Jmat.Real.lambertw, newton loop step

Percentage Accurate: 78.0% → 97.8%

Time: 13.1s

Alternatives: 9

Speedup: 313.0×

• could not determine a ground truth

  1. wj = -6.11827082605207e+88
  2. x = -6.214913175884898e-201

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 78.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 97.8% accurate, 2.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;wj \leq -2.9 \cdot 10^{-6}:\\ \;\;\;\;wj + \frac{1}{\frac{-1 - wj}{wj - \frac{x}{e^{wj}}}}\\ \mathbf{else}:\\ \;\;\;\;x + wj \cdot \left(x \cdot -2 + wj \cdot \left(\left(1 + wj \cdot \left(\left(-1 - x \cdot -3\right) - \left(x \cdot 5 + x \cdot 0.6666666666666666\right)\right)\right) + x \cdot 2.5\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (wj x)
 :precision binary64
 (if (<= wj -2.9e-6)
   (+ wj (/ 1.0 (/ (- -1.0 wj) (- wj (/ x (exp wj))))))
   (+
    x
    (*
     wj
     (+
      (* x -2.0)
      (*
       wj
       (+
        (+
         1.0
         (* wj (- (- -1.0 (* x -3.0)) (+ (* x 5.0) (* x 0.6666666666666666)))))
        (* x 2.5))))))))

C

double code(double wj, double x) {
	double tmp;
	if (wj <= -2.9e-6) {
		tmp = wj + (1.0 / ((-1.0 - wj) / (wj - (x / exp(wj)))));
	} else {
		tmp = x + (wj * ((x * -2.0) + (wj * ((1.0 + (wj * ((-1.0 - (x * -3.0)) - ((x * 5.0) + (x * 0.6666666666666666))))) + (x * 2.5)))));
	}
	return tmp;
}

Fortran

real(8) function code(wj, x)
    real(8), intent (in) :: wj
    real(8), intent (in) :: x
    real(8) :: tmp
    if (wj <= (-2.9d-6)) then
        tmp = wj + (1.0d0 / (((-1.0d0) - wj) / (wj - (x / exp(wj)))))
    else
        tmp = x + (wj * ((x * (-2.0d0)) + (wj * ((1.0d0 + (wj * (((-1.0d0) - (x * (-3.0d0))) - ((x * 5.0d0) + (x * 0.6666666666666666d0))))) + (x * 2.5d0)))))
    end if
    code = tmp
end function

Java

public static double code(double wj, double x) {
	double tmp;
	if (wj <= -2.9e-6) {
		tmp = wj + (1.0 / ((-1.0 - wj) / (wj - (x / Math.exp(wj)))));
	} else {
		tmp = x + (wj * ((x * -2.0) + (wj * ((1.0 + (wj * ((-1.0 - (x * -3.0)) - ((x * 5.0) + (x * 0.6666666666666666))))) + (x * 2.5)))));
	}
	return tmp;
}

Python

def code(wj, x):
	tmp = 0
	if wj <= -2.9e-6:
		tmp = wj + (1.0 / ((-1.0 - wj) / (wj - (x / math.exp(wj)))))
	else:
		tmp = x + (wj * ((x * -2.0) + (wj * ((1.0 + (wj * ((-1.0 - (x * -3.0)) - ((x * 5.0) + (x * 0.6666666666666666))))) + (x * 2.5)))))
	return tmp

Julia

function code(wj, x)
	tmp = 0.0
	if (wj <= -2.9e-6)
		tmp = Float64(wj + Float64(1.0 / Float64(Float64(-1.0 - wj) / Float64(wj - Float64(x / exp(wj))))));
	else
		tmp = Float64(x + Float64(wj * Float64(Float64(x * -2.0) + Float64(wj * Float64(Float64(1.0 + Float64(wj * Float64(Float64(-1.0 - Float64(x * -3.0)) - Float64(Float64(x * 5.0) + Float64(x * 0.6666666666666666))))) + Float64(x * 2.5))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(wj, x)
	tmp = 0.0;
	if (wj <= -2.9e-6)
		tmp = wj + (1.0 / ((-1.0 - wj) / (wj - (x / exp(wj)))));
	else
		tmp = x + (wj * ((x * -2.0) + (wj * ((1.0 + (wj * ((-1.0 - (x * -3.0)) - ((x * 5.0) + (x * 0.6666666666666666))))) + (x * 2.5)))));
	end
	tmp_2 = tmp;
end

Wolfram

code[wj_, x_] := If[LessEqual[wj, -2.9e-6], N[(wj + N[(1.0 / N[(N[(-1.0 - wj), $MachinePrecision] / N[(wj - N[(x / N[Exp[wj], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x + N[(wj * N[(N[(x * -2.0), $MachinePrecision] + N[(wj * N[(N[(1.0 + N[(wj * N[(N[(-1.0 - N[(x * -3.0), $MachinePrecision]), $MachinePrecision] - N[(N[(x * 5.0), $MachinePrecision] + N[(x * 0.6666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(x * 2.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if wj < -2.9000000000000002e-6

   1. Initial program 59.4%

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

   4. Applied egg-rr 100.0%

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

   if -2.9000000000000002e-6 < wj

   1. Initial program 79.6%

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

   3. Taylor expanded in wj around 0

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

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

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

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

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

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

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

      4. metadata-eval N/A

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

      5. +-commutative N/A

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

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

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

   5. Simplified 98.7%

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

4. Final simplification 98.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;wj \leq -2.9 \cdot 10^{-6}:\\ \;\;\;\;wj + \frac{1}{\frac{-1 - wj}{wj - \frac{x}{e^{wj}}}}\\ \mathbf{else}:\\ \;\;\;\;x + wj \cdot \left(x \cdot -2 + wj \cdot \left(\left(1 + wj \cdot \left(\left(-1 - x \cdot -3\right) - \left(x \cdot 5 + x \cdot 0.6666666666666666\right)\right)\right) + x \cdot 2.5\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 97.3% accurate, 2.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;wj \leq -0.0014:\\ \;\;\;\;\frac{x}{e^{wj} \cdot \left(wj + 1\right)}\\ \mathbf{else}:\\ \;\;\;\;x + wj \cdot \left(x \cdot -2 + wj \cdot \left(\left(1 + wj \cdot \left(\left(-1 - x \cdot -3\right) - \left(x \cdot 5 + x \cdot 0.6666666666666666\right)\right)\right) + x \cdot 2.5\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (wj x)
 :precision binary64
 (if (<= wj -0.0014)
   (/ x (* (exp wj) (+ wj 1.0)))
   (+
    x
    (*
     wj
     (+
      (* x -2.0)
      (*
       wj
       (+
        (+
         1.0
         (* wj (- (- -1.0 (* x -3.0)) (+ (* x 5.0) (* x 0.6666666666666666)))))
        (* x 2.5))))))))

C

double code(double wj, double x) {
	double tmp;
	if (wj <= -0.0014) {
		tmp = x / (exp(wj) * (wj + 1.0));
	} else {
		tmp = x + (wj * ((x * -2.0) + (wj * ((1.0 + (wj * ((-1.0 - (x * -3.0)) - ((x * 5.0) + (x * 0.6666666666666666))))) + (x * 2.5)))));
	}
	return tmp;
}

Fortran

real(8) function code(wj, x)
    real(8), intent (in) :: wj
    real(8), intent (in) :: x
    real(8) :: tmp
    if (wj <= (-0.0014d0)) then
        tmp = x / (exp(wj) * (wj + 1.0d0))
    else
        tmp = x + (wj * ((x * (-2.0d0)) + (wj * ((1.0d0 + (wj * (((-1.0d0) - (x * (-3.0d0))) - ((x * 5.0d0) + (x * 0.6666666666666666d0))))) + (x * 2.5d0)))))
    end if
    code = tmp
end function

Java

public static double code(double wj, double x) {
	double tmp;
	if (wj <= -0.0014) {
		tmp = x / (Math.exp(wj) * (wj + 1.0));
	} else {
		tmp = x + (wj * ((x * -2.0) + (wj * ((1.0 + (wj * ((-1.0 - (x * -3.0)) - ((x * 5.0) + (x * 0.6666666666666666))))) + (x * 2.5)))));
	}
	return tmp;
}

Python

def code(wj, x):
	tmp = 0
	if wj <= -0.0014:
		tmp = x / (math.exp(wj) * (wj + 1.0))
	else:
		tmp = x + (wj * ((x * -2.0) + (wj * ((1.0 + (wj * ((-1.0 - (x * -3.0)) - ((x * 5.0) + (x * 0.6666666666666666))))) + (x * 2.5)))))
	return tmp

Julia

function code(wj, x)
	tmp = 0.0
	if (wj <= -0.0014)
		tmp = Float64(x / Float64(exp(wj) * Float64(wj + 1.0)));
	else
		tmp = Float64(x + Float64(wj * Float64(Float64(x * -2.0) + Float64(wj * Float64(Float64(1.0 + Float64(wj * Float64(Float64(-1.0 - Float64(x * -3.0)) - Float64(Float64(x * 5.0) + Float64(x * 0.6666666666666666))))) + Float64(x * 2.5))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(wj, x)
	tmp = 0.0;
	if (wj <= -0.0014)
		tmp = x / (exp(wj) * (wj + 1.0));
	else
		tmp = x + (wj * ((x * -2.0) + (wj * ((1.0 + (wj * ((-1.0 - (x * -3.0)) - ((x * 5.0) + (x * 0.6666666666666666))))) + (x * 2.5)))));
	end
	tmp_2 = tmp;
end

Wolfram

code[wj_, x_] := If[LessEqual[wj, -0.0014], N[(x / N[(N[Exp[wj], $MachinePrecision] * N[(wj + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x + N[(wj * N[(N[(x * -2.0), $MachinePrecision] + N[(wj * N[(N[(1.0 + N[(wj * N[(N[(-1.0 - N[(x * -3.0), $MachinePrecision]), $MachinePrecision] - N[(N[(x * 5.0), $MachinePrecision] + N[(x * 0.6666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(x * 2.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if wj < -0.00139999999999999999

   1. Initial program 59.4%

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

   5. Taylor expanded in x around inf

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

      1. distribute-rgt-in N/A

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

      2. *-lft-identity N/A

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

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

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

      4. *-lft-identity N/A

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

      5. distribute-rgt-in N/A

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

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

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

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

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

      8. +-commutative N/A

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

      9. +-lowering-+.f64 99.7%

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

   7. Simplified 99.7%

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

   if -0.00139999999999999999 < wj

   1. Initial program 79.6%

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

   3. Taylor expanded in wj around 0

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

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

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

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

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

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

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

      4. metadata-eval N/A

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

      5. +-commutative N/A

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

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

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

   5. Simplified 98.7%

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

4. Final simplification 98.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;wj \leq -0.0014:\\ \;\;\;\;\frac{x}{e^{wj} \cdot \left(wj + 1\right)}\\ \mathbf{else}:\\ \;\;\;\;x + wj \cdot \left(x \cdot -2 + wj \cdot \left(\left(1 + wj \cdot \left(\left(-1 - x \cdot -3\right) - \left(x \cdot 5 + x \cdot 0.6666666666666666\right)\right)\right) + x \cdot 2.5\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 96.0% accurate, 10.1× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 79.2%

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

3. Taylor expanded in wj around 0

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

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

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

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

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

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

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

   4. metadata-eval N/A

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

   5. +-commutative N/A

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

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

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

5. Simplified 97.0%

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

6. Final simplification 97.0%

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

Alternative 4: 96.1% accurate, 14.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 79.2%

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

3. Taylor expanded in wj around 0

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

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

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

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

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

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

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

   4. metadata-eval N/A

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

   5. +-commutative N/A

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

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

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

5. Simplified 97.0%

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

6. Taylor expanded in x around inf

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

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

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

   2. sub-neg N/A

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

   3. metadata-eval N/A

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

   4. +-commutative N/A

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

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

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

   6. associate-/l* N/A

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

   7. distribute-lft-out N/A

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

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

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

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

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

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

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

   11. metadata-eval N/A

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

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

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

   13. *-commutative N/A

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

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

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

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

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

   16. --lowering--.f64 96.6%

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

8. Simplified 96.6%

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

Alternative 5: 95.7% accurate, 20.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 79.2%

   \[wj - \frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}} \]
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 96.4%

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

5. Simplified 96.4%

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

Alternative 6: 95.5% accurate, 24.1× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 79.2%

   \[wj - \frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}} \]
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 96.4%

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

5. Simplified 96.4%

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

6. Taylor expanded in x around inf

   \[\leadsto \color{blue}{x \cdot \left(1 + \left(wj \cdot \left(\frac{5}{2} \cdot wj - 2\right) + \frac{{wj}^{2}}{x}\right)\right)} \]
7. Step-by-step derivation

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

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

   2. associate-+r+ N/A

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

   3. +-commutative N/A

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

   4. associate-+l+ N/A

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

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

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

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

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

   7. sub-neg N/A

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

   8. metadata-eval N/A

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

   9. +-commutative N/A

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

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

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

   11. *-commutative N/A

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

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

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

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

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

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

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

   15. unpow2 N/A

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

   16. *-lowering-*.f64 96.4%

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

8. Simplified 96.4%

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

9. Taylor expanded in wj around 0

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

    1. *-commutative N/A

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

    2. *-lowering-*.f64 96.3%

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

11. Simplified 96.3%

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

Alternative 7: 95.1% 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 79.2%

   \[wj - \frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}} \]
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 96.4%

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

5. Simplified 96.4%

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

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 96.1%

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

8. Simplified 96.1%

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

Alternative 8: 84.1% 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 79.2%

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

   1. sub-neg N/A

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

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

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

   3. distribute-rgt1-in N/A

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

   4. associate-/l/ N/A

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

   5. distribute-neg-frac2 N/A

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

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

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

   7. div-sub N/A

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

   8. associate-/l* N/A

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

   9. *-inverses N/A

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

   10. *-rgt-identity N/A

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

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

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

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

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

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

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

   14. +-commutative N/A

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

   15. *-inverses N/A

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

   16. distribute-neg-in N/A

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

3. Simplified 80.4%

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

5. Taylor expanded in wj around 0

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

7. Simplified 83.9%

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

Alternative 9: 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 79.2%

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

   1. sub-neg N/A

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

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

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

   3. distribute-rgt1-in N/A

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

   4. associate-/l/ N/A

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

   5. distribute-neg-frac2 N/A

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

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

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

   7. div-sub N/A

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

   8. associate-/l* N/A

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

   9. *-inverses N/A

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

   10. *-rgt-identity N/A

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

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

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

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

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

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

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

   14. +-commutative N/A

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

   15. *-inverses N/A

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

   16. distribute-neg-in N/A

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

3. Simplified 80.4%

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

5. Taylor expanded in wj around inf

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

7. Simplified 4.3%

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

Developer Target 1: 79.1% 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 2024159 
(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))))))