Jmat.Real.lambertw, newton loop step

Percentage Accurate: 77.3% → 99.5%

Time: 15.0s

Alternatives: 8

Speedup: 313.0×

• could not determine a ground truth

  1. wj = -2.0473821787974078e+147
  2. x = -5.850721659619802e+96

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: 77.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.5% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{-wj}\\ t_1 := wj \cdot e^{wj}\\ \mathbf{if}\;wj + \frac{x - t\_1}{e^{wj} + t\_1} \leq 0.0001:\\ \;\;\;\;x \cdot \left(\frac{t\_0}{wj + 1} + {wj}^{2} \cdot \left(\frac{1}{x} + wj \cdot \left(wj \cdot \left(\frac{1}{x} - \frac{wj}{x}\right) + \frac{-1}{x}\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;wj + x \cdot \left(\frac{wj}{x \cdot \left(-1 - wj\right)} - \frac{t\_0}{-1 - wj}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (wj x)
 :precision binary64
 (let* ((t_0 (exp (- wj))) (t_1 (* wj (exp wj))))
   (if (<= (+ wj (/ (- x t_1) (+ (exp wj) t_1))) 0.0001)
     (*
      x
      (+
       (/ t_0 (+ wj 1.0))
       (*
        (pow wj 2.0)
        (+ (/ 1.0 x) (* wj (+ (* wj (- (/ 1.0 x) (/ wj x))) (/ -1.0 x)))))))
     (+ wj (* x (- (/ wj (* x (- -1.0 wj))) (/ t_0 (- -1.0 wj))))))))

C

double code(double wj, double x) {
	double t_0 = exp(-wj);
	double t_1 = wj * exp(wj);
	double tmp;
	if ((wj + ((x - t_1) / (exp(wj) + t_1))) <= 0.0001) {
		tmp = x * ((t_0 / (wj + 1.0)) + (pow(wj, 2.0) * ((1.0 / x) + (wj * ((wj * ((1.0 / x) - (wj / x))) + (-1.0 / x))))));
	} else {
		tmp = wj + (x * ((wj / (x * (-1.0 - wj))) - (t_0 / (-1.0 - wj))));
	}
	return tmp;
}

Fortran

real(8) function code(wj, x)
    real(8), intent (in) :: wj
    real(8), intent (in) :: x
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = exp(-wj)
    t_1 = wj * exp(wj)
    if ((wj + ((x - t_1) / (exp(wj) + t_1))) <= 0.0001d0) then
        tmp = x * ((t_0 / (wj + 1.0d0)) + ((wj ** 2.0d0) * ((1.0d0 / x) + (wj * ((wj * ((1.0d0 / x) - (wj / x))) + ((-1.0d0) / x))))))
    else
        tmp = wj + (x * ((wj / (x * ((-1.0d0) - wj))) - (t_0 / ((-1.0d0) - wj))))
    end if
    code = tmp
end function

Java

public static double code(double wj, double x) {
	double t_0 = Math.exp(-wj);
	double t_1 = wj * Math.exp(wj);
	double tmp;
	if ((wj + ((x - t_1) / (Math.exp(wj) + t_1))) <= 0.0001) {
		tmp = x * ((t_0 / (wj + 1.0)) + (Math.pow(wj, 2.0) * ((1.0 / x) + (wj * ((wj * ((1.0 / x) - (wj / x))) + (-1.0 / x))))));
	} else {
		tmp = wj + (x * ((wj / (x * (-1.0 - wj))) - (t_0 / (-1.0 - wj))));
	}
	return tmp;
}

Python

def code(wj, x):
	t_0 = math.exp(-wj)
	t_1 = wj * math.exp(wj)
	tmp = 0
	if (wj + ((x - t_1) / (math.exp(wj) + t_1))) <= 0.0001:
		tmp = x * ((t_0 / (wj + 1.0)) + (math.pow(wj, 2.0) * ((1.0 / x) + (wj * ((wj * ((1.0 / x) - (wj / x))) + (-1.0 / x))))))
	else:
		tmp = wj + (x * ((wj / (x * (-1.0 - wj))) - (t_0 / (-1.0 - wj))))
	return tmp

Julia

function code(wj, x)
	t_0 = exp(Float64(-wj))
	t_1 = Float64(wj * exp(wj))
	tmp = 0.0
	if (Float64(wj + Float64(Float64(x - t_1) / Float64(exp(wj) + t_1))) <= 0.0001)
		tmp = Float64(x * Float64(Float64(t_0 / Float64(wj + 1.0)) + Float64((wj ^ 2.0) * Float64(Float64(1.0 / x) + Float64(wj * Float64(Float64(wj * Float64(Float64(1.0 / x) - Float64(wj / x))) + Float64(-1.0 / x)))))));
	else
		tmp = Float64(wj + Float64(x * Float64(Float64(wj / Float64(x * Float64(-1.0 - wj))) - Float64(t_0 / Float64(-1.0 - wj)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(wj, x)
	t_0 = exp(-wj);
	t_1 = wj * exp(wj);
	tmp = 0.0;
	if ((wj + ((x - t_1) / (exp(wj) + t_1))) <= 0.0001)
		tmp = x * ((t_0 / (wj + 1.0)) + ((wj ^ 2.0) * ((1.0 / x) + (wj * ((wj * ((1.0 / x) - (wj / x))) + (-1.0 / x))))));
	else
		tmp = wj + (x * ((wj / (x * (-1.0 - wj))) - (t_0 / (-1.0 - wj))));
	end
	tmp_2 = tmp;
end

Wolfram

code[wj_, x_] := Block[{t$95$0 = N[Exp[(-wj)], $MachinePrecision]}, Block[{t$95$1 = N[(wj * N[Exp[wj], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(wj + N[(N[(x - t$95$1), $MachinePrecision] / N[(N[Exp[wj], $MachinePrecision] + t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 0.0001], N[(x * N[(N[(t$95$0 / N[(wj + 1.0), $MachinePrecision]), $MachinePrecision] + N[(N[Power[wj, 2.0], $MachinePrecision] * N[(N[(1.0 / x), $MachinePrecision] + N[(wj * N[(N[(wj * N[(N[(1.0 / x), $MachinePrecision] - N[(wj / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(-1.0 / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(wj + N[(x * N[(N[(wj / N[(x * N[(-1.0 - wj), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(t$95$0 / 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))))) < 1.00000000000000005e-4

   1. Initial program 72.7%

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

      1. distribute-rgt1-in 72.7%

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

      2. associate-/l/ 72.7%

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

      3. div-sub 72.7%

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

      4. associate-/l* 72.7%

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

      5. *-inverses 72.7%

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

      6. *-rgt-identity 72.7%

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

   3. Simplified 72.7%

      \[\leadsto \color{blue}{wj - \frac{wj - \frac{x}{e^{wj}}}{wj + 1}} \]
   4. Add Preprocessing

   5. Taylor expanded in x around inf 73.6%

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

      1. associate--l+ 85.7%

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

      2. associate-/r* 85.7%

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

      3. exp-neg 85.7%

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

      4. +-commutative 85.7%

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

      5. +-commutative 85.7%

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

   7. Simplified 85.7%

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

   8. Taylor expanded in wj around 0 99.9%

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

   if 1.00000000000000005e-4 < (-.f64 wj (/.f64 (-.f64 (*.f64 wj (exp.f64 wj)) x) (+.f64 (exp.f64 wj) (*.f64 wj (exp.f64 wj)))))

   1. Initial program 94.9%

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

      1. distribute-rgt1-in 94.9%

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

      2. associate-/l/ 94.9%

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

      3. div-sub 94.9%

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

      4. associate-/l* 94.9%

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

      5. *-inverses 99.8%

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

      6. *-rgt-identity 99.8%

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

   3. Simplified 99.8%

      \[\leadsto \color{blue}{wj - \frac{wj - \frac{x}{e^{wj}}}{wj + 1}} \]
   4. Add Preprocessing

   5. Taylor expanded in x around inf 99.8%

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

      1. +-commutative 99.8%

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

      2. associate-/r* 99.9%

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

      3. exp-neg 99.8%

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

      4. +-commutative 99.8%

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

   7. Simplified 99.8%

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

4. Final simplification 99.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;wj + \frac{x - wj \cdot e^{wj}}{e^{wj} + wj \cdot e^{wj}} \leq 0.0001:\\ \;\;\;\;x \cdot \left(\frac{e^{-wj}}{wj + 1} + {wj}^{2} \cdot \left(\frac{1}{x} + wj \cdot \left(wj \cdot \left(\frac{1}{x} - \frac{wj}{x}\right) + \frac{-1}{x}\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;wj + x \cdot \left(\frac{wj}{x \cdot \left(-1 - wj\right)} - \frac{e^{-wj}}{-1 - wj}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 96.5% accurate, 2.5× speedup

Math

\[\begin{array}{l} \\ x \cdot \left(\left(1 + wj \cdot -2\right) + \frac{{wj}^{2} \cdot \left(1 + wj \cdot \left(-1 - wj \cdot \left(wj + -1\right)\right)\right)}{x}\right) \end{array} \]

FPCore

(FPCore (wj x)
 :precision binary64
 (*
  x
  (+
   (+ 1.0 (* wj -2.0))
   (/ (* (pow wj 2.0) (+ 1.0 (* wj (- -1.0 (* wj (+ wj -1.0)))))) x))))

C

double code(double wj, double x) {
	return x * ((1.0 + (wj * -2.0)) + ((pow(wj, 2.0) * (1.0 + (wj * (-1.0 - (wj * (wj + -1.0)))))) / x));
}

Fortran

real(8) function code(wj, x)
    real(8), intent (in) :: wj
    real(8), intent (in) :: x
    code = x * ((1.0d0 + (wj * (-2.0d0))) + (((wj ** 2.0d0) * (1.0d0 + (wj * ((-1.0d0) - (wj * (wj + (-1.0d0))))))) / x))
end function

Java

public static double code(double wj, double x) {
	return x * ((1.0 + (wj * -2.0)) + ((Math.pow(wj, 2.0) * (1.0 + (wj * (-1.0 - (wj * (wj + -1.0)))))) / x));
}

Python

def code(wj, x):
	return x * ((1.0 + (wj * -2.0)) + ((math.pow(wj, 2.0) * (1.0 + (wj * (-1.0 - (wj * (wj + -1.0)))))) / x))

Julia

function code(wj, x)
	return Float64(x * Float64(Float64(1.0 + Float64(wj * -2.0)) + Float64(Float64((wj ^ 2.0) * Float64(1.0 + Float64(wj * Float64(-1.0 - Float64(wj * Float64(wj + -1.0)))))) / x)))
end

MATLAB

function tmp = code(wj, x)
	tmp = x * ((1.0 + (wj * -2.0)) + (((wj ^ 2.0) * (1.0 + (wj * (-1.0 - (wj * (wj + -1.0)))))) / x));
end

Wolfram

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

Derivation

1. Initial program 78.1%

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

   1. distribute-rgt1-in 78.1%

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

   2. associate-/l/ 78.1%

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

   3. div-sub 78.1%

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

   4. associate-/l* 78.1%

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

   5. *-inverses 79.3%

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

   6. *-rgt-identity 79.3%

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

3. Simplified 79.3%

   \[\leadsto \color{blue}{wj - \frac{wj - \frac{x}{e^{wj}}}{wj + 1}} \]
4. Add Preprocessing

5. Taylor expanded in x around inf 80.0%

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

   1. associate--l+ 89.1%

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

   2. associate-/r* 89.1%

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

   3. exp-neg 89.1%

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

   4. +-commutative 89.1%

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

   5. +-commutative 89.1%

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

7. Simplified 89.1%

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

8. Taylor expanded in wj around 0 98.3%

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

9. Taylor expanded in wj around 0 97.3%

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

    1. *-commutative 97.3%

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

11. Simplified 97.3%

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

12. Taylor expanded in x around 0 97.3%

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

13. Final simplification 97.3%

    \[\leadsto x \cdot \left(\left(1 + wj \cdot -2\right) + \frac{{wj}^{2} \cdot \left(1 + wj \cdot \left(-1 - wj \cdot \left(wj + -1\right)\right)\right)}{x}\right) \]
14. Add Preprocessing

Alternative 3: 96.3% accurate, 7.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x \cdot -4 + x \cdot 1.5\\ x + wj \cdot \left(wj \cdot \left(\left(1 + wj \cdot \left(-1 - \left(x \cdot -3 + \left(-2 \cdot t\_0 + x \cdot 0.6666666666666666\right)\right)\right)\right) - t\_0\right) - x \cdot 2\right) \end{array} \end{array} \]

FPCore

(FPCore (wj x)
 :precision binary64
 (let* ((t_0 (+ (* x -4.0) (* x 1.5))))
   (+
    x
    (*
     wj
     (-
      (*
       wj
       (-
        (+
         1.0
         (*
          wj
          (- -1.0 (+ (* x -3.0) (+ (* -2.0 t_0) (* x 0.6666666666666666))))))
        t_0))
      (* x 2.0))))))

C

double code(double wj, double x) {
	double t_0 = (x * -4.0) + (x * 1.5);
	return x + (wj * ((wj * ((1.0 + (wj * (-1.0 - ((x * -3.0) + ((-2.0 * t_0) + (x * 0.6666666666666666)))))) - t_0)) - (x * 2.0)));
}

Fortran

real(8) function code(wj, x)
    real(8), intent (in) :: wj
    real(8), intent (in) :: x
    real(8) :: t_0
    t_0 = (x * (-4.0d0)) + (x * 1.5d0)
    code = x + (wj * ((wj * ((1.0d0 + (wj * ((-1.0d0) - ((x * (-3.0d0)) + (((-2.0d0) * t_0) + (x * 0.6666666666666666d0)))))) - t_0)) - (x * 2.0d0)))
end function

Java

public static double code(double wj, double x) {
	double t_0 = (x * -4.0) + (x * 1.5);
	return x + (wj * ((wj * ((1.0 + (wj * (-1.0 - ((x * -3.0) + ((-2.0 * t_0) + (x * 0.6666666666666666)))))) - t_0)) - (x * 2.0)));
}

Python

def code(wj, x):
	t_0 = (x * -4.0) + (x * 1.5)
	return x + (wj * ((wj * ((1.0 + (wj * (-1.0 - ((x * -3.0) + ((-2.0 * t_0) + (x * 0.6666666666666666)))))) - t_0)) - (x * 2.0)))

Julia

function code(wj, x)
	t_0 = Float64(Float64(x * -4.0) + Float64(x * 1.5))
	return Float64(x + Float64(wj * Float64(Float64(wj * Float64(Float64(1.0 + Float64(wj * Float64(-1.0 - Float64(Float64(x * -3.0) + Float64(Float64(-2.0 * t_0) + Float64(x * 0.6666666666666666)))))) - t_0)) - Float64(x * 2.0))))
end

MATLAB

function tmp = code(wj, x)
	t_0 = (x * -4.0) + (x * 1.5);
	tmp = x + (wj * ((wj * ((1.0 + (wj * (-1.0 - ((x * -3.0) + ((-2.0 * t_0) + (x * 0.6666666666666666)))))) - t_0)) - (x * 2.0)));
end

Wolfram

code[wj_, x_] := Block[{t$95$0 = N[(N[(x * -4.0), $MachinePrecision] + N[(x * 1.5), $MachinePrecision]), $MachinePrecision]}, N[(x + N[(wj * N[(N[(wj * N[(N[(1.0 + N[(wj * N[(-1.0 - N[(N[(x * -3.0), $MachinePrecision] + N[(N[(-2.0 * t$95$0), $MachinePrecision] + N[(x * 0.6666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision]), $MachinePrecision] - N[(x * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Initial program 78.1%

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

   1. distribute-rgt1-in 78.1%

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

   2. associate-/l/ 78.1%

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

   3. div-sub 78.1%

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

   4. associate-/l* 78.1%

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

   5. *-inverses 79.3%

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

   6. *-rgt-identity 79.3%

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

3. Simplified 79.3%

   \[\leadsto \color{blue}{wj - \frac{wj - \frac{x}{e^{wj}}}{wj + 1}} \]
4. Add Preprocessing

5. Taylor expanded in wj around 0 97.2%

   \[\leadsto \color{blue}{x + wj \cdot \left(wj \cdot \left(\left(1 + -1 \cdot \left(wj \cdot \left(1 + \left(-3 \cdot x + \left(-2 \cdot \left(-4 \cdot x + 1.5 \cdot x\right) + 0.6666666666666666 \cdot x\right)\right)\right)\right)\right) - \left(-4 \cdot x + 1.5 \cdot x\right)\right) - 2 \cdot x\right)} \]

6. Final simplification 97.2%

   \[\leadsto x + wj \cdot \left(wj \cdot \left(\left(1 + wj \cdot \left(-1 - \left(x \cdot -3 + \left(-2 \cdot \left(x \cdot -4 + x \cdot 1.5\right) + x \cdot 0.6666666666666666\right)\right)\right)\right) - \left(x \cdot -4 + x \cdot 1.5\right)\right) - x \cdot 2\right) \]
7. Add Preprocessing

Alternative 4: 96.2% accurate, 24.1× speedup

Math

\[\begin{array}{l} \\ x + wj \cdot \left(wj \cdot \left(1 - wj\right) - x \cdot 2\right) \end{array} \]

FPCore

(FPCore (wj x)
 :precision binary64
 (+ x (* wj (- (* wj (- 1.0 wj)) (* x 2.0)))))

C

double code(double wj, double x) {
	return x + (wj * ((wj * (1.0 - wj)) - (x * 2.0)));
}

Fortran

real(8) function code(wj, x)
    real(8), intent (in) :: wj
    real(8), intent (in) :: x
    code = x + (wj * ((wj * (1.0d0 - wj)) - (x * 2.0d0)))
end function

Java

public static double code(double wj, double x) {
	return x + (wj * ((wj * (1.0 - wj)) - (x * 2.0)));
}

Python

def code(wj, x):
	return x + (wj * ((wj * (1.0 - wj)) - (x * 2.0)))

Julia

function code(wj, x)
	return Float64(x + Float64(wj * Float64(Float64(wj * Float64(1.0 - wj)) - Float64(x * 2.0))))
end

MATLAB

function tmp = code(wj, x)
	tmp = x + (wj * ((wj * (1.0 - wj)) - (x * 2.0)));
end

Wolfram

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

Derivation

1. Initial program 78.1%

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

   1. distribute-rgt1-in 78.1%

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

   2. associate-/l/ 78.1%

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

   3. div-sub 78.1%

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

   4. associate-/l* 78.1%

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

   5. *-inverses 79.3%

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

   6. *-rgt-identity 79.3%

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

3. Simplified 79.3%

   \[\leadsto \color{blue}{wj - \frac{wj - \frac{x}{e^{wj}}}{wj + 1}} \]
4. Add Preprocessing

5. Taylor expanded in wj around 0 97.2%

   \[\leadsto \color{blue}{x + wj \cdot \left(wj \cdot \left(\left(1 + -1 \cdot \left(wj \cdot \left(1 + \left(-3 \cdot x + \left(-2 \cdot \left(-4 \cdot x + 1.5 \cdot x\right) + 0.6666666666666666 \cdot x\right)\right)\right)\right)\right) - \left(-4 \cdot x + 1.5 \cdot x\right)\right) - 2 \cdot x\right)} \]

6. Taylor expanded in x around 0 97.0%

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

   1. neg-mul-1 97.0%

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

   2. unsub-neg 97.0%

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

8. Simplified 97.0%

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

9. Final simplification 97.0%

   \[\leadsto x + wj \cdot \left(wj \cdot \left(1 - wj\right) - x \cdot 2\right) \]
10. Add Preprocessing

Alternative 5: 95.7% accurate, 34.8× speedup

Math

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

FPCore

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

C

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

Fortran

real(8) function code(wj, x)
    real(8), intent (in) :: wj
    real(8), intent (in) :: x
    code = x + (wj * (wj - (x * 2.0d0)))
end function

Java

public static double code(double wj, double x) {
	return x + (wj * (wj - (x * 2.0)));
}

Python

def code(wj, x):
	return x + (wj * (wj - (x * 2.0)))

Julia

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

MATLAB

function tmp = code(wj, x)
	tmp = x + (wj * (wj - (x * 2.0)));
end

Wolfram

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

Derivation

1. Initial program 78.1%

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

   1. distribute-rgt1-in 78.1%

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

   2. associate-/l/ 78.1%

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

   3. div-sub 78.1%

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

   4. associate-/l* 78.1%

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

   5. *-inverses 79.3%

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

   6. *-rgt-identity 79.3%

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

3. Simplified 79.3%

   \[\leadsto \color{blue}{wj - \frac{wj - \frac{x}{e^{wj}}}{wj + 1}} \]
4. Add Preprocessing

5. Taylor expanded in wj around 0 97.2%

   \[\leadsto \color{blue}{x + wj \cdot \left(wj \cdot \left(\left(1 + -1 \cdot \left(wj \cdot \left(1 + \left(-3 \cdot x + \left(-2 \cdot \left(-4 \cdot x + 1.5 \cdot x\right) + 0.6666666666666666 \cdot x\right)\right)\right)\right)\right) - \left(-4 \cdot x + 1.5 \cdot x\right)\right) - 2 \cdot x\right)} \]

6. Taylor expanded in x around 0 97.0%

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

   1. neg-mul-1 97.0%

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

   2. unsub-neg 97.0%

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

8. Simplified 97.0%

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

9. Taylor expanded in wj around 0 96.8%

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

10. Final simplification 96.8%

    \[\leadsto x + wj \cdot \left(wj - x \cdot 2\right) \]
11. Add Preprocessing

Alternative 6: 84.0% accurate, 44.7× speedup

Math

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

FPCore

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

C

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

Fortran

real(8) function code(wj, x)
    real(8), intent (in) :: wj
    real(8), intent (in) :: x
    code = x + ((-2.0d0) * (wj * x))
end function

Java

public static double code(double wj, double x) {
	return x + (-2.0 * (wj * x));
}

Python

def code(wj, x):
	return x + (-2.0 * (wj * x))

Julia

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

MATLAB

function tmp = code(wj, x)
	tmp = x + (-2.0 * (wj * x));
end

Wolfram

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

Derivation

1. Initial program 78.1%

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

   1. distribute-rgt1-in 78.1%

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

   2. associate-/l/ 78.1%

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

   3. div-sub 78.1%

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

   4. associate-/l* 78.1%

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

   5. *-inverses 79.3%

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

   6. *-rgt-identity 79.3%

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

3. Simplified 79.3%

   \[\leadsto \color{blue}{wj - \frac{wj - \frac{x}{e^{wj}}}{wj + 1}} \]
4. Add Preprocessing

5. Taylor expanded in wj around 0 85.5%

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

   1. *-commutative 85.5%

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

7. Simplified 85.5%

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

8. Final simplification 85.5%

   \[\leadsto x + -2 \cdot \left(wj \cdot x\right) \]
9. Add Preprocessing

Alternative 7: 83.6% accurate, 313.0× 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 78.1%

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

   1. distribute-rgt1-in 78.1%

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

   2. associate-/l/ 78.1%

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

   3. div-sub 78.1%

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

   4. associate-/l* 78.1%

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

   5. *-inverses 79.3%

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

   6. *-rgt-identity 79.3%

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

3. Simplified 79.3%

   \[\leadsto \color{blue}{wj - \frac{wj - \frac{x}{e^{wj}}}{wj + 1}} \]
4. Add Preprocessing

5. Taylor expanded in wj around 0 85.0%

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

Alternative 8: 4.4% accurate, 313.0× speedup

Math

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

FPCore

(FPCore (wj x) :precision binary64 wj)

C

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

Fortran

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

Java

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

Python

def code(wj, x):
	return wj

Julia

function code(wj, x)
	return wj
end

MATLAB

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

Wolfram

code[wj_, x_] := wj

Derivation

1. Initial program 78.1%

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

   1. distribute-rgt1-in 78.1%

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

   2. associate-/l/ 78.1%

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

   3. div-sub 78.1%

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

   4. associate-/l* 78.1%

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

   5. *-inverses 79.3%

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

   6. *-rgt-identity 79.3%

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

3. Simplified 79.3%

   \[\leadsto \color{blue}{wj - \frac{wj - \frac{x}{e^{wj}}}{wj + 1}} \]
4. Add Preprocessing

5. Taylor expanded in wj around inf 4.3%

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

Developer target: 78.4% accurate, 1.5× 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 2024103 
(FPCore (wj x)
  :name "Jmat.Real.lambertw, newton loop step"
  :precision binary64

  :alt
  (- wj (- (/ wj (+ wj 1.0)) (/ x (+ (exp wj) (* wj (exp wj))))))

  (- wj (/ (- (* wj (exp wj)) x) (+ (exp wj) (* wj (exp wj))))))