Jmat.Real.lambertw, newton loop step

Percentage Accurate: 77.6% → 99.7%

Time: 13.3s

Alternatives: 16

Speedup: 313.0×

• could not determine a ground truth

  1. wj = -2.6690410535507296e+79
  2. x = -6.2195442891105626e-9

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: 77.6% 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: 99.7% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;wj \leq -3.1 \cdot 10^{-6}:\\ \;\;\;\;\left(wj + \frac{x}{e^{wj} \cdot \left(wj + 1\right)}\right) - \frac{wj}{wj + 1}\\ \mathbf{elif}\;wj \leq 5 \cdot 10^{-6}:\\ \;\;\;\;x + \left(-2 \cdot \left(wj \cdot x\right) + \left({wj}^{2} \cdot \left(1 - \left(x \cdot -4 + x \cdot 1.5\right)\right) - {wj}^{3}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;wj + \frac{\frac{x}{e^{wj}} - wj}{wj + 1}\\ \end{array} \end{array} \]

FPCore

(FPCore (wj x)
 :precision binary64
 (if (<= wj -3.1e-6)
   (- (+ wj (/ x (* (exp wj) (+ wj 1.0)))) (/ wj (+ wj 1.0)))
   (if (<= wj 5e-6)
     (+
      x
      (+
       (* -2.0 (* wj x))
       (- (* (pow wj 2.0) (- 1.0 (+ (* x -4.0) (* x 1.5)))) (pow wj 3.0))))
     (+ wj (/ (- (/ x (exp wj)) wj) (+ wj 1.0))))))

C

double code(double wj, double x) {
	double tmp;
	if (wj <= -3.1e-6) {
		tmp = (wj + (x / (exp(wj) * (wj + 1.0)))) - (wj / (wj + 1.0));
	} else if (wj <= 5e-6) {
		tmp = x + ((-2.0 * (wj * x)) + ((pow(wj, 2.0) * (1.0 - ((x * -4.0) + (x * 1.5)))) - pow(wj, 3.0)));
	} else {
		tmp = wj + (((x / exp(wj)) - 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) :: tmp
    if (wj <= (-3.1d-6)) then
        tmp = (wj + (x / (exp(wj) * (wj + 1.0d0)))) - (wj / (wj + 1.0d0))
    else if (wj <= 5d-6) then
        tmp = x + (((-2.0d0) * (wj * x)) + (((wj ** 2.0d0) * (1.0d0 - ((x * (-4.0d0)) + (x * 1.5d0)))) - (wj ** 3.0d0)))
    else
        tmp = wj + (((x / exp(wj)) - wj) / (wj + 1.0d0))
    end if
    code = tmp
end function

Java

public static double code(double wj, double x) {
	double tmp;
	if (wj <= -3.1e-6) {
		tmp = (wj + (x / (Math.exp(wj) * (wj + 1.0)))) - (wj / (wj + 1.0));
	} else if (wj <= 5e-6) {
		tmp = x + ((-2.0 * (wj * x)) + ((Math.pow(wj, 2.0) * (1.0 - ((x * -4.0) + (x * 1.5)))) - Math.pow(wj, 3.0)));
	} else {
		tmp = wj + (((x / Math.exp(wj)) - wj) / (wj + 1.0));
	}
	return tmp;
}

Python

def code(wj, x):
	tmp = 0
	if wj <= -3.1e-6:
		tmp = (wj + (x / (math.exp(wj) * (wj + 1.0)))) - (wj / (wj + 1.0))
	elif wj <= 5e-6:
		tmp = x + ((-2.0 * (wj * x)) + ((math.pow(wj, 2.0) * (1.0 - ((x * -4.0) + (x * 1.5)))) - math.pow(wj, 3.0)))
	else:
		tmp = wj + (((x / math.exp(wj)) - wj) / (wj + 1.0))
	return tmp

Julia

function code(wj, x)
	tmp = 0.0
	if (wj <= -3.1e-6)
		tmp = Float64(Float64(wj + Float64(x / Float64(exp(wj) * Float64(wj + 1.0)))) - Float64(wj / Float64(wj + 1.0)));
	elseif (wj <= 5e-6)
		tmp = Float64(x + Float64(Float64(-2.0 * Float64(wj * x)) + Float64(Float64((wj ^ 2.0) * Float64(1.0 - Float64(Float64(x * -4.0) + Float64(x * 1.5)))) - (wj ^ 3.0))));
	else
		tmp = Float64(wj + Float64(Float64(Float64(x / exp(wj)) - wj) / Float64(wj + 1.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(wj, x)
	tmp = 0.0;
	if (wj <= -3.1e-6)
		tmp = (wj + (x / (exp(wj) * (wj + 1.0)))) - (wj / (wj + 1.0));
	elseif (wj <= 5e-6)
		tmp = x + ((-2.0 * (wj * x)) + (((wj ^ 2.0) * (1.0 - ((x * -4.0) + (x * 1.5)))) - (wj ^ 3.0)));
	else
		tmp = wj + (((x / exp(wj)) - wj) / (wj + 1.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[wj_, x_] := If[LessEqual[wj, -3.1e-6], N[(N[(wj + N[(x / N[(N[Exp[wj], $MachinePrecision] * N[(wj + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(wj / N[(wj + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[wj, 5e-6], N[(x + N[(N[(-2.0 * N[(wj * x), $MachinePrecision]), $MachinePrecision] + N[(N[(N[Power[wj, 2.0], $MachinePrecision] * N[(1.0 - N[(N[(x * -4.0), $MachinePrecision] + N[(x * 1.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[Power[wj, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(wj + N[(N[(N[(x / N[Exp[wj], $MachinePrecision]), $MachinePrecision] - wj), $MachinePrecision] / N[(wj + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

&prev;&pcontext;&pcontext2;&ctx;
1. Add Preprocessing

Alternative 2: 98.6% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := wj \cdot e^{wj}\\ t_1 := x \cdot -4 + x \cdot 1.5\\ \mathbf{if}\;wj + \frac{x - t_0}{e^{wj} + t_0} \leq 1.5 \cdot 10^{-11}:\\ \;\;\;\;x + \left(-2 \cdot \left(wj \cdot x\right) + \left({wj}^{3} \cdot \left(-1 - \left(x \cdot -3 + \left(-2 \cdot t_1 + x \cdot 0.6666666666666666\right)\right)\right) + {wj}^{2} \cdot \left(1 - t_1\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(wj + \frac{x}{e^{wj} \cdot \left(wj + 1\right)}\right) - \frac{wj}{wj + 1}\\ \end{array} \end{array} \]

FPCore

(FPCore (wj x)
 :precision binary64
 (let* ((t_0 (* wj (exp wj))) (t_1 (+ (* x -4.0) (* x 1.5))))
   (if (<= (+ wj (/ (- x t_0) (+ (exp wj) t_0))) 1.5e-11)
     (+
      x
      (+
       (* -2.0 (* wj x))
       (+
        (*
         (pow wj 3.0)
         (- -1.0 (+ (* x -3.0) (+ (* -2.0 t_1) (* x 0.6666666666666666)))))
        (* (pow wj 2.0) (- 1.0 t_1)))))
     (- (+ wj (/ x (* (exp wj) (+ wj 1.0)))) (/ wj (+ wj 1.0))))))

C

double code(double wj, double x) {
	double t_0 = wj * exp(wj);
	double t_1 = (x * -4.0) + (x * 1.5);
	double tmp;
	if ((wj + ((x - t_0) / (exp(wj) + t_0))) <= 1.5e-11) {
		tmp = x + ((-2.0 * (wj * x)) + ((pow(wj, 3.0) * (-1.0 - ((x * -3.0) + ((-2.0 * t_1) + (x * 0.6666666666666666))))) + (pow(wj, 2.0) * (1.0 - t_1))));
	} else {
		tmp = (wj + (x / (exp(wj) * (wj + 1.0)))) - (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) :: t_1
    real(8) :: tmp
    t_0 = wj * exp(wj)
    t_1 = (x * (-4.0d0)) + (x * 1.5d0)
    if ((wj + ((x - t_0) / (exp(wj) + t_0))) <= 1.5d-11) then
        tmp = x + (((-2.0d0) * (wj * x)) + (((wj ** 3.0d0) * ((-1.0d0) - ((x * (-3.0d0)) + (((-2.0d0) * t_1) + (x * 0.6666666666666666d0))))) + ((wj ** 2.0d0) * (1.0d0 - t_1))))
    else
        tmp = (wj + (x / (exp(wj) * (wj + 1.0d0)))) - (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 t_1 = (x * -4.0) + (x * 1.5);
	double tmp;
	if ((wj + ((x - t_0) / (Math.exp(wj) + t_0))) <= 1.5e-11) {
		tmp = x + ((-2.0 * (wj * x)) + ((Math.pow(wj, 3.0) * (-1.0 - ((x * -3.0) + ((-2.0 * t_1) + (x * 0.6666666666666666))))) + (Math.pow(wj, 2.0) * (1.0 - t_1))));
	} else {
		tmp = (wj + (x / (Math.exp(wj) * (wj + 1.0)))) - (wj / (wj + 1.0));
	}
	return tmp;
}

Python

def code(wj, x):
	t_0 = wj * math.exp(wj)
	t_1 = (x * -4.0) + (x * 1.5)
	tmp = 0
	if (wj + ((x - t_0) / (math.exp(wj) + t_0))) <= 1.5e-11:
		tmp = x + ((-2.0 * (wj * x)) + ((math.pow(wj, 3.0) * (-1.0 - ((x * -3.0) + ((-2.0 * t_1) + (x * 0.6666666666666666))))) + (math.pow(wj, 2.0) * (1.0 - t_1))))
	else:
		tmp = (wj + (x / (math.exp(wj) * (wj + 1.0)))) - (wj / (wj + 1.0))
	return tmp

Julia

function code(wj, x)
	t_0 = Float64(wj * exp(wj))
	t_1 = Float64(Float64(x * -4.0) + Float64(x * 1.5))
	tmp = 0.0
	if (Float64(wj + Float64(Float64(x - t_0) / Float64(exp(wj) + t_0))) <= 1.5e-11)
		tmp = Float64(x + Float64(Float64(-2.0 * Float64(wj * x)) + Float64(Float64((wj ^ 3.0) * Float64(-1.0 - Float64(Float64(x * -3.0) + Float64(Float64(-2.0 * t_1) + Float64(x * 0.6666666666666666))))) + Float64((wj ^ 2.0) * Float64(1.0 - t_1)))));
	else
		tmp = Float64(Float64(wj + Float64(x / Float64(exp(wj) * Float64(wj + 1.0)))) - Float64(wj / Float64(wj + 1.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(wj, x)
	t_0 = wj * exp(wj);
	t_1 = (x * -4.0) + (x * 1.5);
	tmp = 0.0;
	if ((wj + ((x - t_0) / (exp(wj) + t_0))) <= 1.5e-11)
		tmp = x + ((-2.0 * (wj * x)) + (((wj ^ 3.0) * (-1.0 - ((x * -3.0) + ((-2.0 * t_1) + (x * 0.6666666666666666))))) + ((wj ^ 2.0) * (1.0 - t_1))));
	else
		tmp = (wj + (x / (exp(wj) * (wj + 1.0)))) - (wj / (wj + 1.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[wj_, x_] := Block[{t$95$0 = N[(wj * N[Exp[wj], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(x * -4.0), $MachinePrecision] + N[(x * 1.5), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(wj + N[(N[(x - t$95$0), $MachinePrecision] / N[(N[Exp[wj], $MachinePrecision] + t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 1.5e-11], N[(x + N[(N[(-2.0 * N[(wj * x), $MachinePrecision]), $MachinePrecision] + N[(N[(N[Power[wj, 3.0], $MachinePrecision] * N[(-1.0 - N[(N[(x * -3.0), $MachinePrecision] + N[(N[(-2.0 * t$95$1), $MachinePrecision] + N[(x * 0.6666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[Power[wj, 2.0], $MachinePrecision] * N[(1.0 - t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(wj + N[(x / N[(N[Exp[wj], $MachinePrecision] * N[(wj + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(wj / N[(wj + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

&prev;&pcontext;&pcontext2;&ctx;
1. Add Preprocessing

Alternative 3: 99.4% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;wj \leq -1.7 \cdot 10^{-8}:\\ \;\;\;\;\left(wj + \frac{x}{e^{wj} \cdot \left(wj + 1\right)}\right) - \frac{wj}{wj + 1}\\ \mathbf{elif}\;wj \leq 3.6 \cdot 10^{-6}:\\ \;\;\;\;x + \left({wj}^{2} - {wj}^{3}\right)\\ \mathbf{else}:\\ \;\;\;\;wj + \frac{\frac{x}{e^{wj}} - wj}{wj + 1}\\ \end{array} \end{array} \]

FPCore

(FPCore (wj x)
 :precision binary64
 (if (<= wj -1.7e-8)
   (- (+ wj (/ x (* (exp wj) (+ wj 1.0)))) (/ wj (+ wj 1.0)))
   (if (<= wj 3.6e-6)
     (+ x (- (pow wj 2.0) (pow wj 3.0)))
     (+ wj (/ (- (/ x (exp wj)) wj) (+ wj 1.0))))))

C

double code(double wj, double x) {
	double tmp;
	if (wj <= -1.7e-8) {
		tmp = (wj + (x / (exp(wj) * (wj + 1.0)))) - (wj / (wj + 1.0));
	} else if (wj <= 3.6e-6) {
		tmp = x + (pow(wj, 2.0) - pow(wj, 3.0));
	} else {
		tmp = wj + (((x / exp(wj)) - 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) :: tmp
    if (wj <= (-1.7d-8)) then
        tmp = (wj + (x / (exp(wj) * (wj + 1.0d0)))) - (wj / (wj + 1.0d0))
    else if (wj <= 3.6d-6) then
        tmp = x + ((wj ** 2.0d0) - (wj ** 3.0d0))
    else
        tmp = wj + (((x / exp(wj)) - wj) / (wj + 1.0d0))
    end if
    code = tmp
end function

Java

public static double code(double wj, double x) {
	double tmp;
	if (wj <= -1.7e-8) {
		tmp = (wj + (x / (Math.exp(wj) * (wj + 1.0)))) - (wj / (wj + 1.0));
	} else if (wj <= 3.6e-6) {
		tmp = x + (Math.pow(wj, 2.0) - Math.pow(wj, 3.0));
	} else {
		tmp = wj + (((x / Math.exp(wj)) - wj) / (wj + 1.0));
	}
	return tmp;
}

Python

def code(wj, x):
	tmp = 0
	if wj <= -1.7e-8:
		tmp = (wj + (x / (math.exp(wj) * (wj + 1.0)))) - (wj / (wj + 1.0))
	elif wj <= 3.6e-6:
		tmp = x + (math.pow(wj, 2.0) - math.pow(wj, 3.0))
	else:
		tmp = wj + (((x / math.exp(wj)) - wj) / (wj + 1.0))
	return tmp

Julia

function code(wj, x)
	tmp = 0.0
	if (wj <= -1.7e-8)
		tmp = Float64(Float64(wj + Float64(x / Float64(exp(wj) * Float64(wj + 1.0)))) - Float64(wj / Float64(wj + 1.0)));
	elseif (wj <= 3.6e-6)
		tmp = Float64(x + Float64((wj ^ 2.0) - (wj ^ 3.0)));
	else
		tmp = Float64(wj + Float64(Float64(Float64(x / exp(wj)) - wj) / Float64(wj + 1.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(wj, x)
	tmp = 0.0;
	if (wj <= -1.7e-8)
		tmp = (wj + (x / (exp(wj) * (wj + 1.0)))) - (wj / (wj + 1.0));
	elseif (wj <= 3.6e-6)
		tmp = x + ((wj ^ 2.0) - (wj ^ 3.0));
	else
		tmp = wj + (((x / exp(wj)) - wj) / (wj + 1.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[wj_, x_] := If[LessEqual[wj, -1.7e-8], N[(N[(wj + N[(x / N[(N[Exp[wj], $MachinePrecision] * N[(wj + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(wj / N[(wj + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[wj, 3.6e-6], N[(x + N[(N[Power[wj, 2.0], $MachinePrecision] - N[Power[wj, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(wj + N[(N[(N[(x / N[Exp[wj], $MachinePrecision]), $MachinePrecision] - wj), $MachinePrecision] / N[(wj + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

&prev;&pcontext;&pcontext2;&ctx;
1. Add Preprocessing

Alternative 4: 99.4% accurate, 2.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;wj \leq -6.7 \cdot 10^{-9}:\\ \;\;\;\;\left(wj + \frac{x}{e^{wj} \cdot \left(wj + 1\right)}\right) - \frac{wj}{wj + 1}\\ \mathbf{elif}\;wj \leq 1.4 \cdot 10^{-8}:\\ \;\;\;\;x + \left(-2 \cdot \left(wj \cdot x\right) + {wj}^{2} \cdot \left(1 - \left(x \cdot -4 + x \cdot 1.5\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;wj + \frac{\frac{x}{e^{wj}} - wj}{wj + 1}\\ \end{array} \end{array} \]

FPCore

(FPCore (wj x)
 :precision binary64
 (if (<= wj -6.7e-9)
   (- (+ wj (/ x (* (exp wj) (+ wj 1.0)))) (/ wj (+ wj 1.0)))
   (if (<= wj 1.4e-8)
     (+
      x
      (+ (* -2.0 (* wj x)) (* (pow wj 2.0) (- 1.0 (+ (* x -4.0) (* x 1.5))))))
     (+ wj (/ (- (/ x (exp wj)) wj) (+ wj 1.0))))))

C

double code(double wj, double x) {
	double tmp;
	if (wj <= -6.7e-9) {
		tmp = (wj + (x / (exp(wj) * (wj + 1.0)))) - (wj / (wj + 1.0));
	} else if (wj <= 1.4e-8) {
		tmp = x + ((-2.0 * (wj * x)) + (pow(wj, 2.0) * (1.0 - ((x * -4.0) + (x * 1.5)))));
	} else {
		tmp = wj + (((x / exp(wj)) - 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) :: tmp
    if (wj <= (-6.7d-9)) then
        tmp = (wj + (x / (exp(wj) * (wj + 1.0d0)))) - (wj / (wj + 1.0d0))
    else if (wj <= 1.4d-8) then
        tmp = x + (((-2.0d0) * (wj * x)) + ((wj ** 2.0d0) * (1.0d0 - ((x * (-4.0d0)) + (x * 1.5d0)))))
    else
        tmp = wj + (((x / exp(wj)) - wj) / (wj + 1.0d0))
    end if
    code = tmp
end function

Java

public static double code(double wj, double x) {
	double tmp;
	if (wj <= -6.7e-9) {
		tmp = (wj + (x / (Math.exp(wj) * (wj + 1.0)))) - (wj / (wj + 1.0));
	} else if (wj <= 1.4e-8) {
		tmp = x + ((-2.0 * (wj * x)) + (Math.pow(wj, 2.0) * (1.0 - ((x * -4.0) + (x * 1.5)))));
	} else {
		tmp = wj + (((x / Math.exp(wj)) - wj) / (wj + 1.0));
	}
	return tmp;
}

Python

def code(wj, x):
	tmp = 0
	if wj <= -6.7e-9:
		tmp = (wj + (x / (math.exp(wj) * (wj + 1.0)))) - (wj / (wj + 1.0))
	elif wj <= 1.4e-8:
		tmp = x + ((-2.0 * (wj * x)) + (math.pow(wj, 2.0) * (1.0 - ((x * -4.0) + (x * 1.5)))))
	else:
		tmp = wj + (((x / math.exp(wj)) - wj) / (wj + 1.0))
	return tmp

Julia

function code(wj, x)
	tmp = 0.0
	if (wj <= -6.7e-9)
		tmp = Float64(Float64(wj + Float64(x / Float64(exp(wj) * Float64(wj + 1.0)))) - Float64(wj / Float64(wj + 1.0)));
	elseif (wj <= 1.4e-8)
		tmp = Float64(x + Float64(Float64(-2.0 * Float64(wj * x)) + Float64((wj ^ 2.0) * Float64(1.0 - Float64(Float64(x * -4.0) + Float64(x * 1.5))))));
	else
		tmp = Float64(wj + Float64(Float64(Float64(x / exp(wj)) - wj) / Float64(wj + 1.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(wj, x)
	tmp = 0.0;
	if (wj <= -6.7e-9)
		tmp = (wj + (x / (exp(wj) * (wj + 1.0)))) - (wj / (wj + 1.0));
	elseif (wj <= 1.4e-8)
		tmp = x + ((-2.0 * (wj * x)) + ((wj ^ 2.0) * (1.0 - ((x * -4.0) + (x * 1.5)))));
	else
		tmp = wj + (((x / exp(wj)) - wj) / (wj + 1.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[wj_, x_] := If[LessEqual[wj, -6.7e-9], N[(N[(wj + N[(x / N[(N[Exp[wj], $MachinePrecision] * N[(wj + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(wj / N[(wj + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[wj, 1.4e-8], N[(x + N[(N[(-2.0 * N[(wj * x), $MachinePrecision]), $MachinePrecision] + N[(N[Power[wj, 2.0], $MachinePrecision] * N[(1.0 - N[(N[(x * -4.0), $MachinePrecision] + N[(x * 1.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(wj + N[(N[(N[(x / N[Exp[wj], $MachinePrecision]), $MachinePrecision] - wj), $MachinePrecision] / N[(wj + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

&prev;&pcontext;&pcontext2;&ctx;
1. Add Preprocessing

Alternative 5: 89.1% accurate, 2.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;wj \leq -5.5 \cdot 10^{-17}:\\ \;\;\;\;\left(wj + \frac{x}{e^{wj} \cdot \left(wj + 1\right)}\right) - \frac{wj}{wj + 1}\\ \mathbf{elif}\;wj \leq 6.5 \cdot 10^{-20}:\\ \;\;\;\;x - {wj}^{3}\\ \mathbf{else}:\\ \;\;\;\;wj + \frac{\frac{x}{e^{wj}} - wj}{wj + 1}\\ \end{array} \end{array} \]

FPCore

(FPCore (wj x)
 :precision binary64
 (if (<= wj -5.5e-17)
   (- (+ wj (/ x (* (exp wj) (+ wj 1.0)))) (/ wj (+ wj 1.0)))
   (if (<= wj 6.5e-20)
     (- x (pow wj 3.0))
     (+ wj (/ (- (/ x (exp wj)) wj) (+ wj 1.0))))))

C

double code(double wj, double x) {
	double tmp;
	if (wj <= -5.5e-17) {
		tmp = (wj + (x / (exp(wj) * (wj + 1.0)))) - (wj / (wj + 1.0));
	} else if (wj <= 6.5e-20) {
		tmp = x - pow(wj, 3.0);
	} else {
		tmp = wj + (((x / exp(wj)) - 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) :: tmp
    if (wj <= (-5.5d-17)) then
        tmp = (wj + (x / (exp(wj) * (wj + 1.0d0)))) - (wj / (wj + 1.0d0))
    else if (wj <= 6.5d-20) then
        tmp = x - (wj ** 3.0d0)
    else
        tmp = wj + (((x / exp(wj)) - wj) / (wj + 1.0d0))
    end if
    code = tmp
end function

Java

public static double code(double wj, double x) {
	double tmp;
	if (wj <= -5.5e-17) {
		tmp = (wj + (x / (Math.exp(wj) * (wj + 1.0)))) - (wj / (wj + 1.0));
	} else if (wj <= 6.5e-20) {
		tmp = x - Math.pow(wj, 3.0);
	} else {
		tmp = wj + (((x / Math.exp(wj)) - wj) / (wj + 1.0));
	}
	return tmp;
}

Python

def code(wj, x):
	tmp = 0
	if wj <= -5.5e-17:
		tmp = (wj + (x / (math.exp(wj) * (wj + 1.0)))) - (wj / (wj + 1.0))
	elif wj <= 6.5e-20:
		tmp = x - math.pow(wj, 3.0)
	else:
		tmp = wj + (((x / math.exp(wj)) - wj) / (wj + 1.0))
	return tmp

Julia

function code(wj, x)
	tmp = 0.0
	if (wj <= -5.5e-17)
		tmp = Float64(Float64(wj + Float64(x / Float64(exp(wj) * Float64(wj + 1.0)))) - Float64(wj / Float64(wj + 1.0)));
	elseif (wj <= 6.5e-20)
		tmp = Float64(x - (wj ^ 3.0));
	else
		tmp = Float64(wj + Float64(Float64(Float64(x / exp(wj)) - wj) / Float64(wj + 1.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(wj, x)
	tmp = 0.0;
	if (wj <= -5.5e-17)
		tmp = (wj + (x / (exp(wj) * (wj + 1.0)))) - (wj / (wj + 1.0));
	elseif (wj <= 6.5e-20)
		tmp = x - (wj ^ 3.0);
	else
		tmp = wj + (((x / exp(wj)) - wj) / (wj + 1.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[wj_, x_] := If[LessEqual[wj, -5.5e-17], N[(N[(wj + N[(x / N[(N[Exp[wj], $MachinePrecision] * N[(wj + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(wj / N[(wj + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[wj, 6.5e-20], N[(x - N[Power[wj, 3.0], $MachinePrecision]), $MachinePrecision], N[(wj + N[(N[(N[(x / N[Exp[wj], $MachinePrecision]), $MachinePrecision] - wj), $MachinePrecision] / N[(wj + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

&prev;&pcontext;&pcontext2;&ctx;
1. Add Preprocessing

Alternative 6: 89.1% accurate, 2.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;wj \leq -5.5 \cdot 10^{-17} \lor \neg \left(wj \leq 5.2 \cdot 10^{-20}\right):\\ \;\;\;\;wj + \frac{\frac{x}{e^{wj}} - wj}{wj + 1}\\ \mathbf{else}:\\ \;\;\;\;x - {wj}^{3}\\ \end{array} \end{array} \]

FPCore

(FPCore (wj x)
 :precision binary64
 (if (or (<= wj -5.5e-17) (not (<= wj 5.2e-20)))
   (+ wj (/ (- (/ x (exp wj)) wj) (+ wj 1.0)))
   (- x (pow wj 3.0))))

C

double code(double wj, double x) {
	double tmp;
	if ((wj <= -5.5e-17) || !(wj <= 5.2e-20)) {
		tmp = wj + (((x / exp(wj)) - wj) / (wj + 1.0));
	} else {
		tmp = x - pow(wj, 3.0);
	}
	return tmp;
}

Fortran

real(8) function code(wj, x)
    real(8), intent (in) :: wj
    real(8), intent (in) :: x
    real(8) :: tmp
    if ((wj <= (-5.5d-17)) .or. (.not. (wj <= 5.2d-20))) then
        tmp = wj + (((x / exp(wj)) - wj) / (wj + 1.0d0))
    else
        tmp = x - (wj ** 3.0d0)
    end if
    code = tmp
end function

Java

public static double code(double wj, double x) {
	double tmp;
	if ((wj <= -5.5e-17) || !(wj <= 5.2e-20)) {
		tmp = wj + (((x / Math.exp(wj)) - wj) / (wj + 1.0));
	} else {
		tmp = x - Math.pow(wj, 3.0);
	}
	return tmp;
}

Python

def code(wj, x):
	tmp = 0
	if (wj <= -5.5e-17) or not (wj <= 5.2e-20):
		tmp = wj + (((x / math.exp(wj)) - wj) / (wj + 1.0))
	else:
		tmp = x - math.pow(wj, 3.0)
	return tmp

Julia

function code(wj, x)
	tmp = 0.0
	if ((wj <= -5.5e-17) || !(wj <= 5.2e-20))
		tmp = Float64(wj + Float64(Float64(Float64(x / exp(wj)) - wj) / Float64(wj + 1.0)));
	else
		tmp = Float64(x - (wj ^ 3.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(wj, x)
	tmp = 0.0;
	if ((wj <= -5.5e-17) || ~((wj <= 5.2e-20)))
		tmp = wj + (((x / exp(wj)) - wj) / (wj + 1.0));
	else
		tmp = x - (wj ^ 3.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[wj_, x_] := If[Or[LessEqual[wj, -5.5e-17], N[Not[LessEqual[wj, 5.2e-20]], $MachinePrecision]], N[(wj + N[(N[(N[(x / N[Exp[wj], $MachinePrecision]), $MachinePrecision] - wj), $MachinePrecision] / N[(wj + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x - N[Power[wj, 3.0], $MachinePrecision]), $MachinePrecision]]

Derivation

&prev;&pcontext;&pcontext2;&ctx;
1. Add Preprocessing

Alternative 7: 87.2% accurate, 2.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;wj \leq 6.2 \cdot 10^{-18}:\\ \;\;\;\;\frac{x}{e^{wj} \cdot \left(wj + 1\right)}\\ \mathbf{else}:\\ \;\;\;\;wj + \frac{\left(x - wj \cdot x\right) - wj}{wj + 1}\\ \end{array} \end{array} \]

FPCore

(FPCore (wj x)
 :precision binary64
 (if (<= wj 6.2e-18)
   (/ x (* (exp wj) (+ wj 1.0)))
   (+ wj (/ (- (- x (* wj x)) wj) (+ wj 1.0)))))

C

double code(double wj, double x) {
	double tmp;
	if (wj <= 6.2e-18) {
		tmp = x / (exp(wj) * (wj + 1.0));
	} else {
		tmp = wj + (((x - (wj * x)) - 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) :: tmp
    if (wj <= 6.2d-18) then
        tmp = x / (exp(wj) * (wj + 1.0d0))
    else
        tmp = wj + (((x - (wj * x)) - wj) / (wj + 1.0d0))
    end if
    code = tmp
end function

Java

public static double code(double wj, double x) {
	double tmp;
	if (wj <= 6.2e-18) {
		tmp = x / (Math.exp(wj) * (wj + 1.0));
	} else {
		tmp = wj + (((x - (wj * x)) - wj) / (wj + 1.0));
	}
	return tmp;
}

Python

def code(wj, x):
	tmp = 0
	if wj <= 6.2e-18:
		tmp = x / (math.exp(wj) * (wj + 1.0))
	else:
		tmp = wj + (((x - (wj * x)) - wj) / (wj + 1.0))
	return tmp

Julia

function code(wj, x)
	tmp = 0.0
	if (wj <= 6.2e-18)
		tmp = Float64(x / Float64(exp(wj) * Float64(wj + 1.0)));
	else
		tmp = Float64(wj + Float64(Float64(Float64(x - Float64(wj * x)) - wj) / Float64(wj + 1.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(wj, x)
	tmp = 0.0;
	if (wj <= 6.2e-18)
		tmp = x / (exp(wj) * (wj + 1.0));
	else
		tmp = wj + (((x - (wj * x)) - wj) / (wj + 1.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[wj_, x_] := If[LessEqual[wj, 6.2e-18], N[(x / N[(N[Exp[wj], $MachinePrecision] * N[(wj + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(wj + N[(N[(N[(x - N[(wj * x), $MachinePrecision]), $MachinePrecision] - wj), $MachinePrecision] / N[(wj + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

&prev;&pcontext;&pcontext2;&ctx;
1. Add Preprocessing

Alternative 8: 86.8% accurate, 2.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;wj \leq -5.5 \cdot 10^{-17} \lor \neg \left(wj \leq 9.2 \cdot 10^{-20}\right):\\ \;\;\;\;wj + \frac{\left(x - wj \cdot x\right) - wj}{wj + 1}\\ \mathbf{else}:\\ \;\;\;\;x - {wj}^{3}\\ \end{array} \end{array} \]

FPCore

(FPCore (wj x)
 :precision binary64
 (if (or (<= wj -5.5e-17) (not (<= wj 9.2e-20)))
   (+ wj (/ (- (- x (* wj x)) wj) (+ wj 1.0)))
   (- x (pow wj 3.0))))

C

double code(double wj, double x) {
	double tmp;
	if ((wj <= -5.5e-17) || !(wj <= 9.2e-20)) {
		tmp = wj + (((x - (wj * x)) - wj) / (wj + 1.0));
	} else {
		tmp = x - pow(wj, 3.0);
	}
	return tmp;
}

Fortran

real(8) function code(wj, x)
    real(8), intent (in) :: wj
    real(8), intent (in) :: x
    real(8) :: tmp
    if ((wj <= (-5.5d-17)) .or. (.not. (wj <= 9.2d-20))) then
        tmp = wj + (((x - (wj * x)) - wj) / (wj + 1.0d0))
    else
        tmp = x - (wj ** 3.0d0)
    end if
    code = tmp
end function

Java

public static double code(double wj, double x) {
	double tmp;
	if ((wj <= -5.5e-17) || !(wj <= 9.2e-20)) {
		tmp = wj + (((x - (wj * x)) - wj) / (wj + 1.0));
	} else {
		tmp = x - Math.pow(wj, 3.0);
	}
	return tmp;
}

Python

def code(wj, x):
	tmp = 0
	if (wj <= -5.5e-17) or not (wj <= 9.2e-20):
		tmp = wj + (((x - (wj * x)) - wj) / (wj + 1.0))
	else:
		tmp = x - math.pow(wj, 3.0)
	return tmp

Julia

function code(wj, x)
	tmp = 0.0
	if ((wj <= -5.5e-17) || !(wj <= 9.2e-20))
		tmp = Float64(wj + Float64(Float64(Float64(x - Float64(wj * x)) - wj) / Float64(wj + 1.0)));
	else
		tmp = Float64(x - (wj ^ 3.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(wj, x)
	tmp = 0.0;
	if ((wj <= -5.5e-17) || ~((wj <= 9.2e-20)))
		tmp = wj + (((x - (wj * x)) - wj) / (wj + 1.0));
	else
		tmp = x - (wj ^ 3.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[wj_, x_] := If[Or[LessEqual[wj, -5.5e-17], N[Not[LessEqual[wj, 9.2e-20]], $MachinePrecision]], N[(wj + N[(N[(N[(x - N[(wj * x), $MachinePrecision]), $MachinePrecision] - wj), $MachinePrecision] / N[(wj + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x - N[Power[wj, 3.0], $MachinePrecision]), $MachinePrecision]]

Derivation

&prev;&pcontext;&pcontext2;&ctx;
1. Add Preprocessing

Alternative 9: 86.8% accurate, 18.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;wj \leq -4 \cdot 10^{-15} \lor \neg \left(wj \leq 1.85 \cdot 10^{-17}\right):\\ \;\;\;\;wj + \frac{\left(x - wj \cdot x\right) - wj}{wj + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{1 + wj \cdot 2}\\ \end{array} \end{array} \]

FPCore

(FPCore (wj x)
 :precision binary64
 (if (or (<= wj -4e-15) (not (<= wj 1.85e-17)))
   (+ wj (/ (- (- x (* wj x)) wj) (+ wj 1.0)))
   (/ x (+ 1.0 (* wj 2.0)))))

C

double code(double wj, double x) {
	double tmp;
	if ((wj <= -4e-15) || !(wj <= 1.85e-17)) {
		tmp = wj + (((x - (wj * x)) - wj) / (wj + 1.0));
	} else {
		tmp = x / (1.0 + (wj * 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 <= (-4d-15)) .or. (.not. (wj <= 1.85d-17))) then
        tmp = wj + (((x - (wj * x)) - wj) / (wj + 1.0d0))
    else
        tmp = x / (1.0d0 + (wj * 2.0d0))
    end if
    code = tmp
end function

Java

public static double code(double wj, double x) {
	double tmp;
	if ((wj <= -4e-15) || !(wj <= 1.85e-17)) {
		tmp = wj + (((x - (wj * x)) - wj) / (wj + 1.0));
	} else {
		tmp = x / (1.0 + (wj * 2.0));
	}
	return tmp;
}

Python

def code(wj, x):
	tmp = 0
	if (wj <= -4e-15) or not (wj <= 1.85e-17):
		tmp = wj + (((x - (wj * x)) - wj) / (wj + 1.0))
	else:
		tmp = x / (1.0 + (wj * 2.0))
	return tmp

Julia

function code(wj, x)
	tmp = 0.0
	if ((wj <= -4e-15) || !(wj <= 1.85e-17))
		tmp = Float64(wj + Float64(Float64(Float64(x - Float64(wj * x)) - wj) / Float64(wj + 1.0)));
	else
		tmp = Float64(x / Float64(1.0 + Float64(wj * 2.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(wj, x)
	tmp = 0.0;
	if ((wj <= -4e-15) || ~((wj <= 1.85e-17)))
		tmp = wj + (((x - (wj * x)) - wj) / (wj + 1.0));
	else
		tmp = x / (1.0 + (wj * 2.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[wj_, x_] := If[Or[LessEqual[wj, -4e-15], N[Not[LessEqual[wj, 1.85e-17]], $MachinePrecision]], N[(wj + N[(N[(N[(x - N[(wj * x), $MachinePrecision]), $MachinePrecision] - wj), $MachinePrecision] / N[(wj + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x / N[(1.0 + N[(wj * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

&prev;&pcontext;&pcontext2;&ctx;
1. Add Preprocessing

Alternative 10: 85.8% accurate, 28.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;wj \leq 6 \cdot 10^{-8}:\\ \;\;\;\;\frac{x}{1 + wj \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;wj + wj \cdot \frac{-1}{wj + 1}\\ \end{array} \end{array} \]

FPCore

(FPCore (wj x)
 :precision binary64
 (if (<= wj 6e-8) (/ x (+ 1.0 (* wj 2.0))) (+ wj (* wj (/ -1.0 (+ wj 1.0))))))

C

double code(double wj, double x) {
	double tmp;
	if (wj <= 6e-8) {
		tmp = x / (1.0 + (wj * 2.0));
	} else {
		tmp = wj + (wj * (-1.0 / (wj + 1.0)));
	}
	return tmp;
}

Fortran

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

Java

public static double code(double wj, double x) {
	double tmp;
	if (wj <= 6e-8) {
		tmp = x / (1.0 + (wj * 2.0));
	} else {
		tmp = wj + (wj * (-1.0 / (wj + 1.0)));
	}
	return tmp;
}

Python

def code(wj, x):
	tmp = 0
	if wj <= 6e-8:
		tmp = x / (1.0 + (wj * 2.0))
	else:
		tmp = wj + (wj * (-1.0 / (wj + 1.0)))
	return tmp

Julia

function code(wj, x)
	tmp = 0.0
	if (wj <= 6e-8)
		tmp = Float64(x / Float64(1.0 + Float64(wj * 2.0)));
	else
		tmp = Float64(wj + Float64(wj * Float64(-1.0 / Float64(wj + 1.0))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(wj, x)
	tmp = 0.0;
	if (wj <= 6e-8)
		tmp = x / (1.0 + (wj * 2.0));
	else
		tmp = wj + (wj * (-1.0 / (wj + 1.0)));
	end
	tmp_2 = tmp;
end

Wolfram

code[wj_, x_] := If[LessEqual[wj, 6e-8], N[(x / N[(1.0 + N[(wj * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(wj + N[(wj * N[(-1.0 / N[(wj + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

&prev;&pcontext;&pcontext2;&ctx;
1. Add Preprocessing

Alternative 11: 85.6% accurate, 28.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;wj \leq 6.4 \cdot 10^{-15}:\\ \;\;\;\;\frac{x}{1 + wj \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\left(wj + x\right) - \frac{wj}{wj + 1}\\ \end{array} \end{array} \]

FPCore

(FPCore (wj x)
 :precision binary64
 (if (<= wj 6.4e-15) (/ x (+ 1.0 (* wj 2.0))) (- (+ wj x) (/ wj (+ wj 1.0)))))

C

double code(double wj, double x) {
	double tmp;
	if (wj <= 6.4e-15) {
		tmp = x / (1.0 + (wj * 2.0));
	} else {
		tmp = (wj + x) - (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) :: tmp
    if (wj <= 6.4d-15) then
        tmp = x / (1.0d0 + (wj * 2.0d0))
    else
        tmp = (wj + x) - (wj / (wj + 1.0d0))
    end if
    code = tmp
end function

Java

public static double code(double wj, double x) {
	double tmp;
	if (wj <= 6.4e-15) {
		tmp = x / (1.0 + (wj * 2.0));
	} else {
		tmp = (wj + x) - (wj / (wj + 1.0));
	}
	return tmp;
}

Python

def code(wj, x):
	tmp = 0
	if wj <= 6.4e-15:
		tmp = x / (1.0 + (wj * 2.0))
	else:
		tmp = (wj + x) - (wj / (wj + 1.0))
	return tmp

Julia

function code(wj, x)
	tmp = 0.0
	if (wj <= 6.4e-15)
		tmp = Float64(x / Float64(1.0 + Float64(wj * 2.0)));
	else
		tmp = Float64(Float64(wj + x) - Float64(wj / Float64(wj + 1.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(wj, x)
	tmp = 0.0;
	if (wj <= 6.4e-15)
		tmp = x / (1.0 + (wj * 2.0));
	else
		tmp = (wj + x) - (wj / (wj + 1.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[wj_, x_] := If[LessEqual[wj, 6.4e-15], N[(x / N[(1.0 + N[(wj * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(wj + x), $MachinePrecision] - N[(wj / N[(wj + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

&prev;&pcontext;&pcontext2;&ctx;
1. Add Preprocessing

Alternative 12: 85.7% accurate, 34.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;wj \leq 4 \cdot 10^{-8}:\\ \;\;\;\;x + -2 \cdot \left(wj \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;wj - \frac{wj}{wj + 1}\\ \end{array} \end{array} \]

FPCore

(FPCore (wj x)
 :precision binary64
 (if (<= wj 4e-8) (+ x (* -2.0 (* wj x))) (- wj (/ wj (+ wj 1.0)))))

C

double code(double wj, double x) {
	double tmp;
	if (wj <= 4e-8) {
		tmp = x + (-2.0 * (wj * x));
	} else {
		tmp = wj - (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) :: tmp
    if (wj <= 4d-8) then
        tmp = x + ((-2.0d0) * (wj * x))
    else
        tmp = wj - (wj / (wj + 1.0d0))
    end if
    code = tmp
end function

Java

public static double code(double wj, double x) {
	double tmp;
	if (wj <= 4e-8) {
		tmp = x + (-2.0 * (wj * x));
	} else {
		tmp = wj - (wj / (wj + 1.0));
	}
	return tmp;
}

Python

def code(wj, x):
	tmp = 0
	if wj <= 4e-8:
		tmp = x + (-2.0 * (wj * x))
	else:
		tmp = wj - (wj / (wj + 1.0))
	return tmp

Julia

function code(wj, x)
	tmp = 0.0
	if (wj <= 4e-8)
		tmp = Float64(x + Float64(-2.0 * Float64(wj * x)));
	else
		tmp = Float64(wj - Float64(wj / Float64(wj + 1.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(wj, x)
	tmp = 0.0;
	if (wj <= 4e-8)
		tmp = x + (-2.0 * (wj * x));
	else
		tmp = wj - (wj / (wj + 1.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[wj_, x_] := If[LessEqual[wj, 4e-8], N[(x + N[(-2.0 * N[(wj * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(wj - N[(wj / N[(wj + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

&prev;&pcontext;&pcontext2;&ctx;
1. Add Preprocessing

Alternative 13: 85.8% accurate, 34.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;wj \leq 7.5 \cdot 10^{-9}:\\ \;\;\;\;\frac{x}{1 + wj \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;wj - \frac{wj}{wj + 1}\\ \end{array} \end{array} \]

FPCore

(FPCore (wj x)
 :precision binary64
 (if (<= wj 7.5e-9) (/ x (+ 1.0 (* wj 2.0))) (- wj (/ wj (+ wj 1.0)))))

C

double code(double wj, double x) {
	double tmp;
	if (wj <= 7.5e-9) {
		tmp = x / (1.0 + (wj * 2.0));
	} else {
		tmp = wj - (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) :: tmp
    if (wj <= 7.5d-9) then
        tmp = x / (1.0d0 + (wj * 2.0d0))
    else
        tmp = wj - (wj / (wj + 1.0d0))
    end if
    code = tmp
end function

Java

public static double code(double wj, double x) {
	double tmp;
	if (wj <= 7.5e-9) {
		tmp = x / (1.0 + (wj * 2.0));
	} else {
		tmp = wj - (wj / (wj + 1.0));
	}
	return tmp;
}

Python

def code(wj, x):
	tmp = 0
	if wj <= 7.5e-9:
		tmp = x / (1.0 + (wj * 2.0))
	else:
		tmp = wj - (wj / (wj + 1.0))
	return tmp

Julia

function code(wj, x)
	tmp = 0.0
	if (wj <= 7.5e-9)
		tmp = Float64(x / Float64(1.0 + Float64(wj * 2.0)));
	else
		tmp = Float64(wj - Float64(wj / Float64(wj + 1.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(wj, x)
	tmp = 0.0;
	if (wj <= 7.5e-9)
		tmp = x / (1.0 + (wj * 2.0));
	else
		tmp = wj - (wj / (wj + 1.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[wj_, x_] := If[LessEqual[wj, 7.5e-9], N[(x / N[(1.0 + N[(wj * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(wj - N[(wj / N[(wj + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

&prev;&pcontext;&pcontext2;&ctx;
1. Add Preprocessing

Alternative 14: 84.3% 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

&prev;&pcontext;&pcontext2;&ctx;
1. Add Preprocessing

Alternative 15: 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

&prev;&pcontext;&pcontext2;&ctx;
1. Add Preprocessing

Alternative 16: 83.8% accurate, 313.0× speedup

Math

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

FPCore

(FPCore (wj x) :precision binary64 x)

C

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

Fortran

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

Java

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

Python

def code(wj, x):
	return x

Julia

function code(wj, x)
	return x
end

MATLAB

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

Wolfram

code[wj_, x_] := x

Derivation

&prev;&pcontext;&pcontext2;&ctx;
1. Add Preprocessing

Developer target: 78.5% 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 2023343 
(FPCore (wj x)
  :name "Jmat.Real.lambertw, newton loop step"
  :precision binary64

  :herbie-target
  (- wj (- (/ wj (+ wj 1.0)) (/ x (+ (exp wj) (* wj (exp wj))))))

  (- wj (/ (- (* wj (exp wj)) x) (+ (exp wj) (* wj (exp wj))))))