frac{x \cdot e^{-wj}}{1 + wj}} \]
   9. Step-by-step derivation

      1. associate-/l* 100.0%

         \[\leadsto \color{blue}{x \cdot \frac{e^{-wj}}{1 + wj}} \]

      2. +-commutative 100.0%

         \[\leadsto x \cdot \frac{e^{-wj}}{\color{blue}{wj + 1}} \]

   10. Simplified 100.0%

       \[\leadsto \color{blue}{x \cdot \frac{e^{-wj}}{wj + 1}} \]

   if 0.0014599999999999999 < wj < 0.0140000000000000003

   1. Initial program 94.8%

      \[wj - \frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0 94.8%

      \[\leadsto \color{blue}{wj - \frac{wj \cdot e^{wj}}{e^{wj} + wj \cdot e^{wj}}} \]
   4. Step-by-step derivation

      1. associate-/l* 94.8%

         \[\leadsto wj - \color{blue}{wj \cdot \frac{e^{wj}}{e^{wj} + wj \cdot e^{wj}}} \]

      2. distribute-rgt1-in 94.8%

         \[\leadsto wj - wj \cdot \frac{e^{wj}}{\color{blue}{\left(wj + 1\right) \cdot e^{wj}}} \]

      3. *-commutative 94.8%

         \[\leadsto wj - wj \cdot \frac{e^{wj}}{\color{blue}{e^{wj} \cdot \left(wj + 1\right)}} \]

   5. Simplified 94.8%

      \[\leadsto \color{blue}{wj - wj \cdot \frac{e^{wj}}{e^{wj} \cdot \left(wj + 1\right)}} \]
3. Recombined 6 regimes into one program.

4. Final simplification 99.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;wj \leq -3.5 \cdot 10^{-6}:\\ \;\;\;\;wj + \frac{wj - \frac{x}{e^{wj}}}{-1 - wj}\\ \mathbf{elif}\;wj \leq -5 \cdot 10^{-32}:\\ \;\;\;\;x + wj \cdot \left({wj}^{2} \cdot \left(\left(\left(-1 - \left(x \cdot -3 + \left(-2 \cdot \left(x \cdot -4 + x \cdot 1.5\right) + x \cdot 0.6666666666666666\right)\right)\right) + \frac{1}{wj}\right) - \left(-4 \cdot \frac{x}{wj} + 1.5 \cdot \frac{x}{wj}\right)\right) - x \cdot 2\right)\\ \mathbf{elif}\;wj \leq 5 \cdot 10^{-83}:\\ \;\;\;\;x - wj \cdot \left(x \cdot 2 - wj\right)\\ \mathbf{elif}\;wj \leq 0.00052:\\ \;\;\;\;x - wj \cdot \left(x \cdot 2 + wj \cdot \left(wj + x \cdot \left(\left(\frac{-1}{x} - wj \cdot -2.6666666666666665\right) - 2.5\right)\right)\right)\\ \mathbf{elif}\;wj \leq 0.00146:\\ \;\;\;\;x \cdot \frac{e^{-wj}}{wj + 1}\\ \mathbf{elif}\;wj \leq 0.014:\\ \;\;\;\;wj + wj \cdot \frac{e^{wj}}{e^{wj} \cdot \left(-1 - wj\right)}\\ \mathbf{else}:\\ \;\;\;\;wj + \frac{wj - \frac{x}{e^{wj}}}{-1 - wj}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 99.6% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := wj + \frac{wj - \frac{x}{e^{wj}}}{-1 - wj}\\ \mathbf{if}\;wj \leq -4.5 \cdot 10^{-6}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;wj \leq -5 \cdot 10^{-32}:\\ \;\;\;\;x + wj \cdot \left({wj}^{2} \cdot \left(\left(\left(-1 - \left(x \cdot -3 + \left(-2 \cdot \left(x \cdot -4 + x \cdot 1.5\right) + x \cdot 0.6666666666666666\right)\right)\right) + \frac{1}{wj}\right) - \left(-4 \cdot \frac{x}{wj} + 1.5 \cdot \frac{x}{wj}\right)\right) - x \cdot 2\right)\\ \mathbf{elif}\;wj \leq 5 \cdot 10^{-83}:\\ \;\;\;\;x - wj \cdot \left(x \cdot 2 - wj\right)\\ \mathbf{elif}\;wj \leq 3.75 \cdot 10^{-6}:\\ \;\;\;\;x - wj \cdot \left(x \cdot 2 + wj \cdot \left(wj + x \cdot \left(\left(\frac{-1}{x} - wj \cdot -2.6666666666666665\right) - 2.5\right)\right)\right)\\ \mathbf{elif}\;wj \leq 0.42:\\ \;\;\;\;x \cdot \left(\left(\frac{wj}{x} - \frac{e^{-wj}}{-1 - wj}\right) + \frac{wj}{x \cdot \left(-1 - wj\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (wj x)
 :precision binary64
 (let* ((t_0 (+ wj (/ (- wj (/ x (exp wj))) (- -1.0 wj)))))
   (if (<= wj -4.5e-6)
     t_0
     (if (<= wj -5e-32)
       (+
        x
        (*
         wj
         (-
          (*
           (pow wj 2.0)
           (-
            (+
             (-
              -1.0
              (+
               (* x -3.0)
               (+ (* -2.0 (+ (* x -4.0) (* x 1.5))) (* x 0.6666666666666666))))
             (/ 1.0 wj))
            (+ (* -4.0 (/ x wj)) (* 1.5 (/ x wj)))))
          (* x 2.0))))
       (if (<= wj 5e-83)
         (- x (* wj (- (* x 2.0) wj)))
         (if (<= wj 3.75e-6)
           (-
            x
            (*
             wj
             (+
              (* x 2.0)
              (*
               wj
               (+
                wj
                (* x (- (- (/ -1.0 x) (* wj -2.6666666666666665)) 2.5)))))))
           (if (<= wj 0.42)
             (*
              x
              (+
               (- (/ wj x) (/ (exp (- wj)) (- -1.0 wj)))
               (/ wj (* x (- -1.0 wj)))))
             t_0)))))))

C

double code(double wj, double x) {
	double t_0 = wj + ((wj - (x / exp(wj))) / (-1.0 - wj));
	double tmp;
	if (wj <= -4.5e-6) {
		tmp = t_0;
	} else if (wj <= -5e-32) {
		tmp = x + (wj * ((pow(wj, 2.0) * (((-1.0 - ((x * -3.0) + ((-2.0 * ((x * -4.0) + (x * 1.5))) + (x * 0.6666666666666666)))) + (1.0 / wj)) - ((-4.0 * (x / wj)) + (1.5 * (x / wj))))) - (x * 2.0)));
	} else if (wj <= 5e-83) {
		tmp = x - (wj * ((x * 2.0) - wj));
	} else if (wj <= 3.75e-6) {
		tmp = x - (wj * ((x * 2.0) + (wj * (wj + (x * (((-1.0 / x) - (wj * -2.6666666666666665)) - 2.5))))));
	} else if (wj <= 0.42) {
		tmp = x * (((wj / x) - (exp(-wj) / (-1.0 - wj))) + (wj / (x * (-1.0 - wj))));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(wj, x)
    real(8), intent (in) :: wj
    real(8), intent (in) :: x
    real(8) :: t_0
    real(8) :: tmp
    t_0 = wj + ((wj - (x / exp(wj))) / ((-1.0d0) - wj))
    if (wj <= (-4.5d-6)) then
        tmp = t_0
    else if (wj <= (-5d-32)) then
        tmp = x + (wj * (((wj ** 2.0d0) * ((((-1.0d0) - ((x * (-3.0d0)) + (((-2.0d0) * ((x * (-4.0d0)) + (x * 1.5d0))) + (x * 0.6666666666666666d0)))) + (1.0d0 / wj)) - (((-4.0d0) * (x / wj)) + (1.5d0 * (x / wj))))) - (x * 2.0d0)))
    else if (wj <= 5d-83) then
        tmp = x - (wj * ((x * 2.0d0) - wj))
    else if (wj <= 3.75d-6) then
        tmp = x - (wj * ((x * 2.0d0) + (wj * (wj + (x * ((((-1.0d0) / x) - (wj * (-2.6666666666666665d0))) - 2.5d0))))))
    else if (wj <= 0.42d0) then
        tmp = x * (((wj / x) - (exp(-wj) / ((-1.0d0) - wj))) + (wj / (x * ((-1.0d0) - wj))))
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double wj, double x) {
	double t_0 = wj + ((wj - (x / Math.exp(wj))) / (-1.0 - wj));
	double tmp;
	if (wj <= -4.5e-6) {
		tmp = t_0;
	} else if (wj <= -5e-32) {
		tmp = x + (wj * ((Math.pow(wj, 2.0) * (((-1.0 - ((x * -3.0) + ((-2.0 * ((x * -4.0) + (x * 1.5))) + (x * 0.6666666666666666)))) + (1.0 / wj)) - ((-4.0 * (x / wj)) + (1.5 * (x / wj))))) - (x * 2.0)));
	} else if (wj <= 5e-83) {
		tmp = x - (wj * ((x * 2.0) - wj));
	} else if (wj <= 3.75e-6) {
		tmp = x - (wj * ((x * 2.0) + (wj * (wj + (x * (((-1.0 / x) - (wj * -2.6666666666666665)) - 2.5))))));
	} else if (wj <= 0.42) {
		tmp = x * (((wj / x) - (Math.exp(-wj) / (-1.0 - wj))) + (wj / (x * (-1.0 - wj))));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(wj, x):
	t_0 = wj + ((wj - (x / math.exp(wj))) / (-1.0 - wj))
	tmp = 0
	if wj <= -4.5e-6:
		tmp = t_0
	elif wj <= -5e-32:
		tmp = x + (wj * ((math.pow(wj, 2.0) * (((-1.0 - ((x * -3.0) + ((-2.0 * ((x * -4.0) + (x * 1.5))) + (x * 0.6666666666666666)))) + (1.0 / wj)) - ((-4.0 * (x / wj)) + (1.5 * (x / wj))))) - (x * 2.0)))
	elif wj <= 5e-83:
		tmp = x - (wj * ((x * 2.0) - wj))
	elif wj <= 3.75e-6:
		tmp = x - (wj * ((x * 2.0) + (wj * (wj + (x * (((-1.0 / x) - (wj * -2.6666666666666665)) - 2.5))))))
	elif wj <= 0.42:
		tmp = x * (((wj / x) - (math.exp(-wj) / (-1.0 - wj))) + (wj / (x * (-1.0 - wj))))
	else:
		tmp = t_0
	return tmp

Julia

function code(wj, x)
	t_0 = Float64(wj + Float64(Float64(wj - Float64(x / exp(wj))) / Float64(-1.0 - wj)))
	tmp = 0.0
	if (wj <= -4.5e-6)
		tmp = t_0;
	elseif (wj <= -5e-32)
		tmp = Float64(x + Float64(wj * Float64(Float64((wj ^ 2.0) * Float64(Float64(Float64(-1.0 - Float64(Float64(x * -3.0) + Float64(Float64(-2.0 * Float64(Float64(x * -4.0) + Float64(x * 1.5))) + Float64(x * 0.6666666666666666)))) + Float64(1.0 / wj)) - Float64(Float64(-4.0 * Float64(x / wj)) + Float64(1.5 * Float64(x / wj))))) - Float64(x * 2.0))));
	elseif (wj <= 5e-83)
		tmp = Float64(x - Float64(wj * Float64(Float64(x * 2.0) - wj)));
	elseif (wj <= 3.75e-6)
		tmp = Float64(x - Float64(wj * Float64(Float64(x * 2.0) + Float64(wj * Float64(wj + Float64(x * Float64(Float64(Float64(-1.0 / x) - Float64(wj * -2.6666666666666665)) - 2.5)))))));
	elseif (wj <= 0.42)
		tmp = Float64(x * Float64(Float64(Float64(wj / x) - Float64(exp(Float64(-wj)) / Float64(-1.0 - wj))) + Float64(wj / Float64(x * Float64(-1.0 - wj)))));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(wj, x)
	t_0 = wj + ((wj - (x / exp(wj))) / (-1.0 - wj));
	tmp = 0.0;
	if (wj <= -4.5e-6)
		tmp = t_0;
	elseif (wj <= -5e-32)
		tmp = x + (wj * (((wj ^ 2.0) * (((-1.0 - ((x * -3.0) + ((-2.0 * ((x * -4.0) + (x * 1.5))) + (x * 0.6666666666666666)))) + (1.0 / wj)) - ((-4.0 * (x / wj)) + (1.5 * (x / wj))))) - (x * 2.0)));
	elseif (wj <= 5e-83)
		tmp = x - (wj * ((x * 2.0) - wj));
	elseif (wj <= 3.75e-6)
		tmp = x - (wj * ((x * 2.0) + (wj * (wj + (x * (((-1.0 / x) - (wj * -2.6666666666666665)) - 2.5))))));
	elseif (wj <= 0.42)
		tmp = x * (((wj / x) - (exp(-wj) / (-1.0 - wj))) + (wj / (x * (-1.0 - wj))));
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[wj_, x_] := Block[{t$95$0 = N[(wj + N[(N[(wj - N[(x / N[Exp[wj], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(-1.0 - wj), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[wj, -4.5e-6], t$95$0, If[LessEqual[wj, -5e-32], N[(x + N[(wj * N[(N[(N[Power[wj, 2.0], $MachinePrecision] * N[(N[(N[(-1.0 - N[(N[(x * -3.0), $MachinePrecision] + N[(N[(-2.0 * N[(N[(x * -4.0), $MachinePrecision] + N[(x * 1.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(x * 0.6666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(1.0 / wj), $MachinePrecision]), $MachinePrecision] - N[(N[(-4.0 * N[(x / wj), $MachinePrecision]), $MachinePrecision] + N[(1.5 * N[(x / wj), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(x * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[wj, 5e-83], N[(x - N[(wj * N[(N[(x * 2.0), $MachinePrecision] - wj), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[wj, 3.75e-6], N[(x - N[(wj * N[(N[(x * 2.0), $MachinePrecision] + N[(wj * N[(wj + N[(x * N[(N[(N[(-1.0 / x), $MachinePrecision] - N[(wj * -2.6666666666666665), $MachinePrecision]), $MachinePrecision] - 2.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[wj, 0.42], N[(x * N[(N[(N[(wj / x), $MachinePrecision] - N[(N[Exp[(-wj)], $MachinePrecision] / N[(-1.0 - wj), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(wj / N[(x * N[(-1.0 - wj), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]]]]]

Derivation

1. Split input into 5 regimes

2. if wj < -4.50000000000000011e-6 or 0.419999999999999984 < wj

   1. Initial program 64.6%

      \[wj - \frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}} \]
   2. Step-by-step derivation

      1. distribute-rgt1-in 89.7%

         \[\leadsto wj - \frac{wj \cdot e^{wj} - x}{\color{blue}{\left(wj + 1\right) \cdot e^{wj}}} \]

      2. *-commutative 89.7%

         \[\leadsto wj - \frac{wj \cdot e^{wj} - x}{\color{blue}{e^{wj} \cdot \left(wj + 1\right)}} \]

      3. associate-/r* 89.7%

         \[\leadsto wj - \color{blue}{\frac{\frac{wj \cdot e^{wj} - x}{e^{wj}}}{wj + 1}} \]

      4. div-sub 64.7%

         \[\leadsto wj - \frac{\color{blue}{\frac{wj \cdot e^{wj}}{e^{wj}} - \frac{x}{e^{wj}}}}{wj + 1} \]

      5. associate-/l* 64.7%

         \[\leadsto wj - \frac{\color{blue}{wj \cdot \frac{e^{wj}}{e^{wj}}} - \frac{x}{e^{wj}}}{wj + 1} \]

      6. *-inverses 98.0%

         \[\leadsto wj - \frac{wj \cdot \color{blue}{1} - \frac{x}{e^{wj}}}{wj + 1} \]

      7. *-rgt-identity 98.0%

         \[\leadsto wj - \frac{\color{blue}{wj} - \frac{x}{e^{wj}}}{wj + 1} \]

   3. Simplified 98.0%

      \[\leadsto \color{blue}{wj - \frac{wj - \frac{x}{e^{wj}}}{wj + 1}} \]
   4. Add Preprocessing

   if -4.50000000000000011e-6 < wj < -5e-32

   1. Initial program 70.8%

      \[wj - \frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}} \]
   2. Step-by-step derivation

      1. distribute-rgt1-in 70.7%

         \[\leadsto wj - \frac{wj \cdot e^{wj} - x}{\color{blue}{\left(wj + 1\right) \cdot e^{wj}}} \]

      2. *-commutative 70.7%

         \[\leadsto wj - \frac{wj \cdot e^{wj} - x}{\color{blue}{e^{wj} \cdot \left(wj + 1\right)}} \]

      3. associate-/r* 70.8%

         \[\leadsto wj - \color{blue}{\frac{\frac{wj \cdot e^{wj} - x}{e^{wj}}}{wj + 1}} \]

      4. div-sub 70.8%

         \[\leadsto wj - \frac{\color{blue}{\frac{wj \cdot e^{wj}}{e^{wj}} - \frac{x}{e^{wj}}}}{wj + 1} \]

      5. associate-/l* 70.8%

         \[\leadsto wj - \frac{\color{blue}{wj \cdot \frac{e^{wj}}{e^{wj}}} - \frac{x}{e^{wj}}}{wj + 1} \]

      6. *-inverses 70.8%

         \[\leadsto wj - \frac{wj \cdot \color{blue}{1} - \frac{x}{e^{wj}}}{wj + 1} \]

      7. *-rgt-identity 70.8%

         \[\leadsto wj - \frac{\color{blue}{wj} - \frac{x}{e^{wj}}}{wj + 1} \]

   3. Simplified 70.8%

      \[\leadsto \color{blue}{wj - \frac{wj - \frac{x}{e^{wj}}}{wj + 1}} \]
   4. Add Preprocessing

   5. Taylor expanded in wj around 0 98.4%

      \[\leadsto \color{blue}{x + wj \cdot \left(wj \cdot \left(\left(1 + -1 \cdot \left(wj \cdot \left(1 + \left(-3 \cdot x + \left(-2 \cdot \left(-4 \cdot x + 1.5 \cdot x\right) + 0.6666666666666666 \cdot x\right)\right)\right)\right)\right) - \left(-4 \cdot x + 1.5 \cdot x\right)\right) - 2 \cdot x\right)} \]

   6. Taylor expanded in wj around inf 98.5%

      \[\leadsto x + wj \cdot \left(\color{blue}{{wj}^{2} \cdot \left(\left(-1 \cdot \left(1 + \left(-3 \cdot x + \left(-2 \cdot \left(-4 \cdot x + 1.5 \cdot x\right) + 0.6666666666666666 \cdot x\right)\right)\right) + \frac{1}{wj}\right) - \left(-4 \cdot \frac{x}{wj} + 1.5 \cdot \frac{x}{wj}\right)\right)} - 2 \cdot x\right) \]

   if -5e-32 < wj < 5e-83

   1. Initial program 81.3%

      \[wj - \frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}} \]
   2. Step-by-step derivation

      1. distribute-rgt1-in 81.3%

         \[\leadsto wj - \frac{wj \cdot e^{wj} - x}{\color{blue}{\left(wj + 1\right) \cdot e^{wj}}} \]

      2. *-commutative 81.3%

         \[\leadsto wj - \frac{wj \cdot e^{wj} - x}{\color{blue}{e^{wj} \cdot \left(wj + 1\right)}} \]

      3. associate-/r* 81.3%

         \[\leadsto wj - \color{blue}{\frac{\frac{wj \cdot e^{wj} - x}{e^{wj}}}{wj + 1}} \]

      4. div-sub 81.3%

         \[\leadsto wj - \frac{\color{blue}{\frac{wj \cdot e^{wj}}{e^{wj}} - \frac{x}{e^{wj}}}}{wj + 1} \]

      5. associate-/l* 81.3%

         \[\leadsto wj - \frac{\color{blue}{wj \cdot \frac{e^{wj}}{e^{wj}}} - \frac{x}{e^{wj}}}{wj + 1} \]

      6. *-inverses 81.3%

         \[\leadsto wj - \frac{wj \cdot \color{blue}{1} - \frac{x}{e^{wj}}}{wj + 1} \]

      7. *-rgt-identity 81.3%

         \[\leadsto wj - \frac{\color{blue}{wj} - \frac{x}{e^{wj}}}{wj + 1} \]

   3. Simplified 81.3%

      \[\leadsto \color{blue}{wj - \frac{wj - \frac{x}{e^{wj}}}{wj + 1}} \]
   4. Add Preprocessing

   5. Taylor expanded in wj around 0 99.6%

      \[\leadsto \color{blue}{x + wj \cdot \left(wj \cdot \left(\left(1 + -1 \cdot \left(wj \cdot \left(1 + \left(-3 \cdot x + \left(-2 \cdot \left(-4 \cdot x + 1.5 \cdot x\right) + 0.6666666666666666 \cdot x\right)\right)\right)\right)\right) - \left(-4 \cdot x + 1.5 \cdot x\right)\right) - 2 \cdot x\right)} \]

   6. Taylor expanded in x around 0 100.0%

      \[\leadsto x + wj \cdot \left(\color{blue}{wj \cdot \left(1 + -1 \cdot wj\right)} - 2 \cdot x\right) \]
   7. Step-by-step derivation

      1. mul-1-neg 100.0%

         \[\leadsto x + wj \cdot \left(wj \cdot \left(1 + \color{blue}{\left(-wj\right)}\right) - 2 \cdot x\right) \]

      2. unsub-neg 100.0%

         \[\leadsto x + wj \cdot \left(wj \cdot \color{blue}{\left(1 - wj\right)} - 2 \cdot x\right) \]

   8. Simplified 100.0%

      \[\leadsto x + wj \cdot \left(\color{blue}{wj \cdot \left(1 - wj\right)} - 2 \cdot x\right) \]

   9. Taylor expanded in wj around 0 100.0%

      \[\leadsto x + wj \cdot \left(\color{blue}{wj} - 2 \cdot x\right) \]

   if 5e-83 < wj < 3.7500000000000001e-6

   1. Initial program 52.2%

      \[wj - \frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}} \]
   2. Step-by-step derivation

      1. distribute-rgt1-in 52.1%

         \[\leadsto wj - \frac{wj \cdot e^{wj} - x}{\color{blue}{\left(wj + 1\right) \cdot e^{wj}}} \]

      2. *-commutative 52.1%

         \[\leadsto wj - \frac{wj \cdot e^{wj} - x}{\color{blue}{e^{wj} \cdot \left(wj + 1\right)}} \]

      3. associate-/r* 52.3%

         \[\leadsto wj - \color{blue}{\frac{\frac{wj \cdot e^{wj} - x}{e^{wj}}}{wj + 1}} \]

      4. div-sub 52.3%

         \[\leadsto wj - \frac{\color{blue}{\frac{wj \cdot e^{wj}}{e^{wj}} - \frac{x}{e^{wj}}}}{wj + 1} \]

      5. associate-/l* 52.3%

         \[\leadsto wj - \frac{\color{blue}{wj \cdot \frac{e^{wj}}{e^{wj}}} - \frac{x}{e^{wj}}}{wj + 1} \]

      6. *-inverses 52.3%

         \[\leadsto wj - \frac{wj \cdot \color{blue}{1} - \frac{x}{e^{wj}}}{wj + 1} \]

      7. *-rgt-identity 52.3%

         \[\leadsto wj - \frac{\color{blue}{wj} - \frac{x}{e^{wj}}}{wj + 1} \]

   3. Simplified 52.3%

      \[\leadsto \color{blue}{wj - \frac{wj - \frac{x}{e^{wj}}}{wj + 1}} \]
   4. Add Preprocessing

   5. Taylor expanded in wj around 0 99.4%

      \[\leadsto \color{blue}{x + wj \cdot \left(wj \cdot \left(\left(1 + -1 \cdot \left(wj \cdot \left(1 + \left(-3 \cdot x + \left(-2 \cdot \left(-4 \cdot x + 1.5 \cdot x\right) + 0.6666666666666666 \cdot x\right)\right)\right)\right)\right) - \left(-4 \cdot x + 1.5 \cdot x\right)\right) - 2 \cdot x\right)} \]

   6. Taylor expanded in x around 0 99.4%

      \[\leadsto x + wj \cdot \left(\color{blue}{\left(wj \cdot \left(x \cdot \left(2.5 + -2.6666666666666665 \cdot wj\right)\right) + wj \cdot \left(1 + -1 \cdot wj\right)\right)} - 2 \cdot x\right) \]
   7. Step-by-step derivation

      1. distribute-lft-out 99.4%

         \[\leadsto x + wj \cdot \left(\color{blue}{wj \cdot \left(x \cdot \left(2.5 + -2.6666666666666665 \cdot wj\right) + \left(1 + -1 \cdot wj\right)\right)} - 2 \cdot x\right) \]

      2. *-commutative 99.4%

         \[\leadsto x + wj \cdot \left(wj \cdot \left(x \cdot \left(2.5 + \color{blue}{wj \cdot -2.6666666666666665}\right) + \left(1 + -1 \cdot wj\right)\right) - 2 \cdot x\right) \]

      3. mul-1-neg 99.4%

         \[\leadsto x + wj \cdot \left(wj \cdot \left(x \cdot \left(2.5 + wj \cdot -2.6666666666666665\right) + \left(1 + \color{blue}{\left(-wj\right)}\right)\right) - 2 \cdot x\right) \]

   8. Simplified 99.4%

      \[\leadsto x + wj \cdot \left(\color{blue}{wj \cdot \left(x \cdot \left(2.5 + wj \cdot -2.6666666666666665\right) + \left(1 + \left(-wj\right)\right)\right)} - 2 \cdot x\right) \]

   9. Taylor expanded in x around 0 99.4%

      \[\leadsto x + wj \cdot \left(wj \cdot \color{blue}{\left(\left(1 + x \cdot \left(2.5 + -2.6666666666666665 \cdot wj\right)\right) - wj\right)} - 2 \cdot x\right) \]

   10. Taylor expanded in x around inf 99.5%

       \[\leadsto x + wj \cdot \left(wj \cdot \left(\color{blue}{x \cdot \left(2.5 + \left(-2.6666666666666665 \cdot wj + \frac{1}{x}\right)\right)} - wj\right) - 2 \cdot x\right) \]

   if 3.7500000000000001e-6 < wj < 0.419999999999999984

   1. Initial program 97.4%

      \[wj - \frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}} \]
   2. Step-by-step derivation

      1. distribute-rgt1-in 96.6%

         \[\leadsto wj - \frac{wj \cdot e^{wj} - x}{\color{blue}{\left(wj + 1\right) \cdot e^{wj}}} \]

      2. *-commutative 96.6%

         \[\leadsto wj - \frac{wj \cdot e^{wj} - x}{\color{blue}{e^{wj} \cdot \left(wj + 1\right)}} \]

      3. associate-/r* 94.7%

         \[\leadsto wj - \color{blue}{\frac{\frac{wj \cdot e^{wj} - x}{e^{wj}}}{wj + 1}} \]

      4. div-sub 94.7%

         \[\leadsto wj - \frac{\color{blue}{\frac{wj \cdot e^{wj}}{e^{wj}} - \frac{x}{e^{wj}}}}{wj + 1} \]

      5. associate-/l* 94.7%

         \[\leadsto wj - \frac{\color{blue}{wj \cdot \frac{e^{wj}}{e^{wj}}} - \frac{x}{e^{wj}}}{wj + 1} \]

      6. *-inverses 94.7%

         \[\leadsto wj - \frac{wj \cdot \color{blue}{1} - \frac{x}{e^{wj}}}{wj + 1} \]

      7. *-rgt-identity 94.7%

         \[\leadsto wj - \frac{\color{blue}{wj} - \frac{x}{e^{wj}}}{wj + 1} \]

   3. Simplified 94.7%

      \[\leadsto \color{blue}{wj - \frac{wj - \frac{x}{e^{wj}}}{wj + 1}} \]
   4. Add Preprocessing

   5. Taylor expanded in x around inf 95.0%

      \[\leadsto \color{blue}{x \cdot \left(\left(\frac{1}{e^{wj} \cdot \left(1 + wj\right)} + \frac{wj}{x}\right) - \frac{wj}{x \cdot \left(1 + wj\right)}\right)} \]
   6. Step-by-step derivation

      1. +-commutative 95.0%

         \[\leadsto x \cdot \left(\color{blue}{\left(\frac{wj}{x} + \frac{1}{e^{wj} \cdot \left(1 + wj\right)}\right)} - \frac{wj}{x \cdot \left(1 + wj\right)}\right) \]

      2. associate-/r* 95.8%

         \[\leadsto x \cdot \left(\left(\frac{wj}{x} + \color{blue}{\frac{\frac{1}{e^{wj}}}{1 + wj}}\right) - \frac{wj}{x \cdot \left(1 + wj\right)}\right) \]

      3. exp-neg 95.8%

         \[\leadsto x \cdot \left(\left(\frac{wj}{x} + \frac{\color{blue}{e^{-wj}}}{1 + wj}\right) - \frac{wj}{x \cdot \left(1 + wj\right)}\right) \]

      4. +-commutative 95.8%

         \[\leadsto x \cdot \left(\left(\frac{wj}{x} + \frac{e^{-wj}}{\color{blue}{wj + 1}}\right) - \frac{wj}{x \cdot \left(1 + wj\right)}\right) \]

      5. +-commutative 95.8%

         \[\leadsto x \cdot \left(\left(\frac{wj}{x} + \frac{e^{-wj}}{wj + 1}\right) - \frac{wj}{x \cdot \color{blue}{\left(wj + 1\right)}}\right) \]

   7. Simplified 95.8%

      \[\leadsto \color{blue}{x \cdot \left(\left(\frac{wj}{x} + \frac{e^{-wj}}{wj + 1}\right) - \frac{wj}{x \cdot \left(wj + 1\right)}\right)} \]
3. Recombined 5 regimes into one program.

4. Final simplification 99.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;wj \leq -4.5 \cdot 10^{-6}:\\ \;\;\;\;wj + \frac{wj - \frac{x}{e^{wj}}}{-1 - wj}\\ \mathbf{elif}\;wj \leq -5 \cdot 10^{-32}:\\ \;\;\;\;x + wj \cdot \left({wj}^{2} \cdot \left(\left(\left(-1 - \left(x \cdot -3 + \left(-2 \cdot \left(x \cdot -4 + x \cdot 1.5\right) + x \cdot 0.6666666666666666\right)\right)\right) + \frac{1}{wj}\right) - \left(-4 \cdot \frac{x}{wj} + 1.5 \cdot \frac{x}{wj}\right)\right) - x \cdot 2\right)\\ \mathbf{elif}\;wj \leq 5 \cdot 10^{-83}:\\ \;\;\;\;x - wj \cdot \left(x \cdot 2 - wj\right)\\ \mathbf{elif}\;wj \leq 3.75 \cdot 10^{-6}:\\ \;\;\;\;x - wj \cdot \left(x \cdot 2 + wj \cdot \left(wj + x \cdot \left(\left(\frac{-1}{x} - wj \cdot -2.6666666666666665\right) - 2.5\right)\right)\right)\\ \mathbf{elif}\;wj \leq 0.42:\\ \;\;\;\;x \cdot \left(\left(\frac{wj}{x} - \frac{e^{-wj}}{-1 - wj}\right) + \frac{wj}{x \cdot \left(-1 - wj\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;wj + \frac{wj - \frac{x}{e^{wj}}}{-1 - wj}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 99.7% accurate, 2.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := wj + \frac{wj - \frac{x}{e^{wj}}}{-1 - wj}\\ t_1 := \left(\frac{-1}{x} - wj \cdot -2.6666666666666665\right) - 2.5\\ \mathbf{if}\;wj \leq -4.5 \cdot 10^{-6}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;wj \leq -5 \cdot 10^{-32}:\\ \;\;\;\;x - wj \cdot \left(x \cdot 2 + wj \cdot \left(x \cdot \left(\frac{wj}{x} + t\_1\right)\right)\right)\\ \mathbf{elif}\;wj \leq 5 \cdot 10^{-83}:\\ \;\;\;\;x - wj \cdot \left(x \cdot 2 - wj\right)\\ \mathbf{elif}\;wj \leq 4.6 \cdot 10^{-6}:\\ \;\;\;\;x - wj \cdot \left(x \cdot 2 + wj \cdot \left(wj + x \cdot t\_1\right)\right)\\ \mathbf{elif}\;wj \leq 0.245:\\ \;\;\;\;x \cdot \left(\left(\frac{wj}{x} - \frac{e^{-wj}}{-1 - wj}\right) + \frac{wj}{x \cdot \left(-1 - wj\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (wj x)
 :precision binary64
 (let* ((t_0 (+ wj (/ (- wj (/ x (exp wj))) (- -1.0 wj))))
        (t_1 (- (- (/ -1.0 x) (* wj -2.6666666666666665)) 2.5)))
   (if (<= wj -4.5e-6)
     t_0
     (if (<= wj -5e-32)
       (- x (* wj (+ (* x 2.0) (* wj (* x (+ (/ wj x) t_1))))))
       (if (<= wj 5e-83)
         (- x (* wj (- (* x 2.0) wj)))
         (if (<= wj 4.6e-6)
           (- x (* wj (+ (* x 2.0) (* wj (+ wj (* x t_1))))))
           (if (<= wj 0.245)
             (*
              x
              (+
               (- (/ wj x) (/ (exp (- wj)) (- -1.0 wj)))
               (/ wj (* x (- -1.0 wj)))))
             t_0)))))))

C

double code(double wj, double x) {
	double t_0 = wj + ((wj - (x / exp(wj))) / (-1.0 - wj));
	double t_1 = ((-1.0 / x) - (wj * -2.6666666666666665)) - 2.5;
	double tmp;
	if (wj <= -4.5e-6) {
		tmp = t_0;
	} else if (wj <= -5e-32) {
		tmp = x - (wj * ((x * 2.0) + (wj * (x * ((wj / x) + t_1)))));
	} else if (wj <= 5e-83) {
		tmp = x - (wj * ((x * 2.0) - wj));
	} else if (wj <= 4.6e-6) {
		tmp = x - (wj * ((x * 2.0) + (wj * (wj + (x * t_1)))));
	} else if (wj <= 0.245) {
		tmp = x * (((wj / x) - (exp(-wj) / (-1.0 - wj))) + (wj / (x * (-1.0 - wj))));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(wj, x)
    real(8), intent (in) :: wj
    real(8), intent (in) :: x
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = wj + ((wj - (x / exp(wj))) / ((-1.0d0) - wj))
    t_1 = (((-1.0d0) / x) - (wj * (-2.6666666666666665d0))) - 2.5d0
    if (wj <= (-4.5d-6)) then
        tmp = t_0
    else if (wj <= (-5d-32)) then
        tmp = x - (wj * ((x * 2.0d0) + (wj * (x * ((wj / x) + t_1)))))
    else if (wj <= 5d-83) then
        tmp = x - (wj * ((x * 2.0d0) - wj))
    else if (wj <= 4.6d-6) then
        tmp = x - (wj * ((x * 2.0d0) + (wj * (wj + (x * t_1)))))
    else if (wj <= 0.245d0) then
        tmp = x * (((wj / x) - (exp(-wj) / ((-1.0d0) - wj))) + (wj / (x * ((-1.0d0) - wj))))
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double wj, double x) {
	double t_0 = wj + ((wj - (x / Math.exp(wj))) / (-1.0 - wj));
	double t_1 = ((-1.0 / x) - (wj * -2.6666666666666665)) - 2.5;
	double tmp;
	if (wj <= -4.5e-6) {
		tmp = t_0;
	} else if (wj <= -5e-32) {
		tmp = x - (wj * ((x * 2.0) + (wj * (x * ((wj / x) + t_1)))));
	} else if (wj <= 5e-83) {
		tmp = x - (wj * ((x * 2.0) - wj));
	} else if (wj <= 4.6e-6) {
		tmp = x - (wj * ((x * 2.0) + (wj * (wj + (x * t_1)))));
	} else if (wj <= 0.245) {
		tmp = x * (((wj / x) - (Math.exp(-wj) / (-1.0 - wj))) + (wj / (x * (-1.0 - wj))));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(wj, x):
	t_0 = wj + ((wj - (x / math.exp(wj))) / (-1.0 - wj))
	t_1 = ((-1.0 / x) - (wj * -2.6666666666666665)) - 2.5
	tmp = 0
	if wj <= -4.5e-6:
		tmp = t_0
	elif wj <= -5e-32:
		tmp = x - (wj * ((x * 2.0) + (wj * (x * ((wj / x) + t_1)))))
	elif wj <= 5e-83:
		tmp = x - (wj * ((x * 2.0) - wj))
	elif wj <= 4.6e-6:
		tmp = x - (wj * ((x * 2.0) + (wj * (wj + (x * t_1)))))
	elif wj <= 0.245:
		tmp = x * (((wj / x) - (math.exp(-wj) / (-1.0 - wj))) + (wj / (x * (-1.0 - wj))))
	else:
		tmp = t_0
	return tmp

Julia

function code(wj, x)
	t_0 = Float64(wj + Float64(Float64(wj - Float64(x / exp(wj))) / Float64(-1.0 - wj)))
	t_1 = Float64(Float64(Float64(-1.0 / x) - Float64(wj * -2.6666666666666665)) - 2.5)
	tmp = 0.0
	if (wj <= -4.5e-6)
		tmp = t_0;
	elseif (wj <= -5e-32)
		tmp = Float64(x - Float64(wj * Float64(Float64(x * 2.0) + Float64(wj * Float64(x * Float64(Float64(wj / x) + t_1))))));
	elseif (wj <= 5e-83)
		tmp = Float64(x - Float64(wj * Float64(Float64(x * 2.0) - wj)));
	elseif (wj <= 4.6e-6)
		tmp = Float64(x - Float64(wj * Float64(Float64(x * 2.0) + Float64(wj * Float64(wj + Float64(x * t_1))))));
	elseif (wj <= 0.245)
		tmp = Float64(x * Float64(Float64(Float64(wj / x) - Float64(exp(Float64(-wj)) / Float64(-1.0 - wj))) + Float64(wj / Float64(x * Float64(-1.0 - wj)))));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(wj, x)
	t_0 = wj + ((wj - (x / exp(wj))) / (-1.0 - wj));
	t_1 = ((-1.0 / x) - (wj * -2.6666666666666665)) - 2.5;
	tmp = 0.0;
	if (wj <= -4.5e-6)
		tmp = t_0;
	elseif (wj <= -5e-32)
		tmp = x - (wj * ((x * 2.0) + (wj * (x * ((wj / x) + t_1)))));
	elseif (wj <= 5e-83)
		tmp = x - (wj * ((x * 2.0) - wj));
	elseif (wj <= 4.6e-6)
		tmp = x - (wj * ((x * 2.0) + (wj * (wj + (x * t_1)))));
	elseif (wj <= 0.245)
		tmp = x * (((wj / x) - (exp(-wj) / (-1.0 - wj))) + (wj / (x * (-1.0 - wj))));
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[wj_, x_] := Block[{t$95$0 = N[(wj + N[(N[(wj - N[(x / N[Exp[wj], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(-1.0 - wj), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[(-1.0 / x), $MachinePrecision] - N[(wj * -2.6666666666666665), $MachinePrecision]), $MachinePrecision] - 2.5), $MachinePrecision]}, If[LessEqual[wj, -4.5e-6], t$95$0, If[LessEqual[wj, -5e-32], N[(x - N[(wj * N[(N[(x * 2.0), $MachinePrecision] + N[(wj * N[(x * N[(N[(wj / x), $MachinePrecision] + t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[wj, 5e-83], N[(x - N[(wj * N[(N[(x * 2.0), $MachinePrecision] - wj), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[wj, 4.6e-6], N[(x - N[(wj * N[(N[(x * 2.0), $MachinePrecision] + N[(wj * N[(wj + N[(x * t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[wj, 0.245], N[(x * N[(N[(N[(wj / x), $MachinePrecision] - N[(N[Exp[(-wj)], $MachinePrecision] / N[(-1.0 - wj), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(wj / N[(x * N[(-1.0 - wj), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]]]]]]

Derivation

1. Split input into 5 regimes

2. if wj < -4.50000000000000011e-6 or 0.245 < wj

   1. Initial program 64.6%

      \[wj - \frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}} \]
   2. Step-by-step derivation

      1. distribute-rgt1-in 89.7%

         \[\leadsto wj - \frac{wj \cdot e^{wj} - x}{\color{blue}{\left(wj + 1\right) \cdot e^{wj}}} \]

      2. *-commutative 89.7%

         \[\leadsto wj - \frac{wj \cdot e^{wj} - x}{\color{blue}{e^{wj} \cdot \left(wj + 1\right)}} \]

      3. associate-/r* 89.7%

         \[\leadsto wj - \color{blue}{\frac{\frac{wj \cdot e^{wj} - x}{e^{wj}}}{wj + 1}} \]

      4. div-sub 64.7%

         \[\leadsto wj - \frac{\color{blue}{\frac{wj \cdot e^{wj}}{e^{wj}} - \frac{x}{e^{wj}}}}{wj + 1} \]

      5. associate-/l* 64.7%

         \[\leadsto wj - \frac{\color{blue}{wj \cdot \frac{e^{wj}}{e^{wj}}} - \frac{x}{e^{wj}}}{wj + 1} \]

      6. *-inverses 98.0%

         \[\leadsto wj - \frac{wj \cdot \color{blue}{1} - \frac{x}{e^{wj}}}{wj + 1} \]

      7. *-rgt-identity 98.0%

         \[\leadsto wj - \frac{\color{blue}{wj} - \frac{x}{e^{wj}}}{wj + 1} \]

   3. Simplified 98.0%

      \[\leadsto \color{blue}{wj - \frac{wj - \frac{x}{e^{wj}}}{wj + 1}} \]
   4. Add Preprocessing

   if -4.50000000000000011e-6 < wj < -5e-32

   1. Initial program 70.8%

      \[wj - \frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}} \]
   2. Step-by-step derivation

      1. distribute-rgt1-in 70.7%

         \[\leadsto wj - \frac{wj \cdot e^{wj} - x}{\color{blue}{\left(wj + 1\right) \cdot e^{wj}}} \]

      2. *-commutative 70.7%

         \[\leadsto wj - \frac{wj \cdot e^{wj} - x}{\color{blue}{e^{wj} \cdot \left(wj + 1\right)}} \]

      3. associate-/r* 70.8%

         \[\leadsto wj - \color{blue}{\frac{\frac{wj \cdot e^{wj} - x}{e^{wj}}}{wj + 1}} \]

      4. div-sub 70.8%

         \[\leadsto wj - \frac{\color{blue}{\frac{wj \cdot e^{wj}}{e^{wj}} - \frac{x}{e^{wj}}}}{wj + 1} \]

      5. associate-/l* 70.8%

         \[\leadsto wj - \frac{\color{blue}{wj \cdot \frac{e^{wj}}{e^{wj}}} - \frac{x}{e^{wj}}}{wj + 1} \]

      6. *-inverses 70.8%

         \[\leadsto wj - \frac{wj \cdot \color{blue}{1} - \frac{x}{e^{wj}}}{wj + 1} \]

      7. *-rgt-identity 70.8%

         \[\leadsto wj - \frac{\color{blue}{wj} - \frac{x}{e^{wj}}}{wj + 1} \]

   3. Simplified 70.8%

      \[\leadsto \color{blue}{wj - \frac{wj - \frac{x}{e^{wj}}}{wj + 1}} \]
   4. Add Preprocessing

   5. Taylor expanded in wj around 0 98.4%

      \[\leadsto \color{blue}{x + wj \cdot \left(wj \cdot \left(\left(1 + -1 \cdot \left(wj \cdot \left(1 + \left(-3 \cdot x + \left(-2 \cdot \left(-4 \cdot x + 1.5 \cdot x\right) + 0.6666666666666666 \cdot x\right)\right)\right)\right)\right) - \left(-4 \cdot x + 1.5 \cdot x\right)\right) - 2 \cdot x\right)} \]

   6. Taylor expanded in x around 0 98.4%

      \[\leadsto x + wj \cdot \left(\color{blue}{\left(wj \cdot \left(x \cdot \left(2.5 + -2.6666666666666665 \cdot wj\right)\right) + wj \cdot \left(1 + -1 \cdot wj\right)\right)} - 2 \cdot x\right) \]
   7. Step-by-step derivation

      1. distribute-lft-out 98.4%

         \[\leadsto x + wj \cdot \left(\color{blue}{wj \cdot \left(x \cdot \left(2.5 + -2.6666666666666665 \cdot wj\right) + \left(1 + -1 \cdot wj\right)\right)} - 2 \cdot x\right) \]

      2. *-commutative 98.4%

         \[\leadsto x + wj \cdot \left(wj \cdot \left(x \cdot \left(2.5 + \color{blue}{wj \cdot -2.6666666666666665}\right) + \left(1 + -1 \cdot wj\right)\right) - 2 \cdot x\right) \]

      3. mul-1-neg 98.4%

         \[\leadsto x + wj \cdot \left(wj \cdot \left(x \cdot \left(2.5 + wj \cdot -2.6666666666666665\right) + \left(1 + \color{blue}{\left(-wj\right)}\right)\right) - 2 \cdot x\right) \]

   8. Simplified 98.4%

      \[\leadsto x + wj \cdot \left(\color{blue}{wj \cdot \left(x \cdot \left(2.5 + wj \cdot -2.6666666666666665\right) + \left(1 + \left(-wj\right)\right)\right)} - 2 \cdot x\right) \]

   9. Taylor expanded in x around inf 98.4%

      \[\leadsto x + wj \cdot \left(wj \cdot \color{blue}{\left(x \cdot \left(\left(2.5 + \left(-2.6666666666666665 \cdot wj + \frac{1}{x}\right)\right) - \frac{wj}{x}\right)\right)} - 2 \cdot x\right) \]

   if -5e-32 < wj < 5e-83

   1. Initial program 81.3%

      \[wj - \frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}} \]
   2. Step-by-step derivation

      1. distribute-rgt1-in 81.3%

         \[\leadsto wj - \frac{wj \cdot e^{wj} - x}{\color{blue}{\left(wj + 1\right) \cdot e^{wj}}} \]

      2. *-commutative 81.3%

         \[\leadsto wj - \frac{wj \cdot e^{wj} - x}{\color{blue}{e^{wj} \cdot \left(wj + 1\right)}} \]

      3. associate-/r* 81.3%

         \[\leadsto wj - \color{blue}{\frac{\frac{wj \cdot e^{wj} - x}{e^{wj}}}{wj + 1}} \]

      4. div-sub 81.3%

         \[\leadsto wj - \frac{\color{blue}{\frac{wj \cdot e^{wj}}{e^{wj}} - \frac{x}{e^{wj}}}}{wj + 1} \]

      5. associate-/l* 81.3%

         \[\leadsto wj - \frac{\color{blue}{wj \cdot \frac{e^{wj}}{e^{wj}}} - \frac{x}{e^{wj}}}{wj + 1} \]

      6. *-inverses 81.3%

         \[\leadsto wj - \frac{wj \cdot \color{blue}{1} - \frac{x}{e^{wj}}}{wj + 1} \]

      7. *-rgt-identity 81.3%

         \[\leadsto wj - \frac{\color{blue}{wj} - \frac{x}{e^{wj}}}{wj + 1} \]

   3. Simplified 81.3%

      \[\leadsto \color{blue}{wj - \frac{wj - \frac{x}{e^{wj}}}{wj + 1}} \]
   4. Add Preprocessing

   5. Taylor expanded in wj around 0 99.6%

      \[\leadsto \color{blue}{x + wj \cdot \left(wj \cdot \left(\left(1 + -1 \cdot \left(wj \cdot \left(1 + \left(-3 \cdot x + \left(-2 \cdot \left(-4 \cdot x + 1.5 \cdot x\right) + 0.6666666666666666 \cdot x\right)\right)\right)\right)\right) - \left(-4 \cdot x + 1.5 \cdot x\right)\right) - 2 \cdot x\right)} \]

   6. Taylor expanded in x around 0 100.0%

      \[\leadsto x + wj \cdot \left(\color{blue}{wj \cdot \left(1 + -1 \cdot wj\right)} - 2 \cdot x\right) \]
   7. Step-by-step derivation

      1. mul-1-neg 100.0%

         \[\leadsto x + wj \cdot \left(wj \cdot \left(1 + \color{blue}{\left(-wj\right)}\right) - 2 \cdot x\right) \]

      2. unsub-neg 100.0%

         \[\leadsto x + wj \cdot \left(wj \cdot \color{blue}{\left(1 - wj\right)} - 2 \cdot x\right) \]

   8. Simplified 100.0%

      \[\leadsto x + wj \cdot \left(\color{blue}{wj \cdot \left(1 - wj\right)} - 2 \cdot x\right) \]

   9. Taylor expanded in wj around 0 100.0%

      \[\leadsto x + wj \cdot \left(\color{blue}{wj} - 2 \cdot x\right) \]

   if 5e-83 < wj < 4.6e-6

   1. Initial program 52.2%

      \[wj - \frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}} \]
   2. Step-by-step derivation

      1. distribute-rgt1-in 52.1%

         \[\leadsto wj - \frac{wj \cdot e^{wj} - x}{\color{blue}{\left(wj + 1\right) \cdot e^{wj}}} \]

      2. *-commutative 52.1%

         \[\leadsto wj - \frac{wj \cdot e^{wj} - x}{\color{blue}{e^{wj} \cdot \left(wj + 1\right)}} \]

      3. associate-/r* 52.3%

         \[\leadsto wj - \color{blue}{\frac{\frac{wj \cdot e^{wj} - x}{e^{wj}}}{wj + 1}} \]

      4. div-sub 52.3%

         \[\leadsto wj - \frac{\color{blue}{\frac{wj \cdot e^{wj}}{e^{wj}} - \frac{x}{e^{wj}}}}{wj + 1} \]

      5. associate-/l* 52.3%

         \[\leadsto wj - \frac{\color{blue}{wj \cdot \frac{e^{wj}}{e^{wj}}} - \frac{x}{e^{wj}}}{wj + 1} \]

      6. *-inverses 52.3%

         \[\leadsto wj - \frac{wj \cdot \color{blue}{1} - \frac{x}{e^{wj}}}{wj + 1} \]

      7. *-rgt-identity 52.3%

         \[\leadsto wj - \frac{\color{blue}{wj} - \frac{x}{e^{wj}}}{wj + 1} \]

   3. Simplified 52.3%

      \[\leadsto \color{blue}{wj - \frac{wj - \frac{x}{e^{wj}}}{wj + 1}} \]
   4. Add Preprocessing

   5. Taylor expanded in wj around 0 99.4%

      \[\leadsto \color{blue}{x + wj \cdot \left(wj \cdot \left(\left(1 + -1 \cdot \left(wj \cdot \left(1 + \left(-3 \cdot x + \left(-2 \cdot \left(-4 \cdot x + 1.5 \cdot x\right) + 0.6666666666666666 \cdot x\right)\right)\right)\right)\right) - \left(-4 \cdot x + 1.5 \cdot x\right)\right) - 2 \cdot x\right)} \]

   6. Taylor expanded in x around 0 99.4%

      \[\leadsto x + wj \cdot \left(\color{blue}{\left(wj \cdot \left(x \cdot \left(2.5 + -2.6666666666666665 \cdot wj\right)\right) + wj \cdot \left(1 + -1 \cdot wj\right)\right)} - 2 \cdot x\right) \]
   7. Step-by-step derivation

      1. distribute-lft-out 99.4%

         \[\leadsto x + wj \cdot \left(\color{blue}{wj \cdot \left(x \cdot \left(2.5 + -2.6666666666666665 \cdot wj\right) + \left(1 + -1 \cdot wj\right)\right)} - 2 \cdot x\right) \]

      2. *-commutative 99.4%

         \[\leadsto x + wj \cdot \left(wj \cdot \left(x \cdot \left(2.5 + \color{blue}{wj \cdot -2.6666666666666665}\right) + \left(1 + -1 \cdot wj\right)\right) - 2 \cdot x\right) \]

      3. mul-1-neg 99.4%

         \[\leadsto x + wj \cdot \left(wj \cdot \left(x \cdot \left(2.5 + wj \cdot -2.6666666666666665\right) + \left(1 + \color{blue}{\left(-wj\right)}\right)\right) - 2 \cdot x\right) \]

   8. Simplified 99.4%

      \[\leadsto x + wj \cdot \left(\color{blue}{wj \cdot \left(x \cdot \left(2.5 + wj \cdot -2.6666666666666665\right) + \left(1 + \left(-wj\right)\right)\right)} - 2 \cdot x\right) \]

   9. Taylor expanded in x around 0 99.4%

      \[\leadsto x + wj \cdot \left(wj \cdot \color{blue}{\left(\left(1 + x \cdot \left(2.5 + -2.6666666666666665 \cdot wj\right)\right) - wj\right)} - 2 \cdot x\right) \]

   10. Taylor expanded in x around inf 99.5%

       \[\leadsto x + wj \cdot \left(wj \cdot \left(\color{blue}{x \cdot \left(2.5 + \left(-2.6666666666666665 \cdot wj + \frac{1}{x}\right)\right)} - wj\right) - 2 \cdot x\right) \]

   if 4.6e-6 < wj < 0.245

   1. Initial program 97.4%

      \[wj - \frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}} \]
   2. Step-by-step derivation

      1. distribute-rgt1-in 96.6%

         \[\leadsto wj - \frac{wj \cdot e^{wj} - x}{\color{blue}{\left(wj + 1\right) \cdot e^{wj}}} \]

      2. *-commutative 96.6%

         \[\leadsto wj - \frac{wj \cdot e^{wj} - x}{\color{blue}{e^{wj} \cdot \left(wj + 1\right)}} \]

      3. associate-/r* 94.7%

         \[\leadsto wj - \color{blue}{\frac{\frac{wj \cdot e^{wj} - x}{e^{wj}}}{wj + 1}} \]

      4. div-sub 94.7%

         \[\leadsto wj - \frac{\color{blue}{\frac{wj \cdot e^{wj}}{e^{wj}} - \frac{x}{e^{wj}}}}{wj + 1} \]

      5. associate-/l* 94.7%

         \[\leadsto wj - \frac{\color{blue}{wj \cdot \frac{e^{wj}}{e^{wj}}} - \frac{x}{e^{wj}}}{wj + 1} \]

      6. *-inverses 94.7%

         \[\leadsto wj - \frac{wj \cdot \color{blue}{1} - \frac{x}{e^{wj}}}{wj + 1} \]

      7. *-rgt-identity 94.7%

         \[\leadsto wj - \frac{\color{blue}{wj} - \frac{x}{e^{wj}}}{wj + 1} \]

   3. Simplified 94.7%

      \[\leadsto \color{blue}{wj - \frac{wj - \frac{x}{e^{wj}}}{wj + 1}} \]
   4. Add Preprocessing

   5. Taylor expanded in x around inf 95.0%

      \[\leadsto \color{blue}{x \cdot \left(\left(\frac{1}{e^{wj} \cdot \left(1 + wj\right)} + \frac{wj}{x}\right) - \frac{wj}{x \cdot \left(1 + wj\right)}\right)} \]
   6. Step-by-step derivation

      1. +-commutative 95.0%

         \[\leadsto x \cdot \left(\color{blue}{\left(\frac{wj}{x} + \frac{1}{e^{wj} \cdot \left(1 + wj\right)}\right)} - \frac{wj}{x \cdot \left(1 + wj\right)}\right) \]

      2. associate-/r* 95.8%

         \[\leadsto x \cdot \left(\left(\frac{wj}{x} + \color{blue}{\frac{\frac{1}{e^{wj}}}{1 + wj}}\right) - \frac{wj}{x \cdot \left(1 + wj\right)}\right) \]

      3. exp-neg 95.8%

         \[\leadsto x \cdot \left(\left(\frac{wj}{x} + \frac{\color{blue}{e^{-wj}}}{1 + wj}\right) - \frac{wj}{x \cdot \left(1 + wj\right)}\right) \]

      4. +-commutative 95.8%

         \[\leadsto x \cdot \left(\left(\frac{wj}{x} + \frac{e^{-wj}}{\color{blue}{wj + 1}}\right) - \frac{wj}{x \cdot \left(1 + wj\right)}\right) \]

      5. +-commutative 95.8%

         \[\leadsto x \cdot \left(\left(\frac{wj}{x} + \frac{e^{-wj}}{wj + 1}\right) - \frac{wj}{x \cdot \color{blue}{\left(wj + 1\right)}}\right) \]

   7. Simplified 95.8%

      \[\leadsto \color{blue}{x \cdot \left(\left(\frac{wj}{x} + \frac{e^{-wj}}{wj + 1}\right) - \frac{wj}{x \cdot \left(wj + 1\right)}\right)} \]
3. Recombined 5 regimes into one program.

4. Final simplification 99.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;wj \leq -4.5 \cdot 10^{-6}:\\ \;\;\;\;wj + \frac{wj - \frac{x}{e^{wj}}}{-1 - wj}\\ \mathbf{elif}\;wj \leq -5 \cdot 10^{-32}:\\ \;\;\;\;x - wj \cdot \left(x \cdot 2 + wj \cdot \left(x \cdot \left(\frac{wj}{x} + \left(\left(\frac{-1}{x} - wj \cdot -2.6666666666666665\right) - 2.5\right)\right)\right)\right)\\ \mathbf{elif}\;wj \leq 5 \cdot 10^{-83}:\\ \;\;\;\;x - wj \cdot \left(x \cdot 2 - wj\right)\\ \mathbf{elif}\;wj \leq 4.6 \cdot 10^{-6}:\\ \;\;\;\;x - wj \cdot \left(x \cdot 2 + wj \cdot \left(wj + x \cdot \left(\left(\frac{-1}{x} - wj \cdot -2.6666666666666665\right) - 2.5\right)\right)\right)\\ \mathbf{elif}\;wj \leq 0.245:\\ \;\;\;\;x \cdot \left(\left(\frac{wj}{x} - \frac{e^{-wj}}{-1 - wj}\right) + \frac{wj}{x \cdot \left(-1 - wj\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;wj + \frac{wj - \frac{x}{e^{wj}}}{-1 - wj}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 99.7% accurate, 2.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := wj + \frac{wj - \frac{x}{e^{wj}}}{-1 - wj}\\ t_1 := \left(\frac{-1}{x} - wj \cdot -2.6666666666666665\right) - 2.5\\ \mathbf{if}\;wj \leq -3.3 \cdot 10^{-6}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;wj \leq -5 \cdot 10^{-32}:\\ \;\;\;\;x - wj \cdot \left(x \cdot 2 + wj \cdot \left(x \cdot \left(\frac{wj}{x} + t\_1\right)\right)\right)\\ \mathbf{elif}\;wj \leq 5 \cdot 10^{-83}:\\ \;\;\;\;x - wj \cdot \left(x \cdot 2 - wj\right)\\ \mathbf{elif}\;wj \leq 0.00052:\\ \;\;\;\;x - wj \cdot \left(x \cdot 2 + wj \cdot \left(wj + x \cdot t\_1\right)\right)\\ \mathbf{elif}\;wj \leq 0.00185:\\ \;\;\;\;x \cdot \frac{e^{-wj}}{wj + 1}\\ \mathbf{elif}\;wj \leq 0.014:\\ \;\;\;\;wj - \frac{wj}{wj + 1}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (wj x)
 :precision binary64
 (let* ((t_0 (+ wj (/ (- wj (/ x (exp wj))) (- -1.0 wj))))
        (t_1 (- (- (/ -1.0 x) (* wj -2.6666666666666665)) 2.5)))
   (if (<= wj -3.3e-6)
     t_0
     (if (<= wj -5e-32)
       (- x (* wj (+ (* x 2.0) (* wj (* x (+ (/ wj x) t_1))))))
       (if (<= wj 5e-83)
         (- x (* wj (- (* x 2.0) wj)))
         (if (<= wj 0.00052)
           (- x (* wj (+ (* x 2.0) (* wj (+ wj (* x t_1))))))
           (if (<= wj 0.00185)
             (* x (/ (exp (- wj)) (+ wj 1.0)))
             (if (<= wj 0.014) (- wj (/ wj (+ wj 1.0))) t_0))))))))

C

double code(double wj, double x) {
	double t_0 = wj + ((wj - (x / exp(wj))) / (-1.0 - wj));
	double t_1 = ((-1.0 / x) - (wj * -2.6666666666666665)) - 2.5;
	double tmp;
	if (wj <= -3.3e-6) {
		tmp = t_0;
	} else if (wj <= -5e-32) {
		tmp = x - (wj * ((x * 2.0) + (wj * (x * ((wj / x) + t_1)))));
	} else if (wj <= 5e-83) {
		tmp = x - (wj * ((x * 2.0) - wj));
	} else if (wj <= 0.00052) {
		tmp = x - (wj * ((x * 2.0) + (wj * (wj + (x * t_1)))));
	} else if (wj <= 0.00185) {
		tmp = x * (exp(-wj) / (wj + 1.0));
	} else if (wj <= 0.014) {
		tmp = wj - (wj / (wj + 1.0));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(wj, x)
    real(8), intent (in) :: wj
    real(8), intent (in) :: x
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = wj + ((wj - (x / exp(wj))) / ((-1.0d0) - wj))
    t_1 = (((-1.0d0) / x) - (wj * (-2.6666666666666665d0))) - 2.5d0
    if (wj <= (-3.3d-6)) then
        tmp = t_0
    else if (wj <= (-5d-32)) then
        tmp = x - (wj * ((x * 2.0d0) + (wj * (x * ((wj / x) + t_1)))))
    else if (wj <= 5d-83) then
        tmp = x - (wj * ((x * 2.0d0) - wj))
    else if (wj <= 0.00052d0) then
        tmp = x - (wj * ((x * 2.0d0) + (wj * (wj + (x * t_1)))))
    else if (wj <= 0.00185d0) then
        tmp = x * (exp(-wj) / (wj + 1.0d0))
    else if (wj <= 0.014d0) then
        tmp = wj - (wj / (wj + 1.0d0))
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double wj, double x) {
	double t_0 = wj + ((wj - (x / Math.exp(wj))) / (-1.0 - wj));
	double t_1 = ((-1.0 / x) - (wj * -2.6666666666666665)) - 2.5;
	double tmp;
	if (wj <= -3.3e-6) {
		tmp = t_0;
	} else if (wj <= -5e-32) {
		tmp = x - (wj * ((x * 2.0) + (wj * (x * ((wj / x) + t_1)))));
	} else if (wj <= 5e-83) {
		tmp = x - (wj * ((x * 2.0) - wj));
	} else if (wj <= 0.00052) {
		tmp = x - (wj * ((x * 2.0) + (wj * (wj + (x * t_1)))));
	} else if (wj <= 0.00185) {
		tmp = x * (Math.exp(-wj) / (wj + 1.0));
	} else if (wj <= 0.014) {
		tmp = wj - (wj / (wj + 1.0));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(wj, x):
	t_0 = wj + ((wj - (x / math.exp(wj))) / (-1.0 - wj))
	t_1 = ((-1.0 / x) - (wj * -2.6666666666666665)) - 2.5
	tmp = 0
	if wj <= -3.3e-6:
		tmp = t_0
	elif wj <= -5e-32:
		tmp = x - (wj * ((x * 2.0) + (wj * (x * ((wj / x) + t_1)))))
	elif wj <= 5e-83:
		tmp = x - (wj * ((x * 2.0) - wj))
	elif wj <= 0.00052:
		tmp = x - (wj * ((x * 2.0) + (wj * (wj + (x * t_1)))))
	elif wj <= 0.00185:
		tmp = x * (math.exp(-wj) / (wj + 1.0))
	elif wj <= 0.014:
		tmp = wj - (wj / (wj + 1.0))
	else:
		tmp = t_0
	return tmp

Julia

function code(wj, x)
	t_0 = Float64(wj + Float64(Float64(wj - Float64(x / exp(wj))) / Float64(-1.0 - wj)))
	t_1 = Float64(Float64(Float64(-1.0 / x) - Float64(wj * -2.6666666666666665)) - 2.5)
	tmp = 0.0
	if (wj <= -3.3e-6)
		tmp = t_0;
	elseif (wj <= -5e-32)
		tmp = Float64(x - Float64(wj * Float64(Float64(x * 2.0) + Float64(wj * Float64(x * Float64(Float64(wj / x) + t_1))))));
	elseif (wj <= 5e-83)
		tmp = Float64(x - Float64(wj * Float64(Float64(x * 2.0) - wj)));
	elseif (wj <= 0.00052)
		tmp = Float64(x - Float64(wj * Float64(Float64(x * 2.0) + Float64(wj * Float64(wj + Float64(x * t_1))))));
	elseif (wj <= 0.00185)
		tmp = Float64(x * Float64(exp(Float64(-wj)) / Float64(wj + 1.0)));
	elseif (wj <= 0.014)
		tmp = Float64(wj - Float64(wj / Float64(wj + 1.0)));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(wj, x)
	t_0 = wj + ((wj - (x / exp(wj))) / (-1.0 - wj));
	t_1 = ((-1.0 / x) - (wj * -2.6666666666666665)) - 2.5;
	tmp = 0.0;
	if (wj <= -3.3e-6)
		tmp = t_0;
	elseif (wj <= -5e-32)
		tmp = x - (wj * ((x * 2.0) + (wj * (x * ((wj / x) + t_1)))));
	elseif (wj <= 5e-83)
		tmp = x - (wj * ((x * 2.0) - wj));
	elseif (wj <= 0.00052)
		tmp = x - (wj * ((x * 2.0) + (wj * (wj + (x * t_1)))));
	elseif (wj <= 0.00185)
		tmp = x * (exp(-wj) / (wj + 1.0));
	elseif (wj <= 0.014)
		tmp = wj - (wj / (wj + 1.0));
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[wj_, x_] := Block[{t$95$0 = N[(wj + N[(N[(wj - N[(x / N[Exp[wj], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(-1.0 - wj), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[(-1.0 / x), $MachinePrecision] - N[(wj * -2.6666666666666665), $MachinePrecision]), $MachinePrecision] - 2.5), $MachinePrecision]}, If[LessEqual[wj, -3.3e-6], t$95$0, If[LessEqual[wj, -5e-32], N[(x - N[(wj * N[(N[(x * 2.0), $MachinePrecision] + N[(wj * N[(x * N[(N[(wj / x), $MachinePrecision] + t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[wj, 5e-83], N[(x - N[(wj * N[(N[(x * 2.0), $MachinePrecision] - wj), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[wj, 0.00052], N[(x - N[(wj * N[(N[(x * 2.0), $MachinePrecision] + N[(wj * N[(wj + N[(x * t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[wj, 0.00185], N[(x * N[(N[Exp[(-wj)], $MachinePrecision] / N[(wj + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[wj, 0.014], N[(wj - N[(wj / N[(wj + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]]]]]]]

Derivation

1. Split input into 6 regimes

2. if wj < -3.30000000000000017e-6 or 0.0140000000000000003 < wj

   1. Initial program 64.6%

      \[wj - \frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}} \]
   2. Step-by-step derivation

      1. distribute-rgt1-in 89.7%

         \[\leadsto wj - \frac{wj \cdot e^{wj} - x}{\color{blue}{\left(wj + 1\right) \cdot e^{wj}}} \]

      2. *-commutative 89.7%

         \[\leadsto wj - \frac{wj \cdot e^{wj} - x}{\color{blue}{e^{wj} \cdot \left(wj + 1\right)}} \]

      3. associate-/r* 89.7%

         \[\leadsto wj - \color{blue}{\frac{\frac{wj \cdot e^{wj} - x}{e^{wj}}}{wj + 1}} \]

      4. div-sub 64.7%

         \[\leadsto wj - \frac{\color{blue}{\frac{wj \cdot e^{wj}}{e^{wj}} - \frac{x}{e^{wj}}}}{wj + 1} \]

      5. associate-/l* 64.7%

         \[\leadsto wj - \frac{\color{blue}{wj \cdot \frac{e^{wj}}{e^{wj}}} - \frac{x}{e^{wj}}}{wj + 1} \]

      6. *-inverses 98.0%

         \[\leadsto wj - \frac{wj \cdot \color{blue}{1} - \frac{x}{e^{wj}}}{wj + 1} \]

      7. *-rgt-identity 98.0%

         \[\leadsto wj - \frac{\color{blue}{wj} - \frac{x}{e^{wj}}}{wj + 1} \]

   3. Simplified 98.0%

      \[\leadsto \color{blue}{wj - \frac{wj - \frac{x}{e^{wj}}}{wj + 1}} \]
   4. Add Preprocessing

   if -3.30000000000000017e-6 < wj < -5e-32

   1. Initial program 70.8%

      \[wj - \frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}} \]
   2. Step-by-step derivation

      1. distribute-rgt1-in 70.7%

         \[\leadsto wj - \frac{wj \cdot e^{wj} - x}{\color{blue}{\left(wj + 1\right) \cdot e^{wj}}} \]

      2. *-commutative 70.7%

         \[\leadsto wj - \frac{wj \cdot e^{wj} - x}{\color{blue}{e^{wj} \cdot \left(wj + 1\right)}} \]

      3. associate-/r* 70.8%

         \[\leadsto wj - \color{blue}{\frac{\frac{wj \cdot e^{wj} - x}{e^{wj}}}{wj + 1}} \]

      4. div-sub 70.8%

         \[\leadsto wj - \frac{\color{blue}{\frac{wj \cdot e^{wj}}{e^{wj}} - \frac{x}{e^{wj}}}}{wj + 1} \]

      5. associate-/l* 70.8%

         \[\leadsto wj - \frac{\color{blue}{wj \cdot \frac{e^{wj}}{e^{wj}}} - \frac{x}{e^{wj}}}{wj + 1} \]

      6. *-inverses 70.8%

         \[\leadsto wj - \frac{wj \cdot \color{blue}{1} - \frac{x}{e^{wj}}}{wj + 1} \]

      7. *-rgt-identity 70.8%

         \[\leadsto wj - \frac{\color{blue}{wj} - \frac{x}{e^{wj}}}{wj + 1} \]

   3. Simplified 70.8%

      \[\leadsto \color{blue}{wj - \frac{wj - \frac{x}{e^{wj}}}{wj + 1}} \]
   4. Add Preprocessing

   5. Taylor expanded in wj around 0 98.4%

      \[\leadsto \color{blue}{x + wj \cdot \left(wj \cdot \left(\left(1 + -1 \cdot \left(wj \cdot \left(1 + \left(-3 \cdot x + \left(-2 \cdot \left(-4 \cdot x + 1.5 \cdot x\right) + 0.6666666666666666 \cdot x\right)\right)\right)\right)\right) - \left(-4 \cdot x + 1.5 \cdot x\right)\right) - 2 \cdot x\right)} \]

   6. Taylor expanded in x around 0 98.4%

      \[\leadsto x + wj \cdot \left(\color{blue}{\left(wj \cdot \left(x \cdot \left(2.5 + -2.6666666666666665 \cdot wj\right)\right) + wj \cdot \left(1 + -1 \cdot wj\right)\right)} - 2 \cdot x\right) \]
   7. Step-by-step derivation

      1. distribute-lft-out 98.4%

         \[\leadsto x + wj \cdot \left(\color{blue}{wj \cdot \left(x \cdot \left(2.5 + -2.6666666666666665 \cdot wj\right) + \left(1 + -1 \cdot wj\right)\right)} - 2 \cdot x\right) \]

      2. *-commutative 98.4%

         \[\leadsto x + wj \cdot \left(wj \cdot \left(x \cdot \left(2.5 + \color{blue}{wj \cdot -2.6666666666666665}\right) + \left(1 + -1 \cdot wj\right)\right) - 2 \cdot x\right) \]

      3. mul-1-neg 98.4%

         \[\leadsto x + wj \cdot \left(wj \cdot \left(x \cdot \left(2.5 + wj \cdot -2.6666666666666665\right) + \left(1 + \color{blue}{\left(-wj\right)}\right)\right) - 2 \cdot x\right) \]

   8. Simplified 98.4%

      \[\leadsto x + wj \cdot \left(\color{blue}{wj \cdot \left(x \cdot \left(2.5 + wj \cdot -2.6666666666666665\right) + \left(1 + \left(-wj\right)\right)\right)} - 2 \cdot x\right) \]

   9. Taylor expanded in x around inf 98.4%

      \[\leadsto x + wj \cdot \left(wj \cdot \color{blue}{\left(x \cdot \left(\left(2.5 + \left(-2.6666666666666665 \cdot wj + \frac{1}{x}\right)\right) - \frac{wj}{x}\right)\right)} - 2 \cdot x\right) \]

   if -5e-32 < wj < 5e-83

   1. Initial program 81.3%

      \[wj - \frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}} \]
   2. Step-by-step derivation

      1. distribute-rgt1-in 81.3%

         \[\leadsto wj - \frac{wj \cdot e^{wj} - x}{\color{blue}{\left(wj + 1\right) \cdot e^{wj}}} \]

      2. *-commutative 81.3%

         \[\leadsto wj - \frac{wj \cdot e^{wj} - x}{\color{blue}{e^{wj} \cdot \left(wj + 1\right)}} \]

      3. associate-/r* 81.3%

         \[\leadsto wj - \color{blue}{\frac{\frac{wj \cdot e^{wj} - x}{e^{wj}}}{wj + 1}} \]

      4. div-sub 81.3%

         \[\leadsto wj - \frac{\color{blue}{\frac{wj \cdot e^{wj}}{e^{wj}} - \frac{x}{e^{wj}}}}{wj + 1} \]

      5. associate-/l* 81.3%

         \[\leadsto wj - \frac{\color{blue}{wj \cdot \frac{e^{wj}}{e^{wj}}} - \frac{x}{e^{wj}}}{wj + 1} \]

      6. *-inverses 81.3%

         \[\leadsto wj - \frac{wj \cdot \color{blue}{1} - \frac{x}{e^{wj}}}{wj + 1} \]

      7. *-rgt-identity 81.3%

         \[\leadsto wj - \frac{\color{blue}{wj} - \frac{x}{e^{wj}}}{wj + 1} \]

   3. Simplified 81.3%

      \[\leadsto \color{blue}{wj - \frac{wj - \frac{x}{e^{wj}}}{wj + 1}} \]
   4. Add Preprocessing

   5. Taylor expanded in wj around 0 99.6%

      \[\leadsto \color{blue}{x + wj \cdot \left(wj \cdot \left(\left(1 + -1 \cdot \left(wj \cdot \left(1 + \left(-3 \cdot x + \left(-2 \cdot \left(-4 \cdot x + 1.5 \cdot x\right) + 0.6666666666666666 \cdot x\right)\right)\right)\right)\right) - \left(-4 \cdot x + 1.5 \cdot x\right)\right) - 2 \cdot x\right)} \]

   6. Taylor expanded in x around 0 100.0%

      \[\leadsto x + wj \cdot \left(\color{blue}{wj \cdot \left(1 + -1 \cdot wj\right)} - 2 \cdot x\right) \]
   7. Step-by-step derivation

      1. mul-1-neg 100.0%

         \[\leadsto x + wj \cdot \left(wj \cdot \left(1 + \color{blue}{\left(-wj\right)}\right) - 2 \cdot x\right) \]

      2. unsub-neg 100.0%

         \[\leadsto x + wj \cdot \left(wj \cdot \color{blue}{\left(1 - wj\right)} - 2 \cdot x\right) \]

   8. Simplified 100.0%

      \[\leadsto x + wj \cdot \left(\color{blue}{wj \cdot \left(1 - wj\right)} - 2 \cdot x\right) \]

   9. Taylor expanded in wj around 0 100.0%

      \[\leadsto x + wj \cdot \left(\color{blue}{wj} - 2 \cdot x\right) \]

   if 5e-83 < wj < 5.19999999999999954e-4

   1. Initial program 52.2%

      \[wj - \frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}} \]
   2. Step-by-step derivation

      1. distribute-rgt1-in 52.1%

         \[\leadsto wj - \frac{wj \cdot e^{wj} - x}{\color{blue}{\left(wj + 1\right) \cdot e^{wj}}} \]

      2. *-commutative 52.1%

         \[\leadsto wj - \frac{wj \cdot e^{wj} - x}{\color{blue}{e^{wj} \cdot \left(wj + 1\right)}} \]

      3. associate-/r* 52.3%

         \[\leadsto wj - \color{blue}{\frac{\frac{wj \cdot e^{wj} - x}{e^{wj}}}{wj + 1}} \]

      4. div-sub 52.3%

         \[\leadsto wj - \frac{\color{blue}{\frac{wj \cdot e^{wj}}{e^{wj}} - \frac{x}{e^{wj}}}}{wj + 1} \]

      5. associate-/l* 52.3%

         \[\leadsto wj - \frac{\color{blue}{wj \cdot \frac{e^{wj}}{e^{wj}}} - \frac{x}{e^{wj}}}{wj + 1} \]

      6. *-inverses 52.3%

         \[\leadsto wj - \frac{wj \cdot \color{blue}{1} - \frac{x}{e^{wj}}}{wj + 1} \]

      7. *-rgt-identity 52.3%

         \[\leadsto wj - \frac{\color{blue}{wj} - \frac{x}{e^{wj}}}{wj + 1} \]

   3. Simplified 52.3%

      \[\leadsto \color{blue}{wj - \frac{wj - \frac{x}{e^{wj}}}{wj + 1}} \]
   4. Add Preprocessing

   5. Taylor expanded in wj around 0 99.4%

      \[\leadsto \color{blue}{x + wj \cdot \left(wj \cdot \left(\left(1 + -1 \cdot \left(wj \cdot \left(1 + \left(-3 \cdot x + \left(-2 \cdot \left(-4 \cdot x + 1.5 \cdot x\right) + 0.6666666666666666 \cdot x\right)\right)\right)\right)\right) - \left(-4 \cdot x + 1.5 \cdot x\right)\right) - 2 \cdot x\right)} \]

   6. Taylor expanded in x around 0 99.4%

      \[\leadsto x + wj \cdot \left(\color{blue}{\left(wj \cdot \left(x \cdot \left(2.5 + -2.6666666666666665 \cdot wj\right)\right) + wj \cdot \left(1 + -1 \cdot wj\right)\right)} - 2 \cdot x\right) \]
   7. Step-by-step derivation

      1. distribute-lft-out 99.4%

         \[\leadsto x + wj \cdot \left(\color{blue}{wj \cdot \left(x \cdot \left(2.5 + -2.6666666666666665 \cdot wj\right) + \left(1 + -1 \cdot wj\right)\right)} - 2 \cdot x\right) \]

      2. *-commutative 99.4%

         \[\leadsto x + wj \cdot \left(wj \cdot \left(x \cdot \left(2.5 + \color{blue}{wj \cdot -2.6666666666666665}\right) + \left(1 + -1 \cdot wj\right)\right) - 2 \cdot x\right) \]

      3. mul-1-neg 99.4%

         \[\leadsto x + wj \cdot \left(wj \cdot \left(x \cdot \left(2.5 + wj \cdot -2.6666666666666665\right) + \left(1 + \color{blue}{\left(-wj\right)}\right)\right) - 2 \cdot x\right) \]

   8. Simplified 99.4%

      \[\leadsto x + wj \cdot \left(\color{blue}{wj \cdot \left(x \cdot \left(2.5 + wj \cdot -2.6666666666666665\right) + \left(1 + \left(-wj\right)\right)\right)} - 2 \cdot x\right) \]

   9. Taylor expanded in x around 0 99.4%

      \[\leadsto x + wj \cdot \left(wj \cdot \color{blue}{\left(\left(1 + x \cdot \left(2.5 + -2.6666666666666665 \cdot wj\right)\right) - wj\right)} - 2 \cdot x\right) \]

   10. Taylor expanded in x around inf 99.5%

       \[\leadsto x + wj \cdot \left(wj \cdot \left(\color{blue}{x \cdot \left(2.5 + \left(-2.6666666666666665 \cdot wj + \frac{1}{x}\right)\right)} - wj\right) - 2 \cdot x\right) \]

   if 5.19999999999999954e-4 < wj < 0.0018500000000000001

   1. Initial program 100.0%

      \[wj - \frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}} \]
   2. Step-by-step derivation

      1. distribute-rgt1-in 98.4%

         \[\leadsto wj - \frac{wj \cdot e^{wj} - x}{\color{blue}{\left(wj + 1\right) \cdot e^{wj}}} \]

      2. *-commutative 98.4%

         \[\leadsto wj - \frac{wj \cdot e^{wj} - x}{\color{blue}{e^{wj} \cdot \left(wj + 1\right)}} \]

      3. associate-/r* 98.4%

         \[\leadsto wj - \color{blue}{\frac{\frac{wj \cdot e^{wj} - x}{e^{wj}}}{wj + 1}} \]

      4. div-sub 98.4%

         \[\leadsto wj - \frac{\color{blue}{\frac{wj \cdot e^{wj}}{e^{wj}} - \frac{x}{e^{wj}}}}{wj + 1} \]

      5. associate-/l* 98.4%

         \[\leadsto wj - \frac{\color{blue}{wj \cdot \frac{e^{wj}}{e^{wj}}} - \frac{x}{e^{wj}}}{wj + 1} \]

      6. *-inverses 98.4%

         \[\leadsto wj - \frac{wj \cdot \color{blue}{1} - \frac{x}{e^{wj}}}{wj + 1} \]

      7. *-rgt-identity 98.4%

         \[\leadsto wj - \frac{\color{blue}{wj} - \frac{x}{e^{wj}}}{wj + 1} \]

   3. Simplified 98.4%

      \[\leadsto \color{blue}{wj - \frac{wj - \frac{x}{e^{wj}}}{wj + 1}} \]
   4. Add Preprocessing

   5. Taylor expanded in x around inf 98.4%

      \[\leadsto wj - \color{blue}{x \cdot \left(\frac{wj}{x \cdot \left(1 + wj\right)} - \frac{1}{e^{wj} \cdot \left(1 + wj\right)}\right)} \]
   6. Step-by-step derivation

      1. +-commutative 98.4%

         \[\leadsto wj - x \cdot \left(\frac{wj}{x \cdot \color{blue}{\left(wj + 1\right)}} - \frac{1}{e^{wj} \cdot \left(1 + wj\right)}\right) \]

      2. associate-/r* 100.0%

         \[\leadsto wj - x \cdot \left(\frac{wj}{x \cdot \left(wj + 1\right)} - \color{blue}{\frac{\frac{1}{e^{wj}}}{1 + wj}}\right) \]

      3. exp-neg 100.0%

         \[\leadsto wj - x \cdot \left(\frac{wj}{x \cdot \left(wj + 1\right)} - \frac{\color{blue}{e^{-wj}}}{1 + wj}\right) \]

      4. +-commutative 100.0%

         \[\leadsto wj - x \cdot \left(\frac{wj}{x \cdot \left(wj + 1\right)} - \frac{e^{-wj}}{\color{blue}{wj + 1}}\right) \]

   7. Simplified 100.0%

      \[\leadsto wj - \color{blue}{x \cdot \left(\frac{wj}{x \cdot \left(wj + 1\right)} - \frac{e^{-wj}}{wj + 1}\right)} \]

   8. Taylor expanded in x around inf 98.4%

      \[\leadsto \color{blue}{\frac{x \cdot e^{-wj}}{1 + wj}} \]
   9. Step-by-step derivation

      1. associate-/l* 100.0%

         \[\leadsto \color{blue}{x \cdot \frac{e^{-wj}}{1 + wj}} \]

      2. +-commutative 100.0%

         \[\leadsto x \cdot \frac{e^{-wj}}{\color{blue}{wj + 1}} \]

   10. Simplified 100.0%

       \[\leadsto \color{blue}{x \cdot \frac{e^{-wj}}{wj + 1}} \]

   if 0.0018500000000000001 < wj < 0.0140000000000000003

   1. Initial program 94.8%

      \[wj - \frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}} \]
   2. Step-by-step derivation

      1. distribute-rgt1-in 94.8%

         \[\leadsto wj - \frac{wj \cdot e^{wj} - x}{\color{blue}{\left(wj + 1\right) \cdot e^{wj}}} \]

      2. *-commutative 94.8%

         \[\leadsto wj - \frac{wj \cdot e^{wj} - x}{\color{blue}{e^{wj} \cdot \left(wj + 1\right)}} \]

      3. associate-/r* 90.9%

         \[\leadsto wj - \color{blue}{\frac{\frac{wj \cdot e^{wj} - x}{e^{wj}}}{wj + 1}} \]

      4. div-sub 90.9%

         \[\leadsto wj - \frac{\color{blue}{\frac{wj \cdot e^{wj}}{e^{wj}} - \frac{x}{e^{wj}}}}{wj + 1} \]

      5. associate-/l* 90.9%

         \[\leadsto wj - \frac{\color{blue}{wj \cdot \frac{e^{wj}}{e^{wj}}} - \frac{x}{e^{wj}}}{wj + 1} \]

      6. *-inverses 90.9%

         \[\leadsto wj - \frac{wj \cdot \color{blue}{1} - \frac{x}{e^{wj}}}{wj + 1} \]

      7. *-rgt-identity 90.9%

         \[\leadsto wj - \frac{\color{blue}{wj} - \frac{x}{e^{wj}}}{wj + 1} \]

   3. Simplified 90.9%

      \[\leadsto \color{blue}{wj - \frac{wj - \frac{x}{e^{wj}}}{wj + 1}} \]
   4. Add Preprocessing

   5. Taylor expanded in x around 0 90.9%

      \[\leadsto \color{blue}{wj - \frac{wj}{1 + wj}} \]
   6. Step-by-step derivation

      1. +-commutative 90.6%

         \[\leadsto \left(wj - \frac{wj}{\color{blue}{wj + 1}}\right) + \mathsf{fma}\left(\frac{-1}{wj + 1}, wj - \frac{x}{e^{wj}}, \frac{wj}{wj + 1}\right) \]

   7. Simplified 90.9%

      \[\leadsto \color{blue}{wj - \frac{wj}{wj + 1}} \]
3. Recombined 6 regimes into one program.

4. Final simplification 99.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;wj \leq -3.3 \cdot 10^{-6}:\\ \;\;\;\;wj + \frac{wj - \frac{x}{e^{wj}}}{-1 - wj}\\ \mathbf{elif}\;wj \leq -5 \cdot 10^{-32}:\\ \;\;\;\;x - wj \cdot \left(x \cdot 2 + wj \cdot \left(x \cdot \left(\frac{wj}{x} + \left(\left(\frac{-1}{x} - wj \cdot -2.6666666666666665\right) - 2.5\right)\right)\right)\right)\\ \mathbf{elif}\;wj \leq 5 \cdot 10^{-83}:\\ \;\;\;\;x - wj \cdot \left(x \cdot 2 - wj\right)\\ \mathbf{elif}\;wj \leq 0.00052:\\ \;\;\;\;x - wj \cdot \left(x \cdot 2 + wj \cdot \left(wj + x \cdot \left(\left(\frac{-1}{x} - wj \cdot -2.6666666666666665\right) - 2.5\right)\right)\right)\\ \mathbf{elif}\;wj \leq 0.00185:\\ \;\;\;\;x \cdot \frac{e^{-wj}}{wj + 1}\\ \mathbf{elif}\;wj \leq 0.014:\\ \;\;\;\;wj - \frac{wj}{wj + 1}\\ \mathbf{else}:\\ \;\;\;\;wj + \frac{wj - \frac{x}{e^{wj}}}{-1 - wj}\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 98.9% accurate, 2.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x}{\left(wj + 1\right) \cdot e^{wj}}\\ \mathbf{if}\;wj \leq -0.018:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;wj \leq 5 \cdot 10^{-83}:\\ \;\;\;\;x + wj \cdot \left(wj \cdot \left(\left(1 + x \cdot \left(2.5 + wj \cdot -2.6666666666666665\right)\right) - wj\right) - x \cdot 2\right)\\ \mathbf{elif}\;wj \leq 0.00052:\\ \;\;\;\;x - wj \cdot \left(x \cdot 2 + wj \cdot \left(wj + x \cdot \left(\left(\frac{-1}{x} - wj \cdot -2.6666666666666665\right) - 2.5\right)\right)\right)\\ \mathbf{elif}\;wj \leq 0.00215:\\ \;\;\;\;x \cdot \frac{e^{-wj}}{wj + 1}\\ \mathbf{elif}\;wj \leq 0.021 \lor \neg \left(wj \leq 3.5\right):\\ \;\;\;\;wj - \frac{wj}{wj + 1}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (wj x)
 :precision binary64
 (let* ((t_0 (/ x (* (+ wj 1.0) (exp wj)))))
   (if (<= wj -0.018)
     t_0
     (if (<= wj 5e-83)
       (+
        x
        (*
         wj
         (-
          (* wj (- (+ 1.0 (* x (+ 2.5 (* wj -2.6666666666666665)))) wj))
          (* x 2.0))))
       (if (<= wj 0.00052)
         (-
          x
          (*
           wj
           (+
            (* x 2.0)
            (*
             wj
             (+ wj (* x (- (- (/ -1.0 x) (* wj -2.6666666666666665)) 2.5)))))))
         (if (<= wj 0.00215)
           (* x (/ (exp (- wj)) (+ wj 1.0)))
           (if (or (<= wj 0.021) (not (<= wj 3.5)))
             (- wj (/ wj (+ wj 1.0)))
             t_0)))))))

C

double code(double wj, double x) {
	double t_0 = x / ((wj + 1.0) * exp(wj));
	double tmp;
	if (wj <= -0.018) {
		tmp = t_0;
	} else if (wj <= 5e-83) {
		tmp = x + (wj * ((wj * ((1.0 + (x * (2.5 + (wj * -2.6666666666666665)))) - wj)) - (x * 2.0)));
	} else if (wj <= 0.00052) {
		tmp = x - (wj * ((x * 2.0) + (wj * (wj + (x * (((-1.0 / x) - (wj * -2.6666666666666665)) - 2.5))))));
	} else if (wj <= 0.00215) {
		tmp = x * (exp(-wj) / (wj + 1.0));
	} else if ((wj <= 0.021) || !(wj <= 3.5)) {
		tmp = wj - (wj / (wj + 1.0));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(wj, x)
    real(8), intent (in) :: wj
    real(8), intent (in) :: x
    real(8) :: t_0
    real(8) :: tmp
    t_0 = x / ((wj + 1.0d0) * exp(wj))
    if (wj <= (-0.018d0)) then
        tmp = t_0
    else if (wj <= 5d-83) then
        tmp = x + (wj * ((wj * ((1.0d0 + (x * (2.5d0 + (wj * (-2.6666666666666665d0))))) - wj)) - (x * 2.0d0)))
    else if (wj <= 0.00052d0) then
        tmp = x - (wj * ((x * 2.0d0) + (wj * (wj + (x * ((((-1.0d0) / x) - (wj * (-2.6666666666666665d0))) - 2.5d0))))))
    else if (wj <= 0.00215d0) then
        tmp = x * (exp(-wj) / (wj + 1.0d0))
    else if ((wj <= 0.021d0) .or. (.not. (wj <= 3.5d0))) then
        tmp = wj - (wj / (wj + 1.0d0))
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double wj, double x) {
	double t_0 = x / ((wj + 1.0) * Math.exp(wj));
	double tmp;
	if (wj <= -0.018) {
		tmp = t_0;
	} else if (wj <= 5e-83) {
		tmp = x + (wj * ((wj * ((1.0 + (x * (2.5 + (wj * -2.6666666666666665)))) - wj)) - (x * 2.0)));
	} else if (wj <= 0.00052) {
		tmp = x - (wj * ((x * 2.0) + (wj * (wj + (x * (((-1.0 / x) - (wj * -2.6666666666666665)) - 2.5))))));
	} else if (wj <= 0.00215) {
		tmp = x * (Math.exp(-wj) / (wj + 1.0));
	} else if ((wj <= 0.021) || !(wj <= 3.5)) {
		tmp = wj - (wj / (wj + 1.0));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(wj, x):
	t_0 = x / ((wj + 1.0) * math.exp(wj))
	tmp = 0
	if wj <= -0.018:
		tmp = t_0
	elif wj <= 5e-83:
		tmp = x + (wj * ((wj * ((1.0 + (x * (2.5 + (wj * -2.6666666666666665)))) - wj)) - (x * 2.0)))
	elif wj <= 0.00052:
		tmp = x - (wj * ((x * 2.0) + (wj * (wj + (x * (((-1.0 / x) - (wj * -2.6666666666666665)) - 2.5))))))
	elif wj <= 0.00215:
		tmp = x * (math.exp(-wj) / (wj + 1.0))
	elif (wj <= 0.021) or not (wj <= 3.5):
		tmp = wj - (wj / (wj + 1.0))
	else:
		tmp = t_0
	return tmp

Julia

function code(wj, x)
	t_0 = Float64(x / Float64(Float64(wj + 1.0) * exp(wj)))
	tmp = 0.0
	if (wj <= -0.018)
		tmp = t_0;
	elseif (wj <= 5e-83)
		tmp = Float64(x + Float64(wj * Float64(Float64(wj * Float64(Float64(1.0 + Float64(x * Float64(2.5 + Float64(wj * -2.6666666666666665)))) - wj)) - Float64(x * 2.0))));
	elseif (wj <= 0.00052)
		tmp = Float64(x - Float64(wj * Float64(Float64(x * 2.0) + Float64(wj * Float64(wj + Float64(x * Float64(Float64(Float64(-1.0 / x) - Float64(wj * -2.6666666666666665)) - 2.5)))))));
	elseif (wj <= 0.00215)
		tmp = Float64(x * Float64(exp(Float64(-wj)) / Float64(wj + 1.0)));
	elseif ((wj <= 0.021) || !(wj <= 3.5))
		tmp = Float64(wj - Float64(wj / Float64(wj + 1.0)));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(wj, x)
	t_0 = x / ((wj + 1.0) * exp(wj));
	tmp = 0.0;
	if (wj <= -0.018)
		tmp = t_0;
	elseif (wj <= 5e-83)
		tmp = x + (wj * ((wj * ((1.0 + (x * (2.5 + (wj * -2.6666666666666665)))) - wj)) - (x * 2.0)));
	elseif (wj <= 0.00052)
		tmp = x - (wj * ((x * 2.0) + (wj * (wj + (x * (((-1.0 / x) - (wj * -2.6666666666666665)) - 2.5))))));
	elseif (wj <= 0.00215)
		tmp = x * (exp(-wj) / (wj + 1.0));
	elseif ((wj <= 0.021) || ~((wj <= 3.5)))
		tmp = wj - (wj / (wj + 1.0));
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[wj_, x_] := Block[{t$95$0 = N[(x / N[(N[(wj + 1.0), $MachinePrecision] * N[Exp[wj], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[wj, -0.018], t$95$0, If[LessEqual[wj, 5e-83], N[(x + N[(wj * N[(N[(wj * N[(N[(1.0 + N[(x * N[(2.5 + N[(wj * -2.6666666666666665), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - wj), $MachinePrecision]), $MachinePrecision] - N[(x * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[wj, 0.00052], N[(x - N[(wj * N[(N[(x * 2.0), $MachinePrecision] + N[(wj * N[(wj + N[(x * N[(N[(N[(-1.0 / x), $MachinePrecision] - N[(wj * -2.6666666666666665), $MachinePrecision]), $MachinePrecision] - 2.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[wj, 0.00215], N[(x * N[(N[Exp[(-wj)], $MachinePrecision] / N[(wj + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[Or[LessEqual[wj, 0.021], N[Not[LessEqual[wj, 3.5]], $MachinePrecision]], N[(wj - N[(wj / N[(wj + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]]]]]

Derivation

1. Split input into 5 regimes

2. if wj < -0.0179999999999999986 or 0.0210000000000000013 < wj < 3.5

   1. Initial program 56.7%

      \[wj - \frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}} \]
   2. Step-by-step derivation

      1. distribute-rgt1-in 100.0%

         \[\leadsto wj - \frac{wj \cdot e^{wj} - x}{\color{blue}{\left(wj + 1\right) \cdot e^{wj}}} \]

      2. *-commutative 100.0%

         \[\leadsto wj - \frac{wj \cdot e^{wj} - x}{\color{blue}{e^{wj} \cdot \left(wj + 1\right)}} \]

      3. associate-/r* 99.8%

         \[\leadsto wj - \color{blue}{\frac{\frac{wj \cdot e^{wj} - x}{e^{wj}}}{wj + 1}} \]

      4. div-sub 56.9%

         \[\leadsto wj - \frac{\color{blue}{\frac{wj \cdot e^{wj}}{e^{wj}} - \frac{x}{e^{wj}}}}{wj + 1} \]

      5. associate-/l* 56.9%

         \[\leadsto wj - \frac{\color{blue}{wj \cdot \frac{e^{wj}}{e^{wj}}} - \frac{x}{e^{wj}}}{wj + 1} \]

      6. *-inverses 99.8%

         \[\leadsto wj - \frac{wj \cdot \color{blue}{1} - \frac{x}{e^{wj}}}{wj + 1} \]

      7. *-rgt-identity 99.8%

         \[\leadsto wj - \frac{\color{blue}{wj} - \frac{x}{e^{wj}}}{wj + 1} \]

   3. Simplified 99.8%

      \[\leadsto \color{blue}{wj - \frac{wj - \frac{x}{e^{wj}}}{wj + 1}} \]
   4. Add Preprocessing

   5. Taylor expanded in x around inf 86.0%

      \[\leadsto \color{blue}{\frac{x}{e^{wj} \cdot \left(1 + wj\right)}} \]
   6. Step-by-step derivation

      1. +-commutative 86.0%

         \[\leadsto \frac{x}{e^{wj} \cdot \color{blue}{\left(wj + 1\right)}} \]

   7. Simplified 86.0%

      \[\leadsto \color{blue}{\frac{x}{e^{wj} \cdot \left(wj + 1\right)}} \]

   if -0.0179999999999999986 < wj < 5e-83

   1. Initial program 80.7%

      \[wj - \frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}} \]
   2. Step-by-step derivation

      1. distribute-rgt1-in 80.7%

         \[\leadsto wj - \frac{wj \cdot e^{wj} - x}{\color{blue}{\left(wj + 1\right) \cdot e^{wj}}} \]

      2. *-commutative 80.7%

         \[\leadsto wj - \frac{wj \cdot e^{wj} - x}{\color{blue}{e^{wj} \cdot \left(wj + 1\right)}} \]

      3. associate-/r* 80.7%

         \[\leadsto wj - \color{blue}{\frac{\frac{wj \cdot e^{wj} - x}{e^{wj}}}{wj + 1}} \]

      4. div-sub 80.7%

         \[\leadsto wj - \frac{\color{blue}{\frac{wj \cdot e^{wj}}{e^{wj}} - \frac{x}{e^{wj}}}}{wj + 1} \]

      5. associate-/l* 80.7%

         \[\leadsto wj - \frac{\color{blue}{wj \cdot \frac{e^{wj}}{e^{wj}}} - \frac{x}{e^{wj}}}{wj + 1} \]

      6. *-inverses 80.7%

         \[\leadsto wj - \frac{wj \cdot \color{blue}{1} - \frac{x}{e^{wj}}}{wj + 1} \]

      7. *-rgt-identity 80.7%

         \[\leadsto wj - \frac{\color{blue}{wj} - \frac{x}{e^{wj}}}{wj + 1} \]

   3. Simplified 80.7%

      \[\leadsto \color{blue}{wj - \frac{wj - \frac{x}{e^{wj}}}{wj + 1}} \]
   4. Add Preprocessing

   5. Taylor expanded in wj around 0 99.2%

      \[\leadsto \color{blue}{x + wj \cdot \left(wj \cdot \left(\left(1 + -1 \cdot \left(wj \cdot \left(1 + \left(-3 \cdot x + \left(-2 \cdot \left(-4 \cdot x + 1.5 \cdot x\right) + 0.6666666666666666 \cdot x\right)\right)\right)\right)\right) - \left(-4 \cdot x + 1.5 \cdot x\right)\right) - 2 \cdot x\right)} \]

   6. Taylor expanded in x around 0 99.6%

      \[\leadsto x + wj \cdot \left(\color{blue}{\left(wj \cdot \left(x \cdot \left(2.5 + -2.6666666666666665 \cdot wj\right)\right) + wj \cdot \left(1 + -1 \cdot wj\right)\right)} - 2 \cdot x\right) \]
   7. Step-by-step derivation

      1. distribute-lft-out 99.6%

         \[\leadsto x + wj \cdot \left(\color{blue}{wj \cdot \left(x \cdot \left(2.5 + -2.6666666666666665 \cdot wj\right) + \left(1 + -1 \cdot wj\right)\right)} - 2 \cdot x\right) \]

      2. *-commutative 99.6%

         \[\leadsto x + wj \cdot \left(wj \cdot \left(x \cdot \left(2.5 + \color{blue}{wj \cdot -2.6666666666666665}\right) + \left(1 + -1 \cdot wj\right)\right) - 2 \cdot x\right) \]

      3. mul-1-neg 99.6%

         \[\leadsto x + wj \cdot \left(wj \cdot \left(x \cdot \left(2.5 + wj \cdot -2.6666666666666665\right) + \left(1 + \color{blue}{\left(-wj\right)}\right)\right) - 2 \cdot x\right) \]

   8. Simplified 99.6%

      \[\leadsto x + wj \cdot \left(\color{blue}{wj \cdot \left(x \cdot \left(2.5 + wj \cdot -2.6666666666666665\right) + \left(1 + \left(-wj\right)\right)\right)} - 2 \cdot x\right) \]

   9. Taylor expanded in x around 0 99.6%

      \[\leadsto x + wj \cdot \left(wj \cdot \color{blue}{\left(\left(1 + x \cdot \left(2.5 + -2.6666666666666665 \cdot wj\right)\right) - wj\right)} - 2 \cdot x\right) \]

   if 5e-83 < wj < 5.19999999999999954e-4

   1. Initial program 52.2%

      \[wj - \frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}} \]
   2. Step-by-step derivation

      1. distribute-rgt1-in 52.1%

         \[\leadsto wj - \frac{wj \cdot e^{wj} - x}{\color{blue}{\left(wj + 1\right) \cdot e^{wj}}} \]

      2. *-commutative 52.1%

         \[\leadsto wj - \frac{wj \cdot e^{wj} - x}{\color{blue}{e^{wj} \cdot \left(wj + 1\right)}} \]

      3. associate-/r* 52.3%

         \[\leadsto wj - \color{blue}{\frac{\frac{wj \cdot e^{wj} - x}{e^{wj}}}{wj + 1}} \]

      4. div-sub 52.3%

         \[\leadsto wj - \frac{\color{blue}{\frac{wj \cdot e^{wj}}{e^{wj}} - \frac{x}{e^{wj}}}}{wj + 1} \]

      5. associate-/l* 52.3%

         \[\leadsto wj - \frac{\color{blue}{wj \cdot \frac{e^{wj}}{e^{wj}}} - \frac{x}{e^{wj}}}{wj + 1} \]

      6. *-inverses 52.3%

         \[\leadsto wj - \frac{wj \cdot \color{blue}{1} - \frac{x}{e^{wj}}}{wj + 1} \]

      7. *-rgt-identity 52.3%

         \[\leadsto wj - \frac{\color{blue}{wj} - \frac{x}{e^{wj}}}{wj + 1} \]

   3. Simplified 52.3%

      \[\leadsto \color{blue}{wj - \frac{wj - \frac{x}{e^{wj}}}{wj + 1}} \]
   4. Add Preprocessing

   5. Taylor expanded in wj around 0 99.4%

      \[\leadsto \color{blue}{x + wj \cdot \left(wj \cdot \left(\left(1 + -1 \cdot \left(wj \cdot \left(1 + \left(-3 \cdot x + \left(-2 \cdot \left(-4 \cdot x + 1.5 \cdot x\right) + 0.6666666666666666 \cdot x\right)\right)\right)\right)\right) - \left(-4 \cdot x + 1.5 \cdot x\right)\right) - 2 \cdot x\right)} \]

   6. Taylor expanded in x around 0 99.4%

      \[\leadsto x + wj \cdot \left(\color{blue}{\left(wj \cdot \left(x \cdot \left(2.5 + -2.6666666666666665 \cdot wj\right)\right) + wj \cdot \left(1 + -1 \cdot wj\right)\right)} - 2 \cdot x\right) \]
   7. Step-by-step derivation

      1. distribute-lft-out 99.4%

         \[\leadsto x + wj \cdot \left(\color{blue}{wj \cdot \left(x \cdot \left(2.5 + -2.6666666666666665 \cdot wj\right) + \left(1 + -1 \cdot wj\right)\right)} - 2 \cdot x\right) \]

      2. *-commutative 99.4%

         \[\leadsto x + wj \cdot \left(wj \cdot \left(x \cdot \left(2.5 + \color{blue}{wj \cdot -2.6666666666666665}\right) + \left(1 + -1 \cdot wj\right)\right) - 2 \cdot x\right) \]

      3. mul-1-neg 99.4%

         \[\leadsto x + wj \cdot \left(wj \cdot \left(x \cdot \left(2.5 + wj \cdot -2.6666666666666665\right) + \left(1 + \color{blue}{\left(-wj\right)}\right)\right) - 2 \cdot x\right) \]

   8. Simplified 99.4%

      \[\leadsto x + wj \cdot \left(\color{blue}{wj \cdot \left(x \cdot \left(2.5 + wj \cdot -2.6666666666666665\right) + \left(1 + \left(-wj\right)\right)\right)} - 2 \cdot x\right) \]

   9. Taylor expanded in x around 0 99.4%

      \[\leadsto x + wj \cdot \left(wj \cdot \color{blue}{\left(\left(1 + x \cdot \left(2.5 + -2.6666666666666665 \cdot wj\right)\right) - wj\right)} - 2 \cdot x\right) \]

   10. Taylor expanded in x around inf 99.5%

       \[\leadsto x + wj \cdot \left(wj \cdot \left(\color{blue}{x \cdot \left(2.5 + \left(-2.6666666666666665 \cdot wj + \frac{1}{x}\right)\right)} - wj\right) - 2 \cdot x\right) \]

   if 5.19999999999999954e-4 < wj < 0.00215

   1. Initial program 100.0%

      \[wj - \frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}} \]
   2. Step-by-step derivation

      1. distribute-rgt1-in 98.4%

         \[\leadsto wj - \frac{wj \cdot e^{wj} - x}{\color{blue}{\left(wj + 1\right) \cdot e^{wj}}} \]

      2. *-commutative 98.4%

         \[\leadsto wj - \frac{wj \cdot e^{wj} - x}{\color{blue}{e^{wj} \cdot \left(wj + 1\right)}} \]

      3. associate-/r* 98.4%

         \[\leadsto wj - \color{blue}{\frac{\frac{wj \cdot e^{wj} - x}{e^{wj}}}{wj + 1}} \]

      4. div-sub 98.4%

         \[\leadsto wj - \frac{\color{blue}{\frac{wj \cdot e^{wj}}{e^{wj}} - \frac{x}{e^{wj}}}}{wj + 1} \]

      5. associate-/l* 98.4%

         \[\leadsto wj - \frac{\color{blue}{wj \cdot \frac{e^{wj}}{e^{wj}}} - \frac{x}{e^{wj}}}{wj + 1} \]

      6. *-inverses 98.4%

         \[\leadsto wj - \frac{wj \cdot \color{blue}{1} - \frac{x}{e^{wj}}}{wj + 1} \]

      7. *-rgt-identity 98.4%

         \[\leadsto wj - \frac{\color{blue}{wj} - \frac{x}{e^{wj}}}{wj + 1} \]

   3. Simplified 98.4%

      \[\leadsto \color{blue}{wj - \frac{wj - \frac{x}{e^{wj}}}{wj + 1}} \]
   4. Add Preprocessing

   5. Taylor expanded in x around inf 98.4%

      \[\leadsto wj - \color{blue}{x \cdot \left(\frac{wj}{x \cdot \left(1 + wj\right)} - \frac{1}{e^{wj} \cdot \left(1 + wj\right)}\right)} \]
   6. Step-by-step derivation

      1. +-commutative 98.4%

         \[\leadsto wj - x \cdot \left(\frac{wj}{x \cdot \color{blue}{\left(wj + 1\right)}} - \frac{1}{e^{wj} \cdot \left(1 + wj\right)}\right) \]

      2. associate-/r* 100.0%

         \[\leadsto wj - x \cdot \left(\frac{wj}{x \cdot \left(wj + 1\right)} - \color{blue}{\frac{\frac{1}{e^{wj}}}{1 + wj}}\right) \]

      3. exp-neg 100.0%

         \[\leadsto wj - x \cdot \left(\frac{wj}{x \cdot \left(wj + 1\right)} - \frac{\color{blue}{e^{-wj}}}{1 + wj}\right) \]

      4. +-commutative 100.0%

         \[\leadsto wj - x \cdot \left(\frac{wj}{x \cdot \left(wj + 1\right)} - \frac{e^{-wj}}{\color{blue}{wj + 1}}\right) \]

   7. Simplified 100.0%

      \[\leadsto wj - \color{blue}{x \cdot \left(\frac{wj}{x \cdot \left(wj + 1\right)} - \frac{e^{-wj}}{wj + 1}\right)} \]

   8. Taylor expanded in x around inf 98.4%

      \[\leadsto \color{blue}{\frac{x \cdot e^{-wj}}{1 + wj}} \]
   9. Step-by-step derivation

      1. associate-/l* 100.0%

         \[\leadsto \color{blue}{x \cdot \frac{e^{-wj}}{1 + wj}} \]

      2. +-commutative 100.0%

         \[\leadsto x \cdot \frac{e^{-wj}}{\color{blue}{wj + 1}} \]

   10. Simplified 100.0%

       \[\leadsto \color{blue}{x \cdot \frac{e^{-wj}}{wj + 1}} \]

   if 0.00215 < wj < 0.0210000000000000013 or 3.5 < wj

   1. Initial program 73.7%

      \[wj - \frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}} \]
   2. Step-by-step derivation

      1. distribute-rgt1-in 73.7%

         \[\leadsto wj - \frac{wj \cdot e^{wj} - x}{\color{blue}{\left(wj + 1\right) \cdot e^{wj}}} \]

      2. *-commutative 73.7%

         \[\leadsto wj - \frac{wj \cdot e^{wj} - x}{\color{blue}{e^{wj} \cdot \left(wj + 1\right)}} \]

      3. associate-/r* 72.7%

         \[\leadsto wj - \color{blue}{\frac{\frac{wj \cdot e^{wj} - x}{e^{wj}}}{wj + 1}} \]

      4. div-sub 72.7%

         \[\leadsto wj - \frac{\color{blue}{\frac{wj \cdot e^{wj}}{e^{wj}} - \frac{x}{e^{wj}}}}{wj + 1} \]

      5. associate-/l* 72.7%

         \[\leadsto wj - \frac{\color{blue}{wj \cdot \frac{e^{wj}}{e^{wj}}} - \frac{x}{e^{wj}}}{wj + 1} \]

      6. *-inverses 97.7%

         \[\leadsto wj - \frac{wj \cdot \color{blue}{1} - \frac{x}{e^{wj}}}{wj + 1} \]

      7. *-rgt-identity 97.7%

         \[\leadsto wj - \frac{\color{blue}{wj} - \frac{x}{e^{wj}}}{wj + 1} \]

   3. Simplified 97.7%

      \[\leadsto \color{blue}{wj - \frac{wj - \frac{x}{e^{wj}}}{wj + 1}} \]
   4. Add Preprocessing

   5. Taylor expanded in x around 0 97.7%

      \[\leadsto \color{blue}{wj - \frac{wj}{1 + wj}} \]
   6. Step-by-step derivation

      1. +-commutative 97.6%

         \[\leadsto \left(wj - \frac{wj}{\color{blue}{wj + 1}}\right) + \mathsf{fma}\left(\frac{-1}{wj + 1}, wj - \frac{x}{e^{wj}}, \frac{wj}{wj + 1}\right) \]

   7. Simplified 97.7%

      \[\leadsto \color{blue}{wj - \frac{wj}{wj + 1}} \]
3. Recombined 5 regimes into one program.

4. Final simplification 99.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;wj \leq -0.018:\\ \;\;\;\;\frac{x}{\left(wj + 1\right) \cdot e^{wj}}\\ \mathbf{elif}\;wj \leq 5 \cdot 10^{-83}:\\ \;\;\;\;x + wj \cdot \left(wj \cdot \left(\left(1 + x \cdot \left(2.5 + wj \cdot -2.6666666666666665\right)\right) - wj\right) - x \cdot 2\right)\\ \mathbf{elif}\;wj \leq 0.00052:\\ \;\;\;\;x - wj \cdot \left(x \cdot 2 + wj \cdot \left(wj + x \cdot \left(\left(\frac{-1}{x} - wj \cdot -2.6666666666666665\right) - 2.5\right)\right)\right)\\ \mathbf{elif}\;wj \leq 0.00215:\\ \;\;\;\;x \cdot \frac{e^{-wj}}{wj + 1}\\ \mathbf{elif}\;wj \leq 0.021 \lor \neg \left(wj \leq 3.5\right):\\ \;\;\;\;wj - \frac{wj}{wj + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\left(wj + 1\right) \cdot e^{wj}}\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 98.9% accurate, 2.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x}{\left(wj + 1\right) \cdot e^{wj}}\\ \mathbf{if}\;wj \leq -0.0074:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;wj \leq 5 \cdot 10^{-83}:\\ \;\;\;\;x + wj \cdot \left(wj \cdot \left(\left(1 + x \cdot \left(2.5 + wj \cdot -2.6666666666666665\right)\right) - wj\right) - x \cdot 2\right)\\ \mathbf{elif}\;wj \leq 0.00052:\\ \;\;\;\;x - wj \cdot \left(x \cdot 2 + wj \cdot \left(wj + x \cdot \left(\left(\frac{-1}{x} - wj \cdot -2.6666666666666665\right) - 2.5\right)\right)\right)\\ \mathbf{elif}\;wj \leq 0.0106 \lor \neg \left(wj \leq 0.16\right) \land wj \leq 4:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;wj - \frac{wj}{wj + 1}\\ \end{array} \end{array} \]

FPCore

(FPCore (wj x)
 :precision binary64
 (let* ((t_0 (/ x (* (+ wj 1.0) (exp wj)))))
   (if (<= wj -0.0074)
     t_0
     (if (<= wj 5e-83)
       (+
        x
        (*
         wj
         (-
          (* wj (- (+ 1.0 (* x (+ 2.5 (* wj -2.6666666666666665)))) wj))
          (* x 2.0))))
       (if (<= wj 0.00052)
         (-
          x
          (*
           wj
           (+
            (* x 2.0)
            (*
             wj
             (+ wj (* x (- (- (/ -1.0 x) (* wj -2.6666666666666665)) 2.5)))))))
         (if (or (<= wj 0.0106) (and (not (<= wj 0.16)) (<= wj 4.0)))
           t_0
           (- wj (/ wj (+ wj 1.0)))))))))

C

double code(double wj, double x) {
	double t_0 = x / ((wj + 1.0) * exp(wj));
	double tmp;
	if (wj <= -0.0074) {
		tmp = t_0;
	} else if (wj <= 5e-83) {
		tmp = x + (wj * ((wj * ((1.0 + (x * (2.5 + (wj * -2.6666666666666665)))) - wj)) - (x * 2.0)));
	} else if (wj <= 0.00052) {
		tmp = x - (wj * ((x * 2.0) + (wj * (wj + (x * (((-1.0 / x) - (wj * -2.6666666666666665)) - 2.5))))));
	} else if ((wj <= 0.0106) || (!(wj <= 0.16) && (wj <= 4.0))) {
		tmp = t_0;
	} else {
		tmp = 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) :: tmp
    t_0 = x / ((wj + 1.0d0) * exp(wj))
    if (wj <= (-0.0074d0)) then
        tmp = t_0
    else if (wj <= 5d-83) then
        tmp = x + (wj * ((wj * ((1.0d0 + (x * (2.5d0 + (wj * (-2.6666666666666665d0))))) - wj)) - (x * 2.0d0)))
    else if (wj <= 0.00052d0) then
        tmp = x - (wj * ((x * 2.0d0) + (wj * (wj + (x * ((((-1.0d0) / x) - (wj * (-2.6666666666666665d0))) - 2.5d0))))))
    else if ((wj <= 0.0106d0) .or. (.not. (wj <= 0.16d0)) .and. (wj <= 4.0d0)) then
        tmp = t_0
    else
        tmp = wj - (wj / (wj + 1.0d0))
    end if
    code = tmp
end function

Java

public static double code(double wj, double x) {
	double t_0 = x / ((wj + 1.0) * Math.exp(wj));
	double tmp;
	if (wj <= -0.0074) {
		tmp = t_0;
	} else if (wj <= 5e-83) {
		tmp = x + (wj * ((wj * ((1.0 + (x * (2.5 + (wj * -2.6666666666666665)))) - wj)) - (x * 2.0)));
	} else if (wj <= 0.00052) {
		tmp = x - (wj * ((x * 2.0) + (wj * (wj + (x * (((-1.0 / x) - (wj * -2.6666666666666665)) - 2.5))))));
	} else if ((wj <= 0.0106) || (!(wj <= 0.16) && (wj <= 4.0))) {
		tmp = t_0;
	} else {
		tmp = wj - (wj / (wj + 1.0));
	}
	return tmp;
}

Python

def code(wj, x):
	t_0 = x / ((wj + 1.0) * math.exp(wj))
	tmp = 0
	if wj <= -0.0074:
		tmp = t_0
	elif wj <= 5e-83:
		tmp = x + (wj * ((wj * ((1.0 + (x * (2.5 + (wj * -2.6666666666666665)))) - wj)) - (x * 2.0)))
	elif wj <= 0.00052:
		tmp = x - (wj * ((x * 2.0) + (wj * (wj + (x * (((-1.0 / x) - (wj * -2.6666666666666665)) - 2.5))))))
	elif (wj <= 0.0106) or (not (wj <= 0.16) and (wj <= 4.0)):
		tmp = t_0
	else:
		tmp = wj - (wj / (wj + 1.0))
	return tmp

Julia

function code(wj, x)
	t_0 = Float64(x / Float64(Float64(wj + 1.0) * exp(wj)))
	tmp = 0.0
	if (wj <= -0.0074)
		tmp = t_0;
	elseif (wj <= 5e-83)
		tmp = Float64(x + Float64(wj * Float64(Float64(wj * Float64(Float64(1.0 + Float64(x * Float64(2.5 + Float64(wj * -2.6666666666666665)))) - wj)) - Float64(x * 2.0))));
	elseif (wj <= 0.00052)
		tmp = Float64(x - Float64(wj * Float64(Float64(x * 2.0) + Float64(wj * Float64(wj + Float64(x * Float64(Float64(Float64(-1.0 / x) - Float64(wj * -2.6666666666666665)) - 2.5)))))));
	elseif ((wj <= 0.0106) || (!(wj <= 0.16) && (wj <= 4.0)))
		tmp = t_0;
	else
		tmp = Float64(wj - Float64(wj / Float64(wj + 1.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(wj, x)
	t_0 = x / ((wj + 1.0) * exp(wj));
	tmp = 0.0;
	if (wj <= -0.0074)
		tmp = t_0;
	elseif (wj <= 5e-83)
		tmp = x + (wj * ((wj * ((1.0 + (x * (2.5 + (wj * -2.6666666666666665)))) - wj)) - (x * 2.0)));
	elseif (wj <= 0.00052)
		tmp = x - (wj * ((x * 2.0) + (wj * (wj + (x * (((-1.0 / x) - (wj * -2.6666666666666665)) - 2.5))))));
	elseif ((wj <= 0.0106) || (~((wj <= 0.16)) && (wj <= 4.0)))
		tmp = t_0;
	else
		tmp = wj - (wj / (wj + 1.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[wj_, x_] := Block[{t$95$0 = N[(x / N[(N[(wj + 1.0), $MachinePrecision] * N[Exp[wj], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[wj, -0.0074], t$95$0, If[LessEqual[wj, 5e-83], N[(x + N[(wj * N[(N[(wj * N[(N[(1.0 + N[(x * N[(2.5 + N[(wj * -2.6666666666666665), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - wj), $MachinePrecision]), $MachinePrecision] - N[(x * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[wj, 0.00052], N[(x - N[(wj * N[(N[(x * 2.0), $MachinePrecision] + N[(wj * N[(wj + N[(x * N[(N[(N[(-1.0 / x), $MachinePrecision] - N[(wj * -2.6666666666666665), $MachinePrecision]), $MachinePrecision] - 2.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[Or[LessEqual[wj, 0.0106], And[N[Not[LessEqual[wj, 0.16]], $MachinePrecision], LessEqual[wj, 4.0]]], t$95$0, N[(wj - N[(wj / N[(wj + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if wj < -0.0074000000000000003 or 5.19999999999999954e-4 < wj < 0.0106 or 0.160000000000000003 < wj < 4

   1. Initial program 62.1%

      \[wj - \frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}} \]
   2. Step-by-step derivation

      1. distribute-rgt1-in 99.8%

         \[\leadsto wj - \frac{wj \cdot e^{wj} - x}{\color{blue}{\left(wj + 1\right) \cdot e^{wj}}} \]

      2. *-commutative 99.8%

         \[\leadsto wj - \frac{wj \cdot e^{wj} - x}{\color{blue}{e^{wj} \cdot \left(wj + 1\right)}} \]

      3. associate-/r* 99.6%

         \[\leadsto wj - \color{blue}{\frac{\frac{wj \cdot e^{wj} - x}{e^{wj}}}{wj + 1}} \]

      4. div-sub 62.1%

         \[\leadsto wj - \frac{\color{blue}{\frac{wj \cdot e^{wj}}{e^{wj}} - \frac{x}{e^{wj}}}}{wj + 1} \]

      5. associate-/l* 62.1%

         \[\leadsto wj - \frac{\color{blue}{wj \cdot \frac{e^{wj}}{e^{wj}}} - \frac{x}{e^{wj}}}{wj + 1} \]

      6. *-inverses 99.6%

         \[\leadsto wj - \frac{wj \cdot \color{blue}{1} - \frac{x}{e^{wj}}}{wj + 1} \]

      7. *-rgt-identity 99.6%

         \[\leadsto wj - \frac{\color{blue}{wj} - \frac{x}{e^{wj}}}{wj + 1} \]

   3. Simplified 99.6%

      \[\leadsto \color{blue}{wj - \frac{wj - \frac{x}{e^{wj}}}{wj + 1}} \]
   4. Add Preprocessing

   5. Taylor expanded in x around inf 87.5%

      \[\leadsto \color{blue}{\frac{x}{e^{wj} \cdot \left(1 + wj\right)}} \]
   6. Step-by-step derivation

      1. +-commutative 87.5%

         \[\leadsto \frac{x}{e^{wj} \cdot \color{blue}{\left(wj + 1\right)}} \]

   7. Simplified 87.5%

      \[\leadsto \color{blue}{\frac{x}{e^{wj} \cdot \left(wj + 1\right)}} \]

   if -0.0074000000000000003 < wj < 5e-83

   1. Initial program 80.7%

      \[wj - \frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}} \]
   2. Step-by-step derivation

      1. distribute-rgt1-in 80.7%

         \[\leadsto wj - \frac{wj \cdot e^{wj} - x}{\color{blue}{\left(wj + 1\right) \cdot e^{wj}}} \]

      2. *-commutative 80.7%

         \[\leadsto wj - \frac{wj \cdot e^{wj} - x}{\color{blue}{e^{wj} \cdot \left(wj + 1\right)}} \]

      3. associate-/r* 80.7%

         \[\leadsto wj - \color{blue}{\frac{\frac{wj \cdot e^{wj} - x}{e^{wj}}}{wj + 1}} \]

      4. div-sub 80.7%

         \[\leadsto wj - \frac{\color{blue}{\frac{wj \cdot e^{wj}}{e^{wj}} - \frac{x}{e^{wj}}}}{wj + 1} \]

      5. associate-/l* 80.7%

         \[\leadsto wj - \frac{\color{blue}{wj \cdot \frac{e^{wj}}{e^{wj}}} - \frac{x}{e^{wj}}}{wj + 1} \]

      6. *-inverses 80.7%

         \[\leadsto wj - \frac{wj \cdot \color{blue}{1} - \frac{x}{e^{wj}}}{wj + 1} \]

      7. *-rgt-identity 80.7%

         \[\leadsto wj - \frac{\color{blue}{wj} - \frac{x}{e^{wj}}}{wj + 1} \]

   3. Simplified 80.7%

      \[\leadsto \color{blue}{wj - \frac{wj - \frac{x}{e^{wj}}}{wj + 1}} \]
   4. Add Preprocessing

   5. Taylor expanded in wj around 0 99.2%

      \[\leadsto \color{blue}{x + wj \cdot \left(wj \cdot \left(\left(1 + -1 \cdot \left(wj \cdot \left(1 + \left(-3 \cdot x + \left(-2 \cdot \left(-4 \cdot x + 1.5 \cdot x\right) + 0.6666666666666666 \cdot x\right)\right)\right)\right)\right) - \left(-4 \cdot x + 1.5 \cdot x\right)\right) - 2 \cdot x\right)} \]

   6. Taylor expanded in x around 0 99.6%

      \[\leadsto x + wj \cdot \left(\color{blue}{\left(wj \cdot \left(x \cdot \left(2.5 + -2.6666666666666665 \cdot wj\right)\right) + wj \cdot \left(1 + -1 \cdot wj\right)\right)} - 2 \cdot x\right) \]
   7. Step-by-step derivation

      1. distribute-lft-out 99.6%

         \[\leadsto x + wj \cdot \left(\color{blue}{wj \cdot \left(x \cdot \left(2.5 + -2.6666666666666665 \cdot wj\right) + \left(1 + -1 \cdot wj\right)\right)} - 2 \cdot x\right) \]

      2. *-commutative 99.6%

         \[\leadsto x + wj \cdot \left(wj \cdot \left(x \cdot \left(2.5 + \color{blue}{wj \cdot -2.6666666666666665}\right) + \left(1 + -1 \cdot wj\right)\right) - 2 \cdot x\right) \]

      3. mul-1-neg 99.6%

         \[\leadsto x + wj \cdot \left(wj \cdot \left(x \cdot \left(2.5 + wj \cdot -2.6666666666666665\right) + \left(1 + \color{blue}{\left(-wj\right)}\right)\right) - 2 \cdot x\right) \]

   8. Simplified 99.6%

      \[\leadsto x + wj \cdot \left(\color{blue}{wj \cdot \left(x \cdot \left(2.5 + wj \cdot -2.6666666666666665\right) + \left(1 + \left(-wj\right)\right)\right)} - 2 \cdot x\right) \]

   9. Taylor expanded in x around 0 99.6%

      \[\leadsto x + wj \cdot \left(wj \cdot \color{blue}{\left(\left(1 + x \cdot \left(2.5 + -2.6666666666666665 \cdot wj\right)\right) - wj\right)} - 2 \cdot x\right) \]

   if 5e-83 < wj < 5.19999999999999954e-4

   1. Initial program 52.2%

      \[wj - \frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}} \]
   2. Step-by-step derivation

      1. distribute-rgt1-in 52.1%

         \[\leadsto wj - \frac{wj \cdot e^{wj} - x}{\color{blue}{\left(wj + 1\right) \cdot e^{wj}}} \]

      2. *-commutative 52.1%

         \[\leadsto wj - \frac{wj \cdot e^{wj} - x}{\color{blue}{e^{wj} \cdot \left(wj + 1\right)}} \]

      3. associate-/r* 52.3%

         \[\leadsto wj - \color{blue}{\frac{\frac{wj \cdot e^{wj} - x}{e^{wj}}}{wj + 1}} \]

      4. div-sub 52.3%

         \[\leadsto wj - \frac{\color{blue}{\frac{wj \cdot e^{wj}}{e^{wj}} - \frac{x}{e^{wj}}}}{wj + 1} \]

      5. associate-/l* 52.3%

         \[\leadsto wj - \frac{\color{blue}{wj \cdot \frac{e^{wj}}{e^{wj}}} - \frac{x}{e^{wj}}}{wj + 1} \]

      6. *-inverses 52.3%

         \[\leadsto wj - \frac{wj \cdot \color{blue}{1} - \frac{x}{e^{wj}}}{wj + 1} \]

      7. *-rgt-identity 52.3%

         \[\leadsto wj - \frac{\color{blue}{wj} - \frac{x}{e^{wj}}}{wj + 1} \]

   3. Simplified 52.3%

      \[\leadsto \color{blue}{wj - \frac{wj - \frac{x}{e^{wj}}}{wj + 1}} \]
   4. Add Preprocessing

   5. Taylor expanded in wj around 0 99.4%

      \[\leadsto \color{blue}{x + wj \cdot \left(wj \cdot \left(\left(1 + -1 \cdot \left(wj \cdot \left(1 + \left(-3 \cdot x + \left(-2 \cdot \left(-4 \cdot x + 1.5 \cdot x\right) + 0.6666666666666666 \cdot x\right)\right)\right)\right)\right) - \left(-4 \cdot x + 1.5 \cdot x\right)\right) - 2 \cdot x\right)} \]

   6. Taylor expanded in x around 0 99.4%

      \[\leadsto x + wj \cdot \left(\color{blue}{\left(wj \cdot \left(x \cdot \left(2.5 + -2.6666666666666665 \cdot wj\right)\right) + wj \cdot \left(1 + -1 \cdot wj\right)\right)} - 2 \cdot x\right) \]
   7. Step-by-step derivation

      1. distribute-lft-out 99.4%

         \[\leadsto x + wj \cdot \left(\color{blue}{wj \cdot \left(x \cdot \left(2.5 + -2.6666666666666665 \cdot wj\right) + \left(1 + -1 \cdot wj\right)\right)} - 2 \cdot x\right) \]

      2. *-commutative 99.4%

         \[\leadsto x + wj \cdot \left(wj \cdot \left(x \cdot \left(2.5 + \color{blue}{wj \cdot -2.6666666666666665}\right) + \left(1 + -1 \cdot wj\right)\right) - 2 \cdot x\right) \]

      3. mul-1-neg 99.4%

         \[\leadsto x + wj \cdot \left(wj \cdot \left(x \cdot \left(2.5 + wj \cdot -2.6666666666666665\right) + \left(1 + \color{blue}{\left(-wj\right)}\right)\right) - 2 \cdot x\right) \]

   8. Simplified 99.4%

      \[\leadsto x + wj \cdot \left(\color{blue}{wj \cdot \left(x \cdot \left(2.5 + wj \cdot -2.6666666666666665\right) + \left(1 + \left(-wj\right)\right)\right)} - 2 \cdot x\right) \]

   9. Taylor expanded in x around 0 99.4%

      \[\leadsto x + wj \cdot \left(wj \cdot \color{blue}{\left(\left(1 + x \cdot \left(2.5 + -2.6666666666666665 \cdot wj\right)\right) - wj\right)} - 2 \cdot x\right) \]

   10. Taylor expanded in x around inf 99.5%

       \[\leadsto x + wj \cdot \left(wj \cdot \left(\color{blue}{x \cdot \left(2.5 + \left(-2.6666666666666665 \cdot wj + \frac{1}{x}\right)\right)} - wj\right) - 2 \cdot x\right) \]

   if 0.0106 < wj < 0.160000000000000003 or 4 < wj

   1. Initial program 73.7%

      \[wj - \frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}} \]
   2. Step-by-step derivation

      1. distribute-rgt1-in 73.7%

         \[\leadsto wj - \frac{wj \cdot e^{wj} - x}{\color{blue}{\left(wj + 1\right) \cdot e^{wj}}} \]

      2. *-commutative 73.7%

         \[\leadsto wj - \frac{wj \cdot e^{wj} - x}{\color{blue}{e^{wj} \cdot \left(wj + 1\right)}} \]

      3. associate-/r* 72.7%

         \[\leadsto wj - \color{blue}{\frac{\frac{wj \cdot e^{wj} - x}{e^{wj}}}{wj + 1}} \]

      4. div-sub 72.7%

         \[\leadsto wj - \frac{\color{blue}{\frac{wj \cdot e^{wj}}{e^{wj}} - \frac{x}{e^{wj}}}}{wj + 1} \]

      5. associate-/l* 72.7%

         \[\leadsto wj - \frac{\color{blue}{wj \cdot \frac{e^{wj}}{e^{wj}}} - \frac{x}{e^{wj}}}{wj + 1} \]

      6. *-inverses 97.7%

         \[\leadsto wj - \frac{wj \cdot \color{blue}{1} - \frac{x}{e^{wj}}}{wj + 1} \]

      7. *-rgt-identity 97.7%

         \[\leadsto wj - \frac{\color{blue}{wj} - \frac{x}{e^{wj}}}{wj + 1} \]

   3. Simplified 97.7%

      \[\leadsto \color{blue}{wj - \frac{wj - \frac{x}{e^{wj}}}{wj + 1}} \]
   4. Add Preprocessing

   5. Taylor expanded in x around 0 97.7%

      \[\leadsto \color{blue}{wj - \frac{wj}{1 + wj}} \]
   6. Step-by-step derivation

      1. +-commutative 97.6%

         \[\leadsto \left(wj - \frac{wj}{\color{blue}{wj + 1}}\right) + \mathsf{fma}\left(\frac{-1}{wj + 1}, wj - \frac{x}{e^{wj}}, \frac{wj}{wj + 1}\right) \]

   7. Simplified 97.7%

      \[\leadsto \color{blue}{wj - \frac{wj}{wj + 1}} \]
3. Recombined 4 regimes into one program.

4. Final simplification 99.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;wj \leq -0.0074:\\ \;\;\;\;\frac{x}{\left(wj + 1\right) \cdot e^{wj}}\\ \mathbf{elif}\;wj \leq 5 \cdot 10^{-83}:\\ \;\;\;\;x + wj \cdot \left(wj \cdot \left(\left(1 + x \cdot \left(2.5 + wj \cdot -2.6666666666666665\right)\right) - wj\right) - x \cdot 2\right)\\ \mathbf{elif}\;wj \leq 0.00052:\\ \;\;\;\;x - wj \cdot \left(x \cdot 2 + wj \cdot \left(wj + x \cdot \left(\left(\frac{-1}{x} - wj \cdot -2.6666666666666665\right) - 2.5\right)\right)\right)\\ \mathbf{elif}\;wj \leq 0.0106 \lor \neg \left(wj \leq 0.16\right) \land wj \leq 4:\\ \;\;\;\;\frac{x}{\left(wj + 1\right) \cdot e^{wj}}\\ \mathbf{else}:\\ \;\;\;\;wj - \frac{wj}{wj + 1}\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 96.4% accurate, 14.9× speedup

Math

\[\begin{array}{l} \\ x + wj \cdot \left(wj \cdot \left(\left(1 + x \cdot \left(2.5 + wj \cdot -2.6666666666666665\right)\right) - wj\right) - x \cdot 2\right) \end{array} \]

FPCore

(FPCore (wj x)
 :precision binary64
 (+
  x
  (*
   wj
   (-
    (* wj (- (+ 1.0 (* x (+ 2.5 (* wj -2.6666666666666665)))) wj))
    (* x 2.0)))))

C

double code(double wj, double x) {
	return x + (wj * ((wj * ((1.0 + (x * (2.5 + (wj * -2.6666666666666665)))) - wj)) - (x * 2.0)));
}

Fortran

real(8) function code(wj, x)
    real(8), intent (in) :: wj
    real(8), intent (in) :: x
    code = x + (wj * ((wj * ((1.0d0 + (x * (2.5d0 + (wj * (-2.6666666666666665d0))))) - wj)) - (x * 2.0d0)))
end function

Java

public static double code(double wj, double x) {
	return x + (wj * ((wj * ((1.0 + (x * (2.5 + (wj * -2.6666666666666665)))) - wj)) - (x * 2.0)));
}

Python

def code(wj, x):
	return x + (wj * ((wj * ((1.0 + (x * (2.5 + (wj * -2.6666666666666665)))) - wj)) - (x * 2.0)))

Julia

function code(wj, x)
	return Float64(x + Float64(wj * Float64(Float64(wj * Float64(Float64(1.0 + Float64(x * Float64(2.5 + Float64(wj * -2.6666666666666665)))) - wj)) - Float64(x * 2.0))))
end

MATLAB

function tmp = code(wj, x)
	tmp = x + (wj * ((wj * ((1.0 + (x * (2.5 + (wj * -2.6666666666666665)))) - wj)) - (x * 2.0)));
end

Wolfram

code[wj_, x_] := N[(x + N[(wj * N[(N[(wj * N[(N[(1.0 + N[(x * N[(2.5 + N[(wj * -2.6666666666666665), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - wj), $MachinePrecision]), $MachinePrecision] - N[(x * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 76.8%

   \[wj - \frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}} \]
2. Step-by-step derivation

   1. distribute-rgt1-in 77.9%

      \[\leadsto wj - \frac{wj \cdot e^{wj} - x}{\color{blue}{\left(wj + 1\right) \cdot e^{wj}}} \]

   2. *-commutative 77.9%

      \[\leadsto wj - \frac{wj \cdot e^{wj} - x}{\color{blue}{e^{wj} \cdot \left(wj + 1\right)}} \]

   3. associate-/r* 77.9%

      \[\leadsto wj - \color{blue}{\frac{\frac{wj \cdot e^{wj} - x}{e^{wj}}}{wj + 1}} \]

   4. div-sub 76.8%

      \[\leadsto wj - \frac{\color{blue}{\frac{wj \cdot e^{wj}}{e^{wj}} - \frac{x}{e^{wj}}}}{wj + 1} \]

   5. associate-/l* 76.8%

      \[\leadsto wj - \frac{\color{blue}{wj \cdot \frac{e^{wj}}{e^{wj}}} - \frac{x}{e^{wj}}}{wj + 1} \]

   6. *-inverses 78.3%

      \[\leadsto wj - \frac{wj \cdot \color{blue}{1} - \frac{x}{e^{wj}}}{wj + 1} \]

   7. *-rgt-identity 78.3%

      \[\leadsto wj - \frac{\color{blue}{wj} - \frac{x}{e^{wj}}}{wj + 1} \]

3. Simplified 78.3%

   \[\leadsto \color{blue}{wj - \frac{wj - \frac{x}{e^{wj}}}{wj + 1}} \]
4. Add Preprocessing

5. Taylor expanded in wj around 0 95.3%

   \[\leadsto \color{blue}{x + wj \cdot \left(wj \cdot \left(\left(1 + -1 \cdot \left(wj \cdot \left(1 + \left(-3 \cdot x + \left(-2 \cdot \left(-4 \cdot x + 1.5 \cdot x\right) + 0.6666666666666666 \cdot x\right)\right)\right)\right)\right) - \left(-4 \cdot x + 1.5 \cdot x\right)\right) - 2 \cdot x\right)} \]

6. Taylor expanded in x around 0 95.6%

   \[\leadsto x + wj \cdot \left(\color{blue}{\left(wj \cdot \left(x \cdot \left(2.5 + -2.6666666666666665 \cdot wj\right)\right) + wj \cdot \left(1 + -1 \cdot wj\right)\right)} - 2 \cdot x\right) \]
7. Step-by-step derivation

   1. distribute-lft-out 95.6%

      \[\leadsto x + wj \cdot \left(\color{blue}{wj \cdot \left(x \cdot \left(2.5 + -2.6666666666666665 \cdot wj\right) + \left(1 + -1 \cdot wj\right)\right)} - 2 \cdot x\right) \]

   2. *-commutative 95.6%

      \[\leadsto x + wj \cdot \left(wj \cdot \left(x \cdot \left(2.5 + \color{blue}{wj \cdot -2.6666666666666665}\right) + \left(1 + -1 \cdot wj\right)\right) - 2 \cdot x\right) \]

   3. mul-1-neg 95.6%

      \[\leadsto x + wj \cdot \left(wj \cdot \left(x \cdot \left(2.5 + wj \cdot -2.6666666666666665\right) + \left(1 + \color{blue}{\left(-wj\right)}\right)\right) - 2 \cdot x\right) \]

8. Simplified 95.6%

   \[\leadsto x + wj \cdot \left(\color{blue}{wj \cdot \left(x \cdot \left(2.5 + wj \cdot -2.6666666666666665\right) + \left(1 + \left(-wj\right)\right)\right)} - 2 \cdot x\right) \]

9. Taylor expanded in x around 0 95.6%

   \[\leadsto x + wj \cdot \left(wj \cdot \color{blue}{\left(\left(1 + x \cdot \left(2.5 + -2.6666666666666665 \cdot wj\right)\right) - wj\right)} - 2 \cdot x\right) \]

10. Final simplification 95.6%

    \[\leadsto x + wj \cdot \left(wj \cdot \left(\left(1 + x \cdot \left(2.5 + wj \cdot -2.6666666666666665\right)\right) - wj\right) - x \cdot 2\right) \]
11. Add Preprocessing

Alternative 16: 96.2% accurate, 24.1× speedup

Math

\[\begin{array}{l} \\ x + wj \cdot \left(wj \cdot \left(1 - wj\right) - x \cdot 2\right) \end{array} \]

FPCore

(FPCore (wj x)
 :precision binary64
 (+ x (* wj (- (* wj (- 1.0 wj)) (* x 2.0)))))

C

double code(double wj, double x) {
	return x + (wj * ((wj * (1.0 - wj)) - (x * 2.0)));
}

Fortran

real(8) function code(wj, x)
    real(8), intent (in) :: wj
    real(8), intent (in) :: x
    code = x + (wj * ((wj * (1.0d0 - wj)) - (x * 2.0d0)))
end function

Java

public static double code(double wj, double x) {
	return x + (wj * ((wj * (1.0 - wj)) - (x * 2.0)));
}

Python

def code(wj, x):
	return x + (wj * ((wj * (1.0 - wj)) - (x * 2.0)))

Julia

function code(wj, x)
	return Float64(x + Float64(wj * Float64(Float64(wj * Float64(1.0 - wj)) - Float64(x * 2.0))))
end

MATLAB

function tmp = code(wj, x)
	tmp = x + (wj * ((wj * (1.0 - wj)) - (x * 2.0)));
end

Wolfram

code[wj_, x_] := N[(x + N[(wj * N[(N[(wj * N[(1.0 - wj), $MachinePrecision]), $MachinePrecision] - N[(x * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 76.8%

   \[wj - \frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}} \]
2. Step-by-step derivation

   1. distribute-rgt1-in 77.9%

      \[\leadsto wj - \frac{wj \cdot e^{wj} - x}{\color{blue}{\left(wj + 1\right) \cdot e^{wj}}} \]

   2. *-commutative 77.9%

      \[\leadsto wj - \frac{wj \cdot e^{wj} - x}{\color{blue}{e^{wj} \cdot \left(wj + 1\right)}} \]

   3. associate-/r* 77.9%

      \[\leadsto wj - \color{blue}{\frac{\frac{wj \cdot e^{wj} - x}{e^{wj}}}{wj + 1}} \]

   4. div-sub 76.8%

      \[\leadsto wj - \frac{\color{blue}{\frac{wj \cdot e^{wj}}{e^{wj}} - \frac{x}{e^{wj}}}}{wj + 1} \]

   5. associate-/l* 76.8%

      \[\leadsto wj - \frac{\color{blue}{wj \cdot \frac{e^{wj}}{e^{wj}}} - \frac{x}{e^{wj}}}{wj + 1} \]

   6. *-inverses 78.3%

      \[\leadsto wj - \frac{wj \cdot \color{blue}{1} - \frac{x}{e^{wj}}}{wj + 1} \]

   7. *-rgt-identity 78.3%

      \[\leadsto wj - \frac{\color{blue}{wj} - \frac{x}{e^{wj}}}{wj + 1} \]

3. Simplified 78.3%

   \[\leadsto \color{blue}{wj - \frac{wj - \frac{x}{e^{wj}}}{wj + 1}} \]
4. Add Preprocessing

5. Taylor expanded in wj around 0 95.3%

   \[\leadsto \color{blue}{x + wj \cdot \left(wj \cdot \left(\left(1 + -1 \cdot \left(wj \cdot \left(1 + \left(-3 \cdot x + \left(-2 \cdot \left(-4 \cdot x + 1.5 \cdot x\right) + 0.6666666666666666 \cdot x\right)\right)\right)\right)\right) - \left(-4 \cdot x + 1.5 \cdot x\right)\right) - 2 \cdot x\right)} \]

6. Taylor expanded in x around 0 95.3%

   \[\leadsto x + wj \cdot \left(\color{blue}{wj \cdot \left(1 + -1 \cdot wj\right)} - 2 \cdot x\right) \]
7. Step-by-step derivation

   1. mul-1-neg 95.3%

      \[\leadsto x + wj \cdot \left(wj \cdot \left(1 + \color{blue}{\left(-wj\right)}\right) - 2 \cdot x\right) \]

   2. unsub-neg 95.3%

      \[\leadsto x + wj \cdot \left(wj \cdot \color{blue}{\left(1 - wj\right)} - 2 \cdot x\right) \]

8. Simplified 95.3%

   \[\leadsto x + wj \cdot \left(\color{blue}{wj \cdot \left(1 - wj\right)} - 2 \cdot x\right) \]

9. Final simplification 95.3%

   \[\leadsto x + wj \cdot \left(wj \cdot \left(1 - wj\right) - x \cdot 2\right) \]
10. Add Preprocessing

Alternative 17: 95.7% accurate, 24.1× speedup

Math

\[\begin{array}{l} \\ x + wj \cdot \left(wj \cdot \left(wj + 1\right) - x \cdot 2\right) \end{array} \]

FPCore

(FPCore (wj x)
 :precision binary64
 (+ x (* wj (- (* wj (+ wj 1.0)) (* x 2.0)))))

C

double code(double wj, double x) {
	return x + (wj * ((wj * (wj + 1.0)) - (x * 2.0)));
}

Fortran

real(8) function code(wj, x)
    real(8), intent (in) :: wj
    real(8), intent (in) :: x
    code = x + (wj * ((wj * (wj + 1.0d0)) - (x * 2.0d0)))
end function

Java

public static double code(double wj, double x) {
	return x + (wj * ((wj * (wj + 1.0)) - (x * 2.0)));
}

Python

def code(wj, x):
	return x + (wj * ((wj * (wj + 1.0)) - (x * 2.0)))

Julia

function code(wj, x)
	return Float64(x + Float64(wj * Float64(Float64(wj * Float64(wj + 1.0)) - Float64(x * 2.0))))
end

MATLAB

function tmp = code(wj, x)
	tmp = x + (wj * ((wj * (wj + 1.0)) - (x * 2.0)));
end

Wolfram

code[wj_, x_] := N[(x + N[(wj * N[(N[(wj * N[(wj + 1.0), $MachinePrecision]), $MachinePrecision] - N[(x * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 76.8%

   \[wj - \frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}} \]
2. Step-by-step derivation

   1. distribute-rgt1-in 77.9%

      \[\leadsto wj - \frac{wj \cdot e^{wj} - x}{\color{blue}{\left(wj + 1\right) \cdot e^{wj}}} \]

   2. *-commutative 77.9%

      \[\leadsto wj - \frac{wj \cdot e^{wj} - x}{\color{blue}{e^{wj} \cdot \left(wj + 1\right)}} \]

   3. associate-/r* 77.9%

      \[\leadsto wj - \color{blue}{\frac{\frac{wj \cdot e^{wj} - x}{e^{wj}}}{wj + 1}} \]

   4. div-sub 76.8%

      \[\leadsto wj - \frac{\color{blue}{\frac{wj \cdot e^{wj}}{e^{wj}} - \frac{x}{e^{wj}}}}{wj + 1} \]

   5. associate-/l* 76.8%

      \[\leadsto wj - \frac{\color{blue}{wj \cdot \frac{e^{wj}}{e^{wj}}} - \frac{x}{e^{wj}}}{wj + 1} \]

   6. *-inverses 78.3%

      \[\leadsto wj - \frac{wj \cdot \color{blue}{1} - \frac{x}{e^{wj}}}{wj + 1} \]

   7. *-rgt-identity 78.3%

      \[\leadsto wj - \frac{\color{blue}{wj} - \frac{x}{e^{wj}}}{wj + 1} \]

3. Simplified 78.3%

   \[\leadsto \color{blue}{wj - \frac{wj - \frac{x}{e^{wj}}}{wj + 1}} \]
4. Add Preprocessing

5. Taylor expanded in wj around 0 95.3%

   \[\leadsto \color{blue}{x + wj \cdot \left(wj \cdot \left(\left(1 + -1 \cdot \left(wj \cdot \left(1 + \left(-3 \cdot x + \left(-2 \cdot \left(-4 \cdot x + 1.5 \cdot x\right) + 0.6666666666666666 \cdot x\right)\right)\right)\right)\right) - \left(-4 \cdot x + 1.5 \cdot x\right)\right) - 2 \cdot x\right)} \]

6. Taylor expanded in x around 0 95.3%

   \[\leadsto x + wj \cdot \left(\color{blue}{wj \cdot \left(1 + -1 \cdot wj\right)} - 2 \cdot x\right) \]
7. Step-by-step derivation

   1. mul-1-neg 95.3%

      \[\leadsto x + wj \cdot \left(wj \cdot \left(1 + \color{blue}{\left(-wj\right)}\right) - 2 \cdot x\right) \]

   2. unsub-neg 95.3%

      \[\leadsto x + wj \cdot \left(wj \cdot \color{blue}{\left(1 - wj\right)} - 2 \cdot x\right) \]

8. Simplified 95.3%

   \[\leadsto x + wj \cdot \left(\color{blue}{wj \cdot \left(1 - wj\right)} - 2 \cdot x\right) \]
9. Step-by-step derivation

   1. sub-neg 95.3%

      \[\leadsto x + wj \cdot \left(wj \cdot \color{blue}{\left(1 + \left(-wj\right)\right)} - 2 \cdot x\right) \]

   2. +-commutative 95.3%

      \[\leadsto x + wj \cdot \left(wj \cdot \color{blue}{\left(\left(-wj\right) + 1\right)} - 2 \cdot x\right) \]

   3. add-sqr-sqrt 51.6%

      \[\leadsto x + wj \cdot \left(wj \cdot \left(\color{blue}{\sqrt{-wj} \cdot \sqrt{-wj}} + 1\right) - 2 \cdot x\right) \]

   4. sqrt-unprod 95.0%

      \[\leadsto x + wj \cdot \left(wj \cdot \left(\color{blue}{\sqrt{\left(-wj\right) \cdot \left(-wj\right)}} + 1\right) - 2 \cdot x\right) \]

   5. sqr-neg 95.0%

      \[\leadsto x + wj \cdot \left(wj \cdot \left(\sqrt{\color{blue}{wj \cdot wj}} + 1\right) - 2 \cdot x\right) \]

   6. sqrt-unprod 43.4%

      \[\leadsto x + wj \cdot \left(wj \cdot \left(\color{blue}{\sqrt{wj} \cdot \sqrt{wj}} + 1\right) - 2 \cdot x\right) \]

   7. add-sqr-sqrt 94.7%

      \[\leadsto x + wj \cdot \left(wj \cdot \left(\color{blue}{wj} + 1\right) - 2 \cdot x\right) \]

   8. pow1 94.7%

      \[\leadsto x + wj \cdot \left(\color{blue}{{\left(wj \cdot \left(wj + 1\right)\right)}^{1}} - 2 \cdot x\right) \]

10. Applied egg-rr 94.7%

    \[\leadsto x + wj \cdot \left(\color{blue}{{\left(wj \cdot \left(wj + 1\right)\right)}^{1}} - 2 \cdot x\right) \]
11. Step-by-step derivation

    1. unpow1 94.7%

       \[\leadsto x + wj \cdot \left(\color{blue}{wj \cdot \left(wj + 1\right)} - 2 \cdot x\right) \]

12. Simplified 94.7%

    \[\leadsto x + wj \cdot \left(\color{blue}{wj \cdot \left(wj + 1\right)} - 2 \cdot x\right) \]

13. Final simplification 94.7%

    \[\leadsto x + wj \cdot \left(wj \cdot \left(wj + 1\right) - x \cdot 2\right) \]
14. Add Preprocessing

Alternative 18: 85.7% accurate, 26.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;wj \leq 3.5 \cdot 10^{-10}:\\ \;\;\;\;x + -2 \cdot \left(wj \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;wj - \frac{wj}{wj + 1}\\ \end{array} \end{array} \]

FPCore

(FPCore (wj x)
 :precision binary64
 (if (<= wj 3.5e-10) (+ x (* -2.0 (* wj x))) (- wj (/ wj (+ wj 1.0)))))

C

double code(double wj, double x) {
	double tmp;
	if (wj <= 3.5e-10) {
		tmp = x + (-2.0 * (wj * x));
	} else {
		tmp = 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) :: tmp
    if (wj <= 3.5d-10) then
        tmp = x + ((-2.0d0) * (wj * x))
    else
        tmp = wj - (wj / (wj + 1.0d0))
    end if
    code = tmp
end function

Java

public static double code(double wj, double x) {
	double tmp;
	if (wj <= 3.5e-10) {
		tmp = x + (-2.0 * (wj * x));
	} else {
		tmp = wj - (wj / (wj + 1.0));
	}
	return tmp;
}

Python

def code(wj, x):
	tmp = 0
	if wj <= 3.5e-10:
		tmp = x + (-2.0 * (wj * x))
	else:
		tmp = wj - (wj / (wj + 1.0))
	return tmp

Julia

function code(wj, x)
	tmp = 0.0
	if (wj <= 3.5e-10)
		tmp = Float64(x + Float64(-2.0 * Float64(wj * x)));
	else
		tmp = Float64(wj - Float64(wj / Float64(wj + 1.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(wj, x)
	tmp = 0.0;
	if (wj <= 3.5e-10)
		tmp = x + (-2.0 * (wj * x));
	else
		tmp = wj - (wj / (wj + 1.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[wj_, x_] := If[LessEqual[wj, 3.5e-10], N[(x + N[(-2.0 * N[(wj * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(wj - N[(wj / N[(wj + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if wj < 3.4999999999999998e-10

   1. Initial program 76.8%

      \[wj - \frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}} \]
   2. Step-by-step derivation

      1. distribute-rgt1-in 78.0%

         \[\leadsto wj - \frac{wj \cdot e^{wj} - x}{\color{blue}{\left(wj + 1\right) \cdot e^{wj}}} \]

      2. *-commutative 78.0%

         \[\leadsto wj - \frac{wj \cdot e^{wj} - x}{\color{blue}{e^{wj} \cdot \left(wj + 1\right)}} \]

      3. associate-/r* 78.0%

         \[\leadsto wj - \color{blue}{\frac{\frac{wj \cdot e^{wj} - x}{e^{wj}}}{wj + 1}} \]

      4. div-sub 76.8%

         \[\leadsto wj - \frac{\color{blue}{\frac{wj \cdot e^{wj}}{e^{wj}} - \frac{x}{e^{wj}}}}{wj + 1} \]

      5. associate-/l* 76.8%

         \[\leadsto wj - \frac{\color{blue}{wj \cdot \frac{e^{wj}}{e^{wj}}} - \frac{x}{e^{wj}}}{wj + 1} \]

      6. *-inverses 78.0%

         \[\leadsto wj - \frac{wj \cdot \color{blue}{1} - \frac{x}{e^{wj}}}{wj + 1} \]

      7. *-rgt-identity 78.0%

         \[\leadsto wj - \frac{\color{blue}{wj} - \frac{x}{e^{wj}}}{wj + 1} \]

   3. Simplified 78.0%

      \[\leadsto \color{blue}{wj - \frac{wj - \frac{x}{e^{wj}}}{wj + 1}} \]
   4. Add Preprocessing

   5. Taylor expanded in wj around 0 81.4%

      \[\leadsto \color{blue}{x + -2 \cdot \left(wj \cdot x\right)} \]
   6. Step-by-step derivation

      1. *-commutative 81.4%

         \[\leadsto x + -2 \cdot \color{blue}{\left(x \cdot wj\right)} \]

   7. Simplified 81.4%

      \[\leadsto \color{blue}{x + -2 \cdot \left(x \cdot wj\right)} \]

   if 3.4999999999999998e-10 < wj

   1. Initial program 76.9%

      \[wj - \frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}} \]
   2. Step-by-step derivation

      1. distribute-rgt1-in 76.7%

         \[\leadsto wj - \frac{wj \cdot e^{wj} - x}{\color{blue}{\left(wj + 1\right) \cdot e^{wj}}} \]

      2. *-commutative 76.7%

         \[\leadsto wj - \frac{wj \cdot e^{wj} - x}{\color{blue}{e^{wj} \cdot \left(wj + 1\right)}} \]

      3. associate-/r* 76.6%

         \[\leadsto wj - \color{blue}{\frac{\frac{wj \cdot e^{wj} - x}{e^{wj}}}{wj + 1}} \]

      4. div-sub 76.6%

         \[\leadsto wj - \frac{\color{blue}{\frac{wj \cdot e^{wj}}{e^{wj}} - \frac{x}{e^{wj}}}}{wj + 1} \]

      5. associate-/l* 76.6%

         \[\leadsto wj - \frac{\color{blue}{wj \cdot \frac{e^{wj}}{e^{wj}}} - \frac{x}{e^{wj}}}{wj + 1} \]

      6. *-inverses 89.1%

         \[\leadsto wj - \frac{wj \cdot \color{blue}{1} - \frac{x}{e^{wj}}}{wj + 1} \]

      7. *-rgt-identity 89.1%

         \[\leadsto wj - \frac{\color{blue}{wj} - \frac{x}{e^{wj}}}{wj + 1} \]

   3. Simplified 89.1%

      \[\leadsto \color{blue}{wj - \frac{wj - \frac{x}{e^{wj}}}{wj + 1}} \]
   4. Add Preprocessing

   5. Taylor expanded in x around 0 64.9%

      \[\leadsto \color{blue}{wj - \frac{wj}{1 + wj}} \]
   6. Step-by-step derivation

      1. +-commutative 88.9%

         \[\leadsto \left(wj - \frac{wj}{\color{blue}{wj + 1}}\right) + \mathsf{fma}\left(\frac{-1}{wj + 1}, wj - \frac{x}{e^{wj}}, \frac{wj}{wj + 1}\right) \]

   7. Simplified 64.9%

      \[\leadsto \color{blue}{wj - \frac{wj}{wj + 1}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 80.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;wj \leq 3.5 \cdot 10^{-10}:\\ \;\;\;\;x + -2 \cdot \left(wj \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;wj - \frac{wj}{wj + 1}\\ \end{array} \]
5. Add Preprocessing

Alternative 19: 95.7% accurate, 34.8× speedup

Math

\[\begin{array}{l} \\ x - wj \cdot \left(x \cdot 2 - wj\right) \end{array} \]

FPCore

(FPCore (wj x) :precision binary64 (- x (* wj (- (* x 2.0) wj))))

C

double code(double wj, double x) {
	return x - (wj * ((x * 2.0) - wj));
}

Fortran

real(8) function code(wj, x)
    real(8), intent (in) :: wj
    real(8), intent (in) :: x
    code = x - (wj * ((x * 2.0d0) - wj))
end function

Java

public static double code(double wj, double x) {
	return x - (wj * ((x * 2.0) - wj));
}

Python

def code(wj, x):
	return x - (wj * ((x * 2.0) - wj))

Julia

function code(wj, x)
	return Float64(x - Float64(wj * Float64(Float64(x * 2.0) - wj)))
end

MATLAB

function tmp = code(wj, x)
	tmp = x - (wj * ((x * 2.0) - wj));
end

Wolfram

code[wj_, x_] := N[(x - N[(wj * N[(N[(x * 2.0), $MachinePrecision] - wj), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 76.8%

   \[wj - \frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}} \]
2. Step-by-step derivation

   1. distribute-rgt1-in 77.9%

      \[\leadsto wj - \frac{wj \cdot e^{wj} - x}{\color{blue}{\left(wj + 1\right) \cdot e^{wj}}} \]

   2. *-commutative 77.9%

      \[\leadsto wj - \frac{wj \cdot e^{wj} - x}{\color{blue}{e^{wj} \cdot \left(wj + 1\right)}} \]

   3. associate-/r* 77.9%

      \[\leadsto wj - \color{blue}{\frac{\frac{wj \cdot e^{wj} - x}{e^{wj}}}{wj + 1}} \]

   4. div-sub 76.8%

      \[\leadsto wj - \frac{\color{blue}{\frac{wj \cdot e^{wj}}{e^{wj}} - \frac{x}{e^{wj}}}}{wj + 1} \]

   5. associate-/l* 76.8%

      \[\leadsto wj - \frac{\color{blue}{wj \cdot \frac{e^{wj}}{e^{wj}}} - \frac{x}{e^{wj}}}{wj + 1} \]

   6. *-inverses 78.3%

      \[\leadsto wj - \frac{wj \cdot \color{blue}{1} - \frac{x}{e^{wj}}}{wj + 1} \]

   7. *-rgt-identity 78.3%

      \[\leadsto wj - \frac{\color{blue}{wj} - \frac{x}{e^{wj}}}{wj + 1} \]

3. Simplified 78.3%

   \[\leadsto \color{blue}{wj - \frac{wj - \frac{x}{e^{wj}}}{wj + 1}} \]
4. Add Preprocessing

5. Taylor expanded in wj around 0 95.3%

   \[\leadsto \color{blue}{x + wj \cdot \left(wj \cdot \left(\left(1 + -1 \cdot \left(wj \cdot \left(1 + \left(-3 \cdot x + \left(-2 \cdot \left(-4 \cdot x + 1.5 \cdot x\right) + 0.6666666666666666 \cdot x\right)\right)\right)\right)\right) - \left(-4 \cdot x + 1.5 \cdot x\right)\right) - 2 \cdot x\right)} \]

6. Taylor expanded in x around 0 95.3%

   \[\leadsto x + wj \cdot \left(\color{blue}{wj \cdot \left(1 + -1 \cdot wj\right)} - 2 \cdot x\right) \]
7. Step-by-step derivation

   1. mul-1-neg 95.3%

      \[\leadsto x + wj \cdot \left(wj \cdot \left(1 + \color{blue}{\left(-wj\right)}\right) - 2 \cdot x\right) \]

   2. unsub-neg 95.3%

      \[\leadsto x + wj \cdot \left(wj \cdot \color{blue}{\left(1 - wj\right)} - 2 \cdot x\right) \]

8. Simplified 95.3%

   \[\leadsto x + wj \cdot \left(\color{blue}{wj \cdot \left(1 - wj\right)} - 2 \cdot x\right) \]

9. Taylor expanded in wj around 0 94.7%

   \[\leadsto x + wj \cdot \left(\color{blue}{wj} - 2 \cdot x\right) \]

10. Final simplification 94.7%

    \[\leadsto x - wj \cdot \left(x \cdot 2 - wj\right) \]
11. Add Preprocessing

Alternative 20: 84.3% accurate, 44.7× speedup

Math

\[\begin{array}{l} \\ x + -2 \cdot \left(wj \cdot x\right) \end{array} \]

FPCore

(FPCore (wj x) :precision binary64 (+ x (* -2.0 (* wj x))))

C

double code(double wj, double x) {
	return x + (-2.0 * (wj * x));
}

Fortran

real(8) function code(wj, x)
    real(8), intent (in) :: wj
    real(8), intent (in) :: x
    code = x + ((-2.0d0) * (wj * x))
end function

Java

public static double code(double wj, double x) {
	return x + (-2.0 * (wj * x));
}

Python

def code(wj, x):
	return x + (-2.0 * (wj * x))

Julia

function code(wj, x)
	return Float64(x + Float64(-2.0 * Float64(wj * x)))
end

MATLAB

function tmp = code(wj, x)
	tmp = x + (-2.0 * (wj * x));
end

Wolfram

code[wj_, x_] := N[(x + N[(-2.0 * N[(wj * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 76.8%

   \[wj - \frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}} \]
2. Step-by-step derivation

   1. distribute-rgt1-in 77.9%

      \[\leadsto wj - \frac{wj \cdot e^{wj} - x}{\color{blue}{\left(wj + 1\right) \cdot e^{wj}}} \]

   2. *-commutative 77.9%

      \[\leadsto wj - \frac{wj \cdot e^{wj} - x}{\color{blue}{e^{wj} \cdot \left(wj + 1\right)}} \]

   3. associate-/r* 77.9%

      \[\leadsto wj - \color{blue}{\frac{\frac{wj \cdot e^{wj} - x}{e^{wj}}}{wj + 1}} \]

   4. div-sub 76.8%

      \[\leadsto wj - \frac{\color{blue}{\frac{wj \cdot e^{wj}}{e^{wj}} - \frac{x}{e^{wj}}}}{wj + 1} \]

   5. associate-/l* 76.8%

      \[\leadsto wj - \frac{\color{blue}{wj \cdot \frac{e^{wj}}{e^{wj}}} - \frac{x}{e^{wj}}}{wj + 1} \]

   6. *-inverses 78.3%

      \[\leadsto wj - \frac{wj \cdot \color{blue}{1} - \frac{x}{e^{wj}}}{wj + 1} \]

   7. *-rgt-identity 78.3%

      \[\leadsto wj - \frac{\color{blue}{wj} - \frac{x}{e^{wj}}}{wj + 1} \]

3. Simplified 78.3%

   \[\leadsto \color{blue}{wj - \frac{wj - \frac{x}{e^{wj}}}{wj + 1}} \]
4. Add Preprocessing

5. Taylor expanded in wj around 0 79.1%

   \[\leadsto \color{blue}{x + -2 \cdot \left(wj \cdot x\right)} \]
6. Step-by-step derivation

   1. *-commutative 79.1%

      \[\leadsto x + -2 \cdot \color{blue}{\left(x \cdot wj\right)} \]

7. Simplified 79.1%

   \[\leadsto \color{blue}{x + -2 \cdot \left(x \cdot wj\right)} \]

8. Final simplification 79.1%

   \[\leadsto x + -2 \cdot \left(wj \cdot x\right) \]
9. Add Preprocessing

Alternative 21: 83.9% 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 76.8%

   \[wj - \frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}} \]
2. Step-by-step derivation

   1. distribute-rgt1-in 77.9%

      \[\leadsto wj - \frac{wj \cdot e^{wj} - x}{\color{blue}{\left(wj + 1\right) \cdot e^{wj}}} \]

   2. *-commutative 77.9%

      \[\leadsto wj - \frac{wj \cdot e^{wj} - x}{\color{blue}{e^{wj} \cdot \left(wj + 1\right)}} \]

   3. associate-/r* 77.9%

      \[\leadsto wj - \color{blue}{\frac{\frac{wj \cdot e^{wj} - x}{e^{wj}}}{wj + 1}} \]

   4. div-sub 76.8%

      \[\leadsto wj - \frac{\color{blue}{\frac{wj \cdot e^{wj}}{e^{wj}} - \frac{x}{e^{wj}}}}{wj + 1} \]

   5. associate-/l* 76.8%

      \[\leadsto wj - \frac{\color{blue}{wj \cdot \frac{e^{wj}}{e^{wj}}} - \frac{x}{e^{wj}}}{wj + 1} \]

   6. *-inverses 78.3%

      \[\leadsto wj - \frac{wj \cdot \color{blue}{1} - \frac{x}{e^{wj}}}{wj + 1} \]

   7. *-rgt-identity 78.3%

      \[\leadsto wj - \frac{\color{blue}{wj} - \frac{x}{e^{wj}}}{wj + 1} \]

3. Simplified 78.3%

   \[\leadsto \color{blue}{wj - \frac{wj - \frac{x}{e^{wj}}}{wj + 1}} \]
4. Add Preprocessing

5. Taylor expanded in wj around 0 78.3%

   \[\leadsto \color{blue}{x} \]
6. Add Preprocessing

Alternative 22: 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 76.8%

   \[wj - \frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}} \]
2. Step-by-step derivation

   1. distribute-rgt1-in 77.9%

      \[\leadsto wj - \frac{wj \cdot e^{wj} - x}{\color{blue}{\left(wj + 1\right) \cdot e^{wj}}} \]

   2. *-commutative 77.9%

      \[\leadsto wj - \frac{wj \cdot e^{wj} - x}{\color{blue}{e^{wj} \cdot \left(wj + 1\right)}} \]

   3. associate-/r* 77.9%

      \[\leadsto wj - \color{blue}{\frac{\frac{wj \cdot e^{wj} - x}{e^{wj}}}{wj + 1}} \]

   4. div-sub 76.8%

      \[\leadsto wj - \frac{\color{blue}{\frac{wj \cdot e^{wj}}{e^{wj}} - \frac{x}{e^{wj}}}}{wj + 1} \]

   5. associate-/l* 76.8%

      \[\leadsto wj - \frac{\color{blue}{wj \cdot \frac{e^{wj}}{e^{wj}}} - \frac{x}{e^{wj}}}{wj + 1} \]

   6. *-inverses 78.3%

      \[\leadsto wj - \frac{wj \cdot \color{blue}{1} - \frac{x}{e^{wj}}}{wj + 1} \]

   7. *-rgt-identity 78.3%

      \[\leadsto wj - \frac{\color{blue}{wj} - \frac{x}{e^{wj}}}{wj + 1} \]

3. Simplified 78.3%

   \[\leadsto \color{blue}{wj - \frac{wj - \frac{x}{e^{wj}}}{wj + 1}} \]
4. Add Preprocessing

5. Taylor expanded in wj around inf 4.4%

   \[\leadsto \color{blue}{wj} \]
6. Add Preprocessing

Developer target: 78.5% 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 2024096 
(FPCore (wj x)
  :name "Jmat.Real.lambertw, newton loop step"
  :precision binary64

  :alt
  (- wj (- (/ wj (+ wj 1.0)) (/ x (+ (exp wj) (* wj (exp wj))))))

  (- wj (/ (- (* wj (exp wj)) x) (+ (exp wj) (* wj (exp wj))))))