Jmat.Real.lambertw, newton loop step

Percentage Accurate: 77.4% → 98.5%

Time: 10.4s

Alternatives: 14

Speedup: 313.0×

• could not determine a ground truth

  1. wj = -8.27836631651987e+233
  2. x = -9.38913082479885e+196

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 77.4% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 98.5% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := wj \cdot e^{wj}\\ \mathbf{if}\;wj + \frac{x - t_0}{e^{wj} + t_0} \leq 1.35 \cdot 10^{-17}:\\ \;\;\;\;wj \cdot wj - \left(wj \cdot \left(x + x\right) + \left({wj}^{3} - x\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{x}{e^{wj}} - wj, \frac{1}{wj + 1}, wj\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (wj x)
 :precision binary64
 (let* ((t_0 (* wj (exp wj))))
   (if (<= (+ wj (/ (- x t_0) (+ (exp wj) t_0))) 1.35e-17)
     (- (* wj wj) (+ (* wj (+ x x)) (- (pow wj 3.0) x)))
     (fma (- (/ x (exp wj)) wj) (/ 1.0 (+ wj 1.0)) wj))))

C

double code(double wj, double x) {
	double t_0 = wj * exp(wj);
	double tmp;
	if ((wj + ((x - t_0) / (exp(wj) + t_0))) <= 1.35e-17) {
		tmp = (wj * wj) - ((wj * (x + x)) + (pow(wj, 3.0) - x));
	} else {
		tmp = fma(((x / exp(wj)) - wj), (1.0 / (wj + 1.0)), wj);
	}
	return tmp;
}

Julia

function code(wj, x)
	t_0 = Float64(wj * exp(wj))
	tmp = 0.0
	if (Float64(wj + Float64(Float64(x - t_0) / Float64(exp(wj) + t_0))) <= 1.35e-17)
		tmp = Float64(Float64(wj * wj) - Float64(Float64(wj * Float64(x + x)) + Float64((wj ^ 3.0) - x)));
	else
		tmp = fma(Float64(Float64(x / exp(wj)) - wj), Float64(1.0 / Float64(wj + 1.0)), wj);
	end
	return tmp
end

Wolfram

code[wj_, x_] := Block[{t$95$0 = N[(wj * N[Exp[wj], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(wj + N[(N[(x - t$95$0), $MachinePrecision] / N[(N[Exp[wj], $MachinePrecision] + t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 1.35e-17], N[(N[(wj * wj), $MachinePrecision] - N[(N[(wj * N[(x + x), $MachinePrecision]), $MachinePrecision] + N[(N[Power[wj, 3.0], $MachinePrecision] - x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(x / N[Exp[wj], $MachinePrecision]), $MachinePrecision] - wj), $MachinePrecision] * N[(1.0 / N[(wj + 1.0), $MachinePrecision]), $MachinePrecision] + wj), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 wj (/.f64 (-.f64 (*.f64 wj (exp.f64 wj)) x) (+.f64 (exp.f64 wj) (*.f64 wj (exp.f64 wj))))) < 1.3500000000000001e-17

   1. Initial program 67.9%

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

      1. sub-neg 67.9%

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

      2. div-sub 67.9%

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

      3. sub-neg 67.9%

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

      4. +-commutative 67.9%

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

      5. distribute-neg-in 67.9%

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

      6. remove-double-neg 67.9%

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

      7. sub-neg 67.9%

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

      8. div-sub 67.9%

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

      9. distribute-rgt1-in 69.0%

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

      10. associate-/l/ 69.0%

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

   3. Simplified 69.0%

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

   4. Taylor expanded in wj around 0 67.9%

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

   5. Taylor expanded in wj around 0 98.9%

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

   6. Taylor expanded in x around 0 98.9%

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

      1. unpow2 98.9%

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

   8. Simplified 98.9%

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

   9. Taylor expanded in x around 0 98.9%

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

       1. mul-1-neg 98.9%

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

   11. Simplified 98.9%

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

   if 1.3500000000000001e-17 < (-.f64 wj (/.f64 (-.f64 (*.f64 wj (exp.f64 wj)) x) (+.f64 (exp.f64 wj) (*.f64 wj (exp.f64 wj)))))

   1. Initial program 95.6%

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

      1. sub-neg 95.6%

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

      2. div-sub 95.6%

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

      3. sub-neg 95.6%

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

      4. +-commutative 95.6%

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

      5. distribute-neg-in 95.6%

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

      6. remove-double-neg 95.6%

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

      7. sub-neg 95.6%

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

      8. div-sub 95.6%

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

      9. distribute-rgt1-in 96.9%

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

      10. associate-/l/ 96.9%

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

   3. Simplified 99.5%

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

      1. +-commutative 99.5%

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

      2. div-inv 99.5%

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

      3. fma-def 99.6%

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

   5. Applied egg-rr 99.6%

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

4. Final simplification 99.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;wj + \frac{x - wj \cdot e^{wj}}{e^{wj} + wj \cdot e^{wj}} \leq 1.35 \cdot 10^{-17}:\\ \;\;\;\;wj \cdot wj - \left(wj \cdot \left(x + x\right) + \left({wj}^{3} - x\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{x}{e^{wj}} - wj, \frac{1}{wj + 1}, wj\right)\\ \end{array} \]

Alternative 2: 96.0% accurate, 2.7× speedup

Math

\[\begin{array}{l} \\ \left(wj \cdot wj + \left(x + -2 \cdot \left(wj \cdot x\right)\right)\right) - {wj}^{3} \end{array} \]

FPCore

(FPCore (wj x)
 :precision binary64
 (- (+ (* wj wj) (+ x (* -2.0 (* wj x)))) (pow wj 3.0)))

C

double code(double wj, double x) {
	return ((wj * wj) + (x + (-2.0 * (wj * x)))) - pow(wj, 3.0);
}

Fortran

real(8) function code(wj, x)
    real(8), intent (in) :: wj
    real(8), intent (in) :: x
    code = ((wj * wj) + (x + ((-2.0d0) * (wj * x)))) - (wj ** 3.0d0)
end function

Java

public static double code(double wj, double x) {
	return ((wj * wj) + (x + (-2.0 * (wj * x)))) - Math.pow(wj, 3.0);
}

Python

def code(wj, x):
	return ((wj * wj) + (x + (-2.0 * (wj * x)))) - math.pow(wj, 3.0)

Julia

function code(wj, x)
	return Float64(Float64(Float64(wj * wj) + Float64(x + Float64(-2.0 * Float64(wj * x)))) - (wj ^ 3.0))
end

MATLAB

function tmp = code(wj, x)
	tmp = ((wj * wj) + (x + (-2.0 * (wj * x)))) - (wj ^ 3.0);
end

Wolfram

code[wj_, x_] := N[(N[(N[(wj * wj), $MachinePrecision] + N[(x + N[(-2.0 * N[(wj * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[Power[wj, 3.0], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 76.1%

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

   1. sub-neg 76.1%

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

   2. div-sub 76.1%

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

   3. sub-neg 76.1%

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

   4. +-commutative 76.1%

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

   5. distribute-neg-in 76.1%

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

   6. remove-double-neg 76.1%

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

   7. sub-neg 76.1%

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

   8. div-sub 76.1%

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

   9. distribute-rgt1-in 77.3%

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

   10. associate-/l/ 77.3%

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

3. Simplified 78.1%

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

4. Taylor expanded in wj around 0 96.9%

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

5. Taylor expanded in x around 0 96.7%

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

   1. unpow2 96.7%

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

7. Simplified 96.7%

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

8. Taylor expanded in x around 0 97.1%

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

9. Final simplification 97.1%

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

Alternative 3: 95.9% accurate, 2.7× speedup

Math

\[\begin{array}{l} \\ wj \cdot wj - \left(wj \cdot \left(x + x\right) + \left({wj}^{3} - x\right)\right) \end{array} \]

FPCore

(FPCore (wj x)
 :precision binary64
 (- (* wj wj) (+ (* wj (+ x x)) (- (pow wj 3.0) x))))

C

double code(double wj, double x) {
	return (wj * wj) - ((wj * (x + x)) + (pow(wj, 3.0) - x));
}

Fortran

real(8) function code(wj, x)
    real(8), intent (in) :: wj
    real(8), intent (in) :: x
    code = (wj * wj) - ((wj * (x + x)) + ((wj ** 3.0d0) - x))
end function

Java

public static double code(double wj, double x) {
	return (wj * wj) - ((wj * (x + x)) + (Math.pow(wj, 3.0) - x));
}

Python

def code(wj, x):
	return (wj * wj) - ((wj * (x + x)) + (math.pow(wj, 3.0) - x))

Julia

function code(wj, x)
	return Float64(Float64(wj * wj) - Float64(Float64(wj * Float64(x + x)) + Float64((wj ^ 3.0) - x)))
end

MATLAB

function tmp = code(wj, x)
	tmp = (wj * wj) - ((wj * (x + x)) + ((wj ^ 3.0) - x));
end

Wolfram

code[wj_, x_] := N[(N[(wj * wj), $MachinePrecision] - N[(N[(wj * N[(x + x), $MachinePrecision]), $MachinePrecision] + N[(N[Power[wj, 3.0], $MachinePrecision] - x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 76.1%

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

   1. sub-neg 76.1%

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

   2. div-sub 76.1%

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

   3. sub-neg 76.1%

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

   4. +-commutative 76.1%

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

   5. distribute-neg-in 76.1%

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

   6. remove-double-neg 76.1%

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

   7. sub-neg 76.1%

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

   8. div-sub 76.1%

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

   9. distribute-rgt1-in 77.3%

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

   10. associate-/l/ 77.3%

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

3. Simplified 78.1%

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

4. Taylor expanded in wj around 0 76.3%

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

5. Taylor expanded in wj around 0 97.1%

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

6. Taylor expanded in x around 0 97.1%

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

   1. unpow2 97.1%

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

8. Simplified 97.1%

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

9. Taylor expanded in x around 0 97.1%

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

    1. mul-1-neg 97.1%

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

11. Simplified 97.1%

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

12. Final simplification 97.1%

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

Alternative 4: 95.6% accurate, 2.8× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 76.1%

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

   1. sub-neg 76.1%

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

   2. div-sub 76.1%

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

   3. sub-neg 76.1%

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

   4. +-commutative 76.1%

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

   5. distribute-neg-in 76.1%

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

   6. remove-double-neg 76.1%

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

   7. sub-neg 76.1%

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

   8. div-sub 76.1%

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

   9. distribute-rgt1-in 77.3%

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

   10. associate-/l/ 77.3%

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

3. Simplified 78.1%

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

4. Taylor expanded in wj around 0 76.3%

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

5. Taylor expanded in wj around 0 97.1%

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

6. Taylor expanded in wj around 0 96.7%

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

   1. associate-+r+ 96.7%

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

   2. remove-double-neg 96.7%

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

   3. unsub-neg 96.7%

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

   4. unpow2 96.7%

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

   5. sub-neg 96.7%

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

   6. neg-mul-1 96.7%

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

   7. remove-double-neg 96.7%

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

   8. associate-*l* 96.7%

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

   9. *-commutative 96.7%

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

   10. distribute-lft-out 96.7%

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

   11. fma-neg 96.7%

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

8. Simplified 96.7%

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

9. Taylor expanded in x around 0 96.7%

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

10. Final simplification 96.7%

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

Alternative 5: 83.8% accurate, 2.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{wj}{wj + 1}\\ \mathbf{if}\;x \leq -1.5 \cdot 10^{-138} \lor \neg \left(x \leq -1.05 \cdot 10^{-225}\right):\\ \;\;\;\;\left(wj - t_0\right) + x \cdot \left(\frac{1}{wj + 1} - t_0\right)\\ \mathbf{else}:\\ \;\;\;\;wj \cdot wj - {wj}^{3}\\ \end{array} \end{array} \]

FPCore

(FPCore (wj x)
 :precision binary64
 (let* ((t_0 (/ wj (+ wj 1.0))))
   (if (or (<= x -1.5e-138) (not (<= x -1.05e-225)))
     (+ (- wj t_0) (* x (- (/ 1.0 (+ wj 1.0)) t_0)))
     (- (* wj wj) (pow wj 3.0)))))

C

double code(double wj, double x) {
	double t_0 = wj / (wj + 1.0);
	double tmp;
	if ((x <= -1.5e-138) || !(x <= -1.05e-225)) {
		tmp = (wj - t_0) + (x * ((1.0 / (wj + 1.0)) - t_0));
	} else {
		tmp = (wj * wj) - pow(wj, 3.0);
	}
	return tmp;
}

Fortran

real(8) function code(wj, x)
    real(8), intent (in) :: wj
    real(8), intent (in) :: x
    real(8) :: t_0
    real(8) :: tmp
    t_0 = wj / (wj + 1.0d0)
    if ((x <= (-1.5d-138)) .or. (.not. (x <= (-1.05d-225)))) then
        tmp = (wj - t_0) + (x * ((1.0d0 / (wj + 1.0d0)) - t_0))
    else
        tmp = (wj * wj) - (wj ** 3.0d0)
    end if
    code = tmp
end function

Java

public static double code(double wj, double x) {
	double t_0 = wj / (wj + 1.0);
	double tmp;
	if ((x <= -1.5e-138) || !(x <= -1.05e-225)) {
		tmp = (wj - t_0) + (x * ((1.0 / (wj + 1.0)) - t_0));
	} else {
		tmp = (wj * wj) - Math.pow(wj, 3.0);
	}
	return tmp;
}

Python

def code(wj, x):
	t_0 = wj / (wj + 1.0)
	tmp = 0
	if (x <= -1.5e-138) or not (x <= -1.05e-225):
		tmp = (wj - t_0) + (x * ((1.0 / (wj + 1.0)) - t_0))
	else:
		tmp = (wj * wj) - math.pow(wj, 3.0)
	return tmp

Julia

function code(wj, x)
	t_0 = Float64(wj / Float64(wj + 1.0))
	tmp = 0.0
	if ((x <= -1.5e-138) || !(x <= -1.05e-225))
		tmp = Float64(Float64(wj - t_0) + Float64(x * Float64(Float64(1.0 / Float64(wj + 1.0)) - t_0)));
	else
		tmp = Float64(Float64(wj * wj) - (wj ^ 3.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(wj, x)
	t_0 = wj / (wj + 1.0);
	tmp = 0.0;
	if ((x <= -1.5e-138) || ~((x <= -1.05e-225)))
		tmp = (wj - t_0) + (x * ((1.0 / (wj + 1.0)) - t_0));
	else
		tmp = (wj * wj) - (wj ^ 3.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[wj_, x_] := Block[{t$95$0 = N[(wj / N[(wj + 1.0), $MachinePrecision]), $MachinePrecision]}, If[Or[LessEqual[x, -1.5e-138], N[Not[LessEqual[x, -1.05e-225]], $MachinePrecision]], N[(N[(wj - t$95$0), $MachinePrecision] + N[(x * N[(N[(1.0 / N[(wj + 1.0), $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(wj * wj), $MachinePrecision] - N[Power[wj, 3.0], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if x < -1.5e-138 or -1.05e-225 < x

   1. Initial program 79.8%

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

      1. sub-neg 79.8%

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

      2. div-sub 79.8%

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

      3. sub-neg 79.8%

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

      4. +-commutative 79.8%

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

      5. distribute-neg-in 79.8%

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

      6. remove-double-neg 79.8%

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

      7. sub-neg 79.8%

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

      8. div-sub 79.8%

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

      9. distribute-rgt1-in 80.7%

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

      10. associate-/l/ 80.7%

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

   3. Simplified 81.5%

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

      1. +-commutative 81.5%

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

      2. div-inv 81.5%

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

      3. fma-def 81.5%

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

   5. Applied egg-rr 81.5%

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

   6. Taylor expanded in wj around 0 80.1%

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

   7. Taylor expanded in x around -inf 80.0%

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

      1. associate-+r+ 91.7%

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

      2. mul-1-neg 91.7%

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

      3. unsub-neg 91.7%

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

      4. +-commutative 91.7%

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

      5. mul-1-neg 91.7%

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

      6. +-commutative 91.7%

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

      7. unsub-neg 91.7%

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

      8. *-commutative 91.7%

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

      9. sub-neg 91.7%

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

      10. +-commutative 91.7%

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

      11. +-commutative 91.7%

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

      12. distribute-neg-frac 91.7%

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

      13. metadata-eval 91.7%

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

   9. Simplified 91.7%

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

   if -1.5e-138 < x < -1.05e-225

   1. Initial program 16.8%

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

      1. sub-neg 16.8%

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

      2. div-sub 16.8%

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

      3. sub-neg 16.8%

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

      4. +-commutative 16.8%

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

      5. distribute-neg-in 16.8%

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

      6. remove-double-neg 16.8%

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

      7. sub-neg 16.8%

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

      8. div-sub 16.8%

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

      9. distribute-rgt1-in 23.5%

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

      10. associate-/l/ 23.5%

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

   3. Simplified 23.5%

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

   4. Taylor expanded in wj around 0 16.8%

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

   5. Taylor expanded in wj around 0 93.4%

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

   6. Taylor expanded in x around 0 68.7%

      \[\leadsto \color{blue}{{wj}^{2} + -1 \cdot {wj}^{3}} \]
   7. Step-by-step derivation

      1. mul-1-neg 68.7%

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

      2. unsub-neg 68.7%

         \[\leadsto \color{blue}{{wj}^{2} - {wj}^{3}} \]

      3. unpow2 68.7%

         \[\leadsto \color{blue}{wj \cdot wj} - {wj}^{3} \]

   8. Simplified 68.7%

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

4. Final simplification 90.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.5 \cdot 10^{-138} \lor \neg \left(x \leq -1.05 \cdot 10^{-225}\right):\\ \;\;\;\;\left(wj - \frac{wj}{wj + 1}\right) + x \cdot \left(\frac{1}{wj + 1} - \frac{wj}{wj + 1}\right)\\ \mathbf{else}:\\ \;\;\;\;wj \cdot wj - {wj}^{3}\\ \end{array} \]

Alternative 6: 95.5% accurate, 2.8× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 76.1%

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

   1. sub-neg 76.1%

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

   2. div-sub 76.1%

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

   3. sub-neg 76.1%

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

   4. +-commutative 76.1%

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

   5. distribute-neg-in 76.1%

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

   6. remove-double-neg 76.1%

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

   7. sub-neg 76.1%

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

   8. div-sub 76.1%

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

   9. distribute-rgt1-in 77.3%

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

   10. associate-/l/ 77.3%

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

3. Simplified 78.1%

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

4. Taylor expanded in wj around 0 76.3%

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

5. Taylor expanded in wj around 0 97.1%

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

6. Taylor expanded in x around 0 97.1%

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

   1. unpow2 97.1%

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

8. Simplified 97.1%

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

9. Taylor expanded in wj around 0 96.7%

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

    1. +-commutative 96.7%

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

    2. fma-def 96.7%

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

    3. neg-mul-1 96.7%

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

    4. unpow2 96.7%

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

11. Simplified 96.7%

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

12. Final simplification 96.7%

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

Alternative 7: 83.7% accurate, 12.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{wj}{wj + 1}\\ \mathbf{if}\;x \leq -1.5 \cdot 10^{-138} \lor \neg \left(x \leq -1.05 \cdot 10^{-225}\right):\\ \;\;\;\;\left(wj - t_0\right) + x \cdot \left(\frac{1}{wj + 1} - t_0\right)\\ \mathbf{else}:\\ \;\;\;\;wj \cdot wj\\ \end{array} \end{array} \]

FPCore

(FPCore (wj x)
 :precision binary64
 (let* ((t_0 (/ wj (+ wj 1.0))))
   (if (or (<= x -1.5e-138) (not (<= x -1.05e-225)))
     (+ (- wj t_0) (* x (- (/ 1.0 (+ wj 1.0)) t_0)))
     (* wj wj))))

C

double code(double wj, double x) {
	double t_0 = wj / (wj + 1.0);
	double tmp;
	if ((x <= -1.5e-138) || !(x <= -1.05e-225)) {
		tmp = (wj - t_0) + (x * ((1.0 / (wj + 1.0)) - t_0));
	} else {
		tmp = wj * wj;
	}
	return tmp;
}

Fortran

real(8) function code(wj, x)
    real(8), intent (in) :: wj
    real(8), intent (in) :: x
    real(8) :: t_0
    real(8) :: tmp
    t_0 = wj / (wj + 1.0d0)
    if ((x <= (-1.5d-138)) .or. (.not. (x <= (-1.05d-225)))) then
        tmp = (wj - t_0) + (x * ((1.0d0 / (wj + 1.0d0)) - t_0))
    else
        tmp = wj * wj
    end if
    code = tmp
end function

Java

public static double code(double wj, double x) {
	double t_0 = wj / (wj + 1.0);
	double tmp;
	if ((x <= -1.5e-138) || !(x <= -1.05e-225)) {
		tmp = (wj - t_0) + (x * ((1.0 / (wj + 1.0)) - t_0));
	} else {
		tmp = wj * wj;
	}
	return tmp;
}

Python

def code(wj, x):
	t_0 = wj / (wj + 1.0)
	tmp = 0
	if (x <= -1.5e-138) or not (x <= -1.05e-225):
		tmp = (wj - t_0) + (x * ((1.0 / (wj + 1.0)) - t_0))
	else:
		tmp = wj * wj
	return tmp

Julia

function code(wj, x)
	t_0 = Float64(wj / Float64(wj + 1.0))
	tmp = 0.0
	if ((x <= -1.5e-138) || !(x <= -1.05e-225))
		tmp = Float64(Float64(wj - t_0) + Float64(x * Float64(Float64(1.0 / Float64(wj + 1.0)) - t_0)));
	else
		tmp = Float64(wj * wj);
	end
	return tmp
end

MATLAB

function tmp_2 = code(wj, x)
	t_0 = wj / (wj + 1.0);
	tmp = 0.0;
	if ((x <= -1.5e-138) || ~((x <= -1.05e-225)))
		tmp = (wj - t_0) + (x * ((1.0 / (wj + 1.0)) - t_0));
	else
		tmp = wj * wj;
	end
	tmp_2 = tmp;
end

Wolfram

code[wj_, x_] := Block[{t$95$0 = N[(wj / N[(wj + 1.0), $MachinePrecision]), $MachinePrecision]}, If[Or[LessEqual[x, -1.5e-138], N[Not[LessEqual[x, -1.05e-225]], $MachinePrecision]], N[(N[(wj - t$95$0), $MachinePrecision] + N[(x * N[(N[(1.0 / N[(wj + 1.0), $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(wj * wj), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if x < -1.5e-138 or -1.05e-225 < x

   1. Initial program 79.8%

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

      1. sub-neg 79.8%

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

      2. div-sub 79.8%

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

      3. sub-neg 79.8%

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

      4. +-commutative 79.8%

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

      5. distribute-neg-in 79.8%

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

      6. remove-double-neg 79.8%

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

      7. sub-neg 79.8%

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

      8. div-sub 79.8%

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

      9. distribute-rgt1-in 80.7%

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

      10. associate-/l/ 80.7%

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

   3. Simplified 81.5%

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

      1. +-commutative 81.5%

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

      2. div-inv 81.5%

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

      3. fma-def 81.5%

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

   5. Applied egg-rr 81.5%

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

   6. Taylor expanded in wj around 0 80.1%

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

   7. Taylor expanded in x around -inf 80.0%

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

      1. associate-+r+ 91.7%

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

      2. mul-1-neg 91.7%

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

      3. unsub-neg 91.7%

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

      4. +-commutative 91.7%

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

      5. mul-1-neg 91.7%

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

      6. +-commutative 91.7%

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

      7. unsub-neg 91.7%

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

      8. *-commutative 91.7%

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

      9. sub-neg 91.7%

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

      10. +-commutative 91.7%

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

      11. +-commutative 91.7%

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

      12. distribute-neg-frac 91.7%

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

      13. metadata-eval 91.7%

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

   9. Simplified 91.7%

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

   if -1.5e-138 < x < -1.05e-225

   1. Initial program 16.8%

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

      1. sub-neg 16.8%

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

      2. div-sub 16.8%

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

      3. sub-neg 16.8%

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

      4. +-commutative 16.8%

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

      5. distribute-neg-in 16.8%

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

      6. remove-double-neg 16.8%

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

      7. sub-neg 16.8%

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

      8. div-sub 16.8%

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

      9. distribute-rgt1-in 23.5%

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

      10. associate-/l/ 23.5%

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

   3. Simplified 23.5%

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

   4. Taylor expanded in wj around 0 16.8%

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

   5. Taylor expanded in wj around 0 93.4%

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

   6. Taylor expanded in wj around 0 93.4%

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

      1. associate-+r+ 93.4%

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

      2. remove-double-neg 93.4%

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

      3. unsub-neg 93.4%

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

      4. unpow2 93.4%

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

      5. sub-neg 93.4%

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

      6. neg-mul-1 93.4%

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

      7. remove-double-neg 93.4%

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

      8. associate-*l* 93.4%

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

      9. *-commutative 93.4%

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

      10. distribute-lft-out 93.4%

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

      11. fma-neg 93.5%

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

   8. Simplified 93.5%

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

   9. Taylor expanded in x around 0 68.7%

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

       1. unpow2 68.7%

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

   11. Simplified 68.7%

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

4. Final simplification 90.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.5 \cdot 10^{-138} \lor \neg \left(x \leq -1.05 \cdot 10^{-225}\right):\\ \;\;\;\;\left(wj - \frac{wj}{wj + 1}\right) + x \cdot \left(\frac{1}{wj + 1} - \frac{wj}{wj + 1}\right)\\ \mathbf{else}:\\ \;\;\;\;wj \cdot wj\\ \end{array} \]

Alternative 8: 80.9% accurate, 20.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.5 \cdot 10^{-138}:\\ \;\;\;\;x \cdot \frac{1 - wj}{wj + 1}\\ \mathbf{elif}\;x \leq -4 \cdot 10^{-282}:\\ \;\;\;\;wj \cdot wj\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(wj \cdot \left(wj \cdot 2 - 2\right) + 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (wj x)
 :precision binary64
 (if (<= x -1.5e-138)
   (* x (/ (- 1.0 wj) (+ wj 1.0)))
   (if (<= x -4e-282) (* wj wj) (* x (+ (* wj (- (* wj 2.0) 2.0)) 1.0)))))

C

double code(double wj, double x) {
	double tmp;
	if (x <= -1.5e-138) {
		tmp = x * ((1.0 - wj) / (wj + 1.0));
	} else if (x <= -4e-282) {
		tmp = wj * wj;
	} else {
		tmp = x * ((wj * ((wj * 2.0) - 2.0)) + 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 <= (-1.5d-138)) then
        tmp = x * ((1.0d0 - wj) / (wj + 1.0d0))
    else if (x <= (-4d-282)) then
        tmp = wj * wj
    else
        tmp = x * ((wj * ((wj * 2.0d0) - 2.0d0)) + 1.0d0)
    end if
    code = tmp
end function

Java

public static double code(double wj, double x) {
	double tmp;
	if (x <= -1.5e-138) {
		tmp = x * ((1.0 - wj) / (wj + 1.0));
	} else if (x <= -4e-282) {
		tmp = wj * wj;
	} else {
		tmp = x * ((wj * ((wj * 2.0) - 2.0)) + 1.0);
	}
	return tmp;
}

Python

def code(wj, x):
	tmp = 0
	if x <= -1.5e-138:
		tmp = x * ((1.0 - wj) / (wj + 1.0))
	elif x <= -4e-282:
		tmp = wj * wj
	else:
		tmp = x * ((wj * ((wj * 2.0) - 2.0)) + 1.0)
	return tmp

Julia

function code(wj, x)
	tmp = 0.0
	if (x <= -1.5e-138)
		tmp = Float64(x * Float64(Float64(1.0 - wj) / Float64(wj + 1.0)));
	elseif (x <= -4e-282)
		tmp = Float64(wj * wj);
	else
		tmp = Float64(x * Float64(Float64(wj * Float64(Float64(wj * 2.0) - 2.0)) + 1.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(wj, x)
	tmp = 0.0;
	if (x <= -1.5e-138)
		tmp = x * ((1.0 - wj) / (wj + 1.0));
	elseif (x <= -4e-282)
		tmp = wj * wj;
	else
		tmp = x * ((wj * ((wj * 2.0) - 2.0)) + 1.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[wj_, x_] := If[LessEqual[x, -1.5e-138], N[(x * N[(N[(1.0 - wj), $MachinePrecision] / N[(wj + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, -4e-282], N[(wj * wj), $MachinePrecision], N[(x * N[(N[(wj * N[(N[(wj * 2.0), $MachinePrecision] - 2.0), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -1.5e-138

   1. Initial program 92.3%

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

      1. sub-neg 92.3%

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

      2. div-sub 92.3%

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

      3. sub-neg 92.3%

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

      4. +-commutative 92.3%

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

      5. distribute-neg-in 92.3%

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

      6. remove-double-neg 92.3%

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

      7. sub-neg 92.3%

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

      8. div-sub 92.3%

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

      9. distribute-rgt1-in 93.3%

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

      10. associate-/l/ 93.3%

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

   3. Simplified 94.4%

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

   4. Taylor expanded in wj around 0 92.3%

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

   5. Taylor expanded in x around inf 95.9%

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

      1. *-commutative 95.9%

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

      2. +-commutative 95.9%

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

      3. mul-1-neg 95.9%

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

      4. unsub-neg 95.9%

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

      5. +-commutative 95.9%

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

      6. +-commutative 95.9%

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

   7. Simplified 95.9%

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

   8. Taylor expanded in x around 0 95.9%

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

      1. *-commutative 95.9%

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

      2. +-commutative 95.9%

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

      3. +-commutative 95.9%

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

      4. div-sub 95.9%

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

   10. Simplified 95.9%

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

   if -1.5e-138 < x < -4.0000000000000001e-282

   1. Initial program 19.4%

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

      1. sub-neg 19.4%

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

      2. div-sub 19.4%

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

      3. sub-neg 19.4%

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

      4. +-commutative 19.4%

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

      5. distribute-neg-in 19.4%

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

      6. remove-double-neg 19.4%

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

      7. sub-neg 19.4%

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

      8. div-sub 19.4%

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

      9. distribute-rgt1-in 22.9%

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

      10. associate-/l/ 22.9%

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

   3. Simplified 22.9%

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

   4. Taylor expanded in wj around 0 19.4%

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

   5. Taylor expanded in wj around 0 93.9%

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

   6. Taylor expanded in wj around 0 93.7%

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

      1. associate-+r+ 93.7%

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

      2. remove-double-neg 93.7%

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

      3. unsub-neg 93.7%

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

      4. unpow2 93.7%

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

      5. sub-neg 93.7%

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

      6. neg-mul-1 93.7%

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

      7. remove-double-neg 93.7%

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

      8. associate-*l* 93.7%

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

      9. *-commutative 93.7%

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

      10. distribute-lft-out 93.7%

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

      11. fma-neg 93.8%

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

   8. Simplified 93.8%

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

   9. Taylor expanded in x around 0 59.2%

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

       1. unpow2 59.2%

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

   11. Simplified 59.2%

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

   if -4.0000000000000001e-282 < x

   1. Initial program 76.7%

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

      1. sub-neg 76.7%

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

      2. div-sub 76.7%

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

      3. sub-neg 76.7%

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

      4. +-commutative 76.7%

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

      5. distribute-neg-in 76.7%

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

      6. remove-double-neg 76.7%

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

      7. sub-neg 76.7%

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

      8. div-sub 76.7%

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

      9. distribute-rgt1-in 77.4%

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

      10. associate-/l/ 77.4%

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

   3. Simplified 78.2%

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

   4. Taylor expanded in wj around 0 77.0%

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

   5. Taylor expanded in wj around 0 97.2%

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

   6. Taylor expanded in wj around 0 97.0%

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

      1. associate-+r+ 97.0%

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

      2. remove-double-neg 97.0%

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

      3. unsub-neg 97.0%

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

      4. unpow2 97.0%

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

      5. sub-neg 97.0%

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

      6. neg-mul-1 97.0%

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

      7. remove-double-neg 97.0%

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

      8. associate-*l* 97.0%

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

      9. *-commutative 97.0%

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

      10. distribute-lft-out 97.0%

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

      11. fma-neg 97.0%

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

   8. Simplified 97.0%

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

   9. Taylor expanded in x around inf 89.6%

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

4. Final simplification 88.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.5 \cdot 10^{-138}:\\ \;\;\;\;x \cdot \frac{1 - wj}{wj + 1}\\ \mathbf{elif}\;x \leq -4 \cdot 10^{-282}:\\ \;\;\;\;wj \cdot wj\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(wj \cdot \left(wj \cdot 2 - 2\right) + 1\right)\\ \end{array} \]

Alternative 9: 81.0% accurate, 23.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.5 \cdot 10^{-138} \lor \neg \left(x \leq -8.5 \cdot 10^{-280}\right):\\ \;\;\;\;x \cdot \frac{1 - wj}{wj + 1}\\ \mathbf{else}:\\ \;\;\;\;wj \cdot wj\\ \end{array} \end{array} \]

FPCore

(FPCore (wj x)
 :precision binary64
 (if (or (<= x -1.5e-138) (not (<= x -8.5e-280)))
   (* x (/ (- 1.0 wj) (+ wj 1.0)))
   (* wj wj)))

C

double code(double wj, double x) {
	double tmp;
	if ((x <= -1.5e-138) || !(x <= -8.5e-280)) {
		tmp = x * ((1.0 - wj) / (wj + 1.0));
	} else {
		tmp = wj * wj;
	}
	return tmp;
}

Fortran

real(8) function code(wj, x)
    real(8), intent (in) :: wj
    real(8), intent (in) :: x
    real(8) :: tmp
    if ((x <= (-1.5d-138)) .or. (.not. (x <= (-8.5d-280)))) then
        tmp = x * ((1.0d0 - wj) / (wj + 1.0d0))
    else
        tmp = wj * wj
    end if
    code = tmp
end function

Java

public static double code(double wj, double x) {
	double tmp;
	if ((x <= -1.5e-138) || !(x <= -8.5e-280)) {
		tmp = x * ((1.0 - wj) / (wj + 1.0));
	} else {
		tmp = wj * wj;
	}
	return tmp;
}

Python

def code(wj, x):
	tmp = 0
	if (x <= -1.5e-138) or not (x <= -8.5e-280):
		tmp = x * ((1.0 - wj) / (wj + 1.0))
	else:
		tmp = wj * wj
	return tmp

Julia

function code(wj, x)
	tmp = 0.0
	if ((x <= -1.5e-138) || !(x <= -8.5e-280))
		tmp = Float64(x * Float64(Float64(1.0 - wj) / Float64(wj + 1.0)));
	else
		tmp = Float64(wj * wj);
	end
	return tmp
end

MATLAB

function tmp_2 = code(wj, x)
	tmp = 0.0;
	if ((x <= -1.5e-138) || ~((x <= -8.5e-280)))
		tmp = x * ((1.0 - wj) / (wj + 1.0));
	else
		tmp = wj * wj;
	end
	tmp_2 = tmp;
end

Wolfram

code[wj_, x_] := If[Or[LessEqual[x, -1.5e-138], N[Not[LessEqual[x, -8.5e-280]], $MachinePrecision]], N[(x * N[(N[(1.0 - wj), $MachinePrecision] / N[(wj + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(wj * wj), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -1.5e-138 or -8.50000000000000037e-280 < x

   1. Initial program 83.1%

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

      1. sub-neg 83.1%

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

      2. div-sub 83.1%

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

      3. sub-neg 83.1%

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

      4. +-commutative 83.1%

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

      5. distribute-neg-in 83.1%

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

      6. remove-double-neg 83.1%

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

      7. sub-neg 83.1%

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

      8. div-sub 83.1%

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

      9. distribute-rgt1-in 84.0%

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

      10. associate-/l/ 84.0%

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

   3. Simplified 84.9%

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

   4. Taylor expanded in wj around 0 83.3%

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

   5. Taylor expanded in x around inf 92.1%

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

      1. *-commutative 92.1%

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

      2. +-commutative 92.1%

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

      3. mul-1-neg 92.1%

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

      4. unsub-neg 92.1%

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

      5. +-commutative 92.1%

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

      6. +-commutative 92.1%

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

   7. Simplified 92.1%

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

   8. Taylor expanded in x around 0 92.1%

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

      1. *-commutative 92.1%

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

      2. +-commutative 92.1%

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

      3. +-commutative 92.1%

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

      4. div-sub 92.1%

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

   10. Simplified 92.1%

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

   if -1.5e-138 < x < -8.50000000000000037e-280

   1. Initial program 19.4%

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

      1. sub-neg 19.4%

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

      2. div-sub 19.4%

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

      3. sub-neg 19.4%

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

      4. +-commutative 19.4%

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

      5. distribute-neg-in 19.4%

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

      6. remove-double-neg 19.4%

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

      7. sub-neg 19.4%

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

      8. div-sub 19.4%

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

      9. distribute-rgt1-in 22.9%

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

      10. associate-/l/ 22.9%

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

   3. Simplified 22.9%

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

   4. Taylor expanded in wj around 0 19.4%

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

   5. Taylor expanded in wj around 0 93.9%

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

   6. Taylor expanded in wj around 0 93.7%

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

      1. associate-+r+ 93.7%

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

      2. remove-double-neg 93.7%

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

      3. unsub-neg 93.7%

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

      4. unpow2 93.7%

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

      5. sub-neg 93.7%

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

      6. neg-mul-1 93.7%

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

      7. remove-double-neg 93.7%

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

      8. associate-*l* 93.7%

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

      9. *-commutative 93.7%

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

      10. distribute-lft-out 93.7%

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

      11. fma-neg 93.8%

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

   8. Simplified 93.8%

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

   9. Taylor expanded in x around 0 59.2%

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

       1. unpow2 59.2%

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

   11. Simplified 59.2%

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

4. Final simplification 88.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.5 \cdot 10^{-138} \lor \neg \left(x \leq -8.5 \cdot 10^{-280}\right):\\ \;\;\;\;x \cdot \frac{1 - wj}{wj + 1}\\ \mathbf{else}:\\ \;\;\;\;wj \cdot wj\\ \end{array} \]

Alternative 10: 80.9% accurate, 28.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.5 \cdot 10^{-138} \lor \neg \left(x \leq -2.6 \cdot 10^{-279}\right):\\ \;\;\;\;x \cdot \left(wj \cdot -2 + 1\right)\\ \mathbf{else}:\\ \;\;\;\;wj \cdot wj\\ \end{array} \end{array} \]

FPCore

(FPCore (wj x)
 :precision binary64
 (if (or (<= x -1.5e-138) (not (<= x -2.6e-279)))
   (* x (+ (* wj -2.0) 1.0))
   (* wj wj)))

C

double code(double wj, double x) {
	double tmp;
	if ((x <= -1.5e-138) || !(x <= -2.6e-279)) {
		tmp = x * ((wj * -2.0) + 1.0);
	} else {
		tmp = wj * wj;
	}
	return tmp;
}

Fortran

real(8) function code(wj, x)
    real(8), intent (in) :: wj
    real(8), intent (in) :: x
    real(8) :: tmp
    if ((x <= (-1.5d-138)) .or. (.not. (x <= (-2.6d-279)))) then
        tmp = x * ((wj * (-2.0d0)) + 1.0d0)
    else
        tmp = wj * wj
    end if
    code = tmp
end function

Java

public static double code(double wj, double x) {
	double tmp;
	if ((x <= -1.5e-138) || !(x <= -2.6e-279)) {
		tmp = x * ((wj * -2.0) + 1.0);
	} else {
		tmp = wj * wj;
	}
	return tmp;
}

Python

def code(wj, x):
	tmp = 0
	if (x <= -1.5e-138) or not (x <= -2.6e-279):
		tmp = x * ((wj * -2.0) + 1.0)
	else:
		tmp = wj * wj
	return tmp

Julia

function code(wj, x)
	tmp = 0.0
	if ((x <= -1.5e-138) || !(x <= -2.6e-279))
		tmp = Float64(x * Float64(Float64(wj * -2.0) + 1.0));
	else
		tmp = Float64(wj * wj);
	end
	return tmp
end

MATLAB

function tmp_2 = code(wj, x)
	tmp = 0.0;
	if ((x <= -1.5e-138) || ~((x <= -2.6e-279)))
		tmp = x * ((wj * -2.0) + 1.0);
	else
		tmp = wj * wj;
	end
	tmp_2 = tmp;
end

Wolfram

code[wj_, x_] := If[Or[LessEqual[x, -1.5e-138], N[Not[LessEqual[x, -2.6e-279]], $MachinePrecision]], N[(x * N[(N[(wj * -2.0), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], N[(wj * wj), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -1.5e-138 or -2.6000000000000002e-279 < x

   1. Initial program 83.1%

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

      1. sub-neg 83.1%

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

      2. div-sub 83.1%

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

      3. sub-neg 83.1%

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

      4. +-commutative 83.1%

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

      5. distribute-neg-in 83.1%

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

      6. remove-double-neg 83.1%

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

      7. sub-neg 83.1%

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

      8. div-sub 83.1%

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

      9. distribute-rgt1-in 84.0%

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

      10. associate-/l/ 84.0%

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

   3. Simplified 84.9%

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

   4. Taylor expanded in wj around 0 92.1%

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

   5. Taylor expanded in x around 0 92.1%

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

   if -1.5e-138 < x < -2.6000000000000002e-279

   1. Initial program 19.4%

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

      1. sub-neg 19.4%

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

      2. div-sub 19.4%

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

      3. sub-neg 19.4%

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

      4. +-commutative 19.4%

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

      5. distribute-neg-in 19.4%

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

      6. remove-double-neg 19.4%

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

      7. sub-neg 19.4%

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

      8. div-sub 19.4%

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

      9. distribute-rgt1-in 22.9%

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

      10. associate-/l/ 22.9%

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

   3. Simplified 22.9%

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

   4. Taylor expanded in wj around 0 19.4%

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

   5. Taylor expanded in wj around 0 93.9%

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

   6. Taylor expanded in wj around 0 93.7%

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

      1. associate-+r+ 93.7%

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

      2. remove-double-neg 93.7%

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

      3. unsub-neg 93.7%

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

      4. unpow2 93.7%

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

      5. sub-neg 93.7%

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

      6. neg-mul-1 93.7%

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

      7. remove-double-neg 93.7%

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

      8. associate-*l* 93.7%

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

      9. *-commutative 93.7%

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

      10. distribute-lft-out 93.7%

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

      11. fma-neg 93.8%

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

   8. Simplified 93.8%

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

   9. Taylor expanded in x around 0 59.2%

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

       1. unpow2 59.2%

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

   11. Simplified 59.2%

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

4. Final simplification 88.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.5 \cdot 10^{-138} \lor \neg \left(x \leq -2.6 \cdot 10^{-279}\right):\\ \;\;\;\;x \cdot \left(wj \cdot -2 + 1\right)\\ \mathbf{else}:\\ \;\;\;\;wj \cdot wj\\ \end{array} \]

Alternative 11: 80.9% accurate, 28.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.5 \cdot 10^{-138}:\\ \;\;\;\;x + -2 \cdot \left(wj \cdot x\right)\\ \mathbf{elif}\;x \leq -2.8 \cdot 10^{-279}:\\ \;\;\;\;wj \cdot wj\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(wj \cdot -2 + 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (wj x)
 :precision binary64
 (if (<= x -1.5e-138)
   (+ x (* -2.0 (* wj x)))
   (if (<= x -2.8e-279) (* wj wj) (* x (+ (* wj -2.0) 1.0)))))

C

double code(double wj, double x) {
	double tmp;
	if (x <= -1.5e-138) {
		tmp = x + (-2.0 * (wj * x));
	} else if (x <= -2.8e-279) {
		tmp = wj * wj;
	} else {
		tmp = x * ((wj * -2.0) + 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 <= (-1.5d-138)) then
        tmp = x + ((-2.0d0) * (wj * x))
    else if (x <= (-2.8d-279)) then
        tmp = wj * wj
    else
        tmp = x * ((wj * (-2.0d0)) + 1.0d0)
    end if
    code = tmp
end function

Java

public static double code(double wj, double x) {
	double tmp;
	if (x <= -1.5e-138) {
		tmp = x + (-2.0 * (wj * x));
	} else if (x <= -2.8e-279) {
		tmp = wj * wj;
	} else {
		tmp = x * ((wj * -2.0) + 1.0);
	}
	return tmp;
}

Python

def code(wj, x):
	tmp = 0
	if x <= -1.5e-138:
		tmp = x + (-2.0 * (wj * x))
	elif x <= -2.8e-279:
		tmp = wj * wj
	else:
		tmp = x * ((wj * -2.0) + 1.0)
	return tmp

Julia

function code(wj, x)
	tmp = 0.0
	if (x <= -1.5e-138)
		tmp = Float64(x + Float64(-2.0 * Float64(wj * x)));
	elseif (x <= -2.8e-279)
		tmp = Float64(wj * wj);
	else
		tmp = Float64(x * Float64(Float64(wj * -2.0) + 1.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(wj, x)
	tmp = 0.0;
	if (x <= -1.5e-138)
		tmp = x + (-2.0 * (wj * x));
	elseif (x <= -2.8e-279)
		tmp = wj * wj;
	else
		tmp = x * ((wj * -2.0) + 1.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[wj_, x_] := If[LessEqual[x, -1.5e-138], N[(x + N[(-2.0 * N[(wj * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, -2.8e-279], N[(wj * wj), $MachinePrecision], N[(x * N[(N[(wj * -2.0), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -1.5e-138

   1. Initial program 92.3%

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

      1. sub-neg 92.3%

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

      2. div-sub 92.3%

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

      3. sub-neg 92.3%

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

      4. +-commutative 92.3%

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

      5. distribute-neg-in 92.3%

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

      6. remove-double-neg 92.3%

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

      7. sub-neg 92.3%

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

      8. div-sub 92.3%

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

      9. distribute-rgt1-in 93.3%

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

      10. associate-/l/ 93.3%

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

   3. Simplified 94.4%

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

   4. Taylor expanded in wj around 0 95.8%

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

   if -1.5e-138 < x < -2.8000000000000001e-279

   1. Initial program 19.4%

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

      1. sub-neg 19.4%

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

      2. div-sub 19.4%

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

      3. sub-neg 19.4%

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

      4. +-commutative 19.4%

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

      5. distribute-neg-in 19.4%

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

      6. remove-double-neg 19.4%

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

      7. sub-neg 19.4%

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

      8. div-sub 19.4%

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

      9. distribute-rgt1-in 22.9%

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

      10. associate-/l/ 22.9%

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

   3. Simplified 22.9%

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

   4. Taylor expanded in wj around 0 19.4%

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

   5. Taylor expanded in wj around 0 93.9%

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

   6. Taylor expanded in wj around 0 93.7%

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

      1. associate-+r+ 93.7%

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

      2. remove-double-neg 93.7%

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

      3. unsub-neg 93.7%

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

      4. unpow2 93.7%

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

      5. sub-neg 93.7%

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

      6. neg-mul-1 93.7%

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

      7. remove-double-neg 93.7%

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

      8. associate-*l* 93.7%

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

      9. *-commutative 93.7%

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

      10. distribute-lft-out 93.7%

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

      11. fma-neg 93.8%

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

   8. Simplified 93.8%

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

   9. Taylor expanded in x around 0 59.2%

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

       1. unpow2 59.2%

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

   11. Simplified 59.2%

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

   if -2.8000000000000001e-279 < x

   1. Initial program 76.7%

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

      1. sub-neg 76.7%

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

      2. div-sub 76.7%

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

      3. sub-neg 76.7%

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

      4. +-commutative 76.7%

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

      5. distribute-neg-in 76.7%

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

      6. remove-double-neg 76.7%

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

      7. sub-neg 76.7%

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

      8. div-sub 76.7%

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

      9. distribute-rgt1-in 77.4%

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

      10. associate-/l/ 77.4%

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

   3. Simplified 78.2%

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

   4. Taylor expanded in wj around 0 89.4%

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

   5. Taylor expanded in x around 0 89.4%

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

4. Final simplification 88.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.5 \cdot 10^{-138}:\\ \;\;\;\;x + -2 \cdot \left(wj \cdot x\right)\\ \mathbf{elif}\;x \leq -2.8 \cdot 10^{-279}:\\ \;\;\;\;wj \cdot wj\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(wj \cdot -2 + 1\right)\\ \end{array} \]

Alternative 12: 80.4% accurate, 43.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.5 \cdot 10^{-138}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq -6 \cdot 10^{-280}:\\ \;\;\;\;wj \cdot wj\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \end{array} \]

FPCore

(FPCore (wj x)
 :precision binary64
 (if (<= x -1.5e-138) x (if (<= x -6e-280) (* wj wj) x)))

C

double code(double wj, double x) {
	double tmp;
	if (x <= -1.5e-138) {
		tmp = x;
	} else if (x <= -6e-280) {
		tmp = wj * wj;
	} else {
		tmp = x;
	}
	return tmp;
}

Fortran

real(8) function code(wj, x)
    real(8), intent (in) :: wj
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= (-1.5d-138)) then
        tmp = x
    else if (x <= (-6d-280)) then
        tmp = wj * wj
    else
        tmp = x
    end if
    code = tmp
end function

Java

public static double code(double wj, double x) {
	double tmp;
	if (x <= -1.5e-138) {
		tmp = x;
	} else if (x <= -6e-280) {
		tmp = wj * wj;
	} else {
		tmp = x;
	}
	return tmp;
}

Python

def code(wj, x):
	tmp = 0
	if x <= -1.5e-138:
		tmp = x
	elif x <= -6e-280:
		tmp = wj * wj
	else:
		tmp = x
	return tmp

Julia

function code(wj, x)
	tmp = 0.0
	if (x <= -1.5e-138)
		tmp = x;
	elseif (x <= -6e-280)
		tmp = Float64(wj * wj);
	else
		tmp = x;
	end
	return tmp
end

MATLAB

function tmp_2 = code(wj, x)
	tmp = 0.0;
	if (x <= -1.5e-138)
		tmp = x;
	elseif (x <= -6e-280)
		tmp = wj * wj;
	else
		tmp = x;
	end
	tmp_2 = tmp;
end

Wolfram

code[wj_, x_] := If[LessEqual[x, -1.5e-138], x, If[LessEqual[x, -6e-280], N[(wj * wj), $MachinePrecision], x]]

Derivation

1. Split input into 2 regimes

2. if x < -1.5e-138 or -5.99999999999999974e-280 < x

   1. Initial program 83.1%

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

      1. sub-neg 83.1%

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

      2. div-sub 83.1%

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

      3. sub-neg 83.1%

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

      4. +-commutative 83.1%

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

      5. distribute-neg-in 83.1%

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

      6. remove-double-neg 83.1%

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

      7. sub-neg 83.1%

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

      8. div-sub 83.1%

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

      9. distribute-rgt1-in 84.0%

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

      10. associate-/l/ 84.0%

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

   3. Simplified 84.9%

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

   4. Taylor expanded in wj around 0 91.6%

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

   if -1.5e-138 < x < -5.99999999999999974e-280

   1. Initial program 19.4%

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

      1. sub-neg 19.4%

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

      2. div-sub 19.4%

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

      3. sub-neg 19.4%

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

      4. +-commutative 19.4%

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

      5. distribute-neg-in 19.4%

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

      6. remove-double-neg 19.4%

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

      7. sub-neg 19.4%

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

      8. div-sub 19.4%

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

      9. distribute-rgt1-in 22.9%

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

      10. associate-/l/ 22.9%

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

   3. Simplified 22.9%

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

   4. Taylor expanded in wj around 0 19.4%

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

   5. Taylor expanded in wj around 0 93.9%

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

   6. Taylor expanded in wj around 0 93.7%

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

      1. associate-+r+ 93.7%

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

      2. remove-double-neg 93.7%

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

      3. unsub-neg 93.7%

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

      4. unpow2 93.7%

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

      5. sub-neg 93.7%

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

      6. neg-mul-1 93.7%

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

      7. remove-double-neg 93.7%

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

      8. associate-*l* 93.7%

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

      9. *-commutative 93.7%

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

      10. distribute-lft-out 93.7%

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

      11. fma-neg 93.8%

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

   8. Simplified 93.8%

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

   9. Taylor expanded in x around 0 59.2%

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

       1. unpow2 59.2%

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

   11. Simplified 59.2%

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

4. Final simplification 88.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.5 \cdot 10^{-138}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq -6 \cdot 10^{-280}:\\ \;\;\;\;wj \cdot wj\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 13: 4.4% accurate, 313.0× speedup

Math

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

FPCore

(FPCore (wj x) :precision binary64 wj)

C

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

Fortran

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

Java

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

Python

def code(wj, x):
	return wj

Julia

function code(wj, x)
	return wj
end

MATLAB

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

Wolfram

code[wj_, x_] := wj

Derivation

1. Initial program 76.1%

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

   1. sub-neg 76.1%

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

   2. div-sub 76.1%

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

   3. sub-neg 76.1%

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

   4. +-commutative 76.1%

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

   5. distribute-neg-in 76.1%

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

   6. remove-double-neg 76.1%

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

   7. sub-neg 76.1%

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

   8. div-sub 76.1%

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

   9. distribute-rgt1-in 77.3%

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

   10. associate-/l/ 77.3%

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

3. Simplified 78.1%

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

4. Taylor expanded in wj around inf 4.4%

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

5. Final simplification 4.4%

   \[\leadsto wj \]

Alternative 14: 83.9% accurate, 313.0× speedup

Math

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

FPCore

(FPCore (wj x) :precision binary64 x)

C

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

Fortran

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

Java

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

Python

def code(wj, x):
	return x

Julia

function code(wj, x)
	return x
end

MATLAB

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

Wolfram

code[wj_, x_] := x

Derivation

1. Initial program 76.1%

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

   1. sub-neg 76.1%

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

   2. div-sub 76.1%

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

   3. sub-neg 76.1%

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

   4. +-commutative 76.1%

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

   5. distribute-neg-in 76.1%

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

   6. remove-double-neg 76.1%

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

   7. sub-neg 76.1%

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

   8. div-sub 76.1%

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

   9. distribute-rgt1-in 77.3%

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

   10. associate-/l/ 77.3%

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

3. Simplified 78.1%

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

4. Taylor expanded in wj around 0 85.3%

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

5. Final simplification 85.3%

   \[\leadsto x \]

Developer target: 78.5% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ wj - \left(\frac{wj}{wj + 1} - \frac{x}{e^{wj} + wj \cdot e^{wj}}\right) \end{array} \]

FPCore

(FPCore (wj x)
 :precision binary64
 (- wj (- (/ wj (+ wj 1.0)) (/ x (+ (exp wj) (* wj (exp wj)))))))

C

double code(double wj, double x) {
	return wj - ((wj / (wj + 1.0)) - (x / (exp(wj) + (wj * exp(wj)))));
}

Fortran

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

Java

public static double code(double wj, double x) {
	return wj - ((wj / (wj + 1.0)) - (x / (Math.exp(wj) + (wj * Math.exp(wj)))));
}

Python

def code(wj, x):
	return wj - ((wj / (wj + 1.0)) - (x / (math.exp(wj) + (wj * math.exp(wj)))))

Julia

function code(wj, x)
	return Float64(wj - Float64(Float64(wj / Float64(wj + 1.0)) - Float64(x / Float64(exp(wj) + Float64(wj * exp(wj))))))
end

MATLAB

function tmp = code(wj, x)
	tmp = wj - ((wj / (wj + 1.0)) - (x / (exp(wj) + (wj * exp(wj)))));
end

Wolfram

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

Reproduce

herbie shell --seed 2023238 
(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))))))