2nthrt (problem 3.4.6)

Percentage Accurate: 54.0% → 92.4%

Time: 41.3s

Alternatives: 16

Speedup: 1.9×

• could not determine a ground truth

  1. x = -2
  2. n = 4.94066e-324

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: 54.0% 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: 92.4% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{if}\;x \leq 6:\\ \;\;\;\;\frac{\log \left(\frac{x}{e^{\mathsf{log1p}\left(x\right) + \frac{\mathsf{fma}\left(0.5, {\left(\mathsf{log1p}\left(x\right)\right)}^{2} - {\log x}^{2}, 0.16666666666666666 \cdot \frac{{\left(\mathsf{log1p}\left(x\right)\right)}^{3} - {\log x}^{3}}{n}\right)}{n}}}\right)}{-n}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\mathsf{fma}\left(t\_0, \frac{\frac{-0.5}{n}}{x}, \frac{t\_0}{n}\right)}{x}\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (let* ((t_0 (pow x (/ 1.0 n))))
   (if (<= x 6.0)
     (/
      (log
       (/
        x
        (exp
         (+
          (log1p x)
          (/
           (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)))))
      (- n))
     (expm1 (log1p (/ (fma t_0 (/ (/ -0.5 n) x) (/ t_0 n)) x))))))

C

double code(double x, double n) {
	double t_0 = pow(x, (1.0 / n));
	double tmp;
	if (x <= 6.0) {
		tmp = log((x / exp((log1p(x) + (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))))) / -n;
	} else {
		tmp = expm1(log1p((fma(t_0, ((-0.5 / n) / x), (t_0 / n)) / x)));
	}
	return tmp;
}

Julia

function code(x, n)
	t_0 = x ^ Float64(1.0 / n)
	tmp = 0.0
	if (x <= 6.0)
		tmp = Float64(log(Float64(x / exp(Float64(log1p(x) + Float64(fma(0.5, Float64((log1p(x) ^ 2.0) - (log(x) ^ 2.0)), Float64(0.16666666666666666 * Float64(Float64((log1p(x) ^ 3.0) - (log(x) ^ 3.0)) / n))) / n))))) / Float64(-n));
	else
		tmp = expm1(log1p(Float64(fma(t_0, Float64(Float64(-0.5 / n) / x), Float64(t_0 / n)) / x)));
	end
	return tmp
end

Wolfram

code[x_, n_] := Block[{t$95$0 = N[Power[x, N[(1.0 / n), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[x, 6.0], N[(N[Log[N[(x / N[Exp[N[(N[Log[1 + x], $MachinePrecision] + 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[(0.16666666666666666 * N[(N[(N[Power[N[Log[1 + x], $MachinePrecision], 3.0], $MachinePrecision] - N[Power[N[Log[x], $MachinePrecision], 3.0], $MachinePrecision]), $MachinePrecision] / n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / n), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / (-n)), $MachinePrecision], N[(Exp[N[Log[1 + N[(N[(t$95$0 * N[(N[(-0.5 / n), $MachinePrecision] / x), $MachinePrecision] + N[(t$95$0 / n), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]], $MachinePrecision]] - 1), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if x < 6

   1. Initial program 43.4%

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

   3. Taylor expanded in n around -inf 79.7%

      \[\leadsto \color{blue}{-1 \cdot \frac{\left(-1 \cdot \log \left(1 + x\right) + -1 \cdot \frac{\left(-1 \cdot \frac{-0.16666666666666666 \cdot {\log \left(1 + x\right)}^{3} - -0.16666666666666666 \cdot {\log x}^{3}}{n} + 0.5 \cdot {\log \left(1 + x\right)}^{2}\right) - 0.5 \cdot {\log x}^{2}}{n}\right) - -1 \cdot \log x}{n}} \]

   5. Simplified 79.7%

      \[\leadsto \color{blue}{\frac{\log x - \left(\mathsf{log1p}\left(x\right) + \frac{0.5 \cdot \left({\left(\mathsf{log1p}\left(x\right)\right)}^{2} - {\log x}^{2}\right) + \frac{0.16666666666666666 \cdot \left({\left(\mathsf{log1p}\left(x\right)\right)}^{3} - {\log x}^{3}\right)}{n}}{n}\right)}{-n}} \]
   6. Step-by-step derivation

      1. add-log-exp 86.3%

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

      2. diff-log 86.3%

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

   7. Applied egg-rr 86.3%

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

   if 6 < x

   1. Initial program 62.2%

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

   3. Taylor expanded in x around inf 81.9%

      \[\leadsto \color{blue}{\frac{\frac{e^{-1 \cdot \frac{\log \left(\frac{1}{x}\right)}{n}}}{n} + \frac{e^{-1 \cdot \frac{\log \left(\frac{1}{x}\right)}{n}} \cdot \left(0.5 \cdot \frac{1}{{n}^{2}} - 0.5 \cdot \frac{1}{n}\right)}{x}}{x}} \]
   4. Step-by-step derivation

   5. Simplified 81.9%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left({x}^{\left(\frac{1}{n}\right)}, \frac{\mathsf{fma}\left(0.5, {n}^{-2}, \frac{-0.5}{n}\right)}{x}, \frac{{x}^{\left(\frac{1}{n}\right)}}{n}\right)}{x}} \]

   6. Taylor expanded in n around inf 98.2%

      \[\leadsto \frac{\mathsf{fma}\left({x}^{\left(\frac{1}{n}\right)}, \color{blue}{\frac{-0.5}{n \cdot x}}, \frac{{x}^{\left(\frac{1}{n}\right)}}{n}\right)}{x} \]
   7. Step-by-step derivation

      1. *-commutative 98.2%

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

   8. Simplified 98.2%

      \[\leadsto \frac{\mathsf{fma}\left({x}^{\left(\frac{1}{n}\right)}, \color{blue}{\frac{-0.5}{x \cdot n}}, \frac{{x}^{\left(\frac{1}{n}\right)}}{n}\right)}{x} \]
   9. Step-by-step derivation

      1. expm1-log1p-u 98.2%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\mathsf{fma}\left({x}^{\left(\frac{1}{n}\right)}, \frac{-0.5}{x \cdot n}, \frac{{x}^{\left(\frac{1}{n}\right)}}{n}\right)}{x}\right)\right)} \]

      2. expm1-undefine 63.0%

         \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{\mathsf{fma}\left({x}^{\left(\frac{1}{n}\right)}, \frac{-0.5}{x \cdot n}, \frac{{x}^{\left(\frac{1}{n}\right)}}{n}\right)}{x}\right)} - 1} \]

   10. Applied egg-rr 63.0%

       \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{\mathsf{fma}\left({x}^{\left(\frac{1}{n}\right)}, \frac{-0.5}{x \cdot n}, \frac{{x}^{\left(\frac{1}{n}\right)}}{n}\right)}{x}\right)} - 1} \]
   11. Step-by-step derivation

       1. expm1-define 98.2%

          \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\mathsf{fma}\left({x}^{\left(\frac{1}{n}\right)}, \frac{-0.5}{x \cdot n}, \frac{{x}^{\left(\frac{1}{n}\right)}}{n}\right)}{x}\right)\right)} \]

       2. *-commutative 98.2%

          \[\leadsto \mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\mathsf{fma}\left({x}^{\left(\frac{1}{n}\right)}, \frac{-0.5}{\color{blue}{n \cdot x}}, \frac{{x}^{\left(\frac{1}{n}\right)}}{n}\right)}{x}\right)\right) \]

       3. associate-/r* 98.2%

          \[\leadsto \mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\mathsf{fma}\left({x}^{\left(\frac{1}{n}\right)}, \color{blue}{\frac{\frac{-0.5}{n}}{x}}, \frac{{x}^{\left(\frac{1}{n}\right)}}{n}\right)}{x}\right)\right) \]

   12. Simplified 98.2%

       \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\mathsf{fma}\left({x}^{\left(\frac{1}{n}\right)}, \frac{\frac{-0.5}{n}}{x}, \frac{{x}^{\left(\frac{1}{n}\right)}}{n}\right)}{x}\right)\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 2: 87.4% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{if}\;x \leq 4:\\ \;\;\;\;\frac{\left(\mathsf{log1p}\left(x\right) + \frac{0.5 \cdot \left({\left(\mathsf{log1p}\left(x\right)\right)}^{2} - {\log x}^{2}\right) + \frac{0.16666666666666666 \cdot \left({\left(\mathsf{log1p}\left(x\right)\right)}^{3} - {\log x}^{3}\right)}{n}}{n}\right) - 2 \cdot \log \left(\sqrt{x}\right)}{n}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\mathsf{fma}\left(t\_0, \frac{\frac{-0.5}{n}}{x}, \frac{t\_0}{n}\right)}{x}\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (let* ((t_0 (pow x (/ 1.0 n))))
   (if (<= x 4.0)
     (/
      (-
       (+
        (log1p x)
        (/
         (+
          (* 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))
       (* 2.0 (log (sqrt x))))
      n)
     (expm1 (log1p (/ (fma t_0 (/ (/ -0.5 n) x) (/ t_0 n)) x))))))

C

double code(double x, double n) {
	double t_0 = pow(x, (1.0 / n));
	double tmp;
	if (x <= 4.0) {
		tmp = ((log1p(x) + (((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)) - (2.0 * log(sqrt(x)))) / n;
	} else {
		tmp = expm1(log1p((fma(t_0, ((-0.5 / n) / x), (t_0 / n)) / x)));
	}
	return tmp;
}

Julia

function code(x, n)
	t_0 = x ^ Float64(1.0 / n)
	tmp = 0.0
	if (x <= 4.0)
		tmp = Float64(Float64(Float64(log1p(x) + Float64(Float64(Float64(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(2.0 * log(sqrt(x)))) / n);
	else
		tmp = expm1(log1p(Float64(fma(t_0, Float64(Float64(-0.5 / n) / x), Float64(t_0 / n)) / x)));
	end
	return tmp
end

Wolfram

code[x_, n_] := Block[{t$95$0 = N[Power[x, N[(1.0 / n), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[x, 4.0], N[(N[(N[(N[Log[1 + x], $MachinePrecision] + 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]), $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]), $MachinePrecision] - N[(2.0 * N[Log[N[Sqrt[x], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / n), $MachinePrecision], N[(Exp[N[Log[1 + N[(N[(t$95$0 * N[(N[(-0.5 / n), $MachinePrecision] / x), $MachinePrecision] + N[(t$95$0 / n), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]], $MachinePrecision]] - 1), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if x < 4

   1. Initial program 43.4%

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

   3. Taylor expanded in n around -inf 79.7%

      \[\leadsto \color{blue}{-1 \cdot \frac{\left(-1 \cdot \log \left(1 + x\right) + -1 \cdot \frac{\left(-1 \cdot \frac{-0.16666666666666666 \cdot {\log \left(1 + x\right)}^{3} - -0.16666666666666666 \cdot {\log x}^{3}}{n} + 0.5 \cdot {\log \left(1 + x\right)}^{2}\right) - 0.5 \cdot {\log x}^{2}}{n}\right) - -1 \cdot \log x}{n}} \]

   5. Simplified 79.7%

      \[\leadsto \color{blue}{\frac{\log x - \left(\mathsf{log1p}\left(x\right) + \frac{0.5 \cdot \left({\left(\mathsf{log1p}\left(x\right)\right)}^{2} - {\log x}^{2}\right) + \frac{0.16666666666666666 \cdot \left({\left(\mathsf{log1p}\left(x\right)\right)}^{3} - {\log x}^{3}\right)}{n}}{n}\right)}{-n}} \]
   6. Step-by-step derivation

      1. add-sqr-sqrt 79.7%

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

      2. log-prod 79.7%

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

   7. Applied egg-rr 79.7%

      \[\leadsto \frac{\color{blue}{\left(\log \left(\sqrt{x}\right) + \log \left(\sqrt{x}\right)\right)} - \left(\mathsf{log1p}\left(x\right) + \frac{0.5 \cdot \left({\left(\mathsf{log1p}\left(x\right)\right)}^{2} - {\log x}^{2}\right) + \frac{0.16666666666666666 \cdot \left({\left(\mathsf{log1p}\left(x\right)\right)}^{3} - {\log x}^{3}\right)}{n}}{n}\right)}{-n} \]
   8. Step-by-step derivation

      1. count-2 79.7%

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

   9. Simplified 79.7%

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

   if 4 < x

   1. Initial program 62.2%

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

   3. Taylor expanded in x around inf 81.9%

      \[\leadsto \color{blue}{\frac{\frac{e^{-1 \cdot \frac{\log \left(\frac{1}{x}\right)}{n}}}{n} + \frac{e^{-1 \cdot \frac{\log \left(\frac{1}{x}\right)}{n}} \cdot \left(0.5 \cdot \frac{1}{{n}^{2}} - 0.5 \cdot \frac{1}{n}\right)}{x}}{x}} \]
   4. Step-by-step derivation

   5. Simplified 81.9%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left({x}^{\left(\frac{1}{n}\right)}, \frac{\mathsf{fma}\left(0.5, {n}^{-2}, \frac{-0.5}{n}\right)}{x}, \frac{{x}^{\left(\frac{1}{n}\right)}}{n}\right)}{x}} \]

   6. Taylor expanded in n around inf 98.2%

      \[\leadsto \frac{\mathsf{fma}\left({x}^{\left(\frac{1}{n}\right)}, \color{blue}{\frac{-0.5}{n \cdot x}}, \frac{{x}^{\left(\frac{1}{n}\right)}}{n}\right)}{x} \]
   7. Step-by-step derivation

      1. *-commutative 98.2%

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

   8. Simplified 98.2%

      \[\leadsto \frac{\mathsf{fma}\left({x}^{\left(\frac{1}{n}\right)}, \color{blue}{\frac{-0.5}{x \cdot n}}, \frac{{x}^{\left(\frac{1}{n}\right)}}{n}\right)}{x} \]
   9. Step-by-step derivation

      1. expm1-log1p-u 98.2%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\mathsf{fma}\left({x}^{\left(\frac{1}{n}\right)}, \frac{-0.5}{x \cdot n}, \frac{{x}^{\left(\frac{1}{n}\right)}}{n}\right)}{x}\right)\right)} \]

      2. expm1-undefine 63.0%

         \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{\mathsf{fma}\left({x}^{\left(\frac{1}{n}\right)}, \frac{-0.5}{x \cdot n}, \frac{{x}^{\left(\frac{1}{n}\right)}}{n}\right)}{x}\right)} - 1} \]

   10. Applied egg-rr 63.0%

       \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{\mathsf{fma}\left({x}^{\left(\frac{1}{n}\right)}, \frac{-0.5}{x \cdot n}, \frac{{x}^{\left(\frac{1}{n}\right)}}{n}\right)}{x}\right)} - 1} \]
   11. Step-by-step derivation

       1. expm1-define 98.2%

          \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\mathsf{fma}\left({x}^{\left(\frac{1}{n}\right)}, \frac{-0.5}{x \cdot n}, \frac{{x}^{\left(\frac{1}{n}\right)}}{n}\right)}{x}\right)\right)} \]

       2. *-commutative 98.2%

          \[\leadsto \mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\mathsf{fma}\left({x}^{\left(\frac{1}{n}\right)}, \frac{-0.5}{\color{blue}{n \cdot x}}, \frac{{x}^{\left(\frac{1}{n}\right)}}{n}\right)}{x}\right)\right) \]

       3. associate-/r* 98.2%

          \[\leadsto \mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\mathsf{fma}\left({x}^{\left(\frac{1}{n}\right)}, \color{blue}{\frac{\frac{-0.5}{n}}{x}}, \frac{{x}^{\left(\frac{1}{n}\right)}}{n}\right)}{x}\right)\right) \]

   12. Simplified 98.2%

       \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\mathsf{fma}\left({x}^{\left(\frac{1}{n}\right)}, \frac{\frac{-0.5}{n}}{x}, \frac{{x}^{\left(\frac{1}{n}\right)}}{n}\right)}{x}\right)\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 88.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 4:\\ \;\;\;\;\frac{\left(\mathsf{log1p}\left(x\right) + \frac{0.5 \cdot \left({\left(\mathsf{log1p}\left(x\right)\right)}^{2} - {\log x}^{2}\right) + \frac{0.16666666666666666 \cdot \left({\left(\mathsf{log1p}\left(x\right)\right)}^{3} - {\log x}^{3}\right)}{n}}{n}\right) - 2 \cdot \log \left(\sqrt{x}\right)}{n}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\mathsf{fma}\left({x}^{\left(\frac{1}{n}\right)}, \frac{\frac{-0.5}{n}}{x}, \frac{{x}^{\left(\frac{1}{n}\right)}}{n}\right)}{x}\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 87.4% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{if}\;x \leq 7.2:\\ \;\;\;\;\frac{\left(\mathsf{log1p}\left(x\right) + \frac{\frac{0.16666666666666666 \cdot \left({\left(\mathsf{log1p}\left(x\right)\right)}^{3} - {\log x}^{3}\right)}{n} + 0.5 \cdot \left({\left(x \cdot \left(x \cdot \left(x \cdot \left(0.3333333333333333 + x \cdot -0.25\right) - 0.5\right) + 1\right)\right)}^{2} - {\log x}^{2}\right)}{n}\right) - \log x}{n}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\mathsf{fma}\left(t\_0, \frac{\frac{-0.5}{n}}{x}, \frac{t\_0}{n}\right)}{x}\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (let* ((t_0 (pow x (/ 1.0 n))))
   (if (<= x 7.2)
     (/
      (-
       (+
        (log1p x)
        (/
         (+
          (/
           (* 0.16666666666666666 (- (pow (log1p x) 3.0) (pow (log x) 3.0)))
           n)
          (*
           0.5
           (-
            (pow
             (*
              x
              (+ (* x (- (* x (+ 0.3333333333333333 (* x -0.25))) 0.5)) 1.0))
             2.0)
            (pow (log x) 2.0))))
         n))
       (log x))
      n)
     (expm1 (log1p (/ (fma t_0 (/ (/ -0.5 n) x) (/ t_0 n)) x))))))

C

double code(double x, double n) {
	double t_0 = pow(x, (1.0 / n));
	double tmp;
	if (x <= 7.2) {
		tmp = ((log1p(x) + ((((0.16666666666666666 * (pow(log1p(x), 3.0) - pow(log(x), 3.0))) / n) + (0.5 * (pow((x * ((x * ((x * (0.3333333333333333 + (x * -0.25))) - 0.5)) + 1.0)), 2.0) - pow(log(x), 2.0)))) / n)) - log(x)) / n;
	} else {
		tmp = expm1(log1p((fma(t_0, ((-0.5 / n) / x), (t_0 / n)) / x)));
	}
	return tmp;
}

Julia

function code(x, n)
	t_0 = x ^ Float64(1.0 / n)
	tmp = 0.0
	if (x <= 7.2)
		tmp = Float64(Float64(Float64(log1p(x) + Float64(Float64(Float64(Float64(0.16666666666666666 * Float64((log1p(x) ^ 3.0) - (log(x) ^ 3.0))) / n) + Float64(0.5 * Float64((Float64(x * Float64(Float64(x * Float64(Float64(x * Float64(0.3333333333333333 + Float64(x * -0.25))) - 0.5)) + 1.0)) ^ 2.0) - (log(x) ^ 2.0)))) / n)) - log(x)) / n);
	else
		tmp = expm1(log1p(Float64(fma(t_0, Float64(Float64(-0.5 / n) / x), Float64(t_0 / n)) / x)));
	end
	return tmp
end

Wolfram

code[x_, n_] := Block[{t$95$0 = N[Power[x, N[(1.0 / n), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[x, 7.2], N[(N[(N[(N[Log[1 + x], $MachinePrecision] + N[(N[(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] + N[(0.5 * N[(N[Power[N[(x * N[(N[(x * N[(N[(x * N[(0.3333333333333333 + N[(x * -0.25), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 0.5), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] - N[Power[N[Log[x], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / n), $MachinePrecision]), $MachinePrecision] - N[Log[x], $MachinePrecision]), $MachinePrecision] / n), $MachinePrecision], N[(Exp[N[Log[1 + N[(N[(t$95$0 * N[(N[(-0.5 / n), $MachinePrecision] / x), $MachinePrecision] + N[(t$95$0 / n), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]], $MachinePrecision]] - 1), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if x < 7.20000000000000018

   1. Initial program 43.4%

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

   3. Taylor expanded in n around -inf 79.7%

      \[\leadsto \color{blue}{-1 \cdot \frac{\left(-1 \cdot \log \left(1 + x\right) + -1 \cdot \frac{\left(-1 \cdot \frac{-0.16666666666666666 \cdot {\log \left(1 + x\right)}^{3} - -0.16666666666666666 \cdot {\log x}^{3}}{n} + 0.5 \cdot {\log \left(1 + x\right)}^{2}\right) - 0.5 \cdot {\log x}^{2}}{n}\right) - -1 \cdot \log x}{n}} \]

   5. Simplified 79.7%

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

   6. Taylor expanded in x around 0 79.7%

      \[\leadsto \frac{\log x - \left(\mathsf{log1p}\left(x\right) + \frac{0.5 \cdot \left({\color{blue}{\left(x \cdot \left(1 + x \cdot \left(x \cdot \left(0.3333333333333333 + -0.25 \cdot x\right) - 0.5\right)\right)\right)}}^{2} - {\log x}^{2}\right) + \frac{0.16666666666666666 \cdot \left({\left(\mathsf{log1p}\left(x\right)\right)}^{3} - {\log x}^{3}\right)}{n}}{n}\right)}{-n} \]

   if 7.20000000000000018 < x

   1. Initial program 62.2%

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

   3. Taylor expanded in x around inf 81.9%

      \[\leadsto \color{blue}{\frac{\frac{e^{-1 \cdot \frac{\log \left(\frac{1}{x}\right)}{n}}}{n} + \frac{e^{-1 \cdot \frac{\log \left(\frac{1}{x}\right)}{n}} \cdot \left(0.5 \cdot \frac{1}{{n}^{2}} - 0.5 \cdot \frac{1}{n}\right)}{x}}{x}} \]
   4. Step-by-step derivation

   5. Simplified 81.9%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left({x}^{\left(\frac{1}{n}\right)}, \frac{\mathsf{fma}\left(0.5, {n}^{-2}, \frac{-0.5}{n}\right)}{x}, \frac{{x}^{\left(\frac{1}{n}\right)}}{n}\right)}{x}} \]

   6. Taylor expanded in n around inf 98.2%

      \[\leadsto \frac{\mathsf{fma}\left({x}^{\left(\frac{1}{n}\right)}, \color{blue}{\frac{-0.5}{n \cdot x}}, \frac{{x}^{\left(\frac{1}{n}\right)}}{n}\right)}{x} \]
   7. Step-by-step derivation

      1. *-commutative 98.2%

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

   8. Simplified 98.2%

      \[\leadsto \frac{\mathsf{fma}\left({x}^{\left(\frac{1}{n}\right)}, \color{blue}{\frac{-0.5}{x \cdot n}}, \frac{{x}^{\left(\frac{1}{n}\right)}}{n}\right)}{x} \]
   9. Step-by-step derivation

      1. expm1-log1p-u 98.2%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\mathsf{fma}\left({x}^{\left(\frac{1}{n}\right)}, \frac{-0.5}{x \cdot n}, \frac{{x}^{\left(\frac{1}{n}\right)}}{n}\right)}{x}\right)\right)} \]

      2. expm1-undefine 63.0%

         \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{\mathsf{fma}\left({x}^{\left(\frac{1}{n}\right)}, \frac{-0.5}{x \cdot n}, \frac{{x}^{\left(\frac{1}{n}\right)}}{n}\right)}{x}\right)} - 1} \]

   10. Applied egg-rr 63.0%

       \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{\mathsf{fma}\left({x}^{\left(\frac{1}{n}\right)}, \frac{-0.5}{x \cdot n}, \frac{{x}^{\left(\frac{1}{n}\right)}}{n}\right)}{x}\right)} - 1} \]
   11. Step-by-step derivation

       1. expm1-define 98.2%

          \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\mathsf{fma}\left({x}^{\left(\frac{1}{n}\right)}, \frac{-0.5}{x \cdot n}, \frac{{x}^{\left(\frac{1}{n}\right)}}{n}\right)}{x}\right)\right)} \]

       2. *-commutative 98.2%

          \[\leadsto \mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\mathsf{fma}\left({x}^{\left(\frac{1}{n}\right)}, \frac{-0.5}{\color{blue}{n \cdot x}}, \frac{{x}^{\left(\frac{1}{n}\right)}}{n}\right)}{x}\right)\right) \]

       3. associate-/r* 98.2%

          \[\leadsto \mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\mathsf{fma}\left({x}^{\left(\frac{1}{n}\right)}, \color{blue}{\frac{\frac{-0.5}{n}}{x}}, \frac{{x}^{\left(\frac{1}{n}\right)}}{n}\right)}{x}\right)\right) \]

   12. Simplified 98.2%

       \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\mathsf{fma}\left({x}^{\left(\frac{1}{n}\right)}, \frac{\frac{-0.5}{n}}{x}, \frac{{x}^{\left(\frac{1}{n}\right)}}{n}\right)}{x}\right)\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 88.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 7.2:\\ \;\;\;\;\frac{\left(\mathsf{log1p}\left(x\right) + \frac{\frac{0.16666666666666666 \cdot \left({\left(\mathsf{log1p}\left(x\right)\right)}^{3} - {\log x}^{3}\right)}{n} + 0.5 \cdot \left({\left(x \cdot \left(x \cdot \left(x \cdot \left(0.3333333333333333 + x \cdot -0.25\right) - 0.5\right) + 1\right)\right)}^{2} - {\log x}^{2}\right)}{n}\right) - \log x}{n}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\mathsf{fma}\left({x}^{\left(\frac{1}{n}\right)}, \frac{\frac{-0.5}{n}}{x}, \frac{{x}^{\left(\frac{1}{n}\right)}}{n}\right)}{x}\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 86.9% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{if}\;x \leq 1:\\ \;\;\;\;\frac{\frac{-0.16666666666666666 \cdot \frac{{\log x}^{3}}{n} + {\log x}^{2} \cdot -0.5}{n} - \log x}{n}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\mathsf{fma}\left(t\_0, \frac{\frac{-0.5}{n}}{x}, \frac{t\_0}{n}\right)}{x}\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (let* ((t_0 (pow x (/ 1.0 n))))
   (if (<= x 1.0)
     (/
      (-
       (/
        (+
         (* -0.16666666666666666 (/ (pow (log x) 3.0) n))
         (* (pow (log x) 2.0) -0.5))
        n)
       (log x))
      n)
     (expm1 (log1p (/ (fma t_0 (/ (/ -0.5 n) x) (/ t_0 n)) x))))))

C

double code(double x, double n) {
	double t_0 = pow(x, (1.0 / n));
	double tmp;
	if (x <= 1.0) {
		tmp = ((((-0.16666666666666666 * (pow(log(x), 3.0) / n)) + (pow(log(x), 2.0) * -0.5)) / n) - log(x)) / n;
	} else {
		tmp = expm1(log1p((fma(t_0, ((-0.5 / n) / x), (t_0 / n)) / x)));
	}
	return tmp;
}

Julia

function code(x, n)
	t_0 = x ^ Float64(1.0 / n)
	tmp = 0.0
	if (x <= 1.0)
		tmp = Float64(Float64(Float64(Float64(Float64(-0.16666666666666666 * Float64((log(x) ^ 3.0) / n)) + Float64((log(x) ^ 2.0) * -0.5)) / n) - log(x)) / n);
	else
		tmp = expm1(log1p(Float64(fma(t_0, Float64(Float64(-0.5 / n) / x), Float64(t_0 / n)) / x)));
	end
	return tmp
end

Wolfram

code[x_, n_] := Block[{t$95$0 = N[Power[x, N[(1.0 / n), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[x, 1.0], N[(N[(N[(N[(N[(-0.16666666666666666 * N[(N[Power[N[Log[x], $MachinePrecision], 3.0], $MachinePrecision] / n), $MachinePrecision]), $MachinePrecision] + N[(N[Power[N[Log[x], $MachinePrecision], 2.0], $MachinePrecision] * -0.5), $MachinePrecision]), $MachinePrecision] / n), $MachinePrecision] - N[Log[x], $MachinePrecision]), $MachinePrecision] / n), $MachinePrecision], N[(Exp[N[Log[1 + N[(N[(t$95$0 * N[(N[(-0.5 / n), $MachinePrecision] / x), $MachinePrecision] + N[(t$95$0 / n), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]], $MachinePrecision]] - 1), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if x < 1

   1. Initial program 43.4%

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

   3. Taylor expanded in x around 0 41.6%

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

      1. *-rgt-identity 41.6%

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

      2. associate-*l/ 41.6%

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

      3. associate-/l* 41.6%

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

      4. exp-to-pow 41.6%

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

   5. Simplified 41.6%

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

   6. Taylor expanded in n around -inf 79.5%

      \[\leadsto \color{blue}{-1 \cdot \frac{-1 \cdot \frac{-0.16666666666666666 \cdot \frac{{\log x}^{3}}{n} - 0.5 \cdot {\log x}^{2}}{n} - -1 \cdot \log x}{n}} \]
   7. Step-by-step derivation

      1. mul-1-neg 79.5%

         \[\leadsto \color{blue}{-\frac{-1 \cdot \frac{-0.16666666666666666 \cdot \frac{{\log x}^{3}}{n} - 0.5 \cdot {\log x}^{2}}{n} - -1 \cdot \log x}{n}} \]

   8. Simplified 79.5%

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

   if 1 < x

   1. Initial program 62.2%

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

   3. Taylor expanded in x around inf 81.9%

      \[\leadsto \color{blue}{\frac{\frac{e^{-1 \cdot \frac{\log \left(\frac{1}{x}\right)}{n}}}{n} + \frac{e^{-1 \cdot \frac{\log \left(\frac{1}{x}\right)}{n}} \cdot \left(0.5 \cdot \frac{1}{{n}^{2}} - 0.5 \cdot \frac{1}{n}\right)}{x}}{x}} \]
   4. Step-by-step derivation

   5. Simplified 81.9%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left({x}^{\left(\frac{1}{n}\right)}, \frac{\mathsf{fma}\left(0.5, {n}^{-2}, \frac{-0.5}{n}\right)}{x}, \frac{{x}^{\left(\frac{1}{n}\right)}}{n}\right)}{x}} \]

   6. Taylor expanded in n around inf 98.2%

      \[\leadsto \frac{\mathsf{fma}\left({x}^{\left(\frac{1}{n}\right)}, \color{blue}{\frac{-0.5}{n \cdot x}}, \frac{{x}^{\left(\frac{1}{n}\right)}}{n}\right)}{x} \]
   7. Step-by-step derivation

      1. *-commutative 98.2%

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

   8. Simplified 98.2%

      \[\leadsto \frac{\mathsf{fma}\left({x}^{\left(\frac{1}{n}\right)}, \color{blue}{\frac{-0.5}{x \cdot n}}, \frac{{x}^{\left(\frac{1}{n}\right)}}{n}\right)}{x} \]
   9. Step-by-step derivation

      1. expm1-log1p-u 98.2%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\mathsf{fma}\left({x}^{\left(\frac{1}{n}\right)}, \frac{-0.5}{x \cdot n}, \frac{{x}^{\left(\frac{1}{n}\right)}}{n}\right)}{x}\right)\right)} \]

      2. expm1-undefine 63.0%

         \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{\mathsf{fma}\left({x}^{\left(\frac{1}{n}\right)}, \frac{-0.5}{x \cdot n}, \frac{{x}^{\left(\frac{1}{n}\right)}}{n}\right)}{x}\right)} - 1} \]

   10. Applied egg-rr 63.0%

       \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{\mathsf{fma}\left({x}^{\left(\frac{1}{n}\right)}, \frac{-0.5}{x \cdot n}, \frac{{x}^{\left(\frac{1}{n}\right)}}{n}\right)}{x}\right)} - 1} \]
   11. Step-by-step derivation

       1. expm1-define 98.2%

          \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\mathsf{fma}\left({x}^{\left(\frac{1}{n}\right)}, \frac{-0.5}{x \cdot n}, \frac{{x}^{\left(\frac{1}{n}\right)}}{n}\right)}{x}\right)\right)} \]

       2. *-commutative 98.2%

          \[\leadsto \mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\mathsf{fma}\left({x}^{\left(\frac{1}{n}\right)}, \frac{-0.5}{\color{blue}{n \cdot x}}, \frac{{x}^{\left(\frac{1}{n}\right)}}{n}\right)}{x}\right)\right) \]

       3. associate-/r* 98.2%

          \[\leadsto \mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\mathsf{fma}\left({x}^{\left(\frac{1}{n}\right)}, \color{blue}{\frac{\frac{-0.5}{n}}{x}}, \frac{{x}^{\left(\frac{1}{n}\right)}}{n}\right)}{x}\right)\right) \]

   12. Simplified 98.2%

       \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\mathsf{fma}\left({x}^{\left(\frac{1}{n}\right)}, \frac{\frac{-0.5}{n}}{x}, \frac{{x}^{\left(\frac{1}{n}\right)}}{n}\right)}{x}\right)\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 88.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 1:\\ \;\;\;\;\frac{\frac{-0.16666666666666666 \cdot \frac{{\log x}^{3}}{n} + {\log x}^{2} \cdot -0.5}{n} - \log x}{n}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\mathsf{fma}\left({x}^{\left(\frac{1}{n}\right)}, \frac{\frac{-0.5}{n}}{x}, \frac{{x}^{\left(\frac{1}{n}\right)}}{n}\right)}{x}\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 85.1% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{if}\;\frac{1}{n} \leq -2 \cdot 10^{-99}:\\ \;\;\;\;\frac{t\_0}{x \cdot n}\\ \mathbf{elif}\;\frac{1}{n} \leq 10:\\ \;\;\;\;\frac{\log \left(x + 1\right) - 2 \cdot \log \left(\sqrt{x}\right)}{n}\\ \mathbf{else}:\\ \;\;\;\;e^{\frac{\mathsf{log1p}\left(x\right)}{n}} - t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (let* ((t_0 (pow x (/ 1.0 n))))
   (if (<= (/ 1.0 n) -2e-99)
     (/ t_0 (* x n))
     (if (<= (/ 1.0 n) 10.0)
       (/ (- (log (+ x 1.0)) (* 2.0 (log (sqrt x)))) n)
       (- (exp (/ (log1p x) n)) t_0)))))

C

double code(double x, double n) {
	double t_0 = pow(x, (1.0 / n));
	double tmp;
	if ((1.0 / n) <= -2e-99) {
		tmp = t_0 / (x * n);
	} else if ((1.0 / n) <= 10.0) {
		tmp = (log((x + 1.0)) - (2.0 * log(sqrt(x)))) / n;
	} else {
		tmp = exp((log1p(x) / n)) - t_0;
	}
	return tmp;
}

Java

public static double code(double x, double n) {
	double t_0 = Math.pow(x, (1.0 / n));
	double tmp;
	if ((1.0 / n) <= -2e-99) {
		tmp = t_0 / (x * n);
	} else if ((1.0 / n) <= 10.0) {
		tmp = (Math.log((x + 1.0)) - (2.0 * Math.log(Math.sqrt(x)))) / n;
	} else {
		tmp = Math.exp((Math.log1p(x) / n)) - t_0;
	}
	return tmp;
}

Python

def code(x, n):
	t_0 = math.pow(x, (1.0 / n))
	tmp = 0
	if (1.0 / n) <= -2e-99:
		tmp = t_0 / (x * n)
	elif (1.0 / n) <= 10.0:
		tmp = (math.log((x + 1.0)) - (2.0 * math.log(math.sqrt(x)))) / n
	else:
		tmp = math.exp((math.log1p(x) / n)) - t_0
	return tmp

Julia

function code(x, n)
	t_0 = x ^ Float64(1.0 / n)
	tmp = 0.0
	if (Float64(1.0 / n) <= -2e-99)
		tmp = Float64(t_0 / Float64(x * n));
	elseif (Float64(1.0 / n) <= 10.0)
		tmp = Float64(Float64(log(Float64(x + 1.0)) - Float64(2.0 * log(sqrt(x)))) / n);
	else
		tmp = Float64(exp(Float64(log1p(x) / n)) - t_0);
	end
	return tmp
end

Wolfram

code[x_, n_] := Block[{t$95$0 = N[Power[x, N[(1.0 / n), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[N[(1.0 / n), $MachinePrecision], -2e-99], N[(t$95$0 / N[(x * n), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 10.0], N[(N[(N[Log[N[(x + 1.0), $MachinePrecision]], $MachinePrecision] - N[(2.0 * N[Log[N[Sqrt[x], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / n), $MachinePrecision], N[(N[Exp[N[(N[Log[1 + x], $MachinePrecision] / n), $MachinePrecision]], $MachinePrecision] - t$95$0), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 #s(literal 1 binary64) n) < -2e-99

   1. Initial program 71.0%

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

   3. Taylor expanded in x around inf 85.2%

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

      1. log-rec 85.2%

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

      2. mul-1-neg 85.2%

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

      3. neg-mul-1 85.2%

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

      4. mul-1-neg 85.2%

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

      5. distribute-frac-neg 85.2%

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

      6. remove-double-neg 85.2%

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

      7. *-rgt-identity 85.2%

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

      8. associate-/l* 85.2%

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

      9. exp-to-pow 85.2%

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

      10. *-commutative 85.2%

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

   5. Simplified 85.2%

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

   if -2e-99 < (/.f64 #s(literal 1 binary64) n) < 10

   1. Initial program 33.8%

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

   3. Taylor expanded in n around -inf 79.9%

      \[\leadsto \color{blue}{-1 \cdot \frac{\left(-1 \cdot \log \left(1 + x\right) + -1 \cdot \frac{\left(-1 \cdot \frac{-0.16666666666666666 \cdot {\log \left(1 + x\right)}^{3} - -0.16666666666666666 \cdot {\log x}^{3}}{n} + 0.5 \cdot {\log \left(1 + x\right)}^{2}\right) - 0.5 \cdot {\log x}^{2}}{n}\right) - -1 \cdot \log x}{n}} \]

   5. Simplified 79.9%

      \[\leadsto \color{blue}{\frac{\log x - \left(\mathsf{log1p}\left(x\right) + \frac{0.5 \cdot \left({\left(\mathsf{log1p}\left(x\right)\right)}^{2} - {\log x}^{2}\right) + \frac{0.16666666666666666 \cdot \left({\left(\mathsf{log1p}\left(x\right)\right)}^{3} - {\log x}^{3}\right)}{n}}{n}\right)}{-n}} \]
   6. Step-by-step derivation

      1. add-sqr-sqrt 79.9%

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

      2. log-prod 79.9%

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

   7. Applied egg-rr 79.9%

      \[\leadsto \frac{\color{blue}{\left(\log \left(\sqrt{x}\right) + \log \left(\sqrt{x}\right)\right)} - \left(\mathsf{log1p}\left(x\right) + \frac{0.5 \cdot \left({\left(\mathsf{log1p}\left(x\right)\right)}^{2} - {\log x}^{2}\right) + \frac{0.16666666666666666 \cdot \left({\left(\mathsf{log1p}\left(x\right)\right)}^{3} - {\log x}^{3}\right)}{n}}{n}\right)}{-n} \]
   8. Step-by-step derivation

      1. count-2 79.9%

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

   9. Simplified 79.9%

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

   10. Taylor expanded in n around inf 79.7%

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

   if 10 < (/.f64 #s(literal 1 binary64) n)

   1. Initial program 58.9%

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

   3. Taylor expanded in n around 0 58.9%

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

      1. log1p-define 100.0%

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

      2. *-rgt-identity 100.0%

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

      3. associate-*l/ 100.0%

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

      4. associate-/l* 100.0%

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

      5. exp-to-pow 100.0%

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

   5. Simplified 100.0%

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

4. Final simplification 84.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{1}{n} \leq -2 \cdot 10^{-99}:\\ \;\;\;\;\frac{{x}^{\left(\frac{1}{n}\right)}}{x \cdot n}\\ \mathbf{elif}\;\frac{1}{n} \leq 10:\\ \;\;\;\;\frac{\log \left(x + 1\right) - 2 \cdot \log \left(\sqrt{x}\right)}{n}\\ \mathbf{else}:\\ \;\;\;\;e^{\frac{\mathsf{log1p}\left(x\right)}{n}} - {x}^{\left(\frac{1}{n}\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 85.1% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{if}\;\frac{1}{n} \leq -2 \cdot 10^{-99}:\\ \;\;\;\;\frac{t\_0}{x \cdot n}\\ \mathbf{elif}\;\frac{1}{n} \leq 10:\\ \;\;\;\;\frac{\mathsf{log1p}\left(x\right) - \log x}{n}\\ \mathbf{else}:\\ \;\;\;\;e^{\frac{\mathsf{log1p}\left(x\right)}{n}} - t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (let* ((t_0 (pow x (/ 1.0 n))))
   (if (<= (/ 1.0 n) -2e-99)
     (/ t_0 (* x n))
     (if (<= (/ 1.0 n) 10.0)
       (/ (- (log1p x) (log x)) n)
       (- (exp (/ (log1p x) n)) t_0)))))

C

double code(double x, double n) {
	double t_0 = pow(x, (1.0 / n));
	double tmp;
	if ((1.0 / n) <= -2e-99) {
		tmp = t_0 / (x * n);
	} else if ((1.0 / n) <= 10.0) {
		tmp = (log1p(x) - log(x)) / n;
	} else {
		tmp = exp((log1p(x) / n)) - t_0;
	}
	return tmp;
}

Java

public static double code(double x, double n) {
	double t_0 = Math.pow(x, (1.0 / n));
	double tmp;
	if ((1.0 / n) <= -2e-99) {
		tmp = t_0 / (x * n);
	} else if ((1.0 / n) <= 10.0) {
		tmp = (Math.log1p(x) - Math.log(x)) / n;
	} else {
		tmp = Math.exp((Math.log1p(x) / n)) - t_0;
	}
	return tmp;
}

Python

def code(x, n):
	t_0 = math.pow(x, (1.0 / n))
	tmp = 0
	if (1.0 / n) <= -2e-99:
		tmp = t_0 / (x * n)
	elif (1.0 / n) <= 10.0:
		tmp = (math.log1p(x) - math.log(x)) / n
	else:
		tmp = math.exp((math.log1p(x) / n)) - t_0
	return tmp

Julia

function code(x, n)
	t_0 = x ^ Float64(1.0 / n)
	tmp = 0.0
	if (Float64(1.0 / n) <= -2e-99)
		tmp = Float64(t_0 / Float64(x * n));
	elseif (Float64(1.0 / n) <= 10.0)
		tmp = Float64(Float64(log1p(x) - log(x)) / n);
	else
		tmp = Float64(exp(Float64(log1p(x) / n)) - t_0);
	end
	return tmp
end

Wolfram

code[x_, n_] := Block[{t$95$0 = N[Power[x, N[(1.0 / n), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[N[(1.0 / n), $MachinePrecision], -2e-99], N[(t$95$0 / N[(x * n), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 10.0], N[(N[(N[Log[1 + x], $MachinePrecision] - N[Log[x], $MachinePrecision]), $MachinePrecision] / n), $MachinePrecision], N[(N[Exp[N[(N[Log[1 + x], $MachinePrecision] / n), $MachinePrecision]], $MachinePrecision] - t$95$0), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 #s(literal 1 binary64) n) < -2e-99

   1. Initial program 71.0%

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

   3. Taylor expanded in x around inf 85.2%

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

      1. log-rec 85.2%

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

      2. mul-1-neg 85.2%

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

      3. neg-mul-1 85.2%

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

      4. mul-1-neg 85.2%

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

      5. distribute-frac-neg 85.2%

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

      6. remove-double-neg 85.2%

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

      7. *-rgt-identity 85.2%

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

      8. associate-/l* 85.2%

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

      9. exp-to-pow 85.2%

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

      10. *-commutative 85.2%

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

   5. Simplified 85.2%

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

   if -2e-99 < (/.f64 #s(literal 1 binary64) n) < 10

   1. Initial program 33.8%

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

   3. Taylor expanded in n around inf 79.6%

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

      1. log1p-define 79.6%

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

   5. Simplified 79.6%

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

   if 10 < (/.f64 #s(literal 1 binary64) n)

   1. Initial program 58.9%

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

   3. Taylor expanded in n around 0 58.9%

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

      1. log1p-define 100.0%

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

      2. *-rgt-identity 100.0%

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

      3. associate-*l/ 100.0%

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

      4. associate-/l* 100.0%

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

      5. exp-to-pow 100.0%

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

   5. Simplified 100.0%

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

Alternative 7: 81.9% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{if}\;\frac{1}{n} \leq -2 \cdot 10^{-99}:\\ \;\;\;\;\frac{t\_0}{x \cdot n}\\ \mathbf{elif}\;\frac{1}{n} \leq 10:\\ \;\;\;\;\frac{\mathsf{log1p}\left(x\right) - \log x}{n}\\ \mathbf{else}:\\ \;\;\;\;\left(x \cdot \left(\frac{1}{n} + x \cdot \left(0.5 \cdot \frac{1}{{n}^{2}} + 0.5 \cdot \frac{-1}{n}\right)\right) + 1\right) - t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (let* ((t_0 (pow x (/ 1.0 n))))
   (if (<= (/ 1.0 n) -2e-99)
     (/ t_0 (* x n))
     (if (<= (/ 1.0 n) 10.0)
       (/ (- (log1p x) (log x)) n)
       (-
        (+
         (*
          x
          (+
           (/ 1.0 n)
           (* x (+ (* 0.5 (/ 1.0 (pow n 2.0))) (* 0.5 (/ -1.0 n))))))
         1.0)
        t_0)))))

C

double code(double x, double n) {
	double t_0 = pow(x, (1.0 / n));
	double tmp;
	if ((1.0 / n) <= -2e-99) {
		tmp = t_0 / (x * n);
	} else if ((1.0 / n) <= 10.0) {
		tmp = (log1p(x) - log(x)) / n;
	} else {
		tmp = ((x * ((1.0 / n) + (x * ((0.5 * (1.0 / pow(n, 2.0))) + (0.5 * (-1.0 / n)))))) + 1.0) - t_0;
	}
	return tmp;
}

Java

public static double code(double x, double n) {
	double t_0 = Math.pow(x, (1.0 / n));
	double tmp;
	if ((1.0 / n) <= -2e-99) {
		tmp = t_0 / (x * n);
	} else if ((1.0 / n) <= 10.0) {
		tmp = (Math.log1p(x) - Math.log(x)) / n;
	} else {
		tmp = ((x * ((1.0 / n) + (x * ((0.5 * (1.0 / Math.pow(n, 2.0))) + (0.5 * (-1.0 / n)))))) + 1.0) - t_0;
	}
	return tmp;
}

Python

def code(x, n):
	t_0 = math.pow(x, (1.0 / n))
	tmp = 0
	if (1.0 / n) <= -2e-99:
		tmp = t_0 / (x * n)
	elif (1.0 / n) <= 10.0:
		tmp = (math.log1p(x) - math.log(x)) / n
	else:
		tmp = ((x * ((1.0 / n) + (x * ((0.5 * (1.0 / math.pow(n, 2.0))) + (0.5 * (-1.0 / n)))))) + 1.0) - t_0
	return tmp

Julia

function code(x, n)
	t_0 = x ^ Float64(1.0 / n)
	tmp = 0.0
	if (Float64(1.0 / n) <= -2e-99)
		tmp = Float64(t_0 / Float64(x * n));
	elseif (Float64(1.0 / n) <= 10.0)
		tmp = Float64(Float64(log1p(x) - log(x)) / n);
	else
		tmp = Float64(Float64(Float64(x * Float64(Float64(1.0 / n) + Float64(x * Float64(Float64(0.5 * Float64(1.0 / (n ^ 2.0))) + Float64(0.5 * Float64(-1.0 / n)))))) + 1.0) - t_0);
	end
	return tmp
end

Wolfram

code[x_, n_] := Block[{t$95$0 = N[Power[x, N[(1.0 / n), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[N[(1.0 / n), $MachinePrecision], -2e-99], N[(t$95$0 / N[(x * n), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 10.0], N[(N[(N[Log[1 + x], $MachinePrecision] - N[Log[x], $MachinePrecision]), $MachinePrecision] / n), $MachinePrecision], N[(N[(N[(x * N[(N[(1.0 / n), $MachinePrecision] + N[(x * N[(N[(0.5 * N[(1.0 / N[Power[n, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(0.5 * N[(-1.0 / n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision] - t$95$0), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 #s(literal 1 binary64) n) < -2e-99

   1. Initial program 71.0%

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

   3. Taylor expanded in x around inf 85.2%

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

      1. log-rec 85.2%

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

      2. mul-1-neg 85.2%

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

      3. neg-mul-1 85.2%

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

      4. mul-1-neg 85.2%

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

      5. distribute-frac-neg 85.2%

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

      6. remove-double-neg 85.2%

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

      7. *-rgt-identity 85.2%

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

      8. associate-/l* 85.2%

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

      9. exp-to-pow 85.2%

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

      10. *-commutative 85.2%

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

   5. Simplified 85.2%

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

   if -2e-99 < (/.f64 #s(literal 1 binary64) n) < 10

   1. Initial program 33.8%

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

   3. Taylor expanded in n around inf 79.6%

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

      1. log1p-define 79.6%

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

   5. Simplified 79.6%

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

   if 10 < (/.f64 #s(literal 1 binary64) n)

   1. Initial program 58.9%

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

   3. Taylor expanded in x around 0 82.4%

      \[\leadsto \color{blue}{\left(1 + x \cdot \left(x \cdot \left(0.5 \cdot \frac{1}{{n}^{2}} - 0.5 \cdot \frac{1}{n}\right) + \frac{1}{n}\right)\right)} - {x}^{\left(\frac{1}{n}\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 82.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{1}{n} \leq -2 \cdot 10^{-99}:\\ \;\;\;\;\frac{{x}^{\left(\frac{1}{n}\right)}}{x \cdot n}\\ \mathbf{elif}\;\frac{1}{n} \leq 10:\\ \;\;\;\;\frac{\mathsf{log1p}\left(x\right) - \log x}{n}\\ \mathbf{else}:\\ \;\;\;\;\left(x \cdot \left(\frac{1}{n} + x \cdot \left(0.5 \cdot \frac{1}{{n}^{2}} + 0.5 \cdot \frac{-1}{n}\right)\right) + 1\right) - {x}^{\left(\frac{1}{n}\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 81.9% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{if}\;\frac{1}{n} \leq -2 \cdot 10^{-99}:\\ \;\;\;\;\frac{t\_0}{x \cdot n}\\ \mathbf{elif}\;\frac{1}{n} \leq 10:\\ \;\;\;\;\frac{\mathsf{log1p}\left(x\right) - \log x}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 2 \cdot 10^{+145}:\\ \;\;\;\;\left(\frac{x}{n} + 1\right) - t\_0\\ \mathbf{else}:\\ \;\;\;\;{\left({\left(\frac{1 - \frac{0.5}{x}}{x \cdot n}\right)}^{3}\right)}^{0.3333333333333333}\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (let* ((t_0 (pow x (/ 1.0 n))))
   (if (<= (/ 1.0 n) -2e-99)
     (/ t_0 (* x n))
     (if (<= (/ 1.0 n) 10.0)
       (/ (- (log1p x) (log x)) n)
       (if (<= (/ 1.0 n) 2e+145)
         (- (+ (/ x n) 1.0) t_0)
         (pow (pow (/ (- 1.0 (/ 0.5 x)) (* x n)) 3.0) 0.3333333333333333))))))

C

double code(double x, double n) {
	double t_0 = pow(x, (1.0 / n));
	double tmp;
	if ((1.0 / n) <= -2e-99) {
		tmp = t_0 / (x * n);
	} else if ((1.0 / n) <= 10.0) {
		tmp = (log1p(x) - log(x)) / n;
	} else if ((1.0 / n) <= 2e+145) {
		tmp = ((x / n) + 1.0) - t_0;
	} else {
		tmp = pow(pow(((1.0 - (0.5 / x)) / (x * n)), 3.0), 0.3333333333333333);
	}
	return tmp;
}

Java

public static double code(double x, double n) {
	double t_0 = Math.pow(x, (1.0 / n));
	double tmp;
	if ((1.0 / n) <= -2e-99) {
		tmp = t_0 / (x * n);
	} else if ((1.0 / n) <= 10.0) {
		tmp = (Math.log1p(x) - Math.log(x)) / n;
	} else if ((1.0 / n) <= 2e+145) {
		tmp = ((x / n) + 1.0) - t_0;
	} else {
		tmp = Math.pow(Math.pow(((1.0 - (0.5 / x)) / (x * n)), 3.0), 0.3333333333333333);
	}
	return tmp;
}

Python

def code(x, n):
	t_0 = math.pow(x, (1.0 / n))
	tmp = 0
	if (1.0 / n) <= -2e-99:
		tmp = t_0 / (x * n)
	elif (1.0 / n) <= 10.0:
		tmp = (math.log1p(x) - math.log(x)) / n
	elif (1.0 / n) <= 2e+145:
		tmp = ((x / n) + 1.0) - t_0
	else:
		tmp = math.pow(math.pow(((1.0 - (0.5 / x)) / (x * n)), 3.0), 0.3333333333333333)
	return tmp

Julia

function code(x, n)
	t_0 = x ^ Float64(1.0 / n)
	tmp = 0.0
	if (Float64(1.0 / n) <= -2e-99)
		tmp = Float64(t_0 / Float64(x * n));
	elseif (Float64(1.0 / n) <= 10.0)
		tmp = Float64(Float64(log1p(x) - log(x)) / n);
	elseif (Float64(1.0 / n) <= 2e+145)
		tmp = Float64(Float64(Float64(x / n) + 1.0) - t_0);
	else
		tmp = (Float64(Float64(1.0 - Float64(0.5 / x)) / Float64(x * n)) ^ 3.0) ^ 0.3333333333333333;
	end
	return tmp
end

Wolfram

code[x_, n_] := Block[{t$95$0 = N[Power[x, N[(1.0 / n), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[N[(1.0 / n), $MachinePrecision], -2e-99], N[(t$95$0 / N[(x * n), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 10.0], N[(N[(N[Log[1 + x], $MachinePrecision] - N[Log[x], $MachinePrecision]), $MachinePrecision] / n), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 2e+145], N[(N[(N[(x / n), $MachinePrecision] + 1.0), $MachinePrecision] - t$95$0), $MachinePrecision], N[Power[N[Power[N[(N[(1.0 - N[(0.5 / x), $MachinePrecision]), $MachinePrecision] / N[(x * n), $MachinePrecision]), $MachinePrecision], 3.0], $MachinePrecision], 0.3333333333333333], $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if (/.f64 #s(literal 1 binary64) n) < -2e-99

   1. Initial program 71.0%

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

   3. Taylor expanded in x around inf 85.2%

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

      1. log-rec 85.2%

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

      2. mul-1-neg 85.2%

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

      3. neg-mul-1 85.2%

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

      4. mul-1-neg 85.2%

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

      5. distribute-frac-neg 85.2%

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

      6. remove-double-neg 85.2%

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

      7. *-rgt-identity 85.2%

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

      8. associate-/l* 85.2%

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

      9. exp-to-pow 85.2%

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

      10. *-commutative 85.2%

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

   5. Simplified 85.2%

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

   if -2e-99 < (/.f64 #s(literal 1 binary64) n) < 10

   1. Initial program 33.8%

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

   3. Taylor expanded in n around inf 79.6%

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

      1. log1p-define 79.6%

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

   5. Simplified 79.6%

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

   if 10 < (/.f64 #s(literal 1 binary64) n) < 2e145

   1. Initial program 83.6%

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

   3. Taylor expanded in x around 0 83.8%

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

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

   1. Initial program 30.8%

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

   3. Taylor expanded in x around inf 0.0%

      \[\leadsto \color{blue}{\frac{\frac{e^{-1 \cdot \frac{\log \left(\frac{1}{x}\right)}{n}}}{n} + \frac{e^{-1 \cdot \frac{\log \left(\frac{1}{x}\right)}{n}} \cdot \left(0.5 \cdot \frac{1}{{n}^{2}} - 0.5 \cdot \frac{1}{n}\right)}{x}}{x}} \]
   4. Step-by-step derivation

   5. Simplified 0.0%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left({x}^{\left(\frac{1}{n}\right)}, \frac{\mathsf{fma}\left(0.5, {n}^{-2}, \frac{-0.5}{n}\right)}{x}, \frac{{x}^{\left(\frac{1}{n}\right)}}{n}\right)}{x}} \]

   6. Taylor expanded in n around inf 0.2%

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

      1. associate-*r/ 0.2%

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

      2. metadata-eval 0.2%

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

      3. *-commutative 0.2%

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

   8. Simplified 0.2%

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

      1. add-cbrt-cube 0.1%

         \[\leadsto \color{blue}{\sqrt[3]{\left(\frac{1 - \frac{0.5}{x}}{x \cdot n} \cdot \frac{1 - \frac{0.5}{x}}{x \cdot n}\right) \cdot \frac{1 - \frac{0.5}{x}}{x \cdot n}}} \]

      2. pow1/3 80.6%

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

      3. pow3 80.6%

         \[\leadsto {\color{blue}{\left({\left(\frac{1 - \frac{0.5}{x}}{x \cdot n}\right)}^{3}\right)}}^{0.3333333333333333} \]

   10. Applied egg-rr 80.6%

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

4. Final simplification 82.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{1}{n} \leq -2 \cdot 10^{-99}:\\ \;\;\;\;\frac{{x}^{\left(\frac{1}{n}\right)}}{x \cdot n}\\ \mathbf{elif}\;\frac{1}{n} \leq 10:\\ \;\;\;\;\frac{\mathsf{log1p}\left(x\right) - \log x}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 2 \cdot 10^{+145}:\\ \;\;\;\;\left(\frac{x}{n} + 1\right) - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{else}:\\ \;\;\;\;{\left({\left(\frac{1 - \frac{0.5}{x}}{x \cdot n}\right)}^{3}\right)}^{0.3333333333333333}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 81.9% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{if}\;\frac{1}{n} \leq -2 \cdot 10^{-99}:\\ \;\;\;\;\frac{t\_0}{x \cdot n}\\ \mathbf{elif}\;\frac{1}{n} \leq 10:\\ \;\;\;\;\frac{\mathsf{log1p}\left(x\right) - \log x}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 2 \cdot 10^{+145}:\\ \;\;\;\;\left(\frac{x}{n} + 1\right) - t\_0\\ \mathbf{else}:\\ \;\;\;\;\mathsf{log1p}\left(\mathsf{expm1}\left(\frac{1}{x \cdot n}\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (let* ((t_0 (pow x (/ 1.0 n))))
   (if (<= (/ 1.0 n) -2e-99)
     (/ t_0 (* x n))
     (if (<= (/ 1.0 n) 10.0)
       (/ (- (log1p x) (log x)) n)
       (if (<= (/ 1.0 n) 2e+145)
         (- (+ (/ x n) 1.0) t_0)
         (log1p (expm1 (/ 1.0 (* x n)))))))))

C

double code(double x, double n) {
	double t_0 = pow(x, (1.0 / n));
	double tmp;
	if ((1.0 / n) <= -2e-99) {
		tmp = t_0 / (x * n);
	} else if ((1.0 / n) <= 10.0) {
		tmp = (log1p(x) - log(x)) / n;
	} else if ((1.0 / n) <= 2e+145) {
		tmp = ((x / n) + 1.0) - t_0;
	} else {
		tmp = log1p(expm1((1.0 / (x * n))));
	}
	return tmp;
}

Java

public static double code(double x, double n) {
	double t_0 = Math.pow(x, (1.0 / n));
	double tmp;
	if ((1.0 / n) <= -2e-99) {
		tmp = t_0 / (x * n);
	} else if ((1.0 / n) <= 10.0) {
		tmp = (Math.log1p(x) - Math.log(x)) / n;
	} else if ((1.0 / n) <= 2e+145) {
		tmp = ((x / n) + 1.0) - t_0;
	} else {
		tmp = Math.log1p(Math.expm1((1.0 / (x * n))));
	}
	return tmp;
}

Python

def code(x, n):
	t_0 = math.pow(x, (1.0 / n))
	tmp = 0
	if (1.0 / n) <= -2e-99:
		tmp = t_0 / (x * n)
	elif (1.0 / n) <= 10.0:
		tmp = (math.log1p(x) - math.log(x)) / n
	elif (1.0 / n) <= 2e+145:
		tmp = ((x / n) + 1.0) - t_0
	else:
		tmp = math.log1p(math.expm1((1.0 / (x * n))))
	return tmp

Julia

function code(x, n)
	t_0 = x ^ Float64(1.0 / n)
	tmp = 0.0
	if (Float64(1.0 / n) <= -2e-99)
		tmp = Float64(t_0 / Float64(x * n));
	elseif (Float64(1.0 / n) <= 10.0)
		tmp = Float64(Float64(log1p(x) - log(x)) / n);
	elseif (Float64(1.0 / n) <= 2e+145)
		tmp = Float64(Float64(Float64(x / n) + 1.0) - t_0);
	else
		tmp = log1p(expm1(Float64(1.0 / Float64(x * n))));
	end
	return tmp
end

Wolfram

code[x_, n_] := Block[{t$95$0 = N[Power[x, N[(1.0 / n), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[N[(1.0 / n), $MachinePrecision], -2e-99], N[(t$95$0 / N[(x * n), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 10.0], N[(N[(N[Log[1 + x], $MachinePrecision] - N[Log[x], $MachinePrecision]), $MachinePrecision] / n), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 2e+145], N[(N[(N[(x / n), $MachinePrecision] + 1.0), $MachinePrecision] - t$95$0), $MachinePrecision], N[Log[1 + N[(Exp[N[(1.0 / N[(x * n), $MachinePrecision]), $MachinePrecision]] - 1), $MachinePrecision]], $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if (/.f64 #s(literal 1 binary64) n) < -2e-99

   1. Initial program 71.0%

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

   3. Taylor expanded in x around inf 85.2%

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

      1. log-rec 85.2%

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

      2. mul-1-neg 85.2%

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

      3. neg-mul-1 85.2%

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

      4. mul-1-neg 85.2%

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

      5. distribute-frac-neg 85.2%

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

      6. remove-double-neg 85.2%

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

      7. *-rgt-identity 85.2%

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

      8. associate-/l* 85.2%

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

      9. exp-to-pow 85.2%

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

      10. *-commutative 85.2%

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

   5. Simplified 85.2%

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

   if -2e-99 < (/.f64 #s(literal 1 binary64) n) < 10

   1. Initial program 33.8%

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

   3. Taylor expanded in n around inf 79.6%

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

      1. log1p-define 79.6%

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

   5. Simplified 79.6%

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

   if 10 < (/.f64 #s(literal 1 binary64) n) < 2e145

   1. Initial program 83.6%

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

   3. Taylor expanded in x around 0 83.8%

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

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

   1. Initial program 30.8%

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

   3. Taylor expanded in x around inf 0.0%

      \[\leadsto \color{blue}{\frac{\frac{e^{-1 \cdot \frac{\log \left(\frac{1}{x}\right)}{n}}}{n} + \frac{e^{-1 \cdot \frac{\log \left(\frac{1}{x}\right)}{n}} \cdot \left(0.5 \cdot \frac{1}{{n}^{2}} - 0.5 \cdot \frac{1}{n}\right)}{x}}{x}} \]
   4. Step-by-step derivation

   5. Simplified 0.0%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left({x}^{\left(\frac{1}{n}\right)}, \frac{\mathsf{fma}\left(0.5, {n}^{-2}, \frac{-0.5}{n}\right)}{x}, \frac{{x}^{\left(\frac{1}{n}\right)}}{n}\right)}{x}} \]

   6. Taylor expanded in n around inf 0.2%

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

      1. associate-*r/ 0.2%

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

      2. metadata-eval 0.2%

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

      3. *-commutative 0.2%

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

   8. Simplified 0.2%

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

   9. Taylor expanded in x around inf 50.0%

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

       1. log1p-expm1-u 80.6%

          \[\leadsto \color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(\frac{1}{x \cdot n}\right)\right)} \]

   11. Applied egg-rr 80.6%

       \[\leadsto \color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(\frac{1}{x \cdot n}\right)\right)} \]
3. Recombined 4 regimes into one program.

4. Final simplification 82.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{1}{n} \leq -2 \cdot 10^{-99}:\\ \;\;\;\;\frac{{x}^{\left(\frac{1}{n}\right)}}{x \cdot n}\\ \mathbf{elif}\;\frac{1}{n} \leq 10:\\ \;\;\;\;\frac{\mathsf{log1p}\left(x\right) - \log x}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 2 \cdot 10^{+145}:\\ \;\;\;\;\left(\frac{x}{n} + 1\right) - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{log1p}\left(\mathsf{expm1}\left(\frac{1}{x \cdot n}\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 70.6% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {x}^{\left(\frac{1}{n}\right)}\\ t_1 := \frac{\log x}{-n}\\ \mathbf{if}\;x \leq 8.2 \cdot 10^{-291}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;x \leq 1.4 \cdot 10^{-221}:\\ \;\;\;\;\left(\frac{x}{n} + 1\right) - t\_0\\ \mathbf{elif}\;x \leq 7 \cdot 10^{-16}:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;\frac{t\_0}{x \cdot n}\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (let* ((t_0 (pow x (/ 1.0 n))) (t_1 (/ (log x) (- n))))
   (if (<= x 8.2e-291)
     t_1
     (if (<= x 1.4e-221)
       (- (+ (/ x n) 1.0) t_0)
       (if (<= x 7e-16) t_1 (/ t_0 (* x n)))))))

C

double code(double x, double n) {
	double t_0 = pow(x, (1.0 / n));
	double t_1 = log(x) / -n;
	double tmp;
	if (x <= 8.2e-291) {
		tmp = t_1;
	} else if (x <= 1.4e-221) {
		tmp = ((x / n) + 1.0) - t_0;
	} else if (x <= 7e-16) {
		tmp = t_1;
	} else {
		tmp = t_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 = log(x) / -n
    if (x <= 8.2d-291) then
        tmp = t_1
    else if (x <= 1.4d-221) then
        tmp = ((x / n) + 1.0d0) - t_0
    else if (x <= 7d-16) then
        tmp = t_1
    else
        tmp = t_0 / (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.log(x) / -n;
	double tmp;
	if (x <= 8.2e-291) {
		tmp = t_1;
	} else if (x <= 1.4e-221) {
		tmp = ((x / n) + 1.0) - t_0;
	} else if (x <= 7e-16) {
		tmp = t_1;
	} else {
		tmp = t_0 / (x * n);
	}
	return tmp;
}

Python

def code(x, n):
	t_0 = math.pow(x, (1.0 / n))
	t_1 = math.log(x) / -n
	tmp = 0
	if x <= 8.2e-291:
		tmp = t_1
	elif x <= 1.4e-221:
		tmp = ((x / n) + 1.0) - t_0
	elif x <= 7e-16:
		tmp = t_1
	else:
		tmp = t_0 / (x * n)
	return tmp

Julia

function code(x, n)
	t_0 = x ^ Float64(1.0 / n)
	t_1 = Float64(log(x) / Float64(-n))
	tmp = 0.0
	if (x <= 8.2e-291)
		tmp = t_1;
	elseif (x <= 1.4e-221)
		tmp = Float64(Float64(Float64(x / n) + 1.0) - t_0);
	elseif (x <= 7e-16)
		tmp = t_1;
	else
		tmp = Float64(t_0 / Float64(x * n));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, n)
	t_0 = x ^ (1.0 / n);
	t_1 = log(x) / -n;
	tmp = 0.0;
	if (x <= 8.2e-291)
		tmp = t_1;
	elseif (x <= 1.4e-221)
		tmp = ((x / n) + 1.0) - t_0;
	elseif (x <= 7e-16)
		tmp = t_1;
	else
		tmp = t_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[Log[x], $MachinePrecision] / (-n)), $MachinePrecision]}, If[LessEqual[x, 8.2e-291], t$95$1, If[LessEqual[x, 1.4e-221], N[(N[(N[(x / n), $MachinePrecision] + 1.0), $MachinePrecision] - t$95$0), $MachinePrecision], If[LessEqual[x, 7e-16], t$95$1, N[(t$95$0 / N[(x * n), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if x < 8.200000000000001e-291 or 1.4000000000000001e-221 < x < 7.00000000000000035e-16

   1. Initial program 31.4%

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

   3. Taylor expanded in x around 0 31.4%

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

      1. *-rgt-identity 31.4%

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

      2. associate-*l/ 31.4%

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

      3. associate-/l* 31.4%

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

      4. exp-to-pow 31.4%

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

   5. Simplified 31.4%

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

   6. Taylor expanded in n around inf 62.2%

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

      1. associate-*r/ 62.2%

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

      2. neg-mul-1 62.2%

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

   8. Simplified 62.2%

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

   if 8.200000000000001e-291 < x < 1.4000000000000001e-221

   1. Initial program 75.3%

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

   3. Taylor expanded in x around 0 76.5%

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

   if 7.00000000000000035e-16 < x

   1. Initial program 63.9%

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

   3. Taylor expanded in x around inf 93.6%

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

      1. log-rec 93.6%

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

      2. mul-1-neg 93.6%

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

      3. neg-mul-1 93.6%

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

      4. mul-1-neg 93.6%

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

      5. distribute-frac-neg 93.6%

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

      6. remove-double-neg 93.6%

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

      7. *-rgt-identity 93.6%

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

      8. associate-/l* 93.6%

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

      9. exp-to-pow 93.7%

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

      10. *-commutative 93.7%

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

   5. Simplified 93.7%

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

4. Final simplification 79.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 8.2 \cdot 10^{-291}:\\ \;\;\;\;\frac{\log x}{-n}\\ \mathbf{elif}\;x \leq 1.4 \cdot 10^{-221}:\\ \;\;\;\;\left(\frac{x}{n} + 1\right) - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{elif}\;x \leq 7 \cdot 10^{-16}:\\ \;\;\;\;\frac{\log x}{-n}\\ \mathbf{else}:\\ \;\;\;\;\frac{{x}^{\left(\frac{1}{n}\right)}}{x \cdot n}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 70.4% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\log x}{-n}\\ t_1 := {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{if}\;x \leq 5.5 \cdot 10^{-291}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq 2.6 \cdot 10^{-222}:\\ \;\;\;\;1 - t\_1\\ \mathbf{elif}\;x \leq 1.2 \cdot 10^{-18}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{t\_1}{x \cdot n}\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (let* ((t_0 (/ (log x) (- n))) (t_1 (pow x (/ 1.0 n))))
   (if (<= x 5.5e-291)
     t_0
     (if (<= x 2.6e-222)
       (- 1.0 t_1)
       (if (<= x 1.2e-18) t_0 (/ t_1 (* x n)))))))

C

double code(double x, double n) {
	double t_0 = log(x) / -n;
	double t_1 = pow(x, (1.0 / n));
	double tmp;
	if (x <= 5.5e-291) {
		tmp = t_0;
	} else if (x <= 2.6e-222) {
		tmp = 1.0 - t_1;
	} else if (x <= 1.2e-18) {
		tmp = t_0;
	} else {
		tmp = t_1 / (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 = log(x) / -n
    t_1 = x ** (1.0d0 / n)
    if (x <= 5.5d-291) then
        tmp = t_0
    else if (x <= 2.6d-222) then
        tmp = 1.0d0 - t_1
    else if (x <= 1.2d-18) then
        tmp = t_0
    else
        tmp = t_1 / (x * n)
    end if
    code = tmp
end function

Java

public static double code(double x, double n) {
	double t_0 = Math.log(x) / -n;
	double t_1 = Math.pow(x, (1.0 / n));
	double tmp;
	if (x <= 5.5e-291) {
		tmp = t_0;
	} else if (x <= 2.6e-222) {
		tmp = 1.0 - t_1;
	} else if (x <= 1.2e-18) {
		tmp = t_0;
	} else {
		tmp = t_1 / (x * n);
	}
	return tmp;
}

Python

def code(x, n):
	t_0 = math.log(x) / -n
	t_1 = math.pow(x, (1.0 / n))
	tmp = 0
	if x <= 5.5e-291:
		tmp = t_0
	elif x <= 2.6e-222:
		tmp = 1.0 - t_1
	elif x <= 1.2e-18:
		tmp = t_0
	else:
		tmp = t_1 / (x * n)
	return tmp

Julia

function code(x, n)
	t_0 = Float64(log(x) / Float64(-n))
	t_1 = x ^ Float64(1.0 / n)
	tmp = 0.0
	if (x <= 5.5e-291)
		tmp = t_0;
	elseif (x <= 2.6e-222)
		tmp = Float64(1.0 - t_1);
	elseif (x <= 1.2e-18)
		tmp = t_0;
	else
		tmp = Float64(t_1 / Float64(x * n));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, n)
	t_0 = log(x) / -n;
	t_1 = x ^ (1.0 / n);
	tmp = 0.0;
	if (x <= 5.5e-291)
		tmp = t_0;
	elseif (x <= 2.6e-222)
		tmp = 1.0 - t_1;
	elseif (x <= 1.2e-18)
		tmp = t_0;
	else
		tmp = t_1 / (x * n);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, n_] := Block[{t$95$0 = N[(N[Log[x], $MachinePrecision] / (-n)), $MachinePrecision]}, Block[{t$95$1 = N[Power[x, N[(1.0 / n), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[x, 5.5e-291], t$95$0, If[LessEqual[x, 2.6e-222], N[(1.0 - t$95$1), $MachinePrecision], If[LessEqual[x, 1.2e-18], t$95$0, N[(t$95$1 / N[(x * n), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if x < 5.5000000000000002e-291 or 2.5999999999999998e-222 < x < 1.19999999999999997e-18

   1. Initial program 31.4%

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

   3. Taylor expanded in x around 0 31.4%

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

      1. *-rgt-identity 31.4%

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

      2. associate-*l/ 31.4%

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

      3. associate-/l* 31.4%

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

      4. exp-to-pow 31.4%

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

   5. Simplified 31.4%

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

   6. Taylor expanded in n around inf 62.2%

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

      1. associate-*r/ 62.2%

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

      2. neg-mul-1 62.2%

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

   8. Simplified 62.2%

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

   if 5.5000000000000002e-291 < x < 2.5999999999999998e-222

   1. Initial program 75.3%

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

   3. Taylor expanded in x around 0 75.3%

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

      1. *-rgt-identity 75.3%

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

      2. associate-*l/ 75.3%

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

      3. associate-/l* 75.3%

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

      4. exp-to-pow 75.3%

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

   5. Simplified 75.3%

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

   if 1.19999999999999997e-18 < x

   1. Initial program 63.9%

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

   3. Taylor expanded in x around inf 93.6%

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

      1. log-rec 93.6%

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

      2. mul-1-neg 93.6%

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

      3. neg-mul-1 93.6%

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

      4. mul-1-neg 93.6%

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

      5. distribute-frac-neg 93.6%

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

      6. remove-double-neg 93.6%

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

      7. *-rgt-identity 93.6%

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

      8. associate-/l* 93.6%

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

      9. exp-to-pow 93.7%

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

      10. *-commutative 93.7%

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

   5. Simplified 93.7%

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

4. Final simplification 78.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 5.5 \cdot 10^{-291}:\\ \;\;\;\;\frac{\log x}{-n}\\ \mathbf{elif}\;x \leq 2.6 \cdot 10^{-222}:\\ \;\;\;\;1 - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{elif}\;x \leq 1.2 \cdot 10^{-18}:\\ \;\;\;\;\frac{\log x}{-n}\\ \mathbf{else}:\\ \;\;\;\;\frac{{x}^{\left(\frac{1}{n}\right)}}{x \cdot n}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 59.7% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\log x}{-n}\\ \mathbf{if}\;x \leq 7 \cdot 10^{-291}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq 2.6 \cdot 10^{-222}:\\ \;\;\;\;1 - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{elif}\;x \leq 0.68:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq 2.3 \cdot 10^{+218}:\\ \;\;\;\;\frac{\frac{\frac{-0.5}{x} + 1}{x}}{n}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{-0.5}{n}}{x \cdot x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (let* ((t_0 (/ (log x) (- n))))
   (if (<= x 7e-291)
     t_0
     (if (<= x 2.6e-222)
       (- 1.0 (pow x (/ 1.0 n)))
       (if (<= x 0.68)
         t_0
         (if (<= x 2.3e+218)
           (/ (/ (+ (/ -0.5 x) 1.0) x) n)
           (/ (/ -0.5 n) (* x x))))))))

C

double code(double x, double n) {
	double t_0 = log(x) / -n;
	double tmp;
	if (x <= 7e-291) {
		tmp = t_0;
	} else if (x <= 2.6e-222) {
		tmp = 1.0 - pow(x, (1.0 / n));
	} else if (x <= 0.68) {
		tmp = t_0;
	} else if (x <= 2.3e+218) {
		tmp = (((-0.5 / x) + 1.0) / x) / n;
	} else {
		tmp = (-0.5 / n) / (x * x);
	}
	return tmp;
}

Fortran

real(8) function code(x, n)
    real(8), intent (in) :: x
    real(8), intent (in) :: n
    real(8) :: t_0
    real(8) :: tmp
    t_0 = log(x) / -n
    if (x <= 7d-291) then
        tmp = t_0
    else if (x <= 2.6d-222) then
        tmp = 1.0d0 - (x ** (1.0d0 / n))
    else if (x <= 0.68d0) then
        tmp = t_0
    else if (x <= 2.3d+218) then
        tmp = ((((-0.5d0) / x) + 1.0d0) / x) / n
    else
        tmp = ((-0.5d0) / n) / (x * x)
    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 <= 7e-291) {
		tmp = t_0;
	} else if (x <= 2.6e-222) {
		tmp = 1.0 - Math.pow(x, (1.0 / n));
	} else if (x <= 0.68) {
		tmp = t_0;
	} else if (x <= 2.3e+218) {
		tmp = (((-0.5 / x) + 1.0) / x) / n;
	} else {
		tmp = (-0.5 / n) / (x * x);
	}
	return tmp;
}

Python

def code(x, n):
	t_0 = math.log(x) / -n
	tmp = 0
	if x <= 7e-291:
		tmp = t_0
	elif x <= 2.6e-222:
		tmp = 1.0 - math.pow(x, (1.0 / n))
	elif x <= 0.68:
		tmp = t_0
	elif x <= 2.3e+218:
		tmp = (((-0.5 / x) + 1.0) / x) / n
	else:
		tmp = (-0.5 / n) / (x * x)
	return tmp

Julia

function code(x, n)
	t_0 = Float64(log(x) / Float64(-n))
	tmp = 0.0
	if (x <= 7e-291)
		tmp = t_0;
	elseif (x <= 2.6e-222)
		tmp = Float64(1.0 - (x ^ Float64(1.0 / n)));
	elseif (x <= 0.68)
		tmp = t_0;
	elseif (x <= 2.3e+218)
		tmp = Float64(Float64(Float64(Float64(-0.5 / x) + 1.0) / x) / n);
	else
		tmp = Float64(Float64(-0.5 / n) / Float64(x * x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, n)
	t_0 = log(x) / -n;
	tmp = 0.0;
	if (x <= 7e-291)
		tmp = t_0;
	elseif (x <= 2.6e-222)
		tmp = 1.0 - (x ^ (1.0 / n));
	elseif (x <= 0.68)
		tmp = t_0;
	elseif (x <= 2.3e+218)
		tmp = (((-0.5 / x) + 1.0) / x) / n;
	else
		tmp = (-0.5 / n) / (x * x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, n_] := Block[{t$95$0 = N[(N[Log[x], $MachinePrecision] / (-n)), $MachinePrecision]}, If[LessEqual[x, 7e-291], t$95$0, If[LessEqual[x, 2.6e-222], N[(1.0 - N[Power[x, N[(1.0 / n), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 0.68], t$95$0, If[LessEqual[x, 2.3e+218], N[(N[(N[(N[(-0.5 / x), $MachinePrecision] + 1.0), $MachinePrecision] / x), $MachinePrecision] / n), $MachinePrecision], N[(N[(-0.5 / n), $MachinePrecision] / N[(x * x), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if x < 6.99999999999999991e-291 or 2.5999999999999998e-222 < x < 0.680000000000000049

   1. Initial program 35.4%

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

   3. Taylor expanded in x around 0 33.1%

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

      1. *-rgt-identity 33.1%

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

      2. associate-*l/ 33.1%

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

      3. associate-/l* 33.1%

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

      4. exp-to-pow 33.1%

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

   5. Simplified 33.1%

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

   6. Taylor expanded in n around inf 58.9%

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

      1. associate-*r/ 58.9%

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

      2. neg-mul-1 58.9%

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

   8. Simplified 58.9%

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

   if 6.99999999999999991e-291 < x < 2.5999999999999998e-222

   1. Initial program 75.3%

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

   3. Taylor expanded in x around 0 75.3%

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

      1. *-rgt-identity 75.3%

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

      2. associate-*l/ 75.3%

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

      3. associate-/l* 75.3%

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

      4. exp-to-pow 75.3%

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

   5. Simplified 75.3%

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

   if 0.680000000000000049 < x < 2.3000000000000001e218

   1. Initial program 46.8%

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

   3. Taylor expanded in x around inf 80.6%

      \[\leadsto \color{blue}{\frac{\frac{e^{-1 \cdot \frac{\log \left(\frac{1}{x}\right)}{n}}}{n} + \frac{e^{-1 \cdot \frac{\log \left(\frac{1}{x}\right)}{n}} \cdot \left(0.5 \cdot \frac{1}{{n}^{2}} - 0.5 \cdot \frac{1}{n}\right)}{x}}{x}} \]
   4. Step-by-step derivation

   5. Simplified 80.6%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left({x}^{\left(\frac{1}{n}\right)}, \frac{\mathsf{fma}\left(0.5, {n}^{-2}, \frac{-0.5}{n}\right)}{x}, \frac{{x}^{\left(\frac{1}{n}\right)}}{n}\right)}{x}} \]

   6. Taylor expanded in n around inf 67.3%

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

      1. associate-*r/ 67.3%

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

      2. metadata-eval 67.3%

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

      3. *-commutative 67.3%

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

   8. Simplified 67.3%

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

   9. Taylor expanded in x around inf 68.0%

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

       1. cancel-sign-sub-inv 68.0%

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

       2. metadata-eval 68.0%

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

       3. *-commutative 68.0%

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

       4. associate-*r/ 68.0%

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

       5. metadata-eval 68.0%

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

       6. associate-/r* 68.0%

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

       7. metadata-eval 68.0%

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

       8. distribute-neg-frac 68.0%

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

       9. metadata-eval 68.0%

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

       10. associate-*r/ 68.0%

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

       11. distribute-frac-neg 68.0%

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

       12. sub-neg 68.0%

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

       13. div-sub 68.0%

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

       14. associate-/r* 67.3%

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

       15. associate-*r/ 67.3%

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

       16. metadata-eval 67.3%

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

       17. *-commutative 67.3%

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

       18. associate-/r* 68.1%

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

   11. Simplified 68.1%

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

   if 2.3000000000000001e218 < x

   1. Initial program 92.6%

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

   3. Taylor expanded in x around inf 84.6%

      \[\leadsto \color{blue}{\frac{\frac{e^{-1 \cdot \frac{\log \left(\frac{1}{x}\right)}{n}}}{n} + \frac{e^{-1 \cdot \frac{\log \left(\frac{1}{x}\right)}{n}} \cdot \left(0.5 \cdot \frac{1}{{n}^{2}} - 0.5 \cdot \frac{1}{n}\right)}{x}}{x}} \]
   4. Step-by-step derivation

   5. Simplified 84.6%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left({x}^{\left(\frac{1}{n}\right)}, \frac{\mathsf{fma}\left(0.5, {n}^{-2}, \frac{-0.5}{n}\right)}{x}, \frac{{x}^{\left(\frac{1}{n}\right)}}{n}\right)}{x}} \]

   6. Taylor expanded in n around inf 70.8%

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

      1. associate-*r/ 70.8%

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

      2. metadata-eval 70.8%

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

      3. *-commutative 70.8%

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

   8. Simplified 70.8%

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

   9. Taylor expanded in x around 0 92.6%

      \[\leadsto \color{blue}{\frac{-0.5}{n \cdot {x}^{2}}} \]
   10. Step-by-step derivation

       1. associate-/r* 92.6%

          \[\leadsto \color{blue}{\frac{\frac{-0.5}{n}}{{x}^{2}}} \]

   11. Simplified 92.6%

       \[\leadsto \color{blue}{\frac{\frac{-0.5}{n}}{{x}^{2}}} \]
   12. Step-by-step derivation

       1. unpow2 92.6%

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

   13. Applied egg-rr 92.6%

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

4. Final simplification 68.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 7 \cdot 10^{-291}:\\ \;\;\;\;\frac{\log x}{-n}\\ \mathbf{elif}\;x \leq 2.6 \cdot 10^{-222}:\\ \;\;\;\;1 - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{elif}\;x \leq 0.68:\\ \;\;\;\;\frac{\log x}{-n}\\ \mathbf{elif}\;x \leq 2.3 \cdot 10^{+218}:\\ \;\;\;\;\frac{\frac{\frac{-0.5}{x} + 1}{x}}{n}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{-0.5}{n}}{x \cdot x}\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 60.0% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 0.68:\\ \;\;\;\;\frac{\log x}{-n}\\ \mathbf{elif}\;x \leq 1.9 \cdot 10^{+218}:\\ \;\;\;\;\frac{\frac{\frac{-0.5}{x} + 1}{x}}{n}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{-0.5}{n}}{x \cdot x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (if (<= x 0.68)
   (/ (log x) (- n))
   (if (<= x 1.9e+218) (/ (/ (+ (/ -0.5 x) 1.0) x) n) (/ (/ -0.5 n) (* x x)))))

C

double code(double x, double n) {
	double tmp;
	if (x <= 0.68) {
		tmp = log(x) / -n;
	} else if (x <= 1.9e+218) {
		tmp = (((-0.5 / x) + 1.0) / x) / n;
	} else {
		tmp = (-0.5 / n) / (x * x);
	}
	return tmp;
}

Fortran

real(8) function code(x, n)
    real(8), intent (in) :: x
    real(8), intent (in) :: n
    real(8) :: tmp
    if (x <= 0.68d0) then
        tmp = log(x) / -n
    else if (x <= 1.9d+218) then
        tmp = ((((-0.5d0) / x) + 1.0d0) / x) / n
    else
        tmp = ((-0.5d0) / n) / (x * x)
    end if
    code = tmp
end function

Java

public static double code(double x, double n) {
	double tmp;
	if (x <= 0.68) {
		tmp = Math.log(x) / -n;
	} else if (x <= 1.9e+218) {
		tmp = (((-0.5 / x) + 1.0) / x) / n;
	} else {
		tmp = (-0.5 / n) / (x * x);
	}
	return tmp;
}

Python

def code(x, n):
	tmp = 0
	if x <= 0.68:
		tmp = math.log(x) / -n
	elif x <= 1.9e+218:
		tmp = (((-0.5 / x) + 1.0) / x) / n
	else:
		tmp = (-0.5 / n) / (x * x)
	return tmp

Julia

function code(x, n)
	tmp = 0.0
	if (x <= 0.68)
		tmp = Float64(log(x) / Float64(-n));
	elseif (x <= 1.9e+218)
		tmp = Float64(Float64(Float64(Float64(-0.5 / x) + 1.0) / x) / n);
	else
		tmp = Float64(Float64(-0.5 / n) / Float64(x * x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, n)
	tmp = 0.0;
	if (x <= 0.68)
		tmp = log(x) / -n;
	elseif (x <= 1.9e+218)
		tmp = (((-0.5 / x) + 1.0) / x) / n;
	else
		tmp = (-0.5 / n) / (x * x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, n_] := If[LessEqual[x, 0.68], N[(N[Log[x], $MachinePrecision] / (-n)), $MachinePrecision], If[LessEqual[x, 1.9e+218], N[(N[(N[(N[(-0.5 / x), $MachinePrecision] + 1.0), $MachinePrecision] / x), $MachinePrecision] / n), $MachinePrecision], N[(N[(-0.5 / n), $MachinePrecision] / N[(x * x), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < 0.680000000000000049

   1. Initial program 43.4%

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

   3. Taylor expanded in x around 0 41.6%

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

      1. *-rgt-identity 41.6%

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

      2. associate-*l/ 41.6%

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

      3. associate-/l* 41.6%

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

      4. exp-to-pow 41.6%

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

   5. Simplified 41.6%

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

   6. Taylor expanded in n around inf 53.5%

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

      1. associate-*r/ 53.5%

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

      2. neg-mul-1 53.5%

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

   8. Simplified 53.5%

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

   if 0.680000000000000049 < x < 1.90000000000000006e218

   1. Initial program 46.8%

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

   3. Taylor expanded in x around inf 80.6%

      \[\leadsto \color{blue}{\frac{\frac{e^{-1 \cdot \frac{\log \left(\frac{1}{x}\right)}{n}}}{n} + \frac{e^{-1 \cdot \frac{\log \left(\frac{1}{x}\right)}{n}} \cdot \left(0.5 \cdot \frac{1}{{n}^{2}} - 0.5 \cdot \frac{1}{n}\right)}{x}}{x}} \]
   4. Step-by-step derivation

   5. Simplified 80.6%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left({x}^{\left(\frac{1}{n}\right)}, \frac{\mathsf{fma}\left(0.5, {n}^{-2}, \frac{-0.5}{n}\right)}{x}, \frac{{x}^{\left(\frac{1}{n}\right)}}{n}\right)}{x}} \]

   6. Taylor expanded in n around inf 67.3%

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

      1. associate-*r/ 67.3%

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

      2. metadata-eval 67.3%

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

      3. *-commutative 67.3%

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

   8. Simplified 67.3%

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

   9. Taylor expanded in x around inf 68.0%

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

       1. cancel-sign-sub-inv 68.0%

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

       2. metadata-eval 68.0%

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

       3. *-commutative 68.0%

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

       4. associate-*r/ 68.0%

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

       5. metadata-eval 68.0%

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

       6. associate-/r* 68.0%

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

       7. metadata-eval 68.0%

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

       8. distribute-neg-frac 68.0%

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

       9. metadata-eval 68.0%

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

       10. associate-*r/ 68.0%

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

       11. distribute-frac-neg 68.0%

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

       12. sub-neg 68.0%

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

       13. div-sub 68.0%

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

       14. associate-/r* 67.3%

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

       15. associate-*r/ 67.3%

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

       16. metadata-eval 67.3%

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

       17. *-commutative 67.3%

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

       18. associate-/r* 68.1%

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

   11. Simplified 68.1%

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

   if 1.90000000000000006e218 < x

   1. Initial program 92.6%

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

   3. Taylor expanded in x around inf 84.6%

      \[\leadsto \color{blue}{\frac{\frac{e^{-1 \cdot \frac{\log \left(\frac{1}{x}\right)}{n}}}{n} + \frac{e^{-1 \cdot \frac{\log \left(\frac{1}{x}\right)}{n}} \cdot \left(0.5 \cdot \frac{1}{{n}^{2}} - 0.5 \cdot \frac{1}{n}\right)}{x}}{x}} \]
   4. Step-by-step derivation

   5. Simplified 84.6%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left({x}^{\left(\frac{1}{n}\right)}, \frac{\mathsf{fma}\left(0.5, {n}^{-2}, \frac{-0.5}{n}\right)}{x}, \frac{{x}^{\left(\frac{1}{n}\right)}}{n}\right)}{x}} \]

   6. Taylor expanded in n around inf 70.8%

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

      1. associate-*r/ 70.8%

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

      2. metadata-eval 70.8%

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

      3. *-commutative 70.8%

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

   8. Simplified 70.8%

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

   9. Taylor expanded in x around 0 92.6%

      \[\leadsto \color{blue}{\frac{-0.5}{n \cdot {x}^{2}}} \]
   10. Step-by-step derivation

       1. associate-/r* 92.6%

          \[\leadsto \color{blue}{\frac{\frac{-0.5}{n}}{{x}^{2}}} \]

   11. Simplified 92.6%

       \[\leadsto \color{blue}{\frac{\frac{-0.5}{n}}{{x}^{2}}} \]
   12. Step-by-step derivation

       1. unpow2 92.6%

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

   13. Applied egg-rr 92.6%

       \[\leadsto \frac{\frac{-0.5}{n}}{\color{blue}{x \cdot x}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 63.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 0.68:\\ \;\;\;\;\frac{\log x}{-n}\\ \mathbf{elif}\;x \leq 1.9 \cdot 10^{+218}:\\ \;\;\;\;\frac{\frac{\frac{-0.5}{x} + 1}{x}}{n}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{-0.5}{n}}{x \cdot x}\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 43.9% accurate, 17.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 2 \cdot 10^{+218}:\\ \;\;\;\;\frac{\frac{1}{n}}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{-0.5}{n}}{x \cdot x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (if (<= x 2e+218) (/ (/ 1.0 n) x) (/ (/ -0.5 n) (* x x))))

C

double code(double x, double n) {
	double tmp;
	if (x <= 2e+218) {
		tmp = (1.0 / n) / x;
	} else {
		tmp = (-0.5 / n) / (x * x);
	}
	return tmp;
}

Fortran

real(8) function code(x, n)
    real(8), intent (in) :: x
    real(8), intent (in) :: n
    real(8) :: tmp
    if (x <= 2d+218) then
        tmp = (1.0d0 / n) / x
    else
        tmp = ((-0.5d0) / n) / (x * x)
    end if
    code = tmp
end function

Java

public static double code(double x, double n) {
	double tmp;
	if (x <= 2e+218) {
		tmp = (1.0 / n) / x;
	} else {
		tmp = (-0.5 / n) / (x * x);
	}
	return tmp;
}

Python

def code(x, n):
	tmp = 0
	if x <= 2e+218:
		tmp = (1.0 / n) / x
	else:
		tmp = (-0.5 / n) / (x * x)
	return tmp

Julia

function code(x, n)
	tmp = 0.0
	if (x <= 2e+218)
		tmp = Float64(Float64(1.0 / n) / x);
	else
		tmp = Float64(Float64(-0.5 / n) / Float64(x * x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, n)
	tmp = 0.0;
	if (x <= 2e+218)
		tmp = (1.0 / n) / x;
	else
		tmp = (-0.5 / n) / (x * x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, n_] := If[LessEqual[x, 2e+218], N[(N[(1.0 / n), $MachinePrecision] / x), $MachinePrecision], N[(N[(-0.5 / n), $MachinePrecision] / N[(x * x), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 2.00000000000000017e218

   1. Initial program 44.6%

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

   3. Taylor expanded in x around inf 29.5%

      \[\leadsto \color{blue}{\frac{\frac{e^{-1 \cdot \frac{\log \left(\frac{1}{x}\right)}{n}}}{n} + \frac{e^{-1 \cdot \frac{\log \left(\frac{1}{x}\right)}{n}} \cdot \left(0.5 \cdot \frac{1}{{n}^{2}} - 0.5 \cdot \frac{1}{n}\right)}{x}}{x}} \]
   4. Step-by-step derivation

   5. Simplified 29.4%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left({x}^{\left(\frac{1}{n}\right)}, \frac{\mathsf{fma}\left(0.5, {n}^{-2}, \frac{-0.5}{n}\right)}{x}, \frac{{x}^{\left(\frac{1}{n}\right)}}{n}\right)}{x}} \]

   6. Taylor expanded in n around inf 24.6%

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

      1. associate-*r/ 24.6%

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

      2. metadata-eval 24.6%

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

      3. *-commutative 24.6%

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

   8. Simplified 24.6%

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

   9. Taylor expanded in x around inf 38.6%

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

       1. associate-/r* 38.9%

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

   11. Simplified 38.9%

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

   if 2.00000000000000017e218 < x

   1. Initial program 92.6%

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

   3. Taylor expanded in x around inf 84.6%

      \[\leadsto \color{blue}{\frac{\frac{e^{-1 \cdot \frac{\log \left(\frac{1}{x}\right)}{n}}}{n} + \frac{e^{-1 \cdot \frac{\log \left(\frac{1}{x}\right)}{n}} \cdot \left(0.5 \cdot \frac{1}{{n}^{2}} - 0.5 \cdot \frac{1}{n}\right)}{x}}{x}} \]
   4. Step-by-step derivation

   5. Simplified 84.6%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left({x}^{\left(\frac{1}{n}\right)}, \frac{\mathsf{fma}\left(0.5, {n}^{-2}, \frac{-0.5}{n}\right)}{x}, \frac{{x}^{\left(\frac{1}{n}\right)}}{n}\right)}{x}} \]

   6. Taylor expanded in n around inf 70.8%

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

      1. associate-*r/ 70.8%

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

      2. metadata-eval 70.8%

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

      3. *-commutative 70.8%

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

   8. Simplified 70.8%

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

   9. Taylor expanded in x around 0 92.6%

      \[\leadsto \color{blue}{\frac{-0.5}{n \cdot {x}^{2}}} \]
   10. Step-by-step derivation

       1. associate-/r* 92.6%

          \[\leadsto \color{blue}{\frac{\frac{-0.5}{n}}{{x}^{2}}} \]

   11. Simplified 92.6%

       \[\leadsto \color{blue}{\frac{\frac{-0.5}{n}}{{x}^{2}}} \]
   12. Step-by-step derivation

       1. unpow2 92.6%

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

   13. Applied egg-rr 92.6%

       \[\leadsto \frac{\frac{-0.5}{n}}{\color{blue}{x \cdot x}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 15: 40.6% accurate, 42.2× speedup

Math

\[\begin{array}{l} \\ \frac{\frac{1}{n}}{x} \end{array} \]

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 51.9%

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

3. Taylor expanded in x around inf 37.9%

   \[\leadsto \color{blue}{\frac{\frac{e^{-1 \cdot \frac{\log \left(\frac{1}{x}\right)}{n}}}{n} + \frac{e^{-1 \cdot \frac{\log \left(\frac{1}{x}\right)}{n}} \cdot \left(0.5 \cdot \frac{1}{{n}^{2}} - 0.5 \cdot \frac{1}{n}\right)}{x}}{x}} \]
4. Step-by-step derivation

5. Simplified 37.8%

   \[\leadsto \color{blue}{\frac{\mathsf{fma}\left({x}^{\left(\frac{1}{n}\right)}, \frac{\mathsf{fma}\left(0.5, {n}^{-2}, \frac{-0.5}{n}\right)}{x}, \frac{{x}^{\left(\frac{1}{n}\right)}}{n}\right)}{x}} \]

6. Taylor expanded in n around inf 31.7%

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

   1. associate-*r/ 31.7%

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

   2. metadata-eval 31.7%

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

   3. *-commutative 31.7%

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

8. Simplified 31.7%

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

9. Taylor expanded in x around inf 43.5%

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

    1. associate-/r* 43.8%

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

11. Simplified 43.8%

    \[\leadsto \color{blue}{\frac{\frac{1}{n}}{x}} \]
12. Add Preprocessing

Alternative 16: 40.2% accurate, 42.2× speedup

Math

\[\begin{array}{l} \\ \frac{1}{x \cdot n} \end{array} \]

FPCore

(FPCore (x n) :precision binary64 (/ 1.0 (* x n)))

C

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

Fortran

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

Java

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

Python

def code(x, n):
	return 1.0 / (x * n)

Julia

function code(x, n)
	return Float64(1.0 / Float64(x * n))
end

MATLAB

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

Wolfram

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

Derivation

1. Initial program 51.9%

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

3. Taylor expanded in x around inf 37.9%

   \[\leadsto \color{blue}{\frac{\frac{e^{-1 \cdot \frac{\log \left(\frac{1}{x}\right)}{n}}}{n} + \frac{e^{-1 \cdot \frac{\log \left(\frac{1}{x}\right)}{n}} \cdot \left(0.5 \cdot \frac{1}{{n}^{2}} - 0.5 \cdot \frac{1}{n}\right)}{x}}{x}} \]
4. Step-by-step derivation

5. Simplified 37.8%

   \[\leadsto \color{blue}{\frac{\mathsf{fma}\left({x}^{\left(\frac{1}{n}\right)}, \frac{\mathsf{fma}\left(0.5, {n}^{-2}, \frac{-0.5}{n}\right)}{x}, \frac{{x}^{\left(\frac{1}{n}\right)}}{n}\right)}{x}} \]

6. Taylor expanded in n around inf 31.7%

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

   1. associate-*r/ 31.7%

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

   2. metadata-eval 31.7%

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

   3. *-commutative 31.7%

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

8. Simplified 31.7%

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

9. Taylor expanded in x around inf 43.5%

   \[\leadsto \frac{\color{blue}{1}}{x \cdot n} \]
10. Add Preprocessing

Reproduce

herbie shell --seed 2024163 
(FPCore (x n)
  :name "2nthrt (problem 3.4.6)"
  :precision binary64
  (- (pow (+ x 1.0) (/ 1.0 n)) (pow x (/ 1.0 n))))