Jmat.Real.lambertw, newton loop step

Percentage Accurate: 77.7% → 96.3%

Time: 23.1s

Alternatives: 11

Speedup: 313.0×

• could not determine a ground truth

  1. wj = -5.6281093849884944e+194
  2. x = 9.740550226098936e+104

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: 77.7% 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: 96.3% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(wj, \mathsf{fma}\left(wj, 1 + \mathsf{fma}\left(-wj, 1 + \mathsf{fma}\left(x, -3, \mathsf{fma}\left(x, 0.6666666666666666, x \cdot 5\right)\right), x \cdot 2.5\right), x \cdot -2\right), x\right) \end{array} \]

FPCore

(FPCore (wj x)
 :precision binary64
 (fma
  wj
  (fma
   wj
   (+
    1.0
    (fma
     (- wj)
     (+ 1.0 (fma x -3.0 (fma x 0.6666666666666666 (* x 5.0))))
     (* x 2.5)))
   (* x -2.0))
  x))

C

double code(double wj, double x) {
	return fma(wj, fma(wj, (1.0 + fma(-wj, (1.0 + fma(x, -3.0, fma(x, 0.6666666666666666, (x * 5.0)))), (x * 2.5))), (x * -2.0)), x);
}

Julia

function code(wj, x)
	return fma(wj, fma(wj, Float64(1.0 + fma(Float64(-wj), Float64(1.0 + fma(x, -3.0, fma(x, 0.6666666666666666, Float64(x * 5.0)))), Float64(x * 2.5))), Float64(x * -2.0)), x)
end

Wolfram

code[wj_, x_] := N[(wj * N[(wj * N[(1.0 + N[((-wj) * N[(1.0 + N[(x * -3.0 + N[(x * 0.6666666666666666 + N[(x * 5.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(x * 2.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(x * -2.0), $MachinePrecision]), $MachinePrecision] + x), $MachinePrecision]

Derivation

1. Initial program 75.6%

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

   1. distribute-rgt1-in 76.3%

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

   2. associate-/l/ 76.3%

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

   3. div-sub 75.6%

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

   4. associate-/l* 75.6%

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

   5. *-inverses 76.7%

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

   6. *-rgt-identity 76.7%

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

3. Simplified 76.7%

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

5. Taylor expanded in wj around 0 97.3%

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

7. Simplified 97.3%

   \[\leadsto \color{blue}{\mathsf{fma}\left(wj, \mathsf{fma}\left(wj, 1 + \mathsf{fma}\left(-wj, 1 + \mathsf{fma}\left(x, -3, \mathsf{fma}\left(x, 0.6666666666666666, x \cdot 5\right)\right), x \cdot 2.5\right), x \cdot -2\right), x\right)} \]

8. Final simplification 97.3%

   \[\leadsto \mathsf{fma}\left(wj, \mathsf{fma}\left(wj, 1 + \mathsf{fma}\left(-wj, 1 + \mathsf{fma}\left(x, -3, \mathsf{fma}\left(x, 0.6666666666666666, x \cdot 5\right)\right), x \cdot 2.5\right), x \cdot -2\right), x\right) \]
9. Add Preprocessing

Alternative 2: 96.3% accurate, 10.1× speedup

Math

\[\begin{array}{l} \\ x + wj \cdot \left(x \cdot -2 + wj \cdot \left(1 + \left(x \cdot 2.5 - wj \cdot \left(1 + \left(x \cdot -3 + \left(x \cdot 0.6666666666666666 + x \cdot 5\right)\right)\right)\right)\right)\right) \end{array} \]

FPCore

(FPCore (wj x)
 :precision binary64
 (+
  x
  (*
   wj
   (+
    (* x -2.0)
    (*
     wj
     (+
      1.0
      (-
       (* x 2.5)
       (*
        wj
        (+ 1.0 (+ (* x -3.0) (+ (* x 0.6666666666666666) (* x 5.0))))))))))))

C

double code(double wj, double x) {
	return x + (wj * ((x * -2.0) + (wj * (1.0 + ((x * 2.5) - (wj * (1.0 + ((x * -3.0) + ((x * 0.6666666666666666) + (x * 5.0))))))))));
}

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 + ((x * 2.5d0) - (wj * (1.0d0 + ((x * (-3.0d0)) + ((x * 0.6666666666666666d0) + (x * 5.0d0))))))))))
end function

Java

public static double code(double wj, double x) {
	return x + (wj * ((x * -2.0) + (wj * (1.0 + ((x * 2.5) - (wj * (1.0 + ((x * -3.0) + ((x * 0.6666666666666666) + (x * 5.0))))))))));
}

Python

def code(wj, x):
	return x + (wj * ((x * -2.0) + (wj * (1.0 + ((x * 2.5) - (wj * (1.0 + ((x * -3.0) + ((x * 0.6666666666666666) + (x * 5.0))))))))))

Julia

function code(wj, x)
	return Float64(x + Float64(wj * Float64(Float64(x * -2.0) + Float64(wj * Float64(1.0 + Float64(Float64(x * 2.5) - Float64(wj * Float64(1.0 + Float64(Float64(x * -3.0) + Float64(Float64(x * 0.6666666666666666) + Float64(x * 5.0)))))))))))
end

MATLAB

function tmp = code(wj, x)
	tmp = x + (wj * ((x * -2.0) + (wj * (1.0 + ((x * 2.5) - (wj * (1.0 + ((x * -3.0) + ((x * 0.6666666666666666) + (x * 5.0))))))))));
end

Wolfram

code[wj_, x_] := N[(x + N[(wj * N[(N[(x * -2.0), $MachinePrecision] + N[(wj * N[(1.0 + N[(N[(x * 2.5), $MachinePrecision] - N[(wj * N[(1.0 + N[(N[(x * -3.0), $MachinePrecision] + N[(N[(x * 0.6666666666666666), $MachinePrecision] + N[(x * 5.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 75.6%

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

   1. distribute-rgt1-in 76.3%

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

   2. associate-/l/ 76.3%

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

   3. div-sub 75.6%

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

   4. associate-/l* 75.6%

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

   5. *-inverses 76.7%

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

   6. *-rgt-identity 76.7%

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

3. Simplified 76.7%

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

5. Taylor expanded in wj around 0 97.3%

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

7. Simplified 97.3%

   \[\leadsto \color{blue}{\mathsf{fma}\left(wj, \mathsf{fma}\left(wj, 1 + \mathsf{fma}\left(-wj, 1 + \mathsf{fma}\left(x, -3, \mathsf{fma}\left(x, 0.6666666666666666, x \cdot 5\right)\right), x \cdot 2.5\right), x \cdot -2\right), x\right)} \]

8. Taylor expanded in wj around 0 97.3%

   \[\leadsto \color{blue}{x + wj \cdot \left(-2 \cdot x + wj \cdot \left(1 + \left(-1 \cdot \left(wj \cdot \left(1 + \left(-3 \cdot x + \left(0.6666666666666666 \cdot x + 5 \cdot x\right)\right)\right)\right) + 2.5 \cdot x\right)\right)\right)} \]

9. Final simplification 97.3%

   \[\leadsto x + wj \cdot \left(x \cdot -2 + wj \cdot \left(1 + \left(x \cdot 2.5 - wj \cdot \left(1 + \left(x \cdot -3 + \left(x \cdot 0.6666666666666666 + x \cdot 5\right)\right)\right)\right)\right)\right) \]
10. Add Preprocessing

Alternative 3: 96.4% accurate, 12.5× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 75.6%

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

   1. distribute-rgt1-in 76.3%

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

   2. associate-/l/ 76.3%

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

   3. div-sub 75.6%

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

   4. associate-/l* 75.6%

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

   5. *-inverses 76.7%

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

   6. *-rgt-identity 76.7%

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

3. Simplified 76.7%

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

5. Taylor expanded in wj around 0 97.3%

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

7. Simplified 97.3%

   \[\leadsto \color{blue}{\mathsf{fma}\left(wj, \mathsf{fma}\left(wj, 1 + \mathsf{fma}\left(-wj, 1 + \mathsf{fma}\left(x, -3, \mathsf{fma}\left(x, 0.6666666666666666, x \cdot 5\right)\right), x \cdot 2.5\right), x \cdot -2\right), x\right)} \]

8. Taylor expanded in wj around 0 97.3%

   \[\leadsto \color{blue}{x + wj \cdot \left(-2 \cdot x + wj \cdot \left(1 + \left(-1 \cdot \left(wj \cdot \left(1 + \left(-3 \cdot x + \left(0.6666666666666666 \cdot x + 5 \cdot x\right)\right)\right)\right) + 2.5 \cdot x\right)\right)\right)} \]

9. Taylor expanded in x around inf 97.3%

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

10. Final simplification 97.3%

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

Alternative 4: 96.1% accurate, 13.6× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 75.6%

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

   1. distribute-rgt1-in 76.3%

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

   2. associate-/l/ 76.3%

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

   3. div-sub 75.6%

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

   4. associate-/l* 75.6%

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

   5. *-inverses 76.7%

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

   6. *-rgt-identity 76.7%

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

3. Simplified 76.7%

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

5. Taylor expanded in wj around 0 74.7%

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

   1. associate-*r* 74.7%

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

   2. neg-mul-1 74.7%

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

   3. distribute-rgt1-in 74.7%

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

   4. +-commutative 74.7%

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

   5. sub-neg 74.7%

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

7. Simplified 74.7%

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

8. Taylor expanded in x around inf 76.4%

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

9. Taylor expanded in wj around 0 97.1%

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

10. Final simplification 97.1%

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

Alternative 5: 96.3% accurate, 14.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 75.6%

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

   1. distribute-rgt1-in 76.3%

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

   2. associate-/l/ 76.3%

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

   3. div-sub 75.6%

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

   4. associate-/l* 75.6%

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

   5. *-inverses 76.7%

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

   6. *-rgt-identity 76.7%

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

3. Simplified 76.7%

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

5. Taylor expanded in wj around 0 97.3%

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

7. Simplified 97.3%

   \[\leadsto \color{blue}{\mathsf{fma}\left(wj, \mathsf{fma}\left(wj, 1 + \mathsf{fma}\left(-wj, 1 + \mathsf{fma}\left(x, -3, \mathsf{fma}\left(x, 0.6666666666666666, x \cdot 5\right)\right), x \cdot 2.5\right), x \cdot -2\right), x\right)} \]

8. Taylor expanded in wj around 0 97.3%

   \[\leadsto \color{blue}{x + wj \cdot \left(-2 \cdot x + wj \cdot \left(1 + \left(-1 \cdot \left(wj \cdot \left(1 + \left(-3 \cdot x + \left(0.6666666666666666 \cdot x + 5 \cdot x\right)\right)\right)\right) + 2.5 \cdot x\right)\right)\right)} \]

9. Taylor expanded in x around inf 97.3%

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

10. Taylor expanded in wj around 0 97.1%

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

    1. *-commutative 97.1%

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

12. Simplified 97.1%

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

13. Final simplification 97.1%

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

Alternative 6: 96.1% 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 75.6%

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

   1. distribute-rgt1-in 76.3%

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

   2. associate-/l/ 76.3%

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

   3. div-sub 75.6%

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

   4. associate-/l* 75.6%

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

   5. *-inverses 76.7%

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

   6. *-rgt-identity 76.7%

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

3. Simplified 76.7%

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

5. Taylor expanded in wj around 0 97.3%

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

7. Simplified 97.3%

   \[\leadsto \color{blue}{\mathsf{fma}\left(wj, \mathsf{fma}\left(wj, 1 + \mathsf{fma}\left(-wj, 1 + \mathsf{fma}\left(x, -3, \mathsf{fma}\left(x, 0.6666666666666666, x \cdot 5\right)\right), x \cdot 2.5\right), x \cdot -2\right), x\right)} \]

8. Taylor expanded in wj around 0 97.3%

   \[\leadsto \color{blue}{x + wj \cdot \left(-2 \cdot x + wj \cdot \left(1 + \left(-1 \cdot \left(wj \cdot \left(1 + \left(-3 \cdot x + \left(0.6666666666666666 \cdot x + 5 \cdot x\right)\right)\right)\right) + 2.5 \cdot x\right)\right)\right)} \]

9. Taylor expanded in x around 0 96.8%

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

    1. mul-1-neg 96.8%

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

    2. sub-neg 96.8%

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

11. Simplified 96.8%

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

12. Final simplification 96.8%

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

Alternative 7: 85.0% 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 75.6%

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

   1. distribute-rgt1-in 76.3%

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

   2. associate-/l/ 76.3%

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

   3. div-sub 75.6%

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

   4. associate-/l* 75.6%

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

   5. *-inverses 76.7%

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

   6. *-rgt-identity 76.7%

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

3. Simplified 76.7%

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

5. Taylor expanded in wj around 0 74.7%

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

   1. associate-*r* 74.7%

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

   2. neg-mul-1 74.7%

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

   3. distribute-rgt1-in 74.7%

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

   4. +-commutative 74.7%

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

   5. sub-neg 74.7%

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

7. Simplified 74.7%

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

8. Taylor expanded in x around inf 84.1%

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

   1. div-sub 84.1%

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

   2. +-commutative 84.1%

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

10. Simplified 84.1%

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

11. Final simplification 84.1%

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

Alternative 8: 85.0% accurate, 34.8× speedup

Math

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

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(Float64(x * 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[(N[(x * N[(1.0 - wj), $MachinePrecision]), $MachinePrecision] / N[(wj + 1.0), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 75.6%

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

   1. distribute-rgt1-in 76.3%

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

   2. associate-/l/ 76.3%

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

   3. div-sub 75.6%

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

   4. associate-/l* 75.6%

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

   5. *-inverses 76.7%

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

   6. *-rgt-identity 76.7%

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

3. Simplified 76.7%

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

5. Taylor expanded in wj around 0 74.7%

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

   1. associate-*r* 74.7%

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

   2. neg-mul-1 74.7%

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

   3. distribute-rgt1-in 74.7%

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

   4. +-commutative 74.7%

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

   5. sub-neg 74.7%

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

7. Simplified 74.7%

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

8. Taylor expanded in x around inf 84.1%

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

   1. div-sub 84.1%

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

   2. +-commutative 84.1%

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

   3. associate-/l* 84.1%

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

10. Simplified 84.1%

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

11. Final simplification 84.1%

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

Alternative 9: 84.9% 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 75.6%

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

   1. distribute-rgt1-in 76.3%

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

   2. associate-/l/ 76.3%

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

   3. div-sub 75.6%

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

   4. associate-/l* 75.6%

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

   5. *-inverses 76.7%

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

   6. *-rgt-identity 76.7%

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

3. Simplified 76.7%

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

5. Taylor expanded in wj around 0 83.9%

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

   1. *-commutative 83.9%

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

7. Simplified 83.9%

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

8. Final simplification 83.9%

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

Alternative 10: 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 75.6%

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

   1. distribute-rgt1-in 76.3%

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

   2. associate-/l/ 76.3%

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

   3. div-sub 75.6%

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

   4. associate-/l* 75.6%

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

   5. *-inverses 76.7%

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

   6. *-rgt-identity 76.7%

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

3. Simplified 76.7%

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

5. Taylor expanded in wj around inf 4.3%

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

6. Final simplification 4.3%

   \[\leadsto wj \]
7. Add Preprocessing

Alternative 11: 84.4% 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 75.6%

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

   1. distribute-rgt1-in 76.3%

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

   2. associate-/l/ 76.3%

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

   3. div-sub 75.6%

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

   4. associate-/l* 75.6%

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

   5. *-inverses 76.7%

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

   6. *-rgt-identity 76.7%

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

3. Simplified 76.7%

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

5. Taylor expanded in wj around 0 83.4%

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

6. Final simplification 83.4%

   \[\leadsto x \]
7. Add Preprocessing

Developer target: 78.6% 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 2024075 
(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))))))