Jmat.Real.lambertw, newton loop step

Percentage Accurate: 78.0% → 98.2%

Time: 14.3s

Alternatives: 9

Speedup: 313.0×

• could not determine a ground truth

  1. wj = -1.1626970055106384e+60
  2. x = -8.610616343153881e+24

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 78.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 98.2% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;wj \leq -2 \cdot 10^{-7}:\\ \;\;\;\;x \cdot \left(\frac{e^{-wj}}{wj + 1} + \left(\mathsf{fma}\left(\frac{-1}{\mathsf{fma}\left(x, wj, x\right)}, wj, \frac{wj}{\mathsf{fma}\left(x, wj, x\right)}\right) + \mathsf{fma}\left(1, \frac{wj}{x}, \frac{wj}{-\mathsf{fma}\left(x, wj, x\right)}\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x - wj \cdot \left(x \cdot 2 + wj \cdot \left(wj + -1\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (wj x)
 :precision binary64
 (if (<= wj -2e-7)
   (*
    x
    (+
     (/ (exp (- wj)) (+ wj 1.0))
     (+
      (fma (/ -1.0 (fma x wj x)) wj (/ wj (fma x wj x)))
      (fma 1.0 (/ wj x) (/ wj (- (fma x wj x)))))))
   (- x (* wj (+ (* x 2.0) (* wj (+ wj -1.0)))))))

C

double code(double wj, double x) {
	double tmp;
	if (wj <= -2e-7) {
		tmp = x * ((exp(-wj) / (wj + 1.0)) + (fma((-1.0 / fma(x, wj, x)), wj, (wj / fma(x, wj, x))) + fma(1.0, (wj / x), (wj / -fma(x, wj, x)))));
	} else {
		tmp = x - (wj * ((x * 2.0) + (wj * (wj + -1.0))));
	}
	return tmp;
}

Julia

function code(wj, x)
	tmp = 0.0
	if (wj <= -2e-7)
		tmp = Float64(x * Float64(Float64(exp(Float64(-wj)) / Float64(wj + 1.0)) + Float64(fma(Float64(-1.0 / fma(x, wj, x)), wj, Float64(wj / fma(x, wj, x))) + fma(1.0, Float64(wj / x), Float64(wj / Float64(-fma(x, wj, x)))))));
	else
		tmp = Float64(x - Float64(wj * Float64(Float64(x * 2.0) + Float64(wj * Float64(wj + -1.0)))));
	end
	return tmp
end

Wolfram

code[wj_, x_] := If[LessEqual[wj, -2e-7], N[(x * N[(N[(N[Exp[(-wj)], $MachinePrecision] / N[(wj + 1.0), $MachinePrecision]), $MachinePrecision] + N[(N[(N[(-1.0 / N[(x * wj + x), $MachinePrecision]), $MachinePrecision] * wj + N[(wj / N[(x * wj + x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(1.0 * N[(wj / x), $MachinePrecision] + N[(wj / (-N[(x * wj + x), $MachinePrecision])), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x - N[(wj * N[(N[(x * 2.0), $MachinePrecision] + N[(wj * N[(wj + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if wj < -1.9999999999999999e-7

   1. Initial program 59.3%

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

      1. distribute-rgt1-in 96.8%

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

      2. associate-/l/ 96.8%

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

      3. div-sub 59.3%

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

      4. associate-/l* 59.3%

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

      5. *-inverses 96.8%

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

      6. *-rgt-identity 96.8%

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

   3. Simplified 96.8%

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

   5. Taylor expanded in x around inf 96.8%

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

      1. associate--l+ 96.8%

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

      2. associate-/r* 96.8%

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

      3. exp-neg 97.1%

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

      4. +-commutative 97.1%

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

      5. +-commutative 97.1%

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

   7. Simplified 97.1%

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

      1. *-un-lft-identity 97.1%

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

      2. div-inv 97.1%

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

      3. prod-diff 98.5%

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

      4. distribute-rgt-in 98.5%

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

      5. *-un-lft-identity 98.5%

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

      6. fma-define 98.5%

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

      7. distribute-rgt-in 98.5%

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

      8. *-un-lft-identity 98.5%

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

      9. fma-define 98.5%

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

      10. distribute-rgt-in 98.5%

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

      11. *-un-lft-identity 98.5%

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

      12. fma-define 98.5%

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

   9. Applied egg-rr 98.5%

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

       1. +-commutative 98.5%

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

       2. distribute-neg-frac 98.5%

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

       3. metadata-eval 98.5%

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

       4. fma-define 98.5%

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

       5. *-commutative 98.5%

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

       6. fma-define 98.5%

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

       7. associate-*l/ 98.5%

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

       8. *-lft-identity 98.5%

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

       9. fma-define 98.5%

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

       10. *-commutative 98.5%

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

       11. fma-define 98.5%

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

       12. neg-mul-1 98.5%

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

       13. associate-*l/ 98.5%

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

       14. *-lft-identity 98.5%

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

       15. fma-define 98.5%

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

       16. distribute-lft1-in 98.5%

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

       17. *-commutative 98.5%

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

       18. +-commutative 98.5%

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

       19. associate-*r/ 98.5%

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

       20. neg-mul-1 98.5%

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

       21. distribute-lft-in 98.5%

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

   11. Simplified 98.5%

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

   if -1.9999999999999999e-7 < wj

   1. Initial program 77.6%

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

      1. distribute-rgt1-in 77.6%

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

      2. associate-/l/ 77.6%

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

      3. div-sub 77.6%

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

      4. associate-/l* 77.6%

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

      5. *-inverses 78.0%

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

      6. *-rgt-identity 78.0%

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

   3. Simplified 78.0%

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

   5. Taylor expanded in wj around 0 99.0%

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

   6. Taylor expanded in x around 0 99.0%

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

      1. neg-mul-1 99.0%

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

      2. unsub-neg 99.0%

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

   8. Simplified 99.0%

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

4. Final simplification 99.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;wj \leq -2 \cdot 10^{-7}:\\ \;\;\;\;x \cdot \left(\frac{e^{-wj}}{wj + 1} + \left(\mathsf{fma}\left(\frac{-1}{\mathsf{fma}\left(x, wj, x\right)}, wj, \frac{wj}{\mathsf{fma}\left(x, wj, x\right)}\right) + \mathsf{fma}\left(1, \frac{wj}{x}, \frac{wj}{-\mathsf{fma}\left(x, wj, x\right)}\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x - wj \cdot \left(x \cdot 2 + wj \cdot \left(wj + -1\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 98.2% accurate, 2.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;wj \leq -2.1 \cdot 10^{-7}:\\ \;\;\;\;x \cdot \left(\frac{e^{-wj}}{wj + 1} + \left(\frac{wj}{x} + \frac{wj}{x \cdot \left(-1 - wj\right)}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x - wj \cdot \left(x \cdot 2 + wj \cdot \left(wj + -1\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (wj x)
 :precision binary64
 (if (<= wj -2.1e-7)
   (* x (+ (/ (exp (- wj)) (+ wj 1.0)) (+ (/ wj x) (/ wj (* x (- -1.0 wj))))))
   (- x (* wj (+ (* x 2.0) (* wj (+ wj -1.0)))))))

C

double code(double wj, double x) {
	double tmp;
	if (wj <= -2.1e-7) {
		tmp = x * ((exp(-wj) / (wj + 1.0)) + ((wj / x) + (wj / (x * (-1.0 - wj)))));
	} else {
		tmp = x - (wj * ((x * 2.0) + (wj * (wj + -1.0))));
	}
	return tmp;
}

Fortran

real(8) function code(wj, x)
    real(8), intent (in) :: wj
    real(8), intent (in) :: x
    real(8) :: tmp
    if (wj <= (-2.1d-7)) then
        tmp = x * ((exp(-wj) / (wj + 1.0d0)) + ((wj / x) + (wj / (x * ((-1.0d0) - wj)))))
    else
        tmp = x - (wj * ((x * 2.0d0) + (wj * (wj + (-1.0d0)))))
    end if
    code = tmp
end function

Java

public static double code(double wj, double x) {
	double tmp;
	if (wj <= -2.1e-7) {
		tmp = x * ((Math.exp(-wj) / (wj + 1.0)) + ((wj / x) + (wj / (x * (-1.0 - wj)))));
	} else {
		tmp = x - (wj * ((x * 2.0) + (wj * (wj + -1.0))));
	}
	return tmp;
}

Python

def code(wj, x):
	tmp = 0
	if wj <= -2.1e-7:
		tmp = x * ((math.exp(-wj) / (wj + 1.0)) + ((wj / x) + (wj / (x * (-1.0 - wj)))))
	else:
		tmp = x - (wj * ((x * 2.0) + (wj * (wj + -1.0))))
	return tmp

Julia

function code(wj, x)
	tmp = 0.0
	if (wj <= -2.1e-7)
		tmp = Float64(x * Float64(Float64(exp(Float64(-wj)) / Float64(wj + 1.0)) + Float64(Float64(wj / x) + Float64(wj / Float64(x * Float64(-1.0 - wj))))));
	else
		tmp = Float64(x - Float64(wj * Float64(Float64(x * 2.0) + Float64(wj * Float64(wj + -1.0)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(wj, x)
	tmp = 0.0;
	if (wj <= -2.1e-7)
		tmp = x * ((exp(-wj) / (wj + 1.0)) + ((wj / x) + (wj / (x * (-1.0 - wj)))));
	else
		tmp = x - (wj * ((x * 2.0) + (wj * (wj + -1.0))));
	end
	tmp_2 = tmp;
end

Wolfram

code[wj_, x_] := If[LessEqual[wj, -2.1e-7], N[(x * N[(N[(N[Exp[(-wj)], $MachinePrecision] / N[(wj + 1.0), $MachinePrecision]), $MachinePrecision] + N[(N[(wj / x), $MachinePrecision] + N[(wj / N[(x * N[(-1.0 - wj), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x - N[(wj * N[(N[(x * 2.0), $MachinePrecision] + N[(wj * N[(wj + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if wj < -2.1e-7

   1. Initial program 59.3%

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

      1. distribute-rgt1-in 96.8%

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

      2. associate-/l/ 96.8%

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

      3. div-sub 59.3%

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

      4. associate-/l* 59.3%

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

      5. *-inverses 96.8%

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

      6. *-rgt-identity 96.8%

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

   3. Simplified 96.8%

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

   5. Taylor expanded in x around inf 96.8%

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

      1. associate--l+ 96.8%

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

      2. associate-/r* 96.8%

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

      3. exp-neg 97.1%

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

      4. +-commutative 97.1%

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

      5. +-commutative 97.1%

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

   7. Simplified 97.1%

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

   if -2.1e-7 < wj

   1. Initial program 77.6%

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

      1. distribute-rgt1-in 77.6%

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

      2. associate-/l/ 77.6%

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

      3. div-sub 77.6%

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

      4. associate-/l* 77.6%

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

      5. *-inverses 78.0%

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

      6. *-rgt-identity 78.0%

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

   3. Simplified 78.0%

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

   5. Taylor expanded in wj around 0 99.0%

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

   6. Taylor expanded in x around 0 99.0%

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

      1. neg-mul-1 99.0%

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

      2. unsub-neg 99.0%

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

   8. Simplified 99.0%

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

4. Final simplification 98.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;wj \leq -2.1 \cdot 10^{-7}:\\ \;\;\;\;x \cdot \left(\frac{e^{-wj}}{wj + 1} + \left(\frac{wj}{x} + \frac{wj}{x \cdot \left(-1 - wj\right)}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x - wj \cdot \left(x \cdot 2 + wj \cdot \left(wj + -1\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 98.3% accurate, 2.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x \cdot -4 + x \cdot 1.5\\ \mathbf{if}\;wj \leq -3.4 \cdot 10^{-6}:\\ \;\;\;\;wj + \frac{wj - \frac{x}{e^{wj}}}{-1 - wj}\\ \mathbf{else}:\\ \;\;\;\;x + wj \cdot \left(wj \cdot \left(\left(1 - wj \cdot \left(1 + \left(x \cdot -3 + \left(-2 \cdot t\_0 + x \cdot 0.6666666666666666\right)\right)\right)\right) - t\_0\right) - x \cdot 2\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (wj x)
 :precision binary64
 (let* ((t_0 (+ (* x -4.0) (* x 1.5))))
   (if (<= wj -3.4e-6)
     (+ wj (/ (- wj (/ x (exp wj))) (- -1.0 wj)))
     (+
      x
      (*
       wj
       (-
        (*
         wj
         (-
          (-
           1.0
           (*
            wj
            (+ 1.0 (+ (* x -3.0) (+ (* -2.0 t_0) (* x 0.6666666666666666))))))
          t_0))
        (* x 2.0)))))))

C

double code(double wj, double x) {
	double t_0 = (x * -4.0) + (x * 1.5);
	double tmp;
	if (wj <= -3.4e-6) {
		tmp = wj + ((wj - (x / exp(wj))) / (-1.0 - wj));
	} else {
		tmp = x + (wj * ((wj * ((1.0 - (wj * (1.0 + ((x * -3.0) + ((-2.0 * t_0) + (x * 0.6666666666666666)))))) - t_0)) - (x * 2.0)));
	}
	return tmp;
}

Fortran

real(8) function code(wj, x)
    real(8), intent (in) :: wj
    real(8), intent (in) :: x
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (x * (-4.0d0)) + (x * 1.5d0)
    if (wj <= (-3.4d-6)) then
        tmp = wj + ((wj - (x / exp(wj))) / ((-1.0d0) - wj))
    else
        tmp = x + (wj * ((wj * ((1.0d0 - (wj * (1.0d0 + ((x * (-3.0d0)) + (((-2.0d0) * t_0) + (x * 0.6666666666666666d0)))))) - t_0)) - (x * 2.0d0)))
    end if
    code = tmp
end function

Java

public static double code(double wj, double x) {
	double t_0 = (x * -4.0) + (x * 1.5);
	double tmp;
	if (wj <= -3.4e-6) {
		tmp = wj + ((wj - (x / Math.exp(wj))) / (-1.0 - wj));
	} else {
		tmp = x + (wj * ((wj * ((1.0 - (wj * (1.0 + ((x * -3.0) + ((-2.0 * t_0) + (x * 0.6666666666666666)))))) - t_0)) - (x * 2.0)));
	}
	return tmp;
}

Python

def code(wj, x):
	t_0 = (x * -4.0) + (x * 1.5)
	tmp = 0
	if wj <= -3.4e-6:
		tmp = wj + ((wj - (x / math.exp(wj))) / (-1.0 - wj))
	else:
		tmp = x + (wj * ((wj * ((1.0 - (wj * (1.0 + ((x * -3.0) + ((-2.0 * t_0) + (x * 0.6666666666666666)))))) - t_0)) - (x * 2.0)))
	return tmp

Julia

function code(wj, x)
	t_0 = Float64(Float64(x * -4.0) + Float64(x * 1.5))
	tmp = 0.0
	if (wj <= -3.4e-6)
		tmp = Float64(wj + Float64(Float64(wj - Float64(x / exp(wj))) / Float64(-1.0 - wj)));
	else
		tmp = Float64(x + Float64(wj * Float64(Float64(wj * Float64(Float64(1.0 - Float64(wj * Float64(1.0 + Float64(Float64(x * -3.0) + Float64(Float64(-2.0 * t_0) + Float64(x * 0.6666666666666666)))))) - t_0)) - Float64(x * 2.0))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(wj, x)
	t_0 = (x * -4.0) + (x * 1.5);
	tmp = 0.0;
	if (wj <= -3.4e-6)
		tmp = wj + ((wj - (x / exp(wj))) / (-1.0 - wj));
	else
		tmp = x + (wj * ((wj * ((1.0 - (wj * (1.0 + ((x * -3.0) + ((-2.0 * t_0) + (x * 0.6666666666666666)))))) - t_0)) - (x * 2.0)));
	end
	tmp_2 = tmp;
end

Wolfram

code[wj_, x_] := Block[{t$95$0 = N[(N[(x * -4.0), $MachinePrecision] + N[(x * 1.5), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[wj, -3.4e-6], N[(wj + N[(N[(wj - N[(x / N[Exp[wj], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(-1.0 - wj), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x + N[(wj * N[(N[(wj * N[(N[(1.0 - N[(wj * N[(1.0 + N[(N[(x * -3.0), $MachinePrecision] + N[(N[(-2.0 * t$95$0), $MachinePrecision] + N[(x * 0.6666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision]), $MachinePrecision] - N[(x * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if wj < -3.40000000000000006e-6

   1. Initial program 53.7%

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

      1. distribute-rgt1-in 96.5%

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

      2. associate-/l/ 96.5%

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

      3. div-sub 53.7%

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

      4. associate-/l* 53.7%

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

      5. *-inverses 96.5%

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

      6. *-rgt-identity 96.5%

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

   3. Simplified 96.5%

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

   if -3.40000000000000006e-6 < wj

   1. Initial program 77.7%

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

      1. distribute-rgt1-in 77.7%

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

      2. associate-/l/ 77.7%

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

      3. div-sub 77.7%

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

      4. associate-/l* 77.7%

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

      5. *-inverses 78.1%

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

      6. *-rgt-identity 78.1%

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

   3. Simplified 78.1%

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

   5. Taylor expanded in wj around 0 99.0%

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

4. Final simplification 98.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;wj \leq -3.4 \cdot 10^{-6}:\\ \;\;\;\;wj + \frac{wj - \frac{x}{e^{wj}}}{-1 - wj}\\ \mathbf{else}:\\ \;\;\;\;x + wj \cdot \left(wj \cdot \left(\left(1 - wj \cdot \left(1 + \left(x \cdot -3 + \left(-2 \cdot \left(x \cdot -4 + x \cdot 1.5\right) + x \cdot 0.6666666666666666\right)\right)\right)\right) - \left(x \cdot -4 + x \cdot 1.5\right)\right) - x \cdot 2\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 96.6% accurate, 7.6× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 77.0%

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

   1. distribute-rgt1-in 78.2%

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

   2. associate-/l/ 78.2%

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

   3. div-sub 77.0%

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

   4. associate-/l* 77.0%

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

   5. *-inverses 78.6%

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

   6. *-rgt-identity 78.6%

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

3. Simplified 78.6%

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

5. Taylor expanded in wj around 0 96.7%

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

6. Final simplification 96.7%

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

Alternative 5: 96.3% accurate, 24.1× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 77.0%

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

   1. distribute-rgt1-in 78.2%

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

   2. associate-/l/ 78.2%

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

   3. div-sub 77.0%

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

   4. associate-/l* 77.0%

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

   5. *-inverses 78.6%

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

   6. *-rgt-identity 78.6%

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

3. Simplified 78.6%

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

5. Taylor expanded in wj around 0 96.7%

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

6. Taylor expanded in x around 0 96.6%

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

   1. neg-mul-1 96.6%

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

   2. unsub-neg 96.6%

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

8. Simplified 96.6%

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

9. Final simplification 96.6%

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

Alternative 6: 95.9% accurate, 34.8× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 77.0%

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

   1. distribute-rgt1-in 78.2%

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

   2. associate-/l/ 78.2%

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

   3. div-sub 77.0%

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

   4. associate-/l* 77.0%

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

   5. *-inverses 78.6%

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

   6. *-rgt-identity 78.6%

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

3. Simplified 78.6%

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

5. Taylor expanded in wj around 0 96.7%

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

6. Taylor expanded in x around 0 96.6%

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

   1. neg-mul-1 96.6%

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

   2. unsub-neg 96.6%

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

8. Simplified 96.6%

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

9. Taylor expanded in wj around 0 96.2%

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

10. Final simplification 96.2%

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

Alternative 7: 84.9% accurate, 44.7× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 77.0%

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

   1. distribute-rgt1-in 78.2%

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

   2. associate-/l/ 78.2%

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

   3. div-sub 77.0%

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

   4. associate-/l* 77.0%

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

   5. *-inverses 78.6%

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

   6. *-rgt-identity 78.6%

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

3. Simplified 78.6%

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

5. Taylor expanded in wj around 0 85.8%

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

   1. *-commutative 85.8%

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

7. Simplified 85.8%

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

8. Final simplification 85.8%

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

Alternative 8: 4.3% accurate, 313.0× speedup

Math

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

FPCore

(FPCore (wj x) :precision binary64 wj)

C

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

Fortran

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

Java

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

Python

def code(wj, x):
	return wj

Julia

function code(wj, x)
	return wj
end

MATLAB

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

Wolfram

code[wj_, x_] := wj

Derivation

1. Initial program 77.0%

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

   1. distribute-rgt1-in 78.2%

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

   2. associate-/l/ 78.2%

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

   3. div-sub 77.0%

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

   4. associate-/l* 77.0%

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

   5. *-inverses 78.6%

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

   6. *-rgt-identity 78.6%

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

3. Simplified 78.6%

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

5. Taylor expanded in wj around inf 4.2%

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

6. Final simplification 4.2%

   \[\leadsto wj \]
7. Add Preprocessing

Alternative 9: 84.3% 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 77.0%

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

   1. distribute-rgt1-in 78.2%

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

   2. associate-/l/ 78.2%

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

   3. div-sub 77.0%

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

   4. associate-/l* 77.0%

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

   5. *-inverses 78.6%

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

   6. *-rgt-identity 78.6%

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

3. Simplified 78.6%

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

5. Taylor expanded in wj around 0 85.4%

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

6. Final simplification 85.4%

   \[\leadsto x \]
7. Add Preprocessing

Developer target: 78.7% 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 2024055 
(FPCore (wj x)
  :name "Jmat.Real.lambertw, newton loop step"
  :precision binary64

  :alt
  (- wj (- (/ wj (+ wj 1.0)) (/ x (+ (exp wj) (* wj (exp wj))))))

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