2nthrt (problem 3.4.6)

Percentage Accurate: 53.2% → 92.6%

Time: 23.7s

Alternatives: 17

Speedup: 1.7×

• could not determine a ground truth

  1. x = -9.73163499735358e-42
  2. n = -2.213006925850301e+44

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: 53.2% 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: 92.6% accurate, 1.0× speedup

Math

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

FPCore

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

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)) / n) / x;
	}
	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)) / n) / x;
	}
	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)) / n) / x
	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)) / n) / x);
	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] / n), $MachinePrecision] / x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 1

   1. Initial program 39.9%

      \[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)} \]
   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 86.5%

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

   if 1 < x

   1. Initial program 71.2%

      \[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)} \]
   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 99.4

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

   5. Applied rewrites 99.4%

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

   7. Applied rewrites 99.5%

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

Alternative 2: 79.1% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {x}^{\left({n}^{-1}\right)}\\ t_1 := {\left(x + 1\right)}^{\left({n}^{-1}\right)} - t\_0\\ \mathbf{if}\;t\_1 \leq -5 \cdot 10^{-7} \lor \neg \left(t\_1 \leq 0\right):\\ \;\;\;\;1 - t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{\log \left(\frac{1 + x}{x}\right)}{n}\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (let* ((t_0 (pow x (pow n -1.0))) (t_1 (- (pow (+ x 1.0) (pow n -1.0)) t_0)))
   (if (or (<= t_1 -5e-7) (not (<= t_1 0.0)))
     (- 1.0 t_0)
     (/ (log (/ (+ 1.0 x) x)) n))))

C

double code(double x, double n) {
	double t_0 = pow(x, pow(n, -1.0));
	double t_1 = pow((x + 1.0), pow(n, -1.0)) - t_0;
	double tmp;
	if ((t_1 <= -5e-7) || !(t_1 <= 0.0)) {
		tmp = 1.0 - t_0;
	} else {
		tmp = log(((1.0 + x) / 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 = ((x + 1.0d0) ** (n ** (-1.0d0))) - t_0
    if ((t_1 <= (-5d-7)) .or. (.not. (t_1 <= 0.0d0))) then
        tmp = 1.0d0 - t_0
    else
        tmp = log(((1.0d0 + x) / 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 = Math.pow((x + 1.0), Math.pow(n, -1.0)) - t_0;
	double tmp;
	if ((t_1 <= -5e-7) || !(t_1 <= 0.0)) {
		tmp = 1.0 - t_0;
	} else {
		tmp = Math.log(((1.0 + x) / x)) / n;
	}
	return tmp;
}

Python

def code(x, n):
	t_0 = math.pow(x, math.pow(n, -1.0))
	t_1 = math.pow((x + 1.0), math.pow(n, -1.0)) - t_0
	tmp = 0
	if (t_1 <= -5e-7) or not (t_1 <= 0.0):
		tmp = 1.0 - t_0
	else:
		tmp = math.log(((1.0 + x) / x)) / n
	return tmp

Julia

function code(x, n)
	t_0 = x ^ (n ^ -1.0)
	t_1 = Float64((Float64(x + 1.0) ^ (n ^ -1.0)) - t_0)
	tmp = 0.0
	if ((t_1 <= -5e-7) || !(t_1 <= 0.0))
		tmp = Float64(1.0 - t_0);
	else
		tmp = Float64(log(Float64(Float64(1.0 + x) / x)) / n);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, n)
	t_0 = x ^ (n ^ -1.0);
	t_1 = ((x + 1.0) ^ (n ^ -1.0)) - t_0;
	tmp = 0.0;
	if ((t_1 <= -5e-7) || ~((t_1 <= 0.0)))
		tmp = 1.0 - t_0;
	else
		tmp = log(((1.0 + x) / 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[Power[N[(x + 1.0), $MachinePrecision], N[Power[n, -1.0], $MachinePrecision]], $MachinePrecision] - t$95$0), $MachinePrecision]}, If[Or[LessEqual[t$95$1, -5e-7], N[Not[LessEqual[t$95$1, 0.0]], $MachinePrecision]], N[(1.0 - t$95$0), $MachinePrecision], N[(N[Log[N[(N[(1.0 + x), $MachinePrecision] / x), $MachinePrecision]], $MachinePrecision] / n), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (pow.f64 (+.f64 x #s(literal 1 binary64)) (/.f64 #s(literal 1 binary64) n)) (pow.f64 x (/.f64 #s(literal 1 binary64) n))) < -4.99999999999999977e-7 or 0.0 < (-.f64 (pow.f64 (+.f64 x #s(literal 1 binary64)) (/.f64 #s(literal 1 binary64) n)) (pow.f64 x (/.f64 #s(literal 1 binary64) n)))

   1. Initial program 74.8%

      \[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)} \]
   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 72.1%

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

   if -4.99999999999999977e-7 < (-.f64 (pow.f64 (+.f64 x #s(literal 1 binary64)) (/.f64 #s(literal 1 binary64) n)) (pow.f64 x (/.f64 #s(literal 1 binary64) n))) < 0.0

   1. Initial program 45.4%

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

   3. Taylor expanded in n around inf

      \[\leadsto \color{blue}{\frac{\log \left(1 + x\right) - \log x}{n}} \]
   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 79.0

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

   5. Applied rewrites 79.0%

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

   7. Applied rewrites 79.7%

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

4. Final simplification 77.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{\left(x + 1\right)}^{\left({n}^{-1}\right)} - {x}^{\left({n}^{-1}\right)} \leq -5 \cdot 10^{-7} \lor \neg \left({\left(x + 1\right)}^{\left({n}^{-1}\right)} - {x}^{\left({n}^{-1}\right)} \leq 0\right):\\ \;\;\;\;1 - {x}^{\left({n}^{-1}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\log \left(\frac{1 + x}{x}\right)}{n}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 82.3% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;{n}^{-1} \leq -1:\\ \;\;\;\;\frac{\frac{{\left(x \cdot x\right)}^{\left(\frac{0.5}{n}\right)}}{x}}{n}\\ \mathbf{elif}\;{n}^{-1} \leq 5 \cdot 10^{-22}:\\ \;\;\;\;\frac{\log \left(\frac{1 + x}{x}\right)}{n}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\frac{\frac{0.5}{n} - 0.5}{n}, x, {n}^{-1}\right), x, 1\right) - {x}^{\left({n}^{-1}\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (if (<= (pow n -1.0) -1.0)
   (/ (/ (pow (* x x) (/ 0.5 n)) x) n)
   (if (<= (pow n -1.0) 5e-22)
     (/ (log (/ (+ 1.0 x) x)) n)
     (-
      (fma (fma (/ (- (/ 0.5 n) 0.5) n) x (pow n -1.0)) x 1.0)
      (pow x (pow n -1.0))))))

C

double code(double x, double n) {
	double tmp;
	if (pow(n, -1.0) <= -1.0) {
		tmp = (pow((x * x), (0.5 / n)) / x) / n;
	} else if (pow(n, -1.0) <= 5e-22) {
		tmp = log(((1.0 + x) / x)) / n;
	} else {
		tmp = fma(fma((((0.5 / n) - 0.5) / n), x, pow(n, -1.0)), x, 1.0) - pow(x, pow(n, -1.0));
	}
	return tmp;
}

Julia

function code(x, n)
	tmp = 0.0
	if ((n ^ -1.0) <= -1.0)
		tmp = Float64(Float64((Float64(x * x) ^ Float64(0.5 / n)) / x) / n);
	elseif ((n ^ -1.0) <= 5e-22)
		tmp = Float64(log(Float64(Float64(1.0 + x) / x)) / n);
	else
		tmp = Float64(fma(fma(Float64(Float64(Float64(0.5 / n) - 0.5) / n), x, (n ^ -1.0)), x, 1.0) - (x ^ (n ^ -1.0)));
	end
	return tmp
end

Wolfram

code[x_, n_] := If[LessEqual[N[Power[n, -1.0], $MachinePrecision], -1.0], N[(N[(N[Power[N[(x * x), $MachinePrecision], N[(0.5 / n), $MachinePrecision]], $MachinePrecision] / x), $MachinePrecision] / n), $MachinePrecision], If[LessEqual[N[Power[n, -1.0], $MachinePrecision], 5e-22], N[(N[Log[N[(N[(1.0 + x), $MachinePrecision] / x), $MachinePrecision]], $MachinePrecision] / n), $MachinePrecision], N[(N[(N[(N[(N[(N[(0.5 / n), $MachinePrecision] - 0.5), $MachinePrecision] / n), $MachinePrecision] * x + N[Power[n, -1.0], $MachinePrecision]), $MachinePrecision] * x + 1.0), $MachinePrecision] - N[Power[x, N[Power[n, -1.0], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 98.5%

      \[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)} \]
   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 99.9

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

   5. Applied rewrites 99.9%

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

   7. Applied rewrites 99.9%

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

   if -1 < (/.f64 #s(literal 1 binary64) n) < 4.99999999999999954e-22

   1. Initial program 32.2%

      \[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)} \]
   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 77.6

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

   5. Applied rewrites 77.6%

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

   7. Applied rewrites 77.9%

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

   if 4.99999999999999954e-22 < (/.f64 #s(literal 1 binary64) n)

   1. Initial program 58.4%

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

      1. lift-pow.f64 N/A

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

      2. pow-to-exp N/A

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

      3. lower-exp.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. un-div-inv N/A

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

      6. lower-/.f64 N/A

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

      7. lift-+.f64 N/A

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

      8. +-commutative N/A

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

      9. lower-log1p.f64 100.0

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

   4. Applied rewrites 100.0%

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

   5. Taylor expanded in x around 0

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

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

   7. Applied rewrites 74.9%

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

4. Final simplification 83.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{n}^{-1} \leq -1:\\ \;\;\;\;\frac{\frac{{\left(x \cdot x\right)}^{\left(\frac{0.5}{n}\right)}}{x}}{n}\\ \mathbf{elif}\;{n}^{-1} \leq 5 \cdot 10^{-22}:\\ \;\;\;\;\frac{\log \left(\frac{1 + x}{x}\right)}{n}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\frac{\frac{0.5}{n} - 0.5}{n}, x, {n}^{-1}\right), x, 1\right) - {x}^{\left({n}^{-1}\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 82.0% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;{n}^{-1} \leq -1:\\ \;\;\;\;\frac{\frac{{\left(x \cdot x\right)}^{\left(\frac{0.5}{n}\right)}}{x}}{n}\\ \mathbf{elif}\;{n}^{-1} \leq 5 \cdot 10^{-22}:\\ \;\;\;\;\frac{\log \left(\frac{1 + x}{x}\right)}{n}\\ \mathbf{elif}\;{n}^{-1} \leq 5 \cdot 10^{+185}:\\ \;\;\;\;\left(\frac{x}{n} + 1\right) - {x}^{\left({n}^{-1}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{n}{x}}{n \cdot n}\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (if (<= (pow n -1.0) -1.0)
   (/ (/ (pow (* x x) (/ 0.5 n)) x) n)
   (if (<= (pow n -1.0) 5e-22)
     (/ (log (/ (+ 1.0 x) x)) n)
     (if (<= (pow n -1.0) 5e+185)
       (- (+ (/ x n) 1.0) (pow x (pow n -1.0)))
       (/ (/ n x) (* n n))))))

C

double code(double x, double n) {
	double tmp;
	if (pow(n, -1.0) <= -1.0) {
		tmp = (pow((x * x), (0.5 / n)) / x) / n;
	} else if (pow(n, -1.0) <= 5e-22) {
		tmp = log(((1.0 + x) / x)) / n;
	} else if (pow(n, -1.0) <= 5e+185) {
		tmp = ((x / n) + 1.0) - pow(x, pow(n, -1.0));
	} else {
		tmp = (n / x) / (n * n);
	}
	return tmp;
}

Fortran

real(8) function code(x, n)
    real(8), intent (in) :: x
    real(8), intent (in) :: n
    real(8) :: tmp
    if ((n ** (-1.0d0)) <= (-1.0d0)) then
        tmp = (((x * x) ** (0.5d0 / n)) / x) / n
    else if ((n ** (-1.0d0)) <= 5d-22) then
        tmp = log(((1.0d0 + x) / x)) / n
    else if ((n ** (-1.0d0)) <= 5d+185) then
        tmp = ((x / n) + 1.0d0) - (x ** (n ** (-1.0d0)))
    else
        tmp = (n / x) / (n * n)
    end if
    code = tmp
end function

Java

public static double code(double x, double n) {
	double tmp;
	if (Math.pow(n, -1.0) <= -1.0) {
		tmp = (Math.pow((x * x), (0.5 / n)) / x) / n;
	} else if (Math.pow(n, -1.0) <= 5e-22) {
		tmp = Math.log(((1.0 + x) / x)) / n;
	} else if (Math.pow(n, -1.0) <= 5e+185) {
		tmp = ((x / n) + 1.0) - Math.pow(x, Math.pow(n, -1.0));
	} else {
		tmp = (n / x) / (n * n);
	}
	return tmp;
}

Python

def code(x, n):
	tmp = 0
	if math.pow(n, -1.0) <= -1.0:
		tmp = (math.pow((x * x), (0.5 / n)) / x) / n
	elif math.pow(n, -1.0) <= 5e-22:
		tmp = math.log(((1.0 + x) / x)) / n
	elif math.pow(n, -1.0) <= 5e+185:
		tmp = ((x / n) + 1.0) - math.pow(x, math.pow(n, -1.0))
	else:
		tmp = (n / x) / (n * n)
	return tmp

Julia

function code(x, n)
	tmp = 0.0
	if ((n ^ -1.0) <= -1.0)
		tmp = Float64(Float64((Float64(x * x) ^ Float64(0.5 / n)) / x) / n);
	elseif ((n ^ -1.0) <= 5e-22)
		tmp = Float64(log(Float64(Float64(1.0 + x) / x)) / n);
	elseif ((n ^ -1.0) <= 5e+185)
		tmp = Float64(Float64(Float64(x / n) + 1.0) - (x ^ (n ^ -1.0)));
	else
		tmp = Float64(Float64(n / x) / Float64(n * n));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, n)
	tmp = 0.0;
	if ((n ^ -1.0) <= -1.0)
		tmp = (((x * x) ^ (0.5 / n)) / x) / n;
	elseif ((n ^ -1.0) <= 5e-22)
		tmp = log(((1.0 + x) / x)) / n;
	elseif ((n ^ -1.0) <= 5e+185)
		tmp = ((x / n) + 1.0) - (x ^ (n ^ -1.0));
	else
		tmp = (n / x) / (n * n);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, n_] := If[LessEqual[N[Power[n, -1.0], $MachinePrecision], -1.0], N[(N[(N[Power[N[(x * x), $MachinePrecision], N[(0.5 / n), $MachinePrecision]], $MachinePrecision] / x), $MachinePrecision] / n), $MachinePrecision], If[LessEqual[N[Power[n, -1.0], $MachinePrecision], 5e-22], N[(N[Log[N[(N[(1.0 + x), $MachinePrecision] / x), $MachinePrecision]], $MachinePrecision] / n), $MachinePrecision], If[LessEqual[N[Power[n, -1.0], $MachinePrecision], 5e+185], N[(N[(N[(x / n), $MachinePrecision] + 1.0), $MachinePrecision] - N[Power[x, N[Power[n, -1.0], $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[(n / x), $MachinePrecision] / N[(n * n), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

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

   1. Initial program 98.5%

      \[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)} \]
   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 99.9

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

   5. Applied rewrites 99.9%

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

   7. Applied rewrites 99.9%

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

   if -1 < (/.f64 #s(literal 1 binary64) n) < 4.99999999999999954e-22

   1. Initial program 32.2%

      \[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)} \]
   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 77.6

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

   5. Applied rewrites 77.6%

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

   7. Applied rewrites 77.9%

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

   if 4.99999999999999954e-22 < (/.f64 #s(literal 1 binary64) n) < 4.9999999999999999e185

   1. Initial program 81.7%

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

   3. Taylor expanded in x around 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 79.8

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

   5. Applied rewrites 79.8%

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

   if 4.9999999999999999e185 < (/.f64 #s(literal 1 binary64) n)

   1. Initial program 27.4%

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

   3. Taylor expanded in n around inf

      \[\leadsto \color{blue}{\frac{\log \left(1 + x\right) - \log x}{n}} \]
   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 6.3

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

   5. Applied rewrites 6.3%

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

   7. Applied rewrites 78.5%

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

   8. Taylor expanded in x around inf

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

   10. Applied rewrites 78.5%

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

4. Final simplification 83.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{n}^{-1} \leq -1:\\ \;\;\;\;\frac{\frac{{\left(x \cdot x\right)}^{\left(\frac{0.5}{n}\right)}}{x}}{n}\\ \mathbf{elif}\;{n}^{-1} \leq 5 \cdot 10^{-22}:\\ \;\;\;\;\frac{\log \left(\frac{1 + x}{x}\right)}{n}\\ \mathbf{elif}\;{n}^{-1} \leq 5 \cdot 10^{+185}:\\ \;\;\;\;\left(\frac{x}{n} + 1\right) - {x}^{\left({n}^{-1}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{n}{x}}{n \cdot n}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 82.0% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {x}^{\left({n}^{-1}\right)}\\ \mathbf{if}\;{n}^{-1} \leq -1:\\ \;\;\;\;\frac{t\_0}{n \cdot x}\\ \mathbf{elif}\;{n}^{-1} \leq 5 \cdot 10^{-22}:\\ \;\;\;\;\frac{\log \left(\frac{1 + x}{x}\right)}{n}\\ \mathbf{elif}\;{n}^{-1} \leq 5 \cdot 10^{+185}:\\ \;\;\;\;\left(\frac{x}{n} + 1\right) - t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{n}{x}}{n \cdot n}\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (let* ((t_0 (pow x (pow n -1.0))))
   (if (<= (pow n -1.0) -1.0)
     (/ t_0 (* n x))
     (if (<= (pow n -1.0) 5e-22)
       (/ (log (/ (+ 1.0 x) x)) n)
       (if (<= (pow n -1.0) 5e+185)
         (- (+ (/ x n) 1.0) t_0)
         (/ (/ n x) (* n n)))))))

C

double code(double x, double n) {
	double t_0 = pow(x, pow(n, -1.0));
	double tmp;
	if (pow(n, -1.0) <= -1.0) {
		tmp = t_0 / (n * x);
	} else if (pow(n, -1.0) <= 5e-22) {
		tmp = log(((1.0 + x) / x)) / n;
	} else if (pow(n, -1.0) <= 5e+185) {
		tmp = ((x / n) + 1.0) - t_0;
	} else {
		tmp = (n / x) / (n * 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)) <= (-1.0d0)) then
        tmp = t_0 / (n * x)
    else if ((n ** (-1.0d0)) <= 5d-22) then
        tmp = log(((1.0d0 + x) / x)) / n
    else if ((n ** (-1.0d0)) <= 5d+185) then
        tmp = ((x / n) + 1.0d0) - t_0
    else
        tmp = (n / x) / (n * 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) <= -1.0) {
		tmp = t_0 / (n * x);
	} else if (Math.pow(n, -1.0) <= 5e-22) {
		tmp = Math.log(((1.0 + x) / x)) / n;
	} else if (Math.pow(n, -1.0) <= 5e+185) {
		tmp = ((x / n) + 1.0) - t_0;
	} else {
		tmp = (n / x) / (n * 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) <= -1.0:
		tmp = t_0 / (n * x)
	elif math.pow(n, -1.0) <= 5e-22:
		tmp = math.log(((1.0 + x) / x)) / n
	elif math.pow(n, -1.0) <= 5e+185:
		tmp = ((x / n) + 1.0) - t_0
	else:
		tmp = (n / x) / (n * n)
	return tmp

Julia

function code(x, n)
	t_0 = x ^ (n ^ -1.0)
	tmp = 0.0
	if ((n ^ -1.0) <= -1.0)
		tmp = Float64(t_0 / Float64(n * x));
	elseif ((n ^ -1.0) <= 5e-22)
		tmp = Float64(log(Float64(Float64(1.0 + x) / x)) / n);
	elseif ((n ^ -1.0) <= 5e+185)
		tmp = Float64(Float64(Float64(x / n) + 1.0) - t_0);
	else
		tmp = Float64(Float64(n / x) / Float64(n * 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) <= -1.0)
		tmp = t_0 / (n * x);
	elseif ((n ^ -1.0) <= 5e-22)
		tmp = log(((1.0 + x) / x)) / n;
	elseif ((n ^ -1.0) <= 5e+185)
		tmp = ((x / n) + 1.0) - t_0;
	else
		tmp = (n / x) / (n * 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], -1.0], N[(t$95$0 / N[(n * x), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[Power[n, -1.0], $MachinePrecision], 5e-22], N[(N[Log[N[(N[(1.0 + x), $MachinePrecision] / x), $MachinePrecision]], $MachinePrecision] / n), $MachinePrecision], If[LessEqual[N[Power[n, -1.0], $MachinePrecision], 5e+185], N[(N[(N[(x / n), $MachinePrecision] + 1.0), $MachinePrecision] - t$95$0), $MachinePrecision], N[(N[(n / x), $MachinePrecision] / N[(n * n), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

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

   1. Initial program 98.5%

      \[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)} \]
   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 99.9

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

   5. Applied rewrites 99.9%

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

   7. Applied rewrites 99.9%

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

   9. Applied rewrites 99.9%

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

   if -1 < (/.f64 #s(literal 1 binary64) n) < 4.99999999999999954e-22

   1. Initial program 32.2%

      \[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)} \]
   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 77.6

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

   5. Applied rewrites 77.6%

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

   7. Applied rewrites 77.9%

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

   if 4.99999999999999954e-22 < (/.f64 #s(literal 1 binary64) n) < 4.9999999999999999e185

   1. Initial program 81.7%

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

   3. Taylor expanded in x around 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 79.8

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

   5. Applied rewrites 79.8%

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

   if 4.9999999999999999e185 < (/.f64 #s(literal 1 binary64) n)

   1. Initial program 27.4%

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

   3. Taylor expanded in n around inf

      \[\leadsto \color{blue}{\frac{\log \left(1 + x\right) - \log x}{n}} \]
   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 6.3

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

   5. Applied rewrites 6.3%

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

   7. Applied rewrites 78.5%

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

   8. Taylor expanded in x around inf

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

   10. Applied rewrites 78.5%

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

4. Final simplification 83.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{n}^{-1} \leq -1:\\ \;\;\;\;\frac{{x}^{\left({n}^{-1}\right)}}{n \cdot x}\\ \mathbf{elif}\;{n}^{-1} \leq 5 \cdot 10^{-22}:\\ \;\;\;\;\frac{\log \left(\frac{1 + x}{x}\right)}{n}\\ \mathbf{elif}\;{n}^{-1} \leq 5 \cdot 10^{+185}:\\ \;\;\;\;\left(\frac{x}{n} + 1\right) - {x}^{\left({n}^{-1}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{n}{x}}{n \cdot n}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 82.0% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {x}^{\left({n}^{-1}\right)}\\ \mathbf{if}\;{n}^{-1} \leq -1:\\ \;\;\;\;\frac{t\_0}{n \cdot x}\\ \mathbf{elif}\;{n}^{-1} \leq 5 \cdot 10^{-22}:\\ \;\;\;\;\frac{\log \left(\frac{1 + x}{x}\right)}{n}\\ \mathbf{elif}\;{n}^{-1} \leq 5 \cdot 10^{+185}:\\ \;\;\;\;\frac{n + x}{n} - t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{n}{x}}{n \cdot n}\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (let* ((t_0 (pow x (pow n -1.0))))
   (if (<= (pow n -1.0) -1.0)
     (/ t_0 (* n x))
     (if (<= (pow n -1.0) 5e-22)
       (/ (log (/ (+ 1.0 x) x)) n)
       (if (<= (pow n -1.0) 5e+185)
         (- (/ (+ n x) n) t_0)
         (/ (/ n x) (* n n)))))))

C

double code(double x, double n) {
	double t_0 = pow(x, pow(n, -1.0));
	double tmp;
	if (pow(n, -1.0) <= -1.0) {
		tmp = t_0 / (n * x);
	} else if (pow(n, -1.0) <= 5e-22) {
		tmp = log(((1.0 + x) / x)) / n;
	} else if (pow(n, -1.0) <= 5e+185) {
		tmp = ((n + x) / n) - t_0;
	} else {
		tmp = (n / x) / (n * 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)) <= (-1.0d0)) then
        tmp = t_0 / (n * x)
    else if ((n ** (-1.0d0)) <= 5d-22) then
        tmp = log(((1.0d0 + x) / x)) / n
    else if ((n ** (-1.0d0)) <= 5d+185) then
        tmp = ((n + x) / n) - t_0
    else
        tmp = (n / x) / (n * 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) <= -1.0) {
		tmp = t_0 / (n * x);
	} else if (Math.pow(n, -1.0) <= 5e-22) {
		tmp = Math.log(((1.0 + x) / x)) / n;
	} else if (Math.pow(n, -1.0) <= 5e+185) {
		tmp = ((n + x) / n) - t_0;
	} else {
		tmp = (n / x) / (n * 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) <= -1.0:
		tmp = t_0 / (n * x)
	elif math.pow(n, -1.0) <= 5e-22:
		tmp = math.log(((1.0 + x) / x)) / n
	elif math.pow(n, -1.0) <= 5e+185:
		tmp = ((n + x) / n) - t_0
	else:
		tmp = (n / x) / (n * n)
	return tmp

Julia

function code(x, n)
	t_0 = x ^ (n ^ -1.0)
	tmp = 0.0
	if ((n ^ -1.0) <= -1.0)
		tmp = Float64(t_0 / Float64(n * x));
	elseif ((n ^ -1.0) <= 5e-22)
		tmp = Float64(log(Float64(Float64(1.0 + x) / x)) / n);
	elseif ((n ^ -1.0) <= 5e+185)
		tmp = Float64(Float64(Float64(n + x) / n) - t_0);
	else
		tmp = Float64(Float64(n / x) / Float64(n * 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) <= -1.0)
		tmp = t_0 / (n * x);
	elseif ((n ^ -1.0) <= 5e-22)
		tmp = log(((1.0 + x) / x)) / n;
	elseif ((n ^ -1.0) <= 5e+185)
		tmp = ((n + x) / n) - t_0;
	else
		tmp = (n / x) / (n * 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], -1.0], N[(t$95$0 / N[(n * x), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[Power[n, -1.0], $MachinePrecision], 5e-22], N[(N[Log[N[(N[(1.0 + x), $MachinePrecision] / x), $MachinePrecision]], $MachinePrecision] / n), $MachinePrecision], If[LessEqual[N[Power[n, -1.0], $MachinePrecision], 5e+185], N[(N[(N[(n + x), $MachinePrecision] / n), $MachinePrecision] - t$95$0), $MachinePrecision], N[(N[(n / x), $MachinePrecision] / N[(n * n), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

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

   1. Initial program 98.5%

      \[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)} \]
   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 99.9

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

   5. Applied rewrites 99.9%

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

   7. Applied rewrites 99.9%

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

   9. Applied rewrites 99.9%

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

   if -1 < (/.f64 #s(literal 1 binary64) n) < 4.99999999999999954e-22

   1. Initial program 32.2%

      \[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)} \]
   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 77.6

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

   5. Applied rewrites 77.6%

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

   7. Applied rewrites 77.9%

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

   if 4.99999999999999954e-22 < (/.f64 #s(literal 1 binary64) n) < 4.9999999999999999e185

   1. Initial program 81.7%

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

   3. Taylor expanded in x around 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 79.8

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

   5. Applied rewrites 79.8%

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

   6. Taylor expanded in n around 0

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

   8. Applied rewrites 79.7%

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

   if 4.9999999999999999e185 < (/.f64 #s(literal 1 binary64) n)

   1. Initial program 27.4%

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

   3. Taylor expanded in n around inf

      \[\leadsto \color{blue}{\frac{\log \left(1 + x\right) - \log x}{n}} \]
   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 6.3

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

   5. Applied rewrites 6.3%

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

   7. Applied rewrites 78.5%

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

   8. Taylor expanded in x around inf

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

   10. Applied rewrites 78.5%

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

4. Final simplification 83.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{n}^{-1} \leq -1:\\ \;\;\;\;\frac{{x}^{\left({n}^{-1}\right)}}{n \cdot x}\\ \mathbf{elif}\;{n}^{-1} \leq 5 \cdot 10^{-22}:\\ \;\;\;\;\frac{\log \left(\frac{1 + x}{x}\right)}{n}\\ \mathbf{elif}\;{n}^{-1} \leq 5 \cdot 10^{+185}:\\ \;\;\;\;\frac{n + x}{n} - {x}^{\left({n}^{-1}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{n}{x}}{n \cdot n}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 82.1% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {x}^{\left({n}^{-1}\right)}\\ \mathbf{if}\;{n}^{-1} \leq -1:\\ \;\;\;\;\frac{t\_0}{n \cdot x}\\ \mathbf{elif}\;{n}^{-1} \leq 5 \cdot 10^{-22}:\\ \;\;\;\;\frac{\log \left(\frac{1 + x}{x}\right)}{n}\\ \mathbf{elif}\;{n}^{-1} \leq 2 \cdot 10^{+163}:\\ \;\;\;\;1 - t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{n}{x}}{n \cdot n}\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (let* ((t_0 (pow x (pow n -1.0))))
   (if (<= (pow n -1.0) -1.0)
     (/ t_0 (* n x))
     (if (<= (pow n -1.0) 5e-22)
       (/ (log (/ (+ 1.0 x) x)) n)
       (if (<= (pow n -1.0) 2e+163) (- 1.0 t_0) (/ (/ n x) (* n n)))))))

C

double code(double x, double n) {
	double t_0 = pow(x, pow(n, -1.0));
	double tmp;
	if (pow(n, -1.0) <= -1.0) {
		tmp = t_0 / (n * x);
	} else if (pow(n, -1.0) <= 5e-22) {
		tmp = log(((1.0 + x) / x)) / n;
	} else if (pow(n, -1.0) <= 2e+163) {
		tmp = 1.0 - t_0;
	} else {
		tmp = (n / x) / (n * 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)) <= (-1.0d0)) then
        tmp = t_0 / (n * x)
    else if ((n ** (-1.0d0)) <= 5d-22) then
        tmp = log(((1.0d0 + x) / x)) / n
    else if ((n ** (-1.0d0)) <= 2d+163) then
        tmp = 1.0d0 - t_0
    else
        tmp = (n / x) / (n * 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) <= -1.0) {
		tmp = t_0 / (n * x);
	} else if (Math.pow(n, -1.0) <= 5e-22) {
		tmp = Math.log(((1.0 + x) / x)) / n;
	} else if (Math.pow(n, -1.0) <= 2e+163) {
		tmp = 1.0 - t_0;
	} else {
		tmp = (n / x) / (n * 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) <= -1.0:
		tmp = t_0 / (n * x)
	elif math.pow(n, -1.0) <= 5e-22:
		tmp = math.log(((1.0 + x) / x)) / n
	elif math.pow(n, -1.0) <= 2e+163:
		tmp = 1.0 - t_0
	else:
		tmp = (n / x) / (n * n)
	return tmp

Julia

function code(x, n)
	t_0 = x ^ (n ^ -1.0)
	tmp = 0.0
	if ((n ^ -1.0) <= -1.0)
		tmp = Float64(t_0 / Float64(n * x));
	elseif ((n ^ -1.0) <= 5e-22)
		tmp = Float64(log(Float64(Float64(1.0 + x) / x)) / n);
	elseif ((n ^ -1.0) <= 2e+163)
		tmp = Float64(1.0 - t_0);
	else
		tmp = Float64(Float64(n / x) / Float64(n * 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) <= -1.0)
		tmp = t_0 / (n * x);
	elseif ((n ^ -1.0) <= 5e-22)
		tmp = log(((1.0 + x) / x)) / n;
	elseif ((n ^ -1.0) <= 2e+163)
		tmp = 1.0 - t_0;
	else
		tmp = (n / x) / (n * 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], -1.0], N[(t$95$0 / N[(n * x), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[Power[n, -1.0], $MachinePrecision], 5e-22], N[(N[Log[N[(N[(1.0 + x), $MachinePrecision] / x), $MachinePrecision]], $MachinePrecision] / n), $MachinePrecision], If[LessEqual[N[Power[n, -1.0], $MachinePrecision], 2e+163], N[(1.0 - t$95$0), $MachinePrecision], N[(N[(n / x), $MachinePrecision] / N[(n * n), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

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

   1. Initial program 98.5%

      \[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)} \]
   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 99.9

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

   5. Applied rewrites 99.9%

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

   7. Applied rewrites 99.9%

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

   9. Applied rewrites 99.9%

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

   if -1 < (/.f64 #s(literal 1 binary64) n) < 4.99999999999999954e-22

   1. Initial program 32.2%

      \[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)} \]
   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 77.6

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

   5. Applied rewrites 77.6%

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

   7. Applied rewrites 77.9%

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

   if 4.99999999999999954e-22 < (/.f64 #s(literal 1 binary64) n) < 1.9999999999999999e163

   1. Initial program 84.5%

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

   3. Taylor expanded in x around 0

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

   5. Applied rewrites 80.1%

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

   if 1.9999999999999999e163 < (/.f64 #s(literal 1 binary64) n)

   1. Initial program 29.8%

      \[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)} \]
   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 6.1

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

   5. Applied rewrites 6.1%

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

   7. Applied rewrites 75.8%

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

   8. Taylor expanded in x around inf

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

   10. Applied rewrites 75.8%

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

4. Final simplification 83.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{n}^{-1} \leq -1:\\ \;\;\;\;\frac{{x}^{\left({n}^{-1}\right)}}{n \cdot x}\\ \mathbf{elif}\;{n}^{-1} \leq 5 \cdot 10^{-22}:\\ \;\;\;\;\frac{\log \left(\frac{1 + x}{x}\right)}{n}\\ \mathbf{elif}\;{n}^{-1} \leq 2 \cdot 10^{+163}:\\ \;\;\;\;1 - {x}^{\left({n}^{-1}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{n}{x}}{n \cdot n}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 51.2% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;{n}^{-1} \leq -20000000:\\ \;\;\;\;\frac{\log 1}{n}\\ \mathbf{elif}\;{n}^{-1} \leq 4 \cdot 10^{+129}:\\ \;\;\;\;\frac{{\left(0.5 + x\right)}^{-1}}{n}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{n}{x}}{n \cdot n}\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (if (<= (pow n -1.0) -20000000.0)
   (/ (log 1.0) n)
   (if (<= (pow n -1.0) 4e+129)
     (/ (pow (+ 0.5 x) -1.0) n)
     (/ (/ n x) (* n n)))))

C

double code(double x, double n) {
	double tmp;
	if (pow(n, -1.0) <= -20000000.0) {
		tmp = log(1.0) / n;
	} else if (pow(n, -1.0) <= 4e+129) {
		tmp = pow((0.5 + x), -1.0) / n;
	} else {
		tmp = (n / x) / (n * n);
	}
	return tmp;
}

Fortran

real(8) function code(x, n)
    real(8), intent (in) :: x
    real(8), intent (in) :: n
    real(8) :: tmp
    if ((n ** (-1.0d0)) <= (-20000000.0d0)) then
        tmp = log(1.0d0) / n
    else if ((n ** (-1.0d0)) <= 4d+129) then
        tmp = ((0.5d0 + x) ** (-1.0d0)) / n
    else
        tmp = (n / x) / (n * n)
    end if
    code = tmp
end function

Java

public static double code(double x, double n) {
	double tmp;
	if (Math.pow(n, -1.0) <= -20000000.0) {
		tmp = Math.log(1.0) / n;
	} else if (Math.pow(n, -1.0) <= 4e+129) {
		tmp = Math.pow((0.5 + x), -1.0) / n;
	} else {
		tmp = (n / x) / (n * n);
	}
	return tmp;
}

Python

def code(x, n):
	tmp = 0
	if math.pow(n, -1.0) <= -20000000.0:
		tmp = math.log(1.0) / n
	elif math.pow(n, -1.0) <= 4e+129:
		tmp = math.pow((0.5 + x), -1.0) / n
	else:
		tmp = (n / x) / (n * n)
	return tmp

Julia

function code(x, n)
	tmp = 0.0
	if ((n ^ -1.0) <= -20000000.0)
		tmp = Float64(log(1.0) / n);
	elseif ((n ^ -1.0) <= 4e+129)
		tmp = Float64((Float64(0.5 + x) ^ -1.0) / n);
	else
		tmp = Float64(Float64(n / x) / Float64(n * n));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, n)
	tmp = 0.0;
	if ((n ^ -1.0) <= -20000000.0)
		tmp = log(1.0) / n;
	elseif ((n ^ -1.0) <= 4e+129)
		tmp = ((0.5 + x) ^ -1.0) / n;
	else
		tmp = (n / x) / (n * n);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, n_] := If[LessEqual[N[Power[n, -1.0], $MachinePrecision], -20000000.0], N[(N[Log[1.0], $MachinePrecision] / n), $MachinePrecision], If[LessEqual[N[Power[n, -1.0], $MachinePrecision], 4e+129], N[(N[Power[N[(0.5 + x), $MachinePrecision], -1.0], $MachinePrecision] / n), $MachinePrecision], N[(N[(n / x), $MachinePrecision] / N[(n * n), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 #s(literal 1 binary64) n) < -2e7

   1. Initial program 100.0%

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

   3. Taylor expanded in n around inf

      \[\leadsto \color{blue}{\frac{\log \left(1 + x\right) - \log x}{n}} \]
   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 52.3

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

   5. Applied rewrites 52.3%

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

   7. Applied rewrites 53.9%

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

   8. Taylor expanded in x around inf

      \[\leadsto \frac{\log 1}{n} \]
   9. Step-by-step derivation

   10. Applied rewrites 61.5%

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

   if -2e7 < (/.f64 #s(literal 1 binary64) n) < 4e129

   1. Initial program 38.7%

      \[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)} \]
   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 69.6

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

   5. Applied rewrites 69.6%

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

   7. Applied rewrites 69.9%

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

   9. Applied rewrites 69.6%

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

   10. Taylor expanded in x around inf

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

   12. Applied rewrites 50.2%

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

   if 4e129 < (/.f64 #s(literal 1 binary64) n)

   1. Initial program 34.4%

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

   3. Taylor expanded in n around inf

      \[\leadsto \color{blue}{\frac{\log \left(1 + x\right) - \log x}{n}} \]
   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 5.8

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

   5. Applied rewrites 5.8%

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

   7. Applied rewrites 61.5%

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

   8. Taylor expanded in x around inf

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

   10. Applied rewrites 61.6%

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

4. Final simplification 54.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{n}^{-1} \leq -20000000:\\ \;\;\;\;\frac{\log 1}{n}\\ \mathbf{elif}\;{n}^{-1} \leq 4 \cdot 10^{+129}:\\ \;\;\;\;\frac{{\left(0.5 + x\right)}^{-1}}{n}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{n}{x}}{n \cdot n}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 48.4% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 3.5 \cdot 10^{+208}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{{x}^{-1}}{n}, \frac{0.3333333333333333}{x} - 0.5, {n}^{-1}\right)}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{-0.5}{x}}{n}}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (if (<= x 3.5e+208)
   (/ (fma (/ (pow x -1.0) n) (- (/ 0.3333333333333333 x) 0.5) (pow n -1.0)) x)
   (/ (/ (/ -0.5 x) n) x)))

C

double code(double x, double n) {
	double tmp;
	if (x <= 3.5e+208) {
		tmp = fma((pow(x, -1.0) / n), ((0.3333333333333333 / x) - 0.5), pow(n, -1.0)) / x;
	} else {
		tmp = ((-0.5 / x) / n) / x;
	}
	return tmp;
}

Julia

function code(x, n)
	tmp = 0.0
	if (x <= 3.5e+208)
		tmp = Float64(fma(Float64((x ^ -1.0) / n), Float64(Float64(0.3333333333333333 / x) - 0.5), (n ^ -1.0)) / x);
	else
		tmp = Float64(Float64(Float64(-0.5 / x) / n) / x);
	end
	return tmp
end

Wolfram

code[x_, n_] := If[LessEqual[x, 3.5e+208], 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[(N[(-0.5 / x), $MachinePrecision] / n), $MachinePrecision] / x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 3.50000000000000016e208

   1. Initial program 46.4%

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

   3. Taylor expanded in n around inf

      \[\leadsto \color{blue}{\frac{\log \left(1 + x\right) - \log x}{n}} \]
   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 52.8

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

   5. Applied rewrites 52.8%

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

   7. Applied rewrites 53.4%

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

   8. Taylor expanded in x around inf

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

   10. Applied rewrites 39.6%

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

   if 3.50000000000000016e208 < x

   1. Initial program 97.5%

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

   3. Taylor expanded in x around inf

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

      1. lower-/.f64 N/A

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

   5. Applied rewrites 88.9%

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

   6. Taylor expanded in n around inf

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

   8. Applied rewrites 70.9%

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

   9. Taylor expanded in x around 0

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

   11. Applied rewrites 90.5%

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

4. Final simplification 46.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 3.5 \cdot 10^{+208}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{{x}^{-1}}{n}, \frac{0.3333333333333333}{x} - 0.5, {n}^{-1}\right)}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{-0.5}{x}}{n}}{x}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 92.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 0.62:\\ \;\;\;\;-\mathsf{expm1}\left(\frac{\log x}{n}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{{x}^{\left({n}^{-1}\right)}}{n}}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (if (<= x 0.62) (- (expm1 (/ (log x) n))) (/ (/ (pow x (pow n -1.0)) n) x)))

C

double code(double x, double n) {
	double tmp;
	if (x <= 0.62) {
		tmp = -expm1((log(x) / n));
	} else {
		tmp = (pow(x, pow(n, -1.0)) / n) / x;
	}
	return tmp;
}

Java

public static double code(double x, double n) {
	double tmp;
	if (x <= 0.62) {
		tmp = -Math.expm1((Math.log(x) / n));
	} else {
		tmp = (Math.pow(x, Math.pow(n, -1.0)) / n) / x;
	}
	return tmp;
}

Python

def code(x, n):
	tmp = 0
	if x <= 0.62:
		tmp = -math.expm1((math.log(x) / n))
	else:
		tmp = (math.pow(x, math.pow(n, -1.0)) / n) / x
	return tmp

Julia

function code(x, n)
	tmp = 0.0
	if (x <= 0.62)
		tmp = Float64(-expm1(Float64(log(x) / n)));
	else
		tmp = Float64(Float64((x ^ (n ^ -1.0)) / n) / x);
	end
	return tmp
end

Wolfram

code[x_, n_] := If[LessEqual[x, 0.62], (-N[(Exp[N[(N[Log[x], $MachinePrecision] / n), $MachinePrecision]] - 1), $MachinePrecision]), N[(N[(N[Power[x, N[Power[n, -1.0], $MachinePrecision]], $MachinePrecision] / n), $MachinePrecision] / x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 0.619999999999999996

   1. Initial program 39.9%

      \[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)} \]
   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 53.9

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

   5. Applied rewrites 53.9%

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

   7. Applied rewrites 53.9%

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

   8. Taylor expanded in x around 0

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

      1. sub-neg N/A

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

      2. +-commutative N/A

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

      3. neg-sub0 N/A

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

      4. remove-double-neg N/A

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

      5. distribute-frac-neg N/A

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

      6. log-rec N/A

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

      7. mul-1-neg N/A

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

      8. associate-+l- N/A

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

      9. lower--.f64 N/A

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

      10. mul-1-neg N/A

          \[\leadsto 0 - \left(e^{\color{blue}{\mathsf{neg}\left(\frac{\log \left(\frac{1}{x}\right)}{n}\right)}} - 1\right) \]

      11. log-rec N/A

          \[\leadsto 0 - \left(e^{\mathsf{neg}\left(\frac{\color{blue}{\mathsf{neg}\left(\log x\right)}}{n}\right)} - 1\right) \]

      12. distribute-frac-neg N/A

          \[\leadsto 0 - \left(e^{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(\frac{\log x}{n}\right)\right)}\right)} - 1\right) \]

      13. remove-double-neg N/A

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

      14. lower-expm1.f64 N/A

          \[\leadsto 0 - \color{blue}{\mathsf{expm1}\left(\frac{\log x}{n}\right)} \]

      15. lower-/.f64 N/A

          \[\leadsto 0 - \mathsf{expm1}\left(\color{blue}{\frac{\log x}{n}}\right) \]

      16. lower-log.f64 85.8

          \[\leadsto 0 - \mathsf{expm1}\left(\frac{\color{blue}{\log x}}{n}\right) \]

   10. Applied rewrites 85.8%

       \[\leadsto \color{blue}{0 - \mathsf{expm1}\left(\frac{\log x}{n}\right)} \]

   if 0.619999999999999996 < x

   1. Initial program 71.2%

      \[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)} \]
   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 99.4

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

   5. Applied rewrites 99.4%

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

   7. Applied rewrites 99.5%

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

4. Final simplification 91.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 0.62:\\ \;\;\;\;-\mathsf{expm1}\left(\frac{\log x}{n}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{{x}^{\left({n}^{-1}\right)}}{n}}{x}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 92.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 0.62:\\ \;\;\;\;-\mathsf{expm1}\left(\frac{\log x}{n}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{{x}^{\left(\mathsf{fma}\left(2, \frac{0.5}{n}, -1\right)\right)}}{n}\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (if (<= x 0.62)
   (- (expm1 (/ (log x) n)))
   (/ (pow x (fma 2.0 (/ 0.5 n) -1.0)) n)))

C

double code(double x, double n) {
	double tmp;
	if (x <= 0.62) {
		tmp = -expm1((log(x) / n));
	} else {
		tmp = pow(x, fma(2.0, (0.5 / n), -1.0)) / n;
	}
	return tmp;
}

Julia

function code(x, n)
	tmp = 0.0
	if (x <= 0.62)
		tmp = Float64(-expm1(Float64(log(x) / n)));
	else
		tmp = Float64((x ^ fma(2.0, Float64(0.5 / n), -1.0)) / n);
	end
	return tmp
end

Wolfram

code[x_, n_] := If[LessEqual[x, 0.62], (-N[(Exp[N[(N[Log[x], $MachinePrecision] / n), $MachinePrecision]] - 1), $MachinePrecision]), N[(N[Power[x, N[(2.0 * N[(0.5 / n), $MachinePrecision] + -1.0), $MachinePrecision]], $MachinePrecision] / n), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 0.619999999999999996

   1. Initial program 39.9%

      \[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)} \]
   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 53.9

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

   5. Applied rewrites 53.9%

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

   7. Applied rewrites 53.9%

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

   8. Taylor expanded in x around 0

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

      1. sub-neg N/A

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

      2. +-commutative N/A

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

      3. neg-sub0 N/A

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

      4. remove-double-neg N/A

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

      5. distribute-frac-neg N/A

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

      6. log-rec N/A

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

      7. mul-1-neg N/A

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

      8. associate-+l- N/A

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

      9. lower--.f64 N/A

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

      10. mul-1-neg N/A

          \[\leadsto 0 - \left(e^{\color{blue}{\mathsf{neg}\left(\frac{\log \left(\frac{1}{x}\right)}{n}\right)}} - 1\right) \]

      11. log-rec N/A

          \[\leadsto 0 - \left(e^{\mathsf{neg}\left(\frac{\color{blue}{\mathsf{neg}\left(\log x\right)}}{n}\right)} - 1\right) \]

      12. distribute-frac-neg N/A

          \[\leadsto 0 - \left(e^{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(\frac{\log x}{n}\right)\right)}\right)} - 1\right) \]

      13. remove-double-neg N/A

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

      14. lower-expm1.f64 N/A

          \[\leadsto 0 - \color{blue}{\mathsf{expm1}\left(\frac{\log x}{n}\right)} \]

      15. lower-/.f64 N/A

          \[\leadsto 0 - \mathsf{expm1}\left(\color{blue}{\frac{\log x}{n}}\right) \]

      16. lower-log.f64 85.8

          \[\leadsto 0 - \mathsf{expm1}\left(\frac{\color{blue}{\log x}}{n}\right) \]

   10. Applied rewrites 85.8%

       \[\leadsto \color{blue}{0 - \mathsf{expm1}\left(\frac{\log x}{n}\right)} \]

   if 0.619999999999999996 < x

   1. Initial program 71.2%

      \[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)} \]
   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 99.4

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

   5. Applied rewrites 99.4%

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

   7. Applied rewrites 99.4%

      \[\leadsto \frac{{x}^{\left(\mathsf{fma}\left(2, \frac{0.5}{n}, -1\right)\right)}}{n} \]
3. Recombined 2 regimes into one program.

4. Final simplification 91.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 0.62:\\ \;\;\;\;-\mathsf{expm1}\left(\frac{\log x}{n}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{{x}^{\left(\mathsf{fma}\left(2, \frac{0.5}{n}, -1\right)\right)}}{n}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 61.3% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 8.8 \cdot 10^{-76}:\\ \;\;\;\;\left(-\log x\right) \cdot {n}^{-1}\\ \mathbf{elif}\;x \leq 1:\\ \;\;\;\;\frac{\left(-n\right) \cdot \log x}{n \cdot n}\\ \mathbf{elif}\;x \leq 540000000000 \lor \neg \left(x \leq 10^{+85}\right):\\ \;\;\;\;\frac{\log 1}{n}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 - \frac{0.5}{x}}{n}}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (if (<= x 8.8e-76)
   (* (- (log x)) (pow n -1.0))
   (if (<= x 1.0)
     (/ (* (- n) (log x)) (* n n))
     (if (or (<= x 540000000000.0) (not (<= x 1e+85)))
       (/ (log 1.0) n)
       (/ (/ (- 1.0 (/ 0.5 x)) n) x)))))

C

double code(double x, double n) {
	double tmp;
	if (x <= 8.8e-76) {
		tmp = -log(x) * pow(n, -1.0);
	} else if (x <= 1.0) {
		tmp = (-n * log(x)) / (n * n);
	} else if ((x <= 540000000000.0) || !(x <= 1e+85)) {
		tmp = log(1.0) / n;
	} else {
		tmp = ((1.0 - (0.5 / x)) / n) / x;
	}
	return tmp;
}

Fortran

real(8) function code(x, n)
    real(8), intent (in) :: x
    real(8), intent (in) :: n
    real(8) :: tmp
    if (x <= 8.8d-76) then
        tmp = -log(x) * (n ** (-1.0d0))
    else if (x <= 1.0d0) then
        tmp = (-n * log(x)) / (n * n)
    else if ((x <= 540000000000.0d0) .or. (.not. (x <= 1d+85))) then
        tmp = log(1.0d0) / n
    else
        tmp = ((1.0d0 - (0.5d0 / x)) / n) / x
    end if
    code = tmp
end function

Java

public static double code(double x, double n) {
	double tmp;
	if (x <= 8.8e-76) {
		tmp = -Math.log(x) * Math.pow(n, -1.0);
	} else if (x <= 1.0) {
		tmp = (-n * Math.log(x)) / (n * n);
	} else if ((x <= 540000000000.0) || !(x <= 1e+85)) {
		tmp = Math.log(1.0) / n;
	} else {
		tmp = ((1.0 - (0.5 / x)) / n) / x;
	}
	return tmp;
}

Python

def code(x, n):
	tmp = 0
	if x <= 8.8e-76:
		tmp = -math.log(x) * math.pow(n, -1.0)
	elif x <= 1.0:
		tmp = (-n * math.log(x)) / (n * n)
	elif (x <= 540000000000.0) or not (x <= 1e+85):
		tmp = math.log(1.0) / n
	else:
		tmp = ((1.0 - (0.5 / x)) / n) / x
	return tmp

Julia

function code(x, n)
	tmp = 0.0
	if (x <= 8.8e-76)
		tmp = Float64(Float64(-log(x)) * (n ^ -1.0));
	elseif (x <= 1.0)
		tmp = Float64(Float64(Float64(-n) * log(x)) / Float64(n * n));
	elseif ((x <= 540000000000.0) || !(x <= 1e+85))
		tmp = Float64(log(1.0) / n);
	else
		tmp = Float64(Float64(Float64(1.0 - Float64(0.5 / x)) / n) / x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, n)
	tmp = 0.0;
	if (x <= 8.8e-76)
		tmp = -log(x) * (n ^ -1.0);
	elseif (x <= 1.0)
		tmp = (-n * log(x)) / (n * n);
	elseif ((x <= 540000000000.0) || ~((x <= 1e+85)))
		tmp = log(1.0) / n;
	else
		tmp = ((1.0 - (0.5 / x)) / n) / x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, n_] := If[LessEqual[x, 8.8e-76], N[((-N[Log[x], $MachinePrecision]) * N[Power[n, -1.0], $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 1.0], N[(N[((-n) * N[Log[x], $MachinePrecision]), $MachinePrecision] / N[(n * n), $MachinePrecision]), $MachinePrecision], If[Or[LessEqual[x, 540000000000.0], N[Not[LessEqual[x, 1e+85]], $MachinePrecision]], N[(N[Log[1.0], $MachinePrecision] / n), $MachinePrecision], N[(N[(N[(1.0 - N[(0.5 / x), $MachinePrecision]), $MachinePrecision] / n), $MachinePrecision] / x), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if x < 8.79999999999999997e-76

   1. Initial program 39.0%

      \[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)} \]
   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 56.7

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

   5. Applied rewrites 56.7%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 56.7%

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

   10. Applied rewrites 56.7%

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

   if 8.79999999999999997e-76 < x < 1

   1. Initial program 43.2%

      \[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)} \]
   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 43.2

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

   5. Applied rewrites 43.2%

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

   7. Applied rewrites 68.0%

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

   8. Taylor expanded in x around 0

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

   10. Applied rewrites 65.5%

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

   if 1 < x < 5.4e11 or 1e85 < x

   1. Initial program 83.3%

      \[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)} \]
   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 76.9

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

   5. Applied rewrites 76.9%

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

   7. Applied rewrites 77.0%

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

   8. Taylor expanded in x around inf

      \[\leadsto \frac{\log 1}{n} \]
   9. Step-by-step derivation

   10. Applied rewrites 83.3%

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

   if 5.4e11 < x < 1e85

   1. Initial program 33.4%

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

   3. Taylor expanded in x around inf

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

      1. lower-/.f64 N/A

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

   5. Applied rewrites 92.0%

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

   6. Taylor expanded in n around inf

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

   8. Applied rewrites 73.6%

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

4. Final simplification 68.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 8.8 \cdot 10^{-76}:\\ \;\;\;\;\left(-\log x\right) \cdot {n}^{-1}\\ \mathbf{elif}\;x \leq 1:\\ \;\;\;\;\frac{\left(-n\right) \cdot \log x}{n \cdot n}\\ \mathbf{elif}\;x \leq 540000000000 \lor \neg \left(x \leq 10^{+85}\right):\\ \;\;\;\;\frac{\log 1}{n}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 - \frac{0.5}{x}}{n}}{x}\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 60.5% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 2.25 \cdot 10^{-27}:\\ \;\;\;\;\frac{-\log x}{n}\\ \mathbf{elif}\;x \leq 1:\\ \;\;\;\;\frac{\frac{n}{x}}{n \cdot n}\\ \mathbf{elif}\;x \leq 540000000000 \lor \neg \left(x \leq 10^{+85}\right):\\ \;\;\;\;\frac{\log 1}{n}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 - \frac{0.5}{x}}{n}}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (if (<= x 2.25e-27)
   (/ (- (log x)) n)
   (if (<= x 1.0)
     (/ (/ n x) (* n n))
     (if (or (<= x 540000000000.0) (not (<= x 1e+85)))
       (/ (log 1.0) n)
       (/ (/ (- 1.0 (/ 0.5 x)) n) x)))))

C

double code(double x, double n) {
	double tmp;
	if (x <= 2.25e-27) {
		tmp = -log(x) / n;
	} else if (x <= 1.0) {
		tmp = (n / x) / (n * n);
	} else if ((x <= 540000000000.0) || !(x <= 1e+85)) {
		tmp = log(1.0) / n;
	} else {
		tmp = ((1.0 - (0.5 / x)) / n) / x;
	}
	return tmp;
}

Fortran

real(8) function code(x, n)
    real(8), intent (in) :: x
    real(8), intent (in) :: n
    real(8) :: tmp
    if (x <= 2.25d-27) then
        tmp = -log(x) / n
    else if (x <= 1.0d0) then
        tmp = (n / x) / (n * n)
    else if ((x <= 540000000000.0d0) .or. (.not. (x <= 1d+85))) then
        tmp = log(1.0d0) / n
    else
        tmp = ((1.0d0 - (0.5d0 / x)) / n) / x
    end if
    code = tmp
end function

Java

public static double code(double x, double n) {
	double tmp;
	if (x <= 2.25e-27) {
		tmp = -Math.log(x) / n;
	} else if (x <= 1.0) {
		tmp = (n / x) / (n * n);
	} else if ((x <= 540000000000.0) || !(x <= 1e+85)) {
		tmp = Math.log(1.0) / n;
	} else {
		tmp = ((1.0 - (0.5 / x)) / n) / x;
	}
	return tmp;
}

Python

def code(x, n):
	tmp = 0
	if x <= 2.25e-27:
		tmp = -math.log(x) / n
	elif x <= 1.0:
		tmp = (n / x) / (n * n)
	elif (x <= 540000000000.0) or not (x <= 1e+85):
		tmp = math.log(1.0) / n
	else:
		tmp = ((1.0 - (0.5 / x)) / n) / x
	return tmp

Julia

function code(x, n)
	tmp = 0.0
	if (x <= 2.25e-27)
		tmp = Float64(Float64(-log(x)) / n);
	elseif (x <= 1.0)
		tmp = Float64(Float64(n / x) / Float64(n * n));
	elseif ((x <= 540000000000.0) || !(x <= 1e+85))
		tmp = Float64(log(1.0) / n);
	else
		tmp = Float64(Float64(Float64(1.0 - Float64(0.5 / x)) / n) / x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, n)
	tmp = 0.0;
	if (x <= 2.25e-27)
		tmp = -log(x) / n;
	elseif (x <= 1.0)
		tmp = (n / x) / (n * n);
	elseif ((x <= 540000000000.0) || ~((x <= 1e+85)))
		tmp = log(1.0) / n;
	else
		tmp = ((1.0 - (0.5 / x)) / n) / x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, n_] := If[LessEqual[x, 2.25e-27], N[((-N[Log[x], $MachinePrecision]) / n), $MachinePrecision], If[LessEqual[x, 1.0], N[(N[(n / x), $MachinePrecision] / N[(n * n), $MachinePrecision]), $MachinePrecision], If[Or[LessEqual[x, 540000000000.0], N[Not[LessEqual[x, 1e+85]], $MachinePrecision]], N[(N[Log[1.0], $MachinePrecision] / n), $MachinePrecision], N[(N[(N[(1.0 - N[(0.5 / x), $MachinePrecision]), $MachinePrecision] / n), $MachinePrecision] / x), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if x < 2.2500000000000001e-27

   1. Initial program 38.1%

      \[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)} \]
   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 56.2

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

   5. Applied rewrites 56.2%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 56.2%

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

   if 2.2500000000000001e-27 < x < 1

   1. Initial program 64.1%

      \[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)} \]
   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 23.6

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

   5. Applied rewrites 23.6%

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

   7. Applied rewrites 79.6%

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

   8. Taylor expanded in x around inf

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

   10. Applied rewrites 63.5%

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

   if 1 < x < 5.4e11 or 1e85 < x

   1. Initial program 83.3%

      \[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)} \]
   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 76.9

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

   5. Applied rewrites 76.9%

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

   7. Applied rewrites 77.0%

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

   8. Taylor expanded in x around inf

      \[\leadsto \frac{\log 1}{n} \]
   9. Step-by-step derivation

   10. Applied rewrites 83.3%

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

   if 5.4e11 < x < 1e85

   1. Initial program 33.4%

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

   3. Taylor expanded in x around inf

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

      1. lower-/.f64 N/A

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

   5. Applied rewrites 92.0%

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

   6. Taylor expanded in n around inf

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

   8. Applied rewrites 73.6%

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

4. Final simplification 67.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 2.25 \cdot 10^{-27}:\\ \;\;\;\;\frac{-\log x}{n}\\ \mathbf{elif}\;x \leq 1:\\ \;\;\;\;\frac{\frac{n}{x}}{n \cdot n}\\ \mathbf{elif}\;x \leq 540000000000 \lor \neg \left(x \leq 10^{+85}\right):\\ \;\;\;\;\frac{\log 1}{n}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 - \frac{0.5}{x}}{n}}{x}\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 47.8% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;n \leq -2 \lor \neg \left(n \leq 7.5 \cdot 10^{-133}\right):\\ \;\;\;\;\frac{{\left(0.5 + x\right)}^{-1}}{n}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{n}{x}}{n \cdot n}\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (if (or (<= n -2.0) (not (<= n 7.5e-133)))
   (/ (pow (+ 0.5 x) -1.0) n)
   (/ (/ n x) (* n n))))

C

double code(double x, double n) {
	double tmp;
	if ((n <= -2.0) || !(n <= 7.5e-133)) {
		tmp = pow((0.5 + x), -1.0) / n;
	} else {
		tmp = (n / x) / (n * n);
	}
	return tmp;
}

Fortran

real(8) function code(x, n)
    real(8), intent (in) :: x
    real(8), intent (in) :: n
    real(8) :: tmp
    if ((n <= (-2.0d0)) .or. (.not. (n <= 7.5d-133))) then
        tmp = ((0.5d0 + x) ** (-1.0d0)) / n
    else
        tmp = (n / x) / (n * n)
    end if
    code = tmp
end function

Java

public static double code(double x, double n) {
	double tmp;
	if ((n <= -2.0) || !(n <= 7.5e-133)) {
		tmp = Math.pow((0.5 + x), -1.0) / n;
	} else {
		tmp = (n / x) / (n * n);
	}
	return tmp;
}

Python

def code(x, n):
	tmp = 0
	if (n <= -2.0) or not (n <= 7.5e-133):
		tmp = math.pow((0.5 + x), -1.0) / n
	else:
		tmp = (n / x) / (n * n)
	return tmp

Julia

function code(x, n)
	tmp = 0.0
	if ((n <= -2.0) || !(n <= 7.5e-133))
		tmp = Float64((Float64(0.5 + x) ^ -1.0) / n);
	else
		tmp = Float64(Float64(n / x) / Float64(n * n));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, n)
	tmp = 0.0;
	if ((n <= -2.0) || ~((n <= 7.5e-133)))
		tmp = ((0.5 + x) ^ -1.0) / n;
	else
		tmp = (n / x) / (n * n);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, n_] := If[Or[LessEqual[n, -2.0], N[Not[LessEqual[n, 7.5e-133]], $MachinePrecision]], N[(N[Power[N[(0.5 + x), $MachinePrecision], -1.0], $MachinePrecision] / n), $MachinePrecision], N[(N[(n / x), $MachinePrecision] / N[(n * n), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if n < -2 or 7.4999999999999999e-133 < n

   1. Initial program 38.5%

      \[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)} \]
   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 70.4

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

   5. Applied rewrites 70.4%

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

   7. Applied rewrites 70.7%

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

   9. Applied rewrites 70.4%

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

   10. Taylor expanded in x around inf

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

   12. Applied rewrites 50.8%

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

   if -2 < n < 7.4999999999999999e-133

   1. Initial program 80.9%

      \[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)} \]
   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 38.5

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

   5. Applied rewrites 38.5%

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

   7. Applied rewrites 50.7%

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

   8. Taylor expanded in x around inf

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

   10. Applied rewrites 42.5%

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

4. Final simplification 47.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq -2 \lor \neg \left(n \leq 7.5 \cdot 10^{-133}\right):\\ \;\;\;\;\frac{{\left(0.5 + x\right)}^{-1}}{n}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{n}{x}}{n \cdot n}\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 41.9% accurate, 1.9× speedup

Math

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

FPCore

(FPCore (x n)
 :precision binary64
 (if (<= n -2.0) (/ (pow (+ 0.5 x) -1.0) n) (/ (pow n -1.0) x)))

C

double code(double x, double n) {
	double tmp;
	if (n <= -2.0) {
		tmp = pow((0.5 + x), -1.0) / n;
	} else {
		tmp = pow(n, -1.0) / x;
	}
	return tmp;
}

Fortran

real(8) function code(x, n)
    real(8), intent (in) :: x
    real(8), intent (in) :: n
    real(8) :: tmp
    if (n <= (-2.0d0)) then
        tmp = ((0.5d0 + x) ** (-1.0d0)) / n
    else
        tmp = (n ** (-1.0d0)) / x
    end if
    code = tmp
end function

Java

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

Python

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

Julia

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

MATLAB

function tmp_2 = code(x, n)
	tmp = 0.0;
	if (n <= -2.0)
		tmp = ((0.5 + x) ^ -1.0) / n;
	else
		tmp = (n ^ -1.0) / x;
	end
	tmp_2 = tmp;
end

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if n < -2

   1. Initial program 32.4%

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

   3. Taylor expanded in n around inf

      \[\leadsto \color{blue}{\frac{\log \left(1 + x\right) - \log x}{n}} \]
   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 77.7

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

   5. Applied rewrites 77.7%

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

   7. Applied rewrites 78.1%

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

   9. Applied rewrites 77.6%

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

   10. Taylor expanded in x around inf

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

   12. Applied rewrites 53.5%

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

   if -2 < n

   1. Initial program 63.6%

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

   3. Taylor expanded in x around inf

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

      1. lower-/.f64 N/A

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

   5. Applied rewrites 33.3%

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

   6. Taylor expanded in n around inf

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

   8. Applied rewrites 20.7%

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

   9. Taylor expanded in x around inf

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

   11. Applied rewrites 35.3%

       \[\leadsto \frac{\frac{1}{n}}{x} \]
3. Recombined 2 regimes into one program.

4. Final simplification 41.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq -2:\\ \;\;\;\;\frac{{\left(0.5 + x\right)}^{-1}}{n}\\ \mathbf{else}:\\ \;\;\;\;\frac{{n}^{-1}}{x}\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 40.4% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 53.6%

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

3. Taylor expanded in x around inf

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

   1. lower-/.f64 N/A

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

5. Applied rewrites 37.8%

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

6. Taylor expanded in n around inf

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

8. Applied rewrites 29.1%

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

9. Taylor expanded in x around inf

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

11. Applied rewrites 39.4%

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

12. Final simplification 39.4%

    \[\leadsto \frac{{n}^{-1}}{x} \]
13. Add Preprocessing

Alternative 17: 48.4% accurate, 4.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 3.5 \cdot 10^{+208}:\\ \;\;\;\;\frac{\frac{\frac{\frac{0.3333333333333333}{x} - 0.5}{x} + 1}{x}}{n}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{-0.5}{x}}{n}}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (if (<= x 3.5e+208)
   (/ (/ (+ (/ (- (/ 0.3333333333333333 x) 0.5) x) 1.0) x) n)
   (/ (/ (/ -0.5 x) n) x)))

C

double code(double x, double n) {
	double tmp;
	if (x <= 3.5e+208) {
		tmp = (((((0.3333333333333333 / x) - 0.5) / x) + 1.0) / x) / n;
	} else {
		tmp = ((-0.5 / x) / n) / x;
	}
	return tmp;
}

Fortran

real(8) function code(x, n)
    real(8), intent (in) :: x
    real(8), intent (in) :: n
    real(8) :: tmp
    if (x <= 3.5d+208) then
        tmp = (((((0.3333333333333333d0 / x) - 0.5d0) / x) + 1.0d0) / x) / n
    else
        tmp = (((-0.5d0) / x) / n) / x
    end if
    code = tmp
end function

Java

public static double code(double x, double n) {
	double tmp;
	if (x <= 3.5e+208) {
		tmp = (((((0.3333333333333333 / x) - 0.5) / x) + 1.0) / x) / n;
	} else {
		tmp = ((-0.5 / x) / n) / x;
	}
	return tmp;
}

Python

def code(x, n):
	tmp = 0
	if x <= 3.5e+208:
		tmp = (((((0.3333333333333333 / x) - 0.5) / x) + 1.0) / x) / n
	else:
		tmp = ((-0.5 / x) / n) / x
	return tmp

Julia

function code(x, n)
	tmp = 0.0
	if (x <= 3.5e+208)
		tmp = Float64(Float64(Float64(Float64(Float64(Float64(0.3333333333333333 / x) - 0.5) / x) + 1.0) / x) / n);
	else
		tmp = Float64(Float64(Float64(-0.5 / x) / n) / x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, n)
	tmp = 0.0;
	if (x <= 3.5e+208)
		tmp = (((((0.3333333333333333 / x) - 0.5) / x) + 1.0) / x) / n;
	else
		tmp = ((-0.5 / x) / n) / x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, n_] := If[LessEqual[x, 3.5e+208], N[(N[(N[(N[(N[(N[(0.3333333333333333 / x), $MachinePrecision] - 0.5), $MachinePrecision] / x), $MachinePrecision] + 1.0), $MachinePrecision] / x), $MachinePrecision] / n), $MachinePrecision], N[(N[(N[(-0.5 / x), $MachinePrecision] / n), $MachinePrecision] / x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 3.50000000000000016e208

   1. Initial program 46.4%

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

   3. Taylor expanded in n around inf

      \[\leadsto \color{blue}{\frac{\log \left(1 + x\right) - \log x}{n}} \]
   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 52.8

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

   5. Applied rewrites 52.8%

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

   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 39.5%

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

   if 3.50000000000000016e208 < x

   1. Initial program 97.5%

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

   3. Taylor expanded in x around inf

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

      1. lower-/.f64 N/A

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

   5. Applied rewrites 88.9%

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

   6. Taylor expanded in n around inf

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

   8. Applied rewrites 70.9%

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

   9. Taylor expanded in x around 0

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

   11. Applied rewrites 90.5%

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

Reproduce

herbie shell --seed 2024318 
(FPCore (x n)
  :name "2nthrt (problem 3.4.6)"
  :precision binary64
  (- (pow (+ x 1.0) (/ 1.0 n)) (pow x (/ 1.0 n))))