2nthrt (problem 3.4.6)

Percentage Accurate: 53.6% → 83.5%

Time: 22.2s

Alternatives: 11

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: 53.6% 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: 83.5% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{1}{n} \leq -1 \cdot 10^{-79}:\\ \;\;\;\;\frac{\frac{\sqrt[3]{{x}^{\left(\frac{3}{n}\right)}}}{n}}{x}\\ \mathbf{elif}\;\frac{1}{n} \leq -5 \cdot 10^{-94}:\\ \;\;\;\;\frac{-\log x}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq -5 \cdot 10^{-129}:\\ \;\;\;\;\frac{\frac{1}{x}}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 5 \cdot 10^{-139}:\\ \;\;\;\;\frac{\mathsf{log1p}\left(x\right) - \log x}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 100:\\ \;\;\;\;\frac{\frac{e^{\frac{\log x}{n}}}{n}}{x}\\ \mathbf{else}:\\ \;\;\;\;e^{\frac{x}{n}} - {x}^{\left(\frac{1}{n}\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (if (<= (/ 1.0 n) -1e-79)
   (/ (/ (cbrt (pow x (/ 3.0 n))) n) x)
   (if (<= (/ 1.0 n) -5e-94)
     (/ (- (log x)) n)
     (if (<= (/ 1.0 n) -5e-129)
       (/ (/ 1.0 x) n)
       (if (<= (/ 1.0 n) 5e-139)
         (/ (- (log1p x) (log x)) n)
         (if (<= (/ 1.0 n) 100.0)
           (/ (/ (exp (/ (log x) n)) n) x)
           (- (exp (/ x n)) (pow x (/ 1.0 n)))))))))

C

double code(double x, double n) {
	double tmp;
	if ((1.0 / n) <= -1e-79) {
		tmp = (cbrt(pow(x, (3.0 / n))) / n) / x;
	} else if ((1.0 / n) <= -5e-94) {
		tmp = -log(x) / n;
	} else if ((1.0 / n) <= -5e-129) {
		tmp = (1.0 / x) / n;
	} else if ((1.0 / n) <= 5e-139) {
		tmp = (log1p(x) - log(x)) / n;
	} else if ((1.0 / n) <= 100.0) {
		tmp = (exp((log(x) / n)) / n) / x;
	} 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-79) {
		tmp = (Math.cbrt(Math.pow(x, (3.0 / n))) / n) / x;
	} else if ((1.0 / n) <= -5e-94) {
		tmp = -Math.log(x) / n;
	} else if ((1.0 / n) <= -5e-129) {
		tmp = (1.0 / x) / n;
	} else if ((1.0 / n) <= 5e-139) {
		tmp = (Math.log1p(x) - Math.log(x)) / n;
	} else if ((1.0 / n) <= 100.0) {
		tmp = (Math.exp((Math.log(x) / n)) / n) / x;
	} 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-79)
		tmp = Float64(Float64(cbrt((x ^ Float64(3.0 / n))) / n) / x);
	elseif (Float64(1.0 / n) <= -5e-94)
		tmp = Float64(Float64(-log(x)) / n);
	elseif (Float64(1.0 / n) <= -5e-129)
		tmp = Float64(Float64(1.0 / x) / n);
	elseif (Float64(1.0 / n) <= 5e-139)
		tmp = Float64(Float64(log1p(x) - log(x)) / n);
	elseif (Float64(1.0 / n) <= 100.0)
		tmp = Float64(Float64(exp(Float64(log(x) / n)) / n) / x);
	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-79], N[(N[(N[Power[N[Power[x, N[(3.0 / n), $MachinePrecision]], $MachinePrecision], 1/3], $MachinePrecision] / n), $MachinePrecision] / x), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], -5e-94], N[((-N[Log[x], $MachinePrecision]) / n), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], -5e-129], N[(N[(1.0 / x), $MachinePrecision] / n), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 5e-139], N[(N[(N[Log[1 + x], $MachinePrecision] - N[Log[x], $MachinePrecision]), $MachinePrecision] / n), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 100.0], N[(N[(N[Exp[N[(N[Log[x], $MachinePrecision] / n), $MachinePrecision]], $MachinePrecision] / n), $MachinePrecision] / x), $MachinePrecision], N[(N[Exp[N[(x / n), $MachinePrecision]], $MachinePrecision] - N[Power[x, N[(1.0 / n), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 6 regimes

2. if (/.f64 1 n) < -1e-79

   1. Initial program 80.6%

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

   2. Taylor expanded in x around inf 92.1%

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

      1. mul-1-neg 92.1%

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

      2. log-rec 92.1%

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

      3. mul-1-neg 92.1%

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

      4. distribute-neg-frac 92.1%

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

      5. mul-1-neg 92.1%

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

      6. remove-double-neg 92.1%

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

      7. *-commutative 92.1%

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

   4. Simplified 92.1%

      \[\leadsto \color{blue}{\frac{e^{\frac{\log x}{n}}}{x \cdot n}} \]
   5. Step-by-step derivation

      1. add-cube-cbrt 91.9%

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

      2. pow3 91.9%

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

      3. associate-/r* 93.0%

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

      4. div-inv 93.0%

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

      5. pow-to-exp 93.0%

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

      6. pow1 93.0%

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

      7. pow-div 93.0%

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

   6. Applied egg-rr 93.0%

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

      1. rem-cube-cbrt 93.1%

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

      2. pow-sub 93.3%

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

      3. pow1 93.3%

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

      4. associate-/l/ 92.1%

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

      5. associate-/r* 93.3%

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

   8. Applied egg-rr 93.3%

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

      1. add-cbrt-cube 93.3%

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

      2. pow3 93.3%

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

      3. pow-pow 93.3%

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

   10. Applied egg-rr 93.3%

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

       1. associate-*l/ 93.3%

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

       2. metadata-eval 93.3%

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

   12. Simplified 93.3%

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

   if -1e-79 < (/.f64 1 n) < -4.9999999999999995e-94

   1. Initial program 3.8%

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

   2. Taylor expanded in x around 0 3.8%

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

   3. Taylor expanded in n around inf 100.0%

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

      1. associate-*r/ 100.0%

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

      2. mul-1-neg 100.0%

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

   5. Simplified 100.0%

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

   if -4.9999999999999995e-94 < (/.f64 1 n) < -5.00000000000000027e-129

   1. Initial program 15.7%

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

   2. Taylor expanded in x around inf 69.6%

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

      1. mul-1-neg 69.6%

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

      2. log-rec 69.6%

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

      3. mul-1-neg 69.6%

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

      4. distribute-neg-frac 69.6%

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

      5. mul-1-neg 69.6%

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

      6. remove-double-neg 69.6%

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

      7. *-commutative 69.6%

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

   4. Simplified 69.6%

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

   5. Taylor expanded in n around inf 69.6%

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

      1. *-commutative 69.6%

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

      2. associate-/r* 73.2%

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

   7. Simplified 73.2%

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

   if -5.00000000000000027e-129 < (/.f64 1 n) < 5.00000000000000034e-139

   1. Initial program 30.8%

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

   2. Taylor expanded in n around inf 86.8%

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

      1. +-rgt-identity 86.8%

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

      2. +-rgt-identity 86.8%

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

      3. log1p-def 86.8%

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

   4. Simplified 86.8%

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

   if 5.00000000000000034e-139 < (/.f64 1 n) < 100

   1. Initial program 30.4%

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

   2. Taylor expanded in x around inf 64.9%

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

      1. mul-1-neg 64.9%

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

      2. log-rec 64.9%

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

      3. mul-1-neg 64.9%

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

      4. distribute-neg-frac 64.9%

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

      5. mul-1-neg 64.9%

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

      6. remove-double-neg 64.9%

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

      7. *-commutative 64.9%

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

   4. Simplified 64.9%

      \[\leadsto \color{blue}{\frac{e^{\frac{\log x}{n}}}{x \cdot n}} \]
   5. Step-by-step derivation

      1. add-cube-cbrt 64.2%

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

      2. pow3 64.3%

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

      3. associate-/r* 68.6%

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

      4. div-inv 68.6%

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

      5. pow-to-exp 68.6%

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

      6. pow1 68.6%

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

      7. pow-div 68.4%

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

   6. Applied egg-rr 68.4%

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

      1. rem-cube-cbrt 68.9%

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

      2. pow-sub 69.2%

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

      3. pow1 69.2%

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

      4. associate-/l/ 64.9%

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

      5. associate-/r* 69.2%

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

   8. Applied egg-rr 69.2%

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

      1. pow1 69.2%

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

      2. pow-to-exp 69.3%

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

      3. pow-exp 69.3%

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

      4. un-div-inv 69.3%

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

   10. Applied egg-rr 69.3%

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

   if 100 < (/.f64 1 n)

   1. Initial program 50.6%

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

   2. Taylor expanded in n around 0 50.6%

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

      1. log1p-def 100.0%

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

   4. Simplified 100.0%

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

   5. Taylor expanded in x around 0 100.0%

      \[\leadsto e^{\color{blue}{\frac{x}{n}}} - {x}^{\left(\frac{1}{n}\right)} \]
3. Recombined 6 regimes into one program.

4. Final simplification 88.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{1}{n} \leq -1 \cdot 10^{-79}:\\ \;\;\;\;\frac{\frac{\sqrt[3]{{x}^{\left(\frac{3}{n}\right)}}}{n}}{x}\\ \mathbf{elif}\;\frac{1}{n} \leq -5 \cdot 10^{-94}:\\ \;\;\;\;\frac{-\log x}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq -5 \cdot 10^{-129}:\\ \;\;\;\;\frac{\frac{1}{x}}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 5 \cdot 10^{-139}:\\ \;\;\;\;\frac{\mathsf{log1p}\left(x\right) - \log x}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 100:\\ \;\;\;\;\frac{\frac{e^{\frac{\log x}{n}}}{n}}{x}\\ \mathbf{else}:\\ \;\;\;\;e^{\frac{x}{n}} - {x}^{\left(\frac{1}{n}\right)}\\ \end{array} \]

Alternative 2: 79.5% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {x}^{\left(\frac{1}{n}\right)}\\ t_1 := \frac{\frac{t_0}{n}}{x}\\ \mathbf{if}\;\frac{1}{n} \leq -1 \cdot 10^{-79}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;\frac{1}{n} \leq -5 \cdot 10^{-94}:\\ \;\;\;\;\frac{-\log x}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq -5 \cdot 10^{-129}:\\ \;\;\;\;\frac{\frac{1}{x}}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 5 \cdot 10^{-139}:\\ \;\;\;\;\frac{\mathsf{log1p}\left(x\right) - \log x}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 100:\\ \;\;\;\;t_1\\ \mathbf{elif}\;\frac{1}{n} \leq 5 \cdot 10^{+179}:\\ \;\;\;\;\left(1 + \frac{x}{n}\right) - t_0\\ \mathbf{else}:\\ \;\;\;\;\sqrt{{\left(n \cdot x\right)}^{-2}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (let* ((t_0 (pow x (/ 1.0 n))) (t_1 (/ (/ t_0 n) x)))
   (if (<= (/ 1.0 n) -1e-79)
     t_1
     (if (<= (/ 1.0 n) -5e-94)
       (/ (- (log x)) n)
       (if (<= (/ 1.0 n) -5e-129)
         (/ (/ 1.0 x) n)
         (if (<= (/ 1.0 n) 5e-139)
           (/ (- (log1p x) (log x)) n)
           (if (<= (/ 1.0 n) 100.0)
             t_1
             (if (<= (/ 1.0 n) 5e+179)
               (- (+ 1.0 (/ x n)) t_0)
               (sqrt (pow (* n x) -2.0))))))))))

C

double code(double x, double n) {
	double t_0 = pow(x, (1.0 / n));
	double t_1 = (t_0 / n) / x;
	double tmp;
	if ((1.0 / n) <= -1e-79) {
		tmp = t_1;
	} else if ((1.0 / n) <= -5e-94) {
		tmp = -log(x) / n;
	} else if ((1.0 / n) <= -5e-129) {
		tmp = (1.0 / x) / n;
	} else if ((1.0 / n) <= 5e-139) {
		tmp = (log1p(x) - log(x)) / n;
	} else if ((1.0 / n) <= 100.0) {
		tmp = t_1;
	} else if ((1.0 / n) <= 5e+179) {
		tmp = (1.0 + (x / n)) - t_0;
	} else {
		tmp = sqrt(pow((n * x), -2.0));
	}
	return tmp;
}

Java

public static double code(double x, double n) {
	double t_0 = Math.pow(x, (1.0 / n));
	double t_1 = (t_0 / n) / x;
	double tmp;
	if ((1.0 / n) <= -1e-79) {
		tmp = t_1;
	} else if ((1.0 / n) <= -5e-94) {
		tmp = -Math.log(x) / n;
	} else if ((1.0 / n) <= -5e-129) {
		tmp = (1.0 / x) / n;
	} else if ((1.0 / n) <= 5e-139) {
		tmp = (Math.log1p(x) - Math.log(x)) / n;
	} else if ((1.0 / n) <= 100.0) {
		tmp = t_1;
	} else if ((1.0 / n) <= 5e+179) {
		tmp = (1.0 + (x / n)) - t_0;
	} else {
		tmp = Math.sqrt(Math.pow((n * x), -2.0));
	}
	return tmp;
}

Python

def code(x, n):
	t_0 = math.pow(x, (1.0 / n))
	t_1 = (t_0 / n) / x
	tmp = 0
	if (1.0 / n) <= -1e-79:
		tmp = t_1
	elif (1.0 / n) <= -5e-94:
		tmp = -math.log(x) / n
	elif (1.0 / n) <= -5e-129:
		tmp = (1.0 / x) / n
	elif (1.0 / n) <= 5e-139:
		tmp = (math.log1p(x) - math.log(x)) / n
	elif (1.0 / n) <= 100.0:
		tmp = t_1
	elif (1.0 / n) <= 5e+179:
		tmp = (1.0 + (x / n)) - t_0
	else:
		tmp = math.sqrt(math.pow((n * x), -2.0))
	return tmp

Julia

function code(x, n)
	t_0 = x ^ Float64(1.0 / n)
	t_1 = Float64(Float64(t_0 / n) / x)
	tmp = 0.0
	if (Float64(1.0 / n) <= -1e-79)
		tmp = t_1;
	elseif (Float64(1.0 / n) <= -5e-94)
		tmp = Float64(Float64(-log(x)) / n);
	elseif (Float64(1.0 / n) <= -5e-129)
		tmp = Float64(Float64(1.0 / x) / n);
	elseif (Float64(1.0 / n) <= 5e-139)
		tmp = Float64(Float64(log1p(x) - log(x)) / n);
	elseif (Float64(1.0 / n) <= 100.0)
		tmp = t_1;
	elseif (Float64(1.0 / n) <= 5e+179)
		tmp = Float64(Float64(1.0 + Float64(x / n)) - t_0);
	else
		tmp = sqrt((Float64(n * x) ^ -2.0));
	end
	return tmp
end

Wolfram

code[x_, n_] := Block[{t$95$0 = N[Power[x, N[(1.0 / n), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(N[(t$95$0 / n), $MachinePrecision] / x), $MachinePrecision]}, If[LessEqual[N[(1.0 / n), $MachinePrecision], -1e-79], t$95$1, If[LessEqual[N[(1.0 / n), $MachinePrecision], -5e-94], N[((-N[Log[x], $MachinePrecision]) / n), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], -5e-129], N[(N[(1.0 / x), $MachinePrecision] / n), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 5e-139], N[(N[(N[Log[1 + x], $MachinePrecision] - N[Log[x], $MachinePrecision]), $MachinePrecision] / n), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 100.0], t$95$1, If[LessEqual[N[(1.0 / n), $MachinePrecision], 5e+179], N[(N[(1.0 + N[(x / n), $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision], N[Sqrt[N[Power[N[(n * x), $MachinePrecision], -2.0], $MachinePrecision]], $MachinePrecision]]]]]]]]]

Derivation

1. Split input into 6 regimes

2. if (/.f64 1 n) < -1e-79 or 5.00000000000000034e-139 < (/.f64 1 n) < 100

   1. Initial program 67.7%

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

   2. Taylor expanded in x around inf 85.2%

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

      1. mul-1-neg 85.2%

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

      2. log-rec 85.2%

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

      3. mul-1-neg 85.2%

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

      4. distribute-neg-frac 85.2%

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

      5. mul-1-neg 85.2%

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

      6. remove-double-neg 85.2%

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

      7. *-commutative 85.2%

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

   4. Simplified 85.2%

      \[\leadsto \color{blue}{\frac{e^{\frac{\log x}{n}}}{x \cdot n}} \]
   5. Step-by-step derivation

      1. add-cube-cbrt 84.8%

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

      2. pow3 84.8%

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

      3. associate-/r* 86.8%

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

      4. div-inv 86.8%

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

      5. pow-to-exp 86.8%

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

      6. pow1 86.8%

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

      7. pow-div 86.7%

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

   6. Applied egg-rr 86.7%

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

      1. rem-cube-cbrt 86.9%

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

      2. pow-sub 87.1%

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

      3. pow1 87.1%

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

      4. associate-/l/ 85.2%

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

      5. associate-/r* 87.1%

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

   8. Applied egg-rr 87.1%

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

   if -1e-79 < (/.f64 1 n) < -4.9999999999999995e-94

   1. Initial program 3.8%

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

   2. Taylor expanded in x around 0 3.8%

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

   3. Taylor expanded in n around inf 100.0%

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

      1. associate-*r/ 100.0%

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

      2. mul-1-neg 100.0%

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

   5. Simplified 100.0%

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

   if -4.9999999999999995e-94 < (/.f64 1 n) < -5.00000000000000027e-129

   1. Initial program 15.7%

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

   2. Taylor expanded in x around inf 69.6%

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

      1. mul-1-neg 69.6%

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

      2. log-rec 69.6%

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

      3. mul-1-neg 69.6%

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

      4. distribute-neg-frac 69.6%

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

      5. mul-1-neg 69.6%

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

      6. remove-double-neg 69.6%

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

      7. *-commutative 69.6%

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

   4. Simplified 69.6%

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

   5. Taylor expanded in n around inf 69.6%

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

      1. *-commutative 69.6%

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

      2. associate-/r* 73.2%

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

   7. Simplified 73.2%

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

   if -5.00000000000000027e-129 < (/.f64 1 n) < 5.00000000000000034e-139

   1. Initial program 30.8%

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

   2. Taylor expanded in n around inf 86.8%

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

      1. +-rgt-identity 86.8%

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

      2. +-rgt-identity 86.8%

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

      3. log1p-def 86.8%

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

   4. Simplified 86.8%

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

   if 100 < (/.f64 1 n) < 5e179

   1. Initial program 73.8%

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

   2. Taylor expanded in x around 0 64.3%

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

   if 5e179 < (/.f64 1 n)

   1. Initial program 22.5%

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

   2. Taylor expanded in x around inf 0.7%

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

      1. mul-1-neg 0.7%

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

      2. log-rec 0.7%

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

      3. mul-1-neg 0.7%

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

      4. distribute-neg-frac 0.7%

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

      5. mul-1-neg 0.7%

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

      6. remove-double-neg 0.7%

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

      7. *-commutative 0.7%

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

   4. Simplified 0.7%

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

   5. Taylor expanded in n around inf 60.2%

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

      1. add-sqr-sqrt 60.2%

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

      2. sqrt-unprod 93.1%

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

      3. inv-pow 93.1%

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

      4. inv-pow 93.1%

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

      5. pow-prod-up 93.1%

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

      6. metadata-eval 93.1%

         \[\leadsto \sqrt{{\left(x \cdot n\right)}^{\color{blue}{-2}}} \]

   7. Applied egg-rr 93.1%

      \[\leadsto \color{blue}{\sqrt{{\left(x \cdot n\right)}^{-2}}} \]
3. Recombined 6 regimes into one program.

4. Final simplification 85.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{1}{n} \leq -1 \cdot 10^{-79}:\\ \;\;\;\;\frac{\frac{{x}^{\left(\frac{1}{n}\right)}}{n}}{x}\\ \mathbf{elif}\;\frac{1}{n} \leq -5 \cdot 10^{-94}:\\ \;\;\;\;\frac{-\log x}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq -5 \cdot 10^{-129}:\\ \;\;\;\;\frac{\frac{1}{x}}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 5 \cdot 10^{-139}:\\ \;\;\;\;\frac{\mathsf{log1p}\left(x\right) - \log x}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 100:\\ \;\;\;\;\frac{\frac{{x}^{\left(\frac{1}{n}\right)}}{n}}{x}\\ \mathbf{elif}\;\frac{1}{n} \leq 5 \cdot 10^{+179}:\\ \;\;\;\;\left(1 + \frac{x}{n}\right) - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{{\left(n \cdot x\right)}^{-2}}\\ \end{array} \]

Alternative 3: 83.5% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {x}^{\left(\frac{1}{n}\right)}\\ t_1 := \frac{\frac{t_0}{n}}{x}\\ \mathbf{if}\;\frac{1}{n} \leq -1 \cdot 10^{-79}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;\frac{1}{n} \leq -5 \cdot 10^{-94}:\\ \;\;\;\;\frac{-\log x}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq -5 \cdot 10^{-129}:\\ \;\;\;\;\frac{\frac{1}{x}}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 5 \cdot 10^{-139}:\\ \;\;\;\;\frac{\mathsf{log1p}\left(x\right) - \log x}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 100:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;e^{\frac{x}{n}} - t_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (let* ((t_0 (pow x (/ 1.0 n))) (t_1 (/ (/ t_0 n) x)))
   (if (<= (/ 1.0 n) -1e-79)
     t_1
     (if (<= (/ 1.0 n) -5e-94)
       (/ (- (log x)) n)
       (if (<= (/ 1.0 n) -5e-129)
         (/ (/ 1.0 x) n)
         (if (<= (/ 1.0 n) 5e-139)
           (/ (- (log1p x) (log x)) n)
           (if (<= (/ 1.0 n) 100.0) t_1 (- (exp (/ x n)) t_0))))))))

C

double code(double x, double n) {
	double t_0 = pow(x, (1.0 / n));
	double t_1 = (t_0 / n) / x;
	double tmp;
	if ((1.0 / n) <= -1e-79) {
		tmp = t_1;
	} else if ((1.0 / n) <= -5e-94) {
		tmp = -log(x) / n;
	} else if ((1.0 / n) <= -5e-129) {
		tmp = (1.0 / x) / n;
	} else if ((1.0 / n) <= 5e-139) {
		tmp = (log1p(x) - log(x)) / n;
	} else if ((1.0 / n) <= 100.0) {
		tmp = t_1;
	} else {
		tmp = exp((x / n)) - t_0;
	}
	return tmp;
}

Java

public static double code(double x, double n) {
	double t_0 = Math.pow(x, (1.0 / n));
	double t_1 = (t_0 / n) / x;
	double tmp;
	if ((1.0 / n) <= -1e-79) {
		tmp = t_1;
	} else if ((1.0 / n) <= -5e-94) {
		tmp = -Math.log(x) / n;
	} else if ((1.0 / n) <= -5e-129) {
		tmp = (1.0 / x) / n;
	} else if ((1.0 / n) <= 5e-139) {
		tmp = (Math.log1p(x) - Math.log(x)) / n;
	} else if ((1.0 / n) <= 100.0) {
		tmp = t_1;
	} else {
		tmp = Math.exp((x / n)) - t_0;
	}
	return tmp;
}

Python

def code(x, n):
	t_0 = math.pow(x, (1.0 / n))
	t_1 = (t_0 / n) / x
	tmp = 0
	if (1.0 / n) <= -1e-79:
		tmp = t_1
	elif (1.0 / n) <= -5e-94:
		tmp = -math.log(x) / n
	elif (1.0 / n) <= -5e-129:
		tmp = (1.0 / x) / n
	elif (1.0 / n) <= 5e-139:
		tmp = (math.log1p(x) - math.log(x)) / n
	elif (1.0 / n) <= 100.0:
		tmp = t_1
	else:
		tmp = math.exp((x / n)) - t_0
	return tmp

Julia

function code(x, n)
	t_0 = x ^ Float64(1.0 / n)
	t_1 = Float64(Float64(t_0 / n) / x)
	tmp = 0.0
	if (Float64(1.0 / n) <= -1e-79)
		tmp = t_1;
	elseif (Float64(1.0 / n) <= -5e-94)
		tmp = Float64(Float64(-log(x)) / n);
	elseif (Float64(1.0 / n) <= -5e-129)
		tmp = Float64(Float64(1.0 / x) / n);
	elseif (Float64(1.0 / n) <= 5e-139)
		tmp = Float64(Float64(log1p(x) - log(x)) / n);
	elseif (Float64(1.0 / n) <= 100.0)
		tmp = t_1;
	else
		tmp = Float64(exp(Float64(x / n)) - t_0);
	end
	return tmp
end

Wolfram

code[x_, n_] := Block[{t$95$0 = N[Power[x, N[(1.0 / n), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(N[(t$95$0 / n), $MachinePrecision] / x), $MachinePrecision]}, If[LessEqual[N[(1.0 / n), $MachinePrecision], -1e-79], t$95$1, If[LessEqual[N[(1.0 / n), $MachinePrecision], -5e-94], N[((-N[Log[x], $MachinePrecision]) / n), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], -5e-129], N[(N[(1.0 / x), $MachinePrecision] / n), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 5e-139], N[(N[(N[Log[1 + x], $MachinePrecision] - N[Log[x], $MachinePrecision]), $MachinePrecision] / n), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 100.0], t$95$1, N[(N[Exp[N[(x / n), $MachinePrecision]], $MachinePrecision] - t$95$0), $MachinePrecision]]]]]]]]

Derivation

1. Split input into 5 regimes

2. if (/.f64 1 n) < -1e-79 or 5.00000000000000034e-139 < (/.f64 1 n) < 100

   1. Initial program 67.7%

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

   2. Taylor expanded in x around inf 85.2%

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

      1. mul-1-neg 85.2%

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

      2. log-rec 85.2%

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

      3. mul-1-neg 85.2%

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

      4. distribute-neg-frac 85.2%

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

      5. mul-1-neg 85.2%

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

      6. remove-double-neg 85.2%

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

      7. *-commutative 85.2%

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

   4. Simplified 85.2%

      \[\leadsto \color{blue}{\frac{e^{\frac{\log x}{n}}}{x \cdot n}} \]
   5. Step-by-step derivation

      1. add-cube-cbrt 84.8%

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

      2. pow3 84.8%

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

      3. associate-/r* 86.8%

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

      4. div-inv 86.8%

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

      5. pow-to-exp 86.8%

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

      6. pow1 86.8%

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

      7. pow-div 86.7%

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

   6. Applied egg-rr 86.7%

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

      1. rem-cube-cbrt 86.9%

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

      2. pow-sub 87.1%

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

      3. pow1 87.1%

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

      4. associate-/l/ 85.2%

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

      5. associate-/r* 87.1%

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

   8. Applied egg-rr 87.1%

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

   if -1e-79 < (/.f64 1 n) < -4.9999999999999995e-94

   1. Initial program 3.8%

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

   2. Taylor expanded in x around 0 3.8%

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

   3. Taylor expanded in n around inf 100.0%

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

      1. associate-*r/ 100.0%

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

      2. mul-1-neg 100.0%

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

   5. Simplified 100.0%

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

   if -4.9999999999999995e-94 < (/.f64 1 n) < -5.00000000000000027e-129

   1. Initial program 15.7%

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

   2. Taylor expanded in x around inf 69.6%

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

      1. mul-1-neg 69.6%

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

      2. log-rec 69.6%

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

      3. mul-1-neg 69.6%

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

      4. distribute-neg-frac 69.6%

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

      5. mul-1-neg 69.6%

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

      6. remove-double-neg 69.6%

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

      7. *-commutative 69.6%

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

   4. Simplified 69.6%

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

   5. Taylor expanded in n around inf 69.6%

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

      1. *-commutative 69.6%

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

      2. associate-/r* 73.2%

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

   7. Simplified 73.2%

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

   if -5.00000000000000027e-129 < (/.f64 1 n) < 5.00000000000000034e-139

   1. Initial program 30.8%

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

   2. Taylor expanded in n around inf 86.8%

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

      1. +-rgt-identity 86.8%

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

      2. +-rgt-identity 86.8%

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

      3. log1p-def 86.8%

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

   4. Simplified 86.8%

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

   if 100 < (/.f64 1 n)

   1. Initial program 50.6%

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

   2. Taylor expanded in n around 0 50.6%

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

      1. log1p-def 100.0%

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

   4. Simplified 100.0%

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

   5. Taylor expanded in x around 0 100.0%

      \[\leadsto e^{\color{blue}{\frac{x}{n}}} - {x}^{\left(\frac{1}{n}\right)} \]
3. Recombined 5 regimes into one program.

4. Final simplification 88.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{1}{n} \leq -1 \cdot 10^{-79}:\\ \;\;\;\;\frac{\frac{{x}^{\left(\frac{1}{n}\right)}}{n}}{x}\\ \mathbf{elif}\;\frac{1}{n} \leq -5 \cdot 10^{-94}:\\ \;\;\;\;\frac{-\log x}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq -5 \cdot 10^{-129}:\\ \;\;\;\;\frac{\frac{1}{x}}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 5 \cdot 10^{-139}:\\ \;\;\;\;\frac{\mathsf{log1p}\left(x\right) - \log x}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 100:\\ \;\;\;\;\frac{\frac{{x}^{\left(\frac{1}{n}\right)}}{n}}{x}\\ \mathbf{else}:\\ \;\;\;\;e^{\frac{x}{n}} - {x}^{\left(\frac{1}{n}\right)}\\ \end{array} \]

Alternative 4: 83.5% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{if}\;\frac{1}{n} \leq -1 \cdot 10^{-79}:\\ \;\;\;\;\frac{\frac{\sqrt[3]{{x}^{\left(\frac{3}{n}\right)}}}{n}}{x}\\ \mathbf{elif}\;\frac{1}{n} \leq -5 \cdot 10^{-94}:\\ \;\;\;\;\frac{-\log x}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq -5 \cdot 10^{-129}:\\ \;\;\;\;\frac{\frac{1}{x}}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 5 \cdot 10^{-139}:\\ \;\;\;\;\frac{\mathsf{log1p}\left(x\right) - \log x}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 100:\\ \;\;\;\;\frac{\frac{t_0}{n}}{x}\\ \mathbf{else}:\\ \;\;\;\;e^{\frac{x}{n}} - t_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (let* ((t_0 (pow x (/ 1.0 n))))
   (if (<= (/ 1.0 n) -1e-79)
     (/ (/ (cbrt (pow x (/ 3.0 n))) n) x)
     (if (<= (/ 1.0 n) -5e-94)
       (/ (- (log x)) n)
       (if (<= (/ 1.0 n) -5e-129)
         (/ (/ 1.0 x) n)
         (if (<= (/ 1.0 n) 5e-139)
           (/ (- (log1p x) (log x)) n)
           (if (<= (/ 1.0 n) 100.0)
             (/ (/ t_0 n) x)
             (- (exp (/ x n)) t_0))))))))

C

double code(double x, double n) {
	double t_0 = pow(x, (1.0 / n));
	double tmp;
	if ((1.0 / n) <= -1e-79) {
		tmp = (cbrt(pow(x, (3.0 / n))) / n) / x;
	} else if ((1.0 / n) <= -5e-94) {
		tmp = -log(x) / n;
	} else if ((1.0 / n) <= -5e-129) {
		tmp = (1.0 / x) / n;
	} else if ((1.0 / n) <= 5e-139) {
		tmp = (log1p(x) - log(x)) / n;
	} else if ((1.0 / n) <= 100.0) {
		tmp = (t_0 / n) / x;
	} else {
		tmp = exp((x / n)) - t_0;
	}
	return tmp;
}

Java

public static double code(double x, double n) {
	double t_0 = Math.pow(x, (1.0 / n));
	double tmp;
	if ((1.0 / n) <= -1e-79) {
		tmp = (Math.cbrt(Math.pow(x, (3.0 / n))) / n) / x;
	} else if ((1.0 / n) <= -5e-94) {
		tmp = -Math.log(x) / n;
	} else if ((1.0 / n) <= -5e-129) {
		tmp = (1.0 / x) / n;
	} else if ((1.0 / n) <= 5e-139) {
		tmp = (Math.log1p(x) - Math.log(x)) / n;
	} else if ((1.0 / n) <= 100.0) {
		tmp = (t_0 / n) / x;
	} else {
		tmp = Math.exp((x / n)) - t_0;
	}
	return tmp;
}

Julia

function code(x, n)
	t_0 = x ^ Float64(1.0 / n)
	tmp = 0.0
	if (Float64(1.0 / n) <= -1e-79)
		tmp = Float64(Float64(cbrt((x ^ Float64(3.0 / n))) / n) / x);
	elseif (Float64(1.0 / n) <= -5e-94)
		tmp = Float64(Float64(-log(x)) / n);
	elseif (Float64(1.0 / n) <= -5e-129)
		tmp = Float64(Float64(1.0 / x) / n);
	elseif (Float64(1.0 / n) <= 5e-139)
		tmp = Float64(Float64(log1p(x) - log(x)) / n);
	elseif (Float64(1.0 / n) <= 100.0)
		tmp = Float64(Float64(t_0 / n) / x);
	else
		tmp = Float64(exp(Float64(x / n)) - t_0);
	end
	return tmp
end

Wolfram

code[x_, n_] := Block[{t$95$0 = N[Power[x, N[(1.0 / n), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[N[(1.0 / n), $MachinePrecision], -1e-79], N[(N[(N[Power[N[Power[x, N[(3.0 / n), $MachinePrecision]], $MachinePrecision], 1/3], $MachinePrecision] / n), $MachinePrecision] / x), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], -5e-94], N[((-N[Log[x], $MachinePrecision]) / n), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], -5e-129], N[(N[(1.0 / x), $MachinePrecision] / n), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 5e-139], N[(N[(N[Log[1 + x], $MachinePrecision] - N[Log[x], $MachinePrecision]), $MachinePrecision] / n), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 100.0], N[(N[(t$95$0 / n), $MachinePrecision] / x), $MachinePrecision], N[(N[Exp[N[(x / n), $MachinePrecision]], $MachinePrecision] - t$95$0), $MachinePrecision]]]]]]]

Derivation

1. Split input into 6 regimes

2. if (/.f64 1 n) < -1e-79

   1. Initial program 80.6%

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

   2. Taylor expanded in x around inf 92.1%

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

      1. mul-1-neg 92.1%

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

      2. log-rec 92.1%

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

      3. mul-1-neg 92.1%

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

      4. distribute-neg-frac 92.1%

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

      5. mul-1-neg 92.1%

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

      6. remove-double-neg 92.1%

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

      7. *-commutative 92.1%

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

   4. Simplified 92.1%

      \[\leadsto \color{blue}{\frac{e^{\frac{\log x}{n}}}{x \cdot n}} \]
   5. Step-by-step derivation

      1. add-cube-cbrt 91.9%

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

      2. pow3 91.9%

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

      3. associate-/r* 93.0%

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

      4. div-inv 93.0%

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

      5. pow-to-exp 93.0%

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

      6. pow1 93.0%

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

      7. pow-div 93.0%

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

   6. Applied egg-rr 93.0%

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

      1. rem-cube-cbrt 93.1%

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

      2. pow-sub 93.3%

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

      3. pow1 93.3%

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

      4. associate-/l/ 92.1%

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

      5. associate-/r* 93.3%

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

   8. Applied egg-rr 93.3%

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

      1. add-cbrt-cube 93.3%

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

      2. pow3 93.3%

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

      3. pow-pow 93.3%

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

   10. Applied egg-rr 93.3%

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

       1. associate-*l/ 93.3%

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

       2. metadata-eval 93.3%

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

   12. Simplified 93.3%

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

   if -1e-79 < (/.f64 1 n) < -4.9999999999999995e-94

   1. Initial program 3.8%

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

   2. Taylor expanded in x around 0 3.8%

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

   3. Taylor expanded in n around inf 100.0%

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

      1. associate-*r/ 100.0%

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

      2. mul-1-neg 100.0%

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

   5. Simplified 100.0%

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

   if -4.9999999999999995e-94 < (/.f64 1 n) < -5.00000000000000027e-129

   1. Initial program 15.7%

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

   2. Taylor expanded in x around inf 69.6%

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

      1. mul-1-neg 69.6%

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

      2. log-rec 69.6%

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

      3. mul-1-neg 69.6%

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

      4. distribute-neg-frac 69.6%

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

      5. mul-1-neg 69.6%

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

      6. remove-double-neg 69.6%

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

      7. *-commutative 69.6%

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

   4. Simplified 69.6%

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

   5. Taylor expanded in n around inf 69.6%

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

      1. *-commutative 69.6%

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

      2. associate-/r* 73.2%

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

   7. Simplified 73.2%

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

   if -5.00000000000000027e-129 < (/.f64 1 n) < 5.00000000000000034e-139

   1. Initial program 30.8%

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

   2. Taylor expanded in n around inf 86.8%

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

      1. +-rgt-identity 86.8%

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

      2. +-rgt-identity 86.8%

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

      3. log1p-def 86.8%

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

   4. Simplified 86.8%

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

   if 5.00000000000000034e-139 < (/.f64 1 n) < 100

   1. Initial program 30.4%

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

   2. Taylor expanded in x around inf 64.9%

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

      1. mul-1-neg 64.9%

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

      2. log-rec 64.9%

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

      3. mul-1-neg 64.9%

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

      4. distribute-neg-frac 64.9%

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

      5. mul-1-neg 64.9%

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

      6. remove-double-neg 64.9%

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

      7. *-commutative 64.9%

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

   4. Simplified 64.9%

      \[\leadsto \color{blue}{\frac{e^{\frac{\log x}{n}}}{x \cdot n}} \]
   5. Step-by-step derivation

      1. add-cube-cbrt 64.2%

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

      2. pow3 64.3%

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

      3. associate-/r* 68.6%

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

      4. div-inv 68.6%

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

      5. pow-to-exp 68.6%

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

      6. pow1 68.6%

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

      7. pow-div 68.4%

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

   6. Applied egg-rr 68.4%

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

      1. rem-cube-cbrt 68.9%

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

      2. pow-sub 69.2%

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

      3. pow1 69.2%

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

      4. associate-/l/ 64.9%

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

      5. associate-/r* 69.2%

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

   8. Applied egg-rr 69.2%

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

   if 100 < (/.f64 1 n)

   1. Initial program 50.6%

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

   2. Taylor expanded in n around 0 50.6%

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

      1. log1p-def 100.0%

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

   4. Simplified 100.0%

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

   5. Taylor expanded in x around 0 100.0%

      \[\leadsto e^{\color{blue}{\frac{x}{n}}} - {x}^{\left(\frac{1}{n}\right)} \]
3. Recombined 6 regimes into one program.

4. Final simplification 88.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{1}{n} \leq -1 \cdot 10^{-79}:\\ \;\;\;\;\frac{\frac{\sqrt[3]{{x}^{\left(\frac{3}{n}\right)}}}{n}}{x}\\ \mathbf{elif}\;\frac{1}{n} \leq -5 \cdot 10^{-94}:\\ \;\;\;\;\frac{-\log x}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq -5 \cdot 10^{-129}:\\ \;\;\;\;\frac{\frac{1}{x}}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 5 \cdot 10^{-139}:\\ \;\;\;\;\frac{\mathsf{log1p}\left(x\right) - \log x}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 100:\\ \;\;\;\;\frac{\frac{{x}^{\left(\frac{1}{n}\right)}}{n}}{x}\\ \mathbf{else}:\\ \;\;\;\;e^{\frac{x}{n}} - {x}^{\left(\frac{1}{n}\right)}\\ \end{array} \]

Alternative 5: 70.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{-\log x}{n}\\ t_1 := {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{if}\;x \leq 9 \cdot 10^{-211}:\\ \;\;\;\;\left(1 + \frac{x}{n}\right) - t_1\\ \mathbf{elif}\;x \leq 7 \cdot 10^{-124}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \leq 1.5 \cdot 10^{-94}:\\ \;\;\;\;\mathsf{log1p}\left(\mathsf{expm1}\left(\frac{1}{n \cdot x}\right)\right)\\ \mathbf{elif}\;x \leq 0.1:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{t_1}{n}}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (let* ((t_0 (/ (- (log x)) n)) (t_1 (pow x (/ 1.0 n))))
   (if (<= x 9e-211)
     (- (+ 1.0 (/ x n)) t_1)
     (if (<= x 7e-124)
       t_0
       (if (<= x 1.5e-94)
         (log1p (expm1 (/ 1.0 (* n x))))
         (if (<= x 0.1) t_0 (/ (/ t_1 n) x)))))))

C

double code(double x, double n) {
	double t_0 = -log(x) / n;
	double t_1 = pow(x, (1.0 / n));
	double tmp;
	if (x <= 9e-211) {
		tmp = (1.0 + (x / n)) - t_1;
	} else if (x <= 7e-124) {
		tmp = t_0;
	} else if (x <= 1.5e-94) {
		tmp = log1p(expm1((1.0 / (n * x))));
	} else if (x <= 0.1) {
		tmp = t_0;
	} else {
		tmp = (t_1 / n) / x;
	}
	return tmp;
}

Java

public static double code(double x, double n) {
	double t_0 = -Math.log(x) / n;
	double t_1 = Math.pow(x, (1.0 / n));
	double tmp;
	if (x <= 9e-211) {
		tmp = (1.0 + (x / n)) - t_1;
	} else if (x <= 7e-124) {
		tmp = t_0;
	} else if (x <= 1.5e-94) {
		tmp = Math.log1p(Math.expm1((1.0 / (n * x))));
	} else if (x <= 0.1) {
		tmp = t_0;
	} else {
		tmp = (t_1 / n) / x;
	}
	return tmp;
}

Python

def code(x, n):
	t_0 = -math.log(x) / n
	t_1 = math.pow(x, (1.0 / n))
	tmp = 0
	if x <= 9e-211:
		tmp = (1.0 + (x / n)) - t_1
	elif x <= 7e-124:
		tmp = t_0
	elif x <= 1.5e-94:
		tmp = math.log1p(math.expm1((1.0 / (n * x))))
	elif x <= 0.1:
		tmp = t_0
	else:
		tmp = (t_1 / n) / x
	return tmp

Julia

function code(x, n)
	t_0 = Float64(Float64(-log(x)) / n)
	t_1 = x ^ Float64(1.0 / n)
	tmp = 0.0
	if (x <= 9e-211)
		tmp = Float64(Float64(1.0 + Float64(x / n)) - t_1);
	elseif (x <= 7e-124)
		tmp = t_0;
	elseif (x <= 1.5e-94)
		tmp = log1p(expm1(Float64(1.0 / Float64(n * x))));
	elseif (x <= 0.1)
		tmp = t_0;
	else
		tmp = Float64(Float64(t_1 / n) / x);
	end
	return tmp
end

Wolfram

code[x_, n_] := Block[{t$95$0 = N[((-N[Log[x], $MachinePrecision]) / n), $MachinePrecision]}, Block[{t$95$1 = N[Power[x, N[(1.0 / n), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[x, 9e-211], N[(N[(1.0 + N[(x / n), $MachinePrecision]), $MachinePrecision] - t$95$1), $MachinePrecision], If[LessEqual[x, 7e-124], t$95$0, If[LessEqual[x, 1.5e-94], N[Log[1 + N[(Exp[N[(1.0 / N[(n * x), $MachinePrecision]), $MachinePrecision]] - 1), $MachinePrecision]], $MachinePrecision], If[LessEqual[x, 0.1], t$95$0, N[(N[(t$95$1 / n), $MachinePrecision] / x), $MachinePrecision]]]]]]]

Derivation

1. Split input into 4 regimes

2. if x < 8.9999999999999997e-211

   1. Initial program 56.0%

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

   2. Taylor expanded in x around 0 56.6%

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

   if 8.9999999999999997e-211 < x < 6.9999999999999997e-124 or 1.5000000000000001e-94 < x < 0.10000000000000001

   1. Initial program 33.3%

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

   2. Taylor expanded in x around 0 28.5%

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

   3. Taylor expanded in n around inf 61.1%

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

      1. associate-*r/ 61.1%

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

      2. mul-1-neg 61.1%

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

   5. Simplified 61.1%

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

   if 6.9999999999999997e-124 < x < 1.5000000000000001e-94

   1. Initial program 38.4%

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

   2. Taylor expanded in x around inf 30.8%

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

      1. mul-1-neg 30.8%

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

      2. log-rec 30.8%

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

      3. mul-1-neg 30.8%

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

      4. distribute-neg-frac 30.8%

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

      5. mul-1-neg 30.8%

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

      6. remove-double-neg 30.8%

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

      7. *-commutative 30.8%

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

   4. Simplified 30.8%

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

   5. Taylor expanded in n around inf 19.5%

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

      1. log1p-expm1-u 65.8%

         \[\leadsto \color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(\frac{1}{x \cdot n}\right)\right)} \]

   7. Applied egg-rr 65.8%

      \[\leadsto \color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(\frac{1}{x \cdot n}\right)\right)} \]

   if 0.10000000000000001 < x

   1. Initial program 62.3%

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

   2. Taylor expanded in x around inf 95.7%

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

      1. mul-1-neg 95.7%

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

      2. log-rec 95.7%

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

      3. mul-1-neg 95.7%

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

      4. distribute-neg-frac 95.7%

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

      5. mul-1-neg 95.7%

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

      6. remove-double-neg 95.7%

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

      7. *-commutative 95.7%

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

   4. Simplified 95.7%

      \[\leadsto \color{blue}{\frac{e^{\frac{\log x}{n}}}{x \cdot n}} \]
   5. Step-by-step derivation

      1. add-cube-cbrt 95.1%

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

      2. pow3 95.1%

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

      3. associate-/r* 97.6%

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

      4. div-inv 97.6%

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

      5. pow-to-exp 97.6%

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

      6. pow1 97.6%

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

      7. pow-div 97.5%

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

   6. Applied egg-rr 97.5%

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

      1. rem-cube-cbrt 98.0%

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

      2. pow-sub 98.2%

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

      3. pow1 98.2%

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

      4. associate-/l/ 95.7%

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

      5. associate-/r* 98.2%

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

   8. Applied egg-rr 98.2%

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

4. Final simplification 79.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 9 \cdot 10^{-211}:\\ \;\;\;\;\left(1 + \frac{x}{n}\right) - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{elif}\;x \leq 7 \cdot 10^{-124}:\\ \;\;\;\;\frac{-\log x}{n}\\ \mathbf{elif}\;x \leq 1.5 \cdot 10^{-94}:\\ \;\;\;\;\mathsf{log1p}\left(\mathsf{expm1}\left(\frac{1}{n \cdot x}\right)\right)\\ \mathbf{elif}\;x \leq 0.1:\\ \;\;\;\;\frac{-\log x}{n}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{{x}^{\left(\frac{1}{n}\right)}}{n}}{x}\\ \end{array} \]

Alternative 6: 71.1% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{if}\;x \leq 2.9 \cdot 10^{-209}:\\ \;\;\;\;1 - t_0\\ \mathbf{elif}\;x \leq 0.0105:\\ \;\;\;\;\frac{-\log x}{n}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{t_0}{n}}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (let* ((t_0 (pow x (/ 1.0 n))))
   (if (<= x 2.9e-209)
     (- 1.0 t_0)
     (if (<= x 0.0105) (/ (- (log x)) n) (/ (/ t_0 n) x)))))

C

double code(double x, double n) {
	double t_0 = pow(x, (1.0 / n));
	double tmp;
	if (x <= 2.9e-209) {
		tmp = 1.0 - t_0;
	} else if (x <= 0.0105) {
		tmp = -log(x) / n;
	} else {
		tmp = (t_0 / n) / x;
	}
	return tmp;
}

Fortran

real(8) function code(x, n)
    real(8), intent (in) :: x
    real(8), intent (in) :: n
    real(8) :: t_0
    real(8) :: tmp
    t_0 = x ** (1.0d0 / n)
    if (x <= 2.9d-209) then
        tmp = 1.0d0 - t_0
    else if (x <= 0.0105d0) then
        tmp = -log(x) / n
    else
        tmp = (t_0 / n) / x
    end if
    code = tmp
end function

Java

public static double code(double x, double n) {
	double t_0 = Math.pow(x, (1.0 / n));
	double tmp;
	if (x <= 2.9e-209) {
		tmp = 1.0 - t_0;
	} else if (x <= 0.0105) {
		tmp = -Math.log(x) / n;
	} else {
		tmp = (t_0 / n) / x;
	}
	return tmp;
}

Python

def code(x, n):
	t_0 = math.pow(x, (1.0 / n))
	tmp = 0
	if x <= 2.9e-209:
		tmp = 1.0 - t_0
	elif x <= 0.0105:
		tmp = -math.log(x) / n
	else:
		tmp = (t_0 / n) / x
	return tmp

Julia

function code(x, n)
	t_0 = x ^ Float64(1.0 / n)
	tmp = 0.0
	if (x <= 2.9e-209)
		tmp = Float64(1.0 - t_0);
	elseif (x <= 0.0105)
		tmp = Float64(Float64(-log(x)) / n);
	else
		tmp = Float64(Float64(t_0 / n) / x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, n)
	t_0 = x ^ (1.0 / n);
	tmp = 0.0;
	if (x <= 2.9e-209)
		tmp = 1.0 - t_0;
	elseif (x <= 0.0105)
		tmp = -log(x) / n;
	else
		tmp = (t_0 / n) / x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, n_] := Block[{t$95$0 = N[Power[x, N[(1.0 / n), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[x, 2.9e-209], N[(1.0 - t$95$0), $MachinePrecision], If[LessEqual[x, 0.0105], N[((-N[Log[x], $MachinePrecision]) / n), $MachinePrecision], N[(N[(t$95$0 / n), $MachinePrecision] / x), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if x < 2.90000000000000026e-209

   1. Initial program 56.0%

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

   2. Taylor expanded in x around 0 56.0%

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

   if 2.90000000000000026e-209 < x < 0.0105000000000000007

   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 30.0%

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

   3. Taylor expanded in n around inf 56.8%

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

      1. associate-*r/ 56.8%

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

      2. mul-1-neg 56.8%

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

   5. Simplified 56.8%

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

   if 0.0105000000000000007 < x

   1. Initial program 62.3%

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

   2. Taylor expanded in x around inf 95.7%

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

      1. mul-1-neg 95.7%

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

      2. log-rec 95.7%

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

      3. mul-1-neg 95.7%

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

      4. distribute-neg-frac 95.7%

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

      5. mul-1-neg 95.7%

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

      6. remove-double-neg 95.7%

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

      7. *-commutative 95.7%

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

   4. Simplified 95.7%

      \[\leadsto \color{blue}{\frac{e^{\frac{\log x}{n}}}{x \cdot n}} \]
   5. Step-by-step derivation

      1. add-cube-cbrt 95.1%

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

      2. pow3 95.1%

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

      3. associate-/r* 97.6%

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

      4. div-inv 97.6%

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

      5. pow-to-exp 97.6%

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

      6. pow1 97.6%

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

      7. pow-div 97.5%

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

   6. Applied egg-rr 97.5%

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

      1. rem-cube-cbrt 98.0%

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

      2. pow-sub 98.2%

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

      3. pow1 98.2%

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

      4. associate-/l/ 95.7%

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

      5. associate-/r* 98.2%

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

   8. Applied egg-rr 98.2%

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

4. Final simplification 77.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 2.9 \cdot 10^{-209}:\\ \;\;\;\;1 - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{elif}\;x \leq 0.0105:\\ \;\;\;\;\frac{-\log x}{n}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{{x}^{\left(\frac{1}{n}\right)}}{n}}{x}\\ \end{array} \]

Alternative 7: 71.3% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{if}\;x \leq 3.4 \cdot 10^{-211}:\\ \;\;\;\;\left(1 + \frac{x}{n}\right) - t_0\\ \mathbf{elif}\;x \leq 0.018:\\ \;\;\;\;\frac{-\log x}{n}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{t_0}{n}}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (let* ((t_0 (pow x (/ 1.0 n))))
   (if (<= x 3.4e-211)
     (- (+ 1.0 (/ x n)) t_0)
     (if (<= x 0.018) (/ (- (log x)) n) (/ (/ t_0 n) x)))))

C

double code(double x, double n) {
	double t_0 = pow(x, (1.0 / n));
	double tmp;
	if (x <= 3.4e-211) {
		tmp = (1.0 + (x / n)) - t_0;
	} else if (x <= 0.018) {
		tmp = -log(x) / n;
	} else {
		tmp = (t_0 / n) / x;
	}
	return tmp;
}

Fortran

real(8) function code(x, n)
    real(8), intent (in) :: x
    real(8), intent (in) :: n
    real(8) :: t_0
    real(8) :: tmp
    t_0 = x ** (1.0d0 / n)
    if (x <= 3.4d-211) then
        tmp = (1.0d0 + (x / n)) - t_0
    else if (x <= 0.018d0) then
        tmp = -log(x) / n
    else
        tmp = (t_0 / n) / x
    end if
    code = tmp
end function

Java

public static double code(double x, double n) {
	double t_0 = Math.pow(x, (1.0 / n));
	double tmp;
	if (x <= 3.4e-211) {
		tmp = (1.0 + (x / n)) - t_0;
	} else if (x <= 0.018) {
		tmp = -Math.log(x) / n;
	} else {
		tmp = (t_0 / n) / x;
	}
	return tmp;
}

Python

def code(x, n):
	t_0 = math.pow(x, (1.0 / n))
	tmp = 0
	if x <= 3.4e-211:
		tmp = (1.0 + (x / n)) - t_0
	elif x <= 0.018:
		tmp = -math.log(x) / n
	else:
		tmp = (t_0 / n) / x
	return tmp

Julia

function code(x, n)
	t_0 = x ^ Float64(1.0 / n)
	tmp = 0.0
	if (x <= 3.4e-211)
		tmp = Float64(Float64(1.0 + Float64(x / n)) - t_0);
	elseif (x <= 0.018)
		tmp = Float64(Float64(-log(x)) / n);
	else
		tmp = Float64(Float64(t_0 / n) / x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, n)
	t_0 = x ^ (1.0 / n);
	tmp = 0.0;
	if (x <= 3.4e-211)
		tmp = (1.0 + (x / n)) - t_0;
	elseif (x <= 0.018)
		tmp = -log(x) / n;
	else
		tmp = (t_0 / n) / x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, n_] := Block[{t$95$0 = N[Power[x, N[(1.0 / n), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[x, 3.4e-211], N[(N[(1.0 + N[(x / n), $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision], If[LessEqual[x, 0.018], N[((-N[Log[x], $MachinePrecision]) / n), $MachinePrecision], N[(N[(t$95$0 / n), $MachinePrecision] / x), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if x < 3.4000000000000001e-211

   1. Initial program 56.0%

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

   2. Taylor expanded in x around 0 56.6%

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

   if 3.4000000000000001e-211 < x < 0.0179999999999999986

   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 30.0%

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

   3. Taylor expanded in n around inf 56.8%

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

      1. associate-*r/ 56.8%

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

      2. mul-1-neg 56.8%

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

   5. Simplified 56.8%

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

   if 0.0179999999999999986 < x

   1. Initial program 62.3%

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

   2. Taylor expanded in x around inf 95.7%

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

      1. mul-1-neg 95.7%

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

      2. log-rec 95.7%

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

      3. mul-1-neg 95.7%

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

      4. distribute-neg-frac 95.7%

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

      5. mul-1-neg 95.7%

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

      6. remove-double-neg 95.7%

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

      7. *-commutative 95.7%

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

   4. Simplified 95.7%

      \[\leadsto \color{blue}{\frac{e^{\frac{\log x}{n}}}{x \cdot n}} \]
   5. Step-by-step derivation

      1. add-cube-cbrt 95.1%

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

      2. pow3 95.1%

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

      3. associate-/r* 97.6%

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

      4. div-inv 97.6%

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

      5. pow-to-exp 97.6%

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

      6. pow1 97.6%

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

      7. pow-div 97.5%

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

   6. Applied egg-rr 97.5%

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

      1. rem-cube-cbrt 98.0%

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

      2. pow-sub 98.2%

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

      3. pow1 98.2%

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

      4. associate-/l/ 95.7%

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

      5. associate-/r* 98.2%

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

   8. Applied egg-rr 98.2%

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

4. Final simplification 77.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 3.4 \cdot 10^{-211}:\\ \;\;\;\;\left(1 + \frac{x}{n}\right) - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{elif}\;x \leq 0.018:\\ \;\;\;\;\frac{-\log x}{n}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{{x}^{\left(\frac{1}{n}\right)}}{n}}{x}\\ \end{array} \]

Alternative 8: 55.7% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{-\log x}{n}\\ \mathbf{if}\;x \leq 5.5 \cdot 10^{-225}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \leq 1.6 \cdot 10^{-212}:\\ \;\;\;\;\frac{1}{n \cdot x}\\ \mathbf{elif}\;x \leq 0.0102:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{x}}{n}\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (let* ((t_0 (/ (- (log x)) n)))
   (if (<= x 5.5e-225)
     t_0
     (if (<= x 1.6e-212)
       (/ 1.0 (* n x))
       (if (<= x 0.0102) t_0 (/ (/ 1.0 x) n))))))

C

double code(double x, double n) {
	double t_0 = -log(x) / n;
	double tmp;
	if (x <= 5.5e-225) {
		tmp = t_0;
	} else if (x <= 1.6e-212) {
		tmp = 1.0 / (n * x);
	} else if (x <= 0.0102) {
		tmp = t_0;
	} else {
		tmp = (1.0 / x) / n;
	}
	return tmp;
}

Fortran

real(8) function code(x, n)
    real(8), intent (in) :: x
    real(8), intent (in) :: n
    real(8) :: t_0
    real(8) :: tmp
    t_0 = -log(x) / n
    if (x <= 5.5d-225) then
        tmp = t_0
    else if (x <= 1.6d-212) then
        tmp = 1.0d0 / (n * x)
    else if (x <= 0.0102d0) then
        tmp = t_0
    else
        tmp = (1.0d0 / x) / n
    end if
    code = tmp
end function

Java

public static double code(double x, double n) {
	double t_0 = -Math.log(x) / n;
	double tmp;
	if (x <= 5.5e-225) {
		tmp = t_0;
	} else if (x <= 1.6e-212) {
		tmp = 1.0 / (n * x);
	} else if (x <= 0.0102) {
		tmp = t_0;
	} else {
		tmp = (1.0 / x) / n;
	}
	return tmp;
}

Python

def code(x, n):
	t_0 = -math.log(x) / n
	tmp = 0
	if x <= 5.5e-225:
		tmp = t_0
	elif x <= 1.6e-212:
		tmp = 1.0 / (n * x)
	elif x <= 0.0102:
		tmp = t_0
	else:
		tmp = (1.0 / x) / n
	return tmp

Julia

function code(x, n)
	t_0 = Float64(Float64(-log(x)) / n)
	tmp = 0.0
	if (x <= 5.5e-225)
		tmp = t_0;
	elseif (x <= 1.6e-212)
		tmp = Float64(1.0 / Float64(n * x));
	elseif (x <= 0.0102)
		tmp = t_0;
	else
		tmp = Float64(Float64(1.0 / x) / n);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, n)
	t_0 = -log(x) / n;
	tmp = 0.0;
	if (x <= 5.5e-225)
		tmp = t_0;
	elseif (x <= 1.6e-212)
		tmp = 1.0 / (n * x);
	elseif (x <= 0.0102)
		tmp = t_0;
	else
		tmp = (1.0 / x) / n;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, n_] := Block[{t$95$0 = N[((-N[Log[x], $MachinePrecision]) / n), $MachinePrecision]}, If[LessEqual[x, 5.5e-225], t$95$0, If[LessEqual[x, 1.6e-212], N[(1.0 / N[(n * x), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 0.0102], t$95$0, N[(N[(1.0 / x), $MachinePrecision] / n), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if x < 5.5000000000000002e-225 or 1.5999999999999999e-212 < x < 0.010200000000000001

   1. Initial program 35.9%

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

   2. Taylor expanded in x around 0 33.5%

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

   3. Taylor expanded in n around inf 56.4%

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

      1. associate-*r/ 56.4%

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

      2. mul-1-neg 56.4%

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

   5. Simplified 56.4%

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

   if 5.5000000000000002e-225 < x < 1.5999999999999999e-212

   1. Initial program 97.1%

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

   2. Taylor expanded in x around inf 85.7%

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

      1. mul-1-neg 85.7%

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

      2. log-rec 85.7%

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

      3. mul-1-neg 85.7%

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

      4. distribute-neg-frac 85.7%

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

      5. mul-1-neg 85.7%

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

      6. remove-double-neg 85.7%

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

      7. *-commutative 85.7%

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

   4. Simplified 85.7%

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

   5. Taylor expanded in n around inf 86.2%

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

   if 0.010200000000000001 < x

   1. Initial program 62.6%

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

   2. Taylor expanded in x around inf 95.0%

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

      1. mul-1-neg 95.0%

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

      2. log-rec 95.0%

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

      3. mul-1-neg 95.0%

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

      4. distribute-neg-frac 95.0%

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

      5. mul-1-neg 95.0%

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

      6. remove-double-neg 95.0%

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

      7. *-commutative 95.0%

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

   4. Simplified 95.0%

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

   5. Taylor expanded in n around inf 58.1%

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

      1. *-commutative 58.1%

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

      2. associate-/r* 60.5%

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

   7. Simplified 60.5%

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

4. Final simplification 59.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 5.5 \cdot 10^{-225}:\\ \;\;\;\;\frac{-\log x}{n}\\ \mathbf{elif}\;x \leq 1.6 \cdot 10^{-212}:\\ \;\;\;\;\frac{1}{n \cdot x}\\ \mathbf{elif}\;x \leq 0.0102:\\ \;\;\;\;\frac{-\log x}{n}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{x}}{n}\\ \end{array} \]

Alternative 9: 56.5% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 6.2 \cdot 10^{-210}:\\ \;\;\;\;1 - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{elif}\;x \leq 0.0102:\\ \;\;\;\;\frac{-\log x}{n}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{x}}{n}\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (if (<= x 6.2e-210)
   (- 1.0 (pow x (/ 1.0 n)))
   (if (<= x 0.0102) (/ (- (log x)) n) (/ (/ 1.0 x) n))))

C

double code(double x, double n) {
	double tmp;
	if (x <= 6.2e-210) {
		tmp = 1.0 - pow(x, (1.0 / n));
	} else if (x <= 0.0102) {
		tmp = -log(x) / n;
	} else {
		tmp = (1.0 / x) / n;
	}
	return tmp;
}

Fortran

real(8) function code(x, n)
    real(8), intent (in) :: x
    real(8), intent (in) :: n
    real(8) :: tmp
    if (x <= 6.2d-210) then
        tmp = 1.0d0 - (x ** (1.0d0 / n))
    else if (x <= 0.0102d0) then
        tmp = -log(x) / n
    else
        tmp = (1.0d0 / x) / n
    end if
    code = tmp
end function

Java

public static double code(double x, double n) {
	double tmp;
	if (x <= 6.2e-210) {
		tmp = 1.0 - Math.pow(x, (1.0 / n));
	} else if (x <= 0.0102) {
		tmp = -Math.log(x) / n;
	} else {
		tmp = (1.0 / x) / n;
	}
	return tmp;
}

Python

def code(x, n):
	tmp = 0
	if x <= 6.2e-210:
		tmp = 1.0 - math.pow(x, (1.0 / n))
	elif x <= 0.0102:
		tmp = -math.log(x) / n
	else:
		tmp = (1.0 / x) / n
	return tmp

Julia

function code(x, n)
	tmp = 0.0
	if (x <= 6.2e-210)
		tmp = Float64(1.0 - (x ^ Float64(1.0 / n)));
	elseif (x <= 0.0102)
		tmp = Float64(Float64(-log(x)) / n);
	else
		tmp = Float64(Float64(1.0 / x) / n);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, n)
	tmp = 0.0;
	if (x <= 6.2e-210)
		tmp = 1.0 - (x ^ (1.0 / n));
	elseif (x <= 0.0102)
		tmp = -log(x) / n;
	else
		tmp = (1.0 / x) / n;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, n_] := If[LessEqual[x, 6.2e-210], N[(1.0 - N[Power[x, N[(1.0 / n), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 0.0102], N[((-N[Log[x], $MachinePrecision]) / n), $MachinePrecision], N[(N[(1.0 / x), $MachinePrecision] / n), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < 6.19999999999999973e-210

   1. Initial program 56.0%

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

   2. Taylor expanded in x around 0 56.0%

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

   if 6.19999999999999973e-210 < x < 0.010200000000000001

   1. Initial program 33.4%

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

   2. Taylor expanded in x around 0 30.3%

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

   3. Taylor expanded in n around inf 57.4%

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

      1. associate-*r/ 57.4%

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

      2. mul-1-neg 57.4%

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

   5. Simplified 57.4%

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

   if 0.010200000000000001 < x

   1. Initial program 62.6%

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

   2. Taylor expanded in x around inf 95.0%

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

      1. mul-1-neg 95.0%

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

      2. log-rec 95.0%

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

      3. mul-1-neg 95.0%

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

      4. distribute-neg-frac 95.0%

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

      5. mul-1-neg 95.0%

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

      6. remove-double-neg 95.0%

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

      7. *-commutative 95.0%

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

   4. Simplified 95.0%

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

   5. Taylor expanded in n around inf 58.1%

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

      1. *-commutative 58.1%

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

      2. associate-/r* 60.5%

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

   7. Simplified 60.5%

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

4. Final simplification 58.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 6.2 \cdot 10^{-210}:\\ \;\;\;\;1 - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{elif}\;x \leq 0.0102:\\ \;\;\;\;\frac{-\log x}{n}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{x}}{n}\\ \end{array} \]

Alternative 10: 40.5% 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 51.0%

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

2. Taylor expanded in x around inf 60.9%

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

   1. mul-1-neg 60.9%

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

   2. log-rec 60.9%

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

   3. mul-1-neg 60.9%

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

   4. distribute-neg-frac 60.9%

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

   5. mul-1-neg 60.9%

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

   6. remove-double-neg 60.9%

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

   7. *-commutative 60.9%

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

4. Simplified 60.9%

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

5. Taylor expanded in n around inf 40.3%

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

6. Final simplification 40.3%

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

Alternative 11: 41.0% accurate, 42.2× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 51.0%

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

2. Taylor expanded in x around inf 60.9%

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

   1. mul-1-neg 60.9%

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

   2. log-rec 60.9%

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

   3. mul-1-neg 60.9%

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

   4. distribute-neg-frac 60.9%

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

   5. mul-1-neg 60.9%

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

   6. remove-double-neg 60.9%

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

   7. *-commutative 60.9%

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

4. Simplified 60.9%

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

5. Taylor expanded in n around inf 40.3%

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

   1. *-commutative 40.3%

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

   2. associate-/r* 41.5%

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

7. Simplified 41.5%

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

8. Final simplification 41.5%

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

Reproduce

herbie shell --seed 2023319 
(FPCore (x n)
  :name "2nthrt (problem 3.4.6)"
  :precision binary64
  (- (pow (+ x 1.0) (/ 1.0 n)) (pow x (/ 1.0 n))))