2nthrt (problem 3.4.6)

Percentage Accurate: 52.8% → 91.8%

Time: 23.1s

Alternatives: 19

Speedup: 1.7×

• could not determine a ground truth

  1. x = -2.644587432418942e-206
  2. n = -1.8582776219073847e-275

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: 52.8% 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: 91.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if x < 1

   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 x around 0

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

      1. associate--l+ N/A

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

      2. +-commutative N/A

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

      3. *-rgt-identity N/A

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

      4. associate-*r/ N/A

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

      5. remove-double-neg N/A

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

      6. mul-1-neg N/A

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

      7. distribute-neg-frac N/A

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

      8. mul-1-neg N/A

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

      9. log-rec N/A

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

      10. mul-1-neg N/A

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

      11. associate-+l- N/A

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

      12. lower--.f64 N/A

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

      13. associate-*r/ N/A

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

      14. *-rgt-identity N/A

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

      15. lower-/.f64 N/A

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

      16. lower-expm1.f64 N/A

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

      17. mul-1-neg N/A

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

   5. Applied rewrites 86.4%

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

   if 1 < x

   1. Initial program 70.1%

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

   3. Taylor expanded in x around inf

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

      1. associate-/l/ N/A

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

      2. lower-/.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. log-rec N/A

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

      5. mul-1-neg N/A

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

      6. associate-*r/ N/A

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

      7. associate-*r* N/A

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

      8. metadata-eval N/A

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

      9. *-commutative N/A

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

      10. associate-/l* N/A

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

      11. exp-to-pow N/A

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

      12. lower-pow.f64 N/A

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

      13. lower-/.f64 99.5

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

   5. Applied rewrites 99.5%

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

   7. Applied rewrites 99.5%

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

Alternative 2: 76.0% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {x}^{\left(\frac{1}{n}\right)}\\ t_1 := {\left(x - -1\right)}^{\left(\frac{1}{n}\right)} - t\_0\\ \mathbf{if}\;t\_1 \leq -5 \cdot 10^{-9}:\\ \;\;\;\;1 - t\_0\\ \mathbf{elif}\;t\_1 \leq 0.001:\\ \;\;\;\;\frac{\log \left(\frac{x - -1}{x}\right)}{n}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{n} - \frac{-0.3333333333333333}{\left(x \cdot x\right) \cdot n}}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (let* ((t_0 (pow x (/ 1.0 n))) (t_1 (- (pow (- x -1.0) (/ 1.0 n)) t_0)))
   (if (<= t_1 -5e-9)
     (- 1.0 t_0)
     (if (<= t_1 0.001)
       (/ (log (/ (- x -1.0) x)) n)
       (/ (- (/ 1.0 n) (/ -0.3333333333333333 (* (* x x) n))) x)))))

C

double code(double x, double n) {
	double t_0 = pow(x, (1.0 / n));
	double t_1 = pow((x - -1.0), (1.0 / n)) - t_0;
	double tmp;
	if (t_1 <= -5e-9) {
		tmp = 1.0 - t_0;
	} else if (t_1 <= 0.001) {
		tmp = log(((x - -1.0) / x)) / n;
	} else {
		tmp = ((1.0 / n) - (-0.3333333333333333 / ((x * x) * n))) / x;
	}
	return tmp;
}

Fortran

real(8) function code(x, n)
    real(8), intent (in) :: x
    real(8), intent (in) :: n
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = x ** (1.0d0 / n)
    t_1 = ((x - (-1.0d0)) ** (1.0d0 / n)) - t_0
    if (t_1 <= (-5d-9)) then
        tmp = 1.0d0 - t_0
    else if (t_1 <= 0.001d0) then
        tmp = log(((x - (-1.0d0)) / x)) / n
    else
        tmp = ((1.0d0 / n) - ((-0.3333333333333333d0) / ((x * x) * n))) / x
    end if
    code = tmp
end function

Java

public static double code(double x, double n) {
	double t_0 = Math.pow(x, (1.0 / n));
	double t_1 = Math.pow((x - -1.0), (1.0 / n)) - t_0;
	double tmp;
	if (t_1 <= -5e-9) {
		tmp = 1.0 - t_0;
	} else if (t_1 <= 0.001) {
		tmp = Math.log(((x - -1.0) / x)) / n;
	} else {
		tmp = ((1.0 / n) - (-0.3333333333333333 / ((x * x) * n))) / x;
	}
	return tmp;
}

Python

def code(x, n):
	t_0 = math.pow(x, (1.0 / n))
	t_1 = math.pow((x - -1.0), (1.0 / n)) - t_0
	tmp = 0
	if t_1 <= -5e-9:
		tmp = 1.0 - t_0
	elif t_1 <= 0.001:
		tmp = math.log(((x - -1.0) / x)) / n
	else:
		tmp = ((1.0 / n) - (-0.3333333333333333 / ((x * x) * n))) / x
	return tmp

Julia

function code(x, n)
	t_0 = x ^ Float64(1.0 / n)
	t_1 = Float64((Float64(x - -1.0) ^ Float64(1.0 / n)) - t_0)
	tmp = 0.0
	if (t_1 <= -5e-9)
		tmp = Float64(1.0 - t_0);
	elseif (t_1 <= 0.001)
		tmp = Float64(log(Float64(Float64(x - -1.0) / x)) / n);
	else
		tmp = Float64(Float64(Float64(1.0 / n) - Float64(-0.3333333333333333 / Float64(Float64(x * x) * n))) / x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, n)
	t_0 = x ^ (1.0 / n);
	t_1 = ((x - -1.0) ^ (1.0 / n)) - t_0;
	tmp = 0.0;
	if (t_1 <= -5e-9)
		tmp = 1.0 - t_0;
	elseif (t_1 <= 0.001)
		tmp = log(((x - -1.0) / x)) / n;
	else
		tmp = ((1.0 / n) - (-0.3333333333333333 / ((x * x) * n))) / x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, n_] := Block[{t$95$0 = N[Power[x, N[(1.0 / n), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(N[Power[N[(x - -1.0), $MachinePrecision], N[(1.0 / n), $MachinePrecision]], $MachinePrecision] - t$95$0), $MachinePrecision]}, If[LessEqual[t$95$1, -5e-9], N[(1.0 - t$95$0), $MachinePrecision], If[LessEqual[t$95$1, 0.001], N[(N[Log[N[(N[(x - -1.0), $MachinePrecision] / x), $MachinePrecision]], $MachinePrecision] / n), $MachinePrecision], N[(N[(N[(1.0 / n), $MachinePrecision] - N[(-0.3333333333333333 / N[(N[(x * x), $MachinePrecision] * n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 99.0%

      \[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)} \]
   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 99.0%

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

   if -5.0000000000000001e-9 < (-.f64 (pow.f64 (+.f64 x #s(literal 1 binary64)) (/.f64 #s(literal 1 binary64) n)) (pow.f64 x (/.f64 #s(literal 1 binary64) n))) < 1e-3

   1. Initial program 46.4%

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

   3. Taylor expanded in n around inf

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

      1. lower-/.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lower-log1p.f64 N/A

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

      4. lower-log.f64 80.4

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

   5. Applied rewrites 80.4%

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

   7. Applied rewrites 80.5%

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

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

   1. Initial program 45.2%

      \[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)} \]
   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 6.0

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

   5. Applied rewrites 6.0%

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

   6. Taylor expanded in x around -inf

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

   8. Applied rewrites 48.9%

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

   9. Taylor expanded in x around 0

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

   11. Applied rewrites 48.9%

       \[\leadsto \frac{\frac{-0.3333333333333333}{\left(x \cdot x\right) \cdot n} - \frac{1}{n}}{-x} \]
3. Recombined 3 regimes into one program.

4. Final simplification 78.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{\left(x - -1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)} \leq -5 \cdot 10^{-9}:\\ \;\;\;\;1 - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{elif}\;{\left(x - -1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)} \leq 0.001:\\ \;\;\;\;\frac{\log \left(\frac{x - -1}{x}\right)}{n}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{n} - \frac{-0.3333333333333333}{\left(x \cdot x\right) \cdot n}}{x}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 82.2% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{if}\;\frac{1}{n} \leq -1 \cdot 10^{-5}:\\ \;\;\;\;\frac{\frac{t\_0}{x}}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 10^{-11}:\\ \;\;\;\;\frac{\log \left(\frac{x - -1}{x}\right)}{n}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\frac{0.5}{n \cdot n} - \frac{0.5}{n}, x, \frac{1}{n}\right), x, 1\right) - t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (let* ((t_0 (pow x (/ 1.0 n))))
   (if (<= (/ 1.0 n) -1e-5)
     (/ (/ t_0 x) n)
     (if (<= (/ 1.0 n) 1e-11)
       (/ (log (/ (- x -1.0) x)) n)
       (- (fma (fma (- (/ 0.5 (* n n)) (/ 0.5 n)) x (/ 1.0 n)) x 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) <= -1e-5) {
		tmp = (t_0 / x) / n;
	} else if ((1.0 / n) <= 1e-11) {
		tmp = log(((x - -1.0) / x)) / n;
	} else {
		tmp = fma(fma(((0.5 / (n * n)) - (0.5 / n)), x, (1.0 / n)), x, 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) <= -1e-5)
		tmp = Float64(Float64(t_0 / x) / n);
	elseif (Float64(1.0 / n) <= 1e-11)
		tmp = Float64(log(Float64(Float64(x - -1.0) / x)) / n);
	else
		tmp = Float64(fma(fma(Float64(Float64(0.5 / Float64(n * n)) - Float64(0.5 / n)), x, Float64(1.0 / n)), x, 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], -1e-5], N[(N[(t$95$0 / x), $MachinePrecision] / n), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 1e-11], N[(N[Log[N[(N[(x - -1.0), $MachinePrecision] / x), $MachinePrecision]], $MachinePrecision] / n), $MachinePrecision], N[(N[(N[(N[(N[(0.5 / N[(n * n), $MachinePrecision]), $MachinePrecision] - N[(0.5 / n), $MachinePrecision]), $MachinePrecision] * x + N[(1.0 / n), $MachinePrecision]), $MachinePrecision] * x + 1.0), $MachinePrecision] - t$95$0), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 #s(literal 1 binary64) n) < -1.00000000000000008e-5

   1. Initial program 97.5%

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

   3. Taylor expanded in x around inf

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

      1. associate-/l/ N/A

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

      2. lower-/.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. log-rec N/A

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

      5. mul-1-neg N/A

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

      6. associate-*r/ N/A

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

      7. associate-*r* N/A

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

      8. metadata-eval N/A

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

      9. *-commutative N/A

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

      10. associate-/l* N/A

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

      11. exp-to-pow N/A

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

      12. lower-pow.f64 N/A

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

      13. lower-/.f64 100.0

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

   5. Applied rewrites 100.0%

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

   if -1.00000000000000008e-5 < (/.f64 #s(literal 1 binary64) n) < 9.99999999999999939e-12

   1. Initial program 31.3%

      \[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)} \]
   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 77.1

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

   5. Applied rewrites 77.1%

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

   7. Applied rewrites 77.1%

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

   if 9.99999999999999939e-12 < (/.f64 #s(literal 1 binary64) n)

   1. Initial program 45.2%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. lower--.f64 N/A

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

      7. associate-*r/ N/A

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

      8. metadata-eval N/A

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

      9. lower-/.f64 N/A

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

      10. unpow2 N/A

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

      11. lower-*.f64 N/A

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

      12. associate-*r/ N/A

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

      13. metadata-eval N/A

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

      14. lower-/.f64 N/A

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

      15. lower-/.f64 70.8

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

   5. Applied rewrites 70.8%

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

4. Final simplification 83.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{1}{n} \leq -1 \cdot 10^{-5}:\\ \;\;\;\;\frac{\frac{{x}^{\left(\frac{1}{n}\right)}}{x}}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 10^{-11}:\\ \;\;\;\;\frac{\log \left(\frac{x - -1}{x}\right)}{n}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\frac{0.5}{n \cdot n} - \frac{0.5}{n}, x, \frac{1}{n}\right), x, 1\right) - {x}^{\left(\frac{1}{n}\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 81.7% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {x}^{\left(\frac{1}{n}\right)}\\ t_1 := \frac{x}{\frac{0.5}{n} - \frac{\frac{0.3333333333333333}{n}}{x}}\\ \mathbf{if}\;\frac{1}{n} \leq -1 \cdot 10^{-5}:\\ \;\;\;\;\frac{\frac{t\_0}{x}}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 10^{-11}:\\ \;\;\;\;\frac{\log \left(\frac{x - -1}{x}\right)}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 3 \cdot 10^{+109}:\\ \;\;\;\;\left(\frac{x}{n} + 1\right) - t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{t\_1 - n}{t\_1 \cdot n}}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (let* ((t_0 (pow x (/ 1.0 n)))
        (t_1 (/ x (- (/ 0.5 n) (/ (/ 0.3333333333333333 n) x)))))
   (if (<= (/ 1.0 n) -1e-5)
     (/ (/ t_0 x) n)
     (if (<= (/ 1.0 n) 1e-11)
       (/ (log (/ (- x -1.0) x)) n)
       (if (<= (/ 1.0 n) 3e+109)
         (- (+ (/ x n) 1.0) t_0)
         (/ (/ (- t_1 n) (* t_1 n)) x))))))

C

double code(double x, double n) {
	double t_0 = pow(x, (1.0 / n));
	double t_1 = x / ((0.5 / n) - ((0.3333333333333333 / n) / x));
	double tmp;
	if ((1.0 / n) <= -1e-5) {
		tmp = (t_0 / x) / n;
	} else if ((1.0 / n) <= 1e-11) {
		tmp = log(((x - -1.0) / x)) / n;
	} else if ((1.0 / n) <= 3e+109) {
		tmp = ((x / n) + 1.0) - t_0;
	} else {
		tmp = ((t_1 - n) / (t_1 * n)) / x;
	}
	return tmp;
}

Fortran

real(8) function code(x, n)
    real(8), intent (in) :: x
    real(8), intent (in) :: n
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = x ** (1.0d0 / n)
    t_1 = x / ((0.5d0 / n) - ((0.3333333333333333d0 / n) / x))
    if ((1.0d0 / n) <= (-1d-5)) then
        tmp = (t_0 / x) / n
    else if ((1.0d0 / n) <= 1d-11) then
        tmp = log(((x - (-1.0d0)) / x)) / n
    else if ((1.0d0 / n) <= 3d+109) then
        tmp = ((x / n) + 1.0d0) - t_0
    else
        tmp = ((t_1 - n) / (t_1 * n)) / x
    end if
    code = tmp
end function

Java

public static double code(double x, double n) {
	double t_0 = Math.pow(x, (1.0 / n));
	double t_1 = x / ((0.5 / n) - ((0.3333333333333333 / n) / x));
	double tmp;
	if ((1.0 / n) <= -1e-5) {
		tmp = (t_0 / x) / n;
	} else if ((1.0 / n) <= 1e-11) {
		tmp = Math.log(((x - -1.0) / x)) / n;
	} else if ((1.0 / n) <= 3e+109) {
		tmp = ((x / n) + 1.0) - t_0;
	} else {
		tmp = ((t_1 - n) / (t_1 * n)) / x;
	}
	return tmp;
}

Python

def code(x, n):
	t_0 = math.pow(x, (1.0 / n))
	t_1 = x / ((0.5 / n) - ((0.3333333333333333 / n) / x))
	tmp = 0
	if (1.0 / n) <= -1e-5:
		tmp = (t_0 / x) / n
	elif (1.0 / n) <= 1e-11:
		tmp = math.log(((x - -1.0) / x)) / n
	elif (1.0 / n) <= 3e+109:
		tmp = ((x / n) + 1.0) - t_0
	else:
		tmp = ((t_1 - n) / (t_1 * n)) / x
	return tmp

Julia

function code(x, n)
	t_0 = x ^ Float64(1.0 / n)
	t_1 = Float64(x / Float64(Float64(0.5 / n) - Float64(Float64(0.3333333333333333 / n) / x)))
	tmp = 0.0
	if (Float64(1.0 / n) <= -1e-5)
		tmp = Float64(Float64(t_0 / x) / n);
	elseif (Float64(1.0 / n) <= 1e-11)
		tmp = Float64(log(Float64(Float64(x - -1.0) / x)) / n);
	elseif (Float64(1.0 / n) <= 3e+109)
		tmp = Float64(Float64(Float64(x / n) + 1.0) - t_0);
	else
		tmp = Float64(Float64(Float64(t_1 - n) / Float64(t_1 * n)) / x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, n)
	t_0 = x ^ (1.0 / n);
	t_1 = x / ((0.5 / n) - ((0.3333333333333333 / n) / x));
	tmp = 0.0;
	if ((1.0 / n) <= -1e-5)
		tmp = (t_0 / x) / n;
	elseif ((1.0 / n) <= 1e-11)
		tmp = log(((x - -1.0) / x)) / n;
	elseif ((1.0 / n) <= 3e+109)
		tmp = ((x / n) + 1.0) - t_0;
	else
		tmp = ((t_1 - n) / (t_1 * n)) / x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, n_] := Block[{t$95$0 = N[Power[x, N[(1.0 / n), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(x / N[(N[(0.5 / n), $MachinePrecision] - N[(N[(0.3333333333333333 / n), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(1.0 / n), $MachinePrecision], -1e-5], N[(N[(t$95$0 / x), $MachinePrecision] / n), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 1e-11], N[(N[Log[N[(N[(x - -1.0), $MachinePrecision] / x), $MachinePrecision]], $MachinePrecision] / n), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 3e+109], N[(N[(N[(x / n), $MachinePrecision] + 1.0), $MachinePrecision] - t$95$0), $MachinePrecision], N[(N[(N[(t$95$1 - n), $MachinePrecision] / N[(t$95$1 * n), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if (/.f64 #s(literal 1 binary64) n) < -1.00000000000000008e-5

   1. Initial program 97.5%

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

   3. Taylor expanded in x around inf

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

      1. associate-/l/ N/A

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

      2. lower-/.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. log-rec N/A

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

      5. mul-1-neg N/A

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

      6. associate-*r/ N/A

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

      7. associate-*r* N/A

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

      8. metadata-eval N/A

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

      9. *-commutative N/A

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

      10. associate-/l* N/A

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

      11. exp-to-pow N/A

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

      12. lower-pow.f64 N/A

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

      13. lower-/.f64 100.0

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

   5. Applied rewrites 100.0%

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

   if -1.00000000000000008e-5 < (/.f64 #s(literal 1 binary64) n) < 9.99999999999999939e-12

   1. Initial program 31.3%

      \[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)} \]
   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 77.1

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

   5. Applied rewrites 77.1%

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

   7. Applied rewrites 77.1%

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

   if 9.99999999999999939e-12 < (/.f64 #s(literal 1 binary64) n) < 3.00000000000000015e109

   1. Initial program 85.2%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. *-rgt-identity N/A

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

      3. associate-*r/ N/A

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

      4. lower-+.f64 N/A

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

      5. associate-*r/ N/A

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

      6. *-rgt-identity N/A

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

      7. lower-/.f64 78.5

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

   5. Applied rewrites 78.5%

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

   if 3.00000000000000015e109 < (/.f64 #s(literal 1 binary64) n)

   1. Initial program 17.2%

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

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

   5. Applied rewrites 6.1%

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

   6. Taylor expanded in x around -inf

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

   8. Applied rewrites 76.1%

      \[\leadsto \frac{\frac{\frac{\frac{0.3333333333333333}{n}}{x} - \frac{0.5}{n}}{-x} - \frac{1}{n}}{\color{blue}{-x}} \]
   9. Step-by-step derivation

   10. Applied rewrites 80.6%

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

4. Final simplification 84.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{1}{n} \leq -1 \cdot 10^{-5}:\\ \;\;\;\;\frac{\frac{{x}^{\left(\frac{1}{n}\right)}}{x}}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 10^{-11}:\\ \;\;\;\;\frac{\log \left(\frac{x - -1}{x}\right)}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 3 \cdot 10^{+109}:\\ \;\;\;\;\left(\frac{x}{n} + 1\right) - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{x}{\frac{0.5}{n} - \frac{\frac{0.3333333333333333}{n}}{x}} - n}{\frac{x}{\frac{0.5}{n} - \frac{\frac{0.3333333333333333}{n}}{x}} \cdot n}}{x}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 81.0% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{if}\;\frac{1}{n} \leq -1 \cdot 10^{-5}:\\ \;\;\;\;\frac{\frac{t\_0}{x}}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 10^{-11}:\\ \;\;\;\;\frac{\log \left(\frac{x - -1}{x}\right)}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 3 \cdot 10^{+109}:\\ \;\;\;\;\left(\frac{x}{n} + 1\right) - t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{-0.3333333333333333}{\left(x \cdot x\right) \cdot n}}{-x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (let* ((t_0 (pow x (/ 1.0 n))))
   (if (<= (/ 1.0 n) -1e-5)
     (/ (/ t_0 x) n)
     (if (<= (/ 1.0 n) 1e-11)
       (/ (log (/ (- x -1.0) x)) n)
       (if (<= (/ 1.0 n) 3e+109)
         (- (+ (/ x n) 1.0) t_0)
         (/ (/ -0.3333333333333333 (* (* x x) n)) (- x)))))))

C

double code(double x, double n) {
	double t_0 = pow(x, (1.0 / n));
	double tmp;
	if ((1.0 / n) <= -1e-5) {
		tmp = (t_0 / x) / n;
	} else if ((1.0 / n) <= 1e-11) {
		tmp = log(((x - -1.0) / x)) / n;
	} else if ((1.0 / n) <= 3e+109) {
		tmp = ((x / n) + 1.0) - t_0;
	} else {
		tmp = (-0.3333333333333333 / ((x * x) * n)) / -x;
	}
	return tmp;
}

Fortran

real(8) function code(x, n)
    real(8), intent (in) :: x
    real(8), intent (in) :: n
    real(8) :: t_0
    real(8) :: tmp
    t_0 = x ** (1.0d0 / n)
    if ((1.0d0 / n) <= (-1d-5)) then
        tmp = (t_0 / x) / n
    else if ((1.0d0 / n) <= 1d-11) then
        tmp = log(((x - (-1.0d0)) / x)) / n
    else if ((1.0d0 / n) <= 3d+109) then
        tmp = ((x / n) + 1.0d0) - t_0
    else
        tmp = ((-0.3333333333333333d0) / ((x * x) * n)) / -x
    end if
    code = tmp
end function

Java

public static double code(double x, double n) {
	double t_0 = Math.pow(x, (1.0 / n));
	double tmp;
	if ((1.0 / n) <= -1e-5) {
		tmp = (t_0 / x) / n;
	} else if ((1.0 / n) <= 1e-11) {
		tmp = Math.log(((x - -1.0) / x)) / n;
	} else if ((1.0 / n) <= 3e+109) {
		tmp = ((x / n) + 1.0) - t_0;
	} else {
		tmp = (-0.3333333333333333 / ((x * x) * n)) / -x;
	}
	return tmp;
}

Python

def code(x, n):
	t_0 = math.pow(x, (1.0 / n))
	tmp = 0
	if (1.0 / n) <= -1e-5:
		tmp = (t_0 / x) / n
	elif (1.0 / n) <= 1e-11:
		tmp = math.log(((x - -1.0) / x)) / n
	elif (1.0 / n) <= 3e+109:
		tmp = ((x / n) + 1.0) - t_0
	else:
		tmp = (-0.3333333333333333 / ((x * x) * n)) / -x
	return tmp

Julia

function code(x, n)
	t_0 = x ^ Float64(1.0 / n)
	tmp = 0.0
	if (Float64(1.0 / n) <= -1e-5)
		tmp = Float64(Float64(t_0 / x) / n);
	elseif (Float64(1.0 / n) <= 1e-11)
		tmp = Float64(log(Float64(Float64(x - -1.0) / x)) / n);
	elseif (Float64(1.0 / n) <= 3e+109)
		tmp = Float64(Float64(Float64(x / n) + 1.0) - t_0);
	else
		tmp = Float64(Float64(-0.3333333333333333 / Float64(Float64(x * x) * n)) / Float64(-x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, n)
	t_0 = x ^ (1.0 / n);
	tmp = 0.0;
	if ((1.0 / n) <= -1e-5)
		tmp = (t_0 / x) / n;
	elseif ((1.0 / n) <= 1e-11)
		tmp = log(((x - -1.0) / x)) / n;
	elseif ((1.0 / n) <= 3e+109)
		tmp = ((x / n) + 1.0) - t_0;
	else
		tmp = (-0.3333333333333333 / ((x * x) * n)) / -x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, n_] := Block[{t$95$0 = N[Power[x, N[(1.0 / n), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[N[(1.0 / n), $MachinePrecision], -1e-5], N[(N[(t$95$0 / x), $MachinePrecision] / n), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 1e-11], N[(N[Log[N[(N[(x - -1.0), $MachinePrecision] / x), $MachinePrecision]], $MachinePrecision] / n), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 3e+109], N[(N[(N[(x / n), $MachinePrecision] + 1.0), $MachinePrecision] - t$95$0), $MachinePrecision], N[(N[(-0.3333333333333333 / N[(N[(x * x), $MachinePrecision] * n), $MachinePrecision]), $MachinePrecision] / (-x)), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if (/.f64 #s(literal 1 binary64) n) < -1.00000000000000008e-5

   1. Initial program 97.5%

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

   3. Taylor expanded in x around inf

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

      1. associate-/l/ N/A

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

      2. lower-/.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. log-rec N/A

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

      5. mul-1-neg N/A

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

      6. associate-*r/ N/A

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

      7. associate-*r* N/A

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

      8. metadata-eval N/A

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

      9. *-commutative N/A

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

      10. associate-/l* N/A

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

      11. exp-to-pow N/A

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

      12. lower-pow.f64 N/A

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

      13. lower-/.f64 100.0

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

   5. Applied rewrites 100.0%

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

   if -1.00000000000000008e-5 < (/.f64 #s(literal 1 binary64) n) < 9.99999999999999939e-12

   1. Initial program 31.3%

      \[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)} \]
   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 77.1

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

   5. Applied rewrites 77.1%

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

   7. Applied rewrites 77.1%

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

   if 9.99999999999999939e-12 < (/.f64 #s(literal 1 binary64) n) < 3.00000000000000015e109

   1. Initial program 85.2%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. *-rgt-identity N/A

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

      3. associate-*r/ N/A

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

      4. lower-+.f64 N/A

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

      5. associate-*r/ N/A

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

      6. *-rgt-identity N/A

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

      7. lower-/.f64 78.5

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

   5. Applied rewrites 78.5%

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

   if 3.00000000000000015e109 < (/.f64 #s(literal 1 binary64) n)

   1. Initial program 17.2%

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

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

   5. Applied rewrites 6.1%

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

   6. Taylor expanded in x around -inf

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

   8. Applied rewrites 76.1%

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

   9. Taylor expanded in x around 0

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

   11. Applied rewrites 76.1%

       \[\leadsto \frac{\frac{-0.3333333333333333}{\left(x \cdot x\right) \cdot n}}{-x} \]
3. Recombined 4 regimes into one program.

4. Final simplification 83.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{1}{n} \leq -1 \cdot 10^{-5}:\\ \;\;\;\;\frac{\frac{{x}^{\left(\frac{1}{n}\right)}}{x}}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 10^{-11}:\\ \;\;\;\;\frac{\log \left(\frac{x - -1}{x}\right)}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 3 \cdot 10^{+109}:\\ \;\;\;\;\left(\frac{x}{n} + 1\right) - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{-0.3333333333333333}{\left(x \cdot x\right) \cdot n}}{-x}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 81.0% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{1}{n} \leq -1 \cdot 10^{-5}:\\ \;\;\;\;\frac{1}{\left({x}^{\left(\frac{-1}{n}\right)} \cdot x\right) \cdot n}\\ \mathbf{elif}\;\frac{1}{n} \leq 10^{-11}:\\ \;\;\;\;\frac{\log \left(\frac{x - -1}{x}\right)}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 3 \cdot 10^{+109}:\\ \;\;\;\;\left(\frac{x}{n} + 1\right) - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{-0.3333333333333333}{\left(x \cdot x\right) \cdot n}}{-x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (if (<= (/ 1.0 n) -1e-5)
   (/ 1.0 (* (* (pow x (/ -1.0 n)) x) n))
   (if (<= (/ 1.0 n) 1e-11)
     (/ (log (/ (- x -1.0) x)) n)
     (if (<= (/ 1.0 n) 3e+109)
       (- (+ (/ x n) 1.0) (pow x (/ 1.0 n)))
       (/ (/ -0.3333333333333333 (* (* x x) n)) (- x))))))

C

double code(double x, double n) {
	double tmp;
	if ((1.0 / n) <= -1e-5) {
		tmp = 1.0 / ((pow(x, (-1.0 / n)) * x) * n);
	} else if ((1.0 / n) <= 1e-11) {
		tmp = log(((x - -1.0) / x)) / n;
	} else if ((1.0 / n) <= 3e+109) {
		tmp = ((x / n) + 1.0) - pow(x, (1.0 / n));
	} else {
		tmp = (-0.3333333333333333 / ((x * x) * n)) / -x;
	}
	return tmp;
}

Fortran

real(8) function code(x, n)
    real(8), intent (in) :: x
    real(8), intent (in) :: n
    real(8) :: tmp
    if ((1.0d0 / n) <= (-1d-5)) then
        tmp = 1.0d0 / (((x ** ((-1.0d0) / n)) * x) * n)
    else if ((1.0d0 / n) <= 1d-11) then
        tmp = log(((x - (-1.0d0)) / x)) / n
    else if ((1.0d0 / n) <= 3d+109) then
        tmp = ((x / n) + 1.0d0) - (x ** (1.0d0 / n))
    else
        tmp = ((-0.3333333333333333d0) / ((x * x) * n)) / -x
    end if
    code = tmp
end function

Java

public static double code(double x, double n) {
	double tmp;
	if ((1.0 / n) <= -1e-5) {
		tmp = 1.0 / ((Math.pow(x, (-1.0 / n)) * x) * n);
	} else if ((1.0 / n) <= 1e-11) {
		tmp = Math.log(((x - -1.0) / x)) / n;
	} else if ((1.0 / n) <= 3e+109) {
		tmp = ((x / n) + 1.0) - Math.pow(x, (1.0 / n));
	} else {
		tmp = (-0.3333333333333333 / ((x * x) * n)) / -x;
	}
	return tmp;
}

Python

def code(x, n):
	tmp = 0
	if (1.0 / n) <= -1e-5:
		tmp = 1.0 / ((math.pow(x, (-1.0 / n)) * x) * n)
	elif (1.0 / n) <= 1e-11:
		tmp = math.log(((x - -1.0) / x)) / n
	elif (1.0 / n) <= 3e+109:
		tmp = ((x / n) + 1.0) - math.pow(x, (1.0 / n))
	else:
		tmp = (-0.3333333333333333 / ((x * x) * n)) / -x
	return tmp

Julia

function code(x, n)
	tmp = 0.0
	if (Float64(1.0 / n) <= -1e-5)
		tmp = Float64(1.0 / Float64(Float64((x ^ Float64(-1.0 / n)) * x) * n));
	elseif (Float64(1.0 / n) <= 1e-11)
		tmp = Float64(log(Float64(Float64(x - -1.0) / x)) / n);
	elseif (Float64(1.0 / n) <= 3e+109)
		tmp = Float64(Float64(Float64(x / n) + 1.0) - (x ^ Float64(1.0 / n)));
	else
		tmp = Float64(Float64(-0.3333333333333333 / Float64(Float64(x * x) * n)) / Float64(-x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, n)
	tmp = 0.0;
	if ((1.0 / n) <= -1e-5)
		tmp = 1.0 / (((x ^ (-1.0 / n)) * x) * n);
	elseif ((1.0 / n) <= 1e-11)
		tmp = log(((x - -1.0) / x)) / n;
	elseif ((1.0 / n) <= 3e+109)
		tmp = ((x / n) + 1.0) - (x ^ (1.0 / n));
	else
		tmp = (-0.3333333333333333 / ((x * x) * n)) / -x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, n_] := If[LessEqual[N[(1.0 / n), $MachinePrecision], -1e-5], N[(1.0 / N[(N[(N[Power[x, N[(-1.0 / n), $MachinePrecision]], $MachinePrecision] * x), $MachinePrecision] * n), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 1e-11], N[(N[Log[N[(N[(x - -1.0), $MachinePrecision] / x), $MachinePrecision]], $MachinePrecision] / n), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 3e+109], N[(N[(N[(x / n), $MachinePrecision] + 1.0), $MachinePrecision] - N[Power[x, N[(1.0 / n), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[(-0.3333333333333333 / N[(N[(x * x), $MachinePrecision] * n), $MachinePrecision]), $MachinePrecision] / (-x)), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if (/.f64 #s(literal 1 binary64) n) < -1.00000000000000008e-5

   1. Initial program 97.5%

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

   3. Taylor expanded in x around inf

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

      1. associate-/l/ N/A

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

      2. lower-/.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. log-rec N/A

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

      5. mul-1-neg N/A

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

      6. associate-*r/ N/A

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

      7. associate-*r* N/A

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

      8. metadata-eval N/A

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

      9. *-commutative N/A

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

      10. associate-/l* N/A

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

      11. exp-to-pow N/A

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

      12. lower-pow.f64 N/A

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

      13. lower-/.f64 100.0

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

   5. Applied rewrites 100.0%

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

   7. Applied rewrites 100.0%

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

   9. Applied rewrites 100.0%

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

   if -1.00000000000000008e-5 < (/.f64 #s(literal 1 binary64) n) < 9.99999999999999939e-12

   1. Initial program 31.3%

      \[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)} \]
   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 77.1

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

   5. Applied rewrites 77.1%

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

   7. Applied rewrites 77.1%

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

   if 9.99999999999999939e-12 < (/.f64 #s(literal 1 binary64) n) < 3.00000000000000015e109

   1. Initial program 85.2%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. *-rgt-identity N/A

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

      3. associate-*r/ N/A

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

      4. lower-+.f64 N/A

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

      5. associate-*r/ N/A

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

      6. *-rgt-identity N/A

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

      7. lower-/.f64 78.5

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

   5. Applied rewrites 78.5%

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

   if 3.00000000000000015e109 < (/.f64 #s(literal 1 binary64) n)

   1. Initial program 17.2%

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

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

   5. Applied rewrites 6.1%

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

   6. Taylor expanded in x around -inf

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

   8. Applied rewrites 76.1%

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

   9. Taylor expanded in x around 0

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

   11. Applied rewrites 76.1%

       \[\leadsto \frac{\frac{-0.3333333333333333}{\left(x \cdot x\right) \cdot n}}{-x} \]
3. Recombined 4 regimes into one program.

4. Final simplification 83.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{1}{n} \leq -1 \cdot 10^{-5}:\\ \;\;\;\;\frac{1}{\left({x}^{\left(\frac{-1}{n}\right)} \cdot x\right) \cdot n}\\ \mathbf{elif}\;\frac{1}{n} \leq 10^{-11}:\\ \;\;\;\;\frac{\log \left(\frac{x - -1}{x}\right)}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 3 \cdot 10^{+109}:\\ \;\;\;\;\left(\frac{x}{n} + 1\right) - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{-0.3333333333333333}{\left(x \cdot x\right) \cdot n}}{-x}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 81.0% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{1}{n} \leq -1 \cdot 10^{-5}:\\ \;\;\;\;\frac{{x}^{\left(\frac{1}{n} - 1\right)}}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 10^{-11}:\\ \;\;\;\;\frac{\log \left(\frac{x - -1}{x}\right)}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 3 \cdot 10^{+109}:\\ \;\;\;\;\left(\frac{x}{n} + 1\right) - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{-0.3333333333333333}{\left(x \cdot x\right) \cdot n}}{-x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (if (<= (/ 1.0 n) -1e-5)
   (/ (pow x (- (/ 1.0 n) 1.0)) n)
   (if (<= (/ 1.0 n) 1e-11)
     (/ (log (/ (- x -1.0) x)) n)
     (if (<= (/ 1.0 n) 3e+109)
       (- (+ (/ x n) 1.0) (pow x (/ 1.0 n)))
       (/ (/ -0.3333333333333333 (* (* x x) n)) (- x))))))

C

double code(double x, double n) {
	double tmp;
	if ((1.0 / n) <= -1e-5) {
		tmp = pow(x, ((1.0 / n) - 1.0)) / n;
	} else if ((1.0 / n) <= 1e-11) {
		tmp = log(((x - -1.0) / x)) / n;
	} else if ((1.0 / n) <= 3e+109) {
		tmp = ((x / n) + 1.0) - pow(x, (1.0 / n));
	} else {
		tmp = (-0.3333333333333333 / ((x * x) * n)) / -x;
	}
	return tmp;
}

Fortran

real(8) function code(x, n)
    real(8), intent (in) :: x
    real(8), intent (in) :: n
    real(8) :: tmp
    if ((1.0d0 / n) <= (-1d-5)) then
        tmp = (x ** ((1.0d0 / n) - 1.0d0)) / n
    else if ((1.0d0 / n) <= 1d-11) then
        tmp = log(((x - (-1.0d0)) / x)) / n
    else if ((1.0d0 / n) <= 3d+109) then
        tmp = ((x / n) + 1.0d0) - (x ** (1.0d0 / n))
    else
        tmp = ((-0.3333333333333333d0) / ((x * x) * n)) / -x
    end if
    code = tmp
end function

Java

public static double code(double x, double n) {
	double tmp;
	if ((1.0 / n) <= -1e-5) {
		tmp = Math.pow(x, ((1.0 / n) - 1.0)) / n;
	} else if ((1.0 / n) <= 1e-11) {
		tmp = Math.log(((x - -1.0) / x)) / n;
	} else if ((1.0 / n) <= 3e+109) {
		tmp = ((x / n) + 1.0) - Math.pow(x, (1.0 / n));
	} else {
		tmp = (-0.3333333333333333 / ((x * x) * n)) / -x;
	}
	return tmp;
}

Python

def code(x, n):
	tmp = 0
	if (1.0 / n) <= -1e-5:
		tmp = math.pow(x, ((1.0 / n) - 1.0)) / n
	elif (1.0 / n) <= 1e-11:
		tmp = math.log(((x - -1.0) / x)) / n
	elif (1.0 / n) <= 3e+109:
		tmp = ((x / n) + 1.0) - math.pow(x, (1.0 / n))
	else:
		tmp = (-0.3333333333333333 / ((x * x) * n)) / -x
	return tmp

Julia

function code(x, n)
	tmp = 0.0
	if (Float64(1.0 / n) <= -1e-5)
		tmp = Float64((x ^ Float64(Float64(1.0 / n) - 1.0)) / n);
	elseif (Float64(1.0 / n) <= 1e-11)
		tmp = Float64(log(Float64(Float64(x - -1.0) / x)) / n);
	elseif (Float64(1.0 / n) <= 3e+109)
		tmp = Float64(Float64(Float64(x / n) + 1.0) - (x ^ Float64(1.0 / n)));
	else
		tmp = Float64(Float64(-0.3333333333333333 / Float64(Float64(x * x) * n)) / Float64(-x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, n)
	tmp = 0.0;
	if ((1.0 / n) <= -1e-5)
		tmp = (x ^ ((1.0 / n) - 1.0)) / n;
	elseif ((1.0 / n) <= 1e-11)
		tmp = log(((x - -1.0) / x)) / n;
	elseif ((1.0 / n) <= 3e+109)
		tmp = ((x / n) + 1.0) - (x ^ (1.0 / n));
	else
		tmp = (-0.3333333333333333 / ((x * x) * n)) / -x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, n_] := If[LessEqual[N[(1.0 / n), $MachinePrecision], -1e-5], N[(N[Power[x, N[(N[(1.0 / n), $MachinePrecision] - 1.0), $MachinePrecision]], $MachinePrecision] / n), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 1e-11], N[(N[Log[N[(N[(x - -1.0), $MachinePrecision] / x), $MachinePrecision]], $MachinePrecision] / n), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 3e+109], N[(N[(N[(x / n), $MachinePrecision] + 1.0), $MachinePrecision] - N[Power[x, N[(1.0 / n), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[(-0.3333333333333333 / N[(N[(x * x), $MachinePrecision] * n), $MachinePrecision]), $MachinePrecision] / (-x)), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if (/.f64 #s(literal 1 binary64) n) < -1.00000000000000008e-5

   1. Initial program 97.5%

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

   3. Taylor expanded in 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 57.7

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

   5. Applied rewrites 57.7%

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

   7. Applied rewrites 57.7%

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

   8. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. associate-/r* N/A

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

      3. lower-/.f64 N/A

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

   10. Applied rewrites 99.8%

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

   if -1.00000000000000008e-5 < (/.f64 #s(literal 1 binary64) n) < 9.99999999999999939e-12

   1. Initial program 31.3%

      \[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)} \]
   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 77.1

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

   5. Applied rewrites 77.1%

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

   7. Applied rewrites 77.1%

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

   if 9.99999999999999939e-12 < (/.f64 #s(literal 1 binary64) n) < 3.00000000000000015e109

   1. Initial program 85.2%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. *-rgt-identity N/A

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

      3. associate-*r/ N/A

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

      4. lower-+.f64 N/A

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

      5. associate-*r/ N/A

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

      6. *-rgt-identity N/A

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

      7. lower-/.f64 78.5

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

   5. Applied rewrites 78.5%

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

   if 3.00000000000000015e109 < (/.f64 #s(literal 1 binary64) n)

   1. Initial program 17.2%

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

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

   5. Applied rewrites 6.1%

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

   6. Taylor expanded in x around -inf

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

   8. Applied rewrites 76.1%

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

   9. Taylor expanded in x around 0

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

   11. Applied rewrites 76.1%

       \[\leadsto \frac{\frac{-0.3333333333333333}{\left(x \cdot x\right) \cdot n}}{-x} \]
3. Recombined 4 regimes into one program.

4. Final simplification 83.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{1}{n} \leq -1 \cdot 10^{-5}:\\ \;\;\;\;\frac{{x}^{\left(\frac{1}{n} - 1\right)}}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 10^{-11}:\\ \;\;\;\;\frac{\log \left(\frac{x - -1}{x}\right)}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 3 \cdot 10^{+109}:\\ \;\;\;\;\left(\frac{x}{n} + 1\right) - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{-0.3333333333333333}{\left(x \cdot x\right) \cdot n}}{-x}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 80.9% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{1}{n} \leq -1 \cdot 10^{-5}:\\ \;\;\;\;\frac{{x}^{\left(\frac{1}{n} - 1\right)}}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 10^{-11}:\\ \;\;\;\;\frac{\log \left(\frac{x - -1}{x}\right)}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 3 \cdot 10^{+109}:\\ \;\;\;\;1 - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{-0.3333333333333333}{\left(x \cdot x\right) \cdot n}}{-x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (if (<= (/ 1.0 n) -1e-5)
   (/ (pow x (- (/ 1.0 n) 1.0)) n)
   (if (<= (/ 1.0 n) 1e-11)
     (/ (log (/ (- x -1.0) x)) n)
     (if (<= (/ 1.0 n) 3e+109)
       (- 1.0 (pow x (/ 1.0 n)))
       (/ (/ -0.3333333333333333 (* (* x x) n)) (- x))))))

C

double code(double x, double n) {
	double tmp;
	if ((1.0 / n) <= -1e-5) {
		tmp = pow(x, ((1.0 / n) - 1.0)) / n;
	} else if ((1.0 / n) <= 1e-11) {
		tmp = log(((x - -1.0) / x)) / n;
	} else if ((1.0 / n) <= 3e+109) {
		tmp = 1.0 - pow(x, (1.0 / n));
	} else {
		tmp = (-0.3333333333333333 / ((x * x) * n)) / -x;
	}
	return tmp;
}

Fortran

real(8) function code(x, n)
    real(8), intent (in) :: x
    real(8), intent (in) :: n
    real(8) :: tmp
    if ((1.0d0 / n) <= (-1d-5)) then
        tmp = (x ** ((1.0d0 / n) - 1.0d0)) / n
    else if ((1.0d0 / n) <= 1d-11) then
        tmp = log(((x - (-1.0d0)) / x)) / n
    else if ((1.0d0 / n) <= 3d+109) then
        tmp = 1.0d0 - (x ** (1.0d0 / n))
    else
        tmp = ((-0.3333333333333333d0) / ((x * x) * n)) / -x
    end if
    code = tmp
end function

Java

public static double code(double x, double n) {
	double tmp;
	if ((1.0 / n) <= -1e-5) {
		tmp = Math.pow(x, ((1.0 / n) - 1.0)) / n;
	} else if ((1.0 / n) <= 1e-11) {
		tmp = Math.log(((x - -1.0) / x)) / n;
	} else if ((1.0 / n) <= 3e+109) {
		tmp = 1.0 - Math.pow(x, (1.0 / n));
	} else {
		tmp = (-0.3333333333333333 / ((x * x) * n)) / -x;
	}
	return tmp;
}

Python

def code(x, n):
	tmp = 0
	if (1.0 / n) <= -1e-5:
		tmp = math.pow(x, ((1.0 / n) - 1.0)) / n
	elif (1.0 / n) <= 1e-11:
		tmp = math.log(((x - -1.0) / x)) / n
	elif (1.0 / n) <= 3e+109:
		tmp = 1.0 - math.pow(x, (1.0 / n))
	else:
		tmp = (-0.3333333333333333 / ((x * x) * n)) / -x
	return tmp

Julia

function code(x, n)
	tmp = 0.0
	if (Float64(1.0 / n) <= -1e-5)
		tmp = Float64((x ^ Float64(Float64(1.0 / n) - 1.0)) / n);
	elseif (Float64(1.0 / n) <= 1e-11)
		tmp = Float64(log(Float64(Float64(x - -1.0) / x)) / n);
	elseif (Float64(1.0 / n) <= 3e+109)
		tmp = Float64(1.0 - (x ^ Float64(1.0 / n)));
	else
		tmp = Float64(Float64(-0.3333333333333333 / Float64(Float64(x * x) * n)) / Float64(-x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, n)
	tmp = 0.0;
	if ((1.0 / n) <= -1e-5)
		tmp = (x ^ ((1.0 / n) - 1.0)) / n;
	elseif ((1.0 / n) <= 1e-11)
		tmp = log(((x - -1.0) / x)) / n;
	elseif ((1.0 / n) <= 3e+109)
		tmp = 1.0 - (x ^ (1.0 / n));
	else
		tmp = (-0.3333333333333333 / ((x * x) * n)) / -x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, n_] := If[LessEqual[N[(1.0 / n), $MachinePrecision], -1e-5], N[(N[Power[x, N[(N[(1.0 / n), $MachinePrecision] - 1.0), $MachinePrecision]], $MachinePrecision] / n), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 1e-11], N[(N[Log[N[(N[(x - -1.0), $MachinePrecision] / x), $MachinePrecision]], $MachinePrecision] / n), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 3e+109], N[(1.0 - N[Power[x, N[(1.0 / n), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[(-0.3333333333333333 / N[(N[(x * x), $MachinePrecision] * n), $MachinePrecision]), $MachinePrecision] / (-x)), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if (/.f64 #s(literal 1 binary64) n) < -1.00000000000000008e-5

   1. Initial program 97.5%

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

   3. Taylor expanded in 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 57.7

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

   5. Applied rewrites 57.7%

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

   7. Applied rewrites 57.7%

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

   8. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. associate-/r* N/A

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

      3. lower-/.f64 N/A

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

   10. Applied rewrites 99.8%

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

   if -1.00000000000000008e-5 < (/.f64 #s(literal 1 binary64) n) < 9.99999999999999939e-12

   1. Initial program 31.3%

      \[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)} \]
   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 77.1

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

   5. Applied rewrites 77.1%

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

   7. Applied rewrites 77.1%

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

   if 9.99999999999999939e-12 < (/.f64 #s(literal 1 binary64) n) < 3.00000000000000015e109

   1. Initial program 85.2%

      \[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)} \]
   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 78.1%

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

   if 3.00000000000000015e109 < (/.f64 #s(literal 1 binary64) n)

   1. Initial program 17.2%

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

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

   5. Applied rewrites 6.1%

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

   6. Taylor expanded in x around -inf

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

   8. Applied rewrites 76.1%

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

   9. Taylor expanded in x around 0

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

   11. Applied rewrites 76.1%

       \[\leadsto \frac{\frac{-0.3333333333333333}{\left(x \cdot x\right) \cdot n}}{-x} \]
3. Recombined 4 regimes into one program.

4. Final simplification 83.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{1}{n} \leq -1 \cdot 10^{-5}:\\ \;\;\;\;\frac{{x}^{\left(\frac{1}{n} - 1\right)}}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 10^{-11}:\\ \;\;\;\;\frac{\log \left(\frac{x - -1}{x}\right)}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 3 \cdot 10^{+109}:\\ \;\;\;\;1 - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{-0.3333333333333333}{\left(x \cdot x\right) \cdot n}}{-x}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 56.8% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 4.1 \cdot 10^{-268}:\\ \;\;\;\;1 - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{elif}\;x \leq 3 \cdot 10^{-119}:\\ \;\;\;\;\frac{-\log x}{n}\\ \mathbf{elif}\;x \leq 2.65 \cdot 10^{-70}:\\ \;\;\;\;\frac{\frac{\frac{0.3333333333333333 \cdot n - \left(n \cdot x\right) \cdot 0.5}{\left(n \cdot x\right) \cdot n}}{x} - \frac{-1}{n}}{x}\\ \mathbf{elif}\;x \leq 1:\\ \;\;\;\;\frac{x - \log x}{n}\\ \mathbf{else}:\\ \;\;\;\;1 - 1\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (if (<= x 4.1e-268)
   (- 1.0 (pow x (/ 1.0 n)))
   (if (<= x 3e-119)
     (/ (- (log x)) n)
     (if (<= x 2.65e-70)
       (/
        (-
         (/ (/ (- (* 0.3333333333333333 n) (* (* n x) 0.5)) (* (* n x) n)) x)
         (/ -1.0 n))
        x)
       (if (<= x 1.0) (/ (- x (log x)) n) (- 1.0 1.0))))))

C

double code(double x, double n) {
	double tmp;
	if (x <= 4.1e-268) {
		tmp = 1.0 - pow(x, (1.0 / n));
	} else if (x <= 3e-119) {
		tmp = -log(x) / n;
	} else if (x <= 2.65e-70) {
		tmp = (((((0.3333333333333333 * n) - ((n * x) * 0.5)) / ((n * x) * n)) / x) - (-1.0 / n)) / x;
	} else if (x <= 1.0) {
		tmp = (x - log(x)) / n;
	} else {
		tmp = 1.0 - 1.0;
	}
	return tmp;
}

Fortran

real(8) function code(x, n)
    real(8), intent (in) :: x
    real(8), intent (in) :: n
    real(8) :: tmp
    if (x <= 4.1d-268) then
        tmp = 1.0d0 - (x ** (1.0d0 / n))
    else if (x <= 3d-119) then
        tmp = -log(x) / n
    else if (x <= 2.65d-70) then
        tmp = (((((0.3333333333333333d0 * n) - ((n * x) * 0.5d0)) / ((n * x) * n)) / x) - ((-1.0d0) / n)) / x
    else if (x <= 1.0d0) then
        tmp = (x - log(x)) / n
    else
        tmp = 1.0d0 - 1.0d0
    end if
    code = tmp
end function

Java

public static double code(double x, double n) {
	double tmp;
	if (x <= 4.1e-268) {
		tmp = 1.0 - Math.pow(x, (1.0 / n));
	} else if (x <= 3e-119) {
		tmp = -Math.log(x) / n;
	} else if (x <= 2.65e-70) {
		tmp = (((((0.3333333333333333 * n) - ((n * x) * 0.5)) / ((n * x) * n)) / x) - (-1.0 / n)) / x;
	} else if (x <= 1.0) {
		tmp = (x - Math.log(x)) / n;
	} else {
		tmp = 1.0 - 1.0;
	}
	return tmp;
}

Python

def code(x, n):
	tmp = 0
	if x <= 4.1e-268:
		tmp = 1.0 - math.pow(x, (1.0 / n))
	elif x <= 3e-119:
		tmp = -math.log(x) / n
	elif x <= 2.65e-70:
		tmp = (((((0.3333333333333333 * n) - ((n * x) * 0.5)) / ((n * x) * n)) / x) - (-1.0 / n)) / x
	elif x <= 1.0:
		tmp = (x - math.log(x)) / n
	else:
		tmp = 1.0 - 1.0
	return tmp

Julia

function code(x, n)
	tmp = 0.0
	if (x <= 4.1e-268)
		tmp = Float64(1.0 - (x ^ Float64(1.0 / n)));
	elseif (x <= 3e-119)
		tmp = Float64(Float64(-log(x)) / n);
	elseif (x <= 2.65e-70)
		tmp = Float64(Float64(Float64(Float64(Float64(Float64(0.3333333333333333 * n) - Float64(Float64(n * x) * 0.5)) / Float64(Float64(n * x) * n)) / x) - Float64(-1.0 / n)) / x);
	elseif (x <= 1.0)
		tmp = Float64(Float64(x - log(x)) / n);
	else
		tmp = Float64(1.0 - 1.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, n)
	tmp = 0.0;
	if (x <= 4.1e-268)
		tmp = 1.0 - (x ^ (1.0 / n));
	elseif (x <= 3e-119)
		tmp = -log(x) / n;
	elseif (x <= 2.65e-70)
		tmp = (((((0.3333333333333333 * n) - ((n * x) * 0.5)) / ((n * x) * n)) / x) - (-1.0 / n)) / x;
	elseif (x <= 1.0)
		tmp = (x - log(x)) / n;
	else
		tmp = 1.0 - 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, n_] := If[LessEqual[x, 4.1e-268], N[(1.0 - N[Power[x, N[(1.0 / n), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 3e-119], N[((-N[Log[x], $MachinePrecision]) / n), $MachinePrecision], If[LessEqual[x, 2.65e-70], N[(N[(N[(N[(N[(N[(0.3333333333333333 * n), $MachinePrecision] - N[(N[(n * x), $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision] / N[(N[(n * x), $MachinePrecision] * n), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision] - N[(-1.0 / n), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision], If[LessEqual[x, 1.0], N[(N[(x - N[Log[x], $MachinePrecision]), $MachinePrecision] / n), $MachinePrecision], N[(1.0 - 1.0), $MachinePrecision]]]]]

Derivation

1. Split input into 5 regimes

2. if x < 4.0999999999999999e-268

   1. Initial program 74.7%

      \[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)} \]
   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 74.7%

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

   if 4.0999999999999999e-268 < x < 3.0000000000000002e-119

   1. Initial program 35.3%

      \[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)} \]
   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 63.0

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

   5. Applied rewrites 63.0%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 63.0%

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

   if 3.0000000000000002e-119 < x < 2.64999999999999992e-70

   1. Initial program 35.8%

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

   3. Taylor expanded in n around inf

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

      1. lower-/.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lower-log1p.f64 N/A

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

      4. lower-log.f64 27.9

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

   5. Applied rewrites 27.9%

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

   6. Taylor expanded in x around -inf

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

   8. Applied rewrites 68.4%

      \[\leadsto \frac{\frac{\frac{\frac{0.3333333333333333}{n}}{x} - \frac{0.5}{n}}{-x} - \frac{1}{n}}{\color{blue}{-x}} \]
   9. Step-by-step derivation

   10. Applied rewrites 68.5%

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

   if 2.64999999999999992e-70 < x < 1

   1. Initial program 25.3%

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

   3. Taylor expanded in n around inf

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

      1. lower-/.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lower-log1p.f64 N/A

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

      4. lower-log.f64 71.5

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

   5. Applied rewrites 71.5%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 69.1%

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

   if 1 < x

   1. Initial program 70.1%

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

   3. Taylor expanded in x around 0

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

   5. Applied rewrites 34.7%

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

   6. Taylor expanded in n around inf

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

   8. Applied rewrites 70.1%

      \[\leadsto 1 - \color{blue}{1} \]
3. Recombined 5 regimes into one program.

4. Final simplification 68.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 4.1 \cdot 10^{-268}:\\ \;\;\;\;1 - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{elif}\;x \leq 3 \cdot 10^{-119}:\\ \;\;\;\;\frac{-\log x}{n}\\ \mathbf{elif}\;x \leq 2.65 \cdot 10^{-70}:\\ \;\;\;\;\frac{\frac{\frac{0.3333333333333333 \cdot n - \left(n \cdot x\right) \cdot 0.5}{\left(n \cdot x\right) \cdot n}}{x} - \frac{-1}{n}}{x}\\ \mathbf{elif}\;x \leq 1:\\ \;\;\;\;\frac{x - \log x}{n}\\ \mathbf{else}:\\ \;\;\;\;1 - 1\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 56.6% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 3 \cdot 10^{-119}:\\ \;\;\;\;\frac{-\log x}{n}\\ \mathbf{elif}\;x \leq 2.65 \cdot 10^{-70}:\\ \;\;\;\;\frac{\frac{\frac{0.3333333333333333 \cdot n - \left(n \cdot x\right) \cdot 0.5}{\left(n \cdot x\right) \cdot n}}{x} - \frac{-1}{n}}{x}\\ \mathbf{elif}\;x \leq 1:\\ \;\;\;\;\frac{x - \log x}{n}\\ \mathbf{else}:\\ \;\;\;\;1 - 1\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (if (<= x 3e-119)
   (/ (- (log x)) n)
   (if (<= x 2.65e-70)
     (/
      (-
       (/ (/ (- (* 0.3333333333333333 n) (* (* n x) 0.5)) (* (* n x) n)) x)
       (/ -1.0 n))
      x)
     (if (<= x 1.0) (/ (- x (log x)) n) (- 1.0 1.0)))))

C

double code(double x, double n) {
	double tmp;
	if (x <= 3e-119) {
		tmp = -log(x) / n;
	} else if (x <= 2.65e-70) {
		tmp = (((((0.3333333333333333 * n) - ((n * x) * 0.5)) / ((n * x) * n)) / x) - (-1.0 / n)) / x;
	} else if (x <= 1.0) {
		tmp = (x - log(x)) / n;
	} else {
		tmp = 1.0 - 1.0;
	}
	return tmp;
}

Fortran

real(8) function code(x, n)
    real(8), intent (in) :: x
    real(8), intent (in) :: n
    real(8) :: tmp
    if (x <= 3d-119) then
        tmp = -log(x) / n
    else if (x <= 2.65d-70) then
        tmp = (((((0.3333333333333333d0 * n) - ((n * x) * 0.5d0)) / ((n * x) * n)) / x) - ((-1.0d0) / n)) / x
    else if (x <= 1.0d0) then
        tmp = (x - log(x)) / n
    else
        tmp = 1.0d0 - 1.0d0
    end if
    code = tmp
end function

Java

public static double code(double x, double n) {
	double tmp;
	if (x <= 3e-119) {
		tmp = -Math.log(x) / n;
	} else if (x <= 2.65e-70) {
		tmp = (((((0.3333333333333333 * n) - ((n * x) * 0.5)) / ((n * x) * n)) / x) - (-1.0 / n)) / x;
	} else if (x <= 1.0) {
		tmp = (x - Math.log(x)) / n;
	} else {
		tmp = 1.0 - 1.0;
	}
	return tmp;
}

Python

def code(x, n):
	tmp = 0
	if x <= 3e-119:
		tmp = -math.log(x) / n
	elif x <= 2.65e-70:
		tmp = (((((0.3333333333333333 * n) - ((n * x) * 0.5)) / ((n * x) * n)) / x) - (-1.0 / n)) / x
	elif x <= 1.0:
		tmp = (x - math.log(x)) / n
	else:
		tmp = 1.0 - 1.0
	return tmp

Julia

function code(x, n)
	tmp = 0.0
	if (x <= 3e-119)
		tmp = Float64(Float64(-log(x)) / n);
	elseif (x <= 2.65e-70)
		tmp = Float64(Float64(Float64(Float64(Float64(Float64(0.3333333333333333 * n) - Float64(Float64(n * x) * 0.5)) / Float64(Float64(n * x) * n)) / x) - Float64(-1.0 / n)) / x);
	elseif (x <= 1.0)
		tmp = Float64(Float64(x - log(x)) / n);
	else
		tmp = Float64(1.0 - 1.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, n)
	tmp = 0.0;
	if (x <= 3e-119)
		tmp = -log(x) / n;
	elseif (x <= 2.65e-70)
		tmp = (((((0.3333333333333333 * n) - ((n * x) * 0.5)) / ((n * x) * n)) / x) - (-1.0 / n)) / x;
	elseif (x <= 1.0)
		tmp = (x - log(x)) / n;
	else
		tmp = 1.0 - 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, n_] := If[LessEqual[x, 3e-119], N[((-N[Log[x], $MachinePrecision]) / n), $MachinePrecision], If[LessEqual[x, 2.65e-70], N[(N[(N[(N[(N[(N[(0.3333333333333333 * n), $MachinePrecision] - N[(N[(n * x), $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision] / N[(N[(n * x), $MachinePrecision] * n), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision] - N[(-1.0 / n), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision], If[LessEqual[x, 1.0], N[(N[(x - N[Log[x], $MachinePrecision]), $MachinePrecision] / n), $MachinePrecision], N[(1.0 - 1.0), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if x < 3.0000000000000002e-119

   1. Initial program 43.0%

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

   3. Taylor expanded in n around inf

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

      1. lower-/.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lower-log1p.f64 N/A

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

      4. lower-log.f64 56.5

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

   5. Applied rewrites 56.5%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 56.5%

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

   if 3.0000000000000002e-119 < x < 2.64999999999999992e-70

   1. Initial program 35.8%

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

   3. Taylor expanded in n around inf

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

      1. lower-/.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lower-log1p.f64 N/A

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

      4. lower-log.f64 27.9

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

   5. Applied rewrites 27.9%

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

   6. Taylor expanded in x around -inf

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

   8. Applied rewrites 68.4%

      \[\leadsto \frac{\frac{\frac{\frac{0.3333333333333333}{n}}{x} - \frac{0.5}{n}}{-x} - \frac{1}{n}}{\color{blue}{-x}} \]
   9. Step-by-step derivation

   10. Applied rewrites 68.5%

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

   if 2.64999999999999992e-70 < x < 1

   1. Initial program 25.3%

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

   3. Taylor expanded in n around inf

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

      1. lower-/.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lower-log1p.f64 N/A

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

      4. lower-log.f64 71.5

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

   5. Applied rewrites 71.5%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 69.1%

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

   if 1 < x

   1. Initial program 70.1%

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

   3. Taylor expanded in x around 0

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

   5. Applied rewrites 34.7%

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

   6. Taylor expanded in n around inf

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

   8. Applied rewrites 70.1%

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

4. Final simplification 65.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 3 \cdot 10^{-119}:\\ \;\;\;\;\frac{-\log x}{n}\\ \mathbf{elif}\;x \leq 2.65 \cdot 10^{-70}:\\ \;\;\;\;\frac{\frac{\frac{0.3333333333333333 \cdot n - \left(n \cdot x\right) \cdot 0.5}{\left(n \cdot x\right) \cdot n}}{x} - \frac{-1}{n}}{x}\\ \mathbf{elif}\;x \leq 1:\\ \;\;\;\;\frac{x - \log x}{n}\\ \mathbf{else}:\\ \;\;\;\;1 - 1\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 56.3% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{-\log x}{n}\\ \mathbf{if}\;x \leq 3 \cdot 10^{-119}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq 2.65 \cdot 10^{-70}:\\ \;\;\;\;\frac{\frac{\frac{0.3333333333333333 \cdot n - \left(n \cdot x\right) \cdot 0.5}{\left(n \cdot x\right) \cdot n}}{x} - \frac{-1}{n}}{x}\\ \mathbf{elif}\;x \leq 1:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;1 - 1\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (let* ((t_0 (/ (- (log x)) n)))
   (if (<= x 3e-119)
     t_0
     (if (<= x 2.65e-70)
       (/
        (-
         (/ (/ (- (* 0.3333333333333333 n) (* (* n x) 0.5)) (* (* n x) n)) x)
         (/ -1.0 n))
        x)
       (if (<= x 1.0) t_0 (- 1.0 1.0))))))

C

double code(double x, double n) {
	double t_0 = -log(x) / n;
	double tmp;
	if (x <= 3e-119) {
		tmp = t_0;
	} else if (x <= 2.65e-70) {
		tmp = (((((0.3333333333333333 * n) - ((n * x) * 0.5)) / ((n * x) * n)) / x) - (-1.0 / n)) / x;
	} else if (x <= 1.0) {
		tmp = t_0;
	} 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) :: t_0
    real(8) :: tmp
    t_0 = -log(x) / n
    if (x <= 3d-119) then
        tmp = t_0
    else if (x <= 2.65d-70) then
        tmp = (((((0.3333333333333333d0 * n) - ((n * x) * 0.5d0)) / ((n * x) * n)) / x) - ((-1.0d0) / n)) / x
    else if (x <= 1.0d0) then
        tmp = t_0
    else
        tmp = 1.0d0 - 1.0d0
    end if
    code = tmp
end function

Java

public static double code(double x, double n) {
	double t_0 = -Math.log(x) / n;
	double tmp;
	if (x <= 3e-119) {
		tmp = t_0;
	} else if (x <= 2.65e-70) {
		tmp = (((((0.3333333333333333 * n) - ((n * x) * 0.5)) / ((n * x) * n)) / x) - (-1.0 / n)) / x;
	} else if (x <= 1.0) {
		tmp = t_0;
	} else {
		tmp = 1.0 - 1.0;
	}
	return tmp;
}

Python

def code(x, n):
	t_0 = -math.log(x) / n
	tmp = 0
	if x <= 3e-119:
		tmp = t_0
	elif x <= 2.65e-70:
		tmp = (((((0.3333333333333333 * n) - ((n * x) * 0.5)) / ((n * x) * n)) / x) - (-1.0 / n)) / x
	elif x <= 1.0:
		tmp = t_0
	else:
		tmp = 1.0 - 1.0
	return tmp

Julia

function code(x, n)
	t_0 = Float64(Float64(-log(x)) / n)
	tmp = 0.0
	if (x <= 3e-119)
		tmp = t_0;
	elseif (x <= 2.65e-70)
		tmp = Float64(Float64(Float64(Float64(Float64(Float64(0.3333333333333333 * n) - Float64(Float64(n * x) * 0.5)) / Float64(Float64(n * x) * n)) / x) - Float64(-1.0 / n)) / x);
	elseif (x <= 1.0)
		tmp = t_0;
	else
		tmp = Float64(1.0 - 1.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, n)
	t_0 = -log(x) / n;
	tmp = 0.0;
	if (x <= 3e-119)
		tmp = t_0;
	elseif (x <= 2.65e-70)
		tmp = (((((0.3333333333333333 * n) - ((n * x) * 0.5)) / ((n * x) * n)) / x) - (-1.0 / n)) / x;
	elseif (x <= 1.0)
		tmp = t_0;
	else
		tmp = 1.0 - 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, n_] := Block[{t$95$0 = N[((-N[Log[x], $MachinePrecision]) / n), $MachinePrecision]}, If[LessEqual[x, 3e-119], t$95$0, If[LessEqual[x, 2.65e-70], N[(N[(N[(N[(N[(N[(0.3333333333333333 * n), $MachinePrecision] - N[(N[(n * x), $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision] / N[(N[(n * x), $MachinePrecision] * n), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision] - N[(-1.0 / n), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision], If[LessEqual[x, 1.0], t$95$0, N[(1.0 - 1.0), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if x < 3.0000000000000002e-119 or 2.64999999999999992e-70 < x < 1

   1. Initial program 38.5%

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

   3. Taylor expanded in n around inf

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

      1. lower-/.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lower-log1p.f64 N/A

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

      4. lower-log.f64 60.3

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

   5. Applied rewrites 60.3%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 58.6%

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

   if 3.0000000000000002e-119 < x < 2.64999999999999992e-70

   1. Initial program 35.8%

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

   3. Taylor expanded in n around inf

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

      1. lower-/.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lower-log1p.f64 N/A

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

      4. lower-log.f64 27.9

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

   5. Applied rewrites 27.9%

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

   6. Taylor expanded in x around -inf

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

   8. Applied rewrites 68.4%

      \[\leadsto \frac{\frac{\frac{\frac{0.3333333333333333}{n}}{x} - \frac{0.5}{n}}{-x} - \frac{1}{n}}{\color{blue}{-x}} \]
   9. Step-by-step derivation

   10. Applied rewrites 68.5%

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

   if 1 < x

   1. Initial program 70.1%

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

   3. Taylor expanded in x around 0

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

   5. Applied rewrites 34.7%

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

   6. Taylor expanded in n around inf

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

   8. Applied rewrites 70.1%

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

4. Final simplification 64.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 3 \cdot 10^{-119}:\\ \;\;\;\;\frac{-\log x}{n}\\ \mathbf{elif}\;x \leq 2.65 \cdot 10^{-70}:\\ \;\;\;\;\frac{\frac{\frac{0.3333333333333333 \cdot n - \left(n \cdot x\right) \cdot 0.5}{\left(n \cdot x\right) \cdot n}}{x} - \frac{-1}{n}}{x}\\ \mathbf{elif}\;x \leq 1:\\ \;\;\;\;\frac{-\log x}{n}\\ \mathbf{else}:\\ \;\;\;\;1 - 1\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 52.2% accurate, 2.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{1}{n} \leq -2.3 \cdot 10^{+87}:\\ \;\;\;\;\frac{\frac{-0.3333333333333333}{\left(x \cdot x\right) \cdot n}}{-x}\\ \mathbf{elif}\;\frac{1}{n} \leq -50000000000:\\ \;\;\;\;1 - 1\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{\frac{1}{x}}{n}, \frac{0.3333333333333333}{x} - 0.5, \frac{1}{n}\right)}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (if (<= (/ 1.0 n) -2.3e+87)
   (/ (/ -0.3333333333333333 (* (* x x) n)) (- x))
   (if (<= (/ 1.0 n) -50000000000.0)
     (- 1.0 1.0)
     (/ (fma (/ (/ 1.0 x) n) (- (/ 0.3333333333333333 x) 0.5) (/ 1.0 n)) x))))

C

double code(double x, double n) {
	double tmp;
	if ((1.0 / n) <= -2.3e+87) {
		tmp = (-0.3333333333333333 / ((x * x) * n)) / -x;
	} else if ((1.0 / n) <= -50000000000.0) {
		tmp = 1.0 - 1.0;
	} else {
		tmp = fma(((1.0 / x) / n), ((0.3333333333333333 / x) - 0.5), (1.0 / n)) / x;
	}
	return tmp;
}

Julia

function code(x, n)
	tmp = 0.0
	if (Float64(1.0 / n) <= -2.3e+87)
		tmp = Float64(Float64(-0.3333333333333333 / Float64(Float64(x * x) * n)) / Float64(-x));
	elseif (Float64(1.0 / n) <= -50000000000.0)
		tmp = Float64(1.0 - 1.0);
	else
		tmp = Float64(fma(Float64(Float64(1.0 / x) / n), Float64(Float64(0.3333333333333333 / x) - 0.5), Float64(1.0 / n)) / x);
	end
	return tmp
end

Wolfram

code[x_, n_] := If[LessEqual[N[(1.0 / n), $MachinePrecision], -2.3e+87], N[(N[(-0.3333333333333333 / N[(N[(x * x), $MachinePrecision] * n), $MachinePrecision]), $MachinePrecision] / (-x)), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], -50000000000.0], N[(1.0 - 1.0), $MachinePrecision], N[(N[(N[(N[(1.0 / x), $MachinePrecision] / n), $MachinePrecision] * N[(N[(0.3333333333333333 / x), $MachinePrecision] - 0.5), $MachinePrecision] + N[(1.0 / n), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 #s(literal 1 binary64) n) < -2.3000000000000002e87

   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 51.1

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

   5. Applied rewrites 51.1%

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

   6. Taylor expanded in x around -inf

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

   8. Applied rewrites 47.7%

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

   9. Taylor expanded in x around 0

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

   11. Applied rewrites 71.3%

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

   if -2.3000000000000002e87 < (/.f64 #s(literal 1 binary64) n) < -5e10

   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 20.7%

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

   6. Taylor expanded in n around inf

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

   8. Applied rewrites 82.1%

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

   if -5e10 < (/.f64 #s(literal 1 binary64) n)

   1. Initial program 34.3%

      \[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)} \]
   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 62.5

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

   5. Applied rewrites 62.5%

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

   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 50.0%

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

   9. Taylor expanded in x around inf

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

   11. Applied rewrites 50.0%

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

Alternative 13: 52.2% accurate, 2.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{1}{n} \leq -2.3 \cdot 10^{+87}:\\ \;\;\;\;\frac{\frac{-0.3333333333333333}{\left(x \cdot x\right) \cdot n}}{-x}\\ \mathbf{elif}\;\frac{1}{n} \leq -50000000000:\\ \;\;\;\;1 - 1\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 - \frac{0.5 - \frac{0.3333333333333333}{x}}{x}}{n}}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (if (<= (/ 1.0 n) -2.3e+87)
   (/ (/ -0.3333333333333333 (* (* x x) n)) (- x))
   (if (<= (/ 1.0 n) -50000000000.0)
     (- 1.0 1.0)
     (/ (/ (- 1.0 (/ (- 0.5 (/ 0.3333333333333333 x)) x)) n) x))))

C

double code(double x, double n) {
	double tmp;
	if ((1.0 / n) <= -2.3e+87) {
		tmp = (-0.3333333333333333 / ((x * x) * n)) / -x;
	} else if ((1.0 / n) <= -50000000000.0) {
		tmp = 1.0 - 1.0;
	} else {
		tmp = ((1.0 - ((0.5 - (0.3333333333333333 / x)) / x)) / n) / x;
	}
	return tmp;
}

Fortran

real(8) function code(x, n)
    real(8), intent (in) :: x
    real(8), intent (in) :: n
    real(8) :: tmp
    if ((1.0d0 / n) <= (-2.3d+87)) then
        tmp = ((-0.3333333333333333d0) / ((x * x) * n)) / -x
    else if ((1.0d0 / n) <= (-50000000000.0d0)) then
        tmp = 1.0d0 - 1.0d0
    else
        tmp = ((1.0d0 - ((0.5d0 - (0.3333333333333333d0 / x)) / x)) / n) / x
    end if
    code = tmp
end function

Java

public static double code(double x, double n) {
	double tmp;
	if ((1.0 / n) <= -2.3e+87) {
		tmp = (-0.3333333333333333 / ((x * x) * n)) / -x;
	} else if ((1.0 / n) <= -50000000000.0) {
		tmp = 1.0 - 1.0;
	} else {
		tmp = ((1.0 - ((0.5 - (0.3333333333333333 / x)) / x)) / n) / x;
	}
	return tmp;
}

Python

def code(x, n):
	tmp = 0
	if (1.0 / n) <= -2.3e+87:
		tmp = (-0.3333333333333333 / ((x * x) * n)) / -x
	elif (1.0 / n) <= -50000000000.0:
		tmp = 1.0 - 1.0
	else:
		tmp = ((1.0 - ((0.5 - (0.3333333333333333 / x)) / x)) / n) / x
	return tmp

Julia

function code(x, n)
	tmp = 0.0
	if (Float64(1.0 / n) <= -2.3e+87)
		tmp = Float64(Float64(-0.3333333333333333 / Float64(Float64(x * x) * n)) / Float64(-x));
	elseif (Float64(1.0 / n) <= -50000000000.0)
		tmp = Float64(1.0 - 1.0);
	else
		tmp = Float64(Float64(Float64(1.0 - Float64(Float64(0.5 - Float64(0.3333333333333333 / x)) / x)) / n) / x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, n)
	tmp = 0.0;
	if ((1.0 / n) <= -2.3e+87)
		tmp = (-0.3333333333333333 / ((x * x) * n)) / -x;
	elseif ((1.0 / n) <= -50000000000.0)
		tmp = 1.0 - 1.0;
	else
		tmp = ((1.0 - ((0.5 - (0.3333333333333333 / x)) / x)) / n) / x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, n_] := If[LessEqual[N[(1.0 / n), $MachinePrecision], -2.3e+87], N[(N[(-0.3333333333333333 / N[(N[(x * x), $MachinePrecision] * n), $MachinePrecision]), $MachinePrecision] / (-x)), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], -50000000000.0], N[(1.0 - 1.0), $MachinePrecision], N[(N[(N[(1.0 - N[(N[(0.5 - N[(0.3333333333333333 / x), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision] / n), $MachinePrecision] / x), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 #s(literal 1 binary64) n) < -2.3000000000000002e87

   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 51.1

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

   5. Applied rewrites 51.1%

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

   6. Taylor expanded in x around -inf

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

   8. Applied rewrites 47.7%

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

   9. Taylor expanded in x around 0

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

   11. Applied rewrites 71.3%

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

   if -2.3000000000000002e87 < (/.f64 #s(literal 1 binary64) n) < -5e10

   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 20.7%

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

   6. Taylor expanded in n around inf

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

   8. Applied rewrites 82.1%

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

   if -5e10 < (/.f64 #s(literal 1 binary64) n)

   1. Initial program 34.3%

      \[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)} \]
   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 62.5

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

   5. Applied rewrites 62.5%

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

   6. Taylor expanded in x around -inf

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

   8. Applied rewrites 50.0%

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

   9. Taylor expanded in n around -inf

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

   11. Applied rewrites 50.0%

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

Alternative 14: 51.8% accurate, 2.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{-0.3333333333333333}{\left(x \cdot x\right) \cdot n}\\ \mathbf{if}\;\frac{1}{n} \leq -2.3 \cdot 10^{+87}:\\ \;\;\;\;\frac{t\_0}{-x}\\ \mathbf{elif}\;\frac{1}{n} \leq -50000000000:\\ \;\;\;\;1 - 1\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{n} - t\_0}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (let* ((t_0 (/ -0.3333333333333333 (* (* x x) n))))
   (if (<= (/ 1.0 n) -2.3e+87)
     (/ t_0 (- x))
     (if (<= (/ 1.0 n) -50000000000.0) (- 1.0 1.0) (/ (- (/ 1.0 n) t_0) x)))))

C

double code(double x, double n) {
	double t_0 = -0.3333333333333333 / ((x * x) * n);
	double tmp;
	if ((1.0 / n) <= -2.3e+87) {
		tmp = t_0 / -x;
	} else if ((1.0 / n) <= -50000000000.0) {
		tmp = 1.0 - 1.0;
	} else {
		tmp = ((1.0 / n) - t_0) / x;
	}
	return tmp;
}

Fortran

real(8) function code(x, n)
    real(8), intent (in) :: x
    real(8), intent (in) :: n
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (-0.3333333333333333d0) / ((x * x) * n)
    if ((1.0d0 / n) <= (-2.3d+87)) then
        tmp = t_0 / -x
    else if ((1.0d0 / n) <= (-50000000000.0d0)) then
        tmp = 1.0d0 - 1.0d0
    else
        tmp = ((1.0d0 / n) - t_0) / x
    end if
    code = tmp
end function

Java

public static double code(double x, double n) {
	double t_0 = -0.3333333333333333 / ((x * x) * n);
	double tmp;
	if ((1.0 / n) <= -2.3e+87) {
		tmp = t_0 / -x;
	} else if ((1.0 / n) <= -50000000000.0) {
		tmp = 1.0 - 1.0;
	} else {
		tmp = ((1.0 / n) - t_0) / x;
	}
	return tmp;
}

Python

def code(x, n):
	t_0 = -0.3333333333333333 / ((x * x) * n)
	tmp = 0
	if (1.0 / n) <= -2.3e+87:
		tmp = t_0 / -x
	elif (1.0 / n) <= -50000000000.0:
		tmp = 1.0 - 1.0
	else:
		tmp = ((1.0 / n) - t_0) / x
	return tmp

Julia

function code(x, n)
	t_0 = Float64(-0.3333333333333333 / Float64(Float64(x * x) * n))
	tmp = 0.0
	if (Float64(1.0 / n) <= -2.3e+87)
		tmp = Float64(t_0 / Float64(-x));
	elseif (Float64(1.0 / n) <= -50000000000.0)
		tmp = Float64(1.0 - 1.0);
	else
		tmp = Float64(Float64(Float64(1.0 / n) - t_0) / x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, n)
	t_0 = -0.3333333333333333 / ((x * x) * n);
	tmp = 0.0;
	if ((1.0 / n) <= -2.3e+87)
		tmp = t_0 / -x;
	elseif ((1.0 / n) <= -50000000000.0)
		tmp = 1.0 - 1.0;
	else
		tmp = ((1.0 / n) - t_0) / x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, n_] := Block[{t$95$0 = N[(-0.3333333333333333 / N[(N[(x * x), $MachinePrecision] * n), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(1.0 / n), $MachinePrecision], -2.3e+87], N[(t$95$0 / (-x)), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], -50000000000.0], N[(1.0 - 1.0), $MachinePrecision], N[(N[(N[(1.0 / n), $MachinePrecision] - t$95$0), $MachinePrecision] / x), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 #s(literal 1 binary64) n) < -2.3000000000000002e87

   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 51.1

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

   5. Applied rewrites 51.1%

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

   6. Taylor expanded in x around -inf

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

   8. Applied rewrites 47.7%

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

   9. Taylor expanded in x around 0

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

   11. Applied rewrites 71.3%

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

   if -2.3000000000000002e87 < (/.f64 #s(literal 1 binary64) n) < -5e10

   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 20.7%

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

   6. Taylor expanded in n around inf

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

   8. Applied rewrites 82.1%

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

   if -5e10 < (/.f64 #s(literal 1 binary64) n)

   1. Initial program 34.3%

      \[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)} \]
   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 62.5

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

   5. Applied rewrites 62.5%

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

   6. Taylor expanded in x around -inf

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

   8. Applied rewrites 50.0%

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

   9. Taylor expanded in x around 0

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

   11. Applied rewrites 49.8%

       \[\leadsto \frac{\frac{-0.3333333333333333}{\left(x \cdot x\right) \cdot n} - \frac{1}{n}}{-x} \]
3. Recombined 3 regimes into one program.

4. Final simplification 56.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{1}{n} \leq -2.3 \cdot 10^{+87}:\\ \;\;\;\;\frac{\frac{-0.3333333333333333}{\left(x \cdot x\right) \cdot n}}{-x}\\ \mathbf{elif}\;\frac{1}{n} \leq -50000000000:\\ \;\;\;\;1 - 1\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{n} - \frac{-0.3333333333333333}{\left(x \cdot x\right) \cdot n}}{x}\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 52.6% accurate, 4.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;n \leq -0.046:\\ \;\;\;\;\frac{-1}{n} \cdot \frac{-1}{x}\\ \mathbf{elif}\;n \leq -1.25 \cdot 10^{-84}:\\ \;\;\;\;1 - 1\\ \mathbf{elif}\;n \leq 3.2 \cdot 10^{-80}:\\ \;\;\;\;\frac{\frac{-0.3333333333333333}{\left(x \cdot x\right) \cdot n}}{-x}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{x}}{n}\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (if (<= n -0.046)
   (* (/ -1.0 n) (/ -1.0 x))
   (if (<= n -1.25e-84)
     (- 1.0 1.0)
     (if (<= n 3.2e-80)
       (/ (/ -0.3333333333333333 (* (* x x) n)) (- x))
       (/ (/ 1.0 x) n)))))

C

double code(double x, double n) {
	double tmp;
	if (n <= -0.046) {
		tmp = (-1.0 / n) * (-1.0 / x);
	} else if (n <= -1.25e-84) {
		tmp = 1.0 - 1.0;
	} else if (n <= 3.2e-80) {
		tmp = (-0.3333333333333333 / ((x * x) * n)) / -x;
	} else {
		tmp = (1.0 / x) / n;
	}
	return tmp;
}

Fortran

real(8) function code(x, n)
    real(8), intent (in) :: x
    real(8), intent (in) :: n
    real(8) :: tmp
    if (n <= (-0.046d0)) then
        tmp = ((-1.0d0) / n) * ((-1.0d0) / x)
    else if (n <= (-1.25d-84)) then
        tmp = 1.0d0 - 1.0d0
    else if (n <= 3.2d-80) then
        tmp = ((-0.3333333333333333d0) / ((x * x) * n)) / -x
    else
        tmp = (1.0d0 / x) / n
    end if
    code = tmp
end function

Java

public static double code(double x, double n) {
	double tmp;
	if (n <= -0.046) {
		tmp = (-1.0 / n) * (-1.0 / x);
	} else if (n <= -1.25e-84) {
		tmp = 1.0 - 1.0;
	} else if (n <= 3.2e-80) {
		tmp = (-0.3333333333333333 / ((x * x) * n)) / -x;
	} else {
		tmp = (1.0 / x) / n;
	}
	return tmp;
}

Python

def code(x, n):
	tmp = 0
	if n <= -0.046:
		tmp = (-1.0 / n) * (-1.0 / x)
	elif n <= -1.25e-84:
		tmp = 1.0 - 1.0
	elif n <= 3.2e-80:
		tmp = (-0.3333333333333333 / ((x * x) * n)) / -x
	else:
		tmp = (1.0 / x) / n
	return tmp

Julia

function code(x, n)
	tmp = 0.0
	if (n <= -0.046)
		tmp = Float64(Float64(-1.0 / n) * Float64(-1.0 / x));
	elseif (n <= -1.25e-84)
		tmp = Float64(1.0 - 1.0);
	elseif (n <= 3.2e-80)
		tmp = Float64(Float64(-0.3333333333333333 / Float64(Float64(x * x) * n)) / Float64(-x));
	else
		tmp = Float64(Float64(1.0 / x) / n);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, n)
	tmp = 0.0;
	if (n <= -0.046)
		tmp = (-1.0 / n) * (-1.0 / x);
	elseif (n <= -1.25e-84)
		tmp = 1.0 - 1.0;
	elseif (n <= 3.2e-80)
		tmp = (-0.3333333333333333 / ((x * x) * n)) / -x;
	else
		tmp = (1.0 / x) / n;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, n_] := If[LessEqual[n, -0.046], N[(N[(-1.0 / n), $MachinePrecision] * N[(-1.0 / x), $MachinePrecision]), $MachinePrecision], If[LessEqual[n, -1.25e-84], N[(1.0 - 1.0), $MachinePrecision], If[LessEqual[n, 3.2e-80], N[(N[(-0.3333333333333333 / N[(N[(x * x), $MachinePrecision] * n), $MachinePrecision]), $MachinePrecision] / (-x)), $MachinePrecision], N[(N[(1.0 / x), $MachinePrecision] / n), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if n < -0.045999999999999999

   1. Initial program 34.3%

      \[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)} \]
   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 72.5

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

   5. Applied rewrites 72.5%

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

   7. Applied rewrites 72.4%

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

   8. Taylor expanded in x around inf

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

   10. Applied rewrites 53.7%

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

   if -0.045999999999999999 < n < -1.25e-84

   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 22.5%

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

   6. Taylor expanded in n around inf

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

   8. Applied rewrites 80.3%

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

   if -1.25e-84 < n < 3.1999999999999999e-80

   1. Initial program 77.2%

      \[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)} \]
   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 37.7

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

   5. Applied rewrites 37.7%

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

   6. Taylor expanded in x around -inf

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

   8. Applied rewrites 51.1%

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

   9. Taylor expanded in x around 0

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

   11. Applied rewrites 68.9%

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

   if 3.1999999999999999e-80 < n

   1. Initial program 35.7%

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

   3. Taylor expanded in x around inf

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

      1. associate-/l/ N/A

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

      2. lower-/.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. log-rec N/A

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

      5. mul-1-neg N/A

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

      6. associate-*r/ N/A

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

      7. associate-*r* N/A

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

      8. metadata-eval N/A

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

      9. *-commutative N/A

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

      10. associate-/l* N/A

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

      11. exp-to-pow N/A

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

      12. lower-pow.f64 N/A

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

      13. lower-/.f64 44.6

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

   5. Applied rewrites 44.6%

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

   6. Taylor expanded in n around inf

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

   8. Applied rewrites 43.7%

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

4. Final simplification 57.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq -0.046:\\ \;\;\;\;\frac{-1}{n} \cdot \frac{-1}{x}\\ \mathbf{elif}\;n \leq -1.25 \cdot 10^{-84}:\\ \;\;\;\;1 - 1\\ \mathbf{elif}\;n \leq 3.2 \cdot 10^{-80}:\\ \;\;\;\;\frac{\frac{-0.3333333333333333}{\left(x \cdot x\right) \cdot n}}{-x}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{x}}{n}\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 46.0% accurate, 6.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;n \leq -0.046:\\ \;\;\;\;\frac{-1}{n} \cdot \frac{-1}{x}\\ \mathbf{elif}\;n \leq -2.8 \cdot 10^{-203}:\\ \;\;\;\;1 - 1\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{x}}{n}\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (if (<= n -0.046)
   (* (/ -1.0 n) (/ -1.0 x))
   (if (<= n -2.8e-203) (- 1.0 1.0) (/ (/ 1.0 x) n))))

C

double code(double x, double n) {
	double tmp;
	if (n <= -0.046) {
		tmp = (-1.0 / n) * (-1.0 / x);
	} else if (n <= -2.8e-203) {
		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 (n <= (-0.046d0)) then
        tmp = ((-1.0d0) / n) * ((-1.0d0) / x)
    else if (n <= (-2.8d-203)) 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 (n <= -0.046) {
		tmp = (-1.0 / n) * (-1.0 / x);
	} else if (n <= -2.8e-203) {
		tmp = 1.0 - 1.0;
	} else {
		tmp = (1.0 / x) / n;
	}
	return tmp;
}

Python

def code(x, n):
	tmp = 0
	if n <= -0.046:
		tmp = (-1.0 / n) * (-1.0 / x)
	elif n <= -2.8e-203:
		tmp = 1.0 - 1.0
	else:
		tmp = (1.0 / x) / n
	return tmp

Julia

function code(x, n)
	tmp = 0.0
	if (n <= -0.046)
		tmp = Float64(Float64(-1.0 / n) * Float64(-1.0 / x));
	elseif (n <= -2.8e-203)
		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 (n <= -0.046)
		tmp = (-1.0 / n) * (-1.0 / x);
	elseif (n <= -2.8e-203)
		tmp = 1.0 - 1.0;
	else
		tmp = (1.0 / x) / n;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, n_] := If[LessEqual[n, -0.046], N[(N[(-1.0 / n), $MachinePrecision] * N[(-1.0 / x), $MachinePrecision]), $MachinePrecision], If[LessEqual[n, -2.8e-203], N[(1.0 - 1.0), $MachinePrecision], N[(N[(1.0 / x), $MachinePrecision] / n), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if n < -0.045999999999999999

   1. Initial program 34.3%

      \[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)} \]
   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 72.5

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

   5. Applied rewrites 72.5%

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

   7. Applied rewrites 72.4%

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

   8. Taylor expanded in x around inf

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

   10. Applied rewrites 53.7%

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

   if -0.045999999999999999 < n < -2.80000000000000022e-203

   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 35.4%

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

   6. Taylor expanded in n around inf

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

   8. Applied rewrites 67.2%

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

   if -2.80000000000000022e-203 < n

   1. Initial program 43.6%

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

   3. Taylor expanded in x around inf

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

      1. associate-/l/ N/A

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

      2. lower-/.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. log-rec N/A

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

      5. mul-1-neg N/A

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

      6. associate-*r/ N/A

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

      7. associate-*r* N/A

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

      8. metadata-eval N/A

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

      9. *-commutative N/A

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

      10. associate-/l* N/A

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

      11. exp-to-pow N/A

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

      12. lower-pow.f64 N/A

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

      13. lower-/.f64 44.1

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

   5. Applied rewrites 44.1%

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

   6. Taylor expanded in n around inf

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

   8. Applied rewrites 46.1%

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

4. Final simplification 52.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq -0.046:\\ \;\;\;\;\frac{-1}{n} \cdot \frac{-1}{x}\\ \mathbf{elif}\;n \leq -2.8 \cdot 10^{-203}:\\ \;\;\;\;1 - 1\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{x}}{n}\\ \end{array} \]
5. Add Preprocessing

Alternative 17: 46.0% accurate, 6.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;n \leq -0.046:\\ \;\;\;\;\frac{\frac{1}{n}}{x}\\ \mathbf{elif}\;n \leq -2.8 \cdot 10^{-203}:\\ \;\;\;\;1 - 1\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{x}}{n}\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (if (<= n -0.046)
   (/ (/ 1.0 n) x)
   (if (<= n -2.8e-203) (- 1.0 1.0) (/ (/ 1.0 x) n))))

C

double code(double x, double n) {
	double tmp;
	if (n <= -0.046) {
		tmp = (1.0 / n) / x;
	} else if (n <= -2.8e-203) {
		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 (n <= (-0.046d0)) then
        tmp = (1.0d0 / n) / x
    else if (n <= (-2.8d-203)) 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 (n <= -0.046) {
		tmp = (1.0 / n) / x;
	} else if (n <= -2.8e-203) {
		tmp = 1.0 - 1.0;
	} else {
		tmp = (1.0 / x) / n;
	}
	return tmp;
}

Python

def code(x, n):
	tmp = 0
	if n <= -0.046:
		tmp = (1.0 / n) / x
	elif n <= -2.8e-203:
		tmp = 1.0 - 1.0
	else:
		tmp = (1.0 / x) / n
	return tmp

Julia

function code(x, n)
	tmp = 0.0
	if (n <= -0.046)
		tmp = Float64(Float64(1.0 / n) / x);
	elseif (n <= -2.8e-203)
		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 (n <= -0.046)
		tmp = (1.0 / n) / x;
	elseif (n <= -2.8e-203)
		tmp = 1.0 - 1.0;
	else
		tmp = (1.0 / x) / n;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, n_] := If[LessEqual[n, -0.046], N[(N[(1.0 / n), $MachinePrecision] / x), $MachinePrecision], If[LessEqual[n, -2.8e-203], N[(1.0 - 1.0), $MachinePrecision], N[(N[(1.0 / x), $MachinePrecision] / n), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if n < -0.045999999999999999

   1. Initial program 34.3%

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

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

      2. lower-/.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. log-rec N/A

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

      5. mul-1-neg N/A

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

      6. associate-*r/ N/A

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

      7. associate-*r* N/A

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

      8. metadata-eval N/A

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

      9. *-commutative N/A

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

      10. associate-/l* N/A

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

      11. exp-to-pow N/A

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

      12. lower-pow.f64 N/A

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

      13. lower-/.f64 58.1

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

   5. Applied rewrites 58.1%

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

   7. Applied rewrites 58.1%

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

   8. Taylor expanded in n around inf

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

   10. Applied rewrites 53.7%

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

   if -0.045999999999999999 < n < -2.80000000000000022e-203

   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 35.4%

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

   6. Taylor expanded in n around inf

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

   8. Applied rewrites 67.2%

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

   if -2.80000000000000022e-203 < n

   1. Initial program 43.6%

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

   3. Taylor expanded in x around inf

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

      1. associate-/l/ N/A

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

      2. lower-/.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. log-rec N/A

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

      5. mul-1-neg N/A

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

      6. associate-*r/ N/A

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

      7. associate-*r* N/A

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

      8. metadata-eval N/A

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

      9. *-commutative N/A

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

      10. associate-/l* N/A

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

      11. exp-to-pow N/A

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

      12. lower-pow.f64 N/A

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

      13. lower-/.f64 44.1

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

   5. Applied rewrites 44.1%

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

   6. Taylor expanded in n around inf

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

   8. Applied rewrites 46.1%

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

Alternative 18: 46.0% accurate, 6.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\frac{1}{n}}{x}\\ \mathbf{if}\;n \leq -0.046:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;n \leq -2.8 \cdot 10^{-203}:\\ \;\;\;\;1 - 1\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (let* ((t_0 (/ (/ 1.0 n) x)))
   (if (<= n -0.046) t_0 (if (<= n -2.8e-203) (- 1.0 1.0) t_0))))

C

double code(double x, double n) {
	double t_0 = (1.0 / n) / x;
	double tmp;
	if (n <= -0.046) {
		tmp = t_0;
	} else if (n <= -2.8e-203) {
		tmp = 1.0 - 1.0;
	} else {
		tmp = t_0;
	}
	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 = (1.0d0 / n) / x
    if (n <= (-0.046d0)) then
        tmp = t_0
    else if (n <= (-2.8d-203)) then
        tmp = 1.0d0 - 1.0d0
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double x, double n) {
	double t_0 = (1.0 / n) / x;
	double tmp;
	if (n <= -0.046) {
		tmp = t_0;
	} else if (n <= -2.8e-203) {
		tmp = 1.0 - 1.0;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, n):
	t_0 = (1.0 / n) / x
	tmp = 0
	if n <= -0.046:
		tmp = t_0
	elif n <= -2.8e-203:
		tmp = 1.0 - 1.0
	else:
		tmp = t_0
	return tmp

Julia

function code(x, n)
	t_0 = Float64(Float64(1.0 / n) / x)
	tmp = 0.0
	if (n <= -0.046)
		tmp = t_0;
	elseif (n <= -2.8e-203)
		tmp = Float64(1.0 - 1.0);
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, n)
	t_0 = (1.0 / n) / x;
	tmp = 0.0;
	if (n <= -0.046)
		tmp = t_0;
	elseif (n <= -2.8e-203)
		tmp = 1.0 - 1.0;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, n_] := Block[{t$95$0 = N[(N[(1.0 / n), $MachinePrecision] / x), $MachinePrecision]}, If[LessEqual[n, -0.046], t$95$0, If[LessEqual[n, -2.8e-203], N[(1.0 - 1.0), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if n < -0.045999999999999999 or -2.80000000000000022e-203 < n

   1. Initial program 40.2%

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

   3. Taylor expanded in x around inf

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

      1. associate-/l/ N/A

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

      2. lower-/.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. log-rec N/A

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

      5. mul-1-neg N/A

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

      6. associate-*r/ N/A

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

      7. associate-*r* N/A

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

      8. metadata-eval N/A

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

      9. *-commutative N/A

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

      10. associate-/l* N/A

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

      11. exp-to-pow N/A

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

      12. lower-pow.f64 N/A

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

      13. lower-/.f64 49.2

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

   5. Applied rewrites 49.2%

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

   7. Applied rewrites 49.2%

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

   8. Taylor expanded in n around inf

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

   10. Applied rewrites 48.9%

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

   if -0.045999999999999999 < n < -2.80000000000000022e-203

   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 35.4%

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

   6. Taylor expanded in n around inf

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

   8. Applied rewrites 67.2%

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

Alternative 19: 30.2% 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 52.8%

   \[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)} \]
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 36.1%

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

6. Taylor expanded in n around inf

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

8. Applied rewrites 34.3%

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

Reproduce

herbie shell --seed 2024283 
(FPCore (x n)
  :name "2nthrt (problem 3.4.6)"
  :precision binary64
  (- (pow (+ x 1.0) (/ 1.0 n)) (pow x (/ 1.0 n))))