Jmat.Real.lambertw, newton loop step

Percentage Accurate: 77.9% → 98.6%

Time: 12.7s

Alternatives: 9

Speedup: 313.0×

• could not determine a ground truth

  1. wj = -2.056869009768565e+32
  2. x = 8.041032435427806e-260

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 77.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 98.6% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := wj \cdot e^{wj}\\ \mathbf{if}\;wj + \frac{x - t\_0}{e^{wj} + t\_0} \leq 2 \cdot 10^{-16}:\\ \;\;\;\;x \cdot \left(1 + wj \cdot \left(\left(wj \cdot \mathsf{fma}\left(wj, -2.6666666666666665, 2.5\right) + \frac{wj - {wj}^{2}}{x}\right) - 2\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(\left(\frac{wj}{x} + \frac{e^{-wj}}{wj + 1}\right) + \frac{wj}{x \cdot \left(-1 - wj\right)}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (wj x)
 :precision binary64
 (let* ((t_0 (* wj (exp wj))))
   (if (<= (+ wj (/ (- x t_0) (+ (exp wj) t_0))) 2e-16)
     (*
      x
      (+
       1.0
       (*
        wj
        (-
         (+ (* wj (fma wj -2.6666666666666665 2.5)) (/ (- wj (pow wj 2.0)) x))
         2.0))))
     (*
      x
      (+ (+ (/ wj x) (/ (exp (- wj)) (+ wj 1.0))) (/ wj (* x (- -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-16) {
		tmp = x * (1.0 + (wj * (((wj * fma(wj, -2.6666666666666665, 2.5)) + ((wj - pow(wj, 2.0)) / x)) - 2.0)));
	} else {
		tmp = x * (((wj / x) + (exp(-wj) / (wj + 1.0))) + (wj / (x * (-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-16)
		tmp = Float64(x * Float64(1.0 + Float64(wj * Float64(Float64(Float64(wj * fma(wj, -2.6666666666666665, 2.5)) + Float64(Float64(wj - (wj ^ 2.0)) / x)) - 2.0))));
	else
		tmp = Float64(x * Float64(Float64(Float64(wj / x) + Float64(exp(Float64(-wj)) / Float64(wj + 1.0))) + Float64(wj / Float64(x * 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-16], N[(x * N[(1.0 + N[(wj * N[(N[(N[(wj * N[(wj * -2.6666666666666665 + 2.5), $MachinePrecision]), $MachinePrecision] + N[(N[(wj - N[Power[wj, 2.0], $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision] - 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x * N[(N[(N[(wj / x), $MachinePrecision] + N[(N[Exp[(-wj)], $MachinePrecision] / N[(wj + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(wj / N[(x * 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))))) < 2e-16

   1. Initial program 71.7%

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

      1. distribute-rgt1-in 72.3%

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

      2. associate-/l/ 72.4%

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

      3. div-sub 71.8%

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

      4. associate-/l* 71.8%

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

      5. *-inverses 72.4%

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

      6. *-rgt-identity 72.4%

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

   3. Simplified 72.4%

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

   5. Taylor expanded in wj around 0 71.6%

      \[\leadsto wj - \frac{wj - \color{blue}{\left(x + wj \cdot \left(wj \cdot \left(-1 \cdot \left(wj \cdot \left(-1 \cdot \left(-1 \cdot x + 0.5 \cdot x\right) + \left(-0.5 \cdot x + 0.16666666666666666 \cdot x\right)\right)\right) - \left(-1 \cdot x + 0.5 \cdot x\right)\right) - x\right)\right)}}{wj + 1} \]

   6. Taylor expanded in x around inf 72.2%

      \[\leadsto \color{blue}{x \cdot \left(\left(\frac{1}{1 + wj} + \left(\frac{wj}{x} + \frac{wj \cdot \left(wj \cdot \left(0.5 + -0.16666666666666666 \cdot wj\right) - 1\right)}{1 + wj}\right)\right) - \frac{wj}{x \cdot \left(1 + wj\right)}\right)} \]

   7. Taylor expanded in wj around 0 99.0%

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

   8. Taylor expanded in x around -inf 99.0%

      \[\leadsto x \cdot \left(1 + wj \cdot \left(\color{blue}{\left(-1 \cdot \frac{wj \cdot \left(wj - 1\right)}{x} + wj \cdot \left(2.5 + -2.6666666666666665 \cdot wj\right)\right)} - 2\right)\right) \]
   9. Step-by-step derivation

      1. +-commutative 99.0%

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

      2. mul-1-neg 99.0%

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

      3. unsub-neg 99.0%

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

      4. +-commutative 99.0%

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

      5. *-commutative 99.0%

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

      6. fma-define 99.0%

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

      7. sub-neg 99.0%

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

      8. metadata-eval 99.0%

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

      9. distribute-rgt-in 99.1%

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

      10. unpow2 99.1%

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

      11. neg-mul-1 99.1%

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

      12. sub-neg 99.1%

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

   10. Simplified 99.1%

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

   if 2e-16 < (-.f64 wj (/.f64 (-.f64 (*.f64 wj (exp.f64 wj)) x) (+.f64 (exp.f64 wj) (*.f64 wj (exp.f64 wj)))))

   1. Initial program 96.8%

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

      1. distribute-rgt1-in 99.4%

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

      2. associate-/l/ 99.4%

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

      3. div-sub 96.8%

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

      4. associate-/l* 96.8%

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

      5. *-inverses 99.4%

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

      6. *-rgt-identity 99.4%

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

   3. Simplified 99.4%

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

   5. Taylor expanded in x around inf 99.4%

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

      1. +-commutative 99.4%

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

      2. associate-/r* 99.4%

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

      3. rec-exp 99.4%

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

      4. +-commutative 99.4%

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

      5. +-commutative 99.4%

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

   7. Simplified 99.4%

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

4. Final simplification 99.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;wj + \frac{x - wj \cdot e^{wj}}{e^{wj} + wj \cdot e^{wj}} \leq 2 \cdot 10^{-16}:\\ \;\;\;\;x \cdot \left(1 + wj \cdot \left(\left(wj \cdot \mathsf{fma}\left(wj, -2.6666666666666665, 2.5\right) + \frac{wj - {wj}^{2}}{x}\right) - 2\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(\left(\frac{wj}{x} + \frac{e^{-wj}}{wj + 1}\right) + \frac{wj}{x \cdot \left(-1 - wj\right)}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 97.9% accurate, 2.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;wj \leq -2.9 \cdot 10^{-6}:\\ \;\;\;\;wj - \frac{wj - \frac{x}{e^{wj}}}{wj + 1}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(1 + wj \cdot \left(wj \cdot \left(2.5 + \left(\frac{1}{x} + wj \cdot \left(\frac{-1}{x} - 2.6666666666666665\right)\right)\right) - 2\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (wj x)
 :precision binary64
 (if (<= wj -2.9e-6)
   (- wj (/ (- wj (/ x (exp wj))) (+ wj 1.0)))
   (*
    x
    (+
     1.0
     (*
      wj
      (-
       (* wj (+ 2.5 (+ (/ 1.0 x) (* wj (- (/ -1.0 x) 2.6666666666666665)))))
       2.0))))))

C

double code(double wj, double x) {
	double tmp;
	if (wj <= -2.9e-6) {
		tmp = wj - ((wj - (x / exp(wj))) / (wj + 1.0));
	} else {
		tmp = x * (1.0 + (wj * ((wj * (2.5 + ((1.0 / x) + (wj * ((-1.0 / x) - 2.6666666666666665))))) - 2.0)));
	}
	return tmp;
}

Fortran

real(8) function code(wj, x)
    real(8), intent (in) :: wj
    real(8), intent (in) :: x
    real(8) :: tmp
    if (wj <= (-2.9d-6)) then
        tmp = wj - ((wj - (x / exp(wj))) / (wj + 1.0d0))
    else
        tmp = x * (1.0d0 + (wj * ((wj * (2.5d0 + ((1.0d0 / x) + (wj * (((-1.0d0) / x) - 2.6666666666666665d0))))) - 2.0d0)))
    end if
    code = tmp
end function

Java

public static double code(double wj, double x) {
	double tmp;
	if (wj <= -2.9e-6) {
		tmp = wj - ((wj - (x / Math.exp(wj))) / (wj + 1.0));
	} else {
		tmp = x * (1.0 + (wj * ((wj * (2.5 + ((1.0 / x) + (wj * ((-1.0 / x) - 2.6666666666666665))))) - 2.0)));
	}
	return tmp;
}

Python

def code(wj, x):
	tmp = 0
	if wj <= -2.9e-6:
		tmp = wj - ((wj - (x / math.exp(wj))) / (wj + 1.0))
	else:
		tmp = x * (1.0 + (wj * ((wj * (2.5 + ((1.0 / x) + (wj * ((-1.0 / x) - 2.6666666666666665))))) - 2.0)))
	return tmp

Julia

function code(wj, x)
	tmp = 0.0
	if (wj <= -2.9e-6)
		tmp = Float64(wj - Float64(Float64(wj - Float64(x / exp(wj))) / Float64(wj + 1.0)));
	else
		tmp = Float64(x * Float64(1.0 + Float64(wj * Float64(Float64(wj * Float64(2.5 + Float64(Float64(1.0 / x) + Float64(wj * Float64(Float64(-1.0 / x) - 2.6666666666666665))))) - 2.0))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(wj, x)
	tmp = 0.0;
	if (wj <= -2.9e-6)
		tmp = wj - ((wj - (x / exp(wj))) / (wj + 1.0));
	else
		tmp = x * (1.0 + (wj * ((wj * (2.5 + ((1.0 / x) + (wj * ((-1.0 / x) - 2.6666666666666665))))) - 2.0)));
	end
	tmp_2 = tmp;
end

Wolfram

code[wj_, x_] := If[LessEqual[wj, -2.9e-6], N[(wj - N[(N[(wj - N[(x / N[Exp[wj], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(wj + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x * N[(1.0 + N[(wj * N[(N[(wj * N[(2.5 + N[(N[(1.0 / x), $MachinePrecision] + N[(wj * N[(N[(-1.0 / x), $MachinePrecision] - 2.6666666666666665), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if wj < -2.9000000000000002e-6

   1. Initial program 46.9%

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

      1. distribute-rgt1-in 96.9%

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

      2. associate-/l/ 96.9%

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

      3. div-sub 46.9%

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

      4. associate-/l* 46.9%

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

      5. *-inverses 96.9%

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

      6. *-rgt-identity 96.9%

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

   3. Simplified 96.9%

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

   if -2.9000000000000002e-6 < wj

   1. Initial program 80.0%

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

      1. distribute-rgt1-in 80.0%

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

      2. associate-/l/ 80.1%

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

      3. div-sub 80.1%

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

      4. associate-/l* 80.1%

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

      5. *-inverses 80.1%

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

      6. *-rgt-identity 80.1%

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

   3. Simplified 80.1%

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

   5. Taylor expanded in wj around 0 80.1%

      \[\leadsto wj - \frac{wj - \color{blue}{\left(x + wj \cdot \left(wj \cdot \left(-1 \cdot \left(wj \cdot \left(-1 \cdot \left(-1 \cdot x + 0.5 \cdot x\right) + \left(-0.5 \cdot x + 0.16666666666666666 \cdot x\right)\right)\right) - \left(-1 \cdot x + 0.5 \cdot x\right)\right) - x\right)\right)}}{wj + 1} \]

   6. Taylor expanded in x around inf 80.6%

      \[\leadsto \color{blue}{x \cdot \left(\left(\frac{1}{1 + wj} + \left(\frac{wj}{x} + \frac{wj \cdot \left(wj \cdot \left(0.5 + -0.16666666666666666 \cdot wj\right) - 1\right)}{1 + wj}\right)\right) - \frac{wj}{x \cdot \left(1 + wj\right)}\right)} \]

   7. Taylor expanded in wj around 0 99.0%

      \[\leadsto x \cdot \color{blue}{\left(1 + wj \cdot \left(wj \cdot \left(2.5 + \left(-1 \cdot \left(wj \cdot \left(2.6666666666666665 + \frac{1}{x}\right)\right) + \frac{1}{x}\right)\right) - 2\right)\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 99.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;wj \leq -2.9 \cdot 10^{-6}:\\ \;\;\;\;wj - \frac{wj - \frac{x}{e^{wj}}}{wj + 1}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(1 + wj \cdot \left(wj \cdot \left(2.5 + \left(\frac{1}{x} + wj \cdot \left(\frac{-1}{x} - 2.6666666666666665\right)\right)\right) - 2\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 97.5% accurate, 2.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;wj \leq -0.002:\\ \;\;\;\;\frac{x}{e^{wj} \cdot \left(wj + 1\right)}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(1 + wj \cdot \left(wj \cdot \left(2.5 + \left(\frac{1}{x} + wj \cdot \left(\frac{-1}{x} - 2.6666666666666665\right)\right)\right) - 2\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (wj x)
 :precision binary64
 (if (<= wj -0.002)
   (/ x (* (exp wj) (+ wj 1.0)))
   (*
    x
    (+
     1.0
     (*
      wj
      (-
       (* wj (+ 2.5 (+ (/ 1.0 x) (* wj (- (/ -1.0 x) 2.6666666666666665)))))
       2.0))))))

C

double code(double wj, double x) {
	double tmp;
	if (wj <= -0.002) {
		tmp = x / (exp(wj) * (wj + 1.0));
	} else {
		tmp = x * (1.0 + (wj * ((wj * (2.5 + ((1.0 / x) + (wj * ((-1.0 / x) - 2.6666666666666665))))) - 2.0)));
	}
	return tmp;
}

Fortran

real(8) function code(wj, x)
    real(8), intent (in) :: wj
    real(8), intent (in) :: x
    real(8) :: tmp
    if (wj <= (-0.002d0)) then
        tmp = x / (exp(wj) * (wj + 1.0d0))
    else
        tmp = x * (1.0d0 + (wj * ((wj * (2.5d0 + ((1.0d0 / x) + (wj * (((-1.0d0) / x) - 2.6666666666666665d0))))) - 2.0d0)))
    end if
    code = tmp
end function

Java

public static double code(double wj, double x) {
	double tmp;
	if (wj <= -0.002) {
		tmp = x / (Math.exp(wj) * (wj + 1.0));
	} else {
		tmp = x * (1.0 + (wj * ((wj * (2.5 + ((1.0 / x) + (wj * ((-1.0 / x) - 2.6666666666666665))))) - 2.0)));
	}
	return tmp;
}

Python

def code(wj, x):
	tmp = 0
	if wj <= -0.002:
		tmp = x / (math.exp(wj) * (wj + 1.0))
	else:
		tmp = x * (1.0 + (wj * ((wj * (2.5 + ((1.0 / x) + (wj * ((-1.0 / x) - 2.6666666666666665))))) - 2.0)))
	return tmp

Julia

function code(wj, x)
	tmp = 0.0
	if (wj <= -0.002)
		tmp = Float64(x / Float64(exp(wj) * Float64(wj + 1.0)));
	else
		tmp = Float64(x * Float64(1.0 + Float64(wj * Float64(Float64(wj * Float64(2.5 + Float64(Float64(1.0 / x) + Float64(wj * Float64(Float64(-1.0 / x) - 2.6666666666666665))))) - 2.0))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(wj, x)
	tmp = 0.0;
	if (wj <= -0.002)
		tmp = x / (exp(wj) * (wj + 1.0));
	else
		tmp = x * (1.0 + (wj * ((wj * (2.5 + ((1.0 / x) + (wj * ((-1.0 / x) - 2.6666666666666665))))) - 2.0)));
	end
	tmp_2 = tmp;
end

Wolfram

code[wj_, x_] := If[LessEqual[wj, -0.002], N[(x / N[(N[Exp[wj], $MachinePrecision] * N[(wj + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x * N[(1.0 + N[(wj * N[(N[(wj * N[(2.5 + N[(N[(1.0 / x), $MachinePrecision] + N[(wj * N[(N[(-1.0 / x), $MachinePrecision] - 2.6666666666666665), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if wj < -2e-3

   1. Initial program 40.0%

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

      1. distribute-rgt1-in 100.0%

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

      2. associate-/l/ 100.0%

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

      3. div-sub 40.0%

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

      4. associate-/l* 40.0%

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

      5. *-inverses 100.0%

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

      6. *-rgt-identity 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in x around inf 100.0%

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

      1. distribute-rgt-in 40.0%

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

      2. *-lft-identity 40.0%

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

      3. distribute-rgt1-in 100.0%

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

      4. *-commutative 100.0%

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

   7. Simplified 100.0%

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

   if -2e-3 < wj

   1. Initial program 80.0%

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

      1. distribute-rgt1-in 80.0%

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

      2. associate-/l/ 80.1%

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

      3. div-sub 80.1%

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

      4. associate-/l* 80.1%

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

      5. *-inverses 80.1%

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

      6. *-rgt-identity 80.1%

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

   3. Simplified 80.1%

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

   5. Taylor expanded in wj around 0 80.1%

      \[\leadsto wj - \frac{wj - \color{blue}{\left(x + wj \cdot \left(wj \cdot \left(-1 \cdot \left(wj \cdot \left(-1 \cdot \left(-1 \cdot x + 0.5 \cdot x\right) + \left(-0.5 \cdot x + 0.16666666666666666 \cdot x\right)\right)\right) - \left(-1 \cdot x + 0.5 \cdot x\right)\right) - x\right)\right)}}{wj + 1} \]

   6. Taylor expanded in x around inf 80.6%

      \[\leadsto \color{blue}{x \cdot \left(\left(\frac{1}{1 + wj} + \left(\frac{wj}{x} + \frac{wj \cdot \left(wj \cdot \left(0.5 + -0.16666666666666666 \cdot wj\right) - 1\right)}{1 + wj}\right)\right) - \frac{wj}{x \cdot \left(1 + wj\right)}\right)} \]

   7. Taylor expanded in wj around 0 98.9%

      \[\leadsto x \cdot \color{blue}{\left(1 + wj \cdot \left(wj \cdot \left(2.5 + \left(-1 \cdot \left(wj \cdot \left(2.6666666666666665 + \frac{1}{x}\right)\right) + \frac{1}{x}\right)\right) - 2\right)\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 98.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;wj \leq -0.002:\\ \;\;\;\;\frac{x}{e^{wj} \cdot \left(wj + 1\right)}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(1 + wj \cdot \left(wj \cdot \left(2.5 + \left(\frac{1}{x} + wj \cdot \left(\frac{-1}{x} - 2.6666666666666665\right)\right)\right) - 2\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 96.0% accurate, 13.6× speedup

Math

\[\begin{array}{l} \\ x \cdot \left(1 + wj \cdot \left(wj \cdot \left(2.5 + \left(\frac{1}{x} + wj \cdot \left(\frac{-1}{x} - 2.6666666666666665\right)\right)\right) - 2\right)\right) \end{array} \]

FPCore

(FPCore (wj x)
 :precision binary64
 (*
  x
  (+
   1.0
   (*
    wj
    (-
     (* wj (+ 2.5 (+ (/ 1.0 x) (* wj (- (/ -1.0 x) 2.6666666666666665)))))
     2.0)))))

C

double code(double wj, double x) {
	return x * (1.0 + (wj * ((wj * (2.5 + ((1.0 / x) + (wj * ((-1.0 / x) - 2.6666666666666665))))) - 2.0)));
}

Fortran

real(8) function code(wj, x)
    real(8), intent (in) :: wj
    real(8), intent (in) :: x
    code = x * (1.0d0 + (wj * ((wj * (2.5d0 + ((1.0d0 / x) + (wj * (((-1.0d0) / x) - 2.6666666666666665d0))))) - 2.0d0)))
end function

Java

public static double code(double wj, double x) {
	return x * (1.0 + (wj * ((wj * (2.5 + ((1.0 / x) + (wj * ((-1.0 / x) - 2.6666666666666665))))) - 2.0)));
}

Python

def code(wj, x):
	return x * (1.0 + (wj * ((wj * (2.5 + ((1.0 / x) + (wj * ((-1.0 / x) - 2.6666666666666665))))) - 2.0)))

Julia

function code(wj, x)
	return Float64(x * Float64(1.0 + Float64(wj * Float64(Float64(wj * Float64(2.5 + Float64(Float64(1.0 / x) + Float64(wj * Float64(Float64(-1.0 / x) - 2.6666666666666665))))) - 2.0))))
end

MATLAB

function tmp = code(wj, x)
	tmp = x * (1.0 + (wj * ((wj * (2.5 + ((1.0 / x) + (wj * ((-1.0 / x) - 2.6666666666666665))))) - 2.0)));
end

Wolfram

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

Derivation

1. Initial program 79.3%

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

   1. distribute-rgt1-in 80.4%

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

   2. associate-/l/ 80.5%

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

   3. div-sub 79.3%

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

   4. associate-/l* 79.3%

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

   5. *-inverses 80.5%

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

   6. *-rgt-identity 80.5%

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

3. Simplified 80.5%

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

5. Taylor expanded in wj around 0 79.0%

   \[\leadsto wj - \frac{wj - \color{blue}{\left(x + wj \cdot \left(wj \cdot \left(-1 \cdot \left(wj \cdot \left(-1 \cdot \left(-1 \cdot x + 0.5 \cdot x\right) + \left(-0.5 \cdot x + 0.16666666666666666 \cdot x\right)\right)\right) - \left(-1 \cdot x + 0.5 \cdot x\right)\right) - x\right)\right)}}{wj + 1} \]

6. Taylor expanded in x around inf 79.5%

   \[\leadsto \color{blue}{x \cdot \left(\left(\frac{1}{1 + wj} + \left(\frac{wj}{x} + \frac{wj \cdot \left(wj \cdot \left(0.5 + -0.16666666666666666 \cdot wj\right) - 1\right)}{1 + wj}\right)\right) - \frac{wj}{x \cdot \left(1 + wj\right)}\right)} \]

7. Taylor expanded in wj around 0 97.3%

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

8. Final simplification 97.3%

   \[\leadsto x \cdot \left(1 + wj \cdot \left(wj \cdot \left(2.5 + \left(\frac{1}{x} + wj \cdot \left(\frac{-1}{x} - 2.6666666666666665\right)\right)\right) - 2\right)\right) \]
9. Add Preprocessing

Alternative 5: 95.8% accurate, 20.9× speedup

Math

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

FPCore

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

C

double code(double wj, double x) {
	return x * (1.0 - (wj * (2.0 + (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 * ((wj + (-1.0d0)) / x)))))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 79.3%

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

   1. distribute-rgt1-in 80.4%

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

   2. associate-/l/ 80.5%

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

   3. div-sub 79.3%

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

   4. associate-/l* 79.3%

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

   5. *-inverses 80.5%

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

   6. *-rgt-identity 80.5%

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

3. Simplified 80.5%

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

5. Taylor expanded in wj around 0 79.0%

   \[\leadsto wj - \frac{wj - \color{blue}{\left(x + wj \cdot \left(wj \cdot \left(-1 \cdot \left(wj \cdot \left(-1 \cdot \left(-1 \cdot x + 0.5 \cdot x\right) + \left(-0.5 \cdot x + 0.16666666666666666 \cdot x\right)\right)\right) - \left(-1 \cdot x + 0.5 \cdot x\right)\right) - x\right)\right)}}{wj + 1} \]

6. Taylor expanded in x around inf 79.5%

   \[\leadsto \color{blue}{x \cdot \left(\left(\frac{1}{1 + wj} + \left(\frac{wj}{x} + \frac{wj \cdot \left(wj \cdot \left(0.5 + -0.16666666666666666 \cdot wj\right) - 1\right)}{1 + wj}\right)\right) - \frac{wj}{x \cdot \left(1 + wj\right)}\right)} \]

7. Taylor expanded in wj around 0 97.3%

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

8. Taylor expanded in x around 0 97.2%

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

   1. associate-/l* 97.2%

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

   2. neg-mul-1 97.2%

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

   3. sub-neg 97.2%

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

10. Simplified 97.2%

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

11. Final simplification 97.2%

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

Alternative 6: 95.9% 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 79.3%

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

   1. distribute-rgt1-in 80.4%

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

   2. associate-/l/ 80.5%

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

   3. div-sub 79.3%

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

   4. associate-/l* 79.3%

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

   5. *-inverses 80.5%

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

   6. *-rgt-identity 80.5%

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

3. Simplified 80.5%

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

5. Taylor expanded in wj around 0 96.6%

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

6. Taylor expanded in x around 0 96.6%

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

   1. mul-1-neg 96.6%

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

   2. unsub-neg 96.6%

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

8. Simplified 96.6%

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

9. Final simplification 96.6%

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

Alternative 7: 95.0% accurate, 62.6× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 79.3%

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

   1. distribute-rgt1-in 80.4%

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

   2. associate-/l/ 80.5%

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

   3. div-sub 79.3%

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

   4. associate-/l* 79.3%

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

   5. *-inverses 80.5%

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

   6. *-rgt-identity 80.5%

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

3. Simplified 80.5%

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

5. Taylor expanded in wj around 0 95.7%

   \[\leadsto \color{blue}{x + wj \cdot \left(wj \cdot \left(1 - \left(-4 \cdot x + 1.5 \cdot x\right)\right) - 2 \cdot x\right)} \]
6. Step-by-step derivation

   1. cancel-sign-sub-inv 95.7%

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

   2. distribute-rgt-out 95.8%

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

   3. metadata-eval 95.8%

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

   4. metadata-eval 95.8%

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

   5. *-commutative 95.8%

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

7. Simplified 95.8%

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

8. Taylor expanded in x around 0 95.8%

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

9. Taylor expanded in wj around inf 95.8%

   \[\leadsto x + wj \cdot \color{blue}{wj} \]
10. Add Preprocessing

Alternative 8: 84.7% 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 79.3%

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

   1. distribute-rgt1-in 80.4%

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

   2. associate-/l/ 80.5%

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

   3. div-sub 79.3%

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

   4. associate-/l* 79.3%

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

   5. *-inverses 80.5%

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

   6. *-rgt-identity 80.5%

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

3. Simplified 80.5%

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

5. Taylor expanded in wj around 0 84.5%

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

Alternative 9: 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 79.3%

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

   1. distribute-rgt1-in 80.4%

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

   2. associate-/l/ 80.5%

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

   3. div-sub 79.3%

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

   4. associate-/l* 79.3%

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

   5. *-inverses 80.5%

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

   6. *-rgt-identity 80.5%

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

3. Simplified 80.5%

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

5. Taylor expanded in wj around inf 3.8%

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

Developer Target 1: 78.9% 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 2024150 
(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))))))