Jmat.Real.lambertw, newton loop step

Percentage Accurate: 77.7% → 99.7%

Time: 18.8s

Alternatives: 14

Speedup: 313.0×

• could not determine a ground truth

  1. wj = -9.575979046246e+143
  2. x = -7.682350656865753e-227

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: 77.7% 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: 99.7% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;wj \leq -3.1 \cdot 10^{-6} \lor \neg \left(wj \leq 8 \cdot 10^{-8}\right):\\ \;\;\;\;wj + \frac{\frac{x}{e^{wj}} - wj}{wj + 1}\\ \mathbf{else}:\\ \;\;\;\;x + \left(-2 \cdot \left(wj \cdot x\right) + \left({wj}^{2} \cdot \left(1 - \left(x \cdot -4 + x \cdot 1.5\right)\right) - {wj}^{3}\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (wj x)
 :precision binary64
 (if (or (<= wj -3.1e-6) (not (<= wj 8e-8)))
   (+ 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)))) (pow wj 3.0))))))

C

double code(double wj, double x) {
	double tmp;
	if ((wj <= -3.1e-6) || !(wj <= 8e-8)) {
		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)))) - 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 <= (-3.1d-6)) .or. (.not. (wj <= 8d-8))) 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)))) - (wj ** 3.0d0)))
    end if
    code = tmp
end function

Java

public static double code(double wj, double x) {
	double tmp;
	if ((wj <= -3.1e-6) || !(wj <= 8e-8)) {
		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)))) - Math.pow(wj, 3.0)));
	}
	return tmp;
}

Python

def code(wj, x):
	tmp = 0
	if (wj <= -3.1e-6) or not (wj <= 8e-8):
		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)))) - math.pow(wj, 3.0)))
	return tmp

Julia

function code(wj, x)
	tmp = 0.0
	if ((wj <= -3.1e-6) || !(wj <= 8e-8))
		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 ^ 2.0) * Float64(1.0 - Float64(Float64(x * -4.0) + Float64(x * 1.5)))) - (wj ^ 3.0))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(wj, x)
	tmp = 0.0;
	if ((wj <= -3.1e-6) || ~((wj <= 8e-8)))
		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)))) - (wj ^ 3.0)));
	end
	tmp_2 = tmp;
end

Wolfram

code[wj_, x_] := If[Or[LessEqual[wj, -3.1e-6], N[Not[LessEqual[wj, 8e-8]], $MachinePrecision]], N[(wj + N[(N[(N[(x / N[Exp[wj], $MachinePrecision]), $MachinePrecision] - wj), $MachinePrecision] / N[(wj + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x + N[(N[(-2.0 * N[(wj * x), $MachinePrecision]), $MachinePrecision] + N[(N[(N[Power[wj, 2.0], $MachinePrecision] * N[(1.0 - N[(N[(x * -4.0), $MachinePrecision] + N[(x * 1.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[Power[wj, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if wj < -3.1e-6 or 8.0000000000000002e-8 < wj

   1. Initial program 73.3%

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

      1. distribute-rgt1-in 82.8%

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

      2. associate-/l/ 83.1%

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

      3. div-sub 73.5%

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

      4. associate-/l* 73.5%

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

      5. *-inverses 97.4%

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

      6. /-rgt-identity 97.4%

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

   3. Simplified 97.4%

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

   if -3.1e-6 < wj < 8.0000000000000002e-8

   1. Initial program 76.4%

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

      1. distribute-rgt1-in 76.4%

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

      2. associate-/l/ 76.4%

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

      3. div-sub 76.4%

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

      4. associate-/l* 76.4%

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

      5. *-inverses 76.4%

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

      6. /-rgt-identity 76.4%

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

   3. Simplified 76.4%

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

   5. Taylor expanded in wj around 0 100.0%

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

   6. Taylor expanded in x around 0 100.0%

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

4. Final simplification 99.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;wj \leq -3.1 \cdot 10^{-6} \lor \neg \left(wj \leq 8 \cdot 10^{-8}\right):\\ \;\;\;\;wj + \frac{\frac{x}{e^{wj}} - wj}{wj + 1}\\ \mathbf{else}:\\ \;\;\;\;x + \left(-2 \cdot \left(wj \cdot x\right) + \left({wj}^{2} \cdot \left(1 - \left(x \cdot -4 + x \cdot 1.5\right)\right) - {wj}^{3}\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 99.4% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;wj \leq -3.1 \cdot 10^{-8} \lor \neg \left(wj \leq 9 \cdot 10^{-11}\right):\\ \;\;\;\;wj + \frac{\frac{x}{e^{wj}} - wj}{wj + 1}\\ \mathbf{else}:\\ \;\;\;\;x + \left({wj}^{2} - {wj}^{3}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (wj x)
 :precision binary64
 (if (or (<= wj -3.1e-8) (not (<= wj 9e-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 <= -3.1e-8) || !(wj <= 9e-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 <= (-3.1d-8)) .or. (.not. (wj <= 9d-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 <= -3.1e-8) || !(wj <= 9e-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 <= -3.1e-8) or not (wj <= 9e-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 <= -3.1e-8) || !(wj <= 9e-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 <= -3.1e-8) || ~((wj <= 9e-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[Or[LessEqual[wj, -3.1e-8], N[Not[LessEqual[wj, 9e-11]], $MachinePrecision]], 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 < -3.1e-8 or 8.9999999999999999e-11 < wj

   1. Initial program 74.6%

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

      1. distribute-rgt1-in 83.6%

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

      2. associate-/l/ 83.8%

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

      3. div-sub 74.8%

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

      4. associate-/l* 74.8%

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

      5. *-inverses 97.5%

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

      6. /-rgt-identity 97.5%

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

   3. Simplified 97.5%

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

   if -3.1e-8 < wj < 8.9999999999999999e-11

   1. Initial program 76.3%

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

      1. distribute-rgt1-in 76.3%

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

      2. associate-/l/ 76.3%

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

      3. div-sub 76.3%

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

      4. associate-/l* 76.3%

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

      5. *-inverses 76.3%

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

      6. /-rgt-identity 76.3%

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

   3. Simplified 76.3%

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

   5. Taylor expanded in wj around 0 100.0%

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

   6. Taylor expanded in x around 0 100.0%

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

   7. Taylor expanded in x around 0 99.9%

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

      1. +-commutative 99.9%

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

      2. neg-mul-1 99.9%

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

      3. unsub-neg 99.9%

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

   9. Simplified 99.9%

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

4. Final simplification 99.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;wj \leq -3.1 \cdot 10^{-8} \lor \neg \left(wj \leq 9 \cdot 10^{-11}\right):\\ \;\;\;\;wj + \frac{\frac{x}{e^{wj}} - wj}{wj + 1}\\ \mathbf{else}:\\ \;\;\;\;x + \left({wj}^{2} - {wj}^{3}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 99.4% accurate, 2.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;wj \leq -2.55 \cdot 10^{-6} \lor \neg \left(wj \leq 1.16 \cdot 10^{-8}\right):\\ \;\;\;\;wj + \frac{\frac{x}{e^{wj}} - wj}{wj + 1}\\ \mathbf{else}:\\ \;\;\;\;x + \left(-2 \cdot \left(wj \cdot x\right) + {wj}^{2} \cdot \left(1 - \left(x \cdot -4 + x \cdot 1.5\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (wj x)
 :precision binary64
 (if (or (<= wj -2.55e-6) (not (<= wj 1.16e-8)))
   (+ 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 <= -2.55e-6) || !(wj <= 1.16e-8)) {
		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 <= (-2.55d-6)) .or. (.not. (wj <= 1.16d-8))) 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 <= -2.55e-6) || !(wj <= 1.16e-8)) {
		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 <= -2.55e-6) or not (wj <= 1.16e-8):
		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 <= -2.55e-6) || !(wj <= 1.16e-8))
		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 <= -2.55e-6) || ~((wj <= 1.16e-8)))
		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[Or[LessEqual[wj, -2.55e-6], N[Not[LessEqual[wj, 1.16e-8]], $MachinePrecision]], 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 < -2.5500000000000001e-6 or 1.15999999999999996e-8 < wj

   1. Initial program 73.3%

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

      1. distribute-rgt1-in 82.8%

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

      2. associate-/l/ 83.1%

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

      3. div-sub 73.5%

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

      4. associate-/l* 73.5%

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

      5. *-inverses 97.4%

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

      6. /-rgt-identity 97.4%

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

   3. Simplified 97.4%

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

   if -2.5500000000000001e-6 < wj < 1.15999999999999996e-8

   1. Initial program 76.4%

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

      1. distribute-rgt1-in 76.4%

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

      2. associate-/l/ 76.4%

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

      3. div-sub 76.4%

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

      4. associate-/l* 76.4%

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

      5. *-inverses 76.4%

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

      6. /-rgt-identity 76.4%

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

   3. Simplified 76.4%

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

   5. Taylor expanded in wj around 0 99.7%

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

4. Final simplification 99.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;wj \leq -2.55 \cdot 10^{-6} \lor \neg \left(wj \leq 1.16 \cdot 10^{-8}\right):\\ \;\;\;\;wj + \frac{\frac{x}{e^{wj}} - wj}{wj + 1}\\ \mathbf{else}:\\ \;\;\;\;x + \left(-2 \cdot \left(wj \cdot x\right) + {wj}^{2} \cdot \left(1 - \left(x \cdot -4 + x \cdot 1.5\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 99.2% accurate, 2.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;wj \leq -6 \cdot 10^{-10} \lor \neg \left(wj \leq 9.5 \cdot 10^{-12}\right):\\ \;\;\;\;wj + \frac{\frac{x}{e^{wj}} - wj}{wj + 1}\\ \mathbf{else}:\\ \;\;\;\;x + {wj}^{2} \cdot \left(1 + x \cdot 2\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (wj x)
 :precision binary64
 (if (or (<= wj -6e-10) (not (<= wj 9.5e-12)))
   (+ wj (/ (- (/ x (exp wj)) wj) (+ wj 1.0)))
   (+ x (* (pow wj 2.0) (+ 1.0 (* x 2.0))))))

C

double code(double wj, double x) {
	double tmp;
	if ((wj <= -6e-10) || !(wj <= 9.5e-12)) {
		tmp = wj + (((x / exp(wj)) - wj) / (wj + 1.0));
	} else {
		tmp = x + (pow(wj, 2.0) * (1.0 + (x * 2.0)));
	}
	return tmp;
}

Fortran

real(8) function code(wj, x)
    real(8), intent (in) :: wj
    real(8), intent (in) :: x
    real(8) :: tmp
    if ((wj <= (-6d-10)) .or. (.not. (wj <= 9.5d-12))) then
        tmp = wj + (((x / exp(wj)) - wj) / (wj + 1.0d0))
    else
        tmp = x + ((wj ** 2.0d0) * (1.0d0 + (x * 2.0d0)))
    end if
    code = tmp
end function

Java

public static double code(double wj, double x) {
	double tmp;
	if ((wj <= -6e-10) || !(wj <= 9.5e-12)) {
		tmp = wj + (((x / Math.exp(wj)) - wj) / (wj + 1.0));
	} else {
		tmp = x + (Math.pow(wj, 2.0) * (1.0 + (x * 2.0)));
	}
	return tmp;
}

Python

def code(wj, x):
	tmp = 0
	if (wj <= -6e-10) or not (wj <= 9.5e-12):
		tmp = wj + (((x / math.exp(wj)) - wj) / (wj + 1.0))
	else:
		tmp = x + (math.pow(wj, 2.0) * (1.0 + (x * 2.0)))
	return tmp

Julia

function code(wj, x)
	tmp = 0.0
	if ((wj <= -6e-10) || !(wj <= 9.5e-12))
		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 * 2.0))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(wj, x)
	tmp = 0.0;
	if ((wj <= -6e-10) || ~((wj <= 9.5e-12)))
		tmp = wj + (((x / exp(wj)) - wj) / (wj + 1.0));
	else
		tmp = x + ((wj ^ 2.0) * (1.0 + (x * 2.0)));
	end
	tmp_2 = tmp;
end

Wolfram

code[wj_, x_] := If[Or[LessEqual[wj, -6e-10], N[Not[LessEqual[wj, 9.5e-12]], $MachinePrecision]], 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 * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if wj < -6e-10 or 9.4999999999999995e-12 < wj

   1. Initial program 74.6%

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

      1. distribute-rgt1-in 83.6%

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

      2. associate-/l/ 83.8%

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

      3. div-sub 74.8%

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

      4. associate-/l* 74.8%

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

      5. *-inverses 97.5%

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

      6. /-rgt-identity 97.5%

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

   3. Simplified 97.5%

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

   if -6e-10 < wj < 9.4999999999999995e-12

   1. Initial program 76.3%

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

      1. distribute-rgt1-in 76.3%

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

      2. associate-/l/ 76.3%

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

      3. div-sub 76.3%

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

      4. associate-/l* 76.3%

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

      5. *-inverses 76.3%

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

      6. /-rgt-identity 76.3%

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

   3. Simplified 76.3%

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

   5. Taylor expanded in wj around 0 76.3%

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

      1. associate-*r* 76.3%

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

      2. neg-mul-1 76.3%

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

      3. distribute-rgt1-in 76.3%

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

      4. +-commutative 76.3%

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

      5. sub-neg 76.3%

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

   7. Simplified 76.3%

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

   8. Taylor expanded in wj around 0 99.7%

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

   9. Taylor expanded in wj around inf 99.7%

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

       1. cancel-sign-sub-inv 99.7%

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

       2. metadata-eval 99.7%

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

       3. *-commutative 99.7%

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

       4. *-rgt-identity 99.7%

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

       5. associate-+r+ 99.7%

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

       6. count-2 99.7%

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

   11. Simplified 99.7%

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

4. Final simplification 99.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;wj \leq -6 \cdot 10^{-10} \lor \neg \left(wj \leq 9.5 \cdot 10^{-12}\right):\\ \;\;\;\;wj + \frac{\frac{x}{e^{wj}} - wj}{wj + 1}\\ \mathbf{else}:\\ \;\;\;\;x + {wj}^{2} \cdot \left(1 + x \cdot 2\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 95.9% accurate, 2.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 1.4 \cdot 10^{-78}:\\ \;\;\;\;x + {wj}^{2} \cdot \left(1 + x \cdot 2\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{e^{wj}}}{wj + 1}\\ \end{array} \end{array} \]

FPCore

(FPCore (wj x)
 :precision binary64
 (if (<= x 1.4e-78)
   (+ x (* (pow wj 2.0) (+ 1.0 (* x 2.0))))
   (/ (/ x (exp wj)) (+ wj 1.0))))

C

double code(double wj, double x) {
	double tmp;
	if (x <= 1.4e-78) {
		tmp = x + (pow(wj, 2.0) * (1.0 + (x * 2.0)));
	} else {
		tmp = (x / exp(wj)) / (wj + 1.0);
	}
	return tmp;
}

Fortran

real(8) function code(wj, x)
    real(8), intent (in) :: wj
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= 1.4d-78) then
        tmp = x + ((wj ** 2.0d0) * (1.0d0 + (x * 2.0d0)))
    else
        tmp = (x / exp(wj)) / (wj + 1.0d0)
    end if
    code = tmp
end function

Java

public static double code(double wj, double x) {
	double tmp;
	if (x <= 1.4e-78) {
		tmp = x + (Math.pow(wj, 2.0) * (1.0 + (x * 2.0)));
	} else {
		tmp = (x / Math.exp(wj)) / (wj + 1.0);
	}
	return tmp;
}

Python

def code(wj, x):
	tmp = 0
	if x <= 1.4e-78:
		tmp = x + (math.pow(wj, 2.0) * (1.0 + (x * 2.0)))
	else:
		tmp = (x / math.exp(wj)) / (wj + 1.0)
	return tmp

Julia

function code(wj, x)
	tmp = 0.0
	if (x <= 1.4e-78)
		tmp = Float64(x + Float64((wj ^ 2.0) * Float64(1.0 + Float64(x * 2.0))));
	else
		tmp = Float64(Float64(x / exp(wj)) / Float64(wj + 1.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(wj, x)
	tmp = 0.0;
	if (x <= 1.4e-78)
		tmp = x + ((wj ^ 2.0) * (1.0 + (x * 2.0)));
	else
		tmp = (x / exp(wj)) / (wj + 1.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[wj_, x_] := If[LessEqual[x, 1.4e-78], N[(x + N[(N[Power[wj, 2.0], $MachinePrecision] * N[(1.0 + N[(x * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x / N[Exp[wj], $MachinePrecision]), $MachinePrecision] / N[(wj + 1.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 1.40000000000000012e-78

   1. Initial program 67.2%

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

      1. distribute-rgt1-in 67.2%

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

      2. associate-/l/ 67.2%

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

      3. div-sub 67.2%

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

      4. associate-/l* 67.2%

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

      5. *-inverses 67.2%

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

      6. /-rgt-identity 67.2%

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

   3. Simplified 67.2%

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

   5. Taylor expanded in wj around 0 66.2%

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

      1. associate-*r* 66.2%

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

      2. neg-mul-1 66.2%

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

      3. distribute-rgt1-in 66.2%

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

      4. +-commutative 66.2%

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

      5. sub-neg 66.2%

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

   7. Simplified 66.2%

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

   8. Taylor expanded in wj around 0 95.9%

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

   9. Taylor expanded in wj around inf 95.5%

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

       1. cancel-sign-sub-inv 95.5%

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

       2. metadata-eval 95.5%

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

       3. *-commutative 95.5%

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

       4. *-rgt-identity 95.5%

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

       5. associate-+r+ 95.5%

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

       6. count-2 95.5%

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

   11. Simplified 95.5%

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

   if 1.40000000000000012e-78 < x

   1. Initial program 93.4%

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

      1. distribute-rgt1-in 95.6%

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

      2. associate-/l/ 95.7%

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

      3. div-sub 93.4%

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

      4. associate-/l* 93.4%

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

      5. *-inverses 99.1%

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

      6. /-rgt-identity 99.1%

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

   3. Simplified 99.1%

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

      1. flip-- 63.9%

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

      2. div-inv 63.6%

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

      3. pow2 63.6%

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

      4. pow2 63.6%

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

   6. Applied egg-rr 63.6%

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

      1. associate-*r/ 63.9%

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

      2. *-rgt-identity 63.9%

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

   8. Simplified 63.9%

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

   9. Taylor expanded in x around inf 94.8%

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

       1. associate-/r* 94.8%

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

       2. +-commutative 94.8%

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

   11. Simplified 94.8%

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

4. Final simplification 95.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 1.4 \cdot 10^{-78}:\\ \;\;\;\;x + {wj}^{2} \cdot \left(1 + x \cdot 2\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{e^{wj}}}{wj + 1}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 86.1% 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 76.2%

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

   1. distribute-rgt1-in 77.0%

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

   2. associate-/l/ 77.0%

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

   3. div-sub 76.2%

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

   4. associate-/l* 76.2%

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

   5. *-inverses 78.1%

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

   6. /-rgt-identity 78.1%

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

3. Simplified 78.1%

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

5. Taylor expanded in x around inf 86.4%

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

   1. +-commutative 86.4%

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

7. Simplified 86.4%

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

8. Final simplification 86.4%

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

Alternative 7: 86.1% accurate, 2.9× speedup

Math

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

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(Float64(x / 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[(N[(x / N[Exp[wj], $MachinePrecision]), $MachinePrecision] / N[(wj + 1.0), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 76.2%

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

   1. distribute-rgt1-in 77.0%

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

   2. associate-/l/ 77.0%

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

   3. div-sub 76.2%

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

   4. associate-/l* 76.2%

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

   5. *-inverses 78.1%

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

   6. /-rgt-identity 78.1%

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

3. Simplified 78.1%

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

   1. flip-- 50.9%

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

   2. div-inv 50.8%

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

   3. pow2 50.8%

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

   4. pow2 50.8%

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

6. Applied egg-rr 50.8%

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

   1. associate-*r/ 50.9%

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

   2. *-rgt-identity 50.9%

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

8. Simplified 50.9%

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

9. Taylor expanded in x around inf 86.4%

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

    1. associate-/r* 86.4%

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

    2. +-commutative 86.4%

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

11. Simplified 86.4%

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

12. Final simplification 86.4%

    \[\leadsto \frac{\frac{x}{e^{wj}}}{wj + 1} \]
13. Add Preprocessing

Alternative 8: 85.3% accurate, 17.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x \cdot \left(1 - wj\right)\\ \mathbf{if}\;wj \leq -6.5 \cdot 10^{-17}:\\ \;\;\;\;wj + \frac{t\_0 - wj}{wj + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{t\_0}{wj + 1}\\ \end{array} \end{array} \]

FPCore

(FPCore (wj x)
 :precision binary64
 (let* ((t_0 (* x (- 1.0 wj))))
   (if (<= wj -6.5e-17) (+ wj (/ (- t_0 wj) (+ wj 1.0))) (/ t_0 (+ wj 1.0)))))

C

double code(double wj, double x) {
	double t_0 = x * (1.0 - wj);
	double tmp;
	if (wj <= -6.5e-17) {
		tmp = wj + ((t_0 - wj) / (wj + 1.0));
	} else {
		tmp = t_0 / (wj + 1.0);
	}
	return tmp;
}

Fortran

real(8) function code(wj, x)
    real(8), intent (in) :: wj
    real(8), intent (in) :: x
    real(8) :: t_0
    real(8) :: tmp
    t_0 = x * (1.0d0 - wj)
    if (wj <= (-6.5d-17)) then
        tmp = wj + ((t_0 - wj) / (wj + 1.0d0))
    else
        tmp = t_0 / (wj + 1.0d0)
    end if
    code = tmp
end function

Java

public static double code(double wj, double x) {
	double t_0 = x * (1.0 - wj);
	double tmp;
	if (wj <= -6.5e-17) {
		tmp = wj + ((t_0 - wj) / (wj + 1.0));
	} else {
		tmp = t_0 / (wj + 1.0);
	}
	return tmp;
}

Python

def code(wj, x):
	t_0 = x * (1.0 - wj)
	tmp = 0
	if wj <= -6.5e-17:
		tmp = wj + ((t_0 - wj) / (wj + 1.0))
	else:
		tmp = t_0 / (wj + 1.0)
	return tmp

Julia

function code(wj, x)
	t_0 = Float64(x * Float64(1.0 - wj))
	tmp = 0.0
	if (wj <= -6.5e-17)
		tmp = Float64(wj + Float64(Float64(t_0 - wj) / Float64(wj + 1.0)));
	else
		tmp = Float64(t_0 / Float64(wj + 1.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(wj, x)
	t_0 = x * (1.0 - wj);
	tmp = 0.0;
	if (wj <= -6.5e-17)
		tmp = wj + ((t_0 - wj) / (wj + 1.0));
	else
		tmp = t_0 / (wj + 1.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[wj_, x_] := Block[{t$95$0 = N[(x * N[(1.0 - wj), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[wj, -6.5e-17], N[(wj + N[(N[(t$95$0 - wj), $MachinePrecision] / N[(wj + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$0 / N[(wj + 1.0), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if wj < -6.4999999999999996e-17

   1. Initial program 78.9%

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

      1. distribute-rgt1-in 93.1%

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

      2. associate-/l/ 93.8%

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

      3. div-sub 79.5%

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

      4. associate-/l* 79.5%

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

      5. *-inverses 93.8%

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

      6. /-rgt-identity 93.8%

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

   3. Simplified 93.8%

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

   5. Taylor expanded in wj around 0 56.2%

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

      1. associate-*r* 56.2%

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

      2. neg-mul-1 56.2%

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

      3. distribute-rgt1-in 56.2%

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

      4. +-commutative 56.2%

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

      5. sub-neg 56.2%

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

   7. Simplified 56.2%

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

   if -6.4999999999999996e-17 < wj

   1. Initial program 76.0%

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

      1. distribute-rgt1-in 76.0%

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

      2. associate-/l/ 76.0%

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

      3. div-sub 76.0%

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

      4. associate-/l* 76.0%

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

      5. *-inverses 77.2%

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

      6. /-rgt-identity 77.2%

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

   3. Simplified 77.2%

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

   5. Taylor expanded in wj around 0 74.9%

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

      1. associate-*r* 74.9%

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

      2. neg-mul-1 74.9%

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

      3. distribute-rgt1-in 74.9%

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

      4. +-commutative 74.9%

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

      5. sub-neg 74.9%

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

   7. Simplified 74.9%

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

   8. Taylor expanded in x around -inf 87.3%

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

4. Final simplification 85.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;wj \leq -6.5 \cdot 10^{-17}:\\ \;\;\;\;wj + \frac{x \cdot \left(1 - wj\right) - wj}{wj + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot \left(1 - wj\right)}{wj + 1}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 84.7% accurate, 22.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;wj \leq -0.000115:\\ \;\;\;\;wj - \frac{wj}{wj + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\frac{wj + 1}{1 - wj}}\\ \end{array} \end{array} \]

FPCore

(FPCore (wj x)
 :precision binary64
 (if (<= wj -0.000115)
   (- wj (/ wj (+ wj 1.0)))
   (/ x (/ (+ wj 1.0) (- 1.0 wj)))))

C

double code(double wj, double x) {
	double tmp;
	if (wj <= -0.000115) {
		tmp = wj - (wj / (wj + 1.0));
	} else {
		tmp = x / ((wj + 1.0) / (1.0 - wj));
	}
	return tmp;
}

Fortran

real(8) function code(wj, x)
    real(8), intent (in) :: wj
    real(8), intent (in) :: x
    real(8) :: tmp
    if (wj <= (-0.000115d0)) then
        tmp = wj - (wj / (wj + 1.0d0))
    else
        tmp = x / ((wj + 1.0d0) / (1.0d0 - wj))
    end if
    code = tmp
end function

Java

public static double code(double wj, double x) {
	double tmp;
	if (wj <= -0.000115) {
		tmp = wj - (wj / (wj + 1.0));
	} else {
		tmp = x / ((wj + 1.0) / (1.0 - wj));
	}
	return tmp;
}

Python

def code(wj, x):
	tmp = 0
	if wj <= -0.000115:
		tmp = wj - (wj / (wj + 1.0))
	else:
		tmp = x / ((wj + 1.0) / (1.0 - wj))
	return tmp

Julia

function code(wj, x)
	tmp = 0.0
	if (wj <= -0.000115)
		tmp = Float64(wj - Float64(wj / Float64(wj + 1.0)));
	else
		tmp = Float64(x / Float64(Float64(wj + 1.0) / Float64(1.0 - wj)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(wj, x)
	tmp = 0.0;
	if (wj <= -0.000115)
		tmp = wj - (wj / (wj + 1.0));
	else
		tmp = x / ((wj + 1.0) / (1.0 - wj));
	end
	tmp_2 = tmp;
end

Wolfram

code[wj_, x_] := If[LessEqual[wj, -0.000115], N[(wj - N[(wj / N[(wj + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x / N[(N[(wj + 1.0), $MachinePrecision] / N[(1.0 - wj), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if wj < -1.15e-4

   1. Initial program 77.5%

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

      1. distribute-rgt1-in 95.7%

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

      2. associate-/l/ 96.2%

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

      3. div-sub 78.0%

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

      4. associate-/l* 78.0%

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

      5. *-inverses 96.2%

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

      6. /-rgt-identity 96.2%

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

   3. Simplified 96.2%

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

   5. Taylor expanded in x around 0 53.2%

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

      1. +-commutative 53.2%

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

   7. Simplified 53.2%

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

   if -1.15e-4 < wj

   1. Initial program 76.1%

      \[wj - \frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}} \]
   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. associate-/l/ 76.1%

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

      3. div-sub 76.1%

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

      4. associate-/l* 76.1%

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

      5. *-inverses 77.3%

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

      6. /-rgt-identity 77.3%

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

   3. Simplified 77.3%

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

   5. Taylor expanded in wj around 0 74.8%

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

      1. associate-*r* 74.8%

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

      2. neg-mul-1 74.8%

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

      3. distribute-rgt1-in 74.8%

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

      4. +-commutative 74.8%

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

      5. sub-neg 74.8%

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

   7. Simplified 74.8%

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

   8. Taylor expanded in x around -inf 86.8%

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

      1. associate-/l* 86.8%

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

      2. +-commutative 86.8%

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

   10. Simplified 86.8%

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

4. Final simplification 85.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;wj \leq -0.000115:\\ \;\;\;\;wj - \frac{wj}{wj + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\frac{wj + 1}{1 - wj}}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 84.7% accurate, 22.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;wj \leq -0.000165:\\ \;\;\;\;wj - \frac{wj}{wj + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot \left(1 - wj\right)}{wj + 1}\\ \end{array} \end{array} \]

FPCore

(FPCore (wj x)
 :precision binary64
 (if (<= wj -0.000165)
   (- wj (/ wj (+ wj 1.0)))
   (/ (* x (- 1.0 wj)) (+ wj 1.0))))

C

double code(double wj, double x) {
	double tmp;
	if (wj <= -0.000165) {
		tmp = wj - (wj / (wj + 1.0));
	} else {
		tmp = (x * (1.0 - wj)) / (wj + 1.0);
	}
	return tmp;
}

Fortran

real(8) function code(wj, x)
    real(8), intent (in) :: wj
    real(8), intent (in) :: x
    real(8) :: tmp
    if (wj <= (-0.000165d0)) then
        tmp = wj - (wj / (wj + 1.0d0))
    else
        tmp = (x * (1.0d0 - wj)) / (wj + 1.0d0)
    end if
    code = tmp
end function

Java

public static double code(double wj, double x) {
	double tmp;
	if (wj <= -0.000165) {
		tmp = wj - (wj / (wj + 1.0));
	} else {
		tmp = (x * (1.0 - wj)) / (wj + 1.0);
	}
	return tmp;
}

Python

def code(wj, x):
	tmp = 0
	if wj <= -0.000165:
		tmp = wj - (wj / (wj + 1.0))
	else:
		tmp = (x * (1.0 - wj)) / (wj + 1.0)
	return tmp

Julia

function code(wj, x)
	tmp = 0.0
	if (wj <= -0.000165)
		tmp = Float64(wj - Float64(wj / Float64(wj + 1.0)));
	else
		tmp = Float64(Float64(x * Float64(1.0 - wj)) / Float64(wj + 1.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(wj, x)
	tmp = 0.0;
	if (wj <= -0.000165)
		tmp = wj - (wj / (wj + 1.0));
	else
		tmp = (x * (1.0 - wj)) / (wj + 1.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[wj_, x_] := If[LessEqual[wj, -0.000165], N[(wj - N[(wj / N[(wj + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x * N[(1.0 - wj), $MachinePrecision]), $MachinePrecision] / N[(wj + 1.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if wj < -1.65e-4

   1. Initial program 77.5%

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

      1. distribute-rgt1-in 95.7%

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

      2. associate-/l/ 96.2%

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

      3. div-sub 78.0%

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

      4. associate-/l* 78.0%

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

      5. *-inverses 96.2%

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

      6. /-rgt-identity 96.2%

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

   3. Simplified 96.2%

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

   5. Taylor expanded in x around 0 53.2%

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

      1. +-commutative 53.2%

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

   7. Simplified 53.2%

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

   if -1.65e-4 < wj

   1. Initial program 76.1%

      \[wj - \frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}} \]
   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. associate-/l/ 76.1%

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

      3. div-sub 76.1%

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

      4. associate-/l* 76.1%

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

      5. *-inverses 77.3%

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

      6. /-rgt-identity 77.3%

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

   3. Simplified 77.3%

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

   5. Taylor expanded in wj around 0 74.8%

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

      1. associate-*r* 74.8%

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

      2. neg-mul-1 74.8%

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

      3. distribute-rgt1-in 74.8%

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

      4. +-commutative 74.8%

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

      5. sub-neg 74.8%

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

   7. Simplified 74.8%

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

   8. Taylor expanded in x around -inf 86.8%

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

4. Final simplification 85.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;wj \leq -0.000165:\\ \;\;\;\;wj - \frac{wj}{wj + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot \left(1 - wj\right)}{wj + 1}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 84.7% accurate, 26.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;wj \leq -9 \cdot 10^{-5}:\\ \;\;\;\;wj - \frac{wj}{wj + 1}\\ \mathbf{else}:\\ \;\;\;\;x + -2 \cdot \left(wj \cdot x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (wj x)
 :precision binary64
 (if (<= wj -9e-5) (- wj (/ wj (+ wj 1.0))) (+ x (* -2.0 (* wj x)))))

C

double code(double wj, double x) {
	double tmp;
	if (wj <= -9e-5) {
		tmp = wj - (wj / (wj + 1.0));
	} else {
		tmp = x + (-2.0 * (wj * x));
	}
	return tmp;
}

Fortran

real(8) function code(wj, x)
    real(8), intent (in) :: wj
    real(8), intent (in) :: x
    real(8) :: tmp
    if (wj <= (-9d-5)) then
        tmp = wj - (wj / (wj + 1.0d0))
    else
        tmp = x + ((-2.0d0) * (wj * x))
    end if
    code = tmp
end function

Java

public static double code(double wj, double x) {
	double tmp;
	if (wj <= -9e-5) {
		tmp = wj - (wj / (wj + 1.0));
	} else {
		tmp = x + (-2.0 * (wj * x));
	}
	return tmp;
}

Python

def code(wj, x):
	tmp = 0
	if wj <= -9e-5:
		tmp = wj - (wj / (wj + 1.0))
	else:
		tmp = x + (-2.0 * (wj * x))
	return tmp

Julia

function code(wj, x)
	tmp = 0.0
	if (wj <= -9e-5)
		tmp = Float64(wj - Float64(wj / Float64(wj + 1.0)));
	else
		tmp = Float64(x + Float64(-2.0 * Float64(wj * x)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(wj, x)
	tmp = 0.0;
	if (wj <= -9e-5)
		tmp = wj - (wj / (wj + 1.0));
	else
		tmp = x + (-2.0 * (wj * x));
	end
	tmp_2 = tmp;
end

Wolfram

code[wj_, x_] := If[LessEqual[wj, -9e-5], N[(wj - N[(wj / N[(wj + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x + N[(-2.0 * N[(wj * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if wj < -9.00000000000000057e-5

   1. Initial program 77.5%

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

      1. distribute-rgt1-in 95.7%

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

      2. associate-/l/ 96.2%

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

      3. div-sub 78.0%

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

      4. associate-/l* 78.0%

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

      5. *-inverses 96.2%

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

      6. /-rgt-identity 96.2%

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

   3. Simplified 96.2%

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

   5. Taylor expanded in x around 0 53.2%

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

      1. +-commutative 53.2%

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

   7. Simplified 53.2%

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

   if -9.00000000000000057e-5 < wj

   1. Initial program 76.1%

      \[wj - \frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}} \]
   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. associate-/l/ 76.1%

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

      3. div-sub 76.1%

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

      4. associate-/l* 76.1%

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

      5. *-inverses 77.3%

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

      6. /-rgt-identity 77.3%

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

   3. Simplified 77.3%

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

   5. Taylor expanded in wj around 0 86.7%

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

      1. *-commutative 86.7%

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

   7. Simplified 86.7%

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

4. Final simplification 85.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;wj \leq -9 \cdot 10^{-5}:\\ \;\;\;\;wj - \frac{wj}{wj + 1}\\ \mathbf{else}:\\ \;\;\;\;x + -2 \cdot \left(wj \cdot x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 84.4% 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 76.2%

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

   1. distribute-rgt1-in 77.0%

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

   2. associate-/l/ 77.0%

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

   3. div-sub 76.2%

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

   4. associate-/l* 76.2%

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

   5. *-inverses 78.1%

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

   6. /-rgt-identity 78.1%

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

3. Simplified 78.1%

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

5. Taylor expanded in wj around 0 83.1%

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

   1. *-commutative 83.1%

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

7. Simplified 83.1%

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

8. Final simplification 83.1%

   \[\leadsto x + -2 \cdot \left(wj \cdot x\right) \]
9. Add Preprocessing

Alternative 13: 4.3% accurate, 313.0× speedup

Math

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

FPCore

(FPCore (wj x) :precision binary64 wj)

C

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

Fortran

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

Java

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

Python

def code(wj, x):
	return wj

Julia

function code(wj, x)
	return wj
end

MATLAB

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

Wolfram

code[wj_, x_] := wj

Derivation

1. Initial program 76.2%

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

   1. distribute-rgt1-in 77.0%

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

   2. associate-/l/ 77.0%

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

   3. div-sub 76.2%

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

   4. associate-/l* 76.2%

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

   5. *-inverses 78.1%

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

   6. /-rgt-identity 78.1%

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

3. Simplified 78.1%

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

5. Taylor expanded in wj around inf 4.3%

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

6. Final simplification 4.3%

   \[\leadsto wj \]
7. Add Preprocessing

Alternative 14: 83.9% accurate, 313.0× speedup

Math

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

FPCore

(FPCore (wj x) :precision binary64 x)

C

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

Fortran

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

Java

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

Python

def code(wj, x):
	return x

Julia

function code(wj, x)
	return x
end

MATLAB

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

Wolfram

code[wj_, x_] := x

Derivation

1. Initial program 76.2%

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

   1. distribute-rgt1-in 77.0%

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

   2. associate-/l/ 77.0%

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

   3. div-sub 76.2%

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

   4. associate-/l* 76.2%

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

   5. *-inverses 78.1%

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

   6. /-rgt-identity 78.1%

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

3. Simplified 78.1%

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

5. Taylor expanded in wj around 0 82.7%

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

6. Final simplification 82.7%

   \[\leadsto x \]
7. Add Preprocessing

Developer target: 78.6% accurate, 1.5× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Reproduce

herbie shell --seed 2024036 
(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))))))