Jmat.Real.lambertw, newton loop step

Percentage Accurate: 77.4% → 99.7%

Time: 19.8s

Alternatives: 13

Speedup: 313.0×

• could not determine a ground truth

  1. wj = -4.2091038317165086e+155
  2. x = 3.508560567941444e+242

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 77.4% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.7% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x}{e^{wj}}\\ t_1 := \frac{-1}{wj + 1}\\ t_2 := wj - t\_0\\ \mathbf{if}\;wj \leq -3.8 \cdot 10^{-6}:\\ \;\;\;\;\left(wj + \frac{wj}{-1 - wj}\right) + \mathsf{fma}\left(t\_1, t\_2, \frac{wj}{wj + 1}\right)\\ \mathbf{elif}\;wj \leq 4.6 \cdot 10^{-6}:\\ \;\;\;\;x + wj \cdot \left(wj \cdot \left(\left(1 + x \cdot \left(2.5 + wj \cdot -2.6666666666666665\right)\right) - wj\right) - x \cdot 2\right)\\ \mathbf{else}:\\ \;\;\;\;\left(wj - {\left(\frac{-1 - wj}{t\_0 - wj}\right)}^{-1}\right) + \mathsf{fma}\left(t\_1, t\_2, \frac{t\_2}{wj + 1}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (wj x)
 :precision binary64
 (let* ((t_0 (/ x (exp wj))) (t_1 (/ -1.0 (+ wj 1.0))) (t_2 (- wj t_0)))
   (if (<= wj -3.8e-6)
     (+ (+ wj (/ wj (- -1.0 wj))) (fma t_1 t_2 (/ wj (+ wj 1.0))))
     (if (<= wj 4.6e-6)
       (+
        x
        (*
         wj
         (-
          (* wj (- (+ 1.0 (* x (+ 2.5 (* wj -2.6666666666666665)))) wj))
          (* x 2.0))))
       (+
        (- wj (pow (/ (- -1.0 wj) (- t_0 wj)) -1.0))
        (fma t_1 t_2 (/ t_2 (+ wj 1.0))))))))

C

double code(double wj, double x) {
	double t_0 = x / exp(wj);
	double t_1 = -1.0 / (wj + 1.0);
	double t_2 = wj - t_0;
	double tmp;
	if (wj <= -3.8e-6) {
		tmp = (wj + (wj / (-1.0 - wj))) + fma(t_1, t_2, (wj / (wj + 1.0)));
	} else if (wj <= 4.6e-6) {
		tmp = x + (wj * ((wj * ((1.0 + (x * (2.5 + (wj * -2.6666666666666665)))) - wj)) - (x * 2.0)));
	} else {
		tmp = (wj - pow(((-1.0 - wj) / (t_0 - wj)), -1.0)) + fma(t_1, t_2, (t_2 / (wj + 1.0)));
	}
	return tmp;
}

Julia

function code(wj, x)
	t_0 = Float64(x / exp(wj))
	t_1 = Float64(-1.0 / Float64(wj + 1.0))
	t_2 = Float64(wj - t_0)
	tmp = 0.0
	if (wj <= -3.8e-6)
		tmp = Float64(Float64(wj + Float64(wj / Float64(-1.0 - wj))) + fma(t_1, t_2, Float64(wj / Float64(wj + 1.0))));
	elseif (wj <= 4.6e-6)
		tmp = Float64(x + Float64(wj * Float64(Float64(wj * Float64(Float64(1.0 + Float64(x * Float64(2.5 + Float64(wj * -2.6666666666666665)))) - wj)) - Float64(x * 2.0))));
	else
		tmp = Float64(Float64(wj - (Float64(Float64(-1.0 - wj) / Float64(t_0 - wj)) ^ -1.0)) + fma(t_1, t_2, Float64(t_2 / Float64(wj + 1.0))));
	end
	return tmp
end

Wolfram

code[wj_, x_] := Block[{t$95$0 = N[(x / N[Exp[wj], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(-1.0 / N[(wj + 1.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(wj - t$95$0), $MachinePrecision]}, If[LessEqual[wj, -3.8e-6], N[(N[(wj + N[(wj / N[(-1.0 - wj), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(t$95$1 * t$95$2 + N[(wj / N[(wj + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[wj, 4.6e-6], N[(x + N[(wj * N[(N[(wj * N[(N[(1.0 + N[(x * N[(2.5 + N[(wj * -2.6666666666666665), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - wj), $MachinePrecision]), $MachinePrecision] - N[(x * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(wj - N[Power[N[(N[(-1.0 - wj), $MachinePrecision] / N[(t$95$0 - wj), $MachinePrecision]), $MachinePrecision], -1.0], $MachinePrecision]), $MachinePrecision] + N[(t$95$1 * t$95$2 + N[(t$95$2 / N[(wj + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if wj < -3.8e-6

   1. Initial program 59.6%

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

      1. distribute-rgt1-in 97.1%

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

      2. associate-/l/ 97.1%

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

      3. div-sub 59.6%

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

      4. associate-/l* 59.6%

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

      5. *-inverses 97.1%

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

      6. *-rgt-identity 97.1%

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

   3. Simplified 97.1%

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

      1. *-un-lft-identity 97.1%

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

      2. div-inv 97.2%

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

      3. prod-diff 59.8%

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

      4. associate-/r/ 59.5%

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

      5. clear-num 59.6%

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

      6. fma-neg 59.6%

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

      7. *-un-lft-identity 59.6%

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

      8. associate-/r/ 59.6%

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

      9. clear-num 59.8%

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

   6. Applied egg-rr 59.8%

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

      1. distribute-neg-frac 59.8%

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

      2. metadata-eval 59.8%

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

   8. Simplified 59.8%

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

   9. Taylor expanded in x around 0 66.8%

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

       1. +-commutative 66.8%

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

   11. Simplified 66.8%

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

   12. Taylor expanded in x around 0 97.1%

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

       1. +-commutative 97.1%

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

   14. Simplified 97.1%

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

   if -3.8e-6 < wj < 4.6e-6

   1. Initial program 77.2%

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

      1. distribute-rgt1-in 77.2%

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

      2. associate-/l/ 77.2%

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

      3. div-sub 77.2%

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

      4. associate-/l* 77.2%

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

      5. *-inverses 77.2%

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

      6. *-rgt-identity 77.2%

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

   3. Simplified 77.2%

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

   5. Taylor expanded in wj around 0 99.5%

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

   6. Taylor expanded in x around 0 99.8%

      \[\leadsto x + wj \cdot \left(\color{blue}{\left(wj \cdot \left(x \cdot \left(2.5 + -2.6666666666666665 \cdot wj\right)\right) + wj \cdot \left(1 + -1 \cdot wj\right)\right)} - 2 \cdot x\right) \]
   7. Step-by-step derivation

      1. distribute-lft-out 99.8%

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

      2. *-commutative 99.8%

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

      3. mul-1-neg 99.8%

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

   8. Simplified 99.8%

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

   9. Taylor expanded in x around 0 99.8%

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

   if 4.6e-6 < wj

   1. Initial program 82.2%

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

      1. distribute-rgt1-in 82.2%

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

      2. associate-/l/ 81.6%

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

      3. div-sub 81.6%

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

      4. associate-/l* 81.6%

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

      5. *-inverses 98.2%

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

      6. *-rgt-identity 98.2%

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

   3. Simplified 98.2%

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

      1. *-un-lft-identity 98.2%

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

      2. div-inv 98.2%

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

      3. prod-diff 98.2%

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

      4. associate-/r/ 99.5%

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

      5. clear-num 98.2%

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

      6. fma-neg 98.2%

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

      7. *-un-lft-identity 98.2%

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

      8. associate-/r/ 97.9%

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

      9. clear-num 98.2%

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

   6. Applied egg-rr 98.2%

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

      1. distribute-neg-frac 98.2%

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

      2. metadata-eval 98.2%

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

   8. Simplified 98.2%

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

      1. clear-num 99.5%

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

      2. inv-pow 99.5%

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

   10. Applied egg-rr 99.5%

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

4. Final simplification 99.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;wj \leq -3.8 \cdot 10^{-6}:\\ \;\;\;\;\left(wj + \frac{wj}{-1 - wj}\right) + \mathsf{fma}\left(\frac{-1}{wj + 1}, wj - \frac{x}{e^{wj}}, \frac{wj}{wj + 1}\right)\\ \mathbf{elif}\;wj \leq 4.6 \cdot 10^{-6}:\\ \;\;\;\;x + wj \cdot \left(wj \cdot \left(\left(1 + x \cdot \left(2.5 + wj \cdot -2.6666666666666665\right)\right) - wj\right) - x \cdot 2\right)\\ \mathbf{else}:\\ \;\;\;\;\left(wj - {\left(\frac{-1 - wj}{\frac{x}{e^{wj}} - wj}\right)}^{-1}\right) + \mathsf{fma}\left(\frac{-1}{wj + 1}, wj - \frac{x}{e^{wj}}, \frac{wj - \frac{x}{e^{wj}}}{wj + 1}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 99.7% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;wj \leq -2.85 \cdot 10^{-6}:\\ \;\;\;\;\left(wj + \frac{wj}{-1 - wj}\right) + \mathsf{fma}\left(\frac{-1}{wj + 1}, wj - \frac{x}{e^{wj}}, \frac{wj}{wj + 1}\right)\\ \mathbf{elif}\;wj \leq 4.6 \cdot 10^{-6}:\\ \;\;\;\;x + wj \cdot \left(wj \cdot \left(\left(1 + x \cdot \left(2.5 + wj \cdot -2.6666666666666665\right)\right) - wj\right) - x \cdot 2\right)\\ \mathbf{else}:\\ \;\;\;\;wj - x \cdot \left(\frac{wj}{\left(wj + 1\right) \cdot x} + \frac{e^{-wj}}{-1 - wj}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (wj x)
 :precision binary64
 (if (<= wj -2.85e-6)
   (+
    (+ wj (/ wj (- -1.0 wj)))
    (fma (/ -1.0 (+ wj 1.0)) (- wj (/ x (exp wj))) (/ wj (+ wj 1.0))))
   (if (<= wj 4.6e-6)
     (+
      x
      (*
       wj
       (-
        (* wj (- (+ 1.0 (* x (+ 2.5 (* wj -2.6666666666666665)))) wj))
        (* x 2.0))))
     (- wj (* x (+ (/ wj (* (+ wj 1.0) x)) (/ (exp (- wj)) (- -1.0 wj))))))))

C

double code(double wj, double x) {
	double tmp;
	if (wj <= -2.85e-6) {
		tmp = (wj + (wj / (-1.0 - wj))) + fma((-1.0 / (wj + 1.0)), (wj - (x / exp(wj))), (wj / (wj + 1.0)));
	} else if (wj <= 4.6e-6) {
		tmp = x + (wj * ((wj * ((1.0 + (x * (2.5 + (wj * -2.6666666666666665)))) - wj)) - (x * 2.0)));
	} else {
		tmp = wj - (x * ((wj / ((wj + 1.0) * x)) + (exp(-wj) / (-1.0 - wj))));
	}
	return tmp;
}

Julia

function code(wj, x)
	tmp = 0.0
	if (wj <= -2.85e-6)
		tmp = Float64(Float64(wj + Float64(wj / Float64(-1.0 - wj))) + fma(Float64(-1.0 / Float64(wj + 1.0)), Float64(wj - Float64(x / exp(wj))), Float64(wj / Float64(wj + 1.0))));
	elseif (wj <= 4.6e-6)
		tmp = Float64(x + Float64(wj * Float64(Float64(wj * Float64(Float64(1.0 + Float64(x * Float64(2.5 + Float64(wj * -2.6666666666666665)))) - wj)) - Float64(x * 2.0))));
	else
		tmp = Float64(wj - Float64(x * Float64(Float64(wj / Float64(Float64(wj + 1.0) * x)) + Float64(exp(Float64(-wj)) / Float64(-1.0 - wj)))));
	end
	return tmp
end

Wolfram

code[wj_, x_] := If[LessEqual[wj, -2.85e-6], N[(N[(wj + N[(wj / N[(-1.0 - wj), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(-1.0 / N[(wj + 1.0), $MachinePrecision]), $MachinePrecision] * N[(wj - N[(x / N[Exp[wj], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(wj / N[(wj + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[wj, 4.6e-6], N[(x + N[(wj * N[(N[(wj * N[(N[(1.0 + N[(x * N[(2.5 + N[(wj * -2.6666666666666665), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - wj), $MachinePrecision]), $MachinePrecision] - N[(x * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(wj - N[(x * N[(N[(wj / N[(N[(wj + 1.0), $MachinePrecision] * x), $MachinePrecision]), $MachinePrecision] + N[(N[Exp[(-wj)], $MachinePrecision] / N[(-1.0 - wj), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if wj < -2.8499999999999998e-6

   1. Initial program 59.6%

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

      1. distribute-rgt1-in 97.1%

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

      2. associate-/l/ 97.1%

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

      3. div-sub 59.6%

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

      4. associate-/l* 59.6%

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

      5. *-inverses 97.1%

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

      6. *-rgt-identity 97.1%

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

   3. Simplified 97.1%

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

      1. *-un-lft-identity 97.1%

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

      2. div-inv 97.2%

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

      3. prod-diff 59.8%

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

      4. associate-/r/ 59.5%

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

      5. clear-num 59.6%

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

      6. fma-neg 59.6%

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

      7. *-un-lft-identity 59.6%

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

      8. associate-/r/ 59.6%

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

      9. clear-num 59.8%

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

   6. Applied egg-rr 59.8%

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

      1. distribute-neg-frac 59.8%

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

      2. metadata-eval 59.8%

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

   8. Simplified 59.8%

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

   9. Taylor expanded in x around 0 66.8%

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

       1. +-commutative 66.8%

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

   11. Simplified 66.8%

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

   12. Taylor expanded in x around 0 97.1%

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

       1. +-commutative 97.1%

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

   14. Simplified 97.1%

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

   if -2.8499999999999998e-6 < wj < 4.6e-6

   1. Initial program 77.2%

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

      1. distribute-rgt1-in 77.2%

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

      2. associate-/l/ 77.2%

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

      3. div-sub 77.2%

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

      4. associate-/l* 77.2%

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

      5. *-inverses 77.2%

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

      6. *-rgt-identity 77.2%

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

   3. Simplified 77.2%

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

   5. Taylor expanded in wj around 0 99.5%

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

   6. Taylor expanded in x around 0 99.8%

      \[\leadsto x + wj \cdot \left(\color{blue}{\left(wj \cdot \left(x \cdot \left(2.5 + -2.6666666666666665 \cdot wj\right)\right) + wj \cdot \left(1 + -1 \cdot wj\right)\right)} - 2 \cdot x\right) \]
   7. Step-by-step derivation

      1. distribute-lft-out 99.8%

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

      2. *-commutative 99.8%

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

      3. mul-1-neg 99.8%

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

   8. Simplified 99.8%

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

   9. Taylor expanded in x around 0 99.8%

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

   if 4.6e-6 < wj

   1. Initial program 82.2%

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

      1. distribute-rgt1-in 82.2%

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

      2. associate-/l/ 81.6%

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

      3. div-sub 81.6%

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

      4. associate-/l* 81.6%

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

      5. *-inverses 98.2%

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

      6. *-rgt-identity 98.2%

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

   3. Simplified 98.2%

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

   5. Taylor expanded in x around inf 98.2%

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

      1. +-commutative 98.2%

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

      2. associate-/r* 98.5%

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

      3. exp-neg 98.2%

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

      4. +-commutative 98.2%

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

   7. Simplified 98.2%

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

4. Final simplification 99.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;wj \leq -2.85 \cdot 10^{-6}:\\ \;\;\;\;\left(wj + \frac{wj}{-1 - wj}\right) + \mathsf{fma}\left(\frac{-1}{wj + 1}, wj - \frac{x}{e^{wj}}, \frac{wj}{wj + 1}\right)\\ \mathbf{elif}\;wj \leq 4.6 \cdot 10^{-6}:\\ \;\;\;\;x + wj \cdot \left(wj \cdot \left(\left(1 + x \cdot \left(2.5 + wj \cdot -2.6666666666666665\right)\right) - wj\right) - x \cdot 2\right)\\ \mathbf{else}:\\ \;\;\;\;wj - x \cdot \left(\frac{wj}{\left(wj + 1\right) \cdot x} + \frac{e^{-wj}}{-1 - wj}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 99.7% accurate, 2.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;wj \leq -4.5 \cdot 10^{-6}:\\ \;\;\;\;wj + \frac{wj - \frac{x}{e^{wj}}}{-1 - wj}\\ \mathbf{elif}\;wj \leq 4.2 \cdot 10^{-6}:\\ \;\;\;\;x + wj \cdot \left(wj \cdot \left(\left(1 + x \cdot \left(2.5 + wj \cdot -2.6666666666666665\right)\right) - wj\right) - x \cdot 2\right)\\ \mathbf{else}:\\ \;\;\;\;wj - x \cdot \left(\frac{wj}{\left(wj + 1\right) \cdot x} + \frac{e^{-wj}}{-1 - wj}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (wj x)
 :precision binary64
 (if (<= wj -4.5e-6)
   (+ wj (/ (- wj (/ x (exp wj))) (- -1.0 wj)))
   (if (<= wj 4.2e-6)
     (+
      x
      (*
       wj
       (-
        (* wj (- (+ 1.0 (* x (+ 2.5 (* wj -2.6666666666666665)))) wj))
        (* x 2.0))))
     (- wj (* x (+ (/ wj (* (+ wj 1.0) x)) (/ (exp (- wj)) (- -1.0 wj))))))))

C

double code(double wj, double x) {
	double tmp;
	if (wj <= -4.5e-6) {
		tmp = wj + ((wj - (x / exp(wj))) / (-1.0 - wj));
	} else if (wj <= 4.2e-6) {
		tmp = x + (wj * ((wj * ((1.0 + (x * (2.5 + (wj * -2.6666666666666665)))) - wj)) - (x * 2.0)));
	} else {
		tmp = wj - (x * ((wj / ((wj + 1.0) * x)) + (exp(-wj) / (-1.0 - wj))));
	}
	return tmp;
}

Fortran

real(8) function code(wj, x)
    real(8), intent (in) :: wj
    real(8), intent (in) :: x
    real(8) :: tmp
    if (wj <= (-4.5d-6)) then
        tmp = wj + ((wj - (x / exp(wj))) / ((-1.0d0) - wj))
    else if (wj <= 4.2d-6) then
        tmp = x + (wj * ((wj * ((1.0d0 + (x * (2.5d0 + (wj * (-2.6666666666666665d0))))) - wj)) - (x * 2.0d0)))
    else
        tmp = wj - (x * ((wj / ((wj + 1.0d0) * x)) + (exp(-wj) / ((-1.0d0) - wj))))
    end if
    code = tmp
end function

Java

public static double code(double wj, double x) {
	double tmp;
	if (wj <= -4.5e-6) {
		tmp = wj + ((wj - (x / Math.exp(wj))) / (-1.0 - wj));
	} else if (wj <= 4.2e-6) {
		tmp = x + (wj * ((wj * ((1.0 + (x * (2.5 + (wj * -2.6666666666666665)))) - wj)) - (x * 2.0)));
	} else {
		tmp = wj - (x * ((wj / ((wj + 1.0) * x)) + (Math.exp(-wj) / (-1.0 - wj))));
	}
	return tmp;
}

Python

def code(wj, x):
	tmp = 0
	if wj <= -4.5e-6:
		tmp = wj + ((wj - (x / math.exp(wj))) / (-1.0 - wj))
	elif wj <= 4.2e-6:
		tmp = x + (wj * ((wj * ((1.0 + (x * (2.5 + (wj * -2.6666666666666665)))) - wj)) - (x * 2.0)))
	else:
		tmp = wj - (x * ((wj / ((wj + 1.0) * x)) + (math.exp(-wj) / (-1.0 - wj))))
	return tmp

Julia

function code(wj, x)
	tmp = 0.0
	if (wj <= -4.5e-6)
		tmp = Float64(wj + Float64(Float64(wj - Float64(x / exp(wj))) / Float64(-1.0 - wj)));
	elseif (wj <= 4.2e-6)
		tmp = Float64(x + Float64(wj * Float64(Float64(wj * Float64(Float64(1.0 + Float64(x * Float64(2.5 + Float64(wj * -2.6666666666666665)))) - wj)) - Float64(x * 2.0))));
	else
		tmp = Float64(wj - Float64(x * Float64(Float64(wj / Float64(Float64(wj + 1.0) * x)) + Float64(exp(Float64(-wj)) / Float64(-1.0 - wj)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(wj, x)
	tmp = 0.0;
	if (wj <= -4.5e-6)
		tmp = wj + ((wj - (x / exp(wj))) / (-1.0 - wj));
	elseif (wj <= 4.2e-6)
		tmp = x + (wj * ((wj * ((1.0 + (x * (2.5 + (wj * -2.6666666666666665)))) - wj)) - (x * 2.0)));
	else
		tmp = wj - (x * ((wj / ((wj + 1.0) * x)) + (exp(-wj) / (-1.0 - wj))));
	end
	tmp_2 = tmp;
end

Wolfram

code[wj_, x_] := If[LessEqual[wj, -4.5e-6], N[(wj + N[(N[(wj - N[(x / N[Exp[wj], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(-1.0 - wj), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[wj, 4.2e-6], N[(x + N[(wj * N[(N[(wj * N[(N[(1.0 + N[(x * N[(2.5 + N[(wj * -2.6666666666666665), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - wj), $MachinePrecision]), $MachinePrecision] - N[(x * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(wj - N[(x * N[(N[(wj / N[(N[(wj + 1.0), $MachinePrecision] * x), $MachinePrecision]), $MachinePrecision] + N[(N[Exp[(-wj)], $MachinePrecision] / N[(-1.0 - wj), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if wj < -4.50000000000000011e-6

   1. Initial program 59.6%

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

      1. distribute-rgt1-in 97.1%

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

      2. associate-/l/ 97.1%

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

      3. div-sub 59.6%

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

      4. associate-/l* 59.6%

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

      5. *-inverses 97.1%

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

      6. *-rgt-identity 97.1%

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

   3. Simplified 97.1%

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

   if -4.50000000000000011e-6 < wj < 4.1999999999999996e-6

   1. Initial program 77.2%

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

      1. distribute-rgt1-in 77.2%

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

      2. associate-/l/ 77.2%

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

      3. div-sub 77.2%

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

      4. associate-/l* 77.2%

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

      5. *-inverses 77.2%

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

      6. *-rgt-identity 77.2%

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

   3. Simplified 77.2%

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

   5. Taylor expanded in wj around 0 99.5%

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

   6. Taylor expanded in x around 0 99.8%

      \[\leadsto x + wj \cdot \left(\color{blue}{\left(wj \cdot \left(x \cdot \left(2.5 + -2.6666666666666665 \cdot wj\right)\right) + wj \cdot \left(1 + -1 \cdot wj\right)\right)} - 2 \cdot x\right) \]
   7. Step-by-step derivation

      1. distribute-lft-out 99.8%

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

      2. *-commutative 99.8%

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

      3. mul-1-neg 99.8%

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

   8. Simplified 99.8%

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

   9. Taylor expanded in x around 0 99.8%

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

   if 4.1999999999999996e-6 < wj

   1. Initial program 82.2%

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

      1. distribute-rgt1-in 82.2%

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

      2. associate-/l/ 81.6%

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

      3. div-sub 81.6%

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

      4. associate-/l* 81.6%

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

      5. *-inverses 98.2%

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

      6. *-rgt-identity 98.2%

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

   3. Simplified 98.2%

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

   5. Taylor expanded in x around inf 98.2%

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

      1. +-commutative 98.2%

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

      2. associate-/r* 98.5%

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

      3. exp-neg 98.2%

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

      4. +-commutative 98.2%

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

   7. Simplified 98.2%

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

4. Final simplification 99.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;wj \leq -4.5 \cdot 10^{-6}:\\ \;\;\;\;wj + \frac{wj - \frac{x}{e^{wj}}}{-1 - wj}\\ \mathbf{elif}\;wj \leq 4.2 \cdot 10^{-6}:\\ \;\;\;\;x + wj \cdot \left(wj \cdot \left(\left(1 + x \cdot \left(2.5 + wj \cdot -2.6666666666666665\right)\right) - wj\right) - x \cdot 2\right)\\ \mathbf{else}:\\ \;\;\;\;wj - x \cdot \left(\frac{wj}{\left(wj + 1\right) \cdot x} + \frac{e^{-wj}}{-1 - wj}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 99.8% accurate, 2.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;wj \leq -4.5 \cdot 10^{-6} \lor \neg \left(wj \leq 4.8 \cdot 10^{-6}\right):\\ \;\;\;\;wj + \frac{wj - \frac{x}{e^{wj}}}{-1 - wj}\\ \mathbf{else}:\\ \;\;\;\;x + wj \cdot \left(wj \cdot \left(\left(1 + x \cdot \left(2.5 + wj \cdot -2.6666666666666665\right)\right) - wj\right) - x \cdot 2\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (wj x)
 :precision binary64
 (if (or (<= wj -4.5e-6) (not (<= wj 4.8e-6)))
   (+ wj (/ (- wj (/ x (exp wj))) (- -1.0 wj)))
   (+
    x
    (*
     wj
     (-
      (* wj (- (+ 1.0 (* x (+ 2.5 (* wj -2.6666666666666665)))) wj))
      (* x 2.0))))))

C

double code(double wj, double x) {
	double tmp;
	if ((wj <= -4.5e-6) || !(wj <= 4.8e-6)) {
		tmp = wj + ((wj - (x / exp(wj))) / (-1.0 - wj));
	} else {
		tmp = x + (wj * ((wj * ((1.0 + (x * (2.5 + (wj * -2.6666666666666665)))) - wj)) - (x * 2.0)));
	}
	return tmp;
}

Fortran

real(8) function code(wj, x)
    real(8), intent (in) :: wj
    real(8), intent (in) :: x
    real(8) :: tmp
    if ((wj <= (-4.5d-6)) .or. (.not. (wj <= 4.8d-6))) then
        tmp = wj + ((wj - (x / exp(wj))) / ((-1.0d0) - wj))
    else
        tmp = x + (wj * ((wj * ((1.0d0 + (x * (2.5d0 + (wj * (-2.6666666666666665d0))))) - wj)) - (x * 2.0d0)))
    end if
    code = tmp
end function

Java

public static double code(double wj, double x) {
	double tmp;
	if ((wj <= -4.5e-6) || !(wj <= 4.8e-6)) {
		tmp = wj + ((wj - (x / Math.exp(wj))) / (-1.0 - wj));
	} else {
		tmp = x + (wj * ((wj * ((1.0 + (x * (2.5 + (wj * -2.6666666666666665)))) - wj)) - (x * 2.0)));
	}
	return tmp;
}

Python

def code(wj, x):
	tmp = 0
	if (wj <= -4.5e-6) or not (wj <= 4.8e-6):
		tmp = wj + ((wj - (x / math.exp(wj))) / (-1.0 - wj))
	else:
		tmp = x + (wj * ((wj * ((1.0 + (x * (2.5 + (wj * -2.6666666666666665)))) - wj)) - (x * 2.0)))
	return tmp

Julia

function code(wj, x)
	tmp = 0.0
	if ((wj <= -4.5e-6) || !(wj <= 4.8e-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(x * Float64(2.5 + Float64(wj * -2.6666666666666665)))) - wj)) - Float64(x * 2.0))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(wj, x)
	tmp = 0.0;
	if ((wj <= -4.5e-6) || ~((wj <= 4.8e-6)))
		tmp = wj + ((wj - (x / exp(wj))) / (-1.0 - wj));
	else
		tmp = x + (wj * ((wj * ((1.0 + (x * (2.5 + (wj * -2.6666666666666665)))) - wj)) - (x * 2.0)));
	end
	tmp_2 = tmp;
end

Wolfram

code[wj_, x_] := If[Or[LessEqual[wj, -4.5e-6], N[Not[LessEqual[wj, 4.8e-6]], $MachinePrecision]], 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[(x * N[(2.5 + N[(wj * -2.6666666666666665), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - wj), $MachinePrecision]), $MachinePrecision] - N[(x * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if wj < -4.50000000000000011e-6 or 4.7999999999999998e-6 < wj

   1. Initial program 69.3%

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

      1. distribute-rgt1-in 90.7%

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

      2. associate-/l/ 90.4%

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

      3. div-sub 69.0%

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

      4. associate-/l* 69.0%

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

      5. *-inverses 97.6%

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

      6. *-rgt-identity 97.6%

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

   3. Simplified 97.6%

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

   if -4.50000000000000011e-6 < wj < 4.7999999999999998e-6

   1. Initial program 77.2%

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

      1. distribute-rgt1-in 77.2%

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

      2. associate-/l/ 77.2%

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

      3. div-sub 77.2%

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

      4. associate-/l* 77.2%

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

      5. *-inverses 77.2%

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

      6. *-rgt-identity 77.2%

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

   3. Simplified 77.2%

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

   5. Taylor expanded in wj around 0 99.5%

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

   6. Taylor expanded in x around 0 99.8%

      \[\leadsto x + wj \cdot \left(\color{blue}{\left(wj \cdot \left(x \cdot \left(2.5 + -2.6666666666666665 \cdot wj\right)\right) + wj \cdot \left(1 + -1 \cdot wj\right)\right)} - 2 \cdot x\right) \]
   7. Step-by-step derivation

      1. distribute-lft-out 99.8%

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

      2. *-commutative 99.8%

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

      3. mul-1-neg 99.8%

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

   8. Simplified 99.8%

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

   9. Taylor expanded in x around 0 99.8%

      \[\leadsto x + wj \cdot \left(wj \cdot \color{blue}{\left(\left(1 + x \cdot \left(2.5 + -2.6666666666666665 \cdot wj\right)\right) - wj\right)} - 2 \cdot x\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 99.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;wj \leq -4.5 \cdot 10^{-6} \lor \neg \left(wj \leq 4.8 \cdot 10^{-6}\right):\\ \;\;\;\;wj + \frac{wj - \frac{x}{e^{wj}}}{-1 - wj}\\ \mathbf{else}:\\ \;\;\;\;x + wj \cdot \left(wj \cdot \left(\left(1 + x \cdot \left(2.5 + wj \cdot -2.6666666666666665\right)\right) - wj\right) - x \cdot 2\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 97.6% accurate, 2.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;wj \leq -0.0112:\\ \;\;\;\;\frac{x}{\left(wj + 1\right) \cdot e^{wj}}\\ \mathbf{else}:\\ \;\;\;\;x + wj \cdot \left(wj \cdot \left(\left(1 + x \cdot \left(2.5 + wj \cdot -2.6666666666666665\right)\right) - wj\right) - x \cdot 2\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (wj x)
 :precision binary64
 (if (<= wj -0.0112)
   (/ x (* (+ wj 1.0) (exp wj)))
   (+
    x
    (*
     wj
     (-
      (* wj (- (+ 1.0 (* x (+ 2.5 (* wj -2.6666666666666665)))) wj))
      (* x 2.0))))))

C

double code(double wj, double x) {
	double tmp;
	if (wj <= -0.0112) {
		tmp = x / ((wj + 1.0) * exp(wj));
	} else {
		tmp = x + (wj * ((wj * ((1.0 + (x * (2.5 + (wj * -2.6666666666666665)))) - wj)) - (x * 2.0)));
	}
	return tmp;
}

Fortran

real(8) function code(wj, x)
    real(8), intent (in) :: wj
    real(8), intent (in) :: x
    real(8) :: tmp
    if (wj <= (-0.0112d0)) then
        tmp = x / ((wj + 1.0d0) * exp(wj))
    else
        tmp = x + (wj * ((wj * ((1.0d0 + (x * (2.5d0 + (wj * (-2.6666666666666665d0))))) - wj)) - (x * 2.0d0)))
    end if
    code = tmp
end function

Java

public static double code(double wj, double x) {
	double tmp;
	if (wj <= -0.0112) {
		tmp = x / ((wj + 1.0) * Math.exp(wj));
	} else {
		tmp = x + (wj * ((wj * ((1.0 + (x * (2.5 + (wj * -2.6666666666666665)))) - wj)) - (x * 2.0)));
	}
	return tmp;
}

Python

def code(wj, x):
	tmp = 0
	if wj <= -0.0112:
		tmp = x / ((wj + 1.0) * math.exp(wj))
	else:
		tmp = x + (wj * ((wj * ((1.0 + (x * (2.5 + (wj * -2.6666666666666665)))) - wj)) - (x * 2.0)))
	return tmp

Julia

function code(wj, x)
	tmp = 0.0
	if (wj <= -0.0112)
		tmp = Float64(x / Float64(Float64(wj + 1.0) * exp(wj)));
	else
		tmp = Float64(x + Float64(wj * Float64(Float64(wj * Float64(Float64(1.0 + Float64(x * Float64(2.5 + Float64(wj * -2.6666666666666665)))) - wj)) - Float64(x * 2.0))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(wj, x)
	tmp = 0.0;
	if (wj <= -0.0112)
		tmp = x / ((wj + 1.0) * exp(wj));
	else
		tmp = x + (wj * ((wj * ((1.0 + (x * (2.5 + (wj * -2.6666666666666665)))) - wj)) - (x * 2.0)));
	end
	tmp_2 = tmp;
end

Wolfram

code[wj_, x_] := If[LessEqual[wj, -0.0112], N[(x / N[(N[(wj + 1.0), $MachinePrecision] * N[Exp[wj], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x + N[(wj * N[(N[(wj * N[(N[(1.0 + N[(x * N[(2.5 + N[(wj * -2.6666666666666665), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - wj), $MachinePrecision]), $MachinePrecision] - N[(x * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if wj < -0.0111999999999999999

   1. Initial program 49.7%

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

      1. distribute-rgt1-in 100.0%

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

      2. associate-/l/ 99.7%

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

      3. div-sub 49.7%

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

      4. associate-/l* 49.7%

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

      5. *-inverses 99.7%

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

      6. *-rgt-identity 99.7%

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

   3. Simplified 99.7%

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

   5. Taylor expanded in x around inf 83.6%

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

      1. +-commutative 83.6%

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

   7. Simplified 83.6%

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

   if -0.0111999999999999999 < wj

   1. Initial program 77.4%

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

      1. distribute-rgt1-in 77.4%

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

      2. associate-/l/ 77.4%

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

      3. div-sub 77.4%

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

      4. associate-/l* 77.4%

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

      5. *-inverses 77.8%

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

      6. *-rgt-identity 77.8%

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

   3. Simplified 77.8%

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

   5. Taylor expanded in wj around 0 97.3%

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

   6. Taylor expanded in x around 0 97.7%

      \[\leadsto x + wj \cdot \left(\color{blue}{\left(wj \cdot \left(x \cdot \left(2.5 + -2.6666666666666665 \cdot wj\right)\right) + wj \cdot \left(1 + -1 \cdot wj\right)\right)} - 2 \cdot x\right) \]
   7. Step-by-step derivation

      1. distribute-lft-out 97.7%

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

      2. *-commutative 97.7%

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

      3. mul-1-neg 97.7%

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

   8. Simplified 97.7%

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

   9. Taylor expanded in x around 0 97.7%

      \[\leadsto x + wj \cdot \left(wj \cdot \color{blue}{\left(\left(1 + x \cdot \left(2.5 + -2.6666666666666665 \cdot wj\right)\right) - wj\right)} - 2 \cdot x\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 97.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;wj \leq -0.0112:\\ \;\;\;\;\frac{x}{\left(wj + 1\right) \cdot e^{wj}}\\ \mathbf{else}:\\ \;\;\;\;x + wj \cdot \left(wj \cdot \left(\left(1 + x \cdot \left(2.5 + wj \cdot -2.6666666666666665\right)\right) - wj\right) - x \cdot 2\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 96.4% accurate, 14.9× speedup

Math

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

FPCore

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

C

double code(double wj, double x) {
	return x + (wj * ((wj * ((1.0 + (x * (2.5 + (wj * -2.6666666666666665)))) - 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 * ((1.0d0 + (x * (2.5d0 + (wj * (-2.6666666666666665d0))))) - wj)) - (x * 2.0d0)))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 76.8%

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

   1. distribute-rgt1-in 77.9%

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

   2. associate-/l/ 77.9%

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

   3. div-sub 76.8%

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

   4. associate-/l* 76.8%

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

   5. *-inverses 78.3%

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

   6. *-rgt-identity 78.3%

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

3. Simplified 78.3%

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

5. Taylor expanded in wj around 0 95.3%

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

6. Taylor expanded in x around 0 95.6%

   \[\leadsto x + wj \cdot \left(\color{blue}{\left(wj \cdot \left(x \cdot \left(2.5 + -2.6666666666666665 \cdot wj\right)\right) + wj \cdot \left(1 + -1 \cdot wj\right)\right)} - 2 \cdot x\right) \]
7. Step-by-step derivation

   1. distribute-lft-out 95.6%

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

   2. *-commutative 95.6%

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

   3. mul-1-neg 95.6%

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

8. Simplified 95.6%

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

9. Taylor expanded in x around 0 95.6%

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

10. Final simplification 95.6%

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

Alternative 7: 96.2% 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 76.8%

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

   1. distribute-rgt1-in 77.9%

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

   2. associate-/l/ 77.9%

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

   3. div-sub 76.8%

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

   4. associate-/l* 76.8%

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

   5. *-inverses 78.3%

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

   6. *-rgt-identity 78.3%

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

3. Simplified 78.3%

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

5. Taylor expanded in wj around 0 95.3%

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

6. Taylor expanded in x around 0 95.3%

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

   1. mul-1-neg 95.3%

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

   2. unsub-neg 95.3%

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

8. Simplified 95.3%

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

9. Final simplification 95.3%

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

Alternative 8: 95.7% accurate, 24.1× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 76.8%

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

   1. distribute-rgt1-in 77.9%

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

   2. associate-/l/ 77.9%

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

   3. div-sub 76.8%

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

   4. associate-/l* 76.8%

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

   5. *-inverses 78.3%

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

   6. *-rgt-identity 78.3%

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

3. Simplified 78.3%

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

5. Taylor expanded in wj around 0 95.3%

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

6. Taylor expanded in x around 0 95.3%

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

   1. mul-1-neg 95.3%

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

   2. unsub-neg 95.3%

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

8. Simplified 95.3%

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

   1. sub-neg 95.3%

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

   2. +-commutative 95.3%

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

   3. add-sqr-sqrt 51.6%

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

   4. sqrt-unprod 95.0%

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

   5. sqr-neg 95.0%

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

   6. sqrt-unprod 43.4%

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

   7. add-sqr-sqrt 94.7%

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

   8. pow1 94.7%

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

10. Applied egg-rr 94.7%

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

    1. unpow1 94.7%

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

12. Simplified 94.7%

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

13. Final simplification 94.7%

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

Alternative 9: 85.7% accurate, 26.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;wj \leq 3.5 \cdot 10^{-10}:\\ \;\;\;\;x + -2 \cdot \left(wj \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;wj + \frac{wj}{-1 - wj}\\ \end{array} \end{array} \]

FPCore

(FPCore (wj x)
 :precision binary64
 (if (<= wj 3.5e-10) (+ x (* -2.0 (* wj x))) (+ wj (/ wj (- -1.0 wj)))))

C

double code(double wj, double x) {
	double tmp;
	if (wj <= 3.5e-10) {
		tmp = x + (-2.0 * (wj * x));
	} else {
		tmp = wj + (wj / (-1.0 - wj));
	}
	return tmp;
}

Fortran

real(8) function code(wj, x)
    real(8), intent (in) :: wj
    real(8), intent (in) :: x
    real(8) :: tmp
    if (wj <= 3.5d-10) then
        tmp = x + ((-2.0d0) * (wj * x))
    else
        tmp = wj + (wj / ((-1.0d0) - wj))
    end if
    code = tmp
end function

Java

public static double code(double wj, double x) {
	double tmp;
	if (wj <= 3.5e-10) {
		tmp = x + (-2.0 * (wj * x));
	} else {
		tmp = wj + (wj / (-1.0 - wj));
	}
	return tmp;
}

Python

def code(wj, x):
	tmp = 0
	if wj <= 3.5e-10:
		tmp = x + (-2.0 * (wj * x))
	else:
		tmp = wj + (wj / (-1.0 - wj))
	return tmp

Julia

function code(wj, x)
	tmp = 0.0
	if (wj <= 3.5e-10)
		tmp = Float64(x + Float64(-2.0 * Float64(wj * x)));
	else
		tmp = Float64(wj + Float64(wj / Float64(-1.0 - wj)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(wj, x)
	tmp = 0.0;
	if (wj <= 3.5e-10)
		tmp = x + (-2.0 * (wj * x));
	else
		tmp = wj + (wj / (-1.0 - wj));
	end
	tmp_2 = tmp;
end

Wolfram

code[wj_, x_] := If[LessEqual[wj, 3.5e-10], N[(x + N[(-2.0 * N[(wj * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(wj + N[(wj / N[(-1.0 - wj), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if wj < 3.4999999999999998e-10

   1. Initial program 76.8%

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

      1. distribute-rgt1-in 78.0%

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

      2. associate-/l/ 78.0%

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

      3. div-sub 76.8%

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

      4. associate-/l* 76.8%

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

      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 81.4%

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

      1. *-commutative 81.4%

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

   7. Simplified 81.4%

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

   if 3.4999999999999998e-10 < wj

   1. Initial program 76.9%

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

      1. distribute-rgt1-in 76.7%

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

      2. associate-/l/ 76.6%

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

      3. div-sub 76.6%

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

      4. associate-/l* 76.6%

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

      5. *-inverses 89.1%

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

      6. *-rgt-identity 89.1%

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

   3. Simplified 89.1%

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

   5. Taylor expanded in x around 0 64.9%

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

      1. +-commutative 88.9%

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

   7. Simplified 64.9%

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

4. Final simplification 80.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;wj \leq 3.5 \cdot 10^{-10}:\\ \;\;\;\;x + -2 \cdot \left(wj \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;wj + \frac{wj}{-1 - wj}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 95.7% 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 76.8%

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

   1. distribute-rgt1-in 77.9%

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

   2. associate-/l/ 77.9%

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

   3. div-sub 76.8%

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

   4. associate-/l* 76.8%

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

   5. *-inverses 78.3%

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

   6. *-rgt-identity 78.3%

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

3. Simplified 78.3%

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

5. Taylor expanded in wj around 0 95.3%

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

6. Taylor expanded in x around 0 95.3%

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

   1. mul-1-neg 95.3%

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

   2. unsub-neg 95.3%

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

8. Simplified 95.3%

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

9. Taylor expanded in wj around 0 94.7%

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

10. Final simplification 94.7%

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

Alternative 11: 84.3% 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 76.8%

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

   1. distribute-rgt1-in 77.9%

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

   2. associate-/l/ 77.9%

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

   3. div-sub 76.8%

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

   4. associate-/l* 76.8%

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

   5. *-inverses 78.3%

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

   6. *-rgt-identity 78.3%

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

3. Simplified 78.3%

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

5. Taylor expanded in wj around 0 79.1%

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

   1. *-commutative 79.1%

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

7. Simplified 79.1%

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

8. Final simplification 79.1%

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

Alternative 12: 83.9% accurate, 313.0× speedup

Math

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

FPCore

(FPCore (wj x) :precision binary64 x)

C

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

Fortran

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

Java

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

Python

def code(wj, x):
	return x

Julia

function code(wj, x)
	return x
end

MATLAB

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

Wolfram

code[wj_, x_] := x

Derivation

1. Initial program 76.8%

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

   1. distribute-rgt1-in 77.9%

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

   2. associate-/l/ 77.9%

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

   3. div-sub 76.8%

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

   4. associate-/l* 76.8%

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

   5. *-inverses 78.3%

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

   6. *-rgt-identity 78.3%

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

3. Simplified 78.3%

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

5. Taylor expanded in wj around 0 78.3%

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

Alternative 13: 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 76.8%

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

   1. distribute-rgt1-in 77.9%

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

   2. associate-/l/ 77.9%

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

   3. div-sub 76.8%

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

   4. associate-/l* 76.8%

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

   5. *-inverses 78.3%

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

   6. *-rgt-identity 78.3%

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

3. Simplified 78.3%

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

5. Taylor expanded in wj around inf 4.4%

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

Developer target: 78.5% accurate, 1.5× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Reproduce

herbie shell --seed 2024096 
(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))))))