Jmat.Real.lambertw, newton loop step

Percentage Accurate: 77.8% → 98.1%

Time: 10.9s

Alternatives: 13

Speedup: 313.0×

• could not determine a ground truth

  1. wj = -4.756516435222462e+113
  2. x = -4.435460935264974e+228

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 77.8% 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: 98.1% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;wj \leq -2.05 \cdot 10^{-7}:\\ \;\;\;\;wj + \frac{x - wj \cdot e^{wj}}{e^{wj} \cdot \left(wj + 1\right)}\\ \mathbf{else}:\\ \;\;\;\;x + \left(-2 \cdot \left(wj \cdot x\right) + \mathsf{fma}\left(wj, wj, -{wj}^{3}\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (wj x)
 :precision binary64
 (if (<= wj -2.05e-7)
   (+ wj (/ (- x (* wj (exp wj))) (* (exp 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 <= -2.05e-7) {
		tmp = wj + ((x - (wj * exp(wj))) / (exp(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 <= -2.05e-7)
		tmp = Float64(wj + Float64(Float64(x - Float64(wj * exp(wj))) / Float64(exp(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, -2.05e-7], N[(wj + N[(N[(x - N[(wj * N[Exp[wj], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[Exp[wj], $MachinePrecision] * N[(wj + 1.0), $MachinePrecision]), $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 < -2.05e-7

   1. Initial program 54.5%

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

      1. distribute-rgt1-in 97.4%

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

      2. *-commutative 97.4%

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

   3. Simplified 97.4%

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

   if -2.05e-7 < wj

   1. Initial program 75.5%

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

      1. div-sub 75.5%

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

      2. distribute-rgt1-in 75.5%

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

      3. times-frac 75.5%

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

      4. *-inverses 75.9%

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

      5. associate-*l/ 75.9%

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

      6. *-rgt-identity 75.9%

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

      7. distribute-rgt1-in 75.9%

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

      8. associate-/l/ 75.9%

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

      9. div-sub 75.9%

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

   3. Simplified 75.9%

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

   4. Taylor expanded in wj around 0 98.9%

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

   5. Taylor expanded in x around 0 98.8%

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

   6. Taylor expanded in x around 0 99.1%

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

      1. +-commutative 99.1%

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

      2. unpow2 99.1%

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

      3. fma-def 99.1%

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

      4. mul-1-neg 99.1%

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

   8. Applied egg-rr 99.1%

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

4. Final simplification 99.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;wj \leq -2.05 \cdot 10^{-7}:\\ \;\;\;\;wj + \frac{x - wj \cdot e^{wj}}{e^{wj} \cdot \left(wj + 1\right)}\\ \mathbf{else}:\\ \;\;\;\;x + \left(-2 \cdot \left(wj \cdot x\right) + \mathsf{fma}\left(wj, wj, -{wj}^{3}\right)\right)\\ \end{array} \]

Alternative 2: 98.1% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;wj \leq -2.6 \cdot 10^{-7}:\\ \;\;\;\;wj + \frac{\frac{x}{e^{wj}} - wj}{wj + 1}\\ \mathbf{else}:\\ \;\;\;\;x + \left(-2 \cdot \left(wj \cdot x\right) + \mathsf{fma}\left(wj, wj, -{wj}^{3}\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (wj x)
 :precision binary64
 (if (<= wj -2.6e-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 <= -2.6e-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 <= -2.6e-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, -2.6e-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 < -2.59999999999999999e-7

   1. Initial program 54.5%

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

      1. div-sub 54.5%

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

      2. distribute-rgt1-in 54.5%

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

      3. times-frac 54.4%

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

      4. *-inverses 54.4%

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

      5. associate-*l/ 54.4%

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

      6. *-rgt-identity 54.4%

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

      7. distribute-rgt1-in 97.3%

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

      8. associate-/l/ 97.3%

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

      9. div-sub 97.3%

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

   3. Simplified 97.3%

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

   if -2.59999999999999999e-7 < wj

   1. Initial program 75.5%

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

      1. div-sub 75.5%

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

      2. distribute-rgt1-in 75.5%

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

      3. times-frac 75.5%

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

      4. *-inverses 75.9%

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

      5. associate-*l/ 75.9%

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

      6. *-rgt-identity 75.9%

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

      7. distribute-rgt1-in 75.9%

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

      8. associate-/l/ 75.9%

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

      9. div-sub 75.9%

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

   3. Simplified 75.9%

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

   4. Taylor expanded in wj around 0 98.9%

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

   5. Taylor expanded in x around 0 98.8%

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

   6. Taylor expanded in x around 0 99.1%

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

      1. +-commutative 99.1%

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

      2. unpow2 99.1%

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

      3. fma-def 99.1%

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

      4. mul-1-neg 99.1%

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

   8. Applied egg-rr 99.1%

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

4. Final simplification 99.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;wj \leq -2.6 \cdot 10^{-7}:\\ \;\;\;\;wj + \frac{\frac{x}{e^{wj}} - wj}{wj + 1}\\ \mathbf{else}:\\ \;\;\;\;x + \left(-2 \cdot \left(wj \cdot x\right) + \mathsf{fma}\left(wj, wj, -{wj}^{3}\right)\right)\\ \end{array} \]

Alternative 3: 97.7% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;wj \leq -8 \cdot 10^{-11}:\\ \;\;\;\;wj + \frac{\frac{x}{e^{wj}} - wj}{wj + 1}\\ \mathbf{else}:\\ \;\;\;\;x + \left({wj}^{2} - {wj}^{3}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (wj x)
 :precision binary64
 (if (<= wj -8e-11)
   (+ wj (/ (- (/ x (exp wj)) wj) (+ wj 1.0)))
   (+ x (- (pow wj 2.0) (pow wj 3.0)))))

C

double code(double wj, double x) {
	double tmp;
	if (wj <= -8e-11) {
		tmp = wj + (((x / exp(wj)) - wj) / (wj + 1.0));
	} else {
		tmp = x + (pow(wj, 2.0) - 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 <= (-8d-11)) then
        tmp = wj + (((x / exp(wj)) - wj) / (wj + 1.0d0))
    else
        tmp = x + ((wj ** 2.0d0) - (wj ** 3.0d0))
    end if
    code = tmp
end function

Java

public static double code(double wj, double x) {
	double tmp;
	if (wj <= -8e-11) {
		tmp = wj + (((x / Math.exp(wj)) - wj) / (wj + 1.0));
	} else {
		tmp = x + (Math.pow(wj, 2.0) - Math.pow(wj, 3.0));
	}
	return tmp;
}

Python

def code(wj, x):
	tmp = 0
	if wj <= -8e-11:
		tmp = wj + (((x / math.exp(wj)) - wj) / (wj + 1.0))
	else:
		tmp = x + (math.pow(wj, 2.0) - math.pow(wj, 3.0))
	return tmp

Julia

function code(wj, x)
	tmp = 0.0
	if (wj <= -8e-11)
		tmp = Float64(wj + Float64(Float64(Float64(x / exp(wj)) - wj) / Float64(wj + 1.0)));
	else
		tmp = Float64(x + Float64((wj ^ 2.0) - (wj ^ 3.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(wj, x)
	tmp = 0.0;
	if (wj <= -8e-11)
		tmp = wj + (((x / exp(wj)) - wj) / (wj + 1.0));
	else
		tmp = x + ((wj ^ 2.0) - (wj ^ 3.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[wj_, x_] := If[LessEqual[wj, -8e-11], N[(wj + N[(N[(N[(x / N[Exp[wj], $MachinePrecision]), $MachinePrecision] - wj), $MachinePrecision] / N[(wj + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x + N[(N[Power[wj, 2.0], $MachinePrecision] - N[Power[wj, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if wj < -7.99999999999999952e-11

   1. Initial program 59.3%

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

      1. div-sub 59.3%

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

      2. distribute-rgt1-in 59.3%

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

      3. times-frac 59.2%

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

      4. *-inverses 59.2%

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

      5. associate-*l/ 59.2%

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

      6. *-rgt-identity 59.2%

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

      7. distribute-rgt1-in 92.6%

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

      8. associate-/l/ 92.6%

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

      9. div-sub 92.6%

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

   3. Simplified 92.6%

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

   if -7.99999999999999952e-11 < wj

   1. Initial program 75.5%

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

      1. div-sub 75.5%

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

      2. distribute-rgt1-in 75.5%

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

      3. times-frac 75.5%

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

      4. *-inverses 75.9%

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

      5. associate-*l/ 75.9%

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

      6. *-rgt-identity 75.9%

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

      7. distribute-rgt1-in 75.9%

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

      8. associate-/l/ 75.9%

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

      9. div-sub 75.9%

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

   3. Simplified 75.9%

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

   4. Taylor expanded in wj around 0 98.9%

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

   5. Taylor expanded in x around 0 98.8%

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

   6. Taylor expanded in x around 0 98.9%

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

      1. +-commutative 98.9%

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

      2. mul-1-neg 98.9%

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

      3. unsub-neg 98.9%

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

   8. Simplified 98.9%

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

4. Final simplification 98.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;wj \leq -8 \cdot 10^{-11}:\\ \;\;\;\;wj + \frac{\frac{x}{e^{wj}} - wj}{wj + 1}\\ \mathbf{else}:\\ \;\;\;\;x + \left({wj}^{2} - {wj}^{3}\right)\\ \end{array} \]

Alternative 4: 97.8% accurate, 2.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;wj \leq -2.65 \cdot 10^{-7}:\\ \;\;\;\;wj + \frac{\frac{x}{e^{wj}} - wj}{wj + 1}\\ \mathbf{else}:\\ \;\;\;\;x + \left(-2 \cdot \left(wj \cdot x\right) + {wj}^{2} \cdot \left(1 - \left(x \cdot -4 + x \cdot 1.5\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (wj x)
 :precision binary64
 (if (<= wj -2.65e-7)
   (+ wj (/ (- (/ x (exp wj)) wj) (+ wj 1.0)))
   (+
    x
    (+ (* -2.0 (* wj x)) (* (pow wj 2.0) (- 1.0 (+ (* x -4.0) (* x 1.5))))))))

C

double code(double wj, double x) {
	double tmp;
	if (wj <= -2.65e-7) {
		tmp = wj + (((x / exp(wj)) - wj) / (wj + 1.0));
	} else {
		tmp = x + ((-2.0 * (wj * x)) + (pow(wj, 2.0) * (1.0 - ((x * -4.0) + (x * 1.5)))));
	}
	return tmp;
}

Fortran

real(8) function code(wj, x)
    real(8), intent (in) :: wj
    real(8), intent (in) :: x
    real(8) :: tmp
    if (wj <= (-2.65d-7)) then
        tmp = wj + (((x / exp(wj)) - wj) / (wj + 1.0d0))
    else
        tmp = x + (((-2.0d0) * (wj * x)) + ((wj ** 2.0d0) * (1.0d0 - ((x * (-4.0d0)) + (x * 1.5d0)))))
    end if
    code = tmp
end function

Java

public static double code(double wj, double x) {
	double tmp;
	if (wj <= -2.65e-7) {
		tmp = wj + (((x / Math.exp(wj)) - wj) / (wj + 1.0));
	} else {
		tmp = x + ((-2.0 * (wj * x)) + (Math.pow(wj, 2.0) * (1.0 - ((x * -4.0) + (x * 1.5)))));
	}
	return tmp;
}

Python

def code(wj, x):
	tmp = 0
	if wj <= -2.65e-7:
		tmp = wj + (((x / math.exp(wj)) - wj) / (wj + 1.0))
	else:
		tmp = x + ((-2.0 * (wj * x)) + (math.pow(wj, 2.0) * (1.0 - ((x * -4.0) + (x * 1.5)))))
	return tmp

Julia

function code(wj, x)
	tmp = 0.0
	if (wj <= -2.65e-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((wj ^ 2.0) * Float64(1.0 - Float64(Float64(x * -4.0) + Float64(x * 1.5))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(wj, x)
	tmp = 0.0;
	if (wj <= -2.65e-7)
		tmp = wj + (((x / exp(wj)) - wj) / (wj + 1.0));
	else
		tmp = x + ((-2.0 * (wj * x)) + ((wj ^ 2.0) * (1.0 - ((x * -4.0) + (x * 1.5)))));
	end
	tmp_2 = tmp;
end

Wolfram

code[wj_, x_] := If[LessEqual[wj, -2.65e-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[Power[wj, 2.0], $MachinePrecision] * N[(1.0 - N[(N[(x * -4.0), $MachinePrecision] + N[(x * 1.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if wj < -2.65e-7

   1. Initial program 47.2%

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

      1. div-sub 47.2%

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

      2. distribute-rgt1-in 47.2%

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

      3. times-frac 47.1%

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

      4. *-inverses 47.1%

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

      5. associate-*l/ 47.1%

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

      6. *-rgt-identity 47.1%

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

      7. distribute-rgt1-in 97.1%

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

      8. associate-/l/ 97.1%

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

      9. div-sub 97.1%

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

   3. Simplified 97.1%

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

   if -2.65e-7 < wj

   1. Initial program 75.6%

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

      1. div-sub 75.6%

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

      2. distribute-rgt1-in 75.6%

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

      3. times-frac 75.6%

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

      4. *-inverses 76.0%

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

      5. associate-*l/ 76.0%

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

      6. *-rgt-identity 76.0%

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

      7. distribute-rgt1-in 76.0%

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

      8. associate-/l/ 76.0%

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

      9. div-sub 76.0%

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

   3. Simplified 76.0%

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

   4. Taylor expanded in wj around 0 98.4%

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

4. Final simplification 98.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;wj \leq -2.65 \cdot 10^{-7}:\\ \;\;\;\;wj + \frac{\frac{x}{e^{wj}} - wj}{wj + 1}\\ \mathbf{else}:\\ \;\;\;\;x + \left(-2 \cdot \left(wj \cdot x\right) + {wj}^{2} \cdot \left(1 - \left(x \cdot -4 + x \cdot 1.5\right)\right)\right)\\ \end{array} \]

Alternative 5: 97.7% accurate, 2.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;wj \leq -3.3 \cdot 10^{-9}:\\ \;\;\;\;wj + \frac{\frac{x}{e^{wj}} - wj}{wj + 1}\\ \mathbf{else}:\\ \;\;\;\;x + \left(-2 \cdot \left(wj \cdot x\right) + {wj}^{2} \cdot \left(1 - x \cdot -4\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (wj x)
 :precision binary64
 (if (<= wj -3.3e-9)
   (+ wj (/ (- (/ x (exp wj)) wj) (+ wj 1.0)))
   (+ x (+ (* -2.0 (* wj x)) (* (pow wj 2.0) (- 1.0 (* x -4.0)))))))

C

double code(double wj, double x) {
	double tmp;
	if (wj <= -3.3e-9) {
		tmp = wj + (((x / exp(wj)) - wj) / (wj + 1.0));
	} else {
		tmp = x + ((-2.0 * (wj * x)) + (pow(wj, 2.0) * (1.0 - (x * -4.0))));
	}
	return tmp;
}

Fortran

real(8) function code(wj, x)
    real(8), intent (in) :: wj
    real(8), intent (in) :: x
    real(8) :: tmp
    if (wj <= (-3.3d-9)) then
        tmp = wj + (((x / exp(wj)) - wj) / (wj + 1.0d0))
    else
        tmp = x + (((-2.0d0) * (wj * x)) + ((wj ** 2.0d0) * (1.0d0 - (x * (-4.0d0)))))
    end if
    code = tmp
end function

Java

public static double code(double wj, double x) {
	double tmp;
	if (wj <= -3.3e-9) {
		tmp = wj + (((x / Math.exp(wj)) - wj) / (wj + 1.0));
	} else {
		tmp = x + ((-2.0 * (wj * x)) + (Math.pow(wj, 2.0) * (1.0 - (x * -4.0))));
	}
	return tmp;
}

Python

def code(wj, x):
	tmp = 0
	if wj <= -3.3e-9:
		tmp = wj + (((x / math.exp(wj)) - wj) / (wj + 1.0))
	else:
		tmp = x + ((-2.0 * (wj * x)) + (math.pow(wj, 2.0) * (1.0 - (x * -4.0))))
	return tmp

Julia

function code(wj, x)
	tmp = 0.0
	if (wj <= -3.3e-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)) + Float64((wj ^ 2.0) * Float64(1.0 - Float64(x * -4.0)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(wj, x)
	tmp = 0.0;
	if (wj <= -3.3e-9)
		tmp = wj + (((x / exp(wj)) - wj) / (wj + 1.0));
	else
		tmp = x + ((-2.0 * (wj * x)) + ((wj ^ 2.0) * (1.0 - (x * -4.0))));
	end
	tmp_2 = tmp;
end

Wolfram

code[wj_, x_] := If[LessEqual[wj, -3.3e-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[(N[Power[wj, 2.0], $MachinePrecision] * N[(1.0 - N[(x * -4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if wj < -3.30000000000000018e-9

   1. Initial program 54.5%

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

      1. div-sub 54.5%

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

      2. distribute-rgt1-in 54.5%

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

      3. times-frac 54.4%

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

      4. *-inverses 54.4%

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

      5. associate-*l/ 54.4%

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

      6. *-rgt-identity 54.4%

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

      7. distribute-rgt1-in 97.3%

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

      8. associate-/l/ 97.3%

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

      9. div-sub 97.3%

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

   3. Simplified 97.3%

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

   if -3.30000000000000018e-9 < wj

   1. Initial program 75.5%

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

      1. distribute-rgt1-in 75.5%

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

      2. *-commutative 75.5%

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

   3. Simplified 75.5%

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

   4. Taylor expanded in wj around 0 75.1%

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

      1. *-commutative 75.1%

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

   6. Simplified 75.1%

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

   7. Taylor expanded in wj around 0 98.4%

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

4. Final simplification 98.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;wj \leq -3.3 \cdot 10^{-9}:\\ \;\;\;\;wj + \frac{\frac{x}{e^{wj}} - wj}{wj + 1}\\ \mathbf{else}:\\ \;\;\;\;x + \left(-2 \cdot \left(wj \cdot x\right) + {wj}^{2} \cdot \left(1 - x \cdot -4\right)\right)\\ \end{array} \]

Alternative 6: 97.4% accurate, 2.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;wj \leq -2.2 \cdot 10^{-11}:\\ \;\;\;\;wj + \frac{\frac{x}{e^{wj}} - wj}{wj + 1}\\ \mathbf{else}:\\ \;\;\;\;x + {wj}^{2} \cdot \left(1 - x \cdot -4\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (wj x)
 :precision binary64
 (if (<= wj -2.2e-11)
   (+ wj (/ (- (/ x (exp wj)) wj) (+ wj 1.0)))
   (+ x (* (pow wj 2.0) (- 1.0 (* x -4.0))))))

C

double code(double wj, double x) {
	double tmp;
	if (wj <= -2.2e-11) {
		tmp = wj + (((x / exp(wj)) - wj) / (wj + 1.0));
	} else {
		tmp = x + (pow(wj, 2.0) * (1.0 - (x * -4.0)));
	}
	return tmp;
}

Fortran

real(8) function code(wj, x)
    real(8), intent (in) :: wj
    real(8), intent (in) :: x
    real(8) :: tmp
    if (wj <= (-2.2d-11)) then
        tmp = wj + (((x / exp(wj)) - wj) / (wj + 1.0d0))
    else
        tmp = x + ((wj ** 2.0d0) * (1.0d0 - (x * (-4.0d0))))
    end if
    code = tmp
end function

Java

public static double code(double wj, double x) {
	double tmp;
	if (wj <= -2.2e-11) {
		tmp = wj + (((x / Math.exp(wj)) - wj) / (wj + 1.0));
	} else {
		tmp = x + (Math.pow(wj, 2.0) * (1.0 - (x * -4.0)));
	}
	return tmp;
}

Python

def code(wj, x):
	tmp = 0
	if wj <= -2.2e-11:
		tmp = wj + (((x / math.exp(wj)) - wj) / (wj + 1.0))
	else:
		tmp = x + (math.pow(wj, 2.0) * (1.0 - (x * -4.0)))
	return tmp

Julia

function code(wj, x)
	tmp = 0.0
	if (wj <= -2.2e-11)
		tmp = Float64(wj + Float64(Float64(Float64(x / exp(wj)) - wj) / Float64(wj + 1.0)));
	else
		tmp = Float64(x + Float64((wj ^ 2.0) * Float64(1.0 - Float64(x * -4.0))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(wj, x)
	tmp = 0.0;
	if (wj <= -2.2e-11)
		tmp = wj + (((x / exp(wj)) - wj) / (wj + 1.0));
	else
		tmp = x + ((wj ^ 2.0) * (1.0 - (x * -4.0)));
	end
	tmp_2 = tmp;
end

Wolfram

code[wj_, x_] := If[LessEqual[wj, -2.2e-11], N[(wj + N[(N[(N[(x / N[Exp[wj], $MachinePrecision]), $MachinePrecision] - wj), $MachinePrecision] / N[(wj + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x + N[(N[Power[wj, 2.0], $MachinePrecision] * N[(1.0 - N[(x * -4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if wj < -2.2000000000000002e-11

   1. Initial program 59.3%

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

      1. div-sub 59.3%

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

      2. distribute-rgt1-in 59.3%

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

      3. times-frac 59.2%

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

      4. *-inverses 59.2%

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

      5. associate-*l/ 59.2%

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

      6. *-rgt-identity 59.2%

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

      7. distribute-rgt1-in 92.6%

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

      8. associate-/l/ 92.6%

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

      9. div-sub 92.6%

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

   3. Simplified 92.6%

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

   if -2.2000000000000002e-11 < wj

   1. Initial program 75.5%

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

      1. distribute-rgt1-in 75.5%

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

      2. *-commutative 75.5%

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

   3. Simplified 75.5%

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

   4. Taylor expanded in wj around 0 75.1%

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

      1. *-commutative 75.1%

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

   6. Simplified 75.1%

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

   7. Taylor expanded in wj around 0 98.5%

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

   8. Taylor expanded in wj around inf 98.3%

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

4. Final simplification 98.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;wj \leq -2.2 \cdot 10^{-11}:\\ \;\;\;\;wj + \frac{\frac{x}{e^{wj}} - wj}{wj + 1}\\ \mathbf{else}:\\ \;\;\;\;x + {wj}^{2} \cdot \left(1 - x \cdot -4\right)\\ \end{array} \]

Alternative 7: 95.4% accurate, 2.8× speedup

Math

\[\begin{array}{l} \\ x + {wj}^{2} \cdot \left(1 - x \cdot -4\right) \end{array} \]

FPCore

(FPCore (wj x) :precision binary64 (+ x (* (pow wj 2.0) (- 1.0 (* x -4.0)))))

C

double code(double wj, double x) {
	return x + (pow(wj, 2.0) * (1.0 - (x * -4.0)));
}

Fortran

real(8) function code(wj, x)
    real(8), intent (in) :: wj
    real(8), intent (in) :: x
    code = x + ((wj ** 2.0d0) * (1.0d0 - (x * (-4.0d0))))
end function

Java

public static double code(double wj, double x) {
	return x + (Math.pow(wj, 2.0) * (1.0 - (x * -4.0)));
}

Python

def code(wj, x):
	return x + (math.pow(wj, 2.0) * (1.0 - (x * -4.0)))

Julia

function code(wj, x)
	return Float64(x + Float64((wj ^ 2.0) * Float64(1.0 - Float64(x * -4.0))))
end

MATLAB

function tmp = code(wj, x)
	tmp = x + ((wj ^ 2.0) * (1.0 - (x * -4.0)));
end

Wolfram

code[wj_, x_] := N[(x + N[(N[Power[wj, 2.0], $MachinePrecision] * N[(1.0 - N[(x * -4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 75.0%

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

   1. distribute-rgt1-in 76.1%

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

   2. *-commutative 76.1%

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

3. Simplified 76.1%

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

4. Taylor expanded in wj around 0 73.9%

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

   1. *-commutative 73.9%

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

6. Simplified 73.9%

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

7. Taylor expanded in wj around 0 96.5%

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

8. Taylor expanded in wj around inf 95.9%

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

9. Final simplification 95.9%

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

Alternative 8: 86.6% accurate, 2.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 75.0%

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

   1. div-sub 75.0%

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

   2. distribute-rgt1-in 75.0%

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

   3. times-frac 75.0%

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

   4. *-inverses 75.4%

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

   5. associate-*l/ 75.4%

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

   6. *-rgt-identity 75.4%

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

   7. distribute-rgt1-in 76.5%

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

   8. associate-/l/ 76.5%

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

   9. div-sub 76.5%

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

3. Simplified 76.5%

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

4. Taylor expanded in x around inf 85.9%

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

   1. +-commutative 85.9%

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

6. Simplified 85.9%

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

7. Final simplification 85.9%

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

Alternative 9: 84.8% accurate, 34.8× speedup

Math

\[\begin{array}{l} \\ \frac{x}{\frac{wj + 1}{1 - wj}} \end{array} \]

FPCore

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

C

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

Fortran

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

Java

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

Python

def code(wj, x):
	return x / ((wj + 1.0) / (1.0 - wj))

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 75.0%

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

   1. div-sub 75.0%

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

   2. distribute-rgt1-in 75.0%

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

   3. times-frac 75.0%

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

   4. *-inverses 75.4%

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

   5. associate-*l/ 75.4%

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

   6. *-rgt-identity 75.4%

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

   7. distribute-rgt1-in 76.5%

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

   8. associate-/l/ 76.5%

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

   9. div-sub 76.5%

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

3. Simplified 76.5%

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

4. Taylor expanded in wj around 0 74.4%

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

   1. mul-1-neg 74.4%

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

   2. unsub-neg 74.4%

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

   3. *-commutative 74.4%

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

6. Simplified 74.4%

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

   1. clear-num 74.1%

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

   2. inv-pow 74.1%

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

   3. +-commutative 74.1%

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

   4. associate--r- 74.1%

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

8. Applied egg-rr 74.1%

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

   1. unpow-1 74.1%

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

   2. associate-+l- 74.1%

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

   3. *-rgt-identity 74.1%

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

   4. distribute-lft-out-- 74.1%

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

10. Simplified 74.1%

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

11. Taylor expanded in x around -inf 84.2%

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

    1. associate-/l* 84.2%

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

    2. +-commutative 84.2%

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

13. Simplified 84.2%

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

14. Final simplification 84.2%

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

Alternative 10: 84.7% 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 75.0%

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

   1. div-sub 75.0%

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

   2. distribute-rgt1-in 75.0%

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

   3. times-frac 75.0%

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

   4. *-inverses 75.4%

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

   5. associate-*l/ 75.4%

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

   6. *-rgt-identity 75.4%

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

   7. distribute-rgt1-in 76.5%

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

   8. associate-/l/ 76.5%

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

   9. div-sub 76.5%

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

3. Simplified 76.5%

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

4. Taylor expanded in wj around 0 84.1%

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

   1. *-commutative 84.1%

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

6. Simplified 84.1%

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

7. Final simplification 84.1%

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

Alternative 11: 84.8% accurate, 44.7× speedup

Math

\[\begin{array}{l} \\ \frac{x}{wj \cdot 2 + 1} \end{array} \]

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 75.0%

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

   1. distribute-rgt1-in 76.1%

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

   2. *-commutative 76.1%

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

3. Simplified 76.1%

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

4. Taylor expanded in wj around 0 73.9%

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

   1. *-commutative 73.9%

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

6. Simplified 73.9%

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

7. Taylor expanded in x around inf 84.1%

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

8. Final simplification 84.1%

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

Alternative 12: 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 75.0%

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

   1. div-sub 75.0%

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

   2. distribute-rgt1-in 75.0%

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

   3. times-frac 75.0%

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

   4. *-inverses 75.4%

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

   5. associate-*l/ 75.4%

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

   6. *-rgt-identity 75.4%

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

   7. distribute-rgt1-in 76.5%

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

   8. associate-/l/ 76.5%

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

   9. div-sub 76.5%

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

3. Simplified 76.5%

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

4. Taylor expanded in wj around inf 3.9%

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

5. Final simplification 3.9%

   \[\leadsto wj \]

Alternative 13: 84.3% accurate, 313.0× speedup

Math

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

FPCore

(FPCore (wj x) :precision binary64 x)

C

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

Fortran

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

Java

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

Python

def code(wj, x):
	return x

Julia

function code(wj, x)
	return x
end

MATLAB

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

Wolfram

code[wj_, x_] := x

Derivation

1. Initial program 75.0%

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

   1. div-sub 75.0%

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

   2. distribute-rgt1-in 75.0%

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

   3. times-frac 75.0%

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

   4. *-inverses 75.4%

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

   5. associate-*l/ 75.4%

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

   6. *-rgt-identity 75.4%

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

   7. distribute-rgt1-in 76.5%

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

   8. associate-/l/ 76.5%

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

   9. div-sub 76.5%

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

3. Simplified 76.5%

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

4. Taylor expanded in wj around 0 83.5%

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

5. Final simplification 83.5%

   \[\leadsto x \]

Developer target: 78.8% 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 2023312 
(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))))))