2nthrt (problem 3.4.6)

Percentage Accurate: 53.5% → 91.9%

Time: 22.7s

Alternatives: 17

Speedup: 1.5×

• could not determine a ground truth

  1. x = -2.141410310023786e-157
  2. n = 6.967214098039158e-60

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 53.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 91.9% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if x < 1

   1. Initial program 48.6%

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

   3. Taylor expanded in x around 0

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

      1. associate--l+ N/A

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

      2. +-commutative N/A

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

      3. *-rgt-identity N/A

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

      4. associate-*r/ N/A

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

      5. remove-double-neg N/A

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

      6. mul-1-neg N/A

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

      7. distribute-neg-frac N/A

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

      8. mul-1-neg N/A

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

      9. log-rec N/A

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

      10. mul-1-neg N/A

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

      11. associate-+l- N/A

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

      12. lower--.f64 N/A

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

      13. associate-*r/ N/A

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

      14. *-rgt-identity N/A

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

      15. lower-/.f64 N/A

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

      16. lower-expm1.f64 N/A

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

      17. mul-1-neg N/A

          \[\leadsto \frac{x}{n} - \mathsf{expm1}\left(\color{blue}{\mathsf{neg}\left(\frac{\log \left(\frac{1}{x}\right)}{n}\right)}\right) \]

   5. Applied rewrites 88.7%

      \[\leadsto \color{blue}{\frac{x}{n} - \mathsf{expm1}\left(\frac{\log x}{n}\right)} \]

   if 1 < x

   1. Initial program 66.1%

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

   3. Taylor expanded in x around inf

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

      1. lower-/.f64 N/A

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

   5. Applied rewrites 81.5%

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

   6. Taylor expanded in n around inf

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

   8. Applied rewrites 99.2%

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

4. Final simplification 93.3%

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

Alternative 2: 78.5% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {x}^{\left({n}^{-1}\right)}\\ t_1 := {\left(x + 1\right)}^{\left({n}^{-1}\right)} - t\_0\\ \mathbf{if}\;t\_1 \leq -0.2 \lor \neg \left(t\_1 \leq 2 \cdot 10^{-9}\right):\\ \;\;\;\;1 - t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{\log \left(\frac{1 + x}{x}\right)}{n}\\ \end{array} \end{array} \]

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 (or (<= t_1 -0.2) (not (<= t_1 2e-9)))
     (- 1.0 t_0)
     (/ (log (/ (+ 1.0 x) x)) n))))

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 <= -0.2) || !(t_1 <= 2e-9)) {
		tmp = 1.0 - t_0;
	} else {
		tmp = log(((1.0 + x) / x)) / n;
	}
	return tmp;
}

Fortran

real(8) function code(x, n)
    real(8), intent (in) :: x
    real(8), intent (in) :: n
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = x ** (n ** (-1.0d0))
    t_1 = ((x + 1.0d0) ** (n ** (-1.0d0))) - t_0
    if ((t_1 <= (-0.2d0)) .or. (.not. (t_1 <= 2d-9))) then
        tmp = 1.0d0 - t_0
    else
        tmp = log(((1.0d0 + x) / x)) / n
    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 <= -0.2) || !(t_1 <= 2e-9)) {
		tmp = 1.0 - t_0;
	} else {
		tmp = Math.log(((1.0 + x) / x)) / n;
	}
	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 <= -0.2) or not (t_1 <= 2e-9):
		tmp = 1.0 - t_0
	else:
		tmp = math.log(((1.0 + x) / x)) / n
	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 <= -0.2) || !(t_1 <= 2e-9))
		tmp = Float64(1.0 - t_0);
	else
		tmp = Float64(log(Float64(Float64(1.0 + x) / x)) / n);
	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 <= -0.2) || ~((t_1 <= 2e-9)))
		tmp = 1.0 - t_0;
	else
		tmp = log(((1.0 + x) / x)) / n;
	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[Or[LessEqual[t$95$1, -0.2], N[Not[LessEqual[t$95$1, 2e-9]], $MachinePrecision]], N[(1.0 - t$95$0), $MachinePrecision], N[(N[Log[N[(N[(1.0 + x), $MachinePrecision] / x), $MachinePrecision]], $MachinePrecision] / n), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (pow.f64 (+.f64 x #s(literal 1 binary64)) (/.f64 #s(literal 1 binary64) n)) (pow.f64 x (/.f64 #s(literal 1 binary64) n))) < -0.20000000000000001 or 2.00000000000000012e-9 < (-.f64 (pow.f64 (+.f64 x #s(literal 1 binary64)) (/.f64 #s(literal 1 binary64) n)) (pow.f64 x (/.f64 #s(literal 1 binary64) n)))

   1. Initial program 80.5%

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

   3. Taylor expanded in x around 0

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

   5. Applied rewrites 80.5%

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

   if -0.20000000000000001 < (-.f64 (pow.f64 (+.f64 x #s(literal 1 binary64)) (/.f64 #s(literal 1 binary64) n)) (pow.f64 x (/.f64 #s(literal 1 binary64) n))) < 2.00000000000000012e-9

   1. Initial program 45.1%

      \[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)} \]
   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 75.6

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

   5. Applied rewrites 75.6%

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

   7. Applied rewrites 75.7%

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

4. Final simplification 77.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{\left(x + 1\right)}^{\left({n}^{-1}\right)} - {x}^{\left({n}^{-1}\right)} \leq -0.2 \lor \neg \left({\left(x + 1\right)}^{\left({n}^{-1}\right)} - {x}^{\left({n}^{-1}\right)} \leq 2 \cdot 10^{-9}\right):\\ \;\;\;\;1 - {x}^{\left({n}^{-1}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\log \left(\frac{1 + x}{x}\right)}{n}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 83.2% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {x}^{\left({n}^{-1}\right)}\\ \mathbf{if}\;{n}^{-1} \leq -4 \cdot 10^{-39}:\\ \;\;\;\;\frac{t\_0}{n \cdot x}\\ \mathbf{elif}\;{n}^{-1} \leq 5 \cdot 10^{-12}:\\ \;\;\;\;\frac{\log \left(\frac{1 + x}{x}\right)}{n}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\frac{\frac{0.5}{n} - 0.5}{n}, x, {n}^{-1}\right), x, 1\right) - t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (let* ((t_0 (pow x (pow n -1.0))))
   (if (<= (pow n -1.0) -4e-39)
     (/ t_0 (* n x))
     (if (<= (pow n -1.0) 5e-12)
       (/ (log (/ (+ 1.0 x) x)) n)
       (- (fma (fma (/ (- (/ 0.5 n) 0.5) 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 tmp;
	if (pow(n, -1.0) <= -4e-39) {
		tmp = t_0 / (n * x);
	} else if (pow(n, -1.0) <= 5e-12) {
		tmp = log(((1.0 + x) / x)) / n;
	} else {
		tmp = fma(fma((((0.5 / n) - 0.5) / n), x, pow(n, -1.0)), x, 1.0) - t_0;
	}
	return tmp;
}

Julia

function code(x, n)
	t_0 = x ^ (n ^ -1.0)
	tmp = 0.0
	if ((n ^ -1.0) <= -4e-39)
		tmp = Float64(t_0 / Float64(n * x));
	elseif ((n ^ -1.0) <= 5e-12)
		tmp = Float64(log(Float64(Float64(1.0 + x) / x)) / n);
	else
		tmp = Float64(fma(fma(Float64(Float64(Float64(0.5 / n) - 0.5) / 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]}, If[LessEqual[N[Power[n, -1.0], $MachinePrecision], -4e-39], N[(t$95$0 / N[(n * x), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[Power[n, -1.0], $MachinePrecision], 5e-12], N[(N[Log[N[(N[(1.0 + x), $MachinePrecision] / x), $MachinePrecision]], $MachinePrecision] / n), $MachinePrecision], N[(N[(N[(N[(N[(N[(0.5 / n), $MachinePrecision] - 0.5), $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 #s(literal 1 binary64) n) < -3.99999999999999972e-39

   1. Initial program 90.2%

      \[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)} \]
   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. associate-/l/ N/A

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

      2. lower-/.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. log-rec N/A

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

      5. mul-1-neg N/A

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

      6. associate-*r/ N/A

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

      7. associate-*r* N/A

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

      8. metadata-eval N/A

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

      9. *-commutative N/A

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

      10. associate-/l* N/A

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

      11. exp-to-pow N/A

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

      12. lower-pow.f64 N/A

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

      13. lower-/.f64 97.4

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

   5. Applied rewrites 97.4%

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

   7. Applied rewrites 97.4%

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

   9. Applied rewrites 97.4%

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

   if -3.99999999999999972e-39 < (/.f64 #s(literal 1 binary64) n) < 4.9999999999999997e-12

   1. Initial program 32.2%

      \[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)} \]
   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 77.0

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

   5. Applied rewrites 77.0%

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

   7. Applied rewrites 77.0%

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

   if 4.9999999999999997e-12 < (/.f64 #s(literal 1 binary64) n)

   1. Initial program 59.7%

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

      1. lift-pow.f64 N/A

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

      2. pow-to-exp N/A

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

      3. lower-exp.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. un-div-inv N/A

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

      6. lower-/.f64 N/A

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

      7. lift-+.f64 N/A

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

      8. +-commutative N/A

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

      9. lower-log1p.f64 100.0

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

   4. Applied rewrites 100.0%

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

   5. Taylor expanded in x around 0

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

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

   7. Applied rewrites 82.5%

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

4. Final simplification 84.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{n}^{-1} \leq -4 \cdot 10^{-39}:\\ \;\;\;\;\frac{{x}^{\left({n}^{-1}\right)}}{n \cdot x}\\ \mathbf{elif}\;{n}^{-1} \leq 5 \cdot 10^{-12}:\\ \;\;\;\;\frac{\log \left(\frac{1 + x}{x}\right)}{n}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\frac{\frac{0.5}{n} - 0.5}{n}, x, {n}^{-1}\right), x, 1\right) - {x}^{\left({n}^{-1}\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 82.6% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {x}^{\left({n}^{-1}\right)}\\ \mathbf{if}\;{n}^{-1} \leq -4 \cdot 10^{-39}:\\ \;\;\;\;\frac{t\_0}{n \cdot x}\\ \mathbf{elif}\;{n}^{-1} \leq 5 \cdot 10^{-12}:\\ \;\;\;\;\frac{\log \left(\frac{1 + x}{x}\right)}{n}\\ \mathbf{elif}\;{n}^{-1} \leq 10^{+196}:\\ \;\;\;\;\left(\frac{x}{n} + 1\right) - t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{0.3333333333333333}{\left(x \cdot x\right) \cdot n}}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (let* ((t_0 (pow x (pow n -1.0))))
   (if (<= (pow n -1.0) -4e-39)
     (/ t_0 (* n x))
     (if (<= (pow n -1.0) 5e-12)
       (/ (log (/ (+ 1.0 x) x)) n)
       (if (<= (pow n -1.0) 1e+196)
         (- (+ (/ x n) 1.0) t_0)
         (/ (/ 0.3333333333333333 (* (* x x) n)) x))))))

C

double code(double x, double n) {
	double t_0 = pow(x, pow(n, -1.0));
	double tmp;
	if (pow(n, -1.0) <= -4e-39) {
		tmp = t_0 / (n * x);
	} else if (pow(n, -1.0) <= 5e-12) {
		tmp = log(((1.0 + x) / x)) / n;
	} else if (pow(n, -1.0) <= 1e+196) {
		tmp = ((x / n) + 1.0) - t_0;
	} else {
		tmp = (0.3333333333333333 / ((x * x) * 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 = x ** (n ** (-1.0d0))
    if ((n ** (-1.0d0)) <= (-4d-39)) then
        tmp = t_0 / (n * x)
    else if ((n ** (-1.0d0)) <= 5d-12) then
        tmp = log(((1.0d0 + x) / x)) / n
    else if ((n ** (-1.0d0)) <= 1d+196) then
        tmp = ((x / n) + 1.0d0) - t_0
    else
        tmp = (0.3333333333333333d0 / ((x * x) * n)) / 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 tmp;
	if (Math.pow(n, -1.0) <= -4e-39) {
		tmp = t_0 / (n * x);
	} else if (Math.pow(n, -1.0) <= 5e-12) {
		tmp = Math.log(((1.0 + x) / x)) / n;
	} else if (Math.pow(n, -1.0) <= 1e+196) {
		tmp = ((x / n) + 1.0) - t_0;
	} else {
		tmp = (0.3333333333333333 / ((x * x) * n)) / x;
	}
	return tmp;
}

Python

def code(x, n):
	t_0 = math.pow(x, math.pow(n, -1.0))
	tmp = 0
	if math.pow(n, -1.0) <= -4e-39:
		tmp = t_0 / (n * x)
	elif math.pow(n, -1.0) <= 5e-12:
		tmp = math.log(((1.0 + x) / x)) / n
	elif math.pow(n, -1.0) <= 1e+196:
		tmp = ((x / n) + 1.0) - t_0
	else:
		tmp = (0.3333333333333333 / ((x * x) * n)) / x
	return tmp

Julia

function code(x, n)
	t_0 = x ^ (n ^ -1.0)
	tmp = 0.0
	if ((n ^ -1.0) <= -4e-39)
		tmp = Float64(t_0 / Float64(n * x));
	elseif ((n ^ -1.0) <= 5e-12)
		tmp = Float64(log(Float64(Float64(1.0 + x) / x)) / n);
	elseif ((n ^ -1.0) <= 1e+196)
		tmp = Float64(Float64(Float64(x / n) + 1.0) - t_0);
	else
		tmp = Float64(Float64(0.3333333333333333 / Float64(Float64(x * x) * n)) / x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, n)
	t_0 = x ^ (n ^ -1.0);
	tmp = 0.0;
	if ((n ^ -1.0) <= -4e-39)
		tmp = t_0 / (n * x);
	elseif ((n ^ -1.0) <= 5e-12)
		tmp = log(((1.0 + x) / x)) / n;
	elseif ((n ^ -1.0) <= 1e+196)
		tmp = ((x / n) + 1.0) - t_0;
	else
		tmp = (0.3333333333333333 / ((x * x) * n)) / x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, n_] := Block[{t$95$0 = N[Power[x, N[Power[n, -1.0], $MachinePrecision]], $MachinePrecision]}, If[LessEqual[N[Power[n, -1.0], $MachinePrecision], -4e-39], N[(t$95$0 / N[(n * x), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[Power[n, -1.0], $MachinePrecision], 5e-12], N[(N[Log[N[(N[(1.0 + x), $MachinePrecision] / x), $MachinePrecision]], $MachinePrecision] / n), $MachinePrecision], If[LessEqual[N[Power[n, -1.0], $MachinePrecision], 1e+196], N[(N[(N[(x / n), $MachinePrecision] + 1.0), $MachinePrecision] - t$95$0), $MachinePrecision], N[(N[(0.3333333333333333 / N[(N[(x * x), $MachinePrecision] * n), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if (/.f64 #s(literal 1 binary64) n) < -3.99999999999999972e-39

   1. Initial program 90.2%

      \[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)} \]
   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. associate-/l/ N/A

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

      2. lower-/.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. log-rec N/A

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

      5. mul-1-neg N/A

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

      6. associate-*r/ N/A

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

      7. associate-*r* N/A

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

      8. metadata-eval N/A

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

      9. *-commutative N/A

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

      10. associate-/l* N/A

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

      11. exp-to-pow N/A

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

      12. lower-pow.f64 N/A

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

      13. lower-/.f64 97.4

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

   5. Applied rewrites 97.4%

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

   7. Applied rewrites 97.4%

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

   9. Applied rewrites 97.4%

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

   if -3.99999999999999972e-39 < (/.f64 #s(literal 1 binary64) n) < 4.9999999999999997e-12

   1. Initial program 32.2%

      \[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)} \]
   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 77.0

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

   5. Applied rewrites 77.0%

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

   7. Applied rewrites 77.0%

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

   if 4.9999999999999997e-12 < (/.f64 #s(literal 1 binary64) n) < 9.9999999999999995e195

   1. Initial program 86.0%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. *-rgt-identity N/A

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

      3. associate-*r/ N/A

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

      4. lower-+.f64 N/A

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

      5. associate-*r/ N/A

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

      6. *-rgt-identity N/A

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

      7. lower-/.f64 87.1

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

   5. Applied rewrites 87.1%

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

   if 9.9999999999999995e195 < (/.f64 #s(literal 1 binary64) n)

   1. Initial program 12.5%

      \[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)} \]
   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 7.3

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

   5. Applied rewrites 7.3%

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

   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 86.2%

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

   9. Taylor expanded in x around 0

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

   11. Applied rewrites 86.2%

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

4. Final simplification 85.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{n}^{-1} \leq -4 \cdot 10^{-39}:\\ \;\;\;\;\frac{{x}^{\left({n}^{-1}\right)}}{n \cdot x}\\ \mathbf{elif}\;{n}^{-1} \leq 5 \cdot 10^{-12}:\\ \;\;\;\;\frac{\log \left(\frac{1 + x}{x}\right)}{n}\\ \mathbf{elif}\;{n}^{-1} \leq 10^{+196}:\\ \;\;\;\;\left(\frac{x}{n} + 1\right) - {x}^{\left({n}^{-1}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{0.3333333333333333}{\left(x \cdot x\right) \cdot n}}{x}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 82.4% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {x}^{\left({n}^{-1}\right)}\\ \mathbf{if}\;{n}^{-1} \leq -4 \cdot 10^{-39}:\\ \;\;\;\;\frac{t\_0}{n \cdot x}\\ \mathbf{elif}\;{n}^{-1} \leq 5 \cdot 10^{-12}:\\ \;\;\;\;\frac{\log \left(\frac{1 + x}{x}\right)}{n}\\ \mathbf{elif}\;{n}^{-1} \leq 10^{+196}:\\ \;\;\;\;1 - t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{0.3333333333333333}{\left(x \cdot x\right) \cdot n}}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (let* ((t_0 (pow x (pow n -1.0))))
   (if (<= (pow n -1.0) -4e-39)
     (/ t_0 (* n x))
     (if (<= (pow n -1.0) 5e-12)
       (/ (log (/ (+ 1.0 x) x)) n)
       (if (<= (pow n -1.0) 1e+196)
         (- 1.0 t_0)
         (/ (/ 0.3333333333333333 (* (* x x) n)) x))))))

C

double code(double x, double n) {
	double t_0 = pow(x, pow(n, -1.0));
	double tmp;
	if (pow(n, -1.0) <= -4e-39) {
		tmp = t_0 / (n * x);
	} else if (pow(n, -1.0) <= 5e-12) {
		tmp = log(((1.0 + x) / x)) / n;
	} else if (pow(n, -1.0) <= 1e+196) {
		tmp = 1.0 - t_0;
	} else {
		tmp = (0.3333333333333333 / ((x * x) * 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 = x ** (n ** (-1.0d0))
    if ((n ** (-1.0d0)) <= (-4d-39)) then
        tmp = t_0 / (n * x)
    else if ((n ** (-1.0d0)) <= 5d-12) then
        tmp = log(((1.0d0 + x) / x)) / n
    else if ((n ** (-1.0d0)) <= 1d+196) then
        tmp = 1.0d0 - t_0
    else
        tmp = (0.3333333333333333d0 / ((x * x) * n)) / 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 tmp;
	if (Math.pow(n, -1.0) <= -4e-39) {
		tmp = t_0 / (n * x);
	} else if (Math.pow(n, -1.0) <= 5e-12) {
		tmp = Math.log(((1.0 + x) / x)) / n;
	} else if (Math.pow(n, -1.0) <= 1e+196) {
		tmp = 1.0 - t_0;
	} else {
		tmp = (0.3333333333333333 / ((x * x) * n)) / x;
	}
	return tmp;
}

Python

def code(x, n):
	t_0 = math.pow(x, math.pow(n, -1.0))
	tmp = 0
	if math.pow(n, -1.0) <= -4e-39:
		tmp = t_0 / (n * x)
	elif math.pow(n, -1.0) <= 5e-12:
		tmp = math.log(((1.0 + x) / x)) / n
	elif math.pow(n, -1.0) <= 1e+196:
		tmp = 1.0 - t_0
	else:
		tmp = (0.3333333333333333 / ((x * x) * n)) / x
	return tmp

Julia

function code(x, n)
	t_0 = x ^ (n ^ -1.0)
	tmp = 0.0
	if ((n ^ -1.0) <= -4e-39)
		tmp = Float64(t_0 / Float64(n * x));
	elseif ((n ^ -1.0) <= 5e-12)
		tmp = Float64(log(Float64(Float64(1.0 + x) / x)) / n);
	elseif ((n ^ -1.0) <= 1e+196)
		tmp = Float64(1.0 - t_0);
	else
		tmp = Float64(Float64(0.3333333333333333 / Float64(Float64(x * x) * n)) / x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, n)
	t_0 = x ^ (n ^ -1.0);
	tmp = 0.0;
	if ((n ^ -1.0) <= -4e-39)
		tmp = t_0 / (n * x);
	elseif ((n ^ -1.0) <= 5e-12)
		tmp = log(((1.0 + x) / x)) / n;
	elseif ((n ^ -1.0) <= 1e+196)
		tmp = 1.0 - t_0;
	else
		tmp = (0.3333333333333333 / ((x * x) * n)) / x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, n_] := Block[{t$95$0 = N[Power[x, N[Power[n, -1.0], $MachinePrecision]], $MachinePrecision]}, If[LessEqual[N[Power[n, -1.0], $MachinePrecision], -4e-39], N[(t$95$0 / N[(n * x), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[Power[n, -1.0], $MachinePrecision], 5e-12], N[(N[Log[N[(N[(1.0 + x), $MachinePrecision] / x), $MachinePrecision]], $MachinePrecision] / n), $MachinePrecision], If[LessEqual[N[Power[n, -1.0], $MachinePrecision], 1e+196], N[(1.0 - t$95$0), $MachinePrecision], N[(N[(0.3333333333333333 / N[(N[(x * x), $MachinePrecision] * n), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if (/.f64 #s(literal 1 binary64) n) < -3.99999999999999972e-39

   1. Initial program 90.2%

      \[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)} \]
   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. associate-/l/ N/A

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

      2. lower-/.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. log-rec N/A

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

      5. mul-1-neg N/A

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

      6. associate-*r/ N/A

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

      7. associate-*r* N/A

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

      8. metadata-eval N/A

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

      9. *-commutative N/A

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

      10. associate-/l* N/A

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

      11. exp-to-pow N/A

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

      12. lower-pow.f64 N/A

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

      13. lower-/.f64 97.4

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

   5. Applied rewrites 97.4%

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

   7. Applied rewrites 97.4%

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

   9. Applied rewrites 97.4%

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

   if -3.99999999999999972e-39 < (/.f64 #s(literal 1 binary64) n) < 4.9999999999999997e-12

   1. Initial program 32.2%

      \[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)} \]
   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 77.0

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

   5. Applied rewrites 77.0%

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

   7. Applied rewrites 77.0%

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

   if 4.9999999999999997e-12 < (/.f64 #s(literal 1 binary64) n) < 9.9999999999999995e195

   1. Initial program 86.0%

      \[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)} \]
   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 86.0%

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

   if 9.9999999999999995e195 < (/.f64 #s(literal 1 binary64) n)

   1. Initial program 12.5%

      \[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)} \]
   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 7.3

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

   5. Applied rewrites 7.3%

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

   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 86.2%

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

   9. Taylor expanded in x around 0

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

   11. Applied rewrites 86.2%

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

4. Final simplification 85.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{n}^{-1} \leq -4 \cdot 10^{-39}:\\ \;\;\;\;\frac{{x}^{\left({n}^{-1}\right)}}{n \cdot x}\\ \mathbf{elif}\;{n}^{-1} \leq 5 \cdot 10^{-12}:\\ \;\;\;\;\frac{\log \left(\frac{1 + x}{x}\right)}{n}\\ \mathbf{elif}\;{n}^{-1} \leq 10^{+196}:\\ \;\;\;\;1 - {x}^{\left({n}^{-1}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{0.3333333333333333}{\left(x \cdot x\right) \cdot n}}{x}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 82.4% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;{n}^{-1} \leq -4 \cdot 10^{-39}:\\ \;\;\;\;\frac{{x}^{\left({n}^{-1} - 1\right)}}{n}\\ \mathbf{elif}\;{n}^{-1} \leq 5 \cdot 10^{-12}:\\ \;\;\;\;\frac{\log \left(\frac{1 + x}{x}\right)}{n}\\ \mathbf{elif}\;{n}^{-1} \leq 10^{+196}:\\ \;\;\;\;1 - {x}^{\left({n}^{-1}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{0.3333333333333333}{\left(x \cdot x\right) \cdot n}}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (if (<= (pow n -1.0) -4e-39)
   (/ (pow x (- (pow n -1.0) 1.0)) n)
   (if (<= (pow n -1.0) 5e-12)
     (/ (log (/ (+ 1.0 x) x)) n)
     (if (<= (pow n -1.0) 1e+196)
       (- 1.0 (pow x (pow n -1.0)))
       (/ (/ 0.3333333333333333 (* (* x x) n)) x)))))

C

double code(double x, double n) {
	double tmp;
	if (pow(n, -1.0) <= -4e-39) {
		tmp = pow(x, (pow(n, -1.0) - 1.0)) / n;
	} else if (pow(n, -1.0) <= 5e-12) {
		tmp = log(((1.0 + x) / x)) / n;
	} else if (pow(n, -1.0) <= 1e+196) {
		tmp = 1.0 - pow(x, pow(n, -1.0));
	} else {
		tmp = (0.3333333333333333 / ((x * x) * 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 ((n ** (-1.0d0)) <= (-4d-39)) then
        tmp = (x ** ((n ** (-1.0d0)) - 1.0d0)) / n
    else if ((n ** (-1.0d0)) <= 5d-12) then
        tmp = log(((1.0d0 + x) / x)) / n
    else if ((n ** (-1.0d0)) <= 1d+196) then
        tmp = 1.0d0 - (x ** (n ** (-1.0d0)))
    else
        tmp = (0.3333333333333333d0 / ((x * x) * n)) / x
    end if
    code = tmp
end function

Java

public static double code(double x, double n) {
	double tmp;
	if (Math.pow(n, -1.0) <= -4e-39) {
		tmp = Math.pow(x, (Math.pow(n, -1.0) - 1.0)) / n;
	} else if (Math.pow(n, -1.0) <= 5e-12) {
		tmp = Math.log(((1.0 + x) / x)) / n;
	} else if (Math.pow(n, -1.0) <= 1e+196) {
		tmp = 1.0 - Math.pow(x, Math.pow(n, -1.0));
	} else {
		tmp = (0.3333333333333333 / ((x * x) * n)) / x;
	}
	return tmp;
}

Python

def code(x, n):
	tmp = 0
	if math.pow(n, -1.0) <= -4e-39:
		tmp = math.pow(x, (math.pow(n, -1.0) - 1.0)) / n
	elif math.pow(n, -1.0) <= 5e-12:
		tmp = math.log(((1.0 + x) / x)) / n
	elif math.pow(n, -1.0) <= 1e+196:
		tmp = 1.0 - math.pow(x, math.pow(n, -1.0))
	else:
		tmp = (0.3333333333333333 / ((x * x) * n)) / x
	return tmp

Julia

function code(x, n)
	tmp = 0.0
	if ((n ^ -1.0) <= -4e-39)
		tmp = Float64((x ^ Float64((n ^ -1.0) - 1.0)) / n);
	elseif ((n ^ -1.0) <= 5e-12)
		tmp = Float64(log(Float64(Float64(1.0 + x) / x)) / n);
	elseif ((n ^ -1.0) <= 1e+196)
		tmp = Float64(1.0 - (x ^ (n ^ -1.0)));
	else
		tmp = Float64(Float64(0.3333333333333333 / Float64(Float64(x * x) * n)) / x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, n)
	tmp = 0.0;
	if ((n ^ -1.0) <= -4e-39)
		tmp = (x ^ ((n ^ -1.0) - 1.0)) / n;
	elseif ((n ^ -1.0) <= 5e-12)
		tmp = log(((1.0 + x) / x)) / n;
	elseif ((n ^ -1.0) <= 1e+196)
		tmp = 1.0 - (x ^ (n ^ -1.0));
	else
		tmp = (0.3333333333333333 / ((x * x) * n)) / x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, n_] := If[LessEqual[N[Power[n, -1.0], $MachinePrecision], -4e-39], N[(N[Power[x, N[(N[Power[n, -1.0], $MachinePrecision] - 1.0), $MachinePrecision]], $MachinePrecision] / n), $MachinePrecision], If[LessEqual[N[Power[n, -1.0], $MachinePrecision], 5e-12], N[(N[Log[N[(N[(1.0 + x), $MachinePrecision] / x), $MachinePrecision]], $MachinePrecision] / n), $MachinePrecision], If[LessEqual[N[Power[n, -1.0], $MachinePrecision], 1e+196], N[(1.0 - N[Power[x, N[Power[n, -1.0], $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[(0.3333333333333333 / N[(N[(x * x), $MachinePrecision] * n), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if (/.f64 #s(literal 1 binary64) n) < -3.99999999999999972e-39

   1. Initial program 90.2%

      \[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)} \]
   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. associate-/l/ N/A

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

      2. lower-/.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. log-rec N/A

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

      5. mul-1-neg N/A

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

      6. associate-*r/ N/A

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

      7. associate-*r* N/A

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

      8. metadata-eval N/A

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

      9. *-commutative N/A

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

      10. associate-/l* N/A

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

      11. exp-to-pow N/A

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

      12. lower-pow.f64 N/A

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

      13. lower-/.f64 97.4

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

   5. Applied rewrites 97.4%

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

   7. Applied rewrites 97.1%

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

   8. Taylor expanded in n around inf

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

   10. Applied rewrites 97.1%

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

   if -3.99999999999999972e-39 < (/.f64 #s(literal 1 binary64) n) < 4.9999999999999997e-12

   1. Initial program 32.2%

      \[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)} \]
   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 77.0

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

   5. Applied rewrites 77.0%

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

   7. Applied rewrites 77.0%

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

   if 4.9999999999999997e-12 < (/.f64 #s(literal 1 binary64) n) < 9.9999999999999995e195

   1. Initial program 86.0%

      \[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)} \]
   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 86.0%

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

   if 9.9999999999999995e195 < (/.f64 #s(literal 1 binary64) n)

   1. Initial program 12.5%

      \[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)} \]
   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 7.3

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

   5. Applied rewrites 7.3%

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

   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 86.2%

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

   9. Taylor expanded in x around 0

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

   11. Applied rewrites 86.2%

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

4. Final simplification 85.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{n}^{-1} \leq -4 \cdot 10^{-39}:\\ \;\;\;\;\frac{{x}^{\left({n}^{-1} - 1\right)}}{n}\\ \mathbf{elif}\;{n}^{-1} \leq 5 \cdot 10^{-12}:\\ \;\;\;\;\frac{\log \left(\frac{1 + x}{x}\right)}{n}\\ \mathbf{elif}\;{n}^{-1} \leq 10^{+196}:\\ \;\;\;\;1 - {x}^{\left({n}^{-1}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{0.3333333333333333}{\left(x \cdot x\right) \cdot n}}{x}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 54.7% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;{n}^{-1} \leq -10000000:\\ \;\;\;\;\frac{\frac{0.3333333333333333}{\left(x \cdot x\right) \cdot n}}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{{x}^{-1}}{n}, \frac{0.3333333333333333}{x} - 0.5, {n}^{-1}\right)}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (if (<= (pow n -1.0) -10000000.0)
   (/ (/ 0.3333333333333333 (* (* x x) n)) x)
   (/
    (fma (/ (pow x -1.0) n) (- (/ 0.3333333333333333 x) 0.5) (pow n -1.0))
    x)))

C

double code(double x, double n) {
	double tmp;
	if (pow(n, -1.0) <= -10000000.0) {
		tmp = (0.3333333333333333 / ((x * x) * n)) / x;
	} else {
		tmp = fma((pow(x, -1.0) / n), ((0.3333333333333333 / x) - 0.5), pow(n, -1.0)) / x;
	}
	return tmp;
}

Julia

function code(x, n)
	tmp = 0.0
	if ((n ^ -1.0) <= -10000000.0)
		tmp = Float64(Float64(0.3333333333333333 / Float64(Float64(x * x) * n)) / x);
	else
		tmp = Float64(fma(Float64((x ^ -1.0) / n), Float64(Float64(0.3333333333333333 / x) - 0.5), (n ^ -1.0)) / x);
	end
	return tmp
end

Wolfram

code[x_, n_] := If[LessEqual[N[Power[n, -1.0], $MachinePrecision], -10000000.0], N[(N[(0.3333333333333333 / N[(N[(x * x), $MachinePrecision] * n), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision], N[(N[(N[(N[Power[x, -1.0], $MachinePrecision] / n), $MachinePrecision] * N[(N[(0.3333333333333333 / x), $MachinePrecision] - 0.5), $MachinePrecision] + N[Power[n, -1.0], $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 #s(literal 1 binary64) n) < -1e7

   1. Initial program 100.0%

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

   3. Taylor expanded in n around inf

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

      1. lower-/.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lower-log1p.f64 N/A

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

      4. lower-log.f64 42.9

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

   5. Applied rewrites 42.9%

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

   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 44.4%

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

   9. Taylor expanded in x around 0

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

   11. Applied rewrites 63.3%

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

   if -1e7 < (/.f64 #s(literal 1 binary64) n)

   1. Initial program 37.2%

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

   3. Taylor expanded in n around inf

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

      1. 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 58.5

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

   5. Applied rewrites 58.5%

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

   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 50.2%

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

   9. Taylor expanded in x around 0

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

   11. Applied rewrites 30.6%

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

   12. Taylor expanded in x around -inf

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

   14. Applied rewrites 50.2%

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

4. Final simplification 54.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{n}^{-1} \leq -10000000:\\ \;\;\;\;\frac{\frac{0.3333333333333333}{\left(x \cdot x\right) \cdot n}}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{{x}^{-1}}{n}, \frac{0.3333333333333333}{x} - 0.5, {n}^{-1}\right)}{x}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 56.0% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;{n}^{-1} \leq -10000000 \lor \neg \left({n}^{-1} \leq 5 \cdot 10^{+75}\right):\\ \;\;\;\;\frac{\frac{0.3333333333333333}{\left(x \cdot x\right) \cdot n}}{x}\\ \mathbf{else}:\\ \;\;\;\;{\left(\mathsf{fma}\left(\frac{n}{x}, 0.5, n\right) \cdot x\right)}^{-1}\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (if (or (<= (pow n -1.0) -10000000.0) (not (<= (pow n -1.0) 5e+75)))
   (/ (/ 0.3333333333333333 (* (* x x) n)) x)
   (pow (* (fma (/ n x) 0.5 n) x) -1.0)))

C

double code(double x, double n) {
	double tmp;
	if ((pow(n, -1.0) <= -10000000.0) || !(pow(n, -1.0) <= 5e+75)) {
		tmp = (0.3333333333333333 / ((x * x) * n)) / x;
	} else {
		tmp = pow((fma((n / x), 0.5, n) * x), -1.0);
	}
	return tmp;
}

Julia

function code(x, n)
	tmp = 0.0
	if (((n ^ -1.0) <= -10000000.0) || !((n ^ -1.0) <= 5e+75))
		tmp = Float64(Float64(0.3333333333333333 / Float64(Float64(x * x) * n)) / x);
	else
		tmp = Float64(fma(Float64(n / x), 0.5, n) * x) ^ -1.0;
	end
	return tmp
end

Wolfram

code[x_, n_] := If[Or[LessEqual[N[Power[n, -1.0], $MachinePrecision], -10000000.0], N[Not[LessEqual[N[Power[n, -1.0], $MachinePrecision], 5e+75]], $MachinePrecision]], N[(N[(0.3333333333333333 / N[(N[(x * x), $MachinePrecision] * n), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision], N[Power[N[(N[(N[(n / x), $MachinePrecision] * 0.5 + n), $MachinePrecision] * x), $MachinePrecision], -1.0], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 #s(literal 1 binary64) n) < -1e7 or 5.0000000000000002e75 < (/.f64 #s(literal 1 binary64) n)

   1. Initial program 85.3%

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

   3. Taylor expanded in n around inf

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

      1. lower-/.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lower-log1p.f64 N/A

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

      4. lower-log.f64 34.0

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

   5. Applied rewrites 34.0%

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

   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 47.6%

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

   9. Taylor expanded in x around 0

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

   11. Applied rewrites 61.9%

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

   if -1e7 < (/.f64 #s(literal 1 binary64) n) < 5.0000000000000002e75

   1. Initial program 36.9%

      \[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)} \]
   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 67.0

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

   5. Applied rewrites 67.0%

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

   7. Applied rewrites 67.0%

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

   8. Taylor expanded in x around inf

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

   10. Applied rewrites 51.5%

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

4. Final simplification 55.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{n}^{-1} \leq -10000000 \lor \neg \left({n}^{-1} \leq 5 \cdot 10^{+75}\right):\\ \;\;\;\;\frac{\frac{0.3333333333333333}{\left(x \cdot x\right) \cdot n}}{x}\\ \mathbf{else}:\\ \;\;\;\;{\left(\mathsf{fma}\left(\frac{n}{x}, 0.5, n\right) \cdot x\right)}^{-1}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 56.7% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x}{\frac{\frac{0.3333333333333333}{x} - 0.5}{n}}\\ \mathbf{if}\;{n}^{-1} \leq -10000000:\\ \;\;\;\;\frac{\frac{0.3333333333333333}{\left(x \cdot x\right) \cdot n}}{x}\\ \mathbf{elif}\;{n}^{-1} \leq 5 \cdot 10^{+75}:\\ \;\;\;\;{\left(\mathsf{fma}\left(\frac{n}{x}, 0.5, n\right) \cdot x\right)}^{-1}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(1, t\_0, n\right)}{n \cdot t\_0}}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (let* ((t_0 (/ x (/ (- (/ 0.3333333333333333 x) 0.5) n))))
   (if (<= (pow n -1.0) -10000000.0)
     (/ (/ 0.3333333333333333 (* (* x x) n)) x)
     (if (<= (pow n -1.0) 5e+75)
       (pow (* (fma (/ n x) 0.5 n) x) -1.0)
       (/ (/ (fma 1.0 t_0 n) (* n t_0)) x)))))

C

double code(double x, double n) {
	double t_0 = x / (((0.3333333333333333 / x) - 0.5) / n);
	double tmp;
	if (pow(n, -1.0) <= -10000000.0) {
		tmp = (0.3333333333333333 / ((x * x) * n)) / x;
	} else if (pow(n, -1.0) <= 5e+75) {
		tmp = pow((fma((n / x), 0.5, n) * x), -1.0);
	} else {
		tmp = (fma(1.0, t_0, n) / (n * t_0)) / x;
	}
	return tmp;
}

Julia

function code(x, n)
	t_0 = Float64(x / Float64(Float64(Float64(0.3333333333333333 / x) - 0.5) / n))
	tmp = 0.0
	if ((n ^ -1.0) <= -10000000.0)
		tmp = Float64(Float64(0.3333333333333333 / Float64(Float64(x * x) * n)) / x);
	elseif ((n ^ -1.0) <= 5e+75)
		tmp = Float64(fma(Float64(n / x), 0.5, n) * x) ^ -1.0;
	else
		tmp = Float64(Float64(fma(1.0, t_0, n) / Float64(n * t_0)) / x);
	end
	return tmp
end

Wolfram

code[x_, n_] := Block[{t$95$0 = N[(x / N[(N[(N[(0.3333333333333333 / x), $MachinePrecision] - 0.5), $MachinePrecision] / n), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[Power[n, -1.0], $MachinePrecision], -10000000.0], N[(N[(0.3333333333333333 / N[(N[(x * x), $MachinePrecision] * n), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision], If[LessEqual[N[Power[n, -1.0], $MachinePrecision], 5e+75], N[Power[N[(N[(N[(n / x), $MachinePrecision] * 0.5 + n), $MachinePrecision] * x), $MachinePrecision], -1.0], $MachinePrecision], N[(N[(N[(1.0 * t$95$0 + n), $MachinePrecision] / N[(n * t$95$0), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 #s(literal 1 binary64) n) < -1e7

   1. Initial program 100.0%

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

   3. Taylor expanded in n around inf

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

      1. lower-/.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lower-log1p.f64 N/A

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

      4. lower-log.f64 42.9

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

   5. Applied rewrites 42.9%

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

   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 44.4%

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

   9. Taylor expanded in x around 0

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

   11. Applied rewrites 63.3%

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

   if -1e7 < (/.f64 #s(literal 1 binary64) n) < 5.0000000000000002e75

   1. Initial program 36.9%

      \[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)} \]
   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 67.0

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

   5. Applied rewrites 67.0%

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

   7. Applied rewrites 67.0%

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

   8. Taylor expanded in x around inf

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

   10. Applied rewrites 51.5%

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

   if 5.0000000000000002e75 < (/.f64 #s(literal 1 binary64) n)

   1. Initial program 39.4%

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

   3. Taylor expanded in n around inf

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

      1. lower-/.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lower-log1p.f64 N/A

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

      4. lower-log.f64 6.1

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

   5. Applied rewrites 6.1%

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

   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 57.7%

      \[\leadsto \frac{\frac{\frac{\frac{0.3333333333333333}{n}}{x} - \frac{0.5}{n}}{x} + \frac{1}{n}}{\color{blue}{x}} \]
   9. Step-by-step derivation

   10. Applied rewrites 61.3%

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

4. Final simplification 56.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{n}^{-1} \leq -10000000:\\ \;\;\;\;\frac{\frac{0.3333333333333333}{\left(x \cdot x\right) \cdot n}}{x}\\ \mathbf{elif}\;{n}^{-1} \leq 5 \cdot 10^{+75}:\\ \;\;\;\;{\left(\mathsf{fma}\left(\frac{n}{x}, 0.5, n\right) \cdot x\right)}^{-1}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(1, \frac{x}{\frac{\frac{0.3333333333333333}{x} - 0.5}{n}}, n\right)}{n \cdot \frac{x}{\frac{\frac{0.3333333333333333}{x} - 0.5}{n}}}}{x}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 55.4% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 2.4 \cdot 10^{-205}:\\ \;\;\;\;\frac{-\log x}{n}\\ \mathbf{elif}\;x \leq 1.1 \cdot 10^{+174}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{{x}^{-1}}{n}, \frac{0.3333333333333333}{x} - 0.5, {n}^{-1}\right)}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{0.3333333333333333}{\left(x \cdot x\right) \cdot n}}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (if (<= x 2.4e-205)
   (/ (- (log x)) n)
   (if (<= x 1.1e+174)
     (/
      (fma (/ (pow x -1.0) n) (- (/ 0.3333333333333333 x) 0.5) (pow n -1.0))
      x)
     (/ (/ 0.3333333333333333 (* (* x x) n)) x))))

C

double code(double x, double n) {
	double tmp;
	if (x <= 2.4e-205) {
		tmp = -log(x) / n;
	} else if (x <= 1.1e+174) {
		tmp = fma((pow(x, -1.0) / n), ((0.3333333333333333 / x) - 0.5), pow(n, -1.0)) / x;
	} else {
		tmp = (0.3333333333333333 / ((x * x) * n)) / x;
	}
	return tmp;
}

Julia

function code(x, n)
	tmp = 0.0
	if (x <= 2.4e-205)
		tmp = Float64(Float64(-log(x)) / n);
	elseif (x <= 1.1e+174)
		tmp = Float64(fma(Float64((x ^ -1.0) / n), Float64(Float64(0.3333333333333333 / x) - 0.5), (n ^ -1.0)) / x);
	else
		tmp = Float64(Float64(0.3333333333333333 / Float64(Float64(x * x) * n)) / x);
	end
	return tmp
end

Wolfram

code[x_, n_] := If[LessEqual[x, 2.4e-205], N[((-N[Log[x], $MachinePrecision]) / n), $MachinePrecision], If[LessEqual[x, 1.1e+174], N[(N[(N[(N[Power[x, -1.0], $MachinePrecision] / n), $MachinePrecision] * N[(N[(0.3333333333333333 / x), $MachinePrecision] - 0.5), $MachinePrecision] + N[Power[n, -1.0], $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision], N[(N[(0.3333333333333333 / N[(N[(x * x), $MachinePrecision] * n), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < 2.4000000000000002e-205

   1. Initial program 44.4%

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

   3. Taylor expanded in n around inf

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

      1. lower-/.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lower-log1p.f64 N/A

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

      4. lower-log.f64 58.1

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

   5. Applied rewrites 58.1%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 58.1%

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

   if 2.4000000000000002e-205 < x < 1.1000000000000001e174

   1. Initial program 52.6%

      \[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)} \]
   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 44.0

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

   5. Applied rewrites 44.0%

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

   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 53.7%

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

   9. Taylor expanded in x around 0

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

   11. Applied rewrites 35.0%

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

   12. Taylor expanded in x around -inf

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

   14. Applied rewrites 53.7%

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

   if 1.1000000000000001e174 < x

   1. Initial program 86.5%

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

   3. Taylor expanded in 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 86.5

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

   5. Applied rewrites 86.5%

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

   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 66.6%

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

   9. Taylor expanded in x around 0

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

   11. Applied rewrites 86.5%

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

4. Final simplification 59.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 2.4 \cdot 10^{-205}:\\ \;\;\;\;\frac{-\log x}{n}\\ \mathbf{elif}\;x \leq 1.1 \cdot 10^{+174}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{{x}^{-1}}{n}, \frac{0.3333333333333333}{x} - 0.5, {n}^{-1}\right)}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{0.3333333333333333}{\left(x \cdot x\right) \cdot n}}{x}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 91.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 1:\\ \;\;\;\;\frac{x}{n} - \mathsf{expm1}\left(\frac{\log x}{n}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{{x}^{\left({n}^{-1}\right)}}{x}}{n}\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (if (<= x 1.0)
   (- (/ x n) (expm1 (/ (log x) n)))
   (/ (/ (pow x (pow n -1.0)) x) n)))

C

double code(double x, double n) {
	double tmp;
	if (x <= 1.0) {
		tmp = (x / n) - expm1((log(x) / n));
	} else {
		tmp = (pow(x, pow(n, -1.0)) / x) / n;
	}
	return tmp;
}

Java

public static double code(double x, double n) {
	double tmp;
	if (x <= 1.0) {
		tmp = (x / n) - Math.expm1((Math.log(x) / n));
	} else {
		tmp = (Math.pow(x, Math.pow(n, -1.0)) / x) / n;
	}
	return tmp;
}

Python

def code(x, n):
	tmp = 0
	if x <= 1.0:
		tmp = (x / n) - math.expm1((math.log(x) / n))
	else:
		tmp = (math.pow(x, math.pow(n, -1.0)) / x) / n
	return tmp

Julia

function code(x, n)
	tmp = 0.0
	if (x <= 1.0)
		tmp = Float64(Float64(x / n) - expm1(Float64(log(x) / n)));
	else
		tmp = Float64(Float64((x ^ (n ^ -1.0)) / x) / n);
	end
	return tmp
end

Wolfram

code[x_, n_] := If[LessEqual[x, 1.0], N[(N[(x / n), $MachinePrecision] - N[(Exp[N[(N[Log[x], $MachinePrecision] / n), $MachinePrecision]] - 1), $MachinePrecision]), $MachinePrecision], N[(N[(N[Power[x, N[Power[n, -1.0], $MachinePrecision]], $MachinePrecision] / x), $MachinePrecision] / n), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 1

   1. Initial program 48.6%

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

   3. Taylor expanded in x around 0

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

      1. associate--l+ N/A

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

      2. +-commutative N/A

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

      3. *-rgt-identity N/A

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

      4. associate-*r/ N/A

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

      5. remove-double-neg N/A

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

      6. mul-1-neg N/A

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

      7. distribute-neg-frac N/A

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

      8. mul-1-neg N/A

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

      9. log-rec N/A

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

      10. mul-1-neg N/A

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

      11. associate-+l- N/A

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

      12. lower--.f64 N/A

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

      13. associate-*r/ N/A

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

      14. *-rgt-identity N/A

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

      15. lower-/.f64 N/A

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

      16. lower-expm1.f64 N/A

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

      17. mul-1-neg N/A

          \[\leadsto \frac{x}{n} - \mathsf{expm1}\left(\color{blue}{\mathsf{neg}\left(\frac{\log \left(\frac{1}{x}\right)}{n}\right)}\right) \]

   5. Applied rewrites 88.7%

      \[\leadsto \color{blue}{\frac{x}{n} - \mathsf{expm1}\left(\frac{\log x}{n}\right)} \]

   if 1 < x

   1. Initial program 66.1%

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

   3. Taylor expanded in x around 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. associate-/l/ N/A

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

      2. lower-/.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. log-rec N/A

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

      5. mul-1-neg N/A

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

      6. associate-*r/ N/A

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

      7. associate-*r* N/A

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

      8. metadata-eval N/A

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

      9. *-commutative N/A

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

      10. associate-/l* N/A

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

      11. exp-to-pow N/A

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

      12. lower-pow.f64 N/A

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

      13. lower-/.f64 98.4

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

   5. Applied rewrites 98.4%

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

4. Final simplification 93.0%

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

Alternative 12: 62.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {\left(\mathsf{fma}\left(\frac{n}{x}, 0.5, n\right) \cdot x\right)}^{-1}\\ \mathbf{if}\;n \leq -6.6 \cdot 10^{+257}:\\ \;\;\;\;\frac{-\log x}{n}\\ \mathbf{elif}\;n \leq -3.85:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;n \leq 4.35 \cdot 10^{-197}:\\ \;\;\;\;\frac{0.3333333333333333}{{x}^{3} \cdot n}\\ \mathbf{elif}\;n \leq 520000000000:\\ \;\;\;\;1 - {x}^{\left({n}^{-1}\right)}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (let* ((t_0 (pow (* (fma (/ n x) 0.5 n) x) -1.0)))
   (if (<= n -6.6e+257)
     (/ (- (log x)) n)
     (if (<= n -3.85)
       t_0
       (if (<= n 4.35e-197)
         (/ 0.3333333333333333 (* (pow x 3.0) n))
         (if (<= n 520000000000.0) (- 1.0 (pow x (pow n -1.0))) t_0))))))

C

double code(double x, double n) {
	double t_0 = pow((fma((n / x), 0.5, n) * x), -1.0);
	double tmp;
	if (n <= -6.6e+257) {
		tmp = -log(x) / n;
	} else if (n <= -3.85) {
		tmp = t_0;
	} else if (n <= 4.35e-197) {
		tmp = 0.3333333333333333 / (pow(x, 3.0) * n);
	} else if (n <= 520000000000.0) {
		tmp = 1.0 - pow(x, pow(n, -1.0));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(x, n)
	t_0 = Float64(fma(Float64(n / x), 0.5, n) * x) ^ -1.0
	tmp = 0.0
	if (n <= -6.6e+257)
		tmp = Float64(Float64(-log(x)) / n);
	elseif (n <= -3.85)
		tmp = t_0;
	elseif (n <= 4.35e-197)
		tmp = Float64(0.3333333333333333 / Float64((x ^ 3.0) * n));
	elseif (n <= 520000000000.0)
		tmp = Float64(1.0 - (x ^ (n ^ -1.0)));
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[x_, n_] := Block[{t$95$0 = N[Power[N[(N[(N[(n / x), $MachinePrecision] * 0.5 + n), $MachinePrecision] * x), $MachinePrecision], -1.0], $MachinePrecision]}, If[LessEqual[n, -6.6e+257], N[((-N[Log[x], $MachinePrecision]) / n), $MachinePrecision], If[LessEqual[n, -3.85], t$95$0, If[LessEqual[n, 4.35e-197], N[(0.3333333333333333 / N[(N[Power[x, 3.0], $MachinePrecision] * n), $MachinePrecision]), $MachinePrecision], If[LessEqual[n, 520000000000.0], N[(1.0 - N[Power[x, N[Power[n, -1.0], $MachinePrecision]], $MachinePrecision]), $MachinePrecision], t$95$0]]]]]

Derivation

1. Split input into 4 regimes

2. if n < -6.6000000000000003e257

   1. Initial program 22.6%

      \[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)} \]
   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 100.0

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

   5. Applied rewrites 100.0%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 86.8%

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

   if -6.6000000000000003e257 < n < -3.85000000000000009 or 5.2e11 < n

   1. Initial program 31.2%

      \[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)} \]
   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 71.5

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

   5. Applied rewrites 71.5%

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

   7. Applied rewrites 71.5%

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

   8. Taylor expanded in x around inf

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

   10. Applied rewrites 58.0%

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

   if -3.85000000000000009 < n < 4.35e-197

   1. Initial program 86.7%

      \[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)} \]
   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 37.5

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

   5. Applied rewrites 37.5%

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

   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 50.8%

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

   9. Taylor expanded in x around 0

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

   11. Applied rewrites 72.9%

       \[\leadsto \frac{0.3333333333333333}{{x}^{3} \cdot \color{blue}{n}} \]

   if 4.35e-197 < n < 5.2e11

   1. Initial program 84.9%

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

   3. Taylor expanded in x around 0

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

   5. Applied rewrites 84.9%

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

4. Final simplification 66.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq -6.6 \cdot 10^{+257}:\\ \;\;\;\;\frac{-\log x}{n}\\ \mathbf{elif}\;n \leq -3.85:\\ \;\;\;\;{\left(\mathsf{fma}\left(\frac{n}{x}, 0.5, n\right) \cdot x\right)}^{-1}\\ \mathbf{elif}\;n \leq 4.35 \cdot 10^{-197}:\\ \;\;\;\;\frac{0.3333333333333333}{{x}^{3} \cdot n}\\ \mathbf{elif}\;n \leq 520000000000:\\ \;\;\;\;1 - {x}^{\left({n}^{-1}\right)}\\ \mathbf{else}:\\ \;\;\;\;{\left(\mathsf{fma}\left(\frac{n}{x}, 0.5, n\right) \cdot x\right)}^{-1}\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 60.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {\left(\mathsf{fma}\left(\frac{n}{x}, 0.5, n\right) \cdot x\right)}^{-1}\\ \mathbf{if}\;n \leq -6.6 \cdot 10^{+257}:\\ \;\;\;\;\frac{-\log x}{n}\\ \mathbf{elif}\;n \leq -3.85:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;n \leq 4.35 \cdot 10^{-197}:\\ \;\;\;\;\frac{\frac{0.3333333333333333}{\left(x \cdot x\right) \cdot n}}{x}\\ \mathbf{elif}\;n \leq 520000000000:\\ \;\;\;\;1 - {x}^{\left({n}^{-1}\right)}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (let* ((t_0 (pow (* (fma (/ n x) 0.5 n) x) -1.0)))
   (if (<= n -6.6e+257)
     (/ (- (log x)) n)
     (if (<= n -3.85)
       t_0
       (if (<= n 4.35e-197)
         (/ (/ 0.3333333333333333 (* (* x x) n)) x)
         (if (<= n 520000000000.0) (- 1.0 (pow x (pow n -1.0))) t_0))))))

C

double code(double x, double n) {
	double t_0 = pow((fma((n / x), 0.5, n) * x), -1.0);
	double tmp;
	if (n <= -6.6e+257) {
		tmp = -log(x) / n;
	} else if (n <= -3.85) {
		tmp = t_0;
	} else if (n <= 4.35e-197) {
		tmp = (0.3333333333333333 / ((x * x) * n)) / x;
	} else if (n <= 520000000000.0) {
		tmp = 1.0 - pow(x, pow(n, -1.0));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(x, n)
	t_0 = Float64(fma(Float64(n / x), 0.5, n) * x) ^ -1.0
	tmp = 0.0
	if (n <= -6.6e+257)
		tmp = Float64(Float64(-log(x)) / n);
	elseif (n <= -3.85)
		tmp = t_0;
	elseif (n <= 4.35e-197)
		tmp = Float64(Float64(0.3333333333333333 / Float64(Float64(x * x) * n)) / x);
	elseif (n <= 520000000000.0)
		tmp = Float64(1.0 - (x ^ (n ^ -1.0)));
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[x_, n_] := Block[{t$95$0 = N[Power[N[(N[(N[(n / x), $MachinePrecision] * 0.5 + n), $MachinePrecision] * x), $MachinePrecision], -1.0], $MachinePrecision]}, If[LessEqual[n, -6.6e+257], N[((-N[Log[x], $MachinePrecision]) / n), $MachinePrecision], If[LessEqual[n, -3.85], t$95$0, If[LessEqual[n, 4.35e-197], N[(N[(0.3333333333333333 / N[(N[(x * x), $MachinePrecision] * n), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision], If[LessEqual[n, 520000000000.0], N[(1.0 - N[Power[x, N[Power[n, -1.0], $MachinePrecision]], $MachinePrecision]), $MachinePrecision], t$95$0]]]]]

Derivation

1. Split input into 4 regimes

2. if n < -6.6000000000000003e257

   1. Initial program 22.6%

      \[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)} \]
   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 100.0

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

   5. Applied rewrites 100.0%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 86.8%

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

   if -6.6000000000000003e257 < n < -3.85000000000000009 or 5.2e11 < n

   1. Initial program 31.2%

      \[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)} \]
   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 71.5

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

   5. Applied rewrites 71.5%

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

   7. Applied rewrites 71.5%

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

   8. Taylor expanded in x around inf

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

   10. Applied rewrites 58.0%

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

   if -3.85000000000000009 < n < 4.35e-197

   1. Initial program 86.7%

      \[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)} \]
   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 37.5

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

   5. Applied rewrites 37.5%

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

   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 50.8%

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

   9. Taylor expanded in x around 0

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

   11. Applied rewrites 66.8%

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

   if 4.35e-197 < n < 5.2e11

   1. Initial program 84.9%

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

   3. Taylor expanded in x around 0

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

   5. Applied rewrites 84.9%

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

4. Final simplification 64.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq -6.6 \cdot 10^{+257}:\\ \;\;\;\;\frac{-\log x}{n}\\ \mathbf{elif}\;n \leq -3.85:\\ \;\;\;\;{\left(\mathsf{fma}\left(\frac{n}{x}, 0.5, n\right) \cdot x\right)}^{-1}\\ \mathbf{elif}\;n \leq 4.35 \cdot 10^{-197}:\\ \;\;\;\;\frac{\frac{0.3333333333333333}{\left(x \cdot x\right) \cdot n}}{x}\\ \mathbf{elif}\;n \leq 520000000000:\\ \;\;\;\;1 - {x}^{\left({n}^{-1}\right)}\\ \mathbf{else}:\\ \;\;\;\;{\left(\mathsf{fma}\left(\frac{n}{x}, 0.5, n\right) \cdot x\right)}^{-1}\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 52.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;{n}^{-1} \leq -10000000:\\ \;\;\;\;\frac{\frac{0.3333333333333333}{\left(x \cdot x\right) \cdot n}}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{{n}^{-1}}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (if (<= (pow n -1.0) -10000000.0)
   (/ (/ 0.3333333333333333 (* (* x x) n)) x)
   (/ (pow n -1.0) x)))

C

double code(double x, double n) {
	double tmp;
	if (pow(n, -1.0) <= -10000000.0) {
		tmp = (0.3333333333333333 / ((x * x) * n)) / x;
	} else {
		tmp = pow(n, -1.0) / x;
	}
	return tmp;
}

Fortran

real(8) function code(x, n)
    real(8), intent (in) :: x
    real(8), intent (in) :: n
    real(8) :: tmp
    if ((n ** (-1.0d0)) <= (-10000000.0d0)) then
        tmp = (0.3333333333333333d0 / ((x * x) * n)) / x
    else
        tmp = (n ** (-1.0d0)) / x
    end if
    code = tmp
end function

Java

public static double code(double x, double n) {
	double tmp;
	if (Math.pow(n, -1.0) <= -10000000.0) {
		tmp = (0.3333333333333333 / ((x * x) * n)) / x;
	} else {
		tmp = Math.pow(n, -1.0) / x;
	}
	return tmp;
}

Python

def code(x, n):
	tmp = 0
	if math.pow(n, -1.0) <= -10000000.0:
		tmp = (0.3333333333333333 / ((x * x) * n)) / x
	else:
		tmp = math.pow(n, -1.0) / x
	return tmp

Julia

function code(x, n)
	tmp = 0.0
	if ((n ^ -1.0) <= -10000000.0)
		tmp = Float64(Float64(0.3333333333333333 / Float64(Float64(x * x) * n)) / x);
	else
		tmp = Float64((n ^ -1.0) / x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, n)
	tmp = 0.0;
	if ((n ^ -1.0) <= -10000000.0)
		tmp = (0.3333333333333333 / ((x * x) * n)) / x;
	else
		tmp = (n ^ -1.0) / x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, n_] := If[LessEqual[N[Power[n, -1.0], $MachinePrecision], -10000000.0], N[(N[(0.3333333333333333 / N[(N[(x * x), $MachinePrecision] * n), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision], N[(N[Power[n, -1.0], $MachinePrecision] / x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 #s(literal 1 binary64) n) < -1e7

   1. Initial program 100.0%

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

   3. Taylor expanded in n around inf

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

      1. lower-/.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lower-log1p.f64 N/A

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

      4. lower-log.f64 42.9

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

   5. Applied rewrites 42.9%

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

   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 44.4%

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

   9. Taylor expanded in x around 0

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

   11. Applied rewrites 63.3%

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

   if -1e7 < (/.f64 #s(literal 1 binary64) n)

   1. Initial program 37.2%

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

   3. Taylor expanded in n around inf

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

      1. 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 58.5

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

   5. Applied rewrites 58.5%

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

   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 50.2%

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

   9. Taylor expanded in x around inf

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

   11. Applied rewrites 49.2%

       \[\leadsto \frac{\frac{1}{n}}{x} \]
3. Recombined 2 regimes into one program.

4. Final simplification 53.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{n}^{-1} \leq -10000000:\\ \;\;\;\;\frac{\frac{0.3333333333333333}{\left(x \cdot x\right) \cdot n}}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{{n}^{-1}}{x}\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 54.7% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;{n}^{-1} \leq -10000000:\\ \;\;\;\;\frac{\frac{0.3333333333333333}{\left(x \cdot x\right) \cdot n}}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{\frac{0.3333333333333333}{x} - 0.5}{x} + 1}{n}}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (if (<= (pow n -1.0) -10000000.0)
   (/ (/ 0.3333333333333333 (* (* x x) n)) x)
   (/ (/ (+ (/ (- (/ 0.3333333333333333 x) 0.5) x) 1.0) n) x)))

C

double code(double x, double n) {
	double tmp;
	if (pow(n, -1.0) <= -10000000.0) {
		tmp = (0.3333333333333333 / ((x * x) * n)) / x;
	} else {
		tmp = (((((0.3333333333333333 / x) - 0.5) / 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 ((n ** (-1.0d0)) <= (-10000000.0d0)) then
        tmp = (0.3333333333333333d0 / ((x * x) * n)) / x
    else
        tmp = (((((0.3333333333333333d0 / x) - 0.5d0) / x) + 1.0d0) / n) / x
    end if
    code = tmp
end function

Java

public static double code(double x, double n) {
	double tmp;
	if (Math.pow(n, -1.0) <= -10000000.0) {
		tmp = (0.3333333333333333 / ((x * x) * n)) / x;
	} else {
		tmp = (((((0.3333333333333333 / x) - 0.5) / x) + 1.0) / n) / x;
	}
	return tmp;
}

Python

def code(x, n):
	tmp = 0
	if math.pow(n, -1.0) <= -10000000.0:
		tmp = (0.3333333333333333 / ((x * x) * n)) / x
	else:
		tmp = (((((0.3333333333333333 / x) - 0.5) / x) + 1.0) / n) / x
	return tmp

Julia

function code(x, n)
	tmp = 0.0
	if ((n ^ -1.0) <= -10000000.0)
		tmp = Float64(Float64(0.3333333333333333 / Float64(Float64(x * x) * n)) / x);
	else
		tmp = Float64(Float64(Float64(Float64(Float64(Float64(0.3333333333333333 / x) - 0.5) / x) + 1.0) / n) / x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, n)
	tmp = 0.0;
	if ((n ^ -1.0) <= -10000000.0)
		tmp = (0.3333333333333333 / ((x * x) * n)) / x;
	else
		tmp = (((((0.3333333333333333 / x) - 0.5) / x) + 1.0) / n) / x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, n_] := If[LessEqual[N[Power[n, -1.0], $MachinePrecision], -10000000.0], N[(N[(0.3333333333333333 / N[(N[(x * x), $MachinePrecision] * n), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision], N[(N[(N[(N[(N[(N[(0.3333333333333333 / x), $MachinePrecision] - 0.5), $MachinePrecision] / x), $MachinePrecision] + 1.0), $MachinePrecision] / n), $MachinePrecision] / x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 #s(literal 1 binary64) n) < -1e7

   1. Initial program 100.0%

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

   3. Taylor expanded in n around inf

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

      1. lower-/.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lower-log1p.f64 N/A

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

      4. lower-log.f64 42.9

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

   5. Applied rewrites 42.9%

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

   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 44.4%

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

   9. Taylor expanded in x around 0

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

   11. Applied rewrites 63.3%

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

   if -1e7 < (/.f64 #s(literal 1 binary64) n)

   1. Initial program 37.2%

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

   3. Taylor expanded in n around inf

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

      1. 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 58.5

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

   5. Applied rewrites 58.5%

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

   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 50.2%

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

   9. Taylor expanded in n around 0

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

   11. Applied rewrites 50.2%

       \[\leadsto \frac{\frac{\frac{\frac{0.3333333333333333}{x} - 0.5}{x} + 1}{n}}{x} \]
3. Recombined 2 regimes into one program.

4. Final simplification 54.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{n}^{-1} \leq -10000000:\\ \;\;\;\;\frac{\frac{0.3333333333333333}{\left(x \cdot x\right) \cdot n}}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{\frac{0.3333333333333333}{x} - 0.5}{x} + 1}{n}}{x}\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 41.0% 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 56.3%

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

3. Taylor expanded in n around inf

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

   1. lower-/.f64 N/A

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

   2. lower--.f64 N/A

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

   3. lower-log1p.f64 N/A

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

   4. lower-log.f64 53.7

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

5. Applied rewrites 53.7%

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

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 48.4%

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

9. Taylor expanded in x around inf

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

11. Applied rewrites 45.1%

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

12. Final simplification 45.1%

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

Alternative 17: 40.3% accurate, 13.6× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 56.3%

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

3. Taylor expanded in x around 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. associate-/l/ N/A

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

   2. lower-/.f64 N/A

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

   3. lower-/.f64 N/A

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

   4. log-rec N/A

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

   5. mul-1-neg N/A

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

   6. associate-*r/ N/A

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

   7. associate-*r* N/A

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

   8. metadata-eval N/A

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

   9. *-commutative N/A

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

   10. associate-/l* N/A

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

   11. exp-to-pow N/A

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

   12. lower-pow.f64 N/A

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

   13. lower-/.f64 61.1

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

5. Applied rewrites 61.1%

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

7. Applied rewrites 60.5%

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

8. Taylor expanded in n around inf

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

10. Applied rewrites 44.7%

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

Reproduce

herbie shell --seed 2024327 
(FPCore (x n)
  :name "2nthrt (problem 3.4.6)"
  :precision binary64
  (- (pow (+ x 1.0) (/ 1.0 n)) (pow x (/ 1.0 n))))