Jmat.Real.lambertw, newton loop step

Percentage Accurate: 78.2% → 96.8%

Time: 8.3s

Alternatives: 8

Speedup: 0.9×

• could not determine a ground truth

  1. wj = 2.808990367376855e+49
  2. x = -8.147236881028251e+69

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: 78.2% 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: 96.8% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{wj} \cdot wj\\ \mathbf{if}\;wj - \frac{t\_0 - x}{e^{wj} + t\_0} \leq 5 \cdot 10^{+302}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-2.6666666666666665, wj, 2.5\right), wj, -2\right), wj, \mathsf{fma}\left(\frac{1 - wj}{x}, wj \cdot wj, 1\right)\right) \cdot x\\ \mathbf{else}:\\ \;\;\;\;wj - \frac{wj}{1 + wj}\\ \end{array} \end{array} \]

FPCore

(FPCore (wj x)
 :precision binary64
 (let* ((t_0 (* (exp wj) wj)))
   (if (<= (- wj (/ (- t_0 x) (+ (exp wj) t_0))) 5e+302)
     (*
      (fma
       (fma (fma -2.6666666666666665 wj 2.5) wj -2.0)
       wj
       (fma (/ (- 1.0 wj) x) (* wj wj) 1.0))
      x)
     (- wj (/ wj (+ 1.0 wj))))))

C

double code(double wj, double x) {
	double t_0 = exp(wj) * wj;
	double tmp;
	if ((wj - ((t_0 - x) / (exp(wj) + t_0))) <= 5e+302) {
		tmp = fma(fma(fma(-2.6666666666666665, wj, 2.5), wj, -2.0), wj, fma(((1.0 - wj) / x), (wj * wj), 1.0)) * x;
	} else {
		tmp = wj - (wj / (1.0 + wj));
	}
	return tmp;
}

Julia

function code(wj, x)
	t_0 = Float64(exp(wj) * wj)
	tmp = 0.0
	if (Float64(wj - Float64(Float64(t_0 - x) / Float64(exp(wj) + t_0))) <= 5e+302)
		tmp = Float64(fma(fma(fma(-2.6666666666666665, wj, 2.5), wj, -2.0), wj, fma(Float64(Float64(1.0 - wj) / x), Float64(wj * wj), 1.0)) * x);
	else
		tmp = Float64(wj - Float64(wj / Float64(1.0 + wj)));
	end
	return tmp
end

Wolfram

code[wj_, x_] := Block[{t$95$0 = N[(N[Exp[wj], $MachinePrecision] * wj), $MachinePrecision]}, If[LessEqual[N[(wj - N[(N[(t$95$0 - x), $MachinePrecision] / N[(N[Exp[wj], $MachinePrecision] + t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 5e+302], N[(N[(N[(N[(-2.6666666666666665 * wj + 2.5), $MachinePrecision] * wj + -2.0), $MachinePrecision] * wj + N[(N[(N[(1.0 - wj), $MachinePrecision] / x), $MachinePrecision] * N[(wj * wj), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] * x), $MachinePrecision], N[(wj - N[(wj / N[(1.0 + 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))))) < 5e302

   1. Initial program 74.6%

      \[wj - \frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}} \]
   2. Add Preprocessing

   3. Taylor expanded in wj around 0

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

   5. Applied rewrites 99.3%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(2.5, x, 1 - \mathsf{fma}\left(0.6666666666666666, x, \mathsf{fma}\left(2, x, 1\right)\right) \cdot wj\right), wj, -2 \cdot x\right), wj, x\right)} \]

   6. Taylor expanded in x around inf

      \[\leadsto x \cdot \color{blue}{\left(1 + \left(wj \cdot \left(wj \cdot \left(\frac{5}{2} - \frac{8}{3} \cdot wj\right) - 2\right) + \frac{{wj}^{2} \cdot \left(1 - wj\right)}{x}\right)\right)} \]
   7. Step-by-step derivation

   8. Applied rewrites 99.7%

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

   if 5e302 < (-.f64 wj (/.f64 (-.f64 (*.f64 wj (exp.f64 wj)) x) (+.f64 (exp.f64 wj) (*.f64 wj (exp.f64 wj)))))

   1. Initial program 0.0%

      \[wj - \frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

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

      1. distribute-rgt1-in N/A

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

      2. +-commutative N/A

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

      3. times-frac N/A

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

      4. *-inverses N/A

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

      5. associate-*l/ N/A

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

      6. *-rgt-identity N/A

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

      7. lower-/.f64 N/A

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

      8. lower-+.f64 83.9

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

   5. Applied rewrites 83.9%

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

4. Final simplification 98.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;wj - \frac{e^{wj} \cdot wj - x}{e^{wj} + e^{wj} \cdot wj} \leq 5 \cdot 10^{+302}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-2.6666666666666665, wj, 2.5\right), wj, -2\right), wj, \mathsf{fma}\left(\frac{1 - wj}{x}, wj \cdot wj, 1\right)\right) \cdot x\\ \mathbf{else}:\\ \;\;\;\;wj - \frac{wj}{1 + wj}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 85.0% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{wj} \cdot wj\\ t_1 := wj - \frac{t\_0 - x}{e^{wj} + t\_0}\\ \mathbf{if}\;t\_1 \leq -1 \cdot 10^{-308}:\\ \;\;\;\;1 \cdot x\\ \mathbf{elif}\;t\_1 \leq 0:\\ \;\;\;\;wj \cdot wj\\ \mathbf{elif}\;t\_1 \leq 5 \cdot 10^{+302}:\\ \;\;\;\;\mathsf{fma}\left(-2, wj, 1\right) \cdot x\\ \mathbf{else}:\\ \;\;\;\;wj - 1\\ \end{array} \end{array} \]

FPCore

(FPCore (wj x)
 :precision binary64
 (let* ((t_0 (* (exp wj) wj)) (t_1 (- wj (/ (- t_0 x) (+ (exp wj) t_0)))))
   (if (<= t_1 -1e-308)
     (* 1.0 x)
     (if (<= t_1 0.0)
       (* wj wj)
       (if (<= t_1 5e+302) (* (fma -2.0 wj 1.0) x) (- wj 1.0))))))

C

double code(double wj, double x) {
	double t_0 = exp(wj) * wj;
	double t_1 = wj - ((t_0 - x) / (exp(wj) + t_0));
	double tmp;
	if (t_1 <= -1e-308) {
		tmp = 1.0 * x;
	} else if (t_1 <= 0.0) {
		tmp = wj * wj;
	} else if (t_1 <= 5e+302) {
		tmp = fma(-2.0, wj, 1.0) * x;
	} else {
		tmp = wj - 1.0;
	}
	return tmp;
}

Julia

function code(wj, x)
	t_0 = Float64(exp(wj) * wj)
	t_1 = Float64(wj - Float64(Float64(t_0 - x) / Float64(exp(wj) + t_0)))
	tmp = 0.0
	if (t_1 <= -1e-308)
		tmp = Float64(1.0 * x);
	elseif (t_1 <= 0.0)
		tmp = Float64(wj * wj);
	elseif (t_1 <= 5e+302)
		tmp = Float64(fma(-2.0, wj, 1.0) * x);
	else
		tmp = Float64(wj - 1.0);
	end
	return tmp
end

Wolfram

code[wj_, x_] := Block[{t$95$0 = N[(N[Exp[wj], $MachinePrecision] * wj), $MachinePrecision]}, Block[{t$95$1 = N[(wj - N[(N[(t$95$0 - x), $MachinePrecision] / N[(N[Exp[wj], $MachinePrecision] + t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -1e-308], N[(1.0 * x), $MachinePrecision], If[LessEqual[t$95$1, 0.0], N[(wj * wj), $MachinePrecision], If[LessEqual[t$95$1, 5e+302], N[(N[(-2.0 * wj + 1.0), $MachinePrecision] * x), $MachinePrecision], N[(wj - 1.0), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if (-.f64 wj (/.f64 (-.f64 (*.f64 wj (exp.f64 wj)) x) (+.f64 (exp.f64 wj) (*.f64 wj (exp.f64 wj))))) < -9.9999999999999991e-309

   1. Initial program 98.7%

      \[wj - \frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}} \]
   2. Add Preprocessing

   3. Taylor expanded in wj around 0

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

   5. Applied rewrites 99.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(2.5, x, 1 - \mathsf{fma}\left(0.6666666666666666, x, \mathsf{fma}\left(2, x, 1\right)\right) \cdot wj\right), wj, -2 \cdot x\right), wj, x\right)} \]

   6. Taylor expanded in x around inf

      \[\leadsto x \cdot \color{blue}{\left(1 + \left(wj \cdot \left(wj \cdot \left(\frac{5}{2} - \frac{8}{3} \cdot wj\right) - 2\right) + \frac{{wj}^{2} \cdot \left(1 - wj\right)}{x}\right)\right)} \]
   7. Step-by-step derivation

   8. Applied rewrites 100.0%

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

   9. Taylor expanded in wj around 0

      \[\leadsto 1 \cdot x \]
   10. Step-by-step derivation

   11. Applied rewrites 99.0%

       \[\leadsto 1 \cdot x \]

   if -9.9999999999999991e-309 < (-.f64 wj (/.f64 (-.f64 (*.f64 wj (exp.f64 wj)) x) (+.f64 (exp.f64 wj) (*.f64 wj (exp.f64 wj))))) < 0.0

   1. Initial program 5.2%

      \[wj - \frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}} \]
   2. Add Preprocessing

   3. Taylor expanded in wj around 0

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

   5. Applied rewrites 100.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(2.5, x, 1 - \mathsf{fma}\left(0.6666666666666666, x, \mathsf{fma}\left(2, x, 1\right)\right) \cdot wj\right), wj, -2 \cdot x\right), wj, x\right)} \]

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 63.8%

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

   9. Taylor expanded in wj around 0

      \[\leadsto {wj}^{2} \]
   10. Step-by-step derivation

   11. Applied rewrites 63.8%

       \[\leadsto wj \cdot wj \]

   if 0.0 < (-.f64 wj (/.f64 (-.f64 (*.f64 wj (exp.f64 wj)) x) (+.f64 (exp.f64 wj) (*.f64 wj (exp.f64 wj))))) < 5e302

   1. Initial program 95.7%

      \[wj - \frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}} \]
   2. Add Preprocessing

   3. Taylor expanded in wj around 0

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

   5. Applied rewrites 99.3%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(2.5, x, 1 - \mathsf{fma}\left(0.6666666666666666, x, \mathsf{fma}\left(2, x, 1\right)\right) \cdot wj\right), wj, -2 \cdot x\right), wj, x\right)} \]

   6. Taylor expanded in wj around 0

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

      1. associate-*r* N/A

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

      2. distribute-rgt1-in N/A

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

      3. +-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. +-commutative N/A

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

      6. lower-fma.f64 91.9

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

   8. Applied rewrites 91.9%

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

   if 5e302 < (-.f64 wj (/.f64 (-.f64 (*.f64 wj (exp.f64 wj)) x) (+.f64 (exp.f64 wj) (*.f64 wj (exp.f64 wj)))))

   1. Initial program 0.0%

      \[wj - \frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}} \]
   2. Add Preprocessing

   3. Taylor expanded in wj around inf

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

   5. Applied rewrites 64.5%

      \[\leadsto wj - \color{blue}{1} \]
3. Recombined 4 regimes into one program.

4. Final simplification 86.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;wj - \frac{e^{wj} \cdot wj - x}{e^{wj} + e^{wj} \cdot wj} \leq -1 \cdot 10^{-308}:\\ \;\;\;\;1 \cdot x\\ \mathbf{elif}\;wj - \frac{e^{wj} \cdot wj - x}{e^{wj} + e^{wj} \cdot wj} \leq 0:\\ \;\;\;\;wj \cdot wj\\ \mathbf{elif}\;wj - \frac{e^{wj} \cdot wj - x}{e^{wj} + e^{wj} \cdot wj} \leq 5 \cdot 10^{+302}:\\ \;\;\;\;\mathsf{fma}\left(-2, wj, 1\right) \cdot x\\ \mathbf{else}:\\ \;\;\;\;wj - 1\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 84.7% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{wj} \cdot wj\\ t_1 := wj - \frac{t\_0 - x}{e^{wj} + t\_0}\\ \mathbf{if}\;t\_1 \leq -1 \cdot 10^{-308}:\\ \;\;\;\;1 \cdot x\\ \mathbf{elif}\;t\_1 \leq 0:\\ \;\;\;\;wj \cdot wj\\ \mathbf{elif}\;t\_1 \leq 5 \cdot 10^{+302}:\\ \;\;\;\;1 \cdot x\\ \mathbf{else}:\\ \;\;\;\;wj - 1\\ \end{array} \end{array} \]

FPCore

(FPCore (wj x)
 :precision binary64
 (let* ((t_0 (* (exp wj) wj)) (t_1 (- wj (/ (- t_0 x) (+ (exp wj) t_0)))))
   (if (<= t_1 -1e-308)
     (* 1.0 x)
     (if (<= t_1 0.0) (* wj wj) (if (<= t_1 5e+302) (* 1.0 x) (- wj 1.0))))))

C

double code(double wj, double x) {
	double t_0 = exp(wj) * wj;
	double t_1 = wj - ((t_0 - x) / (exp(wj) + t_0));
	double tmp;
	if (t_1 <= -1e-308) {
		tmp = 1.0 * x;
	} else if (t_1 <= 0.0) {
		tmp = wj * wj;
	} else if (t_1 <= 5e+302) {
		tmp = 1.0 * x;
	} else {
		tmp = wj - 1.0;
	}
	return tmp;
}

Fortran

real(8) function code(wj, x)
    real(8), intent (in) :: wj
    real(8), intent (in) :: x
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = exp(wj) * wj
    t_1 = wj - ((t_0 - x) / (exp(wj) + t_0))
    if (t_1 <= (-1d-308)) then
        tmp = 1.0d0 * x
    else if (t_1 <= 0.0d0) then
        tmp = wj * wj
    else if (t_1 <= 5d+302) then
        tmp = 1.0d0 * x
    else
        tmp = wj - 1.0d0
    end if
    code = tmp
end function

Java

public static double code(double wj, double x) {
	double t_0 = Math.exp(wj) * wj;
	double t_1 = wj - ((t_0 - x) / (Math.exp(wj) + t_0));
	double tmp;
	if (t_1 <= -1e-308) {
		tmp = 1.0 * x;
	} else if (t_1 <= 0.0) {
		tmp = wj * wj;
	} else if (t_1 <= 5e+302) {
		tmp = 1.0 * x;
	} else {
		tmp = wj - 1.0;
	}
	return tmp;
}

Python

def code(wj, x):
	t_0 = math.exp(wj) * wj
	t_1 = wj - ((t_0 - x) / (math.exp(wj) + t_0))
	tmp = 0
	if t_1 <= -1e-308:
		tmp = 1.0 * x
	elif t_1 <= 0.0:
		tmp = wj * wj
	elif t_1 <= 5e+302:
		tmp = 1.0 * x
	else:
		tmp = wj - 1.0
	return tmp

Julia

function code(wj, x)
	t_0 = Float64(exp(wj) * wj)
	t_1 = Float64(wj - Float64(Float64(t_0 - x) / Float64(exp(wj) + t_0)))
	tmp = 0.0
	if (t_1 <= -1e-308)
		tmp = Float64(1.0 * x);
	elseif (t_1 <= 0.0)
		tmp = Float64(wj * wj);
	elseif (t_1 <= 5e+302)
		tmp = Float64(1.0 * x);
	else
		tmp = Float64(wj - 1.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(wj, x)
	t_0 = exp(wj) * wj;
	t_1 = wj - ((t_0 - x) / (exp(wj) + t_0));
	tmp = 0.0;
	if (t_1 <= -1e-308)
		tmp = 1.0 * x;
	elseif (t_1 <= 0.0)
		tmp = wj * wj;
	elseif (t_1 <= 5e+302)
		tmp = 1.0 * x;
	else
		tmp = wj - 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[wj_, x_] := Block[{t$95$0 = N[(N[Exp[wj], $MachinePrecision] * wj), $MachinePrecision]}, Block[{t$95$1 = N[(wj - N[(N[(t$95$0 - x), $MachinePrecision] / N[(N[Exp[wj], $MachinePrecision] + t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -1e-308], N[(1.0 * x), $MachinePrecision], If[LessEqual[t$95$1, 0.0], N[(wj * wj), $MachinePrecision], If[LessEqual[t$95$1, 5e+302], N[(1.0 * x), $MachinePrecision], N[(wj - 1.0), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if (-.f64 wj (/.f64 (-.f64 (*.f64 wj (exp.f64 wj)) x) (+.f64 (exp.f64 wj) (*.f64 wj (exp.f64 wj))))) < -9.9999999999999991e-309 or 0.0 < (-.f64 wj (/.f64 (-.f64 (*.f64 wj (exp.f64 wj)) x) (+.f64 (exp.f64 wj) (*.f64 wj (exp.f64 wj))))) < 5e302

   1. Initial program 97.3%

      \[wj - \frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}} \]
   2. Add Preprocessing

   3. Taylor expanded in wj around 0

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

   5. Applied rewrites 99.1%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(2.5, x, 1 - \mathsf{fma}\left(0.6666666666666666, x, \mathsf{fma}\left(2, x, 1\right)\right) \cdot wj\right), wj, -2 \cdot x\right), wj, x\right)} \]

   6. Taylor expanded in x around inf

      \[\leadsto x \cdot \color{blue}{\left(1 + \left(wj \cdot \left(wj \cdot \left(\frac{5}{2} - \frac{8}{3} \cdot wj\right) - 2\right) + \frac{{wj}^{2} \cdot \left(1 - wj\right)}{x}\right)\right)} \]
   7. Step-by-step derivation

   8. Applied rewrites 99.7%

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

   9. Taylor expanded in wj around 0

      \[\leadsto 1 \cdot x \]
   10. Step-by-step derivation

   11. Applied rewrites 95.5%

       \[\leadsto 1 \cdot x \]

   if -9.9999999999999991e-309 < (-.f64 wj (/.f64 (-.f64 (*.f64 wj (exp.f64 wj)) x) (+.f64 (exp.f64 wj) (*.f64 wj (exp.f64 wj))))) < 0.0

   1. Initial program 5.2%

      \[wj - \frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}} \]
   2. Add Preprocessing

   3. Taylor expanded in wj around 0

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

   5. Applied rewrites 100.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(2.5, x, 1 - \mathsf{fma}\left(0.6666666666666666, x, \mathsf{fma}\left(2, x, 1\right)\right) \cdot wj\right), wj, -2 \cdot x\right), wj, x\right)} \]

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 63.8%

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

   9. Taylor expanded in wj around 0

      \[\leadsto {wj}^{2} \]
   10. Step-by-step derivation

   11. Applied rewrites 63.8%

       \[\leadsto wj \cdot wj \]

   if 5e302 < (-.f64 wj (/.f64 (-.f64 (*.f64 wj (exp.f64 wj)) x) (+.f64 (exp.f64 wj) (*.f64 wj (exp.f64 wj)))))

   1. Initial program 0.0%

      \[wj - \frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}} \]
   2. Add Preprocessing

   3. Taylor expanded in wj around inf

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

   5. Applied rewrites 64.5%

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

4. Final simplification 86.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;wj - \frac{e^{wj} \cdot wj - x}{e^{wj} + e^{wj} \cdot wj} \leq -1 \cdot 10^{-308}:\\ \;\;\;\;1 \cdot x\\ \mathbf{elif}\;wj - \frac{e^{wj} \cdot wj - x}{e^{wj} + e^{wj} \cdot wj} \leq 0:\\ \;\;\;\;wj \cdot wj\\ \mathbf{elif}\;wj - \frac{e^{wj} \cdot wj - x}{e^{wj} + e^{wj} \cdot wj} \leq 5 \cdot 10^{+302}:\\ \;\;\;\;1 \cdot x\\ \mathbf{else}:\\ \;\;\;\;wj - 1\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 96.8% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{wj} \cdot wj\\ \mathbf{if}\;wj - \frac{t\_0 - x}{e^{wj} + t\_0} \leq 5 \cdot 10^{+302}:\\ \;\;\;\;\mathsf{fma}\left(\left(\frac{1 - wj}{x} \cdot wj + \mathsf{fma}\left(2.5, wj, -2\right)\right) \cdot wj, x, x\right)\\ \mathbf{else}:\\ \;\;\;\;wj - \frac{wj}{1 + wj}\\ \end{array} \end{array} \]

FPCore

(FPCore (wj x)
 :precision binary64
 (let* ((t_0 (* (exp wj) wj)))
   (if (<= (- wj (/ (- t_0 x) (+ (exp wj) t_0))) 5e+302)
     (fma (* (+ (* (/ (- 1.0 wj) x) wj) (fma 2.5 wj -2.0)) wj) x x)
     (- wj (/ wj (+ 1.0 wj))))))

C

double code(double wj, double x) {
	double t_0 = exp(wj) * wj;
	double tmp;
	if ((wj - ((t_0 - x) / (exp(wj) + t_0))) <= 5e+302) {
		tmp = fma((((((1.0 - wj) / x) * wj) + fma(2.5, wj, -2.0)) * wj), x, x);
	} else {
		tmp = wj - (wj / (1.0 + wj));
	}
	return tmp;
}

Julia

function code(wj, x)
	t_0 = Float64(exp(wj) * wj)
	tmp = 0.0
	if (Float64(wj - Float64(Float64(t_0 - x) / Float64(exp(wj) + t_0))) <= 5e+302)
		tmp = fma(Float64(Float64(Float64(Float64(Float64(1.0 - wj) / x) * wj) + fma(2.5, wj, -2.0)) * wj), x, x);
	else
		tmp = Float64(wj - Float64(wj / Float64(1.0 + wj)));
	end
	return tmp
end

Wolfram

code[wj_, x_] := Block[{t$95$0 = N[(N[Exp[wj], $MachinePrecision] * wj), $MachinePrecision]}, If[LessEqual[N[(wj - N[(N[(t$95$0 - x), $MachinePrecision] / N[(N[Exp[wj], $MachinePrecision] + t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 5e+302], N[(N[(N[(N[(N[(N[(1.0 - wj), $MachinePrecision] / x), $MachinePrecision] * wj), $MachinePrecision] + N[(2.5 * wj + -2.0), $MachinePrecision]), $MachinePrecision] * wj), $MachinePrecision] * x + x), $MachinePrecision], N[(wj - N[(wj / N[(1.0 + 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))))) < 5e302

   1. Initial program 74.6%

      \[wj - \frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}} \]
   2. Add Preprocessing

   3. Taylor expanded in wj around 0

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

   5. Applied rewrites 99.3%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(2.5, x, 1 - \mathsf{fma}\left(0.6666666666666666, x, \mathsf{fma}\left(2, x, 1\right)\right) \cdot wj\right), wj, -2 \cdot x\right), wj, x\right)} \]

   6. Taylor expanded in x around inf

      \[\leadsto x \cdot \color{blue}{\left(1 + \left(wj \cdot \left(wj \cdot \left(\frac{5}{2} - \frac{8}{3} \cdot wj\right) - 2\right) + \frac{{wj}^{2} \cdot \left(1 - wj\right)}{x}\right)\right)} \]
   7. Step-by-step derivation

   8. Applied rewrites 99.7%

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

   9. Taylor expanded in wj around 0

      \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\frac{5}{2}, wj, -2\right), wj, \mathsf{fma}\left(\frac{1 - wj}{x}, wj \cdot wj, 1\right)\right) \cdot x \]
   10. Step-by-step derivation

   11. Applied rewrites 99.6%

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

   13. Applied rewrites 99.6%

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

   if 5e302 < (-.f64 wj (/.f64 (-.f64 (*.f64 wj (exp.f64 wj)) x) (+.f64 (exp.f64 wj) (*.f64 wj (exp.f64 wj)))))

   1. Initial program 0.0%

      \[wj - \frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

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

      1. distribute-rgt1-in N/A

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

      2. +-commutative N/A

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

      3. times-frac N/A

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

      4. *-inverses N/A

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

      5. associate-*l/ N/A

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

      6. *-rgt-identity N/A

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

      7. lower-/.f64 N/A

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

      8. lower-+.f64 83.9

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

   5. Applied rewrites 83.9%

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

4. Final simplification 98.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;wj - \frac{e^{wj} \cdot wj - x}{e^{wj} + e^{wj} \cdot wj} \leq 5 \cdot 10^{+302}:\\ \;\;\;\;\mathsf{fma}\left(\left(\frac{1 - wj}{x} \cdot wj + \mathsf{fma}\left(2.5, wj, -2\right)\right) \cdot wj, x, x\right)\\ \mathbf{else}:\\ \;\;\;\;wj - \frac{wj}{1 + wj}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 96.0% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{wj} \cdot wj\\ \mathbf{if}\;wj - \frac{t\_0 - x}{e^{wj} + t\_0} \leq 5 \cdot 10^{+302}:\\ \;\;\;\;\mathsf{fma}\left(\left(1 - wj\right) \cdot wj, wj, x\right)\\ \mathbf{else}:\\ \;\;\;\;wj - \frac{wj}{1 + wj}\\ \end{array} \end{array} \]

FPCore

(FPCore (wj x)
 :precision binary64
 (let* ((t_0 (* (exp wj) wj)))
   (if (<= (- wj (/ (- t_0 x) (+ (exp wj) t_0))) 5e+302)
     (fma (* (- 1.0 wj) wj) wj x)
     (- wj (/ wj (+ 1.0 wj))))))

C

double code(double wj, double x) {
	double t_0 = exp(wj) * wj;
	double tmp;
	if ((wj - ((t_0 - x) / (exp(wj) + t_0))) <= 5e+302) {
		tmp = fma(((1.0 - wj) * wj), wj, x);
	} else {
		tmp = wj - (wj / (1.0 + wj));
	}
	return tmp;
}

Julia

function code(wj, x)
	t_0 = Float64(exp(wj) * wj)
	tmp = 0.0
	if (Float64(wj - Float64(Float64(t_0 - x) / Float64(exp(wj) + t_0))) <= 5e+302)
		tmp = fma(Float64(Float64(1.0 - wj) * wj), wj, x);
	else
		tmp = Float64(wj - Float64(wj / Float64(1.0 + wj)));
	end
	return tmp
end

Wolfram

code[wj_, x_] := Block[{t$95$0 = N[(N[Exp[wj], $MachinePrecision] * wj), $MachinePrecision]}, If[LessEqual[N[(wj - N[(N[(t$95$0 - x), $MachinePrecision] / N[(N[Exp[wj], $MachinePrecision] + t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 5e+302], N[(N[(N[(1.0 - wj), $MachinePrecision] * wj), $MachinePrecision] * wj + x), $MachinePrecision], N[(wj - N[(wj / N[(1.0 + 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))))) < 5e302

   1. Initial program 74.6%

      \[wj - \frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}} \]
   2. Add Preprocessing

   3. Taylor expanded in wj around 0

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

   5. Applied rewrites 99.3%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(2.5, x, 1 - \mathsf{fma}\left(0.6666666666666666, x, \mathsf{fma}\left(2, x, 1\right)\right) \cdot wj\right), wj, -2 \cdot x\right), wj, x\right)} \]

   6. Taylor expanded in x around 0

      \[\leadsto \mathsf{fma}\left(wj \cdot \left(1 - wj\right), wj, x\right) \]
   7. Step-by-step derivation

   8. Applied rewrites 99.1%

      \[\leadsto \mathsf{fma}\left(\left(1 - wj\right) \cdot wj, wj, x\right) \]

   if 5e302 < (-.f64 wj (/.f64 (-.f64 (*.f64 wj (exp.f64 wj)) x) (+.f64 (exp.f64 wj) (*.f64 wj (exp.f64 wj)))))

   1. Initial program 0.0%

      \[wj - \frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

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

      1. distribute-rgt1-in N/A

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

      2. +-commutative N/A

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

      3. times-frac N/A

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

      4. *-inverses N/A

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

      5. associate-*l/ N/A

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

      6. *-rgt-identity N/A

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

      7. lower-/.f64 N/A

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

      8. lower-+.f64 83.9

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

   5. Applied rewrites 83.9%

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

4. Final simplification 98.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;wj - \frac{e^{wj} \cdot wj - x}{e^{wj} + e^{wj} \cdot wj} \leq 5 \cdot 10^{+302}:\\ \;\;\;\;\mathsf{fma}\left(\left(1 - wj\right) \cdot wj, wj, x\right)\\ \mathbf{else}:\\ \;\;\;\;wj - \frac{wj}{1 + wj}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 95.7% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{wj} \cdot wj\\ \mathbf{if}\;wj - \frac{t\_0 - x}{e^{wj} + t\_0} \leq 5 \cdot 10^{+302}:\\ \;\;\;\;\mathsf{fma}\left(\left(1 - wj\right) \cdot wj, wj, x\right)\\ \mathbf{else}:\\ \;\;\;\;wj - 1\\ \end{array} \end{array} \]

FPCore

(FPCore (wj x)
 :precision binary64
 (let* ((t_0 (* (exp wj) wj)))
   (if (<= (- wj (/ (- t_0 x) (+ (exp wj) t_0))) 5e+302)
     (fma (* (- 1.0 wj) wj) wj x)
     (- wj 1.0))))

C

double code(double wj, double x) {
	double t_0 = exp(wj) * wj;
	double tmp;
	if ((wj - ((t_0 - x) / (exp(wj) + t_0))) <= 5e+302) {
		tmp = fma(((1.0 - wj) * wj), wj, x);
	} else {
		tmp = wj - 1.0;
	}
	return tmp;
}

Julia

function code(wj, x)
	t_0 = Float64(exp(wj) * wj)
	tmp = 0.0
	if (Float64(wj - Float64(Float64(t_0 - x) / Float64(exp(wj) + t_0))) <= 5e+302)
		tmp = fma(Float64(Float64(1.0 - wj) * wj), wj, x);
	else
		tmp = Float64(wj - 1.0);
	end
	return tmp
end

Wolfram

code[wj_, x_] := Block[{t$95$0 = N[(N[Exp[wj], $MachinePrecision] * wj), $MachinePrecision]}, If[LessEqual[N[(wj - N[(N[(t$95$0 - x), $MachinePrecision] / N[(N[Exp[wj], $MachinePrecision] + t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 5e+302], N[(N[(N[(1.0 - wj), $MachinePrecision] * wj), $MachinePrecision] * wj + x), $MachinePrecision], N[(wj - 1.0), $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))))) < 5e302

   1. Initial program 74.6%

      \[wj - \frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}} \]
   2. Add Preprocessing

   3. Taylor expanded in wj around 0

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

   5. Applied rewrites 99.3%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(2.5, x, 1 - \mathsf{fma}\left(0.6666666666666666, x, \mathsf{fma}\left(2, x, 1\right)\right) \cdot wj\right), wj, -2 \cdot x\right), wj, x\right)} \]

   6. Taylor expanded in x around 0

      \[\leadsto \mathsf{fma}\left(wj \cdot \left(1 - wj\right), wj, x\right) \]
   7. Step-by-step derivation

   8. Applied rewrites 99.1%

      \[\leadsto \mathsf{fma}\left(\left(1 - wj\right) \cdot wj, wj, x\right) \]

   if 5e302 < (-.f64 wj (/.f64 (-.f64 (*.f64 wj (exp.f64 wj)) x) (+.f64 (exp.f64 wj) (*.f64 wj (exp.f64 wj)))))

   1. Initial program 0.0%

      \[wj - \frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}} \]
   2. Add Preprocessing

   3. Taylor expanded in wj around inf

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

   5. Applied rewrites 64.5%

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

4. Final simplification 97.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;wj - \frac{e^{wj} \cdot wj - x}{e^{wj} + e^{wj} \cdot wj} \leq 5 \cdot 10^{+302}:\\ \;\;\;\;\mathsf{fma}\left(\left(1 - wj\right) \cdot wj, wj, x\right)\\ \mathbf{else}:\\ \;\;\;\;wj - 1\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 14.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{wj} \cdot wj\\ \mathbf{if}\;wj - \frac{t\_0 - x}{e^{wj} + t\_0} \leq 5 \cdot 10^{+302}:\\ \;\;\;\;wj \cdot wj\\ \mathbf{else}:\\ \;\;\;\;wj - 1\\ \end{array} \end{array} \]

FPCore

(FPCore (wj x)
 :precision binary64
 (let* ((t_0 (* (exp wj) wj)))
   (if (<= (- wj (/ (- t_0 x) (+ (exp wj) t_0))) 5e+302)
     (* wj wj)
     (- wj 1.0))))

C

double code(double wj, double x) {
	double t_0 = exp(wj) * wj;
	double tmp;
	if ((wj - ((t_0 - x) / (exp(wj) + t_0))) <= 5e+302) {
		tmp = wj * wj;
	} else {
		tmp = wj - 1.0;
	}
	return tmp;
}

Fortran

real(8) function code(wj, x)
    real(8), intent (in) :: wj
    real(8), intent (in) :: x
    real(8) :: t_0
    real(8) :: tmp
    t_0 = exp(wj) * wj
    if ((wj - ((t_0 - x) / (exp(wj) + t_0))) <= 5d+302) then
        tmp = wj * wj
    else
        tmp = wj - 1.0d0
    end if
    code = tmp
end function

Java

public static double code(double wj, double x) {
	double t_0 = Math.exp(wj) * wj;
	double tmp;
	if ((wj - ((t_0 - x) / (Math.exp(wj) + t_0))) <= 5e+302) {
		tmp = wj * wj;
	} else {
		tmp = wj - 1.0;
	}
	return tmp;
}

Python

def code(wj, x):
	t_0 = math.exp(wj) * wj
	tmp = 0
	if (wj - ((t_0 - x) / (math.exp(wj) + t_0))) <= 5e+302:
		tmp = wj * wj
	else:
		tmp = wj - 1.0
	return tmp

Julia

function code(wj, x)
	t_0 = Float64(exp(wj) * wj)
	tmp = 0.0
	if (Float64(wj - Float64(Float64(t_0 - x) / Float64(exp(wj) + t_0))) <= 5e+302)
		tmp = Float64(wj * wj);
	else
		tmp = Float64(wj - 1.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(wj, x)
	t_0 = exp(wj) * wj;
	tmp = 0.0;
	if ((wj - ((t_0 - x) / (exp(wj) + t_0))) <= 5e+302)
		tmp = wj * wj;
	else
		tmp = wj - 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[wj_, x_] := Block[{t$95$0 = N[(N[Exp[wj], $MachinePrecision] * wj), $MachinePrecision]}, If[LessEqual[N[(wj - N[(N[(t$95$0 - x), $MachinePrecision] / N[(N[Exp[wj], $MachinePrecision] + t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 5e+302], N[(wj * wj), $MachinePrecision], N[(wj - 1.0), $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))))) < 5e302

   1. Initial program 74.6%

      \[wj - \frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}} \]
   2. Add Preprocessing

   3. Taylor expanded in wj around 0

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

   5. Applied rewrites 99.3%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(2.5, x, 1 - \mathsf{fma}\left(0.6666666666666666, x, \mathsf{fma}\left(2, x, 1\right)\right) \cdot wj\right), wj, -2 \cdot x\right), wj, x\right)} \]

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 20.7%

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

   9. Taylor expanded in wj around 0

      \[\leadsto {wj}^{2} \]
   10. Step-by-step derivation

   11. Applied rewrites 20.0%

       \[\leadsto wj \cdot wj \]

   if 5e302 < (-.f64 wj (/.f64 (-.f64 (*.f64 wj (exp.f64 wj)) x) (+.f64 (exp.f64 wj) (*.f64 wj (exp.f64 wj)))))

   1. Initial program 0.0%

      \[wj - \frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}} \]
   2. Add Preprocessing

   3. Taylor expanded in wj around inf

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

   5. Applied rewrites 64.5%

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

4. Final simplification 22.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;wj - \frac{e^{wj} \cdot wj - x}{e^{wj} + e^{wj} \cdot wj} \leq 5 \cdot 10^{+302}:\\ \;\;\;\;wj \cdot wj\\ \mathbf{else}:\\ \;\;\;\;wj - 1\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 4.1% accurate, 82.8× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 71.1%

   \[wj - \frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}} \]
2. Add Preprocessing

3. Taylor expanded in wj around inf

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

5. Applied rewrites 6.0%

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

Developer Target 1: 79.2% accurate, 1.4× 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 2024276 
(FPCore (wj x)
  :name "Jmat.Real.lambertw, newton loop step"
  :precision binary64

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

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