Jmat.Real.lambertw, newton loop step

Percentage Accurate: 78.5% → 98.0%

Time: 9.6s

Alternatives: 9

Speedup: 313.0×

• could not determine a ground truth

  1. wj = -9.339975915731517e+208
  2. x = 1.4784244422467967e+106

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: 78.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 98.0% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;wj \leq -1.15 \cdot 10^{-7}:\\ \;\;\;\;wj + \frac{\frac{x}{e^{wj}} - wj}{wj + 1}\\ \mathbf{else}:\\ \;\;\;\;x + \left(-2 \cdot \left(wj \cdot x\right) + \mathsf{fma}\left(wj, wj, -{wj}^{3}\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (wj x)
 :precision binary64
 (if (<= wj -1.15e-7)
   (+ wj (/ (- (/ x (exp wj)) wj) (+ wj 1.0)))
   (+ x (+ (* -2.0 (* wj x)) (fma wj wj (- (pow wj 3.0)))))))

C

double code(double wj, double x) {
	double tmp;
	if (wj <= -1.15e-7) {
		tmp = wj + (((x / exp(wj)) - wj) / (wj + 1.0));
	} else {
		tmp = x + ((-2.0 * (wj * x)) + fma(wj, wj, -pow(wj, 3.0)));
	}
	return tmp;
}

Julia

function code(wj, x)
	tmp = 0.0
	if (wj <= -1.15e-7)
		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)) + fma(wj, wj, Float64(-(wj ^ 3.0)))));
	end
	return tmp
end

Wolfram

code[wj_, x_] := If[LessEqual[wj, -1.15e-7], 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[(wj * wj + (-N[Power[wj, 3.0], $MachinePrecision])), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if wj < -1.14999999999999997e-7

   1. Initial program 50.0%

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

      1. div-sub 50.0%

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

      2. associate-/l* 50.0%

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

      3. distribute-rgt1-in 50.0%

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

      4. associate-/l* 50.0%

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

      5. *-inverses 50.0%

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

      6. /-rgt-identity 50.0%

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

      7. distribute-rgt1-in 100.0%

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

      8. associate-/l/ 100.0%

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

      9. div-sub 100.0%

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

   3. Simplified 100.0%

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

   if -1.14999999999999997e-7 < wj

   1. Initial program 80.4%

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

      1. div-sub 80.4%

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

      2. associate-/l* 80.4%

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

      3. distribute-rgt1-in 80.4%

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

      4. associate-/l* 80.4%

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

      5. *-inverses 80.8%

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

      6. /-rgt-identity 80.8%

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

      7. distribute-rgt1-in 80.8%

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

      8. associate-/l/ 80.8%

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

      9. div-sub 80.8%

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

   3. Simplified 80.8%

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

   4. Taylor expanded in wj around 0 98.3%

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

   5. Taylor expanded in x around 0 98.0%

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

      1. unpow2 98.0%

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

   7. Simplified 98.0%

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

   8. Taylor expanded in x around 0 98.4%

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

   9. Taylor expanded in wj around 0 98.4%

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

       1. neg-mul-1 98.4%

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

       2. unpow2 98.4%

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

       3. +-commutative 98.4%

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

       4. fma-udef 98.4%

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

   11. Simplified 98.4%

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

4. Final simplification 98.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;wj \leq -1.15 \cdot 10^{-7}:\\ \;\;\;\;wj + \frac{\frac{x}{e^{wj}} - wj}{wj + 1}\\ \mathbf{else}:\\ \;\;\;\;x + \left(-2 \cdot \left(wj \cdot x\right) + \mathsf{fma}\left(wj, wj, -{wj}^{3}\right)\right)\\ \end{array} \]

Alternative 2: 98.0% accurate, 2.7× speedup

Math

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

FPCore

(FPCore (wj x)
 :precision binary64
 (if (<= wj -2e-7)
   (+ wj (/ (- (/ x (exp wj)) wj) (+ wj 1.0)))
   (+ x (+ (* -2.0 (* wj x)) (- (* wj wj) (pow wj 3.0))))))

C

double code(double wj, double x) {
	double tmp;
	if (wj <= -2e-7) {
		tmp = wj + (((x / exp(wj)) - wj) / (wj + 1.0));
	} else {
		tmp = x + ((-2.0 * (wj * x)) + ((wj * wj) - pow(wj, 3.0)));
	}
	return tmp;
}

Fortran

real(8) function code(wj, x)
    real(8), intent (in) :: wj
    real(8), intent (in) :: x
    real(8) :: tmp
    if (wj <= (-2d-7)) then
        tmp = wj + (((x / exp(wj)) - wj) / (wj + 1.0d0))
    else
        tmp = x + (((-2.0d0) * (wj * x)) + ((wj * wj) - (wj ** 3.0d0)))
    end if
    code = tmp
end function

Java

public static double code(double wj, double x) {
	double tmp;
	if (wj <= -2e-7) {
		tmp = wj + (((x / Math.exp(wj)) - wj) / (wj + 1.0));
	} else {
		tmp = x + ((-2.0 * (wj * x)) + ((wj * wj) - Math.pow(wj, 3.0)));
	}
	return tmp;
}

Python

def code(wj, x):
	tmp = 0
	if wj <= -2e-7:
		tmp = wj + (((x / math.exp(wj)) - wj) / (wj + 1.0))
	else:
		tmp = x + ((-2.0 * (wj * x)) + ((wj * wj) - math.pow(wj, 3.0)))
	return tmp

Julia

function code(wj, x)
	tmp = 0.0
	if (wj <= -2e-7)
		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 * wj) - (wj ^ 3.0))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(wj, x)
	tmp = 0.0;
	if (wj <= -2e-7)
		tmp = wj + (((x / exp(wj)) - wj) / (wj + 1.0));
	else
		tmp = x + ((-2.0 * (wj * x)) + ((wj * wj) - (wj ^ 3.0)));
	end
	tmp_2 = tmp;
end

Wolfram

code[wj_, x_] := If[LessEqual[wj, -2e-7], 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[(wj * wj), $MachinePrecision] - N[Power[wj, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if wj < -1.9999999999999999e-7

   1. Initial program 50.0%

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

      1. div-sub 50.0%

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

      2. associate-/l* 50.0%

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

      3. distribute-rgt1-in 50.0%

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

      4. associate-/l* 50.0%

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

      5. *-inverses 50.0%

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

      6. /-rgt-identity 50.0%

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

      7. distribute-rgt1-in 100.0%

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

      8. associate-/l/ 100.0%

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

      9. div-sub 100.0%

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

   3. Simplified 100.0%

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

   if -1.9999999999999999e-7 < wj

   1. Initial program 80.4%

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

      1. div-sub 80.4%

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

      2. associate-/l* 80.4%

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

      3. distribute-rgt1-in 80.4%

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

      4. associate-/l* 80.4%

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

      5. *-inverses 80.8%

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

      6. /-rgt-identity 80.8%

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

      7. distribute-rgt1-in 80.8%

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

      8. associate-/l/ 80.8%

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

      9. div-sub 80.8%

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

   3. Simplified 80.8%

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

   4. Taylor expanded in wj around 0 98.3%

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

   5. Taylor expanded in x around 0 98.0%

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

      1. unpow2 98.0%

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

   7. Simplified 98.0%

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

   8. Taylor expanded in x around 0 98.4%

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

   9. Taylor expanded in wj around 0 98.4%

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

       1. neg-mul-1 98.4%

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

       2. unpow2 98.4%

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

       3. +-commutative 98.4%

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

       4. unsub-neg 98.4%

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

   11. Simplified 98.4%

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

4. Final simplification 98.5%

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

Alternative 3: 97.7% accurate, 2.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;wj \leq -1.55 \cdot 10^{-6}:\\ \;\;\;\;wj + \frac{\frac{x}{e^{wj}} - wj}{wj + 1}\\ \mathbf{else}:\\ \;\;\;\;x + \left(\left(wj \cdot wj\right) \cdot \left(x \cdot 2.5 + 1\right) + x \cdot \left(wj \cdot -2\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (wj x)
 :precision binary64
 (if (<= wj -1.55e-6)
   (+ wj (/ (- (/ x (exp wj)) wj) (+ wj 1.0)))
   (+ x (+ (* (* wj wj) (+ (* x 2.5) 1.0)) (* x (* wj -2.0))))))

C

double code(double wj, double x) {
	double tmp;
	if (wj <= -1.55e-6) {
		tmp = wj + (((x / exp(wj)) - wj) / (wj + 1.0));
	} else {
		tmp = x + (((wj * wj) * ((x * 2.5) + 1.0)) + (x * (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 <= (-1.55d-6)) then
        tmp = wj + (((x / exp(wj)) - wj) / (wj + 1.0d0))
    else
        tmp = x + (((wj * wj) * ((x * 2.5d0) + 1.0d0)) + (x * (wj * (-2.0d0))))
    end if
    code = tmp
end function

Java

public static double code(double wj, double x) {
	double tmp;
	if (wj <= -1.55e-6) {
		tmp = wj + (((x / Math.exp(wj)) - wj) / (wj + 1.0));
	} else {
		tmp = x + (((wj * wj) * ((x * 2.5) + 1.0)) + (x * (wj * -2.0)));
	}
	return tmp;
}

Python

def code(wj, x):
	tmp = 0
	if wj <= -1.55e-6:
		tmp = wj + (((x / math.exp(wj)) - wj) / (wj + 1.0))
	else:
		tmp = x + (((wj * wj) * ((x * 2.5) + 1.0)) + (x * (wj * -2.0)))
	return tmp

Julia

function code(wj, x)
	tmp = 0.0
	if (wj <= -1.55e-6)
		tmp = Float64(wj + Float64(Float64(Float64(x / exp(wj)) - wj) / Float64(wj + 1.0)));
	else
		tmp = Float64(x + Float64(Float64(Float64(wj * wj) * Float64(Float64(x * 2.5) + 1.0)) + Float64(x * Float64(wj * -2.0))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(wj, x)
	tmp = 0.0;
	if (wj <= -1.55e-6)
		tmp = wj + (((x / exp(wj)) - wj) / (wj + 1.0));
	else
		tmp = x + (((wj * wj) * ((x * 2.5) + 1.0)) + (x * (wj * -2.0)));
	end
	tmp_2 = tmp;
end

Wolfram

code[wj_, x_] := If[LessEqual[wj, -1.55e-6], N[(wj + N[(N[(N[(x / N[Exp[wj], $MachinePrecision]), $MachinePrecision] - wj), $MachinePrecision] / N[(wj + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x + N[(N[(N[(wj * wj), $MachinePrecision] * N[(N[(x * 2.5), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] + N[(x * N[(wj * -2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if wj < -1.55e-6

   1. Initial program 42.9%

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

      1. div-sub 42.9%

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

      2. associate-/l* 42.9%

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

      3. distribute-rgt1-in 42.9%

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

      4. associate-/l* 42.9%

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

      5. *-inverses 42.9%

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

      6. /-rgt-identity 42.9%

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

      7. distribute-rgt1-in 100.0%

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

      8. associate-/l/ 100.0%

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

      9. div-sub 100.0%

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

   3. Simplified 100.0%

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

   if -1.55e-6 < wj

   1. Initial program 80.5%

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

      1. div-sub 80.5%

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

      2. associate-/l* 80.5%

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

      3. distribute-rgt1-in 80.5%

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

      4. associate-/l* 80.5%

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

      5. *-inverses 80.9%

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

      6. /-rgt-identity 80.9%

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

      7. distribute-rgt1-in 80.9%

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

      8. associate-/l/ 80.9%

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

      9. div-sub 80.9%

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

   3. Simplified 80.9%

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

   4. Taylor expanded in wj around 0 97.9%

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

      1. +-commutative 97.9%

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

      2. fma-def 97.9%

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

      3. unpow2 97.9%

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

      4. sub-neg 97.9%

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

      5. distribute-rgt-out 98.3%

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

      6. distribute-rgt-neg-in 98.3%

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

      7. metadata-eval 98.3%

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

      8. metadata-eval 98.3%

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

      9. associate-*r* 98.3%

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

      10. *-commutative 98.3%

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

   6. Simplified 98.3%

      \[\leadsto \color{blue}{x + \mathsf{fma}\left(wj \cdot wj, 1 + x \cdot 2.5, x \cdot \left(-2 \cdot wj\right)\right)} \]
   7. Step-by-step derivation

      1. fma-udef 98.3%

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

      2. *-commutative 98.3%

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

   8. Applied egg-rr 98.3%

      \[\leadsto x + \color{blue}{\left(\left(wj \cdot wj\right) \cdot \left(1 + x \cdot 2.5\right) + x \cdot \left(wj \cdot -2\right)\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 98.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;wj \leq -1.55 \cdot 10^{-6}:\\ \;\;\;\;wj + \frac{\frac{x}{e^{wj}} - wj}{wj + 1}\\ \mathbf{else}:\\ \;\;\;\;x + \left(\left(wj \cdot wj\right) \cdot \left(x \cdot 2.5 + 1\right) + x \cdot \left(wj \cdot -2\right)\right)\\ \end{array} \]

Alternative 4: 97.3% accurate, 2.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;wj \leq -3 \cdot 10^{-5}:\\ \;\;\;\;\frac{x}{e^{wj} \cdot \left(wj + 1\right)}\\ \mathbf{else}:\\ \;\;\;\;x + \left(\left(wj \cdot wj\right) \cdot \left(x \cdot 2.5 + 1\right) + x \cdot \left(wj \cdot -2\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (wj x)
 :precision binary64
 (if (<= wj -3e-5)
   (/ x (* (exp wj) (+ wj 1.0)))
   (+ x (+ (* (* wj wj) (+ (* x 2.5) 1.0)) (* x (* wj -2.0))))))

C

double code(double wj, double x) {
	double tmp;
	if (wj <= -3e-5) {
		tmp = x / (exp(wj) * (wj + 1.0));
	} else {
		tmp = x + (((wj * wj) * ((x * 2.5) + 1.0)) + (x * (wj * -2.0)));
	}
	return tmp;
}

Fortran

real(8) function code(wj, x)
    real(8), intent (in) :: wj
    real(8), intent (in) :: x
    real(8) :: tmp
    if (wj <= (-3d-5)) then
        tmp = x / (exp(wj) * (wj + 1.0d0))
    else
        tmp = x + (((wj * wj) * ((x * 2.5d0) + 1.0d0)) + (x * (wj * (-2.0d0))))
    end if
    code = tmp
end function

Java

public static double code(double wj, double x) {
	double tmp;
	if (wj <= -3e-5) {
		tmp = x / (Math.exp(wj) * (wj + 1.0));
	} else {
		tmp = x + (((wj * wj) * ((x * 2.5) + 1.0)) + (x * (wj * -2.0)));
	}
	return tmp;
}

Python

def code(wj, x):
	tmp = 0
	if wj <= -3e-5:
		tmp = x / (math.exp(wj) * (wj + 1.0))
	else:
		tmp = x + (((wj * wj) * ((x * 2.5) + 1.0)) + (x * (wj * -2.0)))
	return tmp

Julia

function code(wj, x)
	tmp = 0.0
	if (wj <= -3e-5)
		tmp = Float64(x / Float64(exp(wj) * Float64(wj + 1.0)));
	else
		tmp = Float64(x + Float64(Float64(Float64(wj * wj) * Float64(Float64(x * 2.5) + 1.0)) + Float64(x * Float64(wj * -2.0))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(wj, x)
	tmp = 0.0;
	if (wj <= -3e-5)
		tmp = x / (exp(wj) * (wj + 1.0));
	else
		tmp = x + (((wj * wj) * ((x * 2.5) + 1.0)) + (x * (wj * -2.0)));
	end
	tmp_2 = tmp;
end

Wolfram

code[wj_, x_] := If[LessEqual[wj, -3e-5], N[(x / N[(N[Exp[wj], $MachinePrecision] * N[(wj + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x + N[(N[(N[(wj * wj), $MachinePrecision] * N[(N[(x * 2.5), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] + N[(x * N[(wj * -2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if wj < -3.00000000000000008e-5

   1. Initial program 42.9%

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

      1. div-sub 42.9%

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

      2. associate-/l* 42.9%

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

      3. distribute-rgt1-in 42.9%

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

      4. associate-/l* 42.9%

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

      5. *-inverses 42.9%

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

      6. /-rgt-identity 42.9%

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

      7. distribute-rgt1-in 100.0%

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

      8. associate-/l/ 100.0%

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

      9. div-sub 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in x around inf 86.4%

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

      1. +-commutative 86.4%

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

   6. Simplified 86.4%

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

   if -3.00000000000000008e-5 < wj

   1. Initial program 80.5%

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

      1. div-sub 80.5%

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

      2. associate-/l* 80.5%

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

      3. distribute-rgt1-in 80.5%

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

      4. associate-/l* 80.5%

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

      5. *-inverses 80.9%

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

      6. /-rgt-identity 80.9%

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

      7. distribute-rgt1-in 80.9%

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

      8. associate-/l/ 80.9%

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

      9. div-sub 80.9%

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

   3. Simplified 80.9%

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

   4. Taylor expanded in wj around 0 97.9%

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

      1. +-commutative 97.9%

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

      2. fma-def 97.9%

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

      3. unpow2 97.9%

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

      4. sub-neg 97.9%

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

      5. distribute-rgt-out 98.3%

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

      6. distribute-rgt-neg-in 98.3%

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

      7. metadata-eval 98.3%

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

      8. metadata-eval 98.3%

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

      9. associate-*r* 98.3%

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

      10. *-commutative 98.3%

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

   6. Simplified 98.3%

      \[\leadsto \color{blue}{x + \mathsf{fma}\left(wj \cdot wj, 1 + x \cdot 2.5, x \cdot \left(-2 \cdot wj\right)\right)} \]
   7. Step-by-step derivation

      1. fma-udef 98.3%

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

      2. *-commutative 98.3%

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

   8. Applied egg-rr 98.3%

      \[\leadsto x + \color{blue}{\left(\left(wj \cdot wj\right) \cdot \left(1 + x \cdot 2.5\right) + x \cdot \left(wj \cdot -2\right)\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 98.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;wj \leq -3 \cdot 10^{-5}:\\ \;\;\;\;\frac{x}{e^{wj} \cdot \left(wj + 1\right)}\\ \mathbf{else}:\\ \;\;\;\;x + \left(\left(wj \cdot wj\right) \cdot \left(x \cdot 2.5 + 1\right) + x \cdot \left(wj \cdot -2\right)\right)\\ \end{array} \]

Alternative 5: 96.0% accurate, 18.4× speedup

Math

\[\begin{array}{l} \\ x + \left(\left(wj \cdot wj\right) \cdot \left(x \cdot 2.5 + 1\right) + x \cdot \left(wj \cdot -2\right)\right) \end{array} \]

FPCore

(FPCore (wj x)
 :precision binary64
 (+ x (+ (* (* wj wj) (+ (* x 2.5) 1.0)) (* x (* wj -2.0)))))

C

double code(double wj, double x) {
	return x + (((wj * wj) * ((x * 2.5) + 1.0)) + (x * (wj * -2.0)));
}

Fortran

real(8) function code(wj, x)
    real(8), intent (in) :: wj
    real(8), intent (in) :: x
    code = x + (((wj * wj) * ((x * 2.5d0) + 1.0d0)) + (x * (wj * (-2.0d0))))
end function

Java

public static double code(double wj, double x) {
	return x + (((wj * wj) * ((x * 2.5) + 1.0)) + (x * (wj * -2.0)));
}

Python

def code(wj, x):
	return x + (((wj * wj) * ((x * 2.5) + 1.0)) + (x * (wj * -2.0)))

Julia

function code(wj, x)
	return Float64(x + Float64(Float64(Float64(wj * wj) * Float64(Float64(x * 2.5) + 1.0)) + Float64(x * Float64(wj * -2.0))))
end

MATLAB

function tmp = code(wj, x)
	tmp = x + (((wj * wj) * ((x * 2.5) + 1.0)) + (x * (wj * -2.0)));
end

Wolfram

code[wj_, x_] := N[(x + N[(N[(N[(wj * wj), $MachinePrecision] * N[(N[(x * 2.5), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] + N[(x * N[(wj * -2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 79.4%

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

   1. div-sub 79.4%

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

   2. associate-/l* 79.4%

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

   3. distribute-rgt1-in 79.4%

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

   4. associate-/l* 79.4%

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

   5. *-inverses 79.8%

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

   6. /-rgt-identity 79.8%

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

   7. distribute-rgt1-in 81.4%

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

   8. associate-/l/ 81.4%

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

   9. div-sub 81.4%

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

3. Simplified 81.4%

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

4. Taylor expanded in wj around 0 95.6%

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

   1. +-commutative 95.6%

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

   2. fma-def 95.6%

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

   3. unpow2 95.6%

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

   4. sub-neg 95.6%

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

   5. distribute-rgt-out 96.0%

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

   6. distribute-rgt-neg-in 96.0%

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

   7. metadata-eval 96.0%

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

   8. metadata-eval 96.0%

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

   9. associate-*r* 96.0%

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

   10. *-commutative 96.0%

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

6. Simplified 96.0%

   \[\leadsto \color{blue}{x + \mathsf{fma}\left(wj \cdot wj, 1 + x \cdot 2.5, x \cdot \left(-2 \cdot wj\right)\right)} \]
7. Step-by-step derivation

   1. fma-udef 96.0%

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

   2. *-commutative 96.0%

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

8. Applied egg-rr 96.0%

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

9. Final simplification 96.0%

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

Alternative 6: 95.9% accurate, 28.5× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 79.4%

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

   1. div-sub 79.4%

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

   2. associate-/l* 79.4%

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

   3. distribute-rgt1-in 79.4%

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

   4. associate-/l* 79.4%

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

   5. *-inverses 79.8%

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

   6. /-rgt-identity 79.8%

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

   7. distribute-rgt1-in 81.4%

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

   8. associate-/l/ 81.4%

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

   9. div-sub 81.4%

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

3. Simplified 81.4%

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

4. Taylor expanded in wj around 0 95.6%

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

   1. +-commutative 95.6%

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

   2. fma-def 95.6%

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

   3. unpow2 95.6%

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

   4. sub-neg 95.6%

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

   5. distribute-rgt-out 96.0%

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

   6. distribute-rgt-neg-in 96.0%

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

   7. metadata-eval 96.0%

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

   8. metadata-eval 96.0%

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

   9. associate-*r* 96.0%

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

   10. *-commutative 96.0%

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

6. Simplified 96.0%

   \[\leadsto \color{blue}{x + \mathsf{fma}\left(wj \cdot wj, 1 + x \cdot 2.5, x \cdot \left(-2 \cdot wj\right)\right)} \]
7. Step-by-step derivation

   1. fma-udef 96.0%

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

   2. *-commutative 96.0%

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

8. Applied egg-rr 96.0%

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

9. Taylor expanded in x around 0 95.6%

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

    1. unpow2 95.6%

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

11. Simplified 95.6%

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

12. Final simplification 95.6%

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

Alternative 7: 85.1% accurate, 44.7× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 79.4%

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

   1. div-sub 79.4%

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

   2. associate-/l* 79.4%

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

   3. distribute-rgt1-in 79.4%

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

   4. associate-/l* 79.4%

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

   5. *-inverses 79.8%

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

   6. /-rgt-identity 79.8%

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

   7. distribute-rgt1-in 81.4%

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

   8. associate-/l/ 81.4%

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

   9. div-sub 81.4%

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

3. Simplified 81.4%

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

4. Taylor expanded in wj around 0 86.9%

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

   1. *-commutative 86.9%

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

6. Simplified 86.9%

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

7. Final simplification 86.9%

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

Alternative 8: 4.4% accurate, 313.0× speedup

Math

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

FPCore

(FPCore (wj x) :precision binary64 wj)

C

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

Fortran

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

Java

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

Python

def code(wj, x):
	return wj

Julia

function code(wj, x)
	return wj
end

MATLAB

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

Wolfram

code[wj_, x_] := wj

Derivation

1. Initial program 79.4%

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

   1. div-sub 79.4%

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

   2. associate-/l* 79.4%

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

   3. distribute-rgt1-in 79.4%

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

   4. associate-/l* 79.4%

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

   5. *-inverses 79.8%

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

   6. /-rgt-identity 79.8%

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

   7. distribute-rgt1-in 81.4%

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

   8. associate-/l/ 81.4%

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

   9. div-sub 81.4%

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

3. Simplified 81.4%

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

4. Taylor expanded in wj around inf 4.0%

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

5. Final simplification 4.0%

   \[\leadsto wj \]

Alternative 9: 84.5% accurate, 313.0× speedup

Math

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

FPCore

(FPCore (wj x) :precision binary64 x)

C

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

Fortran

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

Java

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

Python

def code(wj, x):
	return x

Julia

function code(wj, x)
	return x
end

MATLAB

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

Wolfram

code[wj_, x_] := x

Derivation

1. Initial program 79.4%

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

   1. div-sub 79.4%

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

   2. associate-/l* 79.4%

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

   3. distribute-rgt1-in 79.4%

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

   4. associate-/l* 79.4%

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

   5. *-inverses 79.8%

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

   6. /-rgt-identity 79.8%

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

   7. distribute-rgt1-in 81.4%

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

   8. associate-/l/ 81.4%

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

   9. div-sub 81.4%

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

3. Simplified 81.4%

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

4. Taylor expanded in wj around 0 86.1%

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

5. Final simplification 86.1%

   \[\leadsto x \]

Developer target: 79.6% 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 2023293 
(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))))))