Jmat.Real.lambertw, newton loop step

Percentage Accurate: 78.2% → 97.2%

Time: 21.1s

Alternatives: 9

Speedup: 313.0×

• could not determine a ground truth

  1. wj = -3.513273790779189e+224
  2. x = -1.2220762302192277e-167

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 78.2% 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: 97.2% accurate, 18.4× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[wj_, x_] := N[(x + N[(wj * N[(N[(x * N[(wj / N[(x + N[(x * wj), $MachinePrecision]), $MachinePrecision]), $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 78.9%

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

   2. associate-/l/ 78.8%

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

   3. div-sub 78.0%

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

   4. associate-/l* 78.0%

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

   5. *-inverses 80.0%

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

   6. *-rgt-identity 80.0%

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

3. Simplified 80.0%

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

5. Taylor expanded in wj around 0 96.8%

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

6. Taylor expanded in x around -inf 96.8%

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

   1. mul-1-neg 96.8%

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

   2. *-commutative 96.8%

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

   3. distribute-rgt-neg-in 96.8%

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

   4. +-commutative 96.8%

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

   5. mul-1-neg 96.8%

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

   6. unsub-neg 96.8%

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

   7. fma-neg 96.8%

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

   8. metadata-eval 96.8%

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

   9. *-commutative 96.8%

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

   10. associate-/l* 96.8%

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

   11. mul-1-neg 96.8%

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

   12. sub-neg 96.8%

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

8. Simplified 96.8%

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

9. Taylor expanded in wj around 0 96.7%

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

    1. *-commutative 96.7%

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

11. Simplified 96.7%

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

12. Taylor expanded in x around 0 96.5%

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

    1. associate-*r/ 96.5%

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

    2. neg-mul-1 96.5%

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

    3. distribute-rgt-neg-in 96.5%

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

    4. associate-*l/ 96.5%

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

    5. neg-sub0 96.5%

       \[\leadsto x + wj \cdot \left(\left(\frac{wj}{x} \cdot \color{blue}{\left(0 - \left(1 - wj\right)\right)}\right) \cdot \left(-x\right) - 2 \cdot x\right) \]

    6. associate--r- 96.5%

       \[\leadsto x + wj \cdot \left(\left(\frac{wj}{x} \cdot \color{blue}{\left(\left(0 - 1\right) + wj\right)}\right) \cdot \left(-x\right) - 2 \cdot x\right) \]

    7. metadata-eval 96.5%

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

    8. +-commutative 96.5%

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

14. Simplified 96.5%

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

    1. associate-*l/ 96.5%

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

    2. div-inv 96.5%

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

    3. +-commutative 96.5%

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

    4. associate-*r* 96.5%

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

    5. div-inv 96.5%

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

    6. clear-num 96.5%

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

    7. un-div-inv 96.5%

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

    8. +-commutative 96.5%

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

16. Applied egg-rr 96.5%

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

17. Taylor expanded in wj around 0 98.1%

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

    1. +-commutative 98.1%

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

    2. mul-1-neg 98.1%

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

    3. sub-neg 98.1%

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

    4. associate-*r* 98.1%

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

    5. neg-mul-1 98.1%

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

19. Simplified 98.1%

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

20. Final simplification 98.1%

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

Alternative 2: 95.8% accurate, 24.1× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[wj_, x_] := N[(x - N[(wj * N[(N[(x * 2.0), $MachinePrecision] - N[(wj * N[(1.0 - wj), $MachinePrecision]), $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 78.9%

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

   2. associate-/l/ 78.8%

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

   3. div-sub 78.0%

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

   4. associate-/l* 78.0%

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

   5. *-inverses 80.0%

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

   6. *-rgt-identity 80.0%

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

3. Simplified 80.0%

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

5. Taylor expanded in wj around 0 96.8%

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

6. Taylor expanded in x around 0 96.5%

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

   1. mul-1-neg 96.5%

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

   2. sub-neg 96.5%

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

8. Simplified 96.5%

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

9. Final simplification 96.5%

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

Alternative 3: 95.3% accurate, 34.8× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[wj_, x_] := N[(x - N[(wj * N[(N[(x * 2.0), $MachinePrecision] - wj), $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 78.9%

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

   2. associate-/l/ 78.8%

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

   3. div-sub 78.0%

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

   4. associate-/l* 78.0%

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

   5. *-inverses 80.0%

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

   6. *-rgt-identity 80.0%

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

3. Simplified 80.0%

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

5. Taylor expanded in wj around 0 96.8%

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

6. Taylor expanded in x around 0 96.5%

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

   1. mul-1-neg 96.5%

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

   2. sub-neg 96.5%

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

8. Simplified 96.5%

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

9. Taylor expanded in wj around 0 96.0%

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

10. Final simplification 96.0%

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

Alternative 4: 94.9% 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 78.9%

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

   2. associate-/l/ 78.8%

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

   3. div-sub 78.0%

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

   4. associate-/l* 78.0%

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

   5. *-inverses 80.0%

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

   6. *-rgt-identity 80.0%

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

3. Simplified 80.0%

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

5. Taylor expanded in wj around 0 96.8%

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

6. Taylor expanded in x around 0 96.5%

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

   1. mul-1-neg 96.5%

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

   2. sub-neg 96.5%

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

8. Simplified 96.5%

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

9. Taylor expanded in wj around 0 96.0%

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

    1. sub-neg 96.0%

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

    2. distribute-lft-in 96.0%

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

    3. pow2 96.0%

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

    4. distribute-rgt-neg-in 96.0%

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

    5. add-sqr-sqrt 49.2%

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

    6. sqrt-unprod 75.5%

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

    7. sqr-neg 75.5%

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

    8. sqrt-unprod 46.4%

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

    9. add-sqr-sqrt 95.1%

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

    10. *-commutative 95.1%

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

11. Applied egg-rr 95.1%

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

    1. unpow2 95.1%

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

    2. distribute-lft-in 95.1%

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

    3. *-commutative 95.1%

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

13. Simplified 95.1%

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

14. Final simplification 95.1%

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

Alternative 5: 84.4% accurate, 34.8× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[wj_, x_] := N[(x * N[(N[(1.0 - wj), $MachinePrecision] / N[(wj + 1.0), $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 78.9%

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

   2. associate-/l/ 78.8%

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

   3. div-sub 78.0%

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

   4. associate-/l* 78.0%

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

   5. *-inverses 80.0%

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

   6. *-rgt-identity 80.0%

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

3. Simplified 80.0%

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

5. Taylor expanded in wj around 0 78.3%

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

   1. associate-*r* 78.3%

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

   2. neg-mul-1 78.3%

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

7. Simplified 78.3%

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

8. Taylor expanded in x around inf 86.8%

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

   1. mul-1-neg 86.8%

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

   2. +-commutative 86.8%

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

   3. mul-1-neg 86.8%

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

   4. associate-*r/ 86.8%

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

   5. mul-1-neg 86.8%

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

   6. +-commutative 86.8%

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

   7. distribute-neg-frac 86.8%

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

   8. sub-neg 86.8%

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

   9. +-commutative 86.8%

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

   10. div-sub 86.8%

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

10. Simplified 86.8%

    \[\leadsto \color{blue}{x \cdot \frac{1 - wj}{wj + 1}} \]
11. Add Preprocessing

Alternative 6: 84.3% accurate, 44.7× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[wj_, x_] := N[(x + N[(-2.0 * N[(x * wj), $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 78.9%

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

   2. associate-/l/ 78.8%

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

   3. div-sub 78.0%

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

   4. associate-/l* 78.0%

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

   5. *-inverses 80.0%

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

   6. *-rgt-identity 80.0%

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

3. Simplified 80.0%

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

5. Taylor expanded in wj around 0 86.7%

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

   1. *-commutative 86.7%

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

7. Simplified 86.7%

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

Alternative 7: 84.3% accurate, 44.7× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[wj_, x_] := N[(x * N[(N[(wj * -2.0), $MachinePrecision] + 1.0), $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 78.9%

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

   2. associate-/l/ 78.8%

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

   3. div-sub 78.0%

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

   4. associate-/l* 78.0%

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

   5. *-inverses 80.0%

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

   6. *-rgt-identity 80.0%

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

3. Simplified 80.0%

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

5. Taylor expanded in wj around 0 96.8%

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

6. Taylor expanded in x around 0 96.5%

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

   1. mul-1-neg 96.5%

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

   2. sub-neg 96.5%

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

8. Simplified 96.5%

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

9. Taylor expanded in x around inf 86.7%

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

    1. *-commutative 86.7%

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

11. Simplified 86.7%

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

12. Final simplification 86.7%

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

Alternative 8: 83.8% 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 78.9%

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

   2. associate-/l/ 78.8%

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

   3. div-sub 78.0%

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

   4. associate-/l* 78.0%

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

   5. *-inverses 80.0%

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

   6. *-rgt-identity 80.0%

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

3. Simplified 80.0%

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

5. Taylor expanded in wj around 0 85.9%

   \[\leadsto \color{blue}{x} \]
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 78.1%

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

   1. distribute-rgt1-in 78.9%

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

   2. associate-/l/ 78.8%

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

   3. div-sub 78.0%

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

   4. associate-/l* 78.0%

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

   5. *-inverses 80.0%

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

   6. *-rgt-identity 80.0%

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

3. Simplified 80.0%

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

5. Taylor expanded in wj around inf 4.7%

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

Developer target: 79.2% 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 2024107 
(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))))))