Jmat.Real.lambertw, newton loop step

Percentage Accurate: 77.8% → 98.7%

Time: 12.0s

Alternatives: 12

Speedup: 36.8×

• could not determine a ground truth

  1. wj = -1.3972294521818103e+138
  2. x = -1.646004974134286e+286

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: 77.8% 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

\[\begin{array}{l} \\ \begin{array}{l} t_0 := wj \cdot e^{wj}\\ \mathbf{if}\;wj + \frac{x - t\_0}{e^{wj} + t\_0} \leq 2 \cdot 10^{-13}:\\ \;\;\;\;\mathsf{fma}\left(wj, \mathsf{fma}\left(wj, \mathsf{fma}\left(x, 2.5, 1\right) - \mathsf{fma}\left(wj, \mathsf{fma}\left(x, 0.6666666666666666, x \cdot 2\right), wj\right), x \cdot -2\right), x\right)\\ \mathbf{else}:\\ \;\;\;\;wj + x \cdot \left(\frac{e^{-wj}}{wj + 1} - \frac{wj}{\mathsf{fma}\left(x, wj, x\right)}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (wj x)
 :precision binary64
 (let* ((t_0 (* wj (exp wj))))
   (if (<= (+ wj (/ (- x t_0) (+ (exp wj) t_0))) 2e-13)
     (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-13) {
		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-13)
		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-13], 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))))) < 2.0000000000000001e-13

   1. Initial program 72.4%

      \[wj - \frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}} \]
   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 99.3%

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

   if 2.0000000000000001e-13 < (-.f64 wj (/.f64 (-.f64 (*.f64 wj (exp.f64 wj)) x) (+.f64 (exp.f64 wj) (*.f64 wj (exp.f64 wj)))))

   1. Initial program 94.7%

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

   3. Taylor expanded in 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 99.6%

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

4. Final simplification 99.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;wj + \frac{x - wj \cdot e^{wj}}{e^{wj} + wj \cdot e^{wj}} \leq 2 \cdot 10^{-13}:\\ \;\;\;\;\mathsf{fma}\left(wj, \mathsf{fma}\left(wj, \mathsf{fma}\left(x, 2.5, 1\right) - \mathsf{fma}\left(wj, \mathsf{fma}\left(x, 0.6666666666666666, x \cdot 2\right), wj\right), x \cdot -2\right), x\right)\\ \mathbf{else}:\\ \;\;\;\;wj + x \cdot \left(\frac{e^{-wj}}{wj + 1} - \frac{wj}{\mathsf{fma}\left(x, wj, x\right)}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 81.1% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := wj \cdot e^{wj}\\ t_1 := wj + \frac{x - t\_0}{e^{wj} + t\_0}\\ \mathbf{if}\;t\_1 \leq -5 \cdot 10^{-284}:\\ \;\;\;\;wj + x\\ \mathbf{elif}\;t\_1 \leq 0:\\ \;\;\;\;wj \cdot wj\\ \mathbf{else}:\\ \;\;\;\;wj + x\\ \end{array} \end{array} \]

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 -5e-284) (+ 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 <= -5e-284) {
		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 <= (-5d-284)) 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 <= -5e-284) {
		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 <= -5e-284:
		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 <= -5e-284)
		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 <= -5e-284)
		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, -5e-284], 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))))) < -4.99999999999999973e-284 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 96.1%

      \[wj - \frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}} \]
   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 90.7

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

   5. Simplified 90.7%

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

   if -4.99999999999999973e-284 < (-.f64 wj (/.f64 (-.f64 (*.f64 wj (exp.f64 wj)) x) (+.f64 (exp.f64 wj) (*.f64 wj (exp.f64 wj))))) < 0.0

   1. Initial program 5.7%

      \[wj - \frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}} \]
   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 5.7

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

   5. Simplified 5.7%

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

   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. lower-*.f64 43.6

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

   8. Simplified 43.6%

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

4. Final simplification 82.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;wj + \frac{x - wj \cdot e^{wj}}{e^{wj} + wj \cdot e^{wj}} \leq -5 \cdot 10^{-284}:\\ \;\;\;\;wj + x\\ \mathbf{elif}\;wj + \frac{x - wj \cdot e^{wj}}{e^{wj} + wj \cdot e^{wj}} \leq 0:\\ \;\;\;\;wj \cdot wj\\ \mathbf{else}:\\ \;\;\;\;wj + x\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 98.7% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := wj \cdot e^{wj}\\ \mathbf{if}\;wj + \frac{x - t\_0}{e^{wj} + t\_0} \leq 2 \cdot 10^{-13}:\\ \;\;\;\;\mathsf{fma}\left(wj, \mathsf{fma}\left(wj, \mathsf{fma}\left(x, 2.5, 1\right) - \mathsf{fma}\left(wj, \mathsf{fma}\left(x, 0.6666666666666666, x \cdot 2\right), wj\right), x \cdot -2\right), x\right)\\ \mathbf{else}:\\ \;\;\;\;wj - \mathsf{fma}\left(wj, \frac{1}{wj + 1}, \frac{x}{e^{wj} \cdot \left(-1 - wj\right)}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (wj x)
 :precision binary64
 (let* ((t_0 (* wj (exp wj))))
   (if (<= (+ wj (/ (- x t_0) (+ (exp wj) t_0))) 2e-13)
     (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 (fma wj (/ 1.0 (+ wj 1.0)) (/ x (* (exp wj) (- -1.0 wj))))))))

C

double code(double wj, double x) {
	double t_0 = wj * exp(wj);
	double tmp;
	if ((wj + ((x - t_0) / (exp(wj) + t_0))) <= 2e-13) {
		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 - fma(wj, (1.0 / (wj + 1.0)), (x / (exp(wj) * (-1.0 - wj))));
	}
	return tmp;
}

Julia

function code(wj, x)
	t_0 = Float64(wj * exp(wj))
	tmp = 0.0
	if (Float64(wj + Float64(Float64(x - t_0) / Float64(exp(wj) + t_0))) <= 2e-13)
		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 - fma(wj, Float64(1.0 / Float64(wj + 1.0)), Float64(x / Float64(exp(wj) * Float64(-1.0 - wj)))));
	end
	return tmp
end

Wolfram

code[wj_, x_] := Block[{t$95$0 = N[(wj * N[Exp[wj], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(wj + N[(N[(x - t$95$0), $MachinePrecision] / N[(N[Exp[wj], $MachinePrecision] + t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2e-13], 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[(wj * N[(1.0 / N[(wj + 1.0), $MachinePrecision]), $MachinePrecision] + N[(x / N[(N[Exp[wj], $MachinePrecision] * N[(-1.0 - wj), $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))))) < 2.0000000000000001e-13

   1. Initial program 72.4%

      \[wj - \frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}} \]
   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 99.3%

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

   if 2.0000000000000001e-13 < (-.f64 wj (/.f64 (-.f64 (*.f64 wj (exp.f64 wj)) x) (+.f64 (exp.f64 wj) (*.f64 wj (exp.f64 wj)))))

   1. Initial program 94.7%

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

      1. lift-exp.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-exp.f64 N/A

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

      4. lift-exp.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-+.f64 N/A

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

      7. div-sub N/A

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

      8. sub-neg N/A

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

   4. Applied egg-rr 99.6%

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

4. Final simplification 99.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;wj + \frac{x - wj \cdot e^{wj}}{e^{wj} + wj \cdot e^{wj}} \leq 2 \cdot 10^{-13}:\\ \;\;\;\;\mathsf{fma}\left(wj, \mathsf{fma}\left(wj, \mathsf{fma}\left(x, 2.5, 1\right) - \mathsf{fma}\left(wj, \mathsf{fma}\left(x, 0.6666666666666666, x \cdot 2\right), wj\right), x \cdot -2\right), x\right)\\ \mathbf{else}:\\ \;\;\;\;wj - \mathsf{fma}\left(wj, \frac{1}{wj + 1}, \frac{x}{e^{wj} \cdot \left(-1 - wj\right)}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 97.7% accurate, 2.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;wj \leq -0.0034:\\ \;\;\;\;\frac{x}{e^{wj} \cdot \left(wj + 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(wj, \mathsf{fma}\left(wj, \mathsf{fma}\left(x, 2.5, 1\right) - \mathsf{fma}\left(wj, \mathsf{fma}\left(x, 0.6666666666666666, x \cdot 2\right), wj\right), x \cdot -2\right), x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (wj x)
 :precision binary64
 (if (<= wj -0.0034)
   (/ x (* (exp wj) (+ wj 1.0)))
   (fma
    wj
    (fma
     wj
     (- (fma x 2.5 1.0) (fma wj (fma x 0.6666666666666666 (* x 2.0)) wj))
     (* x -2.0))
    x)))

C

double code(double wj, double x) {
	double tmp;
	if (wj <= -0.0034) {
		tmp = x / (exp(wj) * (wj + 1.0));
	} else {
		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);
	}
	return tmp;
}

Julia

function code(wj, x)
	tmp = 0.0
	if (wj <= -0.0034)
		tmp = Float64(x / Float64(exp(wj) * Float64(wj + 1.0)));
	else
		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);
	end
	return tmp
end

Wolfram

code[wj_, x_] := If[LessEqual[wj, -0.0034], N[(x / N[(N[Exp[wj], $MachinePrecision] * N[(wj + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 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]]

Derivation

1. Split input into 2 regimes

2. if wj < -0.00339999999999999981

   1. Initial program 66.7%

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

   3. Taylor expanded in x around inf

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

      1. lower-/.f64 N/A

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

      2. distribute-rgt1-in N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower-exp.f64 N/A

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

      7. +-commutative N/A

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

      8. lower-+.f64 100.0

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

   5. Simplified 100.0%

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

   if -0.00339999999999999981 < wj

   1. Initial program 79.9%

      \[wj - \frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}} \]
   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 98.4%

      \[\leadsto \color{blue}{\mathsf{fma}\left(wj, \mathsf{fma}\left(wj, \mathsf{fma}\left(x, 2.5, 1\right) - \mathsf{fma}\left(wj, \mathsf{fma}\left(x, 0.6666666666666666, x \cdot 2\right), wj\right), x \cdot -2\right), x\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 5: 96.5% accurate, 7.5× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(wj, \mathsf{fma}\left(wj, \mathsf{fma}\left(x, 2.5, 1\right) - \mathsf{fma}\left(wj, \mathsf{fma}\left(x, 0.6666666666666666, x \cdot 2\right), wj\right), x \cdot -2\right), x\right) \end{array} \]

FPCore

(FPCore (wj x)
 :precision binary64
 (fma
  wj
  (fma
   wj
   (- (fma x 2.5 1.0) (fma wj (fma x 0.6666666666666666 (* x 2.0)) wj))
   (* x -2.0))
  x))

C

double code(double wj, double x) {
	return fma(wj, fma(wj, (fma(x, 2.5, 1.0) - fma(wj, fma(x, 0.6666666666666666, (x * 2.0)), wj)), (x * -2.0)), x);
}

Julia

function code(wj, x)
	return 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)
end

Wolfram

code[wj_, x_] := 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]

Derivation

1. Initial program 79.5%

   \[wj - \frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}} \]
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 96.1%

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

Alternative 6: 96.0% accurate, 17.4× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(wj, \mathsf{fma}\left(x, \mathsf{fma}\left(wj, 2.5, -2\right), wj\right), x\right) \end{array} \]

FPCore

(FPCore (wj x) :precision binary64 (fma wj (fma x (fma wj 2.5 -2.0) wj) x))

C

double code(double wj, double x) {
	return fma(wj, fma(x, fma(wj, 2.5, -2.0), wj), x);
}

Julia

function code(wj, x)
	return fma(wj, fma(x, fma(wj, 2.5, -2.0), wj), x)
end

Wolfram

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

Derivation

1. Initial program 79.5%

   \[wj - \frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}} \]
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 95.7%

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

6. Taylor expanded in x around 0

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

   1. +-commutative N/A

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

   2. lower-fma.f64 N/A

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

   3. sub-neg N/A

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

   4. *-commutative N/A

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

   5. metadata-eval N/A

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

   6. lower-fma.f64 95.7

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

8. Simplified 95.7%

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

Alternative 7: 95.7% 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 79.5%

   \[wj - \frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}} \]
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 95.7%

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

6. Taylor expanded in x around 0

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

   1. +-commutative N/A

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

   2. lower-fma.f64 N/A

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

   3. sub-neg N/A

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

   4. *-commutative N/A

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

   5. metadata-eval N/A

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

   6. lower-fma.f64 95.7

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

8. Simplified 95.7%

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

9. Taylor expanded in wj around 0

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

11. Simplified 95.5%

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

Alternative 8: 84.7% 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 79.5%

   \[wj - \frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}} \]
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 87.4

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

5. Simplified 87.4%

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

Alternative 9: 84.1% accurate, 36.8× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 79.5%

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

   1. lift-exp.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lift-exp.f64 N/A

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

   4. lift-exp.f64 N/A

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

   5. lift-*.f64 N/A

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

   6. lift-+.f64 N/A

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

   7. div-sub N/A

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

   8. sub-neg N/A

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

4. Applied egg-rr 81.1%

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

5. Taylor expanded in wj around 0

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

7. Simplified 77.9%

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

8. Taylor expanded in wj around 0

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

   1. sub-neg N/A

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

   2. mul-1-neg N/A

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

   3. remove-double-neg N/A

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

   4. +-commutative N/A

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

   5. mul-1-neg N/A

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

   6. unsub-neg N/A

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

   7. lower--.f64 N/A

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

   8. lower-*.f64 86.9

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

10. Simplified 86.9%

    \[\leadsto \color{blue}{x - wj \cdot x} \]
11. Add Preprocessing

Alternative 10: 14.0% 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 79.5%

   \[wj - \frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}} \]
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 6.0

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

5. Simplified 6.0%

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

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. lower-*.f64 11.7

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

8. Simplified 11.7%

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

Alternative 11: 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 79.5%

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

3. Taylor expanded in wj around inf

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

5. Simplified 3.9%

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

6. Final simplification 3.9%

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

Alternative 12: 3.3% accurate, 331.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

def code(wj, x):
	return -1.0

Julia

function code(wj, x)
	return -1.0
end

MATLAB

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

Wolfram

code[wj_, x_] := -1.0

Derivation

1. Initial program 79.5%

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

3. Taylor expanded in wj around inf

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

5. Simplified 3.9%

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

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: 78.8% 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 2024207 
(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))))))