Jmat.Real.lambertw, newton loop step

Percentage Accurate: 78.1% → 99.6%

Time: 13.6s

Alternatives: 20

Speedup: 313.0×

• could not determine a ground truth

  1. wj = -3.440911654753403e+21
  2. x = 4.4164015877880217e+123

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 78.1% 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: 99.6% accurate, 2.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;wj \leq -6.8 \cdot 10^{-6}:\\ \;\;\;\;x \cdot \left(\left(\frac{wj}{x} - \frac{\frac{1}{e^{wj}}}{-1 - wj}\right) + \frac{wj}{x} \cdot \frac{1 + wj \cdot \left(wj + -1\right)}{-1 - wj \cdot \left(wj \cdot wj\right)}\right)\\ \mathbf{elif}\;wj \leq 1.3 \cdot 10^{-8}:\\ \;\;\;\;x + wj \cdot \left(x \cdot -2 + wj \cdot \left(\left(1 + wj \cdot \left(\left(-1 - x \cdot 0.6666666666666666\right) - \left(x \cdot -3 + x \cdot 5\right)\right)\right) + x \cdot 2.5\right)\right)\\ \mathbf{else}:\\ \;\;\;\;wj + \frac{1}{-1 - wj} \cdot \left(wj - \frac{x}{e^{wj}}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (wj x)
 :precision binary64
 (if (<= wj -6.8e-6)
   (*
    x
    (+
     (- (/ wj x) (/ (/ 1.0 (exp wj)) (- -1.0 wj)))
     (* (/ wj x) (/ (+ 1.0 (* wj (+ wj -1.0))) (- -1.0 (* wj (* wj wj)))))))
   (if (<= wj 1.3e-8)
     (+
      x
      (*
       wj
       (+
        (* x -2.0)
        (*
         wj
         (+
          (+
           1.0
           (*
            wj
            (- (- -1.0 (* x 0.6666666666666666)) (+ (* x -3.0) (* x 5.0)))))
          (* x 2.5))))))
     (+ wj (* (/ 1.0 (- -1.0 wj)) (- wj (/ x (exp wj))))))))

C

double code(double wj, double x) {
	double tmp;
	if (wj <= -6.8e-6) {
		tmp = x * (((wj / x) - ((1.0 / exp(wj)) / (-1.0 - wj))) + ((wj / x) * ((1.0 + (wj * (wj + -1.0))) / (-1.0 - (wj * (wj * wj))))));
	} else if (wj <= 1.3e-8) {
		tmp = x + (wj * ((x * -2.0) + (wj * ((1.0 + (wj * ((-1.0 - (x * 0.6666666666666666)) - ((x * -3.0) + (x * 5.0))))) + (x * 2.5)))));
	} else {
		tmp = wj + ((1.0 / (-1.0 - wj)) * (wj - (x / exp(wj))));
	}
	return tmp;
}

Fortran

real(8) function code(wj, x)
    real(8), intent (in) :: wj
    real(8), intent (in) :: x
    real(8) :: tmp
    if (wj <= (-6.8d-6)) then
        tmp = x * (((wj / x) - ((1.0d0 / exp(wj)) / ((-1.0d0) - wj))) + ((wj / x) * ((1.0d0 + (wj * (wj + (-1.0d0)))) / ((-1.0d0) - (wj * (wj * wj))))))
    else if (wj <= 1.3d-8) then
        tmp = x + (wj * ((x * (-2.0d0)) + (wj * ((1.0d0 + (wj * (((-1.0d0) - (x * 0.6666666666666666d0)) - ((x * (-3.0d0)) + (x * 5.0d0))))) + (x * 2.5d0)))))
    else
        tmp = wj + ((1.0d0 / ((-1.0d0) - wj)) * (wj - (x / exp(wj))))
    end if
    code = tmp
end function

Java

public static double code(double wj, double x) {
	double tmp;
	if (wj <= -6.8e-6) {
		tmp = x * (((wj / x) - ((1.0 / Math.exp(wj)) / (-1.0 - wj))) + ((wj / x) * ((1.0 + (wj * (wj + -1.0))) / (-1.0 - (wj * (wj * wj))))));
	} else if (wj <= 1.3e-8) {
		tmp = x + (wj * ((x * -2.0) + (wj * ((1.0 + (wj * ((-1.0 - (x * 0.6666666666666666)) - ((x * -3.0) + (x * 5.0))))) + (x * 2.5)))));
	} else {
		tmp = wj + ((1.0 / (-1.0 - wj)) * (wj - (x / Math.exp(wj))));
	}
	return tmp;
}

Python

def code(wj, x):
	tmp = 0
	if wj <= -6.8e-6:
		tmp = x * (((wj / x) - ((1.0 / math.exp(wj)) / (-1.0 - wj))) + ((wj / x) * ((1.0 + (wj * (wj + -1.0))) / (-1.0 - (wj * (wj * wj))))))
	elif wj <= 1.3e-8:
		tmp = x + (wj * ((x * -2.0) + (wj * ((1.0 + (wj * ((-1.0 - (x * 0.6666666666666666)) - ((x * -3.0) + (x * 5.0))))) + (x * 2.5)))))
	else:
		tmp = wj + ((1.0 / (-1.0 - wj)) * (wj - (x / math.exp(wj))))
	return tmp

Julia

function code(wj, x)
	tmp = 0.0
	if (wj <= -6.8e-6)
		tmp = Float64(x * Float64(Float64(Float64(wj / x) - Float64(Float64(1.0 / exp(wj)) / Float64(-1.0 - wj))) + Float64(Float64(wj / x) * Float64(Float64(1.0 + Float64(wj * Float64(wj + -1.0))) / Float64(-1.0 - Float64(wj * Float64(wj * wj)))))));
	elseif (wj <= 1.3e-8)
		tmp = Float64(x + Float64(wj * Float64(Float64(x * -2.0) + Float64(wj * Float64(Float64(1.0 + Float64(wj * Float64(Float64(-1.0 - Float64(x * 0.6666666666666666)) - Float64(Float64(x * -3.0) + Float64(x * 5.0))))) + Float64(x * 2.5))))));
	else
		tmp = Float64(wj + Float64(Float64(1.0 / Float64(-1.0 - wj)) * Float64(wj - Float64(x / exp(wj)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(wj, x)
	tmp = 0.0;
	if (wj <= -6.8e-6)
		tmp = x * (((wj / x) - ((1.0 / exp(wj)) / (-1.0 - wj))) + ((wj / x) * ((1.0 + (wj * (wj + -1.0))) / (-1.0 - (wj * (wj * wj))))));
	elseif (wj <= 1.3e-8)
		tmp = x + (wj * ((x * -2.0) + (wj * ((1.0 + (wj * ((-1.0 - (x * 0.6666666666666666)) - ((x * -3.0) + (x * 5.0))))) + (x * 2.5)))));
	else
		tmp = wj + ((1.0 / (-1.0 - wj)) * (wj - (x / exp(wj))));
	end
	tmp_2 = tmp;
end

Wolfram

code[wj_, x_] := If[LessEqual[wj, -6.8e-6], N[(x * N[(N[(N[(wj / x), $MachinePrecision] - N[(N[(1.0 / N[Exp[wj], $MachinePrecision]), $MachinePrecision] / N[(-1.0 - wj), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(wj / x), $MachinePrecision] * N[(N[(1.0 + N[(wj * N[(wj + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(-1.0 - N[(wj * N[(wj * wj), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[wj, 1.3e-8], N[(x + N[(wj * N[(N[(x * -2.0), $MachinePrecision] + N[(wj * N[(N[(1.0 + N[(wj * N[(N[(-1.0 - N[(x * 0.6666666666666666), $MachinePrecision]), $MachinePrecision] - N[(N[(x * -3.0), $MachinePrecision] + N[(x * 5.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(x * 2.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(wj + N[(N[(1.0 / N[(-1.0 - wj), $MachinePrecision]), $MachinePrecision] * N[(wj - N[(x / N[Exp[wj], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if wj < -6.80000000000000012e-6

   1. Initial program 51.4%

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

      1. sub-neg N/A

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

      2. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(wj, \color{blue}{\left(\mathsf{neg}\left(\frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}}\right)\right)}\right) \]

      3. distribute-rgt1-in N/A

         \[\leadsto \mathsf{+.f64}\left(wj, \left(\mathsf{neg}\left(\frac{wj \cdot e^{wj} - x}{\left(wj + 1\right) \cdot e^{wj}}\right)\right)\right) \]

      4. associate-/l/ N/A

         \[\leadsto \mathsf{+.f64}\left(wj, \left(\mathsf{neg}\left(\frac{\frac{wj \cdot e^{wj} - x}{e^{wj}}}{wj + 1}\right)\right)\right) \]

      5. distribute-neg-frac2 N/A

         \[\leadsto \mathsf{+.f64}\left(wj, \left(\frac{\frac{wj \cdot e^{wj} - x}{e^{wj}}}{\color{blue}{\mathsf{neg}\left(\left(wj + 1\right)\right)}}\right)\right) \]

      6. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(wj, \mathsf{/.f64}\left(\left(\frac{wj \cdot e^{wj} - x}{e^{wj}}\right), \color{blue}{\left(\mathsf{neg}\left(\left(wj + 1\right)\right)\right)}\right)\right) \]

      7. div-sub N/A

         \[\leadsto \mathsf{+.f64}\left(wj, \mathsf{/.f64}\left(\left(\frac{wj \cdot e^{wj}}{e^{wj}} - \frac{x}{e^{wj}}\right), \left(\mathsf{neg}\left(\color{blue}{\left(wj + 1\right)}\right)\right)\right)\right) \]

      8. associate-/l* N/A

         \[\leadsto \mathsf{+.f64}\left(wj, \mathsf{/.f64}\left(\left(wj \cdot \frac{e^{wj}}{e^{wj}} - \frac{x}{e^{wj}}\right), \left(\mathsf{neg}\left(\left(\color{blue}{wj} + 1\right)\right)\right)\right)\right) \]

      9. *-inverses N/A

         \[\leadsto \mathsf{+.f64}\left(wj, \mathsf{/.f64}\left(\left(wj \cdot 1 - \frac{x}{e^{wj}}\right), \left(\mathsf{neg}\left(\left(wj + 1\right)\right)\right)\right)\right) \]

      10. *-rgt-identity N/A

          \[\leadsto \mathsf{+.f64}\left(wj, \mathsf{/.f64}\left(\left(wj - \frac{x}{e^{wj}}\right), \left(\mathsf{neg}\left(\left(\color{blue}{wj} + 1\right)\right)\right)\right)\right) \]

      11. --lowering--.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(wj, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(wj, \left(\frac{x}{e^{wj}}\right)\right), \left(\mathsf{neg}\left(\color{blue}{\left(wj + 1\right)}\right)\right)\right)\right) \]

      12. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(wj, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(wj, \mathsf{/.f64}\left(x, \left(e^{wj}\right)\right)\right), \left(\mathsf{neg}\left(\left(wj + \color{blue}{1}\right)\right)\right)\right)\right) \]

      13. exp-lowering-exp.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(wj, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(wj, \mathsf{/.f64}\left(x, \mathsf{exp.f64}\left(wj\right)\right)\right), \left(\mathsf{neg}\left(\left(wj + 1\right)\right)\right)\right)\right) \]

      14. +-commutative N/A

          \[\leadsto \mathsf{+.f64}\left(wj, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(wj, \mathsf{/.f64}\left(x, \mathsf{exp.f64}\left(wj\right)\right)\right), \left(\mathsf{neg}\left(\left(1 + wj\right)\right)\right)\right)\right) \]

      15. *-inverses N/A

          \[\leadsto \mathsf{+.f64}\left(wj, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(wj, \mathsf{/.f64}\left(x, \mathsf{exp.f64}\left(wj\right)\right)\right), \left(\mathsf{neg}\left(\left(\frac{e^{wj}}{e^{wj}} + wj\right)\right)\right)\right)\right) \]

      16. distribute-neg-in N/A

          \[\leadsto \mathsf{+.f64}\left(wj, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(wj, \mathsf{/.f64}\left(x, \mathsf{exp.f64}\left(wj\right)\right)\right), \left(\left(\mathsf{neg}\left(\frac{e^{wj}}{e^{wj}}\right)\right) + \color{blue}{\left(\mathsf{neg}\left(wj\right)\right)}\right)\right)\right) \]

   3. Simplified 96.1%

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

      1. +-commutative N/A

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

      2. div-sub N/A

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

      3. associate-+l- N/A

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

      4. flip3-- N/A

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

      5. associate-/r/ N/A

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

      6. fmm-def N/A

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

      7. fma-lowering-fma.f64 N/A

         \[\leadsto \mathsf{fma.f64}\left(\left(\frac{wj}{{-1}^{3} - {wj}^{3}}\right), \color{blue}{\left(-1 \cdot -1 + \left(wj \cdot wj + -1 \cdot wj\right)\right)}, \left(\mathsf{neg}\left(\left(\frac{\frac{x}{e^{wj}}}{-1 - wj} - wj\right)\right)\right)\right) \]

   6. Applied egg-rr 96.1%

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

   7. Taylor expanded in x around inf

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

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(x, \color{blue}{\left(-1 \cdot \frac{wj \cdot \left(1 + wj \cdot \left(wj - 1\right)\right)}{x \cdot \left(1 + {wj}^{3}\right)} + \left(\frac{1}{e^{wj} \cdot \left(1 + wj\right)} + \frac{wj}{x}\right)\right)}\right) \]

      2. +-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(x, \left(\left(\frac{1}{e^{wj} \cdot \left(1 + wj\right)} + \frac{wj}{x}\right) + \color{blue}{-1 \cdot \frac{wj \cdot \left(1 + wj \cdot \left(wj - 1\right)\right)}{x \cdot \left(1 + {wj}^{3}\right)}}\right)\right) \]

      3. mul-1-neg N/A

         \[\leadsto \mathsf{*.f64}\left(x, \left(\left(\frac{1}{e^{wj} \cdot \left(1 + wj\right)} + \frac{wj}{x}\right) + \left(\mathsf{neg}\left(\frac{wj \cdot \left(1 + wj \cdot \left(wj - 1\right)\right)}{x \cdot \left(1 + {wj}^{3}\right)}\right)\right)\right)\right) \]

      4. unsub-neg N/A

         \[\leadsto \mathsf{*.f64}\left(x, \left(\left(\frac{1}{e^{wj} \cdot \left(1 + wj\right)} + \frac{wj}{x}\right) - \color{blue}{\frac{wj \cdot \left(1 + wj \cdot \left(wj - 1\right)\right)}{x \cdot \left(1 + {wj}^{3}\right)}}\right)\right) \]

      5. --lowering--.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{\_.f64}\left(\left(\frac{1}{e^{wj} \cdot \left(1 + wj\right)} + \frac{wj}{x}\right), \color{blue}{\left(\frac{wj \cdot \left(1 + wj \cdot \left(wj - 1\right)\right)}{x \cdot \left(1 + {wj}^{3}\right)}\right)}\right)\right) \]

   9. Simplified 96.8%

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

   if -6.80000000000000012e-6 < wj < 1.3000000000000001e-8

   1. Initial program 79.4%

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

   3. Taylor expanded in wj around 0

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

      1. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(x, \color{blue}{\left(wj \cdot \left(wj \cdot \left(\left(1 + -1 \cdot \left(wj \cdot \left(1 + \left(-3 \cdot x + \left(-2 \cdot \left(-4 \cdot x + \frac{3}{2} \cdot x\right) + \frac{2}{3} \cdot x\right)\right)\right)\right)\right) - \left(-4 \cdot x + \frac{3}{2} \cdot x\right)\right) - 2 \cdot x\right)\right)}\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(x, \mathsf{*.f64}\left(wj, \color{blue}{\left(wj \cdot \left(\left(1 + -1 \cdot \left(wj \cdot \left(1 + \left(-3 \cdot x + \left(-2 \cdot \left(-4 \cdot x + \frac{3}{2} \cdot x\right) + \frac{2}{3} \cdot x\right)\right)\right)\right)\right) - \left(-4 \cdot x + \frac{3}{2} \cdot x\right)\right) - 2 \cdot x\right)}\right)\right) \]

      3. cancel-sign-sub-inv N/A

         \[\leadsto \mathsf{+.f64}\left(x, \mathsf{*.f64}\left(wj, \left(wj \cdot \left(\left(1 + -1 \cdot \left(wj \cdot \left(1 + \left(-3 \cdot x + \left(-2 \cdot \left(-4 \cdot x + \frac{3}{2} \cdot x\right) + \frac{2}{3} \cdot x\right)\right)\right)\right)\right) - \left(-4 \cdot x + \frac{3}{2} \cdot x\right)\right) + \color{blue}{\left(\mathsf{neg}\left(2\right)\right) \cdot x}\right)\right)\right) \]

      4. metadata-eval N/A

         \[\leadsto \mathsf{+.f64}\left(x, \mathsf{*.f64}\left(wj, \left(wj \cdot \left(\left(1 + -1 \cdot \left(wj \cdot \left(1 + \left(-3 \cdot x + \left(-2 \cdot \left(-4 \cdot x + \frac{3}{2} \cdot x\right) + \frac{2}{3} \cdot x\right)\right)\right)\right)\right) - \left(-4 \cdot x + \frac{3}{2} \cdot x\right)\right) + -2 \cdot x\right)\right)\right) \]

      5. +-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(x, \mathsf{*.f64}\left(wj, \left(-2 \cdot x + \color{blue}{wj \cdot \left(\left(1 + -1 \cdot \left(wj \cdot \left(1 + \left(-3 \cdot x + \left(-2 \cdot \left(-4 \cdot x + \frac{3}{2} \cdot x\right) + \frac{2}{3} \cdot x\right)\right)\right)\right)\right) - \left(-4 \cdot x + \frac{3}{2} \cdot x\right)\right)}\right)\right)\right) \]

      6. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(x, \mathsf{*.f64}\left(wj, \mathsf{+.f64}\left(\left(-2 \cdot x\right), \color{blue}{\left(wj \cdot \left(\left(1 + -1 \cdot \left(wj \cdot \left(1 + \left(-3 \cdot x + \left(-2 \cdot \left(-4 \cdot x + \frac{3}{2} \cdot x\right) + \frac{2}{3} \cdot x\right)\right)\right)\right)\right) - \left(-4 \cdot x + \frac{3}{2} \cdot x\right)\right)\right)}\right)\right)\right) \]

   5. Simplified 100.0%

      \[\leadsto \color{blue}{x + wj \cdot \left(x \cdot -2 + wj \cdot \left(\left(1 - wj \cdot \left(\left(x \cdot -3 + x \cdot 5\right) + \left(x \cdot 0.6666666666666666 + 1\right)\right)\right) + x \cdot 2.5\right)\right)} \]

   if 1.3000000000000001e-8 < wj

   1. Initial program 49.7%

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

      1. sub-neg N/A

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

      2. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(wj, \color{blue}{\left(\mathsf{neg}\left(\frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}}\right)\right)}\right) \]

      3. distribute-rgt1-in N/A

         \[\leadsto \mathsf{+.f64}\left(wj, \left(\mathsf{neg}\left(\frac{wj \cdot e^{wj} - x}{\left(wj + 1\right) \cdot e^{wj}}\right)\right)\right) \]

      4. associate-/l/ N/A

         \[\leadsto \mathsf{+.f64}\left(wj, \left(\mathsf{neg}\left(\frac{\frac{wj \cdot e^{wj} - x}{e^{wj}}}{wj + 1}\right)\right)\right) \]

      5. distribute-neg-frac2 N/A

         \[\leadsto \mathsf{+.f64}\left(wj, \left(\frac{\frac{wj \cdot e^{wj} - x}{e^{wj}}}{\color{blue}{\mathsf{neg}\left(\left(wj + 1\right)\right)}}\right)\right) \]

      6. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(wj, \mathsf{/.f64}\left(\left(\frac{wj \cdot e^{wj} - x}{e^{wj}}\right), \color{blue}{\left(\mathsf{neg}\left(\left(wj + 1\right)\right)\right)}\right)\right) \]

      7. div-sub N/A

         \[\leadsto \mathsf{+.f64}\left(wj, \mathsf{/.f64}\left(\left(\frac{wj \cdot e^{wj}}{e^{wj}} - \frac{x}{e^{wj}}\right), \left(\mathsf{neg}\left(\color{blue}{\left(wj + 1\right)}\right)\right)\right)\right) \]

      8. associate-/l* N/A

         \[\leadsto \mathsf{+.f64}\left(wj, \mathsf{/.f64}\left(\left(wj \cdot \frac{e^{wj}}{e^{wj}} - \frac{x}{e^{wj}}\right), \left(\mathsf{neg}\left(\left(\color{blue}{wj} + 1\right)\right)\right)\right)\right) \]

      9. *-inverses N/A

         \[\leadsto \mathsf{+.f64}\left(wj, \mathsf{/.f64}\left(\left(wj \cdot 1 - \frac{x}{e^{wj}}\right), \left(\mathsf{neg}\left(\left(wj + 1\right)\right)\right)\right)\right) \]

      10. *-rgt-identity N/A

          \[\leadsto \mathsf{+.f64}\left(wj, \mathsf{/.f64}\left(\left(wj - \frac{x}{e^{wj}}\right), \left(\mathsf{neg}\left(\left(\color{blue}{wj} + 1\right)\right)\right)\right)\right) \]

      11. --lowering--.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(wj, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(wj, \left(\frac{x}{e^{wj}}\right)\right), \left(\mathsf{neg}\left(\color{blue}{\left(wj + 1\right)}\right)\right)\right)\right) \]

      12. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(wj, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(wj, \mathsf{/.f64}\left(x, \left(e^{wj}\right)\right)\right), \left(\mathsf{neg}\left(\left(wj + \color{blue}{1}\right)\right)\right)\right)\right) \]

      13. exp-lowering-exp.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(wj, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(wj, \mathsf{/.f64}\left(x, \mathsf{exp.f64}\left(wj\right)\right)\right), \left(\mathsf{neg}\left(\left(wj + 1\right)\right)\right)\right)\right) \]

      14. +-commutative N/A

          \[\leadsto \mathsf{+.f64}\left(wj, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(wj, \mathsf{/.f64}\left(x, \mathsf{exp.f64}\left(wj\right)\right)\right), \left(\mathsf{neg}\left(\left(1 + wj\right)\right)\right)\right)\right) \]

      15. *-inverses N/A

          \[\leadsto \mathsf{+.f64}\left(wj, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(wj, \mathsf{/.f64}\left(x, \mathsf{exp.f64}\left(wj\right)\right)\right), \left(\mathsf{neg}\left(\left(\frac{e^{wj}}{e^{wj}} + wj\right)\right)\right)\right)\right) \]

      16. distribute-neg-in N/A

          \[\leadsto \mathsf{+.f64}\left(wj, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(wj, \mathsf{/.f64}\left(x, \mathsf{exp.f64}\left(wj\right)\right)\right), \left(\left(\mathsf{neg}\left(\frac{e^{wj}}{e^{wj}}\right)\right) + \color{blue}{\left(\mathsf{neg}\left(wj\right)\right)}\right)\right)\right) \]

   3. Simplified 99.7%

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

      1. div-inv N/A

         \[\leadsto \mathsf{+.f64}\left(wj, \left(\left(wj - \frac{x}{e^{wj}}\right) \cdot \color{blue}{\frac{1}{-1 - wj}}\right)\right) \]

      2. *-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(wj, \left(\frac{1}{-1 - wj} \cdot \color{blue}{\left(wj - \frac{x}{e^{wj}}\right)}\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(wj, \mathsf{*.f64}\left(\left(\frac{1}{-1 - wj}\right), \color{blue}{\left(wj - \frac{x}{e^{wj}}\right)}\right)\right) \]

      4. metadata-eval N/A

         \[\leadsto \mathsf{+.f64}\left(wj, \mathsf{*.f64}\left(\left(\frac{-1 \cdot -1}{-1 - wj}\right), \left(wj - \frac{x}{e^{wj}}\right)\right)\right) \]

      5. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(wj, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\left(-1 \cdot -1\right), \left(-1 - wj\right)\right), \left(\color{blue}{wj} - \frac{x}{e^{wj}}\right)\right)\right) \]

      6. metadata-eval N/A

         \[\leadsto \mathsf{+.f64}\left(wj, \mathsf{*.f64}\left(\mathsf{/.f64}\left(1, \left(-1 - wj\right)\right), \left(wj - \frac{x}{e^{wj}}\right)\right)\right) \]

      7. --lowering--.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(wj, \mathsf{*.f64}\left(\mathsf{/.f64}\left(1, \mathsf{\_.f64}\left(-1, wj\right)\right), \left(wj - \frac{x}{e^{wj}}\right)\right)\right) \]

      8. --lowering--.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(wj, \mathsf{*.f64}\left(\mathsf{/.f64}\left(1, \mathsf{\_.f64}\left(-1, wj\right)\right), \mathsf{\_.f64}\left(wj, \color{blue}{\left(\frac{x}{e^{wj}}\right)}\right)\right)\right) \]

      9. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(wj, \mathsf{*.f64}\left(\mathsf{/.f64}\left(1, \mathsf{\_.f64}\left(-1, wj\right)\right), \mathsf{\_.f64}\left(wj, \mathsf{/.f64}\left(x, \color{blue}{\left(e^{wj}\right)}\right)\right)\right)\right) \]

      10. exp-lowering-exp.f64 100.0%

          \[\leadsto \mathsf{+.f64}\left(wj, \mathsf{*.f64}\left(\mathsf{/.f64}\left(1, \mathsf{\_.f64}\left(-1, wj\right)\right), \mathsf{\_.f64}\left(wj, \mathsf{/.f64}\left(x, \mathsf{exp.f64}\left(wj\right)\right)\right)\right)\right) \]

   6. Applied egg-rr 100.0%

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

4. Final simplification 99.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;wj \leq -6.8 \cdot 10^{-6}:\\ \;\;\;\;x \cdot \left(\left(\frac{wj}{x} - \frac{\frac{1}{e^{wj}}}{-1 - wj}\right) + \frac{wj}{x} \cdot \frac{1 + wj \cdot \left(wj + -1\right)}{-1 - wj \cdot \left(wj \cdot wj\right)}\right)\\ \mathbf{elif}\;wj \leq 1.3 \cdot 10^{-8}:\\ \;\;\;\;x + wj \cdot \left(x \cdot -2 + wj \cdot \left(\left(1 + wj \cdot \left(\left(-1 - x \cdot 0.6666666666666666\right) - \left(x \cdot -3 + x \cdot 5\right)\right)\right) + x \cdot 2.5\right)\right)\\ \mathbf{else}:\\ \;\;\;\;wj + \frac{1}{-1 - wj} \cdot \left(wj - \frac{x}{e^{wj}}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 99.7% accurate, 2.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := wj - \frac{x}{e^{wj}}\\ \mathbf{if}\;wj \leq -6.5 \cdot 10^{-6}:\\ \;\;\;\;wj + \frac{t\_0}{-1 - wj}\\ \mathbf{elif}\;wj \leq 1.3 \cdot 10^{-8}:\\ \;\;\;\;x + wj \cdot \left(x \cdot -2 + wj \cdot \left(\left(1 + wj \cdot \left(\left(-1 - x \cdot 0.6666666666666666\right) - \left(x \cdot -3 + x \cdot 5\right)\right)\right) + x \cdot 2.5\right)\right)\\ \mathbf{else}:\\ \;\;\;\;wj + \frac{1}{-1 - wj} \cdot t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (wj x)
 :precision binary64
 (let* ((t_0 (- wj (/ x (exp wj)))))
   (if (<= wj -6.5e-6)
     (+ wj (/ t_0 (- -1.0 wj)))
     (if (<= wj 1.3e-8)
       (+
        x
        (*
         wj
         (+
          (* x -2.0)
          (*
           wj
           (+
            (+
             1.0
             (*
              wj
              (- (- -1.0 (* x 0.6666666666666666)) (+ (* x -3.0) (* x 5.0)))))
            (* x 2.5))))))
       (+ wj (* (/ 1.0 (- -1.0 wj)) t_0))))))

C

double code(double wj, double x) {
	double t_0 = wj - (x / exp(wj));
	double tmp;
	if (wj <= -6.5e-6) {
		tmp = wj + (t_0 / (-1.0 - wj));
	} else if (wj <= 1.3e-8) {
		tmp = x + (wj * ((x * -2.0) + (wj * ((1.0 + (wj * ((-1.0 - (x * 0.6666666666666666)) - ((x * -3.0) + (x * 5.0))))) + (x * 2.5)))));
	} else {
		tmp = wj + ((1.0 / (-1.0 - wj)) * 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 - (x / exp(wj))
    if (wj <= (-6.5d-6)) then
        tmp = wj + (t_0 / ((-1.0d0) - wj))
    else if (wj <= 1.3d-8) then
        tmp = x + (wj * ((x * (-2.0d0)) + (wj * ((1.0d0 + (wj * (((-1.0d0) - (x * 0.6666666666666666d0)) - ((x * (-3.0d0)) + (x * 5.0d0))))) + (x * 2.5d0)))))
    else
        tmp = wj + ((1.0d0 / ((-1.0d0) - wj)) * t_0)
    end if
    code = tmp
end function

Java

public static double code(double wj, double x) {
	double t_0 = wj - (x / Math.exp(wj));
	double tmp;
	if (wj <= -6.5e-6) {
		tmp = wj + (t_0 / (-1.0 - wj));
	} else if (wj <= 1.3e-8) {
		tmp = x + (wj * ((x * -2.0) + (wj * ((1.0 + (wj * ((-1.0 - (x * 0.6666666666666666)) - ((x * -3.0) + (x * 5.0))))) + (x * 2.5)))));
	} else {
		tmp = wj + ((1.0 / (-1.0 - wj)) * t_0);
	}
	return tmp;
}

Python

def code(wj, x):
	t_0 = wj - (x / math.exp(wj))
	tmp = 0
	if wj <= -6.5e-6:
		tmp = wj + (t_0 / (-1.0 - wj))
	elif wj <= 1.3e-8:
		tmp = x + (wj * ((x * -2.0) + (wj * ((1.0 + (wj * ((-1.0 - (x * 0.6666666666666666)) - ((x * -3.0) + (x * 5.0))))) + (x * 2.5)))))
	else:
		tmp = wj + ((1.0 / (-1.0 - wj)) * t_0)
	return tmp

Julia

function code(wj, x)
	t_0 = Float64(wj - Float64(x / exp(wj)))
	tmp = 0.0
	if (wj <= -6.5e-6)
		tmp = Float64(wj + Float64(t_0 / Float64(-1.0 - wj)));
	elseif (wj <= 1.3e-8)
		tmp = Float64(x + Float64(wj * Float64(Float64(x * -2.0) + Float64(wj * Float64(Float64(1.0 + Float64(wj * Float64(Float64(-1.0 - Float64(x * 0.6666666666666666)) - Float64(Float64(x * -3.0) + Float64(x * 5.0))))) + Float64(x * 2.5))))));
	else
		tmp = Float64(wj + Float64(Float64(1.0 / Float64(-1.0 - wj)) * t_0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(wj, x)
	t_0 = wj - (x / exp(wj));
	tmp = 0.0;
	if (wj <= -6.5e-6)
		tmp = wj + (t_0 / (-1.0 - wj));
	elseif (wj <= 1.3e-8)
		tmp = x + (wj * ((x * -2.0) + (wj * ((1.0 + (wj * ((-1.0 - (x * 0.6666666666666666)) - ((x * -3.0) + (x * 5.0))))) + (x * 2.5)))));
	else
		tmp = wj + ((1.0 / (-1.0 - wj)) * t_0);
	end
	tmp_2 = tmp;
end

Wolfram

code[wj_, x_] := Block[{t$95$0 = N[(wj - N[(x / N[Exp[wj], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[wj, -6.5e-6], N[(wj + N[(t$95$0 / N[(-1.0 - wj), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[wj, 1.3e-8], N[(x + N[(wj * N[(N[(x * -2.0), $MachinePrecision] + N[(wj * N[(N[(1.0 + N[(wj * N[(N[(-1.0 - N[(x * 0.6666666666666666), $MachinePrecision]), $MachinePrecision] - N[(N[(x * -3.0), $MachinePrecision] + N[(x * 5.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(x * 2.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(wj + N[(N[(1.0 / N[(-1.0 - wj), $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if wj < -6.4999999999999996e-6

   1. Initial program 51.4%

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

      1. sub-neg N/A

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

      2. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(wj, \color{blue}{\left(\mathsf{neg}\left(\frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}}\right)\right)}\right) \]

      3. distribute-rgt1-in N/A

         \[\leadsto \mathsf{+.f64}\left(wj, \left(\mathsf{neg}\left(\frac{wj \cdot e^{wj} - x}{\left(wj + 1\right) \cdot e^{wj}}\right)\right)\right) \]

      4. associate-/l/ N/A

         \[\leadsto \mathsf{+.f64}\left(wj, \left(\mathsf{neg}\left(\frac{\frac{wj \cdot e^{wj} - x}{e^{wj}}}{wj + 1}\right)\right)\right) \]

      5. distribute-neg-frac2 N/A

         \[\leadsto \mathsf{+.f64}\left(wj, \left(\frac{\frac{wj \cdot e^{wj} - x}{e^{wj}}}{\color{blue}{\mathsf{neg}\left(\left(wj + 1\right)\right)}}\right)\right) \]

      6. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(wj, \mathsf{/.f64}\left(\left(\frac{wj \cdot e^{wj} - x}{e^{wj}}\right), \color{blue}{\left(\mathsf{neg}\left(\left(wj + 1\right)\right)\right)}\right)\right) \]

      7. div-sub N/A

         \[\leadsto \mathsf{+.f64}\left(wj, \mathsf{/.f64}\left(\left(\frac{wj \cdot e^{wj}}{e^{wj}} - \frac{x}{e^{wj}}\right), \left(\mathsf{neg}\left(\color{blue}{\left(wj + 1\right)}\right)\right)\right)\right) \]

      8. associate-/l* N/A

         \[\leadsto \mathsf{+.f64}\left(wj, \mathsf{/.f64}\left(\left(wj \cdot \frac{e^{wj}}{e^{wj}} - \frac{x}{e^{wj}}\right), \left(\mathsf{neg}\left(\left(\color{blue}{wj} + 1\right)\right)\right)\right)\right) \]

      9. *-inverses N/A

         \[\leadsto \mathsf{+.f64}\left(wj, \mathsf{/.f64}\left(\left(wj \cdot 1 - \frac{x}{e^{wj}}\right), \left(\mathsf{neg}\left(\left(wj + 1\right)\right)\right)\right)\right) \]

      10. *-rgt-identity N/A

          \[\leadsto \mathsf{+.f64}\left(wj, \mathsf{/.f64}\left(\left(wj - \frac{x}{e^{wj}}\right), \left(\mathsf{neg}\left(\left(\color{blue}{wj} + 1\right)\right)\right)\right)\right) \]

      11. --lowering--.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(wj, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(wj, \left(\frac{x}{e^{wj}}\right)\right), \left(\mathsf{neg}\left(\color{blue}{\left(wj + 1\right)}\right)\right)\right)\right) \]

      12. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(wj, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(wj, \mathsf{/.f64}\left(x, \left(e^{wj}\right)\right)\right), \left(\mathsf{neg}\left(\left(wj + \color{blue}{1}\right)\right)\right)\right)\right) \]

      13. exp-lowering-exp.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(wj, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(wj, \mathsf{/.f64}\left(x, \mathsf{exp.f64}\left(wj\right)\right)\right), \left(\mathsf{neg}\left(\left(wj + 1\right)\right)\right)\right)\right) \]

      14. +-commutative N/A

          \[\leadsto \mathsf{+.f64}\left(wj, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(wj, \mathsf{/.f64}\left(x, \mathsf{exp.f64}\left(wj\right)\right)\right), \left(\mathsf{neg}\left(\left(1 + wj\right)\right)\right)\right)\right) \]

      15. *-inverses N/A

          \[\leadsto \mathsf{+.f64}\left(wj, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(wj, \mathsf{/.f64}\left(x, \mathsf{exp.f64}\left(wj\right)\right)\right), \left(\mathsf{neg}\left(\left(\frac{e^{wj}}{e^{wj}} + wj\right)\right)\right)\right)\right) \]

      16. distribute-neg-in N/A

          \[\leadsto \mathsf{+.f64}\left(wj, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(wj, \mathsf{/.f64}\left(x, \mathsf{exp.f64}\left(wj\right)\right)\right), \left(\left(\mathsf{neg}\left(\frac{e^{wj}}{e^{wj}}\right)\right) + \color{blue}{\left(\mathsf{neg}\left(wj\right)\right)}\right)\right)\right) \]

   3. Simplified 96.1%

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

   if -6.4999999999999996e-6 < wj < 1.3000000000000001e-8

   1. Initial program 79.4%

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

   3. Taylor expanded in wj around 0

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

      1. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(x, \color{blue}{\left(wj \cdot \left(wj \cdot \left(\left(1 + -1 \cdot \left(wj \cdot \left(1 + \left(-3 \cdot x + \left(-2 \cdot \left(-4 \cdot x + \frac{3}{2} \cdot x\right) + \frac{2}{3} \cdot x\right)\right)\right)\right)\right) - \left(-4 \cdot x + \frac{3}{2} \cdot x\right)\right) - 2 \cdot x\right)\right)}\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(x, \mathsf{*.f64}\left(wj, \color{blue}{\left(wj \cdot \left(\left(1 + -1 \cdot \left(wj \cdot \left(1 + \left(-3 \cdot x + \left(-2 \cdot \left(-4 \cdot x + \frac{3}{2} \cdot x\right) + \frac{2}{3} \cdot x\right)\right)\right)\right)\right) - \left(-4 \cdot x + \frac{3}{2} \cdot x\right)\right) - 2 \cdot x\right)}\right)\right) \]

      3. cancel-sign-sub-inv N/A

         \[\leadsto \mathsf{+.f64}\left(x, \mathsf{*.f64}\left(wj, \left(wj \cdot \left(\left(1 + -1 \cdot \left(wj \cdot \left(1 + \left(-3 \cdot x + \left(-2 \cdot \left(-4 \cdot x + \frac{3}{2} \cdot x\right) + \frac{2}{3} \cdot x\right)\right)\right)\right)\right) - \left(-4 \cdot x + \frac{3}{2} \cdot x\right)\right) + \color{blue}{\left(\mathsf{neg}\left(2\right)\right) \cdot x}\right)\right)\right) \]

      4. metadata-eval N/A

         \[\leadsto \mathsf{+.f64}\left(x, \mathsf{*.f64}\left(wj, \left(wj \cdot \left(\left(1 + -1 \cdot \left(wj \cdot \left(1 + \left(-3 \cdot x + \left(-2 \cdot \left(-4 \cdot x + \frac{3}{2} \cdot x\right) + \frac{2}{3} \cdot x\right)\right)\right)\right)\right) - \left(-4 \cdot x + \frac{3}{2} \cdot x\right)\right) + -2 \cdot x\right)\right)\right) \]

      5. +-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(x, \mathsf{*.f64}\left(wj, \left(-2 \cdot x + \color{blue}{wj \cdot \left(\left(1 + -1 \cdot \left(wj \cdot \left(1 + \left(-3 \cdot x + \left(-2 \cdot \left(-4 \cdot x + \frac{3}{2} \cdot x\right) + \frac{2}{3} \cdot x\right)\right)\right)\right)\right) - \left(-4 \cdot x + \frac{3}{2} \cdot x\right)\right)}\right)\right)\right) \]

      6. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(x, \mathsf{*.f64}\left(wj, \mathsf{+.f64}\left(\left(-2 \cdot x\right), \color{blue}{\left(wj \cdot \left(\left(1 + -1 \cdot \left(wj \cdot \left(1 + \left(-3 \cdot x + \left(-2 \cdot \left(-4 \cdot x + \frac{3}{2} \cdot x\right) + \frac{2}{3} \cdot x\right)\right)\right)\right)\right) - \left(-4 \cdot x + \frac{3}{2} \cdot x\right)\right)\right)}\right)\right)\right) \]

   5. Simplified 100.0%

      \[\leadsto \color{blue}{x + wj \cdot \left(x \cdot -2 + wj \cdot \left(\left(1 - wj \cdot \left(\left(x \cdot -3 + x \cdot 5\right) + \left(x \cdot 0.6666666666666666 + 1\right)\right)\right) + x \cdot 2.5\right)\right)} \]

   if 1.3000000000000001e-8 < wj

   1. Initial program 49.7%

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

      1. sub-neg N/A

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

      2. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(wj, \color{blue}{\left(\mathsf{neg}\left(\frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}}\right)\right)}\right) \]

      3. distribute-rgt1-in N/A

         \[\leadsto \mathsf{+.f64}\left(wj, \left(\mathsf{neg}\left(\frac{wj \cdot e^{wj} - x}{\left(wj + 1\right) \cdot e^{wj}}\right)\right)\right) \]

      4. associate-/l/ N/A

         \[\leadsto \mathsf{+.f64}\left(wj, \left(\mathsf{neg}\left(\frac{\frac{wj \cdot e^{wj} - x}{e^{wj}}}{wj + 1}\right)\right)\right) \]

      5. distribute-neg-frac2 N/A

         \[\leadsto \mathsf{+.f64}\left(wj, \left(\frac{\frac{wj \cdot e^{wj} - x}{e^{wj}}}{\color{blue}{\mathsf{neg}\left(\left(wj + 1\right)\right)}}\right)\right) \]

      6. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(wj, \mathsf{/.f64}\left(\left(\frac{wj \cdot e^{wj} - x}{e^{wj}}\right), \color{blue}{\left(\mathsf{neg}\left(\left(wj + 1\right)\right)\right)}\right)\right) \]

      7. div-sub N/A

         \[\leadsto \mathsf{+.f64}\left(wj, \mathsf{/.f64}\left(\left(\frac{wj \cdot e^{wj}}{e^{wj}} - \frac{x}{e^{wj}}\right), \left(\mathsf{neg}\left(\color{blue}{\left(wj + 1\right)}\right)\right)\right)\right) \]

      8. associate-/l* N/A

         \[\leadsto \mathsf{+.f64}\left(wj, \mathsf{/.f64}\left(\left(wj \cdot \frac{e^{wj}}{e^{wj}} - \frac{x}{e^{wj}}\right), \left(\mathsf{neg}\left(\left(\color{blue}{wj} + 1\right)\right)\right)\right)\right) \]

      9. *-inverses N/A

         \[\leadsto \mathsf{+.f64}\left(wj, \mathsf{/.f64}\left(\left(wj \cdot 1 - \frac{x}{e^{wj}}\right), \left(\mathsf{neg}\left(\left(wj + 1\right)\right)\right)\right)\right) \]

      10. *-rgt-identity N/A

          \[\leadsto \mathsf{+.f64}\left(wj, \mathsf{/.f64}\left(\left(wj - \frac{x}{e^{wj}}\right), \left(\mathsf{neg}\left(\left(\color{blue}{wj} + 1\right)\right)\right)\right)\right) \]

      11. --lowering--.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(wj, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(wj, \left(\frac{x}{e^{wj}}\right)\right), \left(\mathsf{neg}\left(\color{blue}{\left(wj + 1\right)}\right)\right)\right)\right) \]

      12. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(wj, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(wj, \mathsf{/.f64}\left(x, \left(e^{wj}\right)\right)\right), \left(\mathsf{neg}\left(\left(wj + \color{blue}{1}\right)\right)\right)\right)\right) \]

      13. exp-lowering-exp.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(wj, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(wj, \mathsf{/.f64}\left(x, \mathsf{exp.f64}\left(wj\right)\right)\right), \left(\mathsf{neg}\left(\left(wj + 1\right)\right)\right)\right)\right) \]

      14. +-commutative N/A

          \[\leadsto \mathsf{+.f64}\left(wj, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(wj, \mathsf{/.f64}\left(x, \mathsf{exp.f64}\left(wj\right)\right)\right), \left(\mathsf{neg}\left(\left(1 + wj\right)\right)\right)\right)\right) \]

      15. *-inverses N/A

          \[\leadsto \mathsf{+.f64}\left(wj, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(wj, \mathsf{/.f64}\left(x, \mathsf{exp.f64}\left(wj\right)\right)\right), \left(\mathsf{neg}\left(\left(\frac{e^{wj}}{e^{wj}} + wj\right)\right)\right)\right)\right) \]

      16. distribute-neg-in N/A

          \[\leadsto \mathsf{+.f64}\left(wj, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(wj, \mathsf{/.f64}\left(x, \mathsf{exp.f64}\left(wj\right)\right)\right), \left(\left(\mathsf{neg}\left(\frac{e^{wj}}{e^{wj}}\right)\right) + \color{blue}{\left(\mathsf{neg}\left(wj\right)\right)}\right)\right)\right) \]

   3. Simplified 99.7%

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

      1. div-inv N/A

         \[\leadsto \mathsf{+.f64}\left(wj, \left(\left(wj - \frac{x}{e^{wj}}\right) \cdot \color{blue}{\frac{1}{-1 - wj}}\right)\right) \]

      2. *-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(wj, \left(\frac{1}{-1 - wj} \cdot \color{blue}{\left(wj - \frac{x}{e^{wj}}\right)}\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(wj, \mathsf{*.f64}\left(\left(\frac{1}{-1 - wj}\right), \color{blue}{\left(wj - \frac{x}{e^{wj}}\right)}\right)\right) \]

      4. metadata-eval N/A

         \[\leadsto \mathsf{+.f64}\left(wj, \mathsf{*.f64}\left(\left(\frac{-1 \cdot -1}{-1 - wj}\right), \left(wj - \frac{x}{e^{wj}}\right)\right)\right) \]

      5. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(wj, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\left(-1 \cdot -1\right), \left(-1 - wj\right)\right), \left(\color{blue}{wj} - \frac{x}{e^{wj}}\right)\right)\right) \]

      6. metadata-eval N/A

         \[\leadsto \mathsf{+.f64}\left(wj, \mathsf{*.f64}\left(\mathsf{/.f64}\left(1, \left(-1 - wj\right)\right), \left(wj - \frac{x}{e^{wj}}\right)\right)\right) \]

      7. --lowering--.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(wj, \mathsf{*.f64}\left(\mathsf{/.f64}\left(1, \mathsf{\_.f64}\left(-1, wj\right)\right), \left(wj - \frac{x}{e^{wj}}\right)\right)\right) \]

      8. --lowering--.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(wj, \mathsf{*.f64}\left(\mathsf{/.f64}\left(1, \mathsf{\_.f64}\left(-1, wj\right)\right), \mathsf{\_.f64}\left(wj, \color{blue}{\left(\frac{x}{e^{wj}}\right)}\right)\right)\right) \]

      9. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(wj, \mathsf{*.f64}\left(\mathsf{/.f64}\left(1, \mathsf{\_.f64}\left(-1, wj\right)\right), \mathsf{\_.f64}\left(wj, \mathsf{/.f64}\left(x, \color{blue}{\left(e^{wj}\right)}\right)\right)\right)\right) \]

      10. exp-lowering-exp.f64 100.0%

          \[\leadsto \mathsf{+.f64}\left(wj, \mathsf{*.f64}\left(\mathsf{/.f64}\left(1, \mathsf{\_.f64}\left(-1, wj\right)\right), \mathsf{\_.f64}\left(wj, \mathsf{/.f64}\left(x, \mathsf{exp.f64}\left(wj\right)\right)\right)\right)\right) \]

   6. Applied egg-rr 100.0%

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

4. Final simplification 99.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;wj \leq -6.5 \cdot 10^{-6}:\\ \;\;\;\;wj + \frac{wj - \frac{x}{e^{wj}}}{-1 - wj}\\ \mathbf{elif}\;wj \leq 1.3 \cdot 10^{-8}:\\ \;\;\;\;x + wj \cdot \left(x \cdot -2 + wj \cdot \left(\left(1 + wj \cdot \left(\left(-1 - x \cdot 0.6666666666666666\right) - \left(x \cdot -3 + x \cdot 5\right)\right)\right) + x \cdot 2.5\right)\right)\\ \mathbf{else}:\\ \;\;\;\;wj + \frac{1}{-1 - wj} \cdot \left(wj - \frac{x}{e^{wj}}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 99.7% accurate, 2.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := wj + \frac{wj - \frac{x}{e^{wj}}}{-1 - wj}\\ \mathbf{if}\;wj \leq -6.5 \cdot 10^{-6}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;wj \leq 1.3 \cdot 10^{-8}:\\ \;\;\;\;x + wj \cdot \left(x \cdot -2 + wj \cdot \left(\left(1 + wj \cdot \left(\left(-1 - x \cdot 0.6666666666666666\right) - \left(x \cdot -3 + x \cdot 5\right)\right)\right) + x \cdot 2.5\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (wj x)
 :precision binary64
 (let* ((t_0 (+ wj (/ (- wj (/ x (exp wj))) (- -1.0 wj)))))
   (if (<= wj -6.5e-6)
     t_0
     (if (<= wj 1.3e-8)
       (+
        x
        (*
         wj
         (+
          (* x -2.0)
          (*
           wj
           (+
            (+
             1.0
             (*
              wj
              (- (- -1.0 (* x 0.6666666666666666)) (+ (* x -3.0) (* x 5.0)))))
            (* x 2.5))))))
       t_0))))

C

double code(double wj, double x) {
	double t_0 = wj + ((wj - (x / exp(wj))) / (-1.0 - wj));
	double tmp;
	if (wj <= -6.5e-6) {
		tmp = t_0;
	} else if (wj <= 1.3e-8) {
		tmp = x + (wj * ((x * -2.0) + (wj * ((1.0 + (wj * ((-1.0 - (x * 0.6666666666666666)) - ((x * -3.0) + (x * 5.0))))) + (x * 2.5)))));
	} 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 <= (-6.5d-6)) then
        tmp = t_0
    else if (wj <= 1.3d-8) then
        tmp = x + (wj * ((x * (-2.0d0)) + (wj * ((1.0d0 + (wj * (((-1.0d0) - (x * 0.6666666666666666d0)) - ((x * (-3.0d0)) + (x * 5.0d0))))) + (x * 2.5d0)))))
    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 <= -6.5e-6) {
		tmp = t_0;
	} else if (wj <= 1.3e-8) {
		tmp = x + (wj * ((x * -2.0) + (wj * ((1.0 + (wj * ((-1.0 - (x * 0.6666666666666666)) - ((x * -3.0) + (x * 5.0))))) + (x * 2.5)))));
	} 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 <= -6.5e-6:
		tmp = t_0
	elif wj <= 1.3e-8:
		tmp = x + (wj * ((x * -2.0) + (wj * ((1.0 + (wj * ((-1.0 - (x * 0.6666666666666666)) - ((x * -3.0) + (x * 5.0))))) + (x * 2.5)))))
	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 <= -6.5e-6)
		tmp = t_0;
	elseif (wj <= 1.3e-8)
		tmp = Float64(x + Float64(wj * Float64(Float64(x * -2.0) + Float64(wj * Float64(Float64(1.0 + Float64(wj * Float64(Float64(-1.0 - Float64(x * 0.6666666666666666)) - Float64(Float64(x * -3.0) + Float64(x * 5.0))))) + Float64(x * 2.5))))));
	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 <= -6.5e-6)
		tmp = t_0;
	elseif (wj <= 1.3e-8)
		tmp = x + (wj * ((x * -2.0) + (wj * ((1.0 + (wj * ((-1.0 - (x * 0.6666666666666666)) - ((x * -3.0) + (x * 5.0))))) + (x * 2.5)))));
	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, -6.5e-6], t$95$0, If[LessEqual[wj, 1.3e-8], N[(x + N[(wj * N[(N[(x * -2.0), $MachinePrecision] + N[(wj * N[(N[(1.0 + N[(wj * N[(N[(-1.0 - N[(x * 0.6666666666666666), $MachinePrecision]), $MachinePrecision] - N[(N[(x * -3.0), $MachinePrecision] + N[(x * 5.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(x * 2.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if wj < -6.4999999999999996e-6 or 1.3000000000000001e-8 < wj

   1. Initial program 50.7%

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

      1. sub-neg N/A

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

      2. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(wj, \color{blue}{\left(\mathsf{neg}\left(\frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}}\right)\right)}\right) \]

      3. distribute-rgt1-in N/A

         \[\leadsto \mathsf{+.f64}\left(wj, \left(\mathsf{neg}\left(\frac{wj \cdot e^{wj} - x}{\left(wj + 1\right) \cdot e^{wj}}\right)\right)\right) \]

      4. associate-/l/ N/A

         \[\leadsto \mathsf{+.f64}\left(wj, \left(\mathsf{neg}\left(\frac{\frac{wj \cdot e^{wj} - x}{e^{wj}}}{wj + 1}\right)\right)\right) \]

      5. distribute-neg-frac2 N/A

         \[\leadsto \mathsf{+.f64}\left(wj, \left(\frac{\frac{wj \cdot e^{wj} - x}{e^{wj}}}{\color{blue}{\mathsf{neg}\left(\left(wj + 1\right)\right)}}\right)\right) \]

      6. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(wj, \mathsf{/.f64}\left(\left(\frac{wj \cdot e^{wj} - x}{e^{wj}}\right), \color{blue}{\left(\mathsf{neg}\left(\left(wj + 1\right)\right)\right)}\right)\right) \]

      7. div-sub N/A

         \[\leadsto \mathsf{+.f64}\left(wj, \mathsf{/.f64}\left(\left(\frac{wj \cdot e^{wj}}{e^{wj}} - \frac{x}{e^{wj}}\right), \left(\mathsf{neg}\left(\color{blue}{\left(wj + 1\right)}\right)\right)\right)\right) \]

      8. associate-/l* N/A

         \[\leadsto \mathsf{+.f64}\left(wj, \mathsf{/.f64}\left(\left(wj \cdot \frac{e^{wj}}{e^{wj}} - \frac{x}{e^{wj}}\right), \left(\mathsf{neg}\left(\left(\color{blue}{wj} + 1\right)\right)\right)\right)\right) \]

      9. *-inverses N/A

         \[\leadsto \mathsf{+.f64}\left(wj, \mathsf{/.f64}\left(\left(wj \cdot 1 - \frac{x}{e^{wj}}\right), \left(\mathsf{neg}\left(\left(wj + 1\right)\right)\right)\right)\right) \]

      10. *-rgt-identity N/A

          \[\leadsto \mathsf{+.f64}\left(wj, \mathsf{/.f64}\left(\left(wj - \frac{x}{e^{wj}}\right), \left(\mathsf{neg}\left(\left(\color{blue}{wj} + 1\right)\right)\right)\right)\right) \]

      11. --lowering--.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(wj, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(wj, \left(\frac{x}{e^{wj}}\right)\right), \left(\mathsf{neg}\left(\color{blue}{\left(wj + 1\right)}\right)\right)\right)\right) \]

      12. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(wj, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(wj, \mathsf{/.f64}\left(x, \left(e^{wj}\right)\right)\right), \left(\mathsf{neg}\left(\left(wj + \color{blue}{1}\right)\right)\right)\right)\right) \]

      13. exp-lowering-exp.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(wj, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(wj, \mathsf{/.f64}\left(x, \mathsf{exp.f64}\left(wj\right)\right)\right), \left(\mathsf{neg}\left(\left(wj + 1\right)\right)\right)\right)\right) \]

      14. +-commutative N/A

          \[\leadsto \mathsf{+.f64}\left(wj, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(wj, \mathsf{/.f64}\left(x, \mathsf{exp.f64}\left(wj\right)\right)\right), \left(\mathsf{neg}\left(\left(1 + wj\right)\right)\right)\right)\right) \]

      15. *-inverses N/A

          \[\leadsto \mathsf{+.f64}\left(wj, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(wj, \mathsf{/.f64}\left(x, \mathsf{exp.f64}\left(wj\right)\right)\right), \left(\mathsf{neg}\left(\left(\frac{e^{wj}}{e^{wj}} + wj\right)\right)\right)\right)\right) \]

      16. distribute-neg-in N/A

          \[\leadsto \mathsf{+.f64}\left(wj, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(wj, \mathsf{/.f64}\left(x, \mathsf{exp.f64}\left(wj\right)\right)\right), \left(\left(\mathsf{neg}\left(\frac{e^{wj}}{e^{wj}}\right)\right) + \color{blue}{\left(\mathsf{neg}\left(wj\right)\right)}\right)\right)\right) \]

   3. Simplified 97.5%

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

   if -6.4999999999999996e-6 < wj < 1.3000000000000001e-8

   1. Initial program 79.4%

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

   3. Taylor expanded in wj around 0

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

      1. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(x, \color{blue}{\left(wj \cdot \left(wj \cdot \left(\left(1 + -1 \cdot \left(wj \cdot \left(1 + \left(-3 \cdot x + \left(-2 \cdot \left(-4 \cdot x + \frac{3}{2} \cdot x\right) + \frac{2}{3} \cdot x\right)\right)\right)\right)\right) - \left(-4 \cdot x + \frac{3}{2} \cdot x\right)\right) - 2 \cdot x\right)\right)}\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(x, \mathsf{*.f64}\left(wj, \color{blue}{\left(wj \cdot \left(\left(1 + -1 \cdot \left(wj \cdot \left(1 + \left(-3 \cdot x + \left(-2 \cdot \left(-4 \cdot x + \frac{3}{2} \cdot x\right) + \frac{2}{3} \cdot x\right)\right)\right)\right)\right) - \left(-4 \cdot x + \frac{3}{2} \cdot x\right)\right) - 2 \cdot x\right)}\right)\right) \]

      3. cancel-sign-sub-inv N/A

         \[\leadsto \mathsf{+.f64}\left(x, \mathsf{*.f64}\left(wj, \left(wj \cdot \left(\left(1 + -1 \cdot \left(wj \cdot \left(1 + \left(-3 \cdot x + \left(-2 \cdot \left(-4 \cdot x + \frac{3}{2} \cdot x\right) + \frac{2}{3} \cdot x\right)\right)\right)\right)\right) - \left(-4 \cdot x + \frac{3}{2} \cdot x\right)\right) + \color{blue}{\left(\mathsf{neg}\left(2\right)\right) \cdot x}\right)\right)\right) \]

      4. metadata-eval N/A

         \[\leadsto \mathsf{+.f64}\left(x, \mathsf{*.f64}\left(wj, \left(wj \cdot \left(\left(1 + -1 \cdot \left(wj \cdot \left(1 + \left(-3 \cdot x + \left(-2 \cdot \left(-4 \cdot x + \frac{3}{2} \cdot x\right) + \frac{2}{3} \cdot x\right)\right)\right)\right)\right) - \left(-4 \cdot x + \frac{3}{2} \cdot x\right)\right) + -2 \cdot x\right)\right)\right) \]

      5. +-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(x, \mathsf{*.f64}\left(wj, \left(-2 \cdot x + \color{blue}{wj \cdot \left(\left(1 + -1 \cdot \left(wj \cdot \left(1 + \left(-3 \cdot x + \left(-2 \cdot \left(-4 \cdot x + \frac{3}{2} \cdot x\right) + \frac{2}{3} \cdot x\right)\right)\right)\right)\right) - \left(-4 \cdot x + \frac{3}{2} \cdot x\right)\right)}\right)\right)\right) \]

      6. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(x, \mathsf{*.f64}\left(wj, \mathsf{+.f64}\left(\left(-2 \cdot x\right), \color{blue}{\left(wj \cdot \left(\left(1 + -1 \cdot \left(wj \cdot \left(1 + \left(-3 \cdot x + \left(-2 \cdot \left(-4 \cdot x + \frac{3}{2} \cdot x\right) + \frac{2}{3} \cdot x\right)\right)\right)\right)\right) - \left(-4 \cdot x + \frac{3}{2} \cdot x\right)\right)\right)}\right)\right)\right) \]

   5. Simplified 100.0%

      \[\leadsto \color{blue}{x + wj \cdot \left(x \cdot -2 + wj \cdot \left(\left(1 - wj \cdot \left(\left(x \cdot -3 + x \cdot 5\right) + \left(x \cdot 0.6666666666666666 + 1\right)\right)\right) + x \cdot 2.5\right)\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 99.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;wj \leq -6.5 \cdot 10^{-6}:\\ \;\;\;\;wj + \frac{wj - \frac{x}{e^{wj}}}{-1 - wj}\\ \mathbf{elif}\;wj \leq 1.3 \cdot 10^{-8}:\\ \;\;\;\;x + wj \cdot \left(x \cdot -2 + wj \cdot \left(\left(1 + wj \cdot \left(\left(-1 - x \cdot 0.6666666666666666\right) - \left(x \cdot -3 + x \cdot 5\right)\right)\right) + x \cdot 2.5\right)\right)\\ \mathbf{else}:\\ \;\;\;\;wj + \frac{wj - \frac{x}{e^{wj}}}{-1 - wj}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 97.1% accurate, 8.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;wj \leq 1.3 \cdot 10^{-8}:\\ \;\;\;\;x + wj \cdot \left(x \cdot -2 + wj \cdot \left(\left(1 + wj \cdot \left(\left(-1 - x \cdot 0.6666666666666666\right) - \left(x \cdot -3 + x \cdot 5\right)\right)\right) + x \cdot 2.5\right)\right)\\ \mathbf{else}:\\ \;\;\;\;wj + \frac{wj + \frac{x}{-1 + wj \cdot \left(-1 - wj \cdot \left(0.5 + wj \cdot 0.16666666666666666\right)\right)}}{-1 - wj}\\ \end{array} \end{array} \]

FPCore

(FPCore (wj x)
 :precision binary64
 (if (<= wj 1.3e-8)
   (+
    x
    (*
     wj
     (+
      (* x -2.0)
      (*
       wj
       (+
        (+
         1.0
         (* wj (- (- -1.0 (* x 0.6666666666666666)) (+ (* x -3.0) (* x 5.0)))))
        (* x 2.5))))))
   (+
    wj
    (/
     (+
      wj
      (/ x (+ -1.0 (* wj (- -1.0 (* wj (+ 0.5 (* wj 0.16666666666666666))))))))
     (- -1.0 wj)))))

C

double code(double wj, double x) {
	double tmp;
	if (wj <= 1.3e-8) {
		tmp = x + (wj * ((x * -2.0) + (wj * ((1.0 + (wj * ((-1.0 - (x * 0.6666666666666666)) - ((x * -3.0) + (x * 5.0))))) + (x * 2.5)))));
	} else {
		tmp = wj + ((wj + (x / (-1.0 + (wj * (-1.0 - (wj * (0.5 + (wj * 0.16666666666666666)))))))) / (-1.0 - wj));
	}
	return tmp;
}

Fortran

real(8) function code(wj, x)
    real(8), intent (in) :: wj
    real(8), intent (in) :: x
    real(8) :: tmp
    if (wj <= 1.3d-8) then
        tmp = x + (wj * ((x * (-2.0d0)) + (wj * ((1.0d0 + (wj * (((-1.0d0) - (x * 0.6666666666666666d0)) - ((x * (-3.0d0)) + (x * 5.0d0))))) + (x * 2.5d0)))))
    else
        tmp = wj + ((wj + (x / ((-1.0d0) + (wj * ((-1.0d0) - (wj * (0.5d0 + (wj * 0.16666666666666666d0)))))))) / ((-1.0d0) - wj))
    end if
    code = tmp
end function

Java

public static double code(double wj, double x) {
	double tmp;
	if (wj <= 1.3e-8) {
		tmp = x + (wj * ((x * -2.0) + (wj * ((1.0 + (wj * ((-1.0 - (x * 0.6666666666666666)) - ((x * -3.0) + (x * 5.0))))) + (x * 2.5)))));
	} else {
		tmp = wj + ((wj + (x / (-1.0 + (wj * (-1.0 - (wj * (0.5 + (wj * 0.16666666666666666)))))))) / (-1.0 - wj));
	}
	return tmp;
}

Python

def code(wj, x):
	tmp = 0
	if wj <= 1.3e-8:
		tmp = x + (wj * ((x * -2.0) + (wj * ((1.0 + (wj * ((-1.0 - (x * 0.6666666666666666)) - ((x * -3.0) + (x * 5.0))))) + (x * 2.5)))))
	else:
		tmp = wj + ((wj + (x / (-1.0 + (wj * (-1.0 - (wj * (0.5 + (wj * 0.16666666666666666)))))))) / (-1.0 - wj))
	return tmp

Julia

function code(wj, x)
	tmp = 0.0
	if (wj <= 1.3e-8)
		tmp = Float64(x + Float64(wj * Float64(Float64(x * -2.0) + Float64(wj * Float64(Float64(1.0 + Float64(wj * Float64(Float64(-1.0 - Float64(x * 0.6666666666666666)) - Float64(Float64(x * -3.0) + Float64(x * 5.0))))) + Float64(x * 2.5))))));
	else
		tmp = Float64(wj + Float64(Float64(wj + Float64(x / Float64(-1.0 + Float64(wj * Float64(-1.0 - Float64(wj * Float64(0.5 + Float64(wj * 0.16666666666666666)))))))) / Float64(-1.0 - wj)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(wj, x)
	tmp = 0.0;
	if (wj <= 1.3e-8)
		tmp = x + (wj * ((x * -2.0) + (wj * ((1.0 + (wj * ((-1.0 - (x * 0.6666666666666666)) - ((x * -3.0) + (x * 5.0))))) + (x * 2.5)))));
	else
		tmp = wj + ((wj + (x / (-1.0 + (wj * (-1.0 - (wj * (0.5 + (wj * 0.16666666666666666)))))))) / (-1.0 - wj));
	end
	tmp_2 = tmp;
end

Wolfram

code[wj_, x_] := If[LessEqual[wj, 1.3e-8], N[(x + N[(wj * N[(N[(x * -2.0), $MachinePrecision] + N[(wj * N[(N[(1.0 + N[(wj * N[(N[(-1.0 - N[(x * 0.6666666666666666), $MachinePrecision]), $MachinePrecision] - N[(N[(x * -3.0), $MachinePrecision] + N[(x * 5.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(x * 2.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(wj + N[(N[(wj + N[(x / N[(-1.0 + N[(wj * N[(-1.0 - N[(wj * N[(0.5 + N[(wj * 0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(-1.0 - wj), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if wj < 1.3000000000000001e-8

   1. Initial program 78.4%

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

   3. Taylor expanded in wj around 0

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

      1. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(x, \color{blue}{\left(wj \cdot \left(wj \cdot \left(\left(1 + -1 \cdot \left(wj \cdot \left(1 + \left(-3 \cdot x + \left(-2 \cdot \left(-4 \cdot x + \frac{3}{2} \cdot x\right) + \frac{2}{3} \cdot x\right)\right)\right)\right)\right) - \left(-4 \cdot x + \frac{3}{2} \cdot x\right)\right) - 2 \cdot x\right)\right)}\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(x, \mathsf{*.f64}\left(wj, \color{blue}{\left(wj \cdot \left(\left(1 + -1 \cdot \left(wj \cdot \left(1 + \left(-3 \cdot x + \left(-2 \cdot \left(-4 \cdot x + \frac{3}{2} \cdot x\right) + \frac{2}{3} \cdot x\right)\right)\right)\right)\right) - \left(-4 \cdot x + \frac{3}{2} \cdot x\right)\right) - 2 \cdot x\right)}\right)\right) \]

      3. cancel-sign-sub-inv N/A

         \[\leadsto \mathsf{+.f64}\left(x, \mathsf{*.f64}\left(wj, \left(wj \cdot \left(\left(1 + -1 \cdot \left(wj \cdot \left(1 + \left(-3 \cdot x + \left(-2 \cdot \left(-4 \cdot x + \frac{3}{2} \cdot x\right) + \frac{2}{3} \cdot x\right)\right)\right)\right)\right) - \left(-4 \cdot x + \frac{3}{2} \cdot x\right)\right) + \color{blue}{\left(\mathsf{neg}\left(2\right)\right) \cdot x}\right)\right)\right) \]

      4. metadata-eval N/A

         \[\leadsto \mathsf{+.f64}\left(x, \mathsf{*.f64}\left(wj, \left(wj \cdot \left(\left(1 + -1 \cdot \left(wj \cdot \left(1 + \left(-3 \cdot x + \left(-2 \cdot \left(-4 \cdot x + \frac{3}{2} \cdot x\right) + \frac{2}{3} \cdot x\right)\right)\right)\right)\right) - \left(-4 \cdot x + \frac{3}{2} \cdot x\right)\right) + -2 \cdot x\right)\right)\right) \]

      5. +-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(x, \mathsf{*.f64}\left(wj, \left(-2 \cdot x + \color{blue}{wj \cdot \left(\left(1 + -1 \cdot \left(wj \cdot \left(1 + \left(-3 \cdot x + \left(-2 \cdot \left(-4 \cdot x + \frac{3}{2} \cdot x\right) + \frac{2}{3} \cdot x\right)\right)\right)\right)\right) - \left(-4 \cdot x + \frac{3}{2} \cdot x\right)\right)}\right)\right)\right) \]

      6. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(x, \mathsf{*.f64}\left(wj, \mathsf{+.f64}\left(\left(-2 \cdot x\right), \color{blue}{\left(wj \cdot \left(\left(1 + -1 \cdot \left(wj \cdot \left(1 + \left(-3 \cdot x + \left(-2 \cdot \left(-4 \cdot x + \frac{3}{2} \cdot x\right) + \frac{2}{3} \cdot x\right)\right)\right)\right)\right) - \left(-4 \cdot x + \frac{3}{2} \cdot x\right)\right)\right)}\right)\right)\right) \]

   5. Simplified 97.3%

      \[\leadsto \color{blue}{x + wj \cdot \left(x \cdot -2 + wj \cdot \left(\left(1 - wj \cdot \left(\left(x \cdot -3 + x \cdot 5\right) + \left(x \cdot 0.6666666666666666 + 1\right)\right)\right) + x \cdot 2.5\right)\right)} \]

   if 1.3000000000000001e-8 < wj

   1. Initial program 49.7%

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

      1. sub-neg N/A

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

      2. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(wj, \color{blue}{\left(\mathsf{neg}\left(\frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}}\right)\right)}\right) \]

      3. distribute-rgt1-in N/A

         \[\leadsto \mathsf{+.f64}\left(wj, \left(\mathsf{neg}\left(\frac{wj \cdot e^{wj} - x}{\left(wj + 1\right) \cdot e^{wj}}\right)\right)\right) \]

      4. associate-/l/ N/A

         \[\leadsto \mathsf{+.f64}\left(wj, \left(\mathsf{neg}\left(\frac{\frac{wj \cdot e^{wj} - x}{e^{wj}}}{wj + 1}\right)\right)\right) \]

      5. distribute-neg-frac2 N/A

         \[\leadsto \mathsf{+.f64}\left(wj, \left(\frac{\frac{wj \cdot e^{wj} - x}{e^{wj}}}{\color{blue}{\mathsf{neg}\left(\left(wj + 1\right)\right)}}\right)\right) \]

      6. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(wj, \mathsf{/.f64}\left(\left(\frac{wj \cdot e^{wj} - x}{e^{wj}}\right), \color{blue}{\left(\mathsf{neg}\left(\left(wj + 1\right)\right)\right)}\right)\right) \]

      7. div-sub N/A

         \[\leadsto \mathsf{+.f64}\left(wj, \mathsf{/.f64}\left(\left(\frac{wj \cdot e^{wj}}{e^{wj}} - \frac{x}{e^{wj}}\right), \left(\mathsf{neg}\left(\color{blue}{\left(wj + 1\right)}\right)\right)\right)\right) \]

      8. associate-/l* N/A

         \[\leadsto \mathsf{+.f64}\left(wj, \mathsf{/.f64}\left(\left(wj \cdot \frac{e^{wj}}{e^{wj}} - \frac{x}{e^{wj}}\right), \left(\mathsf{neg}\left(\left(\color{blue}{wj} + 1\right)\right)\right)\right)\right) \]

      9. *-inverses N/A

         \[\leadsto \mathsf{+.f64}\left(wj, \mathsf{/.f64}\left(\left(wj \cdot 1 - \frac{x}{e^{wj}}\right), \left(\mathsf{neg}\left(\left(wj + 1\right)\right)\right)\right)\right) \]

      10. *-rgt-identity N/A

          \[\leadsto \mathsf{+.f64}\left(wj, \mathsf{/.f64}\left(\left(wj - \frac{x}{e^{wj}}\right), \left(\mathsf{neg}\left(\left(\color{blue}{wj} + 1\right)\right)\right)\right)\right) \]

      11. --lowering--.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(wj, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(wj, \left(\frac{x}{e^{wj}}\right)\right), \left(\mathsf{neg}\left(\color{blue}{\left(wj + 1\right)}\right)\right)\right)\right) \]

      12. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(wj, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(wj, \mathsf{/.f64}\left(x, \left(e^{wj}\right)\right)\right), \left(\mathsf{neg}\left(\left(wj + \color{blue}{1}\right)\right)\right)\right)\right) \]

      13. exp-lowering-exp.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(wj, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(wj, \mathsf{/.f64}\left(x, \mathsf{exp.f64}\left(wj\right)\right)\right), \left(\mathsf{neg}\left(\left(wj + 1\right)\right)\right)\right)\right) \]

      14. +-commutative N/A

          \[\leadsto \mathsf{+.f64}\left(wj, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(wj, \mathsf{/.f64}\left(x, \mathsf{exp.f64}\left(wj\right)\right)\right), \left(\mathsf{neg}\left(\left(1 + wj\right)\right)\right)\right)\right) \]

      15. *-inverses N/A

          \[\leadsto \mathsf{+.f64}\left(wj, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(wj, \mathsf{/.f64}\left(x, \mathsf{exp.f64}\left(wj\right)\right)\right), \left(\mathsf{neg}\left(\left(\frac{e^{wj}}{e^{wj}} + wj\right)\right)\right)\right)\right) \]

      16. distribute-neg-in N/A

          \[\leadsto \mathsf{+.f64}\left(wj, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(wj, \mathsf{/.f64}\left(x, \mathsf{exp.f64}\left(wj\right)\right)\right), \left(\left(\mathsf{neg}\left(\frac{e^{wj}}{e^{wj}}\right)\right) + \color{blue}{\left(\mathsf{neg}\left(wj\right)\right)}\right)\right)\right) \]

   3. Simplified 99.7%

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

   5. Taylor expanded in wj around 0

      \[\leadsto \mathsf{+.f64}\left(wj, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(wj, \mathsf{/.f64}\left(x, \color{blue}{\left(1 + wj \cdot \left(1 + wj \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot wj\right)\right)\right)}\right)\right), \mathsf{\_.f64}\left(-1, wj\right)\right)\right) \]
   6. Step-by-step derivation

      1. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(wj, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(wj, \mathsf{/.f64}\left(x, \mathsf{+.f64}\left(1, \left(wj \cdot \left(1 + wj \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot wj\right)\right)\right)\right)\right)\right), \mathsf{\_.f64}\left(-1, wj\right)\right)\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(wj, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(wj, \mathsf{/.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(wj, \left(1 + wj \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot wj\right)\right)\right)\right)\right)\right), \mathsf{\_.f64}\left(-1, wj\right)\right)\right) \]

      3. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(wj, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(wj, \mathsf{/.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(wj, \mathsf{+.f64}\left(1, \left(wj \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot wj\right)\right)\right)\right)\right)\right)\right), \mathsf{\_.f64}\left(-1, wj\right)\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(wj, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(wj, \mathsf{/.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(wj, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(wj, \left(\frac{1}{2} + \frac{1}{6} \cdot wj\right)\right)\right)\right)\right)\right)\right), \mathsf{\_.f64}\left(-1, wj\right)\right)\right) \]

      5. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(wj, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(wj, \mathsf{/.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(wj, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(wj, \mathsf{+.f64}\left(\frac{1}{2}, \left(\frac{1}{6} \cdot wj\right)\right)\right)\right)\right)\right)\right)\right), \mathsf{\_.f64}\left(-1, wj\right)\right)\right) \]

      6. *-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(wj, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(wj, \mathsf{/.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(wj, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(wj, \mathsf{+.f64}\left(\frac{1}{2}, \left(wj \cdot \frac{1}{6}\right)\right)\right)\right)\right)\right)\right)\right), \mathsf{\_.f64}\left(-1, wj\right)\right)\right) \]

      7. *-lowering-*.f64 98.3%

         \[\leadsto \mathsf{+.f64}\left(wj, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(wj, \mathsf{/.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(wj, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(wj, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(wj, \frac{1}{6}\right)\right)\right)\right)\right)\right)\right)\right), \mathsf{\_.f64}\left(-1, wj\right)\right)\right) \]

   7. Simplified 98.3%

      \[\leadsto wj + \frac{wj - \frac{x}{\color{blue}{1 + wj \cdot \left(1 + wj \cdot \left(0.5 + wj \cdot 0.16666666666666666\right)\right)}}}{-1 - wj} \]
3. Recombined 2 regimes into one program.

4. Final simplification 97.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;wj \leq 1.3 \cdot 10^{-8}:\\ \;\;\;\;x + wj \cdot \left(x \cdot -2 + wj \cdot \left(\left(1 + wj \cdot \left(\left(-1 - x \cdot 0.6666666666666666\right) - \left(x \cdot -3 + x \cdot 5\right)\right)\right) + x \cdot 2.5\right)\right)\\ \mathbf{else}:\\ \;\;\;\;wj + \frac{wj + \frac{x}{-1 + wj \cdot \left(-1 - wj \cdot \left(0.5 + wj \cdot 0.16666666666666666\right)\right)}}{-1 - wj}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 97.2% accurate, 10.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;wj \leq 1.3 \cdot 10^{-8}:\\ \;\;\;\;x - x \cdot \left(\left(wj \cdot wj\right) \cdot \frac{wj + -1}{x} - wj \cdot \left(-2 + wj \cdot \left(2.5 + wj \cdot -2.6666666666666665\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;wj + \frac{wj + \frac{x}{-1 + wj \cdot \left(-1 - wj \cdot \left(0.5 + wj \cdot 0.16666666666666666\right)\right)}}{-1 - wj}\\ \end{array} \end{array} \]

FPCore

(FPCore (wj x)
 :precision binary64
 (if (<= wj 1.3e-8)
   (-
    x
    (*
     x
     (-
      (* (* wj wj) (/ (+ wj -1.0) x))
      (* wj (+ -2.0 (* wj (+ 2.5 (* wj -2.6666666666666665))))))))
   (+
    wj
    (/
     (+
      wj
      (/ x (+ -1.0 (* wj (- -1.0 (* wj (+ 0.5 (* wj 0.16666666666666666))))))))
     (- -1.0 wj)))))

C

double code(double wj, double x) {
	double tmp;
	if (wj <= 1.3e-8) {
		tmp = x - (x * (((wj * wj) * ((wj + -1.0) / x)) - (wj * (-2.0 + (wj * (2.5 + (wj * -2.6666666666666665)))))));
	} else {
		tmp = wj + ((wj + (x / (-1.0 + (wj * (-1.0 - (wj * (0.5 + (wj * 0.16666666666666666)))))))) / (-1.0 - wj));
	}
	return tmp;
}

Fortran

real(8) function code(wj, x)
    real(8), intent (in) :: wj
    real(8), intent (in) :: x
    real(8) :: tmp
    if (wj <= 1.3d-8) then
        tmp = x - (x * (((wj * wj) * ((wj + (-1.0d0)) / x)) - (wj * ((-2.0d0) + (wj * (2.5d0 + (wj * (-2.6666666666666665d0))))))))
    else
        tmp = wj + ((wj + (x / ((-1.0d0) + (wj * ((-1.0d0) - (wj * (0.5d0 + (wj * 0.16666666666666666d0)))))))) / ((-1.0d0) - wj))
    end if
    code = tmp
end function

Java

public static double code(double wj, double x) {
	double tmp;
	if (wj <= 1.3e-8) {
		tmp = x - (x * (((wj * wj) * ((wj + -1.0) / x)) - (wj * (-2.0 + (wj * (2.5 + (wj * -2.6666666666666665)))))));
	} else {
		tmp = wj + ((wj + (x / (-1.0 + (wj * (-1.0 - (wj * (0.5 + (wj * 0.16666666666666666)))))))) / (-1.0 - wj));
	}
	return tmp;
}

Python

def code(wj, x):
	tmp = 0
	if wj <= 1.3e-8:
		tmp = x - (x * (((wj * wj) * ((wj + -1.0) / x)) - (wj * (-2.0 + (wj * (2.5 + (wj * -2.6666666666666665)))))))
	else:
		tmp = wj + ((wj + (x / (-1.0 + (wj * (-1.0 - (wj * (0.5 + (wj * 0.16666666666666666)))))))) / (-1.0 - wj))
	return tmp

Julia

function code(wj, x)
	tmp = 0.0
	if (wj <= 1.3e-8)
		tmp = Float64(x - Float64(x * Float64(Float64(Float64(wj * wj) * Float64(Float64(wj + -1.0) / x)) - Float64(wj * Float64(-2.0 + Float64(wj * Float64(2.5 + Float64(wj * -2.6666666666666665))))))));
	else
		tmp = Float64(wj + Float64(Float64(wj + Float64(x / Float64(-1.0 + Float64(wj * Float64(-1.0 - Float64(wj * Float64(0.5 + Float64(wj * 0.16666666666666666)))))))) / Float64(-1.0 - wj)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(wj, x)
	tmp = 0.0;
	if (wj <= 1.3e-8)
		tmp = x - (x * (((wj * wj) * ((wj + -1.0) / x)) - (wj * (-2.0 + (wj * (2.5 + (wj * -2.6666666666666665)))))));
	else
		tmp = wj + ((wj + (x / (-1.0 + (wj * (-1.0 - (wj * (0.5 + (wj * 0.16666666666666666)))))))) / (-1.0 - wj));
	end
	tmp_2 = tmp;
end

Wolfram

code[wj_, x_] := If[LessEqual[wj, 1.3e-8], N[(x - N[(x * N[(N[(N[(wj * wj), $MachinePrecision] * N[(N[(wj + -1.0), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision] - N[(wj * N[(-2.0 + N[(wj * N[(2.5 + N[(wj * -2.6666666666666665), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(wj + N[(N[(wj + N[(x / N[(-1.0 + N[(wj * N[(-1.0 - N[(wj * N[(0.5 + N[(wj * 0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(-1.0 - wj), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if wj < 1.3000000000000001e-8

   1. Initial program 78.4%

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

   3. Taylor expanded in wj around 0

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

      1. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(x, \color{blue}{\left(wj \cdot \left(wj \cdot \left(\left(1 + -1 \cdot \left(wj \cdot \left(1 + \left(-3 \cdot x + \left(-2 \cdot \left(-4 \cdot x + \frac{3}{2} \cdot x\right) + \frac{2}{3} \cdot x\right)\right)\right)\right)\right) - \left(-4 \cdot x + \frac{3}{2} \cdot x\right)\right) - 2 \cdot x\right)\right)}\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(x, \mathsf{*.f64}\left(wj, \color{blue}{\left(wj \cdot \left(\left(1 + -1 \cdot \left(wj \cdot \left(1 + \left(-3 \cdot x + \left(-2 \cdot \left(-4 \cdot x + \frac{3}{2} \cdot x\right) + \frac{2}{3} \cdot x\right)\right)\right)\right)\right) - \left(-4 \cdot x + \frac{3}{2} \cdot x\right)\right) - 2 \cdot x\right)}\right)\right) \]

      3. cancel-sign-sub-inv N/A

         \[\leadsto \mathsf{+.f64}\left(x, \mathsf{*.f64}\left(wj, \left(wj \cdot \left(\left(1 + -1 \cdot \left(wj \cdot \left(1 + \left(-3 \cdot x + \left(-2 \cdot \left(-4 \cdot x + \frac{3}{2} \cdot x\right) + \frac{2}{3} \cdot x\right)\right)\right)\right)\right) - \left(-4 \cdot x + \frac{3}{2} \cdot x\right)\right) + \color{blue}{\left(\mathsf{neg}\left(2\right)\right) \cdot x}\right)\right)\right) \]

      4. metadata-eval N/A

         \[\leadsto \mathsf{+.f64}\left(x, \mathsf{*.f64}\left(wj, \left(wj \cdot \left(\left(1 + -1 \cdot \left(wj \cdot \left(1 + \left(-3 \cdot x + \left(-2 \cdot \left(-4 \cdot x + \frac{3}{2} \cdot x\right) + \frac{2}{3} \cdot x\right)\right)\right)\right)\right) - \left(-4 \cdot x + \frac{3}{2} \cdot x\right)\right) + -2 \cdot x\right)\right)\right) \]

      5. +-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(x, \mathsf{*.f64}\left(wj, \left(-2 \cdot x + \color{blue}{wj \cdot \left(\left(1 + -1 \cdot \left(wj \cdot \left(1 + \left(-3 \cdot x + \left(-2 \cdot \left(-4 \cdot x + \frac{3}{2} \cdot x\right) + \frac{2}{3} \cdot x\right)\right)\right)\right)\right) - \left(-4 \cdot x + \frac{3}{2} \cdot x\right)\right)}\right)\right)\right) \]

      6. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(x, \mathsf{*.f64}\left(wj, \mathsf{+.f64}\left(\left(-2 \cdot x\right), \color{blue}{\left(wj \cdot \left(\left(1 + -1 \cdot \left(wj \cdot \left(1 + \left(-3 \cdot x + \left(-2 \cdot \left(-4 \cdot x + \frac{3}{2} \cdot x\right) + \frac{2}{3} \cdot x\right)\right)\right)\right)\right) - \left(-4 \cdot x + \frac{3}{2} \cdot x\right)\right)\right)}\right)\right)\right) \]

   5. Simplified 97.3%

      \[\leadsto \color{blue}{x + wj \cdot \left(x \cdot -2 + wj \cdot \left(\left(1 - wj \cdot \left(\left(x \cdot -3 + x \cdot 5\right) + \left(x \cdot 0.6666666666666666 + 1\right)\right)\right) + x \cdot 2.5\right)\right)} \]

   6. Taylor expanded in x around inf

      \[\leadsto \mathsf{+.f64}\left(x, \color{blue}{\left(x \cdot \left(wj \cdot \left(wj \cdot \left(\frac{5}{2} - \frac{8}{3} \cdot wj\right) - 2\right) + \frac{{wj}^{2} \cdot \left(1 - wj\right)}{x}\right)\right)}\right) \]
   7. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(x, \mathsf{*.f64}\left(x, \color{blue}{\left(wj \cdot \left(wj \cdot \left(\frac{5}{2} - \frac{8}{3} \cdot wj\right) - 2\right) + \frac{{wj}^{2} \cdot \left(1 - wj\right)}{x}\right)}\right)\right) \]

      2. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\left(wj \cdot \left(wj \cdot \left(\frac{5}{2} - \frac{8}{3} \cdot wj\right) - 2\right)\right), \color{blue}{\left(\frac{{wj}^{2} \cdot \left(1 - wj\right)}{x}\right)}\right)\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\mathsf{*.f64}\left(wj, \left(wj \cdot \left(\frac{5}{2} - \frac{8}{3} \cdot wj\right) - 2\right)\right), \left(\frac{\color{blue}{{wj}^{2} \cdot \left(1 - wj\right)}}{x}\right)\right)\right)\right) \]

      4. sub-neg N/A

         \[\leadsto \mathsf{+.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\mathsf{*.f64}\left(wj, \left(wj \cdot \left(\frac{5}{2} - \frac{8}{3} \cdot wj\right) + \left(\mathsf{neg}\left(2\right)\right)\right)\right), \left(\frac{{wj}^{2} \cdot \color{blue}{\left(1 - wj\right)}}{x}\right)\right)\right)\right) \]

      5. metadata-eval N/A

         \[\leadsto \mathsf{+.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\mathsf{*.f64}\left(wj, \left(wj \cdot \left(\frac{5}{2} - \frac{8}{3} \cdot wj\right) + -2\right)\right), \left(\frac{{wj}^{2} \cdot \left(1 - \color{blue}{wj}\right)}{x}\right)\right)\right)\right) \]

      6. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\mathsf{*.f64}\left(wj, \mathsf{+.f64}\left(\left(wj \cdot \left(\frac{5}{2} - \frac{8}{3} \cdot wj\right)\right), -2\right)\right), \left(\frac{{wj}^{2} \cdot \color{blue}{\left(1 - wj\right)}}{x}\right)\right)\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\mathsf{*.f64}\left(wj, \mathsf{+.f64}\left(\mathsf{*.f64}\left(wj, \left(\frac{5}{2} - \frac{8}{3} \cdot wj\right)\right), -2\right)\right), \left(\frac{{wj}^{2} \cdot \left(\color{blue}{1} - wj\right)}{x}\right)\right)\right)\right) \]

      8. cancel-sign-sub-inv N/A

         \[\leadsto \mathsf{+.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\mathsf{*.f64}\left(wj, \mathsf{+.f64}\left(\mathsf{*.f64}\left(wj, \left(\frac{5}{2} + \left(\mathsf{neg}\left(\frac{8}{3}\right)\right) \cdot wj\right)\right), -2\right)\right), \left(\frac{{wj}^{2} \cdot \left(1 - wj\right)}{x}\right)\right)\right)\right) \]

      9. metadata-eval N/A

         \[\leadsto \mathsf{+.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\mathsf{*.f64}\left(wj, \mathsf{+.f64}\left(\mathsf{*.f64}\left(wj, \left(\frac{5}{2} + \frac{-8}{3} \cdot wj\right)\right), -2\right)\right), \left(\frac{{wj}^{2} \cdot \left(1 - wj\right)}{x}\right)\right)\right)\right) \]

      10. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\mathsf{*.f64}\left(wj, \mathsf{+.f64}\left(\mathsf{*.f64}\left(wj, \mathsf{+.f64}\left(\frac{5}{2}, \left(\frac{-8}{3} \cdot wj\right)\right)\right), -2\right)\right), \left(\frac{{wj}^{2} \cdot \left(1 - wj\right)}{x}\right)\right)\right)\right) \]

      11. *-commutative N/A

          \[\leadsto \mathsf{+.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\mathsf{*.f64}\left(wj, \mathsf{+.f64}\left(\mathsf{*.f64}\left(wj, \mathsf{+.f64}\left(\frac{5}{2}, \left(wj \cdot \frac{-8}{3}\right)\right)\right), -2\right)\right), \left(\frac{{wj}^{2} \cdot \left(1 - wj\right)}{x}\right)\right)\right)\right) \]

      12. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\mathsf{*.f64}\left(wj, \mathsf{+.f64}\left(\mathsf{*.f64}\left(wj, \mathsf{+.f64}\left(\frac{5}{2}, \mathsf{*.f64}\left(wj, \frac{-8}{3}\right)\right)\right), -2\right)\right), \left(\frac{{wj}^{2} \cdot \left(1 - wj\right)}{x}\right)\right)\right)\right) \]

      13. associate-/l* N/A

          \[\leadsto \mathsf{+.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\mathsf{*.f64}\left(wj, \mathsf{+.f64}\left(\mathsf{*.f64}\left(wj, \mathsf{+.f64}\left(\frac{5}{2}, \mathsf{*.f64}\left(wj, \frac{-8}{3}\right)\right)\right), -2\right)\right), \left({wj}^{2} \cdot \color{blue}{\frac{1 - wj}{x}}\right)\right)\right)\right) \]

      14. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\mathsf{*.f64}\left(wj, \mathsf{+.f64}\left(\mathsf{*.f64}\left(wj, \mathsf{+.f64}\left(\frac{5}{2}, \mathsf{*.f64}\left(wj, \frac{-8}{3}\right)\right)\right), -2\right)\right), \mathsf{*.f64}\left(\left({wj}^{2}\right), \color{blue}{\left(\frac{1 - wj}{x}\right)}\right)\right)\right)\right) \]

      15. unpow2 N/A

          \[\leadsto \mathsf{+.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\mathsf{*.f64}\left(wj, \mathsf{+.f64}\left(\mathsf{*.f64}\left(wj, \mathsf{+.f64}\left(\frac{5}{2}, \mathsf{*.f64}\left(wj, \frac{-8}{3}\right)\right)\right), -2\right)\right), \mathsf{*.f64}\left(\left(wj \cdot wj\right), \left(\frac{\color{blue}{1 - wj}}{x}\right)\right)\right)\right)\right) \]

      16. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\mathsf{*.f64}\left(wj, \mathsf{+.f64}\left(\mathsf{*.f64}\left(wj, \mathsf{+.f64}\left(\frac{5}{2}, \mathsf{*.f64}\left(wj, \frac{-8}{3}\right)\right)\right), -2\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(wj, wj\right), \left(\frac{\color{blue}{1 - wj}}{x}\right)\right)\right)\right)\right) \]

      17. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\mathsf{*.f64}\left(wj, \mathsf{+.f64}\left(\mathsf{*.f64}\left(wj, \mathsf{+.f64}\left(\frac{5}{2}, \mathsf{*.f64}\left(wj, \frac{-8}{3}\right)\right)\right), -2\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(wj, wj\right), \mathsf{/.f64}\left(\left(1 - wj\right), \color{blue}{x}\right)\right)\right)\right)\right) \]

      18. --lowering--.f64 97.2%

          \[\leadsto \mathsf{+.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\mathsf{*.f64}\left(wj, \mathsf{+.f64}\left(\mathsf{*.f64}\left(wj, \mathsf{+.f64}\left(\frac{5}{2}, \mathsf{*.f64}\left(wj, \frac{-8}{3}\right)\right)\right), -2\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(wj, wj\right), \mathsf{/.f64}\left(\mathsf{\_.f64}\left(1, wj\right), x\right)\right)\right)\right)\right) \]

   8. Simplified 97.2%

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

   if 1.3000000000000001e-8 < wj

   1. Initial program 49.7%

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

      1. sub-neg N/A

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

      2. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(wj, \color{blue}{\left(\mathsf{neg}\left(\frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}}\right)\right)}\right) \]

      3. distribute-rgt1-in N/A

         \[\leadsto \mathsf{+.f64}\left(wj, \left(\mathsf{neg}\left(\frac{wj \cdot e^{wj} - x}{\left(wj + 1\right) \cdot e^{wj}}\right)\right)\right) \]

      4. associate-/l/ N/A

         \[\leadsto \mathsf{+.f64}\left(wj, \left(\mathsf{neg}\left(\frac{\frac{wj \cdot e^{wj} - x}{e^{wj}}}{wj + 1}\right)\right)\right) \]

      5. distribute-neg-frac2 N/A

         \[\leadsto \mathsf{+.f64}\left(wj, \left(\frac{\frac{wj \cdot e^{wj} - x}{e^{wj}}}{\color{blue}{\mathsf{neg}\left(\left(wj + 1\right)\right)}}\right)\right) \]

      6. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(wj, \mathsf{/.f64}\left(\left(\frac{wj \cdot e^{wj} - x}{e^{wj}}\right), \color{blue}{\left(\mathsf{neg}\left(\left(wj + 1\right)\right)\right)}\right)\right) \]

      7. div-sub N/A

         \[\leadsto \mathsf{+.f64}\left(wj, \mathsf{/.f64}\left(\left(\frac{wj \cdot e^{wj}}{e^{wj}} - \frac{x}{e^{wj}}\right), \left(\mathsf{neg}\left(\color{blue}{\left(wj + 1\right)}\right)\right)\right)\right) \]

      8. associate-/l* N/A

         \[\leadsto \mathsf{+.f64}\left(wj, \mathsf{/.f64}\left(\left(wj \cdot \frac{e^{wj}}{e^{wj}} - \frac{x}{e^{wj}}\right), \left(\mathsf{neg}\left(\left(\color{blue}{wj} + 1\right)\right)\right)\right)\right) \]

      9. *-inverses N/A

         \[\leadsto \mathsf{+.f64}\left(wj, \mathsf{/.f64}\left(\left(wj \cdot 1 - \frac{x}{e^{wj}}\right), \left(\mathsf{neg}\left(\left(wj + 1\right)\right)\right)\right)\right) \]

      10. *-rgt-identity N/A

          \[\leadsto \mathsf{+.f64}\left(wj, \mathsf{/.f64}\left(\left(wj - \frac{x}{e^{wj}}\right), \left(\mathsf{neg}\left(\left(\color{blue}{wj} + 1\right)\right)\right)\right)\right) \]

      11. --lowering--.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(wj, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(wj, \left(\frac{x}{e^{wj}}\right)\right), \left(\mathsf{neg}\left(\color{blue}{\left(wj + 1\right)}\right)\right)\right)\right) \]

      12. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(wj, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(wj, \mathsf{/.f64}\left(x, \left(e^{wj}\right)\right)\right), \left(\mathsf{neg}\left(\left(wj + \color{blue}{1}\right)\right)\right)\right)\right) \]

      13. exp-lowering-exp.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(wj, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(wj, \mathsf{/.f64}\left(x, \mathsf{exp.f64}\left(wj\right)\right)\right), \left(\mathsf{neg}\left(\left(wj + 1\right)\right)\right)\right)\right) \]

      14. +-commutative N/A

          \[\leadsto \mathsf{+.f64}\left(wj, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(wj, \mathsf{/.f64}\left(x, \mathsf{exp.f64}\left(wj\right)\right)\right), \left(\mathsf{neg}\left(\left(1 + wj\right)\right)\right)\right)\right) \]

      15. *-inverses N/A

          \[\leadsto \mathsf{+.f64}\left(wj, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(wj, \mathsf{/.f64}\left(x, \mathsf{exp.f64}\left(wj\right)\right)\right), \left(\mathsf{neg}\left(\left(\frac{e^{wj}}{e^{wj}} + wj\right)\right)\right)\right)\right) \]

      16. distribute-neg-in N/A

          \[\leadsto \mathsf{+.f64}\left(wj, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(wj, \mathsf{/.f64}\left(x, \mathsf{exp.f64}\left(wj\right)\right)\right), \left(\left(\mathsf{neg}\left(\frac{e^{wj}}{e^{wj}}\right)\right) + \color{blue}{\left(\mathsf{neg}\left(wj\right)\right)}\right)\right)\right) \]

   3. Simplified 99.7%

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

   5. Taylor expanded in wj around 0

      \[\leadsto \mathsf{+.f64}\left(wj, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(wj, \mathsf{/.f64}\left(x, \color{blue}{\left(1 + wj \cdot \left(1 + wj \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot wj\right)\right)\right)}\right)\right), \mathsf{\_.f64}\left(-1, wj\right)\right)\right) \]
   6. Step-by-step derivation

      1. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(wj, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(wj, \mathsf{/.f64}\left(x, \mathsf{+.f64}\left(1, \left(wj \cdot \left(1 + wj \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot wj\right)\right)\right)\right)\right)\right), \mathsf{\_.f64}\left(-1, wj\right)\right)\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(wj, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(wj, \mathsf{/.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(wj, \left(1 + wj \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot wj\right)\right)\right)\right)\right)\right), \mathsf{\_.f64}\left(-1, wj\right)\right)\right) \]

      3. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(wj, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(wj, \mathsf{/.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(wj, \mathsf{+.f64}\left(1, \left(wj \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot wj\right)\right)\right)\right)\right)\right)\right), \mathsf{\_.f64}\left(-1, wj\right)\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(wj, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(wj, \mathsf{/.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(wj, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(wj, \left(\frac{1}{2} + \frac{1}{6} \cdot wj\right)\right)\right)\right)\right)\right)\right), \mathsf{\_.f64}\left(-1, wj\right)\right)\right) \]

      5. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(wj, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(wj, \mathsf{/.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(wj, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(wj, \mathsf{+.f64}\left(\frac{1}{2}, \left(\frac{1}{6} \cdot wj\right)\right)\right)\right)\right)\right)\right)\right), \mathsf{\_.f64}\left(-1, wj\right)\right)\right) \]

      6. *-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(wj, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(wj, \mathsf{/.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(wj, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(wj, \mathsf{+.f64}\left(\frac{1}{2}, \left(wj \cdot \frac{1}{6}\right)\right)\right)\right)\right)\right)\right)\right), \mathsf{\_.f64}\left(-1, wj\right)\right)\right) \]

      7. *-lowering-*.f64 98.3%

         \[\leadsto \mathsf{+.f64}\left(wj, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(wj, \mathsf{/.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(wj, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(wj, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(wj, \frac{1}{6}\right)\right)\right)\right)\right)\right)\right)\right), \mathsf{\_.f64}\left(-1, wj\right)\right)\right) \]

   7. Simplified 98.3%

      \[\leadsto wj + \frac{wj - \frac{x}{\color{blue}{1 + wj \cdot \left(1 + wj \cdot \left(0.5 + wj \cdot 0.16666666666666666\right)\right)}}}{-1 - wj} \]
3. Recombined 2 regimes into one program.

4. Final simplification 97.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;wj \leq 1.3 \cdot 10^{-8}:\\ \;\;\;\;x - x \cdot \left(\left(wj \cdot wj\right) \cdot \frac{wj + -1}{x} - wj \cdot \left(-2 + wj \cdot \left(2.5 + wj \cdot -2.6666666666666665\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;wj + \frac{wj + \frac{x}{-1 + wj \cdot \left(-1 - wj \cdot \left(0.5 + wj \cdot 0.16666666666666666\right)\right)}}{-1 - wj}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 97.6% accurate, 10.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;wj \leq -0.00032:\\ \;\;\;\;wj + \frac{\frac{wj \cdot wj - x \cdot x}{wj + x}}{-1 - wj}\\ \mathbf{elif}\;wj \leq 1.3 \cdot 10^{-8}:\\ \;\;\;\;x + wj \cdot \left(x \cdot -2 + wj \cdot \left(1 - wj\right)\right)\\ \mathbf{else}:\\ \;\;\;\;wj + \frac{wj + \frac{x}{-1 + wj \cdot \left(-1 - wj \cdot 0.5\right)}}{-1 - wj}\\ \end{array} \end{array} \]

FPCore

(FPCore (wj x)
 :precision binary64
 (if (<= wj -0.00032)
   (+ wj (/ (/ (- (* wj wj) (* x x)) (+ wj x)) (- -1.0 wj)))
   (if (<= wj 1.3e-8)
     (+ x (* wj (+ (* x -2.0) (* wj (- 1.0 wj)))))
     (+ wj (/ (+ wj (/ x (+ -1.0 (* wj (- -1.0 (* wj 0.5)))))) (- -1.0 wj))))))

C

double code(double wj, double x) {
	double tmp;
	if (wj <= -0.00032) {
		tmp = wj + ((((wj * wj) - (x * x)) / (wj + x)) / (-1.0 - wj));
	} else if (wj <= 1.3e-8) {
		tmp = x + (wj * ((x * -2.0) + (wj * (1.0 - wj))));
	} else {
		tmp = wj + ((wj + (x / (-1.0 + (wj * (-1.0 - (wj * 0.5)))))) / (-1.0 - wj));
	}
	return tmp;
}

Fortran

real(8) function code(wj, x)
    real(8), intent (in) :: wj
    real(8), intent (in) :: x
    real(8) :: tmp
    if (wj <= (-0.00032d0)) then
        tmp = wj + ((((wj * wj) - (x * x)) / (wj + x)) / ((-1.0d0) - wj))
    else if (wj <= 1.3d-8) then
        tmp = x + (wj * ((x * (-2.0d0)) + (wj * (1.0d0 - wj))))
    else
        tmp = wj + ((wj + (x / ((-1.0d0) + (wj * ((-1.0d0) - (wj * 0.5d0)))))) / ((-1.0d0) - wj))
    end if
    code = tmp
end function

Java

public static double code(double wj, double x) {
	double tmp;
	if (wj <= -0.00032) {
		tmp = wj + ((((wj * wj) - (x * x)) / (wj + x)) / (-1.0 - wj));
	} else if (wj <= 1.3e-8) {
		tmp = x + (wj * ((x * -2.0) + (wj * (1.0 - wj))));
	} else {
		tmp = wj + ((wj + (x / (-1.0 + (wj * (-1.0 - (wj * 0.5)))))) / (-1.0 - wj));
	}
	return tmp;
}

Python

def code(wj, x):
	tmp = 0
	if wj <= -0.00032:
		tmp = wj + ((((wj * wj) - (x * x)) / (wj + x)) / (-1.0 - wj))
	elif wj <= 1.3e-8:
		tmp = x + (wj * ((x * -2.0) + (wj * (1.0 - wj))))
	else:
		tmp = wj + ((wj + (x / (-1.0 + (wj * (-1.0 - (wj * 0.5)))))) / (-1.0 - wj))
	return tmp

Julia

function code(wj, x)
	tmp = 0.0
	if (wj <= -0.00032)
		tmp = Float64(wj + Float64(Float64(Float64(Float64(wj * wj) - Float64(x * x)) / Float64(wj + x)) / Float64(-1.0 - wj)));
	elseif (wj <= 1.3e-8)
		tmp = Float64(x + Float64(wj * Float64(Float64(x * -2.0) + Float64(wj * Float64(1.0 - wj)))));
	else
		tmp = Float64(wj + Float64(Float64(wj + Float64(x / Float64(-1.0 + Float64(wj * Float64(-1.0 - Float64(wj * 0.5)))))) / Float64(-1.0 - wj)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(wj, x)
	tmp = 0.0;
	if (wj <= -0.00032)
		tmp = wj + ((((wj * wj) - (x * x)) / (wj + x)) / (-1.0 - wj));
	elseif (wj <= 1.3e-8)
		tmp = x + (wj * ((x * -2.0) + (wj * (1.0 - wj))));
	else
		tmp = wj + ((wj + (x / (-1.0 + (wj * (-1.0 - (wj * 0.5)))))) / (-1.0 - wj));
	end
	tmp_2 = tmp;
end

Wolfram

code[wj_, x_] := If[LessEqual[wj, -0.00032], N[(wj + N[(N[(N[(N[(wj * wj), $MachinePrecision] - N[(x * x), $MachinePrecision]), $MachinePrecision] / N[(wj + x), $MachinePrecision]), $MachinePrecision] / N[(-1.0 - wj), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[wj, 1.3e-8], N[(x + N[(wj * N[(N[(x * -2.0), $MachinePrecision] + N[(wj * N[(1.0 - wj), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(wj + N[(N[(wj + N[(x / N[(-1.0 + N[(wj * N[(-1.0 - N[(wj * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(-1.0 - wj), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if wj < -3.20000000000000026e-4

   1. Initial program 51.4%

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

      1. sub-neg N/A

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

      2. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(wj, \color{blue}{\left(\mathsf{neg}\left(\frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}}\right)\right)}\right) \]

      3. distribute-rgt1-in N/A

         \[\leadsto \mathsf{+.f64}\left(wj, \left(\mathsf{neg}\left(\frac{wj \cdot e^{wj} - x}{\left(wj + 1\right) \cdot e^{wj}}\right)\right)\right) \]

      4. associate-/l/ N/A

         \[\leadsto \mathsf{+.f64}\left(wj, \left(\mathsf{neg}\left(\frac{\frac{wj \cdot e^{wj} - x}{e^{wj}}}{wj + 1}\right)\right)\right) \]

      5. distribute-neg-frac2 N/A

         \[\leadsto \mathsf{+.f64}\left(wj, \left(\frac{\frac{wj \cdot e^{wj} - x}{e^{wj}}}{\color{blue}{\mathsf{neg}\left(\left(wj + 1\right)\right)}}\right)\right) \]

      6. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(wj, \mathsf{/.f64}\left(\left(\frac{wj \cdot e^{wj} - x}{e^{wj}}\right), \color{blue}{\left(\mathsf{neg}\left(\left(wj + 1\right)\right)\right)}\right)\right) \]

      7. div-sub N/A

         \[\leadsto \mathsf{+.f64}\left(wj, \mathsf{/.f64}\left(\left(\frac{wj \cdot e^{wj}}{e^{wj}} - \frac{x}{e^{wj}}\right), \left(\mathsf{neg}\left(\color{blue}{\left(wj + 1\right)}\right)\right)\right)\right) \]

      8. associate-/l* N/A

         \[\leadsto \mathsf{+.f64}\left(wj, \mathsf{/.f64}\left(\left(wj \cdot \frac{e^{wj}}{e^{wj}} - \frac{x}{e^{wj}}\right), \left(\mathsf{neg}\left(\left(\color{blue}{wj} + 1\right)\right)\right)\right)\right) \]

      9. *-inverses N/A

         \[\leadsto \mathsf{+.f64}\left(wj, \mathsf{/.f64}\left(\left(wj \cdot 1 - \frac{x}{e^{wj}}\right), \left(\mathsf{neg}\left(\left(wj + 1\right)\right)\right)\right)\right) \]

      10. *-rgt-identity N/A

          \[\leadsto \mathsf{+.f64}\left(wj, \mathsf{/.f64}\left(\left(wj - \frac{x}{e^{wj}}\right), \left(\mathsf{neg}\left(\left(\color{blue}{wj} + 1\right)\right)\right)\right)\right) \]

      11. --lowering--.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(wj, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(wj, \left(\frac{x}{e^{wj}}\right)\right), \left(\mathsf{neg}\left(\color{blue}{\left(wj + 1\right)}\right)\right)\right)\right) \]

      12. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(wj, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(wj, \mathsf{/.f64}\left(x, \left(e^{wj}\right)\right)\right), \left(\mathsf{neg}\left(\left(wj + \color{blue}{1}\right)\right)\right)\right)\right) \]

      13. exp-lowering-exp.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(wj, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(wj, \mathsf{/.f64}\left(x, \mathsf{exp.f64}\left(wj\right)\right)\right), \left(\mathsf{neg}\left(\left(wj + 1\right)\right)\right)\right)\right) \]

      14. +-commutative N/A

          \[\leadsto \mathsf{+.f64}\left(wj, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(wj, \mathsf{/.f64}\left(x, \mathsf{exp.f64}\left(wj\right)\right)\right), \left(\mathsf{neg}\left(\left(1 + wj\right)\right)\right)\right)\right) \]

      15. *-inverses N/A

          \[\leadsto \mathsf{+.f64}\left(wj, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(wj, \mathsf{/.f64}\left(x, \mathsf{exp.f64}\left(wj\right)\right)\right), \left(\mathsf{neg}\left(\left(\frac{e^{wj}}{e^{wj}} + wj\right)\right)\right)\right)\right) \]

      16. distribute-neg-in N/A

          \[\leadsto \mathsf{+.f64}\left(wj, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(wj, \mathsf{/.f64}\left(x, \mathsf{exp.f64}\left(wj\right)\right)\right), \left(\left(\mathsf{neg}\left(\frac{e^{wj}}{e^{wj}}\right)\right) + \color{blue}{\left(\mathsf{neg}\left(wj\right)\right)}\right)\right)\right) \]

   3. Simplified 96.1%

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

   5. Taylor expanded in wj around 0

      \[\leadsto \mathsf{+.f64}\left(wj, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(wj, \color{blue}{x}\right), \mathsf{\_.f64}\left(-1, wj\right)\right)\right) \]
   6. Step-by-step derivation

   7. Simplified 45.6%

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

      1. flip-- N/A

         \[\leadsto \mathsf{+.f64}\left(wj, \mathsf{/.f64}\left(\left(\frac{wj \cdot wj - x \cdot x}{wj + x}\right), \mathsf{\_.f64}\left(\color{blue}{-1}, wj\right)\right)\right) \]

      2. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(wj, \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(wj \cdot wj - x \cdot x\right), \left(wj + x\right)\right), \mathsf{\_.f64}\left(\color{blue}{-1}, wj\right)\right)\right) \]

      3. --lowering--.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(wj, \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(\left(wj \cdot wj\right), \left(x \cdot x\right)\right), \left(wj + x\right)\right), \mathsf{\_.f64}\left(-1, wj\right)\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(wj, \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(wj, wj\right), \left(x \cdot x\right)\right), \left(wj + x\right)\right), \mathsf{\_.f64}\left(-1, wj\right)\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(wj, \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(wj, wj\right), \mathsf{*.f64}\left(x, x\right)\right), \left(wj + x\right)\right), \mathsf{\_.f64}\left(-1, wj\right)\right)\right) \]

      6. +-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(wj, \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(wj, wj\right), \mathsf{*.f64}\left(x, x\right)\right), \left(x + wj\right)\right), \mathsf{\_.f64}\left(-1, wj\right)\right)\right) \]

      7. +-lowering-+.f64 66.7%

         \[\leadsto \mathsf{+.f64}\left(wj, \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(wj, wj\right), \mathsf{*.f64}\left(x, x\right)\right), \mathsf{+.f64}\left(x, wj\right)\right), \mathsf{\_.f64}\left(-1, wj\right)\right)\right) \]

   9. Applied egg-rr 66.7%

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

   if -3.20000000000000026e-4 < wj < 1.3000000000000001e-8

   1. Initial program 79.4%

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

   3. Taylor expanded in wj around 0

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

      1. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(x, \color{blue}{\left(wj \cdot \left(wj \cdot \left(\left(1 + -1 \cdot \left(wj \cdot \left(1 + \left(-3 \cdot x + \left(-2 \cdot \left(-4 \cdot x + \frac{3}{2} \cdot x\right) + \frac{2}{3} \cdot x\right)\right)\right)\right)\right) - \left(-4 \cdot x + \frac{3}{2} \cdot x\right)\right) - 2 \cdot x\right)\right)}\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(x, \mathsf{*.f64}\left(wj, \color{blue}{\left(wj \cdot \left(\left(1 + -1 \cdot \left(wj \cdot \left(1 + \left(-3 \cdot x + \left(-2 \cdot \left(-4 \cdot x + \frac{3}{2} \cdot x\right) + \frac{2}{3} \cdot x\right)\right)\right)\right)\right) - \left(-4 \cdot x + \frac{3}{2} \cdot x\right)\right) - 2 \cdot x\right)}\right)\right) \]

      3. cancel-sign-sub-inv N/A

         \[\leadsto \mathsf{+.f64}\left(x, \mathsf{*.f64}\left(wj, \left(wj \cdot \left(\left(1 + -1 \cdot \left(wj \cdot \left(1 + \left(-3 \cdot x + \left(-2 \cdot \left(-4 \cdot x + \frac{3}{2} \cdot x\right) + \frac{2}{3} \cdot x\right)\right)\right)\right)\right) - \left(-4 \cdot x + \frac{3}{2} \cdot x\right)\right) + \color{blue}{\left(\mathsf{neg}\left(2\right)\right) \cdot x}\right)\right)\right) \]

      4. metadata-eval N/A

         \[\leadsto \mathsf{+.f64}\left(x, \mathsf{*.f64}\left(wj, \left(wj \cdot \left(\left(1 + -1 \cdot \left(wj \cdot \left(1 + \left(-3 \cdot x + \left(-2 \cdot \left(-4 \cdot x + \frac{3}{2} \cdot x\right) + \frac{2}{3} \cdot x\right)\right)\right)\right)\right) - \left(-4 \cdot x + \frac{3}{2} \cdot x\right)\right) + -2 \cdot x\right)\right)\right) \]

      5. +-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(x, \mathsf{*.f64}\left(wj, \left(-2 \cdot x + \color{blue}{wj \cdot \left(\left(1 + -1 \cdot \left(wj \cdot \left(1 + \left(-3 \cdot x + \left(-2 \cdot \left(-4 \cdot x + \frac{3}{2} \cdot x\right) + \frac{2}{3} \cdot x\right)\right)\right)\right)\right) - \left(-4 \cdot x + \frac{3}{2} \cdot x\right)\right)}\right)\right)\right) \]

      6. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(x, \mathsf{*.f64}\left(wj, \mathsf{+.f64}\left(\left(-2 \cdot x\right), \color{blue}{\left(wj \cdot \left(\left(1 + -1 \cdot \left(wj \cdot \left(1 + \left(-3 \cdot x + \left(-2 \cdot \left(-4 \cdot x + \frac{3}{2} \cdot x\right) + \frac{2}{3} \cdot x\right)\right)\right)\right)\right) - \left(-4 \cdot x + \frac{3}{2} \cdot x\right)\right)\right)}\right)\right)\right) \]

   5. Simplified 100.0%

      \[\leadsto \color{blue}{x + wj \cdot \left(x \cdot -2 + wj \cdot \left(\left(1 - wj \cdot \left(\left(x \cdot -3 + x \cdot 5\right) + \left(x \cdot 0.6666666666666666 + 1\right)\right)\right) + x \cdot 2.5\right)\right)} \]

   6. Taylor expanded in x around 0

      \[\leadsto \mathsf{+.f64}\left(x, \mathsf{*.f64}\left(wj, \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, -2\right), \color{blue}{\left(wj \cdot \left(1 - wj\right)\right)}\right)\right)\right) \]
   7. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(x, \mathsf{*.f64}\left(wj, \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, -2\right), \mathsf{*.f64}\left(wj, \color{blue}{\left(1 - wj\right)}\right)\right)\right)\right) \]

      2. --lowering--.f64 99.8%

         \[\leadsto \mathsf{+.f64}\left(x, \mathsf{*.f64}\left(wj, \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, -2\right), \mathsf{*.f64}\left(wj, \mathsf{\_.f64}\left(1, \color{blue}{wj}\right)\right)\right)\right)\right) \]

   8. Simplified 99.8%

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

   if 1.3000000000000001e-8 < wj

   1. Initial program 49.7%

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

      1. sub-neg N/A

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

      2. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(wj, \color{blue}{\left(\mathsf{neg}\left(\frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}}\right)\right)}\right) \]

      3. distribute-rgt1-in N/A

         \[\leadsto \mathsf{+.f64}\left(wj, \left(\mathsf{neg}\left(\frac{wj \cdot e^{wj} - x}{\left(wj + 1\right) \cdot e^{wj}}\right)\right)\right) \]

      4. associate-/l/ N/A

         \[\leadsto \mathsf{+.f64}\left(wj, \left(\mathsf{neg}\left(\frac{\frac{wj \cdot e^{wj} - x}{e^{wj}}}{wj + 1}\right)\right)\right) \]

      5. distribute-neg-frac2 N/A

         \[\leadsto \mathsf{+.f64}\left(wj, \left(\frac{\frac{wj \cdot e^{wj} - x}{e^{wj}}}{\color{blue}{\mathsf{neg}\left(\left(wj + 1\right)\right)}}\right)\right) \]

      6. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(wj, \mathsf{/.f64}\left(\left(\frac{wj \cdot e^{wj} - x}{e^{wj}}\right), \color{blue}{\left(\mathsf{neg}\left(\left(wj + 1\right)\right)\right)}\right)\right) \]

      7. div-sub N/A

         \[\leadsto \mathsf{+.f64}\left(wj, \mathsf{/.f64}\left(\left(\frac{wj \cdot e^{wj}}{e^{wj}} - \frac{x}{e^{wj}}\right), \left(\mathsf{neg}\left(\color{blue}{\left(wj + 1\right)}\right)\right)\right)\right) \]

      8. associate-/l* N/A

         \[\leadsto \mathsf{+.f64}\left(wj, \mathsf{/.f64}\left(\left(wj \cdot \frac{e^{wj}}{e^{wj}} - \frac{x}{e^{wj}}\right), \left(\mathsf{neg}\left(\left(\color{blue}{wj} + 1\right)\right)\right)\right)\right) \]

      9. *-inverses N/A

         \[\leadsto \mathsf{+.f64}\left(wj, \mathsf{/.f64}\left(\left(wj \cdot 1 - \frac{x}{e^{wj}}\right), \left(\mathsf{neg}\left(\left(wj + 1\right)\right)\right)\right)\right) \]

      10. *-rgt-identity N/A

          \[\leadsto \mathsf{+.f64}\left(wj, \mathsf{/.f64}\left(\left(wj - \frac{x}{e^{wj}}\right), \left(\mathsf{neg}\left(\left(\color{blue}{wj} + 1\right)\right)\right)\right)\right) \]

      11. --lowering--.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(wj, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(wj, \left(\frac{x}{e^{wj}}\right)\right), \left(\mathsf{neg}\left(\color{blue}{\left(wj + 1\right)}\right)\right)\right)\right) \]

      12. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(wj, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(wj, \mathsf{/.f64}\left(x, \left(e^{wj}\right)\right)\right), \left(\mathsf{neg}\left(\left(wj + \color{blue}{1}\right)\right)\right)\right)\right) \]

      13. exp-lowering-exp.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(wj, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(wj, \mathsf{/.f64}\left(x, \mathsf{exp.f64}\left(wj\right)\right)\right), \left(\mathsf{neg}\left(\left(wj + 1\right)\right)\right)\right)\right) \]

      14. +-commutative N/A

          \[\leadsto \mathsf{+.f64}\left(wj, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(wj, \mathsf{/.f64}\left(x, \mathsf{exp.f64}\left(wj\right)\right)\right), \left(\mathsf{neg}\left(\left(1 + wj\right)\right)\right)\right)\right) \]

      15. *-inverses N/A

          \[\leadsto \mathsf{+.f64}\left(wj, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(wj, \mathsf{/.f64}\left(x, \mathsf{exp.f64}\left(wj\right)\right)\right), \left(\mathsf{neg}\left(\left(\frac{e^{wj}}{e^{wj}} + wj\right)\right)\right)\right)\right) \]

      16. distribute-neg-in N/A

          \[\leadsto \mathsf{+.f64}\left(wj, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(wj, \mathsf{/.f64}\left(x, \mathsf{exp.f64}\left(wj\right)\right)\right), \left(\left(\mathsf{neg}\left(\frac{e^{wj}}{e^{wj}}\right)\right) + \color{blue}{\left(\mathsf{neg}\left(wj\right)\right)}\right)\right)\right) \]

   3. Simplified 99.7%

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

   5. Taylor expanded in wj around 0

      \[\leadsto \mathsf{+.f64}\left(wj, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(wj, \mathsf{/.f64}\left(x, \color{blue}{\left(1 + wj \cdot \left(1 + \frac{1}{2} \cdot wj\right)\right)}\right)\right), \mathsf{\_.f64}\left(-1, wj\right)\right)\right) \]
   6. Step-by-step derivation

      1. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(wj, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(wj, \mathsf{/.f64}\left(x, \mathsf{+.f64}\left(1, \left(wj \cdot \left(1 + \frac{1}{2} \cdot wj\right)\right)\right)\right)\right), \mathsf{\_.f64}\left(-1, wj\right)\right)\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(wj, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(wj, \mathsf{/.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(wj, \left(1 + \frac{1}{2} \cdot wj\right)\right)\right)\right)\right), \mathsf{\_.f64}\left(-1, wj\right)\right)\right) \]

      3. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(wj, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(wj, \mathsf{/.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(wj, \mathsf{+.f64}\left(1, \left(\frac{1}{2} \cdot wj\right)\right)\right)\right)\right)\right), \mathsf{\_.f64}\left(-1, wj\right)\right)\right) \]

      4. *-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(wj, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(wj, \mathsf{/.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(wj, \mathsf{+.f64}\left(1, \left(wj \cdot \frac{1}{2}\right)\right)\right)\right)\right)\right), \mathsf{\_.f64}\left(-1, wj\right)\right)\right) \]

      5. *-lowering-*.f64 95.1%

         \[\leadsto \mathsf{+.f64}\left(wj, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(wj, \mathsf{/.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(wj, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(wj, \frac{1}{2}\right)\right)\right)\right)\right)\right), \mathsf{\_.f64}\left(-1, wj\right)\right)\right) \]

   7. Simplified 95.1%

      \[\leadsto wj + \frac{wj - \frac{x}{\color{blue}{1 + wj \cdot \left(1 + wj \cdot 0.5\right)}}}{-1 - wj} \]
3. Recombined 3 regimes into one program.

4. Final simplification 98.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;wj \leq -0.00032:\\ \;\;\;\;wj + \frac{\frac{wj \cdot wj - x \cdot x}{wj + x}}{-1 - wj}\\ \mathbf{elif}\;wj \leq 1.3 \cdot 10^{-8}:\\ \;\;\;\;x + wj \cdot \left(x \cdot -2 + wj \cdot \left(1 - wj\right)\right)\\ \mathbf{else}:\\ \;\;\;\;wj + \frac{wj + \frac{x}{-1 + wj \cdot \left(-1 - wj \cdot 0.5\right)}}{-1 - wj}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 97.1% accurate, 11.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;wj \leq 1.3 \cdot 10^{-8}:\\ \;\;\;\;x + x \cdot \left(wj \cdot \left(-2 + wj \cdot \left(\frac{1}{x} + \left(2.5 + wj \cdot \left(-2.6666666666666665 + \frac{-1}{x}\right)\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;wj + \frac{wj + \frac{x}{-1 + wj \cdot \left(-1 - wj \cdot \left(0.5 + wj \cdot 0.16666666666666666\right)\right)}}{-1 - wj}\\ \end{array} \end{array} \]

FPCore

(FPCore (wj x)
 :precision binary64
 (if (<= wj 1.3e-8)
   (+
    x
    (*
     x
     (*
      wj
      (+
       -2.0
       (*
        wj
        (+ (/ 1.0 x) (+ 2.5 (* wj (+ -2.6666666666666665 (/ -1.0 x))))))))))
   (+
    wj
    (/
     (+
      wj
      (/ x (+ -1.0 (* wj (- -1.0 (* wj (+ 0.5 (* wj 0.16666666666666666))))))))
     (- -1.0 wj)))))

C

double code(double wj, double x) {
	double tmp;
	if (wj <= 1.3e-8) {
		tmp = x + (x * (wj * (-2.0 + (wj * ((1.0 / x) + (2.5 + (wj * (-2.6666666666666665 + (-1.0 / x)))))))));
	} else {
		tmp = wj + ((wj + (x / (-1.0 + (wj * (-1.0 - (wj * (0.5 + (wj * 0.16666666666666666)))))))) / (-1.0 - wj));
	}
	return tmp;
}

Fortran

real(8) function code(wj, x)
    real(8), intent (in) :: wj
    real(8), intent (in) :: x
    real(8) :: tmp
    if (wj <= 1.3d-8) then
        tmp = x + (x * (wj * ((-2.0d0) + (wj * ((1.0d0 / x) + (2.5d0 + (wj * ((-2.6666666666666665d0) + ((-1.0d0) / x)))))))))
    else
        tmp = wj + ((wj + (x / ((-1.0d0) + (wj * ((-1.0d0) - (wj * (0.5d0 + (wj * 0.16666666666666666d0)))))))) / ((-1.0d0) - wj))
    end if
    code = tmp
end function

Java

public static double code(double wj, double x) {
	double tmp;
	if (wj <= 1.3e-8) {
		tmp = x + (x * (wj * (-2.0 + (wj * ((1.0 / x) + (2.5 + (wj * (-2.6666666666666665 + (-1.0 / x)))))))));
	} else {
		tmp = wj + ((wj + (x / (-1.0 + (wj * (-1.0 - (wj * (0.5 + (wj * 0.16666666666666666)))))))) / (-1.0 - wj));
	}
	return tmp;
}

Python

def code(wj, x):
	tmp = 0
	if wj <= 1.3e-8:
		tmp = x + (x * (wj * (-2.0 + (wj * ((1.0 / x) + (2.5 + (wj * (-2.6666666666666665 + (-1.0 / x)))))))))
	else:
		tmp = wj + ((wj + (x / (-1.0 + (wj * (-1.0 - (wj * (0.5 + (wj * 0.16666666666666666)))))))) / (-1.0 - wj))
	return tmp

Julia

function code(wj, x)
	tmp = 0.0
	if (wj <= 1.3e-8)
		tmp = Float64(x + Float64(x * Float64(wj * Float64(-2.0 + Float64(wj * Float64(Float64(1.0 / x) + Float64(2.5 + Float64(wj * Float64(-2.6666666666666665 + Float64(-1.0 / x))))))))));
	else
		tmp = Float64(wj + Float64(Float64(wj + Float64(x / Float64(-1.0 + Float64(wj * Float64(-1.0 - Float64(wj * Float64(0.5 + Float64(wj * 0.16666666666666666)))))))) / Float64(-1.0 - wj)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(wj, x)
	tmp = 0.0;
	if (wj <= 1.3e-8)
		tmp = x + (x * (wj * (-2.0 + (wj * ((1.0 / x) + (2.5 + (wj * (-2.6666666666666665 + (-1.0 / x)))))))));
	else
		tmp = wj + ((wj + (x / (-1.0 + (wj * (-1.0 - (wj * (0.5 + (wj * 0.16666666666666666)))))))) / (-1.0 - wj));
	end
	tmp_2 = tmp;
end

Wolfram

code[wj_, x_] := If[LessEqual[wj, 1.3e-8], N[(x + N[(x * N[(wj * N[(-2.0 + N[(wj * N[(N[(1.0 / x), $MachinePrecision] + N[(2.5 + N[(wj * N[(-2.6666666666666665 + N[(-1.0 / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(wj + N[(N[(wj + N[(x / N[(-1.0 + N[(wj * N[(-1.0 - N[(wj * N[(0.5 + N[(wj * 0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(-1.0 - wj), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if wj < 1.3000000000000001e-8

   1. Initial program 78.4%

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

   3. Taylor expanded in wj around 0

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

      1. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(x, \color{blue}{\left(wj \cdot \left(wj \cdot \left(\left(1 + -1 \cdot \left(wj \cdot \left(1 + \left(-3 \cdot x + \left(-2 \cdot \left(-4 \cdot x + \frac{3}{2} \cdot x\right) + \frac{2}{3} \cdot x\right)\right)\right)\right)\right) - \left(-4 \cdot x + \frac{3}{2} \cdot x\right)\right) - 2 \cdot x\right)\right)}\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(x, \mathsf{*.f64}\left(wj, \color{blue}{\left(wj \cdot \left(\left(1 + -1 \cdot \left(wj \cdot \left(1 + \left(-3 \cdot x + \left(-2 \cdot \left(-4 \cdot x + \frac{3}{2} \cdot x\right) + \frac{2}{3} \cdot x\right)\right)\right)\right)\right) - \left(-4 \cdot x + \frac{3}{2} \cdot x\right)\right) - 2 \cdot x\right)}\right)\right) \]

      3. cancel-sign-sub-inv N/A

         \[\leadsto \mathsf{+.f64}\left(x, \mathsf{*.f64}\left(wj, \left(wj \cdot \left(\left(1 + -1 \cdot \left(wj \cdot \left(1 + \left(-3 \cdot x + \left(-2 \cdot \left(-4 \cdot x + \frac{3}{2} \cdot x\right) + \frac{2}{3} \cdot x\right)\right)\right)\right)\right) - \left(-4 \cdot x + \frac{3}{2} \cdot x\right)\right) + \color{blue}{\left(\mathsf{neg}\left(2\right)\right) \cdot x}\right)\right)\right) \]

      4. metadata-eval N/A

         \[\leadsto \mathsf{+.f64}\left(x, \mathsf{*.f64}\left(wj, \left(wj \cdot \left(\left(1 + -1 \cdot \left(wj \cdot \left(1 + \left(-3 \cdot x + \left(-2 \cdot \left(-4 \cdot x + \frac{3}{2} \cdot x\right) + \frac{2}{3} \cdot x\right)\right)\right)\right)\right) - \left(-4 \cdot x + \frac{3}{2} \cdot x\right)\right) + -2 \cdot x\right)\right)\right) \]

      5. +-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(x, \mathsf{*.f64}\left(wj, \left(-2 \cdot x + \color{blue}{wj \cdot \left(\left(1 + -1 \cdot \left(wj \cdot \left(1 + \left(-3 \cdot x + \left(-2 \cdot \left(-4 \cdot x + \frac{3}{2} \cdot x\right) + \frac{2}{3} \cdot x\right)\right)\right)\right)\right) - \left(-4 \cdot x + \frac{3}{2} \cdot x\right)\right)}\right)\right)\right) \]

      6. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(x, \mathsf{*.f64}\left(wj, \mathsf{+.f64}\left(\left(-2 \cdot x\right), \color{blue}{\left(wj \cdot \left(\left(1 + -1 \cdot \left(wj \cdot \left(1 + \left(-3 \cdot x + \left(-2 \cdot \left(-4 \cdot x + \frac{3}{2} \cdot x\right) + \frac{2}{3} \cdot x\right)\right)\right)\right)\right) - \left(-4 \cdot x + \frac{3}{2} \cdot x\right)\right)\right)}\right)\right)\right) \]

   5. Simplified 97.3%

      \[\leadsto \color{blue}{x + wj \cdot \left(x \cdot -2 + wj \cdot \left(\left(1 - wj \cdot \left(\left(x \cdot -3 + x \cdot 5\right) + \left(x \cdot 0.6666666666666666 + 1\right)\right)\right) + x \cdot 2.5\right)\right)} \]

   6. Taylor expanded in x around inf

      \[\leadsto \mathsf{+.f64}\left(x, \color{blue}{\left(x \cdot \left(wj \cdot \left(wj \cdot \left(\frac{5}{2} - \frac{8}{3} \cdot wj\right) - 2\right) + \frac{{wj}^{2} \cdot \left(1 - wj\right)}{x}\right)\right)}\right) \]
   7. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(x, \mathsf{*.f64}\left(x, \color{blue}{\left(wj \cdot \left(wj \cdot \left(\frac{5}{2} - \frac{8}{3} \cdot wj\right) - 2\right) + \frac{{wj}^{2} \cdot \left(1 - wj\right)}{x}\right)}\right)\right) \]

      2. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\left(wj \cdot \left(wj \cdot \left(\frac{5}{2} - \frac{8}{3} \cdot wj\right) - 2\right)\right), \color{blue}{\left(\frac{{wj}^{2} \cdot \left(1 - wj\right)}{x}\right)}\right)\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\mathsf{*.f64}\left(wj, \left(wj \cdot \left(\frac{5}{2} - \frac{8}{3} \cdot wj\right) - 2\right)\right), \left(\frac{\color{blue}{{wj}^{2} \cdot \left(1 - wj\right)}}{x}\right)\right)\right)\right) \]

      4. sub-neg N/A

         \[\leadsto \mathsf{+.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\mathsf{*.f64}\left(wj, \left(wj \cdot \left(\frac{5}{2} - \frac{8}{3} \cdot wj\right) + \left(\mathsf{neg}\left(2\right)\right)\right)\right), \left(\frac{{wj}^{2} \cdot \color{blue}{\left(1 - wj\right)}}{x}\right)\right)\right)\right) \]

      5. metadata-eval N/A

         \[\leadsto \mathsf{+.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\mathsf{*.f64}\left(wj, \left(wj \cdot \left(\frac{5}{2} - \frac{8}{3} \cdot wj\right) + -2\right)\right), \left(\frac{{wj}^{2} \cdot \left(1 - \color{blue}{wj}\right)}{x}\right)\right)\right)\right) \]

      6. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\mathsf{*.f64}\left(wj, \mathsf{+.f64}\left(\left(wj \cdot \left(\frac{5}{2} - \frac{8}{3} \cdot wj\right)\right), -2\right)\right), \left(\frac{{wj}^{2} \cdot \color{blue}{\left(1 - wj\right)}}{x}\right)\right)\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\mathsf{*.f64}\left(wj, \mathsf{+.f64}\left(\mathsf{*.f64}\left(wj, \left(\frac{5}{2} - \frac{8}{3} \cdot wj\right)\right), -2\right)\right), \left(\frac{{wj}^{2} \cdot \left(\color{blue}{1} - wj\right)}{x}\right)\right)\right)\right) \]

      8. cancel-sign-sub-inv N/A

         \[\leadsto \mathsf{+.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\mathsf{*.f64}\left(wj, \mathsf{+.f64}\left(\mathsf{*.f64}\left(wj, \left(\frac{5}{2} + \left(\mathsf{neg}\left(\frac{8}{3}\right)\right) \cdot wj\right)\right), -2\right)\right), \left(\frac{{wj}^{2} \cdot \left(1 - wj\right)}{x}\right)\right)\right)\right) \]

      9. metadata-eval N/A

         \[\leadsto \mathsf{+.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\mathsf{*.f64}\left(wj, \mathsf{+.f64}\left(\mathsf{*.f64}\left(wj, \left(\frac{5}{2} + \frac{-8}{3} \cdot wj\right)\right), -2\right)\right), \left(\frac{{wj}^{2} \cdot \left(1 - wj\right)}{x}\right)\right)\right)\right) \]

      10. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\mathsf{*.f64}\left(wj, \mathsf{+.f64}\left(\mathsf{*.f64}\left(wj, \mathsf{+.f64}\left(\frac{5}{2}, \left(\frac{-8}{3} \cdot wj\right)\right)\right), -2\right)\right), \left(\frac{{wj}^{2} \cdot \left(1 - wj\right)}{x}\right)\right)\right)\right) \]

      11. *-commutative N/A

          \[\leadsto \mathsf{+.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\mathsf{*.f64}\left(wj, \mathsf{+.f64}\left(\mathsf{*.f64}\left(wj, \mathsf{+.f64}\left(\frac{5}{2}, \left(wj \cdot \frac{-8}{3}\right)\right)\right), -2\right)\right), \left(\frac{{wj}^{2} \cdot \left(1 - wj\right)}{x}\right)\right)\right)\right) \]

      12. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\mathsf{*.f64}\left(wj, \mathsf{+.f64}\left(\mathsf{*.f64}\left(wj, \mathsf{+.f64}\left(\frac{5}{2}, \mathsf{*.f64}\left(wj, \frac{-8}{3}\right)\right)\right), -2\right)\right), \left(\frac{{wj}^{2} \cdot \left(1 - wj\right)}{x}\right)\right)\right)\right) \]

      13. associate-/l* N/A

          \[\leadsto \mathsf{+.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\mathsf{*.f64}\left(wj, \mathsf{+.f64}\left(\mathsf{*.f64}\left(wj, \mathsf{+.f64}\left(\frac{5}{2}, \mathsf{*.f64}\left(wj, \frac{-8}{3}\right)\right)\right), -2\right)\right), \left({wj}^{2} \cdot \color{blue}{\frac{1 - wj}{x}}\right)\right)\right)\right) \]

      14. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\mathsf{*.f64}\left(wj, \mathsf{+.f64}\left(\mathsf{*.f64}\left(wj, \mathsf{+.f64}\left(\frac{5}{2}, \mathsf{*.f64}\left(wj, \frac{-8}{3}\right)\right)\right), -2\right)\right), \mathsf{*.f64}\left(\left({wj}^{2}\right), \color{blue}{\left(\frac{1 - wj}{x}\right)}\right)\right)\right)\right) \]

      15. unpow2 N/A

          \[\leadsto \mathsf{+.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\mathsf{*.f64}\left(wj, \mathsf{+.f64}\left(\mathsf{*.f64}\left(wj, \mathsf{+.f64}\left(\frac{5}{2}, \mathsf{*.f64}\left(wj, \frac{-8}{3}\right)\right)\right), -2\right)\right), \mathsf{*.f64}\left(\left(wj \cdot wj\right), \left(\frac{\color{blue}{1 - wj}}{x}\right)\right)\right)\right)\right) \]

      16. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\mathsf{*.f64}\left(wj, \mathsf{+.f64}\left(\mathsf{*.f64}\left(wj, \mathsf{+.f64}\left(\frac{5}{2}, \mathsf{*.f64}\left(wj, \frac{-8}{3}\right)\right)\right), -2\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(wj, wj\right), \left(\frac{\color{blue}{1 - wj}}{x}\right)\right)\right)\right)\right) \]

      17. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\mathsf{*.f64}\left(wj, \mathsf{+.f64}\left(\mathsf{*.f64}\left(wj, \mathsf{+.f64}\left(\frac{5}{2}, \mathsf{*.f64}\left(wj, \frac{-8}{3}\right)\right)\right), -2\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(wj, wj\right), \mathsf{/.f64}\left(\left(1 - wj\right), \color{blue}{x}\right)\right)\right)\right)\right) \]

      18. --lowering--.f64 97.2%

          \[\leadsto \mathsf{+.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\mathsf{*.f64}\left(wj, \mathsf{+.f64}\left(\mathsf{*.f64}\left(wj, \mathsf{+.f64}\left(\frac{5}{2}, \mathsf{*.f64}\left(wj, \frac{-8}{3}\right)\right)\right), -2\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(wj, wj\right), \mathsf{/.f64}\left(\mathsf{\_.f64}\left(1, wj\right), x\right)\right)\right)\right)\right) \]

   8. Simplified 97.2%

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

   9. Taylor expanded in wj around 0

      \[\leadsto \mathsf{+.f64}\left(x, \mathsf{*.f64}\left(x, \color{blue}{\left(wj \cdot \left(wj \cdot \left(\frac{5}{2} + \left(-1 \cdot \left(wj \cdot \left(\frac{8}{3} + \frac{1}{x}\right)\right) + \frac{1}{x}\right)\right) - 2\right)\right)}\right)\right) \]
   10. Step-by-step derivation

       1. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(wj, \color{blue}{\left(wj \cdot \left(\frac{5}{2} + \left(-1 \cdot \left(wj \cdot \left(\frac{8}{3} + \frac{1}{x}\right)\right) + \frac{1}{x}\right)\right) - 2\right)}\right)\right)\right) \]

       2. sub-neg N/A

          \[\leadsto \mathsf{+.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(wj, \left(wj \cdot \left(\frac{5}{2} + \left(-1 \cdot \left(wj \cdot \left(\frac{8}{3} + \frac{1}{x}\right)\right) + \frac{1}{x}\right)\right) + \color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)\right)\right)\right) \]

       3. metadata-eval N/A

          \[\leadsto \mathsf{+.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(wj, \left(wj \cdot \left(\frac{5}{2} + \left(-1 \cdot \left(wj \cdot \left(\frac{8}{3} + \frac{1}{x}\right)\right) + \frac{1}{x}\right)\right) + -2\right)\right)\right)\right) \]

       4. +-commutative N/A

          \[\leadsto \mathsf{+.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(wj, \left(-2 + \color{blue}{wj \cdot \left(\frac{5}{2} + \left(-1 \cdot \left(wj \cdot \left(\frac{8}{3} + \frac{1}{x}\right)\right) + \frac{1}{x}\right)\right)}\right)\right)\right)\right) \]

       5. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(wj, \mathsf{+.f64}\left(-2, \color{blue}{\left(wj \cdot \left(\frac{5}{2} + \left(-1 \cdot \left(wj \cdot \left(\frac{8}{3} + \frac{1}{x}\right)\right) + \frac{1}{x}\right)\right)\right)}\right)\right)\right)\right) \]

       6. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(wj, \mathsf{+.f64}\left(-2, \mathsf{*.f64}\left(wj, \color{blue}{\left(\frac{5}{2} + \left(-1 \cdot \left(wj \cdot \left(\frac{8}{3} + \frac{1}{x}\right)\right) + \frac{1}{x}\right)\right)}\right)\right)\right)\right)\right) \]

       7. associate-+r+ N/A

          \[\leadsto \mathsf{+.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(wj, \mathsf{+.f64}\left(-2, \mathsf{*.f64}\left(wj, \left(\left(\frac{5}{2} + -1 \cdot \left(wj \cdot \left(\frac{8}{3} + \frac{1}{x}\right)\right)\right) + \color{blue}{\frac{1}{x}}\right)\right)\right)\right)\right)\right) \]

       8. +-commutative N/A

          \[\leadsto \mathsf{+.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(wj, \mathsf{+.f64}\left(-2, \mathsf{*.f64}\left(wj, \left(\frac{1}{x} + \color{blue}{\left(\frac{5}{2} + -1 \cdot \left(wj \cdot \left(\frac{8}{3} + \frac{1}{x}\right)\right)\right)}\right)\right)\right)\right)\right)\right) \]

       9. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(wj, \mathsf{+.f64}\left(-2, \mathsf{*.f64}\left(wj, \mathsf{+.f64}\left(\left(\frac{1}{x}\right), \color{blue}{\left(\frac{5}{2} + -1 \cdot \left(wj \cdot \left(\frac{8}{3} + \frac{1}{x}\right)\right)\right)}\right)\right)\right)\right)\right)\right) \]

       10. /-lowering-/.f64 N/A

           \[\leadsto \mathsf{+.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(wj, \mathsf{+.f64}\left(-2, \mathsf{*.f64}\left(wj, \mathsf{+.f64}\left(\mathsf{/.f64}\left(1, x\right), \left(\color{blue}{\frac{5}{2}} + -1 \cdot \left(wj \cdot \left(\frac{8}{3} + \frac{1}{x}\right)\right)\right)\right)\right)\right)\right)\right)\right) \]

       11. +-lowering-+.f64 N/A

           \[\leadsto \mathsf{+.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(wj, \mathsf{+.f64}\left(-2, \mathsf{*.f64}\left(wj, \mathsf{+.f64}\left(\mathsf{/.f64}\left(1, x\right), \mathsf{+.f64}\left(\frac{5}{2}, \color{blue}{\left(-1 \cdot \left(wj \cdot \left(\frac{8}{3} + \frac{1}{x}\right)\right)\right)}\right)\right)\right)\right)\right)\right)\right) \]

       12. mul-1-neg N/A

           \[\leadsto \mathsf{+.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(wj, \mathsf{+.f64}\left(-2, \mathsf{*.f64}\left(wj, \mathsf{+.f64}\left(\mathsf{/.f64}\left(1, x\right), \mathsf{+.f64}\left(\frac{5}{2}, \left(\mathsf{neg}\left(wj \cdot \left(\frac{8}{3} + \frac{1}{x}\right)\right)\right)\right)\right)\right)\right)\right)\right)\right) \]

       13. distribute-rgt-neg-in N/A

           \[\leadsto \mathsf{+.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(wj, \mathsf{+.f64}\left(-2, \mathsf{*.f64}\left(wj, \mathsf{+.f64}\left(\mathsf{/.f64}\left(1, x\right), \mathsf{+.f64}\left(\frac{5}{2}, \left(wj \cdot \color{blue}{\left(\mathsf{neg}\left(\left(\frac{8}{3} + \frac{1}{x}\right)\right)\right)}\right)\right)\right)\right)\right)\right)\right)\right) \]

       14. *-lowering-*.f64 N/A

           \[\leadsto \mathsf{+.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(wj, \mathsf{+.f64}\left(-2, \mathsf{*.f64}\left(wj, \mathsf{+.f64}\left(\mathsf{/.f64}\left(1, x\right), \mathsf{+.f64}\left(\frac{5}{2}, \mathsf{*.f64}\left(wj, \color{blue}{\left(\mathsf{neg}\left(\left(\frac{8}{3} + \frac{1}{x}\right)\right)\right)}\right)\right)\right)\right)\right)\right)\right)\right) \]

       15. distribute-neg-in N/A

           \[\leadsto \mathsf{+.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(wj, \mathsf{+.f64}\left(-2, \mathsf{*.f64}\left(wj, \mathsf{+.f64}\left(\mathsf{/.f64}\left(1, x\right), \mathsf{+.f64}\left(\frac{5}{2}, \mathsf{*.f64}\left(wj, \left(\left(\mathsf{neg}\left(\frac{8}{3}\right)\right) + \color{blue}{\left(\mathsf{neg}\left(\frac{1}{x}\right)\right)}\right)\right)\right)\right)\right)\right)\right)\right)\right) \]

       16. metadata-eval N/A

           \[\leadsto \mathsf{+.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(wj, \mathsf{+.f64}\left(-2, \mathsf{*.f64}\left(wj, \mathsf{+.f64}\left(\mathsf{/.f64}\left(1, x\right), \mathsf{+.f64}\left(\frac{5}{2}, \mathsf{*.f64}\left(wj, \left(\frac{-8}{3} + \left(\mathsf{neg}\left(\color{blue}{\frac{1}{x}}\right)\right)\right)\right)\right)\right)\right)\right)\right)\right)\right) \]

       17. +-lowering-+.f64 N/A

           \[\leadsto \mathsf{+.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(wj, \mathsf{+.f64}\left(-2, \mathsf{*.f64}\left(wj, \mathsf{+.f64}\left(\mathsf{/.f64}\left(1, x\right), \mathsf{+.f64}\left(\frac{5}{2}, \mathsf{*.f64}\left(wj, \mathsf{+.f64}\left(\frac{-8}{3}, \color{blue}{\left(\mathsf{neg}\left(\frac{1}{x}\right)\right)}\right)\right)\right)\right)\right)\right)\right)\right)\right) \]

       18. distribute-neg-frac N/A

           \[\leadsto \mathsf{+.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(wj, \mathsf{+.f64}\left(-2, \mathsf{*.f64}\left(wj, \mathsf{+.f64}\left(\mathsf{/.f64}\left(1, x\right), \mathsf{+.f64}\left(\frac{5}{2}, \mathsf{*.f64}\left(wj, \mathsf{+.f64}\left(\frac{-8}{3}, \left(\frac{\mathsf{neg}\left(1\right)}{\color{blue}{x}}\right)\right)\right)\right)\right)\right)\right)\right)\right)\right) \]

       19. metadata-eval N/A

           \[\leadsto \mathsf{+.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(wj, \mathsf{+.f64}\left(-2, \mathsf{*.f64}\left(wj, \mathsf{+.f64}\left(\mathsf{/.f64}\left(1, x\right), \mathsf{+.f64}\left(\frac{5}{2}, \mathsf{*.f64}\left(wj, \mathsf{+.f64}\left(\frac{-8}{3}, \left(\frac{-1}{x}\right)\right)\right)\right)\right)\right)\right)\right)\right)\right) \]

       20. /-lowering-/.f64 97.2%

           \[\leadsto \mathsf{+.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(wj, \mathsf{+.f64}\left(-2, \mathsf{*.f64}\left(wj, \mathsf{+.f64}\left(\mathsf{/.f64}\left(1, x\right), \mathsf{+.f64}\left(\frac{5}{2}, \mathsf{*.f64}\left(wj, \mathsf{+.f64}\left(\frac{-8}{3}, \mathsf{/.f64}\left(-1, \color{blue}{x}\right)\right)\right)\right)\right)\right)\right)\right)\right)\right) \]

   11. Simplified 97.2%

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

   if 1.3000000000000001e-8 < wj

   1. Initial program 49.7%

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

      1. sub-neg N/A

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

      2. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(wj, \color{blue}{\left(\mathsf{neg}\left(\frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}}\right)\right)}\right) \]

      3. distribute-rgt1-in N/A

         \[\leadsto \mathsf{+.f64}\left(wj, \left(\mathsf{neg}\left(\frac{wj \cdot e^{wj} - x}{\left(wj + 1\right) \cdot e^{wj}}\right)\right)\right) \]

      4. associate-/l/ N/A

         \[\leadsto \mathsf{+.f64}\left(wj, \left(\mathsf{neg}\left(\frac{\frac{wj \cdot e^{wj} - x}{e^{wj}}}{wj + 1}\right)\right)\right) \]

      5. distribute-neg-frac2 N/A

         \[\leadsto \mathsf{+.f64}\left(wj, \left(\frac{\frac{wj \cdot e^{wj} - x}{e^{wj}}}{\color{blue}{\mathsf{neg}\left(\left(wj + 1\right)\right)}}\right)\right) \]

      6. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(wj, \mathsf{/.f64}\left(\left(\frac{wj \cdot e^{wj} - x}{e^{wj}}\right), \color{blue}{\left(\mathsf{neg}\left(\left(wj + 1\right)\right)\right)}\right)\right) \]

      7. div-sub N/A

         \[\leadsto \mathsf{+.f64}\left(wj, \mathsf{/.f64}\left(\left(\frac{wj \cdot e^{wj}}{e^{wj}} - \frac{x}{e^{wj}}\right), \left(\mathsf{neg}\left(\color{blue}{\left(wj + 1\right)}\right)\right)\right)\right) \]

      8. associate-/l* N/A

         \[\leadsto \mathsf{+.f64}\left(wj, \mathsf{/.f64}\left(\left(wj \cdot \frac{e^{wj}}{e^{wj}} - \frac{x}{e^{wj}}\right), \left(\mathsf{neg}\left(\left(\color{blue}{wj} + 1\right)\right)\right)\right)\right) \]

      9. *-inverses N/A

         \[\leadsto \mathsf{+.f64}\left(wj, \mathsf{/.f64}\left(\left(wj \cdot 1 - \frac{x}{e^{wj}}\right), \left(\mathsf{neg}\left(\left(wj + 1\right)\right)\right)\right)\right) \]

      10. *-rgt-identity N/A

          \[\leadsto \mathsf{+.f64}\left(wj, \mathsf{/.f64}\left(\left(wj - \frac{x}{e^{wj}}\right), \left(\mathsf{neg}\left(\left(\color{blue}{wj} + 1\right)\right)\right)\right)\right) \]

      11. --lowering--.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(wj, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(wj, \left(\frac{x}{e^{wj}}\right)\right), \left(\mathsf{neg}\left(\color{blue}{\left(wj + 1\right)}\right)\right)\right)\right) \]

      12. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(wj, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(wj, \mathsf{/.f64}\left(x, \left(e^{wj}\right)\right)\right), \left(\mathsf{neg}\left(\left(wj + \color{blue}{1}\right)\right)\right)\right)\right) \]

      13. exp-lowering-exp.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(wj, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(wj, \mathsf{/.f64}\left(x, \mathsf{exp.f64}\left(wj\right)\right)\right), \left(\mathsf{neg}\left(\left(wj + 1\right)\right)\right)\right)\right) \]

      14. +-commutative N/A

          \[\leadsto \mathsf{+.f64}\left(wj, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(wj, \mathsf{/.f64}\left(x, \mathsf{exp.f64}\left(wj\right)\right)\right), \left(\mathsf{neg}\left(\left(1 + wj\right)\right)\right)\right)\right) \]

      15. *-inverses N/A

          \[\leadsto \mathsf{+.f64}\left(wj, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(wj, \mathsf{/.f64}\left(x, \mathsf{exp.f64}\left(wj\right)\right)\right), \left(\mathsf{neg}\left(\left(\frac{e^{wj}}{e^{wj}} + wj\right)\right)\right)\right)\right) \]

      16. distribute-neg-in N/A

          \[\leadsto \mathsf{+.f64}\left(wj, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(wj, \mathsf{/.f64}\left(x, \mathsf{exp.f64}\left(wj\right)\right)\right), \left(\left(\mathsf{neg}\left(\frac{e^{wj}}{e^{wj}}\right)\right) + \color{blue}{\left(\mathsf{neg}\left(wj\right)\right)}\right)\right)\right) \]

   3. Simplified 99.7%

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

   5. Taylor expanded in wj around 0

      \[\leadsto \mathsf{+.f64}\left(wj, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(wj, \mathsf{/.f64}\left(x, \color{blue}{\left(1 + wj \cdot \left(1 + wj \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot wj\right)\right)\right)}\right)\right), \mathsf{\_.f64}\left(-1, wj\right)\right)\right) \]
   6. Step-by-step derivation

      1. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(wj, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(wj, \mathsf{/.f64}\left(x, \mathsf{+.f64}\left(1, \left(wj \cdot \left(1 + wj \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot wj\right)\right)\right)\right)\right)\right), \mathsf{\_.f64}\left(-1, wj\right)\right)\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(wj, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(wj, \mathsf{/.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(wj, \left(1 + wj \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot wj\right)\right)\right)\right)\right)\right), \mathsf{\_.f64}\left(-1, wj\right)\right)\right) \]

      3. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(wj, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(wj, \mathsf{/.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(wj, \mathsf{+.f64}\left(1, \left(wj \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot wj\right)\right)\right)\right)\right)\right)\right), \mathsf{\_.f64}\left(-1, wj\right)\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(wj, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(wj, \mathsf{/.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(wj, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(wj, \left(\frac{1}{2} + \frac{1}{6} \cdot wj\right)\right)\right)\right)\right)\right)\right), \mathsf{\_.f64}\left(-1, wj\right)\right)\right) \]

      5. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(wj, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(wj, \mathsf{/.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(wj, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(wj, \mathsf{+.f64}\left(\frac{1}{2}, \left(\frac{1}{6} \cdot wj\right)\right)\right)\right)\right)\right)\right)\right), \mathsf{\_.f64}\left(-1, wj\right)\right)\right) \]

      6. *-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(wj, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(wj, \mathsf{/.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(wj, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(wj, \mathsf{+.f64}\left(\frac{1}{2}, \left(wj \cdot \frac{1}{6}\right)\right)\right)\right)\right)\right)\right)\right), \mathsf{\_.f64}\left(-1, wj\right)\right)\right) \]

      7. *-lowering-*.f64 98.3%

         \[\leadsto \mathsf{+.f64}\left(wj, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(wj, \mathsf{/.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(wj, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(wj, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(wj, \frac{1}{6}\right)\right)\right)\right)\right)\right)\right)\right), \mathsf{\_.f64}\left(-1, wj\right)\right)\right) \]

   7. Simplified 98.3%

      \[\leadsto wj + \frac{wj - \frac{x}{\color{blue}{1 + wj \cdot \left(1 + wj \cdot \left(0.5 + wj \cdot 0.16666666666666666\right)\right)}}}{-1 - wj} \]
3. Recombined 2 regimes into one program.

4. Final simplification 97.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;wj \leq 1.3 \cdot 10^{-8}:\\ \;\;\;\;x + x \cdot \left(wj \cdot \left(-2 + wj \cdot \left(\frac{1}{x} + \left(2.5 + wj \cdot \left(-2.6666666666666665 + \frac{-1}{x}\right)\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;wj + \frac{wj + \frac{x}{-1 + wj \cdot \left(-1 - wj \cdot \left(0.5 + wj \cdot 0.16666666666666666\right)\right)}}{-1 - wj}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 97.1% accurate, 11.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;wj \leq 1.3 \cdot 10^{-8}:\\ \;\;\;\;x + x \cdot \left(wj \cdot \left(-2 + wj \cdot \left(\frac{1}{x} + \left(2.5 - \frac{wj}{x}\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;wj + \frac{wj + \frac{x}{-1 + wj \cdot \left(-1 - wj \cdot \left(0.5 + wj \cdot 0.16666666666666666\right)\right)}}{-1 - wj}\\ \end{array} \end{array} \]

FPCore

(FPCore (wj x)
 :precision binary64
 (if (<= wj 1.3e-8)
   (+ x (* x (* wj (+ -2.0 (* wj (+ (/ 1.0 x) (- 2.5 (/ wj x))))))))
   (+
    wj
    (/
     (+
      wj
      (/ x (+ -1.0 (* wj (- -1.0 (* wj (+ 0.5 (* wj 0.16666666666666666))))))))
     (- -1.0 wj)))))

C

double code(double wj, double x) {
	double tmp;
	if (wj <= 1.3e-8) {
		tmp = x + (x * (wj * (-2.0 + (wj * ((1.0 / x) + (2.5 - (wj / x)))))));
	} else {
		tmp = wj + ((wj + (x / (-1.0 + (wj * (-1.0 - (wj * (0.5 + (wj * 0.16666666666666666)))))))) / (-1.0 - wj));
	}
	return tmp;
}

Fortran

real(8) function code(wj, x)
    real(8), intent (in) :: wj
    real(8), intent (in) :: x
    real(8) :: tmp
    if (wj <= 1.3d-8) then
        tmp = x + (x * (wj * ((-2.0d0) + (wj * ((1.0d0 / x) + (2.5d0 - (wj / x)))))))
    else
        tmp = wj + ((wj + (x / ((-1.0d0) + (wj * ((-1.0d0) - (wj * (0.5d0 + (wj * 0.16666666666666666d0)))))))) / ((-1.0d0) - wj))
    end if
    code = tmp
end function

Java

public static double code(double wj, double x) {
	double tmp;
	if (wj <= 1.3e-8) {
		tmp = x + (x * (wj * (-2.0 + (wj * ((1.0 / x) + (2.5 - (wj / x)))))));
	} else {
		tmp = wj + ((wj + (x / (-1.0 + (wj * (-1.0 - (wj * (0.5 + (wj * 0.16666666666666666)))))))) / (-1.0 - wj));
	}
	return tmp;
}

Python

def code(wj, x):
	tmp = 0
	if wj <= 1.3e-8:
		tmp = x + (x * (wj * (-2.0 + (wj * ((1.0 / x) + (2.5 - (wj / x)))))))
	else:
		tmp = wj + ((wj + (x / (-1.0 + (wj * (-1.0 - (wj * (0.5 + (wj * 0.16666666666666666)))))))) / (-1.0 - wj))
	return tmp

Julia

function code(wj, x)
	tmp = 0.0
	if (wj <= 1.3e-8)
		tmp = Float64(x + Float64(x * Float64(wj * Float64(-2.0 + Float64(wj * Float64(Float64(1.0 / x) + Float64(2.5 - Float64(wj / x))))))));
	else
		tmp = Float64(wj + Float64(Float64(wj + Float64(x / Float64(-1.0 + Float64(wj * Float64(-1.0 - Float64(wj * Float64(0.5 + Float64(wj * 0.16666666666666666)))))))) / Float64(-1.0 - wj)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(wj, x)
	tmp = 0.0;
	if (wj <= 1.3e-8)
		tmp = x + (x * (wj * (-2.0 + (wj * ((1.0 / x) + (2.5 - (wj / x)))))));
	else
		tmp = wj + ((wj + (x / (-1.0 + (wj * (-1.0 - (wj * (0.5 + (wj * 0.16666666666666666)))))))) / (-1.0 - wj));
	end
	tmp_2 = tmp;
end

Wolfram

code[wj_, x_] := If[LessEqual[wj, 1.3e-8], N[(x + N[(x * N[(wj * N[(-2.0 + N[(wj * N[(N[(1.0 / x), $MachinePrecision] + N[(2.5 - N[(wj / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(wj + N[(N[(wj + N[(x / N[(-1.0 + N[(wj * N[(-1.0 - N[(wj * N[(0.5 + N[(wj * 0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(-1.0 - wj), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if wj < 1.3000000000000001e-8

   1. Initial program 78.4%

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

   3. Taylor expanded in wj around 0

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

      1. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(x, \color{blue}{\left(wj \cdot \left(wj \cdot \left(\left(1 + -1 \cdot \left(wj \cdot \left(1 + \left(-3 \cdot x + \left(-2 \cdot \left(-4 \cdot x + \frac{3}{2} \cdot x\right) + \frac{2}{3} \cdot x\right)\right)\right)\right)\right) - \left(-4 \cdot x + \frac{3}{2} \cdot x\right)\right) - 2 \cdot x\right)\right)}\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(x, \mathsf{*.f64}\left(wj, \color{blue}{\left(wj \cdot \left(\left(1 + -1 \cdot \left(wj \cdot \left(1 + \left(-3 \cdot x + \left(-2 \cdot \left(-4 \cdot x + \frac{3}{2} \cdot x\right) + \frac{2}{3} \cdot x\right)\right)\right)\right)\right) - \left(-4 \cdot x + \frac{3}{2} \cdot x\right)\right) - 2 \cdot x\right)}\right)\right) \]

      3. cancel-sign-sub-inv N/A

         \[\leadsto \mathsf{+.f64}\left(x, \mathsf{*.f64}\left(wj, \left(wj \cdot \left(\left(1 + -1 \cdot \left(wj \cdot \left(1 + \left(-3 \cdot x + \left(-2 \cdot \left(-4 \cdot x + \frac{3}{2} \cdot x\right) + \frac{2}{3} \cdot x\right)\right)\right)\right)\right) - \left(-4 \cdot x + \frac{3}{2} \cdot x\right)\right) + \color{blue}{\left(\mathsf{neg}\left(2\right)\right) \cdot x}\right)\right)\right) \]

      4. metadata-eval N/A

         \[\leadsto \mathsf{+.f64}\left(x, \mathsf{*.f64}\left(wj, \left(wj \cdot \left(\left(1 + -1 \cdot \left(wj \cdot \left(1 + \left(-3 \cdot x + \left(-2 \cdot \left(-4 \cdot x + \frac{3}{2} \cdot x\right) + \frac{2}{3} \cdot x\right)\right)\right)\right)\right) - \left(-4 \cdot x + \frac{3}{2} \cdot x\right)\right) + -2 \cdot x\right)\right)\right) \]

      5. +-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(x, \mathsf{*.f64}\left(wj, \left(-2 \cdot x + \color{blue}{wj \cdot \left(\left(1 + -1 \cdot \left(wj \cdot \left(1 + \left(-3 \cdot x + \left(-2 \cdot \left(-4 \cdot x + \frac{3}{2} \cdot x\right) + \frac{2}{3} \cdot x\right)\right)\right)\right)\right) - \left(-4 \cdot x + \frac{3}{2} \cdot x\right)\right)}\right)\right)\right) \]

      6. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(x, \mathsf{*.f64}\left(wj, \mathsf{+.f64}\left(\left(-2 \cdot x\right), \color{blue}{\left(wj \cdot \left(\left(1 + -1 \cdot \left(wj \cdot \left(1 + \left(-3 \cdot x + \left(-2 \cdot \left(-4 \cdot x + \frac{3}{2} \cdot x\right) + \frac{2}{3} \cdot x\right)\right)\right)\right)\right) - \left(-4 \cdot x + \frac{3}{2} \cdot x\right)\right)\right)}\right)\right)\right) \]

   5. Simplified 97.3%

      \[\leadsto \color{blue}{x + wj \cdot \left(x \cdot -2 + wj \cdot \left(\left(1 - wj \cdot \left(\left(x \cdot -3 + x \cdot 5\right) + \left(x \cdot 0.6666666666666666 + 1\right)\right)\right) + x \cdot 2.5\right)\right)} \]

   6. Taylor expanded in x around inf

      \[\leadsto \mathsf{+.f64}\left(x, \color{blue}{\left(x \cdot \left(wj \cdot \left(wj \cdot \left(\frac{5}{2} - \frac{8}{3} \cdot wj\right) - 2\right) + \frac{{wj}^{2} \cdot \left(1 - wj\right)}{x}\right)\right)}\right) \]
   7. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(x, \mathsf{*.f64}\left(x, \color{blue}{\left(wj \cdot \left(wj \cdot \left(\frac{5}{2} - \frac{8}{3} \cdot wj\right) - 2\right) + \frac{{wj}^{2} \cdot \left(1 - wj\right)}{x}\right)}\right)\right) \]

      2. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\left(wj \cdot \left(wj \cdot \left(\frac{5}{2} - \frac{8}{3} \cdot wj\right) - 2\right)\right), \color{blue}{\left(\frac{{wj}^{2} \cdot \left(1 - wj\right)}{x}\right)}\right)\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\mathsf{*.f64}\left(wj, \left(wj \cdot \left(\frac{5}{2} - \frac{8}{3} \cdot wj\right) - 2\right)\right), \left(\frac{\color{blue}{{wj}^{2} \cdot \left(1 - wj\right)}}{x}\right)\right)\right)\right) \]

      4. sub-neg N/A

         \[\leadsto \mathsf{+.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\mathsf{*.f64}\left(wj, \left(wj \cdot \left(\frac{5}{2} - \frac{8}{3} \cdot wj\right) + \left(\mathsf{neg}\left(2\right)\right)\right)\right), \left(\frac{{wj}^{2} \cdot \color{blue}{\left(1 - wj\right)}}{x}\right)\right)\right)\right) \]

      5. metadata-eval N/A

         \[\leadsto \mathsf{+.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\mathsf{*.f64}\left(wj, \left(wj \cdot \left(\frac{5}{2} - \frac{8}{3} \cdot wj\right) + -2\right)\right), \left(\frac{{wj}^{2} \cdot \left(1 - \color{blue}{wj}\right)}{x}\right)\right)\right)\right) \]

      6. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\mathsf{*.f64}\left(wj, \mathsf{+.f64}\left(\left(wj \cdot \left(\frac{5}{2} - \frac{8}{3} \cdot wj\right)\right), -2\right)\right), \left(\frac{{wj}^{2} \cdot \color{blue}{\left(1 - wj\right)}}{x}\right)\right)\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\mathsf{*.f64}\left(wj, \mathsf{+.f64}\left(\mathsf{*.f64}\left(wj, \left(\frac{5}{2} - \frac{8}{3} \cdot wj\right)\right), -2\right)\right), \left(\frac{{wj}^{2} \cdot \left(\color{blue}{1} - wj\right)}{x}\right)\right)\right)\right) \]

      8. cancel-sign-sub-inv N/A

         \[\leadsto \mathsf{+.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\mathsf{*.f64}\left(wj, \mathsf{+.f64}\left(\mathsf{*.f64}\left(wj, \left(\frac{5}{2} + \left(\mathsf{neg}\left(\frac{8}{3}\right)\right) \cdot wj\right)\right), -2\right)\right), \left(\frac{{wj}^{2} \cdot \left(1 - wj\right)}{x}\right)\right)\right)\right) \]

      9. metadata-eval N/A

         \[\leadsto \mathsf{+.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\mathsf{*.f64}\left(wj, \mathsf{+.f64}\left(\mathsf{*.f64}\left(wj, \left(\frac{5}{2} + \frac{-8}{3} \cdot wj\right)\right), -2\right)\right), \left(\frac{{wj}^{2} \cdot \left(1 - wj\right)}{x}\right)\right)\right)\right) \]

      10. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\mathsf{*.f64}\left(wj, \mathsf{+.f64}\left(\mathsf{*.f64}\left(wj, \mathsf{+.f64}\left(\frac{5}{2}, \left(\frac{-8}{3} \cdot wj\right)\right)\right), -2\right)\right), \left(\frac{{wj}^{2} \cdot \left(1 - wj\right)}{x}\right)\right)\right)\right) \]

      11. *-commutative N/A

          \[\leadsto \mathsf{+.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\mathsf{*.f64}\left(wj, \mathsf{+.f64}\left(\mathsf{*.f64}\left(wj, \mathsf{+.f64}\left(\frac{5}{2}, \left(wj \cdot \frac{-8}{3}\right)\right)\right), -2\right)\right), \left(\frac{{wj}^{2} \cdot \left(1 - wj\right)}{x}\right)\right)\right)\right) \]

      12. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\mathsf{*.f64}\left(wj, \mathsf{+.f64}\left(\mathsf{*.f64}\left(wj, \mathsf{+.f64}\left(\frac{5}{2}, \mathsf{*.f64}\left(wj, \frac{-8}{3}\right)\right)\right), -2\right)\right), \left(\frac{{wj}^{2} \cdot \left(1 - wj\right)}{x}\right)\right)\right)\right) \]

      13. associate-/l* N/A

          \[\leadsto \mathsf{+.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\mathsf{*.f64}\left(wj, \mathsf{+.f64}\left(\mathsf{*.f64}\left(wj, \mathsf{+.f64}\left(\frac{5}{2}, \mathsf{*.f64}\left(wj, \frac{-8}{3}\right)\right)\right), -2\right)\right), \left({wj}^{2} \cdot \color{blue}{\frac{1 - wj}{x}}\right)\right)\right)\right) \]

      14. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\mathsf{*.f64}\left(wj, \mathsf{+.f64}\left(\mathsf{*.f64}\left(wj, \mathsf{+.f64}\left(\frac{5}{2}, \mathsf{*.f64}\left(wj, \frac{-8}{3}\right)\right)\right), -2\right)\right), \mathsf{*.f64}\left(\left({wj}^{2}\right), \color{blue}{\left(\frac{1 - wj}{x}\right)}\right)\right)\right)\right) \]

      15. unpow2 N/A

          \[\leadsto \mathsf{+.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\mathsf{*.f64}\left(wj, \mathsf{+.f64}\left(\mathsf{*.f64}\left(wj, \mathsf{+.f64}\left(\frac{5}{2}, \mathsf{*.f64}\left(wj, \frac{-8}{3}\right)\right)\right), -2\right)\right), \mathsf{*.f64}\left(\left(wj \cdot wj\right), \left(\frac{\color{blue}{1 - wj}}{x}\right)\right)\right)\right)\right) \]

      16. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\mathsf{*.f64}\left(wj, \mathsf{+.f64}\left(\mathsf{*.f64}\left(wj, \mathsf{+.f64}\left(\frac{5}{2}, \mathsf{*.f64}\left(wj, \frac{-8}{3}\right)\right)\right), -2\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(wj, wj\right), \left(\frac{\color{blue}{1 - wj}}{x}\right)\right)\right)\right)\right) \]

      17. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\mathsf{*.f64}\left(wj, \mathsf{+.f64}\left(\mathsf{*.f64}\left(wj, \mathsf{+.f64}\left(\frac{5}{2}, \mathsf{*.f64}\left(wj, \frac{-8}{3}\right)\right)\right), -2\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(wj, wj\right), \mathsf{/.f64}\left(\left(1 - wj\right), \color{blue}{x}\right)\right)\right)\right)\right) \]

      18. --lowering--.f64 97.2%

          \[\leadsto \mathsf{+.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\mathsf{*.f64}\left(wj, \mathsf{+.f64}\left(\mathsf{*.f64}\left(wj, \mathsf{+.f64}\left(\frac{5}{2}, \mathsf{*.f64}\left(wj, \frac{-8}{3}\right)\right)\right), -2\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(wj, wj\right), \mathsf{/.f64}\left(\mathsf{\_.f64}\left(1, wj\right), x\right)\right)\right)\right)\right) \]

   8. Simplified 97.2%

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

   9. Taylor expanded in wj around 0

      \[\leadsto \mathsf{+.f64}\left(x, \mathsf{*.f64}\left(x, \color{blue}{\left(wj \cdot \left(wj \cdot \left(\frac{5}{2} + \left(-1 \cdot \left(wj \cdot \left(\frac{8}{3} + \frac{1}{x}\right)\right) + \frac{1}{x}\right)\right) - 2\right)\right)}\right)\right) \]
   10. Step-by-step derivation

       1. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(wj, \color{blue}{\left(wj \cdot \left(\frac{5}{2} + \left(-1 \cdot \left(wj \cdot \left(\frac{8}{3} + \frac{1}{x}\right)\right) + \frac{1}{x}\right)\right) - 2\right)}\right)\right)\right) \]

       2. sub-neg N/A

          \[\leadsto \mathsf{+.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(wj, \left(wj \cdot \left(\frac{5}{2} + \left(-1 \cdot \left(wj \cdot \left(\frac{8}{3} + \frac{1}{x}\right)\right) + \frac{1}{x}\right)\right) + \color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)\right)\right)\right) \]

       3. metadata-eval N/A

          \[\leadsto \mathsf{+.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(wj, \left(wj \cdot \left(\frac{5}{2} + \left(-1 \cdot \left(wj \cdot \left(\frac{8}{3} + \frac{1}{x}\right)\right) + \frac{1}{x}\right)\right) + -2\right)\right)\right)\right) \]

       4. +-commutative N/A

          \[\leadsto \mathsf{+.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(wj, \left(-2 + \color{blue}{wj \cdot \left(\frac{5}{2} + \left(-1 \cdot \left(wj \cdot \left(\frac{8}{3} + \frac{1}{x}\right)\right) + \frac{1}{x}\right)\right)}\right)\right)\right)\right) \]

       5. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(wj, \mathsf{+.f64}\left(-2, \color{blue}{\left(wj \cdot \left(\frac{5}{2} + \left(-1 \cdot \left(wj \cdot \left(\frac{8}{3} + \frac{1}{x}\right)\right) + \frac{1}{x}\right)\right)\right)}\right)\right)\right)\right) \]

       6. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(wj, \mathsf{+.f64}\left(-2, \mathsf{*.f64}\left(wj, \color{blue}{\left(\frac{5}{2} + \left(-1 \cdot \left(wj \cdot \left(\frac{8}{3} + \frac{1}{x}\right)\right) + \frac{1}{x}\right)\right)}\right)\right)\right)\right)\right) \]

       7. associate-+r+ N/A

          \[\leadsto \mathsf{+.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(wj, \mathsf{+.f64}\left(-2, \mathsf{*.f64}\left(wj, \left(\left(\frac{5}{2} + -1 \cdot \left(wj \cdot \left(\frac{8}{3} + \frac{1}{x}\right)\right)\right) + \color{blue}{\frac{1}{x}}\right)\right)\right)\right)\right)\right) \]

       8. +-commutative N/A

          \[\leadsto \mathsf{+.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(wj, \mathsf{+.f64}\left(-2, \mathsf{*.f64}\left(wj, \left(\frac{1}{x} + \color{blue}{\left(\frac{5}{2} + -1 \cdot \left(wj \cdot \left(\frac{8}{3} + \frac{1}{x}\right)\right)\right)}\right)\right)\right)\right)\right)\right) \]

       9. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(wj, \mathsf{+.f64}\left(-2, \mathsf{*.f64}\left(wj, \mathsf{+.f64}\left(\left(\frac{1}{x}\right), \color{blue}{\left(\frac{5}{2} + -1 \cdot \left(wj \cdot \left(\frac{8}{3} + \frac{1}{x}\right)\right)\right)}\right)\right)\right)\right)\right)\right) \]

       10. /-lowering-/.f64 N/A

           \[\leadsto \mathsf{+.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(wj, \mathsf{+.f64}\left(-2, \mathsf{*.f64}\left(wj, \mathsf{+.f64}\left(\mathsf{/.f64}\left(1, x\right), \left(\color{blue}{\frac{5}{2}} + -1 \cdot \left(wj \cdot \left(\frac{8}{3} + \frac{1}{x}\right)\right)\right)\right)\right)\right)\right)\right)\right) \]

       11. +-lowering-+.f64 N/A

           \[\leadsto \mathsf{+.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(wj, \mathsf{+.f64}\left(-2, \mathsf{*.f64}\left(wj, \mathsf{+.f64}\left(\mathsf{/.f64}\left(1, x\right), \mathsf{+.f64}\left(\frac{5}{2}, \color{blue}{\left(-1 \cdot \left(wj \cdot \left(\frac{8}{3} + \frac{1}{x}\right)\right)\right)}\right)\right)\right)\right)\right)\right)\right) \]

       12. mul-1-neg N/A

           \[\leadsto \mathsf{+.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(wj, \mathsf{+.f64}\left(-2, \mathsf{*.f64}\left(wj, \mathsf{+.f64}\left(\mathsf{/.f64}\left(1, x\right), \mathsf{+.f64}\left(\frac{5}{2}, \left(\mathsf{neg}\left(wj \cdot \left(\frac{8}{3} + \frac{1}{x}\right)\right)\right)\right)\right)\right)\right)\right)\right)\right) \]

       13. distribute-rgt-neg-in N/A

           \[\leadsto \mathsf{+.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(wj, \mathsf{+.f64}\left(-2, \mathsf{*.f64}\left(wj, \mathsf{+.f64}\left(\mathsf{/.f64}\left(1, x\right), \mathsf{+.f64}\left(\frac{5}{2}, \left(wj \cdot \color{blue}{\left(\mathsf{neg}\left(\left(\frac{8}{3} + \frac{1}{x}\right)\right)\right)}\right)\right)\right)\right)\right)\right)\right)\right) \]

       14. *-lowering-*.f64 N/A

           \[\leadsto \mathsf{+.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(wj, \mathsf{+.f64}\left(-2, \mathsf{*.f64}\left(wj, \mathsf{+.f64}\left(\mathsf{/.f64}\left(1, x\right), \mathsf{+.f64}\left(\frac{5}{2}, \mathsf{*.f64}\left(wj, \color{blue}{\left(\mathsf{neg}\left(\left(\frac{8}{3} + \frac{1}{x}\right)\right)\right)}\right)\right)\right)\right)\right)\right)\right)\right) \]

       15. distribute-neg-in N/A

           \[\leadsto \mathsf{+.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(wj, \mathsf{+.f64}\left(-2, \mathsf{*.f64}\left(wj, \mathsf{+.f64}\left(\mathsf{/.f64}\left(1, x\right), \mathsf{+.f64}\left(\frac{5}{2}, \mathsf{*.f64}\left(wj, \left(\left(\mathsf{neg}\left(\frac{8}{3}\right)\right) + \color{blue}{\left(\mathsf{neg}\left(\frac{1}{x}\right)\right)}\right)\right)\right)\right)\right)\right)\right)\right)\right) \]

       16. metadata-eval N/A

           \[\leadsto \mathsf{+.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(wj, \mathsf{+.f64}\left(-2, \mathsf{*.f64}\left(wj, \mathsf{+.f64}\left(\mathsf{/.f64}\left(1, x\right), \mathsf{+.f64}\left(\frac{5}{2}, \mathsf{*.f64}\left(wj, \left(\frac{-8}{3} + \left(\mathsf{neg}\left(\color{blue}{\frac{1}{x}}\right)\right)\right)\right)\right)\right)\right)\right)\right)\right)\right) \]

       17. +-lowering-+.f64 N/A

           \[\leadsto \mathsf{+.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(wj, \mathsf{+.f64}\left(-2, \mathsf{*.f64}\left(wj, \mathsf{+.f64}\left(\mathsf{/.f64}\left(1, x\right), \mathsf{+.f64}\left(\frac{5}{2}, \mathsf{*.f64}\left(wj, \mathsf{+.f64}\left(\frac{-8}{3}, \color{blue}{\left(\mathsf{neg}\left(\frac{1}{x}\right)\right)}\right)\right)\right)\right)\right)\right)\right)\right)\right) \]

       18. distribute-neg-frac N/A

           \[\leadsto \mathsf{+.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(wj, \mathsf{+.f64}\left(-2, \mathsf{*.f64}\left(wj, \mathsf{+.f64}\left(\mathsf{/.f64}\left(1, x\right), \mathsf{+.f64}\left(\frac{5}{2}, \mathsf{*.f64}\left(wj, \mathsf{+.f64}\left(\frac{-8}{3}, \left(\frac{\mathsf{neg}\left(1\right)}{\color{blue}{x}}\right)\right)\right)\right)\right)\right)\right)\right)\right)\right) \]

       19. metadata-eval N/A

           \[\leadsto \mathsf{+.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(wj, \mathsf{+.f64}\left(-2, \mathsf{*.f64}\left(wj, \mathsf{+.f64}\left(\mathsf{/.f64}\left(1, x\right), \mathsf{+.f64}\left(\frac{5}{2}, \mathsf{*.f64}\left(wj, \mathsf{+.f64}\left(\frac{-8}{3}, \left(\frac{-1}{x}\right)\right)\right)\right)\right)\right)\right)\right)\right)\right) \]

       20. /-lowering-/.f64 97.2%

           \[\leadsto \mathsf{+.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(wj, \mathsf{+.f64}\left(-2, \mathsf{*.f64}\left(wj, \mathsf{+.f64}\left(\mathsf{/.f64}\left(1, x\right), \mathsf{+.f64}\left(\frac{5}{2}, \mathsf{*.f64}\left(wj, \mathsf{+.f64}\left(\frac{-8}{3}, \mathsf{/.f64}\left(-1, \color{blue}{x}\right)\right)\right)\right)\right)\right)\right)\right)\right)\right) \]

   11. Simplified 97.2%

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

   12. Taylor expanded in x around 0

       \[\leadsto \mathsf{+.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(wj, \mathsf{+.f64}\left(-2, \mathsf{*.f64}\left(wj, \mathsf{+.f64}\left(\mathsf{/.f64}\left(1, x\right), \mathsf{+.f64}\left(\frac{5}{2}, \color{blue}{\left(-1 \cdot \frac{wj}{x}\right)}\right)\right)\right)\right)\right)\right)\right) \]
   13. Step-by-step derivation

       1. mul-1-neg N/A

          \[\leadsto \mathsf{+.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(wj, \mathsf{+.f64}\left(-2, \mathsf{*.f64}\left(wj, \mathsf{+.f64}\left(\mathsf{/.f64}\left(1, x\right), \mathsf{+.f64}\left(\frac{5}{2}, \left(\mathsf{neg}\left(\frac{wj}{x}\right)\right)\right)\right)\right)\right)\right)\right)\right) \]

       2. distribute-neg-frac2 N/A

          \[\leadsto \mathsf{+.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(wj, \mathsf{+.f64}\left(-2, \mathsf{*.f64}\left(wj, \mathsf{+.f64}\left(\mathsf{/.f64}\left(1, x\right), \mathsf{+.f64}\left(\frac{5}{2}, \left(\frac{wj}{\color{blue}{\mathsf{neg}\left(x\right)}}\right)\right)\right)\right)\right)\right)\right)\right) \]

       3. mul-1-neg N/A

          \[\leadsto \mathsf{+.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(wj, \mathsf{+.f64}\left(-2, \mathsf{*.f64}\left(wj, \mathsf{+.f64}\left(\mathsf{/.f64}\left(1, x\right), \mathsf{+.f64}\left(\frac{5}{2}, \left(\frac{wj}{-1 \cdot \color{blue}{x}}\right)\right)\right)\right)\right)\right)\right)\right) \]

       4. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(wj, \mathsf{+.f64}\left(-2, \mathsf{*.f64}\left(wj, \mathsf{+.f64}\left(\mathsf{/.f64}\left(1, x\right), \mathsf{+.f64}\left(\frac{5}{2}, \mathsf{/.f64}\left(wj, \color{blue}{\left(-1 \cdot x\right)}\right)\right)\right)\right)\right)\right)\right)\right) \]

       5. mul-1-neg N/A

          \[\leadsto \mathsf{+.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(wj, \mathsf{+.f64}\left(-2, \mathsf{*.f64}\left(wj, \mathsf{+.f64}\left(\mathsf{/.f64}\left(1, x\right), \mathsf{+.f64}\left(\frac{5}{2}, \mathsf{/.f64}\left(wj, \left(\mathsf{neg}\left(x\right)\right)\right)\right)\right)\right)\right)\right)\right)\right) \]

       6. neg-sub0 N/A

          \[\leadsto \mathsf{+.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(wj, \mathsf{+.f64}\left(-2, \mathsf{*.f64}\left(wj, \mathsf{+.f64}\left(\mathsf{/.f64}\left(1, x\right), \mathsf{+.f64}\left(\frac{5}{2}, \mathsf{/.f64}\left(wj, \left(0 - \color{blue}{x}\right)\right)\right)\right)\right)\right)\right)\right)\right) \]

       7. --lowering--.f64 97.1%

          \[\leadsto \mathsf{+.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(wj, \mathsf{+.f64}\left(-2, \mathsf{*.f64}\left(wj, \mathsf{+.f64}\left(\mathsf{/.f64}\left(1, x\right), \mathsf{+.f64}\left(\frac{5}{2}, \mathsf{/.f64}\left(wj, \mathsf{\_.f64}\left(0, \color{blue}{x}\right)\right)\right)\right)\right)\right)\right)\right)\right) \]

   14. Simplified 97.1%

       \[\leadsto x + x \cdot \left(wj \cdot \left(-2 + wj \cdot \left(\frac{1}{x} + \left(2.5 + \color{blue}{\frac{wj}{0 - x}}\right)\right)\right)\right) \]

   if 1.3000000000000001e-8 < wj

   1. Initial program 49.7%

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

      1. sub-neg N/A

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

      2. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(wj, \color{blue}{\left(\mathsf{neg}\left(\frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}}\right)\right)}\right) \]

      3. distribute-rgt1-in N/A

         \[\leadsto \mathsf{+.f64}\left(wj, \left(\mathsf{neg}\left(\frac{wj \cdot e^{wj} - x}{\left(wj + 1\right) \cdot e^{wj}}\right)\right)\right) \]

      4. associate-/l/ N/A

         \[\leadsto \mathsf{+.f64}\left(wj, \left(\mathsf{neg}\left(\frac{\frac{wj \cdot e^{wj} - x}{e^{wj}}}{wj + 1}\right)\right)\right) \]

      5. distribute-neg-frac2 N/A

         \[\leadsto \mathsf{+.f64}\left(wj, \left(\frac{\frac{wj \cdot e^{wj} - x}{e^{wj}}}{\color{blue}{\mathsf{neg}\left(\left(wj + 1\right)\right)}}\right)\right) \]

      6. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(wj, \mathsf{/.f64}\left(\left(\frac{wj \cdot e^{wj} - x}{e^{wj}}\right), \color{blue}{\left(\mathsf{neg}\left(\left(wj + 1\right)\right)\right)}\right)\right) \]

      7. div-sub N/A

         \[\leadsto \mathsf{+.f64}\left(wj, \mathsf{/.f64}\left(\left(\frac{wj \cdot e^{wj}}{e^{wj}} - \frac{x}{e^{wj}}\right), \left(\mathsf{neg}\left(\color{blue}{\left(wj + 1\right)}\right)\right)\right)\right) \]

      8. associate-/l* N/A

         \[\leadsto \mathsf{+.f64}\left(wj, \mathsf{/.f64}\left(\left(wj \cdot \frac{e^{wj}}{e^{wj}} - \frac{x}{e^{wj}}\right), \left(\mathsf{neg}\left(\left(\color{blue}{wj} + 1\right)\right)\right)\right)\right) \]

      9. *-inverses N/A

         \[\leadsto \mathsf{+.f64}\left(wj, \mathsf{/.f64}\left(\left(wj \cdot 1 - \frac{x}{e^{wj}}\right), \left(\mathsf{neg}\left(\left(wj + 1\right)\right)\right)\right)\right) \]

      10. *-rgt-identity N/A

          \[\leadsto \mathsf{+.f64}\left(wj, \mathsf{/.f64}\left(\left(wj - \frac{x}{e^{wj}}\right), \left(\mathsf{neg}\left(\left(\color{blue}{wj} + 1\right)\right)\right)\right)\right) \]

      11. --lowering--.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(wj, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(wj, \left(\frac{x}{e^{wj}}\right)\right), \left(\mathsf{neg}\left(\color{blue}{\left(wj + 1\right)}\right)\right)\right)\right) \]

      12. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(wj, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(wj, \mathsf{/.f64}\left(x, \left(e^{wj}\right)\right)\right), \left(\mathsf{neg}\left(\left(wj + \color{blue}{1}\right)\right)\right)\right)\right) \]

      13. exp-lowering-exp.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(wj, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(wj, \mathsf{/.f64}\left(x, \mathsf{exp.f64}\left(wj\right)\right)\right), \left(\mathsf{neg}\left(\left(wj + 1\right)\right)\right)\right)\right) \]

      14. +-commutative N/A

          \[\leadsto \mathsf{+.f64}\left(wj, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(wj, \mathsf{/.f64}\left(x, \mathsf{exp.f64}\left(wj\right)\right)\right), \left(\mathsf{neg}\left(\left(1 + wj\right)\right)\right)\right)\right) \]

      15. *-inverses N/A

          \[\leadsto \mathsf{+.f64}\left(wj, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(wj, \mathsf{/.f64}\left(x, \mathsf{exp.f64}\left(wj\right)\right)\right), \left(\mathsf{neg}\left(\left(\frac{e^{wj}}{e^{wj}} + wj\right)\right)\right)\right)\right) \]

      16. distribute-neg-in N/A

          \[\leadsto \mathsf{+.f64}\left(wj, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(wj, \mathsf{/.f64}\left(x, \mathsf{exp.f64}\left(wj\right)\right)\right), \left(\left(\mathsf{neg}\left(\frac{e^{wj}}{e^{wj}}\right)\right) + \color{blue}{\left(\mathsf{neg}\left(wj\right)\right)}\right)\right)\right) \]

   3. Simplified 99.7%

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

   5. Taylor expanded in wj around 0

      \[\leadsto \mathsf{+.f64}\left(wj, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(wj, \mathsf{/.f64}\left(x, \color{blue}{\left(1 + wj \cdot \left(1 + wj \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot wj\right)\right)\right)}\right)\right), \mathsf{\_.f64}\left(-1, wj\right)\right)\right) \]
   6. Step-by-step derivation

      1. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(wj, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(wj, \mathsf{/.f64}\left(x, \mathsf{+.f64}\left(1, \left(wj \cdot \left(1 + wj \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot wj\right)\right)\right)\right)\right)\right), \mathsf{\_.f64}\left(-1, wj\right)\right)\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(wj, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(wj, \mathsf{/.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(wj, \left(1 + wj \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot wj\right)\right)\right)\right)\right)\right), \mathsf{\_.f64}\left(-1, wj\right)\right)\right) \]

      3. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(wj, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(wj, \mathsf{/.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(wj, \mathsf{+.f64}\left(1, \left(wj \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot wj\right)\right)\right)\right)\right)\right)\right), \mathsf{\_.f64}\left(-1, wj\right)\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(wj, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(wj, \mathsf{/.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(wj, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(wj, \left(\frac{1}{2} + \frac{1}{6} \cdot wj\right)\right)\right)\right)\right)\right)\right), \mathsf{\_.f64}\left(-1, wj\right)\right)\right) \]

      5. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(wj, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(wj, \mathsf{/.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(wj, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(wj, \mathsf{+.f64}\left(\frac{1}{2}, \left(\frac{1}{6} \cdot wj\right)\right)\right)\right)\right)\right)\right)\right), \mathsf{\_.f64}\left(-1, wj\right)\right)\right) \]

      6. *-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(wj, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(wj, \mathsf{/.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(wj, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(wj, \mathsf{+.f64}\left(\frac{1}{2}, \left(wj \cdot \frac{1}{6}\right)\right)\right)\right)\right)\right)\right)\right), \mathsf{\_.f64}\left(-1, wj\right)\right)\right) \]

      7. *-lowering-*.f64 98.3%

         \[\leadsto \mathsf{+.f64}\left(wj, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(wj, \mathsf{/.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(wj, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(wj, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(wj, \frac{1}{6}\right)\right)\right)\right)\right)\right)\right)\right), \mathsf{\_.f64}\left(-1, wj\right)\right)\right) \]

   7. Simplified 98.3%

      \[\leadsto wj + \frac{wj - \frac{x}{\color{blue}{1 + wj \cdot \left(1 + wj \cdot \left(0.5 + wj \cdot 0.16666666666666666\right)\right)}}}{-1 - wj} \]
3. Recombined 2 regimes into one program.

4. Final simplification 97.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;wj \leq 1.3 \cdot 10^{-8}:\\ \;\;\;\;x + x \cdot \left(wj \cdot \left(-2 + wj \cdot \left(\frac{1}{x} + \left(2.5 - \frac{wj}{x}\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;wj + \frac{wj + \frac{x}{-1 + wj \cdot \left(-1 - wj \cdot \left(0.5 + wj \cdot 0.16666666666666666\right)\right)}}{-1 - wj}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 97.0% accurate, 13.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;wj \leq 1.3 \cdot 10^{-8}:\\ \;\;\;\;x + x \cdot \left(wj \cdot \left(-2 + wj \cdot \left(\frac{1}{x} + \left(2.5 - \frac{wj}{x}\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;wj + \frac{wj + \frac{x}{-1 + wj \cdot \left(-1 - wj \cdot 0.5\right)}}{-1 - wj}\\ \end{array} \end{array} \]

FPCore

(FPCore (wj x)
 :precision binary64
 (if (<= wj 1.3e-8)
   (+ x (* x (* wj (+ -2.0 (* wj (+ (/ 1.0 x) (- 2.5 (/ wj x))))))))
   (+ wj (/ (+ wj (/ x (+ -1.0 (* wj (- -1.0 (* wj 0.5)))))) (- -1.0 wj)))))

C

double code(double wj, double x) {
	double tmp;
	if (wj <= 1.3e-8) {
		tmp = x + (x * (wj * (-2.0 + (wj * ((1.0 / x) + (2.5 - (wj / x)))))));
	} else {
		tmp = wj + ((wj + (x / (-1.0 + (wj * (-1.0 - (wj * 0.5)))))) / (-1.0 - wj));
	}
	return tmp;
}

Fortran

real(8) function code(wj, x)
    real(8), intent (in) :: wj
    real(8), intent (in) :: x
    real(8) :: tmp
    if (wj <= 1.3d-8) then
        tmp = x + (x * (wj * ((-2.0d0) + (wj * ((1.0d0 / x) + (2.5d0 - (wj / x)))))))
    else
        tmp = wj + ((wj + (x / ((-1.0d0) + (wj * ((-1.0d0) - (wj * 0.5d0)))))) / ((-1.0d0) - wj))
    end if
    code = tmp
end function

Java

public static double code(double wj, double x) {
	double tmp;
	if (wj <= 1.3e-8) {
		tmp = x + (x * (wj * (-2.0 + (wj * ((1.0 / x) + (2.5 - (wj / x)))))));
	} else {
		tmp = wj + ((wj + (x / (-1.0 + (wj * (-1.0 - (wj * 0.5)))))) / (-1.0 - wj));
	}
	return tmp;
}

Python

def code(wj, x):
	tmp = 0
	if wj <= 1.3e-8:
		tmp = x + (x * (wj * (-2.0 + (wj * ((1.0 / x) + (2.5 - (wj / x)))))))
	else:
		tmp = wj + ((wj + (x / (-1.0 + (wj * (-1.0 - (wj * 0.5)))))) / (-1.0 - wj))
	return tmp

Julia

function code(wj, x)
	tmp = 0.0
	if (wj <= 1.3e-8)
		tmp = Float64(x + Float64(x * Float64(wj * Float64(-2.0 + Float64(wj * Float64(Float64(1.0 / x) + Float64(2.5 - Float64(wj / x))))))));
	else
		tmp = Float64(wj + Float64(Float64(wj + Float64(x / Float64(-1.0 + Float64(wj * Float64(-1.0 - Float64(wj * 0.5)))))) / Float64(-1.0 - wj)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(wj, x)
	tmp = 0.0;
	if (wj <= 1.3e-8)
		tmp = x + (x * (wj * (-2.0 + (wj * ((1.0 / x) + (2.5 - (wj / x)))))));
	else
		tmp = wj + ((wj + (x / (-1.0 + (wj * (-1.0 - (wj * 0.5)))))) / (-1.0 - wj));
	end
	tmp_2 = tmp;
end

Wolfram

code[wj_, x_] := If[LessEqual[wj, 1.3e-8], N[(x + N[(x * N[(wj * N[(-2.0 + N[(wj * N[(N[(1.0 / x), $MachinePrecision] + N[(2.5 - N[(wj / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(wj + N[(N[(wj + N[(x / N[(-1.0 + N[(wj * N[(-1.0 - N[(wj * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(-1.0 - wj), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if wj < 1.3000000000000001e-8

   1. Initial program 78.4%

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

   3. Taylor expanded in wj around 0

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

      1. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(x, \color{blue}{\left(wj \cdot \left(wj \cdot \left(\left(1 + -1 \cdot \left(wj \cdot \left(1 + \left(-3 \cdot x + \left(-2 \cdot \left(-4 \cdot x + \frac{3}{2} \cdot x\right) + \frac{2}{3} \cdot x\right)\right)\right)\right)\right) - \left(-4 \cdot x + \frac{3}{2} \cdot x\right)\right) - 2 \cdot x\right)\right)}\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(x, \mathsf{*.f64}\left(wj, \color{blue}{\left(wj \cdot \left(\left(1 + -1 \cdot \left(wj \cdot \left(1 + \left(-3 \cdot x + \left(-2 \cdot \left(-4 \cdot x + \frac{3}{2} \cdot x\right) + \frac{2}{3} \cdot x\right)\right)\right)\right)\right) - \left(-4 \cdot x + \frac{3}{2} \cdot x\right)\right) - 2 \cdot x\right)}\right)\right) \]

      3. cancel-sign-sub-inv N/A

         \[\leadsto \mathsf{+.f64}\left(x, \mathsf{*.f64}\left(wj, \left(wj \cdot \left(\left(1 + -1 \cdot \left(wj \cdot \left(1 + \left(-3 \cdot x + \left(-2 \cdot \left(-4 \cdot x + \frac{3}{2} \cdot x\right) + \frac{2}{3} \cdot x\right)\right)\right)\right)\right) - \left(-4 \cdot x + \frac{3}{2} \cdot x\right)\right) + \color{blue}{\left(\mathsf{neg}\left(2\right)\right) \cdot x}\right)\right)\right) \]

      4. metadata-eval N/A

         \[\leadsto \mathsf{+.f64}\left(x, \mathsf{*.f64}\left(wj, \left(wj \cdot \left(\left(1 + -1 \cdot \left(wj \cdot \left(1 + \left(-3 \cdot x + \left(-2 \cdot \left(-4 \cdot x + \frac{3}{2} \cdot x\right) + \frac{2}{3} \cdot x\right)\right)\right)\right)\right) - \left(-4 \cdot x + \frac{3}{2} \cdot x\right)\right) + -2 \cdot x\right)\right)\right) \]

      5. +-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(x, \mathsf{*.f64}\left(wj, \left(-2 \cdot x + \color{blue}{wj \cdot \left(\left(1 + -1 \cdot \left(wj \cdot \left(1 + \left(-3 \cdot x + \left(-2 \cdot \left(-4 \cdot x + \frac{3}{2} \cdot x\right) + \frac{2}{3} \cdot x\right)\right)\right)\right)\right) - \left(-4 \cdot x + \frac{3}{2} \cdot x\right)\right)}\right)\right)\right) \]

      6. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(x, \mathsf{*.f64}\left(wj, \mathsf{+.f64}\left(\left(-2 \cdot x\right), \color{blue}{\left(wj \cdot \left(\left(1 + -1 \cdot \left(wj \cdot \left(1 + \left(-3 \cdot x + \left(-2 \cdot \left(-4 \cdot x + \frac{3}{2} \cdot x\right) + \frac{2}{3} \cdot x\right)\right)\right)\right)\right) - \left(-4 \cdot x + \frac{3}{2} \cdot x\right)\right)\right)}\right)\right)\right) \]

   5. Simplified 97.3%

      \[\leadsto \color{blue}{x + wj \cdot \left(x \cdot -2 + wj \cdot \left(\left(1 - wj \cdot \left(\left(x \cdot -3 + x \cdot 5\right) + \left(x \cdot 0.6666666666666666 + 1\right)\right)\right) + x \cdot 2.5\right)\right)} \]

   6. Taylor expanded in x around inf

      \[\leadsto \mathsf{+.f64}\left(x, \color{blue}{\left(x \cdot \left(wj \cdot \left(wj \cdot \left(\frac{5}{2} - \frac{8}{3} \cdot wj\right) - 2\right) + \frac{{wj}^{2} \cdot \left(1 - wj\right)}{x}\right)\right)}\right) \]
   7. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(x, \mathsf{*.f64}\left(x, \color{blue}{\left(wj \cdot \left(wj \cdot \left(\frac{5}{2} - \frac{8}{3} \cdot wj\right) - 2\right) + \frac{{wj}^{2} \cdot \left(1 - wj\right)}{x}\right)}\right)\right) \]

      2. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\left(wj \cdot \left(wj \cdot \left(\frac{5}{2} - \frac{8}{3} \cdot wj\right) - 2\right)\right), \color{blue}{\left(\frac{{wj}^{2} \cdot \left(1 - wj\right)}{x}\right)}\right)\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\mathsf{*.f64}\left(wj, \left(wj \cdot \left(\frac{5}{2} - \frac{8}{3} \cdot wj\right) - 2\right)\right), \left(\frac{\color{blue}{{wj}^{2} \cdot \left(1 - wj\right)}}{x}\right)\right)\right)\right) \]

      4. sub-neg N/A

         \[\leadsto \mathsf{+.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\mathsf{*.f64}\left(wj, \left(wj \cdot \left(\frac{5}{2} - \frac{8}{3} \cdot wj\right) + \left(\mathsf{neg}\left(2\right)\right)\right)\right), \left(\frac{{wj}^{2} \cdot \color{blue}{\left(1 - wj\right)}}{x}\right)\right)\right)\right) \]

      5. metadata-eval N/A

         \[\leadsto \mathsf{+.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\mathsf{*.f64}\left(wj, \left(wj \cdot \left(\frac{5}{2} - \frac{8}{3} \cdot wj\right) + -2\right)\right), \left(\frac{{wj}^{2} \cdot \left(1 - \color{blue}{wj}\right)}{x}\right)\right)\right)\right) \]

      6. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\mathsf{*.f64}\left(wj, \mathsf{+.f64}\left(\left(wj \cdot \left(\frac{5}{2} - \frac{8}{3} \cdot wj\right)\right), -2\right)\right), \left(\frac{{wj}^{2} \cdot \color{blue}{\left(1 - wj\right)}}{x}\right)\right)\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\mathsf{*.f64}\left(wj, \mathsf{+.f64}\left(\mathsf{*.f64}\left(wj, \left(\frac{5}{2} - \frac{8}{3} \cdot wj\right)\right), -2\right)\right), \left(\frac{{wj}^{2} \cdot \left(\color{blue}{1} - wj\right)}{x}\right)\right)\right)\right) \]

      8. cancel-sign-sub-inv N/A

         \[\leadsto \mathsf{+.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\mathsf{*.f64}\left(wj, \mathsf{+.f64}\left(\mathsf{*.f64}\left(wj, \left(\frac{5}{2} + \left(\mathsf{neg}\left(\frac{8}{3}\right)\right) \cdot wj\right)\right), -2\right)\right), \left(\frac{{wj}^{2} \cdot \left(1 - wj\right)}{x}\right)\right)\right)\right) \]

      9. metadata-eval N/A

         \[\leadsto \mathsf{+.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\mathsf{*.f64}\left(wj, \mathsf{+.f64}\left(\mathsf{*.f64}\left(wj, \left(\frac{5}{2} + \frac{-8}{3} \cdot wj\right)\right), -2\right)\right), \left(\frac{{wj}^{2} \cdot \left(1 - wj\right)}{x}\right)\right)\right)\right) \]

      10. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\mathsf{*.f64}\left(wj, \mathsf{+.f64}\left(\mathsf{*.f64}\left(wj, \mathsf{+.f64}\left(\frac{5}{2}, \left(\frac{-8}{3} \cdot wj\right)\right)\right), -2\right)\right), \left(\frac{{wj}^{2} \cdot \left(1 - wj\right)}{x}\right)\right)\right)\right) \]

      11. *-commutative N/A

          \[\leadsto \mathsf{+.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\mathsf{*.f64}\left(wj, \mathsf{+.f64}\left(\mathsf{*.f64}\left(wj, \mathsf{+.f64}\left(\frac{5}{2}, \left(wj \cdot \frac{-8}{3}\right)\right)\right), -2\right)\right), \left(\frac{{wj}^{2} \cdot \left(1 - wj\right)}{x}\right)\right)\right)\right) \]

      12. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\mathsf{*.f64}\left(wj, \mathsf{+.f64}\left(\mathsf{*.f64}\left(wj, \mathsf{+.f64}\left(\frac{5}{2}, \mathsf{*.f64}\left(wj, \frac{-8}{3}\right)\right)\right), -2\right)\right), \left(\frac{{wj}^{2} \cdot \left(1 - wj\right)}{x}\right)\right)\right)\right) \]

      13. associate-/l* N/A

          \[\leadsto \mathsf{+.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\mathsf{*.f64}\left(wj, \mathsf{+.f64}\left(\mathsf{*.f64}\left(wj, \mathsf{+.f64}\left(\frac{5}{2}, \mathsf{*.f64}\left(wj, \frac{-8}{3}\right)\right)\right), -2\right)\right), \left({wj}^{2} \cdot \color{blue}{\frac{1 - wj}{x}}\right)\right)\right)\right) \]

      14. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\mathsf{*.f64}\left(wj, \mathsf{+.f64}\left(\mathsf{*.f64}\left(wj, \mathsf{+.f64}\left(\frac{5}{2}, \mathsf{*.f64}\left(wj, \frac{-8}{3}\right)\right)\right), -2\right)\right), \mathsf{*.f64}\left(\left({wj}^{2}\right), \color{blue}{\left(\frac{1 - wj}{x}\right)}\right)\right)\right)\right) \]

      15. unpow2 N/A

          \[\leadsto \mathsf{+.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\mathsf{*.f64}\left(wj, \mathsf{+.f64}\left(\mathsf{*.f64}\left(wj, \mathsf{+.f64}\left(\frac{5}{2}, \mathsf{*.f64}\left(wj, \frac{-8}{3}\right)\right)\right), -2\right)\right), \mathsf{*.f64}\left(\left(wj \cdot wj\right), \left(\frac{\color{blue}{1 - wj}}{x}\right)\right)\right)\right)\right) \]

      16. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\mathsf{*.f64}\left(wj, \mathsf{+.f64}\left(\mathsf{*.f64}\left(wj, \mathsf{+.f64}\left(\frac{5}{2}, \mathsf{*.f64}\left(wj, \frac{-8}{3}\right)\right)\right), -2\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(wj, wj\right), \left(\frac{\color{blue}{1 - wj}}{x}\right)\right)\right)\right)\right) \]

      17. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\mathsf{*.f64}\left(wj, \mathsf{+.f64}\left(\mathsf{*.f64}\left(wj, \mathsf{+.f64}\left(\frac{5}{2}, \mathsf{*.f64}\left(wj, \frac{-8}{3}\right)\right)\right), -2\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(wj, wj\right), \mathsf{/.f64}\left(\left(1 - wj\right), \color{blue}{x}\right)\right)\right)\right)\right) \]

      18. --lowering--.f64 97.2%

          \[\leadsto \mathsf{+.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\mathsf{*.f64}\left(wj, \mathsf{+.f64}\left(\mathsf{*.f64}\left(wj, \mathsf{+.f64}\left(\frac{5}{2}, \mathsf{*.f64}\left(wj, \frac{-8}{3}\right)\right)\right), -2\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(wj, wj\right), \mathsf{/.f64}\left(\mathsf{\_.f64}\left(1, wj\right), x\right)\right)\right)\right)\right) \]

   8. Simplified 97.2%

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

   9. Taylor expanded in wj around 0

      \[\leadsto \mathsf{+.f64}\left(x, \mathsf{*.f64}\left(x, \color{blue}{\left(wj \cdot \left(wj \cdot \left(\frac{5}{2} + \left(-1 \cdot \left(wj \cdot \left(\frac{8}{3} + \frac{1}{x}\right)\right) + \frac{1}{x}\right)\right) - 2\right)\right)}\right)\right) \]
   10. Step-by-step derivation

       1. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(wj, \color{blue}{\left(wj \cdot \left(\frac{5}{2} + \left(-1 \cdot \left(wj \cdot \left(\frac{8}{3} + \frac{1}{x}\right)\right) + \frac{1}{x}\right)\right) - 2\right)}\right)\right)\right) \]

       2. sub-neg N/A

          \[\leadsto \mathsf{+.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(wj, \left(wj \cdot \left(\frac{5}{2} + \left(-1 \cdot \left(wj \cdot \left(\frac{8}{3} + \frac{1}{x}\right)\right) + \frac{1}{x}\right)\right) + \color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)\right)\right)\right) \]

       3. metadata-eval N/A

          \[\leadsto \mathsf{+.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(wj, \left(wj \cdot \left(\frac{5}{2} + \left(-1 \cdot \left(wj \cdot \left(\frac{8}{3} + \frac{1}{x}\right)\right) + \frac{1}{x}\right)\right) + -2\right)\right)\right)\right) \]

       4. +-commutative N/A

          \[\leadsto \mathsf{+.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(wj, \left(-2 + \color{blue}{wj \cdot \left(\frac{5}{2} + \left(-1 \cdot \left(wj \cdot \left(\frac{8}{3} + \frac{1}{x}\right)\right) + \frac{1}{x}\right)\right)}\right)\right)\right)\right) \]

       5. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(wj, \mathsf{+.f64}\left(-2, \color{blue}{\left(wj \cdot \left(\frac{5}{2} + \left(-1 \cdot \left(wj \cdot \left(\frac{8}{3} + \frac{1}{x}\right)\right) + \frac{1}{x}\right)\right)\right)}\right)\right)\right)\right) \]

       6. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(wj, \mathsf{+.f64}\left(-2, \mathsf{*.f64}\left(wj, \color{blue}{\left(\frac{5}{2} + \left(-1 \cdot \left(wj \cdot \left(\frac{8}{3} + \frac{1}{x}\right)\right) + \frac{1}{x}\right)\right)}\right)\right)\right)\right)\right) \]

       7. associate-+r+ N/A

          \[\leadsto \mathsf{+.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(wj, \mathsf{+.f64}\left(-2, \mathsf{*.f64}\left(wj, \left(\left(\frac{5}{2} + -1 \cdot \left(wj \cdot \left(\frac{8}{3} + \frac{1}{x}\right)\right)\right) + \color{blue}{\frac{1}{x}}\right)\right)\right)\right)\right)\right) \]

       8. +-commutative N/A

          \[\leadsto \mathsf{+.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(wj, \mathsf{+.f64}\left(-2, \mathsf{*.f64}\left(wj, \left(\frac{1}{x} + \color{blue}{\left(\frac{5}{2} + -1 \cdot \left(wj \cdot \left(\frac{8}{3} + \frac{1}{x}\right)\right)\right)}\right)\right)\right)\right)\right)\right) \]

       9. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(wj, \mathsf{+.f64}\left(-2, \mathsf{*.f64}\left(wj, \mathsf{+.f64}\left(\left(\frac{1}{x}\right), \color{blue}{\left(\frac{5}{2} + -1 \cdot \left(wj \cdot \left(\frac{8}{3} + \frac{1}{x}\right)\right)\right)}\right)\right)\right)\right)\right)\right) \]

       10. /-lowering-/.f64 N/A

           \[\leadsto \mathsf{+.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(wj, \mathsf{+.f64}\left(-2, \mathsf{*.f64}\left(wj, \mathsf{+.f64}\left(\mathsf{/.f64}\left(1, x\right), \left(\color{blue}{\frac{5}{2}} + -1 \cdot \left(wj \cdot \left(\frac{8}{3} + \frac{1}{x}\right)\right)\right)\right)\right)\right)\right)\right)\right) \]

       11. +-lowering-+.f64 N/A

           \[\leadsto \mathsf{+.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(wj, \mathsf{+.f64}\left(-2, \mathsf{*.f64}\left(wj, \mathsf{+.f64}\left(\mathsf{/.f64}\left(1, x\right), \mathsf{+.f64}\left(\frac{5}{2}, \color{blue}{\left(-1 \cdot \left(wj \cdot \left(\frac{8}{3} + \frac{1}{x}\right)\right)\right)}\right)\right)\right)\right)\right)\right)\right) \]

       12. mul-1-neg N/A

           \[\leadsto \mathsf{+.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(wj, \mathsf{+.f64}\left(-2, \mathsf{*.f64}\left(wj, \mathsf{+.f64}\left(\mathsf{/.f64}\left(1, x\right), \mathsf{+.f64}\left(\frac{5}{2}, \left(\mathsf{neg}\left(wj \cdot \left(\frac{8}{3} + \frac{1}{x}\right)\right)\right)\right)\right)\right)\right)\right)\right)\right) \]

       13. distribute-rgt-neg-in N/A

           \[\leadsto \mathsf{+.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(wj, \mathsf{+.f64}\left(-2, \mathsf{*.f64}\left(wj, \mathsf{+.f64}\left(\mathsf{/.f64}\left(1, x\right), \mathsf{+.f64}\left(\frac{5}{2}, \left(wj \cdot \color{blue}{\left(\mathsf{neg}\left(\left(\frac{8}{3} + \frac{1}{x}\right)\right)\right)}\right)\right)\right)\right)\right)\right)\right)\right) \]

       14. *-lowering-*.f64 N/A

           \[\leadsto \mathsf{+.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(wj, \mathsf{+.f64}\left(-2, \mathsf{*.f64}\left(wj, \mathsf{+.f64}\left(\mathsf{/.f64}\left(1, x\right), \mathsf{+.f64}\left(\frac{5}{2}, \mathsf{*.f64}\left(wj, \color{blue}{\left(\mathsf{neg}\left(\left(\frac{8}{3} + \frac{1}{x}\right)\right)\right)}\right)\right)\right)\right)\right)\right)\right)\right) \]

       15. distribute-neg-in N/A

           \[\leadsto \mathsf{+.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(wj, \mathsf{+.f64}\left(-2, \mathsf{*.f64}\left(wj, \mathsf{+.f64}\left(\mathsf{/.f64}\left(1, x\right), \mathsf{+.f64}\left(\frac{5}{2}, \mathsf{*.f64}\left(wj, \left(\left(\mathsf{neg}\left(\frac{8}{3}\right)\right) + \color{blue}{\left(\mathsf{neg}\left(\frac{1}{x}\right)\right)}\right)\right)\right)\right)\right)\right)\right)\right)\right) \]

       16. metadata-eval N/A

           \[\leadsto \mathsf{+.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(wj, \mathsf{+.f64}\left(-2, \mathsf{*.f64}\left(wj, \mathsf{+.f64}\left(\mathsf{/.f64}\left(1, x\right), \mathsf{+.f64}\left(\frac{5}{2}, \mathsf{*.f64}\left(wj, \left(\frac{-8}{3} + \left(\mathsf{neg}\left(\color{blue}{\frac{1}{x}}\right)\right)\right)\right)\right)\right)\right)\right)\right)\right)\right) \]

       17. +-lowering-+.f64 N/A

           \[\leadsto \mathsf{+.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(wj, \mathsf{+.f64}\left(-2, \mathsf{*.f64}\left(wj, \mathsf{+.f64}\left(\mathsf{/.f64}\left(1, x\right), \mathsf{+.f64}\left(\frac{5}{2}, \mathsf{*.f64}\left(wj, \mathsf{+.f64}\left(\frac{-8}{3}, \color{blue}{\left(\mathsf{neg}\left(\frac{1}{x}\right)\right)}\right)\right)\right)\right)\right)\right)\right)\right)\right) \]

       18. distribute-neg-frac N/A

           \[\leadsto \mathsf{+.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(wj, \mathsf{+.f64}\left(-2, \mathsf{*.f64}\left(wj, \mathsf{+.f64}\left(\mathsf{/.f64}\left(1, x\right), \mathsf{+.f64}\left(\frac{5}{2}, \mathsf{*.f64}\left(wj, \mathsf{+.f64}\left(\frac{-8}{3}, \left(\frac{\mathsf{neg}\left(1\right)}{\color{blue}{x}}\right)\right)\right)\right)\right)\right)\right)\right)\right)\right) \]

       19. metadata-eval N/A

           \[\leadsto \mathsf{+.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(wj, \mathsf{+.f64}\left(-2, \mathsf{*.f64}\left(wj, \mathsf{+.f64}\left(\mathsf{/.f64}\left(1, x\right), \mathsf{+.f64}\left(\frac{5}{2}, \mathsf{*.f64}\left(wj, \mathsf{+.f64}\left(\frac{-8}{3}, \left(\frac{-1}{x}\right)\right)\right)\right)\right)\right)\right)\right)\right)\right) \]

       20. /-lowering-/.f64 97.2%

           \[\leadsto \mathsf{+.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(wj, \mathsf{+.f64}\left(-2, \mathsf{*.f64}\left(wj, \mathsf{+.f64}\left(\mathsf{/.f64}\left(1, x\right), \mathsf{+.f64}\left(\frac{5}{2}, \mathsf{*.f64}\left(wj, \mathsf{+.f64}\left(\frac{-8}{3}, \mathsf{/.f64}\left(-1, \color{blue}{x}\right)\right)\right)\right)\right)\right)\right)\right)\right)\right) \]

   11. Simplified 97.2%

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

   12. Taylor expanded in x around 0

       \[\leadsto \mathsf{+.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(wj, \mathsf{+.f64}\left(-2, \mathsf{*.f64}\left(wj, \mathsf{+.f64}\left(\mathsf{/.f64}\left(1, x\right), \mathsf{+.f64}\left(\frac{5}{2}, \color{blue}{\left(-1 \cdot \frac{wj}{x}\right)}\right)\right)\right)\right)\right)\right)\right) \]
   13. Step-by-step derivation

       1. mul-1-neg N/A

          \[\leadsto \mathsf{+.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(wj, \mathsf{+.f64}\left(-2, \mathsf{*.f64}\left(wj, \mathsf{+.f64}\left(\mathsf{/.f64}\left(1, x\right), \mathsf{+.f64}\left(\frac{5}{2}, \left(\mathsf{neg}\left(\frac{wj}{x}\right)\right)\right)\right)\right)\right)\right)\right)\right) \]

       2. distribute-neg-frac2 N/A

          \[\leadsto \mathsf{+.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(wj, \mathsf{+.f64}\left(-2, \mathsf{*.f64}\left(wj, \mathsf{+.f64}\left(\mathsf{/.f64}\left(1, x\right), \mathsf{+.f64}\left(\frac{5}{2}, \left(\frac{wj}{\color{blue}{\mathsf{neg}\left(x\right)}}\right)\right)\right)\right)\right)\right)\right)\right) \]

       3. mul-1-neg N/A

          \[\leadsto \mathsf{+.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(wj, \mathsf{+.f64}\left(-2, \mathsf{*.f64}\left(wj, \mathsf{+.f64}\left(\mathsf{/.f64}\left(1, x\right), \mathsf{+.f64}\left(\frac{5}{2}, \left(\frac{wj}{-1 \cdot \color{blue}{x}}\right)\right)\right)\right)\right)\right)\right)\right) \]

       4. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(wj, \mathsf{+.f64}\left(-2, \mathsf{*.f64}\left(wj, \mathsf{+.f64}\left(\mathsf{/.f64}\left(1, x\right), \mathsf{+.f64}\left(\frac{5}{2}, \mathsf{/.f64}\left(wj, \color{blue}{\left(-1 \cdot x\right)}\right)\right)\right)\right)\right)\right)\right)\right) \]

       5. mul-1-neg N/A

          \[\leadsto \mathsf{+.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(wj, \mathsf{+.f64}\left(-2, \mathsf{*.f64}\left(wj, \mathsf{+.f64}\left(\mathsf{/.f64}\left(1, x\right), \mathsf{+.f64}\left(\frac{5}{2}, \mathsf{/.f64}\left(wj, \left(\mathsf{neg}\left(x\right)\right)\right)\right)\right)\right)\right)\right)\right)\right) \]

       6. neg-sub0 N/A

          \[\leadsto \mathsf{+.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(wj, \mathsf{+.f64}\left(-2, \mathsf{*.f64}\left(wj, \mathsf{+.f64}\left(\mathsf{/.f64}\left(1, x\right), \mathsf{+.f64}\left(\frac{5}{2}, \mathsf{/.f64}\left(wj, \left(0 - \color{blue}{x}\right)\right)\right)\right)\right)\right)\right)\right)\right) \]

       7. --lowering--.f64 97.1%

          \[\leadsto \mathsf{+.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(wj, \mathsf{+.f64}\left(-2, \mathsf{*.f64}\left(wj, \mathsf{+.f64}\left(\mathsf{/.f64}\left(1, x\right), \mathsf{+.f64}\left(\frac{5}{2}, \mathsf{/.f64}\left(wj, \mathsf{\_.f64}\left(0, \color{blue}{x}\right)\right)\right)\right)\right)\right)\right)\right)\right) \]

   14. Simplified 97.1%

       \[\leadsto x + x \cdot \left(wj \cdot \left(-2 + wj \cdot \left(\frac{1}{x} + \left(2.5 + \color{blue}{\frac{wj}{0 - x}}\right)\right)\right)\right) \]

   if 1.3000000000000001e-8 < wj

   1. Initial program 49.7%

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

      1. sub-neg N/A

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

      2. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(wj, \color{blue}{\left(\mathsf{neg}\left(\frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}}\right)\right)}\right) \]

      3. distribute-rgt1-in N/A

         \[\leadsto \mathsf{+.f64}\left(wj, \left(\mathsf{neg}\left(\frac{wj \cdot e^{wj} - x}{\left(wj + 1\right) \cdot e^{wj}}\right)\right)\right) \]

      4. associate-/l/ N/A

         \[\leadsto \mathsf{+.f64}\left(wj, \left(\mathsf{neg}\left(\frac{\frac{wj \cdot e^{wj} - x}{e^{wj}}}{wj + 1}\right)\right)\right) \]

      5. distribute-neg-frac2 N/A

         \[\leadsto \mathsf{+.f64}\left(wj, \left(\frac{\frac{wj \cdot e^{wj} - x}{e^{wj}}}{\color{blue}{\mathsf{neg}\left(\left(wj + 1\right)\right)}}\right)\right) \]

      6. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(wj, \mathsf{/.f64}\left(\left(\frac{wj \cdot e^{wj} - x}{e^{wj}}\right), \color{blue}{\left(\mathsf{neg}\left(\left(wj + 1\right)\right)\right)}\right)\right) \]

      7. div-sub N/A

         \[\leadsto \mathsf{+.f64}\left(wj, \mathsf{/.f64}\left(\left(\frac{wj \cdot e^{wj}}{e^{wj}} - \frac{x}{e^{wj}}\right), \left(\mathsf{neg}\left(\color{blue}{\left(wj + 1\right)}\right)\right)\right)\right) \]

      8. associate-/l* N/A

         \[\leadsto \mathsf{+.f64}\left(wj, \mathsf{/.f64}\left(\left(wj \cdot \frac{e^{wj}}{e^{wj}} - \frac{x}{e^{wj}}\right), \left(\mathsf{neg}\left(\left(\color{blue}{wj} + 1\right)\right)\right)\right)\right) \]

      9. *-inverses N/A

         \[\leadsto \mathsf{+.f64}\left(wj, \mathsf{/.f64}\left(\left(wj \cdot 1 - \frac{x}{e^{wj}}\right), \left(\mathsf{neg}\left(\left(wj + 1\right)\right)\right)\right)\right) \]

      10. *-rgt-identity N/A

          \[\leadsto \mathsf{+.f64}\left(wj, \mathsf{/.f64}\left(\left(wj - \frac{x}{e^{wj}}\right), \left(\mathsf{neg}\left(\left(\color{blue}{wj} + 1\right)\right)\right)\right)\right) \]

      11. --lowering--.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(wj, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(wj, \left(\frac{x}{e^{wj}}\right)\right), \left(\mathsf{neg}\left(\color{blue}{\left(wj + 1\right)}\right)\right)\right)\right) \]

      12. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(wj, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(wj, \mathsf{/.f64}\left(x, \left(e^{wj}\right)\right)\right), \left(\mathsf{neg}\left(\left(wj + \color{blue}{1}\right)\right)\right)\right)\right) \]

      13. exp-lowering-exp.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(wj, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(wj, \mathsf{/.f64}\left(x, \mathsf{exp.f64}\left(wj\right)\right)\right), \left(\mathsf{neg}\left(\left(wj + 1\right)\right)\right)\right)\right) \]

      14. +-commutative N/A

          \[\leadsto \mathsf{+.f64}\left(wj, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(wj, \mathsf{/.f64}\left(x, \mathsf{exp.f64}\left(wj\right)\right)\right), \left(\mathsf{neg}\left(\left(1 + wj\right)\right)\right)\right)\right) \]

      15. *-inverses N/A

          \[\leadsto \mathsf{+.f64}\left(wj, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(wj, \mathsf{/.f64}\left(x, \mathsf{exp.f64}\left(wj\right)\right)\right), \left(\mathsf{neg}\left(\left(\frac{e^{wj}}{e^{wj}} + wj\right)\right)\right)\right)\right) \]

      16. distribute-neg-in N/A

          \[\leadsto \mathsf{+.f64}\left(wj, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(wj, \mathsf{/.f64}\left(x, \mathsf{exp.f64}\left(wj\right)\right)\right), \left(\left(\mathsf{neg}\left(\frac{e^{wj}}{e^{wj}}\right)\right) + \color{blue}{\left(\mathsf{neg}\left(wj\right)\right)}\right)\right)\right) \]

   3. Simplified 99.7%

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

   5. Taylor expanded in wj around 0

      \[\leadsto \mathsf{+.f64}\left(wj, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(wj, \mathsf{/.f64}\left(x, \color{blue}{\left(1 + wj \cdot \left(1 + \frac{1}{2} \cdot wj\right)\right)}\right)\right), \mathsf{\_.f64}\left(-1, wj\right)\right)\right) \]
   6. Step-by-step derivation

      1. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(wj, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(wj, \mathsf{/.f64}\left(x, \mathsf{+.f64}\left(1, \left(wj \cdot \left(1 + \frac{1}{2} \cdot wj\right)\right)\right)\right)\right), \mathsf{\_.f64}\left(-1, wj\right)\right)\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(wj, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(wj, \mathsf{/.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(wj, \left(1 + \frac{1}{2} \cdot wj\right)\right)\right)\right)\right), \mathsf{\_.f64}\left(-1, wj\right)\right)\right) \]

      3. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(wj, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(wj, \mathsf{/.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(wj, \mathsf{+.f64}\left(1, \left(\frac{1}{2} \cdot wj\right)\right)\right)\right)\right)\right), \mathsf{\_.f64}\left(-1, wj\right)\right)\right) \]

      4. *-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(wj, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(wj, \mathsf{/.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(wj, \mathsf{+.f64}\left(1, \left(wj \cdot \frac{1}{2}\right)\right)\right)\right)\right)\right), \mathsf{\_.f64}\left(-1, wj\right)\right)\right) \]

      5. *-lowering-*.f64 95.1%

         \[\leadsto \mathsf{+.f64}\left(wj, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(wj, \mathsf{/.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(wj, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(wj, \frac{1}{2}\right)\right)\right)\right)\right)\right), \mathsf{\_.f64}\left(-1, wj\right)\right)\right) \]

   7. Simplified 95.1%

      \[\leadsto wj + \frac{wj - \frac{x}{\color{blue}{1 + wj \cdot \left(1 + wj \cdot 0.5\right)}}}{-1 - wj} \]
3. Recombined 2 regimes into one program.

4. Final simplification 97.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;wj \leq 1.3 \cdot 10^{-8}:\\ \;\;\;\;x + x \cdot \left(wj \cdot \left(-2 + wj \cdot \left(\frac{1}{x} + \left(2.5 - \frac{wj}{x}\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;wj + \frac{wj + \frac{x}{-1 + wj \cdot \left(-1 - wj \cdot 0.5\right)}}{-1 - wj}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 97.5% accurate, 13.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;wj \leq -0.000235:\\ \;\;\;\;wj + \frac{\frac{wj \cdot wj - x \cdot x}{wj + x}}{-1 - wj}\\ \mathbf{elif}\;wj \leq 1.3 \cdot 10^{-8}:\\ \;\;\;\;x + wj \cdot \left(x \cdot -2 + wj \cdot \left(1 - wj\right)\right)\\ \mathbf{else}:\\ \;\;\;\;wj + \frac{wj + \frac{x}{-1 - wj}}{-1 - wj}\\ \end{array} \end{array} \]

FPCore

(FPCore (wj x)
 :precision binary64
 (if (<= wj -0.000235)
   (+ wj (/ (/ (- (* wj wj) (* x x)) (+ wj x)) (- -1.0 wj)))
   (if (<= wj 1.3e-8)
     (+ x (* wj (+ (* x -2.0) (* wj (- 1.0 wj)))))
     (+ wj (/ (+ wj (/ x (- -1.0 wj))) (- -1.0 wj))))))

C

double code(double wj, double x) {
	double tmp;
	if (wj <= -0.000235) {
		tmp = wj + ((((wj * wj) - (x * x)) / (wj + x)) / (-1.0 - wj));
	} else if (wj <= 1.3e-8) {
		tmp = x + (wj * ((x * -2.0) + (wj * (1.0 - wj))));
	} else {
		tmp = wj + ((wj + (x / (-1.0 - wj))) / (-1.0 - wj));
	}
	return tmp;
}

Fortran

real(8) function code(wj, x)
    real(8), intent (in) :: wj
    real(8), intent (in) :: x
    real(8) :: tmp
    if (wj <= (-0.000235d0)) then
        tmp = wj + ((((wj * wj) - (x * x)) / (wj + x)) / ((-1.0d0) - wj))
    else if (wj <= 1.3d-8) then
        tmp = x + (wj * ((x * (-2.0d0)) + (wj * (1.0d0 - wj))))
    else
        tmp = wj + ((wj + (x / ((-1.0d0) - wj))) / ((-1.0d0) - wj))
    end if
    code = tmp
end function

Java

public static double code(double wj, double x) {
	double tmp;
	if (wj <= -0.000235) {
		tmp = wj + ((((wj * wj) - (x * x)) / (wj + x)) / (-1.0 - wj));
	} else if (wj <= 1.3e-8) {
		tmp = x + (wj * ((x * -2.0) + (wj * (1.0 - wj))));
	} else {
		tmp = wj + ((wj + (x / (-1.0 - wj))) / (-1.0 - wj));
	}
	return tmp;
}

Python

def code(wj, x):
	tmp = 0
	if wj <= -0.000235:
		tmp = wj + ((((wj * wj) - (x * x)) / (wj + x)) / (-1.0 - wj))
	elif wj <= 1.3e-8:
		tmp = x + (wj * ((x * -2.0) + (wj * (1.0 - wj))))
	else:
		tmp = wj + ((wj + (x / (-1.0 - wj))) / (-1.0 - wj))
	return tmp

Julia

function code(wj, x)
	tmp = 0.0
	if (wj <= -0.000235)
		tmp = Float64(wj + Float64(Float64(Float64(Float64(wj * wj) - Float64(x * x)) / Float64(wj + x)) / Float64(-1.0 - wj)));
	elseif (wj <= 1.3e-8)
		tmp = Float64(x + Float64(wj * Float64(Float64(x * -2.0) + Float64(wj * Float64(1.0 - wj)))));
	else
		tmp = Float64(wj + Float64(Float64(wj + Float64(x / Float64(-1.0 - wj))) / Float64(-1.0 - wj)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(wj, x)
	tmp = 0.0;
	if (wj <= -0.000235)
		tmp = wj + ((((wj * wj) - (x * x)) / (wj + x)) / (-1.0 - wj));
	elseif (wj <= 1.3e-8)
		tmp = x + (wj * ((x * -2.0) + (wj * (1.0 - wj))));
	else
		tmp = wj + ((wj + (x / (-1.0 - wj))) / (-1.0 - wj));
	end
	tmp_2 = tmp;
end

Wolfram

code[wj_, x_] := If[LessEqual[wj, -0.000235], N[(wj + N[(N[(N[(N[(wj * wj), $MachinePrecision] - N[(x * x), $MachinePrecision]), $MachinePrecision] / N[(wj + x), $MachinePrecision]), $MachinePrecision] / N[(-1.0 - wj), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[wj, 1.3e-8], N[(x + N[(wj * N[(N[(x * -2.0), $MachinePrecision] + N[(wj * N[(1.0 - wj), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(wj + N[(N[(wj + N[(x / N[(-1.0 - wj), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(-1.0 - wj), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if wj < -2.34999999999999993e-4

   1. Initial program 51.4%

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

      1. sub-neg N/A

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

      2. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(wj, \color{blue}{\left(\mathsf{neg}\left(\frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}}\right)\right)}\right) \]

      3. distribute-rgt1-in N/A

         \[\leadsto \mathsf{+.f64}\left(wj, \left(\mathsf{neg}\left(\frac{wj \cdot e^{wj} - x}{\left(wj + 1\right) \cdot e^{wj}}\right)\right)\right) \]

      4. associate-/l/ N/A

         \[\leadsto \mathsf{+.f64}\left(wj, \left(\mathsf{neg}\left(\frac{\frac{wj \cdot e^{wj} - x}{e^{wj}}}{wj + 1}\right)\right)\right) \]

      5. distribute-neg-frac2 N/A

         \[\leadsto \mathsf{+.f64}\left(wj, \left(\frac{\frac{wj \cdot e^{wj} - x}{e^{wj}}}{\color{blue}{\mathsf{neg}\left(\left(wj + 1\right)\right)}}\right)\right) \]

      6. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(wj, \mathsf{/.f64}\left(\left(\frac{wj \cdot e^{wj} - x}{e^{wj}}\right), \color{blue}{\left(\mathsf{neg}\left(\left(wj + 1\right)\right)\right)}\right)\right) \]

      7. div-sub N/A

         \[\leadsto \mathsf{+.f64}\left(wj, \mathsf{/.f64}\left(\left(\frac{wj \cdot e^{wj}}{e^{wj}} - \frac{x}{e^{wj}}\right), \left(\mathsf{neg}\left(\color{blue}{\left(wj + 1\right)}\right)\right)\right)\right) \]

      8. associate-/l* N/A

         \[\leadsto \mathsf{+.f64}\left(wj, \mathsf{/.f64}\left(\left(wj \cdot \frac{e^{wj}}{e^{wj}} - \frac{x}{e^{wj}}\right), \left(\mathsf{neg}\left(\left(\color{blue}{wj} + 1\right)\right)\right)\right)\right) \]

      9. *-inverses N/A

         \[\leadsto \mathsf{+.f64}\left(wj, \mathsf{/.f64}\left(\left(wj \cdot 1 - \frac{x}{e^{wj}}\right), \left(\mathsf{neg}\left(\left(wj + 1\right)\right)\right)\right)\right) \]

      10. *-rgt-identity N/A

          \[\leadsto \mathsf{+.f64}\left(wj, \mathsf{/.f64}\left(\left(wj - \frac{x}{e^{wj}}\right), \left(\mathsf{neg}\left(\left(\color{blue}{wj} + 1\right)\right)\right)\right)\right) \]

      11. --lowering--.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(wj, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(wj, \left(\frac{x}{e^{wj}}\right)\right), \left(\mathsf{neg}\left(\color{blue}{\left(wj + 1\right)}\right)\right)\right)\right) \]

      12. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(wj, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(wj, \mathsf{/.f64}\left(x, \left(e^{wj}\right)\right)\right), \left(\mathsf{neg}\left(\left(wj + \color{blue}{1}\right)\right)\right)\right)\right) \]

      13. exp-lowering-exp.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(wj, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(wj, \mathsf{/.f64}\left(x, \mathsf{exp.f64}\left(wj\right)\right)\right), \left(\mathsf{neg}\left(\left(wj + 1\right)\right)\right)\right)\right) \]

      14. +-commutative N/A

          \[\leadsto \mathsf{+.f64}\left(wj, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(wj, \mathsf{/.f64}\left(x, \mathsf{exp.f64}\left(wj\right)\right)\right), \left(\mathsf{neg}\left(\left(1 + wj\right)\right)\right)\right)\right) \]

      15. *-inverses N/A

          \[\leadsto \mathsf{+.f64}\left(wj, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(wj, \mathsf{/.f64}\left(x, \mathsf{exp.f64}\left(wj\right)\right)\right), \left(\mathsf{neg}\left(\left(\frac{e^{wj}}{e^{wj}} + wj\right)\right)\right)\right)\right) \]

      16. distribute-neg-in N/A

          \[\leadsto \mathsf{+.f64}\left(wj, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(wj, \mathsf{/.f64}\left(x, \mathsf{exp.f64}\left(wj\right)\right)\right), \left(\left(\mathsf{neg}\left(\frac{e^{wj}}{e^{wj}}\right)\right) + \color{blue}{\left(\mathsf{neg}\left(wj\right)\right)}\right)\right)\right) \]

   3. Simplified 96.1%

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

   5. Taylor expanded in wj around 0

      \[\leadsto \mathsf{+.f64}\left(wj, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(wj, \color{blue}{x}\right), \mathsf{\_.f64}\left(-1, wj\right)\right)\right) \]
   6. Step-by-step derivation

   7. Simplified 45.6%

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

      1. flip-- N/A

         \[\leadsto \mathsf{+.f64}\left(wj, \mathsf{/.f64}\left(\left(\frac{wj \cdot wj - x \cdot x}{wj + x}\right), \mathsf{\_.f64}\left(\color{blue}{-1}, wj\right)\right)\right) \]

      2. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(wj, \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(wj \cdot wj - x \cdot x\right), \left(wj + x\right)\right), \mathsf{\_.f64}\left(\color{blue}{-1}, wj\right)\right)\right) \]

      3. --lowering--.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(wj, \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(\left(wj \cdot wj\right), \left(x \cdot x\right)\right), \left(wj + x\right)\right), \mathsf{\_.f64}\left(-1, wj\right)\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(wj, \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(wj, wj\right), \left(x \cdot x\right)\right), \left(wj + x\right)\right), \mathsf{\_.f64}\left(-1, wj\right)\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(wj, \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(wj, wj\right), \mathsf{*.f64}\left(x, x\right)\right), \left(wj + x\right)\right), \mathsf{\_.f64}\left(-1, wj\right)\right)\right) \]

      6. +-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(wj, \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(wj, wj\right), \mathsf{*.f64}\left(x, x\right)\right), \left(x + wj\right)\right), \mathsf{\_.f64}\left(-1, wj\right)\right)\right) \]

      7. +-lowering-+.f64 66.7%

         \[\leadsto \mathsf{+.f64}\left(wj, \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(wj, wj\right), \mathsf{*.f64}\left(x, x\right)\right), \mathsf{+.f64}\left(x, wj\right)\right), \mathsf{\_.f64}\left(-1, wj\right)\right)\right) \]

   9. Applied egg-rr 66.7%

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

   if -2.34999999999999993e-4 < wj < 1.3000000000000001e-8

   1. Initial program 79.4%

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

   3. Taylor expanded in wj around 0

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

      1. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(x, \color{blue}{\left(wj \cdot \left(wj \cdot \left(\left(1 + -1 \cdot \left(wj \cdot \left(1 + \left(-3 \cdot x + \left(-2 \cdot \left(-4 \cdot x + \frac{3}{2} \cdot x\right) + \frac{2}{3} \cdot x\right)\right)\right)\right)\right) - \left(-4 \cdot x + \frac{3}{2} \cdot x\right)\right) - 2 \cdot x\right)\right)}\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(x, \mathsf{*.f64}\left(wj, \color{blue}{\left(wj \cdot \left(\left(1 + -1 \cdot \left(wj \cdot \left(1 + \left(-3 \cdot x + \left(-2 \cdot \left(-4 \cdot x + \frac{3}{2} \cdot x\right) + \frac{2}{3} \cdot x\right)\right)\right)\right)\right) - \left(-4 \cdot x + \frac{3}{2} \cdot x\right)\right) - 2 \cdot x\right)}\right)\right) \]

      3. cancel-sign-sub-inv N/A

         \[\leadsto \mathsf{+.f64}\left(x, \mathsf{*.f64}\left(wj, \left(wj \cdot \left(\left(1 + -1 \cdot \left(wj \cdot \left(1 + \left(-3 \cdot x + \left(-2 \cdot \left(-4 \cdot x + \frac{3}{2} \cdot x\right) + \frac{2}{3} \cdot x\right)\right)\right)\right)\right) - \left(-4 \cdot x + \frac{3}{2} \cdot x\right)\right) + \color{blue}{\left(\mathsf{neg}\left(2\right)\right) \cdot x}\right)\right)\right) \]

      4. metadata-eval N/A

         \[\leadsto \mathsf{+.f64}\left(x, \mathsf{*.f64}\left(wj, \left(wj \cdot \left(\left(1 + -1 \cdot \left(wj \cdot \left(1 + \left(-3 \cdot x + \left(-2 \cdot \left(-4 \cdot x + \frac{3}{2} \cdot x\right) + \frac{2}{3} \cdot x\right)\right)\right)\right)\right) - \left(-4 \cdot x + \frac{3}{2} \cdot x\right)\right) + -2 \cdot x\right)\right)\right) \]

      5. +-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(x, \mathsf{*.f64}\left(wj, \left(-2 \cdot x + \color{blue}{wj \cdot \left(\left(1 + -1 \cdot \left(wj \cdot \left(1 + \left(-3 \cdot x + \left(-2 \cdot \left(-4 \cdot x + \frac{3}{2} \cdot x\right) + \frac{2}{3} \cdot x\right)\right)\right)\right)\right) - \left(-4 \cdot x + \frac{3}{2} \cdot x\right)\right)}\right)\right)\right) \]

      6. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(x, \mathsf{*.f64}\left(wj, \mathsf{+.f64}\left(\left(-2 \cdot x\right), \color{blue}{\left(wj \cdot \left(\left(1 + -1 \cdot \left(wj \cdot \left(1 + \left(-3 \cdot x + \left(-2 \cdot \left(-4 \cdot x + \frac{3}{2} \cdot x\right) + \frac{2}{3} \cdot x\right)\right)\right)\right)\right) - \left(-4 \cdot x + \frac{3}{2} \cdot x\right)\right)\right)}\right)\right)\right) \]

   5. Simplified 100.0%

      \[\leadsto \color{blue}{x + wj \cdot \left(x \cdot -2 + wj \cdot \left(\left(1 - wj \cdot \left(\left(x \cdot -3 + x \cdot 5\right) + \left(x \cdot 0.6666666666666666 + 1\right)\right)\right) + x \cdot 2.5\right)\right)} \]

   6. Taylor expanded in x around 0

      \[\leadsto \mathsf{+.f64}\left(x, \mathsf{*.f64}\left(wj, \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, -2\right), \color{blue}{\left(wj \cdot \left(1 - wj\right)\right)}\right)\right)\right) \]
   7. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(x, \mathsf{*.f64}\left(wj, \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, -2\right), \mathsf{*.f64}\left(wj, \color{blue}{\left(1 - wj\right)}\right)\right)\right)\right) \]

      2. --lowering--.f64 99.8%

         \[\leadsto \mathsf{+.f64}\left(x, \mathsf{*.f64}\left(wj, \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, -2\right), \mathsf{*.f64}\left(wj, \mathsf{\_.f64}\left(1, \color{blue}{wj}\right)\right)\right)\right)\right) \]

   8. Simplified 99.8%

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

   if 1.3000000000000001e-8 < wj

   1. Initial program 49.7%

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

      1. sub-neg N/A

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

      2. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(wj, \color{blue}{\left(\mathsf{neg}\left(\frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}}\right)\right)}\right) \]

      3. distribute-rgt1-in N/A

         \[\leadsto \mathsf{+.f64}\left(wj, \left(\mathsf{neg}\left(\frac{wj \cdot e^{wj} - x}{\left(wj + 1\right) \cdot e^{wj}}\right)\right)\right) \]

      4. associate-/l/ N/A

         \[\leadsto \mathsf{+.f64}\left(wj, \left(\mathsf{neg}\left(\frac{\frac{wj \cdot e^{wj} - x}{e^{wj}}}{wj + 1}\right)\right)\right) \]

      5. distribute-neg-frac2 N/A

         \[\leadsto \mathsf{+.f64}\left(wj, \left(\frac{\frac{wj \cdot e^{wj} - x}{e^{wj}}}{\color{blue}{\mathsf{neg}\left(\left(wj + 1\right)\right)}}\right)\right) \]

      6. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(wj, \mathsf{/.f64}\left(\left(\frac{wj \cdot e^{wj} - x}{e^{wj}}\right), \color{blue}{\left(\mathsf{neg}\left(\left(wj + 1\right)\right)\right)}\right)\right) \]

      7. div-sub N/A

         \[\leadsto \mathsf{+.f64}\left(wj, \mathsf{/.f64}\left(\left(\frac{wj \cdot e^{wj}}{e^{wj}} - \frac{x}{e^{wj}}\right), \left(\mathsf{neg}\left(\color{blue}{\left(wj + 1\right)}\right)\right)\right)\right) \]

      8. associate-/l* N/A

         \[\leadsto \mathsf{+.f64}\left(wj, \mathsf{/.f64}\left(\left(wj \cdot \frac{e^{wj}}{e^{wj}} - \frac{x}{e^{wj}}\right), \left(\mathsf{neg}\left(\left(\color{blue}{wj} + 1\right)\right)\right)\right)\right) \]

      9. *-inverses N/A

         \[\leadsto \mathsf{+.f64}\left(wj, \mathsf{/.f64}\left(\left(wj \cdot 1 - \frac{x}{e^{wj}}\right), \left(\mathsf{neg}\left(\left(wj + 1\right)\right)\right)\right)\right) \]

      10. *-rgt-identity N/A

          \[\leadsto \mathsf{+.f64}\left(wj, \mathsf{/.f64}\left(\left(wj - \frac{x}{e^{wj}}\right), \left(\mathsf{neg}\left(\left(\color{blue}{wj} + 1\right)\right)\right)\right)\right) \]

      11. --lowering--.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(wj, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(wj, \left(\frac{x}{e^{wj}}\right)\right), \left(\mathsf{neg}\left(\color{blue}{\left(wj + 1\right)}\right)\right)\right)\right) \]

      12. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(wj, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(wj, \mathsf{/.f64}\left(x, \left(e^{wj}\right)\right)\right), \left(\mathsf{neg}\left(\left(wj + \color{blue}{1}\right)\right)\right)\right)\right) \]

      13. exp-lowering-exp.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(wj, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(wj, \mathsf{/.f64}\left(x, \mathsf{exp.f64}\left(wj\right)\right)\right), \left(\mathsf{neg}\left(\left(wj + 1\right)\right)\right)\right)\right) \]

      14. +-commutative N/A

          \[\leadsto \mathsf{+.f64}\left(wj, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(wj, \mathsf{/.f64}\left(x, \mathsf{exp.f64}\left(wj\right)\right)\right), \left(\mathsf{neg}\left(\left(1 + wj\right)\right)\right)\right)\right) \]

      15. *-inverses N/A

          \[\leadsto \mathsf{+.f64}\left(wj, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(wj, \mathsf{/.f64}\left(x, \mathsf{exp.f64}\left(wj\right)\right)\right), \left(\mathsf{neg}\left(\left(\frac{e^{wj}}{e^{wj}} + wj\right)\right)\right)\right)\right) \]

      16. distribute-neg-in N/A

          \[\leadsto \mathsf{+.f64}\left(wj, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(wj, \mathsf{/.f64}\left(x, \mathsf{exp.f64}\left(wj\right)\right)\right), \left(\left(\mathsf{neg}\left(\frac{e^{wj}}{e^{wj}}\right)\right) + \color{blue}{\left(\mathsf{neg}\left(wj\right)\right)}\right)\right)\right) \]

   3. Simplified 99.7%

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

   5. Taylor expanded in wj around 0

      \[\leadsto \mathsf{+.f64}\left(wj, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(wj, \mathsf{/.f64}\left(x, \color{blue}{\left(1 + wj\right)}\right)\right), \mathsf{\_.f64}\left(-1, wj\right)\right)\right) \]
   6. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(wj, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(wj, \mathsf{/.f64}\left(x, \left(wj + 1\right)\right)\right), \mathsf{\_.f64}\left(-1, wj\right)\right)\right) \]

      2. +-lowering-+.f64 92.0%

         \[\leadsto \mathsf{+.f64}\left(wj, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(wj, \mathsf{/.f64}\left(x, \mathsf{+.f64}\left(wj, 1\right)\right)\right), \mathsf{\_.f64}\left(-1, wj\right)\right)\right) \]

   7. Simplified 92.0%

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

4. Final simplification 98.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;wj \leq -0.000235:\\ \;\;\;\;wj + \frac{\frac{wj \cdot wj - x \cdot x}{wj + x}}{-1 - wj}\\ \mathbf{elif}\;wj \leq 1.3 \cdot 10^{-8}:\\ \;\;\;\;x + wj \cdot \left(x \cdot -2 + wj \cdot \left(1 - wj\right)\right)\\ \mathbf{else}:\\ \;\;\;\;wj + \frac{wj + \frac{x}{-1 - wj}}{-1 - wj}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 85.8% accurate, 16.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := wj + \frac{wj - x}{-1 - wj}\\ \mathbf{if}\;wj \leq -1.15 \cdot 10^{-9}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;wj \leq 2.2 \cdot 10^{-14}:\\ \;\;\;\;x + wj \cdot \left(x \cdot -2\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (wj x)
 :precision binary64
 (let* ((t_0 (+ wj (/ (- wj x) (- -1.0 wj)))))
   (if (<= wj -1.15e-9) t_0 (if (<= wj 2.2e-14) (+ x (* wj (* x -2.0))) t_0))))

C

double code(double wj, double x) {
	double t_0 = wj + ((wj - x) / (-1.0 - wj));
	double tmp;
	if (wj <= -1.15e-9) {
		tmp = t_0;
	} else if (wj <= 2.2e-14) {
		tmp = x + (wj * (x * -2.0));
	} 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 = wj + ((wj - x) / ((-1.0d0) - wj))
    if (wj <= (-1.15d-9)) then
        tmp = t_0
    else if (wj <= 2.2d-14) then
        tmp = x + (wj * (x * (-2.0d0)))
    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) / (-1.0 - wj));
	double tmp;
	if (wj <= -1.15e-9) {
		tmp = t_0;
	} else if (wj <= 2.2e-14) {
		tmp = x + (wj * (x * -2.0));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(wj, x):
	t_0 = wj + ((wj - x) / (-1.0 - wj))
	tmp = 0
	if wj <= -1.15e-9:
		tmp = t_0
	elif wj <= 2.2e-14:
		tmp = x + (wj * (x * -2.0))
	else:
		tmp = t_0
	return tmp

Julia

function code(wj, x)
	t_0 = Float64(wj + Float64(Float64(wj - x) / Float64(-1.0 - wj)))
	tmp = 0.0
	if (wj <= -1.15e-9)
		tmp = t_0;
	elseif (wj <= 2.2e-14)
		tmp = Float64(x + Float64(wj * Float64(x * -2.0)));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(wj, x)
	t_0 = wj + ((wj - x) / (-1.0 - wj));
	tmp = 0.0;
	if (wj <= -1.15e-9)
		tmp = t_0;
	elseif (wj <= 2.2e-14)
		tmp = x + (wj * (x * -2.0));
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[wj_, x_] := Block[{t$95$0 = N[(wj + N[(N[(wj - x), $MachinePrecision] / N[(-1.0 - wj), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[wj, -1.15e-9], t$95$0, If[LessEqual[wj, 2.2e-14], N[(x + N[(wj * N[(x * -2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if wj < -1.15e-9 or 2.2000000000000001e-14 < wj

   1. Initial program 57.3%

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

      1. sub-neg N/A

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

      2. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(wj, \color{blue}{\left(\mathsf{neg}\left(\frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}}\right)\right)}\right) \]

      3. distribute-rgt1-in N/A

         \[\leadsto \mathsf{+.f64}\left(wj, \left(\mathsf{neg}\left(\frac{wj \cdot e^{wj} - x}{\left(wj + 1\right) \cdot e^{wj}}\right)\right)\right) \]

      4. associate-/l/ N/A

         \[\leadsto \mathsf{+.f64}\left(wj, \left(\mathsf{neg}\left(\frac{\frac{wj \cdot e^{wj} - x}{e^{wj}}}{wj + 1}\right)\right)\right) \]

      5. distribute-neg-frac2 N/A

         \[\leadsto \mathsf{+.f64}\left(wj, \left(\frac{\frac{wj \cdot e^{wj} - x}{e^{wj}}}{\color{blue}{\mathsf{neg}\left(\left(wj + 1\right)\right)}}\right)\right) \]

      6. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(wj, \mathsf{/.f64}\left(\left(\frac{wj \cdot e^{wj} - x}{e^{wj}}\right), \color{blue}{\left(\mathsf{neg}\left(\left(wj + 1\right)\right)\right)}\right)\right) \]

      7. div-sub N/A

         \[\leadsto \mathsf{+.f64}\left(wj, \mathsf{/.f64}\left(\left(\frac{wj \cdot e^{wj}}{e^{wj}} - \frac{x}{e^{wj}}\right), \left(\mathsf{neg}\left(\color{blue}{\left(wj + 1\right)}\right)\right)\right)\right) \]

      8. associate-/l* N/A

         \[\leadsto \mathsf{+.f64}\left(wj, \mathsf{/.f64}\left(\left(wj \cdot \frac{e^{wj}}{e^{wj}} - \frac{x}{e^{wj}}\right), \left(\mathsf{neg}\left(\left(\color{blue}{wj} + 1\right)\right)\right)\right)\right) \]

      9. *-inverses N/A

         \[\leadsto \mathsf{+.f64}\left(wj, \mathsf{/.f64}\left(\left(wj \cdot 1 - \frac{x}{e^{wj}}\right), \left(\mathsf{neg}\left(\left(wj + 1\right)\right)\right)\right)\right) \]

      10. *-rgt-identity N/A

          \[\leadsto \mathsf{+.f64}\left(wj, \mathsf{/.f64}\left(\left(wj - \frac{x}{e^{wj}}\right), \left(\mathsf{neg}\left(\left(\color{blue}{wj} + 1\right)\right)\right)\right)\right) \]

      11. --lowering--.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(wj, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(wj, \left(\frac{x}{e^{wj}}\right)\right), \left(\mathsf{neg}\left(\color{blue}{\left(wj + 1\right)}\right)\right)\right)\right) \]

      12. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(wj, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(wj, \mathsf{/.f64}\left(x, \left(e^{wj}\right)\right)\right), \left(\mathsf{neg}\left(\left(wj + \color{blue}{1}\right)\right)\right)\right)\right) \]

      13. exp-lowering-exp.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(wj, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(wj, \mathsf{/.f64}\left(x, \mathsf{exp.f64}\left(wj\right)\right)\right), \left(\mathsf{neg}\left(\left(wj + 1\right)\right)\right)\right)\right) \]

      14. +-commutative N/A

          \[\leadsto \mathsf{+.f64}\left(wj, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(wj, \mathsf{/.f64}\left(x, \mathsf{exp.f64}\left(wj\right)\right)\right), \left(\mathsf{neg}\left(\left(1 + wj\right)\right)\right)\right)\right) \]

      15. *-inverses N/A

          \[\leadsto \mathsf{+.f64}\left(wj, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(wj, \mathsf{/.f64}\left(x, \mathsf{exp.f64}\left(wj\right)\right)\right), \left(\mathsf{neg}\left(\left(\frac{e^{wj}}{e^{wj}} + wj\right)\right)\right)\right)\right) \]

      16. distribute-neg-in N/A

          \[\leadsto \mathsf{+.f64}\left(wj, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(wj, \mathsf{/.f64}\left(x, \mathsf{exp.f64}\left(wj\right)\right)\right), \left(\left(\mathsf{neg}\left(\frac{e^{wj}}{e^{wj}}\right)\right) + \color{blue}{\left(\mathsf{neg}\left(wj\right)\right)}\right)\right)\right) \]

   3. Simplified 90.5%

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

   5. Taylor expanded in wj around 0

      \[\leadsto \mathsf{+.f64}\left(wj, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(wj, \color{blue}{x}\right), \mathsf{\_.f64}\left(-1, wj\right)\right)\right) \]
   6. Step-by-step derivation

   7. Simplified 59.1%

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

   if -1.15e-9 < wj < 2.2000000000000001e-14

   1. Initial program 79.6%

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

   3. Taylor expanded in wj around 0

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

      1. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(x, \color{blue}{\left(wj \cdot \left(wj \cdot \left(\left(1 + -1 \cdot \left(wj \cdot \left(1 + \left(-3 \cdot x + \left(-2 \cdot \left(-4 \cdot x + \frac{3}{2} \cdot x\right) + \frac{2}{3} \cdot x\right)\right)\right)\right)\right) - \left(-4 \cdot x + \frac{3}{2} \cdot x\right)\right) - 2 \cdot x\right)\right)}\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(x, \mathsf{*.f64}\left(wj, \color{blue}{\left(wj \cdot \left(\left(1 + -1 \cdot \left(wj \cdot \left(1 + \left(-3 \cdot x + \left(-2 \cdot \left(-4 \cdot x + \frac{3}{2} \cdot x\right) + \frac{2}{3} \cdot x\right)\right)\right)\right)\right) - \left(-4 \cdot x + \frac{3}{2} \cdot x\right)\right) - 2 \cdot x\right)}\right)\right) \]

      3. cancel-sign-sub-inv N/A

         \[\leadsto \mathsf{+.f64}\left(x, \mathsf{*.f64}\left(wj, \left(wj \cdot \left(\left(1 + -1 \cdot \left(wj \cdot \left(1 + \left(-3 \cdot x + \left(-2 \cdot \left(-4 \cdot x + \frac{3}{2} \cdot x\right) + \frac{2}{3} \cdot x\right)\right)\right)\right)\right) - \left(-4 \cdot x + \frac{3}{2} \cdot x\right)\right) + \color{blue}{\left(\mathsf{neg}\left(2\right)\right) \cdot x}\right)\right)\right) \]

      4. metadata-eval N/A

         \[\leadsto \mathsf{+.f64}\left(x, \mathsf{*.f64}\left(wj, \left(wj \cdot \left(\left(1 + -1 \cdot \left(wj \cdot \left(1 + \left(-3 \cdot x + \left(-2 \cdot \left(-4 \cdot x + \frac{3}{2} \cdot x\right) + \frac{2}{3} \cdot x\right)\right)\right)\right)\right) - \left(-4 \cdot x + \frac{3}{2} \cdot x\right)\right) + -2 \cdot x\right)\right)\right) \]

      5. +-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(x, \mathsf{*.f64}\left(wj, \left(-2 \cdot x + \color{blue}{wj \cdot \left(\left(1 + -1 \cdot \left(wj \cdot \left(1 + \left(-3 \cdot x + \left(-2 \cdot \left(-4 \cdot x + \frac{3}{2} \cdot x\right) + \frac{2}{3} \cdot x\right)\right)\right)\right)\right) - \left(-4 \cdot x + \frac{3}{2} \cdot x\right)\right)}\right)\right)\right) \]

      6. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(x, \mathsf{*.f64}\left(wj, \mathsf{+.f64}\left(\left(-2 \cdot x\right), \color{blue}{\left(wj \cdot \left(\left(1 + -1 \cdot \left(wj \cdot \left(1 + \left(-3 \cdot x + \left(-2 \cdot \left(-4 \cdot x + \frac{3}{2} \cdot x\right) + \frac{2}{3} \cdot x\right)\right)\right)\right)\right) - \left(-4 \cdot x + \frac{3}{2} \cdot x\right)\right)\right)}\right)\right)\right) \]

   5. Simplified 100.0%

      \[\leadsto \color{blue}{x + wj \cdot \left(x \cdot -2 + wj \cdot \left(\left(1 - wj \cdot \left(\left(x \cdot -3 + x \cdot 5\right) + \left(x \cdot 0.6666666666666666 + 1\right)\right)\right) + x \cdot 2.5\right)\right)} \]

   6. Taylor expanded in wj around 0

      \[\leadsto \mathsf{+.f64}\left(x, \color{blue}{\left(-2 \cdot \left(wj \cdot x\right)\right)}\right) \]
   7. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(x, \left(\left(wj \cdot x\right) \cdot \color{blue}{-2}\right)\right) \]

      2. associate-*r* N/A

         \[\leadsto \mathsf{+.f64}\left(x, \left(wj \cdot \color{blue}{\left(x \cdot -2\right)}\right)\right) \]

      3. *-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(x, \left(wj \cdot \left(-2 \cdot \color{blue}{x}\right)\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(x, \mathsf{*.f64}\left(wj, \color{blue}{\left(-2 \cdot x\right)}\right)\right) \]

      5. *-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(x, \mathsf{*.f64}\left(wj, \left(x \cdot \color{blue}{-2}\right)\right)\right) \]

      6. *-lowering-*.f64 91.7%

         \[\leadsto \mathsf{+.f64}\left(x, \mathsf{*.f64}\left(wj, \mathsf{*.f64}\left(x, \color{blue}{-2}\right)\right)\right) \]

   8. Simplified 91.7%

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

Alternative 12: 96.9% accurate, 17.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;wj \leq 1.3 \cdot 10^{-8}:\\ \;\;\;\;x + wj \cdot \left(x \cdot -2 + wj \cdot \left(1 - wj\right)\right)\\ \mathbf{else}:\\ \;\;\;\;wj + \frac{wj + \frac{x}{-1 - wj}}{-1 - wj}\\ \end{array} \end{array} \]

FPCore

(FPCore (wj x)
 :precision binary64
 (if (<= wj 1.3e-8)
   (+ x (* wj (+ (* x -2.0) (* wj (- 1.0 wj)))))
   (+ wj (/ (+ wj (/ x (- -1.0 wj))) (- -1.0 wj)))))

C

double code(double wj, double x) {
	double tmp;
	if (wj <= 1.3e-8) {
		tmp = x + (wj * ((x * -2.0) + (wj * (1.0 - wj))));
	} else {
		tmp = wj + ((wj + (x / (-1.0 - wj))) / (-1.0 - wj));
	}
	return tmp;
}

Fortran

real(8) function code(wj, x)
    real(8), intent (in) :: wj
    real(8), intent (in) :: x
    real(8) :: tmp
    if (wj <= 1.3d-8) then
        tmp = x + (wj * ((x * (-2.0d0)) + (wj * (1.0d0 - wj))))
    else
        tmp = wj + ((wj + (x / ((-1.0d0) - wj))) / ((-1.0d0) - wj))
    end if
    code = tmp
end function

Java

public static double code(double wj, double x) {
	double tmp;
	if (wj <= 1.3e-8) {
		tmp = x + (wj * ((x * -2.0) + (wj * (1.0 - wj))));
	} else {
		tmp = wj + ((wj + (x / (-1.0 - wj))) / (-1.0 - wj));
	}
	return tmp;
}

Python

def code(wj, x):
	tmp = 0
	if wj <= 1.3e-8:
		tmp = x + (wj * ((x * -2.0) + (wj * (1.0 - wj))))
	else:
		tmp = wj + ((wj + (x / (-1.0 - wj))) / (-1.0 - wj))
	return tmp

Julia

function code(wj, x)
	tmp = 0.0
	if (wj <= 1.3e-8)
		tmp = Float64(x + Float64(wj * Float64(Float64(x * -2.0) + Float64(wj * Float64(1.0 - wj)))));
	else
		tmp = Float64(wj + Float64(Float64(wj + Float64(x / Float64(-1.0 - wj))) / Float64(-1.0 - wj)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(wj, x)
	tmp = 0.0;
	if (wj <= 1.3e-8)
		tmp = x + (wj * ((x * -2.0) + (wj * (1.0 - wj))));
	else
		tmp = wj + ((wj + (x / (-1.0 - wj))) / (-1.0 - wj));
	end
	tmp_2 = tmp;
end

Wolfram

code[wj_, x_] := If[LessEqual[wj, 1.3e-8], N[(x + N[(wj * N[(N[(x * -2.0), $MachinePrecision] + N[(wj * N[(1.0 - wj), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(wj + N[(N[(wj + N[(x / N[(-1.0 - wj), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(-1.0 - wj), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if wj < 1.3000000000000001e-8

   1. Initial program 78.4%

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

   3. Taylor expanded in wj around 0

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

      1. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(x, \color{blue}{\left(wj \cdot \left(wj \cdot \left(\left(1 + -1 \cdot \left(wj \cdot \left(1 + \left(-3 \cdot x + \left(-2 \cdot \left(-4 \cdot x + \frac{3}{2} \cdot x\right) + \frac{2}{3} \cdot x\right)\right)\right)\right)\right) - \left(-4 \cdot x + \frac{3}{2} \cdot x\right)\right) - 2 \cdot x\right)\right)}\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(x, \mathsf{*.f64}\left(wj, \color{blue}{\left(wj \cdot \left(\left(1 + -1 \cdot \left(wj \cdot \left(1 + \left(-3 \cdot x + \left(-2 \cdot \left(-4 \cdot x + \frac{3}{2} \cdot x\right) + \frac{2}{3} \cdot x\right)\right)\right)\right)\right) - \left(-4 \cdot x + \frac{3}{2} \cdot x\right)\right) - 2 \cdot x\right)}\right)\right) \]

      3. cancel-sign-sub-inv N/A

         \[\leadsto \mathsf{+.f64}\left(x, \mathsf{*.f64}\left(wj, \left(wj \cdot \left(\left(1 + -1 \cdot \left(wj \cdot \left(1 + \left(-3 \cdot x + \left(-2 \cdot \left(-4 \cdot x + \frac{3}{2} \cdot x\right) + \frac{2}{3} \cdot x\right)\right)\right)\right)\right) - \left(-4 \cdot x + \frac{3}{2} \cdot x\right)\right) + \color{blue}{\left(\mathsf{neg}\left(2\right)\right) \cdot x}\right)\right)\right) \]

      4. metadata-eval N/A

         \[\leadsto \mathsf{+.f64}\left(x, \mathsf{*.f64}\left(wj, \left(wj \cdot \left(\left(1 + -1 \cdot \left(wj \cdot \left(1 + \left(-3 \cdot x + \left(-2 \cdot \left(-4 \cdot x + \frac{3}{2} \cdot x\right) + \frac{2}{3} \cdot x\right)\right)\right)\right)\right) - \left(-4 \cdot x + \frac{3}{2} \cdot x\right)\right) + -2 \cdot x\right)\right)\right) \]

      5. +-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(x, \mathsf{*.f64}\left(wj, \left(-2 \cdot x + \color{blue}{wj \cdot \left(\left(1 + -1 \cdot \left(wj \cdot \left(1 + \left(-3 \cdot x + \left(-2 \cdot \left(-4 \cdot x + \frac{3}{2} \cdot x\right) + \frac{2}{3} \cdot x\right)\right)\right)\right)\right) - \left(-4 \cdot x + \frac{3}{2} \cdot x\right)\right)}\right)\right)\right) \]

      6. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(x, \mathsf{*.f64}\left(wj, \mathsf{+.f64}\left(\left(-2 \cdot x\right), \color{blue}{\left(wj \cdot \left(\left(1 + -1 \cdot \left(wj \cdot \left(1 + \left(-3 \cdot x + \left(-2 \cdot \left(-4 \cdot x + \frac{3}{2} \cdot x\right) + \frac{2}{3} \cdot x\right)\right)\right)\right)\right) - \left(-4 \cdot x + \frac{3}{2} \cdot x\right)\right)\right)}\right)\right)\right) \]

   5. Simplified 97.3%

      \[\leadsto \color{blue}{x + wj \cdot \left(x \cdot -2 + wj \cdot \left(\left(1 - wj \cdot \left(\left(x \cdot -3 + x \cdot 5\right) + \left(x \cdot 0.6666666666666666 + 1\right)\right)\right) + x \cdot 2.5\right)\right)} \]

   6. Taylor expanded in x around 0

      \[\leadsto \mathsf{+.f64}\left(x, \mathsf{*.f64}\left(wj, \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, -2\right), \color{blue}{\left(wj \cdot \left(1 - wj\right)\right)}\right)\right)\right) \]
   7. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(x, \mathsf{*.f64}\left(wj, \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, -2\right), \mathsf{*.f64}\left(wj, \color{blue}{\left(1 - wj\right)}\right)\right)\right)\right) \]

      2. --lowering--.f64 97.0%

         \[\leadsto \mathsf{+.f64}\left(x, \mathsf{*.f64}\left(wj, \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, -2\right), \mathsf{*.f64}\left(wj, \mathsf{\_.f64}\left(1, \color{blue}{wj}\right)\right)\right)\right)\right) \]

   8. Simplified 97.0%

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

   if 1.3000000000000001e-8 < wj

   1. Initial program 49.7%

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

      1. sub-neg N/A

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

      2. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(wj, \color{blue}{\left(\mathsf{neg}\left(\frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}}\right)\right)}\right) \]

      3. distribute-rgt1-in N/A

         \[\leadsto \mathsf{+.f64}\left(wj, \left(\mathsf{neg}\left(\frac{wj \cdot e^{wj} - x}{\left(wj + 1\right) \cdot e^{wj}}\right)\right)\right) \]

      4. associate-/l/ N/A

         \[\leadsto \mathsf{+.f64}\left(wj, \left(\mathsf{neg}\left(\frac{\frac{wj \cdot e^{wj} - x}{e^{wj}}}{wj + 1}\right)\right)\right) \]

      5. distribute-neg-frac2 N/A

         \[\leadsto \mathsf{+.f64}\left(wj, \left(\frac{\frac{wj \cdot e^{wj} - x}{e^{wj}}}{\color{blue}{\mathsf{neg}\left(\left(wj + 1\right)\right)}}\right)\right) \]

      6. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(wj, \mathsf{/.f64}\left(\left(\frac{wj \cdot e^{wj} - x}{e^{wj}}\right), \color{blue}{\left(\mathsf{neg}\left(\left(wj + 1\right)\right)\right)}\right)\right) \]

      7. div-sub N/A

         \[\leadsto \mathsf{+.f64}\left(wj, \mathsf{/.f64}\left(\left(\frac{wj \cdot e^{wj}}{e^{wj}} - \frac{x}{e^{wj}}\right), \left(\mathsf{neg}\left(\color{blue}{\left(wj + 1\right)}\right)\right)\right)\right) \]

      8. associate-/l* N/A

         \[\leadsto \mathsf{+.f64}\left(wj, \mathsf{/.f64}\left(\left(wj \cdot \frac{e^{wj}}{e^{wj}} - \frac{x}{e^{wj}}\right), \left(\mathsf{neg}\left(\left(\color{blue}{wj} + 1\right)\right)\right)\right)\right) \]

      9. *-inverses N/A

         \[\leadsto \mathsf{+.f64}\left(wj, \mathsf{/.f64}\left(\left(wj \cdot 1 - \frac{x}{e^{wj}}\right), \left(\mathsf{neg}\left(\left(wj + 1\right)\right)\right)\right)\right) \]

      10. *-rgt-identity N/A

          \[\leadsto \mathsf{+.f64}\left(wj, \mathsf{/.f64}\left(\left(wj - \frac{x}{e^{wj}}\right), \left(\mathsf{neg}\left(\left(\color{blue}{wj} + 1\right)\right)\right)\right)\right) \]

      11. --lowering--.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(wj, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(wj, \left(\frac{x}{e^{wj}}\right)\right), \left(\mathsf{neg}\left(\color{blue}{\left(wj + 1\right)}\right)\right)\right)\right) \]

      12. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(wj, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(wj, \mathsf{/.f64}\left(x, \left(e^{wj}\right)\right)\right), \left(\mathsf{neg}\left(\left(wj + \color{blue}{1}\right)\right)\right)\right)\right) \]

      13. exp-lowering-exp.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(wj, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(wj, \mathsf{/.f64}\left(x, \mathsf{exp.f64}\left(wj\right)\right)\right), \left(\mathsf{neg}\left(\left(wj + 1\right)\right)\right)\right)\right) \]

      14. +-commutative N/A

          \[\leadsto \mathsf{+.f64}\left(wj, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(wj, \mathsf{/.f64}\left(x, \mathsf{exp.f64}\left(wj\right)\right)\right), \left(\mathsf{neg}\left(\left(1 + wj\right)\right)\right)\right)\right) \]

      15. *-inverses N/A

          \[\leadsto \mathsf{+.f64}\left(wj, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(wj, \mathsf{/.f64}\left(x, \mathsf{exp.f64}\left(wj\right)\right)\right), \left(\mathsf{neg}\left(\left(\frac{e^{wj}}{e^{wj}} + wj\right)\right)\right)\right)\right) \]

      16. distribute-neg-in N/A

          \[\leadsto \mathsf{+.f64}\left(wj, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(wj, \mathsf{/.f64}\left(x, \mathsf{exp.f64}\left(wj\right)\right)\right), \left(\left(\mathsf{neg}\left(\frac{e^{wj}}{e^{wj}}\right)\right) + \color{blue}{\left(\mathsf{neg}\left(wj\right)\right)}\right)\right)\right) \]

   3. Simplified 99.7%

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

   5. Taylor expanded in wj around 0

      \[\leadsto \mathsf{+.f64}\left(wj, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(wj, \mathsf{/.f64}\left(x, \color{blue}{\left(1 + wj\right)}\right)\right), \mathsf{\_.f64}\left(-1, wj\right)\right)\right) \]
   6. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(wj, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(wj, \mathsf{/.f64}\left(x, \left(wj + 1\right)\right)\right), \mathsf{\_.f64}\left(-1, wj\right)\right)\right) \]

      2. +-lowering-+.f64 92.0%

         \[\leadsto \mathsf{+.f64}\left(wj, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(wj, \mathsf{/.f64}\left(x, \mathsf{+.f64}\left(wj, 1\right)\right)\right), \mathsf{\_.f64}\left(-1, wj\right)\right)\right) \]

   7. Simplified 92.0%

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

4. Final simplification 96.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;wj \leq 1.3 \cdot 10^{-8}:\\ \;\;\;\;x + wj \cdot \left(x \cdot -2 + wj \cdot \left(1 - wj\right)\right)\\ \mathbf{else}:\\ \;\;\;\;wj + \frac{wj + \frac{x}{-1 - wj}}{-1 - wj}\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 96.4% accurate, 17.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;wj \leq 5.7 \cdot 10^{-9}:\\ \;\;\;\;x + wj \cdot \left(wj \cdot \left(1 - wj\right)\right)\\ \mathbf{else}:\\ \;\;\;\;wj + \frac{wj + \frac{x}{-1 - wj}}{-1 - wj}\\ \end{array} \end{array} \]

FPCore

(FPCore (wj x)
 :precision binary64
 (if (<= wj 5.7e-9)
   (+ x (* wj (* wj (- 1.0 wj))))
   (+ wj (/ (+ wj (/ x (- -1.0 wj))) (- -1.0 wj)))))

C

double code(double wj, double x) {
	double tmp;
	if (wj <= 5.7e-9) {
		tmp = x + (wj * (wj * (1.0 - wj)));
	} else {
		tmp = wj + ((wj + (x / (-1.0 - wj))) / (-1.0 - wj));
	}
	return tmp;
}

Fortran

real(8) function code(wj, x)
    real(8), intent (in) :: wj
    real(8), intent (in) :: x
    real(8) :: tmp
    if (wj <= 5.7d-9) then
        tmp = x + (wj * (wj * (1.0d0 - wj)))
    else
        tmp = wj + ((wj + (x / ((-1.0d0) - wj))) / ((-1.0d0) - wj))
    end if
    code = tmp
end function

Java

public static double code(double wj, double x) {
	double tmp;
	if (wj <= 5.7e-9) {
		tmp = x + (wj * (wj * (1.0 - wj)));
	} else {
		tmp = wj + ((wj + (x / (-1.0 - wj))) / (-1.0 - wj));
	}
	return tmp;
}

Python

def code(wj, x):
	tmp = 0
	if wj <= 5.7e-9:
		tmp = x + (wj * (wj * (1.0 - wj)))
	else:
		tmp = wj + ((wj + (x / (-1.0 - wj))) / (-1.0 - wj))
	return tmp

Julia

function code(wj, x)
	tmp = 0.0
	if (wj <= 5.7e-9)
		tmp = Float64(x + Float64(wj * Float64(wj * Float64(1.0 - wj))));
	else
		tmp = Float64(wj + Float64(Float64(wj + Float64(x / Float64(-1.0 - wj))) / Float64(-1.0 - wj)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(wj, x)
	tmp = 0.0;
	if (wj <= 5.7e-9)
		tmp = x + (wj * (wj * (1.0 - wj)));
	else
		tmp = wj + ((wj + (x / (-1.0 - wj))) / (-1.0 - wj));
	end
	tmp_2 = tmp;
end

Wolfram

code[wj_, x_] := If[LessEqual[wj, 5.7e-9], N[(x + N[(wj * N[(wj * N[(1.0 - wj), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(wj + N[(N[(wj + N[(x / N[(-1.0 - wj), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(-1.0 - wj), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if wj < 5.6999999999999998e-9

   1. Initial program 78.4%

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

   3. Taylor expanded in wj around 0

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

      1. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(x, \color{blue}{\left(wj \cdot \left(wj \cdot \left(\left(1 + -1 \cdot \left(wj \cdot \left(1 + \left(-3 \cdot x + \left(-2 \cdot \left(-4 \cdot x + \frac{3}{2} \cdot x\right) + \frac{2}{3} \cdot x\right)\right)\right)\right)\right) - \left(-4 \cdot x + \frac{3}{2} \cdot x\right)\right) - 2 \cdot x\right)\right)}\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(x, \mathsf{*.f64}\left(wj, \color{blue}{\left(wj \cdot \left(\left(1 + -1 \cdot \left(wj \cdot \left(1 + \left(-3 \cdot x + \left(-2 \cdot \left(-4 \cdot x + \frac{3}{2} \cdot x\right) + \frac{2}{3} \cdot x\right)\right)\right)\right)\right) - \left(-4 \cdot x + \frac{3}{2} \cdot x\right)\right) - 2 \cdot x\right)}\right)\right) \]

      3. cancel-sign-sub-inv N/A

         \[\leadsto \mathsf{+.f64}\left(x, \mathsf{*.f64}\left(wj, \left(wj \cdot \left(\left(1 + -1 \cdot \left(wj \cdot \left(1 + \left(-3 \cdot x + \left(-2 \cdot \left(-4 \cdot x + \frac{3}{2} \cdot x\right) + \frac{2}{3} \cdot x\right)\right)\right)\right)\right) - \left(-4 \cdot x + \frac{3}{2} \cdot x\right)\right) + \color{blue}{\left(\mathsf{neg}\left(2\right)\right) \cdot x}\right)\right)\right) \]

      4. metadata-eval N/A

         \[\leadsto \mathsf{+.f64}\left(x, \mathsf{*.f64}\left(wj, \left(wj \cdot \left(\left(1 + -1 \cdot \left(wj \cdot \left(1 + \left(-3 \cdot x + \left(-2 \cdot \left(-4 \cdot x + \frac{3}{2} \cdot x\right) + \frac{2}{3} \cdot x\right)\right)\right)\right)\right) - \left(-4 \cdot x + \frac{3}{2} \cdot x\right)\right) + -2 \cdot x\right)\right)\right) \]

      5. +-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(x, \mathsf{*.f64}\left(wj, \left(-2 \cdot x + \color{blue}{wj \cdot \left(\left(1 + -1 \cdot \left(wj \cdot \left(1 + \left(-3 \cdot x + \left(-2 \cdot \left(-4 \cdot x + \frac{3}{2} \cdot x\right) + \frac{2}{3} \cdot x\right)\right)\right)\right)\right) - \left(-4 \cdot x + \frac{3}{2} \cdot x\right)\right)}\right)\right)\right) \]

      6. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(x, \mathsf{*.f64}\left(wj, \mathsf{+.f64}\left(\left(-2 \cdot x\right), \color{blue}{\left(wj \cdot \left(\left(1 + -1 \cdot \left(wj \cdot \left(1 + \left(-3 \cdot x + \left(-2 \cdot \left(-4 \cdot x + \frac{3}{2} \cdot x\right) + \frac{2}{3} \cdot x\right)\right)\right)\right)\right) - \left(-4 \cdot x + \frac{3}{2} \cdot x\right)\right)\right)}\right)\right)\right) \]

   5. Simplified 97.3%

      \[\leadsto \color{blue}{x + wj \cdot \left(x \cdot -2 + wj \cdot \left(\left(1 - wj \cdot \left(\left(x \cdot -3 + x \cdot 5\right) + \left(x \cdot 0.6666666666666666 + 1\right)\right)\right) + x \cdot 2.5\right)\right)} \]

   6. Taylor expanded in x around 0

      \[\leadsto \mathsf{+.f64}\left(x, \color{blue}{\left({wj}^{2} \cdot \left(1 - wj\right)\right)}\right) \]
   7. Step-by-step derivation

      1. unpow2 N/A

         \[\leadsto \mathsf{+.f64}\left(x, \left(\left(wj \cdot wj\right) \cdot \left(\color{blue}{1} - wj\right)\right)\right) \]

      2. associate-*l* N/A

         \[\leadsto \mathsf{+.f64}\left(x, \left(wj \cdot \color{blue}{\left(wj \cdot \left(1 - wj\right)\right)}\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(x, \mathsf{*.f64}\left(wj, \color{blue}{\left(wj \cdot \left(1 - wj\right)\right)}\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(x, \mathsf{*.f64}\left(wj, \mathsf{*.f64}\left(wj, \color{blue}{\left(1 - wj\right)}\right)\right)\right) \]

      5. --lowering--.f64 96.3%

         \[\leadsto \mathsf{+.f64}\left(x, \mathsf{*.f64}\left(wj, \mathsf{*.f64}\left(wj, \mathsf{\_.f64}\left(1, \color{blue}{wj}\right)\right)\right)\right) \]

   8. Simplified 96.3%

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

   if 5.6999999999999998e-9 < wj

   1. Initial program 49.7%

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

      1. sub-neg N/A

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

      2. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(wj, \color{blue}{\left(\mathsf{neg}\left(\frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}}\right)\right)}\right) \]

      3. distribute-rgt1-in N/A

         \[\leadsto \mathsf{+.f64}\left(wj, \left(\mathsf{neg}\left(\frac{wj \cdot e^{wj} - x}{\left(wj + 1\right) \cdot e^{wj}}\right)\right)\right) \]

      4. associate-/l/ N/A

         \[\leadsto \mathsf{+.f64}\left(wj, \left(\mathsf{neg}\left(\frac{\frac{wj \cdot e^{wj} - x}{e^{wj}}}{wj + 1}\right)\right)\right) \]

      5. distribute-neg-frac2 N/A

         \[\leadsto \mathsf{+.f64}\left(wj, \left(\frac{\frac{wj \cdot e^{wj} - x}{e^{wj}}}{\color{blue}{\mathsf{neg}\left(\left(wj + 1\right)\right)}}\right)\right) \]

      6. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(wj, \mathsf{/.f64}\left(\left(\frac{wj \cdot e^{wj} - x}{e^{wj}}\right), \color{blue}{\left(\mathsf{neg}\left(\left(wj + 1\right)\right)\right)}\right)\right) \]

      7. div-sub N/A

         \[\leadsto \mathsf{+.f64}\left(wj, \mathsf{/.f64}\left(\left(\frac{wj \cdot e^{wj}}{e^{wj}} - \frac{x}{e^{wj}}\right), \left(\mathsf{neg}\left(\color{blue}{\left(wj + 1\right)}\right)\right)\right)\right) \]

      8. associate-/l* N/A

         \[\leadsto \mathsf{+.f64}\left(wj, \mathsf{/.f64}\left(\left(wj \cdot \frac{e^{wj}}{e^{wj}} - \frac{x}{e^{wj}}\right), \left(\mathsf{neg}\left(\left(\color{blue}{wj} + 1\right)\right)\right)\right)\right) \]

      9. *-inverses N/A

         \[\leadsto \mathsf{+.f64}\left(wj, \mathsf{/.f64}\left(\left(wj \cdot 1 - \frac{x}{e^{wj}}\right), \left(\mathsf{neg}\left(\left(wj + 1\right)\right)\right)\right)\right) \]

      10. *-rgt-identity N/A

          \[\leadsto \mathsf{+.f64}\left(wj, \mathsf{/.f64}\left(\left(wj - \frac{x}{e^{wj}}\right), \left(\mathsf{neg}\left(\left(\color{blue}{wj} + 1\right)\right)\right)\right)\right) \]

      11. --lowering--.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(wj, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(wj, \left(\frac{x}{e^{wj}}\right)\right), \left(\mathsf{neg}\left(\color{blue}{\left(wj + 1\right)}\right)\right)\right)\right) \]

      12. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(wj, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(wj, \mathsf{/.f64}\left(x, \left(e^{wj}\right)\right)\right), \left(\mathsf{neg}\left(\left(wj + \color{blue}{1}\right)\right)\right)\right)\right) \]

      13. exp-lowering-exp.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(wj, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(wj, \mathsf{/.f64}\left(x, \mathsf{exp.f64}\left(wj\right)\right)\right), \left(\mathsf{neg}\left(\left(wj + 1\right)\right)\right)\right)\right) \]

      14. +-commutative N/A

          \[\leadsto \mathsf{+.f64}\left(wj, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(wj, \mathsf{/.f64}\left(x, \mathsf{exp.f64}\left(wj\right)\right)\right), \left(\mathsf{neg}\left(\left(1 + wj\right)\right)\right)\right)\right) \]

      15. *-inverses N/A

          \[\leadsto \mathsf{+.f64}\left(wj, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(wj, \mathsf{/.f64}\left(x, \mathsf{exp.f64}\left(wj\right)\right)\right), \left(\mathsf{neg}\left(\left(\frac{e^{wj}}{e^{wj}} + wj\right)\right)\right)\right)\right) \]

      16. distribute-neg-in N/A

          \[\leadsto \mathsf{+.f64}\left(wj, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(wj, \mathsf{/.f64}\left(x, \mathsf{exp.f64}\left(wj\right)\right)\right), \left(\left(\mathsf{neg}\left(\frac{e^{wj}}{e^{wj}}\right)\right) + \color{blue}{\left(\mathsf{neg}\left(wj\right)\right)}\right)\right)\right) \]

   3. Simplified 99.7%

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

   5. Taylor expanded in wj around 0

      \[\leadsto \mathsf{+.f64}\left(wj, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(wj, \mathsf{/.f64}\left(x, \color{blue}{\left(1 + wj\right)}\right)\right), \mathsf{\_.f64}\left(-1, wj\right)\right)\right) \]
   6. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(wj, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(wj, \mathsf{/.f64}\left(x, \left(wj + 1\right)\right)\right), \mathsf{\_.f64}\left(-1, wj\right)\right)\right) \]

      2. +-lowering-+.f64 92.0%

         \[\leadsto \mathsf{+.f64}\left(wj, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(wj, \mathsf{/.f64}\left(x, \mathsf{+.f64}\left(wj, 1\right)\right)\right), \mathsf{\_.f64}\left(-1, wj\right)\right)\right) \]

   7. Simplified 92.0%

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

4. Final simplification 96.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;wj \leq 5.7 \cdot 10^{-9}:\\ \;\;\;\;x + wj \cdot \left(wj \cdot \left(1 - wj\right)\right)\\ \mathbf{else}:\\ \;\;\;\;wj + \frac{wj + \frac{x}{-1 - wj}}{-1 - wj}\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 95.3% accurate, 34.8× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 77.7%

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

3. Taylor expanded in wj around 0

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

   1. +-lowering-+.f64 N/A

      \[\leadsto \mathsf{+.f64}\left(x, \color{blue}{\left(wj \cdot \left(wj \cdot \left(\left(1 + -1 \cdot \left(wj \cdot \left(1 + \left(-3 \cdot x + \left(-2 \cdot \left(-4 \cdot x + \frac{3}{2} \cdot x\right) + \frac{2}{3} \cdot x\right)\right)\right)\right)\right) - \left(-4 \cdot x + \frac{3}{2} \cdot x\right)\right) - 2 \cdot x\right)\right)}\right) \]

   2. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{+.f64}\left(x, \mathsf{*.f64}\left(wj, \color{blue}{\left(wj \cdot \left(\left(1 + -1 \cdot \left(wj \cdot \left(1 + \left(-3 \cdot x + \left(-2 \cdot \left(-4 \cdot x + \frac{3}{2} \cdot x\right) + \frac{2}{3} \cdot x\right)\right)\right)\right)\right) - \left(-4 \cdot x + \frac{3}{2} \cdot x\right)\right) - 2 \cdot x\right)}\right)\right) \]

   3. cancel-sign-sub-inv N/A

      \[\leadsto \mathsf{+.f64}\left(x, \mathsf{*.f64}\left(wj, \left(wj \cdot \left(\left(1 + -1 \cdot \left(wj \cdot \left(1 + \left(-3 \cdot x + \left(-2 \cdot \left(-4 \cdot x + \frac{3}{2} \cdot x\right) + \frac{2}{3} \cdot x\right)\right)\right)\right)\right) - \left(-4 \cdot x + \frac{3}{2} \cdot x\right)\right) + \color{blue}{\left(\mathsf{neg}\left(2\right)\right) \cdot x}\right)\right)\right) \]

   4. metadata-eval N/A

      \[\leadsto \mathsf{+.f64}\left(x, \mathsf{*.f64}\left(wj, \left(wj \cdot \left(\left(1 + -1 \cdot \left(wj \cdot \left(1 + \left(-3 \cdot x + \left(-2 \cdot \left(-4 \cdot x + \frac{3}{2} \cdot x\right) + \frac{2}{3} \cdot x\right)\right)\right)\right)\right) - \left(-4 \cdot x + \frac{3}{2} \cdot x\right)\right) + -2 \cdot x\right)\right)\right) \]

   5. +-commutative N/A

      \[\leadsto \mathsf{+.f64}\left(x, \mathsf{*.f64}\left(wj, \left(-2 \cdot x + \color{blue}{wj \cdot \left(\left(1 + -1 \cdot \left(wj \cdot \left(1 + \left(-3 \cdot x + \left(-2 \cdot \left(-4 \cdot x + \frac{3}{2} \cdot x\right) + \frac{2}{3} \cdot x\right)\right)\right)\right)\right) - \left(-4 \cdot x + \frac{3}{2} \cdot x\right)\right)}\right)\right)\right) \]

   6. +-lowering-+.f64 N/A

      \[\leadsto \mathsf{+.f64}\left(x, \mathsf{*.f64}\left(wj, \mathsf{+.f64}\left(\left(-2 \cdot x\right), \color{blue}{\left(wj \cdot \left(\left(1 + -1 \cdot \left(wj \cdot \left(1 + \left(-3 \cdot x + \left(-2 \cdot \left(-4 \cdot x + \frac{3}{2} \cdot x\right) + \frac{2}{3} \cdot x\right)\right)\right)\right)\right) - \left(-4 \cdot x + \frac{3}{2} \cdot x\right)\right)\right)}\right)\right)\right) \]

5. Simplified 95.7%

   \[\leadsto \color{blue}{x + wj \cdot \left(x \cdot -2 + wj \cdot \left(\left(1 - wj \cdot \left(\left(x \cdot -3 + x \cdot 5\right) + \left(x \cdot 0.6666666666666666 + 1\right)\right)\right) + x \cdot 2.5\right)\right)} \]

6. Taylor expanded in x around 0

   \[\leadsto \mathsf{+.f64}\left(x, \color{blue}{\left({wj}^{2} \cdot \left(1 - wj\right)\right)}\right) \]
7. Step-by-step derivation

   1. unpow2 N/A

      \[\leadsto \mathsf{+.f64}\left(x, \left(\left(wj \cdot wj\right) \cdot \left(\color{blue}{1} - wj\right)\right)\right) \]

   2. associate-*l* N/A

      \[\leadsto \mathsf{+.f64}\left(x, \left(wj \cdot \color{blue}{\left(wj \cdot \left(1 - wj\right)\right)}\right)\right) \]

   3. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{+.f64}\left(x, \mathsf{*.f64}\left(wj, \color{blue}{\left(wj \cdot \left(1 - wj\right)\right)}\right)\right) \]

   4. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{+.f64}\left(x, \mathsf{*.f64}\left(wj, \mathsf{*.f64}\left(wj, \color{blue}{\left(1 - wj\right)}\right)\right)\right) \]

   5. --lowering--.f64 94.4%

      \[\leadsto \mathsf{+.f64}\left(x, \mathsf{*.f64}\left(wj, \mathsf{*.f64}\left(wj, \mathsf{\_.f64}\left(1, \color{blue}{wj}\right)\right)\right)\right) \]

8. Simplified 94.4%

   \[\leadsto x + \color{blue}{wj \cdot \left(wj \cdot \left(1 - wj\right)\right)} \]
9. Add Preprocessing

Alternative 15: 83.9% accurate, 44.7× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 77.7%

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

3. Taylor expanded in wj around 0

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

   1. +-lowering-+.f64 N/A

      \[\leadsto \mathsf{+.f64}\left(x, \color{blue}{\left(wj \cdot \left(wj \cdot \left(\left(1 + -1 \cdot \left(wj \cdot \left(1 + \left(-3 \cdot x + \left(-2 \cdot \left(-4 \cdot x + \frac{3}{2} \cdot x\right) + \frac{2}{3} \cdot x\right)\right)\right)\right)\right) - \left(-4 \cdot x + \frac{3}{2} \cdot x\right)\right) - 2 \cdot x\right)\right)}\right) \]

   2. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{+.f64}\left(x, \mathsf{*.f64}\left(wj, \color{blue}{\left(wj \cdot \left(\left(1 + -1 \cdot \left(wj \cdot \left(1 + \left(-3 \cdot x + \left(-2 \cdot \left(-4 \cdot x + \frac{3}{2} \cdot x\right) + \frac{2}{3} \cdot x\right)\right)\right)\right)\right) - \left(-4 \cdot x + \frac{3}{2} \cdot x\right)\right) - 2 \cdot x\right)}\right)\right) \]

   3. cancel-sign-sub-inv N/A

      \[\leadsto \mathsf{+.f64}\left(x, \mathsf{*.f64}\left(wj, \left(wj \cdot \left(\left(1 + -1 \cdot \left(wj \cdot \left(1 + \left(-3 \cdot x + \left(-2 \cdot \left(-4 \cdot x + \frac{3}{2} \cdot x\right) + \frac{2}{3} \cdot x\right)\right)\right)\right)\right) - \left(-4 \cdot x + \frac{3}{2} \cdot x\right)\right) + \color{blue}{\left(\mathsf{neg}\left(2\right)\right) \cdot x}\right)\right)\right) \]

   4. metadata-eval N/A

      \[\leadsto \mathsf{+.f64}\left(x, \mathsf{*.f64}\left(wj, \left(wj \cdot \left(\left(1 + -1 \cdot \left(wj \cdot \left(1 + \left(-3 \cdot x + \left(-2 \cdot \left(-4 \cdot x + \frac{3}{2} \cdot x\right) + \frac{2}{3} \cdot x\right)\right)\right)\right)\right) - \left(-4 \cdot x + \frac{3}{2} \cdot x\right)\right) + -2 \cdot x\right)\right)\right) \]

   5. +-commutative N/A

      \[\leadsto \mathsf{+.f64}\left(x, \mathsf{*.f64}\left(wj, \left(-2 \cdot x + \color{blue}{wj \cdot \left(\left(1 + -1 \cdot \left(wj \cdot \left(1 + \left(-3 \cdot x + \left(-2 \cdot \left(-4 \cdot x + \frac{3}{2} \cdot x\right) + \frac{2}{3} \cdot x\right)\right)\right)\right)\right) - \left(-4 \cdot x + \frac{3}{2} \cdot x\right)\right)}\right)\right)\right) \]

   6. +-lowering-+.f64 N/A

      \[\leadsto \mathsf{+.f64}\left(x, \mathsf{*.f64}\left(wj, \mathsf{+.f64}\left(\left(-2 \cdot x\right), \color{blue}{\left(wj \cdot \left(\left(1 + -1 \cdot \left(wj \cdot \left(1 + \left(-3 \cdot x + \left(-2 \cdot \left(-4 \cdot x + \frac{3}{2} \cdot x\right) + \frac{2}{3} \cdot x\right)\right)\right)\right)\right) - \left(-4 \cdot x + \frac{3}{2} \cdot x\right)\right)\right)}\right)\right)\right) \]

5. Simplified 95.7%

   \[\leadsto \color{blue}{x + wj \cdot \left(x \cdot -2 + wj \cdot \left(\left(1 - wj \cdot \left(\left(x \cdot -3 + x \cdot 5\right) + \left(x \cdot 0.6666666666666666 + 1\right)\right)\right) + x \cdot 2.5\right)\right)} \]

6. Taylor expanded in wj around 0

   \[\leadsto \mathsf{+.f64}\left(x, \color{blue}{\left(-2 \cdot \left(wj \cdot x\right)\right)}\right) \]
7. Step-by-step derivation

   1. *-commutative N/A

      \[\leadsto \mathsf{+.f64}\left(x, \left(\left(wj \cdot x\right) \cdot \color{blue}{-2}\right)\right) \]

   2. associate-*r* N/A

      \[\leadsto \mathsf{+.f64}\left(x, \left(wj \cdot \color{blue}{\left(x \cdot -2\right)}\right)\right) \]

   3. *-commutative N/A

      \[\leadsto \mathsf{+.f64}\left(x, \left(wj \cdot \left(-2 \cdot \color{blue}{x}\right)\right)\right) \]

   4. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{+.f64}\left(x, \mathsf{*.f64}\left(wj, \color{blue}{\left(-2 \cdot x\right)}\right)\right) \]

   5. *-commutative N/A

      \[\leadsto \mathsf{+.f64}\left(x, \mathsf{*.f64}\left(wj, \left(x \cdot \color{blue}{-2}\right)\right)\right) \]

   6. *-lowering-*.f64 85.8%

      \[\leadsto \mathsf{+.f64}\left(x, \mathsf{*.f64}\left(wj, \mathsf{*.f64}\left(x, \color{blue}{-2}\right)\right)\right) \]

8. Simplified 85.8%

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

Alternative 16: 83.9% accurate, 44.7× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 77.7%

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

3. Taylor expanded in wj around 0

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

   1. associate-*r* N/A

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

   2. distribute-rgt1-in N/A

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

   3. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\left(-2 \cdot wj + 1\right), \color{blue}{x}\right) \]

   4. +-lowering-+.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(\left(-2 \cdot wj\right), 1\right), x\right) \]

   5. *-commutative N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(\left(wj \cdot -2\right), 1\right), x\right) \]

   6. *-lowering-*.f64 85.8%

      \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(wj, -2\right), 1\right), x\right) \]

5. Simplified 85.8%

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

6. Final simplification 85.8%

   \[\leadsto x \cdot \left(1 + wj \cdot -2\right) \]
7. Add Preprocessing

Alternative 17: 83.5% accurate, 62.6× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 77.7%

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

   1. sub-neg N/A

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

   2. +-lowering-+.f64 N/A

      \[\leadsto \mathsf{+.f64}\left(wj, \color{blue}{\left(\mathsf{neg}\left(\frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}}\right)\right)}\right) \]

   3. distribute-rgt1-in N/A

      \[\leadsto \mathsf{+.f64}\left(wj, \left(\mathsf{neg}\left(\frac{wj \cdot e^{wj} - x}{\left(wj + 1\right) \cdot e^{wj}}\right)\right)\right) \]

   4. associate-/l/ N/A

      \[\leadsto \mathsf{+.f64}\left(wj, \left(\mathsf{neg}\left(\frac{\frac{wj \cdot e^{wj} - x}{e^{wj}}}{wj + 1}\right)\right)\right) \]

   5. distribute-neg-frac2 N/A

      \[\leadsto \mathsf{+.f64}\left(wj, \left(\frac{\frac{wj \cdot e^{wj} - x}{e^{wj}}}{\color{blue}{\mathsf{neg}\left(\left(wj + 1\right)\right)}}\right)\right) \]

   6. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{+.f64}\left(wj, \mathsf{/.f64}\left(\left(\frac{wj \cdot e^{wj} - x}{e^{wj}}\right), \color{blue}{\left(\mathsf{neg}\left(\left(wj + 1\right)\right)\right)}\right)\right) \]

   7. div-sub N/A

      \[\leadsto \mathsf{+.f64}\left(wj, \mathsf{/.f64}\left(\left(\frac{wj \cdot e^{wj}}{e^{wj}} - \frac{x}{e^{wj}}\right), \left(\mathsf{neg}\left(\color{blue}{\left(wj + 1\right)}\right)\right)\right)\right) \]

   8. associate-/l* N/A

      \[\leadsto \mathsf{+.f64}\left(wj, \mathsf{/.f64}\left(\left(wj \cdot \frac{e^{wj}}{e^{wj}} - \frac{x}{e^{wj}}\right), \left(\mathsf{neg}\left(\left(\color{blue}{wj} + 1\right)\right)\right)\right)\right) \]

   9. *-inverses N/A

      \[\leadsto \mathsf{+.f64}\left(wj, \mathsf{/.f64}\left(\left(wj \cdot 1 - \frac{x}{e^{wj}}\right), \left(\mathsf{neg}\left(\left(wj + 1\right)\right)\right)\right)\right) \]

   10. *-rgt-identity N/A

       \[\leadsto \mathsf{+.f64}\left(wj, \mathsf{/.f64}\left(\left(wj - \frac{x}{e^{wj}}\right), \left(\mathsf{neg}\left(\left(\color{blue}{wj} + 1\right)\right)\right)\right)\right) \]

   11. --lowering--.f64 N/A

       \[\leadsto \mathsf{+.f64}\left(wj, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(wj, \left(\frac{x}{e^{wj}}\right)\right), \left(\mathsf{neg}\left(\color{blue}{\left(wj + 1\right)}\right)\right)\right)\right) \]

   12. /-lowering-/.f64 N/A

       \[\leadsto \mathsf{+.f64}\left(wj, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(wj, \mathsf{/.f64}\left(x, \left(e^{wj}\right)\right)\right), \left(\mathsf{neg}\left(\left(wj + \color{blue}{1}\right)\right)\right)\right)\right) \]

   13. exp-lowering-exp.f64 N/A

       \[\leadsto \mathsf{+.f64}\left(wj, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(wj, \mathsf{/.f64}\left(x, \mathsf{exp.f64}\left(wj\right)\right)\right), \left(\mathsf{neg}\left(\left(wj + 1\right)\right)\right)\right)\right) \]

   14. +-commutative N/A

       \[\leadsto \mathsf{+.f64}\left(wj, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(wj, \mathsf{/.f64}\left(x, \mathsf{exp.f64}\left(wj\right)\right)\right), \left(\mathsf{neg}\left(\left(1 + wj\right)\right)\right)\right)\right) \]

   15. *-inverses N/A

       \[\leadsto \mathsf{+.f64}\left(wj, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(wj, \mathsf{/.f64}\left(x, \mathsf{exp.f64}\left(wj\right)\right)\right), \left(\mathsf{neg}\left(\left(\frac{e^{wj}}{e^{wj}} + wj\right)\right)\right)\right)\right) \]

   16. distribute-neg-in N/A

       \[\leadsto \mathsf{+.f64}\left(wj, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(wj, \mathsf{/.f64}\left(x, \mathsf{exp.f64}\left(wj\right)\right)\right), \left(\left(\mathsf{neg}\left(\frac{e^{wj}}{e^{wj}}\right)\right) + \color{blue}{\left(\mathsf{neg}\left(wj\right)\right)}\right)\right)\right) \]

3. Simplified 80.5%

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

5. Taylor expanded in wj around 0

   \[\leadsto \mathsf{+.f64}\left(wj, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(wj, \color{blue}{x}\right), \mathsf{\_.f64}\left(-1, wj\right)\right)\right) \]
6. Step-by-step derivation

7. Simplified 77.7%

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

8. Taylor expanded in x around inf

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

   1. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(x, \color{blue}{\left(1 + wj\right)}\right) \]

   2. +-commutative N/A

      \[\leadsto \mathsf{/.f64}\left(x, \left(wj + \color{blue}{1}\right)\right) \]

   3. +-lowering-+.f64 85.1%

      \[\leadsto \mathsf{/.f64}\left(x, \mathsf{+.f64}\left(wj, \color{blue}{1}\right)\right) \]

10. Simplified 85.1%

    \[\leadsto \color{blue}{\frac{x}{wj + 1}} \]
11. Add Preprocessing

Alternative 18: 83.4% 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 77.7%

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

   1. sub-neg N/A

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

   2. +-lowering-+.f64 N/A

      \[\leadsto \mathsf{+.f64}\left(wj, \color{blue}{\left(\mathsf{neg}\left(\frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}}\right)\right)}\right) \]

   3. distribute-rgt1-in N/A

      \[\leadsto \mathsf{+.f64}\left(wj, \left(\mathsf{neg}\left(\frac{wj \cdot e^{wj} - x}{\left(wj + 1\right) \cdot e^{wj}}\right)\right)\right) \]

   4. associate-/l/ N/A

      \[\leadsto \mathsf{+.f64}\left(wj, \left(\mathsf{neg}\left(\frac{\frac{wj \cdot e^{wj} - x}{e^{wj}}}{wj + 1}\right)\right)\right) \]

   5. distribute-neg-frac2 N/A

      \[\leadsto \mathsf{+.f64}\left(wj, \left(\frac{\frac{wj \cdot e^{wj} - x}{e^{wj}}}{\color{blue}{\mathsf{neg}\left(\left(wj + 1\right)\right)}}\right)\right) \]

   6. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{+.f64}\left(wj, \mathsf{/.f64}\left(\left(\frac{wj \cdot e^{wj} - x}{e^{wj}}\right), \color{blue}{\left(\mathsf{neg}\left(\left(wj + 1\right)\right)\right)}\right)\right) \]

   7. div-sub N/A

      \[\leadsto \mathsf{+.f64}\left(wj, \mathsf{/.f64}\left(\left(\frac{wj \cdot e^{wj}}{e^{wj}} - \frac{x}{e^{wj}}\right), \left(\mathsf{neg}\left(\color{blue}{\left(wj + 1\right)}\right)\right)\right)\right) \]

   8. associate-/l* N/A

      \[\leadsto \mathsf{+.f64}\left(wj, \mathsf{/.f64}\left(\left(wj \cdot \frac{e^{wj}}{e^{wj}} - \frac{x}{e^{wj}}\right), \left(\mathsf{neg}\left(\left(\color{blue}{wj} + 1\right)\right)\right)\right)\right) \]

   9. *-inverses N/A

      \[\leadsto \mathsf{+.f64}\left(wj, \mathsf{/.f64}\left(\left(wj \cdot 1 - \frac{x}{e^{wj}}\right), \left(\mathsf{neg}\left(\left(wj + 1\right)\right)\right)\right)\right) \]

   10. *-rgt-identity N/A

       \[\leadsto \mathsf{+.f64}\left(wj, \mathsf{/.f64}\left(\left(wj - \frac{x}{e^{wj}}\right), \left(\mathsf{neg}\left(\left(\color{blue}{wj} + 1\right)\right)\right)\right)\right) \]

   11. --lowering--.f64 N/A

       \[\leadsto \mathsf{+.f64}\left(wj, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(wj, \left(\frac{x}{e^{wj}}\right)\right), \left(\mathsf{neg}\left(\color{blue}{\left(wj + 1\right)}\right)\right)\right)\right) \]

   12. /-lowering-/.f64 N/A

       \[\leadsto \mathsf{+.f64}\left(wj, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(wj, \mathsf{/.f64}\left(x, \left(e^{wj}\right)\right)\right), \left(\mathsf{neg}\left(\left(wj + \color{blue}{1}\right)\right)\right)\right)\right) \]

   13. exp-lowering-exp.f64 N/A

       \[\leadsto \mathsf{+.f64}\left(wj, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(wj, \mathsf{/.f64}\left(x, \mathsf{exp.f64}\left(wj\right)\right)\right), \left(\mathsf{neg}\left(\left(wj + 1\right)\right)\right)\right)\right) \]

   14. +-commutative N/A

       \[\leadsto \mathsf{+.f64}\left(wj, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(wj, \mathsf{/.f64}\left(x, \mathsf{exp.f64}\left(wj\right)\right)\right), \left(\mathsf{neg}\left(\left(1 + wj\right)\right)\right)\right)\right) \]

   15. *-inverses N/A

       \[\leadsto \mathsf{+.f64}\left(wj, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(wj, \mathsf{/.f64}\left(x, \mathsf{exp.f64}\left(wj\right)\right)\right), \left(\mathsf{neg}\left(\left(\frac{e^{wj}}{e^{wj}} + wj\right)\right)\right)\right)\right) \]

   16. distribute-neg-in N/A

       \[\leadsto \mathsf{+.f64}\left(wj, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(wj, \mathsf{/.f64}\left(x, \mathsf{exp.f64}\left(wj\right)\right)\right), \left(\left(\mathsf{neg}\left(\frac{e^{wj}}{e^{wj}}\right)\right) + \color{blue}{\left(\mathsf{neg}\left(wj\right)\right)}\right)\right)\right) \]

3. Simplified 80.5%

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

5. Taylor expanded in wj around 0

   \[\leadsto \color{blue}{x} \]
6. Step-by-step derivation

7. Simplified 85.0%

   \[\leadsto \color{blue}{x} \]
8. Add Preprocessing

Alternative 19: 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 77.7%

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

   1. sub-neg N/A

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

   2. +-lowering-+.f64 N/A

      \[\leadsto \mathsf{+.f64}\left(wj, \color{blue}{\left(\mathsf{neg}\left(\frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}}\right)\right)}\right) \]

   3. distribute-rgt1-in N/A

      \[\leadsto \mathsf{+.f64}\left(wj, \left(\mathsf{neg}\left(\frac{wj \cdot e^{wj} - x}{\left(wj + 1\right) \cdot e^{wj}}\right)\right)\right) \]

   4. associate-/l/ N/A

      \[\leadsto \mathsf{+.f64}\left(wj, \left(\mathsf{neg}\left(\frac{\frac{wj \cdot e^{wj} - x}{e^{wj}}}{wj + 1}\right)\right)\right) \]

   5. distribute-neg-frac2 N/A

      \[\leadsto \mathsf{+.f64}\left(wj, \left(\frac{\frac{wj \cdot e^{wj} - x}{e^{wj}}}{\color{blue}{\mathsf{neg}\left(\left(wj + 1\right)\right)}}\right)\right) \]

   6. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{+.f64}\left(wj, \mathsf{/.f64}\left(\left(\frac{wj \cdot e^{wj} - x}{e^{wj}}\right), \color{blue}{\left(\mathsf{neg}\left(\left(wj + 1\right)\right)\right)}\right)\right) \]

   7. div-sub N/A

      \[\leadsto \mathsf{+.f64}\left(wj, \mathsf{/.f64}\left(\left(\frac{wj \cdot e^{wj}}{e^{wj}} - \frac{x}{e^{wj}}\right), \left(\mathsf{neg}\left(\color{blue}{\left(wj + 1\right)}\right)\right)\right)\right) \]

   8. associate-/l* N/A

      \[\leadsto \mathsf{+.f64}\left(wj, \mathsf{/.f64}\left(\left(wj \cdot \frac{e^{wj}}{e^{wj}} - \frac{x}{e^{wj}}\right), \left(\mathsf{neg}\left(\left(\color{blue}{wj} + 1\right)\right)\right)\right)\right) \]

   9. *-inverses N/A

      \[\leadsto \mathsf{+.f64}\left(wj, \mathsf{/.f64}\left(\left(wj \cdot 1 - \frac{x}{e^{wj}}\right), \left(\mathsf{neg}\left(\left(wj + 1\right)\right)\right)\right)\right) \]

   10. *-rgt-identity N/A

       \[\leadsto \mathsf{+.f64}\left(wj, \mathsf{/.f64}\left(\left(wj - \frac{x}{e^{wj}}\right), \left(\mathsf{neg}\left(\left(\color{blue}{wj} + 1\right)\right)\right)\right)\right) \]

   11. --lowering--.f64 N/A

       \[\leadsto \mathsf{+.f64}\left(wj, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(wj, \left(\frac{x}{e^{wj}}\right)\right), \left(\mathsf{neg}\left(\color{blue}{\left(wj + 1\right)}\right)\right)\right)\right) \]

   12. /-lowering-/.f64 N/A

       \[\leadsto \mathsf{+.f64}\left(wj, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(wj, \mathsf{/.f64}\left(x, \left(e^{wj}\right)\right)\right), \left(\mathsf{neg}\left(\left(wj + \color{blue}{1}\right)\right)\right)\right)\right) \]

   13. exp-lowering-exp.f64 N/A

       \[\leadsto \mathsf{+.f64}\left(wj, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(wj, \mathsf{/.f64}\left(x, \mathsf{exp.f64}\left(wj\right)\right)\right), \left(\mathsf{neg}\left(\left(wj + 1\right)\right)\right)\right)\right) \]

   14. +-commutative N/A

       \[\leadsto \mathsf{+.f64}\left(wj, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(wj, \mathsf{/.f64}\left(x, \mathsf{exp.f64}\left(wj\right)\right)\right), \left(\mathsf{neg}\left(\left(1 + wj\right)\right)\right)\right)\right) \]

   15. *-inverses N/A

       \[\leadsto \mathsf{+.f64}\left(wj, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(wj, \mathsf{/.f64}\left(x, \mathsf{exp.f64}\left(wj\right)\right)\right), \left(\mathsf{neg}\left(\left(\frac{e^{wj}}{e^{wj}} + wj\right)\right)\right)\right)\right) \]

   16. distribute-neg-in N/A

       \[\leadsto \mathsf{+.f64}\left(wj, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(wj, \mathsf{/.f64}\left(x, \mathsf{exp.f64}\left(wj\right)\right)\right), \left(\left(\mathsf{neg}\left(\frac{e^{wj}}{e^{wj}}\right)\right) + \color{blue}{\left(\mathsf{neg}\left(wj\right)\right)}\right)\right)\right) \]

3. Simplified 80.5%

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

5. Taylor expanded in wj around inf

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

7. Simplified 4.3%

   \[\leadsto \color{blue}{wj} \]
8. Add Preprocessing

Alternative 20: 3.3% accurate, 313.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

def code(wj, x):
	return -1.0

Julia

function code(wj, x)
	return -1.0
end

MATLAB

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

Wolfram

code[wj_, x_] := -1.0

Derivation

1. Initial program 77.7%

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

3. Taylor expanded in wj around inf

   \[\leadsto \mathsf{\_.f64}\left(wj, \color{blue}{1}\right) \]
4. Step-by-step derivation

5. Simplified 4.5%

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

6. Taylor expanded in wj around 0

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

8. Simplified 3.5%

   \[\leadsto \color{blue}{-1} \]
9. Add Preprocessing

Developer Target 1: 79.2% accurate, 1.5× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Reproduce

herbie shell --seed 2024144 
(FPCore (wj x)
  :name "Jmat.Real.lambertw, newton loop step"
  :precision binary64

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

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