Jmat.Real.lambertw, newton loop step

Percentage Accurate: 78.6% → 97.8%

Time: 11.7s

Alternatives: 8

Speedup: 313.0×

• could not determine a ground truth

  1. wj = -1.2907279350701645e+66
  2. x = -2.40525101489681e+206

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: 78.6% 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: 97.8% accurate, 11.6× speedup

Math

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

FPCore

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

C

double code(double wj, double x) {
	return x + (wj * ((x * ((wj * (2.5 + (wj * -2.6666666666666665))) + (1.0 / ((x + (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 * ((x * ((wj * (2.5d0 + (wj * (-2.6666666666666665d0)))) + (1.0d0 / ((x + (x * wj)) / wj)))) - (x * 2.0d0)))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 79.4%

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

   1. distribute-rgt1-in 79.8%

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

   2. associate-/l/ 79.8%

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

   3. div-sub 79.4%

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

   4. associate-/l* 79.4%

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

   5. *-inverses 80.6%

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

   6. *-rgt-identity 80.6%

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

3. Simplified 80.6%

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

5. Taylor expanded in wj around 0 98.2%

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

6. Taylor expanded in x around inf 98.2%

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

   1. clear-num 98.2%

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

   2. inv-pow 98.2%

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

   3. mul-1-neg 98.2%

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

   4. sub-neg 98.2%

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

8. Applied egg-rr 98.2%

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

   1. unpow-1 98.2%

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

10. Simplified 98.2%

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

11. Taylor expanded in wj around 0 99.1%

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

12. Final simplification 99.1%

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

Alternative 2: 96.4% accurate, 11.6× speedup

Math

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

FPCore

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

C

double code(double wj, double x) {
	return x + (wj * ((x * ((wj * (2.5 + (wj * -2.6666666666666665))) + (1.0 / (x / (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 * ((x * ((wj * (2.5d0 + (wj * (-2.6666666666666665d0)))) + (1.0d0 / (x / (wj * (1.0d0 - wj)))))) - (x * 2.0d0)))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 79.4%

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

   1. distribute-rgt1-in 79.8%

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

   2. associate-/l/ 79.8%

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

   3. div-sub 79.4%

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

   4. associate-/l* 79.4%

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

   5. *-inverses 80.6%

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

   6. *-rgt-identity 80.6%

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

3. Simplified 80.6%

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

5. Taylor expanded in wj around 0 98.2%

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

6. Taylor expanded in x around inf 98.2%

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

   1. clear-num 98.2%

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

   2. inv-pow 98.2%

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

   3. mul-1-neg 98.2%

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

   4. sub-neg 98.2%

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

8. Applied egg-rr 98.2%

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

   1. unpow-1 98.2%

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

10. Simplified 98.2%

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

11. Final simplification 98.2%

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

Alternative 3: 96.1% 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 79.4%

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

   1. distribute-rgt1-in 79.8%

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

   2. associate-/l/ 79.8%

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

   3. div-sub 79.4%

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

   4. associate-/l* 79.4%

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

   5. *-inverses 80.6%

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

   6. *-rgt-identity 80.6%

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

3. Simplified 80.6%

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

5. Taylor expanded in wj around 0 98.2%

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

6. Taylor expanded in x around 0 97.7%

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

   1. mul-1-neg 97.7%

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

   2. sub-neg 97.7%

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

8. Simplified 97.7%

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

9. Final simplification 97.7%

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

Alternative 4: 95.8% 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 79.4%

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

   1. distribute-rgt1-in 79.8%

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

   2. associate-/l/ 79.8%

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

   3. div-sub 79.4%

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

   4. associate-/l* 79.4%

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

   5. *-inverses 80.6%

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

   6. *-rgt-identity 80.6%

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

3. Simplified 80.6%

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

5. Taylor expanded in wj around 0 98.2%

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

6. Taylor expanded in x around 0 97.7%

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

   1. mul-1-neg 97.7%

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

   2. sub-neg 97.7%

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

8. Simplified 97.7%

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

9. Taylor expanded in wj around 0 97.1%

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

10. Final simplification 97.1%

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

Alternative 5: 85.3% accurate, 44.7× speedup

Math

\[\begin{array}{l} \\ \frac{x}{1 + wj \cdot 2} \end{array} \]

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 79.4%

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

   1. distribute-rgt1-in 79.8%

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

   2. associate-/l/ 79.8%

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

   3. div-sub 79.4%

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

   4. associate-/l* 79.4%

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

   5. *-inverses 80.6%

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

   6. *-rgt-identity 80.6%

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

3. Simplified 80.6%

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

5. Taylor expanded in x around inf 85.8%

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

   1. +-commutative 85.8%

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

7. Simplified 85.8%

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

8. Taylor expanded in wj around 0 84.9%

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

   1. *-commutative 84.9%

      \[\leadsto \frac{x}{1 + \color{blue}{wj \cdot 2}} \]

10. Simplified 84.9%

    \[\leadsto \frac{x}{\color{blue}{1 + wj \cdot 2}} \]
11. Add Preprocessing

Alternative 6: 85.3% accurate, 44.7× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 79.4%

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

   1. distribute-rgt1-in 79.8%

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

   2. associate-/l/ 79.8%

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

   3. div-sub 79.4%

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

   4. associate-/l* 79.4%

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

   5. *-inverses 80.6%

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

   6. *-rgt-identity 80.6%

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

3. Simplified 80.6%

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

5. Taylor expanded in wj around 0 84.9%

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

   1. *-commutative 84.9%

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

7. Simplified 84.9%

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

8. Final simplification 84.9%

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

Alternative 7: 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 79.4%

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

   1. distribute-rgt1-in 79.8%

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

   2. associate-/l/ 79.8%

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

   3. div-sub 79.4%

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

   4. associate-/l* 79.4%

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

   5. *-inverses 80.6%

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

   6. *-rgt-identity 80.6%

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

3. Simplified 80.6%

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

5. Taylor expanded in wj around 0 84.2%

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

Alternative 8: 4.4% accurate, 313.0× speedup

Math

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

FPCore

(FPCore (wj x) :precision binary64 wj)

C

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

Fortran

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

Java

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

Python

def code(wj, x):
	return wj

Julia

function code(wj, x)
	return wj
end

MATLAB

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

Wolfram

code[wj_, x_] := wj

Derivation

1. Initial program 79.4%

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

   1. distribute-rgt1-in 79.8%

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

   2. associate-/l/ 79.8%

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

   3. div-sub 79.4%

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

   4. associate-/l* 79.4%

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

   5. *-inverses 80.6%

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

   6. *-rgt-identity 80.6%

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

3. Simplified 80.6%

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

5. Taylor expanded in wj around inf 4.4%

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

Developer Target 1: 79.6% 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 2024121 
(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))))))