Jmat.Real.lambertw, newton loop step

Percentage Accurate: 78.4% → 97.2%

Time: 13.3s

Alternatives: 10

Speedup: 313.0×

• could not determine a ground truth

  1. wj = -3.443890054981751e+282
  2. x = -8.322531777261006e+262

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 78.4% 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: 97.2% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x \cdot -4 + x \cdot 1.5\\ t_1 := wj \cdot e^{wj}\\ \mathbf{if}\;wj + \frac{x - t_1}{e^{wj} + t_1} \leq -2 \cdot 10^{+158}:\\ \;\;\;\;wj + \frac{\frac{x}{e^{wj}} - wj}{wj + 1}\\ \mathbf{else}:\\ \;\;\;\;x + \left(-2 \cdot \left(wj \cdot x\right) + \left({wj}^{3} \cdot \left(-1 - \left(x \cdot -3 + \left(-2 \cdot t_0 + x \cdot 0.6666666666666666\right)\right)\right) + {wj}^{2} \cdot \left(1 - t_0\right)\right)\right)\\ \end{array} \end{array} \]

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))) -2e+158)
     (+ wj (/ (- (/ x (exp wj)) wj) (+ wj 1.0)))
     (+
      x
      (+
       (* -2.0 (* wj x))
       (+
        (*
         (pow wj 3.0)
         (- -1.0 (+ (* x -3.0) (+ (* -2.0 t_0) (* x 0.6666666666666666)))))
        (* (pow wj 2.0) (- 1.0 t_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))) <= -2e+158) {
		tmp = wj + (((x / exp(wj)) - wj) / (wj + 1.0));
	} else {
		tmp = x + ((-2.0 * (wj * x)) + ((pow(wj, 3.0) * (-1.0 - ((x * -3.0) + ((-2.0 * t_0) + (x * 0.6666666666666666))))) + (pow(wj, 2.0) * (1.0 - t_0))));
	}
	return tmp;
}

Fortran

real(8) function code(wj, x)
    real(8), intent (in) :: wj
    real(8), intent (in) :: x
    real(8) :: t_0
    real(8) :: 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))) <= (-2d+158)) then
        tmp = wj + (((x / exp(wj)) - wj) / (wj + 1.0d0))
    else
        tmp = x + (((-2.0d0) * (wj * x)) + (((wj ** 3.0d0) * ((-1.0d0) - ((x * (-3.0d0)) + (((-2.0d0) * t_0) + (x * 0.6666666666666666d0))))) + ((wj ** 2.0d0) * (1.0d0 - t_0))))
    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))) <= -2e+158) {
		tmp = wj + (((x / Math.exp(wj)) - wj) / (wj + 1.0));
	} else {
		tmp = x + ((-2.0 * (wj * x)) + ((Math.pow(wj, 3.0) * (-1.0 - ((x * -3.0) + ((-2.0 * t_0) + (x * 0.6666666666666666))))) + (Math.pow(wj, 2.0) * (1.0 - t_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))) <= -2e+158:
		tmp = wj + (((x / math.exp(wj)) - wj) / (wj + 1.0))
	else:
		tmp = x + ((-2.0 * (wj * x)) + ((math.pow(wj, 3.0) * (-1.0 - ((x * -3.0) + ((-2.0 * t_0) + (x * 0.6666666666666666))))) + (math.pow(wj, 2.0) * (1.0 - t_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))) <= -2e+158)
		tmp = Float64(wj + Float64(Float64(Float64(x / exp(wj)) - wj) / Float64(wj + 1.0)));
	else
		tmp = Float64(x + Float64(Float64(-2.0 * Float64(wj * x)) + Float64(Float64((wj ^ 3.0) * Float64(-1.0 - Float64(Float64(x * -3.0) + Float64(Float64(-2.0 * t_0) + Float64(x * 0.6666666666666666))))) + Float64((wj ^ 2.0) * Float64(1.0 - t_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))) <= -2e+158)
		tmp = wj + (((x / exp(wj)) - wj) / (wj + 1.0));
	else
		tmp = x + ((-2.0 * (wj * x)) + (((wj ^ 3.0) * (-1.0 - ((x * -3.0) + ((-2.0 * t_0) + (x * 0.6666666666666666))))) + ((wj ^ 2.0) * (1.0 - t_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], -2e+158], N[(wj + N[(N[(N[(x / N[Exp[wj], $MachinePrecision]), $MachinePrecision] - wj), $MachinePrecision] / N[(wj + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x + N[(N[(-2.0 * N[(wj * x), $MachinePrecision]), $MachinePrecision] + N[(N[(N[Power[wj, 3.0], $MachinePrecision] * N[(-1.0 - N[(N[(x * -3.0), $MachinePrecision] + N[(N[(-2.0 * t$95$0), $MachinePrecision] + N[(x * 0.6666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[Power[wj, 2.0], $MachinePrecision] * N[(1.0 - t$95$0), $MachinePrecision]), $MachinePrecision]), $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))))) < -1.99999999999999991e158

   1. Initial program 96.8%

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

      1. distribute-rgt1-in 99.9%

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

      2. associate-/l/ 100.0%

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

      3. div-sub 96.9%

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

      4. *-commutative 96.9%

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

      5. associate-*r/ 96.9%

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

      6. *-rgt-identity 96.9%

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

      7. associate-*r/ 96.9%

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

      8. exp-neg 96.9%

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

      9. *-commutative 96.9%

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

      10. associate-*r* 96.9%

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

      11. *-commutative 96.9%

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

      12. exp-neg 96.9%

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

      13. lft-mult-inverse 100.0%

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

      14. *-lft-identity 100.0%

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

   3. Simplified 100.0%

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

   if -1.99999999999999991e158 < (-.f64 wj (/.f64 (-.f64 (*.f64 wj (exp.f64 wj)) x) (+.f64 (exp.f64 wj) (*.f64 wj (exp.f64 wj)))))

   1. Initial program 78.8%

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

      1. distribute-rgt1-in 78.8%

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

      2. associate-/l/ 78.8%

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

      3. div-sub 78.8%

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

      4. *-commutative 78.8%

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

      5. associate-*r/ 78.8%

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

      6. *-rgt-identity 78.8%

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

      7. associate-*r/ 78.8%

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

      8. exp-neg 78.8%

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

      9. *-commutative 78.8%

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

      10. associate-*r* 78.7%

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

      11. *-commutative 78.7%

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

      12. exp-neg 78.8%

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

      13. lft-mult-inverse 78.8%

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

      14. *-lft-identity 78.8%

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

   3. Simplified 78.8%

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

   4. Taylor expanded in wj around 0 98.6%

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

4. Final simplification 98.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;wj + \frac{x - wj \cdot e^{wj}}{e^{wj} + wj \cdot e^{wj}} \leq -2 \cdot 10^{+158}:\\ \;\;\;\;wj + \frac{\frac{x}{e^{wj}} - wj}{wj + 1}\\ \mathbf{else}:\\ \;\;\;\;x + \left(-2 \cdot \left(wj \cdot x\right) + \left({wj}^{3} \cdot \left(-1 - \left(x \cdot -3 + \left(-2 \cdot \left(x \cdot -4 + x \cdot 1.5\right) + x \cdot 0.6666666666666666\right)\right)\right) + {wj}^{2} \cdot \left(1 - \left(x \cdot -4 + x \cdot 1.5\right)\right)\right)\right)\\ \end{array} \]

Alternative 2: 97.8% accurate, 2.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;wj \leq -3.35 \cdot 10^{-9}:\\ \;\;\;\;wj + \frac{\frac{x}{e^{wj}} - wj}{wj + 1}\\ \mathbf{else}:\\ \;\;\;\;x + \left(-2 \cdot \left(wj \cdot x\right) + {wj}^{2}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (wj x)
 :precision binary64
 (if (<= wj -3.35e-9)
   (+ wj (/ (- (/ x (exp wj)) wj) (+ wj 1.0)))
   (+ x (+ (* -2.0 (* wj x)) (pow wj 2.0)))))

C

double code(double wj, double x) {
	double tmp;
	if (wj <= -3.35e-9) {
		tmp = wj + (((x / exp(wj)) - wj) / (wj + 1.0));
	} else {
		tmp = x + ((-2.0 * (wj * x)) + pow(wj, 2.0));
	}
	return tmp;
}

Fortran

real(8) function code(wj, x)
    real(8), intent (in) :: wj
    real(8), intent (in) :: x
    real(8) :: tmp
    if (wj <= (-3.35d-9)) then
        tmp = wj + (((x / exp(wj)) - wj) / (wj + 1.0d0))
    else
        tmp = x + (((-2.0d0) * (wj * x)) + (wj ** 2.0d0))
    end if
    code = tmp
end function

Java

public static double code(double wj, double x) {
	double tmp;
	if (wj <= -3.35e-9) {
		tmp = wj + (((x / Math.exp(wj)) - wj) / (wj + 1.0));
	} else {
		tmp = x + ((-2.0 * (wj * x)) + Math.pow(wj, 2.0));
	}
	return tmp;
}

Python

def code(wj, x):
	tmp = 0
	if wj <= -3.35e-9:
		tmp = wj + (((x / math.exp(wj)) - wj) / (wj + 1.0))
	else:
		tmp = x + ((-2.0 * (wj * x)) + math.pow(wj, 2.0))
	return tmp

Julia

function code(wj, x)
	tmp = 0.0
	if (wj <= -3.35e-9)
		tmp = Float64(wj + Float64(Float64(Float64(x / exp(wj)) - wj) / Float64(wj + 1.0)));
	else
		tmp = Float64(x + Float64(Float64(-2.0 * Float64(wj * x)) + (wj ^ 2.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(wj, x)
	tmp = 0.0;
	if (wj <= -3.35e-9)
		tmp = wj + (((x / exp(wj)) - wj) / (wj + 1.0));
	else
		tmp = x + ((-2.0 * (wj * x)) + (wj ^ 2.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[wj_, x_] := If[LessEqual[wj, -3.35e-9], N[(wj + N[(N[(N[(x / N[Exp[wj], $MachinePrecision]), $MachinePrecision] - wj), $MachinePrecision] / N[(wj + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x + N[(N[(-2.0 * N[(wj * x), $MachinePrecision]), $MachinePrecision] + N[Power[wj, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if wj < -3.34999999999999981e-9

   1. Initial program 81.3%

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

      1. distribute-rgt1-in 98.2%

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

      2. associate-/l/ 98.2%

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

      3. div-sub 81.6%

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

      4. *-commutative 81.6%

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

      5. associate-*r/ 81.6%

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

      6. *-rgt-identity 81.6%

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

      7. associate-*r/ 81.6%

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

      8. exp-neg 81.6%

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

      9. *-commutative 81.6%

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

      10. associate-*r* 81.6%

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

      11. *-commutative 81.6%

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

      12. exp-neg 81.6%

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

      13. lft-mult-inverse 98.2%

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

      14. *-lft-identity 98.2%

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

   3. Simplified 98.2%

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

   if -3.34999999999999981e-9 < wj

   1. Initial program 81.1%

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

      1. distribute-rgt1-in 81.1%

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

      2. associate-/l/ 81.1%

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

      3. div-sub 81.1%

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

      4. *-commutative 81.1%

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

      5. associate-*r/ 81.1%

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

      6. *-rgt-identity 81.1%

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

      7. associate-*r/ 81.1%

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

      8. exp-neg 81.1%

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

      9. *-commutative 81.1%

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

      10. associate-*r* 81.1%

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

      11. *-commutative 81.1%

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

      12. exp-neg 81.1%

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

      13. lft-mult-inverse 81.1%

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

      14. *-lft-identity 81.1%

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

   3. Simplified 81.1%

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

   4. Taylor expanded in wj around 0 98.3%

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

   5. Taylor expanded in x around 0 98.7%

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

4. Final simplification 98.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;wj \leq -3.35 \cdot 10^{-9}:\\ \;\;\;\;wj + \frac{\frac{x}{e^{wj}} - wj}{wj + 1}\\ \mathbf{else}:\\ \;\;\;\;x + \left(-2 \cdot \left(wj \cdot x\right) + {wj}^{2}\right)\\ \end{array} \]

Alternative 3: 95.8% accurate, 2.8× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 81.1%

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

   1. distribute-rgt1-in 81.5%

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

   2. associate-/l/ 81.5%

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

   3. div-sub 81.1%

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

   4. *-commutative 81.1%

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

   5. associate-*r/ 81.1%

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

   6. *-rgt-identity 81.1%

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

   7. associate-*r/ 81.1%

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

   8. exp-neg 81.1%

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

   9. *-commutative 81.1%

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

   10. associate-*r* 81.1%

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

   11. *-commutative 81.1%

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

   12. exp-neg 81.1%

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

   13. lft-mult-inverse 81.5%

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

   14. *-lft-identity 81.5%

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

3. Simplified 81.5%

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

4. Taylor expanded in wj around 0 96.8%

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

5. Taylor expanded in x around 0 97.1%

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

6. Final simplification 97.1%

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

Alternative 4: 95.2% accurate, 3.0× speedup

Math

\[\begin{array}{l} \\ x + {wj}^{2} \end{array} \]

FPCore

(FPCore (wj x) :precision binary64 (+ x (pow wj 2.0)))

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 81.1%

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

   1. distribute-rgt1-in 81.5%

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

   2. associate-/l/ 81.5%

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

   3. div-sub 81.1%

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

   4. *-commutative 81.1%

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

   5. associate-*r/ 81.1%

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

   6. *-rgt-identity 81.1%

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

   7. associate-*r/ 81.1%

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

   8. exp-neg 81.1%

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

   9. *-commutative 81.1%

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

   10. associate-*r* 81.1%

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

   11. *-commutative 81.1%

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

   12. exp-neg 81.1%

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

   13. lft-mult-inverse 81.5%

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

   14. *-lft-identity 81.5%

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

3. Simplified 81.5%

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

4. Taylor expanded in wj around 0 96.8%

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

5. Taylor expanded in x around 0 97.1%

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

6. Taylor expanded in wj around inf 96.7%

   \[\leadsto x + \color{blue}{{wj}^{2}} \]

7. Final simplification 96.7%

   \[\leadsto x + {wj}^{2} \]

Alternative 5: 87.0% accurate, 20.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;wj \leq 2.05 \cdot 10^{-17}:\\ \;\;\;\;\frac{x}{1 + wj \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;wj + \frac{\left(x - wj \cdot x\right) - wj}{wj + 1}\\ \end{array} \end{array} \]

FPCore

(FPCore (wj x)
 :precision binary64
 (if (<= wj 2.05e-17)
   (/ x (+ 1.0 (* wj 2.0)))
   (+ wj (/ (- (- x (* wj x)) wj) (+ wj 1.0)))))

C

double code(double wj, double x) {
	double tmp;
	if (wj <= 2.05e-17) {
		tmp = x / (1.0 + (wj * 2.0));
	} else {
		tmp = wj + (((x - (wj * x)) - 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-17) then
        tmp = x / (1.0d0 + (wj * 2.0d0))
    else
        tmp = wj + (((x - (wj * x)) - 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-17) {
		tmp = x / (1.0 + (wj * 2.0));
	} else {
		tmp = wj + (((x - (wj * x)) - wj) / (wj + 1.0));
	}
	return tmp;
}

Python

def code(wj, x):
	tmp = 0
	if wj <= 2.05e-17:
		tmp = x / (1.0 + (wj * 2.0))
	else:
		tmp = wj + (((x - (wj * x)) - wj) / (wj + 1.0))
	return tmp

Julia

function code(wj, x)
	tmp = 0.0
	if (wj <= 2.05e-17)
		tmp = Float64(x / Float64(1.0 + Float64(wj * 2.0)));
	else
		tmp = Float64(wj + Float64(Float64(Float64(x - Float64(wj * x)) - wj) / Float64(wj + 1.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(wj, x)
	tmp = 0.0;
	if (wj <= 2.05e-17)
		tmp = x / (1.0 + (wj * 2.0));
	else
		tmp = wj + (((x - (wj * x)) - wj) / (wj + 1.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[wj_, x_] := If[LessEqual[wj, 2.05e-17], N[(x / N[(1.0 + N[(wj * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(wj + N[(N[(N[(x - N[(wj * x), $MachinePrecision]), $MachinePrecision] - wj), $MachinePrecision] / N[(wj + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if wj < 2.05e-17

   1. Initial program 82.1%

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

      1. distribute-rgt1-in 82.5%

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

      2. associate-/l/ 82.6%

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

      3. div-sub 82.1%

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

      4. *-commutative 82.1%

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

      5. associate-*r/ 82.1%

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

      6. *-rgt-identity 82.1%

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

      7. associate-*r/ 82.1%

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

      8. exp-neg 82.1%

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

      9. *-commutative 82.1%

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

      10. associate-*r* 82.1%

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

      11. *-commutative 82.1%

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

      12. exp-neg 82.1%

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

      13. lft-mult-inverse 82.6%

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

      14. *-lft-identity 82.6%

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

   3. Simplified 82.6%

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

   4. Taylor expanded in x around inf 89.0%

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

      1. +-commutative 89.0%

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

   6. Simplified 89.0%

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

   7. Taylor expanded in wj around 0 87.6%

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

      1. *-commutative 87.6%

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

   9. Simplified 87.6%

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

   if 2.05e-17 < wj

   1. Initial program 60.5%

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

      1. distribute-rgt1-in 59.9%

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

      2. associate-/l/ 59.9%

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

      3. div-sub 59.9%

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

      4. *-commutative 59.9%

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

      5. associate-*r/ 59.9%

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

      6. *-rgt-identity 59.9%

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

      7. associate-*r/ 59.7%

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

      8. exp-neg 59.9%

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

      9. *-commutative 59.9%

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

      10. associate-*r* 59.4%

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

      11. *-commutative 59.4%

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

      12. exp-neg 59.9%

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

      13. lft-mult-inverse 59.9%

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

      14. *-lft-identity 59.9%

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

   3. Simplified 59.9%

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

   4. Taylor expanded in wj around 0 59.9%

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

      1. *-commutative 59.9%

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

      2. associate-*l* 59.9%

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

      3. *-commutative 59.9%

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

      4. mul-1-neg 59.9%

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

   6. Simplified 59.9%

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

4. Final simplification 86.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;wj \leq 2.05 \cdot 10^{-17}:\\ \;\;\;\;\frac{x}{1 + wj \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;wj + \frac{\left(x - wj \cdot x\right) - wj}{wj + 1}\\ \end{array} \]

Alternative 6: 85.0% accurate, 34.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;wj \leq 1.3 \cdot 10^{-19}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;wj \cdot \left(wj + x \cdot -2\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (wj x)
 :precision binary64
 (if (<= wj 1.3e-19) x (* wj (+ wj (* x -2.0)))))

C

double code(double wj, double x) {
	double tmp;
	if (wj <= 1.3e-19) {
		tmp = x;
	} else {
		tmp = wj * (wj + (x * -2.0));
	}
	return tmp;
}

Fortran

real(8) function code(wj, x)
    real(8), intent (in) :: wj
    real(8), intent (in) :: x
    real(8) :: tmp
    if (wj <= 1.3d-19) then
        tmp = x
    else
        tmp = wj * (wj + (x * (-2.0d0)))
    end if
    code = tmp
end function

Java

public static double code(double wj, double x) {
	double tmp;
	if (wj <= 1.3e-19) {
		tmp = x;
	} else {
		tmp = wj * (wj + (x * -2.0));
	}
	return tmp;
}

Python

def code(wj, x):
	tmp = 0
	if wj <= 1.3e-19:
		tmp = x
	else:
		tmp = wj * (wj + (x * -2.0))
	return tmp

Julia

function code(wj, x)
	tmp = 0.0
	if (wj <= 1.3e-19)
		tmp = x;
	else
		tmp = Float64(wj * Float64(wj + Float64(x * -2.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(wj, x)
	tmp = 0.0;
	if (wj <= 1.3e-19)
		tmp = x;
	else
		tmp = wj * (wj + (x * -2.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[wj_, x_] := If[LessEqual[wj, 1.3e-19], x, N[(wj * N[(wj + N[(x * -2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if wj < 1.30000000000000006e-19

   1. Initial program 82.5%

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

      1. distribute-rgt1-in 82.9%

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

      2. associate-/l/ 82.9%

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

      3. div-sub 82.5%

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

      4. *-commutative 82.5%

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

      5. associate-*r/ 82.5%

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

      6. *-rgt-identity 82.5%

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

      7. associate-*r/ 82.5%

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

      8. exp-neg 82.5%

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

      9. *-commutative 82.5%

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

      10. associate-*r* 82.5%

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

      11. *-commutative 82.5%

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

      12. exp-neg 82.5%

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

      13. lft-mult-inverse 82.9%

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

      14. *-lft-identity 82.9%

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

   3. Simplified 82.9%

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

   4. Taylor expanded in wj around 0 87.6%

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

   if 1.30000000000000006e-19 < wj

   1. Initial program 56.1%

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

      1. distribute-rgt1-in 55.6%

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

      2. associate-/l/ 55.5%

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

      3. div-sub 55.5%

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

      4. *-commutative 55.5%

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

      5. associate-*r/ 55.5%

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

      6. *-rgt-identity 55.5%

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

      7. associate-*r/ 55.4%

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

      8. exp-neg 55.5%

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

      9. *-commutative 55.5%

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

      10. associate-*r* 55.1%

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

      11. *-commutative 55.1%

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

      12. exp-neg 55.5%

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

      13. lft-mult-inverse 55.5%

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

      14. *-lft-identity 55.5%

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

   3. Simplified 55.5%

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

   4. Taylor expanded in wj around 0 76.4%

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

   5. Taylor expanded in x around 0 76.4%

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

   6. Taylor expanded in wj around inf 54.3%

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

      1. +-commutative 54.3%

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

      2. unpow2 54.3%

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

      3. associate-*r* 54.3%

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

      4. *-commutative 54.3%

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

      5. associate-*r* 54.3%

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

      6. distribute-rgt-out 54.3%

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

      7. *-commutative 54.3%

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

   8. Simplified 54.3%

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

4. Final simplification 85.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;wj \leq 1.3 \cdot 10^{-19}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;wj \cdot \left(wj + x \cdot -2\right)\\ \end{array} \]

Alternative 7: 85.4% accurate, 34.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;wj \leq 4.7 \cdot 10^{-21}:\\ \;\;\;\;x + x \cdot \left(wj \cdot -2\right)\\ \mathbf{else}:\\ \;\;\;\;wj \cdot \left(wj + x \cdot -2\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (wj x)
 :precision binary64
 (if (<= wj 4.7e-21) (+ x (* x (* wj -2.0))) (* wj (+ wj (* x -2.0)))))

C

double code(double wj, double x) {
	double tmp;
	if (wj <= 4.7e-21) {
		tmp = x + (x * (wj * -2.0));
	} else {
		tmp = wj * (wj + (x * -2.0));
	}
	return tmp;
}

Fortran

real(8) function code(wj, x)
    real(8), intent (in) :: wj
    real(8), intent (in) :: x
    real(8) :: tmp
    if (wj <= 4.7d-21) then
        tmp = x + (x * (wj * (-2.0d0)))
    else
        tmp = wj * (wj + (x * (-2.0d0)))
    end if
    code = tmp
end function

Java

public static double code(double wj, double x) {
	double tmp;
	if (wj <= 4.7e-21) {
		tmp = x + (x * (wj * -2.0));
	} else {
		tmp = wj * (wj + (x * -2.0));
	}
	return tmp;
}

Python

def code(wj, x):
	tmp = 0
	if wj <= 4.7e-21:
		tmp = x + (x * (wj * -2.0))
	else:
		tmp = wj * (wj + (x * -2.0))
	return tmp

Julia

function code(wj, x)
	tmp = 0.0
	if (wj <= 4.7e-21)
		tmp = Float64(x + Float64(x * Float64(wj * -2.0)));
	else
		tmp = Float64(wj * Float64(wj + Float64(x * -2.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(wj, x)
	tmp = 0.0;
	if (wj <= 4.7e-21)
		tmp = x + (x * (wj * -2.0));
	else
		tmp = wj * (wj + (x * -2.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[wj_, x_] := If[LessEqual[wj, 4.7e-21], N[(x + N[(x * N[(wj * -2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(wj * N[(wj + N[(x * -2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if wj < 4.7000000000000003e-21

   1. Initial program 82.5%

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

      1. distribute-rgt1-in 82.9%

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

      2. associate-/l/ 82.9%

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

      3. div-sub 82.5%

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

      4. *-commutative 82.5%

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

      5. associate-*r/ 82.5%

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

      6. *-rgt-identity 82.5%

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

      7. associate-*r/ 82.5%

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

      8. exp-neg 82.5%

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

      9. *-commutative 82.5%

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

      10. associate-*r* 82.5%

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

      11. *-commutative 82.5%

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

      12. exp-neg 82.5%

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

      13. lft-mult-inverse 82.9%

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

      14. *-lft-identity 82.9%

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

   3. Simplified 82.9%

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

   4. Taylor expanded in wj around 0 87.9%

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

      1. associate-*r* 87.9%

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

      2. *-commutative 87.9%

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

   6. Simplified 87.9%

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

   if 4.7000000000000003e-21 < wj

   1. Initial program 56.1%

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

      1. distribute-rgt1-in 55.6%

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

      2. associate-/l/ 55.5%

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

      3. div-sub 55.5%

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

      4. *-commutative 55.5%

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

      5. associate-*r/ 55.5%

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

      6. *-rgt-identity 55.5%

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

      7. associate-*r/ 55.4%

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

      8. exp-neg 55.5%

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

      9. *-commutative 55.5%

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

      10. associate-*r* 55.1%

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

      11. *-commutative 55.1%

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

      12. exp-neg 55.5%

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

      13. lft-mult-inverse 55.5%

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

      14. *-lft-identity 55.5%

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

   3. Simplified 55.5%

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

   4. Taylor expanded in wj around 0 76.4%

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

   5. Taylor expanded in x around 0 76.4%

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

   6. Taylor expanded in wj around inf 54.3%

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

      1. +-commutative 54.3%

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

      2. unpow2 54.3%

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

      3. associate-*r* 54.3%

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

      4. *-commutative 54.3%

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

      5. associate-*r* 54.3%

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

      6. distribute-rgt-out 54.3%

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

      7. *-commutative 54.3%

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

   8. Simplified 54.3%

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

4. Final simplification 86.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;wj \leq 4.7 \cdot 10^{-21}:\\ \;\;\;\;x + x \cdot \left(wj \cdot -2\right)\\ \mathbf{else}:\\ \;\;\;\;wj \cdot \left(wj + x \cdot -2\right)\\ \end{array} \]

Alternative 8: 85.4% accurate, 34.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;wj \leq 6.2 \cdot 10^{-20}:\\ \;\;\;\;\frac{x}{1 + wj \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;wj \cdot \left(wj + x \cdot -2\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (wj x)
 :precision binary64
 (if (<= wj 6.2e-20) (/ x (+ 1.0 (* wj 2.0))) (* wj (+ wj (* x -2.0)))))

C

double code(double wj, double x) {
	double tmp;
	if (wj <= 6.2e-20) {
		tmp = x / (1.0 + (wj * 2.0));
	} else {
		tmp = wj * (wj + (x * -2.0));
	}
	return tmp;
}

Fortran

real(8) function code(wj, x)
    real(8), intent (in) :: wj
    real(8), intent (in) :: x
    real(8) :: tmp
    if (wj <= 6.2d-20) then
        tmp = x / (1.0d0 + (wj * 2.0d0))
    else
        tmp = wj * (wj + (x * (-2.0d0)))
    end if
    code = tmp
end function

Java

public static double code(double wj, double x) {
	double tmp;
	if (wj <= 6.2e-20) {
		tmp = x / (1.0 + (wj * 2.0));
	} else {
		tmp = wj * (wj + (x * -2.0));
	}
	return tmp;
}

Python

def code(wj, x):
	tmp = 0
	if wj <= 6.2e-20:
		tmp = x / (1.0 + (wj * 2.0))
	else:
		tmp = wj * (wj + (x * -2.0))
	return tmp

Julia

function code(wj, x)
	tmp = 0.0
	if (wj <= 6.2e-20)
		tmp = Float64(x / Float64(1.0 + Float64(wj * 2.0)));
	else
		tmp = Float64(wj * Float64(wj + Float64(x * -2.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(wj, x)
	tmp = 0.0;
	if (wj <= 6.2e-20)
		tmp = x / (1.0 + (wj * 2.0));
	else
		tmp = wj * (wj + (x * -2.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[wj_, x_] := If[LessEqual[wj, 6.2e-20], N[(x / N[(1.0 + N[(wj * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(wj * N[(wj + N[(x * -2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if wj < 6.19999999999999999e-20

   1. Initial program 82.5%

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

      1. distribute-rgt1-in 82.9%

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

      2. associate-/l/ 82.9%

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

      3. div-sub 82.5%

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

      4. *-commutative 82.5%

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

      5. associate-*r/ 82.5%

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

      6. *-rgt-identity 82.5%

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

      7. associate-*r/ 82.5%

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

      8. exp-neg 82.5%

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

      9. *-commutative 82.5%

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

      10. associate-*r* 82.5%

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

      11. *-commutative 82.5%

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

      12. exp-neg 82.5%

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

      13. lft-mult-inverse 82.9%

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

      14. *-lft-identity 82.9%

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

   3. Simplified 82.9%

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

   4. Taylor expanded in x around inf 89.3%

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

      1. +-commutative 89.3%

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

   6. Simplified 89.3%

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

   7. Taylor expanded in wj around 0 88.0%

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

      1. *-commutative 88.0%

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

   9. Simplified 88.0%

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

   if 6.19999999999999999e-20 < wj

   1. Initial program 56.1%

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

      1. distribute-rgt1-in 55.6%

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

      2. associate-/l/ 55.5%

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

      3. div-sub 55.5%

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

      4. *-commutative 55.5%

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

      5. associate-*r/ 55.5%

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

      6. *-rgt-identity 55.5%

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

      7. associate-*r/ 55.4%

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

      8. exp-neg 55.5%

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

      9. *-commutative 55.5%

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

      10. associate-*r* 55.1%

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

      11. *-commutative 55.1%

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

      12. exp-neg 55.5%

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

      13. lft-mult-inverse 55.5%

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

      14. *-lft-identity 55.5%

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

   3. Simplified 55.5%

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

   4. Taylor expanded in wj around 0 76.4%

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

   5. Taylor expanded in x around 0 76.4%

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

   6. Taylor expanded in wj around inf 54.3%

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

      1. +-commutative 54.3%

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

      2. unpow2 54.3%

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

      3. associate-*r* 54.3%

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

      4. *-commutative 54.3%

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

      5. associate-*r* 54.3%

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

      6. distribute-rgt-out 54.3%

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

      7. *-commutative 54.3%

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

   8. Simplified 54.3%

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

4. Final simplification 86.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;wj \leq 6.2 \cdot 10^{-20}:\\ \;\;\;\;\frac{x}{1 + wj \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;wj \cdot \left(wj + x \cdot -2\right)\\ \end{array} \]

Alternative 9: 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 81.1%

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

   1. distribute-rgt1-in 81.5%

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

   2. associate-/l/ 81.5%

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

   3. div-sub 81.1%

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

   4. *-commutative 81.1%

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

   5. associate-*r/ 81.1%

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

   6. *-rgt-identity 81.1%

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

   7. associate-*r/ 81.1%

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

   8. exp-neg 81.1%

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

   9. *-commutative 81.1%

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

   10. associate-*r* 81.1%

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

   11. *-commutative 81.1%

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

   12. exp-neg 81.1%

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

   13. lft-mult-inverse 81.5%

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

   14. *-lft-identity 81.5%

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

3. Simplified 81.5%

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

4. Taylor expanded in wj around inf 4.1%

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

5. Final simplification 4.1%

   \[\leadsto wj \]

Alternative 10: 85.0% 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 81.1%

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

   1. distribute-rgt1-in 81.5%

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

   2. associate-/l/ 81.5%

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

   3. div-sub 81.1%

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

   4. *-commutative 81.1%

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

   5. associate-*r/ 81.1%

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

   6. *-rgt-identity 81.1%

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

   7. associate-*r/ 81.1%

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

   8. exp-neg 81.1%

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

   9. *-commutative 81.1%

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

   10. associate-*r* 81.1%

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

   11. *-commutative 81.1%

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

   12. exp-neg 81.1%

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

   13. lft-mult-inverse 81.5%

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

   14. *-lft-identity 81.5%

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

3. Simplified 81.5%

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

4. Taylor expanded in wj around 0 84.2%

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

5. Final simplification 84.2%

   \[\leadsto x \]

Developer target: 79.4% accurate, 1.5× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Reproduce

herbie shell --seed 2023305 
(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))))))