Jmat.Real.lambertw, newton loop step

Percentage Accurate: 77.7% → 99.8%

Time: 13.4s

Alternatives: 12

Speedup: 313.0×

• could not determine a ground truth

  1. wj = -8.159228384664223e+143
  2. x = 3.2079326040674035e+271

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: 77.7% 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: 99.8% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;wj \leq -4 \cdot 10^{-8}:\\ \;\;\;\;wj - \frac{wj \cdot e^{wj} - x}{e^{wj} \cdot \left(wj + 1\right)}\\ \mathbf{elif}\;wj \leq 4.5 \cdot 10^{-7}:\\ \;\;\;\;x + \left(\mathsf{fma}\left(wj, wj, -{wj}^{3}\right) - wj \cdot \left(x + x\right)\right)\\ \mathbf{else}:\\ \;\;\;\;wj + \frac{1}{wj + 1} \cdot \left(\frac{x}{e^{wj}} - wj\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (wj x)
 :precision binary64
 (if (<= wj -4e-8)
   (- wj (/ (- (* wj (exp wj)) x) (* (exp wj) (+ wj 1.0))))
   (if (<= wj 4.5e-7)
     (+ x (- (fma wj wj (- (pow wj 3.0))) (* wj (+ x x))))
     (+ wj (* (/ 1.0 (+ wj 1.0)) (- (/ x (exp wj)) wj))))))

C

double code(double wj, double x) {
	double tmp;
	if (wj <= -4e-8) {
		tmp = wj - (((wj * exp(wj)) - x) / (exp(wj) * (wj + 1.0)));
	} else if (wj <= 4.5e-7) {
		tmp = x + (fma(wj, wj, -pow(wj, 3.0)) - (wj * (x + x)));
	} else {
		tmp = wj + ((1.0 / (wj + 1.0)) * ((x / exp(wj)) - wj));
	}
	return tmp;
}

Julia

function code(wj, x)
	tmp = 0.0
	if (wj <= -4e-8)
		tmp = Float64(wj - Float64(Float64(Float64(wj * exp(wj)) - x) / Float64(exp(wj) * Float64(wj + 1.0))));
	elseif (wj <= 4.5e-7)
		tmp = Float64(x + Float64(fma(wj, wj, Float64(-(wj ^ 3.0))) - Float64(wj * Float64(x + x))));
	else
		tmp = Float64(wj + Float64(Float64(1.0 / Float64(wj + 1.0)) * Float64(Float64(x / exp(wj)) - wj)));
	end
	return tmp
end

Wolfram

code[wj_, x_] := If[LessEqual[wj, -4e-8], N[(wj - N[(N[(N[(wj * N[Exp[wj], $MachinePrecision]), $MachinePrecision] - x), $MachinePrecision] / N[(N[Exp[wj], $MachinePrecision] * N[(wj + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[wj, 4.5e-7], N[(x + N[(N[(wj * wj + (-N[Power[wj, 3.0], $MachinePrecision])), $MachinePrecision] - N[(wj * N[(x + x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(wj + N[(N[(1.0 / N[(wj + 1.0), $MachinePrecision]), $MachinePrecision] * N[(N[(x / N[Exp[wj], $MachinePrecision]), $MachinePrecision] - wj), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if wj < -4.0000000000000001e-8

   1. Initial program 74.3%

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

      1. distribute-rgt1-in 99.3%

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

   3. Simplified 99.3%

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

   if -4.0000000000000001e-8 < wj < 4.4999999999999998e-7

   1. Initial program 79.9%

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

      1. div-sub 79.9%

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

      2. associate-/l* 80.0%

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

      3. distribute-rgt1-in 80.0%

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

      4. associate-/l* 80.0%

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

      5. *-inverses 80.0%

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

      6. /-rgt-identity 80.0%

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

      7. distribute-rgt1-in 80.0%

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

      8. associate-/l/ 79.9%

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

      9. div-sub 79.9%

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

   3. Simplified 79.9%

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

   4. Taylor expanded in wj around 0 79.9%

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

      1. associate-*r* 79.9%

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

      2. neg-mul-1 79.9%

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

      3. distribute-rgt1-in 79.9%

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

      4. +-commutative 79.9%

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

      5. sub-neg 79.9%

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

   6. Simplified 79.9%

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

   7. Taylor expanded in wj around 0 100.0%

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

   8. Taylor expanded in x around 0 100.0%

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

      1. +-commutative 100.0%

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

      2. mul-1-neg 100.0%

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

      3. unsub-neg 100.0%

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

      4. unpow2 100.0%

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

   10. Simplified 100.0%

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

       1. fma-neg 100.0%

          \[\leadsto x + \left(wj \cdot \left(-1 \cdot x - x\right) + \color{blue}{\mathsf{fma}\left(wj, wj, -{wj}^{3}\right)}\right) \]

   12. Applied egg-rr 100.0%

       \[\leadsto x + \left(wj \cdot \left(-1 \cdot x - x\right) + \color{blue}{\mathsf{fma}\left(wj, wj, -{wj}^{3}\right)}\right) \]

   if 4.4999999999999998e-7 < wj

   1. Initial program 56.9%

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

      1. div-sub 56.9%

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

      2. associate-/l* 56.9%

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

      3. distribute-rgt1-in 56.9%

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

      4. associate-/l* 56.9%

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

      5. *-inverses 99.8%

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

      6. /-rgt-identity 99.8%

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

      7. distribute-rgt1-in 99.8%

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

      8. associate-/l/ 99.8%

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

      9. div-sub 99.8%

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

   3. Simplified 99.8%

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

      1. clear-num 99.8%

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

      2. associate-/r/ 100.0%

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

   5. Applied egg-rr 100.0%

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

4. Final simplification 100.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;wj \leq -4 \cdot 10^{-8}:\\ \;\;\;\;wj - \frac{wj \cdot e^{wj} - x}{e^{wj} \cdot \left(wj + 1\right)}\\ \mathbf{elif}\;wj \leq 4.5 \cdot 10^{-7}:\\ \;\;\;\;x + \left(\mathsf{fma}\left(wj, wj, -{wj}^{3}\right) - wj \cdot \left(x + x\right)\right)\\ \mathbf{else}:\\ \;\;\;\;wj + \frac{1}{wj + 1} \cdot \left(\frac{x}{e^{wj}} - wj\right)\\ \end{array} \]

Alternative 2: 98.8% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := wj \cdot e^{wj}\\ \mathbf{if}\;wj + \frac{x - t_0}{e^{wj} + t_0} \leq 10^{-15}:\\ \;\;\;\;x + \left(\left({wj}^{2} \cdot \left(x + \left(x - -1\right)\right) + {wj}^{3} \cdot \left(\left(-1 - x\right) - x\right)\right) - wj \cdot \left(x + x\right)\right)\\ \mathbf{else}:\\ \;\;\;\;wj + \frac{1}{wj + 1} \cdot \left(\frac{x}{e^{wj}} - wj\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (wj x)
 :precision binary64
 (let* ((t_0 (* wj (exp wj))))
   (if (<= (+ wj (/ (- x t_0) (+ (exp wj) t_0))) 1e-15)
     (+
      x
      (-
       (+ (* (pow wj 2.0) (+ x (- x -1.0))) (* (pow wj 3.0) (- (- -1.0 x) x)))
       (* wj (+ x x))))
     (+ wj (* (/ 1.0 (+ wj 1.0)) (- (/ x (exp wj)) wj))))))

C

double code(double wj, double x) {
	double t_0 = wj * exp(wj);
	double tmp;
	if ((wj + ((x - t_0) / (exp(wj) + t_0))) <= 1e-15) {
		tmp = x + (((pow(wj, 2.0) * (x + (x - -1.0))) + (pow(wj, 3.0) * ((-1.0 - x) - x))) - (wj * (x + x)));
	} else {
		tmp = wj + ((1.0 / (wj + 1.0)) * ((x / exp(wj)) - wj));
	}
	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 = wj * exp(wj)
    if ((wj + ((x - t_0) / (exp(wj) + t_0))) <= 1d-15) then
        tmp = x + ((((wj ** 2.0d0) * (x + (x - (-1.0d0)))) + ((wj ** 3.0d0) * (((-1.0d0) - x) - x))) - (wj * (x + x)))
    else
        tmp = wj + ((1.0d0 / (wj + 1.0d0)) * ((x / exp(wj)) - wj))
    end if
    code = tmp
end function

Java

public static double code(double wj, double x) {
	double t_0 = wj * Math.exp(wj);
	double tmp;
	if ((wj + ((x - t_0) / (Math.exp(wj) + t_0))) <= 1e-15) {
		tmp = x + (((Math.pow(wj, 2.0) * (x + (x - -1.0))) + (Math.pow(wj, 3.0) * ((-1.0 - x) - x))) - (wj * (x + x)));
	} else {
		tmp = wj + ((1.0 / (wj + 1.0)) * ((x / Math.exp(wj)) - wj));
	}
	return tmp;
}

Python

def code(wj, x):
	t_0 = wj * math.exp(wj)
	tmp = 0
	if (wj + ((x - t_0) / (math.exp(wj) + t_0))) <= 1e-15:
		tmp = x + (((math.pow(wj, 2.0) * (x + (x - -1.0))) + (math.pow(wj, 3.0) * ((-1.0 - x) - x))) - (wj * (x + x)))
	else:
		tmp = wj + ((1.0 / (wj + 1.0)) * ((x / math.exp(wj)) - wj))
	return tmp

Julia

function code(wj, x)
	t_0 = Float64(wj * exp(wj))
	tmp = 0.0
	if (Float64(wj + Float64(Float64(x - t_0) / Float64(exp(wj) + t_0))) <= 1e-15)
		tmp = Float64(x + Float64(Float64(Float64((wj ^ 2.0) * Float64(x + Float64(x - -1.0))) + Float64((wj ^ 3.0) * Float64(Float64(-1.0 - x) - x))) - Float64(wj * Float64(x + x))));
	else
		tmp = Float64(wj + Float64(Float64(1.0 / Float64(wj + 1.0)) * Float64(Float64(x / exp(wj)) - wj)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(wj, x)
	t_0 = wj * exp(wj);
	tmp = 0.0;
	if ((wj + ((x - t_0) / (exp(wj) + t_0))) <= 1e-15)
		tmp = x + ((((wj ^ 2.0) * (x + (x - -1.0))) + ((wj ^ 3.0) * ((-1.0 - x) - x))) - (wj * (x + x)));
	else
		tmp = wj + ((1.0 / (wj + 1.0)) * ((x / exp(wj)) - wj));
	end
	tmp_2 = tmp;
end

Wolfram

code[wj_, x_] := Block[{t$95$0 = N[(wj * N[Exp[wj], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(wj + N[(N[(x - t$95$0), $MachinePrecision] / N[(N[Exp[wj], $MachinePrecision] + t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 1e-15], N[(x + N[(N[(N[(N[Power[wj, 2.0], $MachinePrecision] * N[(x + N[(x - -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[Power[wj, 3.0], $MachinePrecision] * N[(N[(-1.0 - x), $MachinePrecision] - x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(wj * N[(x + x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(wj + N[(N[(1.0 / N[(wj + 1.0), $MachinePrecision]), $MachinePrecision] * N[(N[(x / N[Exp[wj], $MachinePrecision]), $MachinePrecision] - wj), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 wj (/.f64 (-.f64 (*.f64 wj (exp.f64 wj)) x) (+.f64 (exp.f64 wj) (*.f64 wj (exp.f64 wj))))) < 1.0000000000000001e-15

   1. Initial program 74.0%

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

      1. div-sub 74.0%

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

      2. associate-/l* 74.0%

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

      3. distribute-rgt1-in 74.0%

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

      4. associate-/l* 74.0%

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

      5. *-inverses 74.0%

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

      6. /-rgt-identity 74.0%

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

      7. distribute-rgt1-in 74.0%

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

      8. associate-/l/ 74.0%

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

      9. div-sub 74.0%

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

   3. Simplified 74.0%

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

   4. Taylor expanded in wj around 0 72.7%

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

      1. associate-*r* 72.7%

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

      2. neg-mul-1 72.7%

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

      3. distribute-rgt1-in 72.7%

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

      4. +-commutative 72.7%

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

      5. sub-neg 72.7%

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

   6. Simplified 72.7%

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

   7. Taylor expanded in wj around 0 98.0%

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

   if 1.0000000000000001e-15 < (-.f64 wj (/.f64 (-.f64 (*.f64 wj (exp.f64 wj)) x) (+.f64 (exp.f64 wj) (*.f64 wj (exp.f64 wj)))))

   1. Initial program 92.8%

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

      1. div-sub 92.8%

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

      2. associate-/l* 92.8%

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

      3. distribute-rgt1-in 92.8%

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

      4. associate-/l* 92.8%

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

      5. *-inverses 97.1%

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

      6. /-rgt-identity 97.1%

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

      7. distribute-rgt1-in 99.9%

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

      8. associate-/l/ 99.9%

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

      9. div-sub 99.9%

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

   3. Simplified 99.9%

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

      1. clear-num 99.7%

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

      2. associate-/r/ 99.9%

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

   5. Applied egg-rr 99.9%

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

4. Final simplification 98.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;wj + \frac{x - wj \cdot e^{wj}}{e^{wj} + wj \cdot e^{wj}} \leq 10^{-15}:\\ \;\;\;\;x + \left(\left({wj}^{2} \cdot \left(x + \left(x - -1\right)\right) + {wj}^{3} \cdot \left(\left(-1 - x\right) - x\right)\right) - wj \cdot \left(x + x\right)\right)\\ \mathbf{else}:\\ \;\;\;\;wj + \frac{1}{wj + 1} \cdot \left(\frac{x}{e^{wj}} - wj\right)\\ \end{array} \]

Alternative 3: 99.8% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;wj \leq -4 \cdot 10^{-8}:\\ \;\;\;\;wj - \frac{wj \cdot e^{wj} - x}{e^{wj} \cdot \left(wj + 1\right)}\\ \mathbf{elif}\;wj \leq 3.75 \cdot 10^{-7}:\\ \;\;\;\;x + \left(\left(wj \cdot wj - {wj}^{3}\right) - wj \cdot \left(x + x\right)\right)\\ \mathbf{else}:\\ \;\;\;\;wj + \frac{1}{wj + 1} \cdot \left(\frac{x}{e^{wj}} - wj\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (wj x)
 :precision binary64
 (if (<= wj -4e-8)
   (- wj (/ (- (* wj (exp wj)) x) (* (exp wj) (+ wj 1.0))))
   (if (<= wj 3.75e-7)
     (+ x (- (- (* wj wj) (pow wj 3.0)) (* wj (+ x x))))
     (+ wj (* (/ 1.0 (+ wj 1.0)) (- (/ x (exp wj)) wj))))))

C

double code(double wj, double x) {
	double tmp;
	if (wj <= -4e-8) {
		tmp = wj - (((wj * exp(wj)) - x) / (exp(wj) * (wj + 1.0)));
	} else if (wj <= 3.75e-7) {
		tmp = x + (((wj * wj) - pow(wj, 3.0)) - (wj * (x + x)));
	} else {
		tmp = wj + ((1.0 / (wj + 1.0)) * ((x / exp(wj)) - 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 <= (-4d-8)) then
        tmp = wj - (((wj * exp(wj)) - x) / (exp(wj) * (wj + 1.0d0)))
    else if (wj <= 3.75d-7) then
        tmp = x + (((wj * wj) - (wj ** 3.0d0)) - (wj * (x + x)))
    else
        tmp = wj + ((1.0d0 / (wj + 1.0d0)) * ((x / exp(wj)) - wj))
    end if
    code = tmp
end function

Java

public static double code(double wj, double x) {
	double tmp;
	if (wj <= -4e-8) {
		tmp = wj - (((wj * Math.exp(wj)) - x) / (Math.exp(wj) * (wj + 1.0)));
	} else if (wj <= 3.75e-7) {
		tmp = x + (((wj * wj) - Math.pow(wj, 3.0)) - (wj * (x + x)));
	} else {
		tmp = wj + ((1.0 / (wj + 1.0)) * ((x / Math.exp(wj)) - wj));
	}
	return tmp;
}

Python

def code(wj, x):
	tmp = 0
	if wj <= -4e-8:
		tmp = wj - (((wj * math.exp(wj)) - x) / (math.exp(wj) * (wj + 1.0)))
	elif wj <= 3.75e-7:
		tmp = x + (((wj * wj) - math.pow(wj, 3.0)) - (wj * (x + x)))
	else:
		tmp = wj + ((1.0 / (wj + 1.0)) * ((x / math.exp(wj)) - wj))
	return tmp

Julia

function code(wj, x)
	tmp = 0.0
	if (wj <= -4e-8)
		tmp = Float64(wj - Float64(Float64(Float64(wj * exp(wj)) - x) / Float64(exp(wj) * Float64(wj + 1.0))));
	elseif (wj <= 3.75e-7)
		tmp = Float64(x + Float64(Float64(Float64(wj * wj) - (wj ^ 3.0)) - Float64(wj * Float64(x + x))));
	else
		tmp = Float64(wj + Float64(Float64(1.0 / Float64(wj + 1.0)) * Float64(Float64(x / exp(wj)) - wj)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(wj, x)
	tmp = 0.0;
	if (wj <= -4e-8)
		tmp = wj - (((wj * exp(wj)) - x) / (exp(wj) * (wj + 1.0)));
	elseif (wj <= 3.75e-7)
		tmp = x + (((wj * wj) - (wj ^ 3.0)) - (wj * (x + x)));
	else
		tmp = wj + ((1.0 / (wj + 1.0)) * ((x / exp(wj)) - wj));
	end
	tmp_2 = tmp;
end

Wolfram

code[wj_, x_] := If[LessEqual[wj, -4e-8], N[(wj - N[(N[(N[(wj * N[Exp[wj], $MachinePrecision]), $MachinePrecision] - x), $MachinePrecision] / N[(N[Exp[wj], $MachinePrecision] * N[(wj + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[wj, 3.75e-7], N[(x + N[(N[(N[(wj * wj), $MachinePrecision] - N[Power[wj, 3.0], $MachinePrecision]), $MachinePrecision] - N[(wj * N[(x + x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(wj + N[(N[(1.0 / N[(wj + 1.0), $MachinePrecision]), $MachinePrecision] * N[(N[(x / N[Exp[wj], $MachinePrecision]), $MachinePrecision] - wj), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if wj < -4.0000000000000001e-8

   1. Initial program 74.3%

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

      1. distribute-rgt1-in 99.3%

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

   3. Simplified 99.3%

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

   if -4.0000000000000001e-8 < wj < 3.7500000000000001e-7

   1. Initial program 79.9%

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

      1. div-sub 79.9%

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

      2. associate-/l* 80.0%

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

      3. distribute-rgt1-in 80.0%

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

      4. associate-/l* 80.0%

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

      5. *-inverses 80.0%

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

      6. /-rgt-identity 80.0%

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

      7. distribute-rgt1-in 80.0%

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

      8. associate-/l/ 79.9%

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

      9. div-sub 79.9%

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

   3. Simplified 79.9%

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

   4. Taylor expanded in wj around 0 79.9%

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

      1. associate-*r* 79.9%

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

      2. neg-mul-1 79.9%

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

      3. distribute-rgt1-in 79.9%

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

      4. +-commutative 79.9%

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

      5. sub-neg 79.9%

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

   6. Simplified 79.9%

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

   7. Taylor expanded in wj around 0 100.0%

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

   8. Taylor expanded in x around 0 100.0%

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

      1. +-commutative 100.0%

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

      2. mul-1-neg 100.0%

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

      3. unsub-neg 100.0%

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

      4. unpow2 100.0%

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

   10. Simplified 100.0%

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

   if 3.7500000000000001e-7 < wj

   1. Initial program 56.9%

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

      1. div-sub 56.9%

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

      2. associate-/l* 56.9%

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

      3. distribute-rgt1-in 56.9%

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

      4. associate-/l* 56.9%

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

      5. *-inverses 99.8%

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

      6. /-rgt-identity 99.8%

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

      7. distribute-rgt1-in 99.8%

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

      8. associate-/l/ 99.8%

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

      9. div-sub 99.8%

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

   3. Simplified 99.8%

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

      1. clear-num 99.8%

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

      2. associate-/r/ 100.0%

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

   5. Applied egg-rr 100.0%

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

4. Final simplification 100.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;wj \leq -4 \cdot 10^{-8}:\\ \;\;\;\;wj - \frac{wj \cdot e^{wj} - x}{e^{wj} \cdot \left(wj + 1\right)}\\ \mathbf{elif}\;wj \leq 3.75 \cdot 10^{-7}:\\ \;\;\;\;x + \left(\left(wj \cdot wj - {wj}^{3}\right) - wj \cdot \left(x + x\right)\right)\\ \mathbf{else}:\\ \;\;\;\;wj + \frac{1}{wj + 1} \cdot \left(\frac{x}{e^{wj}} - wj\right)\\ \end{array} \]

Alternative 4: 99.8% accurate, 2.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x}{e^{wj}} - wj\\ \mathbf{if}\;wj \leq -4 \cdot 10^{-8}:\\ \;\;\;\;wj + \frac{t_0}{wj + 1}\\ \mathbf{elif}\;wj \leq 3 \cdot 10^{-7}:\\ \;\;\;\;x + \left(\left(wj \cdot wj - {wj}^{3}\right) - wj \cdot \left(x + x\right)\right)\\ \mathbf{else}:\\ \;\;\;\;wj + \frac{1}{wj + 1} \cdot t_0\\ \end{array} \end{array} \]

FPCore

(FPCore (wj x)
 :precision binary64
 (let* ((t_0 (- (/ x (exp wj)) wj)))
   (if (<= wj -4e-8)
     (+ wj (/ t_0 (+ wj 1.0)))
     (if (<= wj 3e-7)
       (+ x (- (- (* wj wj) (pow wj 3.0)) (* wj (+ x x))))
       (+ wj (* (/ 1.0 (+ wj 1.0)) t_0))))))

C

double code(double wj, double x) {
	double t_0 = (x / exp(wj)) - wj;
	double tmp;
	if (wj <= -4e-8) {
		tmp = wj + (t_0 / (wj + 1.0));
	} else if (wj <= 3e-7) {
		tmp = x + (((wj * wj) - pow(wj, 3.0)) - (wj * (x + x)));
	} else {
		tmp = wj + ((1.0 / (wj + 1.0)) * t_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 / exp(wj)) - wj
    if (wj <= (-4d-8)) then
        tmp = wj + (t_0 / (wj + 1.0d0))
    else if (wj <= 3d-7) then
        tmp = x + (((wj * wj) - (wj ** 3.0d0)) - (wj * (x + x)))
    else
        tmp = wj + ((1.0d0 / (wj + 1.0d0)) * t_0)
    end if
    code = tmp
end function

Java

public static double code(double wj, double x) {
	double t_0 = (x / Math.exp(wj)) - wj;
	double tmp;
	if (wj <= -4e-8) {
		tmp = wj + (t_0 / (wj + 1.0));
	} else if (wj <= 3e-7) {
		tmp = x + (((wj * wj) - Math.pow(wj, 3.0)) - (wj * (x + x)));
	} else {
		tmp = wj + ((1.0 / (wj + 1.0)) * t_0);
	}
	return tmp;
}

Python

def code(wj, x):
	t_0 = (x / math.exp(wj)) - wj
	tmp = 0
	if wj <= -4e-8:
		tmp = wj + (t_0 / (wj + 1.0))
	elif wj <= 3e-7:
		tmp = x + (((wj * wj) - math.pow(wj, 3.0)) - (wj * (x + x)))
	else:
		tmp = wj + ((1.0 / (wj + 1.0)) * t_0)
	return tmp

Julia

function code(wj, x)
	t_0 = Float64(Float64(x / exp(wj)) - wj)
	tmp = 0.0
	if (wj <= -4e-8)
		tmp = Float64(wj + Float64(t_0 / Float64(wj + 1.0)));
	elseif (wj <= 3e-7)
		tmp = Float64(x + Float64(Float64(Float64(wj * wj) - (wj ^ 3.0)) - Float64(wj * Float64(x + x))));
	else
		tmp = Float64(wj + Float64(Float64(1.0 / Float64(wj + 1.0)) * t_0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(wj, x)
	t_0 = (x / exp(wj)) - wj;
	tmp = 0.0;
	if (wj <= -4e-8)
		tmp = wj + (t_0 / (wj + 1.0));
	elseif (wj <= 3e-7)
		tmp = x + (((wj * wj) - (wj ^ 3.0)) - (wj * (x + x)));
	else
		tmp = wj + ((1.0 / (wj + 1.0)) * t_0);
	end
	tmp_2 = tmp;
end

Wolfram

code[wj_, x_] := Block[{t$95$0 = N[(N[(x / N[Exp[wj], $MachinePrecision]), $MachinePrecision] - wj), $MachinePrecision]}, If[LessEqual[wj, -4e-8], N[(wj + N[(t$95$0 / N[(wj + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[wj, 3e-7], N[(x + N[(N[(N[(wj * wj), $MachinePrecision] - N[Power[wj, 3.0], $MachinePrecision]), $MachinePrecision] - N[(wj * N[(x + x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(wj + N[(N[(1.0 / N[(wj + 1.0), $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if wj < -4.0000000000000001e-8

   1. Initial program 74.3%

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

      1. div-sub 74.3%

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

      2. associate-/l* 74.3%

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

      3. distribute-rgt1-in 74.3%

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

      4. associate-/l* 74.3%

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

      5. *-inverses 74.3%

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

      6. /-rgt-identity 74.3%

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

      7. distribute-rgt1-in 99.3%

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

      8. associate-/l/ 99.1%

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

      9. div-sub 99.1%

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

   3. Simplified 99.1%

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

   if -4.0000000000000001e-8 < wj < 2.9999999999999999e-7

   1. Initial program 79.9%

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

      1. div-sub 79.9%

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

      2. associate-/l* 80.0%

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

      3. distribute-rgt1-in 80.0%

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

      4. associate-/l* 80.0%

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

      5. *-inverses 80.0%

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

      6. /-rgt-identity 80.0%

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

      7. distribute-rgt1-in 80.0%

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

      8. associate-/l/ 79.9%

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

      9. div-sub 79.9%

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

   3. Simplified 79.9%

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

   4. Taylor expanded in wj around 0 79.9%

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

      1. associate-*r* 79.9%

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

      2. neg-mul-1 79.9%

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

      3. distribute-rgt1-in 79.9%

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

      4. +-commutative 79.9%

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

      5. sub-neg 79.9%

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

   6. Simplified 79.9%

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

   7. Taylor expanded in wj around 0 100.0%

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

   8. Taylor expanded in x around 0 100.0%

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

      1. +-commutative 100.0%

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

      2. mul-1-neg 100.0%

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

      3. unsub-neg 100.0%

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

      4. unpow2 100.0%

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

   10. Simplified 100.0%

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

   if 2.9999999999999999e-7 < wj

   1. Initial program 56.9%

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

      1. div-sub 56.9%

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

      2. associate-/l* 56.9%

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

      3. distribute-rgt1-in 56.9%

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

      4. associate-/l* 56.9%

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

      5. *-inverses 99.8%

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

      6. /-rgt-identity 99.8%

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

      7. distribute-rgt1-in 99.8%

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

      8. associate-/l/ 99.8%

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

      9. div-sub 99.8%

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

   3. Simplified 99.8%

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

      1. clear-num 99.8%

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

      2. associate-/r/ 100.0%

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

   5. Applied egg-rr 100.0%

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

4. Final simplification 100.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;wj \leq -4 \cdot 10^{-8}:\\ \;\;\;\;wj + \frac{\frac{x}{e^{wj}} - wj}{wj + 1}\\ \mathbf{elif}\;wj \leq 3 \cdot 10^{-7}:\\ \;\;\;\;x + \left(\left(wj \cdot wj - {wj}^{3}\right) - wj \cdot \left(x + x\right)\right)\\ \mathbf{else}:\\ \;\;\;\;wj + \frac{1}{wj + 1} \cdot \left(\frac{x}{e^{wj}} - wj\right)\\ \end{array} \]

Alternative 5: 99.3% accurate, 2.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x}{e^{wj}} - wj\\ \mathbf{if}\;wj \leq -8 \cdot 10^{-14}:\\ \;\;\;\;wj + \frac{t_0}{wj + 1}\\ \mathbf{elif}\;wj \leq 1.8 \cdot 10^{-9}:\\ \;\;\;\;x + \left(wj \cdot \left(wj + \left(x + x\right)\right) - {wj}^{3}\right)\\ \mathbf{else}:\\ \;\;\;\;wj + \frac{1}{wj + 1} \cdot t_0\\ \end{array} \end{array} \]

FPCore

(FPCore (wj x)
 :precision binary64
 (let* ((t_0 (- (/ x (exp wj)) wj)))
   (if (<= wj -8e-14)
     (+ wj (/ t_0 (+ wj 1.0)))
     (if (<= wj 1.8e-9)
       (+ x (- (* wj (+ wj (+ x x))) (pow wj 3.0)))
       (+ wj (* (/ 1.0 (+ wj 1.0)) t_0))))))

C

double code(double wj, double x) {
	double t_0 = (x / exp(wj)) - wj;
	double tmp;
	if (wj <= -8e-14) {
		tmp = wj + (t_0 / (wj + 1.0));
	} else if (wj <= 1.8e-9) {
		tmp = x + ((wj * (wj + (x + x))) - pow(wj, 3.0));
	} else {
		tmp = wj + ((1.0 / (wj + 1.0)) * t_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 / exp(wj)) - wj
    if (wj <= (-8d-14)) then
        tmp = wj + (t_0 / (wj + 1.0d0))
    else if (wj <= 1.8d-9) then
        tmp = x + ((wj * (wj + (x + x))) - (wj ** 3.0d0))
    else
        tmp = wj + ((1.0d0 / (wj + 1.0d0)) * t_0)
    end if
    code = tmp
end function

Java

public static double code(double wj, double x) {
	double t_0 = (x / Math.exp(wj)) - wj;
	double tmp;
	if (wj <= -8e-14) {
		tmp = wj + (t_0 / (wj + 1.0));
	} else if (wj <= 1.8e-9) {
		tmp = x + ((wj * (wj + (x + x))) - Math.pow(wj, 3.0));
	} else {
		tmp = wj + ((1.0 / (wj + 1.0)) * t_0);
	}
	return tmp;
}

Python

def code(wj, x):
	t_0 = (x / math.exp(wj)) - wj
	tmp = 0
	if wj <= -8e-14:
		tmp = wj + (t_0 / (wj + 1.0))
	elif wj <= 1.8e-9:
		tmp = x + ((wj * (wj + (x + x))) - math.pow(wj, 3.0))
	else:
		tmp = wj + ((1.0 / (wj + 1.0)) * t_0)
	return tmp

Julia

function code(wj, x)
	t_0 = Float64(Float64(x / exp(wj)) - wj)
	tmp = 0.0
	if (wj <= -8e-14)
		tmp = Float64(wj + Float64(t_0 / Float64(wj + 1.0)));
	elseif (wj <= 1.8e-9)
		tmp = Float64(x + Float64(Float64(wj * Float64(wj + Float64(x + x))) - (wj ^ 3.0)));
	else
		tmp = Float64(wj + Float64(Float64(1.0 / Float64(wj + 1.0)) * t_0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(wj, x)
	t_0 = (x / exp(wj)) - wj;
	tmp = 0.0;
	if (wj <= -8e-14)
		tmp = wj + (t_0 / (wj + 1.0));
	elseif (wj <= 1.8e-9)
		tmp = x + ((wj * (wj + (x + x))) - (wj ^ 3.0));
	else
		tmp = wj + ((1.0 / (wj + 1.0)) * t_0);
	end
	tmp_2 = tmp;
end

Wolfram

code[wj_, x_] := Block[{t$95$0 = N[(N[(x / N[Exp[wj], $MachinePrecision]), $MachinePrecision] - wj), $MachinePrecision]}, If[LessEqual[wj, -8e-14], N[(wj + N[(t$95$0 / N[(wj + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[wj, 1.8e-9], N[(x + N[(N[(wj * N[(wj + N[(x + x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[Power[wj, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(wj + N[(N[(1.0 / N[(wj + 1.0), $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if wj < -7.99999999999999999e-14

   1. Initial program 81.2%

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

      1. div-sub 81.2%

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

      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 81.7%

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

      6. /-rgt-identity 81.7%

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

      7. distribute-rgt1-in 97.0%

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

      8. associate-/l/ 96.6%

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

      9. div-sub 96.6%

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

   3. Simplified 96.6%

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

   if -7.99999999999999999e-14 < wj < 1.8e-9

   1. Initial program 79.7%

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

      1. div-sub 79.7%

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

      2. associate-/l* 79.7%

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

      3. distribute-rgt1-in 79.7%

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

      4. associate-/l* 79.7%

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

      5. *-inverses 79.7%

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

      6. /-rgt-identity 79.7%

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

      7. distribute-rgt1-in 79.7%

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

      8. associate-/l/ 79.7%

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

      9. div-sub 79.7%

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

   3. Simplified 79.7%

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

   4. Taylor expanded in wj around 0 79.7%

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

      1. associate-*r* 79.7%

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

      2. neg-mul-1 79.7%

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

      3. distribute-rgt1-in 79.7%

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

      4. +-commutative 79.7%

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

      5. sub-neg 79.7%

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

   6. Simplified 79.7%

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

   7. Taylor expanded in wj around 0 100.0%

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

   8. Taylor expanded in x around 0 100.0%

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

      1. +-commutative 100.0%

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

      2. mul-1-neg 100.0%

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

      3. unsub-neg 100.0%

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

      4. unpow2 100.0%

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

   10. Simplified 100.0%

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

       1. associate-+r- 100.0%

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

       2. fma-def 100.0%

          \[\leadsto x + \left(\color{blue}{\mathsf{fma}\left(wj, -1 \cdot x - x, wj \cdot wj\right)} - {wj}^{3}\right) \]

       3. sub-neg 100.0%

          \[\leadsto x + \left(\mathsf{fma}\left(wj, \color{blue}{-1 \cdot x + \left(-x\right)}, wj \cdot wj\right) - {wj}^{3}\right) \]

       4. add-sqr-sqrt 47.0%

          \[\leadsto x + \left(\mathsf{fma}\left(wj, \color{blue}{\sqrt{-1 \cdot x} \cdot \sqrt{-1 \cdot x}} + \left(-x\right), wj \cdot wj\right) - {wj}^{3}\right) \]

       5. sqrt-unprod 75.0%

          \[\leadsto x + \left(\mathsf{fma}\left(wj, \color{blue}{\sqrt{\left(-1 \cdot x\right) \cdot \left(-1 \cdot x\right)}} + \left(-x\right), wj \cdot wj\right) - {wj}^{3}\right) \]

       6. mul-1-neg 75.0%

          \[\leadsto x + \left(\mathsf{fma}\left(wj, \sqrt{\color{blue}{\left(-x\right)} \cdot \left(-1 \cdot x\right)} + \left(-x\right), wj \cdot wj\right) - {wj}^{3}\right) \]

       7. mul-1-neg 75.0%

          \[\leadsto x + \left(\mathsf{fma}\left(wj, \sqrt{\left(-x\right) \cdot \color{blue}{\left(-x\right)}} + \left(-x\right), wj \cdot wj\right) - {wj}^{3}\right) \]

       8. sqr-neg 75.0%

          \[\leadsto x + \left(\mathsf{fma}\left(wj, \sqrt{\color{blue}{x \cdot x}} + \left(-x\right), wj \cdot wj\right) - {wj}^{3}\right) \]

       9. sqrt-prod 53.0%

          \[\leadsto x + \left(\mathsf{fma}\left(wj, \color{blue}{\sqrt{x} \cdot \sqrt{x}} + \left(-x\right), wj \cdot wj\right) - {wj}^{3}\right) \]

       10. add-sqr-sqrt 100.0%

           \[\leadsto x + \left(\mathsf{fma}\left(wj, \color{blue}{x} + \left(-x\right), wj \cdot wj\right) - {wj}^{3}\right) \]

       11. mul-1-neg 100.0%

           \[\leadsto x + \left(\mathsf{fma}\left(wj, x + \color{blue}{-1 \cdot x}, wj \cdot wj\right) - {wj}^{3}\right) \]

       12. add-sqr-sqrt 47.0%

           \[\leadsto x + \left(\mathsf{fma}\left(wj, x + \color{blue}{\sqrt{-1 \cdot x} \cdot \sqrt{-1 \cdot x}}, wj \cdot wj\right) - {wj}^{3}\right) \]

       13. sqrt-unprod 75.0%

           \[\leadsto x + \left(\mathsf{fma}\left(wj, x + \color{blue}{\sqrt{\left(-1 \cdot x\right) \cdot \left(-1 \cdot x\right)}}, wj \cdot wj\right) - {wj}^{3}\right) \]

       14. mul-1-neg 75.0%

           \[\leadsto x + \left(\mathsf{fma}\left(wj, x + \sqrt{\color{blue}{\left(-x\right)} \cdot \left(-1 \cdot x\right)}, wj \cdot wj\right) - {wj}^{3}\right) \]

       15. mul-1-neg 75.0%

           \[\leadsto x + \left(\mathsf{fma}\left(wj, x + \sqrt{\left(-x\right) \cdot \color{blue}{\left(-x\right)}}, wj \cdot wj\right) - {wj}^{3}\right) \]

       16. sqr-neg 75.0%

           \[\leadsto x + \left(\mathsf{fma}\left(wj, x + \sqrt{\color{blue}{x \cdot x}}, wj \cdot wj\right) - {wj}^{3}\right) \]

       17. sqrt-prod 53.0%

           \[\leadsto x + \left(\mathsf{fma}\left(wj, x + \color{blue}{\sqrt{x} \cdot \sqrt{x}}, wj \cdot wj\right) - {wj}^{3}\right) \]

       18. add-sqr-sqrt 100.0%

           \[\leadsto x + \left(\mathsf{fma}\left(wj, x + \color{blue}{x}, wj \cdot wj\right) - {wj}^{3}\right) \]

   12. Applied egg-rr 100.0%

       \[\leadsto x + \color{blue}{\left(\mathsf{fma}\left(wj, x + x, wj \cdot wj\right) - {wj}^{3}\right)} \]
   13. Step-by-step derivation

       1. fma-udef 100.0%

          \[\leadsto x + \left(\color{blue}{\left(wj \cdot \left(x + x\right) + wj \cdot wj\right)} - {wj}^{3}\right) \]

       2. distribute-lft-out 100.0%

          \[\leadsto x + \left(\color{blue}{wj \cdot \left(\left(x + x\right) + wj\right)} - {wj}^{3}\right) \]

   14. Simplified 100.0%

       \[\leadsto x + \color{blue}{\left(wj \cdot \left(\left(x + x\right) + wj\right) - {wj}^{3}\right)} \]

   if 1.8e-9 < wj

   1. Initial program 56.9%

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

      1. div-sub 56.9%

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

      2. associate-/l* 56.9%

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

      3. distribute-rgt1-in 56.9%

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

      4. associate-/l* 56.9%

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

      5. *-inverses 99.8%

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

      6. /-rgt-identity 99.8%

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

      7. distribute-rgt1-in 99.8%

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

      8. associate-/l/ 99.8%

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

      9. div-sub 99.8%

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

   3. Simplified 99.8%

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

      1. clear-num 99.8%

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

      2. associate-/r/ 100.0%

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

   5. Applied egg-rr 100.0%

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

4. Final simplification 99.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;wj \leq -8 \cdot 10^{-14}:\\ \;\;\;\;wj + \frac{\frac{x}{e^{wj}} - wj}{wj + 1}\\ \mathbf{elif}\;wj \leq 1.8 \cdot 10^{-9}:\\ \;\;\;\;x + \left(wj \cdot \left(wj + \left(x + x\right)\right) - {wj}^{3}\right)\\ \mathbf{else}:\\ \;\;\;\;wj + \frac{1}{wj + 1} \cdot \left(\frac{x}{e^{wj}} - wj\right)\\ \end{array} \]

Alternative 6: 99.3% accurate, 2.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;wj \leq -8 \cdot 10^{-14} \lor \neg \left(wj \leq 2.45 \cdot 10^{-8}\right):\\ \;\;\;\;wj + \frac{\frac{x}{e^{wj}} - wj}{wj + 1}\\ \mathbf{else}:\\ \;\;\;\;x + \left(wj \cdot \left(wj + \left(x + x\right)\right) - {wj}^{3}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (wj x)
 :precision binary64
 (if (or (<= wj -8e-14) (not (<= wj 2.45e-8)))
   (+ wj (/ (- (/ x (exp wj)) wj) (+ wj 1.0)))
   (+ x (- (* wj (+ wj (+ x x))) (pow wj 3.0)))))

C

double code(double wj, double x) {
	double tmp;
	if ((wj <= -8e-14) || !(wj <= 2.45e-8)) {
		tmp = wj + (((x / exp(wj)) - wj) / (wj + 1.0));
	} else {
		tmp = x + ((wj * (wj + (x + x))) - pow(wj, 3.0));
	}
	return tmp;
}

Fortran

real(8) function code(wj, x)
    real(8), intent (in) :: wj
    real(8), intent (in) :: x
    real(8) :: tmp
    if ((wj <= (-8d-14)) .or. (.not. (wj <= 2.45d-8))) then
        tmp = wj + (((x / exp(wj)) - wj) / (wj + 1.0d0))
    else
        tmp = x + ((wj * (wj + (x + x))) - (wj ** 3.0d0))
    end if
    code = tmp
end function

Java

public static double code(double wj, double x) {
	double tmp;
	if ((wj <= -8e-14) || !(wj <= 2.45e-8)) {
		tmp = wj + (((x / Math.exp(wj)) - wj) / (wj + 1.0));
	} else {
		tmp = x + ((wj * (wj + (x + x))) - Math.pow(wj, 3.0));
	}
	return tmp;
}

Python

def code(wj, x):
	tmp = 0
	if (wj <= -8e-14) or not (wj <= 2.45e-8):
		tmp = wj + (((x / math.exp(wj)) - wj) / (wj + 1.0))
	else:
		tmp = x + ((wj * (wj + (x + x))) - math.pow(wj, 3.0))
	return tmp

Julia

function code(wj, x)
	tmp = 0.0
	if ((wj <= -8e-14) || !(wj <= 2.45e-8))
		tmp = Float64(wj + Float64(Float64(Float64(x / exp(wj)) - wj) / Float64(wj + 1.0)));
	else
		tmp = Float64(x + Float64(Float64(wj * Float64(wj + Float64(x + x))) - (wj ^ 3.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(wj, x)
	tmp = 0.0;
	if ((wj <= -8e-14) || ~((wj <= 2.45e-8)))
		tmp = wj + (((x / exp(wj)) - wj) / (wj + 1.0));
	else
		tmp = x + ((wj * (wj + (x + x))) - (wj ^ 3.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[wj_, x_] := If[Or[LessEqual[wj, -8e-14], N[Not[LessEqual[wj, 2.45e-8]], $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[(wj + N[(x + x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[Power[wj, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if wj < -7.99999999999999999e-14 or 2.4500000000000001e-8 < wj

   1. Initial program 72.7%

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

      1. div-sub 72.7%

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

      2. associate-/l* 73.0%

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

      3. distribute-rgt1-in 73.0%

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

      4. associate-/l* 73.0%

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

      5. *-inverses 88.0%

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

      6. /-rgt-identity 88.0%

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

      7. distribute-rgt1-in 98.0%

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

      8. associate-/l/ 97.7%

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

      9. div-sub 97.7%

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

   3. Simplified 97.7%

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

   if -7.99999999999999999e-14 < wj < 2.4500000000000001e-8

   1. Initial program 79.7%

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

      1. div-sub 79.7%

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

      2. associate-/l* 79.7%

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

      3. distribute-rgt1-in 79.7%

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

      4. associate-/l* 79.7%

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

      5. *-inverses 79.7%

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

      6. /-rgt-identity 79.7%

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

      7. distribute-rgt1-in 79.7%

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

      8. associate-/l/ 79.7%

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

      9. div-sub 79.7%

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

   3. Simplified 79.7%

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

   4. Taylor expanded in wj around 0 79.7%

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

      1. associate-*r* 79.7%

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

      2. neg-mul-1 79.7%

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

      3. distribute-rgt1-in 79.7%

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

      4. +-commutative 79.7%

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

      5. sub-neg 79.7%

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

   6. Simplified 79.7%

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

   7. Taylor expanded in wj around 0 100.0%

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

   8. Taylor expanded in x around 0 100.0%

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

      1. +-commutative 100.0%

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

      2. mul-1-neg 100.0%

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

      3. unsub-neg 100.0%

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

      4. unpow2 100.0%

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

   10. Simplified 100.0%

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

       1. associate-+r- 100.0%

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

       2. fma-def 100.0%

          \[\leadsto x + \left(\color{blue}{\mathsf{fma}\left(wj, -1 \cdot x - x, wj \cdot wj\right)} - {wj}^{3}\right) \]

       3. sub-neg 100.0%

          \[\leadsto x + \left(\mathsf{fma}\left(wj, \color{blue}{-1 \cdot x + \left(-x\right)}, wj \cdot wj\right) - {wj}^{3}\right) \]

       4. add-sqr-sqrt 47.0%

          \[\leadsto x + \left(\mathsf{fma}\left(wj, \color{blue}{\sqrt{-1 \cdot x} \cdot \sqrt{-1 \cdot x}} + \left(-x\right), wj \cdot wj\right) - {wj}^{3}\right) \]

       5. sqrt-unprod 75.0%

          \[\leadsto x + \left(\mathsf{fma}\left(wj, \color{blue}{\sqrt{\left(-1 \cdot x\right) \cdot \left(-1 \cdot x\right)}} + \left(-x\right), wj \cdot wj\right) - {wj}^{3}\right) \]

       6. mul-1-neg 75.0%

          \[\leadsto x + \left(\mathsf{fma}\left(wj, \sqrt{\color{blue}{\left(-x\right)} \cdot \left(-1 \cdot x\right)} + \left(-x\right), wj \cdot wj\right) - {wj}^{3}\right) \]

       7. mul-1-neg 75.0%

          \[\leadsto x + \left(\mathsf{fma}\left(wj, \sqrt{\left(-x\right) \cdot \color{blue}{\left(-x\right)}} + \left(-x\right), wj \cdot wj\right) - {wj}^{3}\right) \]

       8. sqr-neg 75.0%

          \[\leadsto x + \left(\mathsf{fma}\left(wj, \sqrt{\color{blue}{x \cdot x}} + \left(-x\right), wj \cdot wj\right) - {wj}^{3}\right) \]

       9. sqrt-prod 53.0%

          \[\leadsto x + \left(\mathsf{fma}\left(wj, \color{blue}{\sqrt{x} \cdot \sqrt{x}} + \left(-x\right), wj \cdot wj\right) - {wj}^{3}\right) \]

       10. add-sqr-sqrt 100.0%

           \[\leadsto x + \left(\mathsf{fma}\left(wj, \color{blue}{x} + \left(-x\right), wj \cdot wj\right) - {wj}^{3}\right) \]

       11. mul-1-neg 100.0%

           \[\leadsto x + \left(\mathsf{fma}\left(wj, x + \color{blue}{-1 \cdot x}, wj \cdot wj\right) - {wj}^{3}\right) \]

       12. add-sqr-sqrt 47.0%

           \[\leadsto x + \left(\mathsf{fma}\left(wj, x + \color{blue}{\sqrt{-1 \cdot x} \cdot \sqrt{-1 \cdot x}}, wj \cdot wj\right) - {wj}^{3}\right) \]

       13. sqrt-unprod 75.0%

           \[\leadsto x + \left(\mathsf{fma}\left(wj, x + \color{blue}{\sqrt{\left(-1 \cdot x\right) \cdot \left(-1 \cdot x\right)}}, wj \cdot wj\right) - {wj}^{3}\right) \]

       14. mul-1-neg 75.0%

           \[\leadsto x + \left(\mathsf{fma}\left(wj, x + \sqrt{\color{blue}{\left(-x\right)} \cdot \left(-1 \cdot x\right)}, wj \cdot wj\right) - {wj}^{3}\right) \]

       15. mul-1-neg 75.0%

           \[\leadsto x + \left(\mathsf{fma}\left(wj, x + \sqrt{\left(-x\right) \cdot \color{blue}{\left(-x\right)}}, wj \cdot wj\right) - {wj}^{3}\right) \]

       16. sqr-neg 75.0%

           \[\leadsto x + \left(\mathsf{fma}\left(wj, x + \sqrt{\color{blue}{x \cdot x}}, wj \cdot wj\right) - {wj}^{3}\right) \]

       17. sqrt-prod 53.0%

           \[\leadsto x + \left(\mathsf{fma}\left(wj, x + \color{blue}{\sqrt{x} \cdot \sqrt{x}}, wj \cdot wj\right) - {wj}^{3}\right) \]

       18. add-sqr-sqrt 100.0%

           \[\leadsto x + \left(\mathsf{fma}\left(wj, x + \color{blue}{x}, wj \cdot wj\right) - {wj}^{3}\right) \]

   12. Applied egg-rr 100.0%

       \[\leadsto x + \color{blue}{\left(\mathsf{fma}\left(wj, x + x, wj \cdot wj\right) - {wj}^{3}\right)} \]
   13. Step-by-step derivation

       1. fma-udef 100.0%

          \[\leadsto x + \left(\color{blue}{\left(wj \cdot \left(x + x\right) + wj \cdot wj\right)} - {wj}^{3}\right) \]

       2. distribute-lft-out 100.0%

          \[\leadsto x + \left(\color{blue}{wj \cdot \left(\left(x + x\right) + wj\right)} - {wj}^{3}\right) \]

   14. Simplified 100.0%

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

4. Final simplification 99.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;wj \leq -8 \cdot 10^{-14} \lor \neg \left(wj \leq 2.45 \cdot 10^{-8}\right):\\ \;\;\;\;wj + \frac{\frac{x}{e^{wj}} - wj}{wj + 1}\\ \mathbf{else}:\\ \;\;\;\;x + \left(wj \cdot \left(wj + \left(x + x\right)\right) - {wj}^{3}\right)\\ \end{array} \]

Alternative 7: 99.5% accurate, 2.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;wj \leq -7.3 \cdot 10^{-9} \lor \neg \left(wj \leq 1.02 \cdot 10^{-8}\right):\\ \;\;\;\;wj + \frac{\frac{x}{e^{wj}} - wj}{wj + 1}\\ \mathbf{else}:\\ \;\;\;\;x - \left(wj \cdot \left(x + x\right) - \left(wj \cdot wj\right) \cdot \left(x + \left(x - -1\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (wj x)
 :precision binary64
 (if (or (<= wj -7.3e-9) (not (<= wj 1.02e-8)))
   (+ wj (/ (- (/ x (exp wj)) wj) (+ wj 1.0)))
   (- x (- (* wj (+ x x)) (* (* wj wj) (+ x (- x -1.0)))))))

C

double code(double wj, double x) {
	double tmp;
	if ((wj <= -7.3e-9) || !(wj <= 1.02e-8)) {
		tmp = wj + (((x / exp(wj)) - wj) / (wj + 1.0));
	} else {
		tmp = x - ((wj * (x + x)) - ((wj * wj) * (x + (x - -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 <= (-7.3d-9)) .or. (.not. (wj <= 1.02d-8))) then
        tmp = wj + (((x / exp(wj)) - wj) / (wj + 1.0d0))
    else
        tmp = x - ((wj * (x + x)) - ((wj * wj) * (x + (x - (-1.0d0)))))
    end if
    code = tmp
end function

Java

public static double code(double wj, double x) {
	double tmp;
	if ((wj <= -7.3e-9) || !(wj <= 1.02e-8)) {
		tmp = wj + (((x / Math.exp(wj)) - wj) / (wj + 1.0));
	} else {
		tmp = x - ((wj * (x + x)) - ((wj * wj) * (x + (x - -1.0))));
	}
	return tmp;
}

Python

def code(wj, x):
	tmp = 0
	if (wj <= -7.3e-9) or not (wj <= 1.02e-8):
		tmp = wj + (((x / math.exp(wj)) - wj) / (wj + 1.0))
	else:
		tmp = x - ((wj * (x + x)) - ((wj * wj) * (x + (x - -1.0))))
	return tmp

Julia

function code(wj, x)
	tmp = 0.0
	if ((wj <= -7.3e-9) || !(wj <= 1.02e-8))
		tmp = Float64(wj + Float64(Float64(Float64(x / exp(wj)) - wj) / Float64(wj + 1.0)));
	else
		tmp = Float64(x - Float64(Float64(wj * Float64(x + x)) - Float64(Float64(wj * wj) * Float64(x + Float64(x - -1.0)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(wj, x)
	tmp = 0.0;
	if ((wj <= -7.3e-9) || ~((wj <= 1.02e-8)))
		tmp = wj + (((x / exp(wj)) - wj) / (wj + 1.0));
	else
		tmp = x - ((wj * (x + x)) - ((wj * wj) * (x + (x - -1.0))));
	end
	tmp_2 = tmp;
end

Wolfram

code[wj_, x_] := If[Or[LessEqual[wj, -7.3e-9], N[Not[LessEqual[wj, 1.02e-8]], $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 + x), $MachinePrecision]), $MachinePrecision] - N[(N[(wj * wj), $MachinePrecision] * N[(x + N[(x - -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if wj < -7.30000000000000002e-9 or 1.02000000000000003e-8 < wj

   1. Initial program 65.9%

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

      1. div-sub 65.9%

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

      2. associate-/l* 66.2%

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

      3. distribute-rgt1-in 66.2%

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

      4. associate-/l* 66.2%

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

      5. *-inverses 85.0%

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

      6. /-rgt-identity 85.0%

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

      7. distribute-rgt1-in 97.5%

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

      8. associate-/l/ 97.4%

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

      9. div-sub 97.4%

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

   3. Simplified 97.4%

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

   if -7.30000000000000002e-9 < wj < 1.02000000000000003e-8

   1. Initial program 80.0%

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

      1. div-sub 80.0%

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

      2. associate-/l* 80.0%

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

      3. distribute-rgt1-in 80.0%

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

      4. associate-/l* 80.0%

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

      5. *-inverses 80.0%

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

      6. /-rgt-identity 80.0%

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

      7. distribute-rgt1-in 80.0%

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

      8. associate-/l/ 80.0%

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

      9. div-sub 80.0%

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

   3. Simplified 80.0%

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

   4. Taylor expanded in wj around 0 80.0%

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

      1. associate-*r* 80.0%

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

      2. neg-mul-1 80.0%

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

      3. distribute-rgt1-in 80.0%

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

      4. +-commutative 80.0%

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

      5. sub-neg 80.0%

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

   6. Simplified 80.0%

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

   7. Taylor expanded in wj around 0 100.0%

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

   8. Taylor expanded in wj around 0 99.8%

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

      1. unpow2 99.8%

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

      2. sub-neg 99.8%

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

      3. neg-mul-1 99.8%

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

      4. remove-double-neg 99.8%

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

      5. +-commutative 99.8%

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

      6. +-commutative 99.8%

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

   10. Simplified 99.8%

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

4. Final simplification 99.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;wj \leq -7.3 \cdot 10^{-9} \lor \neg \left(wj \leq 1.02 \cdot 10^{-8}\right):\\ \;\;\;\;wj + \frac{\frac{x}{e^{wj}} - wj}{wj + 1}\\ \mathbf{else}:\\ \;\;\;\;x - \left(wj \cdot \left(x + x\right) - \left(wj \cdot wj\right) \cdot \left(x + \left(x - -1\right)\right)\right)\\ \end{array} \]

Alternative 8: 95.9% accurate, 18.4× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 79.1%

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

   1. div-sub 79.1%

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

   2. associate-/l* 79.1%

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

   3. distribute-rgt1-in 79.1%

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

   4. associate-/l* 79.1%

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

   5. *-inverses 80.3%

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

   6. /-rgt-identity 80.3%

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

   7. distribute-rgt1-in 81.1%

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

   8. associate-/l/ 81.1%

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

   9. div-sub 81.1%

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

3. Simplified 81.1%

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

4. Taylor expanded in wj around 0 78.9%

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

   1. associate-*r* 78.9%

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

   2. neg-mul-1 78.9%

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

   3. distribute-rgt1-in 78.9%

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

   4. +-commutative 78.9%

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

   5. sub-neg 78.9%

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

6. Simplified 78.9%

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

7. Taylor expanded in wj around 0 95.5%

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

8. Taylor expanded in wj around 0 95.4%

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

   1. unpow2 95.4%

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

   2. sub-neg 95.4%

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

   3. neg-mul-1 95.4%

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

   4. remove-double-neg 95.4%

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

   5. +-commutative 95.4%

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

   6. +-commutative 95.4%

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

10. Simplified 95.4%

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

11. Final simplification 95.4%

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

Alternative 9: 96.1% accurate, 20.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;wj \leq -8 \cdot 10^{-14}:\\ \;\;\;\;wj - \frac{wj + x \cdot \left(wj + -1\right)}{wj + 1}\\ \mathbf{else}:\\ \;\;\;\;x + wj \cdot wj\\ \end{array} \end{array} \]

FPCore

(FPCore (wj x)
 :precision binary64
 (if (<= wj -8e-14)
   (- wj (/ (+ wj (* x (+ wj -1.0))) (+ wj 1.0)))
   (+ x (* wj wj))))

C

double code(double wj, double x) {
	double tmp;
	if (wj <= -8e-14) {
		tmp = wj - ((wj + (x * (wj + -1.0))) / (wj + 1.0));
	} else {
		tmp = x + (wj * 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 <= (-8d-14)) then
        tmp = wj - ((wj + (x * (wj + (-1.0d0)))) / (wj + 1.0d0))
    else
        tmp = x + (wj * wj)
    end if
    code = tmp
end function

Java

public static double code(double wj, double x) {
	double tmp;
	if (wj <= -8e-14) {
		tmp = wj - ((wj + (x * (wj + -1.0))) / (wj + 1.0));
	} else {
		tmp = x + (wj * wj);
	}
	return tmp;
}

Python

def code(wj, x):
	tmp = 0
	if wj <= -8e-14:
		tmp = wj - ((wj + (x * (wj + -1.0))) / (wj + 1.0))
	else:
		tmp = x + (wj * wj)
	return tmp

Julia

function code(wj, x)
	tmp = 0.0
	if (wj <= -8e-14)
		tmp = Float64(wj - Float64(Float64(wj + Float64(x * Float64(wj + -1.0))) / Float64(wj + 1.0)));
	else
		tmp = Float64(x + Float64(wj * wj));
	end
	return tmp
end

MATLAB

function tmp_2 = code(wj, x)
	tmp = 0.0;
	if (wj <= -8e-14)
		tmp = wj - ((wj + (x * (wj + -1.0))) / (wj + 1.0));
	else
		tmp = x + (wj * wj);
	end
	tmp_2 = tmp;
end

Wolfram

code[wj_, x_] := If[LessEqual[wj, -8e-14], N[(wj - N[(N[(wj + N[(x * N[(wj + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(wj + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x + N[(wj * wj), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if wj < -7.99999999999999999e-14

   1. Initial program 81.2%

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

      1. div-sub 81.2%

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

      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 81.7%

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

      6. /-rgt-identity 81.7%

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

      7. distribute-rgt1-in 97.0%

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

      8. associate-/l/ 96.6%

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

      9. div-sub 96.6%

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

   3. Simplified 96.6%

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

   4. Taylor expanded in wj around 0 67.9%

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

      1. associate-*r* 67.9%

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

      2. neg-mul-1 67.9%

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

      3. distribute-rgt1-in 67.8%

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

      4. +-commutative 67.8%

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

      5. sub-neg 67.8%

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

   6. Simplified 67.8%

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

   if -7.99999999999999999e-14 < wj

   1. Initial program 79.0%

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

      1. div-sub 79.0%

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

      2. associate-/l* 79.0%

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

      3. distribute-rgt1-in 79.0%

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

      4. associate-/l* 79.0%

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

      5. *-inverses 80.2%

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

      6. /-rgt-identity 80.2%

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

      7. distribute-rgt1-in 80.2%

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

      8. associate-/l/ 80.2%

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

      9. div-sub 80.2%

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

   3. Simplified 80.2%

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

   4. Taylor expanded in wj around 0 97.5%

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

      1. +-commutative 97.5%

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

      2. fma-def 97.5%

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

      3. unpow2 97.5%

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

      4. sub-neg 97.5%

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

      5. distribute-rgt-out 97.9%

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

      6. distribute-rgt-neg-in 97.9%

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

      7. metadata-eval 97.9%

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

      8. metadata-eval 97.9%

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

      9. associate-*r* 97.9%

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

      10. *-commutative 97.9%

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

   6. Simplified 97.9%

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

   7. Taylor expanded in x around 0 97.5%

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

      1. unpow2 97.5%

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

   9. Simplified 97.5%

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

4. Final simplification 96.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;wj \leq -8 \cdot 10^{-14}:\\ \;\;\;\;wj - \frac{wj + x \cdot \left(wj + -1\right)}{wj + 1}\\ \mathbf{else}:\\ \;\;\;\;x + wj \cdot wj\\ \end{array} \]

Alternative 10: 95.4% accurate, 62.6× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 79.1%

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

   1. div-sub 79.1%

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

   2. associate-/l* 79.1%

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

   3. distribute-rgt1-in 79.1%

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

   4. associate-/l* 79.1%

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

   5. *-inverses 80.3%

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

   6. /-rgt-identity 80.3%

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

   7. distribute-rgt1-in 81.1%

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

   8. associate-/l/ 81.1%

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

   9. div-sub 81.1%

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

3. Simplified 81.1%

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

4. Taylor expanded in wj around 0 95.2%

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

   1. +-commutative 95.2%

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

   2. fma-def 95.2%

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

   3. unpow2 95.2%

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

   4. sub-neg 95.2%

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

   5. distribute-rgt-out 95.6%

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

   6. distribute-rgt-neg-in 95.6%

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

   7. metadata-eval 95.6%

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

   8. metadata-eval 95.6%

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

   9. associate-*r* 95.6%

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

   10. *-commutative 95.6%

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

6. Simplified 95.6%

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

7. Taylor expanded in x around 0 94.5%

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

   1. unpow2 94.5%

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

9. Simplified 94.5%

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

10. Final simplification 94.5%

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

Alternative 11: 4.4% accurate, 313.0× speedup

Math

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

FPCore

(FPCore (wj x) :precision binary64 wj)

C

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

Fortran

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

Java

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

Python

def code(wj, x):
	return wj

Julia

function code(wj, x)
	return wj
end

MATLAB

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

Wolfram

code[wj_, x_] := wj

Derivation

1. Initial program 79.1%

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

   1. div-sub 79.1%

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

   2. associate-/l* 79.1%

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

   3. distribute-rgt1-in 79.1%

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

   4. associate-/l* 79.1%

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

   5. *-inverses 80.3%

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

   6. /-rgt-identity 80.3%

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

   7. distribute-rgt1-in 81.1%

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

   8. associate-/l/ 81.1%

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

   9. div-sub 81.1%

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

3. Simplified 81.1%

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

4. Taylor expanded in wj around inf 4.3%

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

5. Final simplification 4.3%

   \[\leadsto wj \]

Alternative 12: 83.9% accurate, 313.0× speedup

Math

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

FPCore

(FPCore (wj x) :precision binary64 x)

C

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

Fortran

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

Java

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

Python

def code(wj, x):
	return x

Julia

function code(wj, x)
	return x
end

MATLAB

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

Wolfram

code[wj_, x_] := x

Derivation

1. Initial program 79.1%

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

   1. div-sub 79.1%

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

   2. associate-/l* 79.1%

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

   3. distribute-rgt1-in 79.1%

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

   4. associate-/l* 79.1%

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

   5. *-inverses 80.3%

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

   6. /-rgt-identity 80.3%

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

   7. distribute-rgt1-in 81.1%

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

   8. associate-/l/ 81.1%

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

   9. div-sub 81.1%

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

3. Simplified 81.1%

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

4. Taylor expanded in wj around 0 84.9%

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

5. Final simplification 84.9%

   \[\leadsto x \]

Developer target: 78.6% 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 2023287 
(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))))))