Jmat.Real.lambertw, newton loop step

Percentage Accurate: 77.9% → 98.1%

Time: 11.3s

Alternatives: 15

Speedup: 18.4×

• could not determine a ground truth

  1. wj = 7.553936143939574e+95
  2. x = -1.6898827131679362e-61

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: 77.9% 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: 98.1% accurate, 2.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;wj \leq -1.8 \cdot 10^{-11}:\\ \;\;\;\;\frac{x}{e^{wj} \cdot \left(wj + 1\right)}\\ \mathbf{elif}\;wj \leq 2.1 \cdot 10^{-5}:\\ \;\;\;\;\mathsf{fma}\left(wj, wj - wj \cdot wj, x\right)\\ \mathbf{else}:\\ \;\;\;\;wj - x \cdot \frac{wj}{\mathsf{fma}\left(x, wj, x\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (wj x)
 :precision binary64
 (if (<= wj -1.8e-11)
   (/ x (* (exp wj) (+ wj 1.0)))
   (if (<= wj 2.1e-5)
     (fma wj (- wj (* wj wj)) x)
     (- wj (* x (/ wj (fma x wj x)))))))

C

double code(double wj, double x) {
	double tmp;
	if (wj <= -1.8e-11) {
		tmp = x / (exp(wj) * (wj + 1.0));
	} else if (wj <= 2.1e-5) {
		tmp = fma(wj, (wj - (wj * wj)), x);
	} else {
		tmp = wj - (x * (wj / fma(x, wj, x)));
	}
	return tmp;
}

Julia

function code(wj, x)
	tmp = 0.0
	if (wj <= -1.8e-11)
		tmp = Float64(x / Float64(exp(wj) * Float64(wj + 1.0)));
	elseif (wj <= 2.1e-5)
		tmp = fma(wj, Float64(wj - Float64(wj * wj)), x);
	else
		tmp = Float64(wj - Float64(x * Float64(wj / fma(x, wj, x))));
	end
	return tmp
end

Wolfram

code[wj_, x_] := If[LessEqual[wj, -1.8e-11], N[(x / N[(N[Exp[wj], $MachinePrecision] * N[(wj + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[wj, 2.1e-5], N[(wj * N[(wj - N[(wj * wj), $MachinePrecision]), $MachinePrecision] + x), $MachinePrecision], N[(wj - N[(x * N[(wj / N[(x * wj + x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if wj < -1.79999999999999992e-11

   1. Initial program 72.6%

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

   3. Taylor expanded in x around inf

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

      1. lower-/.f64 N/A

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

      2. distribute-rgt1-in N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower-exp.f64 N/A

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

      7. +-commutative N/A

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

      8. lower-+.f64 91.0

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

   5. Applied rewrites 91.0%

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

   if -1.79999999999999992e-11 < wj < 2.09999999999999988e-5

   1. Initial program 76.1%

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

   3. Taylor expanded in wj around 0

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

   5. Applied rewrites 99.5%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 100.0%

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

   if 2.09999999999999988e-5 < wj

   1. Initial program 40.2%

      \[wj - \frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}} \]
   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 96.4%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 79.9%

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

Alternative 2: 98.8% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := wj \cdot e^{wj}\\ \mathbf{if}\;wj + \frac{x - t\_0}{e^{wj} + t\_0} \leq 5 \cdot 10^{-15}:\\ \;\;\;\;\mathsf{fma}\left(wj, \mathsf{fma}\left(wj, \mathsf{fma}\left(x, 2.5, 1\right) - \mathsf{fma}\left(wj, \mathsf{fma}\left(x, 0.6666666666666666, x \cdot 2\right), wj\right), x \cdot -2\right), x\right)\\ \mathbf{else}:\\ \;\;\;\;wj - x \cdot \left(\frac{wj}{\mathsf{fma}\left(x, wj, x\right)} - \frac{e^{-wj}}{wj + 1}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (wj x)
 :precision binary64
 (let* ((t_0 (* wj (exp wj))))
   (if (<= (+ wj (/ (- x t_0) (+ (exp wj) t_0))) 5e-15)
     (fma
      wj
      (fma
       wj
       (- (fma x 2.5 1.0) (fma wj (fma x 0.6666666666666666 (* x 2.0)) wj))
       (* x -2.0))
      x)
     (- wj (* x (- (/ wj (fma x wj x)) (/ (exp (- wj)) (+ wj 1.0))))))))

C

double code(double wj, double x) {
	double t_0 = wj * exp(wj);
	double tmp;
	if ((wj + ((x - t_0) / (exp(wj) + t_0))) <= 5e-15) {
		tmp = fma(wj, fma(wj, (fma(x, 2.5, 1.0) - fma(wj, fma(x, 0.6666666666666666, (x * 2.0)), wj)), (x * -2.0)), x);
	} else {
		tmp = wj - (x * ((wj / fma(x, wj, x)) - (exp(-wj) / (wj + 1.0))));
	}
	return tmp;
}

Julia

function code(wj, x)
	t_0 = Float64(wj * exp(wj))
	tmp = 0.0
	if (Float64(wj + Float64(Float64(x - t_0) / Float64(exp(wj) + t_0))) <= 5e-15)
		tmp = fma(wj, fma(wj, Float64(fma(x, 2.5, 1.0) - fma(wj, fma(x, 0.6666666666666666, Float64(x * 2.0)), wj)), Float64(x * -2.0)), x);
	else
		tmp = Float64(wj - Float64(x * Float64(Float64(wj / fma(x, wj, x)) - Float64(exp(Float64(-wj)) / Float64(wj + 1.0)))));
	end
	return tmp
end

Wolfram

code[wj_, x_] := Block[{t$95$0 = N[(wj * N[Exp[wj], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(wj + N[(N[(x - t$95$0), $MachinePrecision] / N[(N[Exp[wj], $MachinePrecision] + t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 5e-15], N[(wj * N[(wj * N[(N[(x * 2.5 + 1.0), $MachinePrecision] - N[(wj * N[(x * 0.6666666666666666 + N[(x * 2.0), $MachinePrecision]), $MachinePrecision] + wj), $MachinePrecision]), $MachinePrecision] + N[(x * -2.0), $MachinePrecision]), $MachinePrecision] + x), $MachinePrecision], N[(wj - N[(x * N[(N[(wj / N[(x * wj + x), $MachinePrecision]), $MachinePrecision] - N[(N[Exp[(-wj)], $MachinePrecision] / N[(wj + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 wj (/.f64 (-.f64 (*.f64 wj (exp.f64 wj)) x) (+.f64 (exp.f64 wj) (*.f64 wj (exp.f64 wj))))) < 4.99999999999999999e-15

   1. Initial program 69.7%

      \[wj - \frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}} \]
   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 98.2%

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

   if 4.99999999999999999e-15 < (-.f64 wj (/.f64 (-.f64 (*.f64 wj (exp.f64 wj)) x) (+.f64 (exp.f64 wj) (*.f64 wj (exp.f64 wj)))))

   1. Initial program 89.0%

      \[wj - \frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}} \]
   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 99.5%

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

4. Final simplification 98.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;wj + \frac{x - wj \cdot e^{wj}}{e^{wj} + wj \cdot e^{wj}} \leq 5 \cdot 10^{-15}:\\ \;\;\;\;\mathsf{fma}\left(wj, \mathsf{fma}\left(wj, \mathsf{fma}\left(x, 2.5, 1\right) - \mathsf{fma}\left(wj, \mathsf{fma}\left(x, 0.6666666666666666, x \cdot 2\right), wj\right), x \cdot -2\right), x\right)\\ \mathbf{else}:\\ \;\;\;\;wj - x \cdot \left(\frac{wj}{\mathsf{fma}\left(x, wj, x\right)} - \frac{e^{-wj}}{wj + 1}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 98.8% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := wj \cdot e^{wj}\\ \mathbf{if}\;wj + \frac{x - t\_0}{e^{wj} + t\_0} \leq 5 \cdot 10^{-15}:\\ \;\;\;\;\mathsf{fma}\left(wj, \mathsf{fma}\left(wj, \mathsf{fma}\left(x, 2.5, 1\right) - \mathsf{fma}\left(wj, \mathsf{fma}\left(x, 0.6666666666666666, x \cdot 2\right), wj\right), x \cdot -2\right), x\right)\\ \mathbf{else}:\\ \;\;\;\;wj + \left(\frac{x}{e^{wj} \cdot \left(wj + 1\right)} + \frac{wj}{-1 - wj}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (wj x)
 :precision binary64
 (let* ((t_0 (* wj (exp wj))))
   (if (<= (+ wj (/ (- x t_0) (+ (exp wj) t_0))) 5e-15)
     (fma
      wj
      (fma
       wj
       (- (fma x 2.5 1.0) (fma wj (fma x 0.6666666666666666 (* x 2.0)) wj))
       (* x -2.0))
      x)
     (+ wj (+ (/ x (* (exp wj) (+ wj 1.0))) (/ wj (- -1.0 wj)))))))

C

double code(double wj, double x) {
	double t_0 = wj * exp(wj);
	double tmp;
	if ((wj + ((x - t_0) / (exp(wj) + t_0))) <= 5e-15) {
		tmp = fma(wj, fma(wj, (fma(x, 2.5, 1.0) - fma(wj, fma(x, 0.6666666666666666, (x * 2.0)), wj)), (x * -2.0)), x);
	} else {
		tmp = wj + ((x / (exp(wj) * (wj + 1.0))) + (wj / (-1.0 - wj)));
	}
	return tmp;
}

Julia

function code(wj, x)
	t_0 = Float64(wj * exp(wj))
	tmp = 0.0
	if (Float64(wj + Float64(Float64(x - t_0) / Float64(exp(wj) + t_0))) <= 5e-15)
		tmp = fma(wj, fma(wj, Float64(fma(x, 2.5, 1.0) - fma(wj, fma(x, 0.6666666666666666, Float64(x * 2.0)), wj)), Float64(x * -2.0)), x);
	else
		tmp = Float64(wj + Float64(Float64(x / Float64(exp(wj) * Float64(wj + 1.0))) + Float64(wj / Float64(-1.0 - wj))));
	end
	return tmp
end

Wolfram

code[wj_, x_] := Block[{t$95$0 = N[(wj * N[Exp[wj], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(wj + N[(N[(x - t$95$0), $MachinePrecision] / N[(N[Exp[wj], $MachinePrecision] + t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 5e-15], N[(wj * N[(wj * N[(N[(x * 2.5 + 1.0), $MachinePrecision] - N[(wj * N[(x * 0.6666666666666666 + N[(x * 2.0), $MachinePrecision]), $MachinePrecision] + wj), $MachinePrecision]), $MachinePrecision] + N[(x * -2.0), $MachinePrecision]), $MachinePrecision] + x), $MachinePrecision], N[(wj + N[(N[(x / N[(N[Exp[wj], $MachinePrecision] * N[(wj + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(wj / N[(-1.0 - wj), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 wj (/.f64 (-.f64 (*.f64 wj (exp.f64 wj)) x) (+.f64 (exp.f64 wj) (*.f64 wj (exp.f64 wj))))) < 4.99999999999999999e-15

   1. Initial program 69.7%

      \[wj - \frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}} \]
   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 98.2%

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

   if 4.99999999999999999e-15 < (-.f64 wj (/.f64 (-.f64 (*.f64 wj (exp.f64 wj)) x) (+.f64 (exp.f64 wj) (*.f64 wj (exp.f64 wj)))))

   1. Initial program 89.0%

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

      1. lift-/.f64 N/A

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

      2. lift--.f64 N/A

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

      3. div-sub N/A

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

      4. lower--.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-+.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. distribute-rgt1-in N/A

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

      9. times-frac N/A

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

      10. *-inverses N/A

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

      11. associate-*l/ N/A

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

      12. *-rgt-identity N/A

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

      13. lower-/.f64 N/A

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

      14. lower-+.f64 N/A

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

      15. lower-/.f64 96.4

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

      16. lift-+.f64 N/A

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

      17. lift-*.f64 N/A

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

      18. distribute-rgt1-in N/A

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

      19. *-commutative N/A

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

      20. lower-*.f64 N/A

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

      21. lower-+.f64 99.4

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

   4. Applied rewrites 99.4%

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

4. Final simplification 98.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;wj + \frac{x - wj \cdot e^{wj}}{e^{wj} + wj \cdot e^{wj}} \leq 5 \cdot 10^{-15}:\\ \;\;\;\;\mathsf{fma}\left(wj, \mathsf{fma}\left(wj, \mathsf{fma}\left(x, 2.5, 1\right) - \mathsf{fma}\left(wj, \mathsf{fma}\left(x, 0.6666666666666666, x \cdot 2\right), wj\right), x \cdot -2\right), x\right)\\ \mathbf{else}:\\ \;\;\;\;wj + \left(\frac{x}{e^{wj} \cdot \left(wj + 1\right)} + \frac{wj}{-1 - wj}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 97.8% accurate, 6.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;wj \leq 2.1 \cdot 10^{-5}:\\ \;\;\;\;\mathsf{fma}\left(x, \mathsf{fma}\left(wj, \mathsf{fma}\left(wj, \mathsf{fma}\left(wj, -2.6666666666666665, 2.5\right), -2\right), \frac{wj \cdot \left(wj - wj \cdot wj\right)}{x}\right), x\right)\\ \mathbf{else}:\\ \;\;\;\;wj - x \cdot \frac{wj}{\mathsf{fma}\left(x, wj, x\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (wj x)
 :precision binary64
 (if (<= wj 2.1e-5)
   (fma
    x
    (fma
     wj
     (fma wj (fma wj -2.6666666666666665 2.5) -2.0)
     (/ (* wj (- wj (* wj wj))) x))
    x)
   (- wj (* x (/ wj (fma x wj x))))))

C

double code(double wj, double x) {
	double tmp;
	if (wj <= 2.1e-5) {
		tmp = fma(x, fma(wj, fma(wj, fma(wj, -2.6666666666666665, 2.5), -2.0), ((wj * (wj - (wj * wj))) / x)), x);
	} else {
		tmp = wj - (x * (wj / fma(x, wj, x)));
	}
	return tmp;
}

Julia

function code(wj, x)
	tmp = 0.0
	if (wj <= 2.1e-5)
		tmp = fma(x, fma(wj, fma(wj, fma(wj, -2.6666666666666665, 2.5), -2.0), Float64(Float64(wj * Float64(wj - Float64(wj * wj))) / x)), x);
	else
		tmp = Float64(wj - Float64(x * Float64(wj / fma(x, wj, x))));
	end
	return tmp
end

Wolfram

code[wj_, x_] := If[LessEqual[wj, 2.1e-5], N[(x * N[(wj * N[(wj * N[(wj * -2.6666666666666665 + 2.5), $MachinePrecision] + -2.0), $MachinePrecision] + N[(N[(wj * N[(wj - N[(wj * wj), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision] + x), $MachinePrecision], N[(wj - N[(x * N[(wj / N[(x * wj + x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if wj < 2.09999999999999988e-5

   1. Initial program 76.0%

      \[wj - \frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}} \]
   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 96.6%

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

   6. Taylor expanded in x around inf

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

   8. Applied rewrites 97.0%

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

   if 2.09999999999999988e-5 < wj

   1. Initial program 40.2%

      \[wj - \frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}} \]
   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 96.4%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 79.9%

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

Alternative 5: 97.7% accurate, 6.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;wj \leq 2.1 \cdot 10^{-5}:\\ \;\;\;\;\mathsf{fma}\left(x, wj \cdot \left(-2 + \mathsf{fma}\left(wj, 2.5 - \frac{wj}{x}, \frac{wj}{x}\right)\right), x\right)\\ \mathbf{else}:\\ \;\;\;\;wj - x \cdot \frac{wj}{\mathsf{fma}\left(x, wj, x\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (wj x)
 :precision binary64
 (if (<= wj 2.1e-5)
   (fma x (* wj (+ -2.0 (fma wj (- 2.5 (/ wj x)) (/ wj x)))) x)
   (- wj (* x (/ wj (fma x wj x))))))

C

double code(double wj, double x) {
	double tmp;
	if (wj <= 2.1e-5) {
		tmp = fma(x, (wj * (-2.0 + fma(wj, (2.5 - (wj / x)), (wj / x)))), x);
	} else {
		tmp = wj - (x * (wj / fma(x, wj, x)));
	}
	return tmp;
}

Julia

function code(wj, x)
	tmp = 0.0
	if (wj <= 2.1e-5)
		tmp = fma(x, Float64(wj * Float64(-2.0 + fma(wj, Float64(2.5 - Float64(wj / x)), Float64(wj / x)))), x);
	else
		tmp = Float64(wj - Float64(x * Float64(wj / fma(x, wj, x))));
	end
	return tmp
end

Wolfram

code[wj_, x_] := If[LessEqual[wj, 2.1e-5], N[(x * N[(wj * N[(-2.0 + N[(wj * N[(2.5 - N[(wj / x), $MachinePrecision]), $MachinePrecision] + N[(wj / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + x), $MachinePrecision], N[(wj - N[(x * N[(wj / N[(x * wj + x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if wj < 2.09999999999999988e-5

   1. Initial program 76.0%

      \[wj - \frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}} \]
   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 96.6%

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

   6. Taylor expanded in x around inf

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

   8. Applied rewrites 97.0%

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

   9. Taylor expanded in wj around 0

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

   11. Applied rewrites 97.0%

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

   12. Taylor expanded in x around 0

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

   14. Applied rewrites 96.9%

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

   if 2.09999999999999988e-5 < wj

   1. Initial program 40.2%

      \[wj - \frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}} \]
   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 96.4%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 79.9%

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

Alternative 6: 97.7% accurate, 6.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;wj \leq 2.1 \cdot 10^{-5}:\\ \;\;\;\;\mathsf{fma}\left(wj, \mathsf{fma}\left(wj, \mathsf{fma}\left(x, 2.5, 1\right) - \mathsf{fma}\left(wj, \mathsf{fma}\left(x, 0.6666666666666666, x \cdot 2\right), wj\right), x \cdot -2\right), x\right)\\ \mathbf{else}:\\ \;\;\;\;wj - x \cdot \frac{wj}{\mathsf{fma}\left(x, wj, x\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (wj x)
 :precision binary64
 (if (<= wj 2.1e-5)
   (fma
    wj
    (fma
     wj
     (- (fma x 2.5 1.0) (fma wj (fma x 0.6666666666666666 (* x 2.0)) wj))
     (* x -2.0))
    x)
   (- wj (* x (/ wj (fma x wj x))))))

C

double code(double wj, double x) {
	double tmp;
	if (wj <= 2.1e-5) {
		tmp = fma(wj, fma(wj, (fma(x, 2.5, 1.0) - fma(wj, fma(x, 0.6666666666666666, (x * 2.0)), wj)), (x * -2.0)), x);
	} else {
		tmp = wj - (x * (wj / fma(x, wj, x)));
	}
	return tmp;
}

Julia

function code(wj, x)
	tmp = 0.0
	if (wj <= 2.1e-5)
		tmp = fma(wj, fma(wj, Float64(fma(x, 2.5, 1.0) - fma(wj, fma(x, 0.6666666666666666, Float64(x * 2.0)), wj)), Float64(x * -2.0)), x);
	else
		tmp = Float64(wj - Float64(x * Float64(wj / fma(x, wj, x))));
	end
	return tmp
end

Wolfram

code[wj_, x_] := If[LessEqual[wj, 2.1e-5], N[(wj * N[(wj * N[(N[(x * 2.5 + 1.0), $MachinePrecision] - N[(wj * N[(x * 0.6666666666666666 + N[(x * 2.0), $MachinePrecision]), $MachinePrecision] + wj), $MachinePrecision]), $MachinePrecision] + N[(x * -2.0), $MachinePrecision]), $MachinePrecision] + x), $MachinePrecision], N[(wj - N[(x * N[(wj / N[(x * wj + x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if wj < 2.09999999999999988e-5

   1. Initial program 76.0%

      \[wj - \frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}} \]
   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 96.6%

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

   if 2.09999999999999988e-5 < wj

   1. Initial program 40.2%

      \[wj - \frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}} \]
   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 96.4%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 79.9%

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

Alternative 7: 97.1% accurate, 10.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;wj \leq 5 \cdot 10^{-8}:\\ \;\;\;\;\mathsf{fma}\left(wj, \mathsf{fma}\left(x, -2, \mathsf{fma}\left(wj \cdot x, 2.5, wj\right)\right), x\right)\\ \mathbf{else}:\\ \;\;\;\;wj - x \cdot \frac{wj}{\mathsf{fma}\left(x, wj, x\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (wj x)
 :precision binary64
 (if (<= wj 5e-8)
   (fma wj (fma x -2.0 (fma (* wj x) 2.5 wj)) x)
   (- wj (* x (/ wj (fma x wj x))))))

C

double code(double wj, double x) {
	double tmp;
	if (wj <= 5e-8) {
		tmp = fma(wj, fma(x, -2.0, fma((wj * x), 2.5, wj)), x);
	} else {
		tmp = wj - (x * (wj / fma(x, wj, x)));
	}
	return tmp;
}

Julia

function code(wj, x)
	tmp = 0.0
	if (wj <= 5e-8)
		tmp = fma(wj, fma(x, -2.0, fma(Float64(wj * x), 2.5, wj)), x);
	else
		tmp = Float64(wj - Float64(x * Float64(wj / fma(x, wj, x))));
	end
	return tmp
end

Wolfram

code[wj_, x_] := If[LessEqual[wj, 5e-8], N[(wj * N[(x * -2.0 + N[(N[(wj * x), $MachinePrecision] * 2.5 + wj), $MachinePrecision]), $MachinePrecision] + x), $MachinePrecision], N[(wj - N[(x * N[(wj / N[(x * wj + x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if wj < 4.9999999999999998e-8

   1. Initial program 76.0%

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

   3. Taylor expanded in wj around 0

      \[\leadsto \color{blue}{x + wj \cdot \left(wj \cdot \left(1 - \left(-4 \cdot x + \frac{3}{2} \cdot x\right)\right) - 2 \cdot x\right)} \]
   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. lower-fma.f64 N/A

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

   5. Applied rewrites 96.7%

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

   if 4.9999999999999998e-8 < wj

   1. Initial program 42.6%

      \[wj - \frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}} \]
   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 93.3%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 78.4%

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

4. Final simplification 96.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;wj \leq 5 \cdot 10^{-8}:\\ \;\;\;\;\mathsf{fma}\left(wj, \mathsf{fma}\left(x, -2, \mathsf{fma}\left(wj \cdot x, 2.5, wj\right)\right), x\right)\\ \mathbf{else}:\\ \;\;\;\;wj - x \cdot \frac{wj}{\mathsf{fma}\left(x, wj, x\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 97.1% accurate, 11.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;wj \leq 5 \cdot 10^{-8}:\\ \;\;\;\;\mathsf{fma}\left(wj, \mathsf{fma}\left(x, -2, \mathsf{fma}\left(wj \cdot x, 2.5, wj\right)\right), x\right)\\ \mathbf{else}:\\ \;\;\;\;wj - \frac{wj}{wj + 1}\\ \end{array} \end{array} \]

FPCore

(FPCore (wj x)
 :precision binary64
 (if (<= wj 5e-8)
   (fma wj (fma x -2.0 (fma (* wj x) 2.5 wj)) x)
   (- wj (/ wj (+ wj 1.0)))))

C

double code(double wj, double x) {
	double tmp;
	if (wj <= 5e-8) {
		tmp = fma(wj, fma(x, -2.0, fma((wj * x), 2.5, wj)), x);
	} else {
		tmp = wj - (wj / (wj + 1.0));
	}
	return tmp;
}

Julia

function code(wj, x)
	tmp = 0.0
	if (wj <= 5e-8)
		tmp = fma(wj, fma(x, -2.0, fma(Float64(wj * x), 2.5, wj)), x);
	else
		tmp = Float64(wj - Float64(wj / Float64(wj + 1.0)));
	end
	return tmp
end

Wolfram

code[wj_, x_] := If[LessEqual[wj, 5e-8], N[(wj * N[(x * -2.0 + N[(N[(wj * x), $MachinePrecision] * 2.5 + wj), $MachinePrecision]), $MachinePrecision] + x), $MachinePrecision], N[(wj - N[(wj / N[(wj + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if wj < 4.9999999999999998e-8

   1. Initial program 76.0%

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

   3. Taylor expanded in wj around 0

      \[\leadsto \color{blue}{x + wj \cdot \left(wj \cdot \left(1 - \left(-4 \cdot x + \frac{3}{2} \cdot x\right)\right) - 2 \cdot x\right)} \]
   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. lower-fma.f64 N/A

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

   5. Applied rewrites 96.7%

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

   if 4.9999999999999998e-8 < wj

   1. Initial program 42.6%

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

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

      9. lower-+.f64 77.5

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

   5. Applied rewrites 77.5%

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

4. Final simplification 96.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;wj \leq 5 \cdot 10^{-8}:\\ \;\;\;\;\mathsf{fma}\left(wj, \mathsf{fma}\left(x, -2, \mathsf{fma}\left(wj \cdot x, 2.5, wj\right)\right), x\right)\\ \mathbf{else}:\\ \;\;\;\;wj - \frac{wj}{wj + 1}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 84.6% accurate, 12.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;wj \leq 3.3 \cdot 10^{-44}:\\ \;\;\;\;\mathsf{fma}\left(x, wj \cdot -2, x\right)\\ \mathbf{elif}\;wj \leq 0.185:\\ \;\;\;\;wj \cdot \left(wj - wj \cdot wj\right)\\ \mathbf{else}:\\ \;\;\;\;wj - 1\\ \end{array} \end{array} \]

FPCore

(FPCore (wj x)
 :precision binary64
 (if (<= wj 3.3e-44)
   (fma x (* wj -2.0) x)
   (if (<= wj 0.185) (* wj (- wj (* wj wj))) (- wj 1.0))))

C

double code(double wj, double x) {
	double tmp;
	if (wj <= 3.3e-44) {
		tmp = fma(x, (wj * -2.0), x);
	} else if (wj <= 0.185) {
		tmp = wj * (wj - (wj * wj));
	} else {
		tmp = wj - 1.0;
	}
	return tmp;
}

Julia

function code(wj, x)
	tmp = 0.0
	if (wj <= 3.3e-44)
		tmp = fma(x, Float64(wj * -2.0), x);
	elseif (wj <= 0.185)
		tmp = Float64(wj * Float64(wj - Float64(wj * wj)));
	else
		tmp = Float64(wj - 1.0);
	end
	return tmp
end

Wolfram

code[wj_, x_] := If[LessEqual[wj, 3.3e-44], N[(x * N[(wj * -2.0), $MachinePrecision] + x), $MachinePrecision], If[LessEqual[wj, 0.185], N[(wj * N[(wj - N[(wj * wj), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(wj - 1.0), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if wj < 3.30000000000000006e-44

   1. Initial program 78.3%

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

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

      3. *-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 86.4

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

   5. Applied rewrites 86.4%

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

   if 3.30000000000000006e-44 < wj < 0.185

   1. Initial program 44.6%

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

   3. Taylor expanded in wj around 0

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

   5. Applied rewrites 94.8%

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

   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 71.3%

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

   if 0.185 < wj

   1. Initial program 28.6%

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

   3. Taylor expanded in wj around inf

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

   5. Applied rewrites 70.0%

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

Alternative 10: 97.1% accurate, 13.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;wj \leq 2.1 \cdot 10^{-5}:\\ \;\;\;\;\mathsf{fma}\left(wj, wj - wj \cdot wj, x\right)\\ \mathbf{else}:\\ \;\;\;\;wj - \frac{wj}{wj + 1}\\ \end{array} \end{array} \]

FPCore

(FPCore (wj x)
 :precision binary64
 (if (<= wj 2.1e-5) (fma wj (- wj (* wj wj)) x) (- wj (/ wj (+ wj 1.0)))))

C

double code(double wj, double x) {
	double tmp;
	if (wj <= 2.1e-5) {
		tmp = fma(wj, (wj - (wj * wj)), x);
	} else {
		tmp = wj - (wj / (wj + 1.0));
	}
	return tmp;
}

Julia

function code(wj, x)
	tmp = 0.0
	if (wj <= 2.1e-5)
		tmp = fma(wj, Float64(wj - Float64(wj * wj)), x);
	else
		tmp = Float64(wj - Float64(wj / Float64(wj + 1.0)));
	end
	return tmp
end

Wolfram

code[wj_, x_] := If[LessEqual[wj, 2.1e-5], N[(wj * N[(wj - N[(wj * wj), $MachinePrecision]), $MachinePrecision] + x), $MachinePrecision], N[(wj - N[(wj / N[(wj + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if wj < 2.09999999999999988e-5

   1. Initial program 76.0%

      \[wj - \frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}} \]
   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 96.6%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 96.4%

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

   if 2.09999999999999988e-5 < wj

   1. Initial program 40.2%

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

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

      9. lower-+.f64 79.0

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

   5. Applied rewrites 79.0%

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

Alternative 11: 96.6% accurate, 15.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;wj \leq 1.3:\\ \;\;\;\;\mathsf{fma}\left(wj, wj - wj \cdot wj, x\right)\\ \mathbf{else}:\\ \;\;\;\;wj - 1\\ \end{array} \end{array} \]

FPCore

(FPCore (wj x)
 :precision binary64
 (if (<= wj 1.3) (fma wj (- wj (* wj wj)) x) (- wj 1.0)))

C

double code(double wj, double x) {
	double tmp;
	if (wj <= 1.3) {
		tmp = fma(wj, (wj - (wj * wj)), x);
	} else {
		tmp = wj - 1.0;
	}
	return tmp;
}

Julia

function code(wj, x)
	tmp = 0.0
	if (wj <= 1.3)
		tmp = fma(wj, Float64(wj - Float64(wj * wj)), x);
	else
		tmp = Float64(wj - 1.0);
	end
	return tmp
end

Wolfram

code[wj_, x_] := If[LessEqual[wj, 1.3], N[(wj * N[(wj - N[(wj * wj), $MachinePrecision]), $MachinePrecision] + x), $MachinePrecision], N[(wj - 1.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if wj < 1.30000000000000004

   1. Initial program 76.1%

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

   3. Taylor expanded in wj around 0

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

   5. Applied rewrites 96.1%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 95.8%

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

   if 1.30000000000000004 < wj

   1. Initial program 16.7%

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

   3. Taylor expanded in wj around inf

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

   5. Applied rewrites 80.9%

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

Alternative 12: 85.6% accurate, 18.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;wj \leq 0.7:\\ \;\;\;\;\mathsf{fma}\left(x, wj \cdot -2, x\right)\\ \mathbf{else}:\\ \;\;\;\;wj - 1\\ \end{array} \end{array} \]

FPCore

(FPCore (wj x)
 :precision binary64
 (if (<= wj 0.7) (fma x (* wj -2.0) x) (- wj 1.0)))

C

double code(double wj, double x) {
	double tmp;
	if (wj <= 0.7) {
		tmp = fma(x, (wj * -2.0), x);
	} else {
		tmp = wj - 1.0;
	}
	return tmp;
}

Julia

function code(wj, x)
	tmp = 0.0
	if (wj <= 0.7)
		tmp = fma(x, Float64(wj * -2.0), x);
	else
		tmp = Float64(wj - 1.0);
	end
	return tmp
end

Wolfram

code[wj_, x_] := If[LessEqual[wj, 0.7], N[(x * N[(wj * -2.0), $MachinePrecision] + x), $MachinePrecision], N[(wj - 1.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if wj < 0.69999999999999996

   1. Initial program 76.1%

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

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

      3. *-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 82.1

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

   5. Applied rewrites 82.1%

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

   if 0.69999999999999996 < wj

   1. Initial program 16.7%

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

   3. Taylor expanded in wj around inf

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

   5. Applied rewrites 80.9%

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

Alternative 13: 85.1% accurate, 18.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;wj \leq 1.55:\\ \;\;\;\;\frac{x}{1}\\ \mathbf{else}:\\ \;\;\;\;wj - 1\\ \end{array} \end{array} \]

FPCore

(FPCore (wj x) :precision binary64 (if (<= wj 1.55) (/ x 1.0) (- wj 1.0)))

C

double code(double wj, double x) {
	double tmp;
	if (wj <= 1.55) {
		tmp = x / 1.0;
	} else {
		tmp = wj - 1.0;
	}
	return tmp;
}

Fortran

real(8) function code(wj, x)
    real(8), intent (in) :: wj
    real(8), intent (in) :: x
    real(8) :: tmp
    if (wj <= 1.55d0) then
        tmp = x / 1.0d0
    else
        tmp = wj - 1.0d0
    end if
    code = tmp
end function

Java

public static double code(double wj, double x) {
	double tmp;
	if (wj <= 1.55) {
		tmp = x / 1.0;
	} else {
		tmp = wj - 1.0;
	}
	return tmp;
}

Python

def code(wj, x):
	tmp = 0
	if wj <= 1.55:
		tmp = x / 1.0
	else:
		tmp = wj - 1.0
	return tmp

Julia

function code(wj, x)
	tmp = 0.0
	if (wj <= 1.55)
		tmp = Float64(x / 1.0);
	else
		tmp = Float64(wj - 1.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(wj, x)
	tmp = 0.0;
	if (wj <= 1.55)
		tmp = x / 1.0;
	else
		tmp = wj - 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[wj_, x_] := If[LessEqual[wj, 1.55], N[(x / 1.0), $MachinePrecision], N[(wj - 1.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if wj < 1.55000000000000004

   1. Initial program 76.1%

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

   3. Taylor expanded in x around inf

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

      1. lower-/.f64 N/A

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

      2. distribute-rgt1-in N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower-exp.f64 N/A

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

      7. +-commutative N/A

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

      8. lower-+.f64 85.2

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

   5. Applied rewrites 85.2%

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

   6. Taylor expanded in wj around 0

      \[\leadsto \frac{x}{1} \]
   7. Step-by-step derivation

   8. Applied rewrites 81.7%

      \[\leadsto \frac{x}{1} \]

   if 1.55000000000000004 < wj

   1. Initial program 16.7%

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

   3. Taylor expanded in wj around inf

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

   5. Applied rewrites 80.9%

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

Alternative 14: 72.8% accurate, 55.2× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 74.7%

   \[wj - \frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}} \]
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 67.6

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

5. Applied rewrites 67.6%

   \[\leadsto wj - \color{blue}{\left(-x\right)} \]
6. Add Preprocessing

Alternative 15: 4.2% 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 74.7%

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

3. Taylor expanded in wj around inf

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

5. Applied rewrites 5.4%

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

Developer Target 1: 78.9% 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 2024221 
(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))))))