2nthrt (problem 3.4.6)

Percentage Accurate: 54.6% → 97.9%

Time: 33.7s

Alternatives: 17

Speedup: 1.9×

• could not determine a ground truth

  1. x = -3.9601002522433184e+58
  2. n = -4.363429967770047e+28

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: 54.6% 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: 97.9% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{if}\;\frac{1}{n} \leq -1 \cdot 10^{-20}:\\ \;\;\;\;\frac{t\_0}{n \cdot x}\\ \mathbf{elif}\;\frac{1}{n} \leq 0.001:\\ \;\;\;\;\frac{\mathsf{log1p}\left(\frac{1}{x}\right)}{n}\\ \mathbf{else}:\\ \;\;\;\;e^{\frac{\mathsf{log1p}\left(x\right)}{n}} - t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (let* ((t_0 (pow x (/ 1.0 n))))
   (if (<= (/ 1.0 n) -1e-20)
     (/ t_0 (* n x))
     (if (<= (/ 1.0 n) 0.001)
       (/ (log1p (/ 1.0 x)) n)
       (- (exp (/ (log1p 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-20) {
		tmp = t_0 / (n * x);
	} else if ((1.0 / n) <= 0.001) {
		tmp = log1p((1.0 / x)) / n;
	} else {
		tmp = exp((log1p(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-20) {
		tmp = t_0 / (n * x);
	} else if ((1.0 / n) <= 0.001) {
		tmp = Math.log1p((1.0 / x)) / n;
	} else {
		tmp = Math.exp((Math.log1p(x) / n)) - t_0;
	}
	return tmp;
}

Python

def code(x, n):
	t_0 = math.pow(x, (1.0 / n))
	tmp = 0
	if (1.0 / n) <= -1e-20:
		tmp = t_0 / (n * x)
	elif (1.0 / n) <= 0.001:
		tmp = math.log1p((1.0 / x)) / n
	else:
		tmp = math.exp((math.log1p(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-20)
		tmp = Float64(t_0 / Float64(n * x));
	elseif (Float64(1.0 / n) <= 0.001)
		tmp = Float64(log1p(Float64(1.0 / x)) / n);
	else
		tmp = Float64(exp(Float64(log1p(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-20], N[(t$95$0 / N[(n * x), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 0.001], N[(N[Log[1 + N[(1.0 / x), $MachinePrecision]], $MachinePrecision] / n), $MachinePrecision], N[(N[Exp[N[(N[Log[1 + x], $MachinePrecision] / n), $MachinePrecision]], $MachinePrecision] - t$95$0), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 #s(literal 1 binary64) n) < -9.99999999999999945e-21

   1. Initial program 95.3%

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

   3. Taylor expanded in x around inf 98.8%

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

      1. log-rec 98.8%

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

      2. mul-1-neg 98.8%

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

      3. neg-mul-1 98.8%

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

      4. mul-1-neg 98.8%

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

      5. distribute-frac-neg 98.8%

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

      6. remove-double-neg 98.8%

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

      7. *-rgt-identity 98.8%

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

      8. associate-/l* 98.8%

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

      9. exp-to-pow 98.8%

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

      10. *-commutative 98.8%

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

   5. Simplified 98.8%

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

   if -9.99999999999999945e-21 < (/.f64 #s(literal 1 binary64) n) < 1e-3

   1. Initial program 25.3%

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

   3. Taylor expanded in n around inf 79.5%

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

      1. log1p-define 79.5%

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

   5. Simplified 79.5%

      \[\leadsto \color{blue}{\frac{\mathsf{log1p}\left(x\right) - \log x}{n}} \]
   6. Step-by-step derivation

      1. log1p-undefine 79.5%

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

      2. diff-log 79.6%

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

   7. Applied egg-rr 79.6%

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

      1. +-commutative 79.6%

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

   9. Simplified 79.6%

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

   10. Taylor expanded in n around 0 79.6%

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

       1. +-commutative 79.6%

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

       2. *-lft-identity 79.6%

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

       3. associate-*l/ 73.7%

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

       4. distribute-rgt-in 73.7%

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

       5. rgt-mult-inverse 79.6%

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

       6. *-lft-identity 79.6%

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

       7. log1p-define 98.4%

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

   12. Simplified 98.4%

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

   if 1e-3 < (/.f64 #s(literal 1 binary64) n)

   1. Initial program 55.8%

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

   3. Taylor expanded in n around 0 55.8%

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

      1. log1p-define 98.0%

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

      2. *-rgt-identity 98.0%

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

      3. associate-*l/ 98.0%

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

      4. associate-/l* 98.0%

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

      5. exp-to-pow 98.2%

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

   5. Simplified 98.2%

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

4. Final simplification 98.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{1}{n} \leq -1 \cdot 10^{-20}:\\ \;\;\;\;\frac{{x}^{\left(\frac{1}{n}\right)}}{n \cdot x}\\ \mathbf{elif}\;\frac{1}{n} \leq 0.001:\\ \;\;\;\;\frac{\mathsf{log1p}\left(\frac{1}{x}\right)}{n}\\ \mathbf{else}:\\ \;\;\;\;e^{\frac{\mathsf{log1p}\left(x\right)}{n}} - {x}^{\left(\frac{1}{n}\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 93.5% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{if}\;\frac{1}{n} \leq -1 \cdot 10^{-20}:\\ \;\;\;\;\frac{t\_0}{n \cdot x}\\ \mathbf{elif}\;\frac{1}{n} \leq 10:\\ \;\;\;\;\frac{\mathsf{log1p}\left(\frac{1}{x}\right)}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 2 \cdot 10^{+76}:\\ \;\;\;\;\left(1 + \frac{x}{n}\right) - t\_0\\ \mathbf{else}:\\ \;\;\;\;\mathsf{log1p}\left(\mathsf{expm1}\left(\frac{x}{n}\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (let* ((t_0 (pow x (/ 1.0 n))))
   (if (<= (/ 1.0 n) -1e-20)
     (/ t_0 (* n x))
     (if (<= (/ 1.0 n) 10.0)
       (/ (log1p (/ 1.0 x)) n)
       (if (<= (/ 1.0 n) 2e+76)
         (- (+ 1.0 (/ x n)) t_0)
         (log1p (expm1 (/ x n))))))))

C

double code(double x, double n) {
	double t_0 = pow(x, (1.0 / n));
	double tmp;
	if ((1.0 / n) <= -1e-20) {
		tmp = t_0 / (n * x);
	} else if ((1.0 / n) <= 10.0) {
		tmp = log1p((1.0 / x)) / n;
	} else if ((1.0 / n) <= 2e+76) {
		tmp = (1.0 + (x / n)) - t_0;
	} else {
		tmp = log1p(expm1((x / n)));
	}
	return tmp;
}

Java

public static double code(double x, double n) {
	double t_0 = Math.pow(x, (1.0 / n));
	double tmp;
	if ((1.0 / n) <= -1e-20) {
		tmp = t_0 / (n * x);
	} else if ((1.0 / n) <= 10.0) {
		tmp = Math.log1p((1.0 / x)) / n;
	} else if ((1.0 / n) <= 2e+76) {
		tmp = (1.0 + (x / n)) - t_0;
	} else {
		tmp = Math.log1p(Math.expm1((x / n)));
	}
	return tmp;
}

Python

def code(x, n):
	t_0 = math.pow(x, (1.0 / n))
	tmp = 0
	if (1.0 / n) <= -1e-20:
		tmp = t_0 / (n * x)
	elif (1.0 / n) <= 10.0:
		tmp = math.log1p((1.0 / x)) / n
	elif (1.0 / n) <= 2e+76:
		tmp = (1.0 + (x / n)) - t_0
	else:
		tmp = math.log1p(math.expm1((x / n)))
	return tmp

Julia

function code(x, n)
	t_0 = x ^ Float64(1.0 / n)
	tmp = 0.0
	if (Float64(1.0 / n) <= -1e-20)
		tmp = Float64(t_0 / Float64(n * x));
	elseif (Float64(1.0 / n) <= 10.0)
		tmp = Float64(log1p(Float64(1.0 / x)) / n);
	elseif (Float64(1.0 / n) <= 2e+76)
		tmp = Float64(Float64(1.0 + Float64(x / n)) - t_0);
	else
		tmp = log1p(expm1(Float64(x / n)));
	end
	return tmp
end

Wolfram

code[x_, n_] := Block[{t$95$0 = N[Power[x, N[(1.0 / n), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[N[(1.0 / n), $MachinePrecision], -1e-20], N[(t$95$0 / N[(n * x), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 10.0], N[(N[Log[1 + N[(1.0 / x), $MachinePrecision]], $MachinePrecision] / n), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 2e+76], N[(N[(1.0 + N[(x / n), $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision], N[Log[1 + N[(Exp[N[(x / n), $MachinePrecision]] - 1), $MachinePrecision]], $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if (/.f64 #s(literal 1 binary64) n) < -9.99999999999999945e-21

   1. Initial program 95.3%

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

   3. Taylor expanded in x around inf 98.8%

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

      1. log-rec 98.8%

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

      2. mul-1-neg 98.8%

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

      3. neg-mul-1 98.8%

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

      4. mul-1-neg 98.8%

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

      5. distribute-frac-neg 98.8%

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

      6. remove-double-neg 98.8%

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

      7. *-rgt-identity 98.8%

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

      8. associate-/l* 98.8%

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

      9. exp-to-pow 98.8%

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

      10. *-commutative 98.8%

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

   5. Simplified 98.8%

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

   if -9.99999999999999945e-21 < (/.f64 #s(literal 1 binary64) n) < 10

   1. Initial program 25.1%

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

   3. Taylor expanded in n around inf 78.9%

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

      1. log1p-define 78.9%

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

   5. Simplified 78.9%

      \[\leadsto \color{blue}{\frac{\mathsf{log1p}\left(x\right) - \log x}{n}} \]
   6. Step-by-step derivation

      1. log1p-undefine 78.9%

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

      2. diff-log 79.0%

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

   7. Applied egg-rr 79.0%

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

      1. +-commutative 79.0%

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

   9. Simplified 79.0%

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

   10. Taylor expanded in n around 0 79.0%

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

       1. +-commutative 79.0%

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

       2. *-lft-identity 79.0%

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

       3. associate-*l/ 73.2%

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

       4. distribute-rgt-in 73.2%

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

       5. rgt-mult-inverse 79.0%

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

       6. *-lft-identity 79.0%

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

       7. log1p-define 97.7%

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

   12. Simplified 97.7%

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

   if 10 < (/.f64 #s(literal 1 binary64) n) < 2.0000000000000001e76

   1. Initial program 89.1%

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

   3. Taylor expanded in x around 0 92.0%

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

   if 2.0000000000000001e76 < (/.f64 #s(literal 1 binary64) n)

   1. Initial program 44.0%

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

   3. Taylor expanded in x around 0 29.0%

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

   4. Taylor expanded in x around inf 5.4%

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

      1. log1p-expm1-u 76.4%

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

   6. Applied egg-rr 76.4%

      \[\leadsto \color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(\frac{x}{n}\right)\right)} \]
3. Recombined 4 regimes into one program.

4. Final simplification 94.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{1}{n} \leq -1 \cdot 10^{-20}:\\ \;\;\;\;\frac{{x}^{\left(\frac{1}{n}\right)}}{n \cdot x}\\ \mathbf{elif}\;\frac{1}{n} \leq 10:\\ \;\;\;\;\frac{\mathsf{log1p}\left(\frac{1}{x}\right)}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 2 \cdot 10^{+76}:\\ \;\;\;\;\left(1 + \frac{x}{n}\right) - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{log1p}\left(\mathsf{expm1}\left(\frac{x}{n}\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 97.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{if}\;\frac{1}{n} \leq -1 \cdot 10^{-20}:\\ \;\;\;\;\frac{t\_0}{n \cdot x}\\ \mathbf{elif}\;\frac{1}{n} \leq 0.001:\\ \;\;\;\;\frac{\mathsf{log1p}\left(\frac{1}{x}\right)}{n}\\ \mathbf{else}:\\ \;\;\;\;e^{\frac{x}{n}} - t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (let* ((t_0 (pow x (/ 1.0 n))))
   (if (<= (/ 1.0 n) -1e-20)
     (/ t_0 (* n x))
     (if (<= (/ 1.0 n) 0.001) (/ (log1p (/ 1.0 x)) n) (- (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-20) {
		tmp = t_0 / (n * x);
	} else if ((1.0 / n) <= 0.001) {
		tmp = log1p((1.0 / x)) / n;
	} 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-20) {
		tmp = t_0 / (n * x);
	} else if ((1.0 / n) <= 0.001) {
		tmp = Math.log1p((1.0 / x)) / n;
	} else {
		tmp = Math.exp((x / n)) - t_0;
	}
	return tmp;
}

Python

def code(x, n):
	t_0 = math.pow(x, (1.0 / n))
	tmp = 0
	if (1.0 / n) <= -1e-20:
		tmp = t_0 / (n * x)
	elif (1.0 / n) <= 0.001:
		tmp = math.log1p((1.0 / x)) / n
	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-20)
		tmp = Float64(t_0 / Float64(n * x));
	elseif (Float64(1.0 / n) <= 0.001)
		tmp = Float64(log1p(Float64(1.0 / x)) / n);
	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-20], N[(t$95$0 / N[(n * x), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 0.001], N[(N[Log[1 + N[(1.0 / x), $MachinePrecision]], $MachinePrecision] / n), $MachinePrecision], N[(N[Exp[N[(x / n), $MachinePrecision]], $MachinePrecision] - t$95$0), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 #s(literal 1 binary64) n) < -9.99999999999999945e-21

   1. Initial program 95.3%

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

   3. Taylor expanded in x around inf 98.8%

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

      1. log-rec 98.8%

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

      2. mul-1-neg 98.8%

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

      3. neg-mul-1 98.8%

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

      4. mul-1-neg 98.8%

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

      5. distribute-frac-neg 98.8%

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

      6. remove-double-neg 98.8%

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

      7. *-rgt-identity 98.8%

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

      8. associate-/l* 98.8%

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

      9. exp-to-pow 98.8%

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

      10. *-commutative 98.8%

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

   5. Simplified 98.8%

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

   if -9.99999999999999945e-21 < (/.f64 #s(literal 1 binary64) n) < 1e-3

   1. Initial program 25.3%

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

   3. Taylor expanded in n around inf 79.5%

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

      1. log1p-define 79.5%

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

   5. Simplified 79.5%

      \[\leadsto \color{blue}{\frac{\mathsf{log1p}\left(x\right) - \log x}{n}} \]
   6. Step-by-step derivation

      1. log1p-undefine 79.5%

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

      2. diff-log 79.6%

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

   7. Applied egg-rr 79.6%

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

      1. +-commutative 79.6%

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

   9. Simplified 79.6%

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

   10. Taylor expanded in n around 0 79.6%

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

       1. +-commutative 79.6%

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

       2. *-lft-identity 79.6%

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

       3. associate-*l/ 73.7%

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

       4. distribute-rgt-in 73.7%

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

       5. rgt-mult-inverse 79.6%

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

       6. *-lft-identity 79.6%

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

       7. log1p-define 98.4%

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

   12. Simplified 98.4%

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

   if 1e-3 < (/.f64 #s(literal 1 binary64) n)

   1. Initial program 55.8%

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

   3. Taylor expanded in n around 0 55.8%

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

      1. log1p-define 98.0%

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

      2. *-rgt-identity 98.0%

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

      3. associate-*l/ 98.0%

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

      4. associate-/l* 98.0%

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

      5. exp-to-pow 98.2%

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

   5. Simplified 98.2%

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

   6. Taylor expanded in x around 0 98.1%

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

4. Final simplification 98.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{1}{n} \leq -1 \cdot 10^{-20}:\\ \;\;\;\;\frac{{x}^{\left(\frac{1}{n}\right)}}{n \cdot x}\\ \mathbf{elif}\;\frac{1}{n} \leq 0.001:\\ \;\;\;\;\frac{\mathsf{log1p}\left(\frac{1}{x}\right)}{n}\\ \mathbf{else}:\\ \;\;\;\;e^{\frac{x}{n}} - {x}^{\left(\frac{1}{n}\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 93.8% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{if}\;\frac{1}{n} \leq -1 \cdot 10^{-20}:\\ \;\;\;\;\frac{t\_0}{n \cdot x}\\ \mathbf{elif}\;\frac{1}{n} \leq 10:\\ \;\;\;\;\frac{\mathsf{log1p}\left(\frac{1}{x}\right)}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 5 \cdot 10^{+94}:\\ \;\;\;\;\left(1 + \frac{x}{n}\right) - t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{expm1}\left(\frac{1 + \frac{-1 + \frac{1.1666666666666667}{x}}{x}}{x}\right)}{n}\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (let* ((t_0 (pow x (/ 1.0 n))))
   (if (<= (/ 1.0 n) -1e-20)
     (/ t_0 (* n x))
     (if (<= (/ 1.0 n) 10.0)
       (/ (log1p (/ 1.0 x)) n)
       (if (<= (/ 1.0 n) 5e+94)
         (- (+ 1.0 (/ x n)) t_0)
         (/
          (expm1 (/ (+ 1.0 (/ (+ -1.0 (/ 1.1666666666666667 x)) x)) x))
          n))))))

C

double code(double x, double n) {
	double t_0 = pow(x, (1.0 / n));
	double tmp;
	if ((1.0 / n) <= -1e-20) {
		tmp = t_0 / (n * x);
	} else if ((1.0 / n) <= 10.0) {
		tmp = log1p((1.0 / x)) / n;
	} else if ((1.0 / n) <= 5e+94) {
		tmp = (1.0 + (x / n)) - t_0;
	} else {
		tmp = expm1(((1.0 + ((-1.0 + (1.1666666666666667 / x)) / x)) / x)) / n;
	}
	return tmp;
}

Java

public static double code(double x, double n) {
	double t_0 = Math.pow(x, (1.0 / n));
	double tmp;
	if ((1.0 / n) <= -1e-20) {
		tmp = t_0 / (n * x);
	} else if ((1.0 / n) <= 10.0) {
		tmp = Math.log1p((1.0 / x)) / n;
	} else if ((1.0 / n) <= 5e+94) {
		tmp = (1.0 + (x / n)) - t_0;
	} else {
		tmp = Math.expm1(((1.0 + ((-1.0 + (1.1666666666666667 / x)) / x)) / x)) / n;
	}
	return tmp;
}

Python

def code(x, n):
	t_0 = math.pow(x, (1.0 / n))
	tmp = 0
	if (1.0 / n) <= -1e-20:
		tmp = t_0 / (n * x)
	elif (1.0 / n) <= 10.0:
		tmp = math.log1p((1.0 / x)) / n
	elif (1.0 / n) <= 5e+94:
		tmp = (1.0 + (x / n)) - t_0
	else:
		tmp = math.expm1(((1.0 + ((-1.0 + (1.1666666666666667 / x)) / x)) / x)) / n
	return tmp

Julia

function code(x, n)
	t_0 = x ^ Float64(1.0 / n)
	tmp = 0.0
	if (Float64(1.0 / n) <= -1e-20)
		tmp = Float64(t_0 / Float64(n * x));
	elseif (Float64(1.0 / n) <= 10.0)
		tmp = Float64(log1p(Float64(1.0 / x)) / n);
	elseif (Float64(1.0 / n) <= 5e+94)
		tmp = Float64(Float64(1.0 + Float64(x / n)) - t_0);
	else
		tmp = Float64(expm1(Float64(Float64(1.0 + Float64(Float64(-1.0 + Float64(1.1666666666666667 / x)) / x)) / x)) / n);
	end
	return tmp
end

Wolfram

code[x_, n_] := Block[{t$95$0 = N[Power[x, N[(1.0 / n), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[N[(1.0 / n), $MachinePrecision], -1e-20], N[(t$95$0 / N[(n * x), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 10.0], N[(N[Log[1 + N[(1.0 / x), $MachinePrecision]], $MachinePrecision] / n), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 5e+94], N[(N[(1.0 + N[(x / n), $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision], N[(N[(Exp[N[(N[(1.0 + N[(N[(-1.0 + N[(1.1666666666666667 / x), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]] - 1), $MachinePrecision] / n), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if (/.f64 #s(literal 1 binary64) n) < -9.99999999999999945e-21

   1. Initial program 95.3%

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

   3. Taylor expanded in x around inf 98.8%

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

      1. log-rec 98.8%

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

      2. mul-1-neg 98.8%

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

      3. neg-mul-1 98.8%

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

      4. mul-1-neg 98.8%

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

      5. distribute-frac-neg 98.8%

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

      6. remove-double-neg 98.8%

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

      7. *-rgt-identity 98.8%

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

      8. associate-/l* 98.8%

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

      9. exp-to-pow 98.8%

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

      10. *-commutative 98.8%

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

   5. Simplified 98.8%

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

   if -9.99999999999999945e-21 < (/.f64 #s(literal 1 binary64) n) < 10

   1. Initial program 25.1%

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

   3. Taylor expanded in n around inf 78.9%

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

      1. log1p-define 78.9%

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

   5. Simplified 78.9%

      \[\leadsto \color{blue}{\frac{\mathsf{log1p}\left(x\right) - \log x}{n}} \]
   6. Step-by-step derivation

      1. log1p-undefine 78.9%

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

      2. diff-log 79.0%

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

   7. Applied egg-rr 79.0%

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

      1. +-commutative 79.0%

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

   9. Simplified 79.0%

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

   10. Taylor expanded in n around 0 79.0%

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

       1. +-commutative 79.0%

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

       2. *-lft-identity 79.0%

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

       3. associate-*l/ 73.2%

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

       4. distribute-rgt-in 73.2%

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

       5. rgt-mult-inverse 79.0%

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

       6. *-lft-identity 79.0%

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

       7. log1p-define 97.7%

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

   12. Simplified 97.7%

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

   if 10 < (/.f64 #s(literal 1 binary64) n) < 5.0000000000000001e94

   1. Initial program 84.4%

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

   3. Taylor expanded in x around 0 86.9%

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

   if 5.0000000000000001e94 < (/.f64 #s(literal 1 binary64) n)

   1. Initial program 43.6%

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

   3. Taylor expanded in n around inf 8.7%

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

      1. log1p-define 8.7%

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

   5. Simplified 8.7%

      \[\leadsto \color{blue}{\frac{\mathsf{log1p}\left(x\right) - \log x}{n}} \]
   6. Step-by-step derivation

      1. expm1-log1p-u 8.7%

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

   7. Applied egg-rr 8.7%

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

   8. Taylor expanded in x around inf 73.8%

      \[\leadsto \frac{\mathsf{expm1}\left(\color{blue}{\frac{\left(1 + \frac{1.1666666666666667}{{x}^{2}}\right) - \frac{1}{x}}{x}}\right)}{n} \]
   9. Step-by-step derivation

      1. associate--l+ 73.8%

         \[\leadsto \frac{\mathsf{expm1}\left(\frac{\color{blue}{1 + \left(\frac{1.1666666666666667}{{x}^{2}} - \frac{1}{x}\right)}}{x}\right)}{n} \]

      2. unpow2 73.8%

         \[\leadsto \frac{\mathsf{expm1}\left(\frac{1 + \left(\frac{1.1666666666666667}{\color{blue}{x \cdot x}} - \frac{1}{x}\right)}{x}\right)}{n} \]

      3. associate-/r* 73.8%

         \[\leadsto \frac{\mathsf{expm1}\left(\frac{1 + \left(\color{blue}{\frac{\frac{1.1666666666666667}{x}}{x}} - \frac{1}{x}\right)}{x}\right)}{n} \]

      4. metadata-eval 73.8%

         \[\leadsto \frac{\mathsf{expm1}\left(\frac{1 + \left(\frac{\frac{\color{blue}{1.1666666666666667 \cdot 1}}{x}}{x} - \frac{1}{x}\right)}{x}\right)}{n} \]

      5. associate-*r/ 73.8%

         \[\leadsto \frac{\mathsf{expm1}\left(\frac{1 + \left(\frac{\color{blue}{1.1666666666666667 \cdot \frac{1}{x}}}{x} - \frac{1}{x}\right)}{x}\right)}{n} \]

      6. div-sub 73.8%

         \[\leadsto \frac{\mathsf{expm1}\left(\frac{1 + \color{blue}{\frac{1.1666666666666667 \cdot \frac{1}{x} - 1}{x}}}{x}\right)}{n} \]

      7. sub-neg 73.8%

         \[\leadsto \frac{\mathsf{expm1}\left(\frac{1 + \frac{\color{blue}{1.1666666666666667 \cdot \frac{1}{x} + \left(-1\right)}}{x}}{x}\right)}{n} \]

      8. metadata-eval 73.8%

         \[\leadsto \frac{\mathsf{expm1}\left(\frac{1 + \frac{1.1666666666666667 \cdot \frac{1}{x} + \color{blue}{-1}}{x}}{x}\right)}{n} \]

      9. +-commutative 73.8%

         \[\leadsto \frac{\mathsf{expm1}\left(\frac{1 + \frac{\color{blue}{-1 + 1.1666666666666667 \cdot \frac{1}{x}}}{x}}{x}\right)}{n} \]

      10. associate-*r/ 73.8%

          \[\leadsto \frac{\mathsf{expm1}\left(\frac{1 + \frac{-1 + \color{blue}{\frac{1.1666666666666667 \cdot 1}{x}}}{x}}{x}\right)}{n} \]

      11. metadata-eval 73.8%

          \[\leadsto \frac{\mathsf{expm1}\left(\frac{1 + \frac{-1 + \frac{\color{blue}{1.1666666666666667}}{x}}{x}}{x}\right)}{n} \]

   10. Simplified 73.8%

       \[\leadsto \frac{\mathsf{expm1}\left(\color{blue}{\frac{1 + \frac{-1 + \frac{1.1666666666666667}{x}}{x}}{x}}\right)}{n} \]
3. Recombined 4 regimes into one program.

4. Final simplification 94.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{1}{n} \leq -1 \cdot 10^{-20}:\\ \;\;\;\;\frac{{x}^{\left(\frac{1}{n}\right)}}{n \cdot x}\\ \mathbf{elif}\;\frac{1}{n} \leq 10:\\ \;\;\;\;\frac{\mathsf{log1p}\left(\frac{1}{x}\right)}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 5 \cdot 10^{+94}:\\ \;\;\;\;\left(1 + \frac{x}{n}\right) - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{expm1}\left(\frac{1 + \frac{-1 + \frac{1.1666666666666667}{x}}{x}}{x}\right)}{n}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 93.7% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{if}\;\frac{1}{n} \leq -1 \cdot 10^{-20}:\\ \;\;\;\;\frac{t\_0}{n \cdot x}\\ \mathbf{elif}\;\frac{1}{n} \leq 10:\\ \;\;\;\;\frac{\mathsf{log1p}\left(\frac{1}{x}\right)}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 10^{+108}:\\ \;\;\;\;\left(1 + \frac{x}{n}\right) - t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{log1p}\left(x + -1\right)}{-n}\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (let* ((t_0 (pow x (/ 1.0 n))))
   (if (<= (/ 1.0 n) -1e-20)
     (/ t_0 (* n x))
     (if (<= (/ 1.0 n) 10.0)
       (/ (log1p (/ 1.0 x)) n)
       (if (<= (/ 1.0 n) 1e+108)
         (- (+ 1.0 (/ x n)) t_0)
         (/ (log1p (+ x -1.0)) (- n)))))))

C

double code(double x, double n) {
	double t_0 = pow(x, (1.0 / n));
	double tmp;
	if ((1.0 / n) <= -1e-20) {
		tmp = t_0 / (n * x);
	} else if ((1.0 / n) <= 10.0) {
		tmp = log1p((1.0 / x)) / n;
	} else if ((1.0 / n) <= 1e+108) {
		tmp = (1.0 + (x / n)) - t_0;
	} else {
		tmp = log1p((x + -1.0)) / -n;
	}
	return tmp;
}

Java

public static double code(double x, double n) {
	double t_0 = Math.pow(x, (1.0 / n));
	double tmp;
	if ((1.0 / n) <= -1e-20) {
		tmp = t_0 / (n * x);
	} else if ((1.0 / n) <= 10.0) {
		tmp = Math.log1p((1.0 / x)) / n;
	} else if ((1.0 / n) <= 1e+108) {
		tmp = (1.0 + (x / n)) - t_0;
	} else {
		tmp = Math.log1p((x + -1.0)) / -n;
	}
	return tmp;
}

Python

def code(x, n):
	t_0 = math.pow(x, (1.0 / n))
	tmp = 0
	if (1.0 / n) <= -1e-20:
		tmp = t_0 / (n * x)
	elif (1.0 / n) <= 10.0:
		tmp = math.log1p((1.0 / x)) / n
	elif (1.0 / n) <= 1e+108:
		tmp = (1.0 + (x / n)) - t_0
	else:
		tmp = math.log1p((x + -1.0)) / -n
	return tmp

Julia

function code(x, n)
	t_0 = x ^ Float64(1.0 / n)
	tmp = 0.0
	if (Float64(1.0 / n) <= -1e-20)
		tmp = Float64(t_0 / Float64(n * x));
	elseif (Float64(1.0 / n) <= 10.0)
		tmp = Float64(log1p(Float64(1.0 / x)) / n);
	elseif (Float64(1.0 / n) <= 1e+108)
		tmp = Float64(Float64(1.0 + Float64(x / n)) - t_0);
	else
		tmp = Float64(log1p(Float64(x + -1.0)) / Float64(-n));
	end
	return tmp
end

Wolfram

code[x_, n_] := Block[{t$95$0 = N[Power[x, N[(1.0 / n), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[N[(1.0 / n), $MachinePrecision], -1e-20], N[(t$95$0 / N[(n * x), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 10.0], N[(N[Log[1 + N[(1.0 / x), $MachinePrecision]], $MachinePrecision] / n), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 1e+108], N[(N[(1.0 + N[(x / n), $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision], N[(N[Log[1 + N[(x + -1.0), $MachinePrecision]], $MachinePrecision] / (-n)), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if (/.f64 #s(literal 1 binary64) n) < -9.99999999999999945e-21

   1. Initial program 95.3%

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

   3. Taylor expanded in x around inf 98.8%

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

      1. log-rec 98.8%

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

      2. mul-1-neg 98.8%

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

      3. neg-mul-1 98.8%

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

      4. mul-1-neg 98.8%

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

      5. distribute-frac-neg 98.8%

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

      6. remove-double-neg 98.8%

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

      7. *-rgt-identity 98.8%

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

      8. associate-/l* 98.8%

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

      9. exp-to-pow 98.8%

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

      10. *-commutative 98.8%

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

   5. Simplified 98.8%

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

   if -9.99999999999999945e-21 < (/.f64 #s(literal 1 binary64) n) < 10

   1. Initial program 25.1%

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

   3. Taylor expanded in n around inf 78.9%

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

      1. log1p-define 78.9%

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

   5. Simplified 78.9%

      \[\leadsto \color{blue}{\frac{\mathsf{log1p}\left(x\right) - \log x}{n}} \]
   6. Step-by-step derivation

      1. log1p-undefine 78.9%

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

      2. diff-log 79.0%

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

   7. Applied egg-rr 79.0%

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

      1. +-commutative 79.0%

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

   9. Simplified 79.0%

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

   10. Taylor expanded in n around 0 79.0%

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

       1. +-commutative 79.0%

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

       2. *-lft-identity 79.0%

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

       3. associate-*l/ 73.2%

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

       4. distribute-rgt-in 73.2%

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

       5. rgt-mult-inverse 79.0%

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

       6. *-lft-identity 79.0%

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

       7. log1p-define 97.7%

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

   12. Simplified 97.7%

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

   if 10 < (/.f64 #s(literal 1 binary64) n) < 1e108

   1. Initial program 81.8%

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

   3. Taylor expanded in x around 0 78.8%

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

   if 1e108 < (/.f64 #s(literal 1 binary64) n)

   1. Initial program 41.1%

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

   3. Taylor expanded in x around 0 25.0%

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

      1. *-rgt-identity 25.0%

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

      2. associate-*l/ 25.0%

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

      3. associate-/l* 25.0%

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

      4. exp-to-pow 25.0%

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

   5. Simplified 25.0%

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

   6. Taylor expanded in n around inf 9.1%

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

      1. neg-mul-1 9.1%

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

      2. distribute-neg-frac2 9.1%

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

   8. Simplified 9.1%

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

      1. log1p-expm1-u 61.7%

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

      2. expm1-undefine 61.7%

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

      3. add-exp-log 61.7%

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

   10. Applied egg-rr 61.7%

       \[\leadsto \frac{\color{blue}{\mathsf{log1p}\left(x - 1\right)}}{-n} \]
3. Recombined 4 regimes into one program.

4. Final simplification 92.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{1}{n} \leq -1 \cdot 10^{-20}:\\ \;\;\;\;\frac{{x}^{\left(\frac{1}{n}\right)}}{n \cdot x}\\ \mathbf{elif}\;\frac{1}{n} \leq 10:\\ \;\;\;\;\frac{\mathsf{log1p}\left(\frac{1}{x}\right)}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 10^{+108}:\\ \;\;\;\;\left(1 + \frac{x}{n}\right) - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{log1p}\left(x + -1\right)}{-n}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 92.8% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{if}\;\frac{1}{n} \leq -1 \cdot 10^{-20}:\\ \;\;\;\;\frac{t\_0}{n \cdot x}\\ \mathbf{elif}\;\frac{1}{n} \leq 10:\\ \;\;\;\;\frac{\mathsf{log1p}\left(\frac{1}{x}\right)}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 2 \cdot 10^{+76}:\\ \;\;\;\;1 - t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{log1p}\left(x + -1\right)}{-n}\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (let* ((t_0 (pow x (/ 1.0 n))))
   (if (<= (/ 1.0 n) -1e-20)
     (/ t_0 (* n x))
     (if (<= (/ 1.0 n) 10.0)
       (/ (log1p (/ 1.0 x)) n)
       (if (<= (/ 1.0 n) 2e+76) (- 1.0 t_0) (/ (log1p (+ x -1.0)) (- n)))))))

C

double code(double x, double n) {
	double t_0 = pow(x, (1.0 / n));
	double tmp;
	if ((1.0 / n) <= -1e-20) {
		tmp = t_0 / (n * x);
	} else if ((1.0 / n) <= 10.0) {
		tmp = log1p((1.0 / x)) / n;
	} else if ((1.0 / n) <= 2e+76) {
		tmp = 1.0 - t_0;
	} else {
		tmp = log1p((x + -1.0)) / -n;
	}
	return tmp;
}

Java

public static double code(double x, double n) {
	double t_0 = Math.pow(x, (1.0 / n));
	double tmp;
	if ((1.0 / n) <= -1e-20) {
		tmp = t_0 / (n * x);
	} else if ((1.0 / n) <= 10.0) {
		tmp = Math.log1p((1.0 / x)) / n;
	} else if ((1.0 / n) <= 2e+76) {
		tmp = 1.0 - t_0;
	} else {
		tmp = Math.log1p((x + -1.0)) / -n;
	}
	return tmp;
}

Python

def code(x, n):
	t_0 = math.pow(x, (1.0 / n))
	tmp = 0
	if (1.0 / n) <= -1e-20:
		tmp = t_0 / (n * x)
	elif (1.0 / n) <= 10.0:
		tmp = math.log1p((1.0 / x)) / n
	elif (1.0 / n) <= 2e+76:
		tmp = 1.0 - t_0
	else:
		tmp = math.log1p((x + -1.0)) / -n
	return tmp

Julia

function code(x, n)
	t_0 = x ^ Float64(1.0 / n)
	tmp = 0.0
	if (Float64(1.0 / n) <= -1e-20)
		tmp = Float64(t_0 / Float64(n * x));
	elseif (Float64(1.0 / n) <= 10.0)
		tmp = Float64(log1p(Float64(1.0 / x)) / n);
	elseif (Float64(1.0 / n) <= 2e+76)
		tmp = Float64(1.0 - t_0);
	else
		tmp = Float64(log1p(Float64(x + -1.0)) / Float64(-n));
	end
	return tmp
end

Wolfram

code[x_, n_] := Block[{t$95$0 = N[Power[x, N[(1.0 / n), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[N[(1.0 / n), $MachinePrecision], -1e-20], N[(t$95$0 / N[(n * x), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 10.0], N[(N[Log[1 + N[(1.0 / x), $MachinePrecision]], $MachinePrecision] / n), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 2e+76], N[(1.0 - t$95$0), $MachinePrecision], N[(N[Log[1 + N[(x + -1.0), $MachinePrecision]], $MachinePrecision] / (-n)), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if (/.f64 #s(literal 1 binary64) n) < -9.99999999999999945e-21

   1. Initial program 95.3%

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

   3. Taylor expanded in x around inf 98.8%

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

      1. log-rec 98.8%

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

      2. mul-1-neg 98.8%

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

      3. neg-mul-1 98.8%

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

      4. mul-1-neg 98.8%

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

      5. distribute-frac-neg 98.8%

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

      6. remove-double-neg 98.8%

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

      7. *-rgt-identity 98.8%

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

      8. associate-/l* 98.8%

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

      9. exp-to-pow 98.8%

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

      10. *-commutative 98.8%

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

   5. Simplified 98.8%

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

   if -9.99999999999999945e-21 < (/.f64 #s(literal 1 binary64) n) < 10

   1. Initial program 25.1%

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

   3. Taylor expanded in n around inf 78.9%

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

      1. log1p-define 78.9%

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

   5. Simplified 78.9%

      \[\leadsto \color{blue}{\frac{\mathsf{log1p}\left(x\right) - \log x}{n}} \]
   6. Step-by-step derivation

      1. log1p-undefine 78.9%

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

      2. diff-log 79.0%

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

   7. Applied egg-rr 79.0%

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

      1. +-commutative 79.0%

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

   9. Simplified 79.0%

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

   10. Taylor expanded in n around 0 79.0%

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

       1. +-commutative 79.0%

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

       2. *-lft-identity 79.0%

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

       3. associate-*l/ 73.2%

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

       4. distribute-rgt-in 73.2%

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

       5. rgt-mult-inverse 79.0%

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

       6. *-lft-identity 79.0%

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

       7. log1p-define 97.7%

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

   12. Simplified 97.7%

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

   if 10 < (/.f64 #s(literal 1 binary64) n) < 2.0000000000000001e76

   1. Initial program 89.1%

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

   3. Taylor expanded in x around 0 89.1%

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

      1. *-rgt-identity 89.1%

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

      2. associate-*l/ 89.1%

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

      3. associate-/l* 89.1%

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

      4. exp-to-pow 89.1%

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

   5. Simplified 89.1%

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

   if 2.0000000000000001e76 < (/.f64 #s(literal 1 binary64) n)

   1. Initial program 44.0%

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

   3. Taylor expanded in x around 0 27.4%

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

      1. *-rgt-identity 27.4%

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

      2. associate-*l/ 27.4%

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

      3. associate-/l* 27.4%

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

      4. exp-to-pow 27.4%

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

   5. Simplified 27.4%

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

   6. Taylor expanded in n around inf 8.5%

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

      1. neg-mul-1 8.5%

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

      2. distribute-neg-frac2 8.5%

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

   8. Simplified 8.5%

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

      1. log1p-expm1-u 58.9%

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

      2. expm1-undefine 58.9%

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

      3. add-exp-log 58.9%

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

   10. Applied egg-rr 58.9%

       \[\leadsto \frac{\color{blue}{\mathsf{log1p}\left(x - 1\right)}}{-n} \]
3. Recombined 4 regimes into one program.

4. Final simplification 92.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{1}{n} \leq -1 \cdot 10^{-20}:\\ \;\;\;\;\frac{{x}^{\left(\frac{1}{n}\right)}}{n \cdot x}\\ \mathbf{elif}\;\frac{1}{n} \leq 10:\\ \;\;\;\;\frac{\mathsf{log1p}\left(\frac{1}{x}\right)}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 2 \cdot 10^{+76}:\\ \;\;\;\;1 - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{log1p}\left(x + -1\right)}{-n}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 78.9% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{1}{n} \leq -500:\\ \;\;\;\;\frac{1}{n} + \frac{-1}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 10:\\ \;\;\;\;\frac{\mathsf{log1p}\left(\frac{1}{x}\right)}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 2 \cdot 10^{+76}:\\ \;\;\;\;1 - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{log1p}\left(x + -1\right)}{-n}\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (if (<= (/ 1.0 n) -500.0)
   (+ (/ 1.0 n) (/ -1.0 n))
   (if (<= (/ 1.0 n) 10.0)
     (/ (log1p (/ 1.0 x)) n)
     (if (<= (/ 1.0 n) 2e+76)
       (- 1.0 (pow x (/ 1.0 n)))
       (/ (log1p (+ x -1.0)) (- n))))))

C

double code(double x, double n) {
	double tmp;
	if ((1.0 / n) <= -500.0) {
		tmp = (1.0 / n) + (-1.0 / n);
	} else if ((1.0 / n) <= 10.0) {
		tmp = log1p((1.0 / x)) / n;
	} else if ((1.0 / n) <= 2e+76) {
		tmp = 1.0 - pow(x, (1.0 / n));
	} else {
		tmp = log1p((x + -1.0)) / -n;
	}
	return tmp;
}

Java

public static double code(double x, double n) {
	double tmp;
	if ((1.0 / n) <= -500.0) {
		tmp = (1.0 / n) + (-1.0 / n);
	} else if ((1.0 / n) <= 10.0) {
		tmp = Math.log1p((1.0 / x)) / n;
	} else if ((1.0 / n) <= 2e+76) {
		tmp = 1.0 - Math.pow(x, (1.0 / n));
	} else {
		tmp = Math.log1p((x + -1.0)) / -n;
	}
	return tmp;
}

Python

def code(x, n):
	tmp = 0
	if (1.0 / n) <= -500.0:
		tmp = (1.0 / n) + (-1.0 / n)
	elif (1.0 / n) <= 10.0:
		tmp = math.log1p((1.0 / x)) / n
	elif (1.0 / n) <= 2e+76:
		tmp = 1.0 - math.pow(x, (1.0 / n))
	else:
		tmp = math.log1p((x + -1.0)) / -n
	return tmp

Julia

function code(x, n)
	tmp = 0.0
	if (Float64(1.0 / n) <= -500.0)
		tmp = Float64(Float64(1.0 / n) + Float64(-1.0 / n));
	elseif (Float64(1.0 / n) <= 10.0)
		tmp = Float64(log1p(Float64(1.0 / x)) / n);
	elseif (Float64(1.0 / n) <= 2e+76)
		tmp = Float64(1.0 - (x ^ Float64(1.0 / n)));
	else
		tmp = Float64(log1p(Float64(x + -1.0)) / Float64(-n));
	end
	return tmp
end

Wolfram

code[x_, n_] := If[LessEqual[N[(1.0 / n), $MachinePrecision], -500.0], N[(N[(1.0 / n), $MachinePrecision] + N[(-1.0 / n), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 10.0], N[(N[Log[1 + N[(1.0 / x), $MachinePrecision]], $MachinePrecision] / n), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 2e+76], N[(1.0 - N[Power[x, N[(1.0 / n), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[Log[1 + N[(x + -1.0), $MachinePrecision]], $MachinePrecision] / (-n)), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if (/.f64 #s(literal 1 binary64) n) < -500

   1. Initial program 100.0%

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

   3. Taylor expanded in n around inf 64.5%

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

      1. log1p-define 64.5%

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

   5. Simplified 64.5%

      \[\leadsto \color{blue}{\frac{\mathsf{log1p}\left(x\right) - \log x}{n}} \]
   6. Step-by-step derivation

      1. log1p-undefine 64.5%

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

      2. diff-log 64.5%

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

   7. Applied egg-rr 64.5%

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

      1. +-commutative 64.5%

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

   9. Simplified 64.5%

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

       1. log-div 64.5%

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

       2. +-commutative 64.5%

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

       3. log1p-undefine 64.5%

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

       4. expm1-log1p-u 64.5%

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

       5. expm1-undefine 64.5%

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

       6. div-sub 64.5%

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

       7. log1p-undefine 64.5%

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

       8. rem-exp-log 64.5%

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

   11. Applied egg-rr 64.5%

       \[\leadsto \color{blue}{\frac{1 + \left(\mathsf{log1p}\left(x\right) - \log x\right)}{n} - \frac{1}{n}} \]
   12. Step-by-step derivation

       1. sub-neg 64.5%

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

       2. rem-log-exp 64.5%

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

       3. exp-diff 64.5%

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

       4. log1p-define 64.5%

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

       5. +-commutative 64.5%

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

       6. rem-exp-log 3.3%

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

       7. rem-exp-log 64.5%

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

       8. *-lft-identity 64.5%

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

       9. associate-*l/ 59.3%

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

       10. distribute-lft-in 59.3%

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

       11. *-rgt-identity 59.3%

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

       12. lft-mult-inverse 64.5%

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

       13. log1p-define 64.5%

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

       14. distribute-neg-frac 64.5%

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

       15. metadata-eval 64.5%

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

   13. Simplified 64.5%

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

   14. Taylor expanded in x around inf 64.6%

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

   if -500 < (/.f64 #s(literal 1 binary64) n) < 10

   1. Initial program 25.6%

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

   3. Taylor expanded in n around inf 75.6%

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

      1. log1p-define 75.6%

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

   5. Simplified 75.6%

      \[\leadsto \color{blue}{\frac{\mathsf{log1p}\left(x\right) - \log x}{n}} \]
   6. Step-by-step derivation

      1. log1p-undefine 75.6%

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

      2. diff-log 75.7%

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

   7. Applied egg-rr 75.7%

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

      1. +-commutative 75.7%

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

   9. Simplified 75.7%

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

   10. Taylor expanded in n around 0 75.7%

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

       1. +-commutative 75.7%

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

       2. *-lft-identity 75.7%

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

       3. associate-*l/ 70.1%

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

       4. distribute-rgt-in 70.1%

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

       5. rgt-mult-inverse 75.7%

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

       6. *-lft-identity 75.7%

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

       7. log1p-define 95.5%

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

   12. Simplified 95.5%

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

   if 10 < (/.f64 #s(literal 1 binary64) n) < 2.0000000000000001e76

   1. Initial program 89.1%

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

   3. Taylor expanded in x around 0 89.1%

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

      1. *-rgt-identity 89.1%

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

      2. associate-*l/ 89.1%

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

      3. associate-/l* 89.1%

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

      4. exp-to-pow 89.1%

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

   5. Simplified 89.1%

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

   if 2.0000000000000001e76 < (/.f64 #s(literal 1 binary64) n)

   1. Initial program 44.0%

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

   3. Taylor expanded in x around 0 27.4%

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

      1. *-rgt-identity 27.4%

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

      2. associate-*l/ 27.4%

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

      3. associate-/l* 27.4%

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

      4. exp-to-pow 27.4%

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

   5. Simplified 27.4%

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

   6. Taylor expanded in n around inf 8.5%

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

      1. neg-mul-1 8.5%

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

      2. distribute-neg-frac2 8.5%

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

   8. Simplified 8.5%

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

      1. log1p-expm1-u 58.9%

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

      2. expm1-undefine 58.9%

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

      3. add-exp-log 58.9%

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

   10. Applied egg-rr 58.9%

       \[\leadsto \frac{\color{blue}{\mathsf{log1p}\left(x - 1\right)}}{-n} \]
3. Recombined 4 regimes into one program.

4. Final simplification 81.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{1}{n} \leq -500:\\ \;\;\;\;\frac{1}{n} + \frac{-1}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 10:\\ \;\;\;\;\frac{\mathsf{log1p}\left(\frac{1}{x}\right)}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 2 \cdot 10^{+76}:\\ \;\;\;\;1 - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{log1p}\left(x + -1\right)}{-n}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 78.2% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{1}{n} \leq -500:\\ \;\;\;\;\frac{1}{n} + \frac{-1}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 10:\\ \;\;\;\;\frac{\mathsf{log1p}\left(\frac{1}{x}\right)}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 2 \cdot 10^{+76}:\\ \;\;\;\;1 - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 + \frac{-0.5 + \frac{0.3333333333333333}{x}}{x}}{x}}{n}\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (if (<= (/ 1.0 n) -500.0)
   (+ (/ 1.0 n) (/ -1.0 n))
   (if (<= (/ 1.0 n) 10.0)
     (/ (log1p (/ 1.0 x)) n)
     (if (<= (/ 1.0 n) 2e+76)
       (- 1.0 (pow x (/ 1.0 n)))
       (/ (/ (+ 1.0 (/ (+ -0.5 (/ 0.3333333333333333 x)) x)) x) n)))))

C

double code(double x, double n) {
	double tmp;
	if ((1.0 / n) <= -500.0) {
		tmp = (1.0 / n) + (-1.0 / n);
	} else if ((1.0 / n) <= 10.0) {
		tmp = log1p((1.0 / x)) / n;
	} else if ((1.0 / n) <= 2e+76) {
		tmp = 1.0 - pow(x, (1.0 / n));
	} else {
		tmp = ((1.0 + ((-0.5 + (0.3333333333333333 / x)) / x)) / x) / n;
	}
	return tmp;
}

Java

public static double code(double x, double n) {
	double tmp;
	if ((1.0 / n) <= -500.0) {
		tmp = (1.0 / n) + (-1.0 / n);
	} else if ((1.0 / n) <= 10.0) {
		tmp = Math.log1p((1.0 / x)) / n;
	} else if ((1.0 / n) <= 2e+76) {
		tmp = 1.0 - Math.pow(x, (1.0 / n));
	} else {
		tmp = ((1.0 + ((-0.5 + (0.3333333333333333 / x)) / x)) / x) / n;
	}
	return tmp;
}

Python

def code(x, n):
	tmp = 0
	if (1.0 / n) <= -500.0:
		tmp = (1.0 / n) + (-1.0 / n)
	elif (1.0 / n) <= 10.0:
		tmp = math.log1p((1.0 / x)) / n
	elif (1.0 / n) <= 2e+76:
		tmp = 1.0 - math.pow(x, (1.0 / n))
	else:
		tmp = ((1.0 + ((-0.5 + (0.3333333333333333 / x)) / x)) / x) / n
	return tmp

Julia

function code(x, n)
	tmp = 0.0
	if (Float64(1.0 / n) <= -500.0)
		tmp = Float64(Float64(1.0 / n) + Float64(-1.0 / n));
	elseif (Float64(1.0 / n) <= 10.0)
		tmp = Float64(log1p(Float64(1.0 / x)) / n);
	elseif (Float64(1.0 / n) <= 2e+76)
		tmp = Float64(1.0 - (x ^ Float64(1.0 / n)));
	else
		tmp = Float64(Float64(Float64(1.0 + Float64(Float64(-0.5 + Float64(0.3333333333333333 / x)) / x)) / x) / n);
	end
	return tmp
end

Wolfram

code[x_, n_] := If[LessEqual[N[(1.0 / n), $MachinePrecision], -500.0], N[(N[(1.0 / n), $MachinePrecision] + N[(-1.0 / n), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 10.0], N[(N[Log[1 + N[(1.0 / x), $MachinePrecision]], $MachinePrecision] / n), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 2e+76], N[(1.0 - N[Power[x, N[(1.0 / n), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[(N[(1.0 + N[(N[(-0.5 + N[(0.3333333333333333 / x), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision] / n), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if (/.f64 #s(literal 1 binary64) n) < -500

   1. Initial program 100.0%

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

   3. Taylor expanded in n around inf 64.5%

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

      1. log1p-define 64.5%

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

   5. Simplified 64.5%

      \[\leadsto \color{blue}{\frac{\mathsf{log1p}\left(x\right) - \log x}{n}} \]
   6. Step-by-step derivation

      1. log1p-undefine 64.5%

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

      2. diff-log 64.5%

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

   7. Applied egg-rr 64.5%

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

      1. +-commutative 64.5%

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

   9. Simplified 64.5%

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

       1. log-div 64.5%

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

       2. +-commutative 64.5%

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

       3. log1p-undefine 64.5%

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

       4. expm1-log1p-u 64.5%

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

       5. expm1-undefine 64.5%

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

       6. div-sub 64.5%

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

       7. log1p-undefine 64.5%

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

       8. rem-exp-log 64.5%

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

   11. Applied egg-rr 64.5%

       \[\leadsto \color{blue}{\frac{1 + \left(\mathsf{log1p}\left(x\right) - \log x\right)}{n} - \frac{1}{n}} \]
   12. Step-by-step derivation

       1. sub-neg 64.5%

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

       2. rem-log-exp 64.5%

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

       3. exp-diff 64.5%

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

       4. log1p-define 64.5%

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

       5. +-commutative 64.5%

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

       6. rem-exp-log 3.3%

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

       7. rem-exp-log 64.5%

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

       8. *-lft-identity 64.5%

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

       9. associate-*l/ 59.3%

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

       10. distribute-lft-in 59.3%

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

       11. *-rgt-identity 59.3%

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

       12. lft-mult-inverse 64.5%

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

       13. log1p-define 64.5%

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

       14. distribute-neg-frac 64.5%

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

       15. metadata-eval 64.5%

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

   13. Simplified 64.5%

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

   14. Taylor expanded in x around inf 64.6%

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

   if -500 < (/.f64 #s(literal 1 binary64) n) < 10

   1. Initial program 25.6%

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

   3. Taylor expanded in n around inf 75.6%

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

      1. log1p-define 75.6%

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

   5. Simplified 75.6%

      \[\leadsto \color{blue}{\frac{\mathsf{log1p}\left(x\right) - \log x}{n}} \]
   6. Step-by-step derivation

      1. log1p-undefine 75.6%

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

      2. diff-log 75.7%

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

   7. Applied egg-rr 75.7%

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

      1. +-commutative 75.7%

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

   9. Simplified 75.7%

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

   10. Taylor expanded in n around 0 75.7%

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

       1. +-commutative 75.7%

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

       2. *-lft-identity 75.7%

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

       3. associate-*l/ 70.1%

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

       4. distribute-rgt-in 70.1%

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

       5. rgt-mult-inverse 75.7%

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

       6. *-lft-identity 75.7%

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

       7. log1p-define 95.5%

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

   12. Simplified 95.5%

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

   if 10 < (/.f64 #s(literal 1 binary64) n) < 2.0000000000000001e76

   1. Initial program 89.1%

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

   3. Taylor expanded in x around 0 89.1%

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

      1. *-rgt-identity 89.1%

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

      2. associate-*l/ 89.1%

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

      3. associate-/l* 89.1%

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

      4. exp-to-pow 89.1%

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

   5. Simplified 89.1%

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

   if 2.0000000000000001e76 < (/.f64 #s(literal 1 binary64) n)

   1. Initial program 44.0%

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

   3. Taylor expanded in n around inf 8.5%

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

      1. log1p-define 8.5%

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

   5. Simplified 8.5%

      \[\leadsto \color{blue}{\frac{\mathsf{log1p}\left(x\right) - \log x}{n}} \]
   6. Step-by-step derivation

      1. log1p-undefine 8.5%

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

      2. diff-log 8.5%

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

   7. Applied egg-rr 8.5%

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

      1. +-commutative 8.5%

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

   9. Simplified 8.5%

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

   10. Taylor expanded in x around inf 51.2%

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

       1. associate--l+ 51.2%

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

       2. unpow2 51.2%

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

       3. associate-/r* 51.2%

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

       4. metadata-eval 51.2%

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

       5. associate-*r/ 51.2%

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

       6. associate-*r/ 51.2%

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

       7. metadata-eval 51.2%

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

       8. div-sub 51.2%

          \[\leadsto \frac{\frac{1 + \color{blue}{\frac{0.3333333333333333 \cdot \frac{1}{x} - 0.5}{x}}}{x}}{n} \]

       9. sub-neg 51.2%

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

       10. metadata-eval 51.2%

           \[\leadsto \frac{\frac{1 + \frac{0.3333333333333333 \cdot \frac{1}{x} + \color{blue}{-0.5}}{x}}{x}}{n} \]

       11. +-commutative 51.2%

           \[\leadsto \frac{\frac{1 + \frac{\color{blue}{-0.5 + 0.3333333333333333 \cdot \frac{1}{x}}}{x}}{x}}{n} \]

       12. associate-*r/ 51.2%

           \[\leadsto \frac{\frac{1 + \frac{-0.5 + \color{blue}{\frac{0.3333333333333333 \cdot 1}{x}}}{x}}{x}}{n} \]

       13. metadata-eval 51.2%

           \[\leadsto \frac{\frac{1 + \frac{-0.5 + \frac{\color{blue}{0.3333333333333333}}{x}}{x}}{x}}{n} \]

   12. Simplified 51.2%

       \[\leadsto \frac{\color{blue}{\frac{1 + \frac{-0.5 + \frac{0.3333333333333333}{x}}{x}}{x}}}{n} \]
3. Recombined 4 regimes into one program.
4. Add Preprocessing

Alternative 9: 75.2% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{1}{n} \leq -500:\\ \;\;\;\;\frac{1}{n} + \frac{-1}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 2 \cdot 10^{+76}:\\ \;\;\;\;\frac{\mathsf{log1p}\left(\frac{1}{x}\right)}{n}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 + \frac{-0.5 + \frac{0.3333333333333333}{x}}{x}}{x}}{n}\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (if (<= (/ 1.0 n) -500.0)
   (+ (/ 1.0 n) (/ -1.0 n))
   (if (<= (/ 1.0 n) 2e+76)
     (/ (log1p (/ 1.0 x)) n)
     (/ (/ (+ 1.0 (/ (+ -0.5 (/ 0.3333333333333333 x)) x)) x) n))))

C

double code(double x, double n) {
	double tmp;
	if ((1.0 / n) <= -500.0) {
		tmp = (1.0 / n) + (-1.0 / n);
	} else if ((1.0 / n) <= 2e+76) {
		tmp = log1p((1.0 / x)) / n;
	} else {
		tmp = ((1.0 + ((-0.5 + (0.3333333333333333 / x)) / x)) / x) / n;
	}
	return tmp;
}

Java

public static double code(double x, double n) {
	double tmp;
	if ((1.0 / n) <= -500.0) {
		tmp = (1.0 / n) + (-1.0 / n);
	} else if ((1.0 / n) <= 2e+76) {
		tmp = Math.log1p((1.0 / x)) / n;
	} else {
		tmp = ((1.0 + ((-0.5 + (0.3333333333333333 / x)) / x)) / x) / n;
	}
	return tmp;
}

Python

def code(x, n):
	tmp = 0
	if (1.0 / n) <= -500.0:
		tmp = (1.0 / n) + (-1.0 / n)
	elif (1.0 / n) <= 2e+76:
		tmp = math.log1p((1.0 / x)) / n
	else:
		tmp = ((1.0 + ((-0.5 + (0.3333333333333333 / x)) / x)) / x) / n
	return tmp

Julia

function code(x, n)
	tmp = 0.0
	if (Float64(1.0 / n) <= -500.0)
		tmp = Float64(Float64(1.0 / n) + Float64(-1.0 / n));
	elseif (Float64(1.0 / n) <= 2e+76)
		tmp = Float64(log1p(Float64(1.0 / x)) / n);
	else
		tmp = Float64(Float64(Float64(1.0 + Float64(Float64(-0.5 + Float64(0.3333333333333333 / x)) / x)) / x) / n);
	end
	return tmp
end

Wolfram

code[x_, n_] := If[LessEqual[N[(1.0 / n), $MachinePrecision], -500.0], N[(N[(1.0 / n), $MachinePrecision] + N[(-1.0 / n), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 2e+76], N[(N[Log[1 + N[(1.0 / x), $MachinePrecision]], $MachinePrecision] / n), $MachinePrecision], N[(N[(N[(1.0 + N[(N[(-0.5 + N[(0.3333333333333333 / x), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision] / n), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 #s(literal 1 binary64) n) < -500

   1. Initial program 100.0%

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

   3. Taylor expanded in n around inf 64.5%

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

      1. log1p-define 64.5%

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

   5. Simplified 64.5%

      \[\leadsto \color{blue}{\frac{\mathsf{log1p}\left(x\right) - \log x}{n}} \]
   6. Step-by-step derivation

      1. log1p-undefine 64.5%

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

      2. diff-log 64.5%

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

   7. Applied egg-rr 64.5%

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

      1. +-commutative 64.5%

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

   9. Simplified 64.5%

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

       1. log-div 64.5%

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

       2. +-commutative 64.5%

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

       3. log1p-undefine 64.5%

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

       4. expm1-log1p-u 64.5%

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

       5. expm1-undefine 64.5%

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

       6. div-sub 64.5%

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

       7. log1p-undefine 64.5%

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

       8. rem-exp-log 64.5%

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

   11. Applied egg-rr 64.5%

       \[\leadsto \color{blue}{\frac{1 + \left(\mathsf{log1p}\left(x\right) - \log x\right)}{n} - \frac{1}{n}} \]
   12. Step-by-step derivation

       1. sub-neg 64.5%

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

       2. rem-log-exp 64.5%

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

       3. exp-diff 64.5%

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

       4. log1p-define 64.5%

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

       5. +-commutative 64.5%

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

       6. rem-exp-log 3.3%

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

       7. rem-exp-log 64.5%

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

       8. *-lft-identity 64.5%

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

       9. associate-*l/ 59.3%

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

       10. distribute-lft-in 59.3%

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

       11. *-rgt-identity 59.3%

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

       12. lft-mult-inverse 64.5%

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

       13. log1p-define 64.5%

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

       14. distribute-neg-frac 64.5%

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

       15. metadata-eval 64.5%

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

   13. Simplified 64.5%

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

   14. Taylor expanded in x around inf 64.6%

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

   if -500 < (/.f64 #s(literal 1 binary64) n) < 2.0000000000000001e76

   1. Initial program 31.7%

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

   3. Taylor expanded in n around inf 69.1%

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

      1. log1p-define 69.1%

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

   5. Simplified 69.1%

      \[\leadsto \color{blue}{\frac{\mathsf{log1p}\left(x\right) - \log x}{n}} \]
   6. Step-by-step derivation

      1. log1p-undefine 69.1%

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

      2. diff-log 69.1%

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

   7. Applied egg-rr 69.1%

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

      1. +-commutative 69.1%

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

   9. Simplified 69.1%

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

   10. Taylor expanded in n around 0 69.1%

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

       1. +-commutative 69.1%

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

       2. *-lft-identity 69.1%

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

       3. associate-*l/ 64.1%

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

       4. distribute-rgt-in 64.1%

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

       5. rgt-mult-inverse 69.1%

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

       6. *-lft-identity 69.1%

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

       7. log1p-define 87.1%

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

   12. Simplified 87.1%

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

   if 2.0000000000000001e76 < (/.f64 #s(literal 1 binary64) n)

   1. Initial program 44.0%

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

   3. Taylor expanded in n around inf 8.5%

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

      1. log1p-define 8.5%

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

   5. Simplified 8.5%

      \[\leadsto \color{blue}{\frac{\mathsf{log1p}\left(x\right) - \log x}{n}} \]
   6. Step-by-step derivation

      1. log1p-undefine 8.5%

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

      2. diff-log 8.5%

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

   7. Applied egg-rr 8.5%

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

      1. +-commutative 8.5%

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

   9. Simplified 8.5%

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

   10. Taylor expanded in x around inf 51.2%

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

       1. associate--l+ 51.2%

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

       2. unpow2 51.2%

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

       3. associate-/r* 51.2%

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

       4. metadata-eval 51.2%

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

       5. associate-*r/ 51.2%

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

       6. associate-*r/ 51.2%

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

       7. metadata-eval 51.2%

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

       8. div-sub 51.2%

          \[\leadsto \frac{\frac{1 + \color{blue}{\frac{0.3333333333333333 \cdot \frac{1}{x} - 0.5}{x}}}{x}}{n} \]

       9. sub-neg 51.2%

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

       10. metadata-eval 51.2%

           \[\leadsto \frac{\frac{1 + \frac{0.3333333333333333 \cdot \frac{1}{x} + \color{blue}{-0.5}}{x}}{x}}{n} \]

       11. +-commutative 51.2%

           \[\leadsto \frac{\frac{1 + \frac{\color{blue}{-0.5 + 0.3333333333333333 \cdot \frac{1}{x}}}{x}}{x}}{n} \]

       12. associate-*r/ 51.2%

           \[\leadsto \frac{\frac{1 + \frac{-0.5 + \color{blue}{\frac{0.3333333333333333 \cdot 1}{x}}}{x}}{x}}{n} \]

       13. metadata-eval 51.2%

           \[\leadsto \frac{\frac{1 + \frac{-0.5 + \frac{\color{blue}{0.3333333333333333}}{x}}{x}}{x}}{n} \]

   12. Simplified 51.2%

       \[\leadsto \frac{\color{blue}{\frac{1 + \frac{-0.5 + \frac{0.3333333333333333}{x}}{x}}{x}}}{n} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 10: 58.5% accurate, 1.9× speedup

Math

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

FPCore

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

C

double code(double x, double n) {
	double tmp;
	if (x <= 1.0) {
		tmp = log(x) / -n;
	} else {
		tmp = (1.0 / n) + (-1.0 / 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 <= 1.0d0) then
        tmp = log(x) / -n
    else
        tmp = (1.0d0 / n) + ((-1.0d0) / n)
    end if
    code = tmp
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if x < 1

   1. Initial program 40.6%

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

   3. Taylor expanded in x around 0 36.7%

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

      1. *-rgt-identity 36.7%

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

      2. associate-*l/ 36.7%

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

      3. associate-/l* 36.7%

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

      4. exp-to-pow 36.7%

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

   5. Simplified 36.7%

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

   6. Taylor expanded in n around inf 49.7%

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

      1. neg-mul-1 49.7%

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

      2. distribute-neg-frac2 49.7%

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

   8. Simplified 49.7%

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

   if 1 < x

   1. Initial program 71.5%

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

   3. Taylor expanded in n around inf 70.9%

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

      1. log1p-define 70.9%

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

   5. Simplified 70.9%

      \[\leadsto \color{blue}{\frac{\mathsf{log1p}\left(x\right) - \log x}{n}} \]
   6. Step-by-step derivation

      1. log1p-undefine 70.9%

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

      2. diff-log 71.0%

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

   7. Applied egg-rr 71.0%

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

      1. +-commutative 71.0%

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

   9. Simplified 71.0%

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

       1. log-div 70.9%

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

       2. +-commutative 70.9%

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

       3. log1p-undefine 70.9%

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

       4. expm1-log1p-u 70.9%

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

       5. expm1-undefine 70.9%

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

       6. div-sub 70.9%

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

       7. log1p-undefine 70.9%

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

       8. rem-exp-log 70.9%

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

   11. Applied egg-rr 70.9%

       \[\leadsto \color{blue}{\frac{1 + \left(\mathsf{log1p}\left(x\right) - \log x\right)}{n} - \frac{1}{n}} \]
   12. Step-by-step derivation

       1. sub-neg 70.9%

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

       2. rem-log-exp 70.9%

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

       3. exp-diff 70.9%

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

       4. log1p-define 70.9%

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

       5. +-commutative 70.9%

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

       6. rem-exp-log 5.7%

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

       7. rem-exp-log 71.0%

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

       8. *-lft-identity 71.0%

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

       9. associate-*l/ 63.1%

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

       10. distribute-lft-in 63.1%

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

       11. *-rgt-identity 63.1%

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

       12. lft-mult-inverse 71.0%

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

       13. log1p-define 71.0%

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

       14. distribute-neg-frac 71.0%

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

       15. metadata-eval 71.0%

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

   13. Simplified 71.0%

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

   14. Taylor expanded in x around inf 71.5%

       \[\leadsto \color{blue}{\frac{1}{n}} + \frac{-1}{n} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 11: 49.1% accurate, 9.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 1.1 \cdot 10^{+43}:\\ \;\;\;\;\frac{\frac{1}{n} + \frac{\frac{0.3333333333333333}{n \cdot x} + \frac{-0.5}{n}}{x}}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{n} + \frac{-1}{n}\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (if (<= x 1.1e+43)
   (/ (+ (/ 1.0 n) (/ (+ (/ 0.3333333333333333 (* n x)) (/ -0.5 n)) x)) x)
   (+ (/ 1.0 n) (/ -1.0 n))))

C

double code(double x, double n) {
	double tmp;
	if (x <= 1.1e+43) {
		tmp = ((1.0 / n) + (((0.3333333333333333 / (n * x)) + (-0.5 / n)) / x)) / x;
	} else {
		tmp = (1.0 / n) + (-1.0 / 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 <= 1.1d+43) then
        tmp = ((1.0d0 / n) + (((0.3333333333333333d0 / (n * x)) + ((-0.5d0) / n)) / x)) / x
    else
        tmp = (1.0d0 / n) + ((-1.0d0) / n)
    end if
    code = tmp
end function

Java

public static double code(double x, double n) {
	double tmp;
	if (x <= 1.1e+43) {
		tmp = ((1.0 / n) + (((0.3333333333333333 / (n * x)) + (-0.5 / n)) / x)) / x;
	} else {
		tmp = (1.0 / n) + (-1.0 / n);
	}
	return tmp;
}

Python

def code(x, n):
	tmp = 0
	if x <= 1.1e+43:
		tmp = ((1.0 / n) + (((0.3333333333333333 / (n * x)) + (-0.5 / n)) / x)) / x
	else:
		tmp = (1.0 / n) + (-1.0 / n)
	return tmp

Julia

function code(x, n)
	tmp = 0.0
	if (x <= 1.1e+43)
		tmp = Float64(Float64(Float64(1.0 / n) + Float64(Float64(Float64(0.3333333333333333 / Float64(n * x)) + Float64(-0.5 / n)) / x)) / x);
	else
		tmp = Float64(Float64(1.0 / n) + Float64(-1.0 / n));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, n)
	tmp = 0.0;
	if (x <= 1.1e+43)
		tmp = ((1.0 / n) + (((0.3333333333333333 / (n * x)) + (-0.5 / n)) / x)) / x;
	else
		tmp = (1.0 / n) + (-1.0 / n);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, n_] := If[LessEqual[x, 1.1e+43], N[(N[(N[(1.0 / n), $MachinePrecision] + N[(N[(N[(0.3333333333333333 / N[(n * x), $MachinePrecision]), $MachinePrecision] + N[(-0.5 / n), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision], N[(N[(1.0 / n), $MachinePrecision] + N[(-1.0 / n), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 1.1e43

   1. Initial program 39.8%

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

   3. Taylor expanded in n around inf 49.5%

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

      1. log1p-define 49.5%

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

   5. Simplified 49.5%

      \[\leadsto \color{blue}{\frac{\mathsf{log1p}\left(x\right) - \log x}{n}} \]
   6. Step-by-step derivation

      1. log1p-undefine 49.5%

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

      2. diff-log 49.6%

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

   7. Applied egg-rr 49.6%

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

      1. +-commutative 49.6%

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

   9. Simplified 49.6%

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

   10. Taylor expanded in x around inf 15.8%

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

   12. Simplified 33.5%

       \[\leadsto \color{blue}{\frac{\frac{1}{n} + \frac{\frac{0.3333333333333333}{x \cdot n} + \frac{-0.5}{n}}{x}}{x}} \]

   if 1.1e43 < x

   1. Initial program 75.8%

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

   3. Taylor expanded in n around inf 75.8%

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

      1. log1p-define 75.8%

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

   5. Simplified 75.8%

      \[\leadsto \color{blue}{\frac{\mathsf{log1p}\left(x\right) - \log x}{n}} \]
   6. Step-by-step derivation

      1. log1p-undefine 75.8%

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

      2. diff-log 75.8%

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

   7. Applied egg-rr 75.8%

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

      1. +-commutative 75.8%

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

   9. Simplified 75.8%

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

       1. log-div 75.8%

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

       2. +-commutative 75.8%

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

       3. log1p-undefine 75.8%

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

       4. expm1-log1p-u 75.8%

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

       5. expm1-undefine 75.8%

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

       6. div-sub 75.8%

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

       7. log1p-undefine 75.8%

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

       8. rem-exp-log 75.8%

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

   11. Applied egg-rr 75.8%

       \[\leadsto \color{blue}{\frac{1 + \left(\mathsf{log1p}\left(x\right) - \log x\right)}{n} - \frac{1}{n}} \]
   12. Step-by-step derivation

       1. sub-neg 75.8%

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

       2. rem-log-exp 75.8%

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

       3. exp-diff 75.8%

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

       4. log1p-define 75.8%

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

       5. +-commutative 75.8%

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

       6. rem-exp-log 5.6%

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

       7. rem-exp-log 75.8%

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

       8. *-lft-identity 75.8%

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

       9. associate-*l/ 67.2%

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

       10. distribute-lft-in 67.2%

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

       11. *-rgt-identity 67.2%

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

       12. lft-mult-inverse 75.8%

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

       13. log1p-define 75.8%

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

       14. distribute-neg-frac 75.8%

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

       15. metadata-eval 75.8%

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

   13. Simplified 75.8%

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

   14. Taylor expanded in x around inf 75.8%

       \[\leadsto \color{blue}{\frac{1}{n}} + \frac{-1}{n} \]
3. Recombined 2 regimes into one program.

4. Final simplification 49.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 1.1 \cdot 10^{+43}:\\ \;\;\;\;\frac{\frac{1}{n} + \frac{\frac{0.3333333333333333}{n \cdot x} + \frac{-0.5}{n}}{x}}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{n} + \frac{-1}{n}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 49.1% accurate, 11.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 1.7 \cdot 10^{+37}:\\ \;\;\;\;\frac{\frac{1 + \frac{-0.5 + \frac{0.3333333333333333}{x}}{x}}{x}}{n}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{n} + \frac{-1}{n}\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (if (<= x 1.7e+37)
   (/ (/ (+ 1.0 (/ (+ -0.5 (/ 0.3333333333333333 x)) x)) x) n)
   (+ (/ 1.0 n) (/ -1.0 n))))

C

double code(double x, double n) {
	double tmp;
	if (x <= 1.7e+37) {
		tmp = ((1.0 + ((-0.5 + (0.3333333333333333 / x)) / x)) / x) / n;
	} else {
		tmp = (1.0 / n) + (-1.0 / 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 <= 1.7d+37) then
        tmp = ((1.0d0 + (((-0.5d0) + (0.3333333333333333d0 / x)) / x)) / x) / n
    else
        tmp = (1.0d0 / n) + ((-1.0d0) / n)
    end if
    code = tmp
end function

Java

public static double code(double x, double n) {
	double tmp;
	if (x <= 1.7e+37) {
		tmp = ((1.0 + ((-0.5 + (0.3333333333333333 / x)) / x)) / x) / n;
	} else {
		tmp = (1.0 / n) + (-1.0 / n);
	}
	return tmp;
}

Python

def code(x, n):
	tmp = 0
	if x <= 1.7e+37:
		tmp = ((1.0 + ((-0.5 + (0.3333333333333333 / x)) / x)) / x) / n
	else:
		tmp = (1.0 / n) + (-1.0 / n)
	return tmp

Julia

function code(x, n)
	tmp = 0.0
	if (x <= 1.7e+37)
		tmp = Float64(Float64(Float64(1.0 + Float64(Float64(-0.5 + Float64(0.3333333333333333 / x)) / x)) / x) / n);
	else
		tmp = Float64(Float64(1.0 / n) + Float64(-1.0 / n));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, n)
	tmp = 0.0;
	if (x <= 1.7e+37)
		tmp = ((1.0 + ((-0.5 + (0.3333333333333333 / x)) / x)) / x) / n;
	else
		tmp = (1.0 / n) + (-1.0 / n);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, n_] := If[LessEqual[x, 1.7e+37], N[(N[(N[(1.0 + N[(N[(-0.5 + N[(0.3333333333333333 / x), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision] / n), $MachinePrecision], N[(N[(1.0 / n), $MachinePrecision] + N[(-1.0 / n), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 1.70000000000000003e37

   1. Initial program 39.8%

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

   3. Taylor expanded in n around inf 49.5%

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

      1. log1p-define 49.5%

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

   5. Simplified 49.5%

      \[\leadsto \color{blue}{\frac{\mathsf{log1p}\left(x\right) - \log x}{n}} \]
   6. Step-by-step derivation

      1. log1p-undefine 49.5%

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

      2. diff-log 49.6%

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

   7. Applied egg-rr 49.6%

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

      1. +-commutative 49.6%

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

   9. Simplified 49.6%

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

   10. Taylor expanded in x around inf 33.5%

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

       1. associate--l+ 33.5%

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

       2. unpow2 33.5%

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

       3. associate-/r* 33.5%

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

       4. metadata-eval 33.5%

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

       5. associate-*r/ 33.5%

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

       6. associate-*r/ 33.5%

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

       7. metadata-eval 33.5%

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

       8. div-sub 33.5%

          \[\leadsto \frac{\frac{1 + \color{blue}{\frac{0.3333333333333333 \cdot \frac{1}{x} - 0.5}{x}}}{x}}{n} \]

       9. sub-neg 33.5%

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

       10. metadata-eval 33.5%

           \[\leadsto \frac{\frac{1 + \frac{0.3333333333333333 \cdot \frac{1}{x} + \color{blue}{-0.5}}{x}}{x}}{n} \]

       11. +-commutative 33.5%

           \[\leadsto \frac{\frac{1 + \frac{\color{blue}{-0.5 + 0.3333333333333333 \cdot \frac{1}{x}}}{x}}{x}}{n} \]

       12. associate-*r/ 33.5%

           \[\leadsto \frac{\frac{1 + \frac{-0.5 + \color{blue}{\frac{0.3333333333333333 \cdot 1}{x}}}{x}}{x}}{n} \]

       13. metadata-eval 33.5%

           \[\leadsto \frac{\frac{1 + \frac{-0.5 + \frac{\color{blue}{0.3333333333333333}}{x}}{x}}{x}}{n} \]

   12. Simplified 33.5%

       \[\leadsto \frac{\color{blue}{\frac{1 + \frac{-0.5 + \frac{0.3333333333333333}{x}}{x}}{x}}}{n} \]

   if 1.70000000000000003e37 < x

   1. Initial program 75.8%

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

   3. Taylor expanded in n around inf 75.8%

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

      1. log1p-define 75.8%

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

   5. Simplified 75.8%

      \[\leadsto \color{blue}{\frac{\mathsf{log1p}\left(x\right) - \log x}{n}} \]
   6. Step-by-step derivation

      1. log1p-undefine 75.8%

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

      2. diff-log 75.8%

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

   7. Applied egg-rr 75.8%

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

      1. +-commutative 75.8%

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

   9. Simplified 75.8%

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

       1. log-div 75.8%

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

       2. +-commutative 75.8%

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

       3. log1p-undefine 75.8%

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

       4. expm1-log1p-u 75.8%

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

       5. expm1-undefine 75.8%

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

       6. div-sub 75.8%

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

       7. log1p-undefine 75.8%

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

       8. rem-exp-log 75.8%

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

   11. Applied egg-rr 75.8%

       \[\leadsto \color{blue}{\frac{1 + \left(\mathsf{log1p}\left(x\right) - \log x\right)}{n} - \frac{1}{n}} \]
   12. Step-by-step derivation

       1. sub-neg 75.8%

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

       2. rem-log-exp 75.8%

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

       3. exp-diff 75.8%

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

       4. log1p-define 75.8%

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

       5. +-commutative 75.8%

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

       6. rem-exp-log 5.6%

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

       7. rem-exp-log 75.8%

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

       8. *-lft-identity 75.8%

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

       9. associate-*l/ 67.2%

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

       10. distribute-lft-in 67.2%

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

       11. *-rgt-identity 67.2%

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

       12. lft-mult-inverse 75.8%

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

       13. log1p-define 75.8%

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

       14. distribute-neg-frac 75.8%

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

       15. metadata-eval 75.8%

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

   13. Simplified 75.8%

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

   14. Taylor expanded in x around inf 75.8%

       \[\leadsto \color{blue}{\frac{1}{n}} + \frac{-1}{n} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 13: 47.5% accurate, 15.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{1}{n} \leq -500:\\ \;\;\;\;\frac{1}{n} + \frac{-1}{n}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{x}}{n}\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (if (<= (/ 1.0 n) -500.0) (+ (/ 1.0 n) (/ -1.0 n)) (/ (/ 1.0 x) n)))

C

double code(double x, double n) {
	double tmp;
	if ((1.0 / n) <= -500.0) {
		tmp = (1.0 / n) + (-1.0 / 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 ((1.0d0 / n) <= (-500.0d0)) then
        tmp = (1.0d0 / n) + ((-1.0d0) / 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 ((1.0 / n) <= -500.0) {
		tmp = (1.0 / n) + (-1.0 / n);
	} else {
		tmp = (1.0 / x) / n;
	}
	return tmp;
}

Python

def code(x, n):
	tmp = 0
	if (1.0 / n) <= -500.0:
		tmp = (1.0 / n) + (-1.0 / n)
	else:
		tmp = (1.0 / x) / n
	return tmp

Julia

function code(x, n)
	tmp = 0.0
	if (Float64(1.0 / n) <= -500.0)
		tmp = Float64(Float64(1.0 / n) + Float64(-1.0 / n));
	else
		tmp = Float64(Float64(1.0 / x) / n);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, n)
	tmp = 0.0;
	if ((1.0 / n) <= -500.0)
		tmp = (1.0 / n) + (-1.0 / n);
	else
		tmp = (1.0 / x) / n;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, n_] := If[LessEqual[N[(1.0 / n), $MachinePrecision], -500.0], N[(N[(1.0 / n), $MachinePrecision] + N[(-1.0 / n), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 / x), $MachinePrecision] / n), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 #s(literal 1 binary64) n) < -500

   1. Initial program 100.0%

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

   3. Taylor expanded in n around inf 64.5%

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

      1. log1p-define 64.5%

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

   5. Simplified 64.5%

      \[\leadsto \color{blue}{\frac{\mathsf{log1p}\left(x\right) - \log x}{n}} \]
   6. Step-by-step derivation

      1. log1p-undefine 64.5%

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

      2. diff-log 64.5%

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

   7. Applied egg-rr 64.5%

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

      1. +-commutative 64.5%

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

   9. Simplified 64.5%

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

       1. log-div 64.5%

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

       2. +-commutative 64.5%

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

       3. log1p-undefine 64.5%

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

       4. expm1-log1p-u 64.5%

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

       5. expm1-undefine 64.5%

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

       6. div-sub 64.5%

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

       7. log1p-undefine 64.5%

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

       8. rem-exp-log 64.5%

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

   11. Applied egg-rr 64.5%

       \[\leadsto \color{blue}{\frac{1 + \left(\mathsf{log1p}\left(x\right) - \log x\right)}{n} - \frac{1}{n}} \]
   12. Step-by-step derivation

       1. sub-neg 64.5%

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

       2. rem-log-exp 64.5%

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

       3. exp-diff 64.5%

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

       4. log1p-define 64.5%

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

       5. +-commutative 64.5%

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

       6. rem-exp-log 3.3%

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

       7. rem-exp-log 64.5%

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

       8. *-lft-identity 64.5%

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

       9. associate-*l/ 59.3%

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

       10. distribute-lft-in 59.3%

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

       11. *-rgt-identity 59.3%

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

       12. lft-mult-inverse 64.5%

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

       13. log1p-define 64.5%

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

       14. distribute-neg-frac 64.5%

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

       15. metadata-eval 64.5%

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

   13. Simplified 64.5%

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

   14. Taylor expanded in x around inf 64.6%

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

   if -500 < (/.f64 #s(literal 1 binary64) n)

   1. Initial program 34.1%

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

   3. Taylor expanded in n around inf 57.4%

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

      1. log1p-define 57.4%

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

   5. Simplified 57.4%

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

   6. Taylor expanded in x around inf 38.5%

      \[\leadsto \frac{\color{blue}{\frac{1}{x}}}{n} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 14: 40.7% 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 53.4%

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

3. Taylor expanded in n around inf 59.5%

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

   1. log1p-define 59.5%

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

5. Simplified 59.5%

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

6. Taylor expanded in x around inf 35.5%

   \[\leadsto \frac{\color{blue}{\frac{1}{x}}}{n} \]
7. Add Preprocessing

Alternative 15: 40.7% accurate, 42.2× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 53.4%

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

3. Taylor expanded in n around inf 59.5%

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

   1. log1p-define 59.5%

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

5. Simplified 59.5%

   \[\leadsto \color{blue}{\frac{\mathsf{log1p}\left(x\right) - \log x}{n}} \]
6. Step-by-step derivation

   1. log1p-undefine 59.5%

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

   2. diff-log 59.5%

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

7. Applied egg-rr 59.5%

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

   1. +-commutative 59.5%

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

9. Simplified 59.5%

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

10. Taylor expanded in x around inf 34.6%

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

    1. associate-/r* 35.5%

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

12. Simplified 35.5%

    \[\leadsto \color{blue}{\frac{\frac{1}{n}}{x}} \]
13. Add Preprocessing

Alternative 16: 40.3% 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 53.4%

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

3. Taylor expanded in n around inf 59.5%

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

   1. log1p-define 59.5%

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

5. Simplified 59.5%

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

6. Taylor expanded in x around inf 34.6%

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

   1. *-commutative 34.6%

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

8. Simplified 34.6%

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

9. Final simplification 34.6%

   \[\leadsto \frac{1}{n \cdot x} \]
10. Add Preprocessing

Alternative 17: 4.5% accurate, 70.3× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 53.4%

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

3. Taylor expanded in x around 0 27.1%

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

4. Taylor expanded in x around inf 4.4%

   \[\leadsto \color{blue}{\frac{x}{n}} \]
5. Add Preprocessing

Reproduce

herbie shell --seed 2024169 
(FPCore (x n)
  :name "2nthrt (problem 3.4.6)"
  :precision binary64
  (- (pow (+ x 1.0) (/ 1.0 n)) (pow x (/ 1.0 n))))