Jmat.Real.lambertw, newton loop step

Percentage Accurate: 77.7% → 99.0%

Time: 11.2s

Alternatives: 8

Speedup: 313.0×

• could not determine a ground truth

  1. wj = -3.841661781898993e+211
  2. x = 3.342415638758805e-114

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 77.7% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.0% accurate, 2.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;wj \leq -7.2 \cdot 10^{-9}:\\ \;\;\;\;wj + \frac{wj - \frac{x}{e^{wj}}}{-1 - wj}\\ \mathbf{elif}\;wj \leq 0.00052:\\ \;\;\;\;x + wj \cdot \left(x \cdot -2 - wj \cdot \left(-1 - x \cdot 2.5\right)\right)\\ \mathbf{else}:\\ \;\;\;\;wj + \frac{wj}{-1 - wj}\\ \end{array} \end{array} \]

FPCore

(FPCore (wj x)
 :precision binary64
 (if (<= wj -7.2e-9)
   (+ wj (/ (- wj (/ x (exp wj))) (- -1.0 wj)))
   (if (<= wj 0.00052)
     (+ x (* wj (- (* x -2.0) (* wj (- -1.0 (* x 2.5))))))
     (+ wj (/ wj (- -1.0 wj))))))

C

double code(double wj, double x) {
	double tmp;
	if (wj <= -7.2e-9) {
		tmp = wj + ((wj - (x / exp(wj))) / (-1.0 - wj));
	} else if (wj <= 0.00052) {
		tmp = x + (wj * ((x * -2.0) - (wj * (-1.0 - (x * 2.5)))));
	} else {
		tmp = 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) :: tmp
    if (wj <= (-7.2d-9)) then
        tmp = wj + ((wj - (x / exp(wj))) / ((-1.0d0) - wj))
    else if (wj <= 0.00052d0) then
        tmp = x + (wj * ((x * (-2.0d0)) - (wj * ((-1.0d0) - (x * 2.5d0)))))
    else
        tmp = wj + (wj / ((-1.0d0) - wj))
    end if
    code = tmp
end function

Java

public static double code(double wj, double x) {
	double tmp;
	if (wj <= -7.2e-9) {
		tmp = wj + ((wj - (x / Math.exp(wj))) / (-1.0 - wj));
	} else if (wj <= 0.00052) {
		tmp = x + (wj * ((x * -2.0) - (wj * (-1.0 - (x * 2.5)))));
	} else {
		tmp = wj + (wj / (-1.0 - wj));
	}
	return tmp;
}

Python

def code(wj, x):
	tmp = 0
	if wj <= -7.2e-9:
		tmp = wj + ((wj - (x / math.exp(wj))) / (-1.0 - wj))
	elif wj <= 0.00052:
		tmp = x + (wj * ((x * -2.0) - (wj * (-1.0 - (x * 2.5)))))
	else:
		tmp = wj + (wj / (-1.0 - wj))
	return tmp

Julia

function code(wj, x)
	tmp = 0.0
	if (wj <= -7.2e-9)
		tmp = Float64(wj + Float64(Float64(wj - Float64(x / exp(wj))) / Float64(-1.0 - wj)));
	elseif (wj <= 0.00052)
		tmp = Float64(x + Float64(wj * Float64(Float64(x * -2.0) - Float64(wj * Float64(-1.0 - Float64(x * 2.5))))));
	else
		tmp = Float64(wj + Float64(wj / Float64(-1.0 - wj)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(wj, x)
	tmp = 0.0;
	if (wj <= -7.2e-9)
		tmp = wj + ((wj - (x / exp(wj))) / (-1.0 - wj));
	elseif (wj <= 0.00052)
		tmp = x + (wj * ((x * -2.0) - (wj * (-1.0 - (x * 2.5)))));
	else
		tmp = wj + (wj / (-1.0 - wj));
	end
	tmp_2 = tmp;
end

Wolfram

code[wj_, x_] := If[LessEqual[wj, -7.2e-9], N[(wj + N[(N[(wj - N[(x / N[Exp[wj], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(-1.0 - wj), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[wj, 0.00052], 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], N[(wj + N[(wj / N[(-1.0 - wj), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if wj < -7.2e-9

   1. Initial program 66.1%

      \[wj - \frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}} \]
   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 99.7%

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

   if -7.2e-9 < wj < 5.19999999999999954e-4

   1. Initial program 80.3%

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

   3. Taylor expanded in wj around 0

      \[\leadsto \color{blue}{x + wj \cdot \left(wj \cdot \left(1 - \left(-4 \cdot x + \frac{3}{2} \cdot x\right)\right) - 2 \cdot x\right)} \]
   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 99.9%

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

   5. Simplified 99.9%

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

   if 5.19999999999999954e-4 < wj

   1. Initial program 16.7%

      \[wj - \frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}} \]
   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 100.0%

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

   5. Taylor expanded in x around 0

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

      1. mul-1-neg N/A

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

      2. unsub-neg N/A

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

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

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

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

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

      5. +-commutative N/A

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

      6. +-lowering-+.f64 100.0%

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

   7. Simplified 100.0%

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

4. Final simplification 99.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;wj \leq -7.2 \cdot 10^{-9}:\\ \;\;\;\;wj + \frac{wj - \frac{x}{e^{wj}}}{-1 - wj}\\ \mathbf{elif}\;wj \leq 0.00052:\\ \;\;\;\;x + wj \cdot \left(x \cdot -2 - wj \cdot \left(-1 - x \cdot 2.5\right)\right)\\ \mathbf{else}:\\ \;\;\;\;wj + \frac{wj}{-1 - wj}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 98.3% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := wj \cdot e^{wj}\\ \mathbf{if}\;wj + \frac{x - t\_0}{e^{wj} + t\_0} \leq 1.5 \cdot 10^{-22}:\\ \;\;\;\;x + wj \cdot \left(x \cdot -2 - wj \cdot \left(-1 - x \cdot 2.5\right)\right)\\ \mathbf{else}:\\ \;\;\;\;wj + \frac{wj - \frac{x}{e^{wj}}}{1 - wj \cdot wj} \cdot \left(wj + -1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (wj x)
 :precision binary64
 (let* ((t_0 (* wj (exp wj))))
   (if (<= (+ wj (/ (- x t_0) (+ (exp wj) t_0))) 1.5e-22)
     (+ x (* wj (- (* x -2.0) (* wj (- -1.0 (* x 2.5))))))
     (+ wj (* (/ (- wj (/ x (exp wj))) (- 1.0 (* wj wj))) (+ wj -1.0))))))

C

double code(double wj, double x) {
	double t_0 = wj * exp(wj);
	double tmp;
	if ((wj + ((x - t_0) / (exp(wj) + t_0))) <= 1.5e-22) {
		tmp = x + (wj * ((x * -2.0) - (wj * (-1.0 - (x * 2.5)))));
	} else {
		tmp = wj + (((wj - (x / exp(wj))) / (1.0 - (wj * wj))) * (wj + -1.0));
	}
	return tmp;
}

Fortran

real(8) function code(wj, x)
    real(8), intent (in) :: wj
    real(8), intent (in) :: x
    real(8) :: t_0
    real(8) :: tmp
    t_0 = wj * exp(wj)
    if ((wj + ((x - t_0) / (exp(wj) + t_0))) <= 1.5d-22) then
        tmp = x + (wj * ((x * (-2.0d0)) - (wj * ((-1.0d0) - (x * 2.5d0)))))
    else
        tmp = wj + (((wj - (x / exp(wj))) / (1.0d0 - (wj * wj))) * (wj + (-1.0d0)))
    end if
    code = tmp
end function

Java

public static double code(double wj, double x) {
	double t_0 = wj * Math.exp(wj);
	double tmp;
	if ((wj + ((x - t_0) / (Math.exp(wj) + t_0))) <= 1.5e-22) {
		tmp = x + (wj * ((x * -2.0) - (wj * (-1.0 - (x * 2.5)))));
	} else {
		tmp = wj + (((wj - (x / Math.exp(wj))) / (1.0 - (wj * wj))) * (wj + -1.0));
	}
	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))) <= 1.5e-22:
		tmp = x + (wj * ((x * -2.0) - (wj * (-1.0 - (x * 2.5)))))
	else:
		tmp = wj + (((wj - (x / math.exp(wj))) / (1.0 - (wj * wj))) * (wj + -1.0))
	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))) <= 1.5e-22)
		tmp = Float64(x + Float64(wj * Float64(Float64(x * -2.0) - Float64(wj * Float64(-1.0 - Float64(x * 2.5))))));
	else
		tmp = Float64(wj + Float64(Float64(Float64(wj - Float64(x / exp(wj))) / Float64(1.0 - Float64(wj * wj))) * Float64(wj + -1.0)));
	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))) <= 1.5e-22)
		tmp = x + (wj * ((x * -2.0) - (wj * (-1.0 - (x * 2.5)))));
	else
		tmp = wj + (((wj - (x / exp(wj))) / (1.0 - (wj * wj))) * (wj + -1.0));
	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], 1.5e-22], 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], N[(wj + N[(N[(N[(wj - N[(x / N[Exp[wj], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(1.0 - N[(wj * wj), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(wj + -1.0), $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))))) < 1.5e-22

   1. Initial program 73.2%

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

   3. Taylor expanded in wj around 0

      \[\leadsto \color{blue}{x + wj \cdot \left(wj \cdot \left(1 - \left(-4 \cdot x + \frac{3}{2} \cdot x\right)\right) - 2 \cdot x\right)} \]
   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 98.8%

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

   5. Simplified 98.8%

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

   if 1.5e-22 < (-.f64 wj (/.f64 (-.f64 (*.f64 wj (exp.f64 wj)) x) (+.f64 (exp.f64 wj) (*.f64 wj (exp.f64 wj)))))

   1. Initial program 91.7%

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

      1. 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 100.0%

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

      1. flip-- N/A

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

      2. associate-/r/ N/A

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

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

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

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

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

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

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

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

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

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

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

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

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

      9. metadata-eval N/A

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

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

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

      11. +-commutative N/A

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

      12. +-lowering-+.f64 100.0%

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

   6. Applied egg-rr 100.0%

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

4. Final simplification 99.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;wj + \frac{x - wj \cdot e^{wj}}{e^{wj} + wj \cdot e^{wj}} \leq 1.5 \cdot 10^{-22}:\\ \;\;\;\;x + wj \cdot \left(x \cdot -2 - wj \cdot \left(-1 - x \cdot 2.5\right)\right)\\ \mathbf{else}:\\ \;\;\;\;wj + \frac{wj - \frac{x}{e^{wj}}}{1 - wj \cdot wj} \cdot \left(wj + -1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 97.3% accurate, 15.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;wj \leq 0.00155:\\ \;\;\;\;x + wj \cdot \left(x \cdot -2 - wj \cdot \left(-1 - x \cdot 2.5\right)\right)\\ \mathbf{else}:\\ \;\;\;\;wj + \frac{wj}{-1 - wj}\\ \end{array} \end{array} \]

FPCore

(FPCore (wj x)
 :precision binary64
 (if (<= wj 0.00155)
   (+ x (* wj (- (* x -2.0) (* wj (- -1.0 (* x 2.5))))))
   (+ wj (/ wj (- -1.0 wj)))))

C

double code(double wj, double x) {
	double tmp;
	if (wj <= 0.00155) {
		tmp = x + (wj * ((x * -2.0) - (wj * (-1.0 - (x * 2.5)))));
	} else {
		tmp = 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) :: tmp
    if (wj <= 0.00155d0) then
        tmp = x + (wj * ((x * (-2.0d0)) - (wj * ((-1.0d0) - (x * 2.5d0)))))
    else
        tmp = wj + (wj / ((-1.0d0) - wj))
    end if
    code = tmp
end function

Java

public static double code(double wj, double x) {
	double tmp;
	if (wj <= 0.00155) {
		tmp = x + (wj * ((x * -2.0) - (wj * (-1.0 - (x * 2.5)))));
	} else {
		tmp = wj + (wj / (-1.0 - wj));
	}
	return tmp;
}

Python

def code(wj, x):
	tmp = 0
	if wj <= 0.00155:
		tmp = x + (wj * ((x * -2.0) - (wj * (-1.0 - (x * 2.5)))))
	else:
		tmp = wj + (wj / (-1.0 - wj))
	return tmp

Julia

function code(wj, x)
	tmp = 0.0
	if (wj <= 0.00155)
		tmp = Float64(x + Float64(wj * Float64(Float64(x * -2.0) - Float64(wj * Float64(-1.0 - Float64(x * 2.5))))));
	else
		tmp = Float64(wj + Float64(wj / Float64(-1.0 - wj)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(wj, x)
	tmp = 0.0;
	if (wj <= 0.00155)
		tmp = x + (wj * ((x * -2.0) - (wj * (-1.0 - (x * 2.5)))));
	else
		tmp = wj + (wj / (-1.0 - wj));
	end
	tmp_2 = tmp;
end

Wolfram

code[wj_, x_] := If[LessEqual[wj, 0.00155], 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], N[(wj + N[(wj / N[(-1.0 - wj), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if wj < 0.00154999999999999995

   1. Initial program 80.0%

      \[wj - \frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}} \]
   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 97.9%

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

   5. Simplified 97.9%

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

   if 0.00154999999999999995 < wj

   1. Initial program 16.7%

      \[wj - \frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}} \]
   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 100.0%

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

   5. Taylor expanded in x around 0

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

      1. mul-1-neg N/A

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

      2. unsub-neg N/A

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

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

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

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

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

      5. +-commutative N/A

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

      6. +-lowering-+.f64 100.0%

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

   7. Simplified 100.0%

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

4. Final simplification 97.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;wj \leq 0.00155:\\ \;\;\;\;x + wj \cdot \left(x \cdot -2 - wj \cdot \left(-1 - x \cdot 2.5\right)\right)\\ \mathbf{else}:\\ \;\;\;\;wj + \frac{wj}{-1 - wj}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 97.2% accurate, 22.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;wj \leq 0.00078:\\ \;\;\;\;x + wj \cdot \left(wj + x \cdot -2\right)\\ \mathbf{else}:\\ \;\;\;\;wj + \frac{wj}{-1 - wj}\\ \end{array} \end{array} \]

FPCore

(FPCore (wj x)
 :precision binary64
 (if (<= wj 0.00078) (+ x (* wj (+ wj (* x -2.0)))) (+ wj (/ wj (- -1.0 wj)))))

C

double code(double wj, double x) {
	double tmp;
	if (wj <= 0.00078) {
		tmp = x + (wj * (wj + (x * -2.0)));
	} else {
		tmp = 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) :: tmp
    if (wj <= 0.00078d0) then
        tmp = x + (wj * (wj + (x * (-2.0d0))))
    else
        tmp = wj + (wj / ((-1.0d0) - wj))
    end if
    code = tmp
end function

Java

public static double code(double wj, double x) {
	double tmp;
	if (wj <= 0.00078) {
		tmp = x + (wj * (wj + (x * -2.0)));
	} else {
		tmp = wj + (wj / (-1.0 - wj));
	}
	return tmp;
}

Python

def code(wj, x):
	tmp = 0
	if wj <= 0.00078:
		tmp = x + (wj * (wj + (x * -2.0)))
	else:
		tmp = wj + (wj / (-1.0 - wj))
	return tmp

Julia

function code(wj, x)
	tmp = 0.0
	if (wj <= 0.00078)
		tmp = Float64(x + Float64(wj * Float64(wj + Float64(x * -2.0))));
	else
		tmp = Float64(wj + Float64(wj / Float64(-1.0 - wj)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(wj, x)
	tmp = 0.0;
	if (wj <= 0.00078)
		tmp = x + (wj * (wj + (x * -2.0)));
	else
		tmp = wj + (wj / (-1.0 - wj));
	end
	tmp_2 = tmp;
end

Wolfram

code[wj_, x_] := If[LessEqual[wj, 0.00078], N[(x + N[(wj * N[(wj + N[(x * -2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(wj + N[(wj / N[(-1.0 - wj), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if wj < 7.79999999999999986e-4

   1. Initial program 80.0%

      \[wj - \frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}} \]
   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 97.9%

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

   5. Simplified 97.9%

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

   6. Taylor expanded in x around 0

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

   8. Simplified 97.8%

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

   if 7.79999999999999986e-4 < wj

   1. Initial program 16.7%

      \[wj - \frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}} \]
   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 100.0%

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

   5. Taylor expanded in x around 0

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

      1. mul-1-neg N/A

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

      2. unsub-neg N/A

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

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

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

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

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

      5. +-commutative N/A

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

      6. +-lowering-+.f64 100.0%

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

   7. Simplified 100.0%

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

4. Final simplification 97.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;wj \leq 0.00078:\\ \;\;\;\;x + wj \cdot \left(wj + x \cdot -2\right)\\ \mathbf{else}:\\ \;\;\;\;wj + \frac{wj}{-1 - wj}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 96.7% accurate, 26.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;wj \leq 2.3 \cdot 10^{-6}:\\ \;\;\;\;x + wj \cdot wj\\ \mathbf{else}:\\ \;\;\;\;wj + \frac{wj}{-1 - wj}\\ \end{array} \end{array} \]

FPCore

(FPCore (wj x)
 :precision binary64
 (if (<= wj 2.3e-6) (+ x (* wj wj)) (+ wj (/ wj (- -1.0 wj)))))

C

double code(double wj, double x) {
	double tmp;
	if (wj <= 2.3e-6) {
		tmp = x + (wj * wj);
	} else {
		tmp = 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) :: tmp
    if (wj <= 2.3d-6) then
        tmp = x + (wj * wj)
    else
        tmp = wj + (wj / ((-1.0d0) - wj))
    end if
    code = tmp
end function

Java

public static double code(double wj, double x) {
	double tmp;
	if (wj <= 2.3e-6) {
		tmp = x + (wj * wj);
	} else {
		tmp = wj + (wj / (-1.0 - wj));
	}
	return tmp;
}

Python

def code(wj, x):
	tmp = 0
	if wj <= 2.3e-6:
		tmp = x + (wj * wj)
	else:
		tmp = wj + (wj / (-1.0 - wj))
	return tmp

Julia

function code(wj, x)
	tmp = 0.0
	if (wj <= 2.3e-6)
		tmp = Float64(x + Float64(wj * wj));
	else
		tmp = Float64(wj + Float64(wj / Float64(-1.0 - wj)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(wj, x)
	tmp = 0.0;
	if (wj <= 2.3e-6)
		tmp = x + (wj * wj);
	else
		tmp = wj + (wj / (-1.0 - wj));
	end
	tmp_2 = tmp;
end

Wolfram

code[wj_, x_] := If[LessEqual[wj, 2.3e-6], N[(x + N[(wj * wj), $MachinePrecision]), $MachinePrecision], N[(wj + N[(wj / N[(-1.0 - wj), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if wj < 2.3e-6

   1. Initial program 80.0%

      \[wj - \frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}} \]
   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 97.9%

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

   5. Simplified 97.9%

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

   6. Taylor expanded in x around 0

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

      1. unpow2 N/A

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

      2. *-lowering-*.f64 97.1%

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

   8. Simplified 97.1%

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

   if 2.3e-6 < wj

   1. Initial program 16.7%

      \[wj - \frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}} \]
   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 100.0%

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

   5. Taylor expanded in x around 0

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

      1. mul-1-neg N/A

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

      2. unsub-neg N/A

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

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

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

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

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

      5. +-commutative N/A

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

      6. +-lowering-+.f64 100.0%

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

   7. Simplified 100.0%

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

4. Final simplification 97.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;wj \leq 2.3 \cdot 10^{-6}:\\ \;\;\;\;x + wj \cdot wj\\ \mathbf{else}:\\ \;\;\;\;wj + \frac{wj}{-1 - wj}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 95.5% 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 78.5%

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

3. Taylor expanded in wj around 0

   \[\leadsto \color{blue}{x + wj \cdot \left(wj \cdot \left(1 - \left(-4 \cdot x + \frac{3}{2} \cdot x\right)\right) - 2 \cdot x\right)} \]
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. Taylor expanded in x around 0

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

   1. unpow2 N/A

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

   2. *-lowering-*.f64 95.1%

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

8. Simplified 95.1%

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

Alternative 7: 83.8% 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 78.5%

   \[wj - \frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}} \]
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 81.2%

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

5. Taylor expanded in wj around 0

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

7. Simplified 84.9%

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

Alternative 8: 4.4% accurate, 313.0× speedup

Math

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

FPCore

(FPCore (wj x) :precision binary64 wj)

C

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

Fortran

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

Java

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

Python

def code(wj, x):
	return wj

Julia

function code(wj, x)
	return wj
end

MATLAB

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

Wolfram

code[wj_, x_] := wj

Derivation

1. Initial program 78.5%

   \[wj - \frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}} \]
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 81.2%

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

5. Taylor expanded in wj around inf

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

7. Simplified 4.9%

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

Developer Target 1: 78.6% accurate, 1.5× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Reproduce

herbie shell --seed 2024145 
(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))))))