Jmat.Real.lambertw, newton loop step

Percentage Accurate: 78.4% → 98.8%

Time: 12.7s

Alternatives: 10

Speedup: 313.0×

• could not determine a ground truth

  1. wj = 2.0910635163763607e+278
  2. x = -1.3799727760353565e+21

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 78.4% 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: 98.8% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := wj \cdot e^{wj}\\ \mathbf{if}\;wj + \frac{x - t\_0}{e^{wj} + t\_0} \leq 4 \cdot 10^{-18}:\\ \;\;\;\;x + wj \cdot \left(x \cdot -2 + wj \cdot \left(1 - \left(wj + x \cdot \left(-2.5 + wj \cdot 2.6666666666666665\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{e^{wj} \cdot \left(wj + 1\right)} + \left(wj + \frac{wj}{-1 - wj}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (wj x)
 :precision binary64
 (let* ((t_0 (* wj (exp wj))))
   (if (<= (+ wj (/ (- x t_0) (+ (exp wj) t_0))) 4e-18)
     (+
      x
      (*
       wj
       (+
        (* x -2.0)
        (* wj (- 1.0 (+ wj (* x (+ -2.5 (* wj 2.6666666666666665)))))))))
     (+ (/ x (* (exp wj) (+ wj 1.0))) (+ wj (/ wj (- -1.0 wj)))))))

C

double code(double wj, double x) {
	double t_0 = wj * exp(wj);
	double tmp;
	if ((wj + ((x - t_0) / (exp(wj) + t_0))) <= 4e-18) {
		tmp = x + (wj * ((x * -2.0) + (wj * (1.0 - (wj + (x * (-2.5 + (wj * 2.6666666666666665))))))));
	} else {
		tmp = (x / (exp(wj) * (wj + 1.0))) + (wj + (wj / (-1.0 - wj)));
	}
	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 = wj * exp(wj)
    if ((wj + ((x - t_0) / (exp(wj) + t_0))) <= 4d-18) then
        tmp = x + (wj * ((x * (-2.0d0)) + (wj * (1.0d0 - (wj + (x * ((-2.5d0) + (wj * 2.6666666666666665d0))))))))
    else
        tmp = (x / (exp(wj) * (wj + 1.0d0))) + (wj + (wj / ((-1.0d0) - wj)))
    end if
    code = tmp
end function

Java

public static double code(double wj, double x) {
	double t_0 = wj * Math.exp(wj);
	double tmp;
	if ((wj + ((x - t_0) / (Math.exp(wj) + t_0))) <= 4e-18) {
		tmp = x + (wj * ((x * -2.0) + (wj * (1.0 - (wj + (x * (-2.5 + (wj * 2.6666666666666665))))))));
	} else {
		tmp = (x / (Math.exp(wj) * (wj + 1.0))) + (wj + (wj / (-1.0 - wj)));
	}
	return tmp;
}

Python

def code(wj, x):
	t_0 = wj * math.exp(wj)
	tmp = 0
	if (wj + ((x - t_0) / (math.exp(wj) + t_0))) <= 4e-18:
		tmp = x + (wj * ((x * -2.0) + (wj * (1.0 - (wj + (x * (-2.5 + (wj * 2.6666666666666665))))))))
	else:
		tmp = (x / (math.exp(wj) * (wj + 1.0))) + (wj + (wj / (-1.0 - wj)))
	return tmp

Julia

function code(wj, x)
	t_0 = Float64(wj * exp(wj))
	tmp = 0.0
	if (Float64(wj + Float64(Float64(x - t_0) / Float64(exp(wj) + t_0))) <= 4e-18)
		tmp = Float64(x + Float64(wj * Float64(Float64(x * -2.0) + Float64(wj * Float64(1.0 - Float64(wj + Float64(x * Float64(-2.5 + Float64(wj * 2.6666666666666665)))))))));
	else
		tmp = Float64(Float64(x / Float64(exp(wj) * Float64(wj + 1.0))) + Float64(wj + Float64(wj / Float64(-1.0 - wj))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(wj, x)
	t_0 = wj * exp(wj);
	tmp = 0.0;
	if ((wj + ((x - t_0) / (exp(wj) + t_0))) <= 4e-18)
		tmp = x + (wj * ((x * -2.0) + (wj * (1.0 - (wj + (x * (-2.5 + (wj * 2.6666666666666665))))))));
	else
		tmp = (x / (exp(wj) * (wj + 1.0))) + (wj + (wj / (-1.0 - wj)));
	end
	tmp_2 = tmp;
end

Wolfram

code[wj_, x_] := Block[{t$95$0 = N[(wj * N[Exp[wj], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(wj + N[(N[(x - t$95$0), $MachinePrecision] / N[(N[Exp[wj], $MachinePrecision] + t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 4e-18], N[(x + N[(wj * N[(N[(x * -2.0), $MachinePrecision] + N[(wj * N[(1.0 - N[(wj + N[(x * N[(-2.5 + N[(wj * 2.6666666666666665), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x / N[(N[Exp[wj], $MachinePrecision] * N[(wj + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(wj + N[(wj / N[(-1.0 - wj), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 wj (/.f64 (-.f64 (*.f64 wj (exp.f64 wj)) x) (+.f64 (exp.f64 wj) (*.f64 wj (exp.f64 wj))))) < 4.0000000000000003e-18

   1. Initial program 68.3%

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

   3. Taylor expanded in wj around 0

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

   5. Simplified 99.4%

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

   6. Taylor expanded in wj around 0

      \[\leadsto \mathsf{+.f64}\left(x, \mathsf{*.f64}\left(wj, \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, -2\right), \mathsf{*.f64}\left(wj, \mathsf{\_.f64}\left(1, \color{blue}{\left(wj \cdot \left(1 + \left(\frac{2}{3} \cdot x + 2 \cdot x\right)\right) - \frac{5}{2} \cdot x\right)}\right)\right)\right)\right)\right) \]
   7. Step-by-step derivation

      1. distribute-rgt-in N/A

         \[\leadsto \mathsf{+.f64}\left(x, \mathsf{*.f64}\left(wj, \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, -2\right), \mathsf{*.f64}\left(wj, \mathsf{\_.f64}\left(1, \left(\left(1 \cdot wj + \left(\frac{2}{3} \cdot x + 2 \cdot x\right) \cdot wj\right) - \color{blue}{\frac{5}{2}} \cdot x\right)\right)\right)\right)\right)\right) \]

      2. *-lft-identity N/A

         \[\leadsto \mathsf{+.f64}\left(x, \mathsf{*.f64}\left(wj, \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, -2\right), \mathsf{*.f64}\left(wj, \mathsf{\_.f64}\left(1, \left(\left(wj + \left(\frac{2}{3} \cdot x + 2 \cdot x\right) \cdot wj\right) - \frac{5}{2} \cdot x\right)\right)\right)\right)\right)\right) \]

      3. associate--l+ N/A

         \[\leadsto \mathsf{+.f64}\left(x, \mathsf{*.f64}\left(wj, \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, -2\right), \mathsf{*.f64}\left(wj, \mathsf{\_.f64}\left(1, \left(wj + \color{blue}{\left(\left(\frac{2}{3} \cdot x + 2 \cdot x\right) \cdot wj - \frac{5}{2} \cdot x\right)}\right)\right)\right)\right)\right)\right) \]

      4. sub-neg N/A

         \[\leadsto \mathsf{+.f64}\left(x, \mathsf{*.f64}\left(wj, \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, -2\right), \mathsf{*.f64}\left(wj, \mathsf{\_.f64}\left(1, \left(wj + \left(\left(\frac{2}{3} \cdot x + 2 \cdot x\right) \cdot wj + \color{blue}{\left(\mathsf{neg}\left(\frac{5}{2} \cdot x\right)\right)}\right)\right)\right)\right)\right)\right)\right) \]

      5. *-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(x, \mathsf{*.f64}\left(wj, \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, -2\right), \mathsf{*.f64}\left(wj, \mathsf{\_.f64}\left(1, \left(wj + \left(wj \cdot \left(\frac{2}{3} \cdot x + 2 \cdot x\right) + \left(\mathsf{neg}\left(\color{blue}{\frac{5}{2} \cdot x}\right)\right)\right)\right)\right)\right)\right)\right)\right) \]

      6. distribute-rgt-out N/A

         \[\leadsto \mathsf{+.f64}\left(x, \mathsf{*.f64}\left(wj, \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, -2\right), \mathsf{*.f64}\left(wj, \mathsf{\_.f64}\left(1, \left(wj + \left(wj \cdot \left(x \cdot \left(\frac{2}{3} + 2\right)\right) + \left(\mathsf{neg}\left(\frac{5}{2} \cdot \color{blue}{x}\right)\right)\right)\right)\right)\right)\right)\right)\right) \]

      7. metadata-eval N/A

         \[\leadsto \mathsf{+.f64}\left(x, \mathsf{*.f64}\left(wj, \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, -2\right), \mathsf{*.f64}\left(wj, \mathsf{\_.f64}\left(1, \left(wj + \left(wj \cdot \left(x \cdot \frac{8}{3}\right) + \left(\mathsf{neg}\left(\frac{5}{2} \cdot x\right)\right)\right)\right)\right)\right)\right)\right)\right) \]

      8. associate-*r* N/A

         \[\leadsto \mathsf{+.f64}\left(x, \mathsf{*.f64}\left(wj, \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, -2\right), \mathsf{*.f64}\left(wj, \mathsf{\_.f64}\left(1, \left(wj + \left(\left(wj \cdot x\right) \cdot \frac{8}{3} + \left(\mathsf{neg}\left(\color{blue}{\frac{5}{2} \cdot x}\right)\right)\right)\right)\right)\right)\right)\right)\right) \]

      9. *-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(x, \mathsf{*.f64}\left(wj, \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, -2\right), \mathsf{*.f64}\left(wj, \mathsf{\_.f64}\left(1, \left(wj + \left(\frac{8}{3} \cdot \left(wj \cdot x\right) + \left(\mathsf{neg}\left(\color{blue}{\frac{5}{2} \cdot x}\right)\right)\right)\right)\right)\right)\right)\right)\right) \]

      10. metadata-eval N/A

          \[\leadsto \mathsf{+.f64}\left(x, \mathsf{*.f64}\left(wj, \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, -2\right), \mathsf{*.f64}\left(wj, \mathsf{\_.f64}\left(1, \left(wj + \left(\left(\mathsf{neg}\left(\frac{-8}{3}\right)\right) \cdot \left(wj \cdot x\right) + \left(\mathsf{neg}\left(\color{blue}{\frac{5}{2}} \cdot x\right)\right)\right)\right)\right)\right)\right)\right)\right) \]

      11. distribute-lft-neg-in N/A

          \[\leadsto \mathsf{+.f64}\left(x, \mathsf{*.f64}\left(wj, \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, -2\right), \mathsf{*.f64}\left(wj, \mathsf{\_.f64}\left(1, \left(wj + \left(\left(\mathsf{neg}\left(\frac{-8}{3} \cdot \left(wj \cdot x\right)\right)\right) + \left(\mathsf{neg}\left(\color{blue}{\frac{5}{2} \cdot x}\right)\right)\right)\right)\right)\right)\right)\right)\right) \]

      12. associate-*l* N/A

          \[\leadsto \mathsf{+.f64}\left(x, \mathsf{*.f64}\left(wj, \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, -2\right), \mathsf{*.f64}\left(wj, \mathsf{\_.f64}\left(1, \left(wj + \left(\left(\mathsf{neg}\left(\left(\frac{-8}{3} \cdot wj\right) \cdot x\right)\right) + \left(\mathsf{neg}\left(\color{blue}{\frac{5}{2}} \cdot x\right)\right)\right)\right)\right)\right)\right)\right)\right) \]

      13. distribute-neg-in N/A

          \[\leadsto \mathsf{+.f64}\left(x, \mathsf{*.f64}\left(wj, \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, -2\right), \mathsf{*.f64}\left(wj, \mathsf{\_.f64}\left(1, \left(wj + \left(\mathsf{neg}\left(\left(\left(\frac{-8}{3} \cdot wj\right) \cdot x + \frac{5}{2} \cdot x\right)\right)\right)\right)\right)\right)\right)\right)\right) \]

      14. distribute-rgt-in N/A

          \[\leadsto \mathsf{+.f64}\left(x, \mathsf{*.f64}\left(wj, \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, -2\right), \mathsf{*.f64}\left(wj, \mathsf{\_.f64}\left(1, \left(wj + \left(\mathsf{neg}\left(x \cdot \left(\frac{-8}{3} \cdot wj + \frac{5}{2}\right)\right)\right)\right)\right)\right)\right)\right)\right) \]

      15. +-commutative N/A

          \[\leadsto \mathsf{+.f64}\left(x, \mathsf{*.f64}\left(wj, \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, -2\right), \mathsf{*.f64}\left(wj, \mathsf{\_.f64}\left(1, \left(wj + \left(\mathsf{neg}\left(x \cdot \left(\frac{5}{2} + \frac{-8}{3} \cdot wj\right)\right)\right)\right)\right)\right)\right)\right)\right) \]

      16. mul-1-neg N/A

          \[\leadsto \mathsf{+.f64}\left(x, \mathsf{*.f64}\left(wj, \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, -2\right), \mathsf{*.f64}\left(wj, \mathsf{\_.f64}\left(1, \left(wj + -1 \cdot \color{blue}{\left(x \cdot \left(\frac{5}{2} + \frac{-8}{3} \cdot wj\right)\right)}\right)\right)\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{+.f64}\left(x, \mathsf{*.f64}\left(wj, \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, -2\right), \mathsf{*.f64}\left(wj, \mathsf{\_.f64}\left(1, \mathsf{+.f64}\left(wj, \color{blue}{\left(-1 \cdot \left(x \cdot \left(\frac{5}{2} + \frac{-8}{3} \cdot wj\right)\right)\right)}\right)\right)\right)\right)\right)\right) \]

      18. mul-1-neg N/A

          \[\leadsto \mathsf{+.f64}\left(x, \mathsf{*.f64}\left(wj, \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, -2\right), \mathsf{*.f64}\left(wj, \mathsf{\_.f64}\left(1, \mathsf{+.f64}\left(wj, \left(\mathsf{neg}\left(x \cdot \left(\frac{5}{2} + \frac{-8}{3} \cdot wj\right)\right)\right)\right)\right)\right)\right)\right)\right) \]

      19. neg-sub0 N/A

          \[\leadsto \mathsf{+.f64}\left(x, \mathsf{*.f64}\left(wj, \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, -2\right), \mathsf{*.f64}\left(wj, \mathsf{\_.f64}\left(1, \mathsf{+.f64}\left(wj, \left(0 - \color{blue}{x \cdot \left(\frac{5}{2} + \frac{-8}{3} \cdot wj\right)}\right)\right)\right)\right)\right)\right)\right) \]

   8. Simplified 99.4%

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

   if 4.0000000000000003e-18 < (-.f64 wj (/.f64 (-.f64 (*.f64 wj (exp.f64 wj)) x) (+.f64 (exp.f64 wj) (*.f64 wj (exp.f64 wj)))))

   1. Initial program 93.5%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. associate--l+ N/A

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

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

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

      4. /-lowering-/.f64 N/A

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

      5. distribute-rgt1-in N/A

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

      6. +-commutative N/A

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

      7. *-commutative N/A

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

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

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

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

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

      10. +-commutative N/A

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

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

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

      12. distribute-rgt1-in N/A

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

      13. +-commutative N/A

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

      14. times-frac N/A

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

      15. *-inverses N/A

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

      16. associate-*l/ N/A

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

   5. Simplified 99.8%

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

4. Final simplification 99.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;wj + \frac{x - wj \cdot e^{wj}}{e^{wj} + wj \cdot e^{wj}} \leq 4 \cdot 10^{-18}:\\ \;\;\;\;x + wj \cdot \left(x \cdot -2 + wj \cdot \left(1 - \left(wj + x \cdot \left(-2.5 + wj \cdot 2.6666666666666665\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{e^{wj} \cdot \left(wj + 1\right)} + \left(wj + \frac{wj}{-1 - wj}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 96.2% accurate, 14.9× speedup

Math

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

FPCore

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

C

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

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 + (x * ((-2.5d0) + (wj * 2.6666666666666665d0))))))))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 76.1%

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

3. Taylor expanded in wj around 0

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

5. Simplified 97.4%

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

6. Taylor expanded in wj around 0

   \[\leadsto \mathsf{+.f64}\left(x, \mathsf{*.f64}\left(wj, \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, -2\right), \mathsf{*.f64}\left(wj, \mathsf{\_.f64}\left(1, \color{blue}{\left(wj \cdot \left(1 + \left(\frac{2}{3} \cdot x + 2 \cdot x\right)\right) - \frac{5}{2} \cdot x\right)}\right)\right)\right)\right)\right) \]
7. Step-by-step derivation

   1. distribute-rgt-in N/A

      \[\leadsto \mathsf{+.f64}\left(x, \mathsf{*.f64}\left(wj, \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, -2\right), \mathsf{*.f64}\left(wj, \mathsf{\_.f64}\left(1, \left(\left(1 \cdot wj + \left(\frac{2}{3} \cdot x + 2 \cdot x\right) \cdot wj\right) - \color{blue}{\frac{5}{2}} \cdot x\right)\right)\right)\right)\right)\right) \]

   2. *-lft-identity N/A

      \[\leadsto \mathsf{+.f64}\left(x, \mathsf{*.f64}\left(wj, \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, -2\right), \mathsf{*.f64}\left(wj, \mathsf{\_.f64}\left(1, \left(\left(wj + \left(\frac{2}{3} \cdot x + 2 \cdot x\right) \cdot wj\right) - \frac{5}{2} \cdot x\right)\right)\right)\right)\right)\right) \]

   3. associate--l+ N/A

      \[\leadsto \mathsf{+.f64}\left(x, \mathsf{*.f64}\left(wj, \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, -2\right), \mathsf{*.f64}\left(wj, \mathsf{\_.f64}\left(1, \left(wj + \color{blue}{\left(\left(\frac{2}{3} \cdot x + 2 \cdot x\right) \cdot wj - \frac{5}{2} \cdot x\right)}\right)\right)\right)\right)\right)\right) \]

   4. sub-neg N/A

      \[\leadsto \mathsf{+.f64}\left(x, \mathsf{*.f64}\left(wj, \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, -2\right), \mathsf{*.f64}\left(wj, \mathsf{\_.f64}\left(1, \left(wj + \left(\left(\frac{2}{3} \cdot x + 2 \cdot x\right) \cdot wj + \color{blue}{\left(\mathsf{neg}\left(\frac{5}{2} \cdot x\right)\right)}\right)\right)\right)\right)\right)\right)\right) \]

   5. *-commutative N/A

      \[\leadsto \mathsf{+.f64}\left(x, \mathsf{*.f64}\left(wj, \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, -2\right), \mathsf{*.f64}\left(wj, \mathsf{\_.f64}\left(1, \left(wj + \left(wj \cdot \left(\frac{2}{3} \cdot x + 2 \cdot x\right) + \left(\mathsf{neg}\left(\color{blue}{\frac{5}{2} \cdot x}\right)\right)\right)\right)\right)\right)\right)\right)\right) \]

   6. distribute-rgt-out N/A

      \[\leadsto \mathsf{+.f64}\left(x, \mathsf{*.f64}\left(wj, \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, -2\right), \mathsf{*.f64}\left(wj, \mathsf{\_.f64}\left(1, \left(wj + \left(wj \cdot \left(x \cdot \left(\frac{2}{3} + 2\right)\right) + \left(\mathsf{neg}\left(\frac{5}{2} \cdot \color{blue}{x}\right)\right)\right)\right)\right)\right)\right)\right)\right) \]

   7. metadata-eval N/A

      \[\leadsto \mathsf{+.f64}\left(x, \mathsf{*.f64}\left(wj, \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, -2\right), \mathsf{*.f64}\left(wj, \mathsf{\_.f64}\left(1, \left(wj + \left(wj \cdot \left(x \cdot \frac{8}{3}\right) + \left(\mathsf{neg}\left(\frac{5}{2} \cdot x\right)\right)\right)\right)\right)\right)\right)\right)\right) \]

   8. associate-*r* N/A

      \[\leadsto \mathsf{+.f64}\left(x, \mathsf{*.f64}\left(wj, \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, -2\right), \mathsf{*.f64}\left(wj, \mathsf{\_.f64}\left(1, \left(wj + \left(\left(wj \cdot x\right) \cdot \frac{8}{3} + \left(\mathsf{neg}\left(\color{blue}{\frac{5}{2} \cdot x}\right)\right)\right)\right)\right)\right)\right)\right)\right) \]

   9. *-commutative N/A

      \[\leadsto \mathsf{+.f64}\left(x, \mathsf{*.f64}\left(wj, \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, -2\right), \mathsf{*.f64}\left(wj, \mathsf{\_.f64}\left(1, \left(wj + \left(\frac{8}{3} \cdot \left(wj \cdot x\right) + \left(\mathsf{neg}\left(\color{blue}{\frac{5}{2} \cdot x}\right)\right)\right)\right)\right)\right)\right)\right)\right) \]

   10. metadata-eval N/A

       \[\leadsto \mathsf{+.f64}\left(x, \mathsf{*.f64}\left(wj, \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, -2\right), \mathsf{*.f64}\left(wj, \mathsf{\_.f64}\left(1, \left(wj + \left(\left(\mathsf{neg}\left(\frac{-8}{3}\right)\right) \cdot \left(wj \cdot x\right) + \left(\mathsf{neg}\left(\color{blue}{\frac{5}{2}} \cdot x\right)\right)\right)\right)\right)\right)\right)\right)\right) \]

   11. distribute-lft-neg-in N/A

       \[\leadsto \mathsf{+.f64}\left(x, \mathsf{*.f64}\left(wj, \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, -2\right), \mathsf{*.f64}\left(wj, \mathsf{\_.f64}\left(1, \left(wj + \left(\left(\mathsf{neg}\left(\frac{-8}{3} \cdot \left(wj \cdot x\right)\right)\right) + \left(\mathsf{neg}\left(\color{blue}{\frac{5}{2} \cdot x}\right)\right)\right)\right)\right)\right)\right)\right)\right) \]

   12. associate-*l* N/A

       \[\leadsto \mathsf{+.f64}\left(x, \mathsf{*.f64}\left(wj, \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, -2\right), \mathsf{*.f64}\left(wj, \mathsf{\_.f64}\left(1, \left(wj + \left(\left(\mathsf{neg}\left(\left(\frac{-8}{3} \cdot wj\right) \cdot x\right)\right) + \left(\mathsf{neg}\left(\color{blue}{\frac{5}{2}} \cdot x\right)\right)\right)\right)\right)\right)\right)\right)\right) \]

   13. distribute-neg-in N/A

       \[\leadsto \mathsf{+.f64}\left(x, \mathsf{*.f64}\left(wj, \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, -2\right), \mathsf{*.f64}\left(wj, \mathsf{\_.f64}\left(1, \left(wj + \left(\mathsf{neg}\left(\left(\left(\frac{-8}{3} \cdot wj\right) \cdot x + \frac{5}{2} \cdot x\right)\right)\right)\right)\right)\right)\right)\right)\right) \]

   14. distribute-rgt-in N/A

       \[\leadsto \mathsf{+.f64}\left(x, \mathsf{*.f64}\left(wj, \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, -2\right), \mathsf{*.f64}\left(wj, \mathsf{\_.f64}\left(1, \left(wj + \left(\mathsf{neg}\left(x \cdot \left(\frac{-8}{3} \cdot wj + \frac{5}{2}\right)\right)\right)\right)\right)\right)\right)\right)\right) \]

   15. +-commutative N/A

       \[\leadsto \mathsf{+.f64}\left(x, \mathsf{*.f64}\left(wj, \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, -2\right), \mathsf{*.f64}\left(wj, \mathsf{\_.f64}\left(1, \left(wj + \left(\mathsf{neg}\left(x \cdot \left(\frac{5}{2} + \frac{-8}{3} \cdot wj\right)\right)\right)\right)\right)\right)\right)\right)\right) \]

   16. mul-1-neg N/A

       \[\leadsto \mathsf{+.f64}\left(x, \mathsf{*.f64}\left(wj, \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, -2\right), \mathsf{*.f64}\left(wj, \mathsf{\_.f64}\left(1, \left(wj + -1 \cdot \color{blue}{\left(x \cdot \left(\frac{5}{2} + \frac{-8}{3} \cdot wj\right)\right)}\right)\right)\right)\right)\right)\right) \]

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

       \[\leadsto \mathsf{+.f64}\left(x, \mathsf{*.f64}\left(wj, \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, -2\right), \mathsf{*.f64}\left(wj, \mathsf{\_.f64}\left(1, \mathsf{+.f64}\left(wj, \color{blue}{\left(-1 \cdot \left(x \cdot \left(\frac{5}{2} + \frac{-8}{3} \cdot wj\right)\right)\right)}\right)\right)\right)\right)\right)\right) \]

   18. mul-1-neg N/A

       \[\leadsto \mathsf{+.f64}\left(x, \mathsf{*.f64}\left(wj, \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, -2\right), \mathsf{*.f64}\left(wj, \mathsf{\_.f64}\left(1, \mathsf{+.f64}\left(wj, \left(\mathsf{neg}\left(x \cdot \left(\frac{5}{2} + \frac{-8}{3} \cdot wj\right)\right)\right)\right)\right)\right)\right)\right)\right) \]

   19. neg-sub0 N/A

       \[\leadsto \mathsf{+.f64}\left(x, \mathsf{*.f64}\left(wj, \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, -2\right), \mathsf{*.f64}\left(wj, \mathsf{\_.f64}\left(1, \mathsf{+.f64}\left(wj, \left(0 - \color{blue}{x \cdot \left(\frac{5}{2} + \frac{-8}{3} \cdot wj\right)}\right)\right)\right)\right)\right)\right)\right) \]

8. Simplified 97.4%

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

9. Final simplification 97.4%

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

Alternative 3: 96.2% accurate, 18.4× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 76.1%

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

3. Taylor expanded in wj around 0

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

5. Simplified 97.4%

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

6. Taylor expanded in x around 0

   \[\leadsto \mathsf{+.f64}\left(x, \mathsf{*.f64}\left(wj, \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, -2\right), \mathsf{*.f64}\left(wj, \mathsf{\_.f64}\left(1, \mathsf{\_.f64}\left(\color{blue}{wj}, \mathsf{*.f64}\left(x, \frac{5}{2}\right)\right)\right)\right)\right)\right)\right) \]
7. Step-by-step derivation

8. Simplified 97.2%

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

Alternative 4: 96.0% accurate, 24.1× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 76.1%

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

3. Taylor expanded in wj around 0

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

5. Simplified 97.4%

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

6. Taylor expanded in x around 0

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

8. Simplified 97.1%

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

   1. sub-neg N/A

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

   2. distribute-rgt-in N/A

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

   3. *-lft-identity N/A

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

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

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

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

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

   6. neg-sub0 N/A

      \[\leadsto \mathsf{+.f64}\left(x, \mathsf{*.f64}\left(wj, \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, -2\right), \mathsf{+.f64}\left(wj, \mathsf{*.f64}\left(\left(0 - wj\right), wj\right)\right)\right)\right)\right) \]

   7. --lowering--.f64 97.1%

      \[\leadsto \mathsf{+.f64}\left(x, \mathsf{*.f64}\left(wj, \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, -2\right), \mathsf{+.f64}\left(wj, \mathsf{*.f64}\left(\mathsf{\_.f64}\left(0, wj\right), wj\right)\right)\right)\right)\right) \]

10. Applied egg-rr 97.1%

    \[\leadsto x + wj \cdot \left(x \cdot -2 + \color{blue}{\left(wj + \left(0 - wj\right) \cdot wj\right)}\right) \]

11. Final simplification 97.1%

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

Alternative 5: 96.0% accurate, 24.1× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 76.1%

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

3. Taylor expanded in wj around 0

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

5. Simplified 97.4%

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

6. Taylor expanded in x around 0

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

8. Simplified 97.1%

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

Alternative 6: 95.4% accurate, 34.8× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 76.1%

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

3. Taylor expanded in wj around 0

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

5. Simplified 97.4%

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

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. unpow2 N/A

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

   2. associate-*l* N/A

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

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

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

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

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

   5. --lowering--.f64 96.6%

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

8. Simplified 96.6%

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

   1. sub-neg N/A

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

   2. distribute-rgt-in N/A

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

   3. *-lft-identity N/A

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

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

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

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

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

   6. neg-sub0 N/A

      \[\leadsto \mathsf{+.f64}\left(x, \mathsf{*.f64}\left(wj, \mathsf{+.f64}\left(wj, \mathsf{*.f64}\left(\left(0 - wj\right), wj\right)\right)\right)\right) \]

   7. --lowering--.f64 96.6%

      \[\leadsto \mathsf{+.f64}\left(x, \mathsf{*.f64}\left(wj, \mathsf{+.f64}\left(wj, \mathsf{*.f64}\left(\mathsf{\_.f64}\left(0, wj\right), wj\right)\right)\right)\right) \]

10. Applied egg-rr 96.6%

    \[\leadsto x + wj \cdot \color{blue}{\left(wj + \left(0 - wj\right) \cdot wj\right)} \]

11. Final simplification 96.6%

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

Alternative 7: 95.4% accurate, 34.8× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 76.1%

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

3. Taylor expanded in wj around 0

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

5. Simplified 97.4%

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

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. unpow2 N/A

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

   2. associate-*l* N/A

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

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

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

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

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

   5. --lowering--.f64 96.6%

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

8. Simplified 96.6%

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

Alternative 8: 95.0% 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 76.1%

   \[wj - \frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}} \]
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 96.8%

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

5. Simplified 96.8%

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

6. Taylor expanded in x around 0

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

8. Simplified 96.3%

   \[\leadsto x + wj \cdot \color{blue}{wj} \]
9. Add Preprocessing

Alternative 9: 84.7% accurate, 313.0× speedup

Math

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

FPCore

(FPCore (wj x) :precision binary64 x)

C

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

Fortran

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

Java

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

Python

def code(wj, x):
	return x

Julia

function code(wj, x)
	return x
end

MATLAB

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

Wolfram

code[wj_, x_] := x

Derivation

1. Initial program 76.1%

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

3. Taylor expanded in wj around 0

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

5. Simplified 84.7%

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

Alternative 10: 4.5% accurate, 313.0× speedup

Math

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

FPCore

(FPCore (wj x) :precision binary64 wj)

C

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

Fortran

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

Java

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

Python

def code(wj, x):
	return wj

Julia

function code(wj, x)
	return wj
end

MATLAB

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

Wolfram

code[wj_, x_] := wj

Derivation

1. Initial program 76.1%

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

3. Taylor expanded in wj around inf

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

5. Simplified 4.7%

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

Developer Target 1: 79.5% 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 2024191 
(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))))))