2nthrt (problem 3.4.6)

Percentage Accurate: 53.4% → 84.1%

Time: 35.0s

Alternatives: 23

Speedup: 1.9×

• could not determine a ground truth

  1. x = -2
  2. n = 4.94066e-324

Specification

Math

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

FPCore

(FPCore (x n)
 :precision binary64
 (- (pow (+ x 1.0) (/ 1.0 n)) (pow x (/ 1.0 n))))

C

double code(double x, double n) {
	return pow((x + 1.0), (1.0 / n)) - pow(x, (1.0 / n));
}

Fortran

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

Java

public static double code(double x, double n) {
	return Math.pow((x + 1.0), (1.0 / n)) - Math.pow(x, (1.0 / n));
}

Python

def code(x, n):
	return math.pow((x + 1.0), (1.0 / n)) - math.pow(x, (1.0 / n))

Julia

function code(x, n)
	return Float64((Float64(x + 1.0) ^ Float64(1.0 / n)) - (x ^ Float64(1.0 / n)))
end

MATLAB

function tmp = code(x, n)
	tmp = ((x + 1.0) ^ (1.0 / n)) - (x ^ (1.0 / n));
end

Wolfram

code[x_, n_] := N[(N[Power[N[(x + 1.0), $MachinePrecision], N[(1.0 / n), $MachinePrecision]], $MachinePrecision] - N[Power[x, N[(1.0 / n), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Initial Program: 53.4% accurate, 1.0× speedup

Math

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

FPCore

(FPCore (x n)
 :precision binary64
 (- (pow (+ x 1.0) (/ 1.0 n)) (pow x (/ 1.0 n))))

C

double code(double x, double n) {
	return pow((x + 1.0), (1.0 / n)) - pow(x, (1.0 / n));
}

Fortran

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

Java

public static double code(double x, double n) {
	return Math.pow((x + 1.0), (1.0 / n)) - Math.pow(x, (1.0 / n));
}

Python

def code(x, n):
	return math.pow((x + 1.0), (1.0 / n)) - math.pow(x, (1.0 / n))

Julia

function code(x, n)
	return Float64((Float64(x + 1.0) ^ Float64(1.0 / n)) - (x ^ Float64(1.0 / n)))
end

MATLAB

function tmp = code(x, n)
	tmp = ((x + 1.0) ^ (1.0 / n)) - (x ^ (1.0 / n));
end

Wolfram

code[x_, n_] := N[(N[Power[N[(x + 1.0), $MachinePrecision], N[(1.0 / n), $MachinePrecision]], $MachinePrecision] - N[Power[x, N[(1.0 / n), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Alternative 1: 84.1% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \log \left(x + 1\right)\\ \mathbf{if}\;\frac{1}{n} \leq -2 \cdot 10^{-114}:\\ \;\;\;\;\frac{{\left(e^{\frac{\log x}{-n}}\right)}^{-1}}{x \cdot n}\\ \mathbf{elif}\;\frac{1}{n} \leq 5 \cdot 10^{-166}:\\ \;\;\;\;\left(0.5 \cdot \frac{{t_0}^{2}}{{n}^{2}} + \frac{t_0}{n}\right) - \left(\frac{{\log x}^{2}}{{n}^{2}} \cdot 0.5 + \frac{\log x}{n}\right)\\ \mathbf{elif}\;\frac{1}{n} \leq 5 \cdot 10^{-136}:\\ \;\;\;\;\frac{\frac{0.3333333333333333}{{x}^{3}} + \left(\frac{1}{x} - \frac{0.5}{{x}^{2}}\right)}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 5 \cdot 10^{-20}:\\ \;\;\;\;\left(\mathsf{log1p}\left(x\right) - \log x\right) \cdot \frac{1}{n}\\ \mathbf{else}:\\ \;\;\;\;{\left(\sqrt{e^{\frac{\mathsf{log1p}\left(x\right)}{n}} - {x}^{\left(\frac{1}{n}\right)}}\right)}^{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (let* ((t_0 (log (+ x 1.0))))
   (if (<= (/ 1.0 n) -2e-114)
     (/ (pow (exp (/ (log x) (- n))) -1.0) (* x n))
     (if (<= (/ 1.0 n) 5e-166)
       (-
        (+ (* 0.5 (/ (pow t_0 2.0) (pow n 2.0))) (/ t_0 n))
        (+ (* (/ (pow (log x) 2.0) (pow n 2.0)) 0.5) (/ (log x) n)))
       (if (<= (/ 1.0 n) 5e-136)
         (/
          (+
           (/ 0.3333333333333333 (pow x 3.0))
           (- (/ 1.0 x) (/ 0.5 (pow x 2.0))))
          n)
         (if (<= (/ 1.0 n) 5e-20)
           (* (- (log1p x) (log x)) (/ 1.0 n))
           (pow (sqrt (- (exp (/ (log1p x) n)) (pow x (/ 1.0 n)))) 2.0)))))))

C

double code(double x, double n) {
	double t_0 = log((x + 1.0));
	double tmp;
	if ((1.0 / n) <= -2e-114) {
		tmp = pow(exp((log(x) / -n)), -1.0) / (x * n);
	} else if ((1.0 / n) <= 5e-166) {
		tmp = ((0.5 * (pow(t_0, 2.0) / pow(n, 2.0))) + (t_0 / n)) - (((pow(log(x), 2.0) / pow(n, 2.0)) * 0.5) + (log(x) / n));
	} else if ((1.0 / n) <= 5e-136) {
		tmp = ((0.3333333333333333 / pow(x, 3.0)) + ((1.0 / x) - (0.5 / pow(x, 2.0)))) / n;
	} else if ((1.0 / n) <= 5e-20) {
		tmp = (log1p(x) - log(x)) * (1.0 / n);
	} else {
		tmp = pow(sqrt((exp((log1p(x) / n)) - pow(x, (1.0 / n)))), 2.0);
	}
	return tmp;
}

Java

public static double code(double x, double n) {
	double t_0 = Math.log((x + 1.0));
	double tmp;
	if ((1.0 / n) <= -2e-114) {
		tmp = Math.pow(Math.exp((Math.log(x) / -n)), -1.0) / (x * n);
	} else if ((1.0 / n) <= 5e-166) {
		tmp = ((0.5 * (Math.pow(t_0, 2.0) / Math.pow(n, 2.0))) + (t_0 / n)) - (((Math.pow(Math.log(x), 2.0) / Math.pow(n, 2.0)) * 0.5) + (Math.log(x) / n));
	} else if ((1.0 / n) <= 5e-136) {
		tmp = ((0.3333333333333333 / Math.pow(x, 3.0)) + ((1.0 / x) - (0.5 / Math.pow(x, 2.0)))) / n;
	} else if ((1.0 / n) <= 5e-20) {
		tmp = (Math.log1p(x) - Math.log(x)) * (1.0 / n);
	} else {
		tmp = Math.pow(Math.sqrt((Math.exp((Math.log1p(x) / n)) - Math.pow(x, (1.0 / n)))), 2.0);
	}
	return tmp;
}

Python

def code(x, n):
	t_0 = math.log((x + 1.0))
	tmp = 0
	if (1.0 / n) <= -2e-114:
		tmp = math.pow(math.exp((math.log(x) / -n)), -1.0) / (x * n)
	elif (1.0 / n) <= 5e-166:
		tmp = ((0.5 * (math.pow(t_0, 2.0) / math.pow(n, 2.0))) + (t_0 / n)) - (((math.pow(math.log(x), 2.0) / math.pow(n, 2.0)) * 0.5) + (math.log(x) / n))
	elif (1.0 / n) <= 5e-136:
		tmp = ((0.3333333333333333 / math.pow(x, 3.0)) + ((1.0 / x) - (0.5 / math.pow(x, 2.0)))) / n
	elif (1.0 / n) <= 5e-20:
		tmp = (math.log1p(x) - math.log(x)) * (1.0 / n)
	else:
		tmp = math.pow(math.sqrt((math.exp((math.log1p(x) / n)) - math.pow(x, (1.0 / n)))), 2.0)
	return tmp

Julia

function code(x, n)
	t_0 = log(Float64(x + 1.0))
	tmp = 0.0
	if (Float64(1.0 / n) <= -2e-114)
		tmp = Float64((exp(Float64(log(x) / Float64(-n))) ^ -1.0) / Float64(x * n));
	elseif (Float64(1.0 / n) <= 5e-166)
		tmp = Float64(Float64(Float64(0.5 * Float64((t_0 ^ 2.0) / (n ^ 2.0))) + Float64(t_0 / n)) - Float64(Float64(Float64((log(x) ^ 2.0) / (n ^ 2.0)) * 0.5) + Float64(log(x) / n)));
	elseif (Float64(1.0 / n) <= 5e-136)
		tmp = Float64(Float64(Float64(0.3333333333333333 / (x ^ 3.0)) + Float64(Float64(1.0 / x) - Float64(0.5 / (x ^ 2.0)))) / n);
	elseif (Float64(1.0 / n) <= 5e-20)
		tmp = Float64(Float64(log1p(x) - log(x)) * Float64(1.0 / n));
	else
		tmp = sqrt(Float64(exp(Float64(log1p(x) / n)) - (x ^ Float64(1.0 / n)))) ^ 2.0;
	end
	return tmp
end

Wolfram

code[x_, n_] := Block[{t$95$0 = N[Log[N[(x + 1.0), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[N[(1.0 / n), $MachinePrecision], -2e-114], N[(N[Power[N[Exp[N[(N[Log[x], $MachinePrecision] / (-n)), $MachinePrecision]], $MachinePrecision], -1.0], $MachinePrecision] / N[(x * n), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 5e-166], N[(N[(N[(0.5 * N[(N[Power[t$95$0, 2.0], $MachinePrecision] / N[Power[n, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(t$95$0 / n), $MachinePrecision]), $MachinePrecision] - N[(N[(N[(N[Power[N[Log[x], $MachinePrecision], 2.0], $MachinePrecision] / N[Power[n, 2.0], $MachinePrecision]), $MachinePrecision] * 0.5), $MachinePrecision] + N[(N[Log[x], $MachinePrecision] / n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 5e-136], N[(N[(N[(0.3333333333333333 / N[Power[x, 3.0], $MachinePrecision]), $MachinePrecision] + N[(N[(1.0 / x), $MachinePrecision] - N[(0.5 / N[Power[x, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / n), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 5e-20], N[(N[(N[Log[1 + x], $MachinePrecision] - N[Log[x], $MachinePrecision]), $MachinePrecision] * N[(1.0 / n), $MachinePrecision]), $MachinePrecision], N[Power[N[Sqrt[N[(N[Exp[N[(N[Log[1 + x], $MachinePrecision] / n), $MachinePrecision]], $MachinePrecision] - N[Power[x, N[(1.0 / n), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision], 2.0], $MachinePrecision]]]]]]

Derivation

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1. Add Preprocessing

Alternative 2: 85.6% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 3.9:\\ \;\;\;\;\left(\frac{-0.5 \cdot \left(\log x - \mathsf{log1p}\left(x\right)\right)}{n} + \left(-0.125 \cdot \frac{{\log x}^{2}}{{n}^{2}} - 0.020833333333333332 \cdot \frac{{\log x}^{3}}{{n}^{3}}\right)\right) \cdot \left({\left(x + 1\right)}^{\left(\frac{0.5}{n}\right)} + {x}^{\left(\frac{0.5}{n}\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{n} \cdot \frac{{x}^{\left(\frac{1}{n}\right)}}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (if (<= x 3.9)
   (*
    (+
     (/ (* -0.5 (- (log x) (log1p x))) n)
     (-
      (* -0.125 (/ (pow (log x) 2.0) (pow n 2.0)))
      (* 0.020833333333333332 (/ (pow (log x) 3.0) (pow n 3.0)))))
    (+ (pow (+ x 1.0) (/ 0.5 n)) (pow x (/ 0.5 n))))
   (* (/ 1.0 n) (/ (pow x (/ 1.0 n)) x))))

C

double code(double x, double n) {
	double tmp;
	if (x <= 3.9) {
		tmp = (((-0.5 * (log(x) - log1p(x))) / n) + ((-0.125 * (pow(log(x), 2.0) / pow(n, 2.0))) - (0.020833333333333332 * (pow(log(x), 3.0) / pow(n, 3.0))))) * (pow((x + 1.0), (0.5 / n)) + pow(x, (0.5 / n)));
	} else {
		tmp = (1.0 / n) * (pow(x, (1.0 / n)) / x);
	}
	return tmp;
}

Java

public static double code(double x, double n) {
	double tmp;
	if (x <= 3.9) {
		tmp = (((-0.5 * (Math.log(x) - Math.log1p(x))) / n) + ((-0.125 * (Math.pow(Math.log(x), 2.0) / Math.pow(n, 2.0))) - (0.020833333333333332 * (Math.pow(Math.log(x), 3.0) / Math.pow(n, 3.0))))) * (Math.pow((x + 1.0), (0.5 / n)) + Math.pow(x, (0.5 / n)));
	} else {
		tmp = (1.0 / n) * (Math.pow(x, (1.0 / n)) / x);
	}
	return tmp;
}

Python

def code(x, n):
	tmp = 0
	if x <= 3.9:
		tmp = (((-0.5 * (math.log(x) - math.log1p(x))) / n) + ((-0.125 * (math.pow(math.log(x), 2.0) / math.pow(n, 2.0))) - (0.020833333333333332 * (math.pow(math.log(x), 3.0) / math.pow(n, 3.0))))) * (math.pow((x + 1.0), (0.5 / n)) + math.pow(x, (0.5 / n)))
	else:
		tmp = (1.0 / n) * (math.pow(x, (1.0 / n)) / x)
	return tmp

Julia

function code(x, n)
	tmp = 0.0
	if (x <= 3.9)
		tmp = Float64(Float64(Float64(Float64(-0.5 * Float64(log(x) - log1p(x))) / n) + Float64(Float64(-0.125 * Float64((log(x) ^ 2.0) / (n ^ 2.0))) - Float64(0.020833333333333332 * Float64((log(x) ^ 3.0) / (n ^ 3.0))))) * Float64((Float64(x + 1.0) ^ Float64(0.5 / n)) + (x ^ Float64(0.5 / n))));
	else
		tmp = Float64(Float64(1.0 / n) * Float64((x ^ Float64(1.0 / n)) / x));
	end
	return tmp
end

Wolfram

code[x_, n_] := If[LessEqual[x, 3.9], N[(N[(N[(N[(-0.5 * N[(N[Log[x], $MachinePrecision] - N[Log[1 + x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / n), $MachinePrecision] + N[(N[(-0.125 * N[(N[Power[N[Log[x], $MachinePrecision], 2.0], $MachinePrecision] / N[Power[n, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(0.020833333333333332 * N[(N[Power[N[Log[x], $MachinePrecision], 3.0], $MachinePrecision] / N[Power[n, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[Power[N[(x + 1.0), $MachinePrecision], N[(0.5 / n), $MachinePrecision]], $MachinePrecision] + N[Power[x, N[(0.5 / n), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 / n), $MachinePrecision] * N[(N[Power[x, N[(1.0 / n), $MachinePrecision]], $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]]

Derivation

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1. Add Preprocessing

Alternative 3: 85.4% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 2:\\ \;\;\;\;\left(\frac{-0.5 \cdot \left(\log x - \mathsf{log1p}\left(x\right)\right)}{n} + \left(-0.125 \cdot \frac{{\log x}^{2}}{{n}^{2}} - 0.020833333333333332 \cdot \frac{{\log x}^{3}}{{n}^{3}}\right)\right) \cdot \left(1 + {x}^{\left(\frac{0.5}{n}\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{n} \cdot \frac{{x}^{\left(\frac{1}{n}\right)}}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (if (<= x 2.0)
   (*
    (+
     (/ (* -0.5 (- (log x) (log1p x))) n)
     (-
      (* -0.125 (/ (pow (log x) 2.0) (pow n 2.0)))
      (* 0.020833333333333332 (/ (pow (log x) 3.0) (pow n 3.0)))))
    (+ 1.0 (pow x (/ 0.5 n))))
   (* (/ 1.0 n) (/ (pow x (/ 1.0 n)) x))))

C

double code(double x, double n) {
	double tmp;
	if (x <= 2.0) {
		tmp = (((-0.5 * (log(x) - log1p(x))) / n) + ((-0.125 * (pow(log(x), 2.0) / pow(n, 2.0))) - (0.020833333333333332 * (pow(log(x), 3.0) / pow(n, 3.0))))) * (1.0 + pow(x, (0.5 / n)));
	} else {
		tmp = (1.0 / n) * (pow(x, (1.0 / n)) / x);
	}
	return tmp;
}

Java

public static double code(double x, double n) {
	double tmp;
	if (x <= 2.0) {
		tmp = (((-0.5 * (Math.log(x) - Math.log1p(x))) / n) + ((-0.125 * (Math.pow(Math.log(x), 2.0) / Math.pow(n, 2.0))) - (0.020833333333333332 * (Math.pow(Math.log(x), 3.0) / Math.pow(n, 3.0))))) * (1.0 + Math.pow(x, (0.5 / n)));
	} else {
		tmp = (1.0 / n) * (Math.pow(x, (1.0 / n)) / x);
	}
	return tmp;
}

Python

def code(x, n):
	tmp = 0
	if x <= 2.0:
		tmp = (((-0.5 * (math.log(x) - math.log1p(x))) / n) + ((-0.125 * (math.pow(math.log(x), 2.0) / math.pow(n, 2.0))) - (0.020833333333333332 * (math.pow(math.log(x), 3.0) / math.pow(n, 3.0))))) * (1.0 + math.pow(x, (0.5 / n)))
	else:
		tmp = (1.0 / n) * (math.pow(x, (1.0 / n)) / x)
	return tmp

Julia

function code(x, n)
	tmp = 0.0
	if (x <= 2.0)
		tmp = Float64(Float64(Float64(Float64(-0.5 * Float64(log(x) - log1p(x))) / n) + Float64(Float64(-0.125 * Float64((log(x) ^ 2.0) / (n ^ 2.0))) - Float64(0.020833333333333332 * Float64((log(x) ^ 3.0) / (n ^ 3.0))))) * Float64(1.0 + (x ^ Float64(0.5 / n))));
	else
		tmp = Float64(Float64(1.0 / n) * Float64((x ^ Float64(1.0 / n)) / x));
	end
	return tmp
end

Wolfram

code[x_, n_] := If[LessEqual[x, 2.0], N[(N[(N[(N[(-0.5 * N[(N[Log[x], $MachinePrecision] - N[Log[1 + x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / n), $MachinePrecision] + N[(N[(-0.125 * N[(N[Power[N[Log[x], $MachinePrecision], 2.0], $MachinePrecision] / N[Power[n, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(0.020833333333333332 * N[(N[Power[N[Log[x], $MachinePrecision], 3.0], $MachinePrecision] / N[Power[n, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(1.0 + N[Power[x, N[(0.5 / n), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 / n), $MachinePrecision] * N[(N[Power[x, N[(1.0 / n), $MachinePrecision]], $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]]

Derivation

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1. Add Preprocessing

Alternative 4: 84.1% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\mathsf{log1p}\left(x\right)}{n}\\ \mathbf{if}\;\frac{1}{n} \leq -2 \cdot 10^{-114}:\\ \;\;\;\;\frac{{\left(e^{\frac{\log x}{-n}}\right)}^{-1}}{x \cdot n}\\ \mathbf{elif}\;\frac{1}{n} \leq 5 \cdot 10^{-166}:\\ \;\;\;\;t_0 - \frac{\log x}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 5 \cdot 10^{-136}:\\ \;\;\;\;\frac{\frac{0.3333333333333333}{{x}^{3}} + \left(\frac{1}{x} - \frac{0.5}{{x}^{2}}\right)}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 5 \cdot 10^{-20}:\\ \;\;\;\;\left(\mathsf{log1p}\left(x\right) - \log x\right) \cdot \frac{1}{n}\\ \mathbf{else}:\\ \;\;\;\;{\left(\sqrt{e^{t_0} - {x}^{\left(\frac{1}{n}\right)}}\right)}^{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (let* ((t_0 (/ (log1p x) n)))
   (if (<= (/ 1.0 n) -2e-114)
     (/ (pow (exp (/ (log x) (- n))) -1.0) (* x n))
     (if (<= (/ 1.0 n) 5e-166)
       (- t_0 (/ (log x) n))
       (if (<= (/ 1.0 n) 5e-136)
         (/
          (+
           (/ 0.3333333333333333 (pow x 3.0))
           (- (/ 1.0 x) (/ 0.5 (pow x 2.0))))
          n)
         (if (<= (/ 1.0 n) 5e-20)
           (* (- (log1p x) (log x)) (/ 1.0 n))
           (pow (sqrt (- (exp t_0) (pow x (/ 1.0 n)))) 2.0)))))))

C

double code(double x, double n) {
	double t_0 = log1p(x) / n;
	double tmp;
	if ((1.0 / n) <= -2e-114) {
		tmp = pow(exp((log(x) / -n)), -1.0) / (x * n);
	} else if ((1.0 / n) <= 5e-166) {
		tmp = t_0 - (log(x) / n);
	} else if ((1.0 / n) <= 5e-136) {
		tmp = ((0.3333333333333333 / pow(x, 3.0)) + ((1.0 / x) - (0.5 / pow(x, 2.0)))) / n;
	} else if ((1.0 / n) <= 5e-20) {
		tmp = (log1p(x) - log(x)) * (1.0 / n);
	} else {
		tmp = pow(sqrt((exp(t_0) - pow(x, (1.0 / n)))), 2.0);
	}
	return tmp;
}

Java

public static double code(double x, double n) {
	double t_0 = Math.log1p(x) / n;
	double tmp;
	if ((1.0 / n) <= -2e-114) {
		tmp = Math.pow(Math.exp((Math.log(x) / -n)), -1.0) / (x * n);
	} else if ((1.0 / n) <= 5e-166) {
		tmp = t_0 - (Math.log(x) / n);
	} else if ((1.0 / n) <= 5e-136) {
		tmp = ((0.3333333333333333 / Math.pow(x, 3.0)) + ((1.0 / x) - (0.5 / Math.pow(x, 2.0)))) / n;
	} else if ((1.0 / n) <= 5e-20) {
		tmp = (Math.log1p(x) - Math.log(x)) * (1.0 / n);
	} else {
		tmp = Math.pow(Math.sqrt((Math.exp(t_0) - Math.pow(x, (1.0 / n)))), 2.0);
	}
	return tmp;
}

Python

def code(x, n):
	t_0 = math.log1p(x) / n
	tmp = 0
	if (1.0 / n) <= -2e-114:
		tmp = math.pow(math.exp((math.log(x) / -n)), -1.0) / (x * n)
	elif (1.0 / n) <= 5e-166:
		tmp = t_0 - (math.log(x) / n)
	elif (1.0 / n) <= 5e-136:
		tmp = ((0.3333333333333333 / math.pow(x, 3.0)) + ((1.0 / x) - (0.5 / math.pow(x, 2.0)))) / n
	elif (1.0 / n) <= 5e-20:
		tmp = (math.log1p(x) - math.log(x)) * (1.0 / n)
	else:
		tmp = math.pow(math.sqrt((math.exp(t_0) - math.pow(x, (1.0 / n)))), 2.0)
	return tmp

Julia

function code(x, n)
	t_0 = Float64(log1p(x) / n)
	tmp = 0.0
	if (Float64(1.0 / n) <= -2e-114)
		tmp = Float64((exp(Float64(log(x) / Float64(-n))) ^ -1.0) / Float64(x * n));
	elseif (Float64(1.0 / n) <= 5e-166)
		tmp = Float64(t_0 - Float64(log(x) / n));
	elseif (Float64(1.0 / n) <= 5e-136)
		tmp = Float64(Float64(Float64(0.3333333333333333 / (x ^ 3.0)) + Float64(Float64(1.0 / x) - Float64(0.5 / (x ^ 2.0)))) / n);
	elseif (Float64(1.0 / n) <= 5e-20)
		tmp = Float64(Float64(log1p(x) - log(x)) * Float64(1.0 / n));
	else
		tmp = sqrt(Float64(exp(t_0) - (x ^ Float64(1.0 / n)))) ^ 2.0;
	end
	return tmp
end

Wolfram

code[x_, n_] := Block[{t$95$0 = N[(N[Log[1 + x], $MachinePrecision] / n), $MachinePrecision]}, If[LessEqual[N[(1.0 / n), $MachinePrecision], -2e-114], N[(N[Power[N[Exp[N[(N[Log[x], $MachinePrecision] / (-n)), $MachinePrecision]], $MachinePrecision], -1.0], $MachinePrecision] / N[(x * n), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 5e-166], N[(t$95$0 - N[(N[Log[x], $MachinePrecision] / n), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 5e-136], N[(N[(N[(0.3333333333333333 / N[Power[x, 3.0], $MachinePrecision]), $MachinePrecision] + N[(N[(1.0 / x), $MachinePrecision] - N[(0.5 / N[Power[x, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / n), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 5e-20], N[(N[(N[Log[1 + x], $MachinePrecision] - N[Log[x], $MachinePrecision]), $MachinePrecision] * N[(1.0 / n), $MachinePrecision]), $MachinePrecision], N[Power[N[Sqrt[N[(N[Exp[t$95$0], $MachinePrecision] - N[Power[x, N[(1.0 / n), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision], 2.0], $MachinePrecision]]]]]]

Derivation

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1. Add Preprocessing

Alternative 5: 84.1% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\mathsf{log1p}\left(x\right)}{n}\\ t_1 := \frac{\log x}{n}\\ \mathbf{if}\;\frac{1}{n} \leq -2 \cdot 10^{-114}:\\ \;\;\;\;\frac{e^{t_1}}{x \cdot n}\\ \mathbf{elif}\;\frac{1}{n} \leq 5 \cdot 10^{-166}:\\ \;\;\;\;t_0 - t_1\\ \mathbf{elif}\;\frac{1}{n} \leq 5 \cdot 10^{-136}:\\ \;\;\;\;\frac{\frac{0.3333333333333333}{{x}^{3}} + \left(\frac{1}{x} - \frac{0.5}{{x}^{2}}\right)}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 5 \cdot 10^{-20}:\\ \;\;\;\;\left(\mathsf{log1p}\left(x\right) - \log x\right) \cdot \frac{1}{n}\\ \mathbf{else}:\\ \;\;\;\;e^{t_0} - {x}^{\left(\frac{1}{n}\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (let* ((t_0 (/ (log1p x) n)) (t_1 (/ (log x) n)))
   (if (<= (/ 1.0 n) -2e-114)
     (/ (exp t_1) (* x n))
     (if (<= (/ 1.0 n) 5e-166)
       (- t_0 t_1)
       (if (<= (/ 1.0 n) 5e-136)
         (/
          (+
           (/ 0.3333333333333333 (pow x 3.0))
           (- (/ 1.0 x) (/ 0.5 (pow x 2.0))))
          n)
         (if (<= (/ 1.0 n) 5e-20)
           (* (- (log1p x) (log x)) (/ 1.0 n))
           (- (exp t_0) (pow x (/ 1.0 n)))))))))

C

double code(double x, double n) {
	double t_0 = log1p(x) / n;
	double t_1 = log(x) / n;
	double tmp;
	if ((1.0 / n) <= -2e-114) {
		tmp = exp(t_1) / (x * n);
	} else if ((1.0 / n) <= 5e-166) {
		tmp = t_0 - t_1;
	} else if ((1.0 / n) <= 5e-136) {
		tmp = ((0.3333333333333333 / pow(x, 3.0)) + ((1.0 / x) - (0.5 / pow(x, 2.0)))) / n;
	} else if ((1.0 / n) <= 5e-20) {
		tmp = (log1p(x) - log(x)) * (1.0 / n);
	} else {
		tmp = exp(t_0) - pow(x, (1.0 / n));
	}
	return tmp;
}

Java

public static double code(double x, double n) {
	double t_0 = Math.log1p(x) / n;
	double t_1 = Math.log(x) / n;
	double tmp;
	if ((1.0 / n) <= -2e-114) {
		tmp = Math.exp(t_1) / (x * n);
	} else if ((1.0 / n) <= 5e-166) {
		tmp = t_0 - t_1;
	} else if ((1.0 / n) <= 5e-136) {
		tmp = ((0.3333333333333333 / Math.pow(x, 3.0)) + ((1.0 / x) - (0.5 / Math.pow(x, 2.0)))) / n;
	} else if ((1.0 / n) <= 5e-20) {
		tmp = (Math.log1p(x) - Math.log(x)) * (1.0 / n);
	} else {
		tmp = Math.exp(t_0) - Math.pow(x, (1.0 / n));
	}
	return tmp;
}

Python

def code(x, n):
	t_0 = math.log1p(x) / n
	t_1 = math.log(x) / n
	tmp = 0
	if (1.0 / n) <= -2e-114:
		tmp = math.exp(t_1) / (x * n)
	elif (1.0 / n) <= 5e-166:
		tmp = t_0 - t_1
	elif (1.0 / n) <= 5e-136:
		tmp = ((0.3333333333333333 / math.pow(x, 3.0)) + ((1.0 / x) - (0.5 / math.pow(x, 2.0)))) / n
	elif (1.0 / n) <= 5e-20:
		tmp = (math.log1p(x) - math.log(x)) * (1.0 / n)
	else:
		tmp = math.exp(t_0) - math.pow(x, (1.0 / n))
	return tmp

Julia

function code(x, n)
	t_0 = Float64(log1p(x) / n)
	t_1 = Float64(log(x) / n)
	tmp = 0.0
	if (Float64(1.0 / n) <= -2e-114)
		tmp = Float64(exp(t_1) / Float64(x * n));
	elseif (Float64(1.0 / n) <= 5e-166)
		tmp = Float64(t_0 - t_1);
	elseif (Float64(1.0 / n) <= 5e-136)
		tmp = Float64(Float64(Float64(0.3333333333333333 / (x ^ 3.0)) + Float64(Float64(1.0 / x) - Float64(0.5 / (x ^ 2.0)))) / n);
	elseif (Float64(1.0 / n) <= 5e-20)
		tmp = Float64(Float64(log1p(x) - log(x)) * Float64(1.0 / n));
	else
		tmp = Float64(exp(t_0) - (x ^ Float64(1.0 / n)));
	end
	return tmp
end

Wolfram

code[x_, n_] := Block[{t$95$0 = N[(N[Log[1 + x], $MachinePrecision] / n), $MachinePrecision]}, Block[{t$95$1 = N[(N[Log[x], $MachinePrecision] / n), $MachinePrecision]}, If[LessEqual[N[(1.0 / n), $MachinePrecision], -2e-114], N[(N[Exp[t$95$1], $MachinePrecision] / N[(x * n), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 5e-166], N[(t$95$0 - t$95$1), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 5e-136], N[(N[(N[(0.3333333333333333 / N[Power[x, 3.0], $MachinePrecision]), $MachinePrecision] + N[(N[(1.0 / x), $MachinePrecision] - N[(0.5 / N[Power[x, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / n), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 5e-20], N[(N[(N[Log[1 + x], $MachinePrecision] - N[Log[x], $MachinePrecision]), $MachinePrecision] * N[(1.0 / n), $MachinePrecision]), $MachinePrecision], N[(N[Exp[t$95$0], $MachinePrecision] - N[Power[x, N[(1.0 / n), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]]]]

Derivation

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1. Add Preprocessing

Alternative 6: 84.1% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\mathsf{log1p}\left(x\right)}{n}\\ \mathbf{if}\;\frac{1}{n} \leq -2 \cdot 10^{-114}:\\ \;\;\;\;\frac{{\left(e^{\frac{\log x}{-n}}\right)}^{-1}}{x \cdot n}\\ \mathbf{elif}\;\frac{1}{n} \leq 5 \cdot 10^{-166}:\\ \;\;\;\;t_0 - \frac{\log x}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 5 \cdot 10^{-136}:\\ \;\;\;\;\frac{\frac{0.3333333333333333}{{x}^{3}} + \left(\frac{1}{x} - \frac{0.5}{{x}^{2}}\right)}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 5 \cdot 10^{-20}:\\ \;\;\;\;\left(\mathsf{log1p}\left(x\right) - \log x\right) \cdot \frac{1}{n}\\ \mathbf{else}:\\ \;\;\;\;e^{t_0} - {x}^{\left(\frac{1}{n}\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (let* ((t_0 (/ (log1p x) n)))
   (if (<= (/ 1.0 n) -2e-114)
     (/ (pow (exp (/ (log x) (- n))) -1.0) (* x n))
     (if (<= (/ 1.0 n) 5e-166)
       (- t_0 (/ (log x) n))
       (if (<= (/ 1.0 n) 5e-136)
         (/
          (+
           (/ 0.3333333333333333 (pow x 3.0))
           (- (/ 1.0 x) (/ 0.5 (pow x 2.0))))
          n)
         (if (<= (/ 1.0 n) 5e-20)
           (* (- (log1p x) (log x)) (/ 1.0 n))
           (- (exp t_0) (pow x (/ 1.0 n)))))))))

C

double code(double x, double n) {
	double t_0 = log1p(x) / n;
	double tmp;
	if ((1.0 / n) <= -2e-114) {
		tmp = pow(exp((log(x) / -n)), -1.0) / (x * n);
	} else if ((1.0 / n) <= 5e-166) {
		tmp = t_0 - (log(x) / n);
	} else if ((1.0 / n) <= 5e-136) {
		tmp = ((0.3333333333333333 / pow(x, 3.0)) + ((1.0 / x) - (0.5 / pow(x, 2.0)))) / n;
	} else if ((1.0 / n) <= 5e-20) {
		tmp = (log1p(x) - log(x)) * (1.0 / n);
	} else {
		tmp = exp(t_0) - pow(x, (1.0 / n));
	}
	return tmp;
}

Java

public static double code(double x, double n) {
	double t_0 = Math.log1p(x) / n;
	double tmp;
	if ((1.0 / n) <= -2e-114) {
		tmp = Math.pow(Math.exp((Math.log(x) / -n)), -1.0) / (x * n);
	} else if ((1.0 / n) <= 5e-166) {
		tmp = t_0 - (Math.log(x) / n);
	} else if ((1.0 / n) <= 5e-136) {
		tmp = ((0.3333333333333333 / Math.pow(x, 3.0)) + ((1.0 / x) - (0.5 / Math.pow(x, 2.0)))) / n;
	} else if ((1.0 / n) <= 5e-20) {
		tmp = (Math.log1p(x) - Math.log(x)) * (1.0 / n);
	} else {
		tmp = Math.exp(t_0) - Math.pow(x, (1.0 / n));
	}
	return tmp;
}

Python

def code(x, n):
	t_0 = math.log1p(x) / n
	tmp = 0
	if (1.0 / n) <= -2e-114:
		tmp = math.pow(math.exp((math.log(x) / -n)), -1.0) / (x * n)
	elif (1.0 / n) <= 5e-166:
		tmp = t_0 - (math.log(x) / n)
	elif (1.0 / n) <= 5e-136:
		tmp = ((0.3333333333333333 / math.pow(x, 3.0)) + ((1.0 / x) - (0.5 / math.pow(x, 2.0)))) / n
	elif (1.0 / n) <= 5e-20:
		tmp = (math.log1p(x) - math.log(x)) * (1.0 / n)
	else:
		tmp = math.exp(t_0) - math.pow(x, (1.0 / n))
	return tmp

Julia

function code(x, n)
	t_0 = Float64(log1p(x) / n)
	tmp = 0.0
	if (Float64(1.0 / n) <= -2e-114)
		tmp = Float64((exp(Float64(log(x) / Float64(-n))) ^ -1.0) / Float64(x * n));
	elseif (Float64(1.0 / n) <= 5e-166)
		tmp = Float64(t_0 - Float64(log(x) / n));
	elseif (Float64(1.0 / n) <= 5e-136)
		tmp = Float64(Float64(Float64(0.3333333333333333 / (x ^ 3.0)) + Float64(Float64(1.0 / x) - Float64(0.5 / (x ^ 2.0)))) / n);
	elseif (Float64(1.0 / n) <= 5e-20)
		tmp = Float64(Float64(log1p(x) - log(x)) * Float64(1.0 / n));
	else
		tmp = Float64(exp(t_0) - (x ^ Float64(1.0 / n)));
	end
	return tmp
end

Wolfram

code[x_, n_] := Block[{t$95$0 = N[(N[Log[1 + x], $MachinePrecision] / n), $MachinePrecision]}, If[LessEqual[N[(1.0 / n), $MachinePrecision], -2e-114], N[(N[Power[N[Exp[N[(N[Log[x], $MachinePrecision] / (-n)), $MachinePrecision]], $MachinePrecision], -1.0], $MachinePrecision] / N[(x * n), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 5e-166], N[(t$95$0 - N[(N[Log[x], $MachinePrecision] / n), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 5e-136], N[(N[(N[(0.3333333333333333 / N[Power[x, 3.0], $MachinePrecision]), $MachinePrecision] + N[(N[(1.0 / x), $MachinePrecision] - N[(0.5 / N[Power[x, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / n), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 5e-20], N[(N[(N[Log[1 + x], $MachinePrecision] - N[Log[x], $MachinePrecision]), $MachinePrecision] * N[(1.0 / n), $MachinePrecision]), $MachinePrecision], N[(N[Exp[t$95$0], $MachinePrecision] - N[Power[x, N[(1.0 / n), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

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1. Add Preprocessing

Alternative 7: 80.3% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\log x}{n}\\ \mathbf{if}\;\frac{1}{n} \leq -2 \cdot 10^{-114}:\\ \;\;\;\;\frac{e^{t_0}}{x \cdot n}\\ \mathbf{elif}\;\frac{1}{n} \leq 5 \cdot 10^{-166}:\\ \;\;\;\;\frac{\mathsf{log1p}\left(x\right)}{n} - t_0\\ \mathbf{elif}\;\frac{1}{n} \leq 5 \cdot 10^{-136}:\\ \;\;\;\;\frac{\frac{1}{x} - \frac{0.5}{{x}^{2}}}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 5 \cdot 10^{-20}:\\ \;\;\;\;\left(\mathsf{log1p}\left(x\right) - \log x\right) \cdot \frac{1}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 5 \cdot 10^{+183}:\\ \;\;\;\;{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{\frac{1}{x \cdot n}}{x \cdot n}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (let* ((t_0 (/ (log x) n)))
   (if (<= (/ 1.0 n) -2e-114)
     (/ (exp t_0) (* x n))
     (if (<= (/ 1.0 n) 5e-166)
       (- (/ (log1p x) n) t_0)
       (if (<= (/ 1.0 n) 5e-136)
         (/ (- (/ 1.0 x) (/ 0.5 (pow x 2.0))) n)
         (if (<= (/ 1.0 n) 5e-20)
           (* (- (log1p x) (log x)) (/ 1.0 n))
           (if (<= (/ 1.0 n) 5e+183)
             (- (pow (+ x 1.0) (/ 1.0 n)) (pow x (/ 1.0 n)))
             (sqrt (/ (/ 1.0 (* x n)) (* x n))))))))))

C

double code(double x, double n) {
	double t_0 = log(x) / n;
	double tmp;
	if ((1.0 / n) <= -2e-114) {
		tmp = exp(t_0) / (x * n);
	} else if ((1.0 / n) <= 5e-166) {
		tmp = (log1p(x) / n) - t_0;
	} else if ((1.0 / n) <= 5e-136) {
		tmp = ((1.0 / x) - (0.5 / pow(x, 2.0))) / n;
	} else if ((1.0 / n) <= 5e-20) {
		tmp = (log1p(x) - log(x)) * (1.0 / n);
	} else if ((1.0 / n) <= 5e+183) {
		tmp = pow((x + 1.0), (1.0 / n)) - pow(x, (1.0 / n));
	} else {
		tmp = sqrt(((1.0 / (x * n)) / (x * n)));
	}
	return tmp;
}

Java

public static double code(double x, double n) {
	double t_0 = Math.log(x) / n;
	double tmp;
	if ((1.0 / n) <= -2e-114) {
		tmp = Math.exp(t_0) / (x * n);
	} else if ((1.0 / n) <= 5e-166) {
		tmp = (Math.log1p(x) / n) - t_0;
	} else if ((1.0 / n) <= 5e-136) {
		tmp = ((1.0 / x) - (0.5 / Math.pow(x, 2.0))) / n;
	} else if ((1.0 / n) <= 5e-20) {
		tmp = (Math.log1p(x) - Math.log(x)) * (1.0 / n);
	} else if ((1.0 / n) <= 5e+183) {
		tmp = Math.pow((x + 1.0), (1.0 / n)) - Math.pow(x, (1.0 / n));
	} else {
		tmp = Math.sqrt(((1.0 / (x * n)) / (x * n)));
	}
	return tmp;
}

Python

def code(x, n):
	t_0 = math.log(x) / n
	tmp = 0
	if (1.0 / n) <= -2e-114:
		tmp = math.exp(t_0) / (x * n)
	elif (1.0 / n) <= 5e-166:
		tmp = (math.log1p(x) / n) - t_0
	elif (1.0 / n) <= 5e-136:
		tmp = ((1.0 / x) - (0.5 / math.pow(x, 2.0))) / n
	elif (1.0 / n) <= 5e-20:
		tmp = (math.log1p(x) - math.log(x)) * (1.0 / n)
	elif (1.0 / n) <= 5e+183:
		tmp = math.pow((x + 1.0), (1.0 / n)) - math.pow(x, (1.0 / n))
	else:
		tmp = math.sqrt(((1.0 / (x * n)) / (x * n)))
	return tmp

Julia

function code(x, n)
	t_0 = Float64(log(x) / n)
	tmp = 0.0
	if (Float64(1.0 / n) <= -2e-114)
		tmp = Float64(exp(t_0) / Float64(x * n));
	elseif (Float64(1.0 / n) <= 5e-166)
		tmp = Float64(Float64(log1p(x) / n) - t_0);
	elseif (Float64(1.0 / n) <= 5e-136)
		tmp = Float64(Float64(Float64(1.0 / x) - Float64(0.5 / (x ^ 2.0))) / n);
	elseif (Float64(1.0 / n) <= 5e-20)
		tmp = Float64(Float64(log1p(x) - log(x)) * Float64(1.0 / n));
	elseif (Float64(1.0 / n) <= 5e+183)
		tmp = Float64((Float64(x + 1.0) ^ Float64(1.0 / n)) - (x ^ Float64(1.0 / n)));
	else
		tmp = sqrt(Float64(Float64(1.0 / Float64(x * n)) / Float64(x * n)));
	end
	return tmp
end

Wolfram

code[x_, n_] := Block[{t$95$0 = N[(N[Log[x], $MachinePrecision] / n), $MachinePrecision]}, If[LessEqual[N[(1.0 / n), $MachinePrecision], -2e-114], N[(N[Exp[t$95$0], $MachinePrecision] / N[(x * n), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 5e-166], N[(N[(N[Log[1 + x], $MachinePrecision] / n), $MachinePrecision] - t$95$0), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 5e-136], N[(N[(N[(1.0 / x), $MachinePrecision] - N[(0.5 / N[Power[x, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / n), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 5e-20], N[(N[(N[Log[1 + x], $MachinePrecision] - N[Log[x], $MachinePrecision]), $MachinePrecision] * N[(1.0 / n), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 5e+183], N[(N[Power[N[(x + 1.0), $MachinePrecision], N[(1.0 / n), $MachinePrecision]], $MachinePrecision] - N[Power[x, N[(1.0 / n), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[Sqrt[N[(N[(1.0 / N[(x * n), $MachinePrecision]), $MachinePrecision] / N[(x * n), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]]]]]

Derivation

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

Alternative 8: 80.3% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\log x}{n}\\ \mathbf{if}\;\frac{1}{n} \leq -2 \cdot 10^{-114}:\\ \;\;\;\;\frac{e^{t_0}}{x \cdot n}\\ \mathbf{elif}\;\frac{1}{n} \leq 5 \cdot 10^{-166}:\\ \;\;\;\;\frac{\mathsf{log1p}\left(x\right)}{n} - t_0\\ \mathbf{elif}\;\frac{1}{n} \leq 5 \cdot 10^{-136}:\\ \;\;\;\;\frac{\frac{0.3333333333333333}{{x}^{3}} + \left(\frac{1}{x} - \frac{0.5}{{x}^{2}}\right)}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 5 \cdot 10^{-20}:\\ \;\;\;\;\left(\mathsf{log1p}\left(x\right) - \log x\right) \cdot \frac{1}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 5 \cdot 10^{+183}:\\ \;\;\;\;{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{\frac{1}{x \cdot n}}{x \cdot n}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (let* ((t_0 (/ (log x) n)))
   (if (<= (/ 1.0 n) -2e-114)
     (/ (exp t_0) (* x n))
     (if (<= (/ 1.0 n) 5e-166)
       (- (/ (log1p x) n) t_0)
       (if (<= (/ 1.0 n) 5e-136)
         (/
          (+
           (/ 0.3333333333333333 (pow x 3.0))
           (- (/ 1.0 x) (/ 0.5 (pow x 2.0))))
          n)
         (if (<= (/ 1.0 n) 5e-20)
           (* (- (log1p x) (log x)) (/ 1.0 n))
           (if (<= (/ 1.0 n) 5e+183)
             (- (pow (+ x 1.0) (/ 1.0 n)) (pow x (/ 1.0 n)))
             (sqrt (/ (/ 1.0 (* x n)) (* x n))))))))))

C

double code(double x, double n) {
	double t_0 = log(x) / n;
	double tmp;
	if ((1.0 / n) <= -2e-114) {
		tmp = exp(t_0) / (x * n);
	} else if ((1.0 / n) <= 5e-166) {
		tmp = (log1p(x) / n) - t_0;
	} else if ((1.0 / n) <= 5e-136) {
		tmp = ((0.3333333333333333 / pow(x, 3.0)) + ((1.0 / x) - (0.5 / pow(x, 2.0)))) / n;
	} else if ((1.0 / n) <= 5e-20) {
		tmp = (log1p(x) - log(x)) * (1.0 / n);
	} else if ((1.0 / n) <= 5e+183) {
		tmp = pow((x + 1.0), (1.0 / n)) - pow(x, (1.0 / n));
	} else {
		tmp = sqrt(((1.0 / (x * n)) / (x * n)));
	}
	return tmp;
}

Java

public static double code(double x, double n) {
	double t_0 = Math.log(x) / n;
	double tmp;
	if ((1.0 / n) <= -2e-114) {
		tmp = Math.exp(t_0) / (x * n);
	} else if ((1.0 / n) <= 5e-166) {
		tmp = (Math.log1p(x) / n) - t_0;
	} else if ((1.0 / n) <= 5e-136) {
		tmp = ((0.3333333333333333 / Math.pow(x, 3.0)) + ((1.0 / x) - (0.5 / Math.pow(x, 2.0)))) / n;
	} else if ((1.0 / n) <= 5e-20) {
		tmp = (Math.log1p(x) - Math.log(x)) * (1.0 / n);
	} else if ((1.0 / n) <= 5e+183) {
		tmp = Math.pow((x + 1.0), (1.0 / n)) - Math.pow(x, (1.0 / n));
	} else {
		tmp = Math.sqrt(((1.0 / (x * n)) / (x * n)));
	}
	return tmp;
}

Python

def code(x, n):
	t_0 = math.log(x) / n
	tmp = 0
	if (1.0 / n) <= -2e-114:
		tmp = math.exp(t_0) / (x * n)
	elif (1.0 / n) <= 5e-166:
		tmp = (math.log1p(x) / n) - t_0
	elif (1.0 / n) <= 5e-136:
		tmp = ((0.3333333333333333 / math.pow(x, 3.0)) + ((1.0 / x) - (0.5 / math.pow(x, 2.0)))) / n
	elif (1.0 / n) <= 5e-20:
		tmp = (math.log1p(x) - math.log(x)) * (1.0 / n)
	elif (1.0 / n) <= 5e+183:
		tmp = math.pow((x + 1.0), (1.0 / n)) - math.pow(x, (1.0 / n))
	else:
		tmp = math.sqrt(((1.0 / (x * n)) / (x * n)))
	return tmp

Julia

function code(x, n)
	t_0 = Float64(log(x) / n)
	tmp = 0.0
	if (Float64(1.0 / n) <= -2e-114)
		tmp = Float64(exp(t_0) / Float64(x * n));
	elseif (Float64(1.0 / n) <= 5e-166)
		tmp = Float64(Float64(log1p(x) / n) - t_0);
	elseif (Float64(1.0 / n) <= 5e-136)
		tmp = Float64(Float64(Float64(0.3333333333333333 / (x ^ 3.0)) + Float64(Float64(1.0 / x) - Float64(0.5 / (x ^ 2.0)))) / n);
	elseif (Float64(1.0 / n) <= 5e-20)
		tmp = Float64(Float64(log1p(x) - log(x)) * Float64(1.0 / n));
	elseif (Float64(1.0 / n) <= 5e+183)
		tmp = Float64((Float64(x + 1.0) ^ Float64(1.0 / n)) - (x ^ Float64(1.0 / n)));
	else
		tmp = sqrt(Float64(Float64(1.0 / Float64(x * n)) / Float64(x * n)));
	end
	return tmp
end

Wolfram

code[x_, n_] := Block[{t$95$0 = N[(N[Log[x], $MachinePrecision] / n), $MachinePrecision]}, If[LessEqual[N[(1.0 / n), $MachinePrecision], -2e-114], N[(N[Exp[t$95$0], $MachinePrecision] / N[(x * n), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 5e-166], N[(N[(N[Log[1 + x], $MachinePrecision] / n), $MachinePrecision] - t$95$0), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 5e-136], N[(N[(N[(0.3333333333333333 / N[Power[x, 3.0], $MachinePrecision]), $MachinePrecision] + N[(N[(1.0 / x), $MachinePrecision] - N[(0.5 / N[Power[x, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / n), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 5e-20], N[(N[(N[Log[1 + x], $MachinePrecision] - N[Log[x], $MachinePrecision]), $MachinePrecision] * N[(1.0 / n), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 5e+183], N[(N[Power[N[(x + 1.0), $MachinePrecision], N[(1.0 / n), $MachinePrecision]], $MachinePrecision] - N[Power[x, N[(1.0 / n), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[Sqrt[N[(N[(1.0 / N[(x * n), $MachinePrecision]), $MachinePrecision] / N[(x * n), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]]]]]

Derivation

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

Alternative 9: 80.0% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{log1p}\left(x\right) - \log x\\ t_1 := {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{if}\;\frac{1}{n} \leq -2 \cdot 10^{-114}:\\ \;\;\;\;\frac{t_1}{x \cdot n}\\ \mathbf{elif}\;\frac{1}{n} \leq 5 \cdot 10^{-166}:\\ \;\;\;\;\frac{t_0}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 5 \cdot 10^{-136}:\\ \;\;\;\;\frac{\frac{1}{x} - \frac{0.5}{{x}^{2}}}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 5 \cdot 10^{-20}:\\ \;\;\;\;t_0 \cdot \frac{1}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 10^{+199}:\\ \;\;\;\;\left(1 + \frac{x}{n}\right) - t_1\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{\frac{1}{x \cdot n}}{x \cdot n}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (let* ((t_0 (- (log1p x) (log x))) (t_1 (pow x (/ 1.0 n))))
   (if (<= (/ 1.0 n) -2e-114)
     (/ t_1 (* x n))
     (if (<= (/ 1.0 n) 5e-166)
       (/ t_0 n)
       (if (<= (/ 1.0 n) 5e-136)
         (/ (- (/ 1.0 x) (/ 0.5 (pow x 2.0))) n)
         (if (<= (/ 1.0 n) 5e-20)
           (* t_0 (/ 1.0 n))
           (if (<= (/ 1.0 n) 1e+199)
             (- (+ 1.0 (/ x n)) t_1)
             (sqrt (/ (/ 1.0 (* x n)) (* x n))))))))))

C

double code(double x, double n) {
	double t_0 = log1p(x) - log(x);
	double t_1 = pow(x, (1.0 / n));
	double tmp;
	if ((1.0 / n) <= -2e-114) {
		tmp = t_1 / (x * n);
	} else if ((1.0 / n) <= 5e-166) {
		tmp = t_0 / n;
	} else if ((1.0 / n) <= 5e-136) {
		tmp = ((1.0 / x) - (0.5 / pow(x, 2.0))) / n;
	} else if ((1.0 / n) <= 5e-20) {
		tmp = t_0 * (1.0 / n);
	} else if ((1.0 / n) <= 1e+199) {
		tmp = (1.0 + (x / n)) - t_1;
	} else {
		tmp = sqrt(((1.0 / (x * n)) / (x * n)));
	}
	return tmp;
}

Java

public static double code(double x, double n) {
	double t_0 = Math.log1p(x) - Math.log(x);
	double t_1 = Math.pow(x, (1.0 / n));
	double tmp;
	if ((1.0 / n) <= -2e-114) {
		tmp = t_1 / (x * n);
	} else if ((1.0 / n) <= 5e-166) {
		tmp = t_0 / n;
	} else if ((1.0 / n) <= 5e-136) {
		tmp = ((1.0 / x) - (0.5 / Math.pow(x, 2.0))) / n;
	} else if ((1.0 / n) <= 5e-20) {
		tmp = t_0 * (1.0 / n);
	} else if ((1.0 / n) <= 1e+199) {
		tmp = (1.0 + (x / n)) - t_1;
	} else {
		tmp = Math.sqrt(((1.0 / (x * n)) / (x * n)));
	}
	return tmp;
}

Python

def code(x, n):
	t_0 = math.log1p(x) - math.log(x)
	t_1 = math.pow(x, (1.0 / n))
	tmp = 0
	if (1.0 / n) <= -2e-114:
		tmp = t_1 / (x * n)
	elif (1.0 / n) <= 5e-166:
		tmp = t_0 / n
	elif (1.0 / n) <= 5e-136:
		tmp = ((1.0 / x) - (0.5 / math.pow(x, 2.0))) / n
	elif (1.0 / n) <= 5e-20:
		tmp = t_0 * (1.0 / n)
	elif (1.0 / n) <= 1e+199:
		tmp = (1.0 + (x / n)) - t_1
	else:
		tmp = math.sqrt(((1.0 / (x * n)) / (x * n)))
	return tmp

Julia

function code(x, n)
	t_0 = Float64(log1p(x) - log(x))
	t_1 = x ^ Float64(1.0 / n)
	tmp = 0.0
	if (Float64(1.0 / n) <= -2e-114)
		tmp = Float64(t_1 / Float64(x * n));
	elseif (Float64(1.0 / n) <= 5e-166)
		tmp = Float64(t_0 / n);
	elseif (Float64(1.0 / n) <= 5e-136)
		tmp = Float64(Float64(Float64(1.0 / x) - Float64(0.5 / (x ^ 2.0))) / n);
	elseif (Float64(1.0 / n) <= 5e-20)
		tmp = Float64(t_0 * Float64(1.0 / n));
	elseif (Float64(1.0 / n) <= 1e+199)
		tmp = Float64(Float64(1.0 + Float64(x / n)) - t_1);
	else
		tmp = sqrt(Float64(Float64(1.0 / Float64(x * n)) / Float64(x * n)));
	end
	return tmp
end

Wolfram

code[x_, n_] := Block[{t$95$0 = N[(N[Log[1 + x], $MachinePrecision] - N[Log[x], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[Power[x, N[(1.0 / n), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[N[(1.0 / n), $MachinePrecision], -2e-114], N[(t$95$1 / N[(x * n), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 5e-166], N[(t$95$0 / n), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 5e-136], N[(N[(N[(1.0 / x), $MachinePrecision] - N[(0.5 / N[Power[x, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / n), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 5e-20], N[(t$95$0 * N[(1.0 / n), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 1e+199], N[(N[(1.0 + N[(x / n), $MachinePrecision]), $MachinePrecision] - t$95$1), $MachinePrecision], N[Sqrt[N[(N[(1.0 / N[(x * n), $MachinePrecision]), $MachinePrecision] / N[(x * n), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]]]]]]

Derivation

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

Alternative 10: 80.0% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{if}\;\frac{1}{n} \leq -2 \cdot 10^{-114}:\\ \;\;\;\;\frac{t_0}{x \cdot n}\\ \mathbf{elif}\;\frac{1}{n} \leq 5 \cdot 10^{-166}:\\ \;\;\;\;\frac{\mathsf{log1p}\left(x\right)}{n} - \frac{\log x}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 5 \cdot 10^{-136}:\\ \;\;\;\;\frac{\frac{1}{x} - \frac{0.5}{{x}^{2}}}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 5 \cdot 10^{-20}:\\ \;\;\;\;\left(\mathsf{log1p}\left(x\right) - \log x\right) \cdot \frac{1}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 10^{+199}:\\ \;\;\;\;\left(1 + \frac{x}{n}\right) - t_0\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{\frac{1}{x \cdot n}}{x \cdot n}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (let* ((t_0 (pow x (/ 1.0 n))))
   (if (<= (/ 1.0 n) -2e-114)
     (/ t_0 (* x n))
     (if (<= (/ 1.0 n) 5e-166)
       (- (/ (log1p x) n) (/ (log x) n))
       (if (<= (/ 1.0 n) 5e-136)
         (/ (- (/ 1.0 x) (/ 0.5 (pow x 2.0))) n)
         (if (<= (/ 1.0 n) 5e-20)
           (* (- (log1p x) (log x)) (/ 1.0 n))
           (if (<= (/ 1.0 n) 1e+199)
             (- (+ 1.0 (/ x n)) t_0)
             (sqrt (/ (/ 1.0 (* x n)) (* x n))))))))))

C

double code(double x, double n) {
	double t_0 = pow(x, (1.0 / n));
	double tmp;
	if ((1.0 / n) <= -2e-114) {
		tmp = t_0 / (x * n);
	} else if ((1.0 / n) <= 5e-166) {
		tmp = (log1p(x) / n) - (log(x) / n);
	} else if ((1.0 / n) <= 5e-136) {
		tmp = ((1.0 / x) - (0.5 / pow(x, 2.0))) / n;
	} else if ((1.0 / n) <= 5e-20) {
		tmp = (log1p(x) - log(x)) * (1.0 / n);
	} else if ((1.0 / n) <= 1e+199) {
		tmp = (1.0 + (x / n)) - t_0;
	} else {
		tmp = sqrt(((1.0 / (x * n)) / (x * n)));
	}
	return tmp;
}

Java

public static double code(double x, double n) {
	double t_0 = Math.pow(x, (1.0 / n));
	double tmp;
	if ((1.0 / n) <= -2e-114) {
		tmp = t_0 / (x * n);
	} else if ((1.0 / n) <= 5e-166) {
		tmp = (Math.log1p(x) / n) - (Math.log(x) / n);
	} else if ((1.0 / n) <= 5e-136) {
		tmp = ((1.0 / x) - (0.5 / Math.pow(x, 2.0))) / n;
	} else if ((1.0 / n) <= 5e-20) {
		tmp = (Math.log1p(x) - Math.log(x)) * (1.0 / n);
	} else if ((1.0 / n) <= 1e+199) {
		tmp = (1.0 + (x / n)) - t_0;
	} else {
		tmp = Math.sqrt(((1.0 / (x * n)) / (x * n)));
	}
	return tmp;
}

Python

def code(x, n):
	t_0 = math.pow(x, (1.0 / n))
	tmp = 0
	if (1.0 / n) <= -2e-114:
		tmp = t_0 / (x * n)
	elif (1.0 / n) <= 5e-166:
		tmp = (math.log1p(x) / n) - (math.log(x) / n)
	elif (1.0 / n) <= 5e-136:
		tmp = ((1.0 / x) - (0.5 / math.pow(x, 2.0))) / n
	elif (1.0 / n) <= 5e-20:
		tmp = (math.log1p(x) - math.log(x)) * (1.0 / n)
	elif (1.0 / n) <= 1e+199:
		tmp = (1.0 + (x / n)) - t_0
	else:
		tmp = math.sqrt(((1.0 / (x * n)) / (x * n)))
	return tmp

Julia

function code(x, n)
	t_0 = x ^ Float64(1.0 / n)
	tmp = 0.0
	if (Float64(1.0 / n) <= -2e-114)
		tmp = Float64(t_0 / Float64(x * n));
	elseif (Float64(1.0 / n) <= 5e-166)
		tmp = Float64(Float64(log1p(x) / n) - Float64(log(x) / n));
	elseif (Float64(1.0 / n) <= 5e-136)
		tmp = Float64(Float64(Float64(1.0 / x) - Float64(0.5 / (x ^ 2.0))) / n);
	elseif (Float64(1.0 / n) <= 5e-20)
		tmp = Float64(Float64(log1p(x) - log(x)) * Float64(1.0 / n));
	elseif (Float64(1.0 / n) <= 1e+199)
		tmp = Float64(Float64(1.0 + Float64(x / n)) - t_0);
	else
		tmp = sqrt(Float64(Float64(1.0 / Float64(x * n)) / Float64(x * n)));
	end
	return tmp
end

Wolfram

code[x_, n_] := Block[{t$95$0 = N[Power[x, N[(1.0 / n), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[N[(1.0 / n), $MachinePrecision], -2e-114], N[(t$95$0 / N[(x * n), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 5e-166], N[(N[(N[Log[1 + x], $MachinePrecision] / n), $MachinePrecision] - N[(N[Log[x], $MachinePrecision] / n), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 5e-136], N[(N[(N[(1.0 / x), $MachinePrecision] - N[(0.5 / N[Power[x, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / n), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 5e-20], N[(N[(N[Log[1 + x], $MachinePrecision] - N[Log[x], $MachinePrecision]), $MachinePrecision] * N[(1.0 / n), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 1e+199], N[(N[(1.0 + N[(x / n), $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision], N[Sqrt[N[(N[(1.0 / N[(x * n), $MachinePrecision]), $MachinePrecision] / N[(x * n), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]]]]]

Derivation

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

Alternative 11: 80.0% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\log x}{n}\\ \mathbf{if}\;\frac{1}{n} \leq -2 \cdot 10^{-114}:\\ \;\;\;\;\frac{e^{t_0}}{x \cdot n}\\ \mathbf{elif}\;\frac{1}{n} \leq 5 \cdot 10^{-166}:\\ \;\;\;\;\frac{\mathsf{log1p}\left(x\right)}{n} - t_0\\ \mathbf{elif}\;\frac{1}{n} \leq 5 \cdot 10^{-136}:\\ \;\;\;\;\frac{\frac{1}{x} - \frac{0.5}{{x}^{2}}}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 5 \cdot 10^{-20}:\\ \;\;\;\;\left(\mathsf{log1p}\left(x\right) - \log x\right) \cdot \frac{1}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 10^{+199}:\\ \;\;\;\;\left(1 + \frac{x}{n}\right) - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{\frac{1}{x \cdot n}}{x \cdot n}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (let* ((t_0 (/ (log x) n)))
   (if (<= (/ 1.0 n) -2e-114)
     (/ (exp t_0) (* x n))
     (if (<= (/ 1.0 n) 5e-166)
       (- (/ (log1p x) n) t_0)
       (if (<= (/ 1.0 n) 5e-136)
         (/ (- (/ 1.0 x) (/ 0.5 (pow x 2.0))) n)
         (if (<= (/ 1.0 n) 5e-20)
           (* (- (log1p x) (log x)) (/ 1.0 n))
           (if (<= (/ 1.0 n) 1e+199)
             (- (+ 1.0 (/ x n)) (pow x (/ 1.0 n)))
             (sqrt (/ (/ 1.0 (* x n)) (* x n))))))))))

C

double code(double x, double n) {
	double t_0 = log(x) / n;
	double tmp;
	if ((1.0 / n) <= -2e-114) {
		tmp = exp(t_0) / (x * n);
	} else if ((1.0 / n) <= 5e-166) {
		tmp = (log1p(x) / n) - t_0;
	} else if ((1.0 / n) <= 5e-136) {
		tmp = ((1.0 / x) - (0.5 / pow(x, 2.0))) / n;
	} else if ((1.0 / n) <= 5e-20) {
		tmp = (log1p(x) - log(x)) * (1.0 / n);
	} else if ((1.0 / n) <= 1e+199) {
		tmp = (1.0 + (x / n)) - pow(x, (1.0 / n));
	} else {
		tmp = sqrt(((1.0 / (x * n)) / (x * n)));
	}
	return tmp;
}

Java

public static double code(double x, double n) {
	double t_0 = Math.log(x) / n;
	double tmp;
	if ((1.0 / n) <= -2e-114) {
		tmp = Math.exp(t_0) / (x * n);
	} else if ((1.0 / n) <= 5e-166) {
		tmp = (Math.log1p(x) / n) - t_0;
	} else if ((1.0 / n) <= 5e-136) {
		tmp = ((1.0 / x) - (0.5 / Math.pow(x, 2.0))) / n;
	} else if ((1.0 / n) <= 5e-20) {
		tmp = (Math.log1p(x) - Math.log(x)) * (1.0 / n);
	} else if ((1.0 / n) <= 1e+199) {
		tmp = (1.0 + (x / n)) - Math.pow(x, (1.0 / n));
	} else {
		tmp = Math.sqrt(((1.0 / (x * n)) / (x * n)));
	}
	return tmp;
}

Python

def code(x, n):
	t_0 = math.log(x) / n
	tmp = 0
	if (1.0 / n) <= -2e-114:
		tmp = math.exp(t_0) / (x * n)
	elif (1.0 / n) <= 5e-166:
		tmp = (math.log1p(x) / n) - t_0
	elif (1.0 / n) <= 5e-136:
		tmp = ((1.0 / x) - (0.5 / math.pow(x, 2.0))) / n
	elif (1.0 / n) <= 5e-20:
		tmp = (math.log1p(x) - math.log(x)) * (1.0 / n)
	elif (1.0 / n) <= 1e+199:
		tmp = (1.0 + (x / n)) - math.pow(x, (1.0 / n))
	else:
		tmp = math.sqrt(((1.0 / (x * n)) / (x * n)))
	return tmp

Julia

function code(x, n)
	t_0 = Float64(log(x) / n)
	tmp = 0.0
	if (Float64(1.0 / n) <= -2e-114)
		tmp = Float64(exp(t_0) / Float64(x * n));
	elseif (Float64(1.0 / n) <= 5e-166)
		tmp = Float64(Float64(log1p(x) / n) - t_0);
	elseif (Float64(1.0 / n) <= 5e-136)
		tmp = Float64(Float64(Float64(1.0 / x) - Float64(0.5 / (x ^ 2.0))) / n);
	elseif (Float64(1.0 / n) <= 5e-20)
		tmp = Float64(Float64(log1p(x) - log(x)) * Float64(1.0 / n));
	elseif (Float64(1.0 / n) <= 1e+199)
		tmp = Float64(Float64(1.0 + Float64(x / n)) - (x ^ Float64(1.0 / n)));
	else
		tmp = sqrt(Float64(Float64(1.0 / Float64(x * n)) / Float64(x * n)));
	end
	return tmp
end

Wolfram

code[x_, n_] := Block[{t$95$0 = N[(N[Log[x], $MachinePrecision] / n), $MachinePrecision]}, If[LessEqual[N[(1.0 / n), $MachinePrecision], -2e-114], N[(N[Exp[t$95$0], $MachinePrecision] / N[(x * n), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 5e-166], N[(N[(N[Log[1 + x], $MachinePrecision] / n), $MachinePrecision] - t$95$0), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 5e-136], N[(N[(N[(1.0 / x), $MachinePrecision] - N[(0.5 / N[Power[x, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / n), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 5e-20], N[(N[(N[Log[1 + x], $MachinePrecision] - N[Log[x], $MachinePrecision]), $MachinePrecision] * N[(1.0 / n), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 1e+199], N[(N[(1.0 + N[(x / n), $MachinePrecision]), $MachinePrecision] - N[Power[x, N[(1.0 / n), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[Sqrt[N[(N[(1.0 / N[(x * n), $MachinePrecision]), $MachinePrecision] / N[(x * n), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]]]]]

Derivation

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

Alternative 12: 80.0% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {x}^{\left(\frac{1}{n}\right)}\\ t_1 := \frac{\mathsf{log1p}\left(x\right) - \log x}{n}\\ \mathbf{if}\;\frac{1}{n} \leq -2 \cdot 10^{-114}:\\ \;\;\;\;\frac{t_0}{x \cdot n}\\ \mathbf{elif}\;\frac{1}{n} \leq 5 \cdot 10^{-166}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;\frac{1}{n} \leq 5 \cdot 10^{-136}:\\ \;\;\;\;\frac{\frac{1}{x} - \frac{0.5}{{x}^{2}}}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 5 \cdot 10^{-20}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;\frac{1}{n} \leq 10^{+199}:\\ \;\;\;\;\left(1 + \frac{x}{n}\right) - t_0\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{\frac{1}{x \cdot n}}{x \cdot n}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (let* ((t_0 (pow x (/ 1.0 n))) (t_1 (/ (- (log1p x) (log x)) n)))
   (if (<= (/ 1.0 n) -2e-114)
     (/ t_0 (* x n))
     (if (<= (/ 1.0 n) 5e-166)
       t_1
       (if (<= (/ 1.0 n) 5e-136)
         (/ (- (/ 1.0 x) (/ 0.5 (pow x 2.0))) n)
         (if (<= (/ 1.0 n) 5e-20)
           t_1
           (if (<= (/ 1.0 n) 1e+199)
             (- (+ 1.0 (/ x n)) t_0)
             (sqrt (/ (/ 1.0 (* x n)) (* x n))))))))))

C

double code(double x, double n) {
	double t_0 = pow(x, (1.0 / n));
	double t_1 = (log1p(x) - log(x)) / n;
	double tmp;
	if ((1.0 / n) <= -2e-114) {
		tmp = t_0 / (x * n);
	} else if ((1.0 / n) <= 5e-166) {
		tmp = t_1;
	} else if ((1.0 / n) <= 5e-136) {
		tmp = ((1.0 / x) - (0.5 / pow(x, 2.0))) / n;
	} else if ((1.0 / n) <= 5e-20) {
		tmp = t_1;
	} else if ((1.0 / n) <= 1e+199) {
		tmp = (1.0 + (x / n)) - t_0;
	} else {
		tmp = sqrt(((1.0 / (x * n)) / (x * n)));
	}
	return tmp;
}

Java

public static double code(double x, double n) {
	double t_0 = Math.pow(x, (1.0 / n));
	double t_1 = (Math.log1p(x) - Math.log(x)) / n;
	double tmp;
	if ((1.0 / n) <= -2e-114) {
		tmp = t_0 / (x * n);
	} else if ((1.0 / n) <= 5e-166) {
		tmp = t_1;
	} else if ((1.0 / n) <= 5e-136) {
		tmp = ((1.0 / x) - (0.5 / Math.pow(x, 2.0))) / n;
	} else if ((1.0 / n) <= 5e-20) {
		tmp = t_1;
	} else if ((1.0 / n) <= 1e+199) {
		tmp = (1.0 + (x / n)) - t_0;
	} else {
		tmp = Math.sqrt(((1.0 / (x * n)) / (x * n)));
	}
	return tmp;
}

Python

def code(x, n):
	t_0 = math.pow(x, (1.0 / n))
	t_1 = (math.log1p(x) - math.log(x)) / n
	tmp = 0
	if (1.0 / n) <= -2e-114:
		tmp = t_0 / (x * n)
	elif (1.0 / n) <= 5e-166:
		tmp = t_1
	elif (1.0 / n) <= 5e-136:
		tmp = ((1.0 / x) - (0.5 / math.pow(x, 2.0))) / n
	elif (1.0 / n) <= 5e-20:
		tmp = t_1
	elif (1.0 / n) <= 1e+199:
		tmp = (1.0 + (x / n)) - t_0
	else:
		tmp = math.sqrt(((1.0 / (x * n)) / (x * n)))
	return tmp

Julia

function code(x, n)
	t_0 = x ^ Float64(1.0 / n)
	t_1 = Float64(Float64(log1p(x) - log(x)) / n)
	tmp = 0.0
	if (Float64(1.0 / n) <= -2e-114)
		tmp = Float64(t_0 / Float64(x * n));
	elseif (Float64(1.0 / n) <= 5e-166)
		tmp = t_1;
	elseif (Float64(1.0 / n) <= 5e-136)
		tmp = Float64(Float64(Float64(1.0 / x) - Float64(0.5 / (x ^ 2.0))) / n);
	elseif (Float64(1.0 / n) <= 5e-20)
		tmp = t_1;
	elseif (Float64(1.0 / n) <= 1e+199)
		tmp = Float64(Float64(1.0 + Float64(x / n)) - t_0);
	else
		tmp = sqrt(Float64(Float64(1.0 / Float64(x * n)) / Float64(x * n)));
	end
	return tmp
end

Wolfram

code[x_, n_] := Block[{t$95$0 = N[Power[x, N[(1.0 / n), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[Log[1 + x], $MachinePrecision] - N[Log[x], $MachinePrecision]), $MachinePrecision] / n), $MachinePrecision]}, If[LessEqual[N[(1.0 / n), $MachinePrecision], -2e-114], N[(t$95$0 / N[(x * n), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 5e-166], t$95$1, If[LessEqual[N[(1.0 / n), $MachinePrecision], 5e-136], N[(N[(N[(1.0 / x), $MachinePrecision] - N[(0.5 / N[Power[x, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / n), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 5e-20], t$95$1, If[LessEqual[N[(1.0 / n), $MachinePrecision], 1e+199], N[(N[(1.0 + N[(x / n), $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision], N[Sqrt[N[(N[(1.0 / N[(x * n), $MachinePrecision]), $MachinePrecision] / N[(x * n), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]]]]]]

Derivation

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

Alternative 13: 67.6% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{if}\;\frac{1}{n} \leq -4 \cdot 10^{-205}:\\ \;\;\;\;\frac{1}{n} \cdot \frac{t_0}{x}\\ \mathbf{elif}\;\frac{1}{n} \leq 10^{-242}:\\ \;\;\;\;\frac{-\log x}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 4 \cdot 10^{-121}:\\ \;\;\;\;\frac{\frac{1}{x}}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 5 \cdot 10^{-20}:\\ \;\;\;\;\frac{x - \log x}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 10^{+199}:\\ \;\;\;\;\left(1 + \frac{x}{n}\right) - t_0\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{\frac{1}{x \cdot n}}{x \cdot n}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (let* ((t_0 (pow x (/ 1.0 n))))
   (if (<= (/ 1.0 n) -4e-205)
     (* (/ 1.0 n) (/ t_0 x))
     (if (<= (/ 1.0 n) 1e-242)
       (/ (- (log x)) n)
       (if (<= (/ 1.0 n) 4e-121)
         (/ (/ 1.0 x) n)
         (if (<= (/ 1.0 n) 5e-20)
           (/ (- x (log x)) n)
           (if (<= (/ 1.0 n) 1e+199)
             (- (+ 1.0 (/ x n)) t_0)
             (sqrt (/ (/ 1.0 (* x n)) (* x n))))))))))

C

double code(double x, double n) {
	double t_0 = pow(x, (1.0 / n));
	double tmp;
	if ((1.0 / n) <= -4e-205) {
		tmp = (1.0 / n) * (t_0 / x);
	} else if ((1.0 / n) <= 1e-242) {
		tmp = -log(x) / n;
	} else if ((1.0 / n) <= 4e-121) {
		tmp = (1.0 / x) / n;
	} else if ((1.0 / n) <= 5e-20) {
		tmp = (x - log(x)) / n;
	} else if ((1.0 / n) <= 1e+199) {
		tmp = (1.0 + (x / n)) - t_0;
	} else {
		tmp = sqrt(((1.0 / (x * n)) / (x * n)));
	}
	return tmp;
}

Fortran

real(8) function code(x, n)
    real(8), intent (in) :: x
    real(8), intent (in) :: n
    real(8) :: t_0
    real(8) :: tmp
    t_0 = x ** (1.0d0 / n)
    if ((1.0d0 / n) <= (-4d-205)) then
        tmp = (1.0d0 / n) * (t_0 / x)
    else if ((1.0d0 / n) <= 1d-242) then
        tmp = -log(x) / n
    else if ((1.0d0 / n) <= 4d-121) then
        tmp = (1.0d0 / x) / n
    else if ((1.0d0 / n) <= 5d-20) then
        tmp = (x - log(x)) / n
    else if ((1.0d0 / n) <= 1d+199) then
        tmp = (1.0d0 + (x / n)) - t_0
    else
        tmp = sqrt(((1.0d0 / (x * n)) / (x * n)))
    end if
    code = tmp
end function

Java

public static double code(double x, double n) {
	double t_0 = Math.pow(x, (1.0 / n));
	double tmp;
	if ((1.0 / n) <= -4e-205) {
		tmp = (1.0 / n) * (t_0 / x);
	} else if ((1.0 / n) <= 1e-242) {
		tmp = -Math.log(x) / n;
	} else if ((1.0 / n) <= 4e-121) {
		tmp = (1.0 / x) / n;
	} else if ((1.0 / n) <= 5e-20) {
		tmp = (x - Math.log(x)) / n;
	} else if ((1.0 / n) <= 1e+199) {
		tmp = (1.0 + (x / n)) - t_0;
	} else {
		tmp = Math.sqrt(((1.0 / (x * n)) / (x * n)));
	}
	return tmp;
}

Python

def code(x, n):
	t_0 = math.pow(x, (1.0 / n))
	tmp = 0
	if (1.0 / n) <= -4e-205:
		tmp = (1.0 / n) * (t_0 / x)
	elif (1.0 / n) <= 1e-242:
		tmp = -math.log(x) / n
	elif (1.0 / n) <= 4e-121:
		tmp = (1.0 / x) / n
	elif (1.0 / n) <= 5e-20:
		tmp = (x - math.log(x)) / n
	elif (1.0 / n) <= 1e+199:
		tmp = (1.0 + (x / n)) - t_0
	else:
		tmp = math.sqrt(((1.0 / (x * n)) / (x * n)))
	return tmp

Julia

function code(x, n)
	t_0 = x ^ Float64(1.0 / n)
	tmp = 0.0
	if (Float64(1.0 / n) <= -4e-205)
		tmp = Float64(Float64(1.0 / n) * Float64(t_0 / x));
	elseif (Float64(1.0 / n) <= 1e-242)
		tmp = Float64(Float64(-log(x)) / n);
	elseif (Float64(1.0 / n) <= 4e-121)
		tmp = Float64(Float64(1.0 / x) / n);
	elseif (Float64(1.0 / n) <= 5e-20)
		tmp = Float64(Float64(x - log(x)) / n);
	elseif (Float64(1.0 / n) <= 1e+199)
		tmp = Float64(Float64(1.0 + Float64(x / n)) - t_0);
	else
		tmp = sqrt(Float64(Float64(1.0 / Float64(x * n)) / Float64(x * n)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, n)
	t_0 = x ^ (1.0 / n);
	tmp = 0.0;
	if ((1.0 / n) <= -4e-205)
		tmp = (1.0 / n) * (t_0 / x);
	elseif ((1.0 / n) <= 1e-242)
		tmp = -log(x) / n;
	elseif ((1.0 / n) <= 4e-121)
		tmp = (1.0 / x) / n;
	elseif ((1.0 / n) <= 5e-20)
		tmp = (x - log(x)) / n;
	elseif ((1.0 / n) <= 1e+199)
		tmp = (1.0 + (x / n)) - t_0;
	else
		tmp = sqrt(((1.0 / (x * n)) / (x * n)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, n_] := Block[{t$95$0 = N[Power[x, N[(1.0 / n), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[N[(1.0 / n), $MachinePrecision], -4e-205], N[(N[(1.0 / n), $MachinePrecision] * N[(t$95$0 / x), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 1e-242], N[((-N[Log[x], $MachinePrecision]) / n), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 4e-121], N[(N[(1.0 / x), $MachinePrecision] / n), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 5e-20], N[(N[(x - N[Log[x], $MachinePrecision]), $MachinePrecision] / n), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 1e+199], N[(N[(1.0 + N[(x / n), $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision], N[Sqrt[N[(N[(1.0 / N[(x * n), $MachinePrecision]), $MachinePrecision] / N[(x * n), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]]]]]

Derivation

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

Alternative 14: 67.5% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{if}\;\frac{1}{n} \leq -4 \cdot 10^{-205}:\\ \;\;\;\;\frac{1}{n} \cdot \frac{t_0}{x}\\ \mathbf{elif}\;\frac{1}{n} \leq 10^{-242}:\\ \;\;\;\;\frac{-\log x}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 4 \cdot 10^{-121}:\\ \;\;\;\;\frac{\frac{1}{x} - \frac{0.5}{{x}^{2}}}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 5 \cdot 10^{-20}:\\ \;\;\;\;\frac{x - \log x}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 10^{+199}:\\ \;\;\;\;\left(1 + \frac{x}{n}\right) - t_0\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{\frac{1}{x \cdot n}}{x \cdot n}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (let* ((t_0 (pow x (/ 1.0 n))))
   (if (<= (/ 1.0 n) -4e-205)
     (* (/ 1.0 n) (/ t_0 x))
     (if (<= (/ 1.0 n) 1e-242)
       (/ (- (log x)) n)
       (if (<= (/ 1.0 n) 4e-121)
         (/ (- (/ 1.0 x) (/ 0.5 (pow x 2.0))) n)
         (if (<= (/ 1.0 n) 5e-20)
           (/ (- x (log x)) n)
           (if (<= (/ 1.0 n) 1e+199)
             (- (+ 1.0 (/ x n)) t_0)
             (sqrt (/ (/ 1.0 (* x n)) (* x n))))))))))

C

double code(double x, double n) {
	double t_0 = pow(x, (1.0 / n));
	double tmp;
	if ((1.0 / n) <= -4e-205) {
		tmp = (1.0 / n) * (t_0 / x);
	} else if ((1.0 / n) <= 1e-242) {
		tmp = -log(x) / n;
	} else if ((1.0 / n) <= 4e-121) {
		tmp = ((1.0 / x) - (0.5 / pow(x, 2.0))) / n;
	} else if ((1.0 / n) <= 5e-20) {
		tmp = (x - log(x)) / n;
	} else if ((1.0 / n) <= 1e+199) {
		tmp = (1.0 + (x / n)) - t_0;
	} else {
		tmp = sqrt(((1.0 / (x * n)) / (x * n)));
	}
	return tmp;
}

Fortran

real(8) function code(x, n)
    real(8), intent (in) :: x
    real(8), intent (in) :: n
    real(8) :: t_0
    real(8) :: tmp
    t_0 = x ** (1.0d0 / n)
    if ((1.0d0 / n) <= (-4d-205)) then
        tmp = (1.0d0 / n) * (t_0 / x)
    else if ((1.0d0 / n) <= 1d-242) then
        tmp = -log(x) / n
    else if ((1.0d0 / n) <= 4d-121) then
        tmp = ((1.0d0 / x) - (0.5d0 / (x ** 2.0d0))) / n
    else if ((1.0d0 / n) <= 5d-20) then
        tmp = (x - log(x)) / n
    else if ((1.0d0 / n) <= 1d+199) then
        tmp = (1.0d0 + (x / n)) - t_0
    else
        tmp = sqrt(((1.0d0 / (x * n)) / (x * n)))
    end if
    code = tmp
end function

Java

public static double code(double x, double n) {
	double t_0 = Math.pow(x, (1.0 / n));
	double tmp;
	if ((1.0 / n) <= -4e-205) {
		tmp = (1.0 / n) * (t_0 / x);
	} else if ((1.0 / n) <= 1e-242) {
		tmp = -Math.log(x) / n;
	} else if ((1.0 / n) <= 4e-121) {
		tmp = ((1.0 / x) - (0.5 / Math.pow(x, 2.0))) / n;
	} else if ((1.0 / n) <= 5e-20) {
		tmp = (x - Math.log(x)) / n;
	} else if ((1.0 / n) <= 1e+199) {
		tmp = (1.0 + (x / n)) - t_0;
	} else {
		tmp = Math.sqrt(((1.0 / (x * n)) / (x * n)));
	}
	return tmp;
}

Python

def code(x, n):
	t_0 = math.pow(x, (1.0 / n))
	tmp = 0
	if (1.0 / n) <= -4e-205:
		tmp = (1.0 / n) * (t_0 / x)
	elif (1.0 / n) <= 1e-242:
		tmp = -math.log(x) / n
	elif (1.0 / n) <= 4e-121:
		tmp = ((1.0 / x) - (0.5 / math.pow(x, 2.0))) / n
	elif (1.0 / n) <= 5e-20:
		tmp = (x - math.log(x)) / n
	elif (1.0 / n) <= 1e+199:
		tmp = (1.0 + (x / n)) - t_0
	else:
		tmp = math.sqrt(((1.0 / (x * n)) / (x * n)))
	return tmp

Julia

function code(x, n)
	t_0 = x ^ Float64(1.0 / n)
	tmp = 0.0
	if (Float64(1.0 / n) <= -4e-205)
		tmp = Float64(Float64(1.0 / n) * Float64(t_0 / x));
	elseif (Float64(1.0 / n) <= 1e-242)
		tmp = Float64(Float64(-log(x)) / n);
	elseif (Float64(1.0 / n) <= 4e-121)
		tmp = Float64(Float64(Float64(1.0 / x) - Float64(0.5 / (x ^ 2.0))) / n);
	elseif (Float64(1.0 / n) <= 5e-20)
		tmp = Float64(Float64(x - log(x)) / n);
	elseif (Float64(1.0 / n) <= 1e+199)
		tmp = Float64(Float64(1.0 + Float64(x / n)) - t_0);
	else
		tmp = sqrt(Float64(Float64(1.0 / Float64(x * n)) / Float64(x * n)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, n)
	t_0 = x ^ (1.0 / n);
	tmp = 0.0;
	if ((1.0 / n) <= -4e-205)
		tmp = (1.0 / n) * (t_0 / x);
	elseif ((1.0 / n) <= 1e-242)
		tmp = -log(x) / n;
	elseif ((1.0 / n) <= 4e-121)
		tmp = ((1.0 / x) - (0.5 / (x ^ 2.0))) / n;
	elseif ((1.0 / n) <= 5e-20)
		tmp = (x - log(x)) / n;
	elseif ((1.0 / n) <= 1e+199)
		tmp = (1.0 + (x / n)) - t_0;
	else
		tmp = sqrt(((1.0 / (x * n)) / (x * n)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, n_] := Block[{t$95$0 = N[Power[x, N[(1.0 / n), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[N[(1.0 / n), $MachinePrecision], -4e-205], N[(N[(1.0 / n), $MachinePrecision] * N[(t$95$0 / x), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 1e-242], N[((-N[Log[x], $MachinePrecision]) / n), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 4e-121], N[(N[(N[(1.0 / x), $MachinePrecision] - N[(0.5 / N[Power[x, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / n), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 5e-20], N[(N[(x - N[Log[x], $MachinePrecision]), $MachinePrecision] / n), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 1e+199], N[(N[(1.0 + N[(x / n), $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision], N[Sqrt[N[(N[(1.0 / N[(x * n), $MachinePrecision]), $MachinePrecision] / N[(x * n), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]]]]]

Derivation

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

Alternative 15: 67.2% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{if}\;\frac{1}{n} \leq -4 \cdot 10^{-205}:\\ \;\;\;\;\frac{t_0}{x \cdot n}\\ \mathbf{elif}\;\frac{1}{n} \leq 10^{-242}:\\ \;\;\;\;\frac{-\log x}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 4 \cdot 10^{-121}:\\ \;\;\;\;\frac{\frac{1}{x}}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 5 \cdot 10^{-20}:\\ \;\;\;\;\frac{x - \log x}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 5 \cdot 10^{+183}:\\ \;\;\;\;1 - t_0\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{\frac{1}{x \cdot n}}{x \cdot n}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (let* ((t_0 (pow x (/ 1.0 n))))
   (if (<= (/ 1.0 n) -4e-205)
     (/ t_0 (* x n))
     (if (<= (/ 1.0 n) 1e-242)
       (/ (- (log x)) n)
       (if (<= (/ 1.0 n) 4e-121)
         (/ (/ 1.0 x) n)
         (if (<= (/ 1.0 n) 5e-20)
           (/ (- x (log x)) n)
           (if (<= (/ 1.0 n) 5e+183)
             (- 1.0 t_0)
             (sqrt (/ (/ 1.0 (* x n)) (* x n))))))))))

C

double code(double x, double n) {
	double t_0 = pow(x, (1.0 / n));
	double tmp;
	if ((1.0 / n) <= -4e-205) {
		tmp = t_0 / (x * n);
	} else if ((1.0 / n) <= 1e-242) {
		tmp = -log(x) / n;
	} else if ((1.0 / n) <= 4e-121) {
		tmp = (1.0 / x) / n;
	} else if ((1.0 / n) <= 5e-20) {
		tmp = (x - log(x)) / n;
	} else if ((1.0 / n) <= 5e+183) {
		tmp = 1.0 - t_0;
	} else {
		tmp = sqrt(((1.0 / (x * n)) / (x * n)));
	}
	return tmp;
}

Fortran

real(8) function code(x, n)
    real(8), intent (in) :: x
    real(8), intent (in) :: n
    real(8) :: t_0
    real(8) :: tmp
    t_0 = x ** (1.0d0 / n)
    if ((1.0d0 / n) <= (-4d-205)) then
        tmp = t_0 / (x * n)
    else if ((1.0d0 / n) <= 1d-242) then
        tmp = -log(x) / n
    else if ((1.0d0 / n) <= 4d-121) then
        tmp = (1.0d0 / x) / n
    else if ((1.0d0 / n) <= 5d-20) then
        tmp = (x - log(x)) / n
    else if ((1.0d0 / n) <= 5d+183) then
        tmp = 1.0d0 - t_0
    else
        tmp = sqrt(((1.0d0 / (x * n)) / (x * n)))
    end if
    code = tmp
end function

Java

public static double code(double x, double n) {
	double t_0 = Math.pow(x, (1.0 / n));
	double tmp;
	if ((1.0 / n) <= -4e-205) {
		tmp = t_0 / (x * n);
	} else if ((1.0 / n) <= 1e-242) {
		tmp = -Math.log(x) / n;
	} else if ((1.0 / n) <= 4e-121) {
		tmp = (1.0 / x) / n;
	} else if ((1.0 / n) <= 5e-20) {
		tmp = (x - Math.log(x)) / n;
	} else if ((1.0 / n) <= 5e+183) {
		tmp = 1.0 - t_0;
	} else {
		tmp = Math.sqrt(((1.0 / (x * n)) / (x * n)));
	}
	return tmp;
}

Python

def code(x, n):
	t_0 = math.pow(x, (1.0 / n))
	tmp = 0
	if (1.0 / n) <= -4e-205:
		tmp = t_0 / (x * n)
	elif (1.0 / n) <= 1e-242:
		tmp = -math.log(x) / n
	elif (1.0 / n) <= 4e-121:
		tmp = (1.0 / x) / n
	elif (1.0 / n) <= 5e-20:
		tmp = (x - math.log(x)) / n
	elif (1.0 / n) <= 5e+183:
		tmp = 1.0 - t_0
	else:
		tmp = math.sqrt(((1.0 / (x * n)) / (x * n)))
	return tmp

Julia

function code(x, n)
	t_0 = x ^ Float64(1.0 / n)
	tmp = 0.0
	if (Float64(1.0 / n) <= -4e-205)
		tmp = Float64(t_0 / Float64(x * n));
	elseif (Float64(1.0 / n) <= 1e-242)
		tmp = Float64(Float64(-log(x)) / n);
	elseif (Float64(1.0 / n) <= 4e-121)
		tmp = Float64(Float64(1.0 / x) / n);
	elseif (Float64(1.0 / n) <= 5e-20)
		tmp = Float64(Float64(x - log(x)) / n);
	elseif (Float64(1.0 / n) <= 5e+183)
		tmp = Float64(1.0 - t_0);
	else
		tmp = sqrt(Float64(Float64(1.0 / Float64(x * n)) / Float64(x * n)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, n)
	t_0 = x ^ (1.0 / n);
	tmp = 0.0;
	if ((1.0 / n) <= -4e-205)
		tmp = t_0 / (x * n);
	elseif ((1.0 / n) <= 1e-242)
		tmp = -log(x) / n;
	elseif ((1.0 / n) <= 4e-121)
		tmp = (1.0 / x) / n;
	elseif ((1.0 / n) <= 5e-20)
		tmp = (x - log(x)) / n;
	elseif ((1.0 / n) <= 5e+183)
		tmp = 1.0 - t_0;
	else
		tmp = sqrt(((1.0 / (x * n)) / (x * n)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, n_] := Block[{t$95$0 = N[Power[x, N[(1.0 / n), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[N[(1.0 / n), $MachinePrecision], -4e-205], N[(t$95$0 / N[(x * n), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 1e-242], N[((-N[Log[x], $MachinePrecision]) / n), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 4e-121], N[(N[(1.0 / x), $MachinePrecision] / n), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 5e-20], N[(N[(x - N[Log[x], $MachinePrecision]), $MachinePrecision] / n), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 5e+183], N[(1.0 - t$95$0), $MachinePrecision], N[Sqrt[N[(N[(1.0 / N[(x * n), $MachinePrecision]), $MachinePrecision] / N[(x * n), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]]]]]

Derivation

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

Alternative 16: 67.4% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{if}\;\frac{1}{n} \leq -4 \cdot 10^{-205}:\\ \;\;\;\;\frac{1}{n} \cdot \frac{t_0}{x}\\ \mathbf{elif}\;\frac{1}{n} \leq 10^{-242}:\\ \;\;\;\;\frac{-\log x}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 4 \cdot 10^{-121}:\\ \;\;\;\;\frac{\frac{1}{x}}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 5 \cdot 10^{-20}:\\ \;\;\;\;\frac{x - \log x}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 5 \cdot 10^{+183}:\\ \;\;\;\;1 - t_0\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{\frac{1}{x \cdot n}}{x \cdot n}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (let* ((t_0 (pow x (/ 1.0 n))))
   (if (<= (/ 1.0 n) -4e-205)
     (* (/ 1.0 n) (/ t_0 x))
     (if (<= (/ 1.0 n) 1e-242)
       (/ (- (log x)) n)
       (if (<= (/ 1.0 n) 4e-121)
         (/ (/ 1.0 x) n)
         (if (<= (/ 1.0 n) 5e-20)
           (/ (- x (log x)) n)
           (if (<= (/ 1.0 n) 5e+183)
             (- 1.0 t_0)
             (sqrt (/ (/ 1.0 (* x n)) (* x n))))))))))

C

double code(double x, double n) {
	double t_0 = pow(x, (1.0 / n));
	double tmp;
	if ((1.0 / n) <= -4e-205) {
		tmp = (1.0 / n) * (t_0 / x);
	} else if ((1.0 / n) <= 1e-242) {
		tmp = -log(x) / n;
	} else if ((1.0 / n) <= 4e-121) {
		tmp = (1.0 / x) / n;
	} else if ((1.0 / n) <= 5e-20) {
		tmp = (x - log(x)) / n;
	} else if ((1.0 / n) <= 5e+183) {
		tmp = 1.0 - t_0;
	} else {
		tmp = sqrt(((1.0 / (x * n)) / (x * n)));
	}
	return tmp;
}

Fortran

real(8) function code(x, n)
    real(8), intent (in) :: x
    real(8), intent (in) :: n
    real(8) :: t_0
    real(8) :: tmp
    t_0 = x ** (1.0d0 / n)
    if ((1.0d0 / n) <= (-4d-205)) then
        tmp = (1.0d0 / n) * (t_0 / x)
    else if ((1.0d0 / n) <= 1d-242) then
        tmp = -log(x) / n
    else if ((1.0d0 / n) <= 4d-121) then
        tmp = (1.0d0 / x) / n
    else if ((1.0d0 / n) <= 5d-20) then
        tmp = (x - log(x)) / n
    else if ((1.0d0 / n) <= 5d+183) then
        tmp = 1.0d0 - t_0
    else
        tmp = sqrt(((1.0d0 / (x * n)) / (x * n)))
    end if
    code = tmp
end function

Java

public static double code(double x, double n) {
	double t_0 = Math.pow(x, (1.0 / n));
	double tmp;
	if ((1.0 / n) <= -4e-205) {
		tmp = (1.0 / n) * (t_0 / x);
	} else if ((1.0 / n) <= 1e-242) {
		tmp = -Math.log(x) / n;
	} else if ((1.0 / n) <= 4e-121) {
		tmp = (1.0 / x) / n;
	} else if ((1.0 / n) <= 5e-20) {
		tmp = (x - Math.log(x)) / n;
	} else if ((1.0 / n) <= 5e+183) {
		tmp = 1.0 - t_0;
	} else {
		tmp = Math.sqrt(((1.0 / (x * n)) / (x * n)));
	}
	return tmp;
}

Python

def code(x, n):
	t_0 = math.pow(x, (1.0 / n))
	tmp = 0
	if (1.0 / n) <= -4e-205:
		tmp = (1.0 / n) * (t_0 / x)
	elif (1.0 / n) <= 1e-242:
		tmp = -math.log(x) / n
	elif (1.0 / n) <= 4e-121:
		tmp = (1.0 / x) / n
	elif (1.0 / n) <= 5e-20:
		tmp = (x - math.log(x)) / n
	elif (1.0 / n) <= 5e+183:
		tmp = 1.0 - t_0
	else:
		tmp = math.sqrt(((1.0 / (x * n)) / (x * n)))
	return tmp

Julia

function code(x, n)
	t_0 = x ^ Float64(1.0 / n)
	tmp = 0.0
	if (Float64(1.0 / n) <= -4e-205)
		tmp = Float64(Float64(1.0 / n) * Float64(t_0 / x));
	elseif (Float64(1.0 / n) <= 1e-242)
		tmp = Float64(Float64(-log(x)) / n);
	elseif (Float64(1.0 / n) <= 4e-121)
		tmp = Float64(Float64(1.0 / x) / n);
	elseif (Float64(1.0 / n) <= 5e-20)
		tmp = Float64(Float64(x - log(x)) / n);
	elseif (Float64(1.0 / n) <= 5e+183)
		tmp = Float64(1.0 - t_0);
	else
		tmp = sqrt(Float64(Float64(1.0 / Float64(x * n)) / Float64(x * n)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, n)
	t_0 = x ^ (1.0 / n);
	tmp = 0.0;
	if ((1.0 / n) <= -4e-205)
		tmp = (1.0 / n) * (t_0 / x);
	elseif ((1.0 / n) <= 1e-242)
		tmp = -log(x) / n;
	elseif ((1.0 / n) <= 4e-121)
		tmp = (1.0 / x) / n;
	elseif ((1.0 / n) <= 5e-20)
		tmp = (x - log(x)) / n;
	elseif ((1.0 / n) <= 5e+183)
		tmp = 1.0 - t_0;
	else
		tmp = sqrt(((1.0 / (x * n)) / (x * n)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, n_] := Block[{t$95$0 = N[Power[x, N[(1.0 / n), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[N[(1.0 / n), $MachinePrecision], -4e-205], N[(N[(1.0 / n), $MachinePrecision] * N[(t$95$0 / x), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 1e-242], N[((-N[Log[x], $MachinePrecision]) / n), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 4e-121], N[(N[(1.0 / x), $MachinePrecision] / n), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 5e-20], N[(N[(x - N[Log[x], $MachinePrecision]), $MachinePrecision] / n), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 5e+183], N[(1.0 - t$95$0), $MachinePrecision], N[Sqrt[N[(N[(1.0 / N[(x * n), $MachinePrecision]), $MachinePrecision] / N[(x * n), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]]]]]

Derivation

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

Alternative 17: 66.4% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{if}\;\frac{1}{n} \leq -4 \cdot 10^{-205}:\\ \;\;\;\;\frac{t_0}{x \cdot n}\\ \mathbf{elif}\;\frac{1}{n} \leq 10^{-242}:\\ \;\;\;\;\frac{-\log x}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 4 \cdot 10^{-121}:\\ \;\;\;\;\frac{\frac{1}{x}}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 5 \cdot 10^{-20}:\\ \;\;\;\;\frac{x - \log x}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 5 \cdot 10^{+183}:\\ \;\;\;\;1 - t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{x \cdot n}\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (let* ((t_0 (pow x (/ 1.0 n))))
   (if (<= (/ 1.0 n) -4e-205)
     (/ t_0 (* x n))
     (if (<= (/ 1.0 n) 1e-242)
       (/ (- (log x)) n)
       (if (<= (/ 1.0 n) 4e-121)
         (/ (/ 1.0 x) n)
         (if (<= (/ 1.0 n) 5e-20)
           (/ (- x (log x)) n)
           (if (<= (/ 1.0 n) 5e+183) (- 1.0 t_0) (/ 1.0 (* x n)))))))))

C

double code(double x, double n) {
	double t_0 = pow(x, (1.0 / n));
	double tmp;
	if ((1.0 / n) <= -4e-205) {
		tmp = t_0 / (x * n);
	} else if ((1.0 / n) <= 1e-242) {
		tmp = -log(x) / n;
	} else if ((1.0 / n) <= 4e-121) {
		tmp = (1.0 / x) / n;
	} else if ((1.0 / n) <= 5e-20) {
		tmp = (x - log(x)) / n;
	} else if ((1.0 / n) <= 5e+183) {
		tmp = 1.0 - t_0;
	} else {
		tmp = 1.0 / (x * n);
	}
	return tmp;
}

Fortran

real(8) function code(x, n)
    real(8), intent (in) :: x
    real(8), intent (in) :: n
    real(8) :: t_0
    real(8) :: tmp
    t_0 = x ** (1.0d0 / n)
    if ((1.0d0 / n) <= (-4d-205)) then
        tmp = t_0 / (x * n)
    else if ((1.0d0 / n) <= 1d-242) then
        tmp = -log(x) / n
    else if ((1.0d0 / n) <= 4d-121) then
        tmp = (1.0d0 / x) / n
    else if ((1.0d0 / n) <= 5d-20) then
        tmp = (x - log(x)) / n
    else if ((1.0d0 / n) <= 5d+183) then
        tmp = 1.0d0 - t_0
    else
        tmp = 1.0d0 / (x * n)
    end if
    code = tmp
end function

Java

public static double code(double x, double n) {
	double t_0 = Math.pow(x, (1.0 / n));
	double tmp;
	if ((1.0 / n) <= -4e-205) {
		tmp = t_0 / (x * n);
	} else if ((1.0 / n) <= 1e-242) {
		tmp = -Math.log(x) / n;
	} else if ((1.0 / n) <= 4e-121) {
		tmp = (1.0 / x) / n;
	} else if ((1.0 / n) <= 5e-20) {
		tmp = (x - Math.log(x)) / n;
	} else if ((1.0 / n) <= 5e+183) {
		tmp = 1.0 - t_0;
	} else {
		tmp = 1.0 / (x * n);
	}
	return tmp;
}

Python

def code(x, n):
	t_0 = math.pow(x, (1.0 / n))
	tmp = 0
	if (1.0 / n) <= -4e-205:
		tmp = t_0 / (x * n)
	elif (1.0 / n) <= 1e-242:
		tmp = -math.log(x) / n
	elif (1.0 / n) <= 4e-121:
		tmp = (1.0 / x) / n
	elif (1.0 / n) <= 5e-20:
		tmp = (x - math.log(x)) / n
	elif (1.0 / n) <= 5e+183:
		tmp = 1.0 - t_0
	else:
		tmp = 1.0 / (x * n)
	return tmp

Julia

function code(x, n)
	t_0 = x ^ Float64(1.0 / n)
	tmp = 0.0
	if (Float64(1.0 / n) <= -4e-205)
		tmp = Float64(t_0 / Float64(x * n));
	elseif (Float64(1.0 / n) <= 1e-242)
		tmp = Float64(Float64(-log(x)) / n);
	elseif (Float64(1.0 / n) <= 4e-121)
		tmp = Float64(Float64(1.0 / x) / n);
	elseif (Float64(1.0 / n) <= 5e-20)
		tmp = Float64(Float64(x - log(x)) / n);
	elseif (Float64(1.0 / n) <= 5e+183)
		tmp = Float64(1.0 - t_0);
	else
		tmp = Float64(1.0 / Float64(x * n));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, n)
	t_0 = x ^ (1.0 / n);
	tmp = 0.0;
	if ((1.0 / n) <= -4e-205)
		tmp = t_0 / (x * n);
	elseif ((1.0 / n) <= 1e-242)
		tmp = -log(x) / n;
	elseif ((1.0 / n) <= 4e-121)
		tmp = (1.0 / x) / n;
	elseif ((1.0 / n) <= 5e-20)
		tmp = (x - log(x)) / n;
	elseif ((1.0 / n) <= 5e+183)
		tmp = 1.0 - t_0;
	else
		tmp = 1.0 / (x * n);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, n_] := Block[{t$95$0 = N[Power[x, N[(1.0 / n), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[N[(1.0 / n), $MachinePrecision], -4e-205], N[(t$95$0 / N[(x * n), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 1e-242], N[((-N[Log[x], $MachinePrecision]) / n), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 4e-121], N[(N[(1.0 / x), $MachinePrecision] / n), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 5e-20], N[(N[(x - N[Log[x], $MachinePrecision]), $MachinePrecision] / n), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 5e+183], N[(1.0 - t$95$0), $MachinePrecision], N[(1.0 / N[(x * n), $MachinePrecision]), $MachinePrecision]]]]]]]

Derivation

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

Alternative 18: 55.5% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 7 \cdot 10^{-213}:\\ \;\;\;\;\frac{-\log x}{n}\\ \mathbf{elif}\;x \leq 2.3 \cdot 10^{-115}:\\ \;\;\;\;1 - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{elif}\;x \leq 1:\\ \;\;\;\;\frac{x - \log x}{n}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{x}}{n}\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (if (<= x 7e-213)
   (/ (- (log x)) n)
   (if (<= x 2.3e-115)
     (- 1.0 (pow x (/ 1.0 n)))
     (if (<= x 1.0) (/ (- x (log x)) n) (/ (/ 1.0 x) n)))))

C

double code(double x, double n) {
	double tmp;
	if (x <= 7e-213) {
		tmp = -log(x) / n;
	} else if (x <= 2.3e-115) {
		tmp = 1.0 - pow(x, (1.0 / n));
	} else if (x <= 1.0) {
		tmp = (x - log(x)) / n;
	} else {
		tmp = (1.0 / x) / n;
	}
	return tmp;
}

Fortran

real(8) function code(x, n)
    real(8), intent (in) :: x
    real(8), intent (in) :: n
    real(8) :: tmp
    if (x <= 7d-213) then
        tmp = -log(x) / n
    else if (x <= 2.3d-115) then
        tmp = 1.0d0 - (x ** (1.0d0 / n))
    else if (x <= 1.0d0) then
        tmp = (x - log(x)) / n
    else
        tmp = (1.0d0 / x) / n
    end if
    code = tmp
end function

Java

public static double code(double x, double n) {
	double tmp;
	if (x <= 7e-213) {
		tmp = -Math.log(x) / n;
	} else if (x <= 2.3e-115) {
		tmp = 1.0 - Math.pow(x, (1.0 / n));
	} else if (x <= 1.0) {
		tmp = (x - Math.log(x)) / n;
	} else {
		tmp = (1.0 / x) / n;
	}
	return tmp;
}

Python

def code(x, n):
	tmp = 0
	if x <= 7e-213:
		tmp = -math.log(x) / n
	elif x <= 2.3e-115:
		tmp = 1.0 - math.pow(x, (1.0 / n))
	elif x <= 1.0:
		tmp = (x - math.log(x)) / n
	else:
		tmp = (1.0 / x) / n
	return tmp

Julia

function code(x, n)
	tmp = 0.0
	if (x <= 7e-213)
		tmp = Float64(Float64(-log(x)) / n);
	elseif (x <= 2.3e-115)
		tmp = Float64(1.0 - (x ^ Float64(1.0 / n)));
	elseif (x <= 1.0)
		tmp = Float64(Float64(x - log(x)) / n);
	else
		tmp = Float64(Float64(1.0 / x) / n);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, n)
	tmp = 0.0;
	if (x <= 7e-213)
		tmp = -log(x) / n;
	elseif (x <= 2.3e-115)
		tmp = 1.0 - (x ^ (1.0 / n));
	elseif (x <= 1.0)
		tmp = (x - log(x)) / n;
	else
		tmp = (1.0 / x) / n;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, n_] := If[LessEqual[x, 7e-213], N[((-N[Log[x], $MachinePrecision]) / n), $MachinePrecision], If[LessEqual[x, 2.3e-115], N[(1.0 - N[Power[x, N[(1.0 / n), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 1.0], N[(N[(x - N[Log[x], $MachinePrecision]), $MachinePrecision] / n), $MachinePrecision], N[(N[(1.0 / x), $MachinePrecision] / n), $MachinePrecision]]]]

Derivation

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

Alternative 19: 55.2% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{-\log x}{n}\\ \mathbf{if}\;x \leq 2.35 \cdot 10^{-213}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \leq 1.75 \cdot 10^{-115}:\\ \;\;\;\;1 - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{elif}\;x \leq 0.56:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{x}}{n}\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (let* ((t_0 (/ (- (log x)) n)))
   (if (<= x 2.35e-213)
     t_0
     (if (<= x 1.75e-115)
       (- 1.0 (pow x (/ 1.0 n)))
       (if (<= x 0.56) t_0 (/ (/ 1.0 x) n))))))

C

double code(double x, double n) {
	double t_0 = -log(x) / n;
	double tmp;
	if (x <= 2.35e-213) {
		tmp = t_0;
	} else if (x <= 1.75e-115) {
		tmp = 1.0 - pow(x, (1.0 / n));
	} else if (x <= 0.56) {
		tmp = t_0;
	} else {
		tmp = (1.0 / x) / n;
	}
	return tmp;
}

Fortran

real(8) function code(x, n)
    real(8), intent (in) :: x
    real(8), intent (in) :: n
    real(8) :: t_0
    real(8) :: tmp
    t_0 = -log(x) / n
    if (x <= 2.35d-213) then
        tmp = t_0
    else if (x <= 1.75d-115) then
        tmp = 1.0d0 - (x ** (1.0d0 / n))
    else if (x <= 0.56d0) then
        tmp = t_0
    else
        tmp = (1.0d0 / x) / n
    end if
    code = tmp
end function

Java

public static double code(double x, double n) {
	double t_0 = -Math.log(x) / n;
	double tmp;
	if (x <= 2.35e-213) {
		tmp = t_0;
	} else if (x <= 1.75e-115) {
		tmp = 1.0 - Math.pow(x, (1.0 / n));
	} else if (x <= 0.56) {
		tmp = t_0;
	} else {
		tmp = (1.0 / x) / n;
	}
	return tmp;
}

Python

def code(x, n):
	t_0 = -math.log(x) / n
	tmp = 0
	if x <= 2.35e-213:
		tmp = t_0
	elif x <= 1.75e-115:
		tmp = 1.0 - math.pow(x, (1.0 / n))
	elif x <= 0.56:
		tmp = t_0
	else:
		tmp = (1.0 / x) / n
	return tmp

Julia

function code(x, n)
	t_0 = Float64(Float64(-log(x)) / n)
	tmp = 0.0
	if (x <= 2.35e-213)
		tmp = t_0;
	elseif (x <= 1.75e-115)
		tmp = Float64(1.0 - (x ^ Float64(1.0 / n)));
	elseif (x <= 0.56)
		tmp = t_0;
	else
		tmp = Float64(Float64(1.0 / x) / n);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, n)
	t_0 = -log(x) / n;
	tmp = 0.0;
	if (x <= 2.35e-213)
		tmp = t_0;
	elseif (x <= 1.75e-115)
		tmp = 1.0 - (x ^ (1.0 / n));
	elseif (x <= 0.56)
		tmp = t_0;
	else
		tmp = (1.0 / x) / n;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, n_] := Block[{t$95$0 = N[((-N[Log[x], $MachinePrecision]) / n), $MachinePrecision]}, If[LessEqual[x, 2.35e-213], t$95$0, If[LessEqual[x, 1.75e-115], N[(1.0 - N[Power[x, N[(1.0 / n), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 0.56], t$95$0, N[(N[(1.0 / x), $MachinePrecision] / n), $MachinePrecision]]]]]

Derivation

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

Alternative 20: 57.0% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 0.56:\\ \;\;\;\;\frac{-\log x}{n}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{x}}{n}\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (if (<= x 0.56) (/ (- (log x)) n) (/ (/ 1.0 x) n)))

C

double code(double x, double n) {
	double tmp;
	if (x <= 0.56) {
		tmp = -log(x) / n;
	} else {
		tmp = (1.0 / x) / n;
	}
	return tmp;
}

Fortran

real(8) function code(x, n)
    real(8), intent (in) :: x
    real(8), intent (in) :: n
    real(8) :: tmp
    if (x <= 0.56d0) then
        tmp = -log(x) / n
    else
        tmp = (1.0d0 / x) / n
    end if
    code = tmp
end function

Java

public static double code(double x, double n) {
	double tmp;
	if (x <= 0.56) {
		tmp = -Math.log(x) / n;
	} else {
		tmp = (1.0 / x) / n;
	}
	return tmp;
}

Python

def code(x, n):
	tmp = 0
	if x <= 0.56:
		tmp = -math.log(x) / n
	else:
		tmp = (1.0 / x) / n
	return tmp

Julia

function code(x, n)
	tmp = 0.0
	if (x <= 0.56)
		tmp = Float64(Float64(-log(x)) / n);
	else
		tmp = Float64(Float64(1.0 / x) / n);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, n)
	tmp = 0.0;
	if (x <= 0.56)
		tmp = -log(x) / n;
	else
		tmp = (1.0 / x) / n;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, n_] := If[LessEqual[x, 0.56], N[((-N[Log[x], $MachinePrecision]) / n), $MachinePrecision], N[(N[(1.0 / x), $MachinePrecision] / n), $MachinePrecision]]

Derivation

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

Alternative 21: 39.9% accurate, 42.2× speedup

Math

\[\begin{array}{l} \\ \frac{1}{x \cdot n} \end{array} \]

FPCore

(FPCore (x n) :precision binary64 (/ 1.0 (* x n)))

C

double code(double x, double n) {
	return 1.0 / (x * n);
}

Fortran

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

Java

public static double code(double x, double n) {
	return 1.0 / (x * n);
}

Python

def code(x, n):
	return 1.0 / (x * n)

Julia

function code(x, n)
	return Float64(1.0 / Float64(x * n))
end

MATLAB

function tmp = code(x, n)
	tmp = 1.0 / (x * n);
end

Wolfram

code[x_, n_] := N[(1.0 / N[(x * n), $MachinePrecision]), $MachinePrecision]

Derivation

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

Alternative 22: 40.5% accurate, 42.2× speedup

Math

\[\begin{array}{l} \\ \frac{\frac{1}{n}}{x} \end{array} \]

FPCore

(FPCore (x n) :precision binary64 (/ (/ 1.0 n) x))

C

double code(double x, double n) {
	return (1.0 / n) / x;
}

Fortran

real(8) function code(x, n)
    real(8), intent (in) :: x
    real(8), intent (in) :: n
    code = (1.0d0 / n) / x
end function

Java

public static double code(double x, double n) {
	return (1.0 / n) / x;
}

Python

def code(x, n):
	return (1.0 / n) / x

Julia

function code(x, n)
	return Float64(Float64(1.0 / n) / x)
end

MATLAB

function tmp = code(x, n)
	tmp = (1.0 / n) / x;
end

Wolfram

code[x_, n_] := N[(N[(1.0 / n), $MachinePrecision] / x), $MachinePrecision]

Derivation

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

Alternative 23: 40.5% accurate, 42.2× speedup

Math

\[\begin{array}{l} \\ \frac{\frac{1}{x}}{n} \end{array} \]

FPCore

(FPCore (x n) :precision binary64 (/ (/ 1.0 x) n))

C

double code(double x, double n) {
	return (1.0 / x) / n;
}

Fortran

real(8) function code(x, n)
    real(8), intent (in) :: x
    real(8), intent (in) :: n
    code = (1.0d0 / x) / n
end function

Java

public static double code(double x, double n) {
	return (1.0 / x) / n;
}

Python

def code(x, n):
	return (1.0 / x) / n

Julia

function code(x, n)
	return Float64(Float64(1.0 / x) / n)
end

MATLAB

function tmp = code(x, n)
	tmp = (1.0 / x) / n;
end

Wolfram

code[x_, n_] := N[(N[(1.0 / x), $MachinePrecision] / n), $MachinePrecision]

Derivation

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

Reproduce

herbie shell --seed 2024006 
(FPCore (x n)
  :name "2nthrt (problem 3.4.6)"
  :precision binary64
  (- (pow (+ x 1.0) (/ 1.0 n)) (pow x (/ 1.0 n))))