2nthrt (problem 3.4.6)

Percentage Accurate: 53.1% → 86.0%

Time: 25.5s

Alternatives: 22

Speedup: 3.8×

• Could not converge on a ground truth

  1. x = -9.168070044706664e+235
  2. n = -2.0225028884221063e-289

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 53.1% 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: 86.0% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{if}\;\frac{1}{n} \leq -5 \cdot 10^{-71}:\\ \;\;\;\;\frac{t\_0}{n \cdot x}\\ \mathbf{elif}\;\frac{1}{n} \leq 2 \cdot 10^{-11}:\\ \;\;\;\;\frac{\mathsf{fma}\left(0.5, \frac{{\left(\mathsf{log1p}\left(x\right)\right)}^{2} - {\log x}^{2}}{n}, \mathsf{log1p}\left(x\right)\right) - \log x}{n}\\ \mathbf{else}:\\ \;\;\;\;e^{\frac{\mathsf{log1p}\left(x\right)}{n}} - t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (let* ((t_0 (pow x (/ 1.0 n))))
   (if (<= (/ 1.0 n) -5e-71)
     (/ t_0 (* n x))
     (if (<= (/ 1.0 n) 2e-11)
       (/
        (-
         (fma 0.5 (/ (- (pow (log1p x) 2.0) (pow (log x) 2.0)) n) (log1p x))
         (log x))
        n)
       (- (exp (/ (log1p x) n)) t_0)))))

C

double code(double x, double n) {
	double t_0 = pow(x, (1.0 / n));
	double tmp;
	if ((1.0 / n) <= -5e-71) {
		tmp = t_0 / (n * x);
	} else if ((1.0 / n) <= 2e-11) {
		tmp = (fma(0.5, ((pow(log1p(x), 2.0) - pow(log(x), 2.0)) / n), log1p(x)) - log(x)) / n;
	} else {
		tmp = exp((log1p(x) / n)) - t_0;
	}
	return tmp;
}

Julia

function code(x, n)
	t_0 = x ^ Float64(1.0 / n)
	tmp = 0.0
	if (Float64(1.0 / n) <= -5e-71)
		tmp = Float64(t_0 / Float64(n * x));
	elseif (Float64(1.0 / n) <= 2e-11)
		tmp = Float64(Float64(fma(0.5, Float64(Float64((log1p(x) ^ 2.0) - (log(x) ^ 2.0)) / n), log1p(x)) - log(x)) / n);
	else
		tmp = Float64(exp(Float64(log1p(x) / n)) - t_0);
	end
	return tmp
end

Wolfram

code[x_, n_] := Block[{t$95$0 = N[Power[x, N[(1.0 / n), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[N[(1.0 / n), $MachinePrecision], -5e-71], N[(t$95$0 / N[(n * x), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 2e-11], N[(N[(N[(0.5 * N[(N[(N[Power[N[Log[1 + x], $MachinePrecision], 2.0], $MachinePrecision] - N[Power[N[Log[x], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] / n), $MachinePrecision] + N[Log[1 + x], $MachinePrecision]), $MachinePrecision] - N[Log[x], $MachinePrecision]), $MachinePrecision] / n), $MachinePrecision], N[(N[Exp[N[(N[Log[1 + x], $MachinePrecision] / n), $MachinePrecision]], $MachinePrecision] - t$95$0), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 #s(literal 1 binary64) n) < -4.99999999999999998e-71

   1. Initial program 88.2%

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

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

      2. log-rec N/A

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

      3. mul-1-neg N/A

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

      4. associate-*r/ N/A

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

      5. associate-*r* N/A

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

      6. metadata-eval N/A

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

      7. *-commutative N/A

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

      8. associate-/l* N/A

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

      9. exp-to-pow N/A

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

      10. lower-pow.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 94.1

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

   5. Applied rewrites 94.1%

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

   if -4.99999999999999998e-71 < (/.f64 #s(literal 1 binary64) n) < 1.99999999999999988e-11

   1. Initial program 23.7%

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

   3. Taylor expanded in n around inf

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

      1. lower-/.f64 N/A

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

   5. Applied rewrites 84.3%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(0.5, \frac{{\left(\mathsf{log1p}\left(x\right)\right)}^{2} - {\log x}^{2}}{n}, \mathsf{log1p}\left(x\right)\right) - \log x}{n}} \]

   if 1.99999999999999988e-11 < (/.f64 #s(literal 1 binary64) n)

   1. Initial program 52.5%

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

      1. lift-+.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. pow-to-exp N/A

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

      4. lower-exp.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. un-div-inv N/A

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

      7. lower-/.f64 N/A

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

      8. lift-+.f64 N/A

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

      9. +-commutative N/A

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

      10. lower-log1p.f64 95.2

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

   4. Applied rewrites 95.2%

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

4. Final simplification 89.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{1}{n} \leq -5 \cdot 10^{-71}:\\ \;\;\;\;\frac{{x}^{\left(\frac{1}{n}\right)}}{n \cdot x}\\ \mathbf{elif}\;\frac{1}{n} \leq 2 \cdot 10^{-11}:\\ \;\;\;\;\frac{\mathsf{fma}\left(0.5, \frac{{\left(\mathsf{log1p}\left(x\right)\right)}^{2} - {\log x}^{2}}{n}, \mathsf{log1p}\left(x\right)\right) - \log x}{n}\\ \mathbf{else}:\\ \;\;\;\;e^{\frac{\mathsf{log1p}\left(x\right)}{n}} - {x}^{\left(\frac{1}{n}\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 78.2% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {x}^{\left(\frac{1}{n}\right)}\\ t_1 := {\left(1 + x\right)}^{\left(\frac{1}{n}\right)} - t\_0\\ t_2 := 1 - t\_0\\ \mathbf{if}\;t\_1 \leq -0.005:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_1 \leq 5 \cdot 10^{-9}:\\ \;\;\;\;\frac{\log \left(\frac{1 + x}{x}\right)}{n}\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (let* ((t_0 (pow x (/ 1.0 n)))
        (t_1 (- (pow (+ 1.0 x) (/ 1.0 n)) t_0))
        (t_2 (- 1.0 t_0)))
   (if (<= t_1 -0.005)
     t_2
     (if (<= t_1 5e-9) (/ (log (/ (+ 1.0 x) x)) n) t_2))))

C

double code(double x, double n) {
	double t_0 = pow(x, (1.0 / n));
	double t_1 = pow((1.0 + x), (1.0 / n)) - t_0;
	double t_2 = 1.0 - t_0;
	double tmp;
	if (t_1 <= -0.005) {
		tmp = t_2;
	} else if (t_1 <= 5e-9) {
		tmp = log(((1.0 + x) / x)) / n;
	} else {
		tmp = t_2;
	}
	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) :: t_2
    real(8) :: tmp
    t_0 = x ** (1.0d0 / n)
    t_1 = ((1.0d0 + x) ** (1.0d0 / n)) - t_0
    t_2 = 1.0d0 - t_0
    if (t_1 <= (-0.005d0)) then
        tmp = t_2
    else if (t_1 <= 5d-9) then
        tmp = log(((1.0d0 + x) / x)) / n
    else
        tmp = t_2
    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((1.0 + x), (1.0 / n)) - t_0;
	double t_2 = 1.0 - t_0;
	double tmp;
	if (t_1 <= -0.005) {
		tmp = t_2;
	} else if (t_1 <= 5e-9) {
		tmp = Math.log(((1.0 + x) / x)) / n;
	} else {
		tmp = t_2;
	}
	return tmp;
}

Python

def code(x, n):
	t_0 = math.pow(x, (1.0 / n))
	t_1 = math.pow((1.0 + x), (1.0 / n)) - t_0
	t_2 = 1.0 - t_0
	tmp = 0
	if t_1 <= -0.005:
		tmp = t_2
	elif t_1 <= 5e-9:
		tmp = math.log(((1.0 + x) / x)) / n
	else:
		tmp = t_2
	return tmp

Julia

function code(x, n)
	t_0 = x ^ Float64(1.0 / n)
	t_1 = Float64((Float64(1.0 + x) ^ Float64(1.0 / n)) - t_0)
	t_2 = Float64(1.0 - t_0)
	tmp = 0.0
	if (t_1 <= -0.005)
		tmp = t_2;
	elseif (t_1 <= 5e-9)
		tmp = Float64(log(Float64(Float64(1.0 + x) / x)) / n);
	else
		tmp = t_2;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, n)
	t_0 = x ^ (1.0 / n);
	t_1 = ((1.0 + x) ^ (1.0 / n)) - t_0;
	t_2 = 1.0 - t_0;
	tmp = 0.0;
	if (t_1 <= -0.005)
		tmp = t_2;
	elseif (t_1 <= 5e-9)
		tmp = log(((1.0 + x) / x)) / n;
	else
		tmp = t_2;
	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[(1.0 + x), $MachinePrecision], N[(1.0 / n), $MachinePrecision]], $MachinePrecision] - t$95$0), $MachinePrecision]}, Block[{t$95$2 = N[(1.0 - t$95$0), $MachinePrecision]}, If[LessEqual[t$95$1, -0.005], t$95$2, If[LessEqual[t$95$1, 5e-9], N[(N[Log[N[(N[(1.0 + x), $MachinePrecision] / x), $MachinePrecision]], $MachinePrecision] / n), $MachinePrecision], t$95$2]]]]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (pow.f64 (+.f64 x #s(literal 1 binary64)) (/.f64 #s(literal 1 binary64) n)) (pow.f64 x (/.f64 #s(literal 1 binary64) n))) < -0.0050000000000000001 or 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)))

   1. Initial program 79.3%

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

   3. Taylor expanded in x around 0

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

      1. remove-double-neg N/A

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

      2. mul-1-neg N/A

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

      3. distribute-neg-frac N/A

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

      4. mul-1-neg N/A

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

      5. log-rec N/A

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

      6. mul-1-neg N/A

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

      7. lower--.f64 N/A

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

      8. log-rec N/A

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

      9. mul-1-neg N/A

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

      10. associate-*r/ N/A

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

      11. associate-*r* N/A

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

      12. metadata-eval N/A

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

      13. *-commutative N/A

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

      14. associate-/l* N/A

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

      15. exp-to-pow N/A

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

      16. lower-pow.f64 N/A

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

      17. lower-/.f64 77.0

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

   5. Applied rewrites 77.0%

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

   if -0.0050000000000000001 < (-.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 39.1%

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

   3. Taylor expanded in n around inf

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

      1. lower-/.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lower-log1p.f64 N/A

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

      4. lower-log.f64 82.7

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

   5. Applied rewrites 82.7%

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

      1. lift-log1p.f64 N/A

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

      2. lift-log.f64 N/A

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

      3. lift--.f64 N/A

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

      4. lift-/.f64 82.7

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

      5. lift--.f64 N/A

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

      6. lift-log1p.f64 N/A

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

      7. lift-log.f64 N/A

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

      8. diff-log N/A

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

      9. lower-log.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. +-commutative N/A

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

      12. lower-+.f64 82.3

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

   7. Applied rewrites 82.3%

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

4. Final simplification 80.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{\left(1 + x\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)} \leq -0.005:\\ \;\;\;\;1 - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{elif}\;{\left(1 + x\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)} \leq 5 \cdot 10^{-9}:\\ \;\;\;\;\frac{\log \left(\frac{1 + x}{x}\right)}{n}\\ \mathbf{else}:\\ \;\;\;\;1 - {x}^{\left(\frac{1}{n}\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 86.0% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{if}\;\frac{1}{n} \leq -5 \cdot 10^{-71}:\\ \;\;\;\;\frac{t\_0}{n \cdot x}\\ \mathbf{elif}\;\frac{1}{n} \leq 2 \cdot 10^{-11}:\\ \;\;\;\;\frac{\log \left(\frac{1 + x}{x}\right)}{n}\\ \mathbf{else}:\\ \;\;\;\;e^{\frac{\mathsf{log1p}\left(x\right)}{n}} - t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (let* ((t_0 (pow x (/ 1.0 n))))
   (if (<= (/ 1.0 n) -5e-71)
     (/ t_0 (* n x))
     (if (<= (/ 1.0 n) 2e-11)
       (/ (log (/ (+ 1.0 x) x)) n)
       (- (exp (/ (log1p x) n)) t_0)))))

C

double code(double x, double n) {
	double t_0 = pow(x, (1.0 / n));
	double tmp;
	if ((1.0 / n) <= -5e-71) {
		tmp = t_0 / (n * x);
	} else if ((1.0 / n) <= 2e-11) {
		tmp = log(((1.0 + x) / x)) / n;
	} else {
		tmp = exp((log1p(x) / n)) - t_0;
	}
	return tmp;
}

Java

public static double code(double x, double n) {
	double t_0 = Math.pow(x, (1.0 / n));
	double tmp;
	if ((1.0 / n) <= -5e-71) {
		tmp = t_0 / (n * x);
	} else if ((1.0 / n) <= 2e-11) {
		tmp = Math.log(((1.0 + x) / x)) / n;
	} else {
		tmp = Math.exp((Math.log1p(x) / n)) - t_0;
	}
	return tmp;
}

Python

def code(x, n):
	t_0 = math.pow(x, (1.0 / n))
	tmp = 0
	if (1.0 / n) <= -5e-71:
		tmp = t_0 / (n * x)
	elif (1.0 / n) <= 2e-11:
		tmp = math.log(((1.0 + x) / x)) / n
	else:
		tmp = math.exp((math.log1p(x) / n)) - t_0
	return tmp

Julia

function code(x, n)
	t_0 = x ^ Float64(1.0 / n)
	tmp = 0.0
	if (Float64(1.0 / n) <= -5e-71)
		tmp = Float64(t_0 / Float64(n * x));
	elseif (Float64(1.0 / n) <= 2e-11)
		tmp = Float64(log(Float64(Float64(1.0 + x) / x)) / n);
	else
		tmp = Float64(exp(Float64(log1p(x) / n)) - t_0);
	end
	return tmp
end

Wolfram

code[x_, n_] := Block[{t$95$0 = N[Power[x, N[(1.0 / n), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[N[(1.0 / n), $MachinePrecision], -5e-71], N[(t$95$0 / N[(n * x), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 2e-11], N[(N[Log[N[(N[(1.0 + x), $MachinePrecision] / x), $MachinePrecision]], $MachinePrecision] / n), $MachinePrecision], N[(N[Exp[N[(N[Log[1 + x], $MachinePrecision] / n), $MachinePrecision]], $MachinePrecision] - t$95$0), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 #s(literal 1 binary64) n) < -4.99999999999999998e-71

   1. Initial program 88.2%

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

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

      2. log-rec N/A

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

      3. mul-1-neg N/A

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

      4. associate-*r/ N/A

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

      5. associate-*r* N/A

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

      6. metadata-eval N/A

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

      7. *-commutative N/A

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

      8. associate-/l* N/A

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

      9. exp-to-pow N/A

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

      10. lower-pow.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 94.1

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

   5. Applied rewrites 94.1%

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

   if -4.99999999999999998e-71 < (/.f64 #s(literal 1 binary64) n) < 1.99999999999999988e-11

   1. Initial program 23.7%

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

   3. Taylor expanded in n around inf

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

      1. lower-/.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lower-log1p.f64 N/A

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

      4. lower-log.f64 84.0

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

   5. Applied rewrites 84.0%

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

      1. lift-log1p.f64 N/A

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

      2. lift-log.f64 N/A

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

      3. lift--.f64 N/A

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

      4. lift-/.f64 84.0

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

      5. lift--.f64 N/A

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

      6. lift-log1p.f64 N/A

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

      7. lift-log.f64 N/A

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

      8. diff-log N/A

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

      9. lower-log.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. +-commutative N/A

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

      12. lower-+.f64 84.2

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

   7. Applied rewrites 84.2%

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

   if 1.99999999999999988e-11 < (/.f64 #s(literal 1 binary64) n)

   1. Initial program 52.5%

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

      1. lift-+.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. pow-to-exp N/A

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

      4. lower-exp.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. un-div-inv N/A

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

      7. lower-/.f64 N/A

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

      8. lift-+.f64 N/A

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

      9. +-commutative N/A

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

      10. lower-log1p.f64 95.2

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

   4. Applied rewrites 95.2%

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

4. Final simplification 89.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{1}{n} \leq -5 \cdot 10^{-71}:\\ \;\;\;\;\frac{{x}^{\left(\frac{1}{n}\right)}}{n \cdot x}\\ \mathbf{elif}\;\frac{1}{n} \leq 2 \cdot 10^{-11}:\\ \;\;\;\;\frac{\log \left(\frac{1 + x}{x}\right)}{n}\\ \mathbf{else}:\\ \;\;\;\;e^{\frac{\mathsf{log1p}\left(x\right)}{n}} - {x}^{\left(\frac{1}{n}\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 82.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{if}\;\frac{1}{n} \leq -5 \cdot 10^{-71}:\\ \;\;\;\;\frac{t\_0}{n \cdot x}\\ \mathbf{elif}\;\frac{1}{n} \leq 0.05:\\ \;\;\;\;\frac{\log \left(\frac{1 + x}{x}\right)}{n}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x, \frac{\frac{\mathsf{fma}\left(x, \mathsf{fma}\left(-0.5, x, 0.5\right), \frac{x \cdot x}{n} \cdot 0.16666666666666666\right)}{n} - \mathsf{fma}\left(x, \mathsf{fma}\left(x, -0.3333333333333333, 0.5\right), -1\right)}{n}, 1 - t\_0\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (let* ((t_0 (pow x (/ 1.0 n))))
   (if (<= (/ 1.0 n) -5e-71)
     (/ t_0 (* n x))
     (if (<= (/ 1.0 n) 0.05)
       (/ (log (/ (+ 1.0 x) x)) n)
       (fma
        x
        (/
         (-
          (/ (fma x (fma -0.5 x 0.5) (* (/ (* x x) n) 0.16666666666666666)) n)
          (fma x (fma x -0.3333333333333333 0.5) -1.0))
         n)
        (- 1.0 t_0))))))

C

double code(double x, double n) {
	double t_0 = pow(x, (1.0 / n));
	double tmp;
	if ((1.0 / n) <= -5e-71) {
		tmp = t_0 / (n * x);
	} else if ((1.0 / n) <= 0.05) {
		tmp = log(((1.0 + x) / x)) / n;
	} else {
		tmp = fma(x, (((fma(x, fma(-0.5, x, 0.5), (((x * x) / n) * 0.16666666666666666)) / n) - fma(x, fma(x, -0.3333333333333333, 0.5), -1.0)) / n), (1.0 - t_0));
	}
	return tmp;
}

Julia

function code(x, n)
	t_0 = x ^ Float64(1.0 / n)
	tmp = 0.0
	if (Float64(1.0 / n) <= -5e-71)
		tmp = Float64(t_0 / Float64(n * x));
	elseif (Float64(1.0 / n) <= 0.05)
		tmp = Float64(log(Float64(Float64(1.0 + x) / x)) / n);
	else
		tmp = fma(x, Float64(Float64(Float64(fma(x, fma(-0.5, x, 0.5), Float64(Float64(Float64(x * x) / n) * 0.16666666666666666)) / n) - fma(x, fma(x, -0.3333333333333333, 0.5), -1.0)) / n), Float64(1.0 - t_0));
	end
	return tmp
end

Wolfram

code[x_, n_] := Block[{t$95$0 = N[Power[x, N[(1.0 / n), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[N[(1.0 / n), $MachinePrecision], -5e-71], N[(t$95$0 / N[(n * x), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 0.05], N[(N[Log[N[(N[(1.0 + x), $MachinePrecision] / x), $MachinePrecision]], $MachinePrecision] / n), $MachinePrecision], N[(x * N[(N[(N[(N[(x * N[(-0.5 * x + 0.5), $MachinePrecision] + N[(N[(N[(x * x), $MachinePrecision] / n), $MachinePrecision] * 0.16666666666666666), $MachinePrecision]), $MachinePrecision] / n), $MachinePrecision] - N[(x * N[(x * -0.3333333333333333 + 0.5), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision] / n), $MachinePrecision] + N[(1.0 - t$95$0), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 #s(literal 1 binary64) n) < -4.99999999999999998e-71

   1. Initial program 88.2%

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

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

      2. log-rec N/A

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

      3. mul-1-neg N/A

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

      4. associate-*r/ N/A

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

      5. associate-*r* N/A

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

      6. metadata-eval N/A

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

      7. *-commutative N/A

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

      8. associate-/l* N/A

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

      9. exp-to-pow N/A

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

      10. lower-pow.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 94.1

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

   5. Applied rewrites 94.1%

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

   if -4.99999999999999998e-71 < (/.f64 #s(literal 1 binary64) n) < 0.050000000000000003

   1. Initial program 23.4%

      \[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)} \]
   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 82.7

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

   5. Applied rewrites 82.7%

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

      1. lift-log1p.f64 N/A

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

      2. lift-log.f64 N/A

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

      3. lift--.f64 N/A

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

      4. lift-/.f64 82.7

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

      5. lift--.f64 N/A

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

      6. lift-log1p.f64 N/A

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

      7. lift-log.f64 N/A

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

      8. diff-log N/A

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

      9. lower-log.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. +-commutative N/A

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

      12. lower-+.f64 82.9

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

   7. Applied rewrites 82.9%

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

   if 0.050000000000000003 < (/.f64 #s(literal 1 binary64) n)

   1. Initial program 55.1%

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

   3. Taylor expanded in x around 0

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

   5. Applied rewrites 29.6%

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

   6. Taylor expanded in n around -inf

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

      1. mul-1-neg N/A

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

      2. distribute-neg-frac2 N/A

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

      3. lower-/.f64 N/A

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

   8. Applied rewrites 79.8%

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

4. Final simplification 86.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{1}{n} \leq -5 \cdot 10^{-71}:\\ \;\;\;\;\frac{{x}^{\left(\frac{1}{n}\right)}}{n \cdot x}\\ \mathbf{elif}\;\frac{1}{n} \leq 0.05:\\ \;\;\;\;\frac{\log \left(\frac{1 + x}{x}\right)}{n}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x, \frac{\frac{\mathsf{fma}\left(x, \mathsf{fma}\left(-0.5, x, 0.5\right), \frac{x \cdot x}{n} \cdot 0.16666666666666666\right)}{n} - \mathsf{fma}\left(x, \mathsf{fma}\left(x, -0.3333333333333333, 0.5\right), -1\right)}{n}, 1 - {x}^{\left(\frac{1}{n}\right)}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 81.9% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{if}\;\frac{1}{n} \leq -5 \cdot 10^{-71}:\\ \;\;\;\;\frac{t\_0}{n \cdot x}\\ \mathbf{elif}\;\frac{1}{n} \leq 0.05:\\ \;\;\;\;\frac{\log \left(\frac{1 + x}{x}\right)}{n}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x, \frac{\mathsf{fma}\left(x, \mathsf{fma}\left(x, 0.3333333333333333, -0.5\right) + \frac{\mathsf{fma}\left(-0.5, x, 0.5\right)}{n}, 1\right)}{n}, 1 - t\_0\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (let* ((t_0 (pow x (/ 1.0 n))))
   (if (<= (/ 1.0 n) -5e-71)
     (/ t_0 (* n x))
     (if (<= (/ 1.0 n) 0.05)
       (/ (log (/ (+ 1.0 x) x)) n)
       (fma
        x
        (/
         (fma x (+ (fma x 0.3333333333333333 -0.5) (/ (fma -0.5 x 0.5) n)) 1.0)
         n)
        (- 1.0 t_0))))))

C

double code(double x, double n) {
	double t_0 = pow(x, (1.0 / n));
	double tmp;
	if ((1.0 / n) <= -5e-71) {
		tmp = t_0 / (n * x);
	} else if ((1.0 / n) <= 0.05) {
		tmp = log(((1.0 + x) / x)) / n;
	} else {
		tmp = fma(x, (fma(x, (fma(x, 0.3333333333333333, -0.5) + (fma(-0.5, x, 0.5) / n)), 1.0) / n), (1.0 - t_0));
	}
	return tmp;
}

Julia

function code(x, n)
	t_0 = x ^ Float64(1.0 / n)
	tmp = 0.0
	if (Float64(1.0 / n) <= -5e-71)
		tmp = Float64(t_0 / Float64(n * x));
	elseif (Float64(1.0 / n) <= 0.05)
		tmp = Float64(log(Float64(Float64(1.0 + x) / x)) / n);
	else
		tmp = fma(x, Float64(fma(x, Float64(fma(x, 0.3333333333333333, -0.5) + Float64(fma(-0.5, x, 0.5) / n)), 1.0) / n), Float64(1.0 - t_0));
	end
	return tmp
end

Wolfram

code[x_, n_] := Block[{t$95$0 = N[Power[x, N[(1.0 / n), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[N[(1.0 / n), $MachinePrecision], -5e-71], N[(t$95$0 / N[(n * x), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 0.05], N[(N[Log[N[(N[(1.0 + x), $MachinePrecision] / x), $MachinePrecision]], $MachinePrecision] / n), $MachinePrecision], N[(x * N[(N[(x * N[(N[(x * 0.3333333333333333 + -0.5), $MachinePrecision] + N[(N[(-0.5 * x + 0.5), $MachinePrecision] / n), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision] / n), $MachinePrecision] + N[(1.0 - t$95$0), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 #s(literal 1 binary64) n) < -4.99999999999999998e-71

   1. Initial program 88.2%

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

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

      2. log-rec N/A

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

      3. mul-1-neg N/A

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

      4. associate-*r/ N/A

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

      5. associate-*r* N/A

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

      6. metadata-eval N/A

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

      7. *-commutative N/A

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

      8. associate-/l* N/A

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

      9. exp-to-pow N/A

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

      10. lower-pow.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 94.1

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

   5. Applied rewrites 94.1%

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

   if -4.99999999999999998e-71 < (/.f64 #s(literal 1 binary64) n) < 0.050000000000000003

   1. Initial program 23.4%

      \[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)} \]
   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 82.7

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

   5. Applied rewrites 82.7%

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

      1. lift-log1p.f64 N/A

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

      2. lift-log.f64 N/A

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

      3. lift--.f64 N/A

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

      4. lift-/.f64 82.7

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

      5. lift--.f64 N/A

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

      6. lift-log1p.f64 N/A

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

      7. lift-log.f64 N/A

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

      8. diff-log N/A

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

      9. lower-log.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. +-commutative N/A

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

      12. lower-+.f64 82.9

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

   7. Applied rewrites 82.9%

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

   if 0.050000000000000003 < (/.f64 #s(literal 1 binary64) n)

   1. Initial program 55.1%

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

   3. Taylor expanded in x around 0

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

   5. Applied rewrites 29.6%

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

   6. Taylor expanded in n around inf

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

      1. lower-/.f64 N/A

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

      2. +-commutative N/A

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

      3. associate-/l* N/A

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

      4. distribute-lft-out N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-+.f64 N/A

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

      7. sub-neg N/A

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

      8. *-commutative N/A

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

      9. metadata-eval N/A

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

      10. lower-fma.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. +-commutative N/A

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

      13. lower-fma.f64 74.6

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

   8. Applied rewrites 74.6%

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

4. Final simplification 85.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{1}{n} \leq -5 \cdot 10^{-71}:\\ \;\;\;\;\frac{{x}^{\left(\frac{1}{n}\right)}}{n \cdot x}\\ \mathbf{elif}\;\frac{1}{n} \leq 0.05:\\ \;\;\;\;\frac{\log \left(\frac{1 + x}{x}\right)}{n}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x, \frac{\mathsf{fma}\left(x, \mathsf{fma}\left(x, 0.3333333333333333, -0.5\right) + \frac{\mathsf{fma}\left(-0.5, x, 0.5\right)}{n}, 1\right)}{n}, 1 - {x}^{\left(\frac{1}{n}\right)}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 82.3% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{if}\;\frac{1}{n} \leq -5 \cdot 10^{-71}:\\ \;\;\;\;\frac{t\_0}{n \cdot x}\\ \mathbf{elif}\;\frac{1}{n} \leq 0.05:\\ \;\;\;\;\frac{\log \left(\frac{1 + x}{x}\right)}{n}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x, \mathsf{fma}\left(x, \frac{0.5}{n \cdot n}, \frac{1}{n}\right), 1 - t\_0\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (let* ((t_0 (pow x (/ 1.0 n))))
   (if (<= (/ 1.0 n) -5e-71)
     (/ t_0 (* n x))
     (if (<= (/ 1.0 n) 0.05)
       (/ (log (/ (+ 1.0 x) x)) n)
       (fma x (fma x (/ 0.5 (* n n)) (/ 1.0 n)) (- 1.0 t_0))))))

C

double code(double x, double n) {
	double t_0 = pow(x, (1.0 / n));
	double tmp;
	if ((1.0 / n) <= -5e-71) {
		tmp = t_0 / (n * x);
	} else if ((1.0 / n) <= 0.05) {
		tmp = log(((1.0 + x) / x)) / n;
	} else {
		tmp = fma(x, fma(x, (0.5 / (n * n)), (1.0 / n)), (1.0 - t_0));
	}
	return tmp;
}

Julia

function code(x, n)
	t_0 = x ^ Float64(1.0 / n)
	tmp = 0.0
	if (Float64(1.0 / n) <= -5e-71)
		tmp = Float64(t_0 / Float64(n * x));
	elseif (Float64(1.0 / n) <= 0.05)
		tmp = Float64(log(Float64(Float64(1.0 + x) / x)) / n);
	else
		tmp = fma(x, fma(x, Float64(0.5 / Float64(n * n)), Float64(1.0 / n)), Float64(1.0 - t_0));
	end
	return tmp
end

Wolfram

code[x_, n_] := Block[{t$95$0 = N[Power[x, N[(1.0 / n), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[N[(1.0 / n), $MachinePrecision], -5e-71], N[(t$95$0 / N[(n * x), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 0.05], N[(N[Log[N[(N[(1.0 + x), $MachinePrecision] / x), $MachinePrecision]], $MachinePrecision] / n), $MachinePrecision], N[(x * N[(x * N[(0.5 / N[(n * n), $MachinePrecision]), $MachinePrecision] + N[(1.0 / n), $MachinePrecision]), $MachinePrecision] + N[(1.0 - t$95$0), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 #s(literal 1 binary64) n) < -4.99999999999999998e-71

   1. Initial program 88.2%

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

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

      2. log-rec N/A

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

      3. mul-1-neg N/A

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

      4. associate-*r/ N/A

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

      5. associate-*r* N/A

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

      6. metadata-eval N/A

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

      7. *-commutative N/A

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

      8. associate-/l* N/A

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

      9. exp-to-pow N/A

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

      10. lower-pow.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 94.1

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

   5. Applied rewrites 94.1%

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

   if -4.99999999999999998e-71 < (/.f64 #s(literal 1 binary64) n) < 0.050000000000000003

   1. Initial program 23.4%

      \[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)} \]
   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 82.7

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

   5. Applied rewrites 82.7%

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

      1. lift-log1p.f64 N/A

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

      2. lift-log.f64 N/A

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

      3. lift--.f64 N/A

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

      4. lift-/.f64 82.7

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

      5. lift--.f64 N/A

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

      6. lift-log1p.f64 N/A

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

      7. lift-log.f64 N/A

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

      8. diff-log N/A

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

      9. lower-log.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. +-commutative N/A

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

      12. lower-+.f64 82.9

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

   7. Applied rewrites 82.9%

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

   if 0.050000000000000003 < (/.f64 #s(literal 1 binary64) n)

   1. Initial program 55.1%

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

   3. Taylor expanded in x around 0

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

   5. Applied rewrites 29.6%

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

   6. Taylor expanded in n around 0

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

      1. lower-/.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. +-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. cube-mult N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 N/A

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

      10. unpow2 N/A

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

      11. lower-*.f64 64.3

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

   8. Applied rewrites 64.3%

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

   9. Taylor expanded in x around 0

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

       1. *-commutative N/A

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

       2. associate-*l/ N/A

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

       3. associate-*r/ N/A

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

       4. metadata-eval N/A

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

       5. associate-*r/ N/A

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

       6. lower-fma.f64 N/A

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

       7. associate-*r/ N/A

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

       8. metadata-eval N/A

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

       9. lower-/.f64 N/A

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

       10. unpow2 N/A

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

       11. lower-*.f64 N/A

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

       12. lower-/.f64 64.3

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

   11. Applied rewrites 64.3%

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

4. Final simplification 84.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{1}{n} \leq -5 \cdot 10^{-71}:\\ \;\;\;\;\frac{{x}^{\left(\frac{1}{n}\right)}}{n \cdot x}\\ \mathbf{elif}\;\frac{1}{n} \leq 0.05:\\ \;\;\;\;\frac{\log \left(\frac{1 + x}{x}\right)}{n}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x, \mathsf{fma}\left(x, \frac{0.5}{n \cdot n}, \frac{1}{n}\right), 1 - {x}^{\left(\frac{1}{n}\right)}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 81.0% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{if}\;\frac{1}{n} \leq -5 \cdot 10^{-71}:\\ \;\;\;\;\frac{t\_0}{n \cdot x}\\ \mathbf{elif}\;\frac{1}{n} \leq 2 \cdot 10^{-11}:\\ \;\;\;\;\frac{\log \left(\frac{1 + x}{x}\right)}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 2 \cdot 10^{+237}:\\ \;\;\;\;\left(1 + \frac{x}{n}\right) - t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{0.3333333333333333}{x \cdot \left(x \cdot \left(n \cdot x\right)\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (let* ((t_0 (pow x (/ 1.0 n))))
   (if (<= (/ 1.0 n) -5e-71)
     (/ t_0 (* n x))
     (if (<= (/ 1.0 n) 2e-11)
       (/ (log (/ (+ 1.0 x) x)) n)
       (if (<= (/ 1.0 n) 2e+237)
         (- (+ 1.0 (/ x n)) 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) <= -5e-71) {
		tmp = t_0 / (n * x);
	} else if ((1.0 / n) <= 2e-11) {
		tmp = log(((1.0 + x) / x)) / n;
	} else if ((1.0 / n) <= 2e+237) {
		tmp = (1.0 + (x / n)) - 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) <= (-5d-71)) then
        tmp = t_0 / (n * x)
    else if ((1.0d0 / n) <= 2d-11) then
        tmp = log(((1.0d0 + x) / x)) / n
    else if ((1.0d0 / n) <= 2d+237) then
        tmp = (1.0d0 + (x / n)) - 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) <= -5e-71) {
		tmp = t_0 / (n * x);
	} else if ((1.0 / n) <= 2e-11) {
		tmp = Math.log(((1.0 + x) / x)) / n;
	} else if ((1.0 / n) <= 2e+237) {
		tmp = (1.0 + (x / n)) - 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) <= -5e-71:
		tmp = t_0 / (n * x)
	elif (1.0 / n) <= 2e-11:
		tmp = math.log(((1.0 + x) / x)) / n
	elif (1.0 / n) <= 2e+237:
		tmp = (1.0 + (x / n)) - 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) <= -5e-71)
		tmp = Float64(t_0 / Float64(n * x));
	elseif (Float64(1.0 / n) <= 2e-11)
		tmp = Float64(log(Float64(Float64(1.0 + x) / x)) / n);
	elseif (Float64(1.0 / n) <= 2e+237)
		tmp = Float64(Float64(1.0 + Float64(x / n)) - t_0);
	else
		tmp = Float64(0.3333333333333333 / Float64(x * Float64(x * Float64(n * x))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, n)
	t_0 = x ^ (1.0 / n);
	tmp = 0.0;
	if ((1.0 / n) <= -5e-71)
		tmp = t_0 / (n * x);
	elseif ((1.0 / n) <= 2e-11)
		tmp = log(((1.0 + x) / x)) / n;
	elseif ((1.0 / n) <= 2e+237)
		tmp = (1.0 + (x / n)) - 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], -5e-71], N[(t$95$0 / N[(n * x), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 2e-11], N[(N[Log[N[(N[(1.0 + x), $MachinePrecision] / x), $MachinePrecision]], $MachinePrecision] / n), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 2e+237], N[(N[(1.0 + N[(x / n), $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision], N[(0.3333333333333333 / N[(x * N[(x * N[(n * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if (/.f64 #s(literal 1 binary64) n) < -4.99999999999999998e-71

   1. Initial program 88.2%

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

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

      2. log-rec N/A

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

      3. mul-1-neg N/A

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

      4. associate-*r/ N/A

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

      5. associate-*r* N/A

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

      6. metadata-eval N/A

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

      7. *-commutative N/A

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

      8. associate-/l* N/A

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

      9. exp-to-pow N/A

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

      10. lower-pow.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 94.1

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

   5. Applied rewrites 94.1%

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

   if -4.99999999999999998e-71 < (/.f64 #s(literal 1 binary64) n) < 1.99999999999999988e-11

   1. Initial program 23.7%

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

   3. Taylor expanded in n around inf

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

      1. lower-/.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lower-log1p.f64 N/A

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

      4. lower-log.f64 84.0

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

   5. Applied rewrites 84.0%

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

      1. lift-log1p.f64 N/A

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

      2. lift-log.f64 N/A

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

      3. lift--.f64 N/A

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

      4. lift-/.f64 84.0

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

      5. lift--.f64 N/A

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

      6. lift-log1p.f64 N/A

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

      7. lift-log.f64 N/A

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

      8. diff-log N/A

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

      9. lower-log.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. +-commutative N/A

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

      12. lower-+.f64 84.2

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

   7. Applied rewrites 84.2%

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

   if 1.99999999999999988e-11 < (/.f64 #s(literal 1 binary64) n) < 1.99999999999999988e237

   1. Initial program 64.9%

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

   3. Taylor expanded in x around 0

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

      1. *-rgt-identity N/A

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

      2. associate-*r/ N/A

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

      3. lower-+.f64 N/A

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

      4. associate-*r/ N/A

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

      5. *-rgt-identity N/A

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

      6. lower-/.f64 60.0

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

   5. Applied rewrites 60.0%

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

   if 1.99999999999999988e237 < (/.f64 #s(literal 1 binary64) n)

   1. Initial program 3.1%

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

   3. Taylor expanded in n around inf

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

      1. lower-/.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lower-log1p.f64 N/A

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

      4. lower-log.f64 7.8

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

   5. Applied rewrites 7.8%

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

   6. Taylor expanded in x around inf

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

      1. lower-/.f64 N/A

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

   8. Applied rewrites 100.0%

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

   9. Taylor expanded in x around 0

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

       1. lower-/.f64 N/A

          \[\leadsto \color{blue}{\frac{\frac{1}{3}}{n \cdot {x}^{3}}} \]

       2. unpow3 N/A

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

       3. unpow2 N/A

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

       4. associate-*r* N/A

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

       5. *-commutative N/A

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

       6. lower-*.f64 N/A

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

       7. unpow2 N/A

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

       8. associate-*r* N/A

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

       9. *-commutative N/A

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

       10. lower-*.f64 N/A

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

       11. *-commutative N/A

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

       12. lower-*.f64 100.0

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

   11. Applied rewrites 100.0%

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

4. Final simplification 85.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{1}{n} \leq -5 \cdot 10^{-71}:\\ \;\;\;\;\frac{{x}^{\left(\frac{1}{n}\right)}}{n \cdot x}\\ \mathbf{elif}\;\frac{1}{n} \leq 2 \cdot 10^{-11}:\\ \;\;\;\;\frac{\log \left(\frac{1 + x}{x}\right)}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 2 \cdot 10^{+237}:\\ \;\;\;\;\left(1 + \frac{x}{n}\right) - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.3333333333333333}{x \cdot \left(x \cdot \left(n \cdot x\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 80.7% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{if}\;\frac{1}{n} \leq -5 \cdot 10^{-71}:\\ \;\;\;\;\frac{t\_0}{n \cdot x}\\ \mathbf{elif}\;\frac{1}{n} \leq 0.05:\\ \;\;\;\;\frac{\log \left(\frac{1 + x}{x}\right)}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 2 \cdot 10^{+237}:\\ \;\;\;\;1 - t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{0.3333333333333333}{x \cdot \left(x \cdot \left(n \cdot x\right)\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (let* ((t_0 (pow x (/ 1.0 n))))
   (if (<= (/ 1.0 n) -5e-71)
     (/ t_0 (* n x))
     (if (<= (/ 1.0 n) 0.05)
       (/ (log (/ (+ 1.0 x) x)) n)
       (if (<= (/ 1.0 n) 2e+237)
         (- 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) <= -5e-71) {
		tmp = t_0 / (n * x);
	} else if ((1.0 / n) <= 0.05) {
		tmp = log(((1.0 + x) / x)) / n;
	} else if ((1.0 / n) <= 2e+237) {
		tmp = 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) <= (-5d-71)) then
        tmp = t_0 / (n * x)
    else if ((1.0d0 / n) <= 0.05d0) then
        tmp = log(((1.0d0 + x) / x)) / n
    else if ((1.0d0 / n) <= 2d+237) then
        tmp = 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) <= -5e-71) {
		tmp = t_0 / (n * x);
	} else if ((1.0 / n) <= 0.05) {
		tmp = Math.log(((1.0 + x) / x)) / n;
	} else if ((1.0 / n) <= 2e+237) {
		tmp = 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) <= -5e-71:
		tmp = t_0 / (n * x)
	elif (1.0 / n) <= 0.05:
		tmp = math.log(((1.0 + x) / x)) / n
	elif (1.0 / n) <= 2e+237:
		tmp = 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) <= -5e-71)
		tmp = Float64(t_0 / Float64(n * x));
	elseif (Float64(1.0 / n) <= 0.05)
		tmp = Float64(log(Float64(Float64(1.0 + x) / x)) / n);
	elseif (Float64(1.0 / n) <= 2e+237)
		tmp = Float64(1.0 - t_0);
	else
		tmp = Float64(0.3333333333333333 / Float64(x * Float64(x * Float64(n * x))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, n)
	t_0 = x ^ (1.0 / n);
	tmp = 0.0;
	if ((1.0 / n) <= -5e-71)
		tmp = t_0 / (n * x);
	elseif ((1.0 / n) <= 0.05)
		tmp = log(((1.0 + x) / x)) / n;
	elseif ((1.0 / n) <= 2e+237)
		tmp = 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], -5e-71], N[(t$95$0 / N[(n * x), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 0.05], N[(N[Log[N[(N[(1.0 + x), $MachinePrecision] / x), $MachinePrecision]], $MachinePrecision] / n), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 2e+237], N[(1.0 - t$95$0), $MachinePrecision], N[(0.3333333333333333 / N[(x * N[(x * N[(n * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if (/.f64 #s(literal 1 binary64) n) < -4.99999999999999998e-71

   1. Initial program 88.2%

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

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

      2. log-rec N/A

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

      3. mul-1-neg N/A

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

      4. associate-*r/ N/A

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

      5. associate-*r* N/A

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

      6. metadata-eval N/A

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

      7. *-commutative N/A

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

      8. associate-/l* N/A

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

      9. exp-to-pow N/A

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

      10. lower-pow.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 94.1

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

   5. Applied rewrites 94.1%

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

   if -4.99999999999999998e-71 < (/.f64 #s(literal 1 binary64) n) < 0.050000000000000003

   1. Initial program 23.4%

      \[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)} \]
   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 82.7

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

   5. Applied rewrites 82.7%

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

      1. lift-log1p.f64 N/A

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

      2. lift-log.f64 N/A

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

      3. lift--.f64 N/A

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

      4. lift-/.f64 82.7

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

      5. lift--.f64 N/A

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

      6. lift-log1p.f64 N/A

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

      7. lift-log.f64 N/A

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

      8. diff-log N/A

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

      9. lower-log.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. +-commutative N/A

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

      12. lower-+.f64 82.9

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

   7. Applied rewrites 82.9%

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

   if 0.050000000000000003 < (/.f64 #s(literal 1 binary64) n) < 1.99999999999999988e237

   1. Initial program 68.9%

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

   3. Taylor expanded in x around 0

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

      1. remove-double-neg N/A

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

      2. mul-1-neg N/A

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

      3. distribute-neg-frac N/A

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

      4. mul-1-neg N/A

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

      5. log-rec N/A

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

      6. mul-1-neg N/A

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

      7. lower--.f64 N/A

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

      8. log-rec N/A

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

      9. mul-1-neg N/A

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

      10. associate-*r/ N/A

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

      11. associate-*r* N/A

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

      12. metadata-eval N/A

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

      13. *-commutative N/A

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

      14. associate-/l* N/A

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

      15. exp-to-pow N/A

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

      16. lower-pow.f64 N/A

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

      17. lower-/.f64 62.5

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

   5. Applied rewrites 62.5%

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

   if 1.99999999999999988e237 < (/.f64 #s(literal 1 binary64) n)

   1. Initial program 3.1%

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

   3. Taylor expanded in n around inf

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

      1. lower-/.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lower-log1p.f64 N/A

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

      4. lower-log.f64 7.8

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

   5. Applied rewrites 7.8%

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

   6. Taylor expanded in x around inf

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

      1. lower-/.f64 N/A

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

   8. Applied rewrites 100.0%

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

   9. Taylor expanded in x around 0

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

       1. lower-/.f64 N/A

          \[\leadsto \color{blue}{\frac{\frac{1}{3}}{n \cdot {x}^{3}}} \]

       2. unpow3 N/A

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

       3. unpow2 N/A

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

       4. associate-*r* N/A

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

       5. *-commutative N/A

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

       6. lower-*.f64 N/A

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

       7. unpow2 N/A

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

       8. associate-*r* N/A

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

       9. *-commutative N/A

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

       10. lower-*.f64 N/A

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

       11. *-commutative N/A

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

       12. lower-*.f64 100.0

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

   11. Applied rewrites 100.0%

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

4. Final simplification 85.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{1}{n} \leq -5 \cdot 10^{-71}:\\ \;\;\;\;\frac{{x}^{\left(\frac{1}{n}\right)}}{n \cdot x}\\ \mathbf{elif}\;\frac{1}{n} \leq 0.05:\\ \;\;\;\;\frac{\log \left(\frac{1 + x}{x}\right)}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 2 \cdot 10^{+237}:\\ \;\;\;\;1 - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.3333333333333333}{x \cdot \left(x \cdot \left(n \cdot x\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 56.6% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x \cdot \left(x \cdot x\right)\\ \mathbf{if}\;\frac{1}{n} \leq -5000:\\ \;\;\;\;\frac{\frac{0.3333333333333333}{t\_0}}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq -5 \cdot 10^{-71}:\\ \;\;\;\;\frac{\frac{1 + \frac{\left(-0.125 + \frac{0.037037037037037035}{t\_0}\right) \cdot \frac{1}{x}}{\mathsf{fma}\left(\frac{0.3333333333333333}{x}, \frac{0.3333333333333333}{x} - -0.5, 0.25\right)}}{x}}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 2 \cdot 10^{+63}:\\ \;\;\;\;\frac{x - \log x}{n}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x, \mathsf{fma}\left(x, \frac{\mathsf{fma}\left(x, \mathsf{fma}\left(n, -0.5, 0.16666666666666666\right), n \cdot 0.5\right)}{n \cdot \left(n \cdot n\right)}, \frac{1}{n}\right), 1 + -1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (let* ((t_0 (* x (* x x))))
   (if (<= (/ 1.0 n) -5000.0)
     (/ (/ 0.3333333333333333 t_0) n)
     (if (<= (/ 1.0 n) -5e-71)
       (/
        (/
         (+
          1.0
          (/
           (* (+ -0.125 (/ 0.037037037037037035 t_0)) (/ 1.0 x))
           (fma
            (/ 0.3333333333333333 x)
            (- (/ 0.3333333333333333 x) -0.5)
            0.25)))
         x)
        n)
       (if (<= (/ 1.0 n) 2e+63)
         (/ (- x (log x)) n)
         (fma
          x
          (fma
           x
           (/ (fma x (fma n -0.5 0.16666666666666666) (* n 0.5)) (* n (* n n)))
           (/ 1.0 n))
          (+ 1.0 -1.0)))))))

C

double code(double x, double n) {
	double t_0 = x * (x * x);
	double tmp;
	if ((1.0 / n) <= -5000.0) {
		tmp = (0.3333333333333333 / t_0) / n;
	} else if ((1.0 / n) <= -5e-71) {
		tmp = ((1.0 + (((-0.125 + (0.037037037037037035 / t_0)) * (1.0 / x)) / fma((0.3333333333333333 / x), ((0.3333333333333333 / x) - -0.5), 0.25))) / x) / n;
	} else if ((1.0 / n) <= 2e+63) {
		tmp = (x - log(x)) / n;
	} else {
		tmp = fma(x, fma(x, (fma(x, fma(n, -0.5, 0.16666666666666666), (n * 0.5)) / (n * (n * n))), (1.0 / n)), (1.0 + -1.0));
	}
	return tmp;
}

Julia

function code(x, n)
	t_0 = Float64(x * Float64(x * x))
	tmp = 0.0
	if (Float64(1.0 / n) <= -5000.0)
		tmp = Float64(Float64(0.3333333333333333 / t_0) / n);
	elseif (Float64(1.0 / n) <= -5e-71)
		tmp = Float64(Float64(Float64(1.0 + Float64(Float64(Float64(-0.125 + Float64(0.037037037037037035 / t_0)) * Float64(1.0 / x)) / fma(Float64(0.3333333333333333 / x), Float64(Float64(0.3333333333333333 / x) - -0.5), 0.25))) / x) / n);
	elseif (Float64(1.0 / n) <= 2e+63)
		tmp = Float64(Float64(x - log(x)) / n);
	else
		tmp = fma(x, fma(x, Float64(fma(x, fma(n, -0.5, 0.16666666666666666), Float64(n * 0.5)) / Float64(n * Float64(n * n))), Float64(1.0 / n)), Float64(1.0 + -1.0));
	end
	return tmp
end

Wolfram

code[x_, n_] := Block[{t$95$0 = N[(x * N[(x * x), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(1.0 / n), $MachinePrecision], -5000.0], N[(N[(0.3333333333333333 / t$95$0), $MachinePrecision] / n), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], -5e-71], N[(N[(N[(1.0 + N[(N[(N[(-0.125 + N[(0.037037037037037035 / t$95$0), $MachinePrecision]), $MachinePrecision] * N[(1.0 / x), $MachinePrecision]), $MachinePrecision] / N[(N[(0.3333333333333333 / x), $MachinePrecision] * N[(N[(0.3333333333333333 / x), $MachinePrecision] - -0.5), $MachinePrecision] + 0.25), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision] / n), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 2e+63], N[(N[(x - N[Log[x], $MachinePrecision]), $MachinePrecision] / n), $MachinePrecision], N[(x * N[(x * N[(N[(x * N[(n * -0.5 + 0.16666666666666666), $MachinePrecision] + N[(n * 0.5), $MachinePrecision]), $MachinePrecision] / N[(n * N[(n * n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(1.0 / n), $MachinePrecision]), $MachinePrecision] + N[(1.0 + -1.0), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if (/.f64 #s(literal 1 binary64) n) < -5e3

   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 50.9

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

   5. Applied rewrites 50.9%

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

   6. Taylor expanded in x around inf

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

      1. lower-/.f64 N/A

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

   8. Applied rewrites 42.1%

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

   9. Taylor expanded in x around 0

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

       1. lower-/.f64 N/A

          \[\leadsto \frac{\color{blue}{\frac{\frac{1}{3}}{{x}^{3}}}}{n} \]

       2. cube-mult N/A

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

       3. unpow2 N/A

          \[\leadsto \frac{\frac{\frac{1}{3}}{x \cdot \color{blue}{{x}^{2}}}}{n} \]

       4. lower-*.f64 N/A

          \[\leadsto \frac{\frac{\frac{1}{3}}{\color{blue}{x \cdot {x}^{2}}}}{n} \]

       5. unpow2 N/A

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

       6. lower-*.f64 76.2

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

   11. Applied rewrites 76.2%

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

   if -5e3 < (/.f64 #s(literal 1 binary64) n) < -4.99999999999999998e-71

   1. Initial program 29.7%

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

   3. Taylor expanded in n around inf

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

      1. lower-/.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lower-log1p.f64 N/A

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

      4. lower-log.f64 26.3

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

   5. Applied rewrites 26.3%

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

   6. Taylor expanded in x around inf

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

      1. lower-/.f64 N/A

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

   8. Applied rewrites 50.2%

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

      1. lift-/.f64 N/A

         \[\leadsto \frac{\frac{1 + \frac{\frac{-1}{2} + \color{blue}{\frac{\frac{1}{3}}{x}}}{x}}{x}}{n} \]

      2. lift-+.f64 N/A

         \[\leadsto \frac{\frac{1 + \frac{\color{blue}{\frac{-1}{2} + \frac{\frac{1}{3}}{x}}}{x}}{x}}{n} \]

      3. div-inv N/A

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

      4. lift-+.f64 N/A

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

      5. flip3-+ N/A

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

      6. associate-*l/ N/A

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

      7. lower-/.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-+.f64 N/A

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

      10. metadata-eval N/A

          \[\leadsto \frac{\frac{1 + \frac{\left(\color{blue}{\frac{-1}{8}} + {\left(\frac{\frac{1}{3}}{x}\right)}^{3}\right) \cdot \frac{1}{x}}{\frac{-1}{2} \cdot \frac{-1}{2} + \left(\frac{\frac{1}{3}}{x} \cdot \frac{\frac{1}{3}}{x} - \frac{-1}{2} \cdot \frac{\frac{1}{3}}{x}\right)}}{x}}{n} \]

      11. lift-/.f64 N/A

          \[\leadsto \frac{\frac{1 + \frac{\left(\frac{-1}{8} + {\color{blue}{\left(\frac{\frac{1}{3}}{x}\right)}}^{3}\right) \cdot \frac{1}{x}}{\frac{-1}{2} \cdot \frac{-1}{2} + \left(\frac{\frac{1}{3}}{x} \cdot \frac{\frac{1}{3}}{x} - \frac{-1}{2} \cdot \frac{\frac{1}{3}}{x}\right)}}{x}}{n} \]

      12. cube-div N/A

          \[\leadsto \frac{\frac{1 + \frac{\left(\frac{-1}{8} + \color{blue}{\frac{{\frac{1}{3}}^{3}}{{x}^{3}}}\right) \cdot \frac{1}{x}}{\frac{-1}{2} \cdot \frac{-1}{2} + \left(\frac{\frac{1}{3}}{x} \cdot \frac{\frac{1}{3}}{x} - \frac{-1}{2} \cdot \frac{\frac{1}{3}}{x}\right)}}{x}}{n} \]

      13. lower-/.f64 N/A

          \[\leadsto \frac{\frac{1 + \frac{\left(\frac{-1}{8} + \color{blue}{\frac{{\frac{1}{3}}^{3}}{{x}^{3}}}\right) \cdot \frac{1}{x}}{\frac{-1}{2} \cdot \frac{-1}{2} + \left(\frac{\frac{1}{3}}{x} \cdot \frac{\frac{1}{3}}{x} - \frac{-1}{2} \cdot \frac{\frac{1}{3}}{x}\right)}}{x}}{n} \]

      14. metadata-eval N/A

          \[\leadsto \frac{\frac{1 + \frac{\left(\frac{-1}{8} + \frac{\color{blue}{\frac{1}{27}}}{{x}^{3}}\right) \cdot \frac{1}{x}}{\frac{-1}{2} \cdot \frac{-1}{2} + \left(\frac{\frac{1}{3}}{x} \cdot \frac{\frac{1}{3}}{x} - \frac{-1}{2} \cdot \frac{\frac{1}{3}}{x}\right)}}{x}}{n} \]

      15. cube-mult N/A

          \[\leadsto \frac{\frac{1 + \frac{\left(\frac{-1}{8} + \frac{\frac{1}{27}}{\color{blue}{x \cdot \left(x \cdot x\right)}}\right) \cdot \frac{1}{x}}{\frac{-1}{2} \cdot \frac{-1}{2} + \left(\frac{\frac{1}{3}}{x} \cdot \frac{\frac{1}{3}}{x} - \frac{-1}{2} \cdot \frac{\frac{1}{3}}{x}\right)}}{x}}{n} \]

      16. lower-*.f64 N/A

          \[\leadsto \frac{\frac{1 + \frac{\left(\frac{-1}{8} + \frac{\frac{1}{27}}{\color{blue}{x \cdot \left(x \cdot x\right)}}\right) \cdot \frac{1}{x}}{\frac{-1}{2} \cdot \frac{-1}{2} + \left(\frac{\frac{1}{3}}{x} \cdot \frac{\frac{1}{3}}{x} - \frac{-1}{2} \cdot \frac{\frac{1}{3}}{x}\right)}}{x}}{n} \]

      17. lower-*.f64 N/A

          \[\leadsto \frac{\frac{1 + \frac{\left(\frac{-1}{8} + \frac{\frac{1}{27}}{x \cdot \color{blue}{\left(x \cdot x\right)}}\right) \cdot \frac{1}{x}}{\frac{-1}{2} \cdot \frac{-1}{2} + \left(\frac{\frac{1}{3}}{x} \cdot \frac{\frac{1}{3}}{x} - \frac{-1}{2} \cdot \frac{\frac{1}{3}}{x}\right)}}{x}}{n} \]

      18. lower-/.f64 N/A

          \[\leadsto \frac{\frac{1 + \frac{\left(\frac{-1}{8} + \frac{\frac{1}{27}}{x \cdot \left(x \cdot x\right)}\right) \cdot \color{blue}{\frac{1}{x}}}{\frac{-1}{2} \cdot \frac{-1}{2} + \left(\frac{\frac{1}{3}}{x} \cdot \frac{\frac{1}{3}}{x} - \frac{-1}{2} \cdot \frac{\frac{1}{3}}{x}\right)}}{x}}{n} \]

   10. Applied rewrites 54.7%

       \[\leadsto \frac{\frac{1 + \color{blue}{\frac{\left(-0.125 + \frac{0.037037037037037035}{x \cdot \left(x \cdot x\right)}\right) \cdot \frac{1}{x}}{\mathsf{fma}\left(\frac{0.3333333333333333}{x}, \frac{0.3333333333333333}{x} - -0.5, 0.25\right)}}}{x}}{n} \]

   if -4.99999999999999998e-71 < (/.f64 #s(literal 1 binary64) n) < 2.00000000000000012e63

   1. Initial program 25.8%

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

   3. Taylor expanded in x around 0

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

      1. *-rgt-identity N/A

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

      2. associate-*r/ N/A

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

      3. lower-+.f64 N/A

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

      4. associate-*r/ N/A

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

      5. *-rgt-identity N/A

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

      6. lower-/.f64 18.2

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

   5. Applied rewrites 18.2%

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

   6. Taylor expanded in n around inf

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

      1. lower-/.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lower-log.f64 60.9

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

   8. Applied rewrites 60.9%

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

   if 2.00000000000000012e63 < (/.f64 #s(literal 1 binary64) n)

   1. Initial program 51.5%

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

   3. Taylor expanded in x around 0

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

   5. Applied rewrites 19.5%

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

   6. Taylor expanded in n around inf

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

   8. Applied rewrites 16.5%

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

   9. Taylor expanded in n around 0

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

       1. lower-/.f64 N/A

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

       2. +-commutative N/A

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

       3. distribute-lft-in N/A

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

       4. *-commutative N/A

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

       5. associate-+l+ N/A

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

       6. associate-*r* N/A

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

       7. *-commutative N/A

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

       8. distribute-rgt-in N/A

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

       9. +-commutative N/A

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

       10. +-commutative N/A

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

       11. lower-fma.f64 N/A

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

       12. +-commutative N/A

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

       13. *-commutative N/A

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

       14. lower-fma.f64 N/A

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

       15. *-commutative N/A

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

       16. lower-*.f64 N/A

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

       17. cube-mult N/A

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

       18. unpow2 N/A

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

       19. lower-*.f64 N/A

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

       20. unpow2 N/A

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

       21. lower-*.f64 54.8

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

   11. Applied rewrites 54.8%

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

4. Final simplification 64.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{1}{n} \leq -5000:\\ \;\;\;\;\frac{\frac{0.3333333333333333}{x \cdot \left(x \cdot x\right)}}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq -5 \cdot 10^{-71}:\\ \;\;\;\;\frac{\frac{1 + \frac{\left(-0.125 + \frac{0.037037037037037035}{x \cdot \left(x \cdot x\right)}\right) \cdot \frac{1}{x}}{\mathsf{fma}\left(\frac{0.3333333333333333}{x}, \frac{0.3333333333333333}{x} - -0.5, 0.25\right)}}{x}}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 2 \cdot 10^{+63}:\\ \;\;\;\;\frac{x - \log x}{n}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x, \mathsf{fma}\left(x, \frac{\mathsf{fma}\left(x, \mathsf{fma}\left(n, -0.5, 0.16666666666666666\right), n \cdot 0.5\right)}{n \cdot \left(n \cdot n\right)}, \frac{1}{n}\right), 1 + -1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 56.8% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x \cdot \left(x \cdot x\right)\\ \mathbf{if}\;\frac{1}{n} \leq -5000:\\ \;\;\;\;\frac{\frac{0.3333333333333333}{t\_0}}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq -1 \cdot 10^{-77}:\\ \;\;\;\;\frac{\frac{1 + \frac{\left(-0.125 + \frac{0.037037037037037035}{t\_0}\right) \cdot \frac{1}{x}}{\mathsf{fma}\left(\frac{0.3333333333333333}{x}, \frac{0.3333333333333333}{x} - -0.5, 0.25\right)}}{x}}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 2 \cdot 10^{+63}:\\ \;\;\;\;-\frac{\log x}{n}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x, \mathsf{fma}\left(x, \frac{\mathsf{fma}\left(x, \mathsf{fma}\left(n, -0.5, 0.16666666666666666\right), n \cdot 0.5\right)}{n \cdot \left(n \cdot n\right)}, \frac{1}{n}\right), 1 + -1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (let* ((t_0 (* x (* x x))))
   (if (<= (/ 1.0 n) -5000.0)
     (/ (/ 0.3333333333333333 t_0) n)
     (if (<= (/ 1.0 n) -1e-77)
       (/
        (/
         (+
          1.0
          (/
           (* (+ -0.125 (/ 0.037037037037037035 t_0)) (/ 1.0 x))
           (fma
            (/ 0.3333333333333333 x)
            (- (/ 0.3333333333333333 x) -0.5)
            0.25)))
         x)
        n)
       (if (<= (/ 1.0 n) 2e+63)
         (- (/ (log x) n))
         (fma
          x
          (fma
           x
           (/ (fma x (fma n -0.5 0.16666666666666666) (* n 0.5)) (* n (* n n)))
           (/ 1.0 n))
          (+ 1.0 -1.0)))))))

C

double code(double x, double n) {
	double t_0 = x * (x * x);
	double tmp;
	if ((1.0 / n) <= -5000.0) {
		tmp = (0.3333333333333333 / t_0) / n;
	} else if ((1.0 / n) <= -1e-77) {
		tmp = ((1.0 + (((-0.125 + (0.037037037037037035 / t_0)) * (1.0 / x)) / fma((0.3333333333333333 / x), ((0.3333333333333333 / x) - -0.5), 0.25))) / x) / n;
	} else if ((1.0 / n) <= 2e+63) {
		tmp = -(log(x) / n);
	} else {
		tmp = fma(x, fma(x, (fma(x, fma(n, -0.5, 0.16666666666666666), (n * 0.5)) / (n * (n * n))), (1.0 / n)), (1.0 + -1.0));
	}
	return tmp;
}

Julia

function code(x, n)
	t_0 = Float64(x * Float64(x * x))
	tmp = 0.0
	if (Float64(1.0 / n) <= -5000.0)
		tmp = Float64(Float64(0.3333333333333333 / t_0) / n);
	elseif (Float64(1.0 / n) <= -1e-77)
		tmp = Float64(Float64(Float64(1.0 + Float64(Float64(Float64(-0.125 + Float64(0.037037037037037035 / t_0)) * Float64(1.0 / x)) / fma(Float64(0.3333333333333333 / x), Float64(Float64(0.3333333333333333 / x) - -0.5), 0.25))) / x) / n);
	elseif (Float64(1.0 / n) <= 2e+63)
		tmp = Float64(-Float64(log(x) / n));
	else
		tmp = fma(x, fma(x, Float64(fma(x, fma(n, -0.5, 0.16666666666666666), Float64(n * 0.5)) / Float64(n * Float64(n * n))), Float64(1.0 / n)), Float64(1.0 + -1.0));
	end
	return tmp
end

Wolfram

code[x_, n_] := Block[{t$95$0 = N[(x * N[(x * x), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(1.0 / n), $MachinePrecision], -5000.0], N[(N[(0.3333333333333333 / t$95$0), $MachinePrecision] / n), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], -1e-77], N[(N[(N[(1.0 + N[(N[(N[(-0.125 + N[(0.037037037037037035 / t$95$0), $MachinePrecision]), $MachinePrecision] * N[(1.0 / x), $MachinePrecision]), $MachinePrecision] / N[(N[(0.3333333333333333 / x), $MachinePrecision] * N[(N[(0.3333333333333333 / x), $MachinePrecision] - -0.5), $MachinePrecision] + 0.25), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision] / n), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 2e+63], (-N[(N[Log[x], $MachinePrecision] / n), $MachinePrecision]), N[(x * N[(x * N[(N[(x * N[(n * -0.5 + 0.16666666666666666), $MachinePrecision] + N[(n * 0.5), $MachinePrecision]), $MachinePrecision] / N[(n * N[(n * n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(1.0 / n), $MachinePrecision]), $MachinePrecision] + N[(1.0 + -1.0), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if (/.f64 #s(literal 1 binary64) n) < -5e3

   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 50.9

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

   5. Applied rewrites 50.9%

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

   6. Taylor expanded in x around inf

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

      1. lower-/.f64 N/A

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

   8. Applied rewrites 42.1%

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

   9. Taylor expanded in x around 0

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

       1. lower-/.f64 N/A

          \[\leadsto \frac{\color{blue}{\frac{\frac{1}{3}}{{x}^{3}}}}{n} \]

       2. cube-mult N/A

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

       3. unpow2 N/A

          \[\leadsto \frac{\frac{\frac{1}{3}}{x \cdot \color{blue}{{x}^{2}}}}{n} \]

       4. lower-*.f64 N/A

          \[\leadsto \frac{\frac{\frac{1}{3}}{\color{blue}{x \cdot {x}^{2}}}}{n} \]

       5. unpow2 N/A

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

       6. lower-*.f64 76.2

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

   11. Applied rewrites 76.2%

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

   if -5e3 < (/.f64 #s(literal 1 binary64) n) < -9.9999999999999993e-78

   1. Initial program 26.9%

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

   3. Taylor expanded in n around inf

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

      1. lower-/.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lower-log1p.f64 N/A

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

      4. lower-log.f64 31.9

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

   5. Applied rewrites 31.9%

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

   6. Taylor expanded in x around inf

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

      1. lower-/.f64 N/A

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

   8. Applied rewrites 50.3%

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

      1. lift-/.f64 N/A

         \[\leadsto \frac{\frac{1 + \frac{\frac{-1}{2} + \color{blue}{\frac{\frac{1}{3}}{x}}}{x}}{x}}{n} \]

      2. lift-+.f64 N/A

         \[\leadsto \frac{\frac{1 + \frac{\color{blue}{\frac{-1}{2} + \frac{\frac{1}{3}}{x}}}{x}}{x}}{n} \]

      3. div-inv N/A

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

      4. lift-+.f64 N/A

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

      5. flip3-+ N/A

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

      6. associate-*l/ N/A

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

      7. lower-/.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-+.f64 N/A

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

      10. metadata-eval N/A

          \[\leadsto \frac{\frac{1 + \frac{\left(\color{blue}{\frac{-1}{8}} + {\left(\frac{\frac{1}{3}}{x}\right)}^{3}\right) \cdot \frac{1}{x}}{\frac{-1}{2} \cdot \frac{-1}{2} + \left(\frac{\frac{1}{3}}{x} \cdot \frac{\frac{1}{3}}{x} - \frac{-1}{2} \cdot \frac{\frac{1}{3}}{x}\right)}}{x}}{n} \]

      11. lift-/.f64 N/A

          \[\leadsto \frac{\frac{1 + \frac{\left(\frac{-1}{8} + {\color{blue}{\left(\frac{\frac{1}{3}}{x}\right)}}^{3}\right) \cdot \frac{1}{x}}{\frac{-1}{2} \cdot \frac{-1}{2} + \left(\frac{\frac{1}{3}}{x} \cdot \frac{\frac{1}{3}}{x} - \frac{-1}{2} \cdot \frac{\frac{1}{3}}{x}\right)}}{x}}{n} \]

      12. cube-div N/A

          \[\leadsto \frac{\frac{1 + \frac{\left(\frac{-1}{8} + \color{blue}{\frac{{\frac{1}{3}}^{3}}{{x}^{3}}}\right) \cdot \frac{1}{x}}{\frac{-1}{2} \cdot \frac{-1}{2} + \left(\frac{\frac{1}{3}}{x} \cdot \frac{\frac{1}{3}}{x} - \frac{-1}{2} \cdot \frac{\frac{1}{3}}{x}\right)}}{x}}{n} \]

      13. lower-/.f64 N/A

          \[\leadsto \frac{\frac{1 + \frac{\left(\frac{-1}{8} + \color{blue}{\frac{{\frac{1}{3}}^{3}}{{x}^{3}}}\right) \cdot \frac{1}{x}}{\frac{-1}{2} \cdot \frac{-1}{2} + \left(\frac{\frac{1}{3}}{x} \cdot \frac{\frac{1}{3}}{x} - \frac{-1}{2} \cdot \frac{\frac{1}{3}}{x}\right)}}{x}}{n} \]

      14. metadata-eval N/A

          \[\leadsto \frac{\frac{1 + \frac{\left(\frac{-1}{8} + \frac{\color{blue}{\frac{1}{27}}}{{x}^{3}}\right) \cdot \frac{1}{x}}{\frac{-1}{2} \cdot \frac{-1}{2} + \left(\frac{\frac{1}{3}}{x} \cdot \frac{\frac{1}{3}}{x} - \frac{-1}{2} \cdot \frac{\frac{1}{3}}{x}\right)}}{x}}{n} \]

      15. cube-mult N/A

          \[\leadsto \frac{\frac{1 + \frac{\left(\frac{-1}{8} + \frac{\frac{1}{27}}{\color{blue}{x \cdot \left(x \cdot x\right)}}\right) \cdot \frac{1}{x}}{\frac{-1}{2} \cdot \frac{-1}{2} + \left(\frac{\frac{1}{3}}{x} \cdot \frac{\frac{1}{3}}{x} - \frac{-1}{2} \cdot \frac{\frac{1}{3}}{x}\right)}}{x}}{n} \]

      16. lower-*.f64 N/A

          \[\leadsto \frac{\frac{1 + \frac{\left(\frac{-1}{8} + \frac{\frac{1}{27}}{\color{blue}{x \cdot \left(x \cdot x\right)}}\right) \cdot \frac{1}{x}}{\frac{-1}{2} \cdot \frac{-1}{2} + \left(\frac{\frac{1}{3}}{x} \cdot \frac{\frac{1}{3}}{x} - \frac{-1}{2} \cdot \frac{\frac{1}{3}}{x}\right)}}{x}}{n} \]

      17. lower-*.f64 N/A

          \[\leadsto \frac{\frac{1 + \frac{\left(\frac{-1}{8} + \frac{\frac{1}{27}}{x \cdot \color{blue}{\left(x \cdot x\right)}}\right) \cdot \frac{1}{x}}{\frac{-1}{2} \cdot \frac{-1}{2} + \left(\frac{\frac{1}{3}}{x} \cdot \frac{\frac{1}{3}}{x} - \frac{-1}{2} \cdot \frac{\frac{1}{3}}{x}\right)}}{x}}{n} \]

      18. lower-/.f64 N/A

          \[\leadsto \frac{\frac{1 + \frac{\left(\frac{-1}{8} + \frac{\frac{1}{27}}{x \cdot \left(x \cdot x\right)}\right) \cdot \color{blue}{\frac{1}{x}}}{\frac{-1}{2} \cdot \frac{-1}{2} + \left(\frac{\frac{1}{3}}{x} \cdot \frac{\frac{1}{3}}{x} - \frac{-1}{2} \cdot \frac{\frac{1}{3}}{x}\right)}}{x}}{n} \]

   10. Applied rewrites 54.4%

       \[\leadsto \frac{\frac{1 + \color{blue}{\frac{\left(-0.125 + \frac{0.037037037037037035}{x \cdot \left(x \cdot x\right)}\right) \cdot \frac{1}{x}}{\mathsf{fma}\left(\frac{0.3333333333333333}{x}, \frac{0.3333333333333333}{x} - -0.5, 0.25\right)}}}{x}}{n} \]

   if -9.9999999999999993e-78 < (/.f64 #s(literal 1 binary64) n) < 2.00000000000000012e63

   1. Initial program 26.1%

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

   3. Taylor expanded in x around 0

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

      1. remove-double-neg N/A

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

      2. mul-1-neg N/A

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

      3. distribute-neg-frac N/A

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

      4. mul-1-neg N/A

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

      5. log-rec N/A

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

      6. mul-1-neg N/A

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

      7. lower--.f64 N/A

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

      8. log-rec N/A

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

      9. mul-1-neg N/A

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

      10. associate-*r/ N/A

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

      11. associate-*r* N/A

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

      12. metadata-eval N/A

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

      13. *-commutative N/A

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

      14. associate-/l* N/A

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

      15. exp-to-pow N/A

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

      16. lower-pow.f64 N/A

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

      17. lower-/.f64 26.1

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

   5. Applied rewrites 26.1%

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

   6. Taylor expanded in n around inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower-log.f64 60.7

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

   8. Applied rewrites 60.7%

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

   if 2.00000000000000012e63 < (/.f64 #s(literal 1 binary64) n)

   1. Initial program 51.5%

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

   3. Taylor expanded in x around 0

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

   5. Applied rewrites 19.5%

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

   6. Taylor expanded in n around inf

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

   8. Applied rewrites 16.5%

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

   9. Taylor expanded in n around 0

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

       1. lower-/.f64 N/A

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

       2. +-commutative N/A

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

       3. distribute-lft-in N/A

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

       4. *-commutative N/A

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

       5. associate-+l+ N/A

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

       6. associate-*r* N/A

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

       7. *-commutative N/A

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

       8. distribute-rgt-in N/A

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

       9. +-commutative N/A

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

       10. +-commutative N/A

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

       11. lower-fma.f64 N/A

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

       12. +-commutative N/A

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

       13. *-commutative N/A

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

       14. lower-fma.f64 N/A

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

       15. *-commutative N/A

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

       16. lower-*.f64 N/A

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

       17. cube-mult N/A

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

       18. unpow2 N/A

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

       19. lower-*.f64 N/A

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

       20. unpow2 N/A

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

       21. lower-*.f64 54.8

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

   11. Applied rewrites 54.8%

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

4. Final simplification 64.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{1}{n} \leq -5000:\\ \;\;\;\;\frac{\frac{0.3333333333333333}{x \cdot \left(x \cdot x\right)}}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq -1 \cdot 10^{-77}:\\ \;\;\;\;\frac{\frac{1 + \frac{\left(-0.125 + \frac{0.037037037037037035}{x \cdot \left(x \cdot x\right)}\right) \cdot \frac{1}{x}}{\mathsf{fma}\left(\frac{0.3333333333333333}{x}, \frac{0.3333333333333333}{x} - -0.5, 0.25\right)}}{x}}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 2 \cdot 10^{+63}:\\ \;\;\;\;-\frac{\log x}{n}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x, \mathsf{fma}\left(x, \frac{\mathsf{fma}\left(x, \mathsf{fma}\left(n, -0.5, 0.16666666666666666\right), n \cdot 0.5\right)}{n \cdot \left(n \cdot n\right)}, \frac{1}{n}\right), 1 + -1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 59.0% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x - \log x\\ t_1 := x \cdot \left(x \cdot x\right)\\ \mathbf{if}\;n \leq -2.8 \cdot 10^{+70}:\\ \;\;\;\;\frac{t\_0}{n}\\ \mathbf{elif}\;n \leq -0.000118:\\ \;\;\;\;\frac{\frac{1 + \frac{\left(-0.125 + \frac{0.037037037037037035}{t\_1}\right) \cdot \frac{1}{x}}{\mathsf{fma}\left(\frac{0.3333333333333333}{x}, \frac{0.3333333333333333}{x} - -0.5, 0.25\right)}}{x}}{n}\\ \mathbf{elif}\;n \leq 5 \cdot 10^{-238}:\\ \;\;\;\;\frac{\frac{0.3333333333333333}{t\_1}}{n}\\ \mathbf{elif}\;n \leq 36:\\ \;\;\;\;1 - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{elif}\;n \leq 9.5 \cdot 10^{+175}:\\ \;\;\;\;\frac{1}{n} \cdot t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{n} + \left(\frac{0.3333333333333333}{x \cdot \left(n \cdot x\right)} + \frac{-0.5}{n \cdot x}\right)}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (let* ((t_0 (- x (log x))) (t_1 (* x (* x x))))
   (if (<= n -2.8e+70)
     (/ t_0 n)
     (if (<= n -0.000118)
       (/
        (/
         (+
          1.0
          (/
           (* (+ -0.125 (/ 0.037037037037037035 t_1)) (/ 1.0 x))
           (fma
            (/ 0.3333333333333333 x)
            (- (/ 0.3333333333333333 x) -0.5)
            0.25)))
         x)
        n)
       (if (<= n 5e-238)
         (/ (/ 0.3333333333333333 t_1) n)
         (if (<= n 36.0)
           (- 1.0 (pow x (/ 1.0 n)))
           (if (<= n 9.5e+175)
             (* (/ 1.0 n) t_0)
             (/
              (+
               (/ 1.0 n)
               (+ (/ 0.3333333333333333 (* x (* n x))) (/ -0.5 (* n x))))
              x))))))))

C

double code(double x, double n) {
	double t_0 = x - log(x);
	double t_1 = x * (x * x);
	double tmp;
	if (n <= -2.8e+70) {
		tmp = t_0 / n;
	} else if (n <= -0.000118) {
		tmp = ((1.0 + (((-0.125 + (0.037037037037037035 / t_1)) * (1.0 / x)) / fma((0.3333333333333333 / x), ((0.3333333333333333 / x) - -0.5), 0.25))) / x) / n;
	} else if (n <= 5e-238) {
		tmp = (0.3333333333333333 / t_1) / n;
	} else if (n <= 36.0) {
		tmp = 1.0 - pow(x, (1.0 / n));
	} else if (n <= 9.5e+175) {
		tmp = (1.0 / n) * t_0;
	} else {
		tmp = ((1.0 / n) + ((0.3333333333333333 / (x * (n * x))) + (-0.5 / (n * x)))) / x;
	}
	return tmp;
}

Julia

function code(x, n)
	t_0 = Float64(x - log(x))
	t_1 = Float64(x * Float64(x * x))
	tmp = 0.0
	if (n <= -2.8e+70)
		tmp = Float64(t_0 / n);
	elseif (n <= -0.000118)
		tmp = Float64(Float64(Float64(1.0 + Float64(Float64(Float64(-0.125 + Float64(0.037037037037037035 / t_1)) * Float64(1.0 / x)) / fma(Float64(0.3333333333333333 / x), Float64(Float64(0.3333333333333333 / x) - -0.5), 0.25))) / x) / n);
	elseif (n <= 5e-238)
		tmp = Float64(Float64(0.3333333333333333 / t_1) / n);
	elseif (n <= 36.0)
		tmp = Float64(1.0 - (x ^ Float64(1.0 / n)));
	elseif (n <= 9.5e+175)
		tmp = Float64(Float64(1.0 / n) * t_0);
	else
		tmp = Float64(Float64(Float64(1.0 / n) + Float64(Float64(0.3333333333333333 / Float64(x * Float64(n * x))) + Float64(-0.5 / Float64(n * x)))) / x);
	end
	return tmp
end

Wolfram

code[x_, n_] := Block[{t$95$0 = N[(x - N[Log[x], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(x * N[(x * x), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[n, -2.8e+70], N[(t$95$0 / n), $MachinePrecision], If[LessEqual[n, -0.000118], N[(N[(N[(1.0 + N[(N[(N[(-0.125 + N[(0.037037037037037035 / t$95$1), $MachinePrecision]), $MachinePrecision] * N[(1.0 / x), $MachinePrecision]), $MachinePrecision] / N[(N[(0.3333333333333333 / x), $MachinePrecision] * N[(N[(0.3333333333333333 / x), $MachinePrecision] - -0.5), $MachinePrecision] + 0.25), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision] / n), $MachinePrecision], If[LessEqual[n, 5e-238], N[(N[(0.3333333333333333 / t$95$1), $MachinePrecision] / n), $MachinePrecision], If[LessEqual[n, 36.0], N[(1.0 - N[Power[x, N[(1.0 / n), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[n, 9.5e+175], N[(N[(1.0 / n), $MachinePrecision] * t$95$0), $MachinePrecision], N[(N[(N[(1.0 / n), $MachinePrecision] + N[(N[(0.3333333333333333 / N[(x * N[(n * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(-0.5 / N[(n * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]]]]]]]]

Derivation

1. Split input into 6 regimes

2. if n < -2.7999999999999999e70

   1. Initial program 17.9%

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

   3. Taylor expanded in x around 0

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

      1. *-rgt-identity N/A

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

      2. associate-*r/ N/A

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

      3. lower-+.f64 N/A

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

      4. associate-*r/ N/A

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

      5. *-rgt-identity N/A

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

      6. lower-/.f64 13.2

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

   5. Applied rewrites 13.2%

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

   6. Taylor expanded in n around inf

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

      1. lower-/.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lower-log.f64 72.8

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

   8. Applied rewrites 72.8%

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

   if -2.7999999999999999e70 < n < -1.18e-4

   1. Initial program 33.9%

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

   3. Taylor expanded in n around inf

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

      1. lower-/.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lower-log1p.f64 N/A

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

      4. lower-log.f64 24.9

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

   5. Applied rewrites 24.9%

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

   6. Taylor expanded in x around inf

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

      1. lower-/.f64 N/A

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

   8. Applied rewrites 47.5%

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

      1. lift-/.f64 N/A

         \[\leadsto \frac{\frac{1 + \frac{\frac{-1}{2} + \color{blue}{\frac{\frac{1}{3}}{x}}}{x}}{x}}{n} \]

      2. lift-+.f64 N/A

         \[\leadsto \frac{\frac{1 + \frac{\color{blue}{\frac{-1}{2} + \frac{\frac{1}{3}}{x}}}{x}}{x}}{n} \]

      3. div-inv N/A

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

      4. lift-+.f64 N/A

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

      5. flip3-+ N/A

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

      6. associate-*l/ N/A

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

      7. lower-/.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-+.f64 N/A

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

      10. metadata-eval N/A

          \[\leadsto \frac{\frac{1 + \frac{\left(\color{blue}{\frac{-1}{8}} + {\left(\frac{\frac{1}{3}}{x}\right)}^{3}\right) \cdot \frac{1}{x}}{\frac{-1}{2} \cdot \frac{-1}{2} + \left(\frac{\frac{1}{3}}{x} \cdot \frac{\frac{1}{3}}{x} - \frac{-1}{2} \cdot \frac{\frac{1}{3}}{x}\right)}}{x}}{n} \]

      11. lift-/.f64 N/A

          \[\leadsto \frac{\frac{1 + \frac{\left(\frac{-1}{8} + {\color{blue}{\left(\frac{\frac{1}{3}}{x}\right)}}^{3}\right) \cdot \frac{1}{x}}{\frac{-1}{2} \cdot \frac{-1}{2} + \left(\frac{\frac{1}{3}}{x} \cdot \frac{\frac{1}{3}}{x} - \frac{-1}{2} \cdot \frac{\frac{1}{3}}{x}\right)}}{x}}{n} \]

      12. cube-div N/A

          \[\leadsto \frac{\frac{1 + \frac{\left(\frac{-1}{8} + \color{blue}{\frac{{\frac{1}{3}}^{3}}{{x}^{3}}}\right) \cdot \frac{1}{x}}{\frac{-1}{2} \cdot \frac{-1}{2} + \left(\frac{\frac{1}{3}}{x} \cdot \frac{\frac{1}{3}}{x} - \frac{-1}{2} \cdot \frac{\frac{1}{3}}{x}\right)}}{x}}{n} \]

      13. lower-/.f64 N/A

          \[\leadsto \frac{\frac{1 + \frac{\left(\frac{-1}{8} + \color{blue}{\frac{{\frac{1}{3}}^{3}}{{x}^{3}}}\right) \cdot \frac{1}{x}}{\frac{-1}{2} \cdot \frac{-1}{2} + \left(\frac{\frac{1}{3}}{x} \cdot \frac{\frac{1}{3}}{x} - \frac{-1}{2} \cdot \frac{\frac{1}{3}}{x}\right)}}{x}}{n} \]

      14. metadata-eval N/A

          \[\leadsto \frac{\frac{1 + \frac{\left(\frac{-1}{8} + \frac{\color{blue}{\frac{1}{27}}}{{x}^{3}}\right) \cdot \frac{1}{x}}{\frac{-1}{2} \cdot \frac{-1}{2} + \left(\frac{\frac{1}{3}}{x} \cdot \frac{\frac{1}{3}}{x} - \frac{-1}{2} \cdot \frac{\frac{1}{3}}{x}\right)}}{x}}{n} \]

      15. cube-mult N/A

          \[\leadsto \frac{\frac{1 + \frac{\left(\frac{-1}{8} + \frac{\frac{1}{27}}{\color{blue}{x \cdot \left(x \cdot x\right)}}\right) \cdot \frac{1}{x}}{\frac{-1}{2} \cdot \frac{-1}{2} + \left(\frac{\frac{1}{3}}{x} \cdot \frac{\frac{1}{3}}{x} - \frac{-1}{2} \cdot \frac{\frac{1}{3}}{x}\right)}}{x}}{n} \]

      16. lower-*.f64 N/A

          \[\leadsto \frac{\frac{1 + \frac{\left(\frac{-1}{8} + \frac{\frac{1}{27}}{\color{blue}{x \cdot \left(x \cdot x\right)}}\right) \cdot \frac{1}{x}}{\frac{-1}{2} \cdot \frac{-1}{2} + \left(\frac{\frac{1}{3}}{x} \cdot \frac{\frac{1}{3}}{x} - \frac{-1}{2} \cdot \frac{\frac{1}{3}}{x}\right)}}{x}}{n} \]

      17. lower-*.f64 N/A

          \[\leadsto \frac{\frac{1 + \frac{\left(\frac{-1}{8} + \frac{\frac{1}{27}}{x \cdot \color{blue}{\left(x \cdot x\right)}}\right) \cdot \frac{1}{x}}{\frac{-1}{2} \cdot \frac{-1}{2} + \left(\frac{\frac{1}{3}}{x} \cdot \frac{\frac{1}{3}}{x} - \frac{-1}{2} \cdot \frac{\frac{1}{3}}{x}\right)}}{x}}{n} \]

      18. lower-/.f64 N/A

          \[\leadsto \frac{\frac{1 + \frac{\left(\frac{-1}{8} + \frac{\frac{1}{27}}{x \cdot \left(x \cdot x\right)}\right) \cdot \color{blue}{\frac{1}{x}}}{\frac{-1}{2} \cdot \frac{-1}{2} + \left(\frac{\frac{1}{3}}{x} \cdot \frac{\frac{1}{3}}{x} - \frac{-1}{2} \cdot \frac{\frac{1}{3}}{x}\right)}}{x}}{n} \]

   10. Applied rewrites 51.8%

       \[\leadsto \frac{\frac{1 + \color{blue}{\frac{\left(-0.125 + \frac{0.037037037037037035}{x \cdot \left(x \cdot x\right)}\right) \cdot \frac{1}{x}}{\mathsf{fma}\left(\frac{0.3333333333333333}{x}, \frac{0.3333333333333333}{x} - -0.5, 0.25\right)}}}{x}}{n} \]

   if -1.18e-4 < n < 5e-238

   1. Initial program 91.0%

      \[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)} \]
   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 47.5

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

   5. Applied rewrites 47.5%

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

   6. Taylor expanded in x around inf

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

      1. lower-/.f64 N/A

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

   8. Applied rewrites 47.9%

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

   9. Taylor expanded in x around 0

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

       1. lower-/.f64 N/A

          \[\leadsto \frac{\color{blue}{\frac{\frac{1}{3}}{{x}^{3}}}}{n} \]

       2. cube-mult N/A

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

       3. unpow2 N/A

          \[\leadsto \frac{\frac{\frac{1}{3}}{x \cdot \color{blue}{{x}^{2}}}}{n} \]

       4. lower-*.f64 N/A

          \[\leadsto \frac{\frac{\frac{1}{3}}{\color{blue}{x \cdot {x}^{2}}}}{n} \]

       5. unpow2 N/A

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

       6. lower-*.f64 79.3

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

   11. Applied rewrites 79.3%

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

   if 5e-238 < n < 36

   1. Initial program 68.9%

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

   3. Taylor expanded in x around 0

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

      1. remove-double-neg N/A

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

      2. mul-1-neg N/A

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

      3. distribute-neg-frac N/A

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

      4. mul-1-neg N/A

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

      5. log-rec N/A

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

      6. mul-1-neg N/A

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

      7. lower--.f64 N/A

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

      8. log-rec N/A

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

      9. mul-1-neg N/A

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

      10. associate-*r/ N/A

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

      11. associate-*r* N/A

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

      12. metadata-eval N/A

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

      13. *-commutative N/A

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

      14. associate-/l* N/A

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

      15. exp-to-pow N/A

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

      16. lower-pow.f64 N/A

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

      17. lower-/.f64 62.5

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

   5. Applied rewrites 62.5%

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

   if 36 < n < 9.5000000000000006e175

   1. Initial program 18.8%

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

   3. Taylor expanded in x around 0

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

      1. *-rgt-identity N/A

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

      2. associate-*r/ N/A

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

      3. lower-+.f64 N/A

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

      4. associate-*r/ N/A

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

      5. *-rgt-identity N/A

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

      6. lower-/.f64 5.6

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

   5. Applied rewrites 5.6%

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

   6. Taylor expanded in n around inf

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

      1. lower-/.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lower-log.f64 61.3

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

   8. Applied rewrites 61.3%

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

      1. lift-log.f64 N/A

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

      2. lift--.f64 N/A

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

      3. clear-num N/A

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

      4. associate-/r/ N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 61.3

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

   10. Applied rewrites 61.3%

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

   if 9.5000000000000006e175 < n

   1. Initial program 42.0%

      \[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)} \]
   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 81.4

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

   5. Applied rewrites 81.4%

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

   6. Taylor expanded in x around inf

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

   8. Applied rewrites 60.7%

      \[\leadsto \color{blue}{\frac{\frac{1}{n} + \left(\frac{0.3333333333333333}{x \cdot \left(x \cdot n\right)} + \frac{-0.5}{x \cdot n}\right)}{x}} \]
3. Recombined 6 regimes into one program.

4. Final simplification 69.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq -2.8 \cdot 10^{+70}:\\ \;\;\;\;\frac{x - \log x}{n}\\ \mathbf{elif}\;n \leq -0.000118:\\ \;\;\;\;\frac{\frac{1 + \frac{\left(-0.125 + \frac{0.037037037037037035}{x \cdot \left(x \cdot x\right)}\right) \cdot \frac{1}{x}}{\mathsf{fma}\left(\frac{0.3333333333333333}{x}, \frac{0.3333333333333333}{x} - -0.5, 0.25\right)}}{x}}{n}\\ \mathbf{elif}\;n \leq 5 \cdot 10^{-238}:\\ \;\;\;\;\frac{\frac{0.3333333333333333}{x \cdot \left(x \cdot x\right)}}{n}\\ \mathbf{elif}\;n \leq 36:\\ \;\;\;\;1 - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{elif}\;n \leq 9.5 \cdot 10^{+175}:\\ \;\;\;\;\frac{1}{n} \cdot \left(x - \log x\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{n} + \left(\frac{0.3333333333333333}{x \cdot \left(n \cdot x\right)} + \frac{-0.5}{n \cdot x}\right)}{x}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 59.0% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x - \log x}{n}\\ t_1 := x \cdot \left(x \cdot x\right)\\ \mathbf{if}\;n \leq -2.8 \cdot 10^{+70}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;n \leq -0.000118:\\ \;\;\;\;\frac{\frac{1 + \frac{\left(-0.125 + \frac{0.037037037037037035}{t\_1}\right) \cdot \frac{1}{x}}{\mathsf{fma}\left(\frac{0.3333333333333333}{x}, \frac{0.3333333333333333}{x} - -0.5, 0.25\right)}}{x}}{n}\\ \mathbf{elif}\;n \leq 5 \cdot 10^{-238}:\\ \;\;\;\;\frac{\frac{0.3333333333333333}{t\_1}}{n}\\ \mathbf{elif}\;n \leq 36:\\ \;\;\;\;1 - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{elif}\;n \leq 9.5 \cdot 10^{+175}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{n} + \left(\frac{0.3333333333333333}{x \cdot \left(n \cdot x\right)} + \frac{-0.5}{n \cdot x}\right)}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (let* ((t_0 (/ (- x (log x)) n)) (t_1 (* x (* x x))))
   (if (<= n -2.8e+70)
     t_0
     (if (<= n -0.000118)
       (/
        (/
         (+
          1.0
          (/
           (* (+ -0.125 (/ 0.037037037037037035 t_1)) (/ 1.0 x))
           (fma
            (/ 0.3333333333333333 x)
            (- (/ 0.3333333333333333 x) -0.5)
            0.25)))
         x)
        n)
       (if (<= n 5e-238)
         (/ (/ 0.3333333333333333 t_1) n)
         (if (<= n 36.0)
           (- 1.0 (pow x (/ 1.0 n)))
           (if (<= n 9.5e+175)
             t_0
             (/
              (+
               (/ 1.0 n)
               (+ (/ 0.3333333333333333 (* x (* n x))) (/ -0.5 (* n x))))
              x))))))))

C

double code(double x, double n) {
	double t_0 = (x - log(x)) / n;
	double t_1 = x * (x * x);
	double tmp;
	if (n <= -2.8e+70) {
		tmp = t_0;
	} else if (n <= -0.000118) {
		tmp = ((1.0 + (((-0.125 + (0.037037037037037035 / t_1)) * (1.0 / x)) / fma((0.3333333333333333 / x), ((0.3333333333333333 / x) - -0.5), 0.25))) / x) / n;
	} else if (n <= 5e-238) {
		tmp = (0.3333333333333333 / t_1) / n;
	} else if (n <= 36.0) {
		tmp = 1.0 - pow(x, (1.0 / n));
	} else if (n <= 9.5e+175) {
		tmp = t_0;
	} else {
		tmp = ((1.0 / n) + ((0.3333333333333333 / (x * (n * x))) + (-0.5 / (n * x)))) / x;
	}
	return tmp;
}

Julia

function code(x, n)
	t_0 = Float64(Float64(x - log(x)) / n)
	t_1 = Float64(x * Float64(x * x))
	tmp = 0.0
	if (n <= -2.8e+70)
		tmp = t_0;
	elseif (n <= -0.000118)
		tmp = Float64(Float64(Float64(1.0 + Float64(Float64(Float64(-0.125 + Float64(0.037037037037037035 / t_1)) * Float64(1.0 / x)) / fma(Float64(0.3333333333333333 / x), Float64(Float64(0.3333333333333333 / x) - -0.5), 0.25))) / x) / n);
	elseif (n <= 5e-238)
		tmp = Float64(Float64(0.3333333333333333 / t_1) / n);
	elseif (n <= 36.0)
		tmp = Float64(1.0 - (x ^ Float64(1.0 / n)));
	elseif (n <= 9.5e+175)
		tmp = t_0;
	else
		tmp = Float64(Float64(Float64(1.0 / n) + Float64(Float64(0.3333333333333333 / Float64(x * Float64(n * x))) + Float64(-0.5 / Float64(n * x)))) / x);
	end
	return tmp
end

Wolfram

code[x_, n_] := Block[{t$95$0 = N[(N[(x - N[Log[x], $MachinePrecision]), $MachinePrecision] / n), $MachinePrecision]}, Block[{t$95$1 = N[(x * N[(x * x), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[n, -2.8e+70], t$95$0, If[LessEqual[n, -0.000118], N[(N[(N[(1.0 + N[(N[(N[(-0.125 + N[(0.037037037037037035 / t$95$1), $MachinePrecision]), $MachinePrecision] * N[(1.0 / x), $MachinePrecision]), $MachinePrecision] / N[(N[(0.3333333333333333 / x), $MachinePrecision] * N[(N[(0.3333333333333333 / x), $MachinePrecision] - -0.5), $MachinePrecision] + 0.25), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision] / n), $MachinePrecision], If[LessEqual[n, 5e-238], N[(N[(0.3333333333333333 / t$95$1), $MachinePrecision] / n), $MachinePrecision], If[LessEqual[n, 36.0], N[(1.0 - N[Power[x, N[(1.0 / n), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[n, 9.5e+175], t$95$0, N[(N[(N[(1.0 / n), $MachinePrecision] + N[(N[(0.3333333333333333 / N[(x * N[(n * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(-0.5 / N[(n * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]]]]]]]]

Derivation

1. Split input into 5 regimes

2. if n < -2.7999999999999999e70 or 36 < n < 9.5000000000000006e175

   1. Initial program 18.2%

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

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

      2. associate-*r/ N/A

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

      3. lower-+.f64 N/A

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

      4. associate-*r/ N/A

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

      5. *-rgt-identity N/A

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

      6. lower-/.f64 10.8

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

   5. Applied rewrites 10.8%

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

   6. Taylor expanded in n around inf

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

      1. lower-/.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lower-log.f64 69.1

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

   8. Applied rewrites 69.1%

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

   if -2.7999999999999999e70 < n < -1.18e-4

   1. Initial program 33.9%

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

   3. Taylor expanded in n around inf

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

      1. lower-/.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lower-log1p.f64 N/A

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

      4. lower-log.f64 24.9

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

   5. Applied rewrites 24.9%

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

   6. Taylor expanded in x around inf

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

      1. lower-/.f64 N/A

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

   8. Applied rewrites 47.5%

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

      1. lift-/.f64 N/A

         \[\leadsto \frac{\frac{1 + \frac{\frac{-1}{2} + \color{blue}{\frac{\frac{1}{3}}{x}}}{x}}{x}}{n} \]

      2. lift-+.f64 N/A

         \[\leadsto \frac{\frac{1 + \frac{\color{blue}{\frac{-1}{2} + \frac{\frac{1}{3}}{x}}}{x}}{x}}{n} \]

      3. div-inv N/A

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

      4. lift-+.f64 N/A

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

      5. flip3-+ N/A

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

      6. associate-*l/ N/A

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

      7. lower-/.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-+.f64 N/A

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

      10. metadata-eval N/A

          \[\leadsto \frac{\frac{1 + \frac{\left(\color{blue}{\frac{-1}{8}} + {\left(\frac{\frac{1}{3}}{x}\right)}^{3}\right) \cdot \frac{1}{x}}{\frac{-1}{2} \cdot \frac{-1}{2} + \left(\frac{\frac{1}{3}}{x} \cdot \frac{\frac{1}{3}}{x} - \frac{-1}{2} \cdot \frac{\frac{1}{3}}{x}\right)}}{x}}{n} \]

      11. lift-/.f64 N/A

          \[\leadsto \frac{\frac{1 + \frac{\left(\frac{-1}{8} + {\color{blue}{\left(\frac{\frac{1}{3}}{x}\right)}}^{3}\right) \cdot \frac{1}{x}}{\frac{-1}{2} \cdot \frac{-1}{2} + \left(\frac{\frac{1}{3}}{x} \cdot \frac{\frac{1}{3}}{x} - \frac{-1}{2} \cdot \frac{\frac{1}{3}}{x}\right)}}{x}}{n} \]

      12. cube-div N/A

          \[\leadsto \frac{\frac{1 + \frac{\left(\frac{-1}{8} + \color{blue}{\frac{{\frac{1}{3}}^{3}}{{x}^{3}}}\right) \cdot \frac{1}{x}}{\frac{-1}{2} \cdot \frac{-1}{2} + \left(\frac{\frac{1}{3}}{x} \cdot \frac{\frac{1}{3}}{x} - \frac{-1}{2} \cdot \frac{\frac{1}{3}}{x}\right)}}{x}}{n} \]

      13. lower-/.f64 N/A

          \[\leadsto \frac{\frac{1 + \frac{\left(\frac{-1}{8} + \color{blue}{\frac{{\frac{1}{3}}^{3}}{{x}^{3}}}\right) \cdot \frac{1}{x}}{\frac{-1}{2} \cdot \frac{-1}{2} + \left(\frac{\frac{1}{3}}{x} \cdot \frac{\frac{1}{3}}{x} - \frac{-1}{2} \cdot \frac{\frac{1}{3}}{x}\right)}}{x}}{n} \]

      14. metadata-eval N/A

          \[\leadsto \frac{\frac{1 + \frac{\left(\frac{-1}{8} + \frac{\color{blue}{\frac{1}{27}}}{{x}^{3}}\right) \cdot \frac{1}{x}}{\frac{-1}{2} \cdot \frac{-1}{2} + \left(\frac{\frac{1}{3}}{x} \cdot \frac{\frac{1}{3}}{x} - \frac{-1}{2} \cdot \frac{\frac{1}{3}}{x}\right)}}{x}}{n} \]

      15. cube-mult N/A

          \[\leadsto \frac{\frac{1 + \frac{\left(\frac{-1}{8} + \frac{\frac{1}{27}}{\color{blue}{x \cdot \left(x \cdot x\right)}}\right) \cdot \frac{1}{x}}{\frac{-1}{2} \cdot \frac{-1}{2} + \left(\frac{\frac{1}{3}}{x} \cdot \frac{\frac{1}{3}}{x} - \frac{-1}{2} \cdot \frac{\frac{1}{3}}{x}\right)}}{x}}{n} \]

      16. lower-*.f64 N/A

          \[\leadsto \frac{\frac{1 + \frac{\left(\frac{-1}{8} + \frac{\frac{1}{27}}{\color{blue}{x \cdot \left(x \cdot x\right)}}\right) \cdot \frac{1}{x}}{\frac{-1}{2} \cdot \frac{-1}{2} + \left(\frac{\frac{1}{3}}{x} \cdot \frac{\frac{1}{3}}{x} - \frac{-1}{2} \cdot \frac{\frac{1}{3}}{x}\right)}}{x}}{n} \]

      17. lower-*.f64 N/A

          \[\leadsto \frac{\frac{1 + \frac{\left(\frac{-1}{8} + \frac{\frac{1}{27}}{x \cdot \color{blue}{\left(x \cdot x\right)}}\right) \cdot \frac{1}{x}}{\frac{-1}{2} \cdot \frac{-1}{2} + \left(\frac{\frac{1}{3}}{x} \cdot \frac{\frac{1}{3}}{x} - \frac{-1}{2} \cdot \frac{\frac{1}{3}}{x}\right)}}{x}}{n} \]

      18. lower-/.f64 N/A

          \[\leadsto \frac{\frac{1 + \frac{\left(\frac{-1}{8} + \frac{\frac{1}{27}}{x \cdot \left(x \cdot x\right)}\right) \cdot \color{blue}{\frac{1}{x}}}{\frac{-1}{2} \cdot \frac{-1}{2} + \left(\frac{\frac{1}{3}}{x} \cdot \frac{\frac{1}{3}}{x} - \frac{-1}{2} \cdot \frac{\frac{1}{3}}{x}\right)}}{x}}{n} \]

   10. Applied rewrites 51.8%

       \[\leadsto \frac{\frac{1 + \color{blue}{\frac{\left(-0.125 + \frac{0.037037037037037035}{x \cdot \left(x \cdot x\right)}\right) \cdot \frac{1}{x}}{\mathsf{fma}\left(\frac{0.3333333333333333}{x}, \frac{0.3333333333333333}{x} - -0.5, 0.25\right)}}}{x}}{n} \]

   if -1.18e-4 < n < 5e-238

   1. Initial program 91.0%

      \[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)} \]
   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 47.5

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

   5. Applied rewrites 47.5%

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

   6. Taylor expanded in x around inf

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

      1. lower-/.f64 N/A

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

   8. Applied rewrites 47.9%

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

   9. Taylor expanded in x around 0

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

       1. lower-/.f64 N/A

          \[\leadsto \frac{\color{blue}{\frac{\frac{1}{3}}{{x}^{3}}}}{n} \]

       2. cube-mult N/A

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

       3. unpow2 N/A

          \[\leadsto \frac{\frac{\frac{1}{3}}{x \cdot \color{blue}{{x}^{2}}}}{n} \]

       4. lower-*.f64 N/A

          \[\leadsto \frac{\frac{\frac{1}{3}}{\color{blue}{x \cdot {x}^{2}}}}{n} \]

       5. unpow2 N/A

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

       6. lower-*.f64 79.3

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

   11. Applied rewrites 79.3%

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

   if 5e-238 < n < 36

   1. Initial program 68.9%

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

   3. Taylor expanded in x around 0

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

      1. remove-double-neg N/A

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

      2. mul-1-neg N/A

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

      3. distribute-neg-frac N/A

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

      4. mul-1-neg N/A

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

      5. log-rec N/A

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

      6. mul-1-neg N/A

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

      7. lower--.f64 N/A

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

      8. log-rec N/A

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

      9. mul-1-neg N/A

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

      10. associate-*r/ N/A

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

      11. associate-*r* N/A

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

      12. metadata-eval N/A

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

      13. *-commutative N/A

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

      14. associate-/l* N/A

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

      15. exp-to-pow N/A

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

      16. lower-pow.f64 N/A

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

      17. lower-/.f64 62.5

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

   5. Applied rewrites 62.5%

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

   if 9.5000000000000006e175 < n

   1. Initial program 42.0%

      \[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)} \]
   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 81.4

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

   5. Applied rewrites 81.4%

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

   6. Taylor expanded in x around inf

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

   8. Applied rewrites 60.7%

      \[\leadsto \color{blue}{\frac{\frac{1}{n} + \left(\frac{0.3333333333333333}{x \cdot \left(x \cdot n\right)} + \frac{-0.5}{x \cdot n}\right)}{x}} \]
3. Recombined 5 regimes into one program.

4. Final simplification 69.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq -2.8 \cdot 10^{+70}:\\ \;\;\;\;\frac{x - \log x}{n}\\ \mathbf{elif}\;n \leq -0.000118:\\ \;\;\;\;\frac{\frac{1 + \frac{\left(-0.125 + \frac{0.037037037037037035}{x \cdot \left(x \cdot x\right)}\right) \cdot \frac{1}{x}}{\mathsf{fma}\left(\frac{0.3333333333333333}{x}, \frac{0.3333333333333333}{x} - -0.5, 0.25\right)}}{x}}{n}\\ \mathbf{elif}\;n \leq 5 \cdot 10^{-238}:\\ \;\;\;\;\frac{\frac{0.3333333333333333}{x \cdot \left(x \cdot x\right)}}{n}\\ \mathbf{elif}\;n \leq 36:\\ \;\;\;\;1 - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{elif}\;n \leq 9.5 \cdot 10^{+175}:\\ \;\;\;\;\frac{x - \log x}{n}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{n} + \left(\frac{0.3333333333333333}{x \cdot \left(n \cdot x\right)} + \frac{-0.5}{n \cdot x}\right)}{x}\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 56.8% accurate, 2.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{1}{n} \leq -2000:\\ \;\;\;\;\frac{\frac{0.3333333333333333}{x \cdot \left(x \cdot x\right)}}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 0.05:\\ \;\;\;\;\frac{\frac{1}{x}}{n}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x, \mathsf{fma}\left(x, \frac{\mathsf{fma}\left(x, \mathsf{fma}\left(n, -0.5, 0.16666666666666666\right), n \cdot 0.5\right)}{n \cdot \left(n \cdot n\right)}, \frac{1}{n}\right), 1 + -1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (if (<= (/ 1.0 n) -2000.0)
   (/ (/ 0.3333333333333333 (* x (* x x))) n)
   (if (<= (/ 1.0 n) 0.05)
     (/ (/ 1.0 x) n)
     (fma
      x
      (fma
       x
       (/ (fma x (fma n -0.5 0.16666666666666666) (* n 0.5)) (* n (* n n)))
       (/ 1.0 n))
      (+ 1.0 -1.0)))))

C

double code(double x, double n) {
	double tmp;
	if ((1.0 / n) <= -2000.0) {
		tmp = (0.3333333333333333 / (x * (x * x))) / n;
	} else if ((1.0 / n) <= 0.05) {
		tmp = (1.0 / x) / n;
	} else {
		tmp = fma(x, fma(x, (fma(x, fma(n, -0.5, 0.16666666666666666), (n * 0.5)) / (n * (n * n))), (1.0 / n)), (1.0 + -1.0));
	}
	return tmp;
}

Julia

function code(x, n)
	tmp = 0.0
	if (Float64(1.0 / n) <= -2000.0)
		tmp = Float64(Float64(0.3333333333333333 / Float64(x * Float64(x * x))) / n);
	elseif (Float64(1.0 / n) <= 0.05)
		tmp = Float64(Float64(1.0 / x) / n);
	else
		tmp = fma(x, fma(x, Float64(fma(x, fma(n, -0.5, 0.16666666666666666), Float64(n * 0.5)) / Float64(n * Float64(n * n))), Float64(1.0 / n)), Float64(1.0 + -1.0));
	end
	return tmp
end

Wolfram

code[x_, n_] := If[LessEqual[N[(1.0 / n), $MachinePrecision], -2000.0], N[(N[(0.3333333333333333 / N[(x * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / n), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 0.05], N[(N[(1.0 / x), $MachinePrecision] / n), $MachinePrecision], N[(x * N[(x * N[(N[(x * N[(n * -0.5 + 0.16666666666666666), $MachinePrecision] + N[(n * 0.5), $MachinePrecision]), $MachinePrecision] / N[(n * N[(n * n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(1.0 / n), $MachinePrecision]), $MachinePrecision] + N[(1.0 + -1.0), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 #s(literal 1 binary64) n) < -2e3

   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 50.3

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

   5. Applied rewrites 50.3%

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

   6. Taylor expanded in x around inf

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

      1. lower-/.f64 N/A

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

   8. Applied rewrites 41.8%

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

   9. Taylor expanded in x around 0

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

       1. lower-/.f64 N/A

          \[\leadsto \frac{\color{blue}{\frac{\frac{1}{3}}{{x}^{3}}}}{n} \]

       2. cube-mult N/A

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

       3. unpow2 N/A

          \[\leadsto \frac{\frac{\frac{1}{3}}{x \cdot \color{blue}{{x}^{2}}}}{n} \]

       4. lower-*.f64 N/A

          \[\leadsto \frac{\frac{\frac{1}{3}}{\color{blue}{x \cdot {x}^{2}}}}{n} \]

       5. unpow2 N/A

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

       6. lower-*.f64 75.5

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

   11. Applied rewrites 75.5%

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

   if -2e3 < (/.f64 #s(literal 1 binary64) n) < 0.050000000000000003

   1. Initial program 23.6%

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

   3. Taylor expanded in n around inf

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

      1. lower-/.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lower-log1p.f64 N/A

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

      4. lower-log.f64 76.7

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

   5. Applied rewrites 76.7%

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

   6. Taylor expanded in x around inf

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

      1. lower-/.f64 40.0

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

   8. Applied rewrites 40.0%

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

   if 0.050000000000000003 < (/.f64 #s(literal 1 binary64) n)

   1. Initial program 55.1%

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

   3. Taylor expanded in x around 0

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

   5. Applied rewrites 29.6%

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

   6. Taylor expanded in n around inf

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

   8. Applied rewrites 14.9%

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

   9. Taylor expanded in n around 0

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

       1. lower-/.f64 N/A

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

       2. +-commutative N/A

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

       3. distribute-lft-in N/A

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

       4. *-commutative N/A

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

       5. associate-+l+ N/A

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

       6. associate-*r* N/A

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

       7. *-commutative N/A

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

       8. distribute-rgt-in N/A

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

       9. +-commutative N/A

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

       10. +-commutative N/A

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

       11. lower-fma.f64 N/A

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

       12. +-commutative N/A

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

       13. *-commutative N/A

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

       14. lower-fma.f64 N/A

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

       15. *-commutative N/A

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

       16. lower-*.f64 N/A

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

       17. cube-mult N/A

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

       18. unpow2 N/A

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

       19. lower-*.f64 N/A

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

       20. unpow2 N/A

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

       21. lower-*.f64 47.1

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

   11. Applied rewrites 47.1%

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

4. Final simplification 52.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{1}{n} \leq -2000:\\ \;\;\;\;\frac{\frac{0.3333333333333333}{x \cdot \left(x \cdot x\right)}}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 0.05:\\ \;\;\;\;\frac{\frac{1}{x}}{n}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x, \mathsf{fma}\left(x, \frac{\mathsf{fma}\left(x, \mathsf{fma}\left(n, -0.5, 0.16666666666666666\right), n \cdot 0.5\right)}{n \cdot \left(n \cdot n\right)}, \frac{1}{n}\right), 1 + -1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 51.9% accurate, 3.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{0.3333333333333333}{x \cdot \left(x \cdot \left(n \cdot x\right)\right)}\\ \mathbf{if}\;\frac{1}{n} \leq -5 \cdot 10^{+93}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;\frac{1}{n} \leq -1000000000:\\ \;\;\;\;0\\ \mathbf{elif}\;\frac{1}{n} \leq 10^{+26}:\\ \;\;\;\;\frac{\frac{1}{x}}{n}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (let* ((t_0 (/ 0.3333333333333333 (* x (* x (* n x))))))
   (if (<= (/ 1.0 n) -5e+93)
     t_0
     (if (<= (/ 1.0 n) -1000000000.0)
       0.0
       (if (<= (/ 1.0 n) 1e+26) (/ (/ 1.0 x) n) t_0)))))

C

double code(double x, double n) {
	double t_0 = 0.3333333333333333 / (x * (x * (n * x)));
	double tmp;
	if ((1.0 / n) <= -5e+93) {
		tmp = t_0;
	} else if ((1.0 / n) <= -1000000000.0) {
		tmp = 0.0;
	} else if ((1.0 / n) <= 1e+26) {
		tmp = (1.0 / x) / n;
	} 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 = 0.3333333333333333d0 / (x * (x * (n * x)))
    if ((1.0d0 / n) <= (-5d+93)) then
        tmp = t_0
    else if ((1.0d0 / n) <= (-1000000000.0d0)) then
        tmp = 0.0d0
    else if ((1.0d0 / n) <= 1d+26) then
        tmp = (1.0d0 / x) / n
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double x, double n) {
	double t_0 = 0.3333333333333333 / (x * (x * (n * x)));
	double tmp;
	if ((1.0 / n) <= -5e+93) {
		tmp = t_0;
	} else if ((1.0 / n) <= -1000000000.0) {
		tmp = 0.0;
	} else if ((1.0 / n) <= 1e+26) {
		tmp = (1.0 / x) / n;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, n):
	t_0 = 0.3333333333333333 / (x * (x * (n * x)))
	tmp = 0
	if (1.0 / n) <= -5e+93:
		tmp = t_0
	elif (1.0 / n) <= -1000000000.0:
		tmp = 0.0
	elif (1.0 / n) <= 1e+26:
		tmp = (1.0 / x) / n
	else:
		tmp = t_0
	return tmp

Julia

function code(x, n)
	t_0 = Float64(0.3333333333333333 / Float64(x * Float64(x * Float64(n * x))))
	tmp = 0.0
	if (Float64(1.0 / n) <= -5e+93)
		tmp = t_0;
	elseif (Float64(1.0 / n) <= -1000000000.0)
		tmp = 0.0;
	elseif (Float64(1.0 / n) <= 1e+26)
		tmp = Float64(Float64(1.0 / x) / n);
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, n)
	t_0 = 0.3333333333333333 / (x * (x * (n * x)));
	tmp = 0.0;
	if ((1.0 / n) <= -5e+93)
		tmp = t_0;
	elseif ((1.0 / n) <= -1000000000.0)
		tmp = 0.0;
	elseif ((1.0 / n) <= 1e+26)
		tmp = (1.0 / x) / n;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, n_] := Block[{t$95$0 = N[(0.3333333333333333 / N[(x * N[(x * N[(n * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(1.0 / n), $MachinePrecision], -5e+93], t$95$0, If[LessEqual[N[(1.0 / n), $MachinePrecision], -1000000000.0], 0.0, If[LessEqual[N[(1.0 / n), $MachinePrecision], 1e+26], N[(N[(1.0 / x), $MachinePrecision] / n), $MachinePrecision], t$95$0]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 #s(literal 1 binary64) n) < -5.0000000000000001e93 or 1.00000000000000005e26 < (/.f64 #s(literal 1 binary64) n)

   1. Initial program 81.5%

      \[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)} \]
   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 25.8

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

   5. Applied rewrites 25.8%

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

   6. Taylor expanded in x around inf

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

      1. lower-/.f64 N/A

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

   8. Applied rewrites 49.4%

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

   9. Taylor expanded in x around 0

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

       1. lower-/.f64 N/A

          \[\leadsto \color{blue}{\frac{\frac{1}{3}}{n \cdot {x}^{3}}} \]

       2. unpow3 N/A

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

       3. unpow2 N/A

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

       4. associate-*r* N/A

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

       5. *-commutative N/A

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

       6. lower-*.f64 N/A

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

       7. unpow2 N/A

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

       8. associate-*r* N/A

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

       9. *-commutative N/A

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

       10. lower-*.f64 N/A

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

       11. *-commutative N/A

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

       12. lower-*.f64 59.4

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

   11. Applied rewrites 59.4%

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

   if -5.0000000000000001e93 < (/.f64 #s(literal 1 binary64) n) < -1e9

   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

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

      1. remove-double-neg N/A

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

      2. mul-1-neg N/A

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

      3. distribute-neg-frac N/A

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

      4. mul-1-neg N/A

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

      5. log-rec N/A

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

      6. mul-1-neg N/A

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

      7. lower--.f64 N/A

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

      8. log-rec N/A

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

      9. mul-1-neg N/A

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

      10. associate-*r/ N/A

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

      11. associate-*r* N/A

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

      12. metadata-eval N/A

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

      13. *-commutative N/A

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

      14. associate-/l* N/A

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

      15. exp-to-pow N/A

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

      16. lower-pow.f64 N/A

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

      17. lower-/.f64 23.3

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

   5. Applied rewrites 23.3%

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

   6. Taylor expanded in n around inf

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

   8. Applied rewrites 79.5%

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

      1. metadata-eval 79.5

         \[\leadsto \color{blue}{0} \]

   10. Applied rewrites 79.5%

       \[\leadsto \color{blue}{0} \]

   if -1e9 < (/.f64 #s(literal 1 binary64) n) < 1.00000000000000005e26

   1. Initial program 25.8%

      \[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)} \]
   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 74.3

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

   5. Applied rewrites 74.3%

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

   6. Taylor expanded in x around inf

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

      1. lower-/.f64 38.8

         \[\leadsto \frac{\color{blue}{\frac{1}{x}}}{n} \]

   8. Applied rewrites 38.8%

      \[\leadsto \frac{\color{blue}{\frac{1}{x}}}{n} \]
3. Recombined 3 regimes into one program.

4. Final simplification 49.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{1}{n} \leq -5 \cdot 10^{+93}:\\ \;\;\;\;\frac{0.3333333333333333}{x \cdot \left(x \cdot \left(n \cdot x\right)\right)}\\ \mathbf{elif}\;\frac{1}{n} \leq -1000000000:\\ \;\;\;\;0\\ \mathbf{elif}\;\frac{1}{n} \leq 10^{+26}:\\ \;\;\;\;\frac{\frac{1}{x}}{n}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.3333333333333333}{x \cdot \left(x \cdot \left(n \cdot x\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 54.7% accurate, 3.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{1}{n} \leq -2000:\\ \;\;\;\;\frac{\frac{0.3333333333333333}{x \cdot \left(x \cdot x\right)}}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 2000000000:\\ \;\;\;\;\frac{\frac{1}{x}}{n}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x, \frac{\left(x \cdot x\right) \cdot 0.16666666666666666}{n \cdot \left(n \cdot n\right)}, 1 + -1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (if (<= (/ 1.0 n) -2000.0)
   (/ (/ 0.3333333333333333 (* x (* x x))) n)
   (if (<= (/ 1.0 n) 2000000000.0)
     (/ (/ 1.0 x) n)
     (fma x (/ (* (* x x) 0.16666666666666666) (* n (* n n))) (+ 1.0 -1.0)))))

C

double code(double x, double n) {
	double tmp;
	if ((1.0 / n) <= -2000.0) {
		tmp = (0.3333333333333333 / (x * (x * x))) / n;
	} else if ((1.0 / n) <= 2000000000.0) {
		tmp = (1.0 / x) / n;
	} else {
		tmp = fma(x, (((x * x) * 0.16666666666666666) / (n * (n * n))), (1.0 + -1.0));
	}
	return tmp;
}

Julia

function code(x, n)
	tmp = 0.0
	if (Float64(1.0 / n) <= -2000.0)
		tmp = Float64(Float64(0.3333333333333333 / Float64(x * Float64(x * x))) / n);
	elseif (Float64(1.0 / n) <= 2000000000.0)
		tmp = Float64(Float64(1.0 / x) / n);
	else
		tmp = fma(x, Float64(Float64(Float64(x * x) * 0.16666666666666666) / Float64(n * Float64(n * n))), Float64(1.0 + -1.0));
	end
	return tmp
end

Wolfram

code[x_, n_] := If[LessEqual[N[(1.0 / n), $MachinePrecision], -2000.0], N[(N[(0.3333333333333333 / N[(x * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / n), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 2000000000.0], N[(N[(1.0 / x), $MachinePrecision] / n), $MachinePrecision], N[(x * N[(N[(N[(x * x), $MachinePrecision] * 0.16666666666666666), $MachinePrecision] / N[(n * N[(n * n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(1.0 + -1.0), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 #s(literal 1 binary64) n) < -2e3

   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 50.3

         \[\leadsto \frac{\mathsf{log1p}\left(x\right) - \color{blue}{\log x}}{n} \]

   5. Applied rewrites 50.3%

      \[\leadsto \color{blue}{\frac{\mathsf{log1p}\left(x\right) - \log x}{n}} \]

   6. Taylor expanded in x around inf

      \[\leadsto \frac{\color{blue}{\frac{\left(1 + \frac{\frac{1}{3}}{{x}^{2}}\right) - \frac{1}{2} \cdot \frac{1}{x}}{x}}}{n} \]
   7. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{\color{blue}{\frac{\left(1 + \frac{\frac{1}{3}}{{x}^{2}}\right) - \frac{1}{2} \cdot \frac{1}{x}}{x}}}{n} \]

   8. Applied rewrites 41.8%

      \[\leadsto \frac{\color{blue}{\frac{1 + \frac{-0.5 + \frac{0.3333333333333333}{x}}{x}}{x}}}{n} \]

   9. Taylor expanded in x around 0

      \[\leadsto \frac{\color{blue}{\frac{\frac{1}{3}}{{x}^{3}}}}{n} \]
   10. Step-by-step derivation

       1. lower-/.f64 N/A

          \[\leadsto \frac{\color{blue}{\frac{\frac{1}{3}}{{x}^{3}}}}{n} \]

       2. cube-mult N/A

          \[\leadsto \frac{\frac{\frac{1}{3}}{\color{blue}{x \cdot \left(x \cdot x\right)}}}{n} \]

       3. unpow2 N/A

          \[\leadsto \frac{\frac{\frac{1}{3}}{x \cdot \color{blue}{{x}^{2}}}}{n} \]

       4. lower-*.f64 N/A

          \[\leadsto \frac{\frac{\frac{1}{3}}{\color{blue}{x \cdot {x}^{2}}}}{n} \]

       5. unpow2 N/A

          \[\leadsto \frac{\frac{\frac{1}{3}}{x \cdot \color{blue}{\left(x \cdot x\right)}}}{n} \]

       6. lower-*.f64 75.5

          \[\leadsto \frac{\frac{0.3333333333333333}{x \cdot \color{blue}{\left(x \cdot x\right)}}}{n} \]

   11. Applied rewrites 75.5%

       \[\leadsto \frac{\color{blue}{\frac{0.3333333333333333}{x \cdot \left(x \cdot x\right)}}}{n} \]

   if -2e3 < (/.f64 #s(literal 1 binary64) n) < 2e9

   1. Initial program 24.1%

      \[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in n around inf

      \[\leadsto \color{blue}{\frac{\log \left(1 + x\right) - \log x}{n}} \]
   4. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\log \left(1 + x\right) - \log x}{n}} \]

      2. lower--.f64 N/A

         \[\leadsto \frac{\color{blue}{\log \left(1 + x\right) - \log x}}{n} \]

      3. lower-log1p.f64 N/A

         \[\leadsto \frac{\color{blue}{\mathsf{log1p}\left(x\right)} - \log x}{n} \]

      4. lower-log.f64 76.2

         \[\leadsto \frac{\mathsf{log1p}\left(x\right) - \color{blue}{\log x}}{n} \]

   5. Applied rewrites 76.2%

      \[\leadsto \color{blue}{\frac{\mathsf{log1p}\left(x\right) - \log x}{n}} \]

   6. Taylor expanded in x around inf

      \[\leadsto \frac{\color{blue}{\frac{1}{x}}}{n} \]
   7. Step-by-step derivation

      1. lower-/.f64 39.8

         \[\leadsto \frac{\color{blue}{\frac{1}{x}}}{n} \]

   8. Applied rewrites 39.8%

      \[\leadsto \frac{\color{blue}{\frac{1}{x}}}{n} \]

   if 2e9 < (/.f64 #s(literal 1 binary64) n)

   1. Initial program 53.9%

      \[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\left(1 + x \cdot \left(x \cdot \left(\left(\frac{1}{2} \cdot \frac{1}{{n}^{2}} + x \cdot \left(\left(\frac{1}{6} \cdot \frac{1}{{n}^{3}} + \frac{1}{3} \cdot \frac{1}{n}\right) - \frac{1}{2} \cdot \frac{1}{{n}^{2}}\right)\right) - \frac{1}{2} \cdot \frac{1}{n}\right) + \frac{1}{n}\right)\right) - e^{\frac{\log x}{n}}} \]

   5. Applied rewrites 27.7%

      \[\leadsto \color{blue}{\mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, \frac{0.16666666666666666}{n \cdot \left(n \cdot n\right)} + \left(\frac{0.3333333333333333}{n} + \frac{-0.5}{n \cdot n}\right), \frac{0.5}{n \cdot n} + \frac{-0.5}{n}\right), \frac{1}{n}\right), 1 - {x}^{\left(\frac{1}{n}\right)}\right)} \]

   6. Taylor expanded in n around inf

      \[\leadsto \mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, \frac{\frac{1}{6}}{n \cdot \left(n \cdot n\right)} + \left(\frac{\frac{1}{3}}{n} + \frac{\frac{-1}{2}}{n \cdot n}\right), \frac{\frac{1}{2}}{n \cdot n} + \frac{\frac{-1}{2}}{n}\right), \frac{1}{n}\right), 1 - \color{blue}{1}\right) \]
   7. Step-by-step derivation

   8. Applied rewrites 15.1%

      \[\leadsto \mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, \frac{0.16666666666666666}{n \cdot \left(n \cdot n\right)} + \left(\frac{0.3333333333333333}{n} + \frac{-0.5}{n \cdot n}\right), \frac{0.5}{n \cdot n} + \frac{-0.5}{n}\right), \frac{1}{n}\right), 1 - \color{blue}{1}\right) \]

   9. Taylor expanded in n around 0

      \[\leadsto \mathsf{fma}\left(x, \color{blue}{\frac{1}{6} \cdot \frac{{x}^{2}}{{n}^{3}}}, 1 - 1\right) \]
   10. Step-by-step derivation

       1. associate-*r/ N/A

          \[\leadsto \mathsf{fma}\left(x, \color{blue}{\frac{\frac{1}{6} \cdot {x}^{2}}{{n}^{3}}}, 1 - 1\right) \]

       2. lower-/.f64 N/A

          \[\leadsto \mathsf{fma}\left(x, \color{blue}{\frac{\frac{1}{6} \cdot {x}^{2}}{{n}^{3}}}, 1 - 1\right) \]

       3. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(x, \frac{\color{blue}{{x}^{2} \cdot \frac{1}{6}}}{{n}^{3}}, 1 - 1\right) \]

       4. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(x, \frac{\color{blue}{{x}^{2} \cdot \frac{1}{6}}}{{n}^{3}}, 1 - 1\right) \]

       5. unpow2 N/A

          \[\leadsto \mathsf{fma}\left(x, \frac{\color{blue}{\left(x \cdot x\right)} \cdot \frac{1}{6}}{{n}^{3}}, 1 - 1\right) \]

       6. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(x, \frac{\color{blue}{\left(x \cdot x\right)} \cdot \frac{1}{6}}{{n}^{3}}, 1 - 1\right) \]

       7. cube-mult N/A

          \[\leadsto \mathsf{fma}\left(x, \frac{\left(x \cdot x\right) \cdot \frac{1}{6}}{\color{blue}{n \cdot \left(n \cdot n\right)}}, 1 - 1\right) \]

       8. unpow2 N/A

          \[\leadsto \mathsf{fma}\left(x, \frac{\left(x \cdot x\right) \cdot \frac{1}{6}}{n \cdot \color{blue}{{n}^{2}}}, 1 - 1\right) \]

       9. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(x, \frac{\left(x \cdot x\right) \cdot \frac{1}{6}}{\color{blue}{n \cdot {n}^{2}}}, 1 - 1\right) \]

       10. unpow2 N/A

           \[\leadsto \mathsf{fma}\left(x, \frac{\left(x \cdot x\right) \cdot \frac{1}{6}}{n \cdot \color{blue}{\left(n \cdot n\right)}}, 1 - 1\right) \]

       11. lower-*.f64 41.4

           \[\leadsto \mathsf{fma}\left(x, \frac{\left(x \cdot x\right) \cdot 0.16666666666666666}{n \cdot \color{blue}{\left(n \cdot n\right)}}, 1 - 1\right) \]

   11. Applied rewrites 41.4%

       \[\leadsto \mathsf{fma}\left(x, \color{blue}{\frac{\left(x \cdot x\right) \cdot 0.16666666666666666}{n \cdot \left(n \cdot n\right)}}, 1 - 1\right) \]
3. Recombined 3 regimes into one program.

4. Final simplification 51.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{1}{n} \leq -2000:\\ \;\;\;\;\frac{\frac{0.3333333333333333}{x \cdot \left(x \cdot x\right)}}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 2000000000:\\ \;\;\;\;\frac{\frac{1}{x}}{n}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x, \frac{\left(x \cdot x\right) \cdot 0.16666666666666666}{n \cdot \left(n \cdot n\right)}, 1 + -1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 53.0% accurate, 3.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{1}{n} \leq -2000:\\ \;\;\;\;\frac{\frac{0.3333333333333333}{x \cdot \left(x \cdot x\right)}}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 2000000000:\\ \;\;\;\;\frac{\frac{1}{x}}{n}\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot \left(\left(x \cdot x\right) \cdot 0.16666666666666666\right)}{n \cdot \left(n \cdot n\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (if (<= (/ 1.0 n) -2000.0)
   (/ (/ 0.3333333333333333 (* x (* x x))) n)
   (if (<= (/ 1.0 n) 2000000000.0)
     (/ (/ 1.0 x) n)
     (/ (* x (* (* x x) 0.16666666666666666)) (* n (* n n))))))

C

double code(double x, double n) {
	double tmp;
	if ((1.0 / n) <= -2000.0) {
		tmp = (0.3333333333333333 / (x * (x * x))) / n;
	} else if ((1.0 / n) <= 2000000000.0) {
		tmp = (1.0 / x) / n;
	} else {
		tmp = (x * ((x * x) * 0.16666666666666666)) / (n * (n * n));
	}
	return tmp;
}

Fortran

real(8) function code(x, n)
    real(8), intent (in) :: x
    real(8), intent (in) :: n
    real(8) :: tmp
    if ((1.0d0 / n) <= (-2000.0d0)) then
        tmp = (0.3333333333333333d0 / (x * (x * x))) / n
    else if ((1.0d0 / n) <= 2000000000.0d0) then
        tmp = (1.0d0 / x) / n
    else
        tmp = (x * ((x * x) * 0.16666666666666666d0)) / (n * (n * n))
    end if
    code = tmp
end function

Java

public static double code(double x, double n) {
	double tmp;
	if ((1.0 / n) <= -2000.0) {
		tmp = (0.3333333333333333 / (x * (x * x))) / n;
	} else if ((1.0 / n) <= 2000000000.0) {
		tmp = (1.0 / x) / n;
	} else {
		tmp = (x * ((x * x) * 0.16666666666666666)) / (n * (n * n));
	}
	return tmp;
}

Python

def code(x, n):
	tmp = 0
	if (1.0 / n) <= -2000.0:
		tmp = (0.3333333333333333 / (x * (x * x))) / n
	elif (1.0 / n) <= 2000000000.0:
		tmp = (1.0 / x) / n
	else:
		tmp = (x * ((x * x) * 0.16666666666666666)) / (n * (n * n))
	return tmp

Julia

function code(x, n)
	tmp = 0.0
	if (Float64(1.0 / n) <= -2000.0)
		tmp = Float64(Float64(0.3333333333333333 / Float64(x * Float64(x * x))) / n);
	elseif (Float64(1.0 / n) <= 2000000000.0)
		tmp = Float64(Float64(1.0 / x) / n);
	else
		tmp = Float64(Float64(x * Float64(Float64(x * x) * 0.16666666666666666)) / Float64(n * Float64(n * n)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, n)
	tmp = 0.0;
	if ((1.0 / n) <= -2000.0)
		tmp = (0.3333333333333333 / (x * (x * x))) / n;
	elseif ((1.0 / n) <= 2000000000.0)
		tmp = (1.0 / x) / n;
	else
		tmp = (x * ((x * x) * 0.16666666666666666)) / (n * (n * n));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, n_] := If[LessEqual[N[(1.0 / n), $MachinePrecision], -2000.0], N[(N[(0.3333333333333333 / N[(x * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / n), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 2000000000.0], N[(N[(1.0 / x), $MachinePrecision] / n), $MachinePrecision], N[(N[(x * N[(N[(x * x), $MachinePrecision] * 0.16666666666666666), $MachinePrecision]), $MachinePrecision] / N[(n * N[(n * n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 #s(literal 1 binary64) n) < -2e3

   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 50.3

         \[\leadsto \frac{\mathsf{log1p}\left(x\right) - \color{blue}{\log x}}{n} \]

   5. Applied rewrites 50.3%

      \[\leadsto \color{blue}{\frac{\mathsf{log1p}\left(x\right) - \log x}{n}} \]

   6. Taylor expanded in x around inf

      \[\leadsto \frac{\color{blue}{\frac{\left(1 + \frac{\frac{1}{3}}{{x}^{2}}\right) - \frac{1}{2} \cdot \frac{1}{x}}{x}}}{n} \]
   7. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{\color{blue}{\frac{\left(1 + \frac{\frac{1}{3}}{{x}^{2}}\right) - \frac{1}{2} \cdot \frac{1}{x}}{x}}}{n} \]

   8. Applied rewrites 41.8%

      \[\leadsto \frac{\color{blue}{\frac{1 + \frac{-0.5 + \frac{0.3333333333333333}{x}}{x}}{x}}}{n} \]

   9. Taylor expanded in x around 0

      \[\leadsto \frac{\color{blue}{\frac{\frac{1}{3}}{{x}^{3}}}}{n} \]
   10. Step-by-step derivation

       1. lower-/.f64 N/A

          \[\leadsto \frac{\color{blue}{\frac{\frac{1}{3}}{{x}^{3}}}}{n} \]

       2. cube-mult N/A

          \[\leadsto \frac{\frac{\frac{1}{3}}{\color{blue}{x \cdot \left(x \cdot x\right)}}}{n} \]

       3. unpow2 N/A

          \[\leadsto \frac{\frac{\frac{1}{3}}{x \cdot \color{blue}{{x}^{2}}}}{n} \]

       4. lower-*.f64 N/A

          \[\leadsto \frac{\frac{\frac{1}{3}}{\color{blue}{x \cdot {x}^{2}}}}{n} \]

       5. unpow2 N/A

          \[\leadsto \frac{\frac{\frac{1}{3}}{x \cdot \color{blue}{\left(x \cdot x\right)}}}{n} \]

       6. lower-*.f64 75.5

          \[\leadsto \frac{\frac{0.3333333333333333}{x \cdot \color{blue}{\left(x \cdot x\right)}}}{n} \]

   11. Applied rewrites 75.5%

       \[\leadsto \frac{\color{blue}{\frac{0.3333333333333333}{x \cdot \left(x \cdot x\right)}}}{n} \]

   if -2e3 < (/.f64 #s(literal 1 binary64) n) < 2e9

   1. Initial program 24.1%

      \[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in n around inf

      \[\leadsto \color{blue}{\frac{\log \left(1 + x\right) - \log x}{n}} \]
   4. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\log \left(1 + x\right) - \log x}{n}} \]

      2. lower--.f64 N/A

         \[\leadsto \frac{\color{blue}{\log \left(1 + x\right) - \log x}}{n} \]

      3. lower-log1p.f64 N/A

         \[\leadsto \frac{\color{blue}{\mathsf{log1p}\left(x\right)} - \log x}{n} \]

      4. lower-log.f64 76.2

         \[\leadsto \frac{\mathsf{log1p}\left(x\right) - \color{blue}{\log x}}{n} \]

   5. Applied rewrites 76.2%

      \[\leadsto \color{blue}{\frac{\mathsf{log1p}\left(x\right) - \log x}{n}} \]

   6. Taylor expanded in x around inf

      \[\leadsto \frac{\color{blue}{\frac{1}{x}}}{n} \]
   7. Step-by-step derivation

      1. lower-/.f64 39.8

         \[\leadsto \frac{\color{blue}{\frac{1}{x}}}{n} \]

   8. Applied rewrites 39.8%

      \[\leadsto \frac{\color{blue}{\frac{1}{x}}}{n} \]

   if 2e9 < (/.f64 #s(literal 1 binary64) n)

   1. Initial program 53.9%

      \[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\left(1 + x \cdot \left(x \cdot \left(\left(\frac{1}{2} \cdot \frac{1}{{n}^{2}} + x \cdot \left(\left(\frac{1}{6} \cdot \frac{1}{{n}^{3}} + \frac{1}{3} \cdot \frac{1}{n}\right) - \frac{1}{2} \cdot \frac{1}{{n}^{2}}\right)\right) - \frac{1}{2} \cdot \frac{1}{n}\right) + \frac{1}{n}\right)\right) - e^{\frac{\log x}{n}}} \]

   5. Applied rewrites 27.7%

      \[\leadsto \color{blue}{\mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, \frac{0.16666666666666666}{n \cdot \left(n \cdot n\right)} + \left(\frac{0.3333333333333333}{n} + \frac{-0.5}{n \cdot n}\right), \frac{0.5}{n \cdot n} + \frac{-0.5}{n}\right), \frac{1}{n}\right), 1 - {x}^{\left(\frac{1}{n}\right)}\right)} \]

   6. Taylor expanded in n around inf

      \[\leadsto \mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, \frac{\frac{1}{6}}{n \cdot \left(n \cdot n\right)} + \left(\frac{\frac{1}{3}}{n} + \frac{\frac{-1}{2}}{n \cdot n}\right), \frac{\frac{1}{2}}{n \cdot n} + \frac{\frac{-1}{2}}{n}\right), \frac{1}{n}\right), 1 - \color{blue}{1}\right) \]
   7. Step-by-step derivation

   8. Applied rewrites 15.1%

      \[\leadsto \mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, \frac{0.16666666666666666}{n \cdot \left(n \cdot n\right)} + \left(\frac{0.3333333333333333}{n} + \frac{-0.5}{n \cdot n}\right), \frac{0.5}{n \cdot n} + \frac{-0.5}{n}\right), \frac{1}{n}\right), 1 - \color{blue}{1}\right) \]

   9. Taylor expanded in n around 0

      \[\leadsto \color{blue}{\frac{1}{6} \cdot \frac{{x}^{3}}{{n}^{3}}} \]
   10. Step-by-step derivation

       1. associate-*r/ N/A

          \[\leadsto \color{blue}{\frac{\frac{1}{6} \cdot {x}^{3}}{{n}^{3}}} \]

       2. lower-/.f64 N/A

          \[\leadsto \color{blue}{\frac{\frac{1}{6} \cdot {x}^{3}}{{n}^{3}}} \]

       3. *-commutative N/A

          \[\leadsto \frac{\color{blue}{{x}^{3} \cdot \frac{1}{6}}}{{n}^{3}} \]

       4. cube-mult N/A

          \[\leadsto \frac{\color{blue}{\left(x \cdot \left(x \cdot x\right)\right)} \cdot \frac{1}{6}}{{n}^{3}} \]

       5. unpow2 N/A

          \[\leadsto \frac{\left(x \cdot \color{blue}{{x}^{2}}\right) \cdot \frac{1}{6}}{{n}^{3}} \]

       6. associate-*l* N/A

          \[\leadsto \frac{\color{blue}{x \cdot \left({x}^{2} \cdot \frac{1}{6}\right)}}{{n}^{3}} \]

       7. *-commutative N/A

          \[\leadsto \frac{x \cdot \color{blue}{\left(\frac{1}{6} \cdot {x}^{2}\right)}}{{n}^{3}} \]

       8. lower-*.f64 N/A

          \[\leadsto \frac{\color{blue}{x \cdot \left(\frac{1}{6} \cdot {x}^{2}\right)}}{{n}^{3}} \]

       9. *-commutative N/A

          \[\leadsto \frac{x \cdot \color{blue}{\left({x}^{2} \cdot \frac{1}{6}\right)}}{{n}^{3}} \]

       10. lower-*.f64 N/A

           \[\leadsto \frac{x \cdot \color{blue}{\left({x}^{2} \cdot \frac{1}{6}\right)}}{{n}^{3}} \]

       11. unpow2 N/A

           \[\leadsto \frac{x \cdot \left(\color{blue}{\left(x \cdot x\right)} \cdot \frac{1}{6}\right)}{{n}^{3}} \]

       12. lower-*.f64 N/A

           \[\leadsto \frac{x \cdot \left(\color{blue}{\left(x \cdot x\right)} \cdot \frac{1}{6}\right)}{{n}^{3}} \]

       13. cube-mult N/A

           \[\leadsto \frac{x \cdot \left(\left(x \cdot x\right) \cdot \frac{1}{6}\right)}{\color{blue}{n \cdot \left(n \cdot n\right)}} \]

       14. unpow2 N/A

           \[\leadsto \frac{x \cdot \left(\left(x \cdot x\right) \cdot \frac{1}{6}\right)}{n \cdot \color{blue}{{n}^{2}}} \]

       15. lower-*.f64 N/A

           \[\leadsto \frac{x \cdot \left(\left(x \cdot x\right) \cdot \frac{1}{6}\right)}{\color{blue}{n \cdot {n}^{2}}} \]

       16. unpow2 N/A

           \[\leadsto \frac{x \cdot \left(\left(x \cdot x\right) \cdot \frac{1}{6}\right)}{n \cdot \color{blue}{\left(n \cdot n\right)}} \]

       17. lower-*.f64 38.7

           \[\leadsto \frac{x \cdot \left(\left(x \cdot x\right) \cdot 0.16666666666666666\right)}{n \cdot \color{blue}{\left(n \cdot n\right)}} \]

   11. Applied rewrites 38.7%

       \[\leadsto \color{blue}{\frac{x \cdot \left(\left(x \cdot x\right) \cdot 0.16666666666666666\right)}{n \cdot \left(n \cdot n\right)}} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 17: 55.4% accurate, 3.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{1}{n} \leq -1000000000:\\ \;\;\;\;\frac{\frac{0.3333333333333333}{x \cdot \left(x \cdot x\right)}}{n}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 + \frac{-0.5 + \frac{0.3333333333333333}{x}}{x}}{n}}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (if (<= (/ 1.0 n) -1000000000.0)
   (/ (/ 0.3333333333333333 (* x (* x x))) n)
   (/ (/ (+ 1.0 (/ (+ -0.5 (/ 0.3333333333333333 x)) x)) n) x)))

C

double code(double x, double n) {
	double tmp;
	if ((1.0 / n) <= -1000000000.0) {
		tmp = (0.3333333333333333 / (x * (x * x))) / n;
	} 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) <= (-1000000000.0d0)) then
        tmp = (0.3333333333333333d0 / (x * (x * x))) / n
    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) <= -1000000000.0) {
		tmp = (0.3333333333333333 / (x * (x * x))) / n;
	} 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) <= -1000000000.0:
		tmp = (0.3333333333333333 / (x * (x * x))) / n
	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) <= -1000000000.0)
		tmp = Float64(Float64(0.3333333333333333 / Float64(x * Float64(x * x))) / n);
	else
		tmp = Float64(Float64(Float64(1.0 + Float64(Float64(-0.5 + Float64(0.3333333333333333 / x)) / x)) / n) / x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, n)
	tmp = 0.0;
	if ((1.0 / n) <= -1000000000.0)
		tmp = (0.3333333333333333 / (x * (x * x))) / n;
	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], -1000000000.0], N[(N[(0.3333333333333333 / N[(x * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / n), $MachinePrecision], N[(N[(N[(1.0 + N[(N[(-0.5 + N[(0.3333333333333333 / x), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision] / n), $MachinePrecision] / x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 #s(literal 1 binary64) n) < -1e9

   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 51.5

         \[\leadsto \frac{\mathsf{log1p}\left(x\right) - \color{blue}{\log x}}{n} \]

   5. Applied rewrites 51.5%

      \[\leadsto \color{blue}{\frac{\mathsf{log1p}\left(x\right) - \log x}{n}} \]

   6. Taylor expanded in x around inf

      \[\leadsto \frac{\color{blue}{\frac{\left(1 + \frac{\frac{1}{3}}{{x}^{2}}\right) - \frac{1}{2} \cdot \frac{1}{x}}{x}}}{n} \]
   7. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{\color{blue}{\frac{\left(1 + \frac{\frac{1}{3}}{{x}^{2}}\right) - \frac{1}{2} \cdot \frac{1}{x}}{x}}}{n} \]

   8. Applied rewrites 42.6%

      \[\leadsto \frac{\color{blue}{\frac{1 + \frac{-0.5 + \frac{0.3333333333333333}{x}}{x}}{x}}}{n} \]

   9. Taylor expanded in x around 0

      \[\leadsto \frac{\color{blue}{\frac{\frac{1}{3}}{{x}^{3}}}}{n} \]
   10. Step-by-step derivation

       1. lower-/.f64 N/A

          \[\leadsto \frac{\color{blue}{\frac{\frac{1}{3}}{{x}^{3}}}}{n} \]

       2. cube-mult N/A

          \[\leadsto \frac{\frac{\frac{1}{3}}{\color{blue}{x \cdot \left(x \cdot x\right)}}}{n} \]

       3. unpow2 N/A

          \[\leadsto \frac{\frac{\frac{1}{3}}{x \cdot \color{blue}{{x}^{2}}}}{n} \]

       4. lower-*.f64 N/A

          \[\leadsto \frac{\frac{\frac{1}{3}}{\color{blue}{x \cdot {x}^{2}}}}{n} \]

       5. unpow2 N/A

          \[\leadsto \frac{\frac{\frac{1}{3}}{x \cdot \color{blue}{\left(x \cdot x\right)}}}{n} \]

       6. lower-*.f64 77.1

          \[\leadsto \frac{\frac{0.3333333333333333}{x \cdot \color{blue}{\left(x \cdot x\right)}}}{n} \]

   11. Applied rewrites 77.1%

       \[\leadsto \frac{\color{blue}{\frac{0.3333333333333333}{x \cdot \left(x \cdot x\right)}}}{n} \]

   if -1e9 < (/.f64 #s(literal 1 binary64) n)

   1. Initial program 31.2%

      \[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in n around inf

      \[\leadsto \color{blue}{\frac{\log \left(1 + x\right) - \log x}{n}} \]
   4. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\log \left(1 + x\right) - \log x}{n}} \]

      2. lower--.f64 N/A

         \[\leadsto \frac{\color{blue}{\log \left(1 + x\right) - \log x}}{n} \]

      3. lower-log1p.f64 N/A

         \[\leadsto \frac{\color{blue}{\mathsf{log1p}\left(x\right)} - \log x}{n} \]

      4. lower-log.f64 60.8

         \[\leadsto \frac{\mathsf{log1p}\left(x\right) - \color{blue}{\log x}}{n} \]

   5. Applied rewrites 60.8%

      \[\leadsto \color{blue}{\frac{\mathsf{log1p}\left(x\right) - \log x}{n}} \]

   6. Taylor expanded in x around inf

      \[\leadsto \frac{\color{blue}{\frac{\left(1 + \frac{\frac{1}{3}}{{x}^{2}}\right) - \frac{1}{2} \cdot \frac{1}{x}}{x}}}{n} \]
   7. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{\color{blue}{\frac{\left(1 + \frac{\frac{1}{3}}{{x}^{2}}\right) - \frac{1}{2} \cdot \frac{1}{x}}{x}}}{n} \]

   8. Applied rewrites 38.7%

      \[\leadsto \frac{\color{blue}{\frac{1 + \frac{-0.5 + \frac{0.3333333333333333}{x}}{x}}{x}}}{n} \]
   9. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \frac{\frac{1 + \frac{\frac{-1}{2} + \color{blue}{\frac{\frac{1}{3}}{x}}}{x}}{x}}{n} \]

      2. lift-+.f64 N/A

         \[\leadsto \frac{\frac{1 + \frac{\color{blue}{\frac{-1}{2} + \frac{\frac{1}{3}}{x}}}{x}}{x}}{n} \]

      3. lift-/.f64 N/A

         \[\leadsto \frac{\frac{1 + \color{blue}{\frac{\frac{-1}{2} + \frac{\frac{1}{3}}{x}}{x}}}{x}}{n} \]

      4. lift-+.f64 N/A

         \[\leadsto \frac{\frac{\color{blue}{1 + \frac{\frac{-1}{2} + \frac{\frac{1}{3}}{x}}{x}}}{x}}{n} \]

      5. associate-/l/ N/A

         \[\leadsto \color{blue}{\frac{1 + \frac{\frac{-1}{2} + \frac{\frac{1}{3}}{x}}{x}}{n \cdot x}} \]

      6. associate-/r* N/A

         \[\leadsto \color{blue}{\frac{\frac{1 + \frac{\frac{-1}{2} + \frac{\frac{1}{3}}{x}}{x}}{n}}{x}} \]

      7. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\frac{1 + \frac{\frac{-1}{2} + \frac{\frac{1}{3}}{x}}{x}}{n}}{x}} \]

      8. lower-/.f64 38.8

         \[\leadsto \frac{\color{blue}{\frac{1 + \frac{-0.5 + \frac{0.3333333333333333}{x}}{x}}{n}}}{x} \]

   10. Applied rewrites 38.8%

       \[\leadsto \color{blue}{\frac{\frac{1 + \frac{-0.5 + \frac{0.3333333333333333}{x}}{x}}{n}}{x}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 18: 54.9% accurate, 3.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{1}{n} \leq -1000000000:\\ \;\;\;\;\frac{\frac{0.3333333333333333}{x \cdot \left(x \cdot x\right)}}{n}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 + \frac{-0.5 + \frac{0.3333333333333333}{x}}{x}}{n \cdot x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (if (<= (/ 1.0 n) -1000000000.0)
   (/ (/ 0.3333333333333333 (* x (* x x))) n)
   (/ (+ 1.0 (/ (+ -0.5 (/ 0.3333333333333333 x)) x)) (* n x))))

C

double code(double x, double n) {
	double tmp;
	if ((1.0 / n) <= -1000000000.0) {
		tmp = (0.3333333333333333 / (x * (x * x))) / n;
	} 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) <= (-1000000000.0d0)) then
        tmp = (0.3333333333333333d0 / (x * (x * x))) / n
    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) <= -1000000000.0) {
		tmp = (0.3333333333333333 / (x * (x * x))) / n;
	} 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) <= -1000000000.0:
		tmp = (0.3333333333333333 / (x * (x * x))) / n
	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) <= -1000000000.0)
		tmp = Float64(Float64(0.3333333333333333 / Float64(x * Float64(x * x))) / n);
	else
		tmp = Float64(Float64(1.0 + Float64(Float64(-0.5 + Float64(0.3333333333333333 / x)) / x)) / Float64(n * x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, n)
	tmp = 0.0;
	if ((1.0 / n) <= -1000000000.0)
		tmp = (0.3333333333333333 / (x * (x * x))) / n;
	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], -1000000000.0], N[(N[(0.3333333333333333 / N[(x * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / n), $MachinePrecision], N[(N[(1.0 + N[(N[(-0.5 + N[(0.3333333333333333 / x), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision] / N[(n * x), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 #s(literal 1 binary64) n) < -1e9

   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 51.5

         \[\leadsto \frac{\mathsf{log1p}\left(x\right) - \color{blue}{\log x}}{n} \]

   5. Applied rewrites 51.5%

      \[\leadsto \color{blue}{\frac{\mathsf{log1p}\left(x\right) - \log x}{n}} \]

   6. Taylor expanded in x around inf

      \[\leadsto \frac{\color{blue}{\frac{\left(1 + \frac{\frac{1}{3}}{{x}^{2}}\right) - \frac{1}{2} \cdot \frac{1}{x}}{x}}}{n} \]
   7. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{\color{blue}{\frac{\left(1 + \frac{\frac{1}{3}}{{x}^{2}}\right) - \frac{1}{2} \cdot \frac{1}{x}}{x}}}{n} \]

   8. Applied rewrites 42.6%

      \[\leadsto \frac{\color{blue}{\frac{1 + \frac{-0.5 + \frac{0.3333333333333333}{x}}{x}}{x}}}{n} \]

   9. Taylor expanded in x around 0

      \[\leadsto \frac{\color{blue}{\frac{\frac{1}{3}}{{x}^{3}}}}{n} \]
   10. Step-by-step derivation

       1. lower-/.f64 N/A

          \[\leadsto \frac{\color{blue}{\frac{\frac{1}{3}}{{x}^{3}}}}{n} \]

       2. cube-mult N/A

          \[\leadsto \frac{\frac{\frac{1}{3}}{\color{blue}{x \cdot \left(x \cdot x\right)}}}{n} \]

       3. unpow2 N/A

          \[\leadsto \frac{\frac{\frac{1}{3}}{x \cdot \color{blue}{{x}^{2}}}}{n} \]

       4. lower-*.f64 N/A

          \[\leadsto \frac{\frac{\frac{1}{3}}{\color{blue}{x \cdot {x}^{2}}}}{n} \]

       5. unpow2 N/A

          \[\leadsto \frac{\frac{\frac{1}{3}}{x \cdot \color{blue}{\left(x \cdot x\right)}}}{n} \]

       6. lower-*.f64 77.1

          \[\leadsto \frac{\frac{0.3333333333333333}{x \cdot \color{blue}{\left(x \cdot x\right)}}}{n} \]

   11. Applied rewrites 77.1%

       \[\leadsto \frac{\color{blue}{\frac{0.3333333333333333}{x \cdot \left(x \cdot x\right)}}}{n} \]

   if -1e9 < (/.f64 #s(literal 1 binary64) n)

   1. Initial program 31.2%

      \[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in n around inf

      \[\leadsto \color{blue}{\frac{\log \left(1 + x\right) - \log x}{n}} \]
   4. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\log \left(1 + x\right) - \log x}{n}} \]

      2. lower--.f64 N/A

         \[\leadsto \frac{\color{blue}{\log \left(1 + x\right) - \log x}}{n} \]

      3. lower-log1p.f64 N/A

         \[\leadsto \frac{\color{blue}{\mathsf{log1p}\left(x\right)} - \log x}{n} \]

      4. lower-log.f64 60.8

         \[\leadsto \frac{\mathsf{log1p}\left(x\right) - \color{blue}{\log x}}{n} \]

   5. Applied rewrites 60.8%

      \[\leadsto \color{blue}{\frac{\mathsf{log1p}\left(x\right) - \log x}{n}} \]

   6. Taylor expanded in x around inf

      \[\leadsto \frac{\color{blue}{\frac{\left(1 + \frac{\frac{1}{3}}{{x}^{2}}\right) - \frac{1}{2} \cdot \frac{1}{x}}{x}}}{n} \]
   7. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{\color{blue}{\frac{\left(1 + \frac{\frac{1}{3}}{{x}^{2}}\right) - \frac{1}{2} \cdot \frac{1}{x}}{x}}}{n} \]

   8. Applied rewrites 38.7%

      \[\leadsto \frac{\color{blue}{\frac{1 + \frac{-0.5 + \frac{0.3333333333333333}{x}}{x}}{x}}}{n} \]
   9. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \frac{\frac{1 + \frac{\frac{-1}{2} + \color{blue}{\frac{\frac{1}{3}}{x}}}{x}}{x}}{n} \]

      2. lift-+.f64 N/A

         \[\leadsto \frac{\frac{1 + \frac{\color{blue}{\frac{-1}{2} + \frac{\frac{1}{3}}{x}}}{x}}{x}}{n} \]

      3. lift-/.f64 N/A

         \[\leadsto \frac{\frac{1 + \color{blue}{\frac{\frac{-1}{2} + \frac{\frac{1}{3}}{x}}{x}}}{x}}{n} \]

      4. lift-+.f64 N/A

         \[\leadsto \frac{\frac{\color{blue}{1 + \frac{\frac{-1}{2} + \frac{\frac{1}{3}}{x}}{x}}}{x}}{n} \]

      5. associate-/l/ N/A

         \[\leadsto \color{blue}{\frac{1 + \frac{\frac{-1}{2} + \frac{\frac{1}{3}}{x}}{x}}{n \cdot x}} \]

      6. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{1 + \frac{\frac{-1}{2} + \frac{\frac{1}{3}}{x}}{x}}{n \cdot x}} \]

      7. *-commutative N/A

         \[\leadsto \frac{1 + \frac{\frac{-1}{2} + \frac{\frac{1}{3}}{x}}{x}}{\color{blue}{x \cdot n}} \]

      8. lower-*.f64 38.6

         \[\leadsto \frac{1 + \frac{-0.5 + \frac{0.3333333333333333}{x}}{x}}{\color{blue}{x \cdot n}} \]

   10. Applied rewrites 38.6%

       \[\leadsto \color{blue}{\frac{1 + \frac{-0.5 + \frac{0.3333333333333333}{x}}{x}}{x \cdot n}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 50.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{1}{n} \leq -1000000000:\\ \;\;\;\;\frac{\frac{0.3333333333333333}{x \cdot \left(x \cdot x\right)}}{n}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 + \frac{-0.5 + \frac{0.3333333333333333}{x}}{x}}{n \cdot x}\\ \end{array} \]
5. Add Preprocessing

Alternative 19: 55.7% accurate, 3.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{1}{n} \leq -2000:\\ \;\;\;\;\frac{\frac{0.3333333333333333}{x \cdot \left(x \cdot x\right)}}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 10^{+26}:\\ \;\;\;\;\frac{\frac{1}{x}}{n}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.3333333333333333}{x \cdot \left(x \cdot \left(n \cdot x\right)\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (if (<= (/ 1.0 n) -2000.0)
   (/ (/ 0.3333333333333333 (* x (* x x))) n)
   (if (<= (/ 1.0 n) 1e+26)
     (/ (/ 1.0 x) n)
     (/ 0.3333333333333333 (* x (* x (* n x)))))))

C

double code(double x, double n) {
	double tmp;
	if ((1.0 / n) <= -2000.0) {
		tmp = (0.3333333333333333 / (x * (x * x))) / n;
	} else if ((1.0 / n) <= 1e+26) {
		tmp = (1.0 / x) / 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) <= (-2000.0d0)) then
        tmp = (0.3333333333333333d0 / (x * (x * x))) / n
    else if ((1.0d0 / n) <= 1d+26) then
        tmp = (1.0d0 / x) / 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) <= -2000.0) {
		tmp = (0.3333333333333333 / (x * (x * x))) / n;
	} else if ((1.0 / n) <= 1e+26) {
		tmp = (1.0 / x) / n;
	} else {
		tmp = 0.3333333333333333 / (x * (x * (n * x)));
	}
	return tmp;
}

Python

def code(x, n):
	tmp = 0
	if (1.0 / n) <= -2000.0:
		tmp = (0.3333333333333333 / (x * (x * x))) / n
	elif (1.0 / n) <= 1e+26:
		tmp = (1.0 / x) / n
	else:
		tmp = 0.3333333333333333 / (x * (x * (n * x)))
	return tmp

Julia

function code(x, n)
	tmp = 0.0
	if (Float64(1.0 / n) <= -2000.0)
		tmp = Float64(Float64(0.3333333333333333 / Float64(x * Float64(x * x))) / n);
	elseif (Float64(1.0 / n) <= 1e+26)
		tmp = Float64(Float64(1.0 / x) / n);
	else
		tmp = Float64(0.3333333333333333 / Float64(x * Float64(x * Float64(n * x))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, n)
	tmp = 0.0;
	if ((1.0 / n) <= -2000.0)
		tmp = (0.3333333333333333 / (x * (x * x))) / n;
	elseif ((1.0 / n) <= 1e+26)
		tmp = (1.0 / x) / 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], -2000.0], N[(N[(0.3333333333333333 / N[(x * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / n), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 1e+26], N[(N[(1.0 / x), $MachinePrecision] / n), $MachinePrecision], N[(0.3333333333333333 / N[(x * N[(x * N[(n * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 #s(literal 1 binary64) n) < -2e3

   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 50.3

         \[\leadsto \frac{\mathsf{log1p}\left(x\right) - \color{blue}{\log x}}{n} \]

   5. Applied rewrites 50.3%

      \[\leadsto \color{blue}{\frac{\mathsf{log1p}\left(x\right) - \log x}{n}} \]

   6. Taylor expanded in x around inf

      \[\leadsto \frac{\color{blue}{\frac{\left(1 + \frac{\frac{1}{3}}{{x}^{2}}\right) - \frac{1}{2} \cdot \frac{1}{x}}{x}}}{n} \]
   7. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{\color{blue}{\frac{\left(1 + \frac{\frac{1}{3}}{{x}^{2}}\right) - \frac{1}{2} \cdot \frac{1}{x}}{x}}}{n} \]

   8. Applied rewrites 41.8%

      \[\leadsto \frac{\color{blue}{\frac{1 + \frac{-0.5 + \frac{0.3333333333333333}{x}}{x}}{x}}}{n} \]

   9. Taylor expanded in x around 0

      \[\leadsto \frac{\color{blue}{\frac{\frac{1}{3}}{{x}^{3}}}}{n} \]
   10. Step-by-step derivation

       1. lower-/.f64 N/A

          \[\leadsto \frac{\color{blue}{\frac{\frac{1}{3}}{{x}^{3}}}}{n} \]

       2. cube-mult N/A

          \[\leadsto \frac{\frac{\frac{1}{3}}{\color{blue}{x \cdot \left(x \cdot x\right)}}}{n} \]

       3. unpow2 N/A

          \[\leadsto \frac{\frac{\frac{1}{3}}{x \cdot \color{blue}{{x}^{2}}}}{n} \]

       4. lower-*.f64 N/A

          \[\leadsto \frac{\frac{\frac{1}{3}}{\color{blue}{x \cdot {x}^{2}}}}{n} \]

       5. unpow2 N/A

          \[\leadsto \frac{\frac{\frac{1}{3}}{x \cdot \color{blue}{\left(x \cdot x\right)}}}{n} \]

       6. lower-*.f64 75.5

          \[\leadsto \frac{\frac{0.3333333333333333}{x \cdot \color{blue}{\left(x \cdot x\right)}}}{n} \]

   11. Applied rewrites 75.5%

       \[\leadsto \frac{\color{blue}{\frac{0.3333333333333333}{x \cdot \left(x \cdot x\right)}}}{n} \]

   if -2e3 < (/.f64 #s(literal 1 binary64) n) < 1.00000000000000005e26

   1. Initial program 24.8%

      \[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)} \]
   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 75.3

         \[\leadsto \frac{\mathsf{log1p}\left(x\right) - \color{blue}{\log x}}{n} \]

   5. Applied rewrites 75.3%

      \[\leadsto \color{blue}{\frac{\mathsf{log1p}\left(x\right) - \log x}{n}} \]

   6. Taylor expanded in x around inf

      \[\leadsto \frac{\color{blue}{\frac{1}{x}}}{n} \]
   7. Step-by-step derivation

      1. lower-/.f64 39.3

         \[\leadsto \frac{\color{blue}{\frac{1}{x}}}{n} \]

   8. Applied rewrites 39.3%

      \[\leadsto \frac{\color{blue}{\frac{1}{x}}}{n} \]

   if 1.00000000000000005e26 < (/.f64 #s(literal 1 binary64) n)

   1. Initial program 52.9%

      \[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in n around inf

      \[\leadsto \color{blue}{\frac{\log \left(1 + x\right) - \log x}{n}} \]
   4. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\log \left(1 + x\right) - \log x}{n}} \]

      2. lower--.f64 N/A

         \[\leadsto \frac{\color{blue}{\log \left(1 + x\right) - \log x}}{n} \]

      3. lower-log1p.f64 N/A

         \[\leadsto \frac{\color{blue}{\mathsf{log1p}\left(x\right)} - \log x}{n} \]

      4. lower-log.f64 5.5

         \[\leadsto \frac{\mathsf{log1p}\left(x\right) - \color{blue}{\log x}}{n} \]

   5. Applied rewrites 5.5%

      \[\leadsto \color{blue}{\frac{\mathsf{log1p}\left(x\right) - \log x}{n}} \]

   6. Taylor expanded in x around inf

      \[\leadsto \frac{\color{blue}{\frac{\left(1 + \frac{\frac{1}{3}}{{x}^{2}}\right) - \frac{1}{2} \cdot \frac{1}{x}}{x}}}{n} \]
   7. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{\color{blue}{\frac{\left(1 + \frac{\frac{1}{3}}{{x}^{2}}\right) - \frac{1}{2} \cdot \frac{1}{x}}{x}}}{n} \]

   8. Applied rewrites 39.8%

      \[\leadsto \frac{\color{blue}{\frac{1 + \frac{-0.5 + \frac{0.3333333333333333}{x}}{x}}{x}}}{n} \]

   9. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{\frac{1}{3}}{n \cdot {x}^{3}}} \]
   10. Step-by-step derivation

       1. lower-/.f64 N/A

          \[\leadsto \color{blue}{\frac{\frac{1}{3}}{n \cdot {x}^{3}}} \]

       2. unpow3 N/A

          \[\leadsto \frac{\frac{1}{3}}{n \cdot \color{blue}{\left(\left(x \cdot x\right) \cdot x\right)}} \]

       3. unpow2 N/A

          \[\leadsto \frac{\frac{1}{3}}{n \cdot \left(\color{blue}{{x}^{2}} \cdot x\right)} \]

       4. associate-*r* N/A

          \[\leadsto \frac{\frac{1}{3}}{\color{blue}{\left(n \cdot {x}^{2}\right) \cdot x}} \]

       5. *-commutative N/A

          \[\leadsto \frac{\frac{1}{3}}{\color{blue}{x \cdot \left(n \cdot {x}^{2}\right)}} \]

       6. lower-*.f64 N/A

          \[\leadsto \frac{\frac{1}{3}}{\color{blue}{x \cdot \left(n \cdot {x}^{2}\right)}} \]

       7. unpow2 N/A

          \[\leadsto \frac{\frac{1}{3}}{x \cdot \left(n \cdot \color{blue}{\left(x \cdot x\right)}\right)} \]

       8. associate-*r* N/A

          \[\leadsto \frac{\frac{1}{3}}{x \cdot \color{blue}{\left(\left(n \cdot x\right) \cdot x\right)}} \]

       9. *-commutative N/A

          \[\leadsto \frac{\frac{1}{3}}{x \cdot \color{blue}{\left(x \cdot \left(n \cdot x\right)\right)}} \]

       10. lower-*.f64 N/A

           \[\leadsto \frac{\frac{1}{3}}{x \cdot \color{blue}{\left(x \cdot \left(n \cdot x\right)\right)}} \]

       11. *-commutative N/A

           \[\leadsto \frac{\frac{1}{3}}{x \cdot \left(x \cdot \color{blue}{\left(x \cdot n\right)}\right)} \]

       12. lower-*.f64 39.8

           \[\leadsto \frac{0.3333333333333333}{x \cdot \left(x \cdot \color{blue}{\left(x \cdot n\right)}\right)} \]

   11. Applied rewrites 39.8%

       \[\leadsto \color{blue}{\frac{0.3333333333333333}{x \cdot \left(x \cdot \left(x \cdot n\right)\right)}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 50.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{1}{n} \leq -2000:\\ \;\;\;\;\frac{\frac{0.3333333333333333}{x \cdot \left(x \cdot x\right)}}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 10^{+26}:\\ \;\;\;\;\frac{\frac{1}{x}}{n}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.3333333333333333}{x \cdot \left(x \cdot \left(n \cdot x\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 20: 45.2% accurate, 6.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\frac{1}{x}}{n}\\ \mathbf{if}\;n \leq -0.000118:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;n \leq -2.75 \cdot 10^{-129}:\\ \;\;\;\;0\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (let* ((t_0 (/ (/ 1.0 x) n)))
   (if (<= n -0.000118) t_0 (if (<= n -2.75e-129) 0.0 t_0))))

C

double code(double x, double n) {
	double t_0 = (1.0 / x) / n;
	double tmp;
	if (n <= -0.000118) {
		tmp = t_0;
	} else if (n <= -2.75e-129) {
		tmp = 0.0;
	} 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 / x) / n
    if (n <= (-0.000118d0)) then
        tmp = t_0
    else if (n <= (-2.75d-129)) then
        tmp = 0.0d0
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double x, double n) {
	double t_0 = (1.0 / x) / n;
	double tmp;
	if (n <= -0.000118) {
		tmp = t_0;
	} else if (n <= -2.75e-129) {
		tmp = 0.0;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, n):
	t_0 = (1.0 / x) / n
	tmp = 0
	if n <= -0.000118:
		tmp = t_0
	elif n <= -2.75e-129:
		tmp = 0.0
	else:
		tmp = t_0
	return tmp

Julia

function code(x, n)
	t_0 = Float64(Float64(1.0 / x) / n)
	tmp = 0.0
	if (n <= -0.000118)
		tmp = t_0;
	elseif (n <= -2.75e-129)
		tmp = 0.0;
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, n)
	t_0 = (1.0 / x) / n;
	tmp = 0.0;
	if (n <= -0.000118)
		tmp = t_0;
	elseif (n <= -2.75e-129)
		tmp = 0.0;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, n_] := Block[{t$95$0 = N[(N[(1.0 / x), $MachinePrecision] / n), $MachinePrecision]}, If[LessEqual[n, -0.000118], t$95$0, If[LessEqual[n, -2.75e-129], 0.0, t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if n < -1.18e-4 or -2.75000000000000012e-129 < n

   1. Initial program 44.3%

      \[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)} \]
   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 55.9

         \[\leadsto \frac{\mathsf{log1p}\left(x\right) - \color{blue}{\log x}}{n} \]

   5. Applied rewrites 55.9%

      \[\leadsto \color{blue}{\frac{\mathsf{log1p}\left(x\right) - \log x}{n}} \]

   6. Taylor expanded in x around inf

      \[\leadsto \frac{\color{blue}{\frac{1}{x}}}{n} \]
   7. Step-by-step derivation

      1. lower-/.f64 38.5

         \[\leadsto \frac{\color{blue}{\frac{1}{x}}}{n} \]

   8. Applied rewrites 38.5%

      \[\leadsto \frac{\color{blue}{\frac{1}{x}}}{n} \]

   if -1.18e-4 < n < -2.75000000000000012e-129

   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

      \[\leadsto \color{blue}{1 - e^{\frac{\log x}{n}}} \]
   4. Step-by-step derivation

      1. remove-double-neg N/A

         \[\leadsto 1 - e^{\frac{\color{blue}{\mathsf{neg}\left(\left(\mathsf{neg}\left(\log x\right)\right)\right)}}{n}} \]

      2. mul-1-neg N/A

         \[\leadsto 1 - e^{\frac{\mathsf{neg}\left(\color{blue}{-1 \cdot \log x}\right)}{n}} \]

      3. distribute-neg-frac N/A

         \[\leadsto 1 - e^{\color{blue}{\mathsf{neg}\left(\frac{-1 \cdot \log x}{n}\right)}} \]

      4. mul-1-neg N/A

         \[\leadsto 1 - e^{\mathsf{neg}\left(\frac{\color{blue}{\mathsf{neg}\left(\log x\right)}}{n}\right)} \]

      5. log-rec N/A

         \[\leadsto 1 - e^{\mathsf{neg}\left(\frac{\color{blue}{\log \left(\frac{1}{x}\right)}}{n}\right)} \]

      6. mul-1-neg N/A

         \[\leadsto 1 - e^{\color{blue}{-1 \cdot \frac{\log \left(\frac{1}{x}\right)}{n}}} \]

      7. lower--.f64 N/A

         \[\leadsto \color{blue}{1 - e^{-1 \cdot \frac{\log \left(\frac{1}{x}\right)}{n}}} \]

      8. log-rec N/A

         \[\leadsto 1 - e^{-1 \cdot \frac{\color{blue}{\mathsf{neg}\left(\log x\right)}}{n}} \]

      9. mul-1-neg N/A

         \[\leadsto 1 - e^{-1 \cdot \frac{\color{blue}{-1 \cdot \log x}}{n}} \]

      10. associate-*r/ N/A

          \[\leadsto 1 - e^{\color{blue}{\frac{-1 \cdot \left(-1 \cdot \log x\right)}{n}}} \]

      11. associate-*r* N/A

          \[\leadsto 1 - e^{\frac{\color{blue}{\left(-1 \cdot -1\right) \cdot \log x}}{n}} \]

      12. metadata-eval N/A

          \[\leadsto 1 - e^{\frac{\color{blue}{1} \cdot \log x}{n}} \]

      13. *-commutative N/A

          \[\leadsto 1 - e^{\frac{\color{blue}{\log x \cdot 1}}{n}} \]

      14. associate-/l* N/A

          \[\leadsto 1 - e^{\color{blue}{\log x \cdot \frac{1}{n}}} \]

      15. exp-to-pow N/A

          \[\leadsto 1 - \color{blue}{{x}^{\left(\frac{1}{n}\right)}} \]

      16. lower-pow.f64 N/A

          \[\leadsto 1 - \color{blue}{{x}^{\left(\frac{1}{n}\right)}} \]

      17. lower-/.f64 32.7

          \[\leadsto 1 - {x}^{\color{blue}{\left(\frac{1}{n}\right)}} \]

   5. Applied rewrites 32.7%

      \[\leadsto \color{blue}{1 - {x}^{\left(\frac{1}{n}\right)}} \]

   6. Taylor expanded in n around inf

      \[\leadsto 1 - \color{blue}{1} \]
   7. Step-by-step derivation

   8. Applied rewrites 69.9%

      \[\leadsto 1 - \color{blue}{1} \]
   9. Step-by-step derivation

      1. metadata-eval 69.9

         \[\leadsto \color{blue}{0} \]

   10. Applied rewrites 69.9%

       \[\leadsto \color{blue}{0} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 21: 44.6% accurate, 8.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{1}{n \cdot x}\\ \mathbf{if}\;n \leq -0.000118:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;n \leq -2.75 \cdot 10^{-129}:\\ \;\;\;\;0\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (let* ((t_0 (/ 1.0 (* n x))))
   (if (<= n -0.000118) t_0 (if (<= n -2.75e-129) 0.0 t_0))))

C

double code(double x, double n) {
	double t_0 = 1.0 / (n * x);
	double tmp;
	if (n <= -0.000118) {
		tmp = t_0;
	} else if (n <= -2.75e-129) {
		tmp = 0.0;
	} 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 <= (-0.000118d0)) then
        tmp = t_0
    else if (n <= (-2.75d-129)) then
        tmp = 0.0d0
    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 <= -0.000118) {
		tmp = t_0;
	} else if (n <= -2.75e-129) {
		tmp = 0.0;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, n):
	t_0 = 1.0 / (n * x)
	tmp = 0
	if n <= -0.000118:
		tmp = t_0
	elif n <= -2.75e-129:
		tmp = 0.0
	else:
		tmp = t_0
	return tmp

Julia

function code(x, n)
	t_0 = Float64(1.0 / Float64(n * x))
	tmp = 0.0
	if (n <= -0.000118)
		tmp = t_0;
	elseif (n <= -2.75e-129)
		tmp = 0.0;
	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 <= -0.000118)
		tmp = t_0;
	elseif (n <= -2.75e-129)
		tmp = 0.0;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, n_] := Block[{t$95$0 = N[(1.0 / N[(n * x), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[n, -0.000118], t$95$0, If[LessEqual[n, -2.75e-129], 0.0, t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if n < -1.18e-4 or -2.75000000000000012e-129 < n

   1. Initial program 44.3%

      \[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)} \]
   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 55.9

         \[\leadsto \frac{\mathsf{log1p}\left(x\right) - \color{blue}{\log x}}{n} \]

   5. Applied rewrites 55.9%

      \[\leadsto \color{blue}{\frac{\mathsf{log1p}\left(x\right) - \log x}{n}} \]

   6. Taylor expanded in x around inf

      \[\leadsto \color{blue}{\frac{1}{n \cdot x}} \]
   7. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{1}{n \cdot x}} \]

      2. *-commutative N/A

         \[\leadsto \frac{1}{\color{blue}{x \cdot n}} \]

      3. lower-*.f64 38.4

         \[\leadsto \frac{1}{\color{blue}{x \cdot n}} \]

   8. Applied rewrites 38.4%

      \[\leadsto \color{blue}{\frac{1}{x \cdot n}} \]

   if -1.18e-4 < n < -2.75000000000000012e-129

   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

      \[\leadsto \color{blue}{1 - e^{\frac{\log x}{n}}} \]
   4. Step-by-step derivation

      1. remove-double-neg N/A

         \[\leadsto 1 - e^{\frac{\color{blue}{\mathsf{neg}\left(\left(\mathsf{neg}\left(\log x\right)\right)\right)}}{n}} \]

      2. mul-1-neg N/A

         \[\leadsto 1 - e^{\frac{\mathsf{neg}\left(\color{blue}{-1 \cdot \log x}\right)}{n}} \]

      3. distribute-neg-frac N/A

         \[\leadsto 1 - e^{\color{blue}{\mathsf{neg}\left(\frac{-1 \cdot \log x}{n}\right)}} \]

      4. mul-1-neg N/A

         \[\leadsto 1 - e^{\mathsf{neg}\left(\frac{\color{blue}{\mathsf{neg}\left(\log x\right)}}{n}\right)} \]

      5. log-rec N/A

         \[\leadsto 1 - e^{\mathsf{neg}\left(\frac{\color{blue}{\log \left(\frac{1}{x}\right)}}{n}\right)} \]

      6. mul-1-neg N/A

         \[\leadsto 1 - e^{\color{blue}{-1 \cdot \frac{\log \left(\frac{1}{x}\right)}{n}}} \]

      7. lower--.f64 N/A

         \[\leadsto \color{blue}{1 - e^{-1 \cdot \frac{\log \left(\frac{1}{x}\right)}{n}}} \]

      8. log-rec N/A

         \[\leadsto 1 - e^{-1 \cdot \frac{\color{blue}{\mathsf{neg}\left(\log x\right)}}{n}} \]

      9. mul-1-neg N/A

         \[\leadsto 1 - e^{-1 \cdot \frac{\color{blue}{-1 \cdot \log x}}{n}} \]

      10. associate-*r/ N/A

          \[\leadsto 1 - e^{\color{blue}{\frac{-1 \cdot \left(-1 \cdot \log x\right)}{n}}} \]

      11. associate-*r* N/A

          \[\leadsto 1 - e^{\frac{\color{blue}{\left(-1 \cdot -1\right) \cdot \log x}}{n}} \]

      12. metadata-eval N/A

          \[\leadsto 1 - e^{\frac{\color{blue}{1} \cdot \log x}{n}} \]

      13. *-commutative N/A

          \[\leadsto 1 - e^{\frac{\color{blue}{\log x \cdot 1}}{n}} \]

      14. associate-/l* N/A

          \[\leadsto 1 - e^{\color{blue}{\log x \cdot \frac{1}{n}}} \]

      15. exp-to-pow N/A

          \[\leadsto 1 - \color{blue}{{x}^{\left(\frac{1}{n}\right)}} \]

      16. lower-pow.f64 N/A

          \[\leadsto 1 - \color{blue}{{x}^{\left(\frac{1}{n}\right)}} \]

      17. lower-/.f64 32.7

          \[\leadsto 1 - {x}^{\color{blue}{\left(\frac{1}{n}\right)}} \]

   5. Applied rewrites 32.7%

      \[\leadsto \color{blue}{1 - {x}^{\left(\frac{1}{n}\right)}} \]

   6. Taylor expanded in n around inf

      \[\leadsto 1 - \color{blue}{1} \]
   7. Step-by-step derivation

   8. Applied rewrites 69.9%

      \[\leadsto 1 - \color{blue}{1} \]
   9. Step-by-step derivation

      1. metadata-eval 69.9

         \[\leadsto \color{blue}{0} \]

   10. Applied rewrites 69.9%

       \[\leadsto \color{blue}{0} \]
3. Recombined 2 regimes into one program.

4. Final simplification 42.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq -0.000118:\\ \;\;\;\;\frac{1}{n \cdot x}\\ \mathbf{elif}\;n \leq -2.75 \cdot 10^{-129}:\\ \;\;\;\;0\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{n \cdot x}\\ \end{array} \]
5. Add Preprocessing

Alternative 22: 30.5% accurate, 231.0× speedup

Math

\[\begin{array}{l} \\ 0 \end{array} \]

FPCore

(FPCore (x n) :precision binary64 0.0)

C

double code(double x, double n) {
	return 0.0;
}

Fortran

real(8) function code(x, n)
    real(8), intent (in) :: x
    real(8), intent (in) :: n
    code = 0.0d0
end function

Java

public static double code(double x, double n) {
	return 0.0;
}

Python

def code(x, n):
	return 0.0

Julia

function code(x, n)
	return 0.0
end

MATLAB

function tmp = code(x, n)
	tmp = 0.0;
end

Wolfram

code[x_, n_] := 0.0

Derivation

1. Initial program 52.1%

   \[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)} \]
2. Add Preprocessing

3. Taylor expanded in x around 0

   \[\leadsto \color{blue}{1 - e^{\frac{\log x}{n}}} \]
4. Step-by-step derivation

   1. remove-double-neg N/A

      \[\leadsto 1 - e^{\frac{\color{blue}{\mathsf{neg}\left(\left(\mathsf{neg}\left(\log x\right)\right)\right)}}{n}} \]

   2. mul-1-neg N/A

      \[\leadsto 1 - e^{\frac{\mathsf{neg}\left(\color{blue}{-1 \cdot \log x}\right)}{n}} \]

   3. distribute-neg-frac N/A

      \[\leadsto 1 - e^{\color{blue}{\mathsf{neg}\left(\frac{-1 \cdot \log x}{n}\right)}} \]

   4. mul-1-neg N/A

      \[\leadsto 1 - e^{\mathsf{neg}\left(\frac{\color{blue}{\mathsf{neg}\left(\log x\right)}}{n}\right)} \]

   5. log-rec N/A

      \[\leadsto 1 - e^{\mathsf{neg}\left(\frac{\color{blue}{\log \left(\frac{1}{x}\right)}}{n}\right)} \]

   6. mul-1-neg N/A

      \[\leadsto 1 - e^{\color{blue}{-1 \cdot \frac{\log \left(\frac{1}{x}\right)}{n}}} \]

   7. lower--.f64 N/A

      \[\leadsto \color{blue}{1 - e^{-1 \cdot \frac{\log \left(\frac{1}{x}\right)}{n}}} \]

   8. log-rec N/A

      \[\leadsto 1 - e^{-1 \cdot \frac{\color{blue}{\mathsf{neg}\left(\log x\right)}}{n}} \]

   9. mul-1-neg N/A

      \[\leadsto 1 - e^{-1 \cdot \frac{\color{blue}{-1 \cdot \log x}}{n}} \]

   10. associate-*r/ N/A

       \[\leadsto 1 - e^{\color{blue}{\frac{-1 \cdot \left(-1 \cdot \log x\right)}{n}}} \]

   11. associate-*r* N/A

       \[\leadsto 1 - e^{\frac{\color{blue}{\left(-1 \cdot -1\right) \cdot \log x}}{n}} \]

   12. metadata-eval N/A

       \[\leadsto 1 - e^{\frac{\color{blue}{1} \cdot \log x}{n}} \]

   13. *-commutative N/A

       \[\leadsto 1 - e^{\frac{\color{blue}{\log x \cdot 1}}{n}} \]

   14. associate-/l* N/A

       \[\leadsto 1 - e^{\color{blue}{\log x \cdot \frac{1}{n}}} \]

   15. exp-to-pow N/A

       \[\leadsto 1 - \color{blue}{{x}^{\left(\frac{1}{n}\right)}} \]

   16. lower-pow.f64 N/A

       \[\leadsto 1 - \color{blue}{{x}^{\left(\frac{1}{n}\right)}} \]

   17. lower-/.f64 36.9

       \[\leadsto 1 - {x}^{\color{blue}{\left(\frac{1}{n}\right)}} \]

5. Applied rewrites 36.9%

   \[\leadsto \color{blue}{1 - {x}^{\left(\frac{1}{n}\right)}} \]

6. Taylor expanded in n around inf

   \[\leadsto 1 - \color{blue}{1} \]
7. Step-by-step derivation

8. Applied rewrites 26.7%

   \[\leadsto 1 - \color{blue}{1} \]
9. Step-by-step derivation

   1. metadata-eval 26.7

      \[\leadsto \color{blue}{0} \]

10. Applied rewrites 26.7%

    \[\leadsto \color{blue}{0} \]
11. Add Preprocessing

Reproduce

herbie shell --seed 2024214 
(FPCore (x n)
  :name "2nthrt (problem 3.4.6)"
  :precision binary64
  (- (pow (+ x 1.0) (/ 1.0 n)) (pow x (/ 1.0 n))))