Jmat.Real.lambertw, newton loop step

Percentage Accurate: 78.0% → 98.5%

Time: 14.7s

Alternatives: 12

Speedup: 313.0×

• could not determine a ground truth

  1. wj = -7.254600972494622e+146
  2. x = -9.43575480152854e-52

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: 78.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 98.5% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := wj \cdot e^{wj}\\ t_1 := x \cdot -4 + x \cdot 1.5\\ \mathbf{if}\;wj + \frac{x - t\_0}{e^{wj} + t\_0} \leq 10^{-17}:\\ \;\;\;\;x + wj \cdot \left(wj \cdot \left(\left(1 - wj \cdot \left(\left(x \cdot -3 + \left(-2 \cdot t\_1 + x \cdot 0.6666666666666666\right)\right) + 1\right)\right) - t\_1\right) - x \cdot 2\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(\frac{wj - \frac{wj}{wj + 1}}{x} + \frac{e^{-wj}}{wj + 1}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (wj x)
 :precision binary64
 (let* ((t_0 (* wj (exp wj))) (t_1 (+ (* x -4.0) (* x 1.5))))
   (if (<= (+ wj (/ (- x t_0) (+ (exp wj) t_0))) 1e-17)
     (+
      x
      (*
       wj
       (-
        (*
         wj
         (-
          (-
           1.0
           (*
            wj
            (+ (+ (* x -3.0) (+ (* -2.0 t_1) (* x 0.6666666666666666))) 1.0)))
          t_1))
        (* x 2.0))))
     (* x (+ (/ (- wj (/ wj (+ wj 1.0))) x) (/ (exp (- wj)) (+ wj 1.0)))))))

C

double code(double wj, double x) {
	double t_0 = wj * exp(wj);
	double t_1 = (x * -4.0) + (x * 1.5);
	double tmp;
	if ((wj + ((x - t_0) / (exp(wj) + t_0))) <= 1e-17) {
		tmp = x + (wj * ((wj * ((1.0 - (wj * (((x * -3.0) + ((-2.0 * t_1) + (x * 0.6666666666666666))) + 1.0))) - t_1)) - (x * 2.0)));
	} else {
		tmp = x * (((wj - (wj / (wj + 1.0))) / 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) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = wj * exp(wj)
    t_1 = (x * (-4.0d0)) + (x * 1.5d0)
    if ((wj + ((x - t_0) / (exp(wj) + t_0))) <= 1d-17) then
        tmp = x + (wj * ((wj * ((1.0d0 - (wj * (((x * (-3.0d0)) + (((-2.0d0) * t_1) + (x * 0.6666666666666666d0))) + 1.0d0))) - t_1)) - (x * 2.0d0)))
    else
        tmp = x * (((wj - (wj / (wj + 1.0d0))) / x) + (exp(-wj) / (wj + 1.0d0)))
    end if
    code = tmp
end function

Java

public static double code(double wj, double x) {
	double t_0 = wj * Math.exp(wj);
	double t_1 = (x * -4.0) + (x * 1.5);
	double tmp;
	if ((wj + ((x - t_0) / (Math.exp(wj) + t_0))) <= 1e-17) {
		tmp = x + (wj * ((wj * ((1.0 - (wj * (((x * -3.0) + ((-2.0 * t_1) + (x * 0.6666666666666666))) + 1.0))) - t_1)) - (x * 2.0)));
	} else {
		tmp = x * (((wj - (wj / (wj + 1.0))) / x) + (Math.exp(-wj) / (wj + 1.0)));
	}
	return tmp;
}

Python

def code(wj, x):
	t_0 = wj * math.exp(wj)
	t_1 = (x * -4.0) + (x * 1.5)
	tmp = 0
	if (wj + ((x - t_0) / (math.exp(wj) + t_0))) <= 1e-17:
		tmp = x + (wj * ((wj * ((1.0 - (wj * (((x * -3.0) + ((-2.0 * t_1) + (x * 0.6666666666666666))) + 1.0))) - t_1)) - (x * 2.0)))
	else:
		tmp = x * (((wj - (wj / (wj + 1.0))) / x) + (math.exp(-wj) / (wj + 1.0)))
	return tmp

Julia

function code(wj, x)
	t_0 = Float64(wj * exp(wj))
	t_1 = Float64(Float64(x * -4.0) + Float64(x * 1.5))
	tmp = 0.0
	if (Float64(wj + Float64(Float64(x - t_0) / Float64(exp(wj) + t_0))) <= 1e-17)
		tmp = Float64(x + Float64(wj * Float64(Float64(wj * Float64(Float64(1.0 - Float64(wj * Float64(Float64(Float64(x * -3.0) + Float64(Float64(-2.0 * t_1) + Float64(x * 0.6666666666666666))) + 1.0))) - t_1)) - Float64(x * 2.0))));
	else
		tmp = Float64(x * Float64(Float64(Float64(wj - Float64(wj / Float64(wj + 1.0))) / x) + Float64(exp(Float64(-wj)) / Float64(wj + 1.0))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(wj, x)
	t_0 = wj * exp(wj);
	t_1 = (x * -4.0) + (x * 1.5);
	tmp = 0.0;
	if ((wj + ((x - t_0) / (exp(wj) + t_0))) <= 1e-17)
		tmp = x + (wj * ((wj * ((1.0 - (wj * (((x * -3.0) + ((-2.0 * t_1) + (x * 0.6666666666666666))) + 1.0))) - t_1)) - (x * 2.0)));
	else
		tmp = x * (((wj - (wj / (wj + 1.0))) / x) + (exp(-wj) / (wj + 1.0)));
	end
	tmp_2 = tmp;
end

Wolfram

code[wj_, x_] := Block[{t$95$0 = N[(wj * N[Exp[wj], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(x * -4.0), $MachinePrecision] + N[(x * 1.5), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(wj + N[(N[(x - t$95$0), $MachinePrecision] / N[(N[Exp[wj], $MachinePrecision] + t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 1e-17], N[(x + N[(wj * N[(N[(wj * N[(N[(1.0 - N[(wj * N[(N[(N[(x * -3.0), $MachinePrecision] + N[(N[(-2.0 * t$95$1), $MachinePrecision] + N[(x * 0.6666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - t$95$1), $MachinePrecision]), $MachinePrecision] - N[(x * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x * N[(N[(N[(wj - N[(wj / N[(wj + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision] + N[(N[Exp[(-wj)], $MachinePrecision] / N[(wj + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 wj (/.f64 (-.f64 (*.f64 wj (exp.f64 wj)) x) (+.f64 (exp.f64 wj) (*.f64 wj (exp.f64 wj))))) < 1.00000000000000007e-17

   1. Initial program 65.8%

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

      1. distribute-rgt1-in 66.3%

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

      2. associate-/l/ 66.3%

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

      3. div-sub 65.8%

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

      4. associate-/l* 65.8%

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

      5. *-inverses 66.3%

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

      6. *-rgt-identity 66.3%

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

   3. Simplified 66.3%

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

   5. Taylor expanded in wj around 0 99.2%

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

   if 1.00000000000000007e-17 < (-.f64 wj (/.f64 (-.f64 (*.f64 wj (exp.f64 wj)) x) (+.f64 (exp.f64 wj) (*.f64 wj (exp.f64 wj)))))

   1. Initial program 91.7%

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

      1. distribute-rgt1-in 94.3%

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

      2. associate-/l/ 94.2%

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

      3. div-sub 91.7%

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

      4. associate-/l* 91.7%

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

      5. *-inverses 99.3%

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

      6. *-rgt-identity 99.3%

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

   3. Simplified 99.3%

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

   5. Taylor expanded in x around -inf 99.3%

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

      1. associate-*r* 99.3%

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

      2. neg-mul-1 99.3%

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

      3. mul-1-neg 99.3%

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

      4. +-commutative 99.3%

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

      5. associate-/r* 99.3%

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

      6. exp-neg 99.3%

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

      7. +-commutative 99.3%

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

   7. Simplified 99.3%

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

4. Final simplification 99.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;wj + \frac{x - wj \cdot e^{wj}}{e^{wj} + wj \cdot e^{wj}} \leq 10^{-17}:\\ \;\;\;\;x + wj \cdot \left(wj \cdot \left(\left(1 - wj \cdot \left(\left(x \cdot -3 + \left(-2 \cdot \left(x \cdot -4 + x \cdot 1.5\right) + x \cdot 0.6666666666666666\right)\right) + 1\right)\right) - \left(x \cdot -4 + x \cdot 1.5\right)\right) - x \cdot 2\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(\frac{wj - \frac{wj}{wj + 1}}{x} + \frac{e^{-wj}}{wj + 1}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 98.2% accurate, 2.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x \cdot -4 + x \cdot 1.5\\ \mathbf{if}\;wj \leq -3.8 \cdot 10^{-6}:\\ \;\;\;\;wj - \frac{wj - \frac{x}{e^{wj}}}{wj + 1}\\ \mathbf{else}:\\ \;\;\;\;x + wj \cdot \left(wj \cdot \left(\left(1 - wj \cdot \left(\left(x \cdot -3 + \left(-2 \cdot t\_0 + x \cdot 0.6666666666666666\right)\right) + 1\right)\right) - t\_0\right) - x \cdot 2\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (wj x)
 :precision binary64
 (let* ((t_0 (+ (* x -4.0) (* x 1.5))))
   (if (<= wj -3.8e-6)
     (- wj (/ (- wj (/ x (exp wj))) (+ wj 1.0)))
     (+
      x
      (*
       wj
       (-
        (*
         wj
         (-
          (-
           1.0
           (*
            wj
            (+ (+ (* x -3.0) (+ (* -2.0 t_0) (* x 0.6666666666666666))) 1.0)))
          t_0))
        (* x 2.0)))))))

C

double code(double wj, double x) {
	double t_0 = (x * -4.0) + (x * 1.5);
	double tmp;
	if (wj <= -3.8e-6) {
		tmp = wj - ((wj - (x / exp(wj))) / (wj + 1.0));
	} else {
		tmp = x + (wj * ((wj * ((1.0 - (wj * (((x * -3.0) + ((-2.0 * t_0) + (x * 0.6666666666666666))) + 1.0))) - t_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) :: t_0
    real(8) :: tmp
    t_0 = (x * (-4.0d0)) + (x * 1.5d0)
    if (wj <= (-3.8d-6)) then
        tmp = wj - ((wj - (x / exp(wj))) / (wj + 1.0d0))
    else
        tmp = x + (wj * ((wj * ((1.0d0 - (wj * (((x * (-3.0d0)) + (((-2.0d0) * t_0) + (x * 0.6666666666666666d0))) + 1.0d0))) - t_0)) - (x * 2.0d0)))
    end if
    code = tmp
end function

Java

public static double code(double wj, double x) {
	double t_0 = (x * -4.0) + (x * 1.5);
	double tmp;
	if (wj <= -3.8e-6) {
		tmp = wj - ((wj - (x / Math.exp(wj))) / (wj + 1.0));
	} else {
		tmp = x + (wj * ((wj * ((1.0 - (wj * (((x * -3.0) + ((-2.0 * t_0) + (x * 0.6666666666666666))) + 1.0))) - t_0)) - (x * 2.0)));
	}
	return tmp;
}

Python

def code(wj, x):
	t_0 = (x * -4.0) + (x * 1.5)
	tmp = 0
	if wj <= -3.8e-6:
		tmp = wj - ((wj - (x / math.exp(wj))) / (wj + 1.0))
	else:
		tmp = x + (wj * ((wj * ((1.0 - (wj * (((x * -3.0) + ((-2.0 * t_0) + (x * 0.6666666666666666))) + 1.0))) - t_0)) - (x * 2.0)))
	return tmp

Julia

function code(wj, x)
	t_0 = Float64(Float64(x * -4.0) + Float64(x * 1.5))
	tmp = 0.0
	if (wj <= -3.8e-6)
		tmp = Float64(wj - Float64(Float64(wj - Float64(x / exp(wj))) / Float64(wj + 1.0)));
	else
		tmp = Float64(x + Float64(wj * Float64(Float64(wj * Float64(Float64(1.0 - Float64(wj * Float64(Float64(Float64(x * -3.0) + Float64(Float64(-2.0 * t_0) + Float64(x * 0.6666666666666666))) + 1.0))) - t_0)) - Float64(x * 2.0))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(wj, x)
	t_0 = (x * -4.0) + (x * 1.5);
	tmp = 0.0;
	if (wj <= -3.8e-6)
		tmp = wj - ((wj - (x / exp(wj))) / (wj + 1.0));
	else
		tmp = x + (wj * ((wj * ((1.0 - (wj * (((x * -3.0) + ((-2.0 * t_0) + (x * 0.6666666666666666))) + 1.0))) - t_0)) - (x * 2.0)));
	end
	tmp_2 = tmp;
end

Wolfram

code[wj_, x_] := Block[{t$95$0 = N[(N[(x * -4.0), $MachinePrecision] + N[(x * 1.5), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[wj, -3.8e-6], N[(wj - N[(N[(wj - N[(x / N[Exp[wj], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(wj + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x + N[(wj * N[(N[(wj * N[(N[(1.0 - N[(wj * N[(N[(N[(x * -3.0), $MachinePrecision] + N[(N[(-2.0 * t$95$0), $MachinePrecision] + N[(x * 0.6666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision]), $MachinePrecision] - N[(x * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if wj < -3.8e-6

   1. Initial program 61.0%

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

      1. distribute-rgt1-in 94.3%

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

      2. associate-/l/ 94.3%

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

      3. div-sub 60.9%

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

      4. associate-/l* 60.9%

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

      5. *-inverses 94.3%

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

      6. *-rgt-identity 94.3%

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

   3. Simplified 94.3%

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

   if -3.8e-6 < wj

   1. Initial program 74.2%

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

      1. distribute-rgt1-in 74.2%

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

      2. associate-/l/ 74.2%

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

      3. div-sub 74.2%

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

      4. associate-/l* 74.2%

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

      5. *-inverses 75.9%

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

      6. *-rgt-identity 75.9%

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

   3. Simplified 75.9%

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

   5. Taylor expanded in wj around 0 98.4%

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

4. Final simplification 98.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;wj \leq -3.8 \cdot 10^{-6}:\\ \;\;\;\;wj - \frac{wj - \frac{x}{e^{wj}}}{wj + 1}\\ \mathbf{else}:\\ \;\;\;\;x + wj \cdot \left(wj \cdot \left(\left(1 - wj \cdot \left(\left(x \cdot -3 + \left(-2 \cdot \left(x \cdot -4 + x \cdot 1.5\right) + x \cdot 0.6666666666666666\right)\right) + 1\right)\right) - \left(x \cdot -4 + x \cdot 1.5\right)\right) - x \cdot 2\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 96.2% accurate, 7.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x \cdot -4 + x \cdot 1.5\\ x + wj \cdot \left(wj \cdot \left(\left(1 - wj \cdot \left(\left(x \cdot -3 + \left(-2 \cdot t\_0 + x \cdot 0.6666666666666666\right)\right) + 1\right)\right) - t\_0\right) - x \cdot 2\right) \end{array} \end{array} \]

FPCore

(FPCore (wj x)
 :precision binary64
 (let* ((t_0 (+ (* x -4.0) (* x 1.5))))
   (+
    x
    (*
     wj
     (-
      (*
       wj
       (-
        (-
         1.0
         (*
          wj
          (+ (+ (* x -3.0) (+ (* -2.0 t_0) (* x 0.6666666666666666))) 1.0)))
        t_0))
      (* x 2.0))))))

C

double code(double wj, double x) {
	double t_0 = (x * -4.0) + (x * 1.5);
	return x + (wj * ((wj * ((1.0 - (wj * (((x * -3.0) + ((-2.0 * t_0) + (x * 0.6666666666666666))) + 1.0))) - t_0)) - (x * 2.0)));
}

Fortran

real(8) function code(wj, x)
    real(8), intent (in) :: wj
    real(8), intent (in) :: x
    real(8) :: t_0
    t_0 = (x * (-4.0d0)) + (x * 1.5d0)
    code = x + (wj * ((wj * ((1.0d0 - (wj * (((x * (-3.0d0)) + (((-2.0d0) * t_0) + (x * 0.6666666666666666d0))) + 1.0d0))) - t_0)) - (x * 2.0d0)))
end function

Java

public static double code(double wj, double x) {
	double t_0 = (x * -4.0) + (x * 1.5);
	return x + (wj * ((wj * ((1.0 - (wj * (((x * -3.0) + ((-2.0 * t_0) + (x * 0.6666666666666666))) + 1.0))) - t_0)) - (x * 2.0)));
}

Python

def code(wj, x):
	t_0 = (x * -4.0) + (x * 1.5)
	return x + (wj * ((wj * ((1.0 - (wj * (((x * -3.0) + ((-2.0 * t_0) + (x * 0.6666666666666666))) + 1.0))) - t_0)) - (x * 2.0)))

Julia

function code(wj, x)
	t_0 = Float64(Float64(x * -4.0) + Float64(x * 1.5))
	return Float64(x + Float64(wj * Float64(Float64(wj * Float64(Float64(1.0 - Float64(wj * Float64(Float64(Float64(x * -3.0) + Float64(Float64(-2.0 * t_0) + Float64(x * 0.6666666666666666))) + 1.0))) - t_0)) - Float64(x * 2.0))))
end

MATLAB

function tmp = code(wj, x)
	t_0 = (x * -4.0) + (x * 1.5);
	tmp = x + (wj * ((wj * ((1.0 - (wj * (((x * -3.0) + ((-2.0 * t_0) + (x * 0.6666666666666666))) + 1.0))) - t_0)) - (x * 2.0)));
end

Wolfram

code[wj_, x_] := Block[{t$95$0 = N[(N[(x * -4.0), $MachinePrecision] + N[(x * 1.5), $MachinePrecision]), $MachinePrecision]}, N[(x + N[(wj * N[(N[(wj * N[(N[(1.0 - N[(wj * N[(N[(N[(x * -3.0), $MachinePrecision] + N[(N[(-2.0 * t$95$0), $MachinePrecision] + N[(x * 0.6666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision]), $MachinePrecision] - N[(x * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Initial program 73.8%

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

   1. distribute-rgt1-in 75.0%

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

   2. associate-/l/ 75.0%

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

   3. div-sub 73.8%

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

   4. associate-/l* 73.8%

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

   5. *-inverses 76.5%

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

   6. *-rgt-identity 76.5%

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

3. Simplified 76.5%

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

5. Taylor expanded in wj around 0 96.0%

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

6. Final simplification 96.0%

   \[\leadsto x + wj \cdot \left(wj \cdot \left(\left(1 - wj \cdot \left(\left(x \cdot -3 + \left(-2 \cdot \left(x \cdot -4 + x \cdot 1.5\right) + x \cdot 0.6666666666666666\right)\right) + 1\right)\right) - \left(x \cdot -4 + x \cdot 1.5\right)\right) - x \cdot 2\right) \]
7. Add Preprocessing

Alternative 4: 96.3% accurate, 12.5× speedup

Math

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

FPCore

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

C

double code(double wj, double x) {
	return x + (wj * ((x * (wj * (((wj * (-2.6666666666666665 + (-1.0 / x))) + 2.5) + (1.0 / x)))) - (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 * (((wj * ((-2.6666666666666665d0) + ((-1.0d0) / x))) + 2.5d0) + (1.0d0 / x)))) - (x * 2.0d0)))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 73.8%

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

   1. distribute-rgt1-in 75.0%

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

   2. associate-/l/ 75.0%

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

   3. div-sub 73.8%

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

   4. associate-/l* 73.8%

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

   5. *-inverses 76.5%

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

   6. *-rgt-identity 76.5%

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

3. Simplified 76.5%

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

5. Taylor expanded in wj around 0 96.0%

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

6. Taylor expanded in x around inf 96.0%

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

   1. fma-define 96.0%

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

   2. *-commutative 96.0%

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

   3. associate-/l* 95.9%

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

   4. neg-mul-1 95.9%

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

   5. sub-neg 95.9%

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

8. Simplified 95.9%

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

9. Taylor expanded in wj around 0 95.9%

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

    1. +-commutative 95.9%

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

    2. +-commutative 95.9%

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

    3. associate-+l+ 95.9%

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

    4. mul-1-neg 95.9%

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

    5. distribute-rgt-neg-in 95.9%

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

    6. distribute-neg-in 95.9%

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

    7. metadata-eval 95.9%

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

    8. distribute-neg-frac 95.9%

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

    9. metadata-eval 95.9%

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

11. Simplified 95.9%

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

12. Final simplification 95.9%

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

Alternative 5: 85.3% accurate, 19.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;wj \leq -7.6 \cdot 10^{-14}:\\ \;\;\;\;wj - \frac{wj}{wj + 1}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(wj \cdot \left(wj \cdot 2 - 2\right) + 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (wj x)
 :precision binary64
 (if (<= wj -7.6e-14)
   (- wj (/ wj (+ wj 1.0)))
   (* x (+ (* wj (- (* wj 2.0) 2.0)) 1.0))))

C

double code(double wj, double x) {
	double tmp;
	if (wj <= -7.6e-14) {
		tmp = wj - (wj / (wj + 1.0));
	} else {
		tmp = x * ((wj * ((wj * 2.0) - 2.0)) + 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 <= (-7.6d-14)) then
        tmp = wj - (wj / (wj + 1.0d0))
    else
        tmp = x * ((wj * ((wj * 2.0d0) - 2.0d0)) + 1.0d0)
    end if
    code = tmp
end function

Java

public static double code(double wj, double x) {
	double tmp;
	if (wj <= -7.6e-14) {
		tmp = wj - (wj / (wj + 1.0));
	} else {
		tmp = x * ((wj * ((wj * 2.0) - 2.0)) + 1.0);
	}
	return tmp;
}

Python

def code(wj, x):
	tmp = 0
	if wj <= -7.6e-14:
		tmp = wj - (wj / (wj + 1.0))
	else:
		tmp = x * ((wj * ((wj * 2.0) - 2.0)) + 1.0)
	return tmp

Julia

function code(wj, x)
	tmp = 0.0
	if (wj <= -7.6e-14)
		tmp = Float64(wj - Float64(wj / Float64(wj + 1.0)));
	else
		tmp = Float64(x * Float64(Float64(wj * Float64(Float64(wj * 2.0) - 2.0)) + 1.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(wj, x)
	tmp = 0.0;
	if (wj <= -7.6e-14)
		tmp = wj - (wj / (wj + 1.0));
	else
		tmp = x * ((wj * ((wj * 2.0) - 2.0)) + 1.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[wj_, x_] := If[LessEqual[wj, -7.6e-14], N[(wj - N[(wj / N[(wj + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x * N[(N[(wj * N[(N[(wj * 2.0), $MachinePrecision] - 2.0), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if wj < -7.6000000000000004e-14

   1. Initial program 57.7%

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

      1. distribute-rgt1-in 84.9%

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

      2. associate-/l/ 84.9%

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

      3. div-sub 57.6%

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

      4. associate-/l* 57.6%

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

      5. *-inverses 84.9%

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

      6. *-rgt-identity 84.9%

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

   3. Simplified 84.9%

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

   5. Taylor expanded in x around 0 49.3%

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

      1. +-commutative 49.3%

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

   7. Simplified 49.3%

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

   if -7.6000000000000004e-14 < wj

   1. Initial program 74.5%

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

      1. distribute-rgt1-in 74.5%

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

      2. associate-/l/ 74.5%

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

      3. div-sub 74.5%

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

      4. associate-/l* 74.5%

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

      5. *-inverses 76.1%

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

      6. *-rgt-identity 76.1%

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

   3. Simplified 76.1%

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

   5. Taylor expanded in wj around 0 74.3%

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

      1. associate-*r* 74.3%

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

      2. neg-mul-1 74.3%

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

      3. distribute-rgt1-in 74.3%

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

      4. +-commutative 74.3%

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

      5. sub-neg 74.3%

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

   7. Simplified 74.3%

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

   8. Taylor expanded in x around inf 84.5%

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

   9. Taylor expanded in wj around 0 84.5%

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

4. Final simplification 83.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;wj \leq -7.6 \cdot 10^{-14}:\\ \;\;\;\;wj - \frac{wj}{wj + 1}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(wj \cdot \left(wj \cdot 2 - 2\right) + 1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 85.3% accurate, 19.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;wj \leq -7.6 \cdot 10^{-14}:\\ \;\;\;\;x \cdot \frac{wj - \frac{wj}{wj + 1}}{x}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(wj \cdot \left(wj \cdot 2 - 2\right) + 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (wj x)
 :precision binary64
 (if (<= wj -7.6e-14)
   (* x (/ (- wj (/ wj (+ wj 1.0))) x))
   (* x (+ (* wj (- (* wj 2.0) 2.0)) 1.0))))

C

double code(double wj, double x) {
	double tmp;
	if (wj <= -7.6e-14) {
		tmp = x * ((wj - (wj / (wj + 1.0))) / x);
	} else {
		tmp = x * ((wj * ((wj * 2.0) - 2.0)) + 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 <= (-7.6d-14)) then
        tmp = x * ((wj - (wj / (wj + 1.0d0))) / x)
    else
        tmp = x * ((wj * ((wj * 2.0d0) - 2.0d0)) + 1.0d0)
    end if
    code = tmp
end function

Java

public static double code(double wj, double x) {
	double tmp;
	if (wj <= -7.6e-14) {
		tmp = x * ((wj - (wj / (wj + 1.0))) / x);
	} else {
		tmp = x * ((wj * ((wj * 2.0) - 2.0)) + 1.0);
	}
	return tmp;
}

Python

def code(wj, x):
	tmp = 0
	if wj <= -7.6e-14:
		tmp = x * ((wj - (wj / (wj + 1.0))) / x)
	else:
		tmp = x * ((wj * ((wj * 2.0) - 2.0)) + 1.0)
	return tmp

Julia

function code(wj, x)
	tmp = 0.0
	if (wj <= -7.6e-14)
		tmp = Float64(x * Float64(Float64(wj - Float64(wj / Float64(wj + 1.0))) / x));
	else
		tmp = Float64(x * Float64(Float64(wj * Float64(Float64(wj * 2.0) - 2.0)) + 1.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(wj, x)
	tmp = 0.0;
	if (wj <= -7.6e-14)
		tmp = x * ((wj - (wj / (wj + 1.0))) / x);
	else
		tmp = x * ((wj * ((wj * 2.0) - 2.0)) + 1.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[wj_, x_] := If[LessEqual[wj, -7.6e-14], N[(x * N[(N[(wj - N[(wj / N[(wj + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision], N[(x * N[(N[(wj * N[(N[(wj * 2.0), $MachinePrecision] - 2.0), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if wj < -7.6000000000000004e-14

   1. Initial program 57.7%

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

      1. distribute-rgt1-in 84.9%

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

      2. associate-/l/ 84.9%

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

      3. div-sub 57.6%

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

      4. associate-/l* 57.6%

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

      5. *-inverses 84.9%

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

      6. *-rgt-identity 84.9%

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

   3. Simplified 84.9%

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

   5. Taylor expanded in x around inf 84.3%

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

      1. +-commutative 84.3%

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

      2. associate-/r* 84.3%

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

      3. exp-neg 84.3%

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

      4. +-commutative 84.3%

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

      5. +-commutative 84.3%

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

   7. Simplified 84.3%

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

   8. Taylor expanded in x around 0 49.3%

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

      1. +-commutative 49.3%

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

   10. Simplified 49.3%

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

   if -7.6000000000000004e-14 < wj

   1. Initial program 74.5%

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

      1. distribute-rgt1-in 74.5%

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

      2. associate-/l/ 74.5%

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

      3. div-sub 74.5%

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

      4. associate-/l* 74.5%

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

      5. *-inverses 76.1%

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

      6. *-rgt-identity 76.1%

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

   3. Simplified 76.1%

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

   5. Taylor expanded in wj around 0 74.3%

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

      1. associate-*r* 74.3%

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

      2. neg-mul-1 74.3%

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

      3. distribute-rgt1-in 74.3%

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

      4. +-commutative 74.3%

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

      5. sub-neg 74.3%

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

   7. Simplified 74.3%

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

   8. Taylor expanded in x around inf 84.5%

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

   9. Taylor expanded in wj around 0 84.5%

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

4. Final simplification 83.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;wj \leq -7.6 \cdot 10^{-14}:\\ \;\;\;\;x \cdot \frac{wj - \frac{wj}{wj + 1}}{x}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(wj \cdot \left(wj \cdot 2 - 2\right) + 1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 85.2% accurate, 22.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;wj \leq -7.6 \cdot 10^{-14}:\\ \;\;\;\;wj + \frac{wj}{-1 - wj}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{1 - wj}{wj + 1}\\ \end{array} \end{array} \]

FPCore

(FPCore (wj x)
 :precision binary64
 (if (<= wj -7.6e-14)
   (+ wj (/ wj (- -1.0 wj)))
   (* x (/ (- 1.0 wj) (+ wj 1.0)))))

C

double code(double wj, double x) {
	double tmp;
	if (wj <= -7.6e-14) {
		tmp = wj + (wj / (-1.0 - wj));
	} 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 <= (-7.6d-14)) then
        tmp = wj + (wj / ((-1.0d0) - wj))
    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 <= -7.6e-14) {
		tmp = wj + (wj / (-1.0 - wj));
	} else {
		tmp = x * ((1.0 - wj) / (wj + 1.0));
	}
	return tmp;
}

Python

def code(wj, x):
	tmp = 0
	if wj <= -7.6e-14:
		tmp = wj + (wj / (-1.0 - wj))
	else:
		tmp = x * ((1.0 - wj) / (wj + 1.0))
	return tmp

Julia

function code(wj, x)
	tmp = 0.0
	if (wj <= -7.6e-14)
		tmp = Float64(wj + Float64(wj / Float64(-1.0 - wj)));
	else
		tmp = Float64(x * Float64(Float64(1.0 - wj) / Float64(wj + 1.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(wj, x)
	tmp = 0.0;
	if (wj <= -7.6e-14)
		tmp = wj + (wj / (-1.0 - wj));
	else
		tmp = x * ((1.0 - wj) / (wj + 1.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[wj_, x_] := If[LessEqual[wj, -7.6e-14], N[(wj + N[(wj / N[(-1.0 - wj), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x * N[(N[(1.0 - wj), $MachinePrecision] / N[(wj + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if wj < -7.6000000000000004e-14

   1. Initial program 57.7%

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

      1. distribute-rgt1-in 84.9%

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

      2. associate-/l/ 84.9%

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

      3. div-sub 57.6%

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

      4. associate-/l* 57.6%

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

      5. *-inverses 84.9%

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

      6. *-rgt-identity 84.9%

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

   3. Simplified 84.9%

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

   5. Taylor expanded in x around 0 49.3%

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

      1. +-commutative 49.3%

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

   7. Simplified 49.3%

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

   if -7.6000000000000004e-14 < wj

   1. Initial program 74.5%

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

      1. distribute-rgt1-in 74.5%

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

      2. associate-/l/ 74.5%

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

      3. div-sub 74.5%

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

      4. associate-/l* 74.5%

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

      5. *-inverses 76.1%

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

      6. *-rgt-identity 76.1%

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

   3. Simplified 76.1%

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

   5. Taylor expanded in wj around 0 74.3%

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

      1. associate-*r* 74.3%

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

      2. neg-mul-1 74.3%

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

      3. distribute-rgt1-in 74.3%

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

      4. +-commutative 74.3%

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

      5. sub-neg 74.3%

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

   7. Simplified 74.3%

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

   8. Taylor expanded in x around inf 84.5%

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

      1. div-sub 84.5%

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

      2. +-commutative 84.5%

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

   10. Simplified 84.5%

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

4. Final simplification 83.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;wj \leq -7.6 \cdot 10^{-14}:\\ \;\;\;\;wj + \frac{wj}{-1 - wj}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{1 - wj}{wj + 1}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 96.0% 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 73.8%

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

   1. distribute-rgt1-in 75.0%

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

   2. associate-/l/ 75.0%

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

   3. div-sub 73.8%

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

   4. associate-/l* 73.8%

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

   5. *-inverses 76.5%

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

   6. *-rgt-identity 76.5%

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

3. Simplified 76.5%

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

5. Taylor expanded in wj around 0 96.0%

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

6. Taylor expanded in x around 0 95.6%

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

   1. neg-mul-1 95.6%

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

   2. sub-neg 95.6%

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

8. Simplified 95.6%

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

9. Final simplification 95.6%

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

Alternative 9: 85.2% accurate, 26.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;wj \leq -7.6 \cdot 10^{-14}:\\ \;\;\;\;wj + \frac{wj}{-1 - wj}\\ \mathbf{else}:\\ \;\;\;\;x + -2 \cdot \left(wj \cdot x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (wj x)
 :precision binary64
 (if (<= wj -7.6e-14) (+ wj (/ wj (- -1.0 wj))) (+ x (* -2.0 (* wj x)))))

C

double code(double wj, double x) {
	double tmp;
	if (wj <= -7.6e-14) {
		tmp = wj + (wj / (-1.0 - wj));
	} 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 <= (-7.6d-14)) then
        tmp = wj + (wj / ((-1.0d0) - wj))
    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 <= -7.6e-14) {
		tmp = wj + (wj / (-1.0 - wj));
	} else {
		tmp = x + (-2.0 * (wj * x));
	}
	return tmp;
}

Python

def code(wj, x):
	tmp = 0
	if wj <= -7.6e-14:
		tmp = wj + (wj / (-1.0 - wj))
	else:
		tmp = x + (-2.0 * (wj * x))
	return tmp

Julia

function code(wj, x)
	tmp = 0.0
	if (wj <= -7.6e-14)
		tmp = Float64(wj + Float64(wj / Float64(-1.0 - wj)));
	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 <= -7.6e-14)
		tmp = wj + (wj / (-1.0 - wj));
	else
		tmp = x + (-2.0 * (wj * x));
	end
	tmp_2 = tmp;
end

Wolfram

code[wj_, x_] := If[LessEqual[wj, -7.6e-14], N[(wj + N[(wj / N[(-1.0 - wj), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x + N[(-2.0 * N[(wj * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if wj < -7.6000000000000004e-14

   1. Initial program 57.7%

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

      1. distribute-rgt1-in 84.9%

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

      2. associate-/l/ 84.9%

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

      3. div-sub 57.6%

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

      4. associate-/l* 57.6%

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

      5. *-inverses 84.9%

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

      6. *-rgt-identity 84.9%

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

   3. Simplified 84.9%

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

   5. Taylor expanded in x around 0 49.3%

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

      1. +-commutative 49.3%

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

   7. Simplified 49.3%

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

   if -7.6000000000000004e-14 < wj

   1. Initial program 74.5%

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

      1. distribute-rgt1-in 74.5%

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

      2. associate-/l/ 74.5%

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

      3. div-sub 74.5%

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

      4. associate-/l* 74.5%

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

      5. *-inverses 76.1%

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

      6. *-rgt-identity 76.1%

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

   3. Simplified 76.1%

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

   5. Taylor expanded in wj around 0 84.5%

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

      1. *-commutative 84.5%

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

   7. Simplified 84.5%

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

4. Final simplification 83.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;wj \leq -7.6 \cdot 10^{-14}:\\ \;\;\;\;wj + \frac{wj}{-1 - wj}\\ \mathbf{else}:\\ \;\;\;\;x + -2 \cdot \left(wj \cdot x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 84.9% 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 73.8%

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

   1. distribute-rgt1-in 75.0%

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

   2. associate-/l/ 75.0%

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

   3. div-sub 73.8%

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

   4. associate-/l* 73.8%

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

   5. *-inverses 76.5%

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

   6. *-rgt-identity 76.5%

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

3. Simplified 76.5%

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

5. Taylor expanded in wj around 0 81.1%

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

   1. *-commutative 81.1%

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

7. Simplified 81.1%

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

8. Final simplification 81.1%

   \[\leadsto x + -2 \cdot \left(wj \cdot x\right) \]
9. 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 73.8%

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

   1. distribute-rgt1-in 75.0%

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

   2. associate-/l/ 75.0%

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

   3. div-sub 73.8%

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

   4. associate-/l* 73.8%

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

   5. *-inverses 76.5%

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

   6. *-rgt-identity 76.5%

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

3. Simplified 76.5%

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

5. Taylor expanded in wj around inf 4.6%

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

6. Final simplification 4.6%

   \[\leadsto wj \]
7. Add Preprocessing

Alternative 12: 84.4% 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 73.8%

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

   1. distribute-rgt1-in 75.0%

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

   2. associate-/l/ 75.0%

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

   3. div-sub 73.8%

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

   4. associate-/l* 73.8%

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

   5. *-inverses 76.5%

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

   6. *-rgt-identity 76.5%

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

3. Simplified 76.5%

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

5. Taylor expanded in wj around 0 80.9%

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

6. Final simplification 80.9%

   \[\leadsto x \]
7. Add Preprocessing

Developer target: 79.0% 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 2024053 
(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))))))