2nthrt (problem 3.4.6)

Percentage Accurate: 53.3% → 83.2%

Time: 22.3s

Alternatives: 15

Speedup: 1.7×

• could not determine a ground truth

  1. x = 3.155414088588453e+138
  2. n = 3.66918064073382e-135

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: 53.3% 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: 83.2% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{\frac{\log x}{n}}\\ \mathbf{if}\;{n}^{-1} \leq -1 \cdot 10^{-12}:\\ \;\;\;\;\frac{\frac{t\_0}{n}}{x}\\ \mathbf{elif}\;{n}^{-1} \leq 10^{-31}:\\ \;\;\;\;\frac{\mathsf{fma}\left(0.5, \frac{{\left(\mathsf{log1p}\left(x\right)\right)}^{2} - {\log x}^{2}}{n}, \mathsf{log1p}\left(x\right)\right) - \log x}{n}\\ \mathbf{elif}\;{n}^{-1} \leq 10^{-9}:\\ \;\;\;\;\frac{t\_0}{n \cdot x}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\frac{-0.5 + \frac{0.5}{n}}{n}, x, {n}^{-1}\right), x, 1\right) - {x}^{\left({n}^{-1}\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (let* ((t_0 (exp (/ (log x) n))))
   (if (<= (pow n -1.0) -1e-12)
     (/ (/ t_0 n) x)
     (if (<= (pow n -1.0) 1e-31)
       (/
        (-
         (fma 0.5 (/ (- (pow (log1p x) 2.0) (pow (log x) 2.0)) n) (log1p x))
         (log x))
        n)
       (if (<= (pow n -1.0) 1e-9)
         (/ t_0 (* n x))
         (-
          (fma (fma (/ (+ -0.5 (/ 0.5 n)) n) x (pow n -1.0)) x 1.0)
          (pow x (pow n -1.0))))))))

C

double code(double x, double n) {
	double t_0 = exp((log(x) / n));
	double tmp;
	if (pow(n, -1.0) <= -1e-12) {
		tmp = (t_0 / n) / x;
	} else if (pow(n, -1.0) <= 1e-31) {
		tmp = (fma(0.5, ((pow(log1p(x), 2.0) - pow(log(x), 2.0)) / n), log1p(x)) - log(x)) / n;
	} else if (pow(n, -1.0) <= 1e-9) {
		tmp = t_0 / (n * x);
	} else {
		tmp = fma(fma(((-0.5 + (0.5 / n)) / n), x, pow(n, -1.0)), x, 1.0) - pow(x, pow(n, -1.0));
	}
	return tmp;
}

Julia

function code(x, n)
	t_0 = exp(Float64(log(x) / n))
	tmp = 0.0
	if ((n ^ -1.0) <= -1e-12)
		tmp = Float64(Float64(t_0 / n) / x);
	elseif ((n ^ -1.0) <= 1e-31)
		tmp = Float64(Float64(fma(0.5, Float64(Float64((log1p(x) ^ 2.0) - (log(x) ^ 2.0)) / n), log1p(x)) - log(x)) / n);
	elseif ((n ^ -1.0) <= 1e-9)
		tmp = Float64(t_0 / Float64(n * x));
	else
		tmp = Float64(fma(fma(Float64(Float64(-0.5 + Float64(0.5 / n)) / n), x, (n ^ -1.0)), x, 1.0) - (x ^ (n ^ -1.0)));
	end
	return tmp
end

Wolfram

code[x_, n_] := Block[{t$95$0 = N[Exp[N[(N[Log[x], $MachinePrecision] / n), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[N[Power[n, -1.0], $MachinePrecision], -1e-12], N[(N[(t$95$0 / n), $MachinePrecision] / x), $MachinePrecision], If[LessEqual[N[Power[n, -1.0], $MachinePrecision], 1e-31], N[(N[(N[(0.5 * N[(N[(N[Power[N[Log[1 + x], $MachinePrecision], 2.0], $MachinePrecision] - N[Power[N[Log[x], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] / n), $MachinePrecision] + N[Log[1 + x], $MachinePrecision]), $MachinePrecision] - N[Log[x], $MachinePrecision]), $MachinePrecision] / n), $MachinePrecision], If[LessEqual[N[Power[n, -1.0], $MachinePrecision], 1e-9], N[(t$95$0 / N[(n * x), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(N[(-0.5 + N[(0.5 / n), $MachinePrecision]), $MachinePrecision] / n), $MachinePrecision] * x + N[Power[n, -1.0], $MachinePrecision]), $MachinePrecision] * x + 1.0), $MachinePrecision] - N[Power[x, N[Power[n, -1.0], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if (/.f64 #s(literal 1 binary64) n) < -9.9999999999999998e-13

   1. Initial program 94.2%

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

   3. Taylor expanded in x around inf

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

      1. lower-/.f64 N/A

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

      2. log-rec N/A

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

      3. mul-1-neg N/A

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

      4. associate-*r/ N/A

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

      5. associate-*r* N/A

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

      6. metadata-eval N/A

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

      7. *-lft-identity N/A

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

      8. lower-exp.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lower-log.f64 N/A

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

      11. lower-*.f64 98.6

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

   5. Applied rewrites 98.6%

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

   7. Applied rewrites 98.6%

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

   if -9.9999999999999998e-13 < (/.f64 #s(literal 1 binary64) n) < 1e-31

   1. Initial program 28.2%

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

   3. Taylor expanded in n around inf

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

      1. lower-/.f64 N/A

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

   5. Applied rewrites 82.5%

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

   if 1e-31 < (/.f64 #s(literal 1 binary64) n) < 1.00000000000000006e-9

   1. Initial program 5.4%

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

   3. Taylor expanded in x around inf

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

      1. lower-/.f64 N/A

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

      2. log-rec N/A

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

      3. mul-1-neg N/A

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

      4. associate-*r/ N/A

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

      5. associate-*r* N/A

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

      6. metadata-eval N/A

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

      7. *-lft-identity N/A

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

      8. lower-exp.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lower-log.f64 N/A

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

      11. lower-*.f64 99.7

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

   5. Applied rewrites 99.7%

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

   if 1.00000000000000006e-9 < (/.f64 #s(literal 1 binary64) n)

   1. Initial program 45.4%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

   5. Applied rewrites 92.6%

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

4. Final simplification 89.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{n}^{-1} \leq -1 \cdot 10^{-12}:\\ \;\;\;\;\frac{\frac{e^{\frac{\log x}{n}}}{n}}{x}\\ \mathbf{elif}\;{n}^{-1} \leq 10^{-31}:\\ \;\;\;\;\frac{\mathsf{fma}\left(0.5, \frac{{\left(\mathsf{log1p}\left(x\right)\right)}^{2} - {\log x}^{2}}{n}, \mathsf{log1p}\left(x\right)\right) - \log x}{n}\\ \mathbf{elif}\;{n}^{-1} \leq 10^{-9}:\\ \;\;\;\;\frac{e^{\frac{\log x}{n}}}{n \cdot x}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\frac{-0.5 + \frac{0.5}{n}}{n}, x, {n}^{-1}\right), x, 1\right) - {x}^{\left({n}^{-1}\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 81.9% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {x}^{\left({n}^{-1}\right)}\\ t_1 := {\left(x + 1\right)}^{\left({n}^{-1}\right)} - t\_0\\ \mathbf{if}\;t\_1 \leq -0.5:\\ \;\;\;\;1 - t\_0\\ \mathbf{elif}\;t\_1 \leq 0:\\ \;\;\;\;\frac{\log \left(\frac{1 + x}{x}\right)}{n}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\frac{-0.5 + \frac{0.5}{n}}{n}, x, {n}^{-1}\right), x, 1\right) - t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (let* ((t_0 (pow x (pow n -1.0))) (t_1 (- (pow (+ x 1.0) (pow n -1.0)) t_0)))
   (if (<= t_1 -0.5)
     (- 1.0 t_0)
     (if (<= t_1 0.0)
       (/ (log (/ (+ 1.0 x) x)) n)
       (- (fma (fma (/ (+ -0.5 (/ 0.5 n)) n) x (pow n -1.0)) x 1.0) t_0)))))

C

double code(double x, double n) {
	double t_0 = pow(x, pow(n, -1.0));
	double t_1 = pow((x + 1.0), pow(n, -1.0)) - t_0;
	double tmp;
	if (t_1 <= -0.5) {
		tmp = 1.0 - t_0;
	} else if (t_1 <= 0.0) {
		tmp = log(((1.0 + x) / x)) / n;
	} else {
		tmp = fma(fma(((-0.5 + (0.5 / n)) / n), x, pow(n, -1.0)), x, 1.0) - t_0;
	}
	return tmp;
}

Julia

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

Wolfram

code[x_, n_] := Block[{t$95$0 = N[Power[x, N[Power[n, -1.0], $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(N[Power[N[(x + 1.0), $MachinePrecision], N[Power[n, -1.0], $MachinePrecision]], $MachinePrecision] - t$95$0), $MachinePrecision]}, If[LessEqual[t$95$1, -0.5], N[(1.0 - t$95$0), $MachinePrecision], If[LessEqual[t$95$1, 0.0], N[(N[Log[N[(N[(1.0 + x), $MachinePrecision] / x), $MachinePrecision]], $MachinePrecision] / n), $MachinePrecision], N[(N[(N[(N[(N[(-0.5 + N[(0.5 / n), $MachinePrecision]), $MachinePrecision] / n), $MachinePrecision] * x + N[Power[n, -1.0], $MachinePrecision]), $MachinePrecision] * x + 1.0), $MachinePrecision] - t$95$0), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if (-.f64 (pow.f64 (+.f64 x #s(literal 1 binary64)) (/.f64 #s(literal 1 binary64) n)) (pow.f64 x (/.f64 #s(literal 1 binary64) n))) < -0.5

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

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

   5. Applied rewrites 100.0%

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

   if -0.5 < (-.f64 (pow.f64 (+.f64 x #s(literal 1 binary64)) (/.f64 #s(literal 1 binary64) n)) (pow.f64 x (/.f64 #s(literal 1 binary64) n))) < 0.0

   1. Initial program 42.2%

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

   3. Taylor expanded in n around inf

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

      1. lower-/.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lower-log1p.f64 N/A

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

      4. lower-log.f64 80.8

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

   5. Applied rewrites 80.8%

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

   7. Applied rewrites 80.8%

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

   if 0.0 < (-.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 45.4%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

   5. Applied rewrites 92.6%

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

4. Final simplification 85.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{\left(x + 1\right)}^{\left({n}^{-1}\right)} - {x}^{\left({n}^{-1}\right)} \leq -0.5:\\ \;\;\;\;1 - {x}^{\left({n}^{-1}\right)}\\ \mathbf{elif}\;{\left(x + 1\right)}^{\left({n}^{-1}\right)} - {x}^{\left({n}^{-1}\right)} \leq 0:\\ \;\;\;\;\frac{\log \left(\frac{1 + x}{x}\right)}{n}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\frac{-0.5 + \frac{0.5}{n}}{n}, x, {n}^{-1}\right), x, 1\right) - {x}^{\left({n}^{-1}\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 81.7% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {x}^{\left({n}^{-1}\right)}\\ t_1 := {\left(x + 1\right)}^{\left({n}^{-1}\right)} - t\_0\\ \mathbf{if}\;t\_1 \leq -0.5:\\ \;\;\;\;1 - t\_0\\ \mathbf{elif}\;t\_1 \leq 0:\\ \;\;\;\;\frac{\log \left(\frac{1 + x}{x}\right)}{n}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{x}{n \cdot n} \cdot 0.5, x, 1\right) - t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (let* ((t_0 (pow x (pow n -1.0))) (t_1 (- (pow (+ x 1.0) (pow n -1.0)) t_0)))
   (if (<= t_1 -0.5)
     (- 1.0 t_0)
     (if (<= t_1 0.0)
       (/ (log (/ (+ 1.0 x) x)) n)
       (- (fma (* (/ x (* n n)) 0.5) x 1.0) t_0)))))

C

double code(double x, double n) {
	double t_0 = pow(x, pow(n, -1.0));
	double t_1 = pow((x + 1.0), pow(n, -1.0)) - t_0;
	double tmp;
	if (t_1 <= -0.5) {
		tmp = 1.0 - t_0;
	} else if (t_1 <= 0.0) {
		tmp = log(((1.0 + x) / x)) / n;
	} else {
		tmp = fma(((x / (n * n)) * 0.5), x, 1.0) - t_0;
	}
	return tmp;
}

Julia

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

Wolfram

code[x_, n_] := Block[{t$95$0 = N[Power[x, N[Power[n, -1.0], $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(N[Power[N[(x + 1.0), $MachinePrecision], N[Power[n, -1.0], $MachinePrecision]], $MachinePrecision] - t$95$0), $MachinePrecision]}, If[LessEqual[t$95$1, -0.5], N[(1.0 - t$95$0), $MachinePrecision], If[LessEqual[t$95$1, 0.0], N[(N[Log[N[(N[(1.0 + x), $MachinePrecision] / x), $MachinePrecision]], $MachinePrecision] / n), $MachinePrecision], N[(N[(N[(N[(x / N[(n * n), $MachinePrecision]), $MachinePrecision] * 0.5), $MachinePrecision] * x + 1.0), $MachinePrecision] - t$95$0), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if (-.f64 (pow.f64 (+.f64 x #s(literal 1 binary64)) (/.f64 #s(literal 1 binary64) n)) (pow.f64 x (/.f64 #s(literal 1 binary64) n))) < -0.5

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

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

   5. Applied rewrites 100.0%

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

   if -0.5 < (-.f64 (pow.f64 (+.f64 x #s(literal 1 binary64)) (/.f64 #s(literal 1 binary64) n)) (pow.f64 x (/.f64 #s(literal 1 binary64) n))) < 0.0

   1. Initial program 42.2%

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

   3. Taylor expanded in n around inf

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

      1. lower-/.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lower-log1p.f64 N/A

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

      4. lower-log.f64 80.8

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

   5. Applied rewrites 80.8%

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

   7. Applied rewrites 80.8%

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

   if 0.0 < (-.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 45.4%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

   5. Applied rewrites 92.6%

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

   6. Taylor expanded in n around 0

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

   8. Applied rewrites 88.9%

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

4. Final simplification 85.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{\left(x + 1\right)}^{\left({n}^{-1}\right)} - {x}^{\left({n}^{-1}\right)} \leq -0.5:\\ \;\;\;\;1 - {x}^{\left({n}^{-1}\right)}\\ \mathbf{elif}\;{\left(x + 1\right)}^{\left({n}^{-1}\right)} - {x}^{\left({n}^{-1}\right)} \leq 0:\\ \;\;\;\;\frac{\log \left(\frac{1 + x}{x}\right)}{n}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{x}{n \cdot n} \cdot 0.5, x, 1\right) - {x}^{\left({n}^{-1}\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 76.6% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {x}^{\left({n}^{-1}\right)}\\ t_1 := {\left(x + 1\right)}^{\left({n}^{-1}\right)} - t\_0\\ \mathbf{if}\;t\_1 \leq -0.5:\\ \;\;\;\;1 - t\_0\\ \mathbf{elif}\;t\_1 \leq 0.9991227832387742:\\ \;\;\;\;\frac{\log \left(\frac{1 + x}{x}\right)}{n}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{{n}^{-2}}}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (let* ((t_0 (pow x (pow n -1.0))) (t_1 (- (pow (+ x 1.0) (pow n -1.0)) t_0)))
   (if (<= t_1 -0.5)
     (- 1.0 t_0)
     (if (<= t_1 0.9991227832387742)
       (/ (log (/ (+ 1.0 x) x)) n)
       (/ (sqrt (pow n -2.0)) x)))))

C

double code(double x, double n) {
	double t_0 = pow(x, pow(n, -1.0));
	double t_1 = pow((x + 1.0), pow(n, -1.0)) - t_0;
	double tmp;
	if (t_1 <= -0.5) {
		tmp = 1.0 - t_0;
	} else if (t_1 <= 0.9991227832387742) {
		tmp = log(((1.0 + x) / x)) / n;
	} else {
		tmp = sqrt(pow(n, -2.0)) / x;
	}
	return tmp;
}

Fortran

real(8) function code(x, n)
    real(8), intent (in) :: x
    real(8), intent (in) :: n
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = x ** (n ** (-1.0d0))
    t_1 = ((x + 1.0d0) ** (n ** (-1.0d0))) - t_0
    if (t_1 <= (-0.5d0)) then
        tmp = 1.0d0 - t_0
    else if (t_1 <= 0.9991227832387742d0) then
        tmp = log(((1.0d0 + x) / x)) / n
    else
        tmp = sqrt((n ** (-2.0d0))) / x
    end if
    code = tmp
end function

Java

public static double code(double x, double n) {
	double t_0 = Math.pow(x, Math.pow(n, -1.0));
	double t_1 = Math.pow((x + 1.0), Math.pow(n, -1.0)) - t_0;
	double tmp;
	if (t_1 <= -0.5) {
		tmp = 1.0 - t_0;
	} else if (t_1 <= 0.9991227832387742) {
		tmp = Math.log(((1.0 + x) / x)) / n;
	} else {
		tmp = Math.sqrt(Math.pow(n, -2.0)) / x;
	}
	return tmp;
}

Python

def code(x, n):
	t_0 = math.pow(x, math.pow(n, -1.0))
	t_1 = math.pow((x + 1.0), math.pow(n, -1.0)) - t_0
	tmp = 0
	if t_1 <= -0.5:
		tmp = 1.0 - t_0
	elif t_1 <= 0.9991227832387742:
		tmp = math.log(((1.0 + x) / x)) / n
	else:
		tmp = math.sqrt(math.pow(n, -2.0)) / x
	return tmp

Julia

function code(x, n)
	t_0 = x ^ (n ^ -1.0)
	t_1 = Float64((Float64(x + 1.0) ^ (n ^ -1.0)) - t_0)
	tmp = 0.0
	if (t_1 <= -0.5)
		tmp = Float64(1.0 - t_0);
	elseif (t_1 <= 0.9991227832387742)
		tmp = Float64(log(Float64(Float64(1.0 + x) / x)) / n);
	else
		tmp = Float64(sqrt((n ^ -2.0)) / x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, n)
	t_0 = x ^ (n ^ -1.0);
	t_1 = ((x + 1.0) ^ (n ^ -1.0)) - t_0;
	tmp = 0.0;
	if (t_1 <= -0.5)
		tmp = 1.0 - t_0;
	elseif (t_1 <= 0.9991227832387742)
		tmp = log(((1.0 + x) / x)) / n;
	else
		tmp = sqrt((n ^ -2.0)) / x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, n_] := Block[{t$95$0 = N[Power[x, N[Power[n, -1.0], $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(N[Power[N[(x + 1.0), $MachinePrecision], N[Power[n, -1.0], $MachinePrecision]], $MachinePrecision] - t$95$0), $MachinePrecision]}, If[LessEqual[t$95$1, -0.5], N[(1.0 - t$95$0), $MachinePrecision], If[LessEqual[t$95$1, 0.9991227832387742], N[(N[Log[N[(N[(1.0 + x), $MachinePrecision] / x), $MachinePrecision]], $MachinePrecision] / n), $MachinePrecision], N[(N[Sqrt[N[Power[n, -2.0], $MachinePrecision]], $MachinePrecision] / x), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if (-.f64 (pow.f64 (+.f64 x #s(literal 1 binary64)) (/.f64 #s(literal 1 binary64) n)) (pow.f64 x (/.f64 #s(literal 1 binary64) n))) < -0.5

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

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

   5. Applied rewrites 100.0%

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

   if -0.5 < (-.f64 (pow.f64 (+.f64 x #s(literal 1 binary64)) (/.f64 #s(literal 1 binary64) n)) (pow.f64 x (/.f64 #s(literal 1 binary64) n))) < 0.999122783238774237

   1. Initial program 42.5%

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

   3. Taylor expanded in n around inf

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

      1. lower-/.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lower-log1p.f64 N/A

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

      4. lower-log.f64 80.4

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

   5. Applied rewrites 80.4%

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

   7. Applied rewrites 80.4%

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

   if 0.999122783238774237 < (-.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 43.9%

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

   3. Taylor expanded in x around inf

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

      1. lower-/.f64 N/A

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

      2. log-rec N/A

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

      3. mul-1-neg N/A

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

      4. associate-*r/ N/A

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

      5. associate-*r* N/A

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

      6. metadata-eval N/A

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

      7. *-lft-identity N/A

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

      8. lower-exp.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lower-log.f64 N/A

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

      11. lower-*.f64 1.3

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

   5. Applied rewrites 1.3%

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

   6. Taylor expanded in n around inf

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

   8. Applied rewrites 34.2%

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

   10. Applied rewrites 56.0%

       \[\leadsto \frac{\sqrt{{n}^{-2}}}{x} \]
3. Recombined 3 regimes into one program.

4. Final simplification 80.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{\left(x + 1\right)}^{\left({n}^{-1}\right)} - {x}^{\left({n}^{-1}\right)} \leq -0.5:\\ \;\;\;\;1 - {x}^{\left({n}^{-1}\right)}\\ \mathbf{elif}\;{\left(x + 1\right)}^{\left({n}^{-1}\right)} - {x}^{\left({n}^{-1}\right)} \leq 0.9991227832387742:\\ \;\;\;\;\frac{\log \left(\frac{1 + x}{x}\right)}{n}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{{n}^{-2}}}{x}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 83.3% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{\frac{\log x}{n}}\\ \mathbf{if}\;{n}^{-1} \leq -1 \cdot 10^{-12}:\\ \;\;\;\;\frac{\frac{t\_0}{n}}{x}\\ \mathbf{elif}\;{n}^{-1} \leq 10^{-31}:\\ \;\;\;\;\frac{\log \left(\frac{1 + x}{x}\right)}{n}\\ \mathbf{elif}\;{n}^{-1} \leq 10^{-9}:\\ \;\;\;\;\frac{t\_0}{n \cdot x}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\frac{-0.5 + \frac{0.5}{n}}{n}, x, {n}^{-1}\right), x, 1\right) - {x}^{\left({n}^{-1}\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (let* ((t_0 (exp (/ (log x) n))))
   (if (<= (pow n -1.0) -1e-12)
     (/ (/ t_0 n) x)
     (if (<= (pow n -1.0) 1e-31)
       (/ (log (/ (+ 1.0 x) x)) n)
       (if (<= (pow n -1.0) 1e-9)
         (/ t_0 (* n x))
         (-
          (fma (fma (/ (+ -0.5 (/ 0.5 n)) n) x (pow n -1.0)) x 1.0)
          (pow x (pow n -1.0))))))))

C

double code(double x, double n) {
	double t_0 = exp((log(x) / n));
	double tmp;
	if (pow(n, -1.0) <= -1e-12) {
		tmp = (t_0 / n) / x;
	} else if (pow(n, -1.0) <= 1e-31) {
		tmp = log(((1.0 + x) / x)) / n;
	} else if (pow(n, -1.0) <= 1e-9) {
		tmp = t_0 / (n * x);
	} else {
		tmp = fma(fma(((-0.5 + (0.5 / n)) / n), x, pow(n, -1.0)), x, 1.0) - pow(x, pow(n, -1.0));
	}
	return tmp;
}

Julia

function code(x, n)
	t_0 = exp(Float64(log(x) / n))
	tmp = 0.0
	if ((n ^ -1.0) <= -1e-12)
		tmp = Float64(Float64(t_0 / n) / x);
	elseif ((n ^ -1.0) <= 1e-31)
		tmp = Float64(log(Float64(Float64(1.0 + x) / x)) / n);
	elseif ((n ^ -1.0) <= 1e-9)
		tmp = Float64(t_0 / Float64(n * x));
	else
		tmp = Float64(fma(fma(Float64(Float64(-0.5 + Float64(0.5 / n)) / n), x, (n ^ -1.0)), x, 1.0) - (x ^ (n ^ -1.0)));
	end
	return tmp
end

Wolfram

code[x_, n_] := Block[{t$95$0 = N[Exp[N[(N[Log[x], $MachinePrecision] / n), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[N[Power[n, -1.0], $MachinePrecision], -1e-12], N[(N[(t$95$0 / n), $MachinePrecision] / x), $MachinePrecision], If[LessEqual[N[Power[n, -1.0], $MachinePrecision], 1e-31], N[(N[Log[N[(N[(1.0 + x), $MachinePrecision] / x), $MachinePrecision]], $MachinePrecision] / n), $MachinePrecision], If[LessEqual[N[Power[n, -1.0], $MachinePrecision], 1e-9], N[(t$95$0 / N[(n * x), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(N[(-0.5 + N[(0.5 / n), $MachinePrecision]), $MachinePrecision] / n), $MachinePrecision] * x + N[Power[n, -1.0], $MachinePrecision]), $MachinePrecision] * x + 1.0), $MachinePrecision] - N[Power[x, N[Power[n, -1.0], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if (/.f64 #s(literal 1 binary64) n) < -9.9999999999999998e-13

   1. Initial program 94.2%

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

   3. Taylor expanded in x around inf

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

      1. lower-/.f64 N/A

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

      2. log-rec N/A

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

      3. mul-1-neg N/A

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

      4. associate-*r/ N/A

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

      5. associate-*r* N/A

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

      6. metadata-eval N/A

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

      7. *-lft-identity N/A

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

      8. lower-exp.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lower-log.f64 N/A

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

      11. lower-*.f64 98.6

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

   5. Applied rewrites 98.6%

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

   7. Applied rewrites 98.6%

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

   if -9.9999999999999998e-13 < (/.f64 #s(literal 1 binary64) n) < 1e-31

   1. Initial program 28.2%

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

   3. Taylor expanded in n around inf

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

      1. lower-/.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lower-log1p.f64 N/A

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

      4. lower-log.f64 82.2

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

   5. Applied rewrites 82.2%

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

   7. Applied rewrites 82.3%

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

   if 1e-31 < (/.f64 #s(literal 1 binary64) n) < 1.00000000000000006e-9

   1. Initial program 5.4%

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

   3. Taylor expanded in x around inf

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

      1. lower-/.f64 N/A

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

      2. log-rec N/A

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

      3. mul-1-neg N/A

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

      4. associate-*r/ N/A

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

      5. associate-*r* N/A

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

      6. metadata-eval N/A

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

      7. *-lft-identity N/A

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

      8. lower-exp.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lower-log.f64 N/A

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

      11. lower-*.f64 99.7

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

   5. Applied rewrites 99.7%

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

   if 1.00000000000000006e-9 < (/.f64 #s(literal 1 binary64) n)

   1. Initial program 45.4%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

   5. Applied rewrites 92.6%

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

4. Final simplification 89.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{n}^{-1} \leq -1 \cdot 10^{-12}:\\ \;\;\;\;\frac{\frac{e^{\frac{\log x}{n}}}{n}}{x}\\ \mathbf{elif}\;{n}^{-1} \leq 10^{-31}:\\ \;\;\;\;\frac{\log \left(\frac{1 + x}{x}\right)}{n}\\ \mathbf{elif}\;{n}^{-1} \leq 10^{-9}:\\ \;\;\;\;\frac{e^{\frac{\log x}{n}}}{n \cdot x}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\frac{-0.5 + \frac{0.5}{n}}{n}, x, {n}^{-1}\right), x, 1\right) - {x}^{\left({n}^{-1}\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 83.3% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{e^{\frac{\log x}{n}}}{n \cdot x}\\ \mathbf{if}\;{n}^{-1} \leq -1 \cdot 10^{-12}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;{n}^{-1} \leq 10^{-31}:\\ \;\;\;\;\frac{\log \left(\frac{1 + x}{x}\right)}{n}\\ \mathbf{elif}\;{n}^{-1} \leq 10^{-9}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\frac{-0.5 + \frac{0.5}{n}}{n}, x, {n}^{-1}\right), x, 1\right) - {x}^{\left({n}^{-1}\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (let* ((t_0 (/ (exp (/ (log x) n)) (* n x))))
   (if (<= (pow n -1.0) -1e-12)
     t_0
     (if (<= (pow n -1.0) 1e-31)
       (/ (log (/ (+ 1.0 x) x)) n)
       (if (<= (pow n -1.0) 1e-9)
         t_0
         (-
          (fma (fma (/ (+ -0.5 (/ 0.5 n)) n) x (pow n -1.0)) x 1.0)
          (pow x (pow n -1.0))))))))

C

double code(double x, double n) {
	double t_0 = exp((log(x) / n)) / (n * x);
	double tmp;
	if (pow(n, -1.0) <= -1e-12) {
		tmp = t_0;
	} else if (pow(n, -1.0) <= 1e-31) {
		tmp = log(((1.0 + x) / x)) / n;
	} else if (pow(n, -1.0) <= 1e-9) {
		tmp = t_0;
	} else {
		tmp = fma(fma(((-0.5 + (0.5 / n)) / n), x, pow(n, -1.0)), x, 1.0) - pow(x, pow(n, -1.0));
	}
	return tmp;
}

Julia

function code(x, n)
	t_0 = Float64(exp(Float64(log(x) / n)) / Float64(n * x))
	tmp = 0.0
	if ((n ^ -1.0) <= -1e-12)
		tmp = t_0;
	elseif ((n ^ -1.0) <= 1e-31)
		tmp = Float64(log(Float64(Float64(1.0 + x) / x)) / n);
	elseif ((n ^ -1.0) <= 1e-9)
		tmp = t_0;
	else
		tmp = Float64(fma(fma(Float64(Float64(-0.5 + Float64(0.5 / n)) / n), x, (n ^ -1.0)), x, 1.0) - (x ^ (n ^ -1.0)));
	end
	return tmp
end

Wolfram

code[x_, n_] := Block[{t$95$0 = N[(N[Exp[N[(N[Log[x], $MachinePrecision] / n), $MachinePrecision]], $MachinePrecision] / N[(n * x), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[Power[n, -1.0], $MachinePrecision], -1e-12], t$95$0, If[LessEqual[N[Power[n, -1.0], $MachinePrecision], 1e-31], N[(N[Log[N[(N[(1.0 + x), $MachinePrecision] / x), $MachinePrecision]], $MachinePrecision] / n), $MachinePrecision], If[LessEqual[N[Power[n, -1.0], $MachinePrecision], 1e-9], t$95$0, N[(N[(N[(N[(N[(-0.5 + N[(0.5 / n), $MachinePrecision]), $MachinePrecision] / n), $MachinePrecision] * x + N[Power[n, -1.0], $MachinePrecision]), $MachinePrecision] * x + 1.0), $MachinePrecision] - N[Power[x, N[Power[n, -1.0], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 #s(literal 1 binary64) n) < -9.9999999999999998e-13 or 1e-31 < (/.f64 #s(literal 1 binary64) n) < 1.00000000000000006e-9

   1. Initial program 89.2%

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

   3. Taylor expanded in x around inf

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

      1. lower-/.f64 N/A

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

      2. log-rec N/A

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

      3. mul-1-neg N/A

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

      4. associate-*r/ N/A

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

      5. associate-*r* N/A

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

      6. metadata-eval N/A

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

      7. *-lft-identity N/A

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

      8. lower-exp.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lower-log.f64 N/A

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

      11. lower-*.f64 98.7

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

   5. Applied rewrites 98.7%

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

   if -9.9999999999999998e-13 < (/.f64 #s(literal 1 binary64) n) < 1e-31

   1. Initial program 28.2%

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

   3. Taylor expanded in n around inf

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

      1. lower-/.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lower-log1p.f64 N/A

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

      4. lower-log.f64 82.2

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

   5. Applied rewrites 82.2%

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

   7. Applied rewrites 82.3%

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

   if 1.00000000000000006e-9 < (/.f64 #s(literal 1 binary64) n)

   1. Initial program 45.4%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

   5. Applied rewrites 92.6%

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

4. Final simplification 89.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{n}^{-1} \leq -1 \cdot 10^{-12}:\\ \;\;\;\;\frac{e^{\frac{\log x}{n}}}{n \cdot x}\\ \mathbf{elif}\;{n}^{-1} \leq 10^{-31}:\\ \;\;\;\;\frac{\log \left(\frac{1 + x}{x}\right)}{n}\\ \mathbf{elif}\;{n}^{-1} \leq 10^{-9}:\\ \;\;\;\;\frac{e^{\frac{\log x}{n}}}{n \cdot x}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\frac{-0.5 + \frac{0.5}{n}}{n}, x, {n}^{-1}\right), x, 1\right) - {x}^{\left({n}^{-1}\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 53.9% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 1 - {x}^{\left({n}^{-1}\right)}\\ \mathbf{if}\;{n}^{-1} \leq -2 \cdot 10^{-9}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;{n}^{-1} \leq 5 \cdot 10^{-93}:\\ \;\;\;\;\frac{-\log x}{n}\\ \mathbf{elif}\;{n}^{-1} \leq 10^{-9}:\\ \;\;\;\;\frac{{x}^{-1}}{n}\\ \mathbf{elif}\;{n}^{-1} \leq 4 \cdot 10^{+143}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{{n}^{-2}}}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (let* ((t_0 (- 1.0 (pow x (pow n -1.0)))))
   (if (<= (pow n -1.0) -2e-9)
     t_0
     (if (<= (pow n -1.0) 5e-93)
       (/ (- (log x)) n)
       (if (<= (pow n -1.0) 1e-9)
         (/ (pow x -1.0) n)
         (if (<= (pow n -1.0) 4e+143) t_0 (/ (sqrt (pow n -2.0)) x)))))))

C

double code(double x, double n) {
	double t_0 = 1.0 - pow(x, pow(n, -1.0));
	double tmp;
	if (pow(n, -1.0) <= -2e-9) {
		tmp = t_0;
	} else if (pow(n, -1.0) <= 5e-93) {
		tmp = -log(x) / n;
	} else if (pow(n, -1.0) <= 1e-9) {
		tmp = pow(x, -1.0) / n;
	} else if (pow(n, -1.0) <= 4e+143) {
		tmp = t_0;
	} else {
		tmp = sqrt(pow(n, -2.0)) / x;
	}
	return tmp;
}

Fortran

real(8) function code(x, n)
    real(8), intent (in) :: x
    real(8), intent (in) :: n
    real(8) :: t_0
    real(8) :: tmp
    t_0 = 1.0d0 - (x ** (n ** (-1.0d0)))
    if ((n ** (-1.0d0)) <= (-2d-9)) then
        tmp = t_0
    else if ((n ** (-1.0d0)) <= 5d-93) then
        tmp = -log(x) / n
    else if ((n ** (-1.0d0)) <= 1d-9) then
        tmp = (x ** (-1.0d0)) / n
    else if ((n ** (-1.0d0)) <= 4d+143) then
        tmp = t_0
    else
        tmp = sqrt((n ** (-2.0d0))) / x
    end if
    code = tmp
end function

Java

public static double code(double x, double n) {
	double t_0 = 1.0 - Math.pow(x, Math.pow(n, -1.0));
	double tmp;
	if (Math.pow(n, -1.0) <= -2e-9) {
		tmp = t_0;
	} else if (Math.pow(n, -1.0) <= 5e-93) {
		tmp = -Math.log(x) / n;
	} else if (Math.pow(n, -1.0) <= 1e-9) {
		tmp = Math.pow(x, -1.0) / n;
	} else if (Math.pow(n, -1.0) <= 4e+143) {
		tmp = t_0;
	} else {
		tmp = Math.sqrt(Math.pow(n, -2.0)) / x;
	}
	return tmp;
}

Python

def code(x, n):
	t_0 = 1.0 - math.pow(x, math.pow(n, -1.0))
	tmp = 0
	if math.pow(n, -1.0) <= -2e-9:
		tmp = t_0
	elif math.pow(n, -1.0) <= 5e-93:
		tmp = -math.log(x) / n
	elif math.pow(n, -1.0) <= 1e-9:
		tmp = math.pow(x, -1.0) / n
	elif math.pow(n, -1.0) <= 4e+143:
		tmp = t_0
	else:
		tmp = math.sqrt(math.pow(n, -2.0)) / x
	return tmp

Julia

function code(x, n)
	t_0 = Float64(1.0 - (x ^ (n ^ -1.0)))
	tmp = 0.0
	if ((n ^ -1.0) <= -2e-9)
		tmp = t_0;
	elseif ((n ^ -1.0) <= 5e-93)
		tmp = Float64(Float64(-log(x)) / n);
	elseif ((n ^ -1.0) <= 1e-9)
		tmp = Float64((x ^ -1.0) / n);
	elseif ((n ^ -1.0) <= 4e+143)
		tmp = t_0;
	else
		tmp = Float64(sqrt((n ^ -2.0)) / x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, n)
	t_0 = 1.0 - (x ^ (n ^ -1.0));
	tmp = 0.0;
	if ((n ^ -1.0) <= -2e-9)
		tmp = t_0;
	elseif ((n ^ -1.0) <= 5e-93)
		tmp = -log(x) / n;
	elseif ((n ^ -1.0) <= 1e-9)
		tmp = (x ^ -1.0) / n;
	elseif ((n ^ -1.0) <= 4e+143)
		tmp = t_0;
	else
		tmp = sqrt((n ^ -2.0)) / x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, n_] := Block[{t$95$0 = N[(1.0 - N[Power[x, N[Power[n, -1.0], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[Power[n, -1.0], $MachinePrecision], -2e-9], t$95$0, If[LessEqual[N[Power[n, -1.0], $MachinePrecision], 5e-93], N[((-N[Log[x], $MachinePrecision]) / n), $MachinePrecision], If[LessEqual[N[Power[n, -1.0], $MachinePrecision], 1e-9], N[(N[Power[x, -1.0], $MachinePrecision] / n), $MachinePrecision], If[LessEqual[N[Power[n, -1.0], $MachinePrecision], 4e+143], t$95$0, N[(N[Sqrt[N[Power[n, -2.0], $MachinePrecision]], $MachinePrecision] / x), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if (/.f64 #s(literal 1 binary64) n) < -2.00000000000000012e-9 or 1.00000000000000006e-9 < (/.f64 #s(literal 1 binary64) n) < 4.0000000000000001e143

   1. Initial program 95.7%

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

   3. Taylor expanded in x around 0

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

   5. Applied rewrites 57.0%

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

   if -2.00000000000000012e-9 < (/.f64 #s(literal 1 binary64) n) < 4.99999999999999994e-93

   1. Initial program 28.6%

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

   3. Taylor expanded in n around inf

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

      1. lower-/.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lower-log1p.f64 N/A

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

      4. lower-log.f64 82.6

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

   5. Applied rewrites 82.6%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 59.5%

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

   if 4.99999999999999994e-93 < (/.f64 #s(literal 1 binary64) n) < 1.00000000000000006e-9

   1. Initial program 13.0%

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

   3. Taylor expanded in n around inf

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

      1. lower-/.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lower-log1p.f64 N/A

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

      4. lower-log.f64 40.5

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

   5. Applied rewrites 40.5%

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

   6. Taylor expanded in x around inf

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

   8. Applied rewrites 68.7%

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

   if 4.0000000000000001e143 < (/.f64 #s(literal 1 binary64) n)

   1. Initial program 7.7%

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

   3. Taylor expanded in x around inf

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

      1. lower-/.f64 N/A

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

      2. log-rec N/A

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

      3. mul-1-neg N/A

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

      4. associate-*r/ N/A

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

      5. associate-*r* N/A

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

      6. metadata-eval N/A

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

      7. *-lft-identity N/A

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

      8. lower-exp.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lower-log.f64 N/A

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

      11. lower-*.f64 0.7

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

   5. Applied rewrites 0.7%

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

   6. Taylor expanded in n around inf

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

   8. Applied rewrites 57.1%

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

   10. Applied rewrites 95.5%

       \[\leadsto \frac{\sqrt{{n}^{-2}}}{x} \]
3. Recombined 4 regimes into one program.

4. Final simplification 62.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{n}^{-1} \leq -2 \cdot 10^{-9}:\\ \;\;\;\;1 - {x}^{\left({n}^{-1}\right)}\\ \mathbf{elif}\;{n}^{-1} \leq 5 \cdot 10^{-93}:\\ \;\;\;\;\frac{-\log x}{n}\\ \mathbf{elif}\;{n}^{-1} \leq 10^{-9}:\\ \;\;\;\;\frac{{x}^{-1}}{n}\\ \mathbf{elif}\;{n}^{-1} \leq 4 \cdot 10^{+143}:\\ \;\;\;\;1 - {x}^{\left({n}^{-1}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{{n}^{-2}}}{x}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 53.7% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 1 - {x}^{\left({n}^{-1}\right)}\\ \mathbf{if}\;{n}^{-1} \leq -2 \cdot 10^{-9}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;{n}^{-1} \leq 5 \cdot 10^{-93}:\\ \;\;\;\;\frac{-\log x}{n}\\ \mathbf{elif}\;{n}^{-1} \leq 10^{-9}:\\ \;\;\;\;\frac{{x}^{-1}}{n}\\ \mathbf{elif}\;{n}^{-1} \leq 4 \cdot 10^{+143}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{{\left(n \cdot n\right)}^{-0.5}}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (let* ((t_0 (- 1.0 (pow x (pow n -1.0)))))
   (if (<= (pow n -1.0) -2e-9)
     t_0
     (if (<= (pow n -1.0) 5e-93)
       (/ (- (log x)) n)
       (if (<= (pow n -1.0) 1e-9)
         (/ (pow x -1.0) n)
         (if (<= (pow n -1.0) 4e+143) t_0 (/ (pow (* n n) -0.5) x)))))))

C

double code(double x, double n) {
	double t_0 = 1.0 - pow(x, pow(n, -1.0));
	double tmp;
	if (pow(n, -1.0) <= -2e-9) {
		tmp = t_0;
	} else if (pow(n, -1.0) <= 5e-93) {
		tmp = -log(x) / n;
	} else if (pow(n, -1.0) <= 1e-9) {
		tmp = pow(x, -1.0) / n;
	} else if (pow(n, -1.0) <= 4e+143) {
		tmp = t_0;
	} else {
		tmp = pow((n * n), -0.5) / x;
	}
	return tmp;
}

Fortran

real(8) function code(x, n)
    real(8), intent (in) :: x
    real(8), intent (in) :: n
    real(8) :: t_0
    real(8) :: tmp
    t_0 = 1.0d0 - (x ** (n ** (-1.0d0)))
    if ((n ** (-1.0d0)) <= (-2d-9)) then
        tmp = t_0
    else if ((n ** (-1.0d0)) <= 5d-93) then
        tmp = -log(x) / n
    else if ((n ** (-1.0d0)) <= 1d-9) then
        tmp = (x ** (-1.0d0)) / n
    else if ((n ** (-1.0d0)) <= 4d+143) then
        tmp = t_0
    else
        tmp = ((n * n) ** (-0.5d0)) / x
    end if
    code = tmp
end function

Java

public static double code(double x, double n) {
	double t_0 = 1.0 - Math.pow(x, Math.pow(n, -1.0));
	double tmp;
	if (Math.pow(n, -1.0) <= -2e-9) {
		tmp = t_0;
	} else if (Math.pow(n, -1.0) <= 5e-93) {
		tmp = -Math.log(x) / n;
	} else if (Math.pow(n, -1.0) <= 1e-9) {
		tmp = Math.pow(x, -1.0) / n;
	} else if (Math.pow(n, -1.0) <= 4e+143) {
		tmp = t_0;
	} else {
		tmp = Math.pow((n * n), -0.5) / x;
	}
	return tmp;
}

Python

def code(x, n):
	t_0 = 1.0 - math.pow(x, math.pow(n, -1.0))
	tmp = 0
	if math.pow(n, -1.0) <= -2e-9:
		tmp = t_0
	elif math.pow(n, -1.0) <= 5e-93:
		tmp = -math.log(x) / n
	elif math.pow(n, -1.0) <= 1e-9:
		tmp = math.pow(x, -1.0) / n
	elif math.pow(n, -1.0) <= 4e+143:
		tmp = t_0
	else:
		tmp = math.pow((n * n), -0.5) / x
	return tmp

Julia

function code(x, n)
	t_0 = Float64(1.0 - (x ^ (n ^ -1.0)))
	tmp = 0.0
	if ((n ^ -1.0) <= -2e-9)
		tmp = t_0;
	elseif ((n ^ -1.0) <= 5e-93)
		tmp = Float64(Float64(-log(x)) / n);
	elseif ((n ^ -1.0) <= 1e-9)
		tmp = Float64((x ^ -1.0) / n);
	elseif ((n ^ -1.0) <= 4e+143)
		tmp = t_0;
	else
		tmp = Float64((Float64(n * n) ^ -0.5) / x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, n)
	t_0 = 1.0 - (x ^ (n ^ -1.0));
	tmp = 0.0;
	if ((n ^ -1.0) <= -2e-9)
		tmp = t_0;
	elseif ((n ^ -1.0) <= 5e-93)
		tmp = -log(x) / n;
	elseif ((n ^ -1.0) <= 1e-9)
		tmp = (x ^ -1.0) / n;
	elseif ((n ^ -1.0) <= 4e+143)
		tmp = t_0;
	else
		tmp = ((n * n) ^ -0.5) / x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, n_] := Block[{t$95$0 = N[(1.0 - N[Power[x, N[Power[n, -1.0], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[Power[n, -1.0], $MachinePrecision], -2e-9], t$95$0, If[LessEqual[N[Power[n, -1.0], $MachinePrecision], 5e-93], N[((-N[Log[x], $MachinePrecision]) / n), $MachinePrecision], If[LessEqual[N[Power[n, -1.0], $MachinePrecision], 1e-9], N[(N[Power[x, -1.0], $MachinePrecision] / n), $MachinePrecision], If[LessEqual[N[Power[n, -1.0], $MachinePrecision], 4e+143], t$95$0, N[(N[Power[N[(n * n), $MachinePrecision], -0.5], $MachinePrecision] / x), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if (/.f64 #s(literal 1 binary64) n) < -2.00000000000000012e-9 or 1.00000000000000006e-9 < (/.f64 #s(literal 1 binary64) n) < 4.0000000000000001e143

   1. Initial program 95.7%

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

   3. Taylor expanded in x around 0

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

   5. Applied rewrites 57.0%

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

   if -2.00000000000000012e-9 < (/.f64 #s(literal 1 binary64) n) < 4.99999999999999994e-93

   1. Initial program 28.6%

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

   3. Taylor expanded in n around inf

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

      1. lower-/.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lower-log1p.f64 N/A

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

      4. lower-log.f64 82.6

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

   5. Applied rewrites 82.6%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 59.5%

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

   if 4.99999999999999994e-93 < (/.f64 #s(literal 1 binary64) n) < 1.00000000000000006e-9

   1. Initial program 13.0%

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

   3. Taylor expanded in n around inf

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

      1. lower-/.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lower-log1p.f64 N/A

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

      4. lower-log.f64 40.5

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

   5. Applied rewrites 40.5%

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

   6. Taylor expanded in x around inf

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

   8. Applied rewrites 68.7%

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

   if 4.0000000000000001e143 < (/.f64 #s(literal 1 binary64) n)

   1. Initial program 7.7%

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

   3. Taylor expanded in x around inf

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

      1. lower-/.f64 N/A

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

      2. log-rec N/A

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

      3. mul-1-neg N/A

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

      4. associate-*r/ N/A

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

      5. associate-*r* N/A

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

      6. metadata-eval N/A

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

      7. *-lft-identity N/A

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

      8. lower-exp.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lower-log.f64 N/A

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

      11. lower-*.f64 0.7

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

   5. Applied rewrites 0.7%

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

   6. Taylor expanded in n around inf

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

   8. Applied rewrites 57.1%

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

   10. Applied rewrites 91.0%

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

4. Final simplification 61.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{n}^{-1} \leq -2 \cdot 10^{-9}:\\ \;\;\;\;1 - {x}^{\left({n}^{-1}\right)}\\ \mathbf{elif}\;{n}^{-1} \leq 5 \cdot 10^{-93}:\\ \;\;\;\;\frac{-\log x}{n}\\ \mathbf{elif}\;{n}^{-1} \leq 10^{-9}:\\ \;\;\;\;\frac{{x}^{-1}}{n}\\ \mathbf{elif}\;{n}^{-1} \leq 4 \cdot 10^{+143}:\\ \;\;\;\;1 - {x}^{\left({n}^{-1}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{{\left(n \cdot n\right)}^{-0.5}}{x}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 53.4% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 1 - {x}^{\left({n}^{-1}\right)}\\ \mathbf{if}\;{n}^{-1} \leq -2 \cdot 10^{-9}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;{n}^{-1} \leq 5 \cdot 10^{-93}:\\ \;\;\;\;\frac{-\log x}{n}\\ \mathbf{elif}\;{n}^{-1} \leq 10^{-9}:\\ \;\;\;\;\frac{{x}^{-1}}{n}\\ \mathbf{elif}\;{n}^{-1} \leq 4 \cdot 10^{+143}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{0.3333333333333333}{n \cdot x} - \frac{0.5}{n}}{x} - \frac{-1}{n}}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (let* ((t_0 (- 1.0 (pow x (pow n -1.0)))))
   (if (<= (pow n -1.0) -2e-9)
     t_0
     (if (<= (pow n -1.0) 5e-93)
       (/ (- (log x)) n)
       (if (<= (pow n -1.0) 1e-9)
         (/ (pow x -1.0) n)
         (if (<= (pow n -1.0) 4e+143)
           t_0
           (/
            (- (/ (- (/ 0.3333333333333333 (* n x)) (/ 0.5 n)) x) (/ -1.0 n))
            x)))))))

C

double code(double x, double n) {
	double t_0 = 1.0 - pow(x, pow(n, -1.0));
	double tmp;
	if (pow(n, -1.0) <= -2e-9) {
		tmp = t_0;
	} else if (pow(n, -1.0) <= 5e-93) {
		tmp = -log(x) / n;
	} else if (pow(n, -1.0) <= 1e-9) {
		tmp = pow(x, -1.0) / n;
	} else if (pow(n, -1.0) <= 4e+143) {
		tmp = t_0;
	} else {
		tmp = ((((0.3333333333333333 / (n * x)) - (0.5 / n)) / x) - (-1.0 / n)) / x;
	}
	return tmp;
}

Fortran

real(8) function code(x, n)
    real(8), intent (in) :: x
    real(8), intent (in) :: n
    real(8) :: t_0
    real(8) :: tmp
    t_0 = 1.0d0 - (x ** (n ** (-1.0d0)))
    if ((n ** (-1.0d0)) <= (-2d-9)) then
        tmp = t_0
    else if ((n ** (-1.0d0)) <= 5d-93) then
        tmp = -log(x) / n
    else if ((n ** (-1.0d0)) <= 1d-9) then
        tmp = (x ** (-1.0d0)) / n
    else if ((n ** (-1.0d0)) <= 4d+143) then
        tmp = t_0
    else
        tmp = ((((0.3333333333333333d0 / (n * x)) - (0.5d0 / n)) / x) - ((-1.0d0) / n)) / x
    end if
    code = tmp
end function

Java

public static double code(double x, double n) {
	double t_0 = 1.0 - Math.pow(x, Math.pow(n, -1.0));
	double tmp;
	if (Math.pow(n, -1.0) <= -2e-9) {
		tmp = t_0;
	} else if (Math.pow(n, -1.0) <= 5e-93) {
		tmp = -Math.log(x) / n;
	} else if (Math.pow(n, -1.0) <= 1e-9) {
		tmp = Math.pow(x, -1.0) / n;
	} else if (Math.pow(n, -1.0) <= 4e+143) {
		tmp = t_0;
	} else {
		tmp = ((((0.3333333333333333 / (n * x)) - (0.5 / n)) / x) - (-1.0 / n)) / x;
	}
	return tmp;
}

Python

def code(x, n):
	t_0 = 1.0 - math.pow(x, math.pow(n, -1.0))
	tmp = 0
	if math.pow(n, -1.0) <= -2e-9:
		tmp = t_0
	elif math.pow(n, -1.0) <= 5e-93:
		tmp = -math.log(x) / n
	elif math.pow(n, -1.0) <= 1e-9:
		tmp = math.pow(x, -1.0) / n
	elif math.pow(n, -1.0) <= 4e+143:
		tmp = t_0
	else:
		tmp = ((((0.3333333333333333 / (n * x)) - (0.5 / n)) / x) - (-1.0 / n)) / x
	return tmp

Julia

function code(x, n)
	t_0 = Float64(1.0 - (x ^ (n ^ -1.0)))
	tmp = 0.0
	if ((n ^ -1.0) <= -2e-9)
		tmp = t_0;
	elseif ((n ^ -1.0) <= 5e-93)
		tmp = Float64(Float64(-log(x)) / n);
	elseif ((n ^ -1.0) <= 1e-9)
		tmp = Float64((x ^ -1.0) / n);
	elseif ((n ^ -1.0) <= 4e+143)
		tmp = t_0;
	else
		tmp = Float64(Float64(Float64(Float64(Float64(0.3333333333333333 / Float64(n * x)) - Float64(0.5 / n)) / x) - Float64(-1.0 / n)) / x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, n)
	t_0 = 1.0 - (x ^ (n ^ -1.0));
	tmp = 0.0;
	if ((n ^ -1.0) <= -2e-9)
		tmp = t_0;
	elseif ((n ^ -1.0) <= 5e-93)
		tmp = -log(x) / n;
	elseif ((n ^ -1.0) <= 1e-9)
		tmp = (x ^ -1.0) / n;
	elseif ((n ^ -1.0) <= 4e+143)
		tmp = t_0;
	else
		tmp = ((((0.3333333333333333 / (n * x)) - (0.5 / n)) / x) - (-1.0 / n)) / x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, n_] := Block[{t$95$0 = N[(1.0 - N[Power[x, N[Power[n, -1.0], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[Power[n, -1.0], $MachinePrecision], -2e-9], t$95$0, If[LessEqual[N[Power[n, -1.0], $MachinePrecision], 5e-93], N[((-N[Log[x], $MachinePrecision]) / n), $MachinePrecision], If[LessEqual[N[Power[n, -1.0], $MachinePrecision], 1e-9], N[(N[Power[x, -1.0], $MachinePrecision] / n), $MachinePrecision], If[LessEqual[N[Power[n, -1.0], $MachinePrecision], 4e+143], t$95$0, N[(N[(N[(N[(N[(0.3333333333333333 / N[(n * x), $MachinePrecision]), $MachinePrecision] - N[(0.5 / n), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision] - N[(-1.0 / n), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if (/.f64 #s(literal 1 binary64) n) < -2.00000000000000012e-9 or 1.00000000000000006e-9 < (/.f64 #s(literal 1 binary64) n) < 4.0000000000000001e143

   1. Initial program 95.7%

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

   3. Taylor expanded in x around 0

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

   5. Applied rewrites 57.0%

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

   if -2.00000000000000012e-9 < (/.f64 #s(literal 1 binary64) n) < 4.99999999999999994e-93

   1. Initial program 28.6%

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

   3. Taylor expanded in n around inf

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

      1. lower-/.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lower-log1p.f64 N/A

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

      4. lower-log.f64 82.6

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

   5. Applied rewrites 82.6%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 59.5%

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

   if 4.99999999999999994e-93 < (/.f64 #s(literal 1 binary64) n) < 1.00000000000000006e-9

   1. Initial program 13.0%

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

   3. Taylor expanded in n around inf

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

      1. lower-/.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lower-log1p.f64 N/A

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

      4. lower-log.f64 40.5

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

   5. Applied rewrites 40.5%

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

   6. Taylor expanded in x around inf

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

   8. Applied rewrites 68.7%

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

   if 4.0000000000000001e143 < (/.f64 #s(literal 1 binary64) n)

   1. Initial program 7.7%

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

   3. Taylor expanded in n around inf

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

      1. lower-/.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lower-log1p.f64 N/A

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

      4. lower-log.f64 7.8

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

   5. Applied rewrites 7.8%

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

   6. Taylor expanded in x around -inf

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

   8. Applied rewrites 87.0%

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

4. Final simplification 61.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{n}^{-1} \leq -2 \cdot 10^{-9}:\\ \;\;\;\;1 - {x}^{\left({n}^{-1}\right)}\\ \mathbf{elif}\;{n}^{-1} \leq 5 \cdot 10^{-93}:\\ \;\;\;\;\frac{-\log x}{n}\\ \mathbf{elif}\;{n}^{-1} \leq 10^{-9}:\\ \;\;\;\;\frac{{x}^{-1}}{n}\\ \mathbf{elif}\;{n}^{-1} \leq 4 \cdot 10^{+143}:\\ \;\;\;\;1 - {x}^{\left({n}^{-1}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{0.3333333333333333}{n \cdot x} - \frac{0.5}{n}}{x} - \frac{-1}{n}}{x}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 55.6% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 2.75 \cdot 10^{-179} \lor \neg \left(x \leq 3 \cdot 10^{-147} \lor \neg \left(x \leq 2 \cdot 10^{-13}\right)\right):\\ \;\;\;\;\frac{-\log x}{n}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{0.3333333333333333}{n \cdot x} - \frac{0.5}{n}}{x} - \frac{-1}{n}}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (if (or (<= x 2.75e-179) (not (or (<= x 3e-147) (not (<= x 2e-13)))))
   (/ (- (log x)) n)
   (/ (- (/ (- (/ 0.3333333333333333 (* n x)) (/ 0.5 n)) x) (/ -1.0 n)) x)))

C

double code(double x, double n) {
	double tmp;
	if ((x <= 2.75e-179) || !((x <= 3e-147) || !(x <= 2e-13))) {
		tmp = -log(x) / n;
	} else {
		tmp = ((((0.3333333333333333 / (n * x)) - (0.5 / n)) / x) - (-1.0 / n)) / x;
	}
	return tmp;
}

Fortran

real(8) function code(x, n)
    real(8), intent (in) :: x
    real(8), intent (in) :: n
    real(8) :: tmp
    if ((x <= 2.75d-179) .or. (.not. (x <= 3d-147) .or. (.not. (x <= 2d-13)))) then
        tmp = -log(x) / n
    else
        tmp = ((((0.3333333333333333d0 / (n * x)) - (0.5d0 / n)) / x) - ((-1.0d0) / n)) / x
    end if
    code = tmp
end function

Java

public static double code(double x, double n) {
	double tmp;
	if ((x <= 2.75e-179) || !((x <= 3e-147) || !(x <= 2e-13))) {
		tmp = -Math.log(x) / n;
	} else {
		tmp = ((((0.3333333333333333 / (n * x)) - (0.5 / n)) / x) - (-1.0 / n)) / x;
	}
	return tmp;
}

Python

def code(x, n):
	tmp = 0
	if (x <= 2.75e-179) or not ((x <= 3e-147) or not (x <= 2e-13)):
		tmp = -math.log(x) / n
	else:
		tmp = ((((0.3333333333333333 / (n * x)) - (0.5 / n)) / x) - (-1.0 / n)) / x
	return tmp

Julia

function code(x, n)
	tmp = 0.0
	if ((x <= 2.75e-179) || !((x <= 3e-147) || !(x <= 2e-13)))
		tmp = Float64(Float64(-log(x)) / n);
	else
		tmp = Float64(Float64(Float64(Float64(Float64(0.3333333333333333 / Float64(n * x)) - Float64(0.5 / n)) / x) - Float64(-1.0 / n)) / x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, n)
	tmp = 0.0;
	if ((x <= 2.75e-179) || ~(((x <= 3e-147) || ~((x <= 2e-13)))))
		tmp = -log(x) / n;
	else
		tmp = ((((0.3333333333333333 / (n * x)) - (0.5 / n)) / x) - (-1.0 / n)) / x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, n_] := If[Or[LessEqual[x, 2.75e-179], N[Not[Or[LessEqual[x, 3e-147], N[Not[LessEqual[x, 2e-13]], $MachinePrecision]]], $MachinePrecision]], N[((-N[Log[x], $MachinePrecision]) / n), $MachinePrecision], N[(N[(N[(N[(N[(0.3333333333333333 / N[(n * x), $MachinePrecision]), $MachinePrecision] - N[(0.5 / n), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision] - N[(-1.0 / n), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 2.7500000000000001e-179 or 3.0000000000000002e-147 < x < 2.0000000000000001e-13

   1. Initial program 36.1%

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

   3. Taylor expanded in n around inf

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

      1. lower-/.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lower-log1p.f64 N/A

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

      4. lower-log.f64 56.9

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

   5. Applied rewrites 56.9%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 56.9%

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

   if 2.7500000000000001e-179 < x < 3.0000000000000002e-147 or 2.0000000000000001e-13 < x

   1. Initial program 65.9%

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

   3. Taylor expanded in n around inf

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

      1. lower-/.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lower-log1p.f64 N/A

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

      4. lower-log.f64 59.3

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

   5. Applied rewrites 59.3%

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

   6. Taylor expanded in x around -inf

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

   8. Applied rewrites 61.7%

      \[\leadsto -\frac{\left(-\frac{\frac{0.3333333333333333}{n \cdot x} - \frac{0.5}{n}}{x}\right) - \frac{1}{n}}{x} \]
3. Recombined 2 regimes into one program.

4. Final simplification 59.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 2.75 \cdot 10^{-179} \lor \neg \left(x \leq 3 \cdot 10^{-147} \lor \neg \left(x \leq 2 \cdot 10^{-13}\right)\right):\\ \;\;\;\;\frac{-\log x}{n}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{0.3333333333333333}{n \cdot x} - \frac{0.5}{n}}{x} - \frac{-1}{n}}{x}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 40.8% accurate, 2.0× speedup

Math

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

FPCore

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

C

double code(double x, double n) {
	return 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)) / n
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 51.7%

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

3. Taylor expanded in n around inf

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

   1. lower-/.f64 N/A

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

   2. lower--.f64 N/A

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

   3. lower-log1p.f64 N/A

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

   4. lower-log.f64 58.2

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

5. Applied rewrites 58.2%

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

6. Taylor expanded in x around inf

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

8. Applied rewrites 39.6%

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

9. Final simplification 39.6%

   \[\leadsto \frac{{x}^{-1}}{n} \]
10. Add Preprocessing

Alternative 12: 40.8% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 51.7%

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

3. Taylor expanded in x around inf

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

   1. lower-/.f64 N/A

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

   2. log-rec N/A

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

   3. mul-1-neg N/A

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

   4. associate-*r/ N/A

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

   5. associate-*r* N/A

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

   6. metadata-eval N/A

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

   7. *-lft-identity N/A

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

   8. lower-exp.f64 N/A

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

   9. lower-/.f64 N/A

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

   10. lower-log.f64 N/A

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

   11. lower-*.f64 56.1

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

5. Applied rewrites 56.1%

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

6. Taylor expanded in n around inf

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

8. Applied rewrites 39.6%

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

9. Final simplification 39.6%

   \[\leadsto \frac{{n}^{-1}}{x} \]
10. Add Preprocessing

Alternative 13: 40.3% accurate, 2.2× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 51.7%

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

3. Taylor expanded in x around inf

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

   1. lower-/.f64 N/A

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

   2. log-rec N/A

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

   3. mul-1-neg N/A

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

   4. associate-*r/ N/A

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

   5. associate-*r* N/A

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

   6. metadata-eval N/A

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

   7. *-lft-identity N/A

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

   8. lower-exp.f64 N/A

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

   9. lower-/.f64 N/A

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

   10. lower-log.f64 N/A

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

   11. lower-*.f64 56.1

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

5. Applied rewrites 56.1%

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

6. Taylor expanded in n around inf

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

8. Applied rewrites 39.6%

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

10. Applied rewrites 38.9%

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

11. Final simplification 38.9%

    \[\leadsto {\left(n \cdot x\right)}^{-1} \]
12. Add Preprocessing

Alternative 14: 46.3% accurate, 3.4× speedup

Math

\[\begin{array}{l} \\ \frac{\frac{\frac{0.3333333333333333}{n \cdot x} - \frac{0.5}{n}}{x} - \frac{-1}{n}}{x} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (/ (- (/ (- (/ 0.3333333333333333 (* n x)) (/ 0.5 n)) x) (/ -1.0 n)) x))

C

double code(double x, double n) {
	return ((((0.3333333333333333 / (n * x)) - (0.5 / n)) / x) - (-1.0 / n)) / x;
}

Fortran

real(8) function code(x, n)
    real(8), intent (in) :: x
    real(8), intent (in) :: n
    code = ((((0.3333333333333333d0 / (n * x)) - (0.5d0 / n)) / x) - ((-1.0d0) / n)) / x
end function

Java

public static double code(double x, double n) {
	return ((((0.3333333333333333 / (n * x)) - (0.5 / n)) / x) - (-1.0 / n)) / x;
}

Python

def code(x, n):
	return ((((0.3333333333333333 / (n * x)) - (0.5 / n)) / x) - (-1.0 / n)) / x

Julia

function code(x, n)
	return Float64(Float64(Float64(Float64(Float64(0.3333333333333333 / Float64(n * x)) - Float64(0.5 / n)) / x) - Float64(-1.0 / n)) / x)
end

MATLAB

function tmp = code(x, n)
	tmp = ((((0.3333333333333333 / (n * x)) - (0.5 / n)) / x) - (-1.0 / n)) / x;
end

Wolfram

code[x_, n_] := N[(N[(N[(N[(N[(0.3333333333333333 / N[(n * x), $MachinePrecision]), $MachinePrecision] - N[(0.5 / n), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision] - N[(-1.0 / n), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]

Derivation

1. Initial program 51.7%

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

3. Taylor expanded in n around inf

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

   1. lower-/.f64 N/A

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

   2. lower--.f64 N/A

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

   3. lower-log1p.f64 N/A

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

   4. lower-log.f64 58.2

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

5. Applied rewrites 58.2%

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

6. Taylor expanded in x around -inf

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

8. Applied rewrites 46.4%

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

9. Final simplification 46.4%

   \[\leadsto \frac{\frac{\frac{0.3333333333333333}{n \cdot x} - \frac{0.5}{n}}{x} - \frac{-1}{n}}{x} \]
10. Add Preprocessing

Alternative 15: 46.3% accurate, 4.5× speedup

Math

\[\begin{array}{l} \\ \frac{\frac{\frac{\frac{0.3333333333333333}{x} - 0.5}{x} - -1}{x}}{n} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (/ (/ (- (/ (- (/ 0.3333333333333333 x) 0.5) x) -1.0) x) n))

C

double code(double x, double n) {
	return (((((0.3333333333333333 / x) - 0.5) / x) - -1.0) / x) / n;
}

Fortran

real(8) function code(x, n)
    real(8), intent (in) :: x
    real(8), intent (in) :: n
    code = (((((0.3333333333333333d0 / x) - 0.5d0) / x) - (-1.0d0)) / x) / n
end function

Java

public static double code(double x, double n) {
	return (((((0.3333333333333333 / x) - 0.5) / x) - -1.0) / x) / n;
}

Python

def code(x, n):
	return (((((0.3333333333333333 / x) - 0.5) / x) - -1.0) / x) / n

Julia

function code(x, n)
	return Float64(Float64(Float64(Float64(Float64(Float64(0.3333333333333333 / x) - 0.5) / x) - -1.0) / x) / n)
end

MATLAB

function tmp = code(x, n)
	tmp = (((((0.3333333333333333 / x) - 0.5) / x) - -1.0) / x) / n;
end

Wolfram

code[x_, n_] := N[(N[(N[(N[(N[(N[(0.3333333333333333 / x), $MachinePrecision] - 0.5), $MachinePrecision] / x), $MachinePrecision] - -1.0), $MachinePrecision] / x), $MachinePrecision] / n), $MachinePrecision]

Derivation

1. Initial program 51.7%

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

3. Taylor expanded in n around inf

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

   1. lower-/.f64 N/A

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

   2. lower--.f64 N/A

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

   3. lower-log1p.f64 N/A

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

   4. lower-log.f64 58.2

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

5. Applied rewrites 58.2%

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

6. Taylor expanded in x around -inf

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

8. Applied rewrites 46.4%

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

9. Final simplification 46.4%

   \[\leadsto \frac{\frac{\frac{\frac{0.3333333333333333}{x} - 0.5}{x} - -1}{x}}{n} \]
10. Add Preprocessing

Reproduce

herbie shell --seed 2024326 
(FPCore (x n)
  :name "2nthrt (problem 3.4.6)"
  :precision binary64
  (- (pow (+ x 1.0) (/ 1.0 n)) (pow x (/ 1.0 n))))