2nthrt (problem 3.4.6)

Percentage Accurate: 53.5% → 86.1%

Time: 26.8s

Alternatives: 17

Speedup: 1.9×

• could not determine a ground truth

  1. x = 3.962655850681198e+218
  2. n = 9.219559396046477e-306

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 53.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 86.1% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \log \left(\frac{x + 1}{x}\right)\\ \mathbf{if}\;x \leq 420000:\\ \;\;\;\;\frac{t\_0 + \frac{\mathsf{fma}\left(0.5, t\_0 \cdot \log \left(\mathsf{fma}\left(x, x, x\right)\right), \left({\left(\mathsf{log1p}\left(x\right)\right)}^{3} - {\log x}^{3}\right) \cdot \frac{0.16666666666666666}{n}\right)}{n}}{n}\\ \mathbf{else}:\\ \;\;\;\;\frac{{x}^{\left(\frac{1}{n}\right)}}{x \cdot n}\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (let* ((t_0 (log (/ (+ x 1.0) x))))
   (if (<= x 420000.0)
     (/
      (+
       t_0
       (/
        (fma
         0.5
         (* t_0 (log (fma x x x)))
         (*
          (- (pow (log1p x) 3.0) (pow (log x) 3.0))
          (/ 0.16666666666666666 n)))
        n))
      n)
     (/ (pow x (/ 1.0 n)) (* x n)))))

C

double code(double x, double n) {
	double t_0 = log(((x + 1.0) / x));
	double tmp;
	if (x <= 420000.0) {
		tmp = (t_0 + (fma(0.5, (t_0 * log(fma(x, x, x))), ((pow(log1p(x), 3.0) - pow(log(x), 3.0)) * (0.16666666666666666 / n))) / n)) / n;
	} else {
		tmp = pow(x, (1.0 / n)) / (x * n);
	}
	return tmp;
}

Julia

function code(x, n)
	t_0 = log(Float64(Float64(x + 1.0) / x))
	tmp = 0.0
	if (x <= 420000.0)
		tmp = Float64(Float64(t_0 + Float64(fma(0.5, Float64(t_0 * log(fma(x, x, x))), Float64(Float64((log1p(x) ^ 3.0) - (log(x) ^ 3.0)) * Float64(0.16666666666666666 / n))) / n)) / n);
	else
		tmp = Float64((x ^ Float64(1.0 / n)) / Float64(x * n));
	end
	return tmp
end

Wolfram

code[x_, n_] := Block[{t$95$0 = N[Log[N[(N[(x + 1.0), $MachinePrecision] / x), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[x, 420000.0], N[(N[(t$95$0 + N[(N[(0.5 * N[(t$95$0 * N[Log[N[(x * x + x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] + N[(N[(N[Power[N[Log[1 + x], $MachinePrecision], 3.0], $MachinePrecision] - N[Power[N[Log[x], $MachinePrecision], 3.0], $MachinePrecision]), $MachinePrecision] * N[(0.16666666666666666 / n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / n), $MachinePrecision]), $MachinePrecision] / n), $MachinePrecision], N[(N[Power[x, N[(1.0 / n), $MachinePrecision]], $MachinePrecision] / N[(x * n), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if x < 4.2e5

   1. Initial program 36.4%

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

   3. Taylor expanded in n around -inf

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

   5. Simplified 79.7%

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

      1. sub-neg N/A

         \[\leadsto \frac{\color{blue}{\left(\log x + \left(\mathsf{neg}\left(\log \left(1 + x\right)\right)\right)\right)} - \frac{\mathsf{fma}\left(\frac{1}{2}, {\left(\mathsf{log1p}\left(x\right)\right)}^{2} - {\log x}^{2}, \frac{\frac{1}{6} \cdot \left({\left(\mathsf{log1p}\left(x\right)\right)}^{3} - {\log x}^{3}\right)}{n}\right)}{n}}{\mathsf{neg}\left(n\right)} \]

      2. flip-+ N/A

         \[\leadsto \frac{\color{blue}{\frac{\log x \cdot \log x - \left(\mathsf{neg}\left(\log \left(1 + x\right)\right)\right) \cdot \left(\mathsf{neg}\left(\log \left(1 + x\right)\right)\right)}{\log x - \left(\mathsf{neg}\left(\log \left(1 + x\right)\right)\right)}} - \frac{\mathsf{fma}\left(\frac{1}{2}, {\left(\mathsf{log1p}\left(x\right)\right)}^{2} - {\log x}^{2}, \frac{\frac{1}{6} \cdot \left({\left(\mathsf{log1p}\left(x\right)\right)}^{3} - {\log x}^{3}\right)}{n}\right)}{n}}{\mathsf{neg}\left(n\right)} \]

      3. /-lowering-/.f64 N/A

         \[\leadsto \frac{\color{blue}{\frac{\log x \cdot \log x - \left(\mathsf{neg}\left(\log \left(1 + x\right)\right)\right) \cdot \left(\mathsf{neg}\left(\log \left(1 + x\right)\right)\right)}{\log x - \left(\mathsf{neg}\left(\log \left(1 + x\right)\right)\right)}} - \frac{\mathsf{fma}\left(\frac{1}{2}, {\left(\mathsf{log1p}\left(x\right)\right)}^{2} - {\log x}^{2}, \frac{\frac{1}{6} \cdot \left({\left(\mathsf{log1p}\left(x\right)\right)}^{3} - {\log x}^{3}\right)}{n}\right)}{n}}{\mathsf{neg}\left(n\right)} \]

      4. unpow2 N/A

         \[\leadsto \frac{\frac{\color{blue}{{\log x}^{2}} - \left(\mathsf{neg}\left(\log \left(1 + x\right)\right)\right) \cdot \left(\mathsf{neg}\left(\log \left(1 + x\right)\right)\right)}{\log x - \left(\mathsf{neg}\left(\log \left(1 + x\right)\right)\right)} - \frac{\mathsf{fma}\left(\frac{1}{2}, {\left(\mathsf{log1p}\left(x\right)\right)}^{2} - {\log x}^{2}, \frac{\frac{1}{6} \cdot \left({\left(\mathsf{log1p}\left(x\right)\right)}^{3} - {\log x}^{3}\right)}{n}\right)}{n}}{\mathsf{neg}\left(n\right)} \]

      5. --lowering--.f64 N/A

         \[\leadsto \frac{\frac{\color{blue}{{\log x}^{2} - \left(\mathsf{neg}\left(\log \left(1 + x\right)\right)\right) \cdot \left(\mathsf{neg}\left(\log \left(1 + x\right)\right)\right)}}{\log x - \left(\mathsf{neg}\left(\log \left(1 + x\right)\right)\right)} - \frac{\mathsf{fma}\left(\frac{1}{2}, {\left(\mathsf{log1p}\left(x\right)\right)}^{2} - {\log x}^{2}, \frac{\frac{1}{6} \cdot \left({\left(\mathsf{log1p}\left(x\right)\right)}^{3} - {\log x}^{3}\right)}{n}\right)}{n}}{\mathsf{neg}\left(n\right)} \]

      6. pow-lowering-pow.f64 N/A

         \[\leadsto \frac{\frac{\color{blue}{{\log x}^{2}} - \left(\mathsf{neg}\left(\log \left(1 + x\right)\right)\right) \cdot \left(\mathsf{neg}\left(\log \left(1 + x\right)\right)\right)}{\log x - \left(\mathsf{neg}\left(\log \left(1 + x\right)\right)\right)} - \frac{\mathsf{fma}\left(\frac{1}{2}, {\left(\mathsf{log1p}\left(x\right)\right)}^{2} - {\log x}^{2}, \frac{\frac{1}{6} \cdot \left({\left(\mathsf{log1p}\left(x\right)\right)}^{3} - {\log x}^{3}\right)}{n}\right)}{n}}{\mathsf{neg}\left(n\right)} \]

      7. log-lowering-log.f64 N/A

         \[\leadsto \frac{\frac{{\color{blue}{\log x}}^{2} - \left(\mathsf{neg}\left(\log \left(1 + x\right)\right)\right) \cdot \left(\mathsf{neg}\left(\log \left(1 + x\right)\right)\right)}{\log x - \left(\mathsf{neg}\left(\log \left(1 + x\right)\right)\right)} - \frac{\mathsf{fma}\left(\frac{1}{2}, {\left(\mathsf{log1p}\left(x\right)\right)}^{2} - {\log x}^{2}, \frac{\frac{1}{6} \cdot \left({\left(\mathsf{log1p}\left(x\right)\right)}^{3} - {\log x}^{3}\right)}{n}\right)}{n}}{\mathsf{neg}\left(n\right)} \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \frac{\frac{{\log x}^{2} - \color{blue}{\left(\mathsf{neg}\left(\log \left(1 + x\right)\right)\right) \cdot \left(\mathsf{neg}\left(\log \left(1 + x\right)\right)\right)}}{\log x - \left(\mathsf{neg}\left(\log \left(1 + x\right)\right)\right)} - \frac{\mathsf{fma}\left(\frac{1}{2}, {\left(\mathsf{log1p}\left(x\right)\right)}^{2} - {\log x}^{2}, \frac{\frac{1}{6} \cdot \left({\left(\mathsf{log1p}\left(x\right)\right)}^{3} - {\log x}^{3}\right)}{n}\right)}{n}}{\mathsf{neg}\left(n\right)} \]

      9. neg-lowering-neg.f64 N/A

         \[\leadsto \frac{\frac{{\log x}^{2} - \color{blue}{\left(\mathsf{neg}\left(\log \left(1 + x\right)\right)\right)} \cdot \left(\mathsf{neg}\left(\log \left(1 + x\right)\right)\right)}{\log x - \left(\mathsf{neg}\left(\log \left(1 + x\right)\right)\right)} - \frac{\mathsf{fma}\left(\frac{1}{2}, {\left(\mathsf{log1p}\left(x\right)\right)}^{2} - {\log x}^{2}, \frac{\frac{1}{6} \cdot \left({\left(\mathsf{log1p}\left(x\right)\right)}^{3} - {\log x}^{3}\right)}{n}\right)}{n}}{\mathsf{neg}\left(n\right)} \]

      10. accelerator-lowering-log1p.f64 N/A

          \[\leadsto \frac{\frac{{\log x}^{2} - \left(\mathsf{neg}\left(\color{blue}{\mathsf{log1p}\left(x\right)}\right)\right) \cdot \left(\mathsf{neg}\left(\log \left(1 + x\right)\right)\right)}{\log x - \left(\mathsf{neg}\left(\log \left(1 + x\right)\right)\right)} - \frac{\mathsf{fma}\left(\frac{1}{2}, {\left(\mathsf{log1p}\left(x\right)\right)}^{2} - {\log x}^{2}, \frac{\frac{1}{6} \cdot \left({\left(\mathsf{log1p}\left(x\right)\right)}^{3} - {\log x}^{3}\right)}{n}\right)}{n}}{\mathsf{neg}\left(n\right)} \]

      11. neg-lowering-neg.f64 N/A

          \[\leadsto \frac{\frac{{\log x}^{2} - \left(\mathsf{neg}\left(\mathsf{log1p}\left(x\right)\right)\right) \cdot \color{blue}{\left(\mathsf{neg}\left(\log \left(1 + x\right)\right)\right)}}{\log x - \left(\mathsf{neg}\left(\log \left(1 + x\right)\right)\right)} - \frac{\mathsf{fma}\left(\frac{1}{2}, {\left(\mathsf{log1p}\left(x\right)\right)}^{2} - {\log x}^{2}, \frac{\frac{1}{6} \cdot \left({\left(\mathsf{log1p}\left(x\right)\right)}^{3} - {\log x}^{3}\right)}{n}\right)}{n}}{\mathsf{neg}\left(n\right)} \]

      12. accelerator-lowering-log1p.f64 N/A

          \[\leadsto \frac{\frac{{\log x}^{2} - \left(\mathsf{neg}\left(\mathsf{log1p}\left(x\right)\right)\right) \cdot \left(\mathsf{neg}\left(\color{blue}{\mathsf{log1p}\left(x\right)}\right)\right)}{\log x - \left(\mathsf{neg}\left(\log \left(1 + x\right)\right)\right)} - \frac{\mathsf{fma}\left(\frac{1}{2}, {\left(\mathsf{log1p}\left(x\right)\right)}^{2} - {\log x}^{2}, \frac{\frac{1}{6} \cdot \left({\left(\mathsf{log1p}\left(x\right)\right)}^{3} - {\log x}^{3}\right)}{n}\right)}{n}}{\mathsf{neg}\left(n\right)} \]

      13. +-commutative N/A

          \[\leadsto \frac{\frac{{\log x}^{2} - \left(\mathsf{neg}\left(\mathsf{log1p}\left(x\right)\right)\right) \cdot \left(\mathsf{neg}\left(\mathsf{log1p}\left(x\right)\right)\right)}{\log x - \left(\mathsf{neg}\left(\log \color{blue}{\left(x + 1\right)}\right)\right)} - \frac{\mathsf{fma}\left(\frac{1}{2}, {\left(\mathsf{log1p}\left(x\right)\right)}^{2} - {\log x}^{2}, \frac{\frac{1}{6} \cdot \left({\left(\mathsf{log1p}\left(x\right)\right)}^{3} - {\log x}^{3}\right)}{n}\right)}{n}}{\mathsf{neg}\left(n\right)} \]

      14. neg-log N/A

          \[\leadsto \frac{\frac{{\log x}^{2} - \left(\mathsf{neg}\left(\mathsf{log1p}\left(x\right)\right)\right) \cdot \left(\mathsf{neg}\left(\mathsf{log1p}\left(x\right)\right)\right)}{\log x - \color{blue}{\log \left(\frac{1}{x + 1}\right)}} - \frac{\mathsf{fma}\left(\frac{1}{2}, {\left(\mathsf{log1p}\left(x\right)\right)}^{2} - {\log x}^{2}, \frac{\frac{1}{6} \cdot \left({\left(\mathsf{log1p}\left(x\right)\right)}^{3} - {\log x}^{3}\right)}{n}\right)}{n}}{\mathsf{neg}\left(n\right)} \]

   7. Applied egg-rr 79.7%

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

   9. Applied egg-rr 79.7%

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

   if 4.2e5 < x

   1. Initial program 66.0%

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

   3. Taylor expanded in x around inf

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

      1. /-lowering-/.f64 N/A

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

      2. log-rec N/A

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

      3. mul-1-neg N/A

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

      4. associate-*r/ N/A

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

      5. associate-*r* N/A

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

      6. metadata-eval N/A

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

      7. *-commutative N/A

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

      8. associate-/l* N/A

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

      9. exp-to-pow N/A

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

      10. pow-lowering-pow.f64 N/A

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

      11. /-lowering-/.f64 N/A

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

      12. *-commutative N/A

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

      13. *-lowering-*.f64 98.8

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

   5. Simplified 98.8%

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

4. Final simplification 87.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 420000:\\ \;\;\;\;\frac{\log \left(\frac{x + 1}{x}\right) + \frac{\mathsf{fma}\left(0.5, \log \left(\frac{x + 1}{x}\right) \cdot \log \left(\mathsf{fma}\left(x, x, x\right)\right), \left({\left(\mathsf{log1p}\left(x\right)\right)}^{3} - {\log x}^{3}\right) \cdot \frac{0.16666666666666666}{n}\right)}{n}}{n}\\ \mathbf{else}:\\ \;\;\;\;\frac{{x}^{\left(\frac{1}{n}\right)}}{x \cdot n}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 86.1% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 700000:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(0.5, \log \left(\frac{x + 1}{x}\right) \cdot \log \left(x \cdot \left(x + 1\right)\right), \frac{\left({\left(\mathsf{log1p}\left(x\right)\right)}^{3} - {\log x}^{3}\right) \cdot 0.16666666666666666}{n}\right)}{n} - \log \left(\frac{x}{x + 1}\right)}{n}\\ \mathbf{else}:\\ \;\;\;\;\frac{{x}^{\left(\frac{1}{n}\right)}}{x \cdot n}\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (if (<= x 700000.0)
   (/
    (-
     (/
      (fma
       0.5
       (* (log (/ (+ x 1.0) x)) (log (* x (+ x 1.0))))
       (/ (* (- (pow (log1p x) 3.0) (pow (log x) 3.0)) 0.16666666666666666) n))
      n)
     (log (/ x (+ x 1.0))))
    n)
   (/ (pow x (/ 1.0 n)) (* x n))))

C

double code(double x, double n) {
	double tmp;
	if (x <= 700000.0) {
		tmp = ((fma(0.5, (log(((x + 1.0) / x)) * log((x * (x + 1.0)))), (((pow(log1p(x), 3.0) - pow(log(x), 3.0)) * 0.16666666666666666) / n)) / n) - log((x / (x + 1.0)))) / n;
	} else {
		tmp = pow(x, (1.0 / n)) / (x * n);
	}
	return tmp;
}

Julia

function code(x, n)
	tmp = 0.0
	if (x <= 700000.0)
		tmp = Float64(Float64(Float64(fma(0.5, Float64(log(Float64(Float64(x + 1.0) / x)) * log(Float64(x * Float64(x + 1.0)))), Float64(Float64(Float64((log1p(x) ^ 3.0) - (log(x) ^ 3.0)) * 0.16666666666666666) / n)) / n) - log(Float64(x / Float64(x + 1.0)))) / n);
	else
		tmp = Float64((x ^ Float64(1.0 / n)) / Float64(x * n));
	end
	return tmp
end

Wolfram

code[x_, n_] := If[LessEqual[x, 700000.0], N[(N[(N[(N[(0.5 * N[(N[Log[N[(N[(x + 1.0), $MachinePrecision] / x), $MachinePrecision]], $MachinePrecision] * N[Log[N[(x * N[(x + 1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] + N[(N[(N[(N[Power[N[Log[1 + x], $MachinePrecision], 3.0], $MachinePrecision] - N[Power[N[Log[x], $MachinePrecision], 3.0], $MachinePrecision]), $MachinePrecision] * 0.16666666666666666), $MachinePrecision] / n), $MachinePrecision]), $MachinePrecision] / n), $MachinePrecision] - N[Log[N[(x / N[(x + 1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / n), $MachinePrecision], N[(N[Power[x, N[(1.0 / n), $MachinePrecision]], $MachinePrecision] / N[(x * n), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 7e5

   1. Initial program 36.4%

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

   3. Taylor expanded in n around -inf

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

   5. Simplified 79.7%

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

      1. /-lowering-/.f64 N/A

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

   7. Applied egg-rr 79.7%

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

   if 7e5 < x

   1. Initial program 66.0%

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

   3. Taylor expanded in x around inf

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

      1. /-lowering-/.f64 N/A

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

      2. log-rec N/A

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

      3. mul-1-neg N/A

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

      4. associate-*r/ N/A

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

      5. associate-*r* N/A

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

      6. metadata-eval N/A

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

      7. *-commutative N/A

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

      8. associate-/l* N/A

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

      9. exp-to-pow N/A

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

      10. pow-lowering-pow.f64 N/A

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

      11. /-lowering-/.f64 N/A

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

      12. *-commutative N/A

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

      13. *-lowering-*.f64 98.8

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

   5. Simplified 98.8%

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

4. Final simplification 87.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 700000:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(0.5, \log \left(\frac{x + 1}{x}\right) \cdot \log \left(x \cdot \left(x + 1\right)\right), \frac{\left({\left(\mathsf{log1p}\left(x\right)\right)}^{3} - {\log x}^{3}\right) \cdot 0.16666666666666666}{n}\right)}{n} - \log \left(\frac{x}{x + 1}\right)}{n}\\ \mathbf{else}:\\ \;\;\;\;\frac{{x}^{\left(\frac{1}{n}\right)}}{x \cdot n}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 78.3% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {x}^{\left(\frac{1}{n}\right)}\\ t_1 := {\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - t\_0\\ t_2 := 1 - t\_0\\ \mathbf{if}\;t\_1 \leq -0.01:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_1 \leq 5 \cdot 10^{-9}:\\ \;\;\;\;\frac{\log \left(\frac{x + 1}{x}\right)}{n}\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (let* ((t_0 (pow x (/ 1.0 n)))
        (t_1 (- (pow (+ x 1.0) (/ 1.0 n)) t_0))
        (t_2 (- 1.0 t_0)))
   (if (<= t_1 -0.01) t_2 (if (<= t_1 5e-9) (/ (log (/ (+ x 1.0) x)) n) t_2))))

C

double code(double x, double n) {
	double t_0 = pow(x, (1.0 / n));
	double t_1 = pow((x + 1.0), (1.0 / n)) - t_0;
	double t_2 = 1.0 - t_0;
	double tmp;
	if (t_1 <= -0.01) {
		tmp = t_2;
	} else if (t_1 <= 5e-9) {
		tmp = log(((x + 1.0) / x)) / n;
	} else {
		tmp = t_2;
	}
	return tmp;
}

Fortran

real(8) function code(x, n)
    real(8), intent (in) :: x
    real(8), intent (in) :: n
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_0 = x ** (1.0d0 / n)
    t_1 = ((x + 1.0d0) ** (1.0d0 / n)) - t_0
    t_2 = 1.0d0 - t_0
    if (t_1 <= (-0.01d0)) then
        tmp = t_2
    else if (t_1 <= 5d-9) then
        tmp = log(((x + 1.0d0) / x)) / n
    else
        tmp = t_2
    end if
    code = tmp
end function

Java

public static double code(double x, double n) {
	double t_0 = Math.pow(x, (1.0 / n));
	double t_1 = Math.pow((x + 1.0), (1.0 / n)) - t_0;
	double t_2 = 1.0 - t_0;
	double tmp;
	if (t_1 <= -0.01) {
		tmp = t_2;
	} else if (t_1 <= 5e-9) {
		tmp = Math.log(((x + 1.0) / x)) / n;
	} else {
		tmp = t_2;
	}
	return tmp;
}

Python

def code(x, n):
	t_0 = math.pow(x, (1.0 / n))
	t_1 = math.pow((x + 1.0), (1.0 / n)) - t_0
	t_2 = 1.0 - t_0
	tmp = 0
	if t_1 <= -0.01:
		tmp = t_2
	elif t_1 <= 5e-9:
		tmp = math.log(((x + 1.0) / x)) / n
	else:
		tmp = t_2
	return tmp

Julia

function code(x, n)
	t_0 = x ^ Float64(1.0 / n)
	t_1 = Float64((Float64(x + 1.0) ^ Float64(1.0 / n)) - t_0)
	t_2 = Float64(1.0 - t_0)
	tmp = 0.0
	if (t_1 <= -0.01)
		tmp = t_2;
	elseif (t_1 <= 5e-9)
		tmp = Float64(log(Float64(Float64(x + 1.0) / x)) / n);
	else
		tmp = t_2;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, n)
	t_0 = x ^ (1.0 / n);
	t_1 = ((x + 1.0) ^ (1.0 / n)) - t_0;
	t_2 = 1.0 - t_0;
	tmp = 0.0;
	if (t_1 <= -0.01)
		tmp = t_2;
	elseif (t_1 <= 5e-9)
		tmp = log(((x + 1.0) / x)) / n;
	else
		tmp = t_2;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, n_] := Block[{t$95$0 = N[Power[x, N[(1.0 / n), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(N[Power[N[(x + 1.0), $MachinePrecision], N[(1.0 / n), $MachinePrecision]], $MachinePrecision] - t$95$0), $MachinePrecision]}, Block[{t$95$2 = N[(1.0 - t$95$0), $MachinePrecision]}, If[LessEqual[t$95$1, -0.01], t$95$2, If[LessEqual[t$95$1, 5e-9], N[(N[Log[N[(N[(x + 1.0), $MachinePrecision] / x), $MachinePrecision]], $MachinePrecision] / n), $MachinePrecision], t$95$2]]]]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (pow.f64 (+.f64 x #s(literal 1 binary64)) (/.f64 #s(literal 1 binary64) n)) (pow.f64 x (/.f64 #s(literal 1 binary64) n))) < -0.0100000000000000002 or 5.0000000000000001e-9 < (-.f64 (pow.f64 (+.f64 x #s(literal 1 binary64)) (/.f64 #s(literal 1 binary64) n)) (pow.f64 x (/.f64 #s(literal 1 binary64) n)))

   1. Initial program 76.9%

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

   3. Taylor expanded in x around 0

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

      1. remove-double-neg N/A

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

      2. mul-1-neg N/A

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

      3. distribute-neg-frac N/A

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

      4. mul-1-neg N/A

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

      5. log-rec N/A

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

      6. mul-1-neg N/A

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

      7. --lowering--.f64 N/A

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

      8. log-rec N/A

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

      9. mul-1-neg N/A

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

      10. associate-*r/ N/A

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

      11. associate-*r* N/A

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

      12. metadata-eval N/A

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

      13. *-commutative N/A

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

      14. associate-/l* N/A

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

      15. exp-to-pow N/A

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

      16. pow-lowering-pow.f64 N/A

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

      17. /-lowering-/.f64 75.4

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

   5. Simplified 75.4%

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

   if -0.0100000000000000002 < (-.f64 (pow.f64 (+.f64 x #s(literal 1 binary64)) (/.f64 #s(literal 1 binary64) n)) (pow.f64 x (/.f64 #s(literal 1 binary64) n))) < 5.0000000000000001e-9

   1. Initial program 38.1%

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

   3. Taylor expanded in n around inf

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

      1. /-lowering-/.f64 N/A

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

      2. --lowering--.f64 N/A

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

      3. accelerator-lowering-log1p.f64 N/A

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

      4. log-lowering-log.f64 81.0

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

   5. Simplified 81.0%

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

      1. diff-log N/A

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

      2. log-lowering-log.f64 N/A

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

      3. /-lowering-/.f64 N/A

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

      4. +-commutative N/A

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

      5. +-lowering-+.f64 81.2

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

   7. Applied egg-rr 81.2%

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

Alternative 4: 85.6% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 0.24:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(-0.16666666666666666, \frac{{\log x}^{3}}{n}, -0.5 \cdot {\log x}^{2}\right)}{n} - \log x}{n}\\ \mathbf{else}:\\ \;\;\;\;\frac{{x}^{\left(\frac{1}{n}\right)}}{x \cdot n}\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (if (<= x 0.24)
   (/
    (-
     (/
      (fma
       -0.16666666666666666
       (/ (pow (log x) 3.0) n)
       (* -0.5 (pow (log x) 2.0)))
      n)
     (log x))
    n)
   (/ (pow x (/ 1.0 n)) (* x n))))

C

double code(double x, double n) {
	double tmp;
	if (x <= 0.24) {
		tmp = ((fma(-0.16666666666666666, (pow(log(x), 3.0) / n), (-0.5 * pow(log(x), 2.0))) / n) - log(x)) / n;
	} else {
		tmp = pow(x, (1.0 / n)) / (x * n);
	}
	return tmp;
}

Julia

function code(x, n)
	tmp = 0.0
	if (x <= 0.24)
		tmp = Float64(Float64(Float64(fma(-0.16666666666666666, Float64((log(x) ^ 3.0) / n), Float64(-0.5 * (log(x) ^ 2.0))) / n) - log(x)) / n);
	else
		tmp = Float64((x ^ Float64(1.0 / n)) / Float64(x * n));
	end
	return tmp
end

Wolfram

code[x_, n_] := If[LessEqual[x, 0.24], N[(N[(N[(N[(-0.16666666666666666 * N[(N[Power[N[Log[x], $MachinePrecision], 3.0], $MachinePrecision] / n), $MachinePrecision] + N[(-0.5 * N[Power[N[Log[x], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / n), $MachinePrecision] - N[Log[x], $MachinePrecision]), $MachinePrecision] / n), $MachinePrecision], N[(N[Power[x, N[(1.0 / n), $MachinePrecision]], $MachinePrecision] / N[(x * n), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 0.23999999999999999

   1. Initial program 37.0%

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

   3. Taylor expanded in x around 0

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

      1. remove-double-neg N/A

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

      2. mul-1-neg N/A

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

      3. distribute-neg-frac N/A

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

      4. mul-1-neg N/A

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

      5. log-rec N/A

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

      6. mul-1-neg N/A

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

      7. --lowering--.f64 N/A

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

      8. log-rec N/A

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

      9. mul-1-neg N/A

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

      10. associate-*r/ N/A

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

      11. associate-*r* N/A

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

      12. metadata-eval N/A

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

      13. *-commutative N/A

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

      14. associate-/l* N/A

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

      15. exp-to-pow N/A

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

      16. pow-lowering-pow.f64 N/A

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

      17. /-lowering-/.f64 36.4

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

   5. Simplified 36.4%

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

   6. Taylor expanded in n around -inf

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

      1. mul-1-neg N/A

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

      2. distribute-neg-frac2 N/A

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

      3. mul-1-neg N/A

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

      4. /-lowering-/.f64 N/A

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

   8. Simplified 78.0%

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

   if 0.23999999999999999 < x

   1. Initial program 64.2%

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

   3. Taylor expanded in x around inf

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

      1. /-lowering-/.f64 N/A

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

      2. log-rec N/A

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

      3. mul-1-neg N/A

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

      4. associate-*r/ N/A

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

      5. associate-*r* N/A

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

      6. metadata-eval N/A

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

      7. *-commutative N/A

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

      8. associate-/l* N/A

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

      9. exp-to-pow N/A

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

      10. pow-lowering-pow.f64 N/A

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

      11. /-lowering-/.f64 N/A

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

      12. *-commutative N/A

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

      13. *-lowering-*.f64 97.0

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

   5. Simplified 97.0%

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

4. Final simplification 85.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 0.24:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(-0.16666666666666666, \frac{{\log x}^{3}}{n}, -0.5 \cdot {\log x}^{2}\right)}{n} - \log x}{n}\\ \mathbf{else}:\\ \;\;\;\;\frac{{x}^{\left(\frac{1}{n}\right)}}{x \cdot n}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 86.5% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{if}\;\frac{1}{n} \leq -2 \cdot 10^{-28}:\\ \;\;\;\;\frac{t\_0}{x \cdot n}\\ \mathbf{elif}\;\frac{1}{n} \leq 5 \cdot 10^{-11}:\\ \;\;\;\;\frac{\log \left(\frac{x + 1}{x}\right)}{n}\\ \mathbf{else}:\\ \;\;\;\;e^{\frac{\mathsf{log1p}\left(x\right)}{n}} - t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (let* ((t_0 (pow x (/ 1.0 n))))
   (if (<= (/ 1.0 n) -2e-28)
     (/ t_0 (* x n))
     (if (<= (/ 1.0 n) 5e-11)
       (/ (log (/ (+ x 1.0) 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-28) {
		tmp = t_0 / (x * n);
	} else if ((1.0 / n) <= 5e-11) {
		tmp = log(((x + 1.0) / 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-28) {
		tmp = t_0 / (x * n);
	} else if ((1.0 / n) <= 5e-11) {
		tmp = Math.log(((x + 1.0) / 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-28:
		tmp = t_0 / (x * n)
	elif (1.0 / n) <= 5e-11:
		tmp = math.log(((x + 1.0) / 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-28)
		tmp = Float64(t_0 / Float64(x * n));
	elseif (Float64(1.0 / n) <= 5e-11)
		tmp = Float64(log(Float64(Float64(x + 1.0) / 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-28], N[(t$95$0 / N[(x * n), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 5e-11], N[(N[Log[N[(N[(x + 1.0), $MachinePrecision] / x), $MachinePrecision]], $MachinePrecision] / n), $MachinePrecision], N[(N[Exp[N[(N[Log[1 + x], $MachinePrecision] / n), $MachinePrecision]], $MachinePrecision] - t$95$0), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 #s(literal 1 binary64) n) < -1.99999999999999994e-28

   1. Initial program 92.5%

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

   3. Taylor expanded in x around inf

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

      1. /-lowering-/.f64 N/A

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

      2. log-rec N/A

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

      3. mul-1-neg N/A

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

      4. associate-*r/ N/A

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

      5. associate-*r* N/A

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

      6. metadata-eval N/A

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

      7. *-commutative N/A

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

      8. associate-/l* N/A

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

      9. exp-to-pow N/A

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

      10. pow-lowering-pow.f64 N/A

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

      11. /-lowering-/.f64 N/A

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

      12. *-commutative N/A

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

      13. *-lowering-*.f64 96.5

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

   5. Simplified 96.5%

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

   if -1.99999999999999994e-28 < (/.f64 #s(literal 1 binary64) n) < 5.00000000000000018e-11

   1. Initial program 25.0%

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

   3. Taylor expanded in n around inf

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

      1. /-lowering-/.f64 N/A

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

      2. --lowering--.f64 N/A

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

      3. accelerator-lowering-log1p.f64 N/A

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

      4. log-lowering-log.f64 80.1

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

   5. Simplified 80.1%

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

      1. diff-log N/A

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

      2. log-lowering-log.f64 N/A

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

      3. /-lowering-/.f64 N/A

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

      4. +-commutative N/A

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

      5. +-lowering-+.f64 80.3

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

   7. Applied egg-rr 80.3%

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

   if 5.00000000000000018e-11 < (/.f64 #s(literal 1 binary64) n)

   1. Initial program 51.2%

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

      1. pow-to-exp N/A

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

      2. exp-lowering-exp.f64 N/A

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

      3. un-div-inv N/A

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

      4. /-lowering-/.f64 N/A

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

      5. +-commutative N/A

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

      6. accelerator-lowering-log1p.f64 93.8

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

   4. Applied egg-rr 93.8%

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

Alternative 6: 82.6% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{if}\;\frac{1}{n} \leq -2 \cdot 10^{-28}:\\ \;\;\;\;\frac{t\_0}{x \cdot n}\\ \mathbf{elif}\;\frac{1}{n} \leq 5 \cdot 10^{-11}:\\ \;\;\;\;\frac{\log \left(\frac{x + 1}{x}\right)}{n}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x, \mathsf{fma}\left(x, \frac{0.5}{n \cdot n} + \frac{-0.5}{n}, \frac{1}{n}\right), 1\right) - t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (let* ((t_0 (pow x (/ 1.0 n))))
   (if (<= (/ 1.0 n) -2e-28)
     (/ t_0 (* x n))
     (if (<= (/ 1.0 n) 5e-11)
       (/ (log (/ (+ x 1.0) x)) n)
       (- (fma x (fma x (+ (/ 0.5 (* n n)) (/ -0.5 n)) (/ 1.0 n)) 1.0) t_0)))))

C

double code(double x, double n) {
	double t_0 = pow(x, (1.0 / n));
	double tmp;
	if ((1.0 / n) <= -2e-28) {
		tmp = t_0 / (x * n);
	} else if ((1.0 / n) <= 5e-11) {
		tmp = log(((x + 1.0) / x)) / n;
	} else {
		tmp = fma(x, fma(x, ((0.5 / (n * n)) + (-0.5 / n)), (1.0 / n)), 1.0) - t_0;
	}
	return tmp;
}

Julia

function code(x, n)
	t_0 = x ^ Float64(1.0 / n)
	tmp = 0.0
	if (Float64(1.0 / n) <= -2e-28)
		tmp = Float64(t_0 / Float64(x * n));
	elseif (Float64(1.0 / n) <= 5e-11)
		tmp = Float64(log(Float64(Float64(x + 1.0) / x)) / n);
	else
		tmp = Float64(fma(x, fma(x, Float64(Float64(0.5 / Float64(n * n)) + Float64(-0.5 / n)), Float64(1.0 / n)), 1.0) - t_0);
	end
	return tmp
end

Wolfram

code[x_, n_] := Block[{t$95$0 = N[Power[x, N[(1.0 / n), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[N[(1.0 / n), $MachinePrecision], -2e-28], N[(t$95$0 / N[(x * n), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 5e-11], N[(N[Log[N[(N[(x + 1.0), $MachinePrecision] / x), $MachinePrecision]], $MachinePrecision] / n), $MachinePrecision], N[(N[(x * N[(x * N[(N[(0.5 / N[(n * n), $MachinePrecision]), $MachinePrecision] + N[(-0.5 / n), $MachinePrecision]), $MachinePrecision] + N[(1.0 / n), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision] - t$95$0), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 #s(literal 1 binary64) n) < -1.99999999999999994e-28

   1. Initial program 92.5%

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

   3. Taylor expanded in x around inf

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

      1. /-lowering-/.f64 N/A

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

      2. log-rec N/A

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

      3. mul-1-neg N/A

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

      4. associate-*r/ N/A

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

      5. associate-*r* N/A

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

      6. metadata-eval N/A

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

      7. *-commutative N/A

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

      8. associate-/l* N/A

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

      9. exp-to-pow N/A

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

      10. pow-lowering-pow.f64 N/A

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

      11. /-lowering-/.f64 N/A

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

      12. *-commutative N/A

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

      13. *-lowering-*.f64 96.5

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

   5. Simplified 96.5%

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

   if -1.99999999999999994e-28 < (/.f64 #s(literal 1 binary64) n) < 5.00000000000000018e-11

   1. Initial program 25.0%

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

   3. Taylor expanded in n around inf

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

      1. /-lowering-/.f64 N/A

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

      2. --lowering--.f64 N/A

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

      3. accelerator-lowering-log1p.f64 N/A

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

      4. log-lowering-log.f64 80.1

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

   5. Simplified 80.1%

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

      1. diff-log N/A

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

      2. log-lowering-log.f64 N/A

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

      3. /-lowering-/.f64 N/A

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

      4. +-commutative N/A

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

      5. +-lowering-+.f64 80.3

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

   7. Applied egg-rr 80.3%

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

   if 5.00000000000000018e-11 < (/.f64 #s(literal 1 binary64) n)

   1. Initial program 51.2%

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

   3. Taylor expanded in x around 0

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

      1. remove-double-neg N/A

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

      2. mul-1-neg N/A

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

      3. distribute-neg-frac N/A

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

      4. mul-1-neg N/A

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

      5. log-rec N/A

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

      6. mul-1-neg N/A

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

      7. --lowering--.f64 N/A

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

   5. Simplified 66.1%

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

Alternative 7: 82.3% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{if}\;\frac{1}{n} \leq -2 \cdot 10^{-28}:\\ \;\;\;\;\frac{t\_0}{x \cdot n}\\ \mathbf{elif}\;\frac{1}{n} \leq 5 \cdot 10^{-11}:\\ \;\;\;\;\frac{\log \left(\frac{x + 1}{x}\right)}{n}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x, \mathsf{fma}\left(x, \frac{x \cdot 0.16666666666666666}{n \cdot \left(n \cdot n\right)}, \frac{1}{n}\right), 1 - t\_0\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (let* ((t_0 (pow x (/ 1.0 n))))
   (if (<= (/ 1.0 n) -2e-28)
     (/ t_0 (* x n))
     (if (<= (/ 1.0 n) 5e-11)
       (/ (log (/ (+ x 1.0) x)) n)
       (fma
        x
        (fma x (/ (* x 0.16666666666666666) (* n (* n n))) (/ 1.0 n))
        (- 1.0 t_0))))))

C

double code(double x, double n) {
	double t_0 = pow(x, (1.0 / n));
	double tmp;
	if ((1.0 / n) <= -2e-28) {
		tmp = t_0 / (x * n);
	} else if ((1.0 / n) <= 5e-11) {
		tmp = log(((x + 1.0) / x)) / n;
	} else {
		tmp = fma(x, fma(x, ((x * 0.16666666666666666) / (n * (n * n))), (1.0 / n)), (1.0 - t_0));
	}
	return tmp;
}

Julia

function code(x, n)
	t_0 = x ^ Float64(1.0 / n)
	tmp = 0.0
	if (Float64(1.0 / n) <= -2e-28)
		tmp = Float64(t_0 / Float64(x * n));
	elseif (Float64(1.0 / n) <= 5e-11)
		tmp = Float64(log(Float64(Float64(x + 1.0) / x)) / n);
	else
		tmp = fma(x, fma(x, Float64(Float64(x * 0.16666666666666666) / Float64(n * Float64(n * n))), Float64(1.0 / n)), Float64(1.0 - t_0));
	end
	return tmp
end

Wolfram

code[x_, n_] := Block[{t$95$0 = N[Power[x, N[(1.0 / n), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[N[(1.0 / n), $MachinePrecision], -2e-28], N[(t$95$0 / N[(x * n), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 5e-11], N[(N[Log[N[(N[(x + 1.0), $MachinePrecision] / x), $MachinePrecision]], $MachinePrecision] / n), $MachinePrecision], N[(x * N[(x * N[(N[(x * 0.16666666666666666), $MachinePrecision] / N[(n * N[(n * n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(1.0 / n), $MachinePrecision]), $MachinePrecision] + N[(1.0 - t$95$0), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 #s(literal 1 binary64) n) < -1.99999999999999994e-28

   1. Initial program 92.5%

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

   3. Taylor expanded in x around inf

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

      1. /-lowering-/.f64 N/A

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

      2. log-rec N/A

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

      3. mul-1-neg N/A

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

      4. associate-*r/ N/A

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

      5. associate-*r* N/A

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

      6. metadata-eval N/A

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

      7. *-commutative N/A

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

      8. associate-/l* N/A

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

      9. exp-to-pow N/A

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

      10. pow-lowering-pow.f64 N/A

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

      11. /-lowering-/.f64 N/A

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

      12. *-commutative N/A

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

      13. *-lowering-*.f64 96.5

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

   5. Simplified 96.5%

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

   if -1.99999999999999994e-28 < (/.f64 #s(literal 1 binary64) n) < 5.00000000000000018e-11

   1. Initial program 25.0%

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

   3. Taylor expanded in n around inf

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

      1. /-lowering-/.f64 N/A

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

      2. --lowering--.f64 N/A

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

      3. accelerator-lowering-log1p.f64 N/A

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

      4. log-lowering-log.f64 80.1

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

   5. Simplified 80.1%

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

      1. diff-log N/A

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

      2. log-lowering-log.f64 N/A

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

      3. /-lowering-/.f64 N/A

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

      4. +-commutative N/A

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

      5. +-lowering-+.f64 80.3

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

   7. Applied egg-rr 80.3%

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

   if 5.00000000000000018e-11 < (/.f64 #s(literal 1 binary64) n)

   1. Initial program 51.2%

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

   3. Taylor expanded in x around 0

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

   5. Simplified 34.2%

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

   6. Taylor expanded in n around 0

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

      1. associate-*r/ N/A

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

      2. /-lowering-/.f64 N/A

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

      3. *-commutative N/A

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

      4. *-lowering-*.f64 N/A

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

      5. cube-mult N/A

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

      6. unpow2 N/A

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

      7. *-lowering-*.f64 N/A

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

      8. unpow2 N/A

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

      9. *-lowering-*.f64 66.1

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

   8. Simplified 66.1%

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

Alternative 8: 80.7% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{if}\;\frac{1}{n} \leq -2 \cdot 10^{-28}:\\ \;\;\;\;\frac{t\_0}{x \cdot n}\\ \mathbf{elif}\;\frac{1}{n} \leq 5 \cdot 10^{-11}:\\ \;\;\;\;\frac{\log \left(\frac{x + 1}{x}\right)}{n}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x, \frac{0.16666666666666666 \cdot \left(x \cdot x\right)}{n \cdot \left(n \cdot n\right)}, 1 - t\_0\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (let* ((t_0 (pow x (/ 1.0 n))))
   (if (<= (/ 1.0 n) -2e-28)
     (/ t_0 (* x n))
     (if (<= (/ 1.0 n) 5e-11)
       (/ (log (/ (+ x 1.0) x)) n)
       (fma
        x
        (/ (* 0.16666666666666666 (* x x)) (* n (* n 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-28) {
		tmp = t_0 / (x * n);
	} else if ((1.0 / n) <= 5e-11) {
		tmp = log(((x + 1.0) / x)) / n;
	} else {
		tmp = fma(x, ((0.16666666666666666 * (x * x)) / (n * (n * 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-28)
		tmp = Float64(t_0 / Float64(x * n));
	elseif (Float64(1.0 / n) <= 5e-11)
		tmp = Float64(log(Float64(Float64(x + 1.0) / x)) / n);
	else
		tmp = fma(x, Float64(Float64(0.16666666666666666 * Float64(x * x)) / Float64(n * Float64(n * n))), Float64(1.0 - t_0));
	end
	return tmp
end

Wolfram

code[x_, n_] := Block[{t$95$0 = N[Power[x, N[(1.0 / n), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[N[(1.0 / n), $MachinePrecision], -2e-28], N[(t$95$0 / N[(x * n), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 5e-11], N[(N[Log[N[(N[(x + 1.0), $MachinePrecision] / x), $MachinePrecision]], $MachinePrecision] / n), $MachinePrecision], N[(x * N[(N[(0.16666666666666666 * N[(x * x), $MachinePrecision]), $MachinePrecision] / N[(n * N[(n * n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(1.0 - t$95$0), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 #s(literal 1 binary64) n) < -1.99999999999999994e-28

   1. Initial program 92.5%

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

   3. Taylor expanded in x around inf

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

      1. /-lowering-/.f64 N/A

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

      2. log-rec N/A

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

      3. mul-1-neg N/A

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

      4. associate-*r/ N/A

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

      5. associate-*r* N/A

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

      6. metadata-eval N/A

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

      7. *-commutative N/A

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

      8. associate-/l* N/A

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

      9. exp-to-pow N/A

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

      10. pow-lowering-pow.f64 N/A

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

      11. /-lowering-/.f64 N/A

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

      12. *-commutative N/A

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

      13. *-lowering-*.f64 96.5

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

   5. Simplified 96.5%

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

   if -1.99999999999999994e-28 < (/.f64 #s(literal 1 binary64) n) < 5.00000000000000018e-11

   1. Initial program 25.0%

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

   3. Taylor expanded in n around inf

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

      1. /-lowering-/.f64 N/A

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

      2. --lowering--.f64 N/A

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

      3. accelerator-lowering-log1p.f64 N/A

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

      4. log-lowering-log.f64 80.1

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

   5. Simplified 80.1%

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

      1. diff-log N/A

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

      2. log-lowering-log.f64 N/A

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

      3. /-lowering-/.f64 N/A

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

      4. +-commutative N/A

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

      5. +-lowering-+.f64 80.3

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

   7. Applied egg-rr 80.3%

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

   if 5.00000000000000018e-11 < (/.f64 #s(literal 1 binary64) n)

   1. Initial program 51.2%

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

   3. Taylor expanded in x around 0

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

   5. Simplified 34.2%

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

   6. Taylor expanded in n around 0

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

      1. associate-*r/ N/A

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

      2. /-lowering-/.f64 N/A

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

      3. *-lowering-*.f64 N/A

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

      4. unpow2 N/A

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

      5. *-lowering-*.f64 N/A

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

      6. cube-mult N/A

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

      7. unpow2 N/A

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

      8. *-lowering-*.f64 N/A

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

      9. unpow2 N/A

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

      10. *-lowering-*.f64 62.0

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

   8. Simplified 62.0%

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

Alternative 9: 82.5% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{if}\;\frac{1}{n} \leq -2 \cdot 10^{-28}:\\ \;\;\;\;\frac{t\_0}{x \cdot n}\\ \mathbf{elif}\;\frac{1}{n} \leq 5 \cdot 10^{-11}:\\ \;\;\;\;\frac{\log \left(\frac{x + 1}{x}\right)}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 2 \cdot 10^{+123}:\\ \;\;\;\;\left(\frac{x}{n} + 1\right) - t\_0\\ \mathbf{else}:\\ \;\;\;\;x \cdot \mathsf{fma}\left(x, \frac{x \cdot 0.16666666666666666}{n \cdot \left(n \cdot n\right)}, \frac{1}{n}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (let* ((t_0 (pow x (/ 1.0 n))))
   (if (<= (/ 1.0 n) -2e-28)
     (/ t_0 (* x n))
     (if (<= (/ 1.0 n) 5e-11)
       (/ (log (/ (+ x 1.0) x)) n)
       (if (<= (/ 1.0 n) 2e+123)
         (- (+ (/ x n) 1.0) t_0)
         (*
          x
          (fma x (/ (* x 0.16666666666666666) (* n (* n n))) (/ 1.0 n))))))))

C

double code(double x, double n) {
	double t_0 = pow(x, (1.0 / n));
	double tmp;
	if ((1.0 / n) <= -2e-28) {
		tmp = t_0 / (x * n);
	} else if ((1.0 / n) <= 5e-11) {
		tmp = log(((x + 1.0) / x)) / n;
	} else if ((1.0 / n) <= 2e+123) {
		tmp = ((x / n) + 1.0) - t_0;
	} else {
		tmp = x * fma(x, ((x * 0.16666666666666666) / (n * (n * n))), (1.0 / n));
	}
	return tmp;
}

Julia

function code(x, n)
	t_0 = x ^ Float64(1.0 / n)
	tmp = 0.0
	if (Float64(1.0 / n) <= -2e-28)
		tmp = Float64(t_0 / Float64(x * n));
	elseif (Float64(1.0 / n) <= 5e-11)
		tmp = Float64(log(Float64(Float64(x + 1.0) / x)) / n);
	elseif (Float64(1.0 / n) <= 2e+123)
		tmp = Float64(Float64(Float64(x / n) + 1.0) - t_0);
	else
		tmp = Float64(x * fma(x, Float64(Float64(x * 0.16666666666666666) / Float64(n * Float64(n * n))), Float64(1.0 / 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-28], N[(t$95$0 / N[(x * n), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 5e-11], N[(N[Log[N[(N[(x + 1.0), $MachinePrecision] / x), $MachinePrecision]], $MachinePrecision] / n), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 2e+123], N[(N[(N[(x / n), $MachinePrecision] + 1.0), $MachinePrecision] - t$95$0), $MachinePrecision], N[(x * N[(x * N[(N[(x * 0.16666666666666666), $MachinePrecision] / N[(n * N[(n * n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(1.0 / n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if (/.f64 #s(literal 1 binary64) n) < -1.99999999999999994e-28

   1. Initial program 92.5%

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

   3. Taylor expanded in x around inf

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

      1. /-lowering-/.f64 N/A

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

      2. log-rec N/A

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

      3. mul-1-neg N/A

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

      4. associate-*r/ N/A

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

      5. associate-*r* N/A

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

      6. metadata-eval N/A

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

      7. *-commutative N/A

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

      8. associate-/l* N/A

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

      9. exp-to-pow N/A

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

      10. pow-lowering-pow.f64 N/A

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

      11. /-lowering-/.f64 N/A

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

      12. *-commutative N/A

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

      13. *-lowering-*.f64 96.5

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

   5. Simplified 96.5%

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

   if -1.99999999999999994e-28 < (/.f64 #s(literal 1 binary64) n) < 5.00000000000000018e-11

   1. Initial program 25.0%

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

   3. Taylor expanded in n around inf

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

      1. /-lowering-/.f64 N/A

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

      2. --lowering--.f64 N/A

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

      3. accelerator-lowering-log1p.f64 N/A

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

      4. log-lowering-log.f64 80.1

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

   5. Simplified 80.1%

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

      1. diff-log N/A

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

      2. log-lowering-log.f64 N/A

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

      3. /-lowering-/.f64 N/A

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

      4. +-commutative N/A

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

      5. +-lowering-+.f64 80.3

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

   7. Applied egg-rr 80.3%

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

   if 5.00000000000000018e-11 < (/.f64 #s(literal 1 binary64) n) < 1.99999999999999996e123

   1. Initial program 65.2%

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

   3. Taylor expanded in x around 0

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

      1. *-rgt-identity N/A

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

      2. associate-*r/ N/A

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

      3. +-lowering-+.f64 N/A

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

      4. associate-*r/ N/A

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

      5. *-rgt-identity N/A

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

      6. /-lowering-/.f64 66.3

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

   5. Simplified 66.3%

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

   if 1.99999999999999996e123 < (/.f64 #s(literal 1 binary64) n)

   1. Initial program 36.4%

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

   3. Taylor expanded in n around inf

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

   5. Simplified 7.8%

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

   6. Taylor expanded in x around 0

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

      1. *-lowering-*.f64 N/A

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

      2. accelerator-lowering-fma.f64 N/A

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

   8. Simplified 5.9%

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

   9. Taylor expanded in n around 0

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

       1. associate-*r/ N/A

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

       2. /-lowering-/.f64 N/A

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

       3. *-commutative N/A

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

       4. *-lowering-*.f64 N/A

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

       5. cube-mult N/A

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

       6. unpow2 N/A

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

       7. *-lowering-*.f64 N/A

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

       8. unpow2 N/A

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

       9. *-lowering-*.f64 71.5

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

   11. Simplified 71.5%

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

4. Final simplification 83.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{1}{n} \leq -2 \cdot 10^{-28}:\\ \;\;\;\;\frac{{x}^{\left(\frac{1}{n}\right)}}{x \cdot n}\\ \mathbf{elif}\;\frac{1}{n} \leq 5 \cdot 10^{-11}:\\ \;\;\;\;\frac{\log \left(\frac{x + 1}{x}\right)}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 2 \cdot 10^{+123}:\\ \;\;\;\;\left(\frac{x}{n} + 1\right) - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \mathsf{fma}\left(x, \frac{x \cdot 0.16666666666666666}{n \cdot \left(n \cdot n\right)}, \frac{1}{n}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 82.2% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{if}\;\frac{1}{n} \leq -2 \cdot 10^{-28}:\\ \;\;\;\;\frac{t\_0}{x \cdot n}\\ \mathbf{elif}\;\frac{1}{n} \leq 5 \cdot 10^{-11}:\\ \;\;\;\;\frac{\log \left(\frac{x + 1}{x}\right)}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 10^{+113}:\\ \;\;\;\;1 - t\_0\\ \mathbf{else}:\\ \;\;\;\;x \cdot \mathsf{fma}\left(x, \frac{x \cdot 0.16666666666666666}{n \cdot \left(n \cdot n\right)}, \frac{1}{n}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (let* ((t_0 (pow x (/ 1.0 n))))
   (if (<= (/ 1.0 n) -2e-28)
     (/ t_0 (* x n))
     (if (<= (/ 1.0 n) 5e-11)
       (/ (log (/ (+ x 1.0) x)) n)
       (if (<= (/ 1.0 n) 1e+113)
         (- 1.0 t_0)
         (*
          x
          (fma x (/ (* x 0.16666666666666666) (* n (* n n))) (/ 1.0 n))))))))

C

double code(double x, double n) {
	double t_0 = pow(x, (1.0 / n));
	double tmp;
	if ((1.0 / n) <= -2e-28) {
		tmp = t_0 / (x * n);
	} else if ((1.0 / n) <= 5e-11) {
		tmp = log(((x + 1.0) / x)) / n;
	} else if ((1.0 / n) <= 1e+113) {
		tmp = 1.0 - t_0;
	} else {
		tmp = x * fma(x, ((x * 0.16666666666666666) / (n * (n * n))), (1.0 / n));
	}
	return tmp;
}

Julia

function code(x, n)
	t_0 = x ^ Float64(1.0 / n)
	tmp = 0.0
	if (Float64(1.0 / n) <= -2e-28)
		tmp = Float64(t_0 / Float64(x * n));
	elseif (Float64(1.0 / n) <= 5e-11)
		tmp = Float64(log(Float64(Float64(x + 1.0) / x)) / n);
	elseif (Float64(1.0 / n) <= 1e+113)
		tmp = Float64(1.0 - t_0);
	else
		tmp = Float64(x * fma(x, Float64(Float64(x * 0.16666666666666666) / Float64(n * Float64(n * n))), Float64(1.0 / 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-28], N[(t$95$0 / N[(x * n), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 5e-11], N[(N[Log[N[(N[(x + 1.0), $MachinePrecision] / x), $MachinePrecision]], $MachinePrecision] / n), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 1e+113], N[(1.0 - t$95$0), $MachinePrecision], N[(x * N[(x * N[(N[(x * 0.16666666666666666), $MachinePrecision] / N[(n * N[(n * n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(1.0 / n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if (/.f64 #s(literal 1 binary64) n) < -1.99999999999999994e-28

   1. Initial program 92.5%

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

   3. Taylor expanded in x around inf

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

      1. /-lowering-/.f64 N/A

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

      2. log-rec N/A

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

      3. mul-1-neg N/A

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

      4. associate-*r/ N/A

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

      5. associate-*r* N/A

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

      6. metadata-eval N/A

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

      7. *-commutative N/A

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

      8. associate-/l* N/A

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

      9. exp-to-pow N/A

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

      10. pow-lowering-pow.f64 N/A

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

      11. /-lowering-/.f64 N/A

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

      12. *-commutative N/A

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

      13. *-lowering-*.f64 96.5

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

   5. Simplified 96.5%

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

   if -1.99999999999999994e-28 < (/.f64 #s(literal 1 binary64) n) < 5.00000000000000018e-11

   1. Initial program 25.0%

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

   3. Taylor expanded in n around inf

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

      1. /-lowering-/.f64 N/A

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

      2. --lowering--.f64 N/A

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

      3. accelerator-lowering-log1p.f64 N/A

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

      4. log-lowering-log.f64 80.1

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

   5. Simplified 80.1%

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

      1. diff-log N/A

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

      2. log-lowering-log.f64 N/A

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

      3. /-lowering-/.f64 N/A

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

      4. +-commutative N/A

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

      5. +-lowering-+.f64 80.3

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

   7. Applied egg-rr 80.3%

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

   if 5.00000000000000018e-11 < (/.f64 #s(literal 1 binary64) n) < 1e113

   1. Initial program 66.9%

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

   3. Taylor expanded in x around 0

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

      1. remove-double-neg N/A

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

      2. mul-1-neg N/A

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

      3. distribute-neg-frac N/A

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

      4. mul-1-neg N/A

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

      5. log-rec N/A

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

      6. mul-1-neg N/A

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

      7. --lowering--.f64 N/A

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

      8. log-rec N/A

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

      9. mul-1-neg N/A

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

      10. associate-*r/ N/A

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

      11. associate-*r* N/A

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

      12. metadata-eval N/A

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

      13. *-commutative N/A

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

      14. associate-/l* N/A

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

      15. exp-to-pow N/A

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

      16. pow-lowering-pow.f64 N/A

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

      17. /-lowering-/.f64 66.6

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

   5. Simplified 66.6%

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

   if 1e113 < (/.f64 #s(literal 1 binary64) n)

   1. Initial program 38.0%

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

   3. Taylor expanded in n around inf

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

   5. Simplified 7.2%

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

   6. Taylor expanded in x around 0

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

      1. *-lowering-*.f64 N/A

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

      2. accelerator-lowering-fma.f64 N/A

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

   8. Simplified 10.7%

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

   9. Taylor expanded in n around 0

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

       1. associate-*r/ N/A

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

       2. /-lowering-/.f64 N/A

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

       3. *-commutative N/A

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

       4. *-lowering-*.f64 N/A

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

       5. cube-mult N/A

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

       6. unpow2 N/A

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

       7. *-lowering-*.f64 N/A

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

       8. unpow2 N/A

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

       9. *-lowering-*.f64 69.4

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

   11. Simplified 69.4%

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

Alternative 11: 60.7% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 0.9:\\ \;\;\;\;\frac{x - \log x}{n}\\ \mathbf{elif}\;x \leq 10^{+118}:\\ \;\;\;\;\frac{\frac{\frac{-0.5 - \frac{\frac{0.25}{x} + -0.3333333333333333}{x}}{x} + 1}{x}}{n}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (if (<= x 0.9)
   (/ (- x (log x)) n)
   (if (<= x 1e+118)
     (/
      (/ (+ (/ (- -0.5 (/ (+ (/ 0.25 x) -0.3333333333333333) x)) x) 1.0) x)
      n)
     0.0)))

C

double code(double x, double n) {
	double tmp;
	if (x <= 0.9) {
		tmp = (x - log(x)) / n;
	} else if (x <= 1e+118) {
		tmp = ((((-0.5 - (((0.25 / x) + -0.3333333333333333) / x)) / x) + 1.0) / x) / n;
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Fortran

real(8) function code(x, n)
    real(8), intent (in) :: x
    real(8), intent (in) :: n
    real(8) :: tmp
    if (x <= 0.9d0) then
        tmp = (x - log(x)) / n
    else if (x <= 1d+118) then
        tmp = (((((-0.5d0) - (((0.25d0 / x) + (-0.3333333333333333d0)) / x)) / x) + 1.0d0) / x) / n
    else
        tmp = 0.0d0
    end if
    code = tmp
end function

Java

public static double code(double x, double n) {
	double tmp;
	if (x <= 0.9) {
		tmp = (x - Math.log(x)) / n;
	} else if (x <= 1e+118) {
		tmp = ((((-0.5 - (((0.25 / x) + -0.3333333333333333) / x)) / x) + 1.0) / x) / n;
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Python

def code(x, n):
	tmp = 0
	if x <= 0.9:
		tmp = (x - math.log(x)) / n
	elif x <= 1e+118:
		tmp = ((((-0.5 - (((0.25 / x) + -0.3333333333333333) / x)) / x) + 1.0) / x) / n
	else:
		tmp = 0.0
	return tmp

Julia

function code(x, n)
	tmp = 0.0
	if (x <= 0.9)
		tmp = Float64(Float64(x - log(x)) / n);
	elseif (x <= 1e+118)
		tmp = Float64(Float64(Float64(Float64(Float64(-0.5 - Float64(Float64(Float64(0.25 / x) + -0.3333333333333333) / x)) / x) + 1.0) / x) / n);
	else
		tmp = 0.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, n)
	tmp = 0.0;
	if (x <= 0.9)
		tmp = (x - log(x)) / n;
	elseif (x <= 1e+118)
		tmp = ((((-0.5 - (((0.25 / x) + -0.3333333333333333) / x)) / x) + 1.0) / x) / n;
	else
		tmp = 0.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, n_] := If[LessEqual[x, 0.9], N[(N[(x - N[Log[x], $MachinePrecision]), $MachinePrecision] / n), $MachinePrecision], If[LessEqual[x, 1e+118], N[(N[(N[(N[(N[(-0.5 - N[(N[(N[(0.25 / x), $MachinePrecision] + -0.3333333333333333), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision] + 1.0), $MachinePrecision] / x), $MachinePrecision] / n), $MachinePrecision], 0.0]]

Derivation

1. Split input into 3 regimes

2. if x < 0.900000000000000022

   1. Initial program 37.0%

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

   3. Taylor expanded in x around 0

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

      1. *-rgt-identity N/A

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

      2. associate-*r/ N/A

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

      3. +-lowering-+.f64 N/A

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

      4. associate-*r/ N/A

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

      5. *-rgt-identity N/A

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

      6. /-lowering-/.f64 36.8

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

   5. Simplified 36.8%

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

   6. Taylor expanded in n around inf

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

      1. /-lowering-/.f64 N/A

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

      2. --lowering--.f64 N/A

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

      3. log-lowering-log.f64 58.6

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

   8. Simplified 58.6%

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

   if 0.900000000000000022 < x < 9.99999999999999967e117

   1. Initial program 37.7%

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

   3. Taylor expanded in n around inf

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

      1. /-lowering-/.f64 N/A

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

      2. --lowering--.f64 N/A

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

      3. accelerator-lowering-log1p.f64 N/A

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

      4. log-lowering-log.f64 41.9

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

   5. Simplified 41.9%

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

   6. Taylor expanded in x around -inf

      \[\leadsto \frac{\color{blue}{-1 \cdot \frac{-1 \cdot \frac{-1 \cdot \frac{\frac{1}{4} \cdot \frac{1}{x} - \frac{1}{3}}{x} - \frac{1}{2}}{x} - 1}{x}}}{n} \]
   7. Step-by-step derivation

      1. mul-1-neg N/A

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

      2. distribute-neg-frac2 N/A

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

      3. mul-1-neg N/A

         \[\leadsto \frac{\frac{-1 \cdot \frac{-1 \cdot \frac{\frac{1}{4} \cdot \frac{1}{x} - \frac{1}{3}}{x} - \frac{1}{2}}{x} - 1}{\color{blue}{-1 \cdot x}}}{n} \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \frac{\color{blue}{\frac{-1 \cdot \frac{-1 \cdot \frac{\frac{1}{4} \cdot \frac{1}{x} - \frac{1}{3}}{x} - \frac{1}{2}}{x} - 1}{-1 \cdot x}}}{n} \]

   8. Simplified 68.1%

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

   if 9.99999999999999967e117 < x

   1. Initial program 84.4%

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

   3. Taylor expanded in x around 0

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

      1. remove-double-neg N/A

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

      2. mul-1-neg N/A

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

      3. distribute-neg-frac N/A

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

      4. mul-1-neg N/A

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

      5. log-rec N/A

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

      6. mul-1-neg N/A

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

      7. --lowering--.f64 N/A

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

      8. log-rec N/A

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

      9. mul-1-neg N/A

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

      10. associate-*r/ N/A

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

      11. associate-*r* N/A

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

      12. metadata-eval N/A

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

      13. *-commutative N/A

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

      14. associate-/l* N/A

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

      15. exp-to-pow N/A

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

      16. pow-lowering-pow.f64 N/A

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

      17. /-lowering-/.f64 48.0

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

   5. Simplified 48.0%

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

   6. Taylor expanded in n around inf

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

   8. Simplified 84.4%

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

      1. metadata-eval 84.4

         \[\leadsto \color{blue}{0} \]

   10. Applied egg-rr 84.4%

       \[\leadsto \color{blue}{0} \]
3. Recombined 3 regimes into one program.

4. Final simplification 66.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 0.9:\\ \;\;\;\;\frac{x - \log x}{n}\\ \mathbf{elif}\;x \leq 10^{+118}:\\ \;\;\;\;\frac{\frac{\frac{-0.5 - \frac{\frac{0.25}{x} + -0.3333333333333333}{x}}{x} + 1}{x}}{n}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 60.5% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 0.7:\\ \;\;\;\;-\frac{\log x}{n}\\ \mathbf{elif}\;x \leq 7 \cdot 10^{+118}:\\ \;\;\;\;\frac{\frac{\frac{-0.5 - \frac{\frac{0.25}{x} + -0.3333333333333333}{x}}{x} + 1}{x}}{n}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (if (<= x 0.7)
   (- (/ (log x) n))
   (if (<= x 7e+118)
     (/
      (/ (+ (/ (- -0.5 (/ (+ (/ 0.25 x) -0.3333333333333333) x)) x) 1.0) x)
      n)
     0.0)))

C

double code(double x, double n) {
	double tmp;
	if (x <= 0.7) {
		tmp = -(log(x) / n);
	} else if (x <= 7e+118) {
		tmp = ((((-0.5 - (((0.25 / x) + -0.3333333333333333) / x)) / x) + 1.0) / x) / n;
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Fortran

real(8) function code(x, n)
    real(8), intent (in) :: x
    real(8), intent (in) :: n
    real(8) :: tmp
    if (x <= 0.7d0) then
        tmp = -(log(x) / n)
    else if (x <= 7d+118) then
        tmp = (((((-0.5d0) - (((0.25d0 / x) + (-0.3333333333333333d0)) / x)) / x) + 1.0d0) / x) / n
    else
        tmp = 0.0d0
    end if
    code = tmp
end function

Java

public static double code(double x, double n) {
	double tmp;
	if (x <= 0.7) {
		tmp = -(Math.log(x) / n);
	} else if (x <= 7e+118) {
		tmp = ((((-0.5 - (((0.25 / x) + -0.3333333333333333) / x)) / x) + 1.0) / x) / n;
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Python

def code(x, n):
	tmp = 0
	if x <= 0.7:
		tmp = -(math.log(x) / n)
	elif x <= 7e+118:
		tmp = ((((-0.5 - (((0.25 / x) + -0.3333333333333333) / x)) / x) + 1.0) / x) / n
	else:
		tmp = 0.0
	return tmp

Julia

function code(x, n)
	tmp = 0.0
	if (x <= 0.7)
		tmp = Float64(-Float64(log(x) / n));
	elseif (x <= 7e+118)
		tmp = Float64(Float64(Float64(Float64(Float64(-0.5 - Float64(Float64(Float64(0.25 / x) + -0.3333333333333333) / x)) / x) + 1.0) / x) / n);
	else
		tmp = 0.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, n)
	tmp = 0.0;
	if (x <= 0.7)
		tmp = -(log(x) / n);
	elseif (x <= 7e+118)
		tmp = ((((-0.5 - (((0.25 / x) + -0.3333333333333333) / x)) / x) + 1.0) / x) / n;
	else
		tmp = 0.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, n_] := If[LessEqual[x, 0.7], (-N[(N[Log[x], $MachinePrecision] / n), $MachinePrecision]), If[LessEqual[x, 7e+118], N[(N[(N[(N[(N[(-0.5 - N[(N[(N[(0.25 / x), $MachinePrecision] + -0.3333333333333333), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision] + 1.0), $MachinePrecision] / x), $MachinePrecision] / n), $MachinePrecision], 0.0]]

Derivation

1. Split input into 3 regimes

2. if x < 0.69999999999999996

   1. Initial program 37.0%

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

   3. Taylor expanded in x around 0

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

      1. remove-double-neg N/A

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

      2. mul-1-neg N/A

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

      3. distribute-neg-frac N/A

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

      4. mul-1-neg N/A

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

      5. log-rec N/A

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

      6. mul-1-neg N/A

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

      7. --lowering--.f64 N/A

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

      8. log-rec N/A

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

      9. mul-1-neg N/A

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

      10. associate-*r/ N/A

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

      11. associate-*r* N/A

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

      12. metadata-eval N/A

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

      13. *-commutative N/A

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

      14. associate-/l* N/A

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

      15. exp-to-pow N/A

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

      16. pow-lowering-pow.f64 N/A

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

      17. /-lowering-/.f64 36.4

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

   5. Simplified 36.4%

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

   6. Taylor expanded in n around inf

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

      1. mul-1-neg N/A

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

      2. neg-lowering-neg.f64 N/A

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

      3. /-lowering-/.f64 N/A

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

      4. log-lowering-log.f64 57.6

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

   8. Simplified 57.6%

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

   if 0.69999999999999996 < x < 7.00000000000000033e118

   1. Initial program 37.7%

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

   3. Taylor expanded in n around inf

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

      1. /-lowering-/.f64 N/A

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

      2. --lowering--.f64 N/A

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

      3. accelerator-lowering-log1p.f64 N/A

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

      4. log-lowering-log.f64 41.9

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

   5. Simplified 41.9%

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

   6. Taylor expanded in x around -inf

      \[\leadsto \frac{\color{blue}{-1 \cdot \frac{-1 \cdot \frac{-1 \cdot \frac{\frac{1}{4} \cdot \frac{1}{x} - \frac{1}{3}}{x} - \frac{1}{2}}{x} - 1}{x}}}{n} \]
   7. Step-by-step derivation

      1. mul-1-neg N/A

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

      2. distribute-neg-frac2 N/A

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

      3. mul-1-neg N/A

         \[\leadsto \frac{\frac{-1 \cdot \frac{-1 \cdot \frac{\frac{1}{4} \cdot \frac{1}{x} - \frac{1}{3}}{x} - \frac{1}{2}}{x} - 1}{\color{blue}{-1 \cdot x}}}{n} \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \frac{\color{blue}{\frac{-1 \cdot \frac{-1 \cdot \frac{\frac{1}{4} \cdot \frac{1}{x} - \frac{1}{3}}{x} - \frac{1}{2}}{x} - 1}{-1 \cdot x}}}{n} \]

   8. Simplified 68.1%

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

   if 7.00000000000000033e118 < x

   1. Initial program 84.4%

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

   3. Taylor expanded in x around 0

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

      1. remove-double-neg N/A

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

      2. mul-1-neg N/A

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

      3. distribute-neg-frac N/A

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

      4. mul-1-neg N/A

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

      5. log-rec N/A

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

      6. mul-1-neg N/A

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

      7. --lowering--.f64 N/A

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

      8. log-rec N/A

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

      9. mul-1-neg N/A

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

      10. associate-*r/ N/A

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

      11. associate-*r* N/A

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

      12. metadata-eval N/A

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

      13. *-commutative N/A

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

      14. associate-/l* N/A

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

      15. exp-to-pow N/A

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

      16. pow-lowering-pow.f64 N/A

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

      17. /-lowering-/.f64 48.0

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

   5. Simplified 48.0%

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

   6. Taylor expanded in n around inf

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

   8. Simplified 84.4%

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

      1. metadata-eval 84.4

         \[\leadsto \color{blue}{0} \]

   10. Applied egg-rr 84.4%

       \[\leadsto \color{blue}{0} \]
3. Recombined 3 regimes into one program.

4. Final simplification 65.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 0.7:\\ \;\;\;\;-\frac{\log x}{n}\\ \mathbf{elif}\;x \leq 7 \cdot 10^{+118}:\\ \;\;\;\;\frac{\frac{\frac{-0.5 - \frac{\frac{0.25}{x} + -0.3333333333333333}{x}}{x} + 1}{x}}{n}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 50.3% accurate, 4.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 5 \cdot 10^{+117}:\\ \;\;\;\;\frac{\frac{\frac{-0.5 + \frac{0.3333333333333333}{x}}{x} + 1}{x}}{n}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (if (<= x 5e+117)
   (/ (/ (+ (/ (+ -0.5 (/ 0.3333333333333333 x)) x) 1.0) x) n)
   0.0))

C

double code(double x, double n) {
	double tmp;
	if (x <= 5e+117) {
		tmp = ((((-0.5 + (0.3333333333333333 / x)) / x) + 1.0) / x) / n;
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Fortran

real(8) function code(x, n)
    real(8), intent (in) :: x
    real(8), intent (in) :: n
    real(8) :: tmp
    if (x <= 5d+117) then
        tmp = (((((-0.5d0) + (0.3333333333333333d0 / x)) / x) + 1.0d0) / x) / n
    else
        tmp = 0.0d0
    end if
    code = tmp
end function

Java

public static double code(double x, double n) {
	double tmp;
	if (x <= 5e+117) {
		tmp = ((((-0.5 + (0.3333333333333333 / x)) / x) + 1.0) / x) / n;
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Python

def code(x, n):
	tmp = 0
	if x <= 5e+117:
		tmp = ((((-0.5 + (0.3333333333333333 / x)) / x) + 1.0) / x) / n
	else:
		tmp = 0.0
	return tmp

Julia

function code(x, n)
	tmp = 0.0
	if (x <= 5e+117)
		tmp = Float64(Float64(Float64(Float64(Float64(-0.5 + Float64(0.3333333333333333 / x)) / x) + 1.0) / x) / n);
	else
		tmp = 0.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, n)
	tmp = 0.0;
	if (x <= 5e+117)
		tmp = ((((-0.5 + (0.3333333333333333 / x)) / x) + 1.0) / x) / n;
	else
		tmp = 0.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, n_] := If[LessEqual[x, 5e+117], N[(N[(N[(N[(N[(-0.5 + N[(0.3333333333333333 / x), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision] + 1.0), $MachinePrecision] / x), $MachinePrecision] / n), $MachinePrecision], 0.0]

Derivation

1. Split input into 2 regimes

2. if x < 4.99999999999999983e117

   1. Initial program 37.2%

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

   3. Taylor expanded in n around inf

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

      1. /-lowering-/.f64 N/A

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

      2. --lowering--.f64 N/A

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

      3. accelerator-lowering-log1p.f64 N/A

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

      4. log-lowering-log.f64 55.2

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

   5. Simplified 55.2%

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

   6. Taylor expanded in x around -inf

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

      1. mul-1-neg N/A

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

      2. distribute-neg-frac2 N/A

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

      3. mul-1-neg N/A

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

      4. /-lowering-/.f64 N/A

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

   8. Simplified 38.3%

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

   if 4.99999999999999983e117 < x

   1. Initial program 84.4%

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

   3. Taylor expanded in x around 0

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

      1. remove-double-neg N/A

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

      2. mul-1-neg N/A

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

      3. distribute-neg-frac N/A

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

      4. mul-1-neg N/A

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

      5. log-rec N/A

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

      6. mul-1-neg N/A

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

      7. --lowering--.f64 N/A

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

      8. log-rec N/A

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

      9. mul-1-neg N/A

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

      10. associate-*r/ N/A

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

      11. associate-*r* N/A

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

      12. metadata-eval N/A

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

      13. *-commutative N/A

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

      14. associate-/l* N/A

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

      15. exp-to-pow N/A

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

      16. pow-lowering-pow.f64 N/A

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

      17. /-lowering-/.f64 48.0

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

   5. Simplified 48.0%

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

   6. Taylor expanded in n around inf

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

   8. Simplified 84.4%

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

      1. metadata-eval 84.4

         \[\leadsto \color{blue}{0} \]

   10. Applied egg-rr 84.4%

       \[\leadsto \color{blue}{0} \]
3. Recombined 2 regimes into one program.

4. Final simplification 48.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 5 \cdot 10^{+117}:\\ \;\;\;\;\frac{\frac{\frac{-0.5 + \frac{0.3333333333333333}{x}}{x} + 1}{x}}{n}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 49.3% accurate, 4.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 1:\\ \;\;\;\;x \cdot \mathsf{fma}\left(x, \frac{x \cdot 0.16666666666666666}{n \cdot \left(n \cdot n\right)}, \frac{1}{n}\right)\\ \mathbf{elif}\;x \leq 1.8 \cdot 10^{+118}:\\ \;\;\;\;\frac{\frac{\frac{-0.5}{x} + 1}{x}}{n}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (if (<= x 1.0)
   (* x (fma x (/ (* x 0.16666666666666666) (* n (* n n))) (/ 1.0 n)))
   (if (<= x 1.8e+118) (/ (/ (+ (/ -0.5 x) 1.0) x) n) 0.0)))

C

double code(double x, double n) {
	double tmp;
	if (x <= 1.0) {
		tmp = x * fma(x, ((x * 0.16666666666666666) / (n * (n * n))), (1.0 / n));
	} else if (x <= 1.8e+118) {
		tmp = (((-0.5 / x) + 1.0) / x) / n;
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Julia

function code(x, n)
	tmp = 0.0
	if (x <= 1.0)
		tmp = Float64(x * fma(x, Float64(Float64(x * 0.16666666666666666) / Float64(n * Float64(n * n))), Float64(1.0 / n)));
	elseif (x <= 1.8e+118)
		tmp = Float64(Float64(Float64(Float64(-0.5 / x) + 1.0) / x) / n);
	else
		tmp = 0.0;
	end
	return tmp
end

Wolfram

code[x_, n_] := If[LessEqual[x, 1.0], N[(x * N[(x * N[(N[(x * 0.16666666666666666), $MachinePrecision] / N[(n * N[(n * n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(1.0 / n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 1.8e+118], N[(N[(N[(N[(-0.5 / x), $MachinePrecision] + 1.0), $MachinePrecision] / x), $MachinePrecision] / n), $MachinePrecision], 0.0]]

Derivation

1. Split input into 3 regimes

2. if x < 1

   1. Initial program 37.0%

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

   3. Taylor expanded in n around inf

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

   5. Simplified 4.7%

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

   6. Taylor expanded in x around 0

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

      1. *-lowering-*.f64 N/A

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

      2. accelerator-lowering-fma.f64 N/A

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

   8. Simplified 7.7%

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

   9. Taylor expanded in n around 0

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

       1. associate-*r/ N/A

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

       2. /-lowering-/.f64 N/A

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

       3. *-commutative N/A

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

       4. *-lowering-*.f64 N/A

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

       5. cube-mult N/A

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

       6. unpow2 N/A

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

       7. *-lowering-*.f64 N/A

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

       8. unpow2 N/A

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

       9. *-lowering-*.f64 25.6

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

   11. Simplified 25.6%

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

   if 1 < x < 1.8e118

   1. Initial program 37.7%

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

   3. Taylor expanded in n around inf

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

      1. /-lowering-/.f64 N/A

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

      2. --lowering--.f64 N/A

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

      3. accelerator-lowering-log1p.f64 N/A

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

      4. log-lowering-log.f64 41.9

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

   5. Simplified 41.9%

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

   6. Taylor expanded in x around -inf

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

      1. associate-*r/ N/A

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

      2. /-lowering-/.f64 N/A

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

      3. sub-neg N/A

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

      4. metadata-eval N/A

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

      5. distribute-lft-in N/A

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

      6. neg-mul-1 N/A

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

      7. metadata-eval N/A

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

      8. +-lowering-+.f64 N/A

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

      9. associate-*r/ N/A

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

      10. metadata-eval N/A

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

      11. distribute-neg-frac N/A

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

      12. metadata-eval N/A

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

      13. /-lowering-/.f64 66.8

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

   8. Simplified 66.8%

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

   if 1.8e118 < x

   1. Initial program 84.4%

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

   3. Taylor expanded in x around 0

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

      1. remove-double-neg N/A

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

      2. mul-1-neg N/A

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

      3. distribute-neg-frac N/A

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

      4. mul-1-neg N/A

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

      5. log-rec N/A

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

      6. mul-1-neg N/A

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

      7. --lowering--.f64 N/A

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

      8. log-rec N/A

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

      9. mul-1-neg N/A

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

      10. associate-*r/ N/A

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

      11. associate-*r* N/A

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

      12. metadata-eval N/A

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

      13. *-commutative N/A

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

      14. associate-/l* N/A

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

      15. exp-to-pow N/A

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

      16. pow-lowering-pow.f64 N/A

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

      17. /-lowering-/.f64 48.0

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

   5. Simplified 48.0%

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

   6. Taylor expanded in n around inf

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

   8. Simplified 84.4%

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

      1. metadata-eval 84.4

         \[\leadsto \color{blue}{0} \]

   10. Applied egg-rr 84.4%

       \[\leadsto \color{blue}{0} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 15: 47.1% accurate, 5.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{1}{n} \leq -1000:\\ \;\;\;\;0\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{x}}{n}\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (if (<= (/ 1.0 n) -1000.0) 0.0 (/ (/ 1.0 x) n)))

C

double code(double x, double n) {
	double tmp;
	if ((1.0 / n) <= -1000.0) {
		tmp = 0.0;
	} else {
		tmp = (1.0 / x) / n;
	}
	return tmp;
}

Fortran

real(8) function code(x, n)
    real(8), intent (in) :: x
    real(8), intent (in) :: n
    real(8) :: tmp
    if ((1.0d0 / n) <= (-1000.0d0)) then
        tmp = 0.0d0
    else
        tmp = (1.0d0 / x) / n
    end if
    code = tmp
end function

Java

public static double code(double x, double n) {
	double tmp;
	if ((1.0 / n) <= -1000.0) {
		tmp = 0.0;
	} else {
		tmp = (1.0 / x) / n;
	}
	return tmp;
}

Python

def code(x, n):
	tmp = 0
	if (1.0 / n) <= -1000.0:
		tmp = 0.0
	else:
		tmp = (1.0 / x) / n
	return tmp

Julia

function code(x, n)
	tmp = 0.0
	if (Float64(1.0 / n) <= -1000.0)
		tmp = 0.0;
	else
		tmp = Float64(Float64(1.0 / x) / n);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, n)
	tmp = 0.0;
	if ((1.0 / n) <= -1000.0)
		tmp = 0.0;
	else
		tmp = (1.0 / x) / n;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, n_] := If[LessEqual[N[(1.0 / n), $MachinePrecision], -1000.0], 0.0, N[(N[(1.0 / x), $MachinePrecision] / n), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

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

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

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

      1. remove-double-neg N/A

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

      2. mul-1-neg N/A

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

      3. distribute-neg-frac N/A

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

      4. mul-1-neg N/A

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

      5. log-rec N/A

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

      6. mul-1-neg N/A

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

      7. --lowering--.f64 N/A

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

      8. log-rec N/A

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

      9. mul-1-neg N/A

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

      10. associate-*r/ N/A

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

      11. associate-*r* N/A

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

      12. metadata-eval N/A

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

      13. *-commutative N/A

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

      14. associate-/l* N/A

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

      15. exp-to-pow N/A

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

      16. pow-lowering-pow.f64 N/A

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

      17. /-lowering-/.f64 49.4

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

   5. Simplified 49.4%

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

   6. Taylor expanded in n around inf

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

   8. Simplified 53.0%

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

      1. metadata-eval 53.0

         \[\leadsto \color{blue}{0} \]

   10. Applied egg-rr 53.0%

       \[\leadsto \color{blue}{0} \]

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

   1. Initial program 29.7%

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

   3. Taylor expanded in n around inf

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

      1. /-lowering-/.f64 N/A

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

      2. --lowering--.f64 N/A

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

      3. accelerator-lowering-log1p.f64 N/A

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

      4. log-lowering-log.f64 65.0

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

   5. Simplified 65.0%

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

   6. Taylor expanded in x around inf

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

      1. /-lowering-/.f64 41.1

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

   8. Simplified 41.1%

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

Alternative 16: 46.5% accurate, 6.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{1}{n} \leq -1000:\\ \;\;\;\;0\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{x \cdot n}\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (if (<= (/ 1.0 n) -1000.0) 0.0 (/ 1.0 (* x n))))

C

double code(double x, double n) {
	double tmp;
	if ((1.0 / n) <= -1000.0) {
		tmp = 0.0;
	} else {
		tmp = 1.0 / (x * n);
	}
	return tmp;
}

Fortran

real(8) function code(x, n)
    real(8), intent (in) :: x
    real(8), intent (in) :: n
    real(8) :: tmp
    if ((1.0d0 / n) <= (-1000.0d0)) then
        tmp = 0.0d0
    else
        tmp = 1.0d0 / (x * n)
    end if
    code = tmp
end function

Java

public static double code(double x, double n) {
	double tmp;
	if ((1.0 / n) <= -1000.0) {
		tmp = 0.0;
	} else {
		tmp = 1.0 / (x * n);
	}
	return tmp;
}

Python

def code(x, n):
	tmp = 0
	if (1.0 / n) <= -1000.0:
		tmp = 0.0
	else:
		tmp = 1.0 / (x * n)
	return tmp

Julia

function code(x, n)
	tmp = 0.0
	if (Float64(1.0 / n) <= -1000.0)
		tmp = 0.0;
	else
		tmp = Float64(1.0 / Float64(x * n));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, n)
	tmp = 0.0;
	if ((1.0 / n) <= -1000.0)
		tmp = 0.0;
	else
		tmp = 1.0 / (x * n);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, n_] := If[LessEqual[N[(1.0 / n), $MachinePrecision], -1000.0], 0.0, N[(1.0 / N[(x * n), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

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

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

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

      1. remove-double-neg N/A

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

      2. mul-1-neg N/A

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

      3. distribute-neg-frac N/A

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

      4. mul-1-neg N/A

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

      5. log-rec N/A

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

      6. mul-1-neg N/A

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

      7. --lowering--.f64 N/A

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

      8. log-rec N/A

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

      9. mul-1-neg N/A

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

      10. associate-*r/ N/A

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

      11. associate-*r* N/A

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

      12. metadata-eval N/A

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

      13. *-commutative N/A

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

      14. associate-/l* N/A

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

      15. exp-to-pow N/A

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

      16. pow-lowering-pow.f64 N/A

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

      17. /-lowering-/.f64 49.4

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

   5. Simplified 49.4%

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

   6. Taylor expanded in n around inf

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

   8. Simplified 53.0%

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

      1. metadata-eval 53.0

         \[\leadsto \color{blue}{0} \]

   10. Applied egg-rr 53.0%

       \[\leadsto \color{blue}{0} \]

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

   1. Initial program 29.7%

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

   3. Taylor expanded in n around inf

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

      1. /-lowering-/.f64 N/A

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

      2. --lowering--.f64 N/A

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

      3. accelerator-lowering-log1p.f64 N/A

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

      4. log-lowering-log.f64 65.0

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

   5. Simplified 65.0%

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

   6. Taylor expanded in x around inf

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

      1. /-lowering-/.f64 N/A

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

      2. *-commutative N/A

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

      3. *-lowering-*.f64 40.7

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

   8. Simplified 40.7%

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

Alternative 17: 30.9% accurate, 231.0× speedup

Math

\[\begin{array}{l} \\ 0 \end{array} \]

FPCore

(FPCore (x n) :precision binary64 0.0)

C

double code(double x, double n) {
	return 0.0;
}

Fortran

real(8) function code(x, n)
    real(8), intent (in) :: x
    real(8), intent (in) :: n
    code = 0.0d0
end function

Java

public static double code(double x, double n) {
	return 0.0;
}

Python

def code(x, n):
	return 0.0

Julia

function code(x, n)
	return 0.0
end

MATLAB

function tmp = code(x, n)
	tmp = 0.0;
end

Wolfram

code[x_, n_] := 0.0

Derivation

1. Initial program 48.1%

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

3. Taylor expanded in x around 0

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

   1. remove-double-neg N/A

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

   2. mul-1-neg N/A

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

   3. distribute-neg-frac N/A

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

   4. mul-1-neg N/A

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

   5. log-rec N/A

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

   6. mul-1-neg N/A

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

   7. --lowering--.f64 N/A

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

   8. log-rec N/A

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

   9. mul-1-neg N/A

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

   10. associate-*r/ N/A

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

   11. associate-*r* N/A

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

   12. metadata-eval N/A

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

   13. *-commutative N/A

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

   14. associate-/l* N/A

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

   15. exp-to-pow N/A

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

   16. pow-lowering-pow.f64 N/A

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

   17. /-lowering-/.f64 34.4

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

5. Simplified 34.4%

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

6. Taylor expanded in n around inf

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

8. Simplified 28.5%

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

   1. metadata-eval 28.5

      \[\leadsto \color{blue}{0} \]

10. Applied egg-rr 28.5%

    \[\leadsto \color{blue}{0} \]
11. Add Preprocessing

Reproduce

herbie shell --seed 2024199 
(FPCore (x n)
  :name "2nthrt (problem 3.4.6)"
  :precision binary64
  (- (pow (+ x 1.0) (/ 1.0 n)) (pow x (/ 1.0 n))))