Jmat.Real.lambertw, newton loop step

Percentage Accurate: 78.1% → 97.7%

Time: 16.1s

Alternatives: 11

Speedup: 313.0×

• could not determine a ground truth

  1. wj = -9.718738292840118e+181
  2. x = 8.437016341105187e+70

Specification

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := wj \cdot e^{wj}\\ wj - \frac{t\_0 - x}{e^{wj} + t\_0} \end{array} \end{array} \]

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 78.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := wj \cdot e^{wj}\\ wj - \frac{t\_0 - x}{e^{wj} + t\_0} \end{array} \end{array} \]

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 97.7% accurate, 2.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;wj \leq -2.06 \cdot 10^{-7}:\\ \;\;\;\;x \cdot \left(\frac{e^{-wj}}{wj + 1} + \left(\frac{wj}{x} + \frac{wj}{x \cdot \left(-1 - wj\right)}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x + wj \cdot \left(wj \cdot \left(1 - wj\right) - x \cdot 2\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (wj x)
 :precision binary64
 (if (<= wj -2.06e-7)
   (* x (+ (/ (exp (- wj)) (+ wj 1.0)) (+ (/ wj x) (/ wj (* x (- -1.0 wj))))))
   (+ x (* wj (- (* wj (- 1.0 wj)) (* x 2.0))))))

C

double code(double wj, double x) {
	double tmp;
	if (wj <= -2.06e-7) {
		tmp = x * ((exp(-wj) / (wj + 1.0)) + ((wj / x) + (wj / (x * (-1.0 - wj)))));
	} else {
		tmp = x + (wj * ((wj * (1.0 - wj)) - (x * 2.0)));
	}
	return tmp;
}

Fortran

real(8) function code(wj, x)
    real(8), intent (in) :: wj
    real(8), intent (in) :: x
    real(8) :: tmp
    if (wj <= (-2.06d-7)) then
        tmp = x * ((exp(-wj) / (wj + 1.0d0)) + ((wj / x) + (wj / (x * ((-1.0d0) - wj)))))
    else
        tmp = x + (wj * ((wj * (1.0d0 - wj)) - (x * 2.0d0)))
    end if
    code = tmp
end function

Java

public static double code(double wj, double x) {
	double tmp;
	if (wj <= -2.06e-7) {
		tmp = x * ((Math.exp(-wj) / (wj + 1.0)) + ((wj / x) + (wj / (x * (-1.0 - wj)))));
	} else {
		tmp = x + (wj * ((wj * (1.0 - wj)) - (x * 2.0)));
	}
	return tmp;
}

Python

def code(wj, x):
	tmp = 0
	if wj <= -2.06e-7:
		tmp = x * ((math.exp(-wj) / (wj + 1.0)) + ((wj / x) + (wj / (x * (-1.0 - wj)))))
	else:
		tmp = x + (wj * ((wj * (1.0 - wj)) - (x * 2.0)))
	return tmp

Julia

function code(wj, x)
	tmp = 0.0
	if (wj <= -2.06e-7)
		tmp = Float64(x * Float64(Float64(exp(Float64(-wj)) / Float64(wj + 1.0)) + Float64(Float64(wj / x) + Float64(wj / Float64(x * Float64(-1.0 - wj))))));
	else
		tmp = Float64(x + Float64(wj * Float64(Float64(wj * Float64(1.0 - wj)) - Float64(x * 2.0))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(wj, x)
	tmp = 0.0;
	if (wj <= -2.06e-7)
		tmp = x * ((exp(-wj) / (wj + 1.0)) + ((wj / x) + (wj / (x * (-1.0 - wj)))));
	else
		tmp = x + (wj * ((wj * (1.0 - wj)) - (x * 2.0)));
	end
	tmp_2 = tmp;
end

Wolfram

code[wj_, x_] := If[LessEqual[wj, -2.06e-7], N[(x * N[(N[(N[Exp[(-wj)], $MachinePrecision] / N[(wj + 1.0), $MachinePrecision]), $MachinePrecision] + N[(N[(wj / x), $MachinePrecision] + N[(wj / N[(x * N[(-1.0 - wj), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x + N[(wj * N[(N[(wj * N[(1.0 - wj), $MachinePrecision]), $MachinePrecision] - N[(x * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if wj < -2.05999999999999992e-7

   1. Initial program 68.9%

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

      1. distribute-rgt1-in 96.2%

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

      2. associate-/l/ 96.4%

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

      3. div-sub 69.2%

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

      4. associate-/l* 69.2%

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

      5. *-inverses 96.4%

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

      6. *-rgt-identity 96.4%

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

   3. Simplified 96.4%

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

   5. Taylor expanded in x around inf 97.0%

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

      1. associate--l+ 97.0%

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

      2. associate-/r* 97.0%

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

      3. exp-neg 97.1%

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

      4. +-commutative 97.1%

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

      5. +-commutative 97.1%

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

   7. Simplified 97.1%

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

   if -2.05999999999999992e-7 < wj

   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 78.9%

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

      2. associate-/l/ 78.8%

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

      3. div-sub 78.8%

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

      4. associate-/l* 78.8%

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

      5. *-inverses 79.3%

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

      6. *-rgt-identity 79.3%

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

   3. Simplified 79.3%

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

   5. Taylor expanded in wj around 0 99.1%

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

   6. Taylor expanded in x around 0 99.2%

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

      1. neg-mul-1 99.2%

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

      2. sub-neg 99.2%

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

   8. Simplified 99.2%

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

4. Final simplification 99.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;wj \leq -2.06 \cdot 10^{-7}:\\ \;\;\;\;x \cdot \left(\frac{e^{-wj}}{wj + 1} + \left(\frac{wj}{x} + \frac{wj}{x \cdot \left(-1 - wj\right)}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x + wj \cdot \left(wj \cdot \left(1 - wj\right) - x \cdot 2\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 97.7% accurate, 2.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;wj \leq -1.42 \cdot 10^{-7}:\\ \;\;\;\;wj + \frac{wj - \frac{x}{e^{wj}}}{-1 - wj}\\ \mathbf{else}:\\ \;\;\;\;x + wj \cdot \left(wj \cdot \left(1 - wj\right) - x \cdot 2\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (wj x)
 :precision binary64
 (if (<= wj -1.42e-7)
   (+ wj (/ (- wj (/ x (exp wj))) (- -1.0 wj)))
   (+ x (* wj (- (* wj (- 1.0 wj)) (* x 2.0))))))

C

double code(double wj, double x) {
	double tmp;
	if (wj <= -1.42e-7) {
		tmp = wj + ((wj - (x / exp(wj))) / (-1.0 - wj));
	} else {
		tmp = x + (wj * ((wj * (1.0 - wj)) - (x * 2.0)));
	}
	return tmp;
}

Fortran

real(8) function code(wj, x)
    real(8), intent (in) :: wj
    real(8), intent (in) :: x
    real(8) :: tmp
    if (wj <= (-1.42d-7)) then
        tmp = wj + ((wj - (x / exp(wj))) / ((-1.0d0) - wj))
    else
        tmp = x + (wj * ((wj * (1.0d0 - wj)) - (x * 2.0d0)))
    end if
    code = tmp
end function

Java

public static double code(double wj, double x) {
	double tmp;
	if (wj <= -1.42e-7) {
		tmp = wj + ((wj - (x / Math.exp(wj))) / (-1.0 - wj));
	} else {
		tmp = x + (wj * ((wj * (1.0 - wj)) - (x * 2.0)));
	}
	return tmp;
}

Python

def code(wj, x):
	tmp = 0
	if wj <= -1.42e-7:
		tmp = wj + ((wj - (x / math.exp(wj))) / (-1.0 - wj))
	else:
		tmp = x + (wj * ((wj * (1.0 - wj)) - (x * 2.0)))
	return tmp

Julia

function code(wj, x)
	tmp = 0.0
	if (wj <= -1.42e-7)
		tmp = Float64(wj + Float64(Float64(wj - Float64(x / exp(wj))) / Float64(-1.0 - wj)));
	else
		tmp = Float64(x + Float64(wj * Float64(Float64(wj * Float64(1.0 - wj)) - Float64(x * 2.0))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(wj, x)
	tmp = 0.0;
	if (wj <= -1.42e-7)
		tmp = wj + ((wj - (x / exp(wj))) / (-1.0 - wj));
	else
		tmp = x + (wj * ((wj * (1.0 - wj)) - (x * 2.0)));
	end
	tmp_2 = tmp;
end

Wolfram

code[wj_, x_] := If[LessEqual[wj, -1.42e-7], N[(wj + N[(N[(wj - N[(x / N[Exp[wj], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(-1.0 - wj), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x + N[(wj * N[(N[(wj * N[(1.0 - wj), $MachinePrecision]), $MachinePrecision] - N[(x * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if wj < -1.42000000000000001e-7

   1. Initial program 68.9%

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

      1. distribute-rgt1-in 96.2%

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

      2. associate-/l/ 96.4%

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

      3. div-sub 69.2%

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

      4. associate-/l* 69.2%

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

      5. *-inverses 96.4%

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

      6. *-rgt-identity 96.4%

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

   3. Simplified 96.4%

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

   if -1.42000000000000001e-7 < wj

   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 78.9%

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

      2. associate-/l/ 78.8%

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

      3. div-sub 78.8%

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

      4. associate-/l* 78.8%

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

      5. *-inverses 79.3%

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

      6. *-rgt-identity 79.3%

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

   3. Simplified 79.3%

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

   5. Taylor expanded in wj around 0 99.1%

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

   6. Taylor expanded in x around 0 99.2%

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

      1. neg-mul-1 99.2%

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

      2. sub-neg 99.2%

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

   8. Simplified 99.2%

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

4. Final simplification 99.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;wj \leq -1.42 \cdot 10^{-7}:\\ \;\;\;\;wj + \frac{wj - \frac{x}{e^{wj}}}{-1 - wj}\\ \mathbf{else}:\\ \;\;\;\;x + wj \cdot \left(wj \cdot \left(1 - wj\right) - x \cdot 2\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 97.4% accurate, 2.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;wj \leq -0.78:\\ \;\;\;\;\frac{x}{\left(wj + 1\right) \cdot e^{wj}}\\ \mathbf{else}:\\ \;\;\;\;x + wj \cdot \left(x \cdot \left(wj \cdot \left(2.5 - \left(\frac{-1}{x} + wj \cdot \left(\frac{1}{x} - -2.6666666666666665\right)\right)\right)\right) - x \cdot 2\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (wj x)
 :precision binary64
 (if (<= wj -0.78)
   (/ x (* (+ wj 1.0) (exp wj)))
   (+
    x
    (*
     wj
     (-
      (*
       x
       (* wj (- 2.5 (+ (/ -1.0 x) (* wj (- (/ 1.0 x) -2.6666666666666665))))))
      (* x 2.0))))))

C

double code(double wj, double x) {
	double tmp;
	if (wj <= -0.78) {
		tmp = x / ((wj + 1.0) * exp(wj));
	} else {
		tmp = x + (wj * ((x * (wj * (2.5 - ((-1.0 / x) + (wj * ((1.0 / x) - -2.6666666666666665)))))) - (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 <= (-0.78d0)) then
        tmp = x / ((wj + 1.0d0) * exp(wj))
    else
        tmp = x + (wj * ((x * (wj * (2.5d0 - (((-1.0d0) / x) + (wj * ((1.0d0 / x) - (-2.6666666666666665d0))))))) - (x * 2.0d0)))
    end if
    code = tmp
end function

Java

public static double code(double wj, double x) {
	double tmp;
	if (wj <= -0.78) {
		tmp = x / ((wj + 1.0) * Math.exp(wj));
	} else {
		tmp = x + (wj * ((x * (wj * (2.5 - ((-1.0 / x) + (wj * ((1.0 / x) - -2.6666666666666665)))))) - (x * 2.0)));
	}
	return tmp;
}

Python

def code(wj, x):
	tmp = 0
	if wj <= -0.78:
		tmp = x / ((wj + 1.0) * math.exp(wj))
	else:
		tmp = x + (wj * ((x * (wj * (2.5 - ((-1.0 / x) + (wj * ((1.0 / x) - -2.6666666666666665)))))) - (x * 2.0)))
	return tmp

Julia

function code(wj, x)
	tmp = 0.0
	if (wj <= -0.78)
		tmp = Float64(x / Float64(Float64(wj + 1.0) * exp(wj)));
	else
		tmp = Float64(x + Float64(wj * Float64(Float64(x * Float64(wj * Float64(2.5 - Float64(Float64(-1.0 / x) + Float64(wj * Float64(Float64(1.0 / x) - -2.6666666666666665)))))) - Float64(x * 2.0))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(wj, x)
	tmp = 0.0;
	if (wj <= -0.78)
		tmp = x / ((wj + 1.0) * exp(wj));
	else
		tmp = x + (wj * ((x * (wj * (2.5 - ((-1.0 / x) + (wj * ((1.0 / x) - -2.6666666666666665)))))) - (x * 2.0)));
	end
	tmp_2 = tmp;
end

Wolfram

code[wj_, x_] := If[LessEqual[wj, -0.78], N[(x / N[(N[(wj + 1.0), $MachinePrecision] * N[Exp[wj], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x + N[(wj * N[(N[(x * N[(wj * N[(2.5 - N[(N[(-1.0 / x), $MachinePrecision] + N[(wj * N[(N[(1.0 / x), $MachinePrecision] - -2.6666666666666665), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(x * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if wj < -0.78000000000000003

   1. Initial program 40.0%

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

      1. distribute-rgt1-in 99.7%

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

      2. associate-/l/ 100.0%

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

      3. div-sub 40.0%

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

      4. associate-/l* 40.0%

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

      5. *-inverses 100.0%

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

      6. *-rgt-identity 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in x around inf 81.3%

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

      1. +-commutative 81.3%

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

   7. Simplified 81.3%

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

   if -0.78000000000000003 < wj

   1. Initial program 79.2%

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

      1. distribute-rgt1-in 79.2%

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

      2. associate-/l/ 79.2%

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

      3. div-sub 79.2%

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

      4. associate-/l* 79.2%

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

      5. *-inverses 79.6%

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

      6. *-rgt-identity 79.6%

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

   3. Simplified 79.6%

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

   5. Taylor expanded in wj around 0 98.5%

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

   6. Taylor expanded in x around inf 98.5%

      \[\leadsto x + wj \cdot \left(\color{blue}{x \cdot \left(wj \cdot \left(2.5 + -2.6666666666666665 \cdot wj\right) + \frac{wj \cdot \left(1 + -1 \cdot wj\right)}{x}\right)} - 2 \cdot x\right) \]
   7. Step-by-step derivation

      1. associate-/l* 98.5%

         \[\leadsto x + wj \cdot \left(x \cdot \left(wj \cdot \left(2.5 + -2.6666666666666665 \cdot wj\right) + \color{blue}{wj \cdot \frac{1 + -1 \cdot wj}{x}}\right) - 2 \cdot x\right) \]

      2. distribute-lft-out 98.5%

         \[\leadsto x + wj \cdot \left(x \cdot \color{blue}{\left(wj \cdot \left(\left(2.5 + -2.6666666666666665 \cdot wj\right) + \frac{1 + -1 \cdot wj}{x}\right)\right)} - 2 \cdot x\right) \]

      3. +-commutative 98.5%

         \[\leadsto x + wj \cdot \left(x \cdot \left(wj \cdot \left(\color{blue}{\left(-2.6666666666666665 \cdot wj + 2.5\right)} + \frac{1 + -1 \cdot wj}{x}\right)\right) - 2 \cdot x\right) \]

      4. *-commutative 98.5%

         \[\leadsto x + wj \cdot \left(x \cdot \left(wj \cdot \left(\left(\color{blue}{wj \cdot -2.6666666666666665} + 2.5\right) + \frac{1 + -1 \cdot wj}{x}\right)\right) - 2 \cdot x\right) \]

      5. fma-define 98.5%

         \[\leadsto x + wj \cdot \left(x \cdot \left(wj \cdot \left(\color{blue}{\mathsf{fma}\left(wj, -2.6666666666666665, 2.5\right)} + \frac{1 + -1 \cdot wj}{x}\right)\right) - 2 \cdot x\right) \]

      6. neg-mul-1 98.5%

         \[\leadsto x + wj \cdot \left(x \cdot \left(wj \cdot \left(\mathsf{fma}\left(wj, -2.6666666666666665, 2.5\right) + \frac{1 + \color{blue}{\left(-wj\right)}}{x}\right)\right) - 2 \cdot x\right) \]

      7. sub-neg 98.5%

         \[\leadsto x + wj \cdot \left(x \cdot \left(wj \cdot \left(\mathsf{fma}\left(wj, -2.6666666666666665, 2.5\right) + \frac{\color{blue}{1 - wj}}{x}\right)\right) - 2 \cdot x\right) \]

   8. Simplified 98.5%

      \[\leadsto x + wj \cdot \left(\color{blue}{x \cdot \left(wj \cdot \left(\mathsf{fma}\left(wj, -2.6666666666666665, 2.5\right) + \frac{1 - wj}{x}\right)\right)} - 2 \cdot x\right) \]

   9. Taylor expanded in wj around 0 98.5%

      \[\leadsto x + wj \cdot \left(x \cdot \color{blue}{\left(wj \cdot \left(2.5 + \left(-1 \cdot \left(wj \cdot \left(2.6666666666666665 + \frac{1}{x}\right)\right) + \frac{1}{x}\right)\right)\right)} - 2 \cdot x\right) \]
   10. Step-by-step derivation

       1. +-commutative 98.5%

          \[\leadsto x + wj \cdot \left(x \cdot \left(wj \cdot \left(2.5 + \color{blue}{\left(\frac{1}{x} + -1 \cdot \left(wj \cdot \left(2.6666666666666665 + \frac{1}{x}\right)\right)\right)}\right)\right) - 2 \cdot x\right) \]

       2. mul-1-neg 98.5%

          \[\leadsto x + wj \cdot \left(x \cdot \left(wj \cdot \left(2.5 + \left(\frac{1}{x} + \color{blue}{\left(-wj \cdot \left(2.6666666666666665 + \frac{1}{x}\right)\right)}\right)\right)\right) - 2 \cdot x\right) \]

       3. distribute-rgt-neg-in 98.5%

          \[\leadsto x + wj \cdot \left(x \cdot \left(wj \cdot \left(2.5 + \left(\frac{1}{x} + \color{blue}{wj \cdot \left(-\left(2.6666666666666665 + \frac{1}{x}\right)\right)}\right)\right)\right) - 2 \cdot x\right) \]

       4. distribute-neg-in 98.5%

          \[\leadsto x + wj \cdot \left(x \cdot \left(wj \cdot \left(2.5 + \left(\frac{1}{x} + wj \cdot \color{blue}{\left(\left(-2.6666666666666665\right) + \left(-\frac{1}{x}\right)\right)}\right)\right)\right) - 2 \cdot x\right) \]

       5. metadata-eval 98.5%

          \[\leadsto x + wj \cdot \left(x \cdot \left(wj \cdot \left(2.5 + \left(\frac{1}{x} + wj \cdot \left(\color{blue}{-2.6666666666666665} + \left(-\frac{1}{x}\right)\right)\right)\right)\right) - 2 \cdot x\right) \]

       6. distribute-neg-frac 98.5%

          \[\leadsto x + wj \cdot \left(x \cdot \left(wj \cdot \left(2.5 + \left(\frac{1}{x} + wj \cdot \left(-2.6666666666666665 + \color{blue}{\frac{-1}{x}}\right)\right)\right)\right) - 2 \cdot x\right) \]

       7. metadata-eval 98.5%

          \[\leadsto x + wj \cdot \left(x \cdot \left(wj \cdot \left(2.5 + \left(\frac{1}{x} + wj \cdot \left(-2.6666666666666665 + \frac{\color{blue}{-1}}{x}\right)\right)\right)\right) - 2 \cdot x\right) \]

   11. Simplified 98.5%

       \[\leadsto x + wj \cdot \left(x \cdot \color{blue}{\left(wj \cdot \left(2.5 + \left(\frac{1}{x} + wj \cdot \left(-2.6666666666666665 + \frac{-1}{x}\right)\right)\right)\right)} - 2 \cdot x\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 98.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;wj \leq -0.78:\\ \;\;\;\;\frac{x}{\left(wj + 1\right) \cdot e^{wj}}\\ \mathbf{else}:\\ \;\;\;\;x + wj \cdot \left(x \cdot \left(wj \cdot \left(2.5 - \left(\frac{-1}{x} + wj \cdot \left(\frac{1}{x} - -2.6666666666666665\right)\right)\right)\right) - x \cdot 2\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 96.1% accurate, 12.5× speedup

Math

\[\begin{array}{l} \\ x + wj \cdot \left(x \cdot \left(wj \cdot \left(2.5 - \left(\frac{-1}{x} + wj \cdot \left(\frac{1}{x} - -2.6666666666666665\right)\right)\right)\right) - x \cdot 2\right) \end{array} \]

FPCore

(FPCore (wj x)
 :precision binary64
 (+
  x
  (*
   wj
   (-
    (*
     x
     (* wj (- 2.5 (+ (/ -1.0 x) (* wj (- (/ 1.0 x) -2.6666666666666665))))))
    (* x 2.0)))))

C

double code(double wj, double x) {
	return x + (wj * ((x * (wj * (2.5 - ((-1.0 / x) + (wj * ((1.0 / x) - -2.6666666666666665)))))) - (x * 2.0)));
}

Fortran

real(8) function code(wj, x)
    real(8), intent (in) :: wj
    real(8), intent (in) :: x
    code = x + (wj * ((x * (wj * (2.5d0 - (((-1.0d0) / x) + (wj * ((1.0d0 / x) - (-2.6666666666666665d0))))))) - (x * 2.0d0)))
end function

Java

public static double code(double wj, double x) {
	return x + (wj * ((x * (wj * (2.5 - ((-1.0 / x) + (wj * ((1.0 / x) - -2.6666666666666665)))))) - (x * 2.0)));
}

Python

def code(wj, x):
	return x + (wj * ((x * (wj * (2.5 - ((-1.0 / x) + (wj * ((1.0 / x) - -2.6666666666666665)))))) - (x * 2.0)))

Julia

function code(wj, x)
	return Float64(x + Float64(wj * Float64(Float64(x * Float64(wj * Float64(2.5 - Float64(Float64(-1.0 / x) + Float64(wj * Float64(Float64(1.0 / x) - -2.6666666666666665)))))) - Float64(x * 2.0))))
end

MATLAB

function tmp = code(wj, x)
	tmp = x + (wj * ((x * (wj * (2.5 - ((-1.0 / x) + (wj * ((1.0 / x) - -2.6666666666666665)))))) - (x * 2.0)));
end

Wolfram

code[wj_, x_] := N[(x + N[(wj * N[(N[(x * N[(wj * N[(2.5 - N[(N[(-1.0 / x), $MachinePrecision] + N[(wj * N[(N[(1.0 / x), $MachinePrecision] - -2.6666666666666665), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(x * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 78.4%

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

   1. distribute-rgt1-in 79.6%

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

   2. associate-/l/ 79.6%

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

   3. div-sub 78.4%

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

   4. associate-/l* 78.4%

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

   5. *-inverses 80.0%

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

   6. *-rgt-identity 80.0%

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

3. Simplified 80.0%

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

5. Taylor expanded in wj around 0 96.6%

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

6. Taylor expanded in x around inf 96.6%

   \[\leadsto x + wj \cdot \left(\color{blue}{x \cdot \left(wj \cdot \left(2.5 + -2.6666666666666665 \cdot wj\right) + \frac{wj \cdot \left(1 + -1 \cdot wj\right)}{x}\right)} - 2 \cdot x\right) \]
7. Step-by-step derivation

   1. associate-/l* 96.6%

      \[\leadsto x + wj \cdot \left(x \cdot \left(wj \cdot \left(2.5 + -2.6666666666666665 \cdot wj\right) + \color{blue}{wj \cdot \frac{1 + -1 \cdot wj}{x}}\right) - 2 \cdot x\right) \]

   2. distribute-lft-out 96.6%

      \[\leadsto x + wj \cdot \left(x \cdot \color{blue}{\left(wj \cdot \left(\left(2.5 + -2.6666666666666665 \cdot wj\right) + \frac{1 + -1 \cdot wj}{x}\right)\right)} - 2 \cdot x\right) \]

   3. +-commutative 96.6%

      \[\leadsto x + wj \cdot \left(x \cdot \left(wj \cdot \left(\color{blue}{\left(-2.6666666666666665 \cdot wj + 2.5\right)} + \frac{1 + -1 \cdot wj}{x}\right)\right) - 2 \cdot x\right) \]

   4. *-commutative 96.6%

      \[\leadsto x + wj \cdot \left(x \cdot \left(wj \cdot \left(\left(\color{blue}{wj \cdot -2.6666666666666665} + 2.5\right) + \frac{1 + -1 \cdot wj}{x}\right)\right) - 2 \cdot x\right) \]

   5. fma-define 96.6%

      \[\leadsto x + wj \cdot \left(x \cdot \left(wj \cdot \left(\color{blue}{\mathsf{fma}\left(wj, -2.6666666666666665, 2.5\right)} + \frac{1 + -1 \cdot wj}{x}\right)\right) - 2 \cdot x\right) \]

   6. neg-mul-1 96.6%

      \[\leadsto x + wj \cdot \left(x \cdot \left(wj \cdot \left(\mathsf{fma}\left(wj, -2.6666666666666665, 2.5\right) + \frac{1 + \color{blue}{\left(-wj\right)}}{x}\right)\right) - 2 \cdot x\right) \]

   7. sub-neg 96.6%

      \[\leadsto x + wj \cdot \left(x \cdot \left(wj \cdot \left(\mathsf{fma}\left(wj, -2.6666666666666665, 2.5\right) + \frac{\color{blue}{1 - wj}}{x}\right)\right) - 2 \cdot x\right) \]

8. Simplified 96.6%

   \[\leadsto x + wj \cdot \left(\color{blue}{x \cdot \left(wj \cdot \left(\mathsf{fma}\left(wj, -2.6666666666666665, 2.5\right) + \frac{1 - wj}{x}\right)\right)} - 2 \cdot x\right) \]

9. Taylor expanded in wj around 0 96.6%

   \[\leadsto x + wj \cdot \left(x \cdot \color{blue}{\left(wj \cdot \left(2.5 + \left(-1 \cdot \left(wj \cdot \left(2.6666666666666665 + \frac{1}{x}\right)\right) + \frac{1}{x}\right)\right)\right)} - 2 \cdot x\right) \]
10. Step-by-step derivation

    1. +-commutative 96.6%

       \[\leadsto x + wj \cdot \left(x \cdot \left(wj \cdot \left(2.5 + \color{blue}{\left(\frac{1}{x} + -1 \cdot \left(wj \cdot \left(2.6666666666666665 + \frac{1}{x}\right)\right)\right)}\right)\right) - 2 \cdot x\right) \]

    2. mul-1-neg 96.6%

       \[\leadsto x + wj \cdot \left(x \cdot \left(wj \cdot \left(2.5 + \left(\frac{1}{x} + \color{blue}{\left(-wj \cdot \left(2.6666666666666665 + \frac{1}{x}\right)\right)}\right)\right)\right) - 2 \cdot x\right) \]

    3. distribute-rgt-neg-in 96.6%

       \[\leadsto x + wj \cdot \left(x \cdot \left(wj \cdot \left(2.5 + \left(\frac{1}{x} + \color{blue}{wj \cdot \left(-\left(2.6666666666666665 + \frac{1}{x}\right)\right)}\right)\right)\right) - 2 \cdot x\right) \]

    4. distribute-neg-in 96.6%

       \[\leadsto x + wj \cdot \left(x \cdot \left(wj \cdot \left(2.5 + \left(\frac{1}{x} + wj \cdot \color{blue}{\left(\left(-2.6666666666666665\right) + \left(-\frac{1}{x}\right)\right)}\right)\right)\right) - 2 \cdot x\right) \]

    5. metadata-eval 96.6%

       \[\leadsto x + wj \cdot \left(x \cdot \left(wj \cdot \left(2.5 + \left(\frac{1}{x} + wj \cdot \left(\color{blue}{-2.6666666666666665} + \left(-\frac{1}{x}\right)\right)\right)\right)\right) - 2 \cdot x\right) \]

    6. distribute-neg-frac 96.6%

       \[\leadsto x + wj \cdot \left(x \cdot \left(wj \cdot \left(2.5 + \left(\frac{1}{x} + wj \cdot \left(-2.6666666666666665 + \color{blue}{\frac{-1}{x}}\right)\right)\right)\right) - 2 \cdot x\right) \]

    7. metadata-eval 96.6%

       \[\leadsto x + wj \cdot \left(x \cdot \left(wj \cdot \left(2.5 + \left(\frac{1}{x} + wj \cdot \left(-2.6666666666666665 + \frac{\color{blue}{-1}}{x}\right)\right)\right)\right) - 2 \cdot x\right) \]

11. Simplified 96.6%

    \[\leadsto x + wj \cdot \left(x \cdot \color{blue}{\left(wj \cdot \left(2.5 + \left(\frac{1}{x} + wj \cdot \left(-2.6666666666666665 + \frac{-1}{x}\right)\right)\right)\right)} - 2 \cdot x\right) \]

12. Final simplification 96.6%

    \[\leadsto x + wj \cdot \left(x \cdot \left(wj \cdot \left(2.5 - \left(\frac{-1}{x} + wj \cdot \left(\frac{1}{x} - -2.6666666666666665\right)\right)\right)\right) - x \cdot 2\right) \]
13. Add Preprocessing

Alternative 5: 95.9% accurate, 24.1× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 78.4%

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

   1. distribute-rgt1-in 79.6%

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

   2. associate-/l/ 79.6%

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

   3. div-sub 78.4%

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

   4. associate-/l* 78.4%

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

   5. *-inverses 80.0%

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

   6. *-rgt-identity 80.0%

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

3. Simplified 80.0%

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

5. Taylor expanded in wj around 0 96.6%

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

6. Taylor expanded in x around 0 96.2%

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

   1. neg-mul-1 96.2%

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

   2. sub-neg 96.2%

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

8. Simplified 96.2%

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

9. Final simplification 96.2%

   \[\leadsto x + wj \cdot \left(wj \cdot \left(1 - wj\right) - x \cdot 2\right) \]
10. Add Preprocessing

Alternative 6: 95.6% accurate, 34.8× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 78.4%

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

   1. distribute-rgt1-in 79.6%

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

   2. associate-/l/ 79.6%

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

   3. div-sub 78.4%

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

   4. associate-/l* 78.4%

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

   5. *-inverses 80.0%

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

   6. *-rgt-identity 80.0%

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

3. Simplified 80.0%

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

5. Taylor expanded in wj around 0 96.6%

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

6. Taylor expanded in x around inf 96.6%

   \[\leadsto x + wj \cdot \left(\color{blue}{x \cdot \left(wj \cdot \left(2.5 + -2.6666666666666665 \cdot wj\right) + \frac{wj \cdot \left(1 + -1 \cdot wj\right)}{x}\right)} - 2 \cdot x\right) \]
7. Step-by-step derivation

   1. associate-/l* 96.6%

      \[\leadsto x + wj \cdot \left(x \cdot \left(wj \cdot \left(2.5 + -2.6666666666666665 \cdot wj\right) + \color{blue}{wj \cdot \frac{1 + -1 \cdot wj}{x}}\right) - 2 \cdot x\right) \]

   2. distribute-lft-out 96.6%

      \[\leadsto x + wj \cdot \left(x \cdot \color{blue}{\left(wj \cdot \left(\left(2.5 + -2.6666666666666665 \cdot wj\right) + \frac{1 + -1 \cdot wj}{x}\right)\right)} - 2 \cdot x\right) \]

   3. +-commutative 96.6%

      \[\leadsto x + wj \cdot \left(x \cdot \left(wj \cdot \left(\color{blue}{\left(-2.6666666666666665 \cdot wj + 2.5\right)} + \frac{1 + -1 \cdot wj}{x}\right)\right) - 2 \cdot x\right) \]

   4. *-commutative 96.6%

      \[\leadsto x + wj \cdot \left(x \cdot \left(wj \cdot \left(\left(\color{blue}{wj \cdot -2.6666666666666665} + 2.5\right) + \frac{1 + -1 \cdot wj}{x}\right)\right) - 2 \cdot x\right) \]

   5. fma-define 96.6%

      \[\leadsto x + wj \cdot \left(x \cdot \left(wj \cdot \left(\color{blue}{\mathsf{fma}\left(wj, -2.6666666666666665, 2.5\right)} + \frac{1 + -1 \cdot wj}{x}\right)\right) - 2 \cdot x\right) \]

   6. neg-mul-1 96.6%

      \[\leadsto x + wj \cdot \left(x \cdot \left(wj \cdot \left(\mathsf{fma}\left(wj, -2.6666666666666665, 2.5\right) + \frac{1 + \color{blue}{\left(-wj\right)}}{x}\right)\right) - 2 \cdot x\right) \]

   7. sub-neg 96.6%

      \[\leadsto x + wj \cdot \left(x \cdot \left(wj \cdot \left(\mathsf{fma}\left(wj, -2.6666666666666665, 2.5\right) + \frac{\color{blue}{1 - wj}}{x}\right)\right) - 2 \cdot x\right) \]

8. Simplified 96.6%

   \[\leadsto x + wj \cdot \left(\color{blue}{x \cdot \left(wj \cdot \left(\mathsf{fma}\left(wj, -2.6666666666666665, 2.5\right) + \frac{1 - wj}{x}\right)\right)} - 2 \cdot x\right) \]

9. Taylor expanded in x around 0 96.2%

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

    1. associate-*r/ 96.2%

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

11. Simplified 96.2%

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

12. Taylor expanded in wj around 0 95.6%

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

13. Final simplification 95.6%

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

Alternative 7: 84.6% accurate, 34.8× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 78.4%

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

   1. distribute-rgt1-in 79.6%

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

   2. associate-/l/ 79.6%

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

   3. div-sub 78.4%

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

   4. associate-/l* 78.4%

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

   5. *-inverses 80.0%

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

   6. *-rgt-identity 80.0%

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

3. Simplified 80.0%

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

5. Taylor expanded in wj around 0 78.2%

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

   1. associate-*r* 78.2%

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

   2. neg-mul-1 78.2%

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

7. Simplified 78.2%

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

8. Taylor expanded in x around inf 82.7%

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

   1. mul-1-neg 82.7%

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

   2. +-commutative 82.7%

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

   3. sub-neg 82.7%

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

   4. +-commutative 82.7%

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

   5. +-commutative 82.7%

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

   6. div-sub 82.7%

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

   7. +-commutative 82.7%

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

   8. associate-/l* 82.7%

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

   9. *-commutative 82.7%

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

   10. sub-neg 82.7%

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

   11. +-commutative 82.7%

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

   12. neg-mul-1 82.7%

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

   13. fma-undefine 82.7%

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

   14. +-commutative 82.7%

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

   15. associate-/l* 82.7%

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

   16. fma-undefine 82.7%

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

   17. neg-mul-1 82.7%

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

   18. +-commutative 82.7%

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

   19. sub-neg 82.7%

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

10. Simplified 82.7%

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

11. Final simplification 82.7%

    \[\leadsto \frac{x}{-1 - wj} \cdot \left(wj + -1\right) \]
12. Add Preprocessing

Alternative 8: 84.5% accurate, 44.7× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 78.4%

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

   1. distribute-rgt1-in 79.6%

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

   2. associate-/l/ 79.6%

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

   3. div-sub 78.4%

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

   4. associate-/l* 78.4%

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

   5. *-inverses 80.0%

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

   6. *-rgt-identity 80.0%

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

3. Simplified 80.0%

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

5. Taylor expanded in wj around 0 82.5%

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

   1. *-commutative 82.5%

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

7. Simplified 82.5%

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

8. Final simplification 82.5%

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

Alternative 9: 83.9% accurate, 62.6× speedup

Math

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

FPCore

(FPCore (wj x) :precision binary64 (* x (- 1.0 wj)))

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 78.4%

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

   1. distribute-rgt1-in 79.6%

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

   2. associate-/l/ 79.6%

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

   3. div-sub 78.4%

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

   4. associate-/l* 78.4%

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

   5. *-inverses 80.0%

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

   6. *-rgt-identity 80.0%

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

3. Simplified 80.0%

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

5. Taylor expanded in wj around 0 78.2%

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

   1. associate-*r* 78.2%

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

   2. neg-mul-1 78.2%

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

7. Simplified 78.2%

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

8. Taylor expanded in x around inf 82.7%

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

   1. mul-1-neg 82.7%

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

   2. +-commutative 82.7%

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

   3. sub-neg 82.7%

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

   4. +-commutative 82.7%

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

   5. +-commutative 82.7%

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

   6. div-sub 82.7%

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

   7. +-commutative 82.7%

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

   8. associate-/l* 82.7%

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

   9. *-commutative 82.7%

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

   10. sub-neg 82.7%

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

   11. +-commutative 82.7%

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

   12. neg-mul-1 82.7%

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

   13. fma-undefine 82.7%

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

   14. +-commutative 82.7%

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

   15. associate-/l* 82.7%

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

   16. fma-undefine 82.7%

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

   17. neg-mul-1 82.7%

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

   18. +-commutative 82.7%

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

   19. sub-neg 82.7%

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

10. Simplified 82.7%

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

11. Taylor expanded in wj around 0 82.1%

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

12. Final simplification 82.1%

    \[\leadsto x \cdot \left(1 - wj\right) \]
13. Add Preprocessing

Alternative 10: 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 78.4%

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

   1. distribute-rgt1-in 79.6%

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

   2. associate-/l/ 79.6%

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

   3. div-sub 78.4%

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

   4. associate-/l* 78.4%

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

   5. *-inverses 80.0%

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

   6. *-rgt-identity 80.0%

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

3. Simplified 80.0%

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

5. Taylor expanded in wj around 0 82.0%

   \[\leadsto \color{blue}{x} \]
6. Add Preprocessing

Alternative 11: 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 78.4%

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

   1. distribute-rgt1-in 79.6%

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

   2. associate-/l/ 79.6%

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

   3. div-sub 78.4%

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

   4. associate-/l* 78.4%

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

   5. *-inverses 80.0%

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

   6. *-rgt-identity 80.0%

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

3. Simplified 80.0%

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

5. Taylor expanded in wj around inf 3.8%

   \[\leadsto \color{blue}{wj} \]
6. Add Preprocessing

Developer target: 79.2% 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 2024087 
(FPCore (wj x)
  :name "Jmat.Real.lambertw, newton loop step"
  :precision binary64

  :alt
  (- wj (- (/ wj (+ wj 1.0)) (/ x (+ (exp wj) (* wj (exp wj))))))

  (- wj (/ (- (* wj (exp wj)) x) (+ (exp wj) (* wj (exp wj))))))