Jmat.Real.lambertw, newton loop step

Percentage Accurate: 77.9% → 98.7%

Time: 11.5s

Alternatives: 13

Speedup: 331.0×

• could not determine a ground truth

  1. wj = -3.305418235867269e+227
  2. x = -3.692125400845642e+225

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: 98.7% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := wj \cdot e^{wj}\\ \mathbf{if}\;wj + \frac{x - t\_0}{e^{wj} + t\_0} \leq 2 \cdot 10^{-19}:\\ \;\;\;\;\mathsf{fma}\left(wj, \mathsf{fma}\left(wj, \mathsf{fma}\left(x, 2.5, 1\right) - \mathsf{fma}\left(wj, \mathsf{fma}\left(x, 0.6666666666666666, x \cdot 2\right), wj\right), x \cdot -2\right), x\right)\\ \mathbf{else}:\\ \;\;\;\;wj + x \cdot \left(\frac{e^{-wj}}{wj + 1} - \frac{wj}{\mathsf{fma}\left(x, wj, x\right)}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (wj x)
 :precision binary64
 (let* ((t_0 (* wj (exp wj))))
   (if (<= (+ wj (/ (- x t_0) (+ (exp wj) t_0))) 2e-19)
     (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 (fma x wj x))))))))

C

double code(double wj, double x) {
	double t_0 = wj * exp(wj);
	double tmp;
	if ((wj + ((x - t_0) / (exp(wj) + t_0))) <= 2e-19) {
		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 / fma(x, wj, x))));
	}
	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))) <= 2e-19)
		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(exp(Float64(-wj)) / Float64(wj + 1.0)) - Float64(wj / fma(x, wj, x)))));
	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], 2e-19], 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[(N[Exp[(-wj)], $MachinePrecision] / N[(wj + 1.0), $MachinePrecision]), $MachinePrecision] - N[(wj / N[(x * wj + x), $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))))) < 2e-19

   1. Initial program 74.2%

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

   3. Taylor expanded in wj around 0

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

   5. Simplified 99.4%

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

   if 2e-19 < (-.f64 wj (/.f64 (-.f64 (*.f64 wj (exp.f64 wj)) x) (+.f64 (exp.f64 wj) (*.f64 wj (exp.f64 wj)))))

   1. Initial program 93.1%

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

   3. Taylor expanded in x around inf

      \[\leadsto wj - \color{blue}{x \cdot \left(\frac{wj \cdot e^{wj}}{x \cdot \left(e^{wj} + wj \cdot e^{wj}\right)} - \frac{1}{e^{wj} + wj \cdot e^{wj}}\right)} \]
   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. Simplified 100.0%

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

4. Final simplification 99.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;wj + \frac{x - wj \cdot e^{wj}}{e^{wj} + wj \cdot e^{wj}} \leq 2 \cdot 10^{-19}:\\ \;\;\;\;\mathsf{fma}\left(wj, \mathsf{fma}\left(wj, \mathsf{fma}\left(x, 2.5, 1\right) - \mathsf{fma}\left(wj, \mathsf{fma}\left(x, 0.6666666666666666, x \cdot 2\right), wj\right), x \cdot -2\right), x\right)\\ \mathbf{else}:\\ \;\;\;\;wj + x \cdot \left(\frac{e^{-wj}}{wj + 1} - \frac{wj}{\mathsf{fma}\left(x, wj, x\right)}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 97.7% accurate, 6.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;wj \leq 0.052:\\ \;\;\;\;\mathsf{fma}\left(wj, \mathsf{fma}\left(wj, \mathsf{fma}\left(x, 2.5, 1\right) - \mathsf{fma}\left(wj, \mathsf{fma}\left(x, 0.6666666666666666, x \cdot 2\right), wj\right), x \cdot -2\right), x\right)\\ \mathbf{else}:\\ \;\;\;\;wj + \frac{1}{\frac{-1 - wj}{wj}}\\ \end{array} \end{array} \]

FPCore

(FPCore (wj x)
 :precision binary64
 (if (<= wj 0.052)
   (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 (/ 1.0 (/ (- -1.0 wj) wj)))))

C

double code(double wj, double x) {
	double tmp;
	if (wj <= 0.052) {
		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 + (1.0 / ((-1.0 - wj) / wj));
	}
	return tmp;
}

Julia

function code(wj, x)
	tmp = 0.0
	if (wj <= 0.052)
		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(1.0 / Float64(Float64(-1.0 - wj) / wj)));
	end
	return tmp
end

Wolfram

code[wj_, x_] := If[LessEqual[wj, 0.052], 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[(1.0 / N[(N[(-1.0 - wj), $MachinePrecision] / wj), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if wj < 0.0519999999999999976

   1. Initial program 80.7%

      \[wj - \frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}} \]
   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. Simplified 99.1%

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

   if 0.0519999999999999976 < wj

   1. Initial program 33.3%

      \[wj - \frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}} \]
   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. /-lowering-/.f64 N/A

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

      8. +-commutative N/A

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

      9. +-lowering-+.f64 99.7

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

   5. Simplified 99.7%

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

      1. clear-num N/A

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

      2. /-lowering-/.f64 N/A

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

      3. /-lowering-/.f64 N/A

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

      4. +-commutative N/A

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

      5. +-lowering-+.f64 100.0

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

   7. Applied egg-rr 100.0%

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

4. Final simplification 99.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;wj \leq 0.052:\\ \;\;\;\;\mathsf{fma}\left(wj, \mathsf{fma}\left(wj, \mathsf{fma}\left(x, 2.5, 1\right) - \mathsf{fma}\left(wj, \mathsf{fma}\left(x, 0.6666666666666666, x \cdot 2\right), wj\right), x \cdot -2\right), x\right)\\ \mathbf{else}:\\ \;\;\;\;wj + \frac{1}{\frac{-1 - wj}{wj}}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 97.7% accurate, 7.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;wj \leq 0.065:\\ \;\;\;\;\mathsf{fma}\left(x, wj \cdot \mathsf{fma}\left(wj, \mathsf{fma}\left(wj, -2.6666666666666665, 2.5 + \frac{1 - wj}{x}\right), -2\right), x\right)\\ \mathbf{else}:\\ \;\;\;\;wj + \frac{1}{\frac{-1 - wj}{wj}}\\ \end{array} \end{array} \]

FPCore

(FPCore (wj x)
 :precision binary64
 (if (<= wj 0.065)
   (fma
    x
    (* wj (fma wj (fma wj -2.6666666666666665 (+ 2.5 (/ (- 1.0 wj) x))) -2.0))
    x)
   (+ wj (/ 1.0 (/ (- -1.0 wj) wj)))))

C

double code(double wj, double x) {
	double tmp;
	if (wj <= 0.065) {
		tmp = fma(x, (wj * fma(wj, fma(wj, -2.6666666666666665, (2.5 + ((1.0 - wj) / x))), -2.0)), x);
	} else {
		tmp = wj + (1.0 / ((-1.0 - wj) / wj));
	}
	return tmp;
}

Julia

function code(wj, x)
	tmp = 0.0
	if (wj <= 0.065)
		tmp = fma(x, Float64(wj * fma(wj, fma(wj, -2.6666666666666665, Float64(2.5 + Float64(Float64(1.0 - wj) / x))), -2.0)), x);
	else
		tmp = Float64(wj + Float64(1.0 / Float64(Float64(-1.0 - wj) / wj)));
	end
	return tmp
end

Wolfram

code[wj_, x_] := If[LessEqual[wj, 0.065], N[(x * N[(wj * N[(wj * N[(wj * -2.6666666666666665 + N[(2.5 + N[(N[(1.0 - wj), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + -2.0), $MachinePrecision]), $MachinePrecision] + x), $MachinePrecision], N[(wj + N[(1.0 / N[(N[(-1.0 - wj), $MachinePrecision] / wj), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if wj < 0.065000000000000002

   1. Initial program 80.7%

      \[wj - \frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}} \]
   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. Simplified 99.1%

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

   6. Taylor expanded in x around inf

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

   8. Simplified 99.0%

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

   if 0.065000000000000002 < wj

   1. Initial program 33.3%

      \[wj - \frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}} \]
   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. /-lowering-/.f64 N/A

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

      8. +-commutative N/A

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

      9. +-lowering-+.f64 99.7

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

   5. Simplified 99.7%

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

      1. clear-num N/A

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

      2. /-lowering-/.f64 N/A

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

      3. /-lowering-/.f64 N/A

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

      4. +-commutative N/A

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

      5. +-lowering-+.f64 100.0

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

   7. Applied egg-rr 100.0%

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

4. Final simplification 99.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;wj \leq 0.065:\\ \;\;\;\;\mathsf{fma}\left(x, wj \cdot \mathsf{fma}\left(wj, \mathsf{fma}\left(wj, -2.6666666666666665, 2.5 + \frac{1 - wj}{x}\right), -2\right), x\right)\\ \mathbf{else}:\\ \;\;\;\;wj + \frac{1}{\frac{-1 - wj}{wj}}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 97.4% accurate, 9.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;wj \leq 0.004:\\ \;\;\;\;\mathsf{fma}\left(wj, \mathsf{fma}\left(wj, 1 - wj, x \cdot -2\right), x\right)\\ \mathbf{else}:\\ \;\;\;\;wj + \frac{1}{\frac{-1 - wj}{wj}}\\ \end{array} \end{array} \]

FPCore

(FPCore (wj x)
 :precision binary64
 (if (<= wj 0.004)
   (fma wj (fma wj (- 1.0 wj) (* x -2.0)) x)
   (+ wj (/ 1.0 (/ (- -1.0 wj) wj)))))

C

double code(double wj, double x) {
	double tmp;
	if (wj <= 0.004) {
		tmp = fma(wj, fma(wj, (1.0 - wj), (x * -2.0)), x);
	} else {
		tmp = wj + (1.0 / ((-1.0 - wj) / wj));
	}
	return tmp;
}

Julia

function code(wj, x)
	tmp = 0.0
	if (wj <= 0.004)
		tmp = fma(wj, fma(wj, Float64(1.0 - wj), Float64(x * -2.0)), x);
	else
		tmp = Float64(wj + Float64(1.0 / Float64(Float64(-1.0 - wj) / wj)));
	end
	return tmp
end

Wolfram

code[wj_, x_] := If[LessEqual[wj, 0.004], N[(wj * N[(wj * N[(1.0 - wj), $MachinePrecision] + N[(x * -2.0), $MachinePrecision]), $MachinePrecision] + x), $MachinePrecision], N[(wj + N[(1.0 / N[(N[(-1.0 - wj), $MachinePrecision] / wj), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if wj < 0.0040000000000000001

   1. Initial program 80.7%

      \[wj - \frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}} \]
   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. Simplified 99.1%

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

   6. Taylor expanded in x around 0

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

      1. --lowering--.f64 98.8

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

   8. Simplified 98.8%

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

   if 0.0040000000000000001 < wj

   1. Initial program 33.3%

      \[wj - \frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}} \]
   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. /-lowering-/.f64 N/A

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

      8. +-commutative N/A

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

      9. +-lowering-+.f64 99.7

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

   5. Simplified 99.7%

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

      1. clear-num N/A

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

      2. /-lowering-/.f64 N/A

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

      3. /-lowering-/.f64 N/A

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

      4. +-commutative N/A

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

      5. +-lowering-+.f64 100.0

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

   7. Applied egg-rr 100.0%

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

4. Final simplification 98.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;wj \leq 0.004:\\ \;\;\;\;\mathsf{fma}\left(wj, \mathsf{fma}\left(wj, 1 - wj, x \cdot -2\right), x\right)\\ \mathbf{else}:\\ \;\;\;\;wj + \frac{1}{\frac{-1 - wj}{wj}}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 97.4% accurate, 12.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;wj \leq 0.0056:\\ \;\;\;\;\mathsf{fma}\left(wj, \mathsf{fma}\left(wj, 1 - wj, x \cdot -2\right), x\right)\\ \mathbf{else}:\\ \;\;\;\;wj + \frac{wj}{-1 - wj}\\ \end{array} \end{array} \]

FPCore

(FPCore (wj x)
 :precision binary64
 (if (<= wj 0.0056)
   (fma wj (fma wj (- 1.0 wj) (* x -2.0)) x)
   (+ wj (/ wj (- -1.0 wj)))))

C

double code(double wj, double x) {
	double tmp;
	if (wj <= 0.0056) {
		tmp = fma(wj, fma(wj, (1.0 - wj), (x * -2.0)), x);
	} else {
		tmp = wj + (wj / (-1.0 - wj));
	}
	return tmp;
}

Julia

function code(wj, x)
	tmp = 0.0
	if (wj <= 0.0056)
		tmp = fma(wj, fma(wj, Float64(1.0 - wj), Float64(x * -2.0)), x);
	else
		tmp = Float64(wj + Float64(wj / Float64(-1.0 - wj)));
	end
	return tmp
end

Wolfram

code[wj_, x_] := If[LessEqual[wj, 0.0056], N[(wj * N[(wj * N[(1.0 - wj), $MachinePrecision] + N[(x * -2.0), $MachinePrecision]), $MachinePrecision] + x), $MachinePrecision], N[(wj + N[(wj / N[(-1.0 - wj), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if wj < 0.00559999999999999994

   1. Initial program 80.7%

      \[wj - \frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}} \]
   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. Simplified 99.1%

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

   6. Taylor expanded in x around 0

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

      1. --lowering--.f64 98.8

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

   8. Simplified 98.8%

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

   if 0.00559999999999999994 < wj

   1. Initial program 33.3%

      \[wj - \frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}} \]
   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. /-lowering-/.f64 N/A

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

      8. +-commutative N/A

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

      9. +-lowering-+.f64 99.7

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

   5. Simplified 99.7%

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

4. Final simplification 98.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;wj \leq 0.0056:\\ \;\;\;\;\mathsf{fma}\left(wj, \mathsf{fma}\left(wj, 1 - wj, x \cdot -2\right), x\right)\\ \mathbf{else}:\\ \;\;\;\;wj + \frac{wj}{-1 - wj}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 97.1% accurate, 13.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;wj \leq 0.0062:\\ \;\;\;\;\mathsf{fma}\left(wj, \mathsf{fma}\left(x, \mathsf{fma}\left(wj, 2.5, -2\right), wj\right), x\right)\\ \mathbf{else}:\\ \;\;\;\;wj + \frac{wj}{-1 - wj}\\ \end{array} \end{array} \]

FPCore

(FPCore (wj x)
 :precision binary64
 (if (<= wj 0.0062)
   (fma wj (fma x (fma wj 2.5 -2.0) wj) x)
   (+ wj (/ wj (- -1.0 wj)))))

C

double code(double wj, double x) {
	double tmp;
	if (wj <= 0.0062) {
		tmp = fma(wj, fma(x, fma(wj, 2.5, -2.0), wj), x);
	} else {
		tmp = wj + (wj / (-1.0 - wj));
	}
	return tmp;
}

Julia

function code(wj, x)
	tmp = 0.0
	if (wj <= 0.0062)
		tmp = fma(wj, fma(x, fma(wj, 2.5, -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, 0.0062], N[(wj * N[(x * N[(wj * 2.5 + -2.0), $MachinePrecision] + wj), $MachinePrecision] + x), $MachinePrecision], N[(wj + N[(wj / N[(-1.0 - wj), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if wj < 0.00619999999999999978

   1. Initial program 80.7%

      \[wj - \frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}} \]
   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. Simplified 99.1%

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

   6. Taylor expanded in wj around 0

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

      1. +-commutative N/A

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

      2. accelerator-lowering-fma.f64 N/A

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

   8. Simplified 98.6%

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

   if 0.00619999999999999978 < wj

   1. Initial program 33.3%

      \[wj - \frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}} \]
   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. /-lowering-/.f64 N/A

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

      8. +-commutative N/A

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

      9. +-lowering-+.f64 99.7

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

   5. Simplified 99.7%

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

4. Final simplification 98.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;wj \leq 0.0062:\\ \;\;\;\;\mathsf{fma}\left(wj, \mathsf{fma}\left(x, \mathsf{fma}\left(wj, 2.5, -2\right), wj\right), x\right)\\ \mathbf{else}:\\ \;\;\;\;wj + \frac{wj}{-1 - wj}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 96.9% accurate, 13.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;wj \leq 0.0042:\\ \;\;\;\;\mathsf{fma}\left(wj, \mathsf{fma}\left(x, -2, wj\right), x\right)\\ \mathbf{else}:\\ \;\;\;\;wj + \frac{wj}{-1 - wj}\\ \end{array} \end{array} \]

FPCore

(FPCore (wj x)
 :precision binary64
 (if (<= wj 0.0042) (fma wj (fma x -2.0 wj) x) (+ wj (/ wj (- -1.0 wj)))))

C

double code(double wj, double x) {
	double tmp;
	if (wj <= 0.0042) {
		tmp = fma(wj, fma(x, -2.0, 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(wj, fma(x, -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, 0.0042], N[(wj * N[(x * -2.0 + wj), $MachinePrecision] + 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 80.7%

      \[wj - \frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}} \]
   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. Simplified 99.1%

      \[\leadsto \color{blue}{\mathsf{fma}\left(wj, \mathsf{fma}\left(wj, \mathsf{fma}\left(x, 2.5, 1\right) - \mathsf{fma}\left(wj, \mathsf{fma}\left(x, 0.6666666666666666, x \cdot 2\right), wj\right), x \cdot -2\right), x\right)} \]
   6. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \color{blue}{x + wj \cdot \left(wj \cdot \left(\left(x \cdot \frac{5}{2} + 1\right) - \left(wj \cdot \left(x \cdot \frac{2}{3} + x \cdot 2\right) + wj\right)\right) + x \cdot -2\right)} \]

      2. distribute-rgt-in N/A

         \[\leadsto x + \color{blue}{\left(\left(wj \cdot \left(\left(x \cdot \frac{5}{2} + 1\right) - \left(wj \cdot \left(x \cdot \frac{2}{3} + x \cdot 2\right) + wj\right)\right)\right) \cdot wj + \left(x \cdot -2\right) \cdot wj\right)} \]

      3. associate-+r+ N/A

         \[\leadsto \color{blue}{\left(x + \left(wj \cdot \left(\left(x \cdot \frac{5}{2} + 1\right) - \left(wj \cdot \left(x \cdot \frac{2}{3} + x \cdot 2\right) + wj\right)\right)\right) \cdot wj\right) + \left(x \cdot -2\right) \cdot wj} \]

      4. *-commutative N/A

         \[\leadsto \left(x + \left(wj \cdot \left(\left(x \cdot \frac{5}{2} + 1\right) - \left(wj \cdot \left(x \cdot \frac{2}{3} + x \cdot 2\right) + wj\right)\right)\right) \cdot wj\right) + \color{blue}{wj \cdot \left(x \cdot -2\right)} \]

      5. +-lowering-+.f64 N/A

         \[\leadsto \color{blue}{\left(x + \left(wj \cdot \left(\left(x \cdot \frac{5}{2} + 1\right) - \left(wj \cdot \left(x \cdot \frac{2}{3} + x \cdot 2\right) + wj\right)\right)\right) \cdot wj\right) + wj \cdot \left(x \cdot -2\right)} \]

   7. Applied egg-rr 99.1%

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

   8. Taylor expanded in x around 0

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

      1. --lowering--.f64 98.8

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

   10. Simplified 98.8%

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

   11. Taylor expanded in wj around 0

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

       1. +-commutative N/A

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

       2. accelerator-lowering-fma.f64 N/A

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

       3. +-commutative N/A

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

       4. *-commutative N/A

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

       5. accelerator-lowering-fma.f64 98.5

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

   13. Simplified 98.5%

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

   if 0.00419999999999999974 < wj

   1. Initial program 33.3%

      \[wj - \frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}} \]
   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. /-lowering-/.f64 N/A

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

      8. +-commutative N/A

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

      9. +-lowering-+.f64 99.7

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

   5. Simplified 99.7%

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

4. Final simplification 98.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;wj \leq 0.0042:\\ \;\;\;\;\mathsf{fma}\left(wj, \mathsf{fma}\left(x, -2, wj\right), x\right)\\ \mathbf{else}:\\ \;\;\;\;wj + \frac{wj}{-1 - wj}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 95.8% accurate, 25.5× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(wj, \mathsf{fma}\left(x, -2, wj\right), x\right) \end{array} \]

FPCore

(FPCore (wj x) :precision binary64 (fma wj (fma x -2.0 wj) x))

C

double code(double wj, double x) {
	return fma(wj, fma(x, -2.0, wj), x);
}

Julia

function code(wj, x)
	return fma(wj, fma(x, -2.0, wj), x)
end

Wolfram

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

Derivation

1. Initial program 79.6%

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

   \[\leadsto \color{blue}{\mathsf{fma}\left(wj, \mathsf{fma}\left(wj, \mathsf{fma}\left(x, 2.5, 1\right) - \mathsf{fma}\left(wj, \mathsf{fma}\left(x, 0.6666666666666666, x \cdot 2\right), wj\right), x \cdot -2\right), x\right)} \]
6. Step-by-step derivation

   1. +-commutative N/A

      \[\leadsto \color{blue}{x + wj \cdot \left(wj \cdot \left(\left(x \cdot \frac{5}{2} + 1\right) - \left(wj \cdot \left(x \cdot \frac{2}{3} + x \cdot 2\right) + wj\right)\right) + x \cdot -2\right)} \]

   2. distribute-rgt-in N/A

      \[\leadsto x + \color{blue}{\left(\left(wj \cdot \left(\left(x \cdot \frac{5}{2} + 1\right) - \left(wj \cdot \left(x \cdot \frac{2}{3} + x \cdot 2\right) + wj\right)\right)\right) \cdot wj + \left(x \cdot -2\right) \cdot wj\right)} \]

   3. associate-+r+ N/A

      \[\leadsto \color{blue}{\left(x + \left(wj \cdot \left(\left(x \cdot \frac{5}{2} + 1\right) - \left(wj \cdot \left(x \cdot \frac{2}{3} + x \cdot 2\right) + wj\right)\right)\right) \cdot wj\right) + \left(x \cdot -2\right) \cdot wj} \]

   4. *-commutative N/A

      \[\leadsto \left(x + \left(wj \cdot \left(\left(x \cdot \frac{5}{2} + 1\right) - \left(wj \cdot \left(x \cdot \frac{2}{3} + x \cdot 2\right) + wj\right)\right)\right) \cdot wj\right) + \color{blue}{wj \cdot \left(x \cdot -2\right)} \]

   5. +-lowering-+.f64 N/A

      \[\leadsto \color{blue}{\left(x + \left(wj \cdot \left(\left(x \cdot \frac{5}{2} + 1\right) - \left(wj \cdot \left(x \cdot \frac{2}{3} + x \cdot 2\right) + wj\right)\right)\right) \cdot wj\right) + wj \cdot \left(x \cdot -2\right)} \]

7. Applied egg-rr 96.8%

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

8. Taylor expanded in x around 0

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

   1. --lowering--.f64 96.6

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

10. Simplified 96.6%

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

11. Taylor expanded in wj around 0

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

    1. +-commutative N/A

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

    2. accelerator-lowering-fma.f64 N/A

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

    3. +-commutative N/A

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

    4. *-commutative N/A

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

    5. accelerator-lowering-fma.f64 96.4

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

13. Simplified 96.4%

    \[\leadsto \color{blue}{\mathsf{fma}\left(wj, \mathsf{fma}\left(x, -2, wj\right), x\right)} \]
14. Add Preprocessing

Alternative 9: 83.2% accurate, 27.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;wj \leq -6 \cdot 10^{-30}:\\ \;\;\;\;wj \cdot wj\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \end{array} \]

FPCore

(FPCore (wj x) :precision binary64 (if (<= wj -6e-30) (* wj wj) x))

C

double code(double wj, double x) {
	double tmp;
	if (wj <= -6e-30) {
		tmp = wj * wj;
	} else {
		tmp = x;
	}
	return tmp;
}

Fortran

real(8) function code(wj, x)
    real(8), intent (in) :: wj
    real(8), intent (in) :: x
    real(8) :: tmp
    if (wj <= (-6d-30)) then
        tmp = wj * wj
    else
        tmp = x
    end if
    code = tmp
end function

Java

public static double code(double wj, double x) {
	double tmp;
	if (wj <= -6e-30) {
		tmp = wj * wj;
	} else {
		tmp = x;
	}
	return tmp;
}

Python

def code(wj, x):
	tmp = 0
	if wj <= -6e-30:
		tmp = wj * wj
	else:
		tmp = x
	return tmp

Julia

function code(wj, x)
	tmp = 0.0
	if (wj <= -6e-30)
		tmp = Float64(wj * wj);
	else
		tmp = x;
	end
	return tmp
end

MATLAB

function tmp_2 = code(wj, x)
	tmp = 0.0;
	if (wj <= -6e-30)
		tmp = wj * wj;
	else
		tmp = x;
	end
	tmp_2 = tmp;
end

Wolfram

code[wj_, x_] := If[LessEqual[wj, -6e-30], N[(wj * wj), $MachinePrecision], x]

Derivation

1. Split input into 2 regimes

2. if wj < -5.9999999999999998e-30

   1. Initial program 41.3%

      \[wj - \frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}} \]
   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. /-lowering-/.f64 N/A

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

      8. +-commutative N/A

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

      9. +-lowering-+.f64 13.3

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

   5. Simplified 13.3%

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

   6. Taylor expanded in wj around 0

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

      1. unpow2 N/A

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

      2. *-lowering-*.f64 52.0

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

   8. Simplified 52.0%

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

   if -5.9999999999999998e-30 < wj

   1. Initial program 82.3%

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

   3. Taylor expanded in wj around 0

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

   5. Simplified 90.2%

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

Alternative 10: 95.3% accurate, 47.3× speedup

Math

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

FPCore

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

C

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

Julia

function code(wj, x)
	return fma(wj, wj, x)
end

Wolfram

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

Derivation

1. Initial program 79.6%

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

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

6. Taylor expanded in wj around 0

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

   1. +-commutative N/A

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

   2. accelerator-lowering-fma.f64 N/A

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

8. Simplified 96.6%

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

9. Taylor expanded in x around 0

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

11. Simplified 95.8%

    \[\leadsto \mathsf{fma}\left(wj, \color{blue}{wj}, x\right) \]
12. Add Preprocessing

Alternative 11: 83.7% accurate, 331.0× speedup

Math

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

FPCore

(FPCore (wj x) :precision binary64 x)

C

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

Fortran

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

Java

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

Python

def code(wj, x):
	return x

Julia

function code(wj, x)
	return x
end

MATLAB

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

Wolfram

code[wj_, x_] := x

Derivation

1. Initial program 79.6%

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

3. Taylor expanded in wj around 0

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

5. Simplified 85.8%

   \[\leadsto \color{blue}{x} \]
6. Add Preprocessing

Alternative 12: 4.4% accurate, 331.0× speedup

Math

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

FPCore

(FPCore (wj x) :precision binary64 wj)

C

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

Fortran

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

Java

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

Python

def code(wj, x):
	return wj

Julia

function code(wj, x)
	return wj
end

MATLAB

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

Wolfram

code[wj_, x_] := wj

Derivation

1. Initial program 79.6%

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

3. Taylor expanded in wj around inf

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

5. Simplified 4.7%

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

Alternative 13: 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 79.6%

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

3. Taylor expanded in wj around inf

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

5. Simplified 4.8%

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

6. Taylor expanded in wj around 0

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

8. Simplified 3.4%

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

Developer Target 1: 78.8% 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 2024204 
(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))))))