2nthrt (problem 3.4.6)

Percentage Accurate: 53.0% → 92.0%

Time: 24.0s

Alternatives: 17

Speedup: 5.0×

• could not determine a ground truth

  1. x = 1.563576823241593e+264
  2. n = 8.40680479476109e-262

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: 53.0% 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: 92.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if x < 1

   1. Initial program 42.1%

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

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\left(1 + \frac{x}{n}\right) - e^{\frac{\log x}{n}}} \]
   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 91.5%

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

   if 1 < x

   1. Initial program 69.6%

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

   3. Taylor expanded in x around inf

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

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

Alternative 2: 81.6% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{if}\;\frac{1}{n} \leq -1 \cdot 10^{-22}:\\ \;\;\;\;\frac{{x}^{\left(\frac{1 - n}{n}\right)}}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 5 \cdot 10^{-65}:\\ \;\;\;\;\frac{\mathsf{log1p}\left(x\right) - \log x}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 0.004:\\ \;\;\;\;\frac{\frac{t\_0}{x}}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 10^{+232}:\\ \;\;\;\;\left(1 + \frac{x}{n}\right) - t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{n \cdot x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (let* ((t_0 (pow x (/ 1.0 n))))
   (if (<= (/ 1.0 n) -1e-22)
     (/ (pow x (/ (- 1.0 n) n)) n)
     (if (<= (/ 1.0 n) 5e-65)
       (/ (- (log1p x) (log x)) n)
       (if (<= (/ 1.0 n) 0.004)
         (/ (/ t_0 x) n)
         (if (<= (/ 1.0 n) 1e+232)
           (- (+ 1.0 (/ x n)) t_0)
           (/ 1.0 (* n x))))))))

C

double code(double x, double n) {
	double t_0 = pow(x, (1.0 / n));
	double tmp;
	if ((1.0 / n) <= -1e-22) {
		tmp = pow(x, ((1.0 - n) / n)) / n;
	} else if ((1.0 / n) <= 5e-65) {
		tmp = (log1p(x) - log(x)) / n;
	} else if ((1.0 / n) <= 0.004) {
		tmp = (t_0 / x) / n;
	} else if ((1.0 / n) <= 1e+232) {
		tmp = (1.0 + (x / n)) - t_0;
	} else {
		tmp = 1.0 / (n * x);
	}
	return tmp;
}

Java

public static double code(double x, double n) {
	double t_0 = Math.pow(x, (1.0 / n));
	double tmp;
	if ((1.0 / n) <= -1e-22) {
		tmp = Math.pow(x, ((1.0 - n) / n)) / n;
	} else if ((1.0 / n) <= 5e-65) {
		tmp = (Math.log1p(x) - Math.log(x)) / n;
	} else if ((1.0 / n) <= 0.004) {
		tmp = (t_0 / x) / n;
	} else if ((1.0 / n) <= 1e+232) {
		tmp = (1.0 + (x / n)) - t_0;
	} else {
		tmp = 1.0 / (n * x);
	}
	return tmp;
}

Python

def code(x, n):
	t_0 = math.pow(x, (1.0 / n))
	tmp = 0
	if (1.0 / n) <= -1e-22:
		tmp = math.pow(x, ((1.0 - n) / n)) / n
	elif (1.0 / n) <= 5e-65:
		tmp = (math.log1p(x) - math.log(x)) / n
	elif (1.0 / n) <= 0.004:
		tmp = (t_0 / x) / n
	elif (1.0 / n) <= 1e+232:
		tmp = (1.0 + (x / n)) - t_0
	else:
		tmp = 1.0 / (n * x)
	return tmp

Julia

function code(x, n)
	t_0 = x ^ Float64(1.0 / n)
	tmp = 0.0
	if (Float64(1.0 / n) <= -1e-22)
		tmp = Float64((x ^ Float64(Float64(1.0 - n) / n)) / n);
	elseif (Float64(1.0 / n) <= 5e-65)
		tmp = Float64(Float64(log1p(x) - log(x)) / n);
	elseif (Float64(1.0 / n) <= 0.004)
		tmp = Float64(Float64(t_0 / x) / n);
	elseif (Float64(1.0 / n) <= 1e+232)
		tmp = Float64(Float64(1.0 + Float64(x / n)) - t_0);
	else
		tmp = Float64(1.0 / Float64(n * x));
	end
	return tmp
end

Wolfram

code[x_, n_] := Block[{t$95$0 = N[Power[x, N[(1.0 / n), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[N[(1.0 / n), $MachinePrecision], -1e-22], N[(N[Power[x, N[(N[(1.0 - n), $MachinePrecision] / n), $MachinePrecision]], $MachinePrecision] / n), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 5e-65], N[(N[(N[Log[1 + x], $MachinePrecision] - N[Log[x], $MachinePrecision]), $MachinePrecision] / n), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 0.004], N[(N[(t$95$0 / x), $MachinePrecision] / n), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 1e+232], N[(N[(1.0 + N[(x / n), $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision], N[(1.0 / N[(n * x), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 5 regimes

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

   1. Initial program 97.2%

      \[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)} \]
   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 98.0

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

   5. Applied rewrites 98.0%

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

   7. Applied rewrites 98.0%

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

   8. Taylor expanded in n around 0

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

   10. Applied rewrites 98.0%

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

   if -1e-22 < (/.f64 #s(literal 1 binary64) n) < 4.99999999999999983e-65

   1. Initial program 30.8%

      \[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)} \]
   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 83.9

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

   5. Applied rewrites 83.9%

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

   if 4.99999999999999983e-65 < (/.f64 #s(literal 1 binary64) n) < 0.0040000000000000001

   1. Initial program 21.2%

      \[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)} \]
   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 76.2

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

   5. Applied rewrites 76.2%

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

   if 0.0040000000000000001 < (/.f64 #s(literal 1 binary64) n) < 1.00000000000000006e232

   1. Initial program 78.7%

      \[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)} \]
   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 81.1

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

   5. Applied rewrites 81.1%

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

   if 1.00000000000000006e232 < (/.f64 #s(literal 1 binary64) n)

   1. Initial program 3.1%

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

   3. Taylor expanded in n around inf

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

      1. lower-/.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lower-log1p.f64 N/A

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

      4. lower-log.f64 8.3

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

   5. Applied rewrites 8.3%

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

   7. Applied rewrites 8.3%

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

   8. Taylor expanded in x around inf

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

   10. Applied rewrites 100.0%

       \[\leadsto \frac{1}{n \cdot \color{blue}{x}} \]
3. Recombined 5 regimes into one program.

4. Final simplification 87.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{1}{n} \leq -1 \cdot 10^{-22}:\\ \;\;\;\;\frac{{x}^{\left(\frac{1 - n}{n}\right)}}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 5 \cdot 10^{-65}:\\ \;\;\;\;\frac{\mathsf{log1p}\left(x\right) - \log x}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 0.004:\\ \;\;\;\;\frac{\frac{{x}^{\left(\frac{1}{n}\right)}}{x}}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 10^{+232}:\\ \;\;\;\;\left(1 + \frac{x}{n}\right) - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{n \cdot x}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 81.7% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{if}\;\frac{1}{n} \leq -1 \cdot 10^{-22}:\\ \;\;\;\;\frac{{x}^{\left(\frac{1 - n}{n}\right)}}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 5 \cdot 10^{-65}:\\ \;\;\;\;\frac{\log \left(\frac{x - -1}{x}\right)}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 0.004:\\ \;\;\;\;\frac{\frac{t\_0}{x}}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 10^{+232}:\\ \;\;\;\;\left(1 + \frac{x}{n}\right) - t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{n \cdot x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (let* ((t_0 (pow x (/ 1.0 n))))
   (if (<= (/ 1.0 n) -1e-22)
     (/ (pow x (/ (- 1.0 n) n)) n)
     (if (<= (/ 1.0 n) 5e-65)
       (/ (log (/ (- x -1.0) x)) n)
       (if (<= (/ 1.0 n) 0.004)
         (/ (/ t_0 x) n)
         (if (<= (/ 1.0 n) 1e+232)
           (- (+ 1.0 (/ x n)) t_0)
           (/ 1.0 (* n x))))))))

C

double code(double x, double n) {
	double t_0 = pow(x, (1.0 / n));
	double tmp;
	if ((1.0 / n) <= -1e-22) {
		tmp = pow(x, ((1.0 - n) / n)) / n;
	} else if ((1.0 / n) <= 5e-65) {
		tmp = log(((x - -1.0) / x)) / n;
	} else if ((1.0 / n) <= 0.004) {
		tmp = (t_0 / x) / n;
	} else if ((1.0 / n) <= 1e+232) {
		tmp = (1.0 + (x / n)) - t_0;
	} else {
		tmp = 1.0 / (n * x);
	}
	return tmp;
}

Fortran

real(8) function code(x, n)
    real(8), intent (in) :: x
    real(8), intent (in) :: n
    real(8) :: t_0
    real(8) :: tmp
    t_0 = x ** (1.0d0 / n)
    if ((1.0d0 / n) <= (-1d-22)) then
        tmp = (x ** ((1.0d0 - n) / n)) / n
    else if ((1.0d0 / n) <= 5d-65) then
        tmp = log(((x - (-1.0d0)) / x)) / n
    else if ((1.0d0 / n) <= 0.004d0) then
        tmp = (t_0 / x) / n
    else if ((1.0d0 / n) <= 1d+232) then
        tmp = (1.0d0 + (x / n)) - t_0
    else
        tmp = 1.0d0 / (n * x)
    end if
    code = tmp
end function

Java

public static double code(double x, double n) {
	double t_0 = Math.pow(x, (1.0 / n));
	double tmp;
	if ((1.0 / n) <= -1e-22) {
		tmp = Math.pow(x, ((1.0 - n) / n)) / n;
	} else if ((1.0 / n) <= 5e-65) {
		tmp = Math.log(((x - -1.0) / x)) / n;
	} else if ((1.0 / n) <= 0.004) {
		tmp = (t_0 / x) / n;
	} else if ((1.0 / n) <= 1e+232) {
		tmp = (1.0 + (x / n)) - t_0;
	} else {
		tmp = 1.0 / (n * x);
	}
	return tmp;
}

Python

def code(x, n):
	t_0 = math.pow(x, (1.0 / n))
	tmp = 0
	if (1.0 / n) <= -1e-22:
		tmp = math.pow(x, ((1.0 - n) / n)) / n
	elif (1.0 / n) <= 5e-65:
		tmp = math.log(((x - -1.0) / x)) / n
	elif (1.0 / n) <= 0.004:
		tmp = (t_0 / x) / n
	elif (1.0 / n) <= 1e+232:
		tmp = (1.0 + (x / n)) - t_0
	else:
		tmp = 1.0 / (n * x)
	return tmp

Julia

function code(x, n)
	t_0 = x ^ Float64(1.0 / n)
	tmp = 0.0
	if (Float64(1.0 / n) <= -1e-22)
		tmp = Float64((x ^ Float64(Float64(1.0 - n) / n)) / n);
	elseif (Float64(1.0 / n) <= 5e-65)
		tmp = Float64(log(Float64(Float64(x - -1.0) / x)) / n);
	elseif (Float64(1.0 / n) <= 0.004)
		tmp = Float64(Float64(t_0 / x) / n);
	elseif (Float64(1.0 / n) <= 1e+232)
		tmp = Float64(Float64(1.0 + Float64(x / n)) - t_0);
	else
		tmp = Float64(1.0 / Float64(n * x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, n)
	t_0 = x ^ (1.0 / n);
	tmp = 0.0;
	if ((1.0 / n) <= -1e-22)
		tmp = (x ^ ((1.0 - n) / n)) / n;
	elseif ((1.0 / n) <= 5e-65)
		tmp = log(((x - -1.0) / x)) / n;
	elseif ((1.0 / n) <= 0.004)
		tmp = (t_0 / x) / n;
	elseif ((1.0 / n) <= 1e+232)
		tmp = (1.0 + (x / n)) - t_0;
	else
		tmp = 1.0 / (n * x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, n_] := Block[{t$95$0 = N[Power[x, N[(1.0 / n), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[N[(1.0 / n), $MachinePrecision], -1e-22], N[(N[Power[x, N[(N[(1.0 - n), $MachinePrecision] / n), $MachinePrecision]], $MachinePrecision] / n), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 5e-65], N[(N[Log[N[(N[(x - -1.0), $MachinePrecision] / x), $MachinePrecision]], $MachinePrecision] / n), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 0.004], N[(N[(t$95$0 / x), $MachinePrecision] / n), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 1e+232], N[(N[(1.0 + N[(x / n), $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision], N[(1.0 / N[(n * x), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 5 regimes

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

   1. Initial program 97.2%

      \[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)} \]
   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 98.0

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

   5. Applied rewrites 98.0%

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

   7. Applied rewrites 98.0%

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

   8. Taylor expanded in n around 0

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

   10. Applied rewrites 98.0%

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

   if -1e-22 < (/.f64 #s(literal 1 binary64) n) < 4.99999999999999983e-65

   1. Initial program 30.8%

      \[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)} \]
   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 83.9

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

   5. Applied rewrites 83.9%

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

   7. Applied rewrites 83.9%

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

   if 4.99999999999999983e-65 < (/.f64 #s(literal 1 binary64) n) < 0.0040000000000000001

   1. Initial program 21.2%

      \[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)} \]
   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 76.2

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

   5. Applied rewrites 76.2%

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

   if 0.0040000000000000001 < (/.f64 #s(literal 1 binary64) n) < 1.00000000000000006e232

   1. Initial program 78.7%

      \[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)} \]
   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 81.1

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

   5. Applied rewrites 81.1%

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

   if 1.00000000000000006e232 < (/.f64 #s(literal 1 binary64) n)

   1. Initial program 3.1%

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

   3. Taylor expanded in n around inf

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

      1. lower-/.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lower-log1p.f64 N/A

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

      4. lower-log.f64 8.3

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

   5. Applied rewrites 8.3%

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

   7. Applied rewrites 8.3%

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

   8. Taylor expanded in x around inf

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

   10. Applied rewrites 100.0%

       \[\leadsto \frac{1}{n \cdot \color{blue}{x}} \]
3. Recombined 5 regimes into one program.

4. Final simplification 87.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{1}{n} \leq -1 \cdot 10^{-22}:\\ \;\;\;\;\frac{{x}^{\left(\frac{1 - n}{n}\right)}}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 5 \cdot 10^{-65}:\\ \;\;\;\;\frac{\log \left(\frac{x - -1}{x}\right)}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 0.004:\\ \;\;\;\;\frac{\frac{{x}^{\left(\frac{1}{n}\right)}}{x}}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 10^{+232}:\\ \;\;\;\;\left(1 + \frac{x}{n}\right) - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{n \cdot x}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 81.7% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{{x}^{\left(\frac{1 - n}{n}\right)}}{n}\\ \mathbf{if}\;\frac{1}{n} \leq -1 \cdot 10^{-22}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;\frac{1}{n} \leq 5 \cdot 10^{-65}:\\ \;\;\;\;\frac{\log \left(\frac{x - -1}{x}\right)}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 0.0001:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;\frac{1}{n} \leq 10^{+232}:\\ \;\;\;\;\left(1 + \frac{x}{n}\right) - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{n \cdot x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (let* ((t_0 (/ (pow x (/ (- 1.0 n) n)) n)))
   (if (<= (/ 1.0 n) -1e-22)
     t_0
     (if (<= (/ 1.0 n) 5e-65)
       (/ (log (/ (- x -1.0) x)) n)
       (if (<= (/ 1.0 n) 0.0001)
         t_0
         (if (<= (/ 1.0 n) 1e+232)
           (- (+ 1.0 (/ x n)) (pow x (/ 1.0 n)))
           (/ 1.0 (* n x))))))))

C

double code(double x, double n) {
	double t_0 = pow(x, ((1.0 - n) / n)) / n;
	double tmp;
	if ((1.0 / n) <= -1e-22) {
		tmp = t_0;
	} else if ((1.0 / n) <= 5e-65) {
		tmp = log(((x - -1.0) / x)) / n;
	} else if ((1.0 / n) <= 0.0001) {
		tmp = t_0;
	} else if ((1.0 / n) <= 1e+232) {
		tmp = (1.0 + (x / n)) - pow(x, (1.0 / n));
	} else {
		tmp = 1.0 / (n * x);
	}
	return tmp;
}

Fortran

real(8) function code(x, n)
    real(8), intent (in) :: x
    real(8), intent (in) :: n
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (x ** ((1.0d0 - n) / n)) / n
    if ((1.0d0 / n) <= (-1d-22)) then
        tmp = t_0
    else if ((1.0d0 / n) <= 5d-65) then
        tmp = log(((x - (-1.0d0)) / x)) / n
    else if ((1.0d0 / n) <= 0.0001d0) then
        tmp = t_0
    else if ((1.0d0 / n) <= 1d+232) then
        tmp = (1.0d0 + (x / n)) - (x ** (1.0d0 / n))
    else
        tmp = 1.0d0 / (n * x)
    end if
    code = tmp
end function

Java

public static double code(double x, double n) {
	double t_0 = Math.pow(x, ((1.0 - n) / n)) / n;
	double tmp;
	if ((1.0 / n) <= -1e-22) {
		tmp = t_0;
	} else if ((1.0 / n) <= 5e-65) {
		tmp = Math.log(((x - -1.0) / x)) / n;
	} else if ((1.0 / n) <= 0.0001) {
		tmp = t_0;
	} else if ((1.0 / n) <= 1e+232) {
		tmp = (1.0 + (x / n)) - Math.pow(x, (1.0 / n));
	} else {
		tmp = 1.0 / (n * x);
	}
	return tmp;
}

Python

def code(x, n):
	t_0 = math.pow(x, ((1.0 - n) / n)) / n
	tmp = 0
	if (1.0 / n) <= -1e-22:
		tmp = t_0
	elif (1.0 / n) <= 5e-65:
		tmp = math.log(((x - -1.0) / x)) / n
	elif (1.0 / n) <= 0.0001:
		tmp = t_0
	elif (1.0 / n) <= 1e+232:
		tmp = (1.0 + (x / n)) - math.pow(x, (1.0 / n))
	else:
		tmp = 1.0 / (n * x)
	return tmp

Julia

function code(x, n)
	t_0 = Float64((x ^ Float64(Float64(1.0 - n) / n)) / n)
	tmp = 0.0
	if (Float64(1.0 / n) <= -1e-22)
		tmp = t_0;
	elseif (Float64(1.0 / n) <= 5e-65)
		tmp = Float64(log(Float64(Float64(x - -1.0) / x)) / n);
	elseif (Float64(1.0 / n) <= 0.0001)
		tmp = t_0;
	elseif (Float64(1.0 / n) <= 1e+232)
		tmp = Float64(Float64(1.0 + Float64(x / n)) - (x ^ Float64(1.0 / n)));
	else
		tmp = Float64(1.0 / Float64(n * x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, n)
	t_0 = (x ^ ((1.0 - n) / n)) / n;
	tmp = 0.0;
	if ((1.0 / n) <= -1e-22)
		tmp = t_0;
	elseif ((1.0 / n) <= 5e-65)
		tmp = log(((x - -1.0) / x)) / n;
	elseif ((1.0 / n) <= 0.0001)
		tmp = t_0;
	elseif ((1.0 / n) <= 1e+232)
		tmp = (1.0 + (x / n)) - (x ^ (1.0 / n));
	else
		tmp = 1.0 / (n * x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, n_] := Block[{t$95$0 = N[(N[Power[x, N[(N[(1.0 - n), $MachinePrecision] / n), $MachinePrecision]], $MachinePrecision] / n), $MachinePrecision]}, If[LessEqual[N[(1.0 / n), $MachinePrecision], -1e-22], t$95$0, If[LessEqual[N[(1.0 / n), $MachinePrecision], 5e-65], N[(N[Log[N[(N[(x - -1.0), $MachinePrecision] / x), $MachinePrecision]], $MachinePrecision] / n), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 0.0001], t$95$0, If[LessEqual[N[(1.0 / n), $MachinePrecision], 1e+232], N[(N[(1.0 + N[(x / n), $MachinePrecision]), $MachinePrecision] - N[Power[x, N[(1.0 / n), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(n * x), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if (/.f64 #s(literal 1 binary64) n) < -1e-22 or 4.99999999999999983e-65 < (/.f64 #s(literal 1 binary64) n) < 1.00000000000000005e-4

   1. Initial program 84.6%

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

   3. Taylor expanded in x around inf

      \[\leadsto \color{blue}{\frac{e^{-1 \cdot \frac{\log \left(\frac{1}{x}\right)}{n}}}{n \cdot x}} \]
   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 95.0

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

   5. Applied rewrites 95.0%

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

   7. Applied rewrites 94.8%

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

   8. Taylor expanded in n around 0

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

   10. Applied rewrites 94.8%

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

   if -1e-22 < (/.f64 #s(literal 1 binary64) n) < 4.99999999999999983e-65

   1. Initial program 30.8%

      \[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)} \]
   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 83.9

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

   5. Applied rewrites 83.9%

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

   7. Applied rewrites 83.9%

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

   if 1.00000000000000005e-4 < (/.f64 #s(literal 1 binary64) n) < 1.00000000000000006e232

   1. Initial program 76.8%

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

   3. Taylor expanded in x around 0

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

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

   5. Applied rewrites 79.0%

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

   if 1.00000000000000006e232 < (/.f64 #s(literal 1 binary64) n)

   1. Initial program 3.1%

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

   3. Taylor expanded in n around inf

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

      1. lower-/.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lower-log1p.f64 N/A

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

      4. lower-log.f64 8.3

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

   5. Applied rewrites 8.3%

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

   7. Applied rewrites 8.3%

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

   8. Taylor expanded in x around inf

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

   10. Applied rewrites 100.0%

       \[\leadsto \frac{1}{n \cdot \color{blue}{x}} \]
3. Recombined 4 regimes into one program.

4. Final simplification 87.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{1}{n} \leq -1 \cdot 10^{-22}:\\ \;\;\;\;\frac{{x}^{\left(\frac{1 - n}{n}\right)}}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 5 \cdot 10^{-65}:\\ \;\;\;\;\frac{\log \left(\frac{x - -1}{x}\right)}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 0.0001:\\ \;\;\;\;\frac{{x}^{\left(\frac{1 - n}{n}\right)}}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 10^{+232}:\\ \;\;\;\;\left(1 + \frac{x}{n}\right) - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{n \cdot x}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 81.5% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{{x}^{\left(\frac{1 - n}{n}\right)}}{n}\\ \mathbf{if}\;\frac{1}{n} \leq -1 \cdot 10^{-22}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;\frac{1}{n} \leq 5 \cdot 10^{-65}:\\ \;\;\;\;\frac{\log \left(\frac{x - -1}{x}\right)}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 0.0001:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;\frac{1}{n} \leq 10^{+232}:\\ \;\;\;\;1 - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{n \cdot x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (let* ((t_0 (/ (pow x (/ (- 1.0 n) n)) n)))
   (if (<= (/ 1.0 n) -1e-22)
     t_0
     (if (<= (/ 1.0 n) 5e-65)
       (/ (log (/ (- x -1.0) x)) n)
       (if (<= (/ 1.0 n) 0.0001)
         t_0
         (if (<= (/ 1.0 n) 1e+232)
           (- 1.0 (pow x (/ 1.0 n)))
           (/ 1.0 (* n x))))))))

C

double code(double x, double n) {
	double t_0 = pow(x, ((1.0 - n) / n)) / n;
	double tmp;
	if ((1.0 / n) <= -1e-22) {
		tmp = t_0;
	} else if ((1.0 / n) <= 5e-65) {
		tmp = log(((x - -1.0) / x)) / n;
	} else if ((1.0 / n) <= 0.0001) {
		tmp = t_0;
	} else if ((1.0 / n) <= 1e+232) {
		tmp = 1.0 - pow(x, (1.0 / n));
	} else {
		tmp = 1.0 / (n * x);
	}
	return tmp;
}

Fortran

real(8) function code(x, n)
    real(8), intent (in) :: x
    real(8), intent (in) :: n
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (x ** ((1.0d0 - n) / n)) / n
    if ((1.0d0 / n) <= (-1d-22)) then
        tmp = t_0
    else if ((1.0d0 / n) <= 5d-65) then
        tmp = log(((x - (-1.0d0)) / x)) / n
    else if ((1.0d0 / n) <= 0.0001d0) then
        tmp = t_0
    else if ((1.0d0 / n) <= 1d+232) then
        tmp = 1.0d0 - (x ** (1.0d0 / n))
    else
        tmp = 1.0d0 / (n * x)
    end if
    code = tmp
end function

Java

public static double code(double x, double n) {
	double t_0 = Math.pow(x, ((1.0 - n) / n)) / n;
	double tmp;
	if ((1.0 / n) <= -1e-22) {
		tmp = t_0;
	} else if ((1.0 / n) <= 5e-65) {
		tmp = Math.log(((x - -1.0) / x)) / n;
	} else if ((1.0 / n) <= 0.0001) {
		tmp = t_0;
	} else if ((1.0 / n) <= 1e+232) {
		tmp = 1.0 - Math.pow(x, (1.0 / n));
	} else {
		tmp = 1.0 / (n * x);
	}
	return tmp;
}

Python

def code(x, n):
	t_0 = math.pow(x, ((1.0 - n) / n)) / n
	tmp = 0
	if (1.0 / n) <= -1e-22:
		tmp = t_0
	elif (1.0 / n) <= 5e-65:
		tmp = math.log(((x - -1.0) / x)) / n
	elif (1.0 / n) <= 0.0001:
		tmp = t_0
	elif (1.0 / n) <= 1e+232:
		tmp = 1.0 - math.pow(x, (1.0 / n))
	else:
		tmp = 1.0 / (n * x)
	return tmp

Julia

function code(x, n)
	t_0 = Float64((x ^ Float64(Float64(1.0 - n) / n)) / n)
	tmp = 0.0
	if (Float64(1.0 / n) <= -1e-22)
		tmp = t_0;
	elseif (Float64(1.0 / n) <= 5e-65)
		tmp = Float64(log(Float64(Float64(x - -1.0) / x)) / n);
	elseif (Float64(1.0 / n) <= 0.0001)
		tmp = t_0;
	elseif (Float64(1.0 / n) <= 1e+232)
		tmp = Float64(1.0 - (x ^ Float64(1.0 / n)));
	else
		tmp = Float64(1.0 / Float64(n * x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, n)
	t_0 = (x ^ ((1.0 - n) / n)) / n;
	tmp = 0.0;
	if ((1.0 / n) <= -1e-22)
		tmp = t_0;
	elseif ((1.0 / n) <= 5e-65)
		tmp = log(((x - -1.0) / x)) / n;
	elseif ((1.0 / n) <= 0.0001)
		tmp = t_0;
	elseif ((1.0 / n) <= 1e+232)
		tmp = 1.0 - (x ^ (1.0 / n));
	else
		tmp = 1.0 / (n * x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, n_] := Block[{t$95$0 = N[(N[Power[x, N[(N[(1.0 - n), $MachinePrecision] / n), $MachinePrecision]], $MachinePrecision] / n), $MachinePrecision]}, If[LessEqual[N[(1.0 / n), $MachinePrecision], -1e-22], t$95$0, If[LessEqual[N[(1.0 / n), $MachinePrecision], 5e-65], N[(N[Log[N[(N[(x - -1.0), $MachinePrecision] / x), $MachinePrecision]], $MachinePrecision] / n), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 0.0001], t$95$0, If[LessEqual[N[(1.0 / n), $MachinePrecision], 1e+232], N[(1.0 - N[Power[x, N[(1.0 / n), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(n * x), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if (/.f64 #s(literal 1 binary64) n) < -1e-22 or 4.99999999999999983e-65 < (/.f64 #s(literal 1 binary64) n) < 1.00000000000000005e-4

   1. Initial program 84.6%

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

   3. Taylor expanded in x around inf

      \[\leadsto \color{blue}{\frac{e^{-1 \cdot \frac{\log \left(\frac{1}{x}\right)}{n}}}{n \cdot x}} \]
   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 95.0

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

   5. Applied rewrites 95.0%

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

   7. Applied rewrites 94.8%

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

   8. Taylor expanded in n around 0

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

   10. Applied rewrites 94.8%

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

   if -1e-22 < (/.f64 #s(literal 1 binary64) n) < 4.99999999999999983e-65

   1. Initial program 30.8%

      \[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)} \]
   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 83.9

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

   5. Applied rewrites 83.9%

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

   7. Applied rewrites 83.9%

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

   if 1.00000000000000005e-4 < (/.f64 #s(literal 1 binary64) n) < 1.00000000000000006e232

   1. Initial program 76.8%

      \[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)} \]
   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 76.7%

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

   if 1.00000000000000006e232 < (/.f64 #s(literal 1 binary64) n)

   1. Initial program 3.1%

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

   3. Taylor expanded in n around inf

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

      1. lower-/.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lower-log1p.f64 N/A

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

      4. lower-log.f64 8.3

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

   5. Applied rewrites 8.3%

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

   7. Applied rewrites 8.3%

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

   8. Taylor expanded in x around inf

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

   10. Applied rewrites 100.0%

       \[\leadsto \frac{1}{n \cdot \color{blue}{x}} \]
3. Recombined 4 regimes into one program.

4. Final simplification 87.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{1}{n} \leq -1 \cdot 10^{-22}:\\ \;\;\;\;\frac{{x}^{\left(\frac{1 - n}{n}\right)}}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 5 \cdot 10^{-65}:\\ \;\;\;\;\frac{\log \left(\frac{x - -1}{x}\right)}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 0.0001:\\ \;\;\;\;\frac{{x}^{\left(\frac{1 - n}{n}\right)}}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 10^{+232}:\\ \;\;\;\;1 - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{n \cdot x}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 80.7% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{if}\;\frac{1}{n} \leq -1 \cdot 10^{-22}:\\ \;\;\;\;\frac{t\_0}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 5 \cdot 10^{-65}:\\ \;\;\;\;\frac{\log \left(\frac{x - -1}{x}\right)}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 5 \cdot 10^{-13}:\\ \;\;\;\;\frac{\frac{1}{n}}{\mathsf{fma}\left(\frac{0.5}{x}, x, x\right)}\\ \mathbf{elif}\;\frac{1}{n} \leq 10^{+232}:\\ \;\;\;\;1 - t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{n \cdot x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (let* ((t_0 (pow x (/ 1.0 n))))
   (if (<= (/ 1.0 n) -1e-22)
     (/ t_0 n)
     (if (<= (/ 1.0 n) 5e-65)
       (/ (log (/ (- x -1.0) x)) n)
       (if (<= (/ 1.0 n) 5e-13)
         (/ (/ 1.0 n) (fma (/ 0.5 x) x x))
         (if (<= (/ 1.0 n) 1e+232) (- 1.0 t_0) (/ 1.0 (* n x))))))))

C

double code(double x, double n) {
	double t_0 = pow(x, (1.0 / n));
	double tmp;
	if ((1.0 / n) <= -1e-22) {
		tmp = t_0 / n;
	} else if ((1.0 / n) <= 5e-65) {
		tmp = log(((x - -1.0) / x)) / n;
	} else if ((1.0 / n) <= 5e-13) {
		tmp = (1.0 / n) / fma((0.5 / x), x, x);
	} else if ((1.0 / n) <= 1e+232) {
		tmp = 1.0 - t_0;
	} else {
		tmp = 1.0 / (n * x);
	}
	return tmp;
}

Julia

function code(x, n)
	t_0 = x ^ Float64(1.0 / n)
	tmp = 0.0
	if (Float64(1.0 / n) <= -1e-22)
		tmp = Float64(t_0 / n);
	elseif (Float64(1.0 / n) <= 5e-65)
		tmp = Float64(log(Float64(Float64(x - -1.0) / x)) / n);
	elseif (Float64(1.0 / n) <= 5e-13)
		tmp = Float64(Float64(1.0 / n) / fma(Float64(0.5 / x), x, x));
	elseif (Float64(1.0 / n) <= 1e+232)
		tmp = Float64(1.0 - t_0);
	else
		tmp = Float64(1.0 / Float64(n * x));
	end
	return tmp
end

Wolfram

code[x_, n_] := Block[{t$95$0 = N[Power[x, N[(1.0 / n), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[N[(1.0 / n), $MachinePrecision], -1e-22], N[(t$95$0 / n), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 5e-65], N[(N[Log[N[(N[(x - -1.0), $MachinePrecision] / x), $MachinePrecision]], $MachinePrecision] / n), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 5e-13], N[(N[(1.0 / n), $MachinePrecision] / N[(N[(0.5 / x), $MachinePrecision] * x + x), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 1e+232], N[(1.0 - t$95$0), $MachinePrecision], N[(1.0 / N[(n * x), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 5 regimes

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

   1. Initial program 97.2%

      \[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)} \]
   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 98.0

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

   5. Applied rewrites 98.0%

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

   7. Applied rewrites 98.0%

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

   8. Taylor expanded in n around 0

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

   10. Applied rewrites 95.2%

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

   if -1e-22 < (/.f64 #s(literal 1 binary64) n) < 4.99999999999999983e-65

   1. Initial program 30.8%

      \[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)} \]
   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 83.9

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

   5. Applied rewrites 83.9%

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

   7. Applied rewrites 83.9%

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

   if 4.99999999999999983e-65 < (/.f64 #s(literal 1 binary64) n) < 4.9999999999999999e-13

   1. Initial program 13.5%

      \[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)} \]
   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 30.9

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

   5. Applied rewrites 30.9%

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

   7. Applied rewrites 30.9%

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

   9. Applied rewrites 30.9%

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

   10. Taylor expanded in x around inf

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

   12. Applied rewrites 81.6%

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

   if 4.9999999999999999e-13 < (/.f64 #s(literal 1 binary64) n) < 1.00000000000000006e232

   1. Initial program 72.4%

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

   3. Taylor expanded in x around 0

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

   5. Applied rewrites 72.2%

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

   if 1.00000000000000006e232 < (/.f64 #s(literal 1 binary64) n)

   1. Initial program 3.1%

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

   3. Taylor expanded in n around inf

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

      1. lower-/.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lower-log1p.f64 N/A

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

      4. lower-log.f64 8.3

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

   5. Applied rewrites 8.3%

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

   7. Applied rewrites 8.3%

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

   8. Taylor expanded in x around inf

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

   10. Applied rewrites 100.0%

       \[\leadsto \frac{1}{n \cdot \color{blue}{x}} \]
3. Recombined 5 regimes into one program.

4. Final simplification 86.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{1}{n} \leq -1 \cdot 10^{-22}:\\ \;\;\;\;\frac{{x}^{\left(\frac{1}{n}\right)}}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 5 \cdot 10^{-65}:\\ \;\;\;\;\frac{\log \left(\frac{x - -1}{x}\right)}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 5 \cdot 10^{-13}:\\ \;\;\;\;\frac{\frac{1}{n}}{\mathsf{fma}\left(\frac{0.5}{x}, x, x\right)}\\ \mathbf{elif}\;\frac{1}{n} \leq 10^{+232}:\\ \;\;\;\;1 - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{n \cdot x}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 66.9% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{if}\;\frac{1}{n} \leq -1 \cdot 10^{-22}:\\ \;\;\;\;\frac{t\_0}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 5 \cdot 10^{-65}:\\ \;\;\;\;\frac{-\log x}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 5 \cdot 10^{-13}:\\ \;\;\;\;\frac{\frac{1}{n}}{\mathsf{fma}\left(\frac{0.5}{x}, x, x\right)}\\ \mathbf{elif}\;\frac{1}{n} \leq 10^{+232}:\\ \;\;\;\;1 - t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{n \cdot x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (let* ((t_0 (pow x (/ 1.0 n))))
   (if (<= (/ 1.0 n) -1e-22)
     (/ t_0 n)
     (if (<= (/ 1.0 n) 5e-65)
       (/ (- (log x)) n)
       (if (<= (/ 1.0 n) 5e-13)
         (/ (/ 1.0 n) (fma (/ 0.5 x) x x))
         (if (<= (/ 1.0 n) 1e+232) (- 1.0 t_0) (/ 1.0 (* n x))))))))

C

double code(double x, double n) {
	double t_0 = pow(x, (1.0 / n));
	double tmp;
	if ((1.0 / n) <= -1e-22) {
		tmp = t_0 / n;
	} else if ((1.0 / n) <= 5e-65) {
		tmp = -log(x) / n;
	} else if ((1.0 / n) <= 5e-13) {
		tmp = (1.0 / n) / fma((0.5 / x), x, x);
	} else if ((1.0 / n) <= 1e+232) {
		tmp = 1.0 - t_0;
	} else {
		tmp = 1.0 / (n * x);
	}
	return tmp;
}

Julia

function code(x, n)
	t_0 = x ^ Float64(1.0 / n)
	tmp = 0.0
	if (Float64(1.0 / n) <= -1e-22)
		tmp = Float64(t_0 / n);
	elseif (Float64(1.0 / n) <= 5e-65)
		tmp = Float64(Float64(-log(x)) / n);
	elseif (Float64(1.0 / n) <= 5e-13)
		tmp = Float64(Float64(1.0 / n) / fma(Float64(0.5 / x), x, x));
	elseif (Float64(1.0 / n) <= 1e+232)
		tmp = Float64(1.0 - t_0);
	else
		tmp = Float64(1.0 / Float64(n * x));
	end
	return tmp
end

Wolfram

code[x_, n_] := Block[{t$95$0 = N[Power[x, N[(1.0 / n), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[N[(1.0 / n), $MachinePrecision], -1e-22], N[(t$95$0 / n), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 5e-65], N[((-N[Log[x], $MachinePrecision]) / n), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 5e-13], N[(N[(1.0 / n), $MachinePrecision] / N[(N[(0.5 / x), $MachinePrecision] * x + x), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 1e+232], N[(1.0 - t$95$0), $MachinePrecision], N[(1.0 / N[(n * x), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 5 regimes

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

   1. Initial program 97.2%

      \[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)} \]
   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 98.0

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

   5. Applied rewrites 98.0%

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

   7. Applied rewrites 98.0%

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

   8. Taylor expanded in n around 0

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

   10. Applied rewrites 95.2%

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

   if -1e-22 < (/.f64 #s(literal 1 binary64) n) < 4.99999999999999983e-65

   1. Initial program 30.8%

      \[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)} \]
   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 83.9

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

   5. Applied rewrites 83.9%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 58.3%

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

   if 4.99999999999999983e-65 < (/.f64 #s(literal 1 binary64) n) < 4.9999999999999999e-13

   1. Initial program 13.5%

      \[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)} \]
   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 30.9

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

   5. Applied rewrites 30.9%

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

   7. Applied rewrites 30.9%

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

   9. Applied rewrites 30.9%

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

   10. Taylor expanded in x around inf

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

   12. Applied rewrites 81.6%

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

   if 4.9999999999999999e-13 < (/.f64 #s(literal 1 binary64) n) < 1.00000000000000006e232

   1. Initial program 72.4%

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

   3. Taylor expanded in x around 0

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

   5. Applied rewrites 72.2%

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

   if 1.00000000000000006e232 < (/.f64 #s(literal 1 binary64) n)

   1. Initial program 3.1%

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

   3. Taylor expanded in n around inf

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

      1. lower-/.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lower-log1p.f64 N/A

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

      4. lower-log.f64 8.3

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

   5. Applied rewrites 8.3%

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

   7. Applied rewrites 8.3%

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

   8. Taylor expanded in x around inf

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

   10. Applied rewrites 100.0%

       \[\leadsto \frac{1}{n \cdot \color{blue}{x}} \]
3. Recombined 5 regimes into one program.

4. Final simplification 73.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{1}{n} \leq -1 \cdot 10^{-22}:\\ \;\;\;\;\frac{{x}^{\left(\frac{1}{n}\right)}}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 5 \cdot 10^{-65}:\\ \;\;\;\;\frac{-\log x}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 5 \cdot 10^{-13}:\\ \;\;\;\;\frac{\frac{1}{n}}{\mathsf{fma}\left(\frac{0.5}{x}, x, x\right)}\\ \mathbf{elif}\;\frac{1}{n} \leq 10^{+232}:\\ \;\;\;\;1 - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{n \cdot x}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 60.5% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 4.3 \cdot 10^{-212}:\\ \;\;\;\;1 - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{elif}\;x \leq 0.86:\\ \;\;\;\;\frac{x - \log x}{n}\\ \mathbf{elif}\;x \leq 1.4 \cdot 10^{+113}:\\ \;\;\;\;\frac{\frac{1 - \frac{0.5 - \frac{0.3333333333333333}{x}}{x}}{x}}{n}\\ \mathbf{else}:\\ \;\;\;\;1 - 1\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (if (<= x 4.3e-212)
   (- 1.0 (pow x (/ 1.0 n)))
   (if (<= x 0.86)
     (/ (- x (log x)) n)
     (if (<= x 1.4e+113)
       (/ (/ (- 1.0 (/ (- 0.5 (/ 0.3333333333333333 x)) x)) x) n)
       (- 1.0 1.0)))))

C

double code(double x, double n) {
	double tmp;
	if (x <= 4.3e-212) {
		tmp = 1.0 - pow(x, (1.0 / n));
	} else if (x <= 0.86) {
		tmp = (x - log(x)) / n;
	} else if (x <= 1.4e+113) {
		tmp = ((1.0 - ((0.5 - (0.3333333333333333 / x)) / x)) / x) / n;
	} else {
		tmp = 1.0 - 1.0;
	}
	return tmp;
}

Fortran

real(8) function code(x, n)
    real(8), intent (in) :: x
    real(8), intent (in) :: n
    real(8) :: tmp
    if (x <= 4.3d-212) then
        tmp = 1.0d0 - (x ** (1.0d0 / n))
    else if (x <= 0.86d0) then
        tmp = (x - log(x)) / n
    else if (x <= 1.4d+113) then
        tmp = ((1.0d0 - ((0.5d0 - (0.3333333333333333d0 / x)) / x)) / x) / n
    else
        tmp = 1.0d0 - 1.0d0
    end if
    code = tmp
end function

Java

public static double code(double x, double n) {
	double tmp;
	if (x <= 4.3e-212) {
		tmp = 1.0 - Math.pow(x, (1.0 / n));
	} else if (x <= 0.86) {
		tmp = (x - Math.log(x)) / n;
	} else if (x <= 1.4e+113) {
		tmp = ((1.0 - ((0.5 - (0.3333333333333333 / x)) / x)) / x) / n;
	} else {
		tmp = 1.0 - 1.0;
	}
	return tmp;
}

Python

def code(x, n):
	tmp = 0
	if x <= 4.3e-212:
		tmp = 1.0 - math.pow(x, (1.0 / n))
	elif x <= 0.86:
		tmp = (x - math.log(x)) / n
	elif x <= 1.4e+113:
		tmp = ((1.0 - ((0.5 - (0.3333333333333333 / x)) / x)) / x) / n
	else:
		tmp = 1.0 - 1.0
	return tmp

Julia

function code(x, n)
	tmp = 0.0
	if (x <= 4.3e-212)
		tmp = Float64(1.0 - (x ^ Float64(1.0 / n)));
	elseif (x <= 0.86)
		tmp = Float64(Float64(x - log(x)) / n);
	elseif (x <= 1.4e+113)
		tmp = Float64(Float64(Float64(1.0 - Float64(Float64(0.5 - Float64(0.3333333333333333 / x)) / x)) / x) / n);
	else
		tmp = Float64(1.0 - 1.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, n)
	tmp = 0.0;
	if (x <= 4.3e-212)
		tmp = 1.0 - (x ^ (1.0 / n));
	elseif (x <= 0.86)
		tmp = (x - log(x)) / n;
	elseif (x <= 1.4e+113)
		tmp = ((1.0 - ((0.5 - (0.3333333333333333 / x)) / x)) / x) / n;
	else
		tmp = 1.0 - 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, n_] := If[LessEqual[x, 4.3e-212], N[(1.0 - N[Power[x, N[(1.0 / n), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 0.86], N[(N[(x - N[Log[x], $MachinePrecision]), $MachinePrecision] / n), $MachinePrecision], If[LessEqual[x, 1.4e+113], N[(N[(N[(1.0 - N[(N[(0.5 - N[(0.3333333333333333 / x), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision] / n), $MachinePrecision], N[(1.0 - 1.0), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if x < 4.29999999999999974e-212

   1. Initial program 61.9%

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

   3. Taylor expanded in x around 0

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

   5. Applied rewrites 61.9%

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

   if 4.29999999999999974e-212 < x < 0.859999999999999987

   1. Initial program 31.2%

      \[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)} \]
   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 63.4

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

   5. Applied rewrites 63.4%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 63.1%

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

   if 0.859999999999999987 < x < 1.39999999999999999e113

   1. Initial program 46.9%

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

   3. Taylor expanded in x around inf

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

   5. Applied rewrites 80.1%

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

   6. Taylor expanded in n around inf

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

   8. Applied rewrites 63.7%

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

   9. Taylor expanded in n around -inf

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

   11. Applied rewrites 63.8%

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

   if 1.39999999999999999e113 < x

   1. Initial program 83.2%

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

   3. Taylor expanded in x around 0

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

   5. Applied rewrites 46.6%

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

   6. Taylor expanded in n around inf

      \[\leadsto 1 - \color{blue}{1} \]
   7. Step-by-step derivation

   8. Applied rewrites 83.2%

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

Alternative 9: 61.0% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 0.86:\\ \;\;\;\;\frac{x - \log x}{n}\\ \mathbf{elif}\;x \leq 1.4 \cdot 10^{+113}:\\ \;\;\;\;\frac{\frac{1 - \frac{0.5 - \frac{0.3333333333333333}{x}}{x}}{x}}{n}\\ \mathbf{else}:\\ \;\;\;\;1 - 1\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (if (<= x 0.86)
   (/ (- x (log x)) n)
   (if (<= x 1.4e+113)
     (/ (/ (- 1.0 (/ (- 0.5 (/ 0.3333333333333333 x)) x)) x) n)
     (- 1.0 1.0))))

C

double code(double x, double n) {
	double tmp;
	if (x <= 0.86) {
		tmp = (x - log(x)) / n;
	} else if (x <= 1.4e+113) {
		tmp = ((1.0 - ((0.5 - (0.3333333333333333 / x)) / x)) / x) / n;
	} else {
		tmp = 1.0 - 1.0;
	}
	return tmp;
}

Fortran

real(8) function code(x, n)
    real(8), intent (in) :: x
    real(8), intent (in) :: n
    real(8) :: tmp
    if (x <= 0.86d0) then
        tmp = (x - log(x)) / n
    else if (x <= 1.4d+113) then
        tmp = ((1.0d0 - ((0.5d0 - (0.3333333333333333d0 / x)) / x)) / x) / n
    else
        tmp = 1.0d0 - 1.0d0
    end if
    code = tmp
end function

Java

public static double code(double x, double n) {
	double tmp;
	if (x <= 0.86) {
		tmp = (x - Math.log(x)) / n;
	} else if (x <= 1.4e+113) {
		tmp = ((1.0 - ((0.5 - (0.3333333333333333 / x)) / x)) / x) / n;
	} else {
		tmp = 1.0 - 1.0;
	}
	return tmp;
}

Python

def code(x, n):
	tmp = 0
	if x <= 0.86:
		tmp = (x - math.log(x)) / n
	elif x <= 1.4e+113:
		tmp = ((1.0 - ((0.5 - (0.3333333333333333 / x)) / x)) / x) / n
	else:
		tmp = 1.0 - 1.0
	return tmp

Julia

function code(x, n)
	tmp = 0.0
	if (x <= 0.86)
		tmp = Float64(Float64(x - log(x)) / n);
	elseif (x <= 1.4e+113)
		tmp = Float64(Float64(Float64(1.0 - Float64(Float64(0.5 - Float64(0.3333333333333333 / x)) / x)) / x) / n);
	else
		tmp = Float64(1.0 - 1.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, n)
	tmp = 0.0;
	if (x <= 0.86)
		tmp = (x - log(x)) / n;
	elseif (x <= 1.4e+113)
		tmp = ((1.0 - ((0.5 - (0.3333333333333333 / x)) / x)) / x) / n;
	else
		tmp = 1.0 - 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, n_] := If[LessEqual[x, 0.86], N[(N[(x - N[Log[x], $MachinePrecision]), $MachinePrecision] / n), $MachinePrecision], If[LessEqual[x, 1.4e+113], N[(N[(N[(1.0 - N[(N[(0.5 - N[(0.3333333333333333 / x), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision] / n), $MachinePrecision], N[(1.0 - 1.0), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < 0.859999999999999987

   1. Initial program 42.1%

      \[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)} \]
   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 55.3

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

   5. Applied rewrites 55.3%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 55.1%

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

   if 0.859999999999999987 < x < 1.39999999999999999e113

   1. Initial program 46.9%

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

   3. Taylor expanded in x around inf

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

   5. Applied rewrites 80.1%

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

   6. Taylor expanded in n around inf

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

   8. Applied rewrites 63.7%

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

   9. Taylor expanded in n around -inf

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

   11. Applied rewrites 63.8%

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

   if 1.39999999999999999e113 < x

   1. Initial program 83.2%

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

   3. Taylor expanded in x around 0

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

   5. Applied rewrites 46.6%

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

   6. Taylor expanded in n around inf

      \[\leadsto 1 - \color{blue}{1} \]
   7. Step-by-step derivation

   8. Applied rewrites 83.2%

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

Alternative 10: 60.7% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 0.6:\\ \;\;\;\;\frac{-\log x}{n}\\ \mathbf{elif}\;x \leq 1.4 \cdot 10^{+113}:\\ \;\;\;\;\frac{\frac{1 - \frac{0.5 - \frac{0.3333333333333333}{x}}{x}}{x}}{n}\\ \mathbf{else}:\\ \;\;\;\;1 - 1\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (if (<= x 0.6)
   (/ (- (log x)) n)
   (if (<= x 1.4e+113)
     (/ (/ (- 1.0 (/ (- 0.5 (/ 0.3333333333333333 x)) x)) x) n)
     (- 1.0 1.0))))

C

double code(double x, double n) {
	double tmp;
	if (x <= 0.6) {
		tmp = -log(x) / n;
	} else if (x <= 1.4e+113) {
		tmp = ((1.0 - ((0.5 - (0.3333333333333333 / x)) / x)) / x) / n;
	} else {
		tmp = 1.0 - 1.0;
	}
	return tmp;
}

Fortran

real(8) function code(x, n)
    real(8), intent (in) :: x
    real(8), intent (in) :: n
    real(8) :: tmp
    if (x <= 0.6d0) then
        tmp = -log(x) / n
    else if (x <= 1.4d+113) then
        tmp = ((1.0d0 - ((0.5d0 - (0.3333333333333333d0 / x)) / x)) / x) / n
    else
        tmp = 1.0d0 - 1.0d0
    end if
    code = tmp
end function

Java

public static double code(double x, double n) {
	double tmp;
	if (x <= 0.6) {
		tmp = -Math.log(x) / n;
	} else if (x <= 1.4e+113) {
		tmp = ((1.0 - ((0.5 - (0.3333333333333333 / x)) / x)) / x) / n;
	} else {
		tmp = 1.0 - 1.0;
	}
	return tmp;
}

Python

def code(x, n):
	tmp = 0
	if x <= 0.6:
		tmp = -math.log(x) / n
	elif x <= 1.4e+113:
		tmp = ((1.0 - ((0.5 - (0.3333333333333333 / x)) / x)) / x) / n
	else:
		tmp = 1.0 - 1.0
	return tmp

Julia

function code(x, n)
	tmp = 0.0
	if (x <= 0.6)
		tmp = Float64(Float64(-log(x)) / n);
	elseif (x <= 1.4e+113)
		tmp = Float64(Float64(Float64(1.0 - Float64(Float64(0.5 - Float64(0.3333333333333333 / x)) / x)) / x) / n);
	else
		tmp = Float64(1.0 - 1.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, n)
	tmp = 0.0;
	if (x <= 0.6)
		tmp = -log(x) / n;
	elseif (x <= 1.4e+113)
		tmp = ((1.0 - ((0.5 - (0.3333333333333333 / x)) / x)) / x) / n;
	else
		tmp = 1.0 - 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, n_] := If[LessEqual[x, 0.6], N[((-N[Log[x], $MachinePrecision]) / n), $MachinePrecision], If[LessEqual[x, 1.4e+113], N[(N[(N[(1.0 - N[(N[(0.5 - N[(0.3333333333333333 / x), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision] / n), $MachinePrecision], N[(1.0 - 1.0), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < 0.599999999999999978

   1. Initial program 42.1%

      \[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)} \]
   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 55.3

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

   5. Applied rewrites 55.3%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 55.0%

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

   if 0.599999999999999978 < x < 1.39999999999999999e113

   1. Initial program 46.9%

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

   3. Taylor expanded in x around inf

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

   5. Applied rewrites 80.1%

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

   6. Taylor expanded in n around inf

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

   8. Applied rewrites 63.7%

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

   9. Taylor expanded in n around -inf

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

   11. Applied rewrites 63.8%

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

   if 1.39999999999999999e113 < x

   1. Initial program 83.2%

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

   3. Taylor expanded in x around 0

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

   5. Applied rewrites 46.6%

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

   6. Taylor expanded in n around inf

      \[\leadsto 1 - \color{blue}{1} \]
   7. Step-by-step derivation

   8. Applied rewrites 83.2%

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

Alternative 11: 56.3% accurate, 4.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\frac{1}{n}}{\mathsf{fma}\left(\frac{0.5}{x}, x, x\right)}\\ \mathbf{if}\;n \leq -6.2:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;n \leq 8.5 \cdot 10^{-93}:\\ \;\;\;\;\frac{\frac{0.3333333333333333}{x \cdot x}}{n \cdot x}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (let* ((t_0 (/ (/ 1.0 n) (fma (/ 0.5 x) x x))))
   (if (<= n -6.2)
     t_0
     (if (<= n 8.5e-93) (/ (/ 0.3333333333333333 (* x x)) (* n x)) t_0))))

C

double code(double x, double n) {
	double t_0 = (1.0 / n) / fma((0.5 / x), x, x);
	double tmp;
	if (n <= -6.2) {
		tmp = t_0;
	} else if (n <= 8.5e-93) {
		tmp = (0.3333333333333333 / (x * x)) / (n * x);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(x, n)
	t_0 = Float64(Float64(1.0 / n) / fma(Float64(0.5 / x), x, x))
	tmp = 0.0
	if (n <= -6.2)
		tmp = t_0;
	elseif (n <= 8.5e-93)
		tmp = Float64(Float64(0.3333333333333333 / Float64(x * x)) / Float64(n * x));
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[x_, n_] := Block[{t$95$0 = N[(N[(1.0 / n), $MachinePrecision] / N[(N[(0.5 / x), $MachinePrecision] * x + x), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[n, -6.2], t$95$0, If[LessEqual[n, 8.5e-93], N[(N[(0.3333333333333333 / N[(x * x), $MachinePrecision]), $MachinePrecision] / N[(n * x), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if n < -6.20000000000000018 or 8.5000000000000007e-93 < n

   1. Initial program 34.7%

      \[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)} \]
   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 71.6

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

   5. Applied rewrites 71.6%

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

   7. Applied rewrites 71.6%

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

   9. Applied rewrites 71.5%

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

   10. Taylor expanded in x around inf

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

   12. Applied rewrites 50.1%

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

   if -6.20000000000000018 < n < 8.5000000000000007e-93

   1. Initial program 86.6%

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

   3. Taylor expanded in x around inf

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

   5. Applied rewrites 24.1%

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

   6. Taylor expanded in n around inf

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

   8. Applied rewrites 44.2%

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

   9. Taylor expanded in x around 0

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

   11. Applied rewrites 63.9%

       \[\leadsto \frac{\frac{0.3333333333333333}{x \cdot x}}{n \cdot x} \]
3. Recombined 2 regimes into one program.

4. Final simplification 55.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq -6.2:\\ \;\;\;\;\frac{\frac{1}{n}}{\mathsf{fma}\left(\frac{0.5}{x}, x, x\right)}\\ \mathbf{elif}\;n \leq 8.5 \cdot 10^{-93}:\\ \;\;\;\;\frac{\frac{0.3333333333333333}{x \cdot x}}{n \cdot x}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{n}}{\mathsf{fma}\left(\frac{0.5}{x}, x, x\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 51.8% accurate, 4.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{1}{n} \leq -5:\\ \;\;\;\;\frac{\frac{0.3333333333333333}{x \cdot x}}{n \cdot x}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{x}}{n}\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (if (<= (/ 1.0 n) -5.0)
   (/ (/ 0.3333333333333333 (* x x)) (* n x))
   (/ (/ 1.0 x) n)))

C

double code(double x, double n) {
	double tmp;
	if ((1.0 / n) <= -5.0) {
		tmp = (0.3333333333333333 / (x * x)) / (n * x);
	} else {
		tmp = (1.0 / x) / n;
	}
	return tmp;
}

Fortran

real(8) function code(x, n)
    real(8), intent (in) :: x
    real(8), intent (in) :: n
    real(8) :: tmp
    if ((1.0d0 / n) <= (-5.0d0)) then
        tmp = (0.3333333333333333d0 / (x * x)) / (n * x)
    else
        tmp = (1.0d0 / x) / n
    end if
    code = tmp
end function

Java

public static double code(double x, double n) {
	double tmp;
	if ((1.0 / n) <= -5.0) {
		tmp = (0.3333333333333333 / (x * x)) / (n * x);
	} else {
		tmp = (1.0 / x) / n;
	}
	return tmp;
}

Python

def code(x, n):
	tmp = 0
	if (1.0 / n) <= -5.0:
		tmp = (0.3333333333333333 / (x * x)) / (n * x)
	else:
		tmp = (1.0 / x) / n
	return tmp

Julia

function code(x, n)
	tmp = 0.0
	if (Float64(1.0 / n) <= -5.0)
		tmp = Float64(Float64(0.3333333333333333 / Float64(x * x)) / Float64(n * x));
	else
		tmp = Float64(Float64(1.0 / x) / n);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, n)
	tmp = 0.0;
	if ((1.0 / n) <= -5.0)
		tmp = (0.3333333333333333 / (x * x)) / (n * x);
	else
		tmp = (1.0 / x) / n;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, n_] := If[LessEqual[N[(1.0 / n), $MachinePrecision], -5.0], N[(N[(0.3333333333333333 / N[(x * x), $MachinePrecision]), $MachinePrecision] / N[(n * x), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 / x), $MachinePrecision] / n), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

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

   1. Initial program 99.7%

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

   3. Taylor expanded in x around inf

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

   5. Applied rewrites 31.8%

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

   6. Taylor expanded in n around inf

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

   8. Applied rewrites 42.5%

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

   9. Taylor expanded in x around 0

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

   11. Applied rewrites 68.5%

       \[\leadsto \frac{\frac{0.3333333333333333}{x \cdot x}}{n \cdot x} \]

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

   1. Initial program 36.0%

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

   3. Taylor expanded in x around inf

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

      1. associate-/l/ N/A

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

      2. lower-/.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. log-rec N/A

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

      5. mul-1-neg N/A

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

      6. associate-*r/ N/A

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

      7. associate-*r* N/A

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

      8. metadata-eval N/A

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

      9. *-commutative N/A

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

      10. associate-/l* N/A

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

      11. exp-to-pow N/A

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

      12. lower-pow.f64 N/A

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

      13. lower-/.f64 41.2

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

   5. Applied rewrites 41.2%

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

   6. Taylor expanded in n around inf

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

   8. Applied rewrites 44.3%

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

Alternative 13: 54.6% accurate, 5.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{1}{\mathsf{fma}\left(\frac{n}{x}, 0.5, n\right) \cdot x}\\ \mathbf{if}\;n \leq -6.2:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;n \leq 8.5 \cdot 10^{-93}:\\ \;\;\;\;\frac{\frac{0.3333333333333333}{x \cdot x}}{n \cdot x}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (let* ((t_0 (/ 1.0 (* (fma (/ n x) 0.5 n) x))))
   (if (<= n -6.2)
     t_0
     (if (<= n 8.5e-93) (/ (/ 0.3333333333333333 (* x x)) (* n x)) t_0))))

C

double code(double x, double n) {
	double t_0 = 1.0 / (fma((n / x), 0.5, n) * x);
	double tmp;
	if (n <= -6.2) {
		tmp = t_0;
	} else if (n <= 8.5e-93) {
		tmp = (0.3333333333333333 / (x * x)) / (n * x);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(x, n)
	t_0 = Float64(1.0 / Float64(fma(Float64(n / x), 0.5, n) * x))
	tmp = 0.0
	if (n <= -6.2)
		tmp = t_0;
	elseif (n <= 8.5e-93)
		tmp = Float64(Float64(0.3333333333333333 / Float64(x * x)) / Float64(n * x));
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[x_, n_] := Block[{t$95$0 = N[(1.0 / N[(N[(N[(n / x), $MachinePrecision] * 0.5 + n), $MachinePrecision] * x), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[n, -6.2], t$95$0, If[LessEqual[n, 8.5e-93], N[(N[(0.3333333333333333 / N[(x * x), $MachinePrecision]), $MachinePrecision] / N[(n * x), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if n < -6.20000000000000018 or 8.5000000000000007e-93 < n

   1. Initial program 34.7%

      \[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)} \]
   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 71.6

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

   5. Applied rewrites 71.6%

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

   7. Applied rewrites 71.5%

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

   8. Taylor expanded in x around inf

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

   10. Applied rewrites 47.7%

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

   if -6.20000000000000018 < n < 8.5000000000000007e-93

   1. Initial program 86.6%

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

   3. Taylor expanded in x around inf

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

   5. Applied rewrites 24.1%

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

   6. Taylor expanded in n around inf

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

   8. Applied rewrites 44.2%

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

   9. Taylor expanded in x around 0

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

   11. Applied rewrites 63.9%

       \[\leadsto \frac{\frac{0.3333333333333333}{x \cdot x}}{n \cdot x} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 14: 46.5% accurate, 5.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{1}{n} \leq -40000000000:\\ \;\;\;\;1 - 1\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{x}}{n}\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (if (<= (/ 1.0 n) -40000000000.0) (- 1.0 1.0) (/ (/ 1.0 x) n)))

C

double code(double x, double n) {
	double tmp;
	if ((1.0 / n) <= -40000000000.0) {
		tmp = 1.0 - 1.0;
	} else {
		tmp = (1.0 / x) / n;
	}
	return tmp;
}

Fortran

real(8) function code(x, n)
    real(8), intent (in) :: x
    real(8), intent (in) :: n
    real(8) :: tmp
    if ((1.0d0 / n) <= (-40000000000.0d0)) then
        tmp = 1.0d0 - 1.0d0
    else
        tmp = (1.0d0 / x) / n
    end if
    code = tmp
end function

Java

public static double code(double x, double n) {
	double tmp;
	if ((1.0 / n) <= -40000000000.0) {
		tmp = 1.0 - 1.0;
	} else {
		tmp = (1.0 / x) / n;
	}
	return tmp;
}

Python

def code(x, n):
	tmp = 0
	if (1.0 / n) <= -40000000000.0:
		tmp = 1.0 - 1.0
	else:
		tmp = (1.0 / x) / n
	return tmp

Julia

function code(x, n)
	tmp = 0.0
	if (Float64(1.0 / n) <= -40000000000.0)
		tmp = Float64(1.0 - 1.0);
	else
		tmp = Float64(Float64(1.0 / x) / n);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, n)
	tmp = 0.0;
	if ((1.0 / n) <= -40000000000.0)
		tmp = 1.0 - 1.0;
	else
		tmp = (1.0 / x) / n;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, n_] := If[LessEqual[N[(1.0 / n), $MachinePrecision], -40000000000.0], N[(1.0 - 1.0), $MachinePrecision], N[(N[(1.0 / x), $MachinePrecision] / n), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 #s(literal 1 binary64) n) < -4e10

   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 x around 0

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

   5. Applied rewrites 47.4%

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

   6. Taylor expanded in n around inf

      \[\leadsto 1 - \color{blue}{1} \]
   7. Step-by-step derivation

   8. Applied rewrites 55.0%

      \[\leadsto 1 - \color{blue}{1} \]

   if -4e10 < (/.f64 #s(literal 1 binary64) n)

   1. Initial program 36.6%

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

   3. Taylor expanded in x around inf

      \[\leadsto \color{blue}{\frac{e^{-1 \cdot \frac{\log \left(\frac{1}{x}\right)}{n}}}{n \cdot x}} \]
   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 41.5

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

   5. Applied rewrites 41.5%

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

   6. Taylor expanded in n around inf

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

   8. Applied rewrites 43.9%

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

Alternative 15: 46.5% accurate, 5.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{1}{n} \leq -40000000000:\\ \;\;\;\;1 - 1\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{n}}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (if (<= (/ 1.0 n) -40000000000.0) (- 1.0 1.0) (/ (/ 1.0 n) x)))

C

double code(double x, double n) {
	double tmp;
	if ((1.0 / n) <= -40000000000.0) {
		tmp = 1.0 - 1.0;
	} else {
		tmp = (1.0 / n) / x;
	}
	return tmp;
}

Fortran

real(8) function code(x, n)
    real(8), intent (in) :: x
    real(8), intent (in) :: n
    real(8) :: tmp
    if ((1.0d0 / n) <= (-40000000000.0d0)) then
        tmp = 1.0d0 - 1.0d0
    else
        tmp = (1.0d0 / n) / x
    end if
    code = tmp
end function

Java

public static double code(double x, double n) {
	double tmp;
	if ((1.0 / n) <= -40000000000.0) {
		tmp = 1.0 - 1.0;
	} else {
		tmp = (1.0 / n) / x;
	}
	return tmp;
}

Python

def code(x, n):
	tmp = 0
	if (1.0 / n) <= -40000000000.0:
		tmp = 1.0 - 1.0
	else:
		tmp = (1.0 / n) / x
	return tmp

Julia

function code(x, n)
	tmp = 0.0
	if (Float64(1.0 / n) <= -40000000000.0)
		tmp = Float64(1.0 - 1.0);
	else
		tmp = Float64(Float64(1.0 / n) / x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, n)
	tmp = 0.0;
	if ((1.0 / n) <= -40000000000.0)
		tmp = 1.0 - 1.0;
	else
		tmp = (1.0 / n) / x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, n_] := If[LessEqual[N[(1.0 / n), $MachinePrecision], -40000000000.0], N[(1.0 - 1.0), $MachinePrecision], N[(N[(1.0 / n), $MachinePrecision] / x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 #s(literal 1 binary64) n) < -4e10

   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 x around 0

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

   5. Applied rewrites 47.4%

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

   6. Taylor expanded in n around inf

      \[\leadsto 1 - \color{blue}{1} \]
   7. Step-by-step derivation

   8. Applied rewrites 55.0%

      \[\leadsto 1 - \color{blue}{1} \]

   if -4e10 < (/.f64 #s(literal 1 binary64) n)

   1. Initial program 36.6%

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

   3. Taylor expanded in x around inf

      \[\leadsto \color{blue}{\frac{e^{-1 \cdot \frac{\log \left(\frac{1}{x}\right)}{n}}}{n \cdot x}} \]
   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 41.5

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

   5. Applied rewrites 41.5%

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

   6. Taylor expanded in n around inf

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

   8. Applied rewrites 43.9%

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

Alternative 16: 46.0% accurate, 6.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{1}{n} \leq -40000000000:\\ \;\;\;\;1 - 1\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{n \cdot x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (if (<= (/ 1.0 n) -40000000000.0) (- 1.0 1.0) (/ 1.0 (* n x))))

C

double code(double x, double n) {
	double tmp;
	if ((1.0 / n) <= -40000000000.0) {
		tmp = 1.0 - 1.0;
	} else {
		tmp = 1.0 / (n * x);
	}
	return tmp;
}

Fortran

real(8) function code(x, n)
    real(8), intent (in) :: x
    real(8), intent (in) :: n
    real(8) :: tmp
    if ((1.0d0 / n) <= (-40000000000.0d0)) then
        tmp = 1.0d0 - 1.0d0
    else
        tmp = 1.0d0 / (n * x)
    end if
    code = tmp
end function

Java

public static double code(double x, double n) {
	double tmp;
	if ((1.0 / n) <= -40000000000.0) {
		tmp = 1.0 - 1.0;
	} else {
		tmp = 1.0 / (n * x);
	}
	return tmp;
}

Python

def code(x, n):
	tmp = 0
	if (1.0 / n) <= -40000000000.0:
		tmp = 1.0 - 1.0
	else:
		tmp = 1.0 / (n * x)
	return tmp

Julia

function code(x, n)
	tmp = 0.0
	if (Float64(1.0 / n) <= -40000000000.0)
		tmp = Float64(1.0 - 1.0);
	else
		tmp = Float64(1.0 / Float64(n * x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, n)
	tmp = 0.0;
	if ((1.0 / n) <= -40000000000.0)
		tmp = 1.0 - 1.0;
	else
		tmp = 1.0 / (n * x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, n_] := If[LessEqual[N[(1.0 / n), $MachinePrecision], -40000000000.0], N[(1.0 - 1.0), $MachinePrecision], N[(1.0 / N[(n * x), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 #s(literal 1 binary64) n) < -4e10

   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 x around 0

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

   5. Applied rewrites 47.4%

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

   6. Taylor expanded in n around inf

      \[\leadsto 1 - \color{blue}{1} \]
   7. Step-by-step derivation

   8. Applied rewrites 55.0%

      \[\leadsto 1 - \color{blue}{1} \]

   if -4e10 < (/.f64 #s(literal 1 binary64) n)

   1. Initial program 36.6%

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

   3. Taylor expanded in n around inf

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

      1. lower-/.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lower-log1p.f64 N/A

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

      4. lower-log.f64 62.7

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

   5. Applied rewrites 62.7%

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

   7. Applied rewrites 62.6%

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

   8. Taylor expanded in x around inf

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

   10. Applied rewrites 43.7%

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

Alternative 17: 30.8% accurate, 57.8× speedup

Math

\[\begin{array}{l} \\ 1 - 1 \end{array} \]

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_, n_] := N[(1.0 - 1.0), $MachinePrecision]

Derivation

1. Initial program 53.9%

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

3. Taylor expanded in x around 0

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

5. Applied rewrites 39.2%

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

6. Taylor expanded in n around inf

   \[\leadsto 1 - \color{blue}{1} \]
7. Step-by-step derivation

8. Applied rewrites 31.9%

   \[\leadsto 1 - \color{blue}{1} \]
9. Add Preprocessing

Reproduce

herbie shell --seed 2024255 
(FPCore (x n)
  :name "2nthrt (problem 3.4.6)"
  :precision binary64
  (- (pow (+ x 1.0) (/ 1.0 n)) (pow x (/ 1.0 n))))