2nthrt (problem 3.4.6)

Percentage Accurate: 54.1% → 92.3%

Time: 23.4s

Alternatives: 13

Speedup: 1.3×

• Could not converge on a ground truth

  1. x = -2.2542189061644235e+114
  2. n = -7.744982211022988e-119

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 54.1% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 92.3% accurate, 0.9× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if x < 1

   1. Initial program 47.0%

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

   3. Taylor expanded in x around 0

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

   5. Applied rewrites 74.1%

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

   6. Taylor expanded in n around inf

      \[\leadsto \mathsf{fma}\left(x \cdot x, \frac{\frac{-1}{2}}{\color{blue}{n}}, \frac{x}{n} - \mathsf{expm1}\left(\frac{\log x}{n}\right)\right) \]
   7. Step-by-step derivation

   8. Applied rewrites 86.9%

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

   if 1 < x

   1. Initial program 67.0%

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

   3. Taylor expanded in x around inf

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

      1. 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 98.7

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

   5. Applied rewrites 98.7%

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

Alternative 2: 77.2% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {x}^{\left(\frac{1}{n}\right)}\\ t_1 := {\left(1 + x\right)}^{\left(\frac{1}{n}\right)} - t\_0\\ \mathbf{if}\;t\_1 \leq -5000:\\ \;\;\;\;\left(\frac{x}{n} + 1\right) - t\_0\\ \mathbf{elif}\;t\_1 \leq 0.5309801530229148:\\ \;\;\;\;\frac{\log \left(\frac{1 + x}{x}\right)}{n}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\mathsf{fma}\left(-0.5, x, 1\right), n, 0.5 \cdot x\right)}{n \cdot n} \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (let* ((t_0 (pow x (/ 1.0 n))) (t_1 (- (pow (+ 1.0 x) (/ 1.0 n)) t_0)))
   (if (<= t_1 -5000.0)
     (- (+ (/ x n) 1.0) t_0)
     (if (<= t_1 0.5309801530229148)
       (/ (log (/ (+ 1.0 x) x)) n)
       (* (/ (fma (fma -0.5 x 1.0) n (* 0.5 x)) (* n n)) x)))))

C

double code(double x, double n) {
	double t_0 = pow(x, (1.0 / n));
	double t_1 = pow((1.0 + x), (1.0 / n)) - t_0;
	double tmp;
	if (t_1 <= -5000.0) {
		tmp = ((x / n) + 1.0) - t_0;
	} else if (t_1 <= 0.5309801530229148) {
		tmp = log(((1.0 + x) / x)) / n;
	} else {
		tmp = (fma(fma(-0.5, x, 1.0), n, (0.5 * x)) / (n * n)) * x;
	}
	return tmp;
}

Julia

function code(x, n)
	t_0 = x ^ Float64(1.0 / n)
	t_1 = Float64((Float64(1.0 + x) ^ Float64(1.0 / n)) - t_0)
	tmp = 0.0
	if (t_1 <= -5000.0)
		tmp = Float64(Float64(Float64(x / n) + 1.0) - t_0);
	elseif (t_1 <= 0.5309801530229148)
		tmp = Float64(log(Float64(Float64(1.0 + x) / x)) / n);
	else
		tmp = Float64(Float64(fma(fma(-0.5, x, 1.0), n, Float64(0.5 * x)) / Float64(n * n)) * x);
	end
	return tmp
end

Wolfram

code[x_, n_] := Block[{t$95$0 = N[Power[x, N[(1.0 / n), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(N[Power[N[(1.0 + x), $MachinePrecision], N[(1.0 / n), $MachinePrecision]], $MachinePrecision] - t$95$0), $MachinePrecision]}, If[LessEqual[t$95$1, -5000.0], N[(N[(N[(x / n), $MachinePrecision] + 1.0), $MachinePrecision] - t$95$0), $MachinePrecision], If[LessEqual[t$95$1, 0.5309801530229148], N[(N[Log[N[(N[(1.0 + x), $MachinePrecision] / x), $MachinePrecision]], $MachinePrecision] / n), $MachinePrecision], N[(N[(N[(N[(-0.5 * x + 1.0), $MachinePrecision] * n + N[(0.5 * x), $MachinePrecision]), $MachinePrecision] / N[(n * n), $MachinePrecision]), $MachinePrecision] * x), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if (-.f64 (pow.f64 (+.f64 x #s(literal 1 binary64)) (/.f64 #s(literal 1 binary64) n)) (pow.f64 x (/.f64 #s(literal 1 binary64) n))) < -5e3

   1. Initial program 99.9%

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

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\left(1 + \frac{x}{n}\right)} - {x}^{\left(\frac{1}{n}\right)} \]
   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 99.4

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

   5. Applied rewrites 99.4%

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

   if -5e3 < (-.f64 (pow.f64 (+.f64 x #s(literal 1 binary64)) (/.f64 #s(literal 1 binary64) n)) (pow.f64 x (/.f64 #s(literal 1 binary64) n))) < 0.53098015302291479

   1. Initial program 42.5%

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

   3. Taylor expanded in n around inf

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

      1. lower-/.f64 N/A

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

   5. Applied rewrites 80.2%

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

   7. Applied rewrites 80.3%

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

   8. Taylor expanded in n around inf

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

   10. Applied rewrites 80.2%

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

   if 0.53098015302291479 < (-.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 53.4%

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

   3. Taylor expanded in x around 0

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

   5. Applied rewrites 68.6%

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

   6. Taylor expanded in n around 0

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

   8. Applied rewrites 29.2%

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

   10. Applied rewrites 51.1%

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

4. Final simplification 80.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{\left(1 + x\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)} \leq -5000:\\ \;\;\;\;\left(\frac{x}{n} + 1\right) - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{elif}\;{\left(1 + x\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)} \leq 0.5309801530229148:\\ \;\;\;\;\frac{\log \left(\frac{1 + x}{x}\right)}{n}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\mathsf{fma}\left(-0.5, x, 1\right), n, 0.5 \cdot x\right)}{n \cdot n} \cdot x\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 77.2% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {x}^{\left(\frac{1}{n}\right)}\\ t_1 := {\left(1 + x\right)}^{\left(\frac{1}{n}\right)} - t\_0\\ \mathbf{if}\;t\_1 \leq -5000:\\ \;\;\;\;1 - t\_0\\ \mathbf{elif}\;t\_1 \leq 0.5309801530229148:\\ \;\;\;\;\frac{\log \left(\frac{1 + x}{x}\right)}{n}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\mathsf{fma}\left(-0.5, x, 1\right), n, 0.5 \cdot x\right)}{n \cdot n} \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (let* ((t_0 (pow x (/ 1.0 n))) (t_1 (- (pow (+ 1.0 x) (/ 1.0 n)) t_0)))
   (if (<= t_1 -5000.0)
     (- 1.0 t_0)
     (if (<= t_1 0.5309801530229148)
       (/ (log (/ (+ 1.0 x) x)) n)
       (* (/ (fma (fma -0.5 x 1.0) n (* 0.5 x)) (* n n)) x)))))

C

double code(double x, double n) {
	double t_0 = pow(x, (1.0 / n));
	double t_1 = pow((1.0 + x), (1.0 / n)) - t_0;
	double tmp;
	if (t_1 <= -5000.0) {
		tmp = 1.0 - t_0;
	} else if (t_1 <= 0.5309801530229148) {
		tmp = log(((1.0 + x) / x)) / n;
	} else {
		tmp = (fma(fma(-0.5, x, 1.0), n, (0.5 * x)) / (n * n)) * x;
	}
	return tmp;
}

Julia

function code(x, n)
	t_0 = x ^ Float64(1.0 / n)
	t_1 = Float64((Float64(1.0 + x) ^ Float64(1.0 / n)) - t_0)
	tmp = 0.0
	if (t_1 <= -5000.0)
		tmp = Float64(1.0 - t_0);
	elseif (t_1 <= 0.5309801530229148)
		tmp = Float64(log(Float64(Float64(1.0 + x) / x)) / n);
	else
		tmp = Float64(Float64(fma(fma(-0.5, x, 1.0), n, Float64(0.5 * x)) / Float64(n * n)) * x);
	end
	return tmp
end

Wolfram

code[x_, n_] := Block[{t$95$0 = N[Power[x, N[(1.0 / n), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(N[Power[N[(1.0 + x), $MachinePrecision], N[(1.0 / n), $MachinePrecision]], $MachinePrecision] - t$95$0), $MachinePrecision]}, If[LessEqual[t$95$1, -5000.0], N[(1.0 - t$95$0), $MachinePrecision], If[LessEqual[t$95$1, 0.5309801530229148], N[(N[Log[N[(N[(1.0 + x), $MachinePrecision] / x), $MachinePrecision]], $MachinePrecision] / n), $MachinePrecision], N[(N[(N[(N[(-0.5 * x + 1.0), $MachinePrecision] * n + N[(0.5 * x), $MachinePrecision]), $MachinePrecision] / N[(n * n), $MachinePrecision]), $MachinePrecision] * x), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if (-.f64 (pow.f64 (+.f64 x #s(literal 1 binary64)) (/.f64 #s(literal 1 binary64) n)) (pow.f64 x (/.f64 #s(literal 1 binary64) n))) < -5e3

   1. Initial program 99.9%

      \[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)} \]
   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 99.1%

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

   if -5e3 < (-.f64 (pow.f64 (+.f64 x #s(literal 1 binary64)) (/.f64 #s(literal 1 binary64) n)) (pow.f64 x (/.f64 #s(literal 1 binary64) n))) < 0.53098015302291479

   1. Initial program 42.5%

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

   3. Taylor expanded in n around inf

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

      1. lower-/.f64 N/A

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

   5. Applied rewrites 80.2%

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

   7. Applied rewrites 80.3%

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

   8. Taylor expanded in n around inf

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

   10. Applied rewrites 80.2%

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

   if 0.53098015302291479 < (-.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 53.4%

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

   3. Taylor expanded in x around 0

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

   5. Applied rewrites 68.6%

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

   6. Taylor expanded in n around 0

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

   8. Applied rewrites 29.2%

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

   10. Applied rewrites 51.1%

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

4. Final simplification 80.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{\left(1 + x\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)} \leq -5000:\\ \;\;\;\;1 - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{elif}\;{\left(1 + x\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)} \leq 0.5309801530229148:\\ \;\;\;\;\frac{\log \left(\frac{1 + x}{x}\right)}{n}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\mathsf{fma}\left(-0.5, x, 1\right), n, 0.5 \cdot x\right)}{n \cdot n} \cdot x\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 42.9% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;{\left(1 + x\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)} \leq 0.5309801530229148:\\ \;\;\;\;\frac{\frac{1}{x}}{n}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\mathsf{fma}\left(-0.5, x, 1\right), n, 0.5 \cdot x\right)}{n \cdot n} \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (if (<= (- (pow (+ 1.0 x) (/ 1.0 n)) (pow x (/ 1.0 n))) 0.5309801530229148)
   (/ (/ 1.0 x) n)
   (* (/ (fma (fma -0.5 x 1.0) n (* 0.5 x)) (* n n)) x)))

C

double code(double x, double n) {
	double tmp;
	if ((pow((1.0 + x), (1.0 / n)) - pow(x, (1.0 / n))) <= 0.5309801530229148) {
		tmp = (1.0 / x) / n;
	} else {
		tmp = (fma(fma(-0.5, x, 1.0), n, (0.5 * x)) / (n * n)) * x;
	}
	return tmp;
}

Julia

function code(x, n)
	tmp = 0.0
	if (Float64((Float64(1.0 + x) ^ Float64(1.0 / n)) - (x ^ Float64(1.0 / n))) <= 0.5309801530229148)
		tmp = Float64(Float64(1.0 / x) / n);
	else
		tmp = Float64(Float64(fma(fma(-0.5, x, 1.0), n, Float64(0.5 * x)) / Float64(n * n)) * x);
	end
	return tmp
end

Wolfram

code[x_, n_] := If[LessEqual[N[(N[Power[N[(1.0 + x), $MachinePrecision], N[(1.0 / n), $MachinePrecision]], $MachinePrecision] - N[Power[x, N[(1.0 / n), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 0.5309801530229148], N[(N[(1.0 / x), $MachinePrecision] / n), $MachinePrecision], N[(N[(N[(N[(-0.5 * x + 1.0), $MachinePrecision] * n + N[(0.5 * x), $MachinePrecision]), $MachinePrecision] / N[(n * n), $MachinePrecision]), $MachinePrecision] * x), $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))) < 0.53098015302291479

   1. Initial program 55.3%

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

   3. Taylor expanded in n around inf

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

      1. lower-/.f64 N/A

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

   5. Applied rewrites 75.8%

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

   6. Taylor expanded in x around inf

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

      1. lower-/.f64 N/A

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

      2. mul-1-neg N/A

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

      3. distribute-frac-neg N/A

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

      4. log-rec N/A

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

      5. remove-double-neg N/A

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

      6. lower-exp.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower-log.f64 N/A

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

      9. lower-*.f64 68.4

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

   8. Applied rewrites 68.4%

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

   9. Taylor expanded in n around inf

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

   11. Applied rewrites 47.0%

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

   if 0.53098015302291479 < (-.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 53.4%

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

   3. Taylor expanded in x around 0

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

   5. Applied rewrites 68.6%

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

   6. Taylor expanded in n around 0

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

   8. Applied rewrites 29.2%

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

   10. Applied rewrites 51.1%

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

4. Final simplification 47.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{\left(1 + x\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)} \leq 0.5309801530229148:\\ \;\;\;\;\frac{\frac{1}{x}}{n}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\mathsf{fma}\left(-0.5, x, 1\right), n, 0.5 \cdot x\right)}{n \cdot n} \cdot x\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 92.2% accurate, 1.0× speedup

Math

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

FPCore

(FPCore (x n)
 :precision binary64
 (if (<= x 1.0)
   (- (/ x n) (expm1 (/ (log x) n)))
   (/ (/ (pow x (/ 1.0 n)) 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, (1.0 / n)) / 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, (1.0 / n)) / 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, (1.0 / n)) / 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 ^ Float64(1.0 / n)) / 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[(1.0 / n), $MachinePrecision]], $MachinePrecision] / x), $MachinePrecision] / n), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 1

   1. Initial program 47.0%

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

   3. Taylor expanded in x around 0

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

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

   5. Applied rewrites 86.8%

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

   if 1 < x

   1. Initial program 67.0%

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

   3. Taylor expanded in x around inf

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

      1. 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 98.7

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

   5. Applied rewrites 98.7%

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

Alternative 6: 80.8% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{if}\;\frac{1}{n} \leq -5 \cdot 10^{-157}:\\ \;\;\;\;\frac{\frac{t\_0}{x}}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 2 \cdot 10^{-30}:\\ \;\;\;\;\frac{\log \left(\frac{1 + x}{x}\right)}{n}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\frac{0.5}{n \cdot n} - \frac{0.5}{n}, x, \frac{1}{n}\right), x, 1\right) - t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (let* ((t_0 (pow x (/ 1.0 n))))
   (if (<= (/ 1.0 n) -5e-157)
     (/ (/ t_0 x) n)
     (if (<= (/ 1.0 n) 2e-30)
       (/ (log (/ (+ 1.0 x) x)) n)
       (- (fma (fma (- (/ 0.5 (* n n)) (/ 0.5 n)) x (/ 1.0 n)) x 1.0) t_0)))))

C

double code(double x, double n) {
	double t_0 = pow(x, (1.0 / n));
	double tmp;
	if ((1.0 / n) <= -5e-157) {
		tmp = (t_0 / x) / n;
	} else if ((1.0 / n) <= 2e-30) {
		tmp = log(((1.0 + x) / x)) / n;
	} else {
		tmp = fma(fma(((0.5 / (n * n)) - (0.5 / n)), x, (1.0 / n)), x, 1.0) - t_0;
	}
	return tmp;
}

Julia

function code(x, n)
	t_0 = x ^ Float64(1.0 / n)
	tmp = 0.0
	if (Float64(1.0 / n) <= -5e-157)
		tmp = Float64(Float64(t_0 / x) / n);
	elseif (Float64(1.0 / n) <= 2e-30)
		tmp = Float64(log(Float64(Float64(1.0 + x) / x)) / n);
	else
		tmp = Float64(fma(fma(Float64(Float64(0.5 / Float64(n * n)) - Float64(0.5 / n)), x, Float64(1.0 / n)), x, 1.0) - t_0);
	end
	return tmp
end

Wolfram

code[x_, n_] := Block[{t$95$0 = N[Power[x, N[(1.0 / n), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[N[(1.0 / n), $MachinePrecision], -5e-157], N[(N[(t$95$0 / x), $MachinePrecision] / n), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 2e-30], N[(N[Log[N[(N[(1.0 + x), $MachinePrecision] / x), $MachinePrecision]], $MachinePrecision] / n), $MachinePrecision], N[(N[(N[(N[(N[(0.5 / N[(n * n), $MachinePrecision]), $MachinePrecision] - N[(0.5 / n), $MachinePrecision]), $MachinePrecision] * x + N[(1.0 / n), $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) < -5.0000000000000002e-157

   1. Initial program 73.3%

      \[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)} \]
   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 88.0

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

   5. Applied rewrites 88.0%

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

   if -5.0000000000000002e-157 < (/.f64 #s(literal 1 binary64) n) < 2e-30

   1. Initial program 34.4%

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

   3. Taylor expanded in n around inf

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

      1. lower-/.f64 N/A

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

   5. Applied rewrites 86.7%

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

   7. Applied rewrites 86.8%

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

   8. Taylor expanded in n around inf

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

   10. Applied rewrites 86.8%

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

   if 2e-30 < (/.f64 #s(literal 1 binary64) n)

   1. Initial program 54.8%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. lower--.f64 N/A

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

      7. associate-*r/ N/A

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

      8. metadata-eval N/A

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

      9. lower-/.f64 N/A

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

      10. unpow2 N/A

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

      11. lower-*.f64 N/A

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

      12. associate-*r/ N/A

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

      13. metadata-eval N/A

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

      14. lower-/.f64 N/A

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

      15. lower-/.f64 82.1

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

   5. Applied rewrites 82.1%

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

Alternative 7: 53.3% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 1 - {x}^{\left(\frac{1}{n}\right)}\\ t_1 := \frac{\frac{1}{x}}{n}\\ \mathbf{if}\;\frac{1}{n} \leq -5 \cdot 10^{-17}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;\frac{1}{n} \leq -1 \cdot 10^{-302}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;\frac{1}{n} \leq 2 \cdot 10^{-79}:\\ \;\;\;\;\frac{\mathsf{fma}\left(-0.5 \cdot x, x, x - \log x\right)}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 2 \cdot 10^{-30}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;\frac{1}{n} \leq 4 \cdot 10^{+144}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\mathsf{fma}\left(-0.5, x, 1\right), n, 0.5 \cdot x\right)}{n \cdot n} \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (let* ((t_0 (- 1.0 (pow x (/ 1.0 n)))) (t_1 (/ (/ 1.0 x) n)))
   (if (<= (/ 1.0 n) -5e-17)
     t_0
     (if (<= (/ 1.0 n) -1e-302)
       t_1
       (if (<= (/ 1.0 n) 2e-79)
         (/ (fma (* -0.5 x) x (- x (log x))) n)
         (if (<= (/ 1.0 n) 2e-30)
           t_1
           (if (<= (/ 1.0 n) 4e+144)
             t_0
             (* (/ (fma (fma -0.5 x 1.0) n (* 0.5 x)) (* n n)) x))))))))

C

double code(double x, double n) {
	double t_0 = 1.0 - pow(x, (1.0 / n));
	double t_1 = (1.0 / x) / n;
	double tmp;
	if ((1.0 / n) <= -5e-17) {
		tmp = t_0;
	} else if ((1.0 / n) <= -1e-302) {
		tmp = t_1;
	} else if ((1.0 / n) <= 2e-79) {
		tmp = fma((-0.5 * x), x, (x - log(x))) / n;
	} else if ((1.0 / n) <= 2e-30) {
		tmp = t_1;
	} else if ((1.0 / n) <= 4e+144) {
		tmp = t_0;
	} else {
		tmp = (fma(fma(-0.5, x, 1.0), n, (0.5 * x)) / (n * n)) * x;
	}
	return tmp;
}

Julia

function code(x, n)
	t_0 = Float64(1.0 - (x ^ Float64(1.0 / n)))
	t_1 = Float64(Float64(1.0 / x) / n)
	tmp = 0.0
	if (Float64(1.0 / n) <= -5e-17)
		tmp = t_0;
	elseif (Float64(1.0 / n) <= -1e-302)
		tmp = t_1;
	elseif (Float64(1.0 / n) <= 2e-79)
		tmp = Float64(fma(Float64(-0.5 * x), x, Float64(x - log(x))) / n);
	elseif (Float64(1.0 / n) <= 2e-30)
		tmp = t_1;
	elseif (Float64(1.0 / n) <= 4e+144)
		tmp = t_0;
	else
		tmp = Float64(Float64(fma(fma(-0.5, x, 1.0), n, Float64(0.5 * x)) / Float64(n * n)) * x);
	end
	return tmp
end

Wolfram

code[x_, n_] := Block[{t$95$0 = N[(1.0 - N[Power[x, N[(1.0 / n), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(1.0 / x), $MachinePrecision] / n), $MachinePrecision]}, If[LessEqual[N[(1.0 / n), $MachinePrecision], -5e-17], t$95$0, If[LessEqual[N[(1.0 / n), $MachinePrecision], -1e-302], t$95$1, If[LessEqual[N[(1.0 / n), $MachinePrecision], 2e-79], N[(N[(N[(-0.5 * x), $MachinePrecision] * x + N[(x - N[Log[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / n), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 2e-30], t$95$1, If[LessEqual[N[(1.0 / n), $MachinePrecision], 4e+144], t$95$0, N[(N[(N[(N[(-0.5 * x + 1.0), $MachinePrecision] * n + N[(0.5 * x), $MachinePrecision]), $MachinePrecision] / N[(n * n), $MachinePrecision]), $MachinePrecision] * x), $MachinePrecision]]]]]]]]

Derivation

1. Split input into 4 regimes

2. if (/.f64 #s(literal 1 binary64) n) < -4.9999999999999999e-17 or 2e-30 < (/.f64 #s(literal 1 binary64) n) < 4.00000000000000009e144

   1. Initial program 94.7%

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

   3. Taylor expanded in x around 0

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

   5. Applied rewrites 63.7%

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

   if -4.9999999999999999e-17 < (/.f64 #s(literal 1 binary64) n) < -9.9999999999999996e-303 or 2e-79 < (/.f64 #s(literal 1 binary64) n) < 2e-30

   1. Initial program 36.2%

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

   3. Taylor expanded in n around inf

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

      1. lower-/.f64 N/A

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

   5. Applied rewrites 70.3%

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

   6. Taylor expanded in x around inf

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

      1. lower-/.f64 N/A

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

      2. mul-1-neg N/A

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

      3. distribute-frac-neg N/A

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

      4. log-rec N/A

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

      5. remove-double-neg N/A

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

      6. lower-exp.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower-log.f64 N/A

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

      9. lower-*.f64 65.4

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

   8. Applied rewrites 65.4%

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

   9. Taylor expanded in n around inf

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

   11. Applied rewrites 65.9%

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

   if -9.9999999999999996e-303 < (/.f64 #s(literal 1 binary64) n) < 2e-79

   1. Initial program 20.2%

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

   3. Taylor expanded in x around 0

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

   5. Applied rewrites 70.7%

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

   6. Taylor expanded in n around inf

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

   8. Applied rewrites 70.7%

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

   if 4.00000000000000009e144 < (/.f64 #s(literal 1 binary64) n)

   1. Initial program 33.4%

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

   3. Taylor expanded in x around 0

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

   5. Applied rewrites 68.8%

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

   6. Taylor expanded in n around 0

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

   8. Applied rewrites 51.8%

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

   10. Applied rewrites 93.9%

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

4. Final simplification 67.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{1}{n} \leq -5 \cdot 10^{-17}:\\ \;\;\;\;1 - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{elif}\;\frac{1}{n} \leq -1 \cdot 10^{-302}:\\ \;\;\;\;\frac{\frac{1}{x}}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 2 \cdot 10^{-79}:\\ \;\;\;\;\frac{\mathsf{fma}\left(-0.5 \cdot x, x, x - \log x\right)}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 2 \cdot 10^{-30}:\\ \;\;\;\;\frac{\frac{1}{x}}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 4 \cdot 10^{+144}:\\ \;\;\;\;1 - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\mathsf{fma}\left(-0.5, x, 1\right), n, 0.5 \cdot x\right)}{n \cdot n} \cdot x\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 80.7% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{if}\;\frac{1}{n} \leq -5 \cdot 10^{-157}:\\ \;\;\;\;\frac{\frac{t\_0}{x}}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 2 \cdot 10^{-30}:\\ \;\;\;\;\frac{\log \left(\frac{1 + x}{x}\right)}{n}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{\mathsf{fma}\left(\frac{0.5}{n} - 0.5, x, 1\right)}{n}, x, 1\right) - t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (let* ((t_0 (pow x (/ 1.0 n))))
   (if (<= (/ 1.0 n) -5e-157)
     (/ (/ t_0 x) n)
     (if (<= (/ 1.0 n) 2e-30)
       (/ (log (/ (+ 1.0 x) x)) n)
       (- (fma (/ (fma (- (/ 0.5 n) 0.5) x 1.0) n) x 1.0) t_0)))))

C

double code(double x, double n) {
	double t_0 = pow(x, (1.0 / n));
	double tmp;
	if ((1.0 / n) <= -5e-157) {
		tmp = (t_0 / x) / n;
	} else if ((1.0 / n) <= 2e-30) {
		tmp = log(((1.0 + x) / x)) / n;
	} else {
		tmp = fma((fma(((0.5 / n) - 0.5), x, 1.0) / n), x, 1.0) - t_0;
	}
	return tmp;
}

Julia

function code(x, n)
	t_0 = x ^ Float64(1.0 / n)
	tmp = 0.0
	if (Float64(1.0 / n) <= -5e-157)
		tmp = Float64(Float64(t_0 / x) / n);
	elseif (Float64(1.0 / n) <= 2e-30)
		tmp = Float64(log(Float64(Float64(1.0 + x) / x)) / n);
	else
		tmp = Float64(fma(Float64(fma(Float64(Float64(0.5 / n) - 0.5), x, 1.0) / n), x, 1.0) - t_0);
	end
	return tmp
end

Wolfram

code[x_, n_] := Block[{t$95$0 = N[Power[x, N[(1.0 / n), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[N[(1.0 / n), $MachinePrecision], -5e-157], N[(N[(t$95$0 / x), $MachinePrecision] / n), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 2e-30], 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] * x + 1.0), $MachinePrecision] / n), $MachinePrecision] * x + 1.0), $MachinePrecision] - t$95$0), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 #s(literal 1 binary64) n) < -5.0000000000000002e-157

   1. Initial program 73.3%

      \[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)} \]
   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 88.0

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

   5. Applied rewrites 88.0%

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

   if -5.0000000000000002e-157 < (/.f64 #s(literal 1 binary64) n) < 2e-30

   1. Initial program 34.4%

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

   3. Taylor expanded in n around inf

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

      1. lower-/.f64 N/A

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

   5. Applied rewrites 86.7%

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

   7. Applied rewrites 86.8%

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

   8. Taylor expanded in n around inf

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

   10. Applied rewrites 86.8%

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

   if 2e-30 < (/.f64 #s(literal 1 binary64) n)

   1. Initial program 54.8%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. lower--.f64 N/A

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

      7. associate-*r/ N/A

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

      8. metadata-eval N/A

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

      9. lower-/.f64 N/A

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

      10. unpow2 N/A

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

      11. lower-*.f64 N/A

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

      12. associate-*r/ N/A

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

      13. metadata-eval N/A

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

      14. lower-/.f64 N/A

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

      15. lower-/.f64 82.1

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

   5. Applied rewrites 82.1%

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

   6. Taylor expanded in n around inf

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

   8. Applied rewrites 73.2%

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

Alternative 9: 80.4% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{if}\;\frac{1}{n} \leq -5 \cdot 10^{-157}:\\ \;\;\;\;\frac{\frac{t\_0}{x}}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 2 \cdot 10^{-30}:\\ \;\;\;\;\frac{\log \left(\frac{1 + x}{x}\right)}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 4 \cdot 10^{+144}:\\ \;\;\;\;\left(\frac{x}{n} + 1\right) - t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\mathsf{fma}\left(-0.5, x, 1\right), n, 0.5 \cdot x\right)}{n \cdot n} \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (let* ((t_0 (pow x (/ 1.0 n))))
   (if (<= (/ 1.0 n) -5e-157)
     (/ (/ t_0 x) n)
     (if (<= (/ 1.0 n) 2e-30)
       (/ (log (/ (+ 1.0 x) x)) n)
       (if (<= (/ 1.0 n) 4e+144)
         (- (+ (/ x n) 1.0) t_0)
         (* (/ (fma (fma -0.5 x 1.0) n (* 0.5 x)) (* n n)) x))))))

C

double code(double x, double n) {
	double t_0 = pow(x, (1.0 / n));
	double tmp;
	if ((1.0 / n) <= -5e-157) {
		tmp = (t_0 / x) / n;
	} else if ((1.0 / n) <= 2e-30) {
		tmp = log(((1.0 + x) / x)) / n;
	} else if ((1.0 / n) <= 4e+144) {
		tmp = ((x / n) + 1.0) - t_0;
	} else {
		tmp = (fma(fma(-0.5, x, 1.0), n, (0.5 * x)) / (n * n)) * x;
	}
	return tmp;
}

Julia

function code(x, n)
	t_0 = x ^ Float64(1.0 / n)
	tmp = 0.0
	if (Float64(1.0 / n) <= -5e-157)
		tmp = Float64(Float64(t_0 / x) / n);
	elseif (Float64(1.0 / n) <= 2e-30)
		tmp = Float64(log(Float64(Float64(1.0 + x) / x)) / n);
	elseif (Float64(1.0 / n) <= 4e+144)
		tmp = Float64(Float64(Float64(x / n) + 1.0) - t_0);
	else
		tmp = Float64(Float64(fma(fma(-0.5, x, 1.0), n, Float64(0.5 * x)) / Float64(n * n)) * x);
	end
	return tmp
end

Wolfram

code[x_, n_] := Block[{t$95$0 = N[Power[x, N[(1.0 / n), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[N[(1.0 / n), $MachinePrecision], -5e-157], N[(N[(t$95$0 / x), $MachinePrecision] / n), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 2e-30], N[(N[Log[N[(N[(1.0 + x), $MachinePrecision] / x), $MachinePrecision]], $MachinePrecision] / n), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 4e+144], N[(N[(N[(x / n), $MachinePrecision] + 1.0), $MachinePrecision] - t$95$0), $MachinePrecision], N[(N[(N[(N[(-0.5 * x + 1.0), $MachinePrecision] * n + N[(0.5 * x), $MachinePrecision]), $MachinePrecision] / N[(n * n), $MachinePrecision]), $MachinePrecision] * x), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if (/.f64 #s(literal 1 binary64) n) < -5.0000000000000002e-157

   1. Initial program 73.3%

      \[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)} \]
   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 88.0

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

   5. Applied rewrites 88.0%

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

   if -5.0000000000000002e-157 < (/.f64 #s(literal 1 binary64) n) < 2e-30

   1. Initial program 34.4%

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

   3. Taylor expanded in n around inf

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

      1. lower-/.f64 N/A

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

   5. Applied rewrites 86.7%

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

   7. Applied rewrites 86.8%

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

   8. Taylor expanded in n around inf

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

   10. Applied rewrites 86.8%

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

   if 2e-30 < (/.f64 #s(literal 1 binary64) n) < 4.00000000000000009e144

   1. Initial program 76.2%

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

   3. Taylor expanded in x around 0

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

      1. +-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 70.2

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

   5. Applied rewrites 70.2%

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

   if 4.00000000000000009e144 < (/.f64 #s(literal 1 binary64) n)

   1. Initial program 33.4%

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

   3. Taylor expanded in x around 0

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

   5. Applied rewrites 68.8%

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

   6. Taylor expanded in n around 0

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

   8. Applied rewrites 51.8%

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

   10. Applied rewrites 93.9%

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

4. Final simplification 86.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{1}{n} \leq -5 \cdot 10^{-157}:\\ \;\;\;\;\frac{\frac{{x}^{\left(\frac{1}{n}\right)}}{x}}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 2 \cdot 10^{-30}:\\ \;\;\;\;\frac{\log \left(\frac{1 + x}{x}\right)}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 4 \cdot 10^{+144}:\\ \;\;\;\;\left(\frac{x}{n} + 1\right) - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\mathsf{fma}\left(-0.5, x, 1\right), n, 0.5 \cdot x\right)}{n \cdot n} \cdot x\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 80.3% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{1}{n} \leq -2 \cdot 10^{-77}:\\ \;\;\;\;\frac{{\left(x \cdot x\right)}^{\left(\frac{0.5}{n}\right)}}{n \cdot x}\\ \mathbf{elif}\;\frac{1}{n} \leq 2 \cdot 10^{-30}:\\ \;\;\;\;\frac{\log \left(\frac{1 + x}{x}\right)}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 4 \cdot 10^{+144}:\\ \;\;\;\;\left(\frac{x}{n} + 1\right) - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\mathsf{fma}\left(-0.5, x, 1\right), n, 0.5 \cdot x\right)}{n \cdot n} \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (if (<= (/ 1.0 n) -2e-77)
   (/ (pow (* x x) (/ 0.5 n)) (* n x))
   (if (<= (/ 1.0 n) 2e-30)
     (/ (log (/ (+ 1.0 x) x)) n)
     (if (<= (/ 1.0 n) 4e+144)
       (- (+ (/ x n) 1.0) (pow x (/ 1.0 n)))
       (* (/ (fma (fma -0.5 x 1.0) n (* 0.5 x)) (* n n)) x)))))

C

double code(double x, double n) {
	double tmp;
	if ((1.0 / n) <= -2e-77) {
		tmp = pow((x * x), (0.5 / n)) / (n * x);
	} else if ((1.0 / n) <= 2e-30) {
		tmp = log(((1.0 + x) / x)) / n;
	} else if ((1.0 / n) <= 4e+144) {
		tmp = ((x / n) + 1.0) - pow(x, (1.0 / n));
	} else {
		tmp = (fma(fma(-0.5, x, 1.0), n, (0.5 * x)) / (n * n)) * x;
	}
	return tmp;
}

Julia

function code(x, n)
	tmp = 0.0
	if (Float64(1.0 / n) <= -2e-77)
		tmp = Float64((Float64(x * x) ^ Float64(0.5 / n)) / Float64(n * x));
	elseif (Float64(1.0 / n) <= 2e-30)
		tmp = Float64(log(Float64(Float64(1.0 + x) / x)) / n);
	elseif (Float64(1.0 / n) <= 4e+144)
		tmp = Float64(Float64(Float64(x / n) + 1.0) - (x ^ Float64(1.0 / n)));
	else
		tmp = Float64(Float64(fma(fma(-0.5, x, 1.0), n, Float64(0.5 * x)) / Float64(n * n)) * x);
	end
	return tmp
end

Wolfram

code[x_, n_] := If[LessEqual[N[(1.0 / n), $MachinePrecision], -2e-77], N[(N[Power[N[(x * x), $MachinePrecision], N[(0.5 / n), $MachinePrecision]], $MachinePrecision] / N[(n * x), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 2e-30], N[(N[Log[N[(N[(1.0 + x), $MachinePrecision] / x), $MachinePrecision]], $MachinePrecision] / n), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 4e+144], N[(N[(N[(x / n), $MachinePrecision] + 1.0), $MachinePrecision] - N[Power[x, N[(1.0 / n), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(-0.5 * x + 1.0), $MachinePrecision] * n + N[(0.5 * x), $MachinePrecision]), $MachinePrecision] / N[(n * n), $MachinePrecision]), $MachinePrecision] * x), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if (/.f64 #s(literal 1 binary64) n) < -1.9999999999999999e-77

   1. Initial program 84.2%

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

   3. Taylor expanded in n around inf

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

      1. lower-/.f64 N/A

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

   5. Applied rewrites 68.5%

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

   6. Taylor expanded in x around inf

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

      1. lower-/.f64 N/A

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

      2. mul-1-neg N/A

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

      3. distribute-frac-neg N/A

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

      4. log-rec N/A

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

      5. remove-double-neg N/A

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

      6. lower-exp.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower-log.f64 N/A

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

      9. lower-*.f64 93.3

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

   8. Applied rewrites 93.3%

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

   10. Applied rewrites 90.5%

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

   if -1.9999999999999999e-77 < (/.f64 #s(literal 1 binary64) n) < 2e-30

   1. Initial program 32.0%

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

   3. Taylor expanded in n around inf

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

      1. lower-/.f64 N/A

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

   5. Applied rewrites 82.1%

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

   7. Applied rewrites 82.1%

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

   8. Taylor expanded in n around inf

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

   10. Applied rewrites 82.1%

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

   if 2e-30 < (/.f64 #s(literal 1 binary64) n) < 4.00000000000000009e144

   1. Initial program 76.2%

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

   3. Taylor expanded in x around 0

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

      1. +-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 70.2

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

   5. Applied rewrites 70.2%

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

   if 4.00000000000000009e144 < (/.f64 #s(literal 1 binary64) n)

   1. Initial program 33.4%

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

   3. Taylor expanded in x around 0

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

   5. Applied rewrites 68.8%

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

   6. Taylor expanded in n around 0

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

   8. Applied rewrites 51.8%

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

   10. Applied rewrites 93.9%

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

4. Final simplification 85.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{1}{n} \leq -2 \cdot 10^{-77}:\\ \;\;\;\;\frac{{\left(x \cdot x\right)}^{\left(\frac{0.5}{n}\right)}}{n \cdot x}\\ \mathbf{elif}\;\frac{1}{n} \leq 2 \cdot 10^{-30}:\\ \;\;\;\;\frac{\log \left(\frac{1 + x}{x}\right)}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 4 \cdot 10^{+144}:\\ \;\;\;\;\left(\frac{x}{n} + 1\right) - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\mathsf{fma}\left(-0.5, x, 1\right), n, 0.5 \cdot x\right)}{n \cdot n} \cdot x\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 53.8% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 1 - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{if}\;\frac{1}{n} \leq -5 \cdot 10^{-17}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;\frac{1}{n} \leq 2 \cdot 10^{-30}:\\ \;\;\;\;\frac{\frac{1}{x}}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 4 \cdot 10^{+144}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\mathsf{fma}\left(-0.5, x, 1\right), n, 0.5 \cdot x\right)}{n \cdot n} \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (let* ((t_0 (- 1.0 (pow x (/ 1.0 n)))))
   (if (<= (/ 1.0 n) -5e-17)
     t_0
     (if (<= (/ 1.0 n) 2e-30)
       (/ (/ 1.0 x) n)
       (if (<= (/ 1.0 n) 4e+144)
         t_0
         (* (/ (fma (fma -0.5 x 1.0) n (* 0.5 x)) (* n n)) x))))))

C

double code(double x, double n) {
	double t_0 = 1.0 - pow(x, (1.0 / n));
	double tmp;
	if ((1.0 / n) <= -5e-17) {
		tmp = t_0;
	} else if ((1.0 / n) <= 2e-30) {
		tmp = (1.0 / x) / n;
	} else if ((1.0 / n) <= 4e+144) {
		tmp = t_0;
	} else {
		tmp = (fma(fma(-0.5, x, 1.0), n, (0.5 * x)) / (n * n)) * x;
	}
	return tmp;
}

Julia

function code(x, n)
	t_0 = Float64(1.0 - (x ^ Float64(1.0 / n)))
	tmp = 0.0
	if (Float64(1.0 / n) <= -5e-17)
		tmp = t_0;
	elseif (Float64(1.0 / n) <= 2e-30)
		tmp = Float64(Float64(1.0 / x) / n);
	elseif (Float64(1.0 / n) <= 4e+144)
		tmp = t_0;
	else
		tmp = Float64(Float64(fma(fma(-0.5, x, 1.0), n, Float64(0.5 * x)) / Float64(n * n)) * x);
	end
	return tmp
end

Wolfram

code[x_, n_] := Block[{t$95$0 = N[(1.0 - N[Power[x, N[(1.0 / n), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(1.0 / n), $MachinePrecision], -5e-17], t$95$0, If[LessEqual[N[(1.0 / n), $MachinePrecision], 2e-30], N[(N[(1.0 / x), $MachinePrecision] / n), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 4e+144], t$95$0, N[(N[(N[(N[(-0.5 * x + 1.0), $MachinePrecision] * n + N[(0.5 * x), $MachinePrecision]), $MachinePrecision] / N[(n * n), $MachinePrecision]), $MachinePrecision] * x), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 #s(literal 1 binary64) n) < -4.9999999999999999e-17 or 2e-30 < (/.f64 #s(literal 1 binary64) n) < 4.00000000000000009e144

   1. Initial program 94.7%

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

   3. Taylor expanded in x around 0

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

   5. Applied rewrites 63.7%

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

   if -4.9999999999999999e-17 < (/.f64 #s(literal 1 binary64) n) < 2e-30

   1. Initial program 30.1%

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

   3. Taylor expanded in n around inf

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

      1. lower-/.f64 N/A

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

   5. Applied rewrites 76.9%

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

   6. Taylor expanded in x around inf

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

      1. lower-/.f64 N/A

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

      2. mul-1-neg N/A

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

      3. distribute-frac-neg N/A

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

      4. log-rec N/A

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

      5. remove-double-neg N/A

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

      6. lower-exp.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower-log.f64 N/A

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

      9. lower-*.f64 52.6

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

   8. Applied rewrites 52.6%

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

   9. Taylor expanded in n around inf

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

   11. Applied rewrites 52.9%

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

   if 4.00000000000000009e144 < (/.f64 #s(literal 1 binary64) n)

   1. Initial program 33.4%

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

   3. Taylor expanded in x around 0

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

   5. Applied rewrites 68.8%

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

   6. Taylor expanded in n around 0

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

   8. Applied rewrites 51.8%

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

   10. Applied rewrites 93.9%

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

4. Final simplification 59.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{1}{n} \leq -5 \cdot 10^{-17}:\\ \;\;\;\;1 - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{elif}\;\frac{1}{n} \leq 2 \cdot 10^{-30}:\\ \;\;\;\;\frac{\frac{1}{x}}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 4 \cdot 10^{+144}:\\ \;\;\;\;1 - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\mathsf{fma}\left(-0.5, x, 1\right), n, 0.5 \cdot x\right)}{n \cdot n} \cdot x\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 41.4% accurate, 10.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 55.0%

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

3. Taylor expanded in n around inf

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

   1. lower-/.f64 N/A

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

5. Applied rewrites 66.7%

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

6. Taylor expanded in x around inf

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

   1. lower-/.f64 N/A

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

   2. mul-1-neg N/A

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

   3. distribute-frac-neg N/A

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

   4. log-rec N/A

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

   5. remove-double-neg N/A

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

   6. lower-exp.f64 N/A

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

   7. lower-/.f64 N/A

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

   8. lower-log.f64 N/A

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

   9. lower-*.f64 60.3

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

8. Applied rewrites 60.3%

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

9. Taylor expanded in n around inf

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

11. Applied rewrites 45.3%

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

Alternative 13: 4.5% accurate, 19.3× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 55.0%

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

3. Taylor expanded in x around 0

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

5. Applied rewrites 45.0%

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

6. Taylor expanded in n around 0

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

8. Applied rewrites 6.2%

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

9. Taylor expanded in x around 0

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

11. Applied rewrites 4.3%

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

Reproduce

herbie shell --seed 2024284 
(FPCore (x n)
  :name "2nthrt (problem 3.4.6)"
  :precision binary64
  (- (pow (+ x 1.0) (/ 1.0 n)) (pow x (/ 1.0 n))))