Jmat.Real.lambertw, newton loop step

Percentage Accurate: 77.1% → 99.2%

Time: 8.5s

Alternatives: 12

Speedup: 18.4×

• could not determine a ground truth

  1. wj = 3.1363537672157066e+24
  2. x = 3.6518876427161294e-159

Specification

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := wj \cdot e^{wj}\\ wj - \frac{t\_0 - x}{e^{wj} + t\_0} \end{array} \end{array} \]

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 77.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := wj \cdot e^{wj}\\ wj - \frac{t\_0 - x}{e^{wj} + t\_0} \end{array} \end{array} \]

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.2% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;wj \leq -4.1 \cdot 10^{-6}:\\ \;\;\;\;\mathsf{fma}\left(\frac{\mathsf{fma}\left(e^{wj}, wj, -x\right)}{-1 - wj}, e^{-wj}, wj\right)\\ \mathbf{elif}\;wj \leq 0.0048:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(2.5, x, 1 - \mathsf{fma}\left(0.6666666666666666, x, \mathsf{fma}\left(2, x, 1\right)\right) \cdot wj\right), wj, -2 \cdot x\right), wj, x\right)\\ \mathbf{else}:\\ \;\;\;\;wj - \frac{wj}{1 + wj}\\ \end{array} \end{array} \]

FPCore

(FPCore (wj x)
 :precision binary64
 (if (<= wj -4.1e-6)
   (fma (/ (fma (exp wj) wj (- x)) (- -1.0 wj)) (exp (- wj)) wj)
   (if (<= wj 0.0048)
     (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 (+ 1.0 wj))))))

C

double code(double wj, double x) {
	double tmp;
	if (wj <= -4.1e-6) {
		tmp = fma((fma(exp(wj), wj, -x) / (-1.0 - wj)), exp(-wj), wj);
	} else if (wj <= 0.0048) {
		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 / (1.0 + wj));
	}
	return tmp;
}

Julia

function code(wj, x)
	tmp = 0.0
	if (wj <= -4.1e-6)
		tmp = fma(Float64(fma(exp(wj), wj, Float64(-x)) / Float64(-1.0 - wj)), exp(Float64(-wj)), wj);
	elseif (wj <= 0.0048)
		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(wj / Float64(1.0 + wj)));
	end
	return tmp
end

Wolfram

code[wj_, x_] := If[LessEqual[wj, -4.1e-6], N[(N[(N[(N[Exp[wj], $MachinePrecision] * wj + (-x)), $MachinePrecision] / N[(-1.0 - wj), $MachinePrecision]), $MachinePrecision] * N[Exp[(-wj)], $MachinePrecision] + wj), $MachinePrecision], If[LessEqual[wj, 0.0048], 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[(wj / N[(1.0 + wj), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if wj < -4.0999999999999997e-6

   1. Initial program 40.8%

      \[wj - \frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}} \]
   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. *-commutative N/A

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

      11. times-frac N/A

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

      12. lower-fma.f64 N/A

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

   4. Applied rewrites 86.6%

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

      1. lift-fma.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*l/ N/A

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

      4. div-inv N/A

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

      5. lower-fma.f64 N/A

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

   6. Applied rewrites 86.6%

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

   if -4.0999999999999997e-6 < wj < 0.00479999999999999958

   1. Initial program 78.7%

      \[wj - \frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}} \]
   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 99.6%

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

   if 0.00479999999999999958 < wj

   1. Initial program 38.8%

      \[wj - \frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}} \]
   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 79.8

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

   5. Applied rewrites 79.8%

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

4. Final simplification 98.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;wj \leq -4.1 \cdot 10^{-6}:\\ \;\;\;\;\mathsf{fma}\left(\frac{\mathsf{fma}\left(e^{wj}, wj, -x\right)}{-1 - wj}, e^{-wj}, wj\right)\\ \mathbf{elif}\;wj \leq 0.0048:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(2.5, x, 1 - \mathsf{fma}\left(0.6666666666666666, x, \mathsf{fma}\left(2, x, 1\right)\right) \cdot wj\right), wj, -2 \cdot x\right), wj, x\right)\\ \mathbf{else}:\\ \;\;\;\;wj - \frac{wj}{1 + wj}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 81.0% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := wj \cdot e^{wj}\\ t_1 := wj - \frac{t\_0 - x}{e^{wj} + t\_0}\\ \mathbf{if}\;t\_1 \leq -2 \cdot 10^{-267} \lor \neg \left(t\_1 \leq 0\right):\\ \;\;\;\;wj - \left(-x\right)\\ \mathbf{else}:\\ \;\;\;\;wj \cdot wj\\ \end{array} \end{array} \]

FPCore

(FPCore (wj x)
 :precision binary64
 (let* ((t_0 (* wj (exp wj))) (t_1 (- wj (/ (- t_0 x) (+ (exp wj) t_0)))))
   (if (or (<= t_1 -2e-267) (not (<= t_1 0.0))) (- wj (- x)) (* wj wj))))

C

double code(double wj, double x) {
	double t_0 = wj * exp(wj);
	double t_1 = wj - ((t_0 - x) / (exp(wj) + t_0));
	double tmp;
	if ((t_1 <= -2e-267) || !(t_1 <= 0.0)) {
		tmp = wj - -x;
	} 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) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = wj * exp(wj)
    t_1 = wj - ((t_0 - x) / (exp(wj) + t_0))
    if ((t_1 <= (-2d-267)) .or. (.not. (t_1 <= 0.0d0))) then
        tmp = wj - -x
    else
        tmp = wj * wj
    end if
    code = tmp
end function

Java

public static double code(double wj, double x) {
	double t_0 = wj * Math.exp(wj);
	double t_1 = wj - ((t_0 - x) / (Math.exp(wj) + t_0));
	double tmp;
	if ((t_1 <= -2e-267) || !(t_1 <= 0.0)) {
		tmp = wj - -x;
	} else {
		tmp = wj * wj;
	}
	return tmp;
}

Python

def code(wj, x):
	t_0 = wj * math.exp(wj)
	t_1 = wj - ((t_0 - x) / (math.exp(wj) + t_0))
	tmp = 0
	if (t_1 <= -2e-267) or not (t_1 <= 0.0):
		tmp = wj - -x
	else:
		tmp = wj * wj
	return tmp

Julia

function code(wj, x)
	t_0 = Float64(wj * exp(wj))
	t_1 = Float64(wj - Float64(Float64(t_0 - x) / Float64(exp(wj) + t_0)))
	tmp = 0.0
	if ((t_1 <= -2e-267) || !(t_1 <= 0.0))
		tmp = Float64(wj - Float64(-x));
	else
		tmp = Float64(wj * wj);
	end
	return tmp
end

MATLAB

function tmp_2 = code(wj, x)
	t_0 = wj * exp(wj);
	t_1 = wj - ((t_0 - x) / (exp(wj) + t_0));
	tmp = 0.0;
	if ((t_1 <= -2e-267) || ~((t_1 <= 0.0)))
		tmp = wj - -x;
	else
		tmp = wj * wj;
	end
	tmp_2 = tmp;
end

Wolfram

code[wj_, x_] := Block[{t$95$0 = N[(wj * N[Exp[wj], $MachinePrecision]), $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[Or[LessEqual[t$95$1, -2e-267], N[Not[LessEqual[t$95$1, 0.0]], $MachinePrecision]], N[(wj - (-x)), $MachinePrecision], N[(wj * 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))))) < -2e-267 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 92.3%

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

   3. Taylor expanded in wj around 0

      \[\leadsto wj - \color{blue}{-1 \cdot x} \]
   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 85.8

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

   5. Applied rewrites 85.8%

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

   if -2e-267 < (-.f64 wj (/.f64 (-.f64 (*.f64 wj (exp.f64 wj)) x) (+.f64 (exp.f64 wj) (*.f64 wj (exp.f64 wj))))) < 0.0

   1. Initial program 6.0%

      \[wj - \frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}} \]
   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 wj around 0

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

      1. +-commutative N/A

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

   8. Applied rewrites 100.0%

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

   9. Taylor expanded in x around 0

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

   11. Applied rewrites 47.9%

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

4. Final simplification 78.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;wj - \frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}} \leq -2 \cdot 10^{-267} \lor \neg \left(wj - \frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}} \leq 0\right):\\ \;\;\;\;wj - \left(-x\right)\\ \mathbf{else}:\\ \;\;\;\;wj \cdot wj\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 99.2% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;wj \leq -4.6 \cdot 10^{-6}:\\ \;\;\;\;wj - \frac{wj \cdot e^{wj} - x}{\left(1 + wj\right) \cdot e^{wj}}\\ \mathbf{elif}\;wj \leq 0.0048:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(2.5, x, 1 - \mathsf{fma}\left(0.6666666666666666, x, \mathsf{fma}\left(2, x, 1\right)\right) \cdot wj\right), wj, -2 \cdot x\right), wj, x\right)\\ \mathbf{else}:\\ \;\;\;\;wj - \frac{wj}{1 + wj}\\ \end{array} \end{array} \]

FPCore

(FPCore (wj x)
 :precision binary64
 (if (<= wj -4.6e-6)
   (- wj (/ (- (* wj (exp wj)) x) (* (+ 1.0 wj) (exp wj))))
   (if (<= wj 0.0048)
     (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 (+ 1.0 wj))))))

C

double code(double wj, double x) {
	double tmp;
	if (wj <= -4.6e-6) {
		tmp = wj - (((wj * exp(wj)) - x) / ((1.0 + wj) * exp(wj)));
	} else if (wj <= 0.0048) {
		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 / (1.0 + wj));
	}
	return tmp;
}

Julia

function code(wj, x)
	tmp = 0.0
	if (wj <= -4.6e-6)
		tmp = Float64(wj - Float64(Float64(Float64(wj * exp(wj)) - x) / Float64(Float64(1.0 + wj) * exp(wj))));
	elseif (wj <= 0.0048)
		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(wj / Float64(1.0 + wj)));
	end
	return tmp
end

Wolfram

code[wj_, x_] := If[LessEqual[wj, -4.6e-6], N[(wj - N[(N[(N[(wj * N[Exp[wj], $MachinePrecision]), $MachinePrecision] - x), $MachinePrecision] / N[(N[(1.0 + wj), $MachinePrecision] * N[Exp[wj], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[wj, 0.0048], 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[(wj / N[(1.0 + wj), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if wj < -4.6e-6

   1. Initial program 40.8%

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

      1. lift-+.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. distribute-rgt1-in N/A

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

      4. lower-*.f64 N/A

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

      5. +-commutative N/A

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

      6. lower-+.f64 86.5

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

   4. Applied rewrites 86.5%

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

   if -4.6e-6 < wj < 0.00479999999999999958

   1. Initial program 78.7%

      \[wj - \frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}} \]
   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 99.6%

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

   if 0.00479999999999999958 < wj

   1. Initial program 38.8%

      \[wj - \frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}} \]
   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 79.8

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

   5. Applied rewrites 79.8%

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

Alternative 4: 98.8% accurate, 2.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;wj \leq -0.00178:\\ \;\;\;\;\mathsf{fma}\left(\frac{x}{wj - -1}, e^{-wj}, wj\right)\\ \mathbf{elif}\;wj \leq 0.0048:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(2.5, x, 1 - \mathsf{fma}\left(0.6666666666666666, x, \mathsf{fma}\left(2, x, 1\right)\right) \cdot wj\right), wj, -2 \cdot x\right), wj, x\right)\\ \mathbf{else}:\\ \;\;\;\;wj - \frac{wj}{1 + wj}\\ \end{array} \end{array} \]

FPCore

(FPCore (wj x)
 :precision binary64
 (if (<= wj -0.00178)
   (fma (/ x (- wj -1.0)) (exp (- wj)) wj)
   (if (<= wj 0.0048)
     (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 (+ 1.0 wj))))))

C

double code(double wj, double x) {
	double tmp;
	if (wj <= -0.00178) {
		tmp = fma((x / (wj - -1.0)), exp(-wj), wj);
	} else if (wj <= 0.0048) {
		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 / (1.0 + wj));
	}
	return tmp;
}

Julia

function code(wj, x)
	tmp = 0.0
	if (wj <= -0.00178)
		tmp = fma(Float64(x / Float64(wj - -1.0)), exp(Float64(-wj)), wj);
	elseif (wj <= 0.0048)
		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(wj / Float64(1.0 + wj)));
	end
	return tmp
end

Wolfram

code[wj_, x_] := If[LessEqual[wj, -0.00178], N[(N[(x / N[(wj - -1.0), $MachinePrecision]), $MachinePrecision] * N[Exp[(-wj)], $MachinePrecision] + wj), $MachinePrecision], If[LessEqual[wj, 0.0048], 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[(wj / N[(1.0 + wj), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if wj < -0.0017799999999999999

   1. Initial program 28.4%

      \[wj - \frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}} \]
   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. *-commutative N/A

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

      11. times-frac N/A

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

      12. lower-fma.f64 N/A

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

   4. Applied rewrites 86.4%

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

      1. lift-fma.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*l/ N/A

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

      4. div-inv N/A

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

      5. lower-fma.f64 N/A

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

   6. Applied rewrites 86.4%

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

   7. Taylor expanded in x around inf

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

      1. lower-/.f64 N/A

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

      2. +-commutative N/A

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

      3. metadata-eval N/A

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

      4. sub-neg N/A

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

      5. lower--.f64 75.7

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

   9. Applied rewrites 75.7%

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

   if -0.0017799999999999999 < wj < 0.00479999999999999958

   1. Initial program 78.7%

      \[wj - \frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}} \]
   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 99.2%

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

   if 0.00479999999999999958 < wj

   1. Initial program 38.8%

      \[wj - \frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}} \]
   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 79.8

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

   5. Applied rewrites 79.8%

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

Alternative 5: 97.6% accurate, 6.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;wj \leq 0.0048:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(2.5, x, 1 - \mathsf{fma}\left(0.6666666666666666, x, \mathsf{fma}\left(2, x, 1\right)\right) \cdot wj\right), wj, -2 \cdot x\right), wj, x\right)\\ \mathbf{else}:\\ \;\;\;\;wj - \frac{wj}{1 + wj}\\ \end{array} \end{array} \]

FPCore

(FPCore (wj x)
 :precision binary64
 (if (<= wj 0.0048)
   (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 (+ 1.0 wj)))))

C

double code(double wj, double x) {
	double tmp;
	if (wj <= 0.0048) {
		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 / (1.0 + wj));
	}
	return tmp;
}

Julia

function code(wj, x)
	tmp = 0.0
	if (wj <= 0.0048)
		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(wj / Float64(1.0 + wj)));
	end
	return tmp
end

Wolfram

code[wj_, x_] := If[LessEqual[wj, 0.0048], 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[(wj / N[(1.0 + wj), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if wj < 0.00479999999999999958

   1. Initial program 77.3%

      \[wj - \frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}} \]
   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 96.7%

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

   if 0.00479999999999999958 < wj

   1. Initial program 38.8%

      \[wj - \frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}} \]
   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 79.8

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

   5. Applied rewrites 79.8%

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

Alternative 6: 97.1% accurate, 12.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;wj \leq 0.0048:\\ \;\;\;\;\mathsf{fma}\left(wj + x \cdot \mathsf{fma}\left(2.5, wj, -2\right), wj, x\right)\\ \mathbf{else}:\\ \;\;\;\;wj - \frac{wj}{1 + wj}\\ \end{array} \end{array} \]

FPCore

(FPCore (wj x)
 :precision binary64
 (if (<= wj 0.0048)
   (fma (+ wj (* x (fma 2.5 wj -2.0))) wj x)
   (- wj (/ wj (+ 1.0 wj)))))

C

double code(double wj, double x) {
	double tmp;
	if (wj <= 0.0048) {
		tmp = fma((wj + (x * fma(2.5, wj, -2.0))), wj, x);
	} else {
		tmp = wj - (wj / (1.0 + wj));
	}
	return tmp;
}

Julia

function code(wj, x)
	tmp = 0.0
	if (wj <= 0.0048)
		tmp = fma(Float64(wj + Float64(x * fma(2.5, wj, -2.0))), wj, x);
	else
		tmp = Float64(wj - Float64(wj / Float64(1.0 + wj)));
	end
	return tmp
end

Wolfram

code[wj_, x_] := If[LessEqual[wj, 0.0048], N[(N[(wj + N[(x * N[(2.5 * wj + -2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * wj + x), $MachinePrecision], N[(wj - N[(wj / N[(1.0 + wj), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if wj < 0.00479999999999999958

   1. Initial program 77.3%

      \[wj - \frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}} \]
   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 96.7%

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

   6. Taylor expanded in wj around 0

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

      1. +-commutative N/A

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

   8. Applied rewrites 96.0%

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

   if 0.00479999999999999958 < wj

   1. Initial program 38.8%

      \[wj - \frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}} \]
   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 79.8

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

   5. Applied rewrites 79.8%

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

Alternative 7: 96.9% accurate, 13.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;wj \leq 0.0042:\\ \;\;\;\;\mathsf{fma}\left(\left(1 - wj\right) \cdot wj, wj, x\right)\\ \mathbf{else}:\\ \;\;\;\;wj - \frac{wj}{1 + wj}\\ \end{array} \end{array} \]

FPCore

(FPCore (wj x)
 :precision binary64
 (if (<= wj 0.0042) (fma (* (- 1.0 wj) wj) wj x) (- wj (/ wj (+ 1.0 wj)))))

C

double code(double wj, double x) {
	double tmp;
	if (wj <= 0.0042) {
		tmp = fma(((1.0 - wj) * wj), wj, x);
	} else {
		tmp = wj - (wj / (1.0 + wj));
	}
	return tmp;
}

Julia

function code(wj, x)
	tmp = 0.0
	if (wj <= 0.0042)
		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_] := If[LessEqual[wj, 0.0042], 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 wj < 0.00419999999999999974

   1. Initial program 77.3%

      \[wj - \frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}} \]
   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 96.7%

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

   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 95.4%

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

   if 0.00419999999999999974 < wj

   1. Initial program 38.8%

      \[wj - \frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}} \]
   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 79.8

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

   5. Applied rewrites 79.8%

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

Alternative 8: 96.4% accurate, 15.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;wj \leq 1.05:\\ \;\;\;\;\mathsf{fma}\left(\left(1 - wj\right) \cdot wj, wj, x\right)\\ \mathbf{else}:\\ \;\;\;\;-1 + wj\\ \end{array} \end{array} \]

FPCore

(FPCore (wj x)
 :precision binary64
 (if (<= wj 1.05) (fma (* (- 1.0 wj) wj) wj x) (+ -1.0 wj)))

C

double code(double wj, double x) {
	double tmp;
	if (wj <= 1.05) {
		tmp = fma(((1.0 - wj) * wj), wj, x);
	} else {
		tmp = -1.0 + wj;
	}
	return tmp;
}

Julia

function code(wj, x)
	tmp = 0.0
	if (wj <= 1.05)
		tmp = fma(Float64(Float64(1.0 - wj) * wj), wj, x);
	else
		tmp = Float64(-1.0 + wj);
	end
	return tmp
end

Wolfram

code[wj_, x_] := If[LessEqual[wj, 1.05], N[(N[(N[(1.0 - wj), $MachinePrecision] * wj), $MachinePrecision] * wj + x), $MachinePrecision], N[(-1.0 + wj), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if wj < 1.05000000000000004

   1. Initial program 77.3%

      \[wj - \frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}} \]
   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 96.5%

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

   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 95.2%

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

   if 1.05000000000000004 < wj

   1. Initial program 33.2%

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

   3. Taylor expanded in wj around inf

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

      1. sub-neg N/A

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

      2. distribute-rgt-in N/A

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

      3. *-lft-identity N/A

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

      4. distribute-lft-neg-out N/A

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

      5. lft-mult-inverse N/A

         \[\leadsto wj + \left(\mathsf{neg}\left(\color{blue}{1}\right)\right) \]

      6. metadata-eval N/A

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

      7. +-commutative N/A

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

      8. lower-+.f64 53.9

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

   5. Applied rewrites 53.9%

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

Alternative 9: 84.7% accurate, 18.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;wj \leq 0.6:\\ \;\;\;\;\mathsf{fma}\left(x \cdot wj, -2, x\right)\\ \mathbf{else}:\\ \;\;\;\;-1 + wj\\ \end{array} \end{array} \]

FPCore

(FPCore (wj x)
 :precision binary64
 (if (<= wj 0.6) (fma (* x wj) -2.0 x) (+ -1.0 wj)))

C

double code(double wj, double x) {
	double tmp;
	if (wj <= 0.6) {
		tmp = fma((x * wj), -2.0, x);
	} else {
		tmp = -1.0 + wj;
	}
	return tmp;
}

Julia

function code(wj, x)
	tmp = 0.0
	if (wj <= 0.6)
		tmp = fma(Float64(x * wj), -2.0, x);
	else
		tmp = Float64(-1.0 + wj);
	end
	return tmp
end

Wolfram

code[wj_, x_] := If[LessEqual[wj, 0.6], N[(N[(x * wj), $MachinePrecision] * -2.0 + x), $MachinePrecision], N[(-1.0 + wj), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if wj < 0.599999999999999978

   1. Initial program 77.3%

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

   3. Taylor expanded in wj around 0

      \[\leadsto \color{blue}{x + -2 \cdot \left(wj \cdot x\right)} \]
   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. *-commutative N/A

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

      5. lower-*.f64 84.2

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

   5. Applied rewrites 84.2%

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

   if 0.599999999999999978 < wj

   1. Initial program 33.2%

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

   3. Taylor expanded in wj around inf

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

      1. sub-neg N/A

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

      2. distribute-rgt-in N/A

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

      3. *-lft-identity N/A

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

      4. distribute-lft-neg-out N/A

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

      5. lft-mult-inverse N/A

         \[\leadsto wj + \left(\mathsf{neg}\left(\color{blue}{1}\right)\right) \]

      6. metadata-eval N/A

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

      7. +-commutative N/A

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

      8. lower-+.f64 53.9

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

   5. Applied rewrites 53.9%

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

Alternative 10: 15.8% accurate, 27.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -3.6 \cdot 10^{-68}:\\ \;\;\;\;-1 + wj\\ \mathbf{else}:\\ \;\;\;\;wj \cdot wj\\ \end{array} \end{array} \]

FPCore

(FPCore (wj x) :precision binary64 (if (<= x -3.6e-68) (+ -1.0 wj) (* wj wj)))

C

double code(double wj, double x) {
	double tmp;
	if (x <= -3.6e-68) {
		tmp = -1.0 + wj;
	} 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 <= (-3.6d-68)) then
        tmp = (-1.0d0) + wj
    else
        tmp = wj * wj
    end if
    code = tmp
end function

Java

public static double code(double wj, double x) {
	double tmp;
	if (x <= -3.6e-68) {
		tmp = -1.0 + wj;
	} else {
		tmp = wj * wj;
	}
	return tmp;
}

Python

def code(wj, x):
	tmp = 0
	if x <= -3.6e-68:
		tmp = -1.0 + wj
	else:
		tmp = wj * wj
	return tmp

Julia

function code(wj, x)
	tmp = 0.0
	if (x <= -3.6e-68)
		tmp = Float64(-1.0 + wj);
	else
		tmp = Float64(wj * wj);
	end
	return tmp
end

MATLAB

function tmp_2 = code(wj, x)
	tmp = 0.0;
	if (x <= -3.6e-68)
		tmp = -1.0 + wj;
	else
		tmp = wj * wj;
	end
	tmp_2 = tmp;
end

Wolfram

code[wj_, x_] := If[LessEqual[x, -3.6e-68], N[(-1.0 + wj), $MachinePrecision], N[(wj * wj), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -3.60000000000000007e-68

   1. Initial program 94.5%

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

   3. Taylor expanded in wj around inf

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

      1. sub-neg N/A

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

      2. distribute-rgt-in N/A

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

      3. *-lft-identity N/A

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

      4. distribute-lft-neg-out N/A

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

      5. lft-mult-inverse N/A

         \[\leadsto wj + \left(\mathsf{neg}\left(\color{blue}{1}\right)\right) \]

      6. metadata-eval N/A

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

      7. +-commutative N/A

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

      8. lower-+.f64 8.7

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

   5. Applied rewrites 8.7%

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

   if -3.60000000000000007e-68 < x

   1. Initial program 68.3%

      \[wj - \frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}} \]
   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 93.1%

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

   6. Taylor expanded in wj around 0

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

      1. +-commutative N/A

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

   8. Applied rewrites 92.4%

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

   9. Taylor expanded in x around 0

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

   11. Applied rewrites 17.6%

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

Alternative 11: 4.2% accurate, 82.8× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 75.8%

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

3. Taylor expanded in wj around inf

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

   1. sub-neg N/A

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

   2. distribute-rgt-in N/A

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

   3. *-lft-identity N/A

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

   4. distribute-lft-neg-out N/A

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

   5. lft-mult-inverse N/A

      \[\leadsto wj + \left(\mathsf{neg}\left(\color{blue}{1}\right)\right) \]

   6. metadata-eval N/A

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

   7. +-commutative N/A

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

   8. lower-+.f64 5.0

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

5. Applied rewrites 5.0%

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

Alternative 12: 3.3% accurate, 331.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

def code(wj, x):
	return -1.0

Julia

function code(wj, x)
	return -1.0
end

MATLAB

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

Wolfram

code[wj_, x_] := -1.0

Derivation

1. Initial program 75.8%

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

3. Taylor expanded in wj around inf

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

   1. sub-neg N/A

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

   2. distribute-rgt-in N/A

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

   3. *-lft-identity N/A

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

   4. distribute-lft-neg-out N/A

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

   5. lft-mult-inverse N/A

      \[\leadsto wj + \left(\mathsf{neg}\left(\color{blue}{1}\right)\right) \]

   6. metadata-eval N/A

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

   7. +-commutative N/A

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

   8. lower-+.f64 5.0

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

5. Applied rewrites 5.0%

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

6. Taylor expanded in wj around 0

   \[\leadsto -1 \]
7. Step-by-step derivation

8. Applied rewrites 3.1%

   \[\leadsto -1 \]
9. Add Preprocessing

Developer Target 1: 78.2% 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 2024315 
(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))))))