Jmat.Real.lambertw, newton loop step

Percentage Accurate: 78.4% → 98.5%

Time: 8.2s

Alternatives: 12

Speedup: 22.1×

• could not determine a ground truth

  1. wj = 1.2417946412699573e+183
  2. x = -2.7949007957275274e+253

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: 78.4% 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: 98.5% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{wj} \cdot wj\\ \mathbf{if}\;wj - \frac{t\_0 - x}{t\_0 + e^{wj}} \leq 10^{-17}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(2.5, x, 1 - \mathsf{fma}\left(0.6666666666666666, x, \mathsf{fma}\left(2, x, 1\right)\right) \cdot wj\right), wj, -2 \cdot x\right), wj, x\right)\\ \mathbf{else}:\\ \;\;\;\;wj - \left(\frac{\frac{wj}{wj - -1}}{x} - \frac{e^{-wj}}{wj - -1}\right) \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (wj x)
 :precision binary64
 (let* ((t_0 (* (exp wj) wj)))
   (if (<= (- wj (/ (- t_0 x) (+ t_0 (exp wj)))) 1e-17)
     (fma
      (fma
       (fma 2.5 x (- 1.0 (* (fma 0.6666666666666666 x (fma 2.0 x 1.0)) wj)))
       wj
       (* -2.0 x))
      wj
      x)
     (- wj (* (- (/ (/ wj (- wj -1.0)) x) (/ (exp (- wj)) (- wj -1.0))) x)))))

C

double code(double wj, double x) {
	double t_0 = exp(wj) * wj;
	double tmp;
	if ((wj - ((t_0 - x) / (t_0 + exp(wj)))) <= 1e-17) {
		tmp = fma(fma(fma(2.5, x, (1.0 - (fma(0.6666666666666666, x, fma(2.0, x, 1.0)) * wj))), wj, (-2.0 * x)), wj, x);
	} else {
		tmp = wj - ((((wj / (wj - -1.0)) / x) - (exp(-wj) / (wj - -1.0))) * x);
	}
	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(t_0 + exp(wj)))) <= 1e-17)
		tmp = fma(fma(fma(2.5, x, Float64(1.0 - Float64(fma(0.6666666666666666, x, fma(2.0, x, 1.0)) * wj))), wj, Float64(-2.0 * x)), wj, x);
	else
		tmp = Float64(wj - Float64(Float64(Float64(Float64(wj / Float64(wj - -1.0)) / x) - Float64(exp(Float64(-wj)) / Float64(wj - -1.0))) * x));
	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[(t$95$0 + N[Exp[wj], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 1e-17], N[(N[(N[(2.5 * x + N[(1.0 - N[(N[(0.6666666666666666 * x + N[(2.0 * x + 1.0), $MachinePrecision]), $MachinePrecision] * wj), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * wj + N[(-2.0 * x), $MachinePrecision]), $MachinePrecision] * wj + x), $MachinePrecision], N[(wj - N[(N[(N[(N[(wj / N[(wj - -1.0), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision] - N[(N[Exp[(-wj)], $MachinePrecision] / N[(wj - -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * x), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 wj (/.f64 (-.f64 (*.f64 wj (exp.f64 wj)) x) (+.f64 (exp.f64 wj) (*.f64 wj (exp.f64 wj))))) < 1.00000000000000007e-17

   1. Initial program 67.2%

      \[wj - \frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}} \]
   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 99.4%

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

   if 1.00000000000000007e-17 < (-.f64 wj (/.f64 (-.f64 (*.f64 wj (exp.f64 wj)) x) (+.f64 (exp.f64 wj) (*.f64 wj (exp.f64 wj)))))

   1. Initial program 93.3%

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

   3. Taylor expanded in x around inf

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

      1. sub-neg N/A

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

      2. +-commutative N/A

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

      3. neg-sub0 N/A

         \[\leadsto wj - x \cdot \left(\color{blue}{\left(0 - \frac{1}{e^{wj} + wj \cdot e^{wj}}\right)} + \frac{wj \cdot e^{wj}}{x \cdot \left(e^{wj} + wj \cdot e^{wj}\right)}\right) \]

      4. associate-+l- N/A

         \[\leadsto wj - x \cdot \color{blue}{\left(0 - \left(\frac{1}{e^{wj} + wj \cdot e^{wj}} - \frac{wj \cdot e^{wj}}{x \cdot \left(e^{wj} + wj \cdot e^{wj}\right)}\right)\right)} \]

      5. unsub-neg N/A

         \[\leadsto wj - x \cdot \left(0 - \color{blue}{\left(\frac{1}{e^{wj} + wj \cdot e^{wj}} + \left(\mathsf{neg}\left(\frac{wj \cdot e^{wj}}{x \cdot \left(e^{wj} + wj \cdot e^{wj}\right)}\right)\right)\right)}\right) \]

      6. mul-1-neg N/A

         \[\leadsto wj - x \cdot \left(0 - \left(\frac{1}{e^{wj} + wj \cdot e^{wj}} + \color{blue}{-1 \cdot \frac{wj \cdot e^{wj}}{x \cdot \left(e^{wj} + wj \cdot e^{wj}\right)}}\right)\right) \]

      7. +-commutative N/A

         \[\leadsto wj - x \cdot \left(0 - \color{blue}{\left(-1 \cdot \frac{wj \cdot e^{wj}}{x \cdot \left(e^{wj} + wj \cdot e^{wj}\right)} + \frac{1}{e^{wj} + wj \cdot e^{wj}}\right)}\right) \]

   5. Applied rewrites 99.8%

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

4. Final simplification 99.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;wj - \frac{e^{wj} \cdot wj - x}{e^{wj} \cdot wj + e^{wj}} \leq 10^{-17}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(2.5, x, 1 - \mathsf{fma}\left(0.6666666666666666, x, \mathsf{fma}\left(2, x, 1\right)\right) \cdot wj\right), wj, -2 \cdot x\right), wj, x\right)\\ \mathbf{else}:\\ \;\;\;\;wj - \left(\frac{\frac{wj}{wj - -1}}{x} - \frac{e^{-wj}}{wj - -1}\right) \cdot x\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 84.3% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{wj} \cdot wj\\ t_1 := wj - \frac{t\_0 - x}{t\_0 + e^{wj}}\\ \mathbf{if}\;t\_1 \leq -2 \cdot 10^{-270}:\\ \;\;\;\;\mathsf{fma}\left(-2, wj, 1\right) \cdot x\\ \mathbf{elif}\;t\_1 \leq 0:\\ \;\;\;\;wj \cdot wj\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x \cdot wj, -2, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (wj x)
 :precision binary64
 (let* ((t_0 (* (exp wj) wj)) (t_1 (- wj (/ (- t_0 x) (+ t_0 (exp wj))))))
   (if (<= t_1 -2e-270)
     (* (fma -2.0 wj 1.0) x)
     (if (<= t_1 0.0) (* wj wj) (fma (* x wj) -2.0 x)))))

C

double code(double wj, double x) {
	double t_0 = exp(wj) * wj;
	double t_1 = wj - ((t_0 - x) / (t_0 + exp(wj)));
	double tmp;
	if (t_1 <= -2e-270) {
		tmp = fma(-2.0, wj, 1.0) * x;
	} else if (t_1 <= 0.0) {
		tmp = wj * wj;
	} else {
		tmp = fma((x * wj), -2.0, x);
	}
	return tmp;
}

Julia

function code(wj, x)
	t_0 = Float64(exp(wj) * wj)
	t_1 = Float64(wj - Float64(Float64(t_0 - x) / Float64(t_0 + exp(wj))))
	tmp = 0.0
	if (t_1 <= -2e-270)
		tmp = Float64(fma(-2.0, wj, 1.0) * x);
	elseif (t_1 <= 0.0)
		tmp = Float64(wj * wj);
	else
		tmp = fma(Float64(x * wj), -2.0, x);
	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[(t$95$0 + N[Exp[wj], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -2e-270], N[(N[(-2.0 * wj + 1.0), $MachinePrecision] * x), $MachinePrecision], If[LessEqual[t$95$1, 0.0], N[(wj * wj), $MachinePrecision], N[(N[(x * wj), $MachinePrecision] * -2.0 + x), $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))))) < -2.0000000000000001e-270

   1. Initial program 98.8%

      \[wj - \frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}} \]
   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 98.8%

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

   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. lower-*.f64 N/A

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

      4. lower-fma.f64 98.8

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

   8. Applied rewrites 98.8%

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

   if -2.0000000000000001e-270 < (-.f64 wj (/.f64 (-.f64 (*.f64 wj (exp.f64 wj)) x) (+.f64 (exp.f64 wj) (*.f64 wj (exp.f64 wj))))) < 0.0

   1. Initial program 5.3%

      \[wj - \frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}} \]
   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 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 62.9%

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

   9. Taylor expanded in wj around 0

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

   11. Applied rewrites 62.9%

       \[\leadsto wj \cdot wj \]

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

   1. Initial program 92.1%

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

   3. Taylor expanded in wj around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-*.f64 87.9

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

   5. Applied rewrites 87.9%

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

4. Final simplification 85.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;wj - \frac{e^{wj} \cdot wj - x}{e^{wj} \cdot wj + e^{wj}} \leq -2 \cdot 10^{-270}:\\ \;\;\;\;\mathsf{fma}\left(-2, wj, 1\right) \cdot x\\ \mathbf{elif}\;wj - \frac{e^{wj} \cdot wj - x}{e^{wj} \cdot wj + e^{wj}} \leq 0:\\ \;\;\;\;wj \cdot wj\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x \cdot wj, -2, x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 84.3% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{wj} \cdot wj\\ t_1 := wj - \frac{t\_0 - x}{t\_0 + e^{wj}}\\ t_2 := \mathsf{fma}\left(-2, wj, 1\right) \cdot x\\ \mathbf{if}\;t\_1 \leq -2 \cdot 10^{-270}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_1 \leq 0:\\ \;\;\;\;wj \cdot wj\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (wj x)
 :precision binary64
 (let* ((t_0 (* (exp wj) wj))
        (t_1 (- wj (/ (- t_0 x) (+ t_0 (exp wj)))))
        (t_2 (* (fma -2.0 wj 1.0) x)))
   (if (<= t_1 -2e-270) t_2 (if (<= t_1 0.0) (* wj wj) t_2))))

C

double code(double wj, double x) {
	double t_0 = exp(wj) * wj;
	double t_1 = wj - ((t_0 - x) / (t_0 + exp(wj)));
	double t_2 = fma(-2.0, wj, 1.0) * x;
	double tmp;
	if (t_1 <= -2e-270) {
		tmp = t_2;
	} else if (t_1 <= 0.0) {
		tmp = wj * wj;
	} else {
		tmp = t_2;
	}
	return tmp;
}

Julia

function code(wj, x)
	t_0 = Float64(exp(wj) * wj)
	t_1 = Float64(wj - Float64(Float64(t_0 - x) / Float64(t_0 + exp(wj))))
	t_2 = Float64(fma(-2.0, wj, 1.0) * x)
	tmp = 0.0
	if (t_1 <= -2e-270)
		tmp = t_2;
	elseif (t_1 <= 0.0)
		tmp = Float64(wj * wj);
	else
		tmp = t_2;
	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[(t$95$0 + N[Exp[wj], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(-2.0 * wj + 1.0), $MachinePrecision] * x), $MachinePrecision]}, If[LessEqual[t$95$1, -2e-270], t$95$2, If[LessEqual[t$95$1, 0.0], N[(wj * wj), $MachinePrecision], t$95$2]]]]]

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))))) < -2.0000000000000001e-270 or 0.0 < (-.f64 wj (/.f64 (-.f64 (*.f64 wj (exp.f64 wj)) x) (+.f64 (exp.f64 wj) (*.f64 wj (exp.f64 wj)))))

   1. Initial program 94.7%

      \[wj - \frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}} \]
   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 95.8%

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

   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. lower-*.f64 N/A

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

      4. lower-fma.f64 92.2

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

   8. Applied rewrites 92.2%

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

   if -2.0000000000000001e-270 < (-.f64 wj (/.f64 (-.f64 (*.f64 wj (exp.f64 wj)) x) (+.f64 (exp.f64 wj) (*.f64 wj (exp.f64 wj))))) < 0.0

   1. Initial program 5.3%

      \[wj - \frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}} \]
   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 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 62.9%

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

   9. Taylor expanded in wj around 0

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

   11. Applied rewrites 62.9%

       \[\leadsto wj \cdot wj \]
3. Recombined 2 regimes into one program.

4. Final simplification 85.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;wj - \frac{e^{wj} \cdot wj - x}{e^{wj} \cdot wj + e^{wj}} \leq -2 \cdot 10^{-270}:\\ \;\;\;\;\mathsf{fma}\left(-2, wj, 1\right) \cdot x\\ \mathbf{elif}\;wj - \frac{e^{wj} \cdot wj - x}{e^{wj} \cdot wj + e^{wj}} \leq 0:\\ \;\;\;\;wj \cdot wj\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-2, wj, 1\right) \cdot x\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 81.0% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{wj} \cdot wj\\ t_1 := wj - \frac{t\_0 - x}{t\_0 + e^{wj}}\\ t_2 := wj - \left(-x\right)\\ \mathbf{if}\;t\_1 \leq -2 \cdot 10^{-270}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_1 \leq 0:\\ \;\;\;\;wj \cdot wj\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (wj x)
 :precision binary64
 (let* ((t_0 (* (exp wj) wj))
        (t_1 (- wj (/ (- t_0 x) (+ t_0 (exp wj)))))
        (t_2 (- wj (- x))))
   (if (<= t_1 -2e-270) t_2 (if (<= t_1 0.0) (* wj wj) t_2))))

C

double code(double wj, double x) {
	double t_0 = exp(wj) * wj;
	double t_1 = wj - ((t_0 - x) / (t_0 + exp(wj)));
	double t_2 = wj - -x;
	double tmp;
	if (t_1 <= -2e-270) {
		tmp = t_2;
	} else if (t_1 <= 0.0) {
		tmp = wj * wj;
	} else {
		tmp = t_2;
	}
	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) :: t_2
    real(8) :: tmp
    t_0 = exp(wj) * wj
    t_1 = wj - ((t_0 - x) / (t_0 + exp(wj)))
    t_2 = wj - -x
    if (t_1 <= (-2d-270)) then
        tmp = t_2
    else if (t_1 <= 0.0d0) then
        tmp = wj * wj
    else
        tmp = t_2
    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) / (t_0 + Math.exp(wj)));
	double t_2 = wj - -x;
	double tmp;
	if (t_1 <= -2e-270) {
		tmp = t_2;
	} else if (t_1 <= 0.0) {
		tmp = wj * wj;
	} else {
		tmp = t_2;
	}
	return tmp;
}

Python

def code(wj, x):
	t_0 = math.exp(wj) * wj
	t_1 = wj - ((t_0 - x) / (t_0 + math.exp(wj)))
	t_2 = wj - -x
	tmp = 0
	if t_1 <= -2e-270:
		tmp = t_2
	elif t_1 <= 0.0:
		tmp = wj * wj
	else:
		tmp = t_2
	return tmp

Julia

function code(wj, x)
	t_0 = Float64(exp(wj) * wj)
	t_1 = Float64(wj - Float64(Float64(t_0 - x) / Float64(t_0 + exp(wj))))
	t_2 = Float64(wj - Float64(-x))
	tmp = 0.0
	if (t_1 <= -2e-270)
		tmp = t_2;
	elseif (t_1 <= 0.0)
		tmp = Float64(wj * wj);
	else
		tmp = t_2;
	end
	return tmp
end

MATLAB

function tmp_2 = code(wj, x)
	t_0 = exp(wj) * wj;
	t_1 = wj - ((t_0 - x) / (t_0 + exp(wj)));
	t_2 = wj - -x;
	tmp = 0.0;
	if (t_1 <= -2e-270)
		tmp = t_2;
	elseif (t_1 <= 0.0)
		tmp = wj * wj;
	else
		tmp = t_2;
	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[(t$95$0 + N[Exp[wj], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(wj - (-x)), $MachinePrecision]}, If[LessEqual[t$95$1, -2e-270], t$95$2, If[LessEqual[t$95$1, 0.0], N[(wj * wj), $MachinePrecision], t$95$2]]]]]

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))))) < -2.0000000000000001e-270 or 0.0 < (-.f64 wj (/.f64 (-.f64 (*.f64 wj (exp.f64 wj)) x) (+.f64 (exp.f64 wj) (*.f64 wj (exp.f64 wj)))))

   1. Initial program 94.7%

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

   3. Taylor expanded in wj around 0

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

      1. mul-1-neg N/A

         \[\leadsto wj - \color{blue}{\left(\mathsf{neg}\left(x\right)\right)} \]

      2. lower-neg.f64 88.1

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

   5. Applied rewrites 88.1%

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

   if -2.0000000000000001e-270 < (-.f64 wj (/.f64 (-.f64 (*.f64 wj (exp.f64 wj)) x) (+.f64 (exp.f64 wj) (*.f64 wj (exp.f64 wj))))) < 0.0

   1. Initial program 5.3%

      \[wj - \frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}} \]
   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 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 62.9%

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

   9. Taylor expanded in wj around 0

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

   11. Applied rewrites 62.9%

       \[\leadsto wj \cdot wj \]
3. Recombined 2 regimes into one program.

4. Final simplification 82.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;wj - \frac{e^{wj} \cdot wj - x}{e^{wj} \cdot wj + e^{wj}} \leq -2 \cdot 10^{-270}:\\ \;\;\;\;wj - \left(-x\right)\\ \mathbf{elif}\;wj - \frac{e^{wj} \cdot wj - x}{e^{wj} \cdot wj + e^{wj}} \leq 0:\\ \;\;\;\;wj \cdot wj\\ \mathbf{else}:\\ \;\;\;\;wj - \left(-x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 96.7% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{wj} \cdot wj\\ \mathbf{if}\;wj - \frac{t\_0 - x}{t\_0 + e^{wj}} \leq 10^{-17}:\\ \;\;\;\;\mathsf{fma}\left(\left(1 - wj\right) \cdot wj, wj, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{-1}{wj - -1}, \mathsf{fma}\left(\mathsf{fma}\left(-0.5 \cdot x, wj, 1 + x\right), wj, -x\right), wj\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (wj x)
 :precision binary64
 (let* ((t_0 (* (exp wj) wj)))
   (if (<= (- wj (/ (- t_0 x) (+ t_0 (exp wj)))) 1e-17)
     (fma (* (- 1.0 wj) wj) wj x)
     (fma
      (/ -1.0 (- wj -1.0))
      (fma (fma (* -0.5 x) wj (+ 1.0 x)) wj (- x))
      wj))))

C

double code(double wj, double x) {
	double t_0 = exp(wj) * wj;
	double tmp;
	if ((wj - ((t_0 - x) / (t_0 + exp(wj)))) <= 1e-17) {
		tmp = fma(((1.0 - wj) * wj), wj, x);
	} else {
		tmp = fma((-1.0 / (wj - -1.0)), fma(fma((-0.5 * x), wj, (1.0 + x)), wj, -x), 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(t_0 + exp(wj)))) <= 1e-17)
		tmp = fma(Float64(Float64(1.0 - wj) * wj), wj, x);
	else
		tmp = fma(Float64(-1.0 / Float64(wj - -1.0)), fma(fma(Float64(-0.5 * x), wj, Float64(1.0 + x)), wj, Float64(-x)), 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[(t$95$0 + N[Exp[wj], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 1e-17], N[(N[(N[(1.0 - wj), $MachinePrecision] * wj), $MachinePrecision] * wj + x), $MachinePrecision], N[(N[(-1.0 / N[(wj - -1.0), $MachinePrecision]), $MachinePrecision] * N[(N[(N[(-0.5 * x), $MachinePrecision] * wj + N[(1.0 + x), $MachinePrecision]), $MachinePrecision] * wj + (-x)), $MachinePrecision] + wj), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 wj (/.f64 (-.f64 (*.f64 wj (exp.f64 wj)) x) (+.f64 (exp.f64 wj) (*.f64 wj (exp.f64 wj))))) < 1.00000000000000007e-17

   1. Initial program 67.2%

      \[wj - \frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}} \]
   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 99.4%

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

   6. Taylor expanded in wj around 0

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

   8. Applied rewrites 98.6%

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

   9. Taylor expanded in x around 0

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

   11. Applied rewrites 99.4%

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

   if 1.00000000000000007e-17 < (-.f64 wj (/.f64 (-.f64 (*.f64 wj (exp.f64 wj)) x) (+.f64 (exp.f64 wj) (*.f64 wj (exp.f64 wj)))))

   1. Initial program 93.3%

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

      1. lift--.f64 N/A

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

      2. sub-neg N/A

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

      3. +-commutative N/A

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

      4. lift-/.f64 N/A

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

      5. distribute-neg-frac N/A

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

      6. neg-mul-1 N/A

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

      7. lift-+.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. distribute-rgt1-in N/A

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

      10. times-frac N/A

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

      11. lower-fma.f64 N/A

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

   4. Applied rewrites 94.7%

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

   5. Taylor expanded in wj around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. +-commutative N/A

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

      5. associate--l+ N/A

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

      6. distribute-rgt-out-- N/A

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

      7. metadata-eval N/A

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

      8. *-commutative N/A

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

      9. *-commutative N/A

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

      10. lower-fma.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. cancel-sign-sub-inv N/A

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

      13. metadata-eval N/A

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

      14. *-lft-identity N/A

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

      15. +-commutative N/A

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

      16. lower-+.f64 N/A

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

      17. mul-1-neg N/A

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

      18. lower-neg.f64 94.4

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

   7. Applied rewrites 94.4%

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

4. Final simplification 97.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;wj - \frac{e^{wj} \cdot wj - x}{e^{wj} \cdot wj + e^{wj}} \leq 10^{-17}:\\ \;\;\;\;\mathsf{fma}\left(\left(1 - wj\right) \cdot wj, wj, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{-1}{wj - -1}, \mathsf{fma}\left(\mathsf{fma}\left(-0.5 \cdot x, wj, 1 + x\right), wj, -x\right), wj\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 97.4% accurate, 6.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;wj \leq 0.00096:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(2.5, x, 1 - \mathsf{fma}\left(0.6666666666666666, x, \mathsf{fma}\left(2, x, 1\right)\right) \cdot wj\right), wj, -2 \cdot x\right), wj, x\right)\\ \mathbf{else}:\\ \;\;\;\;wj - \frac{1}{wj - -1} \cdot wj\\ \end{array} \end{array} \]

FPCore

(FPCore (wj x)
 :precision binary64
 (if (<= wj 0.00096)
   (fma
    (fma
     (fma 2.5 x (- 1.0 (* (fma 0.6666666666666666 x (fma 2.0 x 1.0)) wj)))
     wj
     (* -2.0 x))
    wj
    x)
   (- wj (* (/ 1.0 (- wj -1.0)) wj))))

C

double code(double wj, double x) {
	double tmp;
	if (wj <= 0.00096) {
		tmp = fma(fma(fma(2.5, x, (1.0 - (fma(0.6666666666666666, x, fma(2.0, x, 1.0)) * wj))), wj, (-2.0 * x)), wj, x);
	} else {
		tmp = wj - ((1.0 / (wj - -1.0)) * wj);
	}
	return tmp;
}

Julia

function code(wj, x)
	tmp = 0.0
	if (wj <= 0.00096)
		tmp = fma(fma(fma(2.5, x, Float64(1.0 - Float64(fma(0.6666666666666666, x, fma(2.0, x, 1.0)) * wj))), wj, Float64(-2.0 * x)), wj, x);
	else
		tmp = Float64(wj - Float64(Float64(1.0 / Float64(wj - -1.0)) * wj));
	end
	return tmp
end

Wolfram

code[wj_, x_] := If[LessEqual[wj, 0.00096], N[(N[(N[(2.5 * x + N[(1.0 - N[(N[(0.6666666666666666 * x + N[(2.0 * x + 1.0), $MachinePrecision]), $MachinePrecision] * wj), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * wj + N[(-2.0 * x), $MachinePrecision]), $MachinePrecision] * wj + x), $MachinePrecision], N[(wj - N[(N[(1.0 / N[(wj - -1.0), $MachinePrecision]), $MachinePrecision] * wj), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if wj < 9.60000000000000024e-4

   1. Initial program 76.2%

      \[wj - \frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}} \]
   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 98.8%

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

   if 9.60000000000000024e-4 < wj

   1. Initial program 30.8%

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

   3. Taylor expanded in wj around inf

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

   5. Applied rewrites 53.9%

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

   6. Taylor expanded in x around 0

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

      1. associate-/l* N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. *-lft-identity N/A

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

      5. distribute-rgt-in N/A

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

      6. associate-/r* N/A

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

      7. *-inverses N/A

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

      8. lower-/.f64 N/A

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

      9. +-commutative N/A

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

      10. lower-+.f64 82.6

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

   8. Applied rewrites 82.6%

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

4. Final simplification 98.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;wj \leq 0.00096:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(2.5, x, 1 - \mathsf{fma}\left(0.6666666666666666, x, \mathsf{fma}\left(2, x, 1\right)\right) \cdot wj\right), wj, -2 \cdot x\right), wj, x\right)\\ \mathbf{else}:\\ \;\;\;\;wj - \frac{1}{wj - -1} \cdot wj\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 96.8% accurate, 11.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;wj \leq 0.00048:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(2.5, x, 1\right), wj, -2 \cdot x\right), wj, x\right)\\ \mathbf{else}:\\ \;\;\;\;wj - \frac{1}{wj - -1} \cdot wj\\ \end{array} \end{array} \]

FPCore

(FPCore (wj x)
 :precision binary64
 (if (<= wj 0.00048)
   (fma (fma (fma 2.5 x 1.0) wj (* -2.0 x)) wj x)
   (- wj (* (/ 1.0 (- wj -1.0)) wj))))

C

double code(double wj, double x) {
	double tmp;
	if (wj <= 0.00048) {
		tmp = fma(fma(fma(2.5, x, 1.0), wj, (-2.0 * x)), wj, x);
	} else {
		tmp = wj - ((1.0 / (wj - -1.0)) * wj);
	}
	return tmp;
}

Julia

function code(wj, x)
	tmp = 0.0
	if (wj <= 0.00048)
		tmp = fma(fma(fma(2.5, x, 1.0), wj, Float64(-2.0 * x)), wj, x);
	else
		tmp = Float64(wj - Float64(Float64(1.0 / Float64(wj - -1.0)) * wj));
	end
	return tmp
end

Wolfram

code[wj_, x_] := If[LessEqual[wj, 0.00048], N[(N[(N[(2.5 * x + 1.0), $MachinePrecision] * wj + N[(-2.0 * x), $MachinePrecision]), $MachinePrecision] * wj + x), $MachinePrecision], N[(wj - N[(N[(1.0 / N[(wj - -1.0), $MachinePrecision]), $MachinePrecision] * wj), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if wj < 4.80000000000000012e-4

   1. Initial program 76.2%

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

   3. Taylor expanded in wj around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. cancel-sign-sub-inv N/A

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

      5. *-commutative N/A

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

      6. metadata-eval N/A

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

      7. lower-fma.f64 N/A

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

      8. sub-neg N/A

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

      9. +-commutative N/A

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

      10. distribute-rgt-out N/A

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

      11. *-commutative N/A

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

      12. distribute-lft-neg-in N/A

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

      13. lower-fma.f64 N/A

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

      14. metadata-eval N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{neg}\left(\color{blue}{\frac{-5}{2}}\right), x, 1\right), wj, -2 \cdot x\right), wj, x\right) \]

      15. metadata-eval N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\color{blue}{\frac{5}{2}}, x, 1\right), wj, -2 \cdot x\right), wj, x\right) \]

      16. lower-*.f64 98.1

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

   5. Applied rewrites 98.1%

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

   if 4.80000000000000012e-4 < wj

   1. Initial program 30.8%

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

   3. Taylor expanded in wj around inf

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

   5. Applied rewrites 53.9%

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

   6. Taylor expanded in x around 0

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

      1. associate-/l* N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. *-lft-identity N/A

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

      5. distribute-rgt-in N/A

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

      6. associate-/r* N/A

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

      7. *-inverses N/A

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

      8. lower-/.f64 N/A

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

      9. +-commutative N/A

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

      10. lower-+.f64 82.6

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

   8. Applied rewrites 82.6%

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

4. Final simplification 97.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;wj \leq 0.00048:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(2.5, x, 1\right), wj, -2 \cdot x\right), wj, x\right)\\ \mathbf{else}:\\ \;\;\;\;wj - \frac{1}{wj - -1} \cdot wj\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 96.6% accurate, 11.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;wj \leq 0.00041:\\ \;\;\;\;\mathsf{fma}\left(\left(1 - wj\right) \cdot wj, wj, x\right)\\ \mathbf{else}:\\ \;\;\;\;wj - \frac{1}{wj - -1} \cdot wj\\ \end{array} \end{array} \]

FPCore

(FPCore (wj x)
 :precision binary64
 (if (<= wj 0.00041)
   (fma (* (- 1.0 wj) wj) wj x)
   (- wj (* (/ 1.0 (- wj -1.0)) wj))))

C

double code(double wj, double x) {
	double tmp;
	if (wj <= 0.00041) {
		tmp = fma(((1.0 - wj) * wj), wj, x);
	} else {
		tmp = wj - ((1.0 / (wj - -1.0)) * wj);
	}
	return tmp;
}

Julia

function code(wj, x)
	tmp = 0.0
	if (wj <= 0.00041)
		tmp = fma(Float64(Float64(1.0 - wj) * wj), wj, x);
	else
		tmp = Float64(wj - Float64(Float64(1.0 / Float64(wj - -1.0)) * wj));
	end
	return tmp
end

Wolfram

code[wj_, x_] := If[LessEqual[wj, 0.00041], N[(N[(N[(1.0 - wj), $MachinePrecision] * wj), $MachinePrecision] * wj + x), $MachinePrecision], N[(wj - N[(N[(1.0 / N[(wj - -1.0), $MachinePrecision]), $MachinePrecision] * wj), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if wj < 4.0999999999999999e-4

   1. Initial program 76.2%

      \[wj - \frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}} \]
   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 98.8%

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

   6. Taylor expanded in wj around 0

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

   8. Applied rewrites 98.1%

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

   9. Taylor expanded in x around 0

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

   11. Applied rewrites 97.9%

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

   if 4.0999999999999999e-4 < wj

   1. Initial program 30.8%

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

   3. Taylor expanded in wj around inf

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

   5. Applied rewrites 53.9%

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

   6. Taylor expanded in x around 0

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

      1. associate-/l* N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. *-lft-identity N/A

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

      5. distribute-rgt-in N/A

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

      6. associate-/r* N/A

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

      7. *-inverses N/A

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

      8. lower-/.f64 N/A

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

      9. +-commutative N/A

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

      10. lower-+.f64 82.6

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

   8. Applied rewrites 82.6%

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

4. Final simplification 97.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;wj \leq 0.00041:\\ \;\;\;\;\mathsf{fma}\left(\left(1 - wj\right) \cdot wj, wj, x\right)\\ \mathbf{else}:\\ \;\;\;\;wj - \frac{1}{wj - -1} \cdot wj\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 96.6% accurate, 13.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;wj \leq 0.00041:\\ \;\;\;\;\mathsf{fma}\left(\left(1 - wj\right) \cdot wj, wj, x\right)\\ \mathbf{else}:\\ \;\;\;\;wj - \frac{wj}{wj - -1}\\ \end{array} \end{array} \]

FPCore

(FPCore (wj x)
 :precision binary64
 (if (<= wj 0.00041) (fma (* (- 1.0 wj) wj) wj x) (- wj (/ wj (- wj -1.0)))))

C

double code(double wj, double x) {
	double tmp;
	if (wj <= 0.00041) {
		tmp = fma(((1.0 - wj) * wj), wj, x);
	} else {
		tmp = wj - (wj / (wj - -1.0));
	}
	return tmp;
}

Julia

function code(wj, x)
	tmp = 0.0
	if (wj <= 0.00041)
		tmp = fma(Float64(Float64(1.0 - wj) * wj), wj, x);
	else
		tmp = Float64(wj - Float64(wj / Float64(wj - -1.0)));
	end
	return tmp
end

Wolfram

code[wj_, x_] := If[LessEqual[wj, 0.00041], N[(N[(N[(1.0 - wj), $MachinePrecision] * wj), $MachinePrecision] * wj + x), $MachinePrecision], N[(wj - N[(wj / N[(wj - -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if wj < 4.0999999999999999e-4

   1. Initial program 76.2%

      \[wj - \frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}} \]
   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 98.8%

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

   6. Taylor expanded in wj around 0

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

   8. Applied rewrites 98.1%

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

   9. Taylor expanded in x around 0

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

   11. Applied rewrites 97.9%

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

   if 4.0999999999999999e-4 < wj

   1. Initial program 30.8%

      \[wj - \frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}} \]
   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 82.1

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

   5. Applied rewrites 82.1%

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

4. Final simplification 97.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;wj \leq 0.00041:\\ \;\;\;\;\mathsf{fma}\left(\left(1 - wj\right) \cdot wj, wj, x\right)\\ \mathbf{else}:\\ \;\;\;\;wj - \frac{wj}{wj - -1}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 95.2% accurate, 22.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 75.2%

   \[wj - \frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}} \]
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 96.8%

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

6. Taylor expanded in wj around 0

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

8. Applied rewrites 96.0%

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

9. Taylor expanded in x around 0

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

11. Applied rewrites 95.9%

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

Alternative 11: 14.6% accurate, 27.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.95 \cdot 10^{-54}:\\ \;\;\;\;wj - 1\\ \mathbf{else}:\\ \;\;\;\;wj \cdot wj\\ \end{array} \end{array} \]

FPCore

(FPCore (wj x) :precision binary64 (if (<= x -1.95e-54) (- wj 1.0) (* wj wj)))

C

double code(double wj, double x) {
	double tmp;
	if (x <= -1.95e-54) {
		tmp = wj - 1.0;
	} else {
		tmp = wj * wj;
	}
	return tmp;
}

Fortran

real(8) function code(wj, x)
    real(8), intent (in) :: wj
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= (-1.95d-54)) then
        tmp = wj - 1.0d0
    else
        tmp = wj * wj
    end if
    code = tmp
end function

Java

public static double code(double wj, double x) {
	double tmp;
	if (x <= -1.95e-54) {
		tmp = wj - 1.0;
	} else {
		tmp = wj * wj;
	}
	return tmp;
}

Python

def code(wj, x):
	tmp = 0
	if x <= -1.95e-54:
		tmp = wj - 1.0
	else:
		tmp = wj * wj
	return tmp

Julia

function code(wj, x)
	tmp = 0.0
	if (x <= -1.95e-54)
		tmp = Float64(wj - 1.0);
	else
		tmp = Float64(wj * wj);
	end
	return tmp
end

MATLAB

function tmp_2 = code(wj, x)
	tmp = 0.0;
	if (x <= -1.95e-54)
		tmp = wj - 1.0;
	else
		tmp = wj * wj;
	end
	tmp_2 = tmp;
end

Wolfram

code[wj_, x_] := If[LessEqual[x, -1.95e-54], N[(wj - 1.0), $MachinePrecision], N[(wj * wj), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -1.95e-54

   1. Initial program 94.0%

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

   3. Taylor expanded in wj around inf

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

   5. Applied rewrites 9.4%

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

   if -1.95e-54 < x

   1. Initial program 69.2%

      \[wj - \frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}} \]
   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 97.2%

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

   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 24.2%

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

   9. Taylor expanded in wj around 0

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

   11. Applied rewrites 23.4%

       \[\leadsto wj \cdot wj \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 12: 4.3% 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 75.2%

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

3. Taylor expanded in wj around inf

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

5. Applied rewrites 4.2%

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

Developer Target 1: 79.4% 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 2024277 
(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))))))