2nthrt (problem 3.4.6)

Percentage Accurate: 53.8% → 85.1%

Time: 25.0s

Alternatives: 20

Speedup: 4.6×

• Could not converge on a ground truth

  1. x = -8.985405270083088e+224
  2. n = -2.888281818567458e-170

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 53.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 85.1% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{if}\;\frac{1}{n} \leq -2 \cdot 10^{-78}:\\ \;\;\;\;\frac{{x}^{\left(\frac{1}{n} + -1\right)}}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 2 \cdot 10^{-85}:\\ \;\;\;\;\frac{\log \left(\frac{1 + x}{x}\right)}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 4000000:\\ \;\;\;\;\frac{t\_0}{n \cdot x}\\ \mathbf{else}:\\ \;\;\;\;e^{\frac{\mathsf{log1p}\left(x\right)}{n}} - t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (let* ((t_0 (pow x (/ 1.0 n))))
   (if (<= (/ 1.0 n) -2e-78)
     (/ (pow x (+ (/ 1.0 n) -1.0)) n)
     (if (<= (/ 1.0 n) 2e-85)
       (/ (log (/ (+ 1.0 x) x)) n)
       (if (<= (/ 1.0 n) 4000000.0)
         (/ t_0 (* n x))
         (- (exp (/ (log1p x) n)) t_0))))))

C

double code(double x, double n) {
	double t_0 = pow(x, (1.0 / n));
	double tmp;
	if ((1.0 / n) <= -2e-78) {
		tmp = pow(x, ((1.0 / n) + -1.0)) / n;
	} else if ((1.0 / n) <= 2e-85) {
		tmp = log(((1.0 + x) / x)) / n;
	} else if ((1.0 / n) <= 4000000.0) {
		tmp = t_0 / (n * x);
	} else {
		tmp = exp((log1p(x) / n)) - t_0;
	}
	return tmp;
}

Java

public static double code(double x, double n) {
	double t_0 = Math.pow(x, (1.0 / n));
	double tmp;
	if ((1.0 / n) <= -2e-78) {
		tmp = Math.pow(x, ((1.0 / n) + -1.0)) / n;
	} else if ((1.0 / n) <= 2e-85) {
		tmp = Math.log(((1.0 + x) / x)) / n;
	} else if ((1.0 / n) <= 4000000.0) {
		tmp = t_0 / (n * x);
	} else {
		tmp = Math.exp((Math.log1p(x) / n)) - t_0;
	}
	return tmp;
}

Python

def code(x, n):
	t_0 = math.pow(x, (1.0 / n))
	tmp = 0
	if (1.0 / n) <= -2e-78:
		tmp = math.pow(x, ((1.0 / n) + -1.0)) / n
	elif (1.0 / n) <= 2e-85:
		tmp = math.log(((1.0 + x) / x)) / n
	elif (1.0 / n) <= 4000000.0:
		tmp = t_0 / (n * x)
	else:
		tmp = math.exp((math.log1p(x) / n)) - t_0
	return tmp

Julia

function code(x, n)
	t_0 = x ^ Float64(1.0 / n)
	tmp = 0.0
	if (Float64(1.0 / n) <= -2e-78)
		tmp = Float64((x ^ Float64(Float64(1.0 / n) + -1.0)) / n);
	elseif (Float64(1.0 / n) <= 2e-85)
		tmp = Float64(log(Float64(Float64(1.0 + x) / x)) / n);
	elseif (Float64(1.0 / n) <= 4000000.0)
		tmp = Float64(t_0 / Float64(n * x));
	else
		tmp = Float64(exp(Float64(log1p(x) / n)) - t_0);
	end
	return tmp
end

Wolfram

code[x_, n_] := Block[{t$95$0 = N[Power[x, N[(1.0 / n), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[N[(1.0 / n), $MachinePrecision], -2e-78], N[(N[Power[x, N[(N[(1.0 / n), $MachinePrecision] + -1.0), $MachinePrecision]], $MachinePrecision] / n), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 2e-85], N[(N[Log[N[(N[(1.0 + x), $MachinePrecision] / x), $MachinePrecision]], $MachinePrecision] / n), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 4000000.0], N[(t$95$0 / N[(n * x), $MachinePrecision]), $MachinePrecision], N[(N[Exp[N[(N[Log[1 + x], $MachinePrecision] / n), $MachinePrecision]], $MachinePrecision] - t$95$0), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if (/.f64 #s(literal 1 binary64) n) < -2e-78

   1. Initial program 83.5%

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

   3. Taylor expanded in x around inf

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

      1. lower-/.f64 N/A

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

      2. log-rec N/A

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

      3. mul-1-neg N/A

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

      4. associate-*r/ N/A

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

      5. associate-*r* N/A

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

      6. metadata-eval N/A

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

      7. *-commutative N/A

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

      8. associate-/l* N/A

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

      9. exp-to-pow N/A

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

      10. lower-pow.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 93.0

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

   5. Applied rewrites 93.0%

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

   7. Applied rewrites 93.1%

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

   if -2e-78 < (/.f64 #s(literal 1 binary64) n) < 2e-85

   1. Initial program 35.9%

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

   3. Taylor expanded in n around inf

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

      1. lower-/.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lower-log1p.f64 N/A

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

      4. lower-log.f64 82.1

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

   5. Applied rewrites 82.1%

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

   7. Applied rewrites 82.4%

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

   if 2e-85 < (/.f64 #s(literal 1 binary64) n) < 4e6

   1. Initial program 21.6%

      \[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)} \]
   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. lower-/.f64 N/A

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

      2. log-rec N/A

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

      3. mul-1-neg N/A

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

      4. associate-*r/ N/A

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

      5. associate-*r* N/A

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

      6. metadata-eval N/A

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

      7. *-commutative N/A

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

      8. associate-/l* N/A

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

      9. exp-to-pow N/A

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

      10. lower-pow.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 81.0

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

   5. Applied rewrites 81.0%

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

   if 4e6 < (/.f64 #s(literal 1 binary64) n)

   1. Initial program 63.7%

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

      1. lift-pow.f64 N/A

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

      2. pow-to-exp N/A

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

      3. lower-exp.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. un-div-inv N/A

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

      6. lower-/.f64 N/A

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

      7. lift-+.f64 N/A

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

      8. +-commutative N/A

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

      9. lower-log1p.f64 99.8

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

   4. Applied rewrites 99.8%

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

4. Final simplification 88.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{1}{n} \leq -2 \cdot 10^{-78}:\\ \;\;\;\;\frac{{x}^{\left(\frac{1}{n} + -1\right)}}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 2 \cdot 10^{-85}:\\ \;\;\;\;\frac{\log \left(\frac{1 + x}{x}\right)}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 4000000:\\ \;\;\;\;\frac{{x}^{\left(\frac{1}{n}\right)}}{n \cdot x}\\ \mathbf{else}:\\ \;\;\;\;e^{\frac{\mathsf{log1p}\left(x\right)}{n}} - {x}^{\left(\frac{1}{n}\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 81.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{if}\;\frac{1}{n} \leq -2 \cdot 10^{-78}:\\ \;\;\;\;\frac{{x}^{\left(\frac{1}{n} + -1\right)}}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 2 \cdot 10^{-85}:\\ \;\;\;\;\frac{\log \left(\frac{1 + x}{x}\right)}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 4000000:\\ \;\;\;\;\frac{t\_0}{n \cdot x}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x, \mathsf{fma}\left(x, \frac{0.5}{n \cdot n} + \frac{-0.5}{n}, \frac{1}{n}\right), 1\right) - t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (let* ((t_0 (pow x (/ 1.0 n))))
   (if (<= (/ 1.0 n) -2e-78)
     (/ (pow x (+ (/ 1.0 n) -1.0)) n)
     (if (<= (/ 1.0 n) 2e-85)
       (/ (log (/ (+ 1.0 x) x)) n)
       (if (<= (/ 1.0 n) 4000000.0)
         (/ t_0 (* n x))
         (-
          (fma x (fma x (+ (/ 0.5 (* n n)) (/ -0.5 n)) (/ 1.0 n)) 1.0)
          t_0))))))

C

double code(double x, double n) {
	double t_0 = pow(x, (1.0 / n));
	double tmp;
	if ((1.0 / n) <= -2e-78) {
		tmp = pow(x, ((1.0 / n) + -1.0)) / n;
	} else if ((1.0 / n) <= 2e-85) {
		tmp = log(((1.0 + x) / x)) / n;
	} else if ((1.0 / n) <= 4000000.0) {
		tmp = t_0 / (n * x);
	} else {
		tmp = fma(x, fma(x, ((0.5 / (n * n)) + (-0.5 / n)), (1.0 / n)), 1.0) - t_0;
	}
	return tmp;
}

Julia

function code(x, n)
	t_0 = x ^ Float64(1.0 / n)
	tmp = 0.0
	if (Float64(1.0 / n) <= -2e-78)
		tmp = Float64((x ^ Float64(Float64(1.0 / n) + -1.0)) / n);
	elseif (Float64(1.0 / n) <= 2e-85)
		tmp = Float64(log(Float64(Float64(1.0 + x) / x)) / n);
	elseif (Float64(1.0 / n) <= 4000000.0)
		tmp = Float64(t_0 / Float64(n * x));
	else
		tmp = Float64(fma(x, fma(x, Float64(Float64(0.5 / Float64(n * n)) + Float64(-0.5 / n)), Float64(1.0 / n)), 1.0) - t_0);
	end
	return tmp
end

Wolfram

code[x_, n_] := Block[{t$95$0 = N[Power[x, N[(1.0 / n), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[N[(1.0 / n), $MachinePrecision], -2e-78], N[(N[Power[x, N[(N[(1.0 / n), $MachinePrecision] + -1.0), $MachinePrecision]], $MachinePrecision] / n), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 2e-85], N[(N[Log[N[(N[(1.0 + x), $MachinePrecision] / x), $MachinePrecision]], $MachinePrecision] / n), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 4000000.0], N[(t$95$0 / N[(n * x), $MachinePrecision]), $MachinePrecision], N[(N[(x * N[(x * N[(N[(0.5 / N[(n * n), $MachinePrecision]), $MachinePrecision] + N[(-0.5 / n), $MachinePrecision]), $MachinePrecision] + N[(1.0 / n), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision] - t$95$0), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if (/.f64 #s(literal 1 binary64) n) < -2e-78

   1. Initial program 83.5%

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

   3. Taylor expanded in x around inf

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

      1. lower-/.f64 N/A

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

      2. log-rec N/A

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

      3. mul-1-neg N/A

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

      4. associate-*r/ N/A

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

      5. associate-*r* N/A

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

      6. metadata-eval N/A

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

      7. *-commutative N/A

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

      8. associate-/l* N/A

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

      9. exp-to-pow N/A

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

      10. lower-pow.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 93.0

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

   5. Applied rewrites 93.0%

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

   7. Applied rewrites 93.1%

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

   if -2e-78 < (/.f64 #s(literal 1 binary64) n) < 2e-85

   1. Initial program 35.9%

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

   3. Taylor expanded in n around inf

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

      1. lower-/.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lower-log1p.f64 N/A

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

      4. lower-log.f64 82.1

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

   5. Applied rewrites 82.1%

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

   7. Applied rewrites 82.4%

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

   if 2e-85 < (/.f64 #s(literal 1 binary64) n) < 4e6

   1. Initial program 21.6%

      \[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)} \]
   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. lower-/.f64 N/A

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

      2. log-rec N/A

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

      3. mul-1-neg N/A

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

      4. associate-*r/ N/A

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

      5. associate-*r* N/A

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

      6. metadata-eval N/A

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

      7. *-commutative N/A

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

      8. associate-/l* N/A

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

      9. exp-to-pow N/A

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

      10. lower-pow.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 81.0

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

   5. Applied rewrites 81.0%

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

   if 4e6 < (/.f64 #s(literal 1 binary64) n)

   1. Initial program 63.7%

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

   3. Taylor expanded in x around 0

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

   5. Applied rewrites 56.2%

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

   6. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. lower-fma.f64 N/A

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

      4. sub-neg N/A

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

      5. lower-+.f64 N/A

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

      6. associate-*r/ N/A

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

      7. metadata-eval N/A

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

      8. lower-/.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 N/A

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

      11. associate-*r/ N/A

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

      12. metadata-eval N/A

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

      13. distribute-neg-frac N/A

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

      14. metadata-eval N/A

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

      15. lower-/.f64 N/A

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

      16. lower-/.f64 75.4

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

   8. Applied rewrites 75.4%

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

4. Final simplification 85.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{1}{n} \leq -2 \cdot 10^{-78}:\\ \;\;\;\;\frac{{x}^{\left(\frac{1}{n} + -1\right)}}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 2 \cdot 10^{-85}:\\ \;\;\;\;\frac{\log \left(\frac{1 + x}{x}\right)}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 4000000:\\ \;\;\;\;\frac{{x}^{\left(\frac{1}{n}\right)}}{n \cdot x}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x, \mathsf{fma}\left(x, \frac{0.5}{n \cdot n} + \frac{-0.5}{n}, \frac{1}{n}\right), 1\right) - {x}^{\left(\frac{1}{n}\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 81.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{if}\;\frac{1}{n} \leq -2 \cdot 10^{-78}:\\ \;\;\;\;\frac{{x}^{\left(\frac{1}{n} + -1\right)}}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 2 \cdot 10^{-85}:\\ \;\;\;\;\frac{\log \left(\frac{1 + x}{x}\right)}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 4000000:\\ \;\;\;\;\frac{t\_0}{n \cdot x}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x, \mathsf{fma}\left(x, \frac{0.5}{n \cdot n} + \frac{-0.5}{n}, \frac{1}{n}\right), 1 - t\_0\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (let* ((t_0 (pow x (/ 1.0 n))))
   (if (<= (/ 1.0 n) -2e-78)
     (/ (pow x (+ (/ 1.0 n) -1.0)) n)
     (if (<= (/ 1.0 n) 2e-85)
       (/ (log (/ (+ 1.0 x) x)) n)
       (if (<= (/ 1.0 n) 4000000.0)
         (/ t_0 (* n x))
         (fma
          x
          (fma x (+ (/ 0.5 (* n n)) (/ -0.5 n)) (/ 1.0 n))
          (- 1.0 t_0)))))))

C

double code(double x, double n) {
	double t_0 = pow(x, (1.0 / n));
	double tmp;
	if ((1.0 / n) <= -2e-78) {
		tmp = pow(x, ((1.0 / n) + -1.0)) / n;
	} else if ((1.0 / n) <= 2e-85) {
		tmp = log(((1.0 + x) / x)) / n;
	} else if ((1.0 / n) <= 4000000.0) {
		tmp = t_0 / (n * x);
	} else {
		tmp = fma(x, fma(x, ((0.5 / (n * n)) + (-0.5 / n)), (1.0 / n)), (1.0 - t_0));
	}
	return tmp;
}

Julia

function code(x, n)
	t_0 = x ^ Float64(1.0 / n)
	tmp = 0.0
	if (Float64(1.0 / n) <= -2e-78)
		tmp = Float64((x ^ Float64(Float64(1.0 / n) + -1.0)) / n);
	elseif (Float64(1.0 / n) <= 2e-85)
		tmp = Float64(log(Float64(Float64(1.0 + x) / x)) / n);
	elseif (Float64(1.0 / n) <= 4000000.0)
		tmp = Float64(t_0 / Float64(n * x));
	else
		tmp = fma(x, fma(x, Float64(Float64(0.5 / Float64(n * n)) + Float64(-0.5 / n)), Float64(1.0 / n)), Float64(1.0 - t_0));
	end
	return tmp
end

Wolfram

code[x_, n_] := Block[{t$95$0 = N[Power[x, N[(1.0 / n), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[N[(1.0 / n), $MachinePrecision], -2e-78], N[(N[Power[x, N[(N[(1.0 / n), $MachinePrecision] + -1.0), $MachinePrecision]], $MachinePrecision] / n), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 2e-85], N[(N[Log[N[(N[(1.0 + x), $MachinePrecision] / x), $MachinePrecision]], $MachinePrecision] / n), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 4000000.0], N[(t$95$0 / N[(n * x), $MachinePrecision]), $MachinePrecision], N[(x * N[(x * N[(N[(0.5 / N[(n * n), $MachinePrecision]), $MachinePrecision] + N[(-0.5 / n), $MachinePrecision]), $MachinePrecision] + N[(1.0 / n), $MachinePrecision]), $MachinePrecision] + N[(1.0 - t$95$0), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if (/.f64 #s(literal 1 binary64) n) < -2e-78

   1. Initial program 83.5%

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

   3. Taylor expanded in x around inf

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

      1. lower-/.f64 N/A

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

      2. log-rec N/A

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

      3. mul-1-neg N/A

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

      4. associate-*r/ N/A

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

      5. associate-*r* N/A

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

      6. metadata-eval N/A

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

      7. *-commutative N/A

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

      8. associate-/l* N/A

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

      9. exp-to-pow N/A

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

      10. lower-pow.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 93.0

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

   5. Applied rewrites 93.0%

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

   7. Applied rewrites 93.1%

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

   if -2e-78 < (/.f64 #s(literal 1 binary64) n) < 2e-85

   1. Initial program 35.9%

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

   3. Taylor expanded in n around inf

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

      1. lower-/.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lower-log1p.f64 N/A

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

      4. lower-log.f64 82.1

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

   5. Applied rewrites 82.1%

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

   7. Applied rewrites 82.4%

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

   if 2e-85 < (/.f64 #s(literal 1 binary64) n) < 4e6

   1. Initial program 21.6%

      \[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)} \]
   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. lower-/.f64 N/A

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

      2. log-rec N/A

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

      3. mul-1-neg N/A

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

      4. associate-*r/ N/A

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

      5. associate-*r* N/A

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

      6. metadata-eval N/A

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

      7. *-commutative N/A

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

      8. associate-/l* N/A

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

      9. exp-to-pow N/A

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

      10. lower-pow.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 81.0

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

   5. Applied rewrites 81.0%

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

   if 4e6 < (/.f64 #s(literal 1 binary64) n)

   1. Initial program 63.7%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. associate--l+ N/A

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

      3. lower-fma.f64 N/A

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

   5. Applied rewrites 75.4%

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

4. Final simplification 85.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{1}{n} \leq -2 \cdot 10^{-78}:\\ \;\;\;\;\frac{{x}^{\left(\frac{1}{n} + -1\right)}}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 2 \cdot 10^{-85}:\\ \;\;\;\;\frac{\log \left(\frac{1 + x}{x}\right)}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 4000000:\\ \;\;\;\;\frac{{x}^{\left(\frac{1}{n}\right)}}{n \cdot x}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x, \mathsf{fma}\left(x, \frac{0.5}{n \cdot n} + \frac{-0.5}{n}, \frac{1}{n}\right), 1 - {x}^{\left(\frac{1}{n}\right)}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 81.3% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{if}\;\frac{1}{n} \leq -2 \cdot 10^{-78}:\\ \;\;\;\;\frac{{x}^{\left(\frac{1}{n} + -1\right)}}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 2 \cdot 10^{-85}:\\ \;\;\;\;\frac{\log \left(\frac{1 + x}{x}\right)}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 4000000:\\ \;\;\;\;\frac{t\_0}{n \cdot x}\\ \mathbf{elif}\;\frac{1}{n} \leq 5 \cdot 10^{+182}:\\ \;\;\;\;\left(1 + \frac{x}{n}\right) - t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{n}{x}}{n \cdot n}\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (let* ((t_0 (pow x (/ 1.0 n))))
   (if (<= (/ 1.0 n) -2e-78)
     (/ (pow x (+ (/ 1.0 n) -1.0)) n)
     (if (<= (/ 1.0 n) 2e-85)
       (/ (log (/ (+ 1.0 x) x)) n)
       (if (<= (/ 1.0 n) 4000000.0)
         (/ t_0 (* n x))
         (if (<= (/ 1.0 n) 5e+182)
           (- (+ 1.0 (/ x n)) t_0)
           (/ (/ n x) (* n n))))))))

C

double code(double x, double n) {
	double t_0 = pow(x, (1.0 / n));
	double tmp;
	if ((1.0 / n) <= -2e-78) {
		tmp = pow(x, ((1.0 / n) + -1.0)) / n;
	} else if ((1.0 / n) <= 2e-85) {
		tmp = log(((1.0 + x) / x)) / n;
	} else if ((1.0 / n) <= 4000000.0) {
		tmp = t_0 / (n * x);
	} else if ((1.0 / n) <= 5e+182) {
		tmp = (1.0 + (x / n)) - t_0;
	} else {
		tmp = (n / x) / (n * n);
	}
	return tmp;
}

Fortran

real(8) function code(x, n)
    real(8), intent (in) :: x
    real(8), intent (in) :: n
    real(8) :: t_0
    real(8) :: tmp
    t_0 = x ** (1.0d0 / n)
    if ((1.0d0 / n) <= (-2d-78)) then
        tmp = (x ** ((1.0d0 / n) + (-1.0d0))) / n
    else if ((1.0d0 / n) <= 2d-85) then
        tmp = log(((1.0d0 + x) / x)) / n
    else if ((1.0d0 / n) <= 4000000.0d0) then
        tmp = t_0 / (n * x)
    else if ((1.0d0 / n) <= 5d+182) then
        tmp = (1.0d0 + (x / n)) - t_0
    else
        tmp = (n / x) / (n * n)
    end if
    code = tmp
end function

Java

public static double code(double x, double n) {
	double t_0 = Math.pow(x, (1.0 / n));
	double tmp;
	if ((1.0 / n) <= -2e-78) {
		tmp = Math.pow(x, ((1.0 / n) + -1.0)) / n;
	} else if ((1.0 / n) <= 2e-85) {
		tmp = Math.log(((1.0 + x) / x)) / n;
	} else if ((1.0 / n) <= 4000000.0) {
		tmp = t_0 / (n * x);
	} else if ((1.0 / n) <= 5e+182) {
		tmp = (1.0 + (x / n)) - t_0;
	} else {
		tmp = (n / x) / (n * n);
	}
	return tmp;
}

Python

def code(x, n):
	t_0 = math.pow(x, (1.0 / n))
	tmp = 0
	if (1.0 / n) <= -2e-78:
		tmp = math.pow(x, ((1.0 / n) + -1.0)) / n
	elif (1.0 / n) <= 2e-85:
		tmp = math.log(((1.0 + x) / x)) / n
	elif (1.0 / n) <= 4000000.0:
		tmp = t_0 / (n * x)
	elif (1.0 / n) <= 5e+182:
		tmp = (1.0 + (x / n)) - t_0
	else:
		tmp = (n / x) / (n * n)
	return tmp

Julia

function code(x, n)
	t_0 = x ^ Float64(1.0 / n)
	tmp = 0.0
	if (Float64(1.0 / n) <= -2e-78)
		tmp = Float64((x ^ Float64(Float64(1.0 / n) + -1.0)) / n);
	elseif (Float64(1.0 / n) <= 2e-85)
		tmp = Float64(log(Float64(Float64(1.0 + x) / x)) / n);
	elseif (Float64(1.0 / n) <= 4000000.0)
		tmp = Float64(t_0 / Float64(n * x));
	elseif (Float64(1.0 / n) <= 5e+182)
		tmp = Float64(Float64(1.0 + Float64(x / n)) - t_0);
	else
		tmp = Float64(Float64(n / x) / Float64(n * n));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, n)
	t_0 = x ^ (1.0 / n);
	tmp = 0.0;
	if ((1.0 / n) <= -2e-78)
		tmp = (x ^ ((1.0 / n) + -1.0)) / n;
	elseif ((1.0 / n) <= 2e-85)
		tmp = log(((1.0 + x) / x)) / n;
	elseif ((1.0 / n) <= 4000000.0)
		tmp = t_0 / (n * x);
	elseif ((1.0 / n) <= 5e+182)
		tmp = (1.0 + (x / n)) - t_0;
	else
		tmp = (n / x) / (n * n);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, n_] := Block[{t$95$0 = N[Power[x, N[(1.0 / n), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[N[(1.0 / n), $MachinePrecision], -2e-78], N[(N[Power[x, N[(N[(1.0 / n), $MachinePrecision] + -1.0), $MachinePrecision]], $MachinePrecision] / n), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 2e-85], N[(N[Log[N[(N[(1.0 + x), $MachinePrecision] / x), $MachinePrecision]], $MachinePrecision] / n), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 4000000.0], N[(t$95$0 / N[(n * x), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 5e+182], N[(N[(1.0 + N[(x / n), $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision], N[(N[(n / x), $MachinePrecision] / N[(n * n), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 5 regimes

2. if (/.f64 #s(literal 1 binary64) n) < -2e-78

   1. Initial program 83.5%

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

   3. Taylor expanded in x around inf

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

      1. lower-/.f64 N/A

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

      2. log-rec N/A

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

      3. mul-1-neg N/A

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

      4. associate-*r/ N/A

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

      5. associate-*r* N/A

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

      6. metadata-eval N/A

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

      7. *-commutative N/A

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

      8. associate-/l* N/A

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

      9. exp-to-pow N/A

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

      10. lower-pow.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 93.0

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

   5. Applied rewrites 93.0%

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

   7. Applied rewrites 93.1%

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

   if -2e-78 < (/.f64 #s(literal 1 binary64) n) < 2e-85

   1. Initial program 35.9%

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

   3. Taylor expanded in n around inf

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

      1. lower-/.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lower-log1p.f64 N/A

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

      4. lower-log.f64 82.1

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

   5. Applied rewrites 82.1%

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

   7. Applied rewrites 82.4%

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

   if 2e-85 < (/.f64 #s(literal 1 binary64) n) < 4e6

   1. Initial program 21.6%

      \[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)} \]
   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. lower-/.f64 N/A

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

      2. log-rec N/A

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

      3. mul-1-neg N/A

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

      4. associate-*r/ N/A

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

      5. associate-*r* N/A

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

      6. metadata-eval N/A

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

      7. *-commutative N/A

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

      8. associate-/l* N/A

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

      9. exp-to-pow N/A

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

      10. lower-pow.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 81.0

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

   5. Applied rewrites 81.0%

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

   if 4e6 < (/.f64 #s(literal 1 binary64) n) < 4.99999999999999973e182

   1. Initial program 79.8%

      \[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)} \]
   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. *-rgt-identity N/A

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

      2. associate-*r/ N/A

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

      3. lower-+.f64 N/A

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

      4. associate-*r/ N/A

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

      5. *-rgt-identity N/A

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

      6. lower-/.f64 75.1

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

   5. Applied rewrites 75.1%

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

   if 4.99999999999999973e182 < (/.f64 #s(literal 1 binary64) n)

   1. Initial program 27.3%

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

   3. Taylor expanded in n around inf

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

      1. lower-/.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lower-log1p.f64 N/A

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

      4. lower-log.f64 22.0

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

   5. Applied rewrites 22.0%

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

   7. Applied rewrites 75.8%

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

   8. Taylor expanded in x around inf

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

   10. Applied rewrites 75.8%

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

4. Final simplification 85.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{1}{n} \leq -2 \cdot 10^{-78}:\\ \;\;\;\;\frac{{x}^{\left(\frac{1}{n} + -1\right)}}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 2 \cdot 10^{-85}:\\ \;\;\;\;\frac{\log \left(\frac{1 + x}{x}\right)}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 4000000:\\ \;\;\;\;\frac{{x}^{\left(\frac{1}{n}\right)}}{n \cdot x}\\ \mathbf{elif}\;\frac{1}{n} \leq 5 \cdot 10^{+182}:\\ \;\;\;\;\left(1 + \frac{x}{n}\right) - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{n}{x}}{n \cdot n}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 81.1% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{if}\;\frac{1}{n} \leq -2 \cdot 10^{-78}:\\ \;\;\;\;\frac{{x}^{\left(\frac{1}{n} + -1\right)}}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 2 \cdot 10^{-85}:\\ \;\;\;\;\frac{\log \left(\frac{1 + x}{x}\right)}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 4000000:\\ \;\;\;\;\frac{t\_0}{n \cdot x}\\ \mathbf{elif}\;\frac{1}{n} \leq 5 \cdot 10^{+182}:\\ \;\;\;\;1 - t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{n}{x}}{n \cdot n}\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (let* ((t_0 (pow x (/ 1.0 n))))
   (if (<= (/ 1.0 n) -2e-78)
     (/ (pow x (+ (/ 1.0 n) -1.0)) n)
     (if (<= (/ 1.0 n) 2e-85)
       (/ (log (/ (+ 1.0 x) x)) n)
       (if (<= (/ 1.0 n) 4000000.0)
         (/ t_0 (* n x))
         (if (<= (/ 1.0 n) 5e+182) (- 1.0 t_0) (/ (/ n x) (* n n))))))))

C

double code(double x, double n) {
	double t_0 = pow(x, (1.0 / n));
	double tmp;
	if ((1.0 / n) <= -2e-78) {
		tmp = pow(x, ((1.0 / n) + -1.0)) / n;
	} else if ((1.0 / n) <= 2e-85) {
		tmp = log(((1.0 + x) / x)) / n;
	} else if ((1.0 / n) <= 4000000.0) {
		tmp = t_0 / (n * x);
	} else if ((1.0 / n) <= 5e+182) {
		tmp = 1.0 - t_0;
	} else {
		tmp = (n / x) / (n * n);
	}
	return tmp;
}

Fortran

real(8) function code(x, n)
    real(8), intent (in) :: x
    real(8), intent (in) :: n
    real(8) :: t_0
    real(8) :: tmp
    t_0 = x ** (1.0d0 / n)
    if ((1.0d0 / n) <= (-2d-78)) then
        tmp = (x ** ((1.0d0 / n) + (-1.0d0))) / n
    else if ((1.0d0 / n) <= 2d-85) then
        tmp = log(((1.0d0 + x) / x)) / n
    else if ((1.0d0 / n) <= 4000000.0d0) then
        tmp = t_0 / (n * x)
    else if ((1.0d0 / n) <= 5d+182) then
        tmp = 1.0d0 - t_0
    else
        tmp = (n / x) / (n * n)
    end if
    code = tmp
end function

Java

public static double code(double x, double n) {
	double t_0 = Math.pow(x, (1.0 / n));
	double tmp;
	if ((1.0 / n) <= -2e-78) {
		tmp = Math.pow(x, ((1.0 / n) + -1.0)) / n;
	} else if ((1.0 / n) <= 2e-85) {
		tmp = Math.log(((1.0 + x) / x)) / n;
	} else if ((1.0 / n) <= 4000000.0) {
		tmp = t_0 / (n * x);
	} else if ((1.0 / n) <= 5e+182) {
		tmp = 1.0 - t_0;
	} else {
		tmp = (n / x) / (n * n);
	}
	return tmp;
}

Python

def code(x, n):
	t_0 = math.pow(x, (1.0 / n))
	tmp = 0
	if (1.0 / n) <= -2e-78:
		tmp = math.pow(x, ((1.0 / n) + -1.0)) / n
	elif (1.0 / n) <= 2e-85:
		tmp = math.log(((1.0 + x) / x)) / n
	elif (1.0 / n) <= 4000000.0:
		tmp = t_0 / (n * x)
	elif (1.0 / n) <= 5e+182:
		tmp = 1.0 - t_0
	else:
		tmp = (n / x) / (n * n)
	return tmp

Julia

function code(x, n)
	t_0 = x ^ Float64(1.0 / n)
	tmp = 0.0
	if (Float64(1.0 / n) <= -2e-78)
		tmp = Float64((x ^ Float64(Float64(1.0 / n) + -1.0)) / n);
	elseif (Float64(1.0 / n) <= 2e-85)
		tmp = Float64(log(Float64(Float64(1.0 + x) / x)) / n);
	elseif (Float64(1.0 / n) <= 4000000.0)
		tmp = Float64(t_0 / Float64(n * x));
	elseif (Float64(1.0 / n) <= 5e+182)
		tmp = Float64(1.0 - t_0);
	else
		tmp = Float64(Float64(n / x) / Float64(n * n));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, n)
	t_0 = x ^ (1.0 / n);
	tmp = 0.0;
	if ((1.0 / n) <= -2e-78)
		tmp = (x ^ ((1.0 / n) + -1.0)) / n;
	elseif ((1.0 / n) <= 2e-85)
		tmp = log(((1.0 + x) / x)) / n;
	elseif ((1.0 / n) <= 4000000.0)
		tmp = t_0 / (n * x);
	elseif ((1.0 / n) <= 5e+182)
		tmp = 1.0 - t_0;
	else
		tmp = (n / x) / (n * n);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, n_] := Block[{t$95$0 = N[Power[x, N[(1.0 / n), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[N[(1.0 / n), $MachinePrecision], -2e-78], N[(N[Power[x, N[(N[(1.0 / n), $MachinePrecision] + -1.0), $MachinePrecision]], $MachinePrecision] / n), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 2e-85], N[(N[Log[N[(N[(1.0 + x), $MachinePrecision] / x), $MachinePrecision]], $MachinePrecision] / n), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 4000000.0], N[(t$95$0 / N[(n * x), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 5e+182], N[(1.0 - t$95$0), $MachinePrecision], N[(N[(n / x), $MachinePrecision] / N[(n * n), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 5 regimes

2. if (/.f64 #s(literal 1 binary64) n) < -2e-78

   1. Initial program 83.5%

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

   3. Taylor expanded in x around inf

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

      1. lower-/.f64 N/A

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

      2. log-rec N/A

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

      3. mul-1-neg N/A

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

      4. associate-*r/ N/A

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

      5. associate-*r* N/A

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

      6. metadata-eval N/A

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

      7. *-commutative N/A

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

      8. associate-/l* N/A

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

      9. exp-to-pow N/A

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

      10. lower-pow.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 93.0

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

   5. Applied rewrites 93.0%

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

   7. Applied rewrites 93.1%

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

   if -2e-78 < (/.f64 #s(literal 1 binary64) n) < 2e-85

   1. Initial program 35.9%

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

   3. Taylor expanded in n around inf

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

      1. lower-/.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lower-log1p.f64 N/A

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

      4. lower-log.f64 82.1

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

   5. Applied rewrites 82.1%

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

   7. Applied rewrites 82.4%

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

   if 2e-85 < (/.f64 #s(literal 1 binary64) n) < 4e6

   1. Initial program 21.6%

      \[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)} \]
   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. lower-/.f64 N/A

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

      2. log-rec N/A

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

      3. mul-1-neg N/A

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

      4. associate-*r/ N/A

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

      5. associate-*r* N/A

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

      6. metadata-eval N/A

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

      7. *-commutative N/A

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

      8. associate-/l* N/A

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

      9. exp-to-pow N/A

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

      10. lower-pow.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 81.0

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

   5. Applied rewrites 81.0%

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

   if 4e6 < (/.f64 #s(literal 1 binary64) n) < 4.99999999999999973e182

   1. Initial program 79.8%

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

   3. Taylor expanded in x around 0

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

   5. Applied rewrites 72.6%

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

   if 4.99999999999999973e182 < (/.f64 #s(literal 1 binary64) n)

   1. Initial program 27.3%

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

   3. Taylor expanded in n around inf

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

      1. lower-/.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lower-log1p.f64 N/A

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

      4. lower-log.f64 22.0

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

   5. Applied rewrites 22.0%

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

   7. Applied rewrites 75.8%

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

   8. Taylor expanded in x around inf

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

   10. Applied rewrites 75.8%

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

4. Final simplification 84.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{1}{n} \leq -2 \cdot 10^{-78}:\\ \;\;\;\;\frac{{x}^{\left(\frac{1}{n} + -1\right)}}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 2 \cdot 10^{-85}:\\ \;\;\;\;\frac{\log \left(\frac{1 + x}{x}\right)}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 4000000:\\ \;\;\;\;\frac{{x}^{\left(\frac{1}{n}\right)}}{n \cdot x}\\ \mathbf{elif}\;\frac{1}{n} \leq 5 \cdot 10^{+182}:\\ \;\;\;\;1 - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{n}{x}}{n \cdot n}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 81.2% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{{x}^{\left(\frac{1}{n} + -1\right)}}{n}\\ \mathbf{if}\;\frac{1}{n} \leq -2 \cdot 10^{-78}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;\frac{1}{n} \leq 2 \cdot 10^{-85}:\\ \;\;\;\;\frac{\log \left(\frac{1 + x}{x}\right)}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 4000000:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;\frac{1}{n} \leq 5 \cdot 10^{+182}:\\ \;\;\;\;1 - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{n}{x}}{n \cdot n}\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (let* ((t_0 (/ (pow x (+ (/ 1.0 n) -1.0)) n)))
   (if (<= (/ 1.0 n) -2e-78)
     t_0
     (if (<= (/ 1.0 n) 2e-85)
       (/ (log (/ (+ 1.0 x) x)) n)
       (if (<= (/ 1.0 n) 4000000.0)
         t_0
         (if (<= (/ 1.0 n) 5e+182)
           (- 1.0 (pow x (/ 1.0 n)))
           (/ (/ n x) (* n n))))))))

C

double code(double x, double n) {
	double t_0 = pow(x, ((1.0 / n) + -1.0)) / n;
	double tmp;
	if ((1.0 / n) <= -2e-78) {
		tmp = t_0;
	} else if ((1.0 / n) <= 2e-85) {
		tmp = log(((1.0 + x) / x)) / n;
	} else if ((1.0 / n) <= 4000000.0) {
		tmp = t_0;
	} else if ((1.0 / n) <= 5e+182) {
		tmp = 1.0 - pow(x, (1.0 / n));
	} else {
		tmp = (n / x) / (n * n);
	}
	return tmp;
}

Fortran

real(8) function code(x, n)
    real(8), intent (in) :: x
    real(8), intent (in) :: n
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (x ** ((1.0d0 / n) + (-1.0d0))) / n
    if ((1.0d0 / n) <= (-2d-78)) then
        tmp = t_0
    else if ((1.0d0 / n) <= 2d-85) then
        tmp = log(((1.0d0 + x) / x)) / n
    else if ((1.0d0 / n) <= 4000000.0d0) then
        tmp = t_0
    else if ((1.0d0 / n) <= 5d+182) then
        tmp = 1.0d0 - (x ** (1.0d0 / n))
    else
        tmp = (n / x) / (n * n)
    end if
    code = tmp
end function

Java

public static double code(double x, double n) {
	double t_0 = Math.pow(x, ((1.0 / n) + -1.0)) / n;
	double tmp;
	if ((1.0 / n) <= -2e-78) {
		tmp = t_0;
	} else if ((1.0 / n) <= 2e-85) {
		tmp = Math.log(((1.0 + x) / x)) / n;
	} else if ((1.0 / n) <= 4000000.0) {
		tmp = t_0;
	} else if ((1.0 / n) <= 5e+182) {
		tmp = 1.0 - Math.pow(x, (1.0 / n));
	} else {
		tmp = (n / x) / (n * n);
	}
	return tmp;
}

Python

def code(x, n):
	t_0 = math.pow(x, ((1.0 / n) + -1.0)) / n
	tmp = 0
	if (1.0 / n) <= -2e-78:
		tmp = t_0
	elif (1.0 / n) <= 2e-85:
		tmp = math.log(((1.0 + x) / x)) / n
	elif (1.0 / n) <= 4000000.0:
		tmp = t_0
	elif (1.0 / n) <= 5e+182:
		tmp = 1.0 - math.pow(x, (1.0 / n))
	else:
		tmp = (n / x) / (n * n)
	return tmp

Julia

function code(x, n)
	t_0 = Float64((x ^ Float64(Float64(1.0 / n) + -1.0)) / n)
	tmp = 0.0
	if (Float64(1.0 / n) <= -2e-78)
		tmp = t_0;
	elseif (Float64(1.0 / n) <= 2e-85)
		tmp = Float64(log(Float64(Float64(1.0 + x) / x)) / n);
	elseif (Float64(1.0 / n) <= 4000000.0)
		tmp = t_0;
	elseif (Float64(1.0 / n) <= 5e+182)
		tmp = Float64(1.0 - (x ^ Float64(1.0 / n)));
	else
		tmp = Float64(Float64(n / x) / Float64(n * n));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, n)
	t_0 = (x ^ ((1.0 / n) + -1.0)) / n;
	tmp = 0.0;
	if ((1.0 / n) <= -2e-78)
		tmp = t_0;
	elseif ((1.0 / n) <= 2e-85)
		tmp = log(((1.0 + x) / x)) / n;
	elseif ((1.0 / n) <= 4000000.0)
		tmp = t_0;
	elseif ((1.0 / n) <= 5e+182)
		tmp = 1.0 - (x ^ (1.0 / n));
	else
		tmp = (n / x) / (n * n);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, n_] := Block[{t$95$0 = N[(N[Power[x, N[(N[(1.0 / n), $MachinePrecision] + -1.0), $MachinePrecision]], $MachinePrecision] / n), $MachinePrecision]}, If[LessEqual[N[(1.0 / n), $MachinePrecision], -2e-78], t$95$0, If[LessEqual[N[(1.0 / n), $MachinePrecision], 2e-85], N[(N[Log[N[(N[(1.0 + x), $MachinePrecision] / x), $MachinePrecision]], $MachinePrecision] / n), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 4000000.0], t$95$0, If[LessEqual[N[(1.0 / n), $MachinePrecision], 5e+182], N[(1.0 - N[Power[x, N[(1.0 / n), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[(n / x), $MachinePrecision] / N[(n * n), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if (/.f64 #s(literal 1 binary64) n) < -2e-78 or 2e-85 < (/.f64 #s(literal 1 binary64) n) < 4e6

   1. Initial program 74.8%

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

   3. Taylor expanded in x around 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. lower-/.f64 N/A

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

      2. log-rec N/A

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

      3. mul-1-neg N/A

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

      4. associate-*r/ N/A

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

      5. associate-*r* N/A

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

      6. metadata-eval N/A

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

      7. *-commutative N/A

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

      8. associate-/l* N/A

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

      9. exp-to-pow N/A

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

      10. lower-pow.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 91.3

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

   5. Applied rewrites 91.3%

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

   7. Applied rewrites 91.2%

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

   if -2e-78 < (/.f64 #s(literal 1 binary64) n) < 2e-85

   1. Initial program 35.9%

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

   3. Taylor expanded in n around inf

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

      1. lower-/.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lower-log1p.f64 N/A

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

      4. lower-log.f64 82.1

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

   5. Applied rewrites 82.1%

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

   7. Applied rewrites 82.4%

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

   if 4e6 < (/.f64 #s(literal 1 binary64) n) < 4.99999999999999973e182

   1. Initial program 79.8%

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

   3. Taylor expanded in x around 0

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

   5. Applied rewrites 72.6%

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

   if 4.99999999999999973e182 < (/.f64 #s(literal 1 binary64) n)

   1. Initial program 27.3%

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

   3. Taylor expanded in n around inf

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

      1. lower-/.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lower-log1p.f64 N/A

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

      4. lower-log.f64 22.0

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

   5. Applied rewrites 22.0%

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

   7. Applied rewrites 75.8%

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

   8. Taylor expanded in x around inf

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

   10. Applied rewrites 75.8%

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

4. Final simplification 84.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{1}{n} \leq -2 \cdot 10^{-78}:\\ \;\;\;\;\frac{{x}^{\left(\frac{1}{n} + -1\right)}}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 2 \cdot 10^{-85}:\\ \;\;\;\;\frac{\log \left(\frac{1 + x}{x}\right)}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 4000000:\\ \;\;\;\;\frac{{x}^{\left(\frac{1}{n} + -1\right)}}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 5 \cdot 10^{+182}:\\ \;\;\;\;1 - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{n}{x}}{n \cdot n}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 81.2% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{if}\;\frac{1}{n} \leq -200000:\\ \;\;\;\;\frac{t\_0}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 2 \cdot 10^{-85}:\\ \;\;\;\;\frac{\log \left(\frac{1 + x}{x}\right)}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 2 \cdot 10^{-8}:\\ \;\;\;\;\frac{1}{x \cdot \mathsf{fma}\left(0.5, \frac{n}{x}, n\right)}\\ \mathbf{elif}\;\frac{1}{n} \leq 5 \cdot 10^{+182}:\\ \;\;\;\;1 - t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{n}{x}}{n \cdot n}\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (let* ((t_0 (pow x (/ 1.0 n))))
   (if (<= (/ 1.0 n) -200000.0)
     (/ t_0 n)
     (if (<= (/ 1.0 n) 2e-85)
       (/ (log (/ (+ 1.0 x) x)) n)
       (if (<= (/ 1.0 n) 2e-8)
         (/ 1.0 (* x (fma 0.5 (/ n x) n)))
         (if (<= (/ 1.0 n) 5e+182) (- 1.0 t_0) (/ (/ n x) (* n n))))))))

C

double code(double x, double n) {
	double t_0 = pow(x, (1.0 / n));
	double tmp;
	if ((1.0 / n) <= -200000.0) {
		tmp = t_0 / n;
	} else if ((1.0 / n) <= 2e-85) {
		tmp = log(((1.0 + x) / x)) / n;
	} else if ((1.0 / n) <= 2e-8) {
		tmp = 1.0 / (x * fma(0.5, (n / x), n));
	} else if ((1.0 / n) <= 5e+182) {
		tmp = 1.0 - t_0;
	} else {
		tmp = (n / x) / (n * n);
	}
	return tmp;
}

Julia

function code(x, n)
	t_0 = x ^ Float64(1.0 / n)
	tmp = 0.0
	if (Float64(1.0 / n) <= -200000.0)
		tmp = Float64(t_0 / n);
	elseif (Float64(1.0 / n) <= 2e-85)
		tmp = Float64(log(Float64(Float64(1.0 + x) / x)) / n);
	elseif (Float64(1.0 / n) <= 2e-8)
		tmp = Float64(1.0 / Float64(x * fma(0.5, Float64(n / x), n)));
	elseif (Float64(1.0 / n) <= 5e+182)
		tmp = Float64(1.0 - t_0);
	else
		tmp = Float64(Float64(n / x) / Float64(n * n));
	end
	return tmp
end

Wolfram

code[x_, n_] := Block[{t$95$0 = N[Power[x, N[(1.0 / n), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[N[(1.0 / n), $MachinePrecision], -200000.0], N[(t$95$0 / n), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 2e-85], N[(N[Log[N[(N[(1.0 + x), $MachinePrecision] / x), $MachinePrecision]], $MachinePrecision] / n), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 2e-8], N[(1.0 / N[(x * N[(0.5 * N[(n / x), $MachinePrecision] + n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 5e+182], N[(1.0 - t$95$0), $MachinePrecision], N[(N[(n / x), $MachinePrecision] / N[(n * n), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 5 regimes

2. if (/.f64 #s(literal 1 binary64) n) < -2e5

   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 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. lower-/.f64 N/A

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

      2. log-rec N/A

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

      3. mul-1-neg N/A

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

      4. associate-*r/ N/A

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

      5. associate-*r* N/A

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

      6. metadata-eval N/A

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

      7. *-commutative N/A

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

      8. associate-/l* N/A

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

      9. exp-to-pow N/A

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

      10. lower-pow.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 100.0

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

   5. Applied rewrites 100.0%

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

   7. Applied rewrites 100.0%

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

   8. Taylor expanded in n around 0

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

   10. Applied rewrites 100.0%

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

   if -2e5 < (/.f64 #s(literal 1 binary64) n) < 2e-85

   1. Initial program 34.6%

      \[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)} \]
   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 78.0

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

   5. Applied rewrites 78.0%

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

   7. Applied rewrites 78.3%

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

   if 2e-85 < (/.f64 #s(literal 1 binary64) n) < 2e-8

   1. Initial program 17.6%

      \[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)} \]
   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 24.9

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

   5. Applied rewrites 24.9%

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

   7. Applied rewrites 24.9%

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

   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 80.1%

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

   if 2e-8 < (/.f64 #s(literal 1 binary64) n) < 4.99999999999999973e182

   1. Initial program 77.6%

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

   3. Taylor expanded in x around 0

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

   5. Applied rewrites 70.9%

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

   if 4.99999999999999973e182 < (/.f64 #s(literal 1 binary64) n)

   1. Initial program 27.3%

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

   3. Taylor expanded in n around inf

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

      1. lower-/.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lower-log1p.f64 N/A

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

      4. lower-log.f64 22.0

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

   5. Applied rewrites 22.0%

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

   7. Applied rewrites 75.8%

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

   8. Taylor expanded in x around inf

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

   10. Applied rewrites 75.8%

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

4. Final simplification 83.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{1}{n} \leq -200000:\\ \;\;\;\;\frac{{x}^{\left(\frac{1}{n}\right)}}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 2 \cdot 10^{-85}:\\ \;\;\;\;\frac{\log \left(\frac{1 + x}{x}\right)}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 2 \cdot 10^{-8}:\\ \;\;\;\;\frac{1}{x \cdot \mathsf{fma}\left(0.5, \frac{n}{x}, n\right)}\\ \mathbf{elif}\;\frac{1}{n} \leq 5 \cdot 10^{+182}:\\ \;\;\;\;1 - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{n}{x}}{n \cdot n}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 69.0% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{if}\;\frac{1}{n} \leq -200000:\\ \;\;\;\;\frac{t\_0}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 2 \cdot 10^{-8}:\\ \;\;\;\;\frac{1}{x \cdot \mathsf{fma}\left(0.5, \frac{n}{x}, n\right)}\\ \mathbf{elif}\;\frac{1}{n} \leq 5 \cdot 10^{+182}:\\ \;\;\;\;1 - t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{n}{x}}{n \cdot n}\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (let* ((t_0 (pow x (/ 1.0 n))))
   (if (<= (/ 1.0 n) -200000.0)
     (/ t_0 n)
     (if (<= (/ 1.0 n) 2e-8)
       (/ 1.0 (* x (fma 0.5 (/ n x) n)))
       (if (<= (/ 1.0 n) 5e+182) (- 1.0 t_0) (/ (/ n x) (* n n)))))))

C

double code(double x, double n) {
	double t_0 = pow(x, (1.0 / n));
	double tmp;
	if ((1.0 / n) <= -200000.0) {
		tmp = t_0 / n;
	} else if ((1.0 / n) <= 2e-8) {
		tmp = 1.0 / (x * fma(0.5, (n / x), n));
	} else if ((1.0 / n) <= 5e+182) {
		tmp = 1.0 - t_0;
	} else {
		tmp = (n / x) / (n * n);
	}
	return tmp;
}

Julia

function code(x, n)
	t_0 = x ^ Float64(1.0 / n)
	tmp = 0.0
	if (Float64(1.0 / n) <= -200000.0)
		tmp = Float64(t_0 / n);
	elseif (Float64(1.0 / n) <= 2e-8)
		tmp = Float64(1.0 / Float64(x * fma(0.5, Float64(n / x), n)));
	elseif (Float64(1.0 / n) <= 5e+182)
		tmp = Float64(1.0 - t_0);
	else
		tmp = Float64(Float64(n / x) / Float64(n * n));
	end
	return tmp
end

Wolfram

code[x_, n_] := Block[{t$95$0 = N[Power[x, N[(1.0 / n), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[N[(1.0 / n), $MachinePrecision], -200000.0], N[(t$95$0 / n), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 2e-8], N[(1.0 / N[(x * N[(0.5 * N[(n / x), $MachinePrecision] + n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 5e+182], N[(1.0 - t$95$0), $MachinePrecision], N[(N[(n / x), $MachinePrecision] / N[(n * n), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if (/.f64 #s(literal 1 binary64) n) < -2e5

   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 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. lower-/.f64 N/A

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

      2. log-rec N/A

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

      3. mul-1-neg N/A

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

      4. associate-*r/ N/A

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

      5. associate-*r* N/A

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

      6. metadata-eval N/A

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

      7. *-commutative N/A

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

      8. associate-/l* N/A

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

      9. exp-to-pow N/A

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

      10. lower-pow.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 100.0

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

   5. Applied rewrites 100.0%

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

   7. Applied rewrites 100.0%

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

   8. Taylor expanded in n around 0

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

   10. Applied rewrites 100.0%

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

   if -2e5 < (/.f64 #s(literal 1 binary64) n) < 2e-8

   1. Initial program 33.1%

      \[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)} \]
   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 73.2

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

   5. Applied rewrites 73.2%

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

   7. Applied rewrites 73.3%

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

   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 59.6%

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

   if 2e-8 < (/.f64 #s(literal 1 binary64) n) < 4.99999999999999973e182

   1. Initial program 77.6%

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

   3. Taylor expanded in x around 0

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

   5. Applied rewrites 70.9%

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

   if 4.99999999999999973e182 < (/.f64 #s(literal 1 binary64) n)

   1. Initial program 27.3%

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

   3. Taylor expanded in n around inf

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

      1. lower-/.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lower-log1p.f64 N/A

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

      4. lower-log.f64 22.0

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

   5. Applied rewrites 22.0%

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

   7. Applied rewrites 75.8%

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

   8. Taylor expanded in x around inf

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

   10. Applied rewrites 75.8%

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

Alternative 9: 62.3% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{1}{n} \leq -200000:\\ \;\;\;\;\frac{\frac{0.3333333333333333}{x \cdot \left(x \cdot x\right)}}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 2 \cdot 10^{-8}:\\ \;\;\;\;\frac{1}{x \cdot \mathsf{fma}\left(0.5, \frac{n}{x}, n\right)}\\ \mathbf{elif}\;\frac{1}{n} \leq 5 \cdot 10^{+182}:\\ \;\;\;\;1 - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{n}{x}}{n \cdot n}\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (if (<= (/ 1.0 n) -200000.0)
   (/ (/ 0.3333333333333333 (* x (* x x))) n)
   (if (<= (/ 1.0 n) 2e-8)
     (/ 1.0 (* x (fma 0.5 (/ n x) n)))
     (if (<= (/ 1.0 n) 5e+182)
       (- 1.0 (pow x (/ 1.0 n)))
       (/ (/ n x) (* n n))))))

C

double code(double x, double n) {
	double tmp;
	if ((1.0 / n) <= -200000.0) {
		tmp = (0.3333333333333333 / (x * (x * x))) / n;
	} else if ((1.0 / n) <= 2e-8) {
		tmp = 1.0 / (x * fma(0.5, (n / x), n));
	} else if ((1.0 / n) <= 5e+182) {
		tmp = 1.0 - pow(x, (1.0 / n));
	} else {
		tmp = (n / x) / (n * n);
	}
	return tmp;
}

Julia

function code(x, n)
	tmp = 0.0
	if (Float64(1.0 / n) <= -200000.0)
		tmp = Float64(Float64(0.3333333333333333 / Float64(x * Float64(x * x))) / n);
	elseif (Float64(1.0 / n) <= 2e-8)
		tmp = Float64(1.0 / Float64(x * fma(0.5, Float64(n / x), n)));
	elseif (Float64(1.0 / n) <= 5e+182)
		tmp = Float64(1.0 - (x ^ Float64(1.0 / n)));
	else
		tmp = Float64(Float64(n / x) / Float64(n * n));
	end
	return tmp
end

Wolfram

code[x_, n_] := If[LessEqual[N[(1.0 / n), $MachinePrecision], -200000.0], N[(N[(0.3333333333333333 / N[(x * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / n), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 2e-8], N[(1.0 / N[(x * N[(0.5 * N[(n / x), $MachinePrecision] + n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 5e+182], N[(1.0 - N[Power[x, N[(1.0 / n), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[(n / x), $MachinePrecision] / N[(n * n), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if (/.f64 #s(literal 1 binary64) n) < -2e5

   1. Initial program 100.0%

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

   3. Taylor expanded in n around inf

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

      1. lower-/.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lower-log1p.f64 N/A

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

      4. lower-log.f64 46.5

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

   5. Applied rewrites 46.5%

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

   6. Taylor expanded in x around inf

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

   8. Applied rewrites 44.6%

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

   9. Taylor expanded in x around 0

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

   11. Applied rewrites 74.6%

       \[\leadsto \frac{\frac{0.3333333333333333}{x \cdot \left(x \cdot x\right)}}{n} \]

   if -2e5 < (/.f64 #s(literal 1 binary64) n) < 2e-8

   1. Initial program 33.1%

      \[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)} \]
   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 73.2

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

   5. Applied rewrites 73.2%

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

   7. Applied rewrites 73.3%

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

   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 59.6%

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

   if 2e-8 < (/.f64 #s(literal 1 binary64) n) < 4.99999999999999973e182

   1. Initial program 77.6%

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

   3. Taylor expanded in x around 0

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

   5. Applied rewrites 70.9%

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

   if 4.99999999999999973e182 < (/.f64 #s(literal 1 binary64) n)

   1. Initial program 27.3%

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

   3. Taylor expanded in n around inf

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

      1. lower-/.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lower-log1p.f64 N/A

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

      4. lower-log.f64 22.0

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

   5. Applied rewrites 22.0%

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

   7. Applied rewrites 75.8%

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

   8. Taylor expanded in x around inf

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

   10. Applied rewrites 75.8%

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

Alternative 10: 60.1% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 4.5 \cdot 10^{-295}:\\ \;\;\;\;\frac{1}{n \cdot x}\\ \mathbf{elif}\;x \leq 0.0078:\\ \;\;\;\;\frac{x - \log x}{n}\\ \mathbf{elif}\;x \leq 1.06 \cdot 10^{+142}:\\ \;\;\;\;\frac{\frac{1 + \frac{-0.5}{x}}{n} + \frac{0.3333333333333333}{x \cdot \left(n \cdot x\right)}}{x}\\ \mathbf{else}:\\ \;\;\;\;1 - 1\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (if (<= x 4.5e-295)
   (/ 1.0 (* n x))
   (if (<= x 0.0078)
     (/ (- x (log x)) n)
     (if (<= x 1.06e+142)
       (/ (+ (/ (+ 1.0 (/ -0.5 x)) n) (/ 0.3333333333333333 (* x (* n x)))) x)
       (- 1.0 1.0)))))

C

double code(double x, double n) {
	double tmp;
	if (x <= 4.5e-295) {
		tmp = 1.0 / (n * x);
	} else if (x <= 0.0078) {
		tmp = (x - log(x)) / n;
	} else if (x <= 1.06e+142) {
		tmp = (((1.0 + (-0.5 / x)) / n) + (0.3333333333333333 / (x * (n * x)))) / x;
	} 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 <= 4.5d-295) then
        tmp = 1.0d0 / (n * x)
    else if (x <= 0.0078d0) then
        tmp = (x - log(x)) / n
    else if (x <= 1.06d+142) then
        tmp = (((1.0d0 + ((-0.5d0) / x)) / n) + (0.3333333333333333d0 / (x * (n * x)))) / x
    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 <= 4.5e-295) {
		tmp = 1.0 / (n * x);
	} else if (x <= 0.0078) {
		tmp = (x - Math.log(x)) / n;
	} else if (x <= 1.06e+142) {
		tmp = (((1.0 + (-0.5 / x)) / n) + (0.3333333333333333 / (x * (n * x)))) / x;
	} else {
		tmp = 1.0 - 1.0;
	}
	return tmp;
}

Python

def code(x, n):
	tmp = 0
	if x <= 4.5e-295:
		tmp = 1.0 / (n * x)
	elif x <= 0.0078:
		tmp = (x - math.log(x)) / n
	elif x <= 1.06e+142:
		tmp = (((1.0 + (-0.5 / x)) / n) + (0.3333333333333333 / (x * (n * x)))) / x
	else:
		tmp = 1.0 - 1.0
	return tmp

Julia

function code(x, n)
	tmp = 0.0
	if (x <= 4.5e-295)
		tmp = Float64(1.0 / Float64(n * x));
	elseif (x <= 0.0078)
		tmp = Float64(Float64(x - log(x)) / n);
	elseif (x <= 1.06e+142)
		tmp = Float64(Float64(Float64(Float64(1.0 + Float64(-0.5 / x)) / n) + Float64(0.3333333333333333 / Float64(x * Float64(n * x)))) / x);
	else
		tmp = Float64(1.0 - 1.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, n)
	tmp = 0.0;
	if (x <= 4.5e-295)
		tmp = 1.0 / (n * x);
	elseif (x <= 0.0078)
		tmp = (x - log(x)) / n;
	elseif (x <= 1.06e+142)
		tmp = (((1.0 + (-0.5 / x)) / n) + (0.3333333333333333 / (x * (n * x)))) / x;
	else
		tmp = 1.0 - 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, n_] := If[LessEqual[x, 4.5e-295], N[(1.0 / N[(n * x), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 0.0078], N[(N[(x - N[Log[x], $MachinePrecision]), $MachinePrecision] / n), $MachinePrecision], If[LessEqual[x, 1.06e+142], N[(N[(N[(N[(1.0 + N[(-0.5 / x), $MachinePrecision]), $MachinePrecision] / n), $MachinePrecision] + N[(0.3333333333333333 / N[(x * N[(n * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision], N[(1.0 - 1.0), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if x < 4.5000000000000002e-295

   1. Initial program 100.0%

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

   3. Taylor expanded in n around inf

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

      1. lower-/.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lower-log1p.f64 N/A

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

      4. lower-log.f64 4.6

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

   5. Applied rewrites 4.6%

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

   6. Taylor expanded in x around inf

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

   8. Applied rewrites 83.9%

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

   if 4.5000000000000002e-295 < x < 0.0077999999999999996

   1. Initial program 46.2%

      \[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)} \]
   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 50.3

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

   5. Applied rewrites 50.3%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 49.8%

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

   if 0.0077999999999999996 < x < 1.06e142

   1. Initial program 44.6%

      \[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)} \]
   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 42.6

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

   5. Applied rewrites 42.6%

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

   6. Taylor expanded in x around inf

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

   8. Applied rewrites 68.7%

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

   if 1.06e142 < x

   1. Initial program 86.7%

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

   3. Taylor expanded in x around 0

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

   5. Applied rewrites 58.7%

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

   6. Taylor expanded in n around inf

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

   8. Applied rewrites 86.7%

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

4. Final simplification 63.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 4.5 \cdot 10^{-295}:\\ \;\;\;\;\frac{1}{n \cdot x}\\ \mathbf{elif}\;x \leq 0.0078:\\ \;\;\;\;\frac{x - \log x}{n}\\ \mathbf{elif}\;x \leq 1.06 \cdot 10^{+142}:\\ \;\;\;\;\frac{\frac{1 + \frac{-0.5}{x}}{n} + \frac{0.3333333333333333}{x \cdot \left(n \cdot x\right)}}{x}\\ \mathbf{else}:\\ \;\;\;\;1 - 1\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 59.9% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 4.5 \cdot 10^{-295}:\\ \;\;\;\;\frac{1}{n \cdot x}\\ \mathbf{elif}\;x \leq 0.0078:\\ \;\;\;\;\frac{-\log x}{n}\\ \mathbf{elif}\;x \leq 1.06 \cdot 10^{+142}:\\ \;\;\;\;\frac{\frac{1 + \frac{-0.5}{x}}{n} + \frac{0.3333333333333333}{x \cdot \left(n \cdot x\right)}}{x}\\ \mathbf{else}:\\ \;\;\;\;1 - 1\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (if (<= x 4.5e-295)
   (/ 1.0 (* n x))
   (if (<= x 0.0078)
     (/ (- (log x)) n)
     (if (<= x 1.06e+142)
       (/ (+ (/ (+ 1.0 (/ -0.5 x)) n) (/ 0.3333333333333333 (* x (* n x)))) x)
       (- 1.0 1.0)))))

C

double code(double x, double n) {
	double tmp;
	if (x <= 4.5e-295) {
		tmp = 1.0 / (n * x);
	} else if (x <= 0.0078) {
		tmp = -log(x) / n;
	} else if (x <= 1.06e+142) {
		tmp = (((1.0 + (-0.5 / x)) / n) + (0.3333333333333333 / (x * (n * x)))) / x;
	} 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 <= 4.5d-295) then
        tmp = 1.0d0 / (n * x)
    else if (x <= 0.0078d0) then
        tmp = -log(x) / n
    else if (x <= 1.06d+142) then
        tmp = (((1.0d0 + ((-0.5d0) / x)) / n) + (0.3333333333333333d0 / (x * (n * x)))) / x
    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 <= 4.5e-295) {
		tmp = 1.0 / (n * x);
	} else if (x <= 0.0078) {
		tmp = -Math.log(x) / n;
	} else if (x <= 1.06e+142) {
		tmp = (((1.0 + (-0.5 / x)) / n) + (0.3333333333333333 / (x * (n * x)))) / x;
	} else {
		tmp = 1.0 - 1.0;
	}
	return tmp;
}

Python

def code(x, n):
	tmp = 0
	if x <= 4.5e-295:
		tmp = 1.0 / (n * x)
	elif x <= 0.0078:
		tmp = -math.log(x) / n
	elif x <= 1.06e+142:
		tmp = (((1.0 + (-0.5 / x)) / n) + (0.3333333333333333 / (x * (n * x)))) / x
	else:
		tmp = 1.0 - 1.0
	return tmp

Julia

function code(x, n)
	tmp = 0.0
	if (x <= 4.5e-295)
		tmp = Float64(1.0 / Float64(n * x));
	elseif (x <= 0.0078)
		tmp = Float64(Float64(-log(x)) / n);
	elseif (x <= 1.06e+142)
		tmp = Float64(Float64(Float64(Float64(1.0 + Float64(-0.5 / x)) / n) + Float64(0.3333333333333333 / Float64(x * Float64(n * x)))) / x);
	else
		tmp = Float64(1.0 - 1.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, n)
	tmp = 0.0;
	if (x <= 4.5e-295)
		tmp = 1.0 / (n * x);
	elseif (x <= 0.0078)
		tmp = -log(x) / n;
	elseif (x <= 1.06e+142)
		tmp = (((1.0 + (-0.5 / x)) / n) + (0.3333333333333333 / (x * (n * x)))) / x;
	else
		tmp = 1.0 - 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, n_] := If[LessEqual[x, 4.5e-295], N[(1.0 / N[(n * x), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 0.0078], N[((-N[Log[x], $MachinePrecision]) / n), $MachinePrecision], If[LessEqual[x, 1.06e+142], N[(N[(N[(N[(1.0 + N[(-0.5 / x), $MachinePrecision]), $MachinePrecision] / n), $MachinePrecision] + N[(0.3333333333333333 / N[(x * N[(n * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision], N[(1.0 - 1.0), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if x < 4.5000000000000002e-295

   1. Initial program 100.0%

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

   3. Taylor expanded in n around inf

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

      1. lower-/.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lower-log1p.f64 N/A

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

      4. lower-log.f64 4.6

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

   5. Applied rewrites 4.6%

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

   6. Taylor expanded in x around inf

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

   8. Applied rewrites 83.9%

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

   if 4.5000000000000002e-295 < x < 0.0077999999999999996

   1. Initial program 46.2%

      \[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)} \]
   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 50.3

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

   5. Applied rewrites 50.3%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 49.4%

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

   if 0.0077999999999999996 < x < 1.06e142

   1. Initial program 44.6%

      \[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)} \]
   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 42.6

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

   5. Applied rewrites 42.6%

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

   6. Taylor expanded in x around inf

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

   8. Applied rewrites 68.7%

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

   if 1.06e142 < x

   1. Initial program 86.7%

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

   3. Taylor expanded in x around 0

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

   5. Applied rewrites 58.7%

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

   6. Taylor expanded in n around inf

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

   8. Applied rewrites 86.7%

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

4. Final simplification 63.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 4.5 \cdot 10^{-295}:\\ \;\;\;\;\frac{1}{n \cdot x}\\ \mathbf{elif}\;x \leq 0.0078:\\ \;\;\;\;\frac{-\log x}{n}\\ \mathbf{elif}\;x \leq 1.06 \cdot 10^{+142}:\\ \;\;\;\;\frac{\frac{1 + \frac{-0.5}{x}}{n} + \frac{0.3333333333333333}{x \cdot \left(n \cdot x\right)}}{x}\\ \mathbf{else}:\\ \;\;\;\;1 - 1\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 58.8% accurate, 2.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{1}{n} \leq -200000:\\ \;\;\;\;\frac{\frac{0.3333333333333333}{x \cdot \left(x \cdot x\right)}}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 5 \cdot 10^{+106}:\\ \;\;\;\;\frac{1}{x \cdot \mathsf{fma}\left(0.5, \frac{n}{x}, n\right)}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x, \mathsf{fma}\left(x, \frac{x \cdot 0.16666666666666666}{n \cdot \left(n \cdot n\right)}, \frac{1}{n}\right), 1\right) - 1\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (if (<= (/ 1.0 n) -200000.0)
   (/ (/ 0.3333333333333333 (* x (* x x))) n)
   (if (<= (/ 1.0 n) 5e+106)
     (/ 1.0 (* x (fma 0.5 (/ n x) n)))
     (-
      (fma x (fma x (/ (* x 0.16666666666666666) (* n (* n n))) (/ 1.0 n)) 1.0)
      1.0))))

C

double code(double x, double n) {
	double tmp;
	if ((1.0 / n) <= -200000.0) {
		tmp = (0.3333333333333333 / (x * (x * x))) / n;
	} else if ((1.0 / n) <= 5e+106) {
		tmp = 1.0 / (x * fma(0.5, (n / x), n));
	} else {
		tmp = fma(x, fma(x, ((x * 0.16666666666666666) / (n * (n * n))), (1.0 / n)), 1.0) - 1.0;
	}
	return tmp;
}

Julia

function code(x, n)
	tmp = 0.0
	if (Float64(1.0 / n) <= -200000.0)
		tmp = Float64(Float64(0.3333333333333333 / Float64(x * Float64(x * x))) / n);
	elseif (Float64(1.0 / n) <= 5e+106)
		tmp = Float64(1.0 / Float64(x * fma(0.5, Float64(n / x), n)));
	else
		tmp = Float64(fma(x, fma(x, Float64(Float64(x * 0.16666666666666666) / Float64(n * Float64(n * n))), Float64(1.0 / n)), 1.0) - 1.0);
	end
	return tmp
end

Wolfram

code[x_, n_] := If[LessEqual[N[(1.0 / n), $MachinePrecision], -200000.0], N[(N[(0.3333333333333333 / N[(x * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / n), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 5e+106], N[(1.0 / N[(x * N[(0.5 * N[(n / x), $MachinePrecision] + n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x * N[(x * N[(N[(x * 0.16666666666666666), $MachinePrecision] / N[(n * N[(n * n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(1.0 / n), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision] - 1.0), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 #s(literal 1 binary64) n) < -2e5

   1. Initial program 100.0%

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

   3. Taylor expanded in n around inf

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

      1. lower-/.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lower-log1p.f64 N/A

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

      4. lower-log.f64 46.5

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

   5. Applied rewrites 46.5%

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

   6. Taylor expanded in x around inf

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

   8. Applied rewrites 44.6%

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

   9. Taylor expanded in x around 0

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

   11. Applied rewrites 74.6%

       \[\leadsto \frac{\frac{0.3333333333333333}{x \cdot \left(x \cdot x\right)}}{n} \]

   if -2e5 < (/.f64 #s(literal 1 binary64) n) < 4.9999999999999998e106

   1. Initial program 38.1%

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

   3. Taylor expanded in n around inf

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

      1. lower-/.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lower-log1p.f64 N/A

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

      4. lower-log.f64 67.0

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

   5. Applied rewrites 67.0%

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

   7. Applied rewrites 67.1%

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

   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 54.6%

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

   if 4.9999999999999998e106 < (/.f64 #s(literal 1 binary64) n)

   1. Initial program 49.2%

      \[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)} \]
   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 41.7%

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

   6. Taylor expanded in n around inf

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

   8. Applied rewrites 2.3%

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

   9. Taylor expanded in x around 0

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

       1. +-commutative N/A

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

       2. lower-fma.f64 N/A

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

   11. Applied rewrites 12.4%

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

   12. Taylor expanded in n around 0

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

   14. Applied rewrites 55.3%

       \[\leadsto \mathsf{fma}\left(x, \mathsf{fma}\left(x, \frac{x \cdot 0.16666666666666666}{\color{blue}{n \cdot \left(n \cdot n\right)}}, \frac{1}{n}\right), 1\right) - 1 \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 13: 57.3% accurate, 3.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{1}{n} \leq -200000:\\ \;\;\;\;\frac{\frac{0.3333333333333333}{x \cdot \left(x \cdot x\right)}}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 5 \cdot 10^{+106}:\\ \;\;\;\;\frac{1}{x \cdot \mathsf{fma}\left(0.5, \frac{n}{x}, n\right)}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x, \frac{\left(x \cdot x\right) \cdot 0.16666666666666666}{n \cdot \left(n \cdot n\right)}, 1\right) - 1\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (if (<= (/ 1.0 n) -200000.0)
   (/ (/ 0.3333333333333333 (* x (* x x))) n)
   (if (<= (/ 1.0 n) 5e+106)
     (/ 1.0 (* x (fma 0.5 (/ n x) n)))
     (- (fma x (/ (* (* x x) 0.16666666666666666) (* n (* n n))) 1.0) 1.0))))

C

double code(double x, double n) {
	double tmp;
	if ((1.0 / n) <= -200000.0) {
		tmp = (0.3333333333333333 / (x * (x * x))) / n;
	} else if ((1.0 / n) <= 5e+106) {
		tmp = 1.0 / (x * fma(0.5, (n / x), n));
	} else {
		tmp = fma(x, (((x * x) * 0.16666666666666666) / (n * (n * n))), 1.0) - 1.0;
	}
	return tmp;
}

Julia

function code(x, n)
	tmp = 0.0
	if (Float64(1.0 / n) <= -200000.0)
		tmp = Float64(Float64(0.3333333333333333 / Float64(x * Float64(x * x))) / n);
	elseif (Float64(1.0 / n) <= 5e+106)
		tmp = Float64(1.0 / Float64(x * fma(0.5, Float64(n / x), n)));
	else
		tmp = Float64(fma(x, Float64(Float64(Float64(x * x) * 0.16666666666666666) / Float64(n * Float64(n * n))), 1.0) - 1.0);
	end
	return tmp
end

Wolfram

code[x_, n_] := If[LessEqual[N[(1.0 / n), $MachinePrecision], -200000.0], N[(N[(0.3333333333333333 / N[(x * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / n), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 5e+106], N[(1.0 / N[(x * N[(0.5 * N[(n / x), $MachinePrecision] + n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x * N[(N[(N[(x * x), $MachinePrecision] * 0.16666666666666666), $MachinePrecision] / N[(n * N[(n * n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision] - 1.0), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 #s(literal 1 binary64) n) < -2e5

   1. Initial program 100.0%

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

   3. Taylor expanded in n around inf

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

      1. lower-/.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lower-log1p.f64 N/A

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

      4. lower-log.f64 46.5

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

   5. Applied rewrites 46.5%

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

   6. Taylor expanded in x around inf

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

   8. Applied rewrites 44.6%

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

   9. Taylor expanded in x around 0

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

   11. Applied rewrites 74.6%

       \[\leadsto \frac{\frac{0.3333333333333333}{x \cdot \left(x \cdot x\right)}}{n} \]

   if -2e5 < (/.f64 #s(literal 1 binary64) n) < 4.9999999999999998e106

   1. Initial program 38.1%

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

   3. Taylor expanded in n around inf

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

      1. lower-/.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lower-log1p.f64 N/A

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

      4. lower-log.f64 67.0

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

   5. Applied rewrites 67.0%

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

   7. Applied rewrites 67.1%

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

   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 54.6%

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

   if 4.9999999999999998e106 < (/.f64 #s(literal 1 binary64) n)

   1. Initial program 49.2%

      \[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)} \]
   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 41.7%

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

   6. Taylor expanded in n around inf

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

   8. Applied rewrites 2.3%

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

   9. Taylor expanded in x around 0

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

       1. +-commutative N/A

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

       2. lower-fma.f64 N/A

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

   11. Applied rewrites 12.4%

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

   12. Taylor expanded in n around 0

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

   14. Applied rewrites 42.5%

       \[\leadsto \mathsf{fma}\left(x, \frac{0.16666666666666666 \cdot \left(x \cdot x\right)}{\color{blue}{n \cdot \left(n \cdot n\right)}}, 1\right) - 1 \]
3. Recombined 3 regimes into one program.

4. Final simplification 58.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{1}{n} \leq -200000:\\ \;\;\;\;\frac{\frac{0.3333333333333333}{x \cdot \left(x \cdot x\right)}}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 5 \cdot 10^{+106}:\\ \;\;\;\;\frac{1}{x \cdot \mathsf{fma}\left(0.5, \frac{n}{x}, n\right)}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x, \frac{\left(x \cdot x\right) \cdot 0.16666666666666666}{n \cdot \left(n \cdot n\right)}, 1\right) - 1\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 57.8% accurate, 3.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\frac{0.3333333333333333}{x \cdot \left(x \cdot x\right)}}{n}\\ \mathbf{if}\;\frac{1}{n} \leq -200000:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;\frac{1}{n} \leq 5 \cdot 10^{+106}:\\ \;\;\;\;\frac{1}{x \cdot \mathsf{fma}\left(0.5, \frac{n}{x}, n\right)}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (let* ((t_0 (/ (/ 0.3333333333333333 (* x (* x x))) n)))
   (if (<= (/ 1.0 n) -200000.0)
     t_0
     (if (<= (/ 1.0 n) 5e+106) (/ 1.0 (* x (fma 0.5 (/ n x) n))) t_0))))

C

double code(double x, double n) {
	double t_0 = (0.3333333333333333 / (x * (x * x))) / n;
	double tmp;
	if ((1.0 / n) <= -200000.0) {
		tmp = t_0;
	} else if ((1.0 / n) <= 5e+106) {
		tmp = 1.0 / (x * fma(0.5, (n / x), n));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(x, n)
	t_0 = Float64(Float64(0.3333333333333333 / Float64(x * Float64(x * x))) / n)
	tmp = 0.0
	if (Float64(1.0 / n) <= -200000.0)
		tmp = t_0;
	elseif (Float64(1.0 / n) <= 5e+106)
		tmp = Float64(1.0 / Float64(x * fma(0.5, Float64(n / x), n)));
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[x_, n_] := Block[{t$95$0 = N[(N[(0.3333333333333333 / N[(x * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / n), $MachinePrecision]}, If[LessEqual[N[(1.0 / n), $MachinePrecision], -200000.0], t$95$0, If[LessEqual[N[(1.0 / n), $MachinePrecision], 5e+106], N[(1.0 / N[(x * N[(0.5 * N[(n / x), $MachinePrecision] + n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 #s(literal 1 binary64) n) < -2e5 or 4.9999999999999998e106 < (/.f64 #s(literal 1 binary64) n)

   1. Initial program 86.4%

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

   3. Taylor expanded in n around inf

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

      1. lower-/.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lower-log1p.f64 N/A

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

      4. lower-log.f64 37.4

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

   5. Applied rewrites 37.4%

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

   6. Taylor expanded in x around inf

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

   8. Applied rewrites 43.6%

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

   9. Taylor expanded in x around 0

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

   11. Applied rewrites 65.5%

       \[\leadsto \frac{\frac{0.3333333333333333}{x \cdot \left(x \cdot x\right)}}{n} \]

   if -2e5 < (/.f64 #s(literal 1 binary64) n) < 4.9999999999999998e106

   1. Initial program 38.1%

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

   3. Taylor expanded in n around inf

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

      1. lower-/.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lower-log1p.f64 N/A

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

      4. lower-log.f64 67.0

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

   5. Applied rewrites 67.0%

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

   7. Applied rewrites 67.1%

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

   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 54.6%

       \[\leadsto \frac{1}{x \cdot \color{blue}{\mathsf{fma}\left(0.5, \frac{n}{x}, n\right)}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 15: 56.0% accurate, 3.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{1}{n} \leq -4 \cdot 10^{+21}:\\ \;\;\;\;\frac{\frac{0.3333333333333333}{x \cdot \left(x \cdot x\right)}}{n}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 + \frac{-0.5 + \frac{0.3333333333333333}{x}}{x}}{x}}{n}\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (if (<= (/ 1.0 n) -4e+21)
   (/ (/ 0.3333333333333333 (* x (* x x))) n)
   (/ (/ (+ 1.0 (/ (+ -0.5 (/ 0.3333333333333333 x)) x)) x) n)))

C

double code(double x, double n) {
	double tmp;
	if ((1.0 / n) <= -4e+21) {
		tmp = (0.3333333333333333 / (x * (x * x))) / n;
	} else {
		tmp = ((1.0 + ((-0.5 + (0.3333333333333333 / x)) / x)) / x) / n;
	}
	return tmp;
}

Fortran

real(8) function code(x, n)
    real(8), intent (in) :: x
    real(8), intent (in) :: n
    real(8) :: tmp
    if ((1.0d0 / n) <= (-4d+21)) then
        tmp = (0.3333333333333333d0 / (x * (x * x))) / n
    else
        tmp = ((1.0d0 + (((-0.5d0) + (0.3333333333333333d0 / x)) / x)) / x) / n
    end if
    code = tmp
end function

Java

public static double code(double x, double n) {
	double tmp;
	if ((1.0 / n) <= -4e+21) {
		tmp = (0.3333333333333333 / (x * (x * x))) / n;
	} else {
		tmp = ((1.0 + ((-0.5 + (0.3333333333333333 / x)) / x)) / x) / n;
	}
	return tmp;
}

Python

def code(x, n):
	tmp = 0
	if (1.0 / n) <= -4e+21:
		tmp = (0.3333333333333333 / (x * (x * x))) / n
	else:
		tmp = ((1.0 + ((-0.5 + (0.3333333333333333 / x)) / x)) / x) / n
	return tmp

Julia

function code(x, n)
	tmp = 0.0
	if (Float64(1.0 / n) <= -4e+21)
		tmp = Float64(Float64(0.3333333333333333 / Float64(x * Float64(x * x))) / n);
	else
		tmp = Float64(Float64(Float64(1.0 + Float64(Float64(-0.5 + Float64(0.3333333333333333 / x)) / x)) / x) / n);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, n)
	tmp = 0.0;
	if ((1.0 / n) <= -4e+21)
		tmp = (0.3333333333333333 / (x * (x * x))) / n;
	else
		tmp = ((1.0 + ((-0.5 + (0.3333333333333333 / x)) / x)) / x) / n;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, n_] := If[LessEqual[N[(1.0 / n), $MachinePrecision], -4e+21], N[(N[(0.3333333333333333 / N[(x * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / n), $MachinePrecision], N[(N[(N[(1.0 + N[(N[(-0.5 + N[(0.3333333333333333 / x), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision] / n), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 #s(literal 1 binary64) n) < -4e21

   1. Initial program 100.0%

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

   3. Taylor expanded in n around inf

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

      1. lower-/.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lower-log1p.f64 N/A

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

      4. lower-log.f64 49.7

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

   5. Applied rewrites 49.7%

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

   6. Taylor expanded in x around inf

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

   8. Applied rewrites 43.2%

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

   9. Taylor expanded in x around 0

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

   11. Applied rewrites 75.4%

       \[\leadsto \frac{\frac{0.3333333333333333}{x \cdot \left(x \cdot x\right)}}{n} \]

   if -4e21 < (/.f64 #s(literal 1 binary64) n)

   1. Initial program 41.3%

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

   3. Taylor expanded in n around inf

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

      1. lower-/.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lower-log1p.f64 N/A

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

      4. lower-log.f64 57.9

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

   5. Applied rewrites 57.9%

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

   6. Taylor expanded in x around inf

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

   8. Applied rewrites 51.6%

      \[\leadsto \frac{\frac{1 + \frac{-0.5 + \frac{0.3333333333333333}{x}}{x}}{x}}{n} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 16: 55.1% accurate, 4.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{1}{n} \leq -200000:\\ \;\;\;\;\frac{\frac{0.3333333333333333}{x \cdot \left(x \cdot x\right)}}{n}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{n}}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (if (<= (/ 1.0 n) -200000.0)
   (/ (/ 0.3333333333333333 (* x (* x x))) n)
   (/ (/ 1.0 n) x)))

C

double code(double x, double n) {
	double tmp;
	if ((1.0 / n) <= -200000.0) {
		tmp = (0.3333333333333333 / (x * (x * x))) / 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) :: tmp
    if ((1.0d0 / n) <= (-200000.0d0)) then
        tmp = (0.3333333333333333d0 / (x * (x * x))) / n
    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) <= -200000.0) {
		tmp = (0.3333333333333333 / (x * (x * x))) / n;
	} else {
		tmp = (1.0 / n) / x;
	}
	return tmp;
}

Python

def code(x, n):
	tmp = 0
	if (1.0 / n) <= -200000.0:
		tmp = (0.3333333333333333 / (x * (x * x))) / n
	else:
		tmp = (1.0 / n) / x
	return tmp

Julia

function code(x, n)
	tmp = 0.0
	if (Float64(1.0 / n) <= -200000.0)
		tmp = Float64(Float64(0.3333333333333333 / Float64(x * Float64(x * x))) / n);
	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) <= -200000.0)
		tmp = (0.3333333333333333 / (x * (x * x))) / n;
	else
		tmp = (1.0 / n) / x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, n_] := If[LessEqual[N[(1.0 / n), $MachinePrecision], -200000.0], N[(N[(0.3333333333333333 / N[(x * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / n), $MachinePrecision], N[(N[(1.0 / n), $MachinePrecision] / x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 #s(literal 1 binary64) n) < -2e5

   1. Initial program 100.0%

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

   3. Taylor expanded in n around inf

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

      1. lower-/.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lower-log1p.f64 N/A

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

      4. lower-log.f64 46.5

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

   5. Applied rewrites 46.5%

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

   6. Taylor expanded in x around inf

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

   8. Applied rewrites 44.6%

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

   9. Taylor expanded in x around 0

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

   11. Applied rewrites 74.6%

       \[\leadsto \frac{\frac{0.3333333333333333}{x \cdot \left(x \cdot x\right)}}{n} \]

   if -2e5 < (/.f64 #s(literal 1 binary64) n)

   1. Initial program 39.7%

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

   3. Taylor expanded in x around inf

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

      1. lower-/.f64 N/A

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

      2. log-rec N/A

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

      3. mul-1-neg N/A

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

      4. associate-*r/ N/A

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

      5. associate-*r* N/A

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

      6. metadata-eval N/A

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

      7. *-commutative N/A

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

      8. associate-/l* N/A

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

      9. exp-to-pow N/A

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

      10. lower-pow.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 45.6

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

   5. Applied rewrites 45.6%

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

   6. Taylor expanded in n around inf

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

   8. Applied rewrites 48.2%

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

   10. Applied rewrites 49.4%

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

Alternative 17: 47.0% accurate, 5.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{1}{n} \leq -4 \cdot 10^{+21}:\\ \;\;\;\;1 - 1\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{n}}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (if (<= (/ 1.0 n) -4e+21) (- 1.0 1.0) (/ (/ 1.0 n) x)))

C

double code(double x, double n) {
	double tmp;
	if ((1.0 / n) <= -4e+21) {
		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) <= (-4d+21)) 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) <= -4e+21) {
		tmp = 1.0 - 1.0;
	} else {
		tmp = (1.0 / n) / x;
	}
	return tmp;
}

Python

def code(x, n):
	tmp = 0
	if (1.0 / n) <= -4e+21:
		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) <= -4e+21)
		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) <= -4e+21)
		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], -4e+21], 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) < -4e21

   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 51.6%

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

   6. Taylor expanded in n around inf

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

   8. Applied rewrites 50.8%

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

   if -4e21 < (/.f64 #s(literal 1 binary64) n)

   1. Initial program 41.3%

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

   3. Taylor expanded in x around inf

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

      1. lower-/.f64 N/A

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

      2. log-rec N/A

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

      3. mul-1-neg N/A

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

      4. associate-*r/ N/A

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

      5. associate-*r* N/A

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

      6. metadata-eval N/A

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

      7. *-commutative N/A

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

      8. associate-/l* N/A

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

      9. exp-to-pow N/A

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

      10. lower-pow.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 47.1

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

   5. Applied rewrites 47.1%

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

   6. Taylor expanded in n around inf

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

   8. Applied rewrites 47.1%

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

   10. Applied rewrites 48.2%

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

Alternative 18: 47.0% accurate, 5.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{1}{n} \leq -4 \cdot 10^{+21}:\\ \;\;\;\;1 - 1\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{x}}{n}\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (if (<= (/ 1.0 n) -4e+21) (- 1.0 1.0) (/ (/ 1.0 x) n)))

C

double code(double x, double n) {
	double tmp;
	if ((1.0 / n) <= -4e+21) {
		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) <= (-4d+21)) 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) <= -4e+21) {
		tmp = 1.0 - 1.0;
	} else {
		tmp = (1.0 / x) / n;
	}
	return tmp;
}

Python

def code(x, n):
	tmp = 0
	if (1.0 / n) <= -4e+21:
		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) <= -4e+21)
		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) <= -4e+21)
		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], -4e+21], 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) < -4e21

   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 51.6%

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

   6. Taylor expanded in n around inf

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

   8. Applied rewrites 50.8%

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

   if -4e21 < (/.f64 #s(literal 1 binary64) n)

   1. Initial program 41.3%

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

   3. Taylor expanded in n around inf

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

      1. lower-/.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lower-log1p.f64 N/A

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

      4. lower-log.f64 57.9

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

   5. Applied rewrites 57.9%

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

   6. Taylor expanded in x around inf

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

   8. Applied rewrites 48.2%

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

Alternative 19: 46.5% accurate, 6.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{1}{n} \leq -4 \cdot 10^{+21}:\\ \;\;\;\;1 - 1\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{n \cdot x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (if (<= (/ 1.0 n) -4e+21) (- 1.0 1.0) (/ 1.0 (* n x))))

C

double code(double x, double n) {
	double tmp;
	if ((1.0 / n) <= -4e+21) {
		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) <= (-4d+21)) 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) <= -4e+21) {
		tmp = 1.0 - 1.0;
	} else {
		tmp = 1.0 / (n * x);
	}
	return tmp;
}

Python

def code(x, n):
	tmp = 0
	if (1.0 / n) <= -4e+21:
		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) <= -4e+21)
		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) <= -4e+21)
		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], -4e+21], 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) < -4e21

   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 51.6%

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

   6. Taylor expanded in n around inf

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

   8. Applied rewrites 50.8%

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

   if -4e21 < (/.f64 #s(literal 1 binary64) n)

   1. Initial program 41.3%

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

   3. Taylor expanded in n around inf

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

      1. lower-/.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lower-log1p.f64 N/A

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

      4. lower-log.f64 57.9

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

   5. Applied rewrites 57.9%

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

   6. Taylor expanded in x around inf

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

   8. Applied rewrites 47.1%

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

4. Final simplification 48.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{1}{n} \leq -4 \cdot 10^{+21}:\\ \;\;\;\;1 - 1\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{n \cdot x}\\ \end{array} \]
5. Add Preprocessing

Alternative 20: 31.3% 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 56.4%

   \[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)} \]
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 42.7%

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

6. Taylor expanded in n around inf

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

8. Applied rewrites 31.8%

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

Reproduce

herbie shell --seed 2024228 
(FPCore (x n)
  :name "2nthrt (problem 3.4.6)"
  :precision binary64
  (- (pow (+ x 1.0) (/ 1.0 n)) (pow x (/ 1.0 n))))