Jmat.Real.lambertw, newton loop step

Percentage Accurate: 52.5% → 99.4%

Time: 28.6s

Alternatives: 19

Speedup: 313.0×

• 25.0% of points produce a very large (infinite) output. You may want to add a precondition.

• could not determine a ground truth

  1. wj = 2.3227803098881875e+90
  2. x = -3.245943116104921e+95

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: 52.5% 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: 99.4% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;wj \leq -2.9 \cdot 10^{-7}:\\ \;\;\;\;wj + \frac{\frac{x}{e^{wj}} - wj}{wj + 1}\\ \mathbf{else}:\\ \;\;\;\;{wj}^{2} \cdot \left(x + \left(x + 1\right)\right) + \left(\left(x + \left(\left(-1 - x\right) - x\right) \cdot {wj}^{3}\right) - wj \cdot \left(x + x\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (wj x)
 :precision binary64
 (if (<= wj -2.9e-7)
   (+ wj (/ (- (/ x (exp wj)) wj) (+ wj 1.0)))
   (+
    (* (pow wj 2.0) (+ x (+ x 1.0)))
    (- (+ x (* (- (- -1.0 x) x) (pow wj 3.0))) (* wj (+ x x))))))

C

double code(double wj, double x) {
	double tmp;
	if (wj <= -2.9e-7) {
		tmp = wj + (((x / exp(wj)) - wj) / (wj + 1.0));
	} else {
		tmp = (pow(wj, 2.0) * (x + (x + 1.0))) + ((x + (((-1.0 - x) - x) * pow(wj, 3.0))) - (wj * (x + 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 <= (-2.9d-7)) then
        tmp = wj + (((x / exp(wj)) - wj) / (wj + 1.0d0))
    else
        tmp = ((wj ** 2.0d0) * (x + (x + 1.0d0))) + ((x + ((((-1.0d0) - x) - x) * (wj ** 3.0d0))) - (wj * (x + x)))
    end if
    code = tmp
end function

Java

public static double code(double wj, double x) {
	double tmp;
	if (wj <= -2.9e-7) {
		tmp = wj + (((x / Math.exp(wj)) - wj) / (wj + 1.0));
	} else {
		tmp = (Math.pow(wj, 2.0) * (x + (x + 1.0))) + ((x + (((-1.0 - x) - x) * Math.pow(wj, 3.0))) - (wj * (x + x)));
	}
	return tmp;
}

Python

def code(wj, x):
	tmp = 0
	if wj <= -2.9e-7:
		tmp = wj + (((x / math.exp(wj)) - wj) / (wj + 1.0))
	else:
		tmp = (math.pow(wj, 2.0) * (x + (x + 1.0))) + ((x + (((-1.0 - x) - x) * math.pow(wj, 3.0))) - (wj * (x + x)))
	return tmp

Julia

function code(wj, x)
	tmp = 0.0
	if (wj <= -2.9e-7)
		tmp = Float64(wj + Float64(Float64(Float64(x / exp(wj)) - wj) / Float64(wj + 1.0)));
	else
		tmp = Float64(Float64((wj ^ 2.0) * Float64(x + Float64(x + 1.0))) + Float64(Float64(x + Float64(Float64(Float64(-1.0 - x) - x) * (wj ^ 3.0))) - Float64(wj * Float64(x + x))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(wj, x)
	tmp = 0.0;
	if (wj <= -2.9e-7)
		tmp = wj + (((x / exp(wj)) - wj) / (wj + 1.0));
	else
		tmp = ((wj ^ 2.0) * (x + (x + 1.0))) + ((x + (((-1.0 - x) - x) * (wj ^ 3.0))) - (wj * (x + x)));
	end
	tmp_2 = tmp;
end

Wolfram

code[wj_, x_] := If[LessEqual[wj, -2.9e-7], N[(wj + N[(N[(N[(x / N[Exp[wj], $MachinePrecision]), $MachinePrecision] - wj), $MachinePrecision] / N[(wj + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[Power[wj, 2.0], $MachinePrecision] * N[(x + N[(x + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(x + N[(N[(N[(-1.0 - x), $MachinePrecision] - x), $MachinePrecision] * N[Power[wj, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(wj * N[(x + x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if wj < -2.8999999999999998e-7

   1. Initial program 2.3%

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

      1. sub-neg 2.3%

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

      2. div-sub 2.3%

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

      3. sub-neg 2.3%

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

      4. +-commutative 2.3%

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

      5. distribute-neg-in 2.3%

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

      6. remove-double-neg 2.3%

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

      7. sub-neg 2.3%

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

      8. div-sub 2.3%

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

      9. distribute-rgt1-in 100.0%

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

      10. associate-/l/ 100.0%

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

   3. Simplified 100.0%

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

   if -2.8999999999999998e-7 < wj

   1. Initial program 81.0%

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

      1. sub-neg 81.0%

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

      2. div-sub 81.0%

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

      3. sub-neg 81.0%

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

      4. +-commutative 81.0%

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

      5. distribute-neg-in 81.0%

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

      6. remove-double-neg 81.0%

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

      7. sub-neg 81.0%

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

      8. div-sub 81.0%

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

      9. distribute-rgt1-in 81.0%

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

      10. associate-/l/ 81.0%

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

   3. Simplified 81.6%

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

   4. Taylor expanded in wj around 0 80.4%

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

   5. Taylor expanded in wj around 0 98.8%

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

4. Final simplification 99.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;wj \leq -2.9 \cdot 10^{-7}:\\ \;\;\;\;wj + \frac{\frac{x}{e^{wj}} - wj}{wj + 1}\\ \mathbf{else}:\\ \;\;\;\;{wj}^{2} \cdot \left(x + \left(x + 1\right)\right) + \left(\left(x + \left(\left(-1 - x\right) - x\right) \cdot {wj}^{3}\right) - wj \cdot \left(x + x\right)\right)\\ \end{array} \]

Alternative 2: 99.2% accurate, 2.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;wj \leq -6.6 \cdot 10^{-9}:\\ \;\;\;\;wj + \frac{\frac{x}{e^{wj}} - wj}{wj + 1}\\ \mathbf{else}:\\ \;\;\;\;{wj}^{2} \cdot \left(1 - \left(x \cdot -4 + x \cdot 1.5\right)\right) + \left(x + -2 \cdot \left(wj \cdot x\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (wj x)
 :precision binary64
 (if (<= wj -6.6e-9)
   (+ wj (/ (- (/ x (exp wj)) wj) (+ wj 1.0)))
   (+
    (* (pow wj 2.0) (- 1.0 (+ (* x -4.0) (* x 1.5))))
    (+ x (* -2.0 (* wj x))))))

C

double code(double wj, double x) {
	double tmp;
	if (wj <= -6.6e-9) {
		tmp = wj + (((x / exp(wj)) - wj) / (wj + 1.0));
	} else {
		tmp = (pow(wj, 2.0) * (1.0 - ((x * -4.0) + (x * 1.5)))) + (x + (-2.0 * (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 <= (-6.6d-9)) then
        tmp = wj + (((x / exp(wj)) - wj) / (wj + 1.0d0))
    else
        tmp = ((wj ** 2.0d0) * (1.0d0 - ((x * (-4.0d0)) + (x * 1.5d0)))) + (x + ((-2.0d0) * (wj * x)))
    end if
    code = tmp
end function

Java

public static double code(double wj, double x) {
	double tmp;
	if (wj <= -6.6e-9) {
		tmp = wj + (((x / Math.exp(wj)) - wj) / (wj + 1.0));
	} else {
		tmp = (Math.pow(wj, 2.0) * (1.0 - ((x * -4.0) + (x * 1.5)))) + (x + (-2.0 * (wj * x)));
	}
	return tmp;
}

Python

def code(wj, x):
	tmp = 0
	if wj <= -6.6e-9:
		tmp = wj + (((x / math.exp(wj)) - wj) / (wj + 1.0))
	else:
		tmp = (math.pow(wj, 2.0) * (1.0 - ((x * -4.0) + (x * 1.5)))) + (x + (-2.0 * (wj * x)))
	return tmp

Julia

function code(wj, x)
	tmp = 0.0
	if (wj <= -6.6e-9)
		tmp = Float64(wj + Float64(Float64(Float64(x / exp(wj)) - wj) / Float64(wj + 1.0)));
	else
		tmp = Float64(Float64((wj ^ 2.0) * Float64(1.0 - Float64(Float64(x * -4.0) + Float64(x * 1.5)))) + Float64(x + Float64(-2.0 * Float64(wj * x))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(wj, x)
	tmp = 0.0;
	if (wj <= -6.6e-9)
		tmp = wj + (((x / exp(wj)) - wj) / (wj + 1.0));
	else
		tmp = ((wj ^ 2.0) * (1.0 - ((x * -4.0) + (x * 1.5)))) + (x + (-2.0 * (wj * x)));
	end
	tmp_2 = tmp;
end

Wolfram

code[wj_, x_] := If[LessEqual[wj, -6.6e-9], N[(wj + N[(N[(N[(x / N[Exp[wj], $MachinePrecision]), $MachinePrecision] - wj), $MachinePrecision] / N[(wj + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[Power[wj, 2.0], $MachinePrecision] * N[(1.0 - N[(N[(x * -4.0), $MachinePrecision] + N[(x * 1.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(x + N[(-2.0 * N[(wj * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if wj < -6.60000000000000037e-9

   1. Initial program 2.3%

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

      1. sub-neg 2.3%

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

      2. div-sub 2.3%

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

      3. sub-neg 2.3%

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

      4. +-commutative 2.3%

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

      5. distribute-neg-in 2.3%

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

      6. remove-double-neg 2.3%

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

      7. sub-neg 2.3%

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

      8. div-sub 2.3%

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

      9. distribute-rgt1-in 100.0%

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

      10. associate-/l/ 100.0%

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

   3. Simplified 100.0%

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

   if -6.60000000000000037e-9 < wj

   1. Initial program 81.0%

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

      1. sub-neg 81.0%

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

      2. div-sub 81.0%

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

      3. sub-neg 81.0%

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

      4. +-commutative 81.0%

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

      5. distribute-neg-in 81.0%

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

      6. remove-double-neg 81.0%

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

      7. sub-neg 81.0%

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

      8. div-sub 81.0%

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

      9. distribute-rgt1-in 81.0%

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

      10. associate-/l/ 81.0%

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

   3. Simplified 81.6%

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

   4. Taylor expanded in wj around 0 98.6%

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

4. Final simplification 99.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;wj \leq -6.6 \cdot 10^{-9}:\\ \;\;\;\;wj + \frac{\frac{x}{e^{wj}} - wj}{wj + 1}\\ \mathbf{else}:\\ \;\;\;\;{wj}^{2} \cdot \left(1 - \left(x \cdot -4 + x \cdot 1.5\right)\right) + \left(x + -2 \cdot \left(wj \cdot x\right)\right)\\ \end{array} \]

Alternative 3: 99.1% accurate, 2.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;wj \leq -3.5 \cdot 10^{-9}:\\ \;\;\;\;wj + \frac{\frac{x}{e^{wj}} - wj}{wj + 1}\\ \mathbf{else}:\\ \;\;\;\;\left(x + -2 \cdot \left(wj \cdot x\right)\right) + {wj}^{2} \cdot \left(1 - x \cdot -4\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (wj x)
 :precision binary64
 (if (<= wj -3.5e-9)
   (+ wj (/ (- (/ x (exp wj)) wj) (+ wj 1.0)))
   (+ (+ x (* -2.0 (* wj x))) (* (pow wj 2.0) (- 1.0 (* x -4.0))))))

C

double code(double wj, double x) {
	double tmp;
	if (wj <= -3.5e-9) {
		tmp = wj + (((x / exp(wj)) - wj) / (wj + 1.0));
	} else {
		tmp = (x + (-2.0 * (wj * x))) + (pow(wj, 2.0) * (1.0 - (x * -4.0)));
	}
	return tmp;
}

Fortran

real(8) function code(wj, x)
    real(8), intent (in) :: wj
    real(8), intent (in) :: x
    real(8) :: tmp
    if (wj <= (-3.5d-9)) then
        tmp = wj + (((x / exp(wj)) - wj) / (wj + 1.0d0))
    else
        tmp = (x + ((-2.0d0) * (wj * x))) + ((wj ** 2.0d0) * (1.0d0 - (x * (-4.0d0))))
    end if
    code = tmp
end function

Java

public static double code(double wj, double x) {
	double tmp;
	if (wj <= -3.5e-9) {
		tmp = wj + (((x / Math.exp(wj)) - wj) / (wj + 1.0));
	} else {
		tmp = (x + (-2.0 * (wj * x))) + (Math.pow(wj, 2.0) * (1.0 - (x * -4.0)));
	}
	return tmp;
}

Python

def code(wj, x):
	tmp = 0
	if wj <= -3.5e-9:
		tmp = wj + (((x / math.exp(wj)) - wj) / (wj + 1.0))
	else:
		tmp = (x + (-2.0 * (wj * x))) + (math.pow(wj, 2.0) * (1.0 - (x * -4.0)))
	return tmp

Julia

function code(wj, x)
	tmp = 0.0
	if (wj <= -3.5e-9)
		tmp = Float64(wj + Float64(Float64(Float64(x / exp(wj)) - wj) / Float64(wj + 1.0)));
	else
		tmp = Float64(Float64(x + Float64(-2.0 * Float64(wj * x))) + Float64((wj ^ 2.0) * Float64(1.0 - Float64(x * -4.0))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(wj, x)
	tmp = 0.0;
	if (wj <= -3.5e-9)
		tmp = wj + (((x / exp(wj)) - wj) / (wj + 1.0));
	else
		tmp = (x + (-2.0 * (wj * x))) + ((wj ^ 2.0) * (1.0 - (x * -4.0)));
	end
	tmp_2 = tmp;
end

Wolfram

code[wj_, x_] := If[LessEqual[wj, -3.5e-9], N[(wj + N[(N[(N[(x / N[Exp[wj], $MachinePrecision]), $MachinePrecision] - wj), $MachinePrecision] / N[(wj + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x + N[(-2.0 * N[(wj * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[Power[wj, 2.0], $MachinePrecision] * N[(1.0 - N[(x * -4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if wj < -3.4999999999999999e-9

   1. Initial program 2.3%

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

      1. sub-neg 2.3%

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

      2. div-sub 2.3%

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

      3. sub-neg 2.3%

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

      4. +-commutative 2.3%

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

      5. distribute-neg-in 2.3%

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

      6. remove-double-neg 2.3%

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

      7. sub-neg 2.3%

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

      8. div-sub 2.3%

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

      9. distribute-rgt1-in 100.0%

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

      10. associate-/l/ 100.0%

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

   3. Simplified 100.0%

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

   if -3.4999999999999999e-9 < wj

   1. Initial program 81.0%

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

      1. distribute-rgt1-in 81.0%

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

   3. Simplified 81.0%

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

   4. Taylor expanded in wj around 0 80.5%

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

      1. *-commutative 80.5%

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

   6. Simplified 80.5%

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

   7. Taylor expanded in wj around 0 98.6%

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

4. Final simplification 99.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;wj \leq -3.5 \cdot 10^{-9}:\\ \;\;\;\;wj + \frac{\frac{x}{e^{wj}} - wj}{wj + 1}\\ \mathbf{else}:\\ \;\;\;\;\left(x + -2 \cdot \left(wj \cdot x\right)\right) + {wj}^{2} \cdot \left(1 - x \cdot -4\right)\\ \end{array} \]

Alternative 4: 99.1% accurate, 2.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;wj \leq -3.5 \cdot 10^{-9}:\\ \;\;\;\;wj + \frac{\frac{x}{e^{wj}} - wj}{wj + 1}\\ \mathbf{else}:\\ \;\;\;\;\left(x + -2 \cdot \left(wj \cdot x\right)\right) + wj \cdot wj\\ \end{array} \end{array} \]

FPCore

(FPCore (wj x)
 :precision binary64
 (if (<= wj -3.5e-9)
   (+ wj (/ (- (/ x (exp wj)) wj) (+ wj 1.0)))
   (+ (+ x (* -2.0 (* wj x))) (* wj wj))))

C

double code(double wj, double x) {
	double tmp;
	if (wj <= -3.5e-9) {
		tmp = wj + (((x / exp(wj)) - wj) / (wj + 1.0));
	} else {
		tmp = (x + (-2.0 * (wj * x))) + (wj * wj);
	}
	return tmp;
}

Fortran

real(8) function code(wj, x)
    real(8), intent (in) :: wj
    real(8), intent (in) :: x
    real(8) :: tmp
    if (wj <= (-3.5d-9)) then
        tmp = wj + (((x / exp(wj)) - wj) / (wj + 1.0d0))
    else
        tmp = (x + ((-2.0d0) * (wj * x))) + (wj * wj)
    end if
    code = tmp
end function

Java

public static double code(double wj, double x) {
	double tmp;
	if (wj <= -3.5e-9) {
		tmp = wj + (((x / Math.exp(wj)) - wj) / (wj + 1.0));
	} else {
		tmp = (x + (-2.0 * (wj * x))) + (wj * wj);
	}
	return tmp;
}

Python

def code(wj, x):
	tmp = 0
	if wj <= -3.5e-9:
		tmp = wj + (((x / math.exp(wj)) - wj) / (wj + 1.0))
	else:
		tmp = (x + (-2.0 * (wj * x))) + (wj * wj)
	return tmp

Julia

function code(wj, x)
	tmp = 0.0
	if (wj <= -3.5e-9)
		tmp = Float64(wj + Float64(Float64(Float64(x / exp(wj)) - wj) / Float64(wj + 1.0)));
	else
		tmp = Float64(Float64(x + Float64(-2.0 * Float64(wj * x))) + Float64(wj * wj));
	end
	return tmp
end

MATLAB

function tmp_2 = code(wj, x)
	tmp = 0.0;
	if (wj <= -3.5e-9)
		tmp = wj + (((x / exp(wj)) - wj) / (wj + 1.0));
	else
		tmp = (x + (-2.0 * (wj * x))) + (wj * wj);
	end
	tmp_2 = tmp;
end

Wolfram

code[wj_, x_] := If[LessEqual[wj, -3.5e-9], N[(wj + N[(N[(N[(x / N[Exp[wj], $MachinePrecision]), $MachinePrecision] - wj), $MachinePrecision] / N[(wj + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x + N[(-2.0 * N[(wj * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(wj * wj), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if wj < -3.4999999999999999e-9

   1. Initial program 2.3%

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

      1. sub-neg 2.3%

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

      2. div-sub 2.3%

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

      3. sub-neg 2.3%

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

      4. +-commutative 2.3%

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

      5. distribute-neg-in 2.3%

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

      6. remove-double-neg 2.3%

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

      7. sub-neg 2.3%

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

      8. div-sub 2.3%

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

      9. distribute-rgt1-in 100.0%

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

      10. associate-/l/ 100.0%

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

   3. Simplified 100.0%

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

   if -3.4999999999999999e-9 < wj

   1. Initial program 81.0%

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

      1. distribute-rgt1-in 81.0%

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

   3. Simplified 81.0%

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

   4. Taylor expanded in wj around 0 80.5%

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

      1. *-commutative 80.5%

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

   6. Simplified 80.5%

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

   7. Taylor expanded in wj around 0 98.6%

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

   8. Taylor expanded in x around 0 98.6%

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

      1. unpow2 98.6%

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

   10. Simplified 98.6%

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

4. Final simplification 99.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;wj \leq -3.5 \cdot 10^{-9}:\\ \;\;\;\;wj + \frac{\frac{x}{e^{wj}} - wj}{wj + 1}\\ \mathbf{else}:\\ \;\;\;\;\left(x + -2 \cdot \left(wj \cdot x\right)\right) + wj \cdot wj\\ \end{array} \]

Alternative 5: 98.8% accurate, 2.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;wj \leq -1.7 \cdot 10^{-5}:\\ \;\;\;\;wj + \frac{\frac{x}{e^{wj}}}{wj + 1}\\ \mathbf{else}:\\ \;\;\;\;\left(x + -2 \cdot \left(wj \cdot x\right)\right) + wj \cdot wj\\ \end{array} \end{array} \]

FPCore

(FPCore (wj x)
 :precision binary64
 (if (<= wj -1.7e-5)
   (+ wj (/ (/ x (exp wj)) (+ wj 1.0)))
   (+ (+ x (* -2.0 (* wj x))) (* wj wj))))

C

double code(double wj, double x) {
	double tmp;
	if (wj <= -1.7e-5) {
		tmp = wj + ((x / exp(wj)) / (wj + 1.0));
	} else {
		tmp = (x + (-2.0 * (wj * x))) + (wj * wj);
	}
	return tmp;
}

Fortran

real(8) function code(wj, x)
    real(8), intent (in) :: wj
    real(8), intent (in) :: x
    real(8) :: tmp
    if (wj <= (-1.7d-5)) then
        tmp = wj + ((x / exp(wj)) / (wj + 1.0d0))
    else
        tmp = (x + ((-2.0d0) * (wj * x))) + (wj * wj)
    end if
    code = tmp
end function

Java

public static double code(double wj, double x) {
	double tmp;
	if (wj <= -1.7e-5) {
		tmp = wj + ((x / Math.exp(wj)) / (wj + 1.0));
	} else {
		tmp = (x + (-2.0 * (wj * x))) + (wj * wj);
	}
	return tmp;
}

Python

def code(wj, x):
	tmp = 0
	if wj <= -1.7e-5:
		tmp = wj + ((x / math.exp(wj)) / (wj + 1.0))
	else:
		tmp = (x + (-2.0 * (wj * x))) + (wj * wj)
	return tmp

Julia

function code(wj, x)
	tmp = 0.0
	if (wj <= -1.7e-5)
		tmp = Float64(wj + Float64(Float64(x / exp(wj)) / Float64(wj + 1.0)));
	else
		tmp = Float64(Float64(x + Float64(-2.0 * Float64(wj * x))) + Float64(wj * wj));
	end
	return tmp
end

MATLAB

function tmp_2 = code(wj, x)
	tmp = 0.0;
	if (wj <= -1.7e-5)
		tmp = wj + ((x / exp(wj)) / (wj + 1.0));
	else
		tmp = (x + (-2.0 * (wj * x))) + (wj * wj);
	end
	tmp_2 = tmp;
end

Wolfram

code[wj_, x_] := If[LessEqual[wj, -1.7e-5], N[(wj + N[(N[(x / N[Exp[wj], $MachinePrecision]), $MachinePrecision] / N[(wj + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x + N[(-2.0 * N[(wj * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(wj * wj), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if wj < -1.7e-5

   1. Initial program 2.3%

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

      1. distribute-rgt1-in 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in x around inf 99.2%

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

      1. associate-*r/ 99.2%

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

      2. +-commutative 99.2%

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

      3. neg-mul-1 99.2%

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

      4. associate-/r* 99.2%

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

   6. Simplified 99.2%

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

   if -1.7e-5 < wj

   1. Initial program 81.0%

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

      1. distribute-rgt1-in 81.0%

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

   3. Simplified 81.0%

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

   4. Taylor expanded in wj around 0 80.5%

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

      1. *-commutative 80.5%

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

   6. Simplified 80.5%

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

   7. Taylor expanded in wj around 0 98.6%

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

   8. Taylor expanded in x around 0 98.6%

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

      1. unpow2 98.6%

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

   10. Simplified 98.6%

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

4. Final simplification 98.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;wj \leq -1.7 \cdot 10^{-5}:\\ \;\;\;\;wj + \frac{\frac{x}{e^{wj}}}{wj + 1}\\ \mathbf{else}:\\ \;\;\;\;\left(x + -2 \cdot \left(wj \cdot x\right)\right) + wj \cdot wj\\ \end{array} \]

Alternative 6: 98.8% accurate, 2.9× speedup

Math

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

FPCore

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

C

double code(double wj, double x) {
	double tmp;
	if (wj <= -8.5e-6) {
		tmp = x / (exp(wj) * (wj + 1.0));
	} else {
		tmp = (x + (-2.0 * (wj * x))) + (wj * wj);
	}
	return tmp;
}

Fortran

real(8) function code(wj, x)
    real(8), intent (in) :: wj
    real(8), intent (in) :: x
    real(8) :: tmp
    if (wj <= (-8.5d-6)) then
        tmp = x / (exp(wj) * (wj + 1.0d0))
    else
        tmp = (x + ((-2.0d0) * (wj * x))) + (wj * wj)
    end if
    code = tmp
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if wj < -8.4999999999999999e-6

   1. Initial program 2.3%

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

      1. sub-neg 2.3%

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

      2. div-sub 2.3%

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

      3. sub-neg 2.3%

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

      4. +-commutative 2.3%

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

      5. distribute-neg-in 2.3%

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

      6. remove-double-neg 2.3%

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

      7. sub-neg 2.3%

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

      8. div-sub 2.3%

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

      9. distribute-rgt1-in 100.0%

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

      10. associate-/l/ 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in x around inf 98.9%

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

   if -8.4999999999999999e-6 < wj

   1. Initial program 81.0%

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

      1. distribute-rgt1-in 81.0%

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

   3. Simplified 81.0%

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

   4. Taylor expanded in wj around 0 80.5%

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

      1. *-commutative 80.5%

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

   6. Simplified 80.5%

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

   7. Taylor expanded in wj around 0 98.6%

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

   8. Taylor expanded in x around 0 98.6%

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

      1. unpow2 98.6%

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

   10. Simplified 98.6%

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

4. Final simplification 98.7%

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

Alternative 7: 98.5% accurate, 2.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if wj < -1

   1. Initial program 1.2%

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

      1. sub-neg 1.2%

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

      2. div-sub 1.1%

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

      3. sub-neg 1.1%

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

      4. +-commutative 1.1%

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

      5. distribute-neg-in 1.1%

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

      6. remove-double-neg 1.1%

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

      7. sub-neg 1.1%

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

      8. div-sub 1.2%

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

      9. distribute-rgt1-in 100.0%

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

      10. associate-/l/ 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in x around inf 98.9%

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

   5. Taylor expanded in wj around inf 98.9%

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

   if -1 < wj

   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%

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

   3. Simplified 81.1%

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

   4. Taylor expanded in wj around 0 80.0%

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

      1. *-commutative 80.0%

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

   6. Simplified 80.0%

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

   7. Taylor expanded in wj around 0 98.1%

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

   8. Taylor expanded in x around 0 98.1%

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

      1. unpow2 98.1%

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

   10. Simplified 98.1%

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

4. Final simplification 98.4%

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

Alternative 8: 75.5% accurate, 18.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := wj + \frac{x \cdot \left(1 - wj\right)}{wj}\\ \mathbf{if}\;wj \leq -1.9 \cdot 10^{+266}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;wj \leq -4.8 \cdot 10^{+193}:\\ \;\;\;\;wj \cdot wj\\ \mathbf{elif}\;wj \leq -1:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\left(x + -2 \cdot \left(wj \cdot x\right)\right) + wj \cdot wj\\ \end{array} \end{array} \]

FPCore

(FPCore (wj x)
 :precision binary64
 (let* ((t_0 (+ wj (/ (* x (- 1.0 wj)) wj))))
   (if (<= wj -1.9e+266)
     t_0
     (if (<= wj -4.8e+193)
       (* wj wj)
       (if (<= wj -1.0) t_0 (+ (+ x (* -2.0 (* wj x))) (* wj wj)))))))

C

double code(double wj, double x) {
	double t_0 = wj + ((x * (1.0 - wj)) / wj);
	double tmp;
	if (wj <= -1.9e+266) {
		tmp = t_0;
	} else if (wj <= -4.8e+193) {
		tmp = wj * wj;
	} else if (wj <= -1.0) {
		tmp = t_0;
	} else {
		tmp = (x + (-2.0 * (wj * x))) + (wj * 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) :: tmp
    t_0 = wj + ((x * (1.0d0 - wj)) / wj)
    if (wj <= (-1.9d+266)) then
        tmp = t_0
    else if (wj <= (-4.8d+193)) then
        tmp = wj * wj
    else if (wj <= (-1.0d0)) then
        tmp = t_0
    else
        tmp = (x + ((-2.0d0) * (wj * x))) + (wj * wj)
    end if
    code = tmp
end function

Java

public static double code(double wj, double x) {
	double t_0 = wj + ((x * (1.0 - wj)) / wj);
	double tmp;
	if (wj <= -1.9e+266) {
		tmp = t_0;
	} else if (wj <= -4.8e+193) {
		tmp = wj * wj;
	} else if (wj <= -1.0) {
		tmp = t_0;
	} else {
		tmp = (x + (-2.0 * (wj * x))) + (wj * wj);
	}
	return tmp;
}

Python

def code(wj, x):
	t_0 = wj + ((x * (1.0 - wj)) / wj)
	tmp = 0
	if wj <= -1.9e+266:
		tmp = t_0
	elif wj <= -4.8e+193:
		tmp = wj * wj
	elif wj <= -1.0:
		tmp = t_0
	else:
		tmp = (x + (-2.0 * (wj * x))) + (wj * wj)
	return tmp

Julia

function code(wj, x)
	t_0 = Float64(wj + Float64(Float64(x * Float64(1.0 - wj)) / wj))
	tmp = 0.0
	if (wj <= -1.9e+266)
		tmp = t_0;
	elseif (wj <= -4.8e+193)
		tmp = Float64(wj * wj);
	elseif (wj <= -1.0)
		tmp = t_0;
	else
		tmp = Float64(Float64(x + Float64(-2.0 * Float64(wj * x))) + Float64(wj * wj));
	end
	return tmp
end

MATLAB

function tmp_2 = code(wj, x)
	t_0 = wj + ((x * (1.0 - wj)) / wj);
	tmp = 0.0;
	if (wj <= -1.9e+266)
		tmp = t_0;
	elseif (wj <= -4.8e+193)
		tmp = wj * wj;
	elseif (wj <= -1.0)
		tmp = t_0;
	else
		tmp = (x + (-2.0 * (wj * x))) + (wj * wj);
	end
	tmp_2 = tmp;
end

Wolfram

code[wj_, x_] := Block[{t$95$0 = N[(wj + N[(N[(x * N[(1.0 - wj), $MachinePrecision]), $MachinePrecision] / wj), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[wj, -1.9e+266], t$95$0, If[LessEqual[wj, -4.8e+193], N[(wj * wj), $MachinePrecision], If[LessEqual[wj, -1.0], t$95$0, N[(N[(x + N[(-2.0 * N[(wj * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(wj * wj), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if wj < -1.8999999999999999e266 or -4.8e193 < wj < -1

   1. Initial program 1.7%

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

      1. sub-neg 1.7%

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

      2. div-sub 1.7%

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

      3. sub-neg 1.7%

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

      4. +-commutative 1.7%

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

      5. distribute-neg-in 1.7%

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

      6. remove-double-neg 1.7%

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

      7. sub-neg 1.7%

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

      8. div-sub 1.7%

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

      9. distribute-rgt1-in 100.0%

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

      10. associate-/l/ 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in wj around 0 37.7%

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

   5. Taylor expanded in x around inf 36.5%

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

      1. associate-/l* 5.0%

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

      2. mul-1-neg 5.0%

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

   7. Simplified 5.0%

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

   8. Taylor expanded in wj around inf 5.0%

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

   9. Taylor expanded in x around 0 36.5%

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

   if -1.8999999999999999e266 < wj < -4.8e193

   1. Initial program 0.0%

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

      1. sub-neg 0.0%

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

      2. div-sub 0.0%

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

      3. sub-neg 0.0%

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

      4. +-commutative 0.0%

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

      5. distribute-neg-in 0.0%

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

      6. remove-double-neg 0.0%

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

      7. sub-neg 0.0%

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

      8. div-sub 0.0%

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

      9. distribute-rgt1-in 100.0%

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

      10. associate-/l/ 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in x around 0 2.5%

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

      1. +-commutative 2.5%

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

   6. Simplified 2.5%

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

   7. Taylor expanded in wj around 0 60.7%

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

      1. unpow2 60.7%

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

   9. Simplified 60.7%

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

   if -1 < wj

   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%

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

   3. Simplified 81.1%

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

   4. Taylor expanded in wj around 0 80.0%

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

      1. *-commutative 80.0%

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

   6. Simplified 80.0%

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

   7. Taylor expanded in wj around 0 98.1%

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

   8. Taylor expanded in x around 0 98.1%

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

      1. unpow2 98.1%

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

   10. Simplified 98.1%

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

4. Final simplification 79.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;wj \leq -1.9 \cdot 10^{+266}:\\ \;\;\;\;wj + \frac{x \cdot \left(1 - wj\right)}{wj}\\ \mathbf{elif}\;wj \leq -4.8 \cdot 10^{+193}:\\ \;\;\;\;wj \cdot wj\\ \mathbf{elif}\;wj \leq -1:\\ \;\;\;\;wj + \frac{x \cdot \left(1 - wj\right)}{wj}\\ \mathbf{else}:\\ \;\;\;\;\left(x + -2 \cdot \left(wj \cdot x\right)\right) + wj \cdot wj\\ \end{array} \]

Alternative 9: 77.5% accurate, 18.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x - wj \cdot x\\ \mathbf{if}\;x \leq -7.6 \cdot 10^{+153}:\\ \;\;\;\;\frac{t_0}{wj + 1}\\ \mathbf{elif}\;x \leq 7.6 \cdot 10^{-127}:\\ \;\;\;\;\left(x + -2 \cdot \left(wj \cdot x\right)\right) + wj \cdot wj\\ \mathbf{else}:\\ \;\;\;\;wj + \frac{t_0 - wj}{wj + 1}\\ \end{array} \end{array} \]

FPCore

(FPCore (wj x)
 :precision binary64
 (let* ((t_0 (- x (* wj x))))
   (if (<= x -7.6e+153)
     (/ t_0 (+ wj 1.0))
     (if (<= x 7.6e-127)
       (+ (+ x (* -2.0 (* wj x))) (* wj wj))
       (+ wj (/ (- t_0 wj) (+ wj 1.0)))))))

C

double code(double wj, double x) {
	double t_0 = x - (wj * x);
	double tmp;
	if (x <= -7.6e+153) {
		tmp = t_0 / (wj + 1.0);
	} else if (x <= 7.6e-127) {
		tmp = (x + (-2.0 * (wj * x))) + (wj * wj);
	} else {
		tmp = wj + ((t_0 - wj) / (wj + 1.0));
	}
	return tmp;
}

Fortran

real(8) function code(wj, x)
    real(8), intent (in) :: wj
    real(8), intent (in) :: x
    real(8) :: t_0
    real(8) :: tmp
    t_0 = x - (wj * x)
    if (x <= (-7.6d+153)) then
        tmp = t_0 / (wj + 1.0d0)
    else if (x <= 7.6d-127) then
        tmp = (x + ((-2.0d0) * (wj * x))) + (wj * wj)
    else
        tmp = wj + ((t_0 - wj) / (wj + 1.0d0))
    end if
    code = tmp
end function

Java

public static double code(double wj, double x) {
	double t_0 = x - (wj * x);
	double tmp;
	if (x <= -7.6e+153) {
		tmp = t_0 / (wj + 1.0);
	} else if (x <= 7.6e-127) {
		tmp = (x + (-2.0 * (wj * x))) + (wj * wj);
	} else {
		tmp = wj + ((t_0 - wj) / (wj + 1.0));
	}
	return tmp;
}

Python

def code(wj, x):
	t_0 = x - (wj * x)
	tmp = 0
	if x <= -7.6e+153:
		tmp = t_0 / (wj + 1.0)
	elif x <= 7.6e-127:
		tmp = (x + (-2.0 * (wj * x))) + (wj * wj)
	else:
		tmp = wj + ((t_0 - wj) / (wj + 1.0))
	return tmp

Julia

function code(wj, x)
	t_0 = Float64(x - Float64(wj * x))
	tmp = 0.0
	if (x <= -7.6e+153)
		tmp = Float64(t_0 / Float64(wj + 1.0));
	elseif (x <= 7.6e-127)
		tmp = Float64(Float64(x + Float64(-2.0 * Float64(wj * x))) + Float64(wj * wj));
	else
		tmp = Float64(wj + Float64(Float64(t_0 - wj) / Float64(wj + 1.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(wj, x)
	t_0 = x - (wj * x);
	tmp = 0.0;
	if (x <= -7.6e+153)
		tmp = t_0 / (wj + 1.0);
	elseif (x <= 7.6e-127)
		tmp = (x + (-2.0 * (wj * x))) + (wj * wj);
	else
		tmp = wj + ((t_0 - wj) / (wj + 1.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[wj_, x_] := Block[{t$95$0 = N[(x - N[(wj * x), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -7.6e+153], N[(t$95$0 / N[(wj + 1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 7.6e-127], N[(N[(x + N[(-2.0 * N[(wj * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(wj * wj), $MachinePrecision]), $MachinePrecision], N[(wj + N[(N[(t$95$0 - wj), $MachinePrecision] / N[(wj + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if x < -7.59999999999999933e153

   1. Initial program 62.5%

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

      1. sub-neg 62.5%

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

      2. div-sub 62.5%

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

      3. sub-neg 62.5%

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

      4. +-commutative 62.5%

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

      5. distribute-neg-in 62.5%

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

      6. remove-double-neg 62.5%

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

      7. sub-neg 62.5%

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

      8. div-sub 62.5%

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

      9. distribute-rgt1-in 96.8%

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

      10. associate-/l/ 96.9%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in x around inf 96.9%

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

      1. *-un-lft-identity 96.9%

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

      2. +-commutative 96.9%

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

      3. times-frac 96.9%

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

      4. +-commutative 96.9%

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

   6. Applied egg-rr 96.9%

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

      1. associate-*l/ 97.0%

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

      2. *-lft-identity 97.0%

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

   8. Simplified 97.0%

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

   9. Taylor expanded in wj around 0 90.9%

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

   if -7.59999999999999933e153 < x < 7.60000000000000005e-127

   1. Initial program 46.9%

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

      1. distribute-rgt1-in 76.4%

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

   3. Simplified 76.4%

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

   4. Taylor expanded in wj around 0 46.8%

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

      1. *-commutative 46.8%

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

   6. Simplified 46.8%

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

   7. Taylor expanded in wj around 0 79.2%

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

   8. Taylor expanded in x around 0 84.0%

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

      1. unpow2 84.0%

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

   10. Simplified 84.0%

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

   if 7.60000000000000005e-127 < x

   1. Initial program 60.6%

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

      1. sub-neg 60.6%

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

      2. div-sub 60.6%

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

      3. sub-neg 60.6%

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

      4. +-commutative 60.6%

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

      5. distribute-neg-in 60.6%

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

      6. remove-double-neg 60.6%

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

      7. sub-neg 60.6%

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

      8. div-sub 60.6%

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

      9. distribute-rgt1-in 99.6%

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

      10. associate-/l/ 99.5%

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

   3. Simplified 99.5%

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

   4. Taylor expanded in wj around 0 77.7%

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

4. Final simplification 82.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -7.6 \cdot 10^{+153}:\\ \;\;\;\;\frac{x - wj \cdot x}{wj + 1}\\ \mathbf{elif}\;x \leq 7.6 \cdot 10^{-127}:\\ \;\;\;\;\left(x + -2 \cdot \left(wj \cdot x\right)\right) + wj \cdot wj\\ \mathbf{else}:\\ \;\;\;\;wj + \frac{\left(x - wj \cdot x\right) - wj}{wj + 1}\\ \end{array} \]

Alternative 10: 67.6% accurate, 20.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := wj + \frac{x \cdot \left(1 - wj\right)}{wj}\\ \mathbf{if}\;wj \leq -2.9 \cdot 10^{+266}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;wj \leq -5.1 \cdot 10^{+191}:\\ \;\;\;\;wj \cdot wj\\ \mathbf{elif}\;wj \leq -1:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;x + -2 \cdot \left(wj \cdot x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (wj x)
 :precision binary64
 (let* ((t_0 (+ wj (/ (* x (- 1.0 wj)) wj))))
   (if (<= wj -2.9e+266)
     t_0
     (if (<= wj -5.1e+191)
       (* wj wj)
       (if (<= wj -1.0) t_0 (+ x (* -2.0 (* wj x))))))))

C

double code(double wj, double x) {
	double t_0 = wj + ((x * (1.0 - wj)) / wj);
	double tmp;
	if (wj <= -2.9e+266) {
		tmp = t_0;
	} else if (wj <= -5.1e+191) {
		tmp = wj * wj;
	} else if (wj <= -1.0) {
		tmp = t_0;
	} else {
		tmp = x + (-2.0 * (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 = wj + ((x * (1.0d0 - wj)) / wj)
    if (wj <= (-2.9d+266)) then
        tmp = t_0
    else if (wj <= (-5.1d+191)) then
        tmp = wj * wj
    else if (wj <= (-1.0d0)) then
        tmp = t_0
    else
        tmp = x + ((-2.0d0) * (wj * x))
    end if
    code = tmp
end function

Java

public static double code(double wj, double x) {
	double t_0 = wj + ((x * (1.0 - wj)) / wj);
	double tmp;
	if (wj <= -2.9e+266) {
		tmp = t_0;
	} else if (wj <= -5.1e+191) {
		tmp = wj * wj;
	} else if (wj <= -1.0) {
		tmp = t_0;
	} else {
		tmp = x + (-2.0 * (wj * x));
	}
	return tmp;
}

Python

def code(wj, x):
	t_0 = wj + ((x * (1.0 - wj)) / wj)
	tmp = 0
	if wj <= -2.9e+266:
		tmp = t_0
	elif wj <= -5.1e+191:
		tmp = wj * wj
	elif wj <= -1.0:
		tmp = t_0
	else:
		tmp = x + (-2.0 * (wj * x))
	return tmp

Julia

function code(wj, x)
	t_0 = Float64(wj + Float64(Float64(x * Float64(1.0 - wj)) / wj))
	tmp = 0.0
	if (wj <= -2.9e+266)
		tmp = t_0;
	elseif (wj <= -5.1e+191)
		tmp = Float64(wj * wj);
	elseif (wj <= -1.0)
		tmp = t_0;
	else
		tmp = Float64(x + Float64(-2.0 * Float64(wj * x)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(wj, x)
	t_0 = wj + ((x * (1.0 - wj)) / wj);
	tmp = 0.0;
	if (wj <= -2.9e+266)
		tmp = t_0;
	elseif (wj <= -5.1e+191)
		tmp = wj * wj;
	elseif (wj <= -1.0)
		tmp = t_0;
	else
		tmp = x + (-2.0 * (wj * x));
	end
	tmp_2 = tmp;
end

Wolfram

code[wj_, x_] := Block[{t$95$0 = N[(wj + N[(N[(x * N[(1.0 - wj), $MachinePrecision]), $MachinePrecision] / wj), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[wj, -2.9e+266], t$95$0, If[LessEqual[wj, -5.1e+191], N[(wj * wj), $MachinePrecision], If[LessEqual[wj, -1.0], t$95$0, N[(x + N[(-2.0 * N[(wj * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if wj < -2.90000000000000017e266 or -5.09999999999999982e191 < wj < -1

   1. Initial program 1.7%

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

      1. sub-neg 1.7%

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

      2. div-sub 1.7%

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

      3. sub-neg 1.7%

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

      4. +-commutative 1.7%

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

      5. distribute-neg-in 1.7%

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

      6. remove-double-neg 1.7%

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

      7. sub-neg 1.7%

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

      8. div-sub 1.7%

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

      9. distribute-rgt1-in 100.0%

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

      10. associate-/l/ 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in wj around 0 37.7%

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

   5. Taylor expanded in x around inf 36.5%

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

      1. associate-/l* 5.0%

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

      2. mul-1-neg 5.0%

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

   7. Simplified 5.0%

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

   8. Taylor expanded in wj around inf 5.0%

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

   9. Taylor expanded in x around 0 36.5%

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

   if -2.90000000000000017e266 < wj < -5.09999999999999982e191

   1. Initial program 0.0%

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

      1. sub-neg 0.0%

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

      2. div-sub 0.0%

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

      3. sub-neg 0.0%

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

      4. +-commutative 0.0%

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

      5. distribute-neg-in 0.0%

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

      6. remove-double-neg 0.0%

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

      7. sub-neg 0.0%

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

      8. div-sub 0.0%

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

      9. distribute-rgt1-in 100.0%

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

      10. associate-/l/ 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in x around 0 2.5%

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

      1. +-commutative 2.5%

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

   6. Simplified 2.5%

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

   7. Taylor expanded in wj around 0 60.7%

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

      1. unpow2 60.7%

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

   9. Simplified 60.7%

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

   if -1 < wj

   1. Initial program 81.1%

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

      1. sub-neg 81.1%

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

      2. div-sub 81.1%

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

      3. sub-neg 81.1%

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

      4. +-commutative 81.1%

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

      5. distribute-neg-in 81.1%

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

      6. remove-double-neg 81.1%

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

      7. sub-neg 81.1%

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

      8. div-sub 81.1%

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

      9. distribute-rgt1-in 81.1%

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

      10. associate-/l/ 81.1%

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

   3. Simplified 81.7%

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

   4. Taylor expanded in wj around 0 87.6%

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

4. Final simplification 72.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;wj \leq -2.9 \cdot 10^{+266}:\\ \;\;\;\;wj + \frac{x \cdot \left(1 - wj\right)}{wj}\\ \mathbf{elif}\;wj \leq -5.1 \cdot 10^{+191}:\\ \;\;\;\;wj \cdot wj\\ \mathbf{elif}\;wj \leq -1:\\ \;\;\;\;wj + \frac{x \cdot \left(1 - wj\right)}{wj}\\ \mathbf{else}:\\ \;\;\;\;x + -2 \cdot \left(wj \cdot x\right)\\ \end{array} \]

Alternative 11: 67.6% accurate, 20.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := wj + \frac{x \cdot \left(1 - wj\right)}{wj}\\ \mathbf{if}\;wj \leq -5.2 \cdot 10^{+265}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;wj \leq -5.2 \cdot 10^{+191}:\\ \;\;\;\;wj \cdot wj\\ \mathbf{elif}\;wj \leq -1:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{1 - wj}{wj + 1}\\ \end{array} \end{array} \]

FPCore

(FPCore (wj x)
 :precision binary64
 (let* ((t_0 (+ wj (/ (* x (- 1.0 wj)) wj))))
   (if (<= wj -5.2e+265)
     t_0
     (if (<= wj -5.2e+191)
       (* wj wj)
       (if (<= wj -1.0) t_0 (* x (/ (- 1.0 wj) (+ wj 1.0))))))))

C

double code(double wj, double x) {
	double t_0 = wj + ((x * (1.0 - wj)) / wj);
	double tmp;
	if (wj <= -5.2e+265) {
		tmp = t_0;
	} else if (wj <= -5.2e+191) {
		tmp = wj * wj;
	} else if (wj <= -1.0) {
		tmp = t_0;
	} else {
		tmp = x * ((1.0 - wj) / (wj + 1.0));
	}
	return tmp;
}

Fortran

real(8) function code(wj, x)
    real(8), intent (in) :: wj
    real(8), intent (in) :: x
    real(8) :: t_0
    real(8) :: tmp
    t_0 = wj + ((x * (1.0d0 - wj)) / wj)
    if (wj <= (-5.2d+265)) then
        tmp = t_0
    else if (wj <= (-5.2d+191)) then
        tmp = wj * wj
    else if (wj <= (-1.0d0)) then
        tmp = t_0
    else
        tmp = x * ((1.0d0 - wj) / (wj + 1.0d0))
    end if
    code = tmp
end function

Java

public static double code(double wj, double x) {
	double t_0 = wj + ((x * (1.0 - wj)) / wj);
	double tmp;
	if (wj <= -5.2e+265) {
		tmp = t_0;
	} else if (wj <= -5.2e+191) {
		tmp = wj * wj;
	} else if (wj <= -1.0) {
		tmp = t_0;
	} else {
		tmp = x * ((1.0 - wj) / (wj + 1.0));
	}
	return tmp;
}

Python

def code(wj, x):
	t_0 = wj + ((x * (1.0 - wj)) / wj)
	tmp = 0
	if wj <= -5.2e+265:
		tmp = t_0
	elif wj <= -5.2e+191:
		tmp = wj * wj
	elif wj <= -1.0:
		tmp = t_0
	else:
		tmp = x * ((1.0 - wj) / (wj + 1.0))
	return tmp

Julia

function code(wj, x)
	t_0 = Float64(wj + Float64(Float64(x * Float64(1.0 - wj)) / wj))
	tmp = 0.0
	if (wj <= -5.2e+265)
		tmp = t_0;
	elseif (wj <= -5.2e+191)
		tmp = Float64(wj * wj);
	elseif (wj <= -1.0)
		tmp = t_0;
	else
		tmp = Float64(x * Float64(Float64(1.0 - wj) / Float64(wj + 1.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(wj, x)
	t_0 = wj + ((x * (1.0 - wj)) / wj);
	tmp = 0.0;
	if (wj <= -5.2e+265)
		tmp = t_0;
	elseif (wj <= -5.2e+191)
		tmp = wj * wj;
	elseif (wj <= -1.0)
		tmp = t_0;
	else
		tmp = x * ((1.0 - wj) / (wj + 1.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[wj_, x_] := Block[{t$95$0 = N[(wj + N[(N[(x * N[(1.0 - wj), $MachinePrecision]), $MachinePrecision] / wj), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[wj, -5.2e+265], t$95$0, If[LessEqual[wj, -5.2e+191], N[(wj * wj), $MachinePrecision], If[LessEqual[wj, -1.0], t$95$0, N[(x * N[(N[(1.0 - wj), $MachinePrecision] / N[(wj + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if wj < -5.2000000000000003e265 or -5.2000000000000001e191 < wj < -1

   1. Initial program 1.7%

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

      1. sub-neg 1.7%

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

      2. div-sub 1.7%

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

      3. sub-neg 1.7%

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

      4. +-commutative 1.7%

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

      5. distribute-neg-in 1.7%

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

      6. remove-double-neg 1.7%

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

      7. sub-neg 1.7%

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

      8. div-sub 1.7%

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

      9. distribute-rgt1-in 100.0%

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

      10. associate-/l/ 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in wj around 0 37.7%

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

   5. Taylor expanded in x around inf 36.5%

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

      1. associate-/l* 5.0%

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

      2. mul-1-neg 5.0%

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

   7. Simplified 5.0%

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

   8. Taylor expanded in wj around inf 5.0%

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

   9. Taylor expanded in x around 0 36.5%

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

   if -5.2000000000000003e265 < wj < -5.2000000000000001e191

   1. Initial program 0.0%

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

      1. sub-neg 0.0%

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

      2. div-sub 0.0%

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

      3. sub-neg 0.0%

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

      4. +-commutative 0.0%

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

      5. distribute-neg-in 0.0%

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

      6. remove-double-neg 0.0%

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

      7. sub-neg 0.0%

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

      8. div-sub 0.0%

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

      9. distribute-rgt1-in 100.0%

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

      10. associate-/l/ 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in x around 0 2.5%

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

      1. +-commutative 2.5%

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

   6. Simplified 2.5%

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

   7. Taylor expanded in wj around 0 60.7%

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

      1. unpow2 60.7%

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

   9. Simplified 60.7%

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

   if -1 < wj

   1. Initial program 81.1%

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

      1. sub-neg 81.1%

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

      2. div-sub 81.1%

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

      3. sub-neg 81.1%

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

      4. +-commutative 81.1%

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

      5. distribute-neg-in 81.1%

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

      6. remove-double-neg 81.1%

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

      7. sub-neg 81.1%

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

      8. div-sub 81.1%

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

      9. distribute-rgt1-in 81.1%

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

      10. associate-/l/ 81.1%

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

   3. Simplified 81.7%

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

   4. Taylor expanded in wj around 0 80.1%

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

   5. Taylor expanded in x around -inf 87.6%

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

      1. mul-1-neg 87.6%

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

      2. *-commutative 87.6%

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

      3. distribute-lft-neg-in 87.6%

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

      4. sub-neg 87.6%

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

      5. *-rgt-identity 87.6%

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

      6. associate-*r/ 87.6%

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

      7. neg-mul-1 87.6%

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

      8. distribute-rgt-out 87.6%

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

   7. Simplified 87.6%

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

      1. distribute-lft-in 87.6%

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

      2. +-commutative 87.6%

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

      3. +-commutative 87.6%

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

   9. Applied egg-rr 87.6%

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

       1. distribute-lft-out 87.6%

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

       2. associate-*l/ 87.6%

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

       3. *-lft-identity 87.6%

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

   11. Simplified 87.6%

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

4. Final simplification 72.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;wj \leq -5.2 \cdot 10^{+265}:\\ \;\;\;\;wj + \frac{x \cdot \left(1 - wj\right)}{wj}\\ \mathbf{elif}\;wj \leq -5.2 \cdot 10^{+191}:\\ \;\;\;\;wj \cdot wj\\ \mathbf{elif}\;wj \leq -1:\\ \;\;\;\;wj + \frac{x \cdot \left(1 - wj\right)}{wj}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{1 - wj}{wj + 1}\\ \end{array} \]

Alternative 12: 78.4% accurate, 20.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -7.6 \cdot 10^{+153} \lor \neg \left(x \leq 5.2 \cdot 10^{-21}\right):\\ \;\;\;\;\frac{x - wj \cdot x}{wj + 1}\\ \mathbf{else}:\\ \;\;\;\;\left(x + -2 \cdot \left(wj \cdot x\right)\right) + wj \cdot wj\\ \end{array} \end{array} \]

FPCore

(FPCore (wj x)
 :precision binary64
 (if (or (<= x -7.6e+153) (not (<= x 5.2e-21)))
   (/ (- x (* wj x)) (+ wj 1.0))
   (+ (+ x (* -2.0 (* wj x))) (* wj wj))))

C

double code(double wj, double x) {
	double tmp;
	if ((x <= -7.6e+153) || !(x <= 5.2e-21)) {
		tmp = (x - (wj * x)) / (wj + 1.0);
	} else {
		tmp = (x + (-2.0 * (wj * x))) + (wj * wj);
	}
	return tmp;
}

Fortran

real(8) function code(wj, x)
    real(8), intent (in) :: wj
    real(8), intent (in) :: x
    real(8) :: tmp
    if ((x <= (-7.6d+153)) .or. (.not. (x <= 5.2d-21))) then
        tmp = (x - (wj * x)) / (wj + 1.0d0)
    else
        tmp = (x + ((-2.0d0) * (wj * x))) + (wj * wj)
    end if
    code = tmp
end function

Java

public static double code(double wj, double x) {
	double tmp;
	if ((x <= -7.6e+153) || !(x <= 5.2e-21)) {
		tmp = (x - (wj * x)) / (wj + 1.0);
	} else {
		tmp = (x + (-2.0 * (wj * x))) + (wj * wj);
	}
	return tmp;
}

Python

def code(wj, x):
	tmp = 0
	if (x <= -7.6e+153) or not (x <= 5.2e-21):
		tmp = (x - (wj * x)) / (wj + 1.0)
	else:
		tmp = (x + (-2.0 * (wj * x))) + (wj * wj)
	return tmp

Julia

function code(wj, x)
	tmp = 0.0
	if ((x <= -7.6e+153) || !(x <= 5.2e-21))
		tmp = Float64(Float64(x - Float64(wj * x)) / Float64(wj + 1.0));
	else
		tmp = Float64(Float64(x + Float64(-2.0 * Float64(wj * x))) + Float64(wj * wj));
	end
	return tmp
end

MATLAB

function tmp_2 = code(wj, x)
	tmp = 0.0;
	if ((x <= -7.6e+153) || ~((x <= 5.2e-21)))
		tmp = (x - (wj * x)) / (wj + 1.0);
	else
		tmp = (x + (-2.0 * (wj * x))) + (wj * wj);
	end
	tmp_2 = tmp;
end

Wolfram

code[wj_, x_] := If[Or[LessEqual[x, -7.6e+153], N[Not[LessEqual[x, 5.2e-21]], $MachinePrecision]], N[(N[(x - N[(wj * x), $MachinePrecision]), $MachinePrecision] / N[(wj + 1.0), $MachinePrecision]), $MachinePrecision], N[(N[(x + N[(-2.0 * N[(wj * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(wj * wj), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -7.59999999999999933e153 or 5.20000000000000035e-21 < x

   1. Initial program 61.8%

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

      1. sub-neg 61.8%

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

      2. div-sub 61.8%

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

      3. sub-neg 61.8%

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

      4. +-commutative 61.8%

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

      5. distribute-neg-in 61.8%

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

      6. remove-double-neg 61.8%

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

      7. sub-neg 61.8%

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

      8. div-sub 61.8%

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

      9. distribute-rgt1-in 99.1%

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

      10. associate-/l/ 99.1%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in x around inf 99.1%

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

      1. *-un-lft-identity 99.1%

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

      2. +-commutative 99.1%

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

      3. times-frac 99.1%

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

      4. +-commutative 99.1%

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

   6. Applied egg-rr 99.1%

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

      1. associate-*l/ 99.1%

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

      2. *-lft-identity 99.1%

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

   8. Simplified 99.1%

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

   9. Taylor expanded in wj around 0 84.5%

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

   if -7.59999999999999933e153 < x < 5.20000000000000035e-21

   1. Initial program 48.0%

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

      1. distribute-rgt1-in 78.9%

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

   3. Simplified 78.9%

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

   4. Taylor expanded in wj around 0 48.1%

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

      1. *-commutative 48.1%

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

   6. Simplified 48.1%

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

   7. Taylor expanded in wj around 0 76.6%

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

   8. Taylor expanded in x around 0 80.8%

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

      1. unpow2 80.8%

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

   10. Simplified 80.8%

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

4. Final simplification 82.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -7.6 \cdot 10^{+153} \lor \neg \left(x \leq 5.2 \cdot 10^{-21}\right):\\ \;\;\;\;\frac{x - wj \cdot x}{wj + 1}\\ \mathbf{else}:\\ \;\;\;\;\left(x + -2 \cdot \left(wj \cdot x\right)\right) + wj \cdot wj\\ \end{array} \]

Alternative 13: 77.2% accurate, 20.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;wj \leq -2.25 \cdot 10^{+266}:\\ \;\;\;\;wj + \frac{x \cdot \left(1 - wj\right)}{wj}\\ \mathbf{elif}\;wj \leq -215:\\ \;\;\;\;wj - \left(wj + \left(wj \cdot wj\right) \cdot \left(wj + -1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(x + -2 \cdot \left(wj \cdot x\right)\right) + wj \cdot wj\\ \end{array} \end{array} \]

FPCore

(FPCore (wj x)
 :precision binary64
 (if (<= wj -2.25e+266)
   (+ wj (/ (* x (- 1.0 wj)) wj))
   (if (<= wj -215.0)
     (- wj (+ wj (* (* wj wj) (+ wj -1.0))))
     (+ (+ x (* -2.0 (* wj x))) (* wj wj)))))

C

double code(double wj, double x) {
	double tmp;
	if (wj <= -2.25e+266) {
		tmp = wj + ((x * (1.0 - wj)) / wj);
	} else if (wj <= -215.0) {
		tmp = wj - (wj + ((wj * wj) * (wj + -1.0)));
	} else {
		tmp = (x + (-2.0 * (wj * x))) + (wj * wj);
	}
	return tmp;
}

Fortran

real(8) function code(wj, x)
    real(8), intent (in) :: wj
    real(8), intent (in) :: x
    real(8) :: tmp
    if (wj <= (-2.25d+266)) then
        tmp = wj + ((x * (1.0d0 - wj)) / wj)
    else if (wj <= (-215.0d0)) then
        tmp = wj - (wj + ((wj * wj) * (wj + (-1.0d0))))
    else
        tmp = (x + ((-2.0d0) * (wj * x))) + (wj * wj)
    end if
    code = tmp
end function

Java

public static double code(double wj, double x) {
	double tmp;
	if (wj <= -2.25e+266) {
		tmp = wj + ((x * (1.0 - wj)) / wj);
	} else if (wj <= -215.0) {
		tmp = wj - (wj + ((wj * wj) * (wj + -1.0)));
	} else {
		tmp = (x + (-2.0 * (wj * x))) + (wj * wj);
	}
	return tmp;
}

Python

def code(wj, x):
	tmp = 0
	if wj <= -2.25e+266:
		tmp = wj + ((x * (1.0 - wj)) / wj)
	elif wj <= -215.0:
		tmp = wj - (wj + ((wj * wj) * (wj + -1.0)))
	else:
		tmp = (x + (-2.0 * (wj * x))) + (wj * wj)
	return tmp

Julia

function code(wj, x)
	tmp = 0.0
	if (wj <= -2.25e+266)
		tmp = Float64(wj + Float64(Float64(x * Float64(1.0 - wj)) / wj));
	elseif (wj <= -215.0)
		tmp = Float64(wj - Float64(wj + Float64(Float64(wj * wj) * Float64(wj + -1.0))));
	else
		tmp = Float64(Float64(x + Float64(-2.0 * Float64(wj * x))) + Float64(wj * wj));
	end
	return tmp
end

MATLAB

function tmp_2 = code(wj, x)
	tmp = 0.0;
	if (wj <= -2.25e+266)
		tmp = wj + ((x * (1.0 - wj)) / wj);
	elseif (wj <= -215.0)
		tmp = wj - (wj + ((wj * wj) * (wj + -1.0)));
	else
		tmp = (x + (-2.0 * (wj * x))) + (wj * wj);
	end
	tmp_2 = tmp;
end

Wolfram

code[wj_, x_] := If[LessEqual[wj, -2.25e+266], N[(wj + N[(N[(x * N[(1.0 - wj), $MachinePrecision]), $MachinePrecision] / wj), $MachinePrecision]), $MachinePrecision], If[LessEqual[wj, -215.0], N[(wj - N[(wj + N[(N[(wj * wj), $MachinePrecision] * N[(wj + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x + N[(-2.0 * N[(wj * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(wj * wj), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if wj < -2.25e266

   1. Initial program 0.0%

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

      1. sub-neg 0.0%

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

      2. div-sub 0.0%

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

      3. sub-neg 0.0%

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

      4. +-commutative 0.0%

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

      5. distribute-neg-in 0.0%

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

      6. remove-double-neg 0.0%

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

      7. sub-neg 0.0%

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

      8. div-sub 0.0%

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

      9. distribute-rgt1-in 100.0%

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

      10. associate-/l/ 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in wj around 0 73.4%

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

   5. Taylor expanded in x around inf 73.4%

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

      1. associate-/l* 5.7%

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

      2. mul-1-neg 5.7%

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

   7. Simplified 5.7%

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

   8. Taylor expanded in wj around inf 5.7%

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

   9. Taylor expanded in x around 0 73.4%

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

   if -2.25e266 < wj < -215

   1. Initial program 0.0%

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

      1. sub-neg 0.0%

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

      2. div-sub 0.0%

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

      3. sub-neg 0.0%

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

      4. +-commutative 0.0%

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

      5. distribute-neg-in 0.0%

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

      6. remove-double-neg 0.0%

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

      7. sub-neg 0.0%

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

      8. div-sub 0.0%

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

      9. distribute-rgt1-in 100.0%

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

      10. associate-/l/ 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in x around 0 2.6%

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

      1. +-commutative 2.6%

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

   6. Simplified 2.6%

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

   7. Taylor expanded in wj around 0 36.7%

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

      1. associate-+r+ 36.7%

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

      2. +-commutative 36.7%

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

      3. cube-mult 36.7%

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

      4. unpow2 36.7%

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

      5. distribute-rgt-out 36.7%

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

      6. unpow2 36.7%

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

      7. +-commutative 36.7%

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

   9. Simplified 36.7%

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

   if -215 < wj

   1. Initial program 81.3%

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

      1. distribute-rgt1-in 81.3%

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

   3. Simplified 81.3%

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

   4. Taylor expanded in wj around 0 79.7%

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

      1. *-commutative 79.7%

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

   6. Simplified 79.7%

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

   7. Taylor expanded in wj around 0 97.6%

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

   8. Taylor expanded in x around 0 97.5%

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

      1. unpow2 97.5%

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

   10. Simplified 97.5%

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

4. Final simplification 78.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;wj \leq -2.25 \cdot 10^{+266}:\\ \;\;\;\;wj + \frac{x \cdot \left(1 - wj\right)}{wj}\\ \mathbf{elif}\;wj \leq -215:\\ \;\;\;\;wj - \left(wj + \left(wj \cdot wj\right) \cdot \left(wj + -1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(x + -2 \cdot \left(wj \cdot x\right)\right) + wj \cdot wj\\ \end{array} \]

Alternative 14: 78.4% accurate, 20.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -7.6 \cdot 10^{+153}:\\ \;\;\;\;\frac{x - wj \cdot x}{wj + 1}\\ \mathbf{elif}\;x \leq 1.85 \cdot 10^{-18}:\\ \;\;\;\;\left(x + -2 \cdot \left(wj \cdot x\right)\right) + wj \cdot wj\\ \mathbf{else}:\\ \;\;\;\;wj + \frac{x \cdot \left(1 - wj\right)}{wj + 1}\\ \end{array} \end{array} \]

FPCore

(FPCore (wj x)
 :precision binary64
 (if (<= x -7.6e+153)
   (/ (- x (* wj x)) (+ wj 1.0))
   (if (<= x 1.85e-18)
     (+ (+ x (* -2.0 (* wj x))) (* wj wj))
     (+ wj (/ (* x (- 1.0 wj)) (+ wj 1.0))))))

C

double code(double wj, double x) {
	double tmp;
	if (x <= -7.6e+153) {
		tmp = (x - (wj * x)) / (wj + 1.0);
	} else if (x <= 1.85e-18) {
		tmp = (x + (-2.0 * (wj * x))) + (wj * wj);
	} else {
		tmp = wj + ((x * (1.0 - wj)) / (wj + 1.0));
	}
	return tmp;
}

Fortran

real(8) function code(wj, x)
    real(8), intent (in) :: wj
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= (-7.6d+153)) then
        tmp = (x - (wj * x)) / (wj + 1.0d0)
    else if (x <= 1.85d-18) then
        tmp = (x + ((-2.0d0) * (wj * x))) + (wj * wj)
    else
        tmp = wj + ((x * (1.0d0 - wj)) / (wj + 1.0d0))
    end if
    code = tmp
end function

Java

public static double code(double wj, double x) {
	double tmp;
	if (x <= -7.6e+153) {
		tmp = (x - (wj * x)) / (wj + 1.0);
	} else if (x <= 1.85e-18) {
		tmp = (x + (-2.0 * (wj * x))) + (wj * wj);
	} else {
		tmp = wj + ((x * (1.0 - wj)) / (wj + 1.0));
	}
	return tmp;
}

Python

def code(wj, x):
	tmp = 0
	if x <= -7.6e+153:
		tmp = (x - (wj * x)) / (wj + 1.0)
	elif x <= 1.85e-18:
		tmp = (x + (-2.0 * (wj * x))) + (wj * wj)
	else:
		tmp = wj + ((x * (1.0 - wj)) / (wj + 1.0))
	return tmp

Julia

function code(wj, x)
	tmp = 0.0
	if (x <= -7.6e+153)
		tmp = Float64(Float64(x - Float64(wj * x)) / Float64(wj + 1.0));
	elseif (x <= 1.85e-18)
		tmp = Float64(Float64(x + Float64(-2.0 * Float64(wj * x))) + Float64(wj * wj));
	else
		tmp = Float64(wj + Float64(Float64(x * Float64(1.0 - wj)) / Float64(wj + 1.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(wj, x)
	tmp = 0.0;
	if (x <= -7.6e+153)
		tmp = (x - (wj * x)) / (wj + 1.0);
	elseif (x <= 1.85e-18)
		tmp = (x + (-2.0 * (wj * x))) + (wj * wj);
	else
		tmp = wj + ((x * (1.0 - wj)) / (wj + 1.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[wj_, x_] := If[LessEqual[x, -7.6e+153], N[(N[(x - N[(wj * x), $MachinePrecision]), $MachinePrecision] / N[(wj + 1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 1.85e-18], N[(N[(x + N[(-2.0 * N[(wj * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(wj * wj), $MachinePrecision]), $MachinePrecision], N[(wj + N[(N[(x * N[(1.0 - wj), $MachinePrecision]), $MachinePrecision] / N[(wj + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -7.59999999999999933e153

   1. Initial program 62.5%

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

      1. sub-neg 62.5%

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

      2. div-sub 62.5%

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

      3. sub-neg 62.5%

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

      4. +-commutative 62.5%

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

      5. distribute-neg-in 62.5%

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

      6. remove-double-neg 62.5%

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

      7. sub-neg 62.5%

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

      8. div-sub 62.5%

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

      9. distribute-rgt1-in 96.8%

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

      10. associate-/l/ 96.9%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in x around inf 96.9%

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

      1. *-un-lft-identity 96.9%

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

      2. +-commutative 96.9%

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

      3. times-frac 96.9%

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

      4. +-commutative 96.9%

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

   6. Applied egg-rr 96.9%

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

      1. associate-*l/ 97.0%

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

      2. *-lft-identity 97.0%

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

   8. Simplified 97.0%

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

   9. Taylor expanded in wj around 0 90.9%

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

   if -7.59999999999999933e153 < x < 1.8500000000000002e-18

   1. Initial program 48.0%

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

      1. distribute-rgt1-in 78.9%

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

   3. Simplified 78.9%

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

   4. Taylor expanded in wj around 0 48.1%

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

      1. *-commutative 48.1%

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

   6. Simplified 48.1%

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

   7. Taylor expanded in wj around 0 76.6%

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

   8. Taylor expanded in x around 0 80.8%

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

      1. unpow2 80.8%

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

   10. Simplified 80.8%

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

   if 1.8500000000000002e-18 < x

   1. Initial program 61.5%

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

      1. sub-neg 61.5%

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

      2. div-sub 61.5%

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

      3. sub-neg 61.5%

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

      4. +-commutative 61.5%

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

      5. distribute-neg-in 61.5%

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

      6. remove-double-neg 61.5%

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

      7. sub-neg 61.5%

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

      8. div-sub 61.5%

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

      9. distribute-rgt1-in 100.0%

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

      10. associate-/l/ 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in wj around 0 82.0%

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

   5. Taylor expanded in x around -inf 82.0%

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

      1. associate-*r/ 82.0%

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

      2. *-commutative 82.0%

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

      3. associate-*r* 82.0%

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

      4. mul-1-neg 82.0%

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

      5. sub-neg 82.0%

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

      6. metadata-eval 82.0%

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

   7. Simplified 82.0%

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

4. Final simplification 82.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -7.6 \cdot 10^{+153}:\\ \;\;\;\;\frac{x - wj \cdot x}{wj + 1}\\ \mathbf{elif}\;x \leq 1.85 \cdot 10^{-18}:\\ \;\;\;\;\left(x + -2 \cdot \left(wj \cdot x\right)\right) + wj \cdot wj\\ \mathbf{else}:\\ \;\;\;\;wj + \frac{x \cdot \left(1 - wj\right)}{wj + 1}\\ \end{array} \]

Alternative 15: 66.8% accurate, 34.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;wj \leq -215:\\ \;\;\;\;wj \cdot wj\\ \mathbf{else}:\\ \;\;\;\;x + -2 \cdot \left(wj \cdot x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (wj x)
 :precision binary64
 (if (<= wj -215.0) (* wj wj) (+ x (* -2.0 (* wj x)))))

C

double code(double wj, double x) {
	double tmp;
	if (wj <= -215.0) {
		tmp = wj * wj;
	} else {
		tmp = x + (-2.0 * (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 <= (-215.0d0)) then
        tmp = wj * wj
    else
        tmp = x + ((-2.0d0) * (wj * x))
    end if
    code = tmp
end function

Java

public static double code(double wj, double x) {
	double tmp;
	if (wj <= -215.0) {
		tmp = wj * wj;
	} else {
		tmp = x + (-2.0 * (wj * x));
	}
	return tmp;
}

Python

def code(wj, x):
	tmp = 0
	if wj <= -215.0:
		tmp = wj * wj
	else:
		tmp = x + (-2.0 * (wj * x))
	return tmp

Julia

function code(wj, x)
	tmp = 0.0
	if (wj <= -215.0)
		tmp = Float64(wj * wj);
	else
		tmp = Float64(x + Float64(-2.0 * Float64(wj * x)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(wj, x)
	tmp = 0.0;
	if (wj <= -215.0)
		tmp = wj * wj;
	else
		tmp = x + (-2.0 * (wj * x));
	end
	tmp_2 = tmp;
end

Wolfram

code[wj_, x_] := If[LessEqual[wj, -215.0], N[(wj * wj), $MachinePrecision], N[(x + N[(-2.0 * N[(wj * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if wj < -215

   1. Initial program 0.0%

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

      1. sub-neg 0.0%

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

      2. div-sub 0.0%

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

      3. sub-neg 0.0%

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

      4. +-commutative 0.0%

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

      5. distribute-neg-in 0.0%

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

      6. remove-double-neg 0.0%

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

      7. sub-neg 0.0%

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

      8. div-sub 0.0%

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

      9. distribute-rgt1-in 100.0%

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

      10. associate-/l/ 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in x around 0 3.0%

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

      1. +-commutative 3.0%

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

   6. Simplified 3.0%

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

   7. Taylor expanded in wj around 0 31.2%

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

      1. unpow2 31.2%

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

   9. Simplified 31.2%

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

   if -215 < wj

   1. Initial program 81.3%

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

      1. sub-neg 81.3%

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

      2. div-sub 81.3%

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

      3. sub-neg 81.3%

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

      4. +-commutative 81.3%

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

      5. distribute-neg-in 81.3%

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

      6. remove-double-neg 81.3%

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

      7. sub-neg 81.3%

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

      8. div-sub 81.3%

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

      9. distribute-rgt1-in 81.3%

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

      10. associate-/l/ 81.2%

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

   3. Simplified 81.8%

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

   4. Taylor expanded in wj around 0 87.1%

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

4. Final simplification 68.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;wj \leq -215:\\ \;\;\;\;wj \cdot wj\\ \mathbf{else}:\\ \;\;\;\;x + -2 \cdot \left(wj \cdot x\right)\\ \end{array} \]

Alternative 16: 66.9% accurate, 34.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;wj \leq -3 \cdot 10^{+98}:\\ \;\;\;\;wj \cdot wj\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{1 + wj \cdot 2}\\ \end{array} \end{array} \]

FPCore

(FPCore (wj x)
 :precision binary64
 (if (<= wj -3e+98) (* wj wj) (/ x (+ 1.0 (* wj 2.0)))))

C

double code(double wj, double x) {
	double tmp;
	if (wj <= -3e+98) {
		tmp = wj * wj;
	} else {
		tmp = x / (1.0 + (wj * 2.0));
	}
	return tmp;
}

Fortran

real(8) function code(wj, x)
    real(8), intent (in) :: wj
    real(8), intent (in) :: x
    real(8) :: tmp
    if (wj <= (-3d+98)) then
        tmp = wj * wj
    else
        tmp = x / (1.0d0 + (wj * 2.0d0))
    end if
    code = tmp
end function

Java

public static double code(double wj, double x) {
	double tmp;
	if (wj <= -3e+98) {
		tmp = wj * wj;
	} else {
		tmp = x / (1.0 + (wj * 2.0));
	}
	return tmp;
}

Python

def code(wj, x):
	tmp = 0
	if wj <= -3e+98:
		tmp = wj * wj
	else:
		tmp = x / (1.0 + (wj * 2.0))
	return tmp

Julia

function code(wj, x)
	tmp = 0.0
	if (wj <= -3e+98)
		tmp = Float64(wj * wj);
	else
		tmp = Float64(x / Float64(1.0 + Float64(wj * 2.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(wj, x)
	tmp = 0.0;
	if (wj <= -3e+98)
		tmp = wj * wj;
	else
		tmp = x / (1.0 + (wj * 2.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[wj_, x_] := If[LessEqual[wj, -3e+98], N[(wj * wj), $MachinePrecision], N[(x / N[(1.0 + N[(wj * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if wj < -3.0000000000000001e98

   1. Initial program 0.0%

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

      1. sub-neg 0.0%

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

      2. div-sub 0.0%

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

      3. sub-neg 0.0%

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

      4. +-commutative 0.0%

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

      5. distribute-neg-in 0.0%

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

      6. remove-double-neg 0.0%

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

      7. sub-neg 0.0%

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

      8. div-sub 0.0%

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

      9. distribute-rgt1-in 100.0%

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

      10. associate-/l/ 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in x around 0 3.1%

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

      1. +-commutative 3.1%

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

   6. Simplified 3.1%

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

   7. Taylor expanded in wj around 0 42.0%

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

      1. unpow2 42.0%

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

   9. Simplified 42.0%

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

   if -3.0000000000000001e98 < wj

   1. Initial program 71.6%

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

      1. sub-neg 71.6%

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

      2. div-sub 71.6%

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

      3. sub-neg 71.6%

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

      4. +-commutative 71.6%

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

      5. distribute-neg-in 71.6%

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

      6. remove-double-neg 71.6%

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

      7. sub-neg 71.6%

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

      8. div-sub 71.6%

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

      9. distribute-rgt1-in 83.5%

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

      10. associate-/l/ 83.4%

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

   3. Simplified 83.9%

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

   4. Taylor expanded in x around inf 89.6%

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

   5. Taylor expanded in wj around 0 77.1%

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

4. Final simplification 68.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;wj \leq -3 \cdot 10^{+98}:\\ \;\;\;\;wj \cdot wj\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{1 + wj \cdot 2}\\ \end{array} \]

Alternative 17: 66.4% accurate, 61.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;wj \leq -215:\\ \;\;\;\;wj \cdot wj\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \end{array} \]

FPCore

(FPCore (wj x) :precision binary64 (if (<= wj -215.0) (* wj wj) x))

C

double code(double wj, double x) {
	double tmp;
	if (wj <= -215.0) {
		tmp = wj * wj;
	} else {
		tmp = x;
	}
	return tmp;
}

Fortran

real(8) function code(wj, x)
    real(8), intent (in) :: wj
    real(8), intent (in) :: x
    real(8) :: tmp
    if (wj <= (-215.0d0)) then
        tmp = wj * wj
    else
        tmp = x
    end if
    code = tmp
end function

Java

public static double code(double wj, double x) {
	double tmp;
	if (wj <= -215.0) {
		tmp = wj * wj;
	} else {
		tmp = x;
	}
	return tmp;
}

Python

def code(wj, x):
	tmp = 0
	if wj <= -215.0:
		tmp = wj * wj
	else:
		tmp = x
	return tmp

Julia

function code(wj, x)
	tmp = 0.0
	if (wj <= -215.0)
		tmp = Float64(wj * wj);
	else
		tmp = x;
	end
	return tmp
end

MATLAB

function tmp_2 = code(wj, x)
	tmp = 0.0;
	if (wj <= -215.0)
		tmp = wj * wj;
	else
		tmp = x;
	end
	tmp_2 = tmp;
end

Wolfram

code[wj_, x_] := If[LessEqual[wj, -215.0], N[(wj * wj), $MachinePrecision], x]

Derivation

1. Split input into 2 regimes

2. if wj < -215

   1. Initial program 0.0%

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

      1. sub-neg 0.0%

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

      2. div-sub 0.0%

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

      3. sub-neg 0.0%

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

      4. +-commutative 0.0%

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

      5. distribute-neg-in 0.0%

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

      6. remove-double-neg 0.0%

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

      7. sub-neg 0.0%

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

      8. div-sub 0.0%

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

      9. distribute-rgt1-in 100.0%

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

      10. associate-/l/ 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in x around 0 3.0%

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

      1. +-commutative 3.0%

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

   6. Simplified 3.0%

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

   7. Taylor expanded in wj around 0 31.2%

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

      1. unpow2 31.2%

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

   9. Simplified 31.2%

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

   if -215 < wj

   1. Initial program 81.3%

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

      1. sub-neg 81.3%

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

      2. div-sub 81.3%

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

      3. sub-neg 81.3%

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

      4. +-commutative 81.3%

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

      5. distribute-neg-in 81.3%

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

      6. remove-double-neg 81.3%

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

      7. sub-neg 81.3%

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

      8. div-sub 81.3%

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

      9. distribute-rgt1-in 81.3%

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

      10. associate-/l/ 81.2%

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

   3. Simplified 81.8%

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

   4. Taylor expanded in wj around 0 86.7%

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

4. Final simplification 68.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;wj \leq -215:\\ \;\;\;\;wj \cdot wj\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 18: 3.6% 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 54.0%

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

   1. sub-neg 54.0%

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

   2. div-sub 54.0%

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

   3. sub-neg 54.0%

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

   4. +-commutative 54.0%

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

   5. distribute-neg-in 54.0%

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

   6. remove-double-neg 54.0%

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

   7. sub-neg 54.0%

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

   8. div-sub 54.0%

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

   9. distribute-rgt1-in 87.5%

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

   10. associate-/l/ 87.5%

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

3. Simplified 87.9%

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

4. Taylor expanded in wj around inf 3.7%

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

5. Final simplification 3.7%

   \[\leadsto wj \]

Alternative 19: 57.2% 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 54.0%

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

   1. sub-neg 54.0%

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

   2. div-sub 54.0%

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

   3. sub-neg 54.0%

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

   4. +-commutative 54.0%

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

   5. distribute-neg-in 54.0%

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

   6. remove-double-neg 54.0%

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

   7. sub-neg 54.0%

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

   8. div-sub 54.0%

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

   9. distribute-rgt1-in 87.5%

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

   10. associate-/l/ 87.5%

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

3. Simplified 87.9%

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

4. Taylor expanded in wj around 0 57.8%

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

5. Final simplification 57.8%

   \[\leadsto x \]

Developer target: 52.6% 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 2023278 
(FPCore (wj x)
  :name "Jmat.Real.lambertw, newton loop step"
  :precision binary64

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

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