2nthrt (problem 3.4.6)

Percentage Accurate: 54.2% → 86.0%

Time: 26.5s

Alternatives: 19

Speedup: 1.7×

• 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: 86.0% accurate, 0.2× speedup

Math

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

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)
   (/ (pow x (/ 1.0 n)) (* x n))))

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 = pow(x, (1.0 / n)) / (x * n);
	}
	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((x ^ Float64(1.0 / n)) / Float64(x * n));
	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[Power[x, N[(1.0 / n), $MachinePrecision]], $MachinePrecision] / N[(x * n), $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%

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

4. Final simplification 87.0%

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

Alternative 2: 77.2% accurate, 0.4× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

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

   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

   7. Applied rewrites 75.0%

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

   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

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

   8. Applied rewrites 0.1%

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

   9. Taylor expanded in n around 0

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

   11. Applied rewrites 0.1%

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

   12. Taylor expanded in x around inf

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

   14. Applied rewrites 42.9%

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

4. Final simplification 73.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)} \leq -2 \cdot 10^{-5}:\\ \;\;\;\;1 - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{elif}\;{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)} \leq 0.9101169306103679:\\ \;\;\;\;\frac{\log \left(\frac{x}{x + 1}\right)}{-n}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{-0.5 + \frac{0.3333333333333333}{x}}{x} + 1}{x \cdot n}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 77.1% accurate, 0.4× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

public static double code(double x, double n) {
	double t_0 = Math.pow(x, (1.0 / n));
	double t_1 = Math.pow((x + 1.0), (1.0 / n)) - t_0;
	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 = (((-0.5 + (0.3333333333333333 / x)) / x) + 1.0) / (x * n);
	}
	return tmp;
}

Python

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

Julia

function code(x, n)
	t_0 = x ^ Float64(1.0 / n)
	t_1 = Float64((Float64(x + 1.0) ^ Float64(1.0 / n)) - t_0)
	tmp = 0.0
	if (t_1 <= -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(Float64(-0.5 + Float64(0.3333333333333333 / x)) / x) + 1.0) / Float64(x * n));
	end
	return tmp
end

MATLAB

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

Wolfram

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

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

   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

   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

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

   8. Applied rewrites 0.1%

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

   9. Taylor expanded in n around 0

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

   11. Applied rewrites 0.1%

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

   12. Taylor expanded in x around inf

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

   14. Applied rewrites 42.9%

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

4. Final simplification 73.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)} \leq -2 \cdot 10^{-5}:\\ \;\;\;\;1 - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{elif}\;{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)} \leq 0.9101169306103679:\\ \;\;\;\;\frac{\log \left(\frac{x + 1}{x}\right)}{n}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{-0.5 + \frac{0.3333333333333333}{x}}{x} + 1}{x \cdot n}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 85.6% accurate, 0.4× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if x < 1

   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 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 77.1%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 76.5%

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

   if 1 < 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%

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

4. Final simplification 86.3%

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

Alternative 5: 85.3% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 0.97:\\ \;\;\;\;\frac{\mathsf{fma}\left(0.5, \frac{{\log x}^{2}}{n}, \log x\right) + \frac{0.16666666666666666 \cdot {\log x}^{3}}{n \cdot n}}{-n}\\ \mathbf{else}:\\ \;\;\;\;\frac{{x}^{\left(\frac{1}{n}\right)}}{x \cdot n}\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (if (<= x 0.97)
   (/
    (+
     (fma 0.5 (/ (pow (log x) 2.0) n) (log x))
     (/ (* 0.16666666666666666 (pow (log x) 3.0)) (* n n)))
    (- n))
   (/ (pow x (/ 1.0 n)) (* x n))))

C

double code(double x, double n) {
	double tmp;
	if (x <= 0.97) {
		tmp = (fma(0.5, (pow(log(x), 2.0) / n), log(x)) + ((0.16666666666666666 * pow(log(x), 3.0)) / (n * n))) / -n;
	} else {
		tmp = pow(x, (1.0 / n)) / (x * n);
	}
	return tmp;
}

Julia

function code(x, n)
	tmp = 0.0
	if (x <= 0.97)
		tmp = Float64(Float64(fma(0.5, Float64((log(x) ^ 2.0) / n), log(x)) + Float64(Float64(0.16666666666666666 * (log(x) ^ 3.0)) / Float64(n * n))) / Float64(-n));
	else
		tmp = Float64((x ^ Float64(1.0 / n)) / Float64(x * n));
	end
	return tmp
end

Wolfram

code[x_, n_] := If[LessEqual[x, 0.97], N[(N[(N[(0.5 * N[(N[Power[N[Log[x], $MachinePrecision], 2.0], $MachinePrecision] / n), $MachinePrecision] + N[Log[x], $MachinePrecision]), $MachinePrecision] + N[(N[(0.16666666666666666 * N[Power[N[Log[x], $MachinePrecision], 3.0], $MachinePrecision]), $MachinePrecision] / N[(n * n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / (-n)), $MachinePrecision], N[(N[Power[x, N[(1.0 / n), $MachinePrecision]], $MachinePrecision] / N[(x * n), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 0.96999999999999997

   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 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 77.1%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 76.2%

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

   if 0.96999999999999997 < 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%

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

4. Final simplification 86.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 0.97:\\ \;\;\;\;\frac{\mathsf{fma}\left(0.5, \frac{{\log x}^{2}}{n}, \log x\right) + \frac{0.16666666666666666 \cdot {\log x}^{3}}{n \cdot n}}{-n}\\ \mathbf{else}:\\ \;\;\;\;\frac{{x}^{\left(\frac{1}{n}\right)}}{x \cdot n}\\ \end{array} \]
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

   7. Applied rewrites 77.5%

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

   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

   7. Applied rewrites 30.2%

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

   8. Taylor expanded in x around inf

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

   10. Applied rewrites 75.2%

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

   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-pow.f64 N/A

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

      2. pow-to-exp N/A

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

      3. lower-exp.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. un-div-inv N/A

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

      6. lower-/.f64 N/A

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

      7. lift-+.f64 N/A

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

      8. +-commutative N/A

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

      9. lower-log1p.f64 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: 82.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{if}\;\frac{1}{n} \leq -4 \cdot 10^{-29}:\\ \;\;\;\;\frac{t\_0}{x \cdot n}\\ \mathbf{elif}\;\frac{1}{n} \leq 10^{-97}:\\ \;\;\;\;\frac{\log \left(\frac{x}{x + 1}\right)}{-n}\\ \mathbf{elif}\;\frac{1}{n} \leq 4 \cdot 10^{-21}:\\ \;\;\;\;\frac{1}{x \cdot \mathsf{fma}\left(0.5, \frac{n}{x}, n\right)}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x, \mathsf{fma}\left(x, \frac{\mathsf{fma}\left(x, 0.16666666666666666, n \cdot \mathsf{fma}\left(x, -0.5, 0.5\right)\right)}{n \cdot \left(n \cdot n\right)}, \frac{1}{n}\right), 1\right) - t\_0\\ \end{array} \end{array} \]

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
            (/
             (fma x 0.16666666666666666 (* n (fma x -0.5 0.5)))
             (* n (* 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, (fma(x, 0.16666666666666666, (n * fma(x, -0.5, 0.5))) / (n * (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(fma(x, 0.16666666666666666, Float64(n * fma(x, -0.5, 0.5))) / Float64(n * 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[(N[(x * 0.16666666666666666 + N[(n * N[(x * -0.5 + 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(n * N[(n * n), $MachinePrecision]), $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

   7. Applied rewrites 77.5%

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

   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

   7. Applied rewrites 35.9%

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

   8. Taylor expanded in x around inf

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

   10. Applied rewrites 69.2%

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

   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

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

   5. Applied rewrites 35.1%

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

   6. Taylor expanded in n around 0

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

   8. Applied rewrites 72.9%

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

4. Final simplification 80.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{1}{n} \leq -4 \cdot 10^{-29}:\\ \;\;\;\;\frac{{x}^{\left(\frac{1}{n}\right)}}{x \cdot n}\\ \mathbf{elif}\;\frac{1}{n} \leq 10^{-97}:\\ \;\;\;\;\frac{\log \left(\frac{x}{x + 1}\right)}{-n}\\ \mathbf{elif}\;\frac{1}{n} \leq 4 \cdot 10^{-21}:\\ \;\;\;\;\frac{1}{x \cdot \mathsf{fma}\left(0.5, \frac{n}{x}, n\right)}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x, \mathsf{fma}\left(x, \frac{\mathsf{fma}\left(x, 0.16666666666666666, n \cdot \mathsf{fma}\left(x, -0.5, 0.5\right)\right)}{n \cdot \left(n \cdot n\right)}, \frac{1}{n}\right), 1\right) - {x}^{\left(\frac{1}{n}\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 82.2% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{if}\;\frac{1}{n} \leq -4 \cdot 10^{-29}:\\ \;\;\;\;\frac{t\_0}{x \cdot n}\\ \mathbf{elif}\;\frac{1}{n} \leq 10^{-97}:\\ \;\;\;\;\frac{\log \left(\frac{x}{x + 1}\right)}{-n}\\ \mathbf{elif}\;\frac{1}{n} \leq 4 \cdot 10^{-21}:\\ \;\;\;\;\frac{1}{x \cdot \mathsf{fma}\left(0.5, \frac{n}{x}, n\right)}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x, \mathsf{fma}\left(x, 0.16666666666666666 \cdot \frac{x}{n \cdot \left(n \cdot n\right)}, \frac{1}{n}\right), 1\right) - t\_0\\ \end{array} \end{array} \]

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 (* 0.16666666666666666 (/ x (* n (* 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, (0.16666666666666666 * (x / (n * (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(0.16666666666666666 * Float64(x / Float64(n * 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[(0.16666666666666666 * N[(x / N[(n * N[(n * n), $MachinePrecision]), $MachinePrecision]), $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

   7. Applied rewrites 77.5%

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

   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

   7. Applied rewrites 35.9%

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

   8. Taylor expanded in x around inf

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

   10. Applied rewrites 69.2%

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

   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

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

   5. Applied rewrites 35.1%

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

   6. Taylor expanded in n around 0

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

   8. Applied rewrites 72.0%

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

4. Final simplification 80.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{1}{n} \leq -4 \cdot 10^{-29}:\\ \;\;\;\;\frac{{x}^{\left(\frac{1}{n}\right)}}{x \cdot n}\\ \mathbf{elif}\;\frac{1}{n} \leq 10^{-97}:\\ \;\;\;\;\frac{\log \left(\frac{x}{x + 1}\right)}{-n}\\ \mathbf{elif}\;\frac{1}{n} \leq 4 \cdot 10^{-21}:\\ \;\;\;\;\frac{1}{x \cdot \mathsf{fma}\left(0.5, \frac{n}{x}, n\right)}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x, \mathsf{fma}\left(x, 0.16666666666666666 \cdot \frac{x}{n \cdot \left(n \cdot n\right)}, \frac{1}{n}\right), 1\right) - {x}^{\left(\frac{1}{n}\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 82.6% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{if}\;\frac{1}{n} \leq -4 \cdot 10^{-29}:\\ \;\;\;\;\frac{t\_0}{x \cdot n}\\ \mathbf{elif}\;\frac{1}{n} \leq 10^{-97}:\\ \;\;\;\;\frac{\log \left(\frac{x}{x + 1}\right)}{-n}\\ \mathbf{elif}\;\frac{1}{n} \leq 4 \cdot 10^{-21}:\\ \;\;\;\;\frac{1}{x \cdot \mathsf{fma}\left(0.5, \frac{n}{x}, n\right)}\\ \mathbf{elif}\;\frac{1}{n} \leq 4 \cdot 10^{+171}:\\ \;\;\;\;\left(\frac{x}{n} + 1\right) - t\_0\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x, \mathsf{fma}\left(x, \frac{0.5}{n \cdot n} - \frac{0.5}{n}, \frac{1}{n}\right), 1\right) - 1\\ \end{array} \end{array} \]

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)))
         (if (<= (/ 1.0 n) 4e+171)
           (- (+ (/ x n) 1.0) t_0)
           (-
            (fma x (fma x (- (/ 0.5 (* n n)) (/ 0.5 n)) (/ 1.0 n)) 1.0)
            1.0)))))))

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 if ((1.0 / n) <= 4e+171) {
		tmp = ((x / n) + 1.0) - t_0;
	} else {
		tmp = fma(x, fma(x, ((0.5 / (n * n)) - (0.5 / n)), (1.0 / n)), 1.0) - 1.0;
	}
	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)));
	elseif (Float64(1.0 / n) <= 4e+171)
		tmp = Float64(Float64(Float64(x / n) + 1.0) - t_0);
	else
		tmp = Float64(fma(x, fma(x, Float64(Float64(0.5 / Float64(n * n)) - Float64(0.5 / n)), Float64(1.0 / n)), 1.0) - 1.0);
	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], If[LessEqual[N[(1.0 / n), $MachinePrecision], 4e+171], N[(N[(N[(x / n), $MachinePrecision] + 1.0), $MachinePrecision] - t$95$0), $MachinePrecision], N[(N[(x * N[(x * N[(N[(0.5 / N[(n * n), $MachinePrecision]), $MachinePrecision] - N[(0.5 / n), $MachinePrecision]), $MachinePrecision] + N[(1.0 / n), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision] - 1.0), $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

   7. Applied rewrites 77.5%

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

   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

   7. Applied rewrites 35.9%

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

   8. Taylor expanded in x around inf

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

   10. Applied rewrites 69.2%

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

   if 3.99999999999999963e-21 < (/.f64 #s(literal 1 binary64) n) < 3.99999999999999982e171

   1. Initial program 63.3%

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

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\left(1 + \frac{x}{n}\right)} - {x}^{\left(\frac{1}{n}\right)} \]
   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 61.1

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

   5. Applied rewrites 61.1%

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

   if 3.99999999999999982e171 < (/.f64 #s(literal 1 binary64) n)

   1. Initial program 29.5%

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

   3. Taylor expanded in x around 0

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

   5. Applied rewrites 25.1%

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

   6. Taylor expanded in n around inf

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

   8. Applied rewrites 2.0%

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

   9. Taylor expanded in x around 0

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

       1. +-commutative N/A

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

       2. lower-fma.f64 N/A

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

       3. lower-fma.f64 N/A

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

       4. lower--.f64 N/A

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

       5. associate-*r/ N/A

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

       6. metadata-eval N/A

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

       7. lower-/.f64 N/A

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

       8. unpow2 N/A

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

       9. lower-*.f64 N/A

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

       10. associate-*r/ N/A

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

       11. metadata-eval N/A

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

       12. lower-/.f64 N/A

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

       13. lower-/.f64 73.6

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

   11. Applied rewrites 73.6%

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

4. Final simplification 79.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{1}{n} \leq -4 \cdot 10^{-29}:\\ \;\;\;\;\frac{{x}^{\left(\frac{1}{n}\right)}}{x \cdot n}\\ \mathbf{elif}\;\frac{1}{n} \leq 10^{-97}:\\ \;\;\;\;\frac{\log \left(\frac{x}{x + 1}\right)}{-n}\\ \mathbf{elif}\;\frac{1}{n} \leq 4 \cdot 10^{-21}:\\ \;\;\;\;\frac{1}{x \cdot \mathsf{fma}\left(0.5, \frac{n}{x}, n\right)}\\ \mathbf{elif}\;\frac{1}{n} \leq 4 \cdot 10^{+171}:\\ \;\;\;\;\left(\frac{x}{n} + 1\right) - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x, \mathsf{fma}\left(x, \frac{0.5}{n \cdot n} - \frac{0.5}{n}, \frac{1}{n}\right), 1\right) - 1\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 81.9% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{if}\;\frac{1}{n} \leq -4 \cdot 10^{-29}:\\ \;\;\;\;\frac{t\_0}{x \cdot n}\\ \mathbf{elif}\;\frac{1}{n} \leq 10^{-97}:\\ \;\;\;\;\frac{\log \left(\frac{x}{x + 1}\right)}{-n}\\ \mathbf{elif}\;\frac{1}{n} \leq 2 \cdot 10^{-34}:\\ \;\;\;\;\frac{1}{x \cdot \mathsf{fma}\left(0.5, \frac{n}{x}, n\right)}\\ \mathbf{elif}\;\frac{1}{n} \leq 4 \cdot 10^{+171}:\\ \;\;\;\;1 - t\_0\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x, \mathsf{fma}\left(x, \frac{0.5}{n \cdot n} - \frac{0.5}{n}, \frac{1}{n}\right), 1\right) - 1\\ \end{array} \end{array} \]

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) 4e+171)
           (- 1.0 t_0)
           (-
            (fma x (fma x (- (/ 0.5 (* n n)) (/ 0.5 n)) (/ 1.0 n)) 1.0)
            1.0)))))))

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) <= 4e+171) {
		tmp = 1.0 - t_0;
	} else {
		tmp = fma(x, fma(x, ((0.5 / (n * n)) - (0.5 / n)), (1.0 / n)), 1.0) - 1.0;
	}
	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) <= 4e+171)
		tmp = Float64(1.0 - t_0);
	else
		tmp = Float64(fma(x, fma(x, Float64(Float64(0.5 / Float64(n * n)) - Float64(0.5 / n)), Float64(1.0 / n)), 1.0) - 1.0);
	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], 4e+171], N[(1.0 - t$95$0), $MachinePrecision], N[(N[(x * N[(x * N[(N[(0.5 / N[(n * n), $MachinePrecision]), $MachinePrecision] - N[(0.5 / n), $MachinePrecision]), $MachinePrecision] + N[(1.0 / n), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision] - 1.0), $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

   7. Applied rewrites 77.5%

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

   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

   7. Applied rewrites 30.2%

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

   8. Taylor expanded in x around inf

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

   10. Applied rewrites 75.2%

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

   if 1.99999999999999986e-34 < (/.f64 #s(literal 1 binary64) n) < 3.99999999999999982e171

   1. Initial program 60.6%

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

   3. Taylor expanded in x around 0

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

   5. Applied rewrites 56.2%

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

   if 3.99999999999999982e171 < (/.f64 #s(literal 1 binary64) n)

   1. Initial program 29.5%

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

   3. Taylor expanded in x around 0

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

   5. Applied rewrites 25.1%

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

   6. Taylor expanded in n around inf

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

   8. Applied rewrites 2.0%

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

   9. Taylor expanded in x around 0

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

       1. +-commutative N/A

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

       2. lower-fma.f64 N/A

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

       3. lower-fma.f64 N/A

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

       4. lower--.f64 N/A

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

       5. associate-*r/ N/A

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

       6. metadata-eval N/A

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

       7. lower-/.f64 N/A

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

       8. unpow2 N/A

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

       9. lower-*.f64 N/A

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

       10. associate-*r/ N/A

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

       11. metadata-eval N/A

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

       12. lower-/.f64 N/A

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

       13. lower-/.f64 73.6

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

   11. Applied rewrites 73.6%

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

4. Final simplification 79.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{1}{n} \leq -4 \cdot 10^{-29}:\\ \;\;\;\;\frac{{x}^{\left(\frac{1}{n}\right)}}{x \cdot n}\\ \mathbf{elif}\;\frac{1}{n} \leq 10^{-97}:\\ \;\;\;\;\frac{\log \left(\frac{x}{x + 1}\right)}{-n}\\ \mathbf{elif}\;\frac{1}{n} \leq 2 \cdot 10^{-34}:\\ \;\;\;\;\frac{1}{x \cdot \mathsf{fma}\left(0.5, \frac{n}{x}, n\right)}\\ \mathbf{elif}\;\frac{1}{n} \leq 4 \cdot 10^{+171}:\\ \;\;\;\;1 - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x, \mathsf{fma}\left(x, \frac{0.5}{n \cdot n} - \frac{0.5}{n}, \frac{1}{n}\right), 1\right) - 1\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 56.6% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 1 - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{if}\;x \leq 1.85 \cdot 10^{-187}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq 1.9 \cdot 10^{-148}:\\ \;\;\;\;\frac{-\log x}{n}\\ \mathbf{elif}\;x \leq 4.2 \cdot 10^{-82}:\\ \;\;\;\;\frac{\frac{-0.5 + \frac{0.3333333333333333}{x}}{x} + 1}{x \cdot n}\\ \mathbf{elif}\;x \leq 0.0145:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq 4.6 \cdot 10^{+128}:\\ \;\;\;\;\frac{1}{x \cdot \left(\left(n - \frac{\mathsf{fma}\left(n, 0.041666666666666664, n \cdot -0.08333333333333333\right)}{x \cdot \left(x \cdot x\right)}\right) + \frac{\mathsf{fma}\left(n, 0.5, \frac{n \cdot -0.08333333333333333}{x}\right)}{x}\right)}\\ \mathbf{else}:\\ \;\;\;\;1 - 1\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (let* ((t_0 (- 1.0 (pow x (/ 1.0 n)))))
   (if (<= x 1.85e-187)
     t_0
     (if (<= x 1.9e-148)
       (/ (- (log x)) n)
       (if (<= x 4.2e-82)
         (/ (+ (/ (+ -0.5 (/ 0.3333333333333333 x)) x) 1.0) (* x n))
         (if (<= x 0.0145)
           t_0
           (if (<= x 4.6e+128)
             (/
              1.0
              (*
               x
               (+
                (-
                 n
                 (/
                  (fma n 0.041666666666666664 (* n -0.08333333333333333))
                  (* x (* x x))))
                (/ (fma n 0.5 (/ (* n -0.08333333333333333) x)) x))))
             (- 1.0 1.0))))))))

C

double code(double x, double n) {
	double t_0 = 1.0 - pow(x, (1.0 / n));
	double tmp;
	if (x <= 1.85e-187) {
		tmp = t_0;
	} else if (x <= 1.9e-148) {
		tmp = -log(x) / n;
	} else if (x <= 4.2e-82) {
		tmp = (((-0.5 + (0.3333333333333333 / x)) / x) + 1.0) / (x * n);
	} else if (x <= 0.0145) {
		tmp = t_0;
	} else if (x <= 4.6e+128) {
		tmp = 1.0 / (x * ((n - (fma(n, 0.041666666666666664, (n * -0.08333333333333333)) / (x * (x * x)))) + (fma(n, 0.5, ((n * -0.08333333333333333) / x)) / x)));
	} else {
		tmp = 1.0 - 1.0;
	}
	return tmp;
}

Julia

function code(x, n)
	t_0 = Float64(1.0 - (x ^ Float64(1.0 / n)))
	tmp = 0.0
	if (x <= 1.85e-187)
		tmp = t_0;
	elseif (x <= 1.9e-148)
		tmp = Float64(Float64(-log(x)) / n);
	elseif (x <= 4.2e-82)
		tmp = Float64(Float64(Float64(Float64(-0.5 + Float64(0.3333333333333333 / x)) / x) + 1.0) / Float64(x * n));
	elseif (x <= 0.0145)
		tmp = t_0;
	elseif (x <= 4.6e+128)
		tmp = Float64(1.0 / Float64(x * Float64(Float64(n - Float64(fma(n, 0.041666666666666664, Float64(n * -0.08333333333333333)) / Float64(x * Float64(x * x)))) + Float64(fma(n, 0.5, Float64(Float64(n * -0.08333333333333333) / x)) / x))));
	else
		tmp = Float64(1.0 - 1.0);
	end
	return tmp
end

Wolfram

code[x_, n_] := Block[{t$95$0 = N[(1.0 - N[Power[x, N[(1.0 / n), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, 1.85e-187], t$95$0, If[LessEqual[x, 1.9e-148], N[((-N[Log[x], $MachinePrecision]) / n), $MachinePrecision], If[LessEqual[x, 4.2e-82], N[(N[(N[(N[(-0.5 + N[(0.3333333333333333 / x), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision] + 1.0), $MachinePrecision] / N[(x * n), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 0.0145], t$95$0, If[LessEqual[x, 4.6e+128], N[(1.0 / N[(x * N[(N[(n - N[(N[(n * 0.041666666666666664 + N[(n * -0.08333333333333333), $MachinePrecision]), $MachinePrecision] / N[(x * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(n * 0.5 + N[(N[(n * -0.08333333333333333), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 - 1.0), $MachinePrecision]]]]]]]

Derivation

1. Split input into 5 regimes

2. if x < 1.85000000000000005e-187 or 4.2000000000000001e-82 < x < 0.0145000000000000007

   1. Initial program 52.4%

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

   3. Taylor expanded in x around 0

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

   5. Applied rewrites 50.4%

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

   if 1.85000000000000005e-187 < x < 1.90000000000000007e-148

   1. Initial program 24.7%

      \[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)} \]
   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 65.6

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

   5. Applied rewrites 65.6%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 65.6%

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

   if 1.90000000000000007e-148 < x < 4.2000000000000001e-82

   1. Initial program 40.5%

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

   3. Taylor expanded in n around inf

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

      1. lower-/.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lower-log1p.f64 N/A

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

      4. lower-log.f64 29.5

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

   5. Applied rewrites 29.5%

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

   6. Taylor expanded in x around inf

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

   8. Applied rewrites 0.3%

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

   9. Taylor expanded in n around 0

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

   11. Applied rewrites 0.3%

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

   12. Taylor expanded in x around inf

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

   14. Applied rewrites 66.3%

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

   if 0.0145000000000000007 < x < 4.59999999999999996e128

   1. Initial program 32.7%

      \[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)} \]
   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 36.3

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

   5. Applied rewrites 36.3%

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

   7. Applied rewrites 37.0%

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

   8. Taylor expanded in x around inf

      \[\leadsto \frac{1}{x \cdot \color{blue}{\left(\left(n + -1 \cdot \frac{\frac{-1}{4} \cdot n + \left(\frac{1}{6} \cdot n + \frac{1}{2} \cdot \left(\frac{-1}{4} \cdot n + \frac{1}{3} \cdot n\right)\right)}{{x}^{3}}\right) - \left(\frac{-1}{2} \cdot \frac{n}{x} + \left(\frac{-1}{4} \cdot \frac{n}{{x}^{2}} + \frac{1}{3} \cdot \frac{n}{{x}^{2}}\right)\right)\right)}} \]

   10. Applied rewrites 80.1%

       \[\leadsto \frac{1}{x \cdot \color{blue}{\left(\left(n - \frac{\mathsf{fma}\left(n, 0.041666666666666664, n \cdot -0.08333333333333333\right)}{x \cdot \left(x \cdot x\right)}\right) + \frac{\mathsf{fma}\left(n, 0.5, \frac{n \cdot -0.08333333333333333}{x}\right)}{x}\right)}} \]

   if 4.59999999999999996e128 < 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

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

   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%

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

4. Final simplification 68.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 1.85 \cdot 10^{-187}:\\ \;\;\;\;1 - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{elif}\;x \leq 1.9 \cdot 10^{-148}:\\ \;\;\;\;\frac{-\log x}{n}\\ \mathbf{elif}\;x \leq 4.2 \cdot 10^{-82}:\\ \;\;\;\;\frac{\frac{-0.5 + \frac{0.3333333333333333}{x}}{x} + 1}{x \cdot n}\\ \mathbf{elif}\;x \leq 0.0145:\\ \;\;\;\;1 - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{elif}\;x \leq 4.6 \cdot 10^{+128}:\\ \;\;\;\;\frac{1}{x \cdot \left(\left(n - \frac{\mathsf{fma}\left(n, 0.041666666666666664, n \cdot -0.08333333333333333\right)}{x \cdot \left(x \cdot x\right)}\right) + \frac{\mathsf{fma}\left(n, 0.5, \frac{n \cdot -0.08333333333333333}{x}\right)}{x}\right)}\\ \mathbf{else}:\\ \;\;\;\;1 - 1\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 59.6% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 1.9 \cdot 10^{-148}:\\ \;\;\;\;\frac{-\log x}{n}\\ \mathbf{elif}\;x \leq 3.2 \cdot 10^{-71}:\\ \;\;\;\;\frac{\frac{-0.5 + \frac{0.3333333333333333}{x}}{x} + 1}{x \cdot n}\\ \mathbf{elif}\;x \leq 0.5:\\ \;\;\;\;\frac{x - \log x}{n}\\ \mathbf{elif}\;x \leq 4.6 \cdot 10^{+128}:\\ \;\;\;\;\frac{1}{x \cdot \left(\left(n - \frac{\mathsf{fma}\left(n, 0.041666666666666664, n \cdot -0.08333333333333333\right)}{x \cdot \left(x \cdot x\right)}\right) + \frac{\mathsf{fma}\left(n, 0.5, \frac{n \cdot -0.08333333333333333}{x}\right)}{x}\right)}\\ \mathbf{else}:\\ \;\;\;\;1 - 1\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (if (<= x 1.9e-148)
   (/ (- (log x)) n)
   (if (<= x 3.2e-71)
     (/ (+ (/ (+ -0.5 (/ 0.3333333333333333 x)) x) 1.0) (* x n))
     (if (<= x 0.5)
       (/ (- x (log x)) n)
       (if (<= x 4.6e+128)
         (/
          1.0
          (*
           x
           (+
            (-
             n
             (/
              (fma n 0.041666666666666664 (* n -0.08333333333333333))
              (* x (* x x))))
            (/ (fma n 0.5 (/ (* n -0.08333333333333333) x)) x))))
         (- 1.0 1.0))))))

C

double code(double x, double n) {
	double tmp;
	if (x <= 1.9e-148) {
		tmp = -log(x) / n;
	} else if (x <= 3.2e-71) {
		tmp = (((-0.5 + (0.3333333333333333 / x)) / x) + 1.0) / (x * n);
	} else if (x <= 0.5) {
		tmp = (x - log(x)) / n;
	} else if (x <= 4.6e+128) {
		tmp = 1.0 / (x * ((n - (fma(n, 0.041666666666666664, (n * -0.08333333333333333)) / (x * (x * x)))) + (fma(n, 0.5, ((n * -0.08333333333333333) / x)) / x)));
	} else {
		tmp = 1.0 - 1.0;
	}
	return tmp;
}

Julia

function code(x, n)
	tmp = 0.0
	if (x <= 1.9e-148)
		tmp = Float64(Float64(-log(x)) / n);
	elseif (x <= 3.2e-71)
		tmp = Float64(Float64(Float64(Float64(-0.5 + Float64(0.3333333333333333 / x)) / x) + 1.0) / Float64(x * n));
	elseif (x <= 0.5)
		tmp = Float64(Float64(x - log(x)) / n);
	elseif (x <= 4.6e+128)
		tmp = Float64(1.0 / Float64(x * Float64(Float64(n - Float64(fma(n, 0.041666666666666664, Float64(n * -0.08333333333333333)) / Float64(x * Float64(x * x)))) + Float64(fma(n, 0.5, Float64(Float64(n * -0.08333333333333333) / x)) / x))));
	else
		tmp = Float64(1.0 - 1.0);
	end
	return tmp
end

Wolfram

code[x_, n_] := If[LessEqual[x, 1.9e-148], N[((-N[Log[x], $MachinePrecision]) / n), $MachinePrecision], If[LessEqual[x, 3.2e-71], N[(N[(N[(N[(-0.5 + N[(0.3333333333333333 / x), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision] + 1.0), $MachinePrecision] / N[(x * n), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 0.5], N[(N[(x - N[Log[x], $MachinePrecision]), $MachinePrecision] / n), $MachinePrecision], If[LessEqual[x, 4.6e+128], N[(1.0 / N[(x * N[(N[(n - N[(N[(n * 0.041666666666666664 + N[(n * -0.08333333333333333), $MachinePrecision]), $MachinePrecision] / N[(x * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(n * 0.5 + N[(N[(n * -0.08333333333333333), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 - 1.0), $MachinePrecision]]]]]

Derivation

1. Split input into 5 regimes

2. if x < 1.90000000000000007e-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

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

   8. Applied rewrites 48.6%

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

   if 1.90000000000000007e-148 < x < 3.1999999999999999e-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

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

   8. Applied rewrites 0.3%

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

   9. Taylor expanded in n around 0

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

   11. Applied rewrites 0.3%

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

   12. Taylor expanded in x around inf

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

   14. Applied rewrites 65.2%

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

   if 3.1999999999999999e-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

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

   8. Applied rewrites 41.8%

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

   if 0.5 < x < 4.59999999999999996e128

   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

   7. Applied rewrites 35.8%

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

   8. Taylor expanded in x around inf

      \[\leadsto \frac{1}{x \cdot \color{blue}{\left(\left(n + -1 \cdot \frac{\frac{-1}{4} \cdot n + \left(\frac{1}{6} \cdot n + \frac{1}{2} \cdot \left(\frac{-1}{4} \cdot n + \frac{1}{3} \cdot n\right)\right)}{{x}^{3}}\right) - \left(\frac{-1}{2} \cdot \frac{n}{x} + \left(\frac{-1}{4} \cdot \frac{n}{{x}^{2}} + \frac{1}{3} \cdot \frac{n}{{x}^{2}}\right)\right)\right)}} \]

   10. Applied rewrites 81.4%

       \[\leadsto \frac{1}{x \cdot \color{blue}{\left(\left(n - \frac{\mathsf{fma}\left(n, 0.041666666666666664, n \cdot -0.08333333333333333\right)}{x \cdot \left(x \cdot x\right)}\right) + \frac{\mathsf{fma}\left(n, 0.5, \frac{n \cdot -0.08333333333333333}{x}\right)}{x}\right)}} \]

   if 4.59999999999999996e128 < 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

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

   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%

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

4. Final simplification 65.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 1.9 \cdot 10^{-148}:\\ \;\;\;\;\frac{-\log x}{n}\\ \mathbf{elif}\;x \leq 3.2 \cdot 10^{-71}:\\ \;\;\;\;\frac{\frac{-0.5 + \frac{0.3333333333333333}{x}}{x} + 1}{x \cdot n}\\ \mathbf{elif}\;x \leq 0.5:\\ \;\;\;\;\frac{x - \log x}{n}\\ \mathbf{elif}\;x \leq 4.6 \cdot 10^{+128}:\\ \;\;\;\;\frac{1}{x \cdot \left(\left(n - \frac{\mathsf{fma}\left(n, 0.041666666666666664, n \cdot -0.08333333333333333\right)}{x \cdot \left(x \cdot x\right)}\right) + \frac{\mathsf{fma}\left(n, 0.5, \frac{n \cdot -0.08333333333333333}{x}\right)}{x}\right)}\\ \mathbf{else}:\\ \;\;\;\;1 - 1\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 58.8% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{-\log x}{n}\\ \mathbf{if}\;x \leq 1.9 \cdot 10^{-148}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq 3.2 \cdot 10^{-71}:\\ \;\;\;\;\frac{\frac{-0.5 + \frac{0.3333333333333333}{x}}{x} + 1}{x \cdot n}\\ \mathbf{elif}\;x \leq 2.7 \cdot 10^{-16}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq 4.6 \cdot 10^{+128}:\\ \;\;\;\;\frac{\frac{1}{n} + \frac{\frac{0.3333333333333333}{x \cdot n} - \frac{0.5}{n}}{x}}{x}\\ \mathbf{else}:\\ \;\;\;\;1 - 1\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (let* ((t_0 (/ (- (log x)) n)))
   (if (<= x 1.9e-148)
     t_0
     (if (<= x 3.2e-71)
       (/ (+ (/ (+ -0.5 (/ 0.3333333333333333 x)) x) 1.0) (* x n))
       (if (<= x 2.7e-16)
         t_0
         (if (<= x 4.6e+128)
           (/
            (+ (/ 1.0 n) (/ (- (/ 0.3333333333333333 (* x n)) (/ 0.5 n)) x))
            x)
           (- 1.0 1.0)))))))

C

double code(double x, double n) {
	double t_0 = -log(x) / n;
	double tmp;
	if (x <= 1.9e-148) {
		tmp = t_0;
	} else if (x <= 3.2e-71) {
		tmp = (((-0.5 + (0.3333333333333333 / x)) / x) + 1.0) / (x * n);
	} else if (x <= 2.7e-16) {
		tmp = t_0;
	} else if (x <= 4.6e+128) {
		tmp = ((1.0 / n) + (((0.3333333333333333 / (x * n)) - (0.5 / n)) / x)) / x;
	} else {
		tmp = 1.0 - 1.0;
	}
	return tmp;
}

Fortran

real(8) function code(x, n)
    real(8), intent (in) :: x
    real(8), intent (in) :: n
    real(8) :: t_0
    real(8) :: tmp
    t_0 = -log(x) / n
    if (x <= 1.9d-148) then
        tmp = t_0
    else if (x <= 3.2d-71) then
        tmp = ((((-0.5d0) + (0.3333333333333333d0 / x)) / x) + 1.0d0) / (x * n)
    else if (x <= 2.7d-16) then
        tmp = t_0
    else if (x <= 4.6d+128) then
        tmp = ((1.0d0 / n) + (((0.3333333333333333d0 / (x * n)) - (0.5d0 / n)) / x)) / x
    else
        tmp = 1.0d0 - 1.0d0
    end if
    code = tmp
end function

Java

public static double code(double x, double n) {
	double t_0 = -Math.log(x) / n;
	double tmp;
	if (x <= 1.9e-148) {
		tmp = t_0;
	} else if (x <= 3.2e-71) {
		tmp = (((-0.5 + (0.3333333333333333 / x)) / x) + 1.0) / (x * n);
	} else if (x <= 2.7e-16) {
		tmp = t_0;
	} else if (x <= 4.6e+128) {
		tmp = ((1.0 / n) + (((0.3333333333333333 / (x * n)) - (0.5 / n)) / x)) / x;
	} else {
		tmp = 1.0 - 1.0;
	}
	return tmp;
}

Python

def code(x, n):
	t_0 = -math.log(x) / n
	tmp = 0
	if x <= 1.9e-148:
		tmp = t_0
	elif x <= 3.2e-71:
		tmp = (((-0.5 + (0.3333333333333333 / x)) / x) + 1.0) / (x * n)
	elif x <= 2.7e-16:
		tmp = t_0
	elif x <= 4.6e+128:
		tmp = ((1.0 / n) + (((0.3333333333333333 / (x * n)) - (0.5 / n)) / x)) / x
	else:
		tmp = 1.0 - 1.0
	return tmp

Julia

function code(x, n)
	t_0 = Float64(Float64(-log(x)) / n)
	tmp = 0.0
	if (x <= 1.9e-148)
		tmp = t_0;
	elseif (x <= 3.2e-71)
		tmp = Float64(Float64(Float64(Float64(-0.5 + Float64(0.3333333333333333 / x)) / x) + 1.0) / Float64(x * n));
	elseif (x <= 2.7e-16)
		tmp = t_0;
	elseif (x <= 4.6e+128)
		tmp = Float64(Float64(Float64(1.0 / n) + Float64(Float64(Float64(0.3333333333333333 / Float64(x * n)) - Float64(0.5 / n)) / x)) / x);
	else
		tmp = Float64(1.0 - 1.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, n)
	t_0 = -log(x) / n;
	tmp = 0.0;
	if (x <= 1.9e-148)
		tmp = t_0;
	elseif (x <= 3.2e-71)
		tmp = (((-0.5 + (0.3333333333333333 / x)) / x) + 1.0) / (x * n);
	elseif (x <= 2.7e-16)
		tmp = t_0;
	elseif (x <= 4.6e+128)
		tmp = ((1.0 / n) + (((0.3333333333333333 / (x * n)) - (0.5 / n)) / x)) / x;
	else
		tmp = 1.0 - 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, n_] := Block[{t$95$0 = N[((-N[Log[x], $MachinePrecision]) / n), $MachinePrecision]}, If[LessEqual[x, 1.9e-148], t$95$0, If[LessEqual[x, 3.2e-71], N[(N[(N[(N[(-0.5 + N[(0.3333333333333333 / x), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision] + 1.0), $MachinePrecision] / N[(x * n), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 2.7e-16], t$95$0, If[LessEqual[x, 4.6e+128], N[(N[(N[(1.0 / n), $MachinePrecision] + N[(N[(N[(0.3333333333333333 / N[(x * n), $MachinePrecision]), $MachinePrecision] - N[(0.5 / n), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision], N[(1.0 - 1.0), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if x < 1.90000000000000007e-148 or 3.1999999999999999e-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

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

   8. Applied rewrites 48.4%

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

   if 1.90000000000000007e-148 < x < 3.1999999999999999e-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

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

   8. Applied rewrites 0.3%

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

   9. Taylor expanded in n around 0

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

   11. Applied rewrites 0.3%

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

   12. Taylor expanded in x around inf

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

   14. Applied rewrites 65.2%

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

   if 2.69999999999999999e-16 < x < 4.59999999999999996e128

   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

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

   8. Applied rewrites 71.7%

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

   9. Taylor expanded in x around -inf

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

   11. Applied rewrites 73.8%

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

   if 4.59999999999999996e128 < 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

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

   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%

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

4. Final simplification 65.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 1.9 \cdot 10^{-148}:\\ \;\;\;\;\frac{-\log x}{n}\\ \mathbf{elif}\;x \leq 3.2 \cdot 10^{-71}:\\ \;\;\;\;\frac{\frac{-0.5 + \frac{0.3333333333333333}{x}}{x} + 1}{x \cdot n}\\ \mathbf{elif}\;x \leq 2.7 \cdot 10^{-16}:\\ \;\;\;\;\frac{-\log x}{n}\\ \mathbf{elif}\;x \leq 4.6 \cdot 10^{+128}:\\ \;\;\;\;\frac{\frac{1}{n} + \frac{\frac{0.3333333333333333}{x \cdot n} - \frac{0.5}{n}}{x}}{x}\\ \mathbf{else}:\\ \;\;\;\;1 - 1\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 50.2% accurate, 3.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 4.6 \cdot 10^{+128}:\\ \;\;\;\;\frac{\frac{1}{n} + \frac{\frac{0.3333333333333333}{x \cdot n} - \frac{0.5}{n}}{x}}{x}\\ \mathbf{else}:\\ \;\;\;\;1 - 1\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (if (<= x 4.6e+128)
   (/ (+ (/ 1.0 n) (/ (- (/ 0.3333333333333333 (* x n)) (/ 0.5 n)) x)) x)
   (- 1.0 1.0)))

C

double code(double x, double n) {
	double tmp;
	if (x <= 4.6e+128) {
		tmp = ((1.0 / n) + (((0.3333333333333333 / (x * n)) - (0.5 / n)) / x)) / x;
	} else {
		tmp = 1.0 - 1.0;
	}
	return tmp;
}

Fortran

real(8) function code(x, n)
    real(8), intent (in) :: x
    real(8), intent (in) :: n
    real(8) :: tmp
    if (x <= 4.6d+128) then
        tmp = ((1.0d0 / n) + (((0.3333333333333333d0 / (x * n)) - (0.5d0 / n)) / x)) / x
    else
        tmp = 1.0d0 - 1.0d0
    end if
    code = tmp
end function

Java

public static double code(double x, double n) {
	double tmp;
	if (x <= 4.6e+128) {
		tmp = ((1.0 / n) + (((0.3333333333333333 / (x * n)) - (0.5 / n)) / x)) / x;
	} else {
		tmp = 1.0 - 1.0;
	}
	return tmp;
}

Python

def code(x, n):
	tmp = 0
	if x <= 4.6e+128:
		tmp = ((1.0 / n) + (((0.3333333333333333 / (x * n)) - (0.5 / n)) / x)) / x
	else:
		tmp = 1.0 - 1.0
	return tmp

Julia

function code(x, n)
	tmp = 0.0
	if (x <= 4.6e+128)
		tmp = Float64(Float64(Float64(1.0 / n) + Float64(Float64(Float64(0.3333333333333333 / Float64(x * n)) - Float64(0.5 / n)) / x)) / x);
	else
		tmp = Float64(1.0 - 1.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, n)
	tmp = 0.0;
	if (x <= 4.6e+128)
		tmp = ((1.0 / n) + (((0.3333333333333333 / (x * n)) - (0.5 / n)) / x)) / x;
	else
		tmp = 1.0 - 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, n_] := If[LessEqual[x, 4.6e+128], N[(N[(N[(1.0 / n), $MachinePrecision] + N[(N[(N[(0.3333333333333333 / N[(x * n), $MachinePrecision]), $MachinePrecision] - N[(0.5 / n), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision], N[(1.0 - 1.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 4.59999999999999996e128

   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

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

   8. Applied rewrites 22.2%

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

   9. Taylor expanded in x around -inf

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

   11. Applied rewrites 47.6%

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

   if 4.59999999999999996e128 < 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

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

   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%

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

4. Final simplification 57.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 4.6 \cdot 10^{+128}:\\ \;\;\;\;\frac{\frac{1}{n} + \frac{\frac{0.3333333333333333}{x \cdot n} - \frac{0.5}{n}}{x}}{x}\\ \mathbf{else}:\\ \;\;\;\;1 - 1\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 50.0% accurate, 4.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 4.6 \cdot 10^{+128}:\\ \;\;\;\;\frac{\frac{-0.5 + \frac{0.3333333333333333}{x}}{x} + 1}{x \cdot n}\\ \mathbf{else}:\\ \;\;\;\;1 - 1\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (if (<= x 4.6e+128)
   (/ (+ (/ (+ -0.5 (/ 0.3333333333333333 x)) x) 1.0) (* x n))
   (- 1.0 1.0)))

C

double code(double x, double n) {
	double tmp;
	if (x <= 4.6e+128) {
		tmp = (((-0.5 + (0.3333333333333333 / x)) / x) + 1.0) / (x * n);
	} else {
		tmp = 1.0 - 1.0;
	}
	return tmp;
}

Fortran

real(8) function code(x, n)
    real(8), intent (in) :: x
    real(8), intent (in) :: n
    real(8) :: tmp
    if (x <= 4.6d+128) then
        tmp = ((((-0.5d0) + (0.3333333333333333d0 / x)) / x) + 1.0d0) / (x * n)
    else
        tmp = 1.0d0 - 1.0d0
    end if
    code = tmp
end function

Java

public static double code(double x, double n) {
	double tmp;
	if (x <= 4.6e+128) {
		tmp = (((-0.5 + (0.3333333333333333 / x)) / x) + 1.0) / (x * n);
	} else {
		tmp = 1.0 - 1.0;
	}
	return tmp;
}

Python

def code(x, n):
	tmp = 0
	if x <= 4.6e+128:
		tmp = (((-0.5 + (0.3333333333333333 / x)) / x) + 1.0) / (x * n)
	else:
		tmp = 1.0 - 1.0
	return tmp

Julia

function code(x, n)
	tmp = 0.0
	if (x <= 4.6e+128)
		tmp = Float64(Float64(Float64(Float64(-0.5 + Float64(0.3333333333333333 / x)) / x) + 1.0) / Float64(x * n));
	else
		tmp = Float64(1.0 - 1.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, n)
	tmp = 0.0;
	if (x <= 4.6e+128)
		tmp = (((-0.5 + (0.3333333333333333 / x)) / x) + 1.0) / (x * n);
	else
		tmp = 1.0 - 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, n_] := If[LessEqual[x, 4.6e+128], N[(N[(N[(N[(-0.5 + N[(0.3333333333333333 / x), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision] + 1.0), $MachinePrecision] / N[(x * n), $MachinePrecision]), $MachinePrecision], N[(1.0 - 1.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 4.59999999999999996e128

   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

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

   8. Applied rewrites 22.2%

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

   9. Taylor expanded in n around 0

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

   11. Applied rewrites 22.2%

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

   12. Taylor expanded in x around inf

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

   14. Applied rewrites 47.6%

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

   if 4.59999999999999996e128 < 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

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

   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%

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

4. Final simplification 57.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 4.6 \cdot 10^{+128}:\\ \;\;\;\;\frac{\frac{-0.5 + \frac{0.3333333333333333}{x}}{x} + 1}{x \cdot n}\\ \mathbf{else}:\\ \;\;\;\;1 - 1\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 45.9% accurate, 5.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;n \leq -0.0075:\\ \;\;\;\;\frac{1}{x \cdot \mathsf{fma}\left(0.5, \frac{n}{x}, n\right)}\\ \mathbf{elif}\;n \leq -1.4 \cdot 10^{-171}:\\ \;\;\;\;1 - 1\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{n}}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (if (<= n -0.0075)
   (/ 1.0 (* x (fma 0.5 (/ n x) n)))
   (if (<= n -1.4e-171) (- 1.0 1.0) (/ (/ 1.0 n) x))))

C

double code(double x, double n) {
	double tmp;
	if (n <= -0.0075) {
		tmp = 1.0 / (x * fma(0.5, (n / x), n));
	} else if (n <= -1.4e-171) {
		tmp = 1.0 - 1.0;
	} else {
		tmp = (1.0 / n) / x;
	}
	return tmp;
}

Julia

function code(x, n)
	tmp = 0.0
	if (n <= -0.0075)
		tmp = Float64(1.0 / Float64(x * fma(0.5, Float64(n / x), n)));
	elseif (n <= -1.4e-171)
		tmp = Float64(1.0 - 1.0);
	else
		tmp = Float64(Float64(1.0 / n) / x);
	end
	return tmp
end

Wolfram

code[x_, n_] := If[LessEqual[n, -0.0075], N[(1.0 / N[(x * N[(0.5 * N[(n / x), $MachinePrecision] + n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[n, -1.4e-171], N[(1.0 - 1.0), $MachinePrecision], N[(N[(1.0 / n), $MachinePrecision] / x), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if n < -0.0074999999999999997

   1. Initial program 37.3%

      \[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)} \]
   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 69.1

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

   5. Applied rewrites 69.1%

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

   7. Applied rewrites 69.2%

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

   8. Taylor expanded in x around inf

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

   10. Applied rewrites 59.9%

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

   if -0.0074999999999999997 < n < -1.40000000000000011e-171

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

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

   5. Applied rewrites 47.4%

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

   6. Taylor expanded in n around inf

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

   8. Applied rewrites 55.0%

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

   if -1.40000000000000011e-171 < n

   1. Initial program 53.7%

      \[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)} \]
   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 41.3

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

   5. Applied rewrites 41.3%

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

   6. Taylor expanded in x around inf

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

   8. Applied rewrites 29.0%

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

   9. Taylor expanded in x around inf

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

   11. Applied rewrites 52.4%

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

Alternative 17: 44.5% accurate, 8.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 4.6 \cdot 10^{+128}:\\ \;\;\;\;\frac{\frac{1}{n}}{x}\\ \mathbf{else}:\\ \;\;\;\;1 - 1\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (if (<= x 4.6e+128) (/ (/ 1.0 n) x) (- 1.0 1.0)))

C

double code(double x, double n) {
	double tmp;
	if (x <= 4.6e+128) {
		tmp = (1.0 / n) / x;
	} else {
		tmp = 1.0 - 1.0;
	}
	return tmp;
}

Fortran

real(8) function code(x, n)
    real(8), intent (in) :: x
    real(8), intent (in) :: n
    real(8) :: tmp
    if (x <= 4.6d+128) then
        tmp = (1.0d0 / n) / x
    else
        tmp = 1.0d0 - 1.0d0
    end if
    code = tmp
end function

Java

public static double code(double x, double n) {
	double tmp;
	if (x <= 4.6e+128) {
		tmp = (1.0 / n) / x;
	} else {
		tmp = 1.0 - 1.0;
	}
	return tmp;
}

Python

def code(x, n):
	tmp = 0
	if x <= 4.6e+128:
		tmp = (1.0 / n) / x
	else:
		tmp = 1.0 - 1.0
	return tmp

Julia

function code(x, n)
	tmp = 0.0
	if (x <= 4.6e+128)
		tmp = Float64(Float64(1.0 / n) / x);
	else
		tmp = Float64(1.0 - 1.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, n)
	tmp = 0.0;
	if (x <= 4.6e+128)
		tmp = (1.0 / n) / x;
	else
		tmp = 1.0 - 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, n_] := If[LessEqual[x, 4.6e+128], N[(N[(1.0 / n), $MachinePrecision] / x), $MachinePrecision], N[(1.0 - 1.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 4.59999999999999996e128

   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

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

   8. Applied rewrites 22.2%

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

   9. Taylor expanded in x around inf

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

   11. Applied rewrites 41.8%

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

   if 4.59999999999999996e128 < 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

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

   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%

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

Alternative 18: 44.3% accurate, 10.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 4.6 \cdot 10^{+128}:\\ \;\;\;\;\frac{1}{x \cdot n}\\ \mathbf{else}:\\ \;\;\;\;1 - 1\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (if (<= x 4.6e+128) (/ 1.0 (* x n)) (- 1.0 1.0)))

C

double code(double x, double n) {
	double tmp;
	if (x <= 4.6e+128) {
		tmp = 1.0 / (x * n);
	} else {
		tmp = 1.0 - 1.0;
	}
	return tmp;
}

Fortran

real(8) function code(x, n)
    real(8), intent (in) :: x
    real(8), intent (in) :: n
    real(8) :: tmp
    if (x <= 4.6d+128) then
        tmp = 1.0d0 / (x * n)
    else
        tmp = 1.0d0 - 1.0d0
    end if
    code = tmp
end function

Java

public static double code(double x, double n) {
	double tmp;
	if (x <= 4.6e+128) {
		tmp = 1.0 / (x * n);
	} else {
		tmp = 1.0 - 1.0;
	}
	return tmp;
}

Python

def code(x, n):
	tmp = 0
	if x <= 4.6e+128:
		tmp = 1.0 / (x * n)
	else:
		tmp = 1.0 - 1.0
	return tmp

Julia

function code(x, n)
	tmp = 0.0
	if (x <= 4.6e+128)
		tmp = Float64(1.0 / Float64(x * n));
	else
		tmp = Float64(1.0 - 1.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, n)
	tmp = 0.0;
	if (x <= 4.6e+128)
		tmp = 1.0 / (x * n);
	else
		tmp = 1.0 - 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, n_] := If[LessEqual[x, 4.6e+128], N[(1.0 / N[(x * n), $MachinePrecision]), $MachinePrecision], N[(1.0 - 1.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 4.59999999999999996e128

   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

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

   8. Applied rewrites 41.8%

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

   if 4.59999999999999996e128 < 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

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

   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%

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

Alternative 19: 31.9% accurate, 57.8× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 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

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

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. 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))))