2nthrt (problem 3.4.6)

Percentage Accurate: 53.8% → 86.3%

Time: 29.2s

Alternatives: 20

Speedup: 2.0×

• 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: 53.8% 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: 86.3% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\log x}{n}\\ \mathbf{if}\;\frac{1}{n} \leq -1 \cdot 10^{-28}:\\ \;\;\;\;\frac{e^{t_0}}{n \cdot x}\\ \mathbf{elif}\;\frac{1}{n} \leq 10^{-9}:\\ \;\;\;\;\mathsf{fma}\left(0.5, \frac{{\left(\mathsf{log1p}\left(x\right)\right)}^{2}}{n \cdot n}, \frac{\mathsf{log1p}\left(x\right)}{n}\right) - \mathsf{fma}\left(0.5, \frac{{\log x}^{2}}{n \cdot n}, t_0\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt[3]{e^{\frac{x}{n} \cdot 3}} - {x}^{\left(\frac{1}{n}\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (let* ((t_0 (/ (log x) n)))
   (if (<= (/ 1.0 n) -1e-28)
     (/ (exp t_0) (* n x))
     (if (<= (/ 1.0 n) 1e-9)
       (-
        (fma 0.5 (/ (pow (log1p x) 2.0) (* n n)) (/ (log1p x) n))
        (fma 0.5 (/ (pow (log x) 2.0) (* n n)) t_0))
       (- (cbrt (exp (* (/ x n) 3.0))) (pow x (/ 1.0 n)))))))

C

double code(double x, double n) {
	double t_0 = log(x) / n;
	double tmp;
	if ((1.0 / n) <= -1e-28) {
		tmp = exp(t_0) / (n * x);
	} else if ((1.0 / n) <= 1e-9) {
		tmp = fma(0.5, (pow(log1p(x), 2.0) / (n * n)), (log1p(x) / n)) - fma(0.5, (pow(log(x), 2.0) / (n * n)), t_0);
	} else {
		tmp = cbrt(exp(((x / n) * 3.0))) - pow(x, (1.0 / n));
	}
	return tmp;
}

Julia

function code(x, n)
	t_0 = Float64(log(x) / n)
	tmp = 0.0
	if (Float64(1.0 / n) <= -1e-28)
		tmp = Float64(exp(t_0) / Float64(n * x));
	elseif (Float64(1.0 / n) <= 1e-9)
		tmp = Float64(fma(0.5, Float64((log1p(x) ^ 2.0) / Float64(n * n)), Float64(log1p(x) / n)) - fma(0.5, Float64((log(x) ^ 2.0) / Float64(n * n)), t_0));
	else
		tmp = Float64(cbrt(exp(Float64(Float64(x / n) * 3.0))) - (x ^ Float64(1.0 / 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], -1e-28], N[(N[Exp[t$95$0], $MachinePrecision] / N[(n * x), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 1e-9], N[(N[(0.5 * N[(N[Power[N[Log[1 + x], $MachinePrecision], 2.0], $MachinePrecision] / N[(n * n), $MachinePrecision]), $MachinePrecision] + N[(N[Log[1 + x], $MachinePrecision] / n), $MachinePrecision]), $MachinePrecision] - N[(0.5 * N[(N[Power[N[Log[x], $MachinePrecision], 2.0], $MachinePrecision] / N[(n * n), $MachinePrecision]), $MachinePrecision] + t$95$0), $MachinePrecision]), $MachinePrecision], N[(N[Power[N[Exp[N[(N[(x / n), $MachinePrecision] * 3.0), $MachinePrecision]], $MachinePrecision], 1/3], $MachinePrecision] - N[Power[x, N[(1.0 / n), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 1 n) < -9.99999999999999971e-29

   1. Initial program 96.3%

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

   2. Taylor expanded in x around inf 97.3%

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

      1. mul-1-neg 97.3%

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

      2. log-rec 97.3%

         \[\leadsto \frac{e^{-\frac{\color{blue}{-\log x}}{n}}}{n \cdot x} \]

      3. distribute-frac-neg 97.3%

         \[\leadsto \frac{e^{-\color{blue}{\left(-\frac{\log x}{n}\right)}}}{n \cdot x} \]

      4. remove-double-neg 97.3%

         \[\leadsto \frac{e^{\color{blue}{\frac{\log x}{n}}}}{n \cdot x} \]

      5. *-commutative 97.3%

         \[\leadsto \frac{e^{\frac{\log x}{n}}}{\color{blue}{x \cdot n}} \]

   4. Simplified 97.3%

      \[\leadsto \color{blue}{\frac{e^{\frac{\log x}{n}}}{x \cdot n}} \]

   if -9.99999999999999971e-29 < (/.f64 1 n) < 1.00000000000000006e-9

   1. Initial program 26.9%

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

   2. Taylor expanded in n around inf 78.2%

      \[\leadsto \color{blue}{\left(0.5 \cdot \frac{{\log \left(1 + x\right)}^{2}}{{n}^{2}} + \frac{\log \left(1 + x\right)}{n}\right) - \left(0.5 \cdot \frac{{\log x}^{2}}{{n}^{2}} + \frac{\log x}{n}\right)} \]
   3. Step-by-step derivation

      1. fma-def 78.2%

         \[\leadsto \color{blue}{\mathsf{fma}\left(0.5, \frac{{\log \left(1 + x\right)}^{2}}{{n}^{2}}, \frac{\log \left(1 + x\right)}{n}\right)} - \left(0.5 \cdot \frac{{\log x}^{2}}{{n}^{2}} + \frac{\log x}{n}\right) \]

      2. log1p-def 78.2%

         \[\leadsto \mathsf{fma}\left(0.5, \frac{{\color{blue}{\left(\mathsf{log1p}\left(x\right)\right)}}^{2}}{{n}^{2}}, \frac{\log \left(1 + x\right)}{n}\right) - \left(0.5 \cdot \frac{{\log x}^{2}}{{n}^{2}} + \frac{\log x}{n}\right) \]

      3. unpow2 78.2%

         \[\leadsto \mathsf{fma}\left(0.5, \frac{{\left(\mathsf{log1p}\left(x\right)\right)}^{2}}{\color{blue}{n \cdot n}}, \frac{\log \left(1 + x\right)}{n}\right) - \left(0.5 \cdot \frac{{\log x}^{2}}{{n}^{2}} + \frac{\log x}{n}\right) \]

      4. log1p-def 78.2%

         \[\leadsto \mathsf{fma}\left(0.5, \frac{{\left(\mathsf{log1p}\left(x\right)\right)}^{2}}{n \cdot n}, \frac{\color{blue}{\mathsf{log1p}\left(x\right)}}{n}\right) - \left(0.5 \cdot \frac{{\log x}^{2}}{{n}^{2}} + \frac{\log x}{n}\right) \]

      5. fma-def 78.2%

         \[\leadsto \mathsf{fma}\left(0.5, \frac{{\left(\mathsf{log1p}\left(x\right)\right)}^{2}}{n \cdot n}, \frac{\mathsf{log1p}\left(x\right)}{n}\right) - \color{blue}{\mathsf{fma}\left(0.5, \frac{{\log x}^{2}}{{n}^{2}}, \frac{\log x}{n}\right)} \]

      6. unpow2 78.2%

         \[\leadsto \mathsf{fma}\left(0.5, \frac{{\left(\mathsf{log1p}\left(x\right)\right)}^{2}}{n \cdot n}, \frac{\mathsf{log1p}\left(x\right)}{n}\right) - \mathsf{fma}\left(0.5, \frac{{\log x}^{2}}{\color{blue}{n \cdot n}}, \frac{\log x}{n}\right) \]

   4. Simplified 78.2%

      \[\leadsto \color{blue}{\mathsf{fma}\left(0.5, \frac{{\left(\mathsf{log1p}\left(x\right)\right)}^{2}}{n \cdot n}, \frac{\mathsf{log1p}\left(x\right)}{n}\right) - \mathsf{fma}\left(0.5, \frac{{\log x}^{2}}{n \cdot n}, \frac{\log x}{n}\right)} \]

   if 1.00000000000000006e-9 < (/.f64 1 n)

   1. Initial program 61.3%

      \[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)} \]
   2. Step-by-step derivation

      1. add-cbrt-cube 61.3%

         \[\leadsto \color{blue}{\sqrt[3]{\left({\left(x + 1\right)}^{\left(\frac{1}{n}\right)} \cdot {\left(x + 1\right)}^{\left(\frac{1}{n}\right)}\right) \cdot {\left(x + 1\right)}^{\left(\frac{1}{n}\right)}}} - {x}^{\left(\frac{1}{n}\right)} \]

      2. pow3 61.3%

         \[\leadsto \sqrt[3]{\color{blue}{{\left({\left(x + 1\right)}^{\left(\frac{1}{n}\right)}\right)}^{3}}} - {x}^{\left(\frac{1}{n}\right)} \]

      3. pow-to-exp 61.3%

         \[\leadsto \sqrt[3]{{\color{blue}{\left(e^{\log \left(x + 1\right) \cdot \frac{1}{n}}\right)}}^{3}} - {x}^{\left(\frac{1}{n}\right)} \]

      4. pow-exp 61.3%

         \[\leadsto \sqrt[3]{\color{blue}{e^{\left(\log \left(x + 1\right) \cdot \frac{1}{n}\right) \cdot 3}}} - {x}^{\left(\frac{1}{n}\right)} \]

      5. un-div-inv 61.3%

         \[\leadsto \sqrt[3]{e^{\color{blue}{\frac{\log \left(x + 1\right)}{n}} \cdot 3}} - {x}^{\left(\frac{1}{n}\right)} \]

      6. +-commutative 61.3%

         \[\leadsto \sqrt[3]{e^{\frac{\log \color{blue}{\left(1 + x\right)}}{n} \cdot 3}} - {x}^{\left(\frac{1}{n}\right)} \]

      7. log1p-udef 99.9%

         \[\leadsto \sqrt[3]{e^{\frac{\color{blue}{\mathsf{log1p}\left(x\right)}}{n} \cdot 3}} - {x}^{\left(\frac{1}{n}\right)} \]

   3. Applied egg-rr 99.9%

      \[\leadsto \color{blue}{\sqrt[3]{e^{\frac{\mathsf{log1p}\left(x\right)}{n} \cdot 3}}} - {x}^{\left(\frac{1}{n}\right)} \]

   4. Taylor expanded in x around 0 99.9%

      \[\leadsto \sqrt[3]{e^{\color{blue}{3 \cdot \frac{x}{n}}}} - {x}^{\left(\frac{1}{n}\right)} \]
   5. Step-by-step derivation

      1. *-commutative 99.9%

         \[\leadsto \sqrt[3]{e^{\color{blue}{\frac{x}{n} \cdot 3}}} - {x}^{\left(\frac{1}{n}\right)} \]

   6. Simplified 99.9%

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

4. Final simplification 87.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{1}{n} \leq -1 \cdot 10^{-28}:\\ \;\;\;\;\frac{e^{\frac{\log x}{n}}}{n \cdot x}\\ \mathbf{elif}\;\frac{1}{n} \leq 10^{-9}:\\ \;\;\;\;\mathsf{fma}\left(0.5, \frac{{\left(\mathsf{log1p}\left(x\right)\right)}^{2}}{n \cdot n}, \frac{\mathsf{log1p}\left(x\right)}{n}\right) - \mathsf{fma}\left(0.5, \frac{{\log x}^{2}}{n \cdot n}, \frac{\log x}{n}\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt[3]{e^{\frac{x}{n} \cdot 3}} - {x}^{\left(\frac{1}{n}\right)}\\ \end{array} \]

Alternative 2: 85.9% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{1}{n} \leq -1 \cdot 10^{-28}:\\ \;\;\;\;\frac{e^{\frac{\log x}{n}}}{n \cdot x}\\ \mathbf{elif}\;\frac{1}{n} \leq 5 \cdot 10^{-22}:\\ \;\;\;\;\frac{\mathsf{log1p}\left(x\right) - \log x}{n}\\ \mathbf{else}:\\ \;\;\;\;\sqrt[3]{e^{\frac{\mathsf{log1p}\left(x\right)}{n} \cdot 3}} - {x}^{\left(\frac{1}{n}\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (if (<= (/ 1.0 n) -1e-28)
   (/ (exp (/ (log x) n)) (* n x))
   (if (<= (/ 1.0 n) 5e-22)
     (/ (- (log1p x) (log x)) n)
     (- (cbrt (exp (* (/ (log1p x) n) 3.0))) (pow x (/ 1.0 n))))))

C

double code(double x, double n) {
	double tmp;
	if ((1.0 / n) <= -1e-28) {
		tmp = exp((log(x) / n)) / (n * x);
	} else if ((1.0 / n) <= 5e-22) {
		tmp = (log1p(x) - log(x)) / n;
	} else {
		tmp = cbrt(exp(((log1p(x) / n) * 3.0))) - pow(x, (1.0 / n));
	}
	return tmp;
}

Java

public static double code(double x, double n) {
	double tmp;
	if ((1.0 / n) <= -1e-28) {
		tmp = Math.exp((Math.log(x) / n)) / (n * x);
	} else if ((1.0 / n) <= 5e-22) {
		tmp = (Math.log1p(x) - Math.log(x)) / n;
	} else {
		tmp = Math.cbrt(Math.exp(((Math.log1p(x) / n) * 3.0))) - Math.pow(x, (1.0 / n));
	}
	return tmp;
}

Julia

function code(x, n)
	tmp = 0.0
	if (Float64(1.0 / n) <= -1e-28)
		tmp = Float64(exp(Float64(log(x) / n)) / Float64(n * x));
	elseif (Float64(1.0 / n) <= 5e-22)
		tmp = Float64(Float64(log1p(x) - log(x)) / n);
	else
		tmp = Float64(cbrt(exp(Float64(Float64(log1p(x) / n) * 3.0))) - (x ^ Float64(1.0 / n)));
	end
	return tmp
end

Wolfram

code[x_, n_] := If[LessEqual[N[(1.0 / n), $MachinePrecision], -1e-28], N[(N[Exp[N[(N[Log[x], $MachinePrecision] / n), $MachinePrecision]], $MachinePrecision] / N[(n * x), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 5e-22], N[(N[(N[Log[1 + x], $MachinePrecision] - N[Log[x], $MachinePrecision]), $MachinePrecision] / n), $MachinePrecision], N[(N[Power[N[Exp[N[(N[(N[Log[1 + x], $MachinePrecision] / n), $MachinePrecision] * 3.0), $MachinePrecision]], $MachinePrecision], 1/3], $MachinePrecision] - N[Power[x, N[(1.0 / n), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 1 n) < -9.99999999999999971e-29

   1. Initial program 96.3%

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

   2. Taylor expanded in x around inf 97.3%

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

      1. mul-1-neg 97.3%

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

      2. log-rec 97.3%

         \[\leadsto \frac{e^{-\frac{\color{blue}{-\log x}}{n}}}{n \cdot x} \]

      3. distribute-frac-neg 97.3%

         \[\leadsto \frac{e^{-\color{blue}{\left(-\frac{\log x}{n}\right)}}}{n \cdot x} \]

      4. remove-double-neg 97.3%

         \[\leadsto \frac{e^{\color{blue}{\frac{\log x}{n}}}}{n \cdot x} \]

      5. *-commutative 97.3%

         \[\leadsto \frac{e^{\frac{\log x}{n}}}{\color{blue}{x \cdot n}} \]

   4. Simplified 97.3%

      \[\leadsto \color{blue}{\frac{e^{\frac{\log x}{n}}}{x \cdot n}} \]

   if -9.99999999999999971e-29 < (/.f64 1 n) < 4.99999999999999954e-22

   1. Initial program 27.0%

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

   2. Taylor expanded in n around inf 79.7%

      \[\leadsto \color{blue}{\frac{\log \left(1 + x\right) - \log x}{n}} \]
   3. Step-by-step derivation

      1. +-rgt-identity 79.7%

         \[\leadsto \frac{\color{blue}{\left(\log \left(1 + x\right) + 0\right)} - \log x}{n} \]

      2. +-rgt-identity 79.7%

         \[\leadsto \frac{\color{blue}{\log \left(1 + x\right)} - \log x}{n} \]

      3. log1p-def 79.7%

         \[\leadsto \frac{\color{blue}{\mathsf{log1p}\left(x\right)} - \log x}{n} \]

   4. Simplified 79.7%

      \[\leadsto \color{blue}{\frac{\mathsf{log1p}\left(x\right) - \log x}{n}} \]

   if 4.99999999999999954e-22 < (/.f64 1 n)

   1. Initial program 58.1%

      \[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)} \]
   2. Step-by-step derivation

      1. add-cbrt-cube 58.1%

         \[\leadsto \color{blue}{\sqrt[3]{\left({\left(x + 1\right)}^{\left(\frac{1}{n}\right)} \cdot {\left(x + 1\right)}^{\left(\frac{1}{n}\right)}\right) \cdot {\left(x + 1\right)}^{\left(\frac{1}{n}\right)}}} - {x}^{\left(\frac{1}{n}\right)} \]

      2. pow3 58.1%

         \[\leadsto \sqrt[3]{\color{blue}{{\left({\left(x + 1\right)}^{\left(\frac{1}{n}\right)}\right)}^{3}}} - {x}^{\left(\frac{1}{n}\right)} \]

      3. pow-to-exp 58.1%

         \[\leadsto \sqrt[3]{{\color{blue}{\left(e^{\log \left(x + 1\right) \cdot \frac{1}{n}}\right)}}^{3}} - {x}^{\left(\frac{1}{n}\right)} \]

      4. pow-exp 58.1%

         \[\leadsto \sqrt[3]{\color{blue}{e^{\left(\log \left(x + 1\right) \cdot \frac{1}{n}\right) \cdot 3}}} - {x}^{\left(\frac{1}{n}\right)} \]

      5. un-div-inv 58.1%

         \[\leadsto \sqrt[3]{e^{\color{blue}{\frac{\log \left(x + 1\right)}{n}} \cdot 3}} - {x}^{\left(\frac{1}{n}\right)} \]

      6. +-commutative 58.1%

         \[\leadsto \sqrt[3]{e^{\frac{\log \color{blue}{\left(1 + x\right)}}{n} \cdot 3}} - {x}^{\left(\frac{1}{n}\right)} \]

      7. log1p-udef 93.6%

         \[\leadsto \sqrt[3]{e^{\frac{\color{blue}{\mathsf{log1p}\left(x\right)}}{n} \cdot 3}} - {x}^{\left(\frac{1}{n}\right)} \]

   3. Applied egg-rr 93.6%

      \[\leadsto \color{blue}{\sqrt[3]{e^{\frac{\mathsf{log1p}\left(x\right)}{n} \cdot 3}}} - {x}^{\left(\frac{1}{n}\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 87.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{1}{n} \leq -1 \cdot 10^{-28}:\\ \;\;\;\;\frac{e^{\frac{\log x}{n}}}{n \cdot x}\\ \mathbf{elif}\;\frac{1}{n} \leq 5 \cdot 10^{-22}:\\ \;\;\;\;\frac{\mathsf{log1p}\left(x\right) - \log x}{n}\\ \mathbf{else}:\\ \;\;\;\;\sqrt[3]{e^{\frac{\mathsf{log1p}\left(x\right)}{n} \cdot 3}} - {x}^{\left(\frac{1}{n}\right)}\\ \end{array} \]

Alternative 3: 86.2% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{1}{n} \leq -1 \cdot 10^{-28}:\\ \;\;\;\;\frac{e^{\frac{\log x}{n}}}{n \cdot x}\\ \mathbf{elif}\;\frac{1}{n} \leq 9.5 \cdot 10^{-10}:\\ \;\;\;\;\frac{\mathsf{log1p}\left(x\right) - \log x}{n}\\ \mathbf{else}:\\ \;\;\;\;\sqrt[3]{e^{\frac{x}{n} \cdot 3}} - {x}^{\left(\frac{1}{n}\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (if (<= (/ 1.0 n) -1e-28)
   (/ (exp (/ (log x) n)) (* n x))
   (if (<= (/ 1.0 n) 9.5e-10)
     (/ (- (log1p x) (log x)) n)
     (- (cbrt (exp (* (/ x n) 3.0))) (pow x (/ 1.0 n))))))

C

double code(double x, double n) {
	double tmp;
	if ((1.0 / n) <= -1e-28) {
		tmp = exp((log(x) / n)) / (n * x);
	} else if ((1.0 / n) <= 9.5e-10) {
		tmp = (log1p(x) - log(x)) / n;
	} else {
		tmp = cbrt(exp(((x / n) * 3.0))) - pow(x, (1.0 / n));
	}
	return tmp;
}

Java

public static double code(double x, double n) {
	double tmp;
	if ((1.0 / n) <= -1e-28) {
		tmp = Math.exp((Math.log(x) / n)) / (n * x);
	} else if ((1.0 / n) <= 9.5e-10) {
		tmp = (Math.log1p(x) - Math.log(x)) / n;
	} else {
		tmp = Math.cbrt(Math.exp(((x / n) * 3.0))) - Math.pow(x, (1.0 / n));
	}
	return tmp;
}

Julia

function code(x, n)
	tmp = 0.0
	if (Float64(1.0 / n) <= -1e-28)
		tmp = Float64(exp(Float64(log(x) / n)) / Float64(n * x));
	elseif (Float64(1.0 / n) <= 9.5e-10)
		tmp = Float64(Float64(log1p(x) - log(x)) / n);
	else
		tmp = Float64(cbrt(exp(Float64(Float64(x / n) * 3.0))) - (x ^ Float64(1.0 / n)));
	end
	return tmp
end

Wolfram

code[x_, n_] := If[LessEqual[N[(1.0 / n), $MachinePrecision], -1e-28], N[(N[Exp[N[(N[Log[x], $MachinePrecision] / n), $MachinePrecision]], $MachinePrecision] / N[(n * x), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 9.5e-10], N[(N[(N[Log[1 + x], $MachinePrecision] - N[Log[x], $MachinePrecision]), $MachinePrecision] / n), $MachinePrecision], N[(N[Power[N[Exp[N[(N[(x / n), $MachinePrecision] * 3.0), $MachinePrecision]], $MachinePrecision], 1/3], $MachinePrecision] - N[Power[x, N[(1.0 / n), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 1 n) < -9.99999999999999971e-29

   1. Initial program 96.3%

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

   2. Taylor expanded in x around inf 97.3%

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

      1. mul-1-neg 97.3%

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

      2. log-rec 97.3%

         \[\leadsto \frac{e^{-\frac{\color{blue}{-\log x}}{n}}}{n \cdot x} \]

      3. distribute-frac-neg 97.3%

         \[\leadsto \frac{e^{-\color{blue}{\left(-\frac{\log x}{n}\right)}}}{n \cdot x} \]

      4. remove-double-neg 97.3%

         \[\leadsto \frac{e^{\color{blue}{\frac{\log x}{n}}}}{n \cdot x} \]

      5. *-commutative 97.3%

         \[\leadsto \frac{e^{\frac{\log x}{n}}}{\color{blue}{x \cdot n}} \]

   4. Simplified 97.3%

      \[\leadsto \color{blue}{\frac{e^{\frac{\log x}{n}}}{x \cdot n}} \]

   if -9.99999999999999971e-29 < (/.f64 1 n) < 9.50000000000000028e-10

   1. Initial program 26.6%

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

   2. Taylor expanded in n around inf 78.1%

      \[\leadsto \color{blue}{\frac{\log \left(1 + x\right) - \log x}{n}} \]
   3. Step-by-step derivation

      1. +-rgt-identity 78.1%

         \[\leadsto \frac{\color{blue}{\left(\log \left(1 + x\right) + 0\right)} - \log x}{n} \]

      2. +-rgt-identity 78.1%

         \[\leadsto \frac{\color{blue}{\log \left(1 + x\right)} - \log x}{n} \]

      3. log1p-def 78.1%

         \[\leadsto \frac{\color{blue}{\mathsf{log1p}\left(x\right)} - \log x}{n} \]

   4. Simplified 78.1%

      \[\leadsto \color{blue}{\frac{\mathsf{log1p}\left(x\right) - \log x}{n}} \]

   if 9.50000000000000028e-10 < (/.f64 1 n)

   1. Initial program 61.4%

      \[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)} \]
   2. Step-by-step derivation

      1. add-cbrt-cube 61.4%

         \[\leadsto \color{blue}{\sqrt[3]{\left({\left(x + 1\right)}^{\left(\frac{1}{n}\right)} \cdot {\left(x + 1\right)}^{\left(\frac{1}{n}\right)}\right) \cdot {\left(x + 1\right)}^{\left(\frac{1}{n}\right)}}} - {x}^{\left(\frac{1}{n}\right)} \]

      2. pow3 61.4%

         \[\leadsto \sqrt[3]{\color{blue}{{\left({\left(x + 1\right)}^{\left(\frac{1}{n}\right)}\right)}^{3}}} - {x}^{\left(\frac{1}{n}\right)} \]

      3. pow-to-exp 61.4%

         \[\leadsto \sqrt[3]{{\color{blue}{\left(e^{\log \left(x + 1\right) \cdot \frac{1}{n}}\right)}}^{3}} - {x}^{\left(\frac{1}{n}\right)} \]

      4. pow-exp 61.4%

         \[\leadsto \sqrt[3]{\color{blue}{e^{\left(\log \left(x + 1\right) \cdot \frac{1}{n}\right) \cdot 3}}} - {x}^{\left(\frac{1}{n}\right)} \]

      5. un-div-inv 61.4%

         \[\leadsto \sqrt[3]{e^{\color{blue}{\frac{\log \left(x + 1\right)}{n}} \cdot 3}} - {x}^{\left(\frac{1}{n}\right)} \]

      6. +-commutative 61.4%

         \[\leadsto \sqrt[3]{e^{\frac{\log \color{blue}{\left(1 + x\right)}}{n} \cdot 3}} - {x}^{\left(\frac{1}{n}\right)} \]

      7. log1p-udef 99.3%

         \[\leadsto \sqrt[3]{e^{\frac{\color{blue}{\mathsf{log1p}\left(x\right)}}{n} \cdot 3}} - {x}^{\left(\frac{1}{n}\right)} \]

   3. Applied egg-rr 99.3%

      \[\leadsto \color{blue}{\sqrt[3]{e^{\frac{\mathsf{log1p}\left(x\right)}{n} \cdot 3}}} - {x}^{\left(\frac{1}{n}\right)} \]

   4. Taylor expanded in x around 0 99.3%

      \[\leadsto \sqrt[3]{e^{\color{blue}{3 \cdot \frac{x}{n}}}} - {x}^{\left(\frac{1}{n}\right)} \]
   5. Step-by-step derivation

      1. *-commutative 99.3%

         \[\leadsto \sqrt[3]{e^{\color{blue}{\frac{x}{n} \cdot 3}}} - {x}^{\left(\frac{1}{n}\right)} \]

   6. Simplified 99.3%

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

4. Final simplification 87.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{1}{n} \leq -1 \cdot 10^{-28}:\\ \;\;\;\;\frac{e^{\frac{\log x}{n}}}{n \cdot x}\\ \mathbf{elif}\;\frac{1}{n} \leq 9.5 \cdot 10^{-10}:\\ \;\;\;\;\frac{\mathsf{log1p}\left(x\right) - \log x}{n}\\ \mathbf{else}:\\ \;\;\;\;\sqrt[3]{e^{\frac{x}{n} \cdot 3}} - {x}^{\left(\frac{1}{n}\right)}\\ \end{array} \]

Alternative 4: 86.0% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{1}{n} \leq -1 \cdot 10^{-28}:\\ \;\;\;\;\frac{e^{\frac{\log x}{n}}}{n \cdot x}\\ \mathbf{elif}\;\frac{1}{n} \leq 5 \cdot 10^{-22}:\\ \;\;\;\;\frac{\mathsf{log1p}\left(x\right) - \log x}{n}\\ \mathbf{else}:\\ \;\;\;\;e^{\frac{\mathsf{log1p}\left(x\right)}{n}} - {x}^{\left(\frac{1}{n}\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (if (<= (/ 1.0 n) -1e-28)
   (/ (exp (/ (log x) n)) (* n x))
   (if (<= (/ 1.0 n) 5e-22)
     (/ (- (log1p x) (log x)) n)
     (- (exp (/ (log1p x) n)) (pow x (/ 1.0 n))))))

C

double code(double x, double n) {
	double tmp;
	if ((1.0 / n) <= -1e-28) {
		tmp = exp((log(x) / n)) / (n * x);
	} else if ((1.0 / n) <= 5e-22) {
		tmp = (log1p(x) - log(x)) / n;
	} else {
		tmp = exp((log1p(x) / n)) - pow(x, (1.0 / n));
	}
	return tmp;
}

Java

public static double code(double x, double n) {
	double tmp;
	if ((1.0 / n) <= -1e-28) {
		tmp = Math.exp((Math.log(x) / n)) / (n * x);
	} else if ((1.0 / n) <= 5e-22) {
		tmp = (Math.log1p(x) - Math.log(x)) / n;
	} else {
		tmp = Math.exp((Math.log1p(x) / n)) - Math.pow(x, (1.0 / n));
	}
	return tmp;
}

Python

def code(x, n):
	tmp = 0
	if (1.0 / n) <= -1e-28:
		tmp = math.exp((math.log(x) / n)) / (n * x)
	elif (1.0 / n) <= 5e-22:
		tmp = (math.log1p(x) - math.log(x)) / n
	else:
		tmp = math.exp((math.log1p(x) / n)) - math.pow(x, (1.0 / n))
	return tmp

Julia

function code(x, n)
	tmp = 0.0
	if (Float64(1.0 / n) <= -1e-28)
		tmp = Float64(exp(Float64(log(x) / n)) / Float64(n * x));
	elseif (Float64(1.0 / n) <= 5e-22)
		tmp = Float64(Float64(log1p(x) - log(x)) / n);
	else
		tmp = Float64(exp(Float64(log1p(x) / n)) - (x ^ Float64(1.0 / n)));
	end
	return tmp
end

Wolfram

code[x_, n_] := If[LessEqual[N[(1.0 / n), $MachinePrecision], -1e-28], N[(N[Exp[N[(N[Log[x], $MachinePrecision] / n), $MachinePrecision]], $MachinePrecision] / N[(n * x), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 5e-22], N[(N[(N[Log[1 + x], $MachinePrecision] - N[Log[x], $MachinePrecision]), $MachinePrecision] / n), $MachinePrecision], N[(N[Exp[N[(N[Log[1 + x], $MachinePrecision] / n), $MachinePrecision]], $MachinePrecision] - N[Power[x, N[(1.0 / n), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 1 n) < -9.99999999999999971e-29

   1. Initial program 96.3%

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

   2. Taylor expanded in x around inf 97.3%

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

      1. mul-1-neg 97.3%

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

      2. log-rec 97.3%

         \[\leadsto \frac{e^{-\frac{\color{blue}{-\log x}}{n}}}{n \cdot x} \]

      3. distribute-frac-neg 97.3%

         \[\leadsto \frac{e^{-\color{blue}{\left(-\frac{\log x}{n}\right)}}}{n \cdot x} \]

      4. remove-double-neg 97.3%

         \[\leadsto \frac{e^{\color{blue}{\frac{\log x}{n}}}}{n \cdot x} \]

      5. *-commutative 97.3%

         \[\leadsto \frac{e^{\frac{\log x}{n}}}{\color{blue}{x \cdot n}} \]

   4. Simplified 97.3%

      \[\leadsto \color{blue}{\frac{e^{\frac{\log x}{n}}}{x \cdot n}} \]

   if -9.99999999999999971e-29 < (/.f64 1 n) < 4.99999999999999954e-22

   1. Initial program 27.0%

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

   2. Taylor expanded in n around inf 79.7%

      \[\leadsto \color{blue}{\frac{\log \left(1 + x\right) - \log x}{n}} \]
   3. Step-by-step derivation

      1. +-rgt-identity 79.7%

         \[\leadsto \frac{\color{blue}{\left(\log \left(1 + x\right) + 0\right)} - \log x}{n} \]

      2. +-rgt-identity 79.7%

         \[\leadsto \frac{\color{blue}{\log \left(1 + x\right)} - \log x}{n} \]

      3. log1p-def 79.7%

         \[\leadsto \frac{\color{blue}{\mathsf{log1p}\left(x\right)} - \log x}{n} \]

   4. Simplified 79.7%

      \[\leadsto \color{blue}{\frac{\mathsf{log1p}\left(x\right) - \log x}{n}} \]

   if 4.99999999999999954e-22 < (/.f64 1 n)

   1. Initial program 58.1%

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

   2. Taylor expanded in n around 0 58.1%

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

      1. log1p-def 93.6%

         \[\leadsto e^{\frac{\color{blue}{\mathsf{log1p}\left(x\right)}}{n}} - {x}^{\left(\frac{1}{n}\right)} \]

   4. Simplified 93.6%

      \[\leadsto \color{blue}{e^{\frac{\mathsf{log1p}\left(x\right)}{n}}} - {x}^{\left(\frac{1}{n}\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 87.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{1}{n} \leq -1 \cdot 10^{-28}:\\ \;\;\;\;\frac{e^{\frac{\log x}{n}}}{n \cdot x}\\ \mathbf{elif}\;\frac{1}{n} \leq 5 \cdot 10^{-22}:\\ \;\;\;\;\frac{\mathsf{log1p}\left(x\right) - \log x}{n}\\ \mathbf{else}:\\ \;\;\;\;e^{\frac{\mathsf{log1p}\left(x\right)}{n}} - {x}^{\left(\frac{1}{n}\right)}\\ \end{array} \]

Alternative 5: 86.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{1}{n} \leq -1 \cdot 10^{-28}:\\ \;\;\;\;\frac{e^{\frac{\log x}{n}}}{n \cdot x}\\ \mathbf{elif}\;\frac{1}{n} \leq 9.5 \cdot 10^{-10}:\\ \;\;\;\;\frac{\mathsf{log1p}\left(x\right) - \log x}{n}\\ \mathbf{else}:\\ \;\;\;\;e^{\frac{x}{n}} - {x}^{\left(\frac{1}{n}\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (if (<= (/ 1.0 n) -1e-28)
   (/ (exp (/ (log x) n)) (* n x))
   (if (<= (/ 1.0 n) 9.5e-10)
     (/ (- (log1p x) (log x)) n)
     (- (exp (/ x n)) (pow x (/ 1.0 n))))))

C

double code(double x, double n) {
	double tmp;
	if ((1.0 / n) <= -1e-28) {
		tmp = exp((log(x) / n)) / (n * x);
	} else if ((1.0 / n) <= 9.5e-10) {
		tmp = (log1p(x) - log(x)) / n;
	} else {
		tmp = exp((x / n)) - pow(x, (1.0 / n));
	}
	return tmp;
}

Java

public static double code(double x, double n) {
	double tmp;
	if ((1.0 / n) <= -1e-28) {
		tmp = Math.exp((Math.log(x) / n)) / (n * x);
	} else if ((1.0 / n) <= 9.5e-10) {
		tmp = (Math.log1p(x) - Math.log(x)) / n;
	} else {
		tmp = Math.exp((x / n)) - Math.pow(x, (1.0 / n));
	}
	return tmp;
}

Python

def code(x, n):
	tmp = 0
	if (1.0 / n) <= -1e-28:
		tmp = math.exp((math.log(x) / n)) / (n * x)
	elif (1.0 / n) <= 9.5e-10:
		tmp = (math.log1p(x) - math.log(x)) / n
	else:
		tmp = math.exp((x / n)) - math.pow(x, (1.0 / n))
	return tmp

Julia

function code(x, n)
	tmp = 0.0
	if (Float64(1.0 / n) <= -1e-28)
		tmp = Float64(exp(Float64(log(x) / n)) / Float64(n * x));
	elseif (Float64(1.0 / n) <= 9.5e-10)
		tmp = Float64(Float64(log1p(x) - log(x)) / n);
	else
		tmp = Float64(exp(Float64(x / n)) - (x ^ Float64(1.0 / n)));
	end
	return tmp
end

Wolfram

code[x_, n_] := If[LessEqual[N[(1.0 / n), $MachinePrecision], -1e-28], N[(N[Exp[N[(N[Log[x], $MachinePrecision] / n), $MachinePrecision]], $MachinePrecision] / N[(n * x), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 9.5e-10], N[(N[(N[Log[1 + x], $MachinePrecision] - N[Log[x], $MachinePrecision]), $MachinePrecision] / n), $MachinePrecision], N[(N[Exp[N[(x / n), $MachinePrecision]], $MachinePrecision] - N[Power[x, N[(1.0 / n), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 1 n) < -9.99999999999999971e-29

   1. Initial program 96.3%

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

   2. Taylor expanded in x around inf 97.3%

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

      1. mul-1-neg 97.3%

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

      2. log-rec 97.3%

         \[\leadsto \frac{e^{-\frac{\color{blue}{-\log x}}{n}}}{n \cdot x} \]

      3. distribute-frac-neg 97.3%

         \[\leadsto \frac{e^{-\color{blue}{\left(-\frac{\log x}{n}\right)}}}{n \cdot x} \]

      4. remove-double-neg 97.3%

         \[\leadsto \frac{e^{\color{blue}{\frac{\log x}{n}}}}{n \cdot x} \]

      5. *-commutative 97.3%

         \[\leadsto \frac{e^{\frac{\log x}{n}}}{\color{blue}{x \cdot n}} \]

   4. Simplified 97.3%

      \[\leadsto \color{blue}{\frac{e^{\frac{\log x}{n}}}{x \cdot n}} \]

   if -9.99999999999999971e-29 < (/.f64 1 n) < 9.50000000000000028e-10

   1. Initial program 26.6%

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

   2. Taylor expanded in n around inf 78.1%

      \[\leadsto \color{blue}{\frac{\log \left(1 + x\right) - \log x}{n}} \]
   3. Step-by-step derivation

      1. +-rgt-identity 78.1%

         \[\leadsto \frac{\color{blue}{\left(\log \left(1 + x\right) + 0\right)} - \log x}{n} \]

      2. +-rgt-identity 78.1%

         \[\leadsto \frac{\color{blue}{\log \left(1 + x\right)} - \log x}{n} \]

      3. log1p-def 78.1%

         \[\leadsto \frac{\color{blue}{\mathsf{log1p}\left(x\right)} - \log x}{n} \]

   4. Simplified 78.1%

      \[\leadsto \color{blue}{\frac{\mathsf{log1p}\left(x\right) - \log x}{n}} \]

   if 9.50000000000000028e-10 < (/.f64 1 n)

   1. Initial program 61.4%

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

   2. Taylor expanded in n around 0 61.4%

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

      1. log1p-def 99.2%

         \[\leadsto e^{\frac{\color{blue}{\mathsf{log1p}\left(x\right)}}{n}} - {x}^{\left(\frac{1}{n}\right)} \]

   4. Simplified 99.2%

      \[\leadsto \color{blue}{e^{\frac{\mathsf{log1p}\left(x\right)}{n}}} - {x}^{\left(\frac{1}{n}\right)} \]

   5. Taylor expanded in x around 0 99.2%

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

4. Final simplification 87.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{1}{n} \leq -1 \cdot 10^{-28}:\\ \;\;\;\;\frac{e^{\frac{\log x}{n}}}{n \cdot x}\\ \mathbf{elif}\;\frac{1}{n} \leq 9.5 \cdot 10^{-10}:\\ \;\;\;\;\frac{\mathsf{log1p}\left(x\right) - \log x}{n}\\ \mathbf{else}:\\ \;\;\;\;e^{\frac{x}{n}} - {x}^{\left(\frac{1}{n}\right)}\\ \end{array} \]

Alternative 6: 81.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{1}{n} \leq -1 \cdot 10^{-28}:\\ \;\;\;\;\frac{e^{\frac{\log x}{n}}}{n \cdot x}\\ \mathbf{elif}\;\frac{1}{n} \leq 9.5 \cdot 10^{-10}:\\ \;\;\;\;\frac{\mathsf{log1p}\left(x\right) - \log x}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 5 \cdot 10^{+118}:\\ \;\;\;\;\left(\left(1 + \frac{x}{n}\right) + \left(x \cdot \frac{x}{n}\right) \cdot \left(-0.5 + \frac{0.5}{n}\right)\right) - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(x \cdot \frac{0.5}{n \cdot n}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (if (<= (/ 1.0 n) -1e-28)
   (/ (exp (/ (log x) n)) (* n x))
   (if (<= (/ 1.0 n) 9.5e-10)
     (/ (- (log1p x) (log x)) n)
     (if (<= (/ 1.0 n) 5e+118)
       (-
        (+ (+ 1.0 (/ x n)) (* (* x (/ x n)) (+ -0.5 (/ 0.5 n))))
        (pow x (/ 1.0 n)))
       (* x (* x (/ 0.5 (* n n))))))))

C

double code(double x, double n) {
	double tmp;
	if ((1.0 / n) <= -1e-28) {
		tmp = exp((log(x) / n)) / (n * x);
	} else if ((1.0 / n) <= 9.5e-10) {
		tmp = (log1p(x) - log(x)) / n;
	} else if ((1.0 / n) <= 5e+118) {
		tmp = ((1.0 + (x / n)) + ((x * (x / n)) * (-0.5 + (0.5 / n)))) - pow(x, (1.0 / n));
	} else {
		tmp = x * (x * (0.5 / (n * n)));
	}
	return tmp;
}

Java

public static double code(double x, double n) {
	double tmp;
	if ((1.0 / n) <= -1e-28) {
		tmp = Math.exp((Math.log(x) / n)) / (n * x);
	} else if ((1.0 / n) <= 9.5e-10) {
		tmp = (Math.log1p(x) - Math.log(x)) / n;
	} else if ((1.0 / n) <= 5e+118) {
		tmp = ((1.0 + (x / n)) + ((x * (x / n)) * (-0.5 + (0.5 / n)))) - Math.pow(x, (1.0 / n));
	} else {
		tmp = x * (x * (0.5 / (n * n)));
	}
	return tmp;
}

Python

def code(x, n):
	tmp = 0
	if (1.0 / n) <= -1e-28:
		tmp = math.exp((math.log(x) / n)) / (n * x)
	elif (1.0 / n) <= 9.5e-10:
		tmp = (math.log1p(x) - math.log(x)) / n
	elif (1.0 / n) <= 5e+118:
		tmp = ((1.0 + (x / n)) + ((x * (x / n)) * (-0.5 + (0.5 / n)))) - math.pow(x, (1.0 / n))
	else:
		tmp = x * (x * (0.5 / (n * n)))
	return tmp

Julia

function code(x, n)
	tmp = 0.0
	if (Float64(1.0 / n) <= -1e-28)
		tmp = Float64(exp(Float64(log(x) / n)) / Float64(n * x));
	elseif (Float64(1.0 / n) <= 9.5e-10)
		tmp = Float64(Float64(log1p(x) - log(x)) / n);
	elseif (Float64(1.0 / n) <= 5e+118)
		tmp = Float64(Float64(Float64(1.0 + Float64(x / n)) + Float64(Float64(x * Float64(x / n)) * Float64(-0.5 + Float64(0.5 / n)))) - (x ^ Float64(1.0 / n)));
	else
		tmp = Float64(x * Float64(x * Float64(0.5 / Float64(n * n))));
	end
	return tmp
end

Wolfram

code[x_, n_] := If[LessEqual[N[(1.0 / n), $MachinePrecision], -1e-28], N[(N[Exp[N[(N[Log[x], $MachinePrecision] / n), $MachinePrecision]], $MachinePrecision] / N[(n * x), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 9.5e-10], N[(N[(N[Log[1 + x], $MachinePrecision] - N[Log[x], $MachinePrecision]), $MachinePrecision] / n), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 5e+118], N[(N[(N[(1.0 + N[(x / n), $MachinePrecision]), $MachinePrecision] + N[(N[(x * N[(x / n), $MachinePrecision]), $MachinePrecision] * N[(-0.5 + N[(0.5 / n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[Power[x, N[(1.0 / n), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(x * N[(x * N[(0.5 / N[(n * n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if (/.f64 1 n) < -9.99999999999999971e-29

   1. Initial program 96.3%

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

   2. Taylor expanded in x around inf 97.3%

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

      1. mul-1-neg 97.3%

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

      2. log-rec 97.3%

         \[\leadsto \frac{e^{-\frac{\color{blue}{-\log x}}{n}}}{n \cdot x} \]

      3. distribute-frac-neg 97.3%

         \[\leadsto \frac{e^{-\color{blue}{\left(-\frac{\log x}{n}\right)}}}{n \cdot x} \]

      4. remove-double-neg 97.3%

         \[\leadsto \frac{e^{\color{blue}{\frac{\log x}{n}}}}{n \cdot x} \]

      5. *-commutative 97.3%

         \[\leadsto \frac{e^{\frac{\log x}{n}}}{\color{blue}{x \cdot n}} \]

   4. Simplified 97.3%

      \[\leadsto \color{blue}{\frac{e^{\frac{\log x}{n}}}{x \cdot n}} \]

   if -9.99999999999999971e-29 < (/.f64 1 n) < 9.50000000000000028e-10

   1. Initial program 26.6%

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

   2. Taylor expanded in n around inf 78.1%

      \[\leadsto \color{blue}{\frac{\log \left(1 + x\right) - \log x}{n}} \]
   3. Step-by-step derivation

      1. +-rgt-identity 78.1%

         \[\leadsto \frac{\color{blue}{\left(\log \left(1 + x\right) + 0\right)} - \log x}{n} \]

      2. +-rgt-identity 78.1%

         \[\leadsto \frac{\color{blue}{\log \left(1 + x\right)} - \log x}{n} \]

      3. log1p-def 78.1%

         \[\leadsto \frac{\color{blue}{\mathsf{log1p}\left(x\right)} - \log x}{n} \]

   4. Simplified 78.1%

      \[\leadsto \color{blue}{\frac{\mathsf{log1p}\left(x\right) - \log x}{n}} \]

   if 9.50000000000000028e-10 < (/.f64 1 n) < 4.99999999999999972e118

   1. Initial program 90.1%

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

   2. Taylor expanded in n around 0 90.1%

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

      1. log1p-def 98.3%

         \[\leadsto e^{\frac{\color{blue}{\mathsf{log1p}\left(x\right)}}{n}} - {x}^{\left(\frac{1}{n}\right)} \]

   4. Simplified 98.3%

      \[\leadsto \color{blue}{e^{\frac{\mathsf{log1p}\left(x\right)}{n}}} - {x}^{\left(\frac{1}{n}\right)} \]

   5. Taylor expanded in x around 0 90.1%

      \[\leadsto \color{blue}{\left(1 + \left({x}^{2} \cdot \left(0.5 \cdot \frac{1}{{n}^{2}} - 0.5 \cdot \frac{1}{n}\right) + \frac{x}{n}\right)\right)} - {x}^{\left(\frac{1}{n}\right)} \]
   6. Step-by-step derivation

      1. +-commutative 90.1%

         \[\leadsto \left(1 + \color{blue}{\left(\frac{x}{n} + {x}^{2} \cdot \left(0.5 \cdot \frac{1}{{n}^{2}} - 0.5 \cdot \frac{1}{n}\right)\right)}\right) - {x}^{\left(\frac{1}{n}\right)} \]

      2. associate-+r+ 90.1%

         \[\leadsto \color{blue}{\left(\left(1 + \frac{x}{n}\right) + {x}^{2} \cdot \left(0.5 \cdot \frac{1}{{n}^{2}} - 0.5 \cdot \frac{1}{n}\right)\right)} - {x}^{\left(\frac{1}{n}\right)} \]

      3. sub-neg 90.1%

         \[\leadsto \left(\left(1 + \frac{x}{n}\right) + {x}^{2} \cdot \color{blue}{\left(0.5 \cdot \frac{1}{{n}^{2}} + \left(-0.5 \cdot \frac{1}{n}\right)\right)}\right) - {x}^{\left(\frac{1}{n}\right)} \]

      4. associate-*r/ 90.1%

         \[\leadsto \left(\left(1 + \frac{x}{n}\right) + {x}^{2} \cdot \left(\color{blue}{\frac{0.5 \cdot 1}{{n}^{2}}} + \left(-0.5 \cdot \frac{1}{n}\right)\right)\right) - {x}^{\left(\frac{1}{n}\right)} \]

      5. metadata-eval 90.1%

         \[\leadsto \left(\left(1 + \frac{x}{n}\right) + {x}^{2} \cdot \left(\frac{\color{blue}{0.5}}{{n}^{2}} + \left(-0.5 \cdot \frac{1}{n}\right)\right)\right) - {x}^{\left(\frac{1}{n}\right)} \]

      6. unpow2 90.1%

         \[\leadsto \left(\left(1 + \frac{x}{n}\right) + {x}^{2} \cdot \left(\frac{0.5}{\color{blue}{n \cdot n}} + \left(-0.5 \cdot \frac{1}{n}\right)\right)\right) - {x}^{\left(\frac{1}{n}\right)} \]

      7. associate-*r/ 90.1%

         \[\leadsto \left(\left(1 + \frac{x}{n}\right) + {x}^{2} \cdot \left(\frac{0.5}{n \cdot n} + \left(-\color{blue}{\frac{0.5 \cdot 1}{n}}\right)\right)\right) - {x}^{\left(\frac{1}{n}\right)} \]

      8. metadata-eval 90.1%

         \[\leadsto \left(\left(1 + \frac{x}{n}\right) + {x}^{2} \cdot \left(\frac{0.5}{n \cdot n} + \left(-\frac{\color{blue}{0.5}}{n}\right)\right)\right) - {x}^{\left(\frac{1}{n}\right)} \]

      9. distribute-neg-frac 90.1%

         \[\leadsto \left(\left(1 + \frac{x}{n}\right) + {x}^{2} \cdot \left(\frac{0.5}{n \cdot n} + \color{blue}{\frac{-0.5}{n}}\right)\right) - {x}^{\left(\frac{1}{n}\right)} \]

      10. metadata-eval 90.1%

          \[\leadsto \left(\left(1 + \frac{x}{n}\right) + {x}^{2} \cdot \left(\frac{0.5}{n \cdot n} + \frac{\color{blue}{-0.5}}{n}\right)\right) - {x}^{\left(\frac{1}{n}\right)} \]

      11. +-commutative 90.1%

          \[\leadsto \left(\left(1 + \frac{x}{n}\right) + {x}^{2} \cdot \color{blue}{\left(\frac{-0.5}{n} + \frac{0.5}{n \cdot n}\right)}\right) - {x}^{\left(\frac{1}{n}\right)} \]

      12. distribute-rgt-in 90.1%

          \[\leadsto \left(\left(1 + \frac{x}{n}\right) + \color{blue}{\left(\frac{-0.5}{n} \cdot {x}^{2} + \frac{0.5}{n \cdot n} \cdot {x}^{2}\right)}\right) - {x}^{\left(\frac{1}{n}\right)} \]

      13. associate-*l/ 90.1%

          \[\leadsto \left(\left(1 + \frac{x}{n}\right) + \left(\color{blue}{\frac{-0.5 \cdot {x}^{2}}{n}} + \frac{0.5}{n \cdot n} \cdot {x}^{2}\right)\right) - {x}^{\left(\frac{1}{n}\right)} \]

      14. associate-*r/ 90.1%

          \[\leadsto \left(\left(1 + \frac{x}{n}\right) + \left(\color{blue}{-0.5 \cdot \frac{{x}^{2}}{n}} + \frac{0.5}{n \cdot n} \cdot {x}^{2}\right)\right) - {x}^{\left(\frac{1}{n}\right)} \]

      15. associate-/r* 90.1%

          \[\leadsto \left(\left(1 + \frac{x}{n}\right) + \left(-0.5 \cdot \frac{{x}^{2}}{n} + \color{blue}{\frac{\frac{0.5}{n}}{n}} \cdot {x}^{2}\right)\right) - {x}^{\left(\frac{1}{n}\right)} \]

   7. Simplified 90.1%

      \[\leadsto \color{blue}{\left(\left(1 + \frac{x}{n}\right) + \left(x \cdot \frac{x}{n}\right) \cdot \left(-0.5 + \frac{0.5}{n}\right)\right)} - {x}^{\left(\frac{1}{n}\right)} \]

   if 4.99999999999999972e118 < (/.f64 1 n)

   1. Initial program 36.2%

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

   2. Taylor expanded in x around 0 56.8%

      \[\leadsto \color{blue}{\left(1 + \left({x}^{2} \cdot \left(0.5 \cdot \frac{1}{{n}^{2}} - 0.5 \cdot \frac{1}{n}\right) + \frac{x}{n}\right)\right)} - {x}^{\left(\frac{1}{n}\right)} \]
   3. Step-by-step derivation

      1. fma-def 56.8%

         \[\leadsto \left(1 + \color{blue}{\mathsf{fma}\left({x}^{2}, 0.5 \cdot \frac{1}{{n}^{2}} - 0.5 \cdot \frac{1}{n}, \frac{x}{n}\right)}\right) - {x}^{\left(\frac{1}{n}\right)} \]

      2. unpow2 56.8%

         \[\leadsto \left(1 + \mathsf{fma}\left(\color{blue}{x \cdot x}, 0.5 \cdot \frac{1}{{n}^{2}} - 0.5 \cdot \frac{1}{n}, \frac{x}{n}\right)\right) - {x}^{\left(\frac{1}{n}\right)} \]

      3. associate-*r/ 56.8%

         \[\leadsto \left(1 + \mathsf{fma}\left(x \cdot x, \color{blue}{\frac{0.5 \cdot 1}{{n}^{2}}} - 0.5 \cdot \frac{1}{n}, \frac{x}{n}\right)\right) - {x}^{\left(\frac{1}{n}\right)} \]

      4. metadata-eval 56.8%

         \[\leadsto \left(1 + \mathsf{fma}\left(x \cdot x, \frac{\color{blue}{0.5}}{{n}^{2}} - 0.5 \cdot \frac{1}{n}, \frac{x}{n}\right)\right) - {x}^{\left(\frac{1}{n}\right)} \]

      5. unpow2 56.8%

         \[\leadsto \left(1 + \mathsf{fma}\left(x \cdot x, \frac{0.5}{\color{blue}{n \cdot n}} - 0.5 \cdot \frac{1}{n}, \frac{x}{n}\right)\right) - {x}^{\left(\frac{1}{n}\right)} \]

      6. associate-*r/ 56.8%

         \[\leadsto \left(1 + \mathsf{fma}\left(x \cdot x, \frac{0.5}{n \cdot n} - \color{blue}{\frac{0.5 \cdot 1}{n}}, \frac{x}{n}\right)\right) - {x}^{\left(\frac{1}{n}\right)} \]

      7. metadata-eval 56.8%

         \[\leadsto \left(1 + \mathsf{fma}\left(x \cdot x, \frac{0.5}{n \cdot n} - \frac{\color{blue}{0.5}}{n}, \frac{x}{n}\right)\right) - {x}^{\left(\frac{1}{n}\right)} \]

   4. Simplified 56.8%

      \[\leadsto \color{blue}{\left(1 + \mathsf{fma}\left(x \cdot x, \frac{0.5}{n \cdot n} - \frac{0.5}{n}, \frac{x}{n}\right)\right)} - {x}^{\left(\frac{1}{n}\right)} \]

   5. Taylor expanded in n around 0 53.0%

      \[\leadsto \color{blue}{0.5 \cdot \frac{{x}^{2}}{{n}^{2}}} \]
   6. Step-by-step derivation

      1. unpow2 53.0%

         \[\leadsto 0.5 \cdot \frac{\color{blue}{x \cdot x}}{{n}^{2}} \]

      2. unpow2 53.0%

         \[\leadsto 0.5 \cdot \frac{x \cdot x}{\color{blue}{n \cdot n}} \]

   7. Simplified 53.0%

      \[\leadsto \color{blue}{0.5 \cdot \frac{x \cdot x}{n \cdot n}} \]

   8. Taylor expanded in x around 0 53.0%

      \[\leadsto \color{blue}{0.5 \cdot \frac{{x}^{2}}{{n}^{2}}} \]
   9. Step-by-step derivation

      1. *-commutative 53.0%

         \[\leadsto \color{blue}{\frac{{x}^{2}}{{n}^{2}} \cdot 0.5} \]

      2. associate-*l/ 53.0%

         \[\leadsto \color{blue}{\frac{{x}^{2} \cdot 0.5}{{n}^{2}}} \]

      3. associate-*r/ 56.8%

         \[\leadsto \color{blue}{{x}^{2} \cdot \frac{0.5}{{n}^{2}}} \]

      4. unpow2 56.8%

         \[\leadsto {x}^{2} \cdot \frac{0.5}{\color{blue}{n \cdot n}} \]

      5. unpow2 56.8%

         \[\leadsto \color{blue}{\left(x \cdot x\right)} \cdot \frac{0.5}{n \cdot n} \]

      6. associate-*l* 73.3%

         \[\leadsto \color{blue}{x \cdot \left(x \cdot \frac{0.5}{n \cdot n}\right)} \]

   10. Simplified 73.3%

       \[\leadsto \color{blue}{x \cdot \left(x \cdot \frac{0.5}{n \cdot n}\right)} \]
3. Recombined 4 regimes into one program.

4. Final simplification 84.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{1}{n} \leq -1 \cdot 10^{-28}:\\ \;\;\;\;\frac{e^{\frac{\log x}{n}}}{n \cdot x}\\ \mathbf{elif}\;\frac{1}{n} \leq 9.5 \cdot 10^{-10}:\\ \;\;\;\;\frac{\mathsf{log1p}\left(x\right) - \log x}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 5 \cdot 10^{+118}:\\ \;\;\;\;\left(\left(1 + \frac{x}{n}\right) + \left(x \cdot \frac{x}{n}\right) \cdot \left(-0.5 + \frac{0.5}{n}\right)\right) - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(x \cdot \frac{0.5}{n \cdot n}\right)\\ \end{array} \]

Alternative 7: 68.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{1}{n} \leq 9.5 \cdot 10^{-10}:\\ \;\;\;\;\frac{\mathsf{log1p}\left(x\right) - \log x}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 5 \cdot 10^{+118}:\\ \;\;\;\;\left(\left(1 + \frac{x}{n}\right) + \left(x \cdot \frac{x}{n}\right) \cdot \left(-0.5 + \frac{0.5}{n}\right)\right) - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(x \cdot \frac{0.5}{n \cdot n}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (if (<= (/ 1.0 n) 9.5e-10)
   (/ (- (log1p x) (log x)) n)
   (if (<= (/ 1.0 n) 5e+118)
     (-
      (+ (+ 1.0 (/ x n)) (* (* x (/ x n)) (+ -0.5 (/ 0.5 n))))
      (pow x (/ 1.0 n)))
     (* x (* x (/ 0.5 (* n n)))))))

C

double code(double x, double n) {
	double tmp;
	if ((1.0 / n) <= 9.5e-10) {
		tmp = (log1p(x) - log(x)) / n;
	} else if ((1.0 / n) <= 5e+118) {
		tmp = ((1.0 + (x / n)) + ((x * (x / n)) * (-0.5 + (0.5 / n)))) - pow(x, (1.0 / n));
	} else {
		tmp = x * (x * (0.5 / (n * n)));
	}
	return tmp;
}

Java

public static double code(double x, double n) {
	double tmp;
	if ((1.0 / n) <= 9.5e-10) {
		tmp = (Math.log1p(x) - Math.log(x)) / n;
	} else if ((1.0 / n) <= 5e+118) {
		tmp = ((1.0 + (x / n)) + ((x * (x / n)) * (-0.5 + (0.5 / n)))) - Math.pow(x, (1.0 / n));
	} else {
		tmp = x * (x * (0.5 / (n * n)));
	}
	return tmp;
}

Python

def code(x, n):
	tmp = 0
	if (1.0 / n) <= 9.5e-10:
		tmp = (math.log1p(x) - math.log(x)) / n
	elif (1.0 / n) <= 5e+118:
		tmp = ((1.0 + (x / n)) + ((x * (x / n)) * (-0.5 + (0.5 / n)))) - math.pow(x, (1.0 / n))
	else:
		tmp = x * (x * (0.5 / (n * n)))
	return tmp

Julia

function code(x, n)
	tmp = 0.0
	if (Float64(1.0 / n) <= 9.5e-10)
		tmp = Float64(Float64(log1p(x) - log(x)) / n);
	elseif (Float64(1.0 / n) <= 5e+118)
		tmp = Float64(Float64(Float64(1.0 + Float64(x / n)) + Float64(Float64(x * Float64(x / n)) * Float64(-0.5 + Float64(0.5 / n)))) - (x ^ Float64(1.0 / n)));
	else
		tmp = Float64(x * Float64(x * Float64(0.5 / Float64(n * n))));
	end
	return tmp
end

Wolfram

code[x_, n_] := If[LessEqual[N[(1.0 / n), $MachinePrecision], 9.5e-10], N[(N[(N[Log[1 + x], $MachinePrecision] - N[Log[x], $MachinePrecision]), $MachinePrecision] / n), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 5e+118], N[(N[(N[(1.0 + N[(x / n), $MachinePrecision]), $MachinePrecision] + N[(N[(x * N[(x / n), $MachinePrecision]), $MachinePrecision] * N[(-0.5 + N[(0.5 / n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[Power[x, N[(1.0 / n), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(x * N[(x * N[(0.5 / N[(n * n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 1 n) < 9.50000000000000028e-10

   1. Initial program 50.2%

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

   2. Taylor expanded in n around inf 69.8%

      \[\leadsto \color{blue}{\frac{\log \left(1 + x\right) - \log x}{n}} \]
   3. Step-by-step derivation

      1. +-rgt-identity 69.8%

         \[\leadsto \frac{\color{blue}{\left(\log \left(1 + x\right) + 0\right)} - \log x}{n} \]

      2. +-rgt-identity 69.8%

         \[\leadsto \frac{\color{blue}{\log \left(1 + x\right)} - \log x}{n} \]

      3. log1p-def 69.8%

         \[\leadsto \frac{\color{blue}{\mathsf{log1p}\left(x\right)} - \log x}{n} \]

   4. Simplified 69.8%

      \[\leadsto \color{blue}{\frac{\mathsf{log1p}\left(x\right) - \log x}{n}} \]

   if 9.50000000000000028e-10 < (/.f64 1 n) < 4.99999999999999972e118

   1. Initial program 90.1%

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

   2. Taylor expanded in n around 0 90.1%

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

      1. log1p-def 98.3%

         \[\leadsto e^{\frac{\color{blue}{\mathsf{log1p}\left(x\right)}}{n}} - {x}^{\left(\frac{1}{n}\right)} \]

   4. Simplified 98.3%

      \[\leadsto \color{blue}{e^{\frac{\mathsf{log1p}\left(x\right)}{n}}} - {x}^{\left(\frac{1}{n}\right)} \]

   5. Taylor expanded in x around 0 90.1%

      \[\leadsto \color{blue}{\left(1 + \left({x}^{2} \cdot \left(0.5 \cdot \frac{1}{{n}^{2}} - 0.5 \cdot \frac{1}{n}\right) + \frac{x}{n}\right)\right)} - {x}^{\left(\frac{1}{n}\right)} \]
   6. Step-by-step derivation

      1. +-commutative 90.1%

         \[\leadsto \left(1 + \color{blue}{\left(\frac{x}{n} + {x}^{2} \cdot \left(0.5 \cdot \frac{1}{{n}^{2}} - 0.5 \cdot \frac{1}{n}\right)\right)}\right) - {x}^{\left(\frac{1}{n}\right)} \]

      2. associate-+r+ 90.1%

         \[\leadsto \color{blue}{\left(\left(1 + \frac{x}{n}\right) + {x}^{2} \cdot \left(0.5 \cdot \frac{1}{{n}^{2}} - 0.5 \cdot \frac{1}{n}\right)\right)} - {x}^{\left(\frac{1}{n}\right)} \]

      3. sub-neg 90.1%

         \[\leadsto \left(\left(1 + \frac{x}{n}\right) + {x}^{2} \cdot \color{blue}{\left(0.5 \cdot \frac{1}{{n}^{2}} + \left(-0.5 \cdot \frac{1}{n}\right)\right)}\right) - {x}^{\left(\frac{1}{n}\right)} \]

      4. associate-*r/ 90.1%

         \[\leadsto \left(\left(1 + \frac{x}{n}\right) + {x}^{2} \cdot \left(\color{blue}{\frac{0.5 \cdot 1}{{n}^{2}}} + \left(-0.5 \cdot \frac{1}{n}\right)\right)\right) - {x}^{\left(\frac{1}{n}\right)} \]

      5. metadata-eval 90.1%

         \[\leadsto \left(\left(1 + \frac{x}{n}\right) + {x}^{2} \cdot \left(\frac{\color{blue}{0.5}}{{n}^{2}} + \left(-0.5 \cdot \frac{1}{n}\right)\right)\right) - {x}^{\left(\frac{1}{n}\right)} \]

      6. unpow2 90.1%

         \[\leadsto \left(\left(1 + \frac{x}{n}\right) + {x}^{2} \cdot \left(\frac{0.5}{\color{blue}{n \cdot n}} + \left(-0.5 \cdot \frac{1}{n}\right)\right)\right) - {x}^{\left(\frac{1}{n}\right)} \]

      7. associate-*r/ 90.1%

         \[\leadsto \left(\left(1 + \frac{x}{n}\right) + {x}^{2} \cdot \left(\frac{0.5}{n \cdot n} + \left(-\color{blue}{\frac{0.5 \cdot 1}{n}}\right)\right)\right) - {x}^{\left(\frac{1}{n}\right)} \]

      8. metadata-eval 90.1%

         \[\leadsto \left(\left(1 + \frac{x}{n}\right) + {x}^{2} \cdot \left(\frac{0.5}{n \cdot n} + \left(-\frac{\color{blue}{0.5}}{n}\right)\right)\right) - {x}^{\left(\frac{1}{n}\right)} \]

      9. distribute-neg-frac 90.1%

         \[\leadsto \left(\left(1 + \frac{x}{n}\right) + {x}^{2} \cdot \left(\frac{0.5}{n \cdot n} + \color{blue}{\frac{-0.5}{n}}\right)\right) - {x}^{\left(\frac{1}{n}\right)} \]

      10. metadata-eval 90.1%

          \[\leadsto \left(\left(1 + \frac{x}{n}\right) + {x}^{2} \cdot \left(\frac{0.5}{n \cdot n} + \frac{\color{blue}{-0.5}}{n}\right)\right) - {x}^{\left(\frac{1}{n}\right)} \]

      11. +-commutative 90.1%

          \[\leadsto \left(\left(1 + \frac{x}{n}\right) + {x}^{2} \cdot \color{blue}{\left(\frac{-0.5}{n} + \frac{0.5}{n \cdot n}\right)}\right) - {x}^{\left(\frac{1}{n}\right)} \]

      12. distribute-rgt-in 90.1%

          \[\leadsto \left(\left(1 + \frac{x}{n}\right) + \color{blue}{\left(\frac{-0.5}{n} \cdot {x}^{2} + \frac{0.5}{n \cdot n} \cdot {x}^{2}\right)}\right) - {x}^{\left(\frac{1}{n}\right)} \]

      13. associate-*l/ 90.1%

          \[\leadsto \left(\left(1 + \frac{x}{n}\right) + \left(\color{blue}{\frac{-0.5 \cdot {x}^{2}}{n}} + \frac{0.5}{n \cdot n} \cdot {x}^{2}\right)\right) - {x}^{\left(\frac{1}{n}\right)} \]

      14. associate-*r/ 90.1%

          \[\leadsto \left(\left(1 + \frac{x}{n}\right) + \left(\color{blue}{-0.5 \cdot \frac{{x}^{2}}{n}} + \frac{0.5}{n \cdot n} \cdot {x}^{2}\right)\right) - {x}^{\left(\frac{1}{n}\right)} \]

      15. associate-/r* 90.1%

          \[\leadsto \left(\left(1 + \frac{x}{n}\right) + \left(-0.5 \cdot \frac{{x}^{2}}{n} + \color{blue}{\frac{\frac{0.5}{n}}{n}} \cdot {x}^{2}\right)\right) - {x}^{\left(\frac{1}{n}\right)} \]

   7. Simplified 90.1%

      \[\leadsto \color{blue}{\left(\left(1 + \frac{x}{n}\right) + \left(x \cdot \frac{x}{n}\right) \cdot \left(-0.5 + \frac{0.5}{n}\right)\right)} - {x}^{\left(\frac{1}{n}\right)} \]

   if 4.99999999999999972e118 < (/.f64 1 n)

   1. Initial program 36.2%

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

   2. Taylor expanded in x around 0 56.8%

      \[\leadsto \color{blue}{\left(1 + \left({x}^{2} \cdot \left(0.5 \cdot \frac{1}{{n}^{2}} - 0.5 \cdot \frac{1}{n}\right) + \frac{x}{n}\right)\right)} - {x}^{\left(\frac{1}{n}\right)} \]
   3. Step-by-step derivation

      1. fma-def 56.8%

         \[\leadsto \left(1 + \color{blue}{\mathsf{fma}\left({x}^{2}, 0.5 \cdot \frac{1}{{n}^{2}} - 0.5 \cdot \frac{1}{n}, \frac{x}{n}\right)}\right) - {x}^{\left(\frac{1}{n}\right)} \]

      2. unpow2 56.8%

         \[\leadsto \left(1 + \mathsf{fma}\left(\color{blue}{x \cdot x}, 0.5 \cdot \frac{1}{{n}^{2}} - 0.5 \cdot \frac{1}{n}, \frac{x}{n}\right)\right) - {x}^{\left(\frac{1}{n}\right)} \]

      3. associate-*r/ 56.8%

         \[\leadsto \left(1 + \mathsf{fma}\left(x \cdot x, \color{blue}{\frac{0.5 \cdot 1}{{n}^{2}}} - 0.5 \cdot \frac{1}{n}, \frac{x}{n}\right)\right) - {x}^{\left(\frac{1}{n}\right)} \]

      4. metadata-eval 56.8%

         \[\leadsto \left(1 + \mathsf{fma}\left(x \cdot x, \frac{\color{blue}{0.5}}{{n}^{2}} - 0.5 \cdot \frac{1}{n}, \frac{x}{n}\right)\right) - {x}^{\left(\frac{1}{n}\right)} \]

      5. unpow2 56.8%

         \[\leadsto \left(1 + \mathsf{fma}\left(x \cdot x, \frac{0.5}{\color{blue}{n \cdot n}} - 0.5 \cdot \frac{1}{n}, \frac{x}{n}\right)\right) - {x}^{\left(\frac{1}{n}\right)} \]

      6. associate-*r/ 56.8%

         \[\leadsto \left(1 + \mathsf{fma}\left(x \cdot x, \frac{0.5}{n \cdot n} - \color{blue}{\frac{0.5 \cdot 1}{n}}, \frac{x}{n}\right)\right) - {x}^{\left(\frac{1}{n}\right)} \]

      7. metadata-eval 56.8%

         \[\leadsto \left(1 + \mathsf{fma}\left(x \cdot x, \frac{0.5}{n \cdot n} - \frac{\color{blue}{0.5}}{n}, \frac{x}{n}\right)\right) - {x}^{\left(\frac{1}{n}\right)} \]

   4. Simplified 56.8%

      \[\leadsto \color{blue}{\left(1 + \mathsf{fma}\left(x \cdot x, \frac{0.5}{n \cdot n} - \frac{0.5}{n}, \frac{x}{n}\right)\right)} - {x}^{\left(\frac{1}{n}\right)} \]

   5. Taylor expanded in n around 0 53.0%

      \[\leadsto \color{blue}{0.5 \cdot \frac{{x}^{2}}{{n}^{2}}} \]
   6. Step-by-step derivation

      1. unpow2 53.0%

         \[\leadsto 0.5 \cdot \frac{\color{blue}{x \cdot x}}{{n}^{2}} \]

      2. unpow2 53.0%

         \[\leadsto 0.5 \cdot \frac{x \cdot x}{\color{blue}{n \cdot n}} \]

   7. Simplified 53.0%

      \[\leadsto \color{blue}{0.5 \cdot \frac{x \cdot x}{n \cdot n}} \]

   8. Taylor expanded in x around 0 53.0%

      \[\leadsto \color{blue}{0.5 \cdot \frac{{x}^{2}}{{n}^{2}}} \]
   9. Step-by-step derivation

      1. *-commutative 53.0%

         \[\leadsto \color{blue}{\frac{{x}^{2}}{{n}^{2}} \cdot 0.5} \]

      2. associate-*l/ 53.0%

         \[\leadsto \color{blue}{\frac{{x}^{2} \cdot 0.5}{{n}^{2}}} \]

      3. associate-*r/ 56.8%

         \[\leadsto \color{blue}{{x}^{2} \cdot \frac{0.5}{{n}^{2}}} \]

      4. unpow2 56.8%

         \[\leadsto {x}^{2} \cdot \frac{0.5}{\color{blue}{n \cdot n}} \]

      5. unpow2 56.8%

         \[\leadsto \color{blue}{\left(x \cdot x\right)} \cdot \frac{0.5}{n \cdot n} \]

      6. associate-*l* 73.3%

         \[\leadsto \color{blue}{x \cdot \left(x \cdot \frac{0.5}{n \cdot n}\right)} \]

   10. Simplified 73.3%

       \[\leadsto \color{blue}{x \cdot \left(x \cdot \frac{0.5}{n \cdot n}\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 71.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{1}{n} \leq 9.5 \cdot 10^{-10}:\\ \;\;\;\;\frac{\mathsf{log1p}\left(x\right) - \log x}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 5 \cdot 10^{+118}:\\ \;\;\;\;\left(\left(1 + \frac{x}{n}\right) + \left(x \cdot \frac{x}{n}\right) \cdot \left(-0.5 + \frac{0.5}{n}\right)\right) - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(x \cdot \frac{0.5}{n \cdot n}\right)\\ \end{array} \]

Alternative 8: 56.7% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 3.3 \cdot 10^{-177}:\\ \;\;\;\;\frac{-\log x}{n}\\ \mathbf{elif}\;x \leq 1.56 \cdot 10^{-155}:\\ \;\;\;\;\left(\left(1 + \frac{x}{n}\right) + \left(x \cdot \frac{x}{n}\right) \cdot \left(-0.5 + \frac{0.5}{n}\right)\right) - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{elif}\;x \leq 3.6 \cdot 10^{-6}:\\ \;\;\;\;\left(x + \left(x \cdot \left(x \cdot -0.5\right) - \log x\right)\right) \cdot \frac{-1}{-n}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{0.3333333333333333}{{x}^{3}} + \left(\frac{1}{x} - \frac{0.5}{x \cdot x}\right)}{n}\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (if (<= x 3.3e-177)
   (/ (- (log x)) n)
   (if (<= x 1.56e-155)
     (-
      (+ (+ 1.0 (/ x n)) (* (* x (/ x n)) (+ -0.5 (/ 0.5 n))))
      (pow x (/ 1.0 n)))
     (if (<= x 3.6e-6)
       (* (+ x (- (* x (* x -0.5)) (log x))) (/ -1.0 (- n)))
       (/
        (+ (/ 0.3333333333333333 (pow x 3.0)) (- (/ 1.0 x) (/ 0.5 (* x x))))
        n)))))

C

double code(double x, double n) {
	double tmp;
	if (x <= 3.3e-177) {
		tmp = -log(x) / n;
	} else if (x <= 1.56e-155) {
		tmp = ((1.0 + (x / n)) + ((x * (x / n)) * (-0.5 + (0.5 / n)))) - pow(x, (1.0 / n));
	} else if (x <= 3.6e-6) {
		tmp = (x + ((x * (x * -0.5)) - log(x))) * (-1.0 / -n);
	} else {
		tmp = ((0.3333333333333333 / pow(x, 3.0)) + ((1.0 / x) - (0.5 / (x * 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 <= 3.3d-177) then
        tmp = -log(x) / n
    else if (x <= 1.56d-155) then
        tmp = ((1.0d0 + (x / n)) + ((x * (x / n)) * ((-0.5d0) + (0.5d0 / n)))) - (x ** (1.0d0 / n))
    else if (x <= 3.6d-6) then
        tmp = (x + ((x * (x * (-0.5d0))) - log(x))) * ((-1.0d0) / -n)
    else
        tmp = ((0.3333333333333333d0 / (x ** 3.0d0)) + ((1.0d0 / x) - (0.5d0 / (x * x)))) / n
    end if
    code = tmp
end function

Java

public static double code(double x, double n) {
	double tmp;
	if (x <= 3.3e-177) {
		tmp = -Math.log(x) / n;
	} else if (x <= 1.56e-155) {
		tmp = ((1.0 + (x / n)) + ((x * (x / n)) * (-0.5 + (0.5 / n)))) - Math.pow(x, (1.0 / n));
	} else if (x <= 3.6e-6) {
		tmp = (x + ((x * (x * -0.5)) - Math.log(x))) * (-1.0 / -n);
	} else {
		tmp = ((0.3333333333333333 / Math.pow(x, 3.0)) + ((1.0 / x) - (0.5 / (x * x)))) / n;
	}
	return tmp;
}

Python

def code(x, n):
	tmp = 0
	if x <= 3.3e-177:
		tmp = -math.log(x) / n
	elif x <= 1.56e-155:
		tmp = ((1.0 + (x / n)) + ((x * (x / n)) * (-0.5 + (0.5 / n)))) - math.pow(x, (1.0 / n))
	elif x <= 3.6e-6:
		tmp = (x + ((x * (x * -0.5)) - math.log(x))) * (-1.0 / -n)
	else:
		tmp = ((0.3333333333333333 / math.pow(x, 3.0)) + ((1.0 / x) - (0.5 / (x * x)))) / n
	return tmp

Julia

function code(x, n)
	tmp = 0.0
	if (x <= 3.3e-177)
		tmp = Float64(Float64(-log(x)) / n);
	elseif (x <= 1.56e-155)
		tmp = Float64(Float64(Float64(1.0 + Float64(x / n)) + Float64(Float64(x * Float64(x / n)) * Float64(-0.5 + Float64(0.5 / n)))) - (x ^ Float64(1.0 / n)));
	elseif (x <= 3.6e-6)
		tmp = Float64(Float64(x + Float64(Float64(x * Float64(x * -0.5)) - log(x))) * Float64(-1.0 / Float64(-n)));
	else
		tmp = Float64(Float64(Float64(0.3333333333333333 / (x ^ 3.0)) + Float64(Float64(1.0 / x) - Float64(0.5 / Float64(x * x)))) / n);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, n)
	tmp = 0.0;
	if (x <= 3.3e-177)
		tmp = -log(x) / n;
	elseif (x <= 1.56e-155)
		tmp = ((1.0 + (x / n)) + ((x * (x / n)) * (-0.5 + (0.5 / n)))) - (x ^ (1.0 / n));
	elseif (x <= 3.6e-6)
		tmp = (x + ((x * (x * -0.5)) - log(x))) * (-1.0 / -n);
	else
		tmp = ((0.3333333333333333 / (x ^ 3.0)) + ((1.0 / x) - (0.5 / (x * x)))) / n;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, n_] := If[LessEqual[x, 3.3e-177], N[((-N[Log[x], $MachinePrecision]) / n), $MachinePrecision], If[LessEqual[x, 1.56e-155], N[(N[(N[(1.0 + N[(x / n), $MachinePrecision]), $MachinePrecision] + N[(N[(x * N[(x / n), $MachinePrecision]), $MachinePrecision] * N[(-0.5 + N[(0.5 / n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[Power[x, N[(1.0 / n), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 3.6e-6], N[(N[(x + N[(N[(x * N[(x * -0.5), $MachinePrecision]), $MachinePrecision] - N[Log[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(-1.0 / (-n)), $MachinePrecision]), $MachinePrecision], N[(N[(N[(0.3333333333333333 / N[Power[x, 3.0], $MachinePrecision]), $MachinePrecision] + N[(N[(1.0 / x), $MachinePrecision] - N[(0.5 / N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / n), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if x < 3.3e-177

   1. Initial program 43.6%

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

   2. Taylor expanded in x around 0 43.6%

      \[\leadsto \color{blue}{1} - {x}^{\left(\frac{1}{n}\right)} \]

   3. Taylor expanded in n around inf 56.5%

      \[\leadsto \color{blue}{-1 \cdot \frac{\log x}{n}} \]
   4. Step-by-step derivation

      1. mul-1-neg 56.5%

         \[\leadsto \color{blue}{-\frac{\log x}{n}} \]

   5. Simplified 56.5%

      \[\leadsto \color{blue}{-\frac{\log x}{n}} \]

   if 3.3e-177 < x < 1.56e-155

   1. Initial program 71.1%

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

   2. Taylor expanded in n around 0 71.1%

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

      1. log1p-def 79.9%

         \[\leadsto e^{\frac{\color{blue}{\mathsf{log1p}\left(x\right)}}{n}} - {x}^{\left(\frac{1}{n}\right)} \]

   4. Simplified 79.9%

      \[\leadsto \color{blue}{e^{\frac{\mathsf{log1p}\left(x\right)}{n}}} - {x}^{\left(\frac{1}{n}\right)} \]

   5. Taylor expanded in x around 0 52.6%

      \[\leadsto \color{blue}{\left(1 + \left({x}^{2} \cdot \left(0.5 \cdot \frac{1}{{n}^{2}} - 0.5 \cdot \frac{1}{n}\right) + \frac{x}{n}\right)\right)} - {x}^{\left(\frac{1}{n}\right)} \]
   6. Step-by-step derivation

      1. +-commutative 52.6%

         \[\leadsto \left(1 + \color{blue}{\left(\frac{x}{n} + {x}^{2} \cdot \left(0.5 \cdot \frac{1}{{n}^{2}} - 0.5 \cdot \frac{1}{n}\right)\right)}\right) - {x}^{\left(\frac{1}{n}\right)} \]

      2. associate-+r+ 52.6%

         \[\leadsto \color{blue}{\left(\left(1 + \frac{x}{n}\right) + {x}^{2} \cdot \left(0.5 \cdot \frac{1}{{n}^{2}} - 0.5 \cdot \frac{1}{n}\right)\right)} - {x}^{\left(\frac{1}{n}\right)} \]

      3. sub-neg 52.6%

         \[\leadsto \left(\left(1 + \frac{x}{n}\right) + {x}^{2} \cdot \color{blue}{\left(0.5 \cdot \frac{1}{{n}^{2}} + \left(-0.5 \cdot \frac{1}{n}\right)\right)}\right) - {x}^{\left(\frac{1}{n}\right)} \]

      4. associate-*r/ 52.6%

         \[\leadsto \left(\left(1 + \frac{x}{n}\right) + {x}^{2} \cdot \left(\color{blue}{\frac{0.5 \cdot 1}{{n}^{2}}} + \left(-0.5 \cdot \frac{1}{n}\right)\right)\right) - {x}^{\left(\frac{1}{n}\right)} \]

      5. metadata-eval 52.6%

         \[\leadsto \left(\left(1 + \frac{x}{n}\right) + {x}^{2} \cdot \left(\frac{\color{blue}{0.5}}{{n}^{2}} + \left(-0.5 \cdot \frac{1}{n}\right)\right)\right) - {x}^{\left(\frac{1}{n}\right)} \]

      6. unpow2 52.6%

         \[\leadsto \left(\left(1 + \frac{x}{n}\right) + {x}^{2} \cdot \left(\frac{0.5}{\color{blue}{n \cdot n}} + \left(-0.5 \cdot \frac{1}{n}\right)\right)\right) - {x}^{\left(\frac{1}{n}\right)} \]

      7. associate-*r/ 52.6%

         \[\leadsto \left(\left(1 + \frac{x}{n}\right) + {x}^{2} \cdot \left(\frac{0.5}{n \cdot n} + \left(-\color{blue}{\frac{0.5 \cdot 1}{n}}\right)\right)\right) - {x}^{\left(\frac{1}{n}\right)} \]

      8. metadata-eval 52.6%

         \[\leadsto \left(\left(1 + \frac{x}{n}\right) + {x}^{2} \cdot \left(\frac{0.5}{n \cdot n} + \left(-\frac{\color{blue}{0.5}}{n}\right)\right)\right) - {x}^{\left(\frac{1}{n}\right)} \]

      9. distribute-neg-frac 52.6%

         \[\leadsto \left(\left(1 + \frac{x}{n}\right) + {x}^{2} \cdot \left(\frac{0.5}{n \cdot n} + \color{blue}{\frac{-0.5}{n}}\right)\right) - {x}^{\left(\frac{1}{n}\right)} \]

      10. metadata-eval 52.6%

          \[\leadsto \left(\left(1 + \frac{x}{n}\right) + {x}^{2} \cdot \left(\frac{0.5}{n \cdot n} + \frac{\color{blue}{-0.5}}{n}\right)\right) - {x}^{\left(\frac{1}{n}\right)} \]

      11. +-commutative 52.6%

          \[\leadsto \left(\left(1 + \frac{x}{n}\right) + {x}^{2} \cdot \color{blue}{\left(\frac{-0.5}{n} + \frac{0.5}{n \cdot n}\right)}\right) - {x}^{\left(\frac{1}{n}\right)} \]

      12. distribute-rgt-in 52.6%

          \[\leadsto \left(\left(1 + \frac{x}{n}\right) + \color{blue}{\left(\frac{-0.5}{n} \cdot {x}^{2} + \frac{0.5}{n \cdot n} \cdot {x}^{2}\right)}\right) - {x}^{\left(\frac{1}{n}\right)} \]

      13. associate-*l/ 52.6%

          \[\leadsto \left(\left(1 + \frac{x}{n}\right) + \left(\color{blue}{\frac{-0.5 \cdot {x}^{2}}{n}} + \frac{0.5}{n \cdot n} \cdot {x}^{2}\right)\right) - {x}^{\left(\frac{1}{n}\right)} \]

      14. associate-*r/ 52.6%

          \[\leadsto \left(\left(1 + \frac{x}{n}\right) + \left(\color{blue}{-0.5 \cdot \frac{{x}^{2}}{n}} + \frac{0.5}{n \cdot n} \cdot {x}^{2}\right)\right) - {x}^{\left(\frac{1}{n}\right)} \]

      15. associate-/r* 52.6%

          \[\leadsto \left(\left(1 + \frac{x}{n}\right) + \left(-0.5 \cdot \frac{{x}^{2}}{n} + \color{blue}{\frac{\frac{0.5}{n}}{n}} \cdot {x}^{2}\right)\right) - {x}^{\left(\frac{1}{n}\right)} \]

   7. Simplified 71.1%

      \[\leadsto \color{blue}{\left(\left(1 + \frac{x}{n}\right) + \left(x \cdot \frac{x}{n}\right) \cdot \left(-0.5 + \frac{0.5}{n}\right)\right)} - {x}^{\left(\frac{1}{n}\right)} \]

   if 1.56e-155 < x < 3.59999999999999984e-6

   1. Initial program 34.1%

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

   2. Taylor expanded in x around 0 37.5%

      \[\leadsto \color{blue}{\left(1 + \left({x}^{2} \cdot \left(0.5 \cdot \frac{1}{{n}^{2}} - 0.5 \cdot \frac{1}{n}\right) + \frac{x}{n}\right)\right)} - {x}^{\left(\frac{1}{n}\right)} \]
   3. Step-by-step derivation

      1. fma-def 37.5%

         \[\leadsto \left(1 + \color{blue}{\mathsf{fma}\left({x}^{2}, 0.5 \cdot \frac{1}{{n}^{2}} - 0.5 \cdot \frac{1}{n}, \frac{x}{n}\right)}\right) - {x}^{\left(\frac{1}{n}\right)} \]

      2. unpow2 37.5%

         \[\leadsto \left(1 + \mathsf{fma}\left(\color{blue}{x \cdot x}, 0.5 \cdot \frac{1}{{n}^{2}} - 0.5 \cdot \frac{1}{n}, \frac{x}{n}\right)\right) - {x}^{\left(\frac{1}{n}\right)} \]

      3. associate-*r/ 37.5%

         \[\leadsto \left(1 + \mathsf{fma}\left(x \cdot x, \color{blue}{\frac{0.5 \cdot 1}{{n}^{2}}} - 0.5 \cdot \frac{1}{n}, \frac{x}{n}\right)\right) - {x}^{\left(\frac{1}{n}\right)} \]

      4. metadata-eval 37.5%

         \[\leadsto \left(1 + \mathsf{fma}\left(x \cdot x, \frac{\color{blue}{0.5}}{{n}^{2}} - 0.5 \cdot \frac{1}{n}, \frac{x}{n}\right)\right) - {x}^{\left(\frac{1}{n}\right)} \]

      5. unpow2 37.5%

         \[\leadsto \left(1 + \mathsf{fma}\left(x \cdot x, \frac{0.5}{\color{blue}{n \cdot n}} - 0.5 \cdot \frac{1}{n}, \frac{x}{n}\right)\right) - {x}^{\left(\frac{1}{n}\right)} \]

      6. associate-*r/ 37.5%

         \[\leadsto \left(1 + \mathsf{fma}\left(x \cdot x, \frac{0.5}{n \cdot n} - \color{blue}{\frac{0.5 \cdot 1}{n}}, \frac{x}{n}\right)\right) - {x}^{\left(\frac{1}{n}\right)} \]

      7. metadata-eval 37.5%

         \[\leadsto \left(1 + \mathsf{fma}\left(x \cdot x, \frac{0.5}{n \cdot n} - \frac{\color{blue}{0.5}}{n}, \frac{x}{n}\right)\right) - {x}^{\left(\frac{1}{n}\right)} \]

   4. Simplified 37.5%

      \[\leadsto \color{blue}{\left(1 + \mathsf{fma}\left(x \cdot x, \frac{0.5}{n \cdot n} - \frac{0.5}{n}, \frac{x}{n}\right)\right)} - {x}^{\left(\frac{1}{n}\right)} \]

   5. Taylor expanded in n around inf 54.7%

      \[\leadsto \color{blue}{\frac{\left(x + -0.5 \cdot {x}^{2}\right) - \log x}{n}} \]
   6. Step-by-step derivation

      1. associate--l+ 54.7%

         \[\leadsto \frac{\color{blue}{x + \left(-0.5 \cdot {x}^{2} - \log x\right)}}{n} \]

      2. *-commutative 54.7%

         \[\leadsto \frac{x + \left(\color{blue}{{x}^{2} \cdot -0.5} - \log x\right)}{n} \]

      3. unpow2 54.7%

         \[\leadsto \frac{x + \left(\color{blue}{\left(x \cdot x\right)} \cdot -0.5 - \log x\right)}{n} \]

   7. Simplified 54.7%

      \[\leadsto \color{blue}{\frac{x + \left(\left(x \cdot x\right) \cdot -0.5 - \log x\right)}{n}} \]
   8. Step-by-step derivation

      1. frac-2neg 54.7%

         \[\leadsto \color{blue}{\frac{-\left(x + \left(\left(x \cdot x\right) \cdot -0.5 - \log x\right)\right)}{-n}} \]

      2. div-inv 54.7%

         \[\leadsto \color{blue}{\left(-\left(x + \left(\left(x \cdot x\right) \cdot -0.5 - \log x\right)\right)\right) \cdot \frac{1}{-n}} \]

      3. associate-*l* 54.7%

         \[\leadsto \left(-\left(x + \left(\color{blue}{x \cdot \left(x \cdot -0.5\right)} - \log x\right)\right)\right) \cdot \frac{1}{-n} \]

   9. Applied egg-rr 54.7%

      \[\leadsto \color{blue}{\left(-\left(x + \left(x \cdot \left(x \cdot -0.5\right) - \log x\right)\right)\right) \cdot \frac{1}{-n}} \]

   if 3.59999999999999984e-6 < x

   1. Initial program 68.5%

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

   2. Taylor expanded in n around inf 65.2%

      \[\leadsto \color{blue}{\frac{\log \left(1 + x\right) - \log x}{n}} \]
   3. Step-by-step derivation

      1. +-rgt-identity 65.2%

         \[\leadsto \frac{\color{blue}{\left(\log \left(1 + x\right) + 0\right)} - \log x}{n} \]

      2. +-rgt-identity 65.2%

         \[\leadsto \frac{\color{blue}{\log \left(1 + x\right)} - \log x}{n} \]

      3. log1p-def 65.2%

         \[\leadsto \frac{\color{blue}{\mathsf{log1p}\left(x\right)} - \log x}{n} \]

   4. Simplified 65.2%

      \[\leadsto \color{blue}{\frac{\mathsf{log1p}\left(x\right) - \log x}{n}} \]

   5. Taylor expanded in x around inf 62.3%

      \[\leadsto \frac{\color{blue}{\left(0.3333333333333333 \cdot \frac{1}{{x}^{3}} + \frac{1}{x}\right) - 0.5 \cdot \frac{1}{{x}^{2}}}}{n} \]
   6. Step-by-step derivation

      1. associate--l+ 62.3%

         \[\leadsto \frac{\color{blue}{0.3333333333333333 \cdot \frac{1}{{x}^{3}} + \left(\frac{1}{x} - 0.5 \cdot \frac{1}{{x}^{2}}\right)}}{n} \]

      2. associate-*r/ 62.3%

         \[\leadsto \frac{\color{blue}{\frac{0.3333333333333333 \cdot 1}{{x}^{3}}} + \left(\frac{1}{x} - 0.5 \cdot \frac{1}{{x}^{2}}\right)}{n} \]

      3. metadata-eval 62.3%

         \[\leadsto \frac{\frac{\color{blue}{0.3333333333333333}}{{x}^{3}} + \left(\frac{1}{x} - 0.5 \cdot \frac{1}{{x}^{2}}\right)}{n} \]

      4. associate-*r/ 62.3%

         \[\leadsto \frac{\frac{0.3333333333333333}{{x}^{3}} + \left(\frac{1}{x} - \color{blue}{\frac{0.5 \cdot 1}{{x}^{2}}}\right)}{n} \]

      5. metadata-eval 62.3%

         \[\leadsto \frac{\frac{0.3333333333333333}{{x}^{3}} + \left(\frac{1}{x} - \frac{\color{blue}{0.5}}{{x}^{2}}\right)}{n} \]

      6. unpow2 62.3%

         \[\leadsto \frac{\frac{0.3333333333333333}{{x}^{3}} + \left(\frac{1}{x} - \frac{0.5}{\color{blue}{x \cdot x}}\right)}{n} \]

   7. Simplified 62.3%

      \[\leadsto \frac{\color{blue}{\frac{0.3333333333333333}{{x}^{3}} + \left(\frac{1}{x} - \frac{0.5}{x \cdot x}\right)}}{n} \]
3. Recombined 4 regimes into one program.

4. Final simplification 59.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 3.3 \cdot 10^{-177}:\\ \;\;\;\;\frac{-\log x}{n}\\ \mathbf{elif}\;x \leq 1.56 \cdot 10^{-155}:\\ \;\;\;\;\left(\left(1 + \frac{x}{n}\right) + \left(x \cdot \frac{x}{n}\right) \cdot \left(-0.5 + \frac{0.5}{n}\right)\right) - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{elif}\;x \leq 3.6 \cdot 10^{-6}:\\ \;\;\;\;\left(x + \left(x \cdot \left(x \cdot -0.5\right) - \log x\right)\right) \cdot \frac{-1}{-n}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{0.3333333333333333}{{x}^{3}} + \left(\frac{1}{x} - \frac{0.5}{x \cdot x}\right)}{n}\\ \end{array} \]

Alternative 9: 56.7% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 5.6 \cdot 10^{-177}:\\ \;\;\;\;\frac{-\log x}{n}\\ \mathbf{elif}\;x \leq 1.25 \cdot 10^{-155}:\\ \;\;\;\;\left(1 + \frac{x}{n}\right) - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{elif}\;x \leq 3.6 \cdot 10^{-6}:\\ \;\;\;\;\left(x + \left(x \cdot \left(x \cdot -0.5\right) - \log x\right)\right) \cdot \frac{-1}{-n}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{0.3333333333333333}{{x}^{3}} + \left(\frac{1}{x} - \frac{0.5}{x \cdot x}\right)}{n}\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (if (<= x 5.6e-177)
   (/ (- (log x)) n)
   (if (<= x 1.25e-155)
     (- (+ 1.0 (/ x n)) (pow x (/ 1.0 n)))
     (if (<= x 3.6e-6)
       (* (+ x (- (* x (* x -0.5)) (log x))) (/ -1.0 (- n)))
       (/
        (+ (/ 0.3333333333333333 (pow x 3.0)) (- (/ 1.0 x) (/ 0.5 (* x x))))
        n)))))

C

double code(double x, double n) {
	double tmp;
	if (x <= 5.6e-177) {
		tmp = -log(x) / n;
	} else if (x <= 1.25e-155) {
		tmp = (1.0 + (x / n)) - pow(x, (1.0 / n));
	} else if (x <= 3.6e-6) {
		tmp = (x + ((x * (x * -0.5)) - log(x))) * (-1.0 / -n);
	} else {
		tmp = ((0.3333333333333333 / pow(x, 3.0)) + ((1.0 / x) - (0.5 / (x * 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 <= 5.6d-177) then
        tmp = -log(x) / n
    else if (x <= 1.25d-155) then
        tmp = (1.0d0 + (x / n)) - (x ** (1.0d0 / n))
    else if (x <= 3.6d-6) then
        tmp = (x + ((x * (x * (-0.5d0))) - log(x))) * ((-1.0d0) / -n)
    else
        tmp = ((0.3333333333333333d0 / (x ** 3.0d0)) + ((1.0d0 / x) - (0.5d0 / (x * x)))) / n
    end if
    code = tmp
end function

Java

public static double code(double x, double n) {
	double tmp;
	if (x <= 5.6e-177) {
		tmp = -Math.log(x) / n;
	} else if (x <= 1.25e-155) {
		tmp = (1.0 + (x / n)) - Math.pow(x, (1.0 / n));
	} else if (x <= 3.6e-6) {
		tmp = (x + ((x * (x * -0.5)) - Math.log(x))) * (-1.0 / -n);
	} else {
		tmp = ((0.3333333333333333 / Math.pow(x, 3.0)) + ((1.0 / x) - (0.5 / (x * x)))) / n;
	}
	return tmp;
}

Python

def code(x, n):
	tmp = 0
	if x <= 5.6e-177:
		tmp = -math.log(x) / n
	elif x <= 1.25e-155:
		tmp = (1.0 + (x / n)) - math.pow(x, (1.0 / n))
	elif x <= 3.6e-6:
		tmp = (x + ((x * (x * -0.5)) - math.log(x))) * (-1.0 / -n)
	else:
		tmp = ((0.3333333333333333 / math.pow(x, 3.0)) + ((1.0 / x) - (0.5 / (x * x)))) / n
	return tmp

Julia

function code(x, n)
	tmp = 0.0
	if (x <= 5.6e-177)
		tmp = Float64(Float64(-log(x)) / n);
	elseif (x <= 1.25e-155)
		tmp = Float64(Float64(1.0 + Float64(x / n)) - (x ^ Float64(1.0 / n)));
	elseif (x <= 3.6e-6)
		tmp = Float64(Float64(x + Float64(Float64(x * Float64(x * -0.5)) - log(x))) * Float64(-1.0 / Float64(-n)));
	else
		tmp = Float64(Float64(Float64(0.3333333333333333 / (x ^ 3.0)) + Float64(Float64(1.0 / x) - Float64(0.5 / Float64(x * x)))) / n);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, n)
	tmp = 0.0;
	if (x <= 5.6e-177)
		tmp = -log(x) / n;
	elseif (x <= 1.25e-155)
		tmp = (1.0 + (x / n)) - (x ^ (1.0 / n));
	elseif (x <= 3.6e-6)
		tmp = (x + ((x * (x * -0.5)) - log(x))) * (-1.0 / -n);
	else
		tmp = ((0.3333333333333333 / (x ^ 3.0)) + ((1.0 / x) - (0.5 / (x * x)))) / n;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, n_] := If[LessEqual[x, 5.6e-177], N[((-N[Log[x], $MachinePrecision]) / n), $MachinePrecision], If[LessEqual[x, 1.25e-155], N[(N[(1.0 + N[(x / n), $MachinePrecision]), $MachinePrecision] - N[Power[x, N[(1.0 / n), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 3.6e-6], N[(N[(x + N[(N[(x * N[(x * -0.5), $MachinePrecision]), $MachinePrecision] - N[Log[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(-1.0 / (-n)), $MachinePrecision]), $MachinePrecision], N[(N[(N[(0.3333333333333333 / N[Power[x, 3.0], $MachinePrecision]), $MachinePrecision] + N[(N[(1.0 / x), $MachinePrecision] - N[(0.5 / N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / n), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if x < 5.59999999999999973e-177

   1. Initial program 43.6%

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

   2. Taylor expanded in x around 0 43.6%

      \[\leadsto \color{blue}{1} - {x}^{\left(\frac{1}{n}\right)} \]

   3. Taylor expanded in n around inf 56.5%

      \[\leadsto \color{blue}{-1 \cdot \frac{\log x}{n}} \]
   4. Step-by-step derivation

      1. mul-1-neg 56.5%

         \[\leadsto \color{blue}{-\frac{\log x}{n}} \]

   5. Simplified 56.5%

      \[\leadsto \color{blue}{-\frac{\log x}{n}} \]

   if 5.59999999999999973e-177 < x < 1.25e-155

   1. Initial program 71.1%

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

   2. Taylor expanded in x around 0 71.1%

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

   if 1.25e-155 < x < 3.59999999999999984e-6

   1. Initial program 34.1%

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

   2. Taylor expanded in x around 0 37.5%

      \[\leadsto \color{blue}{\left(1 + \left({x}^{2} \cdot \left(0.5 \cdot \frac{1}{{n}^{2}} - 0.5 \cdot \frac{1}{n}\right) + \frac{x}{n}\right)\right)} - {x}^{\left(\frac{1}{n}\right)} \]
   3. Step-by-step derivation

      1. fma-def 37.5%

         \[\leadsto \left(1 + \color{blue}{\mathsf{fma}\left({x}^{2}, 0.5 \cdot \frac{1}{{n}^{2}} - 0.5 \cdot \frac{1}{n}, \frac{x}{n}\right)}\right) - {x}^{\left(\frac{1}{n}\right)} \]

      2. unpow2 37.5%

         \[\leadsto \left(1 + \mathsf{fma}\left(\color{blue}{x \cdot x}, 0.5 \cdot \frac{1}{{n}^{2}} - 0.5 \cdot \frac{1}{n}, \frac{x}{n}\right)\right) - {x}^{\left(\frac{1}{n}\right)} \]

      3. associate-*r/ 37.5%

         \[\leadsto \left(1 + \mathsf{fma}\left(x \cdot x, \color{blue}{\frac{0.5 \cdot 1}{{n}^{2}}} - 0.5 \cdot \frac{1}{n}, \frac{x}{n}\right)\right) - {x}^{\left(\frac{1}{n}\right)} \]

      4. metadata-eval 37.5%

         \[\leadsto \left(1 + \mathsf{fma}\left(x \cdot x, \frac{\color{blue}{0.5}}{{n}^{2}} - 0.5 \cdot \frac{1}{n}, \frac{x}{n}\right)\right) - {x}^{\left(\frac{1}{n}\right)} \]

      5. unpow2 37.5%

         \[\leadsto \left(1 + \mathsf{fma}\left(x \cdot x, \frac{0.5}{\color{blue}{n \cdot n}} - 0.5 \cdot \frac{1}{n}, \frac{x}{n}\right)\right) - {x}^{\left(\frac{1}{n}\right)} \]

      6. associate-*r/ 37.5%

         \[\leadsto \left(1 + \mathsf{fma}\left(x \cdot x, \frac{0.5}{n \cdot n} - \color{blue}{\frac{0.5 \cdot 1}{n}}, \frac{x}{n}\right)\right) - {x}^{\left(\frac{1}{n}\right)} \]

      7. metadata-eval 37.5%

         \[\leadsto \left(1 + \mathsf{fma}\left(x \cdot x, \frac{0.5}{n \cdot n} - \frac{\color{blue}{0.5}}{n}, \frac{x}{n}\right)\right) - {x}^{\left(\frac{1}{n}\right)} \]

   4. Simplified 37.5%

      \[\leadsto \color{blue}{\left(1 + \mathsf{fma}\left(x \cdot x, \frac{0.5}{n \cdot n} - \frac{0.5}{n}, \frac{x}{n}\right)\right)} - {x}^{\left(\frac{1}{n}\right)} \]

   5. Taylor expanded in n around inf 54.7%

      \[\leadsto \color{blue}{\frac{\left(x + -0.5 \cdot {x}^{2}\right) - \log x}{n}} \]
   6. Step-by-step derivation

      1. associate--l+ 54.7%

         \[\leadsto \frac{\color{blue}{x + \left(-0.5 \cdot {x}^{2} - \log x\right)}}{n} \]

      2. *-commutative 54.7%

         \[\leadsto \frac{x + \left(\color{blue}{{x}^{2} \cdot -0.5} - \log x\right)}{n} \]

      3. unpow2 54.7%

         \[\leadsto \frac{x + \left(\color{blue}{\left(x \cdot x\right)} \cdot -0.5 - \log x\right)}{n} \]

   7. Simplified 54.7%

      \[\leadsto \color{blue}{\frac{x + \left(\left(x \cdot x\right) \cdot -0.5 - \log x\right)}{n}} \]
   8. Step-by-step derivation

      1. frac-2neg 54.7%

         \[\leadsto \color{blue}{\frac{-\left(x + \left(\left(x \cdot x\right) \cdot -0.5 - \log x\right)\right)}{-n}} \]

      2. div-inv 54.7%

         \[\leadsto \color{blue}{\left(-\left(x + \left(\left(x \cdot x\right) \cdot -0.5 - \log x\right)\right)\right) \cdot \frac{1}{-n}} \]

      3. associate-*l* 54.7%

         \[\leadsto \left(-\left(x + \left(\color{blue}{x \cdot \left(x \cdot -0.5\right)} - \log x\right)\right)\right) \cdot \frac{1}{-n} \]

   9. Applied egg-rr 54.7%

      \[\leadsto \color{blue}{\left(-\left(x + \left(x \cdot \left(x \cdot -0.5\right) - \log x\right)\right)\right) \cdot \frac{1}{-n}} \]

   if 3.59999999999999984e-6 < x

   1. Initial program 68.5%

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

   2. Taylor expanded in n around inf 65.2%

      \[\leadsto \color{blue}{\frac{\log \left(1 + x\right) - \log x}{n}} \]
   3. Step-by-step derivation

      1. +-rgt-identity 65.2%

         \[\leadsto \frac{\color{blue}{\left(\log \left(1 + x\right) + 0\right)} - \log x}{n} \]

      2. +-rgt-identity 65.2%

         \[\leadsto \frac{\color{blue}{\log \left(1 + x\right)} - \log x}{n} \]

      3. log1p-def 65.2%

         \[\leadsto \frac{\color{blue}{\mathsf{log1p}\left(x\right)} - \log x}{n} \]

   4. Simplified 65.2%

      \[\leadsto \color{blue}{\frac{\mathsf{log1p}\left(x\right) - \log x}{n}} \]

   5. Taylor expanded in x around inf 62.3%

      \[\leadsto \frac{\color{blue}{\left(0.3333333333333333 \cdot \frac{1}{{x}^{3}} + \frac{1}{x}\right) - 0.5 \cdot \frac{1}{{x}^{2}}}}{n} \]
   6. Step-by-step derivation

      1. associate--l+ 62.3%

         \[\leadsto \frac{\color{blue}{0.3333333333333333 \cdot \frac{1}{{x}^{3}} + \left(\frac{1}{x} - 0.5 \cdot \frac{1}{{x}^{2}}\right)}}{n} \]

      2. associate-*r/ 62.3%

         \[\leadsto \frac{\color{blue}{\frac{0.3333333333333333 \cdot 1}{{x}^{3}}} + \left(\frac{1}{x} - 0.5 \cdot \frac{1}{{x}^{2}}\right)}{n} \]

      3. metadata-eval 62.3%

         \[\leadsto \frac{\frac{\color{blue}{0.3333333333333333}}{{x}^{3}} + \left(\frac{1}{x} - 0.5 \cdot \frac{1}{{x}^{2}}\right)}{n} \]

      4. associate-*r/ 62.3%

         \[\leadsto \frac{\frac{0.3333333333333333}{{x}^{3}} + \left(\frac{1}{x} - \color{blue}{\frac{0.5 \cdot 1}{{x}^{2}}}\right)}{n} \]

      5. metadata-eval 62.3%

         \[\leadsto \frac{\frac{0.3333333333333333}{{x}^{3}} + \left(\frac{1}{x} - \frac{\color{blue}{0.5}}{{x}^{2}}\right)}{n} \]

      6. unpow2 62.3%

         \[\leadsto \frac{\frac{0.3333333333333333}{{x}^{3}} + \left(\frac{1}{x} - \frac{0.5}{\color{blue}{x \cdot x}}\right)}{n} \]

   7. Simplified 62.3%

      \[\leadsto \frac{\color{blue}{\frac{0.3333333333333333}{{x}^{3}} + \left(\frac{1}{x} - \frac{0.5}{x \cdot x}\right)}}{n} \]
3. Recombined 4 regimes into one program.

4. Final simplification 59.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 5.6 \cdot 10^{-177}:\\ \;\;\;\;\frac{-\log x}{n}\\ \mathbf{elif}\;x \leq 1.25 \cdot 10^{-155}:\\ \;\;\;\;\left(1 + \frac{x}{n}\right) - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{elif}\;x \leq 3.6 \cdot 10^{-6}:\\ \;\;\;\;\left(x + \left(x \cdot \left(x \cdot -0.5\right) - \log x\right)\right) \cdot \frac{-1}{-n}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{0.3333333333333333}{{x}^{3}} + \left(\frac{1}{x} - \frac{0.5}{x \cdot x}\right)}{n}\\ \end{array} \]

Alternative 10: 56.9% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 5.6 \cdot 10^{-177}:\\ \;\;\;\;\frac{-\log x}{n}\\ \mathbf{elif}\;x \leq 2.65 \cdot 10^{-155}:\\ \;\;\;\;\left(1 + \frac{x}{n}\right) - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{elif}\;x \leq 1:\\ \;\;\;\;\left(x + \left(x \cdot \left(x \cdot -0.5\right) - \log x\right)\right) \cdot \frac{-1}{-n}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{n}}{x} - \frac{0.5}{n \cdot \left(x \cdot x\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (if (<= x 5.6e-177)
   (/ (- (log x)) n)
   (if (<= x 2.65e-155)
     (- (+ 1.0 (/ x n)) (pow x (/ 1.0 n)))
     (if (<= x 1.0)
       (* (+ x (- (* x (* x -0.5)) (log x))) (/ -1.0 (- n)))
       (- (/ (/ 1.0 n) x) (/ 0.5 (* n (* x x))))))))

C

double code(double x, double n) {
	double tmp;
	if (x <= 5.6e-177) {
		tmp = -log(x) / n;
	} else if (x <= 2.65e-155) {
		tmp = (1.0 + (x / n)) - pow(x, (1.0 / n));
	} else if (x <= 1.0) {
		tmp = (x + ((x * (x * -0.5)) - log(x))) * (-1.0 / -n);
	} else {
		tmp = ((1.0 / n) / x) - (0.5 / (n * (x * x)));
	}
	return tmp;
}

Fortran

real(8) function code(x, n)
    real(8), intent (in) :: x
    real(8), intent (in) :: n
    real(8) :: tmp
    if (x <= 5.6d-177) then
        tmp = -log(x) / n
    else if (x <= 2.65d-155) then
        tmp = (1.0d0 + (x / n)) - (x ** (1.0d0 / n))
    else if (x <= 1.0d0) then
        tmp = (x + ((x * (x * (-0.5d0))) - log(x))) * ((-1.0d0) / -n)
    else
        tmp = ((1.0d0 / n) / x) - (0.5d0 / (n * (x * x)))
    end if
    code = tmp
end function

Java

public static double code(double x, double n) {
	double tmp;
	if (x <= 5.6e-177) {
		tmp = -Math.log(x) / n;
	} else if (x <= 2.65e-155) {
		tmp = (1.0 + (x / n)) - Math.pow(x, (1.0 / n));
	} else if (x <= 1.0) {
		tmp = (x + ((x * (x * -0.5)) - Math.log(x))) * (-1.0 / -n);
	} else {
		tmp = ((1.0 / n) / x) - (0.5 / (n * (x * x)));
	}
	return tmp;
}

Python

def code(x, n):
	tmp = 0
	if x <= 5.6e-177:
		tmp = -math.log(x) / n
	elif x <= 2.65e-155:
		tmp = (1.0 + (x / n)) - math.pow(x, (1.0 / n))
	elif x <= 1.0:
		tmp = (x + ((x * (x * -0.5)) - math.log(x))) * (-1.0 / -n)
	else:
		tmp = ((1.0 / n) / x) - (0.5 / (n * (x * x)))
	return tmp

Julia

function code(x, n)
	tmp = 0.0
	if (x <= 5.6e-177)
		tmp = Float64(Float64(-log(x)) / n);
	elseif (x <= 2.65e-155)
		tmp = Float64(Float64(1.0 + Float64(x / n)) - (x ^ Float64(1.0 / n)));
	elseif (x <= 1.0)
		tmp = Float64(Float64(x + Float64(Float64(x * Float64(x * -0.5)) - log(x))) * Float64(-1.0 / Float64(-n)));
	else
		tmp = Float64(Float64(Float64(1.0 / n) / x) - Float64(0.5 / Float64(n * Float64(x * x))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, n)
	tmp = 0.0;
	if (x <= 5.6e-177)
		tmp = -log(x) / n;
	elseif (x <= 2.65e-155)
		tmp = (1.0 + (x / n)) - (x ^ (1.0 / n));
	elseif (x <= 1.0)
		tmp = (x + ((x * (x * -0.5)) - log(x))) * (-1.0 / -n);
	else
		tmp = ((1.0 / n) / x) - (0.5 / (n * (x * x)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, n_] := If[LessEqual[x, 5.6e-177], N[((-N[Log[x], $MachinePrecision]) / n), $MachinePrecision], If[LessEqual[x, 2.65e-155], N[(N[(1.0 + N[(x / n), $MachinePrecision]), $MachinePrecision] - N[Power[x, N[(1.0 / n), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 1.0], N[(N[(x + N[(N[(x * N[(x * -0.5), $MachinePrecision]), $MachinePrecision] - N[Log[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(-1.0 / (-n)), $MachinePrecision]), $MachinePrecision], N[(N[(N[(1.0 / n), $MachinePrecision] / x), $MachinePrecision] - N[(0.5 / N[(n * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if x < 5.59999999999999973e-177

   1. Initial program 43.6%

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

   2. Taylor expanded in x around 0 43.6%

      \[\leadsto \color{blue}{1} - {x}^{\left(\frac{1}{n}\right)} \]

   3. Taylor expanded in n around inf 56.5%

      \[\leadsto \color{blue}{-1 \cdot \frac{\log x}{n}} \]
   4. Step-by-step derivation

      1. mul-1-neg 56.5%

         \[\leadsto \color{blue}{-\frac{\log x}{n}} \]

   5. Simplified 56.5%

      \[\leadsto \color{blue}{-\frac{\log x}{n}} \]

   if 5.59999999999999973e-177 < x < 2.6499999999999999e-155

   1. Initial program 71.1%

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

   2. Taylor expanded in x around 0 71.1%

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

   if 2.6499999999999999e-155 < x < 1

   1. Initial program 37.2%

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

   2. Taylor expanded in x around 0 39.4%

      \[\leadsto \color{blue}{\left(1 + \left({x}^{2} \cdot \left(0.5 \cdot \frac{1}{{n}^{2}} - 0.5 \cdot \frac{1}{n}\right) + \frac{x}{n}\right)\right)} - {x}^{\left(\frac{1}{n}\right)} \]
   3. Step-by-step derivation

      1. fma-def 39.4%

         \[\leadsto \left(1 + \color{blue}{\mathsf{fma}\left({x}^{2}, 0.5 \cdot \frac{1}{{n}^{2}} - 0.5 \cdot \frac{1}{n}, \frac{x}{n}\right)}\right) - {x}^{\left(\frac{1}{n}\right)} \]

      2. unpow2 39.4%

         \[\leadsto \left(1 + \mathsf{fma}\left(\color{blue}{x \cdot x}, 0.5 \cdot \frac{1}{{n}^{2}} - 0.5 \cdot \frac{1}{n}, \frac{x}{n}\right)\right) - {x}^{\left(\frac{1}{n}\right)} \]

      3. associate-*r/ 39.4%

         \[\leadsto \left(1 + \mathsf{fma}\left(x \cdot x, \color{blue}{\frac{0.5 \cdot 1}{{n}^{2}}} - 0.5 \cdot \frac{1}{n}, \frac{x}{n}\right)\right) - {x}^{\left(\frac{1}{n}\right)} \]

      4. metadata-eval 39.4%

         \[\leadsto \left(1 + \mathsf{fma}\left(x \cdot x, \frac{\color{blue}{0.5}}{{n}^{2}} - 0.5 \cdot \frac{1}{n}, \frac{x}{n}\right)\right) - {x}^{\left(\frac{1}{n}\right)} \]

      5. unpow2 39.4%

         \[\leadsto \left(1 + \mathsf{fma}\left(x \cdot x, \frac{0.5}{\color{blue}{n \cdot n}} - 0.5 \cdot \frac{1}{n}, \frac{x}{n}\right)\right) - {x}^{\left(\frac{1}{n}\right)} \]

      6. associate-*r/ 39.4%

         \[\leadsto \left(1 + \mathsf{fma}\left(x \cdot x, \frac{0.5}{n \cdot n} - \color{blue}{\frac{0.5 \cdot 1}{n}}, \frac{x}{n}\right)\right) - {x}^{\left(\frac{1}{n}\right)} \]

      7. metadata-eval 39.4%

         \[\leadsto \left(1 + \mathsf{fma}\left(x \cdot x, \frac{0.5}{n \cdot n} - \frac{\color{blue}{0.5}}{n}, \frac{x}{n}\right)\right) - {x}^{\left(\frac{1}{n}\right)} \]

   4. Simplified 39.4%

      \[\leadsto \color{blue}{\left(1 + \mathsf{fma}\left(x \cdot x, \frac{0.5}{n \cdot n} - \frac{0.5}{n}, \frac{x}{n}\right)\right)} - {x}^{\left(\frac{1}{n}\right)} \]

   5. Taylor expanded in n around inf 52.3%

      \[\leadsto \color{blue}{\frac{\left(x + -0.5 \cdot {x}^{2}\right) - \log x}{n}} \]
   6. Step-by-step derivation

      1. associate--l+ 52.3%

         \[\leadsto \frac{\color{blue}{x + \left(-0.5 \cdot {x}^{2} - \log x\right)}}{n} \]

      2. *-commutative 52.3%

         \[\leadsto \frac{x + \left(\color{blue}{{x}^{2} \cdot -0.5} - \log x\right)}{n} \]

      3. unpow2 52.3%

         \[\leadsto \frac{x + \left(\color{blue}{\left(x \cdot x\right)} \cdot -0.5 - \log x\right)}{n} \]

   7. Simplified 52.3%

      \[\leadsto \color{blue}{\frac{x + \left(\left(x \cdot x\right) \cdot -0.5 - \log x\right)}{n}} \]
   8. Step-by-step derivation

      1. frac-2neg 52.3%

         \[\leadsto \color{blue}{\frac{-\left(x + \left(\left(x \cdot x\right) \cdot -0.5 - \log x\right)\right)}{-n}} \]

      2. div-inv 52.3%

         \[\leadsto \color{blue}{\left(-\left(x + \left(\left(x \cdot x\right) \cdot -0.5 - \log x\right)\right)\right) \cdot \frac{1}{-n}} \]

      3. associate-*l* 52.3%

         \[\leadsto \left(-\left(x + \left(\color{blue}{x \cdot \left(x \cdot -0.5\right)} - \log x\right)\right)\right) \cdot \frac{1}{-n} \]

   9. Applied egg-rr 52.3%

      \[\leadsto \color{blue}{\left(-\left(x + \left(x \cdot \left(x \cdot -0.5\right) - \log x\right)\right)\right) \cdot \frac{1}{-n}} \]

   if 1 < x

   1. Initial program 67.3%

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

   2. Taylor expanded in n around inf 67.5%

      \[\leadsto \color{blue}{\frac{\log \left(1 + x\right) - \log x}{n}} \]
   3. Step-by-step derivation

      1. +-rgt-identity 67.5%

         \[\leadsto \frac{\color{blue}{\left(\log \left(1 + x\right) + 0\right)} - \log x}{n} \]

      2. +-rgt-identity 67.5%

         \[\leadsto \frac{\color{blue}{\log \left(1 + x\right)} - \log x}{n} \]

      3. log1p-def 67.5%

         \[\leadsto \frac{\color{blue}{\mathsf{log1p}\left(x\right)} - \log x}{n} \]

   4. Simplified 67.5%

      \[\leadsto \color{blue}{\frac{\mathsf{log1p}\left(x\right) - \log x}{n}} \]

   5. Taylor expanded in x around inf 63.0%

      \[\leadsto \color{blue}{\frac{1}{n \cdot x} - 0.5 \cdot \frac{1}{n \cdot {x}^{2}}} \]
   6. Step-by-step derivation

      1. associate-/r* 64.2%

         \[\leadsto \color{blue}{\frac{\frac{1}{n}}{x}} - 0.5 \cdot \frac{1}{n \cdot {x}^{2}} \]

      2. associate-*r/ 64.2%

         \[\leadsto \frac{\frac{1}{n}}{x} - \color{blue}{\frac{0.5 \cdot 1}{n \cdot {x}^{2}}} \]

      3. metadata-eval 64.2%

         \[\leadsto \frac{\frac{1}{n}}{x} - \frac{\color{blue}{0.5}}{n \cdot {x}^{2}} \]

      4. unpow2 64.2%

         \[\leadsto \frac{\frac{1}{n}}{x} - \frac{0.5}{n \cdot \color{blue}{\left(x \cdot x\right)}} \]

   7. Simplified 64.2%

      \[\leadsto \color{blue}{\frac{\frac{1}{n}}{x} - \frac{0.5}{n \cdot \left(x \cdot x\right)}} \]
3. Recombined 4 regimes into one program.

4. Final simplification 58.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 5.6 \cdot 10^{-177}:\\ \;\;\;\;\frac{-\log x}{n}\\ \mathbf{elif}\;x \leq 2.65 \cdot 10^{-155}:\\ \;\;\;\;\left(1 + \frac{x}{n}\right) - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{elif}\;x \leq 1:\\ \;\;\;\;\left(x + \left(x \cdot \left(x \cdot -0.5\right) - \log x\right)\right) \cdot \frac{-1}{-n}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{n}}{x} - \frac{0.5}{n \cdot \left(x \cdot x\right)}\\ \end{array} \]

Alternative 11: 56.8% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 8.5 \cdot 10^{-178}:\\ \;\;\;\;\frac{-\log x}{n}\\ \mathbf{elif}\;x \leq 1.32 \cdot 10^{-155}:\\ \;\;\;\;\left(1 + \frac{x}{n}\right) - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{elif}\;x \leq 1:\\ \;\;\;\;\frac{x + \left(-0.5 \cdot \left(x \cdot x\right) - \log x\right)}{n}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{n}}{x} - \frac{0.5}{n \cdot \left(x \cdot x\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (if (<= x 8.5e-178)
   (/ (- (log x)) n)
   (if (<= x 1.32e-155)
     (- (+ 1.0 (/ x n)) (pow x (/ 1.0 n)))
     (if (<= x 1.0)
       (/ (+ x (- (* -0.5 (* x x)) (log x))) n)
       (- (/ (/ 1.0 n) x) (/ 0.5 (* n (* x x))))))))

C

double code(double x, double n) {
	double tmp;
	if (x <= 8.5e-178) {
		tmp = -log(x) / n;
	} else if (x <= 1.32e-155) {
		tmp = (1.0 + (x / n)) - pow(x, (1.0 / n));
	} else if (x <= 1.0) {
		tmp = (x + ((-0.5 * (x * x)) - log(x))) / n;
	} else {
		tmp = ((1.0 / n) / x) - (0.5 / (n * (x * x)));
	}
	return tmp;
}

Fortran

real(8) function code(x, n)
    real(8), intent (in) :: x
    real(8), intent (in) :: n
    real(8) :: tmp
    if (x <= 8.5d-178) then
        tmp = -log(x) / n
    else if (x <= 1.32d-155) then
        tmp = (1.0d0 + (x / n)) - (x ** (1.0d0 / n))
    else if (x <= 1.0d0) then
        tmp = (x + (((-0.5d0) * (x * x)) - log(x))) / n
    else
        tmp = ((1.0d0 / n) / x) - (0.5d0 / (n * (x * x)))
    end if
    code = tmp
end function

Java

public static double code(double x, double n) {
	double tmp;
	if (x <= 8.5e-178) {
		tmp = -Math.log(x) / n;
	} else if (x <= 1.32e-155) {
		tmp = (1.0 + (x / n)) - Math.pow(x, (1.0 / n));
	} else if (x <= 1.0) {
		tmp = (x + ((-0.5 * (x * x)) - Math.log(x))) / n;
	} else {
		tmp = ((1.0 / n) / x) - (0.5 / (n * (x * x)));
	}
	return tmp;
}

Python

def code(x, n):
	tmp = 0
	if x <= 8.5e-178:
		tmp = -math.log(x) / n
	elif x <= 1.32e-155:
		tmp = (1.0 + (x / n)) - math.pow(x, (1.0 / n))
	elif x <= 1.0:
		tmp = (x + ((-0.5 * (x * x)) - math.log(x))) / n
	else:
		tmp = ((1.0 / n) / x) - (0.5 / (n * (x * x)))
	return tmp

Julia

function code(x, n)
	tmp = 0.0
	if (x <= 8.5e-178)
		tmp = Float64(Float64(-log(x)) / n);
	elseif (x <= 1.32e-155)
		tmp = Float64(Float64(1.0 + Float64(x / n)) - (x ^ Float64(1.0 / n)));
	elseif (x <= 1.0)
		tmp = Float64(Float64(x + Float64(Float64(-0.5 * Float64(x * x)) - log(x))) / n);
	else
		tmp = Float64(Float64(Float64(1.0 / n) / x) - Float64(0.5 / Float64(n * Float64(x * x))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, n)
	tmp = 0.0;
	if (x <= 8.5e-178)
		tmp = -log(x) / n;
	elseif (x <= 1.32e-155)
		tmp = (1.0 + (x / n)) - (x ^ (1.0 / n));
	elseif (x <= 1.0)
		tmp = (x + ((-0.5 * (x * x)) - log(x))) / n;
	else
		tmp = ((1.0 / n) / x) - (0.5 / (n * (x * x)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, n_] := If[LessEqual[x, 8.5e-178], N[((-N[Log[x], $MachinePrecision]) / n), $MachinePrecision], If[LessEqual[x, 1.32e-155], N[(N[(1.0 + N[(x / n), $MachinePrecision]), $MachinePrecision] - N[Power[x, N[(1.0 / n), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 1.0], N[(N[(x + N[(N[(-0.5 * N[(x * x), $MachinePrecision]), $MachinePrecision] - N[Log[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / n), $MachinePrecision], N[(N[(N[(1.0 / n), $MachinePrecision] / x), $MachinePrecision] - N[(0.5 / N[(n * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if x < 8.5000000000000001e-178

   1. Initial program 43.6%

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

   2. Taylor expanded in x around 0 43.6%

      \[\leadsto \color{blue}{1} - {x}^{\left(\frac{1}{n}\right)} \]

   3. Taylor expanded in n around inf 56.5%

      \[\leadsto \color{blue}{-1 \cdot \frac{\log x}{n}} \]
   4. Step-by-step derivation

      1. mul-1-neg 56.5%

         \[\leadsto \color{blue}{-\frac{\log x}{n}} \]

   5. Simplified 56.5%

      \[\leadsto \color{blue}{-\frac{\log x}{n}} \]

   if 8.5000000000000001e-178 < x < 1.3199999999999999e-155

   1. Initial program 71.1%

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

   2. Taylor expanded in x around 0 71.1%

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

   if 1.3199999999999999e-155 < x < 1

   1. Initial program 37.2%

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

   2. Taylor expanded in x around 0 39.4%

      \[\leadsto \color{blue}{\left(1 + \left({x}^{2} \cdot \left(0.5 \cdot \frac{1}{{n}^{2}} - 0.5 \cdot \frac{1}{n}\right) + \frac{x}{n}\right)\right)} - {x}^{\left(\frac{1}{n}\right)} \]
   3. Step-by-step derivation

      1. fma-def 39.4%

         \[\leadsto \left(1 + \color{blue}{\mathsf{fma}\left({x}^{2}, 0.5 \cdot \frac{1}{{n}^{2}} - 0.5 \cdot \frac{1}{n}, \frac{x}{n}\right)}\right) - {x}^{\left(\frac{1}{n}\right)} \]

      2. unpow2 39.4%

         \[\leadsto \left(1 + \mathsf{fma}\left(\color{blue}{x \cdot x}, 0.5 \cdot \frac{1}{{n}^{2}} - 0.5 \cdot \frac{1}{n}, \frac{x}{n}\right)\right) - {x}^{\left(\frac{1}{n}\right)} \]

      3. associate-*r/ 39.4%

         \[\leadsto \left(1 + \mathsf{fma}\left(x \cdot x, \color{blue}{\frac{0.5 \cdot 1}{{n}^{2}}} - 0.5 \cdot \frac{1}{n}, \frac{x}{n}\right)\right) - {x}^{\left(\frac{1}{n}\right)} \]

      4. metadata-eval 39.4%

         \[\leadsto \left(1 + \mathsf{fma}\left(x \cdot x, \frac{\color{blue}{0.5}}{{n}^{2}} - 0.5 \cdot \frac{1}{n}, \frac{x}{n}\right)\right) - {x}^{\left(\frac{1}{n}\right)} \]

      5. unpow2 39.4%

         \[\leadsto \left(1 + \mathsf{fma}\left(x \cdot x, \frac{0.5}{\color{blue}{n \cdot n}} - 0.5 \cdot \frac{1}{n}, \frac{x}{n}\right)\right) - {x}^{\left(\frac{1}{n}\right)} \]

      6. associate-*r/ 39.4%

         \[\leadsto \left(1 + \mathsf{fma}\left(x \cdot x, \frac{0.5}{n \cdot n} - \color{blue}{\frac{0.5 \cdot 1}{n}}, \frac{x}{n}\right)\right) - {x}^{\left(\frac{1}{n}\right)} \]

      7. metadata-eval 39.4%

         \[\leadsto \left(1 + \mathsf{fma}\left(x \cdot x, \frac{0.5}{n \cdot n} - \frac{\color{blue}{0.5}}{n}, \frac{x}{n}\right)\right) - {x}^{\left(\frac{1}{n}\right)} \]

   4. Simplified 39.4%

      \[\leadsto \color{blue}{\left(1 + \mathsf{fma}\left(x \cdot x, \frac{0.5}{n \cdot n} - \frac{0.5}{n}, \frac{x}{n}\right)\right)} - {x}^{\left(\frac{1}{n}\right)} \]

   5. Taylor expanded in n around inf 52.3%

      \[\leadsto \color{blue}{\frac{\left(x + -0.5 \cdot {x}^{2}\right) - \log x}{n}} \]
   6. Step-by-step derivation

      1. associate--l+ 52.3%

         \[\leadsto \frac{\color{blue}{x + \left(-0.5 \cdot {x}^{2} - \log x\right)}}{n} \]

      2. *-commutative 52.3%

         \[\leadsto \frac{x + \left(\color{blue}{{x}^{2} \cdot -0.5} - \log x\right)}{n} \]

      3. unpow2 52.3%

         \[\leadsto \frac{x + \left(\color{blue}{\left(x \cdot x\right)} \cdot -0.5 - \log x\right)}{n} \]

   7. Simplified 52.3%

      \[\leadsto \color{blue}{\frac{x + \left(\left(x \cdot x\right) \cdot -0.5 - \log x\right)}{n}} \]

   if 1 < x

   1. Initial program 67.3%

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

   2. Taylor expanded in n around inf 67.5%

      \[\leadsto \color{blue}{\frac{\log \left(1 + x\right) - \log x}{n}} \]
   3. Step-by-step derivation

      1. +-rgt-identity 67.5%

         \[\leadsto \frac{\color{blue}{\left(\log \left(1 + x\right) + 0\right)} - \log x}{n} \]

      2. +-rgt-identity 67.5%

         \[\leadsto \frac{\color{blue}{\log \left(1 + x\right)} - \log x}{n} \]

      3. log1p-def 67.5%

         \[\leadsto \frac{\color{blue}{\mathsf{log1p}\left(x\right)} - \log x}{n} \]

   4. Simplified 67.5%

      \[\leadsto \color{blue}{\frac{\mathsf{log1p}\left(x\right) - \log x}{n}} \]

   5. Taylor expanded in x around inf 63.0%

      \[\leadsto \color{blue}{\frac{1}{n \cdot x} - 0.5 \cdot \frac{1}{n \cdot {x}^{2}}} \]
   6. Step-by-step derivation

      1. associate-/r* 64.2%

         \[\leadsto \color{blue}{\frac{\frac{1}{n}}{x}} - 0.5 \cdot \frac{1}{n \cdot {x}^{2}} \]

      2. associate-*r/ 64.2%

         \[\leadsto \frac{\frac{1}{n}}{x} - \color{blue}{\frac{0.5 \cdot 1}{n \cdot {x}^{2}}} \]

      3. metadata-eval 64.2%

         \[\leadsto \frac{\frac{1}{n}}{x} - \frac{\color{blue}{0.5}}{n \cdot {x}^{2}} \]

      4. unpow2 64.2%

         \[\leadsto \frac{\frac{1}{n}}{x} - \frac{0.5}{n \cdot \color{blue}{\left(x \cdot x\right)}} \]

   7. Simplified 64.2%

      \[\leadsto \color{blue}{\frac{\frac{1}{n}}{x} - \frac{0.5}{n \cdot \left(x \cdot x\right)}} \]
3. Recombined 4 regimes into one program.

4. Final simplification 58.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 8.5 \cdot 10^{-178}:\\ \;\;\;\;\frac{-\log x}{n}\\ \mathbf{elif}\;x \leq 1.32 \cdot 10^{-155}:\\ \;\;\;\;\left(1 + \frac{x}{n}\right) - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{elif}\;x \leq 1:\\ \;\;\;\;\frac{x + \left(-0.5 \cdot \left(x \cdot x\right) - \log x\right)}{n}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{n}}{x} - \frac{0.5}{n \cdot \left(x \cdot x\right)}\\ \end{array} \]

Alternative 12: 56.8% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 5.2 \cdot 10^{-177}:\\ \;\;\;\;\frac{-\log x}{n}\\ \mathbf{elif}\;x \leq 1.55 \cdot 10^{-155}:\\ \;\;\;\;\left(1 + \frac{x}{n}\right) - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{elif}\;x \leq 0.95:\\ \;\;\;\;\frac{x - \log x}{n}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{n}}{x} - \frac{0.5}{n \cdot \left(x \cdot x\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (if (<= x 5.2e-177)
   (/ (- (log x)) n)
   (if (<= x 1.55e-155)
     (- (+ 1.0 (/ x n)) (pow x (/ 1.0 n)))
     (if (<= x 0.95)
       (/ (- x (log x)) n)
       (- (/ (/ 1.0 n) x) (/ 0.5 (* n (* x x))))))))

C

double code(double x, double n) {
	double tmp;
	if (x <= 5.2e-177) {
		tmp = -log(x) / n;
	} else if (x <= 1.55e-155) {
		tmp = (1.0 + (x / n)) - pow(x, (1.0 / n));
	} else if (x <= 0.95) {
		tmp = (x - log(x)) / n;
	} else {
		tmp = ((1.0 / n) / x) - (0.5 / (n * (x * x)));
	}
	return tmp;
}

Fortran

real(8) function code(x, n)
    real(8), intent (in) :: x
    real(8), intent (in) :: n
    real(8) :: tmp
    if (x <= 5.2d-177) then
        tmp = -log(x) / n
    else if (x <= 1.55d-155) then
        tmp = (1.0d0 + (x / n)) - (x ** (1.0d0 / n))
    else if (x <= 0.95d0) then
        tmp = (x - log(x)) / n
    else
        tmp = ((1.0d0 / n) / x) - (0.5d0 / (n * (x * x)))
    end if
    code = tmp
end function

Java

public static double code(double x, double n) {
	double tmp;
	if (x <= 5.2e-177) {
		tmp = -Math.log(x) / n;
	} else if (x <= 1.55e-155) {
		tmp = (1.0 + (x / n)) - Math.pow(x, (1.0 / n));
	} else if (x <= 0.95) {
		tmp = (x - Math.log(x)) / n;
	} else {
		tmp = ((1.0 / n) / x) - (0.5 / (n * (x * x)));
	}
	return tmp;
}

Python

def code(x, n):
	tmp = 0
	if x <= 5.2e-177:
		tmp = -math.log(x) / n
	elif x <= 1.55e-155:
		tmp = (1.0 + (x / n)) - math.pow(x, (1.0 / n))
	elif x <= 0.95:
		tmp = (x - math.log(x)) / n
	else:
		tmp = ((1.0 / n) / x) - (0.5 / (n * (x * x)))
	return tmp

Julia

function code(x, n)
	tmp = 0.0
	if (x <= 5.2e-177)
		tmp = Float64(Float64(-log(x)) / n);
	elseif (x <= 1.55e-155)
		tmp = Float64(Float64(1.0 + Float64(x / n)) - (x ^ Float64(1.0 / n)));
	elseif (x <= 0.95)
		tmp = Float64(Float64(x - log(x)) / n);
	else
		tmp = Float64(Float64(Float64(1.0 / n) / x) - Float64(0.5 / Float64(n * Float64(x * x))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, n)
	tmp = 0.0;
	if (x <= 5.2e-177)
		tmp = -log(x) / n;
	elseif (x <= 1.55e-155)
		tmp = (1.0 + (x / n)) - (x ^ (1.0 / n));
	elseif (x <= 0.95)
		tmp = (x - log(x)) / n;
	else
		tmp = ((1.0 / n) / x) - (0.5 / (n * (x * x)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, n_] := If[LessEqual[x, 5.2e-177], N[((-N[Log[x], $MachinePrecision]) / n), $MachinePrecision], If[LessEqual[x, 1.55e-155], N[(N[(1.0 + N[(x / n), $MachinePrecision]), $MachinePrecision] - N[Power[x, N[(1.0 / n), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 0.95], N[(N[(x - N[Log[x], $MachinePrecision]), $MachinePrecision] / n), $MachinePrecision], N[(N[(N[(1.0 / n), $MachinePrecision] / x), $MachinePrecision] - N[(0.5 / N[(n * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if x < 5.2000000000000002e-177

   1. Initial program 43.6%

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

   2. Taylor expanded in x around 0 43.6%

      \[\leadsto \color{blue}{1} - {x}^{\left(\frac{1}{n}\right)} \]

   3. Taylor expanded in n around inf 56.5%

      \[\leadsto \color{blue}{-1 \cdot \frac{\log x}{n}} \]
   4. Step-by-step derivation

      1. mul-1-neg 56.5%

         \[\leadsto \color{blue}{-\frac{\log x}{n}} \]

   5. Simplified 56.5%

      \[\leadsto \color{blue}{-\frac{\log x}{n}} \]

   if 5.2000000000000002e-177 < x < 1.55e-155

   1. Initial program 71.1%

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

   2. Taylor expanded in x around 0 71.1%

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

   if 1.55e-155 < x < 0.94999999999999996

   1. Initial program 37.2%

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

   2. Taylor expanded in x around 0 39.4%

      \[\leadsto \color{blue}{\left(1 + \left({x}^{2} \cdot \left(0.5 \cdot \frac{1}{{n}^{2}} - 0.5 \cdot \frac{1}{n}\right) + \frac{x}{n}\right)\right)} - {x}^{\left(\frac{1}{n}\right)} \]
   3. Step-by-step derivation

      1. fma-def 39.4%

         \[\leadsto \left(1 + \color{blue}{\mathsf{fma}\left({x}^{2}, 0.5 \cdot \frac{1}{{n}^{2}} - 0.5 \cdot \frac{1}{n}, \frac{x}{n}\right)}\right) - {x}^{\left(\frac{1}{n}\right)} \]

      2. unpow2 39.4%

         \[\leadsto \left(1 + \mathsf{fma}\left(\color{blue}{x \cdot x}, 0.5 \cdot \frac{1}{{n}^{2}} - 0.5 \cdot \frac{1}{n}, \frac{x}{n}\right)\right) - {x}^{\left(\frac{1}{n}\right)} \]

      3. associate-*r/ 39.4%

         \[\leadsto \left(1 + \mathsf{fma}\left(x \cdot x, \color{blue}{\frac{0.5 \cdot 1}{{n}^{2}}} - 0.5 \cdot \frac{1}{n}, \frac{x}{n}\right)\right) - {x}^{\left(\frac{1}{n}\right)} \]

      4. metadata-eval 39.4%

         \[\leadsto \left(1 + \mathsf{fma}\left(x \cdot x, \frac{\color{blue}{0.5}}{{n}^{2}} - 0.5 \cdot \frac{1}{n}, \frac{x}{n}\right)\right) - {x}^{\left(\frac{1}{n}\right)} \]

      5. unpow2 39.4%

         \[\leadsto \left(1 + \mathsf{fma}\left(x \cdot x, \frac{0.5}{\color{blue}{n \cdot n}} - 0.5 \cdot \frac{1}{n}, \frac{x}{n}\right)\right) - {x}^{\left(\frac{1}{n}\right)} \]

      6. associate-*r/ 39.4%

         \[\leadsto \left(1 + \mathsf{fma}\left(x \cdot x, \frac{0.5}{n \cdot n} - \color{blue}{\frac{0.5 \cdot 1}{n}}, \frac{x}{n}\right)\right) - {x}^{\left(\frac{1}{n}\right)} \]

      7. metadata-eval 39.4%

         \[\leadsto \left(1 + \mathsf{fma}\left(x \cdot x, \frac{0.5}{n \cdot n} - \frac{\color{blue}{0.5}}{n}, \frac{x}{n}\right)\right) - {x}^{\left(\frac{1}{n}\right)} \]

   4. Simplified 39.4%

      \[\leadsto \color{blue}{\left(1 + \mathsf{fma}\left(x \cdot x, \frac{0.5}{n \cdot n} - \frac{0.5}{n}, \frac{x}{n}\right)\right)} - {x}^{\left(\frac{1}{n}\right)} \]

   5. Taylor expanded in n around inf 52.3%

      \[\leadsto \color{blue}{\frac{\left(x + -0.5 \cdot {x}^{2}\right) - \log x}{n}} \]
   6. Step-by-step derivation

      1. associate--l+ 52.3%

         \[\leadsto \frac{\color{blue}{x + \left(-0.5 \cdot {x}^{2} - \log x\right)}}{n} \]

      2. *-commutative 52.3%

         \[\leadsto \frac{x + \left(\color{blue}{{x}^{2} \cdot -0.5} - \log x\right)}{n} \]

      3. unpow2 52.3%

         \[\leadsto \frac{x + \left(\color{blue}{\left(x \cdot x\right)} \cdot -0.5 - \log x\right)}{n} \]

   7. Simplified 52.3%

      \[\leadsto \color{blue}{\frac{x + \left(\left(x \cdot x\right) \cdot -0.5 - \log x\right)}{n}} \]

   8. Taylor expanded in x around 0 52.2%

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

      1. mul-1-neg 52.2%

         \[\leadsto \frac{x + \color{blue}{\left(-\log x\right)}}{n} \]

      2. unsub-neg 52.2%

         \[\leadsto \frac{\color{blue}{x - \log x}}{n} \]

   10. Simplified 52.2%

       \[\leadsto \frac{\color{blue}{x - \log x}}{n} \]

   if 0.94999999999999996 < x

   1. Initial program 67.3%

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

   2. Taylor expanded in n around inf 67.5%

      \[\leadsto \color{blue}{\frac{\log \left(1 + x\right) - \log x}{n}} \]
   3. Step-by-step derivation

      1. +-rgt-identity 67.5%

         \[\leadsto \frac{\color{blue}{\left(\log \left(1 + x\right) + 0\right)} - \log x}{n} \]

      2. +-rgt-identity 67.5%

         \[\leadsto \frac{\color{blue}{\log \left(1 + x\right)} - \log x}{n} \]

      3. log1p-def 67.5%

         \[\leadsto \frac{\color{blue}{\mathsf{log1p}\left(x\right)} - \log x}{n} \]

   4. Simplified 67.5%

      \[\leadsto \color{blue}{\frac{\mathsf{log1p}\left(x\right) - \log x}{n}} \]

   5. Taylor expanded in x around inf 63.0%

      \[\leadsto \color{blue}{\frac{1}{n \cdot x} - 0.5 \cdot \frac{1}{n \cdot {x}^{2}}} \]
   6. Step-by-step derivation

      1. associate-/r* 64.2%

         \[\leadsto \color{blue}{\frac{\frac{1}{n}}{x}} - 0.5 \cdot \frac{1}{n \cdot {x}^{2}} \]

      2. associate-*r/ 64.2%

         \[\leadsto \frac{\frac{1}{n}}{x} - \color{blue}{\frac{0.5 \cdot 1}{n \cdot {x}^{2}}} \]

      3. metadata-eval 64.2%

         \[\leadsto \frac{\frac{1}{n}}{x} - \frac{\color{blue}{0.5}}{n \cdot {x}^{2}} \]

      4. unpow2 64.2%

         \[\leadsto \frac{\frac{1}{n}}{x} - \frac{0.5}{n \cdot \color{blue}{\left(x \cdot x\right)}} \]

   7. Simplified 64.2%

      \[\leadsto \color{blue}{\frac{\frac{1}{n}}{x} - \frac{0.5}{n \cdot \left(x \cdot x\right)}} \]
3. Recombined 4 regimes into one program.

4. Final simplification 58.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 5.2 \cdot 10^{-177}:\\ \;\;\;\;\frac{-\log x}{n}\\ \mathbf{elif}\;x \leq 1.55 \cdot 10^{-155}:\\ \;\;\;\;\left(1 + \frac{x}{n}\right) - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{elif}\;x \leq 0.95:\\ \;\;\;\;\frac{x - \log x}{n}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{n}}{x} - \frac{0.5}{n \cdot \left(x \cdot x\right)}\\ \end{array} \]

Alternative 13: 56.8% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 3.3 \cdot 10^{-177}:\\ \;\;\;\;\frac{-\log x}{n}\\ \mathbf{elif}\;x \leq 1.25 \cdot 10^{-155}:\\ \;\;\;\;1 - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{elif}\;x \leq 0.95:\\ \;\;\;\;\frac{x - \log x}{n}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{n}}{x} - \frac{0.5}{n \cdot \left(x \cdot x\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (if (<= x 3.3e-177)
   (/ (- (log x)) n)
   (if (<= x 1.25e-155)
     (- 1.0 (pow x (/ 1.0 n)))
     (if (<= x 0.95)
       (/ (- x (log x)) n)
       (- (/ (/ 1.0 n) x) (/ 0.5 (* n (* x x))))))))

C

double code(double x, double n) {
	double tmp;
	if (x <= 3.3e-177) {
		tmp = -log(x) / n;
	} else if (x <= 1.25e-155) {
		tmp = 1.0 - pow(x, (1.0 / n));
	} else if (x <= 0.95) {
		tmp = (x - log(x)) / n;
	} else {
		tmp = ((1.0 / n) / x) - (0.5 / (n * (x * x)));
	}
	return tmp;
}

Fortran

real(8) function code(x, n)
    real(8), intent (in) :: x
    real(8), intent (in) :: n
    real(8) :: tmp
    if (x <= 3.3d-177) then
        tmp = -log(x) / n
    else if (x <= 1.25d-155) then
        tmp = 1.0d0 - (x ** (1.0d0 / n))
    else if (x <= 0.95d0) then
        tmp = (x - log(x)) / n
    else
        tmp = ((1.0d0 / n) / x) - (0.5d0 / (n * (x * x)))
    end if
    code = tmp
end function

Java

public static double code(double x, double n) {
	double tmp;
	if (x <= 3.3e-177) {
		tmp = -Math.log(x) / n;
	} else if (x <= 1.25e-155) {
		tmp = 1.0 - Math.pow(x, (1.0 / n));
	} else if (x <= 0.95) {
		tmp = (x - Math.log(x)) / n;
	} else {
		tmp = ((1.0 / n) / x) - (0.5 / (n * (x * x)));
	}
	return tmp;
}

Python

def code(x, n):
	tmp = 0
	if x <= 3.3e-177:
		tmp = -math.log(x) / n
	elif x <= 1.25e-155:
		tmp = 1.0 - math.pow(x, (1.0 / n))
	elif x <= 0.95:
		tmp = (x - math.log(x)) / n
	else:
		tmp = ((1.0 / n) / x) - (0.5 / (n * (x * x)))
	return tmp

Julia

function code(x, n)
	tmp = 0.0
	if (x <= 3.3e-177)
		tmp = Float64(Float64(-log(x)) / n);
	elseif (x <= 1.25e-155)
		tmp = Float64(1.0 - (x ^ Float64(1.0 / n)));
	elseif (x <= 0.95)
		tmp = Float64(Float64(x - log(x)) / n);
	else
		tmp = Float64(Float64(Float64(1.0 / n) / x) - Float64(0.5 / Float64(n * Float64(x * x))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, n)
	tmp = 0.0;
	if (x <= 3.3e-177)
		tmp = -log(x) / n;
	elseif (x <= 1.25e-155)
		tmp = 1.0 - (x ^ (1.0 / n));
	elseif (x <= 0.95)
		tmp = (x - log(x)) / n;
	else
		tmp = ((1.0 / n) / x) - (0.5 / (n * (x * x)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, n_] := If[LessEqual[x, 3.3e-177], N[((-N[Log[x], $MachinePrecision]) / n), $MachinePrecision], If[LessEqual[x, 1.25e-155], N[(1.0 - N[Power[x, N[(1.0 / n), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 0.95], N[(N[(x - N[Log[x], $MachinePrecision]), $MachinePrecision] / n), $MachinePrecision], N[(N[(N[(1.0 / n), $MachinePrecision] / x), $MachinePrecision] - N[(0.5 / N[(n * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if x < 3.3e-177

   1. Initial program 43.6%

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

   2. Taylor expanded in x around 0 43.6%

      \[\leadsto \color{blue}{1} - {x}^{\left(\frac{1}{n}\right)} \]

   3. Taylor expanded in n around inf 56.5%

      \[\leadsto \color{blue}{-1 \cdot \frac{\log x}{n}} \]
   4. Step-by-step derivation

      1. mul-1-neg 56.5%

         \[\leadsto \color{blue}{-\frac{\log x}{n}} \]

   5. Simplified 56.5%

      \[\leadsto \color{blue}{-\frac{\log x}{n}} \]

   if 3.3e-177 < x < 1.25e-155

   1. Initial program 71.1%

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

   2. Taylor expanded in x around 0 71.1%

      \[\leadsto \color{blue}{1} - {x}^{\left(\frac{1}{n}\right)} \]

   if 1.25e-155 < x < 0.94999999999999996

   1. Initial program 37.2%

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

   2. Taylor expanded in x around 0 39.4%

      \[\leadsto \color{blue}{\left(1 + \left({x}^{2} \cdot \left(0.5 \cdot \frac{1}{{n}^{2}} - 0.5 \cdot \frac{1}{n}\right) + \frac{x}{n}\right)\right)} - {x}^{\left(\frac{1}{n}\right)} \]
   3. Step-by-step derivation

      1. fma-def 39.4%

         \[\leadsto \left(1 + \color{blue}{\mathsf{fma}\left({x}^{2}, 0.5 \cdot \frac{1}{{n}^{2}} - 0.5 \cdot \frac{1}{n}, \frac{x}{n}\right)}\right) - {x}^{\left(\frac{1}{n}\right)} \]

      2. unpow2 39.4%

         \[\leadsto \left(1 + \mathsf{fma}\left(\color{blue}{x \cdot x}, 0.5 \cdot \frac{1}{{n}^{2}} - 0.5 \cdot \frac{1}{n}, \frac{x}{n}\right)\right) - {x}^{\left(\frac{1}{n}\right)} \]

      3. associate-*r/ 39.4%

         \[\leadsto \left(1 + \mathsf{fma}\left(x \cdot x, \color{blue}{\frac{0.5 \cdot 1}{{n}^{2}}} - 0.5 \cdot \frac{1}{n}, \frac{x}{n}\right)\right) - {x}^{\left(\frac{1}{n}\right)} \]

      4. metadata-eval 39.4%

         \[\leadsto \left(1 + \mathsf{fma}\left(x \cdot x, \frac{\color{blue}{0.5}}{{n}^{2}} - 0.5 \cdot \frac{1}{n}, \frac{x}{n}\right)\right) - {x}^{\left(\frac{1}{n}\right)} \]

      5. unpow2 39.4%

         \[\leadsto \left(1 + \mathsf{fma}\left(x \cdot x, \frac{0.5}{\color{blue}{n \cdot n}} - 0.5 \cdot \frac{1}{n}, \frac{x}{n}\right)\right) - {x}^{\left(\frac{1}{n}\right)} \]

      6. associate-*r/ 39.4%

         \[\leadsto \left(1 + \mathsf{fma}\left(x \cdot x, \frac{0.5}{n \cdot n} - \color{blue}{\frac{0.5 \cdot 1}{n}}, \frac{x}{n}\right)\right) - {x}^{\left(\frac{1}{n}\right)} \]

      7. metadata-eval 39.4%

         \[\leadsto \left(1 + \mathsf{fma}\left(x \cdot x, \frac{0.5}{n \cdot n} - \frac{\color{blue}{0.5}}{n}, \frac{x}{n}\right)\right) - {x}^{\left(\frac{1}{n}\right)} \]

   4. Simplified 39.4%

      \[\leadsto \color{blue}{\left(1 + \mathsf{fma}\left(x \cdot x, \frac{0.5}{n \cdot n} - \frac{0.5}{n}, \frac{x}{n}\right)\right)} - {x}^{\left(\frac{1}{n}\right)} \]

   5. Taylor expanded in n around inf 52.3%

      \[\leadsto \color{blue}{\frac{\left(x + -0.5 \cdot {x}^{2}\right) - \log x}{n}} \]
   6. Step-by-step derivation

      1. associate--l+ 52.3%

         \[\leadsto \frac{\color{blue}{x + \left(-0.5 \cdot {x}^{2} - \log x\right)}}{n} \]

      2. *-commutative 52.3%

         \[\leadsto \frac{x + \left(\color{blue}{{x}^{2} \cdot -0.5} - \log x\right)}{n} \]

      3. unpow2 52.3%

         \[\leadsto \frac{x + \left(\color{blue}{\left(x \cdot x\right)} \cdot -0.5 - \log x\right)}{n} \]

   7. Simplified 52.3%

      \[\leadsto \color{blue}{\frac{x + \left(\left(x \cdot x\right) \cdot -0.5 - \log x\right)}{n}} \]

   8. Taylor expanded in x around 0 52.2%

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

      1. mul-1-neg 52.2%

         \[\leadsto \frac{x + \color{blue}{\left(-\log x\right)}}{n} \]

      2. unsub-neg 52.2%

         \[\leadsto \frac{\color{blue}{x - \log x}}{n} \]

   10. Simplified 52.2%

       \[\leadsto \frac{\color{blue}{x - \log x}}{n} \]

   if 0.94999999999999996 < x

   1. Initial program 67.3%

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

   2. Taylor expanded in n around inf 67.5%

      \[\leadsto \color{blue}{\frac{\log \left(1 + x\right) - \log x}{n}} \]
   3. Step-by-step derivation

      1. +-rgt-identity 67.5%

         \[\leadsto \frac{\color{blue}{\left(\log \left(1 + x\right) + 0\right)} - \log x}{n} \]

      2. +-rgt-identity 67.5%

         \[\leadsto \frac{\color{blue}{\log \left(1 + x\right)} - \log x}{n} \]

      3. log1p-def 67.5%

         \[\leadsto \frac{\color{blue}{\mathsf{log1p}\left(x\right)} - \log x}{n} \]

   4. Simplified 67.5%

      \[\leadsto \color{blue}{\frac{\mathsf{log1p}\left(x\right) - \log x}{n}} \]

   5. Taylor expanded in x around inf 63.0%

      \[\leadsto \color{blue}{\frac{1}{n \cdot x} - 0.5 \cdot \frac{1}{n \cdot {x}^{2}}} \]
   6. Step-by-step derivation

      1. associate-/r* 64.2%

         \[\leadsto \color{blue}{\frac{\frac{1}{n}}{x}} - 0.5 \cdot \frac{1}{n \cdot {x}^{2}} \]

      2. associate-*r/ 64.2%

         \[\leadsto \frac{\frac{1}{n}}{x} - \color{blue}{\frac{0.5 \cdot 1}{n \cdot {x}^{2}}} \]

      3. metadata-eval 64.2%

         \[\leadsto \frac{\frac{1}{n}}{x} - \frac{\color{blue}{0.5}}{n \cdot {x}^{2}} \]

      4. unpow2 64.2%

         \[\leadsto \frac{\frac{1}{n}}{x} - \frac{0.5}{n \cdot \color{blue}{\left(x \cdot x\right)}} \]

   7. Simplified 64.2%

      \[\leadsto \color{blue}{\frac{\frac{1}{n}}{x} - \frac{0.5}{n \cdot \left(x \cdot x\right)}} \]
3. Recombined 4 regimes into one program.

4. Final simplification 58.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 3.3 \cdot 10^{-177}:\\ \;\;\;\;\frac{-\log x}{n}\\ \mathbf{elif}\;x \leq 1.25 \cdot 10^{-155}:\\ \;\;\;\;1 - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{elif}\;x \leq 0.95:\\ \;\;\;\;\frac{x - \log x}{n}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{n}}{x} - \frac{0.5}{n \cdot \left(x \cdot x\right)}\\ \end{array} \]

Alternative 14: 57.3% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 0.98:\\ \;\;\;\;\frac{x - \log x}{n}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{n}}{x} - \frac{0.5}{n \cdot \left(x \cdot x\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (if (<= x 0.98)
   (/ (- x (log x)) n)
   (- (/ (/ 1.0 n) x) (/ 0.5 (* n (* x x))))))

C

double code(double x, double n) {
	double tmp;
	if (x <= 0.98) {
		tmp = (x - log(x)) / n;
	} else {
		tmp = ((1.0 / n) / x) - (0.5 / (n * (x * x)));
	}
	return tmp;
}

Fortran

real(8) function code(x, n)
    real(8), intent (in) :: x
    real(8), intent (in) :: n
    real(8) :: tmp
    if (x <= 0.98d0) then
        tmp = (x - log(x)) / n
    else
        tmp = ((1.0d0 / n) / x) - (0.5d0 / (n * (x * x)))
    end if
    code = tmp
end function

Java

public static double code(double x, double n) {
	double tmp;
	if (x <= 0.98) {
		tmp = (x - Math.log(x)) / n;
	} else {
		tmp = ((1.0 / n) / x) - (0.5 / (n * (x * x)));
	}
	return tmp;
}

Python

def code(x, n):
	tmp = 0
	if x <= 0.98:
		tmp = (x - math.log(x)) / n
	else:
		tmp = ((1.0 / n) / x) - (0.5 / (n * (x * x)))
	return tmp

Julia

function code(x, n)
	tmp = 0.0
	if (x <= 0.98)
		tmp = Float64(Float64(x - log(x)) / n);
	else
		tmp = Float64(Float64(Float64(1.0 / n) / x) - Float64(0.5 / Float64(n * Float64(x * x))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, n)
	tmp = 0.0;
	if (x <= 0.98)
		tmp = (x - log(x)) / n;
	else
		tmp = ((1.0 / n) / x) - (0.5 / (n * (x * x)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, n_] := If[LessEqual[x, 0.98], N[(N[(x - N[Log[x], $MachinePrecision]), $MachinePrecision] / n), $MachinePrecision], N[(N[(N[(1.0 / n), $MachinePrecision] / x), $MachinePrecision] - N[(0.5 / N[(n * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 0.97999999999999998

   1. Initial program 42.1%

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

   2. Taylor expanded in x around 0 34.3%

      \[\leadsto \color{blue}{\left(1 + \left({x}^{2} \cdot \left(0.5 \cdot \frac{1}{{n}^{2}} - 0.5 \cdot \frac{1}{n}\right) + \frac{x}{n}\right)\right)} - {x}^{\left(\frac{1}{n}\right)} \]
   3. Step-by-step derivation

      1. fma-def 34.3%

         \[\leadsto \left(1 + \color{blue}{\mathsf{fma}\left({x}^{2}, 0.5 \cdot \frac{1}{{n}^{2}} - 0.5 \cdot \frac{1}{n}, \frac{x}{n}\right)}\right) - {x}^{\left(\frac{1}{n}\right)} \]

      2. unpow2 34.3%

         \[\leadsto \left(1 + \mathsf{fma}\left(\color{blue}{x \cdot x}, 0.5 \cdot \frac{1}{{n}^{2}} - 0.5 \cdot \frac{1}{n}, \frac{x}{n}\right)\right) - {x}^{\left(\frac{1}{n}\right)} \]

      3. associate-*r/ 34.3%

         \[\leadsto \left(1 + \mathsf{fma}\left(x \cdot x, \color{blue}{\frac{0.5 \cdot 1}{{n}^{2}}} - 0.5 \cdot \frac{1}{n}, \frac{x}{n}\right)\right) - {x}^{\left(\frac{1}{n}\right)} \]

      4. metadata-eval 34.3%

         \[\leadsto \left(1 + \mathsf{fma}\left(x \cdot x, \frac{\color{blue}{0.5}}{{n}^{2}} - 0.5 \cdot \frac{1}{n}, \frac{x}{n}\right)\right) - {x}^{\left(\frac{1}{n}\right)} \]

      5. unpow2 34.3%

         \[\leadsto \left(1 + \mathsf{fma}\left(x \cdot x, \frac{0.5}{\color{blue}{n \cdot n}} - 0.5 \cdot \frac{1}{n}, \frac{x}{n}\right)\right) - {x}^{\left(\frac{1}{n}\right)} \]

      6. associate-*r/ 34.3%

         \[\leadsto \left(1 + \mathsf{fma}\left(x \cdot x, \frac{0.5}{n \cdot n} - \color{blue}{\frac{0.5 \cdot 1}{n}}, \frac{x}{n}\right)\right) - {x}^{\left(\frac{1}{n}\right)} \]

      7. metadata-eval 34.3%

         \[\leadsto \left(1 + \mathsf{fma}\left(x \cdot x, \frac{0.5}{n \cdot n} - \frac{\color{blue}{0.5}}{n}, \frac{x}{n}\right)\right) - {x}^{\left(\frac{1}{n}\right)} \]

   4. Simplified 34.3%

      \[\leadsto \color{blue}{\left(1 + \mathsf{fma}\left(x \cdot x, \frac{0.5}{n \cdot n} - \frac{0.5}{n}, \frac{x}{n}\right)\right)} - {x}^{\left(\frac{1}{n}\right)} \]

   5. Taylor expanded in n around inf 52.1%

      \[\leadsto \color{blue}{\frac{\left(x + -0.5 \cdot {x}^{2}\right) - \log x}{n}} \]
   6. Step-by-step derivation

      1. associate--l+ 52.1%

         \[\leadsto \frac{\color{blue}{x + \left(-0.5 \cdot {x}^{2} - \log x\right)}}{n} \]

      2. *-commutative 52.1%

         \[\leadsto \frac{x + \left(\color{blue}{{x}^{2} \cdot -0.5} - \log x\right)}{n} \]

      3. unpow2 52.1%

         \[\leadsto \frac{x + \left(\color{blue}{\left(x \cdot x\right)} \cdot -0.5 - \log x\right)}{n} \]

   7. Simplified 52.1%

      \[\leadsto \color{blue}{\frac{x + \left(\left(x \cdot x\right) \cdot -0.5 - \log x\right)}{n}} \]

   8. Taylor expanded in x around 0 52.0%

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

      1. mul-1-neg 52.0%

         \[\leadsto \frac{x + \color{blue}{\left(-\log x\right)}}{n} \]

      2. unsub-neg 52.0%

         \[\leadsto \frac{\color{blue}{x - \log x}}{n} \]

   10. Simplified 52.0%

       \[\leadsto \frac{\color{blue}{x - \log x}}{n} \]

   if 0.97999999999999998 < x

   1. Initial program 67.3%

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

   2. Taylor expanded in n around inf 67.5%

      \[\leadsto \color{blue}{\frac{\log \left(1 + x\right) - \log x}{n}} \]
   3. Step-by-step derivation

      1. +-rgt-identity 67.5%

         \[\leadsto \frac{\color{blue}{\left(\log \left(1 + x\right) + 0\right)} - \log x}{n} \]

      2. +-rgt-identity 67.5%

         \[\leadsto \frac{\color{blue}{\log \left(1 + x\right)} - \log x}{n} \]

      3. log1p-def 67.5%

         \[\leadsto \frac{\color{blue}{\mathsf{log1p}\left(x\right)} - \log x}{n} \]

   4. Simplified 67.5%

      \[\leadsto \color{blue}{\frac{\mathsf{log1p}\left(x\right) - \log x}{n}} \]

   5. Taylor expanded in x around inf 63.0%

      \[\leadsto \color{blue}{\frac{1}{n \cdot x} - 0.5 \cdot \frac{1}{n \cdot {x}^{2}}} \]
   6. Step-by-step derivation

      1. associate-/r* 64.2%

         \[\leadsto \color{blue}{\frac{\frac{1}{n}}{x}} - 0.5 \cdot \frac{1}{n \cdot {x}^{2}} \]

      2. associate-*r/ 64.2%

         \[\leadsto \frac{\frac{1}{n}}{x} - \color{blue}{\frac{0.5 \cdot 1}{n \cdot {x}^{2}}} \]

      3. metadata-eval 64.2%

         \[\leadsto \frac{\frac{1}{n}}{x} - \frac{\color{blue}{0.5}}{n \cdot {x}^{2}} \]

      4. unpow2 64.2%

         \[\leadsto \frac{\frac{1}{n}}{x} - \frac{0.5}{n \cdot \color{blue}{\left(x \cdot x\right)}} \]

   7. Simplified 64.2%

      \[\leadsto \color{blue}{\frac{\frac{1}{n}}{x} - \frac{0.5}{n \cdot \left(x \cdot x\right)}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 57.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 0.98:\\ \;\;\;\;\frac{x - \log x}{n}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{n}}{x} - \frac{0.5}{n \cdot \left(x \cdot x\right)}\\ \end{array} \]

Alternative 15: 57.1% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 0.66:\\ \;\;\;\;\frac{-\log x}{n}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{n}}{x} - \frac{0.5}{n \cdot \left(x \cdot x\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (if (<= x 0.66) (/ (- (log x)) n) (- (/ (/ 1.0 n) x) (/ 0.5 (* n (* x x))))))

C

double code(double x, double n) {
	double tmp;
	if (x <= 0.66) {
		tmp = -log(x) / n;
	} else {
		tmp = ((1.0 / n) / x) - (0.5 / (n * (x * x)));
	}
	return tmp;
}

Fortran

real(8) function code(x, n)
    real(8), intent (in) :: x
    real(8), intent (in) :: n
    real(8) :: tmp
    if (x <= 0.66d0) then
        tmp = -log(x) / n
    else
        tmp = ((1.0d0 / n) / x) - (0.5d0 / (n * (x * x)))
    end if
    code = tmp
end function

Java

public static double code(double x, double n) {
	double tmp;
	if (x <= 0.66) {
		tmp = -Math.log(x) / n;
	} else {
		tmp = ((1.0 / n) / x) - (0.5 / (n * (x * x)));
	}
	return tmp;
}

Python

def code(x, n):
	tmp = 0
	if x <= 0.66:
		tmp = -math.log(x) / n
	else:
		tmp = ((1.0 / n) / x) - (0.5 / (n * (x * x)))
	return tmp

Julia

function code(x, n)
	tmp = 0.0
	if (x <= 0.66)
		tmp = Float64(Float64(-log(x)) / n);
	else
		tmp = Float64(Float64(Float64(1.0 / n) / x) - Float64(0.5 / Float64(n * Float64(x * x))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, n)
	tmp = 0.0;
	if (x <= 0.66)
		tmp = -log(x) / n;
	else
		tmp = ((1.0 / n) / x) - (0.5 / (n * (x * x)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, n_] := If[LessEqual[x, 0.66], N[((-N[Log[x], $MachinePrecision]) / n), $MachinePrecision], N[(N[(N[(1.0 / n), $MachinePrecision] / x), $MachinePrecision] - N[(0.5 / N[(n * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 0.660000000000000031

   1. Initial program 41.7%

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

   2. Taylor expanded in x around 0 38.5%

      \[\leadsto \color{blue}{1} - {x}^{\left(\frac{1}{n}\right)} \]

   3. Taylor expanded in n around inf 51.7%

      \[\leadsto \color{blue}{-1 \cdot \frac{\log x}{n}} \]
   4. Step-by-step derivation

      1. mul-1-neg 51.7%

         \[\leadsto \color{blue}{-\frac{\log x}{n}} \]

   5. Simplified 51.7%

      \[\leadsto \color{blue}{-\frac{\log x}{n}} \]

   if 0.660000000000000031 < x

   1. Initial program 67.6%

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

   2. Taylor expanded in n around inf 66.9%

      \[\leadsto \color{blue}{\frac{\log \left(1 + x\right) - \log x}{n}} \]
   3. Step-by-step derivation

      1. +-rgt-identity 66.9%

         \[\leadsto \frac{\color{blue}{\left(\log \left(1 + x\right) + 0\right)} - \log x}{n} \]

      2. +-rgt-identity 66.9%

         \[\leadsto \frac{\color{blue}{\log \left(1 + x\right)} - \log x}{n} \]

      3. log1p-def 66.9%

         \[\leadsto \frac{\color{blue}{\mathsf{log1p}\left(x\right)} - \log x}{n} \]

   4. Simplified 66.9%

      \[\leadsto \color{blue}{\frac{\mathsf{log1p}\left(x\right) - \log x}{n}} \]

   5. Taylor expanded in x around inf 62.4%

      \[\leadsto \color{blue}{\frac{1}{n \cdot x} - 0.5 \cdot \frac{1}{n \cdot {x}^{2}}} \]
   6. Step-by-step derivation

      1. associate-/r* 63.6%

         \[\leadsto \color{blue}{\frac{\frac{1}{n}}{x}} - 0.5 \cdot \frac{1}{n \cdot {x}^{2}} \]

      2. associate-*r/ 63.6%

         \[\leadsto \frac{\frac{1}{n}}{x} - \color{blue}{\frac{0.5 \cdot 1}{n \cdot {x}^{2}}} \]

      3. metadata-eval 63.6%

         \[\leadsto \frac{\frac{1}{n}}{x} - \frac{\color{blue}{0.5}}{n \cdot {x}^{2}} \]

      4. unpow2 63.6%

         \[\leadsto \frac{\frac{1}{n}}{x} - \frac{0.5}{n \cdot \color{blue}{\left(x \cdot x\right)}} \]

   7. Simplified 63.6%

      \[\leadsto \color{blue}{\frac{\frac{1}{n}}{x} - \frac{0.5}{n \cdot \left(x \cdot x\right)}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 56.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 0.66:\\ \;\;\;\;\frac{-\log x}{n}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{n}}{x} - \frac{0.5}{n \cdot \left(x \cdot x\right)}\\ \end{array} \]

Alternative 16: 41.8% accurate, 12.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{1}{n} \leq 2 \cdot 10^{+74}:\\ \;\;\;\;\frac{\frac{1}{n}}{x}\\ \mathbf{elif}\;\frac{1}{n} \leq 2 \cdot 10^{+214}:\\ \;\;\;\;0.5 \cdot \frac{x \cdot x}{n \cdot n}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{n \cdot x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (if (<= (/ 1.0 n) 2e+74)
   (/ (/ 1.0 n) x)
   (if (<= (/ 1.0 n) 2e+214) (* 0.5 (/ (* x x) (* n n))) (/ 1.0 (* n x)))))

C

double code(double x, double n) {
	double tmp;
	if ((1.0 / n) <= 2e+74) {
		tmp = (1.0 / n) / x;
	} else if ((1.0 / n) <= 2e+214) {
		tmp = 0.5 * ((x * x) / (n * n));
	} else {
		tmp = 1.0 / (n * x);
	}
	return tmp;
}

Fortran

real(8) function code(x, n)
    real(8), intent (in) :: x
    real(8), intent (in) :: n
    real(8) :: tmp
    if ((1.0d0 / n) <= 2d+74) then
        tmp = (1.0d0 / n) / x
    else if ((1.0d0 / n) <= 2d+214) then
        tmp = 0.5d0 * ((x * x) / (n * n))
    else
        tmp = 1.0d0 / (n * x)
    end if
    code = tmp
end function

Java

public static double code(double x, double n) {
	double tmp;
	if ((1.0 / n) <= 2e+74) {
		tmp = (1.0 / n) / x;
	} else if ((1.0 / n) <= 2e+214) {
		tmp = 0.5 * ((x * x) / (n * n));
	} else {
		tmp = 1.0 / (n * x);
	}
	return tmp;
}

Python

def code(x, n):
	tmp = 0
	if (1.0 / n) <= 2e+74:
		tmp = (1.0 / n) / x
	elif (1.0 / n) <= 2e+214:
		tmp = 0.5 * ((x * x) / (n * n))
	else:
		tmp = 1.0 / (n * x)
	return tmp

Julia

function code(x, n)
	tmp = 0.0
	if (Float64(1.0 / n) <= 2e+74)
		tmp = Float64(Float64(1.0 / n) / x);
	elseif (Float64(1.0 / n) <= 2e+214)
		tmp = Float64(0.5 * Float64(Float64(x * x) / Float64(n * n)));
	else
		tmp = Float64(1.0 / Float64(n * x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, n)
	tmp = 0.0;
	if ((1.0 / n) <= 2e+74)
		tmp = (1.0 / n) / x;
	elseif ((1.0 / n) <= 2e+214)
		tmp = 0.5 * ((x * x) / (n * n));
	else
		tmp = 1.0 / (n * x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, n_] := If[LessEqual[N[(1.0 / n), $MachinePrecision], 2e+74], N[(N[(1.0 / n), $MachinePrecision] / x), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 2e+214], N[(0.5 * N[(N[(x * x), $MachinePrecision] / N[(n * n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(n * x), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 1 n) < 1.9999999999999999e74

   1. Initial program 52.6%

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

   2. Taylor expanded in n around inf 66.4%

      \[\leadsto \color{blue}{\frac{\log \left(1 + x\right) - \log x}{n}} \]
   3. Step-by-step derivation

      1. +-rgt-identity 66.4%

         \[\leadsto \frac{\color{blue}{\left(\log \left(1 + x\right) + 0\right)} - \log x}{n} \]

      2. +-rgt-identity 66.4%

         \[\leadsto \frac{\color{blue}{\log \left(1 + x\right)} - \log x}{n} \]

      3. log1p-def 66.4%

         \[\leadsto \frac{\color{blue}{\mathsf{log1p}\left(x\right)} - \log x}{n} \]

   4. Simplified 66.4%

      \[\leadsto \color{blue}{\frac{\mathsf{log1p}\left(x\right) - \log x}{n}} \]

   5. Taylor expanded in x around inf 39.1%

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

      1. associate-/r* 39.7%

         \[\leadsto \color{blue}{\frac{\frac{1}{n}}{x}} \]

   7. Simplified 39.7%

      \[\leadsto \color{blue}{\frac{\frac{1}{n}}{x}} \]

   if 1.9999999999999999e74 < (/.f64 1 n) < 1.9999999999999999e214

   1. Initial program 61.5%

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

   2. Taylor expanded in x around 0 69.2%

      \[\leadsto \color{blue}{\left(1 + \left({x}^{2} \cdot \left(0.5 \cdot \frac{1}{{n}^{2}} - 0.5 \cdot \frac{1}{n}\right) + \frac{x}{n}\right)\right)} - {x}^{\left(\frac{1}{n}\right)} \]
   3. Step-by-step derivation

      1. fma-def 69.2%

         \[\leadsto \left(1 + \color{blue}{\mathsf{fma}\left({x}^{2}, 0.5 \cdot \frac{1}{{n}^{2}} - 0.5 \cdot \frac{1}{n}, \frac{x}{n}\right)}\right) - {x}^{\left(\frac{1}{n}\right)} \]

      2. unpow2 69.2%

         \[\leadsto \left(1 + \mathsf{fma}\left(\color{blue}{x \cdot x}, 0.5 \cdot \frac{1}{{n}^{2}} - 0.5 \cdot \frac{1}{n}, \frac{x}{n}\right)\right) - {x}^{\left(\frac{1}{n}\right)} \]

      3. associate-*r/ 69.2%

         \[\leadsto \left(1 + \mathsf{fma}\left(x \cdot x, \color{blue}{\frac{0.5 \cdot 1}{{n}^{2}}} - 0.5 \cdot \frac{1}{n}, \frac{x}{n}\right)\right) - {x}^{\left(\frac{1}{n}\right)} \]

      4. metadata-eval 69.2%

         \[\leadsto \left(1 + \mathsf{fma}\left(x \cdot x, \frac{\color{blue}{0.5}}{{n}^{2}} - 0.5 \cdot \frac{1}{n}, \frac{x}{n}\right)\right) - {x}^{\left(\frac{1}{n}\right)} \]

      5. unpow2 69.2%

         \[\leadsto \left(1 + \mathsf{fma}\left(x \cdot x, \frac{0.5}{\color{blue}{n \cdot n}} - 0.5 \cdot \frac{1}{n}, \frac{x}{n}\right)\right) - {x}^{\left(\frac{1}{n}\right)} \]

      6. associate-*r/ 69.2%

         \[\leadsto \left(1 + \mathsf{fma}\left(x \cdot x, \frac{0.5}{n \cdot n} - \color{blue}{\frac{0.5 \cdot 1}{n}}, \frac{x}{n}\right)\right) - {x}^{\left(\frac{1}{n}\right)} \]

      7. metadata-eval 69.2%

         \[\leadsto \left(1 + \mathsf{fma}\left(x \cdot x, \frac{0.5}{n \cdot n} - \frac{\color{blue}{0.5}}{n}, \frac{x}{n}\right)\right) - {x}^{\left(\frac{1}{n}\right)} \]

   4. Simplified 69.2%

      \[\leadsto \color{blue}{\left(1 + \mathsf{fma}\left(x \cdot x, \frac{0.5}{n \cdot n} - \frac{0.5}{n}, \frac{x}{n}\right)\right)} - {x}^{\left(\frac{1}{n}\right)} \]

   5. Taylor expanded in n around 0 34.7%

      \[\leadsto \color{blue}{0.5 \cdot \frac{{x}^{2}}{{n}^{2}}} \]
   6. Step-by-step derivation

      1. unpow2 34.7%

         \[\leadsto 0.5 \cdot \frac{\color{blue}{x \cdot x}}{{n}^{2}} \]

      2. unpow2 34.7%

         \[\leadsto 0.5 \cdot \frac{x \cdot x}{\color{blue}{n \cdot n}} \]

   7. Simplified 34.7%

      \[\leadsto \color{blue}{0.5 \cdot \frac{x \cdot x}{n \cdot n}} \]

   if 1.9999999999999999e214 < (/.f64 1 n)

   1. Initial program 19.7%

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

   2. Taylor expanded in n around inf 8.2%

      \[\leadsto \color{blue}{\frac{\log \left(1 + x\right) - \log x}{n}} \]
   3. Step-by-step derivation

      1. +-rgt-identity 8.2%

         \[\leadsto \frac{\color{blue}{\left(\log \left(1 + x\right) + 0\right)} - \log x}{n} \]

      2. +-rgt-identity 8.2%

         \[\leadsto \frac{\color{blue}{\log \left(1 + x\right)} - \log x}{n} \]

      3. log1p-def 8.2%

         \[\leadsto \frac{\color{blue}{\mathsf{log1p}\left(x\right)} - \log x}{n} \]

   4. Simplified 8.2%

      \[\leadsto \color{blue}{\frac{\mathsf{log1p}\left(x\right) - \log x}{n}} \]

   5. Taylor expanded in x around inf 89.2%

      \[\leadsto \color{blue}{\frac{1}{n \cdot x}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 41.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{1}{n} \leq 2 \cdot 10^{+74}:\\ \;\;\;\;\frac{\frac{1}{n}}{x}\\ \mathbf{elif}\;\frac{1}{n} \leq 2 \cdot 10^{+214}:\\ \;\;\;\;0.5 \cdot \frac{x \cdot x}{n \cdot n}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{n \cdot x}\\ \end{array} \]

Alternative 17: 42.8% accurate, 16.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{1}{n} \leq 2 \cdot 10^{+48}:\\ \;\;\;\;\frac{\frac{1}{n}}{x}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(x \cdot \frac{0.5}{n \cdot n}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (if (<= (/ 1.0 n) 2e+48) (/ (/ 1.0 n) x) (* x (* x (/ 0.5 (* n n))))))

C

double code(double x, double n) {
	double tmp;
	if ((1.0 / n) <= 2e+48) {
		tmp = (1.0 / n) / x;
	} else {
		tmp = x * (x * (0.5 / (n * n)));
	}
	return tmp;
}

Fortran

real(8) function code(x, n)
    real(8), intent (in) :: x
    real(8), intent (in) :: n
    real(8) :: tmp
    if ((1.0d0 / n) <= 2d+48) then
        tmp = (1.0d0 / n) / x
    else
        tmp = x * (x * (0.5d0 / (n * n)))
    end if
    code = tmp
end function

Java

public static double code(double x, double n) {
	double tmp;
	if ((1.0 / n) <= 2e+48) {
		tmp = (1.0 / n) / x;
	} else {
		tmp = x * (x * (0.5 / (n * n)));
	}
	return tmp;
}

Python

def code(x, n):
	tmp = 0
	if (1.0 / n) <= 2e+48:
		tmp = (1.0 / n) / x
	else:
		tmp = x * (x * (0.5 / (n * n)))
	return tmp

Julia

function code(x, n)
	tmp = 0.0
	if (Float64(1.0 / n) <= 2e+48)
		tmp = Float64(Float64(1.0 / n) / x);
	else
		tmp = Float64(x * Float64(x * Float64(0.5 / Float64(n * n))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, n)
	tmp = 0.0;
	if ((1.0 / n) <= 2e+48)
		tmp = (1.0 / n) / x;
	else
		tmp = x * (x * (0.5 / (n * n)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, n_] := If[LessEqual[N[(1.0 / n), $MachinePrecision], 2e+48], N[(N[(1.0 / n), $MachinePrecision] / x), $MachinePrecision], N[(x * N[(x * N[(0.5 / N[(n * n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 1 n) < 2.00000000000000009e48

   1. Initial program 51.9%

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

   2. Taylor expanded in n around inf 67.2%

      \[\leadsto \color{blue}{\frac{\log \left(1 + x\right) - \log x}{n}} \]
   3. Step-by-step derivation

      1. +-rgt-identity 67.2%

         \[\leadsto \frac{\color{blue}{\left(\log \left(1 + x\right) + 0\right)} - \log x}{n} \]

      2. +-rgt-identity 67.2%

         \[\leadsto \frac{\color{blue}{\log \left(1 + x\right)} - \log x}{n} \]

      3. log1p-def 67.2%

         \[\leadsto \frac{\color{blue}{\mathsf{log1p}\left(x\right)} - \log x}{n} \]

   4. Simplified 67.2%

      \[\leadsto \color{blue}{\frac{\mathsf{log1p}\left(x\right) - \log x}{n}} \]

   5. Taylor expanded in x around inf 39.6%

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

      1. associate-/r* 40.2%

         \[\leadsto \color{blue}{\frac{\frac{1}{n}}{x}} \]

   7. Simplified 40.2%

      \[\leadsto \color{blue}{\frac{\frac{1}{n}}{x}} \]

   if 2.00000000000000009e48 < (/.f64 1 n)

   1. Initial program 54.5%

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

   2. Taylor expanded in x around 0 68.4%

      \[\leadsto \color{blue}{\left(1 + \left({x}^{2} \cdot \left(0.5 \cdot \frac{1}{{n}^{2}} - 0.5 \cdot \frac{1}{n}\right) + \frac{x}{n}\right)\right)} - {x}^{\left(\frac{1}{n}\right)} \]
   3. Step-by-step derivation

      1. fma-def 68.4%

         \[\leadsto \left(1 + \color{blue}{\mathsf{fma}\left({x}^{2}, 0.5 \cdot \frac{1}{{n}^{2}} - 0.5 \cdot \frac{1}{n}, \frac{x}{n}\right)}\right) - {x}^{\left(\frac{1}{n}\right)} \]

      2. unpow2 68.4%

         \[\leadsto \left(1 + \mathsf{fma}\left(\color{blue}{x \cdot x}, 0.5 \cdot \frac{1}{{n}^{2}} - 0.5 \cdot \frac{1}{n}, \frac{x}{n}\right)\right) - {x}^{\left(\frac{1}{n}\right)} \]

      3. associate-*r/ 68.4%

         \[\leadsto \left(1 + \mathsf{fma}\left(x \cdot x, \color{blue}{\frac{0.5 \cdot 1}{{n}^{2}}} - 0.5 \cdot \frac{1}{n}, \frac{x}{n}\right)\right) - {x}^{\left(\frac{1}{n}\right)} \]

      4. metadata-eval 68.4%

         \[\leadsto \left(1 + \mathsf{fma}\left(x \cdot x, \frac{\color{blue}{0.5}}{{n}^{2}} - 0.5 \cdot \frac{1}{n}, \frac{x}{n}\right)\right) - {x}^{\left(\frac{1}{n}\right)} \]

      5. unpow2 68.4%

         \[\leadsto \left(1 + \mathsf{fma}\left(x \cdot x, \frac{0.5}{\color{blue}{n \cdot n}} - 0.5 \cdot \frac{1}{n}, \frac{x}{n}\right)\right) - {x}^{\left(\frac{1}{n}\right)} \]

      6. associate-*r/ 68.4%

         \[\leadsto \left(1 + \mathsf{fma}\left(x \cdot x, \frac{0.5}{n \cdot n} - \color{blue}{\frac{0.5 \cdot 1}{n}}, \frac{x}{n}\right)\right) - {x}^{\left(\frac{1}{n}\right)} \]

      7. metadata-eval 68.4%

         \[\leadsto \left(1 + \mathsf{fma}\left(x \cdot x, \frac{0.5}{n \cdot n} - \frac{\color{blue}{0.5}}{n}, \frac{x}{n}\right)\right) - {x}^{\left(\frac{1}{n}\right)} \]

   4. Simplified 68.4%

      \[\leadsto \color{blue}{\left(1 + \mathsf{fma}\left(x \cdot x, \frac{0.5}{n \cdot n} - \frac{0.5}{n}, \frac{x}{n}\right)\right)} - {x}^{\left(\frac{1}{n}\right)} \]

   5. Taylor expanded in n around 0 37.2%

      \[\leadsto \color{blue}{0.5 \cdot \frac{{x}^{2}}{{n}^{2}}} \]
   6. Step-by-step derivation

      1. unpow2 37.2%

         \[\leadsto 0.5 \cdot \frac{\color{blue}{x \cdot x}}{{n}^{2}} \]

      2. unpow2 37.2%

         \[\leadsto 0.5 \cdot \frac{x \cdot x}{\color{blue}{n \cdot n}} \]

   7. Simplified 37.2%

      \[\leadsto \color{blue}{0.5 \cdot \frac{x \cdot x}{n \cdot n}} \]

   8. Taylor expanded in x around 0 37.2%

      \[\leadsto \color{blue}{0.5 \cdot \frac{{x}^{2}}{{n}^{2}}} \]
   9. Step-by-step derivation

      1. *-commutative 37.2%

         \[\leadsto \color{blue}{\frac{{x}^{2}}{{n}^{2}} \cdot 0.5} \]

      2. associate-*l/ 37.2%

         \[\leadsto \color{blue}{\frac{{x}^{2} \cdot 0.5}{{n}^{2}}} \]

      3. associate-*r/ 39.8%

         \[\leadsto \color{blue}{{x}^{2} \cdot \frac{0.5}{{n}^{2}}} \]

      4. unpow2 39.8%

         \[\leadsto {x}^{2} \cdot \frac{0.5}{\color{blue}{n \cdot n}} \]

      5. unpow2 39.8%

         \[\leadsto \color{blue}{\left(x \cdot x\right)} \cdot \frac{0.5}{n \cdot n} \]

      6. associate-*l* 51.0%

         \[\leadsto \color{blue}{x \cdot \left(x \cdot \frac{0.5}{n \cdot n}\right)} \]

   10. Simplified 51.0%

       \[\leadsto \color{blue}{x \cdot \left(x \cdot \frac{0.5}{n \cdot n}\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 41.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{1}{n} \leq 2 \cdot 10^{+48}:\\ \;\;\;\;\frac{\frac{1}{n}}{x}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(x \cdot \frac{0.5}{n \cdot n}\right)\\ \end{array} \]

Alternative 18: 40.6% accurate, 42.2× speedup

Math

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

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(1.0 / Float64(n * x))
end

MATLAB

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

Wolfram

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

Derivation

1. Initial program 52.3%

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

2. Taylor expanded in n around inf 58.4%

   \[\leadsto \color{blue}{\frac{\log \left(1 + x\right) - \log x}{n}} \]
3. Step-by-step derivation

   1. +-rgt-identity 58.4%

      \[\leadsto \frac{\color{blue}{\left(\log \left(1 + x\right) + 0\right)} - \log x}{n} \]

   2. +-rgt-identity 58.4%

      \[\leadsto \frac{\color{blue}{\log \left(1 + x\right)} - \log x}{n} \]

   3. log1p-def 58.4%

      \[\leadsto \frac{\color{blue}{\mathsf{log1p}\left(x\right)} - \log x}{n} \]

4. Simplified 58.4%

   \[\leadsto \color{blue}{\frac{\mathsf{log1p}\left(x\right) - \log x}{n}} \]

5. Taylor expanded in x around inf 38.3%

   \[\leadsto \color{blue}{\frac{1}{n \cdot x}} \]

6. Final simplification 38.3%

   \[\leadsto \frac{1}{n \cdot x} \]

Alternative 19: 41.2% 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

1. Initial program 52.3%

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

2. Taylor expanded in n around inf 58.4%

   \[\leadsto \color{blue}{\frac{\log \left(1 + x\right) - \log x}{n}} \]
3. Step-by-step derivation

   1. +-rgt-identity 58.4%

      \[\leadsto \frac{\color{blue}{\left(\log \left(1 + x\right) + 0\right)} - \log x}{n} \]

   2. +-rgt-identity 58.4%

      \[\leadsto \frac{\color{blue}{\log \left(1 + x\right)} - \log x}{n} \]

   3. log1p-def 58.4%

      \[\leadsto \frac{\color{blue}{\mathsf{log1p}\left(x\right)} - \log x}{n} \]

4. Simplified 58.4%

   \[\leadsto \color{blue}{\frac{\mathsf{log1p}\left(x\right) - \log x}{n}} \]

5. Taylor expanded in x around inf 38.3%

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

   1. associate-/r* 38.8%

      \[\leadsto \color{blue}{\frac{\frac{1}{n}}{x}} \]

7. Simplified 38.8%

   \[\leadsto \color{blue}{\frac{\frac{1}{n}}{x}} \]

8. Final simplification 38.8%

   \[\leadsto \frac{\frac{1}{n}}{x} \]

Alternative 20: 4.5% accurate, 70.3× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 52.3%

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

2. Taylor expanded in x around 0 21.6%

   \[\leadsto \color{blue}{\left(1 + \left({x}^{2} \cdot \left(0.5 \cdot \frac{1}{{n}^{2}} - 0.5 \cdot \frac{1}{n}\right) + \frac{x}{n}\right)\right)} - {x}^{\left(\frac{1}{n}\right)} \]
3. Step-by-step derivation

   1. fma-def 21.6%

      \[\leadsto \left(1 + \color{blue}{\mathsf{fma}\left({x}^{2}, 0.5 \cdot \frac{1}{{n}^{2}} - 0.5 \cdot \frac{1}{n}, \frac{x}{n}\right)}\right) - {x}^{\left(\frac{1}{n}\right)} \]

   2. unpow2 21.6%

      \[\leadsto \left(1 + \mathsf{fma}\left(\color{blue}{x \cdot x}, 0.5 \cdot \frac{1}{{n}^{2}} - 0.5 \cdot \frac{1}{n}, \frac{x}{n}\right)\right) - {x}^{\left(\frac{1}{n}\right)} \]

   3. associate-*r/ 21.6%

      \[\leadsto \left(1 + \mathsf{fma}\left(x \cdot x, \color{blue}{\frac{0.5 \cdot 1}{{n}^{2}}} - 0.5 \cdot \frac{1}{n}, \frac{x}{n}\right)\right) - {x}^{\left(\frac{1}{n}\right)} \]

   4. metadata-eval 21.6%

      \[\leadsto \left(1 + \mathsf{fma}\left(x \cdot x, \frac{\color{blue}{0.5}}{{n}^{2}} - 0.5 \cdot \frac{1}{n}, \frac{x}{n}\right)\right) - {x}^{\left(\frac{1}{n}\right)} \]

   5. unpow2 21.6%

      \[\leadsto \left(1 + \mathsf{fma}\left(x \cdot x, \frac{0.5}{\color{blue}{n \cdot n}} - 0.5 \cdot \frac{1}{n}, \frac{x}{n}\right)\right) - {x}^{\left(\frac{1}{n}\right)} \]

   6. associate-*r/ 21.6%

      \[\leadsto \left(1 + \mathsf{fma}\left(x \cdot x, \frac{0.5}{n \cdot n} - \color{blue}{\frac{0.5 \cdot 1}{n}}, \frac{x}{n}\right)\right) - {x}^{\left(\frac{1}{n}\right)} \]

   7. metadata-eval 21.6%

      \[\leadsto \left(1 + \mathsf{fma}\left(x \cdot x, \frac{0.5}{n \cdot n} - \frac{\color{blue}{0.5}}{n}, \frac{x}{n}\right)\right) - {x}^{\left(\frac{1}{n}\right)} \]

4. Simplified 21.6%

   \[\leadsto \color{blue}{\left(1 + \mathsf{fma}\left(x \cdot x, \frac{0.5}{n \cdot n} - \frac{0.5}{n}, \frac{x}{n}\right)\right)} - {x}^{\left(\frac{1}{n}\right)} \]

5. Taylor expanded in x around inf 8.6%

   \[\leadsto \color{blue}{{x}^{2} \cdot \left(0.5 \cdot \frac{1}{{n}^{2}} - 0.5 \cdot \frac{1}{n}\right) + \frac{x}{n}} \]
6. Step-by-step derivation

   1. fma-def 8.6%

      \[\leadsto \color{blue}{\mathsf{fma}\left({x}^{2}, 0.5 \cdot \frac{1}{{n}^{2}} - 0.5 \cdot \frac{1}{n}, \frac{x}{n}\right)} \]

   2. unpow2 8.6%

      \[\leadsto \mathsf{fma}\left(\color{blue}{x \cdot x}, 0.5 \cdot \frac{1}{{n}^{2}} - 0.5 \cdot \frac{1}{n}, \frac{x}{n}\right) \]

   3. sub-neg 8.6%

      \[\leadsto \mathsf{fma}\left(x \cdot x, \color{blue}{0.5 \cdot \frac{1}{{n}^{2}} + \left(-0.5 \cdot \frac{1}{n}\right)}, \frac{x}{n}\right) \]

   4. associate-*r/ 8.6%

      \[\leadsto \mathsf{fma}\left(x \cdot x, 0.5 \cdot \frac{1}{{n}^{2}} + \left(-\color{blue}{\frac{0.5 \cdot 1}{n}}\right), \frac{x}{n}\right) \]

   5. metadata-eval 8.6%

      \[\leadsto \mathsf{fma}\left(x \cdot x, 0.5 \cdot \frac{1}{{n}^{2}} + \left(-\frac{\color{blue}{0.5}}{n}\right), \frac{x}{n}\right) \]

   6. associate-*r/ 8.6%

      \[\leadsto \mathsf{fma}\left(x \cdot x, \color{blue}{\frac{0.5 \cdot 1}{{n}^{2}}} + \left(-\frac{0.5}{n}\right), \frac{x}{n}\right) \]

   7. metadata-eval 8.6%

      \[\leadsto \mathsf{fma}\left(x \cdot x, \frac{\color{blue}{0.5}}{{n}^{2}} + \left(-\frac{0.5}{n}\right), \frac{x}{n}\right) \]

   8. unpow2 8.6%

      \[\leadsto \mathsf{fma}\left(x \cdot x, \frac{0.5}{\color{blue}{n \cdot n}} + \left(-\frac{0.5}{n}\right), \frac{x}{n}\right) \]

   9. distribute-neg-frac 8.6%

      \[\leadsto \mathsf{fma}\left(x \cdot x, \frac{0.5}{n \cdot n} + \color{blue}{\frac{-0.5}{n}}, \frac{x}{n}\right) \]

   10. metadata-eval 8.6%

       \[\leadsto \mathsf{fma}\left(x \cdot x, \frac{0.5}{n \cdot n} + \frac{\color{blue}{-0.5}}{n}, \frac{x}{n}\right) \]

7. Simplified 8.6%

   \[\leadsto \color{blue}{\mathsf{fma}\left(x \cdot x, \frac{0.5}{n \cdot n} + \frac{-0.5}{n}, \frac{x}{n}\right)} \]

8. Taylor expanded in x around 0 4.6%

   \[\leadsto \color{blue}{\frac{x}{n}} \]

9. Final simplification 4.6%

   \[\leadsto \frac{x}{n} \]

Reproduce

herbie shell --seed 2023275 
(FPCore (x n)
  :name "2nthrt (problem 3.4.6)"
  :precision binary64
  (- (pow (+ x 1.0) (/ 1.0 n)) (pow x (/ 1.0 n))))