Jmat.Real.lambertw, newton loop step

Percentage Accurate: 77.7% → 98.0%

Time: 10.4s

Alternatives: 7

Speedup: 313.0×

• could not determine a ground truth

  1. wj = -2.0280255843952559e+248
  2. x = -2.255581217896515e-14

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: 98.0% accurate, 2.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;wj \leq -5.2 \cdot 10^{-6}:\\ \;\;\;\;wj - \frac{wj - \frac{x}{e^{wj}}}{wj + 1}\\ \mathbf{else}:\\ \;\;\;\;x + wj \cdot \left(wj \cdot \left(x \cdot \left(wj \cdot -2.6666666666666665 + 2.5\right) + \left(1 - wj\right)\right) - x \cdot 2\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (wj x)
 :precision binary64
 (if (<= wj -5.2e-6)
   (- wj (/ (- wj (/ x (exp wj))) (+ wj 1.0)))
   (+
    x
    (*
     wj
     (-
      (* wj (+ (* x (+ (* wj -2.6666666666666665) 2.5)) (- 1.0 wj)))
      (* x 2.0))))))

C

double code(double wj, double x) {
	double tmp;
	if (wj <= -5.2e-6) {
		tmp = wj - ((wj - (x / exp(wj))) / (wj + 1.0));
	} else {
		tmp = x + (wj * ((wj * ((x * ((wj * -2.6666666666666665) + 2.5)) + (1.0 - wj))) - (x * 2.0)));
	}
	return tmp;
}

Fortran

real(8) function code(wj, x)
    real(8), intent (in) :: wj
    real(8), intent (in) :: x
    real(8) :: tmp
    if (wj <= (-5.2d-6)) then
        tmp = wj - ((wj - (x / exp(wj))) / (wj + 1.0d0))
    else
        tmp = x + (wj * ((wj * ((x * ((wj * (-2.6666666666666665d0)) + 2.5d0)) + (1.0d0 - wj))) - (x * 2.0d0)))
    end if
    code = tmp
end function

Java

public static double code(double wj, double x) {
	double tmp;
	if (wj <= -5.2e-6) {
		tmp = wj - ((wj - (x / Math.exp(wj))) / (wj + 1.0));
	} else {
		tmp = x + (wj * ((wj * ((x * ((wj * -2.6666666666666665) + 2.5)) + (1.0 - wj))) - (x * 2.0)));
	}
	return tmp;
}

Python

def code(wj, x):
	tmp = 0
	if wj <= -5.2e-6:
		tmp = wj - ((wj - (x / math.exp(wj))) / (wj + 1.0))
	else:
		tmp = x + (wj * ((wj * ((x * ((wj * -2.6666666666666665) + 2.5)) + (1.0 - wj))) - (x * 2.0)))
	return tmp

Julia

function code(wj, x)
	tmp = 0.0
	if (wj <= -5.2e-6)
		tmp = Float64(wj - Float64(Float64(wj - Float64(x / exp(wj))) / Float64(wj + 1.0)));
	else
		tmp = Float64(x + Float64(wj * Float64(Float64(wj * Float64(Float64(x * Float64(Float64(wj * -2.6666666666666665) + 2.5)) + Float64(1.0 - wj))) - Float64(x * 2.0))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(wj, x)
	tmp = 0.0;
	if (wj <= -5.2e-6)
		tmp = wj - ((wj - (x / exp(wj))) / (wj + 1.0));
	else
		tmp = x + (wj * ((wj * ((x * ((wj * -2.6666666666666665) + 2.5)) + (1.0 - wj))) - (x * 2.0)));
	end
	tmp_2 = tmp;
end

Wolfram

code[wj_, x_] := If[LessEqual[wj, -5.2e-6], N[(wj - N[(N[(wj - N[(x / N[Exp[wj], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(wj + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x + N[(wj * N[(N[(wj * N[(N[(x * N[(N[(wj * -2.6666666666666665), $MachinePrecision] + 2.5), $MachinePrecision]), $MachinePrecision] + N[(1.0 - wj), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(x * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if wj < -5.20000000000000019e-6

   1. Initial program 49.2%

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

      1. distribute-rgt1-in 99.5%

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

      2. associate-/l/ 99.7%

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

      3. div-sub 49.7%

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

      4. associate-/l* 49.7%

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

      5. *-inverses 99.7%

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

      6. *-rgt-identity 99.7%

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

   3. Simplified 99.7%

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

   if -5.20000000000000019e-6 < wj

   1. Initial program 76.2%

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

      1. distribute-rgt1-in 76.2%

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

      2. associate-/l/ 76.2%

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

      3. div-sub 76.2%

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

      4. associate-/l* 76.2%

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

      5. *-inverses 77.4%

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

      6. *-rgt-identity 77.4%

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

   3. Simplified 77.4%

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

   5. Taylor expanded in wj around 0 98.8%

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

   6. Taylor expanded in x around 0 98.8%

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

      1. distribute-lft-out 98.8%

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

      2. +-commutative 98.8%

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

      3. *-commutative 98.8%

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

      4. mul-1-neg 98.8%

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

      5. sub-neg 98.8%

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

   8. Simplified 98.8%

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

4. Final simplification 98.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;wj \leq -5.2 \cdot 10^{-6}:\\ \;\;\;\;wj - \frac{wj - \frac{x}{e^{wj}}}{wj + 1}\\ \mathbf{else}:\\ \;\;\;\;x + wj \cdot \left(wj \cdot \left(x \cdot \left(wj \cdot -2.6666666666666665 + 2.5\right) + \left(1 - wj\right)\right) - x \cdot 2\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 96.2% accurate, 14.9× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 75.5%

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

   1. distribute-rgt1-in 76.7%

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

   2. associate-/l/ 76.7%

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

   3. div-sub 75.5%

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

   4. associate-/l* 75.5%

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

   5. *-inverses 77.9%

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

   6. *-rgt-identity 77.9%

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

3. Simplified 77.9%

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

5. Taylor expanded in wj around 0 96.7%

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

6. Taylor expanded in x around 0 96.7%

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

   1. distribute-lft-out 96.7%

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

   2. +-commutative 96.7%

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

   3. *-commutative 96.7%

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

   4. mul-1-neg 96.7%

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

   5. sub-neg 96.7%

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

8. Simplified 96.7%

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

9. Final simplification 96.7%

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

Alternative 3: 95.9% accurate, 24.1× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 75.5%

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

   1. distribute-rgt1-in 76.7%

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

   2. associate-/l/ 76.7%

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

   3. div-sub 75.5%

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

   4. associate-/l* 75.5%

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

   5. *-inverses 77.9%

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

   6. *-rgt-identity 77.9%

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

3. Simplified 77.9%

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

5. Taylor expanded in wj around 0 96.7%

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

6. Taylor expanded in x around 0 96.7%

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

   1. distribute-lft-out 96.7%

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

   2. +-commutative 96.7%

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

   3. *-commutative 96.7%

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

   4. mul-1-neg 96.7%

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

   5. sub-neg 96.7%

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

8. Simplified 96.7%

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

9. Taylor expanded in x around 0 96.5%

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

10. Final simplification 96.5%

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

Alternative 4: 95.1% accurate, 62.6× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 75.5%

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

   1. distribute-rgt1-in 76.7%

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

   2. associate-/l/ 76.7%

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

   3. div-sub 75.5%

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

   4. associate-/l* 75.5%

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

   5. *-inverses 77.9%

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

   6. *-rgt-identity 77.9%

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

3. Simplified 77.9%

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

5. Taylor expanded in wj around 0 96.7%

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

6. Taylor expanded in wj around 0 96.5%

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

   1. distribute-rgt-out 96.5%

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

   2. metadata-eval 96.5%

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

8. Simplified 96.5%

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

9. Taylor expanded in x around 0 96.0%

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

Alternative 5: 84.1% 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 75.5%

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

   1. distribute-rgt1-in 76.7%

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

   2. associate-/l/ 76.7%

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

   3. div-sub 75.5%

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

   4. associate-/l* 75.5%

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

   5. *-inverses 77.9%

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

   6. *-rgt-identity 77.9%

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

3. Simplified 77.9%

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

5. Taylor expanded in wj around 0 85.8%

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

Alternative 6: 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 75.5%

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

   1. distribute-rgt1-in 76.7%

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

   2. associate-/l/ 76.7%

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

   3. div-sub 75.5%

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

   4. associate-/l* 75.5%

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

   5. *-inverses 77.9%

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

   6. *-rgt-identity 77.9%

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

3. Simplified 77.9%

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

5. Taylor expanded in wj around inf 4.5%

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

Alternative 7: 3.3% accurate, 313.0× speedup

Math

\[\begin{array}{l} \\ -1 \end{array} \]

FPCore

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

C

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

Fortran

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

Java

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

Python

def code(wj, x):
	return -1.0

Julia

function code(wj, x)
	return -1.0
end

MATLAB

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

Wolfram

code[wj_, x_] := -1.0

Derivation

1. Initial program 75.5%

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

   1. distribute-rgt1-in 76.7%

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

   2. associate-/l/ 76.7%

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

   3. div-sub 75.5%

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

   4. associate-/l* 75.5%

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

   5. *-inverses 77.9%

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

   6. *-rgt-identity 77.9%

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

3. Simplified 77.9%

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

5. Taylor expanded in wj around inf 4.4%

   \[\leadsto wj - \color{blue}{1} \]

6. Taylor expanded in wj around 0 3.3%

   \[\leadsto \color{blue}{-1} \]
7. Add Preprocessing

Developer Target 1: 78.7% 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 2024146 
(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))))))