Jmat.Real.lambertw, newton loop step

Percentage Accurate: 77.6% → 99.4%

Time: 10.5s

Alternatives: 9

Speedup: 313.0×

• could not determine a ground truth

  1. wj = -8.109807532009322e+149
  2. x = 2.2999522368616085e-179

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: 77.6% 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: 99.4% accurate, 2.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;wj \leq -4.6 \cdot 10^{-7}:\\ \;\;\;\;wj + \frac{\frac{1}{\frac{e^{wj}}{x}} - wj}{wj + 1}\\ \mathbf{elif}\;wj \leq 8.2 \cdot 10^{-9}:\\ \;\;\;\;x + \left(wj \cdot \left(x \cdot -2\right) - \left(wj \cdot wj\right) \cdot \left(-1 + x \cdot -2.5\right)\right)\\ \mathbf{else}:\\ \;\;\;\;wj - \frac{wj - x \cdot e^{-wj}}{wj + 1}\\ \end{array} \end{array} \]

FPCore

(FPCore (wj x)
 :precision binary64
 (if (<= wj -4.6e-7)
   (+ wj (/ (- (/ 1.0 (/ (exp wj) x)) wj) (+ wj 1.0)))
   (if (<= wj 8.2e-9)
     (+ x (- (* wj (* x -2.0)) (* (* wj wj) (+ -1.0 (* x -2.5)))))
     (- wj (/ (- wj (* x (exp (- wj)))) (+ wj 1.0))))))

C

double code(double wj, double x) {
	double tmp;
	if (wj <= -4.6e-7) {
		tmp = wj + (((1.0 / (exp(wj) / x)) - wj) / (wj + 1.0));
	} else if (wj <= 8.2e-9) {
		tmp = x + ((wj * (x * -2.0)) - ((wj * wj) * (-1.0 + (x * -2.5))));
	} else {
		tmp = wj - ((wj - (x * exp(-wj))) / (wj + 1.0));
	}
	return tmp;
}

Fortran

real(8) function code(wj, x)
    real(8), intent (in) :: wj
    real(8), intent (in) :: x
    real(8) :: tmp
    if (wj <= (-4.6d-7)) then
        tmp = wj + (((1.0d0 / (exp(wj) / x)) - wj) / (wj + 1.0d0))
    else if (wj <= 8.2d-9) then
        tmp = x + ((wj * (x * (-2.0d0))) - ((wj * wj) * ((-1.0d0) + (x * (-2.5d0)))))
    else
        tmp = wj - ((wj - (x * exp(-wj))) / (wj + 1.0d0))
    end if
    code = tmp
end function

Java

public static double code(double wj, double x) {
	double tmp;
	if (wj <= -4.6e-7) {
		tmp = wj + (((1.0 / (Math.exp(wj) / x)) - wj) / (wj + 1.0));
	} else if (wj <= 8.2e-9) {
		tmp = x + ((wj * (x * -2.0)) - ((wj * wj) * (-1.0 + (x * -2.5))));
	} else {
		tmp = wj - ((wj - (x * Math.exp(-wj))) / (wj + 1.0));
	}
	return tmp;
}

Python

def code(wj, x):
	tmp = 0
	if wj <= -4.6e-7:
		tmp = wj + (((1.0 / (math.exp(wj) / x)) - wj) / (wj + 1.0))
	elif wj <= 8.2e-9:
		tmp = x + ((wj * (x * -2.0)) - ((wj * wj) * (-1.0 + (x * -2.5))))
	else:
		tmp = wj - ((wj - (x * math.exp(-wj))) / (wj + 1.0))
	return tmp

Julia

function code(wj, x)
	tmp = 0.0
	if (wj <= -4.6e-7)
		tmp = Float64(wj + Float64(Float64(Float64(1.0 / Float64(exp(wj) / x)) - wj) / Float64(wj + 1.0)));
	elseif (wj <= 8.2e-9)
		tmp = Float64(x + Float64(Float64(wj * Float64(x * -2.0)) - Float64(Float64(wj * wj) * Float64(-1.0 + Float64(x * -2.5)))));
	else
		tmp = Float64(wj - Float64(Float64(wj - Float64(x * exp(Float64(-wj)))) / Float64(wj + 1.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(wj, x)
	tmp = 0.0;
	if (wj <= -4.6e-7)
		tmp = wj + (((1.0 / (exp(wj) / x)) - wj) / (wj + 1.0));
	elseif (wj <= 8.2e-9)
		tmp = x + ((wj * (x * -2.0)) - ((wj * wj) * (-1.0 + (x * -2.5))));
	else
		tmp = wj - ((wj - (x * exp(-wj))) / (wj + 1.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[wj_, x_] := If[LessEqual[wj, -4.6e-7], N[(wj + N[(N[(N[(1.0 / N[(N[Exp[wj], $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision] - wj), $MachinePrecision] / N[(wj + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[wj, 8.2e-9], N[(x + N[(N[(wj * N[(x * -2.0), $MachinePrecision]), $MachinePrecision] - N[(N[(wj * wj), $MachinePrecision] * N[(-1.0 + N[(x * -2.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(wj - N[(N[(wj - N[(x * N[Exp[(-wj)], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(wj + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if wj < -4.5999999999999999e-7

   1. Initial program 76.4%

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

      1. div-sub 76.4%

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

      2. associate-/l* 76.4%

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

      3. distribute-rgt1-in 76.4%

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

      4. associate-/l* 76.4%

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

      5. *-inverses 76.4%

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

      6. /-rgt-identity 76.4%

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

      7. distribute-rgt1-in 98.6%

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

      8. associate-/l/ 98.2%

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

      9. div-sub 98.2%

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

   3. Simplified 98.2%

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

      1. clear-num 98.6%

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

      2. inv-pow 98.6%

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

   5. Applied egg-rr 98.6%

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

      1. unpow-1 98.6%

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

   7. Simplified 98.6%

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

   if -4.5999999999999999e-7 < wj < 8.2000000000000006e-9

   1. Initial program 82.8%

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

      1. div-sub 82.8%

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

      2. associate-/l* 82.8%

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

      3. distribute-rgt1-in 82.8%

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

      4. associate-/l* 82.8%

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

      5. *-inverses 82.8%

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

      6. /-rgt-identity 82.8%

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

      7. distribute-rgt1-in 82.8%

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

      8. associate-/l/ 82.8%

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

      9. div-sub 82.8%

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

   3. Simplified 82.8%

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

      1. clear-num 82.5%

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

      2. inv-pow 82.5%

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

   5. Applied egg-rr 82.5%

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

      1. unpow-1 82.5%

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

   7. Simplified 82.5%

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

   8. Taylor expanded in wj around 0 99.9%

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

      1. +-commutative 99.9%

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

      2. sub-neg 99.9%

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

      3. sub-neg 99.9%

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

      4. distribute-rgt-out 99.9%

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

      5. metadata-eval 99.9%

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

      6. fma-def 99.9%

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

      7. unpow2 99.9%

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

      8. *-commutative 99.9%

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

      9. associate-*l* 99.9%

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

   10. Simplified 99.9%

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

       1. fma-udef 99.9%

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

   12. Applied egg-rr 99.9%

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

   if 8.2000000000000006e-9 < wj

   1. Initial program 51.9%

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

      1. div-sub 51.9%

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

      2. associate-/l* 51.9%

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

      3. distribute-rgt1-in 51.9%

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

      4. associate-/l* 51.9%

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

      5. *-inverses 94.8%

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

      6. /-rgt-identity 94.8%

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

      7. distribute-rgt1-in 94.8%

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

      8. associate-/l/ 94.8%

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

      9. div-sub 94.8%

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

   3. Simplified 94.8%

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

      1. clear-num 95.0%

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

      2. associate-/r/ 95.0%

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

      3. rec-exp 95.0%

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

   5. Applied egg-rr 95.0%

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

4. Final simplification 99.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;wj \leq -4.6 \cdot 10^{-7}:\\ \;\;\;\;wj + \frac{\frac{1}{\frac{e^{wj}}{x}} - wj}{wj + 1}\\ \mathbf{elif}\;wj \leq 8.2 \cdot 10^{-9}:\\ \;\;\;\;x + \left(wj \cdot \left(x \cdot -2\right) - \left(wj \cdot wj\right) \cdot \left(-1 + x \cdot -2.5\right)\right)\\ \mathbf{else}:\\ \;\;\;\;wj - \frac{wj - x \cdot e^{-wj}}{wj + 1}\\ \end{array} \]

Alternative 2: 97.8% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x \cdot -4 + x \cdot 1.5\\ \mathbf{if}\;wj \leq 5 \cdot 10^{-6}:\\ \;\;\;\;x + \left(-2 \cdot \left(wj \cdot x\right) + \left({wj}^{3} \cdot \left(-1 - \left(x \cdot -3 + \left(-2 \cdot t_0 + x \cdot 0.6666666666666666\right)\right)\right) + {wj}^{2} \cdot \left(1 - t_0\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;wj - \frac{wj - x \cdot e^{-wj}}{wj + 1}\\ \end{array} \end{array} \]

FPCore

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

C

double code(double wj, double x) {
	double t_0 = (x * -4.0) + (x * 1.5);
	double tmp;
	if (wj <= 5e-6) {
		tmp = x + ((-2.0 * (wj * x)) + ((pow(wj, 3.0) * (-1.0 - ((x * -3.0) + ((-2.0 * t_0) + (x * 0.6666666666666666))))) + (pow(wj, 2.0) * (1.0 - t_0))));
	} else {
		tmp = wj - ((wj - (x * exp(-wj))) / (wj + 1.0));
	}
	return tmp;
}

Fortran

real(8) function code(wj, x)
    real(8), intent (in) :: wj
    real(8), intent (in) :: x
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (x * (-4.0d0)) + (x * 1.5d0)
    if (wj <= 5d-6) then
        tmp = x + (((-2.0d0) * (wj * x)) + (((wj ** 3.0d0) * ((-1.0d0) - ((x * (-3.0d0)) + (((-2.0d0) * t_0) + (x * 0.6666666666666666d0))))) + ((wj ** 2.0d0) * (1.0d0 - t_0))))
    else
        tmp = wj - ((wj - (x * exp(-wj))) / (wj + 1.0d0))
    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 <= 5e-6) {
		tmp = x + ((-2.0 * (wj * x)) + ((Math.pow(wj, 3.0) * (-1.0 - ((x * -3.0) + ((-2.0 * t_0) + (x * 0.6666666666666666))))) + (Math.pow(wj, 2.0) * (1.0 - t_0))));
	} else {
		tmp = wj - ((wj - (x * Math.exp(-wj))) / (wj + 1.0));
	}
	return tmp;
}

Python

def code(wj, x):
	t_0 = (x * -4.0) + (x * 1.5)
	tmp = 0
	if wj <= 5e-6:
		tmp = x + ((-2.0 * (wj * x)) + ((math.pow(wj, 3.0) * (-1.0 - ((x * -3.0) + ((-2.0 * t_0) + (x * 0.6666666666666666))))) + (math.pow(wj, 2.0) * (1.0 - t_0))))
	else:
		tmp = wj - ((wj - (x * math.exp(-wj))) / (wj + 1.0))
	return tmp

Julia

function code(wj, x)
	t_0 = Float64(Float64(x * -4.0) + Float64(x * 1.5))
	tmp = 0.0
	if (wj <= 5e-6)
		tmp = Float64(x + Float64(Float64(-2.0 * Float64(wj * x)) + Float64(Float64((wj ^ 3.0) * Float64(-1.0 - Float64(Float64(x * -3.0) + Float64(Float64(-2.0 * t_0) + Float64(x * 0.6666666666666666))))) + Float64((wj ^ 2.0) * Float64(1.0 - t_0)))));
	else
		tmp = Float64(wj - Float64(Float64(wj - Float64(x * exp(Float64(-wj)))) / Float64(wj + 1.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(wj, x)
	t_0 = (x * -4.0) + (x * 1.5);
	tmp = 0.0;
	if (wj <= 5e-6)
		tmp = x + ((-2.0 * (wj * x)) + (((wj ^ 3.0) * (-1.0 - ((x * -3.0) + ((-2.0 * t_0) + (x * 0.6666666666666666))))) + ((wj ^ 2.0) * (1.0 - t_0))));
	else
		tmp = wj - ((wj - (x * exp(-wj))) / (wj + 1.0));
	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, 5e-6], N[(x + N[(N[(-2.0 * N[(wj * x), $MachinePrecision]), $MachinePrecision] + N[(N[(N[Power[wj, 3.0], $MachinePrecision] * 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] + N[(N[Power[wj, 2.0], $MachinePrecision] * N[(1.0 - t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(wj - N[(N[(wj - N[(x * N[Exp[(-wj)], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(wj + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if wj < 5.00000000000000041e-6

   1. Initial program 82.5%

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

      1. div-sub 82.5%

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

      2. associate-/l* 82.5%

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

      3. distribute-rgt1-in 82.5%

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

      4. associate-/l* 82.5%

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

      5. *-inverses 82.5%

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

      6. /-rgt-identity 82.5%

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

      7. distribute-rgt1-in 83.3%

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

      8. associate-/l/ 83.3%

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

      9. div-sub 83.3%

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

   3. Simplified 83.3%

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

   4. Taylor expanded in wj around 0 98.4%

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

   if 5.00000000000000041e-6 < wj

   1. Initial program 49.7%

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

      1. div-sub 49.7%

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

      2. associate-/l* 49.7%

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

      3. distribute-rgt1-in 49.7%

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

      4. associate-/l* 49.7%

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

      5. *-inverses 99.7%

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

      6. /-rgt-identity 99.7%

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

      7. distribute-rgt1-in 99.7%

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

      8. associate-/l/ 99.7%

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

      9. div-sub 99.7%

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

   3. Simplified 99.7%

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

      1. clear-num 100.0%

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

      2. associate-/r/ 100.0%

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

      3. rec-exp 100.0%

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

   5. Applied egg-rr 100.0%

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

4. Final simplification 98.5%

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

Alternative 3: 99.4% accurate, 2.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;wj \leq -4.6 \cdot 10^{-7}:\\ \;\;\;\;wj + \frac{\frac{x}{e^{wj}} - wj}{wj + 1}\\ \mathbf{elif}\;wj \leq 7 \cdot 10^{-9}:\\ \;\;\;\;x + \left(wj \cdot \left(x \cdot -2\right) - \left(wj \cdot wj\right) \cdot \left(-1 + x \cdot -2.5\right)\right)\\ \mathbf{else}:\\ \;\;\;\;wj - \frac{wj - x \cdot e^{-wj}}{wj + 1}\\ \end{array} \end{array} \]

FPCore

(FPCore (wj x)
 :precision binary64
 (if (<= wj -4.6e-7)
   (+ wj (/ (- (/ x (exp wj)) wj) (+ wj 1.0)))
   (if (<= wj 7e-9)
     (+ x (- (* wj (* x -2.0)) (* (* wj wj) (+ -1.0 (* x -2.5)))))
     (- wj (/ (- wj (* x (exp (- wj)))) (+ wj 1.0))))))

C

double code(double wj, double x) {
	double tmp;
	if (wj <= -4.6e-7) {
		tmp = wj + (((x / exp(wj)) - wj) / (wj + 1.0));
	} else if (wj <= 7e-9) {
		tmp = x + ((wj * (x * -2.0)) - ((wj * wj) * (-1.0 + (x * -2.5))));
	} else {
		tmp = wj - ((wj - (x * exp(-wj))) / (wj + 1.0));
	}
	return tmp;
}

Fortran

real(8) function code(wj, x)
    real(8), intent (in) :: wj
    real(8), intent (in) :: x
    real(8) :: tmp
    if (wj <= (-4.6d-7)) then
        tmp = wj + (((x / exp(wj)) - wj) / (wj + 1.0d0))
    else if (wj <= 7d-9) then
        tmp = x + ((wj * (x * (-2.0d0))) - ((wj * wj) * ((-1.0d0) + (x * (-2.5d0)))))
    else
        tmp = wj - ((wj - (x * exp(-wj))) / (wj + 1.0d0))
    end if
    code = tmp
end function

Java

public static double code(double wj, double x) {
	double tmp;
	if (wj <= -4.6e-7) {
		tmp = wj + (((x / Math.exp(wj)) - wj) / (wj + 1.0));
	} else if (wj <= 7e-9) {
		tmp = x + ((wj * (x * -2.0)) - ((wj * wj) * (-1.0 + (x * -2.5))));
	} else {
		tmp = wj - ((wj - (x * Math.exp(-wj))) / (wj + 1.0));
	}
	return tmp;
}

Python

def code(wj, x):
	tmp = 0
	if wj <= -4.6e-7:
		tmp = wj + (((x / math.exp(wj)) - wj) / (wj + 1.0))
	elif wj <= 7e-9:
		tmp = x + ((wj * (x * -2.0)) - ((wj * wj) * (-1.0 + (x * -2.5))))
	else:
		tmp = wj - ((wj - (x * math.exp(-wj))) / (wj + 1.0))
	return tmp

Julia

function code(wj, x)
	tmp = 0.0
	if (wj <= -4.6e-7)
		tmp = Float64(wj + Float64(Float64(Float64(x / exp(wj)) - wj) / Float64(wj + 1.0)));
	elseif (wj <= 7e-9)
		tmp = Float64(x + Float64(Float64(wj * Float64(x * -2.0)) - Float64(Float64(wj * wj) * Float64(-1.0 + Float64(x * -2.5)))));
	else
		tmp = Float64(wj - Float64(Float64(wj - Float64(x * exp(Float64(-wj)))) / Float64(wj + 1.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(wj, x)
	tmp = 0.0;
	if (wj <= -4.6e-7)
		tmp = wj + (((x / exp(wj)) - wj) / (wj + 1.0));
	elseif (wj <= 7e-9)
		tmp = x + ((wj * (x * -2.0)) - ((wj * wj) * (-1.0 + (x * -2.5))));
	else
		tmp = wj - ((wj - (x * exp(-wj))) / (wj + 1.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[wj_, x_] := If[LessEqual[wj, -4.6e-7], N[(wj + N[(N[(N[(x / N[Exp[wj], $MachinePrecision]), $MachinePrecision] - wj), $MachinePrecision] / N[(wj + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[wj, 7e-9], N[(x + N[(N[(wj * N[(x * -2.0), $MachinePrecision]), $MachinePrecision] - N[(N[(wj * wj), $MachinePrecision] * N[(-1.0 + N[(x * -2.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(wj - N[(N[(wj - N[(x * N[Exp[(-wj)], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(wj + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if wj < -4.5999999999999999e-7

   1. Initial program 76.4%

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

      1. div-sub 76.4%

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

      2. associate-/l* 76.4%

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

      3. distribute-rgt1-in 76.4%

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

      4. associate-/l* 76.4%

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

      5. *-inverses 76.4%

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

      6. /-rgt-identity 76.4%

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

      7. distribute-rgt1-in 98.6%

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

      8. associate-/l/ 98.2%

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

      9. div-sub 98.2%

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

   3. Simplified 98.2%

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

   if -4.5999999999999999e-7 < wj < 6.9999999999999998e-9

   1. Initial program 82.8%

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

      1. div-sub 82.8%

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

      2. associate-/l* 82.8%

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

      3. distribute-rgt1-in 82.8%

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

      4. associate-/l* 82.8%

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

      5. *-inverses 82.8%

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

      6. /-rgt-identity 82.8%

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

      7. distribute-rgt1-in 82.8%

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

      8. associate-/l/ 82.8%

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

      9. div-sub 82.8%

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

   3. Simplified 82.8%

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

      1. clear-num 82.5%

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

      2. inv-pow 82.5%

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

   5. Applied egg-rr 82.5%

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

      1. unpow-1 82.5%

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

   7. Simplified 82.5%

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

   8. Taylor expanded in wj around 0 99.9%

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

      1. +-commutative 99.9%

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

      2. sub-neg 99.9%

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

      3. sub-neg 99.9%

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

      4. distribute-rgt-out 99.9%

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

      5. metadata-eval 99.9%

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

      6. fma-def 99.9%

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

      7. unpow2 99.9%

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

      8. *-commutative 99.9%

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

      9. associate-*l* 99.9%

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

   10. Simplified 99.9%

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

       1. fma-udef 99.9%

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

   12. Applied egg-rr 99.9%

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

   if 6.9999999999999998e-9 < wj

   1. Initial program 51.9%

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

      1. div-sub 51.9%

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

      2. associate-/l* 51.9%

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

      3. distribute-rgt1-in 51.9%

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

      4. associate-/l* 51.9%

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

      5. *-inverses 94.8%

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

      6. /-rgt-identity 94.8%

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

      7. distribute-rgt1-in 94.8%

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

      8. associate-/l/ 94.8%

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

      9. div-sub 94.8%

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

   3. Simplified 94.8%

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

      1. clear-num 95.0%

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

      2. associate-/r/ 95.0%

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

      3. rec-exp 95.0%

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

   5. Applied egg-rr 95.0%

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

4. Final simplification 99.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;wj \leq -4.6 \cdot 10^{-7}:\\ \;\;\;\;wj + \frac{\frac{x}{e^{wj}} - wj}{wj + 1}\\ \mathbf{elif}\;wj \leq 7 \cdot 10^{-9}:\\ \;\;\;\;x + \left(wj \cdot \left(x \cdot -2\right) - \left(wj \cdot wj\right) \cdot \left(-1 + x \cdot -2.5\right)\right)\\ \mathbf{else}:\\ \;\;\;\;wj - \frac{wj - x \cdot e^{-wj}}{wj + 1}\\ \end{array} \]

Alternative 4: 99.4% accurate, 2.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;wj \leq -4.6 \cdot 10^{-7} \lor \neg \left(wj \leq 7 \cdot 10^{-9}\right):\\ \;\;\;\;wj + \frac{\frac{x}{e^{wj}} - wj}{wj + 1}\\ \mathbf{else}:\\ \;\;\;\;x + \left(wj \cdot \left(x \cdot -2\right) - \left(wj \cdot wj\right) \cdot \left(-1 + x \cdot -2.5\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (wj x)
 :precision binary64
 (if (or (<= wj -4.6e-7) (not (<= wj 7e-9)))
   (+ wj (/ (- (/ x (exp wj)) wj) (+ wj 1.0)))
   (+ x (- (* wj (* x -2.0)) (* (* wj wj) (+ -1.0 (* x -2.5)))))))

C

double code(double wj, double x) {
	double tmp;
	if ((wj <= -4.6e-7) || !(wj <= 7e-9)) {
		tmp = wj + (((x / exp(wj)) - wj) / (wj + 1.0));
	} else {
		tmp = x + ((wj * (x * -2.0)) - ((wj * wj) * (-1.0 + (x * -2.5))));
	}
	return tmp;
}

Fortran

real(8) function code(wj, x)
    real(8), intent (in) :: wj
    real(8), intent (in) :: x
    real(8) :: tmp
    if ((wj <= (-4.6d-7)) .or. (.not. (wj <= 7d-9))) then
        tmp = wj + (((x / exp(wj)) - wj) / (wj + 1.0d0))
    else
        tmp = x + ((wj * (x * (-2.0d0))) - ((wj * wj) * ((-1.0d0) + (x * (-2.5d0)))))
    end if
    code = tmp
end function

Java

public static double code(double wj, double x) {
	double tmp;
	if ((wj <= -4.6e-7) || !(wj <= 7e-9)) {
		tmp = wj + (((x / Math.exp(wj)) - wj) / (wj + 1.0));
	} else {
		tmp = x + ((wj * (x * -2.0)) - ((wj * wj) * (-1.0 + (x * -2.5))));
	}
	return tmp;
}

Python

def code(wj, x):
	tmp = 0
	if (wj <= -4.6e-7) or not (wj <= 7e-9):
		tmp = wj + (((x / math.exp(wj)) - wj) / (wj + 1.0))
	else:
		tmp = x + ((wj * (x * -2.0)) - ((wj * wj) * (-1.0 + (x * -2.5))))
	return tmp

Julia

function code(wj, x)
	tmp = 0.0
	if ((wj <= -4.6e-7) || !(wj <= 7e-9))
		tmp = Float64(wj + Float64(Float64(Float64(x / exp(wj)) - wj) / Float64(wj + 1.0)));
	else
		tmp = Float64(x + Float64(Float64(wj * Float64(x * -2.0)) - Float64(Float64(wj * wj) * Float64(-1.0 + Float64(x * -2.5)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(wj, x)
	tmp = 0.0;
	if ((wj <= -4.6e-7) || ~((wj <= 7e-9)))
		tmp = wj + (((x / exp(wj)) - wj) / (wj + 1.0));
	else
		tmp = x + ((wj * (x * -2.0)) - ((wj * wj) * (-1.0 + (x * -2.5))));
	end
	tmp_2 = tmp;
end

Wolfram

code[wj_, x_] := If[Or[LessEqual[wj, -4.6e-7], N[Not[LessEqual[wj, 7e-9]], $MachinePrecision]], N[(wj + N[(N[(N[(x / N[Exp[wj], $MachinePrecision]), $MachinePrecision] - wj), $MachinePrecision] / N[(wj + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x + N[(N[(wj * N[(x * -2.0), $MachinePrecision]), $MachinePrecision] - N[(N[(wj * wj), $MachinePrecision] * N[(-1.0 + N[(x * -2.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if wj < -4.5999999999999999e-7 or 6.9999999999999998e-9 < wj

   1. Initial program 65.7%

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

      1. div-sub 65.7%

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

      2. associate-/l* 65.7%

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

      3. distribute-rgt1-in 65.7%

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

      4. associate-/l* 65.7%

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

      5. *-inverses 84.4%

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

      6. /-rgt-identity 84.4%

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

      7. distribute-rgt1-in 96.9%

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

      8. associate-/l/ 96.7%

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

      9. div-sub 96.7%

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

   3. Simplified 96.7%

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

   if -4.5999999999999999e-7 < wj < 6.9999999999999998e-9

   1. Initial program 82.8%

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

      1. div-sub 82.8%

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

      2. associate-/l* 82.8%

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

      3. distribute-rgt1-in 82.8%

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

      4. associate-/l* 82.8%

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

      5. *-inverses 82.8%

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

      6. /-rgt-identity 82.8%

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

      7. distribute-rgt1-in 82.8%

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

      8. associate-/l/ 82.8%

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

      9. div-sub 82.8%

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

   3. Simplified 82.8%

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

      1. clear-num 82.5%

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

      2. inv-pow 82.5%

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

   5. Applied egg-rr 82.5%

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

      1. unpow-1 82.5%

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

   7. Simplified 82.5%

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

   8. Taylor expanded in wj around 0 99.9%

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

      1. +-commutative 99.9%

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

      2. sub-neg 99.9%

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

      3. sub-neg 99.9%

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

      4. distribute-rgt-out 99.9%

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

      5. metadata-eval 99.9%

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

      6. fma-def 99.9%

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

      7. unpow2 99.9%

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

      8. *-commutative 99.9%

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

      9. associate-*l* 99.9%

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

   10. Simplified 99.9%

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

       1. fma-udef 99.9%

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

   12. Applied egg-rr 99.9%

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

4. Final simplification 99.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;wj \leq -4.6 \cdot 10^{-7} \lor \neg \left(wj \leq 7 \cdot 10^{-9}\right):\\ \;\;\;\;wj + \frac{\frac{x}{e^{wj}} - wj}{wj + 1}\\ \mathbf{else}:\\ \;\;\;\;x + \left(wj \cdot \left(x \cdot -2\right) - \left(wj \cdot wj\right) \cdot \left(-1 + x \cdot -2.5\right)\right)\\ \end{array} \]

Alternative 5: 95.6% accurate, 18.4× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 81.7%

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

   1. div-sub 81.7%

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

   2. associate-/l* 81.7%

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

   3. distribute-rgt1-in 81.7%

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

   4. associate-/l* 81.7%

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

   5. *-inverses 82.9%

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

   6. /-rgt-identity 82.9%

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

   7. distribute-rgt1-in 83.7%

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

   8. associate-/l/ 83.7%

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

   9. div-sub 83.7%

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

3. Simplified 83.7%

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

   1. clear-num 83.4%

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

   2. inv-pow 83.4%

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

5. Applied egg-rr 83.4%

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

   1. unpow-1 83.4%

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

7. Simplified 83.4%

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

8. Taylor expanded in wj around 0 96.0%

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

   1. +-commutative 96.0%

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

   2. sub-neg 96.0%

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

   3. sub-neg 96.0%

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

   4. distribute-rgt-out 96.0%

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

   5. metadata-eval 96.0%

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

   6. fma-def 96.0%

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

   7. unpow2 96.0%

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

   8. *-commutative 96.0%

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

   9. associate-*l* 96.0%

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

10. Simplified 96.0%

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

    1. fma-udef 96.0%

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

12. Applied egg-rr 96.0%

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

13. Final simplification 96.0%

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

Alternative 6: 95.5% accurate, 28.5× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 81.7%

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

   1. div-sub 81.7%

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

   2. associate-/l* 81.7%

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

   3. distribute-rgt1-in 81.7%

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

   4. associate-/l* 81.7%

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

   5. *-inverses 82.9%

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

   6. /-rgt-identity 82.9%

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

   7. distribute-rgt1-in 83.7%

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

   8. associate-/l/ 83.7%

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

   9. div-sub 83.7%

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

3. Simplified 83.7%

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

   1. clear-num 83.4%

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

   2. inv-pow 83.4%

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

5. Applied egg-rr 83.4%

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

   1. unpow-1 83.4%

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

7. Simplified 83.4%

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

8. Taylor expanded in wj around 0 96.0%

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

   1. +-commutative 96.0%

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

   2. sub-neg 96.0%

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

   3. sub-neg 96.0%

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

   4. distribute-rgt-out 96.0%

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

   5. metadata-eval 96.0%

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

   6. fma-def 96.0%

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

   7. unpow2 96.0%

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

   8. *-commutative 96.0%

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

   9. associate-*l* 96.0%

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

10. Simplified 96.0%

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

    1. fma-udef 96.0%

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

12. Applied egg-rr 96.0%

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

13. Taylor expanded in x around 0 95.4%

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

    1. unpow2 95.4%

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

15. Simplified 95.4%

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

16. Final simplification 95.4%

    \[\leadsto x + \left(wj \cdot wj + wj \cdot \left(x \cdot -2\right)\right) \]

Alternative 7: 95.0% accurate, 62.6× speedup

Math

\[\begin{array}{l} \\ x + wj \cdot wj \end{array} \]

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 81.7%

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

   1. div-sub 81.7%

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

   2. associate-/l* 81.7%

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

   3. distribute-rgt1-in 81.7%

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

   4. associate-/l* 81.7%

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

   5. *-inverses 82.9%

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

   6. /-rgt-identity 82.9%

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

   7. distribute-rgt1-in 83.7%

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

   8. associate-/l/ 83.7%

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

   9. div-sub 83.7%

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

3. Simplified 83.7%

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

   1. clear-num 83.4%

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

   2. inv-pow 83.4%

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

5. Applied egg-rr 83.4%

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

   1. unpow-1 83.4%

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

7. Simplified 83.4%

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

8. Taylor expanded in wj around 0 96.0%

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

   1. +-commutative 96.0%

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

   2. sub-neg 96.0%

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

   3. sub-neg 96.0%

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

   4. distribute-rgt-out 96.0%

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

   5. metadata-eval 96.0%

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

   6. fma-def 96.0%

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

   7. unpow2 96.0%

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

   8. *-commutative 96.0%

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

   9. associate-*l* 96.0%

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

10. Simplified 96.0%

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

11. Taylor expanded in x around 0 94.4%

    \[\leadsto x + \color{blue}{{wj}^{2}} \]
12. Step-by-step derivation

    1. unpow2 94.4%

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

13. Simplified 94.4%

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

14. Final simplification 94.4%

    \[\leadsto x + wj \cdot wj \]

Alternative 8: 4.4% accurate, 313.0× speedup

Math

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

FPCore

(FPCore (wj x) :precision binary64 wj)

C

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

Fortran

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

Java

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

Python

def code(wj, x):
	return wj

Julia

function code(wj, x)
	return wj
end

MATLAB

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

Wolfram

code[wj_, x_] := wj

Derivation

1. Initial program 81.7%

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

   1. div-sub 81.7%

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

   2. associate-/l* 81.7%

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

   3. distribute-rgt1-in 81.7%

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

   4. associate-/l* 81.7%

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

   5. *-inverses 82.9%

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

   6. /-rgt-identity 82.9%

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

   7. distribute-rgt1-in 83.7%

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

   8. associate-/l/ 83.7%

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

   9. div-sub 83.7%

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

3. Simplified 83.7%

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

4. Taylor expanded in wj around inf 4.6%

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

5. Final simplification 4.6%

   \[\leadsto wj \]

Alternative 9: 83.8% accurate, 313.0× speedup

Math

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

FPCore

(FPCore (wj x) :precision binary64 x)

C

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

Fortran

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

Java

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

Python

def code(wj, x):
	return x

Julia

function code(wj, x)
	return x
end

MATLAB

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

Wolfram

code[wj_, x_] := x

Derivation

1. Initial program 81.7%

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

   1. div-sub 81.7%

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

   2. associate-/l* 81.7%

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

   3. distribute-rgt1-in 81.7%

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

   4. associate-/l* 81.7%

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

   5. *-inverses 82.9%

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

   6. /-rgt-identity 82.9%

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

   7. distribute-rgt1-in 83.7%

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

   8. associate-/l/ 83.7%

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

   9. div-sub 83.7%

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

3. Simplified 83.7%

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

4. Taylor expanded in wj around 0 83.4%

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

5. Final simplification 83.4%

   \[\leadsto x \]

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 2023292 
(FPCore (wj x)
  :name "Jmat.Real.lambertw, newton loop step"
  :precision binary64

  :herbie-target
  (- wj (- (/ wj (+ wj 1.0)) (/ x (+ (exp wj) (* wj (exp wj))))))

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