Jmat.Real.lambertw, newton loop step

Percentage Accurate: 77.9% → 98.6%

Time: 9.3s

Alternatives: 9

Speedup: 313.0×

• could not determine a ground truth

  1. wj = -4.256809963496365e+153
  2. x = -9.332734993053553e+275

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 77.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 98.6% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := wj \cdot e^{wj}\\ \mathbf{if}\;wj + \frac{x - t\_0}{e^{wj} + t\_0} \leq 5 \cdot 10^{-15}:\\ \;\;\;\;x \cdot \left(\frac{{wj}^{2} \cdot \left(1 + wj \cdot \left(wj + -1\right)\right)}{x} - \frac{-1}{{\left(wj + 1\right)}^{2}}\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(\frac{wj + \frac{wj}{-1 - wj}}{x} + \frac{e^{-wj}}{wj + 1}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (wj x)
 :precision binary64
 (let* ((t_0 (* wj (exp wj))))
   (if (<= (+ wj (/ (- x t_0) (+ (exp wj) t_0))) 5e-15)
     (*
      x
      (-
       (/ (* (pow wj 2.0) (+ 1.0 (* wj (+ wj -1.0)))) x)
       (/ -1.0 (pow (+ wj 1.0) 2.0))))
     (* x (+ (/ (+ wj (/ wj (- -1.0 wj))) x) (/ (exp (- wj)) (+ wj 1.0)))))))

C

double code(double wj, double x) {
	double t_0 = wj * exp(wj);
	double tmp;
	if ((wj + ((x - t_0) / (exp(wj) + t_0))) <= 5e-15) {
		tmp = x * (((pow(wj, 2.0) * (1.0 + (wj * (wj + -1.0)))) / x) - (-1.0 / pow((wj + 1.0), 2.0)));
	} else {
		tmp = x * (((wj + (wj / (-1.0 - wj))) / x) + (exp(-wj) / (wj + 1.0)));
	}
	return tmp;
}

Fortran

real(8) function code(wj, x)
    real(8), intent (in) :: wj
    real(8), intent (in) :: x
    real(8) :: t_0
    real(8) :: tmp
    t_0 = wj * exp(wj)
    if ((wj + ((x - t_0) / (exp(wj) + t_0))) <= 5d-15) then
        tmp = x * ((((wj ** 2.0d0) * (1.0d0 + (wj * (wj + (-1.0d0))))) / x) - ((-1.0d0) / ((wj + 1.0d0) ** 2.0d0)))
    else
        tmp = x * (((wj + (wj / ((-1.0d0) - wj))) / x) + (exp(-wj) / (wj + 1.0d0)))
    end if
    code = tmp
end function

Java

public static double code(double wj, double x) {
	double t_0 = wj * Math.exp(wj);
	double tmp;
	if ((wj + ((x - t_0) / (Math.exp(wj) + t_0))) <= 5e-15) {
		tmp = x * (((Math.pow(wj, 2.0) * (1.0 + (wj * (wj + -1.0)))) / x) - (-1.0 / Math.pow((wj + 1.0), 2.0)));
	} else {
		tmp = x * (((wj + (wj / (-1.0 - wj))) / x) + (Math.exp(-wj) / (wj + 1.0)));
	}
	return tmp;
}

Python

def code(wj, x):
	t_0 = wj * math.exp(wj)
	tmp = 0
	if (wj + ((x - t_0) / (math.exp(wj) + t_0))) <= 5e-15:
		tmp = x * (((math.pow(wj, 2.0) * (1.0 + (wj * (wj + -1.0)))) / x) - (-1.0 / math.pow((wj + 1.0), 2.0)))
	else:
		tmp = x * (((wj + (wj / (-1.0 - wj))) / x) + (math.exp(-wj) / (wj + 1.0)))
	return tmp

Julia

function code(wj, x)
	t_0 = Float64(wj * exp(wj))
	tmp = 0.0
	if (Float64(wj + Float64(Float64(x - t_0) / Float64(exp(wj) + t_0))) <= 5e-15)
		tmp = Float64(x * Float64(Float64(Float64((wj ^ 2.0) * Float64(1.0 + Float64(wj * Float64(wj + -1.0)))) / x) - Float64(-1.0 / (Float64(wj + 1.0) ^ 2.0))));
	else
		tmp = Float64(x * Float64(Float64(Float64(wj + Float64(wj / Float64(-1.0 - wj))) / x) + Float64(exp(Float64(-wj)) / Float64(wj + 1.0))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(wj, x)
	t_0 = wj * exp(wj);
	tmp = 0.0;
	if ((wj + ((x - t_0) / (exp(wj) + t_0))) <= 5e-15)
		tmp = x * ((((wj ^ 2.0) * (1.0 + (wj * (wj + -1.0)))) / x) - (-1.0 / ((wj + 1.0) ^ 2.0)));
	else
		tmp = x * (((wj + (wj / (-1.0 - wj))) / x) + (exp(-wj) / (wj + 1.0)));
	end
	tmp_2 = tmp;
end

Wolfram

code[wj_, x_] := Block[{t$95$0 = N[(wj * N[Exp[wj], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(wj + N[(N[(x - t$95$0), $MachinePrecision] / N[(N[Exp[wj], $MachinePrecision] + t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 5e-15], N[(x * N[(N[(N[(N[Power[wj, 2.0], $MachinePrecision] * N[(1.0 + N[(wj * N[(wj + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision] - N[(-1.0 / N[Power[N[(wj + 1.0), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x * N[(N[(N[(wj + N[(wj / N[(-1.0 - wj), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision] + N[(N[Exp[(-wj)], $MachinePrecision] / N[(wj + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 wj (/.f64 (-.f64 (*.f64 wj (exp.f64 wj)) x) (+.f64 (exp.f64 wj) (*.f64 wj (exp.f64 wj))))) < 4.99999999999999999e-15

   1. Initial program 67.7%

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

      1. distribute-rgt1-in 68.2%

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

      2. associate-/l/ 68.2%

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

      3. div-sub 67.7%

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

      4. associate-/l* 67.7%

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

      5. *-inverses 68.2%

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

      6. *-rgt-identity 68.2%

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

   3. Simplified 68.2%

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

   5. Taylor expanded in wj around 0 67.7%

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

      1. +-commutative 67.7%

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

   7. Simplified 67.7%

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

   8. Taylor expanded in x around -inf 84.1%

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

      1. associate-*r* 84.1%

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

      2. neg-mul-1 84.1%

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

      3. mul-1-neg 84.1%

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

      4. +-commutative 84.1%

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

      5. +-commutative 84.1%

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

   10. Simplified 84.1%

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

   11. Taylor expanded in wj around 0 98.9%

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

   if 4.99999999999999999e-15 < (-.f64 wj (/.f64 (-.f64 (*.f64 wj (exp.f64 wj)) x) (+.f64 (exp.f64 wj) (*.f64 wj (exp.f64 wj)))))

   1. Initial program 93.7%

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

      1. distribute-rgt1-in 96.7%

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

      2. associate-/l/ 96.7%

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

      3. div-sub 93.7%

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

      4. associate-/l* 93.7%

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

      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

   5. Taylor expanded in x around -inf 99.8%

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

      1. associate-*r* 99.8%

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

      2. neg-mul-1 99.8%

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

      3. mul-1-neg 99.8%

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

      4. +-commutative 99.8%

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

      5. associate-/r* 99.8%

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

      6. rec-exp 99.8%

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

      7. +-commutative 99.8%

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

   7. Simplified 99.8%

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

4. Final simplification 99.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;wj + \frac{x - wj \cdot e^{wj}}{e^{wj} + wj \cdot e^{wj}} \leq 5 \cdot 10^{-15}:\\ \;\;\;\;x \cdot \left(\frac{{wj}^{2} \cdot \left(1 + wj \cdot \left(wj + -1\right)\right)}{x} - \frac{-1}{{\left(wj + 1\right)}^{2}}\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(\frac{wj + \frac{wj}{-1 - wj}}{x} + \frac{e^{-wj}}{wj + 1}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 97.9% accurate, 2.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;wj \leq -3.6 \cdot 10^{-7}:\\ \;\;\;\;x \cdot \left(\frac{wj + \frac{wj}{-1 - wj}}{x} + \frac{e^{-wj}}{wj + 1}\right)\\ \mathbf{else}:\\ \;\;\;\;x + wj \cdot \left(wj \cdot \left(1 - wj\right) - x \cdot 2\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (wj x)
 :precision binary64
 (if (<= wj -3.6e-7)
   (* x (+ (/ (+ wj (/ wj (- -1.0 wj))) x) (/ (exp (- wj)) (+ wj 1.0))))
   (+ x (* wj (- (* wj (- 1.0 wj)) (* x 2.0))))))

C

double code(double wj, double x) {
	double tmp;
	if (wj <= -3.6e-7) {
		tmp = x * (((wj + (wj / (-1.0 - wj))) / x) + (exp(-wj) / (wj + 1.0)));
	} else {
		tmp = x + (wj * ((wj * (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 <= (-3.6d-7)) then
        tmp = x * (((wj + (wj / ((-1.0d0) - wj))) / x) + (exp(-wj) / (wj + 1.0d0)))
    else
        tmp = x + (wj * ((wj * (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 <= -3.6e-7) {
		tmp = x * (((wj + (wj / (-1.0 - wj))) / x) + (Math.exp(-wj) / (wj + 1.0)));
	} else {
		tmp = x + (wj * ((wj * (1.0 - wj)) - (x * 2.0)));
	}
	return tmp;
}

Python

def code(wj, x):
	tmp = 0
	if wj <= -3.6e-7:
		tmp = x * (((wj + (wj / (-1.0 - wj))) / x) + (math.exp(-wj) / (wj + 1.0)))
	else:
		tmp = x + (wj * ((wj * (1.0 - wj)) - (x * 2.0)))
	return tmp

Julia

function code(wj, x)
	tmp = 0.0
	if (wj <= -3.6e-7)
		tmp = Float64(x * Float64(Float64(Float64(wj + Float64(wj / Float64(-1.0 - wj))) / x) + Float64(exp(Float64(-wj)) / Float64(wj + 1.0))));
	else
		tmp = Float64(x + Float64(wj * Float64(Float64(wj * Float64(1.0 - wj)) - Float64(x * 2.0))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(wj, x)
	tmp = 0.0;
	if (wj <= -3.6e-7)
		tmp = x * (((wj + (wj / (-1.0 - wj))) / x) + (exp(-wj) / (wj + 1.0)));
	else
		tmp = x + (wj * ((wj * (1.0 - wj)) - (x * 2.0)));
	end
	tmp_2 = tmp;
end

Wolfram

code[wj_, x_] := If[LessEqual[wj, -3.6e-7], N[(x * N[(N[(N[(wj + N[(wj / N[(-1.0 - wj), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision] + N[(N[Exp[(-wj)], $MachinePrecision] / N[(wj + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x + N[(wj * N[(N[(wj * N[(1.0 - wj), $MachinePrecision]), $MachinePrecision] - N[(x * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if wj < -3.59999999999999994e-7

   1. Initial program 46.1%

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

      1. distribute-rgt1-in 96.1%

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

      2. associate-/l/ 97.1%

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

      3. div-sub 47.1%

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

      4. associate-/l* 47.1%

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

      5. *-inverses 97.1%

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

      6. *-rgt-identity 97.1%

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

   3. Simplified 97.1%

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

   5. Taylor expanded in x around -inf 97.5%

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

      1. associate-*r* 97.5%

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

      2. neg-mul-1 97.5%

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

      3. mul-1-neg 97.5%

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

      4. +-commutative 97.5%

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

      5. associate-/r* 97.5%

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

      6. rec-exp 97.5%

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

      7. +-commutative 97.5%

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

   7. Simplified 97.5%

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

   if -3.59999999999999994e-7 < wj

   1. Initial program 75.2%

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

      1. distribute-rgt1-in 75.2%

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

      2. associate-/l/ 75.2%

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

      3. div-sub 75.2%

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

      4. associate-/l* 75.2%

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

      5. *-inverses 76.0%

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

      6. *-rgt-identity 76.0%

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

   3. Simplified 76.0%

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

   5. Taylor expanded in wj around 0 98.5%

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

   6. Taylor expanded in x around 0 98.5%

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

      1. mul-1-neg 98.5%

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

      2. unsub-neg 98.5%

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

   8. Simplified 98.5%

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

4. Final simplification 98.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;wj \leq -3.6 \cdot 10^{-7}:\\ \;\;\;\;x \cdot \left(\frac{wj + \frac{wj}{-1 - wj}}{x} + \frac{e^{-wj}}{wj + 1}\right)\\ \mathbf{else}:\\ \;\;\;\;x + wj \cdot \left(wj \cdot \left(1 - wj\right) - x \cdot 2\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 97.9% accurate, 2.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;wj \leq -3.7 \cdot 10^{-7}:\\ \;\;\;\;wj + \frac{wj - \frac{x}{e^{wj}}}{-1 - wj}\\ \mathbf{else}:\\ \;\;\;\;x + wj \cdot \left(wj \cdot \left(1 - wj\right) - x \cdot 2\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (wj x)
 :precision binary64
 (if (<= wj -3.7e-7)
   (+ wj (/ (- wj (/ x (exp wj))) (- -1.0 wj)))
   (+ x (* wj (- (* wj (- 1.0 wj)) (* x 2.0))))))

C

double code(double wj, double x) {
	double tmp;
	if (wj <= -3.7e-7) {
		tmp = wj + ((wj - (x / exp(wj))) / (-1.0 - wj));
	} else {
		tmp = x + (wj * ((wj * (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 <= (-3.7d-7)) then
        tmp = wj + ((wj - (x / exp(wj))) / ((-1.0d0) - wj))
    else
        tmp = x + (wj * ((wj * (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 <= -3.7e-7) {
		tmp = wj + ((wj - (x / Math.exp(wj))) / (-1.0 - wj));
	} else {
		tmp = x + (wj * ((wj * (1.0 - wj)) - (x * 2.0)));
	}
	return tmp;
}

Python

def code(wj, x):
	tmp = 0
	if wj <= -3.7e-7:
		tmp = wj + ((wj - (x / math.exp(wj))) / (-1.0 - wj))
	else:
		tmp = x + (wj * ((wj * (1.0 - wj)) - (x * 2.0)))
	return tmp

Julia

function code(wj, x)
	tmp = 0.0
	if (wj <= -3.7e-7)
		tmp = Float64(wj + Float64(Float64(wj - Float64(x / exp(wj))) / Float64(-1.0 - wj)));
	else
		tmp = Float64(x + Float64(wj * Float64(Float64(wj * Float64(1.0 - wj)) - Float64(x * 2.0))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(wj, x)
	tmp = 0.0;
	if (wj <= -3.7e-7)
		tmp = wj + ((wj - (x / exp(wj))) / (-1.0 - wj));
	else
		tmp = x + (wj * ((wj * (1.0 - wj)) - (x * 2.0)));
	end
	tmp_2 = tmp;
end

Wolfram

code[wj_, x_] := If[LessEqual[wj, -3.7e-7], N[(wj + N[(N[(wj - N[(x / N[Exp[wj], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(-1.0 - wj), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x + N[(wj * N[(N[(wj * N[(1.0 - wj), $MachinePrecision]), $MachinePrecision] - N[(x * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if wj < -3.70000000000000004e-7

   1. Initial program 46.1%

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

      1. distribute-rgt1-in 96.1%

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

      2. associate-/l/ 97.1%

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

      3. div-sub 47.1%

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

      4. associate-/l* 47.1%

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

      5. *-inverses 97.1%

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

      6. *-rgt-identity 97.1%

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

   3. Simplified 97.1%

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

   if -3.70000000000000004e-7 < wj

   1. Initial program 75.2%

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

      1. distribute-rgt1-in 75.2%

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

      2. associate-/l/ 75.2%

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

      3. div-sub 75.2%

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

      4. associate-/l* 75.2%

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

      5. *-inverses 76.0%

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

      6. *-rgt-identity 76.0%

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

   3. Simplified 76.0%

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

   5. Taylor expanded in wj around 0 98.5%

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

   6. Taylor expanded in x around 0 98.5%

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

      1. mul-1-neg 98.5%

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

      2. unsub-neg 98.5%

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

   8. Simplified 98.5%

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

4. Final simplification 98.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;wj \leq -3.7 \cdot 10^{-7}:\\ \;\;\;\;wj + \frac{wj - \frac{x}{e^{wj}}}{-1 - wj}\\ \mathbf{else}:\\ \;\;\;\;x + wj \cdot \left(wj \cdot \left(1 - wj\right) - x \cdot 2\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 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 74.5%

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

   1. distribute-rgt1-in 75.7%

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

   2. associate-/l/ 75.7%

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

   3. div-sub 74.5%

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

   4. associate-/l* 74.5%

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

   5. *-inverses 76.5%

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

   6. *-rgt-identity 76.5%

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

3. Simplified 76.5%

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

5. Taylor expanded in wj around 0 96.5%

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

6. Taylor expanded in x around 0 96.5%

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

   1. mul-1-neg 96.5%

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

   2. unsub-neg 96.5%

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

8. Simplified 96.5%

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

9. Final simplification 96.5%

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

Alternative 5: 95.7% accurate, 34.8× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 74.5%

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

   1. distribute-rgt1-in 75.7%

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

   2. associate-/l/ 75.7%

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

   3. div-sub 74.5%

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

   4. associate-/l* 74.5%

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

   5. *-inverses 76.5%

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

   6. *-rgt-identity 76.5%

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

3. Simplified 76.5%

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

5. Taylor expanded in wj around 0 96.5%

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

6. Taylor expanded in x around 0 96.5%

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

   1. mul-1-neg 96.5%

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

   2. unsub-neg 96.5%

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

8. Simplified 96.5%

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

9. Taylor expanded in x around inf 96.5%

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

    1. sub-neg 96.5%

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

    2. associate-/l* 96.5%

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

    3. metadata-eval 96.5%

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

11. Simplified 96.5%

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

12. Taylor expanded in x around 0 96.4%

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

13. Final simplification 96.4%

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

Alternative 6: 84.4% 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 74.5%

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

   1. distribute-rgt1-in 75.7%

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

   2. associate-/l/ 75.7%

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

   3. div-sub 74.5%

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

   4. associate-/l* 74.5%

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

   5. *-inverses 76.5%

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

   6. *-rgt-identity 76.5%

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

3. Simplified 76.5%

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

5. Taylor expanded in wj around 0 84.4%

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

   1. *-commutative 84.4%

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

7. Simplified 84.4%

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

8. Final simplification 84.4%

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

Alternative 7: 83.9% 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 74.5%

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

   1. distribute-rgt1-in 75.7%

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

   2. associate-/l/ 75.7%

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

   3. div-sub 74.5%

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

   4. associate-/l* 74.5%

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

   5. *-inverses 76.5%

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

   6. *-rgt-identity 76.5%

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

3. Simplified 76.5%

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

5. Taylor expanded in wj around 0 84.2%

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

Alternative 8: 4.3% 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 74.5%

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

   1. distribute-rgt1-in 75.7%

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

   2. associate-/l/ 75.7%

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

   3. div-sub 74.5%

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

   4. associate-/l* 74.5%

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

   5. *-inverses 76.5%

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

   6. *-rgt-identity 76.5%

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

3. Simplified 76.5%

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

5. Taylor expanded in wj around inf 4.3%

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

Alternative 9: 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 74.5%

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

   1. distribute-rgt1-in 75.7%

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

   2. associate-/l/ 75.7%

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

   3. div-sub 74.5%

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

   4. associate-/l* 74.5%

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

   5. *-inverses 76.5%

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

   6. *-rgt-identity 76.5%

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

3. Simplified 76.5%

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

5. Taylor expanded in wj around inf 4.1%

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

6. Taylor expanded in wj around 0 3.3%

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

Developer Target 1: 78.8% 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 2024181 
(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))))))