Jmat.Real.lambertw, newton loop step

Percentage Accurate: 77.3% → 98.6%

Time: 13.3s

Alternatives: 13

Speedup: 313.0×

• could not determine a ground truth

  1. wj = -5.396971624905966e+81
  2. x = 8.84743631161465e+211

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: 77.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 98.6% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := wj \cdot e^{wj}\\ \mathbf{if}\;wj - \frac{t_0 - x}{e^{wj} + t_0} \leq 4 \cdot 10^{-12}:\\ \;\;\;\;x + \left(-2 \cdot \left(wj \cdot x\right) + \left({wj}^{2} \cdot \left(1 - \left(x \cdot -4 + x \cdot 1.5\right)\right) - {wj}^{3}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;wj + \frac{1}{wj + 1} \cdot \left(\frac{x}{e^{wj}} - wj\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (wj x)
 :precision binary64
 (let* ((t_0 (* wj (exp wj))))
   (if (<= (- wj (/ (- t_0 x) (+ (exp wj) t_0))) 4e-12)
     (+
      x
      (+
       (* -2.0 (* wj x))
       (- (* (pow wj 2.0) (- 1.0 (+ (* x -4.0) (* x 1.5)))) (pow wj 3.0))))
     (+ 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 - ((t_0 - x) / (exp(wj) + t_0))) <= 4e-12) {
		tmp = x + ((-2.0 * (wj * x)) + ((pow(wj, 2.0) * (1.0 - ((x * -4.0) + (x * 1.5)))) - pow(wj, 3.0)));
	} 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 - ((t_0 - x) / (exp(wj) + t_0))) <= 4d-12) then
        tmp = x + (((-2.0d0) * (wj * x)) + (((wj ** 2.0d0) * (1.0d0 - ((x * (-4.0d0)) + (x * 1.5d0)))) - (wj ** 3.0d0)))
    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 - ((t_0 - x) / (Math.exp(wj) + t_0))) <= 4e-12) {
		tmp = x + ((-2.0 * (wj * x)) + ((Math.pow(wj, 2.0) * (1.0 - ((x * -4.0) + (x * 1.5)))) - Math.pow(wj, 3.0)));
	} 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 - ((t_0 - x) / (math.exp(wj) + t_0))) <= 4e-12:
		tmp = x + ((-2.0 * (wj * x)) + ((math.pow(wj, 2.0) * (1.0 - ((x * -4.0) + (x * 1.5)))) - math.pow(wj, 3.0)))
	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(t_0 - x) / Float64(exp(wj) + t_0))) <= 4e-12)
		tmp = Float64(x + Float64(Float64(-2.0 * Float64(wj * x)) + Float64(Float64((wj ^ 2.0) * Float64(1.0 - Float64(Float64(x * -4.0) + Float64(x * 1.5)))) - (wj ^ 3.0))));
	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 - ((t_0 - x) / (exp(wj) + t_0))) <= 4e-12)
		tmp = x + ((-2.0 * (wj * x)) + (((wj ^ 2.0) * (1.0 - ((x * -4.0) + (x * 1.5)))) - (wj ^ 3.0)));
	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[(t$95$0 - x), $MachinePrecision] / N[(N[Exp[wj], $MachinePrecision] + t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 4e-12], N[(x + N[(N[(-2.0 * N[(wj * x), $MachinePrecision]), $MachinePrecision] + N[(N[(N[Power[wj, 2.0], $MachinePrecision] * N[(1.0 - N[(N[(x * -4.0), $MachinePrecision] + N[(x * 1.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[Power[wj, 3.0], $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))))) < 3.99999999999999992e-12

   1. Initial program 74.2%

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

      1. distribute-rgt1-in 74.7%

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

      2. associate-/l/ 74.7%

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

      3. div-sub 74.2%

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

      4. associate-/l* 74.2%

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

      5. *-inverses 74.7%

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

      6. /-rgt-identity 74.7%

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

   3. Simplified 74.7%

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

   4. Taylor expanded in wj around 0 99.4%

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

   5. Taylor expanded in x around 0 99.4%

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

   if 3.99999999999999992e-12 < (-.f64 wj (/.f64 (-.f64 (*.f64 wj (exp.f64 wj)) x) (+.f64 (exp.f64 wj) (*.f64 wj (exp.f64 wj)))))

   1. Initial program 91.2%

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

      1. distribute-rgt1-in 92.8%

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

      2. associate-/l/ 92.8%

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

      3. div-sub 91.2%

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

      4. associate-/l* 91.2%

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

      5. *-inverses 99.4%

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

      6. /-rgt-identity 99.4%

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

   3. Simplified 99.4%

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

      1. clear-num 99.2%

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

      2. associate-/r/ 99.4%

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

   5. Applied egg-rr 99.4%

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

4. Final simplification 99.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;wj - \frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}} \leq 4 \cdot 10^{-12}:\\ \;\;\;\;x + \left(-2 \cdot \left(wj \cdot x\right) + \left({wj}^{2} \cdot \left(1 - \left(x \cdot -4 + x \cdot 1.5\right)\right) - {wj}^{3}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;wj + \frac{1}{wj + 1} \cdot \left(\frac{x}{e^{wj}} - wj\right)\\ \end{array} \]

Alternative 2: 97.3% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;wj \leq 2.35 \cdot 10^{-5}:\\ \;\;\;\;x + \left(-2 \cdot \left(wj \cdot x\right) + \left({wj}^{2} - {wj}^{3}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;wj - \frac{wj \cdot \left(wj + -1\right)}{\mathsf{fma}\left(wj, wj, -1\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (wj x)
 :precision binary64
 (if (<= wj 2.35e-5)
   (+ x (+ (* -2.0 (* wj x)) (- (pow wj 2.0) (pow wj 3.0))))
   (- wj (/ (* wj (+ wj -1.0)) (fma wj wj -1.0)))))

C

double code(double wj, double x) {
	double tmp;
	if (wj <= 2.35e-5) {
		tmp = x + ((-2.0 * (wj * x)) + (pow(wj, 2.0) - pow(wj, 3.0)));
	} else {
		tmp = wj - ((wj * (wj + -1.0)) / fma(wj, wj, -1.0));
	}
	return tmp;
}

Julia

function code(wj, x)
	tmp = 0.0
	if (wj <= 2.35e-5)
		tmp = Float64(x + Float64(Float64(-2.0 * Float64(wj * x)) + Float64((wj ^ 2.0) - (wj ^ 3.0))));
	else
		tmp = Float64(wj - Float64(Float64(wj * Float64(wj + -1.0)) / fma(wj, wj, -1.0)));
	end
	return tmp
end

Wolfram

code[wj_, x_] := If[LessEqual[wj, 2.35e-5], N[(x + N[(N[(-2.0 * N[(wj * x), $MachinePrecision]), $MachinePrecision] + N[(N[Power[wj, 2.0], $MachinePrecision] - N[Power[wj, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(wj - N[(N[(wj * N[(wj + -1.0), $MachinePrecision]), $MachinePrecision] / N[(wj * wj + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if wj < 2.34999999999999986e-5

   1. Initial program 79.4%

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

      1. distribute-rgt1-in 80.2%

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

      2. associate-/l/ 80.2%

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

      3. div-sub 79.4%

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

      4. associate-/l* 79.4%

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

      5. *-inverses 80.2%

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

      6. /-rgt-identity 80.2%

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

   3. Simplified 80.2%

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

   4. Taylor expanded in wj around 0 98.7%

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

   5. Taylor expanded in x around 0 98.7%

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

   6. Taylor expanded in x around 0 98.6%

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

      1. +-commutative 98.6%

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

      2. mul-1-neg 98.6%

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

      3. unsub-neg 98.6%

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

   8. Simplified 98.6%

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

   if 2.34999999999999986e-5 < wj

   1. Initial program 29.7%

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

      1. distribute-rgt1-in 29.3%

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

      2. associate-/l/ 29.6%

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

      3. div-sub 29.6%

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

      4. associate-/l* 29.6%

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

      5. *-inverses 96.3%

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

      6. /-rgt-identity 96.3%

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

   3. Simplified 96.3%

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

      1. flip-+ 96.4%

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

      2. associate-/r/ 96.4%

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

      3. metadata-eval 96.4%

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

      4. fma-neg 96.4%

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

      5. metadata-eval 96.4%

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

      6. sub-neg 96.4%

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

      7. metadata-eval 96.4%

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

   5. Applied egg-rr 96.4%

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

      1. associate-*l/ 96.4%

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

   7. Simplified 96.4%

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

   8. Taylor expanded in x around 0 96.4%

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

4. Final simplification 98.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;wj \leq 2.35 \cdot 10^{-5}:\\ \;\;\;\;x + \left(-2 \cdot \left(wj \cdot x\right) + \left({wj}^{2} - {wj}^{3}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;wj - \frac{wj \cdot \left(wj + -1\right)}{\mathsf{fma}\left(wj, wj, -1\right)}\\ \end{array} \]

Alternative 3: 97.3% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;wj \leq 2.35 \cdot 10^{-5}:\\ \;\;\;\;x + \left(-2 \cdot \left(wj \cdot x\right) + \mathsf{fma}\left(wj, wj, -{wj}^{3}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;wj - \frac{wj \cdot \left(wj + -1\right)}{\mathsf{fma}\left(wj, wj, -1\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (wj x)
 :precision binary64
 (if (<= wj 2.35e-5)
   (+ x (+ (* -2.0 (* wj x)) (fma wj wj (- (pow wj 3.0)))))
   (- wj (/ (* wj (+ wj -1.0)) (fma wj wj -1.0)))))

C

double code(double wj, double x) {
	double tmp;
	if (wj <= 2.35e-5) {
		tmp = x + ((-2.0 * (wj * x)) + fma(wj, wj, -pow(wj, 3.0)));
	} else {
		tmp = wj - ((wj * (wj + -1.0)) / fma(wj, wj, -1.0));
	}
	return tmp;
}

Julia

function code(wj, x)
	tmp = 0.0
	if (wj <= 2.35e-5)
		tmp = Float64(x + Float64(Float64(-2.0 * Float64(wj * x)) + fma(wj, wj, Float64(-(wj ^ 3.0)))));
	else
		tmp = Float64(wj - Float64(Float64(wj * Float64(wj + -1.0)) / fma(wj, wj, -1.0)));
	end
	return tmp
end

Wolfram

code[wj_, x_] := If[LessEqual[wj, 2.35e-5], N[(x + N[(N[(-2.0 * N[(wj * x), $MachinePrecision]), $MachinePrecision] + N[(wj * wj + (-N[Power[wj, 3.0], $MachinePrecision])), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(wj - N[(N[(wj * N[(wj + -1.0), $MachinePrecision]), $MachinePrecision] / N[(wj * wj + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if wj < 2.34999999999999986e-5

   1. Initial program 79.4%

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

      1. distribute-rgt1-in 80.2%

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

      2. associate-/l/ 80.2%

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

      3. div-sub 79.4%

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

      4. associate-/l* 79.4%

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

      5. *-inverses 80.2%

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

      6. /-rgt-identity 80.2%

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

   3. Simplified 80.2%

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

   4. Taylor expanded in wj around 0 98.7%

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

   5. Taylor expanded in x around 0 98.7%

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

   6. Taylor expanded in x around 0 98.6%

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

      1. +-commutative 98.6%

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

      2. mul-1-neg 98.6%

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

      3. unsub-neg 98.6%

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

   8. Simplified 98.6%

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

      1. unpow2 98.6%

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

      2. fma-neg 98.6%

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

   10. Applied egg-rr 98.6%

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

   if 2.34999999999999986e-5 < wj

   1. Initial program 29.7%

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

      1. distribute-rgt1-in 29.3%

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

      2. associate-/l/ 29.6%

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

      3. div-sub 29.6%

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

      4. associate-/l* 29.6%

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

      5. *-inverses 96.3%

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

      6. /-rgt-identity 96.3%

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

   3. Simplified 96.3%

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

      1. flip-+ 96.4%

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

      2. associate-/r/ 96.4%

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

      3. metadata-eval 96.4%

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

      4. fma-neg 96.4%

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

      5. metadata-eval 96.4%

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

      6. sub-neg 96.4%

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

      7. metadata-eval 96.4%

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

   5. Applied egg-rr 96.4%

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

      1. associate-*l/ 96.4%

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

   7. Simplified 96.4%

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

   8. Taylor expanded in x around 0 96.4%

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

4. Final simplification 98.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;wj \leq 2.35 \cdot 10^{-5}:\\ \;\;\;\;x + \left(-2 \cdot \left(wj \cdot x\right) + \mathsf{fma}\left(wj, wj, -{wj}^{3}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;wj - \frac{wj \cdot \left(wj + -1\right)}{\mathsf{fma}\left(wj, wj, -1\right)}\\ \end{array} \]

Alternative 4: 97.0% accurate, 2.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;wj \leq 2.35 \cdot 10^{-5}:\\ \;\;\;\;x + \left(-2 \cdot \left(wj \cdot x\right) + {wj}^{2} \cdot \left(1 + x \cdot 2.5\right)\right)\\ \mathbf{else}:\\ \;\;\;\;wj - \frac{wj \cdot \left(wj + -1\right)}{\mathsf{fma}\left(wj, wj, -1\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (wj x)
 :precision binary64
 (if (<= wj 2.35e-5)
   (+ x (+ (* -2.0 (* wj x)) (* (pow wj 2.0) (+ 1.0 (* x 2.5)))))
   (- wj (/ (* wj (+ wj -1.0)) (fma wj wj -1.0)))))

C

double code(double wj, double x) {
	double tmp;
	if (wj <= 2.35e-5) {
		tmp = x + ((-2.0 * (wj * x)) + (pow(wj, 2.0) * (1.0 + (x * 2.5))));
	} else {
		tmp = wj - ((wj * (wj + -1.0)) / fma(wj, wj, -1.0));
	}
	return tmp;
}

Julia

function code(wj, x)
	tmp = 0.0
	if (wj <= 2.35e-5)
		tmp = Float64(x + Float64(Float64(-2.0 * Float64(wj * x)) + Float64((wj ^ 2.0) * Float64(1.0 + Float64(x * 2.5)))));
	else
		tmp = Float64(wj - Float64(Float64(wj * Float64(wj + -1.0)) / fma(wj, wj, -1.0)));
	end
	return tmp
end

Wolfram

code[wj_, x_] := If[LessEqual[wj, 2.35e-5], N[(x + N[(N[(-2.0 * N[(wj * x), $MachinePrecision]), $MachinePrecision] + N[(N[Power[wj, 2.0], $MachinePrecision] * N[(1.0 + N[(x * 2.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(wj - N[(N[(wj * N[(wj + -1.0), $MachinePrecision]), $MachinePrecision] / N[(wj * wj + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if wj < 2.34999999999999986e-5

   1. Initial program 79.4%

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

      1. distribute-rgt1-in 80.2%

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

      2. associate-/l/ 80.2%

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

      3. div-sub 79.4%

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

      4. associate-/l* 79.4%

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

      5. *-inverses 80.2%

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

      6. /-rgt-identity 80.2%

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

   3. Simplified 80.2%

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

   4. Taylor expanded in wj around 0 98.7%

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

   5. Taylor expanded in x around 0 98.7%

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

   6. Taylor expanded in wj around 0 98.5%

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

      1. distribute-rgt-out 98.5%

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

      2. metadata-eval 98.5%

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

      3. *-commutative 98.5%

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

      4. cancel-sign-sub-inv 98.5%

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

      5. metadata-eval 98.5%

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

      6. *-commutative 98.5%

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

   8. Simplified 98.5%

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

   if 2.34999999999999986e-5 < wj

   1. Initial program 29.7%

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

      1. distribute-rgt1-in 29.3%

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

      2. associate-/l/ 29.6%

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

      3. div-sub 29.6%

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

      4. associate-/l* 29.6%

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

      5. *-inverses 96.3%

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

      6. /-rgt-identity 96.3%

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

   3. Simplified 96.3%

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

      1. flip-+ 96.4%

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

      2. associate-/r/ 96.4%

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

      3. metadata-eval 96.4%

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

      4. fma-neg 96.4%

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

      5. metadata-eval 96.4%

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

      6. sub-neg 96.4%

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

      7. metadata-eval 96.4%

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

   5. Applied egg-rr 96.4%

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

      1. associate-*l/ 96.4%

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

   7. Simplified 96.4%

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

   8. Taylor expanded in x around 0 96.4%

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

4. Final simplification 98.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;wj \leq 2.35 \cdot 10^{-5}:\\ \;\;\;\;x + \left(-2 \cdot \left(wj \cdot x\right) + {wj}^{2} \cdot \left(1 + x \cdot 2.5\right)\right)\\ \mathbf{else}:\\ \;\;\;\;wj - \frac{wj \cdot \left(wj + -1\right)}{\mathsf{fma}\left(wj, wj, -1\right)}\\ \end{array} \]

Alternative 5: 96.9% accurate, 2.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;wj \leq 7.2 \cdot 10^{-6}:\\ \;\;\;\;x + \left(-2 \cdot \left(wj \cdot x\right) + {wj}^{2}\right)\\ \mathbf{else}:\\ \;\;\;\;wj - \frac{wj \cdot \left(wj + -1\right)}{\mathsf{fma}\left(wj, wj, -1\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (wj x)
 :precision binary64
 (if (<= wj 7.2e-6)
   (+ x (+ (* -2.0 (* wj x)) (pow wj 2.0)))
   (- wj (/ (* wj (+ wj -1.0)) (fma wj wj -1.0)))))

C

double code(double wj, double x) {
	double tmp;
	if (wj <= 7.2e-6) {
		tmp = x + ((-2.0 * (wj * x)) + pow(wj, 2.0));
	} else {
		tmp = wj - ((wj * (wj + -1.0)) / fma(wj, wj, -1.0));
	}
	return tmp;
}

Julia

function code(wj, x)
	tmp = 0.0
	if (wj <= 7.2e-6)
		tmp = Float64(x + Float64(Float64(-2.0 * Float64(wj * x)) + (wj ^ 2.0)));
	else
		tmp = Float64(wj - Float64(Float64(wj * Float64(wj + -1.0)) / fma(wj, wj, -1.0)));
	end
	return tmp
end

Wolfram

code[wj_, x_] := If[LessEqual[wj, 7.2e-6], N[(x + N[(N[(-2.0 * N[(wj * x), $MachinePrecision]), $MachinePrecision] + N[Power[wj, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(wj - N[(N[(wj * N[(wj + -1.0), $MachinePrecision]), $MachinePrecision] / N[(wj * wj + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if wj < 7.19999999999999967e-6

   1. Initial program 79.4%

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

      1. distribute-rgt1-in 80.2%

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

      2. associate-/l/ 80.2%

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

      3. div-sub 79.4%

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

      4. associate-/l* 79.4%

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

      5. *-inverses 80.2%

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

      6. /-rgt-identity 80.2%

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

   3. Simplified 80.2%

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

   4. Taylor expanded in wj around 0 98.7%

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

   5. Taylor expanded in x around 0 98.7%

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

   6. Taylor expanded in x around 0 98.6%

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

      1. +-commutative 98.6%

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

      2. mul-1-neg 98.6%

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

      3. unsub-neg 98.6%

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

   8. Simplified 98.6%

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

   9. Taylor expanded in wj around 0 98.4%

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

   if 7.19999999999999967e-6 < wj

   1. Initial program 29.7%

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

      1. distribute-rgt1-in 29.3%

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

      2. associate-/l/ 29.6%

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

      3. div-sub 29.6%

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

      4. associate-/l* 29.6%

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

      5. *-inverses 96.3%

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

      6. /-rgt-identity 96.3%

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

   3. Simplified 96.3%

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

      1. flip-+ 96.4%

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

      2. associate-/r/ 96.4%

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

      3. metadata-eval 96.4%

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

      4. fma-neg 96.4%

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

      5. metadata-eval 96.4%

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

      6. sub-neg 96.4%

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

      7. metadata-eval 96.4%

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

   5. Applied egg-rr 96.4%

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

      1. associate-*l/ 96.4%

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

   7. Simplified 96.4%

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

   8. Taylor expanded in x around 0 96.4%

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

4. Final simplification 98.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;wj \leq 7.2 \cdot 10^{-6}:\\ \;\;\;\;x + \left(-2 \cdot \left(wj \cdot x\right) + {wj}^{2}\right)\\ \mathbf{else}:\\ \;\;\;\;wj - \frac{wj \cdot \left(wj + -1\right)}{\mathsf{fma}\left(wj, wj, -1\right)}\\ \end{array} \]

Alternative 6: 96.9% accurate, 2.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;wj \leq 2.35 \cdot 10^{-5}:\\ \;\;\;\;x + \left(-2 \cdot \left(wj \cdot x\right) + {wj}^{2}\right)\\ \mathbf{else}:\\ \;\;\;\;wj - \frac{wj}{wj + 1}\\ \end{array} \end{array} \]

FPCore

(FPCore (wj x)
 :precision binary64
 (if (<= wj 2.35e-5)
   (+ x (+ (* -2.0 (* wj x)) (pow wj 2.0)))
   (- wj (/ wj (+ wj 1.0)))))

C

double code(double wj, double x) {
	double tmp;
	if (wj <= 2.35e-5) {
		tmp = x + ((-2.0 * (wj * x)) + pow(wj, 2.0));
	} else {
		tmp = wj - (wj / (wj + 1.0));
	}
	return tmp;
}

Fortran

real(8) function code(wj, x)
    real(8), intent (in) :: wj
    real(8), intent (in) :: x
    real(8) :: tmp
    if (wj <= 2.35d-5) then
        tmp = x + (((-2.0d0) * (wj * x)) + (wj ** 2.0d0))
    else
        tmp = wj - (wj / (wj + 1.0d0))
    end if
    code = tmp
end function

Java

public static double code(double wj, double x) {
	double tmp;
	if (wj <= 2.35e-5) {
		tmp = x + ((-2.0 * (wj * x)) + Math.pow(wj, 2.0));
	} else {
		tmp = wj - (wj / (wj + 1.0));
	}
	return tmp;
}

Python

def code(wj, x):
	tmp = 0
	if wj <= 2.35e-5:
		tmp = x + ((-2.0 * (wj * x)) + math.pow(wj, 2.0))
	else:
		tmp = wj - (wj / (wj + 1.0))
	return tmp

Julia

function code(wj, x)
	tmp = 0.0
	if (wj <= 2.35e-5)
		tmp = Float64(x + Float64(Float64(-2.0 * Float64(wj * x)) + (wj ^ 2.0)));
	else
		tmp = Float64(wj - Float64(wj / Float64(wj + 1.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(wj, x)
	tmp = 0.0;
	if (wj <= 2.35e-5)
		tmp = x + ((-2.0 * (wj * x)) + (wj ^ 2.0));
	else
		tmp = wj - (wj / (wj + 1.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[wj_, x_] := If[LessEqual[wj, 2.35e-5], N[(x + N[(N[(-2.0 * N[(wj * x), $MachinePrecision]), $MachinePrecision] + N[Power[wj, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(wj - N[(wj / N[(wj + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if wj < 2.34999999999999986e-5

   1. Initial program 79.4%

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

      1. distribute-rgt1-in 80.2%

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

      2. associate-/l/ 80.2%

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

      3. div-sub 79.4%

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

      4. associate-/l* 79.4%

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

      5. *-inverses 80.2%

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

      6. /-rgt-identity 80.2%

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

   3. Simplified 80.2%

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

   4. Taylor expanded in wj around 0 98.7%

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

   5. Taylor expanded in x around 0 98.7%

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

   6. Taylor expanded in x around 0 98.6%

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

      1. +-commutative 98.6%

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

      2. mul-1-neg 98.6%

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

      3. unsub-neg 98.6%

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

   8. Simplified 98.6%

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

   9. Taylor expanded in wj around 0 98.4%

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

   if 2.34999999999999986e-5 < wj

   1. Initial program 29.7%

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

      1. distribute-rgt1-in 29.3%

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

      2. associate-/l/ 29.6%

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

      3. div-sub 29.6%

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

      4. associate-/l* 29.6%

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

      5. *-inverses 96.3%

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

      6. /-rgt-identity 96.3%

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

   3. Simplified 96.3%

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

   4. Taylor expanded in x around 0 96.3%

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

      1. +-commutative 96.3%

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

   6. Simplified 96.3%

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

4. Final simplification 98.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;wj \leq 2.35 \cdot 10^{-5}:\\ \;\;\;\;x + \left(-2 \cdot \left(wj \cdot x\right) + {wj}^{2}\right)\\ \mathbf{else}:\\ \;\;\;\;wj - \frac{wj}{wj + 1}\\ \end{array} \]

Alternative 7: 87.1% accurate, 2.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;wj \leq 6.5 \cdot 10^{-17}:\\ \;\;\;\;\frac{1}{wj + 1} \cdot \frac{x}{e^{wj}}\\ \mathbf{else}:\\ \;\;\;\;wj + \frac{x + wj \cdot \left(-1 - x\right)}{wj + 1}\\ \end{array} \end{array} \]

FPCore

(FPCore (wj x)
 :precision binary64
 (if (<= wj 6.5e-17)
   (* (/ 1.0 (+ wj 1.0)) (/ x (exp wj)))
   (+ wj (/ (+ x (* wj (- -1.0 x))) (+ wj 1.0)))))

C

double code(double wj, double x) {
	double tmp;
	if (wj <= 6.5e-17) {
		tmp = (1.0 / (wj + 1.0)) * (x / exp(wj));
	} else {
		tmp = wj + ((x + (wj * (-1.0 - x))) / (wj + 1.0));
	}
	return tmp;
}

Fortran

real(8) function code(wj, x)
    real(8), intent (in) :: wj
    real(8), intent (in) :: x
    real(8) :: tmp
    if (wj <= 6.5d-17) then
        tmp = (1.0d0 / (wj + 1.0d0)) * (x / exp(wj))
    else
        tmp = wj + ((x + (wj * ((-1.0d0) - x))) / (wj + 1.0d0))
    end if
    code = tmp
end function

Java

public static double code(double wj, double x) {
	double tmp;
	if (wj <= 6.5e-17) {
		tmp = (1.0 / (wj + 1.0)) * (x / Math.exp(wj));
	} else {
		tmp = wj + ((x + (wj * (-1.0 - x))) / (wj + 1.0));
	}
	return tmp;
}

Python

def code(wj, x):
	tmp = 0
	if wj <= 6.5e-17:
		tmp = (1.0 / (wj + 1.0)) * (x / math.exp(wj))
	else:
		tmp = wj + ((x + (wj * (-1.0 - x))) / (wj + 1.0))
	return tmp

Julia

function code(wj, x)
	tmp = 0.0
	if (wj <= 6.5e-17)
		tmp = Float64(Float64(1.0 / Float64(wj + 1.0)) * Float64(x / exp(wj)));
	else
		tmp = Float64(wj + Float64(Float64(x + Float64(wj * Float64(-1.0 - x))) / Float64(wj + 1.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(wj, x)
	tmp = 0.0;
	if (wj <= 6.5e-17)
		tmp = (1.0 / (wj + 1.0)) * (x / exp(wj));
	else
		tmp = wj + ((x + (wj * (-1.0 - x))) / (wj + 1.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[wj_, x_] := If[LessEqual[wj, 6.5e-17], N[(N[(1.0 / N[(wj + 1.0), $MachinePrecision]), $MachinePrecision] * N[(x / N[Exp[wj], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(wj + N[(N[(x + N[(wj * N[(-1.0 - x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(wj + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if wj < 6.4999999999999996e-17

   1. Initial program 79.3%

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

      1. distribute-rgt1-in 80.1%

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

      2. associate-/l/ 80.1%

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

      3. div-sub 79.3%

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

      4. associate-/l* 79.3%

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

      5. *-inverses 80.1%

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

      6. /-rgt-identity 80.1%

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

   3. Simplified 80.1%

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

   4. Taylor expanded in x around inf 87.5%

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

      1. associate-/r* 87.5%

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

      2. +-commutative 87.5%

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

   6. Simplified 87.5%

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

      1. div-inv 87.5%

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

   8. Applied egg-rr 87.5%

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

   if 6.4999999999999996e-17 < wj

   1. Initial program 39.6%

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

      1. distribute-rgt1-in 39.1%

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

      2. associate-/l/ 39.5%

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

      3. div-sub 39.5%

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

      4. associate-/l* 39.5%

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

      5. *-inverses 96.6%

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

      6. /-rgt-identity 96.6%

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

   3. Simplified 96.6%

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

   4. Taylor expanded in wj around 0 96.6%

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

4. Final simplification 87.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;wj \leq 6.5 \cdot 10^{-17}:\\ \;\;\;\;\frac{1}{wj + 1} \cdot \frac{x}{e^{wj}}\\ \mathbf{else}:\\ \;\;\;\;wj + \frac{x + wj \cdot \left(-1 - x\right)}{wj + 1}\\ \end{array} \]

Alternative 8: 87.1% accurate, 2.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;wj \leq 7 \cdot 10^{-17}:\\ \;\;\;\;\frac{\frac{x}{e^{wj}}}{wj + 1}\\ \mathbf{else}:\\ \;\;\;\;wj + \frac{x + wj \cdot \left(-1 - x\right)}{wj + 1}\\ \end{array} \end{array} \]

FPCore

(FPCore (wj x)
 :precision binary64
 (if (<= wj 7e-17)
   (/ (/ x (exp wj)) (+ wj 1.0))
   (+ wj (/ (+ x (* wj (- -1.0 x))) (+ wj 1.0)))))

C

double code(double wj, double x) {
	double tmp;
	if (wj <= 7e-17) {
		tmp = (x / exp(wj)) / (wj + 1.0);
	} else {
		tmp = wj + ((x + (wj * (-1.0 - x))) / (wj + 1.0));
	}
	return tmp;
}

Fortran

real(8) function code(wj, x)
    real(8), intent (in) :: wj
    real(8), intent (in) :: x
    real(8) :: tmp
    if (wj <= 7d-17) then
        tmp = (x / exp(wj)) / (wj + 1.0d0)
    else
        tmp = wj + ((x + (wj * ((-1.0d0) - x))) / (wj + 1.0d0))
    end if
    code = tmp
end function

Java

public static double code(double wj, double x) {
	double tmp;
	if (wj <= 7e-17) {
		tmp = (x / Math.exp(wj)) / (wj + 1.0);
	} else {
		tmp = wj + ((x + (wj * (-1.0 - x))) / (wj + 1.0));
	}
	return tmp;
}

Python

def code(wj, x):
	tmp = 0
	if wj <= 7e-17:
		tmp = (x / math.exp(wj)) / (wj + 1.0)
	else:
		tmp = wj + ((x + (wj * (-1.0 - x))) / (wj + 1.0))
	return tmp

Julia

function code(wj, x)
	tmp = 0.0
	if (wj <= 7e-17)
		tmp = Float64(Float64(x / exp(wj)) / Float64(wj + 1.0));
	else
		tmp = Float64(wj + Float64(Float64(x + Float64(wj * Float64(-1.0 - x))) / Float64(wj + 1.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(wj, x)
	tmp = 0.0;
	if (wj <= 7e-17)
		tmp = (x / exp(wj)) / (wj + 1.0);
	else
		tmp = wj + ((x + (wj * (-1.0 - x))) / (wj + 1.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[wj_, x_] := If[LessEqual[wj, 7e-17], N[(N[(x / N[Exp[wj], $MachinePrecision]), $MachinePrecision] / N[(wj + 1.0), $MachinePrecision]), $MachinePrecision], N[(wj + N[(N[(x + N[(wj * N[(-1.0 - x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(wj + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if wj < 7.0000000000000003e-17

   1. Initial program 79.3%

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

      1. distribute-rgt1-in 80.1%

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

      2. associate-/l/ 80.1%

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

      3. div-sub 79.3%

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

      4. associate-/l* 79.3%

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

      5. *-inverses 80.1%

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

      6. /-rgt-identity 80.1%

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

   3. Simplified 80.1%

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

   4. Taylor expanded in x around inf 87.5%

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

      1. associate-/r* 87.5%

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

      2. +-commutative 87.5%

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

   6. Simplified 87.5%

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

   if 7.0000000000000003e-17 < wj

   1. Initial program 39.6%

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

      1. distribute-rgt1-in 39.1%

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

      2. associate-/l/ 39.5%

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

      3. div-sub 39.5%

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

      4. associate-/l* 39.5%

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

      5. *-inverses 96.6%

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

      6. /-rgt-identity 96.6%

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

   3. Simplified 96.6%

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

   4. Taylor expanded in wj around 0 96.6%

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

4. Final simplification 87.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;wj \leq 7 \cdot 10^{-17}:\\ \;\;\;\;\frac{\frac{x}{e^{wj}}}{wj + 1}\\ \mathbf{else}:\\ \;\;\;\;wj + \frac{x + wj \cdot \left(-1 - x\right)}{wj + 1}\\ \end{array} \]

Alternative 9: 85.5% accurate, 2.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;wj \leq 2.4 \cdot 10^{-23}:\\ \;\;\;\;x + {\left(-wj\right)}^{3}\\ \mathbf{else}:\\ \;\;\;\;wj + \frac{x + wj \cdot \left(-1 - x\right)}{wj + 1}\\ \end{array} \end{array} \]

FPCore

(FPCore (wj x)
 :precision binary64
 (if (<= wj 2.4e-23)
   (+ x (pow (- wj) 3.0))
   (+ wj (/ (+ x (* wj (- -1.0 x))) (+ wj 1.0)))))

C

double code(double wj, double x) {
	double tmp;
	if (wj <= 2.4e-23) {
		tmp = x + pow(-wj, 3.0);
	} else {
		tmp = wj + ((x + (wj * (-1.0 - x))) / (wj + 1.0));
	}
	return tmp;
}

Fortran

real(8) function code(wj, x)
    real(8), intent (in) :: wj
    real(8), intent (in) :: x
    real(8) :: tmp
    if (wj <= 2.4d-23) then
        tmp = x + (-wj ** 3.0d0)
    else
        tmp = wj + ((x + (wj * ((-1.0d0) - x))) / (wj + 1.0d0))
    end if
    code = tmp
end function

Java

public static double code(double wj, double x) {
	double tmp;
	if (wj <= 2.4e-23) {
		tmp = x + Math.pow(-wj, 3.0);
	} else {
		tmp = wj + ((x + (wj * (-1.0 - x))) / (wj + 1.0));
	}
	return tmp;
}

Python

def code(wj, x):
	tmp = 0
	if wj <= 2.4e-23:
		tmp = x + math.pow(-wj, 3.0)
	else:
		tmp = wj + ((x + (wj * (-1.0 - x))) / (wj + 1.0))
	return tmp

Julia

function code(wj, x)
	tmp = 0.0
	if (wj <= 2.4e-23)
		tmp = Float64(x + (Float64(-wj) ^ 3.0));
	else
		tmp = Float64(wj + Float64(Float64(x + Float64(wj * Float64(-1.0 - x))) / Float64(wj + 1.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(wj, x)
	tmp = 0.0;
	if (wj <= 2.4e-23)
		tmp = x + (-wj ^ 3.0);
	else
		tmp = wj + ((x + (wj * (-1.0 - x))) / (wj + 1.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[wj_, x_] := If[LessEqual[wj, 2.4e-23], N[(x + N[Power[(-wj), 3.0], $MachinePrecision]), $MachinePrecision], N[(wj + N[(N[(x + N[(wj * N[(-1.0 - x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(wj + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if wj < 2.39999999999999996e-23

   1. Initial program 79.3%

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

      1. distribute-rgt1-in 80.1%

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

      2. associate-/l/ 80.1%

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

      3. div-sub 79.3%

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

      4. associate-/l* 79.3%

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

      5. *-inverses 80.1%

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

      6. /-rgt-identity 80.1%

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

   3. Simplified 80.1%

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

   4. Taylor expanded in wj around 0 98.7%

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

   5. Taylor expanded in x around 0 98.7%

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

   6. Taylor expanded in x around 0 98.6%

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

      1. +-commutative 98.6%

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

      2. mul-1-neg 98.6%

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

      3. unsub-neg 98.6%

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

   8. Simplified 98.6%

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

   9. Taylor expanded in wj around inf 86.5%

      \[\leadsto x + \color{blue}{-1 \cdot {wj}^{3}} \]
   10. Step-by-step derivation

       1. neg-mul-1 86.5%

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

       2. cube-neg 86.5%

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

   11. Simplified 86.5%

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

   if 2.39999999999999996e-23 < wj

   1. Initial program 39.6%

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

      1. distribute-rgt1-in 39.1%

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

      2. associate-/l/ 39.5%

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

      3. div-sub 39.5%

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

      4. associate-/l* 39.5%

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

      5. *-inverses 96.6%

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

      6. /-rgt-identity 96.6%

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

   3. Simplified 96.6%

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

   4. Taylor expanded in wj around 0 96.6%

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

4. Final simplification 86.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;wj \leq 2.4 \cdot 10^{-23}:\\ \;\;\;\;x + {\left(-wj\right)}^{3}\\ \mathbf{else}:\\ \;\;\;\;wj + \frac{x + wj \cdot \left(-1 - x\right)}{wj + 1}\\ \end{array} \]

Alternative 10: 85.6% accurate, 34.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;wj \leq 2.4 \cdot 10^{-7}:\\ \;\;\;\;x \cdot \left(1 + wj \cdot -2\right)\\ \mathbf{else}:\\ \;\;\;\;wj - \frac{wj}{wj + 1}\\ \end{array} \end{array} \]

FPCore

(FPCore (wj x)
 :precision binary64
 (if (<= wj 2.4e-7) (* x (+ 1.0 (* wj -2.0))) (- wj (/ wj (+ wj 1.0)))))

C

double code(double wj, double x) {
	double tmp;
	if (wj <= 2.4e-7) {
		tmp = x * (1.0 + (wj * -2.0));
	} else {
		tmp = wj - (wj / (wj + 1.0));
	}
	return tmp;
}

Fortran

real(8) function code(wj, x)
    real(8), intent (in) :: wj
    real(8), intent (in) :: x
    real(8) :: tmp
    if (wj <= 2.4d-7) then
        tmp = x * (1.0d0 + (wj * (-2.0d0)))
    else
        tmp = wj - (wj / (wj + 1.0d0))
    end if
    code = tmp
end function

Java

public static double code(double wj, double x) {
	double tmp;
	if (wj <= 2.4e-7) {
		tmp = x * (1.0 + (wj * -2.0));
	} else {
		tmp = wj - (wj / (wj + 1.0));
	}
	return tmp;
}

Python

def code(wj, x):
	tmp = 0
	if wj <= 2.4e-7:
		tmp = x * (1.0 + (wj * -2.0))
	else:
		tmp = wj - (wj / (wj + 1.0))
	return tmp

Julia

function code(wj, x)
	tmp = 0.0
	if (wj <= 2.4e-7)
		tmp = Float64(x * Float64(1.0 + Float64(wj * -2.0)));
	else
		tmp = Float64(wj - Float64(wj / Float64(wj + 1.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(wj, x)
	tmp = 0.0;
	if (wj <= 2.4e-7)
		tmp = x * (1.0 + (wj * -2.0));
	else
		tmp = wj - (wj / (wj + 1.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[wj_, x_] := If[LessEqual[wj, 2.4e-7], N[(x * N[(1.0 + N[(wj * -2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(wj - N[(wj / N[(wj + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if wj < 2.39999999999999979e-7

   1. Initial program 79.4%

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

      1. distribute-rgt1-in 80.2%

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

      2. associate-/l/ 80.2%

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

      3. div-sub 79.4%

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

      4. associate-/l* 79.4%

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

      5. *-inverses 80.2%

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

      6. /-rgt-identity 80.2%

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

   3. Simplified 80.2%

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

   4. Taylor expanded in wj around 0 86.4%

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

      1. *-commutative 86.4%

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

   6. Simplified 86.4%

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

   7. Taylor expanded in x around 0 86.4%

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

   if 2.39999999999999979e-7 < wj

   1. Initial program 29.7%

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

      1. distribute-rgt1-in 29.3%

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

      2. associate-/l/ 29.6%

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

      3. div-sub 29.6%

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

      4. associate-/l* 29.6%

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

      5. *-inverses 96.3%

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

      6. /-rgt-identity 96.3%

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

   3. Simplified 96.3%

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

   4. Taylor expanded in x around 0 96.3%

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

      1. +-commutative 96.3%

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

   6. Simplified 96.3%

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

4. Final simplification 86.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;wj \leq 2.4 \cdot 10^{-7}:\\ \;\;\;\;x \cdot \left(1 + wj \cdot -2\right)\\ \mathbf{else}:\\ \;\;\;\;wj - \frac{wj}{wj + 1}\\ \end{array} \]

Alternative 11: 84.3% accurate, 44.7× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 78.2%

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

   1. distribute-rgt1-in 79.0%

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

   2. associate-/l/ 79.0%

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

   3. div-sub 78.2%

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

   4. associate-/l* 78.2%

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

   5. *-inverses 80.6%

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

   6. /-rgt-identity 80.6%

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

3. Simplified 80.6%

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

4. Taylor expanded in wj around 0 84.5%

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

   1. *-commutative 84.5%

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

6. Simplified 84.5%

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

7. Taylor expanded in x around 0 84.5%

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

8. Final simplification 84.5%

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

Alternative 12: 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 78.2%

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

   1. distribute-rgt1-in 79.0%

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

   2. associate-/l/ 79.0%

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

   3. div-sub 78.2%

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

   4. associate-/l* 78.2%

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

   5. *-inverses 80.6%

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

   6. /-rgt-identity 80.6%

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

3. Simplified 80.6%

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

4. Taylor expanded in wj around inf 4.5%

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

5. Final simplification 4.5%

   \[\leadsto wj \]

Alternative 13: 83.8% accurate, 313.0× speedup

Math

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

FPCore

(FPCore (wj x) :precision binary64 x)

C

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

Fortran

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

Java

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

Python

def code(wj, x):
	return x

Julia

function code(wj, x)
	return x
end

MATLAB

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

Wolfram

code[wj_, x_] := x

Derivation

1. Initial program 78.2%

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

   1. distribute-rgt1-in 79.0%

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

   2. associate-/l/ 79.0%

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

   3. div-sub 78.2%

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

   4. associate-/l* 78.2%

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

   5. *-inverses 80.6%

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

   6. /-rgt-identity 80.6%

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

3. Simplified 80.6%

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

4. Taylor expanded in wj around 0 84.2%

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

5. Final simplification 84.2%

   \[\leadsto x \]

Developer target: 78.4% 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 2023318 
(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))))))