Jmat.Real.lambertw, newton loop step

Percentage Accurate: 77.8% → 99.7%

Time: 14.0s

Alternatives: 17

Speedup: 313.0×

• could not determine a ground truth

  1. wj = 9.55044033361264e+197
  2. x = -4.796031333312922e-91

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: 77.8% 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, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x \cdot -4 + x \cdot 1.5\\ t_1 := wj - \frac{x}{e^{wj}}\\ t_2 := wj + \frac{t\_1}{-1 - wj}\\ \mathbf{if}\;wj \leq -2.2 \cdot 10^{-6}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;wj \leq 3.3 \cdot 10^{-5}:\\ \;\;\;\;x + wj \cdot \left(wj \cdot \left(\left(1 + wj \cdot \left(-1 - \left(x \cdot -3 + \left(-2 \cdot t\_0 + x \cdot 0.6666666666666666\right)\right)\right)\right) - t\_0\right) - x \cdot 2\right)\\ \mathbf{else}:\\ \;\;\;\;t\_2 + \mathsf{fma}\left(\frac{-1}{wj + 1}, t\_1, \frac{t\_1}{wj + 1}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (wj x)
 :precision binary64
 (let* ((t_0 (+ (* x -4.0) (* x 1.5)))
        (t_1 (- wj (/ x (exp wj))))
        (t_2 (+ wj (/ t_1 (- -1.0 wj)))))
   (if (<= wj -2.2e-6)
     t_2
     (if (<= wj 3.3e-5)
       (+
        x
        (*
         wj
         (-
          (*
           wj
           (-
            (+
             1.0
             (*
              wj
              (-
               -1.0
               (+ (* x -3.0) (+ (* -2.0 t_0) (* x 0.6666666666666666))))))
            t_0))
          (* x 2.0))))
       (+ t_2 (fma (/ -1.0 (+ wj 1.0)) t_1 (/ t_1 (+ wj 1.0))))))))

C

double code(double wj, double x) {
	double t_0 = (x * -4.0) + (x * 1.5);
	double t_1 = wj - (x / exp(wj));
	double t_2 = wj + (t_1 / (-1.0 - wj));
	double tmp;
	if (wj <= -2.2e-6) {
		tmp = t_2;
	} else if (wj <= 3.3e-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)));
	} else {
		tmp = t_2 + fma((-1.0 / (wj + 1.0)), t_1, (t_1 / (wj + 1.0)));
	}
	return tmp;
}

Julia

function code(wj, x)
	t_0 = Float64(Float64(x * -4.0) + Float64(x * 1.5))
	t_1 = Float64(wj - Float64(x / exp(wj)))
	t_2 = Float64(wj + Float64(t_1 / Float64(-1.0 - wj)))
	tmp = 0.0
	if (wj <= -2.2e-6)
		tmp = t_2;
	elseif (wj <= 3.3e-5)
		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))));
	else
		tmp = Float64(t_2 + fma(Float64(-1.0 / Float64(wj + 1.0)), t_1, Float64(t_1 / Float64(wj + 1.0))));
	end
	return tmp
end

Wolfram

code[wj_, x_] := Block[{t$95$0 = N[(N[(x * -4.0), $MachinePrecision] + N[(x * 1.5), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(wj - N[(x / N[Exp[wj], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(wj + N[(t$95$1 / N[(-1.0 - wj), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[wj, -2.2e-6], t$95$2, If[LessEqual[wj, 3.3e-5], 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], N[(t$95$2 + N[(N[(-1.0 / N[(wj + 1.0), $MachinePrecision]), $MachinePrecision] * t$95$1 + N[(t$95$1 / N[(wj + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if wj < -2.2000000000000001e-6

   1. Initial program 33.3%

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

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

      3. associate-/r* 100.0%

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

      4. div-sub 33.3%

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

      5. associate-/l* 33.3%

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

      6. *-inverses 100.0%

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

      7. *-rgt-identity 100.0%

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

   3. Simplified 100.0%

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

   if -2.2000000000000001e-6 < wj < 3.3000000000000003e-5

   1. Initial program 77.8%

      \[wj - \frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}} \]
   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. *-commutative 77.9%

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

      3. associate-/r* 77.9%

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

      4. div-sub 77.9%

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

      5. associate-/l* 77.9%

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

      6. *-inverses 77.9%

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

      7. *-rgt-identity 77.9%

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

   3. Simplified 77.9%

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

   5. Taylor expanded in wj around 0 99.9%

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

   if 3.3000000000000003e-5 < wj

   1. Initial program 67.7%

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

      1. distribute-rgt1-in 67.7%

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

      2. *-commutative 67.7%

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

      3. associate-/r* 67.7%

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

      4. div-sub 67.7%

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

      5. associate-/l* 67.7%

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

      6. *-inverses 97.7%

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

      7. *-rgt-identity 97.7%

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

   3. Simplified 97.7%

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

      1. *-un-lft-identity 97.7%

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

      2. div-inv 97.7%

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

      3. prod-diff 98.6%

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

      4. associate-/r/ 98.5%

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

      5. clear-num 98.6%

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

      6. fmm-def 98.6%

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

      7. *-un-lft-identity 98.6%

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

      8. associate-/r/ 98.8%

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

      9. clear-num 98.6%

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

   6. Applied egg-rr 98.6%

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

      1. distribute-neg-frac 98.6%

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

      2. metadata-eval 98.6%

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

   8. Simplified 98.6%

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

4. Final simplification 99.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;wj \leq -2.2 \cdot 10^{-6}:\\ \;\;\;\;wj + \frac{wj - \frac{x}{e^{wj}}}{-1 - wj}\\ \mathbf{elif}\;wj \leq 3.3 \cdot 10^{-5}:\\ \;\;\;\;x + wj \cdot \left(wj \cdot \left(\left(1 + wj \cdot \left(-1 - \left(x \cdot -3 + \left(-2 \cdot \left(x \cdot -4 + x \cdot 1.5\right) + x \cdot 0.6666666666666666\right)\right)\right)\right) - \left(x \cdot -4 + x \cdot 1.5\right)\right) - x \cdot 2\right)\\ \mathbf{else}:\\ \;\;\;\;\left(wj + \frac{wj - \frac{x}{e^{wj}}}{-1 - wj}\right) + \mathsf{fma}\left(\frac{-1}{wj + 1}, wj - \frac{x}{e^{wj}}, \frac{wj - \frac{x}{e^{wj}}}{wj + 1}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 99.7% accurate, 2.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x \cdot -4 + x \cdot 1.5\\ \mathbf{if}\;wj \leq -2.2 \cdot 10^{-6}:\\ \;\;\;\;wj + \frac{wj - \frac{x}{e^{wj}}}{-1 - wj}\\ \mathbf{elif}\;wj \leq 3.6 \cdot 10^{-6}:\\ \;\;\;\;x + wj \cdot \left(wj \cdot \left(\left(1 + wj \cdot \left(-1 - \left(x \cdot -3 + \left(-2 \cdot t\_0 + x \cdot 0.6666666666666666\right)\right)\right)\right) - t\_0\right) - x \cdot 2\right)\\ \mathbf{else}:\\ \;\;\;\;wj + \frac{wj - \frac{x}{{e}^{wj}}}{-1 - wj}\\ \end{array} \end{array} \]

FPCore

(FPCore (wj x)
 :precision binary64
 (let* ((t_0 (+ (* x -4.0) (* x 1.5))))
   (if (<= wj -2.2e-6)
     (+ wj (/ (- wj (/ x (exp wj))) (- -1.0 wj)))
     (if (<= wj 3.6e-6)
       (+
        x
        (*
         wj
         (-
          (*
           wj
           (-
            (+
             1.0
             (*
              wj
              (-
               -1.0
               (+ (* x -3.0) (+ (* -2.0 t_0) (* x 0.6666666666666666))))))
            t_0))
          (* x 2.0))))
       (+ wj (/ (- wj (/ x (pow E wj))) (- -1.0 wj)))))))

C

double code(double wj, double x) {
	double t_0 = (x * -4.0) + (x * 1.5);
	double tmp;
	if (wj <= -2.2e-6) {
		tmp = wj + ((wj - (x / exp(wj))) / (-1.0 - wj));
	} else if (wj <= 3.6e-6) {
		tmp = x + (wj * ((wj * ((1.0 + (wj * (-1.0 - ((x * -3.0) + ((-2.0 * t_0) + (x * 0.6666666666666666)))))) - t_0)) - (x * 2.0)));
	} else {
		tmp = wj + ((wj - (x / pow(((double) M_E), wj))) / (-1.0 - wj));
	}
	return tmp;
}

Java

public static double code(double wj, double x) {
	double t_0 = (x * -4.0) + (x * 1.5);
	double tmp;
	if (wj <= -2.2e-6) {
		tmp = wj + ((wj - (x / Math.exp(wj))) / (-1.0 - wj));
	} else if (wj <= 3.6e-6) {
		tmp = x + (wj * ((wj * ((1.0 + (wj * (-1.0 - ((x * -3.0) + ((-2.0 * t_0) + (x * 0.6666666666666666)))))) - t_0)) - (x * 2.0)));
	} else {
		tmp = wj + ((wj - (x / Math.pow(Math.E, wj))) / (-1.0 - wj));
	}
	return tmp;
}

Python

def code(wj, x):
	t_0 = (x * -4.0) + (x * 1.5)
	tmp = 0
	if wj <= -2.2e-6:
		tmp = wj + ((wj - (x / math.exp(wj))) / (-1.0 - wj))
	elif wj <= 3.6e-6:
		tmp = x + (wj * ((wj * ((1.0 + (wj * (-1.0 - ((x * -3.0) + ((-2.0 * t_0) + (x * 0.6666666666666666)))))) - t_0)) - (x * 2.0)))
	else:
		tmp = wj + ((wj - (x / math.pow(math.e, wj))) / (-1.0 - wj))
	return tmp

Julia

function code(wj, x)
	t_0 = Float64(Float64(x * -4.0) + Float64(x * 1.5))
	tmp = 0.0
	if (wj <= -2.2e-6)
		tmp = Float64(wj + Float64(Float64(wj - Float64(x / exp(wj))) / Float64(-1.0 - wj)));
	elseif (wj <= 3.6e-6)
		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))));
	else
		tmp = Float64(wj + Float64(Float64(wj - Float64(x / (exp(1) ^ wj))) / Float64(-1.0 - wj)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(wj, x)
	t_0 = (x * -4.0) + (x * 1.5);
	tmp = 0.0;
	if (wj <= -2.2e-6)
		tmp = wj + ((wj - (x / exp(wj))) / (-1.0 - wj));
	elseif (wj <= 3.6e-6)
		tmp = x + (wj * ((wj * ((1.0 + (wj * (-1.0 - ((x * -3.0) + ((-2.0 * t_0) + (x * 0.6666666666666666)))))) - t_0)) - (x * 2.0)));
	else
		tmp = wj + ((wj - (x / (2.71828182845904523536 ^ wj))) / (-1.0 - wj));
	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, -2.2e-6], N[(wj + N[(N[(wj - N[(x / N[Exp[wj], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(-1.0 - wj), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[wj, 3.6e-6], 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], N[(wj + N[(N[(wj - N[(x / N[Power[E, wj], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(-1.0 - wj), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if wj < -2.2000000000000001e-6

   1. Initial program 33.3%

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

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

      3. associate-/r* 100.0%

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

      4. div-sub 33.3%

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

      5. associate-/l* 33.3%

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

      6. *-inverses 100.0%

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

      7. *-rgt-identity 100.0%

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

   3. Simplified 100.0%

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

   if -2.2000000000000001e-6 < wj < 3.59999999999999984e-6

   1. Initial program 77.8%

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

      1. distribute-rgt1-in 77.8%

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

      2. *-commutative 77.8%

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

      3. associate-/r* 77.8%

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

      4. div-sub 77.8%

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

      5. associate-/l* 77.8%

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

      6. *-inverses 77.8%

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

      7. *-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 99.9%

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

   if 3.59999999999999984e-6 < wj

   1. Initial program 70.5%

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

      1. distribute-rgt1-in 70.5%

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

      2. *-commutative 70.5%

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

      3. associate-/r* 70.7%

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

      4. div-sub 70.7%

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

      5. associate-/l* 70.7%

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

      6. *-inverses 98.0%

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

      7. *-rgt-identity 98.0%

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

   3. Simplified 98.0%

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

      1. *-un-lft-identity 98.0%

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

      2. exp-prod 98.1%

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

   6. Applied egg-rr 98.1%

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

      1. exp-1-e 98.1%

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

   8. Simplified 98.1%

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

4. Final simplification 99.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;wj \leq -2.2 \cdot 10^{-6}:\\ \;\;\;\;wj + \frac{wj - \frac{x}{e^{wj}}}{-1 - wj}\\ \mathbf{elif}\;wj \leq 3.6 \cdot 10^{-6}:\\ \;\;\;\;x + wj \cdot \left(wj \cdot \left(\left(1 + wj \cdot \left(-1 - \left(x \cdot -3 + \left(-2 \cdot \left(x \cdot -4 + x \cdot 1.5\right) + x \cdot 0.6666666666666666\right)\right)\right)\right) - \left(x \cdot -4 + x \cdot 1.5\right)\right) - x \cdot 2\right)\\ \mathbf{else}:\\ \;\;\;\;wj + \frac{wj - \frac{x}{{e}^{wj}}}{-1 - wj}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 99.7% accurate, 2.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x \cdot -4 + x \cdot 1.5\\ \mathbf{if}\;wj \leq -2.2 \cdot 10^{-6} \lor \neg \left(wj \leq 3.6 \cdot 10^{-6}\right):\\ \;\;\;\;wj + \frac{wj - \frac{x}{e^{wj}}}{-1 - wj}\\ \mathbf{else}:\\ \;\;\;\;x + wj \cdot \left(wj \cdot \left(\left(1 + wj \cdot \left(-1 - \left(x \cdot -3 + \left(-2 \cdot t\_0 + x \cdot 0.6666666666666666\right)\right)\right)\right) - t\_0\right) - x \cdot 2\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (wj x)
 :precision binary64
 (let* ((t_0 (+ (* x -4.0) (* x 1.5))))
   (if (or (<= wj -2.2e-6) (not (<= wj 3.6e-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 <= -2.2e-6) || !(wj <= 3.6e-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 <= (-2.2d-6)) .or. (.not. (wj <= 3.6d-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 <= -2.2e-6) || !(wj <= 3.6e-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 <= -2.2e-6) or not (wj <= 3.6e-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 <= -2.2e-6) || !(wj <= 3.6e-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 <= -2.2e-6) || ~((wj <= 3.6e-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[Or[LessEqual[wj, -2.2e-6], N[Not[LessEqual[wj, 3.6e-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[(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 < -2.2000000000000001e-6 or 3.59999999999999984e-6 < wj

   1. Initial program 57.4%

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

      1. distribute-rgt1-in 80.9%

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

      2. *-commutative 80.9%

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

      3. associate-/r* 81.0%

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

      4. div-sub 57.5%

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

      5. associate-/l* 57.5%

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

      6. *-inverses 98.7%

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

      7. *-rgt-identity 98.7%

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

   3. Simplified 98.7%

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

   if -2.2000000000000001e-6 < wj < 3.59999999999999984e-6

   1. Initial program 77.8%

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

      1. distribute-rgt1-in 77.8%

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

      2. *-commutative 77.8%

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

      3. associate-/r* 77.8%

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

      4. div-sub 77.8%

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

      5. associate-/l* 77.8%

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

      6. *-inverses 77.8%

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

      7. *-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 99.9%

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

4. Final simplification 99.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;wj \leq -2.2 \cdot 10^{-6} \lor \neg \left(wj \leq 3.6 \cdot 10^{-6}\right):\\ \;\;\;\;wj + \frac{wj - \frac{x}{e^{wj}}}{-1 - wj}\\ \mathbf{else}:\\ \;\;\;\;x + wj \cdot \left(wj \cdot \left(\left(1 + wj \cdot \left(-1 - \left(x \cdot -3 + \left(-2 \cdot \left(x \cdot -4 + x \cdot 1.5\right) + x \cdot 0.6666666666666666\right)\right)\right)\right) - \left(x \cdot -4 + x \cdot 1.5\right)\right) - x \cdot 2\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 97.6% accurate, 2.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x \cdot -4 + x \cdot 1.5\\ \mathbf{if}\;wj \leq -8.2 \cdot 10^{-5}:\\ \;\;\;\;wj - \frac{\frac{x}{e^{wj}}}{-1 - wj}\\ \mathbf{else}:\\ \;\;\;\;x + wj \cdot \left(wj \cdot \left(\left(1 + wj \cdot \left(-1 - \left(x \cdot -3 + \left(-2 \cdot t\_0 + x \cdot 0.6666666666666666\right)\right)\right)\right) - t\_0\right) - x \cdot 2\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (wj x)
 :precision binary64
 (let* ((t_0 (+ (* x -4.0) (* x 1.5))))
   (if (<= wj -8.2e-5)
     (- 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 <= -8.2e-5) {
		tmp = 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 <= (-8.2d-5)) then
        tmp = 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 <= -8.2e-5) {
		tmp = 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 <= -8.2e-5:
		tmp = 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 <= -8.2e-5)
		tmp = Float64(wj - Float64(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 <= -8.2e-5)
		tmp = 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, -8.2e-5], N[(wj - N[(N[(x / N[Exp[wj], $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 < -8.20000000000000009e-5

   1. Initial program 33.3%

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

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

      3. associate-/r* 100.0%

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

      4. div-sub 33.3%

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

      5. associate-/l* 33.3%

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

      6. *-inverses 100.0%

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

      7. *-rgt-identity 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in x around inf 87.2%

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

      1. associate-*r/ 87.2%

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

      2. neg-mul-1 87.2%

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

   7. Simplified 87.2%

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

   if -8.20000000000000009e-5 < 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. *-commutative 77.4%

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

      3. associate-/r* 77.5%

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

      4. div-sub 77.5%

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

      5. associate-/l* 77.5%

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

      6. *-inverses 78.7%

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

      7. *-rgt-identity 78.7%

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

   3. Simplified 78.7%

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

   5. Taylor expanded in wj around 0 97.4%

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

4. Final simplification 97.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;wj \leq -8.2 \cdot 10^{-5}:\\ \;\;\;\;wj - \frac{\frac{x}{e^{wj}}}{-1 - wj}\\ \mathbf{else}:\\ \;\;\;\;x + wj \cdot \left(wj \cdot \left(\left(1 + wj \cdot \left(-1 - \left(x \cdot -3 + \left(-2 \cdot \left(x \cdot -4 + x \cdot 1.5\right) + x \cdot 0.6666666666666666\right)\right)\right)\right) - \left(x \cdot -4 + x \cdot 1.5\right)\right) - x \cdot 2\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 96.3% 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 76.4%

   \[wj - \frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}} \]
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. *-commutative 78.0%

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

   3. associate-/r* 78.0%

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

   4. div-sub 76.5%

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

   5. associate-/l* 76.5%

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

   6. *-inverses 79.2%

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

   7. *-rgt-identity 79.2%

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

3. Simplified 79.2%

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

5. Taylor expanded in wj around 0 95.6%

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

6. Final simplification 95.6%

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

Alternative 6: 97.2% accurate, 9.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;wj \leq 2.2 \cdot 10^{-6}:\\ \;\;\;\;x + wj \cdot \left(wj \cdot \left(1 - wj\right) - x \cdot 2\right)\\ \mathbf{else}:\\ \;\;\;\;wj + \frac{wj + \frac{1}{\frac{wj \cdot \left(-1 - wj \cdot \left(0.5 + wj \cdot 0.16666666666666666\right)\right)}{x} + \frac{-1}{x}}}{-1 - wj}\\ \end{array} \end{array} \]

FPCore

(FPCore (wj x)
 :precision binary64
 (if (<= wj 2.2e-6)
   (+ x (* wj (- (* wj (- 1.0 wj)) (* x 2.0))))
   (+
    wj
    (/
     (+
      wj
      (/
       1.0
       (+
        (/ (* wj (- -1.0 (* wj (+ 0.5 (* wj 0.16666666666666666))))) x)
        (/ -1.0 x))))
     (- -1.0 wj)))))

C

double code(double wj, double x) {
	double tmp;
	if (wj <= 2.2e-6) {
		tmp = x + (wj * ((wj * (1.0 - wj)) - (x * 2.0)));
	} else {
		tmp = wj + ((wj + (1.0 / (((wj * (-1.0 - (wj * (0.5 + (wj * 0.16666666666666666))))) / x) + (-1.0 / x)))) / (-1.0 - wj));
	}
	return tmp;
}

Fortran

real(8) function code(wj, x)
    real(8), intent (in) :: wj
    real(8), intent (in) :: x
    real(8) :: tmp
    if (wj <= 2.2d-6) then
        tmp = x + (wj * ((wj * (1.0d0 - wj)) - (x * 2.0d0)))
    else
        tmp = wj + ((wj + (1.0d0 / (((wj * ((-1.0d0) - (wj * (0.5d0 + (wj * 0.16666666666666666d0))))) / x) + ((-1.0d0) / x)))) / ((-1.0d0) - wj))
    end if
    code = tmp
end function

Java

public static double code(double wj, double x) {
	double tmp;
	if (wj <= 2.2e-6) {
		tmp = x + (wj * ((wj * (1.0 - wj)) - (x * 2.0)));
	} else {
		tmp = wj + ((wj + (1.0 / (((wj * (-1.0 - (wj * (0.5 + (wj * 0.16666666666666666))))) / x) + (-1.0 / x)))) / (-1.0 - wj));
	}
	return tmp;
}

Python

def code(wj, x):
	tmp = 0
	if wj <= 2.2e-6:
		tmp = x + (wj * ((wj * (1.0 - wj)) - (x * 2.0)))
	else:
		tmp = wj + ((wj + (1.0 / (((wj * (-1.0 - (wj * (0.5 + (wj * 0.16666666666666666))))) / x) + (-1.0 / x)))) / (-1.0 - wj))
	return tmp

Julia

function code(wj, x)
	tmp = 0.0
	if (wj <= 2.2e-6)
		tmp = Float64(x + Float64(wj * Float64(Float64(wj * Float64(1.0 - wj)) - Float64(x * 2.0))));
	else
		tmp = Float64(wj + Float64(Float64(wj + Float64(1.0 / Float64(Float64(Float64(wj * Float64(-1.0 - Float64(wj * Float64(0.5 + Float64(wj * 0.16666666666666666))))) / x) + Float64(-1.0 / x)))) / Float64(-1.0 - wj)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(wj, x)
	tmp = 0.0;
	if (wj <= 2.2e-6)
		tmp = x + (wj * ((wj * (1.0 - wj)) - (x * 2.0)));
	else
		tmp = wj + ((wj + (1.0 / (((wj * (-1.0 - (wj * (0.5 + (wj * 0.16666666666666666))))) / x) + (-1.0 / x)))) / (-1.0 - wj));
	end
	tmp_2 = tmp;
end

Wolfram

code[wj_, x_] := If[LessEqual[wj, 2.2e-6], N[(x + N[(wj * N[(N[(wj * N[(1.0 - wj), $MachinePrecision]), $MachinePrecision] - N[(x * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(wj + N[(N[(wj + N[(1.0 / N[(N[(N[(wj * N[(-1.0 - N[(wj * N[(0.5 + N[(wj * 0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision] + N[(-1.0 / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(-1.0 - wj), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if wj < 2.2000000000000001e-6

   1. Initial program 76.7%

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

      1. distribute-rgt1-in 78.3%

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

      2. *-commutative 78.3%

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

      3. associate-/r* 78.4%

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

      4. div-sub 76.7%

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

      5. associate-/l* 76.7%

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

      6. *-inverses 78.4%

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

      7. *-rgt-identity 78.4%

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

   3. Simplified 78.4%

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

   5. Taylor expanded in wj around 0 97.9%

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

   6. Taylor expanded in x around 0 97.6%

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

      1. neg-mul-1 97.6%

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

      2. sub-neg 97.6%

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

   8. Simplified 97.6%

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

   if 2.2000000000000001e-6 < wj

   1. Initial program 70.5%

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

      1. distribute-rgt1-in 70.5%

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

      2. *-commutative 70.5%

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

      3. associate-/r* 70.7%

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

      4. div-sub 70.7%

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

      5. associate-/l* 70.7%

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

      6. *-inverses 98.0%

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

      7. *-rgt-identity 98.0%

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

   3. Simplified 98.0%

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

      1. clear-num 98.1%

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

      2. inv-pow 98.1%

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

   6. Applied egg-rr 98.1%

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

      1. unpow-1 98.1%

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

   8. Simplified 98.1%

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

   9. Taylor expanded in wj around 0 76.3%

      \[\leadsto wj - \frac{wj - \frac{1}{\color{blue}{wj \cdot \left(wj \cdot \left(0.16666666666666666 \cdot \frac{wj}{x} + 0.5 \cdot \frac{1}{x}\right) + \frac{1}{x}\right) + \frac{1}{x}}}}{wj + 1} \]

   10. Taylor expanded in x around 0 76.3%

       \[\leadsto wj - \frac{wj - \frac{1}{\color{blue}{\frac{wj \cdot \left(1 + wj \cdot \left(0.5 + 0.16666666666666666 \cdot wj\right)\right)}{x}} + \frac{1}{x}}}{wj + 1} \]
3. Recombined 2 regimes into one program.

4. Final simplification 96.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;wj \leq 2.2 \cdot 10^{-6}:\\ \;\;\;\;x + wj \cdot \left(wj \cdot \left(1 - wj\right) - x \cdot 2\right)\\ \mathbf{else}:\\ \;\;\;\;wj + \frac{wj + \frac{1}{\frac{wj \cdot \left(-1 - wj \cdot \left(0.5 + wj \cdot 0.16666666666666666\right)\right)}{x} + \frac{-1}{x}}}{-1 - wj}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 97.2% accurate, 11.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;wj \leq 2.2 \cdot 10^{-6}:\\ \;\;\;\;x + wj \cdot \left(wj \cdot \left(1 - wj\right) - x \cdot 2\right)\\ \mathbf{else}:\\ \;\;\;\;wj + \frac{wj + \frac{x}{-1 + wj \cdot \left(-1 - wj \cdot \left(0.5 + wj \cdot 0.16666666666666666\right)\right)}}{-1 - wj}\\ \end{array} \end{array} \]

FPCore

(FPCore (wj x)
 :precision binary64
 (if (<= wj 2.2e-6)
   (+ x (* wj (- (* wj (- 1.0 wj)) (* x 2.0))))
   (+
    wj
    (/
     (+
      wj
      (/ x (+ -1.0 (* wj (- -1.0 (* wj (+ 0.5 (* wj 0.16666666666666666))))))))
     (- -1.0 wj)))))

C

double code(double wj, double x) {
	double tmp;
	if (wj <= 2.2e-6) {
		tmp = x + (wj * ((wj * (1.0 - wj)) - (x * 2.0)));
	} else {
		tmp = wj + ((wj + (x / (-1.0 + (wj * (-1.0 - (wj * (0.5 + (wj * 0.16666666666666666)))))))) / (-1.0 - wj));
	}
	return tmp;
}

Fortran

real(8) function code(wj, x)
    real(8), intent (in) :: wj
    real(8), intent (in) :: x
    real(8) :: tmp
    if (wj <= 2.2d-6) then
        tmp = x + (wj * ((wj * (1.0d0 - wj)) - (x * 2.0d0)))
    else
        tmp = wj + ((wj + (x / ((-1.0d0) + (wj * ((-1.0d0) - (wj * (0.5d0 + (wj * 0.16666666666666666d0)))))))) / ((-1.0d0) - wj))
    end if
    code = tmp
end function

Java

public static double code(double wj, double x) {
	double tmp;
	if (wj <= 2.2e-6) {
		tmp = x + (wj * ((wj * (1.0 - wj)) - (x * 2.0)));
	} else {
		tmp = wj + ((wj + (x / (-1.0 + (wj * (-1.0 - (wj * (0.5 + (wj * 0.16666666666666666)))))))) / (-1.0 - wj));
	}
	return tmp;
}

Python

def code(wj, x):
	tmp = 0
	if wj <= 2.2e-6:
		tmp = x + (wj * ((wj * (1.0 - wj)) - (x * 2.0)))
	else:
		tmp = wj + ((wj + (x / (-1.0 + (wj * (-1.0 - (wj * (0.5 + (wj * 0.16666666666666666)))))))) / (-1.0 - wj))
	return tmp

Julia

function code(wj, x)
	tmp = 0.0
	if (wj <= 2.2e-6)
		tmp = Float64(x + Float64(wj * Float64(Float64(wj * Float64(1.0 - wj)) - Float64(x * 2.0))));
	else
		tmp = Float64(wj + Float64(Float64(wj + Float64(x / Float64(-1.0 + Float64(wj * Float64(-1.0 - Float64(wj * Float64(0.5 + Float64(wj * 0.16666666666666666)))))))) / Float64(-1.0 - wj)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(wj, x)
	tmp = 0.0;
	if (wj <= 2.2e-6)
		tmp = x + (wj * ((wj * (1.0 - wj)) - (x * 2.0)));
	else
		tmp = wj + ((wj + (x / (-1.0 + (wj * (-1.0 - (wj * (0.5 + (wj * 0.16666666666666666)))))))) / (-1.0 - wj));
	end
	tmp_2 = tmp;
end

Wolfram

code[wj_, x_] := If[LessEqual[wj, 2.2e-6], N[(x + N[(wj * N[(N[(wj * N[(1.0 - wj), $MachinePrecision]), $MachinePrecision] - N[(x * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(wj + N[(N[(wj + N[(x / N[(-1.0 + N[(wj * N[(-1.0 - N[(wj * N[(0.5 + N[(wj * 0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(-1.0 - wj), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if wj < 2.2000000000000001e-6

   1. Initial program 76.7%

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

      1. distribute-rgt1-in 78.3%

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

      2. *-commutative 78.3%

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

      3. associate-/r* 78.4%

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

      4. div-sub 76.7%

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

      5. associate-/l* 76.7%

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

      6. *-inverses 78.4%

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

      7. *-rgt-identity 78.4%

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

   3. Simplified 78.4%

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

   5. Taylor expanded in wj around 0 97.9%

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

   6. Taylor expanded in x around 0 97.6%

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

      1. neg-mul-1 97.6%

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

      2. sub-neg 97.6%

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

   8. Simplified 97.6%

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

   if 2.2000000000000001e-6 < wj

   1. Initial program 70.5%

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

      1. distribute-rgt1-in 70.5%

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

      2. *-commutative 70.5%

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

      3. associate-/r* 70.7%

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

      4. div-sub 70.7%

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

      5. associate-/l* 70.7%

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

      6. *-inverses 98.0%

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

      7. *-rgt-identity 98.0%

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

   3. Simplified 98.0%

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

   5. Taylor expanded in wj around 0 76.3%

      \[\leadsto wj - \frac{wj - \frac{x}{\color{blue}{1 + wj \cdot \left(1 + wj \cdot \left(0.5 + 0.16666666666666666 \cdot wj\right)\right)}}}{wj + 1} \]
   6. Step-by-step derivation

      1. *-commutative 76.3%

         \[\leadsto wj - \frac{wj - \frac{x}{1 + wj \cdot \left(1 + wj \cdot \left(0.5 + \color{blue}{wj \cdot 0.16666666666666666}\right)\right)}}{wj + 1} \]

   7. Simplified 76.3%

      \[\leadsto wj - \frac{wj - \frac{x}{\color{blue}{1 + wj \cdot \left(1 + wj \cdot \left(0.5 + wj \cdot 0.16666666666666666\right)\right)}}}{wj + 1} \]
3. Recombined 2 regimes into one program.

4. Final simplification 96.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;wj \leq 2.2 \cdot 10^{-6}:\\ \;\;\;\;x + wj \cdot \left(wj \cdot \left(1 - wj\right) - x \cdot 2\right)\\ \mathbf{else}:\\ \;\;\;\;wj + \frac{wj + \frac{x}{-1 + wj \cdot \left(-1 - wj \cdot \left(0.5 + wj \cdot 0.16666666666666666\right)\right)}}{-1 - wj}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 97.1% accurate, 13.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;wj \leq 2.2 \cdot 10^{-6}:\\ \;\;\;\;x + wj \cdot \left(wj \cdot \left(1 - wj\right) - x \cdot 2\right)\\ \mathbf{else}:\\ \;\;\;\;wj + \frac{wj + \frac{x}{-1 + wj \cdot \left(-1 - wj \cdot 0.5\right)}}{-1 - wj}\\ \end{array} \end{array} \]

FPCore

(FPCore (wj x)
 :precision binary64
 (if (<= wj 2.2e-6)
   (+ x (* wj (- (* wj (- 1.0 wj)) (* x 2.0))))
   (+ wj (/ (+ wj (/ x (+ -1.0 (* wj (- -1.0 (* wj 0.5)))))) (- -1.0 wj)))))

C

double code(double wj, double x) {
	double tmp;
	if (wj <= 2.2e-6) {
		tmp = x + (wj * ((wj * (1.0 - wj)) - (x * 2.0)));
	} else {
		tmp = wj + ((wj + (x / (-1.0 + (wj * (-1.0 - (wj * 0.5)))))) / (-1.0 - wj));
	}
	return tmp;
}

Fortran

real(8) function code(wj, x)
    real(8), intent (in) :: wj
    real(8), intent (in) :: x
    real(8) :: tmp
    if (wj <= 2.2d-6) then
        tmp = x + (wj * ((wj * (1.0d0 - wj)) - (x * 2.0d0)))
    else
        tmp = wj + ((wj + (x / ((-1.0d0) + (wj * ((-1.0d0) - (wj * 0.5d0)))))) / ((-1.0d0) - wj))
    end if
    code = tmp
end function

Java

public static double code(double wj, double x) {
	double tmp;
	if (wj <= 2.2e-6) {
		tmp = x + (wj * ((wj * (1.0 - wj)) - (x * 2.0)));
	} else {
		tmp = wj + ((wj + (x / (-1.0 + (wj * (-1.0 - (wj * 0.5)))))) / (-1.0 - wj));
	}
	return tmp;
}

Python

def code(wj, x):
	tmp = 0
	if wj <= 2.2e-6:
		tmp = x + (wj * ((wj * (1.0 - wj)) - (x * 2.0)))
	else:
		tmp = wj + ((wj + (x / (-1.0 + (wj * (-1.0 - (wj * 0.5)))))) / (-1.0 - wj))
	return tmp

Julia

function code(wj, x)
	tmp = 0.0
	if (wj <= 2.2e-6)
		tmp = Float64(x + Float64(wj * Float64(Float64(wj * Float64(1.0 - wj)) - Float64(x * 2.0))));
	else
		tmp = Float64(wj + Float64(Float64(wj + Float64(x / Float64(-1.0 + Float64(wj * Float64(-1.0 - Float64(wj * 0.5)))))) / Float64(-1.0 - wj)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(wj, x)
	tmp = 0.0;
	if (wj <= 2.2e-6)
		tmp = x + (wj * ((wj * (1.0 - wj)) - (x * 2.0)));
	else
		tmp = wj + ((wj + (x / (-1.0 + (wj * (-1.0 - (wj * 0.5)))))) / (-1.0 - wj));
	end
	tmp_2 = tmp;
end

Wolfram

code[wj_, x_] := If[LessEqual[wj, 2.2e-6], N[(x + N[(wj * N[(N[(wj * N[(1.0 - wj), $MachinePrecision]), $MachinePrecision] - N[(x * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(wj + N[(N[(wj + N[(x / N[(-1.0 + N[(wj * N[(-1.0 - N[(wj * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(-1.0 - wj), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if wj < 2.2000000000000001e-6

   1. Initial program 76.7%

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

      1. distribute-rgt1-in 78.3%

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

      2. *-commutative 78.3%

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

      3. associate-/r* 78.4%

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

      4. div-sub 76.7%

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

      5. associate-/l* 76.7%

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

      6. *-inverses 78.4%

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

      7. *-rgt-identity 78.4%

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

   3. Simplified 78.4%

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

   5. Taylor expanded in wj around 0 97.9%

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

   6. Taylor expanded in x around 0 97.6%

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

      1. neg-mul-1 97.6%

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

      2. sub-neg 97.6%

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

   8. Simplified 97.6%

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

   if 2.2000000000000001e-6 < wj

   1. Initial program 70.5%

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

      1. distribute-rgt1-in 70.5%

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

      2. *-commutative 70.5%

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

      3. associate-/r* 70.7%

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

      4. div-sub 70.7%

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

      5. associate-/l* 70.7%

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

      6. *-inverses 98.0%

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

      7. *-rgt-identity 98.0%

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

   3. Simplified 98.0%

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

   5. Taylor expanded in wj around 0 70.2%

      \[\leadsto wj - \frac{wj - \frac{x}{\color{blue}{1 + wj \cdot \left(1 + 0.5 \cdot wj\right)}}}{wj + 1} \]
   6. Step-by-step derivation

      1. *-commutative 70.2%

         \[\leadsto wj - \frac{wj - \frac{x}{1 + wj \cdot \left(1 + \color{blue}{wj \cdot 0.5}\right)}}{wj + 1} \]

   7. Simplified 70.2%

      \[\leadsto wj - \frac{wj - \frac{x}{\color{blue}{1 + wj \cdot \left(1 + wj \cdot 0.5\right)}}}{wj + 1} \]
3. Recombined 2 regimes into one program.

4. Final simplification 96.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;wj \leq 2.2 \cdot 10^{-6}:\\ \;\;\;\;x + wj \cdot \left(wj \cdot \left(1 - wj\right) - x \cdot 2\right)\\ \mathbf{else}:\\ \;\;\;\;wj + \frac{wj + \frac{x}{-1 + wj \cdot \left(-1 - wj \cdot 0.5\right)}}{-1 - wj}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 96.0% accurate, 20.9× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 76.4%

   \[wj - \frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}} \]
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. *-commutative 78.0%

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

   3. associate-/r* 78.0%

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

   4. div-sub 76.5%

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

   5. associate-/l* 76.5%

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

   6. *-inverses 79.2%

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

   7. *-rgt-identity 79.2%

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

3. Simplified 79.2%

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

5. Taylor expanded in wj around 0 94.8%

   \[\leadsto \color{blue}{x + wj \cdot \left(wj \cdot \left(1 - \left(-4 \cdot x + 1.5 \cdot x\right)\right) - 2 \cdot x\right)} \]
6. Step-by-step derivation

   1. cancel-sign-sub-inv 94.8%

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

   2. metadata-eval 94.8%

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

   3. distribute-rgt-out 94.8%

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

   4. metadata-eval 94.8%

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

   5. *-commutative 94.8%

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

7. Simplified 94.8%

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

Alternative 10: 96.2% accurate, 24.1× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 76.4%

   \[wj - \frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}} \]
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. *-commutative 78.0%

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

   3. associate-/r* 78.0%

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

   4. div-sub 76.5%

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

   5. associate-/l* 76.5%

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

   6. *-inverses 79.2%

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

   7. *-rgt-identity 79.2%

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

3. Simplified 79.2%

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

5. Taylor expanded in wj around 0 95.6%

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

6. Taylor expanded in x around 0 94.7%

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

   1. neg-mul-1 94.7%

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

   2. sub-neg 94.7%

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

8. Simplified 94.7%

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

9. Final simplification 94.7%

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

Alternative 11: 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 76.4%

   \[wj - \frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}} \]
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. *-commutative 78.0%

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

   3. associate-/r* 78.0%

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

   4. div-sub 76.5%

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

   5. associate-/l* 76.5%

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

   6. *-inverses 79.2%

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

   7. *-rgt-identity 79.2%

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

3. Simplified 79.2%

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

5. Taylor expanded in wj around 0 95.6%

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

6. Taylor expanded in x around 0 94.7%

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

   1. neg-mul-1 94.7%

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

   2. sub-neg 94.7%

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

8. Simplified 94.7%

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

9. Taylor expanded in wj around 0 94.3%

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

    1. *-commutative 94.3%

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

11. Simplified 94.3%

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

Alternative 12: 95.8% accurate, 34.8× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 76.4%

   \[wj - \frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}} \]
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. *-commutative 78.0%

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

   3. associate-/r* 78.0%

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

   4. div-sub 76.5%

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

   5. associate-/l* 76.5%

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

   6. *-inverses 79.2%

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

   7. *-rgt-identity 79.2%

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

3. Simplified 79.2%

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

5. Taylor expanded in wj around 0 95.6%

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

6. Taylor expanded in x around 0 94.7%

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

   1. neg-mul-1 94.7%

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

   2. sub-neg 94.7%

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

8. Simplified 94.7%

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

9. Taylor expanded in x around inf 94.7%

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

    1. sub-neg 94.7%

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

    2. associate-/l* 94.7%

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

    3. metadata-eval 94.7%

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

11. Simplified 94.7%

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

12. Taylor expanded in x around 0 93.9%

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

13. Final simplification 93.9%

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

Alternative 13: 85.1% 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 76.4%

   \[wj - \frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}} \]
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. *-commutative 78.0%

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

   3. associate-/r* 78.0%

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

   4. div-sub 76.5%

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

   5. associate-/l* 76.5%

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

   6. *-inverses 79.2%

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

   7. *-rgt-identity 79.2%

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

3. Simplified 79.2%

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

5. Taylor expanded in wj around 0 81.7%

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

   1. *-commutative 81.7%

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

7. Simplified 81.7%

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

8. Final simplification 81.7%

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

Alternative 14: 85.1% accurate, 44.7× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 76.4%

   \[wj - \frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}} \]
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. *-commutative 78.0%

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

3. Simplified 78.0%

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

5. Taylor expanded in wj around 0 73.2%

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

   1. neg-mul-1 73.2%

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

   2. +-commutative 73.2%

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

   3. unsub-neg 73.2%

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

   4. cancel-sign-sub-inv 73.2%

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

   5. metadata-eval 73.2%

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

   6. *-commutative 73.2%

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

7. Simplified 73.2%

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

8. Taylor expanded in x around 0 81.7%

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

9. Final simplification 81.7%

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

Alternative 15: 84.7% accurate, 313.0× speedup

Math

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

FPCore

(FPCore (wj x) :precision binary64 x)

C

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

Fortran

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

Java

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

Python

def code(wj, x):
	return x

Julia

function code(wj, x)
	return x
end

MATLAB

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

Wolfram

code[wj_, x_] := x

Derivation

1. Initial program 76.4%

   \[wj - \frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}} \]
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. *-commutative 78.0%

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

   3. associate-/r* 78.0%

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

   4. div-sub 76.5%

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

   5. associate-/l* 76.5%

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

   6. *-inverses 79.2%

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

   7. *-rgt-identity 79.2%

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

3. Simplified 79.2%

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

5. Taylor expanded in wj around 0 80.9%

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

Alternative 16: 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 76.4%

   \[wj - \frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}} \]
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. *-commutative 78.0%

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

   3. associate-/r* 78.0%

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

   4. div-sub 76.5%

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

   5. associate-/l* 76.5%

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

   6. *-inverses 79.2%

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

   7. *-rgt-identity 79.2%

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

3. Simplified 79.2%

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

5. Taylor expanded in wj around inf 4.4%

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

Alternative 17: 3.3% accurate, 313.0× speedup

Math

\[\begin{array}{l} \\ -1 \end{array} \]

FPCore

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

C

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

Fortran

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

Java

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

Python

def code(wj, x):
	return -1.0

Julia

function code(wj, x)
	return -1.0
end

MATLAB

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

Wolfram

code[wj_, x_] := -1.0

Derivation

1. Initial program 76.4%

   \[wj - \frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}} \]
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. *-commutative 78.0%

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

   3. associate-/r* 78.0%

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

   4. div-sub 76.5%

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

   5. associate-/l* 76.5%

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

   6. *-inverses 79.2%

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

   7. *-rgt-identity 79.2%

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

3. Simplified 79.2%

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

5. Taylor expanded in wj around inf 4.2%

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

6. Taylor expanded in wj around 0 3.3%

   \[\leadsto \color{blue}{-1} \]
7. Add Preprocessing

Developer Target 1: 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 2024177 
(FPCore (wj x)
  :name "Jmat.Real.lambertw, newton loop step"
  :precision binary64

  :alt
  (! :herbie-platform default (let ((ew (exp wj))) (- wj (- (/ wj (+ wj 1)) (/ x (+ ew (* wj ew)))))))

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