Jmat.Real.lambertw, newton loop step

Percentage Accurate: 77.5% → 99.6%

Time: 17.6s

Alternatives: 20

Speedup: 313.0×

• could not determine a ground truth

  1. wj = -5.3538868362807505e+215
  2. x = 3.434092156723102e+225

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 77.5% 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, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{-wj}\\ t_1 := wj \cdot e^{wj}\\ \mathbf{if}\;wj + \frac{x - t\_1}{e^{wj} + t\_1} \leq 4 \cdot 10^{-9}:\\ \;\;\;\;x \cdot \left(\frac{t\_0}{wj + 1} + {wj}^{2} \cdot \left(\frac{1}{x} - wj \cdot \left(\frac{1}{x} + wj \cdot \left(\frac{wj}{x} + \frac{-1}{x}\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;wj + \frac{wj - x \cdot t\_0}{-1 - wj}\\ \end{array} \end{array} \]

FPCore

(FPCore (wj x)
 :precision binary64
 (let* ((t_0 (exp (- wj))) (t_1 (* wj (exp wj))))
   (if (<= (+ wj (/ (- x t_1) (+ (exp wj) t_1))) 4e-9)
     (*
      x
      (+
       (/ t_0 (+ wj 1.0))
       (*
        (pow wj 2.0)
        (- (/ 1.0 x) (* wj (+ (/ 1.0 x) (* wj (+ (/ wj x) (/ -1.0 x)))))))))
     (+ wj (/ (- wj (* x t_0)) (- -1.0 wj))))))

C

double code(double wj, double x) {
	double t_0 = exp(-wj);
	double t_1 = wj * exp(wj);
	double tmp;
	if ((wj + ((x - t_1) / (exp(wj) + t_1))) <= 4e-9) {
		tmp = x * ((t_0 / (wj + 1.0)) + (pow(wj, 2.0) * ((1.0 / x) - (wj * ((1.0 / x) + (wj * ((wj / x) + (-1.0 / x))))))));
	} else {
		tmp = wj + ((wj - (x * t_0)) / (-1.0 - wj));
	}
	return tmp;
}

Fortran

real(8) function code(wj, x)
    real(8), intent (in) :: wj
    real(8), intent (in) :: x
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = exp(-wj)
    t_1 = wj * exp(wj)
    if ((wj + ((x - t_1) / (exp(wj) + t_1))) <= 4d-9) then
        tmp = x * ((t_0 / (wj + 1.0d0)) + ((wj ** 2.0d0) * ((1.0d0 / x) - (wj * ((1.0d0 / x) + (wj * ((wj / x) + ((-1.0d0) / x))))))))
    else
        tmp = wj + ((wj - (x * t_0)) / ((-1.0d0) - wj))
    end if
    code = tmp
end function

Java

public static double code(double wj, double x) {
	double t_0 = Math.exp(-wj);
	double t_1 = wj * Math.exp(wj);
	double tmp;
	if ((wj + ((x - t_1) / (Math.exp(wj) + t_1))) <= 4e-9) {
		tmp = x * ((t_0 / (wj + 1.0)) + (Math.pow(wj, 2.0) * ((1.0 / x) - (wj * ((1.0 / x) + (wj * ((wj / x) + (-1.0 / x))))))));
	} else {
		tmp = wj + ((wj - (x * t_0)) / (-1.0 - wj));
	}
	return tmp;
}

Python

def code(wj, x):
	t_0 = math.exp(-wj)
	t_1 = wj * math.exp(wj)
	tmp = 0
	if (wj + ((x - t_1) / (math.exp(wj) + t_1))) <= 4e-9:
		tmp = x * ((t_0 / (wj + 1.0)) + (math.pow(wj, 2.0) * ((1.0 / x) - (wj * ((1.0 / x) + (wj * ((wj / x) + (-1.0 / x))))))))
	else:
		tmp = wj + ((wj - (x * t_0)) / (-1.0 - wj))
	return tmp

Julia

function code(wj, x)
	t_0 = exp(Float64(-wj))
	t_1 = Float64(wj * exp(wj))
	tmp = 0.0
	if (Float64(wj + Float64(Float64(x - t_1) / Float64(exp(wj) + t_1))) <= 4e-9)
		tmp = Float64(x * Float64(Float64(t_0 / Float64(wj + 1.0)) + Float64((wj ^ 2.0) * Float64(Float64(1.0 / x) - Float64(wj * Float64(Float64(1.0 / x) + Float64(wj * Float64(Float64(wj / x) + Float64(-1.0 / x)))))))));
	else
		tmp = Float64(wj + Float64(Float64(wj - Float64(x * t_0)) / Float64(-1.0 - wj)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(wj, x)
	t_0 = exp(-wj);
	t_1 = wj * exp(wj);
	tmp = 0.0;
	if ((wj + ((x - t_1) / (exp(wj) + t_1))) <= 4e-9)
		tmp = x * ((t_0 / (wj + 1.0)) + ((wj ^ 2.0) * ((1.0 / x) - (wj * ((1.0 / x) + (wj * ((wj / x) + (-1.0 / x))))))));
	else
		tmp = wj + ((wj - (x * t_0)) / (-1.0 - wj));
	end
	tmp_2 = tmp;
end

Wolfram

code[wj_, x_] := Block[{t$95$0 = N[Exp[(-wj)], $MachinePrecision]}, Block[{t$95$1 = N[(wj * N[Exp[wj], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(wj + N[(N[(x - t$95$1), $MachinePrecision] / N[(N[Exp[wj], $MachinePrecision] + t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 4e-9], N[(x * N[(N[(t$95$0 / N[(wj + 1.0), $MachinePrecision]), $MachinePrecision] + N[(N[Power[wj, 2.0], $MachinePrecision] * N[(N[(1.0 / x), $MachinePrecision] - N[(wj * N[(N[(1.0 / x), $MachinePrecision] + N[(wj * N[(N[(wj / x), $MachinePrecision] + N[(-1.0 / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(wj + N[(N[(wj - N[(x * t$95$0), $MachinePrecision]), $MachinePrecision] / N[(-1.0 - wj), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 wj (/.f64 (-.f64 (*.f64 wj (exp.f64 wj)) x) (+.f64 (exp.f64 wj) (*.f64 wj (exp.f64 wj))))) < 4.00000000000000025e-9

   1. Initial program 72.7%

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

      1. distribute-rgt1-in 73.9%

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

      2. associate-/l/ 73.9%

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

      3. div-sub 72.8%

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

      4. associate-/l* 72.8%

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

      5. *-inverses 73.9%

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

      6. *-rgt-identity 73.9%

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

   3. Simplified 73.9%

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

   5. Taylor expanded in x around inf 74.5%

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

      1. associate--l+ 88.2%

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

      2. associate-/r* 88.2%

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

      3. exp-neg 88.2%

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

      4. +-commutative 88.2%

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

      5. +-commutative 88.2%

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

   7. Simplified 88.2%

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

   8. Taylor expanded in wj around 0 99.9%

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

   if 4.00000000000000025e-9 < (-.f64 wj (/.f64 (-.f64 (*.f64 wj (exp.f64 wj)) x) (+.f64 (exp.f64 wj) (*.f64 wj (exp.f64 wj)))))

   1. Initial program 92.8%

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

      1. distribute-rgt1-in 97.4%

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

      2. associate-/l/ 97.5%

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

      3. div-sub 92.9%

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

      4. associate-/l* 92.9%

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

      5. *-inverses 99.8%

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

      6. *-rgt-identity 99.8%

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

   3. Simplified 99.8%

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

      1. clear-num 99.7%

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

      2. associate-/r/ 99.7%

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

      3. rec-exp 99.8%

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

   6. Applied egg-rr 99.8%

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

4. Final simplification 99.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;wj + \frac{x - wj \cdot e^{wj}}{e^{wj} + wj \cdot e^{wj}} \leq 4 \cdot 10^{-9}:\\ \;\;\;\;x \cdot \left(\frac{e^{-wj}}{wj + 1} + {wj}^{2} \cdot \left(\frac{1}{x} - wj \cdot \left(\frac{1}{x} + wj \cdot \left(\frac{wj}{x} + \frac{-1}{x}\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;wj + \frac{wj - x \cdot e^{-wj}}{-1 - wj}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 99.5% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{e^{-wj}}{wj + 1}\\ \mathbf{if}\;wj \leq 8.8 \cdot 10^{-5}:\\ \;\;\;\;x \cdot \left(t\_0 + {wj}^{2} \cdot \left(\frac{1}{x} + wj \cdot \left(\frac{wj}{x} + \frac{-1}{x}\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(t\_0 + \frac{wj + \frac{wj}{-1 - wj}}{x}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (wj x)
 :precision binary64
 (let* ((t_0 (/ (exp (- wj)) (+ wj 1.0))))
   (if (<= wj 8.8e-5)
     (*
      x
      (+ t_0 (* (pow wj 2.0) (+ (/ 1.0 x) (* wj (+ (/ wj x) (/ -1.0 x)))))))
     (* x (+ t_0 (/ (+ wj (/ wj (- -1.0 wj))) x))))))

C

double code(double wj, double x) {
	double t_0 = exp(-wj) / (wj + 1.0);
	double tmp;
	if (wj <= 8.8e-5) {
		tmp = x * (t_0 + (pow(wj, 2.0) * ((1.0 / x) + (wj * ((wj / x) + (-1.0 / x))))));
	} else {
		tmp = x * (t_0 + ((wj + (wj / (-1.0 - wj))) / x));
	}
	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 = exp(-wj) / (wj + 1.0d0)
    if (wj <= 8.8d-5) then
        tmp = x * (t_0 + ((wj ** 2.0d0) * ((1.0d0 / x) + (wj * ((wj / x) + ((-1.0d0) / x))))))
    else
        tmp = x * (t_0 + ((wj + (wj / ((-1.0d0) - wj))) / x))
    end if
    code = tmp
end function

Java

public static double code(double wj, double x) {
	double t_0 = Math.exp(-wj) / (wj + 1.0);
	double tmp;
	if (wj <= 8.8e-5) {
		tmp = x * (t_0 + (Math.pow(wj, 2.0) * ((1.0 / x) + (wj * ((wj / x) + (-1.0 / x))))));
	} else {
		tmp = x * (t_0 + ((wj + (wj / (-1.0 - wj))) / x));
	}
	return tmp;
}

Python

def code(wj, x):
	t_0 = math.exp(-wj) / (wj + 1.0)
	tmp = 0
	if wj <= 8.8e-5:
		tmp = x * (t_0 + (math.pow(wj, 2.0) * ((1.0 / x) + (wj * ((wj / x) + (-1.0 / x))))))
	else:
		tmp = x * (t_0 + ((wj + (wj / (-1.0 - wj))) / x))
	return tmp

Julia

function code(wj, x)
	t_0 = Float64(exp(Float64(-wj)) / Float64(wj + 1.0))
	tmp = 0.0
	if (wj <= 8.8e-5)
		tmp = Float64(x * Float64(t_0 + Float64((wj ^ 2.0) * Float64(Float64(1.0 / x) + Float64(wj * Float64(Float64(wj / x) + Float64(-1.0 / x)))))));
	else
		tmp = Float64(x * Float64(t_0 + Float64(Float64(wj + Float64(wj / Float64(-1.0 - wj))) / x)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(wj, x)
	t_0 = exp(-wj) / (wj + 1.0);
	tmp = 0.0;
	if (wj <= 8.8e-5)
		tmp = x * (t_0 + ((wj ^ 2.0) * ((1.0 / x) + (wj * ((wj / x) + (-1.0 / x))))));
	else
		tmp = x * (t_0 + ((wj + (wj / (-1.0 - wj))) / x));
	end
	tmp_2 = tmp;
end

Wolfram

code[wj_, x_] := Block[{t$95$0 = N[(N[Exp[(-wj)], $MachinePrecision] / N[(wj + 1.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[wj, 8.8e-5], N[(x * N[(t$95$0 + N[(N[Power[wj, 2.0], $MachinePrecision] * N[(N[(1.0 / x), $MachinePrecision] + N[(wj * N[(N[(wj / x), $MachinePrecision] + N[(-1.0 / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x * N[(t$95$0 + N[(N[(wj + N[(wj / N[(-1.0 - wj), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if wj < 8.7999999999999998e-5

   1. Initial program 79.7%

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

      1. distribute-rgt1-in 82.1%

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

      2. associate-/l/ 82.2%

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

      3. div-sub 79.7%

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

      4. associate-/l* 79.7%

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

      5. *-inverses 82.2%

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

      6. *-rgt-identity 82.2%

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

   3. Simplified 82.2%

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

   5. Taylor expanded in x around inf 82.5%

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

      1. associate--l+ 91.9%

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

      2. associate-/r* 91.9%

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

      3. exp-neg 91.9%

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

      4. +-commutative 91.9%

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

      5. +-commutative 91.9%

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

   7. Simplified 91.9%

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

   8. Taylor expanded in wj around 0 99.2%

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

   if 8.7999999999999998e-5 < wj

   1. Initial program 74.8%

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

      1. distribute-rgt1-in 74.8%

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

      2. associate-/l/ 75.7%

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

      3. div-sub 75.7%

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

      4. associate-/l* 75.7%

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

      5. *-inverses 97.9%

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

      6. *-rgt-identity 97.9%

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

   3. Simplified 97.9%

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

   5. Taylor expanded in x around inf 97.6%

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

      1. associate--l+ 97.8%

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

      2. associate-/r* 97.8%

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

      3. exp-neg 97.8%

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

      4. +-commutative 97.8%

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

      5. +-commutative 97.8%

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

   7. Simplified 97.8%

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

   8. Taylor expanded in x around 0 98.3%

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

      1. +-commutative 98.3%

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

   10. Simplified 98.3%

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

4. Final simplification 99.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;wj \leq 8.8 \cdot 10^{-5}:\\ \;\;\;\;x \cdot \left(\frac{e^{-wj}}{wj + 1} + {wj}^{2} \cdot \left(\frac{1}{x} + wj \cdot \left(\frac{wj}{x} + \frac{-1}{x}\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(\frac{e^{-wj}}{wj + 1} + \frac{wj + \frac{wj}{-1 - wj}}{x}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 98.5% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{e^{-wj}}{wj + 1}\\ \mathbf{if}\;wj \leq -1.2 \cdot 10^{-5}:\\ \;\;\;\;x \cdot \left(t\_0 + \left(wj \cdot \frac{1}{x} - \frac{wj}{\mathsf{fma}\left(wj, x, x\right)}\right)\right)\\ \mathbf{elif}\;wj \leq 8.9 \cdot 10^{-26}:\\ \;\;\;\;x \cdot \left({wj}^{2} \cdot \left(\frac{1}{x} - wj \cdot \left(\frac{1}{x} + wj \cdot \left(\frac{wj}{x} + \frac{-1}{x}\right)\right)\right) + \left(1 + wj \cdot -2\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(t\_0 + \frac{wj + \frac{wj}{-1 - wj}}{x}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (wj x)
 :precision binary64
 (let* ((t_0 (/ (exp (- wj)) (+ wj 1.0))))
   (if (<= wj -1.2e-5)
     (* x (+ t_0 (- (* wj (/ 1.0 x)) (/ wj (fma wj x x)))))
     (if (<= wj 8.9e-26)
       (*
        x
        (+
         (*
          (pow wj 2.0)
          (- (/ 1.0 x) (* wj (+ (/ 1.0 x) (* wj (+ (/ wj x) (/ -1.0 x)))))))
         (+ 1.0 (* wj -2.0))))
       (* x (+ t_0 (/ (+ wj (/ wj (- -1.0 wj))) x)))))))

C

double code(double wj, double x) {
	double t_0 = exp(-wj) / (wj + 1.0);
	double tmp;
	if (wj <= -1.2e-5) {
		tmp = x * (t_0 + ((wj * (1.0 / x)) - (wj / fma(wj, x, x))));
	} else if (wj <= 8.9e-26) {
		tmp = x * ((pow(wj, 2.0) * ((1.0 / x) - (wj * ((1.0 / x) + (wj * ((wj / x) + (-1.0 / x))))))) + (1.0 + (wj * -2.0)));
	} else {
		tmp = x * (t_0 + ((wj + (wj / (-1.0 - wj))) / x));
	}
	return tmp;
}

Julia

function code(wj, x)
	t_0 = Float64(exp(Float64(-wj)) / Float64(wj + 1.0))
	tmp = 0.0
	if (wj <= -1.2e-5)
		tmp = Float64(x * Float64(t_0 + Float64(Float64(wj * Float64(1.0 / x)) - Float64(wj / fma(wj, x, x)))));
	elseif (wj <= 8.9e-26)
		tmp = Float64(x * Float64(Float64((wj ^ 2.0) * Float64(Float64(1.0 / x) - Float64(wj * Float64(Float64(1.0 / x) + Float64(wj * Float64(Float64(wj / x) + Float64(-1.0 / x))))))) + Float64(1.0 + Float64(wj * -2.0))));
	else
		tmp = Float64(x * Float64(t_0 + Float64(Float64(wj + Float64(wj / Float64(-1.0 - wj))) / x)));
	end
	return tmp
end

Wolfram

code[wj_, x_] := Block[{t$95$0 = N[(N[Exp[(-wj)], $MachinePrecision] / N[(wj + 1.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[wj, -1.2e-5], N[(x * N[(t$95$0 + N[(N[(wj * N[(1.0 / x), $MachinePrecision]), $MachinePrecision] - N[(wj / N[(wj * x + x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[wj, 8.9e-26], N[(x * N[(N[(N[Power[wj, 2.0], $MachinePrecision] * N[(N[(1.0 / x), $MachinePrecision] - N[(wj * N[(N[(1.0 / x), $MachinePrecision] + N[(wj * N[(N[(wj / x), $MachinePrecision] + N[(-1.0 / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(1.0 + N[(wj * -2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x * N[(t$95$0 + N[(N[(wj + N[(wj / N[(-1.0 - wj), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if wj < -1.2e-5

   1. Initial program 13.8%

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

      1. distribute-rgt1-in 99.6%

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

      2. associate-/l/ 99.8%

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

      3. div-sub 14.1%

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

      4. associate-/l* 14.1%

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

      5. *-inverses 99.8%

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

      6. *-rgt-identity 99.8%

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

   3. Simplified 99.8%

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

   5. Taylor expanded in x around inf 99.6%

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

      1. associate--l+ 99.6%

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

      2. associate-/r* 99.6%

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

      3. exp-neg 99.6%

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

      4. +-commutative 99.6%

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

      5. +-commutative 99.6%

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

   7. Simplified 99.6%

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

      1. div-inv 100.0%

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

      2. fma-neg 100.0%

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

      3. distribute-rgt-in 100.0%

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

      4. *-un-lft-identity 100.0%

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

      5. fma-define 100.0%

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

   9. Applied egg-rr 100.0%

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

       1. fma-undefine 100.0%

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

       2. unsub-neg 100.0%

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

   11. Simplified 100.0%

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

   if -1.2e-5 < wj < 8.9000000000000001e-26

   1. Initial program 81.1%

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

      1. distribute-rgt1-in 81.1%

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

      2. associate-/l/ 81.1%

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

      3. div-sub 81.1%

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

      4. associate-/l* 81.1%

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

      5. *-inverses 81.1%

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

      6. *-rgt-identity 81.1%

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

   3. Simplified 81.1%

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

   5. Taylor expanded in x around inf 81.5%

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

      1. associate--l+ 91.5%

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

      2. associate-/r* 91.5%

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

      3. exp-neg 91.5%

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

      4. +-commutative 91.5%

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

      5. +-commutative 91.5%

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

   7. Simplified 91.5%

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

   8. Taylor expanded in wj around 0 99.9%

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

   9. Taylor expanded in wj around 0 99.9%

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

       1. *-commutative 99.9%

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

   11. Simplified 99.9%

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

   if 8.9000000000000001e-26 < wj

   1. Initial program 85.3%

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

      1. distribute-rgt1-in 85.3%

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

      2. associate-/l/ 86.0%

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

      3. div-sub 86.0%

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

      4. associate-/l* 86.0%

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

      5. *-inverses 97.7%

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

      6. *-rgt-identity 97.7%

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

   3. Simplified 97.7%

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

   5. Taylor expanded in x around inf 97.5%

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

      1. associate--l+ 97.6%

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

      2. associate-/r* 97.6%

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

      3. exp-neg 97.7%

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

      4. +-commutative 97.7%

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

      5. +-commutative 97.7%

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

   7. Simplified 97.7%

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

   8. Taylor expanded in x around 0 98.0%

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

      1. +-commutative 98.0%

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

   10. Simplified 98.0%

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

4. Final simplification 99.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;wj \leq -1.2 \cdot 10^{-5}:\\ \;\;\;\;x \cdot \left(\frac{e^{-wj}}{wj + 1} + \left(wj \cdot \frac{1}{x} - \frac{wj}{\mathsf{fma}\left(wj, x, x\right)}\right)\right)\\ \mathbf{elif}\;wj \leq 8.9 \cdot 10^{-26}:\\ \;\;\;\;x \cdot \left({wj}^{2} \cdot \left(\frac{1}{x} - wj \cdot \left(\frac{1}{x} + wj \cdot \left(\frac{wj}{x} + \frac{-1}{x}\right)\right)\right) + \left(1 + wj \cdot -2\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(\frac{e^{-wj}}{wj + 1} + \frac{wj + \frac{wj}{-1 - wj}}{x}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 98.5% accurate, 2.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;wj \leq -1.2 \cdot 10^{-5}:\\ \;\;\;\;wj - \frac{\frac{x}{e^{wj}} - wj}{-1 - wj}\\ \mathbf{elif}\;wj \leq 8.9 \cdot 10^{-26}:\\ \;\;\;\;x \cdot \left({wj}^{2} \cdot \left(\frac{1}{x} - wj \cdot \left(\frac{1}{x} + wj \cdot \left(\frac{wj}{x} + \frac{-1}{x}\right)\right)\right) + \left(1 + wj \cdot -2\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(\frac{e^{-wj}}{wj + 1} + \frac{wj + \frac{wj}{-1 - wj}}{x}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (wj x)
 :precision binary64
 (if (<= wj -1.2e-5)
   (- wj (/ (- (/ x (exp wj)) wj) (- -1.0 wj)))
   (if (<= wj 8.9e-26)
     (*
      x
      (+
       (*
        (pow wj 2.0)
        (- (/ 1.0 x) (* wj (+ (/ 1.0 x) (* wj (+ (/ wj x) (/ -1.0 x)))))))
       (+ 1.0 (* wj -2.0))))
     (* x (+ (/ (exp (- wj)) (+ wj 1.0)) (/ (+ wj (/ wj (- -1.0 wj))) x))))))

C

double code(double wj, double x) {
	double tmp;
	if (wj <= -1.2e-5) {
		tmp = wj - (((x / exp(wj)) - wj) / (-1.0 - wj));
	} else if (wj <= 8.9e-26) {
		tmp = x * ((pow(wj, 2.0) * ((1.0 / x) - (wj * ((1.0 / x) + (wj * ((wj / x) + (-1.0 / x))))))) + (1.0 + (wj * -2.0)));
	} else {
		tmp = x * ((exp(-wj) / (wj + 1.0)) + ((wj + (wj / (-1.0 - wj))) / x));
	}
	return tmp;
}

Fortran

real(8) function code(wj, x)
    real(8), intent (in) :: wj
    real(8), intent (in) :: x
    real(8) :: tmp
    if (wj <= (-1.2d-5)) then
        tmp = wj - (((x / exp(wj)) - wj) / ((-1.0d0) - wj))
    else if (wj <= 8.9d-26) then
        tmp = x * (((wj ** 2.0d0) * ((1.0d0 / x) - (wj * ((1.0d0 / x) + (wj * ((wj / x) + ((-1.0d0) / x))))))) + (1.0d0 + (wj * (-2.0d0))))
    else
        tmp = x * ((exp(-wj) / (wj + 1.0d0)) + ((wj + (wj / ((-1.0d0) - wj))) / x))
    end if
    code = tmp
end function

Java

public static double code(double wj, double x) {
	double tmp;
	if (wj <= -1.2e-5) {
		tmp = wj - (((x / Math.exp(wj)) - wj) / (-1.0 - wj));
	} else if (wj <= 8.9e-26) {
		tmp = x * ((Math.pow(wj, 2.0) * ((1.0 / x) - (wj * ((1.0 / x) + (wj * ((wj / x) + (-1.0 / x))))))) + (1.0 + (wj * -2.0)));
	} else {
		tmp = x * ((Math.exp(-wj) / (wj + 1.0)) + ((wj + (wj / (-1.0 - wj))) / x));
	}
	return tmp;
}

Python

def code(wj, x):
	tmp = 0
	if wj <= -1.2e-5:
		tmp = wj - (((x / math.exp(wj)) - wj) / (-1.0 - wj))
	elif wj <= 8.9e-26:
		tmp = x * ((math.pow(wj, 2.0) * ((1.0 / x) - (wj * ((1.0 / x) + (wj * ((wj / x) + (-1.0 / x))))))) + (1.0 + (wj * -2.0)))
	else:
		tmp = x * ((math.exp(-wj) / (wj + 1.0)) + ((wj + (wj / (-1.0 - wj))) / x))
	return tmp

Julia

function code(wj, x)
	tmp = 0.0
	if (wj <= -1.2e-5)
		tmp = Float64(wj - Float64(Float64(Float64(x / exp(wj)) - wj) / Float64(-1.0 - wj)));
	elseif (wj <= 8.9e-26)
		tmp = Float64(x * Float64(Float64((wj ^ 2.0) * Float64(Float64(1.0 / x) - Float64(wj * Float64(Float64(1.0 / x) + Float64(wj * Float64(Float64(wj / x) + Float64(-1.0 / x))))))) + Float64(1.0 + Float64(wj * -2.0))));
	else
		tmp = Float64(x * Float64(Float64(exp(Float64(-wj)) / Float64(wj + 1.0)) + Float64(Float64(wj + Float64(wj / Float64(-1.0 - wj))) / x)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(wj, x)
	tmp = 0.0;
	if (wj <= -1.2e-5)
		tmp = wj - (((x / exp(wj)) - wj) / (-1.0 - wj));
	elseif (wj <= 8.9e-26)
		tmp = x * (((wj ^ 2.0) * ((1.0 / x) - (wj * ((1.0 / x) + (wj * ((wj / x) + (-1.0 / x))))))) + (1.0 + (wj * -2.0)));
	else
		tmp = x * ((exp(-wj) / (wj + 1.0)) + ((wj + (wj / (-1.0 - wj))) / x));
	end
	tmp_2 = tmp;
end

Wolfram

code[wj_, x_] := If[LessEqual[wj, -1.2e-5], N[(wj - N[(N[(N[(x / N[Exp[wj], $MachinePrecision]), $MachinePrecision] - wj), $MachinePrecision] / N[(-1.0 - wj), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[wj, 8.9e-26], N[(x * N[(N[(N[Power[wj, 2.0], $MachinePrecision] * N[(N[(1.0 / x), $MachinePrecision] - N[(wj * N[(N[(1.0 / x), $MachinePrecision] + N[(wj * N[(N[(wj / x), $MachinePrecision] + N[(-1.0 / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(1.0 + N[(wj * -2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x * N[(N[(N[Exp[(-wj)], $MachinePrecision] / N[(wj + 1.0), $MachinePrecision]), $MachinePrecision] + N[(N[(wj + N[(wj / N[(-1.0 - wj), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if wj < -1.2e-5

   1. Initial program 13.8%

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

      1. distribute-rgt1-in 99.6%

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

      2. associate-/l/ 99.8%

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

      3. div-sub 14.1%

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

      4. associate-/l* 14.1%

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

      5. *-inverses 99.8%

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

      6. *-rgt-identity 99.8%

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

   3. Simplified 99.8%

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

   if -1.2e-5 < wj < 8.9000000000000001e-26

   1. Initial program 81.1%

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

      1. distribute-rgt1-in 81.1%

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

      2. associate-/l/ 81.1%

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

      3. div-sub 81.1%

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

      4. associate-/l* 81.1%

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

      5. *-inverses 81.1%

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

      6. *-rgt-identity 81.1%

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

   3. Simplified 81.1%

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

   5. Taylor expanded in x around inf 81.5%

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

      1. associate--l+ 91.5%

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

      2. associate-/r* 91.5%

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

      3. exp-neg 91.5%

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

      4. +-commutative 91.5%

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

      5. +-commutative 91.5%

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

   7. Simplified 91.5%

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

   8. Taylor expanded in wj around 0 99.9%

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

   9. Taylor expanded in wj around 0 99.9%

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

       1. *-commutative 99.9%

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

   11. Simplified 99.9%

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

   if 8.9000000000000001e-26 < wj

   1. Initial program 85.3%

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

      1. distribute-rgt1-in 85.3%

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

      2. associate-/l/ 86.0%

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

      3. div-sub 86.0%

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

      4. associate-/l* 86.0%

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

      5. *-inverses 97.7%

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

      6. *-rgt-identity 97.7%

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

   3. Simplified 97.7%

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

   5. Taylor expanded in x around inf 97.5%

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

      1. associate--l+ 97.6%

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

      2. associate-/r* 97.6%

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

      3. exp-neg 97.7%

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

      4. +-commutative 97.7%

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

      5. +-commutative 97.7%

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

   7. Simplified 97.7%

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

   8. Taylor expanded in x around 0 98.0%

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

      1. +-commutative 98.0%

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

   10. Simplified 98.0%

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

4. Final simplification 99.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;wj \leq -1.2 \cdot 10^{-5}:\\ \;\;\;\;wj - \frac{\frac{x}{e^{wj}} - wj}{-1 - wj}\\ \mathbf{elif}\;wj \leq 8.9 \cdot 10^{-26}:\\ \;\;\;\;x \cdot \left({wj}^{2} \cdot \left(\frac{1}{x} - wj \cdot \left(\frac{1}{x} + wj \cdot \left(\frac{wj}{x} + \frac{-1}{x}\right)\right)\right) + \left(1 + wj \cdot -2\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(\frac{e^{-wj}}{wj + 1} + \frac{wj + \frac{wj}{-1 - wj}}{x}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 98.5% accurate, 2.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;wj \leq -4.2 \cdot 10^{-6}:\\ \;\;\;\;wj - \frac{\frac{x}{e^{wj}} - wj}{-1 - wj}\\ \mathbf{elif}\;wj \leq 8.9 \cdot 10^{-26}:\\ \;\;\;\;x - wj \cdot \left(x + \left(x + wj \cdot \left(x \cdot -2 + \left(-1 + \left(wj \cdot \left(x + \left(1 + \left(x + x \cdot 0.5\right)\right)\right) - x \cdot 0.5\right)\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(\frac{e^{-wj}}{wj + 1} + \frac{wj + \frac{wj}{-1 - wj}}{x}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (wj x)
 :precision binary64
 (if (<= wj -4.2e-6)
   (- wj (/ (- (/ x (exp wj)) wj) (- -1.0 wj)))
   (if (<= wj 8.9e-26)
     (-
      x
      (*
       wj
       (+
        x
        (+
         x
         (*
          wj
          (+
           (* x -2.0)
           (+ -1.0 (- (* wj (+ x (+ 1.0 (+ x (* x 0.5))))) (* x 0.5)))))))))
     (* x (+ (/ (exp (- wj)) (+ wj 1.0)) (/ (+ wj (/ wj (- -1.0 wj))) x))))))

C

double code(double wj, double x) {
	double tmp;
	if (wj <= -4.2e-6) {
		tmp = wj - (((x / exp(wj)) - wj) / (-1.0 - wj));
	} else if (wj <= 8.9e-26) {
		tmp = x - (wj * (x + (x + (wj * ((x * -2.0) + (-1.0 + ((wj * (x + (1.0 + (x + (x * 0.5))))) - (x * 0.5))))))));
	} else {
		tmp = x * ((exp(-wj) / (wj + 1.0)) + ((wj + (wj / (-1.0 - wj))) / x));
	}
	return tmp;
}

Fortran

real(8) function code(wj, x)
    real(8), intent (in) :: wj
    real(8), intent (in) :: x
    real(8) :: tmp
    if (wj <= (-4.2d-6)) then
        tmp = wj - (((x / exp(wj)) - wj) / ((-1.0d0) - wj))
    else if (wj <= 8.9d-26) then
        tmp = x - (wj * (x + (x + (wj * ((x * (-2.0d0)) + ((-1.0d0) + ((wj * (x + (1.0d0 + (x + (x * 0.5d0))))) - (x * 0.5d0))))))))
    else
        tmp = x * ((exp(-wj) / (wj + 1.0d0)) + ((wj + (wj / ((-1.0d0) - wj))) / x))
    end if
    code = tmp
end function

Java

public static double code(double wj, double x) {
	double tmp;
	if (wj <= -4.2e-6) {
		tmp = wj - (((x / Math.exp(wj)) - wj) / (-1.0 - wj));
	} else if (wj <= 8.9e-26) {
		tmp = x - (wj * (x + (x + (wj * ((x * -2.0) + (-1.0 + ((wj * (x + (1.0 + (x + (x * 0.5))))) - (x * 0.5))))))));
	} else {
		tmp = x * ((Math.exp(-wj) / (wj + 1.0)) + ((wj + (wj / (-1.0 - wj))) / x));
	}
	return tmp;
}

Python

def code(wj, x):
	tmp = 0
	if wj <= -4.2e-6:
		tmp = wj - (((x / math.exp(wj)) - wj) / (-1.0 - wj))
	elif wj <= 8.9e-26:
		tmp = x - (wj * (x + (x + (wj * ((x * -2.0) + (-1.0 + ((wj * (x + (1.0 + (x + (x * 0.5))))) - (x * 0.5))))))))
	else:
		tmp = x * ((math.exp(-wj) / (wj + 1.0)) + ((wj + (wj / (-1.0 - wj))) / x))
	return tmp

Julia

function code(wj, x)
	tmp = 0.0
	if (wj <= -4.2e-6)
		tmp = Float64(wj - Float64(Float64(Float64(x / exp(wj)) - wj) / Float64(-1.0 - wj)));
	elseif (wj <= 8.9e-26)
		tmp = Float64(x - Float64(wj * Float64(x + Float64(x + Float64(wj * Float64(Float64(x * -2.0) + Float64(-1.0 + Float64(Float64(wj * Float64(x + Float64(1.0 + Float64(x + Float64(x * 0.5))))) - Float64(x * 0.5)))))))));
	else
		tmp = Float64(x * Float64(Float64(exp(Float64(-wj)) / Float64(wj + 1.0)) + Float64(Float64(wj + Float64(wj / Float64(-1.0 - wj))) / x)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(wj, x)
	tmp = 0.0;
	if (wj <= -4.2e-6)
		tmp = wj - (((x / exp(wj)) - wj) / (-1.0 - wj));
	elseif (wj <= 8.9e-26)
		tmp = x - (wj * (x + (x + (wj * ((x * -2.0) + (-1.0 + ((wj * (x + (1.0 + (x + (x * 0.5))))) - (x * 0.5))))))));
	else
		tmp = x * ((exp(-wj) / (wj + 1.0)) + ((wj + (wj / (-1.0 - wj))) / x));
	end
	tmp_2 = tmp;
end

Wolfram

code[wj_, x_] := If[LessEqual[wj, -4.2e-6], N[(wj - N[(N[(N[(x / N[Exp[wj], $MachinePrecision]), $MachinePrecision] - wj), $MachinePrecision] / N[(-1.0 - wj), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[wj, 8.9e-26], N[(x - N[(wj * N[(x + N[(x + N[(wj * N[(N[(x * -2.0), $MachinePrecision] + N[(-1.0 + N[(N[(wj * N[(x + N[(1.0 + N[(x + N[(x * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(x * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x * N[(N[(N[Exp[(-wj)], $MachinePrecision] / N[(wj + 1.0), $MachinePrecision]), $MachinePrecision] + N[(N[(wj + N[(wj / N[(-1.0 - wj), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if wj < -4.1999999999999996e-6

   1. Initial program 21.6%

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

      1. distribute-rgt1-in 96.6%

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

      2. associate-/l/ 97.0%

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

      3. div-sub 22.0%

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

      4. associate-/l* 22.0%

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

      5. *-inverses 97.0%

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

      6. *-rgt-identity 97.0%

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

   3. Simplified 97.0%

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

   if -4.1999999999999996e-6 < wj < 8.9000000000000001e-26

   1. Initial program 81.1%

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

      1. distribute-rgt1-in 81.1%

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

      2. associate-/l/ 81.1%

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

      3. div-sub 81.1%

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

      4. associate-/l* 81.1%

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

      5. *-inverses 81.1%

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

      6. *-rgt-identity 81.1%

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

   3. Simplified 81.1%

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

   5. Taylor expanded in wj around 0 81.1%

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

      1. associate-*r* 81.1%

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

      2. neg-mul-1 81.1%

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

      3. distribute-rgt-out 81.1%

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

      4. metadata-eval 81.1%

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

   7. Simplified 81.1%

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

   8. Taylor expanded in wj around 0 99.9%

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

   if 8.9000000000000001e-26 < wj

   1. Initial program 85.3%

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

      1. distribute-rgt1-in 85.3%

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

      2. associate-/l/ 86.0%

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

      3. div-sub 86.0%

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

      4. associate-/l* 86.0%

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

      5. *-inverses 97.7%

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

      6. *-rgt-identity 97.7%

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

   3. Simplified 97.7%

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

   5. Taylor expanded in x around inf 97.5%

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

      1. associate--l+ 97.6%

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

      2. associate-/r* 97.6%

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

      3. exp-neg 97.7%

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

      4. +-commutative 97.7%

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

      5. +-commutative 97.7%

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

   7. Simplified 97.7%

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

   8. Taylor expanded in x around 0 98.0%

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

      1. +-commutative 98.0%

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

   10. Simplified 98.0%

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

4. Final simplification 99.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;wj \leq -4.2 \cdot 10^{-6}:\\ \;\;\;\;wj - \frac{\frac{x}{e^{wj}} - wj}{-1 - wj}\\ \mathbf{elif}\;wj \leq 8.9 \cdot 10^{-26}:\\ \;\;\;\;x - wj \cdot \left(x + \left(x + wj \cdot \left(x \cdot -2 + \left(-1 + \left(wj \cdot \left(x + \left(1 + \left(x + x \cdot 0.5\right)\right)\right) - x \cdot 0.5\right)\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(\frac{e^{-wj}}{wj + 1} + \frac{wj + \frac{wj}{-1 - wj}}{x}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 98.4% accurate, 2.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{e^{-wj}}{wj + 1}\\ \mathbf{if}\;wj \leq -1.5 \cdot 10^{-8}:\\ \;\;\;\;x \cdot \left(t\_0 + \left(\frac{wj}{x} + \frac{wj}{x \cdot \left(-1 - wj\right)}\right)\right)\\ \mathbf{elif}\;wj \leq 8.9 \cdot 10^{-26}:\\ \;\;\;\;x + wj \cdot \left(wj \cdot \left(1 - x \cdot -2.5\right) + x \cdot -2\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(t\_0 + \frac{wj + \frac{wj}{-1 - wj}}{x}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (wj x)
 :precision binary64
 (let* ((t_0 (/ (exp (- wj)) (+ wj 1.0))))
   (if (<= wj -1.5e-8)
     (* x (+ t_0 (+ (/ wj x) (/ wj (* x (- -1.0 wj))))))
     (if (<= wj 8.9e-26)
       (+ x (* wj (+ (* wj (- 1.0 (* x -2.5))) (* x -2.0))))
       (* x (+ t_0 (/ (+ wj (/ wj (- -1.0 wj))) x)))))))

C

double code(double wj, double x) {
	double t_0 = exp(-wj) / (wj + 1.0);
	double tmp;
	if (wj <= -1.5e-8) {
		tmp = x * (t_0 + ((wj / x) + (wj / (x * (-1.0 - wj)))));
	} else if (wj <= 8.9e-26) {
		tmp = x + (wj * ((wj * (1.0 - (x * -2.5))) + (x * -2.0)));
	} else {
		tmp = x * (t_0 + ((wj + (wj / (-1.0 - wj))) / x));
	}
	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 = exp(-wj) / (wj + 1.0d0)
    if (wj <= (-1.5d-8)) then
        tmp = x * (t_0 + ((wj / x) + (wj / (x * ((-1.0d0) - wj)))))
    else if (wj <= 8.9d-26) then
        tmp = x + (wj * ((wj * (1.0d0 - (x * (-2.5d0)))) + (x * (-2.0d0))))
    else
        tmp = x * (t_0 + ((wj + (wj / ((-1.0d0) - wj))) / x))
    end if
    code = tmp
end function

Java

public static double code(double wj, double x) {
	double t_0 = Math.exp(-wj) / (wj + 1.0);
	double tmp;
	if (wj <= -1.5e-8) {
		tmp = x * (t_0 + ((wj / x) + (wj / (x * (-1.0 - wj)))));
	} else if (wj <= 8.9e-26) {
		tmp = x + (wj * ((wj * (1.0 - (x * -2.5))) + (x * -2.0)));
	} else {
		tmp = x * (t_0 + ((wj + (wj / (-1.0 - wj))) / x));
	}
	return tmp;
}

Python

def code(wj, x):
	t_0 = math.exp(-wj) / (wj + 1.0)
	tmp = 0
	if wj <= -1.5e-8:
		tmp = x * (t_0 + ((wj / x) + (wj / (x * (-1.0 - wj)))))
	elif wj <= 8.9e-26:
		tmp = x + (wj * ((wj * (1.0 - (x * -2.5))) + (x * -2.0)))
	else:
		tmp = x * (t_0 + ((wj + (wj / (-1.0 - wj))) / x))
	return tmp

Julia

function code(wj, x)
	t_0 = Float64(exp(Float64(-wj)) / Float64(wj + 1.0))
	tmp = 0.0
	if (wj <= -1.5e-8)
		tmp = Float64(x * Float64(t_0 + Float64(Float64(wj / x) + Float64(wj / Float64(x * Float64(-1.0 - wj))))));
	elseif (wj <= 8.9e-26)
		tmp = Float64(x + Float64(wj * Float64(Float64(wj * Float64(1.0 - Float64(x * -2.5))) + Float64(x * -2.0))));
	else
		tmp = Float64(x * Float64(t_0 + Float64(Float64(wj + Float64(wj / Float64(-1.0 - wj))) / x)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(wj, x)
	t_0 = exp(-wj) / (wj + 1.0);
	tmp = 0.0;
	if (wj <= -1.5e-8)
		tmp = x * (t_0 + ((wj / x) + (wj / (x * (-1.0 - wj)))));
	elseif (wj <= 8.9e-26)
		tmp = x + (wj * ((wj * (1.0 - (x * -2.5))) + (x * -2.0)));
	else
		tmp = x * (t_0 + ((wj + (wj / (-1.0 - wj))) / x));
	end
	tmp_2 = tmp;
end

Wolfram

code[wj_, x_] := Block[{t$95$0 = N[(N[Exp[(-wj)], $MachinePrecision] / N[(wj + 1.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[wj, -1.5e-8], N[(x * N[(t$95$0 + N[(N[(wj / x), $MachinePrecision] + N[(wj / N[(x * N[(-1.0 - wj), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[wj, 8.9e-26], N[(x + N[(wj * N[(N[(wj * N[(1.0 - N[(x * -2.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(x * -2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x * N[(t$95$0 + N[(N[(wj + N[(wj / N[(-1.0 - wj), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if wj < -1.49999999999999987e-8

   1. Initial program 27.1%

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

      1. distribute-rgt1-in 93.8%

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

      2. associate-/l/ 93.9%

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

      3. div-sub 27.3%

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

      4. associate-/l* 27.3%

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

      5. *-inverses 93.9%

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

      6. *-rgt-identity 93.9%

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

   3. Simplified 93.9%

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

   5. Taylor expanded in x around inf 95.4%

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

      1. associate--l+ 95.4%

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

      2. associate-/r* 95.4%

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

      3. exp-neg 95.4%

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

      4. +-commutative 95.4%

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

      5. +-commutative 95.4%

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

   7. Simplified 95.4%

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

   if -1.49999999999999987e-8 < wj < 8.9000000000000001e-26

   1. Initial program 81.2%

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

      1. distribute-rgt1-in 81.2%

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

      2. associate-/l/ 81.2%

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

      3. div-sub 81.2%

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

      4. associate-/l* 81.2%

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

      5. *-inverses 81.2%

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

      6. *-rgt-identity 81.2%

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

   3. Simplified 81.2%

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

   5. Taylor expanded in wj around 0 100.0%

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

      1. cancel-sign-sub-inv 100.0%

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

      2. metadata-eval 100.0%

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

      3. distribute-rgt-out 100.0%

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

      4. metadata-eval 100.0%

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

      5. *-commutative 100.0%

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

   7. Simplified 100.0%

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

   if 8.9000000000000001e-26 < wj

   1. Initial program 85.3%

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

      1. distribute-rgt1-in 85.3%

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

      2. associate-/l/ 86.0%

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

      3. div-sub 86.0%

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

      4. associate-/l* 86.0%

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

      5. *-inverses 97.7%

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

      6. *-rgt-identity 97.7%

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

   3. Simplified 97.7%

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

   5. Taylor expanded in x around inf 97.5%

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

      1. associate--l+ 97.6%

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

      2. associate-/r* 97.6%

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

      3. exp-neg 97.7%

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

      4. +-commutative 97.7%

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

      5. +-commutative 97.7%

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

   7. Simplified 97.7%

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

   8. Taylor expanded in x around 0 98.0%

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

      1. +-commutative 98.0%

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

   10. Simplified 98.0%

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

4. Final simplification 99.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;wj \leq -1.5 \cdot 10^{-8}:\\ \;\;\;\;x \cdot \left(\frac{e^{-wj}}{wj + 1} + \left(\frac{wj}{x} + \frac{wj}{x \cdot \left(-1 - wj\right)}\right)\right)\\ \mathbf{elif}\;wj \leq 8.9 \cdot 10^{-26}:\\ \;\;\;\;x + wj \cdot \left(wj \cdot \left(1 - x \cdot -2.5\right) + x \cdot -2\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(\frac{e^{-wj}}{wj + 1} + \frac{wj + \frac{wj}{-1 - wj}}{x}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 99.7% accurate, 2.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;wj \leq -4.2 \cdot 10^{-6}:\\ \;\;\;\;wj - \frac{\frac{x}{e^{wj}} - wj}{-1 - wj}\\ \mathbf{elif}\;wj \leq 4.3 \cdot 10^{-6}:\\ \;\;\;\;x - wj \cdot \left(x + \left(x + wj \cdot \left(x \cdot -2 + \left(-1 + \left(wj \cdot \left(x + \left(1 + \left(x + x \cdot 0.5\right)\right)\right) - x \cdot 0.5\right)\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;wj + \frac{wj - x \cdot e^{-wj}}{-1 - wj}\\ \end{array} \end{array} \]

FPCore

(FPCore (wj x)
 :precision binary64
 (if (<= wj -4.2e-6)
   (- wj (/ (- (/ x (exp wj)) wj) (- -1.0 wj)))
   (if (<= wj 4.3e-6)
     (-
      x
      (*
       wj
       (+
        x
        (+
         x
         (*
          wj
          (+
           (* x -2.0)
           (+ -1.0 (- (* wj (+ x (+ 1.0 (+ x (* x 0.5))))) (* x 0.5)))))))))
     (+ wj (/ (- wj (* x (exp (- wj)))) (- -1.0 wj))))))

C

double code(double wj, double x) {
	double tmp;
	if (wj <= -4.2e-6) {
		tmp = wj - (((x / exp(wj)) - wj) / (-1.0 - wj));
	} else if (wj <= 4.3e-6) {
		tmp = x - (wj * (x + (x + (wj * ((x * -2.0) + (-1.0 + ((wj * (x + (1.0 + (x + (x * 0.5))))) - (x * 0.5))))))));
	} else {
		tmp = wj + ((wj - (x * exp(-wj))) / (-1.0 - wj));
	}
	return tmp;
}

Fortran

real(8) function code(wj, x)
    real(8), intent (in) :: wj
    real(8), intent (in) :: x
    real(8) :: tmp
    if (wj <= (-4.2d-6)) then
        tmp = wj - (((x / exp(wj)) - wj) / ((-1.0d0) - wj))
    else if (wj <= 4.3d-6) then
        tmp = x - (wj * (x + (x + (wj * ((x * (-2.0d0)) + ((-1.0d0) + ((wj * (x + (1.0d0 + (x + (x * 0.5d0))))) - (x * 0.5d0))))))))
    else
        tmp = wj + ((wj - (x * exp(-wj))) / ((-1.0d0) - wj))
    end if
    code = tmp
end function

Java

public static double code(double wj, double x) {
	double tmp;
	if (wj <= -4.2e-6) {
		tmp = wj - (((x / Math.exp(wj)) - wj) / (-1.0 - wj));
	} else if (wj <= 4.3e-6) {
		tmp = x - (wj * (x + (x + (wj * ((x * -2.0) + (-1.0 + ((wj * (x + (1.0 + (x + (x * 0.5))))) - (x * 0.5))))))));
	} else {
		tmp = wj + ((wj - (x * Math.exp(-wj))) / (-1.0 - wj));
	}
	return tmp;
}

Python

def code(wj, x):
	tmp = 0
	if wj <= -4.2e-6:
		tmp = wj - (((x / math.exp(wj)) - wj) / (-1.0 - wj))
	elif wj <= 4.3e-6:
		tmp = x - (wj * (x + (x + (wj * ((x * -2.0) + (-1.0 + ((wj * (x + (1.0 + (x + (x * 0.5))))) - (x * 0.5))))))))
	else:
		tmp = wj + ((wj - (x * math.exp(-wj))) / (-1.0 - wj))
	return tmp

Julia

function code(wj, x)
	tmp = 0.0
	if (wj <= -4.2e-6)
		tmp = Float64(wj - Float64(Float64(Float64(x / exp(wj)) - wj) / Float64(-1.0 - wj)));
	elseif (wj <= 4.3e-6)
		tmp = Float64(x - Float64(wj * Float64(x + Float64(x + Float64(wj * Float64(Float64(x * -2.0) + Float64(-1.0 + Float64(Float64(wj * Float64(x + Float64(1.0 + Float64(x + Float64(x * 0.5))))) - Float64(x * 0.5)))))))));
	else
		tmp = Float64(wj + Float64(Float64(wj - Float64(x * exp(Float64(-wj)))) / Float64(-1.0 - wj)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(wj, x)
	tmp = 0.0;
	if (wj <= -4.2e-6)
		tmp = wj - (((x / exp(wj)) - wj) / (-1.0 - wj));
	elseif (wj <= 4.3e-6)
		tmp = x - (wj * (x + (x + (wj * ((x * -2.0) + (-1.0 + ((wj * (x + (1.0 + (x + (x * 0.5))))) - (x * 0.5))))))));
	else
		tmp = wj + ((wj - (x * exp(-wj))) / (-1.0 - wj));
	end
	tmp_2 = tmp;
end

Wolfram

code[wj_, x_] := If[LessEqual[wj, -4.2e-6], N[(wj - N[(N[(N[(x / N[Exp[wj], $MachinePrecision]), $MachinePrecision] - wj), $MachinePrecision] / N[(-1.0 - wj), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[wj, 4.3e-6], N[(x - N[(wj * N[(x + N[(x + N[(wj * N[(N[(x * -2.0), $MachinePrecision] + N[(-1.0 + N[(N[(wj * N[(x + N[(1.0 + N[(x + N[(x * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(x * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(wj + N[(N[(wj - N[(x * N[Exp[(-wj)], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(-1.0 - wj), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if wj < -4.1999999999999996e-6

   1. Initial program 21.6%

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

      1. distribute-rgt1-in 96.6%

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

      2. associate-/l/ 97.0%

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

      3. div-sub 22.0%

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

      4. associate-/l* 22.0%

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

      5. *-inverses 97.0%

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

      6. *-rgt-identity 97.0%

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

   3. Simplified 97.0%

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

   if -4.1999999999999996e-6 < wj < 4.30000000000000033e-6

   1. Initial program 81.7%

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

      1. distribute-rgt1-in 81.7%

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

      2. associate-/l/ 81.7%

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

      3. div-sub 81.7%

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

      4. associate-/l* 81.7%

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

      5. *-inverses 81.7%

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

      6. *-rgt-identity 81.7%

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

   3. Simplified 81.7%

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

   5. Taylor expanded in wj around 0 81.7%

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

      1. associate-*r* 81.7%

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

      2. neg-mul-1 81.7%

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

      3. distribute-rgt-out 81.7%

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

      4. metadata-eval 81.7%

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

   7. Simplified 81.7%

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

   8. Taylor expanded in wj around 0 99.9%

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

   if 4.30000000000000033e-6 < wj

   1. Initial program 75.2%

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

      1. distribute-rgt1-in 75.2%

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

      2. associate-/l/ 76.3%

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

      3. div-sub 76.3%

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

      4. associate-/l* 76.3%

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

      5. *-inverses 96.3%

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

      6. *-rgt-identity 96.3%

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

   3. Simplified 96.3%

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

      1. clear-num 96.5%

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

      2. associate-/r/ 96.3%

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

      3. rec-exp 96.5%

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

   6. Applied egg-rr 96.5%

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

4. Final simplification 99.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;wj \leq -4.2 \cdot 10^{-6}:\\ \;\;\;\;wj - \frac{\frac{x}{e^{wj}} - wj}{-1 - wj}\\ \mathbf{elif}\;wj \leq 4.3 \cdot 10^{-6}:\\ \;\;\;\;x - wj \cdot \left(x + \left(x + wj \cdot \left(x \cdot -2 + \left(-1 + \left(wj \cdot \left(x + \left(1 + \left(x + x \cdot 0.5\right)\right)\right) - x \cdot 0.5\right)\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;wj + \frac{wj - x \cdot e^{-wj}}{-1 - wj}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 99.7% accurate, 2.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;wj \leq -2.95 \cdot 10^{-6} \lor \neg \left(wj \leq 1.85 \cdot 10^{-6}\right):\\ \;\;\;\;wj - \frac{\frac{x}{e^{wj}} - wj}{-1 - wj}\\ \mathbf{else}:\\ \;\;\;\;x - wj \cdot \left(x + \left(x + wj \cdot \left(x \cdot -2 + \left(-1 + \left(wj \cdot \left(x + \left(1 + \left(x + x \cdot 0.5\right)\right)\right) - x \cdot 0.5\right)\right)\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (wj x)
 :precision binary64
 (if (or (<= wj -2.95e-6) (not (<= wj 1.85e-6)))
   (- wj (/ (- (/ x (exp wj)) wj) (- -1.0 wj)))
   (-
    x
    (*
     wj
     (+
      x
      (+
       x
       (*
        wj
        (+
         (* x -2.0)
         (+ -1.0 (- (* wj (+ x (+ 1.0 (+ x (* x 0.5))))) (* x 0.5)))))))))))

C

double code(double wj, double x) {
	double tmp;
	if ((wj <= -2.95e-6) || !(wj <= 1.85e-6)) {
		tmp = wj - (((x / exp(wj)) - wj) / (-1.0 - wj));
	} else {
		tmp = x - (wj * (x + (x + (wj * ((x * -2.0) + (-1.0 + ((wj * (x + (1.0 + (x + (x * 0.5))))) - (x * 0.5))))))));
	}
	return tmp;
}

Fortran

real(8) function code(wj, x)
    real(8), intent (in) :: wj
    real(8), intent (in) :: x
    real(8) :: tmp
    if ((wj <= (-2.95d-6)) .or. (.not. (wj <= 1.85d-6))) then
        tmp = wj - (((x / exp(wj)) - wj) / ((-1.0d0) - wj))
    else
        tmp = x - (wj * (x + (x + (wj * ((x * (-2.0d0)) + ((-1.0d0) + ((wj * (x + (1.0d0 + (x + (x * 0.5d0))))) - (x * 0.5d0))))))))
    end if
    code = tmp
end function

Java

public static double code(double wj, double x) {
	double tmp;
	if ((wj <= -2.95e-6) || !(wj <= 1.85e-6)) {
		tmp = wj - (((x / Math.exp(wj)) - wj) / (-1.0 - wj));
	} else {
		tmp = x - (wj * (x + (x + (wj * ((x * -2.0) + (-1.0 + ((wj * (x + (1.0 + (x + (x * 0.5))))) - (x * 0.5))))))));
	}
	return tmp;
}

Python

def code(wj, x):
	tmp = 0
	if (wj <= -2.95e-6) or not (wj <= 1.85e-6):
		tmp = wj - (((x / math.exp(wj)) - wj) / (-1.0 - wj))
	else:
		tmp = x - (wj * (x + (x + (wj * ((x * -2.0) + (-1.0 + ((wj * (x + (1.0 + (x + (x * 0.5))))) - (x * 0.5))))))))
	return tmp

Julia

function code(wj, x)
	tmp = 0.0
	if ((wj <= -2.95e-6) || !(wj <= 1.85e-6))
		tmp = Float64(wj - Float64(Float64(Float64(x / exp(wj)) - wj) / Float64(-1.0 - wj)));
	else
		tmp = Float64(x - Float64(wj * Float64(x + Float64(x + Float64(wj * Float64(Float64(x * -2.0) + Float64(-1.0 + Float64(Float64(wj * Float64(x + Float64(1.0 + Float64(x + Float64(x * 0.5))))) - Float64(x * 0.5)))))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(wj, x)
	tmp = 0.0;
	if ((wj <= -2.95e-6) || ~((wj <= 1.85e-6)))
		tmp = wj - (((x / exp(wj)) - wj) / (-1.0 - wj));
	else
		tmp = x - (wj * (x + (x + (wj * ((x * -2.0) + (-1.0 + ((wj * (x + (1.0 + (x + (x * 0.5))))) - (x * 0.5))))))));
	end
	tmp_2 = tmp;
end

Wolfram

code[wj_, x_] := If[Or[LessEqual[wj, -2.95e-6], N[Not[LessEqual[wj, 1.85e-6]], $MachinePrecision]], N[(wj - N[(N[(N[(x / N[Exp[wj], $MachinePrecision]), $MachinePrecision] - wj), $MachinePrecision] / N[(-1.0 - wj), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x - N[(wj * N[(x + N[(x + N[(wj * N[(N[(x * -2.0), $MachinePrecision] + N[(-1.0 + N[(N[(wj * N[(x + N[(1.0 + N[(x + N[(x * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(x * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if wj < -2.95000000000000013e-6 or 1.8500000000000001e-6 < wj

   1. Initial program 53.9%

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

      1. distribute-rgt1-in 85.5%

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

      2. associate-/l/ 86.3%

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

      3. div-sub 54.7%

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

      4. associate-/l* 54.7%

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

      5. *-inverses 96.8%

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

      6. *-rgt-identity 96.8%

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

   3. Simplified 96.8%

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

   if -2.95000000000000013e-6 < wj < 1.8500000000000001e-6

   1. Initial program 81.6%

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

      1. distribute-rgt1-in 81.6%

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

      2. associate-/l/ 81.6%

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

      3. div-sub 81.6%

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

      4. associate-/l* 81.6%

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

      5. *-inverses 81.6%

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

      6. *-rgt-identity 81.6%

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

   3. Simplified 81.6%

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

   5. Taylor expanded in wj around 0 81.6%

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

      1. associate-*r* 81.6%

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

      2. neg-mul-1 81.6%

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

      3. distribute-rgt-out 81.6%

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

      4. metadata-eval 81.6%

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

   7. Simplified 81.6%

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

   8. Taylor expanded in wj around 0 99.9%

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

4. Final simplification 99.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;wj \leq -2.95 \cdot 10^{-6} \lor \neg \left(wj \leq 1.85 \cdot 10^{-6}\right):\\ \;\;\;\;wj - \frac{\frac{x}{e^{wj}} - wj}{-1 - wj}\\ \mathbf{else}:\\ \;\;\;\;x - wj \cdot \left(x + \left(x + wj \cdot \left(x \cdot -2 + \left(-1 + \left(wj \cdot \left(x + \left(1 + \left(x + x \cdot 0.5\right)\right)\right) - x \cdot 0.5\right)\right)\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 97.4% accurate, 2.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;wj \leq -0.165:\\ \;\;\;\;\frac{x}{e^{wj} \cdot \left(wj + 1\right)}\\ \mathbf{elif}\;wj \leq 8.9 \cdot 10^{-26}:\\ \;\;\;\;x - wj \cdot \left(x + \left(x + wj \cdot \left(x \cdot -2 + \left(-1 + \left(wj \cdot \left(x + \left(1 + \left(x + x \cdot 0.5\right)\right)\right) - x \cdot 0.5\right)\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;wj + \frac{\left(x - wj \cdot \left(x + wj \cdot \left(\left(x \cdot 0.5 - x\right) + wj \cdot \left(\left(x - x \cdot 0.5\right) + \left(x \cdot -0.5 + x \cdot 0.16666666666666666\right)\right)\right)\right)\right) - wj}{wj + 1}\\ \end{array} \end{array} \]

FPCore

(FPCore (wj x)
 :precision binary64
 (if (<= wj -0.165)
   (/ x (* (exp wj) (+ wj 1.0)))
   (if (<= wj 8.9e-26)
     (-
      x
      (*
       wj
       (+
        x
        (+
         x
         (*
          wj
          (+
           (* x -2.0)
           (+ -1.0 (- (* wj (+ x (+ 1.0 (+ x (* x 0.5))))) (* x 0.5)))))))))
     (+
      wj
      (/
       (-
        (-
         x
         (*
          wj
          (+
           x
           (*
            wj
            (+
             (- (* x 0.5) x)
             (*
              wj
              (+
               (- x (* x 0.5))
               (+ (* x -0.5) (* x 0.16666666666666666)))))))))
        wj)
       (+ wj 1.0))))))

C

double code(double wj, double x) {
	double tmp;
	if (wj <= -0.165) {
		tmp = x / (exp(wj) * (wj + 1.0));
	} else if (wj <= 8.9e-26) {
		tmp = x - (wj * (x + (x + (wj * ((x * -2.0) + (-1.0 + ((wj * (x + (1.0 + (x + (x * 0.5))))) - (x * 0.5))))))));
	} else {
		tmp = wj + (((x - (wj * (x + (wj * (((x * 0.5) - x) + (wj * ((x - (x * 0.5)) + ((x * -0.5) + (x * 0.16666666666666666))))))))) - wj) / (wj + 1.0));
	}
	return tmp;
}

Fortran

real(8) function code(wj, x)
    real(8), intent (in) :: wj
    real(8), intent (in) :: x
    real(8) :: tmp
    if (wj <= (-0.165d0)) then
        tmp = x / (exp(wj) * (wj + 1.0d0))
    else if (wj <= 8.9d-26) then
        tmp = x - (wj * (x + (x + (wj * ((x * (-2.0d0)) + ((-1.0d0) + ((wj * (x + (1.0d0 + (x + (x * 0.5d0))))) - (x * 0.5d0))))))))
    else
        tmp = wj + (((x - (wj * (x + (wj * (((x * 0.5d0) - x) + (wj * ((x - (x * 0.5d0)) + ((x * (-0.5d0)) + (x * 0.16666666666666666d0))))))))) - wj) / (wj + 1.0d0))
    end if
    code = tmp
end function

Java

public static double code(double wj, double x) {
	double tmp;
	if (wj <= -0.165) {
		tmp = x / (Math.exp(wj) * (wj + 1.0));
	} else if (wj <= 8.9e-26) {
		tmp = x - (wj * (x + (x + (wj * ((x * -2.0) + (-1.0 + ((wj * (x + (1.0 + (x + (x * 0.5))))) - (x * 0.5))))))));
	} else {
		tmp = wj + (((x - (wj * (x + (wj * (((x * 0.5) - x) + (wj * ((x - (x * 0.5)) + ((x * -0.5) + (x * 0.16666666666666666))))))))) - wj) / (wj + 1.0));
	}
	return tmp;
}

Python

def code(wj, x):
	tmp = 0
	if wj <= -0.165:
		tmp = x / (math.exp(wj) * (wj + 1.0))
	elif wj <= 8.9e-26:
		tmp = x - (wj * (x + (x + (wj * ((x * -2.0) + (-1.0 + ((wj * (x + (1.0 + (x + (x * 0.5))))) - (x * 0.5))))))))
	else:
		tmp = wj + (((x - (wj * (x + (wj * (((x * 0.5) - x) + (wj * ((x - (x * 0.5)) + ((x * -0.5) + (x * 0.16666666666666666))))))))) - wj) / (wj + 1.0))
	return tmp

Julia

function code(wj, x)
	tmp = 0.0
	if (wj <= -0.165)
		tmp = Float64(x / Float64(exp(wj) * Float64(wj + 1.0)));
	elseif (wj <= 8.9e-26)
		tmp = Float64(x - Float64(wj * Float64(x + Float64(x + Float64(wj * Float64(Float64(x * -2.0) + Float64(-1.0 + Float64(Float64(wj * Float64(x + Float64(1.0 + Float64(x + Float64(x * 0.5))))) - Float64(x * 0.5)))))))));
	else
		tmp = Float64(wj + Float64(Float64(Float64(x - Float64(wj * Float64(x + Float64(wj * Float64(Float64(Float64(x * 0.5) - x) + Float64(wj * Float64(Float64(x - Float64(x * 0.5)) + Float64(Float64(x * -0.5) + Float64(x * 0.16666666666666666))))))))) - wj) / Float64(wj + 1.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(wj, x)
	tmp = 0.0;
	if (wj <= -0.165)
		tmp = x / (exp(wj) * (wj + 1.0));
	elseif (wj <= 8.9e-26)
		tmp = x - (wj * (x + (x + (wj * ((x * -2.0) + (-1.0 + ((wj * (x + (1.0 + (x + (x * 0.5))))) - (x * 0.5))))))));
	else
		tmp = wj + (((x - (wj * (x + (wj * (((x * 0.5) - x) + (wj * ((x - (x * 0.5)) + ((x * -0.5) + (x * 0.16666666666666666))))))))) - wj) / (wj + 1.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[wj_, x_] := If[LessEqual[wj, -0.165], N[(x / N[(N[Exp[wj], $MachinePrecision] * N[(wj + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[wj, 8.9e-26], N[(x - N[(wj * N[(x + N[(x + N[(wj * N[(N[(x * -2.0), $MachinePrecision] + N[(-1.0 + N[(N[(wj * N[(x + N[(1.0 + N[(x + N[(x * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(x * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(wj + N[(N[(N[(x - N[(wj * N[(x + N[(wj * N[(N[(N[(x * 0.5), $MachinePrecision] - x), $MachinePrecision] + N[(wj * N[(N[(x - N[(x * 0.5), $MachinePrecision]), $MachinePrecision] + N[(N[(x * -0.5), $MachinePrecision] + N[(x * 0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - wj), $MachinePrecision] / N[(wj + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if wj < -0.165000000000000008

   1. Initial program 0.0%

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

      1. distribute-rgt1-in 100.0%

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

      2. associate-/l/ 100.0%

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

      3. div-sub 0.0%

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

      4. associate-/l* 0.0%

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

      5. *-inverses 100.0%

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

      6. *-rgt-identity 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in x around inf 100.0%

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

      1. +-commutative 100.0%

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

   7. Simplified 100.0%

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

   if -0.165000000000000008 < wj < 8.9000000000000001e-26

   1. Initial program 81.2%

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

      1. distribute-rgt1-in 81.2%

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

      2. associate-/l/ 81.2%

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

      3. div-sub 81.2%

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

      4. associate-/l* 81.2%

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

      5. *-inverses 81.2%

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

      6. *-rgt-identity 81.2%

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

   3. Simplified 81.2%

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

   5. Taylor expanded in wj around 0 81.2%

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

      1. associate-*r* 81.2%

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

      2. neg-mul-1 81.2%

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

      3. distribute-rgt-out 81.2%

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

      4. metadata-eval 81.2%

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

   7. Simplified 81.2%

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

   8. Taylor expanded in wj around 0 99.4%

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

   if 8.9000000000000001e-26 < wj

   1. Initial program 85.3%

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

      1. distribute-rgt1-in 85.3%

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

      2. associate-/l/ 86.0%

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

      3. div-sub 86.0%

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

      4. associate-/l* 86.0%

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

      5. *-inverses 97.7%

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

      6. *-rgt-identity 97.7%

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

   3. Simplified 97.7%

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

   5. Taylor expanded in wj around 0 87.0%

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

4. Final simplification 98.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;wj \leq -0.165:\\ \;\;\;\;\frac{x}{e^{wj} \cdot \left(wj + 1\right)}\\ \mathbf{elif}\;wj \leq 8.9 \cdot 10^{-26}:\\ \;\;\;\;x - wj \cdot \left(x + \left(x + wj \cdot \left(x \cdot -2 + \left(-1 + \left(wj \cdot \left(x + \left(1 + \left(x + x \cdot 0.5\right)\right)\right) - x \cdot 0.5\right)\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;wj + \frac{\left(x - wj \cdot \left(x + wj \cdot \left(\left(x \cdot 0.5 - x\right) + wj \cdot \left(\left(x - x \cdot 0.5\right) + \left(x \cdot -0.5 + x \cdot 0.16666666666666666\right)\right)\right)\right)\right) - wj}{wj + 1}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 96.3% accurate, 7.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;wj \leq 8.9 \cdot 10^{-26}:\\ \;\;\;\;x - wj \cdot \left(x + \left(x + wj \cdot \left(x \cdot -2 + \left(-1 + \left(wj \cdot \left(x + \left(1 + \left(x + x \cdot 0.5\right)\right)\right) - x \cdot 0.5\right)\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;wj + \frac{\left(x - wj \cdot \left(x + wj \cdot \left(\left(x \cdot 0.5 - x\right) + wj \cdot \left(\left(x - x \cdot 0.5\right) + \left(x \cdot -0.5 + x \cdot 0.16666666666666666\right)\right)\right)\right)\right) - wj}{wj + 1}\\ \end{array} \end{array} \]

FPCore

(FPCore (wj x)
 :precision binary64
 (if (<= wj 8.9e-26)
   (-
    x
    (*
     wj
     (+
      x
      (+
       x
       (*
        wj
        (+
         (* x -2.0)
         (+ -1.0 (- (* wj (+ x (+ 1.0 (+ x (* x 0.5))))) (* x 0.5)))))))))
   (+
    wj
    (/
     (-
      (-
       x
       (*
        wj
        (+
         x
         (*
          wj
          (+
           (- (* x 0.5) x)
           (*
            wj
            (+ (- x (* x 0.5)) (+ (* x -0.5) (* x 0.16666666666666666)))))))))
      wj)
     (+ wj 1.0)))))

C

double code(double wj, double x) {
	double tmp;
	if (wj <= 8.9e-26) {
		tmp = x - (wj * (x + (x + (wj * ((x * -2.0) + (-1.0 + ((wj * (x + (1.0 + (x + (x * 0.5))))) - (x * 0.5))))))));
	} else {
		tmp = wj + (((x - (wj * (x + (wj * (((x * 0.5) - x) + (wj * ((x - (x * 0.5)) + ((x * -0.5) + (x * 0.16666666666666666))))))))) - wj) / (wj + 1.0));
	}
	return tmp;
}

Fortran

real(8) function code(wj, x)
    real(8), intent (in) :: wj
    real(8), intent (in) :: x
    real(8) :: tmp
    if (wj <= 8.9d-26) then
        tmp = x - (wj * (x + (x + (wj * ((x * (-2.0d0)) + ((-1.0d0) + ((wj * (x + (1.0d0 + (x + (x * 0.5d0))))) - (x * 0.5d0))))))))
    else
        tmp = wj + (((x - (wj * (x + (wj * (((x * 0.5d0) - x) + (wj * ((x - (x * 0.5d0)) + ((x * (-0.5d0)) + (x * 0.16666666666666666d0))))))))) - wj) / (wj + 1.0d0))
    end if
    code = tmp
end function

Java

public static double code(double wj, double x) {
	double tmp;
	if (wj <= 8.9e-26) {
		tmp = x - (wj * (x + (x + (wj * ((x * -2.0) + (-1.0 + ((wj * (x + (1.0 + (x + (x * 0.5))))) - (x * 0.5))))))));
	} else {
		tmp = wj + (((x - (wj * (x + (wj * (((x * 0.5) - x) + (wj * ((x - (x * 0.5)) + ((x * -0.5) + (x * 0.16666666666666666))))))))) - wj) / (wj + 1.0));
	}
	return tmp;
}

Python

def code(wj, x):
	tmp = 0
	if wj <= 8.9e-26:
		tmp = x - (wj * (x + (x + (wj * ((x * -2.0) + (-1.0 + ((wj * (x + (1.0 + (x + (x * 0.5))))) - (x * 0.5))))))))
	else:
		tmp = wj + (((x - (wj * (x + (wj * (((x * 0.5) - x) + (wj * ((x - (x * 0.5)) + ((x * -0.5) + (x * 0.16666666666666666))))))))) - wj) / (wj + 1.0))
	return tmp

Julia

function code(wj, x)
	tmp = 0.0
	if (wj <= 8.9e-26)
		tmp = Float64(x - Float64(wj * Float64(x + Float64(x + Float64(wj * Float64(Float64(x * -2.0) + Float64(-1.0 + Float64(Float64(wj * Float64(x + Float64(1.0 + Float64(x + Float64(x * 0.5))))) - Float64(x * 0.5)))))))));
	else
		tmp = Float64(wj + Float64(Float64(Float64(x - Float64(wj * Float64(x + Float64(wj * Float64(Float64(Float64(x * 0.5) - x) + Float64(wj * Float64(Float64(x - Float64(x * 0.5)) + Float64(Float64(x * -0.5) + Float64(x * 0.16666666666666666))))))))) - wj) / Float64(wj + 1.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(wj, x)
	tmp = 0.0;
	if (wj <= 8.9e-26)
		tmp = x - (wj * (x + (x + (wj * ((x * -2.0) + (-1.0 + ((wj * (x + (1.0 + (x + (x * 0.5))))) - (x * 0.5))))))));
	else
		tmp = wj + (((x - (wj * (x + (wj * (((x * 0.5) - x) + (wj * ((x - (x * 0.5)) + ((x * -0.5) + (x * 0.16666666666666666))))))))) - wj) / (wj + 1.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[wj_, x_] := If[LessEqual[wj, 8.9e-26], N[(x - N[(wj * N[(x + N[(x + N[(wj * N[(N[(x * -2.0), $MachinePrecision] + N[(-1.0 + N[(N[(wj * N[(x + N[(1.0 + N[(x + N[(x * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(x * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(wj + N[(N[(N[(x - N[(wj * N[(x + N[(wj * N[(N[(N[(x * 0.5), $MachinePrecision] - x), $MachinePrecision] + N[(wj * N[(N[(x - N[(x * 0.5), $MachinePrecision]), $MachinePrecision] + N[(N[(x * -0.5), $MachinePrecision] + N[(x * 0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - wj), $MachinePrecision] / N[(wj + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if wj < 8.9000000000000001e-26

   1. Initial program 79.1%

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

      1. distribute-rgt1-in 81.6%

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

      2. associate-/l/ 81.6%

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

      3. div-sub 79.1%

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

      4. associate-/l* 79.1%

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

      5. *-inverses 81.6%

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

      6. *-rgt-identity 81.6%

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

   3. Simplified 81.6%

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

   5. Taylor expanded in wj around 0 79.2%

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

      1. associate-*r* 79.2%

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

      2. neg-mul-1 79.2%

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

      3. distribute-rgt-out 79.2%

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

      4. metadata-eval 79.2%

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

   7. Simplified 79.2%

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

   8. Taylor expanded in wj around 0 97.0%

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

   if 8.9000000000000001e-26 < wj

   1. Initial program 85.3%

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

      1. distribute-rgt1-in 85.3%

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

      2. associate-/l/ 86.0%

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

      3. div-sub 86.0%

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

      4. associate-/l* 86.0%

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

      5. *-inverses 97.7%

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

      6. *-rgt-identity 97.7%

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

   3. Simplified 97.7%

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

   5. Taylor expanded in wj around 0 87.0%

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

4. Final simplification 96.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;wj \leq 8.9 \cdot 10^{-26}:\\ \;\;\;\;x - wj \cdot \left(x + \left(x + wj \cdot \left(x \cdot -2 + \left(-1 + \left(wj \cdot \left(x + \left(1 + \left(x + x \cdot 0.5\right)\right)\right) - x \cdot 0.5\right)\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;wj + \frac{\left(x - wj \cdot \left(x + wj \cdot \left(\left(x \cdot 0.5 - x\right) + wj \cdot \left(\left(x - x \cdot 0.5\right) + \left(x \cdot -0.5 + x \cdot 0.16666666666666666\right)\right)\right)\right)\right) - wj}{wj + 1}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 97.8% accurate, 8.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;wj \leq 4.6 \cdot 10^{-5}:\\ \;\;\;\;x - wj \cdot \left(x + \left(x + wj \cdot \left(x \cdot -2 + \left(-1 + \left(wj \cdot \left(x + \left(1 + \left(x + x \cdot 0.5\right)\right)\right) - x \cdot 0.5\right)\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;wj + \frac{wj}{-1 - wj}\\ \end{array} \end{array} \]

FPCore

(FPCore (wj x)
 :precision binary64
 (if (<= wj 4.6e-5)
   (-
    x
    (*
     wj
     (+
      x
      (+
       x
       (*
        wj
        (+
         (* x -2.0)
         (+ -1.0 (- (* wj (+ x (+ 1.0 (+ x (* x 0.5))))) (* x 0.5)))))))))
   (+ wj (/ wj (- -1.0 wj)))))

C

double code(double wj, double x) {
	double tmp;
	if (wj <= 4.6e-5) {
		tmp = x - (wj * (x + (x + (wj * ((x * -2.0) + (-1.0 + ((wj * (x + (1.0 + (x + (x * 0.5))))) - (x * 0.5))))))));
	} else {
		tmp = wj + (wj / (-1.0 - wj));
	}
	return tmp;
}

Fortran

real(8) function code(wj, x)
    real(8), intent (in) :: wj
    real(8), intent (in) :: x
    real(8) :: tmp
    if (wj <= 4.6d-5) then
        tmp = x - (wj * (x + (x + (wj * ((x * (-2.0d0)) + ((-1.0d0) + ((wj * (x + (1.0d0 + (x + (x * 0.5d0))))) - (x * 0.5d0))))))))
    else
        tmp = wj + (wj / ((-1.0d0) - wj))
    end if
    code = tmp
end function

Java

public static double code(double wj, double x) {
	double tmp;
	if (wj <= 4.6e-5) {
		tmp = x - (wj * (x + (x + (wj * ((x * -2.0) + (-1.0 + ((wj * (x + (1.0 + (x + (x * 0.5))))) - (x * 0.5))))))));
	} else {
		tmp = wj + (wj / (-1.0 - wj));
	}
	return tmp;
}

Python

def code(wj, x):
	tmp = 0
	if wj <= 4.6e-5:
		tmp = x - (wj * (x + (x + (wj * ((x * -2.0) + (-1.0 + ((wj * (x + (1.0 + (x + (x * 0.5))))) - (x * 0.5))))))))
	else:
		tmp = wj + (wj / (-1.0 - wj))
	return tmp

Julia

function code(wj, x)
	tmp = 0.0
	if (wj <= 4.6e-5)
		tmp = Float64(x - Float64(wj * Float64(x + Float64(x + Float64(wj * Float64(Float64(x * -2.0) + Float64(-1.0 + Float64(Float64(wj * Float64(x + Float64(1.0 + Float64(x + Float64(x * 0.5))))) - Float64(x * 0.5)))))))));
	else
		tmp = Float64(wj + Float64(wj / Float64(-1.0 - wj)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(wj, x)
	tmp = 0.0;
	if (wj <= 4.6e-5)
		tmp = x - (wj * (x + (x + (wj * ((x * -2.0) + (-1.0 + ((wj * (x + (1.0 + (x + (x * 0.5))))) - (x * 0.5))))))));
	else
		tmp = wj + (wj / (-1.0 - wj));
	end
	tmp_2 = tmp;
end

Wolfram

code[wj_, x_] := If[LessEqual[wj, 4.6e-5], N[(x - N[(wj * N[(x + N[(x + N[(wj * N[(N[(x * -2.0), $MachinePrecision] + N[(-1.0 + N[(N[(wj * N[(x + N[(1.0 + N[(x + N[(x * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(x * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(wj + N[(wj / N[(-1.0 - wj), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if wj < 4.6e-5

   1. Initial program 79.7%

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

      1. distribute-rgt1-in 82.2%

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

      2. associate-/l/ 82.2%

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

      3. div-sub 79.7%

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

      4. associate-/l* 79.7%

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

      5. *-inverses 82.2%

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

      6. *-rgt-identity 82.2%

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

   3. Simplified 82.2%

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

   5. Taylor expanded in wj around 0 79.8%

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

      1. associate-*r* 79.8%

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

      2. neg-mul-1 79.8%

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

      3. distribute-rgt-out 79.8%

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

      4. metadata-eval 79.8%

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

   7. Simplified 79.8%

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

   8. Taylor expanded in wj around 0 97.0%

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

   if 4.6e-5 < wj

   1. Initial program 75.2%

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

      1. distribute-rgt1-in 75.2%

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

      2. associate-/l/ 76.3%

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

      3. div-sub 76.3%

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

      4. associate-/l* 76.3%

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

      5. *-inverses 96.3%

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

      6. *-rgt-identity 96.3%

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

   3. Simplified 96.3%

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

   5. Taylor expanded in x around 0 76.9%

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

      1. +-commutative 76.9%

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

   7. Simplified 76.9%

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

4. Final simplification 96.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;wj \leq 4.6 \cdot 10^{-5}:\\ \;\;\;\;x - wj \cdot \left(x + \left(x + wj \cdot \left(x \cdot -2 + \left(-1 + \left(wj \cdot \left(x + \left(1 + \left(x + x \cdot 0.5\right)\right)\right) - x \cdot 0.5\right)\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;wj + \frac{wj}{-1 - wj}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 96.2% accurate, 13.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;wj \leq 8.9 \cdot 10^{-26}:\\ \;\;\;\;x - wj \cdot \left(x \cdot 2 + wj \cdot \left(wj + -1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;wj + \frac{wj - \left(x + wj \cdot \left(x \cdot \left(-1 + wj \cdot 0.5\right)\right)\right)}{-1 - wj}\\ \end{array} \end{array} \]

FPCore

(FPCore (wj x)
 :precision binary64
 (if (<= wj 8.9e-26)
   (- x (* wj (+ (* x 2.0) (* wj (+ wj -1.0)))))
   (+ wj (/ (- wj (+ x (* wj (* x (+ -1.0 (* wj 0.5)))))) (- -1.0 wj)))))

C

double code(double wj, double x) {
	double tmp;
	if (wj <= 8.9e-26) {
		tmp = x - (wj * ((x * 2.0) + (wj * (wj + -1.0))));
	} else {
		tmp = wj + ((wj - (x + (wj * (x * (-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 <= 8.9d-26) then
        tmp = x - (wj * ((x * 2.0d0) + (wj * (wj + (-1.0d0)))))
    else
        tmp = wj + ((wj - (x + (wj * (x * ((-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 <= 8.9e-26) {
		tmp = x - (wj * ((x * 2.0) + (wj * (wj + -1.0))));
	} else {
		tmp = wj + ((wj - (x + (wj * (x * (-1.0 + (wj * 0.5)))))) / (-1.0 - wj));
	}
	return tmp;
}

Python

def code(wj, x):
	tmp = 0
	if wj <= 8.9e-26:
		tmp = x - (wj * ((x * 2.0) + (wj * (wj + -1.0))))
	else:
		tmp = wj + ((wj - (x + (wj * (x * (-1.0 + (wj * 0.5)))))) / (-1.0 - wj))
	return tmp

Julia

function code(wj, x)
	tmp = 0.0
	if (wj <= 8.9e-26)
		tmp = Float64(x - Float64(wj * Float64(Float64(x * 2.0) + Float64(wj * Float64(wj + -1.0)))));
	else
		tmp = Float64(wj + Float64(Float64(wj - Float64(x + Float64(wj * Float64(x * 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 <= 8.9e-26)
		tmp = x - (wj * ((x * 2.0) + (wj * (wj + -1.0))));
	else
		tmp = wj + ((wj - (x + (wj * (x * (-1.0 + (wj * 0.5)))))) / (-1.0 - wj));
	end
	tmp_2 = tmp;
end

Wolfram

code[wj_, x_] := If[LessEqual[wj, 8.9e-26], N[(x - N[(wj * N[(N[(x * 2.0), $MachinePrecision] + N[(wj * N[(wj + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(wj + N[(N[(wj - N[(x + N[(wj * N[(x * 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 < 8.9000000000000001e-26

   1. Initial program 79.1%

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

      1. distribute-rgt1-in 81.6%

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

      2. associate-/l/ 81.6%

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

      3. div-sub 79.1%

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

      4. associate-/l* 79.1%

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

      5. *-inverses 81.6%

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

      6. *-rgt-identity 81.6%

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

   3. Simplified 81.6%

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

   5. Taylor expanded in wj around 0 97.0%

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

   6. Taylor expanded in x around 0 96.9%

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

      1. neg-mul-1 96.9%

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

      2. unsub-neg 96.9%

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

   8. Simplified 96.9%

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

   if 8.9000000000000001e-26 < wj

   1. Initial program 85.3%

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

      1. distribute-rgt1-in 85.3%

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

      2. associate-/l/ 86.0%

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

      3. div-sub 86.0%

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

      4. associate-/l* 86.0%

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

      5. *-inverses 97.7%

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

      6. *-rgt-identity 97.7%

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

   3. Simplified 97.7%

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

   5. Taylor expanded in wj around 0 85.5%

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

      1. associate-*r* 85.5%

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

      2. neg-mul-1 85.5%

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

      3. distribute-rgt-out 85.5%

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

      4. metadata-eval 85.5%

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

   7. Simplified 85.5%

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

   8. Taylor expanded in x around 0 85.5%

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

4. Final simplification 96.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;wj \leq 8.9 \cdot 10^{-26}:\\ \;\;\;\;x - wj \cdot \left(x \cdot 2 + wj \cdot \left(wj + -1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;wj + \frac{wj - \left(x + wj \cdot \left(x \cdot \left(-1 + wj \cdot 0.5\right)\right)\right)}{-1 - wj}\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 97.6% accurate, 17.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;wj \leq 4.6 \cdot 10^{-5}:\\ \;\;\;\;x - wj \cdot \left(x \cdot 2 + wj \cdot \left(wj + -1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;wj + \frac{wj}{-1 - wj}\\ \end{array} \end{array} \]

FPCore

(FPCore (wj x)
 :precision binary64
 (if (<= wj 4.6e-5)
   (- x (* wj (+ (* x 2.0) (* wj (+ wj -1.0)))))
   (+ wj (/ wj (- -1.0 wj)))))

C

double code(double wj, double x) {
	double tmp;
	if (wj <= 4.6e-5) {
		tmp = x - (wj * ((x * 2.0) + (wj * (wj + -1.0))));
	} else {
		tmp = wj + (wj / (-1.0 - wj));
	}
	return tmp;
}

Fortran

real(8) function code(wj, x)
    real(8), intent (in) :: wj
    real(8), intent (in) :: x
    real(8) :: tmp
    if (wj <= 4.6d-5) then
        tmp = x - (wj * ((x * 2.0d0) + (wj * (wj + (-1.0d0)))))
    else
        tmp = wj + (wj / ((-1.0d0) - wj))
    end if
    code = tmp
end function

Java

public static double code(double wj, double x) {
	double tmp;
	if (wj <= 4.6e-5) {
		tmp = x - (wj * ((x * 2.0) + (wj * (wj + -1.0))));
	} else {
		tmp = wj + (wj / (-1.0 - wj));
	}
	return tmp;
}

Python

def code(wj, x):
	tmp = 0
	if wj <= 4.6e-5:
		tmp = x - (wj * ((x * 2.0) + (wj * (wj + -1.0))))
	else:
		tmp = wj + (wj / (-1.0 - wj))
	return tmp

Julia

function code(wj, x)
	tmp = 0.0
	if (wj <= 4.6e-5)
		tmp = Float64(x - Float64(wj * Float64(Float64(x * 2.0) + Float64(wj * Float64(wj + -1.0)))));
	else
		tmp = Float64(wj + Float64(wj / Float64(-1.0 - wj)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(wj, x)
	tmp = 0.0;
	if (wj <= 4.6e-5)
		tmp = x - (wj * ((x * 2.0) + (wj * (wj + -1.0))));
	else
		tmp = wj + (wj / (-1.0 - wj));
	end
	tmp_2 = tmp;
end

Wolfram

code[wj_, x_] := If[LessEqual[wj, 4.6e-5], N[(x - N[(wj * N[(N[(x * 2.0), $MachinePrecision] + N[(wj * N[(wj + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(wj + N[(wj / N[(-1.0 - wj), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if wj < 4.6e-5

   1. Initial program 79.7%

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

      1. distribute-rgt1-in 82.2%

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

      2. associate-/l/ 82.2%

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

      3. div-sub 79.7%

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

      4. associate-/l* 79.7%

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

      5. *-inverses 82.2%

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

      6. *-rgt-identity 82.2%

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

   3. Simplified 82.2%

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

   5. Taylor expanded in wj around 0 97.0%

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

   6. Taylor expanded in x around 0 96.9%

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

      1. neg-mul-1 96.9%

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

      2. unsub-neg 96.9%

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

   8. Simplified 96.9%

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

   if 4.6e-5 < wj

   1. Initial program 75.2%

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

      1. distribute-rgt1-in 75.2%

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

      2. associate-/l/ 76.3%

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

      3. div-sub 76.3%

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

      4. associate-/l* 76.3%

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

      5. *-inverses 96.3%

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

      6. *-rgt-identity 96.3%

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

   3. Simplified 96.3%

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

   5. Taylor expanded in x around 0 76.9%

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

      1. +-commutative 76.9%

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

   7. Simplified 76.9%

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

4. Final simplification 96.1%

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

Alternative 14: 85.9% accurate, 19.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;wj \leq 4.6 \cdot 10^{-5}:\\ \;\;\;\;x \cdot \left(1 - wj \cdot \left(2 - wj \cdot 2.5\right)\right)\\ \mathbf{else}:\\ \;\;\;\;wj + \frac{wj}{-1 - wj}\\ \end{array} \end{array} \]

FPCore

(FPCore (wj x)
 :precision binary64
 (if (<= wj 4.6e-5)
   (* x (- 1.0 (* wj (- 2.0 (* wj 2.5)))))
   (+ wj (/ wj (- -1.0 wj)))))

C

double code(double wj, double x) {
	double tmp;
	if (wj <= 4.6e-5) {
		tmp = x * (1.0 - (wj * (2.0 - (wj * 2.5))));
	} else {
		tmp = wj + (wj / (-1.0 - wj));
	}
	return tmp;
}

Fortran

real(8) function code(wj, x)
    real(8), intent (in) :: wj
    real(8), intent (in) :: x
    real(8) :: tmp
    if (wj <= 4.6d-5) then
        tmp = x * (1.0d0 - (wj * (2.0d0 - (wj * 2.5d0))))
    else
        tmp = wj + (wj / ((-1.0d0) - wj))
    end if
    code = tmp
end function

Java

public static double code(double wj, double x) {
	double tmp;
	if (wj <= 4.6e-5) {
		tmp = x * (1.0 - (wj * (2.0 - (wj * 2.5))));
	} else {
		tmp = wj + (wj / (-1.0 - wj));
	}
	return tmp;
}

Python

def code(wj, x):
	tmp = 0
	if wj <= 4.6e-5:
		tmp = x * (1.0 - (wj * (2.0 - (wj * 2.5))))
	else:
		tmp = wj + (wj / (-1.0 - wj))
	return tmp

Julia

function code(wj, x)
	tmp = 0.0
	if (wj <= 4.6e-5)
		tmp = Float64(x * Float64(1.0 - Float64(wj * Float64(2.0 - Float64(wj * 2.5)))));
	else
		tmp = Float64(wj + Float64(wj / Float64(-1.0 - wj)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(wj, x)
	tmp = 0.0;
	if (wj <= 4.6e-5)
		tmp = x * (1.0 - (wj * (2.0 - (wj * 2.5))));
	else
		tmp = wj + (wj / (-1.0 - wj));
	end
	tmp_2 = tmp;
end

Wolfram

code[wj_, x_] := If[LessEqual[wj, 4.6e-5], N[(x * N[(1.0 - N[(wj * N[(2.0 - N[(wj * 2.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(wj + N[(wj / N[(-1.0 - wj), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if wj < 4.6e-5

   1. Initial program 79.7%

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

      1. distribute-rgt1-in 82.2%

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

      2. associate-/l/ 82.2%

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

      3. div-sub 79.7%

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

      4. associate-/l* 79.7%

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

      5. *-inverses 82.2%

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

      6. *-rgt-identity 82.2%

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

   3. Simplified 82.2%

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

   5. Taylor expanded in wj around 0 79.8%

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

      1. associate-*r* 79.8%

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

      2. neg-mul-1 79.8%

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

      3. distribute-rgt-out 79.8%

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

      4. metadata-eval 79.8%

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

   7. Simplified 79.8%

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

   8. Taylor expanded in x around inf 88.6%

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

   9. Taylor expanded in wj around 0 88.5%

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

   if 4.6e-5 < wj

   1. Initial program 75.2%

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

      1. distribute-rgt1-in 75.2%

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

      2. associate-/l/ 76.3%

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

      3. div-sub 76.3%

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

      4. associate-/l* 76.3%

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

      5. *-inverses 96.3%

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

      6. *-rgt-identity 96.3%

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

   3. Simplified 96.3%

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

   5. Taylor expanded in x around 0 76.9%

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

      1. +-commutative 76.9%

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

   7. Simplified 76.9%

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

4. Final simplification 88.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;wj \leq 4.6 \cdot 10^{-5}:\\ \;\;\;\;x \cdot \left(1 - wj \cdot \left(2 - wj \cdot 2.5\right)\right)\\ \mathbf{else}:\\ \;\;\;\;wj + \frac{wj}{-1 - wj}\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 85.7% accurate, 22.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;wj \leq 7.8 \cdot 10^{-6}:\\ \;\;\;\;x \cdot \left(\frac{-1}{-1 - wj} - wj\right)\\ \mathbf{else}:\\ \;\;\;\;wj + \frac{wj}{-1 - wj}\\ \end{array} \end{array} \]

FPCore

(FPCore (wj x)
 :precision binary64
 (if (<= wj 7.8e-6)
   (* x (- (/ -1.0 (- -1.0 wj)) wj))
   (+ wj (/ wj (- -1.0 wj)))))

C

double code(double wj, double x) {
	double tmp;
	if (wj <= 7.8e-6) {
		tmp = x * ((-1.0 / (-1.0 - wj)) - wj);
	} else {
		tmp = wj + (wj / (-1.0 - wj));
	}
	return tmp;
}

Fortran

real(8) function code(wj, x)
    real(8), intent (in) :: wj
    real(8), intent (in) :: x
    real(8) :: tmp
    if (wj <= 7.8d-6) then
        tmp = x * (((-1.0d0) / ((-1.0d0) - wj)) - wj)
    else
        tmp = wj + (wj / ((-1.0d0) - wj))
    end if
    code = tmp
end function

Java

public static double code(double wj, double x) {
	double tmp;
	if (wj <= 7.8e-6) {
		tmp = x * ((-1.0 / (-1.0 - wj)) - wj);
	} else {
		tmp = wj + (wj / (-1.0 - wj));
	}
	return tmp;
}

Python

def code(wj, x):
	tmp = 0
	if wj <= 7.8e-6:
		tmp = x * ((-1.0 / (-1.0 - wj)) - wj)
	else:
		tmp = wj + (wj / (-1.0 - wj))
	return tmp

Julia

function code(wj, x)
	tmp = 0.0
	if (wj <= 7.8e-6)
		tmp = Float64(x * Float64(Float64(-1.0 / Float64(-1.0 - wj)) - wj));
	else
		tmp = Float64(wj + Float64(wj / Float64(-1.0 - wj)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(wj, x)
	tmp = 0.0;
	if (wj <= 7.8e-6)
		tmp = x * ((-1.0 / (-1.0 - wj)) - wj);
	else
		tmp = wj + (wj / (-1.0 - wj));
	end
	tmp_2 = tmp;
end

Wolfram

code[wj_, x_] := If[LessEqual[wj, 7.8e-6], N[(x * N[(N[(-1.0 / N[(-1.0 - wj), $MachinePrecision]), $MachinePrecision] - wj), $MachinePrecision]), $MachinePrecision], N[(wj + N[(wj / N[(-1.0 - wj), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if wj < 7.7999999999999999e-6

   1. Initial program 79.7%

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

      1. distribute-rgt1-in 82.2%

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

      2. associate-/l/ 82.2%

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

      3. div-sub 79.7%

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

      4. associate-/l* 79.7%

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

      5. *-inverses 82.2%

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

      6. *-rgt-identity 82.2%

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

   3. Simplified 82.2%

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

   5. Taylor expanded in wj around 0 79.8%

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

      1. associate-*r* 79.8%

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

      2. neg-mul-1 79.8%

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

      3. distribute-rgt-out 79.8%

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

      4. metadata-eval 79.8%

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

   7. Simplified 79.8%

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

   8. Taylor expanded in x around inf 88.6%

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

   9. Taylor expanded in wj around 0 88.4%

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

       1. neg-mul-1 88.4%

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

   11. Simplified 88.4%

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

   if 7.7999999999999999e-6 < wj

   1. Initial program 75.2%

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

      1. distribute-rgt1-in 75.2%

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

      2. associate-/l/ 76.3%

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

      3. div-sub 76.3%

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

      4. associate-/l* 76.3%

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

      5. *-inverses 96.3%

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

      6. *-rgt-identity 96.3%

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

   3. Simplified 96.3%

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

   5. Taylor expanded in x around 0 76.9%

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

      1. +-commutative 76.9%

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

   7. Simplified 76.9%

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

4. Final simplification 88.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;wj \leq 7.8 \cdot 10^{-6}:\\ \;\;\;\;x \cdot \left(\frac{-1}{-1 - wj} - wj\right)\\ \mathbf{else}:\\ \;\;\;\;wj + \frac{wj}{-1 - wj}\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 85.7% accurate, 26.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;wj \leq 7 \cdot 10^{-6}:\\ \;\;\;\;x + -2 \cdot \left(wj \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;wj + \frac{wj}{-1 - wj}\\ \end{array} \end{array} \]

FPCore

(FPCore (wj x)
 :precision binary64
 (if (<= wj 7e-6) (+ x (* -2.0 (* wj x))) (+ wj (/ wj (- -1.0 wj)))))

C

double code(double wj, double x) {
	double tmp;
	if (wj <= 7e-6) {
		tmp = x + (-2.0 * (wj * x));
	} else {
		tmp = wj + (wj / (-1.0 - wj));
	}
	return tmp;
}

Fortran

real(8) function code(wj, x)
    real(8), intent (in) :: wj
    real(8), intent (in) :: x
    real(8) :: tmp
    if (wj <= 7d-6) then
        tmp = x + ((-2.0d0) * (wj * x))
    else
        tmp = wj + (wj / ((-1.0d0) - wj))
    end if
    code = tmp
end function

Java

public static double code(double wj, double x) {
	double tmp;
	if (wj <= 7e-6) {
		tmp = x + (-2.0 * (wj * x));
	} else {
		tmp = wj + (wj / (-1.0 - wj));
	}
	return tmp;
}

Python

def code(wj, x):
	tmp = 0
	if wj <= 7e-6:
		tmp = x + (-2.0 * (wj * x))
	else:
		tmp = wj + (wj / (-1.0 - wj))
	return tmp

Julia

function code(wj, x)
	tmp = 0.0
	if (wj <= 7e-6)
		tmp = Float64(x + Float64(-2.0 * Float64(wj * x)));
	else
		tmp = Float64(wj + Float64(wj / Float64(-1.0 - wj)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(wj, x)
	tmp = 0.0;
	if (wj <= 7e-6)
		tmp = x + (-2.0 * (wj * x));
	else
		tmp = wj + (wj / (-1.0 - wj));
	end
	tmp_2 = tmp;
end

Wolfram

code[wj_, x_] := If[LessEqual[wj, 7e-6], N[(x + N[(-2.0 * N[(wj * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(wj + N[(wj / N[(-1.0 - wj), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if wj < 6.99999999999999989e-6

   1. Initial program 79.7%

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

      1. distribute-rgt1-in 82.2%

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

      2. associate-/l/ 82.2%

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

      3. div-sub 79.7%

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

      4. associate-/l* 79.7%

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

      5. *-inverses 82.2%

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

      6. *-rgt-identity 82.2%

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

   3. Simplified 82.2%

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

   5. Taylor expanded in wj around 0 88.4%

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

      1. *-commutative 88.4%

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

   7. Simplified 88.4%

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

   if 6.99999999999999989e-6 < wj

   1. Initial program 75.2%

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

      1. distribute-rgt1-in 75.2%

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

      2. associate-/l/ 76.3%

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

      3. div-sub 76.3%

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

      4. associate-/l* 76.3%

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

      5. *-inverses 96.3%

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

      6. *-rgt-identity 96.3%

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

   3. Simplified 96.3%

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

   5. Taylor expanded in x around 0 76.9%

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

      1. +-commutative 76.9%

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

   7. Simplified 76.9%

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

4. Final simplification 88.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;wj \leq 7 \cdot 10^{-6}:\\ \;\;\;\;x + -2 \cdot \left(wj \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;wj + \frac{wj}{-1 - wj}\\ \end{array} \]
5. Add Preprocessing

Alternative 17: 84.3% 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 79.5%

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

   1. distribute-rgt1-in 81.9%

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

   2. associate-/l/ 81.9%

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

   3. div-sub 79.6%

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

   4. associate-/l* 79.6%

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

   5. *-inverses 82.7%

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

   6. *-rgt-identity 82.7%

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

3. Simplified 82.7%

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

5. Taylor expanded in wj around 0 85.4%

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

   1. *-commutative 85.4%

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

7. Simplified 85.4%

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

8. Taylor expanded in x around 0 85.3%

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

   1. *-commutative 85.3%

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

10. Simplified 85.3%

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

11. Final simplification 85.3%

    \[\leadsto x \cdot \left(1 + wj \cdot -2\right) \]
12. Add Preprocessing

Alternative 18: 84.3% accurate, 44.7× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 79.5%

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

   1. distribute-rgt1-in 81.9%

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

   2. associate-/l/ 81.9%

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

   3. div-sub 79.6%

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

   4. associate-/l* 79.6%

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

   5. *-inverses 82.7%

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

   6. *-rgt-identity 82.7%

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

3. Simplified 82.7%

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

5. Taylor expanded in wj around 0 85.4%

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

   1. *-commutative 85.4%

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

7. Simplified 85.4%

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

8. Final simplification 85.4%

   \[\leadsto x + -2 \cdot \left(wj \cdot x\right) \]
9. 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 79.5%

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

   1. distribute-rgt1-in 81.9%

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

   2. associate-/l/ 81.9%

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

   3. div-sub 79.6%

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

   4. associate-/l* 79.6%

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

   5. *-inverses 82.7%

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

   6. *-rgt-identity 82.7%

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

3. Simplified 82.7%

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

5. Taylor expanded in wj around inf 4.2%

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

6. Final simplification 4.2%

   \[\leadsto wj \]
7. Add Preprocessing

Alternative 20: 83.8% 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 79.5%

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

   1. distribute-rgt1-in 81.9%

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

   2. associate-/l/ 81.9%

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

   3. div-sub 79.6%

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

   4. associate-/l* 79.6%

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

   5. *-inverses 82.7%

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

   6. *-rgt-identity 82.7%

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

3. Simplified 82.7%

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

5. Taylor expanded in wj around 0 84.8%

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

6. Final simplification 84.8%

   \[\leadsto x \]
7. Add Preprocessing

Developer target: 78.4% 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 2024080 
(FPCore (wj x)
  :name "Jmat.Real.lambertw, newton loop step"
  :precision binary64

  :alt
  (- wj (- (/ wj (+ wj 1.0)) (/ x (+ (exp wj) (* wj (exp wj))))))

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