Jmat.Real.lambertw, newton loop step

Percentage Accurate: 77.9% → 98.5%

Time: 14.4s

Alternatives: 13

Speedup: 313.0×

• could not determine a ground truth

  1. wj = -2.569635904813445e+151
  2. x = -1.3766248350019757e+219

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 77.9% 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, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := wj - \frac{x}{e^{wj}}\\ t_1 := -1 - wj \cdot \left(wj \cdot wj\right)\\ t_2 := wj \cdot e^{wj}\\ \mathbf{if}\;wj + \frac{x - t\_2}{e^{wj} + t\_2} \leq 10^{-21}:\\ \;\;\;\;x + wj \cdot \left(x \cdot -2 + wj \cdot \left(\left(1 + wj \cdot \left(\left(-1 - x \cdot -3\right) - \left(x \cdot 5 + x \cdot 0.6666666666666666\right)\right)\right) + x \cdot 2.5\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(wj + \frac{t\_0}{t\_1}\right) + \frac{wj \cdot \left(wj + -1\right)}{\frac{t\_1}{t\_0}}\\ \end{array} \end{array} \]

FPCore

(FPCore (wj x)
 :precision binary64
 (let* ((t_0 (- wj (/ x (exp wj))))
        (t_1 (- -1.0 (* wj (* wj wj))))
        (t_2 (* wj (exp wj))))
   (if (<= (+ wj (/ (- x t_2) (+ (exp wj) t_2))) 1e-21)
     (+
      x
      (*
       wj
       (+
        (* x -2.0)
        (*
         wj
         (+
          (+
           1.0
           (*
            wj
            (- (- -1.0 (* x -3.0)) (+ (* x 5.0) (* x 0.6666666666666666)))))
          (* x 2.5))))))
     (+ (+ wj (/ t_0 t_1)) (/ (* wj (+ wj -1.0)) (/ t_1 t_0))))))

C

double code(double wj, double x) {
	double t_0 = wj - (x / exp(wj));
	double t_1 = -1.0 - (wj * (wj * wj));
	double t_2 = wj * exp(wj);
	double tmp;
	if ((wj + ((x - t_2) / (exp(wj) + t_2))) <= 1e-21) {
		tmp = x + (wj * ((x * -2.0) + (wj * ((1.0 + (wj * ((-1.0 - (x * -3.0)) - ((x * 5.0) + (x * 0.6666666666666666))))) + (x * 2.5)))));
	} else {
		tmp = (wj + (t_0 / t_1)) + ((wj * (wj + -1.0)) / (t_1 / t_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) :: t_2
    real(8) :: tmp
    t_0 = wj - (x / exp(wj))
    t_1 = (-1.0d0) - (wj * (wj * wj))
    t_2 = wj * exp(wj)
    if ((wj + ((x - t_2) / (exp(wj) + t_2))) <= 1d-21) then
        tmp = x + (wj * ((x * (-2.0d0)) + (wj * ((1.0d0 + (wj * (((-1.0d0) - (x * (-3.0d0))) - ((x * 5.0d0) + (x * 0.6666666666666666d0))))) + (x * 2.5d0)))))
    else
        tmp = (wj + (t_0 / t_1)) + ((wj * (wj + (-1.0d0))) / (t_1 / t_0))
    end if
    code = tmp
end function

Java

public static double code(double wj, double x) {
	double t_0 = wj - (x / Math.exp(wj));
	double t_1 = -1.0 - (wj * (wj * wj));
	double t_2 = wj * Math.exp(wj);
	double tmp;
	if ((wj + ((x - t_2) / (Math.exp(wj) + t_2))) <= 1e-21) {
		tmp = x + (wj * ((x * -2.0) + (wj * ((1.0 + (wj * ((-1.0 - (x * -3.0)) - ((x * 5.0) + (x * 0.6666666666666666))))) + (x * 2.5)))));
	} else {
		tmp = (wj + (t_0 / t_1)) + ((wj * (wj + -1.0)) / (t_1 / t_0));
	}
	return tmp;
}

Python

def code(wj, x):
	t_0 = wj - (x / math.exp(wj))
	t_1 = -1.0 - (wj * (wj * wj))
	t_2 = wj * math.exp(wj)
	tmp = 0
	if (wj + ((x - t_2) / (math.exp(wj) + t_2))) <= 1e-21:
		tmp = x + (wj * ((x * -2.0) + (wj * ((1.0 + (wj * ((-1.0 - (x * -3.0)) - ((x * 5.0) + (x * 0.6666666666666666))))) + (x * 2.5)))))
	else:
		tmp = (wj + (t_0 / t_1)) + ((wj * (wj + -1.0)) / (t_1 / t_0))
	return tmp

Julia

function code(wj, x)
	t_0 = Float64(wj - Float64(x / exp(wj)))
	t_1 = Float64(-1.0 - Float64(wj * Float64(wj * wj)))
	t_2 = Float64(wj * exp(wj))
	tmp = 0.0
	if (Float64(wj + Float64(Float64(x - t_2) / Float64(exp(wj) + t_2))) <= 1e-21)
		tmp = Float64(x + Float64(wj * Float64(Float64(x * -2.0) + Float64(wj * Float64(Float64(1.0 + Float64(wj * Float64(Float64(-1.0 - Float64(x * -3.0)) - Float64(Float64(x * 5.0) + Float64(x * 0.6666666666666666))))) + Float64(x * 2.5))))));
	else
		tmp = Float64(Float64(wj + Float64(t_0 / t_1)) + Float64(Float64(wj * Float64(wj + -1.0)) / Float64(t_1 / t_0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(wj, x)
	t_0 = wj - (x / exp(wj));
	t_1 = -1.0 - (wj * (wj * wj));
	t_2 = wj * exp(wj);
	tmp = 0.0;
	if ((wj + ((x - t_2) / (exp(wj) + t_2))) <= 1e-21)
		tmp = x + (wj * ((x * -2.0) + (wj * ((1.0 + (wj * ((-1.0 - (x * -3.0)) - ((x * 5.0) + (x * 0.6666666666666666))))) + (x * 2.5)))));
	else
		tmp = (wj + (t_0 / t_1)) + ((wj * (wj + -1.0)) / (t_1 / t_0));
	end
	tmp_2 = tmp;
end

Wolfram

code[wj_, x_] := Block[{t$95$0 = N[(wj - N[(x / N[Exp[wj], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(-1.0 - N[(wj * N[(wj * wj), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(wj * N[Exp[wj], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(wj + N[(N[(x - t$95$2), $MachinePrecision] / N[(N[Exp[wj], $MachinePrecision] + t$95$2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 1e-21], N[(x + N[(wj * N[(N[(x * -2.0), $MachinePrecision] + N[(wj * N[(N[(1.0 + N[(wj * N[(N[(-1.0 - N[(x * -3.0), $MachinePrecision]), $MachinePrecision] - N[(N[(x * 5.0), $MachinePrecision] + N[(x * 0.6666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(x * 2.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(wj + N[(t$95$0 / t$95$1), $MachinePrecision]), $MachinePrecision] + N[(N[(wj * N[(wj + -1.0), $MachinePrecision]), $MachinePrecision] / N[(t$95$1 / t$95$0), $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))))) < 9.99999999999999908e-22

   1. Initial program 69.5%

      \[wj - \frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}} \]
   2. Add Preprocessing

   3. Taylor expanded in wj around 0

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

      1. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(x, \color{blue}{\left(wj \cdot \left(wj \cdot \left(\left(1 + -1 \cdot \left(wj \cdot \left(1 + \left(-3 \cdot x + \left(-2 \cdot \left(-4 \cdot x + \frac{3}{2} \cdot x\right) + \frac{2}{3} \cdot x\right)\right)\right)\right)\right) - \left(-4 \cdot x + \frac{3}{2} \cdot x\right)\right) - 2 \cdot x\right)\right)}\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(x, \mathsf{*.f64}\left(wj, \color{blue}{\left(wj \cdot \left(\left(1 + -1 \cdot \left(wj \cdot \left(1 + \left(-3 \cdot x + \left(-2 \cdot \left(-4 \cdot x + \frac{3}{2} \cdot x\right) + \frac{2}{3} \cdot x\right)\right)\right)\right)\right) - \left(-4 \cdot x + \frac{3}{2} \cdot x\right)\right) - 2 \cdot x\right)}\right)\right) \]

      3. cancel-sign-sub-inv N/A

         \[\leadsto \mathsf{+.f64}\left(x, \mathsf{*.f64}\left(wj, \left(wj \cdot \left(\left(1 + -1 \cdot \left(wj \cdot \left(1 + \left(-3 \cdot x + \left(-2 \cdot \left(-4 \cdot x + \frac{3}{2} \cdot x\right) + \frac{2}{3} \cdot x\right)\right)\right)\right)\right) - \left(-4 \cdot x + \frac{3}{2} \cdot x\right)\right) + \color{blue}{\left(\mathsf{neg}\left(2\right)\right) \cdot x}\right)\right)\right) \]

      4. metadata-eval N/A

         \[\leadsto \mathsf{+.f64}\left(x, \mathsf{*.f64}\left(wj, \left(wj \cdot \left(\left(1 + -1 \cdot \left(wj \cdot \left(1 + \left(-3 \cdot x + \left(-2 \cdot \left(-4 \cdot x + \frac{3}{2} \cdot x\right) + \frac{2}{3} \cdot x\right)\right)\right)\right)\right) - \left(-4 \cdot x + \frac{3}{2} \cdot x\right)\right) + -2 \cdot x\right)\right)\right) \]

      5. +-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(x, \mathsf{*.f64}\left(wj, \left(-2 \cdot x + \color{blue}{wj \cdot \left(\left(1 + -1 \cdot \left(wj \cdot \left(1 + \left(-3 \cdot x + \left(-2 \cdot \left(-4 \cdot x + \frac{3}{2} \cdot x\right) + \frac{2}{3} \cdot x\right)\right)\right)\right)\right) - \left(-4 \cdot x + \frac{3}{2} \cdot x\right)\right)}\right)\right)\right) \]

      6. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(x, \mathsf{*.f64}\left(wj, \mathsf{+.f64}\left(\left(-2 \cdot x\right), \color{blue}{\left(wj \cdot \left(\left(1 + -1 \cdot \left(wj \cdot \left(1 + \left(-3 \cdot x + \left(-2 \cdot \left(-4 \cdot x + \frac{3}{2} \cdot x\right) + \frac{2}{3} \cdot x\right)\right)\right)\right)\right) - \left(-4 \cdot x + \frac{3}{2} \cdot x\right)\right)\right)}\right)\right)\right) \]

   5. Simplified 99.4%

      \[\leadsto \color{blue}{x + wj \cdot \left(x \cdot -2 + wj \cdot \left(\left(1 - wj \cdot \left(\left(x \cdot 5 + x \cdot 0.6666666666666666\right) + \left(1 + x \cdot -3\right)\right)\right) + x \cdot 2.5\right)\right)} \]

   if 9.99999999999999908e-22 < (-.f64 wj (/.f64 (-.f64 (*.f64 wj (exp.f64 wj)) x) (+.f64 (exp.f64 wj) (*.f64 wj (exp.f64 wj)))))

   1. Initial program 94.7%

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

      1. sub-neg N/A

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

      2. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(wj, \color{blue}{\left(\mathsf{neg}\left(\frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}}\right)\right)}\right) \]

      3. distribute-rgt1-in N/A

         \[\leadsto \mathsf{+.f64}\left(wj, \left(\mathsf{neg}\left(\frac{wj \cdot e^{wj} - x}{\left(wj + 1\right) \cdot e^{wj}}\right)\right)\right) \]

      4. associate-/l/ N/A

         \[\leadsto \mathsf{+.f64}\left(wj, \left(\mathsf{neg}\left(\frac{\frac{wj \cdot e^{wj} - x}{e^{wj}}}{wj + 1}\right)\right)\right) \]

      5. distribute-neg-frac2 N/A

         \[\leadsto \mathsf{+.f64}\left(wj, \left(\frac{\frac{wj \cdot e^{wj} - x}{e^{wj}}}{\color{blue}{\mathsf{neg}\left(\left(wj + 1\right)\right)}}\right)\right) \]

      6. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(wj, \mathsf{/.f64}\left(\left(\frac{wj \cdot e^{wj} - x}{e^{wj}}\right), \color{blue}{\left(\mathsf{neg}\left(\left(wj + 1\right)\right)\right)}\right)\right) \]

      7. div-sub N/A

         \[\leadsto \mathsf{+.f64}\left(wj, \mathsf{/.f64}\left(\left(\frac{wj \cdot e^{wj}}{e^{wj}} - \frac{x}{e^{wj}}\right), \left(\mathsf{neg}\left(\color{blue}{\left(wj + 1\right)}\right)\right)\right)\right) \]

      8. associate-/l* N/A

         \[\leadsto \mathsf{+.f64}\left(wj, \mathsf{/.f64}\left(\left(wj \cdot \frac{e^{wj}}{e^{wj}} - \frac{x}{e^{wj}}\right), \left(\mathsf{neg}\left(\left(\color{blue}{wj} + 1\right)\right)\right)\right)\right) \]

      9. *-inverses N/A

         \[\leadsto \mathsf{+.f64}\left(wj, \mathsf{/.f64}\left(\left(wj \cdot 1 - \frac{x}{e^{wj}}\right), \left(\mathsf{neg}\left(\left(wj + 1\right)\right)\right)\right)\right) \]

      10. *-rgt-identity N/A

          \[\leadsto \mathsf{+.f64}\left(wj, \mathsf{/.f64}\left(\left(wj - \frac{x}{e^{wj}}\right), \left(\mathsf{neg}\left(\left(\color{blue}{wj} + 1\right)\right)\right)\right)\right) \]

      11. --lowering--.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(wj, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(wj, \left(\frac{x}{e^{wj}}\right)\right), \left(\mathsf{neg}\left(\color{blue}{\left(wj + 1\right)}\right)\right)\right)\right) \]

      12. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(wj, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(wj, \mathsf{/.f64}\left(x, \left(e^{wj}\right)\right)\right), \left(\mathsf{neg}\left(\left(wj + \color{blue}{1}\right)\right)\right)\right)\right) \]

      13. exp-lowering-exp.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(wj, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(wj, \mathsf{/.f64}\left(x, \mathsf{exp.f64}\left(wj\right)\right)\right), \left(\mathsf{neg}\left(\left(wj + 1\right)\right)\right)\right)\right) \]

      14. +-commutative N/A

          \[\leadsto \mathsf{+.f64}\left(wj, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(wj, \mathsf{/.f64}\left(x, \mathsf{exp.f64}\left(wj\right)\right)\right), \left(\mathsf{neg}\left(\left(1 + wj\right)\right)\right)\right)\right) \]

      15. *-inverses N/A

          \[\leadsto \mathsf{+.f64}\left(wj, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(wj, \mathsf{/.f64}\left(x, \mathsf{exp.f64}\left(wj\right)\right)\right), \left(\mathsf{neg}\left(\left(\frac{e^{wj}}{e^{wj}} + wj\right)\right)\right)\right)\right) \]

      16. distribute-neg-in N/A

          \[\leadsto \mathsf{+.f64}\left(wj, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(wj, \mathsf{/.f64}\left(x, \mathsf{exp.f64}\left(wj\right)\right)\right), \left(\left(\mathsf{neg}\left(\frac{e^{wj}}{e^{wj}}\right)\right) + \color{blue}{\left(\mathsf{neg}\left(wj\right)\right)}\right)\right)\right) \]

   3. Simplified 99.4%

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

      1. +-commutative N/A

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

      2. flip3-- N/A

         \[\leadsto \frac{wj - \frac{x}{e^{wj}}}{\frac{{-1}^{3} - {wj}^{3}}{-1 \cdot -1 + \left(wj \cdot wj + -1 \cdot wj\right)}} + wj \]

      3. associate-/r/ N/A

         \[\leadsto \frac{wj - \frac{x}{e^{wj}}}{{-1}^{3} - {wj}^{3}} \cdot \left(-1 \cdot -1 + \left(wj \cdot wj + -1 \cdot wj\right)\right) + wj \]

      4. fma-define N/A

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

      5. fma-lowering-fma.f64 N/A

         \[\leadsto \mathsf{fma.f64}\left(\left(\frac{wj - \frac{x}{e^{wj}}}{{-1}^{3} - {wj}^{3}}\right), \color{blue}{\left(-1 \cdot -1 + \left(wj \cdot wj + -1 \cdot wj\right)\right)}, wj\right) \]

   6. Applied egg-rr 99.3%

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

      1. +-commutative N/A

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

      2. distribute-lft-in N/A

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

      3. *-rgt-identity N/A

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

      4. associate-+r+ N/A

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

      5. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\left(wj + \frac{wj - \frac{x}{e^{wj}}}{-1 - wj \cdot \left(wj \cdot wj\right)}\right), \color{blue}{\left(\frac{wj - \frac{x}{e^{wj}}}{-1 - wj \cdot \left(wj \cdot wj\right)} \cdot \left(wj \cdot \left(wj + -1\right)\right)\right)}\right) \]

   8. Applied egg-rr 99.4%

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

4. Final simplification 99.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;wj + \frac{x - wj \cdot e^{wj}}{e^{wj} + wj \cdot e^{wj}} \leq 10^{-21}:\\ \;\;\;\;x + wj \cdot \left(x \cdot -2 + wj \cdot \left(\left(1 + wj \cdot \left(\left(-1 - x \cdot -3\right) - \left(x \cdot 5 + x \cdot 0.6666666666666666\right)\right)\right) + x \cdot 2.5\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(wj + \frac{wj - \frac{x}{e^{wj}}}{-1 - wj \cdot \left(wj \cdot wj\right)}\right) + \frac{wj \cdot \left(wj + -1\right)}{\frac{-1 - wj \cdot \left(wj \cdot wj\right)}{wj - \frac{x}{e^{wj}}}}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 98.1% accurate, 2.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;wj \leq -4.6 \cdot 10^{-6}:\\ \;\;\;\;wj + \frac{wj - \frac{x}{e^{wj}}}{-1 - wj}\\ \mathbf{else}:\\ \;\;\;\;x + wj \cdot \left(x \cdot -2 + wj \cdot \left(\left(1 + wj \cdot \left(\left(-1 - x \cdot -3\right) - \left(x \cdot 5 + x \cdot 0.6666666666666666\right)\right)\right) + x \cdot 2.5\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (wj x)
 :precision binary64
 (if (<= wj -4.6e-6)
   (+ wj (/ (- wj (/ x (exp wj))) (- -1.0 wj)))
   (+
    x
    (*
     wj
     (+
      (* x -2.0)
      (*
       wj
       (+
        (+
         1.0
         (* wj (- (- -1.0 (* x -3.0)) (+ (* x 5.0) (* x 0.6666666666666666)))))
        (* x 2.5))))))))

C

double code(double wj, double x) {
	double tmp;
	if (wj <= -4.6e-6) {
		tmp = wj + ((wj - (x / exp(wj))) / (-1.0 - wj));
	} else {
		tmp = x + (wj * ((x * -2.0) + (wj * ((1.0 + (wj * ((-1.0 - (x * -3.0)) - ((x * 5.0) + (x * 0.6666666666666666))))) + (x * 2.5)))));
	}
	return tmp;
}

Fortran

real(8) function code(wj, x)
    real(8), intent (in) :: wj
    real(8), intent (in) :: x
    real(8) :: tmp
    if (wj <= (-4.6d-6)) then
        tmp = wj + ((wj - (x / exp(wj))) / ((-1.0d0) - wj))
    else
        tmp = x + (wj * ((x * (-2.0d0)) + (wj * ((1.0d0 + (wj * (((-1.0d0) - (x * (-3.0d0))) - ((x * 5.0d0) + (x * 0.6666666666666666d0))))) + (x * 2.5d0)))))
    end if
    code = tmp
end function

Java

public static double code(double wj, double x) {
	double tmp;
	if (wj <= -4.6e-6) {
		tmp = wj + ((wj - (x / Math.exp(wj))) / (-1.0 - wj));
	} else {
		tmp = x + (wj * ((x * -2.0) + (wj * ((1.0 + (wj * ((-1.0 - (x * -3.0)) - ((x * 5.0) + (x * 0.6666666666666666))))) + (x * 2.5)))));
	}
	return tmp;
}

Python

def code(wj, x):
	tmp = 0
	if wj <= -4.6e-6:
		tmp = wj + ((wj - (x / math.exp(wj))) / (-1.0 - wj))
	else:
		tmp = x + (wj * ((x * -2.0) + (wj * ((1.0 + (wj * ((-1.0 - (x * -3.0)) - ((x * 5.0) + (x * 0.6666666666666666))))) + (x * 2.5)))))
	return tmp

Julia

function code(wj, x)
	tmp = 0.0
	if (wj <= -4.6e-6)
		tmp = Float64(wj + Float64(Float64(wj - Float64(x / exp(wj))) / Float64(-1.0 - wj)));
	else
		tmp = Float64(x + Float64(wj * Float64(Float64(x * -2.0) + Float64(wj * Float64(Float64(1.0 + Float64(wj * Float64(Float64(-1.0 - Float64(x * -3.0)) - Float64(Float64(x * 5.0) + Float64(x * 0.6666666666666666))))) + Float64(x * 2.5))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(wj, x)
	tmp = 0.0;
	if (wj <= -4.6e-6)
		tmp = wj + ((wj - (x / exp(wj))) / (-1.0 - wj));
	else
		tmp = x + (wj * ((x * -2.0) + (wj * ((1.0 + (wj * ((-1.0 - (x * -3.0)) - ((x * 5.0) + (x * 0.6666666666666666))))) + (x * 2.5)))));
	end
	tmp_2 = tmp;
end

Wolfram

code[wj_, x_] := If[LessEqual[wj, -4.6e-6], 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[(x * -2.0), $MachinePrecision] + N[(wj * N[(N[(1.0 + N[(wj * N[(N[(-1.0 - N[(x * -3.0), $MachinePrecision]), $MachinePrecision] - N[(N[(x * 5.0), $MachinePrecision] + N[(x * 0.6666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(x * 2.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if wj < -4.6e-6

   1. Initial program 71.1%

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

      1. sub-neg N/A

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

      2. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(wj, \color{blue}{\left(\mathsf{neg}\left(\frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}}\right)\right)}\right) \]

      3. distribute-rgt1-in N/A

         \[\leadsto \mathsf{+.f64}\left(wj, \left(\mathsf{neg}\left(\frac{wj \cdot e^{wj} - x}{\left(wj + 1\right) \cdot e^{wj}}\right)\right)\right) \]

      4. associate-/l/ N/A

         \[\leadsto \mathsf{+.f64}\left(wj, \left(\mathsf{neg}\left(\frac{\frac{wj \cdot e^{wj} - x}{e^{wj}}}{wj + 1}\right)\right)\right) \]

      5. distribute-neg-frac2 N/A

         \[\leadsto \mathsf{+.f64}\left(wj, \left(\frac{\frac{wj \cdot e^{wj} - x}{e^{wj}}}{\color{blue}{\mathsf{neg}\left(\left(wj + 1\right)\right)}}\right)\right) \]

      6. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(wj, \mathsf{/.f64}\left(\left(\frac{wj \cdot e^{wj} - x}{e^{wj}}\right), \color{blue}{\left(\mathsf{neg}\left(\left(wj + 1\right)\right)\right)}\right)\right) \]

      7. div-sub N/A

         \[\leadsto \mathsf{+.f64}\left(wj, \mathsf{/.f64}\left(\left(\frac{wj \cdot e^{wj}}{e^{wj}} - \frac{x}{e^{wj}}\right), \left(\mathsf{neg}\left(\color{blue}{\left(wj + 1\right)}\right)\right)\right)\right) \]

      8. associate-/l* N/A

         \[\leadsto \mathsf{+.f64}\left(wj, \mathsf{/.f64}\left(\left(wj \cdot \frac{e^{wj}}{e^{wj}} - \frac{x}{e^{wj}}\right), \left(\mathsf{neg}\left(\left(\color{blue}{wj} + 1\right)\right)\right)\right)\right) \]

      9. *-inverses N/A

         \[\leadsto \mathsf{+.f64}\left(wj, \mathsf{/.f64}\left(\left(wj \cdot 1 - \frac{x}{e^{wj}}\right), \left(\mathsf{neg}\left(\left(wj + 1\right)\right)\right)\right)\right) \]

      10. *-rgt-identity N/A

          \[\leadsto \mathsf{+.f64}\left(wj, \mathsf{/.f64}\left(\left(wj - \frac{x}{e^{wj}}\right), \left(\mathsf{neg}\left(\left(\color{blue}{wj} + 1\right)\right)\right)\right)\right) \]

      11. --lowering--.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(wj, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(wj, \left(\frac{x}{e^{wj}}\right)\right), \left(\mathsf{neg}\left(\color{blue}{\left(wj + 1\right)}\right)\right)\right)\right) \]

      12. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(wj, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(wj, \mathsf{/.f64}\left(x, \left(e^{wj}\right)\right)\right), \left(\mathsf{neg}\left(\left(wj + \color{blue}{1}\right)\right)\right)\right)\right) \]

      13. exp-lowering-exp.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(wj, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(wj, \mathsf{/.f64}\left(x, \mathsf{exp.f64}\left(wj\right)\right)\right), \left(\mathsf{neg}\left(\left(wj + 1\right)\right)\right)\right)\right) \]

      14. +-commutative N/A

          \[\leadsto \mathsf{+.f64}\left(wj, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(wj, \mathsf{/.f64}\left(x, \mathsf{exp.f64}\left(wj\right)\right)\right), \left(\mathsf{neg}\left(\left(1 + wj\right)\right)\right)\right)\right) \]

      15. *-inverses N/A

          \[\leadsto \mathsf{+.f64}\left(wj, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(wj, \mathsf{/.f64}\left(x, \mathsf{exp.f64}\left(wj\right)\right)\right), \left(\mathsf{neg}\left(\left(\frac{e^{wj}}{e^{wj}} + wj\right)\right)\right)\right)\right) \]

      16. distribute-neg-in N/A

          \[\leadsto \mathsf{+.f64}\left(wj, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(wj, \mathsf{/.f64}\left(x, \mathsf{exp.f64}\left(wj\right)\right)\right), \left(\left(\mathsf{neg}\left(\frac{e^{wj}}{e^{wj}}\right)\right) + \color{blue}{\left(\mathsf{neg}\left(wj\right)\right)}\right)\right)\right) \]

   3. Simplified 96.8%

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

   if -4.6e-6 < wj

   1. Initial program 78.3%

      \[wj - \frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}} \]
   2. Add Preprocessing

   3. Taylor expanded in wj around 0

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

      1. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(x, \color{blue}{\left(wj \cdot \left(wj \cdot \left(\left(1 + -1 \cdot \left(wj \cdot \left(1 + \left(-3 \cdot x + \left(-2 \cdot \left(-4 \cdot x + \frac{3}{2} \cdot x\right) + \frac{2}{3} \cdot x\right)\right)\right)\right)\right) - \left(-4 \cdot x + \frac{3}{2} \cdot x\right)\right) - 2 \cdot x\right)\right)}\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(x, \mathsf{*.f64}\left(wj, \color{blue}{\left(wj \cdot \left(\left(1 + -1 \cdot \left(wj \cdot \left(1 + \left(-3 \cdot x + \left(-2 \cdot \left(-4 \cdot x + \frac{3}{2} \cdot x\right) + \frac{2}{3} \cdot x\right)\right)\right)\right)\right) - \left(-4 \cdot x + \frac{3}{2} \cdot x\right)\right) - 2 \cdot x\right)}\right)\right) \]

      3. cancel-sign-sub-inv N/A

         \[\leadsto \mathsf{+.f64}\left(x, \mathsf{*.f64}\left(wj, \left(wj \cdot \left(\left(1 + -1 \cdot \left(wj \cdot \left(1 + \left(-3 \cdot x + \left(-2 \cdot \left(-4 \cdot x + \frac{3}{2} \cdot x\right) + \frac{2}{3} \cdot x\right)\right)\right)\right)\right) - \left(-4 \cdot x + \frac{3}{2} \cdot x\right)\right) + \color{blue}{\left(\mathsf{neg}\left(2\right)\right) \cdot x}\right)\right)\right) \]

      4. metadata-eval N/A

         \[\leadsto \mathsf{+.f64}\left(x, \mathsf{*.f64}\left(wj, \left(wj \cdot \left(\left(1 + -1 \cdot \left(wj \cdot \left(1 + \left(-3 \cdot x + \left(-2 \cdot \left(-4 \cdot x + \frac{3}{2} \cdot x\right) + \frac{2}{3} \cdot x\right)\right)\right)\right)\right) - \left(-4 \cdot x + \frac{3}{2} \cdot x\right)\right) + -2 \cdot x\right)\right)\right) \]

      5. +-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(x, \mathsf{*.f64}\left(wj, \left(-2 \cdot x + \color{blue}{wj \cdot \left(\left(1 + -1 \cdot \left(wj \cdot \left(1 + \left(-3 \cdot x + \left(-2 \cdot \left(-4 \cdot x + \frac{3}{2} \cdot x\right) + \frac{2}{3} \cdot x\right)\right)\right)\right)\right) - \left(-4 \cdot x + \frac{3}{2} \cdot x\right)\right)}\right)\right)\right) \]

      6. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(x, \mathsf{*.f64}\left(wj, \mathsf{+.f64}\left(\left(-2 \cdot x\right), \color{blue}{\left(wj \cdot \left(\left(1 + -1 \cdot \left(wj \cdot \left(1 + \left(-3 \cdot x + \left(-2 \cdot \left(-4 \cdot x + \frac{3}{2} \cdot x\right) + \frac{2}{3} \cdot x\right)\right)\right)\right)\right) - \left(-4 \cdot x + \frac{3}{2} \cdot x\right)\right)\right)}\right)\right)\right) \]

   5. Simplified 98.3%

      \[\leadsto \color{blue}{x + wj \cdot \left(x \cdot -2 + wj \cdot \left(\left(1 - wj \cdot \left(\left(x \cdot 5 + x \cdot 0.6666666666666666\right) + \left(1 + x \cdot -3\right)\right)\right) + x \cdot 2.5\right)\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 98.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;wj \leq -4.6 \cdot 10^{-6}:\\ \;\;\;\;wj + \frac{wj - \frac{x}{e^{wj}}}{-1 - wj}\\ \mathbf{else}:\\ \;\;\;\;x + wj \cdot \left(x \cdot -2 + wj \cdot \left(\left(1 + wj \cdot \left(\left(-1 - x \cdot -3\right) - \left(x \cdot 5 + x \cdot 0.6666666666666666\right)\right)\right) + x \cdot 2.5\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 97.6% accurate, 2.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;wj \leq -0.0023:\\ \;\;\;\;\frac{x}{e^{wj} \cdot \left(wj + 1\right)}\\ \mathbf{else}:\\ \;\;\;\;x + wj \cdot \left(x \cdot -2 + wj \cdot \left(\left(1 + wj \cdot \left(\left(-1 - x \cdot -3\right) - \left(x \cdot 5 + x \cdot 0.6666666666666666\right)\right)\right) + x \cdot 2.5\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (wj x)
 :precision binary64
 (if (<= wj -0.0023)
   (/ x (* (exp wj) (+ wj 1.0)))
   (+
    x
    (*
     wj
     (+
      (* x -2.0)
      (*
       wj
       (+
        (+
         1.0
         (* wj (- (- -1.0 (* x -3.0)) (+ (* x 5.0) (* x 0.6666666666666666)))))
        (* x 2.5))))))))

C

double code(double wj, double x) {
	double tmp;
	if (wj <= -0.0023) {
		tmp = x / (exp(wj) * (wj + 1.0));
	} else {
		tmp = x + (wj * ((x * -2.0) + (wj * ((1.0 + (wj * ((-1.0 - (x * -3.0)) - ((x * 5.0) + (x * 0.6666666666666666))))) + (x * 2.5)))));
	}
	return tmp;
}

Fortran

real(8) function code(wj, x)
    real(8), intent (in) :: wj
    real(8), intent (in) :: x
    real(8) :: tmp
    if (wj <= (-0.0023d0)) then
        tmp = x / (exp(wj) * (wj + 1.0d0))
    else
        tmp = x + (wj * ((x * (-2.0d0)) + (wj * ((1.0d0 + (wj * (((-1.0d0) - (x * (-3.0d0))) - ((x * 5.0d0) + (x * 0.6666666666666666d0))))) + (x * 2.5d0)))))
    end if
    code = tmp
end function

Java

public static double code(double wj, double x) {
	double tmp;
	if (wj <= -0.0023) {
		tmp = x / (Math.exp(wj) * (wj + 1.0));
	} else {
		tmp = x + (wj * ((x * -2.0) + (wj * ((1.0 + (wj * ((-1.0 - (x * -3.0)) - ((x * 5.0) + (x * 0.6666666666666666))))) + (x * 2.5)))));
	}
	return tmp;
}

Python

def code(wj, x):
	tmp = 0
	if wj <= -0.0023:
		tmp = x / (math.exp(wj) * (wj + 1.0))
	else:
		tmp = x + (wj * ((x * -2.0) + (wj * ((1.0 + (wj * ((-1.0 - (x * -3.0)) - ((x * 5.0) + (x * 0.6666666666666666))))) + (x * 2.5)))))
	return tmp

Julia

function code(wj, x)
	tmp = 0.0
	if (wj <= -0.0023)
		tmp = Float64(x / Float64(exp(wj) * Float64(wj + 1.0)));
	else
		tmp = Float64(x + Float64(wj * Float64(Float64(x * -2.0) + Float64(wj * Float64(Float64(1.0 + Float64(wj * Float64(Float64(-1.0 - Float64(x * -3.0)) - Float64(Float64(x * 5.0) + Float64(x * 0.6666666666666666))))) + Float64(x * 2.5))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(wj, x)
	tmp = 0.0;
	if (wj <= -0.0023)
		tmp = x / (exp(wj) * (wj + 1.0));
	else
		tmp = x + (wj * ((x * -2.0) + (wj * ((1.0 + (wj * ((-1.0 - (x * -3.0)) - ((x * 5.0) + (x * 0.6666666666666666))))) + (x * 2.5)))));
	end
	tmp_2 = tmp;
end

Wolfram

code[wj_, x_] := If[LessEqual[wj, -0.0023], N[(x / N[(N[Exp[wj], $MachinePrecision] * N[(wj + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x + N[(wj * N[(N[(x * -2.0), $MachinePrecision] + N[(wj * N[(N[(1.0 + N[(wj * N[(N[(-1.0 - N[(x * -3.0), $MachinePrecision]), $MachinePrecision] - N[(N[(x * 5.0), $MachinePrecision] + N[(x * 0.6666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(x * 2.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if wj < -0.0023

   1. Initial program 65.3%

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

      1. sub-neg N/A

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

      2. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(wj, \color{blue}{\left(\mathsf{neg}\left(\frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}}\right)\right)}\right) \]

      3. distribute-rgt1-in N/A

         \[\leadsto \mathsf{+.f64}\left(wj, \left(\mathsf{neg}\left(\frac{wj \cdot e^{wj} - x}{\left(wj + 1\right) \cdot e^{wj}}\right)\right)\right) \]

      4. associate-/l/ N/A

         \[\leadsto \mathsf{+.f64}\left(wj, \left(\mathsf{neg}\left(\frac{\frac{wj \cdot e^{wj} - x}{e^{wj}}}{wj + 1}\right)\right)\right) \]

      5. distribute-neg-frac2 N/A

         \[\leadsto \mathsf{+.f64}\left(wj, \left(\frac{\frac{wj \cdot e^{wj} - x}{e^{wj}}}{\color{blue}{\mathsf{neg}\left(\left(wj + 1\right)\right)}}\right)\right) \]

      6. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(wj, \mathsf{/.f64}\left(\left(\frac{wj \cdot e^{wj} - x}{e^{wj}}\right), \color{blue}{\left(\mathsf{neg}\left(\left(wj + 1\right)\right)\right)}\right)\right) \]

      7. div-sub N/A

         \[\leadsto \mathsf{+.f64}\left(wj, \mathsf{/.f64}\left(\left(\frac{wj \cdot e^{wj}}{e^{wj}} - \frac{x}{e^{wj}}\right), \left(\mathsf{neg}\left(\color{blue}{\left(wj + 1\right)}\right)\right)\right)\right) \]

      8. associate-/l* N/A

         \[\leadsto \mathsf{+.f64}\left(wj, \mathsf{/.f64}\left(\left(wj \cdot \frac{e^{wj}}{e^{wj}} - \frac{x}{e^{wj}}\right), \left(\mathsf{neg}\left(\left(\color{blue}{wj} + 1\right)\right)\right)\right)\right) \]

      9. *-inverses N/A

         \[\leadsto \mathsf{+.f64}\left(wj, \mathsf{/.f64}\left(\left(wj \cdot 1 - \frac{x}{e^{wj}}\right), \left(\mathsf{neg}\left(\left(wj + 1\right)\right)\right)\right)\right) \]

      10. *-rgt-identity N/A

          \[\leadsto \mathsf{+.f64}\left(wj, \mathsf{/.f64}\left(\left(wj - \frac{x}{e^{wj}}\right), \left(\mathsf{neg}\left(\left(\color{blue}{wj} + 1\right)\right)\right)\right)\right) \]

      11. --lowering--.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(wj, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(wj, \left(\frac{x}{e^{wj}}\right)\right), \left(\mathsf{neg}\left(\color{blue}{\left(wj + 1\right)}\right)\right)\right)\right) \]

      12. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(wj, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(wj, \mathsf{/.f64}\left(x, \left(e^{wj}\right)\right)\right), \left(\mathsf{neg}\left(\left(wj + \color{blue}{1}\right)\right)\right)\right)\right) \]

      13. exp-lowering-exp.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(wj, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(wj, \mathsf{/.f64}\left(x, \mathsf{exp.f64}\left(wj\right)\right)\right), \left(\mathsf{neg}\left(\left(wj + 1\right)\right)\right)\right)\right) \]

      14. +-commutative N/A

          \[\leadsto \mathsf{+.f64}\left(wj, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(wj, \mathsf{/.f64}\left(x, \mathsf{exp.f64}\left(wj\right)\right)\right), \left(\mathsf{neg}\left(\left(1 + wj\right)\right)\right)\right)\right) \]

      15. *-inverses N/A

          \[\leadsto \mathsf{+.f64}\left(wj, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(wj, \mathsf{/.f64}\left(x, \mathsf{exp.f64}\left(wj\right)\right)\right), \left(\mathsf{neg}\left(\left(\frac{e^{wj}}{e^{wj}} + wj\right)\right)\right)\right)\right) \]

      16. distribute-neg-in N/A

          \[\leadsto \mathsf{+.f64}\left(wj, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(wj, \mathsf{/.f64}\left(x, \mathsf{exp.f64}\left(wj\right)\right)\right), \left(\left(\mathsf{neg}\left(\frac{e^{wj}}{e^{wj}}\right)\right) + \color{blue}{\left(\mathsf{neg}\left(wj\right)\right)}\right)\right)\right) \]

   3. Simplified 99.6%

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

   5. Taylor expanded in x around inf

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

      1. distribute-rgt-in N/A

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

      2. *-lft-identity N/A

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

      3. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(x, \color{blue}{\left(e^{wj} + wj \cdot e^{wj}\right)}\right) \]

      4. *-lft-identity N/A

         \[\leadsto \mathsf{/.f64}\left(x, \left(1 \cdot e^{wj} + \color{blue}{wj} \cdot e^{wj}\right)\right) \]

      5. distribute-rgt-in N/A

         \[\leadsto \mathsf{/.f64}\left(x, \left(e^{wj} \cdot \color{blue}{\left(1 + wj\right)}\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(x, \mathsf{*.f64}\left(\left(e^{wj}\right), \color{blue}{\left(1 + wj\right)}\right)\right) \]

      7. exp-lowering-exp.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(x, \mathsf{*.f64}\left(\mathsf{exp.f64}\left(wj\right), \left(\color{blue}{1} + wj\right)\right)\right) \]

      8. +-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(x, \mathsf{*.f64}\left(\mathsf{exp.f64}\left(wj\right), \left(wj + \color{blue}{1}\right)\right)\right) \]

      9. +-lowering-+.f64 83.4%

         \[\leadsto \mathsf{/.f64}\left(x, \mathsf{*.f64}\left(\mathsf{exp.f64}\left(wj\right), \mathsf{+.f64}\left(wj, \color{blue}{1}\right)\right)\right) \]

   7. Simplified 83.4%

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

   if -0.0023 < wj

   1. Initial program 78.3%

      \[wj - \frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}} \]
   2. Add Preprocessing

   3. Taylor expanded in wj around 0

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

      1. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(x, \color{blue}{\left(wj \cdot \left(wj \cdot \left(\left(1 + -1 \cdot \left(wj \cdot \left(1 + \left(-3 \cdot x + \left(-2 \cdot \left(-4 \cdot x + \frac{3}{2} \cdot x\right) + \frac{2}{3} \cdot x\right)\right)\right)\right)\right) - \left(-4 \cdot x + \frac{3}{2} \cdot x\right)\right) - 2 \cdot x\right)\right)}\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(x, \mathsf{*.f64}\left(wj, \color{blue}{\left(wj \cdot \left(\left(1 + -1 \cdot \left(wj \cdot \left(1 + \left(-3 \cdot x + \left(-2 \cdot \left(-4 \cdot x + \frac{3}{2} \cdot x\right) + \frac{2}{3} \cdot x\right)\right)\right)\right)\right) - \left(-4 \cdot x + \frac{3}{2} \cdot x\right)\right) - 2 \cdot x\right)}\right)\right) \]

      3. cancel-sign-sub-inv N/A

         \[\leadsto \mathsf{+.f64}\left(x, \mathsf{*.f64}\left(wj, \left(wj \cdot \left(\left(1 + -1 \cdot \left(wj \cdot \left(1 + \left(-3 \cdot x + \left(-2 \cdot \left(-4 \cdot x + \frac{3}{2} \cdot x\right) + \frac{2}{3} \cdot x\right)\right)\right)\right)\right) - \left(-4 \cdot x + \frac{3}{2} \cdot x\right)\right) + \color{blue}{\left(\mathsf{neg}\left(2\right)\right) \cdot x}\right)\right)\right) \]

      4. metadata-eval N/A

         \[\leadsto \mathsf{+.f64}\left(x, \mathsf{*.f64}\left(wj, \left(wj \cdot \left(\left(1 + -1 \cdot \left(wj \cdot \left(1 + \left(-3 \cdot x + \left(-2 \cdot \left(-4 \cdot x + \frac{3}{2} \cdot x\right) + \frac{2}{3} \cdot x\right)\right)\right)\right)\right) - \left(-4 \cdot x + \frac{3}{2} \cdot x\right)\right) + -2 \cdot x\right)\right)\right) \]

      5. +-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(x, \mathsf{*.f64}\left(wj, \left(-2 \cdot x + \color{blue}{wj \cdot \left(\left(1 + -1 \cdot \left(wj \cdot \left(1 + \left(-3 \cdot x + \left(-2 \cdot \left(-4 \cdot x + \frac{3}{2} \cdot x\right) + \frac{2}{3} \cdot x\right)\right)\right)\right)\right) - \left(-4 \cdot x + \frac{3}{2} \cdot x\right)\right)}\right)\right)\right) \]

      6. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(x, \mathsf{*.f64}\left(wj, \mathsf{+.f64}\left(\left(-2 \cdot x\right), \color{blue}{\left(wj \cdot \left(\left(1 + -1 \cdot \left(wj \cdot \left(1 + \left(-3 \cdot x + \left(-2 \cdot \left(-4 \cdot x + \frac{3}{2} \cdot x\right) + \frac{2}{3} \cdot x\right)\right)\right)\right)\right) - \left(-4 \cdot x + \frac{3}{2} \cdot x\right)\right)\right)}\right)\right)\right) \]

   5. Simplified 98.2%

      \[\leadsto \color{blue}{x + wj \cdot \left(x \cdot -2 + wj \cdot \left(\left(1 - wj \cdot \left(\left(x \cdot 5 + x \cdot 0.6666666666666666\right) + \left(1 + x \cdot -3\right)\right)\right) + x \cdot 2.5\right)\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 97.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;wj \leq -0.0023:\\ \;\;\;\;\frac{x}{e^{wj} \cdot \left(wj + 1\right)}\\ \mathbf{else}:\\ \;\;\;\;x + wj \cdot \left(x \cdot -2 + wj \cdot \left(\left(1 + wj \cdot \left(\left(-1 - x \cdot -3\right) - \left(x \cdot 5 + x \cdot 0.6666666666666666\right)\right)\right) + x \cdot 2.5\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 96.2% accurate, 10.1× speedup

Math

\[\begin{array}{l} \\ x + wj \cdot \left(x \cdot -2 + wj \cdot \left(\left(1 + wj \cdot \left(\left(-1 - x \cdot -3\right) - \left(x \cdot 5 + x \cdot 0.6666666666666666\right)\right)\right) + x \cdot 2.5\right)\right) \end{array} \]

FPCore

(FPCore (wj x)
 :precision binary64
 (+
  x
  (*
   wj
   (+
    (* x -2.0)
    (*
     wj
     (+
      (+
       1.0
       (* wj (- (- -1.0 (* x -3.0)) (+ (* x 5.0) (* x 0.6666666666666666)))))
      (* x 2.5)))))))

C

double code(double wj, double x) {
	return x + (wj * ((x * -2.0) + (wj * ((1.0 + (wj * ((-1.0 - (x * -3.0)) - ((x * 5.0) + (x * 0.6666666666666666))))) + (x * 2.5)))));
}

Fortran

real(8) function code(wj, x)
    real(8), intent (in) :: wj
    real(8), intent (in) :: x
    code = x + (wj * ((x * (-2.0d0)) + (wj * ((1.0d0 + (wj * (((-1.0d0) - (x * (-3.0d0))) - ((x * 5.0d0) + (x * 0.6666666666666666d0))))) + (x * 2.5d0)))))
end function

Java

public static double code(double wj, double x) {
	return x + (wj * ((x * -2.0) + (wj * ((1.0 + (wj * ((-1.0 - (x * -3.0)) - ((x * 5.0) + (x * 0.6666666666666666))))) + (x * 2.5)))));
}

Python

def code(wj, x):
	return x + (wj * ((x * -2.0) + (wj * ((1.0 + (wj * ((-1.0 - (x * -3.0)) - ((x * 5.0) + (x * 0.6666666666666666))))) + (x * 2.5)))))

Julia

function code(wj, x)
	return Float64(x + Float64(wj * Float64(Float64(x * -2.0) + Float64(wj * Float64(Float64(1.0 + Float64(wj * Float64(Float64(-1.0 - Float64(x * -3.0)) - Float64(Float64(x * 5.0) + Float64(x * 0.6666666666666666))))) + Float64(x * 2.5))))))
end

MATLAB

function tmp = code(wj, x)
	tmp = x + (wj * ((x * -2.0) + (wj * ((1.0 + (wj * ((-1.0 - (x * -3.0)) - ((x * 5.0) + (x * 0.6666666666666666))))) + (x * 2.5)))));
end

Wolfram

code[wj_, x_] := N[(x + N[(wj * N[(N[(x * -2.0), $MachinePrecision] + N[(wj * N[(N[(1.0 + N[(wj * N[(N[(-1.0 - N[(x * -3.0), $MachinePrecision]), $MachinePrecision] - N[(N[(x * 5.0), $MachinePrecision] + N[(x * 0.6666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(x * 2.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 78.0%

   \[wj - \frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}} \]
2. Add Preprocessing

3. Taylor expanded in wj around 0

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

   1. +-lowering-+.f64 N/A

      \[\leadsto \mathsf{+.f64}\left(x, \color{blue}{\left(wj \cdot \left(wj \cdot \left(\left(1 + -1 \cdot \left(wj \cdot \left(1 + \left(-3 \cdot x + \left(-2 \cdot \left(-4 \cdot x + \frac{3}{2} \cdot x\right) + \frac{2}{3} \cdot x\right)\right)\right)\right)\right) - \left(-4 \cdot x + \frac{3}{2} \cdot x\right)\right) - 2 \cdot x\right)\right)}\right) \]

   2. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{+.f64}\left(x, \mathsf{*.f64}\left(wj, \color{blue}{\left(wj \cdot \left(\left(1 + -1 \cdot \left(wj \cdot \left(1 + \left(-3 \cdot x + \left(-2 \cdot \left(-4 \cdot x + \frac{3}{2} \cdot x\right) + \frac{2}{3} \cdot x\right)\right)\right)\right)\right) - \left(-4 \cdot x + \frac{3}{2} \cdot x\right)\right) - 2 \cdot x\right)}\right)\right) \]

   3. cancel-sign-sub-inv N/A

      \[\leadsto \mathsf{+.f64}\left(x, \mathsf{*.f64}\left(wj, \left(wj \cdot \left(\left(1 + -1 \cdot \left(wj \cdot \left(1 + \left(-3 \cdot x + \left(-2 \cdot \left(-4 \cdot x + \frac{3}{2} \cdot x\right) + \frac{2}{3} \cdot x\right)\right)\right)\right)\right) - \left(-4 \cdot x + \frac{3}{2} \cdot x\right)\right) + \color{blue}{\left(\mathsf{neg}\left(2\right)\right) \cdot x}\right)\right)\right) \]

   4. metadata-eval N/A

      \[\leadsto \mathsf{+.f64}\left(x, \mathsf{*.f64}\left(wj, \left(wj \cdot \left(\left(1 + -1 \cdot \left(wj \cdot \left(1 + \left(-3 \cdot x + \left(-2 \cdot \left(-4 \cdot x + \frac{3}{2} \cdot x\right) + \frac{2}{3} \cdot x\right)\right)\right)\right)\right) - \left(-4 \cdot x + \frac{3}{2} \cdot x\right)\right) + -2 \cdot x\right)\right)\right) \]

   5. +-commutative N/A

      \[\leadsto \mathsf{+.f64}\left(x, \mathsf{*.f64}\left(wj, \left(-2 \cdot x + \color{blue}{wj \cdot \left(\left(1 + -1 \cdot \left(wj \cdot \left(1 + \left(-3 \cdot x + \left(-2 \cdot \left(-4 \cdot x + \frac{3}{2} \cdot x\right) + \frac{2}{3} \cdot x\right)\right)\right)\right)\right) - \left(-4 \cdot x + \frac{3}{2} \cdot x\right)\right)}\right)\right)\right) \]

   6. +-lowering-+.f64 N/A

      \[\leadsto \mathsf{+.f64}\left(x, \mathsf{*.f64}\left(wj, \mathsf{+.f64}\left(\left(-2 \cdot x\right), \color{blue}{\left(wj \cdot \left(\left(1 + -1 \cdot \left(wj \cdot \left(1 + \left(-3 \cdot x + \left(-2 \cdot \left(-4 \cdot x + \frac{3}{2} \cdot x\right) + \frac{2}{3} \cdot x\right)\right)\right)\right)\right) - \left(-4 \cdot x + \frac{3}{2} \cdot x\right)\right)\right)}\right)\right)\right) \]

5. Simplified 96.3%

   \[\leadsto \color{blue}{x + wj \cdot \left(x \cdot -2 + wj \cdot \left(\left(1 - wj \cdot \left(\left(x \cdot 5 + x \cdot 0.6666666666666666\right) + \left(1 + x \cdot -3\right)\right)\right) + x \cdot 2.5\right)\right)} \]

6. Final simplification 96.3%

   \[\leadsto x + wj \cdot \left(x \cdot -2 + wj \cdot \left(\left(1 + wj \cdot \left(\left(-1 - x \cdot -3\right) - \left(x \cdot 5 + x \cdot 0.6666666666666666\right)\right)\right) + x \cdot 2.5\right)\right) \]
7. Add Preprocessing

Alternative 5: 83.8% accurate, 20.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x \cdot \left(1 - wj\right)\\ \mathbf{if}\;x \leq -4.4 \cdot 10^{-290}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq 1.08 \cdot 10^{-265}:\\ \;\;\;\;wj \cdot wj\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (wj x)
 :precision binary64
 (let* ((t_0 (* x (- 1.0 wj))))
   (if (<= x -4.4e-290) t_0 (if (<= x 1.08e-265) (* wj wj) t_0))))

C

double code(double wj, double x) {
	double t_0 = x * (1.0 - wj);
	double tmp;
	if (x <= -4.4e-290) {
		tmp = t_0;
	} else if (x <= 1.08e-265) {
		tmp = wj * wj;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(wj, x)
    real(8), intent (in) :: wj
    real(8), intent (in) :: x
    real(8) :: t_0
    real(8) :: tmp
    t_0 = x * (1.0d0 - wj)
    if (x <= (-4.4d-290)) then
        tmp = t_0
    else if (x <= 1.08d-265) then
        tmp = wj * wj
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double wj, double x) {
	double t_0 = x * (1.0 - wj);
	double tmp;
	if (x <= -4.4e-290) {
		tmp = t_0;
	} else if (x <= 1.08e-265) {
		tmp = wj * wj;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(wj, x):
	t_0 = x * (1.0 - wj)
	tmp = 0
	if x <= -4.4e-290:
		tmp = t_0
	elif x <= 1.08e-265:
		tmp = wj * wj
	else:
		tmp = t_0
	return tmp

Julia

function code(wj, x)
	t_0 = Float64(x * Float64(1.0 - wj))
	tmp = 0.0
	if (x <= -4.4e-290)
		tmp = t_0;
	elseif (x <= 1.08e-265)
		tmp = Float64(wj * wj);
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(wj, x)
	t_0 = x * (1.0 - wj);
	tmp = 0.0;
	if (x <= -4.4e-290)
		tmp = t_0;
	elseif (x <= 1.08e-265)
		tmp = wj * wj;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[wj_, x_] := Block[{t$95$0 = N[(x * N[(1.0 - wj), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -4.4e-290], t$95$0, If[LessEqual[x, 1.08e-265], N[(wj * wj), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if x < -4.4000000000000002e-290 or 1.07999999999999998e-265 < x

   1. Initial program 82.6%

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

      1. sub-neg N/A

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

      2. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(wj, \color{blue}{\left(\mathsf{neg}\left(\frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}}\right)\right)}\right) \]

      3. distribute-rgt1-in N/A

         \[\leadsto \mathsf{+.f64}\left(wj, \left(\mathsf{neg}\left(\frac{wj \cdot e^{wj} - x}{\left(wj + 1\right) \cdot e^{wj}}\right)\right)\right) \]

      4. associate-/l/ N/A

         \[\leadsto \mathsf{+.f64}\left(wj, \left(\mathsf{neg}\left(\frac{\frac{wj \cdot e^{wj} - x}{e^{wj}}}{wj + 1}\right)\right)\right) \]

      5. distribute-neg-frac2 N/A

         \[\leadsto \mathsf{+.f64}\left(wj, \left(\frac{\frac{wj \cdot e^{wj} - x}{e^{wj}}}{\color{blue}{\mathsf{neg}\left(\left(wj + 1\right)\right)}}\right)\right) \]

      6. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(wj, \mathsf{/.f64}\left(\left(\frac{wj \cdot e^{wj} - x}{e^{wj}}\right), \color{blue}{\left(\mathsf{neg}\left(\left(wj + 1\right)\right)\right)}\right)\right) \]

      7. div-sub N/A

         \[\leadsto \mathsf{+.f64}\left(wj, \mathsf{/.f64}\left(\left(\frac{wj \cdot e^{wj}}{e^{wj}} - \frac{x}{e^{wj}}\right), \left(\mathsf{neg}\left(\color{blue}{\left(wj + 1\right)}\right)\right)\right)\right) \]

      8. associate-/l* N/A

         \[\leadsto \mathsf{+.f64}\left(wj, \mathsf{/.f64}\left(\left(wj \cdot \frac{e^{wj}}{e^{wj}} - \frac{x}{e^{wj}}\right), \left(\mathsf{neg}\left(\left(\color{blue}{wj} + 1\right)\right)\right)\right)\right) \]

      9. *-inverses N/A

         \[\leadsto \mathsf{+.f64}\left(wj, \mathsf{/.f64}\left(\left(wj \cdot 1 - \frac{x}{e^{wj}}\right), \left(\mathsf{neg}\left(\left(wj + 1\right)\right)\right)\right)\right) \]

      10. *-rgt-identity N/A

          \[\leadsto \mathsf{+.f64}\left(wj, \mathsf{/.f64}\left(\left(wj - \frac{x}{e^{wj}}\right), \left(\mathsf{neg}\left(\left(\color{blue}{wj} + 1\right)\right)\right)\right)\right) \]

      11. --lowering--.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(wj, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(wj, \left(\frac{x}{e^{wj}}\right)\right), \left(\mathsf{neg}\left(\color{blue}{\left(wj + 1\right)}\right)\right)\right)\right) \]

      12. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(wj, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(wj, \mathsf{/.f64}\left(x, \left(e^{wj}\right)\right)\right), \left(\mathsf{neg}\left(\left(wj + \color{blue}{1}\right)\right)\right)\right)\right) \]

      13. exp-lowering-exp.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(wj, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(wj, \mathsf{/.f64}\left(x, \mathsf{exp.f64}\left(wj\right)\right)\right), \left(\mathsf{neg}\left(\left(wj + 1\right)\right)\right)\right)\right) \]

      14. +-commutative N/A

          \[\leadsto \mathsf{+.f64}\left(wj, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(wj, \mathsf{/.f64}\left(x, \mathsf{exp.f64}\left(wj\right)\right)\right), \left(\mathsf{neg}\left(\left(1 + wj\right)\right)\right)\right)\right) \]

      15. *-inverses N/A

          \[\leadsto \mathsf{+.f64}\left(wj, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(wj, \mathsf{/.f64}\left(x, \mathsf{exp.f64}\left(wj\right)\right)\right), \left(\mathsf{neg}\left(\left(\frac{e^{wj}}{e^{wj}} + wj\right)\right)\right)\right)\right) \]

      16. distribute-neg-in N/A

          \[\leadsto \mathsf{+.f64}\left(wj, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(wj, \mathsf{/.f64}\left(x, \mathsf{exp.f64}\left(wj\right)\right)\right), \left(\left(\mathsf{neg}\left(\frac{e^{wj}}{e^{wj}}\right)\right) + \color{blue}{\left(\mathsf{neg}\left(wj\right)\right)}\right)\right)\right) \]

   3. Simplified 84.7%

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

   5. Taylor expanded in wj around 0

      \[\leadsto \mathsf{+.f64}\left(wj, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(wj, \color{blue}{x}\right), \mathsf{\_.f64}\left(-1, wj\right)\right)\right) \]
   6. Step-by-step derivation

   7. Simplified 80.2%

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

   8. Taylor expanded in x around inf

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

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(x, \color{blue}{\left(-1 \cdot \frac{wj}{x \cdot \left(1 + wj\right)} + \left(\frac{1}{1 + wj} + \frac{wj}{x}\right)\right)}\right) \]

      2. +-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(x, \left(\left(\frac{1}{1 + wj} + \frac{wj}{x}\right) + \color{blue}{-1 \cdot \frac{wj}{x \cdot \left(1 + wj\right)}}\right)\right) \]

      3. +-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(x, \left(\left(\frac{wj}{x} + \frac{1}{1 + wj}\right) + \color{blue}{-1} \cdot \frac{wj}{x \cdot \left(1 + wj\right)}\right)\right) \]

      4. associate-+l+ N/A

         \[\leadsto \mathsf{*.f64}\left(x, \left(\frac{wj}{x} + \color{blue}{\left(\frac{1}{1 + wj} + -1 \cdot \frac{wj}{x \cdot \left(1 + wj\right)}\right)}\right)\right) \]

      5. +-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(x, \left(\frac{wj}{x} + \left(-1 \cdot \frac{wj}{x \cdot \left(1 + wj\right)} + \color{blue}{\frac{1}{1 + wj}}\right)\right)\right) \]

      6. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\left(\frac{wj}{x}\right), \color{blue}{\left(-1 \cdot \frac{wj}{x \cdot \left(1 + wj\right)} + \frac{1}{1 + wj}\right)}\right)\right) \]

      7. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\mathsf{/.f64}\left(wj, x\right), \left(\color{blue}{-1 \cdot \frac{wj}{x \cdot \left(1 + wj\right)}} + \frac{1}{1 + wj}\right)\right)\right) \]

      8. +-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\mathsf{/.f64}\left(wj, x\right), \left(\frac{1}{1 + wj} + \color{blue}{-1 \cdot \frac{wj}{x \cdot \left(1 + wj\right)}}\right)\right)\right) \]

      9. mul-1-neg N/A

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\mathsf{/.f64}\left(wj, x\right), \left(\frac{1}{1 + wj} + \left(\mathsf{neg}\left(\frac{wj}{x \cdot \left(1 + wj\right)}\right)\right)\right)\right)\right) \]

      10. unsub-neg N/A

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\mathsf{/.f64}\left(wj, x\right), \left(\frac{1}{1 + wj} - \color{blue}{\frac{wj}{x \cdot \left(1 + wj\right)}}\right)\right)\right) \]

      11. --lowering--.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\mathsf{/.f64}\left(wj, x\right), \mathsf{\_.f64}\left(\left(\frac{1}{1 + wj}\right), \color{blue}{\left(\frac{wj}{x \cdot \left(1 + wj\right)}\right)}\right)\right)\right) \]

      12. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\mathsf{/.f64}\left(wj, x\right), \mathsf{\_.f64}\left(\mathsf{/.f64}\left(1, \left(1 + wj\right)\right), \left(\frac{\color{blue}{wj}}{x \cdot \left(1 + wj\right)}\right)\right)\right)\right) \]

      13. +-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\mathsf{/.f64}\left(wj, x\right), \mathsf{\_.f64}\left(\mathsf{/.f64}\left(1, \left(wj + 1\right)\right), \left(\frac{wj}{x \cdot \left(1 + wj\right)}\right)\right)\right)\right) \]

      14. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\mathsf{/.f64}\left(wj, x\right), \mathsf{\_.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(wj, 1\right)\right), \left(\frac{wj}{x \cdot \left(1 + wj\right)}\right)\right)\right)\right) \]

      15. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\mathsf{/.f64}\left(wj, x\right), \mathsf{\_.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(wj, 1\right)\right), \mathsf{/.f64}\left(wj, \color{blue}{\left(x \cdot \left(1 + wj\right)\right)}\right)\right)\right)\right) \]

      16. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\mathsf{/.f64}\left(wj, x\right), \mathsf{\_.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(wj, 1\right)\right), \mathsf{/.f64}\left(wj, \mathsf{*.f64}\left(x, \color{blue}{\left(1 + wj\right)}\right)\right)\right)\right)\right) \]

      17. +-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\mathsf{/.f64}\left(wj, x\right), \mathsf{\_.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(wj, 1\right)\right), \mathsf{/.f64}\left(wj, \mathsf{*.f64}\left(x, \left(wj + \color{blue}{1}\right)\right)\right)\right)\right)\right) \]

      18. +-lowering-+.f64 80.9%

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\mathsf{/.f64}\left(wj, x\right), \mathsf{\_.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(wj, 1\right)\right), \mathsf{/.f64}\left(wj, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(wj, \color{blue}{1}\right)\right)\right)\right)\right)\right) \]

   10. Simplified 80.9%

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

   11. Taylor expanded in wj around 0

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

       1. associate-*r* N/A

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

       2. distribute-rgt1-in N/A

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

       3. +-commutative N/A

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

       4. mul-1-neg N/A

          \[\leadsto \left(1 + \left(\mathsf{neg}\left(wj\right)\right)\right) \cdot x \]

       5. sub-neg N/A

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

       6. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\left(1 - wj\right), \color{blue}{x}\right) \]

       7. --lowering--.f64 87.5%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{\_.f64}\left(1, wj\right), x\right) \]

   13. Simplified 87.5%

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

   if -4.4000000000000002e-290 < x < 1.07999999999999998e-265

   1. Initial program 18.2%

      \[wj - \frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}} \]
   2. Add Preprocessing

   3. Taylor expanded in wj around 0

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

      1. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(x, \color{blue}{\left(wj \cdot \left(wj \cdot \left(\left(1 + -1 \cdot \left(wj \cdot \left(1 + \left(-3 \cdot x + \left(-2 \cdot \left(-4 \cdot x + \frac{3}{2} \cdot x\right) + \frac{2}{3} \cdot x\right)\right)\right)\right)\right) - \left(-4 \cdot x + \frac{3}{2} \cdot x\right)\right) - 2 \cdot x\right)\right)}\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(x, \mathsf{*.f64}\left(wj, \color{blue}{\left(wj \cdot \left(\left(1 + -1 \cdot \left(wj \cdot \left(1 + \left(-3 \cdot x + \left(-2 \cdot \left(-4 \cdot x + \frac{3}{2} \cdot x\right) + \frac{2}{3} \cdot x\right)\right)\right)\right)\right) - \left(-4 \cdot x + \frac{3}{2} \cdot x\right)\right) - 2 \cdot x\right)}\right)\right) \]

      3. cancel-sign-sub-inv N/A

         \[\leadsto \mathsf{+.f64}\left(x, \mathsf{*.f64}\left(wj, \left(wj \cdot \left(\left(1 + -1 \cdot \left(wj \cdot \left(1 + \left(-3 \cdot x + \left(-2 \cdot \left(-4 \cdot x + \frac{3}{2} \cdot x\right) + \frac{2}{3} \cdot x\right)\right)\right)\right)\right) - \left(-4 \cdot x + \frac{3}{2} \cdot x\right)\right) + \color{blue}{\left(\mathsf{neg}\left(2\right)\right) \cdot x}\right)\right)\right) \]

      4. metadata-eval N/A

         \[\leadsto \mathsf{+.f64}\left(x, \mathsf{*.f64}\left(wj, \left(wj \cdot \left(\left(1 + -1 \cdot \left(wj \cdot \left(1 + \left(-3 \cdot x + \left(-2 \cdot \left(-4 \cdot x + \frac{3}{2} \cdot x\right) + \frac{2}{3} \cdot x\right)\right)\right)\right)\right) - \left(-4 \cdot x + \frac{3}{2} \cdot x\right)\right) + -2 \cdot x\right)\right)\right) \]

      5. +-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(x, \mathsf{*.f64}\left(wj, \left(-2 \cdot x + \color{blue}{wj \cdot \left(\left(1 + -1 \cdot \left(wj \cdot \left(1 + \left(-3 \cdot x + \left(-2 \cdot \left(-4 \cdot x + \frac{3}{2} \cdot x\right) + \frac{2}{3} \cdot x\right)\right)\right)\right)\right) - \left(-4 \cdot x + \frac{3}{2} \cdot x\right)\right)}\right)\right)\right) \]

      6. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(x, \mathsf{*.f64}\left(wj, \mathsf{+.f64}\left(\left(-2 \cdot x\right), \color{blue}{\left(wj \cdot \left(\left(1 + -1 \cdot \left(wj \cdot \left(1 + \left(-3 \cdot x + \left(-2 \cdot \left(-4 \cdot x + \frac{3}{2} \cdot x\right) + \frac{2}{3} \cdot x\right)\right)\right)\right)\right) - \left(-4 \cdot x + \frac{3}{2} \cdot x\right)\right)\right)}\right)\right)\right) \]

   5. Simplified 98.4%

      \[\leadsto \color{blue}{x + wj \cdot \left(x \cdot -2 + wj \cdot \left(\left(1 - wj \cdot \left(\left(x \cdot 5 + x \cdot 0.6666666666666666\right) + \left(1 + x \cdot -3\right)\right)\right) + x \cdot 2.5\right)\right)} \]

   6. Taylor expanded in x around 0

      \[\leadsto \mathsf{+.f64}\left(x, \mathsf{*.f64}\left(wj, \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, -2\right), \color{blue}{\left(wj \cdot \left(1 - wj\right)\right)}\right)\right)\right) \]
   7. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(x, \mathsf{*.f64}\left(wj, \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, -2\right), \mathsf{*.f64}\left(wj, \color{blue}{\left(1 - wj\right)}\right)\right)\right)\right) \]

      2. --lowering--.f64 98.4%

         \[\leadsto \mathsf{+.f64}\left(x, \mathsf{*.f64}\left(wj, \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, -2\right), \mathsf{*.f64}\left(wj, \mathsf{\_.f64}\left(1, \color{blue}{wj}\right)\right)\right)\right)\right) \]

   8. Simplified 98.4%

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

   9. Taylor expanded in wj around 0

      \[\leadsto \mathsf{+.f64}\left(x, \mathsf{*.f64}\left(wj, \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, -2\right), \color{blue}{wj}\right)\right)\right) \]
   10. Step-by-step derivation

   11. Simplified 95.2%

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

   12. Taylor expanded in x around 0

       \[\leadsto \color{blue}{{wj}^{2}} \]
   13. Step-by-step derivation

       1. unpow2 N/A

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

       2. *-lowering-*.f64 75.3%

          \[\leadsto \mathsf{*.f64}\left(wj, \color{blue}{wj}\right) \]

   14. Simplified 75.3%

       \[\leadsto \color{blue}{wj \cdot wj} \]
3. Recombined 2 regimes into one program.

4. Final simplification 86.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -4.4 \cdot 10^{-290}:\\ \;\;\;\;x \cdot \left(1 - wj\right)\\ \mathbf{elif}\;x \leq 1.08 \cdot 10^{-265}:\\ \;\;\;\;wj \cdot wj\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(1 - wj\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 95.8% accurate, 20.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 78.0%

   \[wj - \frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}} \]
2. Add Preprocessing

3. Taylor expanded in wj around 0

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

   1. +-lowering-+.f64 N/A

      \[\leadsto \mathsf{+.f64}\left(x, \color{blue}{\left(wj \cdot \left(wj \cdot \left(1 - \left(-4 \cdot x + \frac{3}{2} \cdot x\right)\right) - 2 \cdot x\right)\right)}\right) \]

   2. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{+.f64}\left(x, \mathsf{*.f64}\left(wj, \color{blue}{\left(wj \cdot \left(1 - \left(-4 \cdot x + \frac{3}{2} \cdot x\right)\right) - 2 \cdot x\right)}\right)\right) \]

   3. cancel-sign-sub-inv N/A

      \[\leadsto \mathsf{+.f64}\left(x, \mathsf{*.f64}\left(wj, \left(wj \cdot \left(1 - \left(-4 \cdot x + \frac{3}{2} \cdot x\right)\right) + \color{blue}{\left(\mathsf{neg}\left(2\right)\right) \cdot x}\right)\right)\right) \]

   4. metadata-eval N/A

      \[\leadsto \mathsf{+.f64}\left(x, \mathsf{*.f64}\left(wj, \left(wj \cdot \left(1 - \left(-4 \cdot x + \frac{3}{2} \cdot x\right)\right) + -2 \cdot x\right)\right)\right) \]

   5. +-commutative N/A

      \[\leadsto \mathsf{+.f64}\left(x, \mathsf{*.f64}\left(wj, \left(-2 \cdot x + \color{blue}{wj \cdot \left(1 - \left(-4 \cdot x + \frac{3}{2} \cdot x\right)\right)}\right)\right)\right) \]

   6. +-lowering-+.f64 N/A

      \[\leadsto \mathsf{+.f64}\left(x, \mathsf{*.f64}\left(wj, \mathsf{+.f64}\left(\left(-2 \cdot x\right), \color{blue}{\left(wj \cdot \left(1 - \left(-4 \cdot x + \frac{3}{2} \cdot x\right)\right)\right)}\right)\right)\right) \]

   7. *-commutative N/A

      \[\leadsto \mathsf{+.f64}\left(x, \mathsf{*.f64}\left(wj, \mathsf{+.f64}\left(\left(x \cdot -2\right), \left(\color{blue}{wj} \cdot \left(1 - \left(-4 \cdot x + \frac{3}{2} \cdot x\right)\right)\right)\right)\right)\right) \]

   8. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{+.f64}\left(x, \mathsf{*.f64}\left(wj, \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, -2\right), \left(\color{blue}{wj} \cdot \left(1 - \left(-4 \cdot x + \frac{3}{2} \cdot x\right)\right)\right)\right)\right)\right) \]

   9. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{+.f64}\left(x, \mathsf{*.f64}\left(wj, \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, -2\right), \mathsf{*.f64}\left(wj, \color{blue}{\left(1 - \left(-4 \cdot x + \frac{3}{2} \cdot x\right)\right)}\right)\right)\right)\right) \]

   10. sub-neg N/A

       \[\leadsto \mathsf{+.f64}\left(x, \mathsf{*.f64}\left(wj, \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, -2\right), \mathsf{*.f64}\left(wj, \left(1 + \color{blue}{\left(\mathsf{neg}\left(\left(-4 \cdot x + \frac{3}{2} \cdot x\right)\right)\right)}\right)\right)\right)\right)\right) \]

   11. +-lowering-+.f64 N/A

       \[\leadsto \mathsf{+.f64}\left(x, \mathsf{*.f64}\left(wj, \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, -2\right), \mathsf{*.f64}\left(wj, \mathsf{+.f64}\left(1, \color{blue}{\left(\mathsf{neg}\left(\left(-4 \cdot x + \frac{3}{2} \cdot x\right)\right)\right)}\right)\right)\right)\right)\right) \]

   12. distribute-rgt-out N/A

       \[\leadsto \mathsf{+.f64}\left(x, \mathsf{*.f64}\left(wj, \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, -2\right), \mathsf{*.f64}\left(wj, \mathsf{+.f64}\left(1, \left(\mathsf{neg}\left(x \cdot \left(-4 + \frac{3}{2}\right)\right)\right)\right)\right)\right)\right)\right) \]

   13. distribute-rgt-neg-in N/A

       \[\leadsto \mathsf{+.f64}\left(x, \mathsf{*.f64}\left(wj, \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, -2\right), \mathsf{*.f64}\left(wj, \mathsf{+.f64}\left(1, \left(x \cdot \color{blue}{\left(\mathsf{neg}\left(\left(-4 + \frac{3}{2}\right)\right)\right)}\right)\right)\right)\right)\right)\right) \]

   14. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{+.f64}\left(x, \mathsf{*.f64}\left(wj, \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, -2\right), \mathsf{*.f64}\left(wj, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \color{blue}{\left(\mathsf{neg}\left(\left(-4 + \frac{3}{2}\right)\right)\right)}\right)\right)\right)\right)\right)\right) \]

   15. metadata-eval N/A

       \[\leadsto \mathsf{+.f64}\left(x, \mathsf{*.f64}\left(wj, \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, -2\right), \mathsf{*.f64}\left(wj, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \left(\mathsf{neg}\left(\frac{-5}{2}\right)\right)\right)\right)\right)\right)\right)\right) \]

   16. metadata-eval 95.9%

       \[\leadsto \mathsf{+.f64}\left(x, \mathsf{*.f64}\left(wj, \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, -2\right), \mathsf{*.f64}\left(wj, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \frac{5}{2}\right)\right)\right)\right)\right)\right) \]

5. Simplified 95.9%

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

Alternative 7: 83.8% accurate, 24.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -7.8 \cdot 10^{-291}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq 8.5 \cdot 10^{-264}:\\ \;\;\;\;wj \cdot wj\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \end{array} \]

FPCore

(FPCore (wj x)
 :precision binary64
 (if (<= x -7.8e-291) x (if (<= x 8.5e-264) (* wj wj) x)))

C

double code(double wj, double x) {
	double tmp;
	if (x <= -7.8e-291) {
		tmp = x;
	} else if (x <= 8.5e-264) {
		tmp = wj * wj;
	} else {
		tmp = x;
	}
	return tmp;
}

Fortran

real(8) function code(wj, x)
    real(8), intent (in) :: wj
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= (-7.8d-291)) then
        tmp = x
    else if (x <= 8.5d-264) then
        tmp = wj * wj
    else
        tmp = x
    end if
    code = tmp
end function

Java

public static double code(double wj, double x) {
	double tmp;
	if (x <= -7.8e-291) {
		tmp = x;
	} else if (x <= 8.5e-264) {
		tmp = wj * wj;
	} else {
		tmp = x;
	}
	return tmp;
}

Python

def code(wj, x):
	tmp = 0
	if x <= -7.8e-291:
		tmp = x
	elif x <= 8.5e-264:
		tmp = wj * wj
	else:
		tmp = x
	return tmp

Julia

function code(wj, x)
	tmp = 0.0
	if (x <= -7.8e-291)
		tmp = x;
	elseif (x <= 8.5e-264)
		tmp = Float64(wj * wj);
	else
		tmp = x;
	end
	return tmp
end

MATLAB

function tmp_2 = code(wj, x)
	tmp = 0.0;
	if (x <= -7.8e-291)
		tmp = x;
	elseif (x <= 8.5e-264)
		tmp = wj * wj;
	else
		tmp = x;
	end
	tmp_2 = tmp;
end

Wolfram

code[wj_, x_] := If[LessEqual[x, -7.8e-291], x, If[LessEqual[x, 8.5e-264], N[(wj * wj), $MachinePrecision], x]]

Derivation

1. Split input into 2 regimes

2. if x < -7.80000000000000031e-291 or 8.5000000000000001e-264 < x

   1. Initial program 82.6%

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

      1. sub-neg N/A

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

      2. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(wj, \color{blue}{\left(\mathsf{neg}\left(\frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}}\right)\right)}\right) \]

      3. distribute-rgt1-in N/A

         \[\leadsto \mathsf{+.f64}\left(wj, \left(\mathsf{neg}\left(\frac{wj \cdot e^{wj} - x}{\left(wj + 1\right) \cdot e^{wj}}\right)\right)\right) \]

      4. associate-/l/ N/A

         \[\leadsto \mathsf{+.f64}\left(wj, \left(\mathsf{neg}\left(\frac{\frac{wj \cdot e^{wj} - x}{e^{wj}}}{wj + 1}\right)\right)\right) \]

      5. distribute-neg-frac2 N/A

         \[\leadsto \mathsf{+.f64}\left(wj, \left(\frac{\frac{wj \cdot e^{wj} - x}{e^{wj}}}{\color{blue}{\mathsf{neg}\left(\left(wj + 1\right)\right)}}\right)\right) \]

      6. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(wj, \mathsf{/.f64}\left(\left(\frac{wj \cdot e^{wj} - x}{e^{wj}}\right), \color{blue}{\left(\mathsf{neg}\left(\left(wj + 1\right)\right)\right)}\right)\right) \]

      7. div-sub N/A

         \[\leadsto \mathsf{+.f64}\left(wj, \mathsf{/.f64}\left(\left(\frac{wj \cdot e^{wj}}{e^{wj}} - \frac{x}{e^{wj}}\right), \left(\mathsf{neg}\left(\color{blue}{\left(wj + 1\right)}\right)\right)\right)\right) \]

      8. associate-/l* N/A

         \[\leadsto \mathsf{+.f64}\left(wj, \mathsf{/.f64}\left(\left(wj \cdot \frac{e^{wj}}{e^{wj}} - \frac{x}{e^{wj}}\right), \left(\mathsf{neg}\left(\left(\color{blue}{wj} + 1\right)\right)\right)\right)\right) \]

      9. *-inverses N/A

         \[\leadsto \mathsf{+.f64}\left(wj, \mathsf{/.f64}\left(\left(wj \cdot 1 - \frac{x}{e^{wj}}\right), \left(\mathsf{neg}\left(\left(wj + 1\right)\right)\right)\right)\right) \]

      10. *-rgt-identity N/A

          \[\leadsto \mathsf{+.f64}\left(wj, \mathsf{/.f64}\left(\left(wj - \frac{x}{e^{wj}}\right), \left(\mathsf{neg}\left(\left(\color{blue}{wj} + 1\right)\right)\right)\right)\right) \]

      11. --lowering--.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(wj, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(wj, \left(\frac{x}{e^{wj}}\right)\right), \left(\mathsf{neg}\left(\color{blue}{\left(wj + 1\right)}\right)\right)\right)\right) \]

      12. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(wj, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(wj, \mathsf{/.f64}\left(x, \left(e^{wj}\right)\right)\right), \left(\mathsf{neg}\left(\left(wj + \color{blue}{1}\right)\right)\right)\right)\right) \]

      13. exp-lowering-exp.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(wj, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(wj, \mathsf{/.f64}\left(x, \mathsf{exp.f64}\left(wj\right)\right)\right), \left(\mathsf{neg}\left(\left(wj + 1\right)\right)\right)\right)\right) \]

      14. +-commutative N/A

          \[\leadsto \mathsf{+.f64}\left(wj, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(wj, \mathsf{/.f64}\left(x, \mathsf{exp.f64}\left(wj\right)\right)\right), \left(\mathsf{neg}\left(\left(1 + wj\right)\right)\right)\right)\right) \]

      15. *-inverses N/A

          \[\leadsto \mathsf{+.f64}\left(wj, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(wj, \mathsf{/.f64}\left(x, \mathsf{exp.f64}\left(wj\right)\right)\right), \left(\mathsf{neg}\left(\left(\frac{e^{wj}}{e^{wj}} + wj\right)\right)\right)\right)\right) \]

      16. distribute-neg-in N/A

          \[\leadsto \mathsf{+.f64}\left(wj, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(wj, \mathsf{/.f64}\left(x, \mathsf{exp.f64}\left(wj\right)\right)\right), \left(\left(\mathsf{neg}\left(\frac{e^{wj}}{e^{wj}}\right)\right) + \color{blue}{\left(\mathsf{neg}\left(wj\right)\right)}\right)\right)\right) \]

   3. Simplified 84.7%

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

   5. Taylor expanded in wj around 0

      \[\leadsto \color{blue}{x} \]
   6. Step-by-step derivation

   7. Simplified 87.4%

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

   if -7.80000000000000031e-291 < x < 8.5000000000000001e-264

   1. Initial program 18.2%

      \[wj - \frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}} \]
   2. Add Preprocessing

   3. Taylor expanded in wj around 0

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

      1. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(x, \color{blue}{\left(wj \cdot \left(wj \cdot \left(\left(1 + -1 \cdot \left(wj \cdot \left(1 + \left(-3 \cdot x + \left(-2 \cdot \left(-4 \cdot x + \frac{3}{2} \cdot x\right) + \frac{2}{3} \cdot x\right)\right)\right)\right)\right) - \left(-4 \cdot x + \frac{3}{2} \cdot x\right)\right) - 2 \cdot x\right)\right)}\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(x, \mathsf{*.f64}\left(wj, \color{blue}{\left(wj \cdot \left(\left(1 + -1 \cdot \left(wj \cdot \left(1 + \left(-3 \cdot x + \left(-2 \cdot \left(-4 \cdot x + \frac{3}{2} \cdot x\right) + \frac{2}{3} \cdot x\right)\right)\right)\right)\right) - \left(-4 \cdot x + \frac{3}{2} \cdot x\right)\right) - 2 \cdot x\right)}\right)\right) \]

      3. cancel-sign-sub-inv N/A

         \[\leadsto \mathsf{+.f64}\left(x, \mathsf{*.f64}\left(wj, \left(wj \cdot \left(\left(1 + -1 \cdot \left(wj \cdot \left(1 + \left(-3 \cdot x + \left(-2 \cdot \left(-4 \cdot x + \frac{3}{2} \cdot x\right) + \frac{2}{3} \cdot x\right)\right)\right)\right)\right) - \left(-4 \cdot x + \frac{3}{2} \cdot x\right)\right) + \color{blue}{\left(\mathsf{neg}\left(2\right)\right) \cdot x}\right)\right)\right) \]

      4. metadata-eval N/A

         \[\leadsto \mathsf{+.f64}\left(x, \mathsf{*.f64}\left(wj, \left(wj \cdot \left(\left(1 + -1 \cdot \left(wj \cdot \left(1 + \left(-3 \cdot x + \left(-2 \cdot \left(-4 \cdot x + \frac{3}{2} \cdot x\right) + \frac{2}{3} \cdot x\right)\right)\right)\right)\right) - \left(-4 \cdot x + \frac{3}{2} \cdot x\right)\right) + -2 \cdot x\right)\right)\right) \]

      5. +-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(x, \mathsf{*.f64}\left(wj, \left(-2 \cdot x + \color{blue}{wj \cdot \left(\left(1 + -1 \cdot \left(wj \cdot \left(1 + \left(-3 \cdot x + \left(-2 \cdot \left(-4 \cdot x + \frac{3}{2} \cdot x\right) + \frac{2}{3} \cdot x\right)\right)\right)\right)\right) - \left(-4 \cdot x + \frac{3}{2} \cdot x\right)\right)}\right)\right)\right) \]

      6. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(x, \mathsf{*.f64}\left(wj, \mathsf{+.f64}\left(\left(-2 \cdot x\right), \color{blue}{\left(wj \cdot \left(\left(1 + -1 \cdot \left(wj \cdot \left(1 + \left(-3 \cdot x + \left(-2 \cdot \left(-4 \cdot x + \frac{3}{2} \cdot x\right) + \frac{2}{3} \cdot x\right)\right)\right)\right)\right) - \left(-4 \cdot x + \frac{3}{2} \cdot x\right)\right)\right)}\right)\right)\right) \]

   5. Simplified 98.4%

      \[\leadsto \color{blue}{x + wj \cdot \left(x \cdot -2 + wj \cdot \left(\left(1 - wj \cdot \left(\left(x \cdot 5 + x \cdot 0.6666666666666666\right) + \left(1 + x \cdot -3\right)\right)\right) + x \cdot 2.5\right)\right)} \]

   6. Taylor expanded in x around 0

      \[\leadsto \mathsf{+.f64}\left(x, \mathsf{*.f64}\left(wj, \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, -2\right), \color{blue}{\left(wj \cdot \left(1 - wj\right)\right)}\right)\right)\right) \]
   7. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(x, \mathsf{*.f64}\left(wj, \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, -2\right), \mathsf{*.f64}\left(wj, \color{blue}{\left(1 - wj\right)}\right)\right)\right)\right) \]

      2. --lowering--.f64 98.4%

         \[\leadsto \mathsf{+.f64}\left(x, \mathsf{*.f64}\left(wj, \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, -2\right), \mathsf{*.f64}\left(wj, \mathsf{\_.f64}\left(1, \color{blue}{wj}\right)\right)\right)\right)\right) \]

   8. Simplified 98.4%

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

   9. Taylor expanded in wj around 0

      \[\leadsto \mathsf{+.f64}\left(x, \mathsf{*.f64}\left(wj, \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, -2\right), \color{blue}{wj}\right)\right)\right) \]
   10. Step-by-step derivation

   11. Simplified 95.2%

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

   12. Taylor expanded in x around 0

       \[\leadsto \color{blue}{{wj}^{2}} \]
   13. Step-by-step derivation

       1. unpow2 N/A

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

       2. *-lowering-*.f64 75.3%

          \[\leadsto \mathsf{*.f64}\left(wj, \color{blue}{wj}\right) \]

   14. Simplified 75.3%

       \[\leadsto \color{blue}{wj \cdot wj} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 8: 96.0% accurate, 24.1× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 78.0%

   \[wj - \frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}} \]
2. Add Preprocessing

3. Taylor expanded in wj around 0

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

   1. +-lowering-+.f64 N/A

      \[\leadsto \mathsf{+.f64}\left(x, \color{blue}{\left(wj \cdot \left(wj \cdot \left(\left(1 + -1 \cdot \left(wj \cdot \left(1 + \left(-3 \cdot x + \left(-2 \cdot \left(-4 \cdot x + \frac{3}{2} \cdot x\right) + \frac{2}{3} \cdot x\right)\right)\right)\right)\right) - \left(-4 \cdot x + \frac{3}{2} \cdot x\right)\right) - 2 \cdot x\right)\right)}\right) \]

   2. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{+.f64}\left(x, \mathsf{*.f64}\left(wj, \color{blue}{\left(wj \cdot \left(\left(1 + -1 \cdot \left(wj \cdot \left(1 + \left(-3 \cdot x + \left(-2 \cdot \left(-4 \cdot x + \frac{3}{2} \cdot x\right) + \frac{2}{3} \cdot x\right)\right)\right)\right)\right) - \left(-4 \cdot x + \frac{3}{2} \cdot x\right)\right) - 2 \cdot x\right)}\right)\right) \]

   3. cancel-sign-sub-inv N/A

      \[\leadsto \mathsf{+.f64}\left(x, \mathsf{*.f64}\left(wj, \left(wj \cdot \left(\left(1 + -1 \cdot \left(wj \cdot \left(1 + \left(-3 \cdot x + \left(-2 \cdot \left(-4 \cdot x + \frac{3}{2} \cdot x\right) + \frac{2}{3} \cdot x\right)\right)\right)\right)\right) - \left(-4 \cdot x + \frac{3}{2} \cdot x\right)\right) + \color{blue}{\left(\mathsf{neg}\left(2\right)\right) \cdot x}\right)\right)\right) \]

   4. metadata-eval N/A

      \[\leadsto \mathsf{+.f64}\left(x, \mathsf{*.f64}\left(wj, \left(wj \cdot \left(\left(1 + -1 \cdot \left(wj \cdot \left(1 + \left(-3 \cdot x + \left(-2 \cdot \left(-4 \cdot x + \frac{3}{2} \cdot x\right) + \frac{2}{3} \cdot x\right)\right)\right)\right)\right) - \left(-4 \cdot x + \frac{3}{2} \cdot x\right)\right) + -2 \cdot x\right)\right)\right) \]

   5. +-commutative N/A

      \[\leadsto \mathsf{+.f64}\left(x, \mathsf{*.f64}\left(wj, \left(-2 \cdot x + \color{blue}{wj \cdot \left(\left(1 + -1 \cdot \left(wj \cdot \left(1 + \left(-3 \cdot x + \left(-2 \cdot \left(-4 \cdot x + \frac{3}{2} \cdot x\right) + \frac{2}{3} \cdot x\right)\right)\right)\right)\right) - \left(-4 \cdot x + \frac{3}{2} \cdot x\right)\right)}\right)\right)\right) \]

   6. +-lowering-+.f64 N/A

      \[\leadsto \mathsf{+.f64}\left(x, \mathsf{*.f64}\left(wj, \mathsf{+.f64}\left(\left(-2 \cdot x\right), \color{blue}{\left(wj \cdot \left(\left(1 + -1 \cdot \left(wj \cdot \left(1 + \left(-3 \cdot x + \left(-2 \cdot \left(-4 \cdot x + \frac{3}{2} \cdot x\right) + \frac{2}{3} \cdot x\right)\right)\right)\right)\right) - \left(-4 \cdot x + \frac{3}{2} \cdot x\right)\right)\right)}\right)\right)\right) \]

5. Simplified 96.3%

   \[\leadsto \color{blue}{x + wj \cdot \left(x \cdot -2 + wj \cdot \left(\left(1 - wj \cdot \left(\left(x \cdot 5 + x \cdot 0.6666666666666666\right) + \left(1 + x \cdot -3\right)\right)\right) + x \cdot 2.5\right)\right)} \]

6. Taylor expanded in x around 0

   \[\leadsto \mathsf{+.f64}\left(x, \mathsf{*.f64}\left(wj, \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, -2\right), \color{blue}{\left(wj \cdot \left(1 - wj\right)\right)}\right)\right)\right) \]
7. Step-by-step derivation

   1. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{+.f64}\left(x, \mathsf{*.f64}\left(wj, \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, -2\right), \mathsf{*.f64}\left(wj, \color{blue}{\left(1 - wj\right)}\right)\right)\right)\right) \]

   2. --lowering--.f64 95.8%

      \[\leadsto \mathsf{+.f64}\left(x, \mathsf{*.f64}\left(wj, \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, -2\right), \mathsf{*.f64}\left(wj, \mathsf{\_.f64}\left(1, \color{blue}{wj}\right)\right)\right)\right)\right) \]

8. Simplified 95.8%

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

9. Final simplification 95.8%

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

Alternative 9: 95.6% accurate, 34.8× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 78.0%

   \[wj - \frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}} \]
2. Add Preprocessing

3. Taylor expanded in wj around 0

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

   1. +-lowering-+.f64 N/A

      \[\leadsto \mathsf{+.f64}\left(x, \color{blue}{\left(wj \cdot \left(wj \cdot \left(\left(1 + -1 \cdot \left(wj \cdot \left(1 + \left(-3 \cdot x + \left(-2 \cdot \left(-4 \cdot x + \frac{3}{2} \cdot x\right) + \frac{2}{3} \cdot x\right)\right)\right)\right)\right) - \left(-4 \cdot x + \frac{3}{2} \cdot x\right)\right) - 2 \cdot x\right)\right)}\right) \]

   2. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{+.f64}\left(x, \mathsf{*.f64}\left(wj, \color{blue}{\left(wj \cdot \left(\left(1 + -1 \cdot \left(wj \cdot \left(1 + \left(-3 \cdot x + \left(-2 \cdot \left(-4 \cdot x + \frac{3}{2} \cdot x\right) + \frac{2}{3} \cdot x\right)\right)\right)\right)\right) - \left(-4 \cdot x + \frac{3}{2} \cdot x\right)\right) - 2 \cdot x\right)}\right)\right) \]

   3. cancel-sign-sub-inv N/A

      \[\leadsto \mathsf{+.f64}\left(x, \mathsf{*.f64}\left(wj, \left(wj \cdot \left(\left(1 + -1 \cdot \left(wj \cdot \left(1 + \left(-3 \cdot x + \left(-2 \cdot \left(-4 \cdot x + \frac{3}{2} \cdot x\right) + \frac{2}{3} \cdot x\right)\right)\right)\right)\right) - \left(-4 \cdot x + \frac{3}{2} \cdot x\right)\right) + \color{blue}{\left(\mathsf{neg}\left(2\right)\right) \cdot x}\right)\right)\right) \]

   4. metadata-eval N/A

      \[\leadsto \mathsf{+.f64}\left(x, \mathsf{*.f64}\left(wj, \left(wj \cdot \left(\left(1 + -1 \cdot \left(wj \cdot \left(1 + \left(-3 \cdot x + \left(-2 \cdot \left(-4 \cdot x + \frac{3}{2} \cdot x\right) + \frac{2}{3} \cdot x\right)\right)\right)\right)\right) - \left(-4 \cdot x + \frac{3}{2} \cdot x\right)\right) + -2 \cdot x\right)\right)\right) \]

   5. +-commutative N/A

      \[\leadsto \mathsf{+.f64}\left(x, \mathsf{*.f64}\left(wj, \left(-2 \cdot x + \color{blue}{wj \cdot \left(\left(1 + -1 \cdot \left(wj \cdot \left(1 + \left(-3 \cdot x + \left(-2 \cdot \left(-4 \cdot x + \frac{3}{2} \cdot x\right) + \frac{2}{3} \cdot x\right)\right)\right)\right)\right) - \left(-4 \cdot x + \frac{3}{2} \cdot x\right)\right)}\right)\right)\right) \]

   6. +-lowering-+.f64 N/A

      \[\leadsto \mathsf{+.f64}\left(x, \mathsf{*.f64}\left(wj, \mathsf{+.f64}\left(\left(-2 \cdot x\right), \color{blue}{\left(wj \cdot \left(\left(1 + -1 \cdot \left(wj \cdot \left(1 + \left(-3 \cdot x + \left(-2 \cdot \left(-4 \cdot x + \frac{3}{2} \cdot x\right) + \frac{2}{3} \cdot x\right)\right)\right)\right)\right) - \left(-4 \cdot x + \frac{3}{2} \cdot x\right)\right)\right)}\right)\right)\right) \]

5. Simplified 96.3%

   \[\leadsto \color{blue}{x + wj \cdot \left(x \cdot -2 + wj \cdot \left(\left(1 - wj \cdot \left(\left(x \cdot 5 + x \cdot 0.6666666666666666\right) + \left(1 + x \cdot -3\right)\right)\right) + x \cdot 2.5\right)\right)} \]

6. Taylor expanded in x around 0

   \[\leadsto \mathsf{+.f64}\left(x, \mathsf{*.f64}\left(wj, \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, -2\right), \color{blue}{\left(wj \cdot \left(1 - wj\right)\right)}\right)\right)\right) \]
7. Step-by-step derivation

   1. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{+.f64}\left(x, \mathsf{*.f64}\left(wj, \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, -2\right), \mathsf{*.f64}\left(wj, \color{blue}{\left(1 - wj\right)}\right)\right)\right)\right) \]

   2. --lowering--.f64 95.8%

      \[\leadsto \mathsf{+.f64}\left(x, \mathsf{*.f64}\left(wj, \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, -2\right), \mathsf{*.f64}\left(wj, \mathsf{\_.f64}\left(1, \color{blue}{wj}\right)\right)\right)\right)\right) \]

8. Simplified 95.8%

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

9. Taylor expanded in wj around 0

   \[\leadsto \mathsf{+.f64}\left(x, \mathsf{*.f64}\left(wj, \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, -2\right), \color{blue}{wj}\right)\right)\right) \]
10. Step-by-step derivation

11. Simplified 95.5%

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

12. Final simplification 95.5%

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

Alternative 10: 95.3% accurate, 62.6× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 78.0%

   \[wj - \frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}} \]
2. Add Preprocessing

3. Taylor expanded in wj around 0

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

   1. +-lowering-+.f64 N/A

      \[\leadsto \mathsf{+.f64}\left(x, \color{blue}{\left(wj \cdot \left(wj \cdot \left(\left(1 + -1 \cdot \left(wj \cdot \left(1 + \left(-3 \cdot x + \left(-2 \cdot \left(-4 \cdot x + \frac{3}{2} \cdot x\right) + \frac{2}{3} \cdot x\right)\right)\right)\right)\right) - \left(-4 \cdot x + \frac{3}{2} \cdot x\right)\right) - 2 \cdot x\right)\right)}\right) \]

   2. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{+.f64}\left(x, \mathsf{*.f64}\left(wj, \color{blue}{\left(wj \cdot \left(\left(1 + -1 \cdot \left(wj \cdot \left(1 + \left(-3 \cdot x + \left(-2 \cdot \left(-4 \cdot x + \frac{3}{2} \cdot x\right) + \frac{2}{3} \cdot x\right)\right)\right)\right)\right) - \left(-4 \cdot x + \frac{3}{2} \cdot x\right)\right) - 2 \cdot x\right)}\right)\right) \]

   3. cancel-sign-sub-inv N/A

      \[\leadsto \mathsf{+.f64}\left(x, \mathsf{*.f64}\left(wj, \left(wj \cdot \left(\left(1 + -1 \cdot \left(wj \cdot \left(1 + \left(-3 \cdot x + \left(-2 \cdot \left(-4 \cdot x + \frac{3}{2} \cdot x\right) + \frac{2}{3} \cdot x\right)\right)\right)\right)\right) - \left(-4 \cdot x + \frac{3}{2} \cdot x\right)\right) + \color{blue}{\left(\mathsf{neg}\left(2\right)\right) \cdot x}\right)\right)\right) \]

   4. metadata-eval N/A

      \[\leadsto \mathsf{+.f64}\left(x, \mathsf{*.f64}\left(wj, \left(wj \cdot \left(\left(1 + -1 \cdot \left(wj \cdot \left(1 + \left(-3 \cdot x + \left(-2 \cdot \left(-4 \cdot x + \frac{3}{2} \cdot x\right) + \frac{2}{3} \cdot x\right)\right)\right)\right)\right) - \left(-4 \cdot x + \frac{3}{2} \cdot x\right)\right) + -2 \cdot x\right)\right)\right) \]

   5. +-commutative N/A

      \[\leadsto \mathsf{+.f64}\left(x, \mathsf{*.f64}\left(wj, \left(-2 \cdot x + \color{blue}{wj \cdot \left(\left(1 + -1 \cdot \left(wj \cdot \left(1 + \left(-3 \cdot x + \left(-2 \cdot \left(-4 \cdot x + \frac{3}{2} \cdot x\right) + \frac{2}{3} \cdot x\right)\right)\right)\right)\right) - \left(-4 \cdot x + \frac{3}{2} \cdot x\right)\right)}\right)\right)\right) \]

   6. +-lowering-+.f64 N/A

      \[\leadsto \mathsf{+.f64}\left(x, \mathsf{*.f64}\left(wj, \mathsf{+.f64}\left(\left(-2 \cdot x\right), \color{blue}{\left(wj \cdot \left(\left(1 + -1 \cdot \left(wj \cdot \left(1 + \left(-3 \cdot x + \left(-2 \cdot \left(-4 \cdot x + \frac{3}{2} \cdot x\right) + \frac{2}{3} \cdot x\right)\right)\right)\right)\right) - \left(-4 \cdot x + \frac{3}{2} \cdot x\right)\right)\right)}\right)\right)\right) \]

5. Simplified 96.3%

   \[\leadsto \color{blue}{x + wj \cdot \left(x \cdot -2 + wj \cdot \left(\left(1 - wj \cdot \left(\left(x \cdot 5 + x \cdot 0.6666666666666666\right) + \left(1 + x \cdot -3\right)\right)\right) + x \cdot 2.5\right)\right)} \]

6. Taylor expanded in x around 0

   \[\leadsto \mathsf{+.f64}\left(x, \color{blue}{\left({wj}^{2} \cdot \left(1 - wj\right)\right)}\right) \]
7. Step-by-step derivation

   1. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{+.f64}\left(x, \mathsf{*.f64}\left(\left({wj}^{2}\right), \color{blue}{\left(1 - wj\right)}\right)\right) \]

   2. unpow2 N/A

      \[\leadsto \mathsf{+.f64}\left(x, \mathsf{*.f64}\left(\left(wj \cdot wj\right), \left(\color{blue}{1} - wj\right)\right)\right) \]

   3. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{+.f64}\left(x, \mathsf{*.f64}\left(\mathsf{*.f64}\left(wj, wj\right), \left(\color{blue}{1} - wj\right)\right)\right) \]

   4. --lowering--.f64 94.7%

      \[\leadsto \mathsf{+.f64}\left(x, \mathsf{*.f64}\left(\mathsf{*.f64}\left(wj, wj\right), \mathsf{\_.f64}\left(1, \color{blue}{wj}\right)\right)\right) \]

8. Simplified 94.7%

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

9. Taylor expanded in wj around 0

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

    1. +-commutative N/A

       \[\leadsto {wj}^{2} + \color{blue}{x} \]

    2. +-lowering-+.f64 N/A

       \[\leadsto \mathsf{+.f64}\left(\left({wj}^{2}\right), \color{blue}{x}\right) \]

    3. unpow2 N/A

       \[\leadsto \mathsf{+.f64}\left(\left(wj \cdot wj\right), x\right) \]

    4. *-lowering-*.f64 94.4%

       \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(wj, wj\right), x\right) \]

11. Simplified 94.4%

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

12. Final simplification 94.4%

    \[\leadsto x + wj \cdot wj \]
13. Add Preprocessing

Alternative 11: 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 78.0%

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

   1. sub-neg N/A

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

   2. +-lowering-+.f64 N/A

      \[\leadsto \mathsf{+.f64}\left(wj, \color{blue}{\left(\mathsf{neg}\left(\frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}}\right)\right)}\right) \]

   3. distribute-rgt1-in N/A

      \[\leadsto \mathsf{+.f64}\left(wj, \left(\mathsf{neg}\left(\frac{wj \cdot e^{wj} - x}{\left(wj + 1\right) \cdot e^{wj}}\right)\right)\right) \]

   4. associate-/l/ N/A

      \[\leadsto \mathsf{+.f64}\left(wj, \left(\mathsf{neg}\left(\frac{\frac{wj \cdot e^{wj} - x}{e^{wj}}}{wj + 1}\right)\right)\right) \]

   5. distribute-neg-frac2 N/A

      \[\leadsto \mathsf{+.f64}\left(wj, \left(\frac{\frac{wj \cdot e^{wj} - x}{e^{wj}}}{\color{blue}{\mathsf{neg}\left(\left(wj + 1\right)\right)}}\right)\right) \]

   6. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{+.f64}\left(wj, \mathsf{/.f64}\left(\left(\frac{wj \cdot e^{wj} - x}{e^{wj}}\right), \color{blue}{\left(\mathsf{neg}\left(\left(wj + 1\right)\right)\right)}\right)\right) \]

   7. div-sub N/A

      \[\leadsto \mathsf{+.f64}\left(wj, \mathsf{/.f64}\left(\left(\frac{wj \cdot e^{wj}}{e^{wj}} - \frac{x}{e^{wj}}\right), \left(\mathsf{neg}\left(\color{blue}{\left(wj + 1\right)}\right)\right)\right)\right) \]

   8. associate-/l* N/A

      \[\leadsto \mathsf{+.f64}\left(wj, \mathsf{/.f64}\left(\left(wj \cdot \frac{e^{wj}}{e^{wj}} - \frac{x}{e^{wj}}\right), \left(\mathsf{neg}\left(\left(\color{blue}{wj} + 1\right)\right)\right)\right)\right) \]

   9. *-inverses N/A

      \[\leadsto \mathsf{+.f64}\left(wj, \mathsf{/.f64}\left(\left(wj \cdot 1 - \frac{x}{e^{wj}}\right), \left(\mathsf{neg}\left(\left(wj + 1\right)\right)\right)\right)\right) \]

   10. *-rgt-identity N/A

       \[\leadsto \mathsf{+.f64}\left(wj, \mathsf{/.f64}\left(\left(wj - \frac{x}{e^{wj}}\right), \left(\mathsf{neg}\left(\left(\color{blue}{wj} + 1\right)\right)\right)\right)\right) \]

   11. --lowering--.f64 N/A

       \[\leadsto \mathsf{+.f64}\left(wj, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(wj, \left(\frac{x}{e^{wj}}\right)\right), \left(\mathsf{neg}\left(\color{blue}{\left(wj + 1\right)}\right)\right)\right)\right) \]

   12. /-lowering-/.f64 N/A

       \[\leadsto \mathsf{+.f64}\left(wj, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(wj, \mathsf{/.f64}\left(x, \left(e^{wj}\right)\right)\right), \left(\mathsf{neg}\left(\left(wj + \color{blue}{1}\right)\right)\right)\right)\right) \]

   13. exp-lowering-exp.f64 N/A

       \[\leadsto \mathsf{+.f64}\left(wj, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(wj, \mathsf{/.f64}\left(x, \mathsf{exp.f64}\left(wj\right)\right)\right), \left(\mathsf{neg}\left(\left(wj + 1\right)\right)\right)\right)\right) \]

   14. +-commutative N/A

       \[\leadsto \mathsf{+.f64}\left(wj, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(wj, \mathsf{/.f64}\left(x, \mathsf{exp.f64}\left(wj\right)\right)\right), \left(\mathsf{neg}\left(\left(1 + wj\right)\right)\right)\right)\right) \]

   15. *-inverses N/A

       \[\leadsto \mathsf{+.f64}\left(wj, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(wj, \mathsf{/.f64}\left(x, \mathsf{exp.f64}\left(wj\right)\right)\right), \left(\mathsf{neg}\left(\left(\frac{e^{wj}}{e^{wj}} + wj\right)\right)\right)\right)\right) \]

   16. distribute-neg-in N/A

       \[\leadsto \mathsf{+.f64}\left(wj, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(wj, \mathsf{/.f64}\left(x, \mathsf{exp.f64}\left(wj\right)\right)\right), \left(\left(\mathsf{neg}\left(\frac{e^{wj}}{e^{wj}}\right)\right) + \color{blue}{\left(\mathsf{neg}\left(wj\right)\right)}\right)\right)\right) \]

3. Simplified 80.0%

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

5. Taylor expanded in wj around 0

   \[\leadsto \color{blue}{x} \]
6. Step-by-step derivation

7. Simplified 83.1%

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

Alternative 12: 4.3% accurate, 313.0× speedup

Math

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

FPCore

(FPCore (wj x) :precision binary64 wj)

C

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

Fortran

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

Java

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

Python

def code(wj, x):
	return wj

Julia

function code(wj, x)
	return wj
end

MATLAB

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

Wolfram

code[wj_, x_] := wj

Derivation

1. Initial program 78.0%

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

   1. sub-neg N/A

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

   2. +-lowering-+.f64 N/A

      \[\leadsto \mathsf{+.f64}\left(wj, \color{blue}{\left(\mathsf{neg}\left(\frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}}\right)\right)}\right) \]

   3. distribute-rgt1-in N/A

      \[\leadsto \mathsf{+.f64}\left(wj, \left(\mathsf{neg}\left(\frac{wj \cdot e^{wj} - x}{\left(wj + 1\right) \cdot e^{wj}}\right)\right)\right) \]

   4. associate-/l/ N/A

      \[\leadsto \mathsf{+.f64}\left(wj, \left(\mathsf{neg}\left(\frac{\frac{wj \cdot e^{wj} - x}{e^{wj}}}{wj + 1}\right)\right)\right) \]

   5. distribute-neg-frac2 N/A

      \[\leadsto \mathsf{+.f64}\left(wj, \left(\frac{\frac{wj \cdot e^{wj} - x}{e^{wj}}}{\color{blue}{\mathsf{neg}\left(\left(wj + 1\right)\right)}}\right)\right) \]

   6. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{+.f64}\left(wj, \mathsf{/.f64}\left(\left(\frac{wj \cdot e^{wj} - x}{e^{wj}}\right), \color{blue}{\left(\mathsf{neg}\left(\left(wj + 1\right)\right)\right)}\right)\right) \]

   7. div-sub N/A

      \[\leadsto \mathsf{+.f64}\left(wj, \mathsf{/.f64}\left(\left(\frac{wj \cdot e^{wj}}{e^{wj}} - \frac{x}{e^{wj}}\right), \left(\mathsf{neg}\left(\color{blue}{\left(wj + 1\right)}\right)\right)\right)\right) \]

   8. associate-/l* N/A

      \[\leadsto \mathsf{+.f64}\left(wj, \mathsf{/.f64}\left(\left(wj \cdot \frac{e^{wj}}{e^{wj}} - \frac{x}{e^{wj}}\right), \left(\mathsf{neg}\left(\left(\color{blue}{wj} + 1\right)\right)\right)\right)\right) \]

   9. *-inverses N/A

      \[\leadsto \mathsf{+.f64}\left(wj, \mathsf{/.f64}\left(\left(wj \cdot 1 - \frac{x}{e^{wj}}\right), \left(\mathsf{neg}\left(\left(wj + 1\right)\right)\right)\right)\right) \]

   10. *-rgt-identity N/A

       \[\leadsto \mathsf{+.f64}\left(wj, \mathsf{/.f64}\left(\left(wj - \frac{x}{e^{wj}}\right), \left(\mathsf{neg}\left(\left(\color{blue}{wj} + 1\right)\right)\right)\right)\right) \]

   11. --lowering--.f64 N/A

       \[\leadsto \mathsf{+.f64}\left(wj, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(wj, \left(\frac{x}{e^{wj}}\right)\right), \left(\mathsf{neg}\left(\color{blue}{\left(wj + 1\right)}\right)\right)\right)\right) \]

   12. /-lowering-/.f64 N/A

       \[\leadsto \mathsf{+.f64}\left(wj, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(wj, \mathsf{/.f64}\left(x, \left(e^{wj}\right)\right)\right), \left(\mathsf{neg}\left(\left(wj + \color{blue}{1}\right)\right)\right)\right)\right) \]

   13. exp-lowering-exp.f64 N/A

       \[\leadsto \mathsf{+.f64}\left(wj, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(wj, \mathsf{/.f64}\left(x, \mathsf{exp.f64}\left(wj\right)\right)\right), \left(\mathsf{neg}\left(\left(wj + 1\right)\right)\right)\right)\right) \]

   14. +-commutative N/A

       \[\leadsto \mathsf{+.f64}\left(wj, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(wj, \mathsf{/.f64}\left(x, \mathsf{exp.f64}\left(wj\right)\right)\right), \left(\mathsf{neg}\left(\left(1 + wj\right)\right)\right)\right)\right) \]

   15. *-inverses N/A

       \[\leadsto \mathsf{+.f64}\left(wj, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(wj, \mathsf{/.f64}\left(x, \mathsf{exp.f64}\left(wj\right)\right)\right), \left(\mathsf{neg}\left(\left(\frac{e^{wj}}{e^{wj}} + wj\right)\right)\right)\right)\right) \]

   16. distribute-neg-in N/A

       \[\leadsto \mathsf{+.f64}\left(wj, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(wj, \mathsf{/.f64}\left(x, \mathsf{exp.f64}\left(wj\right)\right)\right), \left(\left(\mathsf{neg}\left(\frac{e^{wj}}{e^{wj}}\right)\right) + \color{blue}{\left(\mathsf{neg}\left(wj\right)\right)}\right)\right)\right) \]

3. Simplified 80.0%

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

5. Taylor expanded in wj around inf

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

7. Simplified 4.4%

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

Alternative 13: 3.3% accurate, 313.0× speedup

Math

\[\begin{array}{l} \\ -1 \end{array} \]

FPCore

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

C

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

Fortran

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

Java

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

Python

def code(wj, x):
	return -1.0

Julia

function code(wj, x)
	return -1.0
end

MATLAB

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

Wolfram

code[wj_, x_] := -1.0

Derivation

1. Initial program 78.0%

   \[wj - \frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}} \]
2. Add Preprocessing

3. Taylor expanded in wj around inf

   \[\leadsto \mathsf{\_.f64}\left(wj, \color{blue}{1}\right) \]
4. Step-by-step derivation

5. Simplified 4.1%

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

6. Taylor expanded in wj around 0

   \[\leadsto \color{blue}{-1} \]
7. Step-by-step derivation

8. Simplified 3.2%

   \[\leadsto \color{blue}{-1} \]
9. Add Preprocessing

Developer Target 1: 78.8% accurate, 1.5× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Reproduce

herbie shell --seed 2024158 
(FPCore (wj x)
  :name "Jmat.Real.lambertw, newton loop step"
  :precision binary64

  :alt
  (! :herbie-platform default (let ((ew (exp wj))) (- wj (- (/ wj (+ wj 1)) (/ x (+ ew (* wj ew)))))))

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