2nthrt (problem 3.4.6)

Percentage Accurate: 53.8% → 92.1%

Time: 23.0s

Alternatives: 20

Speedup: 1.9×

• could not determine a ground truth

  1. x = -5.731918143773236e+28
  2. n = -1.0966074740501992e+209

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 53.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 92.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 1:\\ \;\;\;\;\frac{x}{n} - \mathsf{expm1}\left(\frac{\log x}{n}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{{x}^{\left({n}^{-1}\right)}}{x}}{n}\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (if (<= x 1.0)
   (- (/ x n) (expm1 (/ (log x) n)))
   (/ (/ (pow x (pow n -1.0)) x) n)))

C

double code(double x, double n) {
	double tmp;
	if (x <= 1.0) {
		tmp = (x / n) - expm1((log(x) / n));
	} else {
		tmp = (pow(x, pow(n, -1.0)) / x) / n;
	}
	return tmp;
}

Java

public static double code(double x, double n) {
	double tmp;
	if (x <= 1.0) {
		tmp = (x / n) - Math.expm1((Math.log(x) / n));
	} else {
		tmp = (Math.pow(x, Math.pow(n, -1.0)) / x) / n;
	}
	return tmp;
}

Python

def code(x, n):
	tmp = 0
	if x <= 1.0:
		tmp = (x / n) - math.expm1((math.log(x) / n))
	else:
		tmp = (math.pow(x, math.pow(n, -1.0)) / x) / n
	return tmp

Julia

function code(x, n)
	tmp = 0.0
	if (x <= 1.0)
		tmp = Float64(Float64(x / n) - expm1(Float64(log(x) / n)));
	else
		tmp = Float64(Float64((x ^ (n ^ -1.0)) / x) / n);
	end
	return tmp
end

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if x < 1

   1. Initial program 47.9%

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

   3. Taylor expanded in x around 0

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

      1. associate--l+ N/A

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

      2. +-commutative N/A

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

      3. *-rgt-identity N/A

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

      4. associate-*r/ N/A

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

      5. remove-double-neg N/A

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

      6. mul-1-neg N/A

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

      7. distribute-neg-frac N/A

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

      8. mul-1-neg N/A

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

      9. log-rec N/A

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

      10. mul-1-neg N/A

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

      11. associate-+l- N/A

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

      12. lower--.f64 N/A

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

      13. associate-*r/ N/A

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

      14. *-rgt-identity N/A

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

      15. lower-/.f64 N/A

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

   5. Applied rewrites 93.0%

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

   if 1 < x

   1. Initial program 66.0%

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

   3. Taylor expanded in x around inf

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

      1. associate-/l/ N/A

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

      2. lower-/.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. log-rec N/A

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

      5. mul-1-neg N/A

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

      6. associate-*r/ N/A

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

      7. associate-*r* N/A

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

      8. metadata-eval N/A

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

      9. *-commutative N/A

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

      10. associate-/l* N/A

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

      11. exp-to-pow N/A

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

      12. lower-pow.f64 N/A

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

      13. lower-/.f64 97.2

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

   5. Applied rewrites 97.2%

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

4. Final simplification 95.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 1:\\ \;\;\;\;\frac{x}{n} - \mathsf{expm1}\left(\frac{\log x}{n}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{{x}^{\left({n}^{-1}\right)}}{x}}{n}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 68.2% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{0.3333333333333333}{x} - 0.5\\ t_1 := {x}^{\left({n}^{-1}\right)}\\ t_2 := \frac{\frac{\frac{t\_0}{x} + 1}{x}}{n}\\ \mathbf{if}\;{n}^{-1} \leq -4 \cdot 10^{-10}:\\ \;\;\;\;\frac{t\_1}{n}\\ \mathbf{elif}\;{n}^{-1} \leq -2 \cdot 10^{-199}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;{n}^{-1} \leq -5 \cdot 10^{-257}:\\ \;\;\;\;\frac{x - \log x}{n}\\ \mathbf{elif}\;{n}^{-1} \leq 5 \cdot 10^{-9}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{{x}^{-1}}{n}, t\_0, {n}^{-1}\right)}{x}\\ \mathbf{elif}\;{n}^{-1} \leq 2 \cdot 10^{+201}:\\ \;\;\;\;1 - t\_1\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (let* ((t_0 (- (/ 0.3333333333333333 x) 0.5))
        (t_1 (pow x (pow n -1.0)))
        (t_2 (/ (/ (+ (/ t_0 x) 1.0) x) n)))
   (if (<= (pow n -1.0) -4e-10)
     (/ t_1 n)
     (if (<= (pow n -1.0) -2e-199)
       t_2
       (if (<= (pow n -1.0) -5e-257)
         (/ (- x (log x)) n)
         (if (<= (pow n -1.0) 5e-9)
           (/ (fma (/ (pow x -1.0) n) t_0 (pow n -1.0)) x)
           (if (<= (pow n -1.0) 2e+201) (- 1.0 t_1) t_2)))))))

C

double code(double x, double n) {
	double t_0 = (0.3333333333333333 / x) - 0.5;
	double t_1 = pow(x, pow(n, -1.0));
	double t_2 = (((t_0 / x) + 1.0) / x) / n;
	double tmp;
	if (pow(n, -1.0) <= -4e-10) {
		tmp = t_1 / n;
	} else if (pow(n, -1.0) <= -2e-199) {
		tmp = t_2;
	} else if (pow(n, -1.0) <= -5e-257) {
		tmp = (x - log(x)) / n;
	} else if (pow(n, -1.0) <= 5e-9) {
		tmp = fma((pow(x, -1.0) / n), t_0, pow(n, -1.0)) / x;
	} else if (pow(n, -1.0) <= 2e+201) {
		tmp = 1.0 - t_1;
	} else {
		tmp = t_2;
	}
	return tmp;
}

Julia

function code(x, n)
	t_0 = Float64(Float64(0.3333333333333333 / x) - 0.5)
	t_1 = x ^ (n ^ -1.0)
	t_2 = Float64(Float64(Float64(Float64(t_0 / x) + 1.0) / x) / n)
	tmp = 0.0
	if ((n ^ -1.0) <= -4e-10)
		tmp = Float64(t_1 / n);
	elseif ((n ^ -1.0) <= -2e-199)
		tmp = t_2;
	elseif ((n ^ -1.0) <= -5e-257)
		tmp = Float64(Float64(x - log(x)) / n);
	elseif ((n ^ -1.0) <= 5e-9)
		tmp = Float64(fma(Float64((x ^ -1.0) / n), t_0, (n ^ -1.0)) / x);
	elseif ((n ^ -1.0) <= 2e+201)
		tmp = Float64(1.0 - t_1);
	else
		tmp = t_2;
	end
	return tmp
end

Wolfram

code[x_, n_] := Block[{t$95$0 = N[(N[(0.3333333333333333 / x), $MachinePrecision] - 0.5), $MachinePrecision]}, Block[{t$95$1 = N[Power[x, N[Power[n, -1.0], $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[(N[(N[(N[(t$95$0 / x), $MachinePrecision] + 1.0), $MachinePrecision] / x), $MachinePrecision] / n), $MachinePrecision]}, If[LessEqual[N[Power[n, -1.0], $MachinePrecision], -4e-10], N[(t$95$1 / n), $MachinePrecision], If[LessEqual[N[Power[n, -1.0], $MachinePrecision], -2e-199], t$95$2, If[LessEqual[N[Power[n, -1.0], $MachinePrecision], -5e-257], N[(N[(x - N[Log[x], $MachinePrecision]), $MachinePrecision] / n), $MachinePrecision], If[LessEqual[N[Power[n, -1.0], $MachinePrecision], 5e-9], N[(N[(N[(N[Power[x, -1.0], $MachinePrecision] / n), $MachinePrecision] * t$95$0 + N[Power[n, -1.0], $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision], If[LessEqual[N[Power[n, -1.0], $MachinePrecision], 2e+201], N[(1.0 - t$95$1), $MachinePrecision], t$95$2]]]]]]]]

Derivation

1. Split input into 5 regimes

2. if (/.f64 #s(literal 1 binary64) n) < -4.00000000000000015e-10

   1. Initial program 98.7%

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

   3. Taylor expanded in x around inf

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

      1. associate-/l/ N/A

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

      2. lower-/.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. log-rec N/A

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

      5. mul-1-neg N/A

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

      6. associate-*r/ N/A

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

      7. associate-*r* N/A

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

      8. metadata-eval N/A

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

      9. *-commutative N/A

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

      10. associate-/l* N/A

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

      11. exp-to-pow N/A

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

      12. lower-pow.f64 N/A

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

      13. lower-/.f64 94.3

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

   5. Applied rewrites 94.3%

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

   7. Applied rewrites 94.3%

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

   9. Applied rewrites 94.3%

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

   10. Taylor expanded in n around 0

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

   12. Applied rewrites 94.9%

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

   if -4.00000000000000015e-10 < (/.f64 #s(literal 1 binary64) n) < -1.99999999999999996e-199 or 2.00000000000000008e201 < (/.f64 #s(literal 1 binary64) n)

   1. Initial program 25.0%

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

   3. Taylor expanded in n around inf

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

      1. lower-/.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lower-log1p.f64 N/A

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

      4. lower-log.f64 53.7

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

   5. Applied rewrites 53.7%

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

   6. Taylor expanded in x around inf

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

   8. Applied rewrites 67.9%

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

   if -1.99999999999999996e-199 < (/.f64 #s(literal 1 binary64) n) < -4.99999999999999989e-257

   1. Initial program 22.6%

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

   3. Taylor expanded in n around inf

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

      1. lower-/.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lower-log1p.f64 N/A

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

      4. lower-log.f64 96.0

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

   5. Applied rewrites 96.0%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 74.6%

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

   if -4.99999999999999989e-257 < (/.f64 #s(literal 1 binary64) n) < 5.0000000000000001e-9

   1. Initial program 43.3%

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

   3. Taylor expanded in n around inf

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

      1. lower-/.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lower-log1p.f64 N/A

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

      4. lower-log.f64 79.1

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

   5. Applied rewrites 79.1%

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

   6. Taylor expanded in x around inf

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

   8. Applied rewrites 63.9%

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

   9. Taylor expanded in x around -inf

      \[\leadsto -1 \cdot \color{blue}{\frac{-1 \cdot \frac{\frac{1}{3} \cdot \frac{1}{n \cdot x} - \frac{1}{2} \cdot \frac{1}{n}}{x} - \frac{1}{n}}{x}} \]
   10. Step-by-step derivation

   11. Applied rewrites 64.6%

       \[\leadsto \frac{\mathsf{fma}\left(\frac{\frac{1}{x}}{n}, \frac{0.3333333333333333}{x} - 0.5, \frac{1}{n}\right)}{\color{blue}{x}} \]

   if 5.0000000000000001e-9 < (/.f64 #s(literal 1 binary64) n) < 2.00000000000000008e201

   1. Initial program 93.2%

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

   3. Taylor expanded in x around 0

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

   5. Applied rewrites 93.2%

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

4. Final simplification 76.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{n}^{-1} \leq -4 \cdot 10^{-10}:\\ \;\;\;\;\frac{{x}^{\left({n}^{-1}\right)}}{n}\\ \mathbf{elif}\;{n}^{-1} \leq -2 \cdot 10^{-199}:\\ \;\;\;\;\frac{\frac{\frac{\frac{0.3333333333333333}{x} - 0.5}{x} + 1}{x}}{n}\\ \mathbf{elif}\;{n}^{-1} \leq -5 \cdot 10^{-257}:\\ \;\;\;\;\frac{x - \log x}{n}\\ \mathbf{elif}\;{n}^{-1} \leq 5 \cdot 10^{-9}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{{x}^{-1}}{n}, \frac{0.3333333333333333}{x} - 0.5, {n}^{-1}\right)}{x}\\ \mathbf{elif}\;{n}^{-1} \leq 2 \cdot 10^{+201}:\\ \;\;\;\;1 - {x}^{\left({n}^{-1}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{\frac{0.3333333333333333}{x} - 0.5}{x} + 1}{x}}{n}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 81.3% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {x}^{\left({n}^{-1}\right)}\\ \mathbf{if}\;{n}^{-1} \leq -1 \cdot 10^{-134}:\\ \;\;\;\;\frac{\frac{{\left({x}^{\left(\frac{-1}{n}\right)}\right)}^{-1}}{x}}{n}\\ \mathbf{elif}\;{n}^{-1} \leq 5 \cdot 10^{-53}:\\ \;\;\;\;\frac{\log \left(\frac{x}{1 + x}\right)}{-n}\\ \mathbf{elif}\;{n}^{-1} \leq 5 \cdot 10^{-9}:\\ \;\;\;\;\frac{t\_0}{n \cdot x}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\frac{\mathsf{fma}\left(-0.3333333333333333, x, 0.5\right) - \frac{\mathsf{fma}\left(x, \frac{0.16666666666666666}{n} - 0.5, 0.5\right)}{n}}{-n}, x, {n}^{-1}\right), x, 1\right) - t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (let* ((t_0 (pow x (pow n -1.0))))
   (if (<= (pow n -1.0) -1e-134)
     (/ (/ (pow (pow x (/ -1.0 n)) -1.0) x) n)
     (if (<= (pow n -1.0) 5e-53)
       (/ (log (/ x (+ 1.0 x))) (- n))
       (if (<= (pow n -1.0) 5e-9)
         (/ t_0 (* n x))
         (-
          (fma
           (fma
            (/
             (-
              (fma -0.3333333333333333 x 0.5)
              (/ (fma x (- (/ 0.16666666666666666 n) 0.5) 0.5) n))
             (- n))
            x
            (pow n -1.0))
           x
           1.0)
          t_0))))))

C

double code(double x, double n) {
	double t_0 = pow(x, pow(n, -1.0));
	double tmp;
	if (pow(n, -1.0) <= -1e-134) {
		tmp = (pow(pow(x, (-1.0 / n)), -1.0) / x) / n;
	} else if (pow(n, -1.0) <= 5e-53) {
		tmp = log((x / (1.0 + x))) / -n;
	} else if (pow(n, -1.0) <= 5e-9) {
		tmp = t_0 / (n * x);
	} else {
		tmp = fma(fma(((fma(-0.3333333333333333, x, 0.5) - (fma(x, ((0.16666666666666666 / n) - 0.5), 0.5) / n)) / -n), x, pow(n, -1.0)), x, 1.0) - t_0;
	}
	return tmp;
}

Julia

function code(x, n)
	t_0 = x ^ (n ^ -1.0)
	tmp = 0.0
	if ((n ^ -1.0) <= -1e-134)
		tmp = Float64(Float64(((x ^ Float64(-1.0 / n)) ^ -1.0) / x) / n);
	elseif ((n ^ -1.0) <= 5e-53)
		tmp = Float64(log(Float64(x / Float64(1.0 + x))) / Float64(-n));
	elseif ((n ^ -1.0) <= 5e-9)
		tmp = Float64(t_0 / Float64(n * x));
	else
		tmp = Float64(fma(fma(Float64(Float64(fma(-0.3333333333333333, x, 0.5) - Float64(fma(x, Float64(Float64(0.16666666666666666 / n) - 0.5), 0.5) / n)) / Float64(-n)), x, (n ^ -1.0)), x, 1.0) - t_0);
	end
	return tmp
end

Wolfram

code[x_, n_] := Block[{t$95$0 = N[Power[x, N[Power[n, -1.0], $MachinePrecision]], $MachinePrecision]}, If[LessEqual[N[Power[n, -1.0], $MachinePrecision], -1e-134], N[(N[(N[Power[N[Power[x, N[(-1.0 / n), $MachinePrecision]], $MachinePrecision], -1.0], $MachinePrecision] / x), $MachinePrecision] / n), $MachinePrecision], If[LessEqual[N[Power[n, -1.0], $MachinePrecision], 5e-53], N[(N[Log[N[(x / N[(1.0 + x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / (-n)), $MachinePrecision], If[LessEqual[N[Power[n, -1.0], $MachinePrecision], 5e-9], N[(t$95$0 / N[(n * x), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(N[(N[(-0.3333333333333333 * x + 0.5), $MachinePrecision] - N[(N[(x * N[(N[(0.16666666666666666 / n), $MachinePrecision] - 0.5), $MachinePrecision] + 0.5), $MachinePrecision] / n), $MachinePrecision]), $MachinePrecision] / (-n)), $MachinePrecision] * x + N[Power[n, -1.0], $MachinePrecision]), $MachinePrecision] * x + 1.0), $MachinePrecision] - t$95$0), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if (/.f64 #s(literal 1 binary64) n) < -1.00000000000000004e-134

   1. Initial program 73.9%

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

   3. Taylor expanded in x around inf

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

      1. associate-/l/ N/A

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

      2. lower-/.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. log-rec N/A

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

      5. mul-1-neg N/A

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

      6. associate-*r/ N/A

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

      7. associate-*r* N/A

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

      8. metadata-eval N/A

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

      9. *-commutative N/A

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

      10. associate-/l* N/A

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

      11. exp-to-pow N/A

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

      12. lower-pow.f64 N/A

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

      13. lower-/.f64 84.8

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

   5. Applied rewrites 84.8%

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

   7. Applied rewrites 84.8%

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

   if -1.00000000000000004e-134 < (/.f64 #s(literal 1 binary64) n) < 5e-53

   1. Initial program 40.8%

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

   3. Taylor expanded in n around inf

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

      1. lower-/.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lower-log1p.f64 N/A

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

      4. lower-log.f64 85.1

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

   5. Applied rewrites 85.1%

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

   7. Applied rewrites 85.5%

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

   if 5e-53 < (/.f64 #s(literal 1 binary64) n) < 5.0000000000000001e-9

   1. Initial program 4.9%

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

   3. Taylor expanded in x around inf

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

      1. associate-/l/ N/A

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

      2. lower-/.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. log-rec N/A

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

      5. mul-1-neg N/A

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

      6. associate-*r/ N/A

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

      7. associate-*r* N/A

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

      8. metadata-eval N/A

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

      9. *-commutative N/A

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

      10. associate-/l* N/A

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

      11. exp-to-pow N/A

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

      12. lower-pow.f64 N/A

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

      13. lower-/.f64 85.9

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

   5. Applied rewrites 85.9%

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

   7. Applied rewrites 86.1%

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

   if 5.0000000000000001e-9 < (/.f64 #s(literal 1 binary64) n)

   1. Initial program 73.6%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

         \[\leadsto \color{blue}{\left(x \cdot \left(x \cdot \left(\left(\frac{1}{2} \cdot \frac{1}{{n}^{2}} + x \cdot \left(\left(\frac{1}{6} \cdot \frac{1}{{n}^{3}} + \frac{1}{3} \cdot \frac{1}{n}\right) - \frac{1}{2} \cdot \frac{1}{{n}^{2}}\right)\right) - \frac{1}{2} \cdot \frac{1}{n}\right) + \frac{1}{n}\right) + 1\right)} - {x}^{\left(\frac{1}{n}\right)} \]

      2. *-commutative N/A

         \[\leadsto \left(\color{blue}{\left(x \cdot \left(\left(\frac{1}{2} \cdot \frac{1}{{n}^{2}} + x \cdot \left(\left(\frac{1}{6} \cdot \frac{1}{{n}^{3}} + \frac{1}{3} \cdot \frac{1}{n}\right) - \frac{1}{2} \cdot \frac{1}{{n}^{2}}\right)\right) - \frac{1}{2} \cdot \frac{1}{n}\right) + \frac{1}{n}\right) \cdot x} + 1\right) - {x}^{\left(\frac{1}{n}\right)} \]

      3. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(x \cdot \left(\left(\frac{1}{2} \cdot \frac{1}{{n}^{2}} + x \cdot \left(\left(\frac{1}{6} \cdot \frac{1}{{n}^{3}} + \frac{1}{3} \cdot \frac{1}{n}\right) - \frac{1}{2} \cdot \frac{1}{{n}^{2}}\right)\right) - \frac{1}{2} \cdot \frac{1}{n}\right) + \frac{1}{n}, x, 1\right)} - {x}^{\left(\frac{1}{n}\right)} \]

   5. Applied rewrites 38.1%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\left(\frac{0.3333333333333333}{n} + \frac{0.16666666666666666}{{n}^{3}}\right) - \frac{0.5}{n \cdot n}, x, \frac{0.5}{n \cdot n} - \frac{0.5}{n}\right), x, \frac{1}{n}\right), x, 1\right)} - {x}^{\left(\frac{1}{n}\right)} \]
   6. Step-by-step derivation

   7. Applied rewrites 38.1%

      \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\left(\frac{0.3333333333333333}{n} + \frac{\frac{0.16666666666666666}{n \cdot n}}{n}\right) - \frac{0.5}{n \cdot n}, x, \frac{0.5}{n \cdot n} - \frac{0.5}{n}\right), x, \frac{1}{n}\right), x, 1\right) - {x}^{\left(\frac{1}{n}\right)} \]

   8. Taylor expanded in n around -inf

      \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(-1 \cdot \frac{\frac{1}{2} + \left(-1 \cdot \frac{\frac{1}{2} + \left(\frac{-1}{2} \cdot x + \frac{1}{6} \cdot \frac{x}{n}\right)}{n} + \frac{-1}{3} \cdot x\right)}{n}, x, \frac{1}{n}\right), x, 1\right) - {x}^{\left(\frac{1}{n}\right)} \]
   9. Step-by-step derivation

   10. Applied rewrites 81.7%

       \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\frac{\mathsf{fma}\left(-0.3333333333333333, x, 0.5\right) - \frac{\mathsf{fma}\left(x, \frac{0.16666666666666666}{n} - 0.5, 0.5\right)}{n}}{-n}, x, \frac{1}{n}\right), x, 1\right) - {x}^{\left(\frac{1}{n}\right)} \]
3. Recombined 4 regimes into one program.

4. Final simplification 84.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{n}^{-1} \leq -1 \cdot 10^{-134}:\\ \;\;\;\;\frac{\frac{{\left({x}^{\left(\frac{-1}{n}\right)}\right)}^{-1}}{x}}{n}\\ \mathbf{elif}\;{n}^{-1} \leq 5 \cdot 10^{-53}:\\ \;\;\;\;\frac{\log \left(\frac{x}{1 + x}\right)}{-n}\\ \mathbf{elif}\;{n}^{-1} \leq 5 \cdot 10^{-9}:\\ \;\;\;\;\frac{{x}^{\left({n}^{-1}\right)}}{n \cdot x}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\frac{\mathsf{fma}\left(-0.3333333333333333, x, 0.5\right) - \frac{\mathsf{fma}\left(x, \frac{0.16666666666666666}{n} - 0.5, 0.5\right)}{n}}{-n}, x, {n}^{-1}\right), x, 1\right) - {x}^{\left({n}^{-1}\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 81.4% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {x}^{\left({n}^{-1}\right)}\\ \mathbf{if}\;{n}^{-1} \leq -1 \cdot 10^{-134}:\\ \;\;\;\;\frac{\frac{{\left({x}^{\left(\frac{-1}{n}\right)}\right)}^{-1}}{x}}{n}\\ \mathbf{elif}\;{n}^{-1} \leq 5 \cdot 10^{-53}:\\ \;\;\;\;\frac{\log \left(\frac{x}{1 + x}\right)}{-n}\\ \mathbf{elif}\;{n}^{-1} \leq 5 \cdot 10^{-9}:\\ \;\;\;\;\frac{\frac{t\_0}{x}}{n}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\frac{\mathsf{fma}\left(-0.3333333333333333, x, 0.5\right) - \frac{\mathsf{fma}\left(x, \frac{0.16666666666666666}{n} - 0.5, 0.5\right)}{n}}{-n}, x, {n}^{-1}\right), x, 1\right) - t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (let* ((t_0 (pow x (pow n -1.0))))
   (if (<= (pow n -1.0) -1e-134)
     (/ (/ (pow (pow x (/ -1.0 n)) -1.0) x) n)
     (if (<= (pow n -1.0) 5e-53)
       (/ (log (/ x (+ 1.0 x))) (- n))
       (if (<= (pow n -1.0) 5e-9)
         (/ (/ t_0 x) n)
         (-
          (fma
           (fma
            (/
             (-
              (fma -0.3333333333333333 x 0.5)
              (/ (fma x (- (/ 0.16666666666666666 n) 0.5) 0.5) n))
             (- n))
            x
            (pow n -1.0))
           x
           1.0)
          t_0))))))

C

double code(double x, double n) {
	double t_0 = pow(x, pow(n, -1.0));
	double tmp;
	if (pow(n, -1.0) <= -1e-134) {
		tmp = (pow(pow(x, (-1.0 / n)), -1.0) / x) / n;
	} else if (pow(n, -1.0) <= 5e-53) {
		tmp = log((x / (1.0 + x))) / -n;
	} else if (pow(n, -1.0) <= 5e-9) {
		tmp = (t_0 / x) / n;
	} else {
		tmp = fma(fma(((fma(-0.3333333333333333, x, 0.5) - (fma(x, ((0.16666666666666666 / n) - 0.5), 0.5) / n)) / -n), x, pow(n, -1.0)), x, 1.0) - t_0;
	}
	return tmp;
}

Julia

function code(x, n)
	t_0 = x ^ (n ^ -1.0)
	tmp = 0.0
	if ((n ^ -1.0) <= -1e-134)
		tmp = Float64(Float64(((x ^ Float64(-1.0 / n)) ^ -1.0) / x) / n);
	elseif ((n ^ -1.0) <= 5e-53)
		tmp = Float64(log(Float64(x / Float64(1.0 + x))) / Float64(-n));
	elseif ((n ^ -1.0) <= 5e-9)
		tmp = Float64(Float64(t_0 / x) / n);
	else
		tmp = Float64(fma(fma(Float64(Float64(fma(-0.3333333333333333, x, 0.5) - Float64(fma(x, Float64(Float64(0.16666666666666666 / n) - 0.5), 0.5) / n)) / Float64(-n)), x, (n ^ -1.0)), x, 1.0) - t_0);
	end
	return tmp
end

Wolfram

code[x_, n_] := Block[{t$95$0 = N[Power[x, N[Power[n, -1.0], $MachinePrecision]], $MachinePrecision]}, If[LessEqual[N[Power[n, -1.0], $MachinePrecision], -1e-134], N[(N[(N[Power[N[Power[x, N[(-1.0 / n), $MachinePrecision]], $MachinePrecision], -1.0], $MachinePrecision] / x), $MachinePrecision] / n), $MachinePrecision], If[LessEqual[N[Power[n, -1.0], $MachinePrecision], 5e-53], N[(N[Log[N[(x / N[(1.0 + x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / (-n)), $MachinePrecision], If[LessEqual[N[Power[n, -1.0], $MachinePrecision], 5e-9], N[(N[(t$95$0 / x), $MachinePrecision] / n), $MachinePrecision], N[(N[(N[(N[(N[(N[(-0.3333333333333333 * x + 0.5), $MachinePrecision] - N[(N[(x * N[(N[(0.16666666666666666 / n), $MachinePrecision] - 0.5), $MachinePrecision] + 0.5), $MachinePrecision] / n), $MachinePrecision]), $MachinePrecision] / (-n)), $MachinePrecision] * x + N[Power[n, -1.0], $MachinePrecision]), $MachinePrecision] * x + 1.0), $MachinePrecision] - t$95$0), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if (/.f64 #s(literal 1 binary64) n) < -1.00000000000000004e-134

   1. Initial program 73.9%

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

   3. Taylor expanded in x around inf

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

      1. associate-/l/ N/A

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

      2. lower-/.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. log-rec N/A

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

      5. mul-1-neg N/A

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

      6. associate-*r/ N/A

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

      7. associate-*r* N/A

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

      8. metadata-eval N/A

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

      9. *-commutative N/A

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

      10. associate-/l* N/A

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

      11. exp-to-pow N/A

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

      12. lower-pow.f64 N/A

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

      13. lower-/.f64 84.8

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

   5. Applied rewrites 84.8%

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

   7. Applied rewrites 84.8%

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

   if -1.00000000000000004e-134 < (/.f64 #s(literal 1 binary64) n) < 5e-53

   1. Initial program 40.8%

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

   3. Taylor expanded in n around inf

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

      1. lower-/.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lower-log1p.f64 N/A

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

      4. lower-log.f64 85.1

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

   5. Applied rewrites 85.1%

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

   7. Applied rewrites 85.5%

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

   if 5e-53 < (/.f64 #s(literal 1 binary64) n) < 5.0000000000000001e-9

   1. Initial program 4.9%

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

   3. Taylor expanded in x around inf

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

      1. associate-/l/ N/A

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

      2. lower-/.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. log-rec N/A

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

      5. mul-1-neg N/A

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

      6. associate-*r/ N/A

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

      7. associate-*r* N/A

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

      8. metadata-eval N/A

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

      9. *-commutative N/A

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

      10. associate-/l* N/A

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

      11. exp-to-pow N/A

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

      12. lower-pow.f64 N/A

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

      13. lower-/.f64 85.9

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

   5. Applied rewrites 85.9%

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

   if 5.0000000000000001e-9 < (/.f64 #s(literal 1 binary64) n)

   1. Initial program 73.6%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

         \[\leadsto \color{blue}{\left(x \cdot \left(x \cdot \left(\left(\frac{1}{2} \cdot \frac{1}{{n}^{2}} + x \cdot \left(\left(\frac{1}{6} \cdot \frac{1}{{n}^{3}} + \frac{1}{3} \cdot \frac{1}{n}\right) - \frac{1}{2} \cdot \frac{1}{{n}^{2}}\right)\right) - \frac{1}{2} \cdot \frac{1}{n}\right) + \frac{1}{n}\right) + 1\right)} - {x}^{\left(\frac{1}{n}\right)} \]

      2. *-commutative N/A

         \[\leadsto \left(\color{blue}{\left(x \cdot \left(\left(\frac{1}{2} \cdot \frac{1}{{n}^{2}} + x \cdot \left(\left(\frac{1}{6} \cdot \frac{1}{{n}^{3}} + \frac{1}{3} \cdot \frac{1}{n}\right) - \frac{1}{2} \cdot \frac{1}{{n}^{2}}\right)\right) - \frac{1}{2} \cdot \frac{1}{n}\right) + \frac{1}{n}\right) \cdot x} + 1\right) - {x}^{\left(\frac{1}{n}\right)} \]

      3. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(x \cdot \left(\left(\frac{1}{2} \cdot \frac{1}{{n}^{2}} + x \cdot \left(\left(\frac{1}{6} \cdot \frac{1}{{n}^{3}} + \frac{1}{3} \cdot \frac{1}{n}\right) - \frac{1}{2} \cdot \frac{1}{{n}^{2}}\right)\right) - \frac{1}{2} \cdot \frac{1}{n}\right) + \frac{1}{n}, x, 1\right)} - {x}^{\left(\frac{1}{n}\right)} \]

   5. Applied rewrites 38.1%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\left(\frac{0.3333333333333333}{n} + \frac{0.16666666666666666}{{n}^{3}}\right) - \frac{0.5}{n \cdot n}, x, \frac{0.5}{n \cdot n} - \frac{0.5}{n}\right), x, \frac{1}{n}\right), x, 1\right)} - {x}^{\left(\frac{1}{n}\right)} \]
   6. Step-by-step derivation

   7. Applied rewrites 38.1%

      \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\left(\frac{0.3333333333333333}{n} + \frac{\frac{0.16666666666666666}{n \cdot n}}{n}\right) - \frac{0.5}{n \cdot n}, x, \frac{0.5}{n \cdot n} - \frac{0.5}{n}\right), x, \frac{1}{n}\right), x, 1\right) - {x}^{\left(\frac{1}{n}\right)} \]

   8. Taylor expanded in n around -inf

      \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(-1 \cdot \frac{\frac{1}{2} + \left(-1 \cdot \frac{\frac{1}{2} + \left(\frac{-1}{2} \cdot x + \frac{1}{6} \cdot \frac{x}{n}\right)}{n} + \frac{-1}{3} \cdot x\right)}{n}, x, \frac{1}{n}\right), x, 1\right) - {x}^{\left(\frac{1}{n}\right)} \]
   9. Step-by-step derivation

   10. Applied rewrites 81.7%

       \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\frac{\mathsf{fma}\left(-0.3333333333333333, x, 0.5\right) - \frac{\mathsf{fma}\left(x, \frac{0.16666666666666666}{n} - 0.5, 0.5\right)}{n}}{-n}, x, \frac{1}{n}\right), x, 1\right) - {x}^{\left(\frac{1}{n}\right)} \]
3. Recombined 4 regimes into one program.

4. Final simplification 84.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{n}^{-1} \leq -1 \cdot 10^{-134}:\\ \;\;\;\;\frac{\frac{{\left({x}^{\left(\frac{-1}{n}\right)}\right)}^{-1}}{x}}{n}\\ \mathbf{elif}\;{n}^{-1} \leq 5 \cdot 10^{-53}:\\ \;\;\;\;\frac{\log \left(\frac{x}{1 + x}\right)}{-n}\\ \mathbf{elif}\;{n}^{-1} \leq 5 \cdot 10^{-9}:\\ \;\;\;\;\frac{\frac{{x}^{\left({n}^{-1}\right)}}{x}}{n}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\frac{\mathsf{fma}\left(-0.3333333333333333, x, 0.5\right) - \frac{\mathsf{fma}\left(x, \frac{0.16666666666666666}{n} - 0.5, 0.5\right)}{n}}{-n}, x, {n}^{-1}\right), x, 1\right) - {x}^{\left({n}^{-1}\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 80.8% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {x}^{\left({n}^{-1}\right)}\\ \mathbf{if}\;{n}^{-1} \leq -1 \cdot 10^{-134}:\\ \;\;\;\;\frac{\frac{{\left({x}^{\left(\frac{-1}{n}\right)}\right)}^{-1}}{x}}{n}\\ \mathbf{elif}\;{n}^{-1} \leq 5 \cdot 10^{-53}:\\ \;\;\;\;\frac{\log \left(\frac{x}{1 + x}\right)}{-n}\\ \mathbf{elif}\;{n}^{-1} \leq 5 \cdot 10^{-9}:\\ \;\;\;\;\frac{\frac{t\_0}{x}}{n}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\frac{0.5}{n \cdot n} - \frac{0.5}{n}, x, {n}^{-1}\right), x, 1\right) - t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (let* ((t_0 (pow x (pow n -1.0))))
   (if (<= (pow n -1.0) -1e-134)
     (/ (/ (pow (pow x (/ -1.0 n)) -1.0) x) n)
     (if (<= (pow n -1.0) 5e-53)
       (/ (log (/ x (+ 1.0 x))) (- n))
       (if (<= (pow n -1.0) 5e-9)
         (/ (/ t_0 x) n)
         (-
          (fma (fma (- (/ 0.5 (* n n)) (/ 0.5 n)) x (pow n -1.0)) x 1.0)
          t_0))))))

C

double code(double x, double n) {
	double t_0 = pow(x, pow(n, -1.0));
	double tmp;
	if (pow(n, -1.0) <= -1e-134) {
		tmp = (pow(pow(x, (-1.0 / n)), -1.0) / x) / n;
	} else if (pow(n, -1.0) <= 5e-53) {
		tmp = log((x / (1.0 + x))) / -n;
	} else if (pow(n, -1.0) <= 5e-9) {
		tmp = (t_0 / x) / n;
	} else {
		tmp = fma(fma(((0.5 / (n * n)) - (0.5 / n)), x, pow(n, -1.0)), x, 1.0) - t_0;
	}
	return tmp;
}

Julia

function code(x, n)
	t_0 = x ^ (n ^ -1.0)
	tmp = 0.0
	if ((n ^ -1.0) <= -1e-134)
		tmp = Float64(Float64(((x ^ Float64(-1.0 / n)) ^ -1.0) / x) / n);
	elseif ((n ^ -1.0) <= 5e-53)
		tmp = Float64(log(Float64(x / Float64(1.0 + x))) / Float64(-n));
	elseif ((n ^ -1.0) <= 5e-9)
		tmp = Float64(Float64(t_0 / x) / n);
	else
		tmp = Float64(fma(fma(Float64(Float64(0.5 / Float64(n * n)) - Float64(0.5 / n)), x, (n ^ -1.0)), x, 1.0) - t_0);
	end
	return tmp
end

Wolfram

code[x_, n_] := Block[{t$95$0 = N[Power[x, N[Power[n, -1.0], $MachinePrecision]], $MachinePrecision]}, If[LessEqual[N[Power[n, -1.0], $MachinePrecision], -1e-134], N[(N[(N[Power[N[Power[x, N[(-1.0 / n), $MachinePrecision]], $MachinePrecision], -1.0], $MachinePrecision] / x), $MachinePrecision] / n), $MachinePrecision], If[LessEqual[N[Power[n, -1.0], $MachinePrecision], 5e-53], N[(N[Log[N[(x / N[(1.0 + x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / (-n)), $MachinePrecision], If[LessEqual[N[Power[n, -1.0], $MachinePrecision], 5e-9], N[(N[(t$95$0 / x), $MachinePrecision] / n), $MachinePrecision], N[(N[(N[(N[(N[(0.5 / N[(n * n), $MachinePrecision]), $MachinePrecision] - N[(0.5 / n), $MachinePrecision]), $MachinePrecision] * x + N[Power[n, -1.0], $MachinePrecision]), $MachinePrecision] * x + 1.0), $MachinePrecision] - t$95$0), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if (/.f64 #s(literal 1 binary64) n) < -1.00000000000000004e-134

   1. Initial program 73.9%

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

   3. Taylor expanded in x around inf

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

      1. associate-/l/ N/A

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

      2. lower-/.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. log-rec N/A

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

      5. mul-1-neg N/A

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

      6. associate-*r/ N/A

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

      7. associate-*r* N/A

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

      8. metadata-eval N/A

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

      9. *-commutative N/A

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

      10. associate-/l* N/A

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

      11. exp-to-pow N/A

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

      12. lower-pow.f64 N/A

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

      13. lower-/.f64 84.8

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

   5. Applied rewrites 84.8%

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

   7. Applied rewrites 84.8%

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

   if -1.00000000000000004e-134 < (/.f64 #s(literal 1 binary64) n) < 5e-53

   1. Initial program 40.8%

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

   3. Taylor expanded in n around inf

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

      1. lower-/.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lower-log1p.f64 N/A

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

      4. lower-log.f64 85.1

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

   5. Applied rewrites 85.1%

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

   7. Applied rewrites 85.5%

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

   if 5e-53 < (/.f64 #s(literal 1 binary64) n) < 5.0000000000000001e-9

   1. Initial program 4.9%

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

   3. Taylor expanded in x around inf

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

      1. associate-/l/ N/A

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

      2. lower-/.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. log-rec N/A

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

      5. mul-1-neg N/A

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

      6. associate-*r/ N/A

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

      7. associate-*r* N/A

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

      8. metadata-eval N/A

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

      9. *-commutative N/A

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

      10. associate-/l* N/A

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

      11. exp-to-pow N/A

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

      12. lower-pow.f64 N/A

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

      13. lower-/.f64 85.9

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

   5. Applied rewrites 85.9%

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

   if 5.0000000000000001e-9 < (/.f64 #s(literal 1 binary64) n)

   1. Initial program 73.6%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. lower--.f64 N/A

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

      7. associate-*r/ N/A

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

      8. metadata-eval N/A

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

      9. lower-/.f64 N/A

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

      10. unpow2 N/A

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

      11. lower-*.f64 N/A

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

      12. associate-*r/ N/A

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

      13. metadata-eval N/A

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

      14. lower-/.f64 N/A

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

      15. lower-/.f64 81.2

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

   5. Applied rewrites 81.2%

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

4. Final simplification 84.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{n}^{-1} \leq -1 \cdot 10^{-134}:\\ \;\;\;\;\frac{\frac{{\left({x}^{\left(\frac{-1}{n}\right)}\right)}^{-1}}{x}}{n}\\ \mathbf{elif}\;{n}^{-1} \leq 5 \cdot 10^{-53}:\\ \;\;\;\;\frac{\log \left(\frac{x}{1 + x}\right)}{-n}\\ \mathbf{elif}\;{n}^{-1} \leq 5 \cdot 10^{-9}:\\ \;\;\;\;\frac{\frac{{x}^{\left({n}^{-1}\right)}}{x}}{n}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\frac{0.5}{n \cdot n} - \frac{0.5}{n}, x, {n}^{-1}\right), x, 1\right) - {x}^{\left({n}^{-1}\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 80.3% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {x}^{\left({n}^{-1}\right)}\\ \mathbf{if}\;{n}^{-1} \leq -1 \cdot 10^{-134}:\\ \;\;\;\;\frac{\frac{{\left({x}^{\left(\frac{-1}{n}\right)}\right)}^{-1}}{x}}{n}\\ \mathbf{elif}\;{n}^{-1} \leq 5 \cdot 10^{-53}:\\ \;\;\;\;\frac{\log \left(\frac{x}{1 + x}\right)}{-n}\\ \mathbf{elif}\;{n}^{-1} \leq 5 \cdot 10^{-9}:\\ \;\;\;\;\frac{\frac{t\_0}{x}}{n}\\ \mathbf{elif}\;{n}^{-1} \leq 5 \cdot 10^{+205}:\\ \;\;\;\;\left(\frac{x}{n} + 1\right) - t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{\frac{0.3333333333333333}{x} - 0.5}{x} + 1}{x}}{n}\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (let* ((t_0 (pow x (pow n -1.0))))
   (if (<= (pow n -1.0) -1e-134)
     (/ (/ (pow (pow x (/ -1.0 n)) -1.0) x) n)
     (if (<= (pow n -1.0) 5e-53)
       (/ (log (/ x (+ 1.0 x))) (- n))
       (if (<= (pow n -1.0) 5e-9)
         (/ (/ t_0 x) n)
         (if (<= (pow n -1.0) 5e+205)
           (- (+ (/ x n) 1.0) t_0)
           (/ (/ (+ (/ (- (/ 0.3333333333333333 x) 0.5) x) 1.0) x) n)))))))

C

double code(double x, double n) {
	double t_0 = pow(x, pow(n, -1.0));
	double tmp;
	if (pow(n, -1.0) <= -1e-134) {
		tmp = (pow(pow(x, (-1.0 / n)), -1.0) / x) / n;
	} else if (pow(n, -1.0) <= 5e-53) {
		tmp = log((x / (1.0 + x))) / -n;
	} else if (pow(n, -1.0) <= 5e-9) {
		tmp = (t_0 / x) / n;
	} else if (pow(n, -1.0) <= 5e+205) {
		tmp = ((x / n) + 1.0) - t_0;
	} else {
		tmp = (((((0.3333333333333333 / x) - 0.5) / x) + 1.0) / x) / n;
	}
	return tmp;
}

Fortran

real(8) function code(x, n)
    real(8), intent (in) :: x
    real(8), intent (in) :: n
    real(8) :: t_0
    real(8) :: tmp
    t_0 = x ** (n ** (-1.0d0))
    if ((n ** (-1.0d0)) <= (-1d-134)) then
        tmp = (((x ** ((-1.0d0) / n)) ** (-1.0d0)) / x) / n
    else if ((n ** (-1.0d0)) <= 5d-53) then
        tmp = log((x / (1.0d0 + x))) / -n
    else if ((n ** (-1.0d0)) <= 5d-9) then
        tmp = (t_0 / x) / n
    else if ((n ** (-1.0d0)) <= 5d+205) then
        tmp = ((x / n) + 1.0d0) - t_0
    else
        tmp = (((((0.3333333333333333d0 / x) - 0.5d0) / x) + 1.0d0) / x) / n
    end if
    code = tmp
end function

Java

public static double code(double x, double n) {
	double t_0 = Math.pow(x, Math.pow(n, -1.0));
	double tmp;
	if (Math.pow(n, -1.0) <= -1e-134) {
		tmp = (Math.pow(Math.pow(x, (-1.0 / n)), -1.0) / x) / n;
	} else if (Math.pow(n, -1.0) <= 5e-53) {
		tmp = Math.log((x / (1.0 + x))) / -n;
	} else if (Math.pow(n, -1.0) <= 5e-9) {
		tmp = (t_0 / x) / n;
	} else if (Math.pow(n, -1.0) <= 5e+205) {
		tmp = ((x / n) + 1.0) - t_0;
	} else {
		tmp = (((((0.3333333333333333 / x) - 0.5) / x) + 1.0) / x) / n;
	}
	return tmp;
}

Python

def code(x, n):
	t_0 = math.pow(x, math.pow(n, -1.0))
	tmp = 0
	if math.pow(n, -1.0) <= -1e-134:
		tmp = (math.pow(math.pow(x, (-1.0 / n)), -1.0) / x) / n
	elif math.pow(n, -1.0) <= 5e-53:
		tmp = math.log((x / (1.0 + x))) / -n
	elif math.pow(n, -1.0) <= 5e-9:
		tmp = (t_0 / x) / n
	elif math.pow(n, -1.0) <= 5e+205:
		tmp = ((x / n) + 1.0) - t_0
	else:
		tmp = (((((0.3333333333333333 / x) - 0.5) / x) + 1.0) / x) / n
	return tmp

Julia

function code(x, n)
	t_0 = x ^ (n ^ -1.0)
	tmp = 0.0
	if ((n ^ -1.0) <= -1e-134)
		tmp = Float64(Float64(((x ^ Float64(-1.0 / n)) ^ -1.0) / x) / n);
	elseif ((n ^ -1.0) <= 5e-53)
		tmp = Float64(log(Float64(x / Float64(1.0 + x))) / Float64(-n));
	elseif ((n ^ -1.0) <= 5e-9)
		tmp = Float64(Float64(t_0 / x) / n);
	elseif ((n ^ -1.0) <= 5e+205)
		tmp = Float64(Float64(Float64(x / n) + 1.0) - t_0);
	else
		tmp = Float64(Float64(Float64(Float64(Float64(Float64(0.3333333333333333 / x) - 0.5) / x) + 1.0) / x) / n);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, n)
	t_0 = x ^ (n ^ -1.0);
	tmp = 0.0;
	if ((n ^ -1.0) <= -1e-134)
		tmp = (((x ^ (-1.0 / n)) ^ -1.0) / x) / n;
	elseif ((n ^ -1.0) <= 5e-53)
		tmp = log((x / (1.0 + x))) / -n;
	elseif ((n ^ -1.0) <= 5e-9)
		tmp = (t_0 / x) / n;
	elseif ((n ^ -1.0) <= 5e+205)
		tmp = ((x / n) + 1.0) - t_0;
	else
		tmp = (((((0.3333333333333333 / x) - 0.5) / x) + 1.0) / x) / n;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, n_] := Block[{t$95$0 = N[Power[x, N[Power[n, -1.0], $MachinePrecision]], $MachinePrecision]}, If[LessEqual[N[Power[n, -1.0], $MachinePrecision], -1e-134], N[(N[(N[Power[N[Power[x, N[(-1.0 / n), $MachinePrecision]], $MachinePrecision], -1.0], $MachinePrecision] / x), $MachinePrecision] / n), $MachinePrecision], If[LessEqual[N[Power[n, -1.0], $MachinePrecision], 5e-53], N[(N[Log[N[(x / N[(1.0 + x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / (-n)), $MachinePrecision], If[LessEqual[N[Power[n, -1.0], $MachinePrecision], 5e-9], N[(N[(t$95$0 / x), $MachinePrecision] / n), $MachinePrecision], If[LessEqual[N[Power[n, -1.0], $MachinePrecision], 5e+205], N[(N[(N[(x / n), $MachinePrecision] + 1.0), $MachinePrecision] - t$95$0), $MachinePrecision], N[(N[(N[(N[(N[(N[(0.3333333333333333 / x), $MachinePrecision] - 0.5), $MachinePrecision] / x), $MachinePrecision] + 1.0), $MachinePrecision] / x), $MachinePrecision] / n), $MachinePrecision]]]]]]

Derivation

1. Split input into 5 regimes

2. if (/.f64 #s(literal 1 binary64) n) < -1.00000000000000004e-134

   1. Initial program 73.9%

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

   3. Taylor expanded in x around inf

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

      1. associate-/l/ N/A

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

      2. lower-/.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. log-rec N/A

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

      5. mul-1-neg N/A

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

      6. associate-*r/ N/A

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

      7. associate-*r* N/A

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

      8. metadata-eval N/A

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

      9. *-commutative N/A

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

      10. associate-/l* N/A

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

      11. exp-to-pow N/A

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

      12. lower-pow.f64 N/A

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

      13. lower-/.f64 84.8

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

   5. Applied rewrites 84.8%

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

   7. Applied rewrites 84.8%

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

   if -1.00000000000000004e-134 < (/.f64 #s(literal 1 binary64) n) < 5e-53

   1. Initial program 40.8%

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

   3. Taylor expanded in n around inf

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

      1. lower-/.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lower-log1p.f64 N/A

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

      4. lower-log.f64 85.1

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

   5. Applied rewrites 85.1%

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

   7. Applied rewrites 85.5%

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

   if 5e-53 < (/.f64 #s(literal 1 binary64) n) < 5.0000000000000001e-9

   1. Initial program 4.9%

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

   3. Taylor expanded in x around inf

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

      1. associate-/l/ N/A

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

      2. lower-/.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. log-rec N/A

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

      5. mul-1-neg N/A

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

      6. associate-*r/ N/A

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

      7. associate-*r* N/A

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

      8. metadata-eval N/A

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

      9. *-commutative N/A

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

      10. associate-/l* N/A

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

      11. exp-to-pow N/A

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

      12. lower-pow.f64 N/A

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

      13. lower-/.f64 85.9

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

   5. Applied rewrites 85.9%

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

   if 5.0000000000000001e-9 < (/.f64 #s(literal 1 binary64) n) < 5.0000000000000002e205

   1. Initial program 89.9%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. *-rgt-identity N/A

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

      3. associate-*r/ N/A

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

      4. lower-+.f64 N/A

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

      5. associate-*r/ N/A

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

      6. *-rgt-identity N/A

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

      7. lower-/.f64 90.8

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

   5. Applied rewrites 90.8%

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

   if 5.0000000000000002e205 < (/.f64 #s(literal 1 binary64) n)

   1. Initial program 3.1%

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

   3. Taylor expanded in n around inf

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

      1. lower-/.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lower-log1p.f64 N/A

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

      4. lower-log.f64 7.0

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

   5. Applied rewrites 7.0%

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

   6. Taylor expanded in x around inf

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

   8. Applied rewrites 100.0%

      \[\leadsto \frac{\frac{\frac{\frac{0.3333333333333333}{x} - 0.5}{x} + 1}{x}}{n} \]
3. Recombined 5 regimes into one program.

4. Final simplification 86.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{n}^{-1} \leq -1 \cdot 10^{-134}:\\ \;\;\;\;\frac{\frac{{\left({x}^{\left(\frac{-1}{n}\right)}\right)}^{-1}}{x}}{n}\\ \mathbf{elif}\;{n}^{-1} \leq 5 \cdot 10^{-53}:\\ \;\;\;\;\frac{\log \left(\frac{x}{1 + x}\right)}{-n}\\ \mathbf{elif}\;{n}^{-1} \leq 5 \cdot 10^{-9}:\\ \;\;\;\;\frac{\frac{{x}^{\left({n}^{-1}\right)}}{x}}{n}\\ \mathbf{elif}\;{n}^{-1} \leq 5 \cdot 10^{+205}:\\ \;\;\;\;\left(\frac{x}{n} + 1\right) - {x}^{\left({n}^{-1}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{\frac{0.3333333333333333}{x} - 0.5}{x} + 1}{x}}{n}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 80.3% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {x}^{\left({n}^{-1}\right)}\\ t_1 := \frac{\frac{t\_0}{x}}{n}\\ \mathbf{if}\;{n}^{-1} \leq -1 \cdot 10^{-134}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;{n}^{-1} \leq 5 \cdot 10^{-53}:\\ \;\;\;\;\frac{\log \left(\frac{x}{1 + x}\right)}{-n}\\ \mathbf{elif}\;{n}^{-1} \leq 5 \cdot 10^{-9}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;{n}^{-1} \leq 5 \cdot 10^{+205}:\\ \;\;\;\;\left(\frac{x}{n} + 1\right) - t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{\frac{0.3333333333333333}{x} - 0.5}{x} + 1}{x}}{n}\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (let* ((t_0 (pow x (pow n -1.0))) (t_1 (/ (/ t_0 x) n)))
   (if (<= (pow n -1.0) -1e-134)
     t_1
     (if (<= (pow n -1.0) 5e-53)
       (/ (log (/ x (+ 1.0 x))) (- n))
       (if (<= (pow n -1.0) 5e-9)
         t_1
         (if (<= (pow n -1.0) 5e+205)
           (- (+ (/ x n) 1.0) t_0)
           (/ (/ (+ (/ (- (/ 0.3333333333333333 x) 0.5) x) 1.0) x) n)))))))

C

double code(double x, double n) {
	double t_0 = pow(x, pow(n, -1.0));
	double t_1 = (t_0 / x) / n;
	double tmp;
	if (pow(n, -1.0) <= -1e-134) {
		tmp = t_1;
	} else if (pow(n, -1.0) <= 5e-53) {
		tmp = log((x / (1.0 + x))) / -n;
	} else if (pow(n, -1.0) <= 5e-9) {
		tmp = t_1;
	} else if (pow(n, -1.0) <= 5e+205) {
		tmp = ((x / n) + 1.0) - t_0;
	} else {
		tmp = (((((0.3333333333333333 / x) - 0.5) / x) + 1.0) / x) / n;
	}
	return tmp;
}

Fortran

real(8) function code(x, n)
    real(8), intent (in) :: x
    real(8), intent (in) :: n
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = x ** (n ** (-1.0d0))
    t_1 = (t_0 / x) / n
    if ((n ** (-1.0d0)) <= (-1d-134)) then
        tmp = t_1
    else if ((n ** (-1.0d0)) <= 5d-53) then
        tmp = log((x / (1.0d0 + x))) / -n
    else if ((n ** (-1.0d0)) <= 5d-9) then
        tmp = t_1
    else if ((n ** (-1.0d0)) <= 5d+205) then
        tmp = ((x / n) + 1.0d0) - t_0
    else
        tmp = (((((0.3333333333333333d0 / x) - 0.5d0) / x) + 1.0d0) / x) / n
    end if
    code = tmp
end function

Java

public static double code(double x, double n) {
	double t_0 = Math.pow(x, Math.pow(n, -1.0));
	double t_1 = (t_0 / x) / n;
	double tmp;
	if (Math.pow(n, -1.0) <= -1e-134) {
		tmp = t_1;
	} else if (Math.pow(n, -1.0) <= 5e-53) {
		tmp = Math.log((x / (1.0 + x))) / -n;
	} else if (Math.pow(n, -1.0) <= 5e-9) {
		tmp = t_1;
	} else if (Math.pow(n, -1.0) <= 5e+205) {
		tmp = ((x / n) + 1.0) - t_0;
	} else {
		tmp = (((((0.3333333333333333 / x) - 0.5) / x) + 1.0) / x) / n;
	}
	return tmp;
}

Python

def code(x, n):
	t_0 = math.pow(x, math.pow(n, -1.0))
	t_1 = (t_0 / x) / n
	tmp = 0
	if math.pow(n, -1.0) <= -1e-134:
		tmp = t_1
	elif math.pow(n, -1.0) <= 5e-53:
		tmp = math.log((x / (1.0 + x))) / -n
	elif math.pow(n, -1.0) <= 5e-9:
		tmp = t_1
	elif math.pow(n, -1.0) <= 5e+205:
		tmp = ((x / n) + 1.0) - t_0
	else:
		tmp = (((((0.3333333333333333 / x) - 0.5) / x) + 1.0) / x) / n
	return tmp

Julia

function code(x, n)
	t_0 = x ^ (n ^ -1.0)
	t_1 = Float64(Float64(t_0 / x) / n)
	tmp = 0.0
	if ((n ^ -1.0) <= -1e-134)
		tmp = t_1;
	elseif ((n ^ -1.0) <= 5e-53)
		tmp = Float64(log(Float64(x / Float64(1.0 + x))) / Float64(-n));
	elseif ((n ^ -1.0) <= 5e-9)
		tmp = t_1;
	elseif ((n ^ -1.0) <= 5e+205)
		tmp = Float64(Float64(Float64(x / n) + 1.0) - t_0);
	else
		tmp = Float64(Float64(Float64(Float64(Float64(Float64(0.3333333333333333 / x) - 0.5) / x) + 1.0) / x) / n);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, n)
	t_0 = x ^ (n ^ -1.0);
	t_1 = (t_0 / x) / n;
	tmp = 0.0;
	if ((n ^ -1.0) <= -1e-134)
		tmp = t_1;
	elseif ((n ^ -1.0) <= 5e-53)
		tmp = log((x / (1.0 + x))) / -n;
	elseif ((n ^ -1.0) <= 5e-9)
		tmp = t_1;
	elseif ((n ^ -1.0) <= 5e+205)
		tmp = ((x / n) + 1.0) - t_0;
	else
		tmp = (((((0.3333333333333333 / x) - 0.5) / x) + 1.0) / x) / n;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, n_] := Block[{t$95$0 = N[Power[x, N[Power[n, -1.0], $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(N[(t$95$0 / x), $MachinePrecision] / n), $MachinePrecision]}, If[LessEqual[N[Power[n, -1.0], $MachinePrecision], -1e-134], t$95$1, If[LessEqual[N[Power[n, -1.0], $MachinePrecision], 5e-53], N[(N[Log[N[(x / N[(1.0 + x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / (-n)), $MachinePrecision], If[LessEqual[N[Power[n, -1.0], $MachinePrecision], 5e-9], t$95$1, If[LessEqual[N[Power[n, -1.0], $MachinePrecision], 5e+205], N[(N[(N[(x / n), $MachinePrecision] + 1.0), $MachinePrecision] - t$95$0), $MachinePrecision], N[(N[(N[(N[(N[(N[(0.3333333333333333 / x), $MachinePrecision] - 0.5), $MachinePrecision] / x), $MachinePrecision] + 1.0), $MachinePrecision] / x), $MachinePrecision] / n), $MachinePrecision]]]]]]]

Derivation

1. Split input into 4 regimes

2. if (/.f64 #s(literal 1 binary64) n) < -1.00000000000000004e-134 or 5e-53 < (/.f64 #s(literal 1 binary64) n) < 5.0000000000000001e-9

   1. Initial program 69.2%

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

   3. Taylor expanded in x around inf

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

      1. associate-/l/ N/A

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

      2. lower-/.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. log-rec N/A

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

      5. mul-1-neg N/A

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

      6. associate-*r/ N/A

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

      7. associate-*r* N/A

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

      8. metadata-eval N/A

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

      9. *-commutative N/A

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

      10. associate-/l* N/A

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

      11. exp-to-pow N/A

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

      12. lower-pow.f64 N/A

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

      13. lower-/.f64 84.9

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

   5. Applied rewrites 84.9%

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

   if -1.00000000000000004e-134 < (/.f64 #s(literal 1 binary64) n) < 5e-53

   1. Initial program 40.8%

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

   3. Taylor expanded in n around inf

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

      1. lower-/.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lower-log1p.f64 N/A

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

      4. lower-log.f64 85.1

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

   5. Applied rewrites 85.1%

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

   7. Applied rewrites 85.5%

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

   if 5.0000000000000001e-9 < (/.f64 #s(literal 1 binary64) n) < 5.0000000000000002e205

   1. Initial program 89.9%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. *-rgt-identity N/A

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

      3. associate-*r/ N/A

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

      4. lower-+.f64 N/A

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

      5. associate-*r/ N/A

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

      6. *-rgt-identity N/A

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

      7. lower-/.f64 90.8

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

   5. Applied rewrites 90.8%

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

   if 5.0000000000000002e205 < (/.f64 #s(literal 1 binary64) n)

   1. Initial program 3.1%

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

   3. Taylor expanded in n around inf

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

      1. lower-/.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lower-log1p.f64 N/A

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

      4. lower-log.f64 7.0

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

   5. Applied rewrites 7.0%

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

   6. Taylor expanded in x around inf

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

   8. Applied rewrites 100.0%

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

4. Final simplification 86.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{n}^{-1} \leq -1 \cdot 10^{-134}:\\ \;\;\;\;\frac{\frac{{x}^{\left({n}^{-1}\right)}}{x}}{n}\\ \mathbf{elif}\;{n}^{-1} \leq 5 \cdot 10^{-53}:\\ \;\;\;\;\frac{\log \left(\frac{x}{1 + x}\right)}{-n}\\ \mathbf{elif}\;{n}^{-1} \leq 5 \cdot 10^{-9}:\\ \;\;\;\;\frac{\frac{{x}^{\left({n}^{-1}\right)}}{x}}{n}\\ \mathbf{elif}\;{n}^{-1} \leq 5 \cdot 10^{+205}:\\ \;\;\;\;\left(\frac{x}{n} + 1\right) - {x}^{\left({n}^{-1}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{\frac{0.3333333333333333}{x} - 0.5}{x} + 1}{x}}{n}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 80.1% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {\left(\left({x}^{\left(\frac{-1}{n}\right)} \cdot x\right) \cdot n\right)}^{-1}\\ \mathbf{if}\;{n}^{-1} \leq -1 \cdot 10^{-134}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;{n}^{-1} \leq 5 \cdot 10^{-53}:\\ \;\;\;\;\frac{\log \left(\frac{x}{1 + x}\right)}{-n}\\ \mathbf{elif}\;{n}^{-1} \leq 5 \cdot 10^{-9}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;{n}^{-1} \leq 5 \cdot 10^{+205}:\\ \;\;\;\;\left(\frac{x}{n} + 1\right) - {x}^{\left({n}^{-1}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{\frac{0.3333333333333333}{x} - 0.5}{x} + 1}{x}}{n}\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (let* ((t_0 (pow (* (* (pow x (/ -1.0 n)) x) n) -1.0)))
   (if (<= (pow n -1.0) -1e-134)
     t_0
     (if (<= (pow n -1.0) 5e-53)
       (/ (log (/ x (+ 1.0 x))) (- n))
       (if (<= (pow n -1.0) 5e-9)
         t_0
         (if (<= (pow n -1.0) 5e+205)
           (- (+ (/ x n) 1.0) (pow x (pow n -1.0)))
           (/ (/ (+ (/ (- (/ 0.3333333333333333 x) 0.5) x) 1.0) x) n)))))))

C

double code(double x, double n) {
	double t_0 = pow(((pow(x, (-1.0 / n)) * x) * n), -1.0);
	double tmp;
	if (pow(n, -1.0) <= -1e-134) {
		tmp = t_0;
	} else if (pow(n, -1.0) <= 5e-53) {
		tmp = log((x / (1.0 + x))) / -n;
	} else if (pow(n, -1.0) <= 5e-9) {
		tmp = t_0;
	} else if (pow(n, -1.0) <= 5e+205) {
		tmp = ((x / n) + 1.0) - pow(x, pow(n, -1.0));
	} else {
		tmp = (((((0.3333333333333333 / x) - 0.5) / x) + 1.0) / x) / n;
	}
	return tmp;
}

Fortran

real(8) function code(x, n)
    real(8), intent (in) :: x
    real(8), intent (in) :: n
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (((x ** ((-1.0d0) / n)) * x) * n) ** (-1.0d0)
    if ((n ** (-1.0d0)) <= (-1d-134)) then
        tmp = t_0
    else if ((n ** (-1.0d0)) <= 5d-53) then
        tmp = log((x / (1.0d0 + x))) / -n
    else if ((n ** (-1.0d0)) <= 5d-9) then
        tmp = t_0
    else if ((n ** (-1.0d0)) <= 5d+205) then
        tmp = ((x / n) + 1.0d0) - (x ** (n ** (-1.0d0)))
    else
        tmp = (((((0.3333333333333333d0 / x) - 0.5d0) / x) + 1.0d0) / x) / n
    end if
    code = tmp
end function

Java

public static double code(double x, double n) {
	double t_0 = Math.pow(((Math.pow(x, (-1.0 / n)) * x) * n), -1.0);
	double tmp;
	if (Math.pow(n, -1.0) <= -1e-134) {
		tmp = t_0;
	} else if (Math.pow(n, -1.0) <= 5e-53) {
		tmp = Math.log((x / (1.0 + x))) / -n;
	} else if (Math.pow(n, -1.0) <= 5e-9) {
		tmp = t_0;
	} else if (Math.pow(n, -1.0) <= 5e+205) {
		tmp = ((x / n) + 1.0) - Math.pow(x, Math.pow(n, -1.0));
	} else {
		tmp = (((((0.3333333333333333 / x) - 0.5) / x) + 1.0) / x) / n;
	}
	return tmp;
}

Python

def code(x, n):
	t_0 = math.pow(((math.pow(x, (-1.0 / n)) * x) * n), -1.0)
	tmp = 0
	if math.pow(n, -1.0) <= -1e-134:
		tmp = t_0
	elif math.pow(n, -1.0) <= 5e-53:
		tmp = math.log((x / (1.0 + x))) / -n
	elif math.pow(n, -1.0) <= 5e-9:
		tmp = t_0
	elif math.pow(n, -1.0) <= 5e+205:
		tmp = ((x / n) + 1.0) - math.pow(x, math.pow(n, -1.0))
	else:
		tmp = (((((0.3333333333333333 / x) - 0.5) / x) + 1.0) / x) / n
	return tmp

Julia

function code(x, n)
	t_0 = Float64(Float64((x ^ Float64(-1.0 / n)) * x) * n) ^ -1.0
	tmp = 0.0
	if ((n ^ -1.0) <= -1e-134)
		tmp = t_0;
	elseif ((n ^ -1.0) <= 5e-53)
		tmp = Float64(log(Float64(x / Float64(1.0 + x))) / Float64(-n));
	elseif ((n ^ -1.0) <= 5e-9)
		tmp = t_0;
	elseif ((n ^ -1.0) <= 5e+205)
		tmp = Float64(Float64(Float64(x / n) + 1.0) - (x ^ (n ^ -1.0)));
	else
		tmp = Float64(Float64(Float64(Float64(Float64(Float64(0.3333333333333333 / x) - 0.5) / x) + 1.0) / x) / n);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, n)
	t_0 = (((x ^ (-1.0 / n)) * x) * n) ^ -1.0;
	tmp = 0.0;
	if ((n ^ -1.0) <= -1e-134)
		tmp = t_0;
	elseif ((n ^ -1.0) <= 5e-53)
		tmp = log((x / (1.0 + x))) / -n;
	elseif ((n ^ -1.0) <= 5e-9)
		tmp = t_0;
	elseif ((n ^ -1.0) <= 5e+205)
		tmp = ((x / n) + 1.0) - (x ^ (n ^ -1.0));
	else
		tmp = (((((0.3333333333333333 / x) - 0.5) / x) + 1.0) / x) / n;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, n_] := Block[{t$95$0 = N[Power[N[(N[(N[Power[x, N[(-1.0 / n), $MachinePrecision]], $MachinePrecision] * x), $MachinePrecision] * n), $MachinePrecision], -1.0], $MachinePrecision]}, If[LessEqual[N[Power[n, -1.0], $MachinePrecision], -1e-134], t$95$0, If[LessEqual[N[Power[n, -1.0], $MachinePrecision], 5e-53], N[(N[Log[N[(x / N[(1.0 + x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / (-n)), $MachinePrecision], If[LessEqual[N[Power[n, -1.0], $MachinePrecision], 5e-9], t$95$0, If[LessEqual[N[Power[n, -1.0], $MachinePrecision], 5e+205], N[(N[(N[(x / n), $MachinePrecision] + 1.0), $MachinePrecision] - N[Power[x, N[Power[n, -1.0], $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(N[(N[(0.3333333333333333 / x), $MachinePrecision] - 0.5), $MachinePrecision] / x), $MachinePrecision] + 1.0), $MachinePrecision] / x), $MachinePrecision] / n), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if (/.f64 #s(literal 1 binary64) n) < -1.00000000000000004e-134 or 5e-53 < (/.f64 #s(literal 1 binary64) n) < 5.0000000000000001e-9

   1. Initial program 69.2%

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

   3. Taylor expanded in x around inf

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

      1. associate-/l/ N/A

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

      2. lower-/.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. log-rec N/A

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

      5. mul-1-neg N/A

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

      6. associate-*r/ N/A

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

      7. associate-*r* N/A

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

      8. metadata-eval N/A

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

      9. *-commutative N/A

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

      10. associate-/l* N/A

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

      11. exp-to-pow N/A

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

      12. lower-pow.f64 N/A

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

      13. lower-/.f64 84.9

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

   5. Applied rewrites 84.9%

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

   7. Applied rewrites 84.9%

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

   9. Applied rewrites 84.9%

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

   if -1.00000000000000004e-134 < (/.f64 #s(literal 1 binary64) n) < 5e-53

   1. Initial program 40.8%

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

   3. Taylor expanded in n around inf

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

      1. lower-/.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lower-log1p.f64 N/A

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

      4. lower-log.f64 85.1

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

   5. Applied rewrites 85.1%

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

   7. Applied rewrites 85.5%

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

   if 5.0000000000000001e-9 < (/.f64 #s(literal 1 binary64) n) < 5.0000000000000002e205

   1. Initial program 89.9%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. *-rgt-identity N/A

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

      3. associate-*r/ N/A

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

      4. lower-+.f64 N/A

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

      5. associate-*r/ N/A

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

      6. *-rgt-identity N/A

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

      7. lower-/.f64 90.8

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

   5. Applied rewrites 90.8%

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

   if 5.0000000000000002e205 < (/.f64 #s(literal 1 binary64) n)

   1. Initial program 3.1%

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

   3. Taylor expanded in n around inf

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

      1. lower-/.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lower-log1p.f64 N/A

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

      4. lower-log.f64 7.0

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

   5. Applied rewrites 7.0%

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

   6. Taylor expanded in x around inf

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

   8. Applied rewrites 100.0%

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

4. Final simplification 86.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{n}^{-1} \leq -1 \cdot 10^{-134}:\\ \;\;\;\;{\left(\left({x}^{\left(\frac{-1}{n}\right)} \cdot x\right) \cdot n\right)}^{-1}\\ \mathbf{elif}\;{n}^{-1} \leq 5 \cdot 10^{-53}:\\ \;\;\;\;\frac{\log \left(\frac{x}{1 + x}\right)}{-n}\\ \mathbf{elif}\;{n}^{-1} \leq 5 \cdot 10^{-9}:\\ \;\;\;\;{\left(\left({x}^{\left(\frac{-1}{n}\right)} \cdot x\right) \cdot n\right)}^{-1}\\ \mathbf{elif}\;{n}^{-1} \leq 5 \cdot 10^{+205}:\\ \;\;\;\;\left(\frac{x}{n} + 1\right) - {x}^{\left({n}^{-1}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{\frac{0.3333333333333333}{x} - 0.5}{x} + 1}{x}}{n}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 80.2% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{{x}^{\left(-1 - \frac{-1}{n}\right)}}{n}\\ \mathbf{if}\;{n}^{-1} \leq -1 \cdot 10^{-134}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;{n}^{-1} \leq 5 \cdot 10^{-53}:\\ \;\;\;\;\frac{\log \left(\frac{x}{1 + x}\right)}{-n}\\ \mathbf{elif}\;{n}^{-1} \leq 5 \cdot 10^{-9}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;{n}^{-1} \leq 5 \cdot 10^{+205}:\\ \;\;\;\;\left(\frac{x}{n} + 1\right) - {x}^{\left({n}^{-1}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{\frac{0.3333333333333333}{x} - 0.5}{x} + 1}{x}}{n}\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (let* ((t_0 (/ (pow x (- -1.0 (/ -1.0 n))) n)))
   (if (<= (pow n -1.0) -1e-134)
     t_0
     (if (<= (pow n -1.0) 5e-53)
       (/ (log (/ x (+ 1.0 x))) (- n))
       (if (<= (pow n -1.0) 5e-9)
         t_0
         (if (<= (pow n -1.0) 5e+205)
           (- (+ (/ x n) 1.0) (pow x (pow n -1.0)))
           (/ (/ (+ (/ (- (/ 0.3333333333333333 x) 0.5) x) 1.0) x) n)))))))

C

double code(double x, double n) {
	double t_0 = pow(x, (-1.0 - (-1.0 / n))) / n;
	double tmp;
	if (pow(n, -1.0) <= -1e-134) {
		tmp = t_0;
	} else if (pow(n, -1.0) <= 5e-53) {
		tmp = log((x / (1.0 + x))) / -n;
	} else if (pow(n, -1.0) <= 5e-9) {
		tmp = t_0;
	} else if (pow(n, -1.0) <= 5e+205) {
		tmp = ((x / n) + 1.0) - pow(x, pow(n, -1.0));
	} else {
		tmp = (((((0.3333333333333333 / x) - 0.5) / x) + 1.0) / x) / n;
	}
	return tmp;
}

Fortran

real(8) function code(x, n)
    real(8), intent (in) :: x
    real(8), intent (in) :: n
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (x ** ((-1.0d0) - ((-1.0d0) / n))) / n
    if ((n ** (-1.0d0)) <= (-1d-134)) then
        tmp = t_0
    else if ((n ** (-1.0d0)) <= 5d-53) then
        tmp = log((x / (1.0d0 + x))) / -n
    else if ((n ** (-1.0d0)) <= 5d-9) then
        tmp = t_0
    else if ((n ** (-1.0d0)) <= 5d+205) then
        tmp = ((x / n) + 1.0d0) - (x ** (n ** (-1.0d0)))
    else
        tmp = (((((0.3333333333333333d0 / x) - 0.5d0) / x) + 1.0d0) / x) / n
    end if
    code = tmp
end function

Java

public static double code(double x, double n) {
	double t_0 = Math.pow(x, (-1.0 - (-1.0 / n))) / n;
	double tmp;
	if (Math.pow(n, -1.0) <= -1e-134) {
		tmp = t_0;
	} else if (Math.pow(n, -1.0) <= 5e-53) {
		tmp = Math.log((x / (1.0 + x))) / -n;
	} else if (Math.pow(n, -1.0) <= 5e-9) {
		tmp = t_0;
	} else if (Math.pow(n, -1.0) <= 5e+205) {
		tmp = ((x / n) + 1.0) - Math.pow(x, Math.pow(n, -1.0));
	} else {
		tmp = (((((0.3333333333333333 / x) - 0.5) / x) + 1.0) / x) / n;
	}
	return tmp;
}

Python

def code(x, n):
	t_0 = math.pow(x, (-1.0 - (-1.0 / n))) / n
	tmp = 0
	if math.pow(n, -1.0) <= -1e-134:
		tmp = t_0
	elif math.pow(n, -1.0) <= 5e-53:
		tmp = math.log((x / (1.0 + x))) / -n
	elif math.pow(n, -1.0) <= 5e-9:
		tmp = t_0
	elif math.pow(n, -1.0) <= 5e+205:
		tmp = ((x / n) + 1.0) - math.pow(x, math.pow(n, -1.0))
	else:
		tmp = (((((0.3333333333333333 / x) - 0.5) / x) + 1.0) / x) / n
	return tmp

Julia

function code(x, n)
	t_0 = Float64((x ^ Float64(-1.0 - Float64(-1.0 / n))) / n)
	tmp = 0.0
	if ((n ^ -1.0) <= -1e-134)
		tmp = t_0;
	elseif ((n ^ -1.0) <= 5e-53)
		tmp = Float64(log(Float64(x / Float64(1.0 + x))) / Float64(-n));
	elseif ((n ^ -1.0) <= 5e-9)
		tmp = t_0;
	elseif ((n ^ -1.0) <= 5e+205)
		tmp = Float64(Float64(Float64(x / n) + 1.0) - (x ^ (n ^ -1.0)));
	else
		tmp = Float64(Float64(Float64(Float64(Float64(Float64(0.3333333333333333 / x) - 0.5) / x) + 1.0) / x) / n);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, n)
	t_0 = (x ^ (-1.0 - (-1.0 / n))) / n;
	tmp = 0.0;
	if ((n ^ -1.0) <= -1e-134)
		tmp = t_0;
	elseif ((n ^ -1.0) <= 5e-53)
		tmp = log((x / (1.0 + x))) / -n;
	elseif ((n ^ -1.0) <= 5e-9)
		tmp = t_0;
	elseif ((n ^ -1.0) <= 5e+205)
		tmp = ((x / n) + 1.0) - (x ^ (n ^ -1.0));
	else
		tmp = (((((0.3333333333333333 / x) - 0.5) / x) + 1.0) / x) / n;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, n_] := Block[{t$95$0 = N[(N[Power[x, N[(-1.0 - N[(-1.0 / n), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / n), $MachinePrecision]}, If[LessEqual[N[Power[n, -1.0], $MachinePrecision], -1e-134], t$95$0, If[LessEqual[N[Power[n, -1.0], $MachinePrecision], 5e-53], N[(N[Log[N[(x / N[(1.0 + x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / (-n)), $MachinePrecision], If[LessEqual[N[Power[n, -1.0], $MachinePrecision], 5e-9], t$95$0, If[LessEqual[N[Power[n, -1.0], $MachinePrecision], 5e+205], N[(N[(N[(x / n), $MachinePrecision] + 1.0), $MachinePrecision] - N[Power[x, N[Power[n, -1.0], $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(N[(N[(0.3333333333333333 / x), $MachinePrecision] - 0.5), $MachinePrecision] / x), $MachinePrecision] + 1.0), $MachinePrecision] / x), $MachinePrecision] / n), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if (/.f64 #s(literal 1 binary64) n) < -1.00000000000000004e-134 or 5e-53 < (/.f64 #s(literal 1 binary64) n) < 5.0000000000000001e-9

   1. Initial program 69.2%

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

   3. Taylor expanded in x around inf

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

      1. associate-/l/ N/A

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

      2. lower-/.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. log-rec N/A

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

      5. mul-1-neg N/A

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

      6. associate-*r/ N/A

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

      7. associate-*r* N/A

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

      8. metadata-eval N/A

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

      9. *-commutative N/A

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

      10. associate-/l* N/A

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

      11. exp-to-pow N/A

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

      12. lower-pow.f64 N/A

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

      13. lower-/.f64 84.9

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

   5. Applied rewrites 84.9%

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

   7. Applied rewrites 84.9%

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

   9. Applied rewrites 84.7%

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

   if -1.00000000000000004e-134 < (/.f64 #s(literal 1 binary64) n) < 5e-53

   1. Initial program 40.8%

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

   3. Taylor expanded in n around inf

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

      1. lower-/.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lower-log1p.f64 N/A

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

      4. lower-log.f64 85.1

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

   5. Applied rewrites 85.1%

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

   7. Applied rewrites 85.5%

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

   if 5.0000000000000001e-9 < (/.f64 #s(literal 1 binary64) n) < 5.0000000000000002e205

   1. Initial program 89.9%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. *-rgt-identity N/A

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

      3. associate-*r/ N/A

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

      4. lower-+.f64 N/A

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

      5. associate-*r/ N/A

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

      6. *-rgt-identity N/A

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

      7. lower-/.f64 90.8

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

   5. Applied rewrites 90.8%

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

   if 5.0000000000000002e205 < (/.f64 #s(literal 1 binary64) n)

   1. Initial program 3.1%

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

   3. Taylor expanded in n around inf

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

      1. lower-/.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lower-log1p.f64 N/A

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

      4. lower-log.f64 7.0

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

   5. Applied rewrites 7.0%

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

   6. Taylor expanded in x around inf

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

   8. Applied rewrites 100.0%

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

4. Final simplification 86.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{n}^{-1} \leq -1 \cdot 10^{-134}:\\ \;\;\;\;\frac{{x}^{\left(-1 - \frac{-1}{n}\right)}}{n}\\ \mathbf{elif}\;{n}^{-1} \leq 5 \cdot 10^{-53}:\\ \;\;\;\;\frac{\log \left(\frac{x}{1 + x}\right)}{-n}\\ \mathbf{elif}\;{n}^{-1} \leq 5 \cdot 10^{-9}:\\ \;\;\;\;\frac{{x}^{\left(-1 - \frac{-1}{n}\right)}}{n}\\ \mathbf{elif}\;{n}^{-1} \leq 5 \cdot 10^{+205}:\\ \;\;\;\;\left(\frac{x}{n} + 1\right) - {x}^{\left({n}^{-1}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{\frac{0.3333333333333333}{x} - 0.5}{x} + 1}{x}}{n}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 80.0% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{{x}^{\left(-1 - \frac{-1}{n}\right)}}{n}\\ \mathbf{if}\;{n}^{-1} \leq -1 \cdot 10^{-134}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;{n}^{-1} \leq 5 \cdot 10^{-53}:\\ \;\;\;\;\frac{\log \left(\frac{x}{1 + x}\right)}{-n}\\ \mathbf{elif}\;{n}^{-1} \leq 5 \cdot 10^{-9}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;{n}^{-1} \leq 2 \cdot 10^{+201}:\\ \;\;\;\;1 - {x}^{\left({n}^{-1}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{\frac{0.3333333333333333}{x} - 0.5}{x} + 1}{x}}{n}\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (let* ((t_0 (/ (pow x (- -1.0 (/ -1.0 n))) n)))
   (if (<= (pow n -1.0) -1e-134)
     t_0
     (if (<= (pow n -1.0) 5e-53)
       (/ (log (/ x (+ 1.0 x))) (- n))
       (if (<= (pow n -1.0) 5e-9)
         t_0
         (if (<= (pow n -1.0) 2e+201)
           (- 1.0 (pow x (pow n -1.0)))
           (/ (/ (+ (/ (- (/ 0.3333333333333333 x) 0.5) x) 1.0) x) n)))))))

C

double code(double x, double n) {
	double t_0 = pow(x, (-1.0 - (-1.0 / n))) / n;
	double tmp;
	if (pow(n, -1.0) <= -1e-134) {
		tmp = t_0;
	} else if (pow(n, -1.0) <= 5e-53) {
		tmp = log((x / (1.0 + x))) / -n;
	} else if (pow(n, -1.0) <= 5e-9) {
		tmp = t_0;
	} else if (pow(n, -1.0) <= 2e+201) {
		tmp = 1.0 - pow(x, pow(n, -1.0));
	} else {
		tmp = (((((0.3333333333333333 / x) - 0.5) / x) + 1.0) / x) / n;
	}
	return tmp;
}

Fortran

real(8) function code(x, n)
    real(8), intent (in) :: x
    real(8), intent (in) :: n
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (x ** ((-1.0d0) - ((-1.0d0) / n))) / n
    if ((n ** (-1.0d0)) <= (-1d-134)) then
        tmp = t_0
    else if ((n ** (-1.0d0)) <= 5d-53) then
        tmp = log((x / (1.0d0 + x))) / -n
    else if ((n ** (-1.0d0)) <= 5d-9) then
        tmp = t_0
    else if ((n ** (-1.0d0)) <= 2d+201) then
        tmp = 1.0d0 - (x ** (n ** (-1.0d0)))
    else
        tmp = (((((0.3333333333333333d0 / x) - 0.5d0) / x) + 1.0d0) / x) / n
    end if
    code = tmp
end function

Java

public static double code(double x, double n) {
	double t_0 = Math.pow(x, (-1.0 - (-1.0 / n))) / n;
	double tmp;
	if (Math.pow(n, -1.0) <= -1e-134) {
		tmp = t_0;
	} else if (Math.pow(n, -1.0) <= 5e-53) {
		tmp = Math.log((x / (1.0 + x))) / -n;
	} else if (Math.pow(n, -1.0) <= 5e-9) {
		tmp = t_0;
	} else if (Math.pow(n, -1.0) <= 2e+201) {
		tmp = 1.0 - Math.pow(x, Math.pow(n, -1.0));
	} else {
		tmp = (((((0.3333333333333333 / x) - 0.5) / x) + 1.0) / x) / n;
	}
	return tmp;
}

Python

def code(x, n):
	t_0 = math.pow(x, (-1.0 - (-1.0 / n))) / n
	tmp = 0
	if math.pow(n, -1.0) <= -1e-134:
		tmp = t_0
	elif math.pow(n, -1.0) <= 5e-53:
		tmp = math.log((x / (1.0 + x))) / -n
	elif math.pow(n, -1.0) <= 5e-9:
		tmp = t_0
	elif math.pow(n, -1.0) <= 2e+201:
		tmp = 1.0 - math.pow(x, math.pow(n, -1.0))
	else:
		tmp = (((((0.3333333333333333 / x) - 0.5) / x) + 1.0) / x) / n
	return tmp

Julia

function code(x, n)
	t_0 = Float64((x ^ Float64(-1.0 - Float64(-1.0 / n))) / n)
	tmp = 0.0
	if ((n ^ -1.0) <= -1e-134)
		tmp = t_0;
	elseif ((n ^ -1.0) <= 5e-53)
		tmp = Float64(log(Float64(x / Float64(1.0 + x))) / Float64(-n));
	elseif ((n ^ -1.0) <= 5e-9)
		tmp = t_0;
	elseif ((n ^ -1.0) <= 2e+201)
		tmp = Float64(1.0 - (x ^ (n ^ -1.0)));
	else
		tmp = Float64(Float64(Float64(Float64(Float64(Float64(0.3333333333333333 / x) - 0.5) / x) + 1.0) / x) / n);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, n)
	t_0 = (x ^ (-1.0 - (-1.0 / n))) / n;
	tmp = 0.0;
	if ((n ^ -1.0) <= -1e-134)
		tmp = t_0;
	elseif ((n ^ -1.0) <= 5e-53)
		tmp = log((x / (1.0 + x))) / -n;
	elseif ((n ^ -1.0) <= 5e-9)
		tmp = t_0;
	elseif ((n ^ -1.0) <= 2e+201)
		tmp = 1.0 - (x ^ (n ^ -1.0));
	else
		tmp = (((((0.3333333333333333 / x) - 0.5) / x) + 1.0) / x) / n;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, n_] := Block[{t$95$0 = N[(N[Power[x, N[(-1.0 - N[(-1.0 / n), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / n), $MachinePrecision]}, If[LessEqual[N[Power[n, -1.0], $MachinePrecision], -1e-134], t$95$0, If[LessEqual[N[Power[n, -1.0], $MachinePrecision], 5e-53], N[(N[Log[N[(x / N[(1.0 + x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / (-n)), $MachinePrecision], If[LessEqual[N[Power[n, -1.0], $MachinePrecision], 5e-9], t$95$0, If[LessEqual[N[Power[n, -1.0], $MachinePrecision], 2e+201], N[(1.0 - N[Power[x, N[Power[n, -1.0], $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(N[(N[(0.3333333333333333 / x), $MachinePrecision] - 0.5), $MachinePrecision] / x), $MachinePrecision] + 1.0), $MachinePrecision] / x), $MachinePrecision] / n), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if (/.f64 #s(literal 1 binary64) n) < -1.00000000000000004e-134 or 5e-53 < (/.f64 #s(literal 1 binary64) n) < 5.0000000000000001e-9

   1. Initial program 69.2%

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

   3. Taylor expanded in x around inf

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

      1. associate-/l/ N/A

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

      2. lower-/.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. log-rec N/A

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

      5. mul-1-neg N/A

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

      6. associate-*r/ N/A

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

      7. associate-*r* N/A

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

      8. metadata-eval N/A

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

      9. *-commutative N/A

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

      10. associate-/l* N/A

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

      11. exp-to-pow N/A

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

      12. lower-pow.f64 N/A

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

      13. lower-/.f64 84.9

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

   5. Applied rewrites 84.9%

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

   7. Applied rewrites 84.9%

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

   9. Applied rewrites 84.7%

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

   if -1.00000000000000004e-134 < (/.f64 #s(literal 1 binary64) n) < 5e-53

   1. Initial program 40.8%

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

   3. Taylor expanded in n around inf

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

      1. lower-/.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lower-log1p.f64 N/A

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

      4. lower-log.f64 85.1

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

   5. Applied rewrites 85.1%

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

   7. Applied rewrites 85.5%

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

   if 5.0000000000000001e-9 < (/.f64 #s(literal 1 binary64) n) < 2.00000000000000008e201

   1. Initial program 93.2%

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

   3. Taylor expanded in x around 0

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

   5. Applied rewrites 93.2%

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

   if 2.00000000000000008e201 < (/.f64 #s(literal 1 binary64) n)

   1. Initial program 14.9%

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

   3. Taylor expanded in n around inf

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

      1. lower-/.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lower-log1p.f64 N/A

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

      4. lower-log.f64 6.4

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

   5. Applied rewrites 6.4%

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

   6. Taylor expanded in x around inf

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

   8. Applied rewrites 87.9%

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

4. Final simplification 86.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{n}^{-1} \leq -1 \cdot 10^{-134}:\\ \;\;\;\;\frac{{x}^{\left(-1 - \frac{-1}{n}\right)}}{n}\\ \mathbf{elif}\;{n}^{-1} \leq 5 \cdot 10^{-53}:\\ \;\;\;\;\frac{\log \left(\frac{x}{1 + x}\right)}{-n}\\ \mathbf{elif}\;{n}^{-1} \leq 5 \cdot 10^{-9}:\\ \;\;\;\;\frac{{x}^{\left(-1 - \frac{-1}{n}\right)}}{n}\\ \mathbf{elif}\;{n}^{-1} \leq 2 \cdot 10^{+201}:\\ \;\;\;\;1 - {x}^{\left({n}^{-1}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{\frac{0.3333333333333333}{x} - 0.5}{x} + 1}{x}}{n}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 80.0% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{{x}^{\left(-1 - \frac{-1}{n}\right)}}{n}\\ \mathbf{if}\;{n}^{-1} \leq -1 \cdot 10^{-134}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;{n}^{-1} \leq 5 \cdot 10^{-53}:\\ \;\;\;\;\frac{\log \left(\frac{1 + x}{x}\right)}{n}\\ \mathbf{elif}\;{n}^{-1} \leq 5 \cdot 10^{-9}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;{n}^{-1} \leq 2 \cdot 10^{+201}:\\ \;\;\;\;1 - {x}^{\left({n}^{-1}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{\frac{0.3333333333333333}{x} - 0.5}{x} + 1}{x}}{n}\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (let* ((t_0 (/ (pow x (- -1.0 (/ -1.0 n))) n)))
   (if (<= (pow n -1.0) -1e-134)
     t_0
     (if (<= (pow n -1.0) 5e-53)
       (/ (log (/ (+ 1.0 x) x)) n)
       (if (<= (pow n -1.0) 5e-9)
         t_0
         (if (<= (pow n -1.0) 2e+201)
           (- 1.0 (pow x (pow n -1.0)))
           (/ (/ (+ (/ (- (/ 0.3333333333333333 x) 0.5) x) 1.0) x) n)))))))

C

double code(double x, double n) {
	double t_0 = pow(x, (-1.0 - (-1.0 / n))) / n;
	double tmp;
	if (pow(n, -1.0) <= -1e-134) {
		tmp = t_0;
	} else if (pow(n, -1.0) <= 5e-53) {
		tmp = log(((1.0 + x) / x)) / n;
	} else if (pow(n, -1.0) <= 5e-9) {
		tmp = t_0;
	} else if (pow(n, -1.0) <= 2e+201) {
		tmp = 1.0 - pow(x, pow(n, -1.0));
	} else {
		tmp = (((((0.3333333333333333 / x) - 0.5) / x) + 1.0) / x) / n;
	}
	return tmp;
}

Fortran

real(8) function code(x, n)
    real(8), intent (in) :: x
    real(8), intent (in) :: n
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (x ** ((-1.0d0) - ((-1.0d0) / n))) / n
    if ((n ** (-1.0d0)) <= (-1d-134)) then
        tmp = t_0
    else if ((n ** (-1.0d0)) <= 5d-53) then
        tmp = log(((1.0d0 + x) / x)) / n
    else if ((n ** (-1.0d0)) <= 5d-9) then
        tmp = t_0
    else if ((n ** (-1.0d0)) <= 2d+201) then
        tmp = 1.0d0 - (x ** (n ** (-1.0d0)))
    else
        tmp = (((((0.3333333333333333d0 / x) - 0.5d0) / x) + 1.0d0) / x) / n
    end if
    code = tmp
end function

Java

public static double code(double x, double n) {
	double t_0 = Math.pow(x, (-1.0 - (-1.0 / n))) / n;
	double tmp;
	if (Math.pow(n, -1.0) <= -1e-134) {
		tmp = t_0;
	} else if (Math.pow(n, -1.0) <= 5e-53) {
		tmp = Math.log(((1.0 + x) / x)) / n;
	} else if (Math.pow(n, -1.0) <= 5e-9) {
		tmp = t_0;
	} else if (Math.pow(n, -1.0) <= 2e+201) {
		tmp = 1.0 - Math.pow(x, Math.pow(n, -1.0));
	} else {
		tmp = (((((0.3333333333333333 / x) - 0.5) / x) + 1.0) / x) / n;
	}
	return tmp;
}

Python

def code(x, n):
	t_0 = math.pow(x, (-1.0 - (-1.0 / n))) / n
	tmp = 0
	if math.pow(n, -1.0) <= -1e-134:
		tmp = t_0
	elif math.pow(n, -1.0) <= 5e-53:
		tmp = math.log(((1.0 + x) / x)) / n
	elif math.pow(n, -1.0) <= 5e-9:
		tmp = t_0
	elif math.pow(n, -1.0) <= 2e+201:
		tmp = 1.0 - math.pow(x, math.pow(n, -1.0))
	else:
		tmp = (((((0.3333333333333333 / x) - 0.5) / x) + 1.0) / x) / n
	return tmp

Julia

function code(x, n)
	t_0 = Float64((x ^ Float64(-1.0 - Float64(-1.0 / n))) / n)
	tmp = 0.0
	if ((n ^ -1.0) <= -1e-134)
		tmp = t_0;
	elseif ((n ^ -1.0) <= 5e-53)
		tmp = Float64(log(Float64(Float64(1.0 + x) / x)) / n);
	elseif ((n ^ -1.0) <= 5e-9)
		tmp = t_0;
	elseif ((n ^ -1.0) <= 2e+201)
		tmp = Float64(1.0 - (x ^ (n ^ -1.0)));
	else
		tmp = Float64(Float64(Float64(Float64(Float64(Float64(0.3333333333333333 / x) - 0.5) / x) + 1.0) / x) / n);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, n)
	t_0 = (x ^ (-1.0 - (-1.0 / n))) / n;
	tmp = 0.0;
	if ((n ^ -1.0) <= -1e-134)
		tmp = t_0;
	elseif ((n ^ -1.0) <= 5e-53)
		tmp = log(((1.0 + x) / x)) / n;
	elseif ((n ^ -1.0) <= 5e-9)
		tmp = t_0;
	elseif ((n ^ -1.0) <= 2e+201)
		tmp = 1.0 - (x ^ (n ^ -1.0));
	else
		tmp = (((((0.3333333333333333 / x) - 0.5) / x) + 1.0) / x) / n;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, n_] := Block[{t$95$0 = N[(N[Power[x, N[(-1.0 - N[(-1.0 / n), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / n), $MachinePrecision]}, If[LessEqual[N[Power[n, -1.0], $MachinePrecision], -1e-134], t$95$0, If[LessEqual[N[Power[n, -1.0], $MachinePrecision], 5e-53], N[(N[Log[N[(N[(1.0 + x), $MachinePrecision] / x), $MachinePrecision]], $MachinePrecision] / n), $MachinePrecision], If[LessEqual[N[Power[n, -1.0], $MachinePrecision], 5e-9], t$95$0, If[LessEqual[N[Power[n, -1.0], $MachinePrecision], 2e+201], N[(1.0 - N[Power[x, N[Power[n, -1.0], $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(N[(N[(0.3333333333333333 / x), $MachinePrecision] - 0.5), $MachinePrecision] / x), $MachinePrecision] + 1.0), $MachinePrecision] / x), $MachinePrecision] / n), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if (/.f64 #s(literal 1 binary64) n) < -1.00000000000000004e-134 or 5e-53 < (/.f64 #s(literal 1 binary64) n) < 5.0000000000000001e-9

   1. Initial program 69.2%

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

   3. Taylor expanded in x around inf

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

      1. associate-/l/ N/A

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

      2. lower-/.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. log-rec N/A

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

      5. mul-1-neg N/A

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

      6. associate-*r/ N/A

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

      7. associate-*r* N/A

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

      8. metadata-eval N/A

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

      9. *-commutative N/A

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

      10. associate-/l* N/A

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

      11. exp-to-pow N/A

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

      12. lower-pow.f64 N/A

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

      13. lower-/.f64 84.9

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

   5. Applied rewrites 84.9%

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

   7. Applied rewrites 84.9%

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

   9. Applied rewrites 84.7%

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

   if -1.00000000000000004e-134 < (/.f64 #s(literal 1 binary64) n) < 5e-53

   1. Initial program 40.8%

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

   3. Taylor expanded in n around inf

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

      1. lower-/.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lower-log1p.f64 N/A

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

      4. lower-log.f64 85.1

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

   5. Applied rewrites 85.1%

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

   7. Applied rewrites 85.3%

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

   if 5.0000000000000001e-9 < (/.f64 #s(literal 1 binary64) n) < 2.00000000000000008e201

   1. Initial program 93.2%

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

   3. Taylor expanded in x around 0

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

   5. Applied rewrites 93.2%

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

   if 2.00000000000000008e201 < (/.f64 #s(literal 1 binary64) n)

   1. Initial program 14.9%

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

   3. Taylor expanded in n around inf

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

      1. lower-/.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lower-log1p.f64 N/A

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

      4. lower-log.f64 6.4

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

   5. Applied rewrites 6.4%

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

   6. Taylor expanded in x around inf

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

   8. Applied rewrites 87.9%

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

4. Final simplification 85.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{n}^{-1} \leq -1 \cdot 10^{-134}:\\ \;\;\;\;\frac{{x}^{\left(-1 - \frac{-1}{n}\right)}}{n}\\ \mathbf{elif}\;{n}^{-1} \leq 5 \cdot 10^{-53}:\\ \;\;\;\;\frac{\log \left(\frac{1 + x}{x}\right)}{n}\\ \mathbf{elif}\;{n}^{-1} \leq 5 \cdot 10^{-9}:\\ \;\;\;\;\frac{{x}^{\left(-1 - \frac{-1}{n}\right)}}{n}\\ \mathbf{elif}\;{n}^{-1} \leq 2 \cdot 10^{+201}:\\ \;\;\;\;1 - {x}^{\left({n}^{-1}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{\frac{0.3333333333333333}{x} - 0.5}{x} + 1}{x}}{n}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 81.3% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {x}^{\left({n}^{-1}\right)}\\ t_1 := \frac{0.3333333333333333}{x} - 0.5\\ \mathbf{if}\;{n}^{-1} \leq -1:\\ \;\;\;\;\frac{t\_0}{n}\\ \mathbf{elif}\;{n}^{-1} \leq 5 \cdot 10^{-53}:\\ \;\;\;\;\frac{\log \left(\frac{1 + x}{x}\right)}{n}\\ \mathbf{elif}\;{n}^{-1} \leq 5 \cdot 10^{-9}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{{x}^{-1}}{n}, t\_1, {n}^{-1}\right)}{x}\\ \mathbf{elif}\;{n}^{-1} \leq 2 \cdot 10^{+201}:\\ \;\;\;\;1 - t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{t\_1}{x} + 1}{x}}{n}\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (let* ((t_0 (pow x (pow n -1.0))) (t_1 (- (/ 0.3333333333333333 x) 0.5)))
   (if (<= (pow n -1.0) -1.0)
     (/ t_0 n)
     (if (<= (pow n -1.0) 5e-53)
       (/ (log (/ (+ 1.0 x) x)) n)
       (if (<= (pow n -1.0) 5e-9)
         (/ (fma (/ (pow x -1.0) n) t_1 (pow n -1.0)) x)
         (if (<= (pow n -1.0) 2e+201)
           (- 1.0 t_0)
           (/ (/ (+ (/ t_1 x) 1.0) x) n)))))))

C

double code(double x, double n) {
	double t_0 = pow(x, pow(n, -1.0));
	double t_1 = (0.3333333333333333 / x) - 0.5;
	double tmp;
	if (pow(n, -1.0) <= -1.0) {
		tmp = t_0 / n;
	} else if (pow(n, -1.0) <= 5e-53) {
		tmp = log(((1.0 + x) / x)) / n;
	} else if (pow(n, -1.0) <= 5e-9) {
		tmp = fma((pow(x, -1.0) / n), t_1, pow(n, -1.0)) / x;
	} else if (pow(n, -1.0) <= 2e+201) {
		tmp = 1.0 - t_0;
	} else {
		tmp = (((t_1 / x) + 1.0) / x) / n;
	}
	return tmp;
}

Julia

function code(x, n)
	t_0 = x ^ (n ^ -1.0)
	t_1 = Float64(Float64(0.3333333333333333 / x) - 0.5)
	tmp = 0.0
	if ((n ^ -1.0) <= -1.0)
		tmp = Float64(t_0 / n);
	elseif ((n ^ -1.0) <= 5e-53)
		tmp = Float64(log(Float64(Float64(1.0 + x) / x)) / n);
	elseif ((n ^ -1.0) <= 5e-9)
		tmp = Float64(fma(Float64((x ^ -1.0) / n), t_1, (n ^ -1.0)) / x);
	elseif ((n ^ -1.0) <= 2e+201)
		tmp = Float64(1.0 - t_0);
	else
		tmp = Float64(Float64(Float64(Float64(t_1 / x) + 1.0) / x) / n);
	end
	return tmp
end

Wolfram

code[x_, n_] := Block[{t$95$0 = N[Power[x, N[Power[n, -1.0], $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(N[(0.3333333333333333 / x), $MachinePrecision] - 0.5), $MachinePrecision]}, If[LessEqual[N[Power[n, -1.0], $MachinePrecision], -1.0], N[(t$95$0 / n), $MachinePrecision], If[LessEqual[N[Power[n, -1.0], $MachinePrecision], 5e-53], N[(N[Log[N[(N[(1.0 + x), $MachinePrecision] / x), $MachinePrecision]], $MachinePrecision] / n), $MachinePrecision], If[LessEqual[N[Power[n, -1.0], $MachinePrecision], 5e-9], N[(N[(N[(N[Power[x, -1.0], $MachinePrecision] / n), $MachinePrecision] * t$95$1 + N[Power[n, -1.0], $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision], If[LessEqual[N[Power[n, -1.0], $MachinePrecision], 2e+201], N[(1.0 - t$95$0), $MachinePrecision], N[(N[(N[(N[(t$95$1 / x), $MachinePrecision] + 1.0), $MachinePrecision] / x), $MachinePrecision] / n), $MachinePrecision]]]]]]]

Derivation

1. Split input into 5 regimes

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

   1. Initial program 99.9%

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

   3. Taylor expanded in x around inf

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

      1. associate-/l/ N/A

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

      2. lower-/.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. log-rec N/A

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

      5. mul-1-neg N/A

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

      6. associate-*r/ N/A

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

      7. associate-*r* N/A

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

      8. metadata-eval N/A

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

      9. *-commutative N/A

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

      10. associate-/l* N/A

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

      11. exp-to-pow N/A

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

      12. lower-pow.f64 N/A

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

      13. lower-/.f64 98.5

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

   5. Applied rewrites 98.5%

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

   7. Applied rewrites 98.5%

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

   9. Applied rewrites 98.5%

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

   10. Taylor expanded in n around 0

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

   12. Applied rewrites 98.7%

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

   if -1 < (/.f64 #s(literal 1 binary64) n) < 5e-53

   1. Initial program 37.5%

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

   3. Taylor expanded in n around inf

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

      1. lower-/.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lower-log1p.f64 N/A

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

      4. lower-log.f64 78.5

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

   5. Applied rewrites 78.5%

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

   7. Applied rewrites 78.7%

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

   if 5e-53 < (/.f64 #s(literal 1 binary64) n) < 5.0000000000000001e-9

   1. Initial program 4.9%

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

   3. Taylor expanded in n around inf

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

      1. lower-/.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lower-log1p.f64 N/A

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

      4. lower-log.f64 18.5

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

   5. Applied rewrites 18.5%

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

   6. Taylor expanded in x around inf

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

   8. Applied rewrites 78.2%

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

   9. Taylor expanded in x around -inf

      \[\leadsto -1 \cdot \color{blue}{\frac{-1 \cdot \frac{\frac{1}{3} \cdot \frac{1}{n \cdot x} - \frac{1}{2} \cdot \frac{1}{n}}{x} - \frac{1}{n}}{x}} \]
   10. Step-by-step derivation

   11. Applied rewrites 78.5%

       \[\leadsto \frac{\mathsf{fma}\left(\frac{\frac{1}{x}}{n}, \frac{0.3333333333333333}{x} - 0.5, \frac{1}{n}\right)}{\color{blue}{x}} \]

   if 5.0000000000000001e-9 < (/.f64 #s(literal 1 binary64) n) < 2.00000000000000008e201

   1. Initial program 93.2%

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

   3. Taylor expanded in x around 0

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

   5. Applied rewrites 93.2%

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

   if 2.00000000000000008e201 < (/.f64 #s(literal 1 binary64) n)

   1. Initial program 14.9%

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

   3. Taylor expanded in n around inf

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

      1. lower-/.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lower-log1p.f64 N/A

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

      4. lower-log.f64 6.4

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

   5. Applied rewrites 6.4%

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

   6. Taylor expanded in x around inf

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

   8. Applied rewrites 87.9%

      \[\leadsto \frac{\frac{\frac{\frac{0.3333333333333333}{x} - 0.5}{x} + 1}{x}}{n} \]
3. Recombined 5 regimes into one program.

4. Final simplification 85.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{n}^{-1} \leq -1:\\ \;\;\;\;\frac{{x}^{\left({n}^{-1}\right)}}{n}\\ \mathbf{elif}\;{n}^{-1} \leq 5 \cdot 10^{-53}:\\ \;\;\;\;\frac{\log \left(\frac{1 + x}{x}\right)}{n}\\ \mathbf{elif}\;{n}^{-1} \leq 5 \cdot 10^{-9}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{{x}^{-1}}{n}, \frac{0.3333333333333333}{x} - 0.5, {n}^{-1}\right)}{x}\\ \mathbf{elif}\;{n}^{-1} \leq 2 \cdot 10^{+201}:\\ \;\;\;\;1 - {x}^{\left({n}^{-1}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{\frac{0.3333333333333333}{x} - 0.5}{x} + 1}{x}}{n}\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 51.3% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 4 \cdot 10^{+165}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{{x}^{-1}}{n}, \frac{0.3333333333333333}{x} - 0.5, {n}^{-1}\right)}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{-0.5}{x \cdot x}}{n}\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (if (<= x 4e+165)
   (/ (fma (/ (pow x -1.0) n) (- (/ 0.3333333333333333 x) 0.5) (pow n -1.0)) x)
   (/ (/ -0.5 (* x x)) n)))

C

double code(double x, double n) {
	double tmp;
	if (x <= 4e+165) {
		tmp = fma((pow(x, -1.0) / n), ((0.3333333333333333 / x) - 0.5), pow(n, -1.0)) / x;
	} else {
		tmp = (-0.5 / (x * x)) / n;
	}
	return tmp;
}

Julia

function code(x, n)
	tmp = 0.0
	if (x <= 4e+165)
		tmp = Float64(fma(Float64((x ^ -1.0) / n), Float64(Float64(0.3333333333333333 / x) - 0.5), (n ^ -1.0)) / x);
	else
		tmp = Float64(Float64(-0.5 / Float64(x * x)) / n);
	end
	return tmp
end

Wolfram

code[x_, n_] := If[LessEqual[x, 4e+165], N[(N[(N[(N[Power[x, -1.0], $MachinePrecision] / n), $MachinePrecision] * N[(N[(0.3333333333333333 / x), $MachinePrecision] - 0.5), $MachinePrecision] + N[Power[n, -1.0], $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision], N[(N[(-0.5 / N[(x * x), $MachinePrecision]), $MachinePrecision] / n), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 3.9999999999999996e165

   1. Initial program 47.8%

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

   3. Taylor expanded in n around inf

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

      1. lower-/.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lower-log1p.f64 N/A

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

      4. lower-log.f64 52.2

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

   5. Applied rewrites 52.2%

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

   6. Taylor expanded in x around inf

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

   8. Applied rewrites 39.9%

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

   9. Taylor expanded in x around -inf

      \[\leadsto -1 \cdot \color{blue}{\frac{-1 \cdot \frac{\frac{1}{3} \cdot \frac{1}{n \cdot x} - \frac{1}{2} \cdot \frac{1}{n}}{x} - \frac{1}{n}}{x}} \]
   10. Step-by-step derivation

   11. Applied rewrites 45.1%

       \[\leadsto \frac{\mathsf{fma}\left(\frac{\frac{1}{x}}{n}, \frac{0.3333333333333333}{x} - 0.5, \frac{1}{n}\right)}{\color{blue}{x}} \]

   if 3.9999999999999996e165 < x

   1. Initial program 89.6%

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

   3. Taylor expanded in n around inf

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

      1. lower-/.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lower-log1p.f64 N/A

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

      4. lower-log.f64 89.6

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

   5. Applied rewrites 89.6%

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

   6. Taylor expanded in x around inf

      \[\leadsto \frac{\frac{1 - \frac{1}{2} \cdot \frac{1}{x}}{x}}{n} \]
   7. Step-by-step derivation

   8. Applied rewrites 74.6%

      \[\leadsto \frac{\frac{1 - \frac{0.5}{x}}{x}}{n} \]

   9. Taylor expanded in x around 0

      \[\leadsto \frac{\frac{\frac{-1}{2}}{{x}^{2}}}{n} \]
   10. Step-by-step derivation

   11. Applied rewrites 89.6%

       \[\leadsto \frac{\frac{-0.5}{x \cdot x}}{n} \]
3. Recombined 2 regimes into one program.

4. Final simplification 54.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 4 \cdot 10^{+165}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{{x}^{-1}}{n}, \frac{0.3333333333333333}{x} - 0.5, {n}^{-1}\right)}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{-0.5}{x \cdot x}}{n}\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 60.1% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 1.55 \cdot 10^{-135}:\\ \;\;\;\;1 - {x}^{\left({n}^{-1}\right)}\\ \mathbf{elif}\;x \leq 0.85:\\ \;\;\;\;\frac{x - \log x}{n}\\ \mathbf{elif}\;x \leq 4 \cdot 10^{+165}:\\ \;\;\;\;\frac{\frac{1 + \frac{-0.5 - \frac{\frac{0.25}{x} - 0.3333333333333333}{x}}{x}}{x}}{n}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{-0.5}{x \cdot x}}{n}\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (if (<= x 1.55e-135)
   (- 1.0 (pow x (pow n -1.0)))
   (if (<= x 0.85)
     (/ (- x (log x)) n)
     (if (<= x 4e+165)
       (/
        (/ (+ 1.0 (/ (- -0.5 (/ (- (/ 0.25 x) 0.3333333333333333) x)) x)) x)
        n)
       (/ (/ -0.5 (* x x)) n)))))

C

double code(double x, double n) {
	double tmp;
	if (x <= 1.55e-135) {
		tmp = 1.0 - pow(x, pow(n, -1.0));
	} else if (x <= 0.85) {
		tmp = (x - log(x)) / n;
	} else if (x <= 4e+165) {
		tmp = ((1.0 + ((-0.5 - (((0.25 / x) - 0.3333333333333333) / x)) / x)) / x) / n;
	} else {
		tmp = (-0.5 / (x * x)) / n;
	}
	return tmp;
}

Fortran

real(8) function code(x, n)
    real(8), intent (in) :: x
    real(8), intent (in) :: n
    real(8) :: tmp
    if (x <= 1.55d-135) then
        tmp = 1.0d0 - (x ** (n ** (-1.0d0)))
    else if (x <= 0.85d0) then
        tmp = (x - log(x)) / n
    else if (x <= 4d+165) then
        tmp = ((1.0d0 + (((-0.5d0) - (((0.25d0 / x) - 0.3333333333333333d0) / x)) / x)) / x) / n
    else
        tmp = ((-0.5d0) / (x * x)) / n
    end if
    code = tmp
end function

Java

public static double code(double x, double n) {
	double tmp;
	if (x <= 1.55e-135) {
		tmp = 1.0 - Math.pow(x, Math.pow(n, -1.0));
	} else if (x <= 0.85) {
		tmp = (x - Math.log(x)) / n;
	} else if (x <= 4e+165) {
		tmp = ((1.0 + ((-0.5 - (((0.25 / x) - 0.3333333333333333) / x)) / x)) / x) / n;
	} else {
		tmp = (-0.5 / (x * x)) / n;
	}
	return tmp;
}

Python

def code(x, n):
	tmp = 0
	if x <= 1.55e-135:
		tmp = 1.0 - math.pow(x, math.pow(n, -1.0))
	elif x <= 0.85:
		tmp = (x - math.log(x)) / n
	elif x <= 4e+165:
		tmp = ((1.0 + ((-0.5 - (((0.25 / x) - 0.3333333333333333) / x)) / x)) / x) / n
	else:
		tmp = (-0.5 / (x * x)) / n
	return tmp

Julia

function code(x, n)
	tmp = 0.0
	if (x <= 1.55e-135)
		tmp = Float64(1.0 - (x ^ (n ^ -1.0)));
	elseif (x <= 0.85)
		tmp = Float64(Float64(x - log(x)) / n);
	elseif (x <= 4e+165)
		tmp = Float64(Float64(Float64(1.0 + Float64(Float64(-0.5 - Float64(Float64(Float64(0.25 / x) - 0.3333333333333333) / x)) / x)) / x) / n);
	else
		tmp = Float64(Float64(-0.5 / Float64(x * x)) / n);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, n)
	tmp = 0.0;
	if (x <= 1.55e-135)
		tmp = 1.0 - (x ^ (n ^ -1.0));
	elseif (x <= 0.85)
		tmp = (x - log(x)) / n;
	elseif (x <= 4e+165)
		tmp = ((1.0 + ((-0.5 - (((0.25 / x) - 0.3333333333333333) / x)) / x)) / x) / n;
	else
		tmp = (-0.5 / (x * x)) / n;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, n_] := If[LessEqual[x, 1.55e-135], N[(1.0 - N[Power[x, N[Power[n, -1.0], $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 0.85], N[(N[(x - N[Log[x], $MachinePrecision]), $MachinePrecision] / n), $MachinePrecision], If[LessEqual[x, 4e+165], N[(N[(N[(1.0 + N[(N[(-0.5 - N[(N[(N[(0.25 / x), $MachinePrecision] - 0.3333333333333333), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision] / n), $MachinePrecision], N[(N[(-0.5 / N[(x * x), $MachinePrecision]), $MachinePrecision] / n), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if x < 1.55e-135

   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 x around 0

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

   5. Applied rewrites 55.8%

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

   if 1.55e-135 < x < 0.849999999999999978

   1. Initial program 32.6%

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

   3. Taylor expanded in n around inf

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

      1. lower-/.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lower-log1p.f64 N/A

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

      4. lower-log.f64 66.6

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

   5. Applied rewrites 66.6%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 64.4%

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

   if 0.849999999999999978 < x < 3.9999999999999996e165

   1. Initial program 47.7%

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

   3. Taylor expanded in n around inf

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

      1. lower-/.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lower-log1p.f64 N/A

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

      4. lower-log.f64 52.4

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

   5. Applied rewrites 52.4%

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

   6. Taylor expanded in x around -inf

      \[\leadsto \frac{-1 \cdot \frac{-1 \cdot \frac{-1 \cdot \frac{\frac{1}{4} \cdot \frac{1}{x} - \frac{1}{3}}{x} - \frac{1}{2}}{x} - 1}{x}}{n} \]
   7. Step-by-step derivation

   8. Applied rewrites 77.1%

      \[\leadsto \frac{\frac{-1 - \frac{-0.5 - \frac{\frac{0.25}{x} - 0.3333333333333333}{x}}{x}}{-x}}{n} \]

   if 3.9999999999999996e165 < x

   1. Initial program 89.6%

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

   3. Taylor expanded in n around inf

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

      1. lower-/.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lower-log1p.f64 N/A

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

      4. lower-log.f64 89.6

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

   5. Applied rewrites 89.6%

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

   6. Taylor expanded in x around inf

      \[\leadsto \frac{\frac{1 - \frac{1}{2} \cdot \frac{1}{x}}{x}}{n} \]
   7. Step-by-step derivation

   8. Applied rewrites 74.6%

      \[\leadsto \frac{\frac{1 - \frac{0.5}{x}}{x}}{n} \]

   9. Taylor expanded in x around 0

      \[\leadsto \frac{\frac{\frac{-1}{2}}{{x}^{2}}}{n} \]
   10. Step-by-step derivation

   11. Applied rewrites 89.6%

       \[\leadsto \frac{\frac{-0.5}{x \cdot x}}{n} \]
3. Recombined 4 regimes into one program.

4. Final simplification 70.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 1.55 \cdot 10^{-135}:\\ \;\;\;\;1 - {x}^{\left({n}^{-1}\right)}\\ \mathbf{elif}\;x \leq 0.85:\\ \;\;\;\;\frac{x - \log x}{n}\\ \mathbf{elif}\;x \leq 4 \cdot 10^{+165}:\\ \;\;\;\;\frac{\frac{1 + \frac{-0.5 - \frac{\frac{0.25}{x} - 0.3333333333333333}{x}}{x}}{x}}{n}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{-0.5}{x \cdot x}}{n}\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 61.3% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 0.85:\\ \;\;\;\;\frac{x - \log x}{n}\\ \mathbf{elif}\;x \leq 4 \cdot 10^{+165}:\\ \;\;\;\;\frac{\frac{1 + \frac{-0.5 - \frac{\frac{0.25}{x} - 0.3333333333333333}{x}}{x}}{x}}{n}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{-0.5}{x \cdot x}}{n}\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (if (<= x 0.85)
   (/ (- x (log x)) n)
   (if (<= x 4e+165)
     (/ (/ (+ 1.0 (/ (- -0.5 (/ (- (/ 0.25 x) 0.3333333333333333) x)) x)) x) n)
     (/ (/ -0.5 (* x x)) n))))

C

double code(double x, double n) {
	double tmp;
	if (x <= 0.85) {
		tmp = (x - log(x)) / n;
	} else if (x <= 4e+165) {
		tmp = ((1.0 + ((-0.5 - (((0.25 / x) - 0.3333333333333333) / x)) / x)) / x) / n;
	} else {
		tmp = (-0.5 / (x * x)) / n;
	}
	return tmp;
}

Fortran

real(8) function code(x, n)
    real(8), intent (in) :: x
    real(8), intent (in) :: n
    real(8) :: tmp
    if (x <= 0.85d0) then
        tmp = (x - log(x)) / n
    else if (x <= 4d+165) then
        tmp = ((1.0d0 + (((-0.5d0) - (((0.25d0 / x) - 0.3333333333333333d0) / x)) / x)) / x) / n
    else
        tmp = ((-0.5d0) / (x * x)) / n
    end if
    code = tmp
end function

Java

public static double code(double x, double n) {
	double tmp;
	if (x <= 0.85) {
		tmp = (x - Math.log(x)) / n;
	} else if (x <= 4e+165) {
		tmp = ((1.0 + ((-0.5 - (((0.25 / x) - 0.3333333333333333) / x)) / x)) / x) / n;
	} else {
		tmp = (-0.5 / (x * x)) / n;
	}
	return tmp;
}

Python

def code(x, n):
	tmp = 0
	if x <= 0.85:
		tmp = (x - math.log(x)) / n
	elif x <= 4e+165:
		tmp = ((1.0 + ((-0.5 - (((0.25 / x) - 0.3333333333333333) / x)) / x)) / x) / n
	else:
		tmp = (-0.5 / (x * x)) / n
	return tmp

Julia

function code(x, n)
	tmp = 0.0
	if (x <= 0.85)
		tmp = Float64(Float64(x - log(x)) / n);
	elseif (x <= 4e+165)
		tmp = Float64(Float64(Float64(1.0 + Float64(Float64(-0.5 - Float64(Float64(Float64(0.25 / x) - 0.3333333333333333) / x)) / x)) / x) / n);
	else
		tmp = Float64(Float64(-0.5 / Float64(x * x)) / n);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, n)
	tmp = 0.0;
	if (x <= 0.85)
		tmp = (x - log(x)) / n;
	elseif (x <= 4e+165)
		tmp = ((1.0 + ((-0.5 - (((0.25 / x) - 0.3333333333333333) / x)) / x)) / x) / n;
	else
		tmp = (-0.5 / (x * x)) / n;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, n_] := If[LessEqual[x, 0.85], N[(N[(x - N[Log[x], $MachinePrecision]), $MachinePrecision] / n), $MachinePrecision], If[LessEqual[x, 4e+165], N[(N[(N[(1.0 + N[(N[(-0.5 - N[(N[(N[(0.25 / x), $MachinePrecision] - 0.3333333333333333), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision] / n), $MachinePrecision], N[(N[(-0.5 / N[(x * x), $MachinePrecision]), $MachinePrecision] / n), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < 0.849999999999999978

   1. Initial program 47.9%

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

   3. Taylor expanded in n around inf

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

      1. lower-/.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lower-log1p.f64 N/A

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

      4. lower-log.f64 52.1

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

   5. Applied rewrites 52.1%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 51.3%

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

   if 0.849999999999999978 < x < 3.9999999999999996e165

   1. Initial program 47.7%

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

   3. Taylor expanded in n around inf

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

      1. lower-/.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lower-log1p.f64 N/A

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

      4. lower-log.f64 52.4

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

   5. Applied rewrites 52.4%

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

   6. Taylor expanded in x around -inf

      \[\leadsto \frac{-1 \cdot \frac{-1 \cdot \frac{-1 \cdot \frac{\frac{1}{4} \cdot \frac{1}{x} - \frac{1}{3}}{x} - \frac{1}{2}}{x} - 1}{x}}{n} \]
   7. Step-by-step derivation

   8. Applied rewrites 77.1%

      \[\leadsto \frac{\frac{-1 - \frac{-0.5 - \frac{\frac{0.25}{x} - 0.3333333333333333}{x}}{x}}{-x}}{n} \]

   if 3.9999999999999996e165 < x

   1. Initial program 89.6%

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

   3. Taylor expanded in n around inf

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

      1. lower-/.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lower-log1p.f64 N/A

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

      4. lower-log.f64 89.6

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

   5. Applied rewrites 89.6%

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

   6. Taylor expanded in x around inf

      \[\leadsto \frac{\frac{1 - \frac{1}{2} \cdot \frac{1}{x}}{x}}{n} \]
   7. Step-by-step derivation

   8. Applied rewrites 74.6%

      \[\leadsto \frac{\frac{1 - \frac{0.5}{x}}{x}}{n} \]

   9. Taylor expanded in x around 0

      \[\leadsto \frac{\frac{\frac{-1}{2}}{{x}^{2}}}{n} \]
   10. Step-by-step derivation

   11. Applied rewrites 89.6%

       \[\leadsto \frac{\frac{-0.5}{x \cdot x}}{n} \]
3. Recombined 3 regimes into one program.

4. Final simplification 66.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 0.85:\\ \;\;\;\;\frac{x - \log x}{n}\\ \mathbf{elif}\;x \leq 4 \cdot 10^{+165}:\\ \;\;\;\;\frac{\frac{1 + \frac{-0.5 - \frac{\frac{0.25}{x} - 0.3333333333333333}{x}}{x}}{x}}{n}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{-0.5}{x \cdot x}}{n}\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 61.0% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 0.7:\\ \;\;\;\;\frac{-\log x}{n}\\ \mathbf{elif}\;x \leq 4 \cdot 10^{+165}:\\ \;\;\;\;\frac{\frac{1 + \frac{-0.5 - \frac{\frac{0.25}{x} - 0.3333333333333333}{x}}{x}}{x}}{n}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{-0.5}{x \cdot x}}{n}\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (if (<= x 0.7)
   (/ (- (log x)) n)
   (if (<= x 4e+165)
     (/ (/ (+ 1.0 (/ (- -0.5 (/ (- (/ 0.25 x) 0.3333333333333333) x)) x)) x) n)
     (/ (/ -0.5 (* x x)) n))))

C

double code(double x, double n) {
	double tmp;
	if (x <= 0.7) {
		tmp = -log(x) / n;
	} else if (x <= 4e+165) {
		tmp = ((1.0 + ((-0.5 - (((0.25 / x) - 0.3333333333333333) / x)) / x)) / x) / n;
	} else {
		tmp = (-0.5 / (x * x)) / n;
	}
	return tmp;
}

Fortran

real(8) function code(x, n)
    real(8), intent (in) :: x
    real(8), intent (in) :: n
    real(8) :: tmp
    if (x <= 0.7d0) then
        tmp = -log(x) / n
    else if (x <= 4d+165) then
        tmp = ((1.0d0 + (((-0.5d0) - (((0.25d0 / x) - 0.3333333333333333d0) / x)) / x)) / x) / n
    else
        tmp = ((-0.5d0) / (x * x)) / n
    end if
    code = tmp
end function

Java

public static double code(double x, double n) {
	double tmp;
	if (x <= 0.7) {
		tmp = -Math.log(x) / n;
	} else if (x <= 4e+165) {
		tmp = ((1.0 + ((-0.5 - (((0.25 / x) - 0.3333333333333333) / x)) / x)) / x) / n;
	} else {
		tmp = (-0.5 / (x * x)) / n;
	}
	return tmp;
}

Python

def code(x, n):
	tmp = 0
	if x <= 0.7:
		tmp = -math.log(x) / n
	elif x <= 4e+165:
		tmp = ((1.0 + ((-0.5 - (((0.25 / x) - 0.3333333333333333) / x)) / x)) / x) / n
	else:
		tmp = (-0.5 / (x * x)) / n
	return tmp

Julia

function code(x, n)
	tmp = 0.0
	if (x <= 0.7)
		tmp = Float64(Float64(-log(x)) / n);
	elseif (x <= 4e+165)
		tmp = Float64(Float64(Float64(1.0 + Float64(Float64(-0.5 - Float64(Float64(Float64(0.25 / x) - 0.3333333333333333) / x)) / x)) / x) / n);
	else
		tmp = Float64(Float64(-0.5 / Float64(x * x)) / n);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, n)
	tmp = 0.0;
	if (x <= 0.7)
		tmp = -log(x) / n;
	elseif (x <= 4e+165)
		tmp = ((1.0 + ((-0.5 - (((0.25 / x) - 0.3333333333333333) / x)) / x)) / x) / n;
	else
		tmp = (-0.5 / (x * x)) / n;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, n_] := If[LessEqual[x, 0.7], N[((-N[Log[x], $MachinePrecision]) / n), $MachinePrecision], If[LessEqual[x, 4e+165], N[(N[(N[(1.0 + N[(N[(-0.5 - N[(N[(N[(0.25 / x), $MachinePrecision] - 0.3333333333333333), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision] / n), $MachinePrecision], N[(N[(-0.5 / N[(x * x), $MachinePrecision]), $MachinePrecision] / n), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < 0.69999999999999996

   1. Initial program 47.9%

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

   3. Taylor expanded in n around inf

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

      1. lower-/.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lower-log1p.f64 N/A

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

      4. lower-log.f64 52.1

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

   5. Applied rewrites 52.1%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 51.1%

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

   if 0.69999999999999996 < x < 3.9999999999999996e165

   1. Initial program 47.7%

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

   3. Taylor expanded in n around inf

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

      1. lower-/.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lower-log1p.f64 N/A

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

      4. lower-log.f64 52.4

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

   5. Applied rewrites 52.4%

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

   6. Taylor expanded in x around -inf

      \[\leadsto \frac{-1 \cdot \frac{-1 \cdot \frac{-1 \cdot \frac{\frac{1}{4} \cdot \frac{1}{x} - \frac{1}{3}}{x} - \frac{1}{2}}{x} - 1}{x}}{n} \]
   7. Step-by-step derivation

   8. Applied rewrites 77.1%

      \[\leadsto \frac{\frac{-1 - \frac{-0.5 - \frac{\frac{0.25}{x} - 0.3333333333333333}{x}}{x}}{-x}}{n} \]

   if 3.9999999999999996e165 < x

   1. Initial program 89.6%

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

   3. Taylor expanded in n around inf

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

      1. lower-/.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lower-log1p.f64 N/A

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

      4. lower-log.f64 89.6

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

   5. Applied rewrites 89.6%

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

   6. Taylor expanded in x around inf

      \[\leadsto \frac{\frac{1 - \frac{1}{2} \cdot \frac{1}{x}}{x}}{n} \]
   7. Step-by-step derivation

   8. Applied rewrites 74.6%

      \[\leadsto \frac{\frac{1 - \frac{0.5}{x}}{x}}{n} \]

   9. Taylor expanded in x around 0

      \[\leadsto \frac{\frac{\frac{-1}{2}}{{x}^{2}}}{n} \]
   10. Step-by-step derivation

   11. Applied rewrites 89.6%

       \[\leadsto \frac{\frac{-0.5}{x \cdot x}}{n} \]
3. Recombined 3 regimes into one program.

4. Final simplification 66.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 0.7:\\ \;\;\;\;\frac{-\log x}{n}\\ \mathbf{elif}\;x \leq 4 \cdot 10^{+165}:\\ \;\;\;\;\frac{\frac{1 + \frac{-0.5 - \frac{\frac{0.25}{x} - 0.3333333333333333}{x}}{x}}{x}}{n}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{-0.5}{x \cdot x}}{n}\\ \end{array} \]
5. Add Preprocessing

Alternative 17: 45.5% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 4 \cdot 10^{+165}:\\ \;\;\;\;\frac{{x}^{-1}}{n}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{-0.5}{x \cdot x}}{n}\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (if (<= x 4e+165) (/ (pow x -1.0) n) (/ (/ -0.5 (* x x)) n)))

C

double code(double x, double n) {
	double tmp;
	if (x <= 4e+165) {
		tmp = pow(x, -1.0) / n;
	} else {
		tmp = (-0.5 / (x * x)) / n;
	}
	return tmp;
}

Fortran

real(8) function code(x, n)
    real(8), intent (in) :: x
    real(8), intent (in) :: n
    real(8) :: tmp
    if (x <= 4d+165) then
        tmp = (x ** (-1.0d0)) / n
    else
        tmp = ((-0.5d0) / (x * x)) / n
    end if
    code = tmp
end function

Java

public static double code(double x, double n) {
	double tmp;
	if (x <= 4e+165) {
		tmp = Math.pow(x, -1.0) / n;
	} else {
		tmp = (-0.5 / (x * x)) / n;
	}
	return tmp;
}

Python

def code(x, n):
	tmp = 0
	if x <= 4e+165:
		tmp = math.pow(x, -1.0) / n
	else:
		tmp = (-0.5 / (x * x)) / n
	return tmp

Julia

function code(x, n)
	tmp = 0.0
	if (x <= 4e+165)
		tmp = Float64((x ^ -1.0) / n);
	else
		tmp = Float64(Float64(-0.5 / Float64(x * x)) / n);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, n)
	tmp = 0.0;
	if (x <= 4e+165)
		tmp = (x ^ -1.0) / n;
	else
		tmp = (-0.5 / (x * x)) / n;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, n_] := If[LessEqual[x, 4e+165], N[(N[Power[x, -1.0], $MachinePrecision] / n), $MachinePrecision], N[(N[(-0.5 / N[(x * x), $MachinePrecision]), $MachinePrecision] / n), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 3.9999999999999996e165

   1. Initial program 47.8%

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

   3. Taylor expanded in n around inf

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

      1. lower-/.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lower-log1p.f64 N/A

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

      4. lower-log.f64 52.2

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

   5. Applied rewrites 52.2%

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

   6. Taylor expanded in x around inf

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

   8. Applied rewrites 39.9%

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

   if 3.9999999999999996e165 < x

   1. Initial program 89.6%

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

   3. Taylor expanded in n around inf

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

      1. lower-/.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lower-log1p.f64 N/A

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

      4. lower-log.f64 89.6

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

   5. Applied rewrites 89.6%

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

   6. Taylor expanded in x around inf

      \[\leadsto \frac{\frac{1 - \frac{1}{2} \cdot \frac{1}{x}}{x}}{n} \]
   7. Step-by-step derivation

   8. Applied rewrites 74.6%

      \[\leadsto \frac{\frac{1 - \frac{0.5}{x}}{x}}{n} \]

   9. Taylor expanded in x around 0

      \[\leadsto \frac{\frac{\frac{-1}{2}}{{x}^{2}}}{n} \]
   10. Step-by-step derivation

   11. Applied rewrites 89.6%

       \[\leadsto \frac{\frac{-0.5}{x \cdot x}}{n} \]
3. Recombined 2 regimes into one program.

4. Final simplification 50.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 4 \cdot 10^{+165}:\\ \;\;\;\;\frac{{x}^{-1}}{n}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{-0.5}{x \cdot x}}{n}\\ \end{array} \]
5. Add Preprocessing

Alternative 18: 41.4% accurate, 2.0× speedup

Math

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

FPCore

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

C

double code(double x, double n) {
	return 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)) / n
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 56.5%

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

3. Taylor expanded in n around inf

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

   1. lower-/.f64 N/A

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

   2. lower--.f64 N/A

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

   3. lower-log1p.f64 N/A

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

   4. lower-log.f64 59.9

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

5. Applied rewrites 59.9%

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

6. Taylor expanded in x around inf

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

8. Applied rewrites 47.1%

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

9. Final simplification 47.1%

   \[\leadsto \frac{{x}^{-1}}{n} \]
10. Add Preprocessing

Alternative 19: 40.8% accurate, 2.2× speedup

Math

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

FPCore

(FPCore (x n) :precision binary64 (pow (* n x) -1.0))

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 56.5%

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

3. Taylor expanded in x around inf

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

   1. associate-/l/ N/A

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

   2. lower-/.f64 N/A

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

   3. lower-/.f64 N/A

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

   4. log-rec N/A

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

   5. mul-1-neg N/A

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

   6. associate-*r/ N/A

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

   7. associate-*r* N/A

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

   8. metadata-eval N/A

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

   9. *-commutative N/A

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

   10. associate-/l* N/A

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

   11. exp-to-pow N/A

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

   12. lower-pow.f64 N/A

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

   13. lower-/.f64 61.0

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

5. Applied rewrites 61.0%

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

7. Applied rewrites 61.0%

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

9. Applied rewrites 60.0%

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

10. Taylor expanded in n around inf

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

12. Applied rewrites 46.1%

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

13. Final simplification 46.1%

    \[\leadsto {\left(n \cdot x\right)}^{-1} \]
14. Add Preprocessing

Alternative 20: 51.3% accurate, 4.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 4 \cdot 10^{+165}:\\ \;\;\;\;\frac{\frac{\frac{\frac{0.3333333333333333}{x} - 0.5}{x} + 1}{x}}{n}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{-0.5}{x \cdot x}}{n}\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (if (<= x 4e+165)
   (/ (/ (+ (/ (- (/ 0.3333333333333333 x) 0.5) x) 1.0) x) n)
   (/ (/ -0.5 (* x x)) n)))

C

double code(double x, double n) {
	double tmp;
	if (x <= 4e+165) {
		tmp = (((((0.3333333333333333 / x) - 0.5) / x) + 1.0) / x) / n;
	} else {
		tmp = (-0.5 / (x * x)) / n;
	}
	return tmp;
}

Fortran

real(8) function code(x, n)
    real(8), intent (in) :: x
    real(8), intent (in) :: n
    real(8) :: tmp
    if (x <= 4d+165) then
        tmp = (((((0.3333333333333333d0 / x) - 0.5d0) / x) + 1.0d0) / x) / n
    else
        tmp = ((-0.5d0) / (x * x)) / n
    end if
    code = tmp
end function

Java

public static double code(double x, double n) {
	double tmp;
	if (x <= 4e+165) {
		tmp = (((((0.3333333333333333 / x) - 0.5) / x) + 1.0) / x) / n;
	} else {
		tmp = (-0.5 / (x * x)) / n;
	}
	return tmp;
}

Python

def code(x, n):
	tmp = 0
	if x <= 4e+165:
		tmp = (((((0.3333333333333333 / x) - 0.5) / x) + 1.0) / x) / n
	else:
		tmp = (-0.5 / (x * x)) / n
	return tmp

Julia

function code(x, n)
	tmp = 0.0
	if (x <= 4e+165)
		tmp = Float64(Float64(Float64(Float64(Float64(Float64(0.3333333333333333 / x) - 0.5) / x) + 1.0) / x) / n);
	else
		tmp = Float64(Float64(-0.5 / Float64(x * x)) / n);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, n)
	tmp = 0.0;
	if (x <= 4e+165)
		tmp = (((((0.3333333333333333 / x) - 0.5) / x) + 1.0) / x) / n;
	else
		tmp = (-0.5 / (x * x)) / n;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, n_] := If[LessEqual[x, 4e+165], N[(N[(N[(N[(N[(N[(0.3333333333333333 / x), $MachinePrecision] - 0.5), $MachinePrecision] / x), $MachinePrecision] + 1.0), $MachinePrecision] / x), $MachinePrecision] / n), $MachinePrecision], N[(N[(-0.5 / N[(x * x), $MachinePrecision]), $MachinePrecision] / n), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 3.9999999999999996e165

   1. Initial program 47.8%

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

   3. Taylor expanded in n around inf

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

      1. lower-/.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lower-log1p.f64 N/A

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

      4. lower-log.f64 52.2

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

   5. Applied rewrites 52.2%

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

   6. Taylor expanded in x around inf

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

   8. Applied rewrites 45.0%

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

   if 3.9999999999999996e165 < x

   1. Initial program 89.6%

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

   3. Taylor expanded in n around inf

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

      1. lower-/.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lower-log1p.f64 N/A

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

      4. lower-log.f64 89.6

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

   5. Applied rewrites 89.6%

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

   6. Taylor expanded in x around inf

      \[\leadsto \frac{\frac{1 - \frac{1}{2} \cdot \frac{1}{x}}{x}}{n} \]
   7. Step-by-step derivation

   8. Applied rewrites 74.6%

      \[\leadsto \frac{\frac{1 - \frac{0.5}{x}}{x}}{n} \]

   9. Taylor expanded in x around 0

      \[\leadsto \frac{\frac{\frac{-1}{2}}{{x}^{2}}}{n} \]
   10. Step-by-step derivation

   11. Applied rewrites 89.6%

       \[\leadsto \frac{\frac{-0.5}{x \cdot x}}{n} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Reproduce

herbie shell --seed 2024311 
(FPCore (x n)
  :name "2nthrt (problem 3.4.6)"
  :precision binary64
  (- (pow (+ x 1.0) (/ 1.0 n)) (pow x (/ 1.0 n))))