Jmat.Real.lambertw, newton loop step

Percentage Accurate: 78.1% → 98.8%

Time: 11.1s

Alternatives: 14

Speedup: 331.0×

• could not determine a ground truth

  1. wj = -4.3398719273415456e+160
  2. x = 1.0547875764793873e-99

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 78.1% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 98.8% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := wj \cdot e^{wj}\\ \mathbf{if}\;wj + \frac{x - t\_0}{e^{wj} + t\_0} \leq 10^{-12}:\\ \;\;\;\;\mathsf{fma}\left(wj, \mathsf{fma}\left(wj, \mathsf{fma}\left(x, 2.5, 1\right) - \mathsf{fma}\left(wj, \mathsf{fma}\left(x, 0.6666666666666666, x \cdot 2\right), wj\right), x \cdot -2\right), x\right)\\ \mathbf{else}:\\ \;\;\;\;wj + x \cdot \left(\frac{e^{-wj}}{wj + 1} - \frac{wj}{\mathsf{fma}\left(x, wj, x\right)}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (wj x)
 :precision binary64
 (let* ((t_0 (* wj (exp wj))))
   (if (<= (+ wj (/ (- x t_0) (+ (exp wj) t_0))) 1e-12)
     (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))) <= 1e-12) {
		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))) <= 1e-12)
		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], 1e-12], 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))))) < 9.9999999999999998e-13

   1. Initial program 75.8%

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

   3. Taylor expanded in wj around 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 99.9%

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

   if 9.9999999999999998e-13 < (-.f64 wj (/.f64 (-.f64 (*.f64 wj (exp.f64 wj)) x) (+.f64 (exp.f64 wj) (*.f64 wj (exp.f64 wj)))))

   1. Initial program 95.7%

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

   3. Taylor expanded in 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 99.7%

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

4. Final simplification 99.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;wj + \frac{x - wj \cdot e^{wj}}{e^{wj} + wj \cdot e^{wj}} \leq 10^{-12}:\\ \;\;\;\;\mathsf{fma}\left(wj, \mathsf{fma}\left(wj, \mathsf{fma}\left(x, 2.5, 1\right) - \mathsf{fma}\left(wj, \mathsf{fma}\left(x, 0.6666666666666666, x \cdot 2\right), wj\right), x \cdot -2\right), x\right)\\ \mathbf{else}:\\ \;\;\;\;wj + x \cdot \left(\frac{e^{-wj}}{wj + 1} - \frac{wj}{\mathsf{fma}\left(x, wj, x\right)}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 81.3% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := wj \cdot e^{wj}\\ t_1 := wj + \frac{x - t\_0}{e^{wj} + t\_0}\\ \mathbf{if}\;t\_1 \leq -1 \cdot 10^{-299}:\\ \;\;\;\;wj + x\\ \mathbf{elif}\;t\_1 \leq 0:\\ \;\;\;\;wj \cdot wj\\ \mathbf{else}:\\ \;\;\;\;wj + x\\ \end{array} \end{array} \]

FPCore

(FPCore (wj x)
 :precision binary64
 (let* ((t_0 (* wj (exp wj))) (t_1 (+ wj (/ (- x t_0) (+ (exp wj) t_0)))))
   (if (<= t_1 -1e-299) (+ wj x) (if (<= t_1 0.0) (* wj wj) (+ wj x)))))

C

double code(double wj, double x) {
	double t_0 = wj * exp(wj);
	double t_1 = wj + ((x - t_0) / (exp(wj) + t_0));
	double tmp;
	if (t_1 <= -1e-299) {
		tmp = wj + x;
	} else if (t_1 <= 0.0) {
		tmp = wj * wj;
	} else {
		tmp = wj + x;
	}
	return tmp;
}

Fortran

real(8) function code(wj, x)
    real(8), intent (in) :: wj
    real(8), intent (in) :: x
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = wj * exp(wj)
    t_1 = wj + ((x - t_0) / (exp(wj) + t_0))
    if (t_1 <= (-1d-299)) then
        tmp = wj + x
    else if (t_1 <= 0.0d0) then
        tmp = wj * wj
    else
        tmp = wj + x
    end if
    code = tmp
end function

Java

public static double code(double wj, double x) {
	double t_0 = wj * Math.exp(wj);
	double t_1 = wj + ((x - t_0) / (Math.exp(wj) + t_0));
	double tmp;
	if (t_1 <= -1e-299) {
		tmp = wj + x;
	} else if (t_1 <= 0.0) {
		tmp = wj * wj;
	} else {
		tmp = wj + x;
	}
	return tmp;
}

Python

def code(wj, x):
	t_0 = wj * math.exp(wj)
	t_1 = wj + ((x - t_0) / (math.exp(wj) + t_0))
	tmp = 0
	if t_1 <= -1e-299:
		tmp = wj + x
	elif t_1 <= 0.0:
		tmp = wj * wj
	else:
		tmp = wj + x
	return tmp

Julia

function code(wj, x)
	t_0 = Float64(wj * exp(wj))
	t_1 = Float64(wj + Float64(Float64(x - t_0) / Float64(exp(wj) + t_0)))
	tmp = 0.0
	if (t_1 <= -1e-299)
		tmp = Float64(wj + x);
	elseif (t_1 <= 0.0)
		tmp = Float64(wj * wj);
	else
		tmp = Float64(wj + x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(wj, x)
	t_0 = wj * exp(wj);
	t_1 = wj + ((x - t_0) / (exp(wj) + t_0));
	tmp = 0.0;
	if (t_1 <= -1e-299)
		tmp = wj + x;
	elseif (t_1 <= 0.0)
		tmp = wj * wj;
	else
		tmp = wj + x;
	end
	tmp_2 = tmp;
end

Wolfram

code[wj_, x_] := Block[{t$95$0 = N[(wj * N[Exp[wj], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(wj + N[(N[(x - t$95$0), $MachinePrecision] / N[(N[Exp[wj], $MachinePrecision] + t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -1e-299], N[(wj + x), $MachinePrecision], If[LessEqual[t$95$1, 0.0], N[(wj * wj), $MachinePrecision], N[(wj + x), $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))))) < -9.99999999999999992e-300 or 0.0 < (-.f64 wj (/.f64 (-.f64 (*.f64 wj (exp.f64 wj)) x) (+.f64 (exp.f64 wj) (*.f64 wj (exp.f64 wj)))))

   1. Initial program 97.1%

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

   3. Taylor expanded in wj around 0

      \[\leadsto wj - \color{blue}{-1 \cdot x} \]
   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 91.5

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

   5. Simplified 91.5%

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

   if -9.99999999999999992e-300 < (-.f64 wj (/.f64 (-.f64 (*.f64 wj (exp.f64 wj)) x) (+.f64 (exp.f64 wj) (*.f64 wj (exp.f64 wj))))) < 0.0

   1. Initial program 5.6%

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

   3. Taylor expanded in wj around 0

      \[\leadsto \color{blue}{x + wj \cdot \left(wj \cdot \left(1 - \left(-4 \cdot x + \frac{3}{2} \cdot x\right)\right) - 2 \cdot x\right)} \]
   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. Simplified 100.0%

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

   6. Taylor expanded in x around 0

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

      1. unpow2 N/A

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

      2. lower-*.f64 57.8

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

   8. Simplified 57.8%

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

4. Final simplification 85.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;wj + \frac{x - wj \cdot e^{wj}}{e^{wj} + wj \cdot e^{wj}} \leq -1 \cdot 10^{-299}:\\ \;\;\;\;wj + x\\ \mathbf{elif}\;wj + \frac{x - wj \cdot e^{wj}}{e^{wj} + wj \cdot e^{wj}} \leq 0:\\ \;\;\;\;wj \cdot wj\\ \mathbf{else}:\\ \;\;\;\;wj + x\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 97.7% accurate, 6.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;wj \leq 0.014:\\ \;\;\;\;\mathsf{fma}\left(wj, \mathsf{fma}\left(wj, \mathsf{fma}\left(x, 2.5, 1\right) - \mathsf{fma}\left(wj, \mathsf{fma}\left(x, 0.6666666666666666, x \cdot 2\right), wj\right), x \cdot -2\right), x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-wj, \frac{1}{wj + 1}, wj\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (wj x)
 :precision binary64
 (if (<= wj 0.014)
   (fma
    wj
    (fma
     wj
     (- (fma x 2.5 1.0) (fma wj (fma x 0.6666666666666666 (* x 2.0)) wj))
     (* x -2.0))
    x)
   (fma (- wj) (/ 1.0 (+ wj 1.0)) wj)))

C

double code(double wj, double x) {
	double tmp;
	if (wj <= 0.014) {
		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 = fma(-wj, (1.0 / (wj + 1.0)), wj);
	}
	return tmp;
}

Julia

function code(wj, x)
	tmp = 0.0
	if (wj <= 0.014)
		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 = fma(Float64(-wj), Float64(1.0 / Float64(wj + 1.0)), wj);
	end
	return tmp
end

Wolfram

code[wj_, x_] := If[LessEqual[wj, 0.014], 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[(wj + 1.0), $MachinePrecision]), $MachinePrecision] + wj), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if wj < 0.0140000000000000003

   1. Initial program 82.2%

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

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

   if 0.0140000000000000003 < wj

   1. Initial program 59.1%

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

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

   5. Simplified 98.8%

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

      1. lift-+.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. sub-neg N/A

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

      4. +-commutative N/A

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

      5. lift-/.f64 N/A

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

      6. div-inv N/A

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

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

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

      8. lower-fma.f64 N/A

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

      9. lower-neg.f64 N/A

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

      10. lower-/.f64 99.4

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

   7. Applied egg-rr 99.4%

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

Alternative 4: 97.7% accurate, 6.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;wj \leq 0.014:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(x, 2.5, 1\right) - \mathsf{fma}\left(wj \cdot x, 2.6666666666666665, wj\right), wj \cdot wj, \mathsf{fma}\left(x, wj \cdot -2, x\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-wj, \frac{1}{wj + 1}, wj\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (wj x)
 :precision binary64
 (if (<= wj 0.014)
   (fma
    (- (fma x 2.5 1.0) (fma (* wj x) 2.6666666666666665 wj))
    (* wj wj)
    (fma x (* wj -2.0) x))
   (fma (- wj) (/ 1.0 (+ wj 1.0)) wj)))

C

double code(double wj, double x) {
	double tmp;
	if (wj <= 0.014) {
		tmp = fma((fma(x, 2.5, 1.0) - fma((wj * x), 2.6666666666666665, wj)), (wj * wj), fma(x, (wj * -2.0), x));
	} else {
		tmp = fma(-wj, (1.0 / (wj + 1.0)), wj);
	}
	return tmp;
}

Julia

function code(wj, x)
	tmp = 0.0
	if (wj <= 0.014)
		tmp = fma(Float64(fma(x, 2.5, 1.0) - fma(Float64(wj * x), 2.6666666666666665, wj)), Float64(wj * wj), fma(x, Float64(wj * -2.0), x));
	else
		tmp = fma(Float64(-wj), Float64(1.0 / Float64(wj + 1.0)), wj);
	end
	return tmp
end

Wolfram

code[wj_, x_] := If[LessEqual[wj, 0.014], N[(N[(N[(x * 2.5 + 1.0), $MachinePrecision] - N[(N[(wj * x), $MachinePrecision] * 2.6666666666666665 + wj), $MachinePrecision]), $MachinePrecision] * N[(wj * wj), $MachinePrecision] + N[(x * N[(wj * -2.0), $MachinePrecision] + x), $MachinePrecision]), $MachinePrecision], N[((-wj) * N[(1.0 / N[(wj + 1.0), $MachinePrecision]), $MachinePrecision] + wj), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if wj < 0.0140000000000000003

   1. Initial program 82.2%

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

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

   7. Applied egg-rr 99.2%

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

   if 0.0140000000000000003 < wj

   1. Initial program 59.1%

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

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

   5. Simplified 98.8%

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

      1. lift-+.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. sub-neg N/A

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

      4. +-commutative N/A

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

      5. lift-/.f64 N/A

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

      6. div-inv N/A

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

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

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

      8. lower-fma.f64 N/A

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

      9. lower-neg.f64 N/A

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

      10. lower-/.f64 99.4

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

   7. Applied egg-rr 99.4%

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

4. Final simplification 99.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;wj \leq 0.014:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(x, 2.5, 1\right) - \mathsf{fma}\left(wj \cdot x, 2.6666666666666665, wj\right), wj \cdot wj, \mathsf{fma}\left(x, wj \cdot -2, x\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-wj, \frac{1}{wj + 1}, wj\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 97.4% accurate, 10.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;wj \leq 0.0017:\\ \;\;\;\;\mathsf{fma}\left(1 - wj, wj \cdot wj, \mathsf{fma}\left(x, wj \cdot -2, x\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-wj, \frac{1}{wj + 1}, wj\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (wj x)
 :precision binary64
 (if (<= wj 0.0017)
   (fma (- 1.0 wj) (* wj wj) (fma x (* wj -2.0) x))
   (fma (- wj) (/ 1.0 (+ wj 1.0)) wj)))

C

double code(double wj, double x) {
	double tmp;
	if (wj <= 0.0017) {
		tmp = fma((1.0 - wj), (wj * wj), fma(x, (wj * -2.0), x));
	} else {
		tmp = fma(-wj, (1.0 / (wj + 1.0)), wj);
	}
	return tmp;
}

Julia

function code(wj, x)
	tmp = 0.0
	if (wj <= 0.0017)
		tmp = fma(Float64(1.0 - wj), Float64(wj * wj), fma(x, Float64(wj * -2.0), x));
	else
		tmp = fma(Float64(-wj), Float64(1.0 / Float64(wj + 1.0)), wj);
	end
	return tmp
end

Wolfram

code[wj_, x_] := If[LessEqual[wj, 0.0017], N[(N[(1.0 - wj), $MachinePrecision] * N[(wj * wj), $MachinePrecision] + N[(x * N[(wj * -2.0), $MachinePrecision] + x), $MachinePrecision]), $MachinePrecision], N[((-wj) * N[(1.0 / N[(wj + 1.0), $MachinePrecision]), $MachinePrecision] + wj), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if wj < 0.00169999999999999991

   1. Initial program 82.2%

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

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

   7. Applied egg-rr 99.2%

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

   8. Taylor expanded in x around 0

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

      1. lower--.f64 98.8

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

   10. Simplified 98.8%

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

   if 0.00169999999999999991 < wj

   1. Initial program 59.1%

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

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

   5. Simplified 98.8%

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

      1. lift-+.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. sub-neg N/A

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

      4. +-commutative N/A

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

      5. lift-/.f64 N/A

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

      6. div-inv N/A

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

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

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

      8. lower-fma.f64 N/A

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

      9. lower-neg.f64 N/A

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

      10. lower-/.f64 99.4

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

   7. Applied egg-rr 99.4%

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

Alternative 6: 97.1% accurate, 11.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;wj \leq 0.0017:\\ \;\;\;\;\mathsf{fma}\left(wj, \mathsf{fma}\left(x, -2, \mathsf{fma}\left(wj \cdot x, 2.5, wj\right)\right), x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-wj, \frac{1}{wj + 1}, wj\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (wj x)
 :precision binary64
 (if (<= wj 0.0017)
   (fma wj (fma x -2.0 (fma (* wj x) 2.5 wj)) x)
   (fma (- wj) (/ 1.0 (+ wj 1.0)) wj)))

C

double code(double wj, double x) {
	double tmp;
	if (wj <= 0.0017) {
		tmp = fma(wj, fma(x, -2.0, fma((wj * x), 2.5, wj)), x);
	} else {
		tmp = fma(-wj, (1.0 / (wj + 1.0)), wj);
	}
	return tmp;
}

Julia

function code(wj, x)
	tmp = 0.0
	if (wj <= 0.0017)
		tmp = fma(wj, fma(x, -2.0, fma(Float64(wj * x), 2.5, wj)), x);
	else
		tmp = fma(Float64(-wj), Float64(1.0 / Float64(wj + 1.0)), wj);
	end
	return tmp
end

Wolfram

code[wj_, x_] := If[LessEqual[wj, 0.0017], N[(wj * N[(x * -2.0 + N[(N[(wj * x), $MachinePrecision] * 2.5 + wj), $MachinePrecision]), $MachinePrecision] + x), $MachinePrecision], N[((-wj) * N[(1.0 / N[(wj + 1.0), $MachinePrecision]), $MachinePrecision] + wj), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if wj < 0.00169999999999999991

   1. Initial program 82.2%

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

   3. Taylor expanded in wj around 0

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

      1. +-commutative N/A

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

      2. 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. Simplified 98.7%

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

   if 0.00169999999999999991 < wj

   1. Initial program 59.1%

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

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

   5. Simplified 98.8%

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

      1. lift-+.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. sub-neg N/A

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

      4. +-commutative N/A

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

      5. lift-/.f64 N/A

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

      6. div-inv N/A

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

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

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

      8. lower-fma.f64 N/A

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

      9. lower-neg.f64 N/A

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

      10. lower-/.f64 99.4

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

   7. Applied egg-rr 99.4%

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

4. Final simplification 98.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;wj \leq 0.0017:\\ \;\;\;\;\mathsf{fma}\left(wj, \mathsf{fma}\left(x, -2, \mathsf{fma}\left(wj \cdot x, 2.5, wj\right)\right), x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-wj, \frac{1}{wj + 1}, wj\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 96.9% accurate, 11.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;wj \leq 0.0017:\\ \;\;\;\;\mathsf{fma}\left(wj, \mathsf{fma}\left(x, -2, wj\right), x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-wj, \frac{1}{wj + 1}, wj\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (wj x)
 :precision binary64
 (if (<= wj 0.0017)
   (fma wj (fma x -2.0 wj) x)
   (fma (- wj) (/ 1.0 (+ wj 1.0)) wj)))

C

double code(double wj, double x) {
	double tmp;
	if (wj <= 0.0017) {
		tmp = fma(wj, fma(x, -2.0, wj), x);
	} else {
		tmp = fma(-wj, (1.0 / (wj + 1.0)), wj);
	}
	return tmp;
}

Julia

function code(wj, x)
	tmp = 0.0
	if (wj <= 0.0017)
		tmp = fma(wj, fma(x, -2.0, wj), x);
	else
		tmp = fma(Float64(-wj), Float64(1.0 / Float64(wj + 1.0)), wj);
	end
	return tmp
end

Wolfram

code[wj_, x_] := If[LessEqual[wj, 0.0017], N[(wj * N[(x * -2.0 + wj), $MachinePrecision] + x), $MachinePrecision], N[((-wj) * N[(1.0 / N[(wj + 1.0), $MachinePrecision]), $MachinePrecision] + wj), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if wj < 0.00169999999999999991

   1. Initial program 82.2%

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

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

   7. Applied egg-rr 99.2%

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

   8. Taylor expanded in x around 0

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

      1. lower--.f64 98.8

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

   10. Simplified 98.8%

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

   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. lower-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. lower-fma.f64 98.4

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

   13. Simplified 98.4%

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

   if 0.00169999999999999991 < wj

   1. Initial program 59.1%

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

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

   5. Simplified 98.8%

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

      1. lift-+.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. sub-neg N/A

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

      4. +-commutative N/A

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

      5. lift-/.f64 N/A

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

      6. div-inv N/A

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

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

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

      8. lower-fma.f64 N/A

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

      9. lower-neg.f64 N/A

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

      10. lower-/.f64 99.4

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

   7. Applied egg-rr 99.4%

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

Alternative 8: 96.9% accurate, 13.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;wj \leq 0.0017:\\ \;\;\;\;\mathsf{fma}\left(wj, \mathsf{fma}\left(x, -2, wj\right), x\right)\\ \mathbf{else}:\\ \;\;\;\;wj + \frac{wj}{-1 - wj}\\ \end{array} \end{array} \]

FPCore

(FPCore (wj x)
 :precision binary64
 (if (<= wj 0.0017) (fma wj (fma x -2.0 wj) x) (+ wj (/ wj (- -1.0 wj)))))

C

double code(double wj, double x) {
	double tmp;
	if (wj <= 0.0017) {
		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 <= 0.0017)
		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, 0.0017], 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 < 0.00169999999999999991

   1. Initial program 82.2%

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

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

   7. Applied egg-rr 99.2%

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

   8. Taylor expanded in x around 0

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

      1. lower--.f64 98.8

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

   10. Simplified 98.8%

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

   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. lower-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. lower-fma.f64 98.4

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

   13. Simplified 98.4%

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

   if 0.00169999999999999991 < wj

   1. Initial program 59.1%

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

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

   5. Simplified 98.8%

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

4. Final simplification 98.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;wj \leq 0.0017:\\ \;\;\;\;\mathsf{fma}\left(wj, \mathsf{fma}\left(x, -2, wj\right), x\right)\\ \mathbf{else}:\\ \;\;\;\;wj + \frac{wj}{-1 - wj}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 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 81.7%

   \[wj - \frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}} \]
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 97.5%

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

7. Applied egg-rr 97.5%

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

8. Taylor expanded in x around 0

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

   1. lower--.f64 97.0

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

10. Simplified 97.0%

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

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. lower-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. lower-fma.f64 96.8

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

13. Simplified 96.8%

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

Alternative 10: 85.0% accurate, 27.6× speedup

Math

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

FPCore

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

C

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

Julia

function code(wj, x)
	return fma(x, Float64(wj * -2.0), x)
end

Wolfram

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

Derivation

1. Initial program 81.7%

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

3. Taylor expanded in wj around 0

   \[\leadsto \color{blue}{x + -2 \cdot \left(wj \cdot x\right)} \]
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 86.8

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

5. Simplified 86.8%

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

Alternative 11: 13.6% accurate, 55.2× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 81.7%

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

3. Taylor expanded in wj around 0

   \[\leadsto \color{blue}{x + wj \cdot \left(wj \cdot \left(1 - \left(-4 \cdot x + \frac{3}{2} \cdot x\right)\right) - 2 \cdot x\right)} \]
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. Simplified 97.0%

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

6. Taylor expanded in x around 0

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

   1. unpow2 N/A

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

   2. lower-*.f64 13.1

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

8. Simplified 13.1%

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

Alternative 12: 4.1% accurate, 82.8× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 81.7%

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

3. Taylor expanded in wj around inf

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

5. Simplified 4.0%

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

6. Final simplification 4.0%

   \[\leadsto wj + -1 \]
7. Add Preprocessing

Alternative 13: 84.4% 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 81.7%

   \[wj - \frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}} \]
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 97.5%

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

6. Taylor expanded in x around -inf

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

   1. associate-*r* N/A

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

   2. *-commutative N/A

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

   3. lower-*.f64 N/A

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

8. Simplified 97.4%

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

9. Taylor expanded in wj around 0

   \[\leadsto \color{blue}{-1} \cdot \left(\mathsf{neg}\left(x\right)\right) \]
10. Step-by-step derivation

11. Simplified 86.4%

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

12. Final simplification 86.4%

    \[\leadsto x \]
13. Add Preprocessing

Alternative 14: 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 81.7%

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

3. Taylor expanded in wj around inf

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

5. Simplified 4.0%

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

6. Taylor expanded in wj around 0

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

8. Simplified 3.2%

   \[\leadsto \color{blue}{-1} \]
9. 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 2024208 
(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))))))