Jmat.Real.lambertw, newton loop step

Percentage Accurate: 77.6% → 98.8%

Time: 7.6s

Alternatives: 11

Speedup: 313.0×

• could not determine a ground truth

  1. wj = -3.90519235561825e+260
  2. x = -3.5725981752876704e+200

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: 77.6% 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: 98.8% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x \cdot -4 + x \cdot 1.5\\ t_1 := wj \cdot e^{wj}\\ \mathbf{if}\;wj + \frac{x - t_1}{e^{wj} + t_1} \leq 5 \cdot 10^{-12}:\\ \;\;\;\;{wj}^{3} \cdot \left(\left(\left(-1 - -2 \cdot t_0\right) - x \cdot -3\right) - x \cdot 0.6666666666666666\right) + \left(\left(1 - t_0\right) \cdot {wj}^{2} + \left(x + -2 \cdot \left(wj \cdot x\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;wj + \frac{\frac{x}{e^{wj}} - wj}{wj + 1}\\ \end{array} \end{array} \]

FPCore

(FPCore (wj x)
 :precision binary64
 (let* ((t_0 (+ (* x -4.0) (* x 1.5))) (t_1 (* wj (exp wj))))
   (if (<= (+ wj (/ (- x t_1) (+ (exp wj) t_1))) 5e-12)
     (+
      (*
       (pow wj 3.0)
       (- (- (- -1.0 (* -2.0 t_0)) (* x -3.0)) (* x 0.6666666666666666)))
      (+ (* (- 1.0 t_0) (pow wj 2.0)) (+ x (* -2.0 (* wj x)))))
     (+ wj (/ (- (/ x (exp wj)) wj) (+ wj 1.0))))))

C

double code(double wj, double x) {
	double t_0 = (x * -4.0) + (x * 1.5);
	double t_1 = wj * exp(wj);
	double tmp;
	if ((wj + ((x - t_1) / (exp(wj) + t_1))) <= 5e-12) {
		tmp = (pow(wj, 3.0) * (((-1.0 - (-2.0 * t_0)) - (x * -3.0)) - (x * 0.6666666666666666))) + (((1.0 - t_0) * pow(wj, 2.0)) + (x + (-2.0 * (wj * x))));
	} else {
		tmp = wj + (((x / exp(wj)) - wj) / (wj + 1.0));
	}
	return tmp;
}

Fortran

real(8) function code(wj, x)
    real(8), intent (in) :: wj
    real(8), intent (in) :: x
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = (x * (-4.0d0)) + (x * 1.5d0)
    t_1 = wj * exp(wj)
    if ((wj + ((x - t_1) / (exp(wj) + t_1))) <= 5d-12) then
        tmp = ((wj ** 3.0d0) * ((((-1.0d0) - ((-2.0d0) * t_0)) - (x * (-3.0d0))) - (x * 0.6666666666666666d0))) + (((1.0d0 - t_0) * (wj ** 2.0d0)) + (x + ((-2.0d0) * (wj * x))))
    else
        tmp = wj + (((x / exp(wj)) - wj) / (wj + 1.0d0))
    end if
    code = tmp
end function

Java

public static double code(double wj, double x) {
	double t_0 = (x * -4.0) + (x * 1.5);
	double t_1 = wj * Math.exp(wj);
	double tmp;
	if ((wj + ((x - t_1) / (Math.exp(wj) + t_1))) <= 5e-12) {
		tmp = (Math.pow(wj, 3.0) * (((-1.0 - (-2.0 * t_0)) - (x * -3.0)) - (x * 0.6666666666666666))) + (((1.0 - t_0) * Math.pow(wj, 2.0)) + (x + (-2.0 * (wj * x))));
	} else {
		tmp = wj + (((x / Math.exp(wj)) - wj) / (wj + 1.0));
	}
	return tmp;
}

Python

def code(wj, x):
	t_0 = (x * -4.0) + (x * 1.5)
	t_1 = wj * math.exp(wj)
	tmp = 0
	if (wj + ((x - t_1) / (math.exp(wj) + t_1))) <= 5e-12:
		tmp = (math.pow(wj, 3.0) * (((-1.0 - (-2.0 * t_0)) - (x * -3.0)) - (x * 0.6666666666666666))) + (((1.0 - t_0) * math.pow(wj, 2.0)) + (x + (-2.0 * (wj * x))))
	else:
		tmp = wj + (((x / math.exp(wj)) - wj) / (wj + 1.0))
	return tmp

Julia

function code(wj, x)
	t_0 = Float64(Float64(x * -4.0) + Float64(x * 1.5))
	t_1 = Float64(wj * exp(wj))
	tmp = 0.0
	if (Float64(wj + Float64(Float64(x - t_1) / Float64(exp(wj) + t_1))) <= 5e-12)
		tmp = Float64(Float64((wj ^ 3.0) * Float64(Float64(Float64(-1.0 - Float64(-2.0 * t_0)) - Float64(x * -3.0)) - Float64(x * 0.6666666666666666))) + Float64(Float64(Float64(1.0 - t_0) * (wj ^ 2.0)) + Float64(x + Float64(-2.0 * Float64(wj * x)))));
	else
		tmp = Float64(wj + Float64(Float64(Float64(x / exp(wj)) - wj) / Float64(wj + 1.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(wj, x)
	t_0 = (x * -4.0) + (x * 1.5);
	t_1 = wj * exp(wj);
	tmp = 0.0;
	if ((wj + ((x - t_1) / (exp(wj) + t_1))) <= 5e-12)
		tmp = ((wj ^ 3.0) * (((-1.0 - (-2.0 * t_0)) - (x * -3.0)) - (x * 0.6666666666666666))) + (((1.0 - t_0) * (wj ^ 2.0)) + (x + (-2.0 * (wj * x))));
	else
		tmp = wj + (((x / exp(wj)) - wj) / (wj + 1.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[wj_, x_] := Block[{t$95$0 = N[(N[(x * -4.0), $MachinePrecision] + N[(x * 1.5), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(wj * N[Exp[wj], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(wj + N[(N[(x - t$95$1), $MachinePrecision] / N[(N[Exp[wj], $MachinePrecision] + t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 5e-12], N[(N[(N[Power[wj, 3.0], $MachinePrecision] * N[(N[(N[(-1.0 - N[(-2.0 * t$95$0), $MachinePrecision]), $MachinePrecision] - N[(x * -3.0), $MachinePrecision]), $MachinePrecision] - N[(x * 0.6666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(N[(1.0 - t$95$0), $MachinePrecision] * N[Power[wj, 2.0], $MachinePrecision]), $MachinePrecision] + N[(x + N[(-2.0 * N[(wj * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(wj + N[(N[(N[(x / N[Exp[wj], $MachinePrecision]), $MachinePrecision] - wj), $MachinePrecision] / N[(wj + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 wj (/.f64 (-.f64 (*.f64 wj (exp.f64 wj)) x) (+.f64 (exp.f64 wj) (*.f64 wj (exp.f64 wj))))) < 4.9999999999999997e-12

   1. Initial program 68.5%

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

      1. sub-neg 68.5%

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

      2. div-sub 68.5%

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

      3. sub-neg 68.5%

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

      4. +-commutative 68.5%

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

      5. distribute-neg-in 68.5%

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

      6. remove-double-neg 68.5%

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

      7. sub-neg 68.5%

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

      8. div-sub 68.5%

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

      9. distribute-rgt1-in 68.5%

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

      10. associate-/l/ 68.5%

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

   3. Simplified 68.5%

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

   4. Taylor expanded in wj around 0 99.7%

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

   if 4.9999999999999997e-12 < (-.f64 wj (/.f64 (-.f64 (*.f64 wj (exp.f64 wj)) x) (+.f64 (exp.f64 wj) (*.f64 wj (exp.f64 wj)))))

   1. Initial program 95.3%

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

      1. sub-neg 95.3%

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

      2. div-sub 95.3%

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

      3. sub-neg 95.3%

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

      4. +-commutative 95.3%

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

      5. distribute-neg-in 95.3%

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

      6. remove-double-neg 95.3%

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

      7. sub-neg 95.3%

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

      8. div-sub 95.3%

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

      9. distribute-rgt1-in 96.3%

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

      10. associate-/l/ 96.3%

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

   3. Simplified 99.5%

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

4. Final simplification 99.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;wj + \frac{x - wj \cdot e^{wj}}{e^{wj} + wj \cdot e^{wj}} \leq 5 \cdot 10^{-12}:\\ \;\;\;\;{wj}^{3} \cdot \left(\left(\left(-1 - -2 \cdot \left(x \cdot -4 + x \cdot 1.5\right)\right) - x \cdot -3\right) - x \cdot 0.6666666666666666\right) + \left(\left(1 - \left(x \cdot -4 + x \cdot 1.5\right)\right) \cdot {wj}^{2} + \left(x + -2 \cdot \left(wj \cdot x\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;wj + \frac{\frac{x}{e^{wj}} - wj}{wj + 1}\\ \end{array} \]

Alternative 2: 98.0% accurate, 2.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;wj \leq 1.86 \cdot 10^{-6}:\\ \;\;\;\;\left(\left(x + -2 \cdot \left(wj \cdot x\right)\right) + wj \cdot wj\right) - {wj}^{3}\\ \mathbf{else}:\\ \;\;\;\;wj + \frac{x \cdot e^{-wj} - wj}{wj + 1}\\ \end{array} \end{array} \]

FPCore

(FPCore (wj x)
 :precision binary64
 (if (<= wj 1.86e-6)
   (- (+ (+ x (* -2.0 (* wj x))) (* wj wj)) (pow wj 3.0))
   (+ wj (/ (- (* x (exp (- wj))) wj) (+ wj 1.0)))))

C

double code(double wj, double x) {
	double tmp;
	if (wj <= 1.86e-6) {
		tmp = ((x + (-2.0 * (wj * x))) + (wj * wj)) - pow(wj, 3.0);
	} else {
		tmp = wj + (((x * exp(-wj)) - wj) / (wj + 1.0));
	}
	return tmp;
}

Fortran

real(8) function code(wj, x)
    real(8), intent (in) :: wj
    real(8), intent (in) :: x
    real(8) :: tmp
    if (wj <= 1.86d-6) then
        tmp = ((x + ((-2.0d0) * (wj * x))) + (wj * wj)) - (wj ** 3.0d0)
    else
        tmp = wj + (((x * exp(-wj)) - wj) / (wj + 1.0d0))
    end if
    code = tmp
end function

Java

public static double code(double wj, double x) {
	double tmp;
	if (wj <= 1.86e-6) {
		tmp = ((x + (-2.0 * (wj * x))) + (wj * wj)) - Math.pow(wj, 3.0);
	} else {
		tmp = wj + (((x * Math.exp(-wj)) - wj) / (wj + 1.0));
	}
	return tmp;
}

Python

def code(wj, x):
	tmp = 0
	if wj <= 1.86e-6:
		tmp = ((x + (-2.0 * (wj * x))) + (wj * wj)) - math.pow(wj, 3.0)
	else:
		tmp = wj + (((x * math.exp(-wj)) - wj) / (wj + 1.0))
	return tmp

Julia

function code(wj, x)
	tmp = 0.0
	if (wj <= 1.86e-6)
		tmp = Float64(Float64(Float64(x + Float64(-2.0 * Float64(wj * x))) + Float64(wj * wj)) - (wj ^ 3.0));
	else
		tmp = Float64(wj + Float64(Float64(Float64(x * exp(Float64(-wj))) - wj) / Float64(wj + 1.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(wj, x)
	tmp = 0.0;
	if (wj <= 1.86e-6)
		tmp = ((x + (-2.0 * (wj * x))) + (wj * wj)) - (wj ^ 3.0);
	else
		tmp = wj + (((x * exp(-wj)) - wj) / (wj + 1.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[wj_, x_] := If[LessEqual[wj, 1.86e-6], N[(N[(N[(x + N[(-2.0 * N[(wj * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(wj * wj), $MachinePrecision]), $MachinePrecision] - N[Power[wj, 3.0], $MachinePrecision]), $MachinePrecision], N[(wj + N[(N[(N[(x * N[Exp[(-wj)], $MachinePrecision]), $MachinePrecision] - wj), $MachinePrecision] / N[(wj + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if wj < 1.86e-6

   1. Initial program 79.2%

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

      1. sub-neg 79.2%

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

      2. div-sub 79.2%

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

      3. sub-neg 79.2%

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

      4. +-commutative 79.2%

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

      5. distribute-neg-in 79.2%

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

      6. remove-double-neg 79.2%

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

      7. sub-neg 79.2%

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

      8. div-sub 79.2%

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

      9. distribute-rgt1-in 79.6%

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

      10. associate-/l/ 79.6%

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

   3. Simplified 79.6%

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

   4. Taylor expanded in wj around 0 98.6%

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

   5. Taylor expanded in x around 0 98.6%

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

      1. unpow2 98.6%

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

   7. Simplified 98.6%

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

   8. Taylor expanded in x around 0 99.0%

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

   if 1.86e-6 < wj

   1. Initial program 52.7%

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

      1. sub-neg 52.7%

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

      2. div-sub 52.7%

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

      3. sub-neg 52.7%

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

      4. +-commutative 52.7%

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

      5. distribute-neg-in 52.7%

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

      6. remove-double-neg 52.7%

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

      7. sub-neg 52.7%

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

      8. div-sub 52.7%

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

      9. distribute-rgt1-in 52.7%

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

      10. associate-/l/ 52.7%

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

   3. Simplified 95.6%

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

      1. clear-num 95.6%

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

      2. associate-/r/ 95.6%

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

      3. rec-exp 95.8%

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

   5. Applied egg-rr 95.8%

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

4. Final simplification 98.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;wj \leq 1.86 \cdot 10^{-6}:\\ \;\;\;\;\left(\left(x + -2 \cdot \left(wj \cdot x\right)\right) + wj \cdot wj\right) - {wj}^{3}\\ \mathbf{else}:\\ \;\;\;\;wj + \frac{x \cdot e^{-wj} - wj}{wj + 1}\\ \end{array} \]

Alternative 3: 97.7% accurate, 2.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;wj \leq 8.6 \cdot 10^{-9}:\\ \;\;\;\;{wj}^{2} + \left(x + -2 \cdot \left(wj \cdot x\right)\right)\\ \mathbf{else}:\\ \;\;\;\;wj + \frac{x \cdot e^{-wj} - wj}{wj + 1}\\ \end{array} \end{array} \]

FPCore

(FPCore (wj x)
 :precision binary64
 (if (<= wj 8.6e-9)
   (+ (pow wj 2.0) (+ x (* -2.0 (* wj x))))
   (+ wj (/ (- (* x (exp (- wj))) wj) (+ wj 1.0)))))

C

double code(double wj, double x) {
	double tmp;
	if (wj <= 8.6e-9) {
		tmp = pow(wj, 2.0) + (x + (-2.0 * (wj * x)));
	} else {
		tmp = wj + (((x * exp(-wj)) - wj) / (wj + 1.0));
	}
	return tmp;
}

Fortran

real(8) function code(wj, x)
    real(8), intent (in) :: wj
    real(8), intent (in) :: x
    real(8) :: tmp
    if (wj <= 8.6d-9) then
        tmp = (wj ** 2.0d0) + (x + ((-2.0d0) * (wj * x)))
    else
        tmp = wj + (((x * exp(-wj)) - wj) / (wj + 1.0d0))
    end if
    code = tmp
end function

Java

public static double code(double wj, double x) {
	double tmp;
	if (wj <= 8.6e-9) {
		tmp = Math.pow(wj, 2.0) + (x + (-2.0 * (wj * x)));
	} else {
		tmp = wj + (((x * Math.exp(-wj)) - wj) / (wj + 1.0));
	}
	return tmp;
}

Python

def code(wj, x):
	tmp = 0
	if wj <= 8.6e-9:
		tmp = math.pow(wj, 2.0) + (x + (-2.0 * (wj * x)))
	else:
		tmp = wj + (((x * math.exp(-wj)) - wj) / (wj + 1.0))
	return tmp

Julia

function code(wj, x)
	tmp = 0.0
	if (wj <= 8.6e-9)
		tmp = Float64((wj ^ 2.0) + Float64(x + Float64(-2.0 * Float64(wj * x))));
	else
		tmp = Float64(wj + Float64(Float64(Float64(x * exp(Float64(-wj))) - wj) / Float64(wj + 1.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(wj, x)
	tmp = 0.0;
	if (wj <= 8.6e-9)
		tmp = (wj ^ 2.0) + (x + (-2.0 * (wj * x)));
	else
		tmp = wj + (((x * exp(-wj)) - wj) / (wj + 1.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[wj_, x_] := If[LessEqual[wj, 8.6e-9], N[(N[Power[wj, 2.0], $MachinePrecision] + N[(x + N[(-2.0 * N[(wj * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(wj + N[(N[(N[(x * N[Exp[(-wj)], $MachinePrecision]), $MachinePrecision] - wj), $MachinePrecision] / N[(wj + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if wj < 8.59999999999999925e-9

   1. Initial program 79.2%

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

      1. sub-neg 79.2%

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

      2. div-sub 79.2%

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

      3. sub-neg 79.2%

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

      4. +-commutative 79.2%

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

      5. distribute-neg-in 79.2%

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

      6. remove-double-neg 79.2%

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

      7. sub-neg 79.2%

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

      8. div-sub 79.2%

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

      9. distribute-rgt1-in 79.6%

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

      10. associate-/l/ 79.6%

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

   3. Simplified 79.6%

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

   4. Taylor expanded in wj around 0 98.7%

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

   5. Taylor expanded in x around 0 98.7%

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

      1. unpow2 98.7%

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

   7. Simplified 98.7%

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

   8. Taylor expanded in x around 0 99.1%

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

   9. Taylor expanded in wj around 0 99.0%

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

   if 8.59999999999999925e-9 < wj

   1. Initial program 55.0%

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

      1. sub-neg 55.0%

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

      2. div-sub 55.0%

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

      3. sub-neg 55.0%

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

      4. +-commutative 55.0%

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

      5. distribute-neg-in 55.0%

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

      6. remove-double-neg 55.0%

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

      7. sub-neg 55.0%

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

      8. div-sub 55.0%

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

      9. distribute-rgt1-in 55.0%

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

      10. associate-/l/ 55.3%

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

   3. Simplified 92.8%

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

      1. clear-num 92.8%

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

      2. associate-/r/ 92.8%

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

      3. rec-exp 93.0%

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

   5. Applied egg-rr 93.0%

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

4. Final simplification 98.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;wj \leq 8.6 \cdot 10^{-9}:\\ \;\;\;\;{wj}^{2} + \left(x + -2 \cdot \left(wj \cdot x\right)\right)\\ \mathbf{else}:\\ \;\;\;\;wj + \frac{x \cdot e^{-wj} - wj}{wj + 1}\\ \end{array} \]

Alternative 4: 97.7% accurate, 2.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;wj \leq 8.6 \cdot 10^{-9}:\\ \;\;\;\;{wj}^{2} + \left(x + -2 \cdot \left(wj \cdot x\right)\right)\\ \mathbf{else}:\\ \;\;\;\;wj + \frac{\frac{x}{e^{wj}} - wj}{wj + 1}\\ \end{array} \end{array} \]

FPCore

(FPCore (wj x)
 :precision binary64
 (if (<= wj 8.6e-9)
   (+ (pow wj 2.0) (+ x (* -2.0 (* wj x))))
   (+ wj (/ (- (/ x (exp wj)) wj) (+ wj 1.0)))))

C

double code(double wj, double x) {
	double tmp;
	if (wj <= 8.6e-9) {
		tmp = pow(wj, 2.0) + (x + (-2.0 * (wj * x)));
	} else {
		tmp = wj + (((x / exp(wj)) - wj) / (wj + 1.0));
	}
	return tmp;
}

Fortran

real(8) function code(wj, x)
    real(8), intent (in) :: wj
    real(8), intent (in) :: x
    real(8) :: tmp
    if (wj <= 8.6d-9) then
        tmp = (wj ** 2.0d0) + (x + ((-2.0d0) * (wj * x)))
    else
        tmp = wj + (((x / exp(wj)) - wj) / (wj + 1.0d0))
    end if
    code = tmp
end function

Java

public static double code(double wj, double x) {
	double tmp;
	if (wj <= 8.6e-9) {
		tmp = Math.pow(wj, 2.0) + (x + (-2.0 * (wj * x)));
	} else {
		tmp = wj + (((x / Math.exp(wj)) - wj) / (wj + 1.0));
	}
	return tmp;
}

Python

def code(wj, x):
	tmp = 0
	if wj <= 8.6e-9:
		tmp = math.pow(wj, 2.0) + (x + (-2.0 * (wj * x)))
	else:
		tmp = wj + (((x / math.exp(wj)) - wj) / (wj + 1.0))
	return tmp

Julia

function code(wj, x)
	tmp = 0.0
	if (wj <= 8.6e-9)
		tmp = Float64((wj ^ 2.0) + Float64(x + Float64(-2.0 * Float64(wj * x))));
	else
		tmp = Float64(wj + Float64(Float64(Float64(x / exp(wj)) - wj) / Float64(wj + 1.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(wj, x)
	tmp = 0.0;
	if (wj <= 8.6e-9)
		tmp = (wj ^ 2.0) + (x + (-2.0 * (wj * x)));
	else
		tmp = wj + (((x / exp(wj)) - wj) / (wj + 1.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[wj_, x_] := If[LessEqual[wj, 8.6e-9], N[(N[Power[wj, 2.0], $MachinePrecision] + N[(x + N[(-2.0 * N[(wj * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(wj + N[(N[(N[(x / N[Exp[wj], $MachinePrecision]), $MachinePrecision] - wj), $MachinePrecision] / N[(wj + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if wj < 8.59999999999999925e-9

   1. Initial program 79.2%

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

      1. sub-neg 79.2%

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

      2. div-sub 79.2%

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

      3. sub-neg 79.2%

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

      4. +-commutative 79.2%

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

      5. distribute-neg-in 79.2%

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

      6. remove-double-neg 79.2%

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

      7. sub-neg 79.2%

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

      8. div-sub 79.2%

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

      9. distribute-rgt1-in 79.6%

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

      10. associate-/l/ 79.6%

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

   3. Simplified 79.6%

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

   4. Taylor expanded in wj around 0 98.7%

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

   5. Taylor expanded in x around 0 98.7%

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

      1. unpow2 98.7%

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

   7. Simplified 98.7%

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

   8. Taylor expanded in x around 0 99.1%

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

   9. Taylor expanded in wj around 0 99.0%

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

   if 8.59999999999999925e-9 < wj

   1. Initial program 55.0%

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

      1. sub-neg 55.0%

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

      2. div-sub 55.0%

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

      3. sub-neg 55.0%

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

      4. +-commutative 55.0%

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

      5. distribute-neg-in 55.0%

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

      6. remove-double-neg 55.0%

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

      7. sub-neg 55.0%

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

      8. div-sub 55.0%

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

      9. distribute-rgt1-in 55.0%

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

      10. associate-/l/ 55.3%

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

   3. Simplified 92.8%

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

4. Final simplification 98.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;wj \leq 8.6 \cdot 10^{-9}:\\ \;\;\;\;{wj}^{2} + \left(x + -2 \cdot \left(wj \cdot x\right)\right)\\ \mathbf{else}:\\ \;\;\;\;wj + \frac{\frac{x}{e^{wj}} - wj}{wj + 1}\\ \end{array} \]

Alternative 5: 97.0% accurate, 2.8× speedup

Math

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

FPCore

(FPCore (wj x)
 :precision binary64
 (if (<= wj 1.85e-6)
   (+ (pow wj 2.0) (+ x (* -2.0 (* wj x))))
   (- wj (/ wj (+ wj 1.0)))))

C

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

Fortran

real(8) function code(wj, x)
    real(8), intent (in) :: wj
    real(8), intent (in) :: x
    real(8) :: tmp
    if (wj <= 1.85d-6) then
        tmp = (wj ** 2.0d0) + (x + ((-2.0d0) * (wj * x)))
    else
        tmp = wj - (wj / (wj + 1.0d0))
    end if
    code = tmp
end function

Java

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

Python

def code(wj, x):
	tmp = 0
	if wj <= 1.85e-6:
		tmp = math.pow(wj, 2.0) + (x + (-2.0 * (wj * x)))
	else:
		tmp = wj - (wj / (wj + 1.0))
	return tmp

Julia

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

MATLAB

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

Wolfram

code[wj_, x_] := If[LessEqual[wj, 1.85e-6], N[(N[Power[wj, 2.0], $MachinePrecision] + N[(x + N[(-2.0 * N[(wj * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(wj - N[(wj / N[(wj + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if wj < 1.8500000000000001e-6

   1. Initial program 79.2%

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

      1. sub-neg 79.2%

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

      2. div-sub 79.2%

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

      3. sub-neg 79.2%

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

      4. +-commutative 79.2%

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

      5. distribute-neg-in 79.2%

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

      6. remove-double-neg 79.2%

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

      7. sub-neg 79.2%

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

      8. div-sub 79.2%

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

      9. distribute-rgt1-in 79.6%

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

      10. associate-/l/ 79.6%

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

   3. Simplified 79.6%

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

   4. Taylor expanded in wj around 0 98.7%

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

   5. Taylor expanded in x around 0 98.7%

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

      1. unpow2 98.7%

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

   7. Simplified 98.7%

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

   8. Taylor expanded in x around 0 99.1%

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

   9. Taylor expanded in wj around 0 99.0%

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

   if 1.8500000000000001e-6 < wj

   1. Initial program 55.0%

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

      1. sub-neg 55.0%

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

      2. div-sub 55.0%

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

      3. sub-neg 55.0%

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

      4. +-commutative 55.0%

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

      5. distribute-neg-in 55.0%

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

      6. remove-double-neg 55.0%

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

      7. sub-neg 55.0%

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

      8. div-sub 55.0%

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

      9. distribute-rgt1-in 55.0%

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

      10. associate-/l/ 55.3%

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

   3. Simplified 92.8%

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

   4. Taylor expanded in x around 0 80.6%

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

      1. +-commutative 80.6%

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

   6. Simplified 80.6%

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

4. Final simplification 98.4%

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

Alternative 6: 87.6% accurate, 2.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;wj \leq 2.05 \cdot 10^{-7}:\\ \;\;\;\;\frac{x}{e^{wj} \cdot \left(wj + 1\right)}\\ \mathbf{else}:\\ \;\;\;\;wj - \frac{wj}{wj + 1}\\ \end{array} \end{array} \]

FPCore

(FPCore (wj x)
 :precision binary64
 (if (<= wj 2.05e-7) (/ x (* (exp wj) (+ wj 1.0))) (- wj (/ wj (+ wj 1.0)))))

C

double code(double wj, double x) {
	double tmp;
	if (wj <= 2.05e-7) {
		tmp = x / (exp(wj) * (wj + 1.0));
	} else {
		tmp = wj - (wj / (wj + 1.0));
	}
	return tmp;
}

Fortran

real(8) function code(wj, x)
    real(8), intent (in) :: wj
    real(8), intent (in) :: x
    real(8) :: tmp
    if (wj <= 2.05d-7) then
        tmp = x / (exp(wj) * (wj + 1.0d0))
    else
        tmp = wj - (wj / (wj + 1.0d0))
    end if
    code = tmp
end function

Java

public static double code(double wj, double x) {
	double tmp;
	if (wj <= 2.05e-7) {
		tmp = x / (Math.exp(wj) * (wj + 1.0));
	} else {
		tmp = wj - (wj / (wj + 1.0));
	}
	return tmp;
}

Python

def code(wj, x):
	tmp = 0
	if wj <= 2.05e-7:
		tmp = x / (math.exp(wj) * (wj + 1.0))
	else:
		tmp = wj - (wj / (wj + 1.0))
	return tmp

Julia

function code(wj, x)
	tmp = 0.0
	if (wj <= 2.05e-7)
		tmp = Float64(x / Float64(exp(wj) * Float64(wj + 1.0)));
	else
		tmp = Float64(wj - Float64(wj / Float64(wj + 1.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(wj, x)
	tmp = 0.0;
	if (wj <= 2.05e-7)
		tmp = x / (exp(wj) * (wj + 1.0));
	else
		tmp = wj - (wj / (wj + 1.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[wj_, x_] := If[LessEqual[wj, 2.05e-7], N[(x / N[(N[Exp[wj], $MachinePrecision] * N[(wj + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(wj - N[(wj / N[(wj + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if wj < 2.05e-7

   1. Initial program 79.2%

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

      1. sub-neg 79.2%

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

      2. div-sub 79.2%

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

      3. sub-neg 79.2%

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

      4. +-commutative 79.2%

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

      5. distribute-neg-in 79.2%

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

      6. remove-double-neg 79.2%

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

      7. sub-neg 79.2%

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

      8. div-sub 79.2%

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

      9. distribute-rgt1-in 79.6%

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

      10. associate-/l/ 79.6%

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

   3. Simplified 79.6%

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

   4. Taylor expanded in x around inf 91.9%

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

   if 2.05e-7 < wj

   1. Initial program 55.0%

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

      1. sub-neg 55.0%

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

      2. div-sub 55.0%

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

      3. sub-neg 55.0%

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

      4. +-commutative 55.0%

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

      5. distribute-neg-in 55.0%

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

      6. remove-double-neg 55.0%

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

      7. sub-neg 55.0%

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

      8. div-sub 55.0%

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

      9. distribute-rgt1-in 55.0%

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

      10. associate-/l/ 55.3%

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

   3. Simplified 92.8%

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

   4. Taylor expanded in x around 0 80.6%

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

      1. +-commutative 80.6%

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

   6. Simplified 80.6%

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

4. Final simplification 91.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;wj \leq 2.05 \cdot 10^{-7}:\\ \;\;\;\;\frac{x}{e^{wj} \cdot \left(wj + 1\right)}\\ \mathbf{else}:\\ \;\;\;\;wj - \frac{wj}{wj + 1}\\ \end{array} \]

Alternative 7: 86.4% accurate, 28.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;wj \leq 8.8 \cdot 10^{-7}:\\ \;\;\;\;\frac{x \cdot \left(1 - wj\right)}{wj + 1}\\ \mathbf{else}:\\ \;\;\;\;wj - \frac{wj}{wj + 1}\\ \end{array} \end{array} \]

FPCore

(FPCore (wj x)
 :precision binary64
 (if (<= wj 8.8e-7) (/ (* x (- 1.0 wj)) (+ wj 1.0)) (- wj (/ wj (+ wj 1.0)))))

C

double code(double wj, double x) {
	double tmp;
	if (wj <= 8.8e-7) {
		tmp = (x * (1.0 - wj)) / (wj + 1.0);
	} else {
		tmp = wj - (wj / (wj + 1.0));
	}
	return tmp;
}

Fortran

real(8) function code(wj, x)
    real(8), intent (in) :: wj
    real(8), intent (in) :: x
    real(8) :: tmp
    if (wj <= 8.8d-7) then
        tmp = (x * (1.0d0 - wj)) / (wj + 1.0d0)
    else
        tmp = wj - (wj / (wj + 1.0d0))
    end if
    code = tmp
end function

Java

public static double code(double wj, double x) {
	double tmp;
	if (wj <= 8.8e-7) {
		tmp = (x * (1.0 - wj)) / (wj + 1.0);
	} else {
		tmp = wj - (wj / (wj + 1.0));
	}
	return tmp;
}

Python

def code(wj, x):
	tmp = 0
	if wj <= 8.8e-7:
		tmp = (x * (1.0 - wj)) / (wj + 1.0)
	else:
		tmp = wj - (wj / (wj + 1.0))
	return tmp

Julia

function code(wj, x)
	tmp = 0.0
	if (wj <= 8.8e-7)
		tmp = Float64(Float64(x * Float64(1.0 - wj)) / Float64(wj + 1.0));
	else
		tmp = Float64(wj - Float64(wj / Float64(wj + 1.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(wj, x)
	tmp = 0.0;
	if (wj <= 8.8e-7)
		tmp = (x * (1.0 - wj)) / (wj + 1.0);
	else
		tmp = wj - (wj / (wj + 1.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[wj_, x_] := If[LessEqual[wj, 8.8e-7], N[(N[(x * N[(1.0 - wj), $MachinePrecision]), $MachinePrecision] / N[(wj + 1.0), $MachinePrecision]), $MachinePrecision], N[(wj - N[(wj / N[(wj + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if wj < 8.8000000000000004e-7

   1. Initial program 79.2%

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

      1. sub-neg 79.2%

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

      2. div-sub 79.2%

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

      3. sub-neg 79.2%

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

      4. +-commutative 79.2%

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

      5. distribute-neg-in 79.2%

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

      6. remove-double-neg 79.2%

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

      7. sub-neg 79.2%

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

      8. div-sub 79.2%

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

      9. distribute-rgt1-in 79.6%

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

      10. associate-/l/ 79.6%

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

   3. Simplified 79.6%

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

   4. Taylor expanded in wj around 0 78.8%

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

      1. associate-*r* 78.8%

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

      2. neg-mul-1 78.8%

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

      3. distribute-lft1-in 78.8%

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

      4. +-commutative 78.8%

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

      5. sub-neg 78.8%

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

   6. Simplified 78.8%

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

   7. Taylor expanded in x around -inf 91.3%

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

   if 8.8000000000000004e-7 < wj

   1. Initial program 55.0%

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

      1. sub-neg 55.0%

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

      2. div-sub 55.0%

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

      3. sub-neg 55.0%

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

      4. +-commutative 55.0%

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

      5. distribute-neg-in 55.0%

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

      6. remove-double-neg 55.0%

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

      7. sub-neg 55.0%

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

      8. div-sub 55.0%

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

      9. distribute-rgt1-in 55.0%

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

      10. associate-/l/ 55.3%

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

   3. Simplified 92.8%

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

   4. Taylor expanded in x around 0 80.6%

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

      1. +-commutative 80.6%

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

   6. Simplified 80.6%

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

4. Final simplification 91.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;wj \leq 8.8 \cdot 10^{-7}:\\ \;\;\;\;\frac{x \cdot \left(1 - wj\right)}{wj + 1}\\ \mathbf{else}:\\ \;\;\;\;wj - \frac{wj}{wj + 1}\\ \end{array} \]

Alternative 8: 86.3% accurate, 34.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;wj \leq 1.1 \cdot 10^{-6}:\\ \;\;\;\;x + -2 \cdot \left(wj \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;wj - \frac{wj}{wj + 1}\\ \end{array} \end{array} \]

FPCore

(FPCore (wj x)
 :precision binary64
 (if (<= wj 1.1e-6) (+ x (* -2.0 (* wj x))) (- wj (/ wj (+ wj 1.0)))))

C

double code(double wj, double x) {
	double tmp;
	if (wj <= 1.1e-6) {
		tmp = x + (-2.0 * (wj * x));
	} else {
		tmp = wj - (wj / (wj + 1.0));
	}
	return tmp;
}

Fortran

real(8) function code(wj, x)
    real(8), intent (in) :: wj
    real(8), intent (in) :: x
    real(8) :: tmp
    if (wj <= 1.1d-6) then
        tmp = x + ((-2.0d0) * (wj * x))
    else
        tmp = wj - (wj / (wj + 1.0d0))
    end if
    code = tmp
end function

Java

public static double code(double wj, double x) {
	double tmp;
	if (wj <= 1.1e-6) {
		tmp = x + (-2.0 * (wj * x));
	} else {
		tmp = wj - (wj / (wj + 1.0));
	}
	return tmp;
}

Python

def code(wj, x):
	tmp = 0
	if wj <= 1.1e-6:
		tmp = x + (-2.0 * (wj * x))
	else:
		tmp = wj - (wj / (wj + 1.0))
	return tmp

Julia

function code(wj, x)
	tmp = 0.0
	if (wj <= 1.1e-6)
		tmp = Float64(x + Float64(-2.0 * Float64(wj * x)));
	else
		tmp = Float64(wj - Float64(wj / Float64(wj + 1.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(wj, x)
	tmp = 0.0;
	if (wj <= 1.1e-6)
		tmp = x + (-2.0 * (wj * x));
	else
		tmp = wj - (wj / (wj + 1.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[wj_, x_] := If[LessEqual[wj, 1.1e-6], N[(x + N[(-2.0 * N[(wj * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(wj - N[(wj / N[(wj + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if wj < 1.1000000000000001e-6

   1. Initial program 79.2%

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

      1. sub-neg 79.2%

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

      2. div-sub 79.2%

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

      3. sub-neg 79.2%

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

      4. +-commutative 79.2%

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

      5. distribute-neg-in 79.2%

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

      6. remove-double-neg 79.2%

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

      7. sub-neg 79.2%

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

      8. div-sub 79.2%

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

      9. distribute-rgt1-in 79.6%

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

      10. associate-/l/ 79.6%

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

   3. Simplified 79.6%

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

   4. Taylor expanded in wj around 0 91.3%

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

   if 1.1000000000000001e-6 < wj

   1. Initial program 55.0%

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

      1. sub-neg 55.0%

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

      2. div-sub 55.0%

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

      3. sub-neg 55.0%

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

      4. +-commutative 55.0%

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

      5. distribute-neg-in 55.0%

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

      6. remove-double-neg 55.0%

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

      7. sub-neg 55.0%

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

      8. div-sub 55.0%

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

      9. distribute-rgt1-in 55.0%

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

      10. associate-/l/ 55.3%

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

   3. Simplified 92.8%

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

   4. Taylor expanded in x around 0 80.6%

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

      1. +-commutative 80.6%

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

   6. Simplified 80.6%

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

4. Final simplification 91.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;wj \leq 1.1 \cdot 10^{-6}:\\ \;\;\;\;x + -2 \cdot \left(wj \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;wj - \frac{wj}{wj + 1}\\ \end{array} \]

Alternative 9: 84.9% accurate, 44.7× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 78.4%

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

   1. sub-neg 78.4%

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

   2. div-sub 78.4%

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

   3. sub-neg 78.4%

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

   4. +-commutative 78.4%

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

   5. distribute-neg-in 78.4%

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

   6. remove-double-neg 78.4%

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

   7. sub-neg 78.4%

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

   8. div-sub 78.4%

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

   9. distribute-rgt1-in 78.8%

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

   10. associate-/l/ 78.8%

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

3. Simplified 80.0%

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

4. Taylor expanded in wj around 0 88.8%

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

5. Final simplification 88.8%

   \[\leadsto x + -2 \cdot \left(wj \cdot x\right) \]

Alternative 10: 4.5% 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 78.4%

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

   1. sub-neg 78.4%

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

   2. div-sub 78.4%

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

   3. sub-neg 78.4%

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

   4. +-commutative 78.4%

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

   5. distribute-neg-in 78.4%

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

   6. remove-double-neg 78.4%

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

   7. sub-neg 78.4%

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

   8. div-sub 78.4%

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

   9. distribute-rgt1-in 78.8%

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

   10. associate-/l/ 78.8%

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

3. Simplified 80.0%

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

4. Taylor expanded in wj around inf 4.3%

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

5. Final simplification 4.3%

   \[\leadsto wj \]

Alternative 11: 84.3% 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 78.4%

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

   1. sub-neg 78.4%

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

   2. div-sub 78.4%

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

   3. sub-neg 78.4%

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

   4. +-commutative 78.4%

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

   5. distribute-neg-in 78.4%

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

   6. remove-double-neg 78.4%

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

   7. sub-neg 78.4%

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

   8. div-sub 78.4%

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

   9. distribute-rgt1-in 78.8%

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

   10. associate-/l/ 78.8%

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

3. Simplified 80.0%

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

4. Taylor expanded in wj around 0 88.6%

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

5. Final simplification 88.6%

   \[\leadsto x \]

Developer target: 78.6% accurate, 1.5× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Reproduce

herbie shell --seed 2023189 
(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))))))