2nthrt (problem 3.4.6)

Percentage Accurate: 54.2% → 86.5%

Time: 26.6s

Alternatives: 23

Speedup: 3.4×

• could not determine a ground truth

  1. x = -1.0936638744722639e+148
  2. n = -1.4825148283253816e+149

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: 54.2% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 86.5% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 1650000:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(0.5, {\left(\mathsf{log1p}\left(x\right)\right)}^{2} - {\log x}^{2}, \frac{0.16666666666666666 \cdot \left({\left(\mathsf{log1p}\left(x\right)\right)}^{3} - {\log x}^{3}\right)}{n}\right)}{n} + \left(\mathsf{log1p}\left(x\right) - \log x\right)}{n}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{n} \cdot \frac{{x}^{\left(\frac{1}{n}\right)}}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (if (<= x 1650000.0)
   (/
    (+
     (/
      (fma
       0.5
       (- (pow (log1p x) 2.0) (pow (log x) 2.0))
       (/ (* 0.16666666666666666 (- (pow (log1p x) 3.0) (pow (log x) 3.0))) n))
      n)
     (- (log1p x) (log x)))
    n)
   (* (/ 1.0 n) (/ (pow x (/ 1.0 n)) x))))

C

double code(double x, double n) {
	double tmp;
	if (x <= 1650000.0) {
		tmp = ((fma(0.5, (pow(log1p(x), 2.0) - pow(log(x), 2.0)), ((0.16666666666666666 * (pow(log1p(x), 3.0) - pow(log(x), 3.0))) / n)) / n) + (log1p(x) - log(x))) / n;
	} else {
		tmp = (1.0 / n) * (pow(x, (1.0 / n)) / x);
	}
	return tmp;
}

Julia

function code(x, n)
	tmp = 0.0
	if (x <= 1650000.0)
		tmp = Float64(Float64(Float64(fma(0.5, Float64((log1p(x) ^ 2.0) - (log(x) ^ 2.0)), Float64(Float64(0.16666666666666666 * Float64((log1p(x) ^ 3.0) - (log(x) ^ 3.0))) / n)) / n) + Float64(log1p(x) - log(x))) / n);
	else
		tmp = Float64(Float64(1.0 / n) * Float64((x ^ Float64(1.0 / n)) / x));
	end
	return tmp
end

Wolfram

code[x_, n_] := If[LessEqual[x, 1650000.0], N[(N[(N[(N[(0.5 * N[(N[Power[N[Log[1 + x], $MachinePrecision], 2.0], $MachinePrecision] - N[Power[N[Log[x], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] + N[(N[(0.16666666666666666 * N[(N[Power[N[Log[1 + x], $MachinePrecision], 3.0], $MachinePrecision] - N[Power[N[Log[x], $MachinePrecision], 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / n), $MachinePrecision]), $MachinePrecision] / n), $MachinePrecision] + N[(N[Log[1 + x], $MachinePrecision] - N[Log[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / n), $MachinePrecision], N[(N[(1.0 / n), $MachinePrecision] * N[(N[Power[x, N[(1.0 / n), $MachinePrecision]], $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 1.65e6

   1. Initial program 45.7%

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

   3. Taylor expanded in n around -inf

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

   5. Applied rewrites 77.2%

      \[\leadsto \color{blue}{\frac{\left(\log x - \mathsf{log1p}\left(x\right)\right) - \frac{\mathsf{fma}\left(0.5, {\left(\mathsf{log1p}\left(x\right)\right)}^{2} - {\log x}^{2}, \frac{0.16666666666666666 \cdot \left({\left(\mathsf{log1p}\left(x\right)\right)}^{3} - {\log x}^{3}\right)}{n}\right)}{n}}{-n}} \]

   if 1.65e6 < x

   1. Initial program 64.5%

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

   3. Taylor expanded in x around inf

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

      1. lower-/.f64 N/A

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

      2. log-rec N/A

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

      3. mul-1-neg N/A

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

      4. associate-*r/ N/A

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

      5. associate-*r* N/A

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

      6. metadata-eval N/A

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

      7. *-commutative N/A

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

      8. associate-/l* N/A

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

      9. exp-to-pow N/A

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

      10. lower-pow.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 98.9

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

   5. Applied rewrites 98.9%

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

      1. lift-/.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. frac-2neg N/A

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

      5. neg-mul-1 N/A

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

      8. distribute-rgt-neg-in N/A

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

      9. times-frac N/A

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

      10. frac-2neg N/A

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

      11. metadata-eval N/A

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

      12. lower-*.f64 N/A

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

      13. metadata-eval N/A

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

      14. frac-2neg N/A

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

      15. lower-/.f64 N/A

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

      16. lower-/.f64 N/A

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

      17. lower-neg.f64 99.5

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

   7. Applied rewrites 99.5%

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

4. Final simplification 87.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 1650000:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(0.5, {\left(\mathsf{log1p}\left(x\right)\right)}^{2} - {\log x}^{2}, \frac{0.16666666666666666 \cdot \left({\left(\mathsf{log1p}\left(x\right)\right)}^{3} - {\log x}^{3}\right)}{n}\right)}{n} + \left(\mathsf{log1p}\left(x\right) - \log x\right)}{n}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{n} \cdot \frac{{x}^{\left(\frac{1}{n}\right)}}{x}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 77.8% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {x}^{\left(\frac{1}{n}\right)}\\ t_1 := {\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - t\_0\\ t_2 := n \cdot \left(x \cdot x\right)\\ \mathbf{if}\;t\_1 \leq -2 \cdot 10^{-5}:\\ \;\;\;\;1 - t\_0\\ \mathbf{elif}\;t\_1 \leq 0.9101169306103679:\\ \;\;\;\;\frac{\log \left(\frac{x + 1}{x}\right)}{n}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{n} + \frac{\frac{-0.5}{x} \cdot t\_2 - n \cdot -0.3333333333333333}{n \cdot t\_2}}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (let* ((t_0 (pow x (/ 1.0 n)))
        (t_1 (- (pow (+ x 1.0) (/ 1.0 n)) t_0))
        (t_2 (* n (* x x))))
   (if (<= t_1 -2e-5)
     (- 1.0 t_0)
     (if (<= t_1 0.9101169306103679)
       (/ (log (/ (+ x 1.0) x)) n)
       (/
        (+
         (/ 1.0 n)
         (/ (- (* (/ -0.5 x) t_2) (* n -0.3333333333333333)) (* n t_2)))
        x)))))

C

double code(double x, double n) {
	double t_0 = pow(x, (1.0 / n));
	double t_1 = pow((x + 1.0), (1.0 / n)) - t_0;
	double t_2 = n * (x * x);
	double tmp;
	if (t_1 <= -2e-5) {
		tmp = 1.0 - t_0;
	} else if (t_1 <= 0.9101169306103679) {
		tmp = log(((x + 1.0) / x)) / n;
	} else {
		tmp = ((1.0 / n) + ((((-0.5 / x) * t_2) - (n * -0.3333333333333333)) / (n * t_2))) / x;
	}
	return tmp;
}

Fortran

real(8) function code(x, n)
    real(8), intent (in) :: x
    real(8), intent (in) :: n
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_0 = x ** (1.0d0 / n)
    t_1 = ((x + 1.0d0) ** (1.0d0 / n)) - t_0
    t_2 = n * (x * x)
    if (t_1 <= (-2d-5)) then
        tmp = 1.0d0 - t_0
    else if (t_1 <= 0.9101169306103679d0) then
        tmp = log(((x + 1.0d0) / x)) / n
    else
        tmp = ((1.0d0 / n) + (((((-0.5d0) / x) * t_2) - (n * (-0.3333333333333333d0))) / (n * t_2))) / x
    end if
    code = tmp
end function

Java

public static double code(double x, double n) {
	double t_0 = Math.pow(x, (1.0 / n));
	double t_1 = Math.pow((x + 1.0), (1.0 / n)) - t_0;
	double t_2 = n * (x * x);
	double tmp;
	if (t_1 <= -2e-5) {
		tmp = 1.0 - t_0;
	} else if (t_1 <= 0.9101169306103679) {
		tmp = Math.log(((x + 1.0) / x)) / n;
	} else {
		tmp = ((1.0 / n) + ((((-0.5 / x) * t_2) - (n * -0.3333333333333333)) / (n * t_2))) / x;
	}
	return tmp;
}

Python

def code(x, n):
	t_0 = math.pow(x, (1.0 / n))
	t_1 = math.pow((x + 1.0), (1.0 / n)) - t_0
	t_2 = n * (x * x)
	tmp = 0
	if t_1 <= -2e-5:
		tmp = 1.0 - t_0
	elif t_1 <= 0.9101169306103679:
		tmp = math.log(((x + 1.0) / x)) / n
	else:
		tmp = ((1.0 / n) + ((((-0.5 / x) * t_2) - (n * -0.3333333333333333)) / (n * t_2))) / x
	return tmp

Julia

function code(x, n)
	t_0 = x ^ Float64(1.0 / n)
	t_1 = Float64((Float64(x + 1.0) ^ Float64(1.0 / n)) - t_0)
	t_2 = Float64(n * Float64(x * x))
	tmp = 0.0
	if (t_1 <= -2e-5)
		tmp = Float64(1.0 - t_0);
	elseif (t_1 <= 0.9101169306103679)
		tmp = Float64(log(Float64(Float64(x + 1.0) / x)) / n);
	else
		tmp = Float64(Float64(Float64(1.0 / n) + Float64(Float64(Float64(Float64(-0.5 / x) * t_2) - Float64(n * -0.3333333333333333)) / Float64(n * t_2))) / x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, n)
	t_0 = x ^ (1.0 / n);
	t_1 = ((x + 1.0) ^ (1.0 / n)) - t_0;
	t_2 = n * (x * x);
	tmp = 0.0;
	if (t_1 <= -2e-5)
		tmp = 1.0 - t_0;
	elseif (t_1 <= 0.9101169306103679)
		tmp = log(((x + 1.0) / x)) / n;
	else
		tmp = ((1.0 / n) + ((((-0.5 / x) * t_2) - (n * -0.3333333333333333)) / (n * t_2))) / x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, n_] := Block[{t$95$0 = N[Power[x, N[(1.0 / n), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(N[Power[N[(x + 1.0), $MachinePrecision], N[(1.0 / n), $MachinePrecision]], $MachinePrecision] - t$95$0), $MachinePrecision]}, Block[{t$95$2 = N[(n * N[(x * x), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -2e-5], N[(1.0 - t$95$0), $MachinePrecision], If[LessEqual[t$95$1, 0.9101169306103679], N[(N[Log[N[(N[(x + 1.0), $MachinePrecision] / x), $MachinePrecision]], $MachinePrecision] / n), $MachinePrecision], N[(N[(N[(1.0 / n), $MachinePrecision] + N[(N[(N[(N[(-0.5 / x), $MachinePrecision] * t$95$2), $MachinePrecision] - N[(n * -0.3333333333333333), $MachinePrecision]), $MachinePrecision] / N[(n * t$95$2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if (-.f64 (pow.f64 (+.f64 x #s(literal 1 binary64)) (/.f64 #s(literal 1 binary64) n)) (pow.f64 x (/.f64 #s(literal 1 binary64) n))) < -2.00000000000000016e-5

   1. Initial program 98.8%

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

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{1 - e^{\frac{\log x}{n}}} \]
   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 98.8

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

   5. Applied rewrites 98.8%

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

   if -2.00000000000000016e-5 < (-.f64 (pow.f64 (+.f64 x #s(literal 1 binary64)) (/.f64 #s(literal 1 binary64) n)) (pow.f64 x (/.f64 #s(literal 1 binary64) n))) < 0.910116930610367914

   1. Initial program 45.9%

      \[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)} \]
   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 74.8

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

   5. Applied rewrites 74.8%

      \[\leadsto \color{blue}{\frac{\mathsf{log1p}\left(x\right) - \log x}{n}} \]
   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 74.8

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

      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. +-commutative N/A

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

      11. lift-+.f64 N/A

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

      12. lower-/.f64 75.0

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

   7. Applied rewrites 75.0%

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

   if 0.910116930610367914 < (-.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 44.7%

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

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

   5. Applied rewrites 6.0%

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

   6. Taylor expanded in x around inf

      \[\leadsto \color{blue}{\frac{\left(\frac{1}{n} + \frac{1}{3} \cdot \frac{1}{n \cdot {x}^{2}}\right) - \left(\frac{\frac{1}{2}}{n \cdot x} + \frac{1}{4} \cdot \frac{1}{n \cdot {x}^{3}}\right)}{x}} \]
   7. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\left(\frac{1}{n} + \frac{1}{3} \cdot \frac{1}{n \cdot {x}^{2}}\right) - \left(\frac{\frac{1}{2}}{n \cdot x} + \frac{1}{4} \cdot \frac{1}{n \cdot {x}^{3}}\right)}{x}} \]

   8. Applied rewrites 0.1%

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

   9. Taylor expanded in x around -inf

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

   11. Applied rewrites 11.3%

       \[\leadsto \frac{\color{blue}{\frac{1}{n} + \left(\frac{-0.5}{x \cdot n} - \frac{-0.3333333333333333}{x \cdot \left(x \cdot n\right)}\right)}}{x} \]
   12. Step-by-step derivation

       1. associate-/r* N/A

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

       2. lift-*.f64 N/A

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

       3. lift-*.f64 N/A

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

       4. frac-sub N/A

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

       5. lower-/.f64 N/A

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

       6. lower--.f64 N/A

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

       7. lower-*.f64 N/A

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

       8. lower-/.f64 N/A

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

       9. lift-*.f64 N/A

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

       10. lift-*.f64 N/A

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

       11. associate-*r* N/A

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

       12. pow2 N/A

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

       13. *-commutative N/A

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

       14. lower-*.f64 N/A

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

       15. pow2 N/A

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

       16. lower-*.f64 N/A

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

       17. lower-*.f64 N/A

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

       18. lower-*.f64 47.2

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

       19. lift-*.f64 N/A

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

       20. lift-*.f64 N/A

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

       21. associate-*r* N/A

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

   13. Applied rewrites 47.2%

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

Alternative 3: 85.8% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 0.92:\\ \;\;\;\;\frac{-0.16666666666666666 \cdot \frac{{\log x}^{3}}{n \cdot n} - \mathsf{fma}\left(0.5, \frac{{\log x}^{2}}{n}, \log x\right)}{n}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{n} \cdot \frac{{x}^{\left(\frac{1}{n}\right)}}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (if (<= x 0.92)
   (/
    (-
     (* -0.16666666666666666 (/ (pow (log x) 3.0) (* n n)))
     (fma 0.5 (/ (pow (log x) 2.0) n) (log x)))
    n)
   (* (/ 1.0 n) (/ (pow x (/ 1.0 n)) x))))

C

double code(double x, double n) {
	double tmp;
	if (x <= 0.92) {
		tmp = ((-0.16666666666666666 * (pow(log(x), 3.0) / (n * n))) - fma(0.5, (pow(log(x), 2.0) / n), log(x))) / n;
	} else {
		tmp = (1.0 / n) * (pow(x, (1.0 / n)) / x);
	}
	return tmp;
}

Julia

function code(x, n)
	tmp = 0.0
	if (x <= 0.92)
		tmp = Float64(Float64(Float64(-0.16666666666666666 * Float64((log(x) ^ 3.0) / Float64(n * n))) - fma(0.5, Float64((log(x) ^ 2.0) / n), log(x))) / n);
	else
		tmp = Float64(Float64(1.0 / n) * Float64((x ^ Float64(1.0 / n)) / x));
	end
	return tmp
end

Wolfram

code[x_, n_] := If[LessEqual[x, 0.92], N[(N[(N[(-0.16666666666666666 * N[(N[Power[N[Log[x], $MachinePrecision], 3.0], $MachinePrecision] / N[(n * n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(0.5 * N[(N[Power[N[Log[x], $MachinePrecision], 2.0], $MachinePrecision] / n), $MachinePrecision] + N[Log[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / n), $MachinePrecision], N[(N[(1.0 / n), $MachinePrecision] * N[(N[Power[x, N[(1.0 / n), $MachinePrecision]], $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 0.92000000000000004

   1. Initial program 46.3%

      \[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)} \]
   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 44.9

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

   5. Applied rewrites 44.9%

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

   6. Taylor expanded in n around inf

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

      1. lower-/.f64 N/A

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

   8. Applied rewrites 76.2%

      \[\leadsto \color{blue}{\frac{-0.16666666666666666 \cdot \frac{{\log x}^{3}}{n \cdot n} - \mathsf{fma}\left(0.5, \frac{{\log x}^{2}}{n}, \log x\right)}{n}} \]

   if 0.92000000000000004 < x

   1. Initial program 63.5%

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

   3. Taylor expanded in x around inf

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

      1. lower-/.f64 N/A

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

      2. log-rec N/A

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

      3. mul-1-neg N/A

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

      4. associate-*r/ N/A

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

      5. associate-*r* N/A

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

      6. metadata-eval N/A

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

      7. *-commutative N/A

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

      8. associate-/l* N/A

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

      9. exp-to-pow N/A

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

      10. lower-pow.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 97.8

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

   5. Applied rewrites 97.8%

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

      1. lift-/.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. frac-2neg N/A

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

      5. neg-mul-1 N/A

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

      8. distribute-rgt-neg-in N/A

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

      9. times-frac N/A

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

      10. frac-2neg N/A

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

      11. metadata-eval N/A

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

      12. lower-*.f64 N/A

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

      13. metadata-eval N/A

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

      14. frac-2neg N/A

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

      15. lower-/.f64 N/A

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

      16. lower-/.f64 N/A

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

      17. lower-neg.f64 98.4

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

   7. Applied rewrites 98.4%

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

4. Final simplification 86.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 0.92:\\ \;\;\;\;\frac{-0.16666666666666666 \cdot \frac{{\log x}^{3}}{n \cdot n} - \mathsf{fma}\left(0.5, \frac{{\log x}^{2}}{n}, \log x\right)}{n}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{n} \cdot \frac{{x}^{\left(\frac{1}{n}\right)}}{x}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 86.0% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 0.92:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(-0.16666666666666666, \frac{{\log x}^{3}}{n}, {\log x}^{2} \cdot -0.5\right)}{n} - \log x}{n}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{n} \cdot \frac{{x}^{\left(\frac{1}{n}\right)}}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (if (<= x 0.92)
   (/
    (-
     (/
      (fma
       -0.16666666666666666
       (/ (pow (log x) 3.0) n)
       (* (pow (log x) 2.0) -0.5))
      n)
     (log x))
    n)
   (* (/ 1.0 n) (/ (pow x (/ 1.0 n)) x))))

C

double code(double x, double n) {
	double tmp;
	if (x <= 0.92) {
		tmp = ((fma(-0.16666666666666666, (pow(log(x), 3.0) / n), (pow(log(x), 2.0) * -0.5)) / n) - log(x)) / n;
	} else {
		tmp = (1.0 / n) * (pow(x, (1.0 / n)) / x);
	}
	return tmp;
}

Julia

function code(x, n)
	tmp = 0.0
	if (x <= 0.92)
		tmp = Float64(Float64(Float64(fma(-0.16666666666666666, Float64((log(x) ^ 3.0) / n), Float64((log(x) ^ 2.0) * -0.5)) / n) - log(x)) / n);
	else
		tmp = Float64(Float64(1.0 / n) * Float64((x ^ Float64(1.0 / n)) / x));
	end
	return tmp
end

Wolfram

code[x_, n_] := If[LessEqual[x, 0.92], N[(N[(N[(N[(-0.16666666666666666 * N[(N[Power[N[Log[x], $MachinePrecision], 3.0], $MachinePrecision] / n), $MachinePrecision] + N[(N[Power[N[Log[x], $MachinePrecision], 2.0], $MachinePrecision] * -0.5), $MachinePrecision]), $MachinePrecision] / n), $MachinePrecision] - N[Log[x], $MachinePrecision]), $MachinePrecision] / n), $MachinePrecision], N[(N[(1.0 / n), $MachinePrecision] * N[(N[Power[x, N[(1.0 / n), $MachinePrecision]], $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 0.92000000000000004

   1. Initial program 46.3%

      \[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)} \]
   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 44.9

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

   5. Applied rewrites 44.9%

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

   6. Taylor expanded in n around -inf

      \[\leadsto \color{blue}{-1 \cdot \frac{-1 \cdot \frac{\frac{-1}{6} \cdot \frac{{\log x}^{3}}{n} - \frac{1}{2} \cdot {\log x}^{2}}{n} - -1 \cdot \log x}{n}} \]
   7. Step-by-step derivation

      1. mul-1-neg N/A

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

      2. distribute-neg-frac2 N/A

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

      3. mul-1-neg N/A

         \[\leadsto \frac{-1 \cdot \frac{\frac{-1}{6} \cdot \frac{{\log x}^{3}}{n} - \frac{1}{2} \cdot {\log x}^{2}}{n} - -1 \cdot \log x}{\color{blue}{-1 \cdot n}} \]

      4. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{-1 \cdot \frac{\frac{-1}{6} \cdot \frac{{\log x}^{3}}{n} - \frac{1}{2} \cdot {\log x}^{2}}{n} - -1 \cdot \log x}{-1 \cdot n}} \]

   8. Applied rewrites 76.2%

      \[\leadsto \color{blue}{\frac{\frac{\mathsf{fma}\left(-0.16666666666666666, \frac{{\log x}^{3}}{n}, -0.5 \cdot {\log x}^{2}\right)}{-n} + \log x}{-n}} \]

   if 0.92000000000000004 < x

   1. Initial program 63.5%

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

   3. Taylor expanded in x around inf

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

      1. lower-/.f64 N/A

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

      2. log-rec N/A

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

      3. mul-1-neg N/A

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

      4. associate-*r/ N/A

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

      5. associate-*r* N/A

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

      6. metadata-eval N/A

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

      7. *-commutative N/A

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

      8. associate-/l* N/A

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

      9. exp-to-pow N/A

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

      10. lower-pow.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 97.8

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

   5. Applied rewrites 97.8%

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

      1. lift-/.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. frac-2neg N/A

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

      5. neg-mul-1 N/A

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

      8. distribute-rgt-neg-in N/A

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

      9. times-frac N/A

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

      10. frac-2neg N/A

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

      11. metadata-eval N/A

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

      12. lower-*.f64 N/A

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

      13. metadata-eval N/A

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

      14. frac-2neg N/A

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

      15. lower-/.f64 N/A

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

      16. lower-/.f64 N/A

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

      17. lower-neg.f64 98.4

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

   7. Applied rewrites 98.4%

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

4. Final simplification 86.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 0.92:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(-0.16666666666666666, \frac{{\log x}^{3}}{n}, {\log x}^{2} \cdot -0.5\right)}{n} - \log x}{n}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{n} \cdot \frac{{x}^{\left(\frac{1}{n}\right)}}{x}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 81.9% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{if}\;\frac{1}{n} \leq -4 \cdot 10^{-29}:\\ \;\;\;\;\frac{t\_0}{x \cdot n}\\ \mathbf{elif}\;\frac{1}{n} \leq 10^{-97}:\\ \;\;\;\;\frac{\log \left(\frac{x}{x + 1}\right)}{-n}\\ \mathbf{elif}\;\frac{1}{n} \leq 2 \cdot 10^{-34}:\\ \;\;\;\;\frac{1}{x \cdot \mathsf{fma}\left(0.5, \frac{n}{x}, n\right)}\\ \mathbf{elif}\;\frac{1}{n} \leq 2 \cdot 10^{+145}:\\ \;\;\;\;\mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, \frac{0.16666666666666666}{n \cdot \left(n \cdot n\right)} + \left(\frac{0.3333333333333333}{n} + \frac{-0.5}{n \cdot n}\right), \frac{-0.5 + \frac{0.5}{n}}{n}\right), \frac{1}{n}\right), 1\right) - t\_0\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x, \frac{\mathsf{fma}\left(x, \frac{0.5}{n}, \mathsf{fma}\left(x, -0.5, 1\right)\right)}{n}, 1\right) - t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (let* ((t_0 (pow x (/ 1.0 n))))
   (if (<= (/ 1.0 n) -4e-29)
     (/ t_0 (* x n))
     (if (<= (/ 1.0 n) 1e-97)
       (/ (log (/ x (+ x 1.0))) (- n))
       (if (<= (/ 1.0 n) 2e-34)
         (/ 1.0 (* x (fma 0.5 (/ n x) n)))
         (if (<= (/ 1.0 n) 2e+145)
           (-
            (fma
             x
             (fma
              x
              (fma
               x
               (+
                (/ 0.16666666666666666 (* n (* n n)))
                (+ (/ 0.3333333333333333 n) (/ -0.5 (* n n))))
               (/ (+ -0.5 (/ 0.5 n)) n))
              (/ 1.0 n))
             1.0)
            t_0)
           (- (fma x (/ (fma x (/ 0.5 n) (fma x -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) <= -4e-29) {
		tmp = t_0 / (x * n);
	} else if ((1.0 / n) <= 1e-97) {
		tmp = log((x / (x + 1.0))) / -n;
	} else if ((1.0 / n) <= 2e-34) {
		tmp = 1.0 / (x * fma(0.5, (n / x), n));
	} else if ((1.0 / n) <= 2e+145) {
		tmp = fma(x, fma(x, fma(x, ((0.16666666666666666 / (n * (n * n))) + ((0.3333333333333333 / n) + (-0.5 / (n * n)))), ((-0.5 + (0.5 / n)) / n)), (1.0 / n)), 1.0) - t_0;
	} else {
		tmp = fma(x, (fma(x, (0.5 / n), fma(x, -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) <= -4e-29)
		tmp = Float64(t_0 / Float64(x * n));
	elseif (Float64(1.0 / n) <= 1e-97)
		tmp = Float64(log(Float64(x / Float64(x + 1.0))) / Float64(-n));
	elseif (Float64(1.0 / n) <= 2e-34)
		tmp = Float64(1.0 / Float64(x * fma(0.5, Float64(n / x), n)));
	elseif (Float64(1.0 / n) <= 2e+145)
		tmp = Float64(fma(x, fma(x, fma(x, Float64(Float64(0.16666666666666666 / Float64(n * Float64(n * n))) + Float64(Float64(0.3333333333333333 / n) + Float64(-0.5 / Float64(n * n)))), Float64(Float64(-0.5 + Float64(0.5 / n)) / n)), Float64(1.0 / n)), 1.0) - t_0);
	else
		tmp = Float64(fma(x, Float64(fma(x, Float64(0.5 / n), fma(x, -0.5, 1.0)) / n), 1.0) - t_0);
	end
	return tmp
end

Wolfram

code[x_, n_] := Block[{t$95$0 = N[Power[x, N[(1.0 / n), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[N[(1.0 / n), $MachinePrecision], -4e-29], N[(t$95$0 / N[(x * n), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 1e-97], N[(N[Log[N[(x / N[(x + 1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / (-n)), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 2e-34], N[(1.0 / N[(x * N[(0.5 * N[(n / x), $MachinePrecision] + n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 2e+145], N[(N[(x * N[(x * N[(x * N[(N[(0.16666666666666666 / N[(n * N[(n * n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(0.3333333333333333 / n), $MachinePrecision] + N[(-0.5 / N[(n * n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(-0.5 + N[(0.5 / n), $MachinePrecision]), $MachinePrecision] / n), $MachinePrecision]), $MachinePrecision] + N[(1.0 / n), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision] - t$95$0), $MachinePrecision], N[(N[(x * N[(N[(x * N[(0.5 / n), $MachinePrecision] + N[(x * -0.5 + 1.0), $MachinePrecision]), $MachinePrecision] / n), $MachinePrecision] + 1.0), $MachinePrecision] - t$95$0), $MachinePrecision]]]]]]

Derivation

1. Split input into 5 regimes

2. if (/.f64 #s(literal 1 binary64) n) < -3.99999999999999977e-29

   1. Initial program 88.6%

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

   3. Taylor expanded in x around inf

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

      1. lower-/.f64 N/A

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

      2. log-rec N/A

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

      3. mul-1-neg N/A

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

      4. associate-*r/ N/A

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

      5. associate-*r* N/A

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

      6. metadata-eval N/A

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

      7. *-commutative N/A

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

      8. associate-/l* N/A

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

      9. exp-to-pow N/A

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

      10. lower-pow.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 91.4

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

   5. Applied rewrites 91.4%

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

   if -3.99999999999999977e-29 < (/.f64 #s(literal 1 binary64) n) < 1.00000000000000004e-97

   1. Initial program 39.4%

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

   3. Taylor expanded in n around inf

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

      1. lower-/.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lower-log1p.f64 N/A

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

      4. lower-log.f64 77.3

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

   5. Applied rewrites 77.3%

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

      1. diff-log N/A

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

      2. +-commutative N/A

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

      3. lift-+.f64 N/A

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

      4. clear-num N/A

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

      5. lift-+.f64 N/A

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

      6. +-commutative N/A

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

      7. neg-log N/A

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

      8. diff-log N/A

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

      9. lift-log.f64 N/A

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

      10. lift-log1p.f64 N/A

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

      11. lift--.f64 N/A

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

      12. lower-neg.f64 77.3

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

      13. lift--.f64 N/A

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

      14. lift-log.f64 N/A

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

      15. lift-log1p.f64 N/A

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

      16. diff-log N/A

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

      17. lower-log.f64 N/A

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

      18. +-commutative N/A

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

      19. lift-+.f64 N/A

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

      20. lower-/.f64 77.5

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

   7. Applied rewrites 77.5%

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

   if 1.00000000000000004e-97 < (/.f64 #s(literal 1 binary64) n) < 1.99999999999999986e-34

   1. Initial program 4.0%

      \[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)} \]
   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 30.2

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

   5. Applied rewrites 30.2%

      \[\leadsto \color{blue}{\frac{\mathsf{log1p}\left(x\right) - \log x}{n}} \]
   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. clear-num N/A

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

      5. lower-/.f64 N/A

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

      6. lower-/.f64 30.2

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

      7. lift--.f64 N/A

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

      8. lift-log1p.f64 N/A

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

      9. lift-log.f64 N/A

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

      10. diff-log N/A

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

      11. lower-log.f64 N/A

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

      12. +-commutative N/A

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

      13. lift-+.f64 N/A

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

      14. lower-/.f64 30.2

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

   7. Applied rewrites 30.2%

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

   8. Taylor expanded in x around inf

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

      1. lower-*.f64 N/A

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

      2. +-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-/.f64 75.2

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

   10. Applied rewrites 75.2%

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

   if 1.99999999999999986e-34 < (/.f64 #s(literal 1 binary64) n) < 2e145

   1. Initial program 56.6%

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

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

   5. Taylor expanded in x around 0

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

   7. Applied rewrites 75.7%

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

   if 2e145 < (/.f64 #s(literal 1 binary64) n)

   1. Initial program 35.4%

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

   3. Taylor expanded in x around 0

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

      1. remove-double-neg N/A

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

      2. mul-1-neg N/A

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

      3. distribute-neg-frac N/A

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

      4. mul-1-neg N/A

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

      5. log-rec N/A

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

      6. mul-1-neg N/A

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

      7. lower--.f64 N/A

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

   5. Applied rewrites 67.7%

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

   6. Taylor expanded in n around inf

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

      1. lower-/.f64 N/A

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

      2. associate-+r+ N/A

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

      3. +-commutative N/A

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

      4. associate-*r/ N/A

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

      5. *-commutative N/A

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

      6. associate-/l* N/A

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

      7. metadata-eval N/A

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

      8. associate-*r/ N/A

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

      9. lower-fma.f64 N/A

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

      10. associate-*r/ N/A

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

      11. metadata-eval N/A

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

      12. lower-/.f64 N/A

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

      13. +-commutative N/A

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

      14. *-commutative N/A

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

      15. lower-fma.f64 79.9

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

   8. Applied rewrites 79.9%

      \[\leadsto \mathsf{fma}\left(x, \color{blue}{\frac{\mathsf{fma}\left(x, \frac{0.5}{n}, \mathsf{fma}\left(x, -0.5, 1\right)\right)}{n}}, 1\right) - {x}^{\left(\frac{1}{n}\right)} \]
3. Recombined 5 regimes into one program.

4. Final simplification 81.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{1}{n} \leq -4 \cdot 10^{-29}:\\ \;\;\;\;\frac{{x}^{\left(\frac{1}{n}\right)}}{x \cdot n}\\ \mathbf{elif}\;\frac{1}{n} \leq 10^{-97}:\\ \;\;\;\;\frac{\log \left(\frac{x}{x + 1}\right)}{-n}\\ \mathbf{elif}\;\frac{1}{n} \leq 2 \cdot 10^{-34}:\\ \;\;\;\;\frac{1}{x \cdot \mathsf{fma}\left(0.5, \frac{n}{x}, n\right)}\\ \mathbf{elif}\;\frac{1}{n} \leq 2 \cdot 10^{+145}:\\ \;\;\;\;\mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, \frac{0.16666666666666666}{n \cdot \left(n \cdot n\right)} + \left(\frac{0.3333333333333333}{n} + \frac{-0.5}{n \cdot n}\right), \frac{-0.5 + \frac{0.5}{n}}{n}\right), \frac{1}{n}\right), 1\right) - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x, \frac{\mathsf{fma}\left(x, \frac{0.5}{n}, \mathsf{fma}\left(x, -0.5, 1\right)\right)}{n}, 1\right) - {x}^{\left(\frac{1}{n}\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 85.7% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{if}\;\frac{1}{n} \leq -4 \cdot 10^{-29}:\\ \;\;\;\;\frac{t\_0}{x \cdot n}\\ \mathbf{elif}\;\frac{1}{n} \leq 10^{-97}:\\ \;\;\;\;\frac{\log \left(\frac{x}{x + 1}\right)}{-n}\\ \mathbf{elif}\;\frac{1}{n} \leq 2 \cdot 10^{-34}:\\ \;\;\;\;\frac{1}{x \cdot \mathsf{fma}\left(0.5, \frac{n}{x}, n\right)}\\ \mathbf{else}:\\ \;\;\;\;e^{\frac{x}{n}} - t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (let* ((t_0 (pow x (/ 1.0 n))))
   (if (<= (/ 1.0 n) -4e-29)
     (/ t_0 (* x n))
     (if (<= (/ 1.0 n) 1e-97)
       (/ (log (/ x (+ x 1.0))) (- n))
       (if (<= (/ 1.0 n) 2e-34)
         (/ 1.0 (* x (fma 0.5 (/ n x) n)))
         (- (exp (/ 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) <= -4e-29) {
		tmp = t_0 / (x * n);
	} else if ((1.0 / n) <= 1e-97) {
		tmp = log((x / (x + 1.0))) / -n;
	} else if ((1.0 / n) <= 2e-34) {
		tmp = 1.0 / (x * fma(0.5, (n / x), n));
	} else {
		tmp = exp((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) <= -4e-29)
		tmp = Float64(t_0 / Float64(x * n));
	elseif (Float64(1.0 / n) <= 1e-97)
		tmp = Float64(log(Float64(x / Float64(x + 1.0))) / Float64(-n));
	elseif (Float64(1.0 / n) <= 2e-34)
		tmp = Float64(1.0 / Float64(x * fma(0.5, Float64(n / x), n)));
	else
		tmp = Float64(exp(Float64(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], -4e-29], N[(t$95$0 / N[(x * n), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 1e-97], N[(N[Log[N[(x / N[(x + 1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / (-n)), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 2e-34], N[(1.0 / N[(x * N[(0.5 * N[(n / x), $MachinePrecision] + n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Exp[N[(x / n), $MachinePrecision]], $MachinePrecision] - t$95$0), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if (/.f64 #s(literal 1 binary64) n) < -3.99999999999999977e-29

   1. Initial program 88.6%

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

   3. Taylor expanded in x around inf

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

      1. lower-/.f64 N/A

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

      2. log-rec N/A

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

      3. mul-1-neg N/A

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

      4. associate-*r/ N/A

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

      5. associate-*r* N/A

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

      6. metadata-eval N/A

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

      7. *-commutative N/A

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

      8. associate-/l* N/A

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

      9. exp-to-pow N/A

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

      10. lower-pow.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 91.4

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

   5. Applied rewrites 91.4%

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

   if -3.99999999999999977e-29 < (/.f64 #s(literal 1 binary64) n) < 1.00000000000000004e-97

   1. Initial program 39.4%

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

   3. Taylor expanded in n around inf

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

      1. lower-/.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lower-log1p.f64 N/A

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

      4. lower-log.f64 77.3

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

   5. Applied rewrites 77.3%

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

      1. diff-log N/A

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

      2. +-commutative N/A

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

      3. lift-+.f64 N/A

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

      4. clear-num N/A

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

      5. lift-+.f64 N/A

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

      6. +-commutative N/A

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

      7. neg-log N/A

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

      8. diff-log N/A

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

      9. lift-log.f64 N/A

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

      10. lift-log1p.f64 N/A

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

      11. lift--.f64 N/A

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

      12. lower-neg.f64 77.3

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

      13. lift--.f64 N/A

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

      14. lift-log.f64 N/A

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

      15. lift-log1p.f64 N/A

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

      16. diff-log N/A

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

      17. lower-log.f64 N/A

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

      18. +-commutative N/A

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

      19. lift-+.f64 N/A

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

      20. lower-/.f64 77.5

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

   7. Applied rewrites 77.5%

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

   if 1.00000000000000004e-97 < (/.f64 #s(literal 1 binary64) n) < 1.99999999999999986e-34

   1. Initial program 4.0%

      \[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)} \]
   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 30.2

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

   5. Applied rewrites 30.2%

      \[\leadsto \color{blue}{\frac{\mathsf{log1p}\left(x\right) - \log x}{n}} \]
   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. clear-num N/A

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

      5. lower-/.f64 N/A

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

      6. lower-/.f64 30.2

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

      7. lift--.f64 N/A

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

      8. lift-log1p.f64 N/A

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

      9. lift-log.f64 N/A

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

      10. diff-log N/A

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

      11. lower-log.f64 N/A

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

      12. +-commutative N/A

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

      13. lift-+.f64 N/A

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

      14. lower-/.f64 30.2

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

   7. Applied rewrites 30.2%

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

   8. Taylor expanded in x around inf

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

      1. lower-*.f64 N/A

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

      2. +-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-/.f64 75.2

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

   10. Applied rewrites 75.2%

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

   if 1.99999999999999986e-34 < (/.f64 #s(literal 1 binary64) n)

   1. Initial program 45.0%

      \[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)} \]
   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 97.8

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

   4. Applied rewrites 97.8%

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

   5. Taylor expanded in x around 0

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

      1. lower-/.f64 97.8

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

   7. Applied rewrites 97.8%

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

4. Final simplification 85.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{1}{n} \leq -4 \cdot 10^{-29}:\\ \;\;\;\;\frac{{x}^{\left(\frac{1}{n}\right)}}{x \cdot n}\\ \mathbf{elif}\;\frac{1}{n} \leq 10^{-97}:\\ \;\;\;\;\frac{\log \left(\frac{x}{x + 1}\right)}{-n}\\ \mathbf{elif}\;\frac{1}{n} \leq 2 \cdot 10^{-34}:\\ \;\;\;\;\frac{1}{x \cdot \mathsf{fma}\left(0.5, \frac{n}{x}, n\right)}\\ \mathbf{else}:\\ \;\;\;\;e^{\frac{x}{n}} - {x}^{\left(\frac{1}{n}\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 82.8% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{if}\;\frac{1}{n} \leq -4 \cdot 10^{-29}:\\ \;\;\;\;\frac{t\_0}{x \cdot n}\\ \mathbf{elif}\;\frac{1}{n} \leq 10^{-97}:\\ \;\;\;\;\frac{\log \left(\frac{x}{x + 1}\right)}{-n}\\ \mathbf{elif}\;\frac{1}{n} \leq 4 \cdot 10^{-21}:\\ \;\;\;\;\frac{1}{x \cdot \mathsf{fma}\left(0.5, \frac{n}{x}, n\right)}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x, \frac{\mathsf{fma}\left(x, \frac{0.5}{n}, \mathsf{fma}\left(x, -0.5, 1\right)\right)}{n}, 1\right) - t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (let* ((t_0 (pow x (/ 1.0 n))))
   (if (<= (/ 1.0 n) -4e-29)
     (/ t_0 (* x n))
     (if (<= (/ 1.0 n) 1e-97)
       (/ (log (/ x (+ x 1.0))) (- n))
       (if (<= (/ 1.0 n) 4e-21)
         (/ 1.0 (* x (fma 0.5 (/ n x) n)))
         (- (fma x (/ (fma x (/ 0.5 n) (fma x -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) <= -4e-29) {
		tmp = t_0 / (x * n);
	} else if ((1.0 / n) <= 1e-97) {
		tmp = log((x / (x + 1.0))) / -n;
	} else if ((1.0 / n) <= 4e-21) {
		tmp = 1.0 / (x * fma(0.5, (n / x), n));
	} else {
		tmp = fma(x, (fma(x, (0.5 / n), fma(x, -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) <= -4e-29)
		tmp = Float64(t_0 / Float64(x * n));
	elseif (Float64(1.0 / n) <= 1e-97)
		tmp = Float64(log(Float64(x / Float64(x + 1.0))) / Float64(-n));
	elseif (Float64(1.0 / n) <= 4e-21)
		tmp = Float64(1.0 / Float64(x * fma(0.5, Float64(n / x), n)));
	else
		tmp = Float64(fma(x, Float64(fma(x, Float64(0.5 / n), fma(x, -0.5, 1.0)) / n), 1.0) - t_0);
	end
	return tmp
end

Wolfram

code[x_, n_] := Block[{t$95$0 = N[Power[x, N[(1.0 / n), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[N[(1.0 / n), $MachinePrecision], -4e-29], N[(t$95$0 / N[(x * n), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 1e-97], N[(N[Log[N[(x / N[(x + 1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / (-n)), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 4e-21], N[(1.0 / N[(x * N[(0.5 * N[(n / x), $MachinePrecision] + n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x * N[(N[(x * N[(0.5 / n), $MachinePrecision] + N[(x * -0.5 + 1.0), $MachinePrecision]), $MachinePrecision] / n), $MachinePrecision] + 1.0), $MachinePrecision] - t$95$0), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if (/.f64 #s(literal 1 binary64) n) < -3.99999999999999977e-29

   1. Initial program 88.6%

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

   3. Taylor expanded in x around inf

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

      1. lower-/.f64 N/A

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

      2. log-rec N/A

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

      3. mul-1-neg N/A

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

      4. associate-*r/ N/A

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

      5. associate-*r* N/A

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

      6. metadata-eval N/A

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

      7. *-commutative N/A

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

      8. associate-/l* N/A

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

      9. exp-to-pow N/A

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

      10. lower-pow.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 91.4

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

   5. Applied rewrites 91.4%

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

   if -3.99999999999999977e-29 < (/.f64 #s(literal 1 binary64) n) < 1.00000000000000004e-97

   1. Initial program 39.4%

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

   3. Taylor expanded in n around inf

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

      1. lower-/.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lower-log1p.f64 N/A

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

      4. lower-log.f64 77.3

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

   5. Applied rewrites 77.3%

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

      1. diff-log N/A

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

      2. +-commutative N/A

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

      3. lift-+.f64 N/A

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

      4. clear-num N/A

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

      5. lift-+.f64 N/A

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

      6. +-commutative N/A

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

      7. neg-log N/A

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

      8. diff-log N/A

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

      9. lift-log.f64 N/A

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

      10. lift-log1p.f64 N/A

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

      11. lift--.f64 N/A

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

      12. lower-neg.f64 77.3

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

      13. lift--.f64 N/A

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

      14. lift-log.f64 N/A

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

      15. lift-log1p.f64 N/A

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

      16. diff-log N/A

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

      17. lower-log.f64 N/A

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

      18. +-commutative N/A

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

      19. lift-+.f64 N/A

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

      20. lower-/.f64 77.5

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

   7. Applied rewrites 77.5%

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

   if 1.00000000000000004e-97 < (/.f64 #s(literal 1 binary64) n) < 3.99999999999999963e-21

   1. Initial program 4.0%

      \[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)} \]
   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 35.9

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

   5. Applied rewrites 35.9%

      \[\leadsto \color{blue}{\frac{\mathsf{log1p}\left(x\right) - \log x}{n}} \]
   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. clear-num N/A

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

      5. lower-/.f64 N/A

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

      6. lower-/.f64 35.9

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

      7. lift--.f64 N/A

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

      8. lift-log1p.f64 N/A

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

      9. lift-log.f64 N/A

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

      10. diff-log N/A

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

      11. lower-log.f64 N/A

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

      12. +-commutative N/A

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

      13. lift-+.f64 N/A

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

      14. lower-/.f64 35.9

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

   7. Applied rewrites 35.9%

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

   8. Taylor expanded in x around inf

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

      1. lower-*.f64 N/A

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

      2. +-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-/.f64 69.2

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

   10. Applied rewrites 69.2%

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

   if 3.99999999999999963e-21 < (/.f64 #s(literal 1 binary64) n)

   1. Initial program 46.0%

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

   3. Taylor expanded in x around 0

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

      1. remove-double-neg N/A

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

      2. mul-1-neg N/A

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

      3. distribute-neg-frac N/A

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

      4. mul-1-neg N/A

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

      5. log-rec N/A

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

      6. mul-1-neg N/A

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

      7. lower--.f64 N/A

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

   5. Applied rewrites 63.5%

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

   6. Taylor expanded in n around inf

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

      1. lower-/.f64 N/A

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

      2. associate-+r+ N/A

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

      3. +-commutative N/A

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

      4. associate-*r/ N/A

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

      5. *-commutative N/A

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

      6. associate-/l* N/A

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

      7. metadata-eval N/A

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

      8. associate-*r/ N/A

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

      9. lower-fma.f64 N/A

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

      10. associate-*r/ N/A

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

      11. metadata-eval N/A

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

      12. lower-/.f64 N/A

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

      13. +-commutative N/A

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

      14. *-commutative N/A

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

      15. lower-fma.f64 70.4

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

   8. Applied rewrites 70.4%

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

4. Final simplification 80.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{1}{n} \leq -4 \cdot 10^{-29}:\\ \;\;\;\;\frac{{x}^{\left(\frac{1}{n}\right)}}{x \cdot n}\\ \mathbf{elif}\;\frac{1}{n} \leq 10^{-97}:\\ \;\;\;\;\frac{\log \left(\frac{x}{x + 1}\right)}{-n}\\ \mathbf{elif}\;\frac{1}{n} \leq 4 \cdot 10^{-21}:\\ \;\;\;\;\frac{1}{x \cdot \mathsf{fma}\left(0.5, \frac{n}{x}, n\right)}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x, \frac{\mathsf{fma}\left(x, \frac{0.5}{n}, \mathsf{fma}\left(x, -0.5, 1\right)\right)}{n}, 1\right) - {x}^{\left(\frac{1}{n}\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 82.7% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{if}\;\frac{1}{n} \leq -4 \cdot 10^{-29}:\\ \;\;\;\;\frac{t\_0}{x \cdot n}\\ \mathbf{elif}\;\frac{1}{n} \leq 10^{-97}:\\ \;\;\;\;\frac{\log \left(\frac{x}{x + 1}\right)}{-n}\\ \mathbf{elif}\;\frac{1}{n} \leq 4 \cdot 10^{-21}:\\ \;\;\;\;\frac{1}{x \cdot \mathsf{fma}\left(0.5, \frac{n}{x}, n\right)}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x, \mathsf{fma}\left(x, \frac{0.5}{n \cdot n}, \frac{1}{n}\right), 1\right) - t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (let* ((t_0 (pow x (/ 1.0 n))))
   (if (<= (/ 1.0 n) -4e-29)
     (/ t_0 (* x n))
     (if (<= (/ 1.0 n) 1e-97)
       (/ (log (/ x (+ x 1.0))) (- n))
       (if (<= (/ 1.0 n) 4e-21)
         (/ 1.0 (* x (fma 0.5 (/ n 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) <= -4e-29) {
		tmp = t_0 / (x * n);
	} else if ((1.0 / n) <= 1e-97) {
		tmp = log((x / (x + 1.0))) / -n;
	} else if ((1.0 / n) <= 4e-21) {
		tmp = 1.0 / (x * fma(0.5, (n / 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) <= -4e-29)
		tmp = Float64(t_0 / Float64(x * n));
	elseif (Float64(1.0 / n) <= 1e-97)
		tmp = Float64(log(Float64(x / Float64(x + 1.0))) / Float64(-n));
	elseif (Float64(1.0 / n) <= 4e-21)
		tmp = Float64(1.0 / Float64(x * fma(0.5, Float64(n / x), n)));
	else
		tmp = Float64(fma(x, fma(x, Float64(0.5 / Float64(n * n)), Float64(1.0 / n)), 1.0) - t_0);
	end
	return tmp
end

Wolfram

code[x_, n_] := Block[{t$95$0 = N[Power[x, N[(1.0 / n), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[N[(1.0 / n), $MachinePrecision], -4e-29], N[(t$95$0 / N[(x * n), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 1e-97], N[(N[Log[N[(x / N[(x + 1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / (-n)), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 4e-21], N[(1.0 / N[(x * N[(0.5 * N[(n / x), $MachinePrecision] + n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x * N[(x * N[(0.5 / N[(n * n), $MachinePrecision]), $MachinePrecision] + N[(1.0 / n), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision] - t$95$0), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if (/.f64 #s(literal 1 binary64) n) < -3.99999999999999977e-29

   1. Initial program 88.6%

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

   3. Taylor expanded in x around inf

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

      1. lower-/.f64 N/A

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

      2. log-rec N/A

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

      3. mul-1-neg N/A

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

      4. associate-*r/ N/A

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

      5. associate-*r* N/A

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

      6. metadata-eval N/A

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

      7. *-commutative N/A

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

      8. associate-/l* N/A

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

      9. exp-to-pow N/A

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

      10. lower-pow.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 91.4

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

   5. Applied rewrites 91.4%

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

   if -3.99999999999999977e-29 < (/.f64 #s(literal 1 binary64) n) < 1.00000000000000004e-97

   1. Initial program 39.4%

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

   3. Taylor expanded in n around inf

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

      1. lower-/.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lower-log1p.f64 N/A

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

      4. lower-log.f64 77.3

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

   5. Applied rewrites 77.3%

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

      1. diff-log N/A

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

      2. +-commutative N/A

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

      3. lift-+.f64 N/A

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

      4. clear-num N/A

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

      5. lift-+.f64 N/A

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

      6. +-commutative N/A

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

      7. neg-log N/A

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

      8. diff-log N/A

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

      9. lift-log.f64 N/A

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

      10. lift-log1p.f64 N/A

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

      11. lift--.f64 N/A

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

      12. lower-neg.f64 77.3

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

      13. lift--.f64 N/A

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

      14. lift-log.f64 N/A

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

      15. lift-log1p.f64 N/A

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

      16. diff-log N/A

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

      17. lower-log.f64 N/A

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

      18. +-commutative N/A

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

      19. lift-+.f64 N/A

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

      20. lower-/.f64 77.5

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

   7. Applied rewrites 77.5%

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

   if 1.00000000000000004e-97 < (/.f64 #s(literal 1 binary64) n) < 3.99999999999999963e-21

   1. Initial program 4.0%

      \[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)} \]
   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 35.9

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

   5. Applied rewrites 35.9%

      \[\leadsto \color{blue}{\frac{\mathsf{log1p}\left(x\right) - \log x}{n}} \]
   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. clear-num N/A

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

      5. lower-/.f64 N/A

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

      6. lower-/.f64 35.9

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

      7. lift--.f64 N/A

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

      8. lift-log1p.f64 N/A

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

      9. lift-log.f64 N/A

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

      10. diff-log N/A

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

      11. lower-log.f64 N/A

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

      12. +-commutative N/A

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

      13. lift-+.f64 N/A

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

      14. lower-/.f64 35.9

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

   7. Applied rewrites 35.9%

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

   8. Taylor expanded in x around inf

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

      1. lower-*.f64 N/A

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

      2. +-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-/.f64 69.2

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

   10. Applied rewrites 69.2%

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

   if 3.99999999999999963e-21 < (/.f64 #s(literal 1 binary64) n)

   1. Initial program 46.0%

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

   3. Taylor expanded in x around 0

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

      1. remove-double-neg N/A

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

      2. mul-1-neg N/A

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

      3. distribute-neg-frac N/A

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

      4. mul-1-neg N/A

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

      5. log-rec N/A

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

      6. mul-1-neg N/A

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

      7. lower--.f64 N/A

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

   5. Applied rewrites 63.5%

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

   6. Taylor expanded in n around 0

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

      1. lower-/.f64 N/A

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

      2. unpow2 N/A

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

      3. lower-*.f64 63.5

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

   8. Applied rewrites 63.5%

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

4. Final simplification 79.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{1}{n} \leq -4 \cdot 10^{-29}:\\ \;\;\;\;\frac{{x}^{\left(\frac{1}{n}\right)}}{x \cdot n}\\ \mathbf{elif}\;\frac{1}{n} \leq 10^{-97}:\\ \;\;\;\;\frac{\log \left(\frac{x}{x + 1}\right)}{-n}\\ \mathbf{elif}\;\frac{1}{n} \leq 4 \cdot 10^{-21}:\\ \;\;\;\;\frac{1}{x \cdot \mathsf{fma}\left(0.5, \frac{n}{x}, n\right)}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x, \mathsf{fma}\left(x, \frac{0.5}{n \cdot n}, \frac{1}{n}\right), 1\right) - {x}^{\left(\frac{1}{n}\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 81.7% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {x}^{\left(\frac{1}{n}\right)}\\ t_1 := n \cdot \left(x \cdot x\right)\\ \mathbf{if}\;\frac{1}{n} \leq -4 \cdot 10^{-29}:\\ \;\;\;\;\frac{t\_0}{x \cdot n}\\ \mathbf{elif}\;\frac{1}{n} \leq 10^{-97}:\\ \;\;\;\;\frac{\log \left(\frac{x}{x + 1}\right)}{-n}\\ \mathbf{elif}\;\frac{1}{n} \leq 4 \cdot 10^{-21}:\\ \;\;\;\;\frac{1}{x \cdot \mathsf{fma}\left(0.5, \frac{n}{x}, n\right)}\\ \mathbf{elif}\;\frac{1}{n} \leq 10^{+93}:\\ \;\;\;\;\left(\frac{x}{n} + 1\right) - t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{n} + \frac{\frac{-0.5}{x} \cdot t\_1 - n \cdot -0.3333333333333333}{n \cdot t\_1}}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (let* ((t_0 (pow x (/ 1.0 n))) (t_1 (* n (* x x))))
   (if (<= (/ 1.0 n) -4e-29)
     (/ t_0 (* x n))
     (if (<= (/ 1.0 n) 1e-97)
       (/ (log (/ x (+ x 1.0))) (- n))
       (if (<= (/ 1.0 n) 4e-21)
         (/ 1.0 (* x (fma 0.5 (/ n x) n)))
         (if (<= (/ 1.0 n) 1e+93)
           (- (+ (/ x n) 1.0) t_0)
           (/
            (+
             (/ 1.0 n)
             (/ (- (* (/ -0.5 x) t_1) (* n -0.3333333333333333)) (* n t_1)))
            x)))))))

C

double code(double x, double n) {
	double t_0 = pow(x, (1.0 / n));
	double t_1 = n * (x * x);
	double tmp;
	if ((1.0 / n) <= -4e-29) {
		tmp = t_0 / (x * n);
	} else if ((1.0 / n) <= 1e-97) {
		tmp = log((x / (x + 1.0))) / -n;
	} else if ((1.0 / n) <= 4e-21) {
		tmp = 1.0 / (x * fma(0.5, (n / x), n));
	} else if ((1.0 / n) <= 1e+93) {
		tmp = ((x / n) + 1.0) - t_0;
	} else {
		tmp = ((1.0 / n) + ((((-0.5 / x) * t_1) - (n * -0.3333333333333333)) / (n * t_1))) / x;
	}
	return tmp;
}

Julia

function code(x, n)
	t_0 = x ^ Float64(1.0 / n)
	t_1 = Float64(n * Float64(x * x))
	tmp = 0.0
	if (Float64(1.0 / n) <= -4e-29)
		tmp = Float64(t_0 / Float64(x * n));
	elseif (Float64(1.0 / n) <= 1e-97)
		tmp = Float64(log(Float64(x / Float64(x + 1.0))) / Float64(-n));
	elseif (Float64(1.0 / n) <= 4e-21)
		tmp = Float64(1.0 / Float64(x * fma(0.5, Float64(n / x), n)));
	elseif (Float64(1.0 / n) <= 1e+93)
		tmp = Float64(Float64(Float64(x / n) + 1.0) - t_0);
	else
		tmp = Float64(Float64(Float64(1.0 / n) + Float64(Float64(Float64(Float64(-0.5 / x) * t_1) - Float64(n * -0.3333333333333333)) / Float64(n * t_1))) / x);
	end
	return 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 * N[(x * x), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(1.0 / n), $MachinePrecision], -4e-29], N[(t$95$0 / N[(x * n), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 1e-97], N[(N[Log[N[(x / N[(x + 1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / (-n)), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 4e-21], N[(1.0 / N[(x * N[(0.5 * N[(n / x), $MachinePrecision] + n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 1e+93], N[(N[(N[(x / n), $MachinePrecision] + 1.0), $MachinePrecision] - t$95$0), $MachinePrecision], N[(N[(N[(1.0 / n), $MachinePrecision] + N[(N[(N[(N[(-0.5 / x), $MachinePrecision] * t$95$1), $MachinePrecision] - N[(n * -0.3333333333333333), $MachinePrecision]), $MachinePrecision] / N[(n * t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]]]]]]]

Derivation

1. Split input into 5 regimes

2. if (/.f64 #s(literal 1 binary64) n) < -3.99999999999999977e-29

   1. Initial program 88.6%

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

   3. Taylor expanded in x around inf

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

      1. lower-/.f64 N/A

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

      2. log-rec N/A

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

      3. mul-1-neg N/A

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

      4. associate-*r/ N/A

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

      5. associate-*r* N/A

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

      6. metadata-eval N/A

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

      7. *-commutative N/A

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

      8. associate-/l* N/A

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

      9. exp-to-pow N/A

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

      10. lower-pow.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 91.4

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

   5. Applied rewrites 91.4%

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

   if -3.99999999999999977e-29 < (/.f64 #s(literal 1 binary64) n) < 1.00000000000000004e-97

   1. Initial program 39.4%

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

   3. Taylor expanded in n around inf

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

      1. lower-/.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lower-log1p.f64 N/A

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

      4. lower-log.f64 77.3

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

   5. Applied rewrites 77.3%

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

      1. diff-log N/A

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

      2. +-commutative N/A

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

      3. lift-+.f64 N/A

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

      4. clear-num N/A

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

      5. lift-+.f64 N/A

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

      6. +-commutative N/A

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

      7. neg-log N/A

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

      8. diff-log N/A

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

      9. lift-log.f64 N/A

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

      10. lift-log1p.f64 N/A

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

      11. lift--.f64 N/A

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

      12. lower-neg.f64 77.3

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

      13. lift--.f64 N/A

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

      14. lift-log.f64 N/A

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

      15. lift-log1p.f64 N/A

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

      16. diff-log N/A

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

      17. lower-log.f64 N/A

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

      18. +-commutative N/A

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

      19. lift-+.f64 N/A

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

      20. lower-/.f64 77.5

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

   7. Applied rewrites 77.5%

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

   if 1.00000000000000004e-97 < (/.f64 #s(literal 1 binary64) n) < 3.99999999999999963e-21

   1. Initial program 4.0%

      \[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)} \]
   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 35.9

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

   5. Applied rewrites 35.9%

      \[\leadsto \color{blue}{\frac{\mathsf{log1p}\left(x\right) - \log x}{n}} \]
   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. clear-num N/A

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

      5. lower-/.f64 N/A

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

      6. lower-/.f64 35.9

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

      7. lift--.f64 N/A

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

      8. lift-log1p.f64 N/A

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

      9. lift-log.f64 N/A

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

      10. diff-log N/A

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

      11. lower-log.f64 N/A

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

      12. +-commutative N/A

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

      13. lift-+.f64 N/A

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

      14. lower-/.f64 35.9

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

   7. Applied rewrites 35.9%

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

   8. Taylor expanded in x around inf

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

      1. lower-*.f64 N/A

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

      2. +-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-/.f64 69.2

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

   10. Applied rewrites 69.2%

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

   if 3.99999999999999963e-21 < (/.f64 #s(literal 1 binary64) n) < 1.00000000000000004e93

   1. Initial program 79.5%

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

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\left(1 + \frac{x}{n}\right)} - {x}^{\left(\frac{1}{n}\right)} \]
   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 76.1

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

   5. Applied rewrites 76.1%

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

   if 1.00000000000000004e93 < (/.f64 #s(literal 1 binary64) n)

   1. Initial program 29.8%

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

   3. Taylor expanded in n around inf

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

      1. lower-/.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lower-log1p.f64 N/A

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

      4. lower-log.f64 5.8

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

   5. Applied rewrites 5.8%

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

   6. Taylor expanded in x around inf

      \[\leadsto \color{blue}{\frac{\left(\frac{1}{n} + \frac{1}{3} \cdot \frac{1}{n \cdot {x}^{2}}\right) - \left(\frac{\frac{1}{2}}{n \cdot x} + \frac{1}{4} \cdot \frac{1}{n \cdot {x}^{3}}\right)}{x}} \]
   7. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\left(\frac{1}{n} + \frac{1}{3} \cdot \frac{1}{n \cdot {x}^{2}}\right) - \left(\frac{\frac{1}{2}}{n \cdot x} + \frac{1}{4} \cdot \frac{1}{n \cdot {x}^{3}}\right)}{x}} \]

   8. Applied rewrites 0.0%

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

   9. Taylor expanded in x around -inf

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

   11. Applied rewrites 14.9%

       \[\leadsto \frac{\color{blue}{\frac{1}{n} + \left(\frac{-0.5}{x \cdot n} - \frac{-0.3333333333333333}{x \cdot \left(x \cdot n\right)}\right)}}{x} \]
   12. Step-by-step derivation

       1. associate-/r* N/A

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

       2. lift-*.f64 N/A

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

       3. lift-*.f64 N/A

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

       4. frac-sub N/A

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

       5. lower-/.f64 N/A

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

       6. lower--.f64 N/A

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

       7. lower-*.f64 N/A

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

       8. lower-/.f64 N/A

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

       9. lift-*.f64 N/A

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

       10. lift-*.f64 N/A

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

       11. associate-*r* N/A

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

       12. pow2 N/A

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

       13. *-commutative N/A

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

       14. lower-*.f64 N/A

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

       15. pow2 N/A

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

       16. lower-*.f64 N/A

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

       17. lower-*.f64 N/A

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

       18. lower-*.f64 66.8

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

       19. lift-*.f64 N/A

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

       20. lift-*.f64 N/A

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

       21. associate-*r* N/A

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

   13. Applied rewrites 66.8%

       \[\leadsto \frac{\frac{1}{n} + \color{blue}{\frac{\frac{-0.5}{x} \cdot \left(n \cdot \left(x \cdot x\right)\right) - n \cdot -0.3333333333333333}{n \cdot \left(n \cdot \left(x \cdot x\right)\right)}}}{x} \]
3. Recombined 5 regimes into one program.

4. Final simplification 80.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{1}{n} \leq -4 \cdot 10^{-29}:\\ \;\;\;\;\frac{{x}^{\left(\frac{1}{n}\right)}}{x \cdot n}\\ \mathbf{elif}\;\frac{1}{n} \leq 10^{-97}:\\ \;\;\;\;\frac{\log \left(\frac{x}{x + 1}\right)}{-n}\\ \mathbf{elif}\;\frac{1}{n} \leq 4 \cdot 10^{-21}:\\ \;\;\;\;\frac{1}{x \cdot \mathsf{fma}\left(0.5, \frac{n}{x}, n\right)}\\ \mathbf{elif}\;\frac{1}{n} \leq 10^{+93}:\\ \;\;\;\;\left(\frac{x}{n} + 1\right) - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{n} + \frac{\frac{-0.5}{x} \cdot \left(n \cdot \left(x \cdot x\right)\right) - n \cdot -0.3333333333333333}{n \cdot \left(n \cdot \left(x \cdot x\right)\right)}}{x}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 81.1% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := n \cdot \left(x \cdot x\right)\\ t_1 := {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{if}\;\frac{1}{n} \leq -4 \cdot 10^{-29}:\\ \;\;\;\;\frac{t\_1}{x \cdot n}\\ \mathbf{elif}\;\frac{1}{n} \leq 10^{-97}:\\ \;\;\;\;\frac{\log \left(\frac{x}{x + 1}\right)}{-n}\\ \mathbf{elif}\;\frac{1}{n} \leq 2 \cdot 10^{-34}:\\ \;\;\;\;\frac{1}{x \cdot \mathsf{fma}\left(0.5, \frac{n}{x}, n\right)}\\ \mathbf{elif}\;\frac{1}{n} \leq 10^{+93}:\\ \;\;\;\;1 - t\_1\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{n} + \frac{\frac{-0.5}{x} \cdot t\_0 - n \cdot -0.3333333333333333}{n \cdot t\_0}}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (let* ((t_0 (* n (* x x))) (t_1 (pow x (/ 1.0 n))))
   (if (<= (/ 1.0 n) -4e-29)
     (/ t_1 (* x n))
     (if (<= (/ 1.0 n) 1e-97)
       (/ (log (/ x (+ x 1.0))) (- n))
       (if (<= (/ 1.0 n) 2e-34)
         (/ 1.0 (* x (fma 0.5 (/ n x) n)))
         (if (<= (/ 1.0 n) 1e+93)
           (- 1.0 t_1)
           (/
            (+
             (/ 1.0 n)
             (/ (- (* (/ -0.5 x) t_0) (* n -0.3333333333333333)) (* n t_0)))
            x)))))))

C

double code(double x, double n) {
	double t_0 = n * (x * x);
	double t_1 = pow(x, (1.0 / n));
	double tmp;
	if ((1.0 / n) <= -4e-29) {
		tmp = t_1 / (x * n);
	} else if ((1.0 / n) <= 1e-97) {
		tmp = log((x / (x + 1.0))) / -n;
	} else if ((1.0 / n) <= 2e-34) {
		tmp = 1.0 / (x * fma(0.5, (n / x), n));
	} else if ((1.0 / n) <= 1e+93) {
		tmp = 1.0 - t_1;
	} else {
		tmp = ((1.0 / n) + ((((-0.5 / x) * t_0) - (n * -0.3333333333333333)) / (n * t_0))) / x;
	}
	return tmp;
}

Julia

function code(x, n)
	t_0 = Float64(n * Float64(x * x))
	t_1 = x ^ Float64(1.0 / n)
	tmp = 0.0
	if (Float64(1.0 / n) <= -4e-29)
		tmp = Float64(t_1 / Float64(x * n));
	elseif (Float64(1.0 / n) <= 1e-97)
		tmp = Float64(log(Float64(x / Float64(x + 1.0))) / Float64(-n));
	elseif (Float64(1.0 / n) <= 2e-34)
		tmp = Float64(1.0 / Float64(x * fma(0.5, Float64(n / x), n)));
	elseif (Float64(1.0 / n) <= 1e+93)
		tmp = Float64(1.0 - t_1);
	else
		tmp = Float64(Float64(Float64(1.0 / n) + Float64(Float64(Float64(Float64(-0.5 / x) * t_0) - Float64(n * -0.3333333333333333)) / Float64(n * t_0))) / x);
	end
	return tmp
end

Wolfram

code[x_, n_] := Block[{t$95$0 = N[(n * N[(x * x), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[Power[x, N[(1.0 / n), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[N[(1.0 / n), $MachinePrecision], -4e-29], N[(t$95$1 / N[(x * n), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 1e-97], N[(N[Log[N[(x / N[(x + 1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / (-n)), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 2e-34], N[(1.0 / N[(x * N[(0.5 * N[(n / x), $MachinePrecision] + n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 1e+93], N[(1.0 - t$95$1), $MachinePrecision], N[(N[(N[(1.0 / n), $MachinePrecision] + N[(N[(N[(N[(-0.5 / x), $MachinePrecision] * t$95$0), $MachinePrecision] - N[(n * -0.3333333333333333), $MachinePrecision]), $MachinePrecision] / N[(n * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]]]]]]]

Derivation

1. Split input into 5 regimes

2. if (/.f64 #s(literal 1 binary64) n) < -3.99999999999999977e-29

   1. Initial program 88.6%

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

   3. Taylor expanded in x around inf

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

      1. lower-/.f64 N/A

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

      2. log-rec N/A

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

      3. mul-1-neg N/A

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

      4. associate-*r/ N/A

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

      5. associate-*r* N/A

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

      6. metadata-eval N/A

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

      7. *-commutative N/A

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

      8. associate-/l* N/A

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

      9. exp-to-pow N/A

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

      10. lower-pow.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 91.4

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

   5. Applied rewrites 91.4%

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

   if -3.99999999999999977e-29 < (/.f64 #s(literal 1 binary64) n) < 1.00000000000000004e-97

   1. Initial program 39.4%

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

   3. Taylor expanded in n around inf

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

      1. lower-/.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lower-log1p.f64 N/A

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

      4. lower-log.f64 77.3

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

   5. Applied rewrites 77.3%

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

      1. diff-log N/A

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

      2. +-commutative N/A

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

      3. lift-+.f64 N/A

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

      4. clear-num N/A

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

      5. lift-+.f64 N/A

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

      6. +-commutative N/A

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

      7. neg-log N/A

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

      8. diff-log N/A

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

      9. lift-log.f64 N/A

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

      10. lift-log1p.f64 N/A

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

      11. lift--.f64 N/A

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

      12. lower-neg.f64 77.3

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

      13. lift--.f64 N/A

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

      14. lift-log.f64 N/A

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

      15. lift-log1p.f64 N/A

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

      16. diff-log N/A

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

      17. lower-log.f64 N/A

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

      18. +-commutative N/A

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

      19. lift-+.f64 N/A

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

      20. lower-/.f64 77.5

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

   7. Applied rewrites 77.5%

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

   if 1.00000000000000004e-97 < (/.f64 #s(literal 1 binary64) n) < 1.99999999999999986e-34

   1. Initial program 4.0%

      \[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)} \]
   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 30.2

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

   5. Applied rewrites 30.2%

      \[\leadsto \color{blue}{\frac{\mathsf{log1p}\left(x\right) - \log x}{n}} \]
   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. clear-num N/A

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

      5. lower-/.f64 N/A

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

      6. lower-/.f64 30.2

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

      7. lift--.f64 N/A

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

      8. lift-log1p.f64 N/A

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

      9. lift-log.f64 N/A

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

      10. diff-log N/A

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

      11. lower-log.f64 N/A

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

      12. +-commutative N/A

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

      13. lift-+.f64 N/A

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

      14. lower-/.f64 30.2

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

   7. Applied rewrites 30.2%

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

   8. Taylor expanded in x around inf

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

      1. lower-*.f64 N/A

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

      2. +-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-/.f64 75.2

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

   10. Applied rewrites 75.2%

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

   if 1.99999999999999986e-34 < (/.f64 #s(literal 1 binary64) n) < 1.00000000000000004e93

   1. Initial program 74.5%

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

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{1 - e^{\frac{\log x}{n}}} \]
   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 68.0

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

   5. Applied rewrites 68.0%

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

   if 1.00000000000000004e93 < (/.f64 #s(literal 1 binary64) n)

   1. Initial program 29.8%

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

   3. Taylor expanded in n around inf

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

      1. lower-/.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lower-log1p.f64 N/A

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

      4. lower-log.f64 5.8

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

   5. Applied rewrites 5.8%

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

   6. Taylor expanded in x around inf

      \[\leadsto \color{blue}{\frac{\left(\frac{1}{n} + \frac{1}{3} \cdot \frac{1}{n \cdot {x}^{2}}\right) - \left(\frac{\frac{1}{2}}{n \cdot x} + \frac{1}{4} \cdot \frac{1}{n \cdot {x}^{3}}\right)}{x}} \]
   7. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\left(\frac{1}{n} + \frac{1}{3} \cdot \frac{1}{n \cdot {x}^{2}}\right) - \left(\frac{\frac{1}{2}}{n \cdot x} + \frac{1}{4} \cdot \frac{1}{n \cdot {x}^{3}}\right)}{x}} \]

   8. Applied rewrites 0.0%

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

   9. Taylor expanded in x around -inf

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

   11. Applied rewrites 14.9%

       \[\leadsto \frac{\color{blue}{\frac{1}{n} + \left(\frac{-0.5}{x \cdot n} - \frac{-0.3333333333333333}{x \cdot \left(x \cdot n\right)}\right)}}{x} \]
   12. Step-by-step derivation

       1. associate-/r* N/A

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

       2. lift-*.f64 N/A

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

       3. lift-*.f64 N/A

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

       4. frac-sub N/A

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

       5. lower-/.f64 N/A

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

       6. lower--.f64 N/A

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

       7. lower-*.f64 N/A

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

       8. lower-/.f64 N/A

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

       9. lift-*.f64 N/A

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

       10. lift-*.f64 N/A

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

       11. associate-*r* N/A

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

       12. pow2 N/A

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

       13. *-commutative N/A

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

       14. lower-*.f64 N/A

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

       15. pow2 N/A

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

       16. lower-*.f64 N/A

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

       17. lower-*.f64 N/A

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

       18. lower-*.f64 66.8

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

       19. lift-*.f64 N/A

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

       20. lift-*.f64 N/A

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

       21. associate-*r* N/A

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

   13. Applied rewrites 66.8%

       \[\leadsto \frac{\frac{1}{n} + \color{blue}{\frac{\frac{-0.5}{x} \cdot \left(n \cdot \left(x \cdot x\right)\right) - n \cdot -0.3333333333333333}{n \cdot \left(n \cdot \left(x \cdot x\right)\right)}}}{x} \]
3. Recombined 5 regimes into one program.

4. Final simplification 80.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{1}{n} \leq -4 \cdot 10^{-29}:\\ \;\;\;\;\frac{{x}^{\left(\frac{1}{n}\right)}}{x \cdot n}\\ \mathbf{elif}\;\frac{1}{n} \leq 10^{-97}:\\ \;\;\;\;\frac{\log \left(\frac{x}{x + 1}\right)}{-n}\\ \mathbf{elif}\;\frac{1}{n} \leq 2 \cdot 10^{-34}:\\ \;\;\;\;\frac{1}{x \cdot \mathsf{fma}\left(0.5, \frac{n}{x}, n\right)}\\ \mathbf{elif}\;\frac{1}{n} \leq 10^{+93}:\\ \;\;\;\;1 - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{n} + \frac{\frac{-0.5}{x} \cdot \left(n \cdot \left(x \cdot x\right)\right) - n \cdot -0.3333333333333333}{n \cdot \left(n \cdot \left(x \cdot x\right)\right)}}{x}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 81.0% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := n \cdot \left(x \cdot x\right)\\ \mathbf{if}\;\frac{1}{n} \leq -4 \cdot 10^{-29}:\\ \;\;\;\;\frac{{x}^{\left(-1 + \frac{1}{n}\right)}}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 10^{-97}:\\ \;\;\;\;\frac{\log \left(\frac{x}{x + 1}\right)}{-n}\\ \mathbf{elif}\;\frac{1}{n} \leq 2 \cdot 10^{-34}:\\ \;\;\;\;\frac{1}{x \cdot \mathsf{fma}\left(0.5, \frac{n}{x}, n\right)}\\ \mathbf{elif}\;\frac{1}{n} \leq 10^{+93}:\\ \;\;\;\;1 - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{n} + \frac{\frac{-0.5}{x} \cdot t\_0 - n \cdot -0.3333333333333333}{n \cdot t\_0}}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (let* ((t_0 (* n (* x x))))
   (if (<= (/ 1.0 n) -4e-29)
     (/ (pow x (+ -1.0 (/ 1.0 n))) n)
     (if (<= (/ 1.0 n) 1e-97)
       (/ (log (/ x (+ x 1.0))) (- n))
       (if (<= (/ 1.0 n) 2e-34)
         (/ 1.0 (* x (fma 0.5 (/ n x) n)))
         (if (<= (/ 1.0 n) 1e+93)
           (- 1.0 (pow x (/ 1.0 n)))
           (/
            (+
             (/ 1.0 n)
             (/ (- (* (/ -0.5 x) t_0) (* n -0.3333333333333333)) (* n t_0)))
            x)))))))

C

double code(double x, double n) {
	double t_0 = n * (x * x);
	double tmp;
	if ((1.0 / n) <= -4e-29) {
		tmp = pow(x, (-1.0 + (1.0 / n))) / n;
	} else if ((1.0 / n) <= 1e-97) {
		tmp = log((x / (x + 1.0))) / -n;
	} else if ((1.0 / n) <= 2e-34) {
		tmp = 1.0 / (x * fma(0.5, (n / x), n));
	} else if ((1.0 / n) <= 1e+93) {
		tmp = 1.0 - pow(x, (1.0 / n));
	} else {
		tmp = ((1.0 / n) + ((((-0.5 / x) * t_0) - (n * -0.3333333333333333)) / (n * t_0))) / x;
	}
	return tmp;
}

Julia

function code(x, n)
	t_0 = Float64(n * Float64(x * x))
	tmp = 0.0
	if (Float64(1.0 / n) <= -4e-29)
		tmp = Float64((x ^ Float64(-1.0 + Float64(1.0 / n))) / n);
	elseif (Float64(1.0 / n) <= 1e-97)
		tmp = Float64(log(Float64(x / Float64(x + 1.0))) / Float64(-n));
	elseif (Float64(1.0 / n) <= 2e-34)
		tmp = Float64(1.0 / Float64(x * fma(0.5, Float64(n / x), n)));
	elseif (Float64(1.0 / n) <= 1e+93)
		tmp = Float64(1.0 - (x ^ Float64(1.0 / n)));
	else
		tmp = Float64(Float64(Float64(1.0 / n) + Float64(Float64(Float64(Float64(-0.5 / x) * t_0) - Float64(n * -0.3333333333333333)) / Float64(n * t_0))) / x);
	end
	return tmp
end

Wolfram

code[x_, n_] := Block[{t$95$0 = N[(n * N[(x * x), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(1.0 / n), $MachinePrecision], -4e-29], N[(N[Power[x, N[(-1.0 + N[(1.0 / n), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / n), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 1e-97], N[(N[Log[N[(x / N[(x + 1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / (-n)), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 2e-34], N[(1.0 / N[(x * N[(0.5 * N[(n / x), $MachinePrecision] + n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 1e+93], N[(1.0 - N[Power[x, N[(1.0 / n), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[(N[(1.0 / n), $MachinePrecision] + N[(N[(N[(N[(-0.5 / x), $MachinePrecision] * t$95$0), $MachinePrecision] - N[(n * -0.3333333333333333), $MachinePrecision]), $MachinePrecision] / N[(n * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]]]]]]

Derivation

1. Split input into 5 regimes

2. if (/.f64 #s(literal 1 binary64) n) < -3.99999999999999977e-29

   1. Initial program 88.6%

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

   3. Taylor expanded in x around inf

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

      1. lower-/.f64 N/A

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

      2. log-rec N/A

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

      3. mul-1-neg N/A

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

      4. associate-*r/ N/A

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

      5. associate-*r* N/A

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

      6. metadata-eval N/A

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

      7. *-commutative N/A

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

      8. associate-/l* N/A

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

      9. exp-to-pow N/A

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

      10. lower-pow.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 91.4

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

   5. Applied rewrites 91.4%

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

      1. lift-/.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. associate-/r* N/A

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

      4. lower-/.f64 N/A

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

      5. div-inv N/A

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

      6. lift-pow.f64 N/A

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

      7. inv-pow N/A

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

      8. pow-prod-up N/A

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

      9. lower-pow.f64 N/A

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

      10. lower-+.f64 91.1

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

   7. Applied rewrites 91.1%

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

   if -3.99999999999999977e-29 < (/.f64 #s(literal 1 binary64) n) < 1.00000000000000004e-97

   1. Initial program 39.4%

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

   3. Taylor expanded in n around inf

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

      1. lower-/.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lower-log1p.f64 N/A

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

      4. lower-log.f64 77.3

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

   5. Applied rewrites 77.3%

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

      1. diff-log N/A

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

      2. +-commutative N/A

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

      3. lift-+.f64 N/A

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

      4. clear-num N/A

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

      5. lift-+.f64 N/A

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

      6. +-commutative N/A

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

      7. neg-log N/A

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

      8. diff-log N/A

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

      9. lift-log.f64 N/A

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

      10. lift-log1p.f64 N/A

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

      11. lift--.f64 N/A

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

      12. lower-neg.f64 77.3

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

      13. lift--.f64 N/A

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

      14. lift-log.f64 N/A

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

      15. lift-log1p.f64 N/A

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

      16. diff-log N/A

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

      17. lower-log.f64 N/A

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

      18. +-commutative N/A

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

      19. lift-+.f64 N/A

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

      20. lower-/.f64 77.5

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

   7. Applied rewrites 77.5%

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

   if 1.00000000000000004e-97 < (/.f64 #s(literal 1 binary64) n) < 1.99999999999999986e-34

   1. Initial program 4.0%

      \[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)} \]
   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 30.2

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

   5. Applied rewrites 30.2%

      \[\leadsto \color{blue}{\frac{\mathsf{log1p}\left(x\right) - \log x}{n}} \]
   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. clear-num N/A

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

      5. lower-/.f64 N/A

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

      6. lower-/.f64 30.2

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

      7. lift--.f64 N/A

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

      8. lift-log1p.f64 N/A

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

      9. lift-log.f64 N/A

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

      10. diff-log N/A

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

      11. lower-log.f64 N/A

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

      12. +-commutative N/A

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

      13. lift-+.f64 N/A

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

      14. lower-/.f64 30.2

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

   7. Applied rewrites 30.2%

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

   8. Taylor expanded in x around inf

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

      1. lower-*.f64 N/A

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

      2. +-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-/.f64 75.2

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

   10. Applied rewrites 75.2%

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

   if 1.99999999999999986e-34 < (/.f64 #s(literal 1 binary64) n) < 1.00000000000000004e93

   1. Initial program 74.5%

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

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{1 - e^{\frac{\log x}{n}}} \]
   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 68.0

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

   5. Applied rewrites 68.0%

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

   if 1.00000000000000004e93 < (/.f64 #s(literal 1 binary64) n)

   1. Initial program 29.8%

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

   3. Taylor expanded in n around inf

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

      1. lower-/.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lower-log1p.f64 N/A

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

      4. lower-log.f64 5.8

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

   5. Applied rewrites 5.8%

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

   6. Taylor expanded in x around inf

      \[\leadsto \color{blue}{\frac{\left(\frac{1}{n} + \frac{1}{3} \cdot \frac{1}{n \cdot {x}^{2}}\right) - \left(\frac{\frac{1}{2}}{n \cdot x} + \frac{1}{4} \cdot \frac{1}{n \cdot {x}^{3}}\right)}{x}} \]
   7. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\left(\frac{1}{n} + \frac{1}{3} \cdot \frac{1}{n \cdot {x}^{2}}\right) - \left(\frac{\frac{1}{2}}{n \cdot x} + \frac{1}{4} \cdot \frac{1}{n \cdot {x}^{3}}\right)}{x}} \]

   8. Applied rewrites 0.0%

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

   9. Taylor expanded in x around -inf

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

   11. Applied rewrites 14.9%

       \[\leadsto \frac{\color{blue}{\frac{1}{n} + \left(\frac{-0.5}{x \cdot n} - \frac{-0.3333333333333333}{x \cdot \left(x \cdot n\right)}\right)}}{x} \]
   12. Step-by-step derivation

       1. associate-/r* N/A

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

       2. lift-*.f64 N/A

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

       3. lift-*.f64 N/A

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

       4. frac-sub N/A

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

       5. lower-/.f64 N/A

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

       6. lower--.f64 N/A

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

       7. lower-*.f64 N/A

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

       8. lower-/.f64 N/A

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

       9. lift-*.f64 N/A

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

       10. lift-*.f64 N/A

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

       11. associate-*r* N/A

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

       12. pow2 N/A

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

       13. *-commutative N/A

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

       14. lower-*.f64 N/A

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

       15. pow2 N/A

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

       16. lower-*.f64 N/A

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

       17. lower-*.f64 N/A

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

       18. lower-*.f64 66.8

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

       19. lift-*.f64 N/A

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

       20. lift-*.f64 N/A

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

       21. associate-*r* N/A

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

   13. Applied rewrites 66.8%

       \[\leadsto \frac{\frac{1}{n} + \color{blue}{\frac{\frac{-0.5}{x} \cdot \left(n \cdot \left(x \cdot x\right)\right) - n \cdot -0.3333333333333333}{n \cdot \left(n \cdot \left(x \cdot x\right)\right)}}}{x} \]
3. Recombined 5 regimes into one program.

4. Final simplification 79.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{1}{n} \leq -4 \cdot 10^{-29}:\\ \;\;\;\;\frac{{x}^{\left(-1 + \frac{1}{n}\right)}}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 10^{-97}:\\ \;\;\;\;\frac{\log \left(\frac{x}{x + 1}\right)}{-n}\\ \mathbf{elif}\;\frac{1}{n} \leq 2 \cdot 10^{-34}:\\ \;\;\;\;\frac{1}{x \cdot \mathsf{fma}\left(0.5, \frac{n}{x}, n\right)}\\ \mathbf{elif}\;\frac{1}{n} \leq 10^{+93}:\\ \;\;\;\;1 - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{n} + \frac{\frac{-0.5}{x} \cdot \left(n \cdot \left(x \cdot x\right)\right) - n \cdot -0.3333333333333333}{n \cdot \left(n \cdot \left(x \cdot x\right)\right)}}{x}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 81.0% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := n \cdot \left(x \cdot x\right)\\ \mathbf{if}\;\frac{1}{n} \leq -4 \cdot 10^{-29}:\\ \;\;\;\;\frac{{x}^{\left(-1 + \frac{1}{n}\right)}}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 10^{-97}:\\ \;\;\;\;\frac{\log \left(\frac{x + 1}{x}\right)}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 2 \cdot 10^{-34}:\\ \;\;\;\;\frac{1}{x \cdot \mathsf{fma}\left(0.5, \frac{n}{x}, n\right)}\\ \mathbf{elif}\;\frac{1}{n} \leq 10^{+93}:\\ \;\;\;\;1 - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{n} + \frac{\frac{-0.5}{x} \cdot t\_0 - n \cdot -0.3333333333333333}{n \cdot t\_0}}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (let* ((t_0 (* n (* x x))))
   (if (<= (/ 1.0 n) -4e-29)
     (/ (pow x (+ -1.0 (/ 1.0 n))) n)
     (if (<= (/ 1.0 n) 1e-97)
       (/ (log (/ (+ x 1.0) x)) n)
       (if (<= (/ 1.0 n) 2e-34)
         (/ 1.0 (* x (fma 0.5 (/ n x) n)))
         (if (<= (/ 1.0 n) 1e+93)
           (- 1.0 (pow x (/ 1.0 n)))
           (/
            (+
             (/ 1.0 n)
             (/ (- (* (/ -0.5 x) t_0) (* n -0.3333333333333333)) (* n t_0)))
            x)))))))

C

double code(double x, double n) {
	double t_0 = n * (x * x);
	double tmp;
	if ((1.0 / n) <= -4e-29) {
		tmp = pow(x, (-1.0 + (1.0 / n))) / n;
	} else if ((1.0 / n) <= 1e-97) {
		tmp = log(((x + 1.0) / x)) / n;
	} else if ((1.0 / n) <= 2e-34) {
		tmp = 1.0 / (x * fma(0.5, (n / x), n));
	} else if ((1.0 / n) <= 1e+93) {
		tmp = 1.0 - pow(x, (1.0 / n));
	} else {
		tmp = ((1.0 / n) + ((((-0.5 / x) * t_0) - (n * -0.3333333333333333)) / (n * t_0))) / x;
	}
	return tmp;
}

Julia

function code(x, n)
	t_0 = Float64(n * Float64(x * x))
	tmp = 0.0
	if (Float64(1.0 / n) <= -4e-29)
		tmp = Float64((x ^ Float64(-1.0 + Float64(1.0 / n))) / n);
	elseif (Float64(1.0 / n) <= 1e-97)
		tmp = Float64(log(Float64(Float64(x + 1.0) / x)) / n);
	elseif (Float64(1.0 / n) <= 2e-34)
		tmp = Float64(1.0 / Float64(x * fma(0.5, Float64(n / x), n)));
	elseif (Float64(1.0 / n) <= 1e+93)
		tmp = Float64(1.0 - (x ^ Float64(1.0 / n)));
	else
		tmp = Float64(Float64(Float64(1.0 / n) + Float64(Float64(Float64(Float64(-0.5 / x) * t_0) - Float64(n * -0.3333333333333333)) / Float64(n * t_0))) / x);
	end
	return tmp
end

Wolfram

code[x_, n_] := Block[{t$95$0 = N[(n * N[(x * x), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(1.0 / n), $MachinePrecision], -4e-29], N[(N[Power[x, N[(-1.0 + N[(1.0 / n), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / n), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 1e-97], N[(N[Log[N[(N[(x + 1.0), $MachinePrecision] / x), $MachinePrecision]], $MachinePrecision] / n), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 2e-34], N[(1.0 / N[(x * N[(0.5 * N[(n / x), $MachinePrecision] + n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 1e+93], N[(1.0 - N[Power[x, N[(1.0 / n), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[(N[(1.0 / n), $MachinePrecision] + N[(N[(N[(N[(-0.5 / x), $MachinePrecision] * t$95$0), $MachinePrecision] - N[(n * -0.3333333333333333), $MachinePrecision]), $MachinePrecision] / N[(n * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]]]]]]

Derivation

1. Split input into 5 regimes

2. if (/.f64 #s(literal 1 binary64) n) < -3.99999999999999977e-29

   1. Initial program 88.6%

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

   3. Taylor expanded in x around inf

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

      1. lower-/.f64 N/A

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

      2. log-rec N/A

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

      3. mul-1-neg N/A

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

      4. associate-*r/ N/A

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

      5. associate-*r* N/A

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

      6. metadata-eval N/A

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

      7. *-commutative N/A

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

      8. associate-/l* N/A

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

      9. exp-to-pow N/A

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

      10. lower-pow.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 91.4

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

   5. Applied rewrites 91.4%

      \[\leadsto \color{blue}{\frac{{x}^{\left(\frac{1}{n}\right)}}{x \cdot n}} \]
   6. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \frac{{x}^{\color{blue}{\left(\frac{1}{n}\right)}}}{x \cdot n} \]

      2. lift-pow.f64 N/A

         \[\leadsto \frac{\color{blue}{{x}^{\left(\frac{1}{n}\right)}}}{x \cdot n} \]

      3. associate-/r* N/A

         \[\leadsto \color{blue}{\frac{\frac{{x}^{\left(\frac{1}{n}\right)}}{x}}{n}} \]

      4. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\frac{{x}^{\left(\frac{1}{n}\right)}}{x}}{n}} \]

      5. div-inv N/A

         \[\leadsto \frac{\color{blue}{{x}^{\left(\frac{1}{n}\right)} \cdot \frac{1}{x}}}{n} \]

      6. lift-pow.f64 N/A

         \[\leadsto \frac{\color{blue}{{x}^{\left(\frac{1}{n}\right)}} \cdot \frac{1}{x}}{n} \]

      7. inv-pow N/A

         \[\leadsto \frac{{x}^{\left(\frac{1}{n}\right)} \cdot \color{blue}{{x}^{-1}}}{n} \]

      8. pow-prod-up N/A

         \[\leadsto \frac{\color{blue}{{x}^{\left(\frac{1}{n} + -1\right)}}}{n} \]

      9. lower-pow.f64 N/A

         \[\leadsto \frac{\color{blue}{{x}^{\left(\frac{1}{n} + -1\right)}}}{n} \]

      10. lower-+.f64 91.1

          \[\leadsto \frac{{x}^{\color{blue}{\left(\frac{1}{n} + -1\right)}}}{n} \]

   7. Applied rewrites 91.1%

      \[\leadsto \color{blue}{\frac{{x}^{\left(\frac{1}{n} + -1\right)}}{n}} \]

   if -3.99999999999999977e-29 < (/.f64 #s(literal 1 binary64) n) < 1.00000000000000004e-97

   1. Initial program 39.4%

      \[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in n around inf

      \[\leadsto \color{blue}{\frac{\log \left(1 + x\right) - \log x}{n}} \]
   4. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\log \left(1 + x\right) - \log x}{n}} \]

      2. lower--.f64 N/A

         \[\leadsto \frac{\color{blue}{\log \left(1 + x\right) - \log x}}{n} \]

      3. lower-log1p.f64 N/A

         \[\leadsto \frac{\color{blue}{\mathsf{log1p}\left(x\right)} - \log x}{n} \]

      4. lower-log.f64 77.3

         \[\leadsto \frac{\mathsf{log1p}\left(x\right) - \color{blue}{\log x}}{n} \]

   5. Applied rewrites 77.3%

      \[\leadsto \color{blue}{\frac{\mathsf{log1p}\left(x\right) - \log x}{n}} \]
   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 77.3

         \[\leadsto \color{blue}{\frac{\mathsf{log1p}\left(x\right) - \log x}{n}} \]

      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. +-commutative N/A

          \[\leadsto \frac{\log \left(\frac{\color{blue}{x + 1}}{x}\right)}{n} \]

      11. lift-+.f64 N/A

          \[\leadsto \frac{\log \left(\frac{\color{blue}{x + 1}}{x}\right)}{n} \]

      12. lower-/.f64 77.5

          \[\leadsto \frac{\log \color{blue}{\left(\frac{x + 1}{x}\right)}}{n} \]

   7. Applied rewrites 77.5%

      \[\leadsto \color{blue}{\frac{\log \left(\frac{x + 1}{x}\right)}{n}} \]

   if 1.00000000000000004e-97 < (/.f64 #s(literal 1 binary64) n) < 1.99999999999999986e-34

   1. Initial program 4.0%

      \[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)} \]
   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 30.2

         \[\leadsto \frac{\mathsf{log1p}\left(x\right) - \color{blue}{\log x}}{n} \]

   5. Applied rewrites 30.2%

      \[\leadsto \color{blue}{\frac{\mathsf{log1p}\left(x\right) - \log x}{n}} \]
   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. clear-num N/A

         \[\leadsto \color{blue}{\frac{1}{\frac{n}{\mathsf{log1p}\left(x\right) - \log x}}} \]

      5. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{1}{\frac{n}{\mathsf{log1p}\left(x\right) - \log x}}} \]

      6. lower-/.f64 30.2

         \[\leadsto \frac{1}{\color{blue}{\frac{n}{\mathsf{log1p}\left(x\right) - \log x}}} \]

      7. lift--.f64 N/A

         \[\leadsto \frac{1}{\frac{n}{\color{blue}{\mathsf{log1p}\left(x\right) - \log x}}} \]

      8. lift-log1p.f64 N/A

         \[\leadsto \frac{1}{\frac{n}{\color{blue}{\log \left(1 + x\right)} - \log x}} \]

      9. lift-log.f64 N/A

         \[\leadsto \frac{1}{\frac{n}{\log \left(1 + x\right) - \color{blue}{\log x}}} \]

      10. diff-log N/A

          \[\leadsto \frac{1}{\frac{n}{\color{blue}{\log \left(\frac{1 + x}{x}\right)}}} \]

      11. lower-log.f64 N/A

          \[\leadsto \frac{1}{\frac{n}{\color{blue}{\log \left(\frac{1 + x}{x}\right)}}} \]

      12. +-commutative N/A

          \[\leadsto \frac{1}{\frac{n}{\log \left(\frac{\color{blue}{x + 1}}{x}\right)}} \]

      13. lift-+.f64 N/A

          \[\leadsto \frac{1}{\frac{n}{\log \left(\frac{\color{blue}{x + 1}}{x}\right)}} \]

      14. lower-/.f64 30.2

          \[\leadsto \frac{1}{\frac{n}{\log \color{blue}{\left(\frac{x + 1}{x}\right)}}} \]

   7. Applied rewrites 30.2%

      \[\leadsto \color{blue}{\frac{1}{\frac{n}{\log \left(\frac{x + 1}{x}\right)}}} \]

   8. Taylor expanded in x around inf

      \[\leadsto \frac{1}{\color{blue}{x \cdot \left(n + \frac{1}{2} \cdot \frac{n}{x}\right)}} \]
   9. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \frac{1}{\color{blue}{x \cdot \left(n + \frac{1}{2} \cdot \frac{n}{x}\right)}} \]

      2. +-commutative N/A

         \[\leadsto \frac{1}{x \cdot \color{blue}{\left(\frac{1}{2} \cdot \frac{n}{x} + n\right)}} \]

      3. lower-fma.f64 N/A

         \[\leadsto \frac{1}{x \cdot \color{blue}{\mathsf{fma}\left(\frac{1}{2}, \frac{n}{x}, n\right)}} \]

      4. lower-/.f64 75.2

         \[\leadsto \frac{1}{x \cdot \mathsf{fma}\left(0.5, \color{blue}{\frac{n}{x}}, n\right)} \]

   10. Applied rewrites 75.2%

       \[\leadsto \frac{1}{\color{blue}{x \cdot \mathsf{fma}\left(0.5, \frac{n}{x}, n\right)}} \]

   if 1.99999999999999986e-34 < (/.f64 #s(literal 1 binary64) n) < 1.00000000000000004e93

   1. Initial program 74.5%

      \[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{1 - e^{\frac{\log x}{n}}} \]
   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 68.0

          \[\leadsto 1 - {x}^{\color{blue}{\left(\frac{1}{n}\right)}} \]

   5. Applied rewrites 68.0%

      \[\leadsto \color{blue}{1 - {x}^{\left(\frac{1}{n}\right)}} \]

   if 1.00000000000000004e93 < (/.f64 #s(literal 1 binary64) n)

   1. Initial program 29.8%

      \[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in n around inf

      \[\leadsto \color{blue}{\frac{\log \left(1 + x\right) - \log x}{n}} \]
   4. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\log \left(1 + x\right) - \log x}{n}} \]

      2. lower--.f64 N/A

         \[\leadsto \frac{\color{blue}{\log \left(1 + x\right) - \log x}}{n} \]

      3. lower-log1p.f64 N/A

         \[\leadsto \frac{\color{blue}{\mathsf{log1p}\left(x\right)} - \log x}{n} \]

      4. lower-log.f64 5.8

         \[\leadsto \frac{\mathsf{log1p}\left(x\right) - \color{blue}{\log x}}{n} \]

   5. Applied rewrites 5.8%

      \[\leadsto \color{blue}{\frac{\mathsf{log1p}\left(x\right) - \log x}{n}} \]

   6. Taylor expanded in x around inf

      \[\leadsto \color{blue}{\frac{\left(\frac{1}{n} + \frac{1}{3} \cdot \frac{1}{n \cdot {x}^{2}}\right) - \left(\frac{\frac{1}{2}}{n \cdot x} + \frac{1}{4} \cdot \frac{1}{n \cdot {x}^{3}}\right)}{x}} \]
   7. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\left(\frac{1}{n} + \frac{1}{3} \cdot \frac{1}{n \cdot {x}^{2}}\right) - \left(\frac{\frac{1}{2}}{n \cdot x} + \frac{1}{4} \cdot \frac{1}{n \cdot {x}^{3}}\right)}{x}} \]

   8. Applied rewrites 0.0%

      \[\leadsto \color{blue}{\frac{\left(\frac{1}{n} + \frac{0.3333333333333333}{n \cdot \left(x \cdot x\right)}\right) - \left(\frac{0.5}{x \cdot n} + \frac{0.25}{n \cdot \left(x \cdot \left(x \cdot x\right)\right)}\right)}{x}} \]

   9. Taylor expanded in x around -inf

      \[\leadsto \frac{\color{blue}{-1 \cdot \frac{\frac{1}{2} \cdot \frac{1}{n} - \frac{1}{3} \cdot \frac{1}{n \cdot x}}{x} + \frac{1}{n}}}{x} \]

   11. Applied rewrites 14.9%

       \[\leadsto \frac{\color{blue}{\frac{1}{n} + \left(\frac{-0.5}{x \cdot n} - \frac{-0.3333333333333333}{x \cdot \left(x \cdot n\right)}\right)}}{x} \]
   12. Step-by-step derivation

       1. associate-/r* N/A

          \[\leadsto \frac{\frac{1}{n} + \left(\color{blue}{\frac{\frac{\frac{-1}{2}}{x}}{n}} - \frac{\frac{-1}{3}}{x \cdot \left(x \cdot n\right)}\right)}{x} \]

       2. lift-*.f64 N/A

          \[\leadsto \frac{\frac{1}{n} + \left(\frac{\frac{\frac{-1}{2}}{x}}{n} - \frac{\frac{-1}{3}}{x \cdot \color{blue}{\left(x \cdot n\right)}}\right)}{x} \]

       3. lift-*.f64 N/A

          \[\leadsto \frac{\frac{1}{n} + \left(\frac{\frac{\frac{-1}{2}}{x}}{n} - \frac{\frac{-1}{3}}{\color{blue}{x \cdot \left(x \cdot n\right)}}\right)}{x} \]

       4. frac-sub N/A

          \[\leadsto \frac{\frac{1}{n} + \color{blue}{\frac{\frac{\frac{-1}{2}}{x} \cdot \left(x \cdot \left(x \cdot n\right)\right) - n \cdot \frac{-1}{3}}{n \cdot \left(x \cdot \left(x \cdot n\right)\right)}}}{x} \]

       5. lower-/.f64 N/A

          \[\leadsto \frac{\frac{1}{n} + \color{blue}{\frac{\frac{\frac{-1}{2}}{x} \cdot \left(x \cdot \left(x \cdot n\right)\right) - n \cdot \frac{-1}{3}}{n \cdot \left(x \cdot \left(x \cdot n\right)\right)}}}{x} \]

       6. lower--.f64 N/A

          \[\leadsto \frac{\frac{1}{n} + \frac{\color{blue}{\frac{\frac{-1}{2}}{x} \cdot \left(x \cdot \left(x \cdot n\right)\right) - n \cdot \frac{-1}{3}}}{n \cdot \left(x \cdot \left(x \cdot n\right)\right)}}{x} \]

       7. lower-*.f64 N/A

          \[\leadsto \frac{\frac{1}{n} + \frac{\color{blue}{\frac{\frac{-1}{2}}{x} \cdot \left(x \cdot \left(x \cdot n\right)\right)} - n \cdot \frac{-1}{3}}{n \cdot \left(x \cdot \left(x \cdot n\right)\right)}}{x} \]

       8. lower-/.f64 N/A

          \[\leadsto \frac{\frac{1}{n} + \frac{\color{blue}{\frac{\frac{-1}{2}}{x}} \cdot \left(x \cdot \left(x \cdot n\right)\right) - n \cdot \frac{-1}{3}}{n \cdot \left(x \cdot \left(x \cdot n\right)\right)}}{x} \]

       9. lift-*.f64 N/A

          \[\leadsto \frac{\frac{1}{n} + \frac{\frac{\frac{-1}{2}}{x} \cdot \color{blue}{\left(x \cdot \left(x \cdot n\right)\right)} - n \cdot \frac{-1}{3}}{n \cdot \left(x \cdot \left(x \cdot n\right)\right)}}{x} \]

       10. lift-*.f64 N/A

           \[\leadsto \frac{\frac{1}{n} + \frac{\frac{\frac{-1}{2}}{x} \cdot \left(x \cdot \color{blue}{\left(x \cdot n\right)}\right) - n \cdot \frac{-1}{3}}{n \cdot \left(x \cdot \left(x \cdot n\right)\right)}}{x} \]

       11. associate-*r* N/A

           \[\leadsto \frac{\frac{1}{n} + \frac{\frac{\frac{-1}{2}}{x} \cdot \color{blue}{\left(\left(x \cdot x\right) \cdot n\right)} - n \cdot \frac{-1}{3}}{n \cdot \left(x \cdot \left(x \cdot n\right)\right)}}{x} \]

       12. pow2 N/A

           \[\leadsto \frac{\frac{1}{n} + \frac{\frac{\frac{-1}{2}}{x} \cdot \left(\color{blue}{{x}^{2}} \cdot n\right) - n \cdot \frac{-1}{3}}{n \cdot \left(x \cdot \left(x \cdot n\right)\right)}}{x} \]

       13. *-commutative N/A

           \[\leadsto \frac{\frac{1}{n} + \frac{\frac{\frac{-1}{2}}{x} \cdot \color{blue}{\left(n \cdot {x}^{2}\right)} - n \cdot \frac{-1}{3}}{n \cdot \left(x \cdot \left(x \cdot n\right)\right)}}{x} \]

       14. lower-*.f64 N/A

           \[\leadsto \frac{\frac{1}{n} + \frac{\frac{\frac{-1}{2}}{x} \cdot \color{blue}{\left(n \cdot {x}^{2}\right)} - n \cdot \frac{-1}{3}}{n \cdot \left(x \cdot \left(x \cdot n\right)\right)}}{x} \]

       15. pow2 N/A

           \[\leadsto \frac{\frac{1}{n} + \frac{\frac{\frac{-1}{2}}{x} \cdot \left(n \cdot \color{blue}{\left(x \cdot x\right)}\right) - n \cdot \frac{-1}{3}}{n \cdot \left(x \cdot \left(x \cdot n\right)\right)}}{x} \]

       16. lower-*.f64 N/A

           \[\leadsto \frac{\frac{1}{n} + \frac{\frac{\frac{-1}{2}}{x} \cdot \left(n \cdot \color{blue}{\left(x \cdot x\right)}\right) - n \cdot \frac{-1}{3}}{n \cdot \left(x \cdot \left(x \cdot n\right)\right)}}{x} \]

       17. lower-*.f64 N/A

           \[\leadsto \frac{\frac{1}{n} + \frac{\frac{\frac{-1}{2}}{x} \cdot \left(n \cdot \left(x \cdot x\right)\right) - \color{blue}{n \cdot \frac{-1}{3}}}{n \cdot \left(x \cdot \left(x \cdot n\right)\right)}}{x} \]

       18. lower-*.f64 66.8

           \[\leadsto \frac{\frac{1}{n} + \frac{\frac{-0.5}{x} \cdot \left(n \cdot \left(x \cdot x\right)\right) - n \cdot -0.3333333333333333}{\color{blue}{n \cdot \left(x \cdot \left(x \cdot n\right)\right)}}}{x} \]

       19. lift-*.f64 N/A

           \[\leadsto \frac{\frac{1}{n} + \frac{\frac{\frac{-1}{2}}{x} \cdot \left(n \cdot \left(x \cdot x\right)\right) - n \cdot \frac{-1}{3}}{n \cdot \color{blue}{\left(x \cdot \left(x \cdot n\right)\right)}}}{x} \]

       20. lift-*.f64 N/A

           \[\leadsto \frac{\frac{1}{n} + \frac{\frac{\frac{-1}{2}}{x} \cdot \left(n \cdot \left(x \cdot x\right)\right) - n \cdot \frac{-1}{3}}{n \cdot \left(x \cdot \color{blue}{\left(x \cdot n\right)}\right)}}{x} \]

       21. associate-*r* N/A

           \[\leadsto \frac{\frac{1}{n} + \frac{\frac{\frac{-1}{2}}{x} \cdot \left(n \cdot \left(x \cdot x\right)\right) - n \cdot \frac{-1}{3}}{n \cdot \color{blue}{\left(\left(x \cdot x\right) \cdot n\right)}}}{x} \]

   13. Applied rewrites 66.8%

       \[\leadsto \frac{\frac{1}{n} + \color{blue}{\frac{\frac{-0.5}{x} \cdot \left(n \cdot \left(x \cdot x\right)\right) - n \cdot -0.3333333333333333}{n \cdot \left(n \cdot \left(x \cdot x\right)\right)}}}{x} \]
3. Recombined 5 regimes into one program.

4. Final simplification 79.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{1}{n} \leq -4 \cdot 10^{-29}:\\ \;\;\;\;\frac{{x}^{\left(-1 + \frac{1}{n}\right)}}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 10^{-97}:\\ \;\;\;\;\frac{\log \left(\frac{x + 1}{x}\right)}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 2 \cdot 10^{-34}:\\ \;\;\;\;\frac{1}{x \cdot \mathsf{fma}\left(0.5, \frac{n}{x}, n\right)}\\ \mathbf{elif}\;\frac{1}{n} \leq 10^{+93}:\\ \;\;\;\;1 - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{n} + \frac{\frac{-0.5}{x} \cdot \left(n \cdot \left(x \cdot x\right)\right) - n \cdot -0.3333333333333333}{n \cdot \left(n \cdot \left(x \cdot x\right)\right)}}{x}\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 57.8% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := n \cdot \left(x \cdot x\right)\\ \mathbf{if}\;\frac{1}{n} \leq -100:\\ \;\;\;\;\frac{0.3333333333333333}{x \cdot \left(x \cdot \left(x \cdot n\right)\right)}\\ \mathbf{elif}\;\frac{1}{n} \leq 2 \cdot 10^{-34}:\\ \;\;\;\;\frac{1}{x \cdot \mathsf{fma}\left(0.5, \frac{n}{x}, n\right)}\\ \mathbf{elif}\;\frac{1}{n} \leq 10^{+93}:\\ \;\;\;\;1 - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{n} + \frac{\frac{-0.5}{x} \cdot t\_0 - n \cdot -0.3333333333333333}{n \cdot t\_0}}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (let* ((t_0 (* n (* x x))))
   (if (<= (/ 1.0 n) -100.0)
     (/ 0.3333333333333333 (* x (* x (* x n))))
     (if (<= (/ 1.0 n) 2e-34)
       (/ 1.0 (* x (fma 0.5 (/ n x) n)))
       (if (<= (/ 1.0 n) 1e+93)
         (- 1.0 (pow x (/ 1.0 n)))
         (/
          (+
           (/ 1.0 n)
           (/ (- (* (/ -0.5 x) t_0) (* n -0.3333333333333333)) (* n t_0)))
          x))))))

C

double code(double x, double n) {
	double t_0 = n * (x * x);
	double tmp;
	if ((1.0 / n) <= -100.0) {
		tmp = 0.3333333333333333 / (x * (x * (x * n)));
	} else if ((1.0 / n) <= 2e-34) {
		tmp = 1.0 / (x * fma(0.5, (n / x), n));
	} else if ((1.0 / n) <= 1e+93) {
		tmp = 1.0 - pow(x, (1.0 / n));
	} else {
		tmp = ((1.0 / n) + ((((-0.5 / x) * t_0) - (n * -0.3333333333333333)) / (n * t_0))) / x;
	}
	return tmp;
}

Julia

function code(x, n)
	t_0 = Float64(n * Float64(x * x))
	tmp = 0.0
	if (Float64(1.0 / n) <= -100.0)
		tmp = Float64(0.3333333333333333 / Float64(x * Float64(x * Float64(x * n))));
	elseif (Float64(1.0 / n) <= 2e-34)
		tmp = Float64(1.0 / Float64(x * fma(0.5, Float64(n / x), n)));
	elseif (Float64(1.0 / n) <= 1e+93)
		tmp = Float64(1.0 - (x ^ Float64(1.0 / n)));
	else
		tmp = Float64(Float64(Float64(1.0 / n) + Float64(Float64(Float64(Float64(-0.5 / x) * t_0) - Float64(n * -0.3333333333333333)) / Float64(n * t_0))) / x);
	end
	return tmp
end

Wolfram

code[x_, n_] := Block[{t$95$0 = N[(n * N[(x * x), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(1.0 / n), $MachinePrecision], -100.0], N[(0.3333333333333333 / N[(x * N[(x * N[(x * n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 2e-34], N[(1.0 / N[(x * N[(0.5 * N[(n / x), $MachinePrecision] + n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 1e+93], N[(1.0 - N[Power[x, N[(1.0 / n), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[(N[(1.0 / n), $MachinePrecision] + N[(N[(N[(N[(-0.5 / x), $MachinePrecision] * t$95$0), $MachinePrecision] - N[(n * -0.3333333333333333), $MachinePrecision]), $MachinePrecision] / N[(n * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if (/.f64 #s(literal 1 binary64) n) < -100

   1. Initial program 100.0%

      \[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in n around inf

      \[\leadsto \color{blue}{\frac{\log \left(1 + x\right) - \log x}{n}} \]
   4. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\log \left(1 + x\right) - \log x}{n}} \]

      2. lower--.f64 N/A

         \[\leadsto \frac{\color{blue}{\log \left(1 + x\right) - \log x}}{n} \]

      3. lower-log1p.f64 N/A

         \[\leadsto \frac{\color{blue}{\mathsf{log1p}\left(x\right)} - \log x}{n} \]

      4. lower-log.f64 46.1

         \[\leadsto \frac{\mathsf{log1p}\left(x\right) - \color{blue}{\log x}}{n} \]

   5. Applied rewrites 46.1%

      \[\leadsto \color{blue}{\frac{\mathsf{log1p}\left(x\right) - \log x}{n}} \]

   6. Taylor expanded in x around inf

      \[\leadsto \color{blue}{\frac{\left(\frac{1}{n} + \frac{1}{3} \cdot \frac{1}{n \cdot {x}^{2}}\right) - \left(\frac{\frac{1}{2}}{n \cdot x} + \frac{1}{4} \cdot \frac{1}{n \cdot {x}^{3}}\right)}{x}} \]
   7. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\left(\frac{1}{n} + \frac{1}{3} \cdot \frac{1}{n \cdot {x}^{2}}\right) - \left(\frac{\frac{1}{2}}{n \cdot x} + \frac{1}{4} \cdot \frac{1}{n \cdot {x}^{3}}\right)}{x}} \]

   8. Applied rewrites 1.6%

      \[\leadsto \color{blue}{\frac{\left(\frac{1}{n} + \frac{0.3333333333333333}{n \cdot \left(x \cdot x\right)}\right) - \left(\frac{0.5}{x \cdot n} + \frac{0.25}{n \cdot \left(x \cdot \left(x \cdot x\right)\right)}\right)}{x}} \]

   9. Taylor expanded in x around -inf

      \[\leadsto \frac{\color{blue}{-1 \cdot \frac{\frac{1}{2} \cdot \frac{1}{n} - \frac{1}{3} \cdot \frac{1}{n \cdot x}}{x} + \frac{1}{n}}}{x} \]

   11. Applied rewrites 16.4%

       \[\leadsto \frac{\color{blue}{\frac{1}{n} + \left(\frac{-0.5}{x \cdot n} - \frac{-0.3333333333333333}{x \cdot \left(x \cdot n\right)}\right)}}{x} \]

   12. Taylor expanded in x around 0

       \[\leadsto \color{blue}{\frac{\frac{1}{3}}{n \cdot {x}^{3}}} \]
   13. Step-by-step derivation

       1. lower-/.f64 N/A

          \[\leadsto \color{blue}{\frac{\frac{1}{3}}{n \cdot {x}^{3}}} \]

       2. *-commutative N/A

          \[\leadsto \frac{\frac{1}{3}}{\color{blue}{{x}^{3} \cdot n}} \]

       3. cube-mult N/A

          \[\leadsto \frac{\frac{1}{3}}{\color{blue}{\left(x \cdot \left(x \cdot x\right)\right)} \cdot n} \]

       4. unpow2 N/A

          \[\leadsto \frac{\frac{1}{3}}{\left(x \cdot \color{blue}{{x}^{2}}\right) \cdot n} \]

       5. associate-*l* N/A

          \[\leadsto \frac{\frac{1}{3}}{\color{blue}{x \cdot \left({x}^{2} \cdot n\right)}} \]

       6. *-commutative N/A

          \[\leadsto \frac{\frac{1}{3}}{x \cdot \color{blue}{\left(n \cdot {x}^{2}\right)}} \]

       7. lower-*.f64 N/A

          \[\leadsto \frac{\frac{1}{3}}{\color{blue}{x \cdot \left(n \cdot {x}^{2}\right)}} \]

       8. *-commutative N/A

          \[\leadsto \frac{\frac{1}{3}}{x \cdot \color{blue}{\left({x}^{2} \cdot n\right)}} \]

       9. unpow2 N/A

          \[\leadsto \frac{\frac{1}{3}}{x \cdot \left(\color{blue}{\left(x \cdot x\right)} \cdot n\right)} \]

       10. associate-*l* N/A

           \[\leadsto \frac{\frac{1}{3}}{x \cdot \color{blue}{\left(x \cdot \left(x \cdot n\right)\right)}} \]

       11. *-commutative N/A

           \[\leadsto \frac{\frac{1}{3}}{x \cdot \left(x \cdot \color{blue}{\left(n \cdot x\right)}\right)} \]

       12. lower-*.f64 N/A

           \[\leadsto \frac{\frac{1}{3}}{x \cdot \color{blue}{\left(x \cdot \left(n \cdot x\right)\right)}} \]

       13. *-commutative N/A

           \[\leadsto \frac{\frac{1}{3}}{x \cdot \left(x \cdot \color{blue}{\left(x \cdot n\right)}\right)} \]

       14. lower-*.f64 68.1

           \[\leadsto \frac{0.3333333333333333}{x \cdot \left(x \cdot \color{blue}{\left(x \cdot n\right)}\right)} \]

   14. Applied rewrites 68.1%

       \[\leadsto \color{blue}{\frac{0.3333333333333333}{x \cdot \left(x \cdot \left(x \cdot n\right)\right)}} \]

   if -100 < (/.f64 #s(literal 1 binary64) n) < 1.99999999999999986e-34

   1. Initial program 36.7%

      \[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in n around inf

      \[\leadsto \color{blue}{\frac{\log \left(1 + x\right) - \log x}{n}} \]
   4. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\log \left(1 + x\right) - \log x}{n}} \]

      2. lower--.f64 N/A

         \[\leadsto \frac{\color{blue}{\log \left(1 + x\right) - \log x}}{n} \]

      3. lower-log1p.f64 N/A

         \[\leadsto \frac{\color{blue}{\mathsf{log1p}\left(x\right)} - \log x}{n} \]

      4. lower-log.f64 69.8

         \[\leadsto \frac{\mathsf{log1p}\left(x\right) - \color{blue}{\log x}}{n} \]

   5. Applied rewrites 69.8%

      \[\leadsto \color{blue}{\frac{\mathsf{log1p}\left(x\right) - \log x}{n}} \]
   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. clear-num N/A

         \[\leadsto \color{blue}{\frac{1}{\frac{n}{\mathsf{log1p}\left(x\right) - \log x}}} \]

      5. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{1}{\frac{n}{\mathsf{log1p}\left(x\right) - \log x}}} \]

      6. lower-/.f64 69.7

         \[\leadsto \frac{1}{\color{blue}{\frac{n}{\mathsf{log1p}\left(x\right) - \log x}}} \]

      7. lift--.f64 N/A

         \[\leadsto \frac{1}{\frac{n}{\color{blue}{\mathsf{log1p}\left(x\right) - \log x}}} \]

      8. lift-log1p.f64 N/A

         \[\leadsto \frac{1}{\frac{n}{\color{blue}{\log \left(1 + x\right)} - \log x}} \]

      9. lift-log.f64 N/A

         \[\leadsto \frac{1}{\frac{n}{\log \left(1 + x\right) - \color{blue}{\log x}}} \]

      10. diff-log N/A

          \[\leadsto \frac{1}{\frac{n}{\color{blue}{\log \left(\frac{1 + x}{x}\right)}}} \]

      11. lower-log.f64 N/A

          \[\leadsto \frac{1}{\frac{n}{\color{blue}{\log \left(\frac{1 + x}{x}\right)}}} \]

      12. +-commutative N/A

          \[\leadsto \frac{1}{\frac{n}{\log \left(\frac{\color{blue}{x + 1}}{x}\right)}} \]

      13. lift-+.f64 N/A

          \[\leadsto \frac{1}{\frac{n}{\log \left(\frac{\color{blue}{x + 1}}{x}\right)}} \]

      14. lower-/.f64 70.0

          \[\leadsto \frac{1}{\frac{n}{\log \color{blue}{\left(\frac{x + 1}{x}\right)}}} \]

   7. Applied rewrites 70.0%

      \[\leadsto \color{blue}{\frac{1}{\frac{n}{\log \left(\frac{x + 1}{x}\right)}}} \]

   8. Taylor expanded in x around inf

      \[\leadsto \frac{1}{\color{blue}{x \cdot \left(n + \frac{1}{2} \cdot \frac{n}{x}\right)}} \]
   9. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \frac{1}{\color{blue}{x \cdot \left(n + \frac{1}{2} \cdot \frac{n}{x}\right)}} \]

      2. +-commutative N/A

         \[\leadsto \frac{1}{x \cdot \color{blue}{\left(\frac{1}{2} \cdot \frac{n}{x} + n\right)}} \]

      3. lower-fma.f64 N/A

         \[\leadsto \frac{1}{x \cdot \color{blue}{\mathsf{fma}\left(\frac{1}{2}, \frac{n}{x}, n\right)}} \]

      4. lower-/.f64 63.5

         \[\leadsto \frac{1}{x \cdot \mathsf{fma}\left(0.5, \color{blue}{\frac{n}{x}}, n\right)} \]

   10. Applied rewrites 63.5%

       \[\leadsto \frac{1}{\color{blue}{x \cdot \mathsf{fma}\left(0.5, \frac{n}{x}, n\right)}} \]

   if 1.99999999999999986e-34 < (/.f64 #s(literal 1 binary64) n) < 1.00000000000000004e93

   1. Initial program 74.5%

      \[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{1 - e^{\frac{\log x}{n}}} \]
   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 68.0

          \[\leadsto 1 - {x}^{\color{blue}{\left(\frac{1}{n}\right)}} \]

   5. Applied rewrites 68.0%

      \[\leadsto \color{blue}{1 - {x}^{\left(\frac{1}{n}\right)}} \]

   if 1.00000000000000004e93 < (/.f64 #s(literal 1 binary64) n)

   1. Initial program 29.8%

      \[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in n around inf

      \[\leadsto \color{blue}{\frac{\log \left(1 + x\right) - \log x}{n}} \]
   4. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\log \left(1 + x\right) - \log x}{n}} \]

      2. lower--.f64 N/A

         \[\leadsto \frac{\color{blue}{\log \left(1 + x\right) - \log x}}{n} \]

      3. lower-log1p.f64 N/A

         \[\leadsto \frac{\color{blue}{\mathsf{log1p}\left(x\right)} - \log x}{n} \]

      4. lower-log.f64 5.8

         \[\leadsto \frac{\mathsf{log1p}\left(x\right) - \color{blue}{\log x}}{n} \]

   5. Applied rewrites 5.8%

      \[\leadsto \color{blue}{\frac{\mathsf{log1p}\left(x\right) - \log x}{n}} \]

   6. Taylor expanded in x around inf

      \[\leadsto \color{blue}{\frac{\left(\frac{1}{n} + \frac{1}{3} \cdot \frac{1}{n \cdot {x}^{2}}\right) - \left(\frac{\frac{1}{2}}{n \cdot x} + \frac{1}{4} \cdot \frac{1}{n \cdot {x}^{3}}\right)}{x}} \]
   7. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\left(\frac{1}{n} + \frac{1}{3} \cdot \frac{1}{n \cdot {x}^{2}}\right) - \left(\frac{\frac{1}{2}}{n \cdot x} + \frac{1}{4} \cdot \frac{1}{n \cdot {x}^{3}}\right)}{x}} \]

   8. Applied rewrites 0.0%

      \[\leadsto \color{blue}{\frac{\left(\frac{1}{n} + \frac{0.3333333333333333}{n \cdot \left(x \cdot x\right)}\right) - \left(\frac{0.5}{x \cdot n} + \frac{0.25}{n \cdot \left(x \cdot \left(x \cdot x\right)\right)}\right)}{x}} \]

   9. Taylor expanded in x around -inf

      \[\leadsto \frac{\color{blue}{-1 \cdot \frac{\frac{1}{2} \cdot \frac{1}{n} - \frac{1}{3} \cdot \frac{1}{n \cdot x}}{x} + \frac{1}{n}}}{x} \]

   11. Applied rewrites 14.9%

       \[\leadsto \frac{\color{blue}{\frac{1}{n} + \left(\frac{-0.5}{x \cdot n} - \frac{-0.3333333333333333}{x \cdot \left(x \cdot n\right)}\right)}}{x} \]
   12. Step-by-step derivation

       1. associate-/r* N/A

          \[\leadsto \frac{\frac{1}{n} + \left(\color{blue}{\frac{\frac{\frac{-1}{2}}{x}}{n}} - \frac{\frac{-1}{3}}{x \cdot \left(x \cdot n\right)}\right)}{x} \]

       2. lift-*.f64 N/A

          \[\leadsto \frac{\frac{1}{n} + \left(\frac{\frac{\frac{-1}{2}}{x}}{n} - \frac{\frac{-1}{3}}{x \cdot \color{blue}{\left(x \cdot n\right)}}\right)}{x} \]

       3. lift-*.f64 N/A

          \[\leadsto \frac{\frac{1}{n} + \left(\frac{\frac{\frac{-1}{2}}{x}}{n} - \frac{\frac{-1}{3}}{\color{blue}{x \cdot \left(x \cdot n\right)}}\right)}{x} \]

       4. frac-sub N/A

          \[\leadsto \frac{\frac{1}{n} + \color{blue}{\frac{\frac{\frac{-1}{2}}{x} \cdot \left(x \cdot \left(x \cdot n\right)\right) - n \cdot \frac{-1}{3}}{n \cdot \left(x \cdot \left(x \cdot n\right)\right)}}}{x} \]

       5. lower-/.f64 N/A

          \[\leadsto \frac{\frac{1}{n} + \color{blue}{\frac{\frac{\frac{-1}{2}}{x} \cdot \left(x \cdot \left(x \cdot n\right)\right) - n \cdot \frac{-1}{3}}{n \cdot \left(x \cdot \left(x \cdot n\right)\right)}}}{x} \]

       6. lower--.f64 N/A

          \[\leadsto \frac{\frac{1}{n} + \frac{\color{blue}{\frac{\frac{-1}{2}}{x} \cdot \left(x \cdot \left(x \cdot n\right)\right) - n \cdot \frac{-1}{3}}}{n \cdot \left(x \cdot \left(x \cdot n\right)\right)}}{x} \]

       7. lower-*.f64 N/A

          \[\leadsto \frac{\frac{1}{n} + \frac{\color{blue}{\frac{\frac{-1}{2}}{x} \cdot \left(x \cdot \left(x \cdot n\right)\right)} - n \cdot \frac{-1}{3}}{n \cdot \left(x \cdot \left(x \cdot n\right)\right)}}{x} \]

       8. lower-/.f64 N/A

          \[\leadsto \frac{\frac{1}{n} + \frac{\color{blue}{\frac{\frac{-1}{2}}{x}} \cdot \left(x \cdot \left(x \cdot n\right)\right) - n \cdot \frac{-1}{3}}{n \cdot \left(x \cdot \left(x \cdot n\right)\right)}}{x} \]

       9. lift-*.f64 N/A

          \[\leadsto \frac{\frac{1}{n} + \frac{\frac{\frac{-1}{2}}{x} \cdot \color{blue}{\left(x \cdot \left(x \cdot n\right)\right)} - n \cdot \frac{-1}{3}}{n \cdot \left(x \cdot \left(x \cdot n\right)\right)}}{x} \]

       10. lift-*.f64 N/A

           \[\leadsto \frac{\frac{1}{n} + \frac{\frac{\frac{-1}{2}}{x} \cdot \left(x \cdot \color{blue}{\left(x \cdot n\right)}\right) - n \cdot \frac{-1}{3}}{n \cdot \left(x \cdot \left(x \cdot n\right)\right)}}{x} \]

       11. associate-*r* N/A

           \[\leadsto \frac{\frac{1}{n} + \frac{\frac{\frac{-1}{2}}{x} \cdot \color{blue}{\left(\left(x \cdot x\right) \cdot n\right)} - n \cdot \frac{-1}{3}}{n \cdot \left(x \cdot \left(x \cdot n\right)\right)}}{x} \]

       12. pow2 N/A

           \[\leadsto \frac{\frac{1}{n} + \frac{\frac{\frac{-1}{2}}{x} \cdot \left(\color{blue}{{x}^{2}} \cdot n\right) - n \cdot \frac{-1}{3}}{n \cdot \left(x \cdot \left(x \cdot n\right)\right)}}{x} \]

       13. *-commutative N/A

           \[\leadsto \frac{\frac{1}{n} + \frac{\frac{\frac{-1}{2}}{x} \cdot \color{blue}{\left(n \cdot {x}^{2}\right)} - n \cdot \frac{-1}{3}}{n \cdot \left(x \cdot \left(x \cdot n\right)\right)}}{x} \]

       14. lower-*.f64 N/A

           \[\leadsto \frac{\frac{1}{n} + \frac{\frac{\frac{-1}{2}}{x} \cdot \color{blue}{\left(n \cdot {x}^{2}\right)} - n \cdot \frac{-1}{3}}{n \cdot \left(x \cdot \left(x \cdot n\right)\right)}}{x} \]

       15. pow2 N/A

           \[\leadsto \frac{\frac{1}{n} + \frac{\frac{\frac{-1}{2}}{x} \cdot \left(n \cdot \color{blue}{\left(x \cdot x\right)}\right) - n \cdot \frac{-1}{3}}{n \cdot \left(x \cdot \left(x \cdot n\right)\right)}}{x} \]

       16. lower-*.f64 N/A

           \[\leadsto \frac{\frac{1}{n} + \frac{\frac{\frac{-1}{2}}{x} \cdot \left(n \cdot \color{blue}{\left(x \cdot x\right)}\right) - n \cdot \frac{-1}{3}}{n \cdot \left(x \cdot \left(x \cdot n\right)\right)}}{x} \]

       17. lower-*.f64 N/A

           \[\leadsto \frac{\frac{1}{n} + \frac{\frac{\frac{-1}{2}}{x} \cdot \left(n \cdot \left(x \cdot x\right)\right) - \color{blue}{n \cdot \frac{-1}{3}}}{n \cdot \left(x \cdot \left(x \cdot n\right)\right)}}{x} \]

       18. lower-*.f64 66.8

           \[\leadsto \frac{\frac{1}{n} + \frac{\frac{-0.5}{x} \cdot \left(n \cdot \left(x \cdot x\right)\right) - n \cdot -0.3333333333333333}{\color{blue}{n \cdot \left(x \cdot \left(x \cdot n\right)\right)}}}{x} \]

       19. lift-*.f64 N/A

           \[\leadsto \frac{\frac{1}{n} + \frac{\frac{\frac{-1}{2}}{x} \cdot \left(n \cdot \left(x \cdot x\right)\right) - n \cdot \frac{-1}{3}}{n \cdot \color{blue}{\left(x \cdot \left(x \cdot n\right)\right)}}}{x} \]

       20. lift-*.f64 N/A

           \[\leadsto \frac{\frac{1}{n} + \frac{\frac{\frac{-1}{2}}{x} \cdot \left(n \cdot \left(x \cdot x\right)\right) - n \cdot \frac{-1}{3}}{n \cdot \left(x \cdot \color{blue}{\left(x \cdot n\right)}\right)}}{x} \]

       21. associate-*r* N/A

           \[\leadsto \frac{\frac{1}{n} + \frac{\frac{\frac{-1}{2}}{x} \cdot \left(n \cdot \left(x \cdot x\right)\right) - n \cdot \frac{-1}{3}}{n \cdot \color{blue}{\left(\left(x \cdot x\right) \cdot n\right)}}}{x} \]

   13. Applied rewrites 66.8%

       \[\leadsto \frac{\frac{1}{n} + \color{blue}{\frac{\frac{-0.5}{x} \cdot \left(n \cdot \left(x \cdot x\right)\right) - n \cdot -0.3333333333333333}{n \cdot \left(n \cdot \left(x \cdot x\right)\right)}}}{x} \]
3. Recombined 4 regimes into one program.
4. Add Preprocessing

Alternative 14: 59.7% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 2 \cdot 10^{-148}:\\ \;\;\;\;-\frac{\log x}{n}\\ \mathbf{elif}\;x \leq 3.8 \cdot 10^{-71}:\\ \;\;\;\;\frac{0.3333333333333333}{x \cdot \left(x \cdot \left(x \cdot n\right)\right)}\\ \mathbf{elif}\;x \leq 0.5:\\ \;\;\;\;\frac{x - \log x}{n}\\ \mathbf{elif}\;x \leq 1.1 \cdot 10^{+129}:\\ \;\;\;\;\frac{-1}{x \cdot \left(\left(\frac{\mathsf{fma}\left(n, -0.08333333333333333, n \cdot 0.041666666666666664\right)}{x \cdot \left(x \cdot x\right)} - n\right) - \frac{\mathsf{fma}\left(n, 0.5, \frac{n}{x} \cdot -0.08333333333333333\right)}{x}\right)}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (if (<= x 2e-148)
   (- (/ (log x) n))
   (if (<= x 3.8e-71)
     (/ 0.3333333333333333 (* x (* x (* x n))))
     (if (<= x 0.5)
       (/ (- x (log x)) n)
       (if (<= x 1.1e+129)
         (/
          -1.0
          (*
           x
           (-
            (-
             (/
              (fma n -0.08333333333333333 (* n 0.041666666666666664))
              (* x (* x x)))
             n)
            (/ (fma n 0.5 (* (/ n x) -0.08333333333333333)) x))))
         0.0)))))

C

double code(double x, double n) {
	double tmp;
	if (x <= 2e-148) {
		tmp = -(log(x) / n);
	} else if (x <= 3.8e-71) {
		tmp = 0.3333333333333333 / (x * (x * (x * n)));
	} else if (x <= 0.5) {
		tmp = (x - log(x)) / n;
	} else if (x <= 1.1e+129) {
		tmp = -1.0 / (x * (((fma(n, -0.08333333333333333, (n * 0.041666666666666664)) / (x * (x * x))) - n) - (fma(n, 0.5, ((n / x) * -0.08333333333333333)) / x)));
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Julia

function code(x, n)
	tmp = 0.0
	if (x <= 2e-148)
		tmp = Float64(-Float64(log(x) / n));
	elseif (x <= 3.8e-71)
		tmp = Float64(0.3333333333333333 / Float64(x * Float64(x * Float64(x * n))));
	elseif (x <= 0.5)
		tmp = Float64(Float64(x - log(x)) / n);
	elseif (x <= 1.1e+129)
		tmp = Float64(-1.0 / Float64(x * Float64(Float64(Float64(fma(n, -0.08333333333333333, Float64(n * 0.041666666666666664)) / Float64(x * Float64(x * x))) - n) - Float64(fma(n, 0.5, Float64(Float64(n / x) * -0.08333333333333333)) / x))));
	else
		tmp = 0.0;
	end
	return tmp
end

Wolfram

code[x_, n_] := If[LessEqual[x, 2e-148], (-N[(N[Log[x], $MachinePrecision] / n), $MachinePrecision]), If[LessEqual[x, 3.8e-71], N[(0.3333333333333333 / N[(x * N[(x * N[(x * n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 0.5], N[(N[(x - N[Log[x], $MachinePrecision]), $MachinePrecision] / n), $MachinePrecision], If[LessEqual[x, 1.1e+129], N[(-1.0 / N[(x * N[(N[(N[(N[(n * -0.08333333333333333 + N[(n * 0.041666666666666664), $MachinePrecision]), $MachinePrecision] / N[(x * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - n), $MachinePrecision] - N[(N[(n * 0.5 + N[(N[(n / x), $MachinePrecision] * -0.08333333333333333), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 0.0]]]]

Derivation

1. Split input into 5 regimes

2. if x < 1.99999999999999987e-148

   1. Initial program 46.7%

      \[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)} \]
   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 48.6

         \[\leadsto \frac{\mathsf{log1p}\left(x\right) - \color{blue}{\log x}}{n} \]

   5. Applied rewrites 48.6%

      \[\leadsto \color{blue}{\frac{\mathsf{log1p}\left(x\right) - \log x}{n}} \]

   6. Taylor expanded in x around 0

      \[\leadsto \frac{\color{blue}{-1 \cdot \log x}}{n} \]
   7. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \frac{\color{blue}{\mathsf{neg}\left(\log x\right)}}{n} \]

      2. lower-neg.f64 N/A

         \[\leadsto \frac{\color{blue}{\mathsf{neg}\left(\log x\right)}}{n} \]

      3. lower-log.f64 48.6

         \[\leadsto \frac{-\color{blue}{\log x}}{n} \]

   8. Applied rewrites 48.6%

      \[\leadsto \frac{\color{blue}{-\log x}}{n} \]

   if 1.99999999999999987e-148 < x < 3.79999999999999992e-71

   1. Initial program 45.2%

      \[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in n around inf

      \[\leadsto \color{blue}{\frac{\log \left(1 + x\right) - \log x}{n}} \]
   4. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\log \left(1 + x\right) - \log x}{n}} \]

      2. lower--.f64 N/A

         \[\leadsto \frac{\color{blue}{\log \left(1 + x\right) - \log x}}{n} \]

      3. lower-log1p.f64 N/A

         \[\leadsto \frac{\color{blue}{\mathsf{log1p}\left(x\right)} - \log x}{n} \]

      4. lower-log.f64 27.4

         \[\leadsto \frac{\mathsf{log1p}\left(x\right) - \color{blue}{\log x}}{n} \]

   5. Applied rewrites 27.4%

      \[\leadsto \color{blue}{\frac{\mathsf{log1p}\left(x\right) - \log x}{n}} \]

   6. Taylor expanded in x around inf

      \[\leadsto \color{blue}{\frac{\left(\frac{1}{n} + \frac{1}{3} \cdot \frac{1}{n \cdot {x}^{2}}\right) - \left(\frac{\frac{1}{2}}{n \cdot x} + \frac{1}{4} \cdot \frac{1}{n \cdot {x}^{3}}\right)}{x}} \]
   7. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\left(\frac{1}{n} + \frac{1}{3} \cdot \frac{1}{n \cdot {x}^{2}}\right) - \left(\frac{\frac{1}{2}}{n \cdot x} + \frac{1}{4} \cdot \frac{1}{n \cdot {x}^{3}}\right)}{x}} \]

   8. Applied rewrites 0.3%

      \[\leadsto \color{blue}{\frac{\left(\frac{1}{n} + \frac{0.3333333333333333}{n \cdot \left(x \cdot x\right)}\right) - \left(\frac{0.5}{x \cdot n} + \frac{0.25}{n \cdot \left(x \cdot \left(x \cdot x\right)\right)}\right)}{x}} \]

   9. Taylor expanded in x around -inf

      \[\leadsto \frac{\color{blue}{-1 \cdot \frac{\frac{1}{2} \cdot \frac{1}{n} - \frac{1}{3} \cdot \frac{1}{n \cdot x}}{x} + \frac{1}{n}}}{x} \]

   11. Applied rewrites 33.2%

       \[\leadsto \frac{\color{blue}{\frac{1}{n} + \left(\frac{-0.5}{x \cdot n} - \frac{-0.3333333333333333}{x \cdot \left(x \cdot n\right)}\right)}}{x} \]

   12. Taylor expanded in x around 0

       \[\leadsto \color{blue}{\frac{\frac{1}{3}}{n \cdot {x}^{3}}} \]
   13. Step-by-step derivation

       1. lower-/.f64 N/A

          \[\leadsto \color{blue}{\frac{\frac{1}{3}}{n \cdot {x}^{3}}} \]

       2. *-commutative N/A

          \[\leadsto \frac{\frac{1}{3}}{\color{blue}{{x}^{3} \cdot n}} \]

       3. cube-mult N/A

          \[\leadsto \frac{\frac{1}{3}}{\color{blue}{\left(x \cdot \left(x \cdot x\right)\right)} \cdot n} \]

       4. unpow2 N/A

          \[\leadsto \frac{\frac{1}{3}}{\left(x \cdot \color{blue}{{x}^{2}}\right) \cdot n} \]

       5. associate-*l* N/A

          \[\leadsto \frac{\frac{1}{3}}{\color{blue}{x \cdot \left({x}^{2} \cdot n\right)}} \]

       6. *-commutative N/A

          \[\leadsto \frac{\frac{1}{3}}{x \cdot \color{blue}{\left(n \cdot {x}^{2}\right)}} \]

       7. lower-*.f64 N/A

          \[\leadsto \frac{\frac{1}{3}}{\color{blue}{x \cdot \left(n \cdot {x}^{2}\right)}} \]

       8. *-commutative N/A

          \[\leadsto \frac{\frac{1}{3}}{x \cdot \color{blue}{\left({x}^{2} \cdot n\right)}} \]

       9. unpow2 N/A

          \[\leadsto \frac{\frac{1}{3}}{x \cdot \left(\color{blue}{\left(x \cdot x\right)} \cdot n\right)} \]

       10. associate-*l* N/A

           \[\leadsto \frac{\frac{1}{3}}{x \cdot \color{blue}{\left(x \cdot \left(x \cdot n\right)\right)}} \]

       11. *-commutative N/A

           \[\leadsto \frac{\frac{1}{3}}{x \cdot \left(x \cdot \color{blue}{\left(n \cdot x\right)}\right)} \]

       12. lower-*.f64 N/A

           \[\leadsto \frac{\frac{1}{3}}{x \cdot \color{blue}{\left(x \cdot \left(n \cdot x\right)\right)}} \]

       13. *-commutative N/A

           \[\leadsto \frac{\frac{1}{3}}{x \cdot \left(x \cdot \color{blue}{\left(x \cdot n\right)}\right)} \]

       14. lower-*.f64 65.2

           \[\leadsto \frac{0.3333333333333333}{x \cdot \left(x \cdot \color{blue}{\left(x \cdot n\right)}\right)} \]

   14. Applied rewrites 65.2%

       \[\leadsto \color{blue}{\frac{0.3333333333333333}{x \cdot \left(x \cdot \left(x \cdot n\right)\right)}} \]

   if 3.79999999999999992e-71 < x < 0.5

   1. Initial program 45.9%

      \[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)} \]
   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 44.4

         \[\leadsto \frac{\mathsf{log1p}\left(x\right) - \color{blue}{\log x}}{n} \]

   5. Applied rewrites 44.4%

      \[\leadsto \color{blue}{\frac{\mathsf{log1p}\left(x\right) - \log x}{n}} \]

   6. Taylor expanded in x around 0

      \[\leadsto \frac{\color{blue}{x - \log x}}{n} \]
   7. Step-by-step derivation

      1. lower--.f64 N/A

         \[\leadsto \frac{\color{blue}{x - \log x}}{n} \]

      2. lower-log.f64 41.8

         \[\leadsto \frac{x - \color{blue}{\log x}}{n} \]

   8. Applied rewrites 41.8%

      \[\leadsto \frac{\color{blue}{x - \log x}}{n} \]

   if 0.5 < x < 1.1e129

   1. Initial program 33.2%

      \[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)} \]
   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 35.1

         \[\leadsto \frac{\mathsf{log1p}\left(x\right) - \color{blue}{\log x}}{n} \]

   5. Applied rewrites 35.1%

      \[\leadsto \color{blue}{\frac{\mathsf{log1p}\left(x\right) - \log x}{n}} \]
   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. clear-num N/A

         \[\leadsto \color{blue}{\frac{1}{\frac{n}{\mathsf{log1p}\left(x\right) - \log x}}} \]

      5. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{1}{\frac{n}{\mathsf{log1p}\left(x\right) - \log x}}} \]

      6. lower-/.f64 35.1

         \[\leadsto \frac{1}{\color{blue}{\frac{n}{\mathsf{log1p}\left(x\right) - \log x}}} \]

      7. lift--.f64 N/A

         \[\leadsto \frac{1}{\frac{n}{\color{blue}{\mathsf{log1p}\left(x\right) - \log x}}} \]

      8. lift-log1p.f64 N/A

         \[\leadsto \frac{1}{\frac{n}{\color{blue}{\log \left(1 + x\right)} - \log x}} \]

      9. lift-log.f64 N/A

         \[\leadsto \frac{1}{\frac{n}{\log \left(1 + x\right) - \color{blue}{\log x}}} \]

      10. diff-log N/A

          \[\leadsto \frac{1}{\frac{n}{\color{blue}{\log \left(\frac{1 + x}{x}\right)}}} \]

      11. lower-log.f64 N/A

          \[\leadsto \frac{1}{\frac{n}{\color{blue}{\log \left(\frac{1 + x}{x}\right)}}} \]

      12. +-commutative N/A

          \[\leadsto \frac{1}{\frac{n}{\log \left(\frac{\color{blue}{x + 1}}{x}\right)}} \]

      13. lift-+.f64 N/A

          \[\leadsto \frac{1}{\frac{n}{\log \left(\frac{\color{blue}{x + 1}}{x}\right)}} \]

      14. lower-/.f64 35.8

          \[\leadsto \frac{1}{\frac{n}{\log \color{blue}{\left(\frac{x + 1}{x}\right)}}} \]

   7. Applied rewrites 35.8%

      \[\leadsto \color{blue}{\frac{1}{\frac{n}{\log \left(\frac{x + 1}{x}\right)}}} \]

   8. Taylor expanded in x around inf

      \[\leadsto \frac{1}{\color{blue}{x \cdot \left(\left(n + -1 \cdot \frac{\frac{-1}{4} \cdot n + \left(\frac{1}{6} \cdot n + \frac{1}{2} \cdot \left(\frac{-1}{4} \cdot n + \frac{1}{3} \cdot n\right)\right)}{{x}^{3}}\right) - \left(\frac{-1}{2} \cdot \frac{n}{x} + \left(\frac{-1}{4} \cdot \frac{n}{{x}^{2}} + \frac{1}{3} \cdot \frac{n}{{x}^{2}}\right)\right)\right)}} \]

   10. Applied rewrites 81.4%

       \[\leadsto \frac{1}{\color{blue}{x \cdot \left(\left(n - \frac{\mathsf{fma}\left(n, -0.08333333333333333, n \cdot 0.041666666666666664\right)}{x \cdot \left(x \cdot x\right)}\right) + \frac{\mathsf{fma}\left(n, 0.5, \frac{n}{x} \cdot -0.08333333333333333\right)}{x}\right)}} \]

   if 1.1e129 < x

   1. Initial program 86.5%

      \[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{1 - e^{\frac{\log x}{n}}} \]
   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 58.9

          \[\leadsto 1 - {x}^{\color{blue}{\left(\frac{1}{n}\right)}} \]

   5. Applied rewrites 58.9%

      \[\leadsto \color{blue}{1 - {x}^{\left(\frac{1}{n}\right)}} \]

   6. Taylor expanded in n around inf

      \[\leadsto 1 - \color{blue}{1} \]
   7. Step-by-step derivation

   8. Applied rewrites 86.5%

      \[\leadsto 1 - \color{blue}{1} \]
   9. Step-by-step derivation

      1. metadata-eval 86.5

         \[\leadsto \color{blue}{0} \]

   10. Applied rewrites 86.5%

       \[\leadsto \color{blue}{0} \]
3. Recombined 5 regimes into one program.

4. Final simplification 65.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 2 \cdot 10^{-148}:\\ \;\;\;\;-\frac{\log x}{n}\\ \mathbf{elif}\;x \leq 3.8 \cdot 10^{-71}:\\ \;\;\;\;\frac{0.3333333333333333}{x \cdot \left(x \cdot \left(x \cdot n\right)\right)}\\ \mathbf{elif}\;x \leq 0.5:\\ \;\;\;\;\frac{x - \log x}{n}\\ \mathbf{elif}\;x \leq 1.1 \cdot 10^{+129}:\\ \;\;\;\;\frac{-1}{x \cdot \left(\left(\frac{\mathsf{fma}\left(n, -0.08333333333333333, n \cdot 0.041666666666666664\right)}{x \cdot \left(x \cdot x\right)} - n\right) - \frac{\mathsf{fma}\left(n, 0.5, \frac{n}{x} \cdot -0.08333333333333333\right)}{x}\right)}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 58.9% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := -\frac{\log x}{n}\\ \mathbf{if}\;x \leq 2 \cdot 10^{-148}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq 3.8 \cdot 10^{-71}:\\ \;\;\;\;\frac{0.3333333333333333}{x \cdot \left(x \cdot \left(x \cdot n\right)\right)}\\ \mathbf{elif}\;x \leq 2.7 \cdot 10^{-16}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq 1.45 \cdot 10^{+129}:\\ \;\;\;\;\frac{\frac{\frac{-0.5 + \frac{0.3333333333333333}{x}}{x} + 1}{n}}{x}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (let* ((t_0 (- (/ (log x) n))))
   (if (<= x 2e-148)
     t_0
     (if (<= x 3.8e-71)
       (/ 0.3333333333333333 (* x (* x (* x n))))
       (if (<= x 2.7e-16)
         t_0
         (if (<= x 1.45e+129)
           (/ (/ (+ (/ (+ -0.5 (/ 0.3333333333333333 x)) x) 1.0) n) x)
           0.0))))))

C

double code(double x, double n) {
	double t_0 = -(log(x) / n);
	double tmp;
	if (x <= 2e-148) {
		tmp = t_0;
	} else if (x <= 3.8e-71) {
		tmp = 0.3333333333333333 / (x * (x * (x * n)));
	} else if (x <= 2.7e-16) {
		tmp = t_0;
	} else if (x <= 1.45e+129) {
		tmp = ((((-0.5 + (0.3333333333333333 / x)) / x) + 1.0) / n) / x;
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Fortran

real(8) function code(x, n)
    real(8), intent (in) :: x
    real(8), intent (in) :: n
    real(8) :: t_0
    real(8) :: tmp
    t_0 = -(log(x) / n)
    if (x <= 2d-148) then
        tmp = t_0
    else if (x <= 3.8d-71) then
        tmp = 0.3333333333333333d0 / (x * (x * (x * n)))
    else if (x <= 2.7d-16) then
        tmp = t_0
    else if (x <= 1.45d+129) then
        tmp = (((((-0.5d0) + (0.3333333333333333d0 / x)) / x) + 1.0d0) / n) / x
    else
        tmp = 0.0d0
    end if
    code = tmp
end function

Java

public static double code(double x, double n) {
	double t_0 = -(Math.log(x) / n);
	double tmp;
	if (x <= 2e-148) {
		tmp = t_0;
	} else if (x <= 3.8e-71) {
		tmp = 0.3333333333333333 / (x * (x * (x * n)));
	} else if (x <= 2.7e-16) {
		tmp = t_0;
	} else if (x <= 1.45e+129) {
		tmp = ((((-0.5 + (0.3333333333333333 / x)) / x) + 1.0) / n) / x;
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Python

def code(x, n):
	t_0 = -(math.log(x) / n)
	tmp = 0
	if x <= 2e-148:
		tmp = t_0
	elif x <= 3.8e-71:
		tmp = 0.3333333333333333 / (x * (x * (x * n)))
	elif x <= 2.7e-16:
		tmp = t_0
	elif x <= 1.45e+129:
		tmp = ((((-0.5 + (0.3333333333333333 / x)) / x) + 1.0) / n) / x
	else:
		tmp = 0.0
	return tmp

Julia

function code(x, n)
	t_0 = Float64(-Float64(log(x) / n))
	tmp = 0.0
	if (x <= 2e-148)
		tmp = t_0;
	elseif (x <= 3.8e-71)
		tmp = Float64(0.3333333333333333 / Float64(x * Float64(x * Float64(x * n))));
	elseif (x <= 2.7e-16)
		tmp = t_0;
	elseif (x <= 1.45e+129)
		tmp = Float64(Float64(Float64(Float64(Float64(-0.5 + Float64(0.3333333333333333 / x)) / x) + 1.0) / n) / x);
	else
		tmp = 0.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, n)
	t_0 = -(log(x) / n);
	tmp = 0.0;
	if (x <= 2e-148)
		tmp = t_0;
	elseif (x <= 3.8e-71)
		tmp = 0.3333333333333333 / (x * (x * (x * n)));
	elseif (x <= 2.7e-16)
		tmp = t_0;
	elseif (x <= 1.45e+129)
		tmp = ((((-0.5 + (0.3333333333333333 / x)) / x) + 1.0) / n) / x;
	else
		tmp = 0.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, n_] := Block[{t$95$0 = (-N[(N[Log[x], $MachinePrecision] / n), $MachinePrecision])}, If[LessEqual[x, 2e-148], t$95$0, If[LessEqual[x, 3.8e-71], N[(0.3333333333333333 / N[(x * N[(x * N[(x * n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 2.7e-16], t$95$0, If[LessEqual[x, 1.45e+129], N[(N[(N[(N[(N[(-0.5 + N[(0.3333333333333333 / x), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision] + 1.0), $MachinePrecision] / n), $MachinePrecision] / x), $MachinePrecision], 0.0]]]]]

Derivation

1. Split input into 4 regimes

2. if x < 1.99999999999999987e-148 or 3.79999999999999992e-71 < x < 2.69999999999999999e-16

   1. Initial program 44.7%

      \[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)} \]
   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 48.4

         \[\leadsto \frac{\mathsf{log1p}\left(x\right) - \color{blue}{\log x}}{n} \]

   5. Applied rewrites 48.4%

      \[\leadsto \color{blue}{\frac{\mathsf{log1p}\left(x\right) - \log x}{n}} \]

   6. Taylor expanded in x around 0

      \[\leadsto \frac{\color{blue}{-1 \cdot \log x}}{n} \]
   7. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \frac{\color{blue}{\mathsf{neg}\left(\log x\right)}}{n} \]

      2. lower-neg.f64 N/A

         \[\leadsto \frac{\color{blue}{\mathsf{neg}\left(\log x\right)}}{n} \]

      3. lower-log.f64 48.4

         \[\leadsto \frac{-\color{blue}{\log x}}{n} \]

   8. Applied rewrites 48.4%

      \[\leadsto \frac{\color{blue}{-\log x}}{n} \]

   if 1.99999999999999987e-148 < x < 3.79999999999999992e-71

   1. Initial program 45.2%

      \[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in n around inf

      \[\leadsto \color{blue}{\frac{\log \left(1 + x\right) - \log x}{n}} \]
   4. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\log \left(1 + x\right) - \log x}{n}} \]

      2. lower--.f64 N/A

         \[\leadsto \frac{\color{blue}{\log \left(1 + x\right) - \log x}}{n} \]

      3. lower-log1p.f64 N/A

         \[\leadsto \frac{\color{blue}{\mathsf{log1p}\left(x\right)} - \log x}{n} \]

      4. lower-log.f64 27.4

         \[\leadsto \frac{\mathsf{log1p}\left(x\right) - \color{blue}{\log x}}{n} \]

   5. Applied rewrites 27.4%

      \[\leadsto \color{blue}{\frac{\mathsf{log1p}\left(x\right) - \log x}{n}} \]

   6. Taylor expanded in x around inf

      \[\leadsto \color{blue}{\frac{\left(\frac{1}{n} + \frac{1}{3} \cdot \frac{1}{n \cdot {x}^{2}}\right) - \left(\frac{\frac{1}{2}}{n \cdot x} + \frac{1}{4} \cdot \frac{1}{n \cdot {x}^{3}}\right)}{x}} \]
   7. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\left(\frac{1}{n} + \frac{1}{3} \cdot \frac{1}{n \cdot {x}^{2}}\right) - \left(\frac{\frac{1}{2}}{n \cdot x} + \frac{1}{4} \cdot \frac{1}{n \cdot {x}^{3}}\right)}{x}} \]

   8. Applied rewrites 0.3%

      \[\leadsto \color{blue}{\frac{\left(\frac{1}{n} + \frac{0.3333333333333333}{n \cdot \left(x \cdot x\right)}\right) - \left(\frac{0.5}{x \cdot n} + \frac{0.25}{n \cdot \left(x \cdot \left(x \cdot x\right)\right)}\right)}{x}} \]

   9. Taylor expanded in x around -inf

      \[\leadsto \frac{\color{blue}{-1 \cdot \frac{\frac{1}{2} \cdot \frac{1}{n} - \frac{1}{3} \cdot \frac{1}{n \cdot x}}{x} + \frac{1}{n}}}{x} \]

   11. Applied rewrites 33.2%

       \[\leadsto \frac{\color{blue}{\frac{1}{n} + \left(\frac{-0.5}{x \cdot n} - \frac{-0.3333333333333333}{x \cdot \left(x \cdot n\right)}\right)}}{x} \]

   12. Taylor expanded in x around 0

       \[\leadsto \color{blue}{\frac{\frac{1}{3}}{n \cdot {x}^{3}}} \]
   13. Step-by-step derivation

       1. lower-/.f64 N/A

          \[\leadsto \color{blue}{\frac{\frac{1}{3}}{n \cdot {x}^{3}}} \]

       2. *-commutative N/A

          \[\leadsto \frac{\frac{1}{3}}{\color{blue}{{x}^{3} \cdot n}} \]

       3. cube-mult N/A

          \[\leadsto \frac{\frac{1}{3}}{\color{blue}{\left(x \cdot \left(x \cdot x\right)\right)} \cdot n} \]

       4. unpow2 N/A

          \[\leadsto \frac{\frac{1}{3}}{\left(x \cdot \color{blue}{{x}^{2}}\right) \cdot n} \]

       5. associate-*l* N/A

          \[\leadsto \frac{\frac{1}{3}}{\color{blue}{x \cdot \left({x}^{2} \cdot n\right)}} \]

       6. *-commutative N/A

          \[\leadsto \frac{\frac{1}{3}}{x \cdot \color{blue}{\left(n \cdot {x}^{2}\right)}} \]

       7. lower-*.f64 N/A

          \[\leadsto \frac{\frac{1}{3}}{\color{blue}{x \cdot \left(n \cdot {x}^{2}\right)}} \]

       8. *-commutative N/A

          \[\leadsto \frac{\frac{1}{3}}{x \cdot \color{blue}{\left({x}^{2} \cdot n\right)}} \]

       9. unpow2 N/A

          \[\leadsto \frac{\frac{1}{3}}{x \cdot \left(\color{blue}{\left(x \cdot x\right)} \cdot n\right)} \]

       10. associate-*l* N/A

           \[\leadsto \frac{\frac{1}{3}}{x \cdot \color{blue}{\left(x \cdot \left(x \cdot n\right)\right)}} \]

       11. *-commutative N/A

           \[\leadsto \frac{\frac{1}{3}}{x \cdot \left(x \cdot \color{blue}{\left(n \cdot x\right)}\right)} \]

       12. lower-*.f64 N/A

           \[\leadsto \frac{\frac{1}{3}}{x \cdot \color{blue}{\left(x \cdot \left(n \cdot x\right)\right)}} \]

       13. *-commutative N/A

           \[\leadsto \frac{\frac{1}{3}}{x \cdot \left(x \cdot \color{blue}{\left(x \cdot n\right)}\right)} \]

       14. lower-*.f64 65.2

           \[\leadsto \frac{0.3333333333333333}{x \cdot \left(x \cdot \color{blue}{\left(x \cdot n\right)}\right)} \]

   14. Applied rewrites 65.2%

       \[\leadsto \color{blue}{\frac{0.3333333333333333}{x \cdot \left(x \cdot \left(x \cdot n\right)\right)}} \]

   if 2.69999999999999999e-16 < x < 1.45000000000000001e129

   1. Initial program 38.1%

      \[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in n around inf

      \[\leadsto \color{blue}{\frac{\log \left(1 + x\right) - \log x}{n}} \]
   4. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\log \left(1 + x\right) - \log x}{n}} \]

      2. lower--.f64 N/A

         \[\leadsto \frac{\color{blue}{\log \left(1 + x\right) - \log x}}{n} \]

      3. lower-log1p.f64 N/A

         \[\leadsto \frac{\color{blue}{\mathsf{log1p}\left(x\right)} - \log x}{n} \]

      4. lower-log.f64 34.7

         \[\leadsto \frac{\mathsf{log1p}\left(x\right) - \color{blue}{\log x}}{n} \]

   5. Applied rewrites 34.7%

      \[\leadsto \color{blue}{\frac{\mathsf{log1p}\left(x\right) - \log x}{n}} \]

   6. Taylor expanded in x around inf

      \[\leadsto \color{blue}{\frac{\left(\frac{1}{n} + \frac{1}{3} \cdot \frac{1}{n \cdot {x}^{2}}\right) - \left(\frac{\frac{1}{2}}{n \cdot x} + \frac{1}{4} \cdot \frac{1}{n \cdot {x}^{3}}\right)}{x}} \]
   7. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\left(\frac{1}{n} + \frac{1}{3} \cdot \frac{1}{n \cdot {x}^{2}}\right) - \left(\frac{\frac{1}{2}}{n \cdot x} + \frac{1}{4} \cdot \frac{1}{n \cdot {x}^{3}}\right)}{x}} \]

   8. Applied rewrites 71.7%

      \[\leadsto \color{blue}{\frac{\left(\frac{1}{n} + \frac{0.3333333333333333}{n \cdot \left(x \cdot x\right)}\right) - \left(\frac{0.5}{x \cdot n} + \frac{0.25}{n \cdot \left(x \cdot \left(x \cdot x\right)\right)}\right)}{x}} \]

   9. Taylor expanded in x around -inf

      \[\leadsto \frac{\color{blue}{-1 \cdot \frac{\frac{1}{2} \cdot \frac{1}{n} - \frac{1}{3} \cdot \frac{1}{n \cdot x}}{x} + \frac{1}{n}}}{x} \]

   11. Applied rewrites 73.8%

       \[\leadsto \frac{\color{blue}{\frac{1}{n} + \left(\frac{-0.5}{x \cdot n} - \frac{-0.3333333333333333}{x \cdot \left(x \cdot n\right)}\right)}}{x} \]

   12. Taylor expanded in n around 0

       \[\leadsto \frac{\color{blue}{\frac{\left(1 + \frac{1}{3} \cdot \frac{1}{{x}^{2}}\right) - \frac{1}{2} \cdot \frac{1}{x}}{n}}}{x} \]
   13. Step-by-step derivation

       1. lower-/.f64 N/A

          \[\leadsto \frac{\color{blue}{\frac{\left(1 + \frac{1}{3} \cdot \frac{1}{{x}^{2}}\right) - \frac{1}{2} \cdot \frac{1}{x}}{n}}}{x} \]

   14. Applied rewrites 73.8%

       \[\leadsto \frac{\color{blue}{\frac{1 + \frac{-0.5 + \frac{0.3333333333333333}{x}}{x}}{n}}}{x} \]

   if 1.45000000000000001e129 < x

   1. Initial program 86.5%

      \[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{1 - e^{\frac{\log x}{n}}} \]
   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 58.9

          \[\leadsto 1 - {x}^{\color{blue}{\left(\frac{1}{n}\right)}} \]

   5. Applied rewrites 58.9%

      \[\leadsto \color{blue}{1 - {x}^{\left(\frac{1}{n}\right)}} \]

   6. Taylor expanded in n around inf

      \[\leadsto 1 - \color{blue}{1} \]
   7. Step-by-step derivation

   8. Applied rewrites 86.5%

      \[\leadsto 1 - \color{blue}{1} \]
   9. Step-by-step derivation

      1. metadata-eval 86.5

         \[\leadsto \color{blue}{0} \]

   10. Applied rewrites 86.5%

       \[\leadsto \color{blue}{0} \]
3. Recombined 4 regimes into one program.

4. Final simplification 65.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 2 \cdot 10^{-148}:\\ \;\;\;\;-\frac{\log x}{n}\\ \mathbf{elif}\;x \leq 3.8 \cdot 10^{-71}:\\ \;\;\;\;\frac{0.3333333333333333}{x \cdot \left(x \cdot \left(x \cdot n\right)\right)}\\ \mathbf{elif}\;x \leq 2.7 \cdot 10^{-16}:\\ \;\;\;\;-\frac{\log x}{n}\\ \mathbf{elif}\;x \leq 1.45 \cdot 10^{+129}:\\ \;\;\;\;\frac{\frac{\frac{-0.5 + \frac{0.3333333333333333}{x}}{x} + 1}{n}}{x}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 55.2% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := n \cdot \left(x \cdot x\right)\\ \mathbf{if}\;\frac{1}{n} \leq -100:\\ \;\;\;\;\frac{0.3333333333333333}{x \cdot \left(x \cdot \left(x \cdot n\right)\right)}\\ \mathbf{elif}\;\frac{1}{n} \leq 10^{+93}:\\ \;\;\;\;\frac{1}{x \cdot \mathsf{fma}\left(0.5, \frac{n}{x}, n\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{n} + \frac{\frac{-0.5}{x} \cdot t\_0 - n \cdot -0.3333333333333333}{n \cdot t\_0}}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (let* ((t_0 (* n (* x x))))
   (if (<= (/ 1.0 n) -100.0)
     (/ 0.3333333333333333 (* x (* x (* x n))))
     (if (<= (/ 1.0 n) 1e+93)
       (/ 1.0 (* x (fma 0.5 (/ n x) n)))
       (/
        (+
         (/ 1.0 n)
         (/ (- (* (/ -0.5 x) t_0) (* n -0.3333333333333333)) (* n t_0)))
        x)))))

C

double code(double x, double n) {
	double t_0 = n * (x * x);
	double tmp;
	if ((1.0 / n) <= -100.0) {
		tmp = 0.3333333333333333 / (x * (x * (x * n)));
	} else if ((1.0 / n) <= 1e+93) {
		tmp = 1.0 / (x * fma(0.5, (n / x), n));
	} else {
		tmp = ((1.0 / n) + ((((-0.5 / x) * t_0) - (n * -0.3333333333333333)) / (n * t_0))) / x;
	}
	return tmp;
}

Julia

function code(x, n)
	t_0 = Float64(n * Float64(x * x))
	tmp = 0.0
	if (Float64(1.0 / n) <= -100.0)
		tmp = Float64(0.3333333333333333 / Float64(x * Float64(x * Float64(x * n))));
	elseif (Float64(1.0 / n) <= 1e+93)
		tmp = Float64(1.0 / Float64(x * fma(0.5, Float64(n / x), n)));
	else
		tmp = Float64(Float64(Float64(1.0 / n) + Float64(Float64(Float64(Float64(-0.5 / x) * t_0) - Float64(n * -0.3333333333333333)) / Float64(n * t_0))) / x);
	end
	return tmp
end

Wolfram

code[x_, n_] := Block[{t$95$0 = N[(n * N[(x * x), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(1.0 / n), $MachinePrecision], -100.0], N[(0.3333333333333333 / N[(x * N[(x * N[(x * n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 1e+93], N[(1.0 / N[(x * N[(0.5 * N[(n / x), $MachinePrecision] + n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(1.0 / n), $MachinePrecision] + N[(N[(N[(N[(-0.5 / x), $MachinePrecision] * t$95$0), $MachinePrecision] - N[(n * -0.3333333333333333), $MachinePrecision]), $MachinePrecision] / N[(n * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 #s(literal 1 binary64) n) < -100

   1. Initial program 100.0%

      \[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in n around inf

      \[\leadsto \color{blue}{\frac{\log \left(1 + x\right) - \log x}{n}} \]
   4. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\log \left(1 + x\right) - \log x}{n}} \]

      2. lower--.f64 N/A

         \[\leadsto \frac{\color{blue}{\log \left(1 + x\right) - \log x}}{n} \]

      3. lower-log1p.f64 N/A

         \[\leadsto \frac{\color{blue}{\mathsf{log1p}\left(x\right)} - \log x}{n} \]

      4. lower-log.f64 46.1

         \[\leadsto \frac{\mathsf{log1p}\left(x\right) - \color{blue}{\log x}}{n} \]

   5. Applied rewrites 46.1%

      \[\leadsto \color{blue}{\frac{\mathsf{log1p}\left(x\right) - \log x}{n}} \]

   6. Taylor expanded in x around inf

      \[\leadsto \color{blue}{\frac{\left(\frac{1}{n} + \frac{1}{3} \cdot \frac{1}{n \cdot {x}^{2}}\right) - \left(\frac{\frac{1}{2}}{n \cdot x} + \frac{1}{4} \cdot \frac{1}{n \cdot {x}^{3}}\right)}{x}} \]
   7. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\left(\frac{1}{n} + \frac{1}{3} \cdot \frac{1}{n \cdot {x}^{2}}\right) - \left(\frac{\frac{1}{2}}{n \cdot x} + \frac{1}{4} \cdot \frac{1}{n \cdot {x}^{3}}\right)}{x}} \]

   8. Applied rewrites 1.6%

      \[\leadsto \color{blue}{\frac{\left(\frac{1}{n} + \frac{0.3333333333333333}{n \cdot \left(x \cdot x\right)}\right) - \left(\frac{0.5}{x \cdot n} + \frac{0.25}{n \cdot \left(x \cdot \left(x \cdot x\right)\right)}\right)}{x}} \]

   9. Taylor expanded in x around -inf

      \[\leadsto \frac{\color{blue}{-1 \cdot \frac{\frac{1}{2} \cdot \frac{1}{n} - \frac{1}{3} \cdot \frac{1}{n \cdot x}}{x} + \frac{1}{n}}}{x} \]

   11. Applied rewrites 16.4%

       \[\leadsto \frac{\color{blue}{\frac{1}{n} + \left(\frac{-0.5}{x \cdot n} - \frac{-0.3333333333333333}{x \cdot \left(x \cdot n\right)}\right)}}{x} \]

   12. Taylor expanded in x around 0

       \[\leadsto \color{blue}{\frac{\frac{1}{3}}{n \cdot {x}^{3}}} \]
   13. Step-by-step derivation

       1. lower-/.f64 N/A

          \[\leadsto \color{blue}{\frac{\frac{1}{3}}{n \cdot {x}^{3}}} \]

       2. *-commutative N/A

          \[\leadsto \frac{\frac{1}{3}}{\color{blue}{{x}^{3} \cdot n}} \]

       3. cube-mult N/A

          \[\leadsto \frac{\frac{1}{3}}{\color{blue}{\left(x \cdot \left(x \cdot x\right)\right)} \cdot n} \]

       4. unpow2 N/A

          \[\leadsto \frac{\frac{1}{3}}{\left(x \cdot \color{blue}{{x}^{2}}\right) \cdot n} \]

       5. associate-*l* N/A

          \[\leadsto \frac{\frac{1}{3}}{\color{blue}{x \cdot \left({x}^{2} \cdot n\right)}} \]

       6. *-commutative N/A

          \[\leadsto \frac{\frac{1}{3}}{x \cdot \color{blue}{\left(n \cdot {x}^{2}\right)}} \]

       7. lower-*.f64 N/A

          \[\leadsto \frac{\frac{1}{3}}{\color{blue}{x \cdot \left(n \cdot {x}^{2}\right)}} \]

       8. *-commutative N/A

          \[\leadsto \frac{\frac{1}{3}}{x \cdot \color{blue}{\left({x}^{2} \cdot n\right)}} \]

       9. unpow2 N/A

          \[\leadsto \frac{\frac{1}{3}}{x \cdot \left(\color{blue}{\left(x \cdot x\right)} \cdot n\right)} \]

       10. associate-*l* N/A

           \[\leadsto \frac{\frac{1}{3}}{x \cdot \color{blue}{\left(x \cdot \left(x \cdot n\right)\right)}} \]

       11. *-commutative N/A

           \[\leadsto \frac{\frac{1}{3}}{x \cdot \left(x \cdot \color{blue}{\left(n \cdot x\right)}\right)} \]

       12. lower-*.f64 N/A

           \[\leadsto \frac{\frac{1}{3}}{x \cdot \color{blue}{\left(x \cdot \left(n \cdot x\right)\right)}} \]

       13. *-commutative N/A

           \[\leadsto \frac{\frac{1}{3}}{x \cdot \left(x \cdot \color{blue}{\left(x \cdot n\right)}\right)} \]

       14. lower-*.f64 68.1

           \[\leadsto \frac{0.3333333333333333}{x \cdot \left(x \cdot \color{blue}{\left(x \cdot n\right)}\right)} \]

   14. Applied rewrites 68.1%

       \[\leadsto \color{blue}{\frac{0.3333333333333333}{x \cdot \left(x \cdot \left(x \cdot n\right)\right)}} \]

   if -100 < (/.f64 #s(literal 1 binary64) n) < 1.00000000000000004e93

   1. Initial program 40.2%

      \[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in 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 64.6

         \[\leadsto \frac{\mathsf{log1p}\left(x\right) - \color{blue}{\log x}}{n} \]

   5. Applied rewrites 64.6%

      \[\leadsto \color{blue}{\frac{\mathsf{log1p}\left(x\right) - \log x}{n}} \]
   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. clear-num N/A

         \[\leadsto \color{blue}{\frac{1}{\frac{n}{\mathsf{log1p}\left(x\right) - \log x}}} \]

      5. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{1}{\frac{n}{\mathsf{log1p}\left(x\right) - \log x}}} \]

      6. lower-/.f64 64.5

         \[\leadsto \frac{1}{\color{blue}{\frac{n}{\mathsf{log1p}\left(x\right) - \log x}}} \]

      7. lift--.f64 N/A

         \[\leadsto \frac{1}{\frac{n}{\color{blue}{\mathsf{log1p}\left(x\right) - \log x}}} \]

      8. lift-log1p.f64 N/A

         \[\leadsto \frac{1}{\frac{n}{\color{blue}{\log \left(1 + x\right)} - \log x}} \]

      9. lift-log.f64 N/A

         \[\leadsto \frac{1}{\frac{n}{\log \left(1 + x\right) - \color{blue}{\log x}}} \]

      10. diff-log N/A

          \[\leadsto \frac{1}{\frac{n}{\color{blue}{\log \left(\frac{1 + x}{x}\right)}}} \]

      11. lower-log.f64 N/A

          \[\leadsto \frac{1}{\frac{n}{\color{blue}{\log \left(\frac{1 + x}{x}\right)}}} \]

      12. +-commutative N/A

          \[\leadsto \frac{1}{\frac{n}{\log \left(\frac{\color{blue}{x + 1}}{x}\right)}} \]

      13. lift-+.f64 N/A

          \[\leadsto \frac{1}{\frac{n}{\log \left(\frac{\color{blue}{x + 1}}{x}\right)}} \]

      14. lower-/.f64 64.7

          \[\leadsto \frac{1}{\frac{n}{\log \color{blue}{\left(\frac{x + 1}{x}\right)}}} \]

   7. Applied rewrites 64.7%

      \[\leadsto \color{blue}{\frac{1}{\frac{n}{\log \left(\frac{x + 1}{x}\right)}}} \]

   8. Taylor expanded in x around inf

      \[\leadsto \frac{1}{\color{blue}{x \cdot \left(n + \frac{1}{2} \cdot \frac{n}{x}\right)}} \]
   9. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \frac{1}{\color{blue}{x \cdot \left(n + \frac{1}{2} \cdot \frac{n}{x}\right)}} \]

      2. +-commutative N/A

         \[\leadsto \frac{1}{x \cdot \color{blue}{\left(\frac{1}{2} \cdot \frac{n}{x} + n\right)}} \]

      3. lower-fma.f64 N/A

         \[\leadsto \frac{1}{x \cdot \color{blue}{\mathsf{fma}\left(\frac{1}{2}, \frac{n}{x}, n\right)}} \]

      4. lower-/.f64 58.3

         \[\leadsto \frac{1}{x \cdot \mathsf{fma}\left(0.5, \color{blue}{\frac{n}{x}}, n\right)} \]

   10. Applied rewrites 58.3%

       \[\leadsto \frac{1}{\color{blue}{x \cdot \mathsf{fma}\left(0.5, \frac{n}{x}, n\right)}} \]

   if 1.00000000000000004e93 < (/.f64 #s(literal 1 binary64) n)

   1. Initial program 29.8%

      \[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in n around inf

      \[\leadsto \color{blue}{\frac{\log \left(1 + x\right) - \log x}{n}} \]
   4. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\log \left(1 + x\right) - \log x}{n}} \]

      2. lower--.f64 N/A

         \[\leadsto \frac{\color{blue}{\log \left(1 + x\right) - \log x}}{n} \]

      3. lower-log1p.f64 N/A

         \[\leadsto \frac{\color{blue}{\mathsf{log1p}\left(x\right)} - \log x}{n} \]

      4. lower-log.f64 5.8

         \[\leadsto \frac{\mathsf{log1p}\left(x\right) - \color{blue}{\log x}}{n} \]

   5. Applied rewrites 5.8%

      \[\leadsto \color{blue}{\frac{\mathsf{log1p}\left(x\right) - \log x}{n}} \]

   6. Taylor expanded in x around inf

      \[\leadsto \color{blue}{\frac{\left(\frac{1}{n} + \frac{1}{3} \cdot \frac{1}{n \cdot {x}^{2}}\right) - \left(\frac{\frac{1}{2}}{n \cdot x} + \frac{1}{4} \cdot \frac{1}{n \cdot {x}^{3}}\right)}{x}} \]
   7. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\left(\frac{1}{n} + \frac{1}{3} \cdot \frac{1}{n \cdot {x}^{2}}\right) - \left(\frac{\frac{1}{2}}{n \cdot x} + \frac{1}{4} \cdot \frac{1}{n \cdot {x}^{3}}\right)}{x}} \]

   8. Applied rewrites 0.0%

      \[\leadsto \color{blue}{\frac{\left(\frac{1}{n} + \frac{0.3333333333333333}{n \cdot \left(x \cdot x\right)}\right) - \left(\frac{0.5}{x \cdot n} + \frac{0.25}{n \cdot \left(x \cdot \left(x \cdot x\right)\right)}\right)}{x}} \]

   9. Taylor expanded in x around -inf

      \[\leadsto \frac{\color{blue}{-1 \cdot \frac{\frac{1}{2} \cdot \frac{1}{n} - \frac{1}{3} \cdot \frac{1}{n \cdot x}}{x} + \frac{1}{n}}}{x} \]

   11. Applied rewrites 14.9%

       \[\leadsto \frac{\color{blue}{\frac{1}{n} + \left(\frac{-0.5}{x \cdot n} - \frac{-0.3333333333333333}{x \cdot \left(x \cdot n\right)}\right)}}{x} \]
   12. Step-by-step derivation

       1. associate-/r* N/A

          \[\leadsto \frac{\frac{1}{n} + \left(\color{blue}{\frac{\frac{\frac{-1}{2}}{x}}{n}} - \frac{\frac{-1}{3}}{x \cdot \left(x \cdot n\right)}\right)}{x} \]

       2. lift-*.f64 N/A

          \[\leadsto \frac{\frac{1}{n} + \left(\frac{\frac{\frac{-1}{2}}{x}}{n} - \frac{\frac{-1}{3}}{x \cdot \color{blue}{\left(x \cdot n\right)}}\right)}{x} \]

       3. lift-*.f64 N/A

          \[\leadsto \frac{\frac{1}{n} + \left(\frac{\frac{\frac{-1}{2}}{x}}{n} - \frac{\frac{-1}{3}}{\color{blue}{x \cdot \left(x \cdot n\right)}}\right)}{x} \]

       4. frac-sub N/A

          \[\leadsto \frac{\frac{1}{n} + \color{blue}{\frac{\frac{\frac{-1}{2}}{x} \cdot \left(x \cdot \left(x \cdot n\right)\right) - n \cdot \frac{-1}{3}}{n \cdot \left(x \cdot \left(x \cdot n\right)\right)}}}{x} \]

       5. lower-/.f64 N/A

          \[\leadsto \frac{\frac{1}{n} + \color{blue}{\frac{\frac{\frac{-1}{2}}{x} \cdot \left(x \cdot \left(x \cdot n\right)\right) - n \cdot \frac{-1}{3}}{n \cdot \left(x \cdot \left(x \cdot n\right)\right)}}}{x} \]

       6. lower--.f64 N/A

          \[\leadsto \frac{\frac{1}{n} + \frac{\color{blue}{\frac{\frac{-1}{2}}{x} \cdot \left(x \cdot \left(x \cdot n\right)\right) - n \cdot \frac{-1}{3}}}{n \cdot \left(x \cdot \left(x \cdot n\right)\right)}}{x} \]

       7. lower-*.f64 N/A

          \[\leadsto \frac{\frac{1}{n} + \frac{\color{blue}{\frac{\frac{-1}{2}}{x} \cdot \left(x \cdot \left(x \cdot n\right)\right)} - n \cdot \frac{-1}{3}}{n \cdot \left(x \cdot \left(x \cdot n\right)\right)}}{x} \]

       8. lower-/.f64 N/A

          \[\leadsto \frac{\frac{1}{n} + \frac{\color{blue}{\frac{\frac{-1}{2}}{x}} \cdot \left(x \cdot \left(x \cdot n\right)\right) - n \cdot \frac{-1}{3}}{n \cdot \left(x \cdot \left(x \cdot n\right)\right)}}{x} \]

       9. lift-*.f64 N/A

          \[\leadsto \frac{\frac{1}{n} + \frac{\frac{\frac{-1}{2}}{x} \cdot \color{blue}{\left(x \cdot \left(x \cdot n\right)\right)} - n \cdot \frac{-1}{3}}{n \cdot \left(x \cdot \left(x \cdot n\right)\right)}}{x} \]

       10. lift-*.f64 N/A

           \[\leadsto \frac{\frac{1}{n} + \frac{\frac{\frac{-1}{2}}{x} \cdot \left(x \cdot \color{blue}{\left(x \cdot n\right)}\right) - n \cdot \frac{-1}{3}}{n \cdot \left(x \cdot \left(x \cdot n\right)\right)}}{x} \]

       11. associate-*r* N/A

           \[\leadsto \frac{\frac{1}{n} + \frac{\frac{\frac{-1}{2}}{x} \cdot \color{blue}{\left(\left(x \cdot x\right) \cdot n\right)} - n \cdot \frac{-1}{3}}{n \cdot \left(x \cdot \left(x \cdot n\right)\right)}}{x} \]

       12. pow2 N/A

           \[\leadsto \frac{\frac{1}{n} + \frac{\frac{\frac{-1}{2}}{x} \cdot \left(\color{blue}{{x}^{2}} \cdot n\right) - n \cdot \frac{-1}{3}}{n \cdot \left(x \cdot \left(x \cdot n\right)\right)}}{x} \]

       13. *-commutative N/A

           \[\leadsto \frac{\frac{1}{n} + \frac{\frac{\frac{-1}{2}}{x} \cdot \color{blue}{\left(n \cdot {x}^{2}\right)} - n \cdot \frac{-1}{3}}{n \cdot \left(x \cdot \left(x \cdot n\right)\right)}}{x} \]

       14. lower-*.f64 N/A

           \[\leadsto \frac{\frac{1}{n} + \frac{\frac{\frac{-1}{2}}{x} \cdot \color{blue}{\left(n \cdot {x}^{2}\right)} - n \cdot \frac{-1}{3}}{n \cdot \left(x \cdot \left(x \cdot n\right)\right)}}{x} \]

       15. pow2 N/A

           \[\leadsto \frac{\frac{1}{n} + \frac{\frac{\frac{-1}{2}}{x} \cdot \left(n \cdot \color{blue}{\left(x \cdot x\right)}\right) - n \cdot \frac{-1}{3}}{n \cdot \left(x \cdot \left(x \cdot n\right)\right)}}{x} \]

       16. lower-*.f64 N/A

           \[\leadsto \frac{\frac{1}{n} + \frac{\frac{\frac{-1}{2}}{x} \cdot \left(n \cdot \color{blue}{\left(x \cdot x\right)}\right) - n \cdot \frac{-1}{3}}{n \cdot \left(x \cdot \left(x \cdot n\right)\right)}}{x} \]

       17. lower-*.f64 N/A

           \[\leadsto \frac{\frac{1}{n} + \frac{\frac{\frac{-1}{2}}{x} \cdot \left(n \cdot \left(x \cdot x\right)\right) - \color{blue}{n \cdot \frac{-1}{3}}}{n \cdot \left(x \cdot \left(x \cdot n\right)\right)}}{x} \]

       18. lower-*.f64 66.8

           \[\leadsto \frac{\frac{1}{n} + \frac{\frac{-0.5}{x} \cdot \left(n \cdot \left(x \cdot x\right)\right) - n \cdot -0.3333333333333333}{\color{blue}{n \cdot \left(x \cdot \left(x \cdot n\right)\right)}}}{x} \]

       19. lift-*.f64 N/A

           \[\leadsto \frac{\frac{1}{n} + \frac{\frac{\frac{-1}{2}}{x} \cdot \left(n \cdot \left(x \cdot x\right)\right) - n \cdot \frac{-1}{3}}{n \cdot \color{blue}{\left(x \cdot \left(x \cdot n\right)\right)}}}{x} \]

       20. lift-*.f64 N/A

           \[\leadsto \frac{\frac{1}{n} + \frac{\frac{\frac{-1}{2}}{x} \cdot \left(n \cdot \left(x \cdot x\right)\right) - n \cdot \frac{-1}{3}}{n \cdot \left(x \cdot \color{blue}{\left(x \cdot n\right)}\right)}}{x} \]

       21. associate-*r* N/A

           \[\leadsto \frac{\frac{1}{n} + \frac{\frac{\frac{-1}{2}}{x} \cdot \left(n \cdot \left(x \cdot x\right)\right) - n \cdot \frac{-1}{3}}{n \cdot \color{blue}{\left(\left(x \cdot x\right) \cdot n\right)}}}{x} \]

   13. Applied rewrites 66.8%

       \[\leadsto \frac{\frac{1}{n} + \color{blue}{\frac{\frac{-0.5}{x} \cdot \left(n \cdot \left(x \cdot x\right)\right) - n \cdot -0.3333333333333333}{n \cdot \left(n \cdot \left(x \cdot x\right)\right)}}}{x} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 17: 54.5% accurate, 2.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x \cdot \left(x \cdot n\right)\\ \mathbf{if}\;\frac{1}{n} \leq -100:\\ \;\;\;\;\frac{0.3333333333333333}{x \cdot t\_0}\\ \mathbf{elif}\;\frac{1}{n} \leq 10^{+93}:\\ \;\;\;\;\frac{1}{x \cdot \mathsf{fma}\left(0.5, \frac{n}{x}, n\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{n} + \frac{0.3333333333333333}{t\_0}}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (let* ((t_0 (* x (* x n))))
   (if (<= (/ 1.0 n) -100.0)
     (/ 0.3333333333333333 (* x t_0))
     (if (<= (/ 1.0 n) 1e+93)
       (/ 1.0 (* x (fma 0.5 (/ n x) n)))
       (/ (+ (/ 1.0 n) (/ 0.3333333333333333 t_0)) x)))))

C

double code(double x, double n) {
	double t_0 = x * (x * n);
	double tmp;
	if ((1.0 / n) <= -100.0) {
		tmp = 0.3333333333333333 / (x * t_0);
	} else if ((1.0 / n) <= 1e+93) {
		tmp = 1.0 / (x * fma(0.5, (n / x), n));
	} else {
		tmp = ((1.0 / n) + (0.3333333333333333 / t_0)) / x;
	}
	return tmp;
}

Julia

function code(x, n)
	t_0 = Float64(x * Float64(x * n))
	tmp = 0.0
	if (Float64(1.0 / n) <= -100.0)
		tmp = Float64(0.3333333333333333 / Float64(x * t_0));
	elseif (Float64(1.0 / n) <= 1e+93)
		tmp = Float64(1.0 / Float64(x * fma(0.5, Float64(n / x), n)));
	else
		tmp = Float64(Float64(Float64(1.0 / n) + Float64(0.3333333333333333 / t_0)) / x);
	end
	return tmp
end

Wolfram

code[x_, n_] := Block[{t$95$0 = N[(x * N[(x * n), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(1.0 / n), $MachinePrecision], -100.0], N[(0.3333333333333333 / N[(x * t$95$0), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 1e+93], N[(1.0 / N[(x * N[(0.5 * N[(n / x), $MachinePrecision] + n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(1.0 / n), $MachinePrecision] + N[(0.3333333333333333 / t$95$0), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 #s(literal 1 binary64) n) < -100

   1. Initial program 100.0%

      \[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in n around inf

      \[\leadsto \color{blue}{\frac{\log \left(1 + x\right) - \log x}{n}} \]
   4. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\log \left(1 + x\right) - \log x}{n}} \]

      2. lower--.f64 N/A

         \[\leadsto \frac{\color{blue}{\log \left(1 + x\right) - \log x}}{n} \]

      3. lower-log1p.f64 N/A

         \[\leadsto \frac{\color{blue}{\mathsf{log1p}\left(x\right)} - \log x}{n} \]

      4. lower-log.f64 46.1

         \[\leadsto \frac{\mathsf{log1p}\left(x\right) - \color{blue}{\log x}}{n} \]

   5. Applied rewrites 46.1%

      \[\leadsto \color{blue}{\frac{\mathsf{log1p}\left(x\right) - \log x}{n}} \]

   6. Taylor expanded in x around inf

      \[\leadsto \color{blue}{\frac{\left(\frac{1}{n} + \frac{1}{3} \cdot \frac{1}{n \cdot {x}^{2}}\right) - \left(\frac{\frac{1}{2}}{n \cdot x} + \frac{1}{4} \cdot \frac{1}{n \cdot {x}^{3}}\right)}{x}} \]
   7. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\left(\frac{1}{n} + \frac{1}{3} \cdot \frac{1}{n \cdot {x}^{2}}\right) - \left(\frac{\frac{1}{2}}{n \cdot x} + \frac{1}{4} \cdot \frac{1}{n \cdot {x}^{3}}\right)}{x}} \]

   8. Applied rewrites 1.6%

      \[\leadsto \color{blue}{\frac{\left(\frac{1}{n} + \frac{0.3333333333333333}{n \cdot \left(x \cdot x\right)}\right) - \left(\frac{0.5}{x \cdot n} + \frac{0.25}{n \cdot \left(x \cdot \left(x \cdot x\right)\right)}\right)}{x}} \]

   9. Taylor expanded in x around -inf

      \[\leadsto \frac{\color{blue}{-1 \cdot \frac{\frac{1}{2} \cdot \frac{1}{n} - \frac{1}{3} \cdot \frac{1}{n \cdot x}}{x} + \frac{1}{n}}}{x} \]

   11. Applied rewrites 16.4%

       \[\leadsto \frac{\color{blue}{\frac{1}{n} + \left(\frac{-0.5}{x \cdot n} - \frac{-0.3333333333333333}{x \cdot \left(x \cdot n\right)}\right)}}{x} \]

   12. Taylor expanded in x around 0

       \[\leadsto \color{blue}{\frac{\frac{1}{3}}{n \cdot {x}^{3}}} \]
   13. Step-by-step derivation

       1. lower-/.f64 N/A

          \[\leadsto \color{blue}{\frac{\frac{1}{3}}{n \cdot {x}^{3}}} \]

       2. *-commutative N/A

          \[\leadsto \frac{\frac{1}{3}}{\color{blue}{{x}^{3} \cdot n}} \]

       3. cube-mult N/A

          \[\leadsto \frac{\frac{1}{3}}{\color{blue}{\left(x \cdot \left(x \cdot x\right)\right)} \cdot n} \]

       4. unpow2 N/A

          \[\leadsto \frac{\frac{1}{3}}{\left(x \cdot \color{blue}{{x}^{2}}\right) \cdot n} \]

       5. associate-*l* N/A

          \[\leadsto \frac{\frac{1}{3}}{\color{blue}{x \cdot \left({x}^{2} \cdot n\right)}} \]

       6. *-commutative N/A

          \[\leadsto \frac{\frac{1}{3}}{x \cdot \color{blue}{\left(n \cdot {x}^{2}\right)}} \]

       7. lower-*.f64 N/A

          \[\leadsto \frac{\frac{1}{3}}{\color{blue}{x \cdot \left(n \cdot {x}^{2}\right)}} \]

       8. *-commutative N/A

          \[\leadsto \frac{\frac{1}{3}}{x \cdot \color{blue}{\left({x}^{2} \cdot n\right)}} \]

       9. unpow2 N/A

          \[\leadsto \frac{\frac{1}{3}}{x \cdot \left(\color{blue}{\left(x \cdot x\right)} \cdot n\right)} \]

       10. associate-*l* N/A

           \[\leadsto \frac{\frac{1}{3}}{x \cdot \color{blue}{\left(x \cdot \left(x \cdot n\right)\right)}} \]

       11. *-commutative N/A

           \[\leadsto \frac{\frac{1}{3}}{x \cdot \left(x \cdot \color{blue}{\left(n \cdot x\right)}\right)} \]

       12. lower-*.f64 N/A

           \[\leadsto \frac{\frac{1}{3}}{x \cdot \color{blue}{\left(x \cdot \left(n \cdot x\right)\right)}} \]

       13. *-commutative N/A

           \[\leadsto \frac{\frac{1}{3}}{x \cdot \left(x \cdot \color{blue}{\left(x \cdot n\right)}\right)} \]

       14. lower-*.f64 68.1

           \[\leadsto \frac{0.3333333333333333}{x \cdot \left(x \cdot \color{blue}{\left(x \cdot n\right)}\right)} \]

   14. Applied rewrites 68.1%

       \[\leadsto \color{blue}{\frac{0.3333333333333333}{x \cdot \left(x \cdot \left(x \cdot n\right)\right)}} \]

   if -100 < (/.f64 #s(literal 1 binary64) n) < 1.00000000000000004e93

   1. Initial program 40.2%

      \[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in 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 64.6

         \[\leadsto \frac{\mathsf{log1p}\left(x\right) - \color{blue}{\log x}}{n} \]

   5. Applied rewrites 64.6%

      \[\leadsto \color{blue}{\frac{\mathsf{log1p}\left(x\right) - \log x}{n}} \]
   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. clear-num N/A

         \[\leadsto \color{blue}{\frac{1}{\frac{n}{\mathsf{log1p}\left(x\right) - \log x}}} \]

      5. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{1}{\frac{n}{\mathsf{log1p}\left(x\right) - \log x}}} \]

      6. lower-/.f64 64.5

         \[\leadsto \frac{1}{\color{blue}{\frac{n}{\mathsf{log1p}\left(x\right) - \log x}}} \]

      7. lift--.f64 N/A

         \[\leadsto \frac{1}{\frac{n}{\color{blue}{\mathsf{log1p}\left(x\right) - \log x}}} \]

      8. lift-log1p.f64 N/A

         \[\leadsto \frac{1}{\frac{n}{\color{blue}{\log \left(1 + x\right)} - \log x}} \]

      9. lift-log.f64 N/A

         \[\leadsto \frac{1}{\frac{n}{\log \left(1 + x\right) - \color{blue}{\log x}}} \]

      10. diff-log N/A

          \[\leadsto \frac{1}{\frac{n}{\color{blue}{\log \left(\frac{1 + x}{x}\right)}}} \]

      11. lower-log.f64 N/A

          \[\leadsto \frac{1}{\frac{n}{\color{blue}{\log \left(\frac{1 + x}{x}\right)}}} \]

      12. +-commutative N/A

          \[\leadsto \frac{1}{\frac{n}{\log \left(\frac{\color{blue}{x + 1}}{x}\right)}} \]

      13. lift-+.f64 N/A

          \[\leadsto \frac{1}{\frac{n}{\log \left(\frac{\color{blue}{x + 1}}{x}\right)}} \]

      14. lower-/.f64 64.7

          \[\leadsto \frac{1}{\frac{n}{\log \color{blue}{\left(\frac{x + 1}{x}\right)}}} \]

   7. Applied rewrites 64.7%

      \[\leadsto \color{blue}{\frac{1}{\frac{n}{\log \left(\frac{x + 1}{x}\right)}}} \]

   8. Taylor expanded in x around inf

      \[\leadsto \frac{1}{\color{blue}{x \cdot \left(n + \frac{1}{2} \cdot \frac{n}{x}\right)}} \]
   9. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \frac{1}{\color{blue}{x \cdot \left(n + \frac{1}{2} \cdot \frac{n}{x}\right)}} \]

      2. +-commutative N/A

         \[\leadsto \frac{1}{x \cdot \color{blue}{\left(\frac{1}{2} \cdot \frac{n}{x} + n\right)}} \]

      3. lower-fma.f64 N/A

         \[\leadsto \frac{1}{x \cdot \color{blue}{\mathsf{fma}\left(\frac{1}{2}, \frac{n}{x}, n\right)}} \]

      4. lower-/.f64 58.3

         \[\leadsto \frac{1}{x \cdot \mathsf{fma}\left(0.5, \color{blue}{\frac{n}{x}}, n\right)} \]

   10. Applied rewrites 58.3%

       \[\leadsto \frac{1}{\color{blue}{x \cdot \mathsf{fma}\left(0.5, \frac{n}{x}, n\right)}} \]

   if 1.00000000000000004e93 < (/.f64 #s(literal 1 binary64) n)

   1. Initial program 29.8%

      \[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in n around inf

      \[\leadsto \color{blue}{\frac{\log \left(1 + x\right) - \log x}{n}} \]
   4. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\log \left(1 + x\right) - \log x}{n}} \]

      2. lower--.f64 N/A

         \[\leadsto \frac{\color{blue}{\log \left(1 + x\right) - \log x}}{n} \]

      3. lower-log1p.f64 N/A

         \[\leadsto \frac{\color{blue}{\mathsf{log1p}\left(x\right)} - \log x}{n} \]

      4. lower-log.f64 5.8

         \[\leadsto \frac{\mathsf{log1p}\left(x\right) - \color{blue}{\log x}}{n} \]

   5. Applied rewrites 5.8%

      \[\leadsto \color{blue}{\frac{\mathsf{log1p}\left(x\right) - \log x}{n}} \]

   6. Taylor expanded in x around inf

      \[\leadsto \color{blue}{\frac{\left(\frac{1}{n} + \frac{1}{3} \cdot \frac{1}{n \cdot {x}^{2}}\right) - \left(\frac{\frac{1}{2}}{n \cdot x} + \frac{1}{4} \cdot \frac{1}{n \cdot {x}^{3}}\right)}{x}} \]
   7. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\left(\frac{1}{n} + \frac{1}{3} \cdot \frac{1}{n \cdot {x}^{2}}\right) - \left(\frac{\frac{1}{2}}{n \cdot x} + \frac{1}{4} \cdot \frac{1}{n \cdot {x}^{3}}\right)}{x}} \]

   8. Applied rewrites 0.0%

      \[\leadsto \color{blue}{\frac{\left(\frac{1}{n} + \frac{0.3333333333333333}{n \cdot \left(x \cdot x\right)}\right) - \left(\frac{0.5}{x \cdot n} + \frac{0.25}{n \cdot \left(x \cdot \left(x \cdot x\right)\right)}\right)}{x}} \]

   9. Taylor expanded in x around -inf

      \[\leadsto \frac{\color{blue}{-1 \cdot \frac{\frac{1}{2} \cdot \frac{1}{n} - \frac{1}{3} \cdot \frac{1}{n \cdot x}}{x} + \frac{1}{n}}}{x} \]

   11. Applied rewrites 14.9%

       \[\leadsto \frac{\color{blue}{\frac{1}{n} + \left(\frac{-0.5}{x \cdot n} - \frac{-0.3333333333333333}{x \cdot \left(x \cdot n\right)}\right)}}{x} \]

   12. Taylor expanded in x around 0

       \[\leadsto \frac{\frac{1}{n} + \color{blue}{\frac{\frac{1}{3}}{n \cdot {x}^{2}}}}{x} \]
   13. Step-by-step derivation

       1. lower-/.f64 N/A

          \[\leadsto \frac{\frac{1}{n} + \color{blue}{\frac{\frac{1}{3}}{n \cdot {x}^{2}}}}{x} \]

       2. *-commutative N/A

          \[\leadsto \frac{\frac{1}{n} + \frac{\frac{1}{3}}{\color{blue}{{x}^{2} \cdot n}}}{x} \]

       3. unpow2 N/A

          \[\leadsto \frac{\frac{1}{n} + \frac{\frac{1}{3}}{\color{blue}{\left(x \cdot x\right)} \cdot n}}{x} \]

       4. associate-*l* N/A

          \[\leadsto \frac{\frac{1}{n} + \frac{\frac{1}{3}}{\color{blue}{x \cdot \left(x \cdot n\right)}}}{x} \]

       5. *-commutative N/A

          \[\leadsto \frac{\frac{1}{n} + \frac{\frac{1}{3}}{x \cdot \color{blue}{\left(n \cdot x\right)}}}{x} \]

       6. lower-*.f64 N/A

          \[\leadsto \frac{\frac{1}{n} + \frac{\frac{1}{3}}{\color{blue}{x \cdot \left(n \cdot x\right)}}}{x} \]

       7. *-commutative N/A

          \[\leadsto \frac{\frac{1}{n} + \frac{\frac{1}{3}}{x \cdot \color{blue}{\left(x \cdot n\right)}}}{x} \]

       8. lower-*.f64 60.6

          \[\leadsto \frac{\frac{1}{n} + \frac{0.3333333333333333}{x \cdot \color{blue}{\left(x \cdot n\right)}}}{x} \]

   14. Applied rewrites 60.6%

       \[\leadsto \frac{\frac{1}{n} + \color{blue}{\frac{0.3333333333333333}{x \cdot \left(x \cdot n\right)}}}{x} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 18: 54.5% accurate, 3.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{0.3333333333333333}{x \cdot \left(x \cdot \left(x \cdot n\right)\right)}\\ \mathbf{if}\;\frac{1}{n} \leq -100:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;\frac{1}{n} \leq 10^{+93}:\\ \;\;\;\;\frac{1}{x \cdot \mathsf{fma}\left(0.5, \frac{n}{x}, n\right)}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (let* ((t_0 (/ 0.3333333333333333 (* x (* x (* x n))))))
   (if (<= (/ 1.0 n) -100.0)
     t_0
     (if (<= (/ 1.0 n) 1e+93) (/ 1.0 (* x (fma 0.5 (/ n x) n))) t_0))))

C

double code(double x, double n) {
	double t_0 = 0.3333333333333333 / (x * (x * (x * n)));
	double tmp;
	if ((1.0 / n) <= -100.0) {
		tmp = t_0;
	} else if ((1.0 / n) <= 1e+93) {
		tmp = 1.0 / (x * fma(0.5, (n / x), n));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(x, n)
	t_0 = Float64(0.3333333333333333 / Float64(x * Float64(x * Float64(x * n))))
	tmp = 0.0
	if (Float64(1.0 / n) <= -100.0)
		tmp = t_0;
	elseif (Float64(1.0 / n) <= 1e+93)
		tmp = Float64(1.0 / Float64(x * fma(0.5, Float64(n / x), n)));
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[x_, n_] := Block[{t$95$0 = N[(0.3333333333333333 / N[(x * N[(x * N[(x * n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(1.0 / n), $MachinePrecision], -100.0], t$95$0, If[LessEqual[N[(1.0 / n), $MachinePrecision], 1e+93], N[(1.0 / N[(x * N[(0.5 * N[(n / x), $MachinePrecision] + n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 #s(literal 1 binary64) n) < -100 or 1.00000000000000004e93 < (/.f64 #s(literal 1 binary64) n)

   1. Initial program 78.3%

      \[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)} \]
   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 33.7

         \[\leadsto \frac{\mathsf{log1p}\left(x\right) - \color{blue}{\log x}}{n} \]

   5. Applied rewrites 33.7%

      \[\leadsto \color{blue}{\frac{\mathsf{log1p}\left(x\right) - \log x}{n}} \]

   6. Taylor expanded in x around inf

      \[\leadsto \color{blue}{\frac{\left(\frac{1}{n} + \frac{1}{3} \cdot \frac{1}{n \cdot {x}^{2}}\right) - \left(\frac{\frac{1}{2}}{n \cdot x} + \frac{1}{4} \cdot \frac{1}{n \cdot {x}^{3}}\right)}{x}} \]
   7. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\left(\frac{1}{n} + \frac{1}{3} \cdot \frac{1}{n \cdot {x}^{2}}\right) - \left(\frac{\frac{1}{2}}{n \cdot x} + \frac{1}{4} \cdot \frac{1}{n \cdot {x}^{3}}\right)}{x}} \]

   8. Applied rewrites 1.1%

      \[\leadsto \color{blue}{\frac{\left(\frac{1}{n} + \frac{0.3333333333333333}{n \cdot \left(x \cdot x\right)}\right) - \left(\frac{0.5}{x \cdot n} + \frac{0.25}{n \cdot \left(x \cdot \left(x \cdot x\right)\right)}\right)}{x}} \]

   9. Taylor expanded in x around -inf

      \[\leadsto \frac{\color{blue}{-1 \cdot \frac{\frac{1}{2} \cdot \frac{1}{n} - \frac{1}{3} \cdot \frac{1}{n \cdot x}}{x} + \frac{1}{n}}}{x} \]

   11. Applied rewrites 15.9%

       \[\leadsto \frac{\color{blue}{\frac{1}{n} + \left(\frac{-0.5}{x \cdot n} - \frac{-0.3333333333333333}{x \cdot \left(x \cdot n\right)}\right)}}{x} \]

   12. Taylor expanded in x around 0

       \[\leadsto \color{blue}{\frac{\frac{1}{3}}{n \cdot {x}^{3}}} \]
   13. Step-by-step derivation

       1. lower-/.f64 N/A

          \[\leadsto \color{blue}{\frac{\frac{1}{3}}{n \cdot {x}^{3}}} \]

       2. *-commutative N/A

          \[\leadsto \frac{\frac{1}{3}}{\color{blue}{{x}^{3} \cdot n}} \]

       3. cube-mult N/A

          \[\leadsto \frac{\frac{1}{3}}{\color{blue}{\left(x \cdot \left(x \cdot x\right)\right)} \cdot n} \]

       4. unpow2 N/A

          \[\leadsto \frac{\frac{1}{3}}{\left(x \cdot \color{blue}{{x}^{2}}\right) \cdot n} \]

       5. associate-*l* N/A

          \[\leadsto \frac{\frac{1}{3}}{\color{blue}{x \cdot \left({x}^{2} \cdot n\right)}} \]

       6. *-commutative N/A

          \[\leadsto \frac{\frac{1}{3}}{x \cdot \color{blue}{\left(n \cdot {x}^{2}\right)}} \]

       7. lower-*.f64 N/A

          \[\leadsto \frac{\frac{1}{3}}{\color{blue}{x \cdot \left(n \cdot {x}^{2}\right)}} \]

       8. *-commutative N/A

          \[\leadsto \frac{\frac{1}{3}}{x \cdot \color{blue}{\left({x}^{2} \cdot n\right)}} \]

       9. unpow2 N/A

          \[\leadsto \frac{\frac{1}{3}}{x \cdot \left(\color{blue}{\left(x \cdot x\right)} \cdot n\right)} \]

       10. associate-*l* N/A

           \[\leadsto \frac{\frac{1}{3}}{x \cdot \color{blue}{\left(x \cdot \left(x \cdot n\right)\right)}} \]

       11. *-commutative N/A

           \[\leadsto \frac{\frac{1}{3}}{x \cdot \left(x \cdot \color{blue}{\left(n \cdot x\right)}\right)} \]

       12. lower-*.f64 N/A

           \[\leadsto \frac{\frac{1}{3}}{x \cdot \color{blue}{\left(x \cdot \left(n \cdot x\right)\right)}} \]

       13. *-commutative N/A

           \[\leadsto \frac{\frac{1}{3}}{x \cdot \left(x \cdot \color{blue}{\left(x \cdot n\right)}\right)} \]

       14. lower-*.f64 65.8

           \[\leadsto \frac{0.3333333333333333}{x \cdot \left(x \cdot \color{blue}{\left(x \cdot n\right)}\right)} \]

   14. Applied rewrites 65.8%

       \[\leadsto \color{blue}{\frac{0.3333333333333333}{x \cdot \left(x \cdot \left(x \cdot n\right)\right)}} \]

   if -100 < (/.f64 #s(literal 1 binary64) n) < 1.00000000000000004e93

   1. Initial program 40.2%

      \[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in 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 64.6

         \[\leadsto \frac{\mathsf{log1p}\left(x\right) - \color{blue}{\log x}}{n} \]

   5. Applied rewrites 64.6%

      \[\leadsto \color{blue}{\frac{\mathsf{log1p}\left(x\right) - \log x}{n}} \]
   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. clear-num N/A

         \[\leadsto \color{blue}{\frac{1}{\frac{n}{\mathsf{log1p}\left(x\right) - \log x}}} \]

      5. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{1}{\frac{n}{\mathsf{log1p}\left(x\right) - \log x}}} \]

      6. lower-/.f64 64.5

         \[\leadsto \frac{1}{\color{blue}{\frac{n}{\mathsf{log1p}\left(x\right) - \log x}}} \]

      7. lift--.f64 N/A

         \[\leadsto \frac{1}{\frac{n}{\color{blue}{\mathsf{log1p}\left(x\right) - \log x}}} \]

      8. lift-log1p.f64 N/A

         \[\leadsto \frac{1}{\frac{n}{\color{blue}{\log \left(1 + x\right)} - \log x}} \]

      9. lift-log.f64 N/A

         \[\leadsto \frac{1}{\frac{n}{\log \left(1 + x\right) - \color{blue}{\log x}}} \]

      10. diff-log N/A

          \[\leadsto \frac{1}{\frac{n}{\color{blue}{\log \left(\frac{1 + x}{x}\right)}}} \]

      11. lower-log.f64 N/A

          \[\leadsto \frac{1}{\frac{n}{\color{blue}{\log \left(\frac{1 + x}{x}\right)}}} \]

      12. +-commutative N/A

          \[\leadsto \frac{1}{\frac{n}{\log \left(\frac{\color{blue}{x + 1}}{x}\right)}} \]

      13. lift-+.f64 N/A

          \[\leadsto \frac{1}{\frac{n}{\log \left(\frac{\color{blue}{x + 1}}{x}\right)}} \]

      14. lower-/.f64 64.7

          \[\leadsto \frac{1}{\frac{n}{\log \color{blue}{\left(\frac{x + 1}{x}\right)}}} \]

   7. Applied rewrites 64.7%

      \[\leadsto \color{blue}{\frac{1}{\frac{n}{\log \left(\frac{x + 1}{x}\right)}}} \]

   8. Taylor expanded in x around inf

      \[\leadsto \frac{1}{\color{blue}{x \cdot \left(n + \frac{1}{2} \cdot \frac{n}{x}\right)}} \]
   9. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \frac{1}{\color{blue}{x \cdot \left(n + \frac{1}{2} \cdot \frac{n}{x}\right)}} \]

      2. +-commutative N/A

         \[\leadsto \frac{1}{x \cdot \color{blue}{\left(\frac{1}{2} \cdot \frac{n}{x} + n\right)}} \]

      3. lower-fma.f64 N/A

         \[\leadsto \frac{1}{x \cdot \color{blue}{\mathsf{fma}\left(\frac{1}{2}, \frac{n}{x}, n\right)}} \]

      4. lower-/.f64 58.3

         \[\leadsto \frac{1}{x \cdot \mathsf{fma}\left(0.5, \color{blue}{\frac{n}{x}}, n\right)} \]

   10. Applied rewrites 58.3%

       \[\leadsto \frac{1}{\color{blue}{x \cdot \mathsf{fma}\left(0.5, \frac{n}{x}, n\right)}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 19: 53.0% accurate, 3.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{1}{n} \leq -2 \cdot 10^{-5}:\\ \;\;\;\;\frac{0.3333333333333333}{x \cdot \left(x \cdot \left(x \cdot n\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{-0.5 + \frac{0.3333333333333333}{x}}{x} + 1}{n}}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (if (<= (/ 1.0 n) -2e-5)
   (/ 0.3333333333333333 (* x (* x (* x n))))
   (/ (/ (+ (/ (+ -0.5 (/ 0.3333333333333333 x)) x) 1.0) n) x)))

C

double code(double x, double n) {
	double tmp;
	if ((1.0 / n) <= -2e-5) {
		tmp = 0.3333333333333333 / (x * (x * (x * n)));
	} else {
		tmp = ((((-0.5 + (0.3333333333333333 / x)) / x) + 1.0) / n) / x;
	}
	return tmp;
}

Fortran

real(8) function code(x, n)
    real(8), intent (in) :: x
    real(8), intent (in) :: n
    real(8) :: tmp
    if ((1.0d0 / n) <= (-2d-5)) then
        tmp = 0.3333333333333333d0 / (x * (x * (x * n)))
    else
        tmp = (((((-0.5d0) + (0.3333333333333333d0 / x)) / x) + 1.0d0) / n) / x
    end if
    code = tmp
end function

Java

public static double code(double x, double n) {
	double tmp;
	if ((1.0 / n) <= -2e-5) {
		tmp = 0.3333333333333333 / (x * (x * (x * n)));
	} else {
		tmp = ((((-0.5 + (0.3333333333333333 / x)) / x) + 1.0) / n) / x;
	}
	return tmp;
}

Python

def code(x, n):
	tmp = 0
	if (1.0 / n) <= -2e-5:
		tmp = 0.3333333333333333 / (x * (x * (x * n)))
	else:
		tmp = ((((-0.5 + (0.3333333333333333 / x)) / x) + 1.0) / n) / x
	return tmp

Julia

function code(x, n)
	tmp = 0.0
	if (Float64(1.0 / n) <= -2e-5)
		tmp = Float64(0.3333333333333333 / Float64(x * Float64(x * Float64(x * n))));
	else
		tmp = Float64(Float64(Float64(Float64(Float64(-0.5 + Float64(0.3333333333333333 / x)) / x) + 1.0) / n) / x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, n)
	tmp = 0.0;
	if ((1.0 / n) <= -2e-5)
		tmp = 0.3333333333333333 / (x * (x * (x * n)));
	else
		tmp = ((((-0.5 + (0.3333333333333333 / x)) / x) + 1.0) / n) / x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, n_] := If[LessEqual[N[(1.0 / n), $MachinePrecision], -2e-5], N[(0.3333333333333333 / N[(x * N[(x * N[(x * n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(N[(-0.5 + N[(0.3333333333333333 / x), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision] + 1.0), $MachinePrecision] / n), $MachinePrecision] / x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 #s(literal 1 binary64) n) < -2.00000000000000016e-5

   1. Initial program 99.9%

      \[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)} \]
   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 45.8

         \[\leadsto \frac{\mathsf{log1p}\left(x\right) - \color{blue}{\log x}}{n} \]

   5. Applied rewrites 45.8%

      \[\leadsto \color{blue}{\frac{\mathsf{log1p}\left(x\right) - \log x}{n}} \]

   6. Taylor expanded in x around inf

      \[\leadsto \color{blue}{\frac{\left(\frac{1}{n} + \frac{1}{3} \cdot \frac{1}{n \cdot {x}^{2}}\right) - \left(\frac{\frac{1}{2}}{n \cdot x} + \frac{1}{4} \cdot \frac{1}{n \cdot {x}^{3}}\right)}{x}} \]
   7. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\left(\frac{1}{n} + \frac{1}{3} \cdot \frac{1}{n \cdot {x}^{2}}\right) - \left(\frac{\frac{1}{2}}{n \cdot x} + \frac{1}{4} \cdot \frac{1}{n \cdot {x}^{3}}\right)}{x}} \]

   8. Applied rewrites 1.6%

      \[\leadsto \color{blue}{\frac{\left(\frac{1}{n} + \frac{0.3333333333333333}{n \cdot \left(x \cdot x\right)}\right) - \left(\frac{0.5}{x \cdot n} + \frac{0.25}{n \cdot \left(x \cdot \left(x \cdot x\right)\right)}\right)}{x}} \]

   9. Taylor expanded in x around -inf

      \[\leadsto \frac{\color{blue}{-1 \cdot \frac{\frac{1}{2} \cdot \frac{1}{n} - \frac{1}{3} \cdot \frac{1}{n \cdot x}}{x} + \frac{1}{n}}}{x} \]

   11. Applied rewrites 16.2%

       \[\leadsto \frac{\color{blue}{\frac{1}{n} + \left(\frac{-0.5}{x \cdot n} - \frac{-0.3333333333333333}{x \cdot \left(x \cdot n\right)}\right)}}{x} \]

   12. Taylor expanded in x around 0

       \[\leadsto \color{blue}{\frac{\frac{1}{3}}{n \cdot {x}^{3}}} \]
   13. Step-by-step derivation

       1. lower-/.f64 N/A

          \[\leadsto \color{blue}{\frac{\frac{1}{3}}{n \cdot {x}^{3}}} \]

       2. *-commutative N/A

          \[\leadsto \frac{\frac{1}{3}}{\color{blue}{{x}^{3} \cdot n}} \]

       3. cube-mult N/A

          \[\leadsto \frac{\frac{1}{3}}{\color{blue}{\left(x \cdot \left(x \cdot x\right)\right)} \cdot n} \]

       4. unpow2 N/A

          \[\leadsto \frac{\frac{1}{3}}{\left(x \cdot \color{blue}{{x}^{2}}\right) \cdot n} \]

       5. associate-*l* N/A

          \[\leadsto \frac{\frac{1}{3}}{\color{blue}{x \cdot \left({x}^{2} \cdot n\right)}} \]

       6. *-commutative N/A

          \[\leadsto \frac{\frac{1}{3}}{x \cdot \color{blue}{\left(n \cdot {x}^{2}\right)}} \]

       7. lower-*.f64 N/A

          \[\leadsto \frac{\frac{1}{3}}{\color{blue}{x \cdot \left(n \cdot {x}^{2}\right)}} \]

       8. *-commutative N/A

          \[\leadsto \frac{\frac{1}{3}}{x \cdot \color{blue}{\left({x}^{2} \cdot n\right)}} \]

       9. unpow2 N/A

          \[\leadsto \frac{\frac{1}{3}}{x \cdot \left(\color{blue}{\left(x \cdot x\right)} \cdot n\right)} \]

       10. associate-*l* N/A

           \[\leadsto \frac{\frac{1}{3}}{x \cdot \color{blue}{\left(x \cdot \left(x \cdot n\right)\right)}} \]

       11. *-commutative N/A

           \[\leadsto \frac{\frac{1}{3}}{x \cdot \left(x \cdot \color{blue}{\left(n \cdot x\right)}\right)} \]

       12. lower-*.f64 N/A

           \[\leadsto \frac{\frac{1}{3}}{x \cdot \color{blue}{\left(x \cdot \left(n \cdot x\right)\right)}} \]

       13. *-commutative N/A

           \[\leadsto \frac{\frac{1}{3}}{x \cdot \left(x \cdot \color{blue}{\left(x \cdot n\right)}\right)} \]

       14. lower-*.f64 67.1

           \[\leadsto \frac{0.3333333333333333}{x \cdot \left(x \cdot \color{blue}{\left(x \cdot n\right)}\right)} \]

   14. Applied rewrites 67.1%

       \[\leadsto \color{blue}{\frac{0.3333333333333333}{x \cdot \left(x \cdot \left(x \cdot n\right)\right)}} \]

   if -2.00000000000000016e-5 < (/.f64 #s(literal 1 binary64) n)

   1. Initial program 38.3%

      \[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)} \]
   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 55.8

         \[\leadsto \frac{\mathsf{log1p}\left(x\right) - \color{blue}{\log x}}{n} \]

   5. Applied rewrites 55.8%

      \[\leadsto \color{blue}{\frac{\mathsf{log1p}\left(x\right) - \log x}{n}} \]

   6. Taylor expanded in x around inf

      \[\leadsto \color{blue}{\frac{\left(\frac{1}{n} + \frac{1}{3} \cdot \frac{1}{n \cdot {x}^{2}}\right) - \left(\frac{\frac{1}{2}}{n \cdot x} + \frac{1}{4} \cdot \frac{1}{n \cdot {x}^{3}}\right)}{x}} \]
   7. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\left(\frac{1}{n} + \frac{1}{3} \cdot \frac{1}{n \cdot {x}^{2}}\right) - \left(\frac{\frac{1}{2}}{n \cdot x} + \frac{1}{4} \cdot \frac{1}{n \cdot {x}^{3}}\right)}{x}} \]

   8. Applied rewrites 46.8%

      \[\leadsto \color{blue}{\frac{\left(\frac{1}{n} + \frac{0.3333333333333333}{n \cdot \left(x \cdot x\right)}\right) - \left(\frac{0.5}{x \cdot n} + \frac{0.25}{n \cdot \left(x \cdot \left(x \cdot x\right)\right)}\right)}{x}} \]

   9. Taylor expanded in x around -inf

      \[\leadsto \frac{\color{blue}{-1 \cdot \frac{\frac{1}{2} \cdot \frac{1}{n} - \frac{1}{3} \cdot \frac{1}{n \cdot x}}{x} + \frac{1}{n}}}{x} \]

   11. Applied rewrites 50.1%

       \[\leadsto \frac{\color{blue}{\frac{1}{n} + \left(\frac{-0.5}{x \cdot n} - \frac{-0.3333333333333333}{x \cdot \left(x \cdot n\right)}\right)}}{x} \]

   12. Taylor expanded in n around 0

       \[\leadsto \frac{\color{blue}{\frac{\left(1 + \frac{1}{3} \cdot \frac{1}{{x}^{2}}\right) - \frac{1}{2} \cdot \frac{1}{x}}{n}}}{x} \]
   13. Step-by-step derivation

       1. lower-/.f64 N/A

          \[\leadsto \frac{\color{blue}{\frac{\left(1 + \frac{1}{3} \cdot \frac{1}{{x}^{2}}\right) - \frac{1}{2} \cdot \frac{1}{x}}{n}}}{x} \]

   14. Applied rewrites 57.1%

       \[\leadsto \frac{\color{blue}{\frac{1 + \frac{-0.5 + \frac{0.3333333333333333}{x}}{x}}{n}}}{x} \]
3. Recombined 2 regimes into one program.

4. Final simplification 59.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{1}{n} \leq -2 \cdot 10^{-5}:\\ \;\;\;\;\frac{0.3333333333333333}{x \cdot \left(x \cdot \left(x \cdot n\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{-0.5 + \frac{0.3333333333333333}{x}}{x} + 1}{n}}{x}\\ \end{array} \]
5. Add Preprocessing

Alternative 20: 51.7% accurate, 5.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{1}{n} \leq -100:\\ \;\;\;\;\frac{0.3333333333333333}{x \cdot \left(x \cdot \left(x \cdot n\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{n}}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (if (<= (/ 1.0 n) -100.0)
   (/ 0.3333333333333333 (* x (* x (* x n))))
   (/ (/ 1.0 n) x)))

C

double code(double x, double n) {
	double tmp;
	if ((1.0 / n) <= -100.0) {
		tmp = 0.3333333333333333 / (x * (x * (x * n)));
	} else {
		tmp = (1.0 / n) / x;
	}
	return tmp;
}

Fortran

real(8) function code(x, n)
    real(8), intent (in) :: x
    real(8), intent (in) :: n
    real(8) :: tmp
    if ((1.0d0 / n) <= (-100.0d0)) then
        tmp = 0.3333333333333333d0 / (x * (x * (x * n)))
    else
        tmp = (1.0d0 / n) / x
    end if
    code = tmp
end function

Java

public static double code(double x, double n) {
	double tmp;
	if ((1.0 / n) <= -100.0) {
		tmp = 0.3333333333333333 / (x * (x * (x * n)));
	} else {
		tmp = (1.0 / n) / x;
	}
	return tmp;
}

Python

def code(x, n):
	tmp = 0
	if (1.0 / n) <= -100.0:
		tmp = 0.3333333333333333 / (x * (x * (x * n)))
	else:
		tmp = (1.0 / n) / x
	return tmp

Julia

function code(x, n)
	tmp = 0.0
	if (Float64(1.0 / n) <= -100.0)
		tmp = Float64(0.3333333333333333 / Float64(x * Float64(x * Float64(x * n))));
	else
		tmp = Float64(Float64(1.0 / n) / x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, n)
	tmp = 0.0;
	if ((1.0 / n) <= -100.0)
		tmp = 0.3333333333333333 / (x * (x * (x * n)));
	else
		tmp = (1.0 / n) / x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, n_] := If[LessEqual[N[(1.0 / n), $MachinePrecision], -100.0], N[(0.3333333333333333 / N[(x * N[(x * N[(x * n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 / n), $MachinePrecision] / x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 #s(literal 1 binary64) n) < -100

   1. Initial program 100.0%

      \[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in n around inf

      \[\leadsto \color{blue}{\frac{\log \left(1 + x\right) - \log x}{n}} \]
   4. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\log \left(1 + x\right) - \log x}{n}} \]

      2. lower--.f64 N/A

         \[\leadsto \frac{\color{blue}{\log \left(1 + x\right) - \log x}}{n} \]

      3. lower-log1p.f64 N/A

         \[\leadsto \frac{\color{blue}{\mathsf{log1p}\left(x\right)} - \log x}{n} \]

      4. lower-log.f64 46.1

         \[\leadsto \frac{\mathsf{log1p}\left(x\right) - \color{blue}{\log x}}{n} \]

   5. Applied rewrites 46.1%

      \[\leadsto \color{blue}{\frac{\mathsf{log1p}\left(x\right) - \log x}{n}} \]

   6. Taylor expanded in x around inf

      \[\leadsto \color{blue}{\frac{\left(\frac{1}{n} + \frac{1}{3} \cdot \frac{1}{n \cdot {x}^{2}}\right) - \left(\frac{\frac{1}{2}}{n \cdot x} + \frac{1}{4} \cdot \frac{1}{n \cdot {x}^{3}}\right)}{x}} \]
   7. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\left(\frac{1}{n} + \frac{1}{3} \cdot \frac{1}{n \cdot {x}^{2}}\right) - \left(\frac{\frac{1}{2}}{n \cdot x} + \frac{1}{4} \cdot \frac{1}{n \cdot {x}^{3}}\right)}{x}} \]

   8. Applied rewrites 1.6%

      \[\leadsto \color{blue}{\frac{\left(\frac{1}{n} + \frac{0.3333333333333333}{n \cdot \left(x \cdot x\right)}\right) - \left(\frac{0.5}{x \cdot n} + \frac{0.25}{n \cdot \left(x \cdot \left(x \cdot x\right)\right)}\right)}{x}} \]

   9. Taylor expanded in x around -inf

      \[\leadsto \frac{\color{blue}{-1 \cdot \frac{\frac{1}{2} \cdot \frac{1}{n} - \frac{1}{3} \cdot \frac{1}{n \cdot x}}{x} + \frac{1}{n}}}{x} \]

   11. Applied rewrites 16.4%

       \[\leadsto \frac{\color{blue}{\frac{1}{n} + \left(\frac{-0.5}{x \cdot n} - \frac{-0.3333333333333333}{x \cdot \left(x \cdot n\right)}\right)}}{x} \]

   12. Taylor expanded in x around 0

       \[\leadsto \color{blue}{\frac{\frac{1}{3}}{n \cdot {x}^{3}}} \]
   13. Step-by-step derivation

       1. lower-/.f64 N/A

          \[\leadsto \color{blue}{\frac{\frac{1}{3}}{n \cdot {x}^{3}}} \]

       2. *-commutative N/A

          \[\leadsto \frac{\frac{1}{3}}{\color{blue}{{x}^{3} \cdot n}} \]

       3. cube-mult N/A

          \[\leadsto \frac{\frac{1}{3}}{\color{blue}{\left(x \cdot \left(x \cdot x\right)\right)} \cdot n} \]

       4. unpow2 N/A

          \[\leadsto \frac{\frac{1}{3}}{\left(x \cdot \color{blue}{{x}^{2}}\right) \cdot n} \]

       5. associate-*l* N/A

          \[\leadsto \frac{\frac{1}{3}}{\color{blue}{x \cdot \left({x}^{2} \cdot n\right)}} \]

       6. *-commutative N/A

          \[\leadsto \frac{\frac{1}{3}}{x \cdot \color{blue}{\left(n \cdot {x}^{2}\right)}} \]

       7. lower-*.f64 N/A

          \[\leadsto \frac{\frac{1}{3}}{\color{blue}{x \cdot \left(n \cdot {x}^{2}\right)}} \]

       8. *-commutative N/A

          \[\leadsto \frac{\frac{1}{3}}{x \cdot \color{blue}{\left({x}^{2} \cdot n\right)}} \]

       9. unpow2 N/A

          \[\leadsto \frac{\frac{1}{3}}{x \cdot \left(\color{blue}{\left(x \cdot x\right)} \cdot n\right)} \]

       10. associate-*l* N/A

           \[\leadsto \frac{\frac{1}{3}}{x \cdot \color{blue}{\left(x \cdot \left(x \cdot n\right)\right)}} \]

       11. *-commutative N/A

           \[\leadsto \frac{\frac{1}{3}}{x \cdot \left(x \cdot \color{blue}{\left(n \cdot x\right)}\right)} \]

       12. lower-*.f64 N/A

           \[\leadsto \frac{\frac{1}{3}}{x \cdot \color{blue}{\left(x \cdot \left(n \cdot x\right)\right)}} \]

       13. *-commutative N/A

           \[\leadsto \frac{\frac{1}{3}}{x \cdot \left(x \cdot \color{blue}{\left(x \cdot n\right)}\right)} \]

       14. lower-*.f64 68.1

           \[\leadsto \frac{0.3333333333333333}{x \cdot \left(x \cdot \color{blue}{\left(x \cdot n\right)}\right)} \]

   14. Applied rewrites 68.1%

       \[\leadsto \color{blue}{\frac{0.3333333333333333}{x \cdot \left(x \cdot \left(x \cdot n\right)\right)}} \]

   if -100 < (/.f64 #s(literal 1 binary64) n)

   1. Initial program 38.6%

      \[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)} \]
   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 55.6

         \[\leadsto \frac{\mathsf{log1p}\left(x\right) - \color{blue}{\log x}}{n} \]

   5. Applied rewrites 55.6%

      \[\leadsto \color{blue}{\frac{\mathsf{log1p}\left(x\right) - \log x}{n}} \]

   6. Taylor expanded in x around inf

      \[\leadsto \color{blue}{\frac{\left(\frac{1}{n} + \frac{1}{3} \cdot \frac{1}{n \cdot {x}^{2}}\right) - \left(\frac{\frac{1}{2}}{n \cdot x} + \frac{1}{4} \cdot \frac{1}{n \cdot {x}^{3}}\right)}{x}} \]
   7. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\left(\frac{1}{n} + \frac{1}{3} \cdot \frac{1}{n \cdot {x}^{2}}\right) - \left(\frac{\frac{1}{2}}{n \cdot x} + \frac{1}{4} \cdot \frac{1}{n \cdot {x}^{3}}\right)}{x}} \]

   8. Applied rewrites 46.6%

      \[\leadsto \color{blue}{\frac{\left(\frac{1}{n} + \frac{0.3333333333333333}{n \cdot \left(x \cdot x\right)}\right) - \left(\frac{0.5}{x \cdot n} + \frac{0.25}{n \cdot \left(x \cdot \left(x \cdot x\right)\right)}\right)}{x}} \]

   9. Taylor expanded in x around inf

      \[\leadsto \frac{\color{blue}{\frac{1}{n}}}{x} \]
   10. Step-by-step derivation

       1. lower-/.f64 55.4

          \[\leadsto \frac{\color{blue}{\frac{1}{n}}}{x} \]

   11. Applied rewrites 55.4%

       \[\leadsto \frac{\color{blue}{\frac{1}{n}}}{x} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 21: 44.6% accurate, 8.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 1.45 \cdot 10^{+129}:\\ \;\;\;\;\frac{\frac{1}{n}}{x}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \end{array} \]

FPCore

(FPCore (x n) :precision binary64 (if (<= x 1.45e+129) (/ (/ 1.0 n) x) 0.0))

C

double code(double x, double n) {
	double tmp;
	if (x <= 1.45e+129) {
		tmp = (1.0 / n) / x;
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Fortran

real(8) function code(x, n)
    real(8), intent (in) :: x
    real(8), intent (in) :: n
    real(8) :: tmp
    if (x <= 1.45d+129) then
        tmp = (1.0d0 / n) / x
    else
        tmp = 0.0d0
    end if
    code = tmp
end function

Java

public static double code(double x, double n) {
	double tmp;
	if (x <= 1.45e+129) {
		tmp = (1.0 / n) / x;
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Python

def code(x, n):
	tmp = 0
	if x <= 1.45e+129:
		tmp = (1.0 / n) / x
	else:
		tmp = 0.0
	return tmp

Julia

function code(x, n)
	tmp = 0.0
	if (x <= 1.45e+129)
		tmp = Float64(Float64(1.0 / n) / x);
	else
		tmp = 0.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, n)
	tmp = 0.0;
	if (x <= 1.45e+129)
		tmp = (1.0 / n) / x;
	else
		tmp = 0.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, n_] := If[LessEqual[x, 1.45e+129], N[(N[(1.0 / n), $MachinePrecision] / x), $MachinePrecision], 0.0]

Derivation

1. Split input into 2 regimes

2. if x < 1.45000000000000001e129

   1. Initial program 42.7%

      \[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)} \]
   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 41.4

         \[\leadsto \frac{\mathsf{log1p}\left(x\right) - \color{blue}{\log x}}{n} \]

   5. Applied rewrites 41.4%

      \[\leadsto \color{blue}{\frac{\mathsf{log1p}\left(x\right) - \log x}{n}} \]

   6. Taylor expanded in x around inf

      \[\leadsto \color{blue}{\frac{\left(\frac{1}{n} + \frac{1}{3} \cdot \frac{1}{n \cdot {x}^{2}}\right) - \left(\frac{\frac{1}{2}}{n \cdot x} + \frac{1}{4} \cdot \frac{1}{n \cdot {x}^{3}}\right)}{x}} \]
   7. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\left(\frac{1}{n} + \frac{1}{3} \cdot \frac{1}{n \cdot {x}^{2}}\right) - \left(\frac{\frac{1}{2}}{n \cdot x} + \frac{1}{4} \cdot \frac{1}{n \cdot {x}^{3}}\right)}{x}} \]

   8. Applied rewrites 22.2%

      \[\leadsto \color{blue}{\frac{\left(\frac{1}{n} + \frac{0.3333333333333333}{n \cdot \left(x \cdot x\right)}\right) - \left(\frac{0.5}{x \cdot n} + \frac{0.25}{n \cdot \left(x \cdot \left(x \cdot x\right)\right)}\right)}{x}} \]

   9. Taylor expanded in x around inf

      \[\leadsto \frac{\color{blue}{\frac{1}{n}}}{x} \]
   10. Step-by-step derivation

       1. lower-/.f64 41.8

          \[\leadsto \frac{\color{blue}{\frac{1}{n}}}{x} \]

   11. Applied rewrites 41.8%

       \[\leadsto \frac{\color{blue}{\frac{1}{n}}}{x} \]

   if 1.45000000000000001e129 < x

   1. Initial program 86.5%

      \[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{1 - e^{\frac{\log x}{n}}} \]
   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 58.9

          \[\leadsto 1 - {x}^{\color{blue}{\left(\frac{1}{n}\right)}} \]

   5. Applied rewrites 58.9%

      \[\leadsto \color{blue}{1 - {x}^{\left(\frac{1}{n}\right)}} \]

   6. Taylor expanded in n around inf

      \[\leadsto 1 - \color{blue}{1} \]
   7. Step-by-step derivation

   8. Applied rewrites 86.5%

      \[\leadsto 1 - \color{blue}{1} \]
   9. Step-by-step derivation

      1. metadata-eval 86.5

         \[\leadsto \color{blue}{0} \]

   10. Applied rewrites 86.5%

       \[\leadsto \color{blue}{0} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 22: 44.3% accurate, 10.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 1.1 \cdot 10^{+129}:\\ \;\;\;\;\frac{1}{x \cdot n}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \end{array} \]

FPCore

(FPCore (x n) :precision binary64 (if (<= x 1.1e+129) (/ 1.0 (* x n)) 0.0))

C

double code(double x, double n) {
	double tmp;
	if (x <= 1.1e+129) {
		tmp = 1.0 / (x * n);
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Fortran

real(8) function code(x, n)
    real(8), intent (in) :: x
    real(8), intent (in) :: n
    real(8) :: tmp
    if (x <= 1.1d+129) then
        tmp = 1.0d0 / (x * n)
    else
        tmp = 0.0d0
    end if
    code = tmp
end function

Java

public static double code(double x, double n) {
	double tmp;
	if (x <= 1.1e+129) {
		tmp = 1.0 / (x * n);
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Python

def code(x, n):
	tmp = 0
	if x <= 1.1e+129:
		tmp = 1.0 / (x * n)
	else:
		tmp = 0.0
	return tmp

Julia

function code(x, n)
	tmp = 0.0
	if (x <= 1.1e+129)
		tmp = Float64(1.0 / Float64(x * n));
	else
		tmp = 0.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, n)
	tmp = 0.0;
	if (x <= 1.1e+129)
		tmp = 1.0 / (x * n);
	else
		tmp = 0.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, n_] := If[LessEqual[x, 1.1e+129], N[(1.0 / N[(x * n), $MachinePrecision]), $MachinePrecision], 0.0]

Derivation

1. Split input into 2 regimes

2. if x < 1.1e129

   1. Initial program 42.7%

      \[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)} \]
   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 41.4

         \[\leadsto \frac{\mathsf{log1p}\left(x\right) - \color{blue}{\log x}}{n} \]

   5. Applied rewrites 41.4%

      \[\leadsto \color{blue}{\frac{\mathsf{log1p}\left(x\right) - \log x}{n}} \]

   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 41.8

         \[\leadsto \frac{1}{\color{blue}{x \cdot n}} \]

   8. Applied rewrites 41.8%

      \[\leadsto \color{blue}{\frac{1}{x \cdot n}} \]

   if 1.1e129 < x

   1. Initial program 86.5%

      \[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{1 - e^{\frac{\log x}{n}}} \]
   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 58.9

          \[\leadsto 1 - {x}^{\color{blue}{\left(\frac{1}{n}\right)}} \]

   5. Applied rewrites 58.9%

      \[\leadsto \color{blue}{1 - {x}^{\left(\frac{1}{n}\right)}} \]

   6. Taylor expanded in n around inf

      \[\leadsto 1 - \color{blue}{1} \]
   7. Step-by-step derivation

   8. Applied rewrites 86.5%

      \[\leadsto 1 - \color{blue}{1} \]
   9. Step-by-step derivation

      1. metadata-eval 86.5

         \[\leadsto \color{blue}{0} \]

   10. Applied rewrites 86.5%

       \[\leadsto \color{blue}{0} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 23: 31.9% 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 54.2%

   \[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)} \]
2. Add Preprocessing

3. Taylor expanded in x around 0

   \[\leadsto \color{blue}{1 - e^{\frac{\log x}{n}}} \]
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 42.8

       \[\leadsto 1 - {x}^{\color{blue}{\left(\frac{1}{n}\right)}} \]

5. Applied rewrites 42.8%

   \[\leadsto \color{blue}{1 - {x}^{\left(\frac{1}{n}\right)}} \]

6. Taylor expanded in n around inf

   \[\leadsto 1 - \color{blue}{1} \]
7. Step-by-step derivation

8. Applied rewrites 31.0%

   \[\leadsto 1 - \color{blue}{1} \]
9. Step-by-step derivation

   1. metadata-eval 31.0

      \[\leadsto \color{blue}{0} \]

10. Applied rewrites 31.0%

    \[\leadsto \color{blue}{0} \]
11. Add Preprocessing

Reproduce

herbie shell --seed 2024219 
(FPCore (x n)
  :name "2nthrt (problem 3.4.6)"
  :precision binary64
  (- (pow (+ x 1.0) (/ 1.0 n)) (pow x (/ 1.0 n))))